nirina j.t. santatriniaina

ANNÉE 2015

THÈSE / UNIVERSITÉ DE RENNES 1sous le sceau de l’Université Européenne de Bretagne

pour le grade deDOCTEUR DE L’UNIVERSITÉ DE RENNES 1

Mention : MécaniqueEcole doctorale MATISSE

présentée par

Nirina J.T. SANTATRINIAINApréparée à l’UMR 6625 CNRS - IRMAR

Institut de Recherche Mathématiques de Rennes-EPFL/LBO Laboratoire de Biomécanique Orthopédique de

Lausanne, Suisse

Thermomécanique

des milieux continus:modèles théoriques

et applications au

comportement de

l’hydrogel en ingénierie

biomédicale

Thèse soutenue à Rennesle 06 Octobre 2015devant le jury composé de :

Sophie LANGOUET-PRIGENTD.R à l’Inserm, HDR, Université de Rennes 1/Prési-denteJean-François GANGHOFFERProfesseur à l’Université de Lorraine / rapporteur

Salah NAILIProfesseur à l’Université de Paris-Créteil/rapporteur

Eric DARRIGRANDMcf., HDR à l’Université de Rennes 1/ examinateur

Dominique PIOLETTIProfesseur à l’EPF de Lausanne / co–directeur de thèse

Lalaonirina RAKOTOMANANAProfesseur à l’Université de Rennes 1/directeur de thèse

i

... à mes grands parents,

... à mes parents,

... à mes sœurs,

... à Koloina, merci de faire partie de ma vie!!!.

Remerciements

Ce travail de thèse s’est déroulé dans le cadre d’une collaboration entre le Laboratoire de Biomé-

canique Orthopédique de Lausanne et l’Institut de Recherche Mathématiques de Rennes. Les deux

premiers chapitres sont les fruits de collaboration avec le CEA/ LETI/DTSI/SSURF-Grenoble,

département technologies de silicium et équipe de contamination.

Je tiens à remercier tous ceux qui ont contribué à l’aboutissement de ce travail.

Je voudrais remercier tout particulièrement Lalaonirina Rakotomanana qui m’a dirigé tout au

long de ces trois années de thèse. Il a toujours été disponible avec son petit sourire, à l’écoute de

mes nombreuses questions, et s’est toujours intéressé à l’avancée de mes travaux. J’ai beaucoup

apprécié sa confiance, sa disponibilité et son respect sans faille des délais serrés de relecture des

documents que je lui ai adressés.

De même, je suis particulièrement reconnaissant à monsieur Dominique Pioletti d’avoir co-

encadré ce travail de thèse. Il m’a tout d’abord permis d’intégrer l’équipe calorimétrie en me

proposant un sujet très intéressant sur la calorimétrie et les hydrogels et m’a laissé la liberté de le

réorienter au cours du déroulement de ma thèse. C’est également grâce à son laboratoire LBO que

j’ai eu la chance de travailler avec ses équipes de recherche, ce qui c’est avéré une expérience très

enrichissante sans oublier les financements des déplacements pendant la thèse.

Je remercie "l’équipe contamination" du CEA/LETI de Grenoble: Directeur de Laboratoire

Chystel Deguet et Véronique Carron de m’avoir accuelli dans son laboratoire, Carlos Beitia et

Hervé Fontaine de m’avoir accuelli dans son équipe qui traite la contamination, Agnès Royer et

Alain Presenti de m’avoir formé sur la contamination en général en industrie microelectronique,

je remercie également Thi Quynh Nguyen pour la partie expérimentale et les caractérisations, je

remercie egalement Jonathan Deseure pour ses aides précisuses pour la simulation numériques. Je

n’oublie pas de remercier toute l’équipe "contamination" qui m’a permis de vivre une ambiance

chaleureuse pendant la pause café et le repas: Sylviane Cetre, Ailhas Chrystelle, Jean-Michel

Pedini, Paola Gonzales, Karim Ykach, Thierry Lardin, Ludovic Couture.

Je voudrais remercier meussieurs Salah Naili, Jean-François Ganghoffer et Philippe Buchler

d’avoir accepté de relire cette thèse et d’en être rapporteurs. La version finale de cette thèse a

iii

iv REMERCIEMENTS

bénéficié de leur lecture très attentive et de leurs remarques précieuses.

Je tiens à remercier les membres du jury pour avoir accepté de participer à la soutenance de

cette thèse en commençant par le président pour avoir accepté de présider le jury et je remercie

également tous les membres du jury d’avoir accepté d’assister à la présentation de ce travail.

Un grand remerciement à Eric Darrigrand, Sophie Langouët-Prigent et Mariko Dunseath-Terao

d’avoir accepté d’examiner cette thèse, merci pour l’intérêt que vous avez porté à mes travaux de

recherche.

Je remercie à cette occasion les différentes personnes du Laboratoire de Biomécanique Or-

thopédique de Lausanne en particulier Virginie Kokocinski qui organisait efficacement l’accueil à

Lausanne pendant mon déplacement. Ensuite, je remercie tous les autres membres du Laboratoire

de m’avoir accueilli. Je remercie Arne Vogel et Mohamadreza Nassajian Moghadam pour le tra-

vail réalisé ensemble sur la partie expérimentale de cette thèse. Je remercie aussi Philippe Abdel

Sayed, Sandra Jaccoud, Nasrollahzadeh Mamaghani Naser, Adeliya Latypova et Christoph Anselm

Engelhardt pour l’ambiance chaleureuse pendant la pause café, le repas et le bar.

