Development of a multiscale approach for the …...Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011 Figure 3. Fonctions de corrélation à trois points pour un composite

2011/10

Development of a multiscale approach for

the characterization and modeling of heterogeneous materials : (Application to polymer nanocomposites)

Majid BANIASSADI

Université de Strasbourg-CNRS

Institut de Mécanique des Fluides et des Solides

UNIVERSITE DE STRASBOURG

École Doctorale Mathématiques, Sciences de l'Information et de l'Ingénieur

Institut de Mécanique des Fluides et des Solides

THÈSE

présentée pour obtenir le grade de:

Docteur de l’Université de Strasbourg

Discipline : Mécanique des matériaux

Spécialité : Micromécanique

par

Majid BANIASSADI

Development of a multiscale approach for the characterization and modeling of heterogeneous materials : Application to polymer nanocomposites

Soutenue publiquement le 19 Décembre 2011

Membres du jury

Directeur de thèse : Prof. Saïd AHZI, Université de Strasbourg Co-Directeur de thèse : Prof. René MULLER, Université de Strasbourg Rapporteur externe : Prof. Moussa NAïT ABDELAZIZ, École Polytechnique de l'Université de Lille Rapporteur externe : Prof. Sébastien MERCIER, Université Paul Verlaine-Metz

Examinateur : Prof. Hamid GARMESTANI, Georgia Institute of Technology, Atlanta-USA Examinateur : Prof. Abdel-Mjid NOURREDDINE, Université de Strasbourg Examinateur : Dr. David RUCH, Centre de Recherche Public Henri Tudor, Luxembourg

Invité : Prof. Yves REMOND, Université de Strasbourg Invité : Prof. Madjid FATHI, University of Siegen, Siegen-Germany Invité : Dr. Valérie TONIAZZO , Centre de Recherche Public Henri Tudor, Luxembourg

Nom du Laboratoire: IMFS N° de l’Unité FRE 3240

This thesis is dedicated to my parents

(Parvaneh Khadiv & Mahmoud Baniassadi)

for their love, endless support and encouragement.

ACKNOWLEDGMENT

I would like to express my gratitude to all those who gave me the possibility to complete this

thesis. I want to express my sincere gratitude to my advisor, Professor Said Ahzi, who

throughout my doctoral studies has contributed with excellent scientific support and

encouragement to commence and achieve this work. I have furthermore to thank my co-advisor,

Prof. Muller, who supported me with excellent scientific help, particularly in the domain of

polymers. I wish also to express my deepest gratitude to Prof. Garmestani for his valuable ideas

and suggestions and fruitful discussions. His encouragements have been a major reason for me to

start and advance this work. I am deeply indebted to Prof. Remond for being an inspiration for

me and providing an all-out support during these years.

I am also grateful to the Department of Advanced Materials and Structures from the Public

Research Center - Henri Tudor for the excellent technical support and FNR-Luxembourg for the

financial support, and also to my dear colleagues from AMS HT especially Dr. Ruch, director of

AMS HT, Dr. Toniazzo, Dr. Laachachi, Dr. Addiego and Dr. Hassouna. Special thanks go to

Prof. Fathi from the University of Siegen, Prof . Patlazhan from the University of Moscow and

Prof. Gracio from the University of Aveiro for encouraging me to follow the academic research,

and to Prof. Nait-Abdelaziz from the University of Lille and Prof. Mercier from the University of

Metz for their hints and suggestions.

I am heartily thankful to my greatest source of inspiration, my parents, who have always been

there for me, understanding and unconditionally supportive of my endeavors as I pursued this

goal.

Special thanks are reserved for Mr. Ghazavizadeh and Mr. Mortazavi from IMFS, Mrs. Amani

from Georgia Tech, Dr. Li from PNNL, and Mrs. Sheidaei from Michigan State University, Mr.

Kaboli from the University of Strasbourg, Mr. Safdari from Virginia Tech, Mr. Wen, Mr. Barth,

Mr. Nierenberger, Mr. Wang , Mrs. Lhadi, Dr. Joulaee, Mr. Essa and Dr. Mguil from IMFS,

Mrs. Morais, Mrs. Vergnat, Dr. Angotti and Mr. Delgado-Rangel from AMS HT, Mr. Etesami

and all my friends and colleagues from IMFS and AMS HT for helping me to follow my

researches.

ix

Contents

Résumé .......................................................................................................................................... 13

Abstract ......................................................................................................................................... 21

Introduction ................................................................................................................................... 25

References ............................................................................................................................. 32

Chapter I ...................................................................................................................................... 35

Literature Survey ....................................................................................................................... 37

I.1. Random heterogeneous material ..................................................................................... 39

I.2. Two-Point Probability Functions .................................................................................... 39

I.3. Two-Point Cluster Functions .......................................................................................... 41

I.5. Approximation of higher order correlation functions ..................................................... 44

I.6. Homogenization methods for effective properties .......................................................... 45

I.7. Assumption of Statistical Continuum Mechanics .......................................................... 46

I.8. Reconstruction ................................................................................................................. 47

I. References .......................................................................................................................... 51

Chapter II .................................................................................................................................... 53

Using SAXS Approach to Calculate Two-Point Correlation Function..................................... 55

II.1. Introduction .................................................................................................................... 57

II.2. Correlation between SAXS data and two-point correlation functions ........................... 58

II.3. Structural characterization ............................................................................................. 61

II.4. Conclusion ..................................................................................................................... 66

II. References ......................................................................................................................... 67

Chapter III ................................................................................................................................... 69

New Approximate Solution for N-Point Correlation Functions for Heterogeneous Materials . 71

III.1. Introduction .................................................................................................................. 72

III. 2. Approximation of tree-point correlation functions ...................................................... 75

III. 3. Approximation of four-point correlation function....................................................... 80

III. 4. Approximation of N-point correlation function .......................................................... 85

III. 5. Results ......................................................................................................................... 86

III. 6. Conclusion ................................................................................................................... 95

x

III. References ....................................................................................................................... 96

Chapter IV ................................................................................................................................... 99

A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials Using Two-Point Correlation Functions .............................................................................................................. 101

IV. 1. Introduction ............................................................................................................... 103

IV. 2. Development of a Monte Carlo reconstruction methodology ................................... 106

IV. 3. Optimization of the statistical correlation functions ................................................. 117

IV. 4. Three-phase solid oxide fuel cell anode microstructure ............................................ 119

IV. 5. Reconstruction of multiphase heterogeneous materials ............................................ 120

IV. 6. Conclusion ................................................................................................................. 127

IV. References ..................................................................................................................... 128

Chapter V .................................................................................................................................. 131

Homogenization of Mechanical and Thermal Behavior of Nanocomposites Using Strong Contrast ................................................................................................................................... 133

V.1. Introduction................................................................................................................. 135

V.2. Computer generated model ......................................................................................... 137

V.3. Thermal conductivity .................................................................................................. 139

V.4. Mechanical model ....................................................................................................... 140

V.5. Experimental part........................................................................................................ 144

V.6. Results and discussion ................................................................................................. 146

V.7. Conclusion ................................................................................................................... 150

V. References....................................................................................................................... 151

Chapter VI ................................................................................................................................. 155

Three-dimensional Reconstruction and Homogenization of Heterogeneous Materials Using Statistical Correlation Functions and FEM ............................................................................. 157

VI.1. Introduction ................................................................................................................ 159

VI.2. Reconstruction of heterogeneous materials using two-point cluster function .......... 160

VI.3. Statistical characterization of microstructures ........................................................... 163

VI.4. FEM characterization of multiphase heterogeneous materials ................................... 163

VI.5. Result and discussion ................................................................................................. 165

VI.6. Conclusion .................................................................................................................. 172

VI. References ..................................................................................................................... 173

xi

Conclusion and Future Work ...................................................................................................... 175

Appendix .................................................................................................................................... 179

Appendix A ............................................................................................................................. 181

Appendix B ............................................................................................................................. 184

Résumé

12 Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011

Résumé


Résumé

Résumé


Résumé


Les fonctions de corrélation à deux points sont une catégorie de descripteurs statistiques bien

connus pour décrire théoriquement la morphologie et les relations entre morphologie et

propriétés d’un matériau.

Nous approfondissons dans ce travail les connaissances liées à l’application des fonctions de

corrélation à deux points pour la reconstruction et l’homogénéisation de matériaux composites.

Plus particulièrement, les fonctions de corrélation à deux points ont été déterminées à partir de

données expérimentales provenant de différentes techniques comme la microscopie électronique

à balayage (MEB) ou à transmission (TEM), la diffusion des rayons X aux petits angles (SAXS),

et de la méthode de Monte-Carlo. Dans une première application, nous avons exploité des

données SAXS provenant de la caractérisation d’un composite polymère à deux phases. Pour

cela, une matrice polystyrène (PS) chargée de nanoparticules d’oxyde de zirconium (ZrO2) a été

sélectionné. Par ailleurs, la morphologie de ce matériau a été observée par MEB au moyen du

mode de détection transmission (STEM). L’évolution de l’intensité des rayons X diffusés I en

fonction du vecteur d’onde h est représentée à la figure 1 dans le cas du PS chargé de 3 % et 5 %

en poids de ZrO2. Les fonctions de corrélation à deux points pour les composites PS-ZrO2 (3 %

et 5% de charges) sont montrées à la Figure 2.

.

Figure 1. Intensité des rayons X diffusés I en fonction du vecteur d’onde h dans le cas de la nanopoudre ZrO2 et des nanocomposites PS-ZrO2 (3 % et 5% de charges) (avec correction

d’absorption et élimination du fond continu)

Résumé


Figure 2. Fonctions de corrélation à deux points des composites PS-ZrO2 (3 % et 5% de charges)

Afin d’augmenter la précision de l’approche continuum statistique, des fonctions de corrélation

de plus grand ordre doivent en principe être déterminées. Ainsi, une nouvelle méthodologie

d’approximation a été développée pour obtenir des fonctions de corrélation à N-points dans le

cas de microstructures hétérogènes de matériaux sans gradient fonctionnel (FGM). Des fonctions

de probabilité conditionnelle ont été utilisées pour formuler l’approximation théorique proposée.

Dans cette approximation, des fonctions de pondération ont été considérées pour connecter des

sous-ensembles de fonctions de corrélation d’ordre N-1 et estimer la totalité des ensembles de

fonctions de corrélation d’ordre N. Dans le cas de l’approximation des fonctions de corrélation

d’ordre 3 et 4, de simples fonctions de pondération ont été utilisées. Les résultats provenant de

cette nouvelle approximation, dans le cas des fonctions de corrélation à trois points, ont été

comparés à la fonction de probabilité réelle déterminée à partir d’une microstructure

tridimensionnelle à trois phases générée par ordinateur. Cette reconstruction tridimensionnelle a

été obtenue à partir d’une microstructure bidimensionnelle (résultant d’images MEB) d’un

matériau à trois phases. Cette comparaison a prouvé que notre nouvelle approximation est

capable de décrire des fonctions statistiques de corrélation de plus grand ordre et ce, avec une

grande précision. La comparaison ente les fonctions de corrélation à trois points simulées et

approximés est montrée à la figure 3 dans le cas de phases (noire-noire-noire).

Résumé


Figure 3. Fonctions de corrélation à trois points pour un composite à trois phases. Les corrélation à trois points sont montées pour les phases (noire-noire-noire).

Les fonctions à deux points provenant de différentes techniques ont été calculées et exploitées

pour reconstruire la microstructure de systèmes hétérogènes. Une nouvelle méthodologie Monte-

Carlo a été développée comme moyen de reconstruction tridimensionnel (3D) de la

microstructure de matériaux hétérogènes, sur la base de fonctions statistiques à deux points.

L’aspect le plus pertinent de la méthodologie de reconstruction présentée est sa capacité de

réalisée des reconstructions 3D à partir d’image MEB 2D pour un système à trois phases,

extrapolable à un système à N phases. La reconstruction tridimensionnelle d’un système

hétérogène a été exploitée pour prédire le seuil de percolation de matériaux hétérogènes. Des

micrographies MEB d’une anode constituée de trois phases et utilisée dans les piles à

combustible à oxyde solide (rouge : nickel, bleu : ZYS, noir : vides), et l’image de l’anode

reconstruite selon trois directions sont respectivement montrées aux figures 4 et 5.

Résumé


Figure 4. Micrographie MEB de la microstructure d’une anode constituée de trois phases et utilisée dans les piles à combustible à oxyde solide (rouge : nickel, bleu : ZYS, noir : vides)

(a)

(b)

Figure 5. a) Volume reconstruit d’une microstructure d’anode, b) Sections du volume selon l’épaisseur (rouge : nickel, bleu : ZYS, noir : vides).

Enfin, la théorie continuum statistique a été utilisée pour prédire la conductivité thermique effective et le

module élastique effectif d’un composite polymère. Pour cela, nous avons proposé l’utilisation de la

théorie continuum statistique à fort contraste pour prédire les propriétés élastique et thermique effectives

d’un nanocomposite. En particulier, des échantillons de nanocomposites isotropes contenant des

monofeuillets d’argile orientés de manière aléatoire ont été générés et utilisés pour calculer les fonctions

de corrélation statistique à partir de notre modèle. L’orientation, la forme et la distribution spatiale des

nanoargiles ont été pris en compte à travers les fonctions statistiques de corrélation à deux et trois points.

Résumé


Ces fonctions de corrélation ont été exploitées pour calculer les propriétés thermiques et élastiques

effectives du nanocomposite. Pour valider notre approche théorique, nous avons réalisé des mesures

expérimentales de ces propriétés dans le cas de nanocomposites polyamide/nanoargile avec des

concentration en nanoparticules d’argile de 1 %, 3 % et 5 %. Les résultats de la simulation ont montré que

la rigidité effective de la matrice est significativement augmentée par l’ajout d’une faible quantité de

feuillets d’argile exfoliés. La conductivité thermique effective et le module élastique effectif ont été

comparés avec nos résultats théoriques. Une bonne corrélation entre expérience et simulation a été

obtenue dans le cas de la conductivité thermique. L’effet de l’ajout de nanoargiles sur les propriétés

thermiques et mécaniques effectives du nanocomposite polymère chargé d’argile a été étudié à l’aide des

approches théoriques et expérimentales. Toutefois, dans ce travail de recherche, le module élastique prédit

est supérieur au module élastique expérimental, ce qui peut être dû à la présence de morphologies

intercalées pour des taux d’argile élevés et à l’anisotropie des propriétés des nanoargiles. Par rapport à la

matrice vierge de polyamide, les résultats théoriques et expérimentaux montrent une augmentation de la

conductivité thermique effective et du module élastique effectif du composite en fonction de la fraction

volumique de nanoargile. L’évolution du module élastique simulé et expérimental avec la température est

représentée à la figure 6 pour la matrice PA vierge et ses composites avec OMMT (1%, 3% et 5 %). La

comparaison entre la conductivité thermique expérimentale et théorique du PA et de ses nanocomposites

avec OMMT est quant à elle montrée à la Figure 7.

Figure 6. Module élastique expérimental et théorique d’un composite à deux phases en fonction de la température T pour le PA vierge et ses composites avec OMMT (1%, 3% et 5 % en poids)

Résumé


Figure 7. Comparaison entre la conductivité thermique expérimentale et théorique du PA et de ses nanocomposites avec OMMT

Abstract


Abstract

Abstract


Abstract


Microstructural two-point correlation functions are a well-known class of statistical descriptors

that can be used to describe the morphology and the microstructure-properties relationship. A

comprehensive study has been performed for the use of these correlation functions for the

reconstruction and homogenization in nano-composite materials. Two-point correlation functions

are measured from different techniques such as microscopy (SEM or TEM), small X-Ray

scattering (SAXS) and Monte Carlo simulations. In our study, SAXS data is used to calculate

Two-Point correlation function correlation for two phase polymer composite. The selected

material is polystyrene (PS) filled with zirconium oxide nanoparticles (ZrO2). The

nanocomposite morphology was first examined by scanning transmission electron microscopy

(STEM) and SAXS.

Higher order correlation functions must be calculated or measured to increase the precision of

the statistical continuum approach. To achieve this aim, a new approximation methodology is

utilized to obtain N-point correlation functions for multiphase heterogeneous materials. The two-

point functions measured by different techniques have been exploited to reconstruct the

microstructure of heterogeneous media. A new Monte Carlo methodology is also developed as a

mean for three-dimensional (3D) reconstruction of the microstructure of heterogeneous

materials, based on two-point statistical functions. The salient feature of the presented

reconstruction methodology is the ability to realize the 3D microstructure from its 2D SEM

image for a three-phase medium extendable to n-phase media. Three dimensional reconstruction

of heterogeneous media have been exploited to predict percolation of heterogamous materials. In

this study, the reconstruction methodology is used to reconstruct 3D microstructures of a three-

phase anode structure in a solid oxide fuel cell (SOFC) from a 2D SEM micrograph.

Finally, Statistical continuum theory is used to predict the effective thermal conductivity and

elastic modulus of polymer composites. Two-point and three-point probability functions as

statistical descriptor of inclusions have been exploited to solve strong contrast homogenization

for effective thermal conductivity and elastic modulus properties of nanoclay based polymer

composites and computer generated microstructure. To validate our modeling approach, we

conducted several experimental measurements and FEM calculation.

Introduction


Introduction


Introduction

Introduction


Introduction


Development of advanced microstructure reconstruction methodologies is essential to access a

variety of analytical information associated with complexities in the microstructure of multi-

phase materials. Several experimental and theoretical techniques such as X-ray computed

tomography (CT), scanning and computer generated micrographs have been used to obtain a

sequence of two-dimensional (2D) images that can be further reconstructed in a 3D space.

However, due to cost of sample preparation processes, simulation methods are often more

applicable in reconstruction of heterogeneous microstructures in different areas [1-7].

Using lower-order statistical correlation functions, Torquato [8] established the reconstruction of

one- and two-dimensional microstructures with short-range order using stochastic optimization.

However, he later showed that the lower-order correlation functions cannot solely represent a

two-phase heterogeneous material and therefore more than one solution may exist for a specific

low-order correlation function [8]. Sheehan and Torquato [9] later introduced more orientations

in the correlation functions to effectively eliminate the effect of artificial anisotropy. In the case

of multi-phase materials, Kröner [10] and Beran [11] have developed statistical mathematical

formulations to link correlation functions to properties in multiphase materials. Using higher-

order correlation functions, one can account for the contribution of shape and geometry effects

[8].

Torquato [12] also developed a new hybrid stochastic reconstruction technique for reconstruction

of three-dimensional (3D) random media by using the information from the lineal path function

and the two-point correlation functions during the nucleation annealing process. Different

optimization techniques such as simulated annealing and maximum entropy have been applied in

order to improve the reconstruction procedure [13]. In addition to 3D reconstruction processes

based on probability functions, these functions can be used to account for more details of

microstructure heterogeneities and for the relationships between microstructure, local and

effective properties of multi-phase materials. The effective properties can be obtained via

perturbation expansions [14, 15]. One general approach for the prediction of the effective

properties of a two-phase material with properties of each phase near the average ones is called

“weak-contrast” expansion. However, in materials with a high degree of contrast between the

properties of their phases, “strong-contrast” theory is applied. Brown [16] suggested an

expansion for effective dielectric property of two-phase heterogeneous materials. This

Introduction


expansion for perturbation homogenization was modified and extended for elasticity by Torquato

[17] for two-phase materials and later the solution was extended to multi-phase materials by

others [15, 18]. Several numerical methods can be used to obtain the effective thermal/electrical

conductivity as well as effective elastic properties of multiphase composites of complex

geometries containing arbitrary oriented inhomogeneities [19-21].

In this thesis, statistical correlation functions have been exploited to reconstruct microstructurs

and to develop a multiscale homogenization approach. Two-point correlation functions are the

lowest order of the correlation functions that can describe the morphology and the microstructure

properties relationships. Two-point correlation functions can be measured using SAXS data or

SEM/TEM images for different microstructures. Monte Carlo simulation is a numerical

technique that is capable of predicting two-point or higher order correlation functions. Higher

order correlation functions can be approximated using lower order of correlation functions. In

this study, a new approximation has been developed to predict the higher order correlation

functions based on the lower order ones which efficiently facilitate the characterization of the

effective properties. In this research work, a new Monte Carlo methodology is developed and

implemented as a mean for three-dimensional (3D) reconstruction of multi-phase

microstructures, based on two-point statistical functions.

Finally, Statistical continuum theory of strong contrast has been exploited to predict effective

thermal and elastic properties of two phase heterogeneous materials using two-point and three-

point correlation functions. To validate our modeling approach, we also conducted experimental

measurements and FEM simulations.

The details of each of the 6 chapters are provided in the following. we should note that these

chapters are reproduced from our published paper in international journals.

Chapter 1 consist of literature survey where we briefly present what is statistical descriptor of

heterogeneous materials and then we consider Monte Carlo simulation to predict the statistical

correlation function of heterogeneous materials. We also briefly present homogenization

methods for the effective properties. At the end of the chapter, we give the definitions for

reconstruction of heterogeneous materials and we explain the annealing reconstruction

technique.

