Contributions to Science by Institut d'Estudis Catalans

VOLUME 11 | ISSUE 2 | DECEMBER 2015

Thematic issue on

Non-equilibrium physics Ignacio Pagonabarraga, FÃ¨lix Ritort (eds)

Volume 11 | Issue 2 | December 2015

OPEN ACCESS JOURNAL

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FRONT COVER

INSIDE PICTURES

Non-equilibrium bacterial growth. The photo graph shows a growing bacterial colony in a non-equilibrium state during growth. In the photograph, a colony of bacteria (Bacillus sp.) shows the formation of vortex during the growth of the colony and the circulation of bacterial cells both inside and in the surface of the colony. The photograph has been taken by scientific photographer biologist Rubén Duro using a light microscope at low magnification. Duro is also the editorial office’s managing coordinator of the issue. This issue of Contributions to Science, coordinated by Profs. Ignasi Pagonabarraga and Fèlix Ritort is devoted to the non-equilibrium systems, the most fascinating field of physics. The journal considers it is appropriated for this topic to be accompanied by Prof. Douglas Zook’s beautiful photographs of ephimeral images as viewed reflected by the crystal of a window.

Prof. Douglas Zook views the world with the mind of a scientist and the soul of an artist. Zook, has been until his recent retirement professor of science education and global ecology at the Boston University, Boston, MA. He is passionate about nature, conservation, and symbiosis. Several photographs presented in this issue were exhibited at the Botanical Garden Spring Festival in Kraków, Poland, as well as at the Jagiellonian University in Kraków, and galleries there and in Boston. He realized that while most reflections on window surfaces keep hidden, they can under certain specific conditions become mysterious masterpieces of Nature. Windows reflections shown in this issue are from from Kraków, Prague and Boston. All of the images were captured from sections of window panes in these cities. No color changes or other photoshop alterations were made.

BACK COVER In this issue we bring Pere Vieta i Gibert (1779– 1856). Contributions to Science, issue by issue, offers in its back cover a small homage to people of the Catalan Countries who dedicated their inventiveness, wisdom and time to their passion, and who are universally known. We have published already a short biography of Rafael Guastavino Moreno (València 1842– 1908), Ramon Casanova Danés (Campdevànol, Catalonia; 1892–1968), Margalida Comas Camps (Alaior, Minorca 1892–1972) and Ramon Casas (Barcelona 1866–1932).

Volume 11 | Issue 2 | December 2015

Editorial Board

EDITOR-IN-CHIEF Ricard Guerrero

Biological Sciences Section, IEC

ASSOCIATE EDITOR Salvador Alegret

ASSOCIATE EDITOR Ramon Gomis

Science and Technology Section, IEC

Biological Sciences Section, IEC

EDITORIAL BOARD The Science and Technology and Biological Sciences Sections:

Joaquim Agulló, Technical University of Catalonia • Josep Amat, Technical University of Catalonia • Francesc Asensi, University of Valencia • Damià Barceló, Spanish National Research Council (Barcelona) • Carles Bas, Institute of Marine Sciences-CSIC (Barcelona) • Pilar Bayer, University of Barcelona • Xavier Bellés, Spanish National Research Council (Barcelona) • Jaume Bertranpetit, Pompeu Fabra University (Barcelona) • Eduard Bonet, ESADE (Barcelona) • Joaquim Casal, Technical University of Catalonia • Alícia Casals, Technical University of Catalonia • Josep Castells, University of Barcelona • Jacint Corbella, University of Barcelona • Jordi Corominas, Technical University of Catalonia • Michel Delseny, University of Perpinyà • Josep M. Domènech, Autonomous University of Barcelona • Mercè Durfort, University of Barcelona • Marta Estrada, Institute of Marine Sciences-CSIC (Barcelona) • Gabriel Ferraté, Technical University of Catalonia • Ramon Folch, Institute for Catalan Studies • Màrius Foz, Autonomous University of Barcelona • Jesús A. Garcia-Sevilla, University of the Balearic Islands • Lluís Garcia-Sevilla, Autonomous University of Barcelona • Joan Genescà, National Autonomous University of Mexico • Evarist Giné, University of Connecticut (USA) • Joan Girbau, Autonomous University of Barcelona • Pilar González-Duarte, Autonomous University of Barcelona • Francesc González-Sastre, Autonomous University of Barcelona • Joaquim Gosálbez, University of Barcelona • Albert Gras, University of Alacant • Gonzalo Halffter, National Polytechnic Institute (Mexico) • Lluís Jofre, Technical University of Catalonia • Joan Jofre, University of Barcelona • David Jou, Autonomous University of Barcelona • Ramon Lapiedra, University of Valencia • Àngel Llàcer, Hospital Clinic of Valencia • Josep Enric Llebot, Autonomous University of Barcelona • Jordi Lleonart, Spanish National Research Council (Barcelona) • Xavier Llimona, University of Barcelona • Antoni Lloret, Institute for Catalan Studies • Abel Mariné, University of Barcelona • Joan Massagué, Memorial Sloan-Kettering Cancer Center, New York (USA) • Federico Mayor-Zaragoza, Foundation for a Culture of Peace (Madrid) • Adélio Machado, University of Porto (Portugal) • Gabriel Navarro, University of Valencia • Jaume Pagès, Technical University of Catalonia • Ramon Parés, University of Barcelona • Àngel Pellicer, New York University (USA) • Juli Peretó, University of Valencia • F. Xavier Pi-Sunyer, Harvard University (USA) • Norberto Piccinini, Politecnico di Torino (Italy) • Jaume Porta, University of Lleida • Pere Puigdomènech, Spanish National Research Council (Barcelona) • Jorge-Óscar Rabassa, National University of La Plata (Argentina) • Pere Roca, University of Barcelona • Joan Rodés, University of Barcelona • Joandomènec Ros, University of Barcelona • Claude Roux, University of Aix-Marseille III (France) • Pere Santanach, University of Barcelona • Francesc Serra, Autonomous University of Barcelona • David Serrat, University of Barcelona • Boris P. Sobolev, Russian Academy of Sciences, Moscow, Russia • Carles Solà, Autonomous University of Barcelona • Joan Antoni Solans, Technical University of Catalonia • Rolf Tarrach, University of Luxembourg • Jaume Terradas, Autonomous University of Barcelona • Antoni Torre, Obra Cultural de l’Alguer • Josep Vaquer, University of Barcelona • Josep Vigo, University of Barcelona • Miquel Vilardell, Autonomous University of Barcelona • Jordi Vives, Hospital Clinic of Barcelona

Volume 11 | Issue 2 | December 2015

©DZook

Contents

FOREWORD / PRESENTATION OF THE ISSUE Pagonabarraga I, Ritort F

125

Non-equilibrium, the most fascinating field of physics

NON-EQUILIBRIUM THERMODYNAMICS Jona-Lasinio G

127

Understanding non-equilibrium: A challenge for the future

Jou D, Casas-Vázquez J

131

Heat transfer and thermodynamics: A foundational problem in classical thermodynamics and in contemporary non-equilibrium thermodynamics

Ritort F

137

The physics of small systems: From energy to information

Rubi JM

147

Mesoscopic non-equilibrium thermodynamics

Baró J, Planes A, Vives E

153

Avalanche dynamics in driven materials

Palassini M

163

Electron glasses

Reguera D

173

Nucleation phenomena: The non-equilibrium kinetics of phase change

MATERIALS

SOFT MATTER Faraudo J, Aguilella-Arzo M

181

Ion transport through biological channels

Ortín J

189

Non-equilibrium dynamics of fluids in disordered media

Claret J, Ignés-Mullol J, Sagués F

199

Chiral selection under swirling: From a concept to its realization in soft-matter self-assembly

BIOPHYSICS Díaz-Guilera A, Pérez-Vicente CJ

207

Synchronization

Frigola D, Sancho JM, Ibañes M

215

On the principles of multicellular organism development

Soriano J, Casademunt J

225

Neuronal cultures: The brain’s complexity and non-equilibrium physics, all in a dish

HISTORICAL CORNER Puig-Pla C

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Pere Vieta (1779–1856), promoter of free public teaching of physics in Catalonia

PERSPECTIVES Zook D

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Reflections: The enduring symbiosis between art and science

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ANNUAL INDEXES (Volumes 7, 8, 9: years 2011, 2012, 2013)

FOREWORD/PRESENTATION OF THE ISSUE Institut d’Estudis Catalans, Barcelona, Catalonia

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CONTRIB SCI 11:125-126 (2015) doi:10.2436/20.7010.01.221

Non-equilibrium, the most fascinating field of physics Ignacio Pagonabarraga1,2 Fèlix Ritort1 Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barce lona, Catalonia. 2Universitat de Barcelona Institute of Complex Systems (UBICS), Barcelona, Catalonia © Douglas Zook. http://www.douglaszookphotography.com

Correspondence: Ignasi Pagonabarraga ipagonabarraga@ub.edu Fèlix Ritort ritort@ub.edu

This thematic issue of Contributions to Science is devoted to one of the most fascinating fields in modern science, the study of non-equilibrium systems. Nonequilibrium pervades nature. From a galaxy to a bacterium, all systems are out of equilibrium to some extent. The understanding of non-equilibrium phenomena has been a persistent goal in physics since the emergence of thermodynamics back in the 19th century, its influential role far reaching other fields such as chemistry and biology. Non-equilibrium physics is key in many widespread disciplines such as astrophysics, atmospheric sciences, condensed matter physics, chemical physics, economy, evolutionary biology to cite a few. Physicists might say that a system is out of equilibrium when there are net currents of any physical conserved quantity such as energy, charge, momentum, etc. In contrast, equilibrium systems are those where such currents vanish. Since our universe is continuously expanding, strictly speaking everything contained in it is out of equilibrium, the equilibrium assumption being only an approximation (and a very good one in many cases). However, and despite of its importance, our current understanding of non-equilibrium, albeit large, still remains incomplete. An underlying unified picture is still missing, fundamental results being still seen as partial results describing phenomena under specific conditions (e.g., the fluctuation-dissipation theorem). If it were not for the sole existence of the second law of thermodynamics we could say that fundamental laws for non-equilibrium are still to be discovered. In view of these facts we found timely to setup a special issue for Contributions to Science gathering papers on a few selected topics currently investigated by physicists in the Catalan community. The papers cover different areas of the nonequilibrium science such as non-linear phenomena, disordered systems, statistical thermodynamics, fluid dynamics, materials science, biophysics and neuroscience. Nowadays physics has reached such a degree of maturity to become a highly

Keywords: non-equilibrium physics · disordered systems · statistical thermodynamics · fluid dynamics · materials science · biophysics and neuroscience ISSN (print): 1575-6343 e-ISSN: 2013-410X

CONTRIBUTIONS to SCIENCE 11:125-126 (2015)

Foreword

interdisciplinary science, being applicable to the most diverse problems and contexts. In turn, we are witnessing a situation where physics steadily delves inside the most unexpected niches of knowledge that, until a few years ago, remained unfit to it. There is a long way ahead of us for such interdisciplinary quest and new unexpected fundamental discoveries in physics might emerge in the coming years. If this holds true then non-equilibrium physics surely remains the right area to grind. We hope this recollection of selected articles in this exciting area will contribute to stimulate

further scientific discussions and collaborations within our small but highly active research community. It remains to us acknowledging those who have made possible this special issue, particularly physicist and historian of science Dr. Emma Sallent Del Colombo and Prof. Ricard Guerrero, the editor-in-chief of Contributions to Science. Their willingness and concerted effort in preparing this volume are definitely contributing to the visibility of one of the most active research areas in the Catalan physics community.

About the images on the first page of the articles in this issue. Articles of this thematic issue of Contributions to Science show in their first page one photograph made by Prof. Douglas Zook (Boston University, Boston, MA, USA) from his book Earth Gazes Back. Those fourteen ephemeral images were reflections on glass windows of different landscapes in the cities of Kraków, Prague and Boston. Multiple angles of light beams dance across the air, reflecting the ever-changing reality around us. The editors of this issue believe that those photographs represent very appropriately the ever-changing reality conceptualized by one of the most fascinating fields in modern science: the “Non-equilibrium physics”. (See also the article “Reflections: The enduring symbiosis between art and science,” by D. Zook, on pages 249-251 of this issue [http://revistes.iec.cat/index.php/CtS/article/view/142178/141126]. This thematic issue can be unloaded in ISSUU format and the individual articles can be found in the Institute for Catalan Studies journals’ repository [www.cat-science.cat; http://revistes.iec.cat/contributions].

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CONTRIBUTIONS to SCIENCE 11:125-126 (2015)

NON-EQUILIBRIUM THERMODYNAMICS Institut d’Estudis Catalans, Barcelona, Catalonia

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CONTRIB SCI 11:127-130(2015) doi:10.2436/20.7010.01.222

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Understanding non-equilibrium: a challenge for the future Giovanni Jona-Lasinio Dipartimento di Fisica e INFN, Università di Roma “La Sapienza”, Roma, Italy

Summary. Far from equilibrium behavior is ubiquitous. Indeed most of the processes that characterize energy flow occur far from equilibrium. These range from very large systems, such as weather patterns or ocean currents that remain far from equilibrium owing to an influx of energy, to biological structures. Away-from-equilibrium processes occur on time scales ranging from nanoseconds to millennia. A difficulty of non-equilibrium physics is that usual thermodynamic functions of state like entropy or free energy do not generalize easily. The study of rare fluctuations of thermodynamic variables like densities or currents in stationary states has led to the identification of thermodynamic functions relevant in far from equilibrium situations. For a wide class of systems, called diffusive systems, it has been possible to develop a comprehensive unified theory, known as Macroscopic Fluctuation Theory, with considerable predictive power. Much remains to be done. Besides the purely scientific motivation, the challenge is to deal effectively with basic issues facing humanity like energy problems, climate control, understanding living matter. [Contrib Sci 11(2): 127-130 (2015)] Correspondence: Giovanni Jona-Lasinio Dipartimento di Fisica e INFN Università di Roma “La Sapienza” Piazzale A. Moro 2 Roma 00185, Italy E-mail: gianni.jona@roma1.infn.it

Introduction Understanding non-equilibrium is considered a basic challenge in an authoritative report issued in 2007 by the Department of Energy of the United States with the significant title Directing Matter and Energy: Five Challenges for Science and the Imagination [1]. Non-equilibrium is the fifth challenge! Far from equilibrium behavior is ubiquitous. Indeed (we

freely quote from this report), most of the processes that characterize energy flow occur far from equilibrium. These range from very large systems, such as weather patterns or ocean currents that remain far from equilibrium owing to an influx of energy (in this case very large amounts of heat), to biological structures from humans to horseflies whose very existence requires the maintenance of non-equilibrium conditions through the consumption of energy. Non-equi-

Keywords: equilibrium-non-equilibrium · entropy · fluctuations · stationary states · diffusive states ISSN (print): 1575-6343 e-ISSN: 2013-410X

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understanding non-equilibrium

librium includes processes that store energy by pushing ions against a gradient and complex networks that carry out, for example, metabolic processes. Away-from-equilibrium processes occur on time scales ranging from nanoseconds to millennia. The relevance of the subject emphasized in this report was brought to the attention of the physics community in 2008 in an article appeared in Physics Today, [2], a wellknown journal of the American Institute of Physics. Despite the pervasiveness of non-equilibrium systems and processes, most of the current understanding of physical and biological systems is based on equilibrium concepts even when they are not strictly applicable. The concepts of equilibrium and non-equilibrium belong to mechanics and thermodynamics. Here we are mainly interested in the thermodynamic point of view. Even when we deal with microscopic systems as in biological matter or nanostructures in advanced technology, the number of atoms or molecules involved is so large that thermodynamics and a statistical point of view provide the appropriate approach. Classical thermodynamics deals with states of a system, possibly in contact with an environment, where no flow of energy is present. These states either do not change in time (equilibrium) or change very slowly so that they can be described by a sequence of equilibrium states. This is the notion of quasi static or reversible transformation. However, as emphasized in a well-known textbook on thermodynamics [3], to define in a precise way this notion we need to go beyond equilibrium: “A quasi-static process is thus defined in terms of a dense succession of equilibrium states. It is to be stressed that a quasi-static process therefore is an idealized concept, quite distinct from a real physical process, for a real process always involves nonequilibrium intermediate states having no representation in the thermodynamic configuration space. Furthermore, a quasistatic process, in contrast to a real process, does not involve considerations of rates, velocities or time. The quasi-static process simply is an ordered succession of equilibrium states, whereas a real process is a temporal succession of equilibrium and non-equilibrium states.” In spite of the inadequacy in clarifying its own foundations, classical thermodynamics has been very successful. Its principles were formulated in the 19th century at the time of the industrial revolution and provided the basis for conceiving and producing the necessary engines and mawww.cat-science.cat

chines. The first principle, recognizing that heat is a form of energy, states that 1. in a transformation of a system the variation of its energy is equal to the sum of the mechanical work and the heat exchanged. Several formulations have been given for the second principle and we choose the one which is convenient for the ensuing discussion even though more abstract. 2. There is a quantity called entropy which in a transformation of an isolated system (universe) can either remain constant or increase. An equilibrium state is therefore a state of maximum entropy. There is also a third principle which is of a more special character but will not be relevant in the following. Entropy is a somewhat mysterious concept and when reading the original papers of its inventor, Robert Clausius, the first reaction is “How did he think of it?”. Actually entropy was introduced by Clausius as a quantitative characterization of the equivalence of two transformations, i.e., transformations which can be substituted one to the other as they produce exactly the same effects. Its deep meaning was discovered by Boltzmann by taking a statistical point of view. He realized that the entropy is related in a simple way to the probability of a macroscopic state. A macroscopic state is characterized by few parameters like energy, density, temperature, pressure, etc. and many different microscopic atomic configurations are compatible with them. The probability of a macroscopic state is proportional to the number of the microscopic configurations compatible with these parameters. The more microscopic states correspond to the macroscopic parameters, the higher is the probability. Entropy is in fact the logarithm of such a probability times a universal constant. Intuitively, either a system stays in equilibrium, which in Boltzmann’s view has maximal probability, or evolves spontaneously towards a more probable state. This is the reason why entropy increases. One may think of entropy as a measure of the microscopic complexity. There is also a similar definition of entropy in information theory which measures the content of information of a message. A story says that such a terminology was suggested to Shannon, the inventor of the theory, by the famous mathematician John Von Neumann with the comment: “You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statisti128

CONTRIBUTIONS to SCIENCE 11:127-130 (2015)

Jona-Lasinio

cal mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage.” For systems out of equilibrium it does not exist yet a macroscopic description of a scope comparable with equilibrium thermodynamics. In non-equilibrium one has to cope with a variety of phenomena much greater than in equilibrium. From a conceptual point of view the non-equilibrium situations closest to equilibrium are the stationary non-equilibrium states which describe a steady flow through some system. Simple examples are the heat flow in an iron rod whose endpoints are thermostated at different temperatures or the stationary flow of electrical current in a given potential difference. For such states the fluctuations exhibit novel and rich features with respect to the equilibrium situation. For example, as experimentally observed, the space correlations of the density extend to macroscopic distances, which mean that the fluctuations of the density in different points of the system are not independent. Stationary states appear as a natural generalization of equilibrium and the first important question is to what extent can we formulate a thermodynamics for these states based on general principles. The first difficulty encountered for such a program is that there is no obvious definition of basic thermodynamic concepts as free energy or entropy in states far from equilibrium. In the examples above energy flows through the bar or the wire and there is dissipation of heat in the environment, which is usually called production of entropy. In other words it is not easy to define entropy for the bar or the wire but it is natural to define entropy production in terms of the increase of the entropy of the environment which is supposed approximately in equilibrium. We have considered so far two non-equilibrium situations. As we have remarked during a thermodynamic transformation, even if quasi-static, a system necessarily goes out of equilibrium. Another situation is provided by stationary states. Progress in the last twenty years has in fact concerned mainly these situations. Exact relations not restricted to quasistatic transformations have been established. A thermodynamics of stationary states has started to emerge. The leitmotiv has been the study of fluctuations within a variety of approaches and physical models. Fluctuations of thermodynamic variables in equilibrium due to the atomic structure of matter is an old subject to which Einstein devoted

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a series of basic papers showing the strict connection between fluctuations and thermodynamic functions like free energy and entropy. In the last two decades important relationships have been established which are known under the heading Fluctuation Theorems. These refer mainly to microscopic systems characterized by a finite but arbitrarily large number of degrees of freedom, performing any transformation between equilibrium states. In particular these relationships permit to recover for example the variation of the free energy in a transformation by measuring the probability distribution of the fluctuations of the work done on a system over a certain interval of time. For a not too technical introduction see [4]. As an example of interesting application to biology of fluctuation theorems let us mention recent work on the statistical physics of phenomena like self-replication and adaptation [5]. In these papers it is argued that in many particle systems in contact with the environment there is a tendency towards self-organized states that form through increased dissipation and suppression of fluctuations. The authors suggest that these findings may lead to questioning a strictly Darwinian view of evolution. The study of rare fluctuations of thermodynamic variables like densities or currents in stationary states has led to the identification of thermodynamic functions relevant in far from equilibrium situations. As one may expect such functions do not depend only on the state of the system but also on parameters describing the interaction with the environment. Besides a quantity which plays a role analogous to the free energy in equilibrium, other functions relevant for the thermodynamics of currents have been discovered. For a wide class of systems, called diffusive systems (the case of the bar or the electric wire above is included but also many biological processes), it has been possible to develop a comprehensive unified theory, known as Macroscopic Fluctuation Theory, with considerable predictive power [6]. For example, besides the already mentioned long range correlations, phase transitions possible in non-equilibrium but impossible in equilibrium are predicted or phase transitions in current fluctuations which may be relevant in view of minimizing the dissipation of energy. It is important to realize that, due to the variety of nonequilibrium phenomena, the generalization of a thermodynamic notion may depend on the case in study. An interesting example is provided by the so called active matter, typically biological matter like a “gas’’ of bacteria in a

CONTRIBUTIONS to SCIENCE 11:127-130 (2015)

understanding non-equilibrium

fluid. A bacterium can be considered as a self-propelled particle. In this case the pressure in general does not satisfy an equation of state as it depends on the interaction of the bacteria with the confining walls [7]. Looking back on the progress of the last two decades we are definitely more confident in the possibility of understanding the many manifestations of non-equilibrium. Actually it is a very interesting time to work in this field both theoretically and experimentally as much has still to be done. Furthermore, besides the purely scientific motivation, the goal is to deal effectively with basic issues facing humanity like energy problems, climate control, understanding living matter. Competing interests. None declared.

References 1. Bertini L, De Sole A, Gabrielli D, Jona-Lasinio G, Landim C (2015) Macroscopic fluctuation theory. Rev Mod Phys 87:593 arXiv:1404.6466 2. Callen H (1985) Thermodynamics and an introduction to thermostatistics, 2nd ed. John Wiley & sons, New York 3. England J (2013) Statistical physics of self-replication, J Chem Phys 139: 121923 doi:10.1063/1.4818538 4. Fleming GR, Ratner MA (2008) Grand challenges in basic energy sciences. Physics Today 61:7-28 5. Jarzynski C (2008) Nonequilibrium work relations: foundations and applications. Eur Phys J B 64:331-340 6. Jona-Lasinio G (2015) Large deviations and the Boltzmann entropy formula. Braz J Prob Statistics 29:494-501 arXiv:1411.1250 7. Perunov N, Marsland R, England J (2016) Statistical physics of adaptation. Phys Rev X 6:021036 arXiv:1412.1875 8. Solon AP, Fily Y, Baskaran A, Cates ME, Kafri Y, Kardar M, Tailleur J (2015) Pressure is not a state function for generic active fluids. Nature Physics 11:673 arXiv:1412.3952 9. U.S. Department of Energy (2007) Directing Matter and Energy: Five Challenges for Science and the Imagination. Report December 20

About the image on the first page of this article. This photograph was made by Prof. Douglas Zook (Boston University) for his book Earth Gazes Back [www.douglaszookphotography.com]. See the article “Reflections: The enduring symbiosis between art and science,” by D. Zook, on pages 249-251 of this issue [http://revistes.iec.cat/index.php/CtS/article/view/142178/141126]. This thematic issue on “Non-equilibrium physics” can be unloaded in ISSUU format and the individual articles can be found in the Institute for Catalan Studies journals’ repository [www.cat-science.cat; http://revistes.iec. cat/contributions].

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NON-EQUILIBRIUM THERMODYNAMICS Institut d’Estudis Catalans, Barcelona, Catalonia

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CONTRIB SCI 11:131-136 (2015) doi:10.2436/20.7010.01.223

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Heat transfer and thermodynamics: A foundational problem in classical thermodynamics and in contemporary non-equilibrium thermodynamics

David Jou,1,2,* José Casas-Vázquez1 Departament de Física, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Catalonia. 2Institute for Catalan Studies, Barcelona, Catalonia 1

*Correspondence: David Jou Departament de Física Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona, Catalonia

Summary. The search for generalized heat transport equations describing Fourier’s diffusive regime is a frontier in nanoscale technology and energy management, together with thermal waves, Ziman regime, phonon hydrodynamics, and ballistic heat transport, as well as their respective transitions. Here we discuss the close connection between this search and another, much less known aspect, namely, the exploration of new forms of entropy and of the second law of thermodynamics for fast and steep perturbations and high values of heat flux, which would make generalized transport equations compatible with the second law. We also draw several analogies between this situation and the confluence of Fourier and Carnot theories that resulted in a general formulation during the foundational period of thermodynamics. [Contrib Sci 11(2): 131-136 (2015)]

E-mail: David.Jou@uab.cat

Introduction Research on heat transport in nanoscopic systems and the formulation of a thermodynamic framework that accommodates them are current frontiers in non-equilibrium physics. There are several analogies with the foundational period of thermodynamics in the 1820s. During that time, Jean-Baptiste Joseph Fourier (1768–1830) published the Theorie analytique de la chaleur (1821) [13], setting the foundations for the

mathematical description of heat conduction, and Nicolas Léonard Sadi Carnot (1796–1832) published his Réflexions sur la puissance motrice du feu et sur des machines propres à développer cette puissance (1824) [4], which is considered the foundational cornerstone of thermodynamics. Although, at least initially, heat transport and heat engines were considered to be largely unrelated, they were soon united under the more encompassing statements of classical thermodynamics, developed in the 1850s.

Keywords: non-equilibrium thermodynamics · nanoscale heat transfer · local equilibrium · extended thermodynamics · non-equilibrium entropy ISSN (print): 1575-6343 e-ISSN: 2013-410X

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Heat transfer and thermodynamics

In this article, we compare those early challenges and achievements with the current problems faced by physicists and engineers working in the field of heat transfer and thermodynamics. These problems extend beyond Fourier’s law and local-equilibrium thermodynamics and we emphasize some of their conceptual aspects, especially how the search for generalized heat transport equations has been a powerful stimulus for the exploration of the foundations of thermodynamics. This effort has led to the recognition of the confluence of heat transport and thermodynamics, which for the most part had been ignored due to the dominance of more practical and urgent topics related to heat transport over subtler and more abstract aspects, such as a revised definition of entropy.

Heat transfer and thermodynamics in the early days of thermodynamics Calorimetry and the analysis of heat transfer greatly preceded the formulation of thermodynamics. Experimental and conceptual research on specific and latent forms of heat was conducted even before the caloric theory was formulated by Antoine-Laurent de Lavoisier (1743–1794) in the Traité élémentaire de chimie (1789) [20]. There, he proposed a hypothetical subtle and weightless matter, the caloric, as the substance of heat, moving from higher to lower temperatures or accumulating in some hidden form in phase transitions. The caloric theory was so direct, so intuitive, and so inspiring and fruitful that it was difficult to overturn and an alternative, more realistic theory would not be proposed until the end of the 1840s. In 1701, well before the caloric theory, Isaac Newton (1642–1727) had proposed a law of heat transfer. It stated that the rate of heat transfer between two bodies at different empirical temperatures was proportional to the difference of the temperatures. Newton’s reasoning stemmed from his interest in alchemy and, in particular, in the fusion temperature of several metals and alloys. Of course, at that time this was a subject beyond the frontiers of thermometric measurement. Instead, by comparing the time that it took for a mass of material to cool from its fusion point to the environmental temperature, Newton was able to estimate, albeit indirectly and very imprecisely, the fusion temperatures he was searching for. Newton’s law provided a first theoretical framework for the description of heat exchange. Incidentally, it also had an influence on Buffon’s method for estimating the age of the Earth, by studying the cooling time of a hot iron sphere and www.cat-science.cat

then extrapolating the result to a sphere the size of our planet. Fourier’s mathematical description of heat transfer inside, rather than between different bodies was a stimulus for mathematics, physics, and philosophy. From a mathematical point of view, solving the temperature evolution equation was the motivation underlying the Fourier transformations which, despite their early rejection by some mathematicians, became a very relevant tool in mathematics and mathematical physics. In physics, they were a source of inspiration for Fick’s diffusion law and Ohm’s electrical transport law. They also led to extensive experimental research on the transport properties of many materials and stimulated the development of the kinetic theory of gases proposed by Rudolf Julius Emmanuel Clausius (1822–1888), James Clerk Maxwell (1831–1879), and Ludwig Edward Boltzmann (1844–1906). In the case of Boltzmann, one of the aims of his studies was to obtain detailed insights into transport coefficients. In natural philosophy, the Fourier transformations provided a mathematical elegance and rigorous framework for a paradigmatic irreversible phenomenon. They stood in contrast to Newton’s mathematical framework for reversible mechanics and, especially, gravitational phenomena, which had been a paradigm of the relation between Eternal Mathematics and celestial motion. Heat transport theory does not necessarily need nor imply an explicit and full-fledged thermodynamic framework. In fact, both Fourier and Carnot used the caloric theory, assuming the materiality of heat, but without general statements about its behavior. For instance, heat transport theory provided a dynamic equation for heat transport, one that obeyed the apparently trivial, although actually very deep condition that heat goes spontaneously from higher to lower temperatures. However, it did not state this requirement as a general law of nature, applicable to predictions beyond heat transfer. Instead, the main object of Carnot’s theory, stimulated by the industrial revolution fostered by steam machines, was to explore the maximum efficiency of heat engines and the conversion of heat into work. Carnot largely ignored the rate of heat transfer, which was the problem analyzed by Fourier. Indeed, although Carnot cited the works of many scientists, he did not mention Fourier—further evidence that the problems of heat transfer and thermodynamics were considered to be unrelated. In contrast to the rapid success of Fourier’s work, Carnot’s went unnoticed until it was discovered by Benoit Paul Émil Clapeyron (1799–1864) in 1835, several years after Carnot’s death. In 1848 it was used by William Thomson (later Lord 132

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Jou and Casas-Vázquez

Kelvin) (1824–1907) to define absolute temperature. The discovery by Julius von Mayer (1814–1878) and James Prescott Joule (1818–1889) that heat is not a material but a form of energy exchange led to the formulation of the two general laws of thermodynamics, in 1850. In that year, Clausius and Kelvin formulated their respective versions of the second law in terms of the framework of heat engines established by Carnot, but without his assumption of heat conservation. The results of both Carnot and Fourier survived this deep conceptual change, but Carnot’s theory was to be reformulated. With Clausius’ statement of the second law, heat transfer became an essential topic of thermodynamics, not from the point of view of the rate of exchange, but regarding the directionality of the exchange. Half a century after Fourier’s law on heat conduction, Josef Stefan (1835–1893) proposed an equation for heat radiation that was complemented by Boltzmann’s description of the relation between this law and Maxwell’s electromagnetic theory. Radiative heat exchange became a topic of interest in astrophysics, metallurgy, and electric bulbs. By 1900, it had led to quantum theory. Since then, heat transport, in its different forms (conduction, convection, and radiation), has been a classical topic in physics, engineering, geophysics, meteorology, and the life sciences.

Heat transport beyond a diffusive regime: Revolutions in transport theory In the last two decades of the 20th century, the field of heat transport underwent a genuine revolution, with enlarged domains of applicability and the appreciation of new regimes and phenomena, where Fourier’s theory is no longer applicable. From a microscopic perspective, Fourier’s law is valid in the diffusive regime, i.e., when there are many collisions between heat carriers, but not when the frequency of collisions between heat carriers and the boundaries of the container become comparable to or higher than the frequencies of the collisions amongst the heat carriers themselves. The domain of validity of Fourier’s law is described by the socalled Knudsen number, defined as the ratio between the mean free path of the heat carriers and the characteristic size of the system. When the Knudsen number is very small, collisions amongst particles dominate, the regime is diffusive, and Fourier’s law is valid. When the Knudsen number is >1, the regime is ballistic, i.e., the particles move between opposite boundaries without experiencing collisions with other particles. At the beginning of the 20th century there was www.cat-science.cat

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great interest in transport theory for rarefied gases, a topic in which the contributions of Martin Knudsen (1871–1949) were especially relevant. The topic of transport theory developed, in part, as an extension of the kinetic theory of gases in rarefied situations. In the 1950s and 1960s, rapid advances in astronautics revived interest in rarefied gases because a relevant part of the re-entry of satellites takes place in rarefied regions of the atmosphere. In this case, the usual hydrodynamics—valid, like Fourier’s equation, when collisions amongst particles have a dominant effect, namely, for small values of the Knudsen number—are no longer applicable. An extreme situation of very rarefied gases occurs in mechanics, because the effects of the collisions of particles with an object are such that each collision may be considered a single mechanical event. In the 1940s, heat transfer theory was stimulated by the observation of second sound in superfluid liquid helium. The wave propagation nature of heat, instead of the usual diffusive pattern, came as a surprise, but it did not deeply influence heat transport theory because it was considered a peculiar behavior restricted to a special physical system characterized by macroscopic coherent quantum properties. The problem of thermal waves was put in a more general perspective, albeit merely a theoretical one, by the work of Carlo Cattaneo (1911–1979) and Pierre Vernotte (1898–1970) at the end of the 1940s. Both were searching for a finite speed of propagation for thermal pulses or high-frequency thermal waves [7,8,19,34,35,40]. In the 1990s, nanotechnology emerged as a technological frontier, with a huge economic impact. In many nanoscopic systems, especially at low temperatures, the size of the system is of the order of the phonon mean free path, or even smaller, and, in some cases it is smaller than electron mean free path. In these situations, the classical transport equations for heat, electricity, and thermoelectricity are no longer valid. Usually, the situation in nanosystems lies somewhere between diffusive and ballistic, which adds to the complexity of the problem. In considering heat transport in nanosystems, several non-Fourier regimes must be taken into account. On the one hand, even in the case of Fourier’s law, thermal conductivity is no longer a purely material property; rather, it also depends on the size of the system (the radius of nanowires or the thickness of thin layers, for instance) [6,15,18,38]. Furthermore, there are typical non-Fourier regimes: heat waves, the Ziman regime, phonon hydrodynamics, and ballistic transport. These are usually dealt with from microscopic perspectives, based on kinetic equations, or by ab initio computer simulations. With respect to transport theory, one aim CONTRIBUTIONS to SCIENCE 11:131-136 (2015)

Heat transfer and thermodynamics

is to formulate generalized transport equations at a mesoscopic level, such that the different, above-mentioned regimes can be described by a single equation [2,3,10,11,24,27]. Heat transport in nanosystems plays a role in three contemporary industrial revolutions. Ones of them is the miniaturization leading to more powerful computers, but also to the need for computer refrigeration because of the large amount of heat that is dissipated in tiny spaces as many miniaturized devices accumulate; however, refrigeration becomes more difficult at miniaturized scales because the effective thermal conductivity is reduced with respect to the bulk values. A second revolution is in energy management, basically through photovoltaic and thermoelectric effects, which may be more efficient at nanoscales than in bulk. The third revolution is in material sciences, in which nanostructures such as superlattices, carbon nanotubes, graphene, nanoporous materials, and silicon nanowires are expected to play relevant roles in heat transport, whether for insulation and refrigeration or for delicate phonon control in heat rectification and thermal transistors and commuters in the emerging area of phononics [18].

Generalized heat transport and the frontiers of thermodynamics: TempÂ erature and entropy In general, heat transport is approached mainly as a mathematical theory, with a dynamic equation whose mathematical solutions are obtained under certain boundary and initial conditions, independent of thermodynamic considerations, and then compared to observations. However, when dealing with generalized equations for heat transport, a generalization of thermodynamics cannot truly be avoided because of two problems: the physical meaning of temperature in fast processes, small systems, and far from equilibrium, and the definition of entropy and the statement of the second law of thermodynamics.

Zeroth principle and temperature Out of equilibrium, the zeroth principle of thermodynamics is no longer valid, neither in general nor in the particular, but it is relevant in the case of steady states. Intuitively, it is easy to understand that, since far from equilibrium no energy equipartition is expected, all the theoretical or operational definitions or measurements of temperature involving the interacwww.cat-science.cat

tion with different sets of degrees of freedom will lead to different values of the corresponding temperature. Thus, for instance, the kinetic temperature may depend on the direction, the average potential energy, or other definitions. Since different degrees of freedom may have different temperatures, their respective contribution to the heat flux may differ; it could even be that for some degrees of freedom heat flux goes, as usual, from a higher to a lower temperature whereas for other degrees of freedom heat flux is in the opposite direction [5,14,21,31]. Because different measurement methods explore different aspects of the system, we must be aware of the deep meaning of temperature obtained by each one.

Second principle and entropy One of the most fruitful and versatile statements of the second law is in terms of entropy: the entropy of the final equilibrium state should be equal to or higher than the entropy of the initial equilibrium state, after some internal constraints on the system are removed. This statement considers only the entropy of the initial and final equilibrium states, but does not refer to what happens during the intervening process. This is not a trivial issue because classical entropy is a function of state defined only for equilibrium states [23]. Attempts to generalize thermodynamics to non-equilibrium states usually invoke the local-equilibrium hypothesis, assuming that locallyâ&#x20AC;&#x2022;in sufficiently small regionsâ&#x20AC;&#x2022;the system is in thermodynamic equilibrium, although globally it may be far from equilibrium, by having, for instance, strong temperature or pressure gradients. This assumption allows entropy to be defined at a local level. From the balance equation for the local-equilibrium entropy, and assuming the local-equilibrium version of the entropy flux, an expression is obtained for local entropy production per unit time and volume. The second law is thus stated in terms of the positive definite character of entropy production. This statement is clearly more restrictive than the classical statement of the second law because it requires not only that the final entropy is greater than or equal to the initial one, but also that entropy always increases, at any time and in any volume. This is incompatible with equations allowing for thermal waves, in which heat may flow during short periods from lower to higher temperature, implying a negative entropy production, or in equations of phonon hydrodynamics, in which axial heat transport may lead to heat transport from lower to higher temperatures in small regions, i.e., 134

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those comparable in size to the phonon mean free path. Thus, generalized heat transport equations, even if they successfully describe experimental observations, may in some aspects be at odds with the local-equilibrium formulation of the second law. In this case there are several possibilities. One is to work with the integrated local-entropy production, which requires that only its integral from the initial to the final state is positive. Another is to define the generalized entropies whose positive production is compatible with generalized heat transfer equations. The advantage of the first formulation is that it does not introduce new restrictions on the equations beyond those of classical equilibrium thermodynamics; thus, while no new hypotheses are made, less information is gained about the system. The advantage of the second one is that it strives for more detailed versions of heat transport and of entropy and for insights into the relation between macroscopic and microscopic approaches. Several possibilities to describe systems beyond local equilibrium have been examined [23]. In particular, our group at the Autonomous University of Barcelona (Universitat AutĂ˛noma de Barcelona, UAB) has been working in so-called extended thermodynamics, in which several fluxes intervening in the system are considered as additional independent variables (heat flux, diffusion flux, electric current, viscous pressure tensor, and corresponding higher-order fluxes). Both the entropy and the entropy flux depend on all fluxes, in addition to classical variables [1,12,16â&#x20AC;&#x201C;18,28,33]. The generalized transport equations are the evolution equations of the fluxes, which are subject to the conjecture that local generalized entropy production must be positive at any time and at any point. The thermodynamic formulation may be carried out as an extension of either classical irreversible thermodynamics (in extended irreversible thermodynamics) [16,18,21] or rational thermodynamics (in rational extended thermodynamics) [28,37]. The ensuing generalized transport equations for heat flux (and for other fluxes) are able to address the different regimes of heat transport, i.e., heat waves, the Ziman regime, phonon hydrodynamics, and the ballistic regime, as well as intermediate regimes, without violating the tentative requirement of positive entropy production. The main results in this field can be summarized as follows: (i) the relaxational terms in the transport equations (namely, terms in the first-order time derivative of the fluxes) correspond to second-order contributions of the respective fluxes to the extended entropy; (ii) the presence of non-local terms (namely, terms in the Laplacians or gradients of the fluxes) is related to the second-order contributions of the www.cat-science.cat

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fluxes to the entropy flux; (iii) the corresponding contributions to the Gibbs equation of the extended entropy have the form of an intensive quantity times the differential of an extensive one; the intensive one may be compared to the Legendre multipliers used in information-theory approaches (or similar ones) to statistical descriptions of non-equilibrium steady states; (iv) the combination of linearized equations for higher-order fluxes leads to a hierarchy of equations yielding a continued-fraction expansion of the thermal conductivity in terms of the wave vector or the Knudsen number, which describes the transition from diffusive to ballistic regimes [2,9,22,36]; (v) the analysis of higher-order fluxes allows for a multilevel mesoscopic description, and thus for studies of the effects of the elimination of a set of fast variables in order to project the dynamics on slower variables, depending on the time rate of the perturbations or experiments [18,29]; and (vi) fluctuations of the fluxes around equilibrium or non-equilibrium steady state are described by the second differential of the generalized entropy.

Conclusions The search for generalized heat transport equations (and other transport equations) is not only driven by a practical need to improve material engineering and energy management, but has also been a stimulus to explore the frontiers of non-equilibrium thermodynamics, going beyond local equilibrium approximations. Although in many situations nonequilibrium contributions to local entropy are small, in some circumstances they may be relevant, especially from a conceptual point of view [23,24]. A closely related frontier in miniaturization is the thermodynamics of small systems [30,32,39], which are also related to the Knudsen number, because what makes a system small is not its size but the number of particles it contains, and the relation between the rate of its internal collisions and collisions with the walls, or between the relaxation time and the characteristic rate of energy transfer with the outside. Systems as small as atomic nuclei have been considered as hydrodynamic and thermodynamic systems, at least around equilibrium, because they are so dense that the mean free path is smaller than the size of the system. However, in relativistic nuclear collisions, this assumption breaks down, because the energy transferred by the fast collision to the nuclei may be comparable to the average energy of each nucleus, such that nucleons inside the nuclei do not have enough time to equilibrate their energy. Nowadays, CONTRIBUTIONS to SCIENCE 11:131-136 (2015)

Heat transfer and thermodynamics

fluxes of people, goods, and capital are very intense and have become ballistic, in the sense that all of them may go from one place to another, distant place without almost no interaction with their respective counterparts along the way. The high values of these fluxes is such that, on some occasions, the rate of exchange of people, capital, or goods is much faster than the time it takes to for the respective situation to equilibrate (from a social or a political aspect). This may bring the system very far from local equilibrium, leading in some cases to sociological and cultural conflicts. Acknowledgements. We acknowledge the financial support of the Spanish Ministry of Economy and Competitiveness under grant FIS201233099 and of the Direcció General de Recerca of the Government of Catalonia under grant 2009 SGR 00164. Competing interests. None declared.

References 1. Albers B, Wilmanski K (2015) Continuum thermodynamics Part II: Applications and examples. World Scientific: Singapore 2. Álvarez FX, Jou D (2007) Memory and nonlocal effects in heat transport. From diffusive to ballistic regimes. Appl Phys Lett 90:083109 doi:10.1063/1.2645110 3. Álvarez FX, Jou D, Sellitto A (2009) Phonon hydrodynamics and phononboundary scattering in nanosystems. J Appl Phys 105:014317 doi:10.1063/1.3056136 4. Carnot S (1824) Réflexions sur la puissance motrice du feu et sur des machines propres à développer cette puissance. Librairie Bachelier, Paris 5. Casas-Vázquez J, Jou D (2003) Temperature in nonequilibrium states: a review of open problems and current proposals. Rep Prog Phys 66:19372023 6. Chen G (2005) Nanoscale energy transport and conversion. Oxford University Press, New York 7. Cimmelli VA (2009) Different thermodynamic theories and different heat conduction laws. J Non-Equilib Thermodyn 34:299-333 doi:10.1515/JNETDY.2009.016 8. Cimmelli VA, Sellitto A, Jou D (2010) Non-equilibrium temperatures, heat waves and non-linear heat transport equations. Phys Rev B 81:054301 doi:10.1103/PhysRevB.81.054301 9. Cimmelli VA, Jou D, Ruggeri T, Van P (2014) Entropy principle and recent results in non-equilibrium theories. Electronic open journal, Entropy 16:1756-1807 doi:10.3390/e16031756 10. de Tomás C, Cantarero A, Lopeandía AF, Álvarez FX (2014) Thermal conductivity of group-iv semiconductors from a kinetic-collective model. Proc Roy Soc A 470:20140371 doi:10.1098/rspa.2014.0371 11. Dong Y, Cao B-Y, Guo Z-Y (2011) Generalized heat conduction laws based on thermomass theory and phonon hydrodynamics. J Appl Phys 110:063504 doi:10.1063/1.3634113 12. Eu BC (1992) Kinetic theory and irreversible thermodynamics. Wiley, New York 13. Fourier JB (1822) Théorie analytique de la chaleur. Editeur F. Didot, Paris 14. Hatano T, Jou D (2003) Measuring temperature of forced oscillators. Phys Rev E 67:026121

15. Huang B-W, Hsiao T-K, Lin K-H, Chiou D-W, Chang C-W (2015) Lengthdependent thermal transport and ballistic thermal conduction. AIP Advances 5:053202 doi:10.1063/1.4914584 16. Jou D, Casas-Vázquez J, Criado-Sancho M (2011) Thermodynamics of fluids under flow. 2nd ed, Springer, Berlin 17. Jou D, Casas-Vázquez J, Criado-Sancho M (2003) Thermodynamics and dynamics of flowing polymer solutions and blends. Contrib Sci 2:315332 18. Jou D, Restuccia L (2011) Mesoscopic transport equations and contemporary thermodynamics. Contemporary Physics 52:465-474 doi:10.108 0/00107514.2011.595596 19. Kovacs R, Van P (2015) Generalized heat conduction in heat pulse experiments. Int J Heat Mass Transfer 83:613-620 doi:10.1016/j.ijheatmasstransfer.2014.12.045 20. Lavoisier A (1789) Traité élémentaire de chimie, présenté dans un ordre nouveau et d’après les découvertes modernes. 2 vols, Cuchet libraire, Paris 21. Lebon G, Jou D, Casas-Vázquez J (1992) Questions and answers about a thermodynamic theory of the third type. Contemporary Physics 33:41-51 22. Lebon G, Jou D, Casas-Vázquez J, Muschik W (1998) Weakly nonlocal and nonlinear heat transport. J. Non-Equilib Thermodyn 23:176-191 23. Lebon G., Jou D, Casas-Vázquez J (2008) Understanding nonequilibrium thermodynamics: foundations, applications, frontiers. Springer, Berlin 24. Lebon G (2014) Heat conduction at micro and nanoscales: a review through the prism of extended irreversible thermodynamics. J NonEquilib Thermodyn 39:35-59 doi:10.1515/jnetdy-2013-0029 25. Luzzi R, Vasconcellos AR, Casas-Vázquez J, Jou D (1997) Characterization and measurement of a nonequilibrium temperature-like variable in irreversible thermodynamics. Physica A 234:699-714 26. Luzzi R, Vasconcellos AR, Ramos JS (2002) Predictive statistical mechanics: a non-equilibrium ensemble formalism. Kluwer, Dordrecht 27. Ma Y (2012) Size-dependent thermal conductivity in nanosystems based on non-Fourier heat transfer. Appl Phys Lett 101:211905 doi:10.1063/1.4767337 28. Müller I, Ruggeri T (1998) Rational extended thermodynamics. Springer, New York 29. Pavelka M, Klika V, Grmela M (2014) Time reversal in non equilibrium thermodynamics. Phys Rev E 90:062131 doi:10.1103/PhysRevE.90.062131 30. Reguera D, Rubi JM, Vilar JMG (2005) The mesoscopic dynamics of thermodynamic systems. J Phys Chem B 109:21502-21515 doi:10.1021/ jp052904i 31. Ritort F (2005) Resonant nonequilibrium temperatures. J Phys Chem B 109:6787-6792 doi:10.1021/jp045380f 32. Ritort F (2007) The nonequilibrium thermodynamics of small systems. Comptes Rendus Acad Sci Physique 8:528-539 33. Sieniutycz S, Salomon P (eds.) (1992) Extended thermodynamics systems. Taylor and Francis, New York 34. Sieniutycz S (1994) Conservation laws in variational thermo-hydrodynamics. Springer, Berlin 35. Straughan B (2011) Heat waves. Springer, Berlin 36. Struchtrup H (2005) Macroscopic transport equations for rarefied gas flows. Springer, Berlin 37. Truesdell C (1969) Rational thermodynamics. MacGraw Hill, New York 38. Tzou DY (2004) Macro-to-microscale heat transfer: The lagging behaviour. 2nd ed, Wiley, New York 39. Vilar JMG, Rubi JM (2001) Thermodynamics “beyond” local equilibrium. Proc Nat Acad Sci USA 98:11081-11084 doi:10.1073/pnas.191360398 40. Zhang Z (2007) Nano/microscale heat transfer. McGraw Hill, New York

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OPENAACCESS

CONTRIB SCI 11:137-146 (2015) doi:10.2436/20.7010.01.224

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The physics of small systems: From energy to information Fèlix Ritort1,2 Small Biosystems Lab, Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona, Catalonia. 2Ciber-BBN, Instituto de Salud Carlos III, Madrid, Spain 1

Summary. The focus of this review is the recent developments in the non-equilibrium physics of small systems. Special emphasis is placed on singlemolecule experiments and their contribution to expanding our current understanding of fundamental concepts, such as temperature, energy, entropy, and information. [Contrib Sci 11(2): 137-146 (2015)]

Correspondence: Fèlix Ritort Departament de Física de la Matèria Condensada Universitat de Barcelona Martí i Franquès, 1 08028 Barcelona, Catalonia Tel.:+34-934035869 E-mail: ritort@ub.edu

Non-equilibrium, dissipation, and entropy production Non-equilibrium conditions pervade nature. From a waterfall to a star, from a microbe to a human being, all natural systems are intrinsically non-equilibrium, as equilibrium systems are only an approximate description of what we observe (but indeed a very good one in some cases). What is the key signature of non-equilibrium systems? In general, we can say that a system is out of equilibrium when there are net currents across the system of any conserved quantity, such as mass, charge,

momentum, and energy. For example, consider a block of mass sitting on the floor. If we move the block by pulling on it, then frictional forces between the block and the floor arise that heat up the block at the contact area. A net amount of energy in the form of heat then flows from the block to the floor. The block is out of equilibrium. In yet another example, an electric current flowing through a resistance (e.g., a metal) heats up the resistance due to the frictional forces generated by the collisions between the moving electrons and the metal atoms of the resistance. The heat generated in the resistance is then dissipated to the environment, resulting in a net heat flux.

Keywords: Brownian motion · Nyquist noise · stochastic thermodynamics · Maxwell demon · glassy systems ISSN (print): 1575-6343 e-ISSN: 2013-410X

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Physics of small systems

In both examples, an external agent (a mechanical force for the block and an electric field for the current) exerts power on the system that results in the delivery of heat Q to the environment and a corresponding increase in entropy given by Q/T, where T is the temperature of the environment. Physicists define this increase as positive entropy production. Broadly speaking, non-equilibrium systems are those in which entropy is produced whereas for equilibrium systems entropy production equals zero. Entropy production is strictly positive and the heat produced in the previous examples is always dissipated to the surroundings. If the opposite is observed, for example, if heat spontaneously flows from the environment to the resistance, generating a net electric current, entropy production is then negative. Such events are rare, their occurrence being interpreted as evidence that somewhere else in the universe entropy must have been produced to compensate for the decrease, because the global balance of entropy production in the universe is always positive. However, closer examination shows more complex behaviors than heat flowing from hot to cold. Suppose we have an experimental device capable of measuring how much heat is delivered to the surroundings during a given amount of time. If the time window were long enough, then we would observe that average entropy production is always positive. However, if the time window was decreased below a characteristic timescale, then entropy production would sometimes be positive and other times negative, meaning that heat ocassionally flows from the colder surrounding to the hotter body. In this case, the amount of heat transferred fluctuates from measurement to measurement, not only in magnitude but also in sign. The shorter the time window, the stronger the heat fluctuations and the more probable such negative events are. But what is the origin of these fluctuations? In 1827, Robert Brown, a botanist well-known for his detailed descriptions of the cell nucleus and cytoplasm and for his contributions to the taxonomy of plants, made an important discovery. During microscopy observations of the grains of pollen of a plant suspended in water, he noticed that their motion was erratic and unpredictable, as if the grains were alive. It was only after the validation of the atomic hypothesis at the beginning of the 20th century that it became clear that what Brown had observed was the effect of the stochastic or random collisions of the molecules of water against the grains of pollen. Bombarded from all directions, the suspended grains of pollen jiggled erratically. These conclusions were supported by the Smoluchovsky-Einstein theory of Brownian motion, in 1905. Later experiments conducted by Jean-Baptiste Perrin on diffusive colloidal particles provided the final www.cat-science.cat

proof. Perrin was also able to accurately estimate Avogadroâ&#x20AC;&#x2122;s number using physical methods alone; his results agreed with those reported by chemists. Statistical physics, the appropriate theoretical ground for thermodynamics, builds on the atomistic nature of matter and the probabilistic nature of heat and work. Today, not only has the probabilistic feature been acknowledged by scientists, it is also fundamental to our current understanding of non-equilibrium thermodynamics. When an electric current flows through a resistance, heat is generated (and entropy produced). However, the moving cloud of electrons experiences Brownian forces that lead to voltage fluctuations across the resistance. This effect is commonly known as Nyquist noise and was first experimentally observed by John Bertrand Johnson at Bell Labs in 1926. For time windows that are comparable to the decorrelation time of the voltage signal, the heat and entropy produced are positive most of the times (the cloud moves in the direction of the electric field), but occasionally heat and entropy production are negative (the electron cloud moves against the field). The latter are rare events, marked by the flow of heat from the environment to the resistance and its conversion into work to move the electron cloud against the field. It is important to stress that, despite large fluctuations, average entropy production is always positive, which ultimately constitutes the core of the second law of thermodynamics. Under which conditions are fluctuations in entropy production experimentally observable? For a system of N degrees of freedom, extensive quantities, such as entropy production or the energy content, grow linearly with N while the size of the fluctuations scale according to N based on the law of large numbers. The relative magnitude of these fluctuations then decays as 1 N , indicating that they are measurable if N it is not too large. According to the equipartition law, each degree of freedom contributes to the average energy by an amount roughly equal to kBT, where kB is the Boltzmann constant and T is the temperature. Entropy production and energy fluctuations are measurable if the energies involved are a few kBT, meaning that the energies delivered to the system by the external agent are not too high and are comparable to the average kinetic energy carried by the colliding molecules in the thermal environment (on the order of kBT). These are the socalled small systems and the branch of physics devoted to the study of the energy transformation processes in them under non-equilibrium conditions is referred as the non-equilibrium thermodynamics of small systems, or stochastic thermodynamics [4,13,18,21]. Examples of small systems that have been experimentally 138

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Contrib Sci

Box 1. Optical tweezers are based on the principle of the conservation of linear momentum, by which a microscopic transparent object (e.g., a polystyrene or silica bead) with an index of refraction higher than that of the surrounding medium deflects an incoming light ray, thus exerting a net force on the object. An optical trap for manipulating single molecules is produced by focusing an infrared beam inside a fluidics chamber, optically trapping a micrometer-sized bead, and measuring either the deflected light using position-sensitive detectors or the beadâ&#x20AC;&#x2122;s position with a CCD camera or back focal plane interferometry. Pulling experiments use dumbbells made of a molecule tethered between two beads (Fig. 1). In singletrap setups, one bead is immobilized in a pipette by air suction, and the other is captured in an optical trap that measures the force exerted on the molecule. By moving the optical trap relative to the pipette we can record the so-called force-distance curve.