J’adresse mes remerciements à Monsieur Bachir Bekka Directeur du laboratoire IRMAR et à

Monsieur Roger Lewandowski responsable de l’équipe mécanique pour m’avoir accueilli dans le

laboratoire et intégrer dans leur équipe.

Je remercie Benjamin Boutin, Fabrice Mahé, Eric Darrigrand et Nicolas Crouseilles de l’équipe

d’analyse numérique de l’Institut de Recherche Mathématique de Rennes, Fulgence Razafimahery

de l’équipe mécanique pour les échanges que nous avons eue et je suis reconnaissant pour le temps

que vous avez consacré pour moi.

Je remercie également la Fondation Rennes 1 qui était la fondation des fondations de m’avoir

accompagné pour la venue en France et tout au long de mes cursus en particuliers Sophie Langouët-

Prigent, Nolwenn Saget et Johanne Beauclair.

Je tiens à remercier aussi les responsables administratives pour leur travail efficace pour le

bon déroulement des missions et les autres démarches administratives, je cite quelques noms;

Elodie Cottrel, Anne-Joelle Chauvin, Marie-Aude Verger, Chantal Halet, Carole Wosiak, Virginie

Kokocinski. Je remercie Patrick Perez et Olivier Garo pour les assistances techniques sans faille.

Pour les déplacements fréquents, je remercie l’Ecole Doctorale MATISSE, UEB/CDI, Région de

Bretagne pour les aides à la mobilité sortante.

Pour finir je tiens à remercier à tout ce qui ont contribué de près ou de loin à l’aboutissement

de ma thèse spécialement Guillaume et Thomas de m’avoir accueilli au bureau 213 au tout début

de la thèse, Kodjo et Elise qui m’ont accueilli au bureau 434 (les séances de musculations et mots

fléchés), tous mes amis en particulier Alexandre, Romain, Christophe, Loubna, Celik, Richard,

v

Hasina, bienvenue et bon courage à Maria (434) et tous les doctorants de l’IRMAR et du LBO

pour les bons moments passés tout au long de ces années.

En dernier lieu, j’aimerais adresser mes remerciements les plus chaleureux à mes parents et mes

sœurs pour les soutiens et les encouragements pendant les coups de fil magiques tout au long de

ce travail de thèse notamment dans les moments les plus difficiles.

vi REMERCIEMENTS

Résumé

Résumé – Dans la première partie on propose un outil mathématique pour traiter les condi-

tions aux limites dynamiques d’un problème couplé d’EDP. La simulation avec des conditions aux

limites dynamiques nécessite quelques fois une condition de "switch" en temps des conditions aux

limites de Dirichlet en Neumann. La méthode numérique (StDN) a été validée avec des mesures

expérimentales pour le cas de la contamination croisée en industrie micro-électronique. Cet outil

sera utilisé par la suite pour simuler le phénomène de « self-heating » dans les polymères et les

hydrogels sous sollicitations dynamiques. Dans la deuxième partie, on s’intéresse à la modélisation

du phénomène de self-heating dans les polymères, les hydrogels et les tissus biologiques. D’abord,

nous nous sommes focalisés sur la modélisation de la loi constitutive de l’hydrogel de type HEMA-

EGDMA. Nous avons utilisé la théorie des invariants polynomiaux pour définir la loi constitutive du

matériau. Ensuite, nous avons mis en place un modèle théorique en thermomécanique couplée d’un

milieu continu classique pour analyser la production de chaleur dans ce matériau. Deux potentiels

thermodynamiques ont été proposés et identifiés avec les mesures expérimentales. Une nouvelle

forme d’équation du mouvement non-linéaire et couplée a été obtenue (un système d’équations aux

dérivées partielles parabolique et hyperbolique non-linéaire couplé avec des conditions aux limites

dynamiques). Dans la troisième partie, une méthode numérique des équations thermomécaniques

(couplage parabolique-hyperbolique) pour les modèles a été utilisée. Cette étape nous a permis, en-

tre autres, de résoudre ce système couplé. La méthode est basée sur la méthode des éléments finis.

Divers résultats expérimentaux obtenus sur ce phénomène de self-heating sont présentés dans ce

travail suivi d’une étude de corrélations des résultats théoriques et expérimentaux. Dans la dernière

partie de ce travail, ces divers résultats sont repris et leurs conséquences sur la modélisation du

comportement de l’hydrogel naturel utilisé dans le domaine biomédical sont discutées.

Mots clés – Thermomécanique, self-heating, hydrogel, tissus biologiques, EDP, couplage

parabolique-hyperbolique, calorimétrie, caractérisations, méthodes numériques.

vii

viii RÉSUMÉ

Abstract

Abstract –In the first part, we propose a mathematical tool for treating the dynamic bound-

ary conditions. The simulation within dynamic boundary condition requires sometimes “switch”

condition in time of the Dirichlet to Neumann boundary condition (StDN). We propose a nu-

merical method validated with experimental measurements for the case of cross-contamination in

microelectronics industry. This tool will be used to compute self-heating in the polymers and

hydrogels under dynamic loading. In the second part we focus on modeling the self-heating phe-

nomenon in polymers, hydrogels and biological tissues. We develop constitutive law of the hydrogel

type HEMA-EGDMA, focusing on the heat effects (dissipation) in this material. Then we set up

a theoretical model of coupled thermo-mechanical classic continuum for a better understanding of

the heat production in this media. We use polynomial invariants theory to define the constitutive

law of the media. Two original thermodynamic potentials are proposed. Original non-linear and

coupled governing equations were obtained and identified with the experimental measurements

