PhD thesis by dr Marko Samec by Marko Samec

Kompleksnost porozne strukture gradbenih materialov

The Complexity of Porous Structure of Building Materials

doktorska disertacija

PhD thesis

Marko Samec

mentor: prof. dr. Dean Korošak

co-‐mentor: prof. dr. Sacha J. Mooney

Hvala... |Thank you.. dr. Dean Korošak – Za mentorstvo v najboljšem pomenu besede. Za potrpežljivost, dobre nasvete, vsestransko pomoč in prijateljstvo v času doktorskega študija. Za predstavitev sveta znanosti, raziskovanja in širjenje znanj v tujini. Vse to so neprecenljive izkušnje, ki so me oblikovala tako na osebnem kot profesionalnem nivoju. dr. Danijel Rebolj, dr. Bruno Cvikl in dr. Borut Zalar – V prvem letu študija ste name naredili ogromen vtis. Če ne bi bilo vas, se skoraj zagotovo ne bi podal na akademsko pot z željo, da bom nekoč vsaj delno tako dober učitelj in akademik kot je vsak izmed vas, vsak na svoj način. Radmilo Perunović – RSP – Za prijateljstvo, širjenje obzorij, ustvarjalnosti in duha. Zelo mi je žal, da nisi dočakal tega dne. Kljub temu pa bo del tebe za vedno ostal v naših srcih in tvoja neprecenljiva zapuščina ne bo nikoli pozabljena. mami in očetu – Za brezpogojno ljubezen, podporo in razumevanje. Za nudenje življenjskih pogojev, v katerih sem se lahko brez distrakcij osredotočal na študij. Za zgled, kaj je najpomembnejše za srečo. Urški – Za uresničitev sreče.Za ljubezen. Za podporo pri še tako nemogočih idejah. Da sprejemaš in ceniš to, kar sem. Javni agenciji za raziskovalno dejavnost Republike Slovenije – Za financiranje doktorskega študija in raziskovalnega usposabljanja v okviru programa mladih raziskovalcev. Javni sklad Republike Slovenije za razvoj kadrov in štipendije – Za sofinanciranje raziskovalnega usposabljanja v Veliki Britaniji. dr. Sacha Mooney – For all the support, guidance and warm welcome during my stays in Nottingham. colleagues at University of Nottingham – For accepting me into their informal environment and thus making me feel less homesick. For the help with X-‐CT image analysis and visualizations (Thanks, Craig Sturrock!).

Izjava

Izjavljam, da je doktorska disertacija rezultat mojega raziskovalnega dela.

Statement

I declare this phd thesis is the result of my research work.

Marko Samec

Kazalo | Contents povzetek v slovenskem jeziku | abstract in slovene 1 2

Uvod ............................................................................................................................................................. I Teorija in model .......................................................................................................................................... III 2.1 Kompleksne mreže ............................................................................................................................. III 2.1.1 Teorija kompleksnih mrež ......................................................................................................... VI 2.1.2 Temeljni koncepti teorije kompleksnih mrež ........................................................................... VII 2.2 Opis porozne strukture gradbenih materialov z uporabo kompleksnih mrež .................................... IX 2.3 Transport v poroznih snoveh ............................................................................................................. XII 2.3.1 Osnove ionske prevodnosti ....................................................................................................... XII 2.3.2 Gibanje delca v kompleksni porozni snovi .............................................................................. XIV 3 Eksperimenti ........................................................................................................................................... XVII 3.1 Dielektrična spektroskopija ........................................................................................................... XVIII 3.2 Rentgenska računalniška tomografija ............................................................................................ XXV 4 Diskusija ................................................................................................................................................... XXX 5 Zaključki .............................................................................................................................................. XXXVII PhD thesis | doktorska disertacija 1 2

Introduction ................................................................................................................................................. 1 Theory and model ........................................................................................................................................ 3 2.1 Complex networks ............................................................................................................................... 3 2.1.1 Background and history of network science ............................................................................... 5 2.1.2 Introduction to complex network theory ................................................................................... 8 2.1.3 Fundamental concepts in complex network theory ................................................................. 10 2.2 Using complex network to describe porous structure of building materials .................................... 14 2.3 Slow kinetics in porous media ........................................................................................................... 18 2.3.1 Basics of ion conduction ........................................................................................................... 19 2.3.2 Particle in complex porous matter ............................................................................................ 23 3 Experiments ............................................................................................................................................... 31 3.1 Specimens ......................................................................................................................................... 32 3.2 Dielectric spectroscopy ..................................................................................................................... 34 3.2.1 Measuring cell ........................................................................................................................... 45 3.2.2 Measuring device ...................................................................................................................... 46 3.2.3 Measurements .......................................................................................................................... 47 3.3 X-‐ray computed tomography ............................................................................................................ 48 3.3.1 How X-‐ray inspection system works / X-‐ray production ........................................................... 49 3.3.2 Principles of CT sample scanning .............................................................................................. 53 3.3.3 Tomography system and image acquiring ................................................................................ 56 3.3.4 Image analysis ........................................................................................................................... 56 3.4 Data analysis ..................................................................................................................................... 59 3.4.1 Analysis of data acquired from dielectric spectroscopy ........................................................... 59 3.4.2 Analysis of data acquired from X-‐ray CT ................................................................................... 70 4 Discussion .................................................................................................................................................. 78 5 Conclusions ................................................................................................................................................ 99 6 Appendix .................................................................................................................................................. 104 7 Bibliography ............................................................................................................................................. 113 Strokovni življenjepis ....................................................................................................................................... 124 CV ..................................................................................................................................................................... 126

DOKTORSKA Kompleksnost porozne strukture DISERTACIJA gradbenih materialov

povzetek v slovenskem jeziku

Marko Samec

mentor: prof. dr. Dean Korošak

somentor: prof. dr. Sacha J. Mooney

1 Uvod V času podnebnih sprememb in nujnosti zmanjševanja izpustov toplogrednih plinov so naravni gradbeni materiali, predvsem gline z različnimi primesmi, nizkoogljična alternativa pri izbiri gradiva in so lokalno dostopni. Za zagotavljanje ustreznega bioklimatskega bivanjskega okolja je nujno potrebno poznavanje in razumevanje lastnosti ovoja stavbe iz naravnih gradbenih materialov, še posebej povezanosti strukture ter transportnih lastnosti. Gradbeni materiali so večinoma heterogene porozne snovi, transport snovi (denimo vode in kontaminantov) pa je difuzijski. Zanimivo je, da doslej vpliv sprememb porozne strukture pri naravnih gradbenih materialih na transportne lastnosti ni bil deležen širše pozornosti. V tem delu je predstavljen nov pristop za raziskavo povezanosti med strukturo poroznih snovi in dinamiko transportnih pojavov, ki temelji na povezani uporabi kompleksnih mrež za opis strukture povezanosti por in frakcijskega diferencialnega računa za opis mikroskopske dinamike v takih kompleksnih sistemih. Teza izhaja iz aktualnih spoznanj o možnostih opisa porozne strukture

2001) z ustrezno kvantifikacijo povezanosti na eni strani ter prav tako aktualnih ugotovitvah o

Poglavje: Uvod

naravnih gradbenih materialov (zemljin stabiliziranih s cementom) kot kompleksne mreže por (Blunt,

anomalni dinamiki transporta snovi v poroznih snoveh na drugi (Metzler & Klafter, 2004). Uporabljena je bila kombinacija teoretičnih in eksperimentalnih metod. Da bi razumeli medsebojno povezanost transporta in strukture, smo strukturo vzorcev raziskovali s pomočjo rentgenske računalniške tomografije, ki omogoča 3D analizo porozne strukture. Z analizo binarnih slik tomografskih prerezov vzorcev smo postavili model mreže povezanosti posameznih por, razvrstili vzorce glede na kompleksnost porozne strukture in raziskovali, kakšen vpliv ima topologija na transportne lastnosti. Z metodo nizkofrekvenčne dielektrične spektroskopije smo ugotavljali, kakšne so električne lastnosti vzorcev in izpeljali splošni model, ki opisuje dinamiko ionov v porozni snovi. Opravljeno delo temelji na predhodnih raziskavah na obeh področjih – strukture in dinamike v porozni snovi – v sklopu raziskovalnih skupin mentorja prof. dr. Deana Korošaka in somentorja prof. dr. Sache J. Mooneyja. Del raziskav je bil opravljen na vrhunski poskusni napravi za 3D rentgensko računalniško tomografijo na Oddelku za okoljske znanosti, Univerza v Nottinghamu, Velika Britanija. Eksperimente dielektrične spektroskopije smo opravili na Katedri za aplikativno fiziko, Fakulteta za Gradbeništvo, Univerza v Mariboru, Slovenija. Celotna doktorska disertacija je napisana v angleškem jeziku, ta del v slovenskem jeziku pa je povzetek bistvenih delov disertacije. Povzetek v slovenskem jeziku (kot tudi celotna disertacija v angleškem jeziku), je sestavljen iz petih poglavij, ki se delijo na podpoglavja. V tem uvodu so razloženi temeljni koncepti kompleksnih mrež in dinamike v poroznih snoveh. V drugem poglavju so opisane eksperimentalne metode in pridobljeni rezultati. V tretjem poglavju sta predstavljena

Poglavje: Uvod

računski model in teorija, na kateri model temelji. Četrto poglavje vsebuje diskusijo o rezultatih in

ključnih ugotovitvah. V petem poglavju so povzeti glavni zaključki in ugotovitve.

2 Teorija in model Porozna snov sestoji iz ogrodja ali skeleta sestavljenega iz trdnine, in praznin ali por, običajno napolnjenih s tekočino (kapljevino ali plinom). Porozna snov poseduje določene lastnosti (na primer, poroznost, natezno in tlačno trdnost, električno prevodnost), določitev le-‐teh pa je kompleksna naloga. Mnogo naravnih (zemljina, kamnine, les, kosti, itd.) in umetnih (cement, keraka, itd.) snovi je poroznih, in če jih kot obravnavamo kot takšne, lahko z uporabo načinov in metod, opisanih v nadaljevanju, racionaliziramo mnogo njihovih pomembnih lastnosti.

2.1 Kompleksne mreže Študij kompleksnih mrež je desetletje staro raziskovalno področje, katerega uporabnost je našla pot

sociologijo, ekologijo, epidemiologijo in nevroznanostjo (Watts & Strogatz, 1998), (Albert & Barabasi, 2002), (Barabasi, 2009), (Zimmer, 2010). Osnove študija kompleksnih mrež izhajajo iz veje matematike, imenovane teorija grafov, ki preučuje lastnosti povezav v mrežni strukturi. Eden prvih

Poglavje: Teorija in model

v mnogih znanstvenih disciplinah, vključno z računalništvom, fiziko osnovnih delcev, ekonomijo,

III

zapisov s tega področja je Eulerjev članek iz leta 1741 o sedmih königsberških mostovih (Euler, 1741).

Slika 2-‐1: Štirje deli mesta Königsberg so bili povezani s sedmimi mostovi. Meščani so skušali najti rešitev, kako se sprehoditi po mestu tako, da bi vsak most prečkali samo enkrat. Euler je dokazal, problem nima rešitve. (Wikipedia), (Žerovnik, 2005)

Pomemben mejnik v razvoju kompleksnih mrež se je zgodil sredi prejšnjega stoletja, ko so Paul Erdős in Alfred Rényi (Erdős & Rényi, 1959) ter istočasno, a neodvisno, Edgar Gilbert (Gilbert, 1959) predstavili model za generiranje naključnih grafov1 (angl. random graph). V zadnju času se znanosti o mrežah osredotočajo na razvoj tehnik in modelov za opis različnih mrežnih topologij (Slika 2-‐2). Pozornost se vedno bolj preusmerja iz analize malih grafov in lastnosti posameznih vozlišč na študij statističnih lastnosti obsežnih grafov (Newman, 2003) in prostorskih mrež (Barthélemy, 2010). Zbiranje in analiza podatkov vedno večjih grafov je mogoča zahvaljujoč

Poglavje: Teorija in model

razvoju računalnikov in komunikacijskih omrežij.

Naključni graf dobimo, če nizu vozlišč naključno dodajamo povezave med njimi.

c) b)

plen med različnimi živalskimi vrstami v sladkovodnem jezeru (Martinez, 1991). (c) Mreža sodelovanja med znanstveniki na zasebnem raziskovalnem inštitutu(Girvan & Newman, 2002).

Poglavje: Teorija in model

Slika 2-‐2: Primeri različnih mrež. (a) Vizualizacija različnih poti skozi del spleta(Opte Project). (b) Mreža interakcij plenilec-‐

2.1.1

Teorija kompleksnih mrež

Kompleksne sisteme, ki so sestavljeni iz številnih povezanih enot, lahko lažje razumemo ter jih analiziramo, če jih predstavimo kot omrežja. Spoznanja zadnjih let kažejo, da topologije mnogih realnih mrež ni mogoče dobro opisati s klasično teorijo grafov. Iskanje novih topologij je vodilo k aktivnemu področju kompleksnih mrež in razvoju novih pristopov ter teorij (Havlin, 2008). Mreža ali graf je v matematiki struktura, ki predstavlja množico objektov, ki so med seboj povezani z vezmi. Običajno se graf v enostavni matematični abstrakciji prikaže kot množica točk (vozlišč); vezi, ki povezujejo določene točke, pa se imenujejo povezave. Vendar obstaja veliko primerov, ko so mreže bolj kompleksne. Kompleksna mreža je mreža z netrivialnimi topološkimi značilnostmi. Vozlišča in povezave lahko imajo različne lastnosti. Povezave so lahko usmerjene ali neusmerjene. Vozlišča in povezave so lahko neenakovredne in imajo imajo različne uteži (Slika 2-‐3). (a)

(b)

(c)

Slika 2-‐3: Primeri različnih vrst grafov s šestimi vozlišči in sedmimi povezavami. (a) Enostaven graf z enakimi vozlišči in povezavami. (b) Utežen graf z različnimi utežmi vozlišč in povezav. (c) Usmerjen graf, kjer ima vsaka povezava določeno smer. (Newman, 2003)

V zadnji letih sta bila največ zanimanja deležna dva tipa kompleksnih mrež: mreže majhnih svetov

Poglavje: Teorija in model

(angl. small-‐world networks) in skalno neodvisne mreže (angl. scale-‐free networks). Za vsak tip

veljajo določene lastnosti: kratke razdalje2 in grupiranje za prvi tip ter sledenje porazdelitve stopenj povezav potenčnemu zakonu (angl. power law) za drugi tip. Preučevanje skalno neodvisnih mrež je v zadnjem času še posebej aktualno, saj je dokazano, da je mnogo realnih mrež ravno takšnega tipa, 2

Razdalja med dvema vozliščema je najkrajša pot med njima.

kot na primer, svetovni splet, internet, mreža znanstvenih citatov in socialnih omrežij (Barabasi, 2009).

2.1.2

Temeljni koncepti teorije kompleksnih mrež

Neusmerjen in neutežen graf z N vozlišči lahko opišemo z matriko N x N, kjer je nediagonalni člen aij povezava med vozliščem i in j (člen ima vrednost 1, če sta vozlišči povezani, sicer je vrednost 0), diagonalni člen aii pa je je število zank na vozlišču i. Stopnja vozlišča ki (angl. node degree) je število povezav, ki jih ima vozlišče i s sosednjimi vozlišči: ! !!" .

!! =

(1)

Porazdelitev stopenj P(k) (angl. degree distribution) je definirana kot verjetnost, da ima naključno izbrano vozlišče stopnjo k. Porazdelitev stopenj vozlišč se pogosto uporablja za kategorizacijo topološke strukture mrež. Povprečna stopnja vozlišča je

! =

! ! ! ! ! .

(2)

Veliko mrež, ki jih najdemo v resničnem svetu, posedujejo značilno organizacijo, ki jo opiše potenčna odvisnost porazdelitev stopenj (Slika 2-‐4): !(!)~! !! ,

(3)

kjer je ! konstanta v mejah 2 < ! < 3 (Boccaletti et al., 2006). Takšne mreže imenujemo skalno

Poglavje: Teorija in model

neodvisne mreže(Barabási & Albert, 1999).

VII

porazdelitev stopenj P(k)

veliko vozlišč z majhnim številom povezav

nekaj vozlišč z velikim številom povezav

0 0

število povezav k

Slika 2-‐4: Porazdelitve stopenj povezav po potenčnem zakonu je značilna lastnost skalno neodvisnih mrež.

Določeni primeri realnih mrež imajo širok rep porazdelitve stopenj; le-‐te je zaradi velikega šuma težje analizirati. V tem primeru podatke stopenj predstavimo s kumulativno porazdelitvijo stopenj

! ! ! ,

! ! =

(4)

! ! !!

ki predstavlja verjetnost, da je stopnja večja ali enaka k. V številnih sistemih v realnem svetu je verjetnost, da je vozlišče s stopnjo k povezano z vozliščem s stopnjo k’, odvisno od stopnje k; takšna mreža je korelacijska (angl. correlated). Korelacijske lastnosti mreže so izražene s pogojno verjetnostjo ! (!′|!), da obstaja povezava med vozliščema s stopnjama k in k’. Izračun povprečne stopnje sosedov vozilišča s stopnjo k

! ! ! ! ! |!

!!! =

(5)

Poglavje: Teorija in model

uporabimo za klasifikacijo mrež glede na asortativnost (angl. assortativity) 3 . Če !!! (!) narašča

VIII

pomeni, da se vozlišča z visoko stopnjo v glavnem povezujejo z vozlišči, ki imajo prav tako visoko stopnjo. Takšna mreža je asortativna; če !!! (!) pada, je mreža neasortativna, saj so vozlišča z visoko stopnjo povezana predvsem z vozlišči, ki imajo nizko stopnjo (Error! Reference source not found.). 3

Asortativnost opisuje ali se med seboj v glavnem povezujejo podobna ali različna vozlišča glede na njihovo stopnjo.

2.2 Opis porozne strukture gradbenih materialov z uporabo kompleksnih mrež Ideja za uporabo teorije mrež za opis porozne arhitekture gradbenih materialov je zagotoviti nov način razumevanja kompleksnosti teh snovi skozi preučevanje organizacije velikega obsega porozne strukture. Predstavljena sta dva nedavno razvita mrežna modela organizacije por v zemljinah (Mooney & Korošak, 2009), (Santiago et al., 2008), (Cárdenas et al., 2010). Izkaže se namreč, da je topologija porozne arhitekture v zemljinah in drugih gradbenih materialih (kot je cementna pasta) podobna. Obe metodi pokažeta skalno neodvisno topologijo zemljin s potenčno odvisno porazdelitvijo stopenj, vsaka s svojim načinom konstrukcije kompleksne mreže por. Vzemimo niz N por, ki predstavljajo vozlišča v mreži. Vozlišča so središča por in povezave med njimi obstajajo v odvisnosti od verjetnosti, ki zavisi od lastnosti mreže – lege in velikosti por. Vozlišča so porazdeljena v D-‐dimenzionalnem prostoru z gostoto !(!). Vsaka pora ima dodeljene stanje s (velikost pore), ki opisuje lastnosti vozlišča. Stanja vozlišč so porazdeljena glede na verjetnost P(s) (porazdelitev velikosti por). Mejni mrežni model (angl. threshold network model) (Mooney & Korošak, 2009) narekuje, da sta dva vozlišča s stanji si, sj (pori i in j z določenima velikostima) povezani, če !! !! ! > ! , !!"

(6)

kjer mejna vrednost ! > 0 kontrolira število povezav v mreži (Slika 2-‐5); če ! = 0 dobimo polni graf,

kakšen pomen ima razdalja med vozlišči. S to metodo dobimo statično mrežo z N vozlišči povezanimi z M povezavami s povprečno stopnjo vozlišča ! =

!! !

Poglavje: Teorija in model

! kjer obstajajo vse možne povezave. !!" = !! − !! je razdalja med porama in parameter m opisuje,

Slika 2-‐5: Razvoj mreže z uporabo statičnega mrežnega modela. Ko manjšamo mejni parameter ε, število povezav v mreži raste (iz leve proti desni). Sestavljanje mreže je postavljeno na dvodimenzionalno tomografsko sliko zemljine za lažjo predstavo. (Mooney & Korošak, 2009)

V modelu preferenčne povezanosti (angl. preferential attachment model) (Santiago et al., 2008), (Cárdenas et al., 2010) kompleksna mreža por raste z zaporednim dodajanjem in povezovanjem novih vozlišč v mrežo. Verjetnost, da je dodano vozlišče j (naključno dodano na točki !! ) s stanjem sj , dobljeno iz porazdelitve P(s), povezano, je sorazmerno s produktom stopnje obstoječega vozlišča ki in afinitetne funkcije (angl. affinity function) σ(i,j) !!" ~!! ! !, ! .

(7)

Afinitetna funkcija zavisi od stanja novega vozlišča in razdalje do obstoječega vozlišča:

! !, ! =

!!!

! , !!"

(8)

kjer je b parameter, ki opiše, kakšen pomen ima velikost pore na proces dodajanja. Ko je doseženo željeno število vozlišč N, mreža preneha rasti. V obeh opisanih modelih, statičnem mrežnem modelu (SN, anlg. static network model) in rastočem

Poglavje: Teorija in model

mrežnem modelu (EN, angl. evolving network model), lahko heterogenost mreže prilagajamo s spreminjanjem parametrov b in m. Oba modela kažeta podobno topologijo porozne strukture v

zemljinah s potenčno odvisnostjo porazdelitve ! (!)~! !! , ki kaže na skalno neodvisne topološke značilnosti.

Slika 2-‐6: Mrežni predstavitvi porozne strukture zemljin generirani z SN modelom (zgoraj) in EN modelom (spodaj) z številom povezav in m ≈ 1. Površina vozlišča ustreza stopnji vozlišča, torej številu povezav. Obe mreži kažeta podobno topologijo, kjer bolj povezana vozlišča (večje točke) med seboj načeloma niso direktno povezana, ampak se povezujejo preko vozlišč z nižjo stopnjo (manjše točke). Takšna struktura nakazuje, da sta obe mreži korelacijski (porazdelitev stopenj povezanosti je podrobneje razdelana v Diskusiji).

Poglavje: Teorija in model

enakimi vhodnimi podatki, pridobljenimi iz tomografskega posnetka na sliki Slika 2-‐5 in enakimi parametri modela: enakim

2.3 Transport v poroznih snoveh Stanje trdnih materialov še zdaleč ni statično. Zaradi toplotne energije atomi vibrirajo, med seboj trkajo in izmenjujejo energijo s sosednjimi atomi in okolico. Če atom pridobi dovolj energije, lahko migrira. Ko nosilci naboja 4 migrirajo zaradi gradienta kemijskega potenciala 5 govorimo o difuziji, kadar pa je migracija posledica gradienta električnega potenciala, pa gre za električno prevodnost. V poroznih gradbenih materialih, kot sta cement in glina, so nosilci naboja ioni v vodi, ki je prisotna v porah. V nasprotju z električno prevodnostjo, ko se naboji gibajo na velikih razdaljah, je dielektrični odziv posledica gibanja kratkega dosega pod vplivom zunanjega električnega polja. Zunanje električno polje povzroči, da se pozitivni in negativni naboji ločijo, polarizirajo. V zadnjem desetletju je bila teorija poroznih snovi deležna pomembnega napredka glede razumevanja temeljnih principov in razvojem modelov na različnih področjih tehnike in biomehanike, vključno z mehaniko tal (de Boer, 2000), znanostjo o materialih (Sánchez et al., 2008), okoljsko mehaniko (Kaczmarek et al., 1997) (Gaucher & Blanc, 2006), gradbeno fiziko (de Boer, 2005), dentalno medicino (Leskovec et al., 2005) in drugih področjih.

2.3.1

Osnove ionske prevodnosti

Gibanje ionov v nehomogenih trdnih snoveh je različno od prevajanja elektronov v kristalnih snoveh. Gibanje ionov lahko opišemo z ionskimi skoki med mesti, ločenimi z energijskimi ovirami (Dyre et al., 2009). Ion lahko migrira, če ima zadostno energijo, da premaga oviro (Slika 2-‐7). Če je ΔE energijska

Poglavje: Teorija in model

ovira, T temperatura in kB Boltzmannova konstanta, je verjetnost, da bo preskok iona uspešen,

XII

približna exp(−!!/!! !) . Po kratkem času ioni premagajo samo manjše energijske ovire. S povečanjem časovnega intervala, lahko ioni prepotujejo daljše razdalje, saj premagajo več ovir. 4

Nosilec naboja je prosti (mobilni in nevezan) delec, ki nosi električni naboj. V kovinah so nosilci naboja elektroni; v ionskih razstopinah, na primer slana voda, so nosilci naboja ioni (kationi+ in anioni-‐). 5 Kemijski potencial je termodinamska spremenljivka, ki meri, za koliko se poveča prosta entalpija homogeno porazdeljene snovi, če termodinamskemu sistemu dodamo en mol te snovi. (Gibbs, 1876), (Baierlein, 2001)

Časovno kratkoročno dinamiko karakterizirajo skoki »naprej-‐nazaj« med sosednjimi mesti, medtem ko časovno dolgoročno dinamiko karakterizira naključno gibanje (angl. random walk), transport ionov čez velike razdalje ki ga lahko makroskopsko opišemo z difuzijsko dinamiko (Dyre, 1988).

Slika 2-‐7: Enodimenzijski prikaz ionskih skokov v nehomogeni trdni snovi. Puščice nakazujejo poskuse ionskih preskokov, ki so večinoma neuspešni in v tem priemeru ioni ostanejo v minimumu, ki so ga skušali zapustiti (Dyre et al., 2009).

Lastnosti difuzijskega gibanja ionov opisuje frekvenčno odvisna specifična prevodnost:

! ! =

!!! ! ! , !! !

(9)

kjer je q osnovni naboj, n gostota ionov, !! Boltzmannova konstanta, T temperatura in D(ω) difuzijski koeficient, ki opisuje lastnosti transporta delca v snovi in je podan z izrazom (Scher & Lax, 1973) ! 1 ! ! = − !! ! !!"# ! ! !" , 6 !

(10)

kjer je ! ! kvadratna vrednost odmika iona. Za normalno difuzijo je značlno, da je kvadrat pomika sorazmeren s časom ! ! ∝ !, medtem ko se anomalna difuzija zapiše kot ! ! ∝ ! ! , kjer ! ≠ 1. Za

prevodnosti mešanic gline in vode ! ∝ ! ! . Površinska difuzija ionov je subdifuzivna, sorazmerje časovno odvisnega difuzijskega koeficienta pa je !(!) ∝ !" !! . Opisano nakazuje, da so eksperimentalno pridobljene dielektrične lastnosti rezultat anomalne difuzije, ki jo karakterizira časovno odvisen difuzijski koeficient, kar je pogosto opaženo v kompleksnih sistemih (Campos et al.,

Poglavje: Teorija in model

! = 1 − ! dobimo iz zgornjih enačb izraz, ki velja v nizko frekvenčnem območju specifične

XIII

2004). Sisteme, ki kažejo počasno anomalno difuzijo, ali subdifuzijo, je možno opisati z uporabo frakcijskih diferencialnih enačb6 (Sokolov et al., 2002), (Tarasov, 2008), (Tarasov, 2008). Frakcijski Liouvillov integral (Oldham & Spanier, 1974), (Sokolov et al., 2002) je definiran kot

!!! ! ! =

1 Γ α

! !′ !"′ . !!! !! ! − !′

(11)

1 ℱ! ! . !" !

(12)

Po Fourierjevi transformaciji ℱ dobimo

ℱ!!! ! ! =

S frakcijsko difuzijsko enačbo lahko opišemo kompleksne sisteme z anomalnim obnašanjem približno enako kot enostavnejše sisteme. Naključno gibanje in difuzija služita kot vmesnik med kinematiko na eni strani in frakcijskim integralom in odvodom na drugi. Z naključnim gibanjem lahko preučujemo tudi kompleksne mreže (Gallos, 2004). Združitev teorije kompleksnih mrež in kinematike v porozni snovi tako vodi do boljšega razumevanja procesov, ki se vršijo v porozni strukturi materialov.

2.3.2

Gibanje delca v kompleksni porozni snovi

Enačbo, ki opisuje gibanje iona v zunanjem električnem polju ! ! , zapišemo:

!!(!) = −!!! ! − !!!! ! ! + !" ! , !"

(13)

kjer je ! masa naboja, ! koeficient upora, !! karakteristična frekvenca in ! naboj. Člen na levi strani enačbe je pospešek, prvi člen na desni strani je sila upora zaradi viskoznosti, drugi člen opisuje

Poglavje: Teorija in model

harmonsko silo, tretji pa silo zaradi zunanjega polja ! .

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Frakcijska dinamika je področje fizike, matematike in ekonomije, ki preučuje obnašanje objektov in sistemov, ki so opisani s pomočjo integracije ali odvajanja frakcijskega reda z metodami frakcijskega računa.

Pokazalo se je (Metzler & Klafter, 2000), da lahko stohastično gibanje delcev v kompleksnem sistemu opišemo s frakcijsko Langevinovo enačbo, kar vodi do zveze med odmikom in hitrostjo delca, izražene z Weylovim frakcijskim operatorjem !!!!!!! !" !! = !!!! ! . !" !!! !! !

(14)

!! je čakalni čas, ki opiše dogodek ujetja iona na površini snovi, !! pa je čas gibanja iona v pori. Po Fourierjevi transformaciji zgornje enačbe gibanja in ob upoštevanju izraza za gostoto električnega toka ! ! = ! ! !(!) dobimo splošni model za frekvenčno odvisno električno prevodnost:

! ! =

!! , 1 + !"!! + !"!! !!