Fig. 1. Schematics of pulling experiments. (A) Dual- and single-trap setups. (B) Force-distance curves for a DNA molecule pulled in a dual-trap setup. The forces measured at the two traps are of equal magnitude but opposite sign. (C) Mechanically unfolding and folding DNA hairpins. The force jumps correspond to unfolding (red) and folding (blue) transitions.

studied over the past several years are the current flowing across a resistance, a colloidal microsphere captured in an optical trap, a biological molecule with two or more conformational states, and a single-electron transistor to cite a few. In general, the extensiveness property of energy is related to the size of the system and to the measurement time; therefore, fluctuations should be difficult to observe in macroscopic systems and over times that are so long that large deviations from the average become exceedingly rare. However, technological developments over the past two decades in the fields of micro- and nanotechnologies and the development of high-temporal resolution cameras, microfluidics devices, photomultipliers, and photodetectors for light detection have enormously expanded our ability to measure these phenomena. Entropy production and energy fluctuation measurements in non-equilibrium systems are now accessible to the experimentalist and our opportunities to expand our knowledge in this exciting field are steadily growing.

Small systems and single-molecule experiments Biology investigates all aspects of living organisms whereas traditionally physics has largely ignored the study of the matwww.cat-science.cat

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ter of living systems. Yet, except for its complexity, living matter is the same as ordinary matter. It was mainly 19th century chemists who appreciated the importance of understanding living matter, through the newborn discipline of biochemistry. However, concepts such as space, time, force, and energy are not only fundamental quantities in physics they are also central to biology. Thus, the field of biophysics applies the concepts and techniques of physics to study living beings. Physical techniques, such as X-ray diffraction, nuclear magnetic resonance, and electron microscopy, are among those that have contributed to the recent revolution in biology. Conversely, many physicists are now using biological systems as the basis of physical models to test and scrutinize new physical theories. A prominent example of this trend is the recent developments in single-molecule biophysics, in which individual biological molecules are manipulated one at a time, not only to unravel the most complex biomolecular reactions but also to discover new aspects of biological organization or even to challenge physical theories in statistical mechanics. Biological matter is intrinsically soft, with weak molecular forces (electrostatic, hydrophobic, etc.) responsible for its thermodynamic stability. Moreover, typical energies involved in remodeling processes fall in the kBT range, at the level of thermal noise (Box 1). This means that living matter is subject to CONTRIBUTIONS to SCIENCE 11:137-146 (2015)

Physics of small systems

strong fluctuations due to the comparable magnitudes of weak interacting forces and the Brownian forces present in an aqueous environment. This feature distinguishes biological matter from ordinary matter and makes the former an ideal “playground” to investigate non-equilibrium phenomena. The possibility of manipulating one molecule at a time offers exciting prospects to acquire valuable information about molecular processes [17]. Atomic force microscopy and magnetic and optical tweezers are commonly employed techniques (Box 1) that enable measurements of the elastic properties of biological polymers, the thermodynamic and kinetic stability of molecular folders (nucleic acid structures, proteins), DNA-protein and DNA-peptide interactions (e.g., intercalation, condensation, aggregation phenomena), proteinprotein interactions (e.g., ligand-receptor binding), and molecular motors (e.g., cellular transport, DNA-RNA polymerases, ATPases and proton pumps, viral packaging motors, topoisomerases, helicases). For example, it is nowadays possible to attach a DNA molecule between two beads and, by pulling on its two phosphate strands, to unzip it (Fig. 1). By varying the pulling speed, both the force and the work distributions can be measured, which provides valuable information about the folding process (free energy, kinetic rates, and folding pathways). Single-molecule biophysics is not restricted to investigations of the most complex biomolecular processes (such as segregation of the DNA chromatids during cellular division to cite a remarkable example), it can also be used to study and characterize the mechanical properties in single cells. For example, micrometer-sized beads can be passively attached to the cellular cytoskeleton via integrin receptors located at the cellular membrane and the power spectrum of the position of the bead then measured using time-resolved fluctuation spectroscopy. This type of measurements provides valuable information about the viscoelastic properties of the cell. This novel type of cellular characterization is called mechanical phenotyping, and explorations using this approach are just beginning.

Fluctuation theorems: where do we stand? According to the second law of thermodynamics, any irreversible transformation produces entropy and therefore increases the total entropy of the universe. For example, when a spoon is used to gently stir a cup of coffee, the work exerted www.cat-science.cat

by the spoon is dissipated in the form of heat. However, as previously explained for the case of electric current flowing across a resistance, this statement only holds on average and is apparent in macroscopic systems or for very long times. In small systems, Brownian forces introduce large fluctuations in measurable quantities, such as work or heat, and actual entropy production values vary across repetitions of the same experiment. Fluctuation theorems (FTs) quantify the occurrence of negative entropy production events relative to positive ones. Let us suppose that a system in a non-equilibrium state produces or consumes a given amount of entropy St along a trajectory or path of time t. In general, FTs obey a simple mathematical relation of the following type [9,10]: P ( St )

P ( − St )

= e kB

(1)

with St the entropy production during time t, P its probability distribution, and kB is the Boltzmann constant. This very simple relation tells us two things. First, positive St trajectories are exponentially more probable than negative St ones, the overall probability of negative St trajectories being exponentially suppressed both over time and increasing system size. For a macroscopic system in which the number of degrees of freedom is of the order of Avogadro’s number (~1023), the extensiveness property of St shows how such negative events are severely penalized with exceedingly small probabilities 23 (on the order of e −10 ). Second, by rewriting the previous expression as e − S / k P( St= ) P(− St ) and integrating over St we ( − St / k B ) get 〈e 〉 =1 (the probability P is normalized to 1) where <…> denotes an average over many repetitions of the same experiment. A result of this type is often known as the Jarzynski equality; it was first obtained by G. Kochkov and B. Kuzovlev in 1991 and later derived by Chris Jarzynski in 1997 in a different non-equilibrium setting. As we explain below, Jarzynski was also the first to recognize the importance of this equality for free energy calculations, a result with practical implications. A corollary of the Jarzynski equality is that 〈 St 〉 ≥ 0 , a result that we recognize as the second law of thermodynamics. The beauty of Eq. (1) and the Jarzynski equality lies in the fact that the second law does not appear to be an inequality, but an equality instead. There are three major categories of non-equilibrium systems: transient systems, steady-state systems, and aging systems. Roughly speaking, a suitable form of entropy production S always exists. In transient systems (initially in thermal equilibrium, but driven out of equilibrium by the action of time-dependent external forces), this quantity corresponds t

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Box 2. One of the main advantages of transient fluctuation theorems (FTs) is the possibility to determine free energy differences from irreversible work measurements. A long-held postulate of thermodynamics, that the equilibrium free energy difference between two states can only be measured through the work produced in a reversible transformation, is now disputed. The Jarzynski equality allows us to extract free energy differences from irreversible transformations [8]. By repeatedly measuring the irreversible work exerted upon the system, the free energy difference can be extracted by e( − G kBT or ÄG = exponentially averaging the work values, 〈 eÄ( −W/ / kBT) ) 〉 = −kBTlog (〈e( −W / kBT ) 〉 ) . The average must be determined from an infinite number of repetitions of the same experiment, which is unfeasible. Due to the exponential character of the average, the Jarzynski equality is strongly biased for a finite number of experiments N; however, it is possible to extract accurate estimates of ∆G by exploiting the dependence of the free energy estimator on N. An extension of the Jarzynski equality is the FT by Crooks [9] (given by Eq.1 with St equal to Wdis / T ), which allows determination of the free energies of native structures from bi-directional pulling experiments, i.e., by combining unfolding (forward) and folding (reverse) work measurements. A 2005 experiment [10] demonstrated how molecular free energy differences in RNA molecules could be determined from irreversible pulling experiments.

Fig. 2. (A) Pulling curves of a 20-basepair DNA hairpin. The molecule unravels around 15 pN. (B) Forward and reverse work distributions at three pulling rates: 1 (blue), 5 (green), and 15 (red) pN/s. Distributions cross at Work=DG for all pulling rates. (C) Log-normal plot of the ratio between the forward and reverse work distributions.

Different research groups worldwide have applied the technique to extract free energy differences in nucleic acids and proteins, all examples of molecular transformations driven by intramolecular forces (Fig. 2). The domain of applicability of fluctuation theorems has been also extended to extract free energies of kinetic states (i.e., states that are metastable such as intermediate and misfolded states) and intermolecular interactions such as ligand binding reactions [1]. Theoretical studies and experiments show how fluctuation relations are applicable to a wide range of systems in varied conditions: from high to low dissipation, from short to long times or from weakly to strongly interacting systems. In the purely physics domain it has been successfully applied to mesoscopic systems such as beads in optical traps, electric resistances, single electron transistors, Bose-Einstein condensates, etc…. Time will show the overall implications of this fascinating result.

to the amount of dissipated work Wdis divided by the temperature T, S = Wdis / T . On the other hand, Wdis equals the total work W exerted by the external forces minus the free energy difference ∆G , W= W – ∆G ; ∆G also equals the dis reversible work or the work performed on the system under quasi-static conditions, i.e., in an infinitely slow transformation. In steady-state systems (driven to a stationary state by the action of time-dependent or non-conservative forces, such as the previous example of the electric current flowing through a resistance), the entropy production S equals the work done by the external agent divided by the temperature, S = W / T . Finally, in aging systems (relaxational systems that equilibrate over very long-time scales), the heat Q released to the environment during the relaxation process is the key www.cat-science.cat

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quantity, S = Q / T . Transient FTs have been used to recover free energy differences through the Jarzynski equality 〈 e( − St / kB ) 〉 = 〈 e( −Wdis / kBT ) 〉 = 1 or 〈 e −W / kBT 〉 =e −∆G / kBT , by exponentially averaging the work over many experiments (Box 2). Beyond recovering free energy differences the steady-state FT might be applicable to molecular motors in making inferences about the properties of their mechanochemical cycles (see below). The experimental measurement of work fluctuations in molecular systems irreversibly pulled by mechanical forces not only has practical advantages, but might also provide new perspectives that contribute to our understanding of living matter. Indeed, one might hypothesize that the marvelous complexity and efficiency of molecular systems in biology CONTRIBUTIONS to SCIENCE 11:137-146 (2015)

Physics of small systems

(and biological organisms in general) are the result of an evolutionary process that has taken advantage of such large energy fluctuations in a way that is yet unknown to us. In fact, by rectifying thermal fluctuations, molecular motors have reached astonishing large efficiencies, observable in many enzymatic reactions (such as translocating motors in the cell powered by ATP hydrolysis or energy conversion by light-harvesting complexes in photosynthetic reactions), that have yet to be paralleled by human-designed systems. The importance of the weak forces (hydrogen bonds, electrostatic, hydrophobic) responsible for remodeling events at the molecular and cellular level in low-energy processes at the level of thermal noise suggests that fluctuations and large deviations have played crucial roles during molecular evolution.

From energy to information: thermo dynamic inference In 1867, James Clerk Maxwell, the Scottish scientist who unified electricity and magnetism, proposed a thought experiment to violate the second law of thermodynamics. Maxwell imagined a very small intelligent being endowed with free will, and fine enough tactile and perceptive organization to give him the faculty of observing and influencing individual molecules of matter [3]. How a Maxwell demon operates is shown in Fig. 3 (left). In Maxwell’s thought experiment, two chambers of a gas kept at equal temperatures are separated by an adiabatic wall with a small hole and a gate that can be opened and closed by the demon. By observing the speed of the individual molecules, the demon selectively opens and closes the gate to separate fast from slow molecules creating a net temperature difference between the two chambers. The demon can do this effortlessly, without the expenditure of work, thereby violating the second law. There have been several attempts to exorcise the Maxwell demon, but the definitive resolution of the paradox came from the theory of computing. In the 1960s, Rolf Landauer, from IBM, demonstrated that, by recording the information, the Maxwell demon’s system never returns to its original state unless information is erased after each observation in a cycle. However, the erasure of information increases the overall entropy, ultimately restoring the validity of the second law. The Maxwell demon can be experimentally realized in the Szilard engine (Fig. 3, right). A demon observes the position of a single particle in a gas chamber in contact with a thermal bath at temperature T. When the particle occupies one of the halves of the chamber, a movable wall and a pulley mechawww.cat-science.cat

nism capable of pulling a weight are implemented in the middle of the chamber. The molecule then pushes against the wall, lifting up the weight. When the full volume of the chamber is finally restored, a cycle has been completed and the process starts anew. Along each cycle heat is transferred from the bath to the system and fully converted into work, violating the second law. The maximum amount of work that can be extracted per cycle, Wmax , equals the net heat transferred from the bath, Wmax = Q = T ∆S = k BT log(2) . This value of Wmax equals the maximum work that can be extracted by a single-bit Szilard engine in the classical regime, often called the Landauer limit. The Szilard engine, as an example of a Maxwell demon, can be experimentally realized in small systems such as quantum systems, electronic devices and, more recently, in single molecules. As expected in small systems, there are large fluctuations in the amount of work that can be extracted along a cycle. In the convention that work W extracted from the system is negative (whereas delivered work is positive) this amounts to saying that W ≥ −Wmax = −k BT log(2) , or that the average extracted work per cycle cannot exceed the Landauer limit. Alongside these developments, FTs have now been generally extended to the case in which there is information feedback. A system is driven out of equilibrium by the action of an external agent; however, the non-equilibrium protocol is changed depending on the outcome of one (or more) measurements taken at specific times (discrete time feedback) or when a continuously monitored observable fulfills a specific condition (continuous time feedback). In this situation, the Jarzynski equality 〈e( −Wdis / kBT ) 〉 =1 becomes 〈e[ −(Wdis − I / kBT )] 〉 =1 , where I is a new quantity called information that is directly related to the feedback protocol [13]. The convex property of the exponential function immediately leads to 〈Wdis 〉 ≥ −k BT 〈 I 〉 or 〈W 〉 ≥ ∆G − k BT 〈 I 〉 , meaning that the minimum amount of work exerted during a transformation in the presence of feedback can be less than ∆G . For cyclic protocols where ∆G = 0, feedback enables the extraction, on average, of a maximum amount of (negative) work equal to 〈W 〉 max ≥ −k BT 〈 I 〉 , with 〈 I 〉 ≥ 0 . The mathematical expression for the path-dependent information I is in general complicated and depends on the specific feedback protocol. However, for the simple one-bit Szilard engine case one has 〈 I 〉 =log(2) . Information-to-energy conversion experiments show that energy and information are highly related quantities, one does not proceed without the other. Another twist in this exciting field is provided by thermodynamic inference, or the possibility to extract useful information about a non-equilib142

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Fig. 3. (A) Maxwell demon. A small being (green) generates a temperature gradient without the expenditure of work. (B) Szilard engine. The same small being fully converts heat into work without any further change.

rium system by imposing the validity of the FT [16]. Let us consider a DNA molecule tethered between two beads in a dual-trap optical tweezers setup. Dual traps are useful for the experimentalist because of their high resolution and reduced instrumental drift. The setup can be used to repeatedly pull a molecule along a cycle by moving one optical trap while the other remains at rest (Fig. 4). The bead captured in the moving trap is then dragged through water, where it is subject to Stokes friction. The total work W exerted upon the system as measured by the force recorded in the moving trap has two contributions: the work exerted to stretch the DNA molecule and the work required to move the bead in the fluid against frictional forces. The total dissipated work D along a cycle satisfies Crooks’ FT [Eq. (1)], with S = D / T . However, some dual-trap setups cannot measure the force in the moving trap but only in the trap that remains at rest during the pulling experiment. As the bead in the trap at rest hardly moves along the pulling cycle, the Stokes friction force experienced by the bead is smaller in that trap than in the moving trap (cyan arrows in Fig. 4). The dissipated work D’ extracted from the force measured in the trap at rest is missing a dissipative component and does not satisfy the FT of Eq. (1). We might say that devices measuring the force in the trap at rest ignore an essential part of the total work (accounted for in the moving trap instead), resulting in a partial work measurement. In other words, these devices cannot be used to measure full entropy production in a non-equilibrium experiment or to test the validity of FTs. However, one can impose the validity of the FT to infer, from only partial work measurements in the trap at www.cat-science.cat

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rest, the full work distribution one would measure in the moving trap. In the case of the DNA molecule shown in Fig. 4, this is accomplished by shifting the partial work distribution P(D’) by a constant value ∆ , which is equal to the average dissipation of the center of mass in the dumbbell along a cycle. We call this thermodynamic inference: it is the procedure by which we infer the full entropy production distribution in a non-equilibrium experiment from partial work (D’) measurements. Thermodynamic inference is therefore a powerful concept that is applicable whenever full entropy production in a non-equilibrium system is not measurable, either because it cannot be experimentally accessed or because it has unexpected hidden contributions. It might be used in a wide range of situations, such as extracting the free energies of (hidden) kinetic states in single molecules or in obtaining useful information about the mechanochemical cycle in ATP-powered machines. It might be also used to quantify randomness in heterogeneous molecular ensembles, such as protein sequences exhibiting a multiplicity of native states or sequence ensembles of nucleic acids and proteins generated in molecular evolution experiments. Finally we stress the importance of measuring work fluctuations in gauging the power of the thermodynamic inference approach. The sole measurement of the average partial work precludes inference of the average total work, as it is not possible to impose the validity of the FT using only measured average values. In this regard, thermodynamic inference is a key feature of small systems and is without parallel in irreversible thermodynamics of macroscopic systems. CONTRIBUTIONS to SCIENCE 11:137-146 (2015)

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Fig. 4. Thermodynamic inference. (A) Schematics of a dual trap with moving trap A at speed λ° and trap B at rest in the water frame. Cyan arrows indicate drag forces on each bead. The energy must be defined in terms of the configurational variables yA,yB with respect to the frame at rest rather than the moving-frame variables xA,xB. (B) Dissipated work D,D’ distributions measured for forces recorded in traps A and B respectively when pulling a DNA molecule at different speeds along a cycle. Filled symbols are the P( x) and open symbols are P(− x)e ( x / kBT ) with x = D(green), D’(blue). Only D satisfies the FT Eq.(1).

Lessons from glassy systems: the effective temperature The validity of Eq. (1) relies on properly accounting for the full entropy production St. However, as previously explained, sometimes only a partial entropy production measurement St’ is feasible. With the use of several simplifying assumptions [2] one can prove that St’ satisfies a modified version of the FT shown in Eq. (2): P ( St' )

P ( − St' )

St' kB

(2)

with x a dimensionless quantity. Equation (2) is often called an x-FT and it differs from Eq. (1) only by the pre-factor x in the exponent. In general, the x-FT is not an exact result, but it does hold in the Gaussian regime exemplified by the abovedescribed DNA molecule pulled in a dual-trap setup, where 〈 S ' t 〉 = x〈 St 〉 . The inequality 〈 S ' t 〉 ≤ 〈 St 〉 then leads to x < 1. What is the physical interpretation of x? According to the previous relation, x is the fraction of the total entropy production measured in the trap at rest. In other words, if x is < 1, then a part of the total entropy production is missing in the measurement. A very interesting connection has recently emerged in the context of glassy systems. Glasses and spin glasses are systems containing a high degree of structural disorder. They www.cat-science.cat

relax to equilibrium extremely slowly after a quench (i.e., a very fast change in external parameters, such as temperature and volume). This slow relaxation implies an extremely low entropy production rate. In many respects the amount of dissipation falls within the small systems regime, where fluctuations and large deviations from the average behavior are characteristic features. Interestingly, however, it has been demonstrated that glassy systems fulfill an x-FT of the type shown in Eq. (2), with x being equal to the so-called fluctuation-dissipation ratio describing violations of the fluctuationdissipation theorem that have been interpreted with the notion of an effective temperature [6]. Roughly speaking, the effective temperature Teff quantifies how fast correlations decay in the glassy state vs. in a system thermally equilibrated at the quenching temperature. Because of the large viscosity of supercooled glasses (in may change by twenty orders of magnitude in a narrow temperature range), values of Teff are often too high (thousands of Kelvins in many cases). What is the physical significance of these strikingly high temperatures? For glassy systems characterized by a single relaxational timescale one finds x = T / Teff ; therefore the value of x in glasses is expected to be small. We hypothesize that the physical meaning of x and Teff should not be considered from an energetic viewpoint but rather from an informational one. The low values of x (and the correspondingly high values of Teff) should be interpreted in terms of missing information in 144

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the aging state. In other words, the average entropy production commonly measured in aging experiments 〈 S ' t 〉 is just a tiny fraction (x) of the total entropy produced 〈 St 〉 . An informational theoretical interpretation of the fluctuation-dissipation ratio should also be possible in other non-equilibrium contexts, e.g., in non-equilibrium steady states that violate the fluctuation dissipation theorem and show the emergence of effective temperatures [8].

The role of information in biology In 1944, Erwin Schrödinger published an enlightening monograph titled “What is life?”, in which he wrote [20]: The large and important and very much discussed question is: How can the events in space and time which take place within the spatial boundary of a living organism be accounted for by physics and chemistry? We accept that living beings do not violate fundamental laws of physics. However, we also immediately recognize that living beings are very special. They seem to circumvent or mock the laws of physics as we understand them: a stone will fall because it is acted upon by gravity, whereas birds fly whenever they feel the need to do so. Biologists refer to this behavior as teleonomy, which recognizes that living beings have their own agenda; that is, they move, jump, play, eat, reproduce, plan, shop, do business, carry out research, etc. [15]. Unlike ordinary matter, living beings are always part of a population and an ecological niche. The physicist might call this an ensemble of individuals. Biological populations evolve under the rules of Darwinian selection, in which those individuals that best respond to the pressures of their environment succeed over those less well able to do so. Darwinian evolution rests on a dynamics of a very special kind, in which mutations and selective amplifications of the fittest species determine the evolving phenotypes. In the eyes of a physicist, evolving populations produce a startling non-stationary state in which basic thermodynamics concepts such as energy, matter, entropy, and information are intertwined in a complex and undecipherable manner [11]. Living matter has two features: it is heterogeneous and soft. Because cell populations are intrinsically heterogeneous, experiments that aim to reveal their features are difficult to reproduce: the same strain, the same environmental conditions, etc., often produces different outcomes. Hete rogeneity is not only restricted to cells, it is also present at the molecular level. Myoglobin, the oxygen carrier protein in the muscle tissue of vertebrates, is known to fold into a heterogeneous set of different native structures, all them able to www.cat-science.cat

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bind oxygen. Other proteins (e.g., enzymes and polyclonal antibodies in the immune system) are also able to assume a multiplicity of native states. In addition, living matter is soft and actively subjected to remodeling. Embedded in noisy aqueous environments, the forces inside biological structures must be strong enough to keep them stable and, at the same time, weak enough for the structures to continuously remodel in response to changing environments. It is not by chance that the fundamental energy scale of statistical physicists equals that of biochemists (1 kBT = 0.6kcal/mol at 298K). In fact, fundamental biological forces operate at the edge of the thermal noise level and the stabilizing free energies of macromolecules in tissues are on the order of a few kcal/mol or kBT. As enthalpy (H) and entropy (TS) contributions are typically much higher, such low free energies can be achieved only by a fine compensation between both contributions. Molecular pathways in the cell have been finely tuned to operate within a narrow range of conditions, and they are extremely sensitive to small variations in the environment. The task of precisely determining the energies involved in molecular pathways is a daunting one, due to the many endogenous and exogenous factors that are beyond control, molecular heterogeneity being a relevant one. In this setting, inference may offer a powerful approach to deepening our knowledge of biological systems. The enormous amount of information needed to describe biological systems and the sensitivity of these systems to various noise sources and heterogeneous disorder may favor inference reasoning over deductive knowledge. Inference represents knowledge that is not accessible by direct measurement; it is knowledge consistently implied by the validity of physical laws. Whether inference is the correct tool with which to face the new challenges in biology remains to be seen.

Closing remarks Energy, entropy, and information are the three main driving forces underlying the remodeling of biological matter. While thermodynamic processes in ordinary matter are driven by free-energy minimization (i.e., competition between energy and entropy), living matter seems to be predominantly governed by information flows across many different organizational and stratification levels, leading to complex integrated biological cells and organisms and resulting in what has been dubbed as “molecular vitalism” [14]. How the non-equilibrium physics of small systems can contribute to furthering our understanding of the marvelous attributes of living matter CONTRIBUTIONS to SCIENCE 11:137-146 (2015)

Physics of small systems

remains to be determined. Nonetheless, the experiments and theories developed in statistical physics over the past decades have demonstrated the prominent role of information, a quantity that physicists generally identify with entropy but which may be a more general one when used to explain the emergent complexity of biological matter. One of the most appealing features of the Jarzynski equality and fluctuation relations is that they allow us to recover the second law of thermodynamics as a particular case of a more general mathematical equality. This raises the intriguing question whether the second law is ultimately a conservation law rather than an inequality, paralleling the mathematical equality represented by the first law of thermodynamics. From our perspective statistical mechanics and biophysics are intimately related disciplines; in the latter, biological systems are used to investigate non-equilibrium phenomena and, perhaps, to uncover new physical laws as well. The non-equilibrium physics of small systems might thereby represent the first step in finally unraveling new secrets of nature. Competing interests. None declared.

References 1. Alemany A, Mossa A, Junier I, Ritort F (2012) Experimental free-energy measurements of kinetic molecular states using fluctuation theorems. Nature Physics 8:688-694 doi:10.1038/nphys2375 2. Alemany A, Ribezzi-Crivellari M, Ritort F (2015) From free energy measurements to thermodynamic inference in non-equilibrium small systems. New J. Phys. 17:075009 3. Bennett CH (1987) Demons, engines and the second law. Scientific American 257:108-116 4. Bustamante C, Liphardt J, Ritort F (2005) The non-equilibrium thermodynamics of small systems. Phys. Today 58:43-48

5. Collin D, Ritort F, Jarzynski C, Smith SB, Tinoco I, Bustamante C (2005) Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies. Nature 437:231-234 doi:10.1038/nature04061 6. Crisanti A, Picco M, Ritort F (2013) Fluctuation relation for weakly ergodic aging systems. Phys. Rev. Lett. 110:080601 doi:10.1103/PhysRevLett.110.080601 7. Crooks GE (1999) Entropy production fluctuation theorem and the nonequilibrium work relation for free-energy differences. Phys. Rev. E 60:2721-2726 8. Dieterich E, Camunas-Soler J, Ribezzi-Crivellari M, Seifert M, Ritort F (2015) Single-molecule measurement of the effective temperature in non-equilibrium steady states. Nat. Phys. 11:971-977 9. Evans DJ, Searles D (2002) The fluctuation theorem. Adv. Phys. 51:15291585 10. Gallavotti G, Cohen EGD (1995) Dynamical ensembles in non-equilibrium statistical mechanics. Phys. Rev. Lett. 74:2694-2697 doi:10.1103/ PhysRevLett.74.2694 11. Goldenfeld N, Woese C (2011) Life is Physics. Annu Rev Condens. Matter Phys. 2:375-399 12. Jarzynski C (1997) Non-equilibrium equality for free-energy differences. Phys Rev Lett 78: 2690 doi:10.1103/PhysRevLett.78.2690 13. Jarzynski C (2011) Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale. Ann Rev Condens Matter Phys. 2:329-351 doi:10.1146/annurev-conmatphys-062910-140506 14. Kirschner M, Gerhart J, Mitchison T (2000) Molecular “vitalism”. Cell 100:79-88 doi:10.1016/S0092-8674(00)81685-2 15. Pross A (2012) What is life? Oxford University Press 16. Ribezzi-Crivellari M, Ritort F (2014) Free-energy inference from partial work measurements in small systems. Proc Natl Acad Sci 111:E33863394 doi:10.1073/pnas.1320006111 17. Ritort F (2006) Single molecule experiments in biological physics: methods and applications. Journal of Physics C (Condensed Matter) 18: R531R583 doi:10.1088/0953-8984/18/32/R01 18. Ritort F (2008) Non-equilibrium fluctuations in small systems: from physics to biology. Adv Chem Phys 137:31-123 19. Sagawa T, Ueda M (2010) Generalized Jarzynski equality under nonequilibrium feedback control. Phys.Rev.Lett. 104:090602 doi:10.1103/ PhysRevLett.104.090602 20. Schrödinger E (1944) What is life? Cambridge University Press 21. Seifert U (2012) Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep Prog Phys 75:126001 doi:10.1088/00344885/75/12/126001

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CONTRIB SCI 11:147-151 (2015) doi:10.2436/20.7010.01.225

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Mesoscopic non-equilibrium thermodynamics

J. Miguel Rubi Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona, Catalonia

Summary. Autonomous microsystems, such as biomolecules, molecular motors, nanomotors, and active particles, are functionally dependent on nanoscale energy conversion mechanisms. The non-equilibrium processes taking place in those systems are strongly influenced by the presence of fluctuations. Contributions to the free energy that vanish in the infinite particle number limit cannot be neglected and may exert an important influence in the dynamics of the system. We show that, in spite of these features, non-equilibrium thermodynamics applies. A rigorous theoretical foundation that accounts for the statistical nature of mesoscale systems over short time scales, “mesoscopic non-equilibrium thermodynamics”, is currently being developed and offers a promising framework for interpreting future experiments in chemistry and biochemistry. [Contrib Sci 11(2): 147-151 (2015)] Correspondence: J. Miguel Rubi Departament de Física de la Matèria Condensada Universitat de Barcelona Martí i Franquès, 1 08028 Barcelona, Catalonia Tel.: +34-934021168 E-mail: mrubi@ub.edu

Non-equilibrium thermodynamics Irreversible processes taking place in large-scale systems are well described by non-equilibrium thermodynamics [1]. The theory applies to a general description of these systems but ignores their molecular nature and assumes that they behave as a continuum medium. Earlier efforts to develop this theory started from the concept of local equilibrium states. Although a system may not be in equilibrium,

individual pieces of it can be. For instance, imagine stirring a cocktail with a swizzle stick. The equilibrium is disturbed by the motion of the stick but can still be found if one looks closely at small pockets of fluid, which retain their internal coherence. These small regions are able to reach equilibrium if the forces acting on the system are not too large and if its properties do not change by large amounts over small distances. Concepts such as temperature and entropy apply to these islands of equilibrium, although their numerical

Keywords: non-equilibrium · nanoscale-mesoscale · fluctuations · thermodynamics · free energy ISSN (print): 1575-6343 e-ISSN: 2013-410X

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Mesoscopic non-equilibrium thermodynamics

values may vary from island to island. For instance, when How small is “small”? the ends of a metal bar are heated, heat flows through the bar toward the other end. The temperature difference With the development of experimental techniques, such as between the ends of the bar acts as a force driving the the X-ray laser, that are able to describe processes taking place heat flow, or flux, along the bar. A similar phenomenon over very short time scales (of the order of the femtosecond) occurs with a drop of ink in water. The difference in the [4], it has become possible to describe intimate mechanisms ink concentration is the driving force that makes the ink in proteins, biomolecules, molecular motors, self-assembled invade the host liquid until it becomes uniformly colored. structures, and nanocolloids. But to reach this limit raises These forces are linear: the heat flux is proportional to the question: is thermodynamics, originally formulated to the temperature difference and the particle flux to the deal with macroscopic systems containing a large number of concentration difference. This proportionality holds even molecules, even valid at these small observational scales? when the forces acting on the system are strong. Also, in Thermodynamics applies to infinite systems containing many turbulent flows, the internal stresses in the fluid are an infinite number of particles with a constant density—a proportional to the velocity gradients. For these cases, Lars situation referred to as the thermodynamic limit [5]. Basic Onsager (1903–1976) and others formulated a theory of concepts, such as heat, temperature, entropy, free energy, and non-equilibrium thermodynamics and showed that the thermodynamic principles, have been proposed for systems second law of thermodynamics continues to apply [2]. that fulfill this limit. Non-equilibrium thermodynamics Under these conditions, the description of the system does also considers these concepts and assimilates the system not depend on its size, and scaling up the size does not lead to into a continuum medium without molecular structure. new behaviors. Irreversible processes taking place in a small In this description, the volume elements are considered vessel or in an industrial plant can therefore be analyzed using thermodynamic systems since they still contain a large the same conceptual framework. But when scaling down the number of particles. size of the system, its molecular nature becomes manifest; In many cases, the system is characterized in terms of coarse-graining is no longer valid due to fluctuations in the its size. Macroscopic systems that fulfill the thermodynamic quantities used in its description. A very different scenario is limit obey the laws of thermodynamics and can thus be then reached, in which non-equilibrium thermodynamics only studied using a thermodynamics approach. One would then describe the evolution of the mean values of those quantities be tempted to think that small-scale systems should obey but does not completely characterize their actual values. In other types of laws. But a classification based on size can systems such as small clusters and biomolecules, fluctuations be incomplete or even misleading. When one talks of small that are small in large-scale systems can be so large that they systems a key question arises: how small is “small?” become the dominant factor in their evolution. The so-called Small-scale systems can be better described through mesostructures, defined as entities with sizes in between the extensivity (a mathematical property) of the relevant those of particles and objects, are examples of small systems variables. Consider a system with a given free energy resulting undergoing assembly, impingement, and pattern formation from the contribution of the different degrees of freedom. processes in which fluctuations may play a very important A reduction of the size of the system entails a diminution role. Knowledge of the functionality of molecular motors of the number of degrees of freedom. The thermodynamic (the small engines present in many biological systems) and description of small systems exhibits peculiar features, the ability to manipulate matter at small scales to improve as pointed out in the classical book by Terrell Leslie Hill its performance—which constitutes the basic objective of (1917–2014) Thermodynamics of small systems [5]. Since nanoscience and nanotechnology—require a thermodynamic in those systems the number of particles is not infinite, the characterization of the system [3]. free energy may contain contributions not present when Non-equilibrium thermodynamics is restricted to the the number of particles becomes very large. For example, linear domain of fluxes and thermodynamic forces [1]; in a small cluster composed of N particles, the free energy consequently, it is unable to completely characterize F contains, in addition to the volume term, a surface nonlinear transformations, in particular activated processes contribution proportional to N 2/3 . It can be expressed as such as chemical reactions, nucleation, or self-assembly = F Nf (T , P ) + N 2/3 g (T , P ) , where f is the free energy per processes and, in general, abrupt changes in the state of a unit volume and g is a function of the temperature T and system described by a nonlinear equation. the pressure P . When the number of particles is very large www.cat-science.cat

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and the system becomes of macroscopic size, the surface contribution is negligible and the free energy is simply Nf .

Far-from-equilibrium microsystems The evolution of the state of a system is the result of the actuation of two competing factors. The presence of external forces, or imposed gradients of the relevant quantities, drives the system from an initial equilibrium state to a non-equilibrium state. Collisions between particles tend to restore the initial equilibrium situation. A force applied at the ends of a RNA molecule progressively breaks down the bonds of the molecule such that it adopts different conformations [6]. The addition of one of the components of a biochemical cycle alters the chemical equilibrium of a system, removing the state leading to a stationary state [7]. When the force applied is very intense and the collisions are unable to restore the initial equilibrium conditions, is it possible to thermodynamically describe this system? An analysis based on the solution of the Boltzmann equation for reactive gases by means of a Chapmann-Enskog expansion [8] enables one to describe the transition from equilibrium to non-equilibrium states. The law of mass action, the expression of entropy production as the product of the reaction rate and the affinity, and the detailed-balance principle cease to be valid when the system is very far from equilibrium. But there are cases in which the departures from equilibrium are less drastic. Application of a temperature difference at the ends of a metal bar induces a heat flux along the bar that is linear with respect to the temperature gradient. This linearity holds even for very large gradients. Experiments performed in a nanomotor that moves along a carbon nanotube under the influence of a temperature difference generated by an external current [9] have shown that, in spite of the very large gradients coming into play, of the order of 1 K/nm, the driving force is still linear along the gradient. This phenomenon also occurs regarding the orientation of non-polar molecules by means of a thermal gradient [10]. Quantities such as the temperature or the pressure of a system that are directly related to the collisions between the constituent particles relax very fast whereas the density associated with conformational changes involving many particles relax much slower. Molecular dynamic simulations have shown that temperature relaxation in a protein, determined from the average kinetic energy of www.cat-science.cat

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the constituent atoms, takes place in the range of tens of picoseconds [11]. Thus, although the system is very small, of nanometric size, because it contains a sufficiently large number of particles it reaches local equilibrium very rapidly, which allows its thermodynamic description. The transition towards non-equilibrium states relies on the nature of the irreversible process that takes place. A fluid at rest starts to move under the action of a pressure difference although the perturbation caused by that difference may be very small. By contrast, a chemical reaction needs a minimum amount of energy to proceed. Whereas the linear approximation is valid for many transport processes, such as heat conduction and mass diffusion, even in the presence of large gradients [12], it is not appropriate for activated processes in which the system immediately enters the nonlinear domain. Non-equilibrium thermodynamics does not provides a complete description of activated processes. But, is there a general framework within which both types of processes can be treated?

Mesoscopic non-equilibrium thermoÂ dynamics When the system is very small, the chaotic jumble of molecular motions that dictates its behavior causes the systemâ&#x20AC;&#x2122;s properties to vary wildly over short distances and time intervals. Processes taking place in small systems, such as the condensation of water vapor and the transport of ions through a protein channel in a cell membrane, are dominated by such fluctuations, during which temperature and entropy cease to be well-defined quantities. Does the failure of the non-equilibrium thermodynamics theory in these instances imply the failure of the second law? We have shown that many of the problems are eliminated with a change of perspective [13]. Our perception of abruptness depends on the time scale we use to observe the respective processes. If we analyze one of the seemingly instantaneous chemical processes in slow motion, we would see a gradual transformation, as if we were watching a pat of butter melting in the sun. When the process is viewed frame by frame, the changes are not abrupt. The trick is to track the intermediate stages of the reaction using a new set of variables beyond those of classical thermodynamics (Fig. 1). Within this expanded framework, the system remains in local thermodynamic equilibrium throughout the process. These additional variables enrich the behavior of the system. They define a landscape of energy that the system rambles through CONTRIBUTIONS to SCIENCE 11:147-151 (2015)

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Mesoscopic non-equilibrium thermodynamics

Fig. 1. Activated process of a non-linear nature can be treated using a mesoscopic non-equilibrium thermodynamics approach. (A) Self-assembly. (B) Protein folding. (C) Over short-time scales, the transition can be viewed as diffusion through a potential barrier G that depends on an internal coordinate.

like a backpacker in the mountains. Valleys correspond to a dip in energy, sometimes involving molecular chaos, other times molecular order. The system can settle into one valley and then be kicked by external forces into another. If it is in the grasp of chaos, it can break away from disorder and find order, or vice-versa. Important biochemical reactions that require microseconds to milliseconds to run to completion are composed of more elementary steps on the sub-nanosecond scale. Ultrafast events, on the time scale of femptoseconds to picoseconds, occur in photo-initiated reactions, such as photosynthesis, vision, and DNA repair, and as fluctuations during slower reactions, which effectively pre-sample the relevant energy landscape. Next, consider the problem of fluctuations. Does thermodynamics fail when systems are excessively small? A simple example shows that the answer is no. If we toss a coin only a few times, it could be that, by chance, we get a series of heads. But if we flip the coin many times, the result www.cat-science.cat

reliably approaches an average. Nature flips coins quite often. A few particles moving around in a container collide only occasionally and can maintain large velocity differences among themselves. But even in a seemingly small system, the number of particles is much larger, so collisions are much more frequent and the speed of the particles is brought down to an average (if slightly fluctuating) value. Although a few isolated events may show completely unpredictable behavior, a multitude of events shows a certain regularity. Therefore, quantities such as density can fluctuate but remain predictable overall. For this reason, the second law continues to rule over the world of the small. In general, the reduction of the observational time and length scales of a system usually entails an increase in the number of non-equilibrated degrees of freedom [13]. Those degrees of freedom may, for example, represent the position and velocity of a single particle, the orientation of a magnetic moment, the size of a macromolecule, the number of particles that integrate self-assembled structures, or any internal coordinate or order parameter whose values define the state of the system in a phase space. The characterization at the mesoscopic level of the state of the system follows from the knowledge of the probability distribution defined in that space. To bring the system to a state characterized by a given value of the coordinates, we need to exert a certain amount of work, which is related to the different thermodynamic potentials by imposing the constraints that define those potentials. For instance, for the case of constant temperature, volume, and number of particles, the minimum work corresponds to the Helmholtz free energy. The theory assumes local equilibrium in the space of the degrees of freedom, with a probability distribution that undergoes a diffusion process in the space of the internal degrees of freedom. The probability current is obtained from the entropy production in the coordinate space, which follows from the Gibbs entropyâ&#x20AC;&#x201D;in the same way as Fickâ&#x20AC;&#x2122;s law is obtained in non-equilibrium thermodynamics [1]. The kinetic equation for the evolution of the probability density follows from the probability conservation law after substituting the obtained expression for the diffusion current. Mesoscopic non-equilibrium thermodynamics provides a simple and direct method to determine the dynamics of a system from its equilibrium properties, obtained from the equilibrium probability density. It has been used to analyze the many different non-equilibrium processes taking place in small-scale systems. In particular, the theory has been used to study nucleation and self-assembly processes [14], transport 150

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through ion channels with the presence of entropic forces [15], polymer crystallization in the presence of gradients [16], active transport in biological membranes [17], diffusion in confined systems [18], and near-field thermodynamics [19]. The formulation of the theory and its applications are described in [10,20]. Acknowledgements. I would like to thank J.M.G. Vilar, D. Reguera and A. Pérez-Madrid for fruitful discussions. I acknowledge financial support from Generalitat de Catalunya under the program ICREA Academia. Competing interests. None declared.

References 1. Barreiro A, Rurali R, Hernández ER, Moser J, Pichler T, Forró L, Bachtold A (2008) Subnanometer motion of cargoes driven by thermal gradients along carbon nanotubes. Science 320:775-778 doi:10.1126/ science.1155559 2. Callen HB (1985) Thermodynamics and an Introduction to Thermostatistics. John Wiley and Sons, New York 3. de Groot SR, Mazur P (1984) Non-equilibrium thermodynamics. Dover, New York 4. Hill TL (1989) Free energy transduction and biochemical cycle kinetics. Dover, New York 5. Hill TL (1994) Thermodynamics of small systems. Ed. Dover, New York doi:10.1007/978-1-4612-3558-3 6. Kjelstrup S, Rubi JM, Bedeaux D (2005) Active transport: a kinetic description based on thermodynamic grounds. J Theor Biol 234:7-12 doi:10.1016/j.jtbi.2004.11.001 7. Latella I, Pérez-Madrid A, Lapas LC, Rubi JM (2014) Near-field thermodynamics: Useful work, efficiency, and energy harvesting. J Appl Phys 115:124307 doi:10.1063/1.4869744

8. Lervik A, Bresme F, Kjelstrup S, Bedeaux D, Rubi JM (2010) Heat transfer in protein-water interfaces. Phys Chem Chem Phys 12:1610-1617 doi:10.1039/b918607g 9. Paul Sherrer Institute (2009) Ultrafast phenomena at the nanoscale: Science opportunities at the SwissFEL X-ray Laser. PSI Bericht 09-10 B.D. Patterson (ed) 10. Reguera D, Rubi JM, Vilar JMG (2005) The mesoscopic dynamics of thermodynamic systems. J Phys Chem B 109:21502-21515 doi:10.1021/ jp052904i 11. Reguera D, Luque A, Burada,PS, Schmid G, Rubi JM, Hänggi P (2012) Entropic splitter for particle separation. Phys Rev Lett 108:020604(1-4) doi:10.1103/PhysRevLett.108.020604 12. Reguera D, Rubi JM (2001) Kinetic equations for diffusion in the presence of entropic barriers. Phys Rev E 64:061106 (1-8) doi:10.1103/ PhysRevE.64.061106 13. Reguera D, Rubi JM (2001) Non-equilibrium translational-rotational effects in nucleation. J Chem Phys 115:7100-7106 14. Reguera D, Rubi JM (2003) Nucleation in inhomogeneous media. II. Nucleation in a shear flow. J Chem Phys 119:9888-9893 doi:10.1063/1.1614777 15. Römer F, Bresme F, Muscatello J, Bedeaux D, Rubi JM (2012) Thermomolecular orientation of nonpolar fluids. Phys Rev Lett 108:105901(1-4) doi:10.1103/PhysRevLett.108.105901 16. Ross J, Mazur P (1961) Some deductions from a formal statistical mechanical theory of chemical kinetics. J Chem Phys 35:19-28 doi:10.1063/1.1731889 17. Rubi JM (2008) The long arm of the second law. Sci Am 299:62-67 doi:10.1038/scientificamerican1108-62 18. Rubi JM, Bedeaux D, Kjelstrup S (2006) Thermodynamics for singlemolecule stretching experiments. J Phys Chem B 110:12733-12737 19. Santamaria-Holek I, Reguera D, Rubi JM (2013) Carbon-nanotube-based motor driven by a thermal gradient. J Phys Chem C Phys 117(6):31093113 doi:10.1021/jp311028e 20. Vilar JMG, Rubi JM (2001) Thermodynamics “beyond” local equilibrium. Proc Nat Acad Sci 98:11081-11084 doi:10.1073/pnas.191360398

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CONTRIB SCI 11:153-162 (2015) doi:10.2436/20.7010.01.226

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Avalanche dynamics in driven materials

Jordi Baró1, Antoni Planes1, Eduard Vives1,2* Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona, Catalonia. 2Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Catalonia 1

*Correspondence: Eduard Vives Departament de Física de la Matèria Condensada Universitat de Barcelona Martí i Franquès, 1 08028 Barcelona, Catalonia Tel. +34-934021586 E-mail: eduard@ecm.ub.edu

Summary. Phase transitions in equilibrium have traditionally been classified as first-order or second-order (critical). The essential difference between the two is whether the order parameter exhibits a discontinuous or a continuous behavior at the transition point. In the second half of the last century, second-order phase transitions were extensively studied. Concepts such as the lack of characteristic scales, divergence of correlation length, criticality, critical exponents, and universality were established. Very powerful techniques, such as the renormalization group approach, were developed as well. In the last 20 years, the focus has been on first-order phase transitions (FOPTs). Theoretically, systems slowly driven across a FOPT exhibit an equilibrium behavior with a single discontinuity of the order parameter. However, even when driven very slowly, they often evolve following a non-equilibrium metastable trajectory. This trajectory, instead of consisting of a single macroscopic discontinuity, exhibits many small discontinuities, or “avalanches,” with sizes ranging from the microscopic to the macroscopic. This behavior is an example of the “avalanche dynamics” discussed herein. The essential difference that distinguishes this behavior from other non-equilibrium intermittent dynamics is the lack of characteristic scales. This is why the term “critical” is applied to these systems, despite the fact that they undergo a FOPT. For this phenomenon to occur, two ingredients are needed: quenched-in disorder and athermal behavior, a consequence of low thermal fluctuations. However, “avalanche dynamics” is not limited to systems with FOPTs but may also occur in heterogeneous systems irreversibly driven by an instability. A second example discussed in this article is the case of the mechanical failure of porous materials under compression, for which disorder and athermal behavior play crucial roles. [Contrib Sci 11(2): 153-162 (2015)]

Matter is organized as structures with different properties on both the macroscopic and the mesoscopic scale, the so-called phases. The relevant macroscopic properties are typically measured by extensive thermodynamic variables, such as volume (V), magnetization (M), polarization (P), and strain (ε).

A phase transition consists of a microscopic reorganization that alters some of these macroscopic properties of matter [12,23]. This reorganization can be interpreted as competition between three terms: internal energy ( U ), entropy ( S ), and the energy due to external forces. The three terms de-

Keywords: non-equilibrium first-order phase transitions · metastable behavior · hysteresis · avalanche dynamics · compression of porous materials ISSN (print): 1575-6343 e-ISSN: 2013-410X

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fine the Gibbs free energy ( G ). For instance, for a system that interacts with the surroundings only through hydrostatic pressure ( p ) and temperature ( T ), G can be written as: G = U ‒ TS ‒ (‒pV)

(1)

In general, there can be many terms like the one defined by Eq. (1), which is related to the work performed by external forces. Such terms are always written as the product of an intensive variable, whether pressure (‒p), magnetic field ( H ), electric field (E), or stress (σ), and its conjugated extensive variable, i.e.: ( − pV ), HM , EP , or σε . Under slowly changing external conditions, systems relax to the equilibrium state, at which G is at an absolute minimum. For instance, an increase of T (at constant p ) may cause the system to choose a state with higher entropy ( S ) and lower internal energy (U ). These changes usually lead to a smooth response of the system, except at first-order phase transitions (FOPTs), as in the case of liquid–vapor phase transitions, in which entropy, energy, and volume undergo a sharp macroscopic change (related to the exchange of latent heat). A second example is an increase of p at constant T , which may cause the system to choose a state with a lower volume (V ) and lower entropy ( S ), as occurs abruptly during condensation. Therefore, FOPTs can be induced by changes in temperature or external forces. FOPTs are also seen in the singular behavior of thermodynamic response functions related to derivatives of extensive properties: specific heat ( dU / dT ), compressibility ( dV / dT ), susceptibilities ( dM / dH and dP / dE ), and elastic moduli ( d ε / dσ ). FOPTs are widespread in nature. Some are very familiar, such as the freezing and boiling of water; others are related to changes in the crystallographic structure of solids or to the magnetic field induced magnetization switching of iron (below a given temperature, referred to as the Curie temperature, Tc ). Some phase transitions involve more exotic properties, such as superconductivity or superfluidity. Future technological innovation will require an understanding of both natural and man-made materials and of the methods to achieve their control. The development of increasingly smaller sensors and actuators awaits detailed knowledge of the equilibrium and out-of-equilibrium behaviors of materials driven by external forces and fields. Materials that respond with large variations in their order parameter when driven by relatively small forces are precisely those that exhibit a FOPT. Historically, the first motors were based on the expansion and contraction of gases. Nevertheless, motors only became truly powerful when the liquid-vapor www.cat-science.cat

FOPT was exploited in the steam engine. Similarly, today’s sensors and actuators are designed using ferroic or multiferroic materials involving FOPTs. Thus, both knowledge of the discontinuous dynamics of these systems during the transition and the ability to control this process are of extreme importance in the development of applications.