(non-linear parabolic-hyperbolic system with the dynamic boundary condition). In the third part,

numerical methods were used to solve thermo-mechanical formalism for the model. This step

deals with a numerical method of a coupled partial differential equation system of the self-heating

(parabolic-hyperbolic coupling). Then, is step allows us, among other things, to propose an ap-

propriate numerical methods to solve this system. The numerical method is based on the finite

element methods. Numerous experimental results on the self-heating phenomenon are presented in

this work together with correlations studies between the theoretical and experimental results. In

the last part of the thesis, these various results will be presented and their impact on the modeling

of the behavior of the natural hydrogel used in the biomedical field will be discussed.

Keywords – Thermomechanics, self-heating, hydrogels, biological tissues, PDEs, parabolic-

hyperbolic coupling, characterizations, numerical methods.

ix

x ABSTRACT

Notations

To start, let us introduce some table of global notations for introducing notations, differential

operators, writing convention, physical quantities, acronyms and used contractions which will be

used in this work.

Remark 0.1. The Einstein notation of sum over repeated indices is used throughout this work.

Partial derivative of the quantity T with respect to the space components are denoted by ∂T/∂Xik :=

T,ik .

Writing Convention / Convention d’écriture

a : scalar / scalaire

a : vector / vecteur (a)i = ai, i := 1 · · ·n

A : second order tensor/ tenseur du second ordre (A)ij = Aij , i, j := 1 · · ·n

A : fourth order tensor/ tenseur du quatrième ordre (A)ijkl = Aijkl, i, j, k, l := 1 · · ·n

I : second order identity tensor/ tenseur identité du second ordre Iij = δij ,

δij = 0 if i 6= j, δij = 1 if i = j,

I : fourth order identity tensor/ tenseur identité du quatrième ordre Iijkl = δijδkl

Operators / Operateurs

∇ : gradient operator / gradient, (∇A)i1,··· ,ik,ik+1= (Ai1,··· ,ik),ik+1

∇.(· · · ) : Eulerian divergence operator / divergence Eulerienne, (∇.A)i1,··· ,ik = (Ai1,··· ,ik,ik+1),ik+1

Div(· · · ) : Lagrangian divergence / divergence Langrangienne, (DivA)i1,··· ,ik = (Ai1,··· ,ik,ik+1),ik+1

X : differentiation of X with respect time variable / dérivation temporelle de X

· : single contraction / simple contraction, (A ·B)i1,··· ,ih,j1,··· ,jk = Ai1,··· ,ih+1Bih+1,j1··· ,jk

× : vector product / produit vectoriel

(· · · )(· · · ) : scalar product / produit scalaire AB = Ai1,··· ,ikBi1,··· ,ik

⊗ : tensor product / produit tensoriel, (u⊗ v)ij = uivj , (A⊗B)ijkl = AijBkl

(· · · )T : transpose of (...) / transposé de (...), (Aij)T = Aji

xi

xii NOTATIONS

: : double contractions / double contractions,

(A : B)i1,··· ,ih,j1,··· ,jk = Ai1,··· ,ih+1,ih+2Bih+2,ih+1,j1··· ,jk

det(· · · ) : determinant of (· · · ) / déterminant de (· · · )

tr(· · · ) : or I : (· · · ) trace of (· · · ), I : A = Aii =∑n

i=1Aii

Global notations / Notations Globales

B : reference configuration / configuration de référence

S : current configuration / configuration actuelle

∂Bi : boundary of B / bord du milieu B

Γi : parts of ∂Bi / une partie de ∂Bi

n : normal unit outward vector/ vecteur unitaire normal sortant à Γi

x : Eulerian position of the material in point S / position du point material dans S

X : Lagrangian position of the material in point B / position du point material dans B

Md : set of second order square matrix (d× d) / matrice carré d’ordre deux

O : full orthogonal group / groupe orthogonal

g : symmetry group / groupe des symétries

d : space dimension / dimension de l’espace

Rd : d-dimensional Euclidean space

ϕ : piecewise C1 diffeomorphisms

∆t : time step / pas de temps

Notations in chapters 2, 3 / Notations dans les chapitres 2 , 3

∇.(· · · ) : divergence operator / opérateur divergence

∇(· · · ) : gradient operator / opérateur gradient

t, ti : time and the characteristic time for the process i / temps- caractéristiques i

h0 : Henry constant / constante d’Henry

u : contaminant velocity / vitesse du contaminant

q1 : source in the polymer s / source dans le polymer s

q2 : source in the contaminant g / source dans le contaminant g

Ds(T ) : non-isothermal diffusion coefficient in polymer s

Dg(T ) : non-isothermal diffusion coefficient in the contaminant g

T : temperature / température

D0s : reference diffusion coefficient in the polymer

xiii

D0g : reference diffusion coefficient in the FOUP’s atmosphere

Cs : concentration in polymer / concentration dans le polymer

Cg : concentration in internal FOUP’s atmosphere / concentration dans l’atmosphère

δCs : test function of the concentration in polymer / fonction test

δCg : test function of the concentration in internal FOUP’s atmosphere / fonction test