(15)

kjer sta !! in !! izražena s karakterističnima časoma !! in !! in !! = !!! !!! !! . Model smo preverili s preliminarnimi meritvami na vzorcih vlažne gline (Tabela 2-‐1, Slika 2-‐8, Slika 2-‐9). Tabela 2-‐1: Vrednosti parametrov uporabljenih v izračunu realnega in imaginarnega dela prevodnosti prikazane na spodnjih dveh slikah.

parametri σ0 [A/Vm]

vlažnost gline 36 %

56 %

0,43

0,725 -‐8

-‐8

τ1 [s]

0,9 * 10

1,45 * 10

τ2 [s]

3,2 * 10

-‐4

2,1 * 10

0,67

-‐4

Poglavje: Teorija in model

Slika 2-‐8: Izmerjen (točke) in izračunan (krivulji) realni del frekvenčno odvisne električne prevodnosti za vzorec kaolinita 36 % (spodnja krivulja) in 56 % (zgornja krivulja) vlažnosti (Samec et al., 2007).

Slika 2-‐9: Izmerjen (točke) in izračunan (krivulji) imaginarni del frekvenčno odvisne električne prevodnosti za vzorec kaolinita 36 % (spodnja krivulja) in 56 % (zgornja krivulja) vlažnosti (Samec et al., 2007).

Poglavje: Teorija in model

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3 Eksperimenti Pri raziskavi povezave med kompleksnostjo porozne strukture in dinamiko ionov smo uporabili eksperimentalni metodi rentgensko računalniško tomografijo (X-‐CT, angl. X-‐ray computed tomography) in nizkofrekvenčno dielektrično spektroskopijo. Na ta način smo ugotavljali, kakšna je mikrostruktura poroznih gradbenih materialov in kakšen je njihov dielektrični odziv. Za vzorce gradbenih materialov smo uporabili cement in glino, zaradi širokega spektra uporabe obeh materialov v gradbeništvu. Fizikalne lastnosti obeh materialov so bile v zadnjem času deležne obširnih preiskav (Fossum, 2000), (Allen et al., 2007). Pri eksperimentih smo uporabljali štiri skupine vzorcev pri različnih vlažnostih: čista glina, čisti cement, glino in cement v plasteh ter glino, stabilizirano s cementom (seznam vseh uporabljenih vzorcev je prikazan v tabeli Tabela 3-‐1). Kot glino smo za vzorce uporabili kaolinit KGa-‐1b, pridobljen od The Clay Minerals Society, katerega glavna sestavina je mineral kaolinit, vsebuje pa tudi majhne količine kovin (Ca, K, Na, Mg). Meja

mešanjem kaolinita in destilirane vode.

Poglavje: Eksperimenti

plastičnosti vzorca je pri 25,9 % vlažnosti, meja židkosti pa pri 40,1 % vlažnosti. Zunanja specifična

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površina zrn vzorca je 10 m2/g, specifična teža pa 2,6. Vzorce mešanic gline in vode smo dobili z

Za cementne vzorce smo uporabili cement z oznako CEM 42.5 R po standardu EN 197-‐1 (2000). Oznaka pomeni, da gre za portlandski cement, ki vsebuje vsaj 95 % mineralov cementnega klinkerja, tlačne trdnosti 42,5 MPa po 28 dneh z visoko zgodnjo trdnostjo (R). Pri vzorcih stabilizirane gline smo zmešali suhi cement in glino (od tega 6 % masni delež cementa) ter dodali destilirano vodo (30 % od skupne mase cementa in gline). Tabela 3-‐1: Recepture vzorcev uporabljenih pri eksperimentih. no.

ime vzorca

vlažnost kaolinita

cement

kaolinit

cement

vsebnost vode

[%wt]

[w/c faktor]

[% wt]

[w/(c+k) faktor

clay_0

clay_25

clay_50

clay_75

clay_100

100

clay_300

300

cement_0.3

0,3

cement_0.4

0,4

cement_0.5

0,5

CemKaol01

0,4

3.1 Dielektrična spektroskopija Karakterizacija transporta v poroznih snoveh je netrivialna naloga, tako eksperimentalno, kot tudi teoretično (Lounev et al., 2002), (Sanabria & Miller, 2006), (Rotenberg, B. et al., 2005). Številne raziskave kažejo na anomalne lastnosti difuzijskega transporta v naravnih materialih, ki je posledica

Poglavje: Eksperimenti

heterogenosti snovi (Hunt, 2005). Po drugi strani so bili odkriti določeni anomalni pojavi transporta z

XVIII

analizo dinamike naključnega gibanja (angl. random walk) (Gallos, 2004) in prevodnosti istosmernega električnega toka (Lopez et al., 2005) na kompleksnih mrežah.

Pri preučevanju dinamike ionov v poroznih gradbenih materialih smo uporabili eksperimentalno metodo dielektrične spektroskopije. Dielektriče meritve temeljijo na merjenju napetosti in toka med parom elektrod, s čimer je mogoče ugotoviti prevodnost in kapacitivnost med elektrodama. Dielektrična spektroskopija (imenovana tudi impedančna spektroskopija) je metoda za merjenje dielektričnih lastnosti snovi v odvisnosti od frekvence. Temelji na interakciji zunanjega električnega polja z električnim dipolnim momentom vzorca (Volkov & Prokhorov, 2003), (Lambert, 2008). Dielektrične lastnosti se med materiali razlikujejo in so odvisne od temperature, frekvence električnega polja, vlage, kristalne strukture in drugih zunanjih faktorjev (Barsoum, 1997), (Hager & Domszy, 2004). Ko v kondenzator7 vstavimo dielektrik, napetost med ploščama ploščatega kondenzatorja pade (Slika 3-‐1). Kapaciteta kondenzatorja C se je zaradi dielektrika povečala na

! = !! !!

! = !! !! , !

(16)

kjer je !! influenčna konstanta z vrednostjo 8,85 x 10-‐12 F/m, A površina ploščina plošče in d razmik med ploščama kondenzatorja ter !! relativna dielektrična konstanta, ki je razmerje med dielektričnostjo8 in influenčno konstanto !! = ! !! . Ko dielektrik vstavimo v električno polje, se naboji polarizirajo (Slika 3-‐2). Električna polarizacija P je premo sorazmerna električnemu polju E : ! = !! !" ,

(17)

kjer je ! električna suspeptibilnost, ki jo lahko izračunamo neposredno iz relativne dielektrične

Kondenzator je element, sestavljen iz ploščatih elektrod, ki lahko shranjuje energijo v obliki električnega polja. Njegova fizikalna količina je kapacitivnost. 8 Dielektričnost je brezrazsežna količina, ki opisuje obnašanje dielektrika v električnem polju.

Poglavje: Eksperimenti

konstante ! = !! − 1 .

XIX

Slika 3-‐1: Kondenzator pred in po vstavitvi dielektrika (Breuer, 1993). Če prazni kondenzator priključimo na električno napetost, se bo v kondenzatorju shranil naboj. Ko med plošči kondenzatorja vstavimo dielektrik, se kapacitivnost kondenzatorja poveča.

(a) (b)

II.

(a) (b) Slika 3-‐2: Mehanizem elektronske in ionske polarizacije: I. Elektronska polarizacija nastane zaradi premika težišča

Poglavje: Eksperimenti

elektronskega oblaka glede na jedro atoma (a) v ravnotežnem stanju težišče elektronskega oblaka sovpada s težiščem jedra, (b) ob prisotnosti zunanjega električnega polja se težišči pozitivno in negativno nabitih delov razmakneta; II. Ionska

polarizacija je pomik pozitivnih in negativnih ionov proti negativni oz. pozitivni elektrodi (a) lega ionov v ravnovesju, (b) ob prisotnosti električnega polja težišči negativnih nabojev ne sovpada več s težiščem pozitivnih nabojev. (Barsoum, 1997)

Vpeljemo gostoto električnega polja, ki je sestavljena iz polarizacije in gostote električnega polja v vakuumu: !(!) = !! !(!) + !(!) .

(18)

V splošnem je odziv na zunanjo motnjo odvisen od časa, saj sprememba potrebuje nekaj časa, da pride do izraza. S Fourierjevo transformacijo9 enačbe gostote električnega polja dobimo: ! ! = ! ! ! ! ,

(19)

kjer je ! kotna frekvenca priključne napetosti. Frekvenčno odvisna dielektrična funkcija ! ! je kompleksna količina. Realni del ! ! je povezan kapacitivnim vedenjem, oziroma sposobnostjo materiala, da se polarizira zaradi zunanjega električnega polja, medtem ko je imaginarni del ! ! ′ povezan z disipacijo energije10. Dielektrično funkcijo zapišemo kot ! ! = ! ! ! − ! !! ! .

(20)

Ko dielektrik izpostavimo električnemu polju, naboji potrebujejo določen čas, da se polarizirajo; prav tako se dielektrik šele v določenem času povrne v začetno stanje, če električno polje izključimo. Pojav časovnega zamika dielektrične konstante se imenuje dielektrična relaksacija, ki jo je vpeljal Peter Debye leta 1913 (Williams, 1975). To relaksacijo pogosto opišemo z Debyjevo enačbo (Debye, 1913):

! ! = !! +

∆! , 1 + !"#

(21)

kjer je !! dielektričnost pri visokofrekvenčni limiti, ∆! = !! − !! , kjer je !! statična dielektričnost,

redkokdaj opazimo klasični Debyjev odziv. Spekter dielektričnosti za različne materiale opišemo s 9

Fourierjeva transformacija je operacija, ki funkcijo prestavi iz časovnega v frekvenčni prostor. Disipacija je pojav pri katerem energija iz telesa prehaja na okolico, navadno zaradi trenja ali turbulence. Energija se pretvori v toploto, zato se poviša temperatura sistema. Takšnemu sistemu pravimo disipativni sistem. 10

Poglavje: Eksperimenti

ko frekvenca limitira proti 0, in ! karakteristični relaksacijski čas snovi. Vendar pri eksperimentih

XXI

frakcijsko potenčno odvisnostjo od frekvence ! ∝ ! ! in jo za različne materiale enostavno prikažemo z dielektrično susceptibilnostjo ! ! = ! ! ! − ! !! ! , kar imenujemo zakon univerzalnega odziva (Jonscher, 1983), (Jonscher, 1999). Meritve smo opravljali na Fakulteti za gradbeništvo UM z impedančnim analizatorjem QuadTech 1920 Precision LCR meter, ki omogoča avtomatsko merjenje kapacitete C in prevodnosti G v frekvenčnem območju od 20 Hz do 1 MHz. Merilna celica je bil ploščati kondenzator v obliki valja, sestavljen iz dveh kovinskih elektrod s površino A = 1,89 cm2. Zgornja elektroda kondenzatorja je bila pomična, da smo celico lahko izpolnili z vzorcem. Razdaljo med elektrodama d smo izmerili, ko je bila celica napolnjena z vzorcem in v primeru cementnih vzorcev pred vsako meritvijo (zaradi krčenja cementa). Razdalja med elektrodama d se je spreminjala glede na količino vzorca v celici. Za pridobitev primerljivih rezultatov med različnimi vzorci, neodvisne od razporeditev elektrod (torej razdaljo d), smo najprej izračunali kapaciteto praznega kondenzatorja !! = !! ! ! in nato specifično prevodnost

!=!

!! , !!

(22)

Iz meritev smo določili dielektrično funkcijo, ki je sestavljena iz realnega

!! =

! , !!

(23)

! , !! !

(24)

in imaginarnega dela dielektrične funkcije

Poglavje: Eksperimenti

! !! =

XXII

kjer je C izmerjena kapaciteta, G izmerjena prevodnost, !! influenčna konstanta in ω kotna frekvenca priključne napetosti (! = 2!").

Analizo vzorcev smo prilagodili lastnostim, značilnim za določen material. Pri glini nas je zanimalo, kako se električne lastnosti spreminjajo v odvisnosti od vlažnosti vzorca, medtem ko nas je pri vzorcih, ki so vsebovali cement, zanimal električni odziv v različnih časih hidratacije. Podatke meritev cementnih vzorcev smo vzeli iz diplomske naloge Gorana Mandžuke (Mandžuka, 2007).

7,E-‐03 0% 25% 50%

6,E-‐03

75% 100% 300%

speciﬁčna prevodnost σ [S/m]

5,E-‐03

4,E-‐03

3,E-‐03

2,E-‐03

1,E-‐03

0,E+00 100

1000

10000

100000

1000000

frekvenca ν [Hz]

Slika 3-‐3 Frekvenčna odvisnost specifične prevodnosti vzorca kaolinita pri različnih vsebnostih vlage.

Poglavje: Eksperimenti

XXIII

1,2

0 1 4 8 15 1

0 4

speciﬁčna prevodnost σ [S/m]

0,8

0,6

0,4 1 0 4

0,2

15 30

Poglavje: Eksperimenti

XXIV

100

1000 frekvenca ν [Hz]

10000

100000

1000000

Slika 3-‐4: Frekvenčna odvisnost specifične prevodnosti cementne paste z w/c faktorjem 0,3 (vzorec cement_0.3) pri različnih hidratacijskih časih (v urah). Izmerjeni podatki so poznačeni s točkami; črte, ki točke povezujejo, so narisane zaradi boljšega pregleda.

3.2 Rentgenska računalniška tomografija Rentgenska računalniška tomografija (X-‐CT, angl. X-‐ray computed tomography) je nedestruktivna metoda za slikanje heterogenih snovi, ki nam omogoča 3D vizualizacijo geometrije sistema. Od odkritja v sedemdesetih letih prejšnjega stoletja za medicinske namene (Beckmann, 2006), je sledila uporaba tehnike X-‐CT v drugih znanostih in industriji, med drugim v pedologiji (Petrovic et al., 1981), znanosti o materialih (Baruchel et al., 2000) in živilski industriji (Lim & Barigou, 2004). Pri tomografskem slikanju iz različnih smeri usmerimo rentgenske žarke (ali žarke X, angl. x-‐rays) na opazovani vzorec in merimo njihovo absorbcijo. Absorbcija rentgenskih žarkov, oziroma njihov prehod skozi snov, je odvisna od gostote in debeline snovi. Ko oslabljeni rentgenski žarki zapustijo vzorec, padejo na detektor, ki zazna intenziteto oslabljenih žarkov (Slika 3-‐5). Nastane slika, na kateri so snovi iz gostejših materialov (npr. železo) svetlejši kot manj gosti (npr. voda, zrak) (Slika 3-‐6). Končna slika nastane s pretvorbo rentgenskih žarkov v vidno svetlobo. Računalnik po posebnem algoritmu na osnovi dobljenih podatkov ustvari sliko po plasteh.

Poglavje: Eksperimenti

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Slika 3-‐5: X-‐CT sistem je sestavljen iz izvora rentgenskih žarkov in detekorja. Med zajemanjem slik preiskovani vzorec rotira okoli svoje osi za polni krog v korakih manjših od 1°C. (phoenix|x-‐ray GE Sensing & Inspection Technologies, 2009)

merilna celica

pora

trdna snov

Slika 3-‐6: Dvodimenzionalna tomografska slika vzorca stabilizirane gline.

X-‐CT vzorcev smo opravili na School of Biosciences, University of Nottingham, Velika Britanija, z napravo Phoenix Nanotom 180NF (GE Sensing and Inspection Technologies) z energijo pospeševanja elektronov 100 kV in električnim tokom 90 µA. Med rotacijo vzorca za 360 stopinj je bilo zajetih 1400 kotnih projekcij. Slike projekcij so nastale kot povprečje treh slik s časom osvetlitve 500 ms. Ločljivost slik je bila 2000 x 2000 pikslov in prostorska ločljivost volumske enote (voksel) 8 µm x 8 µm x 8 µm. Niz slik je bil pripravljen s programskim orodjem Volume Graphics VGStudioMax V2.0(VolumeGraphics, 2010). Črno-‐bele slike nizov posameznih vzorcev smo obdelali s programsko opremo Imaje J V1.43u (Rasband, 2010), da smo lahko izolirali pore (). Gre za netrivialen postopek, ki zahteva ustrezno

Poglavje: Eksperimenti

subjektivno presojo pri analizi vsakega vzorca posebej, pri katerem z uporabo različnih orodij, filtrov

XXVI

in algoritmov iz preiskovanega vzorca izločimo trdne snovi in izoliramo pore. Iz analize tako pridobljenih binarnih slik pridobimo podatke o poroznosti, velikosti in obliki por (angl. circularity). Površine manjše od 5 pikslov so bile izločene iz analize.

tomografski posnetek

obrezan in umerjen posnetek

prilagojen kontrast

algoritem za določitev praga por

binarna slika

Slika 3-‐7: Postopek analize tomografskih prerezov vzorcev.

Poglavje: Eksperimenti

XXVII

Tabela 3-‐2: Analiza meritev vseh vzorcev. Iz pridobljenih podatkov po plasteh smo za vsak vzorec izračunali povprečne vrednosti za velikosti por, obseg, poroznost vzorca in oblikovni faktor (angl. circularity). vzorec

velikost por

obseg por

poroznost

oblikovni faktor

[mm ]

[mm]

[%]

cement 0 h

0,000946

0,130280

4,70

0,69937

cement 2 h

0,000945

0,123051

2,70

0,72173

cement 24 h

0,000923

0,127010

4,11

0,70707

clay 0 h

0,003331

0,223884

0,83

0,68927

clay 2 h

0,004022

0,247481

1,00

0,68024

clay 24 h

0,003497

0,233920

1,01

0,66495

SC 0 days

0,004285

0,241832

2,56

0,66450

SC 14 days

0,002455

0,191862

2,89

0,68797

SC 28 days

0,003884

0,252968

2,78

0,67895

Poroznost je lastnost, s katero označujemo prisotnost praznih prostorov v snovi. Definirana je kot razmerje med prostornino por in celotno prostornino vzorca. Na binarnih slikah si lahko poroznost predstavljamo kot delež črnih pikslov. Oblikovni faktor (angl. circularity) je merilo za kompaktnost in je definirana kot:

obl. f. = 4!

površina obseg !

(25)

Za popolni krog je vrednost 1; bolj kot je oblika podovgovata, bolj se vrednost približuje 0. Na spodnjih grafih je prikazano, kako se porazdelitev por spreminja z globino v odvisnosti časa

Poglavje: Eksperimenti

hidratacije.

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poroznost [%] 0

0 cement 0 h cement 2 h

0,2

cement 24 h

globina [mm]

0,4

0,6

0,8

1,2

Slika 3-‐8: Porazdelitev por po višini za cement pri različnih hidratacijskih časih.

poroznost [%]

5 4 3 2 1 0 2 čas hidratacije [h]

Slika 3-‐9: Spreminjanje povprečne poroznosti v času hidratacije.

Poglavje: Eksperimenti

XXIX

4 Diskusija Začnimo s predstavitvijo modela, s katerim lahko pojasnimo možen izvor porazdelitev potenčne odvisnosti čakalnih časov podanih z !! !(!)~ !!! !

(26)

in prikazom, kako so parametri Cole-‐Cole modela odvisni od lastnosti sistema. Cole-‐Cole model kompleksne specifične prevodnosti zapišemo:

! ! = !!" + !!! − !!"

!"!!! ! , 1 + !"!!! !

(27)

kjer je relaksacijski čas

!!! = !! (!! !! )!!/! = !!

!" ! sinh

!!! !"

!!/!

(28)

Poglavje: Diskusija

Iz zgornje enačbe lahko izrazimo ! parameter Cole-‐Cole modela:

XXX

!=−

2ln (!! !! ) . ! ln ( !! ) !!

(29)

Naključno izbran delec, ki se giblje med dvema točkama v porozni snovi, je na svoji poti deležen naključnih časovnih zamikov, ki so posledica interakcij med delcem in površino večjega delca snovi. Karakteristični čas zamikov ! (!) je odvisen od efektivne dolžine večjega delca L. L je spremenljivka s verjetnostjo porazdelitve !! (!), ki je odvisna od mikrostrukture porozne snovi. Obravnavamo model prekrivajočih sfer, pri katere dvofazni sistem sestoji iz enakih sfer z radijem R, in za verjetnost porazdelitve !! (!) izberemo porazdelitv dolžine tetiv (Torquado & Lu, 1993), tako da je !! (!)!" verjetnost, da je na segmentu dolžine med L in L+dL trdna snov: !! ! ∝ (1 − !)! ≡ ! !!" ,

(30)

kjer je ! poroznost. Karakteristični čas ! ! v naključni strukturi je (Bunde et al., 1986):

! ! ∝

1+! ! ≡ ! !!" , 1−!

(31)

kjer je 0 < ! < 1 particijski koeficient, topološki element, ki opiše kako potujoči delec odvaja od minimalne poti. V našem primeru, ko je ion deležen ujemanja in sprostitve ob površini por topološki element f izboljša verjetnost ujetja in zmanjša verjetnost sprostitve. Z uporabo zgornjih enačb lahko izračunamo porazdelitev čakalnih časov. Ker je ! ! = !! ! !"/!" velja: ! ! ~! ! !!!/! ,

(32)

kjer je ! /! = ln (1 − !)/ln ((1 − !)/(1 + !)). Če primerjamo z enačbo (26), vidimo, da je Cole-‐ Cole parameter ! : ln (1 − !) . 1−! ln 1+!

(33)

porozni snovi. Exponent ! je odvisen od poroznosti oziroma velikosti por sistema in ločenosti

Poglavje: Diskusija

XXXI

Pridobili smo funkcijsko odvisnost parametrov ! in !!! Cole-‐Cole modela na podlagi podrobnosti o

matrike. Relaksacijski čas !!! je odvisen od velikosti por, velikosti glinenih zrn, slanosti elektrolita v porah in medravninskega potenciala med nabitimi površinami minerala. Na naslednjih slikah je prikazana odvisnost Cole-‐Cole parametrov od parametrov snovi.

-‐3

Slika 4-‐1: Odvisnost Cole-‐Cole parametra ! od relaksacijskega časa !!! po enačbi (28). Slanost je nastavljena na !! = 10 M, medravninski potencial !! = 10 mV in velikost por != 3. (Samec et al., to be published)

Poglavje: Diskusija

XXXII

Slika 4-‐2: Cole-‐Cole parameter ! kot funkcija velikosti por !. Particijski koeficient je ! = 0,8. (Samec et al., to be published)

Pri eksperimentih pridobljene spektre specifične prevodnosti vzorcev lahko dobro opišemo z izrazom podanim v enačbi (15). Slika 4-‐3 kaže, kako se dielektrični spektri cementnih vzorcev ločijo na dve skupini glede na hidratacijski čas.

Slika 4-‐3: Normaliziran nizkofrekvenčni del specifične prevodnosti: prazni simboli – zgodnji časi hidratacije 0 h – 12 h, polni simboli – pozni časi hidratacije 30 h – 179 h. Črte prikazujejo izračun iz enačbe (15) za vrednosti α=0.82 za zgodnji in α=0,6 za pozni čas hidratacije.

V nasprotju s cementnimi vzorci, vzorci gline in cementa ne kažejo očitne ločitve med zgodnjim in

Slika 4-‐4: Normaliziran nizkofrekvenčni del specifične prevodnost vzorca gline in cementa (vzorec CemKaol01).

Poglavje: Diskusija

poznim časom hidratacije, temveč se spekter prevodnosti razteza med celotnim območjem.

XXXIII

Strukturo poroznih gradbenih materialov smo opisali z mejnim mrežnim modelom (enačba (6)) in preferenčnim modelom (enačba (7)). S potenčno odvisnostjo porazdelitve velikosti por !(!) ∝ ! !!

(34)

SN model predvideva skalno neodvisno organizacijo porozne strukture zemljin s porazdelitvijo stopenj ! ! ∝ ! !! . Za mreži (Slika 2-‐6) smo izračunali odgovarjajoče kumulativne porazdelitve stopenj

! ! =

(35)

! ! ! !"′

za m ≈ 1 in ! ≫ 1 v obeh primerih. Vidimo, da iz obeh modelov dobimo mrežo por z heterogeno, skalno neodvisno organizacijo za nizke vrednosti parametra m (prostorska lega vozlišč igra manj pomembno vlogo) z ! = 2 in bolj kompaktno, naključno mrežo por za velike vrednosti m.

Slika 4-‐5: Kumulativna porazdelitev stopenj za SN (levo) in EN (desno) model za majhne in visoke vrednosti parametra m. Eksponent za skalno neodvisno porazdelitev stopenj za majhno vrednost m (ravna črta na diagramih) je ! = 2. (Korošak &

Poglavje: Diskusija

Mooney, 2011)

Primer naključnega sprehoda na dvodimenzionalni tomografski sliki je predstavljen na sliki Slika 4-‐6 z

XXXIV

eksponentom porazdelitve velikosti por ! = 1,56. Naključno gibajoči delec se lahko premika samo

od pore do pore (črne površine na sliki). Povprečni (normalizirani) kvadratni odmik v odvisnosti od časa je prikazan na sliki Slika 4-‐7 (zgornja krivulja). Za primerjavo je prikazan naključni sprehod na naključni sliki, generirani z enako poroznostjo kot slika zemljine – ravna črta na sliki Slika 4-‐7. Naključni sprehod na sliki zemljine kaže subdifuzivno obnašanje !(!)! ∝ ! ! , kjer ! ≈ 0,7.

Slika 4-‐6: Naključni sprehod na dvodimenzionalnem tomografskem posnetku. Naključno gibajoči delec se lahko premika samo od pore (črne površine). (Samec et al., 2010)

prikazuje naključno gibajoči delec na naključni sliki, ki je generirana na podlagi enake poroznosti kot je poroznost vzorca na sliki. (Samec et al., 2010)

Poglavje: Diskusija

Slika 4-‐7: Normalizirana povprečni kvadratni odmik v odvisnosti od časa za naključni sprehod na sliki Slika 4-‐6; ravna črta

XXXV

Predlagana je ocena kompleksnosti porozne strukture s pomočjo meritev korelacij povezav med vozlišči. Z izračunano kompleksnostjo ℎ različnih vzorcev zemljin (z enakimi mrežnimi parametri) je pokazano, da je tovrstno merilo kompleksnosti občutljivo na razlike med poroznimi arhitekturami v zemljinah. Tabela 4-‐1: Kompleksnot h za različne vzorce zemljin z različnimi eksponenti porazdelitve velikosti por ! pri ! = !! /2. (Samec et al., to be published)

Poglavje: Diskusija

XXXVI

1,4

1,5

1,6

2,85231

3,06864

3,48277

5 Zaključki Ob trenutni tendenci vse večje potrebe po zmanjšanju porabe energije in izboljševanju notranjega bivalnega udobja imajo zemljine, stabilizirane s cementom, velik potencial pri iskanju trajnostne alternative. Za zagotovitev visoke ravni bioklimatskega udobja je nujno poznati in razumeti strukturo in transportne procese skozi ovoj stavbe, kar je bistvenega pomena za reševanje gradbenofizikalnih problemov pri prenosu vlage in toplote, interakcijah med stavbo in tlemi ter transportu kontaminantov v bližini odpadkov, kjer je prenos vlage v zemljini lahko vir onesnaževanja. Glavni zaključek doktorske disertacije je dokaz, da obstaja povezava med porozno strukturo in dinamiko nabitih delcev v porozni strukturi. To je bilo doseženo z izvedbo dveh eksperimentov: porozna struktura poroznih gradbenih materialov (vzorci zemljin, glin, cementov ter mešanic cementa in gline) je bila raziskana z uporabo najsodobnejše rentgenske računalniške mikrotomografije, medtem ko je bila dinamika nabitih delcev v vzorcih preučevana z

Z analizo dvodimenzionalnih tomografskih posnetkov so bili pridobljeni podatki o legi in velikosti por ter lokalni in celotni poroznosti vzorcev. Za analizirane slike je bila določena porazdelitev in pokazano, da v vzorcih cementa in gline, stabilizirane s cementom, sledijo potenčni odvisnosti z

Poglavje: Zaključki

nizkofrekvenčno dielektrično spektroskopijo.

XXXVII

eksponentom med 1 in 2. Izmerjen je bil dielektrični spekter vzorcev gline različnih vlažnosti in cementa z različnim vodocementnim faktorjem. Vsi pridobljeni spektri frekvenčno odvisne specifične prevodnosti kažejo podobne lastnosti: naraščanje specifične prevodnosti v nizkem frekvenčnem območju (100 Hz – 10 kHz) in padanje v področju visokih frekvenc (10 kHz – 1 MHz). Preučevane povezave med porami v kompleksnih mrežah kažejo potenčno odvisnost porazdelitev stopenj, kar kaže na skalno neodvisno topologijo preučevanih materialov. Z metodo naključnega sprehoda in multifraktalne analize tomografskih slik zemljin smo pokazali, da je topologija mreže odvisna od fraktalnih lastnosti snovi. Predlagana je ocena kompleksnosti porozne strukture s pomočjo meritev korelacij povezav med vozlišči, s čimer lahko povežemo eksponent porazdelitve velikosti por in kompleksnost stukrute poroznih snovi. Pričakujemo, da bomo predstavljene raziskave nadaljevali in razširili v vsaj treh smereh: povezati izdledke lastnosti transporta nabitega delca z mikrostrukturo glin in podobnih materialov upoštevaje elektrokemijo (Dufreche et al., 2010), (Rotenberg et al., 2010), raziskovati vpliv procesa anomalne difuzije na termo-‐fizikalne lastnosti ter prenos vlage in snovi v poroznih materialih (Hall & Allinson, 2009), (Hall & Djerbib, 2004), (Hall, 2010) za izboljšanje energetske učinkovitosti in toplotnega ugodja z optimizacijo gradbenih materialov iz zemljine. Omeniti je potrebno, da pri raziskavah nismo uporabili identičnih vzorcev kot v in (Hall & Allinson, 2008), (Hall & Allinson, 2009) zaradi omejitev pri eksperimentih in pripravi vzorcev, vendar imajo vzorci podobno mikroskopsko sestavo, uporaba razvitih modelov kompleksnih mrež porozne arhitekture zemljin pri razumevanju

Poglavje: Zaključki

določenih funkcij v zemljinah, kot na primer samoorganizacija mikrobov (Young & Crawford,

XXXVIII

2004). Kot prvi, preliminarni korak v tej smeri smo izračunali kompleksnost h v odvisnosti od časa za tri različne tipe zemljin, ki vsebujejo mikrobe. Slika 5-‐1 kaže, da kompleksnost v prvih

petih mesecih od naselitve mikrobov pada, nato pa začne naraščati. Z vidika strukture rastoče mreže se zdi, da v prvih mesecih aktivnosti mikrobov, porozna struktura postane bolj homogena, kasneje pa začne naraščati heterogenost.