Landau theory of FOPTs A simple framework that provides an understanding of the occurrence of FOPTs is the Landau theory [10]. It also provides a basic description of the hysteretic behavior of athermal systems and of the occurrence of avalanche dynamics in disordered-athermal systems. The theory was originally developed for continuous phase transitions but it was soon realized that, in some cases, it was also suitable to describe FOPTs. Firstly, the relevant order parameter for the transition, i.e., the extensive variable that exhibits macroscopic discontinuity, must be identified. Secondly, the free energy must be expanded in power series in the order parameter, including only the terms allowed by the symmetry of the problem. As an example, the case of a uniaxial ferromagnet under an applied external field is shown in Fig. 1. The two intensive control variables are the temperature ( T ) and the external field along the z-axis ( H ). Here, the order parameter is the magnetization ( M ), which exhibits a discontinuity at the transition, below the Curie temperature ( Tc ). The phase diagram (Fig. 1, left) shows a FOPT (dashed line) exactly at H = 0. Due to symmetry reasons, the two ferromagnetic phases, with M > 0 and M < 0, are completely equivalent when subjected to an inversion operation. The FOPT line ends at the critical point, at Tc . Let us assume that we drive the system at low enough temperatures, by changing H from a very large to a very low value, as indicated by the blue arrow in the phase diagram of Fig. 1. The expansion of the Gibbs energy in terms of the magnetization will be: G ( M= ) AM 2 + bM 4 + ... − HM

(2)

Note that, taking into account symmetry considerations, in Eq. (2) only even terms are allowed, except for the term − HM already discussed in the previous section. This is because symmetry between the two ferromagnetic phases is broken in the presence of an external field. In general, the coefficients in the expansion A , b … depend on temperature. To a first approximation these dependences can be ig154

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Fig. 1. (A) Schematic representation of the phase diagram of a uniaxial ferromagnet under athermal conditions. The dashed line is the equilibrium first-order phase transition (FOPT); the continuous lines indicates the metastability limits, in which the transition occurs in the perfectly athermal case. The two colors (blue and red) correspond to the two directions of the field variation: from +∞ to –∞ (blue) or from –∞ to +∞ (red). (B) Free energy [G(M)] for different field values as indicated by the legend. The blue dot represents the metastable evolution of the system when the field is changed from +∞ to –∞. (C) Magnetization vs. field [M(H)], in which hysteresis depends on the direction of the field variation. The dashed lines indicate the position of the jumps when a certain degree of thermal fluctuations is allowed.

nored, except for parameter A . This first term in the expansion is expected to vary rapidly near the critical temperature Tc . Its dependence on temperature is usually assumed to be of the form: = A a (T − Tc )

(3)

Note that in Eq. (3), A vanishes exactly at Tc ; it is negative for T < Tc and positive for T > Tc . The panel in the center of Fig. 1 shows the behavior of the Landau free energy [ G ( M ) ] for values of the field ( H ) that range from very large and positive (Fig. 1, top) to very large and negative (Fig. 1, bottom). Note the existence of two wells precisely due to the fact that T < Tc and A is negative. Above Tc ( A > 0), the Landau free energy will exhibit a unique well and the FOPT will not occur. Note also that the effect of the external field [given by the last linear term in the expansion−(HM ) is simply to tilt the double-well function. As explained in the Introduction, the system chooses the equilibrium state that corresponds to the global minimum of G ( M ) . For positive fields the equilibrium value of M will correspond to the well on the right hand side ( M > 0) and for negative fields to the well on the left hand side ( M < 0). At H = 0, the two wells have the same depth at symmetric positions ± M 0 . The system will therefore display a discontinuous change of magnetization ( ∆M = 2 M 0 ) as the field www.cat-science.cat

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moves from positive to negative values. The panel on the right in Fig. 1 shows the corresponding behavior of the magnetization as discussed. The equilibrium trajectory is shown by a thick line, displaying the discontinuity at H = 0. At exactly H = 0, the two phases (corresponding to the two symmetric wells) will coexist, giving rise to a heterogeneous microstructure. This simple version of the Landau theory cannot provide a detailed description of the coexisting state. The details will be determined by the system’s shape and surfaces and by interaction terms such as those of demagnetizing fields (in ferromagnets) or elastic forces (in ferroelastics). An important factor that must be taken into account in order to understand avalanche dynamics is the role of thermal fluctuations. For a system to follow an equilibrium trajectory, microscopic thermal fluctuations are needed because they allow temporal and spatial deviations of the energy from its exact value at the minimum. The system, therefore, is able to explore the phase space and jump over energetic barriers. This ability is essential in the vicinity of a FOPT for the system to abandon a local minimum and jump into a global minimum. This can be understood from the central panel in Fig. 1. As soon as the field ( H ) becomes infinitesimally negative, the system should jump towards the well on the left hand side. There are several scenarios in which thermal fluctuations CONTRIBUTIONS to SCIENCE 11:153-162 (2015)

Avalanche dynamics

become irrelevant. In some cases the reasons are kinetic: the system is externally driven very rapidly compared to the time needed to explore the phase space. In other cases, the existence of long-range forces (of an elastic, magnetic, or electrical nature) may enormously increase the size of the energetic barriers separating the minima of the free energy. In both cases the consequence is that the system will not follow an equilibrium path and will remain trapped in local (metastable) minima, as shown by the blue dots in the central panel of Fig. 1. This gives rise to hysteretic (out-of-equilibrium) behavior [3], an effect well known to occur in many FOPTs. For instance, liquid water at atmospheric pressure may become undercooled below 0ยบC and freeze (phase change) at much lower temperatures. The right-hand panel in Fig. 1 also shows the trajectories that the system will follow in the case of its having a completely athermal character (thin continuous lines with arrows). In this case, the transition to the absolute minimum only occurs when the local minima disappear. Note that the transitions occur at values of the field ( H ) that depend on whether the field is increased or decreased. The difference between these two fields provides a measure of hysteresis. In general, if the athermal character is not strictly ideal, the system may display the transition when the local minimum is shallow enough. In this case hysteresis will depend on the rate at which the external field is driven (dashed lines in the right-hand panel of Fig. 1). In the ideal athermal case, hysteresis becomes rate independent. In the phase diagram, the transitions will not occur when the equilibrium FOPT line at H = 0 is crossed but, instead, when the so-called metastability limits are exceeded. In the left-hand panel in Fig. 1, the continuous blue and red lines show the metastability limits corresponding to trajectories that decrease or increase the field, respectively. We now discuss the role of the second factor that is needwww.cat-science.cat

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Fig. 2. (A) Schematic representation of the free energy [G(M)] for a system with a FOPT and quenched disorder. (B) Corresponding hysteresis loops exhibiting avalanches associated with the jumps from one metastable minimum to the next.

ed for avalanche dynamics to occur: disorder. Real materials are never perfect. Even crystals exhibit inhomogeneities in the form of dislocations, vacancies, lack of stoichiometry, and impurities. One of the consequences of disorder is that the free energy landscape of the system is much more complex than the free energy in the case of only two wells, as depicted in Fig. 1. Different parts of the system may exhibit different degrees of metastability with correspondingly different values of the local magnetizations. Of course, under these conditions, the correct description of the system cannot be provided with a free energy function of a simple scalar order parameter such as the magnetization ( M ), but requires a description using a functional of an order parameter field [ m( x) ], such that its integral over the whole system volume gives M . The free energy functional will then include terms coupling the local disorder with m( x) . Nevertheless, for the purpose of simplicity, such as the example in the central panel of Fig. 1 we project the free energy functional into a function of the average magnetization (< M >). Therefore, the consequence of the existence of inhomogeneities is that the free energy displays multiple small wells separated by free energy barriers between the two phases with positive and negative magnetization, as shown in the left-hand panel of Fig. 2. If we now combine both the athermal behavior and the existence of disorder, the metastable trajectory of the system will consist of a series of small jumps which take place every time the system relaxes from a local metastable minimum and falls into a new one. A schematic representation is shown in the right-hand panel of Fig. 2. These jumps are avalanches. Their characteristic property is that, even if the system is driven very smoothly, the response of the system to the applied field is intermittent, in which periods without activity alternate with periods characterized by very fast changes in the order parameter. 156

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Fig. 3. Direct measurement of the avalanches in the hysteresis loop in a Pr0.6Ca0.4Mn0.96Ga0.04O3 sample. The botton panel shows an enlargement of the region indicated by the dotted rectangle. (From [9]).

Figure 4 shows two examples of avalanches detected in ferromagnets [20] and ferroelastic materials. A similar behavior occurs in superconductors (vortex avalanches) [8] and ferroelectric materials (polarization avalanches). Avalanche sequences are often seen to be reproducible when the system is cycled many times through the transition. This indicates that the disorder is rather stationary (quenched disorder) and the system is highly athermal. However, as we discuss below, this is not always true. The experimental observation that has allowed avalanche dynamics to be classified within the paradigm of out-of-equilibrium critical phenomena [19] is related to the statistical analysis of the properties of individual avalanches. For each jump event different properties can be measured, such as the amount of energy relaxed, the change in the order parameter, the duration of the avalanche, and the avalanche size. When these properties are measured for a large number of avalanches (for instance, those recorded during the whole transition), there is a significant difference compared to the statistics of standard measurements of physical properties. In general, due to the central limit theorem, the expectation is that the physical measurements will be Gaussian distributed, centered around an average value and with a certain standard deviation. However, avalanche properties are distributed according to probability densities with much fatter tails. Their distributions are typically described using a power-law probability density (Eq. 4): p ( x ) ~ x −α

Avalanche dynamics In general, avalanches are microscopic and localized in small regions of the material. Direct measurement of the discontinuities in the hysteresis loops is difficult, but it has been achieved, for instance, in the case of ferromagnets [9]. An example is given in Fig. 3. More generally, avalanches are detected by experimental methods that are sensitive to the time derivative of the order parameter. In the case of ferromagnetic materials, fast variations of magnetization can be detected by measuring the induced voltage in a coil. The avalanches, in this case, produce so-called Barkhausen noise [7]. In the example of ferroelastic materials exhibiting structural FOPTs for which the order parameter is a component of the strain tensor, the avalanches can be detected by the acoustic emission technique [21], in which rapid variations of strain in certain regions of the material induce the emission of ultrasounds that can be detected on the surface by the appropriate transducers. www.cat-science.cat

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(4)

This indicates that the average values of such properties are meaningless in characterizing the system (i.e., they depend on the observation window) and that the only characteristic parameter is the power-law exponent α . Examples of such distributions for the case of Barkhausen noise in ferromagnets [6] or acoustic emission in ferroelastic materials [5] are shown in Fig. 5. Note that different materials exhibit the same values of the power-law exponents (universality) and that families (or classes) of materials can be identified. This lack of characteristic scales indicates that systems displaying avalanche dynamics diverge in their spatial and/or temporal correlations, overriding any other microscopically relevant scale. This is a clear indication that, in a generalized parameter space (using renormalization group language), the systems are located in the close vicinity of a dynamic critical point and the critical exponents are expected to be universal. This hypothesis has been reinforced in studies of the evolution of the avalanche size distribution when some athermal CONTRIBUTIONS to SCIENCE 11:153-162 (2015)

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Avalanche dynamics

Fig. 4. Indirect measurement of avalanches by (A) Barkhausen noise detection in a ferromagnetic transition in a Fe-Ni-Co alloy [20] and (B) acoustic emission detection in a ferroelastic (martensitic) transition in a Cu-Zn-Al alloy.

ferroelastic systems are cycled many times across a FOPT [14]. An example of the experimental results obtained with a Cu-Al-Mn alloy exhibiting a structural phase transition is shown in Fig. 6. For some of these materials, the disorder (dislocations, etc.) is not totally quenched in. When the system is driven through the transition, simultaneous with the avalanche response, the disorder is slightly modified. Thus, the avalanche sequences are not exactly reproducible when one cycle is compared to the next. It has been observed that as-cast samples that cross the FOPT for the first time do not

exhibit a power-law distribution of avalanche sizes, but rather an exponentially dampened distribution (Eq. 5): −α − ë x (5) p( x) ~ x e In the subsequent cycles through the transition, the cutoff parameter λ evolves towards zero, as shown in Fig. 6. After a certain number of cycles, the system reaches the true power-law distribution of avalanches characterized by the exponent α , which becomes quite stable. In other words,

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Fig. 5. Distribution of avalanche sizes in (A) ferromagnetic transitions [6] and (B) ferroelastic (martensitic) transitions [5]. The histograms, in log-log scale, show the lack of a characteristic scale (power-law behavior). The data were obtained from different alloys and thus reveal a certain degree of universality.

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during the first few cycles, the system self-organizes its disorder and finds an optimal non-equilibrium path (avalanche sequence) that exhibits “critical” properties. This confirms the attractive character of the “critical” state.

Models Apart from the above-mentioned thermodynamic description in terms of macroscopic variables ( M ) or mesoscopic fields [ m( x) ], a number of microscopic models of avalanche dynamics within FOPT have been proposed. The first was the random field Ising model (RFIM) with athermal dynamics, introduced by J.P. Sethna and co-workers for the study of hysteresis and metastablity [17]. The RFIM is a modification of the standard ferromagnetic Ising model with an external field ( H ) that describes a FOPT. On the sites of a regular lattice spin variables, Si , are defined whose values are ± 1. The Hamiltonian of the system is: n.n

H= − ∑ Si S j − H ∑ Si − ∑ hi Si i, j

(6)

where the first sum extends over all the pairs of nearest neighbors. Apart from the first two standard terms in Eq. (6), there is a new, last term that accounts for the local quenched disorder [Gaussian-distributed quenched random fields hi )], and the use of zero-temperature local relaxation dynamics. This is an athermal mechanism for the relaxation of the model when www.cat-science.cat

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Fig. 6. Power-law exponent α and the exponential correction parameter λ for a Cu-Al-Mn alloy [14] as a function of the number of cycles through the ferroelastic (martensitic) transition. The data show the evolution of the disorder towards a self-organized situation, in which avalanches are power-law distributed (λ = 0).

H is varied; it consists of flipping spins individually, as soon as they decrease the energy of the system. The model has been solved numerically within the mean-field approximation, using Bethe lattices, for different spatial dimensions and via Renormalization Group approaches. Some details are still not fully understood, but the model well describes the occurrence of avalanche dynamics and the critical power-law distribution of avalanche sizes [18]. After the RFIM was introduced, other models, including nucleation and the growth of many coexisting domains, were formulated, with similar properties [22]. Another set of models, focusing on the dynamics of a unique interface that separates the stable from the metastable phase, has been proposed. These models, designed to understand many out-of-equilibrium systems, collect ideas from the study of the pinning-depining transitions of an elastic line and the self-organized criticality theory proposed by P. Bak [1]. In these models, the external field ( H ) provides the main driving force, but it should be compensated for by the elastic terms, which tend to reduce the interface curvature and the pinning forces due to the interaction of the interface with the local quenched disorder. The main difference between these interface dynamics models and the RFIM is that, in the former, the avalanches turn out to be “critical” (power-law distributed) irrespective of the amount of disorder, whereas for the RFIM the system becomes critical only for a precise value of disorder (critical disorder). Finally, it should be noted that none of these microscopic CONTRIBUTIONS to SCIENCE 11:153-162 (2015)

Avalanche dynamics

models is able to account for the self-organization in the evolution of disorder. Only very recently, PĂŠrez-Reche and collaborators [15,16] were able to unify the above-mentioned models and to offer an explanation for the evolution towards a critical self-organized path.

Porous materials under compression In general, material fracture under external stress cannot be explained by the paradigm of FOPT and does not show avalanche dynamics. One of the main difficulties is that, because fracture is irreversible, it is difficult to identify concepts such as an underlying equilibrium free energy, metastability, and hysteresis in the process. Nevertheless, in the case of heterogeneous materials (amorphous materials, porous materials, granular materials, etc.) recent work [4] has demonstrated the importance of models for dynamic phase transitions. It is not our aim to discuss this topic here, but we will show that, in the case of highly porous materials, failure under compression does indeed exhibit avalanche dynamics and that this process has many similarities with other non-equilibrium failure processes, including earthquakes. Thus, avalanche dynamics could be a very general paradigm for understanding the physical mechanisms behind processes at very different scales. The failure of porous materials under compression has recently received much attention, due to its relevance in forecasting the collapse of natural and artificial structures, www.cat-science.cat

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Fig. 7. Experimental results corresponding to the uniaxial compression of a porous SiO2 sample (Vycor). (A) Force vs. time, (B) sample height vs. time, and (C) acoustic emission activity rate (in events per seconds). Note the logarithmic scale in the bottom panel.

whether mines, buildings, or bones. Because material failure is heralded by precursor activity, interest lies in whether or not this activity can be used for prediction. Laboratory experiments have been carried out using many natural minerals and artificial materials. Here we describe, as an example, the results reported for Vycor [2,11], a material that consists of an interconnected quartz skeleton (SiO2) with 40% porosity. The average pore diameter is < 10 nm. Materials at room temperature are placed between two plates and a compressional force is applied. The experiment can be optionally performed by imposing a lateral pressure or by controlling the speed of the plates (strain driven) or the force rate (stress driven). Here we focus on stress-driven experiments without lateral pressure. Acoustic emission sensors are embedded in the compression plates and detect ultrasonic events, just as seismographs detect earthquakes. Figure 6 shows a typical experiment using Vycor. While the force is monotonously increased, the length of the sample, measured by a laser extensometer, decreases in intermittent steps until a large drop occurs, corresponding to the failure of the sample. Acoustic emission activity (number of detected events per unit time) exhibits peaks of high activity not totally correlated with the decreases in sample height. The random process is therefore not homogeneous in time, but has a rate that varies from 10â&#x20AC;&#x201C;2 to 103 events per second. Furthermore, acoustic activity continues after sample collapse, indicating that the behavior of the materialâ&#x20AC;&#x2122;s debris is similar to that of the intact sample. Events (avalanches) are 160

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Fig. 8. Distribution of avalanche energies for the experiment shown in Fig. 7. The different histograms correspond to different time windows, as indicated by the legend. The data show the robust power-law behavior (lack of characteristic scale), extending 5–6 decades and independent of the analysed time window.

separated by waiting periods that range from 10–4 to 105 s. Analysis of the energies of each single event shows a powerlaw distribution of energies (Eq. 7): p ( E ) ~ E −ε

(7)

that spans eight orders of magnitude, as shown in Fig. 7. Despite the fact that the avalanche rate fluctuates enormously, the distribution p ( E ) is quite stationary and has a very welldetermined exponent ( ε = 1.39). Thus, the intermittent failure processes exhibit a lack of characteristic scales that allows them to be classified as an example of avalanche dynamics. As mentioned above, the statistics of acoustic emission events exhibit many similarities with those of seismological problems. The power-law distribution of energies, despite the large differences in energy scales, is nothing more than a Gutenberg–Richter law describing earthquake magnitude. However, the similarities go far beyond this distribution. Statistical techniques for the analysis of waiting times, correlations between avalanches, and the existence of aftershocks have revealed unexpected equivalences between the two phenomena. This opens up the possibility that an underwww.cat-science.cat

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standing of the avalanche dynamics during the compression of porous materials in the lab (labquakes) may, in the future, improve our understanding of earthquakes in the Earth’s crust. Acknowledgements. The authors acknowledge the Spanish Ministry of Economy and Competitiveness for financial support (grant numbers MAT2013-40590-P and MAT2015-69777-REDT). Competing interests. None declared.

References 1. Bak P, Tang C, Wiesenfeld K (1987) Self-organized criticality: An explanation of the 1/f noise. Phys Rev Lett 59:381 doi:10.1103/PhysRevLett.59.381 2. Baró J, Corral A, Illa X, Planes A, Salje EKH, Schranz W, Soto-Parra DE, Vives E (2013) Statistical similarity between the compression of a porous material and earthquakes. Phys Rev Lett 110:088702 doi:10.1103/ PhysRevLett.110.088702 3. Bertotti G (1998) Hysteresis in magnetism: for physicists, materials scientists and engineers. Academic Press, London, UK doi:10.1016/S0042207X(99)80004-9

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4. Bonamy D, Bouchaud E (2011) Failure of heterogeneous materials: a dynamic phase transition? Phys Rep 498:1-44 doi:10.1016/j.physrep.2010.07.006 5. Carrillo L, Mañosa L, Ortín J, Planes A, Vives E (1998) Experimental evidence for universality of acoustic emission avalanche distributions during structural transitions. Phys Rev Lett 81:1889 doi:10.1103/PhysRevLett.81.1889 6. Durin G, Zapperi S (2000) Scaling exponents for barkhausen avalanches in polycrystalline and amorphous ferromagnets. Phys Rev Lett 84:4705 doi:10.1038/nphys1884 7. Durin G, Zapperi S (2006) The Barkhausen effect. In: Bertotti G, Mayergoyz ID (eds) The Science of hysteresis, Vol. II:181-267. Academic Press, Oxford, UK 8. Field S, Witt J, Nori F, Ling X (1995) Superconducting vortex avalanches. Phys Rev Lett 74:1206 doi:10.1103/PhysRevLett.74.1206 9. Hardy V, Majumdar S, Lees MR, Paul DMcK, Yaicle C, Hervieu M (2004) Power-law distribution of avalanche sizes in the field-driven transformation of a phase-separated oxide. Phys Rev B 70:104423 doi:10.1103/ PhysRevB.70.104423 10. Lifshitz EM, Pitaevskii LP (1976) Landau and Lifshitz course of theoretical physics. Vol. 5, Statistical Physics, 3rd ed., part 1, Chapter XIV, Pergamon Press, Oxford, UK 11. Nataf GF, Castillo-Villa PO, Baró J, Illa X, Vives E, Planes A, Salje EKH (2014) Avalanches in compressed porous SiO2-based materials. Phys Rev E 90:022405 doi: 10.1103/PhysRevE.90.022405 12. Papon P, Leblond J, Meijer PHE (2006) The physics of phase transitions. 2nd edition, Springer Verlag, Berlin, Heidelberg, Germany ISBN-10 3-540-33389-4 13. Pérez-Reche FJ, Vives E, Mañosa L, Planes A (2001) Athermal character of structural phase transitions. Phys Rev Lett 87:195701 doi:10.1103/ PhysRevLett.87.195701

14. Pérez-Reche FJ, Stipcich M, Vives E, Mañosa Ll, Planes A, Morin M (2004) Kinetics of martensitic transitions in Cu-Al-Mn under thermal cycling: Analysis at multiple length scales. Phys Rev B 69:064101 doi:10.1103/ PhysRevB.69.064101 15. Pérez-Reche FJ, Truskinovsky L, Zanzotto G (2007) Training-induced criticality in martensites. Phys Rev Lett 99:075501 doi:10.1103/PhysRevLett.99.075501 16. Pérez-Reche FJ, Truskinovsky L, Zanzotto G (2008) Driving-induced crossover: from classical criticality to self-organized criticality. Phys Rev Lett 101:230601 doi:10.1103/PhysRevLett.101.230601 17. Sethna JP, Dahmen K, Kartha S, Krumhansl JA, Roberts BW, Shore JD (1993) Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transformations. Phys Rev Lett 70:3347 doi:10.1103/PhysRevLett.70.3347 18. Sethna JP, Dahmen KA, Myers CR (2001) Crackling noise. Nature 410: 242-250 doi:10.1038/35065675 19. Sornette D (2000) Critical phenomena in natural sciences. Springer Verlag, Berlin, Germany doi:10.1007/3-540-33182-4 20. Urbach JS, Madison RC, Markert JT (1995) Interface depinning, self-organized criticality, and the Barkhausen effect. Phys Rev Lett 75:276 doi:10.1103/PhysRevLett.75.276 21. Vives E, Ortín J, Mañosa L, Ràfols I, Pérez-Magrané R, Planes A (1994) Distributions of avalanches in martensitic transformations. Phys Rev Lett 72:1694 doi:10.1103/PhysRevLett.72.1694 22. Vives E, Goicoechea J, Ortín J, Planes A (1995) Universality in models for disorder-induced phase transitions. Phys Rev E 52:R5 doi:10.1103/PhysRevE.52.R5 23. Yeomans JM (1992) Statistical mechanics of phase transitions. Oxford Science Publications, Clarendon Press, Oxford, UK doi:10.1080/00107517808210882

About the image on the first page of this article. This photograph was made by Prof. Douglas Zook (Boston University) for his book Earth Gazes Back [www.douglaszookphotography.com]. See the article “Reflections: The enduring symbiosis between art and science,” by D. Zook, on pages 249-251 of this issue [http://revistes.iec.cat/index.php/CtS/article/view/142178/141126]. This thematic issue on “Non-equilibrium physics” can be unloaded in ISSUU format and the individual articles can be found in the Institute for Catalan Studies journals’ repository [www. cat-science.cat; http://revistes.iec.cat/contributions].

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CONTRIB SCI 11:163-171 (2015) doi:10.2436/20.7010.01.227

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Electron glasses

Matteo Palassini Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona, Catalonia. Universitat de Barcelona Institute of Complex Systems (UBICS), Barcelona, Catalonia

Summary. Electron glasses are disordered insulators with long-range interactions that exhibit slow non-equilibrium electronic transport effects. After introducing the basic physics of disordered insulators and hopping conduction, I briefly review the experimental evidence of glassy dynamics in these systems and some of the theoretical work aimed at understanding its origin. Similarities and differences with structural glasses and with another glass of electronic origin, the spin glass, are pointed out. [Contrib Sci 11(2): 163-171 (2015)]

Correspondence: Matteo Palassini Departament de Física de la Matèria Condensada, Universitat de Barcelona Martí i Franquès, 1 08028 Barcelona, Catalonia E-mail: palassini@ub.edu

Introduction In condensed matter physics, the term glass denotes a variety of systems that, due to their very slow dynamics, fail to reach thermal equilibrium on any reasonable experimental time scale. Structural glasses (such as the silica glass in windows), produced by quickly cooling a liquid so that it cannot crystallize, its molecules becoming trapped in an amorphous configuration, are the best known example.

This article is a non-technical introduction to a different type of glass in which electrons, rather than molecules, display a slow collective dynamics. Because of their light mass, one usually thinks of electrons as fast particles. However, experimental studies of electronic transport in various kinds of disordered insulators have shown that their electrical conductivity can relax over huge time scales, up to 20 orders of magnitude larger than the microscopic relaxation time [1,6,12,13,22,24,30,33]. Understanding the origin of the slow

Keywords: glassy dynamics · structural glasses · spin glass · disordered insulators · hopping conduction ISSN (print): 1575-6343 e-ISSN: 2013-410X

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Electron glasses

dynamics in these materials, collectively known as electron glasses1 [1,8,30] or as Coulomb glasses, remains an outstanding challenge. Below I briefly recall the main predictions of the band theory of solids, and then discuss how these are modified by disorder and by the interactions between electrons. I then briefly review the hopping conduction mechanism in disordered insulators, some of the experimental evidence of glassy relaxation, and recent theoretical work aimed ad understanding it.

Normal metals and band insulators The band theory of solids predicts whether a crystalline solid is a metal or an insulator, based on its chemical composition. In a perfect crystal, the allowed energy levels of non-interacting electrons form bands separated by forbidden gaps; at an absolute temperature of zero, the electrons fill the levels from the bottom up. If some bands are filled only partially, the electrons near the Fermi energy EF (the energy of the highest occupied level) move freely through the crystal as waves. This defines metallic behavior. At temperature T > 0 , crystal vibrations (phonons) scatter the electron waves in random directions, thus reducing the electrical conductivity σ (T ) as the temperature is increased. Partially filled bands arise if each crystal unit cell contains an odd number of electrons, or if two bands overlap. If neither of these conditions is met, then all bands are either empty or full and, since quantum mechanics forbids a net displacement of electrons in a full band, σ (T ) vanishes as T → 0 , which is the defining property of an insulator. As the temperature is increased, more and more electrons in the valence band (the full band of highest energy) receive enough energy from the crystal vibrations to overcome the band gap Eg , allowing them to populate the empty conduction band above it (Fig.1). This gives rise to a strong increase of the conductivity with temperature, σ (T ) ∝ exp(− Eg / 2k BT ) , where k B is the Boltzmann constant. Nervertheless, the conductivity remains negligible compared to metals (for example, at room temperature it is 20 orders of magnitude smaller in diamond than in copper).

Disorder and interactions: the metalinsulator transition Band theory is remarkably successful even though it neglects two key features of real solids: disorder, which breaks the translational symmetry of perfect crystals, and the repulsive Coulomb interaction between electrons. In metals, the average interaction energy between conduction electrons is not negligible compared to their kinetic energy. Nevertheless, Landau’s theory of Fermi liquids showed that, in normal metals, the correlations between electrons are sufficiently weak that they do not destroy metallic conductivity, thanks to efficient screening2 and to the Pauli exclusion principle. The main effect of interactions is to cause electrons to scatter each other, reducing σ (T ) at low temperature. Certain materials, however, do not behave as Fermi liquids, an example being high-critical-temperature cuprate superconductors. Another notable example is represented by Mott insulators, which should be metallic according to band theory but behave as insulators because the Coulomb repulsion prevents an electron from traveling to a site occupied by another electron, thus splitting the metallic band into a lower full band and an upper empty band. A sharp transition from insulator to metal, the Mott transition [18], occurs when the width of both bands becomes large compared with the intra-site Coulomb energy and thus the bands merge. The transition can be induced by applying external pressure to affect a stronger overlap, thus broadening the bands. A certain amount of disorder, in the form of impurities or crystal defects, is unavoidable in crystals. Furthermore, inherently disordered, non-crystalline materials are found in many technological applications. In these systems, disorder creates a random potential that scatters the electron waves. If the random potential is small compared with the electron kinetic energy, then metallic behavior is preserved: the conductivity, albeit reduced by disorder, remains finite down to zero temperature. On the other hand, Anderson [2] predicted that for strong enough disorder the electrons become trapped in small regions of typical size ξ (the localization length). The system then behaves as an Anderson insulator:

1 Electron glasses were a major topic of the 15th International Conference on Transport in Interacting Disordered Systems (TIDS15), organized by the author in Sant Feliu de Guíxols (Girona) in September 2013. The proceedings are published in Ref. [25]. 2 Self-screening of the Coulomb interaction is based on the fact that electrons push away other mobile electrons and are thus surrounded by a net positive charge. In metals, due to the high mobility of electrons, the electrostatic potential created by a point charge decays exponentially with distance as V (r ) ∝ exp(– r / l ) , where l is the screening length, rather than as the inverse of the distance as in the unscreened case.

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Fig. 1. On the right, a pictorial representation of the energy levels in a lightly doped, n-type compensated semiconductor in the Anderson insulating regime. The abscissa represents position (drawn in one dimension) and the ordinate represents energy. Full circles represent neutral donors and charged acceptors. Empty circles represent donors that have lost an electron to a nearby acceptor (dotted yellow line). Eg is the energy gap between the valence band and the conduction band. The acceptors act as “spectators” and do not participate in conduction, but they create a random potential of width W . When W is large compared to the quantum mechanical bandwidth, the states in the impurity band are localized and the electronic wave functions decay exponentially over a characteristic distance ξ . Electrons can tunnel from occupied to empty donors at distance r and energy difference ∆E . Also shown is the upper donor impurity band, separated by the lower band by the intra-site Coulomb energy U . The density of electronic levels g (ε ) is represented schematically on the left: the depletion near the Fermi energy EF is the Coulomb gap.

the conductivity vanishes at T = 0 because localized electrons cannot diffuse. A sharp transition from metal to insulator, the Anderson transition, occurs upon reducing the electron concentration (thus decreasing the kinetic energy) or upon increasing the disorder [18]. Loosely speaking, the transition occurs when the mean free path of the conducting electrons (the average distance they travel between two scattering events) becomes comparable to their wavelength. The Mott and Anderson transitions, driven by interactions and disorder, respectively, are examples of zero-temperature quantum phase transitions.

Hopping conduction in disordered insulators At finite temperature, Anderson insulators conduct electricity via quantum-mechanical tunneling of the electrons between localization sites. This conduction mechanism, called hopping, is made possible by crystal vibrations that emit or absorb the energy difference (∆E ) between the initial and final electronic states. The transition rate of a single electron tunneling between two sites separated by distance r is proportional to exp(−2r / ξ − ∆E / k BT ) for ∆E > 0 , where the www.cat-science.cat

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exponential dependence on r arises from the overlap between the wave functions of the two states. Mott [19] predicted that as T is lowered, the most probable hops take place between sites with increasingly small ∆E and increasingly large r , a mechanism called variable-range hopping [18,32]. Consequently, when the long-range Coulomb interaction can be neglected, the conductivity follows the Mott law σ (T ) ∝ exp[−(TM / T )1/( d +1) ] , where d is the system dimensionality and TM is a characteristic temperature that depends on ξ and on the density of electronic levels at the Fermi energy (provided the density does not deviate much from this value throughout the hopping energy range). Due to the low mobility of electrons, the Coulomb interaction is poorly screened at low temperature and thus, unlike in metals, it retains its long-range character, inducing strong correlations in the motion of the electrons. The best known correlation effect is the Coulomb gap, a reduction of the density of levels near the Fermi energy [9,29], which in turn reduces the conductivity (Fig. 1). Efros and Shklovskii [9] argued that at T = 0 the singleparticle density of levels g (ε ) must vanish proportionally to d −1 ε − EF or faster as ε → EF , and predicted that if this bound is saturated, the conductivity follows σ (T ) ∝ exp[−(TES / T )1/2 ] , where TES is a universal temperaCONTRIBUTIONS to SCIENCE 11:163-171 (2015)

Electron glasses

Doped crystalline semiconductors. In a semiconductor (a band insulator with a relatively small band gap) doped with donor impurities3, the ground-state energy levels of electrons bound to the impurities form a narrow “impurity band” below the conduction band [18,32] (Fig. 1). At room temperature, most impurities are ionized by thermal fluctuations, thus populating levels in the conduction band. These levels being delocalized, conduction occurs in the same way as in metals. At temperatures so low that ionization is negligible, conduction occurs only within the impurity band, which is half-filled since each impurity contributes one electron and two spin-degenerate levels. (An identical picture holds if the impurities are acceptors, except that the impurity band is near the valence band and charge is carried by holes instead of electrons.) The width of the impurity band arises from two facts: i) quantum broadening spreads the levels by an amount I b that increases upon reducing the average distance between impurities; ii) because of the random spatial distribution, the impurities experience a random potential which spreads the levels by an amount W . In the impurity conduction regime, the only relevant energy scales are I b , W , the thermal energy (k BT ) , and the intra-site Coulomb repulsion energy (U ) between two electrons bound to the same impurity. If only impurities of one type (e.g., donors) are present, they remain electrically neutral and thus the random potential is typically small, i.e., W ˂˂ U . For I b ˃˃ U the impurity

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ture that depends only on ξ . A change from the Mott law to the Efros-Shklovskii law has been since been confirmed experimentally for a wide variety of materials. The original argument of Efros and Shklovskii [9] considered only single-electron hops that leave the rest of the system unperturbed. However, a full understanding of hopping conduction requires the consideration of multi-electron hops (i.e., the simultaneus tunneling of more than one electron), and of sequential hops induced by the Coulomb interaction after a hop [14,29]. In particular, it has been argued that sequential hops cause the density of levels to vanish faster than quadratically in three dimensions [5], and that many-electron transitions are responsible for glassy non-equilibrium relaxation [1,15,24]. Next, I describe several types of Anderson insulators exhibiting hopping conductivity.

Fig. 2. Pictorial representation of the density of levels in an amorphous semiconductor. The dashed lines represent the mobility edges in the tails of the conduction (valence) band, below (above) which states are localized.

states are delocalized; thus, conductivity is metallic in the sense that σ (T = 0) > 0 . By decreasing the impurity concentration, a Mott transition occurs as the ratio I b / U falls below a certain threshold. In the Mott insulating regime, hopping conduction is not possible since there are no empty donor sites [18]. On the other hand, if both donors and acceptors are present, for example with a slightly higher concentration of donors (n-type compensated semiconductor), then each acceptor captures an electron from a nearby donor and becomes charged (Fig. 1). The charged impurities create a strong random potential, and typically W >> U. If the impurity concentration is so high that I b ˃˃ W , the states are delocalized and conduction is metallic. Upon reducing the concentration, an Anderson localization transition occurs at a certain threshold for the ratio I b / W . In the Anderson insulating regime, pho��

3 Donors (resp. acceptors) are atoms with one more (less) valence electron than the semiconductor atoms; thus, they “donate” an electron (hole) that, if the impurity is isolated, remains weakly bound to the impurity core, forming a hydrogen-like "atom" whose radius can be much larger than the crystal lattice spacing. Examples are phosphorous (boron) in a silicon crystal.

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Fig. 3. Scheme of the field effect transistor experimental setup used in conductance experiments. The Anderson insulator under study is deposited on an insulator layer in contact with a metal (the gate). The conductance is measured by applying a voltage between the source and drain electrodes attached to the sample. A voltatge Vg between the gate and the source allows a change in the density of the carriers in the sample. Capacitance experiments use a similar setup except that no source-drain voltage is applied and the current between the gate and the insulator is measured as a function of time.

non-assisted hopping from occupied to empty donor sites can take place. Amorphous semiconductors. These are covalent structural glasses prepared by rapid cooling of a melt or by deposition. Although they lack translational symmetry, a band structure still exists [18], but tails of localized states appear in the band gap, separated from the extended states by a mobility edge (Fig. 2). If the Fermi energy falls within the localized region, the system is an Anderson insulator and conduction takes place via variable-range hopping between localization sites. A metal-insulator transition can be driven by pressure, an increase in the carrier concentration, or thermal annealing. Granular metals. They are produced by embedding nanometric metallic grains in an insulating material. Electronic transport takes place via tunneling between grains. Below a certain average inter-grain separation, conduction is metallic even if the metal does not percolate the sample. Above this threshold, conduction is of the hopping type. Discontinuous metallic films, consisting of metallic islands deposited on a substrate, behave similarly [30]. Two dimensional systems. In field effect transistors (Fig. 3), the charge carriers of a semiconductor can be confined in a narrow 2D layer at an interface with an insulator. The interfacial electronic states display a mobility edge, and by tuning the carrier concentration the Fermi energy can be brought in the localized region, giving rise to a 2D Anderson insulator. Homogeneous ultrathin metallic films, in which lowww.cat-science.cat

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calization occurs below a certain thickness, are another example of a 2D Anderson insulator. Self-assembled arrays of metallic/semiconÂ ductor nanocrystals. These consist of nanometric grains coated with an insulating material that self-assemble in a colloidal solution. Variable-range hopping due to tunneling between grains has been experimentally observed [16]. These materials are currently of great interest as an alternative to semiconductor technology for electronic devices.

Glassy behavior The idea that Anderson insulators with long-range interactions may exhibit glassy dynamics was first put forward in theoretical work by Davies et al. [8], who, by analogy with spin glasses, coined the term â&#x20AC;&#x153;electron glassâ&#x20AC;? for these systems. Spin glasses are disordered magnetic materials in which the interaction between two given magnetic moments (spins) can be, depending on their positions in the solid, either ferromagnetic or antiferromagnetic, favoring a parallel or antiparallel orientation of the spins [34]. Because of the random mixture of signs, the spins cannot satisfy all interactions simultaneously, a fact referred to as frustration. The interplay of disorder and interaction gives rise to frustration in electron glasses as well: disorder tends to push the electrons to sites where the random potential is large, while interactions tend to push them away from each other. In both spin and electron glasses, frustration induces a large number of metastable states (i.e., local energy minima) in which the CONTRIBUTIONS to SCIENCE 11:163-171 (2015)

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Electron glasses

system can become trapped for a very long time. In this respect, electron and spin glasses differ from structural glasses, since in the latter there is no external disorder and slow dynamics is generated only by interactions. Early observations of slow relaxation in semiconductors [17] were followed by the groundbreaking experiments of Ovadyahu and collaborators on amorphous and crystalline indium oxide [6,24]. These experiments typically use a field effect transistor setup (Fig. 3) and monitor the time evolution of the conductance after the system has been pushed out of equilibrium, for example by a sudden change in temperature or gate voltage. Figure 4 shows representative results obtained with the following protocol [24]: the sample is cooled at liquid helium temperature ( T = 4.11K ) and kept at a gate voltage Vg = Vg 1 for a time t1 . The gate voltage is then switched to Vg = Vg 2 for a “waiting time” tw , after which it is finally switched back to Vg = Vg 1 . At both changes of the gate voltage, the conductance increases abruptly and then slowly relaxes. The figure shows the conductance relaxation after the second change, where t is the time passed since the change. A few observations can be made: (i) The rapid conductance increase occurs regardless of whether Vg was increased or decreased (i.e., whether electrons are injected or removed), unlike in ordinary semiconductors in which the conductance is monotonic in Vg . (ii) The relaxation curve depends on tw , showing that the system has not equilibrated during the time tw , even when the latter is large. This is referred to as “aging” in the glass literature. (iii) tw enters only via the ratio t / tw (“simple www.cat-science.cat

Fig. 4. Experimental results using the protocol described in the text for a film of crystalline indium oxide with a thickness of 5 nm, evaporated on a 140-µm-thick cover-glass coated on the back with a gold film. Here t1 = 6 days, Vg 1 = 50V , Vg 2 = −50V . The plot shows the relative change in conductance ∆G (t ) / G (tW ) where ∆G (t ) = G (t + tW ) − G (tW ) , as a function of t / tw . Data for several values of tw are shown together. Figure adapted with permission from [24]. Copyright of the American Physical Society.

aging” or “full aging.”) (iv) the relaxation follows a logarithmic law ∆G (t ) ≈ log(1 + t / tw ) for several decades of time. A similar behavior has been observed in amorphous indium oxide, thin beryllium films, granular aluminum [12], and discontinuous metallic films [13], also for much longer waiting times. (Recent reviews can be found in [1,22,30]). A rapid increase of the conductance followed by logarithmic relaxation has been reported with other types of perturbations, such as rapid cooling, the application of strong electric fields, and exposure to infrared radiation [23,30]. More complex protocols have been used as well [12,24], in particular to investigate how the system “remembers” its previous history over long time scales (the so called memory effect) [1]. An important observation from these studies is that the relaxation time t (defined, for example, as the time necessary for G (t ) to decrease to half its peak value after a perturbation) increases rapidly with the carrier concentration [33]. This fact provides strong evidence that glassiness is an intrinsic property of the electron dynamics and is not due to extrinsic relaxing elements, such as slowly moving charges in the substrate or structural rearrangements. It might also explain [22] why glassy effects of the type described above have not been observed in lightly doped semiconductors, which have a much lower carrier density than all the other Anderson insulators discussed above. Another crucial finding is that τ does not depend appreciably on temperature, in contrast to both structural glasses and spin glasses, in which this dependence is strong4. A likely explanation lies in the fact that in structural and spin glasses 168

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the dynamics is thermally activated (i.e., the transition rate between two configurations is proportional to exp(−∆E / k BT ) ), while in electron glasses it depends on both quantum tunneling and thermal activation. The temperature independence of t thus suggests that quantum tunneling plays a key role in glassy relaxation. Expert readers may note that the time protocol discussed above is very similar to the “isothermal remanent magnetization” protocol used in spin glass experiments [7]. Aging is observed in spin glasses as well, but the relaxation is not logarithmic and the dependence on t and tw is more complicated. The reason for this different behavior of spin and electron glasses is not fully understood.

Recent theoretical work The origin of slow non-equilibrium relaxation in electron glasses is still actively debated, and it is beyond the scope of this introductory paper to discuss the many theoretical ideas that have been put forward [30]. Only a few aspects are mentioned below. Many-electron transitions. Because the Coulomb gap strongly affects the stationary conductivity, the slow nonequilibrium relaxation of the conductivity is sometimes attributed to the slow formation of the Coulomb gap after having been disrupted by a perturbation. Recently, by means of kinetic Monte Carlo simulations of transport in a model electron glass in which only single-electron transitions are allowed, we found [10] that after an abrupt temperature quench the conductivity reaches a stationary value σ eq (T ) on a time scale of order t M ≈ (T 2σ eq (T )) −1 (the Maxwell relaxation time), and that indeed the Coulomb gap forms on the same time scale. However, the measured t M is at most 10−6 s , many orders of magnitude smaller than the relaxation time observed experimentally in a comparable temperature range. This suggests that glassy dynamics depends on the simultaneous tunneling of more than on electron [1,15]. Multielectron transitions are slow processes, as the transition rate decays exponentially with the total distance traveled by the electrons. The most probable transitions involve small clusters of sites, thus they do not participate directly in conduction, which occurs via long single-electron transitions. Howe-

ver, it has been conjectured [1,15] that when these clusters relax, they lower the conductivity by affecting the conduction path via the Coulomb interaction. Search for a thermodynamic phase transition. A long-debated question in spin glass physics concerns the existence of a thermodynamic transition to a low temperature “spin glass phase” [34], in which each spin points in a preferential orientation (unlike in the paramagnetic phase in which spins spend, on average, the same time in all orientations), but the pattern of orientations, dictated by the interactions, has a random appearance. It is now well established [3,26] that in three dimensions and in the absence of an external magnetic field the spin glass phase exists, although its precise nature is still debated. In their seminal paper, Davies et al. [8] conjectured the existence of a similar phase in electron glasses. A mean-field theory [20,28], inspired by a mathematical similarity between electron glass models and long-range spin glass models, predicted a transition to a “marginally stable” low temperature phase in three dimensions. A consequence of marginal stability is that the density of levels goes to zero quadratically at the Fermi energy, saturating the Efros-Shklovskii bound [9]. By means of large-scale equilibrium Monte Carlo simulations, however, we found convincing evidence against the existence of a phase transition down to very low temperatures [11]. Furthermore, using energy minimization computations with large system sizes we determined that density of levels in the Coulomb gap vanishes faster than quadratically [27], confirming some earlier findings (references can be found in [27]). Avalanches and nonlinear screening. Screening in a system of localized electrons is very different than in a metal [4]. Because charge is discrete, weak electric fields can penetrate over large distances, a phenomenon referred to as nonlinear screening. Capacitance experiments using a metalinsulator-semiconductor structure (Fig. 3) found evidence that the charge injected in an Anderson insulator through a gate voltage moves very slowly from the “top” to the “bottom” of the sample [17]. Motivated by these observations, we recently investigated numerically the “avalanches” created by inserting or displacing a charge in an electron glass [27]. Both types of per-

4 The relaxation time can be measured in structural glasses from the relaxation of the viscosity after a temperature quench or a stress, and in spin glasses from the response of the magnetization to a change in temperature or magnetic field.

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turbation cause some electrons to tunnel to other sites, which in turn can create further hops, in a cascade process. We found that the avalanche size has a scale-free probability distribution p ( S ) ≈ S −t exp(− S / Sc ) , with a cutoff diverging with the linear size of the sample as Sc ≈ L / r0 , where r0 is the characteristic size of the soft dipoles (i.e., electron-hole pairs with low excitation energy). Furthermore, we showed that the avalanche is well described by a branching process in which each electron hop induces in average another hop. Recent experiments on 3D indium-oxide samples are consistent with the hypothesis that a change in gate voltage induces avalanche-like rearrangements in the whole sample [23]. Scale-free avalanches have been observed before in a wide variety of systems ranging from earthquakes to paper crumpling [31]. Recently, it was argued that, under certain conditions, they are a universal consequence of the existence of a Coulomb gap (or similar gaps in related systems) stemming from the long-range interaction [21].

Conclusions Electron glasses show a remarkable non-equilibrium phenomenology resulting from the interplay of disorder and electron-electron interactions. Conductance and capacitance experiments allow to measure the responses of electron glasses to a variety of excitations (temperature, electric field, charge injection, electromagnetic radiation), thus offering an ideal testing ground to investigate the properties of glasses in general. The basic mechanism leading to slow relaxation in electron glasses is as yet unknown. Its elucidation will probably require theoretical efforts combining concepts from the statistical physics of disordered systems, quantum condensed-matter physics, and non-equilibrium statistical physics, as well as powerful computational algorithms. Quantum phenomena beyond tunneling, such as Anderson orthogonality catastrophe and many-body localization, not discussed here, may play an important role.

Acknowledgements. I thank Martin Goethe, Ezequiel Ferrero, and Alejandro Kolton for stimulating collaborations, and Ariel Amir, Joakim Bergli, Alexei Efros, Markus Müller, Miguel Ortuño, Zvi Ovadyahu, Mike Pollak, Brian Skinner, Andrés Somoza, and Boris Shklovskii for illuminating discussions. This work was supported by MINECO (FIS2012-38266-C02-02 and PRI-AIBAR-2011-1206) and AGAUR (2012 BE 00850). Competing interests. None declared

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References 1. Amir A, Oreg Y, Imry Y (2011) Electron glass dynamics. Annu Rev Condens Matter Phys 2:235 2. Anderson PW (1958) Absence of diffusion in certain random lattices. Phys Rev 109:1492 3. Ballesteros HG, et al. (2000) Critical behavior of the three-dimensional Ising spin glass. Phys Rev B 62:14237 4. Baranovskii SD, Shklovskii BI, Efros AL (1984) Screening in a system with localized electrons. Sov Phys JETP 60:1031-1036 5. Baranovskii SD, Shklovskii BI, Efros AL (1980), Elementary excitations in disordered systems with localized electrons. Sov Phys JETP 51:199-207 6. Ben-Chorin M, Ovadyahu Z, Pollak M (1993) Non-equilibrium transport and slow relaxation in hopping conductivity. Phys Rev B Condens Matter 48:15025-15034 7. Bouchaud J-P, et al. (1997) Out of equilibrium dynamics in spin glasses and other glassy systems. In: Young AP (ed) Spin glasses and random fields. World Scientific, Singapore 8. Davies JH, Lee PA, Rice TM (1982) Electron glass. Phys Rev Lett 49:758 9. Efros AL, Shklovskii BI (1975) Coulomb gap and low temperature conductivity of disordered systems. J Phys C: Solid State Phys 8 L49 doi:10.1088/0022-3719/8/4/003 10. Ferrero EE, Kolton AB, Palassini M (2014) Parallel kinetic Monte Carlo simulation of Coulomb glasses. AIP Conf Proc 1610:71-76 doi:10.1063/1.4893513 11. Goethe M, Palassini M (2009) Phase diagram, correlation gap, and critical properties of the Coulomb glass. Phys Rev Lett 103:045702 doi:10.1103/PhysRevLett.103.045702 12. Grenet T, Delahaye J, Sabra M, Gay F (2007) Anomalous electric-field effect and glassy behaviour in granular aluminium thin films: electron glass? Eur Phys J B 56:183-197 doi:10.1140/epjb/e2007-00109-4 13. Havdala T, Eisenbach A, Frydman A (2012) Ulta-slow relaxation in discontinuous-film based electron glasses. Euro Phys. Lett. 98:67006 14. Knotek ML, Pollak M (1974) Correlation effects in hopping conduction: A treatment in terms of multielectron transitions. Phys Rev B 9:664 15. Kozub VI, Galperin YM, Vinokur V, Burin AL (2008) Memory effects in transport through a hopping insulator: Understanding two-dip experiments. Phys Rev B 78:132201 16. Liu H, Pourret A, Guyot-Sionnest P (2010) Mott and Efros-Shklovskii variable range hopping in CdSe quantum dots films. ACS Nano 4:52115216 doi:10.1021/nn101376u 17. Monroe D, Gossard AC, English JH, Golding B, Haemmerle WH (1987) Long-lived Coulomb gap in a compensated semiconductor—the electron glass. Phys Rev Lett 59:1148 doi:10.1103/PhysRevLett.59.1148 18. Mott NF (1993) Conduction in non-crystalline materials, 2nd ed. Oxford University Press, Oxford, UK 19. Mott NF (1968) Conduction in glasses containing transition metal ions. J Non-Crystal Solids 1, 1 20. Müller M, Ioffe LB (2004) Glass transition and the Coulomb gap in electron glasses. Phys Rev Lett 93:256403 doi:10.1103/PhysRevLett.93.256403 21. Müller M, Wyart M (2015) Marginal stability in structural, spin, and electron glasses. Annu Rev Condens Matter Phys 6:177-200 doi:10.1146/annurev-conmatphys-031214-014614 22. �� Ovadyahu Z (2013) Interacting Anderson insulators: The intrinsic electron glass. C R Physique 14:700-711 23. Ovadyahu Z (2014) Electron glass in a three-dimensional system. Phys Rev B 90:054204 doi:10.1103/PhysRevB.90.054204 24. Ovadyahu Z, Pollak M (2003) History dependent relaxation and the energy scale of correlations in the electron glass. Phys Rev B 68:184204 doi:10.1103/PhysRev B 68:184204

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25. Palassini M (ed) (2014) Proceedings of the 15th International Conference on Transport in Interacting Disordered Systems, AIP Conf Proc 1610 26. Palassini M, Caracciolo S (1999) Universal finite-size scaling functions in the 3D Ising spin glass. Phys Rev Lett 82:5128 doi:10.1103/PhysRevLett.82.5128 27. Palassini M, Goethe M (2012) Elementary excitations and avalanches in the Coulomb glass. J Phys: Conf Ser 376:012009 doi:10.1088/17426596/376/1/012009 28. Pankov S, Dobrosavljevic V (2005) Nonlinear screening theory of the Coulomb glass. Phys Rev Lett 94:046402 doi:10.1103/PhysRevLett.94.046402 29. Pollak M (1970), Effect of carrier-carrier interactions on some transport properties in disordered semiconductors. Discuss Faraday Soc 50:13-19

30. Pollak M, Ortuño M, Frydman A (2013) The electron glass. Cambridge University Press. Cambridge, UK 31. Sethna JP, Dahmen KA, Myers CR (2001) Crackling noise. Nature 410:242-250 doi:10.1038/35065675 32. Shklovskii BI, Efros AL (1984) Electronic properties of doped semiconductors. Springer-Verlag Berlin Heidelberg doi:10.1007/978-3-66202403-4 33. Vaknin A, Ovadyahu Z, Pollak M (1998) Evidence for interactions in nonergodic electronic transport. Phys Rev Lett 81:669 doi:10.1103/ PhysRevLett.81.669 34. Young AP (ed) (1997) Spin glasses and random fields. World Scientific, Singapore

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CONTRIB SCI 11:173-180 (2015) doi:10.2436/20.7010.01.228

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Nucleation phenomena: The non-equilibrium kinetics of phase change David Reguera Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona, Catalonia

Summary. The molecules and atoms that comprise matter have the tendency to join in different aggregation states called phases. How these atoms and molecules manage to shift between these different states is one of the most fascinating processes in physics. These phase transitions are commonly controlled and triggered by a non-equilibrium physical mechanism, called nucleation, that describes the formation of the first seeds of the new phase. Nucleation is behind many phenomena of utmost scientific and technological interest, ranging from nuclear phenomena and biological assembly to galaxy formation. However, due to its rare non-equilibrium nature, it is still one of the few classical problems that remain incompletely understood. Indeed, deviations between theoretical predictions and experiments can reach several orders of magnitude. In this article, we review the essential aspects of nucleation and the challenges it poses to current research. [Contrib Sci 11(2): 173-180 (2015)] *Correspondence: David Reguera Departament de Física de la Matèria Condensada Universitat de Barcelona, Martí i Franquès, 1 08028 Barcelona, Catalonia Tel. +34-934039214; E-mail: dreguera@ub.edu

Introduction Matter appears in nature in different aggregation states called phases. A familiar example is water, whose molecules can be in a gaseous (vapor) state, glued together to form a liquid, or trapped in a static arrangement in a solid phase (ice). The state of a particular substance is the result of a delicate balance between thermal agitation, which tends to set molecules free, and attractive molecular interactions, which

bind molecules together. This balance sensitively depends on the thermodynamic ambient conditions, especially on the temperature, which controls the strength of the thermal agitation, and the density or pressure, which determines the average proximity between molecules. These parameters can be used to construct a phase diagram that defines the equilibrium phase, i.e., the most stable state of a particular substance under the given conditions. As an example, Fig. 1 reproduces the phase diagram of water. The solid lines indi-

Keywords: phase transition · non-equilibrium · nucleation · metastable states · water ISSN (print): 1575-6343 e-ISSN: 2013-410X

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Fig. 1. Phase diagram of water. The most stable phase of water is shown as a function of pressure and temperature. The solid lines mark the boundaries between different phases. The triple point, i.e., the temperature and pressure at which the three phases of water can coexist, and the critical point, where the phase boundary between the liquid and the vapor vanishes, are also indicated.

cate the coexistence boundaries that define the frontiers between different equilibrium states. For instance, at ambient pressure, water prefers to be in a vapor state at temperatures above 100°C but forms ice at temperatures below 0°C. However, the division between states is not perfectly sharp. It is well known that purified water can be held as a liquid almost indefinitely at –10°C without freezing; and a gas can be compressed several times its equilibrium pressure before the first liquid drop suddenly appears [6]. These special situations, in which a system persists for long periods of time in a phase that is not its equilibrium state, are generally called metastable states. The ultimate reason why water can be kept in super cooled or supersaturated metastable states is the presence of an energetic barrier that hinders spontaneous transitions between different phases. The initial and crucial step required to overcome this barrier and to trigger a phase transformation is the generation of a small embryo, or nucleus, of the new phase within the bulk metastable substance. This fundamental mechanism of phase transformation is known as nucleation and it constitutes a central problem in many areas of scientific and technical interest [1,6,15,16]. Condenwww.cat-science.cat

sation (liquid drop formation in a supersaturated gas), boiling (vapor formation in a superheated liquid), cavitation (bubble formation in a stretched fluid), and crystallization are perhaps the most common examples of nucleation processes. But nucleation also plays a decisive role in very different fields of science and technology. In the atmospheric sciences, the nucleation of water droplets, ice crystals, or aerosols (liquid droplets suspended in a gas) in the atmosphere is a fundamental issue in weather forecasting and climate change [22]. Practical uses of nucleation include the induction or prevention of precipitation and hail by cloud seeding [5] and the collection of liquid water from moist air in airwells. Examples of biomedical interest encompass the cryopreservation of embryos and human tissues, the bio-mineralization of bone, teeth, and shells, the formation of kidney stones or uric acid crystals in gout, and protein aggregation/ crystallization [13,17] that underlies many diseases, such as Alzheimer, sickle-cell anemia, and cataract formation. At the industrial level, control of the nucleation stage is an important requirement in the fabrication of novel and advanced materials. The stability of pharmaceutical compounds, damage to manmade materials (e.g., propellers) or tissues by cavitation, the explosive boiling of vapors, the performance of motor engines and turbines, and the extraction of oil and gas are among the many industrial problems involving different aspects of nucleation. Further examples can be cited at all scales ranging from nuclear phenomena to the formation of planets and galaxies. But how well do we understand nucleation? Can we predict and control its outcome? As will become clear in the following, despite significant efforts, we are still very far from being able to accurately predict the occurrence of nucleation, not even in the simplest cases. In fact, the errors in predictions are not just a few percent, but often span many orders of magnitude. In this article, we discuss the particular features of nucleation that give rise to this discrepancy. For simplicity, we focus on the simplest case, that of homogeneous condensation. But the same considerations can be applied to more complex phenomena, such as protein crystallization and the self-assembly of viruses [33].

