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 :
Co-Directeur de thèse :
Prof. Saïd AHZI, Université de Strasbourg
Prof. René MULLER, Université de Strasbourg
Rapporteur externe :
Rapporteur externe :
Prof. Moussa NAïT ABDELAZIZ, École Polytechnique de l'Université de Lille
Prof. Sébastien MERCIER, Université Paul Verlaine-Metz
Examinateur :
Examinateur :
Examinateur :
Prof. Hamid GARMESTANI, Georgia Institute of Technology, Atlanta-USA
Prof. Abdel-Mjid NOURREDDINE, Université de Strasbourg
Dr. David RUCH, Centre de Recherche Public Henri Tudor, Luxembourg
Invité :
Invité :
Invité :
Prof. Yves REMOND, Université de Strasbourg
Prof. Madjid FATHI, University of Siegen, Siegen-Germany
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.
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
ix
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
x
Conclusion and Future Work ...................................................................................................... 175
Appendix .................................................................................................................................... 179
Appendix A ............................................................................................................................. 181
Appendix B ............................................................................................................................. 184
xi
Résumé
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Résumé
Résumé
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Résumé
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
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)
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
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).
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
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 MonteCarlo 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.
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
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.
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
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)
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Résumé
Figure 7. Comparaison entre la conductivité thermique expérimentale et théorique du PA et de
ses nanocomposites avec OMMT
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Abstract
Abstract
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Abstract
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
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 twopoint 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 threephase 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.
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Introduction
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Introduction
Introduction
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Introduction
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Introduction
Development of advanced microstructure reconstruction methodologies is essential to access a
variety of analytical information associated with complexities in the microstructure of multiphase 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 higherorder 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
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
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 threepoint 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.
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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 threedimensional 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.
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
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 threepoint 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
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
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.
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Introduction
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porous media using image analysis and Fourier transforms. Journal of Petroleum Science and
Engineering. 1998;21(3-4):273-283.
[4] Pierret A, Capowiez Y, Belzunces L, Moran CJ. 3D reconstruction and quantification of macropores
using X-ray computed tomography and image analysis. Geoderma. 2002;106(3-4):247-271.
[5] Sundararaghavan V, Zabaras N. Classification and reconstruction of three-dimensional
microstructures using support vector machines. Computational Materials Science. 2005;32(2):223-239.
[6] 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.
[7] Tariel V, Jeulin D, Fanget A, Contesse G. 3D multiscale segmentation of granular materials. Image
Anal Stereol. 2011;27(1):23-28.
[8] Torquato S. Random heterogeneous materials : microstructure and macroscopic properties. New York:
Springer; 2002.
[9] Sheehan N, Torquato S. Generating microstructures with specified correlation functions. Journal of
Applied Physics. 2001;89(1):53-60.
[10] Kroner E. Bounds for effective elastic moduli of disordered materials,. J Mech Phys Solids
1977;25:137-155.
[11] Beran MJ. Statistical continuum theories. New York: Interscience Publishers; 1968.
[12] Yeong CLY, Torquato S. Reconstructing random media. PHYSICAL REVIEW E. 1998;57(1):495506.
[13] Manwart C, Hilfer R. Reconstruction of random media using Monte Carlo methods. PHYSICAL
REVIEW E. 1999;59(5):5596.
[14] Fullwood DT, Adams BL, Kalidindi SR. A strong contrast homogenization formulation for multiphase anisotropic materials. Journal of the Mechanics and Physics of Solids. 2008;56(6):2287-2297.
[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.
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
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.
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Chapter 1: Literature Survey
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Chapter 1: Literature Survey
Chapter I
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Chapter 1: Literature Survey
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter 1: Literature Survey
Literature Survey
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Chapter 1: Literature Survey
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter 1: 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 ( P1i ) is defined as the
volume fraction through the following relation, as M (the total number) is increased to infinity
P1i
Mi
M
vi
(1)
M of
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
¦Vi
i 1
2
Vtotal and
¦v
i
1
(2)
i 1
I.2. Two-Point Probability Functions
Now assign a vector ‫ݎ‬Ԧstarting 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 ( P212 ¨ r ¸ , P2 21 ¨ r ¸ , P211 ¨ r ¸ and P2 22 ¨ r ¸ ) defined as [1]:
© ¹
© ¹
© ¹
© ¹
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter 1: Literature Survey
­
®r
M M of ¯
§ ·
P2ij ¨ r ¸
© ¹
M ij
§
· §
·½
r 2 r1 , ¨ r1  Mi ¸ ˆ ¨ r 2  M j ¸ ¾
©
¹ ©
¹¿
(3)
where, Mij are the number of vectors with the beginning in phase-i ( I i ) and the end in phase-j (
I j ). Eq. (3) defines a joint probability distribution function for the occurrence of events
constructed by two points (‫ݎ‬Ԧଵ and‫ݎ‬Ԧଶ ) as the beginning and end of a vector ‫ݎ‬Ԧ when it is randomly
inserted in a microstructure. The two-point function can be defined based on two other
probability functions such that [1]:
­§
·§
·½
§
·
P
Probability
b bili ®¨ r1  Mi ¸ ¨ r 2  M j ¸ ¾ P
Probability
b bili
b l ¨ r 2 M ¸
¹©
¹¿
©
¹j
¯©
§ ·
P2ij ¨ r ¸
© ¹
(4)
The first term on the right hand side is a conditional probability function. At very large distances,
rÆf, 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:
­§
·§
·§
·½
§
·
Probability ®¨ r o f ¸ ¨ r1  Mi ¸ ¨ r 2  M j ¸ ¾ P
Probability
b b
bili
l ¨ r1  Mi ¸ (5)
¹©
¹¿
©
¹
¹©
¯©
The two-point function will then reduce to [1]:
§
·
P2ij ¨ r , r o f ¸
©
¹
§
·
Probability ¨ r1  Mi ¸ Probability(r 2  M j )
©
¹
(6)
or,
lim
r of
P2ij r
Q iQ j
(7)
For the case of a two-point function in a two phase composite, we have symmetry for non FGM
microstructure [1]:
P2ij r
P2ji r
(8)
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter 1: Literature Survey
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 ( P211 , P222 , P233 , P212 , P221 , P213 , P231 , P223 , P232 ).
Due to normality conditions the following equations are satisfied:
¦ ¦ P r ij
2
1
(9)
i 1,3 j 1,3
¦ P r ij
2
vi
(10)
j 1,3
¦ P r ij
2
vj
(11)
i 1,3
Satisfying all three conditions for a three-phase composite ( i , j{1,2,3}) and knowing that the
probability functions are symmetric ( P2ij = P2ji ) results in the important conclusion that only three
of the nine probabilities are independent variables. For instance, we can choose P211 or (P11), P212
or (P12), and P222 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) P2C ii (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.
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Chapter 1: Literature Survey
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
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Chapter 1: Literature Survey
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,...,q
Pn
(x1, x 2 ,..., x n ) Pr obability(x1  Phase(q) ˆ x 2  Phase(q) ˆ ... ˆ x n  Phase(q))
(12)
Where, xi is the vector position of the points in the microstructure.
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter 1: Literature Survey
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:
1 ii
1 ii
iii
P3 x1 , x2 , x3 # P2 x1 , x3 P2 x1 , x2 2
2
(13)
Garmestani et al. [6] also proposed another approximation for three point correlation functions:
§
x1 x3
iii
P3 x1 , x2 , x3 # ¨
¨ x1 x3 x2 x3
©
·
§ xx
2 3
¸ P ii x , x ¨
2
2 3
¸
¨ x1 x3 x2 x3
¹
©
·
¸ P ii x , x (14)
¸ 2 1 3
¹
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.
§§
x1 x3
P3 x1 , x2 , x3 # ¨ ¨
¨¨ x x x x
2 3
©© 1 3
iii
·
§
x2 x3
¸ P ii x , x ¨
¸ 2 2 3 ¨ x1 x3 x2 x3
¹
©
· P ii x , x
·
¸ P ii x , x ¸ 2 2 3 ¸ 2 1 3 ¸ P2ii 0 ¹
¹
(15)
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter 1: Literature Survey
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 v Ke .G
(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
Average generalized flux
Average generalized intensity
G
Ke
F
Thermal conductivity
Heat flux
Temperature gradient
Electrical conductivity
Electric current
Electric field
Magnetic permeability
Magnetic induction
Magnetic field
To estimate the bulk properties of such heterogeneous materials, multiscale homogenization
approaches are utilized. The multiscale homogenization techniques might be well categorized
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter 1: Literature Survey
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 HashinShtrikman 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]:
c
c( x)
1
V
³
V
c( x)dV
¦ c( x)
(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.