Slika 5-‐1: Časovni razvoj kompleksnosti v treh različnih tipih zemljn naseljenih z mikrobi. Prikazana je kompleksnost v odvisnosti od časa za tri različne tipe zemljin.

Poglavje: Zaključki

XXXIX

PhD The Complexity of Porous Structure THESIS of Building Materials

Marko Samec

mentor: prof. dr. Dean Korošak

co-‐mentor: prof. dr. Sacha J. Mooney

List of Figures Figure 2-‐1: Example of small network with eight vertices or nodes, and ten connections between them, called edges. ........ 4 Figure 2-‐2: Königsberg was set on both sides of river, and included two large islands which were connected to each other and the mainland by seven bridges. The problem was to find a walk through the city that would cross each bridge only once. Euler proved that the problem has no solution. (Wikipedia) ..................................................... 5 Figure 2-‐3: Sociogram of relationships amongst and between children in two schools; boys (the triangles) and girls (the circles)(Moreno, 1933). You can see the relationships amongst the groups, the subgroups which form, and the connections between the two groups. Moreno said 'with these charts, we have the opportunity to grasp the myriad of networks of human relations and at the same time, view any part or portion which we desire to relate to or distinguish.' At the top right we see two girls isolated from the rest. Moreno and those he was working with predicted these two girls would run away, and they did. People who are isolated have yet to find their true 'companions', the group and people they accept and who accept them. .............................................. 6 Figure 2-‐4: Examples of different networks. (a) Visualization of the various routes through a portion of the internet(Opte Project). (b) A food web of predator-‐prey interactions between species in a freshwater lake (Martinez, 1991). (c) The network of collaborations between scientists at a private research institution (Girvan & Newman, 2002). ................................................................................................................................................................................. 7 Figure 2-‐5: Examples of simple networks. (a) Ring, a connected graph in which each node is connected to exactly two other nodes. (b) Lattice, a graph in which the nodes are placed on a grid and the neighbouring nodes are connected by an edge. (c) Tree, a connected graph which contains no circles. (d) Full graph with every possible link realized. (Érdi & Csárdi) ........................................................................................................................................... 8 Figure 2-‐6: Examples of different types of networks (a) An undirected and unweighted network with a single type of node and a single type of edge. (b) Weighted network with varying node and edge weights. (c) Directed network in which each edge has a specified direction. (Newman, 2003) ................................................................................. 9 Figure 2-‐7: An example of a network with 6 nodes and its adjacency matrix. ........................................................................ 11 Figure 2-‐8: Power law degree distribution of a scale-‐free network. ........................................................................................ 12 Figure 2-‐9: Log-‐log power law degree distribution of a scale-‐free network. .......................................................................... 12 Figure 2-‐10: knn distribution for two real-‐world networks (Lee et al., 2006). The top network is dissasortative, since the slope is negative; on the other hand the bottom one is assortative, since the slope is positive. ......................... 13 Figure 2-‐11: Network development using static network model. As the threshold parameter ε is lowered, the number of links in the network increases (from left to right). The formation of the network is overlain on the 2-‐D X-‐ray CT image of soil for clarity. (Mooney & Korošak, 2009) ............................................................................................. 16

Figure 2-‐12: Network representation of soil pore structure obtained by SN model (top) and EN model (bottom) from the same set of data derived from soil image shown in Figure 2-‐11 and with equal model parameters: equal number of links and m ≈ 1. The area of node corresponds to the node degree, i.e. the number of links attached. Both networks show similar topology where the well connected nodes (larger dots) are not directly linked, but are preferably linked through nodes with small degrees (smaller dots) (Samec et al, 2010). This type of network structure indicates that both networks are correlated (degree distributions are considered in more detail in Discussion. ............................................................................................................................................................. 17 Figure 2-‐13: One dimensional illustration of ion jumps in disordered landscape on short, intermediate and long time scales. The arrows mark attempted jumps which are mostly unsuccessful and the ion ends back in the minimum it tried to leave. (Dyre et al., 2009) .......................................................................................................................... 20 Figure 2-‐14: Triple layer model of near mineral surface electrostatics and electrochemistry (Leroy & Revill, 2004). .......... 26 Figure 2-‐15: Measured (open dots) and calculated (solid lines) real part of frequency dependent conductivity of kaolinitic clay samples at 36 % (lower curve) and 56 % (upper curve) water content (Samec et al., 2007). ........................ 29 Figure 2-‐16: Measured (open dots) and calculated (solid lines) imaginary part of frequency dependent conductivity of kaolinitic clay samples at 36 % (lower curve) and 56 % (upper curve) water content (Samec et al., 2007). ........ 29 Figure 2-‐17: Low-‐frequency spectra (Lounev et al., 2002). ...................................................................................................... 30 Figure 3-‐1: Multiphase specimen showing layers of cement and clay. .................................................................................... 33 Figure 3-‐2: Capacitor before and after dielectric is introduced (Breuer, 1993). When a voltage is applied to a parallel-‐plate capacitor in vacuum, the capacitor will store charge. When a dielectric is placed between the plates, a charge stored on the parallel plates is greater. ................................................................................................................ 36 Figure 3-‐3: Functional dependence of Q on applied voltage. Slope of the line is related to the dielectric constant of the material. (Barsoum, 1997) ..................................................................................................................................... 37 Figure 3-‐4: Definition of an electric dipole moment; ! is a vector pointing from the negative to the positive charge. The electric dipole moment indicates the orientation of the dipole (positions of the charges) and has nothing to do with the direction of the field originating the cause of dielectric formation. ....................................................... 38 Figure 3-‐5: Electronic and ionic polarization mechanisms: I. Electronic polarization occurs when the electron cloud is displaced relative to the nucleus it is surrounding (a) at Equilibrium, i.e., in the absence of an external electric field, (b) in the presence of an external electric field; II. Ionic polarization, the displacement of positive and negative ions toward the negative and positive electrodes, respectively (a) ion positions at equilibrium, (b) when an external electric field is applied, the centre of negative charge is no longer coincident with the centre of positive charge. (Barsoum, 1997) ..................................................................................................................... 39 Figure 3-‐6: Schematic characterization of three types of porous geomaterials (Revil & Cosenza, 2010). .............................. 44 Figure 3-‐7: Sketch of plan parallel capacitor, filled with the sample. ...................................................................................... 46 Figure 3-‐8: Measured real part of the dielectric function and the numerically computed values from measured conductance (Kramers-‐Krönig transform). ................................................................................................................................. 47 Figure 3-‐9: Measuring cell, filled with sample, connected to the instrument. ........................................................................ 48 Figure 3-‐10: Principle of Beer-‐Lambert law. Object A is of the same thickness as object B but yield a higher absorption due to higher density or higher atomic number. Object C consists of the same material as object B but absorbs less radiation than the thicker object B. ...................................................................................................................... 49 Figure 3-‐11: The electromagnetic spectrum. X-‐rays have a photon energy ranging from 1.2 to 240 keV (www.phoenix-‐ xray.com). .............................................................................................................................................................. 50 Figure 3-‐12: Detail sketch of X-‐ray source. Electrons are emitted from a heated filament and accelerated towards the anode by the potential difference and enter through a hole in the anode into magnetic lens which focuses the electron beam to a small spot on a target (www.phoenix-‐xray.com). ................................................................................ 51 Figure 3-‐13: X-‐ray inspection system consists of the X-‐ray tube and the detector. Remotely controllable manipulating unit allows positioning the specimen within the beam (www.phoenix-‐xray.com). ..................................................... 51 Figure 3-‐14: A 2-‐D image acquired from X-‐ray CT representing stabilized clay sample. .......................................................... 52 Figure 3-‐15: The image intensifier consists of scintillator input window, photocathode, electro-‐optics, luminescent screen and an output window. (www.phoenix-‐xray.com). ............................................................................................... 53 Figure 3-‐16: Operating principle of X-‐ray sample scanning. X-‐rays pass through a specimen and hit a detector, where two dimensional X-‐ray image is created. (www.phoenix-‐xray.com). .......................................................................... 54

Figure 3-‐17: Voxel resolution variables (www.phoenix-‐xray.com). ......................................................................................... 55 Figure 3-‐18: Model of volume data set of 6 x 6 x 6 voxels. The voxel is associated with a grey value and has a size of V = pixel size / magnification. ...................................................................................................................................... 55 Figure 3-‐19: From image scanning to 3-‐D pore space quantification. ..................................................................................... 56 Figure 3-‐20: Image analysis procedure. a) Greyscale image of the specimen is acquired using X-‐ray CT; b) original image is cropped and scaled; c) contrast is enchanced to visible separate pore space from solid matrix; d) automatic Yen threshold algorithm was used to determine pore space; e) the result is a binary image which is further analysed for different properties. ........................................................................................................................................ 58 Figure 3-‐21: The range of negative values of the real part of dielectric function for clay-‐water system from our measurement (open dots) and the data (closed dots) obtained from Carrier and Soga (Carrier & Soga, 1999). . 61 Figure 3-‐22: Measured conductivity spectra of clay specimens with various water content levels. ....................................... 62 Figure 3-‐23: Real part of dielectric function (calculated from measured capacity) spectra of clay specimens with various water content levels. Measurements are marked with points – lines connecting the markers are plotted for better overview. .................................................................................................................................................... 63 Figure 3-‐24: Detailed display of real part of dielectric function spectra of clay in the range of negative values of capacity (same measurements as in Figure 3-‐23). Measurements are marked with points – lines connecting the markers are plotted for better overview. ........................................................................................................................... 64 Figure 3-‐25: Imaginary part of dielectric function (calculated from measured capacity) spectra of clay at different water content levels. Measurements are marked with points – lines connecting the markers are plotted for better overview. ............................................................................................................................................................... 65 Figure 3-‐26: Measured conductivity spectra of multiphase clay-‐cement water system (specimen CemKaol01) at different hydration times (in hours). Measurements are marked with points – lines connecting the markers are plotted for better overview. .............................................................................................................................................. 66 Figure 3-‐27: Measured conductivity spectra of cement paste with 0,3 w/c ratio (specimen cement_0.3) at different hydration times (in hours). Measurements are marked with points – lines connecting the markers are plotted for better overview. .............................................................................................................................................. 67 Figure 3-‐28: Measured conductivity spectra of cement paste with 0,4 w/c ratio (specimen cement_0.4) at different hydration times (in hours). Measurements are marked with points – lines connecting the markers are plotted for better overview. .............................................................................................................................................. 68 Figure 3-‐29: Measured conductivity spectra of cement paste with 0,5 w/c ratio (specimen cement_0.5) at different hydration times (in hours). Measurements are marked with points – lines connecting the markers are plotted for better overview. .............................................................................................................................................. 69 Figure 3-‐30: Distribution of porosity in depth for cement in different hydration times. ........................................................ 71 Figure 3-‐31: Total porosity to hydration times of cement. Error bars indicate standard error. .............................................. 71 Figure 3-‐32: Distribution of porosity in depth for clay in different sorption times. ................................................................. 72 Figure 3-‐33: Total porosity to sorption times of clay. Error bars indicate standard error. ...................................................... 72 Figure 3-‐34: Distribution of porosity in depth for stabilized clay in different hydration times. .............................................. 73 Figure 3-‐35: Total porosity to hydration times of stabilized clay. Error bars indicate standard error. .................................... 74 Figure 3-‐36: Two-‐dimensional representation of soil porous structure. Binary image showing pore space (black areas) and soil matrix (white areas). ....................................................................................................................................... 74 Figure 3-‐37: Images of three different soil pore structures. These images were taken outside the purpose of this thesis, but are used here to demonstrate how complexity of the network of soil porous structure is quantified (see chapter 4). Image courtesy of Department of Environmental Sciences, University of Nottingham. ................................. 75 Figure 3-‐38: Three-‐dimensional visualization of the soil pore architecture from stack of X-‐ray CT scans. The pore space is shown while the soil matrix is here transparent. The height of the specimen shown here is 16 mm and the diameter is 14 mm. .............................................................................................................................................. 75 Figure 3-‐39: Cumulative size distribution (left), determined from X-‐CT image (right) for clay, cement and stabilized clay (from top to bottom, respectively) of different porosity φ. All samples show scale-‐free behaviour of size distribution (solid line) with the exponent α=1,8-‐2. ............................................................................................. 77

Figure 4-‐1: Imaginary part of the complex conductivity obtained using eq. (39) with the parameters: !""= 10−4, !"#= 0,04 A/Vm, !ℎ! = 0,67 A/Vm, ! = 0,67. The open dots are experimental points for pure kaolinite measured with dielectric spectroscopy. (Samec et al., to be published). Data from: Mesure de la réponse PPS (Polarisation Provoquée Spectrale) de mélanges artificiels argilo-‐sableux non consolidés Gonca et al. (private comm.) ........ 79 −4

Figure 4-‐2: Real and imaginary part of the complex conductivity obtained using eq.(39) with the parameters: !""= 1,5 10 , !"#= 0,04 A/Vm, !ℎ! = 0,67 A/Vm, ! = 0,67. The open dots are experimental points for pure kaolinite measured with dielectric spectroscopy. (Samec et al., to be published) .............................................................. 79 Figure 4-‐3: Schematic of the chord-‐length measurements for a cross section of a two-‐phase random medium. The chords are defined by the intersection of lines with the two-‐phase interface. (Torquado & Lu, 1993) ........................... 80 Figure 4-‐4: Dependence of Cole-‐Cole parameter ! on relaxation time !"" according to eq. (37). The salinity was set to !"= -‐3 10 M, the midplane potential was !"= 10 mV, the pore size was != 3. (Samec et al., to be published) ........... 82 Figure 4-‐5: Cole-‐Cole exponent vs relaxation time (Feldman et al., 2002). ............................................................................ 83 Figure 4-‐6: Cole-‐Cole parameter ! as a function of pore size !. The partition coefficient was set to ! = 0,8. (Samec et al., to be published) ......................................................................................................................................................... 83 Figure 4-‐7: Cole-‐Cole relaxation time as a function of pore size. The partition coefficient was set to ! = 0,8. The salinity was -‐3 set to !"= 10 M, the midplane potential was !"= 10 mV. (Samec et al., to be published) ............................... 84 Figure 4-‐8: Cole-‐Cole relaxation time as a function of pore electrolyte salinity. The pore size of the system was set to size != 25 nm, the midplane potential was !"= 10 mV. (Samec et al., to be published) ............................................ 84 Figure 4-‐9: Scaled low-‐frequency part of conductivity: open symbols -‐ early hydration times 0 h -‐12 h, solid symbols -‐ late hydration times 30 h -‐179 h. ................................................................................................................................. 85 Figure 4-‐10: Normalized low-‐frequency part of conductivity: open symbols -‐ early hydration times 0 h -‐12 h, solid symbols -‐ late hydration times 30 h -‐179 h. Solid lines were computed with eq.(41) using α=0.82 for early and α=0,6 late hydration times. .................................................................................................................................................... 85 Figure 4-‐11: Clay-‐cement system low-‐frequency normalized conductivity spectra (sample CemKaol01). ............................ 86 Figure 4-‐12: Scaling exponent dependence on hydration time. Open dots – NMR data (Blinc et al., 1988), solid triangles – dielectric spectroscopy data (Korošak et al., 2010). ............................................................................................. 86 Figure 4-‐13: Cumulative degree distributions of SN (left) and EN (right) models for small and large values of the parameter m. The scaling exponent of the scale-‐free degree distribution obtained for small m (straight line in plots) is ! = 2. The fits are power-‐laws (straight lines) and cummulative Poissonian distributions (Samec et al, 2010). . 88 Figure 4-‐14: Two-‐point correlation function and its normalized version computed for the soil image shown in Figure 3-‐36. (Korošak & Mooney, 2011) ................................................................................................................................... 89 Figure 4-‐15: Binary image obtained with growing network algorithm simulating soil pore structure at early (upeer left) and late (lower right) stage of computation. The porosities are: 0.09, 0.11, 0.16, 0.18. ............................................. 91 Figure 4-‐16: Two point correlations for simulated binary soil images (left) and normalized correlation function of the two simulated binary soil image from Figure 4-‐17 (right). The correlation functions on the left panel correspond to porosities 0.09, 0.11, 0.16, 0.18 from bottom to top. The two fitted normalized correlation function correspond to porosities 0.1, and 0.18. .................................................................................................................................... 91 Figure 4-‐17: Random walk on the 2D X-‐CT soil image. The random walker is allowed to move only from pore to pore (black areas). (Samec et al., 2010) ................................................................................................................................... 92 Figure 4-‐18: Mean square distance as a function of time and normalized mean square distance as a function of time for random walks on soil images; straight line represents random walker on the random image, generated to have the same porosity as the soil image, presented as straight line. (Samec et al., 2010) .......................................... 93 Figure 4-‐19: Mean squared distance as a function of time and normalized mean squared distance (inset) as a function of time for random walks on scale-‐free networks with different degree distribution exponents (Gallos, 2004). .... 94 Figure 4-‐21: Correlation matrices for a smaller geographical threshold network. (Samec et al, 2010) .................................. 98 Figure 5-‐1: Time evolution of complexity in soil-‐microbe system for three soil samples. Plotted is the complexity vs time for all three soil samples (CL -‐ , LS -‐ , SL -‐ ). The same set of parameters was used and ! set to: ! = !" − 1 (! is the exponent of the pore size distribution). ....................................................................................................... 103

List of Tables Table 2-‐1: Values of the parameter used to calculate the real and imaginary parts of the conductivity shown in Figure 2-‐15 and Figure 2-‐16. .................................................................................................................................................... 28 Table 3-‐1: Mix designs of specimens used for research .......................................................................................................... 34 Table 3-‐2: The ratios of real part of dielectric function ε’ and conductivity σ for the same sample of clay_300 measured at two different values of d (d1 = 18,1 mm, d2 = 6,6 mmi). All ratios σd1 / σd2 have the same magnitude, while some εd1 and εd2 differ more than 3-‐times, indicating the conductivity σ is meaningful quantity for sample comparison of electrical characteristics. ................................................................................................................................... 60 Table 3-‐3: Data analysis of different specimens. ..................................................................................................................... 70 Table 4-‐1: Complexity h of several soil pore structures with different scaling exponents of their pore size distributions at ! = !"/2. (Samec et al, 2010) ............................................................................................................................ 97 Table 6-‐1: Measured conductivity spectra of clay specimens with various water content levels (see Figure 3-‐22 for graphic presentation of the results). ................................................................................................................................ 104 Table 6-‐2: Real part of dielectric function (calculated from measured capacity) spectra of clay specimens with various water content levels (see Figures Figure 3-‐23 and Figure 3-‐24 for graphic presentation of the results). ..................... 105 Table 6-‐3: Imaginary part of dielectric function (calculated from measured capacity) spectra of clay specimens with various water content levels (see Figure 3-‐25 for graphic presentation of the results). ................................................. 105 Table 6-‐4: Measured conductivity spectra of multiphase clay-‐cement water system (specimen CemKaol01) at hydration times from 0 to 6,75 hours (see Figure 3-‐26 for graphic presentation of the results). ....................................... 106 Table 6-‐5: Measured conductivity spectra of multiphase clay-‐cement water system (specimen CemKaol01) at hydration times above 7 hours (see Figure 3-‐26 for graphic presentation of the results). ................................................. 106 Table 6-‐6: Measured conductivity spectra of cement paste with 0,3 w/c ratio (specimen cement_0.3) at different hydration times (see Figure 3-‐27 for graphic presentation of the results). ......................................................................... 107 Table 6-‐7: Measured conductivity spectra of cement paste with 0,4 w/c ratio (specimen cement_0.4) at different hydration times (see Figure 3-‐28 for graphic presentation of the results). ......................................................................... 107 Table 6-‐8: Measured conductivity spectra of cement paste with 0,5 w/c ratio (specimen cement_0.5) at different hydration times (see Figure 3-‐29 for graphic presentation of the results). ......................................................................... 108 Table 6-‐9: Distribution of porosity in depth for cement in different hydration times (see Figure 3-‐30 and Figure 3-‐31 for graphic presentation of the results) and clay in different sorption times (Figure 3-‐32 and Figure 3-‐33). .......... 109

Table 6-‐10: Distribution of porosity in depth for stabilized clay in different hydration times (see Figure 3-‐34 and Figure 3-‐35 for graphic presentation of the results). ............................................................................................................. 110

1 Introduction Today, facing the climate change and the need to reduce the release of greenhouse gases into the atmosphere, the locally available natural building materials provide a low-‐carbon alternative. However, to ensure a proper bioclimatic indoor environment it is critically important to know and understand the properties of the building envelope made from natural materials, especially the correlation between the structure and transport processes. Building materials are mostly heterogeneous porous media, and the transport mechanism of substances (such as water and contaminants) is diffusion. Surprisingly though, the effect of changes in the natural building material porous architecture on thermal conductivity and moisture content has not yet received wide attention. In this work a novel approach towards the link between the structure and dynamics in porous media transport phenomena is presented, using complex networks to describe the structure of pore connectivity and fractional calculus for the description of dynamics on microscopic level in such

science has found its way in soil science because of growing need to seek for new approaches to understand how porous structure affects soil function and vice versa. This thesis builds on current knowledge of the possibilities of describing porous structure of natural building materials (soils

Chapter: Introduction

complex system. With increasing anthropogenic demands on soil resources (Lal, 2007), network

stabilized with cement) as a complex network of pores (Blunt, 2001) with the relevant quantification of connectivity on the one hand, as well as current findings on the dynamics of anomalous transport in porous materials (Metzler & Klafter, 2004). A combination of theoretical and experimental methods was used. Complex networks were constructed using image analysis of 2D slices and 3D reconstructions of X-‐ ray computed tomography of the samples. This enabled us to connect the porous structure of the sample with the complex network topology, to classify the samples with respect to complexity of the porous structure, and to investigate the effect of the topology on the transport properties thus giving us the clues for optimization. Low-‐frequency dielectric spectroscopy was used to obtain electrical characteristics of the samples and to derive a generalized model describing ion dynamics in porous media. The thesis builds on previous research in both areas, structure and dynamics in porous media, organized in research groups of supervisor prof. dr. Dean Korošak and co-‐supervisor prof. dr. Sacha J. Mooney. The research methods were partially carried out on the cutting-‐edge experimental device for 3D X-‐ray tomography at the Department of Environmental Sciences, University of Nottingham, United Kingdom. The dielectric spectroscopy experiments were conducted at the Chair of Applied Physics, Faculty of Civil Engineering, University of Maribor, Slovenia. The thesis is organized into five sections and subsections. In Chapter 2 we present basic concepts of complex networks and dynamics in porous media, both being underlying theme of the thesis. A

Chapter: Introduction

model and the fundamental theoryis introduced. In Chapter 3 we describe experimental methods

and reports on acquired results. Chapter 4 provides a general discussion of the results and our key findings. In Chapter 5 we draw the major conclusions from all the performed research.

2 Theory and model A porous media is a material containing pores or voids. The pores are typically filled with fluids – liquid or gas. The skeletal portion of the material, usually a solid, is often called matrix or frame. Porous matter is most often characterised by its porosity. Other properties (e.g., permeability, tensile strength and electrical conductivity) can be determined from the properties of its constituents, porosity and pores structure, but such a derivation is usually complex. Many natural (soils, rocks, wood, bones, etc.) and man-‐made (cement, ceramics, etc.) substances can be considered as porous matter. Treating them as such, many of their important properties can be rationalized.

The study of complex networks is a decade old research field whose applications reach many scientific disciplines including computer science, particle physics, economics, sociology, ecology, epidemiology and neuroscience (Watts & Strogatz, 1998), (Albert & Barabasi, 2002), (Barabasi, 2009), (Zimmer, 2010). In general, network theory concerns itself with the study of large graphs as a

Chapter: Theory and model

2.1 Complex networks

representation of relations between objects (Figure 2-‐1). This young discipline was born of interest in studies of networks such as the internet, social networks and biological networks. Inspired by these kind of network structures, researchers are seeking to discover common principles, algorithms and tools to understand network behaviour. An interesting example of argumentation why studying networks is important comes from medicine. One of the most important discoveries in cancer research, the most studied human illness, is the p53 gene11 which is a tumor suppressor. Mutations in p53 usually results in cancer. Despite its important role, fixing the p53 alone does not lead to cure cancer. Volgestein, Lane and Levine discovered that “one has to understand the p53 network which is just like the internet. A small subset of proteins is highly connected and controls the activity of a large number of other proteins” (Volgestein et al., 2000). That means if we understand the protein itself, we know which proteins causes cancer. But this is not enough; we need to understand the whole network.

Figure 2-‐1: Example of small network with eight vertices or nodes, and ten connections between them, called edges.

The network study is very active in biology and medicine ever since and is also rapidly evolving in other sciences and engineering fields. For example Volgestein, Lane and Levine discovery has an

Chapter: Theory and model

analogy in traffic engineering – it is not enough to fix one junction in a road to optimize a whole

traffic network.

P53 was identified in 1979 by Lionel Crawford, David P. Lane, Arnold Levine and Lloyd Old

2.1.1

Background and history of network science

Fundamentals of network study come from graph theory. The famous paper Seven Bridges of Königsberg written by Leonhard Euler in 1735 is the earliest known paper in this field (Euler, 1741). Euler’s mathematical description of vertices and edges was the foundation of mathematical graph theory that studies the properties of relations in a network structure (Figure 2-‐2).

Figure 2-‐2: Königsberg was set on both sides of river, and included two large islands which were connected to each other and the mainland by seven bridges. The problem was to find a walk through the city that would cross each bridge only once. Euler proved that the problem has no solution. (Wikipedia)

In the 1933 Jacob Moreno developed a sociogram (Figure 2-‐3), the representation of a group of elementary school students, linking friends together. This network representation of social structure was found so intriguing that it was presented in The New York Times (Moreno, 1933). The sociogram found many applications and has grown into the field of social network analysis. Probalistic theory of graphs was started in 1950’s with papers on random graphs by Paul Erdős and Alfred Rényi (Erdős & Rényi, 1959) and independently by Edgar Gilbert (Gilbert, 1959). They

vertices and adding edges between them at random. Different random graph models produce different probability distributions on graphs. The models can be used to prove the existence of graphs satisfying various properties, or to provide a rigorous definition of what it means for a property to hold for almost all graphs.

Chapter: Theory and model

introduced models for generating random graphs. A random graph is obtained by starting with set of

Figure 2-‐3: Sociogram of relationships amongst and between children in two schools; boys (the triangles) and girls (the circles)(Moreno, 1933). You can see the relationships amongst the groups, the subgroups which form, and the connections between the two groups. Moreno said 'with these charts, we have the opportunity to grasp the myriad of networks of human relations and at the same time, view any part or portion which we desire to relate to or distinguish.' At the top right we see two girls isolated from the rest. Moreno and those he was working with predicted these two girls would run away, and they did. People who are isolated have yet to find their true 'companions', the group and people they accept and who accept them.

More recently other parts of network sciences have focused on developing a variety of techniques and models to help describe different network topologies (Figure 2-‐4). The focus shifted from analysis of single small graphs and the properties of individual vertices within such graphs to consideration of large-‐scale statistical properties of graphs (Newman, 2003) and, very recently, of spatial networks (Barthélemy, 2010). This new approach was made possible by the availability of computers and communication networks allowing gathering and analyzing data on a larger scale

Chapter: Theory and model

than previously possible.

c) b)

Figure 2-‐4: Examples of different networks. (a) Visualization of the various routes through a portion of the internet(Opte Project). (b) A food web of predator-‐prey interactions between species in a freshwater lake (Martinez, 1991). (c) The network of collaborations between scientists at a private research institution (Girvan & Newman, 2002).

