Phd Thesis: phase equilibria in polymer systems by paricaud

UNDERSTANDING THE FLUID PHASE BEHAVIOUR OF POLYMER SYSTEMS WITH THE SAFT THEORY

PATRICE PARICAUD

Ph.D. IMPERIAL COLLEGE OF LONDON 2003

UNDERSTANDING THE FLUID PHASE BEHAVIOUR OF POLYMER SYSTEMS WITH THE SAFT THEORY

Patrice Paricaud

January 24, 2003

Department of Chemical Engineering and Chemical Technology Imperial College of Science, Technology, and Medicine London SW7 3BY A thesis submitted for the degree of Doctor of Philosophy of the University of London and for the Diploma of Membership of Imperial College

To Sandra and my family.

Abstract

Fluid phase equilibria in polymer systems are of crucial practical importance for polymer process design. In this work, a theoretical study is first undertaken for model molecules. The features governing the phase equilibria in purely repulsive systems of hard spheres and hard sphere chains are first examined, using a simple description based on the Wertheim TPT1 theory. For hard spheres of the same size as the segments of the chain, the system is fully miscible. When the hard spheres are made larger than the chain segments, the system demixes into low and high-density regions. This phase separation is driven by depletion interactions which corresponds to an effective attraction between the large spherical particles. The mechanism of phase separation in polymer solutions is completely different. For simple systems in which the size and energy parameters of the spherical molecules and the chain segments are all equivalent, a region of liquid-liquid coexistence is found when the chain length is increased, and the system exhibits a lower critical solution temperature (LCST). The nature of LCST behaviour is studied using the Wertheim TPT1 description and a mean-field treatment of the attractions. The effect of chain length polydispersity is also examined using continuous thermodynamics, and the results are compared with those for a discrete ternary mixture. Real systems of alkanes and polyethylene have been also examined, using the SAFT-VR theory. The n-alkanes and polymer molecules are modelled as chains of spherical segments. The segment-segment interactions are characterized by a squarewell potential. The phase equilibria of polyethylene solutions can be understood as an extreme case of a binary mixture of a short and a very long n-alkane. Appropriate comparisons are made with experiments in the discussion and excellent agreement is found for adsorption of gases in polyethylene, and cloud curves.

Acknowledgments

The person that I am most grateful to is probably Prof. George Jackson. I would like first to thank him for welcoming me, and helping me so much to set up in London. He has also brought me a lot knowledge in Statistical Mechanics and in many other topics. I thank him for his great generosity, enthusiasm and friendship. I would like also to thank very much Dr Amparo Galindo for her precious help, for all her encouragement and kindness. I thank both George and her for letting me participating to international conferences, and for all the delicious meals at ”La Bouchee” and ”de Mario”. I also thank Dr Szabolcs Varga very much, for his clever advices, efficiency at work, and especially for his friendship and the unforgettable holidays I spent with his family in Hungary. I am very grateful to my colleague Guy Gloor, for his permanent good mood, for having helped me a lot in computing, and above all for having saved my life by repairing my computer which was about to die just before the end of this PhD. I would like to thank Mario Franco and Birju Patel for all the good times spent in the office or around a glass, and my great Mexican flatmates: Alex, Javier, and Carlos, for their joy of life and the great Latin parties. I would like also thank the Modelling Research Group of BP Chemicals based in Lavera for funding a studentship. I would like to end with some French words addressed to my relatives. Je voudrais remercier de tout mon coeur mes parents a ` qui je dois tout, ma grand-mère, ma soeur Isabelle et son mari Jérome, Jean-Louis, et toute la famille Rakoto, pour avoir toujours été l` a, m’avoir soutenu et encouragé dans les moments difficiles. Enfin, je remercie Sandra mon épouse, pour son amour, sa gentillesse, son attention, et pour toute la joie de vivre qu’elle apporte a ` ma vie.

Contents 1 Introduction

1.1

Fluid Phase Equilibria in Pure Component Systems . . . . . . . . .

1.2

Fluid Phase Equilibria in Binary Mixtures . . . . . . . . . . . . . . .

1.2.1

Type I Phase Behaviour . . . . . . . . . . . . . . . . . . . . .

1.2.2

Type II Phase Behaviour . . . . . . . . . . . . . . . . . . . .

1.2.3

Type V Phase Behaviour . . . . . . . . . . . . . . . . . . . .

1.2.4

Type IV Phase Behaviour . . . . . . . . . . . . . . . . . . . .

1.2.5

Type III Phase Behaviour . . . . . . . . . . . . . . . . . . . .

1.2.6

Type VI Phase Behaviour . . . . . . . . . . . . . . . . . . . .

1.2.7

Gibbs Phase Rule and Properties of Mixing . . . . . . . . . .

Phase Behaviour in Polymer-Solvent Systems . . . . . . . . . . . . .

1.3.1

Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.2

Binodal and spinodal curves . . . . . . . . . . . . . . . . . . .

1.3.3

The Effect of Polydispersity . . . . . . . . . . . . . . . . . . .

Equations of State for Polymer Melts and Polymer Solutions . . . .

1.4.1

The Flory-Huggins-Staverman Theory . . . . . . . . . . . . .

1.4.2

Compressible Lattice Models . . . . . . . . . . . . . . . . . .

1.4.3

Continuous Systems: Tangent Spheres Models

. . . . . . . .

1.5

Chain Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.6

Remaining Questions and Theme of the Thesis . . . . . . . . . . . .

1.3

1.4

2 LCST in Polymer Solutions 2.1

Wertheim Association Theory . . . . . . . . . . . . . . . . . . . . . .

2.1.1

TPT1 Theory for Chain Formation and Polymers . . . . . . .

2.1.2

Chain Contribution Derived from the Cavity Function . . . .

2.2

SAFT-HS Theory for Polymer Solution . . . . . . . . . . . . . . . .

2.3

Phase Diagrams in Attractive Hard Spheres and Chain Binary Mixtures 91 4

CONTENTS 2.4

Properties of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4.1

Ideal Gas Mixture . . . . . . . . . . . . . . . . . . . . . . . .

2.4.2

Polymer Solution Modelled with Flory-Huggins Theory . . . 100

2.4.3

Polymer Solution Described by TPT1 Theory . . . . . . . . . 102

2.5

Density Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

2.6

Associating Hard Spheres . . . . . . . . . . . . . . . . . . . . . . . . 112 2.6.1

Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 112

2.6.2

Properties of Mixing . . . . . . . . . . . . . . . . . . . . . . . 113

2.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

2.8

Appendix A: Algorithm to Solve LLE in Polymer Solution . . . . . . 119

2.9

Appendix B: Mixture of Associating Hard Spheres . . . . . . . . . . 123

3 Demixing in Colloids + Polymer Systems

126

3.1

The Depletion effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

3.2

Wertheim TPT1 Approach for Colloid-Polymer Systems . . . . . . . 134

3.3

Spinodal and Binodal Curves . . . . . . . . . . . . . . . . . . . . . . 138

3.4

Properties of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3.5

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4 The Effect of Polydispersity In Polymer Systems 4.1

4.2

4.3

4.4

151

Polymer-Solvent System: Discrete Distribution . . . . . . . . . . . . 156 4.1.1

Density Free Energy . . . . . . . . . . . . . . . . . . . . . . . 159

4.1.2

Chemical Potentials . . . . . . . . . . . . . . . . . . . . . . . 163

4.1.3

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.1.4

Cloud and Shadow Curve Calculation . . . . . . . . . . . . . 165

Polymer-Solvent Systems: Continuous Distributions . . . . . . . . . 168 4.2.1

Density Free Energy . . . . . . . . . . . . . . . . . . . . . . . 169

4.2.2

Chemical Potential and Pressure . . . . . . . . . . . . . . . . 171

4.2.3

Cloud and Shadow Curve Calculation . . . . . . . . . . . . . 172

Moment Method Applied to Polymer-Solvent Systems . . . . . . . . 174 4.3.1

Moment Free Energy . . . . . . . . . . . . . . . . . . . . . . . 174

4.3.2

Moment Chemical Potentials . . . . . . . . . . . . . . . . . . 178

4.3.3

Cloud and Shadow Curve . . . . . . . . . . . . . . . . . . . . 179

Ternary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.4.1

Cloud and shadow curves . . . . . . . . . . . . . . . . . . . . 182

CONTENTS

4.4.2

Ternary Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 185

4.4.3

Cloud and Shadow Curves Obtained with Schulz-Flory Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

4.5

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

5 Polyethylene + Hydrocarbon Systems 5.1

5.2

5.3

5.4

5.5

5.6

198

SAFT-VR Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.1.1

Molecular Model . . . . . . . . . . . . . . . . . . . . . . . . . 203

5.1.2

Calculation of the Helmholtz Free Energy . . . . . . . . . . . 205

5.1.3

Combining Rules . . . . . . . . . . . . . . . . . . . . . . . . . 210

Modelling of Pure Components . . . . . . . . . . . . . . . . . . . . . 211 5.2.1

n-Alkanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

5.2.2

Îą-Olefins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5.2.3

Other Pure Compounds . . . . . . . . . . . . . . . . . . . . . 224

Modelling n-Alkane + linear Polyethylene Systems . . . . . . . . . . 227 5.3.1

Pentane + Polyethylene . . . . . . . . . . . . . . . . . . . . . 227

5.3.2

Influence of the Polymer Parameters on Cloud Curves . . . . 230

5.3.3

Other alkanes + Polyethylene . . . . . . . . . . . . . . . . . . 234

Solubility of Gases in Amorphous Polyethylene . . . . . . . . . . . . 235 5.4.1

Small Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

5.4.2

Solubility of Hydrogen in Polyethylene . . . . . . . . . . . . . 239

Crystallinity of Polyethylene . . . . . . . . . . . . . . . . . . . . . . . 243 5.5.1

Experimental Measurements of Crystallinity . . . . . . . . . . 244

5.5.2

Floryâ&#x20AC;&#x2122;s Theory of the Fusion Behaviour of Copolymers . . . . 246

5.5.3

Modelling of Polyethylene Crystallinity . . . . . . . . . . . . 252

Solubility of Gases in Semi-Crystalline Polyethylene . . . . . . . . . 257 5.6.1

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

5.6.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

5.7

Effects of co-absorption . . . . . . . . . . . . . . . . . . . . . . . . . 261

5.8

Conclusion

6 Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 266

List of Figures 1.1

Schematic representation of a pressure-temperature P T diagram of a pure component. The continuous boundary represents the vapourpressure curve, the long-dashed boundary represents the melting curve, and the dotted boundary represents the sublimation curve. The circle denotes the vapour-liquid critical point and the square denotes the triple point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

The six main types of behaviour for binary mixtures shown as pressuretemperature P T projections of the phase diagram. The continuous curves represent the vapour pressure curves of pure components 1 and 2 and the three phase lines. The dashed curves represent critical lines. The triangles are used to represent upper and lower critical end points, and the circles the critical points of the two pure components

1.3

Pressure-temperature P T projection of the phase diagram of a type I binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components. The dashed curves represent critical lines. The circles denote the critical points of the pure components. 30

1.4

a) Pressure-composition P x and b) temperature-composition T x slices of the phase diagrams obtained for a type I binary mixture. The temperatures (T1 , T2 , T3 ) and pressures (P1 , P2 ) correspond to slices shown on figure 1.3. The continuous curves represent coexistence boundaries. The circles denote vapour-liquid critical points. The dashed-dotted line represents a tie line.

. . . . . . . . . . . . . . . .

LIST OF FIGURES 1.5

Pressure-temperature P T projections of the phase diagram characteristic of a system exhibiting type II phase behaviour. The continuous curves represent the vapor-pressure curve of the two pure components and the three-phase line. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangle denotes the UCEP. . . . . . . . . . . . . . . . . . .

1.6

a) Pressure-composition P x (at temperature T1 ) and b) temperaturecomposition T x (at pressure P1 ) slices of the phase diagrams obtained for a type II binary mixture. The regions of vapour-liquid and liquidliquid equilibria are labelled on the figure. The continuous curves represent the coexistence curves. The circles denote critical points (vapour-liquid and UCST) . . . . . . . . . . . . . . . . . . . . . . . .

1.7

Pressure-temperature P T projections of the phase diagram of a type V binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent critical lines. The circles denote the critical points of the pure components, and the triangles denote the UCEP and LCEP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.8

Pressure-composition P x slices of the phase diagram at a) a temperature T1 , and b) a temperature T2 , corresponding to a type V binary mixture. The continuous curves represent the coexistence curves and three phase lines. The circle denotes a critical point (UCSP), the triangle denotes the UCEP. . . . . . . . . . . . . . . . . . . . . . . .

1.9

Pressure-temperature P T projection of the phase diagram of a type IV binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase lines. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangles denote the UCEPs and LCEP. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.10 Pressure-composition P x slices of the phase diagram at constant temperatures a) T1 , and b) T2 of a type IV binary mixture. The continuous curves represent coexistence curves and three-phase lines. The circles denote critical points (vapour-liquid and UCSP). The triangle denotes a UCEP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

LIST OF FIGURES 1.11 Temperature-composition T x slice of the diagram at a constant pressure P1 of a type IV binary mixture. The continuous curves represent coexistence curves and the three-phase line. The circles denote critical points (vapour-liquid, LCST and UCST). . . . . . . . . . . . . . . .

1.12 Pressure-temperature P T projections of the phase diagram of type III. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangle denotes the UCEP. . . . . . .

1.13 Temperature-composition T x slice of the phase diagram at a): a pressure P1 , and b) a pressure P2 , corresponding to a type III behaviour. The continuous curves represent the coexistence curves and the threephase line. The circles denote critical points (vapour-liquid, LCST and UCST). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.14 Pressure-temperature P T projections of the phase diagram of type VI binary mixtures. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent critical lines. The circles denote pure component critical points, and the triangles denote the LCEP and UCEP. 39 1.15 Temperature-composition T x slice of the phase diagrams (at a pressure P1 ) obtained for a type VI binary mixture. The continous curves represent the coexistence curves. The circles denote critical points (vapour-liquid, LCST and UCST). . . . . . . . . . . . . . . . . . . .

1.16 Molar Gibbs free energy of mixing for a binary mixture at given T and P , as a function of the mole fraction x of one of the two component. The points A and D represent the binodal points, and B and C the spinodal points. The dotted line corresponds to the common tangent, and the dotted-dashed lines denote the compositions of the two phases Îą and Î˛ in coexistence.

. . . . . . . . . . . . . . . . . . . . . . . . .

1.17 Schematic representation of different kinds of polyethylene. a) High Density Polyethylene (HDPE), b) Low Density Polyethylene(LDPE), c) Linear Low Density Polyethylene (LLDPE) . . . . . . . . . . . . .

1.18 Atomic representation of some polyolefines (from Lipson et al. [27]). Each black circle represents a carbon atom. . . . . . . . . . . . . . .

LIST OF FIGURES

1.19 Temperature-composition T x slices of the phase diagrams of a) a type IV binary mixture corresponding to figure 1.9, and b) a type IV monodisperse polymer solution. The symbols X and W refer to mole and weight fractions. The continuous curves represent coexistence curves and three-phase lines. The circles denote critical points. The dashed-dotted lines represent the limits of the T x diagram. . . . . .

1.20 Pressure-composition P w slice of the phase diagram of a polymer solution in the case of monodisperse polymer. The white circle denotes the critical point (UCSP). The black circles represent the binodal points 50 1.21 Pressure-composition P w phase diagram of a polydisperse polymer solution, where the global composition of the sample W0 is higher than the critical composition. The white circle denotes the critical point. The black circles represent the coexistence points. The thick continuous curve represents the cloud curve. The dashed line is the shadow curve. The thin continuous curves represent the coexistence curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.22 Pressure-composition P w phase diagram of a polydisperse polymer solution, where the global composition of the sample W0 is lower than the critical composition. The white circle denotes the critical point. The black circles represent the coexistence points. The thick continuous curve represents the cloud curve. The dashed line is the shadow curve. The thin continuous curves represent the coexistence curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.23 Lattice model used in the Flory-Huggins theory to represent a mixture of chain molecules of 10 segments and a monomeric solvent. The white circles denote solvent units. The black circles denote polymer units. 1.24 Polymer molecule and the end-to-end distance RE . . . . . . . . . . 2.1

55 66

Top: Schema of an associating sphere with one site. Bottom: mixture of associating spheres and formation of chains . . . . . . . . . . . . .

LIST OF FIGURES 2.2

P T pressure-temperature diagrams for binary mixtures of hard spheres + chains of m2 tangent hard spheres of same diameter, and same mean field energy per segment, with a) m2 = 2, m2 = 6, b) m2 = 7, and c) m2 = 100. The continuous lines are the vapour-pressure curves of the pure components and the three phase line. The dashed lines are critical lines. The white circles are the critical points of the pure components. The white triangles are LCEP and UCEP. . . . . . . .

2.3

Global phase diagram for the binary mixture of two hard sphere chains of respectively m1 and m2 segments. The solid line delimits regions of the diagram where different types of phase behaviour are encountered. It represents all the type V binary mixtures for which the critical point of the pure shorter chain, LCEP and UCEP are confused. . . . . . .

2.4

P T pressure-temperature diagram for a binary mixture of hard spheres + chains of 10 tangent hard spheres of same diameter and mean field energy per segment (system (4)). The continuous lines are vapourpressure curves of the pure components. The dashed lines are critical lines. The white circles are critical points of the pure components. The white triangles are LCEP and UCEP joined by the 3-phase line. The dash-dot lines denote constant temperature or pressure slices. .

2.5

T x temperature-composition diagrams, corresponding to the P T diagram of figure 2 obtained for the binary mixture of hard spheres + chains of 10 tangent hard spheres with mean field attractions (see figure 2), where xs,2 is the fraction of polymer segments. The considered pressure slices are at P ∗ = a) P1∗ , b) P2∗ , and c) P3∗ . The continuous lines are coexistence curves. The white circles are critical points (UCST and LCST) of the binary mixture. . . . . . . . . . . .

2.6

Enlarged T x temperature-composition diagram corresponding to the T x diagram at pressure P2∗ shown in figure 3 b), obtained for the binary mixture of hard spheres + chains of 10 tangent hard spheres with mean field attractions. The continuous lines are coexistence curves. The white circles are critical points (UCST and LCST) of the binary mixture. The dashed line is the three phase line. The dashdot line denotes a constant temperature slice at T2∗ in the liquid-liquid immiscibility region. . . . . . . . . . . . . . . . . . . . . . . . . . . .

LIST OF FIGURES 2.7

Reduced Gibbs free energy of mixing and its second derivative obtained for different model systems at constant pressure P2∗ and temperatures T1∗ (a), c)) and T2∗ (b), d)). The thin dash line corresponds to an ideal gas mixture (system(1)). The thick dash line corresponds to a mixture of one monomeric solvent and a polymer of 10 segments, and is obtained from Flory-Huggins theory (system (2)). The thin continuous line corresponds to a binary mixture of hard spheres and a chain of 10 tangent hard spheres modelled with SAFT-HS (system (3)). The thick continuous line corresponds to the same binary mixture as system (3) but with mean field segment-segment attractions (system (4)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

2.8

Reduced properties of mixing (a) internal energy, b) enthalpy, c) entropy, d) molar volume) obtained for a binary mixture of hard spheres and a chain of 10 tangent hard spheres with mean field attractions (system (4)), at constant pressure P2∗ and at temperature T1∗ in the stable region, and at temperature T2∗ in the liquid-liquid immiscibility region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

2.9

a) Reduced density obtained for system (4) at constant pressure P 2∗ and temperatures T1∗ and T2∗ . b) Isotherms obtained for the same system at temperature T2∗ in the demixing region, for different chain mole fractions x2 : the thick continuous line , and white triangles, correspond to x2 = 0 (pure solvent), the thin continuous line, and white diamonds, correspond to x2 = 0.01, and the dashed line and white squares correspond to x2 = 0.05. . . . . . . . . . . . . . . . . . 108

2.10 a) repulsive and b) attractive parts of the pressure, at temperature T 2∗ , corresponding to the isotherms shown in figure 7 b) for different chain mole fractions x2 : the thick continuous lines correspond to x2 = 0 (pure solvent), the thin continuous lines correspond to x2 = 0.01 and the dashed line correspond to x2 = 0.05.

. . . . . . . . . . . . . . . 109

2.11 Schematic representation the binary mixture of hard spheres and chains to represent the evolution of the density as a function of the composition, at fixed temperature and pressure. Grey spheres are solvent molecules and black spheres are chain segments. . . . . . . . 111

LIST OF FIGURES 2.12 T x temperature-composition diagram at pressure P4∗ for a binary mixture of equal-size hard spheres, with mean field attractions α11 = α22 , α12 = 0, and unlike site-site association. The continuous line is the liquid-liquid coexistence curve as a function of the mole fraction x2 of component 2, the dash line is the total mole fraction XT of bonded molecules in the coexistent phases with respect to temperature. The white circles are UCST and LCST.

. . . . . . . . . . . . . . . . . . 113

2.13 Reduced properties of mixing (a) internal energy, b) enthalpy, c) entropy, d) molar volume) obtained for a binary mixture of equal-size hard spheres, with mean field attractions α11 = α22 , α12 = 0 , and unlike site-site association, at constant pressure P2∗ and temperatures T3∗ , T4∗ and T5∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 2.14 Reduced density obtained for a binary mixture of equal-size hard spheres, with mean field attractions α11 = α22 , α12 = 0, and unlike site-site association, at constant pressure P2∗ and at temperatures T3∗ , T4∗ and T5∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 2.15 Evolution of the chemical potential of component 1 at fixed temperature T and pressure P , as a function of composition x2 in a demixing binary polymer-solvent system. The circles denote binodal points, and the square denotes a metastable point. . . . . . . . . . . . . . . 123 3.1

Representation of the depletion effect in colloid-polymer system. Colloidal particles are represented by grey spheres, and polymer segments are represented by black spheres.The centers of mass of the polymer coils of diameter 2Rg are excluded from a volume represented by a sphere of diameter 2 (Rg + RC ) (dotted-dashed line), for each colloidal particles of diameter σC = 2RC ( see figure a)). However, when the colloid particles are close to each other (see figure b)), the excluded volumes related to the colloids overlap, the total excluded volume is decreased, and the coils particles can then move in a larger volume. The overlapping regions are represented in black. In the SAFT (TPT1) theory, the polymer chain is modelled as a fully flexible chain of segments of diameter σP << σC . . . . . . . . . . . . . . . . . . . 133

LIST OF FIGURES 3.2

The pressure-composition representation of the spinodal curves of the fluid-fluid phase equilibria for athermal binary mixtures of colloids (hard spheres of diameter σC ) and flexible polymers (chains formed from tangent hard spheres of diameter σP ) determined from the Wertheim TPT1 approach. Results are presented for systems with a fixed polymer-segment to colloid diameter ratio of d = σP /σC = 0.06 for polymer chain lengths ranging from m = 15 up to 1000. The 3 /kT , and x represents the reduced pressure is defined as P ∗ = P σC P

mole fraction of the polymers. . . . . . . . . . . . . . . . . . . . . . . 140 3.3

The critical packing fraction of the fluid-fluid phase equilibria for athermal binary mixtures of colloids (hard spheres of diameter σC ) and flexible polymers (chains formed from tangent hard spheres of diameter σP ) determined from the Wertheim TPT1 approach. Results are presented for systems with varying polymer-segment to colloid diameter ratio of d = σP /σC = 0.06 and polymer chain length m. The packing fraction is defined as η = πρ∗ 1 − xP + xP md3 /6. . . . . . 141

3.4

The dependence of the critical properties on the polymer-segment to colloid diameter ratio d = σP /σC for athermal binary mixtures of colloids (hard spheres of diameter σC ) and flexible polymers (chains formed from tangent hard spheres of diameter σP ) determined from the Wertheim TPT1 approach. The critical packing fraction (continuous curves) and composition (dashed curves) are shown in a), and the critical pressure in b) for selected values of the polymer chain length m. η = πρ∗ 1 − xP + xP md3 /6 and , and xP represents the ¡

mole fraction of polymer. . . . . . . . . . . . . . . . . . . . . . . . . 143 3.5

Maximum value of diameter ratio d = σP /σC for the demixing transition of athermal binary mixtures of colloids (hard spheres of diameter σC ) and flexible polymers (chains formed from tangent hard spheres of diameter sP) determined from the Wertheim TPT1 approach as a function of the chain length m. The term I-I demixing is used to denote a phase separation between to isotropic fluid states. . . . . . 145

LIST OF FIGURES 3.6

The temperature-composition phase diagram of the fluid-fluid phase equilibria for an athermal binary mixture of colloids (hard spheres of diameter σC ) and flexible polymers (chains formed from tangent hard spheres of diameter σP ) determined from the Wertheim TPT1 approach. The parameters characterizing the system are a polymersegment to colloid diameter ratio of d = σP /σC = 0.06 and a polymer chain length of m = 500. The reduced temperature is defined as the 3 , and x reciprocal of the reduced pressure T ∗ = 1/P ∗ = kT / P σC P

represents the mole fraction of the polymers. . . . . . . . . . . . . . 146 3.7

The thermodynamic properties of mixing for an athermal binary mixture of colloids (hard spheres of diameter σC ) and flexible polymers (chains formed from tangent hard spheres of diameter σP ) determined from the Wertheim TPT1 approach. The parameters characterizing the system are a polymer-segment to colloid diameter ratio of d = σP /σC = 0.06 and a polymer chain length of m = 500. ∗ = ∆G /N kT , b) entropy The reduced a) Gibbs free energy ∆gm m

∆s∗m = ∆Sm /N k , c) enthalpy ∆h∗m = ∆Sm /N kT , and d) volume ∗ = ∆V /σ 3 of mixing are plotted for mixed (P ∗ = 0.8) and a ∆vm m C

demixed (P ∗ = 1.2) states; the ideal entropy of mixing is also shown 3 /kT , as a dashed curve. The reduced pressure is defined as P ∗ = P σC

and xP represents the mole fraction of polymer. . . . . . . . . . . . . 148 4.1

Continuous chain length distribution function in terms of weight fraction W , and pseudocomponents, from Browarzik et al. [263] . . . . . 153

4.2

Discrete parent chain length distribution functions in terms of weight fraction, for polymer (1) (black bars), and polymer (2) (white bars).

182

LIST OF FIGURES 4.3

Cloud and shadow curves calculated with the SAFT-HS equation for the ternary system solvent + polymer (1) (chain m2 = 66.7 + chain m3 = 400). The dashed dotted curves are binodal curves of the binary systems: a) solvent + chain m = 66.7 , b) solvent + chain m = 100, and c) solvent + chain m = 400. The thick continuous curve represents the cloud curve of the ternary system, and the dotted curve represents the shadow curve. The circles denote the critical points (LCST); wpoly = w2 + w3 = 1 − w1 is the total polymer weight fraction where w1 , w2 and w3 are the weight fractions of the solvent, and the chain molecules 2 and 3, respectively. . . . . . . . . . . . . . 183

4.4

Cloud and shadow curves calculated with the SAFT-HS equation for the ternary system solvent + polymer (1) (chain m2 = 66.7 + chain m3 = 400). The think continuous curve represents the cloud curve of the ternary system, and the dotted curve represents the shadow curve. The circles denote the critical points (LCST); wpoly = w2 + w3 = 1 − w1 is the total weight fraction of polymer, where w1 , w2 and w3 are the weight fractions of the solvent, and the chain molecules 2 and 3, respectively. The dashed and dotted lines represent constant temperature slices corresponding to the ternary diagrams depicted in figure 4.5. The white squares denote cloud points and the black squares are the corresponding shadow points. . . . . . . . . . . . . . 185

4.5

Ternary diagrams in weight fractions, at constant pressure P ∗ = 0.001, and temperatures a) T1∗ = 0.0839, b) T2∗ = 0.0794, and c) T3∗ = 0.0787) corresponding to the cloud and shadow curves of the ternary system solvent + polymer (1) (chain m2 = 66.7 + chain m3 = 400) shown in figure 4.4. The thick and continuous curves represent the coexistence curves in the ternary mixture, and the circles denote the critical point (LCST). The thin and continuous lines are tie lines. The dashed and dotted lines represent a fixed ratio of composition between chain 2, and chain 3, corresponding to the parent distribution W (m2 )(0) = 0.6, W (m3 )(0) = 0.4. The white squares denote cloud points and the black squares are the corresponding shadow points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

LIST OF FIGURES 4.6

Cloud and shadow curves calculated with the SAFT-HS equation for the ternary system solvent + polymer (2) (chain m2 = 88.8 + chain m3 = 1000). The think continuous curve represents the cloud curve of the ternary system, and the dotted curve represents the shadow curve. The circles denote the critical points (LCST). The white square is a three phase point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

4.7

Number average chain length in the shadow phase at fixed pressure P â&#x2C6;&#x2014; = 0.001, as a function of the polymer weight fraction in the shadow phase, for the ternary systems solvent + polymer (1), and solvent + polymer (2). The circles denote critical points, and the square denotes a three phase point. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

4.8

Parent Schulz-Flory distributions in terms of weight fractions with a number-average chain length hmi(0) n = 100 and polydispersity indexes Ip = 2, Ip = 15. The corresponding cloud and shadow curves are depicted in figure 4.9. . . . . . . . . . . . . . . . . . . . . . . . . . . 194

4.9

Cloud (continuous) and shadow (dotted) curves for the system solvent + polydisperse polymer, obtained with the Schulz-Flory distributions depicted in figure 4.8 and various polydispersity indexes. The dashed-dotted curve represent the binodal obtained for a monodisperse polymer-solvent system (Ip = 1). The circles denote critical points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

4.10 Number average chain length in the shadow phase, as a function of polymer weight fraction in the shadow phase, at fixed pressure P â&#x2C6;&#x2014; = 0.001, calculated for the system solvent + polydisperse polymer by using a Schulz-Flory feed distribution with various polydispersity indexes. The circles denote critical points. The number-average chain length of the feed distribution is hmin(Cl) = 100 for both cases, and the two curves correspond to those in figure 4.9 . . . . . . . . . . . . 195

LIST OF FIGURES

4.11 Cloud (continuous) and shadow (dotted) curves for the systems solvent + polydisperse polymer by using a Schulz-Flory feed distribution (thick lines), and for the ternary system solvent + polymer (1) (thin curves). The circles denote critical points. In both systems, the number-average chain length of the feed distribution is hmin(Cl) = 100, and the index of polydispersity is Ip = 2. . . . . . . . . . . . . . . . . 196 5.1

Potential models which can be used in the SAFT-VR theory to model segment-segment interactions. ² and λ are the depth and the range of the potential, respectively. . . . . . . . . . . . . . . . . . . . . . . 204

5.2

General molecular model used in the SAFT theory. In this case, the molecules comprised five spherical segments of diameter σ with four association sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

5.3

Examples of models used for common molecules in the SAFT theory. The n-alkanes are represented by a flexible chain of segments without association sites. Water is represented by a spherical core with 4 sites to represent the hydrogen bonds. . . . . . . . . . . . . . . . . . . . . 205

5.4

Saturated densities a) and vapour pressures b) for methane: comparison between experimental data (circles) from reference [382] with the SAFT-HS (thin lines) and SAFT-VR (thick lines) theories after fitting of parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

5.5

Vapour pressures of the n-alkanes from C1 to C8 , compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) vapour pressure curve. b) Clausius-Clapeyron representation. . . 214

5.6

Vapour-liquid coexistence curves of the n-alkanes from C1 to C8 compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) coexistence volumes. . . . . . 215

5.7

Vapour-liquid coexistence curves of the n-alkanes from C12 to C28 , compared with the SAFT-VR predictions. The circles represent the experimental data [373,383], the continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) vapour pressures in Clausius-Clapeyron representation. . . . . . . . . . . . . . . . . . . . 216

LIST OF FIGURES 5.8

SAFT-VR parameters: a) λ ; b) σ; c) and ² of the n-alkanes as a function of the molecular weight, obtained after fitting. Simple linear correlations have been used to extrapolate the parameters to longer alkanes and polyethylene. The circles denote the values of the parameters fitted on vapour pressures and saturated densities, and the continuous lines represent the correlations (5.32). . . . . . . . . . 220

5.9

Vapour-liquid coexistence curves obtained for the n-alkanes C16 , C24 , and C48 . The circles represent experimental data [373, 383]. The crosses represent Monte-Carlo simulation data [384]. The continuous curves correspond to the SAFT-VR predictions, and the parameters of the long n-alkanes are calculated with the correlations (5.32). . . . 221

5.10 Vapour pressures of ethylene and α-olefins (propene, 1-butene, 1hexene), compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) vapour pressure curve. b) Clausius Clapeyron representation. . . . . . . . . . . . . . . . . . . . . . . . . 222 5.11 Vapour-liquid coexistence curves of of ethylene and α-olefins (propene, 1-butene, 1-hexene) compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) coexistence volumes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5.12 Vapour pressures for CO2 , H2 O, and HF compared with the SAFTVR predictions. The circles represent the experimental data [386], [387], [388]. The continuous curves correspond to the SAFT-VR approach. a) vapour pressure curves. b) Clausius Clapeyron representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 5.13 Vapour-liquid coexistence curves for CO2 , H2 O, and HF compared with the SAFT-VR predictions. The circles represent the experimental data [386], [387], [388]. The continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) coexistence volumes. 226

LIST OF FIGURES

5.14 Pressure-temperature P T projection of the phase diagram obtained for the binary mixtures pentane + long alkanes (n-C36 ) and pentane + polyethylene(PE) of molecular weights 1, 10 and 100 kg mol−1 , predicted with SAFT-VR. The white triangles denote calculated upper and lower critical end points. The continuous curves are the vapour pressure curve of the pure compounds. The dashed curves are critical lines. The dashed-dotted lines represent the temperature range where fluid phase are stable: at temperatures below about 400K, crystallisation occurs, and at temperatures above 650K, the alkyl chains decompose. No cross interaction parameter is used (kij = 0). . . . . 228 5.15 a) Vapour-liquid equilibria (bubble point curves) obtained with SAFTVR for the mixture pentane polyethylene (LDPE, M W = 76 kg.mol−1 ), at temperatures T = 150.5◦ C and T = 201◦ C. b) Vapour liquid and liquid-liquid equilibria obtained at T = 201◦ C for the mixture npentane + polyethylene (LDPE, M W = 76kg.mol−1). W pentane is pentane weight fraction. No cross interaction parameter is used (kij = 0). The calculated vapour liquid coexistence curve (continuous curves) is compared with experimental data [389] (black circles). The while circle denotes the calculated UCSP. The dashed curves represent the three phase line. . . . . . . . . . . . . . . . . . . . . . . 229 5.16 Cloud curves calculated with SAFT-VR at several pressures for the mixture n-pentane+polyethylene (M W = 108 kg mol−1 ) and compared to experimental data [360] (circles). No cross interaction parameter is used. kij = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 232 5.17 Cloud curves calculated at pressure P = 5 MPa with SAFT-VR for the mixtures n-pentane + polyethylene(MW= 16.4 kg mol−1 ) and pentane + polyethylene(MW = 2.150 kg mol−1 ), and compared with experimental data [360] (circles). No cross interaction parameter is used (kij = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 5.18 Influence of the parameter ²22 of the polymer on the cloud curve calculated with SAFT-VR at pressure P = 10 MPa for the mixtures pentane + polyethylene (M W = 108 kg mol−1 ), and compared with experimental data [360]. (circles). A small change of ²22 give rise to a shift of 5 K for the LCST. . . . . . . . . . . . . . . . . . . . . . . . 233

LIST OF FIGURES 5.19 Schematic representation of the attractions between solvent-polymer and polymer-polymer segments. The attraction energy for solventpolymer interactions evolve with a factor m2 ²12 , while the attraction energy for polymer-polymer interactions evolve with a factor m22 ²22 .

233

5.20 Effect of the chain length of the n-alkane on LCST, for solutions of nalkane + polyethylene (M W = 140 kg.mol−1 ). The predictions of the SAFT-VR theory (black square) and the BYG theory (white circles) are compared with the experimental data [390], [351], [391], [392] (circles). No cross interaction parameter is used (kij = 0). . . . . . . 234 5.21 Solubility of pentane in amorphous LDPE (M W = 76 kg mol−1 ). SAFT-VR predictions (continuous curves are compared with experimental data [389] (black circles). No cross interaction parameter is used (kij = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 5.22 Solubility of pentane in polyethylene calculated with SAFT-VR at T = 150.5◦ C, for different molecular weights of the polyethylene. No cross interaction parameter is used (kij = 0). The thick line represents the absorption curve of pentane in an infinitely long and linear polyethylene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 5.23 Solubility of ethylene in amorphous LDPE (M W = 248 kg mol−1 ). Comparison of experimental data [393] (T = 126◦ C, squares, and T = 155◦ C, circles) and SAFT-VR predictions (continuous lines). One (²)

cross interaction parameter k12 = 0.075 is used for all temperatures.

238

5.24 Solubility of Nitrogen (N2 ) in amorphous HDPE (M W = 111 kg mol−1 ). Comparison of experimental data [394] (T = 160◦ C, squares, and T = 300◦ C, circles) and SAFT-VR predictions (continuous lines).One (²)

cross interaction parameter k12 = 0.15 is used for all temperatures. . 238 5.25 Pressure composition P x diagram of the binary system H2 + propane. The symbols denote the VLE experimental data from the DECHEMA series [395]: squares: T= 223 K; crosses: T = 298 K; circles: T = 348K. The continuous curves represent the predictions of SAFTVR without any cross interaction parameter (opt 1), and the dotted lines represent the predictions of SAFT-VR with the cross interaction (²)

(λ)

parameters kij = −0.0524, kij = −0.1963 (opt 2). . . . . . . . . . . 241

LIST OF FIGURES 5.26 Pressure composition P x diagram of the binary system H2 + ndecane. The symbols denote the VLE experimental data from Sebastian et al. series [396]: squares: T = 462.45 K; crosses: T = 503.35 K; circles: T = 583.45 K. The continuous curves represent the predictions of SAFT-VR without any cross interaction parameter (opt 1), and the dotted lines represent the predictions of SAFT-VR with the (²)

(λ)

cross interaction parameters kij = −0.0524, kij = −0.1963(opt 2). . 241 5.27 Pressure composition P x diagram of the binary system H2 + nhexadecane. The symbols denote the VLE experimental data from Lin et al. [397]: squares: T = 461.65 K; crosses: T = 542.25 K; circles: T = 622.85 K; triangles: T = 664.05 K. The continuous curves represent the predictions of SAFT-VR without any cross interaction parameter (opt 1), and the dotted lines represent the predictions (²)

of SAFT-VR with the cross interaction parameters kij = −0.0524, (λ)

kij = −0.1963 (opt 2).

. . . . . . . . . . . . . . . . . . . . . . . . . 242

5.28 Solubility of hydrogen in HDPE (M W = 100 kg.mol−1 ), at T = 120◦ C and T = 180◦ C. The hydrogen parameters of optimisation 1 (²)

(λ)

are used. No cross interaction parameter is used (kij = 0, kij = 0)

242

5.29 Schematic representation of the structure a semi-crystalline polymer. The crystallites have an average thickness hζi.

. . . . . . . . . . . . 244

5.30 Crystallinity of various polyethylene samples (HDPE, LDPE, LLDPE) as a function of their densities. The experimental data are from McKenna [327], Jordens et al. [328], Moore et al. [336], Starck et al. [330], and the continuous line corresponds to the equation (5.36) with ρc = 1.005 g. mol−1 and ρa = 0.862 g mol−1 . . . . . . . . . . . 245 5.31 Schematic representation of alternating and block-type copolymers. Thick lines represent crystallisable A units, and thin lines represent noncrystallisable B units. Block-like copolymers is more likely to crystallise than alternating-type copolymers, with the same mole fraction XA of crystallisable A units. . . . . . . . . . . . . . . . . . . . . . . . 247

LIST OF FIGURES

5.32 a) Melting point Tm and b) probability p that a crystallisable A unit is followed by another A units, for polyethylene samples, as a function of the experimental crystallinity wcrys,25 at 25◦ C. In figure a), the white circles denote experimental melting points (references [327,330]) for PE samples made with metallocen (Me) catalysts, and the black circles denote experimental melting points for PE samples made with Ziegler-Natta (ZN) catalysts. In figure b), the circles (white for Me, black for ZN catalysts) correspond to fitted p parameters of equation (5.60) on experimental crystallinity vs temperatures curves shown in figures 5.33 a) and b). . . . . . . . . . . . . . . . . . . . . . . . . . . 255 5.33 Crystallinity wcrys as a function of temperature T , for metallocen a) and Ziegler-Natta polyethylenes b). The symbols represent the experimental data from references [327, 330] (see the corresponding papers for the meaning of the names of the PE samples). The continuous curves represent the predictions of the modified Flory equation (5.60) where wcrys,25 is the experimental value, and Tm and p are given by the correlations (5.61) and (5.62). . . . . . . . . . . . . . . . . . . . . 256 5.34 Schematic representation of the absorption of gas in semi-crystalline polyethylene. The grey spheres represent gas molecules and black spheres represent polymer segments. The black zones denote crystallites. It is assumed that the gas only absorbs in the amorphous regions, and that the polymer molecules are either completely amorphous or totally crystallised. . . . . . . . . . . . . . . . . . . . . . . . 258 5.35 Solubility of a) 1-butene and b) 1-hexene in semi-crystalline HDPE (M W = 11.49 kg.mol−1 , wcrys,25 = 0.702) at different temperatures. Symbols represent experimental data [336] and continuous lines represent SAFT-VR predictions. . . . . . . . . . . . . . . . . . . . . . . 259 5.36 Solubility of a) 1-butene and b) 1-hexene in semi-crystalline LDPE (M W = 22.01 kg.mol−1 , wcrys,25 = 0.504) at different temperatures. Symbols represent experimental data [336] and continuous lines represent SAFT-VR predictions. . . . . . . . . . . . . . . . . . . . . . . 260 5.37 Pressure composition diagram at temperature T = 100◦ C of the binary 1-butene + 1-hexene. The circles denote experimental data [400] and the continuous lines represent SAFT-VR predictions. . . . . . . 262

LIST OF FIGURES

5.38 Solubility of 1-butene (gas 1) in amorphous HDPE (M W = 100 kg. mol−1 ) at temperature T = 150◦ C, at fixed partial pressure of 1butene PnC4 = = 0.05 MPa, as a function of the partial pressure of a second gas (nitrogen or 1-hexene). . . . . . . . . . . . . . . . . . . . 262 5.39 Ternary diagrams of the system 1-butene + 1-hexene + amorphous HDPE (M W = 100 kg. mol−1 ) calculated with SAFT-VR at temperature T = 460 K, and pressures P = 1 MPa a) and P = 3 MPa b). The continuous line represent coexistence curves. The dashed lines represent tie lines. The squares denote calculated three phase points. 263

Chapter 1

Introduction The aim of the work presented in this thesis is to understand and model the fluid phase behaviour of polymer systems including both polymer-solvent and polymercolloid mixtures, paying a special attention to polymer solutions. Polymers are molecules of high molecular weight, consisting of monomer segments linked together by covalent bonds. Before the Second World War, the economic impact of polymers was weak, but since then, the development and use of polymers has increased constantly with rate of about ten percent per year; at the moment, the annual production of polymers currently stands at over one hundred million of tons. A knowledge of fluid phase behaviour is essential for the efficient design of both production and separation in chemical processes which make a significant contribution to global economics. An example of the economic impact of simple separation processes is that in 1989 more than 40,000 distillation columns were in operation in the USA accounting for 3% of the total energy consumption. Much effort has been devoted in the last decade to develop the technologies for supercritical polymer processing. It is very difficult, expensive, and time consuming to carry out experimental measurements of fluid phase equilibria in polymer systems, which means that accurate thermodynamic models that can be used to predict the fluid phase equilibria of polymer solutions out of the range of the experimental data are essential for industrial applications. The understanding of the phase behaviour of polymer-solvent systems is central to polymer processing. For example, exothermic polymerisation is usually carried out in a single phase to facilitate heat removal and ensure good temperature control. Occasionally, however, two phase polymerisation may be useful: low-density polyethylene produced via a two phase process exhibits less polydispersity and branching, and hence, better film properties. The use of solvents is essentially in both polymer production and purification and in processing. 25

CHAPTER 1. INTRODUCTION

Polymer-solvent systems present a very rich phase behaviour, which is often not well understood. The size asymmetry of the components in polymer systems leads to a very complicated phase behaviour, even when the components are chemically similar. Polymer-colloid systems exhibit also a very interesting and complex phase behaviour, which are often completely different than that of polymer solutions. Such systems are of great industrial interest with applications in the stabilisation of colloidal systems in paint and food production and in the purification of proteins and virus. Colloids are particles which are much larger than the solvent molecules or the monomeric segments making up a polymer, but small enough to neglect the effect of sedimentation due to gravity, and to show significant Brownian motion. The study of simple model colloid-polymer mixture represent a beautiful extension of the â&#x20AC;&#x153;colloid as atomâ&#x20AC;? approach. The physical properties of colloidal systems is dominated by the so-called depletion effect of Asakura and Oosawa, [1, 2]. The addition of polymer molecules to a colloidal suspension induces an effective attraction between the colloidal particles, which in turn can give rise to a phase separation into colloid rich and colloid poor phases. Although the focus of the work presented in this thesis is mainly on understanding and predicting the fluid phase equilibria in polymer solutions, a detailed discussion on the depletion effect and the thermodynamics of polymer-colloid systems is given in chapter 3, in order to compare and clearly distinguish the behaviour of polymer-solvent and polymer-colloid systems. Before discussing the fluid phase equilibria of polymer systems, it is worth recalling some of the general features of the fluid phase equilibria of pure components and mixtures. The fluid phase behaviour of polymer-solvent mixtures turn out to be a particular case of that exhibited by mixtures of simpler, smaller molecules.

1.1

Fluid Phase Equilibria in Pure Component Systems

The usual phase behaviour of a pure component exhibiting the three states of the matter (solid, liquid and gas) is illustrated in figure 1. In this simple description orientationally and positionally ordered states such as nematic and smectic liquid crystalline phases or metastable glassy states are acknowledged but not treated. The solid, liquid and gas regions regions of stability are bounded by the phase boundary lines, along which two phases coexist. The vapour-pressure curve which

CHAPTER 1. INTRODUCTION

Figure 1.1: Schematic representation of a pressure-temperature P T diagram of a pure component. The continuous boundary represents the vapour-pressure curve, the long-dashed boundary represents the melting curve, and the dotted boundary represents the sublimation curve. The circle denotes the vapour-liquid critical point and the square denotes the triple point. represented the coexistence between a vapour and a liquid state extends from the triple point to the critical point. No upper limit for the melting curve has ever been observed; the difference in symmetry of the liquid (positionally disordered) and solid (positionally ordered) can be used an as explanation of this feature. The vapour-solid sublimation equilibrium tends to a limiting value at absolute zero temperature. The three curves meet at the triple point, where the vapour, liquid and solid phases are simultaneously at equilibrium. In this work, we are primarily interested in the fluid regions of the phase diagram which are delimited by the vapour-pressure curve. The vapour-pressure curve of a pure component is only a function of the temperature, according to the Gibbs phase rule. If the gas is compressed isothermally, at a temperature below the critical point, the pressure rises until the vapour pressure is reached and the first droplet of liquid is formed. The critical state of the fluid is the point at which the densities of the vapour and liquid become identical. The critical point is characterised by a point of inflection on the isotherm and corresponds to the thermodynamic relation

P > 0,

∂P ∂v

= 0, Tc

∂2P ∂v 2

= 0, Tc

∂3P ∂v 3

> 0, Tc

where P is the pressure, Tc the critical temperature, and v the molar volume.

(1.1)

CHAPTER 1. INTRODUCTION

1.2

Fluid Phase Equilibria in Binary Mixtures

Because of the great variety of phase behaviour exhibited in practice, it is convenient to classify the different types of phase diagrams in some way. Scott and van Konynenburg [3] have proposed a useful classification in of fluid phase equilibria into six types of behaviour, and showed that the van der Waals equation of state [4] is able to predict five of the six types of behaviour. A general discussion of the types of phase equilibria exhibited by mixtures be found in the following references: [5, 6]. A vast amount of experimental data for binary and ternary systems can, of course, be found in the literature; the reader is directed to the extensive reviews of Dohrn et al. [7, 8]. The phase diagrams presented in this section (from figure 1.2 to figure 1.15) do not represent actual experimental data or calculations, but just schematic representations used to exemplify the different types of behaviour. Similar figures corresponding to the phase behaviour of model systems and real mixtures will be presented in subsequent chapters. The six main types of fluid phase behaviour in binary mixtures are summarised as pressure-temperature P T diagrams in figure 1.2.

Figure 1.2: The six main types of behaviour for binary mixtures shown as pressuretemperature P T projections of the phase diagram. The continuous curves represent the vapour pressure curves of pure components 1 and 2 and the three phase lines. The dashed curves represent critical lines. The triangles are used to represent upper and lower critical end points, and the circles the critical points of the two pure components Note that azeotropic behaviour is not considered in this discussion. In order to explain the progression from one type of phase behaviour to another, the different

CHAPTER 1. INTRODUCTION

types of phase behaviour are discussed in the following order: type I, II, V, IV, III, and VI). In the thesis we use the term liquid to denote a dense fluid phase and the term gas to denote a low-density fluid; vapour is used to refer to a gas phase in coexistence with a liquid phase.

1.2.1

Type I Phase Behaviour

In the case of type I fluid phase behaviour, the gas-liquid critical line is seen to be continuous, and there is no liquid-liquid immiscibility. The two components tend to be similar in size, are of the same chemical type, and have similar critical properties in such systems. Some examples of mixtures which exhibit type I phase behaviour include argon + krypton, methane + ethane, methane + nitrogen, and carbon dioxide + oxygen. The components giving rise to type I behaviour in mixtures are usually non-polar. Mixtures of chemically similar substances of the same chemical series, such as the n-alkanes, deviate from type I phase behaviour when the size of the two compoenents become significantly different. In the case of binary mixtures of methane with n-alkanes, type I is observed for methane + ethane, + propane, + n-butane and + n-pentane. A change from type I to type V phase behaviour is first observed for methane + n-hexane [5, 9]. With ethane as the more volatile component, the transition from type I to Type V first occurs in the mixture with n-nonodecane. In the case of propane, the transition occurs in the range n-C 40 and n-C50 but the phase diagrams are complicated with a appearance of solid phases as the freezing point of the heavier component moves to higher temperatures. Type V fluid phase behavior is discussed in greater detail in section 1.2.3. Typical pressure-composition P x and temperature-composition T x slices of the fluid phase equilibria of a type I binary mixture (figure 1.3) are shown in figure 1.4. The continuous curves represent the vapour-liquid coexistence boundaries. In the case of the P x slice at the temperature T3 , component 1 is supercritical and the system exhibits a vapour-liquid critical point. The two components are supercritical at the pressure P2 shown on T x sections and the system exhibits two vapour-liquid critical points. The Gibbs phase rule (see section 1.2.7) requires that in a binary mixture the number of degree of freedom is 2 when two phases coexist. Therefore, vapour-liquid equilibrium is defined when both the temperature and the pressure are fixed.