Introduction


In chapter 2, capability of the statistical continuum approach is directly linked to statistical

information of microstructure. Two-point correlation functions are the lowest order of

correlation functions that can describe the morphology and the microstructure-properties

relationship. In this chapter, SAXS data is used to calculate two-point correlation function

correlation for two phase polymer composite. The selected material is polystyrene (PS) filled

with zirconium oxide nanoparticles (ZrO2).

In chapter 3, higher order correlation functions must be calculated or measured to increase the

precision of the statistical continuum approach. To achieve this aim a new approximation

methodology is utilized to obtain N-point correlation functions for non-FGM (functional graded

materials) heterogeneous microstructures. Conditional probability functions are used to

formulate the proposed theoretical approximation. In this approximation, weight functions are

used to connect subsets of (N-1)-Point correlation functions to estimate the full set of N-Point

correlation function. For the approximation of three and four point correlation functions, simple

weight functions have been introduced. The results from this new approximation, for three-point

probability functions, are compared to the real probability functions calculated from a computer

generated three-phase reconstructed microstructure in three-dimensional space. This three-

dimensional reconstruction was based on an experimental two-dimensional microstructure (SEM

image) of a three-phase material. This comparison proves that our new comprehensive

approximation is capable of describing higher order statistical correlation functions with the

needed accuracy.

In chapter 4, a new Monte Carlo methodology is developed as a mean for three-dimensional

(3D) reconstruction of the microstructure, based on two-point statistical functions. The salient

feature of the presented reconstruction methodology is the ability to realize the 3D

microstructure from its 2D SEM image for a three-phase medium extendable to n-phase media.

In the realization procedure, different phases of the heterogeneous medium are represented by

different cells which are allowed to grow. The growth of cells, however, are controlled via

several optimization parameters related to rotation, shrinkage, translation, distribution and

growth rates of the cells. Indeed, the proposed realization algorithm can be categorized as a

member of dynamic programming methods and is designed so comprehensive that can realize

any desired microstructure.

Introduction


To be more specific, at first the initial 2D image is successfully reconstructed and then the final

optimization parameters are used as the initial values for the initiation of the 3D reconstruction

algorithm. This work presents a novel hybrid stochastic methodology based on the colony and

kinetic algorithm for the simulation of the virtual microstructure. The simulation procedure

involves repeated realizations where each realization in turn consists of nucleation and growth of

cells. For each of the subsequent realizations, the controlling parameters get updated by

minimization of an objective function (OF) at the end of the preceding realization. Here, the OF

is defined based on the two-point correlation functions from the simulated and real

microstructures. The kinetic growth algorithm is established on the cellular automata approach

which facilitates the simulation procedure. Comparison of the two-point correlation functions

from different sections of the final 3D reconstructed microstructure with the initial real

microstructure shows a satisfactory agreement which confirms the proposed methodology.

In chapter 5, we propose the use of strong contrast statistical continuum theory to predict the

effective elastic and thermal properties of nanocomposites. Three-dimensional isotropic

nanocomposite samples with randomly oriented monolayer nanoclay s are computer generated

and used to calculate the statistical correlation functions of the realized model. The nanoclay

orientation, shape and spatial distribution are taken into account through two-point and three-

point probability functions. These correlation functions have been exploited to calculate effective

thermal and elastic properties of the nanocomposite. To validate our modeling approach, we

conducted experimental measurements of these properties for Nanoclay/Polyamide

nanocomposites with concentrations of 1, 3 and 5 wt. % of nanoclay particles. The simulation

results have shown that effective stiffness can be increased significantly with small amounts of particle

concentration for the exfoliated clay monolayers.

The predicted effective conductivity and elastic modulus have been compared to our

experimental results. Effective thermal conductivity shows satisfactory agreement with

experimental data. The effects of nanoclay additives on the effective mechanical and thermal

properties of nanoclay based polymer composites have been investigated using experimental and

simulation analyses. In this research however, the predicted results for elastic modulus

overestimate the experimental data, which might be due to the increasing intercalated structure

for high concentration of nanofiller and to anisotropic properties of nanoclay. Relative to the

Introduction


pure polyamide matrix, both the modeling and the experiments show an increase of the effective

thermal conductivity and effective elastic modulus of the composite as a function of the nanoclay

volume fraction.

In chapter 6, the previously developed reconstruction methodology (in chapter 4) is extended

to three-dimensional reconstruction of a three-phase microstructure, based on two-point

correlation functions and two-point cluster functions. The reconstruction process has been

implemented based on hybrid stochastic methodology for simulating the virtual microstructure.

While different phases of the heterogeneous medium are represented by different cells, growth of

these cells is controlled by optimizing parameters such as rotation, shrinkage, translation,

distribution and growth rates of the cells. Based on the reconstructed microstructure, finite

element method (FEM) was used to compute the effective elastic modulus and effective thermal

conductivity. In addition, the statistical approach based on two-point correlation functions and

our proposed approximation of three point correlation functions (Derived in chapter 3 ) was

also used to directly estimate the effective properties of the generated microstructures. Good

agreement between the predicted results from FEM analysis and statistical methods was found

which confirms the efficiency of the statistical methods for the prediction of thermo-mechanical

properties of three-phase composites. Our results from statistical approach were also compared

to the case of the previous(existing) three-point correlation approximation [22]. This comparison

shows that our new approximation yields better results.

Finally, to conclude this thesis, general conclusions and remarks are reported. Some

perspectives and suggestions for the continuity for this research work are exposed.

Introduction


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[15] Tewari A, Gokhale AM, Spowart JE, Miracle DB. Quantitative characterization of spatial clustering in three-dimensional microstructures using two-point correlation functions. Acta Materialia. 2004;52(2):307-319.

[16] Brown JWF. Solid Mixture Permittivities. The Journal of Chemical Physics. 1955;23(8):1514-1517.

[17] Torquato S. Effective stiffness tensor of composite media--I. Exact series expansions. Journal of the Mechanics and Physics of Solids. 1997;45(9):1421-1448.

Introduction


[18] Mikdam A, Makradi A, Ahzi S, Garmestani H, Li DS, Remond Y. Effective conductivity in isotropic heterogeneous media using a strong-contrast statistical continuum theory. Journal of the Mechanics and Physics of Solids. 2009;57(1):76-86.

[19] Giraud A, Gruescu C, Do DP, Homand F, Kondo D. Effective thermal conductivity of transversely isotropic media with arbitrary oriented ellipsoïdal inhomogeneities. International Journal of Solids and Structures. 2007;44(9):2627-2647.

[20] Spanos PD, Kontsos A. A multiscale Monte Carlo finite element method for determining mechanical properties of polymer nanocomposites. Probabilistic Engineering Mechanics. 2008;23(4):456-470.

[21] Wang M, Pan N. Elastic property of multiphase composites with random microstructures. Journal of Computational Physics. 2009;228(16):5978-5988.

[22] Mikdam A, Makradi A, Ahzi S, Garmestani H, Li DS, Remond Y. A new approximation for the three-point probability function. International Journal of Solids and Structures. 2009;46(21):3782-3787.

Chapter 1: Literature Survey




Chapter I





Literature Survey





I.1. Random heterogeneous material

A random heterogeneous material is a class of materials which is composed of different materials

or states, such as a composite and a polycrystals. “Microscopic” length scale is much larger than

the molecular scale but much smaller than the characteristic length of the macroscopic sample

.The heterogeneous material can be supposed as a continuum on the microscopic scale, and

therefore its effective properties can be defined [1].

Statistical methods, using correlation functions, are one of the most practical and powerful

approaches to estimate properties of heterogeneous materials [1]. Properties of materials can be

approximated by using different order of statistical correlation functions [1-3]. In multiphase

materials, the first order correlation functions represent volume fractions of different phases and

do not describe any information about the distribution and morphology of phases [1].

If M-number of random points are inserted within a given microstructure and the number of

points in phase-i is counted as Mi, the one-point probability function ( 1iP ) is defined as the

volume fraction through the following relation, as M (the total number) is increased to infinity

1M

i ii

MP vM ��

� � (1)

where Vi is the volume of phase i (Φi), Vtotal is the total volume and vi is the volume fraction of

phase i. Clearly, for two phases microstructure:

2 2

1 1

1i total ii i

V V and v� �

� �� (2)

I.2. Two-Point Probability Functions

Now assign a vector starting at each of the random points in a heterogeneous microstructure.

Depending on whether the beginning and the end of these vectors fall within phase-1 or phase-2,

there will be four different probabilities ( 122P r� �� , 21

2P r� �� , 11

2P r� �� and 22

2P r� �� ) defined as [1]:



2 1 1 22 ,ijiji j

M

MP r r r r r r

M� �

��

��

� � � ��

ijMijij �� (3)

where, Mij are the number of vectors with the beginning in phase-i ( ) and the end in phase-j (

). Eq. (3) defines a joint probability distribution function for the occurrence of events

constructed by two points ( and ) as the beginning and end of a vector when it is randomly

inserted in a microstructure. The two-point function can be defined based on two other

probability functions such that [1]:

1 2 22ij

i jj

P r Probability r r Probability r� � � ��

��

� ��

P b bili �� b b l

� ��P b bili� ��

(4)

The first term on the right hand side is a conditional probability function. At very large distances,

r��, the probability of occurrence of the beginning point does not affect the end point and the

two points become uncorrelated or statistically independent and the conditional probability

function reduces to a one-point correlation function:

1 2 1i j iProbability r r r Probability r� � ��

� � ��

�� b b l� ��P b bili� ��

(5)

The two-point function will then reduce to [1]:

1 22 , ( )iji jP r r Probability r Probability r� ��

( )(� � � (6)

or,

� �2lim ijj

riP r ��

�� (7)

For the case of a two-point function in a two phase composite, we have symmetry for non FGM

microstructure [1]:

� � � �2 2ij jiP r P r�� ji� r�ji�

(8)

� i

� j



For a three-phase composite, the indices (i, j) in the probability functions representation extend

to three and as a result we have nine probabilities ( 112P , 22

2P , 332P , 12

2P , 212P ,

132P , 31

2P , 232P , 32

2P ).

Due to normality conditions the following equations are satisfied:

� �21,3 1,3

1ij

i jP r

� �

�� 1r �

(9)

� �21,3

iji

jP r v

�

�� r v�

(10)

� �2

1,3

ijj

iP r v

�

�� r v�

(11)

Satisfying all three conditions for a three-phase composite ( i , j�{1,2,3}) and knowing that the

probability functions are symmetric ( 2ijP = 2

jiP ) results in the important conclusion that only three

of the nine probabilities are independent variables. For instance, we can choose 112P or (P11), 12

2P

or (P12), and 222P or (P22) as the three probability parameters.

I.3. Two-Point Cluster Functions

Two-point cluster function is the other microstructure descriptor of heterogeneous materials

which can reflect more precise information for heterogeneous materials [4].The two-point cluster

function (TPCCF) 2 ( )C iiP r� ) is the probability of finding both points (starting and ending point of

vector ( r )) in the same cluster of one of the phase (i). This quantity is a useful signature of the

microstructure as it reflects clustering information. Incorporation of such information in addition

to the lower-order two-point cluster functions have led to the formulation of rigorous bounds on

transport and mechanical properties of two-phase media [1, 4].

I.4. Monte Carlo simulation of Correlation functions

The one-point probability function of the phase p is defined by the probability of occurrence of

random points in this phase [1]. Therefore, one-point correlation function for each phase

indicates the volume fraction of this particular phase. Convergence to the real volume fraction by

the soft core algorithm (allowing for penetrable inclusions) is one of the advantages of using

Eq.1 for randomly distributed penetrable inclusions.



Two-point correlation functions are determined based on the probability of occurrence of the

head and tail of each vector in a particular phase. For example for the nanoclay polymer

composites, there exist exactly two states, phase-1 (polymer matrix) and phase-2 (nanoclay

particles). Therefore, four different configurations of Two-point correlation functions are

obtained. These should satisfy normality conditions which results in the important conclusion

that only one of the four functions is independent (See Fig. 2) .

the Monte Carlo estimation of Two-point correlation function are acquired by assigning large

number of random vectors within the generated microstructure and examining the number

fraction of the sets (of vectors) which satisfy the different types of correlation functions .

Fig. 2. Two-point correlation functions for three composites with 3 wt% of nanoclay.

Three-point correlation functions for phase P can be interpreted as the probability that three

points at positions x1, x2, x3 are found in phase P. The vectors x2-x1, x3-x1 and x3-x2 are invariant

by translation and just depend on the relative positions of the points [1]. Thus, the three-point

correlation functions can also be interpreted as the probability of finding three points in a certain



triangular configuration as shown in Fig. 3, This interpretation can be generalized for N-point

correlation functions [1].

Fig. 3.Vectors for Three-point correlation function

Statistical homogenization techniques are limited by the use of explicit equations for calculating

governing multiple integral solutions. Therefore, the direct Monte Carlo approach cannot be used

to achieve a fast algorithm to estimate the effective properties of heterogeneous materials.

Generally, N-point correlation functions are defined as probability of occurrence of N-points

which are invariant relative to a fixed position in desired phases. The expression of these

functions for a given phase q can be written as [1]:

q,...,qn 1 2 n 1 2 nP (x ,x ,..., x ) Pr obability(x Phase(q) x Phase(q) ... x Phase(q))� � � � � � � (12)

Where, xi is the vector position of the points in the microstructure.



I.5. Approximation of higher order correlation functions

More detailed morphological description of heterogeneous materials is obtained by using higher

order correlation functions. Measuring higher order correlation functions is difficult because of

the increase of the number of independent variables to define correlation functions. For instance,

the approximation of three-point correlation functions using two-point correlation functions is

one of the best possible approach for calculating three point correlation functions.

Several simple analytical approximations were reported for three point correlation functions.

Adams [5] proposed an approximation of three point correlation functions using two-point

probability functions:

� � � � � �3 1 2 3 2 1 3 2 1 21 1, , , ,2 2

iii ii iiP x x x P x x P x x� � (13)

Garmestani et al. [6] also proposed another approximation for three point correlation functions:

� � � � � �1 3 2 33 1 2 3 2 2 3 2 1 3

1 3 2 3 1 3 2 3

, , , ,iii ii iix x x xP x x x P x x P x x

x x x x x x x x

� � � ��

�x x �x x�2 �2 �iiP ��2 �ii � 3 ii � �1 3 2 3x x11 x x2PP � � 2 3iiii � �1 3 2 3

� � �� x x x xP x x� � �

� �2 2 32 2 3� �2 2 3� �2 2 322 2 3� �x x x x x x x x

P x x� �,� �2 2 32 2� �P x x� � � �x x x xx x x x x xx x x x x xx x x x

(14)

These two approximations do not satisfy all normalization relations. Mikdam et al. [7]. proposed

a new approximation for two phase materials that satisfies the normalization relations.

� � � � � � � ��

1 3 2 3 2 2 33 1 2 3 2 2 3 2 1 3

21 3 2 3 1 3 2 3

,, , , ,

0

iiiii ii ii

ii

x x x x P x xP x x x P x x P x x

Px x x x x x x x

� ��

x x x x�iiP �ii31 3 2 3� �

x x x x1 3 231 3 23 2P x xx x� �ii � �1 3 2 3� ��x x x xx x x x

P x x� � �2 �2 �P ��2 �P �P x x� �P x x� �� 2 2 32 2� �2 2 3� �2 2 322 2� �x x x x x x x x

P x x� �,,� �2 2 32 2� �P x x� �x x x xx x x x x xx x x x x xx x x x

(15)



Fig. 4. Schematic representation of vectors for approximation of three point correlation functions

I.6. Homogenization methods for effective properties

The effective property Ke is defined by a relationship between an average of a generalized local

flux F and an average of a generalized local intensity G [1]:

.�F K Ge (16)

Table 1 summarizes the average local flux F and the average local intensity G for some physical

linear problems like conductivity, magnetic permeability, elastic moduli, viscosity and fluid

permeability.

Table 1 F, G and Ke for different physical problems [1]

General effective property

Ke

Average generalized flux

F

Average generalized intensity G

Thermal conductivity

Electrical conductivity

Magnetic permeability

Heat flux

Electric current

Magnetic induction

Temperature gradient

Electric field

Magnetic field

To estimate the bulk properties of such heterogeneous materials, multiscale homogenization

approaches are utilized. The multiscale homogenization techniques might be well categorized



into the following six classes: statistical methods such as strong-contrast [2, 3], inclusion-based

methods such as self-consistent or Mori-Tanaka [8], numerical methods such as finite element

analysis and asymptotic methods [9], variational/energy based methods such as Hashin-

Shtrikman bounds [10], and empirical/semi-empirical methods such as Halpin-Tsai and classical

upper and lower bounds (Voigt–Reuss) [11]. Here, we specifically turn our attention to the

statistical continuum mechanics of strong-contrast which, although difficult to implement, is

applicable to any form of micro-structural inhomogeneity and relies heavily on the statistical

information of the microstructure reflected in the correlation functions. In other words, to predict

the effective properties of heterogeneous media with a high degree of contrast between the

properties of phases and indistinguishable morphology of phases, strong-contrast approach is

highly suitable [1]. As pointed out earlier, one of the well-known applications of n-point

correlation functions can be found in properties characterization. For this, exact perturbation

expansions are used to predict the effective stiffness/thermal properties of a macroscopically

isotropic two phase composite media. Manipulating integral equations for the local “cavity”

strain field and polarization leads to finding series’ expansions for the effective stiffness tensor

or thermal tensor [1]. Unlike the classical homogenization methods the statistical approach

accounts not only for the interactions between the phases but also for the distribution of the

phases [1].

I.7. Assumption of Statistical Continuum Mechanics

Statistical information of the microstructure can be used to predict the effective properties.

There are some assumption for the samples and the domains as follows:

A. All the random variables of the heterogeneous media such as stress, strain, stiffness,... have

to obey the ergodic hypothes therefore the ensemble average of each variable can be defined

as follows [1]:

1( ) ( ) ( )V

c c x c x dV c xV

� � ��

(17)

B. Distribution of the considered property over the particles of the media is assumed statistically

homogenous. This assumption doesn’t prevent using the heterogeneous microstructures.



Since the microstructure can be heterogeneous in each section however to calculate the

overall elastic properties the microstructure is assumed to be statistically homogenous.

C. The considered bodies which are infinite in extent are assumed to be in equilibrium condition

at each point.

I.8. Reconstruction

Experimental and numerical Reconstruction of heterogeneous materials to get an accurate

structure can be used to characterize and optimized heterogeneous materials. there are different

experimental techniques such as x-ray tomography or focused ion beam/scanning electron

microscopy (FIB/SEM) which are used to reconstruct three dimensional microstructures. For

numerical reconstruction, statistical information are extracted from the microstructure of the

considered heterogeneous material and can be used to reconstruct three dimensional

microstructures [1, 12-17].

I.8.1 X-Ray Computed Tomography

X-Ray Computed Tomography is a non-destructive technique that can be utilized to reconstruct

micro-heterogeneous materials such as metal matrix composites. In this technique, X-ray beams

hits a rotating sample and two-dimensional projections are recorded using a detector in the

other side of the sample (see Fig. 5) [15, 17].

Fig. 5. Principle of x-ray tomography [17]



In classical tomography (attenuation tomography), three dimensional reconstruction is

performed by combining the two dimensional projections. This technique has some limitation

such as [15]:

� Resolution limited to about 1000-2000x the object cross-section diameter;

� Blurring of material boundaries;

� Weak attenuation contrasts for imaging ;

� Complicated data acquisition and interpretation due to the image artifacts (beam

hardening);

� Large data volumes and difficulty of visualization and analysis

However, this technique has several strengths such as [15]:

� Non-destructive 3D imaging

� Easy sample preparation required

� Extraction of sub-voxel level details.

I.8.2 FIB/SEM

FEI's DualBeam™ (FIB/SEM) systems are used for 3D microscopy and reconstruction of

micro-and-nano-composites. For this purpose, dual-beam FIB/SEM is utilized to obtain

microscopic two-dimensional (2D) SEM images in x–y plane by sectioning the specimen from

the surface in the vertical direction along z axis (see Fig. 6). Using Auto Slice and View software

(FEI Co.) serial-sectioning, SEM slices are stitched together to perform reconstruction. The dual-

beam FIB/SEM is composed of ion beam which allows milling of the surface while the imaging

is conducted by the electron gun [12].



Fig. 6. Principle of FIB/SEM [16]

I.8.3 Reconstruction using statistical descriptor (computer realization)

The reconstruction of random media using limited microstructural information (correlation

functions) is one the intriguing inverse problem in engineering. Various reconstruction

techniques have been developed to generate realizations with lower-order correlation functions

[13, 14]. In what follows, we briefly explain one of the most popular reconstruction approaches

which was developed using annealing optimization technique [1, 18]. Using a set of correlation

functions, partial information of heterogeneous media can be provided. This information can be

used to reconstruct and characterize random media. Generally, in a reconstruction procedure, we

would like to generate a microstructure with specified set of two-point correlation functions.

Numerical reconstruction of heterogeneous media can be utilized to solve an optimization

problem for a random generated microstructure. Monte Carlo reconstruction, using annealing

technique is an optimization technique that can be used to reconstruct heterogeneous materials

[13, 14, 18]. In this method, at the first step, a random image are generated with the same

volume fraction of target sample then annealing optimization technique is used to move pixel of

each phase for minimizing error between correlation function of realized model and sample.