δT : test function of the temperature / fonction test pour la température

C0 : concentration on the wafer surface / concentration sur la surface du wafer

r : heat source / source de chaleur

Ng0 : inlet concentration flux / flux de concentration

H(t− ε) : Heaviside function with delay ε, 0 if t < ε, 1 if t > ε / fonction Heaviside

(Ωs) : polymer subdomain / sous-domaine polymer

(Ωg) : contaminant subdomain / sous-domaine contaminant

(Γi) : denote the boundary of the domain i / bord du domaine i

ρs : density of the polymer / densité du polymer

cs : specific heat of the polymer / chaleur spécifique du polymer

κs : heat conductivity constant of the polymer / conductivité thermique du polymer

(*) : variable with the temperature effect using Arrhenius law

E : activation energy / énergie d’activation

(· · · )c or (· · · )c : notation for the contamination process / notation pour la phase de contamination

(· · · )d or (· · · )d : notation for the decontamination process / notation pour la phase de nettoyage

(· · · )p or (· · · )p : notation for the purge process / notation pour la phase de purge

Notations in chapter 4 / Notations dans le chapitre 4

Notations for kinematics / Notations pour la cinématique

u : displacement vector / vecteur déplacement

v : velocity vector / vecteur vitesse

θ : temperature in B / température dans B

δu : virtual displacement vector / déplacement virtuel

δv : virtual velocity vector / vitesse virtuelle

δθ : virtual temperature / température virtuelle

I : identity matrix / matrice identité

Ii : invariant of the strain tensor / invariant (i = 1, 2, 3)

Jj : mixed invariant of the strain tensor / invariant mixte

xiv NOTATIONS

Measures of stress-strain / Mesures des contraintes et déformation

σ : Cauchy stress tensor / contrainte de Cauchy

σe : elastic part of Cauchy stress tensor / contrainte de Cauchy

σv : viscous part of Cauchy stress tensor / contrainte de Cauchy

E : Green-Lagrange strain tensor / tenseur de déformation de Green-Lagrange

C : Cauchy-Green strain tensor / tenseur de déformation de Cauchy-Green

P : first Piola-Kirchhoff’ stress tensor / premier tenseur de Piola-Kirchhoff

S : second Piola-Kirchhoff’ stress tensor/ second tenseur de Piola-Kirchhoff

Se : elastic part of the second Piola-Kirchhoff’ stress tensor

Sv : viscous part of the second Piola-Kirchhoff’ stress tensor

F : deformation gradient / gradient de déformation

J : local variation of volume / variation locale de volume, J = det(F)

Physical quantities / Quantités physiques

λ, µ : Lamé’ constants / constantes de Lamé

η : viscosity coefficient / coefficient de viscosité

θ0 : initial temperature in B / température initiale de B

κ : thermal conductivity coefficient of B / conductivity thermique de B

α : thermal expansion coefficient of B / coefficient de dilatation de B

cv : specific heat coefficient of B / chaleur spécifique de B

s : entropy density / entropie

e : internal energy / énergie interne

ψ : Helmholtz’ free energy / energie libre d’Helmohltz

χ : dissipation potential / potentiel de dissipation

ρ : mass density of B / masse volumique de B

ρb : Eulerian body force / force volumique Eulerienne

ρB : Lagrangian body force / force volumique Lagrangienne

Q : heat flux / flux de chaleur

ρr : Eulerian heat source / source de chaleur Eulerienne

ρR : Lagrangian heat source / source de chaleur Lagrangienne

φ : cross-link density/densité de réticulation

λ′, µ′ : viscosity coefficients / coefficients de viscosité

η1, η2 : viscosity coefficients / coefficients de viscosité

xv

Global acronyms / Acronymes globaux

B.C : Boundary condition / condition aux limites

I.C : Initial condition / condition initiale

REV : Representative Elementary Volume / Volume Elémentaire Représentatif

StDN : Switch Dirchlet-Neumann / condition de switch en temps Dirichlet-Neumann

Acronyms in chapters 2, 3 / Acronymes dans les chapitres 2,

3

AMCs : Airborne Molecular Contamination

FOUP : Frount Unified Pods

AFM : Atomic force Microscopy / Microscope à Force Atomique

Acronyms in chapter 4 / Acronymes dans le chapitre 4

MCDA : micro-calorimètre à déformation adiabatique / adiadatic micro-calorimeter

MCDI : micro-calorimètre à déformation isotherme / isothermal micro-calorimeter

HEMA : hydroexyethyl métacrylate / hydroexyethyl metacrylate

EGDMA : ethylene glycol di-métacrylate / ethylene glycol di-metacrylate

xvi NOTATIONS

Contents

Remerciements iii

Résumé vii

Abstract ix

Notations xi

1 Introduction générale 1

1.1 Hydrogel à haute dissipation et ses applications . . . . . . . . . . . . . . . . . . . . 5

1.2 Modèle thermomécanique et "self-heating" dans les tissus biologiques et hydrogel . 6

1.3 Couplage parabolique-hyperbolique . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Organisation du travail de thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Contribution de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Plan de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Switch conditions for coupled system of PDEs. 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Problem statements and model settings . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Experimental measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.2 Cross-contamination model . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Switch condition on time StDN of the boundary conditions . . . . . . . . . . . . . 25

2.4 Identification of the physical constants . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Model of contamination process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Model of purging and outgassing process . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Model of decontamination and cleaning process . . . . . . . . . . . . . . . . . . . . 31