CHAPTER 1. INTRODUCTION

Figure 1.3: Pressure-temperature P T projection of the phase diagram of a type I binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components. The dashed curves represent critical lines. The circles denote the critical points of the pure components.

Figure 1.4: a) Pressure-composition P x and b) temperature-composition T x slices of the phase diagrams obtained for a type I binary mixture. The temperatures (T1 , T2 , T3 ) and pressures (P1 , P2 ) correspond to slices shown on figure 1.3. The continuous curves represent coexistence boundaries. The circles denote vapourliquid critical points. The dashed-dotted line represents a tie line.

CHAPTER 1. INTRODUCTION

The compositions of the vapour and the liquid phase in equilibrium are represented by the dew point curve and the bubble point curve, respectively, and the coexistence points on the two curves are joined by the tie lines (figure 1.4 b)).

1.2.2

Type II Phase Behaviour

Type II fluid phase behaviour is differentiated from type I behaviour by an additional region of liquid-liquid immiscibility at low temperature; a three phase line extends from the upper critical end point (UCEP) (figure 1.5) to low temperatures and pressures. The pressure-composition P x diagram at low temperature (figure 1.6 a)), is seen to exhibit a three phase line together with separate regions of liquid-liquid and vapour-liquid equilibria. The three phase line disappears at temperatures and pressures above the UCEP. The region of liquid-liquid coexistence can be seen in the constant pressure temperature-composition T x slice (figure 1.6 b)). This region of immiscibility ends at a liquid-liquid critical point often referred to as the upper critical solution temperature (UCST). It is likely that many type I mixtures would conform to type II behaviour at sufficient low temperature, if the presence of a solid phase did not first intervene. Some examples of mixtures which exhibit type II phase behaviour are carbon dioxide + n-octane, + n-decane, + n-undecane, + 2-octanol, in which the UCSTs increases with applied pressure.

CHAPTER 1. INTRODUCTION

Figure 1.5: Pressure-temperature P T projections of the phase diagram characteristic of a system exhibiting type II phase behaviour. The continuous curves represent the vapor-pressure curve of the two pure components and the three-phase line. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangle denotes the UCEP.

Figure 1.6: a) Pressure-composition P x (at temperature T1 ) and b) temperaturecomposition T x (at pressure P1 ) slices of the phase diagrams obtained for a type II binary mixture. The regions of vapour-liquid and liquid-liquid equilibria are labelled on the figure. The continuous curves represent the coexistence curves. The circles denote critical points (vapour-liquid and UCST)

CHAPTER 1. INTRODUCTION

1.2.3

Type V Phase Behaviour

The pressure-temperature P T projection of a type V system is represented schematically in Figure 1.7. This type of behaviour is characterised by discontinuous gasliquid critical lines and a three-phase line, which is bounded by lower and a upper critical end points (LCEP and UCEP). Typical pressure-composition P x diagrams corresponding to type V phase behaviour are shown in figure 1.8. At a temperature T1 which is just at the LCEP, the diagram is identical to that of a type I system (figure 1.8 a),1.6 a)). For temperatures between the LCEP and UCEP (see figure 1.8 b)), a region of liquid-liquid immiscibility appears, ending at a critical point at high pressure. The critical points corresponding to that region of liquid-liquid immiscibility are called upper critical solution pressures (UCSPs) in the case of a constant temperature slice, and lower critical solution temperatures (LCSTs) in the case of a constant pressure slice. Type V phase behaviour is experimentally observed for the systems methane + n-alkanes from n-hexane (methane + n-hexane [10], methane + n-eicosane [11]). There are, however, reasons to believe that some of those systems are in reality characterised by a type IV phase behaviour, as we explain in the next section. The type V behaviour and the thermodynamics of LCSTs is discussed in details in the chapter 2, using a molecular theory.

Figure 1.7: Pressure-temperature P T projections of the phase diagram of a type V binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent critical lines. The circles denote the critical points of the pure components, and the triangles denote the UCEP and LCEP.

CHAPTER 1. INTRODUCTION

Figure 1.8: Pressure-composition P x slices of the phase diagram at a) a temperature T1 , and b) a temperature T2 , corresponding to a type V binary mixture. The continuous curves represent the coexistence curves and three phase lines. The circle denotes a critical point (UCSP), the triangle denotes the UCEP.

1.2.4

Type IV Phase Behaviour

The difference between type V and type IV fluid phase behaviour (figure 1.9) is the additional presence of a liquid-liquid critical line at low temperature (comparable to that found in type II behaviour). The corresponding P x and T x diagrams are represented in figure 1.10. Two constant temperature pressure-composition P x slices are shown in figures 1.10 a) and b). The separate regions corresponding to vapour-liquid and liquid-liquid equilibrium seen at constant temperature T1 merge into a single region of fluid-fluid equilibrium at a temperature T2 , which is just at the second UCEP. Two regions of liquid-liquid immiscibility can be seen in the T x representation of figure 1.10 c): the liquid-liquid coexistence at low temperature terminates at an upper critical solution temperature (UCST). The liquid-liquid coexistence at higher temperature ends at a lower critical solution temperature (LCST). The critical point seen at higher temperature is the usual liquid-vapour critical point. A number of examples of binary mixtures which exhibit type IV are known. It is commonly thought that some type V systems should in fact be characterised as type IV; the additional low-temperature liquid-liquid critical line seen in type IV systems is thought to always exist and be masked by the appearance of solidification. The system carbon dioxide + n-tridecane [12], is an example of a system exhibiting type IV behaviour.

CHAPTER 1. INTRODUCTION

Figure 1.9: Pressure-temperature P T projection of the phase diagram of a type IV binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase lines. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangles denote the UCEPs and LCEP.

Figure 1.10: Pressure-composition P x slices of the phase diagram at constant temperatures a) T1 , and b) T2 of a type IV binary mixture. The continuous curves represent coexistence curves and three-phase lines. The circles denote critical points (vapour-liquid and UCSP). The triangle denotes a UCEP.

CHAPTER 1. INTRODUCTION

Figure 1.11: Temperature-composition T x slice of the diagram at a constant pressure P1 of a type IV binary mixture. The continuous curves represent coexistence curves and the three-phase line. The circles denote critical points (vapour-liquid, LCST and UCST).

1.2.5

Type III Phase Behaviour

Type III behaviour can be thought of as a type IV behaviour with extreme liquidliquid immiscibility. In type III phase behaviour, the two liquid-liquid critical lines seen in type IV phase behaviour join. The UCEP at low temperature and the LCEP of type IV also merge so that only a UCEP remains. Typical pressure-temperature P T projections and temperature-composition T x slices of the phase diagram of a type III binary mixture are shown in the figures 1.12 and 1.13. Two regions of liquid-liquid immiscibility can be seen in T x slices at high pressure above the critical pressure of the two pure components (figure 1.13 b)). Three different critical points are seen in this slice: one LCST, one UCST and one usual vapour-liquid critical point. At a lower pressure which is still above the critical pressure of the two pure components (figure 1.13 a)), the two regions of liquid-liquid immiscibility merge, to form only one region of fluid-fluid immiscibility which ends at a vapour-liquid critical point; at lower temperatures the coexistence is essentially between two high density liquid phases, and as the temperature is increased (above the critical value of the more volatile component) the density of one of the phases decreased rapidly so that the phase behaviour essentially represents vapour-liquid equilibria. Type III binary mixtures are relatively common. An example of a system which exhibits type III behaviour is the mixture of carbon dioxide and tetradecane [13â&#x20AC;&#x201C;15].

CHAPTER 1. INTRODUCTION

Figure 1.12: Pressure-temperature P T projections of the phase diagram of type III. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangle denotes the UCEP. Type III behaviour occurs when the immiscibility of two components is sufficiently large. Mixtures of methane + long-alkanes should behave like mixtures of carbon dioxide + n-alkanes with a change from type IV to type III behaviour. The change of fluid phase behaviour from type II to type IV and the change type IV to type III is continuous. When the two pure components become more and more asymmetric in size, the two regions of liquid-liquid immiscibility become more extensive until the three-phase lines merge. Such a transition in behaviour has not been seen experimentally for the n-alkane series as the critical lines at low temperatures are hidden by the appearance of solid phases. [6]. The binary mixtures water + n-alkane are other examples of type III binary mixtures. Those systems have been recently discussed experimentally by de Loos et al. [16], and modelled with the SAFT theory approach by Galindo et al. [17, 18].

CHAPTER 1. INTRODUCTION

Figure 1.13: Temperature-composition T x slice of the phase diagram at a): a pressure P1 , and b) a pressure P2 , corresponding to a type III behaviour. The continuous curves represent the coexistence curves and the three-phase line. The circles denote critical points (vapour-liquid, LCST and UCST).

1.2.6

Type VI Phase Behaviour

Type VI behaviour is quite similar to type II, but in this case the liquid-liquid immiscibility disappears again at low temperatures characterised by a locus of LCSTs terminating on the three phase line at a LCEP (see figure 1.14 a). This re-entrant miscibility (also referred to as closed-loop liquid-liquid immiscibility) is the main characteristic of type VI phase behaviour in the temperature-composition representation (see figure 1.14 b)). This type of phase behaviour was first reported for the nicotine + water system in 1904 [19], and is also seen when aliphatic or aromatic alcohols, amines, ethers or ketones are mixed with water or alcohols [5]. Some common examples of type VI binary mixtures are water + 2-butanone [20], water + 2-butanol [20], or water +2-butoxyethanol [21, 22]. It is clear that a common feature of all of these systems is the possibility of hydrogen bonding interactions between the components. Type VI behaviour is only found in mixtures of chemically complex substances where one or both of the pure components exhibit self hydrogen bonding, and in the mixture, there is a strong hydrogen bonding between the two components. Water is usually one of the two components. The regions of liquid-liquid immiscibility occur at low temperatures, well removed from the regions corresponding to the vapour-liquid critical lines. At low temperatures (below the LCST) the strong directional attractive interactions (hydrogen bonding type) give rise to favorable heats of solution and a miscible liquid

CHAPTER 1. INTRODUCTION

state; as the temperature is raised above the LCST the hydrogen bonds break and the liquid becomes unstable leading to demixing. The thermodynamic contributions which give rise to LCSTs in mixtures of low molecular weight hydrogen-bonding molecules are very different to those leading to LCSTs in polymer-solvent systems. All the thermodynamics of the type VI behaviour is described is chapter 2 by using the SAFT approach, and a comparison between LCSTs in type V and VI systems is made. The effect of pressure on UCSTs and LCSTs in type V and VI systems has been studied systematically by Timmermans [23,24]. More recently, Franck and Schneider [25, 26] have made excellent reviews in this area. The phase behaviour in type VI binary mixture is often complicated by the presence of azeotropic lines which we will not discuss here.

Figure 1.14: Pressure-temperature P T projections of the phase diagram of type VI binary mixtures. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent critical lines. The circles denote pure component critical points, and the triangles denote the LCEP and UCEP.

CHAPTER 1. INTRODUCTION

Figure 1.15: Temperature-composition T x slice of the phase diagrams (at a pressure P1 ) obtained for a type VI binary mixture. The continous curves represent the coexistence curves. The circles denote critical points (vapour-liquid, LCST and UCST).

1.2.7

Gibbs Phase Rule and Properties of Mixing

A system of n components with ϕ phases in equilibrium is fully described by 2+ϕ(n− 1) variables: the pressure P , the temperature T and the ϕ(n − 1) independent mole fractions xi,k of component i in all phases k = 1, .., ϕ. The equilibrium conditions correspond to the equality of the pressure, the temperature, and the n(ϕ−1) relations for the equality of the chemical potential µi of each component i in all phases 1, .., ϕ, as µαi = µβi = . . . = µϕ i , for i = 1, . . . , n.

(1.2)

The Gibbs phase rule expressed as the number of degrees of freedom F is

= 2 + ϕ(n − 1) − n(ϕ − 1)

= n + 2 − ϕ.

(1.3)

If we apply the Gibbs phase rule (1.3) to a binary mixture(n = 2), the number of degree of freedom F is 2 if two phases are in coexistence (ϕ = 2). As a result, the equilibrium point is defined by arbitrarily fixing two variables, such as the temperature and the pressure. The phase rule is particularly useful in understanding the phase equilibria of multicomponent systems.

CHAPTER 1. INTRODUCTION

The Gibbs energy of mixing is a useful thermodynamic variable to discuss the different contributions which cause phase separation. In a binary mixture of components 1 and 2, the molar Gibbs free energy of mixing ∆gm , at given temperature T and pressure P , is defined as:

∆gm = g − x1 g1 − x2 g2 ,

(1.4)

where g is the molar Gibbs free energy at (T , P ) of the mixture of composition (x 1 , x2 ), and g1 and g2 the molar Gibbs free energies of the pure components at (T , P ). The two necessary requirements to form an homogenous solution in a binary mixture are [5] ∆gm ≤ 0, Ã

∂ 2 ∆gm ∂x2

(1.5)

> 0.

(1.6)

T,P

If one of the conditions (1.5) and (1.6) is not satisfied, the liquid is not stable at (T , P , x), and the system demixes into two coexisting liquid phases. The Gibbs free energy of mixing is related to the other properties of mixing via the thermodynamic relation

∆gm = ∆hm − T ∆sm ,

(1.7)

where ∆hm and ∆sm are the molar enthalpy and entropy of mixing. According to the relations (1.5), (1.5) and (1.7), a positive enthalpy of mixing induces demixing, whereas a positive entropy of mixing induces mixing; phase separation is governed by a competition between enthalpic and entropic effects. The evolution of the Gibbs free energy of mixing ∆gm as a function of the composition, for a demixing binary system at fixed (T , P ), is shown schematically in figure 1.16. The coexistence point at (T, P ) between phases α and β is given by the common tangent of the curve ∆gm (x). The common tangent criteria comes from the equality of the chemical potentials of the components in both phases, and the Gibbs-Duhem condition. In figure 1.16, the points A and D are coexistence or so-called binodal points, and the corresponding coexistence compositions are xα and xβ . The points of inflection of the curve, B and C, represent the spinodal points which are determined with the following relation

CHAPTER 1. INTRODUCTION

Figure 1.16: Molar Gibbs free energy of mixing for a binary mixture at given T and P , as a function of the mole fraction x of one of the two component. The points A and D represent the binodal points, and B and C the spinodal points. The dotted line corresponds to the common tangent, and the dotted-dashed lines denote the compositions of the two phases α and β in coexistence.

∂ 2 ∆gm ∂x2

= 0.

(1.8)

T,P

The second derivative of ∆gm with respect to composition is negative between points B and C, corresponding to the region of material or diffusion instability. The regions A − B and C − D where the derivative is positive correspond to metastable regions.

1.3

Phase Behaviour in Polymer-Solvent Systems

Polymers are molecule of hight molecular weight, consisted of monomers linked to each other by covalent bounds. There are two kinds of polymers: - The homopolymers are consisted of the repetition of one monomer. The degree of polymerisation x represents the number of monomer units on the macromolecule. - The copolymers are consisted of the repetition of several monomers. Copolymers prepared from bifunctional monomers can be subdivided further into four main categories:

• Statistical copolymers where the distribition of the tow monomers in the chain is essentially random, but influenced by the individual monomer reactivities.

CHAPTER 1. INTRODUCTION —ABBAAABBABABA—

• Alternating copolymers with a regular placement along the chain.

—ABABABABABAB—

• Block copolymers consisted of substantial sequences or blocks of monomers.

—AAAAAAABBBBBBBBAAAAAAA—

• Graft copolymers in which blocks of one monomer are grafted on the backbone of the other as branches:

—AAAAAAAAAAAAA— B

The structure of polymers depends on the functionality of the monomers. There are three different structures : - linear polymers ( for instance the high density polyethylene HDPE) - branched polymers ( for instance the low density polyethylene LDPE) - resins which are consisted of 3 dimension networks. The first are obtained from bi-functional monomers, the others from monomers of functionality > 2. The linear and branched polymers are soluble in organic solvents, thermoplastic, and can be melted. The resins are insoluble and can not be melted.

CHAPTER 1. INTRODUCTION

Polyethylene (PE) is the most popular plastic in the world. This is the polymer that makes grocery bags, shampoo bottles, pipes, children’s toys. It has a very simple structure, the simplest of all commercial polymers. A molecule of linear polyethylene is nothing more than a long chain of carbon atoms, with two hydrogen atoms attached to each carbon atom, and can be considered as a very long linear alkane. There are several kinds of polyethylene (figure 1.17). Their properties differ resulting from variations in structure. • High density polyethylene (HDPE) has linear chains with a melting point Tm in the range 130 ˚C < Tm <140˚C, and is normally produced with molecular weights in the range of 200,000 to 500,000 g/mol. Its density is in the range 0.94-0.96 g.cm−3 . • Low density polyethylene (LDPE) contains both short and long chain branches and has typically a melting point of 105˚C< Tm <115˚C. Its density is about 0.92 g.cm−3 • Polyethylene with molecular weights of three to six million is referred to as ultra-high molecular weight polyethylene, or UHMWPE. It is prepared for making very strong fibre. • Linear low density polyethylene (LLDPE) is produced by copolymerizing ethylene with an alpha-olefin and has a melting point which is intermediate between HDPE and LDPE. The linear polyethylene molecule is the simplest polyolefin, composed of a chain of carbon atoms with a tetrahedral geometry. For carbons in all trans geometry, the most energetically stableconfiguration resembles a zig-zag. If the carbons are all in trans position, the most stable one, the molecule looks like a zig-zag chain :

CHAPTER 1. INTRODUCTION

Figure 1.17: Schematic representation of different kinds of polyethylene. a) High Density Polyethylene (HDPE), b) Low Density Polyethylene(LDPE), c) Linear Low Density Polyethylene (LLDPE) For a typical molar mass M 1.6 Ă&#x2014; 105 g/mol, the chains consists of 10 000 carbon

atoms, and assuming a tetrahedral angle of 109â&#x2014;Ś and a carbon-carbon bond length

of 0.154nm, the chain would be about 1260nm long and 0.3nm diameter. In reality, however, the polymer is not in an all trans configuration and some portions of the chain can rotate into a gauche geometry. The distribution of trans and gauche geometries is a function of the temperature and the relative stability of these states. Because of the rotation about the carbon bonds, the chain is in a state of perpetual motion. The speed of this moves is a function of the temperature and dictates many of the physical characteristic of the polymer. As a consequence a polymer is better represented as a coiled ball than an extended rod. Branched polyethylene is often made by free radical vinyl polymerization of ethylene. Linear polyethylene is made by a more complicated and expensive procedure called Ziegler-Natta polymerization. UHMWPE is made using metallocene catalysis polymerization. LLDPE is produced using an olefin as comonomer (butene, 4-methyl-1-pentene, hexene, heptene or octene) and sold under the trade name Innovex. Polypropylene is an other variety of polyolefin which has two methyl branches per repeated units, and can be either head-to-tail (PP), or head-to-head(hhPP). The co-polymer poly(ethyl-propylene) (PEP) has one branch per repeat unit, while polyisobuthylene(PIB) is the most heavily branched. In figure 1.18 is represented the type of branching of the polyolefins PE, PP, hhPP, PEP and PIB. Polymer-solvent mixtures are multicomponent systems, because of the natural molecular weight polydispersity of the polymer. There are countless possible polymersolvent binary mixtures in practice. Fortunately, the general features of the phase

CHAPTER 1. INTRODUCTION

Figure 1.18: Atomic representation of some polyolefines (from Lipson et al. [27]). Each black circle represents a carbon atom. behaviour observed in these mixtures can be characterised using the classification of Scott and van Konynenburg principally in terms of three types of behaviour (types III, IV, V), as will be shown in this section. The phase diagram of polymer solutions are best understood by comparing them to the fluid phase equilibria of binary mixtures. The reader is directed to general reviews of polymer solution fluid phase equilibria [28–30] for further details. One can first assume that the polymer is mono-disperse. The effect of polydispersity is discussed later in this section. The fluid phase equilibria of a mixture of a mono-disperse polymer and a solvent can be understood as an extreme representation of that of a binary mixture which differ greatly in size. Such systems exhibit features corresponding to type III, IV or type V phase behaviours (see section 1.2).

1.3.1

Phase Diagram

For polymer solutions, the phase behaviour is usually represented as temperaturecomposition or pressure-composition phase diagrams (T x, P x), and the composition is usually expressed as the weight fraction of polymer. The whole range of P T fluid phase equilibria of a polymer solution is usually not shown because the vapour pressure of the polymer is very low and polymers tend to decompose at high temperatures. A schematic representation of T x slices of a type IV binary mixture (for instance a mixture of two n-alkanes of highly differing molecular weight, see figure 1.9, 1.11) and of a polymer solution (for instance polystyrene in cyclohexane) are depicted in figures 1.19 a) and b). An analogy can clearly be made between the two phase diagrams. At high temperatures, the phase equilibrium corresponds to a vapour-liquid equilibrium (V − L1 ) between an essentially pure solvent gas phase and a liquid phase consisting of the solvent ’adsorbed’ in the polymer. In the T x diagram 1.19 b), the dew point curves are confused with the y axes, since there

CHAPTER 1. INTRODUCTION

are essentially no polymer molecules in the gas phases. As the temperature is decreased close to the boiling point of the solvent, a vapour-liquid-liquid three-phase coexistence region is reached.

Figure 1.19: Temperature-composition T x slices of the phase diagrams of a) a type IV binary mixture corresponding to figure 1.9, and b) a type IV monodisperse polymer solution. The symbols X and W refer to mole and weight fractions. The continuous curves represent coexistence curves and three-phase lines. The circles denote critical points. The dashed-dotted lines represent the limits of the T x diagram. Below the three-phase line, the system exhibits liquid-liquid equilibrium between a polymer-rich (L2 ) and a polymer-poor (L1 ) phase which has a low-temperature bound at a critical point, the lower critical solution temperature (LCST). The liquidliquid coexistence curve for the polymer-solvent mixture is very asymmetric compared to the type V behaviour exhibited by binary mixtures of shorter n-alkanes, and the concentrations of the polymer in the solvent-rich phases are very low. The liquid-liquid critical points are shifted to the solvent rich region. Lower critical solution temperatures were first observed in polymer systems by Malcolm and Rowlinson [31] for aqueous solutions of polyethylene oxide (in which the chain length and polar interactions play a role), and by Freeman and Rolwinson [9] for apolar solutions of hydrocarbon polymers in a hydrocarbon solvent. Extensive liquid-liquid separation is observed when the solvent and polymer are very dissimilar (either in terms of their molecular weight or in terms of their interactions). A second region of vapour-liquid equilibria (V â&#x2C6;&#x2019; L1 ) which starts at the three phase line and ends at the boiling point of the pure solvent can be seen in the T x diagram 1.19 a) of the type V n-alkane binary mixture. It is however impossible to study this region

CHAPTER 1. INTRODUCTION

experimentally in polymer solutions, and the three-phase line temperature is about the same as the boiling point of the pure solvent. For a system with unfavourable unlike interactions, a second liquid-liquid immiscibility at low temperatures can appear (c.f. type II and IV behaviour), which ends at an upper critical solution temperature (UCST). In many polymer systems, this low temperature region of liquid-liquid immiscibility is masked by the solidification or crystallisation of the polymer. It has been suggested that the UCST critical line depicted in the P T projection of the phase diagram always exists but is metastable with respect to solid phases [6]. When the UCST critical line is inside the crystalline region, type V fluid phase behaviour is observed instead of type IV. Both boundaries of liquid-liquid immiscibility delimit the transition between transparent homogeneous phases and opaque (or cloudy) two-phase regions, often called cloud curves in the field of polymer science. It has been found experimentally that at a given pressure, the UCST increases with increasing molecular weight of the polymer, while the LCST is seen to decrease [29]. As the degree of asymmetry between the polymer and the solvent becomes larger, the two regions of liquid-liquid immiscibility expand and can merge. In this case, the behaviour can change from type IV to type III phase behaviour. The effect of the pressure and the molecular weight of the polymer on the UCST and LCST will be discussed in more detail for polyethylene/n-alkane systems in chapter 5 .. As was mentioned earlier the LCST in mixtures comprising relatively small molecules which hydrogen bond is driven by an increasing enthalpic contribution at low temperatures. The presence of LCSTs in polymer-solvent systems is caused by a very different balance between entropic and enthalpic contributions that can be attributed to density changes [32, 33]. A lack of understanding of this balance often leads to some confusion about the nature of fluid phase equilibria in polymer solutions. A common explanation for the LCST behaviour polymer blends is that the entropy of mixing is unusually small and that differences in compressibility (again due to density effects) of the two components is the driving force for the phase separation [34]. In the fist chapter, we examine the relevant thermodynamic contributions that give rise to the liquid phase instability for both polymer-solvent systems and mixtures of molecules with hydrogen-bonding interactions, by examining prototype systems within the context of a simplified version of the statistical associating fluid theory (SAFT) [35, 36].

CHAPTER 1. INTRODUCTION

1.3.2

Binodal and spinodal curves

In polymer systems, phase separation occurs over the entire region A − D shown in figure 1.16 (chapter 1), but the mechanism for microscopic separation in the unstable region B − C is different from that in the metastable regions A − B and C − D. In the metastable region, the mixture must overcome a potential barrier before it can lower its free energy by phase separation. The nuclei of the new phase must form and grow into dispersed droplets in a continuous matrix of the other phase. This mechanism is called the “nucleation and growth” process [37]. In the unstable region, no potential barrier exists and quenching into this region leads to ”spinodal decomposition”. The spinodal curve is the locus of points for which the second derivative of the Gibbs free energy is zero (c. f. equation (1.8)). The development of phase separation by spinodal decomposition can be thought of as a growth in a three-dimensional wave profile of polymer concentration. The process has a characteristic ”wavelength” and develops a ribbon-like morphology. In the unstable region where spinodal decomposition occurs, small fluctuations in composition trigger phase separation which proceeds rapidly. Hence, it is important to know the boundaries of the binodal and spinodal regions because each process tends to produce its own distinctive morphology of the phase-separated regions. Nucleation and growth produces spherical domains of the minority phase whereas, spinodal decomposition leads to a co-continuous ”ribbon” structure which can lead to desirable mechanical properties.

1.3.3

The Effect of Polydispersity

As was mentioned earlier, polymer molecules are rarely of the same length, and the polymerisation reaction leads to varying degrees of size polydispersity. Polydispersity influences the physical properties of the polymer to differing degrees. In the case of the fluid phase equilibria of polymer solution, the effect may be considerable especially for liquid-liquid equilibria. A polymer is composed of an almost infinite number of types of chain molecules, and according to the Gibbs phase rule (1.3), the number of possible phases in coexistence may also be infinite. Fortunately this is not the case, and the number of fluid phases encountered in polydisperse polymer+ solvent system are usually the same as those encountered when the polymer is assumed to be monodisperse. In polydisperse polymer-solvent systems, the transition point

CHAPTER 1. INTRODUCTION

from a transparent single phase to an opaque two liquid phase system, at either a UCST or a LCST, is also called the cloud point [38, 39]. However, this point no longer represents a binodal but a saturation point of a multicomponent system. The effect of polydispersity can be discussed by examining the pressure-composition P x diagrams of two polymer solutions, one where the polymer is monodisperse, and the other where the polymer is polydisperse. We will not give a lot of details in this section, as a general study of the polydispersty effect is given in chapter 4 of this thesis. The liquid-liquid equilibrium of a binary mixture of a monodisperse polymer and a solvent is represented in figure 1.20. Tie lines join the two coexistence points on the binodal curve. The critical point, in this case an upper critical solution pressure or UCSP, is located at the maximum of the binodal curve. The diagram contains only one coexistence curve which separates the one-fluid phase region from the two-fluid phase region. In other words, if the temperature and pressure are fixed, the compositions of the two coexistent phases do no depend on the initial global composition of the sample.

Figure 1.20: Pressure-composition P w slice of the phase diagram of a polymer solution in the case of monodisperse polymer. The white circle denotes the critical point (UCSP). The black circles represent the binodal points Two schematic P w diagrams obtained for a polydisperse polymer in a solvent are shown in figures 1.3.3 and 1.22. The binodal curve splits into three kinds of curves: a cloud point curve, a shadow curve, and an infinite number of coexistence curves. To

CHAPTER 1. INTRODUCTION

understand the meaning of these curves, we can a discuss an isothermal decrease in pressure along an isopleth. Let W0 designate the global weight fraction of polymer in a sample. Starting in the homogeneous region at high pressure, we progressively decrease the pressure along the isopleth W = W0 . The phase remains homogenous until the cloud curve is reached. The cloud curve is the point where the first droplet of the second phase appears. The composition of this first droplet is then given by the shadow curve. If we keep decreasing the pressure, the compositions of the two phases are not longer determined by the cloud and the shadow curves, but by two coexistence curves. A separate pair of coexistence curves is obtained for each isopleth W0 . The ensemble of coexistence curves are then divided into two branches beginning at corresponding points, at same pressure, on the cloud curve and on the shadow curve. The critical point in this diagram (UCSP) is no longer at the maximum of the cloud curve, but is shifted to higher overall concentrations of polymer, at the intersection of the shadow and the cloud curves. If the pressure decrease is carried out for an initial global composition W0 equal to the critical composition, a single coexistence curve is found which is delimited by the shadow and the cloud curves. Polydispersity gives rise to a similar type representation in temperature-composition T w diagrams for liquid-liquid equilibria. For further details about the effects of the polydispersity in phase equilibria, see the recent reviews of Browarzik and Kehlen [40], Chapman [41] and Sollich [42]. In chapter 4, a complete discussion of the effect of polydispersity on the cloud curve is given by using the SAFT equation, and the numerical methods to calculate phase equilibria in such systems are explained.

CHAPTER 1. INTRODUCTION

Figure 1.21: Pressure-composition P w phase diagram of a polydisperse polymer solution, where the global composition of the sample W0 is higher than the critical composition. The white circle denotes the critical point. The black circles represent the coexistence points. The thick continuous curve represents the cloud curve. The dashed line is the shadow curve. The thin continuous curves represent the coexistence curves.

Figure 1.22: Pressure-composition P w phase diagram of a polydisperse polymer solution, where the global composition of the sample W0 is lower than the critical composition. The white circle denotes the critical point. The black circles represent the coexistence points. The thick continuous curve represents the cloud curve. The dashed line is the shadow curve. The thin continuous curves represent the coexistence curves.

CHAPTER 1. INTRODUCTION

In chapter 4, the reader can find a general literature review about continuous thermodynamics and phase equilibria calculation in polydisperse system. A general study of the effect of polydispersity in both polymer-solvent and polymer-colloidal systems is also given in chapter 4, and the reader can find a detailed explanation of cloud and shadow curves.

1.4

Equations of State for Polymer Melts and Polymer Solutions

Since the forties, a large number of different theories have been developed for chain molecules. Most theories have been extended to mixtures and can treat both polymer melts (liquid state) and polymer solutions. Excellent reviews have been made by Curro [43] and Rogers [44] for equations of state for polymers in the fluid state. Curroâ&#x20AC;&#x2122;s [43] review emphasizes the theoretical developments of compressible lattice models, bases on hole or cell theories, whereas Rogersâ&#x20AC;&#x2122; contribution [44] focusses on the ability of some equations to fit the PVT data of polymer melts. A very general review is also given by Lambert et al. [45] on both the equations of states and the fluid phase behaviour of polymer systems. The equations of state for chain molecules can be classified into two main categories: lattice approaches and continuum approaches. The use of lattice models is well-established in the polymer community, however, recent advances is statistical mechanics has enabled the development of accurate theories for continuum models of polymer. Since various approximations are used in treating the lattice and continuum models, it is often difficult to compare the theoretical foundations of both types of models. On one hand, lattice models assumes that the fluid adopts a discrete solid-like structure, and simple analytical expressions can then be derived. On the other, continuum models, built from the development of integral equation technics and perturbations theories [46], may require a numerical solution at some stage. Continuum models provides a more realistic structure for a fluid, and can deal with a much larger range of densities than lattice models. A very interesting comparisons between lattice and continuum models from a theoretical point of view has recently been made by Lipson et al. [47]. In this section, we discuss the theoretical basis of the most widely used lattice and continuum models. Further details on the statistical associating fluid theory (SAFT) developed by Chapman, and co-workers [35, 36] are given in chapter 2.

CHAPTER 1. INTRODUCTION

1.4.1

The Flory-Huggins-Staverman Theory

. The earliest molecular description of polymer systems from a theoretical point of view is due to Flory [48] and to Huggins [49] (see Flory’s seminal text [50] for the usual notation). It is less commonly recognised that an identical expression for the entropy of mixing in a polymer solution was derived independently in the same year by Staverman and van Santen [51]; this may after all not be that surprising as the world was at war at the time, and the paper reports work undertaken and published in the Netherlands. In the Flory-Huggins theory, the polymer molecules are treated as composed of m adjacent units, where each unit occupies one cell of the lattice. The solvent molecule is represented by a single unit (m = 1) or in some cases a small number of units (<< m). All the cells of the lattice are occupied by either a polymer unit, or a solvent unit (this means that the fluid is incompressible). To calculate the entropy of mixing of the system, the number Wp of possible configurations of putting N1 polymer molecules in a square lattice of N cells are counted (c. f., figure 1.23). The empty cells are then filled by N2 solvent units such that mN1 + N2 = N . After placing a first molecule i of length m into the lattice, the number of vacant cells left for the next molecules is N − m as the segments can not overlap on the same cell. The probability pi to place the first segment of a second chain molecule i + 1 is then

pi =

N −m . N

(1.9)

Let z be the co-ordination number, i.e. the number of neighbour cells of a given cell. The probability of placing the second segment of the chain molecule i + 1 is approximately zpi , assuming pi has not changed after placing the fist segment, and the probability to place the third segment is (z − 1)pi . Hence it can be shown (see [50]) that the number Wp of ways to put N1 chain molecules of length m is

Wp =

1 (N/m)! z − 1 N2 ! (N1 /m)! N ·

¸N2 (m−1)

(1.10)

where z is the coordination number, i.e., the number of neighbouring cells for one cell. The entropy of the mixture is given by

CHAPTER 1. INTRODUCTION

S M = ln Wp .

(1.11)

It is possible to show that the expression of the molar entropy of mixing ∆Sm of this system is a function of the packing fractions ηi : ∆Sm = −N1 ln η1 − N2 ln η2 , k where k is the Boltzmann constant, η1 =

N1 N1 +mN2

(1.12)

and η2 = 1 − η1 ; η1 and η2 are

the fractions of lattice cells occupied by the polymer units, and solvent molecules.

Figure 1.23: Lattice model used in the Flory-Huggins theory to represent a mixture of chain molecules of 10 segments and a monomeric solvent. The white circles denote solvent units. The black circles denote polymer units. The enthalpy of mixing is calculated by assuming that the units in the lattices interact only with their nearest neighbours. If −²11 , −²22 , and −²12 are taken to represent the interaction energy between polymer-polymer units, solvent-solvent units, and polymer-solvent units, the enthalpy of mixing of the system is then given by

∆Hm = kT χN1 η1 , where χ is a parameter defined as

(1.13)

CHAPTER 1. INTRODUCTION

χ=−

z∆²12 , kT

(1.14)

with 1 ∆²12 = ²12 − (²11 + ²22 ). 2

(1.15)

The Gibbs free energy of mixing of this system is then obtained by ∆Gm = ∆Hm − T ∆Sm .

(1.16)

The main advantage of the Flory-Huggins approach is the simplicity of its expressions. However, its main drawback is to treat the structure of the fluid as a solid (a discrete representation of the segment positions), and to only consider neighbouring interactions. It is also unable to take into account density variations and pressure effects, since all the lattice cells are fully occupied by the solvent and the polymer units. The entropy of mixing is always positive within the Flory-Huggins description, and therefore always favours a mixed state. For certain values of χ corresponding to unfavourable interactions, the enthalpy of mixing becomes sufficiently positive and the system exhibits liquid-liquid immiscibility characterised by a UCST at lower temperatures. It is also possible to describe polymer solutions with an LCST by introducing an empirical temperature, composition, and even pressure (or density) dependence on the χ energy parameter (e.g., [52–56]); unfortunately, an understanding of the fundamental nature of the phase equilibria in such systems is lost in this type of phenomenological approach where χ now represents a free energy rather than a balance of the segment interactions.

1.4.2

Compressible Lattice Models

The original Flory-Huggins lattice model corresponds to an incompressible system, and as a consequence volume changes cannot be taken into account. The compressibility of the system can be taken into account in various ways, either by assuming that the segments can move and vibrate around their equilibrium position in the lattice (lattice cell models), or by using partially filled lattices and treating the holes as a third component (lattice hole models).

CHAPTER 1. INTRODUCTION

Lattice Cell Theory The compressibility of the polymer system can be incorporated by using a generalised van der Waals partition function (which includes the volume dependence) together with a cell model description [57]. In this context the approach of Prigogine [58] is often followed where the partition function of the chain is described in terms of its internal degrees of freedom (c factor); the problem with this type of approach is that there is no clear link between the empirical c parameter which is used to describe the chain and the molecular features of the chain. Patterson [32, 33] recognised early on that volume effects are responsible for the phase separation associated with an LCST in polymeric systems. Using the Prigogine-Flory cell theory [58], in which free-volume effects are taken into account in the χ parameter, Patterson was able to model the phase behaviour of polymersolvent systems exhibiting LCSTs. As the temperature is increased towards the critical point of the solvent, the solvent expands rapidly from liquid-like to gaslike densities, while the polymer remains essentially in a liquid like state. The increasing difference in compressibility between the solvent and the polymer induces a negative contribution in the volume of mixing ∆Vm . This decrease in volume results in negative contributions in both ∆Hm and ∆Sm due to the closer average proximity of the molecules. At sufficiently high temperatures the entropy change dominates the overall balance between ∆Hm and ∆Sm resulting in a less favourable ∆Gm which promotes demixing (also see Folie and Radosz, [30]). Studies of polymer solutions with compressible lattice models are not restricted to the Prigogine-Flory cell theory. The perturbed hard chain theory (PHCT) of Prausnitz and co-workers [59] is also based on the Prigogine partition function [58]. The PHCT expressions have been subsequently simplified by Kim et al. [60], and the resulting SPHCT equation of state is the version that tends to be used to describe mixtures of chain molecules. A good example of the use of the PHCT approach in describing LCST in polyethylene solutions in hydrocarbons is the work of de Loos et al. [61]. Lattice Hole theory An alternative and simpler way to include the effects of compressibility in the FloryHuggins lattice model is to treat partially filled lattices resulting in a lattice-fluid

CHAPTER 1. INTRODUCTION

equation of state. The Sanchez-Lacombe [62–64] equation of state for chain-like fluids is essentially a compressible Flory-Huggins lattice model, in which volume effects are incorporated via vacant lattice sites. The description of polymer solutions with the Sanchez-Lacombe equation of state can lead to both LCST and UCST behaviour depending on the intermolecular interactions. The applications of these types of lattice models (including the quasichemical lattice approximations of Guggenheim) to the thermodynamics and fluid phase equilibria of polymer systems are too numerous to discuss here; the reader is directed to reviews of the area for details [29, 65]; S. M. Lambert et al. in [64]. The Sanchez-Lacombe equation of state [62] is one of the most widespread in industry. This approach is more accurate than the Flory-Huggins equation, but requires the use of binary parameters fitted to experimental data to predict accurately the behaviour of polymer solutions. It is important to note also that the parameter χ which characterises the cross interactions between the polymer and the solvent, often depends on the concentration of the components for a satisfactory treatment. Moreover, some systems (polyethylene oxide - water) exhibit strong anisotropic interactions (such as hydrogen bonds) which play an important role in the fluid phase behaviour. In the original formulation the Sanchez-Lacombe equation is unable to model polymer systems with such interactions. BYG Theory A recent lattice model derived from the Green-Yvon-Born (BYG) integral equation [66–69], has been successfully used to model polymer solutions and polymer blends [70]. The BYG theory includes an explicit treatment of the correlations between the segments, and has been very successful in describing the thermodynamics of polymer solutions and blends. Luettmer-Strathmann and Lipson [71] have used a lattice BYG description to examine fluid phase separation and LCSTs in mixtures of alkanes and polyethylene, the generic system which is the focus of this PhD. This BYG approach provides a good representation of cloud curves and excess properties of polyethylene systems, and can be used to take into account branching on the polymer to treat polymer blends. In chapter 5, we compare the results of the BGY lattice theory with the SAFT-VR continuum theory for the prediction of cloud curves in polyethylene systems. The advantage of the BYG lattice approach is that it can also be used to obtain information about the overall dimension of the polymer

CHAPTER 1. INTRODUCTION

chain and it is readily generalised to provide a continuum treatment of polymer systems [47, 72]. A continuum description is often preferable as the definition of density is rather nebulous in lattice models, and as a consequence the density (and pressure) dependence of the thermodynamic functions is often inadequate and can lead to the wrong conclusions about the nature of the phase behaviour especially for molecules with segments of very different size.

1.4.3

Continuous Systems: Tangent Spheres Models

In the 1980s, a number of continuum statistical mechanical theories were developed to describe the thermodynamics and structure of hard-chain fluids at the molecular level. Such hard-chain models, although very simple, incorporate the essential features of polymer molecules, namely, excluded volume effects and chain connectivity. An accurate description of chains of hard-spherical segments is particularly important as the model can be used as the reference in perturbation theories. The chain molecules of interest in our contribution consists of tangentially bonded hard-sphere segments, and are assumed to be fully flexible. Generalised Flory Dimer Theory Dickman and Hall were the first to present an equation of state for a freely jointed hard-sphere chains [73], which corresponds to an extension of the Flory-Huggins treatment of the configuration probabilities of chain molecules to continuous space. This approach is referred to as the generalised Flory theory (GF). In the GF theory, the pressure P of a system of N chain molecules of m tangentially bonded hard spheres is given by the so-called osmotic equation of state [73], as

P µEX AEX 1 =1+ − = 1 − ln pm (η) + ρkT kT N KT η

η 0

ln pm (x)dx

(1.17)

where T is the temperature, k the Boltzmann constant, µEX the excess chemical potential of the chain molecule of length m, AEX the excess Helmholtz free energy of the system, ρ = N/V and η the number density and the packing fraction of the system, pm the insertion probability, defined as the probability of inserting a chain of m segments in a system of N chains and of packing fraction η. One can note that the insertion probability is directly linked to the excess chemical potential µ EX

CHAPTER 1. INTRODUCTION as pm (η) = e−µ

EX /KT

. If the chain is formed of m tangentially bonded spheres

of diameter σ, the packing fraction is defined as η = (π/6)ρmσ 3 . To obtain the analytical expression for the equation of state, one must find the expression for the insertion probability. Assuming a random insertion of m monomers into a fluid of monomers, the probability pm can be estimated in a manner similar to that in the original Flory-Huggins lattice theory [73], as v (m)/ve (1)

pm = p1e

(1.18)

where p1 is the insertion probability of a monomer in a system of N monomers of volume V and at temperature T , and where ve (m) and ve (1) are the excluded volumes between two m-mers and two spheres of same diameter, respectively. The excluded volume ve (1) between two hard spheres of same diameter σ is the volume of a sphere of diameter 2σ

ve (1) =

4π 3 σ . 3

(1.19)

The excluded volume ve (m) between two chains of m spheres can be estimated as

ve (m) = ve (3) + (m − 3) (ve (3) − ve (2)) ,

(1.20)

where the exact expressions of excluded volume of the dimer and the trimer are given by

ve (2) =

9π 3 σ 4

(1.21)

ve (3) = 0.982605σ 3 .

Inserting the expression (1.18) for the insertion probability into the osmotic pressure equation (1.17), one obtain the GF equations of state for chain molecules as

P ve (m) =1+ (Z1 − 1) , ρkT ve (1)

(1.22)

where Z1 is the compressibility factor of the hard sphere system, which can be calculated with the Carnahan-Starling equation [74]. Dickman and Hall compared the

CHAPTER 1. INTRODUCTION

GF equation with Monte Carlo simulation results [75], and although the agreement was reasonable there was some room for improvement. The deficiency of the GF equations comes from the assumption of randomly inserting m monomers into a fluid of monomers without taking into account properly the connectivity of the chain. Honnell and Hall [76] developed the Generalised Flory dimer (GF-D) theory which is based on the properties of the monomer and dimer fluid. In this approach, the insertion probability pm of a chain of m monomers into a fluid of m-mers is evaluated as a function of both the insertion probability p1 of a monomer into a fluid of monomers, and the insertion probability p2 of a dimer into a fluid of dimers. In the GF-D theory, the probability pm is given by

pm = p 1

p2 p1

¶(ve (m)−ve (1))(ve (2)−ve (1))

(1.23)

After inserting equation (1.23) into equation (1.17), the GF-D equation of state is obtained as

ve (m) − ve (1) ve (m) − ve (2) P = Z2 − Z1 , ρkT ve (2) − ve (1) ve (2) − ve (1)

(1.24)

where Z1 and Z2 are the compressibility factors of the monomer and the dimer fluid. Honnell and Hall used the Carnahan and Starling [74] hard-sphere expression for Z1 , and the Tildesley and Street [77] hard-dimer equation of state for Z2 . Excellent agreement is found between the GF-D theory and Monte Carlo simulation results [78]. Both GF and GF-D theories are for pure repulsive hard sphere chains. Yethiraj and Hall [79] extended the GF-D theory to chains of square-well attractive segments, and compared it with the predictions with Monte Carlo simulation data [80]. The GF-D theory has also been generalised to deal with chains of heteronuclear segments [81, 82]. TPT1 Approach and the Statistical Associating Fluid Theory (SAFT) A refreshing alternative to the Flory-Huggins approach was developed by Wertheim [83] based on the free energy of an associating fluid; the approach provides an excellent representation of the structure and thermodynamic properties of chain

CHAPTER 1. INTRODUCTION

molecules, and in its various forms has become the most widespread and versatile theory in the description of the thermodynamic properties of complex fluids. Wertheim used a first-order thermodynamic perturbation theory (TPT1) that he developed for associating fluids [84–87] to obtain an analytical expression for the contribution to the free energy of a two-site associating fluid simply in terms of the structure of a monomer hard-sphere reference system and the average chain length < m > of the polydisperse mixture of chain aggregates that are formed on association. ACHAIN = −N kT (< m > −1) ln g HS (σ),

(1.25)

where g HS (sigma) is the contact value of the radial distribution function which can be obtained from the Carnahan and Starling equation. We will develop the TPT1 theory in details in chapter 2. An identical expression for the monodisperse hard-sphere chain fluid was derived by Chapman et al. [35] by considering a mixture of associating hard spheres in the limit of complete association. Such a description has been at the core of the development of equations of state for chains formed from monomeric segments such as SAFT in its numerous incarnations. It is interesting to note that very similar relationships have been obtained for the contribution to the free energy in forming a single chain in a reference monomeric solvent (corresponding to infinite dilution) HS of expressed in terms of logarithm of the n-body cavity distribution function ym

the reference monomeric system (e.g., see [88–94]: HS (σ12 . . . σm−1,m ) ' −N kT (m − 1) ln g HS (σ). ACHAIN = −N kT lnym

(1.26)

We gives a detailed description of that derivation in Chapter 2, sections 2.1.1 and 2.1.2. The Wertheim description is not restricted to chains of hard-core segments and a large number of studies have been devoted to developing an equation of state for chains formed from Lennard-Jones, Square-Well or Yukawa segments. The reader is direct to the excellent review of M¨ uller and Gubbins [95, 96] for a more complete discussion of the most recent developments. The SAFT equation of state with its TPT1 description of the chain has now been widely used to describe the thermodynamics and phase equilibria in polymer fluid phase equilibria including gas adsorption, polymer solutions, and blends (see [29,45,65,97,98]). Of direct relevance to our contribution are the SAFT studies of fluid phase equilibria of binary mixtures of short- and long-chain hydrocarbons (prototypes of polymer solutions): the

CHAPTER 1. INTRODUCTION

SAFT-VR expressions [99, 100] for chain molecules with potentials of variable range has been used with the Lorentz-Berthelot combining rules to describe the LCST behaviour in a binary mixture of methane and n-hexane (the first such homologous pair to exhibit the behaviour) [101]; more recently, Gregorowicz and de Loos [102] used both the Huang and Radosz version of SAFT and the Peng-Robinson equation of state to examine UCST and LCST behaviour in asymmetric binary mixtures of hydrocarbons, showing that the extent liquid-liquid immiscibility increases with the asymmetry in molecular weight (as is found experimentally). The original TPT1 treatment of Wertheim essentially involves a linear approximation for the n-body distribution so that only the hard-sphere pair distribution function at contact is required to describe the free energy of a chain of tangent hard spheres [83]. The TPT1 versions of the theory give a good description of chain molecules, but the adequacy of the approximation tends to decrease as the molecules get longer. There have been a number of extensions of the approach to give a more accurate representation of the many-body effects. The thermodynamic perturbation theory of Wertheim can be examined at second order (TPT2) by including the triplet distribution function [83] (e.g., see [103–107]. Alternatively, one can start with a dimeric (or oligomeric) reference fluid and construct the chain in terms of dimers (or oligomers) as building blocks, which implicitly incorporates three and four-body correlations into the theory. This approach is used in the dimer versions of the SAFT equation of state (SAFT-D), which is found to provide an excellent description of chains formed from Leonard-Jones and square-well segments [106, 108–113]. The dimer versions of the theory are found to perform slightly better than the secondorder TPT2 description [106], and have been shown to be equivalent to the generalised Flory two-fluid approaches [114] (see [115]). Trimer versions of the theory have also appeared (e.g., see [116]. The TPT1 theory of Wertheim can also be improved by simply including the correct virial coefficients of the entire chain molecule which by definition includes the many-body segment-segment interactions [117–124]. An excellent review of virial equations of state derived from statistical mechanics has recently been made by Trusler [125]. Integral Equations Approach to Chain Molecules Before we conclude our overview of the general approaches that are used to describe the thermodynamics of chain fluids it is important to mention that the many-body

CHAPTER 1. INTRODUCTION

interactions in such systems can also be treated directly by considering average structural information about the chain. Standard integral equation techniques can be used to determine the correlations between the segments forming the chains and this information can in turn be used to obtain the thermodynamic properties (see [126] for a review). The extension of the reference interaction site model (RISM) approach [127, 128], originally developed for small molecules formed from spherical segments, to oligomeric molecules and polymers (PRISM) [129–132] has not been as widely taken up in studies of fluid phase equilibria of chain molecules. This is probably due to the fact that even though the PRISM theory provides a good representation of the segment-segment structure of the chain fluid, the description of the thermodynamics can be rather poor (e.g., see [133]). The solution of the Percus-Yevick (PY) integral equation for the structure of a chain fluid turns out to provide a better description of the thermodynamics and phase equilibria of such systems. Chiew [134] derived an accurate equation of state for hard-sphere chains by using the Percus-Yevick closure together with a number of chain connectivity constraints which closely resemble those of Baxter [135, 136] in the adhesive hardsphere approach. Chang and Sandler [137] obtained a very similar description of the structure of hard-sphere chains (with almost identical expressions for the average sphere-sphere contact values) by solving the Wertheim integral equation theory [86, 87]. A recent example of the use of integral equation theories is the work with the polymer mean spherical approximation (PMSA) to obtain the thermodynamic properties of chains of Yukawa spheres [138]. The PY theory of Chiew for athermal chains forms the basis of a number of equations of state for chain molecules [113, 116, 139–145]. In particular, the perturbed hard sphere chain (PHSC) equation of state of Song and co-workers has been used to describe the UCST and LCST cloud-curve behaviour in models of polymer solutions [141]. The PHSC approach for continuum chain fluids incorporates Chiew’s PY description of the structure and should not be confused with the PHCT mentioned earlier which is based on the Prigogine c factor. The attractive interactions are treated at the mean-field level of van der Waals but where the effective size and energy parameters (a and b) are assumed to be temperature dependent, the form of which is empirically determined from experimental data for argon and methane and then scaled to account for the non-spherical nature of the molecules. Regions of liquid-liquid immiscibility bounded at lower temperatures by an LCST are found

CHAPTER 1. INTRODUCTION

even for the case of monomer-polymer systems with symmetric interactions (segments and attractive interactions of the same magnitude); no UCST is found in this case so the system corresponds to type V behaviour in the nomenclature of Scott and van Konynenburg. In Chapter 2, we examine the same prototype of a polymer solution in our contribution which corresponds to a system with a Flory-Huggins χ parameter of zero; the Flory-Huggins lattice description for a system with χ = 0 would lead one to the conclusion that the polymer is completely miscible in the solvent. Song et al. show that the degree of liquid-liquid immiscibility increases with the chain length of the polymer, and, as Patterson [33] concluded in his studies with a compressible lattice model, the LCST is found to be driven by the asymmetric nature of the change in density (volume effect) of the two components and the mixture when the temperature is varied. When an asymmetry is introduced in the segment interaction energies (by considering different like interactions and/or a weaker unlike interaction) the system can exhibit an additional region of liquid-liquid immiscibility at ’lower’ temperatures below an UCST (corresponding to type IV behaviour in the Scott and van Konynenburg language); for sufficiently asymmetric values of the interactions the UCST and LCST merge at lower pressures to give a cloud curve with a characteristic hourglass shape (type III behaviour). Unfortunately, the temperature dependence that Song et al. use for the segment size and, more particularly, the energy parameter introduces an additional temperature dependence into the reference and mean-field terms of the free energy which complicate the analysis as to the microscopic nature of the liquid-liquid demixing; it is important to recall that the LCST is governed by the magnitude of the change in density with temperature. We shall address this point later.