An initial random configuration is generated until the one point function is similar to the target

sample. Then, an initial “temperature” is selected considering periodic boundary conditions and a

correlation function is calculated for this configuration. The result are then been compared to the



original target correlation function. Two pixels with different phases are chosen at random then

swapped; ensuring the volume fraction of each phase is preserved. Then, the same correlation

functions are calculated and the Mean Square Error (Error) is compared to the corresponding

correlation functions. In this method, the Metropolis algorithm is chosen as the acceptance

criterion for the pixel interchange and P is the acceptance probability for the pixel interchange as

follows:

(18)

Where ΔError=Errornew−Errorold and function of T will be defined base on step of annealing

solution. This process is repeated until the convergence to the target correlation functions.



I. References

[1] Torquato S. Random heterogeneous materials : microstructure and macroscopic properties. New York ; London: Springer; 2002.


[3] Pham DC, Torquato S. Strong-contrast expansions and approximations for the effective conductivity of isotropic multiphase composites. Journal of Applied Physics. 2003;94(10):6591-6602.

[4] Jiao Y, Stillinger FH, Torquato S. A superior descriptor of random textures and its predictive capacity. Proceedings of the National Academy of Sciences. 2009;106(42):17634-17639.

[5] Adams BL, Canova GR, Molinari A. A Statistical Formulation of Viscoplastic Behavior in Heterogeneous Polycrystals. Textures and Microstructures. 1989;11(1):57-71.

[6] Garmestani H, Lin S, Adams BL, Ahzi S. Statistical continuum theory for large plastic deformation of polycrystalline materials. Journal of the Mechanics and Physics of Solids. 2001;49(3):589-607.


[8] Nemat-Nasser S, Hori M. Micromechanics : overall properties of heterogeneous materials. 2nd rev. ed. Amsterdam ; New York: Elsevier; 1999.

[9] Dumont JP, Ladeveze P, Poss M, Remond Y. Damage mechanics for 3-D composites. Composite Structures. 1987;8(2):119-141.

[10] Hori M, Munasighe S. Generalized Hashin-Shtrikman variational principle for boundary-value problem of linear and non-linear heterogeneous body. Mechanics of Materials. 1999;31(7):471-486.

[11] Affdl JCH, Kardos JL. The Halpin-Tsai equations: A review. Polymer Engineering & Science. 1976;16(5):344-352.

[12] Edward R, Principe L. How to Use FIB-SEM Data for 3-D Reconstruction. 2005.

[13] Jiao Y, Stillinger FH, Torquato S. Modeling heterogeneous materials via two-point correlation functions: Basic principles. Physical Review E. 2007;76(3):031110.

[14] Jiao Y, Stillinger FH, Torquato S. Modeling heterogeneous materials via two-point correlation functions. II. Algorithmic details and applications. Physical Review E. 2008;77(3):031135.

[15] Ketcham R. X-ray Computed Tomography (CT). 2011.

[16] Reuteler J. Introduction to FIB-SEM.



[17] Merle P. X-Ray Computed Tomography on Metal Matrix Composites. Vienna University of Technolog: Insitute of Materials Science and Testing 2000.

[18] Yeong CLY, Torquato S. Reconstructing random media. Physical Review E. 1998;57(1):495.

Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function


Chapter II





Using SAXS Approach to Calculate

Two-Point Correlation Function:

(Application to Polystyrene/Zirconia

Nanocomposite)





II.1. Introduction

Statistical continuum theory correlates the morphology of microstructures to physical properties

of heterogeneous materials through correlation functions. In this framework, statistical n-point

correlation functions provide a mathematical representation of heterogeneous materials

morphology [1]. Particularly, one-point correlation functions gives information about the volume

fraction of each constituent (phase) of the heterogeneous material [1]. The distribution,

orientation and shape of the heterogeneous material phases are described by two-point or higher

order correlation functions, which can be in general determined from appropriate microstructure

measurements [2]. These measurements must be representative of the material morphology, i.e.

the experimental information must reflect all the variation of phase distribution within the

material. The heterogeneity, introduced through the polymer-based nanocomposites, can be

represented by: i) the overall distribution of the nanoparticles within the polymer matrix and ii)

the local heterogeneity of the nanoparticles which is called dispersion state [3]. What dictates the

material properties is actually the dispersion state of the nanoparticles. We therefore consider

that the dispersion of the nanoparticles within polymer matrix is the key distribution parameter to

take into account in the statistical theory.

To have information about nanoparticles dispersion, transmission electron microscopy (TEM) or

x-ray scattering can be used [4]. However, in the case of TEM analysis, the TEM images are

only relevant when the entire dispersion gradients of the nanoparticles are represented [5].

Particularly, uniform nanoparticles dispersion is not usually achieved. In this case, the

microstructure is characterized by a mixture of single particles and aggregates containing more

than one particle (aggregation). Note that, the nanoparticles aggregate size can reach several

hundred nanometers depending on the nanoparticle size, processing method and the chemical

interactions between the nanoparticle and the matrix. Therefore, the calculated correlation

functions strongly depend on the magnification at which the TEM images are recorded. Using a

high magnification, the correlation function will be dictated by the position within the

heterogeneous material where the microscopy images are taken (e.g. whether the TEM images

are chosen to include aggregates or not) [6]. In other words, the resolution can be high but the

representative area (or volume) is much larger that the selected image [7]. On the contrary, using

a low magnification, more representative information about the dispersion of the nanoparticles



will be obtained. In this case, the statistics are high but the resolution is low. As an alternative,

dispersion state of nanoparticles in the polymer-based nanocomposites can be characterized by

small-angle x-ray scattering (SAXS) measurement [8]. SAXS is an easy and fast method that is

applied to a volume of the order of several cubic millimeters (high statistics) without

compromising the resolution. The obtained scattering signal of the nanoparticles reflects the size

distribution and shape of the nanoparticles (form factor) and their position with respect to each

other (structure factor). For example, a high dispersion state of the nanoparticles within a

polymer matrix will be characterized by an average particle size near that of a single particle and

eventually a homogeneous interparticle distance. SAXS signal can be consequently exploited to

calculate two-point correlation functions with a high accuracy since it produces a very accurate

representation of the material morphology [9-12].

In this work, SAXS data is exploited to calculate two-point correlation function correlation for

two phase polymer composite. The selected material is polystyrene (PS) filled with zirconium

oxide nanoparticles (ZrO2). The nanocomposite morphology was first examined by scanning

transmission electron microscopy (STEM) and SAXS. The two-point correlation functions were

then calculated from SAXS measurements, while the three-point correlation functions can be

approximated [13] from two-point correlation functions relation .

II.2. Correlation between SAXS data and two-point correlation functions

Small-angle x-ray scattering technique relies on electron density scattered from heterogeneities

particles whose size typically ranges between 1 and 1000 nm, depending on the equipment

configuration [14-16]. The scattered intensity depends on the difference between a local

electronic density from the scattered heterogeneities and its surrounding, which can be

represented by an average density . The local fluctuation ! of the electron density can be

defined as follows:

!� � . (1)

Assuming a statistically isotropic system with no long-range order, a correlation function that

considers the amplitude of the density fluctuations can be defined as:



� � 2" ! � ! !A Br (2)

where A and B are two distinct points in the medium represented by the vectors 1r , 2r , 2 1r r r� �

and � �r" is the characteristic or autocorrelation function depending on the position r. � �r" can

be defined as follows:

� � 1 2( ) ( )r r r" � ! ! (3)

For random distribution of heterogeneities, the autocorrelation function � �r" satisfy the

following conditions: � � 20r" � � ! and � � 0r" �� . It is convenient to define the auto-

covariance of phase-1 for a statistically homogeneous media as [1] :

� � 11 21 2 2 1( ) ( ) ( )r r r P r" � ! ! � �� (4)

where 1� is the volume fraction of phase 1 (fillers) and 112 ( )P r is the two-point probability

function. Recalling that � �r is the number of electrons per unit volume, a volume element dV

at position r will contain � �r dV # electrons.

The intensity of the x-ray scattering I as a function of the scattering vector h over the entire

volume V is given by the following Fourier integral [17] :

1 2 1 2 1 2 1 2( ) ( ) ( ) ( ) ( )ihr ihr

V

I h dV dV r r e r r e drdr� �� h) �� (5)

Summing all pairs with the same relative distance, then integrating over all relative distances,

seems to be a logical course. An autocorrelation function can be defined as:

21 1 2( ) ( ) ( )r dV r r $ ��2 ( ) $2 ( )) �� (6)

which allows to rewrite � �I h as:

2 ihrI(h) dV (r)e�� h) dVdV�� 2 ihr(r)e2 (7)



implying that the intensity distribution in h or reciprocal space, is uniquely determined by the

structure of the density field. Considering statistical isotropy, Debye [9, 10] proved that

sin( )ihr hrehr

� � (8)

As a result, the average scattering intensity reduces to:

2 2 sin( )( ) 4 ( ) hrI h r dr rhr

� % & � 2 sin(( )2 (2

hr (9)

Recalling the autocorrelation function, " , the above equation can be rewritten:

2 20

sin( )( ) 4 ( ) hrI h Vn r dr rhr

� % & "� (10)

where 0n is the mean density of electrons. Or,

22 2 0

0

1 sin( )( ) ( )2

hrr I h h dhVn hr

�" �

% � (11)

here, 0n is a constant. Using equation (4), the equation (11) can be rewritten as follow:

11 2 22 1 2 2 0

0

1 sin( )( ) ( ) ( )2

hrr P r I h h dhVn hr

�" � ��

% � (12)

where � �112P r represents the two-point probability correlation function which measures the

spatial distribution of the heterogeneities (phase-1) in the matrix (phase-2). � �112P r should verify

the following condition:

� ��

112 1

2112 1

when 0

when

P r r

P r r

� � �

� � ��. (13)

The second condition in equation (13) is an indicator of the degree of homogeneity of the

distribution of heterogeneities in the matrix (i.e. if the second condition is not verified then the

distribution of the heterogeneities are not homogeneous in the matrix).



II.3. Structural characterization

3.1. Materials

The polymer matrix of the studied nanocomposite, polystyrene (PS), was supplied by Scientific

Polymer Products Inc. It has a molecular weight of about 120,000 g/mol. As for the zirconium

oxide (ZrO2) nanofiller, it was provided by Sigma Aldrich under the reference # 544760 (average

particle size < 100 nm according to the datasheet). The specific surface area (measured by the

Brunauer, Emmett, and Teller method) and the density of ZrO2 were 25 m2/g and 5.89 g/cm3,

respectively.

II.3.2. Preparation of the nanocomposites

All nanocomposites were prepared by melt mixing. The following material systems were

extruded by means of a micro-compounder DSM (reference Xplore 15 mL): neat PS, PS + 1 wt.

% ZrO2, PS + 3 wt. % ZrO2, and PS + 5 wt. % ZrO2. During this procedure, each system was

compounded during 5 minutes at 230°C with a screw co-rotating speed of 200 rpm. To avoid

oxidation phenomena, the extrusion was carried out under argon gas. The produced materials

were extruded cylinders, 5 mm in diameter. Thermogravimetric analysis was performed after the

processing step to measure the effective amount of nanoparticles within PS. The results indicated

that the amount of nanoparticles used for the processing are preserved.

II.3.3. Scanning transmission electron microscopy

We performed a structural characterization of the nanocomposite by scanning transmission

electron microscopy (STEM) to verify the presence of aggregates within the polymer matrix.

STEM analyses of PS-ZrO2 nanocomposites were carried out using a scanning electron

microscope FEI Quanta FEG 200 apparatus at 7 kV. The STEM samples were ultra-thin films

(70 nm-thick) that were prepared with a Leica EM FC6 cryo-ultra-microtome at 25°C using a

trimming diamond blade.

Fig. 1 shows some tendency to aggregation whatever the amount of filler. The size of the

aggregates is much less than 200 nm except for very few cases for which the size of the

aggregates is in the micrometric range. The tendency to aggregation can be explained by the fact

that the particles were not coated, which does not enable to increase the interaction between the



oxide particle and the polymer matrix. It is also thought that the used micro-compounder do not

enable to reach an optimal dispersion state. This can be explained by the geometry of the

extrusion screws that does not permit to obtain a high enough elongational flow to completely

break up the aggregates into primary particles. However, the low beam energy of the scanning

electron microscope does not enable to observe the local distribution state of the nanoparticles,

i.e. the dispersion state, with the transmission mode. To characterize this local distribution of the

particles, small-angle x-ray scattering technique was employed.

Fig. 1. STEM micrographs at two magnifications, 5 000 (a, c and e) and 50 000 (b, d and f), of the composites PS +1 wt. % ZrO2 (a and b), PS + 3 wt. % ZrO2 (c and d), and PS + 5 wt. % ZrO2

(e and f)



II.3.4. Small-angle x-ray scattering

Small-angle x-ray scattering tests were performed by means of a Panalytical X’Pert Pro MPD

device to study the scattering signal of the nanoparticles within the polymer matrix, and hence to

obtain physical and structural information. Particularly, the analysis of the scattering signal

enables to characterize the size and shape of the particles (form factor) as well as their relative

ordering (structure factor) [12]. The radiation used ' =1.54 Å (Cu K(), at 45 kV and 40 mA was

generated by an x-ray tube operating at 40 kV and 45 mA. A focused parallel mirror and a

PIXcel detector were employed with specific slits in order to obtain the highest resolution at

small-angle. Also, to attract strong signal from the nanoparticles, the background noise from the

SAXS curves, i.e. the scattering curve of neat PS, was systematically subtracted. The last

treatment of the curves consisted of assessing the intensity of primary beam that passes through

the nanoparticles (absorption correction). The scattering intensity I is plotted as a function of

scattering vector h = (4π/') sin (θ) where θ is the scattering angle. Each SAXS test was repeated

on three specimens.

Fig. 2 shows representative scattered intensity I(h) of ZrO2 characterized alone (as-received

powder), and characterized within PS matrix (investigated amounts: 3 and 5 wt. %). A high

reproducibility of I(h) curves was found for each material. It is to be noted that no scattering

signal was obtained for 1 wt. % of ZrO2 within PS, and hence the scattering curve of this system

was not plotted in Fig. 2. This is certainly due to the resolution of the x-ray scattering equipment

that does not enable to characterize such a low content of particles (1 wt. %) within a polymer

matrix. For the other investigated ZrO2 amounts, no long-range Bragg peak is noted on the

scattering curves, indicating no ordering of ZrO2 nanoparticles and hence no interaction strength

between the particles as for example Van-der-Waals or hard-sphere interactions. Consequently,

the scattering signal of the nanoparticles is only induced by the form factor. The initial parts of

I(h) curves, below h = 0.07 nm-1, show a continuous decrease of the intensity with h that

suggests some large aggregate. This observation is in line with the aggregation tendency noted

on STEM images (Fig. 1). It is to be noted that in the case of a well-dispersed system, an initial

"plateau" of scattering intensity at very low h would have been noted, followed by a gradual

decrease of the intensity with h. Above h = 0.07 nm-1, the presence of some oscillation indicates

the presence of scattering objects that have a relative uniform size. These objects are most



probably single particles and small aggregates constituted by few single particles. By means of

Guinier plot (ln I as a function of h2), we found a radius of gyration Rg (deduced from the slope

of ln(I)-h2 curve) of about 13.5 nm for as-received ZrO2, 5% of ZrO2 in PS and 3 % of ZrO2 in

PS. Considering ZrO2 nanoparticles as having a spherical shape, the average particle size

(diameter), deduced from the relationship 2×Rg×(5/3)0.5= 34.8 nm. Despite the presence of some

big aggregates in the micrometric range (Fig. 1), we considered that the most representative

information about the distribution of the particles is provided by SAXS. I(h) curves (Fig. 2) are

hence used for the calculation of the two-point correlation functions.

Fig. 2. The scattered intensity I as a function of scattering vector h for ZrO2 nanopowder and PS-ZrO2 composites (3 and 5 wt %) (background- and absorption-corrected curves) .

The two-point probability functions representing the distribution of the ZrO2 nanoparticles

within the PS matrix are calculated using equation (12) and reported in Fig. 3. Note that since



SAXS diagram of 1 % ZrO2 in PS do not show any signal from the nanoparticles, � �112P r curve

presents negative values and does not verify the second condition in equation (13). Therefore

� �112P r for 1 wt. % ZrO2 cannot be exploited to calculate the physical properties of the

nanocomposite. However, the two-point probability function � �12P r , for 3 and 5 wt. % ZrO2 (see

Fig. 3) verifies the limits given in equation (13).

Fig. 3. Two-point correlation functions TPCF for PS-ZrO2 composites (3 and 5 wt. % of ZrO2)



II.4. Conclusion

Polystyrene (PS) nanocomposites were produced by melt mixing technique using zirconium

oxide (ZrO2) as fillers. The spatial dispersion of nanoparticles within the polymer matrix was

characterized by STEM and SAXS measurements. A non-uniform dispersion of the nanoparticles

within the polymer matrix with a tendency to aggregation is obtained. The SAXS signals are

used to calculate the correlation functions that represent the spatial dispersion of the

nanoparticles considered as the key distribution parameter in such heterogeneous materials. The

calculated correlation functions can be used in conjunction with the strong contrast version of the

statistical continuum theory to predict the effective mechanical and thermal properties for both 3

and 5 wt. % ZrO2..



II. References

[1] Torquato S, Haslach HW. Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Applied Mechanics Reviews. 2002;55(4):B62-B63.

[2] Jiao Y, Stillinger FH, Torquato S. Modeling heterogeneous materials via two-point correlation functions: Basic principles. Physical Review E. 2007;76(3):031110.

[3] Alexandre M, Dubois P. Polymer-layered silicate nanocomposites: preparation, properties and uses of a new class of materials. Materials Science and Engineering: R: Reports. 2000;28(1-2):1-63.

[4] Kashiwagi T, Harris RH, Zhang X, Briber RM, Cipriano BH, Raghavan SR, et al. Flame retardant mechanism of polyamide 6-clay nanocomposites. Polymer. 2004;45(3):881-891.

[5] Kashiwagi T, Fagan J, Douglas JF, Yamamoto K, Heckert AN, Leigh SD, et al. Relationship between dispersion metric and properties of PMMA/SWNT nanocomposites. Polymer. 2007;48(16):4855-4866.

[6] Li DS, Baniassadi M, Garmestani H, Ahzi S, Reda Taha MM, Ruch D. 3D Reconstruction of Carbon Nanotube Composite Microstructure Using Correlation Functions. journal of computational and theoretical nanoscience. 2010;7(8):1462-1468.

[7] Lingaiah S, Sadler R, Ibeh C, Shivakumar K. A method of visualization of inorganic nanoparticles dispersion in nanocomposites. Composites Part B: Engineering. 2008;39(1):196-201.

[8] Bandyopadhyay J, Sinha Ray S. The quantitative analysis of nano-clay dispersion in polymer nanocomposites by small angle X-ray scattering combined with electron microscopy. Polymer. 2010;51(6):1437-1449.

[9] Debye P, Anderson HR. The correlations Function and Its Application. JOURNAL OF APPLIED PHYSICS. 1957;28(6):4.

[10] Debye P, Anderson HR, Brumberger H. Scattering by an Inhomogeneous Solid 2. The Correlations Function and Its Application. JOURNAL OF APPLIED PHYSICS. 1957;28(6):679-683.

[11] Frisch HL, Stillinger FH. Contribution to the Statistical Geometric Basis of Radiation Scattering. The Journal of Chemical Physics. 1963;38(9):2200-2207.

[12] Gunier A, Fournet G. Small Angle Scattering of X-Rays., New York: John Wiley; 1955.


[14] Brumberger H. Modern Aspects of Small-Angle Scattering. Boston: Kluwer Academic Publishers; 1995.



[15] Feigin LA, Svergun DI. Structure Analysis by Small-Angle X-ray and Neutron Scattering. New York Plenum Press; 1987.

[16] Cullity BD, Stock SR. Elements of X-ray Diffraction, . New Jersey: Prentice Hall; 2001.

[17] Glatter O, Kratky O. Small Angle X-ray Scattering. New York: : Academic Press; 1982.