Chapter: Theory and model

2.1.2

Introduction to complex network theory

Complex systems which consist of many interacting entities can be analyzed and better understood using a network representation. In recent years it was realized that the topology of many real networks cannot be presented using classical graph theory leading to many open questions. The finding of the new topology led to the emergence of an active field of complex networks where new suitable theories and approaches are developed (Havlin, 2008). This section is only an overview of basic ideas about complex networks. There are some excellent review papers and books that describe complex network theory in detail (Strogatz, 2001), (Albert & Barabasi, 2002), (Newman, 2003), (Barabási, 2003). a)

Figure 2-‐5: Examples of simple networks. (a) Ring, a connected graph in which each node is connected to exactly two other nodes. (b) Lattice, a graph in which the nodes are placed on a grid and the neighbouring nodes are connected by an edge. (c) Tree, a connected graph which contains no circles. (d) Full graph with every possible link realized. (Érdi & Csárdi)

Network or graph is a simple object consisting of nodes or vertices connected with links or edges. But there are many ways where networks tend to be more complex than this. A complex network is

Chapter: Theory and model

a network with non-‐trivial topological features that do not occur in simple networks (Figure 2-‐5). For

instance, there may be different type of nodes or edges in a network. Nodes and edges may have a variety of properties associated with them. Edges can be directed, where the order of the nodes i and j connected with a link (i,j) is important (the link points in only one direction, from node i to node j). Networks composed of directed edges are called directed networks. On the other hand, if

the order of connected nodes is unimportant, then such a network is undirected. Furthermore, there are weighted networks (when nodes or edges have different weights), or unweighted networks (each node or edge in the network is equally important) (Figure 2-‐6). (a)

(b)

(c)

Figure 2-‐6: Examples of different types of networks (a) An undirected and unweighted network with a single type of node and a single type of edge. (b) Weighted network with varying node and edge weights. (c) Directed network in which each edge has a specified direction. (Newman, 2003)

The number of edges connected to a node is called a degree of the node. For example, on Figure 2-‐6 (a) the degree of the node in the middle is four. In general, the degree is not necessarily equal to the number of nodes adjacent to a node, since there may be more than one edge between any two nodes. A node in directed graph has an in-‐degree (number of incoming edges) and out-‐degree (number of outgoing edges). A graph where each node is connected to every other node is a complete graph. A geodesic path or distance is the shortest path between two nodes, there may exist more than one geodesic paths between two nodes. The length, measured in number of edges,

Two much studied types of complex networks in recent years are small-‐world networks and scale-‐ free networks. Both are characterized by specific structural features – short geodesic paths and

Chapter: Theory and model

of the longest distance between any two nodes is the diameter of a network.

clustering12 for former and power-‐law degree distribution for latter. In small-‐world network two arbitrary nodes are connected by only few edges, i.e. diameter of the corresponding graph is small. This kind of network is named a small-‐world network by analogy with the small-‐world phenomenon, popularly known as six degrees of separation13. Small-‐world properties are found in many real-‐world networks, e.g. road maps, food chains, electric power grids, networks of brain neurons, voter networks and social impact networks. A network is called scale-‐free if its degree distribution (the probability that a randomly selected node has a certain degree) follows power law, a particular mathematical function. The power law implies that the degree distribution of these networks has no characteristic scale, i.e. the functional form is retained under the change of scale, hence the term scale-‐free networks. In scale-‐free networks, some nodes (a small number of them) have a degree that is orders of magnitude larger than the average – these vertices are called hubs. The study of scale-‐free networks is the focus of a great deal of attention due to the fact that many real-‐world networks appear to be scale-‐free, including World Wide Web, the internet, citation networks and some social networks.

2.1.3

Fundamental concepts in complex network theory

An unweighted and undirected network with N nodes can be described by N x N adjacency matrix, where non-‐diagonal matrix element aij is the edge from node i to node j (the element equals 1 if there is a link between nodes i and j or 0 otherwise), and the diagonal entry aii is the link (loop) from

Chapter: Theory and model

node to itself. The degree of a node ki is the number of edges the node i has to other nodes:

Clustering means the presence of a heightened number of triangles in the network – sets of three vertices each of which is connected to each of the others. It can be quantified by defining a clustering coefficient C, a measure of degree to which nodes in a graph tend to cluster together (Newman, 2003). 13 “Six degrees of separation” refers to the idea that everyone is on average approximately six steps away from any other person on Earth. In 1967, Stanley Milgram carried out famous experiment in which group of people were asked to forward letters to a friend whom they thought would bring the letter closer to final individual. The average path length to reach a designated target individual was around six (Barabási, 2003). This is one of the first demonstrations of the small-‐world effect, the fact that most nodes are connected by a short path through the network.

!! =

! !!" .

(1)

Figure 2-‐7: An example of a network with 6 nodes and its adjacency matrix.

The degree distribution, defined as the probability that a randomly chosen node will have a degree k, is given by P(k). Node degrees distribution is often used to characterize topological structure of the network. The mean degree of the network ! is

! =

! ! ! ! ! .

(2)

As discussed earlier, many real-‐world networks are discovered to poses a particular network organization characterized by a power-‐law degree distribution (Figure 2-‐8, Figure 2-‐9) !(!)~! !! ,

(3)

where ! is a constant usually in the range 2 < ! < 3 (Boccaletti et al., 2006). Such networks are termed scale-‐free networks(Barabási & Albert, 1999), although it is only their degree distribution that is scale-‐free. While some networks display and exponential tail, the functional form of ! (!)

as Erdős-‐Rényi networks) (Albert & Barabasi, 2002).

Chapter: Theory and model

often deviates from the Poissonian degree distribution expected for random networks (also known

degree distribu‹on P(k)

many nodes with only a few links

a few hubs with large number of links

0 0

number of links k

log (degree distribu‹on P(k))

Figure 2-‐8: Power law degree distribution of a scale-‐free network.

many nodes with only a few links

a few hubs with large number of links

1 0,025

log (number of links k)

Figure 2-‐9: Log-‐log power law degree distribution of a scale-‐free network.

Many real-‐world networks have fat-‐tailed degree distribution which is difficult to analyze, due to

Chapter: Theory and model

rather strong noise because one rarely has enough data to get good statistics in the tail. In such case an alternative way of presenting degree data is to consider cumulative degree distribution

! ! ! ,

! ! = ! ! !!

(4)

which is the probability that the degree is greater than or equal to k. The cumulative distribution reduces the noise in the tail. Furthermore, in many real-‐world systems the probability of a node with degree k to link to a node with degree k’ depends on degree k; such network is correlated. The correlation properties of the network are expressed with conditional probability ! (!′|!) that an edge of node degree k points to a node with degree k’. The computation of average degree of neighbours of a node with degree k (5)

! ! ! ! ! |!

!!! = !!

is used to classify networks by assortativity14. If !!! (!) is increasing, the network is assortative, since it shows that a nodes of high degree tend to connect to nodes of high degree; if !!! (!) is decreasing, the network is dissasortative, since nodes nodes of high degree connect to nodes of lower degree (Figure 2-‐10).

slope is negative; on the other hand the bottom one is assortative, since the slope is positive.

Assorativity refers to a preference for a network's nodes to attach to others that are similar or different in terms of node's degree.

Chapter: Theory and model

Figure 2-‐10: knn distribution for two real-‐world networks (Lee et al., 2006). The top network is dissasortative, since the

2.2 Using complex network to describe porous structure of building materials From the early network models (Fatt, 1956) many different approaches to describe and model the complex structure of soils have been proposed. They include applications of fractal theory (Posadas et al., 2003), Boolean random sets (Horgan & Ball, 2006), cellular automata (Prosperini & Perugini, 2007) and newer network models (Vogel & Roth, 2001). The development of modelling porous networks advanced based on the support of 3D imaging techniques of porous media, such as X-‐ray CT, and the powerful computational techniques to extract the pore space topology from soil images (Ioannidis & Chatzis, 2000), (Liang et al., 2000). Pore network models are generally based on idealised pore topology often coupled with percolation theory (Berkowitz & Ewing, 1998),(Hunt & Ewing, 2009) using fixed grid lattices (Vogel & Roth, 2001) with a defined connectivity function (Vogel, 1997). This approach while describing the essential physics of the random porous media neglects the possible spatial correlations and the long-‐range heterogeneities (Stanley & Meakin, 1988), (Posadas et al., 2003), (Dathe et al., 2006), (Tarquis et al., 2009). To capture both short-‐ and long-‐range effects in pore network models a new approach for considering the porous architecture of soils using complex random networks has recently been suggested (Santiago et al., 2008), (Mooney & Korošak, 2009), (Cárdenas et al., 2010). Even though the last decade has witnessed a fast advancement in the field of network theory, geographical effects or spatial embedding that significantly constrains the network topology and plays an

Chapter: Theory and model

important role in many real world systems, including soils, has not yet received the wide attention as it has in some other areas (Yook et al., 2002), (Masuda et al., 2005), (Bianconi et al., 2009). The idea to use network theory for describing porous architecture of building materials is to provide a new way to understand the complexity of such media through the study of large scale organization

of pore structure. Two recently developed complex network models of pore organization of soils are presented (Mooney & Korošak, 2009), (Santiago et al., 2008), (Cárdenas et al., 2010) as it turns out the topology of porous architecture of soils and other building materials (like cement paste) is similar. Both methods reveal scale-‐free topology of soils with power law degree distribution (eq. (3)), but use different ways to construct the complex pore networks. Let’s take a set of N pores representing the nodes of the network. The nodes of the network are the centres of the pores and the edges between nodes are created with the probability depending on the properties of the pores – pore positions and sizes. The nodes are distributed in D-‐dimensional space and to each node a state s (pore size) is assigned describing the properties of the node. The node states are distributed with a probability distribution P(s), the pore size distribution. Threshold network model (Mooney & Korošak, 2009) says that two nodes with states si, sj (pores i and j with specific sizes) are connected if !! !! ! > ! , !!"

(6)

where the threshold value ! > 0 controls the number of links in the network (Figure 2-‐11); if ! = 0 ! we get a complete graph. !!" = !! − !! is the distance between pores and the parameter m

measures the importance of the distance between nodes. This method results in a static network of N nodes connected with M links with the mean degree of the network ! =

!! !

Chapter: Theory and model

Figure 2-‐11: Network development using static network model. As the threshold parameter ε is lowered, the number of links in the network increases (from left to right). The formation of the network is overlain on the 2-‐D X-‐ray CT image of soil for clarity. (Mooney & Korošak, 2009)

In heterogeneous preferential attachment model (Santiago et al., 2008), (Cárdenas et al., 2010) the complex pore network grows by sequentially adding new nodes to the network and linking them to the existing ones. The probability to connect the new node j (added randomly at point !! ) with a state sj extracted from distribution P(s),is proportional to the product of the degree of the existing node ki and the affinity function σ(i,j) !!" ~!! ! !, ! .

(7)

The affinity function depends on the state of the new node and the distance to the existing node:

! !, ! =

!!!

! , !!"

(8)

where b is a parameter that measures the importance of the pore sizes in the attachment process. When the desired number of the nodes N is reached, growing of the network stops.

Chapter: Theory and model

In models described above, the static network model (SN) and the evolving network model (EN), the network heterogeneity can be adjusted by changing the parameters b and m. As shown in the chapter Discussion, both models predict similar topological networks of pore structure in soils with power law degree distribution ! (!)~! !! revealing scale-‐free topology of soils (Samec et al, 2010).

Chapter: Theory and model

2.3 Slow kinetics in porous media The state of material, yet solid, is far from static. Thermal energy keeps the atoms vibrating, bumping into each other and exchanging energy with their neighbours and surroundings. Sometimes, an atom will gain sufficient energy to migrate. When charge carriers15 are induced to migrate under the influence of a chemical potential16 gradient, we talk about diffusion, and electrical conductivity in case of migration under electric potential gradient. In porous building materials, like cement and clay, the particles that carry electric charge are ions dissolved in water which is present in the pore space. In contrast to electrical conductivity, which deals with long-‐range motion of charge carriers, the dielectric response results from short-‐range motion of these carriers under the influence of externally applied electric field. The application of an electric field to material results in a separation of its positive and negative charges. This separation of charge is called polarization, defined as the finite displacement of bound charges of a dielectric in response to an applied electric field, or the orientation of their molecular dipoles if the latter exist (Barsoum, 1997). In the last decade the theory of porous matter has made an important progress concerning fundamentals and the development of models in various fields of engineering and biomechanics, including soil mechanics (de Boer, 2000), material science (Sánchez et al., 2008), environmental mechanics (Kaczmarek et al., 1997), (Gaucher & Blanc, 2006), building physics (de Boer, 2005),

Chapter: Theory and model

dental medicine (Leskovec et al., 2005) and many others.

Charge carrier denotes a free (mobile and unbound) particle carrying an electric charge. In metals, charge carriers are electrons; in ionic solutions, such as salt water, the charge carriers are ions (cations+ and anions-‐). 16 Chemical potential is a thermodynamic variable that measures the increase of the free enthalpy of homogeneously distributed matter with respect to change in the amount of component added to the thermodynamic system. In simple terms, it is analogue to electric and gravitational potential, utilizing the same idea that force fields are the cause of things to move, e.g. charges, masses, or chemicals. (Gibbs, 1876) (Baierlein, 2001)

2.3.1

Basics of ion conduction

Here we give an only brief overview of basic ideas of ion transport in disordered solids. Excellent review papers exist (Jonscher, 1999), (Dyre & Schroder, 2000), (Dyre et al., 2009) that describe ion conduction in disordered matter in detail. Motion of ions in disordered solid is far different from electronic conduction in crystalline solids. Ion motion is described semiclassically rather than by quantum mechanical description of motion of electron. Motion of ion can be described by ion jumps between charge compensating sites (Dyre et al., 2009). Moving ions produce an electrical current because they carry charge. If ion has enough energy it may migrate along the irregular potential energy landscape, which contains a distribution of effective depths and barrier heights (Figure 2-‐13). If the barrier is ΔE, T is the temperature and kB is Boltzmann’s constant, the probability of a successful jump is roughly exp (−!!/!! !). The effective energies result from binding energies at residence sites and saddle point energies between them, and they are influenced by interactions between the ions. On short time scales only the smallest barriers are overcame. With increasing time scale, ions can explore larger parts of space as higher barriers are surmounted. In more than one dimension the highest barrier is determined by percolation theory17. Short-‐time dynamics is characterized by a back and forth motion over limited ranges, while the long-‐time dynamics is characterized by random walks resulting in long-‐range ion

Percolation theory is a general mathematical theory of connectivity and transport in geometrically complex systems. (King et al., 2002)

Chapter: Theory and model

transport, macroscopically described by diffusive dynamics (Dyre, 1988).

Short time scales

Intermediate time scales

Long time scales Figure 2-‐13: One dimensional illustration of ion jumps in disordered landscape on short, intermediate and long time scales. The arrows mark attempted jumps which are mostly unsuccessful and the ion ends back in the minimum it tried to leave. (Dyre et al., 2009)

Chapter: Theory and model

When an electrical potential difference is applied to material, its movable charges transport, giving rise to an electric current. The conductivity σ is defined as the ratio of the current density j to the magnitude of the electric field E:

! ! = . !

(9)

Suppose ions with charge q are subjected to an electric field E. The result of an applied electric field is the force ! = !" ,

(10)

which is experienced by each ion, resulting in an average drift velocity v in the field direction. The ion mobility µ is defined by ! ! = . !

(11)

If the number of mobile ions is nmob, the current density is given by ! = ! !!"# ! ,

(12)

and we obtain the following expression for dc conductivity defined in eq. (10): ! = ! !!"# ! .

(13)

This equation tells us that the conductivity is proportional to the ion charge, to the number of mobile ions and to how easily an ion is moved through the solid. Application of fluctuation-‐ dissipation theorem18 implies the following expression for frequency dependent conductivity:

! ! =

!!! ! ! , !! !

(14)

where n is ion density and D(ω) the diffusion coefficient, which describes the properties of particle transport in matter and is given by (Scher & Lax, 1973) (15)

Fluctation-‐dissipatin theorem (FDT) is a powerful tool in statistical physics for predicting the behaviour of non-‐equilibrium thermodynamical systems. These systems involve the irreversible dissipation of energy into heat from their reversible thermal fluctuations at thermodynamic equilibrium.

Chapter: Theory and model

! 1 ! ! = − !! ! !!"# ! ! !" , 6 !

where ! ! is the mean square displacement of ion. Normal diffusion is characterized by the mean-‐ square displacement that is linear in time ! ! ∝ !, while anomalously diffusing particles display mean square displacement of the ! ! ∝ ! ! , where ! ≠ 1. For ! = 1 − !, eqs. (15) and (16) yields the expression for the low frequency limit of the ion conductivity in clay-‐water mixture ! ∝ ! ! . The ion surface diffusion transport is therefore subdiffusive, and could be characterized with the time-‐ dependent diffusion coefficient !(!) ∝ !" !! . This indicates that the experimentally observed dielectric properties result from anomalous ion transport characterized with time-‐dependent diffusion coefficient often observed in complex systems (Campos et al., 2004). Systems exhibiting anomalously slow diffusion, or subdiffusion, are usefully described by fractional differential equations19 (Sokolov et al., 2002), (Tarasov, 2008), (Tarasov, 2008). The fractional Liouville integral (Oldham & Spanier, 1974), (Sokolov et al., 2002) is defined by

!!! ! ! =

1 Γ α

! !′ !"′ . !!! !! ! − !′

(16)

The Fourier transform ℱ of this integral is given by the relation

ℱ!!! ! ! =

1 ℱ! ! . !" !

(17)

Fractional diffusion equation generalize Fick’s second law and the Fokker-‐Planck equation by taking into account power-‐law memory effects of fractal properties, such as trapping of charge carriers in

Chapter: Theory and model

amorphous matter. Such generalized diffusion equations describe complex systems with anomalous behaviour in much the same way as simpler systems. Random walks20 and diffusion serve as an interface between kinetics on one hand and derivatives and integrals of fractional order on the 19

Fractional dynamics is a field of study in physics, mechanics, mathematics, and economics studying the behavior of objects and systems that are described by using integrations and differentiation of fractional orders, by methods of fractional calculus. 20 Random walk is a random process consisting of a sequence of discrete steps of fixed length.

other. Furthermore, using random walk we can study complex networks (Gallos, 2004). Thus, combining complex network theory and kinetics in porous media leads to better understanding of processes taking place in porous architecture of materials.

2.3.2

Particle in complex porous matter

The mobility of ions reflects the long-‐time average ion displacement after many jumps influenced by charged mineral surface and pore fluid. If we consider some ions are very tightly bound (vibrating in a potential-‐energy minimum defined by the surrounding matrix) while other ions are quite mobile (jumping between different minima matter), the density of mobile ions is lower than the total ion density. The thing is, tightly bound ions sooner or later become mobile and mobile ions sooner or later become trapped. In the long run, all ions are equivalent, meaning the motion of ion consists of trapping events near the mineral surface with the characteristic waiting time scale τt followed by the Langevin motion21 in the pore space with an average time τr . We write the waiting times to depend on the particle size a:

!! =

! , !!!

(18)

and an average time of motion in the pore space to depend on the average pore size r:

!! =

! , !!!

(19)

Consider the motion of a particle through a medium. As a result of friction with the medium, the particle will be slowed down (i.e., its kinetic energy will be dissipated by heating up the medium). The motion of such particle is described by generalized Langevin equation. In Langevin dynamics apporach, dynamics of molecular systems is being modeled by simplified models while the omitted degrees of freedom are accounted by the use of stachastic differential equations (SDE).

Chapter: Theory and model

where vth is thermal velocity of the ion defined as

!!! =

!! ! , !

(20)

with m denoting the mass of the ion. We estimate particle size a and average pore size r using the specific surface area As, the density ρ, and porosity φ of the medium as: 6 , !! !

(21)

φ . (1 − φ)!! !

(22)

and

The stochastic motion of particle consisting of trapped and released events can be studied as random walk (RW), a mathematical formalisation for a series of sequential movements in which the direction and size of each move is randomly determined (Barber & Ninham, 1970). Since the term was introduced a century ago (Pearson, 1905), the RW has been extensively studied ever since. When using RW to study motion of ions, it was consistently shown that stochastic properties of random walker leads to slow kinetics of the particle or to macroscopically observed subdiffusion when the probability distribution of the waiting times exhibits power-‐law tail !! !(!)~ !!! . !

(23)

It has been shown (Metzler & Klafter, 2000) that averaging over microscopic Langevin dynamics and trapping events in such case results in the memory relation between the mean position and mean

Chapter: Theory and model

velocity: !" !! = !!!! ! , !" !!! !! ! where !!!!!!! is the Weyl fractional operator of order α defined as:

(24)

!!! ! ! = !!!!

1 Γ 1−!

! − ! !! !(!)!! ,

(25)

where Γ is the gamma function, defined as a function of integer n: Γ ! = ! − 1 ! .

(26)

The effect of mineral surface on the ions is twofold. Not only does it provide the mechanism for trapping the ion near the surface, the electric field of the charged surface acts on the ion moving in the pore space. The dynamics of the ion is determined by the drag force in the pore space, which is proportional to mass of the ion m, and drag force coefficient γ: !! = −!!! ,

(27)

and the overlapping electric fields of the charged mineral surfaces. In the triple layer model (TLM) (Figure 2-‐14) of near mineral surface electrostatics and electrochemistry (Leroy & Revill, 2004) there is a diffusive region placed between the compact Stern layer and the free electrolyte layer. The overlapping potential is in the first approximation a quadratic function of the distance:

!! ! = !! +

!!! !!! ! sinh ! . !!! !"

(28)

Here !! is the potential at the midplane between two mineral surfaces, !! is the salinity of the pore electrolyte, ! is the static dielectric constant of the pore electrolyte and ! is elementary charge.

Chapter: Theory and model

Figure 2-‐14: Triple layer model of near mineral surface electrostatics and electrochemistry (Leroy & Revill, 2004).

The overlapping potential leads to the harmonic restoring force

!! = −

!"!! = −!!!! ! . !"

(29)

Since we are interested in low frequency conductivity let there exist an external ac field ! ! = ! ! exp !"# acting on the ions. The equation of motion for ion in external field ! ! is: !!(!) ! = −!!(!) − !!! !(!) + !(!) . !" !

(30)

After applying Fourier transform to equation of motion we have:

Chapter: Theory and model

!! ! + !!! !(!) =

(31)

Using the memory relation between the Fourier components of position and velocity from eq. (25)

! !(!) . !

!" ! ! ! =

!! ! ! , !!!

(32)

we obtain: !"!! ! ! ! ! + !!! ! ! = !(!) , ! ! !!

(33)

where friction coefficient was set to ! = 1/!! . Finally, we obtain the expression for the current:

! = !"# ! =

! ! ! !"!!! ! !(!) . ! 1 + !"!!! !

(34)

The frequency dependent conductivity is thus

!(!) = !!

!"!!! ! . 1 + !"!!! !

(35)

Here the relaxation time !!! was introduced as

!!! = !! (!! !! )!!/! = !!

!" ! sinh

!!! !"

!!/!

(36)

where ! is the inverse Debye length22:

2! ! !! . !!! !"

(37)

Adding the dc part of the conductivity in eq. (36) we have Cole-‐Cole model for the complex conductivity:

! ! = !!" + (!!! − !!" )

!"!!! ! . 1 + !"!!! !

(38)

!=−

2ln (!! !! ) . ! ln ( !! ) !!

(39)

Debye length is the scale over which mobile charge carriers scren out electric fields in conductor. In other words, it is a distance over which significant charge separation can occur.

Chapter: Theory and model

Using the eq. (37) we can equivalently express the ! parameter of the Cole-‐Cole model as

The above represented derivation of frequency dependent conductivity depends on the assumption of the specific form of the probability distribution of waiting times. Extending to the complete frequency range we can obtain the generalized model for the frequency dependent conductivity (Samec et al., 2007):

! ! =

!! , 1 + !"!! + !"!! !!

(40)

where the constants !! , and !! can be expressed in terms of previously introduced characteristic times. We have tested the derived expression on preliminary measurements of clay-‐water systems (Table 2-‐1, Figure 2-‐15, Figure 2-‐16). Table 2-‐1: Values of the parameter used to calculate the real and imaginary parts of the conductivity shown in Figure 2-‐15 and Figure 2-‐16.

parameters σ0 [A/Vm]

Chapter: Theory and model

water content 36 %

56 %

0,43

0,725 -‐8

-‐8

τ1 [s]

0,9 * 10

1,45 * 10

τ2 [s]

3,2 * 10

-‐4

2,1 * 10

0,67

-‐4

Figure 2-‐15: Measured (open dots) and calculated (solid lines) real part of frequency dependent conductivity of kaolinitic clay samples at 36 % (lower curve) and 56 % (upper curve) water content (Samec et al., 2007).

Figure 2-‐16: Measured (open dots) and calculated (solid lines) imaginary part of frequency dependent conductivity of kaolinitic clay samples at 36 % (lower curve) and 56 % (upper curve) water content (Samec et al., 2007).

to 92 % were studied in (Lounev et al., 2002). An example of the obtained spectra are shown in Figure 2-‐17.

Chapter: Theory and model

The low-‐frequency part of pure kaolinit and sand samples at different moisture contents from 12 %

Figure 2-‐17: Low-‐frequency spectra (Lounev et al., 2002).

In this work (Lounev et al., 2002) were the low-‐frequency spectra described as a sum of two terms describing the response of ions in bound and free water layers of clay-‐water system: ! ! = !(!")! + !(!")! ,

(41)

where !, !, !, ! are the parameters of the model. The value of parameter describing the free water part is ! = 1 almost consistently through the whole moisture content range, while the parameter describing the bound water response ! is, for the water content 33 % -‐ 53 % (similar to the range of our data analysis presented in Figure 2-‐15, Figure 2-‐16 and Table 2-‐1, in the range 0.64-‐0.67. This is consistent with our findings where the low-‐frequency limit in our model follows from eq. (41):

Chapter: Theory and model

! ! ∝ (!")! .

(42)

3 Experiments What is the correlation between dynamical processes taking place in porous material and its structure? Using two methods we devised a way to look on the matter from multiple viewpoints: the first method being dielectric spectroscopy and X-‐ray computed tomography the second. This way we established how building materials behave to dielectric response and what is the microstructure of such materials. The sample building materials used in this research were cement and clay due to wide scope of application of both substances in the field of civil engineering. Clay, being significant constituent of soils (amongst sand and silt), has been used in construction of dwelling structures for centuries. A half of the world’s population lives or work in an earth building. It has long been speculated that earth materials have the ability to provide superior levels of indoor thermal comfort mostly because of their moisture storage potential, vapour permeability and specific heat capacity. Cement is one of the most common used building materials. It is an essential

However its application is not limited only to concrete structures. Because of the unique properties of concrete its value has also been acknowledged in production of “modified” earth building materials. Stabilised compressed earth materials are made using graded soils with the addition on a

Chapter: Experiments

ingredient for concrete thus making it the indispensable material for many building structures.

hydraulic binder (e.g. Portland cement) and either statically or dynamically compacted into moulds to form compressed earth bricks, or monolithically inside formwork to create rammed earth walls. (Hall & Allinson, 2010) The bricks can be built with using ordinary bricklaying trade practices and offer a number of advantages over conventional masonry bricks such as: §

the ability buffer indoor relative humidity and air temperature,

§

the reduction/removal of problems associated with condensation and mould growth, and

§

reduction in embodied energy through use of local soils and/or recycled material or demolition waste.

3.1 Specimens Clay and cement are both interesting and important complex materials whose physical properties and microstructure have recently been in the focus of intense research (Fossum, 2000), (Allen et al., 2007). Four major series of specimens were included in our research: pure clay, pure cement, clay and cement in layers and clay stabilized with cement (see Table 3-‐1 for the list of specimens used). The gravimetric water content was altered to control the saturation of the porous media. The clay sample used was source clay Kaolin KGa-‐1b purchased from The Clay Minerals Society with the primary constituent mineral kaolinite, containing also small amounts of metals (Ca, K, Na, Mg). The plastic and liquid limits of the sample were at 25,9 % and 40,1 % water content respectively. The

Chapter: Experiments

specific surface area of the clay sample was 10 m2/g and its solid specific gravity 2,6. The frequency dependence of the conductance and capacitance was determined first for dry sample, and then for wet samples. Wet samples were obtained with addition of distilled water to the clay.

Cement paste specimens were prepared of cement designated as CEM 42.5 R according to standard EN 197-‐1 (2000). Designation stands for Portland cement containing at least 95 wt % of clinker minerals, having compressive strength of at least 42,5 MPa at the age of 28 but not exceeding 62,5 MPa. R denotes a rapid development of strength in the early age of hydration. Three sets of cement paste were prepared having w/c ratios 0,3, 0,4 and 0,5. In case of multiphase specimens two different ways of sample preparation were used. In one case wet clay with gravimetric water content 30 % was inserted in the measuring cell. Cement paste with w/c ratio 0,4 was prepared separately and inserted in the measuring cell on top of the layer of wet clay (Fig. 2-‐1). In case of stabilized clay, dry clay and dry cement (6 % weight ratio of cement) were mixed together and distilled water was added; gravimetric water content of clay-‐cement sample was 30 %. For all samples containing cement, cement paste was inserted into the measuring cell immediately after mixing and the first record of capacitance C and conductivity G was made ~ 5 min after mixing.

Figure 3-‐1: Multiphase specimen showing layers of cement and clay.

Chapter: Experiments

The follow-‐up records were made in selected time intervals up to 456 h after mixing.