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Chapter 1: Literature Survey
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]
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter 1: Literature Survey
In classical tomography (attenuation tomography), three dimensional reconstruction is
performed by combining the two dimensional projections. This technique has some limitation
such as [15]:
x
Resolution limited to about 1000-2000x the object cross-section diameter;
x
Blurring of material boundaries;
x
Weak attenuation contrasts for imaging ;
x
Complicated data acquisition and interpretation due to the image artifacts (beam
hardening);
x
Large data volumes and difficulty of visualization and analysis
However, this technique has several strengths such as [15]:
x
Non-destructive 3D imaging
x
Easy sample preparation required
x
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 dualbeam FIB/SEM is composed of ion beam which allows milling of the surface while the imaging
is conducted by the electron gun [12].
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Chapter 1: Literature Survey
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
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Chapter 1: Literature Survey
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:
ܲሺ‫ݎ݋ݎݎܧ‬௢௟ௗ ՜ ‫ݎ݋ݎݎܧ‬௡௘௪ ሻ ൌ ቐ
ͳǡ
݁
οಶೝೝ೚ೝ
ି
೅
οா௥௥௢௥
்
οா௥௥௢௥
்
൏Ͳ
(18)
൒ Ͳ
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.
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter 1: Literature Survey
I. References
[1] Torquato S. Random heterogeneous materials : microstructure and macroscopic properties.
New York ; London: Springer; 2002.
[2] 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.
[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):65916602.
[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.
[7] 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.
[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 boundaryvalue 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.
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Chapter 1: Literature Survey
[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.
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
Chapter II
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
Using SAXS Approach to Calculate
Two-Point Correlation Function:
(Application to Polystyrene/Zirconia
Nanocomposite)
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
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
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
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 U from the scattered heterogeneities and its surrounding, which can be
represented by an average density U . The local fluctuation K of the electron density can be
defined as follows:
K UU .
(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:
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
J r K2
KAKB
(2)
where A and B are two distinct points in the medium represented by the vectors r1 , r2 , r
r2 r1
and J r is the characteristic or autocorrelation function depending on the position r. J r can
be defined as follows:
J r K(r1 )K(r2 )
(3)
For random distribution of heterogeneities, the autocorrelation function J r satisfy the
following conditions: J r 0 K and J r o f 0 . It is convenient to define the auto2
covariance of phase-1 for a statistically homogeneous media as [1] :
J r K(r1 )K(r2 )
P211 (r ) I12
(4)
where I1 is the volume fraction of phase 1 (fillers) and P211 (r ) is the two-point probability
function. Recalling that U r is the number of electrons per unit volume, a volume element
dV
at position r will contain U r u 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] :
I (h
h))
³³³ ³³³ dV dV U(r )U(r )e
1
2
1
ihr
2
³³ U(r )U(r )e
1
2
ihr
dr1dr2 (5)
V
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:
U2 (r ) { ³³³ dV1U(r1 )U(r2 )
(6)
which allows to rewrite I h as:
I(h)
h)
³³³ dVU (r)e
2
ihr
(7)
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
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(hr )
hr
eihr
(8)
As a result, the average scattering intensity reduces to:
³ 4Sr dr ˜UU (r )
2
I ( h)
2
sin(hr )
hr
(9)
Recalling the autocorrelation function, J , the above equation can be rewritten:
I (h) Vn02 ³ 4Sr 2 dr ˜ J(r )
sin(hr )
hr
(10)
where n0 is the mean density of electrons. Or,
J (r )
1
2S Vn02
2
³
f
0
I ( h)
sin(hr ) 2
h dh
hr
(11)
here, n0 is a constant. Using equation (4), the equation (11) can be rewritten as follow:
J (r )
11
where P2
1
2S Vn02
P211 (r ) I12
r represents
2
³
f
0
I (h)
sin(hr ) 2
h dh
hr
(12)
the two-point probability correlation function which measures the
11
spatial distribution of the heterogeneities (phase-1) in the matrix (phase-2). P2
r should verify
the following condition:
P211 r I1
P211 r I1 when
2
r
0
when r o f
.
(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).
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
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
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
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)
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
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 O =1.54 Å (Cu KD), 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/O) 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
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
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
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
11
SAXS diagram of 1 % ZrO2 in PS do not show any signal from the nanoparticles, P2
r curve
presents negative values and does not verify the second condition in equation (13). Therefore
P211 r for 1 wt. % ZrO2 cannot be exploited to calculate the physical properties of the
nanocomposite. However, the two-point probability function P2 r , for 3 and 5 wt. % ZrO2 (see
1
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)
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
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..
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
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(12):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):196201.
[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.
[13] 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.
[14] Brumberger H. Modern Aspects of Small-Angle Scattering. Boston: Kluwer Academic
Publishers; 1995.
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Chapter II: Using SAXS Approach to Calculate Two-Point Correlation Function
[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.
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
Chapter III
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
New Approximate Solution for
N-Point Correlation Functions
for Heterogeneous Materials
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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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 threedimensional 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 coworkers [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 twophase 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
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
utilized for heterogeneous materials with small variation in the multi-phase properties. A strongcontrast 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.
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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, Npoint 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 (N1)-point correlation functions. For instance, the set of X of points or events (x1,x2, …xN), the
subsets of X are given below:
X=ሼ‫ݔ‬ଵ , ‫ݔ‬ଶ ,…., ‫ݔ‬௡ ሽ
(1)
ܷܵ‫ܵܶܧܵܤ‬ሺܺሻ ൌ ሼሼ‫ݔ‬௜ ሽǡ ሼ‫ݔ‬ଵ , ‫ݔ‬ଶ ,…., ‫ݔ‬௡ିଵ ሽ ǡ ሼ‫ݔ‬ଵ , ‫ݔ‬ଶ ,…., ‫ݔ‬௡ ሽ ǡ ǥ ሽ(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 ሺ‫ݔ‬ଵ ‫ݔ ת‬ଶ ‫ ת‬ǥ ‫ݔ ת‬ே ሻ):
CN (‫ݔ‬ଵ , ‫ݔ‬ଶ , ǥ‫ݔ‬ே ) ൌ ܲሺ‫ݔ‬ଵ ‫ݔ ת‬ଶ ‫ ת‬ǥ ‫ݔ ת‬ே ሻ
(3)
Here, ܲሺ‫ݔ‬ଵ ‫ݔ ת‬ଶ ‫ ת‬ǥ ‫ݔ ת‬ே ሻ represents the probability of the eventሺ‫ݔ‬ଵ ‫ݔ ת‬ଶ ‫ ת‬ǥ ‫ݔ ת‬ே ሻ. For
simplicity, the following properties of this correlation function are shown for the case of N=3
C3 (‫ݔ‬ଵ , ‫ݔ‬ଶ , ‫ݔ‬ଷ ) = C3 (‫ݔ‬ଶ , ‫ݔ‬ଵ , ‫ݔ‬ଷ ) = C3 (‫ݔ‬ଵ , ‫ݔ‬ଷ , ‫ݔ‬ଶ )ൌ ܲሺ‫ݔ‬ଵ ‫ݔ ת‬ଶ ‫ݔ ת‬ଷ ሻ
(4)
C3 (‫ݔ‬ଵ , ‫ݔ‬ଶ , ‫ݔ‬ଶ ) = ܲሺ‫ݔ‬ଵ ‫ݔ ת‬ଶ ‫ݔ ת‬ଶ ሻ= C2 (‫ݔ‬ଵ , ‫ݔ‬ଶ )
(5a)
C3 (‫ݔ‬ଶ ,‫ݔ‬ଷ ,‫ݔ‬ଶ ) = ܲሺ‫ݔ‬ଶ ‫ݔ ת‬ଶ ‫ݔ ת‬ଷ ሻ = C2 (‫ݔ‬ଶ , ‫ݔ‬ଷ )
(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.