1.5

Chain Dimension

The nature of the liquid phase separation in polymer-solvent systems has also been examined in the context of changes in the dimension of the polymer chain close to the phase transition. The dimension of the polymer chain is characterised with the 2 > and the radius of gyration < R2 > (see mean-square end-to-end distance < RE G

figure 1.24). The polymer dimension is known to change both with the density of

CHAPTER 1. INTRODUCTION

the medium (whether it be the pure polymer melt or the solvent) and the quality of the solvent.

Figure 1.24: Polymer molecule and the end-to-end distance RE The importance of the chain dimension is that it can be used as a measure of the compatibility of the polymer with a solvent: polymers adopt a more extended configuration in a ’good’ solvent and a more collapsed configuration in a ’bad’ solvent [146]. Polymer collapse was first predicted by Stockmayer [147] and later confirmed experimentally [148, 149]. One should point out that chain collapse does not necessarily imply phase separation, but clearly the two are closely related. Dijkstra and co-workers [150, 151] have reported the results of Monte Carlo simulations for purely repulsive (athermal) lattice models where the solvent and segments making up the polymer are of the same size (symmetric). The polymer is found to collapse at high solvent densities (solvent packing fractions in excess of 70 %) leading Dijkstra and co-workers to the conclusion that a fluid-fluid demixing transition is possible in these athermal polymer solutions; the very high densities required for chain collapse in these lattice systems are much larger than the solidification limit of about 50 % found in continuum systems and should be treated with caution. On using a different discretisation of the lattice, Luna-Barcenas et al. [152] were unable to reproduce the chain collapse of Dijkstra and co-workers for the athermal polymersolvent systems with segments of the same size, but these authors still predict that chain collapse would be found in the continuum limit. The continuum simulations of Escobedo and de Pablo [153] for chains of tangent hard spheres in a hard-sphere solvent of the same segment diameter indicate that the chains contract moderately with increasing solvent density, with a collapse transition seen at only very high packing fractions of 48 % (which could again be metastable relative to a solid state);

CHAPTER 1. INTRODUCTION

these authors find only a small effect of the diameter difference between the solvent and the segments making up the chain. A similar moderate contraction for a hardsphere polymer solution was also found by Grayce [154]. The evidence of a polymer collapse transition (and subsequent phase separation) in these simulation studies of athermal polymer-solvent systems with segments of the same size is certainly not conclusive. In a very interesting paper, Taylor and Lipson [155] used the BYG approach to examine symmetric hard-sphere polymer-solvent systems: as well as providing a good representation of the scaling exponent for the radius of gyration of the polymer, such a description accurately describes the compression of the dimension of the polymer chain with increasing solvent density. Taylor and Lipson show that the chain contraction with increasing solvent density is almost identical to what one would expect in the pure polymer melt, and is thus consistent with a polymer in a good solvent which would not undergo a demixing transition; this is confirmed by the positive value of the second virial coefficient obtained for the segment-segment solvation potential which indicates an expanded chain. Although this suggests that the athermal symmetrical polymer-solvent system does not exhibit fluid phase separation, there is clear evidence of the existence of an “entropy” driven demixing transition for athermal systems in which the solvent diameter is considerably larger than that of the segments making up the chain (colloid-polymer system) [156–164] here the mechanism of phase separation is due to the depletion interaction of Asakura and Oosawa, which is very different to that in polymer solutions. The incorporation of attractive interactions changes the picture completely: in this case a clear chain collapse is found even for a symmetric system in which the solvent-solvent, solvent-segment and segment-segment size and interaction energies are all equivalent [165–170]. The very interesting study of chain collapse on the approach of the LCST by Luna-Barcenas et al. [169] for Lennard-Jones chains in a Lennard-Jones solvent with the same size and interaction parameters is particularly relevant to our work; below the LCST the polymer is found in an expanded state, while above the LCST the polymer exists in a collapsed state in the solvent rich phase. The results of the simulation studies for this symmetric system is consistent with the theoretical findings of Song et al. [141] which indicate that such a system would exhibit LCST fluid phase equilibria.

CHAPTER 1. INTRODUCTION

1.6

Remaining Questions and Theme of the Thesis

After this rather long , but, considering the large volume of work expended on the topic, necessary introduction, it is time to summarise the main conclusions with regard to fluid-fluid phase separation and LCST behaviour in polymer solutions. In the case of an incompressible Flory-Huggins lattice model no liquid-liquid phase separation is predicted for the system with symmetrical interactions (corresponding to a Flory-Huggins parameter χ = 0); an asymmetry which produces as sufficiently unfavourable χ parameter can give rise to UCST behaviour but a specific temperature (and/or density) dependence has to be introduced in order to describe an LCST. Patterson’s [33] studies of the thermodynamics of mixing of polymer solutions with a compressible lattice model indicate that LCST behaviour is due to density (compressibility) effects as the temperature is increased towards the critical point of the solvent. However, a certain number of crucial questions have not been answered: if we consider first a purely repulsive (athermal) binary mixture of spheres and chain of bonded spheres of same size, can we find fluid-fluid phase separation in such an athermal systems on increasing the chain length? If now we consider the same simple system, but with equivalent (symmetric) sphere-sphere interaction energies (χ = 0), does this system exhibit liquid-liquid phase separation corresponding to LCST behaviour as suggested by Patterson and by Song et al.? At what chain length would one expect its solution in a spherical solvent to exhibit a demixing instability in the liquid phase? What are the dominant contributions in terms of the thermodynamic functions of mixing, and what is the microscopic mechanism of the density effect? Is UCST behaviour possible in such a system with symmetric attractive interactions? In a second chapter, we will attempt to answer these questions, using the Wertheim TPT1 theory with mean-field attractions (SAFT-HS, [171]), and hope to improve the general understanding of the subtle and beguiling phase behaviour exhibited by polymer solutions. Following on from the work of Song et al., we also determine the thermodynamic properties of mixing for a system of spherical molecules of the same size with directional interactions that give rise to LCST and type VI behaviour. As expected the mechanism for phase separation in such systems is very different to that in polymer solutions. The focus of Chapter 3 is on colloid-polymer systems which exhibit a completly different phase behvaiour than polymer-solution, although the molecular models con-

CHAPTER 1. INTRODUCTION

sidered are quite similar to those in polymer solution. Indeed, the colloid-polymer system corresponds to the case where the size difference between the spheres and the spherical segments making up the chain is very large. The colloid is represented by a very large hard sphere, and the diameter ratio between colloid particles and chain segments is about 1%. The system is assumed to be purely repulsive, and we use the TPT1 theory to calculate the physical properties and the phase equilibria of the system. Our approach is different from the usual treatment of polymer-colloid systems in the sense that segment-segment and colloid-segment interactions are treated at the same microscopic level; the well-known Asakura-Oosawa (AO) [1] approximations which involve an ideal chain inducing an effective attractive interaction between the colloids are not used. The effect on the fluid phase equilibria of varying both the chain length and the diameter of segments in the polymer are determined from an analysis of the spinodal instability. The thermodynamic properties of mixing for such a system are also determined to provide an explanation of the demixing in terms of the various thermodynamic contributions, and compare the observed behaviour with that of polymer solutions. The effect of the polydispersity of the polymer molecular weight is examined in the fourth Chapter for the models of polymer solutions. It is found that polydispersity can have a big effect, and can lead to an added complexity in the fluid phase behaviour, depending on the molecular weight distribution. We use both a discrete and a continuous representation of the polydispersity, and we find that the approaches give very similar results. The last chapter deals with the modelling of real polymer-solvent systems. A more sophisticated version of SAFT is used (SAFT-VR) [99], which is based on the Barker and Henderson pertubation theory [46, 172, 173]. Predictions are first made for the solubility of gases in polyethylene which essentially corresponds to vapourliquid equilibria. It is found that the co-adsorbtion of gases in a polymer exhibit a synergy: the solubility of small gas molecules such as methane or nitrogen, is enhanced by the presence of a longer molecule such as an n-alkane in the gas phase. This synergy has already been observed both experimentally [174,175], and by Monte Carlo simulation [176â&#x20AC;&#x201C;178], but remained unexplained. In the fifth chapter a detailed explanation of the co-absortion effect is made, and the SAFT-VR predictions are compared with both experimental and Monte Carlo simulation results. Liquid-liquid immiscibility and cloud curves in polyethylene/n-alkane solutions are also examined

CHAPTER 1. INTRODUCTION

with SAFT-VR approach, and a good representation of the experimental data is presented. Finally the effect of polydispersity in the VLE and LLE of the real polymer systems is discussed.

Chapter 2

LCST in Polymer Solutions Pattersonâ&#x20AC;&#x2122;s [33] has shown with the Prigogine Lattice cell model [64] that LCST behaviour in polymer solution is due free volume and density (compressibility) effects as the temperature is just below the critical temperature of the solvent: he explains that the phase separation in polymer systems is due to the difference in compressibility factor between the solvent and the polymer, which leads to a negative volume of mixing and to a less favourable (more positive) free energy of mixing. However, the combined effects of repulsive and attractive interactions are difficult to decouple with such lattice approaches. The work of Song et al. [141] confirms the conclusion of Patterson with a continuum equation of state based on the Chiew PY description of the chain [64], and shows that the density of the solvent rich phase falls sharply on increasing the temperature above the LCST. In their work Song et al. do not examine the possibility of liquid-liquid demixing in the purely repulsive system, and the temperature dependence that is introduced for the segment size and energy interactions complicates the analysis. The conflicting findings of chain collapse in purely repulsive polymer-solvent systems with segments of the same size add to the confusion in the understanding of the separate roles of the repulsive and attractive interactions in liquid-liquid demixing. It is clear, however, that the attractive interactions play a more important part in the case of polymer solutions (but not necessarily in polymer-colloid systems). In this study we have chosen to use the Wertheim TPT1 description [35, 83] of the repulsive interactions of chain systems. This choice is made because the TPT1 description has been extensively compared with simulation and is the core of the SAFT approach which is widely used in describing the fluid phase equilibria of real polymeric systems. We start by examining model polymer solutions of hard spheres

CHAPTER 2. LCST IN POLYMER SOLUTIONS

and hard-sphere chains of the same diameter (symmetric segments). Is fluid-fluid phase separation possible in such athermal systems with symmetric segment sizes on increasing the chain length? In the same fashion as Patterson we examine the thermodynamic functions (volume, enthalpy, entropy and Gibbs function) of mixing to answer this question. We then investigate the role of the attractive interactions by incorporating sphere-sphere attractive interactions at the simplest mean-level of van der Waals; in this case the attractive interactions do not introduce an additional temperature dependence into the free energy (cf. the approach of Song et al.). The simplest system where like and unlike sphere-sphere interaction energies are equal is examined (χ = 0). Does such a system exhibit liquid-liquid phase separation with LCST behaviour, as suggested by Patterson and by Song et al.? If so, what are the driving forces of the phase separation, and and what is the microscopic mechanism of the density effect? What is happening in the case the system is purely repulsive (no attractions). Before discussing phase equilibria, one recall Wertheim’s association and temperature perturbation at first order (TPT1) theories, which are the foundations of the SAFT equation of state. We also show how Boubl´ık [90] derived an identical expression of the contribution to the free energy in forming a dimer in a reference monomeric solvent (corresponding to infinite dilution) expressed in terms of logarithm of the 2-body cavity distribution function y2HS of the reference monomeric system. To answer to those questions, we calculate the properties of mixing of a polymer-solvent system with the SAFT-HS theory [35], and compare enthalpic and entropic effects. We then show that the LCST behaviour encountered in polymer solutions, is completely different than that encountered in associating systems.

2.1

Wertheim Association Theory

In the early 1980s, Wertheim developed a theory to describe the thermodynamic properties of fluids with association interactions [84–87]. In the Wertheim approach, the intermolecular-pair potential is taken to be the sum of a repulsive reference part φR (e.g. the hard sphere repulsive core) and a number of short range association sites which interact via a potential φA . If the molecules are modelled as repulsive spherical cores, the potential φR is ∞ if the molecules overlap, and zero otherwise.

CHAPTER 2. LCST IN POLYMER SOLUTIONS

The total interaction potential φij (12) between molecule 1 and 2 of component i and j can be written as: φij (12) = φij R (12) +

a∈Γ(i)

φij Aab (12),

(2.1)

a∈Γ(j)

where Γ(i) is the set of sites on the molecule of component i. The notation (12) denotes the relative positions and orientations of the molecule 1 and 2. In the case of molecules with one site, each molecule A can bond only to one other molecule to form a dimer AA: A+A* ) AA.

(2.2)

We denote the number density of monomers as ρ0 , the number density of dimers as ρD , and the total number density of the system as ρ. The equilibrium constant K(ρ, ρ0 , T ) for the dimerisation is given by:

K(ρ, ρ0 , T ) =

ρD . ρ20

(2.3)

The mass action equation for this dimerisation gives a separate equation: ρ = ρ 0 + 2ρD . Hence equation (2.3) becomes ρ = ρ0 + 2ρ20 K(ρ, ρ0 , T ).

(2.4)

In a dimerising ideal gas (i.e., in the low density limit), the equilibrium constant K is only a function of the temperature according to rigorous statistical mechanics [179], and is given by:

K0 (T ) = lim K(ρ, ρ0 , T ) = ρ→0

1 2

eR (12)f (12)d(12),

(2.5)

where eR (12) = exp[−βφR (12)], and f (12) = [exp φA (12) − 1] is the Mayer function. In other words, the constant K is proportional to a Boltzmann factor, and could be written in a more common way:

ln K0 (T ) =

∆rH ∆rS − . RT R

(2.6)

When the expression of K is substituted into the mass action equation, we have

CHAPTER 2. LCST IN POLYMER SOLUTIONS

ρ = ρ0 + ρ20

eR (12)f (12)d(12).

(2.7)

Wertheim showed that, for a dense associating fluid where only dimers can form, the total number density ρ can be expressed exactly as

ρ = ρ0 + ρ20

g00 (12)f (12)d(12),

(2.8)

where g00 (12) is the monomer-monomer distribution function. The Boltzmann factor eR (12) has been replaced by g00 (12). The remaining problem is that g00 (12) is not known. The thermodynamic perturbation theory up to first order (TPT1) is obtained by replacing g00 in equation (2.8) by the distribution function of the reference fluid gR . This approximation provides a good representation of the equilibrium even when a large fraction of molecules are bonded [180]. The chemical potential of a pure ideal gas is given by

IDEAL

∂A ∂N

T,V

= −kT ln ρΛ3 ,

(2.9)

where Λ is the thermal de Broglie wavelength, and the pressure is given by pIDEAL = kT ρ.

(2.10)

In the dimerising ideal gas, the monomer and dimer species are in equilibrium and their chemical potential must be equal. :

IDEAL µ0IDEAL = µD

kT ln ρ0 Λ3

(2.11) ´

= kT ln ρD Λ3 .

The pressure of this ideal gas is given by

pV = Nspecies kT =⇒ p = kT (ρ0 + ρD ) ,

(2.12)

and the free energy of the system is given by the thermodynamic relation A = N µ − pV , so that

CHAPTER 2. LCST IN POLYMER SOLUTIONS

IDEAL ³ ´ ρ +ρ AD 0 D = ln ρ0 Λ3 − . N kT ρ

(2.13)

The excess association contribution due to dimer formation is then given by:

AASSOC N kT

IDEAL AD AIDEAL − N kT N kT

= ln ρ0 Λ3 − = ln

(2.14)

³ ´ ρ0 + ρ D − ln ρΛ3 + 1 ρ

ρ0 ρ0 + ρ D − + 1. ρ ρ

In terms of the monomer density ρ0 and the total density ρ = ρ0 + 2ρD , we have IDEAL ρ0 1 ρ0 1 AD = ln − + . N kT ρ 2 ρ 2

(2.15)

If the fraction of free monomers in the system is denoted by X = ρ0 /ρ, the contribution to the free energy due to association is given by

ASSOC AD X 1 = ln X − + . N kT 2 2

(2.16)

Using the relation given in equation (2.7) between ρ and ρ0 , the fraction of molecules not bonded is obtained as

1 , 1 + ρX∆

(2.17)

where

∆=

eR (12)f (12)d(12).

(2.18)

In a non-ideal dimerising system, the Boltzmann factor eR (12) is replaced by the radial distribution function gR (12) of the reference system. The exact expression would include the distribution function g00 (12) of the associating system. The TPT1 approximation can easily be extended to a mixture with species having multiple associating sites giving [181], [35].

CHAPTER 2. LCST IN POLYMER SOLUTIONS

¶ si µ n X AASSOC. X si Xa,i = + , xi ln Xa,i − N kT 2 2 a=1 i=1 #

(2.19)

where xi is the molar fraction of the component i, Xa,i is the fraction of molecules of type i not bonded at a site a, si is the number of sites on the molecule of type i. At this level of approximation the various sites are treated as independent. The fractions Xa,i are calculated from the relation

Xa,i =

j=1

1 b=1 ρxj Xb,j , ∆a,b,i,j

P sj

(2.20)

where ρ is the total number density and ∆a,b,i,j is given by R ∆a,b,i,j = Ka,b,i,j fa,b,i,j gij (σij ).

(2.21)

. Since bonding at one site is independent from bonding at any other site in a given molecule, the total fraction Xi of free monomers of type i can be obtained as

Xi =

si Y

Xa,i .

(2.22)

2.1.1

TPT1 Theory for Chain Formation and Polymers

In the limit of complete association, the monomeric segments form a chain of segments. In the case of a component with one associating site, the Wertheim theory provides an equation of state for the hard-dumbbell fluid [182]. For a system with two bonding sites, an expression for a flexible chain of tangent spherical segments can be obtained [83]. In this approach, the molecules are modelled as hard spherical segments of diameter σ. The site on the molecule is placed at a distance rd from the center of the molecule, and it has a corresponding “bonding volume” modelled as a sphere of diameter rc (figure 2.1). Wertheim derived the expression for the density of aggregated species. When the energy is infinite and the range of the association is zero the system corresponds to a fully bonded chain fluid. The mass action equation for the total density of segments ρ in terms of the density of undissociated monomers ρ0 and dimers ρD is given by

CHAPTER 2. LCST IN POLYMER SOLUTIONS

ρ = ρ0 + ρD ⇒ ρ = ρ0 + ρ20 K HB f HB g HS (σ).

(2.23)

Here, f HB is the Mayer function for the association HB, with f HB = [exp(²H B/kT )− 1], ² is the association energy, and K HB is the bonding volume. g HS (σ) is the value at contact of the pair distribution function of the hard sphere system which has been taken as the reference fluid here. In terms of the fraction of non-bonded molecules X = ρ0 /ρ, equation (2.23) can be expressed as 1 = X + ρX 2 ∆HB

(2.24)

where ∆HB = K HB f HB g HS (σ) is the function which characterises the association in the fluid. The expression can be written as a quadratic equation ρX 2 ∆ + X − 1 = 0.

(2.25)

In order to form a chain, the bonding energy tends to infinity (f HB → ∞) while the bonding volume tends to 0. (K → 0). In this limit ∆ → 0 and lim X = 0.

(2.26)

∆→∞

The compressibility factor is obtained by differentiating the Helmholtz free energy (equation (2.16)) with respect to the density:

ZsASSOC.

∂X pV = = ρs Ns kT ∂ρs µ

T,V

1 1 , − X 2 ¶

(2.27)

where Ns is the total number of spherical segments in the system (whereas N will be used to denote the total number of chain molecules). Similarly, ρs = Ns /V is the density of segments in the system. If m represents the number of spherical segments per molecule, then

Ns = mN.

(2.28)

The derivative of the mass action equation (2.25) with respect to the density of segments ρs is

CHAPTER 2. LCST IN POLYMER SOLUTIONS

´ ∂ ³ ρs ∆X 2 + X − 1 = 0 ∂ρs

ρs

∂∆ 2 ∂X ∂X X + ∆X 2 + ρs 2X ∆+ ∂ρs ∂ρs ∂ρs

(2.29)

= 0,

and

∂∆ −X 2 (∆ + ρs ∂ρ ) ∂X s = . ∂ρs ρs 2X∆ + 1

(2.30)

The expression of the compressibility factor can therefore be written as

ZsASSOC.

= ρs

¶ ∂∆ # µ −X 2 (∆ + ρs ∂ρ ) 1 1 s − ρs 2X∆ + 1 X 2

ρs X 2 ∂∆ . = − ∆ + ρs 2 ∂ρs ¶

(2.31)

The compressibility of the fully associated dimer is then given by

ZsD

= lim Z

ASSOC.

X→0

ρs X 2 ∆ ρs ∂∆ = lim − 1+ X→0 2 ∆ ∂ρs µ

¶#

(2.32)

From equation (2.25), we have:

X 2∆ =

1−X , ρs

(2.33)

and since in the limit of complete association, X → 0 and lim X 2 ∆ =

X→0

1 , ρs

(2.34)

if we substitute these results into equation (2.32), we obtain the compressibility factor in the limit of complete association as

ρs ∂∆ 1 . lim ZsASSOC. = − − X→0 2 2∆ ∂ρs

(2.35)

CHAPTER 2. LCST IN POLYMER SOLUTIONS When the expression for ∆ is substituted in the above equation, we have

lim

X→0

ZsASSOC.

ZsD

1 =− 2

∂ ln g HS (σ) 1 + ρs ∂ρs

(2.36)

The compressibility factor of a system of N fully bonded hard dumbbell molecules is

ZD =

pV N kT

= mZs

(m = 2)

(2.37)

∂ ln g HS (σ) = − 1+ρ ∂ρ

We can calculate the contribution to the pressure due to association by subtracting the ideal contribution. The excess contribution Z D,EX due to association is then given by:

Z D,EX

= Z D − lim Z D ρ→0

∂ ln g HS (σ) = − 1+ρ ∂ρ = −ρ

∂ ln g HS (σ) . ∂ρ

(2.38)

The excess contribution to the Helmholtz free energy due to dimer formation is obtained by integrating the equation (2.38) with respect to ρ as:

AD,EX = − ln g HS (σ). N kT

(2.39)

The expression for a fully bonded chain of m segments can be obtained in the same way, by taking the limit of infinite association bonds. Consider a mixture of Ns spheres of two types: spheres A with only one bonding site, and spheres B with two bonding sites (figure 2.1). All the spheres have the same diameter σ. The stoichiometry of the initial mixture must be consistent with the formation of full chain: (m − 2)/Ns spheres must be of type B, and 2/Ns of them must be of type A.

CHAPTER 2. LCST IN POLYMER SOLUTIONS

Figure 2.1: Top: Schema of an associating sphere with one site. Bottom: mixture of associating spheres and formation of chains We can consider each sphere in the mixture as a different component and guarantee that sphere 1 is bonded with sphere 2, sphere 2 with sphere 3,... Sites A can bond only to sites B, and sites B can bond to sites A and B. As a result no dimers can be formed. The compressibility factor due to association of the system is given by:

ZsASSOC.

m X

ρs x i

" Ã si X ∂Xa,i 1

∂ρs

a=1

Xa,i

1 − 2

(2.40)

where the index i labels sphere i. Xa,i is the fraction of spheres i non-bonded at the site a. By taking into account the stoichiometry of the initial mixture and the specification of the bonds between the site A and B, the compressibility factor can be written in this form:

ZsASSOC.

2(m − 1) ∂X = ρs m ∂ρs ·

1 1 − X 2

¶¸

(2.41)

where X is the fraction of molecules non-bonded given by

. 1 X∆AB ρs m

(2.42)

By taking the limit of infinite association, the compressibility factor of the chain system is obtained as

CHAPTER 2. LCST IN POLYMER SOLUTIONS

ZsCHAIN

= lim Z

ASSOC.

X→0

2m − 2 ∂X = lim ρs m X→0 ∂ρs ·

1 1 − X 2

¶¸

(2.43)

as before, the compressibility factor is written as

ZsASSOC.

ρs ∂∆ m − 1 ρs 2 X ∆ 1+ , =− m m ∆ ∂ρs µ

(2.44)

so that in the limit of infinite association

lim

X→0

ZsASSOC.

ZsCHAIN

m−1 ρs ∂∆ = − 1+ m ∆ ∂ρs µ

m−1 = − m

∂ ln g HS (σ) 1 + ρs ∂ρs

(2.45) !

The compressibility factor for N chains formed of m segments is Z CHAIN . The excess compressibility factor due to the chain formation is then :

Z CHAIN

= m ZsCHAIN − lim ZsCHAIN ρ→0

= mZ CHAIN + (m − 1) = −(m − 1)ρ

∂ ln g HS (σ) . ρ

(2.46)

The corresponding excess Helmholtz free energy is then given by

ACHAIN = −(m − 1) ln g HS (σ). N kT

(2.47)

This expression has been obtained for a hard sphere reference system. The equilibrium expression can be obtained for any kind of reference system for which the monomer radial distribution function g is known. The TPT1 theory of Wertheim has certain limitations. First of all, it does not take into account the angle between the vectors from the center of the sphere to the various sites. Moreover, intramolecular bonding interactions have not been taken into account. These limitations can be overcome by taking into account higher order terms, or by including

CHAPTER 2. LCST IN POLYMER SOLUTIONS

additional graphs, such as ring graphs. Sear and Jackson [183], [184], and Chapman et al. [185], [186] have studied the effect of the intramolecular association and the formation of rings.

2.1.2

Chain Contribution Derived from the Cavity Function

Boubl´ık [90] has determined an expression identical to TPT1 of the excess free energy of an infinitely dilute dumbbell fluid in a mixture of hard spheres, as a function of the cavity function y. We briefly discuss this approach by considering a closed system where the number of particles N , the volume V and the temperature T are fixed. The probability of finding particle 1 in volume element dr1 at a position r1 , particle 2 in volume element dr2 at a position r2 , ..., particle N in volume element drN at a position rN is given by :

ρ(N ) (rN ) =

exp −U (rN )/kT dr1 ...drN ZN

(2.48)

where U (r N ) is the energy of the N particle system, and ZN is the usual configuration integral (partition function) given by

ZN (V, T ) =

...

exp −βU (r N ) dr1 ...drN .

(2.49)

For a subset of n particles from the total N , the probability of finding particle 1 in volume element dr1 at a position r1 , particle 2 in volume element dr2 at a position r2 , ..., particle n in volume element drn at a position rn is given by: h

... exp −βU (r N ) drn+1 ...drN N! (n) n ρ (r ) = , (N − n)! ZN R

where the factor

N! (N −n)!

(2.50)

is the number of ways of choosing n particles from the total

N . The extent to which the structure of the fluid deviates from complete randomness is related to the correlation function g n (r(n) ) which is defined as ρ(n) (rn ) = ρ(r1 )ρ(rn )...ρ(rn )g (n) (rn ),

(2.51)

where ρ(r1 ) is the probability of finding particle 1 at position r1 , integrated over the other particles, ..., ρ(rn ) the probability of finding particle n at position rn .

CHAPTER 2. LCST IN POLYMER SOLUTIONS

In a homogenous fluid, all the probabilities ρ(r1 ), ..., ρ(rn ) are equal, in this case equation (2.51) can be written as ρ(n) (rn ) = ρn g (n) (rn ).

(2.52)

And the correlation function g (n) (rn ) given by

exp −βU (r N ) drn+1 ...drN N! (n) n g (r ) = . (N − n)! ρn ZN R

If the total number of particles N is much greater than n, the factor

(2.53) N! (N −n)!

approximate to N n . The above expression simplifies to

g n (r(n) ) =

V n exp −βU (r N ) drn+1 ...drN R

(2.54)

The n body cavity function y (n) (rn ) is defined as

y (n) (rn ) = exp [βU (r n )] g n (r(n) )

(2.55) i

V n exp −βU (r N −n ) drn+1 ...drN R

and the pair cavity function is given by

y (2) (r1 , r2 ) =

V 2 exp −βU (r N −2 ) dr3 ...drN R

(2.56)

Consider that particle 1 and 2 form a dimer to give a system of N-2 monomers and one dimer D (there are N − 1 particles in total). The configurational integral of this system is given by

ZN −2,1D (V, T ) =

exp −βU (r N −2,1D ) dr1D , dr3 ...drN .

(2.57)

Note that this integral has fewer degrees of freedom than the integral of equation 2.49, as the energy U depends only on the relative positions. Integrating over dr 1D , R

( dr1D = V ), we obtain

CHAPTER 2. LCST IN POLYMER SOLUTIONS

ZN −2,1D (V, T ) = V

(2.58)

(2.59)

exp −βU (r N −2 ) dr3 ...drN

ZN (V, T )y (2)(r1 ,r2 ) . V

The free energy of an ideal gas of N particles is

IDEAL AN

= −kT ln

Λ−3N V N N!

while the total free energy AN corresponding to the configurational integral ZN is given by

AN = −kT ln

Λ−3N ZN N!

(2.60)

The excess free energy AEX N is thus given by

IDEAL = −kT ln AEX N = AN − AN

ZN (V, T ) , VN ¶

(2.61)

and the excess free energy of the system of N − 2 monomers and one dimer can be similarly obtained as

AEX N −2,1D = −kT ln

ZN −2,1D (V, T ) . V N −1 ¶

(2.62)

Therefore, the change of free energy due to the formation of one dimer can be expressed as

∆A1D =

AEX (N −2,1D)

−

AEX N

Z(N −2,D) V = −kT ln ZN µ

(2.63)

Substituting equation (2.58) into equation (2.63), we obtain: ³

∆A1D = −kT ln y (2) (r1 , r2 ) .

(2.64)

CHAPTER 2. LCST IN POLYMER SOLUTIONS h

For the hard sphere system, exp −βU (r N ) = 0 for overlapping configurations, and the free energy due to the formation of dimers can be written in term of the contact value of the pair distribution function of the hard sphere reference system: ³

HS(2) (σ) . ∆AHS 1D = −kT ln g

(2.65)

If we assume that the free energy change for the formation of additional dimers is identical to that of the infinitely dilute system, we can write the free energy of N D dimers as ³

HS(2) (σ) . ∆AHS ND = −ND kT ln g

(2.66)

A similar expression for the free energy can be obtained for the formation of chains of m tangent hard spheres. Consider a system of N particles, m of which form a chain, the excess free energy for such a system composed of N − m monomers and 1 chain (N − m + 1 species) is AEX N −m,1C

ZN −m,1C ) = −kT ln . V N −m+1 µ

(2.67)

The contribution due to the formation of one chain is given by

∆AN −m,1C = −kT ln

ZN −m,1C V m−1 ZN

(2.68)

Following similar steps as for dimer formation, the configurational integral of this system can be given in terms of the m-body cavity function y (m) :

ZN −m,1C (V, T ) = V

exp −βU (r N −m ) drm+1 ...drN

ZN (V, T )y (m) (r(m) ) V

(2.69)

m−1

The resulting contribution due to chain formation can be written as ³

∆AN −m,1C = −kT ln y (m) (r(m) ) ,

(2.70)

CHAPTER 2. LCST IN POLYMER SOLUTIONS and in the case of the formation of Nc chains of tangent hard spheres: ³

∆A = −Nc kT ln g HS(m) (σ) ,

(2.71)

where g HS(m) is the m-body pair distribution function of the hard sphere system at contact. A chain of m tangent spheres consists of m − 1 contacts. By assuming that each pair contact occurs with equivalent probability, we obtain: g HS(m) (σ) = (g HS(2) )m−1 .

(2.72)

This corresponds to the so-called linear approximation. The free energy due to the formation of Nc chains becomes : ∆A = −Nc (m − 1)kT ln g HS (σ),

(2.73)

hence, for a system of N chains formed of tangent hard spheres of diameter σ, the excess free energy due to the formation of the chains is

ACHAIN = −(m − 1) ln g HS (σ). N kT

(2.74)

This equation is the same as the expression derived from TPT1 theory (equation (2.47)). The chain contribution can be combined to the Carnahan-Starling expression for hard spheres to obtain the SAFT-HS theory [35]. We now describe the SAFT theory for a binary mixture of a chain molecule and a spherical solvents.

2.2

SAFT-HS Theory for Polymer Solution

We now consider a prototype polymer-solvent system corresponds to a binary mixture, where component 1 is represented as a hard sphere of diameter σ, and component 2 representing a monodisperse polymer, is modelled as a flexible chain formed from m2 tangent hard spheres of equal diameter σ. The attractive dispersion interactions are treated at the mean-field level of van der Waals, with an equal segment energy parameter α for each segment. As discussed in the introductory section, the thermodynamic properties of the chain molecules can be described using the theory of Wertheim [182], [83], [35]. We refer to the use of a mean-field description of the attractive interactions with the Wertheim chain contribution as the SAFT-HS

CHAPTER 2. LCST IN POLYMER SOLUTIONS

approach [17]. In this simplified version of the statistical associating fluid theory (SAFT) the Helmholtz free energy A for the mixture can be expressed as [17]

A AIDEAL AM ON O. ACHAIN = + + , N kT N kT N kT N kT

(2.75)

where N is the total number of molecules, T is the temperature, and k the Boltzmann constant. The separate terms are defined in the following discussion both for a n component mixture of chain molecules of different length, and for our symmetric model polymer solution of monomers and chain molecules. The ideal contribution is given by the usual expressions in terms of the the mole fractions and number densities of each component:

AIDEAL N kT

n X i=1

xi ln ρi νi − 1

= x1 ln ρ1 ν1 + x2 ln ρ2 ν2 − 1,

(2.76)

where xi = Ni /N is the mole fraction, ρi = Ni /V is the molecular number density (Ni is the number of molecules and V is the total volume), and νi is the thermal de Broglie volume of component i (incorporating the translational and rotational kinetic contributions). The contribution due to the interactions between the spherical monomeric segments AM ON O can be separated into a repulsive hard-sphere term AHS and an attractive mean-field term AM F :

AHS AM F AM ON O. = + . N kT N kT N kT

(2.77)

An accurate representation of the repulsive term of a multicomponent mixture of hard spheres, AHS , is given by the expression of Boubl´ık [187] and Mansoori et al. [188] as

AHS 6 = N kT πρ

"Ã

ζ23 ζ23 3ζ1 ζ2 + − ζ0 ln(1 − ζ3 ) + , 2 (1 − ζ3 ) ζ3 (1 − ζ3 )2 ζ3

(2.78)

where ρ = N/V is the total molecular number density of the system. The reduced densities ζl are defined as

CHAPTER 2. LCST IN POLYMER SOLUTIONS

ζl =

n πρ X xi mi σil , 6 i=1

(2.79)

where mi represents the number of spherical segments of the chain molecule i and σi the segment diameter. The equation (4.20) can be also written in terms of the segment density as

ζl =

n πρs X xs,i σil , 6 i=1

(2.80)

where ρs = Ns /V is the total number density of segments in the system, and xs,i = Ns,i /Ns is the mole fraction of segments of molecule i. In our case, all of the segments are of the same diameter σ, and component 1 comprises a single spherical segment (m1 = 1). As a result, expression (4.19) reduces to the well known Carnahan and Starling equation [74]:

4η − 3η 2 AHS , = (x1 + x2 m2 ) N kT (1 − η)2

(2.81)

where η = ζ3 is the packing fraction of the mixture, and where we have used the relation Ns = N (x1 + x2 m2 ) to express the free energy per molecule N rather than per spherical segment Ns . In the case of our model, the radial segment-segment distribution function at contact (which is required to describe the free energy contribution due to the bonding of the segments in a chain) can be obtained from the Carnahan and Starling relation as [189]

g HS (σ) =

1 − η/2 . (1 − η)3

(2.82)

The attractive term for the interaction between the monomeric segments in the mixture is written using the van de Waals one-fluid theory [5]. n n X AM F ρ X αij xi xj mi mj , =− N kT kT i=1 j=1

(2.83)

where αij is the van der Waals attractive constant (integrated attractive energy) for the interaction between segments of components i and j. For our symmetric

CHAPTER 2. LCST IN POLYMER SOLUTIONS

polymer model, the mean-field energies αij = α are equivalent for all segmentsegment interactions, and since m1 = 1, equation (4.23) is simply given by AM F ρ = − α (x1 + x2 m2 )2 . N kT kT

(2.84)

The change in the Helmholtz free energy in forming chains of hard-spherical segments from a reference system of hard spheres ACHAIN can be obtained as a special limit of the theory of Wertheim for associating systems [35, 83, 182]

ACHAIN N kT

= −

n X i=1

xi (mi − 1) ln g HS (σi )

(2.85)

= −x2 (m2 − 1) ln g HS (σ) . (2.86) In our particular case the chain contribution only describes the polymer molecule (component 2). The other thermodynamic properties such as the pressure P , the chemical potential of each component µi , the internal energy U , enthalpy H and Gibbs function G can be obtained from the total Helmholtz free energy using the standard thermodynamic relations [5]: P = − −

∂(A/T ) ∂(1/T ) V,N ,

H = U + PV , G =

∂A ∂V T,N ,

µi =

∂A , ∂Ni V,T,N j6=i

U =

µi Ni . In describing our model system, it is

useful to define a reduced temperature T ∗ = kT b/α, a reduced pressure P ∗ = P b2 /α, a reduced density ρ∗ = ρσ 3 or packing fraction η = ρs b, and a reduced volume v ∗ 1/ρ∗ , where b = πσ 3 /6 is the volume of a spherical segment. These reduced variables are used throughout our work. A reduced Gibbs free energy g ∗ of the mixture at a given temperature T , pressure P , and composition x can also be defined as

g∗ =

G 1 = (x1 µ1 + x2 µ2 ) , N kT kT

(2.87)

where G is the total Gibbs free energy of the system, and µ1 and µ2 the chemical potentials of the two components. The reduced internal energy of this system u ∗ is determined from:

CHAPTER 2. LCST IN POLYMER SOLUTIONS

U g = = −T N kT ∗

∂a∗ ∂T

(2.88)

V,x

where a∗ = A/N kT is the reduced Helmholtz free energy. The reduced entropy s∗ and enthalpy h∗ are calculated from

s∗ =

S = u∗ − a∗ , Nk

(2.89)

H = g ∗ + s∗ . N kT

(2.90)

and

h∗ =

In examining the stability or otherwise of a given phase in a mixture it is often more convenient to focus on the thermodynamic mixing functions. The reduced Gibbs ∗ is defined as free energy of mixing ∆gm

∗ ∆gm (T, P ) = g ∗ (T, P ) − x1 g1∗ (T, P ) − x2 g2∗ (T, P ),

(2.91)

where g1∗ and g2∗ are the reduced Gibbs free energies of the two pure components at the same conditions T and P . The reduced internal energy, volume, enthalpy ∆h∗m and entropy ∆s∗m of mixing are defined in the same way (c.f. equation (2.91)): ∆u∗m (T, P ) = u∗ (T, P ) − x1 u∗1 (T, P ) − x2 u∗2 (T, P ),

(2.92)

∗ ∆vm (T, P ) = v ∗ (T, P ) − x1 v1∗ (T, P ) − x2 v2∗ (T, P ),

(2.93)

∆h∗m (T, P ) = h∗ (T, P ) − x1 h∗1 (T, P ) − x2 h∗2 (T, P ),

(2.94)

= ∆u∗m − P ∗ ∆u∗m /T ∗ ,

(2.95)

and ∆s∗m (T, P ) = s∗ (T, P ) − x1 s∗1 (T, P ) − x2 s∗2 (T, P ).

(2.96)

The Gibbs free energy of mixing is sensitive to both the enthalpic and entropic mixing contributions through the relation

CHAPTER 2. LCST IN POLYMER SOLUTIONS

∗ ∆gm = ∆h∗m − ∆s∗m .

(2.97)

, which is the reduced form The conditions for the stability of a single phase in the mixture are given by [5] ∗ < 0, ∆gm

(2.98)

and

∗ ∂ 2 ∆gm ∂x2

= T,P

∂ 2 ∆g ∗ ∂x2

> 0.

(2.99)

T,P

If either one of these conditions is not satisfied for a given mole fraction x, the mixture is not stable and separates (demixes) into two liquid phases. In the following section we use the thermodynamic functions of mixing to examine the stability of the fluid phase in model polymer solutions described with the Wertheim TPT1 theory.

2.3

Phase Diagrams in Attractive Hard Spheres and Chain Binary Mixtures

The phase behaviour of binary mixtures of short and long chain molecules is extremely sensitive to the segment interactions and to the asymmetry in the chain lengths of the two components. Depending of the difference in chain length and interactions, this type of system can exhibit type I, II, III, IV or V phase behaviour, in the classification scheme of Scott and van Konynenburg, [3] which can be used to characterise binary mixtures in terms of the P T projections of the P T x surface ( see Chapter 1 Introduction, section 1.2). We have calculated the fluid phase equilibria for our model polymer solutions using the SAFT-HS expressions by numerically solving the conditions of equality of temperature, pressure, and chemical potential of each component in each phase. The P T projections of the fluid phase equilibria are shown in figure 2.2. Here, the first component is represented as spherical and the second as a chain of tangent spheres, with equal size and energy interaction parameters.

CHAPTER 2. LCST IN POLYMER SOLUTIONS

Figure 2.2: P T pressure-temperature diagrams for binary mixtures of hard spheres + chains of m2 tangent hard spheres of same diameter, and same mean field energy per segment, with a) m2 = 2, m2 = 6, b) m2 = 7, and c) m2 = 100. The continuous lines are the vapour-pressure curves of the pure components and the three phase line. The dashed lines are critical lines. The white circles are the critical points of the pure components. The white triangles are LCEP and UCEP.

CHAPTER 2. LCST IN POLYMER SOLUTIONS

For the binary mixtures comprising molecules with a shorter chain length (m 2 = 2 to 6) the two components are completely miscible corresponding to a type I behaviour (see Chapter 1, section 1.2.1) : a gas-liquid critical curve extends continuously from the critical point of one component to the critical point of the other (see figure 2.2 a)). As the chain length of component 2 is increased its critical temperature is seen to increase while its critical pressure decreases. When the critical temperatures of the monomeric solvent and the chain fluid become sufficiently different, an instability in the liquid phase appears close to the critical point of the solvent. For molecules with chain lengths m2 > 6, our model â&#x20AC;&#x153;polymerâ&#x20AC;? solutions are found to exhibit type V phase behaviour (the P T projection for the mixtures with m2 = 7 and m2 = 100 are shown in the figures 2.2 b) and 1 c)). Type V behaviour (see chapter 1, section 1.2.3) is characterised by a region of liquid-liquid immiscibility close to the critical point of the more volatile (smaller) component. A liquid-liquidvapour three-phase curve is associated with this region of immiscibility, and it is bounded by upper and lower critical end points (UCEP and LCEP). The threephase curve and critical end points can be clearly seen in the inset of figure 2.2 b) for the system with m2 = 7.

Figure 2.3: Global phase diagram for the binary mixture of two hard sphere chains of respectively m1 and m2 segments. The solid line delimits regions of the diagram where different types of phase behaviour are encountered. It represents all the type V binary mixtures for which the critical point of the pure shorter chain, LCEP and UCEP are confused.

CHAPTER 2. LCST IN POLYMER SOLUTIONS

The “gas-liquid” critical curve is no longer continuous, but has two branches; the locus of “liquid-liquid” critical points emerging from the LCEP correspond to the LCSTs of the system at increasing pressures. As the chain length of component 2 is increased, the region of liquid-liquid immiscibility increases (figure 2.2 c)); a chain of m = 100 can already be taken to represent a low molecular weight polymer (M W > 1 kg mol−1 ). The predictions of the Wertheim TPT1 theory with a van der Waals (temperature independent) attractive contribution thus confirm the finding of Song et al. [141] that such a system with symmetric interactions (corresponding to a Flory-Huggins parameter of χ = 0) will exhibit LCST behaviour. Furthermore, the symmetric system is not found to exhibit UCST behaviour at lower temperatures (which would correspond to type IV behaviour in the Scott and van Konynenburg nomenclature, cf chapter 1, section 1.2.4) and type V behaviour is always seen regardless of the chain length of the polymer. As Patterson [33] has shown in his studies with a compressible lattice model, UCST behaviour is possible when the system has unfavourable attractive interactions (χ > 0), but we will not be examining models with asymmetric attractive interactions in this contribution. As we have shown, the limit of instability of the liquid phase for our model binary mixtures with symmetrical interactions occurs at m2 = 7 for the m1 = 1. When component 1 is a dimer (m1 = 2) the first sign of liquid-liquid immiscibility occurs when the second component has a chain length of m2 = 15; in the case of m1 = 3 we find that second component must have a chain length of at least m2 = 24 for the system to be immiscible. This information about the limiting behaviour is collected in figure 2.3; it is rather surprising to find a very simple power law m2 = 7.2346m1.0955 between the chain length of the shorter component and that 1 of the longer component which first gives rise to an instability in the liquid phase. Care should be taken, however, in drawing a direct relationship between the chain length in our model mixtures with unrealistic symmetric interactions and the chain length of the alkanes. In modelling n-alkanes with the SAFT approach, the spherical segments are taken to represent a group of atoms (united atom approach), and a parameterisation m = (C − 1)/3 + 1 is often used to relate the m to the number of carbon atoms C in the n-alkane chain [190]. This relation has proved to be useful in previous accurate descriptions of the fluid phase n-alkane molecules [99], and it has been shown to be adequate for very long alkanes [191] and polyethylene systems [98, 192]. In the case of binary systems of methane with its n-alkane homologues,

CHAPTER 2. LCST IN POLYMER SOLUTIONS

type V fluid phase behaviour is first observed experimentally in the mixture of methane + n-hexane [5]; methane and n-pentane are miscible in all proportions and their mixture corresponds to type I phase behaviour. It is encouraging to find that one is able to provide an excellent description of type V behaviour for methane + n-hexane with the SAFT-VR approach, which incorporates the Wertheim TPT1 expressions, simply by using Lorentz-Berthelot combining rules for the unlike interactions [193]; the liquid-liquid immiscibility appears over a small (ca. 15 degree) temperature range close to the critical point of methane. When the chain length of the shorter component is increased, the limit of instability of the liquid phase occurs for progressively longer chain lengths of the second component. For example, the first member of the n-alkane homologous series mixture which exhibits liquidliquid immiscibility with ethane is n-C19 H40 , while propane + n-C39 H80 , is the first immiscible binary mixture in the case propane. The fluid phase behaviour of alkane + polyethylene polymer solutions is, in essence, an extreme case of the phase behaviour exhibited by the methane + nhexane system ( see in chapter 1, section 1.3.1 the description of polymer solution T x diagrams). The cloud curve in polymer solutions corresponds to type V liquidliquid immiscibility which is highly distorted due to the difference in size of the two components; the vapour regions are hidden close to the solvent axis as the polymer is so involatile. The main aim of our work is to examine in detail the unusual region of liquid-liquid immiscibility that characterises these systems. In particular, we focus on the thermodynamic contributions that lead to the lower critical solution temperatures in polymer solutions, and make comparisons with other systems in which LCST behaviour is governed by enthalpic contributions. In order to display the main features of the behaviour, it is convenient to examine a prototype polymer-solvent system with a less extreme chain length difference. The pressure-temperature P T phase diagram for a binary “polymer” solution with m1 = 1 and m2 = 10 is shown in figure 2.4. As expected from our earlier discussion, the phase diagram corresponds to type V phase behaviour. The coexisting phases are better seen from an examination of the temperature-composition T x slices of this diagram at constant pressures. Three such slices are presented in figure 2.5; these are also indicated in figure 3. In the case of the lowest pressure P 1∗ = 4 × 10−4 (figure 2.5 a)), a large region of vapour-liquid equilibrium (VLE) can be seen from the boiling point of component 1 (m1 = 1) to that of component 2 (m2 = 10).

CHAPTER 2. LCST IN POLYMER SOLUTIONS

For significant part of the phase diagram, the dew-point curve (vapour branch) is almost indistinguishable from the y axis indicating that there are essentially no chain molecules in the vapour phase; this is due to the low vapour pressure of the chain molecule with m2 = 10, and it is even more dramatic for longer chains. For mixtures of spherical solvents with long chain molecules, the phase behaviour is very skewed.