Table 3-‐1: Mix designs of specimens used for research no.

sample name

kaolinite water

cement

kaolinite

cement

content [%wt]

[w/c ratio]

[% wt]

water content

clay_0

clay_25

clay_50

clay_75

clay_100

100

clay_300

300

cement_0.3

0,3

cement_0.4

0,4

cement_0.5

0,5

CemKaol01

0,4

3.2 Dielectric spectroscopy Characterization of porous materials is not a trivial task experimentally as well as theoretically (Lounev et al., 2002), (Sanabria & Miller, 2006), (Rotenberg, B. et al., 2005). Numerous studies have already demonstrated and analyzed the anomalous properties of the diffusion transport in natural porous media caused by its heterogeneous nature (Hunt, 2005). On the other hand, certain anomalous features of the transport were also recently discovered studying random walks (Gallos, 2004) and dc electrical conductance (Lopez et al., 2005) on complex networks. Suitable technique to probe ion dynamical processes is dielectric spectroscopy (DS). Sometimes also

Chapter: Experiments

called impedance spectroscopy, this experimental method measures dielectric properties over a range of frequencies, therefore the frequency response of the system is revealed. It is based on the interaction of an external electric field with the electric dipole moment of the sample, often

expressed by permittivity23. Permittivity or dielectric constant is often used to compare electric properties of different matter. It is also extensively used in the soil physics for measuring water content of soils (Campbell, 1990), (Hallikainen et al., 2007). Dielectric spectroscopy is based on the interaction of an external field with the dielectric dipole moment of the sample (Volkov & Prokhorov, 2003), (Lambert, 2008). The dielectric properties differ a lot between materials and are a function of temperature, frequency of applied field, humidity, crystal structure and other external factors (Barsoum, 1997), (Hager & Domszy, 2004). When discussing dielectric spectroscopy it is imperative to understand basic theory of dielectrics. The measuring of dielectric properties with a method of dielectric spectroscopy is based on the measurement of voltage and current between a pair of electrodes. This enables the identification of conductivity and capacitance between the electrodes. When dielectric is placed in an electric field, electric current does not flow through the material (as in conductors). Electric charges only slightly shift from their equilibrium positions causing dielectric polarization: the positive charges within dielectric are displaced in the direction of the electric field and negative charges shift in the opposite direction. This slight separation of charge is known as the polarization. It creates an internal electric field that partly compensates the external field inside the dielectric (Britannica, 2011). The dielectric property of material is expressed by dielectric constant. A common example to enlighten dielectric principles, is a material inside a capacitor24. Consider two metallic parallel plates of area A separated by a distance d before and after placing dielectric

Permittivity is the measure of how much resistance is encountered when forming an electric field in a matter. In other words, it is a measure of how electric field affects a dielectric material. Permittivity is determined by the ability of a material to polarize in response to the field and thereby reduce the total electric field inside the material. 24 Capacitor is a device for storing electrical energy, consisting of a pair of conductors in close proximity and insulated from each other by dielectric medium, discovered independetly by Ewald Georg von Kleist and Pieter van Musschenbroek in 1745 and 1746 respectively with a capacitor called a Leyden jar (Williams, 1999).

Chapter: Experiments

between the plates (Figure 3-‐2).

vacuum

empty capacitor in vacuum

dielectric placed between the plates

dielectric

Figure 3-‐2: Capacitor before and after dielectric is introduced (Breuer, 1993). When a voltage is applied to a parallel-‐plate capacitor in vacuum, the capacitor will store charge. When a dielectric is placed between the plates, a charge stored on the parallel plates is greater.

If the charges on the plates are +Q and -‐Q, and V gives the voltage between the plates, then the capacitance is given by:

! . !

(43)

Plotting Q versus V should yield a straight line () and the slope of the line is the capacitance !! of the parallel plates in vacuum is given by: ! !! = !! , !

(44)

where !! is vacuum permittivity, equal to 8,85 x 10-‐12 F/m. If a dielectric is introduced between plates of capacitor, the capacitance ! is:

Chapter: Experiments

! ! = ! , !

where ! is dielectric constant, also called permittivity, a measure of how electric field affects, and is affected by, a dielectric medium.

(45)

Figure 3-‐3: Functional dependence of Q on applied voltage. Slope of the line is related to the dielectric constant of the material. (Barsoum, 1997)

Usually, when comparing dielectrics, relative permittivity, is used:

!! =

! . !!

(46)

Since !! is always smaller than ! , the minimum value of !! is 1. Combining eqs. (46) and (47), the capacitance of the metal plates separated by the dielectric is

! = !! !!

! = !! !! . !

(47)

Relative dielectric constant !! is dimensionless number that measures the charge-‐storing capacity, relative to that stored in vacuum. The presence of an electric field E in a dielectric material causes the bound charges in the material (atomic nuclei and their electrons) to slightly separate, inducing a local electric dipole moment. When the electric field is removed the material returns to original state and the time required to do

dielectric material polarizes in response to an electric field and is defined as the constant of proportionality relating an electric field E to the induced dielectric polarization density P such that ! = !! !" .

(48)

Chapter: Experiments

so is called relaxation time. The electric susceptibility ! is a quantitative measure of how easily

The susceptibility in related to relative permittivity of the medium by (49)

! = !! − 1 . The electric displacement field D is defined as

(50)

! = !! ! + ! .

The electric dipole moment is a measure of the separation of positive and negative charges. It is defined as (51)

! = ! ! ,

where ! is the displacement vector pointing from the negative to the positive charge (Figure 3-‐4).

Figure 3-‐4: Definition of an electric dipole moment; ! is a vector pointing from the negative to the positive charge. The electric dipole moment indicates the orientation of the dipole (positions of the charges) and has nothing to do with the direction of the field originating the cause of dielectric formation.

If N is a number of dipoles per unit volume it can be shown: ! = ! ! .

(52)

!"# = !! − 1 , !! !

(53)

Combining eqs. (49) and (53) yields

Chapter: Experiments

which is essential for understanding the dielectric response of the material. It states that the greater the distance between the charges of a dipole for a given applied field, the greater the relative

dielectric constant. In other words, the greater dielectric constant, the more polarizable a medium is. In solids, different charged entities can polarize (e.g., electrons, protons, cations, anions) so basically most important polarization mechanisms are electronic, ionic and dipolar polarization. The first two mechanisms are shown in Figure 3-‐5. In contrast to electronic and ionic polarization, dipolar polarization results from permanent dipoles, asymmetric bonds between atoms, which maintain polarization without the presence of external electric field (Lambert, 2008).

(a)

(b)

II.

Figure 3-‐5: Electronic and ionic polarization mechanisms: I. Electronic polarization occurs when the electron cloud is displaced relative to the nucleus it is surrounding (a) at Equilibrium, i.e., in the absence of an external electric field, (b) in the presence of an external electric field; II. Ionic polarization, the displacement of positive and negative ions toward the negative and positive electrodes, respectively (a) ion positions at equilibrium, (b) when an external electric field is applied, the centre of negative charge is no longer coincident with the centre of positive charge. (Barsoum, 1997)

Chapter: Experiments

(a) (b)

In general, dielectric response of the material to applied external field is nonlocal in time. This means the charges need some time to polarize. Thus, the more general formulation of polarization density as a function of time is: !

!(!) = !!

! ! − ! ! ! ! ! !"′ .

(54)

The function ! ! − ! ! is called non-‐local time function of the dielectric response to applied electric field. As the polarization is not instantaneous, the observation time must be longer than time which corresponds to the cause of the response (thus integration from −∞ to t). It is more convenient to take the Fourier transform25 and rewrite polarization density as a function of frequency: (55)

! ! = !! ! ! ! ! , Similar, after Fourier transforming the electric displacement field is

(56)

! ! = ! ! ! ! ,

where ! is frequency of applied field. Frequency-‐dependent dielectric function introduced in eq. (57) is !

! ! =1−

! ! ! !"# !" = 1 +

! ! ! !"# !" .

(57)

Frequency-‐dependent dielectric function is a complex quantity. The real part ! ! is related to the capacitive behaviour or ability of the material to be polarized by an external electric field, while the

Chapter: Experiments

imaginary part ! ! ′ is related to energy dissipation26. The dielectric function is simply written

Fourier transform is the operation that transforms one complex-‐valued function of a real variable into another. The original signal depends on time (time domain representation of the signal), whereas the Fourier transform depends on frequency (frequency domain representation of signal).

! ! = ! ! ! − ! !! ! .

(58)

The real and imaginary part of the dielectric function are not independent functions but are connected through Kramers-‐Kronig relations (Landau et al., 1984):

! ! ! = !! +

! 2 !! !! ! ! !" . ! ! !! − !!

(59)

Dielectrics exhibit a momentary delay in the dielectric constant, known as dielectric relaxation introduced by chemist Peter Debye in 1913 (Williams, 1975). The cause of this phenomenon is in the fact that when the field is applied, or released, a finite time is required for the charges to come to their equilibrium orientation because there is a viscous resistance to these rotary motions. This relaxation is often described with permittivity as a function of frequency (Debye, 1913). The dielectric relaxation response of ideal systems can be described by the Debye equation:

! ! = !! +

∆! , 1 + !"#

(60)

where !! is the permittivity at the high frequency limit, ∆! = !! − !! where !! is the static, low frequency permittivity, and ! is the characteristic relaxation time of the matter. However, classical Debye behaviour is hardly ever observed experimentally. Instead different dielectric spectra for various matter are described by the power laws over extended frequency ranges (Jonscher, 1983), (Jonscher, 1999). This fractional power-‐law frequency dependence is easily displayed in terms of the

universal response. The behaviour

Dissipation embodies a concept of dynamical system where mechanical models lose energy over time, typically from friction or turbulence. Energy converts into heat and so the temperature of the system is raised. Such systems are called dissipative systems.

Chapter: Experiments

dielectric susceptibility ! ! = ! ! ! − ! !! ! for all kind of substances, which is called a law of

! ! ! ~! !!! ,

! !! ! ~! !!! ,

(61)

0 < ! < 1, ! ≫ !! , and ! ! 0 − ! ! ! ~! ! ,

! !! ! ~! ! ,

(62)

0 < ! < 1, ! ≪ !! , where ! ! 0 is the static polarization and !! the loss-‐peak frequency, is observed over wide frequency range. The above expressions serve as the definition of the universal response behaviour. For the region ! ≫ !! , the universal fractional power law (eq. (62)) can be presented in the following form (Tarasov, 2008): ! ! = !! !" !! ,

0 < ! < 1

(63)

with a positive constant !! and ! = 1 − !. The polarization density can be written as ! !, ! = ℱ !! ! !, !

= !! !! ℱ !!

= !! ℱ !! ! ! ! !, !

(64)

!" !! ! !, !

where ! !, ! is Fourier transform ℱ of ! !, ! . Eq. (65) can be represented by integrals of arbitrary order ! = 1 − !. The fractional power law (eq. (64)) results ! !, ! = !! !! !!! ! !, ! ,

0 < ! < 1

(65)

Chapter: Experiments

For the region ! ≪ !! , the universal fractional power law can be presented as

! ! = ! 0 − !! !" ! ,

0 < ! < 1

(66)

with a positive constant !! , ! 0 , and ! = !.This law can be presented by the fractional Liouville !

derivative denoted by !! (Oldham & Spanier, 1974), (Sokolov et al., 2002). The differential operator !

!! of order ! is defined by the equation

!! ! ! =

! ! !!! ! ! ! !!! !

1 !! ! ! ! ! !! ! × ! , Γ k−β !! !! ! − ! ! !!!!!

(67)

(! − 1 < ! < !)

The Fourier transform of this derivative is given by !

(68)

ℱ!! ! ! = !" ! ℱ! ! . As a result, the fractional power law (eq. (67)) gives the polarization density in the form !

! !, ! = !! ! 0 ! !, ! − !! !! !! ! !, ! ,

0 < ! < 1 .

(69)

Eqs. (66) and (70) are considered to be the universal laws for the time domain (Tarasov, 2008). The analysis of low-‐frequency dielectric response (or moving ion polarization effects at low frequencies) in porous geomaterials is a complex task due to various polarization mechanisms that occur at frequencies below 1 MHz. There are four main polarization mechanisms in this range: the Maxwell-‐Wagner polarization, the polarization of the diffuse layer, the polarization of the Stern layer, and the membrane polarization(Revil & Cosenza, 2010). All four mechanisms coexist in colloidal suspensions (Figure 3-‐6 A) approximately corresponding to the very high water content limit in our case, in granular matter and for macroscopic scale representation only Maxwell-‐Wagner

approach here, we do not attempt to distinguish and extract these different polarization mechanisms contributions from measured data, but instead present a model based on effective single ion dynamics (see model description in Theory and model and Discussion sections of this thesis).

Chapter: Experiments

and membrane polarization contribute to the dielectric response (Figure 3-‐6 B and C). In our

Figure 3-‐6: Schematic characterization of three types of porous geomaterials (Revil & Cosenza, 2010).

When measuring dielectric properties of matter a measuring cell is used. It consists of two metal plan parallel electrodes and non-‐conducting casing. Electric field is established between electrodes, resulting electric potential and current between plates are measured. In the dielectric spectroscopy experiment the admittance as a function of frequency of the sample is measured from which the complex dielectric function ! = ! ! + !! !! is then deduced. The measured capacitance and conductance of the sample give the real and imaginary part of the dielectric

Chapter: Experiments

function: ! ! = ! ! !! ,

(70)

The values of capacity C and conductance G, which are then used in the calculation of dielectric function, are standardised to the values independent of electrodes arrangement. This is done by

! ! = ! !! !"! ,

calculating the cell constant. In case of plate capacitor cell constant is a value of capacity of empty capacitor C0: ! !! = !! , !

(71)

where A is area of plates in m2, d is the distance between the plates in m and ε0 is vacuum permittivity. In the next step we can calculate the real part of dielectric function, which is dimensionless number and is independent of the geometry of the system:

!! =

! , !!

(72)

where ε’ is the real part of dielectric function and C measured capacity. The imaginary part of dielectric function is a function of electric conductance G:

! !! ! =

! , !! !

(73)

where ω is angular frequency of connected voltage (! = 2!").

3.2.1

Measuring cell

Specimens were loaded into the measuring cell, which was specially designed for the purpose of our research. The measuring cell is a capacitor in the form of cylinder, consisting of two metal plates – electrodes, in the form of a circle with a diameter r = 15,5 mm. Each plate is round with the area A = 1,89 cm2. The upper plate is movable, which allows the measuring cell to be filled with a sample. The

Chapter: Experiments

distance between the plates d was measured after the cell was being filled.

Figure 3-‐7: Sketch of plan parallel capacitor, filled with the sample.

3.2.2

Measuring device

Dielectric spectroscopy measurements were performed using instrumentation QuadTech 1920 Precision LCR Meter at the Chair of Applied Physics, Faculty of Civil Engineering, University of Maribor, Slovenia. The device enables automatic measurements of capacity and conductivity of the sample in the frequency range from 20 Hz to 1 MHz and has the amplitude between 20 mV and 1 V. To check the consistency of the measuring set-‐up we have numerically computed the Kramers-‐ Krönig relations from the measured data. Figure 3-‐8 shows the results of the comparison between

Chapter: Experiments

the measured real part of dielectric function and computed real part from the imaginary one (for the clay-‐water sample at 32 % water content).

Figure 3-‐8: Measured real part of the dielectric function and the numerically computed values from measured conductance (Kramers-‐Krönig transform).

3.2.3

Measurements

The measuring cell filled with the sample was connected to the measuring device, which was set with appropriate constant parameters. Initially, measuring cell was connected to the device using cables. Few measurements were done and we soon discovered that the cables impedance is large and depends on cables’ length and position therefore results shown were unreal or unsatisfactory for further research (e.g., obtained results from multiple measurements of identical sample were too diverse to estimate the difference as negligible error). The problem was not resolved even with device calibration, because the position of the cables should be constant for all further measurements, which is practically impossible. As a solution to this problem a special fitting was obtained that attaches to the measuring device

smaller than cables’ resistance, the results obtained after calibration were correct, because they were representation of only actual events in the sample. All was set for starting with the main measurements.

Chapter: Experiments

connector. Since the fitting was fixed in the same position and the resistance of such connector is

The capacity and conductivity in the low-‐frequency range was measured; in terms of frequencies: 20 Hz, 50 Hz, 100 Hz, 200 Hz, 500 Hz, 1 MHz, 2 MHz, 5 MHz, 10 MHz, 20 MHz, 50 MHz, 500 MHz, and 1 GHz. The room temperature was measured with a thermometer. Measured results were written in the tables.

Figure 3-‐9: Measuring cell, filled with sample, connected to the instrument.

3.3 X-‐ray computed tomography X-‐ray computed tomography (CT) is a non-‐destructive method for visualizing heterogeneous

Chapter: Experiments

materials and for obtaining digital information on their 3-‐D geometries and properties. Since introduction of X-‐ray CT systems in 1970s (Beckmann, 2006) by Godfrey Hounsfield27 and Allan McLeod Cormack in medical sciences, the application of the technique to numerous other scientific 27

It has been claimed that thanks to the success of The Beatles, EMI could fund research and build early CT scanners (in fact called the “EMI-‐Scanner”) for medical use. (Whittington Hospital, 2005)

fields and industry followed, such as soil science(Petrovic et al., 1981), materials science(Baruchel et al., 2000), metrology(Garant, 2010), and food industry(Lim & Barigou, 2004). Tomography imaging consists of directing X-‐rays at an object from multiple orientations and measuring the decrease in intensity along a series of linear paths. This decrease is characterized by Beer-‐Lambert law, which relates the absorption of light to the properties of the material through which the light is travelling. This means intensity reduction is a function of X-‐ray energy, path, length, and material linear attenuation coefficient (Figure 3-‐10). A specialized algorithm is then used to reconstruct the distribution of X-‐ray attenuation in the volume being imaged.

Figure 3-‐10: Principle of Beer-‐Lambert law. Object A is of the same thickness as object B but yield a higher absorption due to higher density or higher atomic number. Object C consists of the same material as object B but absorbs less radiation than the thicker object B.

3.3.1

How X-‐ray inspection system works / X-‐ray production

To fully understand how X-‐ray CT operation works – starting from X-‐rays and ending with 3D model of sample – we have to start at the beginning. X-‐rays are a highly-‐energetic form of electromagnetic radiation with a wavelength in the range of 0.01 to 10 nanometers, approximately 1000 to 1,000,000 times smaller than the wavelength of light (Figure 3-‐11). Due to their being highly energetic, X-‐rays

Chapter: Experiments

are able to pass through matter.

Figure 3-‐11: The electromagnetic spectrum. X-‐rays have a photon energy ranging from 1.2 to 240 keV (www.phoenix-‐ xray.com).

The production of X-‐rays takes place in X-‐ray source, also referred to as the X-‐ray tube (Figure 3-‐12). This is the most important system component in determining the overall quality of the image acquired in the X-‐ray CT process(Webb, 2002). The X-‐ray tube has two electrodes: a negatively charged cathode, which consists of tungsten filament and acts as the electron source, and a positively charged anode. An electric current from power source passes through tungsten filament, causing it to heat up. When the temperature reaches ~2200°C the thermal energy absorbed by the tungsten atoms allows a small number of electrons to move away from the metallic surface, a process called thermionic emission. The large positive voltage applied to the anode causes these free electrons to accelerate toward the anode, due to the potential difference between filament and anode. Electrons enter through a hole in the anode and the beam is then directed into an electromagnetic lens which focuses the electron beam to a small spot of a few microns in diameter on the target. The target consists of thin layer of tungsten deposited on a plate of light metal which

Chapter: Experiments

also serves as an exit window for the X-‐ray radiation. In the tungsten layer the electrons are abruptly decelerated whereby X-‐rays are generated.

Figure 3-‐12: Detail sketch of X-‐ray source. Electrons are emitted from a heated filament and accelerated towards the anode by the potential difference and enter through a hole in the anode into magnetic lens which focuses the electron beam to a small spot on a target (www.phoenix-‐xray.com).

The specimen to be inspected is placed between X-‐ray tube and the detecting device (Figure 3-‐13). The specimen’s position can be manipulated in all three dimensional axis. The z-‐axis describes the line between tube and detector. Moving the specimen along z-‐axis closer or further away from X-‐ray

Figure 3-‐13: X-‐ray inspection system consists of the X-‐ray tube and the detector. Remotely controllable manipulating unit allows positioning the specimen within the beam (www.phoenix-‐xray.com).

Chapter: Experiments

source affects geometric magnification.

When X-‐rays pass through a specimen, part of the X-‐ray spectrum is absorbed. The higher the density or the thicker the matter is, the more X-‐rays are absorbed and do not pass through. The ones that manage to pass through the matter strike a detector and thus an image is created, where X-‐rays appear in different shades of gray according to their intensity. Parts of the matter that consist of more dense materials such as lead are displayed darker than low density materials such as water or air. The final image is created in image intensifier (Figure 3-‐15). After X-‐ray strike the scintillating layer of the input screen of the image intensifier, they are transformed into optically visible rays of light. From the photocathode this light releases a beam of electrons carrying the image information. The electrons are accelerated and focused by electric fields onto a phosphor screen where a bright and visible image is produced which can be recorded(phoenix|x-‐ray GE Sensing & Inspection Technologies, 2009).

measuring cell

pore space

Chapter: Experiments

solid space

Figure 3-‐14: A 2-‐D image acquired from X-‐ray CT representing stabilized clay sample.

Figure 3-‐15: The image intensifier consists of scintillator input window, photocathode, electro-‐optics, luminescent screen and an output window. (www.phoenix-‐xray.com).

3.3.2

Principles of CT sample scanning

Generating three-‐dimensional images using CT starts with the acquisition of a series of two dimensional X-‐ray images while progressively rotating the sample step by step through a full 360° rotation at increments of less than 1° per step (Figure 3-‐16). These projections contain information on the position and density of absorbing object features within the sample. This data set is then used for the numerical reconstruction of the volumetric data. The final image is displayed as a series of sectional images or a 3-‐D image. In order to reproduce an accurate reconstruction of the volumetric data, the entire sample

completely displayed in each projection or line profile captured in the acquisition process. Hence, the magnification is limited by sample diameter d and detector width D.

Chapter: Experiments

depth/diameter must remain within the beam cone of radiation so that the full diameter is

Figure 3-‐16: Operating principle of X-‐ray sample scanning. X-‐rays pass through a specimen and hit a detector, where two dimensional X-‐ray image is created. (www.phoenix-‐xray.com).

Resolution V in the X-‐ray tomography image equals the voxel size of the reconstructed volume data set and is defined as:

! !

(74)

where P is detector and M geometric magnification. M is defined as the ratio:

!"" !"#

(75)

where FDD is focus-‐detector-‐distance and FOD focus-‐object-‐distance. Parameters affecting geometric magnification are sample diameter d and size of the effective detector surface D:

!!"# =

! !

(76)

Chapter: Experiments

The smallest voxel size possible, and therefore the best resolution obtained, is defined as:

!" !

(77)

Figure 3-‐17: Voxel resolution depends on focus-‐object-‐distance, focus-‐detector-‐distance, sample diameter and size of the effective detector surface (www.phoenix-‐xray.com).

A CT image is usually called a slice, as it corresponds to what the specimen being analysed would look like if it were sliced along a plane. A CT slice corresponds to a certain thickness of the specimen being scanned; so, a CT slice image is composed of voxels (volume elements), unlike a typical digital image composed of pixels (picture elements). A complete volumetric representation of an object is obtained by acquiring a contiguous set of CT slices. The CT image shows different materials as different shades of gray resulting from X-‐ray attenuation which reflects the proportion of X-‐rays scattered or absorbed as they pass through each voxel. X-‐ray attenuation is primarily a function of X-‐ ray energy and the density and composition of the material being imaged (Ketcham & Carlson,

Figure 3-‐18: Model of volume data set of 6 x 6 x 6 voxels. The voxel is associated with a grey value and has a size of V = pixel size / magnification.

Chapter: Experiments

2001).

3.3.3

Tomography system and image acquiring

Clay-‐cement-‐water samples 28 were scanned using a Phoenix Nanotom 180NF (GE Sensing and Inspection Technologies) with electron acceleration energy of 100kV and current of 90µA. 1440 angular projections were collected over 360 degree rotation. Projection images were the mean of 3 images each with a 500ms exposure time. The resolution of the scanners output device was 2000 x 2000 pixels and spatial resolution of each volume unit (voxel) was 8 µm x 8 µm x 8 µm. Image stacks were prepared in Volume Graphics VGStudioMax V2.0 (VolumeGraphics, 2010). The scanning was conducted at the Department of Environmental Sciences, School of Biosciences, University of Nottingham, United Kingdom.

1. sample scanning

2. image processing

3. 3-‐D visualization of pore space

Figure 3-‐19: From image scanning to 3-‐D pore space quantification.

3.3.4

Image analysis

Collection of images – image stacks – acquired at scanning were 2000 x 2000 x 1800 pixels in size

Chapter: Experiments

(cca. 1.1 GB). Each frame within this was 2000 x 2000 pixels which provided a resolution of 8 microns per pixel. Individual grey-‐level image stacks for each specimen were processed using public domain software ImageJ V1.43u (Rasband, 2010). Image sequences varied between samples due to different

Specimens CemKaol01 and SC were scanned at three different hydration times.

specimen size, and images taken from the centre of each sequence were used for image analysis. Image manipulation was performed in ImageJ to isolate pore space (Figure 3-‐20). This involved cropping and rescaling of each sample to exclude the area outside the specimen’s measuring cell. Contrast enhancement was applied to normalize all slices and consequently reduce the effect of large differences in grey-‐level of same particle in two neighbouring slices. This unwanted effect arises because the power of X-‐ray beam is not constant throughout the process of CT scanning of the sample. In order to separate pores from solid matrix automatic Yen threshold algorithm (Yen et al., 1995) was used, because it showed the most consistant and meaningful threshold assessment compared to other automatic algorithms. It is significant to use the same automatic threshold algorithm in all the analysis to get the comparable results between different specimens.

Chapter: Experiments

a) acquired image

b) cropped and scaled image

c) contrast enhancement

d) automatic threshold algorithm

Chapter: Experiments

e) binary image

Figure 3-‐20: Image analysis procedure. a) Greyscale image of the specimen is acquired using X-‐ray CT; b) original image is cropped and scaled; c) contrast is enchanced to visible separate pore space from solid matrix; d) automatic Yen threshold algorithm was used to determine pore space; e) the result is a binary image which is further analysed for different properties.

3.4 Data analysis The data obtained by the methods described in the previous section was analyzed and the results of dielectric spectroscopy and X-‐ray CT are shown in the subsections bellow.

3.4.1

Analysis of data acquired from dielectric spectroscopy

The electrical characteristics of the specimens placed between two planparallel electrodes (area S = 1,89 cm2, distance d = 5,5 – 10,6 mm) were obtained in the experiment as conductance G and capacity C in frequency range from 20 Hz to 1 MHz. To lose the effect of changeable distance between the electrodes (caused by different volume of the samples) we recalculated the measured data to obtain comparable values. From conductance G we derived conductivity σ:

!=!

!! !!

(78)

where G is measured conductance, ε0 is vacuum permittivity with value 8,854 x 10-‐12 A s/(V m) and C0 capacity of empty capacitor (= !! !/!). Measured capacity C is used to calculate the real part of dielectric function ε’ (= !/!! ). We measured electrical characteristic of the same sample with different values of distance d to see how distance affects measured quantities. It turns out that from both derived quantities, real part of dielectric function ε’ and conductivity σ the latter is meaningful quantity for comparing samples when the distance d is not the same among different samples (Table 3-‐2).

results of conductivity σ, real and imaginary part of dielectric function, ε’ and ε’’ respectively, are shown in figures bellow. Different types of specimens were analyzed separately. When measuring clay, we were interested how electrical properties vary as a function of water content (shown in Figure 3-‐22, Figure 3-‐23, Figure 3-‐24 and Figure 3-‐25) whereas for specimens containing cement

Chapter: Experiments

Finally, we calculated imaginary part of dielectric function ε’’ (= !/(! ∙ !! ) , where ! = 2! !). The

electric response at different hydration time was measured (shown in Figure 3-‐26, Figure 3-‐27, Figure 3-‐28, Figure 3-‐29). Data for measurements of cement samples was collected from previous experiments conducted in our lab. Table 3-‐2: The ratios of real part of dielectric function ε’ and conductivity σ for the same sample of clay_300 measured at two different values of d (d1 = 18,1 mm, d2 = 6,6 mmi). All ratios σd1 / σd2 have the same magnitude, while some εd1 and εd2 differ more than 3-‐times, indicating the conductivity σ is meaningful quantity for sample comparison of electrical characteristics. frequency ν

σd1

ε d1

σ d2

ε d2

σd1/σd2 ·∙100%

εd1/εd2 ·∙100%

[Hz]

[S/m]

3,44E-‐03

2,83E+05

3,53E-‐03

8,79E+05

97,48

310,36

3,55E-‐03

8,12E+04

3,99E-‐03

2,63E+05

89,01

323,37

100

3,58E-‐03

2,69E+04

4,30E-‐03

9,50E+04

83,16

352,93

200

3,62E-‐03

9,90E+03

4,48E-‐03

3,36E+04

80,77

339,14

500

3,65E-‐03

1,99E+03

4,65E-‐03

6,82E+03

78,56

342,84

1000

3,68E-‐03

2,85E+02

4,72E-‐03

9,19E+02

78,01

321,82

2000

3,69E-‐03

-‐1,31E+03

4,76E-‐03

-‐1,38E+03

77,63

105,78

5000

3,65E-‐03

-‐1,86E+03

4,72E-‐03

-‐2,42E+03

77,19

130,17

10000

3,34E-‐03

-‐1,84E+03

4,34E-‐03

-‐2,41E+03

76,96

131,27

20000

2,52E-‐03

-‐1,30E+03

3,19E-‐03

-‐1,73E+03

79,00

133,09

50000

1,31E-‐03

-‐4,09E+02

1,71E-‐03

-‐5,36E+02

76,99

131,19

100000

9,31E-‐04

-‐1,09E+02

1,19E-‐03

-‐1,46E+02

78,55

133,58

200000

7,89E-‐04

-‐2,16E+01

1,02E-‐03

-‐2,75E+01

77,56

127,08

500000

7,94E-‐04

2,07E+00

9,44E-‐04

5,44E+00

84,10

263,46

1000000

9,49E-‐04

7,47E+00

1,07E-‐03

8,40E+00

88,93

112,40

All of the conductivity spectra show similar features: the increasing part for very low frequencies (< 10 kHZ), reaching the plateau of maximum values (10 kHz < ω < 30 kHz), decreasing part in the intermediate range (30 kHz < ω < 1 MHz) and again the increasing part into the high frequency

Chapter: Experiments

region. The large increase of capacitance (or equivalently real part of the dielectric function) is often ascribed to electrode polarization effect in electrolyte solutions and colloidal suspensions. This is usually modelled with the equivalent circuit containing combination of frequency dependent

capacitors and resistors with the high frequency ! !! decay. A detailed analysis of the electrokinetics model for colloidal suspension given by Hollingsworth and Saville (Hollingsworth & Saville, 2003) also yielded the same power law, ! !! for the electrode polarization. On the other hand it was shown (Cirkel et al., 1997) that the real part of the dielectric function decays as ! !!/! when the influence of the migrating charges in a finite space on the electric field is considered. Also it was found that the real part of the dielectric function starts to deviate from the high frequency value at crossover frequency !! ∝ !! !/! which scales with the high-‐frequency conductivity. Another prominent feature of the measured data is the range of negative values of the real part of the dielectric function. The generalized conductivity model describes this effect well as displayed in Figure 3-‐21.

measurement (open dots) and the data (closed dots) obtained from Carrier and Soga (Carrier & Soga, 1999).