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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, ଵଶ , which will allow us
to express the 2-point correlation function in terms of the 1-point probability functions in the
following multiplicative decomposition:
‫ ܥ‬2ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ ൌ ܲሺ‫ݔ‬ଵ ‫ݔ ת‬ଶ ሻ ൌ ܹଵଶ ‫ ܥ כ‬1ሺ‫ݔ‬ଵ )‫ ܥ כ‬1ሺ‫ݔ‬ଶ ሻ ൌ ܹଵଶ ‫ܲ כ‬ሺ‫ݔ‬ଵ )‫ܲ כ‬ሺ‫ݔ‬ଶ ሻ
(6)
The dependency factor is a function of the vector length:
ܹଵଶ ൌ ݂ሺȁ‫ݎ‬ȁሻ
(7)
For very large r, the independence of events x1 and x2 yields the following:
݈݅݉ȁ௥ȁ՜ஶ ݂ሺȁ‫ݎ‬ȁሻ ൌ ͳ
(8)
In addition, for very small length of r (ȁ”ȁ ՜ Ͳሻǡif the two events x1 and x2 are compatible we
have
݈݅݉ȁ௥ȁ՜଴ ݂ሺȁ‫ݎ‬ȁሻ ൌ
ଵ
௉ሺ௫భ ሻ
(9)
However, if the two events are incompatible we have:
݈݅݉ȁ௥ȁ՜଴ ݂ሺȁ‫ݎ‬ȁሻ ൌ Ͳ
(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
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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:
ܲሺ‫ݔ‬ଶ ȁ‫ݔ‬ଵ ሻ ൌ
௉ሺ௫భ ‫ת‬௫మ ሻ
ܲሺ‫ݔ‬ଷ ȁ‫ݔ‬ଵ ሻ ൌ
௉ሺ௫భ ‫ת‬௫య ሻ
௉ሺ௫భ ሻ
௉ሺ௫భ ሻ
(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
(ܹଵଷ , ܹଶଷ and ܹଷଷ ) 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:
ܹଵଷ ‫ܲ כ‬ሺ‫ݔ‬ଵ ሻ ‫ܲ כ‬ሺ‫ݔ‬ଶ ȁ‫ݔ‬ଵ ሻ ‫ܲ כ‬ሺ‫ݔ‬ଷ ȁ‫ݔ‬ଵ ሻ
(13)
77
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
Similarly, the probability of occurrence of x2 followed by x1 and then x3 is :
ܹଶଷ ‫ܲ כ‬ሺ‫ݔ‬ଶ ሻ ‫ܲ כ‬ሺ‫ݔ‬ଵ ȁ‫ݔ‬ଶ ሻ ‫ܲ כ‬ሺ‫ݔ‬ଷ ȁ‫ݔ‬ଶ ሻ
(14)
Finally, the probability of occurrence of x3 followed by x1 and then x2 is given by :
ܹଷଷ ‫ܲ כ‬ሺ‫ݔ‬ଷ ሻ ‫ܲ כ‬ሺ‫ݔ‬ଵ ȁ‫ݔ‬ଷ ሻ ‫ܲ כ‬ሺ‫ݔ‬ଶ ȁ‫ݔ‬ଷ ሻ
(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:
‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ܲሺ‫ݔ‬ଵ ‫ݔ ת‬ଶ ‫ݔ ת‬ଷ ሻ ൎ ܹଵଷ ‫ܲ כ‬ሺ‫ݔ‬ଵ ሻ ‫ܲ כ‬ሺ‫ݔ‬ଶ ȁ‫ݔ‬ଵ ሻ ‫ܲ כ‬ሺ‫ݔ‬ଷ ȁ‫ݔ‬ଵ ሻ ൅ ܹଶଷ ‫ܲ כ‬ሺ‫ݔ‬ଶ ሻ ‫כ‬
ܲሺ‫ݔ‬ଷ ȁ‫ݔ‬ଶ ሻ ‫ܲ כ‬ሺ‫ݔ‬ଵ ȁ‫ݔ‬ଶ ሻ ൅ ܹଷଷ ‫ܲ כ‬ሺ‫ݔ‬ଷ ሻ ‫ܲ כ‬ሺ‫ݔ‬ଵ ȁ‫ݔ‬ଷ ሻ ‫ܲ כ‬ሺ‫ݔ‬ଶ ȁ‫ݔ‬ଷ ሻ
(16)
We can then write:
‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൎ ܹଵଷ ‫כ‬
஼మ ሺ௫భ ǡ௫మ ሻ‫כ‬஼మ ሺ௫య ǡ௫భ ሻ
஼భ ሺ௫భ ሻ
൅ ܹଶଷ ‫כ‬
஼మ ሺ௫భ ǡ௫మ ሻ‫כ‬஼మ ሺ௫మ ǡ௫య ሻ
஼భ ሺ௫మ ሻ
൅ ܹଷଷ ‫כ‬
஼మ ሺ௫య ǡ௫మ ሻ‫כ‬஼మ ሺ௫య ǡ௫భ ሻ
஼భ ሺ௫య ሻ
(17)
The weight functions ܹ௔௕ can now be calculated using the boundary conditions: The first
boundary condition is:
݈݅݉௫భ ՜ஶ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ
(18)
where, x1՜ λ is meant to satisfy the following conditions: |r12ȁ ՜ λ and |r13ȁ ՜ λ. Therefore
݈݅݉௫భ ՜ஶ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ܹଵଷ ‫כ‬
஼భ ሺ௫భ ሻ‫כ‬஼భ ሺ௫మ ሻ‫כ‬஼భ ሺ௫య ሻ‫כ‬஼భ ሺ௫భ ሻ
஼భ ሺ௫భ ሻ
஼భ ሺ௫భ ሻ‫כ‬஼భ ሺ௫య ሻ‫כ‬஼మ ሺ௫మ ǡ௫య ሻ
஼భ ሺ௫య ሻ
൅ ܹଶଷ ‫כ‬
஼భ ሺ௫భ ሻ‫כ‬஼భ ሺ௫మ ሻ‫כ‬஼మ ሺ௫మ ǡ௫య ሻ
஼భ ሺ௫మ ሻ
൅ ܹଷଷ ‫כ‬
(19)
Applying this boundary condition we get:
‫ݔ‬ଵ ՜ λܹଵଷ ൌ Ͳ
(20)
Similarly, for šଶ and šଷ we get:
‫ݔ‬ଶ ՜ λܹଶଷ ൌ Ͳ
‫ݔ‬ଷ ՜ λܹଷଷ ൌ Ͳ
(21)
(22)
78
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
The second boundary condition is:
݈݅݉ ௫భ ՜ஶ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ
(23)
௫మ ՜ஶ
௫య ՜ஶ
From equality condition for left side and right side of Eq. (17) we have:
‫ݔ‬ଵ ՜ λǡ ‫ݔ‬ଶ ՜ λǡ ‫ݔ‬ଷ ՜ λܹଵଷ ൅ ܹଶଷ ൅ ܹଷଷ ൌ ͳ
(24)
Third boundary condition:
݈݅݉௫భ ՜௫మ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ‫ܥ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ
(25)
This yields:
݈݅݉௫భ ՜௫మ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ܹଵଷ ‫כ‬
஼మ ሺ௫య ǡ௫మ ሻ‫כ‬஼మ ሺ௫య ǡ௫మ ሻ
஼భ ሺ௫య ሻ
஼మ ሺ௫మ ǡ௫మ ሻ‫כ‬஼మ ሺ௫య ǡ௫మ ሻ
஼భ ሺ௫మ ሻ
൅ ܹଶଷ ‫כ‬
஼మ ሺ௫మ ǡ௫మ ሻ‫כ‬஼మ ሺ௫మ ǡ௫య ሻ
஼భ ሺ௫మ ሻ
൅ ܹଷଷ ‫כ‬
(26)
From this boundary conditions and using Eq. (17) we get:
‫ݔ‬ଵ ՜ ‫ݔ‬ଶ ܹଷଷ ൌ Ͳ
(27)
By applying similar methodology for ‫ݔ‬ଶ and ‫ݔ‬ଷ we obtain:
‫ݔ‬ଵ ՜ ‫ݔ‬ଷ ܹଶଷ ൌ Ͳ
(28)
‫ݔ‬ଶ ՜ ‫ݔ‬ଷ ܹଵଷ ൌ Ͳ
(29)
Therefore, necessary conditions for weight function are referred as follows (see details in
additional Appendix A):
‫ݔ‬௜ ՜ λܹ௜ଷ ൌ Ͳǡ ܹ௝ଷ ് Ͳ݂‫݅ ് ݆ݎ݋‬
(30)
‫ݔ‬௜ ՜ λሺ݅ ൌ ͳǡ ǥ ͵ሻǡσ ܹ௜ଷ ൌ ͳ
(31)
‫ݔ‬௜ ՜ ‫ݔ‬௝ ሺ݂‫݆ ് ݅ݎ݋‬ሻǡܹ௞ଷ ൌ Ͳǡ ݇ ് ݅ܽ݊݀݇ ് ݆
(32)
79
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
By assuming ܹ௜ଷ = 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:
ோଵ
ܹଵଷ ൌ ோଵାோଶାோଷ
ܹଶଷ ൌ ோଵାோଶାோଷ
ܹଷଷ ൌ ோଵାோଶାோଷ
(33)
ோଶ
(34)
ோଷ
(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.