Chapter III: New Approximate Solution for N-Point Correlation Functions for Heterogeneous Materials


Chapter III





New Approximate Solution for

N-Point Correlation Functions

for Heterogeneous Materials





III.1. Introduction

Description and characterization of heterogeneous systems have become of extreme importance

to scientists during the past decades. Many techniques have been developed to realise three-

dimensional descriptions of heterogeneous systems [1]. Statistical continuum mechanics

provides a robust alternative for the reconstruction and characterization techniques of

heterogeneous systems. The reconstruction techniques have been empowered by the

development of numerous simulation methodologies in recent years. Anisotropic features,

orientation distribution, shape and geometrical features can be extracted from statistical

correlation functions. Yeong and Torquato [1, 2] have initiated the study of microstructure

reconstruction using correlation functions. Random heterogeneous materials were reconstructed

from low order correlation functions via stochastic optimization annealing techniques. Different

types of microstructures were investigated to examine the limitations of the reconstruction

techniques to include short-range order. An exact mathematical formulation of the reconstruction

algorithm was presented by Yeong and Toquarto [1, 2]. In a recent work, Garmestani and co-

workers [3] have developed a new Monte Carlo (MC) methodology using Colony and kinetic

growth algorithm. This approach have been developed to reconstruct the microstructure of two-

phase composites using statistical correlation functions [3]. This was recently extended by

Baniassadi and co-workers [4] to three-dimensional multiphase composites, specifically applied

to planar section solid oxide fuel cell materials, to develop three-dimensional microstructures. Li

and co-workers[5] have presented a novel Monte Carlo technique by incorporating geometry,

distribution and waviness of virtual nanotube fillers for the reconstruction of Carbon Nanotube

(CNT) polymer composites. In this approach, the nanotubes were described as a chain of links

and the reconstruction was performed by the optimization of the waviness, geometry and

preferential distribution of CNTs.

Characterization of mechanical, magnetic, electrical and thermal properties can be performed

directly from descriptors such as N-point statistics. Different statistical continuum approaches

(weak-contrast and strong-contrast) have been developed to account for the material

heterogeneity through probability functions (Kröner [6]; Beran [7]; Phan-Thien and Milton [8];

Dederichs and Zeller [9] , Willis [10]; McCoy [11]; Torquato [1, 12, 13]; Sen and Torquato

[14]). Weak contrast technique is based on perturbation from the average property and can be



utilized for heterogeneous materials with small variation in the multi-phase properties. A strong-

contrast expansion for two-phase isotropic media was developed by Brown [15] for the effective

conductivity that resulted in convergent integrals. Other scientists, such as Torquato [1] and

Fullwood [16], developed this method for n-dimensional space and anisotropic multiphase

heterogeneous materials. N-point correlation functions have a long history in important science

and engineering applications going back to the invention of X-ray scattering and diffraction early

last century. Statistical information in the form of pair-correlation functions can be extracted by

using scattering data [17, 18]. Small angle X-ray scattering technique has been used to get

information on the distribution of inclusions and dispersion of particles [19] . Corson [20] has

developed methodologies linking properties of two-phase structures to the experimentally

calculated two-, and three-point probability functions. In this approach the probability functions

are assumed to be isotropic. In 1987, Adams et al. [21] introduced a set of two-point probability

functions based on spherical harmonics. The spectral technique was used to account for

orientation and point-to-point correlations in the microstructure. Garmestani and others [22-27]

have later extended the statistical continuum approach to both composites and polycrystalline

materials using two-point functions. Mikdam et al. [28] have developed an approximation for the

3-point correlation functions based on two-point functions. In other researches, Mikdam et al.

[29] and Baniassadi et al. [17, 30] have applied the strong-contrast formulation to predict the

effective electrical and thermal conductivity of a two-phase composite material where the

distribution, shape and orientation of the two phases are taken into account using two-point and

three-point correlation functions.

In the present work, we propose to use the conditional probability to derive a comprehensive

formulation of the N-point correlation functions for multiphase non FGM heterogeneous

materials. The approximation of the used probabilities and the use of the boundary conditions

allowed us to derive a new and broad approximation of the N-point probability functions. We

show the capability of this new approach by comparing our predicted results to results from the

computed real probability functions (for a computer generated microstructure) for three-point

correlation functions.



III. 2. Approximation of tree-point correlation functions

III. 2.1. Decomposition of Higher Order Statistics

Higher order correlations can incorporate more details of the morphology of the secondary

phases. Theoretically, a unique microstructure can be reconstructed by using an infinite order

correlation function. In statistical mechanics formulations, it is necessary to exploit higher order

correlation functions for a better identification of heterogeneous systems. In the current work, N-

point correlation functions have been approximated by use of (N-1)-point correlation functions.

To obtain this approximation, N-point correlation function was partitioned into N subsets of (N-

1)-point correlation functions. For instance, the set of X of points or events (x1,x2, …xN), the

subsets of X are given below:

X= , ,…., (1)

, ,…., , ,…., (2)

In this work, we denote by CN(x1, x2, ….xN) the N-point probability function for the occurrence of

the point (x1, x2, ….xN) in a desired phase (occurrence of the event ):

CN ( , , ) (3)

Here, represents the probability of the event . For

simplicity, the following properties of this correlation function are shown for the case of N=3

C3 ( , , ) = C3 ( , , ) = C3 ( , , ) (4)

C3 ( , , ) = = C2 ( , ) (5a)

C3 ( , , ) = = C2 ( , ) (5b)

III. 2.2. Decomposition of two-point correlation functions

Two-point correlation function is the probability of finding the beginning and ending points of a

random vector with length r in a desired phase. According to the probability theory, two

compatible events can be independent or dependent under favorable conditions. Dependency and

independency of the two events in a heterogamous system depend on the length of the vector r.



This means that for very small r (very small in comparison to the RVE dimension), the

probabilities of the occurrence of two events x1 and x2 are within the correlation limit (or

dependant). However, for very large values of r these probabilities will be independent.

For a general formulation, we introduce the dependency weight factor, , which will allow us

to express the 2-point correlation function in terms of the 1-point probability functions in the

following multiplicative decomposition:

2 1 ) 1 ) (6)

The dependency factor is a function of the vector length:

(7)

For very large r, the independence of events x1 and x2 yields the following:

(8)

In addition, for very small length of r ( if the two events x1 and x2 are compatible we

have

(9)

However, if the two events are incompatible we have:

(10)

Note that the indices in the dependency weight factor represent the order of the correlation

function (upper index b) and the number indicating each of factors needed (lower index a).

III. 2. 3. Decomposition of three-point correlation functions

A decomposition methodology is presented here to represent and estimate three-point correlation

functions by use of two-point correlation functions. A full set of information for the two-point

correlation functions must be available for the correct representation of the 3-point functions.

First, the set of points (x1,x2,x3) is selected in a heterogeneous system and an analysis is

performed according to conditional probability. Fig. 1 illustrates the three random points



representing the set (x1,x2,x3). Assume the occurrence of the event x1 (point x1 is found in

specified phase). The probabilities of finding second and third points (x2 and x3) in the same

phase are given by the following conditional probabilities:

(11)

(12)

Fig. 1.Three random points selected to calculate the three-point correlation function

The three-point probability function for the occurrence of the event (x1,x2,x3) is equal to the sum

of the probabilities of the following possible events. In this, we introduce the dependency factors

( , and ) used to formulate our proposed approximation for the three-point correlation

functions:

The probability of occurrence of x1 followed by x2 and then x3 can be expressed as:

(13)



Similarly, the probability of occurrence of x2 followed by x1 and then x3 is :

(14)

Finally, the probability of occurrence of x3 followed by x1 and then x2 is given by :

(15)

Therefore, the three-point correlation function C3(x1,x2,x3) is given by the following

approximation which adds the above probability approximations of the three possible events:

(16)

We can then write:

(17)

The weight functions can now be calculated using the boundary conditions: The first

boundary condition is:

(18)

where, x1 is meant to satisfy the following conditions: |r12 and |r13 . Therefore

(19)

Applying this boundary condition we get:

(20)

Similarly, for and we get:

(21)

(22)



The second boundary condition is:

(23)

From equality condition for left side and right side of Eq. (17) we have:

(24)

Third boundary condition:

(25)

This yields:

(26)

From this boundary conditions and using Eq. (17) we get:

(27)

By applying similar methodology for and we obtain:

(28)

(29)

Therefore, necessary conditions for weight function are referred as follows (see details in

additional Appendix A):

(30)

(31)

(32)



By assuming = fi (R1, R2, R3) where R1, R2 and R2 are the lengths of the radii between the

three random points shown in Fig. 1. One choice for the weight functions that verifies all of the

above boundary conditions is given by the proposed following radii ratios:

(33)

(34)

(35)

III. 3. Approximation of four-point correlation function

We consider four random points arranged as a tetrahedron which encompasses a sphere of the

radius Ri with the following four-point probability function (Fig. 2):

Fig. 2. Four random points selected to calculate the four-point correlation function

Similarly to the development of the three-point correlation functions in the previous section, the

four-point correlation function has been approximated using three and two-point correlation

functions.



(36)

Or,

(37)

The weight functions are calculated using boundary conditions:

The first boundary condition is:

(38)

This limit can be written as:



(39)

Applying the boundary condition we get:

(40)

Similarly for , and , we obtain

(41)

(42)

(43)

The second boundary condition is:

(44)

We can also write:



(45)

Applying this boundary condition leads to:

(46)

Third boundary condition:

(47)



(48)

The boundary conditions in Eq. (47) require the following conditions:

(49)

Similarly, for and we get:

(50)

(51)

And finally:

(52)

Therefore, necessary conditions for weight functions are obtained as (see details in additional

Appendix B):

(53)

(54)

(55)

Assuming that are function of area fractions of the tetrahedron faces (areas) in Fig. 2, all

boundary condition can be shown to be satisfied through the following ratios:

(56)

(57)

(58)

(59)



where, is the area of the side of the tetrahedron encompassing the three points

.

III. 4. Approximation of N-point correlation function

The methodology for deriving the approximations in previous sections can be extended to N-

point correlations which unfortunately yields a lengthy procedure for N>3. Thus, we limit our

analysis to the following brief general description of the methodology:

(60)

where are the dependency weight functions. In the formulation above, , is

defined as the subset of (N-1)- points that include xi as a member of the subset. The weight

functions must satisfy the following limiting boundary conditions:

(61)

(62)

(63)

In the next section, we will present numerical results but only for three-point correlation

functions.



III. 5. Results

III. 5.1. Approximation of Three-Point correlation functions for a three-dimensional

reconstructed microstructure

In this section, the numerical verification of the above approximations is conducted to show the

accuracy and the limitations of this methodology. Here, we chose to show results only for N=3.

For this, a numerical Monte Carlo program was constructed and used to calculate different 3-

point statistical functions for a three-dimensional reconstructed microstructure. Monte Carlo

methodology is used to reconstruct 3D microstructures of a three-phase anode structure in a solid

oxide fuel cell from an experimental 2D SEM micrograph (see Fig. 3) [4]. The three phases

shown on the SEM micrograph in Fig. 3 are nickel, yttria-stabilized zirconia (YSZ) and voids

[4].

Fig. 3- SEM micrographs of a three-phase Anode microstructure of Solid Oxide Fuel Cell [4] (red: Nickel, blue: YSZ, Black: voids)



The methodology for the reconstruction is based on a two-point statistical function as a

microstructure descriptor. Colony and Kinetic Growth algorithms are used to enable the

realization process based on an optimization methodology described in the next chapter (see also

[4]). The generated 3-D reconstruction of the microstructure is shown in Fig. 4.

(a)

(b)

Fig. 4. a) Three-dimensional reconstructed image of the Anode microstructure b) several sections through the depth of the 3D microstructure (red: Nickel, blue: YSZ, Black: voids) . Phase 1

(Blue color) Phase 2 (Red color) Phase 3 (black color) [4]

For a three-phase composite, we have nine two-point probabilities. Due to normality conditions

and knowing that the probability functions are symmetric the number of independent two-point

correlation functions reduce to three. For instance, we can choose C2(red-red), C2(black-black),

and C2(red-black), as the three probability parameters. As an example, the diagrams of the three

independent two-point correlation functions for the anode microstructure of Fig. 3 are shown in

Fig. 5, 6 and 7. These results are finally used to approximate three-point correlation functions

using Eq. (17).



Fig. 5. Two-point correlation functions (TPCF) for the three-phase composite. Here we show the 2-point correlation function for the red-red phases

Fig. 6.Two-point correlation functions for the three-phase composite. Here we show the 2-point correlation function for the black-black phases



Fig. 7. Two-point correlation functions for the three-phase composite. Here we show the 2-point correlation function for the red- black phases

Three-point correlation functions were estimated by the use of Monte Carlo theory. In this

approach a large number of vectors have been generated randomly within the 3D microstructure

and the probability of the occurrence of desired events were calculated. The results show very

good agreement between numerical Monte Carlo simulation of the real sample and the

approximation method. The three-point correlation functions based on the two selected vectors

(R12=(9.62)i+(9.62)j+(9.62)k (constant length), R13=xi (varying length with ))

originating from a random point are calculated while the length of one and the angle between the

two remain unchanged.

In Fig. 8 the simulations are performed for the probability of occurrence of the three points (X1,

X2 and X3) in phase 1 (red phase). The result is plotted against the length of one vector (R13)

while satisfying the conditions above.



Fig. 8. Three-point correlation functions for the three-phase composite. The corresponding 3-

point correlations are shown for the (red- red-red) phases (average error = 0.046).

In Fig. 9 the simulations has been carried out to approximate probability of occurrence of the

three points (X1, X2 and X3) in phase 3 (black phase).



Fig. 9. Three-point correlation functions for the three-phase composite. The corresponding 3-point correlations are shown for the (black- black-black) phases (average error = 0.049).

The corresponding three-point correlation functions plotted in Fig. 10 and Fig. 11 show that the

approximation based on the methodologies described here match fairly well the simulated

correlations calculated from the three dimensional reconstructed microstructure.



Fig. 10. Three-point correlation functions for the three-phase composite. The corresponding 3-point correlations are shown for the (red- black-red) phases( average error = 0.052).

Fig. 11. Three-point correlation functions for the three-phase composite. The corresponding 3-point correlations are shown for the (red- black-blue) phases (average error = 0.066).



Three- Point correlation functions have been simulated and approximated for a variety of vector

lengths from 10 to 400 unit lengths of the representative volume element (2000 units) which is

adapted to the convergence range of three point correlation functions for reconstructed RVE. The

average errors are reported in Fig. 12 for a large amount of data (more than 50000 three-point

correlation functions) and various types of three-point correlation functions.

We note that the error was calculated using the following equation where THPCF represents the

three point correlation function,

(64)

Fig. 12. Average error for various types of three-point correlation functions

III. 5.2. Approximation of three-point correlation functions for computer generated of

hard-sphere microstructure

Three-dimensional isotropic virtual samples with randomly distributed hard spheres are

generated and used to calculate the statistical two-point correlation functions of high density

spheres. In this study, Three-point correlation functions have been approximated using two-point

correlation functions which are calculated using Monte-Carlo simulations. The sphere geometry



is defined by a radius and center of spheres. The center of the spheres has been allocated

randomly inside a cubic volume. In the next step, two-point correlation functions are determined

based on the probability of occurrence of the head and tail of each vector in a particular phase

(spheres). The size of representative volume element (RVE) has been verified using

convergence of the two-point correlation function [31]. Three- point correlation functions –

THPCP (sphere-sphere-sphere) have been simulated and approximated using Eq. (17) for a large

amount of vectors and the errors are reported via average length of these vectors in Fig. 13. In

this work, we have studied more than 40000 three-point correlation functions with different

magnitude of vector lengths from 10 to 400 of unit lengths of cubic RVE, we note that the

chosen RVE dimensions was 1000 in this simulation; the error has been calculated using Eq.

(64). Although we see a large dispersion of the error in Fig. 13, we must note that the average

error has been found to be equal to eight percent.

Fig. 13. Error of Three-point correlation between the simulated and approximation

for the (sphere, sphere, sphere) phases



III. 6. Conclusion

In the present study, a new formulation is proposed to obtain a relation between the higher and

lower order correlation functions for heterogeneous materials. The approximation was developed

using the conditional probability theory and the formulation is valid for multiphase

heterogeneous materials. Comparison between the three-point correlation functions computed

from a 3D reconstructed microstructure and from the proposed approximation shows satisfactory

agreement. The compared results confirm the capability of our proposed approximation scheme

to estimate N-point correlation functions using the information from the lower order (N-1)-point

correlation functions. In future work, the authors would like to incorporate two-point cluster

functions as suitable descriptor of microstructures [32] to find more precise approximation. An

investigation of different type of weight functions needs also to be conducted.



III. References


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[17] Baniassadi M, Addiego F, Laachachi A, Ahzi S, Garmestani H, Hassouna F, et al. Using SAXS approach to estimate thermal conductivity of polystyrene/zirconia nanocomposite by exploiting strong contrast technique. Acta Materialia. 2011;59(7):2742-2748.

[18] Debye P, Anderson HR. The correlations Function and Its Application. Journal of Applied Physics. 1957;28(6):4.

[19] Glatter O, Kratky O. Small angle X-ray scattering. London: Academic; 1982.

[20] Corson PB. Correlation functions for predicting properties of heterogeneous materials. II. Empirical construction of spatial correlation functions for two phase solids. J Applied Physics. 1974;45(b).

[21] Adams BL, Morris PR, Wang TT, Willden KS, Wright SI. Description of orientation coherence in polycrystalline materials. Acta Metallurgica. 1987;35(12):2935-2946.

[22] Garmestani H, Lin S, Adams BL. Statistical continuum theory for inelastic behavior of a two-phase medium. International Journal of Plasticity. 1998;14(8):719-731.

[23] Garmestani H, Lin S, Adams BL, Ahzi S. Statistical continuum theory for large plastic deformation of polycrystalline materials. Journal of the Mechanics and Physics of Solids. 2001;49(3):589-607.

[24] Gokhale AM, Tewari A, Garmestani H. Constraints on microstructural two-point correlation functions. Scripta Materialia. 2005;53(8):989-993.

[25] Li DS, Saheli G, Khaleel M, Garmestani H. Microstructure optimization in fuel cell electrodes using materials design. CMC-Computers Materials & Continua. 2006; 4(1)::11.

[26] Lin S, Garmestani H, Adams B. The evolution of probability functions in an inelasticly deforming two-phase medium. International Journal of Solids and Structures. 2000;37(3):423-434.

[27] Saheli G, Garmestani H, Adams BL. Microstructure design of a two phase composite using two-point correlation functions. Journal of Computer-Aided Materials Design. 2004;11(2):103-115.


[29] Mikdam A, Makradi A, Ahzi S, Garmestani H, Li DS, Remond Y. Effective conductivity in isotropic heterogeneous media using a strong-contrast statistical continuum theory. Journal of the Mechanics and Physics of Solids. 2009;57(1):76-86.

[30] Baniassadi M, Laachachi A, Makradi A, Belouettar S, Ruch D, Muller R, et al. Statistical continuum theory for the effective conductivity of carbon nanotubes filled polymer composites. Thermochimica Acta. 2011;520(1-2):33-37.



[31] Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D. Determination of the size of the representative volume element for random composites: statistical and numerical approach. International Journal of Solids and Structures. 2003;40(13-14):3647-3679.

[32] Jiao Y, Stillinger FH, Torquato S. A superior descriptor of random textures and its predictive capacity. Proceedings of the National Academy of Sciences. 2009;106(42):17634-17639.

Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials Using Two-Point Correlation Functions


Chapter IV





A New Monte Carlo Solution for

Reconstruction of Heterogeneous

Materials Using Two-Point

Correlation Functions:

(Application to Three-Phase Solid Oxide

Fuel Cell Anode Microstructure)





IV. 1. Introduction

There is a growing need for a mathematical linkage between microstructure and some of the

important properties in materials [1-3]. Such a linkage can provide the means to design

microstructures with optimum properties [4, 5] . Representation of microstructures based on n-

point correlation functions has a long history going back to the discovery of x-ray scattering and

the understanding that the result of scattering provides statistical information in the form of pair-

correlation functions [6]. The community of small angle scattering has a rich history of

developing structure functions to get information about the microstructure in the form of particle

size and distributions. More recently, reconstruction methodologies based on two-point functions

have evolved as a challenging problem [7]. Yeong and Torquato [8] introduced a stochastic

optimization technique that enables one to generate realizations of heterogeneous materials from

a prescribed set of correlation functions. They have provided examples of realizable two-point

correlation functions and introduced a set of analytical basis functions for their representations.

They have presented an exact mathematical formulation of the reconstruction algorithm. Jiao

and co-workers [9] has also shown that the two-point functions alone cannot completely specify

a two-phase heterogeneous material. As a result they have developed an efficient and isotropy-

preserving lattice-point algorithm to generate realizations of materials.

Kröner [10, 11] and Beran [12] have developed statistical mathematical formulations to link

correlation functions to properties in multiphase materials. Analytical techniques based on one-

point probability have a significant drawback in that important characteristics such as shape and

geometry are not considered. Thus, to determine the contribution of shape and distribution

effects, higher order probability functions must be developed.

Corson [13-15] was among the first to attempt to incorporate shape and geometry effects by

using an experimental form of the two- and three-point probability functions. In this formulation,

Corson assumes that the probability functions are independent of orientation. In 1987, Adams

introduced a set of two-point probability functions based on spherical harmonics [16] . The

harmonics were used to account for orientation and point-to-point correlation in the

microstructure. Garmestani and co-workers later extended the statistical continuum approach to

both composites and polycrystalline materials using two-point functions [4, 17-24].