The problem of condensation To better illustrate the physical nature of the nucleation process let us focus on a simple and well-known example: the condensation of a vapor. In most practical instances, condensation, as well as the majority of phase transitions, 174

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Fig. 2. Mechanisms of phase transition. A representative phase diagram in terms of the pressure (P) and volume (V) of a simple fluid. The green line shows an isotherm, showing the values of P as a function of the volume of the system for a given constant temperature (T). Psat(T) is the value of the saturation pressure at this particular temperature, at which the liquid and vapor phases coexist. The red line is the coexistence curve of the liquid (left branch) and the vapor (right branch) at all temperatures, ending at the critical point. The blue line is the spinodal, below which the fluid is mechanically unstable (indicated by the dotted lines) and is transformed into a new phase by a spontaneous mechanism known as spinodal decomposition. Nucleation is the mechanism of phase transition between the coexistence curve and the spinodal, where the system is in a metastable state (represented by the dashed lines in the isotherm).

preferentially occurs in the presence of surfaces on impurities, by a mechanism known as heterogeneous nucleation. However, in the following we describe the simplest situation, that of homogeneous condensation occurring in the bulk of a pure substance. Figure 2 shows a representative phase diagram of a simple model fluid in terms of pressure and volume. The green line indicates the pressure of the system as a function of the volume that it occupies at a fixed temperature. At large volumes, the system is in the vapor state. As the volume is reduced, the pressure increases up to a point at which the liquid phase begins to be more favorable. At this so-called saturation pressure, the vapor can coexist with the liquid, whose properties are defined by the left branch of the green curve. If the volume is further reduced, the vapor becomes supersaturated and remains in a metastable state (indicated by the dashed green line) until the phase transformation takes place. If we trace the values of the saturation pressure at different temperatures, we obtain the curve representing the coexistence of the liquid and the vapor, indicated by the red solid line in Fig. 2. The blue line represents the spinodal, which is the locus of points at which the mechanical stability requirement of a positive isothermal compressibility (i.e., that the volume of a system

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becomes smaller as more pressure is applied to it) is first violated. These two lines delimit the two basic mechanisms of a phase transition. Below the spinodal line, the vapor phase is thermodynamically unstable, and the phase transformation proceeds spontaneously by a barrierless mechanism known as spinodal decomposition. In the region between the spinodal and the coexistence lines, the system is metastable and the phase transformation occurs by the nucleation of the first embryos of the new phase and their subsequent growth. The degree of metastability of a vapor is usually measured in terms of the supersaturation, defined as the ratio between the actual pressure of the vapor and the saturation pressure. As mentioned in the Introduction, a metastable phase can persist over long periods of time. This feature reflects the fact that the development of the new phase must surmount an energy barrier. This barrier becomes infinity at the coexistence curve and vanishes at the spinodal and is thus strongly dependent on temperature and supersaturation. The origin of this barrier and the means by which it is overcome are explained below. In a supersaturated vapor, thermal agitation of the molecules induces density fluctuations. These fluctuations generate small aggregates of molecules (clusters) with proper-

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Fig. 3. Nucleation barrier. Schematic representation of the free energy of liquid cluster formation as a function of the number of molecules contained by the cluster (red line). This free energy landscape is derived from two contributions: a volume term (green line), proportional to the number of molecules and to the difference in chemical potentials between the old and the new phases, and a surface term (blue line) associated with the cost of forming a new interface between the incipient liquid and the vapor phase. The sum of these two terms gives rise to a barrier. The maximum is located at the critical cluster size n* and its height is known as the nucleation barrier â&#x2C6;&#x2020;G*. Liquid clusters smaller than the critical size will slide down the barrier until they disappear, whereas clusters larger than the critical size will tend to grow spontaneously.

ties similar to those of the stable phase, in our case, tiny liquid droplets. The formation of these small droplets is favored by an energy decrease associated with a more stable liquid phase under the given conditions. This energy gain is proportional to the number of molecules or the volume of the cluster. However, the construction of the new phase from the bulk metastable phase requires the creation of an interface between the two phases, which implies an energetic cost proportional to the droplet surface. The competition between these two effects gives rise to the nucleation barrier, as illustrated in Fig. 3. Surface effects are dominant for small clusters and hence tiny droplets tend to disintegrate. However, there is a characteristic size at which volume effects override surface contributions and clusters tend to grow spontaneously. This size, signaling the top of the free energy of cluster formation, is known as the critical size and the energy required in its formation constitutes the nucleation barrier. The rate at which critical-sized embryos are formed is the nucleation rate and its prediction is one of the major goals of nucleation theories. www.cat-science.cat

Although the first studies of phase equilibrium and metastability of undercooled substances date back to the investigations of Fahrenheit in the 18th century [7], the study of the kinetics of nucleation was initiated by the pioneering work of Volmer and Weber [23] in 1926 and of Farkas in 1927 [8]. The field was subsequently developed by the contributions of Becker and DĂśring [3], Frenkel [11], and Zeldovich [34], among others. These investigations collectively gave rise to classical nucleation theory (CNT) [1,6,15,16]. According to CNT, the initial stages of droplet formation are modeled as a sort of chemical reaction in which a cluster of a particular number of molecules grows or shrinks by the addition or loss of one molecule at a time. Thus, a balance equation can be formulated describing the evolution over time of the cluster population of a given size in the system. The relationship involves two different size-dependent parameters: the rate at which molecules attach to a cluster and the rate at which they evaporate from it. The rate of attachment can be rather accurately quantified using kinetic theory of gases [19], as the rate of thermal collisions. However, the detachment rate depends very sensitively on the arrangement and interactions between molecules and is thus hard to model. CNT circumvents this problem by resorting to detailed balance considerations. At equilibrium conditions, to maintain a balance in the population of cluster sizes, the rates of attachment and detachment should be connected to the differences in the free energies of clusters of different sizes. Accordingly, the unknown detachment rate can be expressed as a function of both the attachment rate and the free energy of formation of a cluster of any given size. CNT therefore rephrases a complicated kinetic problem in simpler equilibrium thermodynamic terms. The free energy of formation of a tiny liquid droplet is then evaluated using crude approximations. The first one is to consider the vapor phase as nearly ideal, i.e., neglecting the interactions between its molecules. Next, the incipient liquid drop is modeled as a tiny spherical drop with a sharp interface and the same thermodynamic properties as the macroscopic liquid. With these assumptions the critical cluster size and the height of the nucleation barrier can be evaluated using just three simple variables: the surface tension, the saturation pressure, and the density of the liquid, assumed to be incompressible. Knowing the height of the barrier, the nucleation rate (i.e., the rate at which critically sized clusters form per unit 176

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volume and unit time) can be evaluated as the rate of jumping over the nucleation barrier. When this barrier is high, as is the case in most practical instances, the rate turns out to be proportional to the exponential of the barrier height, which is expressed in units of the characteristic thermal energy kBT, where kB is Boltzmannâ&#x20AC;&#x2122;s constant. Therefore, CNT provides a very simple expression to predict the rate at which any phase transition occurs, using simple expressions and thermodynamic parameters that are available for most substances.

How good is CNT? CNT has dominated our understanding of nucleation during the last several decades. The secret of its success is its striking simplicity and the initially reasonable agreement between experimental results and the theoryâ&#x20AC;&#x2122;s predictions for the metastability limits of the majority of substances. With the recent development of new and very accurate experimental techniques able to measure actual nucleation rates, the molecular details of nucleation are slowly being revealed. Nowadays, the real-time experimental observation of the appearance and growth of nuclei and small crystals at the nanoscale [12,32] is becoming feasible. At the same time, in the context of condensation, experiments using different techniques, such as thermal diffusion chambers, expansion chambers, and nozzles, have provided accurate measurements of the homogeneous nucleation rates of different substances, including water and alcohols [18]. In these experiments, the vapor is quickly supersaturated either by a temperature gradient or by a fast expansion, and the rate of appearance of the new phase is monitored by optical techniques. The results have revealed the shortcomings of CNT, in particular its common tendency to incorrectly predict the temperature dependence of nucleation rates. CNT generally overpredicts the rates at high temperatures and underpredicts the values at low temperatures. But the most serious problem is that these discrepancies are not a small factor or percentage, but can reach many orders of magnitude (Fig. 4). Perhaps the most dramatic example is the case of argon condensation. Argon is a noble gas with a nearly ideal behavior and its equilibrium properties can be reasonably described by simple intermolecular potentials. It was therefore expected to provide the perfect test of the validity of CNT. However, as shown in Fig. 4, the results of recent experiments have exposed our lack of understanding of nucleation [14]. The discrepancies between CNT predictions and the measured rates can reach more than 20 orders of magniwww.cat-science.cat

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tude! This is metaphorically equivalent to the difference between predicting flooding and not having a drop of rain throughout the entire age of the universe. This is a record hard to beat in any scientific discipline manifesting the peculiar non-equilibrium characteristics of nucleation.

Why does CNT fail? Classical nucleation theory uses macroscopic and equilibrium arguments to describe a non-equilibrium process that occurs in most cases at nanometer scales. Approaching the problem from this perspective has been the cause of long-standing controversies and of the misunderstanding of several key concepts. In addition, as evidenced by modern experimental and simulation techniques, the simplified scheme used in CNT is insufficient to accurately characterize the process of nucleation. Since CNT involves many crude approximations, there are many potential sources of error. Most of them are related to the so-called capillary approximation, which considers nucleating clusters as homogeneous spherical drops with a sharp interface and the same properties as the bulk liquid, including the surface tension of its planar interface. But the real interface is not sharp; rather, there is a relatively smooth change in properties from the liquid phase to the vapor. In addition, the properties of small droplets, for instance their density, may differ from those of the bulk macroscopic fluid. Another important factor is the potential influence of non-isothermal effects and temperature fluctuations. During the condensation process, whenever a molecule is incorporated into a liquid cluster the amount of latent heat that is released may be significant enough to heat it up. Since the nucleation rate is exponentially sensitive to temperature, temperature variations in the cluster will alter nucleation rates considerably. To properly thermalize the experimental system, a second inert gas, called a carrier gas, is usually employed. Its main role is to get rid of this extra heat. However, the potential influence of the pressure of this carrier gas is also questionable, since in some experiments it seemed to influence the nucleation rate. Other non-equilibrium effects, related, for instance, to the presence of temperature, density, or velocity gradients, may also lead to important effects. There have also been several theoretical concerns regarding self-consistency, the proper accounting of translational and rotational degrees of freedom, and the inability of CNT to account for the existence of the spinodal, (i.e., in which the barrier towards the formation of the new phase comCONTRIBUTIONS to SCIENCE 11:173-180 (2015)

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Fig. 4. Failure of CNT. Plot of the ratio of the nucleation rates of argon and nitrogen as measured in a cryogenic pulse chamber and the predictions by CNT, as a function of the inverse of temperature. The deviations in both cases range from 10 to 25 orders of magnitude; a clear temperature trend is also evident.

pletely vanishes) [20]. An accurate check of these factors within an experimental system is not easily performed. There are several features of nucleation that complicate the testing of hypotheses and the formulation of accurate predictions. The first is its non-equilibrium nature, given that nucleation is essentially a kinetic process that occurs at out-of-equilibrium conditions. The second is the activated nature of the process, which implies that the rate depends exponentially on the barrier; this makes nucleation extraordinarily sensitive to small variations of the parameters involved and thus very difficult to control and measure experimentally. Third, the appearance of the critical cluster that initiates a phase transition is a random process. Finally, the most important entity in nucleation, the tiny embryos of the new phase, are completely unstable, of nanometric dimensions, and typically contain very few molecules. Thus, an accurate measurement of their properties is very difficult to achieve. Experiments are hard to perform and thus have largely been done in substances of practical interest, such as water, alcohols, and sulfuric acid. Although their results have provided invaluable information about real nucleation rates, the extremely small time and length scales involved in nucleation have limited our capabilities to characterize the microscopic details of this process. www.cat-science.cat

At the theoretical level, there have been many important developments aimed at correcting and overcoming the limitations of CNT. Since the free energy barrier is the dominant factor in the nucleation rate, most theoretical work has focused on calculating accurately the equilibrium energy of critical nucleus formation. Theories aimed at improving the capillarity approximation and thus providing a more realistic description of the properties of nucleating clusters have achieved promising results. In particular, density functional theory [21], which describes the free energy of a cluster in terms of a smooth density profile, has solved some of the inconsistencies of CNT. However, this approach often requires the use of accurate functionals and intermolecular potentials, which are not available, not even for common substances such as water. Many phenomenological theories have tried to correct the nucleation barrier by incorporating extra terms. Together with newly formulated kinetic theories they have met with different degrees of success. Unfortunately, many developments that seem to correct some of the limitations or inconsistencies of CNT turn out to worsen the predictions (increase the deviations) with respect to the experimental results.

The challenge of understanding and accurately predicting nucleation Simulations are increasingly becoming an impressive tool to characterize the rate of appearance and the structure of nucleation events at the molecular scale [2]. The development and application of novel simulation algorithms has shed light on the nucleation rates of different substances, in addition to providing direct access to the molecular and thermodynamic details of the respective processes. Simulations mimic the motion of molecules using a variety of different methods [10]. They include molecular dynamic simulations, which essentially solve Newtonâ&#x20AC;&#x2122;s equations of motion for all molecules in the system, and Monte Carlo simulations, which explore different molecular configurations by proposing random displacements. Given the rare and stochastic nature of nucleation, more sophisticated techniques, such as umbrella sampling [2] and transition path sampling [4], have been developed to cover longer time scales. We recently developed an accurate simulation method to analyze the real non-equilibrium kinetics of nucleation. Our method was first applied to molecular dynamics simulations, because they best reproduce the dynamics of the process avoiding artifacts. However, molecular dynamics simulations 178

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are restricted to conditions characterized by very high supersaturations, which are often far from experimentally achievable. This limitation is due to the stochastic nature of the process, that requires good statistics, and the short time scales that can be achieved in simulations (typically on the order of microseconds). At these very high supersaturations, very small critical cluster sizes and nucleation barriers are expected; thus, the discrepancies should be larger and the limitations of CNT even more evident. To obtain accurate results, we developed a series of techniques based on the concept of mean first-passage time (MFPT). Our approach enabled the accurate evaluation of nucleation rates and critical cluster sizes [29] as well as a kinetic reconstruction of the full free energy landscape of cluster formation [26]. In addition, we analyzed and quantified the importance of finite size effects and different thermostating procedures [27,28,31], which we then used to design efficient simulation in terms of computational cost. Equipped with these techniques, we performed molecular dynamics simulations to explore the quantitative influence of several controversial aspects regarding the accuracy of CNT [30]. One question we sought to answer is the influence on nucleation rates of temperature fluctuations and the heating up of the nucleating embryos due to the unavoidable release of latent heat [28]. Although the proper definition of temperature and its fluctuations remains controversial for small systems, the important conclusion of our studies on vapor condensation, at least for non-associating vapors, is that the impact of temperature fluctuations follows the classical predictions of non-isothermal nucleation [9] and does not dramatically alter the nucleation rates. We were also able to unravel, both theoretically and using simulations, the controversial “pressure effect” associated with the thermalizing carrier gas and to accurately describe its influence on the rates of nucleation [25]. Finally, we also looked carefully at how nucleation takes place at extreme supersaturations, at the crossover between nucleation and spinodal decomposition [24]. The results of our studies suggest that, despite its crude approximations and simplifications, CNT turns out to be not very far off in its predictions. Amazingly, in most cases CNT is able to fairly accurately predict the number of molecules in the critical cluster, even for critical clusters containing as few as five or ten molecules, in which case all macroscopic thermodynamic assumptions are doomed to fail. In addition, while our simulations confirm that CNT incorrectly predicts the height of the nucleation barrier, it is off only by a constant, which depends solely on the temperature. However, www.cat-science.cat

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given the exponential dependence of nucleation rates on barrier height, even a single constant can yield a prediction that is off by several orders of magnitude. If the discrepancy lies solely with a temperature-dependent constant, what is the physical origin of this constant and how can CNT be corrected accordingly? This is still an unsolved question, but there are several encouraging clues regarding its answer. The incorporation of fluctuations, which are important for small clusters, and curvature corrections of the surface tension seem to be the most promising routes. Further investigations are underway. They seek to shed light on this fundamental process and to solve this classical problem and thereby open the door to understanding and thus controlling nucleation in more complicated situations. Acknowledgements. The work was partially supported by the Spanish MINECO through Grant No. FIS2011-22603. Competing interests. None declared.

References 1. Abraham FF (1974) Homogenous nucleation theory: the pretransition theory of vapor condensation. Academic Press, New York 2. Auer S, Frenkel D (2001) Prediction of absolute crystal-nucleation rate in hard-sphere colloids. Nature 409:1020-1023 doi:10.1038/35059035 3. Becker R, Döring W (1935) Kinetic theory for nucleation of supersaturated structures. Ann Phys 24:719-752 (in German) 4. Bolhuis PG, Chandler D, Dellago C, Geissler PL (2002) Transition path sampling: throwing ropes over rough mountain passes, in the dark. Annu Rev Phys Chem 53:291-318 doi:10.1146/annurev.physchem.53.082301.113146 5. Bruintjes RT (1999) A review of cloud seeding experiments to enhance precipitation and some new prospects. Bull Am Meteorol Soc 80:805820 doi:10.1175/1520-0477(1999)080<0805:AROCSE>2.0.CO;2 6. Debenedetti PG (1996) Metastable Liquids: concepts and principles. Princeton Univ. Press, Princeton, NJ 7. Fahrenheit DG (1724) Experimenta & observationes de congelatione aquae in vacuo factae a d. g. Fahrenheit. R. S. S. Philos Trans R Soc London 33:78-84 8. Farkas I (1927) Velocity of nucleation in supersaturated vapors. Z Phys Chem 125:236-242 (in German) 9. Feder J, Russell K, Lothe J, Pound G (1966) Homogeneous nucleation and growth of droplets in vapours. Adv Phys 15:111-178 doi:10.1080/00018736600101264 10. Frenkel D, Smit B (2002) Understanding molecular simulation: from algorithms to applications. Academic Press, San Diego, CA 11. Frenkel J (1955) Kinetic theory of liquids. Dover, New York 12. Gasser U, Weeks ERR, Schofield A, et al. (2001) Real-space imaging of nucleation and growth in colloidal crystallization. Science 292:258-262 doi:10.1126/science.1058457 13. Gunton JD, Shiryayev A, Pagan DL (2007) Kinetic pathways to crystallization and disease. Cambridge University Press, Cambridge, UK

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14. Iland K, Wölk J, Strey R, Kashchiev D (2007) Argon nucleation in a cryogenic nucleation pulse chamber. J Chem Phys 127:154506 doi:10.1063/1.2764486 15. Kalikmanov VI (2013) Nucleation Theory. doi:10.1007/978-90-4813643-8 16. Kashchiev D (2000) Nucleation: basic theory with applications. Butterworth-Heinemann, Oxford, UK 17. Kelton KF, Greer AL (2010) Nucleation in condensed matter. Elsevier, Amsterdam, the Netherlands 18. Laaksonen A, Talanquer V, Oxtoby DW (1995) Nucleation - measurements, theory, and atmospheric applications. Annu Rev Phys Chem 46: 489-524 doi:10.1146/annurev.pc.46.100195.002421 19. Lifshitz EM, Pitaevskii LP (1981) Physical kinetics. Pergamon Press, Oxford, UK 20. Oxtoby DW (1998) Nucleation of First-Order Phase Transitions. Acc Chem Res 31:91–97 doi:10.1021/ar9702278 21. Oxtoby DW, Evans R (1988) Nonclassical nucleation theory for the gas– liquid transition. J Chem Phys 89:7521-7530 22. Seinfeld JH, Pandis SN (2006) Atmospheric chemistry and physics: from air pollution to climate change. John Wiley & Sons, New Jersey 23. Volmer M, Weber A (1926) A nuclei formation in supersaturated states. Z Phys Chem 119:277-301 (in German) 24. Wedekind J, Chkonia G, Wölk J, et al. (2009) Crossover from nucleation to spinodal decomposition in a condensing vapor. J Chem Phys 131: 114506 doi:10.1063/1.3204448 25. Wedekind J, Hyvärinen A-P, Brus D, Reguera D (2008) Unraveling the “Pressure Effect” in Nucleation. Phys Rev Lett 101:125703 doi:10.1103/PhysRevLett.101.125703

26. Wedekind J, Reguera D (2008) Kinetic reconstruction of the free-energy landscape. J Phys Chem B 112:11060-11063 doi:10.1021/jp804014h 27. Wedekind J, Reguera D, Strey R (2006) Finite-size effects in simulations of nucleation. J Chem Phys 125:214505 doi:10.1063/1.2402167 28. Wedekind J, Reguera D, Strey R (2007) Influence of thermostats and carrier gas on simulations of nucleation. J Chem Phys 127:064501 doi:10.1063/1.2752154 29. Wedekind J, Strey R, Reguera D (2007) New method to analyze simulations of activated processes. J Chem Phys 126:134103 doi:10.1063/1.2713401 30. Wedekind J, Wölk J, Reguera D, Strey R (2007) Nucleation rate isotherms of argon from molecular dynamics simulations. J Chem Phys 127:154515 doi:10.1063/1.2784122 31. Wilhelmsen Ø, Reguera D (2015) Evaluation of finite-size effects in cavitation and droplet formation. J Chem Phys 142:064703 doi:10.1063/1.4907367 32. Yau ST, Vekilov PG (2000) Quasi-planar nucleus structure in apoferritin crystallization. Nature 406:494-497 doi:10.1038/35020035 33. Zandi R, van der Schoot P, Reguera D, et al. (2006) Classical nucleation theory of virus capsids. Biophys J 90:1939-1948 doi:10.1529/biophysj.105.072975 34. Zeldovich JB (1943) On the theory of new phase formation: cavitation. Acta Physicochim URSS 18:1-22

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SOFT MATTER Institut d’Estudis Catalans, Barcelona, Catalonia

OPENAACCESS

CONTRIB SCI 11:181-188 (2015) doi:10.2436/20.7010.01.229

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Ion transport through biological channels Jordi Faraudo,1,* Marcel Aguilella-Arzo2 Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Bellaterra, Barcelona, Catalonia. 2Grup de Biofísica, Departament de Física, Universitat Jaume I, Castelló de la Plana

Summary. The transport of ions across single-molecule protein nanochannels is important both in the biological context and in proposed nanotechnological applications. Here we discuss these systems from the perspective of non-equilibrium physics, and in particular, whether the concepts underlying the physics of diffusive and electrokinetic transport can be employed to predict and understand these systems. [Contrib Sci 11(2): 181-188 (2015)]

*Correspondence: Jordi Faraudo Institut de Ciència de Materials de Barcelona (ICMAB-CSIC) Til·lers, s/n Campus de la UAB 08193 Bellaterra, Barcelona, Catalonia Tel. +34-935801853 E-mail: jfaraudo@icmab.es

A brief introduction to biological ion channels All living cells are immersed in a solution of salts and separated from the external environment by the cell membrane. To precisely regulate the entry and exit of ions and other molecules, cells are equipped with structures that control ion passage bidirectionally. These remarkable nanostructures are made up of proteins that make subtle use of the principles of non-equilibrium physics to achieve their essential

functions. There are two different types of ion transport across the cell membrane: active transport by ion pumps or ion transporters requires energy input from the cell, whereas passive transport across selective ion channels occurs without the direct consumption of energy. Active transport acts against the natural flow of ions. A classical example is the sodium-potassium pump, discovered in 1957 by Jens Christian Skou (Nobel Prize in Chemistry in 1997). This pump is responsible for maintaining a high concentration of K+ ions and a relatively low concentration of Na+

Keywords: ionic transport · protein channels · non-equilibrium physics · Poisson-Nernst-Planck equation · molecular dynamics ISSN (print): 1575-6343 e-ISSN: 2013-410X

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Ion transport

Fig. 1. Lateral view of the central region (the so-called selectivity filter) of the potassium channel (KSCA) of Streptomyces lividans (Protein Data Bank accession code: 1BL8). (A) Cartoon view of the channel inserted in the cell membrane (indicated by horizontal lines). There are three potassium ions (blue spheres) in the channel, together with an oxygen atom from a hydration water molecule (red sphere), as found in the structure obtained by X-ray. The red arrow indicates the transmembrane direction followed by the ions. (B) Surface representation of the potassium channel using a 0.7Ă&#x2026; probe radius. The orientation and color code are the same as in the left panel. To visualize the narrow permeation pathway available to ions, a cut has been made near the permeation pathway, in a plane perpendicular to the observerâ&#x20AC;&#x2122;s view.

ions inside the cell. In each cycle, the pump transports two K+ from the exterior into the cell and releases three intracellular Na+ to the exterior. During this exchange, energy is supplied by the hydrolysis of one ATP molecule. The pump is responsible for maintaining a concentration gradient of sodium and potassium ions across the cell membrane. This gradient is essential for many biological functions and for establishing the resting electrical potential of the cell membrane. Active transport accounts for a substantial part of the energy budget of animal cells, but a full understanding of the energetic requirements of active transport remains elusive [18,19]. In passive transport, ions flow across ion channels by following the spontaneous flow dictated by gradients of concentration and electrostatic potential [17]. Ion channels are proteins embedded in the cell membrane; they form hydrophilic channels connecting the extracellular and intracellular environment. The importance of ion channels can be appreciated by considering that a significant fraction of DNA-encoded proteins are ion channels, which is the reason for the many diseases linked to their abnormal functioning (known as channelopathies [14]). An important property of ion channels is their selectivity: only certain ions are transported www.cat-science.cat

across the channel. For example, only K+ ions can significantly cross potassium channels, a family of channels widely found among organisms [16]. Figure 1 shows an example of the potassium channel KSCA from Streptomyces lividans. This extremely narrow protein channel allows the passage of ions in single file. Other ion channels are less specific as they are selective for particular kinds of ions, for example, cations or anions. This class of channels includes outer membrane bacterial porin F (OmpF) from Escherichia coli (Fig. 2). OmpF is a relatively wide channel in that it allows the simultaneous permeation of both cations and anions (in hydrated form) as well as the entry of relatively large molecules, such as antibiotics and polymers. In the case of monovalent electrolyte solutions (KCl, NaCl, LiCl, CsCl) the channel has slight cationic selectivity, i.e., the current from cations is larger than that from anions [1,2]. Interestingly, the channel blocks the passage of multivalent cations (such as Mg2+ or La3+), in which case the current is due only to anions [3]. These two channels are representative of the narrow and wide ionic channels that have been extensively studied, both experimentally and theoretically. The basic transport concepts emerging from these studies are the subject of this article. 182

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Fig. 2. Trimeric OmpF protein channel from Escherichia coli (Protein Data Bank accession code: 2OMF). (A) Top view of a cartoon representation of the trimeric protein channel. (B) Surface representation of one of the channels of the trimeric protein. The sizes of the protein and the narrow constriction zone are shown.

Molecular dynamics simulations of ion channels Molecular dynamics (MD) simulations consist of solving numerically (using a computer) the Newtonian equations of motion in a system made up of a large number of atoms, taking into account their mutual interactions and external thermodynamic constraints (such as the presence of a barostat or a thermostat) and/or external fields (such as an applied electric field). In the case of ion channels, this technique is relevant as long as the 3D structure of the channel and the atomic details thereof are available. Due to the impressive advances in protein crystallography, atomistically resolved structures are available for a large number of channels (at the time of this writing, the Protein Data Bank database has 3882 entries for ion channels). These can be investigated in MD simulations, as a kind of computational microscope, to obtain dynamic images of the respective systems and their transport processes at the atomic level. Typical simulation scales are of the order of 105 atoms, tens of nm, and hundreds of ns [10]. These scales allow studies of ion-channel interactions, ion kinetics, and calculations of the conductivity properties of the channels. However, because these simulations require huge amounts of computer power, they cannot be employed to exhaustively analyze the behavior of a protein channel under different conditions (for example, differwww.cat-science.cat

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ent ion concentrations). Rather, these MD simulations are, at least thus far, limited to investigating specific, fundamental questions. More extensive calculations can then be performed by combining the results of MD simulations with theoretical approaches or by employing other simulations that use less precise but also less computationally demanding techniques, such as Brownian dynamic simulations [8]. The results of these MD simulations provide insights into the physical mechanisms of ion transport across ion channels.

A minimal model of an ion channel To gain a better understanding of the physical properties of ion channels, it is instructive to consider a minimalistic model (Fig. 3), in which the ion channel is modeled as a cylindrical pore of known radius a through a low dielectric medium of width L (the membrane). The low dielectric constant of the membrane (Îľr~2â&#x20AC;&#x201C;3) presents a strong barrier to the passage of electrical charges, so that ions are forced to cross through the pores. In our simplified channel model, selectivity is achieved primarily by the fixed charges anchored to the channel. These electric charges exert repulsive forces on ions of the same sign, reducing their relative numbers inside the channel, a property known as electrostatic exclusion. The opposite occurs with charges of opposite sign, and their concentration inCONTRIBUTIONS to SCIENCE 11:181-188 (2015)

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Ion transport

Fig. 3. A simple model of an ionic channel in equilibrium and in the presence of concentration and electric potential gradients. (A) A low dielectric region representing the cell membrane (gray area) is traversed by an aqueous pore channel in equilibrium with an ionic solution. The ions in the solution region are represented by blue (negative charge) and red (positive charge) spheres with a charge sign. Some of the negative charges are anchored to the pore wall (blue anchored spheres), causing ion selection inside the pore channel and thus exclusion (co-ions) or favoring (counter-ions) of ions from the ionic solution, according to their electric charge. This leads to a passive charge selectivity of the channel. (B) Same as (A) but now in the presence of concentration and electrostatic potential gradients. A concentration gradient through the system creates a net flow of ions from the side with a higher concentration to the side with a lower concentration (the light red arrow in the pore region). An electric field that is axially applied through the system (light yellow arrow) causes the positive and negative ions to move in opposite directions (green arrows).

side the channel increases (Fig. 3A). If this channel is placed in contact with an electrolyte-containing aqueous solution (e.g., KCl), then in the equilibrium state the concentrations of anions www.cat-science.cat

and cations will be different (Fig. 3A). Due to the negative charge of the channel, there will be an excess of K+ ions over Cl– ions in order to compensate for the fixed charge present in the channel walls. Thus, a positively charged channel becomes an anion-selective channel, and a negatively charged channel a cation-selective channel. However, the effect of the wall charges is only relevant at distances sufficiently small from the charged wall and declines rapidly with distance. The passive transport of ions can be induced by a concentration gradient, an electrostatic potential difference, or both (Fig. 3B). Cells are characterized by substantial concentration gradients of ions, such as K+, Na+, and Cl–, and membrane potentials of the order of a few hundreds of mV are common. In the laboratory, larger concentration and/or potential gradients can be easily induced by the experimenter. From a physical standpoint, passive transport through ion channels can be described, in the continuum approach, as the movement of ion species in response to the electrochemical potential gradient. The concentration gradient induces an ion flux in the direction of the low concentration, proportional to the concentration gradient and to the diffusion coefficient D, as dictated by Fick’s law of diffusive transport. The presence of an electrostatic potential difference gives the ions a drift velocity in the same or in the opposite direction of the electric field, depending on whether the ion is positively or negatively charged and proportional to the gradient of the electrostatic potential. Mathematically, the electrodiffusion of an ion species is described by the Nernst-Planck equation (Eq. 1): (1) where c is the local concentration of electrolyte, D is the diffusion coefficient, e is the elementary charge, k B is the Boltzman constant, T is the absolute temperature, and z is the valence of the ion species. The Nernst-Planck equation contains two terms: the first corresponds to diffusion and is also known as the Fick equation, while the second describes the motion of charged ions under the influence of electric fields. Eq. (1), together with the Poisson equation (Eq. 2) ∇(ε∇ϕ ) =– ρ

(2)

describing the connection between the electrostatic potential ( ϕ ) and its sources, i.e., the electric charge density ( ρ ), forms a closed system of equations known as the Poisson-NernstPlanck equations (PNP). The surface charge density σ of the channel walls also enters into the PNP equations as a boundary condition. Before the advent of computers, the system de184

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scribed by the PNP equations was solved by applying various simplifying approximations, such as the existence of a constant electric field across the membrane (sometimes referred to as the Goldman hypothesis); the important theoretical results obtained in this way are still widely used today. One of the most famous equations is that of Goldman-Hodgkin-Katz (GHK), for the resting potential across a membrane [11]. The advent of computers allowed the numerical resolution of the differential equations comprising the PNP equations. Today, it is possible to numerically solve the equations for a system such as the one described in Fig. 3B using a standard personal computer. Real channels are more complex than the simple illustration provided in Fig. 3 (compare, for example, Fig. 3 with Figs. 1 and 2). Nonetheless, Eqs. (1) and (2) can be solved such that they include all the structural information of the protein in its full three dimensions, without a substantial increase in the computational cost. However, there are other simplifications present in the PNP equations that have prompted the search for fixes for some of the resulting shortcomings. Among these fixes are the inclusion of the finite size of the ions [15], as steric effects (although there already are generalizations to incorporate correlations between ions), and improved numerical algorithms based on finite element, finite volume, etc. [6]. The continuum approach, implicit in Eqs. (1) and (2), assumes that the system under study is much larger than the typical distance between its elementary constituents, that is, the atoms, ions, and molecules forming the system, and thus can be described mathematically by fields. This approach, however, may be a bit harsh, given that ion channels have dimensions in the nanometer scale, which is only an order of magnitude greater than the size of many atomic species. The question is how this complexity affects the validity of the basic physical principles and to what extent it is relevant for the prediction of transport. This is discussed in the following sections, using results from the PNP equations and from MD simulations.

The physics of transport in wide ion channels OmpF provides a useful model of a wide ion channel (Fig. 2). Like other wide channels it has a net charge (at pH 7, the charge of an OmpF pore is about –11e), due to the complex arrangement of the local positive and negative charges from amino acids. Some of these charges are located at the surface while others are buried inside the protein but still influwww.cat-science.cat

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ence the ions. In these channels, the electrostatic exclusion mechanism described schematically in Fig. 3A is operative. For example, computer simulations [10,13] have shown that in the presence of 1 M KCl the inner aspect of the OmpF channel has an average of 7 Cl– anions and 11.8 K+ cations (Table 1). In other words, outside the channel, there is one K+ cation for each Cl– anion but inside the channel there is an average of 1.68 cations (K+) for each Cl– ion. Under an applied voltage, the contribution of K+ to the current is larger than that of Cl–, but the ratio is 1.4, which is smaller than expected based on the channel population. This is due to a greater reduction of the K+ mobility inside the channel. The concept that the excess of cations inside the channel is due to electrostatic compensation of the protein charge (charge neutralization) can be further tested by considering mutants of this protein channel. In the OmpF-CC mutant, two negatively charged amino acids (one aspartic and one glutamic acid) present inside the channel are replaced by two neutral cysteine amino acids. In the OmpF-RR mutant, these two negatively charged amino acids are replaced by positively charged arginines. Therefore, the protein charge changes by +2e in OmpF-CC and by +4e in OmpF-RR (vs. the wild type OmpF channel). Our simulations [10] indicate that the charge due to ions inside the mutant channels changes to compensate for these alterations and maintain electroneutrality (Table 1). Inside the OmpF-CC channel, there is an increase of Cl– and a reduction of K+ ions to compensate for the increase of +2e of the channel. In OmpF-RR, the increase of +4e is compensated by a concomitant increase in the Cl– population inside the channel (Table 1). Interestingly, for both mutants the current attributable to K+ is extremely low and almost all ion transport is due to Cl– (with a flux of Cl– similar to that observed for the wild type). In addition to their effect on the total charge (which determines the population of ions), the small changes in the channel wall that are caused by these mutations also strongly impact the mobility of the K+ ions inside the channel, which is severely reduced. This effect illustrates that the simple relation between the static and dynamic case (compare Fig. 3a and Fig. 3b) is lost in a protein channel. A further illustration of these complexities of ionic channels is seen in the behavior of the channels in response to multivalent cations. In the presence of 1 M MgCl2, there is again an excess of cations over anions inside the channel. In bulk solution, there is one Mg2+ cation for each two Cl– anions, whereas inside the channel there are 1.44 Mg2+ cations for each two Cl– anions. However, because these Mg2+ cations are tightly bound to certain anionic amino acids, their mobility is extremely low and almost all the current is due to anions [3]. CONTRIBUTIONS to SCIENCE 11:181-188 (2015)

Ion transport

Table 1. Charge balance inside the OmpF ion channel as computed by molecular dynamics simulations (data from [10]). Qc is the total charge of a (monomer) channel at neutral pH. The charge balance is the difference in charge between the mutant proteins and the wild type (∆Qc) and their corresponding difference in ionic charge inside the channel (∆Qions). All charges are given in units of the elementary charge e Number of ions inside the channel

Charge balance

Cl–

K+

∆Qc

∆Qions

OmpF (wild type)

–11

7.0 ± 0.1

11.8 ± 0.1

–

CC mutant

–9

7.73 ± 0.03

11.0 ± 0.5

–1.5 ± 0.7

RR mutant

–7

10.13 ± 0.03

11.5 ± 0.5

–3.4 ± 0.7

fusion theory described in the previous section assumes that ion transport can be described by the superposition of a diffusive motion (characterized by the diffusion coefficient D) and an electrostatic drift resulting from the applied voltage difference. However, given the complex interactions between ions and channel walls, the motion of atoms is likely to be more complicated. For example, how do we know that the diffusion coefficient D of a given ion inside the channel is equal to the diffusion coefficient measured for this same ion in a simple electrolyte solution? This question has been addressed in many studies, whose results suggest that the diffusion coefficient inside the channel is lower than the diffusion coefficient of the same ion in bulk electrolyte solutions. Our own simulations indicate that, inside the OmpF channel, the diffusion coefficients of K+ and Cl– ions are substantially lower than their

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Another complexity of ion channels, which is missing in the simple picture provided in Fig. 3, is that not only the total charge of the channel matters for electrostatic exclusion but also the fact that the walls of the channel have patches or regions of opposite charge. This charge distribution creates welldefined regions in which cations or anions are excluded. For example, in computer simulations of OmpF, [10,13], K+ and Cl– ions are distributed inside the channel, occupying well-defined regions guided by the charge distribution: cations are located near walls with anionic amino acids and anions near walls with cationic amino acids (Fig. 4). Recently, these highly local electrostatic exclusion effects have been observed experimentally using anomalous X-ray diffraction [9]. Another interesting aspect that can be explored by computer simulations is the nature of the motion of ions inside the protein channel. As noted above, the basic electrodif-

Fig. 4. Different regions occupied by ions inside the OmpF channel (shown as a light blue cage) according to molecular dynamics simulations [10]. (A) region occupied by Cl– ions (shown in orange); (B) region occupied by K+ cations (shown in gray). The charged amino acids (positive Arg and negative Glu) responsible for the ionic distribution (shown in van der Waals representation) are indicated by arrows.

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bulk values [5] (by a factor of about 4 in the wider regions of the channel and by a factor of about 10 in the constriction located in the center of the channel). In addition, we were able to show that the drift of the ions in the direction of the applied voltage is strongly influenced by an additional force, due to changes in the geometry of the channel. This is an entropic effect, predicted by non-equilibrium thermodynamics [20], in which ions are dragged toward regions with larger cross-sectional areas. These atomic-level studies can be complemented by examining several macroscopic aspects. For example, the transport of ions under an applied voltage is usually ohmic in ionic channels, which means that the current intensity ( I ) and the applied voltage difference ( ∆V ) follow Ohm’s law: ∆V = I × R , where R is the resistance. In general, it is customary in this type of application to report the conductance, G = 1/ R . The conductance of a channel for a given concentration of electrolyte solution can be calculated. For example, using MD simulations, after the application of an external voltage the flow of ions can be counted to determine both the stationary current I and the conductance G [10]. These simulations yield reasonably accurate results. For example, in the presence of 1 M KCl, simulations predict [10] a conductance of 2.7 nanoSiemens (nS), in agreement with experimental results. Other quantities of interest for these channels are more difficult to predict from MD simulations, but they can be predicted with the help of PNP equations. A particularly important example is the reversal potential (RP), which is defined experimentally as follows. In a channel under a concentration gradient, the flow of ions will behave according to Fick’s law. The RP is defined as the (external) electrostatic potential difference that has to be applied to counteract the effect of this concentration gradient and to obtain a net current of zero. This is difficult to determine directly from simulations, because it requires the maintenance of a concentration gradient (which is difficult in simulations) and the testing of different voltages to find the RP. However, this is an example of a quantity that can be obtained as a combination of results from MD simulations and a 3D version of the PNP equations that accounts for the detailed channel geometry. From the MD simulations, the binding sites of ions can be determined and suitable values for the diffusion coefficients of ions inside the channel proposed. This information can be entered into a PNP calculation. Figure 5 provides an example of this type of calculation, made for the OmpF channel in the presence of KCl. There is excellent agreement with the experimental results. www.cat-science.cat

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Fig. 5. Reversal potential of the OmpF channel at neutral pH (pH = 7) at different “trans” (extracellular) side KCl concentrations. The concentration on the “cis” (intracellular; negative z coordinates in the Protein Data Bank coordinates) side is maintained to yield a ccis/ctrans ratio of 0.2. The blue line shows the experimental data and the orange points the theoretical data obtained through numerical solutions of the 3D PNP equations [Eqs. (1) and (2) in the text]. The full 3D structure, obtained from the Protein Data Bank (code: 2OMF) was used as input.

The physics of transport in narrow channels As in the case of wide channels, the basic concepts describing the mechanism of ion transport through narrow channels, such as the potassium channel shown in Fig. 2, need further refinement. Ions cross the channel of very narrow pores in single file, with fewer ions occupying the narrower part of the channel, which acts as a selectivity filter. The transport mechanism in these narrow channels combines the presence of binding sites (due to strong, attractive, and highly specific ion-channel interactions) and ion-ion repulsion. For example, in the case of the KcsA K+ channel of Fig. 1, conduction proceeds as follows [4]: Inside the channel, there are five specific sites for K+ ions. The channel has two states, one with two K+ ions inside the selectivity filter of the channel and another with three K+ ions (this latter state is shown in Fig. 1). In the state with two K+ ions, the outer K+ ion is adsorbed in a deep free energy well near the exit, where it is trapped. As a new ion enters the channel, the three ions became located as shown in Fig. 1 and the strong repulsion of the two other K+ ions forces the outer K+ ion to leave the channel. Interestingly, without the two other K+ ions, this outer K+ can remain adsorbed indefinitely. Inside the selectivity filter, the K+ ions are hydrated by a single water molecule (Fig. 1) and full hydration is recovered at the more exterior site. During this very rapid (about 108 ions/s) vacancy transport, the flexibility CONTRIBUTIONS to SCIENCE 11:181-188 (2015)

Ion transport

of the protein plays a substantial role, as a rigid protein is unable to correctly accommodate the unsolvated K+ ions and their transitions between adjacent states. Mechanisms similar to that of the potassium channel have been determined for other channels, where transport proceeds single file and involves strong ion-channel affinity and ion-ion repulsion. For example, in the transport of Cl– ions through the CmCLC transporter, a chloride ion channel [7], a single Cl– ion has a strong affinity for a binding site at the central region of the channel and thus remains adsorbed. The entrance of a second Cl– induces both a strong repulsion to the previously adsorbed Cl– ion and the deprotonation of a particular amino acid. The released proton is transported to the exterior of the cell and the two Cl– ions are transported toward the interior.

Conclusions Biophysical studies of transport in ion channels have pushed the concepts of nonequilibrium physics to their limits in describing transport processes. In these systems, gradients are extremely large (a 100-mV drop along a 4-nm-thick membrane results in an electric field of 2.5 × 107 V/m) and transport processes occur in extremely narrow regions (such as the narrow constriction zone of OmpF, which is < 1 nm across). Nonetheless, approaches based on diffusion coefficients and classical electrodiffusion theory are still useful. They are complemented by novel techniques such as MD simulations with atomistic resolution, which have also revealed the limits of applicability of non-equilibrium physics. Using these physical techniques will aid biologists in elucidating the relation between the structural details of the proteins and their biophysical mechanisms of operation, the structure-function relationship. Acknowledgements. This work was supported by the Spanish Government (grant FIS2011-1305 1-E) and of University Jaume I grant P1·1B2012-16. All images of protein structures have been made using the free software Visual Molecular Dynamics software [12].

Competing interests. None declared.

References 1. Aguilella VM, Alcaraz A (2008) The ionic selectivity of large protein ion channels. Contrib to Sci 4:11-19 doi:10.2436/20.7010.01.311 2. Aguilella-Arzo M, García-Celma JJ, Cervera J, Alcaraz A, Aguilella VM (2007) Electrostatic properties and macroscopic electrodiffusion in OmpF porin and mutants. Bioelectrochemistry 70:320-327doi:10.1016/ j.bioelechem.2006.04.005 3. Aguilella-Arzo M., Calero C, Faraudo J (2010) Simulation of electrokinetics at the nanoscale: inversion of selectivity in a bio-nanochannel. Soft Matter 6:6079 doi:10.1039/C0SM00904K 4. Bernèche S, Roux B (2001) Energetics of ion conduction through the K+ channel. Nature. 414:73-7 doi:10.1038/35102067 5. Calero C, Faraudo J, Aguilella-Arzo M (2011) First-passage-time analysis of atomic-resolution simulations of the ionic transport in a bacterial porin. Phys Rev E 83:1–12 doi:10.1103/PhysRevE.83.021908 6. Chaudhry JH, Comer J, Aksimentiev A, Olson LN (2014) A stabilized finite element method for modified poisson-nernst-planck equations to determine ion flow through a nanopore. Commun Comput Phys 15 doi:10.4208/cicp.101112.100413a. 7. Cheng MH, Coalson RD (2012) Molecular dynamics investigation of Cl– and water transport through a eukaryotic CLC transporter. Biophys J 102:1363-71 doi:10.1016/j.bpj.2012.01.056 8. Chung S-H, Corry B (2005) Three computational methods for studying permeation, selectivity and dynamics in biological ion channels. Soft Matter 1:417 doi:10.1039/B512455G 9. Dhakshnamoorthy B, Raychaudhury S, Blachowicz L, Roux B (2010) Cation-selective pathway of OmpF porin revealed by anomalous X-ray diffraction. J Mol Biol 396:293-300 doi:10.1016/j.jmb.2009.11.042 10. Faraudo J, Calero C, Aguilella-Arzo M (2010) Ionic partition and transport in multi-ionic channels: a molecular dynamics simulation study of the OmpF bacterial porin. Biophys J 99:2107-2115 doi:10.1016/j.bpj.2010.07.058 11. Hille B (2001) Ion channels of excitable membranes. 3rd edition. Sinauer Associates 2001-07-01 12. Humphrey W, Dalke A, Schulten K (1996) VMD: Visual molecular dynamics. J Mol Graph 14:33-38 doi:10.1016/0263-7855(96)00018-5 13. Im W, Roux B (2002) Ion permeation and selectivity of ompf porin: a theoretical study based on molecular dynamics, brownian dynamics, and continuum electrodiffusion theory. J Mol Biol 322:851-869 doi:10.1016/S0022-2836(02)00778-7 14. Kim J-B (2014) Channelopathies. Korean J Pediatr 57:1-18 15. Koumanov A, Zachariae U, Engelhardt H, Karshikoff A (2003) Improved 3D continuum calculations of ion flux through membrane channels. Eur Biophys J 32:689-702 doi:10.1007/s00249-003-0330-y 16. Littleton JT, Ganetzky B (2000) Ion channels and synaptic organization. Neuron 26:35-43 doi:10.1016/S0896-6273(00)81135-6 17. Lodish H, Berk A, Zipursky SL, Matsudaira P, Baltimore D, Darnell J (2000) Cotransport by symporters and antiporters. In: Molecular Cell Biology 4th ed. W. H. Freeman, New York, NY, USA 18. Pollack GH (2001) Cells, gels and the engines of life: A new, unifying approach to cell function. Ebner and Sons Publ., Seattle, WA, USA 19. Qian H, Autzen HE (2012). A little engine that could: ATP-powered electrical battery and heater inside cells. Biophys J 103:1409-10 doi:10.1016/j.bpj.2012.08.047 20. Reguera D, Schmid G, Burada P, Rubí J, Reimann P, Hänggi P (2006) Entropic transport: kinetics, scaling, and control mechanisms. Phys Rev Lett 96:130603 doi:10.1103/PhysRevLett.96.130603

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SOFT MATTER Institut d’Estudis Catalans, Barcelona, Catalonia

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Non-equilibrium dynamics of fluids in disordered media

Jordi Ortín Grup de Física No Linial, Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona, Catalonia. Universitat de Barcelona Institute of Complex Systems (UBICS), Barcelona, Catalonia

Summary. Fluid flows in disordered media are present in natural and industrial processes such as soil irrigation and secondary oil recovery. These flows display complex spatial and temporal non-equilibrium dynamics arising from the heterogeneities of the medium. Average magnitudes are not sufficient to allow the complexity of their dynamics to be captured. Deeper insight can be gained from a scaledependent statistical analysis of the fluctuations. Here we introduce the basic laws governing fluid flows in disordered media. Focusing on two-fluid displacements with a well-defined interface, we discuss several non-equilibrium dynamic features that include scale-invariance, avalanches, non-Gaussian fluctuations, and intermittency. [Contrib Sci 11(2): 189-198 (2015)] Correspondence: Jordi Ortín Departament de Física de la Matèria Condensada Universitat de Barcelona Martí i Franquès, 1 08028 Barcelona, Catalonia Tel. +34 934 021 189 E-mail: jordi.ortin@ub.edu

Introduction Fluid flow through disordered media occurs in geological, agricultural, and industrial processes of great importance. Crude oil and gas, for instance, is present in large natural reservoirs and impregnates porous rocks. In secondary oil recovery, water or gas is injected into the oil reservoir in order to displace the oil through the medium and drive it to the production wellbore 1. Underground water similarly flows through geological formations consisting of porous solids, granular materials, and fractured rocks. Flow in either

porous or fractured media is, thus, a central topic in the fields of petrology and hydrology. It is also highly relevant in challenging industrial processes, such as the filtering of chemicals and contaminants. The complexity of flows in disordered media arises from the heterogeneous structure of these media. The relevant features encompass a very wide range of spatial scales. The smallest correspond to pore sizes, typically ranging from 1 nm (micropores) to > 50 nm (macropores). Flow at pore scales is described by the classical hydrodynamic equations of mass continuity and momentum conservation. A continuum

Keywords: fluid flows in disordered media · Darcy’s law · drainage and imbibition · scale invariance · fluctuations and intermittency ISSN (print): 1575-6343 e-ISSN: 2013-410X

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description, on a coarser scale called the Darcy scale, can be obtained by proper averaging over a representative volume of the pore space. In averaging, the assumption of a random distribution of pore-scale properties is usually not sufficiently accurate. The presence of extended spatial correlations in natural porous formations (e.g., the clustering of low- and high-porosity areas, or the strong correlation between the volume of the throats and the average volume of the pores to which they are connected) must be taken into account. Flow at this scale is described by the equations of continuum mechanics of porous media, of which Darcy’s law (discussed below) plays a central role. The final scale of interest is the field scale, which can extend over kilometers. In this case, the large-scale heterogeneities in the physical properties of the medium are an important consideration. Consistently relating the micro-scale flow quantities to their corresponding Darcy-scale variables is key to understanding transport and transport-controlled processes such as chemical reactions in porous media [20]. Here we focus on two-phase fluid displacements in disordered media. In this kind of flow, a fluid originally residing in the disordered medium is displaced by a second, invading fluid. There is a well-defined (although eventually highly-distorted) interface separating the two fluids. Secondary oil recovery, soil irrigation, and several other relevant fluid displacements belong to this family of flows. As discussed below, both the morphology and the dynamics of the interface are dramatically influenced by the randomness inherent to the heterogeneous structure of the disordered medium. In considering the different features of this problem, several advanced concepts of non-equilibrium statistical physics are encountered, including scale invariance, complex correlations, anomalous fluctuations, and intermittency.