2.8 Finite element approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.9 Computation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.10 Main results, findings and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 44

xvii

xviii CONTENTS

2.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Dynamic boundary conditions for coupled system of PDEs. 57

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2 Physical problem and experiment procedure . . . . . . . . . . . . . . . . . . . . . . 60

3.3 Mathematical settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4 Mathematical model with temperature effect . . . . . . . . . . . . . . . . . . . . . 64

3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4.2 Mathematical model using Arrhenius law . . . . . . . . . . . . . . . . . . . 65

3.4.3 Mathematical model using heat equation . . . . . . . . . . . . . . . . . . . 71

3.5 Applications of the model in industrial processes . . . . . . . . . . . . . . . . . . . 78

3.5.1 Heat effect on contamination process . . . . . . . . . . . . . . . . . . . . . . 78

3.5.2 Heat effect on outgassing process . . . . . . . . . . . . . . . . . . . . . . . . 79

3.5.3 Heat effect on decontamination process . . . . . . . . . . . . . . . . . . . . 80

3.6 Computation order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.7 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4 Experimental identification of self-heating in HEMA-EGDMA. 95

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 Microcalorimetric test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4 2D and 1D approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.5 Identification of the model parameters . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.5.1 Cost functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.5.2 Computation, splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.6 Numerical approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.6.1 Governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.6.2 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.6.3 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.6.4 Numerical approximations for local self-heating . . . . . . . . . . . . . . . . 114

4.6.5 Numerical approximations for non-local self-heating . . . . . . . . . . . . . 115

4.7 Experimental and numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.7.1 Influence of the cross-link density on the self-heating . . . . . . . . . . . . 118

4.7.2 Dissipation in function of frequency and cross-link density . . . . . . . . . . 119

4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

CONTENTS xix

4.9 Nonlinear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.9.1 Thermodynamic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.9.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5 Conclusion générale 125

Perspectives 129

Appendix A

Continuum thermomechanics 131

A-1 Strain and stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A-2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

A-3 Constitutive laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

xx CONTENTS

Chapter 1

Introduction générale

Le développement de nouveaux matériaux pour les traitements thérapeutiques (implant, matériau

de remplacement, pansement) dans le domaine biomédical exige des nouvelles méthodes de car-

actérisation de ces matériaux pour optimiser leurs propriétés, leurs tenues en service et leurs bio-

compatibilités avec les tissus vivants [1]. De plus, la rapidité de l’évolution technologique dans la

fabrication des matériaux utilisés dans le domaine de la biomécanique orthopédique implique des

méthodes rigoureuses pour assurer les bons fonctionnements de ces implants [4], [5]. D’une part,

les matériaux utilisés sont soumis à plusieurs sollicitations physico-chimiques répétées (sollicita-

tion mécanique, thermique, chimique) durant leur utilisation [1], [6], [8], [23]. D’autre part, ces

sollicitations induisent des changements de propriétés du matériau qui pourraient être fatals pour

le patient (le matériau, due à la fatigue, devient toxique ou sur le plan biomédical, incompatible)

[13], [14], [15], [16]. Enfin, le développement de nouveaux materiaux pourrait être bénéfique pour

d’autres exploitations bien ciblées tel le largage des médicaments actionné par le changement de

température [7] par exemple.

Cependant, la réponse aux sollicitations de chaque matériau a une forte liaison avec ses pro-

priétés et ses constituants, le choix de ces matériaux est donc basé sur ses propriétés et sa bio-

compatibilité [7]. L’application des sollicitations mécaniques répétées sur les tissus biologiques

(mouvement du sujet, poids du corps) et les matériaux polymères engendre un phénomène d’auto-

échauffement couramment appelé "self-heating" [7], [8]. Le "self-heating", dans les matériaux

solides, appelé aussi "frottement interne" est la propriété que possèdent ces matériaux soumis à

des contraintes mécaniques cycliques, d’absorber de l’énergie en transformant l’énergie mécanique

en chaleur [7], [8]. Il est caractérisé à l’échelle microscopique par le fait que les éléments consti-

tutifs du matériau occupent initialement des positions relatives moyennes bien définies [9], [10].

Ces éléments entrent en mouvement pendant la sollicitation (mouvement, rupture des chaînes con-

stituants le matériau, réarrangement structural) à cause d’une déformation irréversible induisant

1

2 CHAPTER 1. INTRODUCTION GÉNÉRALE

ainsi une dissipation d’énergie [9], [12]. Ce phénomène se manifeste aussi bien en petites qu’en

grandes déformations dans le cas irréversible mais également dans le domaine élastique si elles sont

accompagnées d’une dissipation visqueuse [7], [8].

Dans cette optique, l’étude du phénomène de « self-heating » est un domaine de recherche

à la fois ouvert et complexe car il demande des connaissances issues de domaines distincts de la

physique, mécanique et thermique et même de la biologie. Bien qu’un nombre considérable de

publications aient déjà traité ce sujet, les interprétations ne sont toutefois pas si bien définies dans

les tissus biologiques et les matériaux polymères. Sans prétendre à l’exhaustivité, on peut citer

un certain nombre de publications dont [7], [8], [28], [32], [44] qui traitent ce sujet. Cependant,

des expériences sur les phénomènes de "self-heating" dans les polymères et composites se sont

beaucoup développées au cours de ces dernières années [7], [8], [44].