Chapter: Experiments

Figure 3-‐21: The range of negative values of the real part of dielectric function for clay-‐water system from our

7,E-‐03

0% 25% 50% 75%

6,E-‐03

100% 300%

conduc<vity σ [S/m]

5,E-‐03

4,E-‐03

3,E-‐03

2,E-‐03

1,E-‐03

0,E+00 10

100

1000

10000

100000

Chapter: Experiments

frequency ν [Hz]

Figure 3-‐22: Measured conductivity spectra of clay specimens with various water content levels.

1000000

1,E+06

0% 50%

25%

1,E+06

50% 75% 100%

9,E+05

300%

8,E+05

7,E+05

real part of dielectric func<on ε'

75% 6,E+05

100%

5,E+05

4,E+05

3,E+05 300% 2,E+05

25%

1,E+05

0% 0,E+00

-‐1,E+05

100

1000

10000

100000

1000000

frequency ν [Hz]

Figure 3-‐23: Real part of dielectric function (calculated from measured capacity) spectra of clay specimens with various water content levels. Measurements are marked with points – lines connecting the markers are plotted for better overview.

Chapter: Experiments

5,E+02

0% 0,E+00 10

100

1000

10000

100000

1000000 frequency ν [Hz]

25%

real part of dielectric func<on ε'

-‐5,E+02

-‐1,E+03

-‐2,E+03

300%

0% 25% -‐3,E+03

100%

50%

75%

75% 100% 300%

50%

Chapter: Experiments

-‐3,E+03

Figure 3-‐24: Detailed display of real part of dielectric function spectra of clay in the range of negative values of capacity (same measurements as in Figure 3-‐23). Measurements are marked with points – lines connecting the markers are plotted for better overview.

1,E+07 0% 75% 50% 100% 300%

25% 50% 75%

1,E+06

100% 25%

300%

1,E+05

imaginary part of dielectric func<on ε''

1,E+04

1,E+03

1,E+02

100% 300%

0% 1,E+01

1,E+00 100

25%

1000

10000

100000

1000000

10000000 0%

1,E-‐01

Figure 3-‐25: Imaginary part of dielectric function (calculated from measured capacity) spectra of clay at different water content levels. Measurements are marked with points – lines connecting the markers are plotted for better overview.

Chapter: Experiments

frekquency ω [Hz]

0,8

0 0,25 0,5 1 2 3,42 7,75 10 26 120 144 94,33 288

3,42

0,7

3,42 7,75 7,75 2 0,6

10 3,42 2 2

3,42

7,75 10 2 3,42

10 7,75

0,5 conduc<vity σ [S/m]

7,75 10

3,42

0,4

7,75 2 10

26 26

1 26

0,3 3,42 2 7,75 10 0,2

1 26

26 1

0,5

0,1

2 1 0,5 0,25 26 10 3,42 94,33 0 7,75 120 144 288

0 10

2 0,25 3,42 10 94,33 0 26 7,75 1 94,33 0,5 0,25 0 94,33 120 0 120 120 144 144 144 288 288 288 100

0,5

0,25 0,25

26 0,5 0,25

0 94,33 0 94,33 94,33

0,25 0

7,75 3,42 7,75 2 1 10

94,33 0

120 120

0 94,33 120

3,42 1 26 0,5

0,25

120

0,5

0,25

0,5

7,75 2 10

120

94,33 120

144 144 144

144 144

288 288 288

1000

10000

144

3,42 2 7,75 10

3,42 3,42 7,75 7,75 10 10 2 2

0,5 1 1 0,25 26 1 1 26 26 26 0,5 0,5 0 0,25 0,5 0,5 0,25 94,33 0 0,25 0,25 0 0 0 120 94,33 94,33 94,33 94,33 120 120 120 120 144 144 144 144 288 288 144 288 288 288 100000 1000000

Chapter: Experiments

frequency ν [Hz]

Figure 3-‐26: Measured conductivity spectra of multiphase clay-‐cement water system (specimen CemKaol01) at different hydration times (in hours). Measurements are marked with points – lines connecting the markers are plotted for better overview.

1,2

0 1 4 8 15 1

0 4

conduc<vity σ [S/m]

0,8

0,6

0,4 1 0 4

0,2

15 30

100

1000 frequency ν [Hz]

10000

100000

1000000

Figure 3-‐27: Measured conductivity spectra of cement paste with 0,3 w/c ratio (specimen cement_0.3) at different hydration times (in hours). Measurements are marked with points – lines connecting the markers are plotted for better overview.

Chapter: Experiments

0,9

0 1

4 8

0,8

15 30

0 0,7 4 8

0,6

conduc<vity σ [S/m]

0,5 0

4 0,4

15 0,3

8 0,2

0,1 30 30 0 10

100

1000

10000

100000

1000000

Chapter: Experiments

frequency ν [Hz]

Figure 3-‐28: Measured conductivity spectra of cement paste with 0,4 w/c ratio (specimen cement_0.4) at different hydration times (in hours). Measurements are marked with points – lines connecting the markers are plotted for better overview.

0,9

0 1 4

0,8

15 30

0,7

0,6

conduc<vity σ [S/m]

0,5 0 1

0,4

0,3 15

0,2

0,1 30

30 0 100

1000

10000

100000

1000000

frequency ν [Hz]

Figure 3-‐29: Measured conductivity spectra of cement paste with 0,5 w/c ratio (specimen cement_0.5) at different hydration times (in hours). Measurements are marked with points – lines connecting the markers are plotted for better overview.

Chapter: Experiments

3.4.2

Analysis of data acquired from X-‐ray CT

Binary images were analysed and data for porosity, pore size and circularity was obtained. Areas smaller than 5 pixels (=336 µm2) were excluded from the analysis. Data for each slice of any specimen was recorded and average values were calculated (Table 3-‐3). Table 3-‐3: Data analysis of different specimens. sample

average pore size

perimeter

porosity

circularity

[µm ]

[mm]

[%]

cement 0 h

946

0,130280

4,70

0,69937

cement 2 h

945

0,123051

2,70

0,72173

cement 24 h

923

0,127010

4,11

0,70707

clay 0 h

3331

0,223884

0,83

0,68927

clay 2 h

4022

0,247481

1,00

0,68024

clay 24 h

3497

0,233920

1,01

0,66495

SC 0 days

4285

0,241832

2,56

0,66450

SC 14 days

2455

0,191862

2,89

0,68797

SC 28 days

3884

0,252968

2,78

0,67895

Porosity is a measure of the pore space in a material; it is a fraction of the volume of voids over the total volume, between 0 and 1 or as percentage between 0 and 100 %. When analysing images acquired with X-‐ray CT this means that the porosity or area fraction is the percentage of black pixels in the binary image. Circularity, or roundness, is a dimensionless quantity representing the degree to which a shape is compact. It is defined as:

Chapter: Experiments

circ. = 4!

(79)

A value of 1.0 indicates a perfect circle. As the value approaches 0.0, it indicates an increasingly elongated shape. Following graphs show distribution of porosity in depth for different specimens obtained from X-‐ray CT data.

area perimeter !

porosity [%] 0

0 cement 0 h cement 2 h

0,2

cement 24 h

depth [mm]

0,4

0,6

0,8

1,2 Figure 3-‐30: Distribution of porosity in depth for cement in different hydration times.

porosity [%]

5 4 3 2

2 ‹me [h]

Figure 3-‐31: Total porosity to hydration times of cement. Error bars indicate standard error.

Chapter: Experiments

porosity [%] 0

0 clay 0 h 0,1

clay 2 h clay 24 h

0,2

depth [mm]

0,3 0,4 0,5 0,6 0,7 0,8 0,9 Figure 3-‐32: Distribution of porosity in depth for clay in different sorption times.

porosity [%]

5 4 3 2

Chapter: Experiments

2 ‹me [h]

Figure 3-‐33: Total porosity to sorption times of clay. Error bars indicate standard error.

porosity [%] 0

0 SC 0 days SC 14 days

SC 28 days

depth [mm]

10 Figure 3-‐34: Distribution of porosity in depth for stabilized clay in different hydration times.

Chapter: Experiments

porosity [%]

5 4 3 2 1 0

‹me [day]

Figure 3-‐35: Total porosity to hydration times of stabilized clay. Error bars indicate standard error.

Figure 3-‐36: Two-‐dimensional representation of soil porous structure. Binary image showing pore space (black areas) and soil matrix (white areas).

Chapter: Experiments

Figure 3-‐37: Images of three different soil pore structures. These images were taken outside the purpose of this thesis, but are used here to demonstrate how complexity of the network of soil porous structure is quantified (see chapter 4). Image courtesy of Department of Environmental Sciences, University of Nottingham.

Figure 3-‐38: Three-‐dimensional visualization of the soil pore architecture from stack of X-‐ray CT scans. The pore space is shown while the soil matrix is here transparent. The height of the specimen shown here is 16 mm and the diameter is 14 mm.

Chapter: Experiments

In Figure 3-‐39 we show examples of 2D X-‐CT soil structure images and the corresponding normalized cumulative pore size distribution

! ! =

! ! ! !"′

(1)

extracted from the image analysis showing scaling behaviour with scaling exponent in the interval 1,8 ≤ ! ≤ 2. The pore size distributions obtained from a set of contrasting soil samples (mainly different textures from sands to clays) all follow similar scaling law ! (!) ∝ ! !! (Mooney & Korošak, 2009), (Cárdenas et al., 2010), with the scaling exponents in the interval 1 < ! < 2.

Chapter: Experiments

CLAY

φ = 0,9 %

CEMENT

φ = 1,8 %

φ = 4,2 %

STABILIZED CLAY

φ = 2,4 %

φ = 4,2 %

Figure 3-‐39: Cumulative size distribution (left), determined from X-‐CT image (right) for clay, cement and stabilized clay (from top to bottom, respectively) of different porosity φ. All samples show scale-‐free behaviour of size distribution (solid line) with the exponent α=1,8-‐2.

Chapter: Experiments

4 Discussion We will start the discussion with introducing a physical model through which we can explain the possible origin of the power-‐law distribution of waiting times given by eq. (24) and show how the parameters of the Cole-‐Cole model (eq. (39)) are governed by the properties of the system. Then, we will compare experimental data from dielectric spectroscopy with the generalized conductivity model (eq. (41)). Further, we will analyse X-‐ray CT experimental data using network models introduced in subchapter 2.2. We start with the idea introduced by Bunde et al. (Bunde et al., 1986) in the context of biased diffusion in random medium. Imagine a particle moving between two points chosen at random in porous medium experiencing random delays along its minimal path caused by the interaction of the moving particle with the surface of the large particle. The characteristic time of the delay ! (!) depends on the effective length of the large particle L. Let L be a random variable with the

Chapter: Discussion

probability distribution !! (!) that depends on the microstructure of the porous medium. The length scale we use to scale the effective length is defined by the first moment of !! (!). Here we consider the overlapping spheres model in which the random two-‐phase medium consists

Figure 4-‐1: Imaginary part of the complex conductivity obtained using eq. (39) with the parameters: !!! = 10−4, !!" = 0,04 A/Vm, !!! = 0,67 A/Vm, ! = 0,67. The open dots are experimental points for pure kaolinite measured with dielectric spectroscopy. (Samec et al., to be published). Data from: Mesure de la réponse PPS (Polarisation Provoquée Spectrale) de mélanges artificiels argilo-‐sableux non consolidés Gonca et al. (private comm.)

−4

Figure 4-‐2: Real and imaginary part of the complex conductivity obtained using eq.(39) with the parameters: !!! = 1,5 10 , !!" = 0,04 A/Vm, !!! = 0,67 A/Vm, ! = 0,67. The open dots are experimental points for pure kaolinite measured with dielectric spectroscopy. (Samec et al., to be published)

Chapter: Discussion

Figure 4-‐3: Schematic of the chord-‐length measurements for a cross section of a two-‐phase random medium. The chords are defined by the intersection of lines with the two-‐phase interface. (Torquado & Lu, 1993)

of identical spheres of radius R and choose for the probability distribution !! (!) the chord-‐length distribution introduced by Lu and Torquato (Torquado & Lu, 1993) thus !! (!)!" is the probability of finding a length segment between L and L+dL lying in the solid phase, the ends of the length segment being at the interfaces between the two phases: !! ! ∝ (1 − !)! ≡ ! !!" ,

(2)

where ! is the porosity. Bunde et al. calculated the characteristic time ! ! for biased diffusion in random structures: 1+! ! ! ! ∝ ≡ ! !!" . 1−!

(3)

Chapter: Discussion

Here 0 < ! < 1 is the topological bias driving the walker away from the minimal path. In our picture of an ion experiencing trapping and releasing events near the pore surface the effect of topological bias is to enhance the probability for the ion to stay trapped and to decrease the probability to be

released. We set the topological bias is in our case set equal to the partition coefficient of the TLM (Leroy & Revill, 2004) defined as the ratio of the number of counter ions in the Stern layer to total number of counter ions. Thus the values of the partition coefficient close to 1 indicate large influence of the mineral surface on the motion of ions. Using the above equations we can calculate the distribution of waiting times. Since ! ! = !! ! !"/!" we have: ! ! ~! ! !!!/! ,

(4)

where !/! = ln (1 − !)/ln ((1 − !)/(1 + !)). Comparing with eq. (24) we see that Cole-‐Cole parameter ! is:

ln (1 − !) . 1−! ln 1+!

(5)

We have derived the functional dependence of the parameters ! and !!! of the Cole-‐Cole model on the details of the porous medium in a closed form without any free parameters. The exponent ! depends on the porosity or the pore size of the system and on the partition coefficient of the mineral surface electric field structure. The relaxation time !!! depends on the pore size, clay particle size, salinity of the pore electrolyte and the midplane potential bewteen charged mineral surfaces. Following figures displays the dependence of the Cole-‐Cole parameter on the medium

Chapter: Discussion

properties.

Figure 4-‐4: Dependence of Cole-‐Cole parameter ! on relaxation time !!! according to eq. (37). The salinity was set to -‐3

!! = 10 M, the midplane potential was !! = 10 mV, the pore size was != 3. (Samec et al., to be published)

We can compare the obtained depence of the Cole-‐Cole parameter on relation time derived from our model:

!=−

2 ln !! !! , ! ln !! !!

(6)

with the result obtained from the self-‐similar (fractal) properties of complex materials (Feldman et al., 2002):

!=−

! ln !! !!! , ! 2 ln !! !!

(7)

Chapter: Discussion

where d is the fractal dimension of the system, !! is the characteristic frequeny for diffusion process and !! is the cutoff time. Although the expressions are not identical it is interesting to note that the similar description of overall dependence of the Cole-‐Cole exponent on relaxation time (Figure 4-‐4 and Figure 4-‐5).

Figure 4-‐5: Cole-‐Cole exponent vs relaxation time (Feldman et al., 2002).

Figure 4-‐6: Cole-‐Cole parameter ! as a function of pore size !. The partition coefficient was set to ! = 0,8. (Samec et al., to

Chapter: Discussion

be published)

Figure 4-‐7: Cole-‐Cole relaxation time as a function of pore size. The partition coefficient was set to ! = 0,8. The salinity was -‐3

set to !! = 10 M, the midplane potential was !! = 10 mV. (Samec et al., to be published)

Figure 4-‐8: Cole-‐Cole relaxation time as a function of pore electrolyte salinity. The pore size of the system was set to size

Chapter: Discussion

!= 25 nm, the midplane potential was !! = 10 mV. (Samec et al., to be published)

The experimentally observed conductivity spectra of clay-‐water mixtures, hydrating cement paste and clay-‐cement systems can also be well described with the expression given in eq. (41). From Figure 4-‐9 and Figure 4-‐10 it seems that the dielectric spectra of hydrating cements separate in two distintc groups depending on the hydration times.

Figure 4-‐9: Scaled low-‐frequency part of conductivity: open symbols -‐ early hydration times 0 h -‐12 h, solid symbols -‐ late hydration times 30 h -‐179 h.

Figure 4-‐10: Normalized low-‐frequency part of conductivity: open symbols -‐ early hydration times 0 h -‐12 h, solid symbols -‐ late hydration times 30 h -‐179 h. Solid lines were computed with eq.(41) using α=0.82 for early and α=0,6 late hydration times.

Chapter: Discussion

In contrast to cement samples, the clay-‐cement mixtures do not show clear separation into early and late hydration periods, but their conductivity spectra rather span the whole band of values.

Figure 4-‐11: Clay-‐cement system low-‐frequency normalized conductivity spectra (sample CemKaol01).

We can follow the time evolution of the scaling exponent by analyzing the measured cement paste conductivity spectra over the whole hydration time. In Figure 4-‐12 we show the time evolution of the scaling exponent obtained from the analysis of dielectric spectra compared with the results of NMR study (Blinc et al., 1988). We can see a clear transition between early and late hydration time

Chapter: Discussion

detected in both experiments.

Figure 4-‐12: Scaling exponent dependence on hydration time. Open dots – NMR data (Blinc et al., 1988), solid triangles – dielectric spectroscopy data (Korošak et al., 2010).

Let’s look now at the the porous structure of building materials from complex network models point of view. In Figure 3-‐39 we showed examples of 2D X-‐CT soil structure images and the corresponding normalized cumulative pore size distribution, and extracted from the (in part) observed scaling behaviour the scaling exponent in the interval 1,8 ≤ ! ≤ 2. The pore size distribution therefore follows the power-‐law form !(!) ∝ ! !! .

(8)

In the SN model, this type of pore size distribution yield the scale-‐free organization of soil pore structure with the folllowing degree distribution ! ! ∝ ! !! .

(9)

The scaling exponent of the degree distribution can in this network model be analytically calculated (Masuda et al., 2005) and is

! =1+

so γ=[2,3] when

! !!!

!(! − 1) , !

(10)

≤ ! ≤ 2!/(! − 1).

Here the degreee distribution exponent depends on the parameters of the model !, ! , and the euclidean dimension of the space D. We show the computed cummulative degree distributions for

! ! =

! ! ! !"′

(11)

both, SN and EN, network models in Figure 4-‐13. We have chosen two specific cases: m ≈ 1 and

significant role, the networks are scale-‐free with ! = 2 , while for for large values of m the networks have compact, geometrical (random) structure with Poissonian degree distribution.

Chapter: Discussion

! ≫ 1. For low values of the parameter m where the spatial positions of the nodes play less

Figure 4-‐13: Cumulative degree distributions of SN (left) and EN (right) models for small and large values of the parameter m. The scaling exponent of the scale-‐free degree distribution obtained for small m (straight line in plots) is ! = 2. The fits are power-‐laws (straight lines) and cummulative Poissonian distributions (Samec et al, 2010).

SN and EN soil network models are built on data obtained from image analysis (to determine the positions and areas of the pores) of the soil structure. We can also search for possible correlations in the structure of soil images and try to relate them to the derived network structure. The correlation properties of statistical heterogeneous porous medium are best described by the two-‐point correlation function !! !! , !! . First we define the characteristic function h of the pore space

ℎ ! =

1, if ! at pore . 0, otherwise

(1)

The correlation function is then defined as the average

Chapter: Discussion

!! !! , !! = ℎ !! ℎ !! .

Given !! !! , !! , the porosity ! is obtained as ! = !! 0 , and the limit lim!→∞ !! ! = ! ! is also valid (Yeong & Torquato, 1998a), (Yeong & Torquato, 1998b). For the isotropic systems !! !! , !!

(2)

depends only on the distance ! = !! − !! . The normalized version of the two-‐point correlation function ! ! is

! ! =

!! ! − ! ! . ! 1−!

(3)

Now we have ! 0 = 1 and lim!→∞ ! ! = 0. In Figure 4-‐14 we show the computed two-‐point correlation function and the normalized correlation function for the soil sample image shown in Figure 3-‐36. The computation is performed by averaging over two perpendicular directions of the image. The computed correlation function in Figure 4-‐14 can be well described with a power-‐law correlation function for a fractal medium (Freltoft et al., 1986) (4)

! ! ∝ ! ! !!!! ! !!/ϛ ,

where ! , !! , ! are the Euclidean dimension, the fractal dimension and the cutoff length. For the analyzed image the fractal dimension was found to be !!≈ 1,7.

Figure 4-‐14: Two-‐point correlation function and its normalized version computed for the soil image shown in Figure 3-‐36. (Korošak & Mooney, 2011)

Chapter: Discussion

Using the described image correlation analysis we can compare the statistical properties of the real soil images and the images created with a network mechanism described above. Figure 4-‐15 displays the evolution of the binary image at four instances of simulation time. We constructed them using the heterogeneous preferential attachment mechanism setting b=0 so the new nodes of the network had equal intrinsic weights. The algorithm to produce the image was as follows: we start with the empty 2D array representing the soil image. In each step of the computation a new node j is introduced into the 2D array (by randomly chosen coordinates) and connected to a randomly chosen existing node i with the probability given by the network model. A pixel representing pore space is then set in the image array and the link is drawn in the network model. The algorithm is stopped when the desired porosity of the simulated soil structure is reached. The porosities of the presented images are ! = 0.09, 0.11, 0.16, and 0.18 ( from upper left to lower right panel in Figure 4-‐15). Figure 4-‐16 shows the two-‐point correlations for all four cases shown and the normalized correlation functions for first and last image. The fractal dimensions obtained from fitting the normalized correlation functions are ! − !! = 0,6 for the first and ! − !! = 0,5 for the last image. The cutoff length of the correlation function increased from ϛ = 15 to ϛ = 18 from the first to the last image. This might point to the possible changes of the fractal structure of soils with increasing porosity and demonstrates the statistical similarity between images of real soil samples and the ones generated by network models.

Chapter: Discussion

Figure 4-‐15: Binary image obtained with growing network algorithm simulating soil pore structure at early (upeer left) and late (lower right) stage of computation. The porosities are: 0.09, 0.11, 0.16, 0.18.

o o 0.1

o o o o o o oo o o oo oooo o o o oo o o oo o oo oooo o o oo ooo o o ooooo o o oo o o o oo o o oo oooo oooo o o oo oooo ooooo

o o

S 2H r L

0.05

0.02

0.01 1

10 r

simulated binary soil image from Figure 4-‐15 (right). The correlation functions on the left panel correspond to porosities 0.09, 0.11, 0.16, 0.18 from bottom to top. The two fitted normalized correlation function correspond to porosities 0.1, and 0.18.

Chapter: Discussion

Figure 4-‐16: Two point correlations for simulated binary soil images (left) and normalized correlation function of the two

Yet another way to explore the properties of the porous structure is to perform numerical experiments with random walks on binary soil images. An example of the random walk on the 2D X-‐ CT soil image is presented in Figure 4-‐17. The random walker is allowed to move only from pore to pore (black areas on the image) with a prescribed step length (chosen so that the walker performs enough steps and explores all parts of the image). The (normalized) mean square distance as a function of time is given in Figure 4-‐18 (upper curve). For comparison the random walk on the random image, generated to have the same porosity as the soil image, was performed and produced a straight line in Figure 4-‐18 as expected. The random walk on the soil image show sublinear behaviour !(!)! ∝ ! ! with ! ≈ 0,7.

Chapter: Discussion

Figure 4-‐17: Random walk on the 2D X-‐CT soil image. The random walker is allowed to move only from pore to pore (black areas). (Samec et al., 2010)

Figure 4-‐18: Mean square distance as a function of time and normalized mean square distance as a function of time for random walks on soil images; straight line represents random walker on the random image, generated to have the same porosity as the soil image, presented as straight line. (Samec et al., 2010)

We note here that the sublinear time dependence of mean squared distance of the random walker corresponds to subdiffusive transport process in the porous structure as detected in the dielectric spectra and desribed using fractional calculus approach earlier in this and previous chapters. Furthermore, the studies of random walks on complex (scale-‐free) networks (Gallos, 2004) have also shown that the random walk on these structures is anomalous (see Figure 4-‐19), changing from super-‐ to sub-‐diffusive regime with the increasing value of the degree distribution scaling exponent. The crossover from super-‐ to sub-‐linear behavior seems to be at ! = 3.

Chapter: Discussion

Figure 4-‐19: Mean squared distance as a function of time and normalized mean squared distance (inset) as a function of time for random walks on scale-‐free networks with different degree distribution exponents (Gallos, 2004).

In the SN network model we have ! ≥ 3 when ! ≥ 2!/(! − 1). So to correspond to observed sub-‐ linear regime of transport the network models should have the network parameter m set large enough leading to a more compact network structure. From the shape of the correlation function we expect that the probability to find a pore ! away from the chosen pore is a power-‐law (up to a cutoff length) with the scaling exponent !! − !. In the SN network is the the probability that two nodes at distance ! are connected also given by a power-‐law ! ! ∝ ! !! !!!

(5)

(Masuda et al., 2005). The probability that two pores at the distance ! pores are linked in the network sense will then also fall off with the distance as the power-‐law ! !δ with the scaling

Chapter: Discussion

exponent

! = ! − !! + ! ! − 1

(6)

The value of the scaling parameter ! plays a crucial role and determines the network structure when we start to add long-‐range links with the probability ! !δ to the ordered network such as a lattice (dimension D). If δ < 2! (Petermann & De Los Rios, 2006) then the resulting network is a small world while to navigate the network optimally the scaling paramter should be ! = ! (Kleinberg, 2000). These observations might also be valid for other network types besides lattices so in the SN network model the optimal m is !! , !−1

(7)

! + !! . !−1

(8)

!!"# = and the critical

!! =

We note that these two limiting cases correspond to !!"# =

! !!!

, !! =

!! !!!

with Df -‐> D. To

further explore the effect of the network parameter m on node correlations we calculated the complexity of the pore network based on node-‐node link correlations (Anastasiadis et al., 2005), (Claussen, 2007), (Claussen, 2007). The complexity measure is here defined as the entropy of node-‐ node link correlations that are given with the matrix elements !!,! ! counting the number of links between nodes with degrees k and k' in the network: !

(9)

!!" !!! ! !!! ! ! ! ≠ !′ !!,! ! =

!,!!! !

1 2

, !!" !!! ! !!! ! ! ! = !′

!,!!!

links two nodes with the degree diﬀerence Δ!, !!! is then constructed from the node-‐node link correlation matrix !!,! ! with:

Chapter: Discussion

where !!" are the elements of the adjacency matrix. The probability that a randomly chosen edge to

!!! =

!!"# !!! !!,!!!! !!! . !!"# !! !!"# !!! !!,!!!! !!!! !!!

(10)

Here is the denominator equal to M (total number of edges) and the numerator gives the number of edges connecting node pairs with the degree diﬀerence Δ! for any two ! , ! ! . Finally, the complexity is defined as an entropy measure of !!! and measures how widely are the degree diﬀerences of connected node pairs in the network distributed: !!"# !!

ℎ=−

(11) !!! log !!! .

!!!!

A network with a large h would have many links that connect nodes of various degrees, while networks with low h will have mostly links connecting node pairs of almost the same degree resembling a lattice-‐like network or a network close to a complete graph. We should observe this transition of pore network complexity ℎ(!) from large to smaller values as a function of the network parameter m. The correlation matrices for a smaller geographical threshold network are shown in Figure 4-‐21 for ! ≪ 1 (Fig. a), ! = 1 (Fig. b) and ! ≫ 1 (Fig. c). Again, we see that with the increasing value of the parameter m the network changes from disassortative (nodes with different degrees preferably connected) to assortative type. To quantify the complexity h of the soil pore structure represented with the network, we computed the entropy of the normalized diagonal sums of the correlation matrix !!" . Finally, we have computed the complexity of several soil pore structures with different scaling exponents of their pore size distributions at ! = !! /2. The results are displayed in Table 4-‐1.

Chapter: Discussion

Table 4-‐1: Complexity h of several soil pore structures with different scaling exponents of their pore size distributions at ! = !! /2. (Samec et al, 2010)

1,4

1,5

1,6

2,85231

3,06864

3,48277

Results of the computed soil pore structure complexity shown in Table 4-‐1 indicate that the complexity measure h based on the complex network construction might be senstitive to differences in soil structure through dependence on pore arrangement and distribution. The influence of the network parameter m on the correlation properties of the network is presented in Figure 4-‐21 where we have plotted the values of the correlation matrix !!,! ! for three different cases: ! ≪ 1, ! = 1 and ! ≫ 1. The high correlation values are clustering along the diagonal (nodes with similar number of neighbors are mostly connected) as we increase the network parameter.