Figure 2.4: P T pressure-temperature diagram for a binary mixture of hard spheres + chains of 10 tangent hard spheres of same diameter and mean field energy per segment (system (4)). The continuous lines are vapour-pressure curves of the pure components. The dashed lines are critical lines. The white circles are critical points of the pure components. The white triangles are LCEP and UCEP joined by the 3-phase line. The dash-dot lines denote constant temperature or pressure slices. The pressure indicated by P3∗ = 0.010 in figure 3 is above the critical points of both pure components, and in these cases, a region of vapour-liquid equilibrium is seen bounded by two critical points (see figure 2.5 c)). At high pressures such as P 3∗ , the term “fluid-fluid” is more convenient than “vapour-liquid” in designating the coexistent phases, as their densities are higher than the usual gas densities in the neighbourhood of the LCST. It is also important to note that the high temperature fluid-fluid critical points are essentially part of the vapour-liquid critical curve which meets the critical point of the chain molecules, and are very different in nature from the UCST behaviour found in systems of partially miscible liquids. This can lead to some confusion about the existence (or otherwise) of so-called closed-loop behaviour in such systems (e.g., see [169]; as we will show later “low”- temperature closed-loop

CHAPTER 2. LCST IN POLYMER SOLUTIONS

behaviour with its corresponding LCST and UCST is a consequence of directional interactions such as hydrogen bonding [5].

Figure 2.5: T x temperature-composition diagrams, corresponding to the P T diagram of figure 2 obtained for the binary mixture of hard spheres + chains of 10 tangent hard spheres with mean field attractions (see figure 2), where x s,2 is the fraction of polymer segments. The considered pressure slices are at P ∗ = a) P1∗ , b) P2∗ , and c) P3∗ . The continuous lines are coexistence curves. The white circles are critical points (UCST and LCST) of the binary mixture.

CHAPTER 2. LCST IN POLYMER SOLUTIONS

At the intermediate pressure P2∗ = 0.0037 (see figures 2.5 b) and 2.6), an extensive VLE region is still present, with a vapour-liquid critical point observed at high temperatures and high concentrations of component 2. In practice the critical point is at such high temperatures that the long chain alkanes tend to decompose before this point is reached. A second, very narrow, vapour-liquid region is observed close to pure component 1 (see the enlarged T x diagram in figure 2.6)). Furthermore, a region of immiscibility between two liquid phases, one richer in the spherical solvent and one richer in the chain, is also observed in the low-temperature part of figure 2.6. This liquid-liquid equilibrium region is bounded at higher temperatures where it meets the vapour-liquid coexistence region at a three phase line, and has a lower temperature bound at an LCST. This region of liquid-liquid immiscibility and LCST behaviour is observed only when the difference in chain length between the “solvent” and the “polymer” is large enough; for our model polymer solution with symmetric interactions the first binary system to exhibit this instability is the m1 = 1 solvent with an m2 = 7 tangent chain. The extent of the liquid-liquid immiscibility increases for longer polymer chains.

Figure 2.6: Enlarged T x temperature-composition diagram corresponding to the T x diagram at pressure P2∗ shown in figure 3 b), obtained for the binary mixture of hard spheres + chains of 10 tangent hard spheres with mean field attractions. The continuous lines are coexistence curves. The white circles are critical points (UCST and LCST) of the binary mixture. The dashed line is the three phase line. The dash-dot line denotes a constant temperature slice at T2∗ in the liquid-liquid immiscibility region.

CHAPTER 2. LCST IN POLYMER SOLUTIONS

2.4

Properties of Mixing

As we have already mentioned in the introduction this type of fluid phase separation may at first appear surprising considering that the molecules are made up of segments of the same size which interact with the same segment-segment mean-field energy (χ = 0). We now follow on from the ideas of Patterson [33] and examine the molar Gibbs free energy of mixing (see Chapter 1, section 1.2.7) in order to understand the nature of such a phase separation by comparing the enthalpic and entropic contributions. To discuss the molar properties of mixing, it is useful to reduce them by kT . Particular attention is paid to the effect of the chain on the density of the solvent (the so-called compressibility effect). Four systems are considered in order to assess the contributions leading to the liquid-liquid phase separation in our model polymer solutions.

2.4.1

Ideal Gas Mixture

System 1 corresponds to a mixture of ideal molecules. One should recall that demixing never occurs for such a system. For the ideal gas mixture, the Gibbs energy of mixing is always negative and only comprises an entropic contribution of mixing ∗ = −∆s∗ and ∆h∗ = 0): (∆gm m m ∗ ∆gm = x1 ln x1 + x2 ln x2 ,

(2.100)

so that

∗ ∂ 2 ∆gm ∂x2

= T,P

1 . x1 x2

(2.101)

∗ is symmetrical about x = 1/2 with a minimum value of For ideal mixing ∆gm

ln 2 ≈ 0.69 (see figures 2.7 a) and b)). The second derivative of the Gibbs function with composition is always positive, and is also symmetrical about x = 1/2 with a minimum at 4 (see figures 2.7 c) and d)).

100

CHAPTER 2. LCST IN POLYMER SOLUTIONS

Figure 2.7: Reduced Gibbs free energy of mixing and its second derivative obtained for different model systems at constant pressure P2∗ and temperatures T1∗ (a), c)) and T2∗ (b), d)). The thin dash line corresponds to an ideal gas mixture (system(1)). The thick dash line corresponds to a mixture of one monomeric solvent and a polymer of 10 segments, and is obtained from Flory-Huggins theory (system (2)). The thin continuous line corresponds to a binary mixture of hard spheres and a chain of 10 tangent hard spheres modelled with SAFT-HS (system (3)). The thick continuous line corresponds to the same binary mixture as system (3) but with mean field segment-segment attractions (system (4)).

2.4.2

Polymer Solution Modelled with Flory-Huggins Theory

System 2 corresponds to the lattice model of Flory and Huggins [50] for solventpolymer systems. In this model, the solvent is represented by a single lattice site, and the polymer by a chain occupying m2 = 10 adjacent lattice sites. Pairwise additivity is assumed, but only nearest neighbours interact. As for the continuum SAFT-HS model described earlier, the solvent-solvent, solvent-polymer, and polymer-polymer segment attractive interactions are all set to be equal to a value α. The original model developed by Flory and Huggins corresponds to a fully-occupied lattice, and so one can not treat pressure or density changes. The reduced entropy of mixing ∆s∗m only depends on the composition, and is given by [50] ∆s∗m = −x1 ln φ1 − x2 ln φ2 ,

(2.102)

with the volume fractions defined as

φ1 =

x1 , x1 + m 2 x2

(2.103)

101

CHAPTER 2. LCST IN POLYMER SOLUTIONS and

φ2 = 1 − φ 1 =

m2 x2 . x1 + m 2 x2

(2.104)

The enthalpy of mixing ∆h∗m (which for the Flory-Huggins incompressible lattice model also corresponds to the internal energy of mixing ∆u∗m ) is calculated by assuming that the lattice sites interact only with z nearest neighbours, such that ∆h∗m = χx1 φ2 ,

(2.105)

where in our notation the interaction parameter χ is defined as

χ=−

z∆α12 , kT

(2.106)

with 1 ∆α12 = α12 − (α11 + α22 ). 2

(2.107)

In our case all the mean-field energies αij are taken to be equal, so that the χ parameter and the enthalpy of mixing are zero. This means that the Gibbs free energy of mixing is always positive and is only governed by the entropic term: ∗ ∆gm = −∆s∗m = x1 ln φ1 + x2 ln φ2 .

(2.108)

The second derivative of the reduced Flory-Huggins Gibbs free energy with respect to composition,

∗ ∂ 2 ∆gm ∂x2

= T,P

(m2 − 1)2 1 + , x1 x2 (x1 + m2 x2 )2

(2.109)

is then always positive, and even greater than that of the ideal gas mixture (see figures 2.7 c) and 2.7 d)). As a result demixing is never predicted for the FloryHuggins fully occupied lattice model when χ = 0, regardless of the chain length m 2 of the polymer. As for the ideal mixture the reduced Gibbs function of mixing (and its second derivative) is independent of the temperature (athermal). It should be noted that for certain unfavourable values of χ > 0, liquid-liquid separation leading

CHAPTER 2. LCST IN POLYMER SOLUTIONS

102

to an upper critical solution temperature (UCST) can be observed for the FloryHuggins model [194]. This liquid-liquid phase separation is driven by the enthalpy of mixing, as was discussed in the introduction.

2.4.3

Polymer Solution Described by TPT1 Theory

We now examine a continuum versions of the Flory-Huggins system with χ = 0. System 3 is taken to represent a binary mixture of hard spheres of diameter σ and chains formed from m2 = 10 tangent hard spherical segments of equal diameter σ. Only repulsive interactions are taken into account in the first instance (i.e., χ = 0). The mixture is modelled with the the SAFT-HS (Wertheim TPT1) expressions as given in section 2 (equation (4.18)), without dispersive attractions (A M F = 0). This system is athermal, so that the phase equilibria depends only on the composition and the density. Before we discuss the results for the purely repulsive system 3, we define system 4 as the full (repulsive and attractive) model of the solvent-polymer mixture described in section 2.3 with m1 = 1 and m2 = 10. As discussed earlier, segment-segment attractive dispersion interactions are incorporated at the meanfield level of van der Waals, where all of the attractive interactions are of the same magnitude α; the Flory-Huggins parameter is also zero χ = 0 in this case. The P T projection of the phase behaviour, and constant pressure T x slices for this system have already been presented in figures 2.4, 2.5 and 2.6. It is clear that the fluid phase equilibria of the full system 4 exhibits a region of instability in the liquid phase and that the behaviour is both temperature and pressure dependent. The nature of this instability will become clearer from an inspection of the thermodynamic functions of mixing. In the case of the continuum systems 3 and 4, it is not possible to write ∗ , ∆s∗ and ∆h∗ . The equation of state is more down analytical expressions for ∆gm m m

conveniently analysed in terms of the Helmholtz free energy. The liquid density root of the pressure has to be solved for a given composition and temperature, and the second derivative of the Gibbs free energy with respect to composition can be determined analytically using the standard thermodynamic relations [5]:

∂ 2g∗ ∂x2

= T,P

∂ 2 a∗ ∂x2

T,v ∗

−

´2 ∂ 2 a∗ ∂x∂v ∗ T ³ 2 ∗´ ∂ a ∂v ∗2 T,x ³

(2.110)

103

CHAPTER 2. LCST IN POLYMER SOLUTIONS

∂ 2 a∗ ∂x2

T,ρ

−

´2 ∂ 2 a∗ ∂x∂ρ T ³ 2 ∗´ . ³ ∗´ ∂a + ρ ∂∂ρa2 2 ∂ρ T,x T,x

∗ for the four systems (ideal, FloryThe reduced Gibbs free energy of mixing ∆gm

Huggins, repulsive system, and the full polymer solution model) are depicted in figures 2.7 a) and b), at a reduced pressure of P2∗ = 0.0037 and at reduced temperatures T1∗ = 0.050 and T2∗ = 0.090 (cf. figures 2.4 and 2.6). The temperatures and pressure are chosen so that T1∗ is below the LCST of the full model (corresponding to a mixed state), and T2∗ is above the LCST (corresponding to a demixed state). The corresponding second derivatives of the Gibbs free energy with composition ¡

∂ 2 g ∗ /∂x2

T,P

are shown in figures 2.7 c) and 2.7 d). The reduced Gibbs free energy

of mixing is always negative for the all the systems studied (see figures 2.7 a)). At the temperature T1∗ below the LCST, the liquid phase of the four systems is stable over the entire composition range, as confirmed by a positive value of the second derivative of the Gibbs function depicted in figure 2.7 b). A liquid phase instability is found for system 4 (the full model incorporating attractive interactions) at a temperature T2∗ (above the LCST): negative values of the second derivative of the Gibbs function can be seen in figure 2.7 d) for this system at low mole fractions of the chain molecules, which is consistent with the regions of liquid-liquid phase separation exhibited by the system (see figures 2.4, 2.5, and 2.6)). It is much more difficult to observe a change of curvature in the Gibbs free energy of mixing of system 4 (cf. figure 2.7 b)), which is the feature of the immiscibility. The solutions of the equation

∂ 2g∗ ∂x2

= 0,

(2.111)

T,P

at the reduced temperature T2∗ = 0.090 and pressure P2∗ = 0.0037 give the spinodal compositions, i.e., the limits of instability, as x2 = 0.006 and x2 = 0.016. L2 1 Note that the binodal (phase coexistence) compositions xL 2 = 0.004 and x2 = 0.021

are slightly different from these values (in terms of the mole fractions of segments xs,2 depicted in figure 2.6 the coexistence values are 0.039 and 0.177). Using the Wertheim TPT1 expression for the free energy of chain molecules together with a simple mean-field description of the attractive interactions we have

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104

thus confirmed that liquid-liquid immiscibility with a corresponding LCST is exhibited by a system in which the size and energy segment parameters are all equal (χ = 0), when the chain is of sufficient length (m2 > 6). What happens in the case of a purely repulsive system? The Gibbs free energy of mixing and its second derivative with composition for the model mixture of a hard-sphere solvent and chains of m2 = 10 tangentially bonded hard segments of the same size (system 3) are also shown in figure 2.7. The purely repulsive system 3 does not show any evidence of ∗ < 0 and ∂ 2 g ∗ ∂x2 demixing with ∆gm

T,P

> 0 at the two temperatures studied

(note that in the case of such an athermal system the important parameter is the ratio T ∗ /P ∗ ). In order to confirm that the liquid phase is stable in the repulsive case, we have carried out a global minimisation of the second derivative of the Gibbs function for system 3 over the entire composition range (0 < x2 < 1) for fluid-like packing fractions(0 < η < 0.5). No negative values of the second derivative of the Gibbs free energy are found. This is true even for systems with polymer chain lengths of up to m2 = 1 × 105 . From this analysis one can be confident that liquidliquid demixing is not observed for polymer-solvent systems of hard spheres and hard-sphere chains with segments of the same size regardless of the chain length, albeit within a Wertheim TPT1 description. This finding is in keeping with the results of Taylor and Lipson [155] who showed that in the case of such a system the change in the dimension of a polymer with the solvent density is consistent with that of a polymer in a good solvent. The lack of liquid-liquid immiscibility in these fully repulsive models of polymer solutions with segments of the same size may not at first appear very surprising. However, as was mentioned in the introduction, fluid phase separation is possible in purely repulsive systems of hard spheres and hard-sphere chains when the hard spheres are of a much larger diameter than the spherical segments making up the chain. In this case the instability of the liquid phase is due to the Asakura and Oosawa depletion interaction (an effective attractive interaction between the larger spherical particles), and has been confirmed by theoretical and simulation studies of model colloid-polymer systems [156–161,163,164,164,195]. We will not discuss the possibility of fluid phase separation in athermal mixtures of hard spheres and hard-sphere chains of different hard-sphere diameter as the focus here is on polymer solutions and not on colloidal systems; the limiting diameter ratio and chain length which lead to a liquid phase instability in polymer-colloid systems are studied in a separate contribution using the Wertheim TPT1 description [196].

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The important role played by the attractive interactions in the instability of the liquid phase for our simple models of polymer solutions is clear from a comparison of the Gibbs free energy of mixing and its second derivative with composition shown in figure 2.7 for system 3 (repulsive model) and system 4 (full repulsive-attractive model). We should also note that although our Flory-Huggins model (system 2) incorporates attractive interactions, no liquid-liquid immiscibility is predicted in this case. This is because the Flory-Huggins model represents a fully occupied (incompressible) lattice, and the description does not incorporate density and pressure changes. It is therefore also clear that though attractive interactions offer the key to understanding the phase separation, changes in density also play an important part in the appearance of LCSTs in polymer solutions. In order to study the thermodynamics of mixing and demixing close to the LCST in more detail, we have determined some of the other important mixing functions for the full model (system 4) using the SAFT-HS description presented in section 2 (see figure 2.8). The negative values of the reduced internal energy of mixing ∆u ∗m presented in figure 2.8 a) are a direct consequence of the attractive contribution to the free energy AM F (equation 4.24). As the temperature is increased to T2∗ , ∆u∗m becomes more negative; we will show later that this is due to an increase in density (decrease in molar volume) of the solvent on addition of the polymer chain which is seen at higher temperatures. The reduced enthalpy of mixing ∆h∗m = ∗ /T ∗ is shown in figure 2.8 b), and is found to be very similar in form ∆u∗m + P ∗ ∆vm ∗ is significantly and magnitude to ∆u∗m : while the reduced volume of mixing ∆vm

more negative at T2∗ than at T1∗ (see figure 2.8 d)). The negative sign of volumes of mixing obtained here is in agreement with MC simulation [197] and experimental studies of mixture of methane + tetracosane [198]. The reduced pressure is small (∼ 10−4 ), so that the contribution due to the compression of the system (P ∆vm term) is small. The consequence of this is that the enthalpy of mixing is always negative, and becomes even more negative as the temperature is increased above the LCST. In other words the enthalpy of mixing is characterised by favourable (attractive) interactions between the solvent and the polymer, and does not favour a demixed liquid state. The unfavourable entropy of mixing turns out to be the driving force of the instability of the liquid phase as the temperature is increased above the LCST. studied (see figure 2.8 c)). In the case of a temperature T 1∗ well below the LCST (mixed state), the entropy of mixing is positive over the entire

106

CHAPTER 2. LCST IN POLYMER SOLUTIONS composition range; it is dominated by the ideal contribution to mixing −

i (xi ln xi ),

which for an equimolar mixture ∼ 0.69. When the system is at a temperature T 2∗ above the LCST (demixed state), there is a range of compositions close to the pure monomeric solvent for which the entropy of mixing is negative ∆s∗m < 0. This negative contribution of the entropy of mixing leads to a positive (less favourable) ∗ = ∆h∗ − ∆s∗ , and to the change in contribution to the free energy of mixing ∆gm m m

curvature of the Gibbs function observed for the solvent rich states (see figure 2.7 d)) which is signature of the instability of the liquid phase.

Figure 2.8: Reduced properties of mixing (a) internal energy, b) enthalpy, c) entropy, d) molar volume) obtained for a binary mixture of hard spheres and a chain of 10 tangent hard spheres with mean field attractions (system (4)), at constant pressure P2∗ and at temperature T1∗ in the stable region, and at temperature T2∗ in the liquidliquid immiscibility region. Let us now examine the entropic contributions in more detail. In our simple mean-field (temperature independent) description of the attractive interactions, the contribution of the attractions to the Helmholtz free energy and the internal energy are identical (cf. equation (2.88)). This in turn means that there is no attractive contribution to the entropy (cf. equation (2.89)). The entropy of mixing ∆s ∗m is governed by two terms: an ideal mixing term , and a repulsive term due to the hard-sphere and chain contributions. The ideal entropy of mixing term is always positive and favours mixing. The change of sign of ∆s∗m is thus due to the repulsive contribution, which is very sensitive to changes in the density of the mixture. A degree of caution is necessary at this point as the change in density itself will turn

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out to be the result of a balance between the repulsive and attractive interactions.

2.5

Density Changes

The reduced liquid density ρ∗ of the mixture has been determined at a constant pressure of P2∗ for temperatures above (T1∗ ) and below (T2∗ ) the LCST, and is displayed in figure 2.9 a). It is important to note that no vapour roots were found for the conditions studied. At the lower temperature T1∗ , the reduced density of the solvent ρ∗ is relatively high and decreases monotonically as the composition of the chain molecule is increased in the mixture. This gives rise to a small negative volume of mixing (see figure 2.8 d)). By contrast for the higher temperature T2∗ (above the LCST) the reduced density of the solvent is lower than at the denser state T 1∗ , and a maximum is observed with composition for low mole fractions of the chain molecule; this composition range corresponds to that where the demixing occurs. As a conse∗ quence, the reduced volume of the mixture decreases and the volume of mixing ∆vm

is more negative at T2∗ than at T1∗ especially for low mole fractions of polymer(see figure 2.8 d)). For these low mole fractions of chain molecules, ρ∗ essentially represents the density of the solvent. The presence of this maximum in density can be used to explain why the entropy of mixing changes sign at low-polymer compositions: the entropy decreases as the mixture becomes denser and more contracted (the free volume decreases), turning positive again when the density drops. One could now ask the question: Why is there a maximum in the density in such a system in the first place? An examination of the pressure-density isotherms of the polymer solution for different mole fractions of the mixture can help in the understanding of this effect. Three T2∗ isotherms corresponding to mixtures with mole fractions of x2 = 0 (pure solvent), x2 = 0.01 (maximum in density shown in figure 2.9 a)), and x2 = 0.05 are depicted in figure 2.9 b). The reduced pressure P ∗ can be separated into two ∗ due to the repulsive interactions which incorporates the terms: a positive part Prep ∗ ideal, hard-sphere, and hard-sphere chain contributions; and a negative part P att

corresponding to the attractive contribution.

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108

Figure 2.9: a) Reduced density obtained for system (4) at constant pressure P 2∗ and temperatures T1∗ and T2∗ . b) Isotherms obtained for the same system at temperature T2∗ in the demixing region, for different chain mole fractions x2 : the thick continuous line , and white triangles, correspond to x2 = 0 (pure solvent), the thin continuous line, and white diamonds, correspond to x2 = 0.01, and the dashed line and white squares correspond to x2 = 0.05.

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109

Figure 2.10: a) repulsive and b) attractive parts of the pressure, at temperature T2â&#x2C6;&#x2014; , corresponding to the isotherms shown in figure 7 b) for different chain mole fractions x2 : the thick continuous lines correspond to x2 = 0 (pure solvent), the thin continuous lines correspond to x2 = 0.01 and the dashed line correspond to x2 = 0.05.

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110

As the mole fraction of chain molecules in the mixture is increased, the repulsive â&#x2C6;&#x2014; becomes more positive, corresponding to more repulsive interpressure term Prep

actions in the system (see figure 2.10 a)); the opposite is seen in the case of the â&#x2C6;&#x2014; which becomes more negative on the addition of the polymer attractive term Patt

(see figure 2.10 b)). On increasing the mole fraction of the polymer, the repulsive interactions increasingly dominate at higher densities, while the attractive play an increasingly important role at lower densities. The competition between these two terms gives rise to the different isotherms shown in figure 2.9 b); the isotherms are seen to have progressively steeper slopes. The liquid roots of the isotherms for the fixed pressure of P2â&#x2C6;&#x2014; are also shown in figure 2.9 b); on addition of polymer the density root is first seen to increase (diamond for system with x2 = 0.01) from the pure solvent value (triangle, x2 = 0), and then decreases again (square, x2 = 0.05), giving rise to the maximum in the density root (see figure 2.9 a)). This is summarised in a pictorial fashion in figure 2.11. For a higher density solvent (lower temperature), the addition of polymer (at fixed pressure) leads to a decrease in the density of the mixture due to the dominant repulsive contribution. When a small amount of polymer is added to a solvent of lower density (on increasing the temperature towards the critical point of the pure solvent), the density of the mixture first increases due to the dominant attractive contribution; however, when polymer is progressively added to this system the effect is then reversed and the density decreases as the repulsive interactions again become dominant. The large chain-length asymmetry between the monomeric solvent and the polymer is responsible for these rather peculiar density variations. For low-density (high-temperature) states, the presence of small amounts of polymer in the solvent cause a marked increase in the density. As a consequence of the increase in confinement the entropy of the mixture decreases and the system becomes unstable leading to a demixed state. At temperatures below the LCST such an increase in density is not seen, and the liquid phase is stable over the entire composition range.

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111

Figure 2.11: Schematic representation the binary mixture of hard spheres and chains to represent the evolution of the density as a function of the composition, at fixed temperature and pressure. Grey spheres are solvent molecules and black spheres are chain segments. In this work all of the segment-segment attractive interactions are chosen to be equal, and no specific interactions (e.g., hydrogen bonding) are incorporated to promote demixing between the monomer and the polymer. A very different class of systems exhibit liquid-liquid immiscibility bounded at lower temperatures by an LCST. These systems almost invariably involve aqueous solutions, and hydrogenbonding directional interactions are the key to the behaviour in this case [5]. The first reports of LCST behaviour were almost 100 years ago for the mixtures nicotine + water [19], and water + butan-2-ol [199]; such mixtures present closed-loop regions of liquid-liquid immiscibility (also referred to as re-entrant miscibility) bounded at high temperatures by a UCST and at low temperatures by an LCST. In the nomenclature of Scott and van Konynenburg this corresponds to type VI phase behaviour. By contrast to the polymer solutions examined earlier, the asymmetry in size of the two components is not responsible for the demixing observed in these

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112

systems with directional interactions. Instead, the large enthalpic contributions due the hydrogen-bonding attractive interactions which are favoured at low temperatures are responsible for the re-entrant miscibility.

2.6

Associating Hard Spheres

As a final exercise we take a leaf out of the paper of Song et al. [141] and examine the thermodynamics of mixing of molecules with directional attractive interactions, making comparisons with the polymer-solution thermodynamics where appropriate. It is useful to consider a symmetrical binary mixture of equal-size hard spheres (m 1 = 1 and m2 = 1) with dispersive interactions between like species (α11 = α22 = α), but no dispersive interactions between unlike species (α12 = 0). A highly directional interaction between molecules 1 and 2 is also included; the interaction is modelled simply as a short-range off-center square-well site, characterised by a site-site energy ²AB and bonding volume KAB . A full description of the free energy of association for this model is given in the Appendix B; the Wertheim TPT1 description which was originally developed to treat the thermodynamics of systems with directional interactions is used to describe the association in the symmetrical mixture. The dispersive attractive interactions are again treated at the mean-field level (cf. SAFTHS equation of state presented in section 2 without the chain contributions). The SAFT-HS description has already been used to examine the fluid phase behaviour of this simple model of an associating mixture [200], and closed-loop and LCST behaviour was found for certain values of the site-site parameters. We determine the phase coexistence of the mixture using a set of parameters (²∗AB = ²AB /(α/kb) = 1.1 ∗ and KAB = KAB /σ 3 = 5.55938 × 10−4 ) that give rise to the re-entrant behaviour,

and examine in detail the contributions to the Gibbs free energy which determine the stability of the phases.

2.6.1

Phase Diagram

A temperature-composition T x slice of the phase behaviour for a reduced pressure of P4∗ = 4.588 × 10−2 is shown in figure 2.12. The system exhibits a closed-loop liquid-

liquid immiscibility region with a UCST at T ∗ = 0.16 and a LCST at T ∗ = 0.13. In this particular case, the phase boundaries are symmetrical with respect to the equimolar composition (x = 0.5), since the interactions between the molecules 1

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113

and 2 are identical. The gas-liquid coexistence regions, which are also found in this system, are not seen over this range of temperatures [200]. The inherent liquid-liquid immiscibility in this system is due to the lack of mean-field attractive interactions between unlike species. Also shown on the figure is the total fraction of molecules bonded in the coexisting liquid phases. The degree of association is seen to increase dramatically close to the LCST, and can be used to explain the occurrence of LCST behaviour in this case [201, 202].

Figure 2.12: T x temperature-composition diagram at pressure P 4∗ for a binary mixture of equal-size hard spheres, with mean field attractions α11 = α22 , α12 = 0, and unlike site-site association. The continuous line is the liquid-liquid coexistence curve as a function of the mole fraction x2 of component 2, the dash line is the total mole fraction XT of bonded molecules in the coexistent phases with respect to temperature. The white circles are UCST and LCST.

2.6.2

Properties of Mixing

The thermodynamic properties of mixing are determined for the pressure P 4∗ = 4.588 × 10−2 at three temperatures (see figure 2.13): T1∗ = 0.128 is just below

the LCST and corresponds to a homogeneous mixed liquid, T2∗ = 0.143 is inside the

liquid-liquid immiscibility region and corresponds to a demixed state, and T3∗ = 0.167 is just above the UCST where the liquid phase is stable again (these states are

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114

labelled in figure 2.12). As for the phase diagram, the mixing functions are sym∗ is negative metrical in composition. The reduced Gibbs free energy of mixing ∆gm

over the entire composition range for the three temperatures studied (see figure 2.13 ∗ is always found to have a single minia)). For the mixed states T1∗ and T3∗ , ∆gm

mum, characterising a stable liquid for all compositions. However, in the case of the ∗ can be seen, demixed state T2∗ , a change from negative to positive curvature in ∆gm

indicating that ∂ 2 g ∗ /∂x2 ¡

T,P

> 0; this means that the mixture is unstable over a

relatively wide range of compositions. In the case of this mixture with symmetrical interactions, the coexistence compositions at T2∗ are found at the two minima of ∗ (x = 0.2, and x = 0.8); in other words the binodal and spinodal composi∆gm 2 2

tions are identical for this system. In comparing the mixing properties at the three temperatures it is evident that the greatest differences occur in the region of the intermediate compositions associated with the demixed states.

Figure 2.13: Reduced properties of mixing (a) internal energy, b) enthalpy, c) entropy, d) molar volume) obtained for a binary mixture of equal-size hard spheres, with mean field attractions α11 = α22 , α12 = 0 , and unlike site-site association, at constant pressure P2∗ and temperatures T3∗ , T4∗ and T5∗ .

CHAPTER 2. LCST IN POLYMER SOLUTIONS

115

The main contributions to the enthalpy of mixing are due to the mean-field dispersion and association interactions, both of which are attractive. As there are no unlike dispersion interactions, the dispersive contribution to the enthalpy of mixing is always positive, which is consistent with the unfavourable unlike interactions. By contrast, the association contribution to the enthalpy of mixing is always negative, indicating favourable unlike interactions. At high temperatures the degree of association is low (e.g., from figure 2.12 we can see that less than 20% of the molecules are associated at T3∗ ) and the enthalpy of mixing is dominated by the positive dispersive contribution (see T3∗ in figure 2.13 b)). As the temperature is decreased the degree of association increases sharply (cf. Figure 11), and the negative associative term makes an increasing contribution so that the enthalpy of mixing will decrease, eventually becoming negative below the LCST (see T1∗ in figure 2.13 b)). This is very different from what is found for the polymer-solvent system, where the enthalpy of mixing is always positive (even below the LCST) as a consequence of the negative volume of mixing due to the increase in density of the solvent on addition of small amounts of polymer. The entropy of mixing ∆s∗m of the system with directional interactions can be separated in three contributions: the ideal part, a term due to the repulsive interactions, and a term due to the association. At a high temperature (T3∗ ), the ideal entropy of mixing dominates, and ∆s∗m is very positive (see figure 2.13 c)). As the temperature is decreased, the degree of association and the number of bonded molecules increases. As a consequence the number of free species in the mixture decreases (with more 1-2 dimers formed), resulting in a decrease in the entropy of the mixture and the entropy of mixing ∆s∗m with decreasing temperature; for low enough temperatures ∆s∗m becomes negative (see T1∗ in figure 2.13 c)) over a wide range of compositions. ∗ is positive (see figure 2.13 d)) which is consistent with The volume of mixing ∆vm

the unfavourable unlike interactions due to the lack of dispersive attractions between ∗ decreases with decreasing temperature indicating unlike. The magnitude of ∆vm

that the unlike interactions become more favourable due to the increasing molecular association associations (figure 2.9a)). In contrast to the behaviour of the polymersolvent system, the density of the equimolar mixture is always lower than that of the pure components (see figure 2.14). In the case of the system with directional interactions the negative entropy of mixing is due to the decrease in the number

CHAPTER 2. LCST IN POLYMER SOLUTIONS

116

of free species as a consequence of molecular association, and not to the density changes in the mixture.

Figure 2.14: Reduced density obtained for a binary mixture of equal-size hard spheres, with mean field attractions α11 = α22 , α12 = 0, and unlike site-site association, at constant pressure P2∗ and at temperatures T3∗ , T4∗ and T5∗ . We can thus summarise the LCST and UCST behaviour of the associating mixture as follows: at the intermediate temperature (T2∗ ), liquid-liquid phase separation occurs due to the weak attractive interactions between unlike species driven by the unfavourable enthalpy of mixing (∆h∗m > 0). In the case of high temperature states such as T3∗ (above the UCST), the positive entropy of mixing, which is essentially the ideal contribution, more than compensates for the positive enthalpy of mixing, ∗ < 0 and convex, so that the resulting Gibbs free energy of mixing is negative ∆gm

corresponds to a stable mixture. At the low temperature T1∗ (below the LCST), the entropy of mixing becomes negative for over a wide range of compositions as the molecular association reduces the number of free species; the negative entropy thus favours a demixed mixture in these conditions. However, the enthalpy of mixing is also negative due to the energetically favourable site-site associations, and dominates the unfavourable entropic contribution. As a consequence the mixing below the LCST is driven by the enthalpic contribution to the Gibbs free energy of mixing which becomes convex again convex corresponding to a stable mixture. and the

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117

mixture is stable. Accordingly, the nature of LCST behaviour is completely different from that for the model polymer-solvent system, which was driven by entropic effects due to changes in the density of the mixture.

2.7

Conclusions

The main aim of this chapter is an attempt to provide a better understanding of the beguiling LCST cloud-curve behaviour exhibited by polymer-solvent systems. The fluid phase behaviour and thermodynamics of mixing is examined for simple models of polymer solutions in which the solvent is represented as a spherical monomer and the polymer as a tangent chain of the monomeric segments; here the diameter of the monomeric segments and the energy of the attractive interactions between them are taken to be the same (symmetrical interactions). The repulsive contributions in the system are treated using the Carnahan and Starling description for the hardsphere segments and a Wertheim TPT1 treatment of the hard-sphere chain, while the attractive interactions are treated at the simple mean-field level of van der Waals. We show that liquid-liquid immiscibility with a corresponding LCST is exhibited by the simple model with symmetrical interactions for chains of seven or more monomeric segments; as inferred by Patterson [33] in his studies with a compressible lattice model, no UCST is seen for the system with symmetrical interaction χ = 0. In the case of non-spherical solvents, we also show that there is a simple power law representing the limiting chain length of the polymer which is immiscible in the shorter chain solvent. The system with attractive interactions (athermal system) is of particular interest. We use the Wertheim TPT1 description to examine a mixture of monomeric hard spheres and chains of hard monomeric segments of the same size, concluding that there is no fluid phase demixing in the athermal system regardless of the chain length. This is consistent with the work of Taylor and Lipson [155] who have shown that for such an athermal system with segments of the same size the hard spheres behave as “good” solvent. The possibility of an instability in the liquid phase of athermal solvent-polymer systems in which the solvent is much larger than the segments making up the polymer (a simple model of polymer-colloid systems) is not addressed in this work, and is the theme of a separate contribution [196]. As far as the mechanism of phase separation in polymer solutions is concerned we show that attractive interactions provide the key. We quantify Patterson’s explana-

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118

tion of the free-volume (density) effects by examining the thermodynamic properties of mixing of our simple solvent-polymer model with symmetrical segmental repulsive and attractive interactions. The enthalpy of mixing is always negative in these mixtures owing to the increase in the attractive interactions in the mixture due to the addition of polymer; this indicates that the enthalpic contribution would always favour a mixed state. There is, however, a competing entropic contribution which gives rise to a demixed liquid state above the LCST. At low temperatures the solvent is at a relatively high density and the addition of polymer leads to a decrease in the density of the mixture (due to dominant repulsive contributions), with a corresponding large and positive entropy of mixing; at the lower temperatures both the entropic and enthalpic contributions to the free energy thus favour a mixed liquid state. On increasing the temperature the density of the solvent decreases. Above the LCST the addition of small amounts of polymer to the solvent leads to an increase in the density (due to dominant attractive contributions). This gives rise to a large decrease in the volume of mixing, and a corresponding negative entropy of mixing for low compositions of polymer due to the decrease in the free volume of the mixture. For progressively larger compositions of polymer the trend is reversed and the density is again seen to decrease, leading to a positive entropy of mixing at higher mole fractions of polymer. The negative entropy of mixing exhibit by the mixture for small compositions of polymer dominates the negative enthalpic contribution and gives rise to a change in curvature in the Gibbs function of the mixture which gives rise to the instability of the liquid and to demixing. The LCST behaviour in polymer solutions is thus due to an unfavourable entropic contribution for low mole fractions of polymer due to subtle density effects (contraction). It is interesting to note at this stage that one is able to explain the thermodynamics of fluid phase behaviour in polymer solutions without the need to treat explicitly the dimensions of the polymer (or the entropy of the chain conformations). In the TPT1 description a rather drastic approximation is made for the many-body correlation function as we show in section 2.1.1 and 2.1.2. In approximating the m-body correlation function as a product of m â&#x2C6;&#x2019; 1 identical pair hard-sphere distribution functions at contact (cf. equations (2.47) and (2.72)), one has lost all information about the correlations between different segments and the dimension of the chain. This does not mean that the polymer dimension does not change on changing the state of the polymer-solvent mixture from a mixed to a demixed state;

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119

as was discussed in the introduction the radius of gyration of the polymer is a very sensitive function of both the temperature and density of the system. The point that we are trying to bring across is that the fluid phase separation in polymer solutions is governed by the thermodynamics of the interactions between the solvent and the segments making up the polymer at a relatively short length scale. The chain dimension responds to the changes in the state of the system but does not drive the phase separation. We have also compared the thermodynamics of liquid-liquid phase and LCST behaviour in polymer solutions with that found in mixtures with directional interactions (corresponding to re-entrant miscibility). In the latter case the immiscibility is due to unfavourable enthalpic contributions at moderate temperatures; although the entropy of mixing is less than that of an ideal system the contribution is still positive in this case. As the temperature falls below the LCST the enthalpy of the mixture becomes more negative due to the favourable directional interactions on association. As a consequence of the association the number of free species in solution decreases and the entropy of mixing becomes negative and unfavourable. The enthalpic contribution is, however, the dominant contribution to the free energy and the mixture is found to be stable below the LCST. By contrast to the polymer solutions where the LCST behaviour was found to be driven by entropic effects due to density changes, the LCST behaviour in mixtures of molecules of similar size with directional interactions is governed by the dominant enthalpy of mixing at lower temperatures. In the next chapter, we treat the case of a pure repulsive binary mixture big hard spheres representing colloids, and chains of much smaller segments representing a polymer. If TPT1 predicts phase separation in such an athermal system, and if so, what are the causes of the demixing?

2.8

Appendix A: Algorithm to Solve LLE in Polymer Solution

Liquid-Liquid immiscibility and cloud point are particularly difficult to calculate in mixtures of very asymmetric molecules such as polymer solutions, since the composition of polymer molecules in the solvent-rich phase can be very low. The usual substitution methods developed by Michelsen [203â&#x20AC;&#x201C;205], and widely used in industry for saturated pressure, temperature, or flash calculation in multicomponent systems,

120

CHAPTER 2. LCST IN POLYMER SOLUTIONS

rarely converge for the calculation of liquid-liquid immiscibility in polymer-solvent systems, even by using very good guesses for the compositions of the phases in coexistence. This is a convergence mathematical problem [206], [207] which comes from the very non-ideality of the system, and the strong dependance on compositions of (β)

(α)

the equilibrium constants Ki = xi /xi

(β)

where xi

(α)

and xi

are the mole fraction

of component i in respectively the phases α and β in coexistence. A very good algorithm for phase stability and Flash calculation has been recently developed by Xu al [208] for the SAFT equation. This numerical method, based on interval Newton/generalised bisection methods, has been mathematically proved to be perfectly reliable to carry out tangent plane analysis and phase equilibria. However, that method is very slow, quite complex to set up, so that it is only used to check the numerical results of other and less reliable algorithms used in industry, when the system is very complex (multicomponent, multiphase). Koak and Heidemann [207] use a modified substitution method which involves both a ”damping” factor in the substitution iterations and a variable searching in logarithm space, to enable convergence. Such method can be very useful to calculate liquid-liquid immiscibility in polymer-solvent systems, but requires proper values of damping factors to assure convergence. Furthermore, the ”dumping” factor + substitution method always fails closed to the critical point of the mixture (LCST) and for very low concentration of polymer molecules in the solvent-rich phase. To avoid such failing problems, Chen et al. use a combination of Newton-Raphson technics and dumping method [206], where the dumping method is used as a initial step for the Newton-Raphson iterations. Block algebra is also used by Chen et al. to simplify the inversion of Jacobian matrix of the Newton-Raphson method. In this work, we have developed, in collaboration with the Process Center Engineering of Imperial College, a complete different approach to solve liquid-liquid immiscibility in polymer solution at fixed temperature and pressure. Our method is exclusively applicable to binary mixtures, however we are not interested here in systems involving more than two components. The method is fast and does not need any guess. The reliability of our method relies on the systematic use of the bisection method [209] for finding the root of a monotone non-linear function in a given interval. The algorithm is based on the following stability criteria for binary mixture: if a binary mixture of component 1 and component 2, of composition (x 1 ,

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CHAPTER 2. LCST IN POLYMER SOLUTIONS

x2 ) is stable at temperature T and pressure P , the chemical potentials µ1 and µ2 of components 1 and 2 are such as

∂µ1 ∂x2

T,P

(x2 ) 6= 0,

∂µ2 ∀x2 ∈ [0, 1] , ∂x2

T,P

(x2 ) 6= 0

∀x2 ∈ [0, 1] ,

µ µ

(2.112)

Equations (2.112) are not independent since they are related by the Gibbs-Duhem equation

∂µ1 ∂x2

+ x2 T,P

∂µ2 ∂x2

= 0.

(2.113)

T,P

Equations (2.112) mean that the chemical potentials at fixed T, P must be monotone functions of the mole fraction x2 = 1 − x1 of component 2. It the system demixes into to liquid phases α and β, then

∃!x02 , ∃!x002

∈ [0, 1] =

∂µ1 ∂x2

∂µ2 ∂x2

< 0,

T,P

¢ x02

∂µ1 ∂x2

∂µ2 ∂x2

T,P

x002

T,P

x002 = 0,

(2.114)

with

∂ 2 µ1 ∂x2 2

T,P

¢ x02

¢ x002

> 0,

∂ 2 µ2 ∂x2 2

T,P

x02 > 0,

(2.115)

x002 < 0 ¢

If x02 < x002 , the compositions of the two liquid phases α and β in coexistence, respectively x2,α and x2,β , are such as

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CHAPTER 2. LCST IN POLYMER SOLUTIONS

0 < x2,α < x02 and x002 < x2,β < 1.

(2.116)

In the case x02 > x002 , we have 0 < x2,α < x002 and x02 < x2,β < 1. (α)

As a result, the intervals containing x2

(β)

(2.117)

are well defined if x02 and x002 are

and x2

known, and do not overlap. If demixing occurs, the function

∂µ2 ∂x2 T,P

= f (x2 ) has

two roots x02 and x002 according to equation (2.114). One can apply the well-known Newton-Raphson method [209] which has to converge to a certain root (x 02 or x002 ) of f (x2 ) in the demixing case, for any initial guess taken in the interval ]0, 1[. If the Newton-Raphson method converges to x02 , one can find for sure the second root x002 of f (x2 ) by using the bisection method [209] on the intervals ]0, x02 [ and ]x002 , 1[. Let assume that the second root we find with the bisection method belongs to ]x 02 , 1[, so that x002 > x02 , and that the roots x02 and x002 correspond respectively to the maximum µ1 max and the minimum µ1 min of the chemical potential µ1 of component 1 (see illustration in figure 2.15). The equilibrium conditions are

(β)

µ1 x 2

(α)

= µ 1 x2

(α)

= µ 2 x2

µ2 x 2

(2.118) .

The non-linear system of two equations (2.118) can be reduced to a unique nonlinear equation which can be solved with the perfectly reliable bisection method. The non-linear equation is obtained by using the chemical potential µ1 as a space variable, and is given by ³

(α)

(β)

g(µ1 ) = µ2 x2 (µ1 ) − µ2 x2 (µ1 ) = 0, (α)

(2.119)

(β)

where x2 (µ1 ) and x2 (µ1 ) are the roots of the function h(x2 ) = µ1 (x2 )−µ1 , found by bisection method in the intervals ]0, x02 [ and ]x002 , 1[. One can note that equation (2.119) is equivalent the Maxwell relation for binary mixtures which says that the two grey areas in figure 2.15 have to be equal at equilibrium. The root µ 1 coex of

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equation (2.119) can be solved by bisection method in the interval ]µ1 min , µ1 max [, (α)

(β)

and the corresponding roots x2 (µ1 coex ) and x2 (µ1 coex ) are the composition of the coexistence phases. In polymer solution, one should search for the composition (α)

(β)

roots x2 (µ1 ) and x2 (µ1 ) in logarithm space as suggested by Koak et al, because the composition of the polymer in the solvent-rich phase can be very low with very different order of magnitude.

Figure 2.15: Evolution of the chemical potential of component 1 at fixed temperature T and pressure P , as a function of composition x2 in a demixing binary polymer-solvent system. The circles denote binodal points, and the square denotes a metastable point.

2.9

Appendix B: Mixture of Associating Hard Spheres

Here, we give the expressions of the contribution to the free energy due to association for the binary mixture of hard spheres, of equal diameter σ, interacting with mean field energies α11 = α22 between like species, but with no mean field attraction between unlike species, i.e. α12 = 0. Highly directional attractive forces are introduced between unlike species in the form of off-center, square-well bonding sites. Molecules corresponding to component 1 are modelled with one association site labelled A, while molecules corresponding to component 2 have one association site labelled B. Only AB interactions are allowed. When the two sites A and B are closer than a cut-off distance rcAB , there is an attractive interaction with a well

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CHAPTER 2. LCST IN POLYMER SOLUTIONS

depth energy of ²AB , giving rise to 1-2 dimers. The Helmholtz free energy for this system is then given by:

A AIDEAL AHS AM F AASSOC = + + + , N kT N kT N kT N kT N kT

(2.120)

The contribution due to the ideal AIDEAL , repulsive hard-sphere AHS (Carnahan and Starling), and attractive mean-field AM F contributions are obtained as special cases of the expressions presented in section 2. The general expression for A ASSOC is derived from the Wertheim TPT1 treatment [35, 84–87] as ¶ si µ n X X Xa,i si AASSOC ln Xa,i − xi = + , N kT 2 2 a=1 i=1 "

(2.121)

where n is the number of species, si the number of sites on species i, and Xa,i the mole fraction of molecules of type i not bonded at the site a. In the case of the binary mixture studied here, expression (2.121) reduces to [200]

XA AASSOC = x1 ln XA − N kT 2 µ

+ x2

XB ln XB − 2

+ 1,

(2.122)

where XA is the fraction of molecules of type 1 not bonded at site A, and XB molecules of type 2 not bonded at site B. is the fractions of molecules not bonded are obtained from the mass action equations as

XA =

1 1 + ρx2 XA ∆AB

XB =

1 , 1 + ρx2 XB ∆AB

where ∆AB characterises the AB interaction, and is given by HS ∆AB = g12 (σ)FAB KAB .

(2.123)

HS (σ), is The radial distribution function at contact between molecules 1 and 2, g 12

given by equation (4.26), since the two molecules are of equal size. The function FAB = exp (²AB /kT ) − 1 is the Mayer function of the attractive square-well site-site

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CHAPTER 2. LCST IN POLYMER SOLUTIONS

interaction of depth ²AB , and KAB [36] is the bonding volume. Since the number of bonded sites A bonded must be equal to the number of bonded sites B, XA and XB are related by [200]

XB = 1 −

x1 (1 − XA ) . x2

(2.124)

On substitution of equation (2.124) into equation (2.123), the fraction of free sites XA can be determined analytically by solving the quadratic equation

2 XA

x2 − x 1 1 + x1 ρx1 ∆AB

XA −

1 = 0. ρx1 ∆AB

(2.125)

On determining the fractions of molecules not bonded the free energy due to association can then be determined directly from equation (2.122), and the total Helmholtz free energy can then be calculated by combining all the expressions in equation (2.120).

Chapter 3

Demixing in Colloids + Polymer Systems The theme of this chapter is an examination of the nature of fluid phase separation in purely repulsive models of colloid-polymer systems. In the case of mixtures of simple molecules which interact with dispersive attractive interactions, an unfavourable enthalpy of mixing due to relatively weak unlike interactions is responsible for liquidliquid phase separation; the long established thermodynamic explanation for the effect is that the unfavourable (positive) enthalpic contribution to the free energy increasingly dominates the favourable entropy of mixing as the temperature is lowered, leading to an instability of the fluid relative to a demixed state [5]. On the other hand, there is a compelling body of more recent experimental and theoretical work which provides unequivocal evidence for ”entropy driven” fluid phase separation in purely repulsive, athermal, systems (e.g., see the reviews by Frenkel [210, 211] . It is widely accepted that fluid phase separation can occur in binary mixtures of hard repulsive particles if the shapes or the sizes of the components are very different. In these athermal systems the internal energy of the system is zero so that the only contribution to the Helmholtz free energy is entropic; it is important to recognize that in the case of phase transitions at constant pressure an enthalpic contribution to the Gibbs free energy may also be important. Contrary to some preliminary theoretical predictions [212, 213] a simple binary mixture of large and small hard spheres does not undergo a phase transition between two fluid phases, irrespective of the diameter ratio, as it is found to be metastable with respect to a first-order fluid-solid transition [214–216]. Fluid phase separation is, however, possible in multicomponent (polydisperse) hard-sphere mixtures for appropriate choices of the size distributions [217–219].