Dans cette étude, nous nous focalisons sur l’impact de ce phénomène dans le domaine de la

biomécanique, en particulier dans les tissus biologiques et les hydrogels pour des applications

cliniques. Dans le cas des tissus biologiques tels que le tendon et le cartilage, une augmentation

de la température est observée sous une activité physique intense e.g [8]. Cette augmentation est

favorisée dans les tissus avasculaires car le sang ne transporte pas et ne repartit pas la chaleur

générée par la dissipation visqueuse dans ces tissus e.g [7]. De plus, cartilages articulaires et

tendons sont constamment soumis à des sollicitations cycliques e.g [8].

En effet, la propriété viscoélastique du cartilage (du genou par exemple), sous sollicitation

mécanique cyclique (poids du corps lorsque le sujet est en mouvement) induit une augmentation

locale de la température au sein du matériau [7], [8], [15]. Au fil du temps, ces tissus peuvent subir

des usures (érosion, écaillage, etc..) qui provoquent des traumatismes irréversibles et aussi dû à

l’avascularisation des ces tissus dont la capacité de régénération cellulaire est très faible [17], [18],

[19]. Dans cette thèse qui est focalisée sur les matériaux de remplacement, les propriétés physiques,

mécaniques et thermiques peuvent être reproduites avec les hydrogels en agissant sur la réticulation

du réseau constituant ces derniers [29],[30], [34], [35], [37], [39].

Le “self-heating” est un phénomène naturel, classique, couramment observé qui se développe au

sein d’un volume de matière solide viscoélastique soumis à des sollicitations cycliques extérieures

donc à des déformations de natures dynamiques. Ces sollicitations extérieures donnent naissance

à un phénomène de dissipation ou de frottement interne dans le volume. Ensuite, cette dissipation

visqueuse donne naissance, à son tour, une source de chaleur interne dans le volume du matériau.

Le “self-heating” est donc un effet causé par l’énergie de dissipation visqueuse (le comportement

viscoélastique induit une énergie de dissipation qui se transforme en chaleur) dans le volume de

matériau soumis à des charges harmoniques. Cependant, cette production de chaleur est fortement

3

liée aux types de matériaux (polymères, métaux), aux propriétés du matériau (cristallin, non

cristallin) sollicité et à la nature même des sollicitations subies. Les phénomènes de dissipation

peuvent être de natures diverses, complexes et pourraient être dus aux réarrangements structurels

ou à des ruptures de liaisons (physiques ou chimiques) des chaînes polymériques.

Face à cette pluralité de connaissance requise, un point de vue clairement défini doit être

envisagé pour entamer l’étude du phénomène "self-heating" dans les matériaux polymères et les

tissus biologiques. Bien que quelques travaux traitent avec des méthodes de caractérisations (en

particulier par la méthode calorimétrique) la quantification de cette production de chaleur dans les

matériaux polymères e.g [7], [8], [20], [16], [44], l’analyse de ce phénomène est loin d’être achevée

et on n’a pas à disposition un modèle théorique permettant de bien décrire ce phénomène.

Ainsi, un modèle théorique pertinent devrait permettre de comprendre et de mieux exploiter ou

éviter ce phénomène. D’une part, ces outils théoriques permettent d’accéder à d’autres informations

qu’on ne peut ni observer facilement ni quantifier expérimentalement, ensuite d’analyser l’influence

de chaque phénomène mis en jeu. D’autre part, ils servent à identifier et à quantifier les paramètres

jouant un rôle très important dans la production de chaleur (augmentation ou diminution de la

production).

Dans cette optique, on peut exploiter la connaissance de ces paramètres soit pour augmenter

l’effet qui favorise la production de chaleur dans un but particulier en agissant sur les paramètres

favorisants (influençants) la production de chaleur. Avec cette approche, on peut par exemple

retarder l’effet de relargage des médicaments de manière contrôlée sous l’effet d’une contrainte

mécanique dynamique comme dans [7], [8], [21]. On pourrait au contraire diminuer les effets du

self-heating afin d’éviter le “thermal-failure” (endommagement provoqué par l’augmentation de la

température induite par la contrainte mécanique dynamique) dans l’échantillon [22], [19]. Pour

certains matériaux, la variation de température peut influencer ou engendrer des défauts dans le

matériau (changement de caractéristiques, de propriétés, rupture des chaînes) [15].

Pour déterminer les paramètres dans la loi constitutive via les potentiels thermodynamiques,

il faut mesurer simultanément l’énergie mécanique et thermique mises en jeu pendant l’essai. En

général, un microcalorimètre à déformation est le type d’appareil que l’on utilise lorsqu’on souhaite

avoir une mesure fine et sensible d’un échange ou de production de chaleur dans un échantillon

sollicité avec des charges mécaniques dynamiques. On dit "à déformation" car le calorimètre

est constitué d’une partie mécanique qui sert à augmenter la température de l’échantillon par

déformations cycliques à fréquence variable. L’application de cette déformation dynamique est

l’un des moyens les plus efficaces pour augmenter la température dans un échantillon, dans le but

d’étudier l’effet de variation de température et ses transformations. La partie mécanique pilotée


par ordinateur permet d’appliquer les contraintes mécaniques cycliques dans l’échantillon. Le

calorimètre, permet de mesurer la variation de température provoquée par l’excitation mécanique

(mesure de la production de chaleur sous forme de température). Avec ce système on peut mesurer

simultanément l’énergie mécanique et la production de chaleur dans l’échantillon par l’intermédiaire

de la mesure de la température.