Torquato and co-workers have developed a procedure for the realization for a two-phase media

using stochastic optimization techniques [9]. A stochastic reconstruction technique was used to

generate random heterogeneous media with specified correlation functions. An optimization

technique was applied to the two orthogonal directions and the autocorrelation functions for the

generated two orthogonal sets are then calculated between these two sets [25]. The comparison is

then used as a means for the reconstruction methodology by examining autocorrelation functions

that display no appreciable short-range order [25]. Elsewhere, Torquato tried to develop a new

methodology to reconstruct 3D random media by using the information from 2D sections [8]. In

this methodology, a hybrid stochastic reconstruction technique was developed for the

optimization of the lineal-path function and the two-point correlation functions during nucleation

annealing technique [8]. In most of the numerical setup reviewed above, the simulated annealing

methodology was used to reconstruct the random media while in our proposed reconstruction

algorithm, the realization procedure is implemented using several optimization parameters which

controls the overall reconstruction of heterogeneity.

Heterogeneity can be observed in a wide range of natural and artificial substances [26].

Heterogeneity can be recognized in a material system by the local measurements of particle

orientation and size distribution. Two mechanisms of nucleation and grain growth are examples

of processing controlling the development of heterogeneities. Heterogeneity can take place

during casting (as a result of nucleation) and crystallographic grain orientation distribution

during grain growth [26]. It is clear that by using the grain growth as a function of time and

morphology a certain level of heterogeneity can be developed. Inspired by the two mechanisms

of nucleation and grain growth, we founded our proposed algorithm of heterogeneity

reconstruction on three steps: generation, distribution and growth of cells. For illustration, Table

1 lists the technical equivalent of the three steps for two metallurgical processes.



Table 1- Different steps of heterogeneity generation in two metallurgical processes

Sim. Steps

Process

Cell generation

Cell distribution Growth of cells

Casting Nucleation Nucleation rate Grain growth

Powder metallurgy

Powder

(particles)

Packing Sintering

In the present study, a cellular automata approach [27] was utilized to implement the kinetic

growth of cells. The cellular automaton model used for kinetic growth of cells is similar to the

Eden fractal model employed as an efficient tool to simulate some natural spatiotemporal

phenomena [27]. It has been noted that the grain boundaries (boundary morphology) in

heterogeneous materials look highly like fractalian geometries [28] .

In this study, the Monte Carlo simulation is the primary modeling tool for the development of the

realization methodology. Our Monte Carlo approaches rely on the definition of important

parameters that affect nucleation and grain growth as parts of a kinetic growth model. The

microstructure is then evolved and optimized by manipulating the prescribed parameters of the

model through an objective function (OF) minimization for the statistical correlation function.

In a previous work [29], we have developed a two-dimensional reconstruction methodology for

two-phase composite materials. Under this methodology, random realizations are generated

using statistical correlation functions based on the Monte Carlo simulation. The microstructures

are then explored and modified by mimicking the natural processes of materials synthesis to

predict the final realization. A kinetic growth model [27] was combined with a colony algorithm

based on the Monte Carlo methodology. The present work concentrates on the 3D realizations as

compared to our previous 2D-based work [29]. A three-phase anode microstructure of a solid

oxide fuel cell is considered, which increased the order of the statistical representation.



IV. 2. Development of a Monte Carlo reconstruction methodology

A new algorithm is presented based on Monte Carlo methodology for the reconstruction of

microstructures using two-point statistical functions [30]. The realization process includes three

steps: 1) generation, 2) distribution, and 3) growth of cells. Here, cells (or alternately grains or

particles) refer to initial geometries assigned to each phase before the growth step. During the

initial microstructure generation, basic cells are created from the random nucleation points and

then the growth occurs as in crystalline grain growth in real materials [31, 32] . After distribution

of nucleation points and assignment of basic cell geometries, the growth of cells starts according

to the cellular automaton approach. The three steps of realization algorithm are repeated

continuously to satisfy the optimization parameters until an adequately realistic microstructure is

developed as compared statistically to the true microstructure. It is worth noting that in various

steps of algorithm execution, several controlling parameters are developed that facilitate

minimization of the objective function (OF) which is an index of successful realization.

Before the 3D realization process, the microstructure of interest is reconstructed in 2D using the

planar basic cells, as depicted schematically in Fig. 3. First of all, a sufficiently fine 2D grid is

produced. Then for each phase and based on their associated volume fractions, a number of basic

cells of arbitrary geometries representing the rough initial shape of existing phases are placed at

some random nucleation points. Then these entities are allowed to grow in the next step. Fig. 3

illustrates the growth of three typical cells after being generated in several evolutionary stages.

Afterwards, the procedures of basic cells distributions, examining the volume fractions and

growth continue until the cells meet each other and the grid is filled.



Stage (1)

Stage (2)

Stage (3)

Stage (4)

Stage (5)

Stage (6)

Stage (7)

Stage (8)

Fig. 3. Step-by-step growth of three typical cells in a 2D grid

During simulations, it was observed that simulation results are insensitive to the rough initial

geometry of the basic cell. Additionally, the computer code was designed such that overlapping

of dissimilar basic cells is avoided. Furthermore, the distribution form of basic cells, or, more

precisely, the closeness or clustering of similar basic cells is controlled by colony algorithm

detailed in subsection 2.2.

At the end of a 2D reconstruction, the objective function (OF) which is defined based on the

three independent two-point correlation functions as

� � � � � �2 2 211 11 12 12 22 222 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )real sim real sim real simOF P P P P P P� � � � � � (1)

where the subscripts real and sim indicate, respectively, the relevant values from the real and

simulated microstructures, is evaluated numerically. For the subsequent reconstructions in Fig. 4,

the optimization parameters such as shrinkage of basic cells, growth factors in the X- and Y-

direction, parameters of the colony algorithm, rotation angles of basic cells and so on are varied

such that the objective function of Eq.(1) is minimized. The procedure of reconstruction and

optimization is repeated until the objective function takes a sufficiently close to zero value and



meanwhile less than the Monte Carlo (M-C) repeat error. This repeat error depends on the

microstructure.



Fig. 4. Basic steps in the realization algorithm (OF = objective function; MC=Monte Carlo)

Growth of cells

Compute the two-point correlations (From

Experimental SEM Images)

Compute the two-point correlations (Simulation)

NO

OF < M-C repeat error

Reconstruction is done.

Generation of cells

Optimization

Start Reconstruction

Distribution of cells

YES



IV. 2.1. 3D cell generation

After successful reconstruction of 2D microstructure, certain simulation parameters including

optimum growth factors in the X- and Y-direction, colony parameters and shrinkage factor are

inherited by the 3D realization algorithm. For 3D generation of basic cells, the 2D cell can be

extruded to form a 3D one based on the extrusion shape function:

� �,Z h x y� �� (2)

where � refers to the phase of interest. Some typical simple forms of the function h are listed in

Table 2. In this work, however, different but constant extrusion functions leading to cubic basic

cells were used.

Table 2- Typical mathematical forms for extrusion shape function

shape Equation

Ellipsoid 2 22 2

2 2(( ( )) )i

x yZ c ka b� �� ) � � a, b, c and k( ) are constants for each phase

Torus � �2

2 2 2( ( )) ( )Z k R x y� � �� ) � � k( ) and R( ) are constant for each phase

Cube ( )Z ak� �� ) a and k( ) are constant for each phase

The cells are then allowed to undergo sort of a local shrinkage through a shrinkage function, S,

defined as:

� ��

1 1

2 2

3 3

, , , ,, , , ,, , , ,

f x y z pS f x y z p

f x y z p

***

+ ,- .� - .- ./ 0

(3)

Where x , y and z are cartesian coordinate inside the extrusion shape define by Z� .

The mathematical forms of fi can be, for example, based on simple polynomial functions. The

dependency of the transformation matrix on local Cartesian coordinates can be used to develop a



methodology for the 3D simulation. In the matrix above 0 1*1 1 is a random variable and ip

is the optimization parameter satisfying 0 1ip2 2 . Each of the three components of the S vector

takes values from the interval3 40,1 , hence the term shrinkage function. In this work, the simple

forms of i if p*� were selected to represent the shrinkage function that only scales down the

initial basic cell.

Local rotation of basic cells is another operation that can be performed to achieve optimum

reconstruction. The three local rotation matrices are represented by the following:

4 4

4 4

1 0 00 cos( ( , )) sin( ( , ))0 sin( ( , )) cos( ( , ))

x x x

x x

Q p pp p

5 * 5 *5 * 5 *

+ ,- .� �- .- ./ 0

(4)

5 5

5 5

cos( ( , )) 0 sin( ( , ))0 1 0

sin( ( , )) 0 cos( ( , ))

y y

y

y y

p pQ

p p

5 * 5 *

5 * 5 *

+ ,�- .� - .- ./ 0

(5)

6 6

6 6

cos( ( , )) sin( ( , )) 0sin( ( , )) cos( ( , )) 0

0 0 1

z z

z z z

p pQ p p

5 * 5 *5 * 5 *

�+ ,- .� - .- ./ 0

(6)

where rotation angles, , ,x y z5 5 5 , depend on the random parameter, * , and the optimization

factor, ip . The mathematical form of the rotation angles may be represented by

, , 2x y z ip5 %*� (7)

with * and ip satisfying the same conditions that they have in Eq. (3).



IV. 2.2. Cell distribution

The subsection of cell distribution consists of two parts: distribution of cells’ centers and the

relative positioning of identical cells. For the first part a random generator function was defined

to calculate the Xc ,Yc, and Zc coordinates for the initial position of the cells in the Monte Carlo

simulation.

7( , )CX F p*� (8)

8( , )CY F p*� (9)

9( , )CZ F p*� (10)

where * and ip have the same definitions as in Eq. (3) or (7) and F can assume different

forms depending on the expertise of the user. One possible form of dependency, for example, can

be represented as

ipF L*� (11)

where L is the dimensional length of the 3D grid in the X-, Y-, or Z-direction depending on the

coordinate under consideration. Here, we have used the simple linear form of Eq.(11) , F=Lβ.

For the second part, the analysis can be performed according to the desired model whether the

overlapping or penetration of identical phases is allowed or not. In other words, the models can

allow for coalescence of the particles (cells) using the colony algorithm resulting in

agglomeration or can allow for the model to remain devoid of any agglomeration of particles

using the contactless procedure. The flow diagram provided in Fig. 5 helps to better understand

the distribution procedure. By generating a cell, if the simulated volume fraction of the

corresponding phase is lower than the input volume fraction, then the center of the particle is

relocated using Eq. (8),(9) and Eq.(10). If coalescence is allowed and the new cell overlaps with

another similar one, the new cell is placed at the generated coordinates otherwise the next

condition is checked. This new condition, discussed in detail in the following paragraph, controls

the state of bundling or clustering of homogenous cells. If this last condition is not satisfied it



means that the location of the new cell is far from the regions of space occupied by similar

particles and there is no similar entity in its neighborhood. On the contrary, if the conditional

term is satisfied it means that the new cell is going to be located in the neighborhood of some

other similar particle(s) and has an adverse effect on the minimization of the objective function.

Therefore it should be rejected and a newer center location (coordinates) be generated. Different

alternate coordinates are selected until this criterion is met.



Fig. 5. Algorithm for cell distribution.



In the colony algorithm, one possible form of conditional statement is

� � � � � �exp where niC C* � * *6 7 7 � (12)

where 0 1*1 1 is a random variable, i� is the volume fraction of the phase of interest, and ,n 7 are

two optimization parameters. Indeed, by changing the colony parameter n (power of the bundling

distribution function) and Ω (input probability criterion), the clustering rate of cells can be monitored. On

the right hand side of the above inequality, the proposed exponential form guarantees the stability of the

algorithm.

IV. 2. 3. Cell growth

For implementation of the final step of the realization process, i.e. the cell growth, the well

known cellular automaton approach (CA) is utilized [33]. The model has the potential for being

used in computability theory (mathematical logic), physics, theoretical biology and

microstructural reconstruction. The concept is explored on a grid of sites with each site capable

of assuming a finite number of states. By assigning an initial state to each site of the grid, the

following process can be generated (or the growth of the grid) according to the states of the

neighboring sites along with a few growth rules which are usually similar for all sites. Concisely,

a cellular automaton consists of a site space with a neighborhood relation, a set of states and a

local transition function.

The neighborhood relation considered in this work is of Neumann type (Fig. 6). In Neumann

neighborhood for a 3D lattice, six adjacent sites on top, bottom, right, left, front and back of a

central site are regarded as its neighbors whose states contribute to the determination of the

subsequent growth state of the grid.



Fig. 6. Von Neumann neighborhood relation in a 3D grid of sites

Indeed, the growth argument applies only to the sites on the exterior layer of each grain. The

transition or update function exploited to predict the directional growth can be either

deterministic or stochastic and can be applied either synchronously or asynchronously. Here

stochastic transition functions are chosen whereby the model is updated synchronously. Given

the Neumann neighborhood, six directional transition functions are suggested corresponding to

six directions/neighbors around each site. For every site, six conditional statements are checked

in the following way:

� �, 0i i ip p8 * *� � 6 (13)

Here, i=1,2, …6 , β ( 0 1*1 1 ) is a random variable and ip is an optimization parameter

(0≤pi≤1). If the condition (25) is satisfied, the growth continues in that direction by one site

provided not already occupied. Then the procedure continues to examine the other directions and

other sites on the exterior layer.

The adopted kinetic growth model can be regarded as an extended version of the Eden fractal

algorithm [27] used in biology and chemistry for describing the growth of bacterial colonies and

deposition of materials. The current proposed growth methodology not only does not suffer from

the instability issues but also it is capable of allowing growth in any preferential orientation

which is useful when simulating anisotropic materials. This is because of the introduction of



optimization parameters in the present algorithm that allow control over the growth of all cells

individually.

IV. 3. Optimization of the statistical correlation functions

In this work, material’s heterogeneity is represented by statistical distribution functions. A

hypothetical statistical function is optimized and compared to the experimental statistical

distribution functions. Stochastic optimization methodologies incorporate probabilistic (random)

elements, either in the input data (the objective function, the constraints, etc.), or in the algorithm

itself (through random parameters, etc.) or both [34]. By applying different optimization

parameters to the simulations, a minimum error is achieved through minimization of the

objective function that is constructed from the comparison of the two-point correlation function

of the experimental and simulated images. A direct simple search optimization technique [34]

was used for finding the minimum objective function. The optimization technique was applied in

two stages: first step is used to extract the optimization factor for a 2D image (rotation factor in

Z axis, shrinkage factor in the XY plane, colony factors, grain growth factors in the XY plane. In

the second step, the optimization and other parameters (rotation about the X and Y axis, grain

growth in the XZ or YZ plane) are used as initial input parameters for the 3D reconstruction. One

of the main advantages of this technique is the decreased time of optimization.

IV. 3.1. Percolation

Percolation analysis is one of the most complicated and time-consuming computational

methodologies in engineering. Percolation algorithms are used to exploit the continuity of

objects and morphologies that are affected by certain properties and processes. Many different

types of algorithms are presented to solve percolation problem, but some of them are not

efficient and others are only useful for specific tasks [27]. As one of the important applications

of percolation analysis in the realization and reconstruction methodologies for a heterogeneous

microstructure, it is usually necessary to check percolation of the different phases during cell

generation. In every step of the percolation, the continuity of cells is checked and the number of

cells are recalculated for the entire cluster. The knowledge of the percolation cluster numbers



[35], allows other higher order statistical correlation cluster functions to be recalculated. The

percolated phase can then be shown as one color for graphical representation.

In this procedure, the boundary of the percolated regions is calculated for the simulated

microstructures. A new Monte Carlo methodology for percolation is used to examine the extent

of the clustering process in the heterogeneous material (Fig. 7). In this model, every cell is

assigned a number (cluster number) that evolves through the cell growth process. A random node

is selected and for every node a minimum cluster value of neighboring nodes will be assigned as

shown in Fig. 6. This process is repeated until percolation is completed. This algorithm is very

simple and it converges very quickly. The simulation processes for the percolation in this

approach occur simultaneously for all cells and phases.

Fig. 7. Algorithm of percolation based on the Monte Carlo methodology.

Yes

No

Save Cluster Number

Percolation completed?

Allocate minimum cluster numbre to all neighbor’s site

Calculate minimum value of cluster number of phases

Select Random Site in Grid Phase Network



IV. 4. Three-phase solid oxide fuel cell anode microstructure

Performance and properties of solid oxide fuel cell are determined by microstructure of

components, just like most other engineering materials. For example, our previous studies

revealed that the degradation mechanism in fuel cell anode depends on anode support

microstructure [36]. It is very important to understand the relationship between microstructure

and properties. Verification of modeling performance requires the capability of microstructure

reconstruction. In this study we developed a new microstructure reconstruction method and

applied on fuel cell anode. The anode microstructure of a solid oxide fuel cell (SOFCs) is

presented in Fig. 8. Due to its functionality and operational environment requirements, SOFC

anodes must have high catalytic activity for hydrogen oxidation, high electronic conductivity,

and sufficient open porosity for unimpeded transport of gaseous reactants and products. SOFCs

must also be stable at SOFC operating temperatures in reducing environments. The material of

choice for long-term stability, chemical and mechanical compatibility with the YSZ electrolyte

and low cost is Ni-YSZ cermet [37, 38] (see Fig. 8). The nickel serves as an electrochemical

catalyst and electronic conductor. The YSZ provides mechanical strength, inhibits coarsening of

the nickel particles, provides porosity for gaseous transport to the electrolyte, and yields an

anode material with a coefficient of thermal expansion (CTE) that is similar to that of the YSZ

electrolyte [39]. Within the porous structure of the anode material, nickel particles typically

protrude from the YSZ substrate into the pores. The line at which the three phases (nickel, YSZ,

and porosity) meet is referred to as the as the triple-phase boundary (TPB). In the active part of

the anode, near the electrolyte, the active species converge for the electrochemical reaction at the

TPB. Pathways must be provided to transport the species to the TPB in order for it to be active.

Electrons are conducted through the nickel, the oxide ions are conducted within the YSZ and

hydrogen gas flows through the porosity to the TPB.

Some investigators have observed degradation in electrochemical performance during testing

with Ni-YSZ anodes. In 1996, Iwata [40] fabricated a roughly 3-mm-thick anode by mixing and

cold pressing 8-YSZ (8 mol% yttrium doped zirconium) and nickel-oxide (NiO) powders. An 8-

YSZ electrolyte was then deposited to the anode substrate by plasma spray (to ~200 µm

thickness). Iwata performed duration tests of 211 hr at 927 °C, and 1015 hr at 1008 °C with cells



made of these anode/electrolyte layers. Both tests exhibited anode performance degradation

apparently proportional to the duration and experienced temperatures. Clearly, the features of the

anode microstructure can have a significant influence on its long-term electrochemical

performance. The goal of this study is to develop a methodology to reconstruct the three-phase

microstructure of the SOFC anode to facilitate the subsequent performance and degradation

studies.

Fig. 8. SEM micrographs of a three-phase Anode microstructure of Solid Oxide Fuel Cell (red: Nickel, blue: YSZ, Black: voids)

IV. 5. Reconstruction of multiphase heterogeneous materials

A three-phase anode microstructure of solid oxide fuel cells (SOFC) is considered for the

reconstruction methodologies introduced above. The three constituents of this anode are Nickel,

YSZ and voids (see Fig. 8). The methodology uses the two-point correlation functions calculated

from the 2D SEM micrographs as an input to produce different 2D and 3D realizations of the

microstructure with special attention to the percolation of the porous media.

For illustration of the proposed methodology, Fig. 9a shows the phase distribution for a

computer-generated three-phase composite (red, green, and white) with a 20% volume fraction

for red phase, 20% for the green phase and 60% for the white phase. This microstructure is

simulated to examine the reproducibility of the details of the microstructure represented by the

two-point correlation functions (Fig. 9b). This is accomplished by using the same first-order



statistics and input simulation parameters for both the red and green phases in this trial

realization. The results for the two-point correlation functions (P11 for red-red and P22 for green-

green) in Fig 9b show that the realization of the red and green phases are statistically

indistinguishable Thus, we can conclude that the proposed methodology well controlled by the

input parameters of the Monte Carlo algorithm.

(a)

(b)

Fig. 9. a) 2D simulation image of a three-phase microstructure (red, green and white) with 20% for red and green and 60% white. b) the corresponding 2D probability statistics

(TPCF = 2-point correlation function)

After the above illustration based on a numerical 2D microstructure, now we consider the real

microstructure of the SOFC anode. Fig. 10 shows the 2D SEM micrograph of the anode

microstructure and the corresponding two separate 2D realizations. The two-point correlations

calculated from the SEM micrograph are used as initial input for the realizations in Fig. 10b and

c. The corresponding two-point correlation functions plotted in Fig. 10d-e show that the

realizations based on the methodologies described here match fairly well the original correlations

calculated from the SEM micrograph.

0

0.05

0.1

0.15

0.2

0.25

0 200 400 600 800 1000r

TP

CF

RED PHASEGREEN PHASE



(a)

(b)

(c)

(d)

(e)

(f)

Fig. 10. 2D realizations for an experimental image and comparison of the two-point correlation functions (TPCF). a) the 2D SEM micrograph for the anode microstructure (from Fig 1). b)

realization-1 c) realization-2 d) the 2-point correlation function ( 112P or P11) for the red-phase, e)

the 2-point correlation function ( 222P or P22) black phase f) 2-point correlation (black-red) function

( 122P or P12)

The 2D reconstruction requires simulation parameters for cell generation, nucleation, and growth

that are calculated during the optimization process to arrive at a final microstructure. These

parameters along with the input two-point statistical functions are now used as input parameters

for the 3D realizations. Fig. 11 presents four 2D sections through the depth of the 3D realizations

for the input three-phase anode microstructure. For this realization we have used the 2D

microstructure in Fig. 10b. Table 3 and 4 show the final simulation parameters for the 3-D

reconstruction.