126

CHAPTER 3. DEMIXING IN COLLOIDS + POLYMER SYSTEMS

127

The case of colloid-polymer mixtures, where to a reasonable level of approximation the colloidal particles can be thought of as large hard spheres and the polymers as chains of small hard-sphere segments, is more complex, and perhaps more interesting. Such an athermal system can exhibit phase separation into colloid-rich and polymer-rich fluid phases (the review by Poon and Pusey [220] provides an excellent introduction to the topic). This type of phase behaviour has now been studied experimentally in a variety of systems ranging from comparatively simple latex-polystyrene and silica-polydimethylsiloxane colloidal dispersions to biological systems containing proteins or DNA (e.g., see references [220â&#x20AC;&#x201C;234]). The tendency of the system to reduce the excluded volume interaction between the polymer coils and the colloidal spheres, which then leads to a larger free-volume (translational) entropy, induces an effective attractive (depletion) interaction between the colloidal spheres; this attractive interaction in turn leads to a fluid phase separation into lowand high-concentration colloidal phases which closely resembles the vapour-liquid transition in simple molecules with dispersive interactions. The nature of the effective interaction was first described by Asakura and Oosawa [1, 195], and then by Vrij [2] and Joanny et al. [235]. In the case of polymers with overall dimensions which are small compared to the colloid, the colloid-colloid attractive interaction is pair-wise additive [157,160] (and as a consequence is easy to treat theoretically); for polymer dimensions which are of the order of the size of the colloid or larger, pairwise additivity can no longer be assumed [157, 236]. Depending on the relative size and the concentration of the polymer, the effective attractive interaction between the colloids can be tuned resulting in a rich variety of phases, including both fluid and solid states. It was recognised early on that the polymers-polymer interaction can be neglected at a first level of approximation (ideal chain), and a qualitative description of the phase behaviour of polymer-colloid mixtures can be achieved solely with the incorporation of the colloid-colloid and colloid-polymer hard-body excluded volume interactions [1,195]. Moreover, if the colloid-polymer interaction is modelled by treating the polymer coil as an effective hard sphere, which corresponds to the so-called Asakura-Oosawa (AO) model, a knowledge of the polymer-colloid size ratio q = Rg /RC (where Rg is the polymer radius of gyration and RC is the radius of the colloid) is sufficient to describe the main features of the phase behaviour. The first theoretical studies of the global phase diagram of the AO model were undertaken by Gast et al. [237] and by Vincent and co-workers [238, 239] for mixtures of

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colloidal hard spheres and non-interacting polymer in a background solvent using a standard perturbation theory for the depletion potential (see reference [240] for a review of these types of theoretical approaches). The partitioning of the polymeric component was not considered in these preliminary studies, but was later incorporated into the description of the AO colloid-polymer model by Lekkerkerker et al. [156] who employed a free-volume theory; this provides the correct description of the fluid-fluid-solid three-phase coexistence region. In the case of short polymers and large colloidal particles corresponding to q < 1 (ideal polymer limit), the free-volume treatment of the AO model provides a good representation of the simulation data [157, 160, 241], and has been validated by an exact one-dimensional description [242â&#x20AC;&#x201C;244]. As the size of the polymer increases relative to the colloid (q > 1) the validity of the approximations inherent in the AO becomes questionable; this does not of course imply that the segments making up the chain are larger than the colloid only that the overall polymer dimension is large. For long polymer chains the interactions between polymer chains are no longer negligible, and the overall shape of the polymer is not globular as far as its interaction with colloid is concerned. Sear [161] has addressed the latter problem by extending the AO treatment to non-interacting chains with dimensions that are larger than the colloidal hard spheres (q > 1) . In his approach the ideal polymer is described as a number of spherical blobs nb obtained by scaling 2 ); the the polymer dimension with the size of the colloidal particle (nb â&#x2C6;ź Rg2 /RC

polymer-colloid interaction can then be treated as an excluded volume interaction between a hard-sphere colloid and a chain of nb hard-core spherical blobs of the same diameter as the colloid. Sear treats the contribution of the colloid-polymer interaction to the free energy by using the first-order perturbation theory (TPT1) of Wertheim [35, 83]; we will also use the Wertheim TPT1 description but in our study the polymer-polymer interactions are included explicitly. Sear is thus able to describe the non-spherical nature of the colloid-polymer interaction in the case of long chains for which the standard AO model is inadequate. As with the usual AO model, the system exhibits a fluid-fluid demixing transition into colloid-rich and polymer-rich phases [161]; the main finding of the work is that the critical density of the colloid decreases and tends to zero as the length of the polymer is increased. We shall return to this point later in the discussion. Schmidt and Fuchs [245] have also examined the case of polymer dimensions which are larger than the colloid

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by replacing the hard-spherical repulsion model between the polymer coil and the colloid by a repulsive step function potential to allow for the possibility of chain configurations where the polymers are close to the colloid. The main drawback of this model (and that of Sear [161]) is that the interaction between the polymers is assumed to be ideal. Up until this point we have only discussed the description of colloid-polymer systems where the polymer is treated as a non-interacting chain. There has also been recent effort in incorporating the polymer-polymer interactions into the treatment of such systems (the reader is directed to the recent review by Fuchs and Schweizer [246] for more details, where a particular emphasis is placed on the integral equation approaches). Warren et al. [247] were one of the first to examine the effect of polymer-polymer interactions on the phase behaviour of polymer-colloid systems using a perturbation theory around the θ-point conditions. Integral equation approaches have also been used extensively to treat the thermodynamic and structural properties of colloid-polymer systems [246, 248]. Sear [249] has recently used his rescaling approach (representing the polymer which is larger than the colloid as a chain of spherical blobs) to take into account polymer-polymer, polymer-colloid, and colloid-colloid repulsive interactions at the level of the second virial coefficient within a Flory-Huggins-type theory; a discussion of the effect of varying the solvent quality on the phase behaviour is also made. The incorporation of polymer-polymer interactions, even at the crude second virial level, leads to the prediction that the density of the colloids at the critical point tends to a finite value as the polymer is made longer, a contrary finding to that of studies with non-interacting polymers [161]. Although the qualitative trends obtained in simulation studies [250] are described correctly with the Flory-Huggins-like approach, the critical densities are not in close agreement with the simulation data; this is most likely due to the truncation of the free energy at the level of the second virial coefficient which essentially means that the treatment will only be valid at low densities. Another particularly appealing approach referred to as the ”polymer as soft colloids” model is worth a separate mention [251–254]: the polymer-polymer interactions are incorporated by using an effective potential acting between the centres of mass of the chains which enables large scale simulations of the system. This coarse-graining method has been recently employed by Bolhius et al. [162] to study the phase behaviour of colloid-polymer mixtures. Simulations of self avoiding walks were undertaken for pure polymers and

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130

their mixtures with a single colloidal hard sphere to compute the polymer-polymer and polymer-colloid centre of mass radial distribution functions; the structural information can then be inverted by using the hypernetted chain integral equations to determine the effective polymer-polymer and polymer-colloid interaction potentials, which are parameterised in terms of a sum of Gaussians or exponentials. The spherically symmetric effective interactions are thus obtained for a series of polymer dimensions (q) and concentrations, and used to simulate the full phase behaviour of the colloid-polymer system. As expected, the effect of the polymer-polymer interactions is small for the systems of short chains (q = 0.34) and there is close agreement with the results of the AO model and free-volume theory [156]. When the polymers dimension is increased the effect of the polymer-polymer interactions becomes more important and significant qualitative differences are seen with the AO description. As has been predicted by Aarts et al. [163] with a recent extension of the free-volume theory to systems of interacting polymers, the extent of the fluid-fluid immiscibility is found to decrease on incorporating the polymer-polymer interactions. Bolhius et al. [162] also show that the colloid density at the critical point is found to be insensitive to the length of the polymer in agreement with the Flory-Huggins findings of Sear [249]. The main disadvantage of the use of these types spherically symmetric effective interactions is that the approach will be inadequate in describing polymer chains which are large compared to the colloid and that it is restricted to moderate densities of polymer. Before we outline the main goals of our study it is important to acknowledge the work on the effect of the spherical particles on the dimension of a polymer coil. It is well recognised that a polymers adopts a more extended configuration in a ”good” solvent and a more collapsed configuration in a ”bad” solvent (which can lead to a demixed system) [146–149]. The collapse of the polymer dimension does not necessarily imply phase separation but is clearly closely related to it. The evidence of a polymer collapse transition in simulation studies for purely repulsive (athermal) models where the solvent and segments making up the polymer are of the same size is rather inconclusive [150–152,154,255]; one should point out that these are models of polymer solutions and not colloid-polymer systems where the colloid is much larger than the monomeric segments making up the polymer. Taylor and Lipson [155] have used the Born-Yvon-Green integral equation approach for hard-sphere/hardsphere chain models of polymer solutions in which the spherical segments are all

CHAPTER 3. DEMIXING IN COLLOIDS + POLYMER SYSTEMS

131

the same size to show that the chain contraction with increasing solvent density is almost identical to what one would expect in the pure polymer melt; this is consistent with a polymer in a good solvent which would not undergo a fluid phase transition. The lack of a demixing transition in such an athermal system has been corroborated by studies of the fluid phase equilibria of model polymer solutions with the Wertheim TPT1 description [256] (see chapter 2 for further details): the homogeneous fluid mixture is stable and does not demix regardless of the length of the hard-sphere chain; in this case the fluid phase separation is driven by the polymer-solvent attractive interactions [256]. As was mentioned earlier there is clear evidence of the existence of a demixing transition for athermal systems in which the diameter of the spherical particle (in our case the colloid) is considerably larger than that of the segments making up the chain. For these athermal models of colloid-polymer systems a collapse of the polymer dimension is unambiguously observed when the concentration of the colloidal hard-spheres is increased [157, 159]. The situation is completely different when attractive interactions are present: chain collapse is found for a symmetric model in which the sphere-sphere, spheresegment and segment-segment size and interaction energies are all equivalent [165– 170], implying liquid-liquid immiscibility in such a system. Less work has been on the incorporation of attractive interactions in models of colloid-polymer systems in which the sphere-segment size ratio is large (e.g., see reference [257]). We do not discuss the effect of attractive interactions on the phase behaviour further, as the focus of this chapter is on the fluid-phase separation of athermal hard-core models of both the colloid and the polymer. In our contribution we study the fluid phase equilibria of mixtures of large hardspheres (colloid) and chains formed from smaller hard-sphere segments (polymer). The density range over which one would find fluid-solid phase transitions in such systems is not considered in the first instance. In contrast to the other theoretical work mentioned earlier we study a ”microscopic” model of the polymer in which both the polymer-polymer and polymer-colloid excluded volume interaction are treated at the level of the monomeric segments making up the chain. The polymer dimension (radius of gyration) does not enter into the description in an explicit manner. In principle this means that our approach can deal with the situation in which the polymer is smaller or larger than the colloidal particle. The Wertheim TPT1 approach [35,83] is used to describe the polymer-polymer and polymer-colloid contributions to the

CHAPTER 3. DEMIXING IN COLLOIDS + POLYMER SYSTEMS

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free energy; this choice is made as the theory provides an excellent description of the equation of state of chains of tangent hard spheres and mixtures [35, 75, 76]. By using a spinodal stability analysis similar to that employed in an examination of isotropic-isotropic demixing in mixtures of cylindrical particles of different length and shape (diameter) [164], we examine the global fluid phase diagram of the model colloid-polymer system. The precise chain-segment to colloid diameter ratio that gives rise to an instability in the fluid (and leads to fluid-fluid demixing) is determined for varying lengths of the polymer chain. The full fluid-phase behaviour (binodals) are also determined for the system. We relate our findings to the existing theoretical predictions, but leave any direct comparison with experimental data to a future publication. In addition an analysis of the thermodynamics of mixing in our athermal colloid-polymer model is made to highlight the roles of the entropic and enthalpic contributions to the free energy in such systems.

3.1

The Depletion effect

The so-called depletion effect enables to explain why demixing occurs in colloidpolymer system: the addition of small amount of polymer molecules in a system of colloidal particles induce an effective attraction between the colloidal particles. The phenomenon is due to free volume effects, and is illustrated in figure 3.1: the polymer molecules take the shape of a coil of average diameter Rg called radius of gyration. This radius depends not only on the polymer chain length, but also on the packing fraction and the composition of the system. In the case where the radius of gyration is smaller than the radius of the colloids (q = Rg /RC < 1), the depletion effect can be explained as follows: the center of mass of the polymer molecules is excluded from coming closer than a certain distance to the surface of the colloidal particle . Each colloid is surrounded by an excluded volume zone for the polymer molecules (see figure 3.1 a)). When the colloidal particles approach each other (see figure 3.1 b)), the excluded volume zones may overlap so that the global excluded volume for the polymer molecules decrease. As a result, the configurational and ideal entropy of the polymer increase when the colloidal particle are closer to each other, and the number of possible states in bigger (the free energy of the system is lower).

CHAPTER 3. DEMIXING IN COLLOIDS + POLYMER SYSTEMS

133

Figure 3.1: Representation of the depletion effect in colloid-polymer system. Colloidal particles are represented by grey spheres, and polymer segments are represented by black spheres.The centers of mass of the polymer coils of diameter 2R g are excluded from a volume represented by a sphere of diameter 2 (Rg + RC ) (dotteddashed line), for each colloidal particles of diameter ﾏイ = 2RC ( see figure a)). However, when the colloid particles are close to each other (see figure b)), the excluded volumes related to the colloids overlap, the total excluded volume is decreased, and the coils particles can then move in a larger volume. The overlapping regions are represented in black. In the SAFT (TPT1) theory, the polymer chain is modelled as a fully flexible chain of segments of diameter ﾏケ << ﾏイ

CHAPTER 3. DEMIXING IN COLLOIDS + POLYMER SYSTEMS

134

The presence of polymer in a system of colloid particle induce an effective attraction between the colloids, as the clustering of colloids in a bath of polymer molecules gives rise to a decrease of the excluded volume for the polymer coils. An other way to explain the depletion effect is by using osmotic pressure arguments: when the colloid particles are close to each other, the polymer coils can not penetrate in the intersticial regions. This results in a lack of polymer molecules in that region, and in an effective attraction or depletion between the colloid particles. As discussed in the introduction, the common way to treat theoretically the depletion effect is to assume a pure-component system of colloids, where the contributions of the polymer molecules are replaced by an effective attractive potential between the colloids. This attractive potential induces a vapour-liquid transition for the pseudo-pure colloid system. In our approach, we do not model the polymer at the coil scale, but at the chain segment scale, and it is not necessary to know the radius of gyration Rg .

3.2

Wertheim TPT1 Approach for Colloid-Polymer Systems

We use the Wertheim thermodynamic perturbation theory (TPT1) [35, 83] to study the global fluid phase behaviour of the binary mixture of colloids and polymer chains as it provides a direct insight into the effect of varying the size of the segment diameter and the chain length. As was mentioned earlier the theory enables one to treat the system at the level of the monomeric segments. Both the colloidal particles and the segments making up the chain are modelled as a hard spheres of diameters σC (colloid) and σP (polymer segment). The chains are assumed to be formed from m tangentially bonded segments, but are otherwise completely flexible. The interaction between the colloidal hard-sphere and a polymer hard-sphere segment is taken to be additive, i.e., σCP = (σC + σP )/2. The pair potential Φij between the same and different species can be written in a compact form as

Φij (rij ) =

(

∞ if rij ≤ σij 0 if rij > σij ,

(3.1)

where i, j = P, C and rij is the distance between the centers of the two spheres (colloid-colloid, colloid-monomer and monomer-monomer). Within the TPT1 ap-

CHAPTER 3. DEMIXING IN COLLOIDS + POLYMER SYSTEMS

135

proach the Helmoltz free energy of this binary colloid-polymer mixture can be written as a sum of ideal, hard-sphere , and chain formation terms, in the same way as in the polymer-solvent case (see chapter 2, equation (4.12)) :

A AIDEAL AHS ACHAIN = + + , N kT N kT N kT N kT

(3.2)

where T is the temperature, k the Boltzmann’s constant, and N is the number of particles (sum of the NC colloids and NP polymers). The ideal free energy of the binary mixture is given by , X AIDEAL = xi (ln ρ − 1 + ln νi + ln xi ) , N kT i=C,P

(3.3)

where ρ = N/V is the number density, xi is the mole fraction of the component i, and νi the de Broglie volume which takes into account the translational (and rotational in the case of the polymer) contributions of the kinetic energy of each component; though the precise form of the de Broglie volume is not important in studies of phase equilibria it is included for completeness. The ideal entropy of mixing is seen as the last term of equation (3.3). As in the polymer-solvent case (see chapter 2, equation (4.19)), the repulsive free energy term for a fluid mixture of hard spheres of different diameter is accurately described by the simple expression of Boubl´ık [187] and Mansoori et al. [188] as

AHS 6 = N kT πρ

"Ã

3ζ1 ζ2 ζ23 ζ23 ln(1 − ζ ) + − ζ , + 3 0 (1 − ζ3 ) ζ3 (1 − ζ3 )2 ζ32

(3.4)

In the case of our colloid-polymer binary mixture the reduced densities are defined as

ζl =

´ π ³ l + mxP σPl , ρ xC σC 6

(3.5)

where l = 0 to 3; the chain length is included in the definition of the reduced density as the number of monomeric segments making up the chain is given by NSEC = mNP = mxP N . The total packing fraction of the system is given by 3 + mx σ 3 . The Boubl´ ık-Mansoori description has been shown η = ζ3 = π6 ρ xC σC P P

to provide an accurate representation of the equation of state for binary mixtures of

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136

hard spheres with significantly different diameter (corresponding to diameter ratios of up to 20:1) over a wide range of compositions [258]; this is particularly relevant to our model colloid-polymer systems where the diameter of the polymer segment is much smaller than the colloid. The expression reduces to the accurate CarnahanStarling [74] equation for the hard spheres in pure fluid limits, as shown in chapter 2, equation (2.81). Wertheim [83] obtained an expression for the free energy of the chain fluid simply in terms of the structure of the monomer hard-sphere reference fluid and the average chain length of the poly-disperse mixture of chain aggregates that are formed on association. An identical expression for the free energy of the mono-disperse hardsphere chain fluid can be obtained by considering a mixture of associating hard spheres in the limit of complete association [35] (see chapter 2); Assuming a linear approximation of the m-body distribution function (see chapter 2, section 2.1 for further details), the chain contribution is given by

ACHAIN = −xP (m − 1) ln gPHS P (σP ) , N kT

(3.6)

where the radial distribution function at contact gPHS P (σP ) between monomers, in a system of monomers + colloids, is obtained from the Boubl´ık [187] and Mansoori et al. [188] equation (3.4) as

gPHS P (σP ) =

σP2 3σP ζ2 ζ22 1 + . + 1 − ζ3 2 (1 − ζ3 )2 2 (1 − ζ3 )3

(3.7)

As discussed in the previous chapter, the significance of the linear approximation to the m-body distribution function is that one loses information about the structure of the chain, including the lack of an explicit description of the end-to-end vector and the radius of gyration which are used as measures of the dimension of the polymer. This does not mean, however, that the polymer dimension is absent from the Wertheim TPT1 description. When the TPT1 free energy is expanded as a virial series in density, the resulting virial coefficients are found to be dependent on the chain length m. The TPT1 second virial coefficient turns out to be a linear function of m [117, 118, 124, 259]; this corresponds to the correct scaling for long linear chains [259], but not for fully flexible chains in a good solvent where the second virial coefficient scales as ∼ m1.5 [117, 118, 124, 146]. A simple correction can

137

CHAPTER 3. DEMIXING IN COLLOIDS + POLYMER SYSTEMS

be added to the TPT1 treatment to account for the correct second-virial coefficient of the fully flexible chains, though only a slight improvement in the description of the equation of state is expected [117, 118, 124]. The ultimate consequence of this for our colloid-polymer model is that both the polymer-polymer and polymer-colloid TPT1 excluded volumes will depend on the chain length, i.e., the dimension of the polymer. The Wertheim TPT1 theory described here was developed to deal with the fluid state of mixtures of chain molecules, and no description of the solid states was incorporated into the treatment. Previous work with athermal colloid-polymer systems has shown that there are regions corresponding to fluid-solid equilibria which are particularly extensive for colloid-polymer systems of short chains at high densities of the colloid (e.g., see [156, 162, 163, 246]). Although there has been some work on the fluid-solid equilibria of flexible chains of tangent hard spheres within an TPT1-like treatment [260], no general TPT1 description is available at this stage for mixtures of large hard-spheres and hard-sphere chains. As with some of the other studies one could assume that the polymer is not playing an important role in determining the structure of the solid, but this will only be valid for short polymers with segments that are much smaller than the colloid. In this work we focus on the effect of the polymer chain length (m) and the segment-colloid size asymmetry (characterized by the diameter ratio d = σ p /σC which should not be confused with the size ratio q defined earlier in this chapter) on the fluid phase behaviour, avoiding the high density states where one would expect fluid-solid transitions (or at the very least highlighting states where a stable solid phase would pre-empt the fluid-fluid coexistence). On examining the TPT1 expressions (3.2) to (3.7) that make up the free energy of the athermal colloidpolymer system, it is clear that the thermodynamic properties of the mixture at a given composition (the mole fraction of the polymer x = xP ) and density (reduced number density ρ∗ = ρσC ) only depend on the number of segments in the chain m and the polymer-colloid diameter ratio d. One can thus examine the global fluid phase behaviour entirely in terms of this two parameters. Having described the analytical form of the Helmholtz free energy for our athermal colloid-polymer model, the TPT1 expressions can easily be used to determine the thermodynamics and the fluid phase equilibria of the system. The other thermodynamic properties are obtained through the standard relations [5]: the pressure P = −

∂A ∂V N,T

;

CHAPTER 3. DEMIXING IN COLLOIDS + POLYMER SYSTEMS the chemical potential of each component i µi = −

∂A ∂T N,V

∂A , ∂V N j6=i ,V,T

138

the entropy S =

, the internal energy U = A + T S, the enthalpy H = U + P V , and the

Gibbs free energy G = A + P V (the last three expressions corresponding to the appropriate Legendre transformations).

3.3

Spinodal and Binodal Curves

The simplest method to search for molecular parameters (d, m) which give rise to an instability of the mixed (homogeneous) fluid state relative to a demixed state in the binary colloid-polymer hard-core mixture is based on a determination of the spinodal. Reduced properties of mixing can be defined the same way as in chapter 2. (see chapter 1, equations (1.4) to (1.7), and chapter 2, equations (2.91) to (2.97)). The molar Gibbs function g ∗ is the natural thermodynamic function for phase equilibria in mixtures at constant pressure. The solution of equation (2.98) determines the spinodal curve of the phase diagram, which meets the binodal (phase coexistence) curve at the critical point. In order to locate the critical points of the phase separation, the third derivative of the molar Gibbs free energy with respect to composition must also vanish and the critical point of the binary mixture is determined by solving the system of equation:

∂ 2 ∆g ∗ ∂x2

T,P

∂ 3 ∆g ∗ ∂x3

T,P

(3.8)

For given values of the molecular parameters (d, m) of the colloid-polymer system, the system of equations (3.8) together determines the critical density and composition of the fluid-fluid demixing transition if it exists. The two conditions are solved numerically by expressing them analytically in terms of the appropriate volume and composition derivatives of the Helmholtz free energy [5]. From previous studies on athermal mixtures of colloids and polymers and of other hard-body cylindrical molecules [164], it is well recognized that the size and

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the shape of the two components have to be very different for fluid-fluid demixing to be observed. In such systems the mixing of the two components is associated with an unfavourable unlike packing (excluded volume) entropy. As an initial step we search for the first sign of an instability in the fluid phase of the colloid-polymer mixture for systems with an increasing polymer chain length (m = 15 to 1000 segments), but with a fixed segment/colloid diameter ratio (d = 0.06). If one employs the usual polymer scaling ideas [146], the radius of gyration of a polymer in a good solvent scales as Rg ∼ m1/2 , which would correspond to a polymer/colloid dimension ratio of q = Rg /RC ∼ m1/2 d in terms of our parameters. In the case of the systems with chains of length m = 500 and 1000, the overall dimension of the polymer is larger than the colloid (q ∼ 1.2 and q ∼ 1.9, respectively), while for systems with the polymer dimension is smaller than that of the colloid with q < 1 (q ∼ 0.85 for chains of length m = 200 , and q ∼ 0.23 for m = 15). With this choice of diameter ratio d = 0.06 and set of chain lengths m one is thus able to cover systems of both short and long polymers relative to the colloid. The spinodal boundaries denoting the limit of stability of the colloid-polymer mixture with d = 0.06 are shown in figure 3.2 for polymer chains of varying length. The extent of the fluid-fluid demixing is seen to increase with increasing polymer ∗ = P σ 3 /kT and the critical packing chain length. The reduced critical pressure Pcr cr C

fraction ncr corresponding to the minima of the spinodal curves both decrease with increasing chain length. As the chain length is increased, the composition at critical point also moves from the polymer-rich region of the phase diagram to the colloidrich region. This means that for systems with short chains a ”colloidal vapourliquid” transition is seen between a colloid-poor and a colloid-rich phase (see inset of figure 3.2), while a ”polymeric vapour-liquid” transition between a polymer-poor and a polymer-rich phase can be observed for long chains. In other words, the short polymers act as the ”depletion agent” and mediate the phase separation between the colloids in one limit, while the colloids play the role of the ”depletion agent” in the phase separation of long polymers in the other. This reversal in the role of the colloid and the polymer has not been highlighted previously. For polymer chains of moderate length (e.g., 100 < m < 200) the phase behaviour is more symmetric in composition and neither component can strictly be referred to as the ”depletion agent”; one should not use the analogy with a ”vapour-liquid” transition for the fluid phase behaviour in this case. An additional consequence of this is that neither the

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polymer nor the colloid should strictly be taken as the perturbative component to describe the phase separation of the other component in the semi-grand ensemble. It is also interesting to note that the critical pressure and packing fraction (c.f. figure 3.3) do not tend to zero in the limit of a very long polymer chain (m → ∞), but converge to finite values: in the case of the colloid-polymer system with d = 0.06 the infinite chain limiting values of the critical point are and . This limiting behaviour is has also been found in the studies of Sear [249] and Bolhuis et al. [162] which explicitly include the polymer-polymer interactions; when the polymer is treated as an ideal chain the critical density is found to vanish for infinite chain lengths of the polymer [35].

Figure 3.2: The pressure-composition representation of the spinodal curves of the fluid-fluid phase equilibria for athermal binary mixtures of colloids (hard spheres of diameter σC ) and flexible polymers (chains formed from tangent hard spheres of diameter σP ) determined from the Wertheim TPT1 approach. Results are presented for systems with a fixed polymer-segment to colloid diameter ratio of d = σ P /σC = 0.06 for polymer chain lengths ranging from m = 15 up to 1000. The reduced 3 /kT , and x represents the mole fraction of the pressure is defined as P ∗ = P σC P polymers.

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Figure 3.3: The critical packing fraction of the fluid-fluid phase equilibria for athermal binary mixtures of colloids (hard spheres of diameter σC ) and flexible polymers (chains formed from tangent hard spheres of diameter σP ) determined from the Wertheim TPT1 approach. Results are presented for systems with varying polymersegment to colloid diameter ratio of d = σP /σ¡C = 0.06 and polymer chain length ¢ m. The packing fraction is defined as η = πρ∗ 1 − xP + xP md3 /6. We now examine the effect of varying the segment to colloid diameter ratio on the stabilization of fluid-fluid demixing transition. By simultaneously solving the two conditions for criticality (system of equations (3.8)), the critical properties (packing fraction, mole fraction, and pressure) are determined for the colloid-polymer mixture with a given chain length m and diameter ratio d. The chain-length dependence of the critical packing fraction for selected diameter ratios are shown in figure 3.3. In all cases a fluid demixing transition is not likely to be found for systems of short chains (m < 15), because this would correspond to states with a very high packing fraction (ηcr > 0.5 at the critical point) where one would expect the solid phases to be stable. On the other hand, the critical packing fraction ηcr is seen to decrease rapidly with increasing chain length, and as observed previously, it tends to a limiting value at large m. The long-chain limit of ηcr is seen to decrease when the diameter ratio is decreased. For the system of polymers with the smallest segments (d = 0.001), the

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critical packing fraction decays more slowly with chain length, and is always found to be higher, than that of the system with d = 0.005 over the range of m shown in the figure. This finding suggest that a decrease in the polymer segment size promotes the demixing transition up to a certain value of d, but that any further decrease in the segment size results in a stabilisation of the mixed state with a corresponding increase in the critical density. This result is not surprising because as d is made very small (for a fixed chain length m) the polymer can be thought of as an ideal gas; there is no fluid phase demixing transition in the limiting ideal-gas + hard-sphere mixture, though there is the usual fluid-solid phase transition at high packing fraction [261]. The dependence of the critical packing fraction ηcr and composition xcr on the diameter ratio is plotted in figure 3.4 a) for polymers of short, moderate, and long length (m = 10, 100, and 1000). The observation that the segment size cannot be too low or too high in order for the system to exhibit a stable region of fluid phase separation, either in the polymer-rich (small d) or in the colloid rich (large d) regions, is clearer to see in this representation. For values of the critical packing fraction above 50% one would expect the appearance of solid phases. In the case of the larger diameter ratios (d > 0.2), the critical point of the mixture moves deep into the solid region; it is likely that a first-order fluid-solid transition between two colloid-rich phases pre-empts the ”polymeric vapour-liquid” transition, because the pure hardsphere colloids freezes at around η = 0.5. However, for systems with small values of d (d < 0.01), the critical packing fraction does not increase in the same way as for systems with large d. In this case it is probable that there is a ”colloidal vapoursolid” transition because the critical pressure is seen to diverge for very small values of d (see figure 3.4 b)). This is also consistent with the phase diagram of the AO model for small polymers where the size of the polymer (characterised by the radius of gyration) must exceed a certain value (q > 0.3) to stabilise the ”colloidal vapourliquid” transition with respect to the ”colloidal vapour-solid” transition [156, 162].

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Figure 3.4: The dependence of the critical properties on the polymer-segment to colloid diameter ratio d = σP /σC for athermal binary mixtures of colloids (hard spheres of diameter σC ) and flexible polymers (chains formed from tangent hard spheres of diameter σP ) determined from the Wertheim TPT1 approach. The critical packing fraction (continuous curves) and composition (dashed curves) are shown in a), and the ¡ critical pressure ¢ in b) for selected values of the polymer chain length m. ∗ 3 η = πρ 1 − xP + xP md /6 and , and xP represents the mole fraction of polymer.

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On the basis of the above findings it would be desirable to determine the precise molecular parameters (d, m) for which we would expect our polymer-colloids system to exhibit fluid-phase separation. This is highly nontrivial task because the solid phase is not incorporated into the theory. In this work we make no attempt to determine the Asakura-Oosawa limit (minimum size of the segments making up the polymer characterised by d), but the protein limit (maximum size of the polymeric segments) can be predicted in the light of knowledge of colloidal solidification. A pure hard-sphere system (colloid) undergoes a first order fluid-solid transition at η F = 0.494 and ηS = 0.545 [210, 211], so a ”polymeric vapour-liquid” transition (which occurs in the high-colloid mole fraction region of the phase diagram) is not likely for the molecular parameters resulting in critical packing fraction higher than about 50%. To determine the maximum size of the monomers for which the system still exhibits a stable fluid-fluid demixing transition, we assume that the critical packing max = 0.5) fraction can not be higher than 50%. By adding this new constraint (ηcr

to the conditions (3.8), the upper limit of the diameter ratio d can be determined for a system with a given chain length m. In figure 3.5 we show that the maximum value of the diameter of the segment increases as the polymer chain in made longer. Furthermore, it is clear that the diameter of the segments making up the change is always smaller than the diameter of the colloid (d < 1). This also supports previous findings [155,256] that model athermal polymer solutions of hard spheres and chains of hard sphere segments of the same size (d = 1) does not exhibit a fluid phase instability and the corresponding demixing. In the case of polymer solutions the attractive interactions play the key role in promoting the fluid-phase immiscibility (cloud curves, see chapter 2) [256]. The limiting diameter ratio below which one would expect the athermal colloid-polymer model to exhibit fluid phase separation appears to go to a limiting value for infinitely long chains d∞ ∼ 0.22. This result is consistent with the simulation study of a single athermal hard-sphere polymer in a hard-sphere solvent by Suen et al. [159] where the system with d = 0.2 exhibits a collapse transition at a packing fraction 44% which points to a demixed state. The promotion of a collapse transition for such systems with a larger hard-sphere solvent (colloid), corresponding to a decreasing value of d, has also been noted [152]. As we mentioned earlier, the lower AO limit (minimum size of the polymer segment corresponding to a lower value of d) is not shown in figure 3.5 because of the lack of a proper stability condition to include the solidification of the polymer-rich phase.

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Figure 3.5: Maximum value of diameter ratio d = σP /σC for the demixing transition of athermal binary mixtures of colloids (hard spheres of diameter σC ) and flexible polymers (chains formed from tangent hard spheres of diameter sP) determined from the Wertheim TPT1 approach as a function of the chain length m. The term I-I demixing is used to denote a phase separation between to isotropic fluid states.

3.4

Properties of Mixing

We end our discussion with an examination of the thermodynamics of mixing for the athermal colloid-polymer model. For the purpose of comparison with the fluid-fluid phase equilibria exhibited by simple mixtures [5], it is useful to display the phase separation of the colloid-polymer system in the ”temperature”-composition representation. A reduced temperature can be defined in terms of the reciprocal of the 3 , noting that in an athermal system only reduced pressure as T ∗ = 1/P ∗ = kT /P σC

the temperature-pressure ratio is the important variable. The fluid-phase equilibria (binodal curve) for the colloid-polymer mixture with m = 500 and d = 0.06, determined by ensuring that the pressure and chemical potentials of each component in each phase are equal, is represented in this fashion in figure 3.6. At temperatures above the upper critical solution temperature (UCST), which correspond to the low pressure states, the mixed fluid phase of the system is stable. When the temperature is lowered below the UCST (in this case T ∗ = 0.98), corresponding to increasing

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the pressure above P ∗ = 1.02, the mixture exhibits fluid-fluid phase separation into colloid-rich and polymer-rich phases. One can now ask the question: What is the main thermodynamic contribution leading to such a fluid phase separation? In order to answer this type of question it is common to determine the thermodynamic functions of mixing. The changes in the Gibbs free energy, entropy, enthalpy, and volume on mixing of the colloid and polymer components for a composition xC and xP at a constant pressure P are defined in the usual way (see chapter 1, equations (1.4) to (1.7), and chapter 2, equations (2.91) to (2.97)).

Figure 3.6: The temperature-composition phase diagram of the fluid-fluid phase equilibria for an athermal binary mixture of colloids (hard spheres of diameter σ C ) and flexible polymers (chains formed from tangent hard spheres of diameter σ P ) determined from the Wertheim TPT1 approach. The parameters characterizing the system are a polymer-segment to colloid diameter ratio of d = σP /σC = 0.06 and a polymer chain length of m = 500. The reduced temperature is defined as the ¡ ¢ 3 , and x represents the reciprocal of the reduced pressure T ∗ = 1/P ∗ = kT / P σC P mole fraction of the polymers.

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∗ of the colloid-polymer system The reduced Gibbs free energy of mixing ∆gm

with m = 500 and d = 0.06 is shown in figure 3.7 a) for a reduced temperature T ∗ = 1.25 (P ∗ = 0.8) above and a temperature T ∗ = 0.833 (P ∗ = 1.2) below the critical point. In the case of P ∗ = 0.8, the state above the UCST the Gibbs free energy of mixing is always negative and does not exhibit a change in curvature over the entire composition range indicating a stable miscible mixture. For the state below the UCST, the Gibbs function remains negative but a change in curvature can be seen in the colloid-rich region of the diagram; the second derivative of the Gibbs free energy with composition becomes negative in this region indicating an instability in the fluid mixture which leads to demixing (the spinodal point is that at which the curvature is first seen to become negative). The entropy of mixing of the colloid-polymer system for these two (mixed and demixed) states is represented in figure 3.7 b). The first rather surprising observation is that the entropy of mixing is always positive for the athermal colloid-polymer system; for the mixed state (P ∗ = 0.8) it is virtually indistinguishable from the ideal entropy of mixing, and though the entropy of mixing is seen to drop for the demixed state (P ∗ = 0.8) it is still significantly positive. The consequence of this is that the entropy of mixing in this system favours the mixed state, and care should be taken when one refers to ”entropy driven” transitions in athermal colloid-polymer systems. So what is responsible for the phase separation? The answer of course lies in the enthalpy of mixing which constitutes the other contribution to the free energy of a system at ∗ = ∆h∗ − ∆s∗ ). It is clear from constant pressure (∆Gm = ∆Hm − T ∆Sm , or ∆gm m m

figure 3.7 c) that the enthalpy of mixing is large (in comparative terms) and positive over the entire composition range both for the mixed and demixed states, with a slight asymmetry in the curves towards the colloid-rich part of the diagram. The importance of such an enthalpic contribution to the free energy in these athermal mixtures has already been recognised [75, 76]. As the temperature is lowered (or in this case the pressure increased) the contribution of the enthalpy to the Gibbs free energy of mixing dominates the entropic contribution and mixing becomes unstable relative to fluid phase separation. This is the same mechanism that is responsible for liquid-liquid immiscibility in simple systems [5] (c.f., figure 3.6). The positive contribution of the enthalpy of mixing gives rise to the change in curvature of the Gibbs free energy in the colloid region of the diagram, and to the subsequent phase separation (common tangent condition).

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Figure 3.7: The thermodynamic properties of mixing for an athermal binary mixture of colloids (hard spheres of diameter σC ) and flexible polymers (chains formed from tangent hard spheres of diameter σP ) determined from the Wertheim TPT1 approach. The parameters characterizing the system are a polymer-segment to colloid diameter ratio of d = σP /σC = 0.06 and a polymer chain length of m = 500. The ∗ = ∆G /N kT , b) entropy ∆s∗ = ∆S /N k , c) reduced a) Gibbs free energy ∆gm m m m ∗ = ∆V /σ 3 of mixing are plotted ∗ enthalpy ∆hm = ∆Sm /N kT , and d) volume ∆vm m C for mixed (P ∗ = 0.8) and a demixed (P ∗ = 1.2) states; the ideal entropy of mixing 3 /kT , is also shown as a dashed curve. The reduced pressure is defined as P ∗ = P σC and xP represents the mole fraction of polymer. The remaining question to answer is the reason for the large unfavourable (positive) enthalpy of mixing in the athermal colloid-polymer system. The change in the internal energy of an athermal system is zero (∆u∗m = 0) so the only contribution to the enthalpy is due to the change in volume of mixing (∆h∗m = ∆u∗m + P ∗ ∆v ∗ = P ∗ ∆v ∗ ). This means that the volume of mixing in the system must be positive, as indeed can be seen from figure 3.7 d). The large positive volume of mixing leads to a large positive enthalpy of mixing which in turn contributes to an unfavourable Gibbs free energy of mixing. The large volume of mixing in the system is due to the large colloid-polymer excluded volume interaction: when the colloid and the polymer are mixed at constant volume the increase in the excluded volume interactions leads to

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a larger pressure; in order to maintain the pressure constant the mixture will tend to increase its volume and minimise the unfavourable excluded volume interactions. Of course, the maximisation of the free volume (minimisation of the excluded volume) is an entropic effect, but as we have shown for the athermal colloid-polymer system this leads to an unfavourable enthalpy of mixing not an unfavourable entropy of mixing. The Gibbs function represents all the entropic contributions of the system and its surroundings, and every spontaneous process is driven by an increase in the total entropy.

3.5

Conclusion

In this chapter, we have presented the results of a study of the molecular conditions which give rise to fluid phase separation in binary mixture of hard colloids and flexible polymers. By using the Wertheim TPT1 theory, it is possible to treat both components at the ”microscopic” level of the segments making up the polymer, i.e., not only the length of the polymer but the diameter of the segment is a relevant variable. As well as a description of the colloid-colloid and colloid-polymer interactions, this approach allows for the excluded volume effect between polymers to be taken into consideration. One advantage of the TPT1 approach is that one does not explicitly require any information about the dimension of the polymer, which can change abruptly both with density and composition. The incorporation of polymer-polymer interactions leads to significant differences with the theoretical predictions for ideal chains, not only in the protein-limit (large polymers and small colloids) but also in the AO-limit (polymers which are much smaller than the colloids). In common with other recent studies [162, 249] we find that the treatment of the polymer-polymer interactions suggest that the critical density for fluid-phase separation tends to a finite value as the chain length is increased. In addition, there also appears to be a maximum in the polymer segment-colloid diameter ratio of about d = σP /σC ∼ 0.2 above which the athermal colloid-polymer system is not expected to exhibit fluid phase separation regardless of the chain length of the polymer. We also show that an unfavourable enthalpy of mixing (due to the large colloid-polymer excluded volume interaction) is responsible the fluid-phase separation in our athermal colloid polymer system; the entropy of mixing in this case is always positive and favours a mixed state, so that a more careful use of the term ”entropy driven” phase transition is

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recommended for such systems. In the case of polymer solutions the segments making up the polymer are of a similar size to the solvent and attractive interactions play a crucial role. The mechanism for fluid phase separation is completely different in polymer solution (chapter 2): the volume of mixing can be negative in regions of the phase diagram corresponding to dilute solutions of polymer, which gives rise to a negative (unfavorable) entropic contribution (see chapter 2 or [256]). We hope that our analysis of the nature of fluid phase separation in colloid-polymer systems has provided a slightly different perspective to this well studied area. In future work, we plan to incorporate both the effect of the attractive interactions to make quantitative comparisons with experimental measurements of phase separation in colloid-polymer mixtures. An attempt will also be made to treat the solid phases in such systems. In the next chapter, we present the effect of polydispersity on the demixing in both polymer-solvent and polymer-colloid systems. In the case of colloidal systems, we treat both the polydispersity of the polymer chain length, and the polydispersity of the colloid diameter.

Chapter 4

The Effect of Polydispersity In Polymer Systems Polydisperse systems are encountered in many different industrial applications, and quantitative predictions of their thermodynamic properties are required for process design. Some common examples of polydisperse systems are crude oils and condensate gases, colloidal particles of polydisperse size such as droplets of oils in milk, and, more relevant to the focus of this thesis, polymer systems. There are two main difficulties in determining the phase equilibria of a polydisperse system. Firstly, polydisperse systems usually involve a very large number of different species. As a result, a large number of equations (corresponding to the equality of the chemical potentials of each species in all phases) have to be simultaneously solved. In that case, usual numerical methods such as the isothermal flash of Michelsen [203â&#x20AC;&#x201C;205] converge slowly or fail. The other problem with polydisperse systems is that the experimental characterisation of the polydispersity does not provide the mole fractions of each individual species, but a continuous distribution of mole fractions. The true boiling point (TBP) curve, used in petroleum industry, is an example of such continuous distribution curve. The TBP curve is obtained by distillation of a polydisperse mixture (crude oil) using a distillation column with many different plates and a high reflux ratio. The boiling point temperature equivalent to the temperature at the top of the column is plotted as a function of the weight fraction distilled. A differentiation of the weight fraction distilled with respect to the boiling point provides the TBP curve, which is a boiling point distribution function. Another example of continuous distribution obtained experimentally is the molecular weight distribution function of a polymer, which is determined by gel permeation chromatography.

151

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS152 The normalised distribution functions provide the mole fraction of the species as a function of a certain property (boiling point in the case of TBP curve, or molecular weight in the case of polymers). There is an inconsistency between continuous distributions of species and conventional thermodynamics which involves discrete mole fractions corresponding to each individual component. The term continuous thermodynamics is then used to deal with continuous mole fraction distribution functions. Excellent reviews for phase equilibria calculation in polydisperse systems have been proposed by Cotterman and Prausnitz [262], and recently by Browarzik and Kehlen [263], and Sollich [42]. There are two ways to calculate phase equilibria in polydisperse systems. A first approach to treat continuous thermodynamics is to use pseudocomponents. The pseudocomponent method consists in dividing the distribution function into an arbitrary number of k intervals, as illustrated in figure 4.1. The mole fraction x k of the pseudocomponent k is equal to the integral of the distribution function over the interval k. In petroleum industry, the term ”cut” is used to call such intervals of components. After a true boiling point curve experiment is undertaken, each cut Cn is determined as the sum of all the components with boiling points bounded by the boiling points of the two succesive n-alkanes Cn−1 and Cn . In practice, standard TBP distillation is never performed beyond the C19 cut so that the C20 cut is collected at the bottom of the column and defined as the heaviest pseudocomponent. Traditional thermodynamics and numerical methods for flash calculation can be applied to pseudocomponent systems with low computer-time requirements, as shown by Pedersen [264]. Pedersen and co-workers [264–267] have used the Soave-RedlichKwong equation of state [268], and pseudocomponents to calculate vapour-liquid phase equilibria of heavy oils and gas condensates. An accurate prediction of experimental data was obtained with dew, bubble points and flash calculation by using 6 pseudocomponents to represent 80 components. The physical properties of each pseudocomponent, such as critical point and acentric factor, are calculated as an average of the properties of the components represented by the pseudocomponent. The main problem to represent a polydisperse petroleum fluid is that the compositions of heaviest cuts of a petroleum fluid are usually unknown. However, it is possible to represented each cut by a mixture of three well-defined components: an alkane, a cyclane, and an aromatics, as suggested by Jaubert and Neau [269]. Because of the imperfect knowledge of the fluid composition especially of the heaviest cut, the

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Figure 4.1: Continuous chain length distribution function in terms of weight fraction W , and pseudocomponents, from Browarzik et al. [263] . common practice is to fit the parameters of a pseudocomponent representing the heaviest cut on experimental data characterising the petroleum fluid. The compositions of the components representing the lighter cuts can be determined in order to satisfy the experimental value of both the density and the molecular weight of each cut [269]. The heaviest cut C20 can be represented by a single pseudo-component whose physical properties are directly fitted to experimental measurements (molecular weight and density of the cut C20 , crude oil PVT data). Such method is used by Avaulle et al. [270] to represent the properties of gas condensates. However, the tuning of parameters is always a difficult problem, and may yield errors in experimental data which are not included in the fitting [271]. An other way to represent the heaviest cut of petroleum fluids is to use discrete distribution functions of welldefined components (n-alkanes and aromatics) [272]. The shape of the distributions can be fitted to PVT data of the petroleum fluid [272], and improvement can be made by fitting both molecular weights, densities of heavy cuts, and PVT data, as shown by Jaubert et al. [273]. The second way to deal with continuous thermodynamics is to use the experimental continuous distribution function itself in the expressions of the thermodynamic model. The free energy of the system then becomes a functional of the distribution function. The main advantage of applying continuous thermodynamics is that analytical expressions can often be obtained for bubble and dew point, cloud and shadow curve calculations, or stability criteria. Continuous distribution functions

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS154 were first applied in the thirties by Katz and Brown [274], to calculate vapour-liquid equilibria of a polydisperse systems with Raoult’s Law. They replaced the sum terms over all the mole fractions by an integral with respect to the continuous distribution of mole fractions. Many other authors have used continuous thermodynamics for vapour-liquid equilibria of petroleum fluids with cubic equations of states (e.g., see [275–281]). Depending on the thermodynamic model used, there are many ways to calculate phase equilibria with a continuous distribution. Assuming an analytical form of the feed distribution function, phase equilibria can often be solved analytically: Cotterman et al. [282] use a gamma distribution function as the feed distribution to calculate dew and bubble points, and showed that the calculated distribution function of the new phase is also a gamma function. Kehlen and R¨atzsch [283] calculated cloud and shadow curves of polymer systems by using a Schulz-Flory [284,285] distribution as the parent distribution. Analytical expressions were obtained by the same authors with bivariate Stockmayer distributions [286] to describe polydispersity both in terms of molecular weight and co-monomers. Flash calculations are more difficult to carry out with continuous distributions, and usually no analytical expressions are possible. Cotterman and Prausnitz [287] proposed a moment method for flash calculation, and Due and Mansoori [288] developed a similar method called the equilibrium-ratio technique. They developed a method based on minimisation of the Gibbs free energy with respect the moment of the distribution, and applied it with the Peng-Robinson equation [289] to mixtures of hydrocarbons. Mansoori et al. [290] proposed the equality-chemical-potential technique which gives similar results as the Gibbs minimisation technique. According to the continuous parent distribution function used, the expression may not be analytical and integration may have to be carried out numerically. Quadrature methods [209] are then very useful to calculate the integrals numerically and to perform phase equilibria with any type of feed distribution functions. Cotterman and Prausnitz [291] and Chou and Prausnitz [292] applied quadrature methods to flash calculation. This method consists in reducing the continuous distribution function to a given number of pseudocomponents. Shibata et al. [293] have used a specific quadrature according to the shape of the feed distribution function, and derived a general Gaussian quadrature method to calculate integrals defined for intervals other than [0, ∞].

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS155 It is also important to determine stability criteria for critical and spinodal points in polydisperse systems, especially in the case of polymer systems. It is indeed important to determine the spinodal curves, as mentioned in chapter 1, section 1.3.2. Browarzik and Kehlen [294] developed a criteria for continuous polydisperse systems, to calculate spinodal and critical points. The method was extended to equations of state by Browarzik and Kehlen [295]. Hu and Prausnitz [296] derived similar equations for spinodal and critical point calculations and generalised the method for mixtures of discrete compounds and polydisperse species (semi-continuous systems). Rochocz et al. [297] developed expressions of stability criteria for continuous and semi-continuous systems based on Heidemann and Khalil’s [298] stability criteria for discrete mixtures. Browarzik and Kowalewski [299] derived a stability criteria for mixtures of two discrete compounds and a polydisperse polymer, and applied it to polystyrene/cyclohexane/carbon dioxide systems with the Sako-Wu-Prausnitz equation of state [300]. The problem of liquid-liquid immiscibility in polymer-solvent systems has been treated by many authors, each of them using their own thermodynamic model and numerical methods. Schulz [301] was the first to use continuous thermodynamics for polymer systems. One should also mention papers of Koningsveld [302, 303], Okamoto and Sekikawa [304], Tung [305], Huggins [306], Baysal and Stockmayer ˇ [307], Solk [308–310], and Kamide et al. [311]. Koningsveld [302, 303] studied polydispersity in polymer systems by looking at ternary systems where the polymer is ˇ represented by two chain molecules. Solk [308–310] did a general study of the effect of polydispersity on the liquid-liquid immiscibility in polymer systems. He showed that that the shape of the feed distribution function has a big influence on the cloud and shadow curves, the critical points, and that wide distributions can give rises to three-phase points corresponding to an equilibrium between three-phases. Similarly, de Loos et al. [312,313] studied experimentally and theoretically the influence of the shape of the distribution function on critical and three-phase points in ethylenepolyethylene systems. More recently, Koak and Heidemann [314, 315] developed an algorithm to calculate phase boundaries in polymer-solvent systems with the Sanchez-Lacombe equation of state. Later, Phoenix and Heidemann [316] derived a robust algorithm to calculate cloud and shadow curves, and used a Schulz-Flory (gamma function) [284,285] and log-normal (or Wesslau) feed distribution functions

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS156 to describe polyethylene/n-alkane systems. They showed that the divergent lognormal distribution give rise to the same kind of three-phase points as those found ˇ by Solk and de Loos et al., but not the Schulz-Flory distribution. Recently, Choi and Bae [317] have applied continuous thermodynamics to describe polystyrenecyclohexane systems. Jog and Chapman [318] proposed an algorithm based on the quadrature method [287, 291, 292] to determine cloud and shadow curve with a SAFT-type equation of state. Gross and Sadowski [319] applied a similar method with the PC-SAFT equation of state to successfully predict the liquid-liquid immiscibility in different polymer systems. The aim of our work is first to calculate liquid-liquid immiscibility in polymersolvent systems with the SAFT-HS (Wertheim TPT1) approach, and to provide a detailed explanation of the cloud and shadow curves in terms of ternary diagrams. The different methods to treat polydispersity will also be compared. A number of questions will be addressed: Can a simple ternary system of a solvent and two polymer molecules as proposed by Koningsveld [302,303] give rise to the same kind of cloud and shadow curves as a continuous system? Is the shape of the distribution for ˇ a fixed polydispersity index important as mentioned by Solk [308–310]? In the first part of this chapter a slightly different algorithm from Jog and Chapman [318] (but equivalent in terms of equations) to calculate cloud and shadow curves for discrete systems is described. The algorithm is also extended to treat continuous systems. In the second part, the moment method [320–322] is described and shown to be equivalent to our algorithm. We then apply our algorithm for a discrete distribution and the moment method for a continuous distribution to calculate phase equilibria in ternary and continuous polymer systems, respectively.