Darcy’s law and permeability Henry Darcy, a pioneer in hydrology studies, carried out systematic experiments on the flow of water through beds

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Fig. 1. Sketch of the apparatus used by Henry Darcy to verify his law Q= − A (κ / µ ) ( P − P ) / L , where κ and µ are the hydraulic permeability of the medium and the dynamic viscosity of the fluid, respectively. The background image is reproduced from Darcy’s book [4]. 2

of sands. In his monograph “Les fontaines publiques de la ville de Dijon” [4], published in 1856, he established that the discharge rate (volume per unit time) of fluid flowing through a long cylinder filled with sand was proportional to the pressure drop between the two ends of the cylinder (inlet and outlet). This linear relationship is analogous to Ohm’s law of electricity or Fourier’s law of heat conduction. However, like those other phenomenological relations, it has limited validity. Darcy’s law applies to creeping flows, for which the Reynolds number (Re) is < 1 2. Apart from geometrical factors, the proportionality coefficient in Darcy’s law is the ratio of an intrinsic property of the medium, its hydraulic permeability, to an intrinsic property of the fluid, its dynamic viscosity. The concept of hydraulic permeability makes sense only on a coarse-grained spatial scale (the Darcy scale), above the pore scale. Moreover, to have constant permeability a medium will

1 This technique can result in the recovery of 20–40% of the original oil in place. This is a significant improvement over primary recovery, in which crude oil is driven into the wellbore by the combined action of the natural pressure of the reservoir and gravity, in which case only about 10% of a reservoir’s original oil is typically recovered in place. 2 The Reynolds number Re is a dimensionless ratio of inertial to viscous forces. It is typically used to distinguish between laminar (low Re) and turbulent (high Re) flows. 3 In a medium with a permeability of 1 darcy, a pressure gradient of 1 atm/cm on water (dynamic viscosity 1 mPa·s) produces a discharge rate of 1 cm3/s through a cross-section of 1 cm2.

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have to exhibit rather homogeneous and isotropic porosity on the Darcy scale. In geology and petroleum engineering permeability is measured in darcy units, with 1 darcy equal to 0.9869233 (µm)2. Typical permeabilities of soils are 105 darcy for gravel, 1 darcy for sand, and 0.01 darcy for granite 3. Even though Darcy derived his law phenomenologically, on the basis of his own experimental observations, it can be derived also from the Navier-Stokes equations of fluid mechanics as applied to stationary, creeping, incompressible flow, assuming that the viscous resisting force depends linearly on the fluid velocity. Analytic calculations can be carried out on simple geometries. Thus, an analysis of laminar flow through a long cylindrical capillary of radius R shows that the permeability of the capillary is given by R 2 / 8 . Similarly the hydraulic permeability of a Hele-Shaw cell, consisting of two large parallel plates separated by a narrow gap spacing b , is given by b 2 / 12 . Combining simple geometries of this kind it is possible to derive approximate expressions of the hydraulic permeability of more complex materials, such as bundles of capillaries and beds of closely packed spheres. This approach to modeling the geometrical complexity of actual disordered media is an active field of research (Fig. 1).

Two-phase displacements Many flows of interest involve the presence of an interface separating two different phases. Oil displacement by water in secondary oil recovery is an important example of two-phase displacement in a porous medium, in which the dynamics of the front between the two immiscible phases determine the efficiency of the recovery process. Printing, coating, impregnation, soil irrigation, and the rise of sap in plants are also examples of two-phase displacements. If a cube of sugar or a biscuit is dipped in a cup of coffee or tea, the liquid quickly invades the pore spaces of the solid material, displacing the air initially present4. The physical phenomenon behind this process of spontaneous fluid invasion, which seems to defy Earth’s gravitational attraction, is capillarity. The interface separating the invading fluid from the displaced air touches the walls of the material on each pore. Depending on the relative interfacial energies of the three phases present (solid, liquid, and gas), there is a non-zero capillary force acting on the fluid menisci on

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Fig. 2. Capillary rise and fall of water and mercury in glass. Source: Wikimedia commons, created by M. Woland.

each pore and throat of the material. In the case of tea or coffee invading a cube of sugar or a biscuit, this force drives the liquid into the medium. With mercury in glass, however, the capillary force would drive the menisci in the opposite direction, because glass is not wetted by liquid mercury, i.e., the contact angle is >90o (Fig. 2).The dynamics of the invasion process indeed depend on the relative ability of the two fluids (displacing and displaced) to wet the walls of the disordered medium. If the invading fluid preferentially wets the medium, the displacement is favored by capillary forces and is referred to as imbibition (from the Latin verb imbibere, to drink in). Conversely, when the preferentially wetting fluid is the displaced one, capillary forces oppose the displacement of the menisci inside the pores. The corresponding process is called drainage. It plays a central role in hydrology when surface and subsurface water are removed, either naturally or artificially, from an area thus preventing erosion and the leaching of nutrients. A second important issue is the stability of the interface separating two fluids. The displacement may be either stable or unstable, depending on the relative viscosity of the fluids involved. The displacement is stable when the displacing fluid is more viscous than the displaced fluid. Because of

4 Len Fisher won the Ig Nobel prize in Physics in 1999 for his experiments on the optimal way of dipping a biscuit into tea or coffee, an experience that enhances flavor release by up to ten fold. For a full account, see L. Fisher (1999) Nature 397:469.

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ratios: the capillary number, Ca = µ 2 v / γ , which compares the strength of viscous and interfacial forces, and the viscosity ratio M = µ 2 / µ1 . Here µ1 and µ 2 are the dynamic viscosities of displaced and displacing fluid, v the average velocity of the front, and γ its interfacial tension. Viscous fingers are formed when M < 1 , covering a band of wavelengths that widens with Ca . Saffman and Taylor also performed experiments in which air (of negligible viscosity) slowly displaced a viscous fluid, and thereby confirmed the predictions of their linear stability analysis. They found that a dynamic competition between air fingers of different sizes that formed at the onset finally gave rise to a stationary single finger that occupied half of the cell width at large Ca . Their seminal work has been the starting point for a great deal of research activity on all possible aspects of viscous fingering and pattern formation resulting from this interfacial instability [2] (Fig. 3). Differences in the wettability and viscosity of the fluids therefore provide a rationale for classifying two-phase fluid displacements in disordered media, as shown in Table 1. Using quasi-two-dimensional transparent micromodels of porous media, Lenormand, Zarcone, and others explored these scenarios systematically and compiled their main findings in schematic phase diagrams, with Ca and M as the controlling parameters [13]. Very slow drainage is governed by capillary fingering. A fraction of the available channels are invaded by the lesswetting fluid, following the order dictated by the values of the capillary pressure jump across the meniscus in each channel. This sequential invasion is well described by the model of invasion percolation, and the resulting pattern is a self-similar fractal. At larger Ca , unstable drainage results in a highly ramified pattern of invaded pores, also a self-similar fractal but of smaller fractal dimension. Its morphology and growth dynamics correspond to a process of diffusion limited aggregation, which describes the pattern formed by a growing unstable interface in an external field that obeys Laplace’s equation (Fig. 4). Finally, stable drainage displacements at large Ca produce compact patterns. Fast imbibition displacements lead to the same morphologies as fast drainage, because the dynamic contact angle increases with Ca and makes the injected

Fig. 3. Viscous fingering in channel and radial geometries (schematic).

capillary pressure fluctuations at the pore scale, the front at large scales is slightly irregular, but front disturbances cannot increase because the viscous pressure gradient on the side of the displacing fluid is larger than on the side of the displaced fluid. When the displacing fluid is less viscous than the displaced fluid (e.g., when water displaces oil) the situation is the opposite. Small front disturbances become amplified and rapidly growing fingers of the displacing fluid invade the displaced fluid, limiting the effectiveness of the displacement process. This is a very serious difficulty in secondary oil recovery and has motivated many research efforts [11]. The interfacial instability leading to viscous fingering is known as Saffman-Taylor instability [19]. In 1958, these authors studied two-phase displacements in a Hele-Shaw cell. The narrow gap of the cell results in high-friction (inertialess) bulk fluid motion that follows Darcy’s law, in analogy with flow in a disordered medium. Performing a linear stability analysis of the unperturbed front between the two fluids, Saffman and Taylor found that the interfacial tension along the front always damped infinitesimal perturbations of small wavelength; but, depending on the relative viscosity of the two fluids, viscous pressure could either damp or amplify infinitesimal perturbations of large wavelength. The linear stability of the front was thus controlled by two dimensionless

Table 1. Classification of two-phase fluid displacements based on differences in the wettability and viscosity of the fluids Fluid 2 ⇒ Fluid 1

µ2 < µ1

µ2 > µ1

Less wetting ⇒ More wetting

Unstable drainage

Stable drainage

More wetting ⇒ Less wetting

Unstable imbibition

Stable imbibition

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fluid invade the central part of the channels, as in drainage. However, slow imbibition displacements depend on the pore-to-channel aspect ratio. For a large aspect ratio, the injected fluid first wets the walls of the channels and then progressively invades the network by a sequence of collapses in the channels. At even slower displacements, the invasion mechanism is the flow of a precursor film ahead of the menisci, so that apparently disconnected filled channels may appear anywhere. For a small aspect ratio, invasion at moderate Ca takes place by a sequence of channel by channel pore invasions that lead to a faceted domain whose shape is dictated by the underlying network topology. Finally, very slow displacements combine this invasion mechanism with the flow of the precursor film, so that compact clusters appear anywhere in the network. Imbibition is therefore complicated by the underlying geometry of the model porous medium and by a new mechanism of invasion, the flow of a precursor film, at very low flow rates.

Dynamics of porous media

capillary

invasion

An often-studied scenario of stable imbibition is the invasion of a porous medium by a liquid that preferentially wets the walls and displaces the air initially present, as in the case of the biscuit dipped in coffee. This process occurs spontaneously and is driven by capillary forces. An equation for the average position of the invading front versus time can be derived in a continuous (hydrodynamic) framework at the Darcy scale. The equation is obtained by combining Darcy’s law for the viscous pressure drop in the bulk flow with the average capillary pressure that acts on the different menisci, assuming a constant contact angle between the invading fluid and the inner walls of the porous medium. The solution h (t ) is known as Washburn’s law. In the absence of gravity (e.g., for horizontal displacements), this law predicts that the average position of the imbibition front grows as the square root of time, h ~ t 1/2 , so that the front slows down in time but never stops. When gravity resists capillary invasion, there is a crossover between this behavior at early times and an exponential slowing down at late times that finally brings the front to rest, although theoretically over an infinite time. The average position reached by the front is called Jurin’s height. It depends on the surface tension of the liquid-air interface, the contact angle of the liquid with the surface of the material, the density of the liquid, and the permeability of the porous medium. www.cat-science.cat

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Fig. 4. Pattern of diffusion-limited aggregation obtained by allowing incident particles to perform a random walk and finally adhere to the aggregate. Different colors indicate different arrival times of the random walkers. Source: Wikimedia commons, public domain.

Washburn’s law provides a very good description of capillary-driven fluid invasion of simple capillaries. Traditionally it has also been considered as an appropriate description of capillary invasion of disordered media, since it well describes the fast stages of invasion, up to the dramatic slowing down theoretically associated with approaching Jurin’s height. Several years ago, however, capillary rise in porous media over very long durations was investigated. Using vertical cylindrical columns packed with glass spheres, several groups [6,12] showed that Washburn behavior was indeed observed in the initial stages of invasion, which lasted a few minutes. The slow advancement of the front at the end of this Washburn invasion, however, switched to a motion of smallamplitude jumps on the pore scale that continued for hours. The average position followed a power law over time, rather than the predicted exponential dynamics, and did not seem to approach an equilibrium height asymptotically (Fig. 5). The reasons for these unexpected observations remained elusive until recently, when a new theory of capillary rise in disordered media was proposed. This theory considers the different modes of motion that menisci go through on the pore scale, in the framework of a macroscopic (Darcy scale) description. Three main modes are considered. The first is a wetting mode, which describes the motion of the contact line on the pore scale driven by capillary forces, essentially in the CONTRIBUTIONS to SCIENCE 11:189-198 (2015)

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Washburn regime, with a shape that is practically unchanged but with a velocity-dependent contact angle. The second is a threshold mode, which describes the local pinning of the contact line upon reaching a pore, followed by deformation of the meniscus until the contact angle reaches a critical value at which the contact line can resume its motion. These two modes alone reproduce the essential features of Washburn dynamics, although the final height is attained in finite time and depends only on the threshold value of the contact angle [23]. The third mode, called subcritical depinning, introduces a new mechanism of motion. On the pore scale there are pressure fluctuations, due to the mutual influence of menisci that are in different modes of motion at a given moment. These fluctuations are unimportant when the local pressure is capable of pushing the meniscus through the threshold mode. But when this is not the case the front remains locally pinned, so that the effectiveness of the fluctuations grows over time until, finally, the interface may depin subcritically due to random fluctuations. This mode of motion, responsible for the long-term behavior of capillary invasion, is associated with avalanches of the invading menisci that had been observed experimentally indeed in the long-term regime of capillary rise. The theory, moreover, predicts that capillary rise eventually comes to a halt, albeit at very long times that fall outside the range of currently available measurements [22].

Kinetic roughening Interfacial growth driven by competing forces at different length scales is known to result in rough interfaces that www.cat-science.cat

Fig. 5. Capillary rise of water in a vertical cylinder packed with glass beads. Measurements of the average position of the imbibition front vs. time are represented by different symbols corresponding to different bead diameters. Lines represent the predictions of a theory that considers three different modes of motion of the invading menisci. Adapted from [22].

exhibit scale-invariant properties. The morphology of the interface looks the same (i.e., it has the same statistical properties) at different magnifications, at least within a wide range. Scale-invariance is ubiquitous in nature (Fig. 6). Stable imbibition fronts in disordered media are subject to the competing influence of surface tension, viscous pressure drop, capillary pressure fluctuations, and permeability variations. While surface tension and viscous pressure drop keep the front smooth at different length scales, capillary pressure fluctuations and permeability variations distort the front. The result is that an initially smooth front undergoes a kinetic roughening process [1], in which front fluctuations grow over time due to the progressive correlation of different points 1/ z in the front, with increasing correlation length ξ c ~ t . The rootmean-square fluctuations of the front position grow over time according to W ~ t β , until they saturate. The resulting front is scale-invariant and verifies W ~ Lα , where L is the lateral size of the system. The exponents α , β , and z are called the roughness, growth, and dynamic exponent, respectively, and they satisfy the scaling relation α / β = z . Since α is usually a non-integer, the rough front is a self-affine fractal object. This is called Family-Vicsek scaling, the simplest possible scenario of scale-invariant growth. In a pioneering work, Rubio et al. [18] found that it applied to stable imbibition displacements in twodimensional models of porous media consisting of Hele-Shaw cells packed with glass beads. The actual values of the scaling exponents are important because, following the concept of scale invariance in equilibrium critical phenomena, there are solid arguments to believe that the long-term, large-scale asymptotic behavior of growing interfaces does not depend on the microscopic details of the systems under study but 194

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Fig. 6. The concept of scale-invariance is exemplified here by Romanesco broccoli, shown at two magnifications. Public domain photographs by John Walker.

only on general properties, such as the space dimensionality and the interaction range. Hence a few basic models are sufficient to identify universality classes of kinetic roughening and their corresponding scaling exponents. This framework is well established for interfacial growth problems with local interactions. However, stable imbibition displacements in disordered media are intrinsically non-local: the dynamics of one point of the front depend on all other points, because of mass continuity. Accordingly, in spite of serious theoretical efforts [8–10,15], it is not yet clear to which universality class of kinetic roughening this problem belongs and which scaling exponents define it. The Family-Vicsek scaling scenario can be extended by considering the way in which front fluctuations scale with the lateral size of the window of observation, i.e., W ~ α . If they scale in the same way as they do with the lateral system size L , Family-Vicsek scaling is recovered. If not, three additional scaling scenarios (defined by the values of five loc

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Fig. 7. Stable imbibition fronts in a laboratory model of an open fracture with persistent disorder in the displacement direction (top), and scale-invariant growth of the interfacial fluctuations with time (bottom) both for the whole system size (main plot) and for a measuring window 256 times smaller (inset). Figures courtesy of J. Soriano.

scaling exponents that verify two scaling relations) are possible [17]. These anomalous scaling scenarios appear in interfacial problems in which the mean local slope of the front diverges in time, thus introducing a new correlation length in the growth direction [14]. A few years ago we showed that tailoring the disorder properties of the medium can lead to stable imbibition displacements with anomalous kinetic roughening [24,25]. The experiments were carried out in a laboratory model of an open fracture, a Hele-Shaw cell with quenched disorder consisting of random dichotomic variations in the gap thickness. When this disorder is persistent in the direction of displacement, it gives rise to rough fronts with very large local slopes and, consequently, to anomalous scaling (Fig. 7).

Avalanches, non-Gaussian velocity fluc tuations and intermittency Fluid invasion of porous and fractured media at low velocity (low Ca ) is dominated by fluctuations in capillary CONTRIBUTIONS to SCIENCE 11:189-198 (2015)

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Fig. 8. Probability distribution of areas spanned by local velocity bursts in the course of slow stable imbibition displacements, at constant flow rate, in our model open fracture. The main plot shows the rescaled data and a powerlaw fit with an exponential upper cutoff. The inset shows that this upper cutoff depends on  through the imposed velocity v . c

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pressure arising from the disorder of the medium. The slow advancement of the front takes place by localized velocity bursts or avalanches. Avalanches represent the dominant mode of motion in the late stages of capillary rise in porous media, as discussed earlier, and they are also at the origin of

the scale-invariant properties of invasion fronts. In slow stable imbibition displacements, avalanches are triggered by capillary pressure fluctuations and suppressed by the interfacial tension of the invading front and by fluid viscosity. In the case of a porous medium modeled by a Hele-Shaw cell packed with glass beads, Dougherty and Carle [7] showed that the areas swept by the avalanches followed an exponential distribution with the characteristic size of the pore spaces. The burst dynamics in porous media are therefore controlled by pore-scale dynamics. This is in striking contrast to the case of an open fracture, where the length scale  c = κ / Ca ~ (µ v) –1/2 at which the two damping mechanisms (interfacial tension and fluid viscosity) cross over plays a very relevant role. Considering an initially flat front, the lateral correlation length grows in time until it reaches  c , and the front then reaches a statistically stationary state of saturated roughness. In this stationary state, the motion of the front is composed of local avalanches with a very wide distribution of lateral sizes, from the lower cutoff imposed by the characteristic size of the disorder to an upper cutoff given by  c . This upper cutoff may be tuned by controlling the average velocity of the front, v . As v approaches zero  c diverges and so does the lateral correlation length, revealing that the limit of zero velocity (pinning) corresponds to a nonequilibrium critical point. Near this point relevant quantities

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Fig. 9. Normalized statistical distributions of V (the  average velocity over a window of lateral size  ) of an imbibition front that invades our model open fracture at constant flow rate. The results correspond to two windows of observation,  = L (system size) and  = L / 40 , and two driving velocities, which combine to yield four different ratios  /  . Symbols are the experimental values, and the solid lines the generalized Gumbel distributions with the same skewness. Figure courtesy of R. Planet. c

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such as lateral size, area and duration of local velocity bursts should be scale invariant; that is, they will follow power-law probability distributions over a very wide range of values, within the upper and lower bounds imposed by the cutoffs just discussed. An example of the probability distribution of avalanche amplitudes in slow imbibition displacements at constant flow rate, based on our model open fracture with random (non persistent) dichotomic gap thickness [21], is shown in Fig. 8. Since local velocities of contiguous points in the front are laterally correlated up to a distance  c , additional information about the dynamics of slow imbibition in our model open fracture can be obtained from scale-dependent statistics. The mean velocity of the front within a window of observation of lateral size  can be computed by taking the spatial average of the local velocities over a lateral distance  . The statistical distribution of this new quantity, V , is sensitively dependent on how  compares with  c . When , the normalized V statistical distribution of  is heavily skewed towards values above the ensemble average and clearly non-Gaussian (Fig. 9). Remarkably, the experimental distributions are accurately represented by generalized Gumbel distributions of the same skewness, with no other fitting parameters. The origin of this behavior is the fact that, for  ≤  c , V is an average over strongly correlated local velocities. For larger windows of observation or larger capillary numbers, for which  /  c > 1 , a Gaussian distribution of V is progressively attained as the average involves more and more uncorrelated local velocities, in correspondence with the result of the central limit theorem. The skewness is controlled by the ratio  /  c , which can be thought of as counting the effective number of independent degrees of freedom of the invading front [16]. Furthermore, the velocity V is intermittent. This refers to the presence of anomalous temporal correlations, such that periods of low velocities and small accelerations alternate with periods of very large velocities and highly fluctuating accelerations (Fig. 10). Intermittency has been a key concept in hydrodynamic turbulence. It also has been found numerically in the Lagrangian velocities of fluid particles flowing through porous media [5]. A characteristic signature of intermittency is the observation of fat-tailed probability distributions of the mean-velocity increments ∆V (τ )= V ( t +τ ) −V ( t ) . The scale-dependent analysis is now carried out in terms of the spatial scale  , introduced earlier, and a new temporal scale τ . Our results for slow imbibition displacements in laboratory models of open fractures [3] show that the intermittent dynamics of the front are controlled again by the lateral correlation length of the front through the ratio  /  c , as before, but also www.cat-science.cat

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Fig. 10. Experimental results for slow stable imbibition in our model open fracture. Top: Mean front velocity on scale  ( = L / 4 ) and its corresponding acceleration. Bottom: Statistical distributions of velocity increments ∆V (τ ) for experimental conditions spanning a wide range of Ca , organized in terms of the same ratios  /  c and τ / τ c . Distributions are shifted arbitrarily for visual clarity. The dashed curve represents a Gaussian distribution. Figure courtesy of X. Clotet.

by a new time scale in the direction of invasion, τ c , such that intermittency depends only on  /  c and τ / τ c (Fig. 10). Not surprisingly, experimental results show that τ c =  d / v , where  d is the characteristic extent of the disorder in the direction of front advancement. Since both  c and τ c vanish at high flow rates, intermittency is present only in slow imbibition displacements, dominated by capillary pressure fluctuations.

Conclusions Understanding the complex spatiotemporal dynamics of fluid flows in disordered media is relevant for natural and industrial processes of importance, such as soil irrigation, water filtering, and secondary oil recovery. Progress in this direction is being achieved through the combination CONTRIBUTIONS to SCIENCE 11:189-198 (2015)

Fluids in disordered media

of hydrodynamic models and analytical tools from nonequilibrium statistical physics. Acknowledgements. My work on the dynamics of flows in disordered media has been developed in close collaboration with my Ph.D. students J. Soriano, R. Planet, and X. Clotet, and my colleagues A. Hernández-Machado, J. Casademunt, J.M. López, M.A. Rodríguez, and K.J. Måløy. I acknowledge S. Santucci in particular for our fruitful collaborations on this subject. This research is currently funded by Generalitat de Catalunya and MINECO (Spain), through projects 2014-SGR-878 and 2013-41144-P, respectively. Competing interests. None declared

References 1. Alava M, Rost M, Dubé M (2004) Imbibition in disordered media. Advances in Physics 53:83-175 2. Casademunt J (2004) Viscous fingering as a paradigm of interfacial pattern formation: recent results and new challenges. Chaos 14:809824 3. Clotet X, Ortín J, Santucci S (2014) Disorder-Induced capillary bursts control intermittency in slow imbibition. Phys Rev Lett 113:074501 doi:10.1103/PhysRevLett.113.074501 4. Darcy H (1978) Les fontaines publiques de la ville de Dijon. Dalmont, Paris 5. de Anna P, Le Borgne T, Dentz M, Tartakovsky AM, Bolster D, Davy P (2013) Flow intermittency, dispersion, and correlated continuous time random walks in porous media. Phys Rev Lett 110:184502 6. Delker T, Pengra DB, Wong P-z (1996) Interface pinning and the dynamics of capillary rise in porous media. Phys Rev Lett 76:2902-2905 doi:10.1103/PhysRevLett.76.2902 7. Dougherty A, Carle N (1998) Distribution of avalanches in interfacial motion in a porous medium. Phys Rev E 58:2889-2893 doi:10.1103/ PhysRevE.58.2889 8. Dubé M, Rost M, Elder KR, Alava M, Majaniemi S, Ala-Nissila T (1999) Liquid conservation and nonlocal interface dynamics in imbibition. Phys Rev Lett 83:1628-1631 doi:10.1103/PhysRevLett.83.1628 9. Ganessan V, Brenner H (1998) Dynamics of two-phase fluid interfaces in random porous media. Phys Rev Lett 81:578-581 doi:10.1103/ PhysRevLett.81.578 10. Hernández-Machado A, Soriano J, Lacasta AM, Rodríguez MA, RamírezPiscina L, Ortín J (2001) Interface roughening in Hele-Shaw flows with quenched disorder: Experimental and theoretical results. Europhys Lett 55:194-200 doi:10.1209/epl/i2001-00399-6

11. Homsy GM (1987) Viscous fingering in porous media. Annu Rev Fluid Mech 19:271-311 doi:10.1146/annurev.fl.19.010187.001415 12. Lago M, Araujo M (2001) Capillary rise in porous media. J Colloid Interface Sci 234:35-43 doi:10.1006/jcis.2000.7241 13. Lenormand R (1990) Liquids in porous media. J Phys Cond Mat 2:A79-A88 doi:10.1088/0953-8984/2/S/008 14. López JM (1998) Scaling approach to calculate critical exponents in anomalous surface roughening. Phys Rev Lett 83:4594-4597 doi:10.1103/PhysRevLett.83.4594 15. Pauné E, Casademunt J (2003) Kinetic roughening in two-phase fluid flow through a random Hele-Shaw cell. Phys Rev Lett 90:144504 doi:10.1103/PhysRevLett.90.144504 16. Planet R, Santucci S, Ortín J (2009) Avalanches and non-Gaussian fluctuations of the global velocity of imbibition fronts. Phys Rev Lett 102:094502 doi:10.1103/PhysRevLett.102.094502 17. Ramasco JJ, López JM, Rodríguez MA (2000) Generic dynamic scaling in kinetic roughening. Phys Rev Lett 84:2199-2202 doi:10.1103/ PhysRevLett.84.2199 18. Rubio MA, Edwards CA, Dougherty A, Gollub JP (1989) Self-affine fractal interfaces from immiscible displacements in porous media. Phys Rev Lett 63:1685-1688 19. Saffman PG, Taylor GI (1958) The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc Royal Soc London A 245:312-329 doi:10.1098/rspa.1958.0085 20. Sahimi M (2011) Flow and transport in porous media and fractured rock: from classical methods to modern approaches. 2nd edition. Wiley–VCH, Weinheim, Germany 21. Santucci S, Planet R, Måløy KJ, Ortín J (2011) Avalanches of imbibition fronts: Towards critical pinning. Europhys Lett 94:46005 doi:10.1209/0295-5075/94/46005 22. Shikhmurzaev YD, Sprittles JE (2012) Anomalous dynamics of capillary rise in porous media. Phys Rev E 86:016306 doi:10.1103/ PhysRevE.86.016306 23. Shikhmurzaev YD, Sprittles JE (2012) Wetting front dynamics in an isotropic porous medium. J Fluid Mech 694:399-407 24. Soriano J, Mercier A, Planet R, Hernández-Machado A, Rodríguez MA, Ortín J (2005) Anomalous roughening of viscous fluid fronts in spontaneous imbibition. Phys Rev Lett 95:104501 doi:10.1103/ PhysRevLett.95.104501 25. Soriano J, Ramasco JJ, Rodríguez MA, Hernández-Machado A, Ortín J (2002) Anomalous roughening of Hele-Shaw flows with quenched disorder. Phys Rev Lett 89:026102 doi:10.1103/PhysRevLett.89.026102

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SOFT MATTER Institut d’Estudis Catalans, Barcelona, Catalonia

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Chiral selection under swirling: From a concept to its realization in soft-matter self-assembly Josep Claret, Jordi Ignés-Mullol, Francesc Sagués*

Departament de Ciència de Materials i Química Física, Facultat de Química, Univer sitat de Barcelona, Barcelona, Catalonia Summary. Chirality, i.e., the absence of mirror symmetry, is evident in nature and has profound implications in the field of Biology but also in Materials Science. The aim of this contribution is: (i) to concisely present the basic concepts and definitions related to chirality and (ii) to briefly summarize our own research into the induction of chiral selection under vortical motion during the self-assembly of soft-matter aggregates. [Contrib Sci 11(2): 199-205 (2015)]

*Correspondence: Francesc Sagués Departament de Ciència de Materials i Química Física Facultat de Química Universitat de Barcelona Martí i Franquès, 1 08028 Barcelona, Catalonia E-mail: f.sagues@ub.edu

A brief introduction to chirality: thoughts, concepts and definitions Chirality is a pervasive concept in nature and is a feature of structures and phenomena at disparate scales, with relevance for disciplines ranging from Cosmology to the Biological Sciences. The world, as we know it, is intrinsically asymmetric. Indeed, we directly quote the opening sentences of G.H. Wagnière in his book On Chirality and the Universal Asymmetry [26]: “That the world is asymmetric is a trivial observation. How can the world be anything but asymmetric?

If asymmetry is a fundamental property of the universe, then a basic question has yet to be satisfactorily answered. According to most laws of physics, any asymmetric object or any asymmetrically moving system could, in principle, exist with equal probability as the corresponding mirror object or mirror system. The laws of gravitation and electromagnetism together with the strong interactions governing the structure of atomic nuclei predict that image and mirror image should have exactly the same energy and, consequently, the same probability of occurring. So why are we living in the world as it is, and not in the mirror world? Or why do the actual world

Keywords: chirality · enantiomorphic · vortical stirring · helicity · soft matter ISSN (print): 1575-6343 e-ISSN: 2013-410X

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Chiral selection

and the mirror world not coexist and be simultaneously perceptible?” Probably the deeper and more challenging questions on chirality refer to the homochirality of life. In the following, we thus briefly focus first on this context before turning to scenarios of chirality that are closer to our experience in Physical Chemistry. One of the most trivial ways to experience chirality in life is by observing our hands. They look similar, certainly, but they are not identical. In fact, they are in-plane mirror images. Objects that are distinct from, that is, not superimposable onto, their mirror images are referred to as chiral, while any macroscopic chiral object and its mirror image can be enantiomorphic (enantiomeric). Now let us consider the elementary components that living entities are built from. The essential molecules on which life, as we know it today, is based, such as DNA, RNA, proteins, and sugars, are all chiral. Moreover, one of the most striking observations in Biology is that this biological or biomolecular homochirality is the same everywhere, whether in a virus or in a human cell. Quoting again Wagnière: “Suppose that in the universe there exists an as yet unknown planet…, and on which atmosphere and climatic conditions resemble those on Earth. In the course of history of this planet, life chemically resembling that on Earth has evolved. Will these living organisms be molecularly homochiral to those on Earth? In other words, is the biological homochirality the direct consequence of universal and thus fundamental laws or is it due to chance―whatever that means—and thus mainly depends on local conditions?” When referring to molecules, one is obligated to first adopt a chemical perspective. In fact, and from a purely thermodynamics standpoint, one could easily conclude that a chiral molecule and its enantiomer should basically have the same energy. Under conditions of thermodynamic equilibrium, both forms should thus display equal existing probabilities. This would be exactly true if we ignored subtle effects rooted in elementary particles theories. To be precise, the so-called parity-violating weak forces would subtly break the aforementioned balance, favoring one chiral form over the other. However, whether that biological homochirality arises from such elementary particle interaction effects is unclear, and nowadays mostly disregarded. We are thus led, and probably will be for many years, to look for new responses to the intriguing question about the origin of homochirality in life. But let us skip this challenging question to concentrate on chirality as it appears in non-living matter. A series of concepts and a brief historical account of how this concept has become rooted in general sciences are probably worth reviewing in the first part of this contribution. www.cat-science.cat

Although chirality is a term more often associated with Chemistry than with Physics, physicists are also acquainted with it, since the implications extend over a disparity of problems covering many different and very actively investigated disciplines. We have already mentioned the fundamental question of parity violation at the level of weak interactions among elementary particles. The history dates back to 1927, when E.P. Wigner [27] formulated the principle of conservation of parity according to which “all interactions in nature are invariant with respect to space inversion.” With the discovery of new interactions, this statement was questioned theoretically by T.D. Lee and C.N. Yang in 1956 [12]. Only one year later, parity violation was confirmed in a brilliant experiment by C. S. Wu et al. [28]. Closer to our times and interests, chirality is often mentioned when referring to Materials Science, for example, in relation to supramolecular assemblies [15], soft-matterbased systems [9,23], or to Biophysics and Developmental Biology [3,8,24], among others.

Definitions of chirality The term chiral was adopted by Lord Kelvin in 1904, when he stated “I call any geometrical figure, or group of points, chiral, and say it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself “[22]. Mathematically formulated, one could use a definition based on group theory, by specifying the criterion for an object to display chirality as the absence of any rotation-reflection (or improper rotation) axes Sn [10]. This supposes, in particular, the lack of planes of reflection and inversion points, the more natural symmetries of achiral objects. Notice also that all totally asymmetric bodies, as well as those objects displaying only rotation symmetry axes Cn, are chiral [10]. With a more modern perspective, the concept of chirality has been extended to encompass not only static objects, but also time-dependent physical vector fields eventually related to translational or rotational dynamic processes. In this respect and already more than 25 years ago, L.D. Barron [2] proposed to unambiguously distinguish between “true” and “false” chirality, by establishing that true chirality is exhibited by systems that exist in two enantiomeric states that are inter-converted by space inversion, but not by time reversal combined with any proper spatial rotation. In fact, this last requirement is what enforces true chirality in a broad sense. For example, false chirality is exhibited by co-linear electric and magnetic fields, while true chirality appears when a 200

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magnetic field is applied parallel to the propagation direction of a light beam of arbitrary polarization.

Louis Pasteur’s dissymmetric forces A particularly important discovery that advanced the modern history of chirality in the natural sciences was that of the French chemist and microbiologist Louis Pasteur (1822– 1895). In the mid-19th century he was working with solutions of sodium ammonium tartrate that were non-optically active; that is, they did not rotate the plane of a polarized light. After recrystallization, he found, however, that the crystals could be distinctively distinguished in terms of their chirality. In solutions of one type of crystals, the plane of polarization was rotated to one side; the reverse was true with the enantiomorphic crystals. Thus, one could say that Pasteur was able to resolve a racemic mixture, containing equal amounts of both chiral forms, into its pure enatioselective components. Referring back to Wagnière [26], it is clear that Pasteur was fortunate to have dealt with a striking example of racemic resolution, since very often racemic solutions also form racemic crystals, in which case resolution must be attempted using other, more sophisticated procedures. It is also worth remembering at this point that Pasteur was the first to propose a theory to explain the origin of biomolecular homochiralty. He did so by conceiving of the existence of chiral, or what he referred to as dissymmetric, forces [14]. In spite of its speculative origin, the idea of chiral forces as conjectured by Pasteur has by now gained wide recognition. In this regard, one should bear in mind that in the absence of chiral influences, achiral compounds can be transformed only into other achiral compounds, a racemic mixture, or an enantiopure component with equal probability of the two enantiomers. This conserved symmetry can, in principle, only be broken by using chiral components (catalysts, promoters, or chemical modifiers), by applying special sorts of confinements that may favor chiral arrangements from atomic to meso-scales (oriented reacting surfaces, aligning boundaries, or similar controlling boundaries), or, finally, by the application of the aforementioned chiral forces. According to Guijarro et al. [10], the list of such chiral forces is rather limited in nature but also in laboratory experiments. Based on what we know today, in this category we can above all include the action of a circularly polarized light, as it is probably the most representative influence. A well-known example of its practical utilization as a chiral selector within a chemical context is the asymmetric www.cat-science.cat

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photodegradation of racemic leucine [7]. The two remaining effects are much less widely recognized: (1) the magnetochiral influence [20], mentioned above as the combination of a static magnetic field with a co-linearly applied light beam of arbitrary polarization, and (2) the vortical stirring effect. The demonstration of the chiral nature of swirling has an interesting history of its own. In fact, as quoted in Guijarro et al. [10], Pasteur himself was determined to find the “cosmic force of dissymmetry” inspired by the motion of the Earth. He therefore tried, albeit in vain, to induce selection in the dextro- or levo-optical rotation of chemically synthesized molecules by conducting reactions in a centrifuge. These attempts were long considered as complete nonsense and Pasteur’s efforts were totally disregarded. This skepticism was not totally unfounded, since from a superficial point of view one could sustain that stirring a fluid does nothing but impart a simple rotary motion to it. However, the situation turns out to be somewhat more subtle. Vortices in fluids are the more familiar signatures of turbulence and are thus associated with complex, or more precisely, chaotic hydrodynamic motions. Using dynamic concepts, we can say that vortical flows result from the superposition of circular and translational motions. The first component or, to be more precise, its associated angular momentum, is represented by a time-odd, axial (unchanged under space inversion P) vector, whereas the translational momentum is clearly timeodd but polar (reversed under the operation P). Because this combination is endowed with helicity, vortices are true chiral dynamic structures. In the following, we restrict our discussion to the role of vortical stirring in chiral selection, since this has been the main goal of our recent research on chiral effects.

Vortical stirring as a selection me chanism for the chirality of soft aggregates The possibility to break the chiral symmetry of solid materials when crystallized under stirring has been widely recognized since it was first observed nearly 25 years ago for supersaturated solutions of sodium chlorate [11]. More recently, similar observations were published for the same system when initial solutions containing both enantiomers were ground with glass beads [25]. In the case of soft materials, the possibility is even more intriguing. In 2001, members of our group coauthored with J. M. Ribó a study demonstrating not only signatures of CONTRIBUTIONS to SCIENCE 11:199-205 (2015)

Chiral selection

induced chirality under stirring, but what turned out to be the first experimental evidence of chiral selection conditions [19]. Here, the term chiral selection has to be understood in the sense that in addition to the realization of chiral symmetry-breaking, more importantly, the effective sign of the desired enatiomeric excess can be chosen at will, according to the desired handedness, right or left. Note that in this respect chiral selection, as compared to pure chiral symmetry-breaking in which both signs are a priori equally realized, is a different, deeper, and broader scenario for the spontaneous emergence of chirality. The investigated system consisted of the self-assembly of homoassociates of achiral porphyrin derivatives during rotatory evaporation. For that particular system it was demonstrated that the helicity sign of the circular dichroism spectra of the dissolved aggregates correlated, within statistical confidence levels, with the handedness of the applied stirring, either clockwise or counterclockwise. Evidence for the formation of helical bundles following this association process was obtained a few years later from atomic force microscopy images [6]. A more recent discussion pointed out the need to better discern the intrinsic chirality of the aggregates [21], while other authors revisited the question by considering the reversible or dynamic nature of chirality [5,29]. The need to extend the scenario first reported by J. M. Ribó and coworkers and to exclude any artifact arising from the employed observation technique led us to devise a different experimental approach. We intended to prove chiral selection of soft-matter condensates, by employing more direct in situ characterization techniques that would make any further interpretation of the measurements unnecessary. Our research is summarized in four consecutive publications [4,16–18]. All of them refer essentially to the same experimental setup but address three complementary aspects. These are briefly summarized below.

Chiral-symmetry selection in interfacial systems induced by vortical flows The chosen system refers to surfactant monolayers spread at the air/water interface under moderate stirring of the underlying aqueous subphase. This scenario intrinsically differs in many respects from the one considered by Ribó et al. First, in this case chiral selection is sought at the level of a soft interface lying on the boundary separating two different phases, while flow shearing is applied only to one of them. We thus searched for the possibility that a genuine www.cat-science.cat

three-dimensional centimeter-scale stirring mechanism, when applied to the water subphase, could be effectively imprinted into a two-dimensional monomolecular-thick monolayer on top of it. In short, this would be a signature of an effective scaling-down of chirality encompassing different dimensions and disparate length-scales. On the other hand, observation of the reflectivity patterns displayed at the interface, here realized via simple polarized-light (Brewster angle) microscopy (BAM), would provide a direct measure of the orientations of the amphiphilic molecules residing at the monolayer, in which case the chiral symmetry-breaking measure would be free of any ambiguity. The chosen system was a Langmuir monolayer of a photosensitive azobenzene derivative (Fig. 1). This system is particularly interesting because, under suitable illumination conditions, a very slowly evolving interfacial emulsion forms at the aqueous-air interface; at the same time, micron-size circular domains (like two-dimensional droplets) enriched with the trans isomer progressively coarsen, while embedded in a continuous and unstructured mostly cis phase. The selfassembled circular domains display a long-range orientational order of the stretched trans azo-molecules, a feature that BAM distinctively resolves in so-called bend-like textures: Tilted molecules organize tangentially to the droplet’s contour around a central singularity or defect of positive topological charge. This system intrinsically supposes a symmetry-breaking scenario. However, as expected, domains of either sign, i.e., clockwise and counterclockwise bends, are equally realized, since the azobenzene molecule is itself achiral. Observations change drastically when the monolayer is led to evolve under gentle stirring of the aqueous subphase [16,17] or in the presence of chiral dopants [4,18]. Following this intervention, the previous delicate balance of equally realized chiralities is strikingly broken. To elucidate the mechanism of chiral selection, we first need to understand how circular bend domains, with resolved but random orientational (clockwise or counterclockwise) chirality, are formed. The answer can be found by following the spontaneous evolution of an initially pure cis component into a mixed cis-trans monolayer that, eventually, will feature the desired bend domains. Unlike cis isomers, the geometry of the trans azobenzene amphiphile favors supramolecular assembly. Moreover, the two isomeric forms perform very differently as surfactants. As mentioned above, this leads to their spontaneous phase separation at the air/water interface and to the initial formation of submicron agglomerates with elliptical, rather than circular, shape. The reason why these domains are not circular at the 202

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Fig. 1. A droplet of a chloroform solution of the azoben zene derivative is spread at the air/water interface. The solution is irradiated with UV light prior to spreading in order to maximize the presence of the cis isomer. Brewster angle microscopy (BAM) observation reveals the formation of segregated domains rich in the trans isomer. Inside the domains, surfactant molecules orga nize with long-range orientational order, as interpreted from BAM image analysis. The coalescence of antipar allel elliptical domains results in the formation of circu lar domains whose bend texture has an orientational chirality, correlated with the arrangement of the dou blet of the parent elliptical domains. (Details are pro vided in [16]).

very beginning is because the assembled trans amphiphiles organize with their hydrophobic tails mostly aligned along the same direction. This in turn promotes the stretching of the two-dimensional droplets into ellipses. Moreover, since polar heads and hydrophobic tails in the aligned trans amphiphiles are clearly different, the way they assemble at either tip of an elliptical domain differs as well. This is an important result that helps in understanding the mechanism leading to the coupling of subphase stirring and the orientational chirality of monolayer domains. BAM imaging unambiguously shows towards which tip of the ellipse the hydrophobic tails are tilted. Just as oil droplets suspended in water tend to coalesce spontaneously, elliptical domains surrounded by the cis phase in a monolayer of our azobenzene amphiphile tend to merge, prompted by the tendency to decrease the length of the contact region between the two dissimilar monolayer phases. Remarkably, the outcome of this process depends on the relative orientation of the ellipses in a merging pair. If the two elliptical domains are initially parallel, that is, if the hydrophobic tails inside either ellipse are tilted towards the same direction, the resulting domain after coalescence will be a larger ellipse with the same inner configuration as the two parent domains. Conversely, if the two elliptical domains are antiparallel, the coalescence results in a circular domain with a bend inner texture, that is, with a resolved orientational www.cat-science.cat

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chirality. The resulting handedness depends on the spatial arrangement of the parent domains, which constitute a chiral doublet. If the northern-most domain is aligned eastwards, then the outcome is a domain with clockwise handedness. If the southern-most domain is aligned eastwards, the resulting domain will be counterclockwise. Since the elliptical domains are randomly oriented, on average the same number of bend domains of either handedness is formed. The above scenario is drastically altered when the monolayer is spread and is allowed to develop on a stirred, instead of a quiescent, air/water interface. After a period of incubation that can be as short as a few minutes, stirring is stopped and the configuration of the monolayer is monitored under BAM observation. The result is that the population of chiral bend domains is biased towards a handedness that is correlated with the sign of stirring. We quantify this result by defining an enantiomeric excess parameter, eeCW , as the statistically meaningful difference between the fraction of domains with clockwise handedness and the fraction with the opposite one. Although the difference is hardly measurable at low stirring rates (racemic mixture), >90% of the domains have the same handedness as the stirring vortex for rates of about 1000 rpm [16,17] (Fig. 2). From this intriguing observation we can ask: What is the nature of the mechanism that leads to this chiral selection? We propose a description that is based on the asymmetric CONTRIBUTIONS to SCIENCE 11:199-205 (2015)

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Chiral selection

Fig. 2. Chiral selection as a function of the stirring rate is quantified by means of an enantiomeric excess parameter, eeCW , obtained by exploring a statistically meaningful number of bend domains. The orientational chirality individual bend domains can be directly assessed from BAM imaging by comparing them with the motifs shown in the figure. (Details are provided in [16,17]).

Fig. 3. Coupling between chiral dopants and vortical stirring in the chiral selection process. Stirring rates in the range â&#x20AC;&#x201C;1600 rpm (counterclockwise) to +1600 rpm (clockwise) are explored for monolayers including 6% S dopant (red circles), 6% R dopant (blue squares), and a 3%/3% racemic mixture of S and R dopants (black triangles). Solid lines are fit to a kinetic model. (Details are provided in [18]).

nature of the elliptical domains detailed above. When two antiparallel domains merge to generate a chiral bend domain, the configuration of the ordered surfactant molecules undergoes a dramatic rearrangement, which can be described using the framework of creation and annihilation of topological defects that is often invoked when describing the dynamics of soft materials, most notably liquid crystals [13]. Suffice it to say in the present case that this rearrangement is faster under stirring when the sign of the vortex coincides with the handedness of the doublet of antiparallel ellipses, thus with the handedness of the resulting bend domain. Although bend domains of either chirality may form, those with the same handedness as the underlying vortex are kinetically favored and therefore comprise the majority [16]. Less dramatic than the physical chiral selection described above is the possibility to achieve a similar effect by doping the surfactant monolayer with a chiral dopant whose molecular structure is similar to that of the original amphiphile. In this case, we chose custom-synthesized azobenzene surfactants with a chiral center of either sign. When the monolayer is prepared, as explained above, over a quiescent subphase and doped with a racemic mixture of the dopants, the result is a racemic mixture of chiral bend domains. That is, eeCW = 0 . On the other hand, an imbalance of either chiral dopant results in the formation of a monolayer with the majority of bend domains of the selected handedness. Just as above, we can provide a mechanism for the chiral selection, which it is now of chemical origin. Once again, the clue is found using

BAM image analysis. We observe that, when the chiral dopant has a significant enantiomeric excess, the evolved elliptical domains are distorted, adopting a shape like a kidney bean. In fact, opposite signs of the enantiomeric excess of the chiral dopant result in bean-like domains with enantiomorphic (mirror-image) configurations. Because of their shape, merging encounters of two antiparallel bean-like domains are dissimilar when they lead to a clockwise, rather than a counterclockwise, bend domain. This enhances the kinetics of coalescence into one particular type of bend domain, resulting in the observed enantiomeric excess. A definitive and very valuable advantage of working with this chemically doped system is that it also permits confrontations with chemical and physical chirality-inducing effects when the chemically modified monolayer is allowed to evolve under stirring conditions. The two chiral influences are forced to compete and situations of improved or reversed chiral sign selection are observed [18] (Fig. 3). The result is remarkable: The two chiral influences couple such that at low stirring rates the chemical effect dominates, while at high stirring rates the physical force determines the selection. The single main result that evidences this coupling is the existence of a compensation point indicating the existence of a subphase stirring rate that exactly compensates the chiral influence of the chiral dopant of opposite sign. A similar chiral selection of bend domains can be achieved when a non-amphiphilic chiral solute is present in the subphase during monolayer incubation. We have performed experiments

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in which different concentrations of amino acids (available in either chirality) are dissolved in the water subphase prior to monolayer spreading. The underlying physical mechanism is less clear in this case. Nevertheless, the chiral environment present in the water subphase may couple with the surfactant configuration during the domain rearrangement following ellipse coalescence, resulting in the kinetic dissymmetry that leads to chiral selection [4]. No effects of stirring have been investigated yet for this particular situation. In summary, in the second part of this contribution we presented a soft-matter system in which orientational chirality, as it appears in domains of self-assembled surfactants, can be selected using either a chemical influence (chiral dopants) or a physical effect (vortical stirring). From this modest perspective and by advancing into the selection control of the chirality exhibited during the self-organization of soft materials, our aim is to provide new perspectives on the emergence of chirality in Nature. These same perspectives could be eventually invoked to solve the intriguing mystery of the biological homochirality observed in the living universe. Competing interests. None declared

References 1. Barron LD (1986) Symmetry and molecular chirality. Chem Soc Rev 15:189-223 doi:10.1039/CS9861500189 2. Barron LD (1986) True and false chirality and parity violation. Chem Phys Lett 123:423-427 doi:10.1016/0009-2614(86)80035-5 3. Berg HC (2004) E. coli in motion. Springer-Verlag, New York 4. Dong H, Ignés-Mullol J, Claret J, Pérez L, Pinazo A, Sagués F (2014) Interfacial chiral selection by bulk species. Chemistry 20:7396-7401 doi:10.1002/chem.201400248 5. D’Urso A, Randazzo R, Lo Faro L, Purrello R (2010) Vortexes and nanoscale chirality. Angew Chem Int 49:108-112 doi:10.1002/anie.200903543 6. Escudero C, Crusats J, Díez-Pérez I, El-Hachemi Z, Ribó JM (2006) Folding and hydrodynamic forces in J-aggregates of 5-phenyl-10,15,20-tris(4sulfophenyl)porphyrin. Angew Chem Int 45:8032-8035 doi:10.1002/ ange.200603182 7. Flores JJ, Bonner A, Massey GA (1977) Asymmetric photolysis of (Rs)Leucine with circularly polarized uv light. J Am Chem Soc 99:3622-3625 8. Fürthauer S, Strempel M, Grill W, Julicher F (2012) Active chiral fluids. Eu Phys J E 35:89 9. Gibaud T, Barry E, Zakhary MJ, Henglin M, Ward A, Yang Y, Berciu C, Oldenbourg R, Hagan MF, Nicastro D, Meyer RB, Dogic Z (2012) Reconfigurable selfassembly through chiral control of interfacial tension. Nature 481:348351 doi:10.1038/nature10769 10. Guijarro A, Yus M (2009) The origin of chirality in the molecules of life. A

revision from awareness to the current theories and perspectives of this unsolved problem. RSC Publishing, Cambridge, UK 11. Kondepudi DK, Kaufman RJ, Singh N (1990) Chiral symmetry-breaking in sodium-chlorate crystallization. Science 250:975-976 doi:10.1002/ crat.2170300714 12. Lee TD, Yang CN (1956) Question of parity conservation in weak interactions. Phys Rev 104:254-258 doi:10.1103/PhysRev.104.254 13. Oswald P, Pieranski P (2005) Nematic and cholesteric liquid crystals: concepts and physical properties illustrated by experiments. Taylor & Francis, Boca Raton, FL, USA 14. Pasteur L (1861) Leçons de chimie proffesseés en 1860. Hachette, Ed., Paris 15. Pérez-García L, Amabilino DB (2007) Spontaneous resolution, whence and whither: from enantiomorphic solids to chiral liquid crystals, monolayers and macro- and supra-molecular polymers and assemblies. Chem Soc Rev 36:941-967 16. Petit-Garrido N, Ignés-Mullol J, Claret J, Sagués F (2009) Chiral selection by interfacial shearing of self-assembled achiral molecules. Phys Rev Lett 103:237802 17. Petit-Garrido N, Claret J, Ignés-Mullol J, Farrera JA, Sagués F (2012) Chiral-symmetry selection in soft monolayers under vortical flow. Chemistry 18:3975-3980 doi:10.1002/chem.201102358 18. Petit-Garrido N, Claret J, Ignés-Mullol J, Sagués F (2012) Stirring competes with chemical induction in chiral selection of soft matter aggregates. Nat Commun 3:1001 doi:10.1038/ncomms1987 19. Ribó JM, Crusats J, Sagués F, Claret J, Rubires R (2001) Chiral sign induction by vortices during the formation of mesophases in stirred solutions. Science 292:2063-2066 doi:10.1126/science.1060835 20. Rikken GLJA, Raupach E (2000) Enantioselective magnetochiral photochemistry. Nature 405:932-935 doi:10.1038/35016043 21. Spada GP (2008) Alignment by the convective and vortex flow of achiral self-assembled fibers induces strong circular dichroism effects. Angew Chem Int 47:636-638 doi:10.1002/anie.200704602 22. Thomson WH, Lord Kelvin (1904) Baltimore Lectures. C. J. C. a. Sons, Ed., London 23. Tortora L, Lavrentovich OD (2011) Chiral symmetry breaking by spatial confinement in tactoidal droplets of lyotropic chromonic liquid crystals. Proc Natl Acad Sci USA 108:5163-5168 doi:10.1073/pnas.1100087108 24. Vandenberg LN, Levin M (2009) Perspectives and open problems in the early phases of left-right patterning. Semin Cell Dev Biol 20:456-463 doi:10.1016/j.semcdb.2008.11.010 25. Viedma C (2005) Chiral symmetry breaking during crystallization: complete chiral purity induced by nonlinear autocatalysis and recycling. Phys Rev Lett 94:065504 doi:10.1103/PhysRevLett.94.065504 26. Wagnière GH (2007) On chirality and the universal asymmetry: reflections on image and mirror image. Wiley-VCH Ed., New Jersey 27. Wigner EP (1927) Einige folgerungen aus der schrödingerschen theorie für die termstrukturen. Zeitschrift Fur Physik 43:624-652 doi:10.1007/ BF01397327 28. Wu CS, Ambler E, Hayward RW, Hoppes DD, Hudson RP (1957) Experimental test of parity conservation in beta-decay. Phys Rev 105:1413-1415 doi:10.1103/PhysRev.105.1413 29. Yamaguchi T, Kimura T, Matsuda H, Aida T (2004) Macroscopic spinning chirality memorized in spin-coated films of spatially designed dendritic zinc porphyrin J-aggregates. Angew Chem Int 43:6350-6355 doi:10.1002/anie.200461431

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Synchronization

Albert Díaz-Guilera,* Conrad J. Pérez-Vicente © Douglas Zook. http://www.douglaszookphotography.com

Departament de Fisica de la Matèria Condensada, Universitat de Barcelona, Barcelona, Catalonia. Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Catalonia