Il existe deux types de calorimètre à déformation, microcalorimètre isotherme et microcalorimètre

adiabatique. Le choix de microcalorimètre dépend du type d’essais et des domaines d’applications

pour la production de chaleur dans l’échantillon. Dans notre cas, nous utiliserons le microcalorimètre

à déformation adiabatique. Ce microcalorimètre nous permet de mesurer la variation de la tem-

pérature dans l’échantillon durant un essai.

En parallèle, une autre méthode basée sur la modélisation thermomécanique semble très intéres-

sante à développer pour apporter un éclairage sur ce phénomène de "self-heating". Les difficultés

liées à cette modélisation se situent principalement au stade de la définition des lois constitutives

du matériau (polymères, hydrogels, cartilage).

Pour mieux comprendre ce phénomène de dissipation conduisant au "self-heating", nous dévelop-

pons les lois constitutives thermomécanique du milieu étudié [10], [12], [57]-[60], [61]-[62], [63]-[64].

Ces modèles doivent vérifier les principes fondamentaux de la thermodynamique e.g [10], [12],

[76]. En général, le phénomène de dissipation est multi-physique faisant intervenir une interaction

entre la thermique et la mécanique [10], [12]. Cependant, comme les matériaux polymères sont

des matériaux très complexes, nous utiliserons dans un premiers temps, la théorie classique des

matériaux standards généralisés e.g [10], [12]. Les propriétés thermomécaniques de ces matéri-

aux peuvent varier en fonction du type d’élaboration (polymérisation), de la composition et des

domaines d’utilisations. Ainsi, les possibilités de réorientations et réarrangements de la structure

interne dans le milieu par la relaxation des segments de chaîne définissent directement les propriétés

thermomécaniques du matériau.

Ces processus de fabrication engendrent deux types de milieux notamment les milieux à effets

locaux et les milieux à effets non-locaux. Dans ce dernier cas, il serait possible d’introduire dans

un deuxième temps un modèle théorique avec les effets non-locaux en considérant que le milieu

est un milieu à gradient (ce type de milieu représente surtout le cas des hydrogels). Cependant,

on peut aussi introduire la théorie des matériaux faiblement continus e.g [125], [126], qui est une

théorie pertinente pour traiter des milieux à gradient avec ou sans endommagement e.g [132], [134],

[137], [138]. Cette dernière méthode permet aussi de modéliser et d’évaluer les "thermal-failure"

dans les matériaux thermosensible (ce sont des défauts et changements de propriétés induites par

le changement de température dans le milieu).

1.1. HYDROGEL À HAUTE DISSIPATION ET SES APPLICATIONS 5

En introduisant les lois de comportement constitutives dans les équations de conservation

de quantité de mouvement, de masse et d’énergie, les équations mathématiques gouvernant ce

phénomène thermomécanique constitue un système d’équations aux dérivées partielles non linéaires

couplées. Ce système est constitué d’une équation aux dérivées partielles hyperbolique pour le mod-

èle de propagation d’ondes mécaniques et d’une équation aux dérivées partielles parabolique pour

la propagation de chaleur e.g [10], [12], [76]. Ces équations sont couplées par des termes sources no-

tamment la production de chaleur dans l’équation parabolique dues aux chargements mécaniques.

Ensuite, elles sont couplées par une source de contrainte thermique dans l’équation hyperbolique

due à la variation de la température dans l’échantillon. Enfin, elles sont couplées implicitement

avec les équations d’évolution relatives aux autres variables internes. Le système obtenu comporte

aussi des conditions aux limites dynamiques. On fait appel aux méthodes numériques pour ré-

soudre le problème. La complexité de ce système repose en plus sur la forme générale des lois

constitutives du matériau étudié. Par conséquent, plus la loi de comportement est compliquée plus

on se retrouve avec un système d’équations difficile à résoudre.

Dans cette optique, peu de littérature traite le couplage parabolique/hyperbolique pour un

modèle physiquement admissible (cf. section 1.3) e.g [167], [170], [174]. Quelques articles traitent du

problème de couplage mais avec des termes qu’on ne peut pas obtenir en appliquant les équations de

conservation c’est-à-dire que le modèle mathématique est un modèle non physique ou un problème

qui n’est pas admissible physiquement (cf. section 1.3). Par conséquent, nous devons développer

une méthode originale de résolution numérique du problème.

Avant de présenter la contribution et le plan de la thèse, nous exposons une brève étude bib-

liographique qui s’articule en trois paragraphes. Dans le premier paragraphe, nous présentons les

hydrogels et ses applications dans le domaine biomédical (cf. section 1.1). Ensuite, nous nous

focalisons sur le self-heating pour les tissus biologiques et hydrogels (cf. section 1.2). Le dernier

paragraphe, traite la mise en œuvre numérique du couplage parabolique-hyperbolique (cf. section

1.3).

1.1 Hydrogel à haute dissipation et ses applications

Certaines propriétés physiques, mécaniques et thermiques de quelques tissus biologiques avascular-

isés tels que le cartilage (du genou par exemple) peuvent être reproduites avec les hydrogels [21],

[24], [33], [39]. Elles peuvent être obtenues en agissant sur la densité de réticulation, leur consti-

tuant et la polymérisation du réseau constituant ces matériaux, [7], [8], [29], [41]. En plus, de ces

propriétés physico-chimiques reproduites, certains hydrogels sont biocompatibles et biodegradables

[25], [27], [29], [30], [33], [34], [35], [51].