‫ܥ‬ସ ൫‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷǡ ‫ݔ‬ସ ൯ ൌ ܲሺ‫ݔ‬ଵ ‫ݔ ת‬ଶ ‫ݔ ת‬ଷ ‫ݔ ת‬ସ ሻ ൎ
80
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
ܹଵସ ‫ܲ כ‬ሺ‫ݔ‬ଵ ሻ ‫ܲ כ‬൫ሺ‫ݔ‬ଶ ‫ݔ ת‬ଷ ሻหሺ‫ݔ‬ଵ ‫ݔ ת‬ଷ ሻ൯ ‫ܲ כ‬൫ሺ‫ݔ‬ସ ‫ݔ ת‬ଶ ሻหሺ‫ݔ‬ଵ ‫ݔ ת‬ଶ ሻ൯ ‫ܲ כ‬൫ሺ‫ݔ‬ସ ‫ݔ ת‬ଷ ሻหሺ‫ݔ‬ଵ ‫ݔ ת‬ସ ሻ൯
൅ܹଶସ ‫ܲ כ‬ሺ‫ݔ‬ଶ ሻ ‫ܲ כ‬൫ሺ‫ݔ‬ଷ ‫ݔ ת‬ଵ ሻหሺ‫ݔ‬ଶ ‫ݔ ת‬ଵ ሻ൯ ‫ܲ כ‬൫ሺ‫ݔ‬ଵ ‫ݔ ת‬ସ ሻหሺ‫ݔ‬ଶ ‫ݔ ת‬ସ ሻ൯ ‫ܲ כ‬൫ሺ‫ݔ‬ଷ ‫ݔ ת‬ସ ሻหሺ‫ݔ‬ଶ ‫ݔ ת‬ଷ ሻ൯
൅ܹଷସ ‫ܲ כ‬ሺ‫ݔ‬ଷ ሻ ‫ܲ כ‬൫ሺ‫ݔ‬ଵ ‫ݔ ת‬ଶ ሻหሺ‫ݔ‬ଷ ‫ݔ ת‬ଶ ሻ൯ ‫ܲ כ‬൫ሺ‫ݔ‬ସ ‫ݔ ת‬ଵ ሻหሺ‫ݔ‬ଷ ‫ݔ ת‬ଵ ሻ൯ ‫ܲ כ‬൫ሺ‫ݔ‬ସ ‫ݔ ת‬ଶ ሻหሺ‫ݔ‬ଷ ‫ݔ ת‬ସ ሻ൯
൅ܹସସ ‫ܲ כ‬ሺ‫ݔ‬ସ ሻ ‫ܲ כ‬൫ሺ‫ݔ‬ଶ ‫ݔ ת‬ଵ ሻหሺ‫ݔ‬ସ ‫ݔ ת‬ଵ ሻ൯ ‫ܲ כ‬൫ሺ‫ݔ‬ଷ ‫ݔ ת‬ଵ ሻหሺ‫ݔ‬ସ ‫ݔ ת‬ଵ ሻ൯ ‫כ‬
ܲ൫ሺ‫ݔ‬ଶ ‫ݔ ת‬ଷ ሻหሺ‫ݔ‬ସ ‫ݔ ת‬ଶ ሻ൯
(36)
Or,
‫ܥ‬ସ ൫‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷǡ ‫ݔ‬ସ ൯ ൌ
ଵସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ସ ሻ
ଶସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଵ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଵ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
ଷସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଵ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
ସସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଵ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ
‫ܥ‬ଶ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଷ ሻ
(37)
The weight functions ܹ௔௕ are calculated using boundary conditions:
The first boundary condition is:
݈݅݉௫భ ՜ஶ ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
(38)
This limit can be written as:
81
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
݈݅݉ ‫ܥ‬ସ ൫‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷǡ ‫ݔ‬ସ ൯
௫భ ՜ஶ
ൌ ܹଵସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ
‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
ܹଶସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
ܹଷସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
ܹସସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଷ ሻ
(39)
Applying the boundary condition we get:
‫ݔ‬ଵ ՜ λܹଵସ ൌ Ͳ
(40)
Similarly for šଶ , šଷ and šସ , we obtain
‫ݔ‬ଶ ՜ λܹଶସ ൌ Ͳ
(41)
‫ݔ‬ଷ ՜ λܹଷସ ൌ Ͳ
(42)
‫ݔ‬ସ ՜ λܹସସ ൌ Ͳ
(43)
The second boundary condition is:
݈݅݉௫భ ՜ஶ ‫ܥ‬ସ ൫‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷǡ ‫ݔ‬ସ ൯ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
௫మ ՜ஶ
௫య ՜ஶ
௫ర ՜ஶ (44)
We can also write:
݈݅݉௫భ ՜ஶ ‫ܥ‬ସ ൫‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷǡ ‫ݔ‬ସ ൯ ൌ
௫మ ՜ஶ
௫య ՜ஶ
௫ర ՜ஶ
82
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
ܹଵସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ
൅ܹଶସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ
‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ
൅ܹଷସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ
‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ
‫ܥ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
൅ܹସସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
(45)
Applying this boundary condition leads to:
‫ݔ‬ଵ ՜ λǡ ‫ݔ‬ଶ ՜ λǡ ‫ݔ‬ଷ ՜ λ‫ݔ‬ସ ՜ λܹଵସ ൅ ܹଶସ ൅ ܹଷସ ൅ ܹସସ ൌ ͳ
(46)
Third boundary condition:
݈݅݉௫భ ՜௫మ ‫ܥ‬ସ ൫‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷǡ ‫ݔ‬ସ ൯ ൌ ‫ܥ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
(47)
݈݅݉ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ൌ
௫భ ՜௫మ
ଵସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
ଶସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
ଷସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
ସସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ
‫ܥ‬ଶ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଷ ሻ
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
(48)
The boundary conditions in Eq. (47) require the following conditions:
‫ݔ‬ଵ ՜ ‫ݔ‬ଶ ܹଷସ ൌ Ͳܹܽ݊݀ସସ ൌ Ͳ
(49)
Similarly, for šଶ and šଷ we get:
‫ݔ‬ଵ ՜ ‫ݔ‬ଷ ܹଶସ ൌ Ͳܹܽ݊݀ସସ ൌ Ͳ
(50)
‫ݔ‬ଶ ՜ ‫ݔ‬ଷ ܹଵସ ൌ Ͳܹܽ݊݀ସସ ൌ Ͳ
(51)
And finally:
‫ݔ‬ଷ ՜ ‫ݔ‬ସ ܹଵସ ൌ Ͳܹܽ݊݀ଶସ ൌ Ͳ
(52)
Therefore, necessary conditions for weight functions are obtained as (see details in additional
Appendix B):
‫ݔ‬௜ ՜ λܹ௜ସ ൌ Ͳǡ ܹ௝ସ ് Ͳ݂‫݅ ് ݆ݎ݋‬
(53)
‫ݔ‬௜ ՜ λሺ݅ ൌ ͳǡ ǥ Ͷሻǡ σ ܹ௜ସ ൌ ͳ
(54)
‫ݔ‬௜ ՜ ‫ݔ‬௝ ሺ݂‫݆ ് ݅ݎ݋‬ሻǡܹ௞ସ ൌ Ͳ݂‫݆ ് ݇݀݊ܽ݅ ് ݇ݎ݋‬
(55)
Assuming that ୧ସ 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:
ܹ௟ସ ൌ
ܹଶସ ൌ
ܹଷସ ൌ
ܹସସ ൌ
஺ሺ௫మ ǡ௫య ǡ௫ర ሻ
ሻା஺ሺ௫
஺ሺ௫భ ǡ௫మ ǡ௫య
మ ǡ௫య ǡ௫ర ሻା஺ሺ௫మ ǡ௫భ ǡ௫ర ሻା஺ሺ௫భ ǡ௫య ǡ௫ర ሻ
஺ሺ௫భ ǡ௫య ǡ௫ర ሻ
஺ሺ௫భ ǡ௫మ ǡ௫య ሻା஺ሺ௫మ ǡ௫య ǡ௫ర ሻା஺ሺ௫మ ǡ௫భ ǡ௫ర ሻା஺ሺ௫భ ǡ௫య ǡ௫ర ሻ
஺ሺ௫మ ǡ௫భ ǡ௫ర ሻ
஺ሺ௫భ ǡ௫మ ǡ௫య ሻା஺ሺ௫మ ǡ௫య ǡ௫ర ሻା஺ሺ௫మ ǡ௫భ ǡ௫ర ሻା஺ሺ௫భ ǡ௫య ǡ௫ర ሻ
஺ሺ௫భ ǡ௫మ ǡ௫య ሻ
஺ሺ௫భ ǡ௫మ ǡ௫య ሻା஺ሺ௫మ ǡ௫య ǡ௫ర ሻା஺ሺ௫మ ǡ௫భ ǡ௫ర ሻା஺ሺ௫భ ǡ௫య ǡ௫ర ሻ
(56)
(57)
(58)
(59)
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
where, ‫ܣ‬൫‫ݔ‬୧ ǡ ‫ݔ‬୨ ǡ ‫ݔ‬୩ ൯ is the area of the side of the tetrahedron encompassing the three points
‫ݔ‬୧ ǡ ‫ݔ‬୨ ǡ ‫ݔ‬୩ .