Summary. In this paper we analyze the dynamics of two different models of oscillators. The most relevant aspect of both models is that synchronization emerges spontaneously as a natural stationary state and therefore may be a starting point for understanding complex patterns where exact timing plays a relevant role. However, the physical mechanisms leading to this temporal coherence are quite different in the two models, evidence of the richness of dynamic behavior in real systems. We are still far from a complete understanding of the whole process. The effect of the topology on the dynamics, the effect of mobility, the effect of disorder, etc., are all very important in biological, physical, and social environments and are the current focus of research in the field. [Contrib Sci 11(2): 207-214 (2015)] *Correspondence: Albert Díaz-Guilera Departament de Física de la Matèria Condensada Universitat de Barcelona Martí i Franquès, 1 08028 Barcelona, Catalonia Tel. +34-934021167 E-mail: albert.diaz@ub.edu

Introduction A 2012 issue of Scientific American (October 2012 for the English version, December 2012 for the Spanish Investigación y Ciencia) contained an inspiring paper entitled “How cells communicate?” The article analyzed recent experiments that used multi-recording devices to simultaneously record the activity of several neurons. The novel technology and clever setup allowed the authors to perform measurements with a previously unattainable accuracy and thus provided very

useful information about how we learn from experience. The main message was that precise timing plays a crucial role in certain perception tasks. One example is the auditory system, in which the arrival of a signal in just a few milliseconds is enough to allow discrimination between right and left and to determine the origin of the sound’s source. To perform this task, cells (neurons) synchronize their activity via a remarkable process. The visual system is another illustrative example. Neurons capable of detecting features are distributed over different areas of the visual cortex. These neurons process

Keywords: oscillator · Kuramoto model · phase transition · pulse-coupled oscillators · connectivity ISSN (print): 1575-6343 e-ISSN: 2013-410X

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Synchronization

information from a restricted region of the visual field and then integrate it through a complex, dynamic process that allows the detection of objects, their separation from the background, the identification of their characteristics, etc. Together, these tasks give rise to cognition. Experiments performed in the primary visual cortex of cats showed that some stimuli induce a correlation between the firing patterns of simultaneously recorded neurons, suggesting that certain global properties of stimuli can be identified through correlations in the temporal firing of different neurons. These oscillatory patterns may reflect the organized, temporally structured activity often associated with synchronous firing. This phenomenology is not restricted to information processing in the brain. Synchronization is observed in biological, chemical, physical, and social systems and it has attracted the interest of scientists for centuries [9]. A paradigmatic example is the synchronous flashing of the fireflies found in some South Asian forests. At night, myriads of fireflies hover over the bushes. Suddenly, several of them start emitting flashes of light. Initially, they do so without any coherence, but after a short period of time the whole swarm is flashing in unison, creating one of the most striking visual effects ever seen. Another spectacular example concerns a group of metronomes. The reader can take advantage of the resources of the World Wide Web to watch several videos displaying the effect. Mechanical metronomes (those typically used in music) are placed on a table, each one with a random initial condition so that the global beat is incoherent. After a short transitory period (a couple of minutes) they adjust their relative phases so that they become temporally closer and closer, finally reaching a dynamic state in which the orchestra beats in unison in perfect synchrony. These are a few of the many examples from well-studied systems, but there are plenty of others in which the synchronized activity among members of a given population is the result of an emergent cooperative process. Several questions arise immediately. When we consider concepts such as timing or synchronization, what are we talking about? Is it possible to precisely define the concept synchronized state? What is the physical mechanism that gives rise to it? These questions are by no means trivial and significant research efforts have been devoted to answering them. Arthur Winfree was one of the first scientists to seek an answer to the first question. He combined concepts from biology and mathematics to construct a theoretical framework in which formal ideas could be converted to mathematical www.cat-science.cat

modeling. His book â&#x20AC;&#x153;The Geometry of Biological Timeâ&#x20AC;? [11] is a summary of his seminal work. Winfree showed that there are many different ways to entrain two or more physical or biological entities. Phase locking, frequency locking, partial synchronization, total synchronization, m:n entrainment, etc., are different dynamic states characterized by a coherent temporal behavior among members of a population. Thus, synchronization is one of the most well studied emergent properties of complex systems, and it has remained so in different areas. The mechanisms leading to these remarkable dynamic states can be analyzed from a physical perspective. There are two fundamental issues that deserve special attention. The first concerns the units themselves. They are usually considered as oscillators, either autonomous (keeping a natural rhythm on their own, such as pacemakers do), or exogenous (in which the oscillatory patterns are triggered by external stimuli). The second concerns the interaction between units, i.e., the mechanism by which they exchange information, which depends on the nature of the system under study. For instance, metronomes synchronize through a physical or mechanical mechanism (vibrations and movement of the air column), brain neurons do so through a combination of chemical and physical elements (such as electric currents plus synaptic neurotransmitters), whereas fireflies use flashes of light and visual communication. But we can also analyze synchronization in a more general way. Scientists working in this field tend to classify synchronization in two different major categories: diffusive and pulse-coupled systems. In the first case, the interaction between members of a population is considered a continuous time function, while the second (which concerns typically excitable systems) is characterized by a non-continuous, non-linear, time function, which makes the problem much more difficult to tackle, at least from a theoretical point of view. In this paper, we analyze two models extensively studied in the last years, each representing one of the aforementioned categories, and look at the precise mechanisms leading to synchronization. Although the stable attractor of the dynamics is the same (the synchronized state), the way it is reached substantially differs between the two models, thus providing useful insights on how information is processed in real systems. The two models are introduced below. The Kuramoto model is the typical paradigm and the most extensively studied example of phase oscillators. Given a population of oscillators, it can be shown that, under certain conditions that affect the intensity of the coupling, 208

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Fig. 1. Phase evolution of the oscillator’s phase. When oscillator j fires at time tj the phase of oscillator i advances by an amount that depends on its own phase.

the whole system can be treated as if only a globally limited Two oscillators cycle acts as an attractor of the dynamics. When this happens only one degree of freedom is needed to characterize the To understand the emergence of synchronization in a set state of a given oscillator, its phase. Kuramoto [6] realized of dynamic systems, we first consider the simplest case of that just two elements are needed to obtain a non-trivial two oscillators [5]. Let us start with the “integrate and fire” collective behavior. He assumed that every oscillator has a oscillator model using two units whose phase evolves at a natural frequency, acquired from a random distribution; in constant speed (equal to 1, without loss of generality): the absence of coupling, each runs incoherently. In addition, dj1,2 =1 he considered that the oscillators interact with each other dt through a non-linear function that depends on the phase difference between each pair. This type of coupling tends The phase of each oscillator increases linearly in time until to synchronize the population. Therefore, in the complete one of them reaches the threshold (assumed to be equal to model there is a tradeoff between two ingredients: if the 1) at which point the oscillator “fires,” thus resetting its phase distribution of frequencies is wide enough compared to to 0 but sending a signal to the other oscillator that produces the intensity of the coupling, there is no synchronization. In a sudden change in its phase (Fig. 1). the opposite case, above a critical value of the coupling an Schematically we can write the evolution of the phases of emergent collective behavior arises and a fraction of the total the oscillators as follows: population becomes synchronized (phase locked). A closer Osc 1 1 → 0 → 1 – j – ∆(j) analysis of this model is presented later on. firing driving The other system to be considered is an “integrate and Osc 2 j → j + ∆(j) → 1 fire” oscillator, which is a standard approach to excitable, pulse-coupled units. In the simplest description, it is assumed Initially, oscillator one has a phase equal to 1. After firing that a phase defined between [0,1] evolves in time with a and resetting, followed by a driving, the phase of the second constant velocity. When the phase reaches the upper value oscillator is equal to 1. This evolution allows us to identify a j = 1 , the unit fires, sending a signal to its neighbors, upon transformation: which it resets to j = 0 . When a unit receives the pulse, it j → 1 − j − ∆(j ) changes its inner state according to a so-called phase response curve. The particular shape of this function depends on the We can then ask whether there exists a phase that system under consideration (for instance, it is quite different represents a fixed point, i.e., a phase that is invariant after in cardiac vs. liver cells). We examine the dynamic behavior this transformation: of this system in the following sections, starting with just two j * 1 – j * – ∆(j *) units and then extending the calculation to a population of= N oscillators. Just two simple conditions are enough to ensure the www.cat-science.cat

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existence and unicity of the fixed point of the transformation: (a) If ∆(j ) is bounded ( ∆(j ) ˂ 1) and the function is continuous, then the fixed point exists; (b) If the derivative ∆ '(j ) > −2 for all values of the argument, then the fixed point is unique. The existence of the fixed point is not sufficient to understand the collective behavior of the two oscillators. Instead, we must consider the stability of the fixed point; that is, rather than starting at the fixed point of the transformation, we start at a slightly different value, = j j * +δ . The behavior of this transformation is shown in Fig. 2. In the first case (positive derivative) the transformed phase is farther from the fixed point (on the opposite side) than the original phase. This means that the fixed point is unstable and acts as a repeller. On the other hand, after a half-cycle transformation, when the derivative is negative, the transformed phase is closer to the fixed point; hence the fixed point is an attractor. Counterintuitively, the synchronization of the two oscillators lies in the fact that the fixed point is a repeller, such that after every transformation of the transformed phase is further and further from the fixed point, until it reaches the value 0 or 1 (from a periodic phase point of view, these two values are identical), in which case the two oscillators remain “synchronized” with exactly the same phase forever. However, if the fixed point was stable, after www.cat-science.cat

Fig. 2. Stability analysis of the fixed point.

every transformation the phase becomes closer and closer to the fixed point, thus remaining out of phase with respect to the other oscillator. In the synchronization of two pulse-coupled oscillators, the synchronization mechanism comes from the instability of the fixed point. This is, however, not always necessary; for instance, when the two oscillators are described by continuous equations in time, as happens in the Kuramoto model: d ji ωi + K sin (j j − ji ) = dt

= i, j 1, 2; j ≠ i

In this case there are two terms, the first corresponds to the natural frequency of every oscillator, and the second is the coupling term. When K = 0 , the two oscillators are independent and will never synchronize. However, for increasing values of K there is some amount of phase entrainment between the two units, as can be easily deduced. We now introduce two auxiliary variables j1 − j 2 , y = j1 + j2 ) , for which the equations of (x = motion are: dx = ω − ω − K sin x 1 2 dt 210

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Fig. 3. Phase evolution for two oscillators coupled according to the Kuramoto model, for different values of the coupling constant.

dy = ω1 + ω2 dt and from which we can conclude that when K < | ω1 – ω2 | , the term on the right hand side is always positive or negative, meaning that x is an ever growing or ever decreasing function of time; therefore, the phases of the oscillators will tend to be apart. However, when K >| ω1 − ω2 | a steady state is reached, such that:  ω − ω2 sin  1 (j1 − j2 ) t→ →∞  K −1

N oscillators with global coupling We start by considering the original Kuramoto model [1,6], in which the phase of each oscillator evolves in time according to:

 , 

meaning that the phase difference tends to a constant value in time, i.e., the two oscillators will become entrained (Fig. 3). It is worth noting that when considering synchronization two situations must be distinguished: (a) strong synchronization, when all units have identical phases and frequencies; (b) www.cat-science.cat

weak synchronization (also known as phase entrainment), when the frequencies are identical, but the phases keep a constant difference in time Having understood this simple setting, let us now analyze a globally coupled system.

211

N d ji K ωi + = sin (j j − ji ) dt N j =1

∑

i = 1,..., N where ωi denotes the natural frequency picked up from a CONTRIBUTIONS to SCIENCE 11:207-214 (2015)

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given probability distribution g (ω ) . In this model, because all the units are mutually interconnected, the factor K / N ensures a proper scaling behavior within the thermodynamic limit (large N ). The basic concepts introduced in the previous section are still valid: there is a relationship between K and the frequency distribution that determines the appearance of a transition from an initially incoherent state to another, partially synchronized one. However, because finding this particular value is much more complicated, we need to resort to statistical mechanics, a discipline of physics able to deal with large populations. The first step is to define an appropriate order parameter. Landau [7] proposed this approach to characterize phase transitions. It is known that symmetry breaking underlies a phase transition and the order parameter helps to identify when it occurs, since in one phase the order parameter is 0 while in the other it takes a non-vanishing value. For the Kuramoto model, the order parameter takes the form: 1 N ij reiψ = ∑e j N j =1 where r (t ) with 0 < r (t ) < 1 measures the coherence of the oscillator population, and ψ ( t ) is the average phase. With this definition, the dynamic equation becomes: d ji ωi + K r sin (ψ − ji ) = i = 1,..., N dt In the limit of infinitely many oscillators, they will be distributed with a probability density ρ (j , ω ,t ) , such that: π +∞

∫ ∫ e ρ (j , ω , t ) g ( ω ) d j d ω

reiψ =

− π −∞

An appropriate mathematical treatment of this probability density leads to: r = Kr

π /2

∫ (cos j ) g ( Kr sin j ) dj 2

− π /2

This equation always has the trivial solution r = 0 , corresponding to incoherence, ρ = 1/ ( 2π ) , which means that the phase of the oscillators is uniformly distributed over the circle. However, it also has a second branch of solutions, corresponding to the partially synchronized phase and satisfying: 1= K

π /2

∫ (cos j ) g ( Kr sin j ) dj 2

− π /2

This branch bifurcates continuously from r = 0 at the value K = K c , obtained by setting r = 0 , which yields www.cat-science.cat

K c = 2 / π g ( 0 )  , where g (0) is simply the distribution of frequencies evaluated at 0 . Kuramoto was the first to devise this formula and the argument leading to it. Regarding a population of N “integrate and fire” oscillators, the analytical procedure to elucidate the attractor of the dynamics is quite involved, but the basic idea is simple, and, again, we can use the previous example of two oscillators. The key element is to realize that when two units fire simultaneously, they keep firing in unison forever. In mathematical terms, this is called an absorption, which technically is equivalent to considering that the number of independent oscillating units is reduced. It can be shown that the probability of finding two not-absorbed units when t → ∞ tends to zero, confirming that the final attractor of the dynamics is the fully synchronized state [8].

New paradigms In the previous sections, we focused on two special limits, one in which connectivity is minimal and the other in which it is maximal. The former is a good example of simple mathematics that can be solved exactly, helping us to understand the emergence of certain collective behaviors. The latter is complicated from a mathematical point of view, but the fact that all units are connected allows certain approximations. The first step to obtaining more realistic structures is to consider regular settings, for instance, rings in one dimension, planes in two dimensions, and, in general, hypercubic lattices in any dimension. In this case, the previous approximations are not valid and new theories are needed. From a purely phenomenological point of view, the main findings include the potential appearance of new structures. Synchronization is indeed possible under certain circumstances, but other phenomena, such as phase-locking (in which effective frequencies are identical but phases are not), emerge. For instance, in pulse-coupled oscillators, when the fixed points are repellers, synchronization emerges, as is the case in reduced and all-to-all connectivities; but if the fixed points are attractors, different local structures are possible. Nature and society are similarly organized, forming structures that are far from regularly connected and thus unlike those described previously. This also affects how complex systems can become synchronized. In a recent review [2], we presented a detailed overview of the different aspects that complex topologies represent for synchronization. Here, we summarize the main implications for the emergence of synchronization. 212

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Fig. 4. The left to right movement over time of Kuramoto identical oscillators along a square. The colors correspond to phases in the interval [0,2π]. The Venn diagrams group oscillators that are already synchronized.

The modern theory of complex networks is intimately intertwined with synchronization. In their seminal paper, published in 1998 [10], Watts and Strogatz introduced a simple model structure (the small world network). This structure was originally considered as a necessary ingredient in the problem of synchronization of cricket chirps, since they show a higher than expected degree of synchronization with the chirps of distant peers, as if they were “connected.” Unlike other models, the Watts and Strogatz model uses this as an initial setup and adds long-range random links between units, which makes the effective distances between units decrease substantially. This is but one of the many cases in science in which a proposed model makes a remarkable contribution to a field other than the one it was originally developed for. In their paper, Watts and Strogatz noted a number of systems in which the connectivity patterns could be mapped using their model, showing, simultaneously, the effect of reducing the average distance between nodes (as they appear in random graphs) but also of keeping the local degree of clustering. And what is the effect of these new models of synchronization? Many researchers have turned their attention to the features of synchronization. Qualitatively, it can be stated that, indeed, the decrease in the average distance makes the units interact more strongly, thus enhancing synchronization. On the other hand, local irregularities prevent the emergence of certain heterogeneous structures. The paper by Watts and Strogatz was followed shortly thereafter by a seminal paper on complex network science authored by Barabasi and Albert [4], which recognized that some real-world networks are even more “complex.” The newly recognized feature was that the distribution of degrees www.cat-science.cat

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was not close to the classical one, and a clear power-law decay with no characteristic scales was demonstrated. The implication of these so-called scale-free networks was that there are many nodes with a small number of connections, but some of them are highly connected, forming hubs. In this case, the focus on synchronization was directed to the role played by the different types of nodes, classified according to their topological properties, such as the degree or the different types of centrality. It is clear, however, that there must be an intrinsic relation between topological scales and the dynamic evolution of the synchronization process. We previously showed that a system composed of identical Kuramoto oscillators evolving from random initial conditions towards the only attractor, the synchronized state, produces phase correlations (which act as a kind of local-order parameter) that are the dynamic consequence of the topological distribution of the network [3]. Last, but not least, an additional degree of complexity arises for networks that, having their own dynamic rules, evolve with time. This time dependence can have different origins; one that is quite easy to understand corresponds to the motions of the units. Thus, when units move very fast synchronization is enhanced (Fig. 4), whereas when their motion is very slow, they also reach the final synchronized state, but over a much longer scale and with a different mechanism. For intermediate velocities, the compromise between the two mechanisms can produce undesirable (or desirable) consequences and disable synchronization.

Competing interests. None declared.

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References 1. Acebrón JA, Bonilla LL, Pérez-Vicente CJ, Ritort F, Spigler R (2005) The Kuramoto model: A simple paradigm for synchronization phenomena. Rev Mod Phys 77:137 doi:10.1103/RevModPhys.77.137 2. Arenas A, Díaz-Guilera A, Kurths J, Moreno Y, Zhou C (2008) Synchronization in complex networks. Physics Reports 469:93-153 doi:10.1016/j.physrep.2008.09.002 3. Arenas A, Díaz-Guilera A, Pérez-Vicente CJ (2006) Synchronization reveals topological scales in complex networks. Phys Rev Lett 96:114102 doi:10.1103/PhysRevLett.96.114102 4. Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286:509-512 doi:10.1126/science.286.5439.509 5. Díaz-Guilera A, Arenas A, Corral A, Pérez-Vicente CJ (1997) Stability

of spatio-temporal structures in a lattice model of pulse-coupled oscillators. Physica D 103:419-429 doi:10.1016/S0167-2789(96)00274-6 6. Kuramoto Y (2003) Chemical oscillations, waves, and turbulence. Dover Publications, Mineola NY doi:10.1007/978-3-642-69689-3 7. Landau LD, Lifshitz EM (1980) Statistical Physics. Vol. 5 (3rd ed). Butterworth-Heinemann, Oxford, UK 8. Mirollo RE, Strogatz SH (1990) Synchronization of Pulse-Coupled Biological Oscillators. SIAM J Appl Math 50:1645-1662 doi:10.1137/0150098 9. Strogatz SH (2003) Sync: the emerging science of spontaneous order. Hyperion, New York 10. Watts DJ, Strogatz SH (1998) Collective dynamics of “small-world” networks. Nature 393:440-442 doi:10.1038/30918 11. Winfree AT (2003) The geometry of biological time. Springer-Verlag, Berlin, Germany

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On the principles of multicellular organism development David Frigola,* José M. Sancho, Marta Ibañes

Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barce lona, Catalonia. Universitat de Barcelona Institute of Complex Systems (UBICS), Barcelona, Catalonia

Summary. Non-equilibrium physics has traditionally dealt mostly with inanimate matter. Yet, in the last decades there has been increasing interest in understanding living systems from this perspective. One example is using the framework and tools of non-equilibrium statistical mechanics and nonlinear physics to study how living organisms composed of many differentiated cells develop from a single initial cell. The dynamic process of multicellular organism development is out of equilibrium, in that it consumes and dissipates energy. It also involves the formation of many precise and complex structures. Herein we review some of the paradigms being used that focus on how these multicellular structures initially emerge at the molecular level. [Contrib Sci 11(2):215-223 (2015)] *Correspondence: David Frigola Departament de Física de la Matèria Condensada Universitat de Barcelona Martí i Franqués 1 08028 Barcelona, Catalonia E-mail: frigola@ecm.ub.edu

A historical overview An example of the beauty and complexity of Nature is the development of multicellular organisms. Animals and plants develop from a single cell, which through division gives rise to all the cells of the organism. During development, these cells become distinct in an organized and precise manner to robustly form complex structures such as organs. How does this occur? What are the principles behind it? Many physicists are now engaged in investigations of multicellular

organism development, with the aim of understanding how it proceeds and finding its fundamental principles. Resolving these questions is expected to help shed light on more applied challenges ranging from biomedical issues, such as embryonic malformations and cancer, to agricultural issues, such as the optimization of crop growth. However, the quest for underlying principles is still in its own early developmental stage, and an immense universe of knowledge lies ahead. In the following, we consider some of the ideas and insights that appeared early on and that have influenced current research.

Keywords: development · morphogen · attractor · non-equilibrium · differentiation · self-organization · lateral inhibition · multistep signaling · ultrasensitive response · bistability · fluctuation ISSN (print): 1575-6343 e-ISSN: 2013-410X

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Multicellular organism development

Over 70 years ago, Conrad H. Waddington used the metaphor that during development cells roll down through valleys that bifurcate [25], having to choose what to become at each bifurcation. This metaphor for cell differentiation is now commonly used, with a free energy landscape of which Waddingtonâ&#x20AC;&#x2122;s valleys are the minima. Alan Turing, very well known for his contributions to computer science, proposed, in a seminal work published in 1952, that patterns arising during development might be the natural output of chemical reactions between molecules that diffuse with different diffusion coefficients across space [23]. That chemical systems can form spatiotemporal patterns, in which the concentrations of molecules are organized in space and time, was proven later in well-controlled chemical and physical assays. However, these were not linked to multicellular organism development, but instead drove intense research in the field of nonlinear dynamics. The relevance of this mechanism, known as Turingâ&#x20AC;&#x2122;s instability, in the context of development is now appreciated but still debated [15,18,19]. In 1970, the Nobel Laureate Francis Crick proposed that the diffusion of molecules could create gradients across developing tissues [4]. These gradients could convey to the cells the positional information that Lewis Wolpert had already proposed [26], guiding them in their further development. This is the morphogen gradient paradigm, which has dominated research on patterning in developmental biology. The finding that numerous molecules form gradients during development and that the gradients themselves are relevant for the development of different tissues has led to many other complex questions: How does the gradient form? How is it sensed? And what information from the gradient does the cell use? Stuart Kauffmann showed that the interactions between genes strongly restrict the possible cell types [14]. In this case, cell types are understood as the attractors of the dynamics of genetic interactions. At present, deciphering the large gene regulatory and signaling networks and their dynamics in a developing cell is an intense field of research. These conceptual frameworks, i.e., bifurcations to produce changes of cell types, self-organization out of equilibrium and cell types as attractors, were mathematically formulated and developed. However, in the last decades of the 20th century, the use of mathematical formulations to understand development became unpopular because they failed at describing and predicting patterns. The result was a split between developmental biologists and physicists/ mathematicians [16]. More recently, however, knowledge of www.cat-science.cat

which biological molecules participate in development, the ability to manipulate them, and their spatial and temporal resolution, have increased dramatically. At the same time, important progress has been made in non-equilibrium statistical mechanics, dissipative systems, complex systems, nonlinear dynamics, and networks, accompanied by an extraordinary increase in computational power. As a result, interdisciplinary research involving both physicists and biologists has become more common and the advantages to this approach are now acknowledged [20]. Thus, we are in an exceptional position to embrace the challenge to understand development and the principles behind it.

Patterning the embryo A crucial step in understanding how multicellular organisms develop is to unravel how cells become distinct in a coordinated and organized manner. In the language of developmental biology, this can be rephrased as how a cell attains a specific fate into which it ultimately differentiates. Two main mechanisms have been proposed for coordinated cell differentiation in tissues. One mechanism is through positional information, proposed by Lewis Wolpert as mentioned above [26]: the fate of a cell is a readout of its spatial localization from a reference system (Fig. 1A). Cells read the information of where they are located and differentiate accordingly. Gradients of molecules, previously referred to as morphogens (we retain this term here for convenience), have been proposed to confer such positional information. The origin of the reference system is the source where the morphogen is produced. The amount or concentration of the morphogen decays as the distance from the source increases and thereby conveys positional information to the cell. This information can be conferred to cells through molecules that become activated at distinct thresholds of morphogen concentrations (Fig. 1A). There are multiple proteins that have been shown to be distributed along gradients in different developing embryos and that seem to convey positional information. Specifically, if the gradient is altered, the fate of the cells changes accordingly (Fig. 1B). This is the case, for instance, for the protein Bicoid, which forms a gradient along the anterior-posterior axis of the embryo during the very early stages of insect development, including that of the fruit fly Drosophila [10]. The region where Bicoid is at high concentration becomes the head of the fly. The other proposed mechanism is that cells become distinct only because of coupling. This is an example of 216

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Fig. 1. Pattern formation mechanisms that rely on diffusion along an extracellular medium. (A–B) Morphogen gradient mechanism. The leftmost, green cell generates a molecule that acts as a morphogen. The molecule diffuses to the right and generates a gradient, as shown in the curve above the cells. The amount of morphogen sensed by each cell conveys positional information to it. There are two thresholds, one at concentration 1 and another at concentration 10, and cells differentiate depending on whether the concentration is above or below these thresholds. In A there is extensive diffusion, as indicated by the larger curvy arrow. In B, there is less diffusion, altering the gradient and the position of cell types accordingly. (C–D) Turing pattern mechanism. Two or more chemicals that diffuse and react are needed to establish a pattern. In C the pattern for certain values of the parameters is shown. In D, when diffusion is modified so is the pattern and the corresponding cell fates. (Note that the term “morphogen” is no longer used with the mechanism shown in C and D. We use the term here because it was introduced by Turing precisely in this context).

self-organization in which a structure or order emerges spontaneously because of the interactions between elements. Coupled dynamics enable the emergence of robust proportions and periodic distributions of cell types. In contrast with the positional information mechanism, coupling does not drive a specific cell type in a given spatial position. In a developing organism this self-organization can happen in different ways. The first one corresponds to the dynamics Alan Turing studied [23]. When chemicals initially distributed homogeneously throughout a given space react and diffuse, they form heterogeneous distributions. Because the reactants diffuse with different diffusion coefficients, tiny small random fluctuations in the reactant concentrations become amplified, such that the homogeneous state becomes destabilized. This happens for a wide range of diffusion coefficients and reaction kinetics. It is an example of a non-equilibrium pattern formation process, in which the www.cat-science.cat

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balance between antagonistic processes, such as driving and dissipation, results in the formation of non-homogeneous structures [5]. Thus, for instance, periodic stationary distributions of the molecules can emerge. Cells produce proteins, which react and diffuse in the extracellular space. Accordingly, when a periodic pattern of protein distributions emerges from these dynamics, some cells end up producing or sensing large amounts of proteins while others do not. Therefore, cells become distinct (Fig. 1C,D). A change in the spatial interactions, as in the diffusion coefficient, results in relevant changes of the molecular pattern being formed. Accordingly, the pattern, if periodic and stationary, can change its periodicity (Fig. 1D). Empirical evidence that such a mechanism can drive the formation of the digits in vertebrates has recently been provided [21]. The digits form from an initial rather two-dimensional round palette. In this palette a stripe-like pattern emerges that divides it into two CONTRIBUTIONS to SCIENCE 11:215-223 (2015)

Multicellular organism development

intercalating regions: interdigital and digital regions. In this specific case the reaction-diffusion mechanism does not act in isolation but it is coupled to a positional information mechanism. Another way that could drive the differentiation of cells in a self-organized manner but does not require the transport of a molecule is through direct cell-to-cell contact. In this case, cells interact through molecules present on the cell membrane that, upon binding, send signals to the cell nucleus. An example of this is lateral inhibition with feedback [3]. In this case, the signal a cell receives arises from protein ligands in adjacent cells and it decreases the amount of ligand in the cell. Thus, a cell that has more ligand than its neighboring cells, even if the difference is very small, will reduce their amount of ligand and, at the same time, increase its own ligand production by preventing inhibition by those neighbors. Ultimately, the cell with an initially very small excess of ligand will end up with a relatively large amount of that ligand, while ligand in neighboring cells will be almost completely eliminated. This type of interaction underlies the specification of neurons, for instance. In the 1970s, Meinhardt and Gierer proposed a theory for biological pattern formation based on two elements: (1) self-activation and (2) long-range inhibition [9]. Turinglike reaction-diffusion dynamics and lateral inhibition with feedback can both be understood in terms of these two elements. Moreover, self-activation evidences a key aspect in the dynamics of coupled elements that drive patterning: nonlinearities. All these self-organizing interacting dynamics drive the emergence of robust proportions and periodic distributions of cell types. In this mechanism based on coupling, the cell types arise in a coordinated manner but, unlike in the positional information mechanism, it does not enable the robust specification of a cell type in a given spatial position. Nevertheless, if spatial asymmetric cues are added to interacting dynamics, then spatial precision can arise as well. It is worth noting that how the pattern will be modified when the elements driving it are altered can be predicted by constructing mathematical and computational models of the dynamics. The resulting predictions can then be used to test whether assumptions regarding the mechanism of patterning are correct, by comparing the predicted results with the empirically derived data. This task is nowadays common routinely done but it has not always been so easily possible. Now we can propose which specific molecules are acting and, in several cases, we can experimentally see how their distribution changes over time and space with detailed www.cat-science.cat

resolution. Manipulations of the interactions and reactions and how the molecular distribution changes accordingly can now be done and the results measured. The mechanisms described herein assume that, in terms of their patterning, cells can be described by only a few relevant molecules. The role of cell dynamics and the particular mechanical forces that are active are not taken into account. This simplification is valid in some circumstances, especially when the dynamics that control the molecular concentrations are much faster than those of the cell. Many efforts are being done on the role of mechanical forces in shaping developing multicellular organisms, which are not reviewed herein. A challenge that remains is to determine how mechanical forces and the dynamics of the molecular components that direct cell signaling or impinge on gene regulation are coupled to each other.

Nonlinear responses We have discussed how molecular gradients can confer positional information, in which each cell type is dictated by a threshold, cell-type-dependent, morphogen concentration. In Fig. 1A, cell type “blue” is induced above a morphogen concentration of 10 (arbitrary units), whereas cell type “white” is induced above a morphogen concentration of 1. Yet, is this type of threshold response possible in biological systems? It is, thanks to ultrasensitivity. As opposed to a gradual or linear response, in which the relative changes in input (signal) and output (response) are equal, an ultrasensitive response is that in which a small relative change in the signal generates a very large (relative) response. Since a cellular response usually saturates (i.e., when the input signal is large enough, the response no longer changes), an ultrasensitive response in cells can translate to a threshold or “all-or-nothing” response (Fig. 2A). But how is this ultrasensitivity achieved by cells? A variety of mechanisms have been elucidated through mathematics and then experimentally demonstrated [27]. A few of them are summarized in Fig. 2 and reviewed in [27]. “Zero-order ultrasensitivity” was the first of these mechanisms to be proposed, in 1981 [11]. In this mechanism, an enzyme covalently modifies a protein (covalent modification is a common regulatory mechanism in which a molecule such as a phosphate or methyl group is bound to a protein by an enzyme), and an opposing enzyme restores the protein to its unmodified state. When both enzymes are working at saturation, a small change in the amount of one of them can 218

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Bistability A positive feedback loop can also enable bistability, i.e., two different responses to the same input (mathematically, the equation that represents the system has two stable solutions instead of one). In other words, genetically identical cells exposed to the same environmental conditions can be in two different states and hence become two distinct cell types. An example of bistability in development occurs in the vulval development of the hermaphroditic nematode worm Caenorhabditis elegans [10,12]. Before this egg-laying organ is formed, two adjacent cells, which can be labeled 1 and 2, for instance, become distinct from each other based on their position in the embryo. One becomes an anchor cell (AC) and the other a ventral uterine (VU) cell. Each cell has a 50% probability of becoming an AC. Hence, under the same conditions two states can arise, with 50% probability each: (AC,VU) or (VU,AC), in which the first term within the parentheses denotes the type acquired by cell 1, and the second term refers to cell 2. In this case, the bistability of these two states arises through a positive feedback that involves the above-described lateral inhibition with feedback. Nonlinearities are essential for this bistability. Figure 3 provides an example of this case and shows how a mathematical model of the interactions can help us to understand and visualize this process. www.cat-science.cat

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produce a large change in the proportion of the modified and unmodified proteins, thus enabling an ultrasensitive response (Fig. 2B). Another, very common mechanism is multistep signaling, in which an element representing the signal, or proportional to the signal intensity, acts on two or more elements that independently affect the strength of the response. An example is a signal that acts on two different steps of the modification of a protein that will ultimately turn it into its active form. The multiplied effect elicits an ultrasensitive behavior. Mathematically, the repeated effect of the signal is represented as multiplicative terms that can raise it up to the power of the number of points at which the signal affects the system independently (Fig. 2D). Direct or indirect self-activation, also known as positive feedback, can drive ultrasensitive responses as well. Positive feedback occurs, for instance, when a protein binds to its own DNA promoter to boost its own transcription (autoactivation), or when a protein inhibits the production of its inhibitor (mutual inhibition) (Fig. 2F).

Fig. 2. Mechanisms that generate ultrasensitivity. (A) A signal-response function showing ultrasensitivity and an all-or-nothing response, as shown in panels B, D, E, F. (B) Zero-order ultrasensitivity. As explained in the text, the purple enzyme, corresponding to signal S, enhances the covalent modification of the red protein, while the yellow enzyme mediates its demodification. The modified protein amount corresponds to the response R. (C) Molecular titration. Free molecule A (corresponding to or activating a response R) can be sequestered by B, which is present in very large amounts. Molecule A exhibits an ultrasensitive response to changes in its production. There are free A molecules only when their production level surpasses the sequestering effect. At this threshold, the amount of A suddenly increases. This behavior is not like that shown in A, because its response does not saturate. (D) Multistep signaling. The signal S, or some element proportional to it, aids in two different steps of the modification of a protein that will ultimately assume its active form, which then enacts response R. Its effect is multiplied and can elicit an ultrasensitive response. (E) Cooperative binding. A receptor, in green, has several binding sites for the same ligand, the amount of which corresponds to signal strength S. If full occupancy of the receptorâ&#x20AC;&#x2122;s binding sites is needed to elicit a response R, or if each occupied site increases the chance that a new ligand will bind (thicker arrows indicate larger amounts of bound ligand), ultrasensitivity arises. (F) Positive feedback loop. A signal S (here a blue enzyme) activates a protein (in red). This active protein elicits response R, but it can also bind to DNA and enhance the production of its own unmodified form. This increases the amount of substrate upon which the signal can act, multiplying its effect and making the response ultrasensitive. These mechansims are reviewed in [27].

Fluctuations As we have seen, cells have mechanisms to process signals coming from neighboring cells and from their surroundings that can yield precise results. However, these signals cannot CONTRIBUTIONS to SCIENCE 11:215-223 (2015)

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Fig. 3. Bistability. (A) Lateral inhibition with feedback. The ligand in cell 1 inhibits the ligand in neighboring cell 2 and vice versa, establishing positive feedback. Inhibition is represented by the blunt arrows. There is a 50% probability for the (AC,VU) outcome and 50% for the (VU,CA) outcome, determined by which cell achieves a high or low amount of ligand. (B) The equation that governs the temporal evolution of a ligand in cell i (1 or 2). dli / dt is the time derivative of concentration li and represents its changes over time. The production term g (l ) decreases nonlinearly when l j (the ligand in the other cell) increases. (C) Phase diagram of this two-cell system. Each point corresponds to a unique pair of l1 − l2 values. The evolution of either one is fully determined and shown by the blue arrows of the vector field. The red and blue dashed lines are called nullclines and correspond to the points at which the time derivative, i.e., the rate of change, for the ligand at one of the cells (blue for cell 1 and red for 2) is zero. At the points where the nullclines cross both derivatives have the values of zero, so the system, if unperturbed, will not move away from them. Because of the nonlinearity of the nullclines, there are three of these points; if they were not nonlinear, there would only be one such point. Of these, the black points are stable states: when the system is at one of them, it will return to it after a small perturbation (this state is therefore also called an attractor). Indeed, all trajectories starting in the purple half of the portrait (called the basin of attraction) will evolve towards the (AC,VU) stable state at the bottom left (one such trajectory is shown in black). Similarly, the green area is the basin of attraction for the (VU,AC) stable state. The orange point represents a state with intermediate values of ligand for both cells, as shown in gray, that is not stable. A small perturbation from this state can lead the system away from it and to one of the stable solutions. The scenario in B was obtained from simulations performed by Juan Camilo Luna-Escalante, Dept. of Condensed Matter Physics, University of Barcelona). The data are used with permission. j

be sensed with perfect precision due to the physical laws that govern molecular dynamics [2]. These signals, and the proteins that process them, consist of discrete molecules that jiggle around, embedded in the thermal bath of the cytoplasm. This aqueous medium is crowded with many moving molecules such as proteins. Some molecules move stochastically without a preferred direction, because of thermal forces coming from collisions with water molecules. Others, such as molecular motors, move directionally using electrochemical forces. Several of these electrochemical reactions have associated energies (such as the energy required for some reactions to start, or the energy required to break specific chemical bonds) comparable to the thermal energy of the medium. Therefore the stochastic “jiggling” of molecules can spontaneously activate reactions or break chemical bonds. These fluctuations also affect the production and degradation of different proteins in the cell, which stochas tically vary in time. This could not be directly observed until the very recent advances in the spatial and temporal resolution of fluorescence microscopy techniques. Before that (but also only recently), temporal fluctuations in the www.cat-science.cat

amount of specific molecules could only be inferred from the heterogeneous amounts found among genetically identical cells in the same environment. Even though fluctuations are a common object of study in non-equilibrium and statistical physics, our direct knowledge of the motion and fluctuations of particles embedded in the crowded medium of a cell is still incipient. Yet, with the advent of nanotechnologies we are entering a new era in which it will be possible to characterize the motions of and fluctuations in cellular components.

Fluctuations and cell decisions Because fluctuations are ubiquitous in the cell, they must somehow be relevant to an understanding of all cellular processes, including those in the previously mentioned examples of morphogen diffusion, cellular sensing of these molecules, and the related signaling processes. The exquisite precision and regularity of developmental processes indicates that cells can cope with this variability, or perhaps even profit from it. One obvious way of avoiding the effect of fluctuations is by producing large amounts of molecules to minimize 220

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Fig. 4. Stochastic switching. (A) A model of a bistable system. The black continuous line is the energy landscape of the system; the bottoms of the two wells are the stable states. The blue circle represents the system at one of these states, and the blue arrows the fluctuations, which can drive the system to higher energies. If the fluctuations are large enough, or the energy barrier (â&#x2C6;&#x2020;U ) low enough, the system can jump to the leftmost well and switch states. (B) Time evolution of the amounts of a protein for a single cell in two different cases. There are two clearly defined states, a high concentration state at 270 protein copies and a low concentration state at 50 copies. The cells switch from one state to the other. Note that the transitions are very fast and that the system spends most of its time around one of the two stable states. (C) Evolution over time of the concentrations of a protein of interest in a cell culture, as shown in a histogram. When a subpopulation in one of the states from an originally bistable population is separated and left to evolve over time, stochastic switching allows the recovery of the two states. The cells in a population are shown on the right. Note how one cell may switch states more than one time. Panels B and C are simulations of a mutual inhibition system, simulated through the Gillespie algorithm, which allows exact simulations based on the theoretical description of discrete stochastic systems in the form of master equations.

their effects. This is not always worthwhile, or possible. For instance, when a cell receives a fluctuating signal that it cannot control, how can it cope with the fluctuations? One way to buffer fluctuations is to respond to the amount of signaling molecules received only during an interval of time [2]. This corresponds to an integration over time of the number of molecules, the result of which is much less variable than the number of molecules at any given time. Therefore, the input to which the cell responds is not the highly fluctuating number of molecules but the much more constant total number of molecules received per unit time. This time integration is performed, for instance, by bacteria to sense the level of nutrients in their environment [2]. It also is the mechanism proposed for developing embryos, in the cellular response to morphogen gradients [6]. In cases in which cells respond too rapidly compared to the time interval that would be required for integration to filter out fluctuations, the additional interactions of neighboring www.cat-science.cat

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cells may reinforce the correct cell decision and increase its robustness [13]. There are several examples of biological systems that profit from fluctuations [7]. Most of them are in unicellular rather than multicellular developing organisms, but their existence can suggest that fluctuations may also be used during development. For instance, fluctuations enable wide-ranging heterogeneity between genetically identical cells in the same environment. This heterogeneity can be beneficial when the environment changes rapidly and the cellular response is heterogeneous. If this heterogeneous population of cells comprises different cell types that respond differently, then when the environment changes some of the cell types may die while others will prevail. Because of this heterogeneous response to environmental change, the cell population persists, providing a benefit. This is known as bethedging (the colony of cells hedges its bets instead of putting â&#x20AC;&#x153;all of its eggs in one basketâ&#x20AC;?) and has been described in CONTRIBUTIONS to SCIENCE 11:215-223 (2015)

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different types of bacteria [24]. Fluctuations in the number of molecules can drive large heterogeneities among cells in different ways. One is through positive feedback, which can drive the molecule to be present at either high or low concentrations. These two concentration states can be understood, at least conceptually, as free energy minima and are separated by an energy barrier [1]. Dissipation drives the molecular concentration to reach one of these two states and remain there forever after. Which concentration is achieved depends on the initial state; that is, on which concentration was present initially. This scenario changes when we take into account that there are fluctuations. They provide the energy required to surpass the energy barrier that separates the states, allowing a switch from a low to a high concentration or vice versa (Fig. 4). An example of heterogeneous cell populations comes from experiments using mouse embryonic stem cells (ESC), which in culture express pluripotency factor NANOG in a highly stochastic manner [8]. NANOG allows ESCs to selfrenew and to maintain their pluripotency. When NANOG levels of individual ESCs in a culture are measured, the distribution of values is very broad. If cells with, for instance, low NANOG expression are selected, separated from the others, and allowed to divide over time, measurements show that the very broad distribution of NANOG concentrations is eventually recovered. Hence, some cells, despite initially being in the low NANOG concentration state, have clearly switched and now express very high concentrations of NANOG. Whether this stochastic switching corresponds to bistable or other type of dynamics is a current topic of research. A role of fluctuations in multicellular development has been proposed for cells that need to establish a pattern that is not spatially ordered but, instead, only needs to preserve certain proportions of different types of cells, randomly spaced around the tissue. A stochastic decision mechanism has been proposed for processes such as the differentiation of different photoreceptors in the retina of humans and flies, or of olfactory cells in the mouse [17]. Mice have 1000 olfactory proteins, with only one expressed in any given cell to avoid sensory confusion. Hence, initially equivalent cells become distinct, reaching one of 1000 different states. This has been proposed to be accomplished by the activation of one olfactory protein type stochastically and subsequent inhibition of all the other remaining types of olfactory proteins. In addition, fluctuations of molecular components can be expected to trigger patterns arising from interacting self-organizing dynamics such as reaction-diffusion and lateral inhibition. www.cat-science.cat

Our knowledge on the effect and role of fluctuations in developmental processes is still limited. However, research in physics over the last few decades has evidenced that nonlinear systems can take advantage of fluctuations [22]. Thus, it is to be expected that developing organisms, which exhibit highly nonlinear dynamics and are subject to fluctuations, profit from them as well. The concepts and tools to study this topic have already been developed by physicists and biologists, and the results should soon be available.

Conclusions The development of multicellular organisms is subject to the physical laws that govern Nature. It is indeed because cells live out of equilibrium that they are able to create the myriad of rich and complex structures that form multicellular organisms. Insights have been gained into some of the molecular gene regulatory and signaling mechanisms used by cells in the spatially and temporally coordinated processes that allow them to become distinct in an organized and reproducible manner. These processes require nonlinear responses and dynamics. Previously, development was mostly understood as a succession of stationary states and many aspects were described through averages over many cells. However, we now have strong evidence that development is a highly dynamic process and that cellular dynamics are strongly stochastic. Although many technical limitations to advancing our knowledge remain, new data are expected that will reveal the highly complex and dynamic nature of developing organisms. As physicists, we expect to continue to work together with biologists to define the principles that govern multicellular organism development. Competing interests. None declared Acknowledgements. The authors acknowledge financial support by project FIS2012-37655-C02-02 by the Spanish Ministry de Economy and Competitiveness and 2014SGR878 from the Generalitat de Catalunya.

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16. Lawrence P (2004) Theoretical embryology: a route to extinction? Curr Biol 17:R7-8 doi:10.1016/j.cub.2003.12.010 17. Losick R, Desplan C (2008) Stochasticity and cell fate. Science 320:65-68 doi:10.1126/science.1147888 18. Mahalwar P, Walderich B, Singh AP, Nüsslein-Volhard C (2014) Local reorganization of xanthophores fine-tunes and colors the striped pattern of zebrafish. Science 345:1362-1364 doi:10.1126/science.1254837 19. Marcon, L, Sharpe, J (2012) Turing patterns in development: what about the horse part? Curr Opin Genet Dev 22:578-584 doi:10.1016/j. gde.2012.11.013 20. Perrimon N, Barkai N (2011) The era of systems developmental biology. Curr Opin Genet Dev 21:681-683 doi:10.1016/j.gde.2011.10.004 21. Raspopovic J, Marcon L, Russo L, Sharpe J (2014) Digit patterning is controlled by a Bmpt-Sox9-Wnt Turing network modulated by morphogen gradients. Science 345:566-570 doi:10.1126/science.1252960 22. Sagués F, Sancho JM, García-Ojalvo J (2007) Spatiotemporal order out of noise. Rev Mod Phys 79:829-882 doi:10.1103/RevModPhys.79.829 23. Turing AM (1952) The chemical basis of morphogenesis. Phil Trans R Soc London B 237:37-72 doi:10.1098/rstb.1952.0012 24. Veening JW, Smits WK, Kuipers OP (2008) Bistability, epigenetics, and bet-hedging in bacteria. Annu Rev Microbiol 62:193-210. 25. Waddington CH (1942) The epigenotype. Endeavour 1:18-20 26. Wolpert L (1969) Positional information and the spatial pattern of cellular differentiation. J Theor Biol 25:1-47 doi:10.1016/S0022-5193(69)80016-0 27. Zhang Q, Bhattacharya S, Andersen ME (2013) Ultrasensitive response motifs: basic amplifiers in molecular signalling networks. Open Biology 3:130031 doi:10.1098/rsob.130031

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BIOPHYSICS Institut d’Estudis Catalans, Barcelona, Catalonia

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CONTRIB SCI 11:225-235 (2015) doi:10.2436/20.7010.01.234

Neuronal cultures: The brain’s complexity and non-equilibrium physics, all in a dish Jordi Soriano,* Jaume Casademunt Departament de Físca de la Matèria Condensada, Universitat de Barcelona, Barcelona, Catalonia. Universitat de Barcelona Institute of Complex Systems (UBICS), Barcelona, Catalonia

*Correspondence: Jordi Soriano Departament de Físca de la Matèria Condensada Universitat de Barcelona 08028 Barcelona, Catalonia Tel.: +34-934020554 E-mail: jordi.soriano@ub.edu

Summary. Neuronal networks—and the brain in particular—are out of equilibrium systems in which neurons are the interacting elements. These cells are coupled through physical connections and complex biochemical processes. Moreover, they are capable of self-organization and shape a rich repertoire of spatiotemporal patterns and dynamic states. An elegant yet powerful experimental tool to investigate and describe the features of neuronal networks is neuronal cultures, in which neurons are extracted from brain tissue, dissociated, and cultured in an appropriate environment. Here we introduce the difficulties in understanding the complexity of the brain and its dynamics. We then present the fundamental concepts—from a statistical and non-linear physics viewpoint— needed to describe neurons and networks. These concepts lay the foundations needed to discuss recent models of brain dynamics. We then introduce neuronal cultures, highlighting their enormous potential as accessible and controllable living neuronal networks. Finally, we show how neuronal cultures, and their physical modeling, constitute a remarkable platform to investigate fascinating questions in the non-equilibrium physics of the brain and to provide new insights to advance the treatment of neurological disorders. [Contrib Sci 11(2):225-235 (2015)]

Approaching the brain’s complexity through model systems Understanding the brain is not only a scientific curiosity. It is the path to understanding ourselves, our social behavior, and the modification of both upon the brain’s malfunction. This quest recently fostered the establishment of two grand international brain enterprises, the “Human Brain Project” [23,47] and the “BRAIN Initiative” [19,23,47], supported, respectively, by the European Commission and

the US government. The scope of these projects, “a bold new research effort to revolutionize our understanding of the human mind” in President Obama’s words, is to bridge basic neuron-to-neuron interactions with brain function and cognition, which could ultimately shed light on new approaches to the treatment of neurological disorders, one of the largest burdens of our aging society. The two projects consider different yet complementary strategies: the “Human Brain Project” aspires to build a meticulous computer simulation of the human brain, while the “BRAIN Initiative”

Keywords: connectivity · neuronal culture · network models · noise · spontaneous activity ISSN (print): 1575-6343 e-ISSN: 2013-410X

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has targeted the development of new technologies aimed at the simultaneous monitoring of all neurons in the brain. Although both actions have a collaborative vision and facilitate interdisciplinary research, the scientific community has recently raised serious concerns as to the validity of some of the proposed methodologies and their goals. For instance, if the “Human Brain Project” is based on a simulation of 1011 neurons and 1014 connections, how sensitive would it be to the simulation parameters themselves? Or, would the observed activity patterns reflect actual brain functions? The debate surrounding these grand projects points out the importance of tackling the brain using different scales and approaches, to first comprehend general mechanisms before investigating the precise molecular details. Indeed, the brain exhibits a number of reliable features (such as widespread rhythmic activity or synchronization across distant areas) that suggest the involvement of robust mechanisms. These must be sufficiently stable in the presence of perturbations or fluctuations in the biochemical environment, but sufficiently flexible to allow for the processing of information and to respond to stimuli. In this context, two fundamental questions arise: How can these mechanisms be investigated in an accessible manner? What principles govern the emergence of collective behavior in neuronal networks? Reducing the brain’s complexity to unveil general mecha nisms has both experimental and theoretical aspects. Experimentally, one can start by analyzing living neuronal networks of gradually larger size and richness. The nematode worm Caenorhabditis elegans and the freshwater fish Danio rerio (zebrafish) constitute the two most well-explored “simple” model organisms used in brain research. C. elegans has 302 neurons spread along its body and its set of 7000 connections (connectome) has been mapped out in its entirety [41]. Recent experiments have managed to simultaneously record the activity in all of the worm’s neurons [27] and to even perturb the activity of some of them. Zebrafish have their neurons organized in a brain; although its precise connectome is still not fully drawn, in recent experiments the activity of most of the 100,000 neurons of a larval-stage zebrafish were simultaneously recorded [1]. In addition to these highly valuable model organisms, the need for systems that can be more readily controlled and accessed has put neuronal cultures in the frontline of tunable (and living) complex systems [31]. Indeed, the small size of neuronal cultures, as well as their preparation in controlled environments, has greatly facilitated their manipulation, monitoring, and analysis. Theoretically, the brain—and its “simplified” analogs— can be approached by developing models with different levels www.cat-science.cat

of physico-mathematical complexity and biological accuracy, depending on the scale of the system under study and the particular problem to be addressed. Indeed, the neuronal assembly and its set of input and output connections configure a network whose ultimate dynamic traits depend on three major agents: the neurons themselves (intrinsic neuronal firing properties), the layout of connections (connectivity), and the inherent fluctuations in this biological system (noise). The beauty of nonlinear physics is that it provides remarkable tools to describe these agents and to delineate their operation and mutual interaction. These interacting agents are then used to reproduce the characteristics of the observed activity patterns, predict their behavior, expose universal mechanisms, and even uncover hidden processes. The resources at hand are extensive and include: (i) the use of biophysical models to describe neurons [16,21]; (ii) graphtheoretical tools from statistical physics and mathematics to describe the connectivity map [4,5]; (iii) dynamic systems approaches to render the organization and stability of activity patterns [22]; and (iv) fluctuation theory to account for the effects of noise [12,24,30].

Neuronal cultures Neuronal cultures [11,25] are typically prepared by first isolating a fragment of neuronal tissue from a specific brain region, for instance, the hippocampus or cortex of embryonic mice or rats. The neurons are then dissociated and seeded over culturing substrates such as glass (Fig. 1A), effectively establishing within a few days a de novo network rich in spontaneous activity. The size and shape of the culture can be controlled by different means (such as hollow masks, Fig. 1A), which allows the establishment of multiple cultures in the same well and thus their simultaneous access (Fig. 1B) [26]. These types of preparations are the expertise of our laboratory in Barcelona [26,36,37]. A detail of a typical neuronal culture is shown in Fig. 1B. Neurons appear as spherical objects whose connections are so dense and entangled that they cannot be resolved. Measuring activity in these cultures is obviously the first step towards understanding their modus operandi. Two main techniques are used to record neuronal activity: fluorescence calcium imaging [17] and electrodes [33]. Calcium techniques (Fig. 1C,D) are based on the use of fluorescent probes to detect the influx of calcium ions upon neuronal firing. Active neurons appear as bright 226

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Fig. 1. Neuronal cultures. (A) Schematic representation of the culture protocol. Neuronal growth is supported by a substrate (typically glass), which may be accompanied by a hollow mask or other structure to control the size and shape of the network. Dissociated neurons are homogeneously plated over the substrate in the presence of chemicals that support their development and then cultured for several days. After 1–2 weeks in vitro, the neurons have reconnected and shaped a new network with rich spontaneous activity. (B) A single glass coverslip 13 mm in diameter may easily contain a single network with ~105 neurons, or a number of small networks with a typical density of 1000 neurons/mm2. A fluorescence camera attached to an optical system can access in its field of view one or more small cultures. (C) Detail of a small region of a culture. Individual neurons appear as circular objects 10 µm in diameter. (D) Corresponding fluorescence image. Bright spots are firing neurons. (E) Neurons cultured over a substrate that contains a grid of electrodes (black structures in the image), which directly record the electrical activity of the neurons. (F) Patterned culture in which neurons grow at the crevices of a mold. The scale bars in C–F are 100 µm. (Figures adapted from [11,26]).

spots whose fluorescence intensity is proportional to the number of firings elicited by the neurons. Electrode-based techniques (Fig. 1E) directly measure the electrical activity of neurons. Although the latter methods offer higher temporal resolution and sensitivity, the number of neurons that can be accessed is limited by the number of electrodes, which impedes the study of large networks. Calcium imaging, by contrast, is limited only by the optical recording system. Current technology has made possible the simultaneous www.cat-science.cat

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recording of thousands of neurons, both in vitro [26,37] and in vivo [1]. Neuronal cultures prepared under conditions in which the neurons uniformly cover the substrate and connect equally in any direction are termed homogeneous (Fig. 1C– E). Conversely, patterned cultures are those in which the position of the neurons and their connectivity are in some way dictated, thus allowing for complex configurations or neuronal circuits with specific characteristics [13,45]. One of CONTRIBUTIONS to SCIENCE 11:225-235 (2015)

Neuronal cultures

the many possibilities is the use of topographical molds (Fig. 1F), in which neurons grow along either the bottom or the top of a two-level pattern, giving rise to a highly anisotropic and inhomogeneous culture. In general, the interest of patterned cultures for Physics and Neuroscience is twofold. On the one hand, they allow the design of simple circuits with known properties from which basic questions, such as the propagation of activity [14,15] and information coding [13,14], can be investigated. On the other hand, and within a more general perspective, patterned networks allow the controlled study of activity-connectivity relationships. This “engineering” procedure is not possible in intact, native brain tissue since the major structural paths are genetically dictated and therefore hardwired. Patterning in cultures adds guidance and strong spatial anisotropies, effectively shaping different connectivity layouts and activity patterns.