4.1

Polymer-Solvent System: Discrete Distribution

We consider a prototype polymer-solvent system of N molecules, where the solvent is represented by a spherical molecule of diameter σ, and the polymer is a mixture of n − 1 types of chain molecules of various chain lengths. Each polymer molecule i is represented as a chain of mi hard spherical segments of diameter equal to the diameter σ of the solvent. The solvent molecules and polymer spherical segments interact with the same mean field energy α, similar to the “symmetric” binary systems studied in Chapter 2, section 2.2.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS157 In our study, the polymer is polydisperse only in terms of chain length. Note that in the case of heteropolymers formed from the co-polymerisation of two different monomers, two other kinds of polydispersity are possible: polydispersity in the global ratio between the two monomers in the chain, and polydispersity due to the sequence of the monomers along the chain. In this work, we do not treat the effect of the polydispersity due to co-polymerisation, but this could be the topic of future studies. In order to characterise the polydispersity of chain length, a normalised and discrete length distribution function X in terms of mole fraction can be defined as

X(mi ) =

xpoly =

xi xpoly n X i=2

for i = 2, .., n

(4.1)

xi = 1 − x 1 ,

where xi is the mole fraction of chain molecules i of length mi in the system, and xpoly = 1−x1 the sum of the mole fractions of all polymer molecules. The normalised length distribution function in terms of weight fractions W , corresponding to X, is given by

W (mi ) =

wpoly =

wi for i = 2, .., n wpoly n X i=2

(4.2)

wi = 1 − w 1 ,

where wi is the weight fraction of chain molecules of length mi in the system, wpoly the total weight fraction of chain molecules, and w1 the weight fraction of solvent molecules. The two distribution functions W and X are related by the relation

or equivalently by

X(mi )mi , W (mi ) = Pn i=2 X(mi )mi

W (mi )/mi . X(mi ) = Pn i=2 W (mi )/mi

(4.3)

(4.4)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS158 For a real polymer, the molecular weight distribution functions in terms of mole and mass fractions would be identical to the length distribution functions X and W in our approach, since in the SAFT theory, the molecular weight Mi of a linear polymer molecule is directly proportional to the chain length mi with a factor 両 = mi /Mi common for all chain lengths of the polymer. We choose for instance 両 ' 0.238 for very long n-alkanes and linear polyethylene (cf., last chapter). The molecular weight distribution function of a polymer is determined experimentally in terms of weight fractions (distribution W ) by gel permeation chromatography. The corresponding distribution function X in terms of mole fraction, which is useful for phase equilibria calculation, can then be obtained with equation (4.4). The shape of the distribution function X is characterised by its moments hXik which are defined as

hXik =

n X

X(mi )mi k .

(4.5)

i=2

The zero moment hXi0 of the distribution X is equal to 1 since X is normalised. The first moment hXi1 of the distribution X is also the number average chain length hmin = hXi1 . In practice, the number average molecular weight hM in = hmin /両 is measured in experiments that determine the number of particles, such as the determination of the osmotic pressure or the direct counting of particles using electron microscopy. A weight average chain length hmiw is defined as the first moment hW i1 of the weight fraction distribution W , or as the ratio of the second and first moments of the mole fraction distribution X, and is given by n X

n hXi2 X(mi )m2i W (mi )mi = hmiw = hW i1 = . = Pi=2 n hXi1 i=2 X(mi )mi i=2

(4.6)

The weight-average molecular weight hM iw = hmiw /両 is measured in experiments that determine the mass of molecules, such as sedimentation or light scattering. The breadth of the distribution can be gauged by establishing the heterogeneity or polydispersity index Ip defined as

Ip =

hmiw hXi2 . = hmin hXi1 2

(4.7)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS159 If all chain molecules have the same chain length, the polymer is monodisperse and Ip = 1. The numerical value of Ip is usually around 2 in real systems, but both larger and smaller values can be obtained experimentally. For phase equilibria calculations, it is useful to define a density distribution ρm of chain length as

ρm (mi ) = ρi = ρxi = ρxpoly X(mi ),

(4.8)

where ρi is the number density of component i, and ρ = N/V the total number density of the system of volume V . The distribution ρm is not normalised, and its moments hρm ik , referred to as “density moments” [42] are related to the moments hXik of the mole fraction distribution X by

hρm ik =

n X

ρm (mi )mi k

(4.9)

i=2

= ρxpoly hXik .

The zero and first moments of ρm are related to xpoly and hmin via the relations

hρm i0 = ρxpoly

(4.10)

hρm i1 = ρxpoly hmin ,

and the density distribution can be expressed as

ρm (mi ) = hρm i0 X(mi ).

(4.11)

It can be shown that the residual part of the SAFT free energy can be expressed in terms of a finite number of density moments, and that phase equilibria can be solved in this case (cf. reference [318]).

4.1.1

Density Free Energy

In the system of interest, the attractive dispersion interactions are treated at the mean-field level of van der Waals, with a segment energy parameter α which is equal for each segment (and for the solvent). As is usual within a SAFT desciption

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS160 the contribution to the free energy due to the formation of chains are treated with Wertheim’s TPT1 approach [35, 83, 182]. The SAFT-HS equation can be written in terms of the Helmholtz free energy A by separating each contribution as [35]

A AIDEAL AM ON O. ACHAIN = + + , N kT N kT N kT N kT

(4.12)

where N is the total number of molecules, T is the temperature and k the Boltzmann constant. For polydisperse system it is convenient to use the density free energy f = A/V = ρA/N and express f in terms of the number densities ρi to determine the chemical potential µi of the component i via the relation

µi =

∂f ∂ρi

(4.13)

T,ρj ,j6=i

The density free energy f can also be expressed by separating each contribution as

f A f IDEAL f M ON O. f CHAIN =ρ = + + . kT N kT kT kT kT

(4.14)

The ideal term is given by n X f IDEAL =ρ xi [ln (ρxi νi ) − 1] , kT i=1

(4.15)

where νi is the thermal de Broglie volume of component i (incorporating the translational and rotational kinetic contributions). The terms νi which do not contribute to phase equilibria calculation, can be removed from equation (4.15) to give n f IDEAL X ρi [ln (ρi ) − 1] + constant. = kT i=1

(4.16)

Applying the definition (4.8) of the density distribution of chain length ρm , one can write the ideal density free energy in terms of the density of solvent ρ1 and the density distribution ρm as n X f IDEAL = ρ1 [ln (ρ1 ) − 1] + ρm (mi ) [ln (ρm (mi )) − 1] . kT i=2

(4.17)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS161 The monomer-monomer contribution f M ON O can be separated into a repulsive term f HS and an attractive mean field term f M F as

f M ON O f HS f MF = + . kT kT kT

(4.18)

The repulsive contribution of a multicomponent mixture of hard spheres, f HS , is described by Boubl´ık [187] and Mansoori et al. [188] as

f HS 6 = kT π

"Ã

3ζ1 ζ2 ζ23 ζ23 ln(1 − ζ ) + − ζ + , 3 0 (1 − ζ3 ) ζ3 (1 − ζ3 )2 ζ32

(4.19)

where the reduced densities ζl are defined as

n π X xi mi σil ρ 6 i=1

ζl =

(4.20)

n πX ρi mi σil , 6 i=1

and mi is the number of segments of chain molecule i and segment diameter σi . Since all chain segments and solvent molecules have the same diameter σ, and component 1 is modelled as spherical (m1 = 1), the reduced densities can be simplified by applying definitions (4.8) and (4.9) as

ζl = =

n X π l ρm (mi )mi σ ρ1 + 6 i=2

(4.21)

π l σ (ρ1 + hρm i1 ) . 6

Since all segments have the same diameter σ, expression (4.19) reduces to that of Carnahan-Starling equation (see Hansen and McDonald [189])

f HS kT

n X i=1

ρi m i

4η − 3η 2 (1 − η)2

(4.22)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS162

= (ρ1 + hρm i1 )

4η − 3η 2 , (1 − η)2

where η = ζ3 is the packing fraction of the mixture. The attractive term in the mixture is written using the usual van der Waals one-fluid mixing rule [5] n X n 1 X f MF =− αij ρi ρj mi mj . kT kT i=1 j=1

(4.23)

In our case, the mean-field energies αij are all equal to α, and m1 = 1. If we define a reduced temperature T ∗ = kT /α, equation (4.23) can be written as

f MF kT





n n X n X X 1  2 ρm (mi )ρm (mj )mi mj (4.24) ρm (mi )mi + = − ∗ ρ1 + 2ρ1 T i=2 j=2 i=2

= −

´ 1 ³ 2 2 ρ + 2ρ hρ i + hρ i . 1 m m 1 1 1 T∗

The chain contribution f CHAIN due to the formation of chains of segments is obtained from Wertheim’s TPT1 expression [35, 83, 182] as n X f CHAIN ρi (mi − 1) ln gii (σi ) , =− kT i=2

(4.25)

where gii (σ) is the radial distribution function at contact between two hard sphere segments of type i in a mixture of hard spheres. Since all segments have the same diameter σ, the radial distribution function at contact gii (σ) is the radial distribution function of a pure hard-sphere system g HS (σ) derived from the Carnahan-Starling expression [74], and is given by [189]

g HS (σ) =

1 − η/2 . (1 − η)3

(4.26)

Hence, the radial distribution function g HS (σ) has the same value for all chain molecules, and equation (4.25) can be reduced to

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS163

f CHAIN kT

= −

n X i=2

ρm (mi ) (mi − 1) ln g HS (σ)

(4.27)

= − [hρm i1 − hρm i0 ] ln g HS (σ) .

It can be seen from equations (4.22), (4.24) and (4.27), that the residual part of the density free energy, f RES /kT = (f − f IDEAL )/kT , is only a function of the reduced temperature T ∗ , the density of solvent ρ1 , and the zero and first density

moments hρm i0 and hρm i1 of the density distribution of the chain length ρm . As a result, the residual density free energy depends on the length density distribution function only via the first density moment. Sollich [322] refers to such polydisperse systems as “truncable”, i.e., systems for which the residual density free energy can be expressed as a function of a finite number of density moments. Phase equilibria calculation in such systems can be easily solved, as we show in the next sections. We use here a different notation from Sollich’s papers to avoid any confusion with the usual notations in SAFT theory.

4.1.2

Chemical Potentials

To solve the phase equilibria, it is necessary to determine the chemical potential µ i of each compoenent i from the expression of the free energy (cf. equation (4.14)). Applying the thermodynamic relation (4.13), the ideal part of the chemical potentials of each molecule i is obtained as

µiIDEAL = ln ρi . kT

(4.28)

The residual chemical potential µRES of the solvent molecule is obtained by applying 1 relation (4.13) to equations (4.21), (4.22), (4.24) and (4.27), as

µRES 1 = kT

∂f RES /kT ∂ρ1

(4.29)

hρm i0 hρm i1

The analytical expression of the residual chemical potential of the solvent molecule can be easily obtained via equation (4.29) since the residual density free energy

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS164 f RES /N kT has been expressed in terms of the density ρ1 of the solvent (cf. equations (4.22), (4.24) and (4.27)). The residual chemical potential of the chain molecule i is obtained by applying equation (4.29) and the chain rule, as

µRES i kT

∂f RES /kT ∂ρi

∂f RES /kT ∂hρm i0

∂f RES /kT + ∂hρm i1

(4.30) ρj ,j6=i

ρ1 ,hρm i1

ρ1 ,hρm i0

∂hρm i0 ∂ρi µ

ρj ,j6=i

∂hρm i1 ∂ρi

. ρj ,j6=i

Sollich [320, 322] defines the residual “moment chemical potentials” hµi RES , as k hµiRES k = kT

∂f RES /kT ∂hρm ik

(4.31)

ρ1 ,hρm ij6=k

From the definition of the density moments (4.9),

∂hρm ik ∂ρi

= mk .

(4.32)

ρj ,j6=i

Combining equations (4.30) through (4.32), the residual chemical potential µ RES of i a chain molecule i 6= 1 can be expressed as ´ µRES 1 ³ RES i . hµi0 + mi hµiRES = 1 kT kT

(4.33)

The residual chemical potential of a chain molecule in the SAFT approach is a linear function of the chain length at fixed volume V , composition, and temperature T . Note that it would be the same if the polymeric segments had different diameters from that of the solvent molecule.

4.1.3

Pressure

The pressure of the system of hard spheres and polydisperse chains of spheres can be obtained from the thermodynamic relation

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS165

P = ρkT − f RES +

n X

ρi µRES , i

(4.34)

i=1

Inserting equation (4.33) into (4.34), the pressure can be expressed in terms of the solvent density ρ1 and the density moments hρm i0 and hρm i1 as

= ρkT − f RES + ρ1 µ1 +

n X i=1

ρm (mi ) hµiRES + mi hµiRES 0 1

(4.35)

= ρkT − f RES + ρ1 µ1 + hρm i0 hµiRES + hρm i1 hµiRES . 0 1

4.1.4

Cloud and Shadow Curve Calculation

A polymer consists of a very large number of components differing in molecular weight. According to the Gibbs phase rule, the number of phases in equilibrium may also be very large. Fortunately, in polydisperse polymer systems the coexistence of only two phases often occurs. In this work, only two-phase equilibria calculations will be considered. In the polymer community, the transition from one transparent single phase to an opaque two-phase system is called a cloud point. This point is basically a saturation point of a multicomponent mixture. It is the point where the first droplet of the second phase appears. The composition given by the cloud curve is equal to the global composition of the sample, and the corresponding length distribution function of chain molecules is called “parent” of “feed” distribution. The shadow point gives the composition of the first droplet of the second phase. The cloud curve is often determined experimentally at fixed pressure, and the usual temperature-composition diagrams are then obtained. The cloud and shadow phases are respectively labelled Cl and Sh for convenience. The vector composition of the cloud phase Cl, x(Cl) is equal to the fixed global composition of the sample x(0) , as the amount of the shadow phase Sh is negligible compared to phase Cl. The vector composition of the shadow phase x(Sh) is unknown at this stage. Length distribution functions can be defined for both phases, in the same way as in equations (4.1), (4.2) and (4.8). The mole fraction distribution X (Cl) of the cloud phase is equal to the

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS166 fixed parent distribution X (0) , while the mole fraction distribution X (Sh) of the shadow phase is initially unknown. To calculate the cloud and shadow points at given pressure Pf ixed and global composition x(0) = x(Cl) of the sample, the following equilibrium conditions (equality of pressure and chemical potentials in both phases) have to be satisfied:

P (Cl) = Pf ixed

(4.36)

P (Sh) = Pf ixed

(4.37)

(Cl)

(Sh)

µi

= µi

(4.38)

Combining the equations (4.28) and (4.33), and the definition of the density distribution (4.8) , equation (4.38) for the equality of chemical potential of each chain molecules i has the following form

RES(Cl)

(Cl) kT ln ρm (mi ) + hµi0

RES(Cl)

RES(Sh)

= kT ln ρ(Sh) m (mi ) + hµi0 (Cl)

where ρm

(Sh)

and ρm

(4.39)

+ mi hµi1

RES(Sh)

+ mi hµi1

are the density length distributions of the cloud and the shadow RES(Cl)

phase respectively, and (hµi0

RES(Cl)

, hµi1

RES(Sh)

), and ( hµi0

RES(Sh)

, hµi1

) the

(zero, first) residual moment chemical potentials in phases Cl and Sh defined in equation (4.31). By factorising the terms in mi , and taking the exponential of each term of equation (4.39), the density distributions of the two phase Cl and Sh in coexistence can be related as

(Cl) ρ(Sh) m (mi ) = ρm (mi )e

RES(Cl)

hµi0

RES(Sh)

−hµi0

/kT

RES(Cl)

hµi1

RES(Sh)

−hµi1

mi /kT

(4.40) Equation (4.40) can also be written in terms of mole fraction distributions as

(Cl)

(Sh)

(mi ) = X

(Cl)

(mi )

xpoly ρ(Cl) (Sh) xpoly ρ(Sh)

RES(Cl)

hµi0

RES(Sh)

−hµi0

/kT

RES(Cl)

hµi1

RES(Sh)

−hµi1

mi /kT

(4.41)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS167 (Sh)

(Cl)

where xpoly and xpoly are the mole fractions of the polymer molecules, and ρ(Cl) and ρ(Sh) the densities of phases Cl and Sh. Since the free enery of the chain molecules only depend on the zero and first moments of the distribution, the n − 1 equations (4.40) are not mathematically independent: they can be reduced to only 2 independent equations (4.42) and (4.43) which satisfy the definitions of the zero (Sh)

and first (hρm i0

(Sh)

hρm i0

n X

ρ(Sh) m (mi )

n X

) density moments of phase Sh

(4.42)

i−2

i=2

= e

(Sh) hρm i1

(Sh)

and hρm i1

(Cl) ρm (mi )e

RES(Cl)

hµi0

RES(Cl)

hµi0

RES(Sh)

−hµi0

n X

ρ(Sh) m (mi )mi

n X

RES(Sh)

−hµi0

/kT

n X i=2

/kT

RES(Cl)

hµi1

RES(Cl)

hµi1 (Cl) ρm (mi )e

RES(Sh)

−hµi1

RES(Sh)

−hµi1

mi /kT

(4.43)

i−2

i=2

= e

RES(Cl)

hµi0 (Cl) ρm (mi )mi e

RES(Cl)

hµi0

RES(Sh)

−hµi0

/kT

RES(Sh)

−hµi0

n X i=2

/kT

(Cl) ρm (mi )mi e

RES(Cl)

hµi1

RES(Cl)

hµi1

RES(Sh)

−hµi1

RES(Sh)

−hµi1

mi /kT

The conditions (4.42) and (4.43) are sufficient to assure the equality of the chemical potentials of each chain molecule in both phases. Indeed, if the temperature T , the density of solvent, and the density moments of both phases Cl and Sh are known, and if (4.42) and (4.43) are satisfied, the n−1 relations (4.40) are all satisfied and the (Sh)

distribution function ρm

is unique. Hence, solving the cloud point calculation at a

fixed pressure Pf ixed consists in solving a non-linear system of five equations and five unknown variables. The equations are the equality of the pressure of both phases to the fixed pressure Pf ixed (equation 4.35), the equality of the chemical potential of the

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS168 solvent in both phases (equation (1.29)), and the two equations (4.42) and (4.43). Concerning the variables of the system of equations, the density moments are useful variables to derive the expression of the free energy and the chemical potentials, but the variables xpoly and hmin are more convenient to explain the physics of the system and have a well-defined definition range. We then choose as variables of (Sh)

the non-linear system of equations the mole fraction xpoly of polymer molecules in phase Sh (related to the zero moment), the number-average chain length hmi (Sh) n (related to the first moment), the densities ρ(Cl) and ρ(Sh) of the two phases, and the saturated temperature T . The variables (ρ, xpoly , hmin ) can be expressed in terms of the density moments (hρm i0 , hρm i1 ) via equation (4.10). The composition of the cloud phase Cl is fixed, so that the length distribution (Cl)

X (Cl) , the number average hmin(Cl) and the mole fraction of polymer xpoly in phase Cl are known. The system of non-linear equations can be solved by using the minpack algorithm based on the Levenberg-Marquard algorithm [209]. The initial guesses for the variables are the values obtained from the cloud point of the corresponding binary mixture where the polymer is mono-disperse. The cloud point of the binary mixture is calculated by using a specific algorithm for cloud point calculation, which is explained in detail in chapter 2, section 2.8.

4.2

Polymer-Solvent Systems: Continuous Distributions

The real molecular weight distribution function of a polymer is always discrete since the polymer consists of a very large but finite number of chain molecules. However, it is a good approximation to consider the molecular weight distribution as continuous since the molecular weight of the monomer is very small compared to the number average molecular weight and the width of the distribution. Whereas the density distribution in the discrete case is a sum of delta functions corresponding to the density of each pure chain molecule of length mi , the density distribution in the continuous case represents the density of a continuum of species as a function of the chain length . Polymer distribution functions have specific shapes which depend on the mechanism of the polymerisation reaction. For instance, the Flory molecular weight distribution function describes the polydispersity of a step-growth polymerisation model

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS169 which is based on two assumptions: polymers form linear chains through the reaction of bi-functional molecules, and reactivity is not a function of chain length, and therefore completely random. An other well-known function is the Schultz-Flory distribution which models high-conversion chain-growth polymerisation. In this case, the rate of chain growth polymerisation is known to be a function of viscosity which are in turn a function of chain length. The distribution functions are determined experimentally in terms of weight fraction by gel permeation chromatography. However, it is necessary to know the corresponding mole fraction distributions to perform phase equilibria calculations. We consider a polymer solution, where the solvent molecule is modelled as a hard sphere of diameter σ (component 1), and the polymer as a mixture of an infinite number of chain molecules with segments of the same diameter σ but of different lengths: the distribution of chain length is continuous. The analytical expressions of the free energy, chemical potentials and pressure for a continuous distribution can easily be obtained by replacing the term

i=2

by an integral

dm in the expressions

obtained in the previous sections for a discrete distribution. The density moments hρm ik in the continuous case are defined as hρm ik =

dmρm (m)mk ,

(4.44)

which is the obvious continuous version of equation (4.9), and the density distribution is given by

ρm (m) = hρm i0 X(m).

4.2.1

(4.45)

Density Free Energy

The density free energy of the continuous system is

f f IDEAL f RES = + , kT kT kT

(4.46)

where the ideal part of the free energy is given by

f IDEAL = ρ1 [ln ρ1 − 1] + kT

dmρm (m) [ln ρm (m) − 1] .

(4.47)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS170 The residual free energy f res is a functional of the density distribution of chains ρm , and can be written as the sum of the different contributions as

f RES f HS f MF f CHAIN = + + . kT kT kT kT

(4.48)

The hard sphere term is again given by the Boubl´ık [187] and Mansoori et al. [188] expression (cf. equation (4.19)), where the reduced densities ζl are defined as

π l σ ρ1 + 6 µ

ζl =

dmρm (m)m

(4.49)

π l σ (ρ1 + hρm i1 ) . 6

The attractive contribution (cf. equations (4.23) and (4.24)) for a continuous density distribution becomes

f MF kT

= −

1 ρ1 2 + 2ρ1 T∗

= −

´ 1 ³ 2 2 . ρ + 2ρ hρ i + hρ i 1 m m 1 1 T∗ 1

dmρm (m)m +

0 dm0 ρm (m)ρm (m0 )mm(4.50)

The expression (4.27) of the chain contribution in the discrete case, can be written in the continuous case as

f CHAIN kT

= −

·Z

dmρm (m) (m − 1) ln g HS (σ)

(4.51)

= − [hρm i1 − hρm i0 ] ln g HS (σ) .

The residual free energy is now a functional of the density distribution ρm . However, as in the case of a discrete distribution, the residual free energy only depends on finite number of density moments of the continuous density distribution (the zero and first density moments hρm i0 and hρm i1 to be precise), and the system is still “truncable”.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS171

4.2.2

Chemical Potential and Pressure

The chemical potential of the solvent molecules can be obtained as in the discrete case, via equation (4.13). The chemical potential of a chain molecule of length m is obtained by using the mathematical theorems of variational calculus. A detailed description of the properties of functionals is given by Hansen and McDonald [189]. The variation laws for functionals can usually be obtained from the corresponding derivatives of the discrete case. The ideal part of the chemical potential of the chain molecule of length is given by

µiIDEAL = kT

∂ ( dmρm [ln ρm (m) − 1]) ∂ρm (m)

= ln ρm (m).

(4.52)

ρ1

As in the discrete case, the residual chemical potential can be obtained via the chain rule as

µRES (m) kT

∂f RES /kT ∂ρm (m)

∂f RES /kT ∂hρm i0

∂f RES /kT + ∂hρm i1

(4.53) ρ1

ρ1 ,hρm i1

ρ1 ,hρm i0

∂hρm i0 ∂ρm (m) µ

∂hρm i1 ∂ρm (m)

ρ1

. ρ1

The residual “moment chemical potentials” hµik >RES , can be defined as hµiRES k = kT

∂f RES /kT ∂hρm ik

(4.54)

ρ1 ,hρm ij6=k

From the definition of the density moments (4.44) and using the properties of functionals,

∂hρm ik ∂ρm (m)

= mk .

(4.55)

ρ1

Combining equations (4.53) and (4.55), the residual chemical potential µ RES of a i chain molecule i 6= 1 can be expressed as a

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS172

´ µRES (m) 1 ³ RES . hµi0 + mi hµiRES = 1 kT kT

(4.56)

The residual chemical potential is a linear function of the chain length m as in the discrete case (cf., equation (4.33)). The pressure is given by

= ρkT − f RES + ρ1 µ1 +

dmρm (m)µRES (m)

= ρkT − f RES + ρ1 µ1 +

dmρm (m) hµiRES + mhµiRES 0 1

(4.57)

= ρkT − f RES + ρ1 µ1 + hρm i0 · hµiRES + hρm i1 · hµiRES . 0 1

4.2.3

Cloud and Shadow Curve Calculation

As for a discrete distribution, the phase equilibria for a continuous system is solved by imposing the equality of the pressure of both phases to a fixed pressure Pf ixed (cf. equations (4.37) and (4.38)), and the equality of chemical potentials in the two phases. The following equation must be satisfied for all values of m belonging to (0)

the interval of definition of the parent density distribution ρm equal to the density (Cl)

distribution ρm

of the cloud phase Cl µ(Cl) (m) = µ(Sh) (m),

(4.58)

where µ(Cl) (m) and µ(Sh) (m) are the chemical potentials of the chain molecule m in the cloud phase Cl and shadow phase Sh. Inserting equations (4.52) and (4.56) into (4.58), and rearranging the expression as in the discrete case (cf. equation (4.40), (Sh)

the density distribution ρm

of the shadow phase can be expressed as a function of (Cl)

the density distribution of the cloud phase ρm

ρ(Sh) m (m)

RES(Cl)

where (hµi0

RES(Cl)

hµi0 (Cl) (m)e ρm

RES(Cl)

, hµi1

RES(Sh)

−hµi0

/kT

RES(Sh)

), and ( hµi0

(equal to the parent distribution)

RES(Cl)

hµi1

RES(Sh)

−hµi1

m/kT

, (4.59)

, hµiRES 1,Sh ) are the (zero, first) resid-

ual moment chemical potentials in phases Cl and Sh defined in equation (4.54). The

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS173 (Sh)

density distribution ρm

of the shadow phase has to satisfy the two following con-

straints (4.60), (4.61) corresponding to the definitions of the zero and first moments (Sh)

hρm i0

(Sh) hρm i0

(Sh)

and hρm i0

= e

(Sh)

hρm i1

= e

dmρ(Sh) m (m)

(4.60)

(Cl) ρm (m)e

RES(Cl)

hµi0

RES(Cl)

hµi0

RES(Sh)

−hµi0

/kT

RES(Sh)

−hµi0

/kT

RES(Cl)

hµi1

RES(Cl)

hµi1 (Cl) ρm (m)e

RES(Sh)

−hµi1

RES(Sh)

−hµi1

m/kT

dmρ(Sh) m (m)m

RES(Sh)

−hµi0

(4.61) RES(Cl)

hµi0 (Cl) ρm (mi )mi e

RES(Cl)

hµi0

/kT

RES(Sh)

−hµi0

/kT

(Cl) dm ρm (m)me

RES(Cl)

hµi1

RES(Cl)

hµi1

RES(Sh)

−hµi1

RES(Sh)

−hµi1

m/kT

As in the discrete case (see section 4.1.4), solving the cloud-point calculation at a fixed pressure Pf ixed consists in solving a non-linear system of five equations and five unknown variables. The equations are the equality of the pressure of both phases to the fixed pressure Pf ixed (equations (4.37) and (4.38)), the equality of the chemical potential of the solvent in both phases, and the two equations (4.60) (Sh)

and (4.61). The variables are the mole fraction xpoly of polymer molecules and the number average chain length hmi(Sh) in the shadow phase, the densities ρ(Cl) n (Sh)

and ρ(Sh) of the two phases, and the saturated temperature T . The variables xpoly , hmi(Sh) , ρ(Cl) and ρ(Sh ) are directly related to the density moments as in the discrete n case (cf. equation (4.10). Since the cloud and shadow curve calculation consists in determining only the density moments of the density distribution for each phase in coexistence, one should try to express the pressure and the free energy only in terms

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS174 of those moments. Warren [321], and Sollich et al. [320] have precisely derived such expressions and developed the so-called “moment method”.

4.3

Moment Method Applied to Polymer-Solvent Systems

The moment method represents a powerful alternative tool to calculate phase equilibria in polydisperse systems. The method was developed independently by Sollich et al. [320] and by Warren [321] from two different approaches to end up with the same conclusions. A detailed description of the method is given in reference [322]. The method can only be applied with continuous density distributions. The main feature of the method is to express the ideal part of the free energy f IDEAL as a function of a finite number M of density moments hρm ik defined in equation (4.44). In this way, the total free energy f would be a function of only the M density moments. hρm ik . By considering one pseudo-component k corresponding to each moment density hρm ik appearing in the expression of the total free energy, the phase equilibria can be solved with the usual tangent-plane analysis by imposing the equality of each “moment chemical potential” hµik of pseudo-component k in both phases. The“moment chemical potential” are defined as the derivatives of the density free energy f with respect to the density moment hρm ik (see equation (1.31)). The moment method is very useful to get analytical expressions for the determination of multiple critical points and stability criteria. It gives exact results for cloud and shadow curves as we show later in this section, but only an approximate result in the case of multi-phase flash calculation. However in that case, the moment method becomes more and more accurate as the number of density moments used in the expression of the free energy f is increased (see [322] for further details).

4.3.1

Moment Free Energy

The moment method is built as follows: in the case of our polymer-solvent system described in section 4.2, the ideal density free energy f IDEAL is given by equation (4.47). One can add to the ideal term f IDEAL a term −kT

dmρm (m) ln R(m)

where R(m) is a continuous function of m independent of the density distribution ρm . This additional term linear in density does not affect the phase equilibria since its derivative with respect to ρm

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS175

∂ dmρm (m) ln R(m) kT = kT ln R(m), ∂ρm (m)

(4.62)

does not depend on ρm . We will explain the role of this term later in the discussion. The ideal free energy becomes

f IDEAL = ρ1 [ln ρ1 − 1] + kT

ρm (m) dmρm (m) ln −1 . R(m) ·

(4.63)

As was shown earlier, the total density free energy f is a functional of the density distribution ρm . If the system is “truncable”, the residual free energy f res can be expressed in terms of a finite number of moments (zero and first moment in the case of a polymer solution within a SAFT description). The ideal term f IDEAL depends on the same density moments, but also on an infinite number of higher moments because of the logarithm term in the expression of f IDEAL . At fixed temperature T , volume V , and composition, the total density free energy of the system has to be minimised in order to perform phase equilibria or salability analysis. Therefore, if f is expressed in terms of a finite number M of density moments hρm ik , f has to be a minimum for fixed values of the M moments. The ensemble of moments hρm ik includes at least the moments appearing in the expression of the residual free energy (hρm i0 , and hρm i1 in our case). The problem consists in finding the density distribution function ρm which minimises the total free energy f , with the M constraints given by the definition of the M fixed density moments (equation (4.44)). The minimisation criteria is obtained by using M Lagrange multipliers λ k , as

∂f /kT ∂ρm (m)

ρ1 ,hρm ik

∂ + ∂ρm (m)

X k

λk hρm ik −

dmρm (m)m

¶!

= 0. ρ1 ,hρm ik

(4.64) Applying the rules for functionals, the variation of the ideal free energy is,

∂f IDEAL /kT ∂ρm (m)

ρ1 ,hρm ik

ρm (m) = ln . R(m) µ

(4.65)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS176 Since the residual free energy f res only depends on a finite number of density moments included in the M fixed moments,

∂f RES /kT ∂ρm (m)

= 0.

(4.66)

ρ1 ,hρm ik

Inserting (4.65), (4.66), and (4.55) into (4.64), the minimisation criteria becomes

ρm (m) R(m)

−

λk mk = 0.

(4.67)

Therefore, the density distribution which minimises the density free energy for fixed values of the density moments hρm ik has the form ρm (m) = R(m)e

λ k mk

(4.68)

and the corresponding density moments hρm ik are given by hρm ii =

dm mi R(m)e

λ k mk

(4.69)

Equation (4.68) represents a family of density distributions, generated by R(m) and (Cl)

by the Lagrange multipliers λk . The exact density distributions ρm

(Sh)

and ρm

(cf.

equation (4.59)) of the cloud and shadow curves must be included in this family. On inserting equations (4.68), (4.69), and (4.44) into equation (4.63), the ideal term of the free energy can be written as

f IDEAL IDEAL = ρ1 [ln ρ1 − 1] − Smom , kT

(4.70)

IDEAL is given by where the “ideal moment entropy” Smom

IDEAL Smom = hρm i0 −

X i

λi hρm ii

(4.71)

The total “moment” free energy fmom is equal to X f RES fmom = ρ1 [ln ρ1 − 1] + . λi hρm ii − hρm i0 + kT kT i

(4.72)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS177 The free energy fmom depends only on the solvent density ρ1 and the moments, as the Lagrange multipliers implicitly depend on the moments by inverting equation (4.69); fmom is also called “projected moment free energy” [320,322] as it corresponds mathematically to a projection of the free energy of the system on a hypersurface of density defined by the fixed density moments hρm ik . The Lagrange multipliers λk are related to the density moments hρm ik by equation (4.69). The zero density moment hρm i0 is a function of the λk for fixed R(m), and can be considered as a differential of the λk as

dhρm i0 =

X µ ∂hρm i ¶ 0

∂λi

dλi

(4.73)

λj6=i

From equation (4.69),

∂hρm i0 ∂λi

= λj6=i

Ã R

∂ dm R(m)e ∂λi

λ k mk

λj6=i

= hρm ii .

(4.74)

Equation (4.73) then becomes

dhρm i0 =

X i

hρm ii dλi .

(4.75)

IDEAL can be seen as a On inspecting equation (4.71) the ideal moment entropy Smom

Lagrange transform of the zero density moment hρm i0 and

IDEAL dSmom = dhρm i0 − d

X i

= −

X i

hρm ii dλi −

X i

hρm idλi

X i

hρm ii dλi −

(4.76) X i

λi dhρm ii

λi dhρm ii .

As a result,

IDEAL ∂Smom ∂hρm ik

hρm ij6=k

= −λk .

(4.77)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS178

4.3.2

Moment Chemical Potentials

As we saw in the previous sections 4.1.2 and 4.2.2, one can define the “moment chemical potentials” hµii as the derivatives of the moment density free energy fmom with respect to the density moments

hµik = kT

∂f /kT ∂hρm ik

(4.78)

ρ1 ,hρm ij6=k

where the residual moment chemical potentials hµiRES are defined in (4.54). The k ideal chemical potentials can be obtained from (4.78), (4.70) and (4.77) as

hµikIDEAL = kT

∂f IDEAL /kT ∂hρm ik

= λk .

(4.79)

ρ1 ,hρm ij6=k

The residual moment chemical potentials hµik are given by (4.54). The pressure P obtained from the moment free energy is given by

= −fmom + ρ1 µ1 +

X k

hµik hρm ik

= −kT ρ1 [ln ρ1 − 1] − kT + kT ρ1 ln ρ1 +

X k

(4.80)

λk hρm ik + kT hρm i0 − f RES

hρm ik kT λk + hµiRES . k

After simplification, and since ρ1 +hρm i0 = ρ, the expression of the pressure becomes P = ρkT − f RES + ρ1 µ1 +

X k

hρm ik hµiRES . k

(4.81)

Expressions (4.72), (4.79), and (4.81) are general for any “truncable” systems. In the case of our continuous polymer-solvent system described in section 4.2, only the zero and first density moments and chemical potentials have to be considered in the sum of equation (4.81). Expression (4.81) then becomes . P = ρkT − f RES + ρ1 µ1 + hρm i0 · hµiRES + hρm i1 · hµiRES 0 1

(4.82)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS179 which is equivalent to the expression (4.57). Hence the pressure derived from the moment free energy and moment chemical potentials gives the exact pressure of the system for fixed temperature, density, and composition (see [322] for further details).

4.3.3

Cloud and Shadow Curve (Cl)

(Sh)

The density distributions ρm

and ρm

of phases Cl and Sh corresponding re-

spectively to the cloud and shadow points, must belong to the family of density distributions generated by (4.68). The mole fraction distribution function X (Cl) of the cloud phase is known and equal to the parent distribution X (0) as seen in section (Cl)

4.1.4, and is related to ρm

(Cl)

by ρm

= ρ(Cl) X (Cl) , where ρ(Cl) is the unknown den-

sity of phase Cl. One can choose any function for the distribution R(m) common for all density distributions of the family defined by (4.68) without changing the phase equilibria, as mentioned in section 4.3.1. To assure that the parent density distri(Cl)(m)

bution belongs to the family (4.68), one should choose R(m) = ρm

common for

all density distributions of the family (4.68) (see [322]). The density distributions (Sh)

(Cl)

and ρm

(Cl)

and λk

ρm λk

(Sh)

are then defined by the corresponding vectors of Lagrange multipliers and

(Cl) (Cl) e = ρm ρm

(Cl) ρ(Sh) = ρm e m

(Cl)

Equation (4.83) is satisfied if and only if λk (Sh)

the vector λk

(Cl)

λk

(Sh)

(4.83) .

(4.84)

= 0 for all k. One must only determine

to solve the phase equilibria. The moment free energy at fixed

pressure has to be minimised at equilibrium and it can be shown [322] that the equilibrium conditions are the equality of each moment chemical potential in both phases

(Cl)

hµik (Cl)

kT λk

RES(Cl)

+ hµik

(Sh)

(4.85)

= hµik

(Sh)

= kT λk

RES(Sh)

+ hµik

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS180 (Cl)

Since λk

= 0, equation (4.85) becomes (Sh)

λk

RES(Cl)

= hµik

RES(Sh)

− hµik

/kT.

(4.86)

By substituting (4.86) into (4.84), the distribution of the shadow phase at equilibrium is obtained as

(Cl) e = ρm ρ(Sh) m

P ³ k

RES(Cl)

hµik

RES(Sh)

−hµik

mk /kT

(4.87)

In the case of our polymer solution, equation (4.87) is written only in terms of the zero and first density as

(Cl) ρ(Sh) m (m) = ρm (m)e

RES(Cl)

hµi0

RES(Sh)

−hµi0

/kT

RES(Cl)

hµi1

RES(Sh)

−hµi1

m/kT

, (4.88)

which is identical to expression (4.59). The two equilibrium conditions (4.60) and (4.61) of the exact solution are satisfied in the moment method via the relations given by equation (4.69). The moment method consists then in expressing the (Sh)

Lagrange multipliers λk

(Cl)

in terms of the density moments hρm ik

(Sh)

and hρm ik

via the relations (4.69), and solving the non-linear system of five equations (equality of the pressure of both phases to the fixed pressure Pf ixed , equality of the chemical potential of the solvent in both phase, and relations (4.85) for the zero and first moment chemical potential). The variables of this system are the same as in the exact case (see section 4.1.4). The moment method is mathematically equivalent to the exact solution of section 4.1.4, and it does not bring any improvement in terms of numerical solving. However, its main advantage is to bring a simplification in the analytical expressions, notably in the determination of stability criteria and critical points. The reader can find in reference [322] a detailed description of the matrices and determinants for stability analysis in polydisperse systems.

4.4

Ternary Systems

Following Koningsveld’s work [302,303], we consider a polymer-solvent system where the solvent (component 1) is represented by a hard sphere (m1 = 1) of diameter σ and the “polydisperse” polymer is a mixture of two chain molecules (components 2

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS181 and 3) composed of m2 and m3 spherical segments of same diameter σ. This is the simplest polydisperse system with a discrete distribution and we can study the effect of polydispersity on the liquid-liquid immiscibility, and explain the shapes of cloud and shadow curves by analysing the corresponding ternary diagrams at constant temperature T and pressure P . The polymer is assumed to have a polydispersity index Ip = 2, and a number average chain length hmin = 100. The chain lengths m2 and m3 , and the parent mole fraction distribution X (0) are related with Ip and hmin via equations (4.5) and (4.7). The chain lengths m2 and m3 must satisfy

X (0) (m2 )m2 + X (0) (m3 )m3 = hmin = 100

(4.89)

X (0) (m2 )m2 2 + X (0) (m3 )m3 2 = Ip hmin = 200.

We consider two different “polydisperse” polymers (1) and (2) composed of the two chain molecules 2 and 3, and satisfying the relations (4.89). The corresponding parent distribution functions X (mole fractions), W (weight fractions) and chain lengths m2 , m3 are given in table 1. The discrete weight fraction distributions of the two polymers are shown in figure 4.2. In the case of polymer (1) there is more of the longer chains, while for polymer (2) the shorter chains predominate. A more asymmetrical distribution has been chosen for polymer (2). Table 3.1: Chain lengths, and mole and weight fraction parent distribution functions for polymer (1) and (2), such that hmi(0) n = 100 and Ip = 2.

m2 m3 X (0) (m2 ) X (0) (m3 ) W (0) (m2 ) W (0) (m3 )

polymer (1)

polymer (2)

66.7 400 0.9 0.1 0.6 0.4

88.8 1000 0.988 0.012 0.878 0.122

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS182

Figure 4.2: Discrete parent chain length distribution functions in terms of weight fraction, for polymer (1) (black bars), and polymer (2) (white bars).

4.4.1

Cloud and shadow curves

We first focus on the polymer (1) + solvent system and analyse the effect of polydispersity on the liquid-liquid immiscibility with an LCST. In chapter 2, the reader can find a detailed explanation of the LCST behaviour occurring in polymer systems. We re-define a reduced temperature as T ∗ = kbT /α (not to be confused with that defined in the preceding sections) and a reduced pressure as P ∗ = P b2 /α where b = πσ 3 /6 is the volume of a spherical segment. We calculate the cloud and shadow curves (T x diagram) of four different polymer systems, at the same reduced pressure P ∗ = 0.001. The first system is a binary mixture of a spherical solvent and a monodisperse polymer of chain length m = 100. The second system is a ternary mixture of the same spherical solvent, and two chain molecules of length m2 = 66.7 and m3 = 400. The last two systems are the binary solvent + monodisperse chain of length m2 = 66.7, and the binary solvent + monodisperse chain of length m3 = 400. In the four systems, the segments of the chain molecules all have the same diameter σ which is equal to the diameter of the solvent, and all the sphere-sphere attractive interactions correspond to the same mean-field energy Cl. The curves obtained for the four systems are shown in figure 4.3. The binodal curves of the binary mixtures were determined by using the algorithm described in chapter 2, Anpendix A.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS183

0.090

0.085

T* 0.080

0.075 0.0

0.1

wpoly

0.2

0.3

Figure 4.3: Cloud and shadow curves calculated with the SAFT-HS equation for the ternary system solvent + polymer (1) (chain m2 = 66.7 + chain m3 = 400). The dashed dotted curves are binodal curves of the binary systems: a) solvent + chain m = 66.7 , b) solvent + chain m = 100, and c) solvent + chain m = 400. The thick continuous curve represents the cloud curve of the ternary system, and the dotted curve represents the shadow curve. The circles denote the critical points (LCST); wpoly = w2 + w3 = 1 â&#x2C6;&#x2019; w1 is the total polymer weight fraction where w1 , w2 and w3 are the weight fractions of the solvent, and the chain molecules 2 and 3, respectively. For the ternary mixture, we used the method for discrete system, described at the beginning of this chapter in section 4.1. We recall that in the case of the binary systems, the cloud and shadow curves are identical, and correspond to the binodal curves. For the polydisperse ternary system, the cloud curve is different from the shadow curves, and different from the binodal curves (see chapter 1, section 1.3.3 for further explanation). From an inspection the binodal curves obtained for the three binary systems (solvent + chain m = 100), (solvent + chain m = 66.7, solvent + chain m = 400), depicted in figure 4.3, one can see that increasing the chain length of the polymer molecule extends the region of liquid-liquid immiscibility region: the LCST is found at lower temperatures. As mentioned in chapter 2, the LCST behaviour is due to density changes leading to unfavourable entropic effects, which are enhanced as the chain length of the polymer is increased. The critical point (LCST) is always at the minimum of the binodal curves for the binary systems. Furthermore, as the length of the chain is increased, the binodal curves are more

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS184 skewed and the LCST is shifted to lower weight fractions of polymer. Concerning the ternary system, two different curves can be seen: the cloud curve which gives the first temperature where phase separation occurs in the system at a fixed global composition of the sample, and the shadow curve which gives the composition of the first droplet of the new phase (see chapter 1, section 1.3.3). The polydispersity induces an increase in the extent of liquid-liquid immiscibility when compared with the mono-disperse system (solvent + chain m = 100), and makes the cloud curve more skewed. This result is surprising as the mole fraction of short chain molecules X (0) (m2 ) is much bigger than the mole fraction of long chain molecules X (0) (m3 ) in the parent distribution (see table 3.1). One can conclude that the longer chain molecules dominate the liquid-liquid demixing in the system, and that only a small presence of long chain molecules in a system of short chains + solvent can enhance the liquid-liquid immiscibility significantly. Furthermore, the critical point is no longer observed at the minimum of the cloud curve, but is shifted to higher weight fractions of polymer from the minimum of the curve; the critical point is at the intersection point of the cloud and shadow curves. The point critical point of the ternary system is located however almost at the same weight fraction of polymer as in the monodisperse case. The cloud and shadow curves appear to tend to the binodals of the binary mixture in the high temperature regions of the phase diagram. If the global weight fraction of polymer molecules is below the critical point, the cloud curve corresponds to a polymer-poor phase, while the shadow curve is the polymer-rich phase, and vice et versa. For a global weight fraction wpoly of polymer fixed below the critical point and closed to the pure solvent region, the shadow curve appear to tend to the binodal of the binary mixture (solvent + chain m = 400). This can be explained by the fact that at low temperature, the demixing is dominated by the presence of the long chain molecules (component 3), and the polymer-rich phase (shadow phase) predominatly consists of the longest chains (component 3), so that the number average hmi m (Sh) in the shadow phase is very close to m3 = 400. The shortest chains do not tend to demix with the solvent at that temperature, and remain in the solvent rich phase (cloud phase). However, at higher temperature and for a weight fraction wpoly above the critical point, the cloud curve tends to the binodal of the binary system (solvent + chain

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS185 m = 100), while the corresponding shadow curve appears to follow the binodal of the binary (solvent + chain m = 66.7). At higher temperature, both chain molecules (Cl) 2 and 3 demix with the solvent. The number average chain length hmim of the

polymer-rich cloud phase is 100, so that the cloud curve follows the binodal of the system (solvent + chain m = 100). The long chain molecules 3 are almost absent from the solvent-rich phase at that temperature, and the shadow phase consists mainly the shorter chain molecules 2: the shadow curve then follows the binodal of the system (solvent + chain m = 66.7), and the number average hmim (Sh) is close to m2 = 66.7.

4.4.2

Ternary Diagrams

Figure 4.4: Cloud and shadow curves calculated with the SAFT-HS equation for the ternary system solvent + polymer (1) (chain m2 = 66.7 + chain m3 = 400). The think continuous curve represents the cloud curve of the ternary system, and the dotted curve represents the shadow curve. The circles denote the critical points (LCST); wpoly = w2 + w3 = 1 â&#x2C6;&#x2019; w1 is the total weight fraction of polymer, where w1 , w2 and w3 are the weight fractions of the solvent, and the chain molecules 2 and 3, respectively. The dashed and dotted lines represent constant temperature slices corresponding to the ternary diagrams depicted in figure 4.5. The white squares denote cloud points and the black squares are the corresponding shadow points.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS186

Figure 4.5: Ternary diagrams in weight fractions, at constant pressure P ∗ = 0.001, and temperatures a) T1∗ = 0.0839, b) T2∗ = 0.0794, and c) T3∗ = 0.0787) corresponding to the cloud and shadow curves of the ternary system solvent + polymer (1) (chain m2 = 66.7 + chain m3 = 400) shown in figure 4.4. The thick and continuous curves represent the coexistence curves in the ternary mixture, and the circles denote the critical point (LCST). The thin and continuous lines are tie lines. The dashed and dotted lines represent a fixed ratio of composition between chain 2, and chain 3, corresponding to the parent distribution W (m2 )(0) = 0.6, W (m3 )(0) = 0.4. The white squares denote cloud points and the black squares are the corresponding shadow points.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS187 To explain further details of the nature of the cloud and shadow curves, one can study the corresponding ternary diagrams, at different temperatures and at constant pressure P ∗ = 0.001. We consider there different constant temperature slices (T 1∗ = 0.0839, T2∗ = 0.0794, T3∗ = 0.0787), all at the pressure of P ∗ = 0.001, denoted by the dotted and dashed lines in the temperature composition T w diagram of the ternary mixture solvent + polymer (1) (see figure 4.4). The three ternary diagrams are depicted in figure 4.5. The coexistence curves of the ternary diagrams were determined not by using the algorithm for cloud and shadow curves, but by solving the equality of the chemical potential of all the species (solvent, chain 2, chain 3) at constant temperature and pressure. The temperature slices are chosen such that T1∗ is above the critical point, T2∗ is the temperature of the critical point, and T3∗ is the temperature at the minimum of the cloud and shadow curves ( see figure 4.4). It can be seen in figure 4.4 that there are two different cloud points and two corresponding shadow points at temperature T1∗ : one cloud point corresponds to a polymer-rich phase, and the other to a polymer-poor phase. The corresponding ternary diagram is shown is figure 4.5 a). The dashed-dotted line on this diagram corresponds to a fixed length distribution function of the polymer (1), which is equal to the parent distribution: on that line, the ratio between the weight fractions of the chain molecules 2 and 3 is fixed to W (0) (m2 )/W (0) (m3 ). This line crosses the coexistence curve of the ternary system twice, and the intersection points correspond to the two cloud points of figure 4.4 at the temperature T1∗ . The shadow points are given by the tie lines connecting the corresponding cloud points (see the ternary diagram 4.5 a)). At temperature T2∗ , there are still two cloud and shadow points as can be seen in figure 4.4, however one cloud point is identical to its corresponding shadow point. At that point, the cloud and shadow curves cross, the densities and compositions of both phases become equal. This point is therefore a critical point. In the ternary diagram shown in figure 4.5 b), the dashed-dotted line corresponding to a fixed parent distribution crosses the coexistence curve twice at two cloud points. However, one of the cloud points is also the critical point of the ternary system at that temperature. T3∗ is the lowest temperature where demixing can occur in the ternary system solvent + polymer (1) at pressure P ∗ = 0.001. At that temperature, there is only one cloud point and one corresponding shadow point. These two points are the minimum of the cloud and shadow curves.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS188

0.090

0.085

T* 0.080

0.075 0.00

0.10

wpoly

0.20

0.30

Figure 4.6: Cloud and shadow curves calculated with the SAFT-HS equation for the ternary system solvent + polymer (2) (chain m2 = 88.8 + chain m3 = 1000). The think continuous curve represents the cloud curve of the ternary system, and the dotted curve represents the shadow curve. The circles denote the critical points (LCST). The white square is a three phase point. In the ternary phase diagram at temperature T3â&#x2C6;&#x2014; (figure 4.5 c)), the dasheddotted is tangent to the coexistence curve since it crosses it only once. The tie line going through the tangent point provides the corresponding shadow point on the coexistence curve. The study of the ternary system solvent + polymer (1) enables us to discuss the effect of polydispersity: the shape of the cloud and shadow curves obtained for the ternary system are very similar to those obtained experimentally (see [313]) when the molecular weight distribution function is not too wide. The polydispersity of chain length has a big effect on the liquid-liquid immiscibility at fixed numberaverage chain length. As the index of polydispersity Ip increases, the liquid-liquid immiscibility region becomes more extensive. In a polydisperse system, the presence of longest chains appears to dictate the shape of the shadow curves at low temperatures below the critical point even if their weight fraction is low in the system, while at temperatures above the critical point, the shortest chains govern the shape of the shadow curves.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS189 As shown in figure 4.7, the shadow phase corresponding to a polymer-rich phase (at temperature below the critical point) mainly consists of the longest chains, while the shadow phase corresponding to a solvent rich phase mainly consists of the shortest chains (at temperature above the critical point).The cloud curve of the ternary system tends to the binodal of the corresponding monodisperse binary system for hight weight fractions of polymer, and the minimum of the cloud curve is shifted to low temperatures from the binodal curve. ˇ Some time ago now Solk [308–310] observed that a very asymmetric distribution can lead to equilibria between three liquid phases. To study the effect of the shape of ˇ the chain length distribution function and confirm Solk results which were obtained with a lattice model very different from SAFT, we consider a second ternary system solvent + polymer (2) (solvent + chain m2 = 88.8 + chain m3 = 1000), where polymer (2) has the same number-average chain length hmin = 100 and same polydispersity index Ip = 2 as polymer (1), but a much more asymmetric distribution function (see the distribution functions of polymers (1) and (2) in figure 4.2.) The temperature composition diagram of the ternary system solvent + polymer (2) is shown in figure 4.6. The cloud curve and especially the shadow curve have a different shape than the cloud and shadow curves obtained for the first ternary system, although polymer (1) and polymer (2) have same hmin and Ip . The cloud curve exhibits a break at a temperature higher than the critical point (see the white square on figure 4.6). At that point, the cloud curve is continuous, but its derivative with respect to the weight fraction of polymer wpoly is clearly discontinuous. The corresponding shadow curve is continuous, but exhibits an inflection point at the same temperature. This behaviour is explain by the occurrence of three-phase equilibria, ˇ as has been mentioned by Solk [308–310], de Loos et al. [313] and Phoenix and Heidemann [316], and the break in the cloud curve corresponds to a three phase point. ˇ Solk performed a full study with the Flory-Huggins theory of the relative position between the critical point and the three phase point, by using continuous distributions. We confirm here his results with a simple ternary mixture using the SAFT-HS equation of state. Three phase equilibria occurs in polymer systems with very asymmetrical distributions, because of the large difference of size between the shortest and the longest chain molecules. According to our relation m3 = 7.2346m1.0955 2 between the chain length of the shorter component and that of the longer component which first gives rise to type V behaviour (see chapter2, figure 2.3), the binary

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS190 mixture chain m2 = 88.8 + chain m3 = 1000 exhibits a type V behaviour: the two chain molecules demix close to the critical point of chain 2. The liquid-liquid immiscibility between both chains is enhanced by the presence of the solvent. In the temperature-composition diagram of the ternary system solvent + polymer (2) (figure 4.6), we can see that the cloud curve consists of two stable liquid-liquid branches intersecting at the three phase point. In figure 4.7, it can be seen that the shadow phase rich in polymer consists mainly of the longest chains, while the shadow phase rich in solvent molecules consists mainly of the shortest chains. In the case of a very asymmetrical distribution function, the evolution of the number-average chain length of the shadow phase is quite complicated.