III. 4. Approximation of N-point correlation function
The methodology for deriving the approximations in previous sections can be extended to Npoint correlations which unfortunately yields a lengthy procedure for N>3. Thus, we limit our
analysis to the following brief general description of the methodology:
೙షభ
ቀ
ቁ
‫ܥ‬௡ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ǥ Ǥ ǡ ‫ݔ‬௡ ሻ ൌ σ௡௜ୀଵሺ୧୬ ‫ כ‬
೙షమ ஼
ς೗సభ
ሺ೙షభሻ ൫௫౟ ǡǥǤǡ௫ሺ೙షభሻ ൯
೙షభ
ቀ
ቁ
೙షయ ஼
ς೗సభ
ሺ೙షమሻ ൫௫౟ ǡǥǤǡ௫ሺ೙షమሻ ൯
೙షభ
ቀ
ቁ
‫כ‬
೙షర ஼
ς೗సభ
ሺ೙షయሻ ൫௫౟ ǡǥǤǡ௫ሺ೙షయሻ ൯
೙షభ
ቀ
ቁ
‫ כ‬ǥሻ
೙షఱ ஼
ς೗సభ
ሺ೙షరሻ ൫௫౟ ǡǥǤǡ௫ሺ೙షరሻ ൯
(60)
where ୧୬ are the dependency weight functions. In the formulation above, ൫‫ݔ‬୫ ǡ ǥ š୧ ǥ ǡ ‫ݔ‬୮ ൯, 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:
‫ݔ‬௜ ՜ λܹ௜௡ ൌ Ͳǡ ܹ௝௡ ് Ͳ݂‫݅ ് ݆ݎ݋‬
(61)
‫ݔ‬௜ ՜ λሺ݅ ൌ ͳǡ ǥ ݊ሻǡ σ ܹ௜௡ ൌ ͳ
(62)
‫ݔ‬௜ ՜ ‫ݔ‬௝ ሺ݂‫݆ ് ݅ݎ݋‬ሻǡܹ௞௡ ൌ Ͳ݂‫݆ ് ݇݀݊ܽ݅ ് ݈݈݇ܽݎ݋‬
(63)
In the next section, we will present numerical results but only for three-point correlation
functions.
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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 3point 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)
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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).
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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 ‫ ݔ‬ൌ ȁܴଵଷ ȁ))
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.
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
Fig. 8. Three-point correlation functions for the three-phase composite. The corresponding 3point 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).
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
Fig. 9. Three-point correlation functions for the three-phase composite. The corresponding 3point 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.
91
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
Fig. 10. Three-point correlation functions for the three-phase composite. The corresponding 3point 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 3point correlations are shown for the (red- black-blue) phases (average error = 0.066).
92
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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,
‫ ݎ݋ݎݎܧ‬ൌ
ห்ு௉஼ிሺ௥ሻೞ೔೘ೠ೗ೌ೟೔೚೙ ି்ு௉஼ிሺ௥ሻೌ೛೛ೝ೚ೣ೔೘ೌ೟೔೚೙ ห
்ு௉஼ிሺ௥ሻೞ೔೘ೠ೗ೌ೟೔೚೙
(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
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
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.
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Chapter III: New Approximate Solution for N-Point Correlation Functions for
Heterogeneous Materials
III. References
[1] Torquato S. Random heterogeneous materials : microstructure and macroscopic properties.
New York: Springer; 2002.
[2] Yeong CLY, Torquato S. Reconstructing random media. Physical Review E. 1998;57(1):495.
[3] Garmestani H, Baniassadi M, Li DS, Fathi M, Ahzi S. Semi-inverse Monte Carlo
reconstruction of two-phase heterogeneous material using two-point functions. IJTAMM.
2009;1(2):6.
[4] Baniassadi M, Garmestani H, Li DS, Ahzi S, Khaleel M, Sun X. Three-phase solid oxide fuel
cell anode microstructure realization using two-point correlation functions. Acta Materialia.
2011;59(1):30-43.
[5] 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;7:1462-1468.
[6] Kröner E. Modeling Small Deformation in Polycrystals. Amsterdam: Elsevier; 1986.
[7] Beran MJ. Statistical continuum theories. New York: Interscience Publishers; 1968.
[8] Phan-Thien N, Milton GW. New Bounds on the Effective Thermal Conductivity of N-Phase
Materials. Proceedings of the Royal Society of London Series A, Mathematical and Physical
Sciences. 1982;380(1779):333-348.
[9] Dederichs PH, Zeller R. Variational treatment of the elastic constants of disordered materials.
Zeitschrift für Physik A Hadrons and Nuclei. 1973;259(2):103-116.
[10] Willis JR. Variational and related methods for the overall properties of composites. Adv
Appl Mech. 1981;21:78.
[11] McCoy JJ. On the calculation of bulk properties of heterogeneous materials. Appl Math.
1979;37:13.
[12] Torquato S. Effective electrical conductivity of two-phase disordered composite media. J
Appl Phys. 1985;58(10):8.
[13] 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.
[14] Sen AK, Torquato S. Effective conductivity of anisotropic two-phase composite media.
Physical Review B. 1989;39(7):4504.
[15] Brown JWF. Solid mixture permittivities. The Journal of Chemical Physics. 1955;23 (8):4.
[16] Fullwood DT, Adams BL, Kalidindi SR. A strong contrast homogenization formulation for
multi-phase anisotropic materials. Journal of the Mechanics and Physics of Solids.
2008;56(6):2287-2297.
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Heterogeneous Materials
[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):423434.
[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):103115.
[28] 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.
[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.
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Heterogeneous Materials
[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.
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[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):1763417639.
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Using Two-Point Correlation Functions
Chapter IV
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Using Two-Point Correlation Functions
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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)
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Using Two-Point Correlation Functions
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 npoint 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 paircorrelation 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 isotropypreserving 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 onepoint 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].
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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.
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
Table 1- Different steps of heterogeneity generation in two metallurgical processes
Sim. Steps
Cell generation
Cell distribution
Growth of cells
Casting
Nucleation
Nucleation rate
Grain growth
Powder
metallurgy
Powder
Packing
Sintering
Process
(particles)
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.
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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.
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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
OF
(P
11
2
real
)
( P211 ) sim ( P212 )real ( P212 ) sim ( P222 ) real ( P222 ) sim 2
2
2
(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 Ydirection, 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
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meanwhile less than the Monte Carlo (M-C) repeat error. This repeat error depends on the
microstructure.
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Start
Reconstruction
Generation of cells
Distribution of cells
Growth of cells
Compute the two-point
correlations (Simulation)
Optimization
NO
Compute the two-point
correlations (From
Experimental SEM Images)
OF < M-C
repeat error
YES
Reconstruction is
done.
Fig. 4. Basic steps in the realization algorithm (OF = objective function; MC=Monte Carlo)
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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:
ZM
hM x, y (2)
where M 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
r c 2 ((k (M )) 2 ZM
Torus
r (k (M ))2 R(M ) x 2 y 2
ZM
Cube
ZM
x2
y2
) a, b, c and k(߮) are constants for each phase
a 2 i b2
rak (M )
2
k(߮) and R(߮) are constant for each phase
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:
S
ª f1 x, y, z, E , p1 º
«
»
« f 2 x, y, z , E , p2 »
«¬ f3 x, y, z , E , p3 »¼
(3)
Where x , y and z are cartesian coordinate inside the extrusion shape define by ZM .
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
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methodology for the 3D simulation. In the matrix above 0 E 1 is a random variable and pi
is the optimization parameter satisfying 0 d pi d 1 . Each of the three components of the S vector
takes values from the interval > 0,1@ , hence the term shrinkage function. In this work, the simple
forms of fi
E pi 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:
Qx
0
0
ª1
º
«0 cos(T ( E , p )) sin(T ( E , p )) »
x
4
x
4 »
«
«¬0 sin(T x ( E , p4 )) cos(T x ( E , p4 )) »¼
(4)
Qy
ªcos(T y ( E , p5 )) 0 sin(T y ( E , p5 )) º
«
»
0
1
0
«
»
« sin(T y ( E , p5 )) 0 cos(T y ( E , p5 )) »
¬
¼
(5)
Qz
ªcos(T z ( E , p6 )) sin(T z ( E , p6 )) 0 º
« sin(T ( E , p )) cos(T ( E , p )) 0 »
z
6
z
6
«
»
«¬
0
0
1 »¼
(6)
where rotation angles, T x ,T y ,T z , depend on the random parameter, E , and the optimization
factor, pi . The mathematical form of the rotation angles may be represented by
T x, y , z
2SE pi
(7)
with E and pi satisfying the same conditions that they have in Eq. (3).
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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.
XC
F (E , p7 )
(8)
YC
F ( E , p8 )
(9)
ZC
F ( E , p9 )
(10)
where E and pi 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
F
LE pi
(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
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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.
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Fig. 5. Algorithm for cell distribution.
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
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In the colony algorithm, one possible form of conditional statement is
C E ! : exp :Ii where
C E En
(12)
where 0 E 1 is a random variable, Ii is the volume fraction of the phase of interest, and n, : 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.