Chapter: Discussion

a) m≪1

b) ! = 1

Chapter: Discussion

c) ! ≫ 1

Figure 4-‐20: Correlation matrices for a smaller geographical threshold network. (Samec et al, 2010)

5 Conclusions In times of growing need to reduce energy consumption and improve indoor living comfort stabilised earth building materials have major potential to be a sustainable alternative. In order to provide superior levels of bioclimatic indoor environment it is vital to know and understand the structure and transport processes through the building envelope which are of paramount importance for solving problems in building physics dealing with moist transport in building's envelope, the building-‐ ground interaction, and in transport of contaminants in the vicinity of the repositories where the transfer of moist (in vapor or fluid phase) through soil can be the source of contamination. The key result of this thesis is the demonstration that there is a correlation between the large scale structure of the pore space and the properties of the motion of charged particles through the pore space. This was achieved by conducting and analyzing the results of two experiments: the structure of pore space of selected porous materials (soil samples, clays, cements, clay-‐cement mixtures) was

particles in the samples was probed using low-‐frequency dielectric spectroscopy. It iwas shown that the microscopic motion of the ions in a complex environment of clay-‐water system can be described with fractional dynamics leading to subdiffusive behaviour. A physical

Chapter: Conclusions

investigated using state-‐of-‐the-‐art X-‐ray computed microtomography, while the dynamics of charged

model for explaining the power law distribution of waiting times was introduced showing how the parameters of the model depend on the structural properties of the porous system: the relaxation time and high frequency conductivity depend on the material properties of the samples, the only free parameter is the scaling exponent of the distribution of waiting times. X-‐ray computed tomography and complex network models were used to determine the large-‐scale topological organization of the pore structure. The image analysis of 2D X-‐CT image slices was used to obtain pore positions and sizes and the determination of the local as well as total porosity of the samples. Pore size distributions were determined for the analyzed images and it was shown that in soil samples, clay-‐cement mixtures and hydrated cements they follow the power-‐law form with the scaling exponent between 1 and 2. The pore-‐to-‐pore connections in complex network of studied system follow a power law degree distribution, revealing the apparent scale-‐free topology of studied materials. It was shown that topology of the network depends on fractal properties of the media obtained from random walk and multifractal analysis of X-‐ray computed tomography soil images. Finally, the entropy of node-‐node link correlations was used to quantify the complexity of soil porous architecture allowing us to relate the scaling exponent of the pore size distribution and the complexity of the network of soil porous structure. The work done is a combination of experimental research, using two experimental methods, and modelling of analysed data using advanced and expanded network models to model pore structure and generalized conductivity model.

Chapter: Conclusions

The dielectric spectra were measured at several moisture content of clay samples and several hydration ratios of cements. All obtained spectra of frequency depedent conductivity shared similar features: the increasing conductivity with frequency at the low end of the frequency range (100 Hz – 10 kHz) and decreasing high frequency part (10 kHz -‐ 1 MHz).

100

To explain the observed features in frequency dependent conductivity a model for ions moving in a complex porous space was used in which the surface of the solid matrix plays a double role. It provides the trapping mechanism for ions near the charged surface and causes the electric field that influences the moving of the ion in the pore space. The triple layer model of near mineral surface electrostatics was used to describe the overlapping potential of the charged surfaces. The one particle model was obtained by the averaging over microscopic dynamics with trapping events with power-‐law distributed waiting times. The specific form of the waiting times distribution introduced fractional dynamics into the one particle equation of motion which was shown to lead to the Cole-‐ Cole form of the frequency dependence of the conductivity. The scaling exponent of the waiting times distribution was shown to depend on the Cole-‐Cole relaxation time or the link between the microscopic properties of motion and the phenomenological parameters of the Cole-‐Cole model was found. Furthermore, the possible origin of the power-‐law distribution of waiting times itself was suggested by combining the biased diffusion and the overlapping spheres models of random media. The scaling exponent of the waiting times distribution was found to depend on the total porosity and on the ratio between the density of ions in the Stern layer near the charged surface and the total density of ions. Complex network models in which the nodes of the network are the centers of pores and the links are drawn according to the specified rules were used to describe the structure of pore space. The treshold network model was further expanded and compared to the growing network model built with the same data. Both models showed that at the large scale the porous space is organized as a scale-‐free network. The dependence of the degree distribution scaling exponent is given in terms of

between the two-‐point correlation function of the porous structure and the dependence of the probability to connect two pores on their distance were compared. This was used to show that the critical value of the network parameter can be related to the fractal dimension of the porous structure as determined from the image (box counting method).

Chapter: Conclusions

the scaling exponent of the pore size distribution and a network parameter. Also the similarities

101

To correlate the dynamics of particles moving in pore space to the properties of the porous structure the numerical simulation of random walk was performed on the 2D X-‐CT images. The time (steps) dependence of the mean squared distance showed sublinear behavior indicating the subdiffusive dynamics consistent with the power-‐law waiting time distribution used in derivation the frequency dependent conducitivity. Using network model description, the quantification of the porous structure complexity is suggested using entropy based measure of node-‐node link correlations. By computing the complexity of different soil samples (using the same network parameters) we have shown that this kind of complexity measure is sensitive to differences in soil porous architecture. We expect to continue and exted the research presented here along at least three lines: §

to link the findings of charged particle transport properties with the microstructures of clays and similar multi-‐scale materials, taking the full electrochemistry into account (Dufreche et al., 2010), (Rotenberg et al., 2010),

§

to study the impact of anomalous diffusion processes on macro scale thermo-‐physical properties and heat and mass transport in porous materials (Hall & Allinson, 2009), (Hall & Djerbib, 2004), (Hall, 2010) for improvement of energy efficiency and thermal comfort by optimizing earth building materials. It should be stressed that the samples used in our research were not identical to stabilized rammed earth materials used in (Hall & Allinson, 2008), (Hall & Allinson, 2009) due to the constraints of the experimental set-‐up and sample preparation, but both samples have similar microscopical composition,

Chapter: Conclusions

§

102

to use the developed complex network models of soil porous architecture to understand given soil functions such as the self-‐organization of soil-‐microbe complex (Young & Crawford, 2004). As the first, preliminary step in this direction we have computed the

complexity measure h for three different soil types containing population of microbes as a function of time (Figure 5-‐1).

Figure 5-‐1: Time evolution of complexity in soil-‐microbe system for three soil samples. Plotted is the complexity vs time for all three soil samples (CL -‐ , LS -‐ , SL -‐ ). The same set of parameters was used and ! set to: ! =

! !!!

(! is the exponent of

the pore size distribution).

The complexity seems to decrese in the first 5 months after the introduction of microbes into soil, but then starting to increase. From the view point of the evolving pore network structure it seems that in the first months after the start of microbial activity the soil pore structure moves to more homogeneous one, while in later times the heterogeneity of the porous structure gradually

We hope that the findings discriminating between the physical and biological perturbations information on connection would be key in elucidating the microbial effects.

Chapter: Conclusions

increases.

103

6 Appendix Experimental results that were not presented in detail in chapter 3 Experiments (only the graphic presentations were given) are introduced in the following tables. Table 6-‐1: Measured conductivity spectra of clay specimens with various water content levels (see Figure 3-‐22 for graphic presentation of the results). ν [Hz]

σ [S/m] for various water content levels 0 %

25 %

50 %

75 %

100 %

300 %

2,57E-‐08

6,15E-‐04

4,73E-‐03

5,09E-‐03

4,69E-‐03

3,62E-‐03

1,16E-‐06

6,88E-‐04

5,14E-‐03

5,37E-‐03

4,78E-‐03

3,74E-‐03

100

1,09E-‐08

7,35E-‐04

5,46E-‐03

5,33E-‐03

4,95E-‐03

3,77E-‐03

200

4,25E-‐09

7,78E-‐04

5,61E-‐03

5,41E-‐03

5,00E-‐03

3,83E-‐03

500

3,86E-‐08

8,22E-‐04

5,80E-‐03

5,53E-‐03

5,13E-‐03

3,90E-‐03

1000

6,04E-‐08

8,66E-‐04

5,96E-‐03

5,61E-‐03

5,21E-‐03

3,96E-‐03

2000

1,95E-‐07

9,04E-‐04

6,09E-‐03

5,72E-‐03

5,30E-‐03

4,01E-‐03

5000

5,53E-‐07

9,39E-‐04

6,15E-‐03

5,72E-‐03

5,30E-‐03

3,99E-‐03

10000

1,72E-‐06

9,12E-‐04

5,77E-‐03

5,33E-‐03

4,95E-‐03

3,69E-‐03

20000

4,36E-‐06

7,40E-‐04

4,45E-‐03

4,19E-‐03

3,73E-‐03

2,77E-‐03

Chapter: Appendix

104

50000

6,00E-‐06

4,55E-‐04

2,36E-‐03

2,22E-‐03

1,97E-‐03

1,43E-‐03

100000

3,17E-‐06

3,47E-‐04

1,65E-‐03

1,54E-‐03

1,35E-‐03

1,00E-‐03

200000

1,97E-‐06

3,12E-‐04

1,44E-‐03

1,34E-‐03

1,17E-‐03

8,71E-‐04

500000

5,96E-‐06

3,91E-‐04

1,53E-‐03

1,40E-‐03

1,23E-‐03

8,54E-‐04

1000000

2,15E-‐05

4,55E-‐04

1,65E-‐03

1,45E-‐03

1,29E-‐03

1,01E-‐03

Table 6-‐2: Real part of dielectric function (calculated from measured capacity) spectra of clay specimens with various water content levels (see Figures Figure 3-‐23 and Figure 3-‐24 for graphic presentation of the results). ν [Hz]

ε’ for various water content levels 0 %

25 %

50 %

75 %

100 %

300 %

-‐4,06E+01

9,53E+04

1,03E+06

6,14E+05

4,76E+05

2,35E+05

-‐5,67E+02

2,88E+04

2,68E+05

1,63E+05

1,22E+05

4,12E+04

100

1,84E+01

1,29E+04

1,00E+05

5,25E+04

4,36E+04

2,24E+04

200

8,04E+00

5,65E+03

3,80E+04

2,15E+04

1,75E+04

8,61E+03

500

6,50E+00

1,86E+03

1,08E+04

6,37E+03

4,70E+03

1,82E+03

1000

5,09E+00

7,03E+02

3,13E+03

1,39E+03

9,50E+02

-‐3,31E+02

2000

4,37E+00

1,34E+02

-‐4,09E+02

-‐9,61E+02

-‐1,16E+03

-‐1,39E+03

5000

4,08E+00

-‐2,37E+02

-‐2,39E+03

-‐2,36E+03

-‐2,40E+03

-‐2,04E+03

10000

2,86E+00

-‐3,42E+02

-‐2,80E+03

-‐2,70E+03

-‐2,55E+03

-‐2,02E+03

20000

6,06E-‐01

-‐2,80E+02

-‐2,11E+03

-‐2,03E+03

-‐1,84E+03

-‐1,40E+03

50000

-‐2,14E+00

-‐9,46E+01

-‐6,90E+02

-‐6,54E+02

-‐5,93E+02

-‐4,40E+02

100000

-‐2,50E+00

-‐2,38E+01

-‐1,92E+02

-‐1,81E+02

-‐1,63E+02

-‐1,17E+02

200000

-‐2,63E+00

-‐9,63E-‐01

-‐3,70E+01

-‐3,43E+01

-‐3,11E+01

-‐2,15E+01

500000

-‐2,56E+00

5,52E+00

4,73E+00

3,59E-‐01

2,60E+00

5,51E+00

1000000

-‐1,84E+00

5,06E+00

8,57E+00

7,17E+00

8,87E+00

1,11E+01

Table 6-‐3: Imaginary part of dielectric function (calculated from measured capacity) spectra of clay specimens with various water content levels (see Figure 3-‐25 for graphic presentation of the results). ε’’ for various water content levels 0 %

25 %

50 %

75 %

100 %

300 %

2,30E+01

5,52E+05

4,24E+06

4,57E+06

4,21E+06

3,25E+06

4,17E+02

2,47E+05

1,84E+06

1,93E+06

1,72E+06

1,34E+06

100

1,95E+00

1,32E+05

9,79E+05

9,56E+05

8,89E+05

6,76E+05

200

3,81E-‐01

6,98E+04

5,04E+05

4,85E+05

4,48E+05

3,44E+05

500

1,39E+00

2,95E+04

2,08E+05

1,98E+05

1,84E+05

1,40E+05

1000

1,08E+00

1,55E+04

1,07E+05

1,01E+05

9,36E+04

7,11E+04

2000

1,75E+00

8,11E+03

5,46E+04

5,14E+04

4,76E+04

3,59E+04

5000

1,98E+00

3,37E+03

2,21E+04

2,05E+04

1,90E+04

1,43E+04

10000

3,08E+00

1,64E+03

1,04E+04

9,56E+03

8,89E+03

6,61E+03

20000

3,91E+00

6,64E+02

3,99E+03

3,75E+03

3,35E+03

2,48E+03

50000

2,15E+00

1,63E+02

8,49E+02

7,98E+02

7,08E+02

5,14E+02

100000

5,69E-‐01

6,22E+01

2,96E+02

2,77E+02

2,43E+02

1,79E+02

200000

1,77E-‐01

2,80E+01

1,30E+02

1,20E+02

1,05E+02

7,81E+01

500000

2,14E-‐01

1,40E+01

5,50E+01

5,02E+01

4,43E+01

3,06E+01

1000000

3,85E-‐01

8,16E+00

2,96E+01

2,59E+01

2,31E+01

1,81E+01

Chapter: Appendix

ν [Hz]

105

Table 6-‐4: Measured conductivity spectra of multiphase clay-‐cement water system (specimen CemKaol01) at hydration times from 0 to 6,75 hours (see Figure 3-‐26 for graphic presentation of the results).

ν [Hz]

σ [S/m] for different hydration times (in hours) 0

0,25

0,5

0,75

2,58

3,42

4,1

6,75

3,71E-‐02

4,32E-‐02

4,45E-‐02

4,55E-‐02

4,61E-‐02

4,70E-‐02

3,85E-‐02

3,99E-‐02

3,80E-‐02

4,36E-‐02

6,64E-‐02

9,31E-‐02

1,02E-‐01

1,13E-‐01

1,12E-‐01

1,43E-‐01

1,29E-‐01

1,27E-‐01

1,02E-‐01

1,35E-‐01

100

9,36E-‐02

1,36E-‐01

1,62E-‐01

1,77E-‐01

1,96E-‐01

2,55E-‐01

2,22E-‐01

2,71E-‐01

2,31E-‐01

2,53E-‐01

200

1,24E-‐01

1,83E-‐01

2,22E-‐01

2,48E-‐01

2,81E-‐01

3,84E-‐01

4,06E-‐01

4,59E-‐01

3,84E-‐01

4,11E-‐01

500

1,46E-‐01

2,13E-‐01

2,71E-‐01

3,12E-‐01

3,31E-‐01

5,39E-‐01

5,58E-‐01

5,49E-‐01

6,32E-‐01

6,27E-‐01

1000

1,63E-‐01

2,37E-‐01

2,85E-‐01

3,46E-‐01

3,91E-‐01

5,39E-‐01

6,22E-‐01

7,05E-‐01

6,32E-‐01

6,87E-‐01

2000

1,77E-‐01

2,38E-‐01

3,03E-‐01

3,54E-‐01

3,84E-‐01

6,18E-‐01

6,78E-‐01

7,70E-‐01

6,82E-‐01

7,47E-‐01

5000

1,74E-‐01

2,44E-‐01

3,05E-‐01

3,43E-‐01

3,89E-‐01

5,39E-‐01

6,41E-‐01

6,82E-‐01

5,81E-‐01

6,50E-‐01

10000

1,59E-‐01

2,19E-‐01

2,72E-‐01

2,95E-‐01

3,53E-‐01

4,14E-‐01

5,39E-‐01

5,16E-‐01

5,35E-‐01

5,95E-‐01

20000

1,30E-‐01

1,84E-‐01

2,29E-‐01

2,42E-‐01

2,74E-‐01

4,29E-‐01

4,75E-‐01

5,53E-‐01

4,98E-‐01

6,09E-‐01

50000

6,64E-‐02

8,25E-‐02

1,01E-‐01

1,18E-‐01

1,67E-‐01

2,33E-‐01

2,85E-‐01

2,90E-‐01

2,68E-‐01

2,50E-‐01

100000

4,56E-‐02

6,18E-‐02

7,01E-‐02

8,16E-‐02

9,36E-‐02

1,85E-‐01

1,72E-‐01

1,98E-‐01

1,84E-‐01

1,63E-‐01

200000

3,86E-‐02

5,12E-‐02

6,22E-‐02

7,33E-‐02

7,56E-‐02

1,35E-‐01

1,26E-‐01

1,48E-‐01

1,53E-‐01

1,66E-‐01

500000

4,03E-‐02

4,98E-‐02

6,13E-‐02

7,19E-‐02

8,67E-‐02

1,22E-‐01

1,50E-‐01

1,60E-‐01

1,66E-‐01

1,48E-‐01

1000000

4,02E-‐02

5,26E-‐02

6,73E-‐02

7,10E-‐02

8,21E-‐02

1,24E-‐01

1,55E-‐01

1,60E-‐01

1,70E-‐01

1,54E-‐01

Table 6-‐5: Measured conductivity spectra of multiphase clay-‐cement water system (specimen CemKaol01) at hydration times above 7 hours (see Figure 3-‐26 for graphic presentation of the results). ν [Hz]

Chapter: Appendix

106

σ [S/m] for different hydration times (in hours) 7,75

8,75

94,33

120

144

168

288

3,67E-‐02

3,83E-‐02

4,06E-‐02

3,97E-‐02

4,10E-‐02

4,48E-‐02

3,73E-‐02

2,13E-‐02

1,16E-‐02

8,11E-‐03

3,38E-‐03

1,22E-‐01

1,25E-‐01

1,14E-‐01

1,24E-‐01

1,01E-‐01

7,70E-‐02

3,74E-‐02

1,74E-‐02

1,24E-‐02

5,07E-‐03

100

2,40E-‐01

2,28E-‐01

2,31E-‐01

2,19E-‐01

1,91E-‐01

1,48E-‐01

1,09E-‐01

5,39E-‐02

2,33E-‐02

1,66E-‐02

6,69E-‐03

200

3,86E-‐01

3,66E-‐01

3,51E-‐01

3,72E-‐01

2,86E-‐01

1,85E-‐01

1,24E-‐01

7,15E-‐02

3,00E-‐02

2,16E-‐02

8,44E-‐03

500

4,98E-‐01

5,03E-‐01

5,07E-‐01

4,53E-‐01

3,09E-‐01

2,03E-‐01

1,42E-‐01

8,71E-‐02

3,96E-‐02

2,80E-‐02

1,06E-‐02

1000

6,50E-‐01

5,21E-‐01

5,86E-‐01

4,75E-‐01

3,33E-‐01

1,99E-‐01

1,44E-‐01

9,36E-‐02

4,55E-‐02

3,19E-‐02

1,18E-‐02

2000

6,27E-‐01

5,90E-‐01

5,53E-‐01

5,35E-‐01

3,49E-‐01

2,08E-‐01

1,48E-‐01

9,87E-‐02

4,98E-‐02

3,43E-‐02

1,27E-‐02

5000

5,49E-‐01

5,35E-‐01

5,39E-‐01

5,12E-‐01

3,39E-‐01

2,06E-‐01

1,44E-‐01

1,00E-‐01

5,21E-‐02

3,59E-‐02

1,30E-‐02

10000

4,80E-‐01

5,30E-‐01

4,70E-‐01

4,44E-‐01

2,90E-‐01

1,85E-‐01

1,32E-‐01

9,18E-‐02

4,80E-‐02

3,37E-‐02

1,22E-‐02

20000

4,47E-‐01

4,66E-‐01

4,12E-‐01

3,96E-‐01

2,31E-‐01

1,43E-‐01

1,05E-‐01

6,82E-‐02

3,75E-‐02

2,55E-‐02

9,27E-‐03

50000

1,96E-‐01

2,01E-‐01

1,57E-‐01

1,78E-‐01

1,31E-‐01

7,15E-‐02

5,07E-‐02

3,54E-‐02

1,88E-‐02

1,28E-‐02

4,66E-‐03

100000

1,87E-‐01

1,79E-‐01

1,36E-‐01

1,35E-‐01

8,48E-‐02

4,98E-‐02

3,50E-‐02

2,28E-‐02

1,23E-‐02

8,85E-‐03

3,10E-‐03

200000

1,26E-‐01

1,18E-‐01

1,13E-‐01

1,05E-‐01

7,05E-‐02

4,27E-‐02

3,04E-‐02

1,95E-‐02

1,05E-‐02

7,56E-‐03

2,69E-‐03

500000

1,36E-‐01

1,49E-‐01

1,30E-‐01

1,20E-‐01

7,42E-‐02

4,56E-‐02

3,15E-‐02

2,00E-‐02

1,02E-‐02

7,70E-‐03

2,72E-‐03

1000000

1,48E-‐01

1,39E-‐01

1,25E-‐01

1,20E-‐01

7,47E-‐02

4,27E-‐02

3,11E-‐02

2,00E-‐02

1,06E-‐02

7,88E-‐03

2,80E-‐03

Table 6-‐6: Measured conductivity spectra of cement paste with 0,3 w/c ratio (specimen cement_0.3) at different hydration times (see Figure 3-‐27 for graphic presentation of the results). ν [Hz]

σ [S/m] for different hydration times (in hours) 0

20 1,99E-‐02 2,02E-‐02 1,95E-‐02 1,92E-‐02 1,84E-‐02 1,83E-‐02 1,78E-‐02 1,88E-‐02 2,28E-‐02 2,39E-‐02 2,42E-‐02 2,38E-‐02 2,03E-‐02 50 4,91E-‐02 5,01E-‐02 5,04E-‐02 5,21E-‐02 4,72E-‐02 5,04E-‐02 4,51E-‐02 4,99E-‐02 5,17E-‐02 4,99E-‐02 4,78E-‐02 4,45E-‐02 3,25E-‐02 100 1,01E-‐01 1,06E-‐01 1,10E-‐01 1,16E-‐01 9,70E-‐02 1,08E-‐01 9,64E-‐02 9,23E-‐02 7,97E-‐02 7,17E-‐02 6,54E-‐02 5,91E-‐02 4,03E-‐02 200 2,11E-‐01 2,14E-‐01 2,31E-‐01 2,30E-‐01 1,84E-‐01 1,95E-‐01 1,75E-‐01 1,51E-‐01 1,04E-‐01 7,31E-‐02 7,82E-‐02 6,84E-‐02 4,54E-‐02 500 3,96E-‐01 4,60E-‐01 4,77E-‐01 4,74E-‐01 3,52E-‐01 3,33E-‐01 2,83E-‐01 2,07E-‐01 1,25E-‐01 1,01E-‐01 8,72E-‐02 7,55E-‐02 4,96E-‐02 1000 6,90E-‐01 7,61E-‐01 6,77E-‐01 7,28E-‐01 4,98E-‐01 4,56E-‐01 3,55E-‐01 2,31E-‐01 1,31E-‐01 1,05E-‐01 9,14E-‐02 7,79E-‐02 5,20E-‐02 2000 9,17E-‐01 9,91E-‐01 8,63E-‐01 8,97E-‐01 5,80E-‐01 4,75E-‐01 3,85E-‐01 2,56E-‐01 1,35E-‐01 1,10E-‐01 9,38E-‐02 8,03E-‐02 5,34E-‐02 5000 9,17E-‐01 9,31E-‐01 9,81E-‐01 8,56E-‐01 6,15E-‐01 4,82E-‐01 4,12E-‐01 2,55E-‐01 1,35E-‐01 1,08E-‐01 8,36E-‐02 8,00E-‐02 5,28E-‐02 10000 6,46E-‐01 6,56E-‐01 7,28E-‐01 7,48E-‐01 5,17E-‐01 4,53E-‐01 3,31E-‐01 2,20E-‐01 1,20E-‐01 9,73E-‐02 6,75E-‐02 7,17E-‐02 4,81E-‐02 20000 6,90E-‐01 5,41E-‐01 5,89E-‐01 6,19E-‐01 4,85E-‐01 4,50E-‐01 3,02E-‐01 1,91E-‐01 1,01E-‐01 7,76E-‐02 3,37E-‐02 5,58E-‐02 3,70E-‐02 50000 6,67E-‐01 5,08E-‐01 4,87E-‐01 5,18E-‐01 2,87E-‐01 4,18E-‐01 1,87E-‐01 1,05E-‐01 4,96E-‐02 3,82E-‐02 2,22E-‐02 2,76E-‐02 1,83E-‐02 100000 2,99E-‐01 2,83E-‐01 3,62E-‐01 2,62E-‐01 2,08E-‐01 2,65E-‐01 1,13E-‐01 6,90E-‐02 3,31E-‐02 2,57E-‐02 1,89E-‐02 1,90E-‐02 1,24E-‐02 200000 2,06E-‐01 4,03E-‐01 2,18E-‐01 2,36E-‐01 1,82E-‐01 1,44E-‐01 9,97E-‐02 5,91E-‐02 2,91E-‐02 1,33E-‐02 1,93E-‐02 1,61E-‐02 1,07E-‐02 500000 3,42E-‐01 3,96E-‐01 4,03E-‐01 3,04E-‐01 2,09E-‐01 1,45E-‐01 9,61E-‐02 5,82E-‐02 2,90E-‐02 2,28E-‐02 2,02E-‐02 1,59E-‐02 1,08E-‐02 1000000 3,07E-‐01 3,72E-‐01 3,79E-‐01 3,15E-‐01 2,03E-‐01 1,45E-‐01 9,94E-‐02 5,85E-‐02 3,02E-‐02 2,34E-‐02 2,02E-‐02 1,69E-‐02 1,16E-‐02

Table 6-‐7: Measured conductivity spectra of cement paste with 0,4 w/c ratio (specimen cement_0.4) at different hydration times (see Figure 3-‐28 for graphic presentation of the results). ν [Hz]

σ [S/m] for different hydration times (in hours) 0

20 1,07E-‐02 1,06E-‐02 1,06E-‐02 7,31E-‐03 1,05E-‐02 1,00E-‐02 9,42E-‐03 9,27E-‐03 9,20E-‐03 8,54E-‐03 8,31E-‐03 8,33E-‐03 3,67E-‐03 50 2,19E-‐02 2,37E-‐02 2,30E-‐02 1,71E-‐02 2,43E-‐02 2,33E-‐02 2,20E-‐02 2,33E-‐02 2,00E-‐02 1,75E-‐02 1,50E-‐02 1,54E-‐02 5,49E-‐03 100 3,34E-‐02 4,33E-‐02 4,39E-‐02 3,34E-‐02 4,68E-‐02 4,61E-‐02 4,62E-‐02 4,91E-‐02 3,67E-‐02 2,74E-‐02 2,15E-‐02 2,20E-‐02 7,33E-‐03 200 6,57E-‐02 8,03E-‐02 8,18E-‐02 6,81E-‐02 9,51E-‐02 8,86E-‐02 9,76E-‐02 1,06E-‐01 6,41E-‐02 4,04E-‐02 2,87E-‐02 2,93E-‐02 9,34E-‐03 500 1,47E-‐01 1,91E-‐01 2,25E-‐01 1,72E-‐01 2,24E-‐01 2,14E-‐01 2,30E-‐01 2,29E-‐01 1,16E-‐01 5,75E-‐02 3,80E-‐02 3,80E-‐02 1,21E-‐02 1000 2,64E-‐01 3,37E-‐01 4,45E-‐01 3,11E-‐01 4,37E-‐01 4,07E-‐01 3,74E-‐01 3,23E-‐01 1,55E-‐01 7,07E-‐02 4,46E-‐02 4,40E-‐02 1,41E-‐02 2000 4,81E-‐01 6,21E-‐01 6,18E-‐01 4,75E-‐01 6,69E-‐01 7,57E-‐01 5,05E-‐01 4,06E-‐01 1,84E-‐01 8,12E-‐02 5,09E-‐02 4,91E-‐02 1,59E-‐02 5000 7,64E-‐01 8,45E-‐01 9,32E-‐01 6,96E-‐01 7,93E-‐01 6,94E-‐01 5,64E-‐01 4,25E-‐01 2,07E-‐01 9,10E-‐02 5,54E-‐02 5,33E-‐02 1,79E-‐02 10000 7,49E-‐01 7,23E-‐01 8,39E-‐01 6,33E-‐01 6,97E-‐01 5,96E-‐01 5,22E-‐01 3,40E-‐01 1,93E-‐01 8,68E-‐02 5,38E-‐02 5,17E-‐02 1,79E-‐02 20000 7,85E-‐01 7,82E-‐01 6,30E-‐01 5,37E-‐01 6,46E-‐01 5,76E-‐01 4,71E-‐01 3,83E-‐01 1,79E-‐01 7,54E-‐02 4,62E-‐02 4,27E-‐02 1,46E-‐02 50000 2,73E-‐01 3,11E-‐01 4,27E-‐01 5,28E-‐01 3,27E-‐01 1,96E-‐01 2,08E-‐01 1,88E-‐01 9,76E-‐02 4,01E-‐02 2,52E-‐02 2,32E-‐02 8,39E-‐03

200000 2,71E-‐01 3,73E-‐01 1,53E-‐01 3,05E-‐01 1,49E-‐01 2,36E-‐01 1,87E-‐01 1,15E-‐01 5,64E-‐02 2,45E-‐02 1,56E-‐02 1,42E-‐02 5,27E-‐03 500000 4,21E-‐01 4,72E-‐01 3,31E-‐01 3,52E-‐01 3,41E-‐01 2,27E-‐01 1,90E-‐01 1,23E-‐01 5,83E-‐02 2,50E-‐02 1,64E-‐02 1,48E-‐02 5,72E-‐03 1000000 4,78E-‐01 5,79E-‐01 6,42E-‐01 4,18E-‐01 3,24E-‐01 2,46E-‐01 1,91E-‐01 1,22E-‐01 5,96E-‐02 2,77E-‐02 1,88E-‐02 1,65E-‐02 7,23E-‐03