Figure 4.7: Number average chain length in the shadow phase at fixed pressure P â&#x2C6;&#x2014; = 0.001, as a function of the polymer weight fraction in the shadow phase, for the ternary systems solvent + polymer (1), and solvent + polymer (2). The circles denote critical points, and the square denotes a three phase point.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS191

4.4.3

Cloud and Shadow Curves Obtained with Schulz-Flory Distributions

We now consider a polymer-solvent system where the solvent is again represented by a hard sphere of diameter σ, and the polymer by an infinite number of chain molecules of different lengths. In this work, we use a Schulz-Flory function [284,285] for the parent distribution function X (0) of chain length. The Schulz-Flory function depends on two parameters a and b: a is called the degree of coupling between polymer chain and b is related to the number average chain length hmin as b = a/hmin . The index of polydispersity Ip of the Schulz-Flory distribution is related to the parameter a as Ip = (a + 1)/a. The parent Schulz-Flory distribution X (0) in terms of mole fractions, which is equal to the length distribution X (Cl) in the cloud phase, is given by

X (0) (m) = X (Cl) (m) =

ba ma−1 e−bm , Γ(a)

(4.90)

where Γ is the gamma function defined as

Γ(a) =

∞

dx xa−1 e−x .

(4.91)

The corresponding parent distribution W (0) in terms of weight fractions is given by

W (0) (m) = W (Cl) (m) =

ba+1 ma e−bm . Γ(a + 1)

(4.92)

Following equation (4.45), the parent density distribution is equal to the density distribution of the cloud phase Cl,

(Cl)

(Cl) ρ(0) m (m) = ρm (m) = hρm i0

ba ma−1 e−bm . Γ(a)

(4.93)

The two distributions are defined in the interval [0, ∞[. They are both normalised since

∞ 0

dm ma−1 e−bm = b−a Γ(a).

(4.94)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS192 It would be more accurate to use a lower boundary of 1 instead of 0 in the integral, since the minimum number of segments is physically 1. However, using a lower boundary equal to 0 does not make any significant change to the cloud curve calculation, and enables one to simplify the analytical expressions of the integrals. The moment method described in section 4.3 is applied to calculate the cloud and shadow curves of the polydisperse polymer solution with the Schulz-Flory distribution function. The moment free energy of this system can be separated into an ideal part, and a residual part (see equation (4.72)); both parts can be expressed in solely terms of the density of solvent, the temperature, and the zero and first density moments of the density distribution. The expression of the residual moment free energy is given in section 4.1.2, equation (4.29). The ideal part of the moment free energy is given by equation (4.70). One must express the Lagrange multipliers (Sh)

λ0

(Sh)

and λ1

(Sh)

and hρm i1

(Cl)

as a function of the density moments hρm i0

(Cl)

, hρm i1

(Sh)

, hρm i0

(Cl)

by inverting equations (4.69). The Lagrange multipliers λ0

in the

cloud phase are equal to zero, as shown in section 4.3.3. Using equations (4.84) and (Sh)

(4.93), the density distribution ρm

of the shadow phase can be expressed in terms

of the density distribution of the cloud phase as

(Sh)

(Cl) λ0 ρ(Sh) = ρm e m

(Cl) hρm i0

(Sh)

+λ1

(Sh) λ +m ba ma−1 e 0 Γ(a)

(4.95) ³

(Sh)

λ1

−b

It is shown in equation (4.95) that the length distribution function of the shadow phase is also a Schulz-Flory distribution. The length distribution in the shadow phase does not have the same number average as the parent distribution because (Cl)

the factor of m in the exponential is different for both distributions ρm

(Sh)

and ρm

(see equations (4.93) and (4.95)). However, the polydispersity index Ip is conserved, as the power of m equal to a − 1 is the same for both distributions. Consequently, the SAFT theory combined with the moment method predicts that phase separation in polymer systems with a Schulz-Flory distribution gives rise to a different average chain length in both phases, but that the width of the distribution is not affected. This result is in agreement with those of Cotterman [282] who found that if the

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS193 distribution of the feed is a gamma function then the distribution of the shadow phase is also a gamma function. Applying equations (4.94), (4.95), and (4.69), the (Sh)

zero and first density moments hρm i0

(Sh)

and hρm i1

of the shadow phase can be

expressed as [322]

(Sh) hρm i0

(Sh) hρm i1

(Cl) (Sh) hρm i0 eλ0

(Cl) (Sh) hρm i1 eλ0

1 (Sh)

1 − bλ1 1

(Sh)

1 − bλ1

!a !a+1

(4.96)

By rearranging equation (4.96), the Lagrange multipliers of the shadow phase are given by

(Sh)

λ0

(Sh) λ1

(Sh)

= (a + 1) ln

1 b

hρm i0

(Cl)

hρm i0

(Sh)

1−

hρm i0

(Sh)

hρm i1

(Sh)

− a ln (Cl)

/hρm i0

(Cl)

/hρm i1

hρm i1

(Cl)

hρm i1 !

(4.97)

One can then substitute equation (4.97) into (4.79) and (4.85), and determine the cloud and shadow point at a fixed pressure Pf ixed by solving the non-linear system of the following five equations: equalities of the zero and first moment chemical potential in both phases (see equation (4.85)), equality of the pressure (see equation (4.80)) of both phases to the fixed pressure Pf ixed , and equality of the chemical potential of the solvent in both phases). The five variables of the system of equations (Sh)

are the temperature T , the densities ρ(Cl) , ρ(Sh) , the polymer mole fraction xpoly

and the average length hmi(Sh) in the shadow phase. A Newton-Raphson method n can be used, and the variables can be guessed by using the values obtained with the corresponding mono-disperse system and the algorithm described in chapter 2, Appendix A. The cloud and shadow curves have been determined in this way for various indexes of polydispersity (Ip = 2, and Ip = 15). The corresponding parent distribution

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS194 function are shown in figure 4.8. As can be seen in figure 4.9, the liquid-liquid immiscibility region expands as the polydispersity is increased, and the critical point is shifted to lower polymer weight fractions. The number-average chain length hmi (Sh) n of the shadow phase as a function of the composition of the shadow phase are shown in figure 4.10: as in the discrete case, the shadow phase rich in polymer molecules consists mainly of long chain molecules, while the shadow phase rich in solvent molecules consists mainly of short molecules. The evolution of the number average chain length hmi(Sh) in the shadow phase is however very different from the disn crete case: hmi(Sh) tends to infinity as the shadow phase becomes richer in polymer, n while hmi(Sh) tends to the longest possible chain length in the discrete case. We can n also compare the cloud and shadow curves obtained for a Schulz-Flory distributions with the cloud and shadow curves obtained with the ternary system solvent + polymer (1). The polymer in both systems has the same number average chain length hnin = 100, and the same index of polydispersity Ip = 2.

Figure 4.8: Parent Schulz-Flory distributions in terms of weight fractions with a number-average chain length hmi(0) n = 100 and polydispersity indexes Ip = 2, Ip = 15. The corresponding cloud and shadow curves are depicted in figure 4.9.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS195

Figure 4.9: Cloud (continuous) and shadow (dotted) curves for the system solvent + polydisperse polymer, obtained with the Schulz-Flory distributions depicted in figure 4.8 and various polydispersity indexes. The dashed-dotted curve represent the binodal obtained for a mono-disperse polymer-solvent system (Ip = 1). The circles denote critical points.

Figure 4.10: Number average chain length in the shadow phase, as a function of polymer weight fraction in the shadow phase, at fixed pressure P â&#x2C6;&#x2014; = 0.001, calculated for the system solvent + polydisperse polymer by using a Schulz-Flory feed distribution with various polydispersity indexes. The circles denote critical points. The number-average chain length of the feed distribution is hmin(Cl) = 100 for both cases, and the two curves correspond to those in figure 4.9 .

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS196

0.090

0.085

T* 0.080

0.075 0.0

0.1

wpoly

0.2

0.3

Figure 4.11: Cloud (continuous) and shadow (dotted) curves for the systems solvent + polydisperse polymer by using a Schulz-Flory feed distribution (thick lines), and for the ternary system solvent + polymer (1) (thin curves). The circles denote critical points. In both systems, the number-average chain length of the feed distribution is hmin(Cl) = 100, and the index of polydispersity is Ip = 2. For an inspection of figure 4.11 it can be seen that the cloud curve obtained with a continuous Schulz distribution is very similar to the cloud curve obtained for the discrete ternary system. However, the shadow curve in the continuous case has a smoother evolution. One can however conclude that a simple ternary system is sufficient to predict the cloud curve reasonably accurately. However, the shape of the shadow curves depend much on the shape of the feed distribution function.

4.5

Conclusion

In this chapter, we present three different methods to calculate phase equilibria in polydisperse polymer solutions with SAFT-type equations of states. The first method, developed by Jog and Chapman [318] enables to calculate cloud and shadow curves for discrete chain length distributions. The number of equations to be solved is always of five and does not depend on the number of pseudo-component used to represent the chain length distribution. The second method is an extension of Jog and Chapman method to treat phase equilibria with continuous distributions.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS197 The third method, called moment method and developed by Sollich and co workers [320â&#x20AC;&#x201C;322], is described and compared with the two first methods. The moment method can be applied only to continuous distributions. We first use the first method to calculate phase equilibria of a simple ternary system where the polymer is represented by two chain molecules. Two ternary systems corresponding to two different polymers (polymer (1) and polymer (2)) have been discussed. Both polymers have the same number average chain length and polydispersity index. By analysing the cloud and shadow curves obtained for these systems, we show that the shape of the distribution can have a big effect on the cloud and shadow curves, and that three phase equilibria may appear if the distribution is very broad. Moreover, we show that the liquid-liquid immiscibility is governed by the presence of the longest chain molecules: only a small presence of long chain molecules in the mixture give rise to an expansion of the liquid-liquid immiscibility region. We have used the moment method to calculate cloud and shadow curves with Schulz-Flory chain length distributions. There phase coexistence is never found for such distributions, confirming previous studies [283, 316]. Furthermore, we compare the results obtained with continuous and discrete distributions (ternary systems), and we show that the simple ternary system (solvent + polymer (1)) exhibits very similar cloud and shadow curves as these obtained with Schulz-Flory distributions. In a near future, we will provide results obtained for polydisperse polymer-colloid systems.

Chapter 5

Polyethylene + Hydrocarbon Systems It is important to know the phase behaviour of polymer systems in order to design and optimise chemical processes. For over fifty years, numerous theories (derived for lattice or tangent-sphere models) have been used to describe the phase equilibria of real polymer systems. A summary of all the various approaches is given in chapter 1. In this last chapter, we mainly deal with the phase equilibria of hydrocarbons and polyethylene (PE) mixtures as polyethylene is the simplest and most wide spread polymer. There are three common types of polyethylene: highdensity polyethylene (HDP E), made from the Ziegler-Natta homopolymerisation of ethylene; low-density polyethylene (LDPE), made by high-pressure free radical homopolymerisation of ethylene; and linear low-density polyethylene (LLDPE), made from the co-polymerisation of ethylene with one Îą-ofelin (1-butene, 1-hexene, or 1-octene). The thermodynamic experimental studies on polymer systems can be classified into three different classes: Firstly, experiments can be carried out on pure polymer samples to measure their melting points, crystallinities, or specific volumes (PVT data). Excellent introductions to polymer crystallisation can be found in the texts by Mandelkern and by Sharples [323, 324]. Experimental studies on polymer crystallisation are very numerous; the reader can find the most recent experimental studies on high- and low-density polyethylene (HDPE and LDPE) and poly(ethylene-olefins) (or linear low-density polyethylene, LLDPE) crystallisations in various references [325â&#x20AC;&#x201C;330]. Theoretical studies are however very rare and most of the approaches are empirical. It is indeed very difficult to predict with accuracy the fusion properties of polymers as they depend on many different factors (co-monomer composition, molecular

198

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

199

and isomeric structure, branching, polydispersity, lamellar thickness, thermal history of the sample) which are difficult to characterise. Furthermore, the accuracy of the experimental measurements of crystallinity is of order 5 to 10 %. The only available theory to describe the crystallinity of polymers at equilibrium (after an infinite crystallisation time) was derived almost 50 years ago by Flory [331] who proposed an analytical expression for the melting point and the crystallinity of an ideal co-polymer. Flory’s theory provides a good qualitative description of the melting point of polymers as a function of the co-monomers composition, and of the evolution of the crystallinity with temperature. Sanchez and Eby [332] improved Flory’s expression for the melting points of co-polymers by introducing some extra parameters to take into account defects (branching) and chain ends. Kim et al. [333] have recently tested the adequacy of the Flory [331] and the Sanchez and Eby [332] expressions to predict the melting points of LLDPE samples. They conclude that Sanchez and Eby [332] theory can give better results than Flory’s theory if the parameters characterising the defects are fitted on experimental data. However, their study is restricted to poly(ethylene-octene). Crist and Howard [334] tested Flory’s theory on both melting point and crystallinity data of poly(ethylene-butene) samples, and conclude that Flory’s equation overestimates the crystallinity. In this chapter, we propose a semi-empirical model based on Flory’s theory, to predict the melting points and crystallinities of any polyethylenes and poly(ethylene-olefins). The model only requires the crystallinity of the sample at 25◦ C. Another type of experiment carried out in polymer systems are measurements of the solubility of gases in the polymer. It is important to know the solubility of gases in polyethylene, to design low-pressure separation processes, and polymerisation reactors in the gas phase. During the reaction, polymer particles are suspended by a gas flow, and the monomers must sorb into the polymer sample and diffuse through the grain to reach the catalytic sites [335]. The rate of the reaction is directly proportional to the solubility of the monomer at the active site. Furthermore, it is important to know the solubility of gases in polymer in order to eliminate as much as possible of the remaining solvent from the polymer for safety and economic reasons [336]: fires have been experienced in numerous plants involved in polymer production due to the adsorbed flammable gas in the polymer [337, 338]; it is also good practice to recycle as much as of the monomer gas which has not reacted as possible.

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

200

The experimental studies of absorption of gases in polyethylene are scarce as they are expensive and time consuming. Some researchers have also recently studied these systems by Monte Carlo simulation [177, 178, 339–341], or integral equation theories [342]. It is however necessary to use an accurate, predictive, and fast approach to calculate the solubility of gases in polymer in order to design reactors and separation processes. The solubility is a function of temperature and composition of the gas phase. The term ”adsorption” is often used in an industrial context to denote the quantity of gases present in the polymer sample, as the adsorption experiments are usually carried out at temperatures under the melting point of the polymer (the polymer sample is often a powder). However, the term ”adsorption” is not relevant in this case, as an adsorption process occurs when the gas molecules sorb only at the surface of a solid. The sorption of gases in a polymer is a bulk process as the gas molecules can diffuse through the amorphous parts of the polymer sample, and the terms ”absorption” or ”solubility” are more appropriate. The crystalline parts of the polymer sample, also called ”crystallites”, behave as a barrier against the diffusion of the gas molecules in the sample. As a result, it is usually assumed that the gas can not penetrate the crystallites for steric reasons, and absorb only in the amorphous regions [327, 336, 339, 343]. In order to quantify the total absorption of gases in the polymer, it is hence necessary to know the crystallinity of the polymer sample as a function of temperature. If the temperature of the absorption experiment is above the melting point of the polymer sample, the crystallinity is zero and the gas absorbs in the whole sample. The amorphous regions present a liquid-like structure, and one can model the solubility of gases in polyethylene as a vapour-liquid equilibria between a gas phase where no polymer molecules are present, and a liquid phase consisting of gases molecules absorbing into an amorphous polymer matrix. Two main difficulties can be encountered in the modelling of phase equilibria in gas-polymer systems: firstly, the absorption of gases in a semi-crystalline polymer is basically a three-phase equilibria between a vapour phase, a liquid phase of amorphous polymer, and a solid phase of crystallites. The assumption that gas molecules absorb only in the amorphous regions may be too crude: if the gas molecules are small enough, they can penetrate into the crystalline lattice and the solubility of gases in the crystallites will not be negligible. Furthermore, the presence of gas in the polymer sample decreases the melting point of the pure polymer due to cryoscopic effects [344–346] and in turn the crystallinity decreases. The assumptions that the

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201

gas molecules only absorb in the amorphous regions, or that the crystallinity does not change in presence of gas, can both lead to an underestimate of the total absorption. The adequacy of these two assumptions will be addressed in this chapter by modelling polyethylene + gas systems with the SAFT-VR approach. The second problem in polymer-gas absorption is when two or more gases absorb simultaneously in the same polymer sample. It has been observed by some experimentalists (e.g., see [336, 347]) that the solubility of a gas in a polymer sample at a given partial pressure, may or may not be enhanced by the presence of a second gas in the vapour phase, depending on the nature of this second gas. For instance, the solubility of methane in polyethylene, at a fixed partial pressure of methane, is enhanced in the presence of ethylene in the gas phase, while the solubility of methane does not change in the presence of nitrogen [347]. The co-absorption problem is crucial in the case of the copolymerisation of ethylene + Îą-olefins to make LLDPE: the solubilities of the co-monomers in the polyethylene grains determine the rate of reaction for each monomer, and then the final composition of the co-monomers in the polymer chains. This type of synergy has been confirmed by Monte Carlo simulation [177,339]; however, the effect remains unclear and is sometimes explained in terms of diffusionnal and swelling mechanisms [339]. An explanation of the cosorption synergy is proposed in this chapter by discussing the interactions between molecules. Another more specific problem in the general area of absorption of gases in polymer is the modelling of vapour-liquid equilibria in hydrogen (H2 ) + hydrocarbon (n-alkanes) and hydrogen + polyethylene systems. Industrial processes such as petroleum refining, coal conversion, enhanced oil recovery and supercritical separation have a great demand for phase equilibria data of very asymmetric compounds such as hydrogen and n-alkanes. The VLE of such systems can not be predicted by using simple mixing rules combined with the usual engineering cubic equations of state (PR [289] or SRK [268]). The reason for this is that hydrogen must be treated at the quantum mechanical level in order to take into account the various vibration modes at low temperatures [179]. One approach to model hydrogen systems is to use quantum Monte-Carlo technics [348] and specific intermolecular potentials. However, Monte-Carlo simulations are computational intensive and an equation of state is more useful in the design and optimisation of chemical processes. To calculate phase equilibria of hydrogen systems with an equations of state, some authors have

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used the pure parameter of hydrogen obtained from the critical properties, and very specific mixing rules [349]. An other way is to use simple one-fluid mixing rules, but a temperature dependent attractive parameter for hydrogen [350]. Here, we are interested in modelling the solubility of hydrogen in polyethylene only at temperature around 400 K, well above the critical point of hydrogen. To model such a system one has to determine the most reliable parameters for hydrogen at these temperatures, and use simple mixing rules to allow an extrapolation from short n-alkanes to very long n-alkanes and polyethylene. One can determine the optimal SAFT-VR parameters of hydrogen at high temperatures directly by fitting to vapour-liquid experimental data of hydrogen+ n-alkane binary mixtures. As is shown in this chapter, a very good accuracy can be obtain with this method in the predictions of both bubble and dew points, without using any cross-interaction parameters k ij . A last type of phase equilibria experiment which is often carried out on polyethylenehydrocarbon systems is the determination of cloud curves and liquid-liquid immiscibility regions. The experimental studies are very numerous in this case, and the reader can find a rich source of experimental data in the review of Kirby and McHugh [29]. One should also mention papers of Hamada et al [351], de Loos and co-workers [61, 312, 313, 352, 353], and Kiran and co-workers [354â&#x20AC;&#x201C;361] in this context. It has been seen shown [362] that the region of liquid-liquid immiscibility becomes more extensive with decreasing pressure, decreasing number of carbon atoms of the n-alkane solvent, and increasing molecular weight of the polyethylene. Many authors have already modelled cloud points in polyethylene + n-alkane systems: some authors use binary cross-interaction parameters [207, 316, 363, 364], while others use the simple Lorentz-Berthelot combining rules [70,144,319,359,365]. No new modelling of cloud curves is presented in this thesis, and instead we test the adequacy of the SAFT-VR theory to predict liquid-liquid immiscibility in polyethylene + n-alkane systems, without the use of binary cross-interaction parameters. Cloud-point modelling is indeed a very good test of the theory as the results are very sensitive to the molecular parameters which are used to represent the polymer. Some predictions of of cloud points in polyethylene + n-alkanes systems with the SAFT-VR equation of state are presented in this chapter, and comparisons will be made with the predictions of the BYG theory [27, 70]. The polyethylene molecules are modelled as a very long n-alkanes and the SAFT parameters of the polymer are

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obtained by using simple linear relationships with molecular weight, derived from the n-alkanes parameters. In a first section, the main expressions of the SAFT-VR theory are presented. The predictions for the vapour-liquid phase equilibria in pure compounds, hydrocarbon binary mixtures, and polyethylene + hydrocarbons are then shown. The cloud curves in polyethylene + n-alkane systems obtained from the SAFT-VR approach are then presented. We then propose a model based on Flory’s theory to predict the crystallinity of polyethylene, and show the prediction of solubilities of gases in semi-crystalline polyethylene. We finally discuss the synergy in the co-absorption of different monomers in polyethylene.

5.1

SAFT-VR Theory

In this section, the focus is on the statistical associating fluid theory for potentials of variable Range (SAFT-VR) theory [99], and a description is given for each of the terms in the equation. As was mentioned in the earlier chapters, the statistical associating fluid theory (SAFT) was developed by Chapman and co workers [366], [171]. The theory is based on a molecular model derived from statistical mechanics, and has been used to predict successfully the properties of a large variety of compounds and mixtures. The main advantage of the SAFT theory is its ability to take into account the non-sphericity of molecules, and to model short-range directional interactions in associating fluids such as water. The mean-field version of SAFT (SAFT-HS) is described in details in chapter 2. The most sophisticated version of SAFT, SAFT-VR, takes into account the structure of the fluid by incorporating the radial distribution function in the attractive term, and an additional non-conformal parameter is used to describe attractive interactions of variable range.

5.1.1

Molecular Model

In the SAFT-VR theory, the molecules are modelled as a flexible chains formed from m tangent spherical segments. Each segment of the chain i has the same diameter σi , but segments belonging to different species can have different diameters, contrary to the prototype model discussed in chapter 2. The dispersive interactions between the segments are modelled with an intermolecular potential (square well, Sutherland, Yukawa or Mie potentials; see figure 5.1) of variable range λ and depth

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204

². In this work, the square-well potential has been used to describe the intermolecular interactions of real molecules. Although the shape of this potential is rather unrealistic, it enables one to simplify dramatically the expressions of the free energy, and obtain a very description of the phase equilibria.

Figure 5.1: Potential models which can be used in the SAFT-VR theory to model segment-segment interactions. ² and λ are the depth and the range of the potential, respectively. Anisotropic attractions such as hydrogen bonds can also be modelled in the SAFT-VR approach by incorporating a number of short-range associating sites on the molecule. A typical molecular model used in the SAFT approach is depicted figure 5.2. In this case the molecule is made of 6 segments of diameter σ and 4 different association sites. In figure 5.3, models used to represent some common molecules are also shown. Methane is simply represented by one spherical segment since the methane molecule is almost spherical, while butane is represented by two tangent spherical segments.

Figure 5.2: General molecular model used in the SAFT theory. In this case, the molecules comprised five spherical segments of diameter σ with four association sites.

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Note that the number of spherical segments is not equal to the number of carbon atoms in the n-alkane series, but represents the effective non-sphericity of the molecule (see section 5.2.1). Water is also considered to be a spherical molecule. The hydrogen bonds are modelled by four sites: two sites represent the two hydrogen atoms, and two represent the two electron pairs of the oxygen atom.

Figure 5.3: Examples of models used for common molecules in the SAFT theory. The n-alkanes are represented by a flexible chain of segments without association sites. Water is represented by a spherical core with 4 sites to represent the hydrogen bonds.

5.1.2

Calculation of the Helmholtz Free Energy

In this section, we summarise the main expressions of the SAFT-VR equation for mixtures with the square-well potential. The reader can find further details in references [99,100]. Here, we assume that the segments belonging to different types of chain molecules can have various diameters and can interact with different potentials. However, the segments belonging to the same molecule have all equivalent. For mixtures of chain molecules composed of hard spherical segments with attractive interactions and associating sites, the expression of the Helmholtz free energy can be written in four separate contributions as

AIDEAL AM ON O. ACHAIN AASSOC A = + + + . N kT N kT N kT N kT N kT

(5.1)

Note that the associating term AASSOC , which is derived from Wertheimâ&#x20AC;&#x2122;s theory, is not required in the description of alkane mixtures and polyethylene-alkane systems

206

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as the n-alkane molecules only interact through van der Waals dispersive forces. The ideal contribution to the free energy is given as a sum over all species i in the mixture:

AIDEAL = N kT

n X i=1

xi ln ρi νi − 1,

(5.2)

where xi = Ni /N is the mole fraction of component i, ρi = Ni /V is the molecular number density, Ni is the number of molecules of type i, Λi is the thermal de Broglie volume (incorporating the translational and rotational kinetic contributions), and V is the total volume of the system. AM ON O. is the contribution due to the interactions between the spherical monomeric segments, which is given by

AM ON O. N kT

= (

xi mi )

xi mi )aM ,

i=1

= (

AM Ns kT

(5.3)

i=1

where mi is the number of spherical segments in a chain of species i, and Ns is the total number of segments in the system. The monomer free energy per segment of the mixture, aM , is obtained from the high-temperature perturbation theory of Barker and Henderson [46, 172, 173] with a hard-sphere reference system:

aM = aHS +

a2 a1 + + ..., kT (kT )2

(5.4)

The expression of the free energy contribution for a multicomponent mixture of hard spheres, aHS , is described by Boubl´ık [187] and Mansoori et al. [188] as a generalisation of the Carnahan and Starling repulsive term [74]:

6 = πρs

"Ã

ζ23 ζ23 3ζ1 ζ2 + − ζ . ln(1 − ζ ) + 0 3 (1 − ζ3 ) ζ3 (1 − ζ3 )2 ζ32

(5.5)

Here, ρs = Ns /V is the total number density of spherical segments, Ns the total number of segments, V the volume of the system, and ζl are the reduced densities defined by

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

n πρs X xs,i σiil , 6 i=1

ζl =

207

(5.6)

where σii is the diameter of the spherical segments of chain i, and xs,i is the mole fraction of segments of type i in the mixture. Notice that ζ3 = η. The first term a1 is the mean-attractive energy and is obtained from the sum of the partial terms corresponding to each type of pair attractive interaction:

a1 =

n n X X

xs,i xs,j aij 1,

(5.7)

i=1 j=1

where

aij 1 = −2πρs ²ij

λij σij σij

2 HS rij gij [rij ; ζ3 ] drij ,

(5.8)

HS is the radial pair distribution function for a mixture of hard spheres. where gij

Using the mean-value theorem, the expression for a1 is obtained in terms of the contact value of g HS at an effective packing fraction ζ3ef f : h

V DW HS gij σij ; ζ3ef f , a1 = −ρs Σi Σj xs,i xs,j αij

(5.9)

where the van der Waals attractive constant is obtained as 3 V DW = 2πεij σij (λ3ij − 1)/3. αij

(5.10)

The parameters εij and λij are respectively the depth and the range of the square well potential for i−j segment interactions. σij defines the contact distance between spheres i and j. The expression of this potential (square-well) is given by

Φij (r) =

   +∞

−εij   0

if r < σij , if σij ≤ r < λij σij , if r ≥ λij σij .

(5.11)

The van der Waals (one-fluid) mixing rule is used in this work. The radial distribution function of the mixture g HS is obtained as the radial distribution function

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208

of a hypothetical pure fluid of packing fraction ζxef f . The expression for a1 is then written as h

V DW HS a1 = −ρs Σi Σj xs,i xs,j αij g0 σx ; ζxef f ,

(5.12)

where g0HS is the radial distribution function at contact for a hard-sphere system, and its expression can be obtained from the Carnahan and Starling repulsive term as

1 − ζxef f /2 g0HS = ³ ´3 . 1 − ζxef f

(5.13)

The effective packing fraction ζxef f in the one-fluid description is given by: ζxef f (ζx , λij ) = c1 (λij )ζx + c2 (λij )ζx2 + c3 (λij )ζx3 ,

(5.14)

where

ζx =

πρs 3 σ , 6 x

(5.15)

with σx3 =

XX i

3 xs,i xs,j σij .

(5.16)

The parameters c1 , c2 , c3 are given in terms of the following matrices: c1 2.25855  c2  =  −0.669270 c3 10.1576 





−1.50349 0.249434 1 1.40049 −0.827739   λij  . λ2ij −15.0427 5.30827 



(5.17)

The coefficients of this matrix have been obtained by integrating equation (5.8) for a pure fluid with the radial distribution function g HS (r) of the hard sphere system given by the expression of Malijevski and Labik [367]. The second perturbation term a2 is the fluctuation term in the Baker and Henderson perturbation theory. It is evaluated using the local compressibility approximation for mixtures :

a2 =

n n X X

i=1 j=1

xs,i xs,j aij 2,

(5.18)

209

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS where aij 2 corresponds to each of the partial fluctuation terms defined as 1 HS ∂aij 1 aij = K ² ρ . ij s 2 2 ∂ρs

(5.19)

K HS is the hard-sphere isothermal compressibility factor of Percus-Yevick [368] and is given by

K HS =

ζ0 (1 − ζ3

ζ0 (1 − ζ3 )4 . + 6ζ1 ζ2 (1 − ζ3 ) + 9ζ23

(5.20)

Finally, the contribution to the free energy due to chain formation is expressed in terms of the contact value of the background correlation function (or cavity function) of the unbounded monomer fluid y SW as n X ACHAIN xi (mi − 1) ln yiiSW (σii ), =− N kT i=1

(5.21)

where yiiSW (σii ) = giiSW (σii ) exp(−β²ii ) and β = 1/kT . yiiSW (σii ) is obtained from the high-temperature expansion of giiSW (σii ) as giiSW (σii ) = giiHS (σii ) + β²ii g1SW (σii ),

(5.22)

where the term g1SW (σii ) is obtained from a self-consistency between with the pressure equation (Clausius theorem [189]) and the density derivative of the Helmholtz free energy:

giiSW [σii ; ζ3 ] = giiHS [σii ; ζ3 ] + βεii

giiHS [σii ; ζ3ef f ]

(λ3ii

− 1)

∂giiHS [σii ; ζ3ef f ] ∂ζ3ef f

∂ζ ef f λii ∂ζ3ef f − ζ3 3 3 ∂λii ∂ζ3

HS [σ ; ζ ef f ] is The contact value of the hard sphere pair distribution function gij ij 3

obtained from the expression of Boub´ık [187] as

HS gij [σij ; ζ3ef f ] =

1 1 − ζ3ef f

Dij ζ3ef f (1 − ζ3ef f )2

Dij ζ3ef f

´2

(1 − ζ3ef f )3

(5.23)

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210

where n σii σjj xs,i σii2 Pi=1 Dij = , σii + σjj ni=1 xs,i σii3

(5.24)

and where another effective packing fraction ζ3ef f different from ζxef f is defined as ζ3ef f (ζ3 , λij ) = c1 (λij )ζ3 + c2 (λij )ζ32 + c3 (λij )ζ33 ,

(5.25)

The coefficients c1 , c2 , and c3 are the same as those used in equation 5.17.

5.1.3

Combining Rules

In order to evaluate the unlike interaction parameters (also called cross parameters) in the mixture, the standard Lorentz-Berthelot combining rules can be used:

σij =

σii σjj , 2

(5.26)

²ij =

√ ²ii ²jj .

(5.27)

A simple arithmetic combining rule is used for λij :

λij =

λii σii λjj σjj . σii + σjj

(5.28)

The first combining rule (equation (5.26)) is exact since all the segments are hard spherical cores. The second combining rule (equation (5.27)) is an approximation which is reliable when the two compounds interact with only dispersive forces and are chemically similar (such as n-alkanes). The third combining rule (equation (5.28)) is seen to be reliable for n-alkane mixtures. If the two last combining rules do no give (²)

(λ)

satisfactory results, binary adjustable parameters kij and kij can be used, and the relations 5.27 and 5.28 become √ (²) ²ii ²jj (1 − kij )

(5.29)

λii σii λjj σjj (λ) (1 − kij ) σii + σjj

(5.30)

²ij =

λij =

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS (²)

211

(λ)

The parameters kij and kij can be fitted to experimental data of a binary mixture and then transferred to other binary mixtures of similar compounds.

5.2

Modelling of Pure Components

We recall that in the SAFT-VR approach each pure component is characterised by a set of molecular parameters: m is the number of spherical segments which form the chain molecule; each segment belonging to the same chain has the same diameter σ, and interacts with a square well potential of depth ² and range λ. The pure components which only interact with van de Waals dispersive forces like the n-alkanes are usually characterised by these four parameters: m, σ, ², and λ. Some components like water (H2 O) or hydrogen fluoride (HF) exhibit anisotropic associations which can be modelled by a certain number of associating sites, according to the theory of Wertheim (see chapter 2). Each type of bonds is characterised by a bonding volume KHB (or cut-off distance rc , see chapter 2, figure 2.1) and an association energy ²HB . Water is thus characterised by 6 parameters in the SAFT-VR approach, m, σ, ², λ, ²HB , rc .

5.2.1

n-Alkanes

We have modelled the n-alkanes as a flexible chain of m spherical segments of diameter σ [99]. Methane can be considered as the first member of the n-alkane series. The molecule of methane has spherical shape and is represented by only one spherical segment (m = 1). For the longer n-alkanes, we use a simple semi-empirical relationship (equation 5.31 ) between the number of carbon atoms C and the number of segments in the chain m, which was found in previous work [99, 369]:

m = 1 + 1/3 (C − 1)

(5.31)

As a result, butane (C = 4) is modelled by only 2 tangent hard spheres (see chapter 3, figure 5.3). In the SAFT-VR theory, the parameter m does not have to take integer values: for instance, the number of segments representing the molecule of ethane is methane = 1/3 ' 1.333.

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212

Determination of the Molecular Parameters Following previous work [99, 369], we have fixed m according to the relation 5.31, and the other molecular parameters σ, ² and λ are determined to provide an optimum description of experimental vapour pressure and saturated liquid densities. Experimental data are available from the triple point up to the critical point for the first members of the n-alkane series. However, such experimental data become increasingly difficult to find for alkanes heavier than decane because hydrocarbons are thermally unstable at a temperature above 650K. Although some data are available in the literature up to hexatriacontane (n-C36 ) [370, 371], they are not very reliable, as the purity of the n-alkanes used was not very high. New techniques are have been used to determine accurately vapour pressures and critical points of the long alkanes [372, 373]. Monte Carlo simulation data on n-alkanes are reliable and can be used as pseudo-experimental data [374–378]. We have determined the molecular parameters of the n-alkanes up to C28 , and the fitting was carried out with the well-known simplex method of Nelder and Mead [379], coupled with an ”annealing” method [209] which introduces some randomness in the parameters searching and enables one to find the global minima of the objective function. The results are presented in various projections slices of the P V T surface: temperature vs saturated molar volume (T V ), temperature vs saturated density (T ρ), vapour pressure vs temperature (P T ), and Clausius Clapeyron diagram ln P vs 1/T . The vapourpressure curves shown in the diagram (ln P vs 1/T ) are almost linear, according to the Clapeyron relation. It is clear from the figures 5.4 a) and b) that the SAFT-VR approach provides a net improvement over the Carnahan and Starling equation of state + mean-field attractions (equivalent to SAFT-HS with m = 1, see chapter 2 for the description of theory), for both saturated liquid densities (figure 5.4 a)) and vapour pressures (figure 5.4 b)). In general, very good agreement with experimental vapour pressure and saturated densities is obtained with the SAFT-VR approach for all of the n-alkanes up to C28 (figure 5.5 to figure 5.7). The critical regions are in general less well represented. The overvaluation of the critical temperature and pressure is a common drawback of most equations of state , and a specific treatment based on normalisation group theory [64, 380, 381] is required to obtain a better agreement in the critical region.

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213

Figure 5.4: Saturated densities a) and vapour pressures b) for methane: comparison between experimental data (circles) from reference [382] with the SAFT-HS (thin lines) and SAFT-VR (thick lines) theories after fitting of parameters.

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214

Figure 5.5: Vapour pressures of the n-alkanes from C1 to C8 , compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) vapour pressure curve. b) Clausius-Clapeyron representation.

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215

Figure 5.6: Vapour-liquid coexistence curves of the n-alkanes from C1 to C8 compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) coexistence volumes.

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216

Figure 5.7: Vapour-liquid coexistence curves of the n-alkanes from C12 to C28 , compared with the SAFT-VR predictions. The circles represent the experimental data [373, 383], the continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) vapour pressures in Clausius-Clapeyron representation.

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217

The parameters of the n-alkanes obtained after optimisation are shown in table 5.1. Table 5.1. SAFT-VR square-well parameters of the n-alkanes obtained after optimisation.

Compound

MW/(g.mol−1 )

CH4 C2 H6 C3 H8 C4 H10 C5 H12 C6 H14 C7 H16 C8 H18 C9 H20 C10 H22 C11 H24 C12 H26 C13 H28 C14 H30 C15 H32 C16 H34 C17 H36 C18 H38 C19 H40 C20 H42 C24 H50 C28 H58

16.04 30.07 44.10 58.12 72.15 86.18 100.20 114.23 128.26 142.28 156.31 170.33 184.36 198.39 212.41 226.44 240.47 254.49 268.52 282.55 338.65 394.76

1 1.3333 1.6667 2 2.3333 2.6667 3 3.3333 3.6667 4 4.3333 4.6667 5 5.3333 5.6667 6 6.3333 6.6667 7 7.3333 8.6667 10

σ (˚ A)

²/k (K)

1.4479 1.4233 1.4537 1.4922 1.5060 1.5492 1.5574 1.5751 1.5745 1.5925 1.5854 1.6101 1.6479 1.6023 1.5978 1.6325 1.6091 1.6565 1.6625 1.6637 1.6819 1.6542

3.6847 3.8115 3.8899 3.9332 3.9430 3.9396 3.9567 3.9455 3.9635 3.9675 3.9775 3.9663 3.9583 3.9745 3.9964 3.9810 3.9954 3.9562 3.9713 3.9726 3.9809 3.9896

167.30 249.19 260.91 259.56 264.37 251.66 253.28 249.52 251.53 247.08 252.65 243.03 227.31 249.74 252.87 237.33 249.76 228.81 226.31 227.07 220.00 233.50

Long n-Alkanes Since experimental vapour pressure and saturated densities are not available in the literature for the long n-alkanes and polyethylene, it is necessary to find a method to evaluate the SAFT-VR parameters of those compounds. The polyethylene molecule can be considered as a very long n-alkane and it can be also modelled as a flexible chain of m spherical segments of the same diameter σ. The chain length m is calculated with the same linear relation as that used for the short n-alkanes (equation 5.31. Following the previous work of McCabe et al. [191], we have calculated the SAFT-VR parameters for the long n-alkanes and polyethylene by using simple relations as a function of the molecular weight, obtained for the short alkanes. We

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218

can be confident that such an extrapolation will be valid because the parameters are based on a clear molecular description. Very good agreement is obtained between the correlations and the SAFT-VR parameters of the short alkanes obtained after fitting (see figures 5.8). However, we observe that the well depth ² exhibits more fluctuations than the other parameters and the correlation for this parameter is more difficult to determine (figure 5.8 (d)). The analytical expressions of the correlations are:

m = 0.02376M W + 0.6188

(5.32)

mλ = 0.04024M W + 0.6570 mσ 3 = 1.53212M W + 30.753 mε/k = 5.46587M W + 194.263,

where M W is the molecular weight of the alkane. The parameters apart from the chain length m tend to a finite limit when M W tends to ∞, They do not vary much with the molecular weight for very long polymeric chains. The limits of the parameters can be easily determined. For instance, the limit value of λ for very high molecular weight is given by

lim

M W →∞

= =

lim

M W →∞

(mλ) m

(5.33)

0.04024 ' 1.694 0.02376

The limits of the other parameters are obtained as

lim

M W →∞

lim

²/k ' 230.04 K

M W →∞

(5.34)

σ ' 4.010 ˚ A.

Note that the first correlation (5.32) becomes m ' ξMi , where ξ = 0.02376, when the molecular weight of the long n-alkane is very high. The number of segments is

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219

then proportional to the molecular weight, as already mentioned in chapter 3, section 4.1. We have tested the SAFT-VR correlations by comparing with the Monte-Carlo simulation data obtained for the pure long-alkanes C16 , C24 , and C48 [384].

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220

Figure 5.8: SAFT-VR parameters: a) λ ; b) σ; c) and ² of the n-alkanes as a function of the molecular weight, obtained after fitting. Simple linear correlations have been used to extrapolate the parameters to longer alkanes and polyethylene. The circles denote the values of the parameters fitted on vapour pressures and saturated densities, and the continuous lines represent the correlations (5.32).

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221

Figure 5.9: Vapour-liquid coexistence curves obtained for the n-alkanes C 16 , C24 , and C48 . The circles represent experimental data [373, 383]. The crosses represent Monte-Carlo simulation data [384]. The continuous curves correspond to the SAFTVR predictions, and the parameters of the long n-alkanes are calculated with the correlations (5.32). . The coexistence densities are well represented, but the critical point is overestimated as for short n-alkanes. One should note that there might also be some errors in the simulation data around the critical point. The coexistence curves obtained with the correlations for the n-alkanes C16 and C24 (see figure 5.9) are very similar to those obtained with the fitted parameters (see figure 5.7).

5.2.2

α-Olefins

Ethylene and α-olefins (propene, 1-butene, 1-hexene) are the monomers used in the polymerisation of polyethylene (only ethylene for HDPE and LDPE, and ethylene + α-olefins for LLDPE). It is then necessary to calculate the phase equilibria of mixtures of polyethylene + olefins for process design. The SAFT-VR parameters of ethylene and α-olefins have been determined by fitting experimental vapour pressures and coexistence densities of pure compounds. The results are shown in figure 5.10 to 5.10. We use the same relation (5.31) between the number of segments m and the number of carbon atoms of the olefins. Good agreement is obtained with both experimental densities and vapour pressures. The parameters for the olefins are given in table 5.2.

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Figure 5.10: Vapour pressures of ethylene and Îą-olefins (propene, 1-butene, 1hexene), compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) vapour pressure curve. b) Clausius Clapeyron representation.

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Figure 5.11: Vapour-liquid coexistence curves of of ethylene and Îą-olefins (propene, 1-butene, 1-hexene) compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) coexistence volumes.

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Table 5.2. SAFT-VR square-well parameters of the α-olefins obtained after optimisation.

5.2.3

Compound

MW/(g.mol−1 )

ethylene propene 1-butene 1-hexene

28.05 42.08 56.11 84.16

1.333 1.667 2.000 2.667

σ (˚ A)

²/k (K)

1.4432 1.4465 1.5564 1.6244

3.6627 3.7839 3.7706 3.82590

222.17 259.80 228.49 217.78

Other Pure Compounds

The SAFT-VR parameters of other pure components have also been optimised as a test of the SAFT-VR theory and the optimisation algorithm. We have modelled nitrogen (N2 ), carbon dioxide (CO2 ), hydrogen fluoride (HF) and water (H2 O), which exhibit very different intermolecular interactions and thermodynamic properties. Nitrogen is modelled as a “dumbbell” (m = 1.3) and only dispersive forces are taken into account via square well potentials. CO2 is treated as an elongated molecule (m = 2) without any association sites. Water is represented as a spherical molecule with four association sites (2 0, 2 H) to model the hydrogen bonds (see figure 5.3): 2 sites model the hydrogen atoms and 2 sites model the electron loan pairs on the oxygen atom. Hydrogen fluoride is also modelled as a spherical molecule with three association sites (2 F, 1 H) which allow for hydrogen fluorine hydrogen bonds. The coexistence curves of the pure components are shown in figures 5.12 and 5.13. Good agreement with the experimental data is also obtained on both coexistence densities and vapour pressure, except in the critical region. One can note in figure 5.13 that the gaseous molar volumes of HF are overestimated by SAFT-VR. This is due to the fact that HF can form rings in the gas phase due to hydrogen bonding; a Wertheim theory taking into account ring formation could be used to improve the prediction of the gaseous volumes and the vapour pressures, as suggested by Galindo et al. [385]. The square-well SAFT-VR parameters obtained after fitting are presented in the table 5.3

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Figure 5.12: Vapour pressures for CO2 , H2 O, and HF compared with the SAFTVR predictions. The circles represent the experimental data [386], [387], [388]. The continuous curves correspond to the SAFT-VR approach. a) vapour pressure curves. b) Clausius Clapeyron representations.

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Figure 5.13: Vapour-liquid coexistence curves for CO2 , H2 O, and HF compared with the SAFT-VR predictions. The circles represent the experimental data [386], [387], [388]. The continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) coexistence volumes.

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Table 5.3 SAFT-VR square-well parameters obtained after fitting for water, hydrogen fluoride and carbon dioxide. The parameters ²HB and KHB are site-site association energy and bonding volume respectively.

Compound

MW/(g.mol−1 )

σ (˚ A)

²/k (K)

²HB /k(K)

KHB (˚ A3 )

N2 CO2 H2 O (4 sites) HF (3 sites)

28.01 44.01 18.02 20.01

1.3 2.0 1.0 1.0

1.534 1.516 1.800 1.483

3.194 2.786 3.036 2.856

84.53 179.27 253.30 168.82

0 0 1365.92 1782.11

0 0 1.020 9.443

5.3

Modelling n-Alkane + linear Polyethylene Systems

A lot of experimental data are available in the literature for polyethylene + short alkanes solutions, though most of the experiments are quite old, as polyethylene was one of the first synthetic polymers. One problem with polyethylene is that it is not well-defined in terms of branching and polydispersity, and one has to be careful with the reliability of the experimental data. As was shown in chapter 3 the polydispersity of the polymer can have a great influence on the fluid phase behaviour, especially the liquid-liquid equilibria. Polydispersity and branching can differ from one polymer to another. Most of the predictions of cloud curves in the literature have been carried out with the use of cross interaction parameters kij fitted to the experimental data, and generally depending on temperature. Good agreement is usually obtained with this kind method [207, 316, 363, 364]. However, the dependence on temperature of the cross parameters is physically suspect as a true intermolecular potential should not depend on temperature, and transferable parameters which are independent of temperature are usually required for truly predictive applications. In our approach, we take the same parameters for low-density (LDPE) and high-density (HDPE) polyethylenes from the correlations (5.32).

5.3.1

Pentane + Polyethylene

The ability of the SAFT-VR approach combined with the relations (5.32) to predict the fluid phase equilibria of polyethylene solutions is first examined by using the

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combining rules of Lorentz and Berthelot (no cross interaction parameters). We first discuss n-pentane + polyethylene systems for which extensive and reliable experimental data are available [355, 359, 360, 389]. The pressure-temperature projection of the phase diagram of binary mixture of pentane + long n-alkanes and pentane + monodisperse linear polyethylene are calculated first. The results are shown in the figure 5.14.