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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:
\ i E , pi pi E ! 0
(13)
Here, i=1,2, …6 , β ( 0 E 1 ) is a random variable and pi 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
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Using Two-Point Correlation Functions
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
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[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.
Select Random Site in Grid Phase
Network
Calculate minimum value of
cluster number of phases
Allocate minimum cluster
numbre to all neighbor’s site
No
Percolation
completed?
Yes
Save Cluster Number
Fig. 7. Algorithm of percolation based on the Monte Carlo methodology.
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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 8YSZ 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
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
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
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
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 greengreen) 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.
0.25
RED PHASE
0.2
GREEN PHASE
TPCF
0.15
0.1
0.05
0
0
200
400
600
800
1000
r
(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.
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
(a)
(b)
TPCF P11
TPCF P11
P 12
0.4
0.25
0.09
0.35
EXPRIMENTAL RESULT
0.2
SIMULATION1
0.3
0.15
0.1
0.05
EXPERIMENTAL
0.08
SIMULATION1
0.07
0.25
0.2
0.15
0.06
0.05
0.04
0.03
0.02
0.1
0.01
0.05
0
EXPERIMENTAL
Simulation1
Simulation2
SIMULATION2
TPCF(P12)
TPCF(PHASE2)
SIMULATION2
TPCF(PHASE1)
(c)
0
0
200
400
r
(d)
600
800
0
0
100
200
300
400
r
500
(e)
600
700
800
0
200
400 r
600
800
(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 ( P211 or P11) for the red-phase, e)
the 2-point correlation function ( P222 or P22) black phase f) 2-point correlation (black-red) function
( P212 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.
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
Table 3- RVE properties
Pixel
RVE
Type
Cube
X Dimension
205
Y Dimension
154
Z Dimension
116
Table 4- Reconstruction parameters
Cell
Shrinkage
Cube
ª0.5º
«0.5»
« »
¬«0.5¼»
S
Rotation
Distribution
Colony
F=Lβ
disabled
Cell growth
P4=1
P5=1
\ i E ,.001 .001 E ! 0
P6=1
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
(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
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. 13ac. 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.
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
(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
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
0.25
EXPRIMENTAL RESULT
SIMULATION
TPCF(PHASE1)
0.2
0.15
0.1
0.05
0
0
200
400
r
600
800
(a)
0.4
EXPERIMENTAL
RESULT
TPCF(PHASE2)
0.35
SIMULATION RESULT
0.3
0.25
0.2
0.15
0.1
0.05
0
0
200
400
r
600
800
(b)
0.09
TPCF(P1-2)
0.08
0.07
0.06
0.05
0.04
EXPERIMENTAL
RESULT
0.03
SIMULATION RESULT
0.02
0.01
0
0
400
r
200
600
800
(c)
Fig. 13. Comparison of the two-point correlation functions from
the experimental and the 3D realizations. a) 2-point correlation
functions P211 or P11 (red-red, phase 1), b) P222 or P22 for the
porous phase (black-black, phase 2), c) P212 or P12 (black-red)
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
(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.
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
IV. References
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Kirkwood-Salsburg hierarchies. J Chem Phys. 1983;78:3062-3072.
[2] Torquato S, Stell G. Microstructure of two-phase random media. I. The n-point probability
functions. J Chem Phys. 1982;77:2071-2077.
[3] Torquato S. Random heterogeneous materials : microstructure and macroscopic properties.
New York: Springer; 2002.
[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.
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[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.
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[7] Liang ZR, Fernandes CP, Magnani FS, Philippi PC. A reconstruction technique for threedimensional porous media using image analysis and Fourier transforms. Journal of Petroleum
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[8] Yeong CLY, Torquato S. Reconstructing random media. PHYSICAL REVIEW E.
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[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
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[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.
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
[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 twophase 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.
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Chapter IV: A New Monte Carlo Solution for Reconstruction of Heterogeneous Materials
Using Two-Point Correlation Functions
[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).
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
Chapter V
131
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
132
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
Homogenization of Mechanical and
Thermal Behavior of Nanocomposites
Using Strong Contrast:
(Application to Nanoclay Based Polymer
Nanocomposites)
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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
135
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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.
136
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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
137
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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].
­
®
¯M
T
cos
2S v
1
(1)
2u 1
Where T  >0, 2S > and M  > 0, S @ are spherical coordinates as shown in Fig..2 and where u, v are
random variables belonging to @0,1>
M
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.
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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 Oe of the nanocomposite is determined using the strong-contrast formulation of the
statistical continuum theory [27]:
^Oe O I ` .^Oe 2O I `
-1
R
R
ª S S 1, 2 S S 1 S S 2 º R
1
1
3
I
O
« 2
» M 1, 2 d2
R³
S
S
S
S1 1 S1 2 E SR S1 1
«¬
»¼
1
S
S
ª S S (1, 2,3)
S 2 (1, 2) S 2 (2,3) º R
2
2
R
3
OR d E SR ³³ « S
S
» M 1, 2 ˜ M 2,3 d2d3 ....
S
S
S
«¬ S1 1 S1 2 S1 1 S1 2 S1 3 »¼
(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, O is the reference conductivity,
R
M (1, 2) is a second-order tensor defined below, and E SR is the polarizability:
R
ESR
OS O R
OS d 1 O R
(3)
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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.
R
The second order tensor M (1, 2) is defined by:
R
M (1, 2)
1
3tt I
: OR x1 x2 3
where : is the total solid angle contained in a 3-dimensional sphere and t
(4)
( x1 x2 ) s
. S 1 ,
x1 x2 1
S2 1, 2 and S3 1, 2,3 are the probability functions that contain the microstructure information.
s
s
The one-point probability function, S (1) , is the volume fraction of the nanoparticles. The twoS
1
point probability function, S (1, 2) , is calculated from the Monte Carlo simulation. The three-point
S
2
probability function, S3 1, 2,3 , is calculated from the following analytical approximation [30]:
s
ª
º S p 2,3
x1 x2
x1 x3
p
p
«
S3 1, 2,3 #
S 2 1,3
S 2 1
1, 2 » 2 p
1 3 +
« x1 x2 x1 x3
» S1 1
x1 x2 x1 x3
¬
¼
p
(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, strongcontrast 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]:
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
M p2 L( q ) : ª¬ L(eq ) º¼
f
1
M p I ¦ Bn( p )
(6)
n 2
Where MP is the volume fraction of phase P and I is the fourth-order identity tensor,
I
ijkl
º
1ª
G G
«G G
»
ik
il
il
jk
¼
2¬
(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:
B2( p)
³H d 2U
( q)
(1, 2) ª¬ S2( p) (1, 2) M p2 º¼
(8)
n2
§ 1 ·
B
d 2...³ dnU ( q ) (1, 2) : U ( q ) (2,3)
(1) ¨
¨ M p ¸¸ ³
©
¹
( p)
U (q)(n 1, n)' n (1,..., n),
n t 3,
( p)
n
n
r
x1 x2 ,
t
r
r
(9)
(10)
In Eq. (6) the effective tensor L(eq ) is given by :
L(eq )
^C
e
^
`
C ( q ) ` I A( q ) : ª¬Ce C ( q ) º¼
1
(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 '(np) (1,..., n) is a position-dependent determinant that is calculated using N-point correlation
function for a given phase p by:
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( p)
S2 (1, 2)
( P)
S3 (1, 2,3)
( p)
S1 (2)
( p)
S2 (2,3)
0
0
0
0
' (np ) (1,..., n)
(12)
( p)
Sn1 (1, 2,..., n 1)
( p)
Sn2 (2,3,..., n 1)
( p)
( p)
Sn (1, 2,..., n) Sn1 (2,3,..., n)
( p)
S2 (n 2, n 1)
( p)
S3 (n 2, n 1)
( p)
S1 ( n 1)
( p)
S2 (n 1, n)
The tensor U is calculated based on the position-dependent fourth-order H(r) and the related
tensor for phase q, L (q):
(q)
U ijkl
(r )
q)
(q)
L(ijmn
H mnkl
(r )
­° ª
ºG
(d 2)Gq
(q)
ª¬ dK q 2(d 1)Gq º¼ ® « K pq P pq » ij H mmkl
(r )
d
(
K
2
G
)
d
q
q
°¯ ¬«
¼»
(d 2)Gq
(q)
P pq H ijkl
(r )`
d ( K q 2Gq )
(13)
Where d is space dimension and the tensor H( r ) is the symmetrized double gradient tensor [28]
which is given below:
(q)
(r )
H ijkl
1
1
d
2: ª¬ dK q 2(d 1)Gq º¼ r
ªD qG ijG kl d ªG ik G il G ilG jk º dD q ªG ij tk tl G kl ti t j º ¬
¼
¬
¼
¬
d (d D q )
(14)
2
ª¬G ik t j tl G il t j tk G ik ti tl G ij ti tk º¼ d (d 2)D q ti t j tk tl º¼
The constant tensor for phase q is expressed as:
L( q )
ª
º
(d 2)Gq
ª¬ dKq 2(d 1)Gq ¼º « k pq /h P pq / s »
d ( Kq 2Gq )
¬«
¼»
(15)
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
Where d is space dimension and where k pq and P pq are introduced as bulk and shear modulus
polarizabilities, K q and Gq 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]. k pq and P pq are given by the following relations:
K p Kq
4
K p Gq
3
k pq
P
G p Gq
pq
Gp Gq ª¬3K q / 2 4Gq / d º¼
(16)
(17)
K q 2Gq
For macroscopically isotropic media, Eq. (6) can be simplified as [29]:
ª k pq
M p2 «
/h «¬ keq
f
P pq º
/ s » M p I ¦ Bn( p )
Peq »¼
n 2
(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:
ªk
º
P
M p2 « pq / h pq / s » M p I B2( p ) B3( p )
Peq »¼
¬« keq
B2( p)
B3( p)
³H d 2U
( q)
(1, 2) ª¬ S2( p) (1, 2) M p2 º¼
§ 1 ·
( q)
( q)
( p)
¨¨ ¸¸ ³ d 2...³ dnU (1, 2) : U (2,3)'3 (1,...,3)
© Mp ¹
' 3( p ) (1,..., 3)
S 2( p ) (1, 2)
S1( p ) (2)
S3( P ) (1, 2,3)
S 2( p ) (2,3)
(19)
(20)
(21)
(22)
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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).