Spontaneous activity in cultures Examples of recorded activity in neuronal cultures are shown in Fig. 2. In a homogenous culture (Fig. 2A), each neuron is selected as a region of interest to extract its fluorescent trace, i.e., the brightness of the neuron over time. As depicted in Fig. 2B, the fluorescence intensity averaged over the population (3000 neurons in this particular preparation) is characterized by quasi-periodic episodes of high activity combined with silent intervals. By inspecting each neuronal trace, one observes that these activity events encompass all the neurons (Fig. 2B, yellow box), shaping what is known as a “network burst.” The analysis of these bursts reveals several very interesting features of the network and the mechanisms that control spontaneous activity [26], a problem that we address in more detail below. The fluorescence traces also reveal sporadic, asynchronous firing events (red arrowheads). These firings are also of interest, since they convey information on single neuron-to-neuron interactions that possibly reflect a direct physical connection between those neurons [35]. In patterned cultures, the inclusion of strong inhomogeneities in the connectivity induces very different dynamics. A simple yet informative preparation—and the focus of research in our laboratory [36]—consists of shaping aggregates of neurons that connect to one another (Fig. 2C, top). Each aggregate can be treated as an “effective neuron,” which greatly reduces the number of dynamic elements in the network (Fig. 2C, bottom). These networks exhibit www.cat-science.cat

spontaneous activity patterns with rich spatiotemporal variety (Fig. 2D). Additionally, some of the connections between aggregates are directly visible, which makes the system a very attractive one to study the interplay between activity and connectivity, or as a model system to investigate network resilience to damage [36].

Neuron models A neuron can be viewed as a “black box” able to receive inputs from other neurons, finally generating an output if the number of received inputs within a short time window is sufficiently large. The key variable for the representation of a firing neuron is the membrane potential, which changes according to whether the inputs are excitatory or inhibitory. The former increases the membrane potential, while the latter decreases it. The timing between inputs, their strength, and their ultimate integration by the membrane are important aspects that shape neuronal responses and the dynamics of the network. It is precisely the complex processing of the neuronal inputs, as well as the on/off nature of the outputs, that confers a neuron with its nonlinear behavior. Describing a neuron in detail is a mathematically difficult task, and a serious challenge when thousands of neurons are coupled together to assemble a meaningful network. However, what often interests physicists and neuroscientists is the collective action of the networks, in which case a very detailed biological description of the individual neurons is not necessary. This allows the use of relatively simple models with phenomenological parameters adequate to the scale and characteristics of the system under study [11,21,26]. The most popular and intuitive model is the “integrateand-fire” (IF) model [21], in which the membrane potential increases with the number of inputs until it reaches a threshold, at which point the neuron “fires” (i.e., generates an action potential) and the membrane potential is reset to the resting state. The action potential travels along the axon to finally become the input of the connected neurons. The IF model uses a single equation and four parameters and can be easily modified and extended to reproduce sufficiently well the behavior of most neurons. One of these modifications defines the so-called Izhikevich model [20,21], which includes an additional variable that accounts for the recovery of the membrane potential, giving rise to a set of two coupled differential equations with four parameters. A significant jump in biological detail (and mathematical 228

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Fig. 2. Fluorescence imaging and neuronal network activity. (A) Fluorescence image of a small region of a homogeneous neuronal culture. Each neuron (bright spots) is associated with a region of interest (square box) from which time traces of neuronal activity are extracted. (B) The analysis of spontaneous activity in a typical experiment provides the average fluorescence signal (corresponding to the activity of ~1000 neurons) and all the individual single neuron traces. The yellow box depicts a network burst; the red arrowheads indicate scattered neuronal firings. (C) Patterned culture formed by interconnected islands of tightly packed neurons. Top: a detail of the network; bottom: a fluorescence image of the entire network during activity. The yellow arrowhead indicates a connection between clusters. (D) Corresponding network spontaneous activity. Top trace: the average fluorescence signal of the network along the recording; bottom plot: the activity of the network, with each dot indicating the occurrence of a firing in one of the islands. Neuronal islands tend to fire in groups with rich variety in the number of participating islands (yellow boxes). (Figures adapted from [36,37]).

complexity) is obtained with the Hodgkin-Huxley model [18], which describes the variations in the membrane potential together with the activation or inactivation of sodium and potassium currents. This model was introduced already in 1952 to explain the generation and propagation of action potentials in the axon of the giant squid. The model was easily adapted to different neural systems, and thus within a short time became a fundamental tool in theoretical and computational neuroscience. Indeed, its developers were awarded the Nobel Prize in Physiology or Medicine. The model consists of four main equations and tens of parameters www.cat-science.cat

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which, when properly adjusted, allow the reproduction of all the different forms of spiking neurons [21]. However, the complexity of the model has made it impractical for studying systems with a large number of neurons, and therefore the simpler models outlined aboveâ&#x20AC;&#x201D;which are actually reduced versions of the Hodgkin-Huxley modelâ&#x20AC;&#x201D;are largely used, despite the loss of detail. Because collective phenomena are dominant in large neuronal assemblies, simple neuronal descriptions suffice to remarkably well reproduce the major dynamic traits of a neuronal network (e.g., the spontaneous activity patterns shown in Fig. 2B [26]). CONTRIBUTIONS to SCIENCE 11:225-235 (2015)

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Network models Like the streets and traffic lights in a city, the layout of neuronal connections and the adequate balance between excitation and inhibition are crucial for the normal operation of brain circuits. In epilepsy, for instance, an imbalance towards an excess of excitation brings the brain to a state of abnormal synchronization that totally disrupts its operability. The description of the connectivity of networks has significantly advanced in the last two decades due to the enormous progress made in Network Science, which studies the statistical properties of complex networks in systems as diverse as the Internet, social relations, protein interactions, transportation, and neuronal tissues [4,5]. The identification of nodes and links is easy, for instance, in the case of airports and air routes, but when approaching the brain and neuronal networks it becomes more complicated. On the one hand, in neuroscience a link can be structural or functional. Structural links are the actual physical connections between the neurons, while the functional ones are statistical correlations between the activity patterns of any two neurons. For instance, the activity traces of Fig. 2B show that neurons #1 and #3 fire together systematically and are therefore strongly correlated, i.e., they are linked in a functional manner. Since functional links reflect the flow of information across the neuronal circuit, they must be related to the underlying structural connectivity. How similar the functional network is to the structural one is an elegant problem and an entire research topic by itself [28]. On the other hand, the ability to identify nodes and links greatly decreases with the size of the network under study. A few neurons in a dish can be well monitored and their structural connections can even be resolved to some extent. This type of experiment, performed in our laboratory in Barcelona, can aid in uncovering the relation between structure and function [36]. At the scale of the brain, however, its sheer size and the limits of instrumentation impose the use of parcellations that contain thousands or millions of neurons. These parcellations are then the nodes, while correlations between those parcellations upon their activity define the corresponding functional links. Although the exact connectivity map between the neurons in the brain is unknown, there is a wealth of data describing major structural paths and interconnections between brain areas [38,44]. Again, the comparison of these maps may provide very important information and greatly help in understanding the brain and its alteration by various diseases. The fundamental property that describes a given network, www.cat-science.cat

whether structural or functional, is the degree distribution p(k), i.e., the probability that a node has k connections. The properties of this distribution delineate the important features of the system that it represents. For instance, let us consider a “toy network” in which a set of neurons are deposited over a flat surface. With no restrictions or guidance of any kind, all neurons will essentially connect to their neighbors, in which case the resulting p(k) distribution is close to a Gaussian distribution, with a mean indicating the typical average connectivity of a neuron in the network and a width that reflects the inherent variability in the number of connections across neurons. A signal generated at one end of this network will advance towards the other end in several steps, since it has to pass locally from neuron to neuron. This process can be well studied in the framework of statistical physics using concepts from percolation theory and criticality [7,32]. Conceptually, a network percolates if there is at least one path of connected neurons that bonds both ends of the network. If the neurons are highly connected, there will be several of these paths. The number of possible paths will rapidly diminish if the connectivity decreases. At a particular value of connectivity, percolation will no longer be possible, effectively breaking the circuit apart into small, disconnected islands. This value of connectivity defines a critical point that separates the connected from the fragmented layouts, and its study from an experimental perspective can shed light on interesting features of the structure and resilience of a neuronal network [32]. The toy network introduced above can be explored further to tackle other powerful concepts. If a few neurons in this network are allowed to form connections with very distant ones, then not only does the distribution of connections change, but the dynamics and percolative aspects are entirely reshaped. These “shortcuts” advance the information much faster and even synchronize a large number of neurons. They confer upon the network a “small-world” property [43], meaning that any neuron can be connected to any other through relatively few steps. C. elegans is an example of a known living neuronal network exhibiting this feature [39,41]. Again using this toy network, one can now imagine that neurons with these long-range connections have many more connections than other neurons and serve as true “hubs” that route information flow. These hubs may connect to one another to shape a structural core (or “rich club,” in network language) that provides robustness to the network in the case of a random failure of nodes. Of the 302 neurons 230

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of C. elegans, 11 have a much higher connectivity than the rest and form this type of structural core [39]. The failure of these neurons compromises the functionally of the entire system, while the loss of any other neuron causes relatively small damage since the core holds the network together. This example provides just a glimpse of the potential of network theory. The true virtue of network theory, however, is that with minor modifications its theoretical framework can be applied to very different systems. In particular, it has revolutionized in just a decade our view of the healthy and diseased human brain [2,34,40]. The “small world” property, for instance, is believed to enable optimal cognitive functions at a low wiring cost, while the existence of “hubs” has been ascribed to efficient neuronal signaling, the integration of information, and communication. The alteration or loss of these topological traits in the brain has been extensively investigated, as it accounts for the damage in disorders such as Alzheimer’s disease, Parkinson’s disease, and multiple sclerosis.

Noise in neuronal systems Noise is a remarkable example of a paradigm shift in neuroscience. Fluctuations initially considered as undesired were finally appreciated as a fundamental mechanism in the dynamics of neurons and neuronal circuits [24]. Generally speaking, noise can be viewed as the random fluctuations inherent in any physical system with a large number of degrees of freedom. In neuronal systems, at molecular and cellular scales, thermal fluctuations or variations in the biochemical environment often suffice to produce spontaneous neuronal firings that can propagate and be amplified throughout the network. At a macroscopic level, these firings shape the network noise, i.e., trains of low-amplitude random spikes that bombard neurons and circuits. An illustrative case of the benefits of noise is “stochastic resonance” [46]. A neuron that receives a subthreshold input that varies in time (for instance in the form of a sine wave) cannot fire. However, the addition of noise of appropriate amplitude might induce firing just at the peak of the subthreshold signal. Since noise is ubiquitous in the network, the existence of a “resonator” greatly facilitates the amplification of small signals, and their detection. This is indeed the mechanism that the brains of predators use to detect very small perturbations in the environment and that signal the approach of a prey [29]. Stochastic resonance is just one of the several mechanisms www.cat-science.cat

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in which the participation of noise is fundamental. For this reason, there is a tendency nowadays to introduce the more general term “stochastic facilitation” [24] to account for all possible mechanisms through a unified framework. Experiments and theoretical models show that, even in the absence of a periodic subthreshold signal, stochastic facilitation not only enhances the firing of neurons but also facilitates the generation of precisely timed (clock-like) trains of spikes, the synchronization of large populations of neurons, and the generation of oscillatory activity across brain areas. Precise timing and synchronization are pivotal since they confer reliability to key brain circuits, for instance, those involved in stimuli processing or motor coordination. Reliability is also a fundamental ingredient in the coding of information, its representation, and its recall through memory [12]. The stochastic facilitation framework is grounded in important physical concepts from dynamic systems theory, which by themselves have greatly helped to understand the exquisite dynamic repertoire of neuronal networks [42]. An example is the “attractor” concept, in which the dynamics of a neuronal network evolve towards a bounded number of states without much sensitivity to the initial conditions. At the other extreme, “chaotic” neuronal circuits are those with high sensitivity to the initial conditions; their dynamics are highly variable and unpredictable. Indeed, a fascinating feature of the brain is that some neuronal circuits exhibit striking reliability while others function at the verge of chaos. Chaotic operation seems to facilitate a quick response to stimuli [42] and may even drive complex cognitive tasks such as imagination.

The importance of spontaneous acti vity and its modeling Living neuronal networks are active. Although this may seem obvious, a common and remarkable feature of all living neuronal networks is their rich dynamics, evoked by external stimuli or occurring spontaneously. Neuronal network dynamics range from the scattered firing of a few neurons to massive synchronous activations, giving rise to waves of activity or sustained oscillations with broad spatiotemporal characteristics. In the mammalian brain, correlated activity covers temporal scales of roughly 4 orders of magnitude, typically from 2 ms to 20 s, and encompasses a few tens to millions of neurons [6]. Additionally, the interplay between intrinsic neuronal dynamics and circuit connectivity is so CONTRIBUTIONS to SCIENCE 11:225-235 (2015)

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flexible and versatile that several patterns of activity can coexist within the same network. The repertoire of rhythms and their relative importance also depend on the state of the brain, i.e., at rest, sleeping, or performing specific tasks, and reflect the intimate relation of these rhythms to brain function, actually linking single-neuron activity to behavior. Within the broad dynamic repertoire of the brain, spontaneous activity is a central feature [3,6]. This activity is not a “trivial random activity,” as assumed for decades, but comprises well-structured dynamic patterns that are pivotal for the functioning of brain circuits. Spontaneous activity appears early during embryonic development and participates in the formation and interconnectivity of the developing neuronal circuits. At early postnatal stages, the combination of evoked activity (from sensory inputs) and spontaneous activity refines the young neuronal circuits to master complex tasks, such as motor coordination and the processing of visual information. In the adult brain, spontaneous activity takes part in input selection, information processing, memory consolidation and retrieval, and several other actions [6]. Despite continuous advances, the mechanisms that initiate and maintain spontaneous activity in neuronal circuits are still poorly understood, both from a physiological perspective and in modeling scenarios. However, circuits as diverse as the retina, the spinal cord, the cortex, thin slices of brain tissue, and the cultures of dissociated neurons described above exhibit some sort of sustained spontaneous activity patterns. This hints at the existence of universal mechanisms that robustly drive any neuronal network towards the generation of these structured, spatiotemporal, spontaneous discharges. To tackle this elegant and important neuroscience paradigm, different scales and systems are being investigated, from measurements and modeling at the brain level to more accessible and controllable in vitro preparations in the form of neuronal cultures. In the human brain, spontaneous activity is often referred to as the resting state [9], i.e., the basal activity in the brain in the absence of stimuli and the conductance of specific tasks. Recent studies, particularly those led by G. Deco in Barcelona, have shown that the resting state exhibits a series of properties that reflect pivotal aspects of the functioning of the brain and its complexity [9,10]. These studies have been framed in the context of a non-linear physical model to unveil its key elements and mechanisms [8]. The model considers a set of brain areas, each of them formed by an ensemble of interconnected excitatory and inhibitory neurons and whose dynamics follow realistic IF descriptions. The dynamics in www.cat-science.cat

each single brain area are completed with a background noise that stimulates activity in the neuronal population and whose structure is similar to the one observed in actual brain measurements. The non-linear nature of the IF model and the coupling between neurons and the noise ultimately settles the dynamics of each brain area in a stationary “attractor” state. The different brain areas are then interconnected following real, precisely measured structural maps and the dynamics of the entire “brain” are then investigated as a function of the coupling strength between brain areas. This model shows that when the coupling is too weak, different brain areas fire independently in a low-firing regime. When the coupling is too strong, the entire brain fires synchronously, i.e., in an “epileptiform” manner with no spatiotemporal structure. For intermediate couplings, a multi-stability scenario emerges, characterized by the coexistence of many attractors, each of them a focus of high neuronal activity. This activity propagates across different areas in the brain, ultimately shaping a resting state with a rich spatiotemporal structure and a functional connectivity that is very similar to the one measured in humans. This study by Deco’s group is not only elegant from a physical perspective, but also enlightening in its neuroscience implications. First, the resting state is tightly linked to the structural connectivity of the brain, so that strong variations in the repertoire of activity patterns indicate important circuit anomalies, for instance due to disease. Second, the spatiotemporal structure of the resting state emerges as a subtle interplay between three key elements, none of them expendable: brain circuitry, local neuronal dynamics, and noise. Hence, the resting state reflects the dynamic capabilities of the brain and its capacity to respond to stimuli.

Noise focusing: addressing sponta neous activity in cultures A complementary approach to understand the mechanisms that initiate spontaneous activity is being pursued by our group in Barcelona, in studies of neuronal cultures. One major aim is to elucidate the physical basis of the emergence of coherent spontaneous activity in relatively simple and controlled networks [26]. Specifically, we have studied the transition from the completely random firing of neurons at early stages of culture development to the first signature of coordinated collective action, in the form of synchronous and almost periodic bursts of activity by the entire network (Figs. 232

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Fig. 3. Initiation and propagation of activity in homogeneous neuronal cultures. (A) Typical average fluorescence trace of a neuronal culture, limited to the first 10 min of recording. The recording corresponded to a circular culture 13 mm in diameter containing on the order of 100,000 neurons. The trace shows the average signal of all the neurons and illustrates the existence of regimes of high coherent activity (bursts, peaks of fluorescence) combined with intervals of almost no activity. The bursts marked with arrowheads are those analyzed in (B). (B) Examples of the initiation and propagation of activity in the same culture. The encircled values indicate the burst number along the recording. The color plots show the propagation of activity, which approaches a circular wave advancing at 50â&#x20AC;&#x201C;60 mm/s. The gray circles mark the region where the burst initiated. Bursts #3, #11, and #26 initiated in completely different areas. Bursts #3 and #28 essentially started at the same location and displayed almost identical characteristics, which were also shared by ~80% of the recorded bursts. (C) Probability distribution function of initiation, highlighting the existence of a strong focus of activity at the bottom-left corner, i.e., the area where activity initiated in most of the cases. (See [26] for a detailed explanation).

2B and 3A). This phenomenon is very robust but nevertheless defies intuition, as it is a self-organized process that functions without the guidance of an internal clock or pacemaker, not even the presence of specialized leader neurons coordinating the process, as proven by computer simulations with identical neurons mimicking the cultured networks. The mechanism that enables the spontaneous generation of this coherent pulsation of large numbers of randomly connected neurons subject to random firing was recently discovered [26] through experiments using calcium imaging techniques in neuronal cultures, combined with a detailed in silico model of simulated networks. The first important experimental observation in our study of neuronal cultures was that the global bursts were mediated by the fast propagation of an excitation wave through the culture that had not been resolved before (Fig. 3B). Once the waves were resolved in space and time, we observed that they were always initiated at a few localized spots, characteristic of www.cat-science.cat

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each culture (Fig. 3C). The striking observation was that the nucleation of waves at different spots occurred in a completely random sequence, even though the bursts occurred nearly periodically in time. The phenomenon was thus noise-driven, but the period between bursts had to be fixed by an intrinsic time scale of recovery of the synaptic connections. These results were explained on the basis of a new mechanism that we called â&#x20AC;&#x153;noise focusing.â&#x20AC;? The basic idea is that in a network of IF elements, the spontaneous (random) firing of neurons propagates its influence through the network connections such that it is strongly amplified both by the nonlinear dynamics of the network nodes and by the multiplicity of paths that connect two neurons. A single spontaneous firing may thus induce a cascade of activity in a group of neurons. The result is that the network endows the background activity of spontaneous firing with a nontrivial spatiotemporal structure that is not simply related to the specific connectivity of the network but CONTRIBUTIONS to SCIENCE 11:225-235 (2015)

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involves a complex interplay between topology and dynamics. The background activity is composed of the superposition of avalanches of activity of all sizes following a power law distribution. A careful statistical analysis of this structured background activity reveals an effective functional network with a scale-free degree distribution, even though the structural network has a Gaussian degree distribution [26]. Within this framework, the selection of nucleation spots is the result of highly inhomogeneous and anisotropic mechanisms of noise amplification introduced by the IF dynamics in a highly clustered network. This local amplification is very sensitive to the detailed wiring of the network. The nucleation sites can then be seen as the sinks of the averaged noise flow, that is, those points at the confluence of paths of high noise amplification. The a priori homogenous primary source of noise, that is, the spontaneous random firing of the neurons, is propagated and amplified, resulting in a strongly localized concentration of noise-generated activity at some specific spots. This spatiotemporal concentration of the background activity is what we call “noise focusing.” As a basal physical phenomenon this should be generically present in neuronal networks unless specific regulation or other, stronger effects are taming, shaping, or preventing this activity. Moreover, this model nicely illustrates how physical phenomena can shape a situation in which biological blueprints must adapt and specific biochemical and genetic regulation must operate to build up the complex architecture of real neuronal tissues.

Conclusions The comprehension of the human brain has long fascinated humanity as much as the structure of Nature and its governing laws. However, what initially began as an exclusive task of neurobiologists has evolved in just a century towards a highly interdisciplinary field of research in which Nonlinear Physics and Network Science are major contributors. These branches of knowledge have provided a wealth of resources and modeling tools that, together with relatively simple experimental systems, have revolutionized our vision of the main agents that shape the dynamics of the brain and its exquisite complexity. The concepts presented in this brief review are just a fraction of all the potential that Physics can offer to studies of brain function. Indeed, if the 21st century is to be the “century of the brain,” we firmly believe that Physics will play a central role in it. Competing interests. None declared.

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36. Teller S, Granell C, De Domenico M, Soriano J, Gómez S, Arenas A (2014) Emergence of assortative mixing between clusters of cultured neurons. PLoS Comput Biol 10:e1003796 doi:10.1371/journal.pcbi.1003796 37. Tibau E, Valencia M, Soriano J (2013) Identification of neuronal network properties from the spectral analysis of calcium imaging signals in neuronal cultures. Front Neural Circuits 7:199 doi:10.3389/ fncir.2013.00199 38. Toga AW, Thompson PM, Mori S, Amunts K, Zilles K (2006) Towards multimodal atlases of the human brain. Nat Rev Neurosci 7:952-966 doi:10.1038/nrn2012 39. Towlson EK, Vértes PE, Ahnert SE, Schafer WR, Bullmore ET (2013) The rich club of the C. elegans neuronal connectome. J Neurosci 33:63806387 doi:10.1523/JNEUROSCI.3784-12.2013 40. Van den Heuvel MP, Sporns O (2013) Network hubs in the human brain. Trends Cogn Sci 17:683-696 doi:10.1016/j.tics.2013.09.012 41. Varshney LR, Chen BL, Paniagua E, Hall DH, Chklovskii DB (2011) Structural properties of the Caenorhabditis elegans neuronal network. PLoS Comput Biol 7:e1001066 doi:10.1371/journal.pcbi.1001066 42. Vogels TP, Rajan K, Abbott LF (2005) Neural network dynamics. Annu Rev Neurosci 28:357-376 doi:10.1146/annurev.neuro.28.061604.135637 43. Watts DJ, Strogatz SH (1998) Collective dynamics of “small-world” networks. Nature 393:440-442 doi:10.1038/30918 44. Wedeen VJ, Rosene DL, Wang R, Dai G, Mortazavi F, Hagmann P, Kaas JH, Tseng W-YI (2012). The geometric structure of the brain fiber pathways. Science 335:1628-1634 doi:10.1126/science.1215280 45. Wheeler BC, Brewer GJ (2010) Designing neural networks in culture: Experiments are described for controlled growth, of nerve cells taken from rats, in predesigned geometrical patterns on laboratory culture dishes. Proc IEEE Inst Electr Electron Eng 98:398-406 doi:10.1109/ JPROC.2009.2039029 46. Wiesenfeld K, Moss F (1995) Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs. Nature 373:33-36 doi:10.1038/373033a0 47. Yuste R, Church GM (2014) The new century of the brain. Sci Am 310: 38-45 doi:10.1038/scientificamerican0314-38

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HISTORICAL CORNER Institut d’Estudis Catalans, Barcelona, Catalonia

OPENAACCESS

CONTRIB SCI 11:237-247 (2015) doi:10.2436/20.7010.01.235

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Pere Vieta (1779–1856), promoter of free public teaching of physics in Catalonia Carles Puig-Pla Research Center of History of Technology, School of Engineering (ETSEIB-UPC), Technical University of Catalonia, Barcelona, Catalonia

Fig. 1. Portrait of Pere Vieta i Gibert, 1779– 1856. (From Palomeque, 1979). Correspondence: Carles Puig-Pla Research Center of History of Technology School of Engineering (ETSEIB- UPC) Technical University of Catalonia Av. Diagonal, 647 08028 Barcelona, Catalonia Tel: +34-934017782

Summary. Free public teaching of physics in Catalonia started in the early 19th century, even if the interest in experimental physics goes back to the 18th century, where this discipline was discussed at various learned societies. The first chair of Physics in Barcelona was not a university chair but that of the Junta de Comerç de Barcelona (Trade Board of Barcelona), which had several scientific-technical Schools. In fact, at that time, Barcelona had no university, because it had been supressed by King Felipe V after the War of the Spanish Succession (ended in 1714). The promoter of free public teaching of experimental physics was Pere (Pedro) Vieta i Gibert (1779–1856), who was the first professor of that subject both at the School of the Trade Board and at the University of Barcelona, once it was restored in 1842. Vieta, who was a surgeon in the Army, combined his two professions and his interest in meteorology, he having recorded meteorological observations in Barcelona for many years. Many of his students were influential people in the scientific, intellectual, political and economic history of the 19th century in Catalonia and Spain. [Contrib Sci 11:237-247 (2015)]

E-mail: carles.puig@upc.edu

The interest in experimental physics in Barcelona goes back to the 18th century and, in particular, to the creation of the “Conferencia Físico-Matemática Experimental” (Expe rimental Physico-Mathematical Conference) in 1764, which became the “Real Academia de Ciencias Naturales y Artes” (Royal Academy of Natural Sciences and Arts) in 1770 [39]. At those times, there was no university in Barcelona, because it had been supressed by a Royal Decree of King

Felipe V of May 11, 1717, and first public, free School dedicated explicitly to experimental physics was that of the Junta de Comerç de Barcelona (Trade Board of Barcelona), which was founded in 1814. Its promoter and first professor was Pere (Pedro) Vieta i Gibert (1779–1856) who, some years later, when the University of Barcelona was restored in 1842, became the first professor of physics at the Faculty of Philosophy (Fig. 1).

Keywords: Pere Vieta (1779–1856) · experimental physics’ teaching · Trade Board of Barcelona (mid 19th c.) · University of Barcelona (mid 19th c.) · Royal Academy of Natural Sciences and Arts (mid 19th c.) ISSN (print): 1575-6343 e-ISSN: 2013-410X

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Pere Vieta (1779–1856)

Fig. 2. The Elementos de Física Experimental (Elements of Experimental Physics) (Vol. 1, 1804, and Vol.2, 1815) by Antoni Cibat.

Origins and early education of Pere Vieta Pere Vieta i Gibert (Pedro Vieta) (1779–1856) was born in 1779 in Sant Andreu de Llavaneres, in the Catalan region of Maresme, near the city of Barcelona. He was a son of surgeon Antoni Vieta, and was a nephew of Didac Vieta (Diego Vieta), benefited presbyter of Santa María del Mar—the second most important church in Barcelona—, and professor of theology and rector of the University of Barcelona [41]. We have little information about his childhood. In the funeral praise, made by his friend the lawyer Pau Valls i Bonet (1814–1888) in 1857, Valls simply asserted that his friend “had received a very careful early education”. And he assumed that, like others at that time, Vieta was introduced to knowledge with “Nebrija, Torrella and Goudin, with the advancements of Rosselli and Palmira archbishop” authors, all of them, with whom he undertook studies in medical sciences [42]. In fact, we know that Vieta was trained at the Real Colegio de Cirugía de Barcelona (Royal College of Surgery of Barcelona). This was the most important educational institution in the health sector in Catalonia at the time as well as a paradigm of the modernization of the Spanish surgery during the 18th century. There Vieta was influenced by professor Antoni San German i Tort (1755–1833), and Antoni Cibat i Arnautó (1771– 1811), among other mentors. www.cat-science.cat

Of the above mentioned Vieta mentors, Antoni Cibat i Arnautó, deserves special attention. Born in Cistella, in the Catalan region of Alt Empordà, Cibat, after having studied at the Royal College of Surgery of Barcelona from 1788 to 1792, went to Scotland to study physics, chemistry, medicine and surgery, earning a doctorate at the Marischal College, Aberdeen. His latinized name (Antonius Cibatet Auranto) appears in the list of diplomas awarded by that university from January 1792 to January 1793 [28]. In 1795, Cibat, being an honorary member of the Guy’s Hospital Medical Physical Academy in London, he became a member of the Royal Academy of Natural Sciences and Arts of Barcelona (Fig. 2). In 1795, some decrees restructured surgery Colleges. New chairs were created, including that of experimental physics, which was included in the curriculum of the Royal College of Surgery of Barcelona. Vieta became acquainted with experimental physics by Cibat, who was appointed professor of experimental physics and began teaching this discipline. Cibat taught there from 1796 to 1806, when the chair was suppressed. Vieta must have excelled in experimental physics classes because in the absence of Cibat, he replaced him (Fig. 3).

The years of the Peninsular War At the beginning of the 19th century, the army needed “practitioners” (i.e., surgeon assistants), especially during 238

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Fig. 3. Apparatus for the decomposition of water. Illustration from the book Elementos de Física Experimental (Elements of Experimental Physics) by Antoni Cibat (vol. 2, 1815).

wartime. The Royal College of Surgery usually sent student volunteers that returned to the College when the military campaign was over [22]. Vieta served as a practitioner in military hospitals of the army under the command of surgeon Antoni San German (1755–1833). On 4 February 1803, Vieta was granted with the military status and was allowed to use the uniform of second assistant of the Army Medical Corps. On the 24th of October, 1804 he graduated in Medical Surgery and on November 11 that same year he was appointed Surgeon of the First Infantry Battalion of the Bourbon line. At the end of 1806, Vieta sat for an official examination to win the chair of experimental physics at the Real Seminario de Nobles (Royal Seminary for Noblemen) in Madrid, which had been vacant since the death of professor Juan Manuel Pérez that same year. Pérez had been the holder of that chair since José Moñino y Redondo (1728–1808), count of Floridablanca and Chief Minister of King Charles III, offered him the position in 1783. Vieta passed the examination and was the second in the shortlist of three candidates that was submitted to the King. However, on the14th of February, 1807 another candidated, Liborio Pelleport, won the chair [23]. In 1808 the Peninsular War (1808–1814, “Guerra de la Independencia” in Spanish, “Guerra del Francès”, in Catalan) began, and on July 27 that year, Vieta was appointed first assistant of the army top brass. During the war he was sent to Mequinenza, in Aragon, by the General Captain of the Principality of Catalonia, Theodor von Reding (1755–1809). There he served as Surgery Consultant. He was present at the siege of Tarragona (1811) [4] in which, apparently, he was taken prisoner. Later, in 1813, he was assigned to the Hospital of Vic [42].

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After the war, the Trade Board of Barcelona (Junta de Comerç de Barcelona) reopened its scientific-technical Schools (Navigation, Fine Arts, Business, Shorthand, Chemistry applied to Arts, and Mechanics), which had been closed during the war [1]. In particular, in 1814 the School of Mechanics of Francesc Santponç i Roca (1756–1821), introducer of the double-effect steam engine in Barcelona, reopened and Vieta enrolled. That same year, Vieta, who was first assistant surgeon at the Royal Army, requested the Trade Board to establish new experimental physics studies and suggested that they should appoint himself professor. He defended the necessity and importance of experimental physics and assured that, in his classes, experiments would be abundant because he was convinced that only experience could prove the truth or the falsity of facts stated [43]. The Trade Board agreed to establish the new free School of Experimental Physics and appointed Vieta professor “for during the will of the Trade Board”. The opening of the College had been announced for the 29th of October, 1814, but it opened a day before on October 28 in the building of the Llotja (building where commercial transactions were made) of Barcelona [12], where Vieta read the inaugural speech [43]. Classes were scheduled from 11 am to 12 am, to assure there would be light enough for optical experiments (Fig. 4). In May 1815, Vieta applied to join the Royal Academy of Natural Sciences and Arts of Barcelona, where he presented an entry report on aurora borealis (northern lights) [44]. That report was related to questions that someone who signed DFS (DFS might have stood for “Don Francisco Sanpons”) had addressed him through the Diario de Barcelona newspaper. There were questions about luminous meteors observations

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Pere Vieta (1779–1856)

Fig. 4. The Llotja building, the Board’s headquarters. (Antoni Roca Sallent engraving, 1842).

made in Barcelona over several days in November 1814 [2]. Vieta was admitted in the Optics Section of the Academy on June 14, after a favorable report from Joan Francesc Bahí (1775–1841), who acted as censor. A month later, on July 15, when he took office as a member of the Academy, he was elected director of the Optics Section.

Army and professorship, a difficult marriage The obligations of the military status of Vieta repeatedly interfered with his teaching activities. This already happened during the first academic year; as a surgeon of the first battalion of the Royal Regiment of Sapper-Miners, allocated in Alcalá de Henares, Madrid, he applied for a leave of abscence to remain in Barcelona. After other similar situations, and before the alternative of giving up his position in the army or leaving the chair of physics, the Trade Board finally granted him the chair “in perpetuity” on 17 June 1816, provided he asked the King (Ferdinand the Seventh), through the Minister of War, a permit to leave active duty [30]. On August 9 that same year, he became a member of the “Real Academia de Buenas Letras de Barcelona” (Royal Academy of Letters), where he had read some Consideraciones históricas físicas (Physical Historical Considerations) (April 20, 1816) and where he also read a Memoria sobre la sequía de los veinte y cuatro años (Report on the twenty-and-four-year draught) [14]. In www.cat-science.cat

that report he proved, by means of physical and geological arguments, the impossibility of a supposed twenty-four-year drought in Spain, which many historians defended [13]. For his lectures, in addition to using the Elementos de geometría (Elements of Geometry) by Father Roger Martin (translated from the Institutions mathématiques by Francesc Santponç i Roca, 1756–1821), Vieta used the Elementos de Física Experimental (Elements of Experimental Physics) by Antoni Cibat as a textbook [8,9]. The use of Cibat’s book is corroborated by some handwritten notes, dated from 1814, found by the author of this biography [32]. Cibat’s work was aimed to surgeons, but rather, Vieta translated into Spanish the Traité complet et élémentaire de physique (Complete and Elementary Treatise of Physics) in three volumes, of the French physicist Antoine Libes (1752–1832), teacher at lycées of Paris and, later, lecturer at the École Centrale. This work was formally structured in topics close to physics as we understand it today, a discipline that was consolidated in the first decades of the 19th century (Fig. 5). Vieta paid out of his pocket 18,000 reales for the printing of the translated book. In 1818 the first edition of this translation appeared under the title: Tratado de física completo y elemental [21]. In addition to translating it, Vieta expanded it. In the years 1821, 1827–28 and 1838 other editions of the book were released [40]; in the study plan of 1824 this book was the selected textbook in Spain by royal provision. 240

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At the Royal Academy of Natural Sciences and Arts, Vieta presented a Memoria sobre la doble refracción (Report on double refraction) on 20 May 1818, and on July he was elected director of the section of Optics and Cosmography. Since then, he was reelected in this position every year until the forced closure of the Academy in 1824 during the Década Ominosa (Ominous Decade, 1823–1833, the last ten years of the reign of King Ferdinand VII, a period of absolutism in the government of Spain) [36]. In 1819, Vieta earned his degree in Medicine [15]. At that time, the Trade Board of Barcelona published the first technical journal in Catalonia, Memorias de Agricultura y Artes (Reports on Agriculture and Arts), published between 1815 and 1821 [33]. The authors were lecturers from the schools of Botany, Chemistry and Mechanics of the Trade Board of Barcelona. It was in that journal that Vieta refuted the objections about the weight of the air that the French surgeon Hyacinthe Bodélio had published five months earlier [45]. During the Liberal Triennium (1820–1823, a three-year period of liberal government in Spain), Vieta was appointed corresponding member by the Academia General de Ciencias, Bellas Letras y Nobles Artes (General Academy of Sciences, Beautiful Letters and Noble Arts) of Cordoba on March 13,1821 [25,42]. The summer that year, when there was an epidemic outbreak of yellow fever in Barcelona, Vieta remained in the city and assisted the Trade Board’s employees and their families [6]. In 1822 he participated in the first and ephemeral attempt to restore the University of Barcelona—closed by the Bourbon King Felipe V after the War of the Spanish Succession—and the Government appointed him professor of Physics on 22 November [34,16]. At that time, on June 9, 1922, he was also appointed member of the Imperiale e Reale Accademia Economico-Agraria dei Georgofili di Firenze (Imperial and Royal Academy of Agricultural Economics of the Agriculturalists of Florence). After Francesc Santponç death, in 1821, the School of Mechanics of the Trade Board of Barcelona was closed. In 1824, Vieta suggested to the Trade Board to compensate for the absence of the chair of Mechanics by broadening the lessons of the rudiments of statics and hydrostatics and increasing the number of experiments related to these disciplines. Moreover, from 1827 to 1843 Vieta continued his meteorological observations in Barcelona. Francesc Salvà i Campillo (1751–1828), a physician from Barcelona also known for his contributions to meteorology and telegraphy, had started collecting meteorological data from his home and would published them in the Diario de Barcelona newspaper www.cat-science.cat

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Fig. 5. The Spanish Traité complet et élémentaire de physique (Complete and Elementary Treatise of Physics) of Antoine Libes, translated by Pere Vieta.

from 1780 to 1824 [18]. Vieta moved the observation point to the headquarters of that newspaper [3]. At the School of Experimental Physics, Vieta could have a cabinet with a considerable amount of instruments thanks to the monthly financial contributions of the Trade Board of Barcelona. In 1830, Vieta himself requested to the Trade Board various tools that were located at the Trade Board’s headquarters, the Llotja building, to promote a factory of scientific equipment: “a factory to produce physics, mechanics, mathematics equipment” because this was, in his opinion, “the kind of manufacture not found in Spain” [32]. On April 6, 1831 he was promoted to Deputy Director of the Army Surgeons Corps of the Catalonia District. A few months later, in January 1832, he had to travel to the castle of San Fernando, in Figueres (Girona province), to try to alleviate the intermittent fevers affecting the garrison of that fortresse. He wrote a report Sobre las fiebres intermitentes (On intermittent fevers), preserved in the Academy of Medicine of Barcelona [14]. According to Joan Corminas (1800–1854), contemporary to Vieta, Vieta wrote “in a medical journal of Barcelona” a report on the tercianes (intermittent fevers) prevalent in the castles of Figueres and Hostalric” following the instructions he received from the Government [11]. CONTRIBUTIONS to SCIENCE 11:237-247 (2015)

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Pere Vieta (1779–1856)

Fig. 6. The pamphlet Reflexiones físico-geológicas sobre fuentes ascendentes ó artificiales (Physical-geological reflections on ascending or artificial water sources) published by Pere Vieta and Josep Roura in 1835

The Trade Board of Barcelona found it contradictory to the agreement they had reached with Vieta in 1816 that his new position forced him to be absent from the city. The Trade Board, however, given the merits Vieta had acquired after eighteen years as professor of Physics, accepted that during Vieta’s absences there was a substitute lecturer in class, provided he notified his absences beforehand. The substitute lecturer, proposed by Vieta himself, was his disciple Joaquim Balcells i Pasqual (1807–1879) [6]. Vieta was a member of the Royal Academy of Medicine of Barcelona, where he pronounced his induction speech on June 17, 1833 [7]. When, in 1834, there was an epidemic of cholera in Barcelona, Vieta took the lead of the Cholera’s Hospital in the city. At the Royal Academy of Medicine and Surgery, he was a member of several committees. In 1835, he participated in the commission of Topografía (Topography) along with Joan Lopez, Fèlix Janer and Joan Isidor de Bahí, and also as a member of the commission of Aguas y Baños minerales (Mineral baths and waters) with the two latter and with Joan Ribot and Francesc Carbonell i Bravo [19]. He made outstanding contributions to produce a medical topography of Catalonia [7]. www.cat-science.cat

At those times Vieta collaborated with Josep Roura i Estrada (1797–1860), who, like himself, was a member of the Royal Academy of Natural Sciences and Arts of Barcelona and was a lecturer at the School of Chemistry applied to the Arts of the Trade Board of Barcelona. Both published a pamphlet in 1835: Reflexiones físico-geológicas sobre fuentes ascendentes ó artificiales, con motivo del pozo taladrado que mandó abrir la Real Junta de Comercio de este principado (Physical-geological reflections on ascending or artificial water sources, on the occasion of the drilled well bored following the commission of the Royal Trade Board of Barcelona of this Principality) [46]. In this work, they analyzed the interest of finding and using groundwater. That work was based on a detailed analysis of the drilling a borehole in Les Corts de Sarrià—by then a village adjacent to Barcelona, nowadays one of its neighborhoods—in 1834. The drilling works had been conducted by Hilarión Bordeje i Piña (1792–1869), professor of the School of Machinery of the Trade Board of Barcelona. Vieta and Roura proposed a practical method to make the artesian wells profitable and advantageous (Fig. 6). Vieta’s interest in issues related to geology and upwelling groundwater was also reflected in the offer he made to the Royal Academy of Natural Sciences and Arts (October 21, 1835) of several samples of sand from the mouths opened following the March 21 earthquake in Torrevieja, near the city of Alicante, and a bottle of water thrown by one of those mouth, which he accompanied with the analysis made by the academician Agustí Yáñez i Girona (1789–1857) [36]. During the First Carlist War (1833–1840) Vieta was mobilized and took a leading role as a surgeon. Towards the end of the war, in 1839, that the experience he had gained allowed him to write drug formulary for military hospitals [15]. Since 1835, his relationship with the Trade Board deteriorated due to his absences from the classes. He often was late for the classes, and in October 1835 was absent without no leave. Vieta and the Trade Board recriminated to each other. Given its experience, on 21 November 1835 he was appointed Major-Doctor Surgeon of the Army of the North [36], and in early 1836 he joined that Army. Given the situation of Civil War, the Trade Board agreed about Vieta’s leave, and he was temporarily replaced by Joan Agell i Torrents (1809–1868). However, Vieta was warned that he should return to work after the end of the academic year, otherwise, his chair would be declared vacant. Vieta appealed to the Ministry against the decision of the Trade Board. Through a Royal Order of 20 July 1837, Queen Isabel II (1830–1904) agreed to Vieta’s request of retaining the ownership of the chair of Physics of the Trade Board of Barcelona throughout 242

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Fig. 7. The ancient Convent of Carmen, where the University of Barcelona was provisionally set up in 1838.

the time he would be in the Army. At that time Vieta was Deputy Inspector of the Army Surgeon Corps at the Army operating in the North [26].

The University of Barcelona and the Trade Board of Barcelona Schools In the early 19th century there was no University in Barcelona since it had been suppressed after the War of the Spanish Succession by Royal Decree signed by Felipe V on May 11, 1717 [38]. The city was living decisive moments in the attempt to recover its University. The academic year 1836–1837 the Estudis Generals (Studia Generalia) were restored. In 1837 the process of provisionally restoring the University of Barcelona started; a process that would culminate in 1842. The Trade Board suppressed the chair of Experimental Physics, which was included in the studies of the University, and decided to create another chair of Física Experimental aplicada a la indústria (Experimental Physics applied to industry) that finally would be of Física aplicada a les Arts (Physics applied to Arts). After a series of appeals, proposals and counterproposals, Vieta finally had to choose one of both chairs because he could not receive two salaries. In June 1838 opted for the University chair of Physics (who ran temporarily) and left the Trade Board (Fig. 7). Educational changes and the confrontations with Vieta paralyzed the teaching of Physics of the Trade Board from 1837–1838 to 1839–1840 academic years. The School of Experimental Physics ended so a period (1814–1837) during www.cat-science.cat

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which the School had been the busiest of all scientific, tech nical educational centers of the Trade Board of Barcelona [31]. In October 1840, the Governmental Trade Board of the province separated from their chairs the rector and several professors of the University of Barcelona, including Vieta, presumably due to their hostility to the progressist ideology [26]. Finally, when the First Carlist War was over, in the academic year 1840–1841, the School of Physics applied to Arts of the Trade Board of Barcelona opened. Vieta presented a program to the Trade Board and the Ministry resolved that he would be the lecturer until he obtained the chair at the University. At the beginning of November 1840, the first volume of the third edition of Élements de physique expérimentale et météorologie (Elements of Experimental Physics and Meteorology), which Vieta had translated into Spanish was already in press. The author, Claude Pouillet (1790–1868), was professor of Physics applied to Arts at the Conservatoire des Arts et Métiers of Paris and professor of Physics, after Louis Joseph Gay-Lussac (1778–1850) and Pierre Louis Dulong (1785–1838), at the Faculty of Sciences of that city. Vieta dedicated this translation to the Trade Board. In August 1841, he had already translated the second volume, and published Pouillet’s work with the title of Elementos de Física Esperimental [sic] y de Meteorología [29]. He added a report on the daguerrotype [42]. This textbook was his reference text for classes from the academic year 1840–1841 until his resignation in 1844 (Fig. 8). Vieta grew an interest in fires and how to estinguish them. In the public examination of Experimental Physics held on July 3 and 4, 1835, his inaugural lecture dealt with CONTRIBUTIONS to SCIENCE 11:237-247 (2015)

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Pere Vieta (1779–1856)

Fig. 8. The book Elementos de Física Esperimental y de Meteorología by Claude Pouillet, which Vieta translated into Spanish (Vol. 1 and Vol. 2, 1841).

this issue. On October 25, 1840, he discussed it again when he read a report at the Royal Academy of Natural Sciences and Arts entitled: Reflecsiones [sic] físico-químicas sobre incendios (Physicochemical reflexions on fires). He showed, with scientific arguments, that the hypothesis defended by various authors that stated that the water in steam state intensified ignition was false. He also warned about the danger of making openings in the rooms to leave the smoke go out, because this would favor combustion [47]. He also dealt with the solar eclipse in 1842. Thus, on March 16, 1843, as a director of the Physical-Chemical Section of the Royal Academy of Natural Sciences and Arts, he presented a Memoria sobre el eclipse de Sol acaecido en 8 de julio del año próximo pasado (Report on the solar eclipse having taken place on July 8 last year). He combined his dedication to the Royal Academy of Natural Sciences and Arts and his lectures at the School of Physics of the Trade Board with activities related to medicine. On May 9, 1841, at the initiative of the Saint-Simonian physician Pere Mata i Fontanet (1811–1877) the Sociedad Médica de Emulación (Medical Society of Emulation) of Barcelona was founded, with the aim of dealing with all branches of knowledge about the art of healing, under the motto “mutual instruction, fraternity, scientific progress” [37], and Vieta was one of its members [20]. On January 3, 1842, www.cat-science.cat

he imparted the inaugural lecture of that society at the Royal Academy of Medicine Apuntes acerca de la catarata (Notes on the cataract). He described the disease, the symptoms and the methods to operate on a cataract, indicating the changes that he had introduced [48]. Between 1842 and 1843 he became interested in phrenology, that Marià Cubí i Soler (1801–1875) had disseminated, first with some prevention and later with a better disposition, clearly describing the objectives of that new discipline—now regarded as obsolete—through contributions to the Diario de Barcelona newspaper, where he signed as M.M. [24]. Since 1843 and for the three consecutive biennia (1843–1848) Vieta was presided of the Royal Academy of Medicine, though officially he was Vice-President because the presidency de iure was in Madrid [15]. In 1844 he read a Memoria médico-manicómica ó sean observaciones médicas sobre los dementes (Report on medical-insane asylum issues i.e., medical observations on the insane) [14]. In early 1844, Vieta was a member of the Barcelona City Council, provisionally chaired by the Major Josep Bertran i Ros (1795–1855). In March that same year, a Royal Order returned to Vieta the chair of Physics of the University [26]. However, until August 31 he only held the chair of the Trade Board of Barcelona because he could not hold two chairs at the same time. In September 1844, Vieta notified the 244

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Trade Board that he was forced to resign his professorship in Physics applied to the Arts because it had recovered his chair at the University. During the Moderate Decade (May 1844–July 1854), with the power in hands of the Moderate Party, education was restructured in Spain. Pedro José Pidal (1799–1865), Governance Minister, enacted the so-called Plan Pidal (Pidal’s Plan) (September 17, 1845), which centralized and uniformized education. On October 7, 1845, Vieta was appointed Dean of the School of Philosophy at the University of Barcelona, and on June 2, 1846, he obtained a doctoral degree in Sciences. In the mid 1840’s Barcelona was experiencing a remark able industrial development, which generated a great con cern about the influence of steam engines and chemical industries on public health. In early 1845, Vieta was a member of a commission of the Royal Academy of Natural Sciences and Arts that had to assess and report about this issue to the City Council [5]. His reputation as both physician and physicist was widely recognized. He was the President of the Royal Academy of Natural Sciences and Arts (1846– 1847) and on August 31, 1847 he was appointed director of the Physicochemical Sciences Section for the academic year 1848. By then he was a member of the Sociedad Económica Barcelonesa de Amigos del País (Barcelonian Economic Society of Friends of the Country), of which he became the director on 1849. The following year, on April 11, he was elected member of the Provincial Academy of Sciences and Letters of the Balearic Islands [42]. At the University of Barcelona, Vieta taught Física y nociones de química (Physics and Chemistry concepts) and Ampliación de Física (Advanced Physics). The programs of both subjects, which he prepared and published for the 1847–1848 academic year have been preserved [49,50]. Due to the death of the rector of the University of Barcelona, Joaquim Rey i Esteve (1775–1850), in January 1850, Vieta, who was Dean of the School of Philosophy, was appointed Rector-in-Office until March 7, when a new rector, Mariano Antonio Collado (1796–1853), was appointed. In late August that year, 1850, a new university curriculum was approved. A Royal Order (September 10, 1850) regulated that no student would be accepted that had not studied advanced algebra and analytic geometry. A few weeks later, another Royal Order (September 26, 1850) established that the textbooks for Advanced Physics would be: Tratado de Física Experimental y meteorología (Treaty of Experimental Physics and Meteorology) by Claude Pouillet, translated by Vieta, Curso completo de Física Experimental (Complete Course www.cat-science.cat

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Puig-Pla

Fig. 9. The painting with the idealized portrait of the lithotomist Frère Côme (1703–1781) that Vieta gave to the Royal Academy of Medicine (from Corbella, 2010).

of Experimental Physics) by Fernando Santos Castro (1809– 1890) and the Tratado elemental de Física (Elementary Treatise of Physics) by Cesar Despretz (1791–1863), translated into Spanish [27].

Facing the end About the last years of Vieta’s life, we know that in 1851 he gave to the Royal Academy of Medicine a painting with the portrait of the lithotomist Jean Baseilhac (1703–1781), known as Frère Côme, which is kept in the Turró’s room of that institution [10]. The following years, Vieta received the recognition of institutions that had accepted him as a member, such as the Instituto Agrícola Catalán de San Isidro (Catalan Agricultural Institute of San Isidro) (December 10, 1852), the Asociación de Socorro y Protección de la clase obrera (Association for the Relief and Protection of the Working Class) (December 14, 1853) or the philanthropic Instituto de África (Institute of Africa) to abolish slavery (March 20, 1854) [42]. On April 27, 1854, Vieta requested retirement from work and he was granted it. Two and half years later, he died (Fig. 9). According to the announcement that his son-in-law, CONTRIBUTIONS to SCIENCE 11:237-247 (2015)

Pere Vieta (1779–1856)

Narcís Gay (1819–1872), sent to the Royal Academy of Natural Sciences and Arts, Vieta died on October 7, 1856 at 3:30 am. The following day (October 8, 1856) at 11 am the funeral was held in the Church of Santa Maria del Pi and his remains were buried in the cemetery of the Parish Church of Sants [17]. Through Experimental Physics classes, Vieta influenced many students. Some youngsters of the first years including Joaquim Llaró i Vidal (1796–1824), Bonaventura Carles Aribau i Farriols (1798–1862) and Ignasi Santponç i Barba (1795–1846), who adored his mentor, were among the founders of the Philosophical Society (1815–1821). The members of that society were later involved in the birth of Catalan romanticism [35]. Moreover, a considerable number of Vieta students played major roles in various fields of the scientific, cultural life of Catalonia and Spain in the mid 19th century. Several of them were devoted to physics or chemistry, while others were lawyers, economists, botanists, physicians, businessmen, philosophers, mathematicians or entomologists. Many were influential members, and with responsibilities, in scientific or literary academies, or professors at different universities around Spain. For example, among Vieta’s disciples we can find Laureà Figuerola (1816–1903), who became Minister for Finance and promoter of the creation of the “peseta” (the currency used in Spain before the euro was adopted); Marià de la Pau Graells (1809–1898), who became senator of the kingdom, and Joan Agell i Torrents (1809–1868), who was a rector of the University of Barcelona. The reason for all this was that, at the time of Vieta, experimental physics became a symbol of modernity and was considered a basic ingredient for education [32]. Competing interests. None declared.