Chapter: Appendix

100000 2,28E-‐01 2,91E-‐01 1,90E-‐01 2,44E-‐01 2,49E-‐01 2,77E-‐01 1,97E-‐01 1,45E-‐01 6,41E-‐02 2,74E-‐02 1,79E-‐02 1,62E-‐02 5,96E-‐03

107

Table 6-‐8: Measured conductivity spectra of cement paste with 0,5 w/c ratio (specimen cement_0.5) at different hydration times (see Figure 3-‐29 for graphic presentation of the results). ν [Hz]

σ [S/m] for different hydration times (in hours) 0

20 6,64E-‐03 7,48E-‐03 7,24E-‐03 7,27E-‐03 6,74E-‐03 6,18E-‐03 5,03E-‐03 4,83E-‐03 5,86E-‐03 6,27E-‐03 8,02E-‐03 6,65E-‐03 5,18E-‐03 50 1,46E-‐02 1,67E-‐02 1,60E-‐02 1,71E-‐02 1,53E-‐02 1,45E-‐02 1,21E-‐02 1,14E-‐02 1,36E-‐02 1,30E-‐02 1,44E-‐02 1,26E-‐02 8,17E-‐03 100 2,73E-‐02 3,15E-‐02 3,20E-‐02 3,32E-‐02 2,98E-‐02 2,82E-‐02 2,43E-‐02 2,29E-‐02 2,45E-‐02 2,12E-‐02 2,06E-‐02 1,86E-‐02 1,07E-‐02 200 5,48E-‐02 6,38E-‐02 6,35E-‐02 6,78E-‐02 6,37E-‐02 5,88E-‐02 5,03E-‐02 4,43E-‐02 4,04E-‐02 3,15E-‐02 2,74E-‐02 2,53E-‐02 1,32E-‐02 500 1,36E-‐01 1,60E-‐01 1,60E-‐01 1,60E-‐01 1,52E-‐01 1,47E-‐01 1,26E-‐01 9,73E-‐02 6,65E-‐02 4,75E-‐02 3,63E-‐02 3,53E-‐02 1,65E-‐02 1000 2,08E-‐01 2,89E-‐01 3,00E-‐01 3,00E-‐01 2,79E-‐01 2,80E-‐01 2,14E-‐01 1,49E-‐01 8,58E-‐02 6,17E-‐02 4,24E-‐02 4,39E-‐02 1,90E-‐02 2000 3,87E-‐01 4,59E-‐01 5,11E-‐01 4,70E-‐01 4,33E-‐01 3,99E-‐01 3,38E-‐01 2,09E-‐01 1,09E-‐01 7,77E-‐02 4,75E-‐02 5,15E-‐02 2,12E-‐02 5000 6,72E-‐01 6,78E-‐01 7,30E-‐01 6,72E-‐01 6,53E-‐01 6,26E-‐01 4,34E-‐01 2,80E-‐01 1,32E-‐01 9,75E-‐02 5,28E-‐02 6,07E-‐02 2,34E-‐02 10000 5,89E-‐01 7,65E-‐01 7,53E-‐01 6,84E-‐01 5,67E-‐01 5,56E-‐01 4,53E-‐01 2,64E-‐01 1,33E-‐01 1,03E-‐01 5,13E-‐02 6,24E-‐02 2,31E-‐02 20000 5,11E-‐01 5,45E-‐01 9,41E-‐01 5,77E-‐01 5,37E-‐01 5,48E-‐01 6,66E-‐01 2,63E-‐01 1,17E-‐01 8,86E-‐02 4,29E-‐02 5,15E-‐02 1,87E-‐02 50000 4,73E-‐01 3,41E-‐01 3,72E-‐01 3,58E-‐01 3,20E-‐01 3,17E-‐01 3,27E-‐01 1,70E-‐01 6,73E-‐02 5,08E-‐02 2,44E-‐02 2,84E-‐02 1,05E-‐02 100000 2,55E-‐01 1,95E-‐01 1,84E-‐01 2,76E-‐01 2,37E-‐01 1,97E-‐01 1,78E-‐01 1,06E-‐01 4,54E-‐02 3,20E-‐02 1,71E-‐02 1,99E-‐02 7,33E-‐03 200000 1,95E-‐01 3,29E-‐01 2,72E-‐01 3,12E-‐01 2,43E-‐01 1,87E-‐01 1,77E-‐01 8,70E-‐02 3,81E-‐02 2,84E-‐02 1,50E-‐02 1,72E-‐02 6,45E-‐03 500000 3,67E-‐01 3,38E-‐01 4,18E-‐01 3,64E-‐01 2,88E-‐01 2,53E-‐01 1,80E-‐01 9,65E-‐02 3,96E-‐02 2,87E-‐02 1,56E-‐02 1,80E-‐02 7,08E-‐03 1000000 3,90E-‐01 4,53E-‐01 4,96E-‐01 3,95E-‐01 3,25E-‐01 2,57E-‐01 1,86E-‐01 1,00E-‐01 4,19E-‐02 3,05E-‐02 1,81E-‐02 1,89E-‐02 8,27E-‐03

Chapter: Appendix

108

Table 6-‐9: Distribution of porosity in depth for cement in different hydration times (see Figure 3-‐30 and Figure 3-‐31 for

depth [mm]

porosity [%] of cement

porosity [%] of clay

for different hydration times (in hours)

for different sorption times (in hours)

3,3

3,6

4,5

0,7

0,8

0,7

0,024

3,2

5,3

0,7

0,8

0,048

3,2

4,6

0,7

0,072

3,1

4,5

0,7

0,8

0,096

3,2

2,9

4,6

0,7

0,9

0,12

3,2

4,6

0,8

0,7

0,9

0,144

3,2

3,1

5,3

0,7

0,8

0,168

3,2

4,6

0,7

0,8

0,9

0,192

3,2

2,5

4,4

0,8

0,9

1,1

0,216

3,2

2,8

4,5

0,7

0,9

1,1

0,24

2,8

4,3

0,8

0,9

1,1

0,264

2,7

3,7

0,8

0,9

0,288

2,5

2,1

3,2

0,9

0,312

1,9

2,2

3,3

0,9

0,336

2,3

3,5

0,9

1,1

0,36

1,8

3,1

0,9

1,1

0,384

1,8

1,9

2,6

0,9

1,1

1,2

0,408

1,7

1,6

2,7

0,9

1,3

0,432

1,8

1,7

2,4

0,9

1,3

0,456

1,8

1,6

2,3

0,9

1,1

0,48

1,4

2,4

0,9

1,2

1,1

0,504

1,6

2,8

0,9

1,2

0,528

1,9

2,7

1,1

0,9

0,552

1,9

3,2

1,2

0,576

2,1

0,9

1,2

0,6

2,5

2,1

3,5

0,9

1,2

0,624

2,6

2,3

4,2

0,9

1,3

0,648

2,3

4,1

0,9

1,2

0,672

3,4

2,7

4,1

0,9

1,2

0,9

0,696

3,3

2,7

4,1

0,9

1,1

0,72

3,9

2,6

4,6

0,9

1,1

0,744

3,8

4,5

0,9

1,1

0,768

3,8

4,6

0,8

1,1

0,792

3,7

2,9

4,7

0,8

1,1

0,816

3,1

2,5

4,7

0,8

0,84

3,7

3,9

0,864

3,1

4,7

0,888

2,7

2,4

0,912

3,3

2,5

4,8

0,936

3,7

2,9

4,1

0,96

3,1

2,5

0,984

3,1

4,1

1,008

3,8

4,1

1,032

3,7

4,7

1,056

3,6

3,5

4,8

1,08

3,5

3,4

4,8

1,104

4,2

4,6

1,128

4,2

5,6

1,152

5,1

4,8

5,6

1,176

4,5

4,6

5,6

AVERAGE

3,05

2,70

4,11

0,84

1,00

1,01

Chapter: Appendix

graphic presentation of the results) and clay in different sorption times (Figure 3-‐32 and Figure 3-‐33).

109

Table 6-‐10: Distribution of porosity in depth for stabilized clay in different hydration times (see Figure 3-‐34 and Figure 3-‐35 for graphic presentation of the results).

depth [mm]

Chapter: Appendix

110

0,000 0,024 0,048 0,072 0,096 0,120 0,144 0,168 0,192 0,216 0,240 0,264 0,288 0,312 0,336 0,360 0,384 0,408 0,432 0,456 0,480 0,504 0,528 0,552 0,576 0,600 0,624 0,648 0,672 0,696 0,720 0,744 0,768 0,792 0,816 0,840 0,864 0,888 0,912 0,936 0,960 0,984 1,008 1,032 1,056 1,080 1,104 1,128 1,152 1,176 1,200 1,224 1,248 1,272 1,296 1,320 1,344 1,368 1,392 1,416 1,440 1,464 1,488 1,512 1,536 1,560 1,584 1,608 1,632 1,656 1,680 1,704

porosity [%] of stabilized clay for different hydration times (in days) 0

2,6 2,5 2,8 2,9 2,6 2,7 2,7 2,6 2,6 2,4 2,5 2,6 2,8 3,1 2,9 3,1 3,1 3 3,3 3,1 2,8 2,3 2,4 2,3 2,6 2,7 2,7 2,6 2,6 2,6 2,8 2,9 2,7 2,7 2,6 2,9 2,8 2,8 2,9 2,9 2,8 2,8 3,2 2,9 2,9 2,9 2,9 3,1 3,1 3,5 3,5 3,1 3,4 3,3 3,3 3,1 3 2,9 3,2 3,1 3 3,2 3,1 3 3 2,9 2,8 2,7 2,9 3 2,9 2,6

2,9 3,1 3 3 3,1 3 2,9 2,8 2,7 2,7 2,7 2,9 2,6 2,6 2,8 2,9 3 2,8 2,8 2,9 2,6 2,3 2,5 2,6 2,7 3,1 2,9 2,7 2,7 2,6 2,8 2,5 2,5 2,6 2,4 2,2 2,1 2,5 2,5 2,3 2,2 2,4 2,6 3,5 3,5 2,9 2,7 2,6 2,2 2,4 2,9 2,9 2,6 3,7 3,6 3,1 2,8 2,4 2 2,4 2,8 2,3 2,5 2,6 2,4 2,6 2,3 2,6 3,2 2,9 3,1 2,9

3,1 3,7 3,3 3,1 2,8 2,8 3,1 3,2 3 3,2 2,8 2,6 2,8 3,1 3 2,8 2,5 2,7 2,7 3 2,9 3,2 3,4 3,9 4 4 4,1 3,3 3,3 3,1 2,9 2,8 2,9 3,1 3,1 3 2,9 2,9 3 3,1 2,8 2,9 3,1 3 3 2,6 2,6 2,8 2,8 2,6 2,6 2,5 2,6 2,7 3 3 3 2,9 2,8 2,8 3 2,9 2,5 2,5 2,6 2,7 3,1 3,1 3,1 3,2 3,3 3,6

depth [mm]

1,728 1,752 1,776 1,800 1,824 1,848 1,872 1,896 1,920 1,944 1,968 1,992 2,016 2,040 2,064 2,088 2,112 2,136 2,160 2,184 2,208 2,232 2,256 2,280 2,304 2,328 2,352 2,376 2,400 2,424 2,448 2,472 2,496 2,520 2,544 2,568 2,592 2,616 2,640 2,664 2,688 2,712 2,736 2,760 2,784 2,808 2,832 2,856 2,880 2,904 2,928 2,952 2,976 3,000 3,024 3,048 3,072 3,096 3,120 3,144 3,168 3,192 3,216 3,240 3,264 3,288 3,312 3,336 3,360 3,384 3,408 3,432

porosity [%] of stabilized clay for different hydration times (in days) 0

2,5 2,5 2,6 2,5 2,6 2,3 2 2 2,2 2,4 2,5 2,6 2,8 2,8 2,8 2,8 2,9 2,9 2,8 2,8 2,9 2,9 3 3 3,2 3,2 3,4 3,1 2,5 2,7 2,7 2,7 2,8 3 2,9 3 2,9 2,8 2,8 3 2,9 3,2 3 2,8 3,1 2,9 2,9 3 2,9 2,7 2,8 2,8 3 2,7 2,7 2,6 2,8 3 3,5 3,9 3,4 3 2,8 2,9 2,5 2,6 3 3 2,7 2,5 2,6 2,6

2,7 2,5 2,5 2,7 2,7 2,7 3 2,8 2,8 2,7 2,7 3,2 2,8 2,6 2,4 2,6 2,8 2,6 2,9 2,8 2,7 2,6 2,4 2,8 2,9 2,9 2,5 2,4 2,4 2,4 2,4 2,3 2,2 2,7 2,5 2,5 2,8 3,1 2,9 2,6 2,7 2,8 2,6 2,5 2,6 2,6 2,9 3 2,9 3 2,8 2,8 2,8 2,9 3,2 3 3 2,8 2,8 2,8 2,9 2,8 2,3 2,4 2,7 2,6 2,9 2,8 2,7 2,9 3 3,1

3,7 3,8 3,7 3,6 3,4 2,9 2,4 2,5 2,2 2,3 2,4 2,6 2,7 2,8 2,8 3 3,3 3,1 2,8 2,8 2,6 2,7 2,7 2,7 2,8 2,7 3 3 2,7 2,7 2,7 2,5 2,4 2,6 2,3 2,2 2,3 2,4 2,4 2,5 2,4 2,3 2,4 2,5 2,4 2,6 2,9 2,9 2,4 2,5 2,8 2,9 3,2 3,5 3,2 2,8 2,6 2,8 2,7 2,7 2,6 2,7 2,9 2,7 2,8 2,7 3,3 3,2 3,1 3 2,4 2,4

3,456 3,480 3,504 3,528 3,552 3,576 3,600 3,624 3,648 3,672 3,696 3,720 3,744 3,768 3,792 3,816 3,840 3,864 3,888 3,912 3,936 3,960 3,984 4,008 4,032 4,056 4,080 4,104 4,128 4,152 4,176 4,200 4,224 4,248 4,272 4,296 4,320 4,344 4,368 4,392 4,416 4,440 4,464 4,488 4,512 4,536 4,560 4,584 4,608 4,632 4,656 4,680 4,704 4,728 4,752 4,776 4,800 4,824 4,848 4,872 4,896 4,920 4,944 4,968 4,992 5,016 5,040 5,064 5,088 5,112 5,136 5,160 5,184 5,208 5,232 5,256 5,280

porosity [%] of stabilized clay for different hydration times (in days) 0

2,4 2,6 3 3 3,1 3,2 3,2 3,2 3,1 3,2 3,2 3,4 3,3 2,8 2,6 2,9 3 3,2 3,3 3,3 3,4 3,7 3,5 3,2 3 2,9 3 3 3 2,9 3 3,3 3,3 3,3 3,3 3,4 3,1 2,7 2,8 2,6 2,5 2,8 2,7 2,6 2,8 2,7 2,6 2,4 2,6 2,5 2,8 2,9 2,9 2,9 2,8 2,9 2,6 2,5 2,6 2,8 2,8 2,8 3 2,9 2,8 2,8 2,9 3 3 2,7 2,7 3,1 3 3 3,2 3 3

3,1 3,1 2,8 2,8 3,2 2,9 2,6 2,7 3 3 3 2,8 2,8 3 2,9 3 2,5 2,7 3 2,6 2,7 2,5 2,5 2,7 3 2,9 2,8 2,8 3,4 3 2,7 2,6 2,5 2,5 3,2 3,4 3,2 2,9 2,9 2,6 2,7 2,9 2,7 2,5 2,6 2,7 2,4 2,6 2,5 2,7 2,6 2,6 2,6 2,8 3,1 3,5 3 3,1 2,9 3 3,4 3,3 3,4 3 3,1 3 3 3,2 3,2 3 2,7 2,6 2,8 2,7 2,5 2,5 2,6

2,6 2,4 2,5 2,7 2,5 2,6 2,5 2,5 2,4 2,3 2,3 2,4 2,5 2,3 2,7 2,6 2,3 2,3 2,6 2,9 2,9 3 3,5 2,7 2,9 2,7 2,6 3,3 4,2 3,1 2,8 2,8 2,8 3,3 3 3,2 4,1 3,5 3,1 3 3 2,7 3,2 2,9 2,9 2,7 2,6 2,4 2,5 2,4 2,8 3,2 3,5 3,9 2,6 2,4 2,5 2,4 2,7 2,4 2,4 2,2 2,2 2,3 2,1 2,2 2,4 2,2 2,2 2,1 2,2 2,3 2,3 2,6 2,6 2,7 2,5

depth [mm]

5,304 5,328 5,352 5,376 5,400 5,424 5,448 5,472 5,496 5,520 5,544 5,568 5,592 5,616 5,640 5,664 5,688 5,712 5,736 5,760 5,784 5,808 5,832 5,856 5,880 5,904 5,928 5,952 5,976 6,000 6,024 6,048 6,072 6,096 6,120 6,144 6,168 6,192 6,216 6,240 6,264 6,288 6,312 6,336 6,360 6,384 6,408 6,432 6,456 6,480 6,504 6,528 6,552 6,576 6,600 6,624 6,648 6,672 6,696 6,720 6,744 6,768 6,792 6,816 6,840 6,864 6,888 6,912 6,936 6,960 6,984 7,008 7,032 7,056 7,080 7,104 7,128

porosity [%] of stabilized clay for different hydration times (in days) 0

3,1 3 3 2,9 2,9 3,1 2,8 2,6 2,6 2,8 2,7 2,8 2,6 2,5 2,7 2,8 2,6 2,6 2,6 2,5 2,6 2,7 2,7 2,4 2,5 2,5 2,6 3,1 3,1 2,9 3,3 3,4 3,2 2,7 2,8 2,9 2,6 2,7 2,7 2,6 3 3 3 2,8 2,5 2,4 2,5 2,6 2,6 2,7 2,6 2,4 2,3 2,5 2,7 2,7 3,2 2,9 2,8 2,8 2,5 2,4 2,6 2,7 2,7 2,9 2,5 2,4 2,7 2,8 2,4 2,5 2,6 2,8 3,3 3,3 2,8

2,8 2,7 2,9 3,1 2,8 3 3,5 3,4 3,3 3,1 3,3 3,1 3,2 3,3 3,3 3,4 3,3 3,2 3,1 2,9 2,9 2,9 2,9 3,1 3 3 3,5 3,1 2,9 3,2 3,2 3,2 3,3 2,8 2,7 2,6 2,9 3,2 3,3 2,9 3 2,9 2,6 2,9 3 2,7 3,1 2,8 2,7 3,1 3,3 3,7 4,4 3,2 3,2 2,9 2,7 2,9 2,7 2,6 2,8 2,9 2,7 2,8 2,8 2,9 3,3 3,3 3,1 2,6 2,8 3,1 3,5 3,1 3,3 3,6 3,8

2,6 2,7 2,9 2,7 2,6 2,4 2,7 2,5 2,1 1,9 1,9 1,7 1,7 1,8 2,1 2,4 2,4 2,2 2,3 2,3 2,2 2 1,9 1,7 1,9 1,9 1,9 2 2 2 1,8 1,9 1,7 1,9 2 1,9 1,9 1,9 2 2,1 2,1 2,1 2 2,2 2,3 2,2 2,3 2,3 2,5 2,7 3 3,1 3,4 3,2 3 2,9 2,8 2,9 2,8 2,6 2,2 2 1,8 1,9 1,9 2,1 2,1 1,9 1,8 2 2,1 2,2 2,1 2 1,9 2,1 2,1

Chapter: Appendix

depth [mm]

111

depth [mm]

7,152 7,176 7,200 7,224 7,248 7,272 7,296 7,320 7,344 7,368 7,392 7,416 7,440 7,464 7,488 7,512 7,536 7,560 7,584 7,608 7,632 7,656 7,680 7,704 7,728 7,752 7,776 7,800 7,824 7,848 7,872 7,896 7,920 7,944 7,968 7,992 8,016 8,040 8,064 8,088 8,112 8,136 8,160 8,184 8,208 8,232 8,256 8,280 8,304 8,328 8,352 8,376

Chapter: Appendix

112

porosity [%] of stabilized clay for different hydration times (in days) 0

2,9 3 2,5 2,4 2,6 2,8 2,6 2,7 2,9 2,8 2,6 2,6 2,4 2,1 2,1 2,4 2,4 2,5 2,8 2,9 2,9 2,7 2,7 2,4 2,5 2,6 2,8 2,6 2,4 2,5 2,3 2,6 2,6 2,8 2,8 2,6 2,8 3,1 3,1 2,6 2,5 2,6 2,5 2,7 2,6 2,7 2,9 2,9 2,5 2,3 2,3 2,5

3,3 3,2 2,9 3 2,7 3,1 2,7 3,1 3,1 3 3,5 3,8 3,4 3,2 4 3,4 3,9 3,7 2,9 2,9 3,3 3,1 3,1 3,3 3 2,7 2,7 2,9 2,5 2,6 2,7 3,1 3,1 3 3 2,5 2,8 3 3,1 2,9 2,8 2,7 2,8 3,2 3 3,1 3,2 2,7 2,4 2,8 2,8 2,7

2 2,1 2 2 2 2,1 2,3 2,2 2,5 2,5 2,5 2,4 2,2 2,2 2,2 2,2 2 2 1,9 1,9 1,8 2,1 2,1 2,2 1,9 1,7 1,9 2,2 2,2 2,1 2,1 2,1 2,3 2,4 2,7 2,4 2,3 2,4 2,2 1,9 1,9 1,8 1,9 1,9 1,9 1,8 1,9 2 2,2 2,3 2,5 2,4

depth [mm]

8,400 8,424 8,448 8,472 8,496 8,520 8,544 8,568 8,592 8,616 8,640 8,664 8,688 8,712 8,736 8,760 8,784 8,808 8,832 8,856 8,880 8,904 8,928 8,952 8,976 9,000 9,024 9,048 9,072 9,096 9,120 9,144 9,168 9,192 9,216 9,240 9,264 9,288 9,312 9,336 9,360 9,384 9,408 9,432 9,456 9,480 9,504 9,528 9,552 9,576 average

porosity [%] of stabilized clay for different hydration times (in days) 0

2,7 2,6 2,7 2,4 2,6 2,4 2,4 2,6 2,5 2,3 2,3 2,2 2,2 2,3 2,6 2,8 2,7 2,7 2,5 2,4 2,5 2,3 2,3 2,3 2,3 2,5 2,6 2,4 2,9 3 2,6 2,6 2,7 3 2,4 2,5 2,4 2,5 2,4 2,5 2,9 3,7 3,2 3,2 2,6 2,6 2,6 2,9 3,1 2,8 2,78

2,7 2,8 2,8 2,5 2,9 2,8 3,4 3,4 3,4 2,8 2,8 3,4 3,7 4,1 3,4 2,6 2,7 2,9 3,6 3,5 3,1 3,1 3 3 2,7 2,6 2,7 3,6 3,3 3,3 2,9 2,9 3,2 3 2,8 2,6 2,4 2,8 2,7 3,2 3 2,8 3,4 3,3 3 2,9 3 2,4 2,3 2,5 2,89

2,3 2,3 2,4 2,6 2,7 2,7 2,6 2,6 2,6 2,7 2,8 2,7 2,9 3,1 3,1 3,1 3,2 3,3 3,2 3,1 3,1 3 2,7 2,4 2,3 2,2 2,2 2,2 2,1 1,9 2 2 1,9 1,8 1,8 2,1 2,3 2,2 1,9 2 2 2 2,1 2 2 2 1,9 1,7 1,7 1,8 2,56

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Strokovni življenjepis Marko Samec je leta 2001 opravil gimnazijsko maturo z odlično oceno na Gimnaziji Celje – Center. Po končani gimnaziji se je vpisal na univerzitetni študij gradbeništva na Fakulteti za gradbeništvo Univerze v Mariboru. Leta 2005 je prejel Rektorjevo nagrado kot najboljši študent v generaciji. Naslednje leto je z diplomo zaključil dodiplomski študij in si pridobil naziv univerzitetni diplomirani inženir gradbeništva. Še istega leta se je vpisal na enovit doktorski študij gradbeništva na Fakulteti za gradbeništvo v Mariboru. Od leta 2007 je kot mladi raziskovalec pod mentorstvom prof. dr. Deana Korošaka vključen v program financiranja podiplomskega študija in raziskovalnega usposabljanja Agencije za raziskovalno dejavnost Republike Slovenije. V času podiplomskega študija se je raziskovalno usposabljal tudi na Univerzi v Nottinghamu v Veliki Britaniji. Pod mentorstvom prof. dr. Sache J. Mooneya, ki je somentor pri njegovem doktorskem delu, je delal na School of Biosciences Environmental Science Division. Pri raziskovalnem delu se osredotoča na povezave med strukturo poroznih snovi in dinamiko transportnih pojavov. Z eksperimentalnimi metodami – predvsem rentgensko računalniško tomografijo in nizkofrekvenčno dielektrično spektroskopijo – analizo binarnih slik tomografskih prerezov vzorcev, teorijo kompleksnih mrež in aktualnimi spoznanji o anomalni dinamiki transporta

Curriculum Vitae

snovi v poroznih snoveh raziskuje odnos med kompleksnostjo porozne strukture in dinamiko ionov v nasičenih in nenasičenih komponentah gradbenih materialov.

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IME IN PRIIMEK Marko Samec IZOBRAZBA univ. dipl. inž. grad. DATUM IN KRAJ ROJSTVA 13. 10. 1982, Maribor DRŽAVLJANSTVO slovensko KONTAKTNI PODATKI Bevkova ulica 5, 3240 Šmarje pri Jelšah +386 41 353037 marko.samec@gmail.com ZNANJE TUJIH JEZIKOV angleški hrvaški nemški (pasivno) VOZNIŠKI IZPIT B, C, E, G, H IZOBRAŽEVANJE 2009 – raziskovalno usposabljanje na Univerzi v Nottinghamu, Velika Britanija 2007 – mladi raziskovalec pod mentorstvom prof. dr. Deana Korošaka 2006 – vpisan na enovit doktorski študij, Univerza v Mariboru, Fakulteta za gradbeništvo 2006 – diplomiral z oceno 10; diplomska naloga z naslovom »Raziskava lastnosti mešanic gline in vode z dielektrično spektroskopijo«, mentor prof. dr. Dean Korošak 2005 – rektorjeva nagrada za najboljšega študenta v vpisni generaciji 2001 do 2006 – univerzitetni študij gradbeništva, Univerza v Mariboru, Fakulteta za gradbeništvo; povprečna ocena vseh izpitov 8,54 1997 do 2001 – Gimnazija Celje Center 1989 do 1997 – Osnovna šola Šmarje pri Jelšah HABILITACIJA 2007 – asistent za predmete »gradbena fizika«, »meritve mehanskih količin«, »numerične metode in linearno programiranje« BIBLIOGRAFIJA http://splet02.izum.si/cobiss/bibliography?code=29567

2005 do 2007 – prodekan za študentska vprašanja na Fakulteti za gradbeništvo, Univerza v Mariboru 2002 do 2007 – počitniška dela ter praksa v gradbenem podjetju Vajcer & Samec

Curriculum Vitae

DELOVNE IZKUŠNJE od 2007 zaposlen kot raziskovalec na Katedri za aplikativno fiziko, Univerza v Mariboru, Fakulteta za gradbeništvo

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CV Marko Samec graduated from high school Gimnazija Celje – Center in 2001. In the same year he enrolled in the University of Maribor, Faculty of Civil Engineering. In 2005 he received Rector’s Award for academic excellence. In the following year he finished his undergraduate studies and received Bachelor’s degree in civil engineering. In the same year he enrolled in PhD studies at the Faculty of Civil Engineering in Maribor. Since 2007 he is included in the funding program of postgraduate study and research training of Slovenian Research Agency as a young researcher under the supervision of prof. dr. Dean Korošak. During his PhD studies he has also received research training at the University of Nottingham in United Kingdom. He worked at the School of Biosciences, Environmental Science Division under the supervision of prof. dr. Sacha J. Mooney, his PhD co-‐ mentor. His research work is focused on the link between the structure of porous media and dynamics of transport phenomena. With the use of experimental methods – especially X-‐ray computed tomography and low-‐frequency dielectric spectroscopy – analysis of binary images of sections of the

Curriculum Vitae

samples, complex network theory and the current findings on the dynamics of anomalous transport in porous matter he explores the correlation between the complexity of the porous structure and dynamics of ions in the saturated and unsaturated components of building materials.

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NAME Marko SURNAME Samec GENDER male DATE OF BIRTH 13. 10. 1982 NATIONALITY slovenian CONTACT INFORMATION Bevkova 5, 3240 Šmarje pri Jelšah, Slovenia +386 41 353037 marko.samec@gmail.com LANGUAGES slovenian english croatian german (passive) DRIVING LICENSE B, C, E, G, H EDUCATION 2009 – research training at University of Nottingham, United Kingdom 2007 – young researcher under the supervision of dr. Dean Korošak 2006 – PhD study, University of Maribor, Faculty of civil engineering 2005 – rector's award for academic excellence 2001 -‐ 2006 – graduate study, University of Maribor, Faculty of civil engineering 1997 -‐ 2001 – high school Gimnazija Celje – Center 1989 -‐ 1997 – primary school OŠ Šmarje pri Jelšah TEACHING since 2007 – assistant for courses »building physics«, »measurements of mechanical quantities«, »numerical methods and linear programming« BIBLIOGRAPHY http://splet02.izum.si/cobiss/bibliography?code=29567 WORKING EXPERIENCE since 2007 – researcher at Chair of Applied Physics, Faculty of Civil Engineering, University of Maribor 2005 -‐ 2007 – vice dean for student affairs, Faculty of Civil Engineering, University of Maribor

Curriculum Vitae

2002 do 2007 – practical training at construction company Vajcer & Samec

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