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
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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 9.7010.40 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 planeparallel 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
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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).
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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
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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.
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
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.
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Using Strong Contrast
V. References
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Chen B, Evans JRG. Elastic moduli of clay platelets. Scripta Materialia 2006;54:1581.
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Chapter V: Homogenization of Mechanical and Thermal Behavior of Nanocomposites
Using Strong Contrast
[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
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Mura T. Micromechanics of defects in solids. Lancaster: Nijhoff, 1987.
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modeling and characterization of the effective properties in starch-based nano-biocomposites.
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coated inclusions: Applications to tetragonal array of spheroids. Journal of Applied Physics
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Using Strong Contrast
[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
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[31] Weisstein EW. Sphere Point Picking., vol. 2010: From MathWorld--A Wolfram Web
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[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
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[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
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
Chapter VI
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
Three-dimensional Reconstruction and
Homogenization of Heterogeneous
Materials Using Statistical Correlation
Functions and FEM
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
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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, 710].
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
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Materials Using Statistical Correlation Functions and FEM
contrast statistical method, based on two-point and three-point correlation functions. The twopoint 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 ( P2ij ) and two-point
cluster functions ( P2c ii ) as follows:
OF
(P )
ij
2 real
2
3
( P2ij ) sim ¦ ( P2ii )real ( P2ii ) sim ¦ ( P2c ii ) real ( P2cii ) sim
2
i 1
2
i 1
2
(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
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Materials Using Statistical Correlation Functions and FEM
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 twopoint 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.
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
Fig. 1. Basic steps in the realization algorithm (OF = objective function; MC=Monte Carlo)
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
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).
ª
º S p x , x x1 x2
x1 x3
p
p
«
S x , x +
S 2 x1 , x2 » 2 p 2 3
S3 x1 , x2 , x3 #
« x1 x2 x1 x3 2 1 3
» S1 x1 x1 x2 x1 x3
¬
¼
p
§
x2 x3
p
S3 x1 , x2 , x3 # ¨
¨ x1 x2 x1 x3 x2 x3
©
§
x1 x3
¨
¨ x1 x2 x1 x3 x2 x3
©
·
§
x1 x2
¸S p x , x S p x , x + ¨
¸ 2 1 3 2 1 2 ¨ x1 x2 x1 x3 x2 x3
¹
©
(2)
·
¸S p x , x S p x , x ¸ 2 2 3 2 1 2
¹
·
¸S p x , x S p x , x ¸ 2 1 3 2 2 3
¹
(3)
Here, S 2 x1 , x2 and S3 x1 , x2 , x3 are the two and three point correlation functions,
p
p
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
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
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.
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
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 ( P211 or (P11) for red-red and P222 or (P22) for
black-black and P212 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
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
(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
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
0.12
0.1
TPCF (Microstructure 1)
TPCF
0.08
TPCF (Microstructure 2)
0.06
0.04
0.02
0
0
100
200
300
r
400
500
(a)
0.7
0.6
TPCF (Microstructure 1)
TPCF
0.5
TPCF (Microstructure 2)
0.4
0.3
0.2
0.1
0
0
100
r
200
300
400
500
(b)
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
TPCF
TPCF (Microstructure 1)
TPCF (Microstructure 2)
0
100
200
r
300
400
500
(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.
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
0.12
TPCCF (Microstructure 1)
TPCCF (Microstructure 2)
0.1
TPCCF
0.08
0.06
0.04
0.02
0
0
50
100
150
200
250
r
300
350
400
450
500
Fig. 5. Two-point cluster function P2c 11 (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.
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
(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
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
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
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
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
Sample 2
Sample 3
(Thermal conductivity
and elastic modules)
(Thermal conductivity
and elastic modules)
(Thermal conductivity
and elastic modules)
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)
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
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.
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Chapter VI: Three-dimensional Reconstruction and Homogenization of Heterogeneous
Materials Using Statistical Correlation Functions and FEM
VI. References
[1] Fullwood DT, Adams BL, Kalidindi SR. A strong contrast homogenization formulation for
multi-phase anisotropic materials. Journal of the Mechanics and Physics of Solids.
2008;56(6):2287-2297.
[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):65916602.
[4] 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.
[5] Torquato S. Random heterogeneous materials : microstructure and macroscopic properties.
New York: Springer; 2002.
[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 threedimensional 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.
[11] 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.
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Conclusion and Future Work
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Conclusion and Future Work
Conclusion
and
Future Work
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Conclusion and Future Work
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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
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Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
Conclusion and Future Work
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.
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Appendix
Appendix
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Appendix
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Appendix
Appendix A: Verification of the Boundary Conditions for the Approximated Three-Point
Probability Function:
In this section, different limiting conditions (‫ݔ‬ଵ ՜ λǡ ǥ ሻare examined.