References 1. Barca FX, Bernat P, Pont M, Puig-Pla C (coords.) (2009) Fàbrica, taller, laboratori. La Junta de Comerç de Barcelona: ciència i tècnica per a la indústria i el comerç (1769–1851). Cambra Oficial de Comerç, Indústria i Navegació de Barcelona, Barcelona 2. Barca FX (2014) Una llum vermella apareguda a la posta de Sol. In: Bernat P (ed.) (2014) Astres i Meteors. Edicions Talaiots. Palmanova (Calvià, Mallorca), pp. 49-71 3. Barriendos M, Peña JC, Martín Vide J (1998) La calibración instrumental de registros climáticos documentales. Aproximación metodológica a resolución anual para el caso de la precipitación en Barcelona (1521–1989). Investigaciones Geográficas, 20: 99-117 doi:10.14198/ INGEO1998.20.01

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4. Blanch A (1861) Cataluña. Historia de la Guerra de la Independencia en el antiguo Principado. Imprenta y librería politécnica de Tomás Gorchs, Barcelona, vol. 2 5. Bouza, J (1992) Una visión progressista del desarrollo urbano: el informe sobre vapores de la Academia de Ciencias de Barcelona. In: Capel, H, López Piñero JM, Pardo, J (eds.) (1992) Ciencia e ideología en la ciudad. Generalitat Valenciana, Conselleria d’Obres Públiques, Urbanisme i Transports, Valencia, vol II, pp 139-150 6. Carrera Pujal J (1957) La enseñanza profesional en Barcelona en los siglos xviii y xix. Bosch, Barcelona 7. Calbet JM, Corbella J (1983) Diccionari biogràfic de metges catalans, Fundació Salvador Vives Casajuana / Seminari Pere Mata. Universitat de Barcelona, Barcelona, vol 3 8. Cibat A (1804) Elementos de física experimental. Imprenta de Brusi y Ferrer, Barcelona, vol. 1 9. Cibat A (1815) Elementos de física experimental. Antonio Brusi, Barcelona, vol. 2 10. Corbella J (2010) L’obra de Jean Baseilhac, més conegut com Fra Cosme (1703–1781). Rev R Acad Med Catalunya 25:73-76 11. Corminas J (1849) Suplemento à las Memorias para ayudar a formar un diccionario crítico de los escritores catalanes...que en 1836...publicó Félix Torres Amat. Imprenta de Arnaiz, Burgos 12. Diario de Barcelona (1814, 27th September) 13. Diccionario Geográfico Universal dedicado a la Reina Nuestra Señora (1831). Imprenta de José Torner, Barcelona, p. 710 14. Elías de Molins A (1889) Diccionario biográfico y bibliográfico de escritores y artistas catalanes del siglo XIX. Imprenta de Fidel Giró, Barcelona, vol.II, 753-754 15. Escudé M (2008) Pere Vieta i Gibert, president de la Reial Acadèmia de Medicina de Barcelona (1843–1848). Rev R Acad Med Catalunya 23:28 16. El Pensamiento de la Nación (1846, 3rd June) 17. Expedient de Pere Vieta (Pere Vieta dossier). RACAB Archive. 18. Fontseré E (1930) Pròleg. In: Febrer, J (1930) Atlas pluviomètric de Catalunya: 9-16. Institució Patxot, Barcelona 19. Gaceta de Madrid (1835, 25th April & 22nd June) 20. Gil Pérez JI (2001) La obra de Cayetano Garviso (1807–post.1871). Cirujano vasco-navarro liberal de América. Publicacions del Seminari Pere Mata de la Universitat de Barcelona, Barcelona 21. Libes A (1818) Tratado de física completo y elemental presentado bajo un nuevo orden con los descubrimientos modernos. Impr. de Antonio Brusi, Barcelona [Translator: Pedro Vieta] 22. Massons JM (1993) Francesc Puig (1720–1797) i els cirurgians del seu temps. PPU, Barcelona 23. Moreno A (1988) Una ciencia en cuarentena. La física académica en España. CSIC, Madrid 24. Nofre D (2004) En els marges de la ciència? Frenologia i mesmerisme en una cultura industrial, Barcelona 1842–1845. Universitat Autònoma de Barcelona, Barcelona (Ph.D. thesis). Available at: http://ddd.uab.cat/ pub/trerecpro/2004/hdl_2072_5116/TR_David_Nofre_Mateo.pdf 25. Noticia de la Academia de Ciencias, Bellas Letras y Nobles Artes, de esta Ciudad de Córdoba (1847). Imprenta de D. Juan Manté, Córdoba 26. Palomeque A (1974) Los estudios universitarios en Cataluña bajo la reacción absolutista y el triunfo liberal hasta la reforma de Pidal (1824– 1845). Publicaciones de la cátedra de Historia universal. Departamento de Historia contemporánea. Universidad de Barcelona, Barcelona 27. Palomeque A (1979) La Universidad de Barcelona desde el Plan Pidal de 1845 a la Ley Moyano de 1857. Ediciones de la Universidad de Barcelona, Barcelona 28. Pérez N (2007) Anatomia, química i física experimental al Reial Col·legi de Cirurgia de Barcelona (1760–1808). Universitat Autònoma de Barcelona, Barcelona (Ph.D. thesis) Available at: http://ddd.uab.cat/ pub/tesis/2007/tdx-1203107-162239/npp1de1.pdf

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29. Pouillet C (1841) Elementos de Física Esperimental y de Meteorologia, Imprenta de Brusi, Barcelona, 2 vol [Translator: Pedro Vieta] 30. Puig-Pla C (2000) De la física experimental a la física industrial. Anàlisi d’una càtedra barcelonina (1814–1851). Quaderns d’Història de l’Enginyeria IV:119-172 31. Puig-Pla C (2001) La escuela de física experimental de Barcelona: primera época (1814–1836). In: Álvarez Lires M et al. (eds.) Estudios de Historia das Ciencias e das Técnicas, vol.2:1015-1026. Diputación Provincial, Pontevedra 32. Puig Pla C (2006) Física Tècnica i Il·lustració a Catalunya. La cultura de la utilitat: assimilar, divulgar, aprofitar. Universitat Autònoma de Barcelona, Departament de Física, Barcelona (Ph.D. thesis) Available at: http://ddd.uab.cat/pub/tesis/2006/tdx-1106107-172655/cpp1de1.pdf 33. Puig-Pla C (2009) L’influence française dans les premiers périodiques scientifiques et techniques espagnols. Les Memorias de agricultura y artes (1815–1821). In: Bret P, Chatzis K, Perez, L (eds.) La presse et les periodiques techniques en Europe 1750–1950:51:70. L’Harmattan, París 34. Puig-Pla C (2014) Establecimiento de las primeras cátedras científicas de la Universidad de Barcelona durante el Trienio Liberal. In: Blanco A (ed.) Enseñanza e Historia de las Ciencias y de las Técnicas: Orientación, Metodologías y Perspectivas: 113-119. SEHCYT, Barcelona 35. Puig-Pla C (in press) La Societat Filosòfica de Barcelona 1815–1821. In: Almirall: portal de cultura i pensament del segle XIX [on line] 36. Real Academia de Ciencias y Artes (1906–07) Nómina del personal académico, Barcelona: 113-125. A. López Robert, impresor, Barcelona 37. Reglamento de la Sociedad Médica de Emulación de Barcelona (1841). Imprenta de A. Albert, Barcelona 38. Reseña histórica y guía descriptiva de la Universidad (1929). Universidad de Barcelona, Barcelona 39. Roca A (2014) Reial Academia de Ciències i Arts de Barcelona (1764– 2014). Reial Acadèmia de Ciències i Arts de Barcelona / Diputació de Barcelona, Barcelona 40. Sales J (2011) La química a la Universitat de Barcelona. Facultat de Química, Publicaciones y Ediciones de la Universitat de Barcelona, Barcelona

41. Torres Amat F (1836) Memorias para ayudar a formar un diccionario crítico de los escritores catalanes. Imprenta de J. Verdaguer, Barcelona 42. Valls P (1857) Elogio fúnebre del M.I.S. Dr. D. Pedro Vieta y Gibert: leído por el socio Pablo Valls y Bonet en la sesión pública celebrada por la Sociedad Económica Barcelonesa de Amigos del País en 29 de junio de 1857. Imprenta de Joaquín Bosch, Barcelona 43. Vieta P (1814) Discurso inaugural que en la abertura de la cátedra de física experimental establecida en esta ciudad por la Real Junta de Comercio del Principado de Cataluña dixo Don Pedro Vieta catedrático de la misma. Oficina de Antonio Brusi, Barcelona 44. Vieta P (1815) Auroras boreales. Diferencia que hay entre ellas y los fenómenos de luz observados en plagas ecuatoriales de la atmósfera [manuscript report read on April 11, 1815], RACAB Archive 75.28 (CF. 20) 45. Vieta P (1819) Refutacion á las objeciones de Mr. Bodelio contra la pesantez del aire, insertadas en el número de este periódico del mes de setiembre. Memorias de Agricultura y Artes 8:81-85 46. Vieta P, Roura J (1835) Reflexiones físico-geológicas sobre fuentes ascendentes ó artificiales. Imprenta de los Herederos de Roca, Barcelona 47. Vieta P (1841) Reflecsiones físico-químicas, sobre incendios. Memoria leída por el socio Dr. D. Pedro Vieta en la sesión pública de 25 de octubre de 1840. Boletín de la Real Academia de Ciencias y Artes de Barcelona 11:72-75 48. Vieta P (1842) Apuntes acerca de la catarata, leídos en la sesión pública de la academia de medicina y cirugía de Barcelona; por el Dr. D. Pedro Vieta (estracto de dicha memoria). Boletín de Medicina, Cirujia y Farmacia 85:103 49. Vieta P (1847) Asignatura de física y nociones de química: programa que ha formado el profesor de dicha asignatura D. Pedro Vieta para la enseñanza de la misma en el curso de 1847 á 1848. Tomas Gorchs, Barcelona 50. Vieta P (1847) Asignatura de ampliación de física: programa que ha formado el profesor de dicha asignatura D. Pedro Vieta para la enseñanza de la misma en el curso de 1847 á 1848. Tomas Gorchs, Barcelona

See on the back cover the photograph of Pere Vieta i Gibert (1779–1856) and in page A2 a brief comment about him. This thematic issue on “Non-equilibrium physics” can be unloaded in ISSUU format and the individual articles can be found in the Institute for Catalan Studies journals’ repository [www.cat-science.cat; http://revistes.iec.cat/contributions].

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PERSPECTIVES Institut d’Estudis Catalans, Barcelona, Catalonia

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CONTRIB SCI 11:249-251 (2015) doi:10.2436/20.7010.01.236

Reflections: The enduring sym biosis between art and science Douglas Zook

Global Ecology Education Initiative, Boston University, Boston, MA, USA

Summary. We have all seen reflections in glass windows every day. We take

Correspondence: Douglas Zook

it for granted. Ever changing angles of light beams dance across the atmosphere bringing reflections of the reality all around us. But in exploring the uniquely charming and magnificent stare miasto (old city) of Kraków in Southern Poland, as well as in Prague and Boston, I have come to realize that the place where silica and simbeams meet –window glass- can be the homes of nature’s elusive poets and poetry... that silica, oxygen, and light combine to often reveal a side of earth, of nature, of us, that is hidden, beautiful, and mysterious. I have known this reflection world existed at the water’s surface, especially on a wind-less sunny day. But its’ expression in the compound silica dioxide, most familiar to us as “window glass”, may be the most revealing. Formed from unimaginable heat emissions of massive stars or supernovae in deep time, these elements actually make up much of our planet’s crust. One can see the reflective properties of silicon in rocks containing quartz minerals. These crystals with origins far in the universe are now commonly a part of windows, framing our lives by protecting and giving us essential and even profound connections to the outside, including our life-giving sun-star. [Contrib Sci 11(2): 249-251 (2015)]

E-mail: dzook@bu.edu

My 200-year old “mentor,” the great polymath Alexander von Humboldt, is reported to have spoken these words just prior to passing away at age 89, “How glorious these sunbeams are…!” When reaching glass surfaces such as windows, these persistent rays journeying from our nearest star and on an amazing journey through the spheres – exo, thermo, meso, strato, and tropo -- penetrate, reflect, refract, and scatter. My “Earth Gazes Back” photographic project captures and celebrates a few of these infinite number of ever-changing expressions, several of which are included in this article collection. We all certainly know the often stimulating display when

sunlight meets water surfaces, and then becomes a part of our artistic sensibilities, even gravitating toward what became a new 19th century art form, Impressionism. But in the grand Renaissance architecture city of Kraków, Poland, a couple hours before sunset one day, I realized that while most reflections on window surfaces keep subdued or hidden, they can under certain specific conditions become mysterious masterpieces of Nature. That ride of rays through the spheres results in a special expression, here and there interacting with the second most abundant element in the earth’s crust, silicon, a key constituent of glass. When I glanced my initial photographs of distant window surfaces of Kraków on my laptop later, I con-

Keywords: art and science symbiosis · windows reflections · ephemerous reality

ISSN (print): 1575-6343 e-ISSN: 2013-410X

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Reflections

cluded that sometimes the reflections I had discovered are far more beautiful, poetic, and mysterious than the reality that they reflect! To my delight, I found that only minimal adjustments for contrast and intensity were necessary to make the window come to life in often profound ways that defy analyses of angles, origins, source. Indeed, at first it was commonplace for me to turn around and see behind me structures, pipes, trees, gardens, fence, roof, clouds – whatever – that were apparently being reflected, often like a soft abstract painting. I quickly decided to abandon this fruitless venture and simply enjoy the outcome, eventually producing prints for on-line enjoyment [http://www.douglaszookphotography.com] and exhibitions in special venues, such as the historic 14th century Collegium Maius cellarium gallery/ café at Jagiellonian University in Kraków in 2014. While intriguing reflections can be noticed and photographed from exterior window surfaces anywhere, there are certain conditions that can enhance Nature’s expression. For example, much of Kraków is a United Nations Educational Scientific and Cultural Organization (UNESCO World Heritage site, meaning that its historic architecture, particularly from the 13th through the 19th century, must be preserved in perpetuity. This material used for the many gothic, renaissance, and baroque buildings, ornamentations, edifices, sculptures is substantially linked to the ancient geology of southern Poland, including Kraków, wherein significant periods of geologic time featured shallow warm seas dominated by coral, sponge, and red algal reefs. These reefs and other related life forms contributed massive calcium carbonate or limestone that eventually accumulated through orogeny as layered terrestrial masses. These have been partially mined by humans and became the often extraordinary pliable building material which in the hands of famed architects and designers in Italy and Poland resulted in this brilliant medieval cityscape display. Moreover, the unique city features a 4.5 km arboretum-like park, the Planty, which surrounds the oldest region of the city. This synergy of current nature via its trees and ancient nature via human-constructed remnants of ancient life contribute significantly to unique reflections. This can be further enhanced when some of the glass used in windows dates back many decades or even more than century, allowing chemical “imperfections” in the glass to further dazzle the incoming usually indirect sunbeams. This glass window history is mirrored somewhat in another extraordinary city where I have captured some unique reflections, Prague. That said, while Boston has some European flavor it has a different landscape and urban expreswww.cat-science.cat

sion that in some districts and times of day resonates with provocative reflections. Despite the fact that there have been some window reflection images done over the years, in my most egocentric moments, I don’t find any like these and thus I end up perhaps falsely concluding that I have discovered a new art form—one where the earth and its spectacular biosphere is gazing back, revealing in special moments its beauty and mystery. Consequently, I feel that these ephemerous images of mine can proudly introduce the excellent scholarly written articles in this issue of Contributions to Science.

About the author Douglas Zook is a biologist and directs the Global Ecology Education Initiative [http://dpzook.wix. com/geei] based currently at Boston University. In over three decades, his time extending to the present usually is spent teaching and researching how human beings can renew efforts to be more compatible with the planet that sustains us. He has also guided the development of hundreds of students who seek to become innovative biosphere-conscious science teachers through the Science Education Program at Boston University’s School of Education. His photographic work can be viewed at http://www.douglaszookphotography.com. He can be reached at dpzook@gmail.com.

Prof. Douglas Zook, Boston University.

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Reflections Photography: Earth Gazes Back. Window reflections from from Kraków (K), Prague (P) and Boston (B). No color changes or other photoshop alterations were made. Names and pages between parentheses indicate the authors and initial pages of the articles in this issue of Contributions to Science [http://www.cat-science.cat], vol. 11, issue 2. Photo 1-K (see Pagonabarraga, Ritort, p. 125); Photo 2-P (see Jou, Casas-Vázquez, p. 131); Photo 3-K (see Ritort, p. 137); Photo 4-B (see Rubi, p. 147); Photo 5-B (see Baró, Planes, Vives, p. 153); Photo 6-K (see Palassini, p. 163); Photo 7-K (see Faraudo, Aguilella-Arzo, p. 181); Photo 8-K (see Ortín, p. 189); Photo 9-K (see Reguera, p. 173); Photo 10-K (see Díaz-Guilera, Pérez-Vicente, p. 207); Photo 11-K (see Claret, Ignés-Mullol, Sagués, p. 199); Photo 12-K (Frigola, Sancho, Ibañes, p. 215); Photo 13-K (see Jona-Lasinio, p. 127); Photo 14-K (see Soriano, Casademunt, p. 225). www.cat-science.cat

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INDEXES VOLUMES 7, 8, 9 CONTRIBUTIONS to SCIENCE 11 (2015) Institut d’Estudis Catalans, Barcelona, Catalonia ISSN: 1575-6343 www.cat-science.cat

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Contents Index · Volumes 7, 8, 9 (years 2011, 2012, 2013) Alegret S → Some salmon-colored keywords regarding various aspects of chemistry, 7: 71 doi:10.2436/20.7010.01.111 Alsina C → Professor Pere Pi Calleja (1907– 1986), 7: 85 doi:10.2436/20.7010.01.113 Asensi Botet F → Fighting against smallpox around the world. The vaccination expeditions of Xavier de Balmis (1803–1806) and Josep Salvany (1803–1810), 8: 99 doi:10.2436/20.7010.01.140 Aymerich MS → Franco R Aymerich M → Presentation, 8: 137 doi:10.2436/20.7010.01.145 Ballabrera-Poy J → Salat J Beato M → What is our level of knowledge about the genome today?, 8: 155 doi:10.2436/20.7010.01.149 Berlanga M → Guerrero R Bolufer P → Science and technology in the 20th century as seen through the journal Ibérica (1914–2003), 7: 185 doi:10.2436/20.7010.01.125 Bradley RS → Natural archives, changing climates, 7: 21 doi:10.2436/20.7010.01.104 Bradley RS → What can we learn from past warm periods?, 8: 53 doi:10.2436/20.7010.01.134 Bradley RS → Where do we stand on global warming?, 7: 45 doi:10.2436/20.7010.01.107 Buceta J → Multidisciplinary approaches towards compartmentalization in development: Dorsoventral boundary formation of the Drosophila wing disc as a case study, 9: 57 doi:10.2436/20.7010.01.164 Calisto BM → The race to resolve the atomic structures of the ribosome. On the Nobel Prize in Chemistry awarded to Venkatraman Ramakrishnan, Thomas A. Steitz, and Ada E. Yonath, 7: 125 doi:10.2436/20.7010.01.117 Camarasa JM → Roca-Rosell A Camí J → Bioethical challenges in personalised medicine, 8: 171 doi:10.2436/20.7010.01.152 Cardona P-J → Will personalized medicine be the key to eradicating TB?, 8: 181 doi:10.2436/20.7010.01.154 Casadesús J → Bacterial pathogenesis as an imperfect symbiosis, 9: 51 doi:10.2436/20.7010.01.163 Casanovas, O → Jiménez-Valerio G Castellà i Clave A → Presentation, 8: 137 doi:10.2436/20.7010.01.145

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Castilla JC → Conservation and socialecological systems in the 21st century of the Anthropocene era, 8: 11 doi:10.2436/20.7010.01.129 Chica C → Latindex: A tool to extend the dissemination of scientific publications and to improve their quality assessment, 9: 151 doi:10.2436/20.7010.01.174 Ciurana J → Ros J Ciurana J → Foreword, 9: 113 doi:10.2436/20.7010.01.171 Clotet J → First International Conference of Biology of Catalonia (CIBICAT), ‘Global questions on advanced biology’ (Barcelona, 9–12 July, 2012), 9: 43 doi:10.2436/20.7010.01.162 Cope D → Forty Years On, 8: 121 doi:10.2436/20.7010.01.143 de Gispert N → Foreword, 8: 119 doi:10.2436/20.7010.01.142 de Solà-Morales O → Sustainability of personalised medicine, 8: 149 doi:10.2436/20.7010.01.148 Domínguez M → Gozzer S Domínguez García F → CAPCIT: The Advisory Board of the Parliament of Catalonia for Science and Technology, 8: 131 doi:10.2436/20.7010.01.144 Escalas Llimona R → Temperament and tuning of early 19th century Hispanic keyboard instruments: A study of the monochord integrated into a fortepiano made by Francisco Fernández (1828), 9: 75 doi:10.2436/20.7010.01.166 Esteller M → Forecasting limits in personalized medicine, 8: 145 doi:10.2436/20.7010.01.147 Fernández P → Salat J Fita I→ Calisto BM Folch R → The immediate future: Challenges and scales, 7: 51 doi:10.2436/20.7010.01.108 Franco R → Smart cell-surface receptors: On the 2012 Nobel Prize in Chemistry, awarded to Robert J. Lefkowitz and Brian K. Kobilka, 9: 25 doi:10.2436/20.7010.01.160 García-Lladó A → Ciència magazine, first period (1926–1933): A project for the recovery and dissemination of Catalan scientific heritage, 9: 169 doi:10.2436/20.7010.01.176 Genescà-Sitjes M → Ibérica magazine (1913–

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2004) and the Ebro Observatory, 9: 159 doi:10.2436/20.7010.01.175 Giner S → Foreword, 8: 09 Giner S → Piedmont and Catalonia: The unification of Italy and Spain. Some comparative remarks, 7: 171 doi:10.2436/20.7010.01.123 Gonzàlez Sastre F → Foreword, 7: 101 doi:10.2436/20.7010.01.114 Gozzer S → Global climate change in the Spanish media: How the conservative press portrayed Al Gore’s initiative, 7: 65 doi:10.2436/20.7010.01.110 Granados A → Challenges for industry developers, 8: 167 doi:10.2436/20.7010.01.151 Guerrero R → An integrated ecogenetical study of minimal ecosystems: The microbial mats of Ebro Delta and the Camargue (Western Mediterranean), 9: 117 doi:10.2436/20.7010.01.172 Guerrero R → Conclusions, 8: 187 doi:10.2436/20.7010.01.155 Guerrero R → Piqueras M Hahn E → Martínez-Francés V Herms A → The CCD sensor: A semiconductor circuit for capturing images. On the Nobel Prize in Physics awarded to Charles Kuen Kao, Willard S. Boyle, and George E. Smith (II), 7: 117 doi:10.2436/20.7010.01.116 Jiménez-Valerio G → Anti-angiogenic therapy for cancer and mechanisms of tumor resistance, 9: 67 doi:10.2436/20.7010.01.165 Juan i Otero M → Dendritic cells (CD) and their Toll-like receptors (TLR): Vital elements at the core of all individual immune responses. On the Nobel Prize in Physiology or Medicine 2011 awarded to Bruce A. Beutler, Jules A. Hoffmann, and Ralf M. Steinman, 8: 61 doi:10.2436/20.7010.01.135 Juan-Vicedo J → Martínez-Francés V Levin SA → Evolution at the ecosystem level: On the evolution of ecosystem patterns, 7: 11 doi:10.2436/20.7010.01.102 Llebot JE → Can we be confident with climate models?, 7: 27 doi:10.2436/20.7010.01.105 Lleonart J → The history of Scientia Marina, 7: 175 doi:10.2436/20.7010.01.124 Llimona X → Professor Creu Casas i Sicart (1913–2007), 8: 107 doi:10.2436/20.7010.01.141 Llorca J → Energy from hydrogen. Hydrogen

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volumes 7,8, 9 (2011, 2012, 2013)

from renewable fuels for portable applications, 7: 57 doi:10.2436/20.7010.01.109 Lovelock JE → Climate change on a live Earth, 7: 17 doi:10.2436/20.7010.01.103 Luttikhuizen F → Professor Ignasi Ponseti i Vives (1914–2009), 7: 205 doi:10.2436/20.7010.01.128 March Noguera J → Science of the Balearic Islands. A collection on the past that looks toward the future, 7: 191 doi:10.2436/20.7010.01.126 Marco J → The role of autobiography, biography, and history in the works of Mario Vargas Llosa. On the Nobel Prize in Literature awarded to Mario Vargas Llosa, 7: 155 doi:10.2436/20.7010.01.121 Martínez J → Salat J Martínez-Francés V → Ethnobotanical study of the sages used in tradicional Valencian medicine and as essential oil: Characterization of an endemic Salvia and its contribution to local development, 8: 77 doi:10.2436/20.7010.01.137 Martínez-Vidal A → García-Lladó A Massó E → The accelerated universe. On the Nobel Prize in Physics 2011 awarded to Saul Perlmutter, Brian P. Schmidt, and Adam G. Riess, 8: 69 doi:10.2436/20.7010.01.136 Mena FX → Companies, markets, and management of common property. On the Nobel Prize in Economics awarded to Elinor Ostrom and Oliver E. Williamson, 7: 141 doi:10.2436/20.7010.01.119 Molina T → The theme of Earth Day and the social perception of what is really happening to our planet, 8: 33 doi:10.2436/20.7010.01.131 Mompart J → The Gedankenexperimente of quantum mechanics become reality: On the 2012 Nobel Prize in Physics, awarded to Serge Haroche and David J. Wineland, 9: 33 doi:10.2436/20.7010.01.161 Montero-Pich O → García-Lladó A Murià JM → A transition from indigenous to European technology in colonial Mexico: The case of tequila, 8: 93 doi:10.2436/20.7010.01.139 Nair P → The United Nations University Institute on Globalization, Culture and Mobility (UNU-GCM) in Barcelona: Mission and vision, 9: 101 doi:10.2436/20.7010.01.168 Olivar MP → Lleonart J Omedes A → Piqueras M Petrus JL → Daniel Simberloff: Creative and devastating, 9: 5 doi:10.2436/20.7010.01.157 Piniella JF → Crystallography and the Nobel

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Prizes: On the 2011 Nobel Prize in Chemistry, awarded to Dan Shechtman, 9: 17 doi:10.2436/20.7010.01.159 Piqueras M → David Cardús (1922–2003), the physician of the space, 9: 183 doi:10.2436/20.7010.01.178 Piqueras M → Ramon Casanova (1892– 1968) and the pulse jet engine, 9: 195 doi:10.2436/20.7010.01.179 Piqueras M → The American dream of Rafael Guastavino (1842–1908), 9: 109 doi:10.2436/20.7010.01.170 Piqueras M → The Museu Blau, a natural history museum for the 21st century, 8: 85 doi:10.2436/20.7010.01.138 Plasència A → Global health challenges and personalised medicine, 8: 175 doi:10.2436/20.7010.01.153 Puche C → The Institute for Catalan Studies and the International Women’s Day, 2006–2013, 9: 107 doi:10.2436/20.7010.01.169 Ríos S → Martínez-Francés V Roca-Rosell A → The Foundation of the Sciences Section on the Institute for Catalan Studies (1911) and its early years, 7: 195 doi:10.2436/20.7010.01.127 Ros J → Biodiversity: Origin, function and threats, 7: 37 doi:10.2436/20.7010.01.106 Ros J → Rachel Carson, sensitive and perceptive interpreter of nature, 8: 23 doi:10.2436/20.7010.01.130 Rovira L → Carhus Plus+: A classification of social science and humanities journals on the basis of international visibility standards, 9: 141 doi:10.2436/20.7010.01.173 Ryan C → Margalida Comas Camps (18921972): Scientist and science educator, 7: 77 doi:10.2436/20.7010.01.112 Salas E → Complex diseases: the relationship between genetic and sociocultural factors in the risk of disease, 8: 161 doi:10.2436/20.7010.01.150 Salat J → The contribution of the Barcelona World Race to improved ocean surface information. A validation of the SMOS remotely sensed salinity, 9: 89 doi:10.2436/20.7010.01.167 Salvador K → Salat J Santaló J → Changing the perception of our own nature. On the Nobel Prize in Physiology or Medicine awarded to Robert G. Edwards, 7: 149 doi:10.2436/20.7010.01.120 Serrat D → Foreword, 7: 9 Serrat D → Gonzàlez Sastre F Siguan M → Writing with the eyes. On the Nobel Prize in Literature awarded to Herta Müller, 7: 131 doi:10.2436/20.7010.01.118

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Simberloff D → Biological invasions: Much progress plus several controversies, 9: 7 doi:10.2436/20.7010.01.158 Simó R → Sea and sky. The marine biosphere as an agent of change, 8: 47 doi:10.2436/20.7010.01.133 Suriñach E → Recent large earthquakes from a geophysical perspective, 8: 41 doi:10.2436/20.7010.01.132 Tomàs Salvà M → Activities of the Royal Academy of Medicine of the Balearic Islands, 9: 199 doi:10.2436/20.7010.01.180 Tort Ll → Foreword, 9: 1 doi:10.2436/20.7010.01.156 Tugores Ques J → Unemployment and other challenges. On the Nobel Prize in Economics awarded to Peter A. Diamond, Dale T. Mortensen and Christopher A. Pissarides, 7: 163 doi:10.2436/20.7010.01.122 Tusell L → Telomeres, the beginning(s) of the end. On the Nobel Prize in Physiology or Medicine awarded to Elizabeth H. Blackburn, Carol W. Greider, and Jack W. Szostak, 7: 101 doi:10.2436/20.7010.01.114 Umbert M → Salat J Vallmitjana M → Ciència magazine, second period (1980–1991): Recovering normality for the Catalan scientific language, 9: 177 doi:10.2436/20.7010.01.177 Vallmitjana S → Transmission of light by fibers for optical communication. On the Nobel Prize in Physics awarded to Charles Kuen Kao, Willard S. Boyle, and George E. Smith (I), 7: 109 doi:10.2436/20.7010.01.115 Vendrell M → Personalized medicine: needs, challenges, and considerations, 8: 139 doi:10.2436/20.7010.01.146 Vila R → Martínez-Francés V Zarzoso A → García-Lladó A

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Authors Index · Volumes 7, 8, 9 (years 2011, 2012, 2013) Alegret S → 7: 71 Alsina C → 7: 85 Asensi Botet F → 8: 99 Aymerich MS → 9: 25 Aymerich, M → 8: 137 Ballabrera-Poy J → 9: 89 Beato M → 8: 155 Berlanga M → 9: 117 Bolufer P → 7: 185 Bradley RS → 7: 21, 45; 8: 53 Buceta J → 9: 57 Calisto BM → 7: 125 Camarasa JM → 7: 195 Camí J → 8: 171 Cardona P-J → 8: 181 Casadesús J → 9: 51 Casanovas, O → 9: 67 Castellà i Clavé A → 8: 137 Castilla JC → 8: 11 Chica C → 9: 151 Ciurana J → 9: 113 Clotet J → 9: 43 Cope D → 8: 121 de Gispert N → 8: 119 de Solà-Morales O → 8: 149 Domínguez M → 7: 65 Domínguez García F → 8: 131 Escalas Llimona R → 9: 75 Esteller M → 8: 145 Fernández P → 9: 89 Fita I→ 7: 125 Folch R → 7: 51 Franco R → 9: 25

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García-Lladó A → 9: 169 Genescà-Sitjes M → 9: 159 Giner S → 7: 171; 8: 09 Gonzàlez Sastre F → 7: 101 Gozzer S → 7: 65 Granados A → 8: 167 Guerrero R → 8: 85, 187; 9: 117 Hahn E → 8: 77 Herms A → 7: 117 Jiménez-Valerio G → 9: 67 Juan i Otero M → 8: 61 Juan-Vicedo J → 8: 77 Levin SA → 7: 11 Llebot JE → 7: 27 Lleonart J → 7: 175 Llimona X → 8: 107 Llorca J → 7: 57 Lovelock JE → 7: 17 Luttikhuizen F → 7: 205 March Noguera J → 7: 191 Marco J → 7: 155 Martínez J → 9: 89 Martínez-Francés V → 8: 77 Martínez-Vidal A → 9: 169 Massó E → 8: 69 Mena FX → 7: 141 Molina T → 8: 33 Mompart J → 9: 33 Montero-Pich O → 9: 169 Murià JM → 8: 93

Petrus JL → 9: 5 Piniella JF → 9: 17 Piqueras M → 8: 85; 9: 109, 183, 195 Plasència A → 8: 175 Puche C → 9: 107 Ríos S → 8: 77 Roca-Rosell A → 7: 195 Ros J → 7: 37; 8: 23 Rovira L → 9: 141 Ryan C → 7: 77 Salas E → 8: 161 Salat J → 9: 89 Salvador K → 9: 89 Santaló J → 7: 149 Serrat D → 7: 9, 101 Siguan M → 7: 131 Simberloff D → 9: 7 Simó R → 8: 47 Suriñach E → 8: 41 Tomàs Salvà M → 9: 199 Tort Ll → 9: 1 Tugores Ques J → 7: 163 Tusell L → 7: 101 Umbert M → 9: 89 Vallmitjana M → 9: 177 Vallmitjana S → 7: 109 Vendrell M → 8: 139 Vila R → 8: 77 Zarzoso A → 9: 169

Nair P → 9: 101 Olivar MP → 7: 175 Omedes A → 8: 85

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Keywords Index · Volumes 7, 8, 9 (years 2011, 2012, 2013) Acquired resistance → 9: 67 Adenosine receptors → 9: 25 Adrenergic receptors → 9: 25 Advisory Board of the Parliament of Catalonia for Science and Technology (CAPCIT) → 8: 121, 131 Aerosols → 8: 47 Aging population → 8: 139 Al Gore → 7: 65 Albedo → 8: 47 An Inconvenient Truth → 7: 65 Anthropocene era → 8: 11 Anthropogenic climate change → 7: 27 Anti-angiogenic therapy → 9: 67 Archeological remains → 7: 21 Artic Oscillation → 7: 45 Article citation analysis → 9: 141 Assisted reproduction techniques → 7: 149 Attenuation → 7: 109 Autobiography → 7: 131 Autobiographycal realism → 7: 155 Barcelona Music Museum → 9: 75 Bell Labs → 7: 117 Biocides → 8: 23 Biodiversity → 7: 37 Bioethics → 7: 149 Biological control → 9: 7 Biological invasion → 9: 7 Biomarkers → 8: 145, 161 Biomechanics → 9: 57 BRCA1 → 8: 145 Cancer → 8: 145 Carcinogenesis → 7: 101 Cardio InCode → 8: 161 Cardiovascular disease → 8: 161 Carhus Plus+ → 9: 141 Catalan scientific-medical-technological lexicon → 9: 169 Catalan → 9: 177 Catalanism → 9: 169 Catalyst → 7: 57 Categorical challenges → 7: 51 Cavity quantum electrodynamics → 9: 33 CCD sensor → 7: 117 Centres for therapeutic innovation (CTI) → 8: 139 Charge transfer → 7: 117 Chemical industry → 8: 23 Chile → 8: 11 Chromatographic identification → 8: 77 Climate change → 7: 17, 65; 8: 33 Climate models → 7: 27 Climate skepticism → 7: 65 Climate system → 7: 27 Cloud formation → 8: 47 Coastal regions → 8: 53 Common-pool resources → 7: 141

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Compartimentalization → 9: 57 Complex diseases → 8: 161 Conservation → 8: 11 Cosmological constant → 8: 69 Cosmology → 8: 69 Cost-efficacy ratio → 7: 51 Crystal structure → 9: 17 Culture → 9: 101, 177 Dark energy → 8: 69 DDT → 8: 23 Dendritic cells → 8: 61 Developmental biology → 9: 57 Digital photography and video → 7: 117 Directly observed therapy-short course (DOTS) → 8: 181 Dispersion → 7: 109 Disruptive innovations → 8: 139 Diversity → 9: 101 DNA methylation → 8: 145 DNA regulation → 8: 155 Drug pricing mechanisms → 8: 149 Earliest ecosystems → 9: 117 Earth System Science → 7: 17 Earthquakes → 8: 41 Ebro Observatory → 9: 159 Ecodiversity → 7: 37 Ecological and evolutionary dynamics → 7: 11 Ecology → 8: 23 Economic governance → 7: 141 Economics of organizations → 7: 141 Ecosystem impact → 9: 7 Ecosystems science → 7: 11 Eemian interglacial → 8: 53 Efficiency → 7: 51 Electron diffraction → 9: 17 Energy → 7: 57 Environmental ethics → 8: 11 Epigenetics → 8: 145, 155 Epigenomics → 8: 145 Eradication → 9: 7 Ethnobotany → 8: 77 European Parlamentary Technology Assessment (EPTA) → 8: 121, 131 Evaluation of scientific journals → 9: 151 Evolution → 9: 51 Extensively drug resistant TB (XDR-TB) → 8: 181 Fishery → 8: 11 Flow of genetic information → 8: 155 Formalism → 7: 155 Fortepiano → 9: 75 Francisco Fernández (1766–1852) → 9: 75 Frictions → 7: 163 Gaia theory → 7: 17

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Gaia → 8: 47 Gene expression → 8: 155 Gene regulatory networks → 9: 57 Genetic code → 7: 125 Genetic counselling → 8: 171 Genetic discrimination → 8: 171 Genetic networks → 8: 155 Genetic risk factors → 8: 161 Genetic testing → 8: 171 Geophysics → 8: 41 German-minority in Romania → 7: 131 Global health → 8: 175 Global sustainability → 7: 51 Global warming → 7: 17, 45; 8: 33, 53 Globalization → 9: 101 Governance → 8: 11 G-protein-coupled receptors → 9: 25 Health equity → 8: 175 Health industry pressures → 8: 167 Health technology assessment (HTA) → 8: 167 Healthcare systems → 8: 139 Host susceptibility → 9: 51 Human pathogens → 9: 51 Humanities and social sciences evaluation → 9: 141 Hurricanes → 8: 53 Hybridization → 9: 7 Hydrogen → 7: 57 Ibérica magazine → 9: 159 Ice cores → 7: 21 In vitro fertilization → 7: 149 Inflammation → 8: 61 Innate immunity → 8: 61 Institutional economics → 7: 141 Intergovernmental Panel on Climate Change (IPCC) → 7: 17, 27, 45; 8: 33, 53 International Year of Biodiversity → 7: 37 Intrinsic resistance → 9: 67 Journalism → 7: 155 Knowledge society → 7: 51 Lag time → 9: 7 Lake sediments → 7: 21 Landscape of the disposed → 7: 131 Latin American dictatorships → 7: 155 Latindex criteria → 9: 151 Latindex system → 9: 151 Levels of greenhause gases→ 7: 45 LTA4H gene polymorphisms → 8: 181 Maintenance management → 9: 7 Management → 8: 11 Margalef, Ramon → 7: 11; 8: 11 Marine regulation → 8: 47

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volumes 7, 8, 9 (2011, 2012, 2013)

Matching → 7: 163 Medicinal ethnobotany → 8: 77 MGMT → 8: 145 Microbial mats → 9: 117 Microreactor → 7: 57 Migration → 9: 101 Minimal ecosystems → 9: 117 Mobility → 9: 101 Monochord → 9: 75 Multiple drug resistant TB (MDR-TB) → 8: 181 Musical temperament → 9: 75 Natural archives → 7: 21 Neoplasia → 7: 101 Objectivism → 7: 155 Ocean circumnavigation → 9: 89 Ocean races → 9: 89 Office of Technology Assessment (OTA) → 8: 121, 131 Optical fibers → 7: 109 Optical networks → 7: 117 Oral language → 7: 155 Orphan drugs → 8: 149 Paleoclimatology → 7: 21; 8: 53 Parliamentary Office of Science and Technology (POST) → 8: 121 Parliamentary Technology Assessment (PTA) → 8: 131 Personalised medicine → 8: 139, 145, 149, 161, 167, 171, 175, 181 Perspectivism → 7: 155 Pesticides → 8: 23 Pharmacogenetics → 8: 149 Phenological changes → 7: 45 Photodetection → 7: 117 Plankton → 8: 47 Pollution → 8: 23 Popular science magazine → 9: 177 Popular science → 9: 169 Population biology → 7: 11 Population diversity and dynamics → 9: 117 Poverty-related diseases → 8: 175 Power structures → 7: 155 Professional identity → 9: 169 Progeria → 7: 101 Prokaryotic diversity → 9: 117 Protein synthesis → 7: 125 Quantum mechanics → 9: 33 Quantum optics → 9: 33 Quasicrystals → 9: 17

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Rare diseases → 8: 167 Receptor heteromers → 9: 25 Reflection → 7: 109 Research → 9: 177 Ribosome → 7: 125 Risk factors → 8: 171 Romanticism → 9: 75 Saccharomyces cerevisiae → 7: 101 Salvia → 8: 77 Scalar levels → 7: 51 Science and technology popularization → 9: 159 Science assessment → 9: 151 Science audiences → 9: 169 Science → 9: 177 Science-medical-technology journalism → 9: 169 Scientific dissemination → 9: 151 Scientific journals → 9: 141 Scientific popularization → 8: 23 Sea surface temperature and salinity → 9: 89 Searching → 7: 163 Seismic records → 8: 41 Seismology → 8: 41 Ships of opportunity → 9: 89 SLCO1B1 gene polymorphism → 8: 181 SMOS → 9: 89 Social and global perception → 8: 33 Social networks → 8: 33 Social responsibility → 8: 167 Social-ecologycal systems → 8: 11 Societal effects → 7: 21 Society of Jesus → 9: 159 Sociocultural risk factors → 8: 161 Solid state arrays → 7: 117 Spanish columnists → 7: 65 Spanish media → 7: 65 Stalagmites → 7: 21 Stromal cells → 9: 67 Structured biocenoses → 9: 117 Supernova → 8: 69 Surrealism → 7: 155 Sustainability → 7: 11, 37; 8: 11, 149 Symbiosis → 9: 51 Systems biology → 9: 57

TLR → 8: 61 Toll → 8: 61 Topological association domains → 8: 155 Transaction cost economics → 7: 141 Trapping and cooling of ions → 9: 33 Tree rings → 7: 21 Tumor cells → 9: 67 Tuning → 9: 75 Unemployment → 7: 163 Vacancies → 7: 163 Valencia region → 8: 77 Value of socio-environmental services → 7: 51 Visual language → 7: 131 X-ray crystallography → 7: 125

Technology assessment → 8: 121 Tectonic plates → 8: 41 Telecommunications → 7: 109 Telomerase → 7: 101 Telomeres → 7: 101 Tessellations → 9: 17 Tetrahymena thermophila → 7: 101

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Paraules clau · Index volums 7, 8, 9 (anys 2011, 2012, 2013) Aerosols → 8: 47 Afinació 9: 75 Al Gore → 7: 65 Albedo → 8: 47 An Inconvenient Truth (Una veritat incòmoda) → 7: 65 Anàlisi de cites d’articles científics → 9: 141 Anells dels arbres → 7: 21 Any Internacional de la Biodiversitat → 7: 37 Arxius naturals → 7: 21 Assessorament Científic i Tecnològic als Parlaments (PTA) → 8: 121, 131 Assessorament de tecnologia sanitària (HTA) → 8: 167 Assessorament genètic → 8: 171 Assessorament tecnològic → 8: 121 Atenuació → 7: 109 Atur → 7: 163 Autobiografia → 7: 131 Avaluació d’humanitats i ciències socials → 9: 141 Avaluació de la ciència → 9: 151 Avaluació de revistes científiques → 9: 151 Biocenosis estructurades → 9: 117 Biocides → 8: 23 Biodiversitat → 7: 37 Bioètica → 7: 149 Biologia de poblacions → 7: 11 Biologia del desenvolupament → 9: 57 Biologia dels sistemes → 9: 57 Biomarcadors → 8: 145, 161 Biomecànica → 9: 57 BRCA1 → 8: 145 Càncer → 8: 145 Canvi climàtic antropogènic → 7: 27 Canvi climàtic → 7: 17, 65; 8: 33 Captura i refredament d’ions → 9: 33 Carcinogènesis → 7: 101 Cardio InCode → 8: 161 Carhus Plus+ → 9: 141 Català → 9: 177 Catalanisme → 9: 169 Catalitzador → 7: 57 Cèl·lules de l’estroma → 9: 67 Cèl·lules dendrítiques → 8: 61 Cèl·lules tumorals → 9: 67 Centres d’innovació terapèutica (CTI) → 8: 139 Cerca d’ocupació → 7: 163 Ciència del sistema terrestre → 7: 17 Ciència dels ecosistemes → 7: 11 Ciència → 9: 177 Circumnavegació oceànica → 9: 89 Codi genètic → 7: 125 Coincidència → 7: 163 Columnistes a Espanya → 7: 65

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Companyia de Jesús → 9: 159 Compartimentació → 9: 57 Consell Assessor del Parlament sobre Ciència i Tecnologia (CAPCIT) → 8: 121, 131 Conservació → 8: 11 Constant cosmològica → 8: 69 Contaminació → 8: 23 Control biològic → 9: 7 Cosmologia → 8: 69 Cristal·lografia de raigs X → 7: 125 Criteris Latindex → 9: 151 Cultura → 9: 101, 177 DDT → 8: 23 Desfasament temporal → 9: 7 Dictadures llatinoamericanes → 7: 155 Difracció d’electrons → 9: 17 Difusió científica → 9: 151 Dimensions escalars → 7: 51 Dinàmica ecològica i evolutiva → 7: 11 Discriminació genètica → 8: 171 Dispersió → 7: 109 Diversitat i dinàmica de poblacions → 9: 117 Diversitat procariota → 9: 117 Diversitat → 9: 101 Divulgació científica i tecnològica → 9: 159 Divulgació científica → 8: 23; 9: 169 Dominis d’associació topològica → 8: 155 Ecodiversitat → 7: 37 Ecologia → 8: 23 Economia de les organitzacions → 7: 141 Economia dels costos de transaccions → 7: 141 Economia institucional → 7: 141 Ecosistemes mínims → 9: 117 Ecosistemes primitius → 9: 117 Efectes socials → 7: 21 Eficiència → 7: 51 Electrodinàmica quàntica en cavitats → 9: 33 Energia fosca → 8: 69 Energia → 7: 57 Envelliment de la població → 8: 139 Epigenètica → 8: 145, 155 Epigenòmica → 8: 145 Equitat en salut → 8: 175 Era antropocènica → 8: 11 Eradicació → 9: 7 Escalfament global → 7: 17, 45; 8: 33, 53 Escepticisme climàtic → 7: 65 Estalagmites → 7: 21 Estructura cristal·lina → 9: 17 Estructures de poder → 7: 155 Ètica ambiental → 8: 11 Etnobotànica → 8: 77 Evolució → 9: 51 Expressió gènica → 8: 155

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Factors de risc genètics → 8: 161 Factors de risc socioculturals → 8: 161 Factors de risc → 8: 171 Farmacogenètica → 8: 149 Fertilització in vitro → 7: 149 Fibra òptica → 7: 109 Flux d’informació genètica → 8: 155 Formació de núvols → 8: 47 Formalisme → 7: 155 Fortepiano → 9: 75 Fotodetecció → 7: 117 Fotografia i vídeo digitals → 7: 117 Francisco Fernández (1766–1852) → 9: 75 Friccions → 7: 163 Gaia → 8: 47 Geofísica → 8: 41 Gestió del manteniment → 9: 7 Gestió → 8: 11 Globalització → 9: 101 Governança econòmica → 7: 141 Governança → 8: 11 Grup Intergovernamental d’Experts sobre el Canvi Climàtic (GIECC) → 7: 17, 27, 45; 8: 3 Grup IPCC → 8: 53 Heteròmers de receptors → 9: 25 Hibridació → 9: 7 Hidrogen → 7: 57 Huracans → 8: 53 Identificació cromatogràfica → 8: 77 Identitat professional → 9: 169 Immunitat innata → 8: 61 Impacte a l’ecosistema → 9: 7 Indústria química → 8: 23 Inflamació → 8: 61 Innovacions disruptives → 8: 139 Interglacial Riss-Wurm → 8: 53 Invasió biològica → 9: 7 Laboratoris Bell → 7: 117 Lèxic científic-mèdic-tècnic català → 9: 169 Llenguatge oral → 7: 155 Llenguatge visual → 7: 131 Malaltia cardiovascular → 8: 161 Malalties complexes → 8: 161 Malalties rares → 8: 167 Malalties relacionades amb la probresa → 8: 175 Margalef, Ramon → 7: 11; 8: 11 Matrius d’estat sòlid → 7: 117 Mecànica quàntica → 9: 33 Mecanismes de fixació dels preus dels medica-

CONTRIBUTIONS to SCIENCE 11 (2015) XX-XX

volumes 7, 8, 9 (2011, 2012, 2013)

ments → 8: 149 Medicaments orfes → 8: 149 Medicina personalitzada → 8: 139, 145, 149, 161, 167, 171, 175, 181 Medicinal → 8: 77 Metilació del DNA → 8: 145 MGMT → 8: 145 Microreactor → 7: 57 Migració → 9: 101 Minoria alemanya a Romania → 7: 131 Mitjans de comunicació espanyols → 7: 65 Mobilitat → 9: 101 Models climàtics → 7: 27 Monocordi → 9: 75 Museu de la Música de Barcelona → 9: 75 Neoplàsia → 7: 101 Nivells dels gasos d’efecte hivernacle → 7: 45 Nuclis de gel → 7: 21 Objectivisme → 7: 155 Observatori de l’Ebre → 9: 159 Oficina d’Assessorament Tecnològic (OTA) → 8: 121, 131 Oficina Parlamentària de Ciència i Tecnologia (POST) → 8: 121 Òptica quàntica → 9: 33 Oscil·lació àrtica → 7: 45 País Valencià → 8: 77 Paisatge dels desposseïts → 7: 131 Paleoclimatologia → 7: 21; 8: 53 Patògens humans → 9: 51 Percepció social i global → 8: 33 Periodisme científic-mèdic-tècnic → 9: 169 Periodisme → 7: 155 Perspectivisme → 7: 155 Pesca → 8: 11 Plaguicides → 8: 23 Plàncton → 8: 47 Plaques tectòniques → 8: 41 Polimorfisme gen LTA4H → 8: 181 Polimorfisme gen SLCO1B1 → 8: 181 Pressions de la industria sanitària → 8: 167 Progèria → 7: 101 Proves genètiques → 8: 171 Públics de la ciència → 9: 169

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Quasicristalls → 9: 17 Realisme autobiogràfic → 7: 155 Receptor acoblat a proteïnes G → 9: 25 Receptors adrenèrgics → 9: 25 Receptors d’adenosina → 9: 25 Recerca → 9: 177 Recursos comuns → 7: 141 Reflexió → 7: 109 Regates oceàniques → 9: 89 Registres sísmics → 8: 41 Regulació del DNA → 8: 155 Regulació marina → 8: 47 Relació cost-eficàcia → 7: 51 Reptes categòrics → 7: 51 Resistència adquirida → 9: 67 Resistència intrínseca → 9: 67 Responsabilitat social → 8: 167 Restes arqueològiques → 7: 21 Revista de divulgació científica → 9: 177 Revista Ibérica → 9: 159 Revistes científiques → 9: 141 Ribosoma → 7: 125 Romanticisme → 9: 75 Saccharomyces cerevisiae → 7: 101 Salut global → 8: 175 Salvia → 8: 77 Sediments lacustres → 7: 21 Sensor CCD → 7: 117 Simbiosi → 9: 51 Síntesi proteica → 7: 125 Sismologia → 8: 41 Sistema climàtic → 7: 27 Sistema Latindex → 9: 151 Sistemes de salut → 8: 139 Sistemes socioecològics → 8: 11 SMOS → 9: 89 Societat del coneixement → 7: 51 Sostenibilitat → 7: 11, 37; 8: 11, 149 Sostenibilitat global → 7: 51 Supernova → 8: 69 Surrealisme → 7: 155 Susceptibilitat de l’hoste → 9: 51

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Tapissos microbians → 9: 117 Tècniques de reproducció assistida → 7: 149 Telecomunicació → 7: 109 Telomerasa → 7: 101 Telòmers → 7: 101 Temperament musical → 9: 75 Temperatura i salinitat de la superfície marina → 9: 89 Teoria de Gaia → 7: 17 Teràpia antiangiogènica → 9: 67 Teràpia d’observació directa de breu duració (DOTS) → 8: 181 Terratrèmols → 8: 41 Tessel·lacions → 9: 17 Tetrahymena thermophila → 7: 101 TLR → 8: 61 Toll → 8: 61 Transferència de càrrega → 7: 117 Tuberculosi extremament resistent (XDR-TB) → 8: 181 Tuberculosi multiresistent (MDR-TB) → 8: 181 Vacants → 7: 163 Vaixells d’observació d’oportunitat → 9: 89 Valors dels serveis sòcio-ambientals → 7: 51 Xarxa Europea d’Assessorament Tecnològic als Parlaments (EPTA) → 8: 121, 131 Xarxes de regulació gèniques → 9: 57 Xarxes genètiques → 8: 155 Xarxes òptiques → 7: 117 Xarxes socials → 8: 33 Xile → 8: 11 Zones costaneres → 8: 53

CONTRIBUTIONS to SCIENCE 11 (2015) XX-XX

OPENAACCESS

Institute for Catalan Studies

The Institute for Catalan Studies (IEC), academy of sciences and humanities, founded in 1907, is the top academic corporation of the territories of Catalan language and culture, and has been a full member of the International Academic Union since 1922. The IEC has 186 full or emeritus members from throughout the linguistic territory, and 72 corresponding members that represent our institution’s relations with the international scientific community, and has 28 filial societies of all fields of knowledge, with a total membership of around 10,000 across the whole territory. In addition, 111 local research centres also belong to it, and this shows how well grounded the research community is, throughout our cultural territory. The IEC is the central institution in the Catalan cultural world. It was set up in 1907 at the initiative of the Diputació de Barcelona to “establish here scientific study centres specialising and working not just in education, but in producing science and aiding research.” In the following years, the Institute set up its various science departments. The Philology Department, directed by Pompeu Fabra, played a key role in establishing the rules of the Catalan language.

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