TPCF P11

0

0.05

0.1

0.15

0.2

0.25

0 200 400 600 800r

TPC

F(P

HA

SE

1)

EXPRIMENTAL RESULTSIMULATION1SIMULATION2

TPCF P11

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 100 200 300 400 500 600 700 800r

TPC

F(P

HA

SE

2)

EXPERIMENTALSIMULATION1SIMULATION2

P 12

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 200 400 600 800r

TPC

F(P

12)

EXPERIMENTALSimulation1Simulation2



Table 3- RVE properties Pixel RVE Type Cube

X Dimension 205 Y Dimension 154 Z Dimension 116

Table 4- Reconstruction parameters

Cell Shrinkage Rotation Distribution Colony Cell growth

Cube

0.50.50.5

S+ ,- .� - .- ./ 0

P4=1

F=Lβ

disabled

� �,.001 .001 0i8 * *� � 6

P5=1

P6=1



The boundaries of the percolated regions of the porous phase for the 3D realization are identified

for one of the 2D sections (Fig. 12a) and is shown in Fig. 11b. The three independent two-point

correlation functions are compared with the original experimental SEM micrograph in Fig. 13a-

c. The results show that the methodologies adopted here can produce microstructures with the

same statistical information based on two-point statistics in a 3D microstructure. The 3D

microstructure is then plotted from the 2D sections and shown in Fig. 14-a, b.

(a)

(b)

(d)

(e)

Fig. 11. 2D sections in the z-direction of the 3D image for the reconstructed microstructure. a) Layer close to the bottom surface , b) Layer in the middle area, c) Layer middle and top d) Layer

close to the top surface



(a)

(b)

Fig. 12. a) A 2D section of the 3D image for the reconstructed microstructure (black=porosity); b) the corresponding percolation of voids (porosity) showing the percolation clusters by similar

colors other than white



(a)

(b)

(c)

Fig. 13. Comparison of the two-point correlation functions from the experimental and the 3D realizations. a) 2-point correlation functions 11

2P or P11 (red-red, phase 1), b) 222P or P22 for the

porous phase (black-black, phase 2), c) 122P or P12 (black-red)

0

0.05

0.1

0.15

0.2

0.25

0 200 400 600 800r

TPCF

(PHA

SE1)

EXPRIMENTAL RESULTSIMULATION

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 200 400 600 800r

TPCF

(PHA

SE2)

EXPERIMENTALRESULTSIMULATION RESULT

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 200 400 600 800 r

TPC

F(P1

-2)

EXPERIMENTAL RESULT SIMULATION RESULT



(a)

(b)

Fig. 14: a) Three-dimensional reconstructed image of the Anode microstructure. b) several sections through the depth of the 3D microstructure (red: Nickel, blue: YSZ, Black: voids).

IV. 6. Conclusion

A Monte Carlo methodology is used to reconstruct 3D microstructures of a three-phase anode

structure in a solid oxide fuel cell (SOFC) from a 2D SEM micrograph. The methodology is

based on two-point statistical functions as microstructure descriptors. The realization uses a

hybrid stochastic reconstruction technique for the optimization of the two-point correlation

functions during different 3D realizations. Colony and kinetic growth algorithms (cellular

automata) are used to enable the realization process based on an optimization methodology. The

main challenge in the 3D reconstruction is the degree of complexity due to the increased number

of microstructure parameters as compared to 2D realization. Another important aspect of the new

methodology is the establishment of a simple numerical routine to examine the percolation of

desired phases relevant to fuel cell technology [35]. Comparison of the two-point correlation

functions from different sections of the final 3D reconstructed microstructure with the initial real

microstructure shows good agreement. This supports the capability of our proposed methodology

to reconstruct 3D microstructure from an experimental 2D SEM result.



IV. References

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[2] Torquato S, Stell G. Microstructure of two-phase random media. I. The n-point probability functions. J Chem Phys. 1982;77:2071-2077.


[4] Saheli G, Garmestani H, Adams BL. Microstructure Design of a Two Phase Composite Using Two-point Correlation Functions. international Journal of Computer Aided Design. 2004;11(2-3):103 - 115.

[5] Adams BL, Lyon M, Henrie B, Kalidindi SR, Garmestani H. Spectral integration of microstructure and design Textures Of Materials, ed. 2002;408-4:487-492.

[6] Debye P, Anderson HR, Brumberger H. Scattering by an Inhomogeneous Solid 2. The Correlations Function and Its Application. JOURNAL OF APPLIED PHYSICS. 1957;28(6):679-683.

[7] Liang ZR, Fernandes CP, Magnani FS, Philippi PC. A reconstruction technique for three-dimensional porous media using image analysis and Fourier transforms. Journal of Petroleum Science and Engineering. 1998;21:273–283.

[8] Yeong CLY, Torquato S. Reconstructing random media. PHYSICAL REVIEW E. 1998;57(1):495-506.

[9] Jiao Y, Stillinger FH, Torquato S. Modeling heterogeneous materials via two-point correlation functions: Basic principles. PHYSICAL REVIEW E. 2007;76.

[10] Kröner E. Statistical Continuum Mechanics. Wien: Springer Verlag; 1972.

[11] Kroner E. Bounds for effective elastic moduli of disordered materials,. J Mech Phys Solids 1977;25:137-155.

[12] Beran MJ. Statistical continuum theories. New York,: Interscience Publishers; 1968.

[13] Corson PB. Correlation Functions for Predicting Properties of Heterogeneous Materials. I. Experimental Measurement of Spatial Correlation Functions in Multiphase Solids. J Applied Physics. 1976;45(a):3159-3164.

[14] Corson PB. Correlation functions for predicting properties of heterogeneous materials. II. Empirical construction of spatial correlation functions for two phase solids. J Applied Physics. 1974;45(b):3165.



[15] Corson PB. Correlation functions for predicting properties of heterogeneous materials. I. Experimental measurement of spatial correlation functions in multiphase solids. J Applied Physics. 1974;45(a): 3159.

[16] Adams BL, Morris PR, Wang TT. Description of Orientation Coherence in Polycrystalline Materials. Acta Metall. 1987:2935-2946.

[17] Lin S, Garmestani H, Adams B. The Evolution of Probability Functions in an Inelastically Deforming Two-Phase Medium. International journal of solids and structures. 2000;37(2):423.

[18] Lin S, Adams BL, Garmestani H. Statistical continuum theory for inelastic behavior of two-phase medium. International Journal of Plasticity. 1998;17(8):719-731

[19] Li DS, Saheli G, Khaleel M, Garmestani H. Quantitative Prediction of Effective Conductivity in Anisotropic Heterogeneous Media Using Two–point Correlation Functions Computational Materials Science. Computational Materials Science. 2006;38(1):45-50.

[20] Li DS, Saheli G, Khaleel M, Garmestani H. Microstructure optimization in fuel cell electrodes using materials design. CMC-COMPUTERS MATERIALS & CONTINUA. 2006;4(1):31-42.

[21] Li D, Garmestani D. Microstructure Sensitive Design and Quantitative Prediction of Effective Conductivity in Fuel Cell Design. In: Khan A, editor. The 13th International Symposium on Plasticity and Its Current Applications, Alaska.2007.

[22] Gokhale AM, Tewari A, Garmestani H. Constraints on Microstructural Two-Point Correlation Functions. SCRIPTA MATERIALIA. 2005;53(8):989-993

[23] Garmestani H, Lin S, Adams B, Ahzi S. Statistical Continuum Theory for Texture Evolution of Polycrystals. Journal of the Mechanics and physics of Solids. 2001;49:589-607.

[24] Garmestani H, Lin S, Adams B, Ahzi S. Statistical Continuum Mechanics Analysis of an Elastic Two-Isotropic-Phase Composite Material. Composites: Part B. 2000;31:39-46.

[25] Sheehan N, Torquato S. Generating microstructures with specified correlation functions. JOURNAL OF APPLIED PHYSICS. 2001;89(11):53-60.

[26] Riosa PR, Siciliano F, Sandimc HRZ, Plautd RL, Padilhad AF. Nucleation and Growth During Recrystallization. Materials Research. 2005;8(3):14.

[27] Gould H, Tobochnik J, Wolfgang C. An Introduction to Computer Simulation Methods: Applications to Physical Systems (3rd Edition): Addison-Wesley Longman Publishing Co., Inc. Boston, MA, USA 2005.

[28] Cao Q-z, On P-z. Fractal Interfaces in Heterogeneous Eden-like Growth. PHYSICAL REVIEW LETTERS. 1991;67(1):4.



[29] Garmestani H, Baniassadi M, Li DS, Fathi M, Ahzi S. Semi-inverse Monte Carlo reconstruction of two-phase heterogeneous material using two-point functions. International Journal of Theoretical and Applied Multiscale Mechanics. 2009;1:134-149.

[30] Kalos MH, Whitlock PA. Monte-Carlo Methods: WIiley-VCH Verlag GmbH & Co. KGaA; 2004.

[31] Blikstein P, Tschiptschin AP. Monte Carlo Simulation of Grain Growth. Materials Research. 1999;2(3):4.

[32] El-Khozondar R, El-Khozondar H, Gottstein G, Rollet A. Microstructural Simulation of Grain Growth in Two-phase Polycrystalline Materials. Egypt J Solids,. 2006;29(1):35-47.

[33] Ilachinski A. Cellular automata Texte imprimÂe a discrete universe. Singapore: World Scientific; 2001.

[34] Spall JC. Introduction to stochastic search and optimization: Wiley- Interscience; 2003.

[35] Asiaei S, Khatibi AA, Baniasadi M, Safdari M. Effects of Carbon Nanotubes Geometrical Distribution on Electrical Percolation of Nanocomposites: A Comprehensive Approach. Journal of Reinforced Plastics and Composites. 2009:0731684408100701.

[36] Liu W, Sun X, Pederson LR, Marina OA, Khaleel MA. Effect of nickel-phosphorus interactions on structural integrity of anode-supported solid oxide fuel cells. Journal of Power Sources. 2010;195(21):7140-7145.

[37] Zhu WZ, Deevi SC. A Review of the Status of Anode Materials for Solid Oxide Fuel Cells. Materials and Science Engineering. 2003;A362:228-239.

[38] Fuel Cell Handbook. Morgantown, West Virginia: U.S. Department of Energy; 2004.

[39] Holtappels P, Vogt U, Graule. T. Ceramic Materials for Advanced Solid Oxide Fuel Cells. Advanced Engineering Materials. 2005;75(5):292-302.

[40] Iwata T. Characterization of Ni-YSZ Anode Degradation for Substrate-Type Solid Oxide Fuel Cells. J Electrochemical Society. 1996;143(5).

Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites Using Strong Contrast


Chapter V





Homogenization of Mechanical and

Thermal Behavior of Nanocomposites

Using Strong Contrast:

(Application to Nanoclay Based Polymer

Nanocomposites)





V.1. Introduction

The improvement of mechanical, thermal, gas barrier and fire resistance properties of organic

polymer materials is a major concern, particularly in the domains of transportation, building

construction, and electrical engineering. Polymer nanocomposites often exhibit physical and

chemical properties dramatically different from the corresponding pure polymers. Numerous and

recent studies have shown the interest of the use of clay nanoparticles (above all modified

montmorillonites) as nanofillers for several polymers [1, 2]. The usual volume fraction of clay

that has been used is in the range of 5 to 10 wt% organo-modified montmorillonite. The reasons

are the high aspect ratio (more than 1000), the high surface area (more than 750 m2/g) and the

high modulus of these lamellar nanoparticles (170 GPa). Depending upon the processing

conditions and characteristics of both the polymer matrix and organoclay, the in-situ dispersion

of organoclay inside the host polymer by melt blending can be more or less achieved, leading to

intercalated or exfoliated nanocomposites.

Recently most of the researches about layered silicates are focused especially on

montmorillonites (MMT), as the reinforcing phase due to availability and versatility of these

types of nano fillers [3]. Depending on the process conditions and on the polymer/nanofiller

affinity, The layered silicates dispersed into the polymer matrix can be observed in different

states of intercalation and/or exfoliation [4]. The best performances are commonly achieved with

the exfoliated structures [5]. Besides that, the insertion of clay materials into a polymer matrix

led to a significant decrease of the diffusion coefficient of various gases into the composites [3,

6].

Over the last few years, development of computer engineering, and of numerical methods for

molecular dynamics simulations, allowed a detailed study of the structure of nano-objects and

their thermomechanical properties,which is in general difficult or even impossible to study by

other methods. Among different modeling techniques, Molecular Dynamics (MD) are now

becoming standard means for the simulation of matter at the molecular scale [7]. Now-a-days

MD is considered as the most realistic simulation technique as well as an alternative to



experiment in atomic scale science [8]. Recent studies verified that material properties acquired

from MD simulations could be efficiently utilized in order to perform homogenization for

effective thermal and mechanical properties of the nanocomposite material [9]. In MD

simulations the structure is usually considered to be free of any impurities and defects, which

leads to an upper bound of the experimental results for the modulus and the thermal conductivity

[8, 10]. Particularly, molecular dynamic (MD) methods have been actively used to study

montmorillonite lamellar structures and intercalate in the interlamellars space[11-13].

Several homogeneization methods have been used in the literature to predict effective properties

of nanocomposite properties. For instance, the effective mechanical properties of such

nanocomposites have been investigated using inclusion-based theories which call for the Eshelby

solution for ellipsoidal inclusions in a homogeneous medium [14-22]. For example, the

generalized Mori-Tanka model has been exploited to predict the effective elastic modulus of the

starch/clay nanobiocomposites [23]. Similarly, the effective thermal conductivity of composites

with ellipsoidal inclusions have been widely considered using various micromechanical models

in the literature[24, 25].

In this work, we used a strong contrast [26-29] multiscale statistical method to predict the

overall modulus and thermal conductivity of montmorillonite polymer based nanocomposites. To

take into account the geometrical information on inclusions and their distribution in the matrix, a

statistical continuum approach has been developed based on statistical correlation functions [29].

In this study two-point and three-point correlation functions have been taken into account to

describe the microstructure. Using Monte Carlo simulation, two-point correlation functions of

the realized nanostructures have been extracted and in a following step three point correlation

functions have been estimated based on the previously determined two-point correlation

functions [30]. From the two-point and three-point correlation functions, the effective thermal

conductivity of the nanocomposites was calculated using a strong contrast expansion. To validate

our proposed statistical approach, we conducted experimental tests to measure both the elastic

and thermal properties for polyamide/MMT nanocomposites with 1, 3 and 5 wt% of

nanoparticles. We then compared our simulate results to the experimental one.



V.2. Computer generated model

In this research, Three-dimensional isotropic virtual samples with randomly oriented disks as

mono layer nanoclays are generated and used to calculate the statistical two-point correlation

functions of the realized model. These statistical correlation functions have been utilized as

nanostructure descriptor to approximate the strong contrast solution for thermal and mechanical

properties of nanocomposites. In this solution two-point and three-point correlation functions

have been exploited as input function to solve the strong contrast equations for the effective

thermal and elastic properties. In this study, three point correlation functions have been

approximated using two-point correlation functions which are calculated using computer

generated sample for nanocomposite nanostructures(see Fig. 1).

Fig. 1. Two-point correlation function



An exfoliated nanoclay is created as the set of two parallel random surfaces with a specified

distance equal the thickness of the nanoclay particles. The disk geometry is defined by a normal

vector to the nanoparticles surface. The center of the disk has been allocated randomly inside a

cubic volume. Then the normal vector is specified by random homogeneous functions given

below which surveys uniformly on the surface of a sphere [31].

� �1

2cos 2 1

vu

5 %� �

� � � ��

(1)

Where 3 30,25 %� and 3 40,� %� are spherical coordinates as shown in Fig..2 and where ,u v are

random variables belonging to 4 30,1 �

In this simulation, the soft-core algorithm is used to generate nanoclay particles which allows

for penetration [32]. Thus a new plate of the nanoclay is randomly placed somewhere in the unit

cell regardless of the ones already present. In other words, regions of space may be occupied by

more than one nanoclay. However, the reason for using the soft core approach is its simplicity

and its reduced computational time. Besides, by using this algorithm, one can simulate nearly

every volume fraction of nanoclay in the composites.



Fig. 2. spherical coordinate of normal vector

V.3. Thermal conductivity

To evaluate the effective conductivity at macroscopically anisotropic two-phase composites, the

strong-contrast expansion approach has been further improved by establishing an integral

equation for the cavity intensity field [29]. The nth order tensorial expansions are expressed in

terms of integrals over products of certain tensorial fields and a determinant of N-point statistical

correlation functions which make the integrals convergent for the infinite volume limit. Owing to

the procedure of solving the integral equations which produces absolutely convergent integrals,

no additional renormalization analysis is needed. Another salient aspect of this expansion is that

when truncated, at finite order, they give reasonably accurate estimates at rather all

concentrations even though the contrast between the conductivities is high.

Assuming isotropic properties of the PA matrix and nanoclay particles, the effective conductivity

tensor ''e of the nanocomposite is determined using the strong-contrast formulation of the

statistical continuum theory [27]:

9 : 9 :� �

� � � � � ��

� �

� � � � � � � � � ��

-1 2 1 1

1 1 1

2 2 3 2 2

1 1 1 1 1

1,2 1 21. 2 3 1,2 d21 1 2

(1,2,3) (1,2) (2,3)1,2 2,3 d2d3 ....

1 2 1 2 3

S S SS S S

S S Sd

S S S S S

*

*

' ' ' ' '

'

+ ,�� - .

- ./ 0+ ,

� � & �- .- ./ 0

�

��

I I I M

M M

e e

S S S

R

R R RS S S

SR

S S SR R

R SR S S S S S

(2)

Here, we have adopted the shorthand notation consisting in representing x1, x2, x3 by 1 and 2, 3

respectively. In Eq. (2), I is the second-order identity tensor, 'R is the reference conductivity,

(1, 2)M R is a second-order tensor defined below, and *SR is the polarizability:

� �S R

SRS Rd 1

�* �

� �' '

' ' (3)



The subscript R stands for the reference phase, which is chosen here to be the nanoclay phase,

and the subscript/superscript S stands for the PA matrix.

The second order tensor (1, 2)M R is defined by:

� �3

1 2

1(1,2) 3x x'

��7 �

tt IM R

R (4)

where 7 is the total solid angle contained in a 3-dimensional sphere and 1 2

1 2

( )x xx x�

��

t . � �s1S 1 ,

� �s2S 1,2 and � �s

3S 1,2,3 are the probability functions that contain the microstructure information.

The one-point probability function, 1

S (1)S , is the volume fraction of the nanoparticles. The two-

point probability function,2

S (1, 2)S , is calculated from the Monte Carlo simulation. The three-point

probability function, � �s3S 1,2,3 , is calculated from the following analytical approximation [30]:

� � � � � � � ��

1 2 1 3 23 2 2

11 2 1 3 1 2 1 3

2,31,2,3 1,3 1,2

1

pp p p

p

x x x x SS S S

Sx x x x x x x x

+ ,- .�- .� �/ 0

x x x xx x x x�1 2 1 3 1�p1 2 1 3� �

x x x x1 2 12 11 31 3� �1 2 1 3� �1 3� �1 2 1 3� �x x x x

� � �2 2� � 1�2 2� �1,31,3� �22 � �x xx x

1�1,31,3� �x x x xx x x x x xx x x x x xx x x x

+ (5)

Fig. 4 defines the variables used in this approximation in local coordinates.

V.4. Mechanical model

Exact perturbation series (weak-contrast expansions) are valid for two phase media with small

variation of effective conductivity and elastic moduli of composites [29]. In general, strong-

contrast expansions take a larger radius of convergence than weak-contrast expansion for the

same reference properties. The statistical theory of strong contrast has been used to determine the

effective stiffness tensor of macroscopically isotropic two-phase composites. In this approach, an

integral equation for the strain field leads to an exact series expansions for the effective stiffness

tensor of two-phase composite media. In this method, N-point correlation functions show up in

the final equations that characterize the microstructure. The general term of the expansion for a

reference phase q is written as follows [28]:



12 ( ) ( ) ( )

2:q q p

p e p nn

L L I B� ��

�

+ , � �/ 0 �

(6)

Where P� is the volume fraction of phase P and I is the fourth-order identity tensor,

12

Iijkl ik il il jk; ; ; ;+ ,- ./ 0

� �

(7)

In Eq. (6) the tensor coefficients (Bn ) are the following integrals over products of the U Tensors

and the Sn represent the N-point correlation functions for phase P:

( ) ( ) ( ) 22 22 (1,2) (1,2)p q p

pB d U S<

�+ ,� �/ 0�

(8)

2

( ) ( ) ( )

( )

1(1 ) 2... (1,2) : (2,3)

( )( 1, ) (1,..., ), 3,

n

p n q qn

p

pn

B d dnU U

U q n n n n

�

��

� � � � �

� = >

� �

(9)

1 2 , rr x x tr

� � �

(10)

In Eq. (6) the effective tensor ( )qeL is given by :

9 :9 : 1( ) ( ) ( ) ( ):q q q qe e eL C C I A C C

�+ ,� � � �/ 0

(11)

Where Ce is the effective stiffness tensor, Cq is the stiffness tensor of the reference phase and A(q)

is a forth order constant tensor [28].