A.1
First, we consider the case: šଵ ՜ λǣ
݈݅݉ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ
௫భ ՜ஶ
ሺܹଵଷ ൌ Ͳሻ ‫כ‬
ሺܹଷଷ ൌ Ǥͷሻ ‫כ‬
‫ܥ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଵ ሻ
‫ܥ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ
൅ ሺܹଶଷ ൌ Ǥͷሻ ‫כ‬
൅
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ
‫ܥ‬ଵ ሺ‫ݔ‬ଶ ሻ
஼మ ሺ௫య ǡ௫మ ሻ‫כ‬஼మ ሺ௫య ǡ௫భ ሻ
஼భ ሺ௫య ሻ
(A.1)
݈݅݉ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ
௫భ ՜ஶ
ሺǤ ͷሻ ‫כ‬
஼భ ሺ௫భ ሻ‫כ‬஼భ ሺ௫మ ሻ‫כ‬஼మ ሺ௫మ ǡ௫య ሻ
஼భ ሺ௫మ ሻ
൅ ሺǤͷሻ ‫כ‬
஼మ ሺ௫మ ǡ௫య ሻ‫כ‬஼భ ሺ௫భ ሻ‫כ‬஼భ ሺ௫య ሻ
஼భ ሺ௫య ሻ
݈݅݉௫భ ՜ஶ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ
(A.2)
(A.3)
Similarly for ‫ݔ‬ଶ ՜ λ:
݈݅݉௫మ ՜ஶ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଷ ሻ
(A.4)
And when šଷ ՜ λ
݈݅݉௫య ՜ஶ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ
A.2
(A.5)
Considering the case: šଵ ՜ šଶ
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Appendix
݈݅݉௫భ ՜௫మ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ሺܹଵଷ ൌ ͲǤͷሻ ‫כ‬
ሺܹଷଷ ൌ Ͳሻ ‫כ‬
஼మ ሺ௫భ ǡ௫భ ሻ‫כ‬஼మ ሺ௫య ǡ௫మ ሻ
஼భ ሺ௫భ ሻ
൅ ሺܹଶଷ ൌ Ǥͷሻ ‫כ‬
஼మ ሺ௫భ ǡ௫భ ሻ‫כ‬஼మ ሺ௫మ ǡ௫య ሻ
஼భ ሺ௫భ ሻ
஼మ ሺ௫య ǡ௫భ ሻ‫כ‬஼మ ሺ௫య ǡ௫భ ሻ
஼భ ሺ௫య ሻ
൅
(A.6)
݈݅݉௫భ ՜௫మ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ‫ܥ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ
(A.7)
Similarly, we have:
݈݅݉௫మ ՜௫య ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ‫ܥ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଵ ሻ
(A.8)
݈݅݉௫య ՜௫భ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ‫ܥ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ
(A.9)
A.3
Now, consider the case: š୧ ՜ λሺ‹ ൌ ͳǡʹǡ͵ሻǣ
݈݅݉ ௫భ ՜ஶ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ܹଵଷ ‫כ‬
௫మ ՜ஶ
௫య ՜ஶ
ܹଷଷ ‫כ‬
A.4
஼భ ሺ௫భ ሻ‫כ‬஼భ ሺ௫మ ሻ‫כ‬஼భ ሺ௫మ ሻ‫כ‬஼భ ሺ௫య ሻ
ൌ
஼భ ሺ௫య ሻ
஼భ ሺ௫భ ሻ‫כ‬஼భ ሺ௫మ ሻ‫כ‬஼భ ሺ௫య ሻ‫כ‬஼భ ሺ௫భ ሻ
஼భ ሺ௫భ ሻ
൅ ܹଶଷ ‫כ‬
஼భ ሺ௫భ ሻ‫כ‬஼భ ሺ௫మ ሻ‫כ‬஼భ ሺ௫మ ሻ‫כ‬஼భ ሺ௫య ሻ
஼భ ሺ௫మ ሻ
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ
൅
(A.10)
Finally, let’s consider the case: š୧ ՜ š୨ ሺ‹ ൌ ͳǡʹǡ͵ሻƒ†ሺŒ ൌ ͳǡʹǡ͵ሻǣ
݈݅݉ோ௔՜଴ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ܹଵଷ ‫כ‬
஼మ ሺ௫భ ǡ௫భ ሻ‫כ‬஼మ ሺ௫భ ǡ௫భ ሻ
஼భ ሺ௫భ ሻ
൅ ܹଶଷ ‫כ‬
஼మ ሺ௫భ ǡ௫భ ሻ‫כ‬஼మ ሺ௫భ ǡ௫భ ሻ
஼భ ሺ௫భ ሻ
஼మ ሺ௫భ ǡ௫భ ሻ‫כ‬஼మ ሺ௫భ ǡ௫భ ሻ
஼భ ሺ௫భ ሻ
൅ ܹଷଷ ‫כ‬
(A.11)
And therefore, we have:
݈݅݉ ௫భ ՜௫మ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ
(A.12)
௫య ՜௫మ
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:
݈݅݉௫భ ՜௫మ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ܹଵଷ ‫כ‬
஼మ ሺ௫య ǡ௫మ ሻ‫כ‬஼మ ሺ௫య ǡ௫భ ሻ
஼భ ሺ௫య ሻ
஼మ ሺ௫భ ǡ௫మ ሻ‫כ‬஼మ ሺ௫య ǡ௫భ ሻ
஼భ ሺ௫భ ሻ
൅ ܹଶଷ ‫כ‬
ൌͲ
஼మ ሺ௫భ ǡ௫మ ሻ‫כ‬஼మ ሺ௫మ ǡ௫య ሻ
஼భ ሺ௫మ ሻ
൅ ܹଷଷ ‫כ‬
(A.13)
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Therefore we have:
݈݅݉௫భ ՜௫మ ‫ܥ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ ൌ Ͳ
(A.14)
and finally:
݈݅݉௫భ ՜௫మ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ Ͳ
(A.15)
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Appendix B: Verification of the Boundary Conditions for the Approximated Four-Point
Probability Function:
In this section, different limiting conditions (šଵ ՜ λǡ ǥ ሻare probed.
B.1
First, we consider the case: ‫ݔ‬ଵ ՜ λǣ
݈݅݉ ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ൌ
௫భ ՜ஶ
ሺܹଵସ ൌ Ͳሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
ܹଶସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
ܹଷସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
ସସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫כ‬
஼భ ሺ௫భ ሻ‫כ‬஼మ ሺ௫ర ǡ௫మ ሻ‫כ‬஼భ ሺ௫భ ሻ‫כ‬஼మ ሺ௫ర ǡ௫య ሻ‫כ‬஼య ሺ௫ర ǡ௫మ ǡ௫య ሻ
஼భ ሺ௫భ ሻ‫כ‬஼భ ሺ௫ర ሻ‫כ‬஼మ ሺ௫ర ǡ௫మ ሻ‫כ‬஼మ ሺ௫ర ǡ௫య ሻ
(B.1)
Using boundary conditions of Eq. (54 and 55),
We have:
ܹଵସ ൌ Ͳ
(B.2)
ܹଶସ ൅ ܹଷସ ൅ ܹସସ ൌ ͳ
(B.3)
By substituting weight functions and simplifying Eq. (B.1), we get:
݈݅݉௫భ ՜ஶ ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
(B.4)
Similarly, we have:
݈݅݉௫మ ՜ஶ ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ
(B.5)
݈݅݉௫య ՜ஶ ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
(B.6)
݈݅݉௫ర ՜ஶ ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ
(B.7)
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B.2
Considering the case: šଵ ՜ šଶ
݈݅݉ ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ൌ
௫భ ՜௫మ
ܹସସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫כ‬
ܹଵସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
ܹଶସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
ܹଷସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
஼య ሺ௫ర ǡ௫మ ǡ௫మ ሻ‫כ‬஼య ሺ௫ర ǡ௫మ ǡ௫య ሻ‫כ‬஼య ሺ௫ర ǡ௫మ ǡ௫య ሻ
஼మ ሺ௫ర ǡ௫మ ሻ‫כ‬஼మ ሺ௫ర ǡ௫మ ሻ‫כ‬஼మ ሺ௫ర ǡ௫య ሻ
݈݅݉௫భ ՜௫మ ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ=‫ܥ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
(B.8)
(B.9)
Similarly, we have:
݈݅݉௫మ ՜௫య ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ=‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
(B.10)
݈݅݉௫య ՜௫ర ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ=‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
(B.11)
݈݅݉௫ర ՜௫భ ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ=‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ
(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:
݈݅݉ ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ൌ
௫భ ՜௫మ
ଵସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ସ ሻ
ଶସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଵ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଵ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ
ଷସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫כ‬
‫ܥ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଷ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଵ ǡ ‫ݔ‬ସ ሻ
൅
‫ܥ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଶ ሺ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ
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ܹସସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫כ‬
஼య ሺ௫ర ǡ௫భ ǡ௫మ ሻ‫כ‬஼య ሺ௫ర ǡ௫భ ǡ௫య ሻ‫כ‬஼య ሺ௫ర ǡ௫మ ǡ௫య ሻ
஼మ ሺ௫ర ǡ௫భ ሻ‫כ‬஼మ ሺ௫ర ǡ௫మ ሻ‫כ‬஼మ ሺ௫ర ǡ௫య ሻ
(B.13)
By substituting three point correlations function using Eq. (18) in Eq. (39) and calculating limit,
we have:
݈݅݉௫భ ՜௫మ ‫ܥ‬ସ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ሻ ൌ ‫ܥ‬ଷ ሺ‫ݔ‬ସ ǡ ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ሻ
B.3
(B.14)
Now, consider the case: ‫ݔ‬௜ ՜ λሺ݅ ൌ ͳǡʹǡ͵ǡͶሻǣ
݈݅݉௫భ ՜ஶ ‫ܥ‬ସ ൫‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷǡ ‫ݔ‬ସ ൯ ൌ
௫మ ՜ஶ
௫య ՜ஶ
௫ర ՜ஶ
ܹଵସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ
൅ܹଶସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ
‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ
൅ܹଷସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ
‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ
‫ܥ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
൅ܹସସ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
‫כ‬
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ସ ሻ
ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଵ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଶ ሻ ‫ܥ כ‬ଵ ሺ‫ݔ‬ଷ ሻ‫ܥ‬ଵ ሺ‫ݔ‬ସ ሻ
(B.15)
B.4
Finally, let’s consider the case: ‫ݔ‬௜ ՜ ‫ݔ‬௝ ሺ݅ ൌ ͳǡʹǡ͵ǡͶሻܽ݊݀ሺ݆ ൌ ͳǡʹǡ͵ǡͶሻǣ
݈݅݉ோ௔՜଴ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ ݈݅݉௫భ ՜௫మ ‫ܥ‬ସ ൫‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷǡ ‫ݔ‬ସ ൯ ൌ ݈݅݉௫ర ՜௫మ ‫ܥ‬ଷ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ሻ ൌ
௫య ՜௫మ
௫ర ՜௫మ
݈݅݉௫ర ՜௫మ ‫ܥ‬ଶ ሺ‫ݔ‬ଶ ǡ ‫ݔ‬ସ ሻ ൌ ‫ܥ‬ଵ ሺ‫ݔ‬ଶ ሻ
௫య ՜௫మ
(B.16)
186
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
187
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
188
Majid BANIASSADI, University of Strasbourg, Strasbourg, 2011
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.