Figure 5.14: Pressure-temperature P T projection of the phase diagram obtained for the binary mixtures pentane + long alkanes (n-C36 ) and pentane + polyethylene(PE) of molecular weights 1, 10 and 100 kg molâ&#x2C6;&#x2019;1 , predicted with SAFT-VR. The white triangles denote calculated upper and lower critical end points. The continuous curves are the vapour pressure curve of the pure compounds. The dashed curves are critical lines. The dashed-dotted lines represent the temperature range where fluid phase are stable: at temperatures below about 400K, crystallisation occurs, and at temperatures above 650K, the alkyl chains decompose. No cross interaction parameter is used (kij = 0). The P T diagram 5.14 of the pentane+PE mixture is similar to those obtained for the prototype polymer solutions discussed in Chapter2 (see figures 2.2 and 2.4). However, the diagram is limited at temperatures below 400K where crystallisation occurs, and above 650K where the alkyl chains decompose. Pentane + polyethylene systems do not exhibit a change from type IV to type III behaviour (see chapter 1 for a description of the various types) contrary to what is found in methane-polyethylene systems. Type IV is always observed, even for very high molecular weight of the polyethylene (500 kg molâ&#x2C6;&#x2019;1 ). As the asymmetry of the two compounds is increased, the region of liquid-liquid immiscibility becomes more extensive, but the two critical lines never meet. The liquid-liquid critical line at low temperature is predicted at temperature around 40 K (not shown in figure 5.14).

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Figure 5.15: a) Vapour-liquid equilibria (bubble point curves) obtained with SAFTVR for the mixture pentane polyethylene (LDPE, M W = 76 kg.mol−1 ), at temperatures T = 150.5◦ C and T = 201◦ C. b) Vapour liquid and liquid-liquid equilibria obtained at T = 201◦ C for the mixture n-pentane + polyethylene (LDPE, M W = 76kg.mol−1). W pentane is pentane weight fraction. No cross interaction parameter is used (kij = 0). The calculated vapour liquid coexistence curve (continuous curves) is compared with experimental data [389] (black circles). The while circle denotes the calculated UCSP. The dashed curves represent the three phase line.

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The lower critical end points appears to tend to a limiting temperature as the molecular weight of the polyethylene is increased. These results are in agreement with experimental data since UCST behavior is never observed in practice for pentane-polyethylene systems. Following the work of McCabe et al. [192], we have modelled both vapour-liquid and liquid-liquid equilibria for pentane + polyethylene systems. It is assume that there are no polymer molecules in vapour phases. As a result, the polymer chain length polydispersity has no effect on the vapour-liquid coexistence curve. The polymer is assumed to be monodisperse and the experimental number average molecular weight hM W in is used. A pressure composition diagram obtained for the system n-pentane + low density polyethylene (LDPE) of molecular weight 76 kg mol−1 is represented in figure 5.15. The temperatures considered (150.5 and 201◦ C) are above the melting point of polyethylene so that the polymer is completely amorphous. Only the bubble point curves are shown in the diagram as the dew point curve are confused with the x axes. Good agreement with the experimental vapour-liquid experimental data [389] is found all temperatures (see figure 5.15 a)). We observe that at temperature 201 ◦ C K, the vapour pressure is slightly overestimated. In figure 5.15 b), the liquid-liquid coexistence predicted with SAFT-VR is also shown. This curves ends at an upper critical solution pressure (UCSP). We have also compared the predictions with the liquid-liquid equilibria data of Xiong et al [360]. The results are shown in the figures 5.16-5.17. Here again the polyethylene is assumed to be monodisperse as the experimental polydispersity index is close to 1. Good agreement is obtained with the cloud curves: SAFT-VT can be used to accurately predict the influence of pressure and polyethylene molecular weight on the cloud curves and LCSTs. We recall that no cross parameters are used here. Some deviations can however be noted and the correlations (5.32) could be slightly modified to improve the predictions of the LCSTs. To do so, the correlations (5.32) could be determined by fitting simultaneously pure n-alkane data (vapour pressures and saturated densities), P V T data of polyethylene, and cloud curves of PE + n-alkane systems. We plan to follow this type of approach in future work.

5.3.2

Influence of the Polymer Parameters on Cloud Curves

It can be seen in figure 5.18 that a tiny change in the depth of the potential ² 22 for the polymer segments has a big effect on the cloud curves. In figure 5.19 we

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represent the different segment-segment attractions between solvent-polymer and polymer-polymer molecules. The overall attractive energy of for a solvent-molecule interaction is proportional of m²12 , while the energy for a polymer-polymer interaction is proportional of m2 ²22 . The chain length m is around 1000, a small changes in ²22 and ²12 makes a big change on the total attractive energy and thus on the phase behaviour. This result has been also observed by Lipson et al. [27, 70]

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Figure 5.16: Cloud curves calculated with SAFT-VR at several pressures for the mixture n-pentane+polyethylene (M W = 108 kg mol−1 ) and compared to experimental data [360] (circles). No cross interaction parameter is used. k ij = 0

Figure 5.17: Cloud curves calculated at pressure P = 5 MPa with SAFT-VR for the mixtures n-pentane + polyethylene(MW= 16.4 kg mol−1 ) and pentane + polyethylene(MW = 2.150 kg mol−1 ), and compared with experimental data [360] (circles). No cross interaction parameter is used (kij = 0).

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Figure 5.18: Influence of the parameter ²22 of the polymer on the cloud curve calculated with SAFT-VR at pressure P = 10 MPa for the mixtures pentane + polyethylene (M W = 108 kg mol−1 ), and compared with experimental data [360]. (circles). A small change of ²22 give rise to a shift of 5 K for the LCST.

Figure 5.19: Schematic representation of the attractions between solvent-polymer and polymer-polymer segments. The attraction energy for solvent-polymer interactions evolve with a factor m2 ²12 , while the attraction energy for polymer-polymer interactions evolve with a factor m22 ²22 .

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234

Other alkanes + Polyethylene

Finally, we have examined other n-alkanes solvents and observed the effect of the molecular weight of the solvent on the LCSTs. As the solvent becomes heavier, it is more compatible with polyethylene and the LCST is increased. We have compared the LCSTs calculated with SAFT-VR and with the BYG theory [70]. One can see from 5.20 that the SAFT-VR approach gives a better description of the LCST than the BYG for short alkanes, however the results are less satisfactory for long alkanes. The deviations can be due to the presence of polydispersity, branching or that our correlations are not adequate enough.

Figure 5.20: Effect of the chain length of the n-alkane on LCST, for solutions of n-alkane + polyethylene (M W = 140 kg.molâ&#x2C6;&#x2019;1 ). The predictions of the SAFT-VR theory (black square) and the BYG theory (white circles) are compared with the experimental data [390], [351], [391], [392] (circles). No cross interaction parameter is used (kij = 0).

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5.4

Solubility of Gases in Amorphous Polyethylene

We focus now on the vapour-liquid equilibria in polyethylene + gas systems. The solubility Sol of a gas in a polymer is defined as the mass of absorbed gas divided by the mass of polyethylene (in percent), and is given by

Sol = 100

massgas wgas = 100 , masspoly wpoly

(5.35)

where wgas and wpoly are the weight fractions of gas and polymer in the liquid phase. The absorption curve (solubility vs Pressure) corresponds to the bubble point curve of the vapour-liquid equilibria (Pressure vs gas weight fraction) for the gas and the polymer. The corresponding absorption curve to the pressure composition diagram of the system pentane + LDPE is shown in figure 5.21. It can be seen in figure 5.22 that the solubility of gas decreases as the molecular weight of polyethylene is increased. This is due to the increase of incompatibility between the gas and the polymer as the difference in size is increased (see chapter 2 for further explanations). It can be also observed that the absorption curve tends to a limiting curve as the molecular weight of polyethylene tends to infinity. The limiting values of solubility are reached for molecular weights higher than about 5 kg molâ&#x2C6;&#x2019;1 . When the chain length is large enough, the solubility of the gas is no longer affected, as the gas molecules interact with polymer at the level of the polymer segments.

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Figure 5.21: Solubility of pentane in amorphous LDPE (M W = 76 kg mol−1 ). SAFT-VR predictions (continuous curves are compared with experimental data [389] (black circles). No cross interaction parameter is used (kij = 0).

Figure 5.22: Solubility of pentane in polyethylene calculated with SAFT-VR at T = 150.5◦ C, for different molecular weights of the polyethylene. No cross interaction parameter is used (kij = 0). The thick line represents the absorption curve of pentane in an infinitely long and linear polyethylene.

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5.4.1

Small Gases

The solubility of some small gas molecules (ethylene, and nitrogen) in amorphous polyethylene have been calculated with SAFT-VR. To predict the experimental data, one requires some cross interaction parameters. The mixing rules for small molecules deviate from from Lorenz-Berthelot combining rules because the difference between the ² parameters of the small gases and polyethylene is important. However, thye knowledge of a single cross interaction parameter is sufficient for each gas to predict (²)

the solubility over the entire range of temperatures. We have used k12 = 0.075 for (²)

the system ethylene + PE and k12 = 0.15 for the system N2 + polyethylene. The absorption curves are well predicted by SAFT-VR for both gases. It can be seen in figures 5.21, 5.23 and 5.24, that the solubility of the gas is increased if the gas is less volatile (or longer). The solubilities are in this order : SolnC5 > SolC2 = > SolN2 . Usually the solubility of the gas decreases with increasing temperature (see pentane + PE in figure 5.21, and ethylene + PE in 5.23), as the gas becomes more volatile with increasing temperature. Moreover, the absorption curve is more and more concave as the gas becomes less and less volatile. This feature is analogue the phenomenon of condensation which occurs in the adsorption of gas on the surface of solids. The liquid phase is a mixture of pentane and PE. As pentane molecules absorb in this mixture, they encounter more and more pentane molecules. Since the pentane-pentane interactions are attractive and more favorable than the polymer segment - pentane interactions, there is a synergy effect and the absorption curve has an increasing exponential shape. However, the ethylene-ethylene interactions are not as attractive as the pentane-pentane interactions, and the synergy effect is less pronounced: the solubility curve of ethylene is almost a straight line (see figure 5.23) and Henry’s law could be applied. At high temperatures, the nitrogen-nitrogen interactions are almost purely repulsive (low ²N2 ). In this case the synergy is inverse and the curve is convex (see figure 5.24) .

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Figure 5.23: Solubility of ethylene in amorphous LDPE (M W = 248 kg mol−1 ). Comparison of experimental data [393] (T = 126◦ C, squares, and T = 155◦ C, circles) and SAFT-VR predictions (continuous lines). One cross interaction parameter (²) k12 = 0.075 is used for all temperatures.

Figure 5.24: Solubility of Nitrogen (N2 ) in amorphous HDPE (M W = 111 kg mol−1 ). Comparison of experimental data [394] (T = 160◦ C, squares, and T = 300◦ C, circles) and SAFT-VR predictions (continuous lines).One cross interaction (²) parameter k12 = 0.15 is used for all temperatures.

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An other particularity of the solubility of nitrogen in polyethylene is observed experimentally and well predicted by SAFT-VR, where the solubility of N2 in PE is seen to decrease with decreasing temperature (see figure 5.24), contrary to the solubility of pentane and ethylene in PE. This result can be explained as follows: at high temperatures, the nitrogen-nitrogen are almost purely repulsive (nitrogen behaves almost as a hard dumbbell) so that the volatility of nitrogen does not increase much as the temperature is increased. Due to the ideal entropy of mixing, the nitrogen molecules will tend to mix more with polyethylene at higher temperatures. Consequently, the solubility of nitrogen increases with increasing temperature. These high temperatures correspond to the right side of the pressure temperature P T diagram of the binary mixture N2 + PE : in this region of the P T diagram, the vapour-liquid critical pressure of the mixture N2 + PE decreases as the temperature is increased, and this is consistent with the evolution of nitrogen solubility. Similar results are obtained for the solubility of hydrogen (H2 ) in n-alkanes and PE (see next section, figures 5.25 to 5.28 ).

5.4.2

Solubility of Hydrogen in Polyethylene

It is necessary to determine the SAFT-VR parameters of pure hydrogen H 2 in order to predict the absorption of hydrogen in polyethylene. Within the SAFT approach one is not able to model hydrogen with only one set of parameters, since for the hydrogen molecule, the quantum effects due to the vibration modes of the H-H bond are not negligible at low temperatures below the critical point of H 2 . As a result, one can not obtain reliable parameters for H2 by fitting vapour pressures and saturated liquid densities. Hoverer, at higher temperatures, the quantum effects can be neglected. One could then optimised the SAFT parameters (m, ², σ, λ) for P V T data of pure hydrogen at high temperatures. However, at these temperatures, hydrogen almost behaves as an ideal gas and many different sets of parameters (m, ², σ, λ) would be consistent with the P V T experimental data. In order to obtain reliable parameters for hydrogen at high temperatures, an other way is to fit the pure hydrogen parameters on vapour-liquid equilibria data of binary mixtures. The binary mixtures H2 + n-alkanes is one of the most convenient mixtures that we can use in this regard, as we would like to predict the solubility of H2 in polyethylene. We have used the VLE experimental data of binary systems (H2 + propane [395], H2 + n-decane [396], H2 + n-hexadecane [397]) and carried out two kinds of optimisation:

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• The parameters (m, ², σ, λ) of pure hydrogen are optimised for the VLE data of H2 + n − alkane binary mixtures, without any binary cross interaction (²)

(λ)

parameters (see equations (5.29), (5.30), kij = 0, kij = 0). • The parameters (m, ², σ, λ) of pure hydrogen and the cross interaction param(²)

(λ)

eters (kij , and kij ) are both optimised for the VLE data of H2 + n − alkane (²)

(λ)

binary mixture. Only one set of parameters kij and kij is used for all binary mixtures and temperatures considered. The results are shown in figures 5.25, 5.26 and 5.27 and the optimised parameters are given in table 5.4. Good agreement can be obtained on both bubble an dew point curves, for all temperatures and binary mixtures. It can be seen that the use (²)

(λ)

of binary parameters kij and kij does not improve much the results, but the pure hydrogen parameters are quite different in the two cases. As a result, we decide (²)

(λ)

not to use any binary parameter (kij = 0, kij

= 0) in order to have a more

predictive model. Our approach enables us to obtained similar accuracy as other methods (complex mixing rules with binary parameters sometimes depending on temperatures and on the number of carbon atoms of the n-alkane, see [349,350,398]). However, as we do not use any cross interaction parameters, our method is more predictive and can be extrapolated with confidence to predict the absorption of hydrogen in polyethylene. The pure parameters of hydrogen obtained in the first optimisation are very similar to those obtained by Wang et al. [348] by path integral simulation with a potential different from square-well. This result confirms the validity of our approach. We use the parameters of optimisation 1 to predict the solubility of hydrogen in polyethylene (see figure 5.28). It is seen that the solubility of H2 is increased if the temperature is increased, as for the solubility of N2 in HDPE (cf. figure 5.24).

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Figure 5.25: Pressure composition P x diagram of the binary system H 2 + propane. The symbols denote the VLE experimental data from the DECHEMA series [395]: squares: T= 223 K; crosses: T = 298 K; circles: T = 348K. The continuous curves represent the predictions of SAFT-VR without any cross interaction parameter (opt 1), and the dotted lines represent the predictions of SAFT-VR with the cross inter(²) (λ) action parameters kij = −0.0524, kij = −0.1963 (opt 2).

Figure 5.26: Pressure composition P x diagram of the binary system H 2 + n-decane. The symbols denote the VLE experimental data from Sebastian et al. series [396]: squares: T = 462.45 K; crosses: T = 503.35 K; circles: T = 583.45 K. The continuous curves represent the predictions of SAFT-VR without any cross interaction parameter (opt 1), and the dotted lines represent the predictions of SAFT-VR with (²) (λ) the cross interaction parameters kij = −0.0524, kij = −0.1963(opt 2).

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Figure 5.27: Pressure composition P x diagram of the binary system H 2 + nhexadecane. The symbols denote the VLE experimental data from Lin et al. [397]: squares: T = 461.65 K; crosses: T = 542.25 K; circles: T = 622.85 K; triangles: T = 664.05 K. The continuous curves represent the predictions of SAFT-VR without any cross interaction parameter (opt 1), and the dotted lines represent (²) the predictions of SAFT-VR with the cross interaction parameters k ij = −0.0524, (λ)

kij = −0.1963 (opt 2).

Figure 5.28: Solubility of hydrogen in HDPE (M W = 100 kg.mol−1 ), at T = 120◦ C and T = 180◦ C. The hydrogen parameters of optimisation 1 are used. No cross (²) (λ) interaction parameter is used (kij = 0, kij = 0)

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Table 5.4. SAFT-VR square-well parameters for pure gas hydrogen and binary cross interaction parameters with n-alkanes (optimisation 2) obtained after optimisation on the VLE data of the binaries (H2 + propane, H2 + n-decane, H2 + n-hexadecane).

Opt 1 Opt 2

5.5

(²)

(λ)

σ (˚ A)

²/k (K)

kij

1.0 1.0

1.800 1.594

3.1503 3.1930

37.0182 17.9868

0.0 -0.0524

0.0 -0.1963

Crystallinity of Polyethylene

As for other polymers, polyethylene does not exhibit the crystallisation behaviour of a simple pure component. The introduction into the polymeric chain of units that differ chemically, stereo-chemically, or structurally from the predominant chain repeating units, imposes restrictions on the crystallisation and fusion process. According to the Gibbs phase rule (see chapter 1, section 1.2.7) applied to a pure component, there is only one melting point Tm corresponding to a given pressure P ; if the temperature T is just below Tm , the equilibrium state pure component sample is totally solid, and at temperature T just above the fusion temperature T m the component is totally liquid. A polymer is not a pure component, but a mixture of compounds varying in size, and chemical structures. Moreover, some elements of the polymer chain can not crystallise, such as alkyl branches, as they can not arrange themselves into a solid structure (in the usual timescale of the experiment). One can define a melting point Tm for the polymer sample similar to the melting point of a pure component, such that the polymer sample is totally amorphous at temperatures T > Tm . However, at temperature T < Tm , only a fraction wcyrs of the polymer exists as crystallised units; this is referred to as the crystallinity of the polymer. Under this melting point, the polymer sample is semi-crystalline, as it consists of amorphous regions which exhibit a liquid-like structure, and ordered crystallised regions called crystallites (see figure 5.29). Polyethylene crystallises in lamellar structures with a certain average thickness hζi. The lamellar thickness hζi affects the melting point Tm due to surface effects [323]: the melting temperature decreases as the average lamellar thickness decreases according to the ThompsonGibbs equation.

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Figure 5.29: Schematic representation of the structure a semi-crystalline polymer. The crystallites have an average thickness hζi.

5.5.1

Experimental Measurements of Crystallinity

A summary of the various techniques to measure the crystallinity of a polymer sample is given in reference [324]. The main experimental techniques include: Dilatometry The most widely used property is the density of the polymer sample, for the reason that it can be measured simply and with great accuracy. The polymer densities can be measured with a dilatometer. The density of the crystallites can be as much as 10% greater than that of the amorphous regions. As the density measurements are very precise, the crystallinity wcrys can be obtained with the relation [399]:

wcrys =

ρ − ρ a ρc , ρc − ρ a ρ

(5.36)

where ρ, ρc , and ρa are the densities in (g cm−3 ) of the polymer sample, the crystallites, and the amorphous regions respectively. The relation (5.36) is exact as it is derived from a mass balance equation; the problem consists in determining accurately the densities ρc and ρa . In the case of polyethylene systems, these two densities may vary from one sample to another, however, it is a good approximation to use the same densities ρc and ρa for all kinds of polyethylene samples (HDPE, LDPE, LLDPE). We propose slightly different values from those of Yiagopoulos [399] for the densities ρc and ρa by fitting ρc and ρa to experimental data involving different types of

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polyethylene (published data by [327, 328, 330, 336]). The crystallinities were measured by DSC or X-rays (see the next sections for further explanations about these techniques). We obtained ρc = 1.005 g mol−1 and ρa = 0.862 g mol−1 . It can be seen from figure 5.30 that the agreement between the experimental data and equation (5.36) is good. Some significant deviations (max 12%) are however observed for certain polymer samples, which can be due to the assumption that ρc and ρa have the same values for all polyethylenes, or due to some errors in the measurements as the accuracy of the DSC technique is about 10%.

Figure 5.30: Crystallinity of various polyethylene samples (HDPE, LDPE, LLDPE) as a function of their densities. The experimental data are from McKenna [327], Jordens et al. [328], Moore et al. [336], Starck et al. [330], and the continuous line corresponds to the equation (5.36) with ρc = 1.005 g. mol−1 and ρa = 0.862 g mol−1 .

X-ray Diffraction The scattering of X-rays by a polymer sample gives rise to diffraction rings. The amorphous regions give rise to broad diffraction regions, while crystallites give rise to diffraction peaks. If the broad diffraction regions due to amorphous regions is distinguishable from the crystallite diffraction peaks, then the ratio of the two intensities provides a measure of the ratio between amorphous and crystalline regions. This technique is widely used and can measure crystallinities ranging from 0.2 to 0.8.

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Differential Thermal Analysis Differential Thermal Analysis, also called Dynamic Scanning Calorimetry (DSC), is a versatile technique for a rapid characterisation of several aspects of polymer crystallisation and melting. The DSC method consists in measuring the heat flow received by a sample as the temperature is increased. The curve of the heat flow vs temperature provides several peaks, and the highest peak corresponds to the melting point of the sample. Knowing the enthalpy of fusion ∆Hu of the polymer units (for polyethylene, ∆Hu ' 296 J/g. [330, 333, 334]), the crystallinity can be obtained by calculating the integral of the curve of the heat flow/ temperature. Improved DSC techniques can be used to measure crystallinity as a function of temperature [327], [330].

5.5.2

Flory’s Theory of the Fusion Behaviour of Copolymers

The most robust theory in the literature to predict the crystallinity of polymer samples at equilibrium as a function of temperature was developed fifty year ago by Flory [331]. Although this theory is old, tests of the theory are very scarce in the literature [333, 334], since accurate techniques to measure crystallinities as a function of temperature have only been developed recently [327], [330]. Flory’s idea is based on the following argument: the crystallinity wcrys represents the weight fraction of crystallised polymer in the sample. As the mass of a polymer molecule is proportional to the number of units in the chain (the proportionality factor is the molecular weight of the monomer), the crystallinity also corresponds to the total fraction of crystallised polymer units, or to the probability that a given polymer unit is crystallised. Flory developed then his theory by evaluating this probability as a function of temperature and some properties of the copolymer. In Flory’s approach, one can call a ”copolymer” any polymer containing different types of units along the main chain, such as alkyl branches. For instance, HDPE and LDPE which are made with solely with ethylene, can be treated in this approach as a pseudo-copolymer, since they contain both crystallisable ethylene groups and noncrystallisable units such as alkyl branches along the chain. Following Flory’s development, we consider a model polymer which contains crystallisable units designated as ”A” units, and noncrystallisable units designated as ”B” units. We define the crystallinity wcrys as the mass of crystallised polymer in the sample. wcrys is

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then equal to the fraction of crystallised polymer units at equilibrium. One should not confuse the fraction of crystallisable units A XA = 1 â&#x2C6;&#x2019; XB in the chain, which is the fraction the polymer units susceptible to crystallise, with the fraction of crystallised units equal to the crystallinity wcrys . In the case of polyethylene, A units would represent ethylene groups while B units represent branches along the chain. It may be reasonable to assume that polymer samples containing a high fraction X A of crystallisable A units are likely to have at equilibrium a larger crystallinity wcrys , however this is not always the case. The frequency of A units along the chain is also an important factor.

Figure 5.31: Schematic representation of alternating and block-type copolymers. Thick lines represent crystallisable A units, and thin lines represent noncrystallisable B units. Block-like copolymers is more likely to crystallise than alternating-type copolymers, with the same mole fraction XA of crystallisable A units. As is illustrated in figure 5.31, a block-like copolymer has a higher crystallinity than an alternative-like copolymer, as the probability that a block of consecutive A units is crystallised increases with an increase in the number of consecutive A units in the block. One can define the probability p that a given A unit in the chain is followed by another A unit along the same chain (in any direction). The probability p is used as an average, independently on the direction and the position of the A unit chosen in the chain. This probability p provides a quantitative differentiation between random, alternative, and block-type copolymers (see chapter 1): for random-like copolymers, p is equal to the fraction of A units p = XA ; for block-type copolymers,

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p greatly exceeds XA , as the A units are more likely to be in blocks; for alternating types of copolymers, p is around zero as each A unit is surrounded by two B units. In Flory’s theory, the longitudinal development of the crystallites is restricted by the occurrence of noncrystallisable B units along the polymer chain. The lateral development is governed by the availability or concentration of sequences of suitable length in the residual melt, and by the decrease in free energy that occurs when a sequence of ζ A units is transferred from the amorphous to the crystalline phase. A quantitative formulation is developed by relating to the melt composition the probability Pζ that a given A unit in the amorphous phase is located within a sequence of at least ζ such A units. Let wζ be the probability that a unit chosen at random from the amorphous regions is an A unit, and also that this A unit belongs to a block of exactly ζ A units. It can be shown [323, 331] that the probabilities P ζ and wζ are related by

wζ = ζ (Pζ − 2Pζ+1 + Pζ+2 ) .

(5.37)

The quantities Pζ and wζ can be related to the constitution of both the initially molten polymer, and to the partially crystalline one. In the completely molten copolymer, prior to any crystallisation, the initial probability wζ0 is given by

wζ0 =

NA,ζ , N

(5.38)

where NA,ζ is the numbers of A units in the molten polymer belonging to blocks (or sequences) containing exactly ζ consecutive A units, and N the total number of units (A + B). Let νζ0 be the number of sequences of ζ A units in the molten polymer. As NA,ζ = ζνζ0 , and as NA = XA N where NA and XA are the total number of mole fraction of A units respectively, wζ0 can be written as

wζ0 =

XA ζνζ0 . NA

(5.39)

By assuming that the probability p of an A unit being succeeded by another A unit is independent of the number of preceding A units in the sequence, one can write equation (5.39) as

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

wζ0 =

XA ζ (1 − p)2 pζ . p

249

(5.40)

By combining equations (5.37) and (5.40), the probability Pζ0 that in the initial state a given A unit belongs to a sequence of at least ζ units can be written as Pζ0 = XA pζ−1 .

(5.41)

Equation (5.41) means that the probability Pζ0 is equal to the product of the probability XA of choosing a A units beyond the N total units, and the probability that this given A unit is followed by at least ζ − 1 successive A units. When crystallisa-

tion occurs and thermodynamic equilibrium is maintained, the probability Pζeq that a given A unit in the amorphous regions is located in a sequence of at least ζ units is equal to the Boltzmann factor of the change of energy due to the fusion of a block of ζ units (similar to the acceptance probability of a move in an NVT Monte Carlo simulation), and is given by

Pζeq

∆Fζ = exp − RT µ

(5.42)

where ∆Fζ is the free energy of fusion per mole of sequence of ζ units from a mole of crystallites of ζ A units. ∆Fζ can be expressed as

∆Fζ = ζ∆Fu − 2σe ,

(5.43)

where ∆Fu = ∆Hu −T ∆Su is the free energy of fusion per mole of units and σe is the surface free energy at the crystallite ends per mole of sequences. For polyethylene, σe /R = 1190.7 K where R is the ideal gas constant (R = 8, 314411 J. mol −1 . K−1 ); 0 where T 0 is the melting point of an ideal ∆Hu /R = 996.34K, and ∆Su = ∆Hu /Tm m

totally crystallisable polyethylene (see references [330, 333, 334]). Equation (5.42) can be expressed as

Pζeq = where

1 exp (−ζθ) , D

(5.44)

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

∆Hu θ= R

1 1 − 0 , T Tm

250

(5.45)

(5.46)

and

2σe D = exp − RT µ

Equations (5.37) and (5.44 ) can then be combined to give wζeq = ζD−1 (1 − exp((−θ))2 exp (−ζθ) ,

(5.47)

which is the fraction of molten sequences of ζ A units. The necessary and sufficient condition that blocks containing at least ζ A units crystallise is Pζ0 > Pζeq .

(5.48)

There is a critical length ζcrit under which the blocks are not crystallised. This critical length corresponds to , Pζ0crit = Pζeq crit

(5.49)

so that ζcrit is expressed as

ζcrit = −

ln (DXA /p) + 2 ln (1 − p) / 1 − eθ θ + ln p

´´

(5.50)

The inequality (5.48) can also be expressed as

XA ζ 1 p > e−θζ . p D

(5.51)

Except for copolymers exhibiting a high tendency to for alternation, 1/D is greater than XA /p. Thus the inequality (5.51) becomes

θ > − ln p, or

(5.52)

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

1 R 1 − 0 >− ln p, T Tm ∆Hu

251

(5.53)

Equation (5.52) gives the limiting temperature where crystallisation can occur. The melting point Tm is then 1 R 1 − 0 =− ln p, Tm Tm ∆Hu

(5.54)

Since XA is usually closed to p, one can neglect the term XA /p in equation (5.50) (the term ln (DXA /p) does not affect much the final crystallinity, as shown in [323]), and equation (5.50) becomes

ζcrit = −

ln D + 2 ln (1 − p) / 1 − eθ θ + ln p

´´

(5.55)

0 = ∆H /∆S is the melting temperature of the homopolymer. The temperature Tm u u

This temperature can be considered as a reference melting point including all the effects (molecular weight, lamellar thickness [323]) different from the copolymer 0 from the expressions, one can combine equations (5.54), effects. To remove Tm

(5.55) and (5.45) to give

∆Hu θ= R

1 1 − T Tm

− ln p,

(5.56)

and

ζcrit = −

ln D + 2 ln (1 − p) / 1 − eθ ∆Hu R

1 T

−

1 Tm

´´

(5.57)

It can be seen in equation (5.57) that only very long sequence of A units can crystallise if the temperature T is close to the melting point. Let wζcris be the fraction (concentration) of sequence of ζ A units which are crystallised, then wζcris = wζ0 − wζeq .

(5.58)

252

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

Applying the properties of the mathematical series, the total fraction of crystallised units wcris (crystallinity) can be obtained as

wcris (T ) =

∞ X

wζcris

(5.59)

ζ=ζcrit

5.5.3

XA e−θ p 1 1 − (1 − p)2 pζcrit 2 2 + ζcrit 1 − p − 1 − e−θ −θ p (1 − p) (1 − e ) µ

¶#

Modelling of Polyethylene Crystallinity

The main difficulty to model the crystallinity of polyethylene if that the exact mole fraction of crystallisable units XA and probability p are can not be measured. More other, additional effects due to defects, lamellar thickness, molecular weight, and 0 [332–334]. One could chain ends must be included via the reference temperature Tm 0 is equal to the melting temperature of neglect all these effects, and assume that Tm

an infinitely long linear polyethylene which is about 145 ◦ C [194,333]. One also could assume that polyethylene is a random copolymer (p = XA ), and that only ethylene group along the main chain crystallise. Thus XA = 1 − Xbr , where Xbr = XB is the fraction of branches characterised by the number of CH3 groups in the chain which can be measured experimentally. Such approach has been tested recently [333, 334] and it has been shown that these assumptions give rise to an overestimation of both crystallinity and melting point. Sanchez and Eby [332] derived an expression of the melting point taking into account defects, chain ends and lamellar thickness effects. The Sanchez-Eby equation enables to get better predictions of the melting point [333], but some parameters have to be fitted on experimental data, and the prediction capability are restraint. This theory also requires the knowledge of the experimental value of the fraction XA (or XB ) of ethylene units which is not always available. Moreover, the assumption that polyethylene is a random-type copolymer (i.e. p = XA ) may be too crude for block-type polymers: it has been seen experimentally that Ziegler-Natta (ZN) catalysts lead to block-type copolymers, while other catalysts like metallocens (Me) give rise to random-like copolymers [326, 329, 330]. In that case, p ≥ XA depending on the catalyst used. We have then developed a model based on Flory’s theory, to predict the melting point and crystallinity of any polyethylene with the minimum of experimental data.

253

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As mentioned by experimentalists ( [61]), the density ρ25 of the polyethylene sample measured at 25◦ C, 1 atm, characterises the degree of branching of the polymer. The density ρ25 is about 1 g.cm−3 for a totally crystallised linear polyethylene, and about 0.86 g.cm−3 for a complete amorphous polyethylene. For HDPE, ρ25 is about 0.96 g.cm−3 and for LDPE and LLDPE, ρ25 is about 0.92 g.cm−3 . ρ25 is the related to the crystallinity wcris,25 at 25◦ C via equation (5.36). Hence wcris,25 also characterises the branching of the polyethylene. Following those ideas, we develop some correlations of the melting point and of the parameters XA , p as a function of wcris,25 only.

Two different correla-

tions have been developed according to the type of catalyst used (ZN and Me), for various polyethylenes (HDPE, LDPE, LLDPE made from different olefins). The mole fraction XA can be removed from equation (5.59) such that the relation wcris (T = 25) = wcris,25 is satisfied. Equation (5.59) then becomes

wcris (T )

ζcrit

ζcrit,25

wcrys,25 p Ã

÷ p

p e−θ 1 1 − 2 2 + ζcrit 1 − p − 1 − e−θ −θ (1 − p) (1 − e ) µ

e−θ25 p + ζcrit25 − (1 − p)2 (1 − e−θ25 )2

¶#

1 1 − 1 − p 1 − e−θ25

(5.60) ¶#!

where wcrys,25 is the experimental value of the crystallinity at 25◦ C (which can be evaluated by equation (5.36) if unknown), and where θ25 and ζcrit,25 are given by equations (5.56) and (5.57) respectively at temperature T = 25◦ C . The melting point Tm in ◦ C is given by

for catalyst ZN

2 Tm = 13.689wcrys,25 + 5.015wcrys,25 + 124.33

for catalyst Me

2 Tm = −81.498wcrys,25 + 163.3wcrys,25 + 63.415

(5.61)

Both correlations go through the point (wcrys,25 = 1., Tm = 145◦ C) corresponding to an ideal infinity long and linear polyethylene. The probability p i given by

for catalyst ZN

2 + 0.1397wcrys,25 + 0.9142 p = −0.0538wcrys,25

(5.62)

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS for catalyst Me

254

2 p = â&#x2C6;&#x2019;0.0581wcrys,25 + 0.1279wcrys,25 + 0.9303

For both catalysts, the probability p is equal to 1 for an ideal infinitely long and linear polyethylene which totally crystallises. The melting point Tm and probability p that a A unit is followed by an other A unit along the chain are depicted in figures 5.32 a) and b). It can be seen that both melting Tm and probability p are higher for ZieglerNatta polyethylenes, than for metallocen polyethylenes. This result is in agreement with experimental observations [326, 329, 330]: the Ziegler-Natta polyethylene are block-type copolymers (higher p) and crystallises more (higher Tm ) than metallocen polyethylene which are more random-type copolymers (lower p).

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255

Figure 5.32: a) Melting point Tm and b) probability p that a crystallisable A unit is followed by another A units, for polyethylene samples, as a function of the experimental crystallinity wcrys,25 at 25â&#x2014;Ś C. In figure a), the white circles denote experimental melting points (references [327, 330]) for PE samples made with metallocen (Me) catalysts, and the black circles denote experimental melting points for PE samples made with Ziegler-Natta (ZN) catalysts. In figure b), the circles (white for Me, black for ZN catalysts) correspond to fitted p parameters of equation (5.60) on experimental crystallinity vs temperatures curves shown in figures 5.33 a) and b).

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256

Figure 5.33: Crystallinity wcrys as a function of temperature T , for metallocen a) and Ziegler-Natta polyethylenes b). The symbols represent the experimental data from references [327,330] (see the corresponding papers for the meaning of the names of the PE samples). The continuous curves represent the predictions of the modified Flory equation (5.60) where wcrys,25 is the experimental value, and Tm and p are given by the correlations (5.61) and (5.62).

257

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

As can be seen in figures 5.33 a) and b), our model can predict very well the crystallinity of a large variety of polyethylenes (ZN or Me PE: HDPE, LDPE, or LLDPE made from butene, hexene or octene). Both qualitative and quantitative agreement is obtained with experimental data: at low temperatures, the crystallinity tends to a finite crystallinity, decreases exponentially as the temperature is increased, then tends to zero as the temperature approaches the melting point Tm . The only necessary parameter is the experimental value of the crystallinity of the sample at 25 ◦ C. If that data is unknown, it can be evaluated by equation (5.36) within 10 % of deviation if the density at 25◦ C is known.

5.6

Solubility of Gases in Semi-Crystalline Polyethylene

At temperatures below the melting point of polyethylene (about 130◦ C), the polyethylene sample is semi-crystalline. The phase behaviour then becomes more complex as there is an equilibrium between three phases (vapour, liquid amorphous polymer, and solid crystallites). It has been observed that the solubility of gases is a linear function of crystallinity [327, 336]. Once can assume that the crystallites behave as steric barriers against the diffusion of the gas molecules so that the gas only absorb in the amorphous regions. One also assume that the polymer molecules are either totally amorphous, or totally crystallised, i.e. there are not partial crystallised blocks along the chain. The latter assumption must be accurate as we showed before that the chain length does not affect much the solubility when the molecular weight is high. We also assume that the presence of gas in the polymer sample does not affect crystallinity and melting point (no cryoscopic effects). A schema representing our model of the system gas + semi-crystalline polyethylene is given in figure 5.34.

5.6.1

Model

The solubility is calculated by

(am)

Sol = 100

wgas wgas = 100 (1 − wcrys (T )) (am) , wpoly w

(5.63)

poly

where wgas and wpoly are the global weight fractions of gas and polymer molecules (am)

in the sample, and wgas

(am)

and wpoly are the weight fractions of gas and poly-

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

258

mer molecules in the amorphous phase at equilibrium, calculated with SAFT-VR; wcrys (T ) is the crystallinity of the pure polymer sample at temperature T , calculated with equation (5.60).

Figure 5.34: Schematic representation of the absorption of gas in semi-crystalline polyethylene. The grey spheres represent gas molecules and black spheres represent polymer segments. The black zones denote crystallites. It is assumed that the gas only absorbs in the amorphous regions, and that the polymer molecules are either completely amorphous or totally crystallised.

5.6.2

Results

We have predicted the recent data of Moore et al. [336] of absorption ofÎą-olefins in semi-crystalline HDPE and LDPE. The absorption curves of 1-butene and 1-hexene in HDPE and LDPE are displayed in figures 5.35 and 5.36 respectively. In reference [336], the experimental values of crystallinity at 25â&#x2014;Ś C are wcrys,25 = 0.702 for the high density polyethylene and wcrys,25 = 0.504 for the low density polyethylene. For HDPE, we have used the correlations for ZN catalyst and for LDPE, we have used the correlations for Me catalysts (equations (5.61) and (5.62)). The crystallinities of both polymer samples have been calculated at the temperatures of the absorption curves with equation (5.60). It can be seen that SAFT-VR combined with our model for crystallinity can predict very well the solubility of olefins in polyethylene for the entire range of temperatures.

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

259

Figure 5.35: Solubility of a) 1-butene and b) 1-hexene in semi-crystalline HDPE (M W = 11.49 kg.molâ&#x2C6;&#x2019;1 , wcrys,25 = 0.702) at different temperatures. Symbols represent experimental data [336] and continuous lines represent SAFT-VR predictions.

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

260

Figure 5.36: Solubility of a) 1-butene and b) 1-hexene in semi-crystalline LDPE (M W = 22.01 kg.molâ&#x2C6;&#x2019;1 , wcrys,25 = 0.504) at different temperatures. Symbols represent experimental data [336] and continuous lines represent SAFT-VR predictions.

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261

The solubility of 1-hexene in LDPE is however underestimated at high pressures (figure 5.36 b)). The deviations may be due to cryoscopic effects: the presence of gas in the polymer sample decreases the melting point and crystallinity of the polymer sample, thus the absorption of the gas is increased. In further work, we are planning to take into account these effects by following some recent approaches [345, 346] to calculate solid-liquid equilibria in polyethylene solutions.

5.7

Effects of co-absorption

The co-absorption effects represent an important industrial problems, as the solubilities of the different gas in polyethylene determine the rate of the polymerisation reaction and the final composition of the monomers included in the polymer chain. We consider here the ternary systems 1-butene + nitrogen +HDPE, and 1-butene + 1-hexene + HDPE. The binary system 1-butene + 1-hexene almost behaves as an ideal mixture. Hence there is no need on cross interaction parameters between 1-butene and 1-hexene to predict accurately the experimental VLE data [400] of that system, as shown in figure 5.37. We also assume that no cross interaction parameters are needed for the binary system 1-butene + N2 . The influence on the solubility of 1-butene in of the presence of an other gas (nitrogen or 1-hexene) is shown in figure 5.38. The partial pressure of 1-butene is fixed, and the partial pressure of the second gas is increased. Very different behaviour are observed depending on the nature of the second: the presence of nitrogen hardly affects the solubility of 1-butene in PE (the solubility slightly decreases), while with increasing partial pressure of 1-hexene, the solubility of 1-butene is dramatically increased. Such effects have been observed experimentally (citeLi69) or predicted by MC simulation [177,339], but have not been explained in terms of the intermolecular interactions.

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

262

Figure 5.37: Pressure composition diagram at temperature T = 100 ◦ C of the binary 1-butene + 1-hexene. The circles denote experimental data [400] and the continuous lines represent SAFT-VR predictions.

Figure 5.38: Solubility of 1-butene (gas 1) in amorphous HDPE (M W = 100 kg. mol−1 ) at temperature T = 150◦ C, at fixed partial pressure of 1-butene PnC4 = = 0.05 MPa, as a function of the partial pressure of a second gas (nitrogen or 1-hexene).

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

263

Figure 5.39: Ternary diagrams of the system 1-butene + 1-hexene + amorphous HDPE (M W = 100 kg. molâ&#x2C6;&#x2019;1 ) calculated with SAFT-VR at temperature T = 460 K, and pressures P = 1 MPa a) and P = 3 MPa b). The continuous line represent coexistence curves. The dashed lines represent tie lines. The squares denote calculated three phase points.

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

264

We give here a simple explanation in terms on thermodynamic interactions: in the system 1-butene + nitrogen + PE, nitrogen behaves almost as an ideal gas at T = 460K and the interactions nitrogen - nitrogen and 1-butene - nitrogen are repulsive. Due to the ideal entropy of mixing, N2 absorbs slightly in PE. The presence of N2 in PE then decreases a little the solubility of 1-butene in PE as 1butene - nitrogen interactions are unfavorable. In the case of the system 1-butene + 1-hexene + PE, the interactions 1-butene - 1-hexene are attractive (almost similar to the interactions 1-butene - 1-butene). As a result, the presence of 1-hexene in PE gives rise to an increase of the solubility of butene and the absorption curve is concave (see section 5.4.1 for further explanations about the concavity of the curve). Ternary diagrams of the system 1-butene + 1-hexene + amorphous HDPE have been calculated with SAFT-VR at T = 460 K, and pressures P = 1 MPa (figure 5.39 a)) and P = 3 MPa (figure 5.39 b)). At low pressures, a broad region of vapourliquid region can be seen and the solubility of both olefins in PE are low. However when the pressure is increased, the diagram changes dramatically as liquid-liquid immiscibility occurs. The solubility of the gas 1-hexene in PE is increased a lot. There is liquid-liquid coexistence between 1-butene and PE and the concentration of 1-butene in both liquid is very high. Such diagrams could be used to design separation processes as the weight fractions of a given gas can change dramatically by changing pressure.

5.8

Conclusion

We have used the SAFT-VR theory to model the vapour pressures are saturated densities of a large variety of pure components (Hydrocarbons and associating compounds). By using simple extrapolations with molecular weight, we have obtained the SAFT-VR parameters of polyethylene from those of the n-alkanes. Good agreement with experimental data is obtained for the absorption of gases in amorphous polyethylene. The absorption of the gas hydrogen in polyethylene can also be predicted, by fitting the SAFT parameters of hydrogen on vapour-liquid experimental data of binary mixtures of hydrogen + n-alkanes. The results obtained for the cloud curves in polyethylene + n-alkane systems are also satisfactory. This confirms the capability of the SAFT theory to extrapolate the parameters from one system to an other. This extrapolation can be made

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

265

with confidence as the SAFT parameters have physical meanings. One could however improve the predictions of the cloud curves by fitting the SAFT parameters of polyethylene directly on experimental cloud points, and change the relation between the number of segments m and the molecular weight of polyethylene. The liquid-liquid region is indeed very sensitive to the parameters, and one could change slightly the correlations (5.32) to get better agreements with the experimental cloud curves for polyethylene + n-alkane systems. Polyethylene is semi-crystalline when the temperature if below its melting point. In this case, the crystallinity of polyethylene has to be taken into account. We developed an accurate model based on Floryâ&#x20AC;&#x2122;s theory of copolymer crystallinity to predict the crystallinity of any kind of polyethylene as a function of temperature. This model just requires one data of crystallinity at 25â&#x2014;Ś C. Applying this model and the SAFT-VR theory, good predictions can be obtained on the absorption of olefins in polyethylene by assuming that the gas molecule only absorb in the amorphous regions of the polyethylene sample. One could bring improvement to the model by taking into account cryoscopic effects [345, 346]. We also have shown that coabsorption may exhibit strong synergy effects which can be explained by a difference of interactions between the absorbed gases.

Chapter 6

Conclusion After having discussed in a first chapter the different types of phase behaviour encountered in binary mixtures, prototypes of polymer + solvent systems have been studied in order to provide a better understanding of the liquid-liquid immiscibility and LCST behaviour if these mixtures. The phase behaviour and the properties of mixing are examined for simple models of polymer solutions in which the solvent is represented as a hard sphere and the polymer as a chain of tangent spherical segments of same diameter. The TPT1 description of the free energy originally developed by Wertheim is used to treat the hard-sphere chain molecules, and the attractive interactions are described at the simple mean-field level of van der Waals. The diameter and attractive energies are taken to be the same for all segments (symmetrical interactions). One of the main findings presented in the thesis is that attractive interactions play a key role in the fluid phase demixing, and the purely athermal system is shown not to exhibit not exhibit any phase separation. A region of liquid-liquid immiscibility region can be seen at temperatures just below the critical temperature of the pure solvent. The enthalpy of mixing is always negative and decreases as the temperature is increased. The entropy of mixing is positive at low temperature, then becomes negative (for mixtures with low compositions of polymer)at high temperature and favour demixing. The liquid-liquid phase separation in these model polymer-solvent systems corresponds to an LCST which is an entropy driven process. The decrease of entropy at temperature closed to the critical temperature of the solvent can be explained in terms of density effects and a negative volume of mixing. The subtle density changes (contraction) which give rise to demixing occur under specific conditions ( see chapter 2) , and are due to a complex balance between attractive and repulsive terms in the equations of state. The main point is that the demixing is governed by the thermodynamics of the interactions 266

CHAPTER 6. CONCLUSION

267

between the solvent and polymer molecules in terms of the segments, and not by the chain dimension. A comparison is also made between the LCSTs seen in polymer solutions and the LCSTs seen in mixtures of associating compounds. In the latter case, the liquid-liquid phase separation is due to unfavorable dispersive attractions between unlike species, and the LCST behaviour is explained by the presence of an associative attraction (of hydrogen bonding type) between unlike species which is dominant at low temperatures. The nature of the fluid phase separation in colloid-polymer systems is examined in a third chapter, using the Wertheim TPT1 description. Although the common approach to treat colloid-polymer systems is to used scaling theories which involves a knowledge of the chain dimension, we show here that it is possible to treat both components at the “microscopic” level. The colloid particles are represented by a large hard sphere and the polymer by a chain of tangent hard-spherical segments of diameter much smaller than the colloid particles. Contrary to the polymer solutions examined in chapter 2, the colloid-polymer system is treated as athermal (no attractions). The main conclusion is that the incorporation of repulsive interactions between polymer-polymer segments leads to significant differences with the theoretical predictions for ideal chains (which is the common Asakura-Oosawa model): the critical density for fluid-phase separation tends to a finite value as the chain length is increased if these interactions are taken into account, while the critical density tends to zero for ideal chains. This result is in agreement with recent studies which require information about the chain dimension. [162,249]. We have also calculated the properties of mixing of this system. As the system is athermal, the term “entropy driven” phase separation is often used in the literature to describe the demixing in the colloid-polymer fluid. One should, however, apply caution in the use of this term as it is found that an unfavourable enthalpy of mixing ∆Hm = P ∆V (due to the difference in excluded volume between the polymer and colloid particles) is responsible for the fluid-phase separation. The fluid phase separation in colloid-polymer systems is thus due to a completely different mechanism than in polymer solutions where attractive interactions are central. In future work it would be interesting to study the effect of incorporating attractive interactions in the description of the fluid-phase behaviour in colloid-polymers systems, and discuss the limits of the various regions of fluid-fluid immiscibility. A proper treatment of the fluid-solid phase transition is also necessary at this stage.

CHAPTER 6. CONCLUSION

268

Polymer are naturally polydisperse in terms of molecular weight, and polydispersity may affect a lot the phase behaviour. We present three different methods to calculate phase equilibria in polydisperse polymer solutions with a SAFT-type equation of state. The first method is based on Jog and Chapman [318] work and can be applied to discrete chain length distributions. The second method is an extension of the first one to calculate cloud and shadow curves with continuous distributions. The third method, developed by Sollich and co workers [320â&#x20AC;&#x201C;322], can be applied to continuous distribution and is compared with the two first methods. We discuss the effect of polydispersity on the liquid-liquid immiscibility. In agreement with previous studies, we found that polydispersity enhances the phase separation, and that the shape of the distribution has a big effect on the phase equilibria. We then compare the results obtained with discrete and continuous distributions, and we show that a simple ternary system (solvent + polymer (1)) exhibits very similar cloud and shadow curves as these obtained with Schulz-Flory distributions. We are planning to calculate fluid phase equilibria in polydisperse polymer-colloid systems. Finally, the vapour-liquid and liquid-liquid fluid phase equilibria of HDPE and LDPE polyethylene as very long n-alkane molecules has been modelled with the more sophisticated (and realistic) SAFT-VR description, without taking into account any branching. The molecular parameters of the pure polymer are extrapolated from those of the first members of the n-alkane series, with some correlations as a function of molecular weight. The SAFT-VR approach can qualitatively predict the effects of the various parameters (polymer contribution, nature and concentration of the solvent, temperature and pressure) on both vapour-liquid and liquid-liquid equilibria very successfully, without any temperature dependent cross-interaction parameters. Good predictions of the liquid-liquid regions (cloud curves) and LCSTs are obtained for polyethylene and n-alkane systems. The SAFT-VR approach can also be used to accurately predict the absorption of gases in amorphous polyethylene, which is essentially treated as vapour-liquid equilibria A model based on Floryâ&#x20AC;&#x2122;s work to predict the crystallinity of polyethylene is also developed. By combining this model with the SAFT-VR equation of state, one is able to predict the absorption of olefins in semi-crystalline polyethylene samples. In future work we are planning to extend our approach to other polymer systems which involves more complicated molecular interactions, such as polyethylene oxide - water solutions. We also intend to improve the theoretical approach by using the

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dimer version of the SAFT equation and extend the Wertheimâ&#x20AC;&#x2122;s association theory to higher order, in order to take into account branching.

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