Here ( )(1,..., )pn n= is a position-dependent determinant that is calculated using N-point correlation

function for a given phase p by:



( )

( ) ( )(1,2) (2) 0 02 1( ) ( )(1,2,3) (2,3) 0 03 2

(1,..., )

( ) ( ) ( ) ( )(1,2,..., 1) (2,3,..., 1) ( 2, 1) ( 1)1 2 2 1( ) ( ) ( ) ( )(1,2,..., ) (2,3,..., ) ( 2, 1) ( 1,1 3 2

pn

p pS S

P pS S

n

p p p pS n S n S n n S nn np p p pS n S n S n n S nn n

= �

� � � � ��

� � ��

2((((

3(((( )n

(12)

The tensor U is calculated based on the position-dependent fourth-order H(r) and the related

tensor for phase q, L (q):

:

( ) ( ) ( )

( )

( )

( ) ( )

( 2)2( 1) ( )

( 2 )

( 2)( )

( 2 )

q q qijkl ijmn mnkl

q ij qq q pq pq mmkl

q q

q qpq ijkl

q q

U r L H r

d GdK d G K H r

d K G d

d GH r

d K G

;?

?

�

+ ,�@+ ,� � � �- .�/ 0 �- .@/ 0��

��

(13)

Where d is space dimension and the tensor H( r ) is the symmetrized double gradient tensor [28]

which is given below:

( ) 1 1( )2 2( 1)

( )2

( 2)

qijkl d

q q

qq ij kl ik il il jk q ij k l kl i j

ik j l il j k ik i l ij i k q i j k l

H rrdK d G

d dd d t t t t

t t t t t t t t d d t t t t

(( ; ; ; ; ; ; ( ; ;

; ; ; ; (

�+ ,7 � �/ 0

�+ + , + ,� � � � �/ 0 / 0/

+ , ,� � � � �/ 0 0

(14)

The constant tensor for phase q is expressed as:

( ) ( 2)2( 1)

( 2 )qq

q q pq h pq sq q

d GL dK d G k

d K G?

+ ,�+ ,� � � A � A- ./ 0 �- ./ 0

(15)



Where d is space dimension and where pqk and pq? are introduced as bulk and shear modulus

polarizabilities, qK and qG are respectively the bulk modulus and the shear modulus of the

reference phase and Λh and Λs are the fourth-order hydrostatic and shear projection tensors

[29]. pqk and pq? are given by the following relations:

43

p qpq

p q

K Kk

K G

��

� (16)

3 / 2 4 /2

pq

p q

q q qp

q q

G GG K G d

GK G

?�

�+ ,�/ 0�

�

(17)

For macroscopically isotropic media, Eq. (6) can be simplified as [29]:

2 ( )

2

pq pq pp h s p n

neq eq

kI B

k?

� �?

�

�

+ ,A � A � �- .

- ./ 0�

(18)

In this work, the calculations have been performed for the first and second terms of B and other

terms have been neglected because of the complexity of the calculations:

2 ( ) ( )2 3

pq pq p pp h s p

eq eq

kI B B

k?

� �?

+ ,A � A � � �- .

- ./ 0 (19)

( ) ( ) ( ) 22 22 (1,2) (1,2)p q p

pB d U S<

�+ ,� �/ 0�

(20)

( ) ( ) ( ) ( )3 3

1 2... (1,2) : (2,3) (1,...,3)p q q p

p

B d dnU U��

� =� � �

� �

(21)

( ) ( )2 1( )

3 ( ) ( )3 2

(1, 2) (2)(1,...,3)

(1, 2,3) (2,3)

p pp

P p

S SS S

= �

(22)



We recall that tor three-point correlation functions, we are using the analytical approximation in

Eq. (5)(see Fig. 3).

Fig. 3. Representation of vectors in spherical coordinate

V.5. Experimental part

V.5.1. Materials

The polyamide (PA) resin (viscosity 35p, at 240°C) was supplied by Scientific Polymer. The PA

density was 0.99 g.cm-3 (at 23°C). The filler was a commercial organo-modified

montmorillonite, Cloisite 30B (OMMT) and was purchased from Southern Clay Co. The

modifier was methyl bis-2-hydroxyethyl tallow ammonium and its concentration was 90 meq per

100 g of clay. This treatment leads to a good dispersion in the polar polymer matrix and allows

preparing intercalated or exfoliated nanocomposites. The density of organo-modified

montmorillonite was 1.98 g.cm-3 (at 23°C).



V.5.2. Nanocomposites preparation

PA and OMMT were first dried at 80°C during 4 hours. PA-OMMT nanocomposites were then

prepared by melt-mixing, the molten PA pellets and the OMMT at different weight fractions of

clay, using a co-rotating twin-screw extruder (DSM Xplore), at 180°C for 5 min, with a rotation

speed of 150 rpm. The investigated weight fractions of OMMT in PA nanocomposites were 0, 1,

3 and 5 wt%.

V.5.3. Transmission electron microscopy

Transmission electron microscopy (TEM) analyses of PA-OMMT nanocomposites were carried

out using a LEO 922 apparatus at 200 kV. The ultrathin films (70 nm thick) were prepared with a

LEICA EM FC6 cryo-ultramicrotome at 25 °C.

V.5.4. Mechanical properties

The evaluation of the mechanical properties of PA and its nanocomposites was carried out using

a Dynamic Mechanical Analyzer (DMA 242C-Netzsch). Storage (E’) and loss (E’’) modulus

were measured as a function of temperature (-175 °C to +70 °C) with a dynamic temperature

ramp sweep at 2 °K.min-1. Measurements were performed using the single cantilever bending

mode at a frequency of 1 Hz. The storage modulus is the elastic response to deformation,

whereas the loss modulus is the dissipative response corresponding to the energy lost during the

cyclic deformation of the material. All DMA samples were pressed and cut in the form of 9.70-

10.40 mm-long, 1.15-1.47 mm-thick and 4.95-5.9 mm-wide specimens. To check the

reproducibility of the experimental data and to ensure their consistency, 3 specimens were tested

for each formulation.

V.5.5. Laser flash

Thermal diffusivity and thermal conductivity of studied materials were measured by the laser

flash method. This technique entails heating the front side of a small, usually disk-shaped plane-

parallel sample by a short (≤ 1ms) laser pulse. The temperature rise on the rear surface is

measured versus time using an infrared detector. All samples were coated on both faces with a

very thin layer of colloidal graphite. The thermal diffusivity a(T) values can then be converted to



thermal conductivity λ(T) by using the specific heat Cp(T) and bulk density ρ(T) of studied

material according to:

λ(T) = ρ(T) • Cp(T) • a(T) (23)

The samples in the shape of discs, 12 mm in diameter and 1 mm in thickness were prepared by

compression molding. The measurements were carried out from room temperature to 100 °C

under an argon flow. Three samples were tested for each system and the uncertainty for the

determination of thermal diffusivity was evaluated to ±3%.

V.6. Results and discussion

V.6.1. Thermal conductivity

Thermal conductivity of neat PA decreases from its room temperature value of 0.127 W.m-1.K-1

with increasing temperature (see Fig. 4). In our calculation, the thermal conductivity of nanoclay

particles has been estimated using a semi-inverse strong contrast approach [33] for the

compressed powder sample at about 0.55(W.m-1.K-1). We have neglected the effect of

temperature on this property. We analyzed the thermal conductivity for PA/nanoclay with 1, 3

and 5 wt%. The corresponding volume fractions are obtained from the two-point correlation

functions (see Fig. 1) as 0.55%, 1.6% and 2.5%, respectively.

Our results show that the addition of nanoclay leads to an increase in thermal conductivity of PA.

Moreover, the higher the amount of nanoclay, the higher the thermal conductivity becomes. As

shown in Fig. 4, the thermal conductivity of the PA-OMMT composites predicted using the

strong contrast approach fits quite well with the experimental results. The simulated curves are

not smooth because we used non-smooth experimental data of conductivity for pure polymer as a

function of temperature (see Fig. 4).



Fig. 4. Comparison between experimental and simulation thermal conductivity of PA and its nanocomposites with OMMT

V.6.2. Thermo-mechanical properties

Since montmorillonite can be used for improving thermal stability, it is important that it does not

dramatically deteriorate the mechanical properties (stiffness). To predict the elastic modulus of

the composite, values of the elastic modulus of nanoclay found in the literature [11, 18] were

used, Enanoclay= 176 Gpa [11, 18]. The elastic modulus of the PA matrix is shown in Fig. 5 as

function of temperature.

Fig. 5 shows the effect of the nanoclay on the mechanical properties (storage modulus E’)

obtained by DMA measurements as well as those obtained using statistical continuum theory. At

room temperature, PA exhibits a significant storage modulus (E’25°C = 550 MPa. The addition of



1-5 wt.% nanoparticles did not have any impact on E’25°C. Below 0°C, the values of E’ of the

composites containing 1 or 3 wt. % nanoclay are similar. However E’ increases by ~20% when

5wt.% clay is added to PA.

It is found that E’ of the composites predicted by our simulations fit well with the experimental

data for 1 wt. % . However, simulated values of E’, for the composites containing more than 1

wt. % nanoclay, are unfortunately higher than the experimental ones for the same composition.

In the next, we will attempt to explain these discrepancies.

Fig. 5. Experimental and simulated elastic modulus of two phases composite as a function of temperature T for neat PA and its composites with OMMT (1, 3 and 5 wt. %).

TEM analyses of the PA-OMMT nanocomposites were performed in order to investigate the

distribution and the dispersion of OMMT into the PA matrix. Fig. 6 shows two TEM images for

two different nanofiller contents, 3 and 5wt%. The images show decreasing exfoliation state of

nanofillers with increasing volume fraction of the fillers.



The discrepancies between the experimental and the theoretical results or the elastic modulus can

be explained by the dispersion of the nanoclays in PA matrix. Indeed, the statistical continuum

theory calculations assume that nanoclays in PA are in exfoliated state (Fig. 7). On the other

hand, the experimental results showed that the nanoclays are in exfoliated state in the composites

PA - 1wt. % OMMT and in both exfoliated-intercalated state in the composites PA – 3 wt. %

OMMT and PA – 5 wt. % OMMT.

Fig. 6.TEM micrographs of PA-3%OMMT and PA-5%OMMT nanocomposite

Fig. 7. Polymer/clay nanocomposite morphologies



V.7. Conclusion

In the present study, the effects of nanoclay additives on the effective mechanical and thermal

properties of nanoclay based polymer composites have been investigated using both

experimental and simulation analysis. In the present study, statistical continuum theory is used to

predict the effective thermal conductivity and elastic modules of nanoclay based polymer

composites.

In this research, Monte Carlo simulations have been performed to find two-point probability

functions of each phase. Two-point and three-point probability functions, as statistical

descriptors of inclusions(fillers) distribution have been used to solve strong contrast

homogenization for the effective thermal and mechanical properties of nanoclay based polymer

composites. The predicted thermal conductivity results have shown satisfactory agreement with

experimental data. However, the predicted effective elastic modulus results for high

concentration of nanoclay overestimate the experimental data. This discrepancy is probably due

to increasing intercalated structure of nanoclay for high nanofiller concentrations.



V. References

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[2] Matadi R, Gueguen,O., Ahzi, S.,Gracio, J. , Muller, R. , Ruch, D. . Investigation of the Stiffness and Yield Behaviour of Melt-Intercalated Poly(methyl methacrylate)/Organoclay Nanocomposites: Characterisation and Modelling. Journal of Nanoscience and Nanotechnology 2010;10:2956.

[3] Sinha Ray S, Okamoto M. Polymer/layered silicate nanocomposites: a review from preparation to processing. Progress in Polymer Science 2003;28:1539.

[4] Vaia RA, Giannelis EP. Lattice Model of Polymer Melt Intercalation in Organically-Modified Layered Silicates. Macromolecules 1997;30:7990.

[5] Pollet E, Delcourt C, Alexandre M, Dubois P. Organic-Inorganic Nanohybrids Obtained by Sequential Copolymerization of ε-Caprolactone and L,L-Lactide from Activated Clay Surface. Macromolecular Chemistry and Physics 2004;205:2235.

[6] Yoshitsugu Kojima, Arimitsu Usuki, Masaya Kawasumi, Akane Okada, Yoshiaki Fukushima, Toshio Kurauchi, Kamigaito O. Mechanical properties of nylon 6-clay hybrid. J.Mater. Res 1993; 8 5.

[7] Haile JM. Molecular dynamics simulation : elementary methods. New York: Wiley, 1992.

[8] Mortazavi B, Khatibi, A. A., Politis, C. Molecular Dynamics Investigation of Loading Rate Effects on Mechanical-Failure Behaviour of FCC Metals. Journal of Computational and Theoretical Nanoscience 2009;6:644.

[9] Afaghi Khatibi A, Mortazavi B. A Study on the Nanoindentation Behaviour of Single Crystal Silicon Using Hybrid MD-FE Method. Advanced Materials Research 2008;32:3.

[10] Komanduri R, Chandrasekaran N, Raff LM. Molecular dynamics (MD) simulation of uniaxial tension of some single-crystal cubic metals at nanolevel. International Journal of Mechanical Sciences 2001;43:2237.

[11] Chen B, Evans JRG. Elastic moduli of clay platelets. Scripta Materialia 2006;54:1581.

[12] Hackett E, Manias E, Giannelis EP. Computer Simulation Studies of PEO/Layer Silicate Nanocomposites. Chemistry of Materials 2000;12:2161.

[13] Suter JL, Boek ES, Sprik M. Adsorption of a Sodium Ion on a Smectite Clay from Constrained Ab Initio Molecular Dynamics Simulations. The Journal of Physical Chemistry C 2008;112:18832.

[14] Benveniste Y. A new approach to the application of Mori-Tanaka's theory in composite materials. Mechanics of Materials 1987;6:147.



[15] Eshelby JD. The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 1957;241:376.

[16] Hori M, Nemat-Nasser S. Double-inclusion model and overall moduli of multi-phase composites. Mechanics of Materials 1993;14:189.

[17] Hu GK, Weng GJ. The connections between the double-inclusion model and the Ponte Castaneda-Willis, Mori-Tanaka, and Kuster-Toksoz models. Mechanics of Materials 2000;32:495.

[18] Luo J-J, Daniel IM. Characterization and modeling of mechanical behavior of polymer/clay nanocomposites. Composites Science and Technology 2003;63:1607.

[19] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica 1973;21:571.

[20] Mura T. Micromechanics of defects in solids. Lancaster: Nijhoff, 1987.

[21] Nemat-Nasser S, Hori M. Micromechanics : overall properties of heterogeneous materials. Amsterdam ; New York: Elsevier, 1999.

[22] Sheng N, Boyce MC, Parks DM, Rutledge GC, Abes JI, Cohen RE. Multiscale micromechanical modeling of polymer/clay nanocomposites and the effective clay particle. Polymer 2004;45:487.

[23] Chivrac F, Gueguen O, Pollet E, Ahzi S, Makradi A, Averous L. Micromechanical modeling and characterization of the effective properties in starch-based nano-biocomposites. Acta Biomaterialia 2008;4:1707.

[24] Mercier S, Molinari A, El Mouden M. Thermal conductivity of composite material with coated inclusions: Applications to tetragonal array of spheroids. Journal of Applied Physics 2000;87:3511.

[25] Milhans J, Ahzi S, Garmestani H, Khaleel MA, Sun X, Koeppel BJ. Modeling of the effective elastic and thermal properties of glass-ceramic solid oxide fuel cell seal materials. Materials & Design 2009;30:1667.

[26] Pham DC, Torquato S. Strong-contrast expansions and approximations for the effective conductivity of isotropic multiphase composites. Journal of Applied Physics 2003;94:6591.

[27] Sen AK, Torquato S. Effective conductivity of anisotropic two-phase composite media. Physical Review B 1989;39:4504.

[28] Torquato S. Effective stiffness tensor of composite media--I. Exact series expansions. Journal of the Mechanics and Physics of Solids 1997;45:1421.

[29] Torquato S. Random heterogeneous materials : microstructure and macroscopic properties. New York ; London: Springer, 2002.



[30] Mikdam A, Makradi A, Ahzi S, Garmestani H, Li DS, Remond Y. A new approximation for the three-point probability function. International Journal of Solids and Structures 2009;46:3782.

[31] Weisstein EW. Sphere Point Picking., vol. 2010: From MathWorld--A Wolfram Web Resource.

[32] Ghazavizadeh A, Baniassadi M, Safdari M, Ataei AA, Ahzi S, Grácio J, Patlazhan S, Ruch D. Evaluating the effect of mechanical loading on the electrical percolation threshold of carbon nanotube reinforced polymers: A 3D Monte-Carlo study. Journal of Computational and Theoretical Nanoscience.

[33] Baniassadi M, Addiego F, Laachachi A, Ahzi S, Garmestani H, Hassouna F, Makradi A, Toniazzo V, Ruch D. Using SAXS approach to estimate thermal conductivity of polystyrene/zirconia nanocomposite by exploiting strong contrast technique. Acta Materialia 2011;59:2742.

Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous Materials Using Statistical Correlation Functions and FEM




Chapter VI





Three-dimensional Reconstruction and

Homogenization of Heterogeneous

Materials Using Statistical Correlation

Functions and FEM





VI.1. Introduction

Heterogeneous media are abundantly found in a wide range of synthetic materials such as

composites or natural materials such as living tissues. As a microstructural descriptor of

heterogeneous materials, statistical correlation functions are among the most efficient ones.

Mechanical, thermal, electrical and in general physical properties characterization of

heterogeneous materials can be realized directly by means of such descriptors which are further

known under the general designation of N-point correlation functions [1-5].

TPCFs are the basic statistical functions required to evaluate the effective/homogenized

properties of micro/nanostructures. Homogenization approaches developed based on statistical

continuum mechanics such as weak-contrast or strong-contrast approach are able to evaluate the

effective properties through n-point correlation functions. Multi-phase heterogeneous materials

with slight variation of properties are closely simulated by applying weak-contrast expansions.

For the case of large differences between the properties of phases, strong-contrast technique is

the suitable one for physical characterization purposes [1, 5, 6].

Micro/nanostructural reconstruction is another equally valuable application area of TPCFs

besides physical properties characterization. Statistical continuum mechanics can be exploited to

provide a robust alternative to X-ray tomography for the reconstruction of heterogeneous materials.

Statistical reconstruction of heterogeneous media has become an intriguing inverse problem

which has found application in various fields of engineering and biology to obtain 3D realization

from the lower order correlation functions. Reconstruction using TPCFs is much simpler and less

expensive than the other rival methods such as X-ray tomography or stitching technique [5, 7-

10].

In this chapter, we extend our previously developed reconstruction methodology (In chapter 4) to

3D microstructure reconstruction based on two-point correlation functions and two-point cluster

functions. Using a hybrid stochastic methodology for simulating the virtual microstructure,

growth of the phases represented by different cells is controlled by optimizing parameters such

as rotation, shrinkage, translation, distribution and growth rates of the cells. We used the finite

element method (FEM) to predict the effective thermo-mechanical properties such as the elastic

modulus and thermal conductivity of the reconstructed microstructure. We also used the strong



contrast statistical method, based on two-point and three-point correlation functions. The two-

point correlation functions are calculated from the computer generated microstructure. For the

three-point correlation functions, we used two approximations, the existing approximation of

Mikdam et al. [11] and our proposed new approximation detailed in Chapter 3. Comparison of

the results from both approaches and FEM simulations show that our new approximation, for the

tree-point correlation functions, gives a better agreement with the FEM results.

VI.2. Reconstruction of heterogeneous materials using two-point cluster function (TPCCF)

The previously developed algorithm based on Monte Carlo methodology for the reconstruction

of microstructures using two-point correlation functions is now extended by the use of an

additional microstructure descriptor, the two-point cluster functions. In the next, we briefly

summarize the reconstruction methodology. The realization process includes three steps: 1)

generation, 2) distribution, and 3) growth of cells. Here, cells (or alternately grains or particles)

refer to initial geometries assigned to each phase before the growth step. During the initial

microstructure generation, basic cells are created from the random nucleation points and then the

growth occurs as in crystalline grain growth in real materials. After distribution of nucleation

points and assignment of basic cell geometries, the growth of cells starts according to the cellular

automaton approach. The three steps of realization algorithm are repeated continuously to satisfy

the optimization parameters until an adequately realistic microstructure is developed as

compared statistically to the true microstructure. It is worth noting that in various steps of

algorithm execution; several controlling parameters are developed that facilitate minimization of

the objective function (OF) which is an index of successful realization. This objective function is

defined based on the three independent two-point correlation functions ( 2ijP ) and two-point

cluster functions (2

c iiP � ) as follows:

� � � � � �2 2 2 2

2 32 22

2 21 1

( ) ( ) ( ) ( ) ( ) ( )ij ij ii ii c ii c iireal sim real sim real sim

i iOF P P P P P P� �

� �

� � � � � ��

(1)

where the subscripts real and sim indicate, respectively, the values from the real and

reconstructed microstructures. The procedure of reconstruction and optimization is repeated until



the objective function takes a value that is of the same order as the Monte Carlo (M-C) repeat

error.

The material heterogeneity is represented by statistical two-point correlation functions and two-

point cluster functions. Hypothetical statistical functions are optimized and compared to the

intial statistical functions of the sample microstructure. Stochastic optimization methodologies

incorporate probabilistic (random) elements, either in the input data (the object function, the

constraints, etc.), or in the algorithm itself (through random parameters, etc.) or both . By

applying different optimization parameters to the simulations, a minimum error is achieved

through minimization of the objective function (Eq. 1) that is constructed from the comparison of

the two-point correlation function and two-point cluster functions of the sample and simulated

(realization) microstructures. A direct simple search optimization technique was used for finding

the minimum objective function. Fig. 1 depicts a schematic of the extended reconstruction

algorithm. We recall that two-point cluster function is the probability of finding both beginning

and ending points of a random vector in the same phase and same cluster.



Fig. 1. Basic steps in the realization algorithm (OF = objective function; MC=Monte Carlo)



VI.3.Statistical characterization of microstructures

Exact perturbation expansions were used to predict the effective elastic modulus and thermal

conductivity of two phase heterogeneous materials [5] . In this chapter , we have compared the

effective properties of heterogeneous materials for two different approximation of three-point

correlation functions. The first approximation (see Eq. 2 below) has been developed by Mikdam

[11] and the second approximation has been proposed in the third chapter of this dissertation (see

Eq. 3 below).

� � � � � � � ��

1 2 1 3 2 2 33 1 2 3 2 1 3 2 1 2

1 11 2 1 3 1 2 1 3

,, , ,

pp p p

p

x x x x S x xS x x x S x x S x x

S xx x x x x x x x

+ ,- .�- .� �/ 0

x x x xx x x x�1 2 1 3� � p1 2 1 3� �

x x x x1 2 12 1S x x S� � x�p pp � �1 2 1 3� �x x x x

� � �2 1 3 2� � �2 1 3 2� � �,,� � �2 1 3 22 1 3 2� � �x xx x

S x x S x� � �,,� � �x x x xx x x x x xx x x x x xx x x x

, +

(2)

� � � � � � � � � �

� � � �

2 3 1 23 1 2 3 2 1 3 2 1 2 2 2 3 2 1 2

1 2 1 3 2 3 1 2 1 3 2 3

1 32 1 3 2 2 3

1 2 1 3 2 3

, , , , ,

, ,

p p p p p

p p

x x x xS x x x S x x S x x S x x S x x

x x x x x x x x x x x x

x xS x x S x x

x x x x x x

� � � ��

� ��

� � �x x x x�2 �2 �

p �S ��2 �p �2 3 1 2� � � �2 3 1 2� � � �

x x x x2 3 13 1S x x S x x� � � �p ppp p� � � �2 3 1 2� � � ��x x x x

� � � � �2 1 3 2 1 22 1 3 2 1 2� � � �2 1 3 2 1 23 2 12 1 3 2 1 2� � � �S x x S x x� � � �, ,, ,, ,� � � �2 1 3 2 1 23 2 12 1 3 2 12 1 3 2 1 2� � � �

x x x x x x x x x x x xS x x S x xS x x S x xS x x S x xS x x S x x� � � � � �x x x x x x x xx x x x x x x x x xx x x x x x x x x xx x x x x x x x 1

x x�2 �

p �S �2 �p �1 3x x1

��x x

x x x x x x

x xx x x xx x x xx x

, +

(3)

Here, � �2 1 2,pS x x and � �3 1 2 3,pS x x x, are the two and three point correlation functions,

respectively. The effective conductivity and elastic modulus of the composite material can be

determined using the strong-contrast formulation of the statistical continuum theory considering

the isotropic properties of the phases.

VI.4. FEM characterization of multiphase heterogeneous materials

The computer generated sample and the 3D reconstructed microstructure based on two-point

correlation functions and two-point cluster functions are used for our FEM characterization.

Finite Element simulations were carried out using ABAQUS/Standard (Version 6.10). Due to the

extensive computationally time, only ten layers of the real specimens were included in the

modeling. For the purpose of thermal modeling, the specimen was meshed using eight-node

linear heat transfer brick (DC3D8-type) elements. For the mechanical modeling, the eight-node



linear brick, 3D stress with reduced integration (C3D8R-type) elements, were used. Each mesh

element was assigned to the corresponding phase.

Fig. 2. Finite element illustration and boundary condition of computer generated and reconstructed microstructure (Left: computer generated and right: reconstructed microstructures),

for thermal and mechanical loading.

In order to obtain the thermal conductivity of the specimen, constant heating surface heat flux

was applied to a plane in the X direction while cooling surface heat flux equal to the cooling

heating flux was applied in the opposite surface. In this way, steady state heat transfer criteria

will be fully observed and by averaging the temperatures in each surface, the created temperature

gradient as a function of distance in the specimen can be evaluated. The loading condition for

the thermal and mechanical models are illustrated in Fig. 2. Using a one-dimensional form of the

Fourier law, the thermal conductivity of the specimens was obtained. In order to obtain the

elastic modulus of the specimen, a small strain was applied to the loading surface in its normal

direction while the opposite surface was fixed only in its normal direction. By summation of the

reaction forces in the fixed surface, the applied stress was calculated. Then, the elastic modulus

of the specimen was obtained using Hook's law.



VI.5. Result and discussion

The computer-generated three-phase sample is assumed to contain 10% volume fraction of red

phase, 30% of the green phase and 60% of the black phase (see Table 1 and Fig. 3). This

computer-generated three-phase sample is reconstructed and imported to the ABAQUS package

for the FEM characterization. Fig. 3 shows 2D sections of three arbitrary layers taken from

through-the-depth of the corresponding reconstructed microstructure in the ABAQUS package.

These sections are arbitrarily chosen from the top, middle and bottom parts of the 3D

reconstructed domain.

The corresponding two-point correlation functions ( 112P or (P11) for red-red and 22

2P or (P22) for

black-black and 122P or (P12) for red-black ) are calculated for both computer-generated and

reconstructed microstructures shown in Fig. 4. As shown in this Fig., there is a good agreement

between the two-point correlation functions of the reconstructed and computer-generated

microstructures. The reconstruction process is performed based on the two-point correlation and

the two-point cluster functions which had been extracted from computer generated

microstructure. To check the validity of the reconstruction process, two-point cluster function for

non-percolated phase (red-red) is calculated and shown in Fig. 5. Good agreement between the

calculated two-point cluster functions for the two microstructures is obtained which strongly

confirms the validity of the reconstruction process.

Table 1.Phases properties

Phase Number Phase 1 Phase 2 Phase 3

Volume Percent 60% 10% 30%

Phase color black red blue



(a) Layer 25 (b) Layer 75 (c) Layer 125

Fig. 3. 2D arbitrary sections in the z-direction of the 3D reconstructed microstructure. a) Layer close to the bottom surface , b) Layer in the middle area, c) Layer close to the top surface



(a)

(b)

(c)

Fig. 4. a) Two-point correlation function (P11) for the red-phase, b) two-point correlation function (P22) for the black-phase c) two-point correlation function (P12) for the black-red

phases for the computer generated and reconstructed microstructures.

0

0.02

0.04

0.06

0.08

0.1

0.12

0 100 200 300 400 500

TPCF

r

TPCF (Microstructure 1)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 100 200 300 400 500

TPCF

r



0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.1

0 100 200 300 400 500

TPCF

r

TPCF (Microstructure 1) TPCF (Microstructure 2)



Fig. 5. Two-point cluster function 2

11cP � (TPCCF) for the red-phase,

The boundaries of the percolated regions of different phases are identified for one of the phases

(red-phase) in 2D section shown in (Fig. 6a) in which the phase percolation is less than the

percolation threshold. The percolated aggregates have been recognized using different colors in

Fig. 6b. In Fig. 6b and c, wide percolated clusters have been observed in the cut section images.

As the other two phases are intrinsically percolated and their corresponding two-point correlation

functions and two-point cluster functions are identical, there was no need to analyze the

percolation in these phases.

0

0.02

0.04

0.06

0.08

0.1

0.12

0 50 100 150 200 250 300 350 400 450 500

TPCC

F

r

TPCCF (Microstructure 1) TPCCF (Microstructure 2)



(a)

(b)

(c)

(d)

Fig. 6. a) An arbitrary 2D section of the 3D reconstructed microstructure (black=porosity); b ,c ,d) the corresponding percolation of voids (porosity) showing the percolation clusters by similar

colors other than white



In Fig. 7, the temperatures and Von Mises stress contours are represented for different cases for

the local properties. The normalized properties (both thermal and elastic) for the three phases are

taken (1,10,1) for case 1, (1,1,10) for case 2 ,(1,10,10) for case 3. Table 2 summarizes values of

the phase properties assumed for different target samples. As it can be clearly seen from Fig. 7,

there is fine agreement between these three cases with respect of the obtained fields of

temperature and stress values. As a result, the differences in the obtained thermal conductivity

and elastic modulus of the two microstructures (sample and reconstructed) are less than 1%

error. Elastic properties and thermal conductivity of these microstructures have been compared

using strong contrast (with existing and new approximation) and FEM analysis of 3-D

reconstructed microstructures (cases).

Fig. 7. Temperatures and Mises stress contours (Left : computer generated and right: reconstructed microstructure)

The elastic properties for the three target samples are shown in Fig. 8 (left). The FEM results

show very good agreement with strong contrast results which were obtained using statistical

correlation functions of the microstructure along with both existing and new approximation for



the three-point correlation functions. Similarly, thermal conductivity for the samples has been

calculated using FEM analysis of 3-D reconstructed microstructure and strong contrast

technique. A small gap has been observed between the results obtained from the FEM and strong

contrast method (Fig. 8 (right)). We note that the results with our new approximation are much

closer to the FEM simulations. This shows the validity of the proposed statistical

homogenization technique for three-phase heterogeneous materials and our approximation.

Table 2. Phase’s properties

Sample Sample 1

(Thermal conductivity and elastic modules)

Sample 2


Sample 3


Phase 1 1 1 1

Phase 2 10 1 10

Phase 3 10 10 1

Fig 8. Elastic module of reconstructed microstructure using FEM and Strong contrast technique (first approximation and second approximation)(left), Thermal conductivity of reconstructed microstructure

using FEM and strong contrast technique (first approximation and second approximation).(right)



VI.6. Conclusion

A Monte Carlo methodology is developed to reconstruct 3D microstructures of a three-phase

microstructure. Two-point correlation functions and two-point cluster functions are used as

microstructure descriptors in the reconstruction procedure. Using a hybrid stochastic

reconstruction technique, optimization of the function during different 3D realizations is

performed repeatedly. The main challenge in the 3D reconstruction is incorporating two-point

cluster function as complimentary statistical descriptor to perform reconstruction technique.

Comparison of the two-point correlation functions from different sections of the final 3D

reconstructed microstructure with the initial computer generated microstructure (sample

microstructure) shows good agreement. In addition, we have shown that the thermo-mechanical

properties of the generated and reconstructed microstructures are close by means of FEM

simulations. This supports the capability of our proposed methodology to reconstruct 3D

microstructure. We have also used the statistical homogenization technique to compute the

effective elastic and thermal properties. The comparison of the results with those of the FEM

simulations shows a fairly good agreement. This agreement between the two approaches suggest

that the statistical approach is a reliable approach, particularly when the new approximation for

the tree-point correlation functions is used.



VI. References


[2] Kröner E. Statistical Continuum Mechanics. Wien: Springer-Verlag, ; 1977.

[3] Pham DC, Torquato S. Strong-contrast expansions and approximations for the effective conductivity of isotropic multiphase composites. Journal of Applied Physics. 2003;94(10):6591-6602.



[6] Wang M, Pan N. Elastic property of multiphase composites with random microstructures. Journal of Computational Physics. 2009;228(16):5978-5988.

[7] Bochenek B, Pyrz R. Reconstruction of random microstructures--a stochastic optimization problem. Computational Materials Science. 2004;31(1-2):93-112.

[8] Liang ZR, Fernandes CP, Magnani FS, Philippi PC. A reconstruction technique for three-dimensional porous media using image analysis and Fourier transforms. Journal of Petroleum Science and Engineering. 1998;21(3-4):273-283.

[9] Manwart C, Hilfer R. Reconstruction of random media using Monte Carlo methods. Physical Review E. 1999;59(5):5596.

[10] Talukdar MS, Torsaeter O. Reconstruction of chalk pore networks from 2D backscatter electron micrographs using a simulated annealing technique. Journal of Petroleum Science and Engineering. 2002;33(4):265-282.


Conclusion and Future Work




Conclusion

and

Future Work





In this study , statistical two point correlation functions as microstructure descriptors of

heterogeneous media has been utilized to reconstruct the microstructures and homogenization

thermal conductivity and elastic modulus of nanocomposites. different techniques such as Monte

Carlo, SAXS data analysis and image processing of TEM/SEM images were exploited to

calculated two point correlation functions. in future work , we are looking to extract statistical

correlation functions using SAXS data analysis of anisotropic multiphase heterogeneous

materials.

Due to the complexity of calculating higher order correlation functions, in this research a new

novel formulation has been proposed to obtain a relation between the higher and lower order

correlation functions for heterogeneous materials using the conditional probability theory. This

approximation is valid for N-Point correlation functions of multiphase heterogeneous materials .

Comparison between the three-point correlation functions from the final 3D reconstructed

microstructure and the approximate correlation functions shows satisfactory agreement.

In future work, we would like to extend the weight functions of approximation to achieve

optimum solution for N-Point correlation functions. Statistical two point correlation functions

can be exploited to realize two or three dimensional microstructure of heterogamous materials.

In this research work, a new Monte Carlo methodology is developed to reconstruct 3D

microstructures of a N-phase microstructure. Two-point statistical functions are used as

microstructure descriptors in the reconstruction procedure. Using a hybrid stochastic

reconstruction technique, optimization of the two-point correlation functions during different 3D

realizations is performed repeatedly.

The main challenge in the 3D reconstruction is the possibility to incorporate other statistical

descriptors similar two-point cluster function and lineal path functions as complimentary

statistical descriptor to perform reconstruction technique.

In the final step of this research project, statistical two-point correlation functions were used to

homogenize thermal conductivity and elastic modulus of isotropic nanocomposite. For this,

strong contrast homogenization approach was used. One advantage of this approach is to take

into account the details of the microstructure which plays a very important role on the physical



properties of materials. For this purpose, two-point probability functions was calculated using

Monte Carlo technique to represent the distribution, shape and orientation of nanofillers

(inclusions). In this approach , three point correlation functions have been estimated using the

two point correlation functions then the effective thermal conductivity and elastic modulus of

nanocomposite was calculated using strong contrast approach. It will be interesting to calculate

four-point and five-point correlation function using the new developed approximation in future

study for seeing the influence of the higher order correlation functions on the effective

properties of heterogeneous materials. in future work , we would like to extend the numerical

solution for calculating stiffness tensor and thermal conductivity tensor of multiphase

anisotropic heterogonous media.

Appendix


Appendix

Appendix


Appendix


Appendix A: Verification of the Boundary Conditions for the Approximated Three-Point

Probability Function:

In this section, different limiting conditions ( are examined.

A.1 First, we consider the case:

(A.1)

(A.2)

(A.3)

Similarly for :

(A.4)

And when

(A.5)

A.2 Considering the case:

Appendix


(A.6)

(A.7)

Similarly, we have:

(A.8)

(A.9)

A.3 Now, consider the case:

(A.10)

A.4 Finally, let’s consider the case:

(A.11)

And therefore, we have:

(A.12)

This approximation is also valid for incompatible events (e.g. when x1 and x2 fall in two different

phases) because in the limit, the terms containing correlation functions vanish to zero. For

example in the case of incompatible event for x1 and x2 we have:

(A.13)

Appendix


Therefore we have:

(A.14)

and finally:

(A.15)

Appendix


Appendix B: Verification of the Boundary Conditions for the Approximated Four-Point

Probability Function:

In this section, different limiting conditions ( are probed.

B.1 First, we consider the case:

(B.1)

Using boundary conditions of Eq. (54 and 55),

We have:

(B.2)

(B.3)

By substituting weight functions and simplifying Eq. (B.1), we get:

(B.4)

Similarly, we have:

(B.5)

(B.6)

(B.7)

Appendix


B.2 Considering the case:

(B.8)

= (B.9)

Similarly, we have:

= (B.10)

= (B.11)

= (B.12)

This approximation is also valid for incompatible events (e.g. when x1 and x2 fall in two different

phases) because in the limit, the terms containing correlation functions vanish to zero. For

example in the case of incompatible event for x1 and x2 we have:

Appendix


(B.13)

By substituting three point correlations function using Eq. (18) in Eq. (39) and calculating limit,

we have:

(B.14)

B.3 Now, consider the case:

(B.15)

B.4 Finally, let’s consider the case:

(B.16)



Development of a multiscale approach for the characterization and modelling of heterogeneous materials : Application to polymer nanocomposites

In this research, a comprehensive study has been performed in the use of two-point correlation functions for reconstruction and homogenization in nano-composite materials. Two-point correlation functions are measured from different techniques such as microscopy (SEM or TEM), scattering and Monte Carlo simulations. Higher order correlation functions must be calculated or measured to increase the precision of the statistical continuum approach. To achieve this aim a new approximation methodology is utilized to obtain N-point correlation functions for multiphase heterogeneous materials. The two-point functions from different techniques have been measured and exploited to reconstruct microstructure of heterogeneous media. A new Monte Carlo methodology is developed as a mean for three-dimensional (3D) reconstruction, of the microstructure of heterogeneous materials, based on two-point statistical functions. The salient feature of the presented reconstruction methodology is the ability to realize the 3D microstructure from its 2D SEM image for a three-phase medium extendable to n-phase media. Three dimensional reconstruction of heterogeneous media have been exploited to predict percolation of heterogamous materials. Finally, Statistical continuum theory is used to predict the effective thermal conductivity and elastic modulus of polymer composites. Two-point and three-point probability functions as statistical descriptor of inclusions have been exploited to solve strong contrast homogenization for effective thermal and mechanical properties of nanoclay based polymer composites. To validate our modeling approach, we conducted several experimental measurements for nanoclay/polymer of composite. Comparison of our predictions with the experimental results led to a good agreement. this allows us to conclude that the proposed methodlogy is accurate.

Développement d'une approche multi-échelle pour la caractérisation et la modélisation des matériaux hétérogènes: Application aux polymères nanocomposites

Dans ce projet de recherche, une étude approfondie a été effectuée en utilisant des fonctions de corrélation à deux points pour la reconstruction et l'homogénéisation de nano-matériaux composites. Ces fonctions de corrélation à deux points sont mesurées à l’aide de différentes techniques telles que la microscopie (MEB ou TEM), la diffraction des rayons X et les simulations de type Monte Carlo. Des fonctions de corrélation d'ordre supérieur doivent être calculées ou mesurées si l’on souhaite augmenter la précision de l'approche de la méthode statistique. Pour atteindre cet objectif, une nouvelle méthodologie d’approximation est utilisée pour obtenir des fonctions de corrélation à N-points pour les matériaux hétérogènes multiphasiques. Les fonctions à deux points ont été mesurées à partir de techniques différentes et exploitées pour reconstituer la microstructure des milieux hétérogènes. Dans la suite de ce travail, une nouvelle méthodologie Monte Carlo est développée comme outil pour la reconstruction en trois dimensions (3D) de la microstructure des matériaux hétérogènes, fondée sur les fonctions statistiques à deux points (TPFC). La caractéristique principale de la méthodologie de reconstruction présentée ici est la capacité de réaliser la microstructure 3D à partir de son image SEM 2D pour un milieu à trois phases extensible à n-phases. Trois reconstructions tridimensionnelles des milieux hétérogènes ont été exploitées pour prédire la percolation des matériaux hétérogames. Enfin, la théorie de la statistique des milieux continus est utilisée pour prédire la conductivité thermique effective ainsi que le module d'élasticité des composites polymères. Des fonctions de probabilité à deux points et à trois points, utilisées comme descripteurs statistiques des inclusions (renfort) ont été exploitées pour résoudre le problème de l’homogénéisation à fort contraste des propriétés thermiques et mécaniques effectives des matériaux composites à base de polymère/ nano-argile. Pour valider notre approche de modélisation, nous avons mené plusieurs mesures expérimentales pour les composites polymère/nanoargile. La comparaison de nos prédictions avec les résultats expérimentaux ont conduit à un bon accord ce qui confirme la qualité et la précision de la méthodologie proposée.

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Development of a multiscale approach for the …...Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011 Figure 3. Fonctions de corrélation à trois points pour un composite