Un hydrogel est un réseau constitué de chaînes de polymères hydrophiles interconnectées et que

ce réseau est gonflé par l’eau [24], [28], [29]. On distingue deux types d’hydrogels: les hydrogels

chimiques (l’interconnexion du réseau se fait par des liaisons covalentes) et les hydrogels physiques

(l’interconnexion du réseau se fait par les interactions de Van der Walls) [24], [29]. Les hydrogels

peuvent êtres obtenus par polymérisation avec plusieurs réticulations [30]. De plus, il y a plusieurs

type de réticulation d’hydrogels selon l’utilisations et l’applications [30], [36]. Ainsi, chaque rétic-

ulation et ses pourcentages en solvant définissent leur fonction, leurs propriétés physico-chimiques

et leur domaine d’application en fonction des besoins [35], [37].

En effet, il est possible de changer en fonction des besoins la densité de réticulation de l’hydrogel

aussi de le mélanger avec d’autres types de réticulation pour avoir une fonction complexe, [39], [38],

[40]. Quelques auteurs proposent également différentes méthodes de polymérisation de l’hydrogel,

pour une utilisation bien ciblée [41], [42], [43], [44]. Voici quelques exemples d’utilisation de

l’hydrogels dans le domaine biomédical; utilisation comme un gel de substitution injectable dans les

tissus osseux [45], utilisation comme biosenseur [46], utilisation comme membranes artificielles [47],

utilisation comme organe artificiel [48], utilisation comme outils d’administrations des médicaments

[49], [50], utilisations comme support de cellules pour le cartilage [51], [7].

Dans le cadre de ce travail, nous utiliserons l’hydrogel HEMA-EGDMA ou Hydroxyethyl

Metacrylate-Ethylène Glycol Dimethacrylate avec un certain pourcentage en eau [7]. Cet hydrogel

possède une propriété dissipative qui pourrait être bénéfique pour des utilisations biomédicales

en particulier une méthode d’administration des médicaments contrôlée par une variation de la

température [7], [74], [75], [8].

Afin de nous placer dans le contexte de la thermomécanique des milieux continus, nous présen-

tons une brève étude bibliographique dans la section suivante (cf. section 1.2). Cette étude

s’articule autour du phénomène "self-heating" en général et sur les tissus biologiques.

1.2 Modèle thermomécanique et "self-heating" dans les tissus

biologiques et hydrogel

Pour la modélisation (tissus biologiques et les hydrogels, en particulier de type HEMA-EGDMA),

nous allons utiliser la théorie des matériaux standards généralisés pour formuler le phénomène de

"self-heating". Nous introduisons une brève étude bibliographique sur cette théorie en général, et

ensuite nous nous focalisons sur les tissus biologiques. Pour les deux cas, nous résumons quelques

travaux sur le cas des matériaux fibreux et non fibreux.

La thermomécanique rationnelle des milieux continus classiques a été introduite par Truesdell,

Colleman, Noll, Toupin et al., elle est basée sur la théorie des processus thermodynamiques

1.2. MODÈLE THERMOMÉCANIQUE ET "SELF-HEATING" DANS LES TISSUS BIOLOGIQUES ET HYDROGEL7

irréversibles [120], [10], [12], [82], [86], [93], [105]. Puis quelques travaux traitent et font des

extensions sur la thermomécanique des milieux continus pour les matériaux standards généralisées

en introduisant dans la loi de comportement les variables de plasticité et d’endommagement [127],

[128]. Le premier problème de la modélisation et simulation des tissus vivants reposent sur la

recherche des lois de comportement des tissus concernés [9], [68], [133]. A partir des années 1950,

l’utilisation des invariants a été étudiée pour définir les lois constitutives des matériaux. Ces

invariants sont basés sur l’introduction des tenseurs structuraux qui représentent les milieux étudiés.

Parmi les nombreux auteurs s’inscrivant dans la ligne de cette étude, nous citons Rivlin et al. [52],

Spencer et al. [53], [54], [55], [56], Boelher et al. [57], [58], [59], [60], Wang et al. [61], [62],

Peng et al. [63], [64], Schroder et al. [65], [66], Liu et al. [67]. Pour modéliser les milieux

complexes, l’anisotropie est représentée par l’introduction d’un tenseur structurel et d’un invariant

mixte qui couple ce tenseur avec celui de Cauchy-Green (Spencer, [56], Boelher et al., [57]). Ces

formulations sont le plus utilisées pour modéliser les tissues biologiques. Dans le cadre de ce travail,

nous utiliserons aussi ce type de formulation pour la modélisation.

La densité d’énergie et la loi de comportement principale modélisant les tissus biologiques mous

sont souvent viscoélastiques et hyper-viscoélastiques en grande transformation, nous nous référons

aux travaux effectués par Rakotomanana, Pioletti et al. pour le cas des tissus biologiques mous

[9], [68]. Pour une loi constitutive viscoélastique dépendant de la température, la modélisation se

fait en utilisant des variables internes [158], [159], [160]. Une loi de comportement de ce type a

été établie par Pioletti et al. [9], [68]. Une autre forme de loi constitutive non-linéaire munie

d’une relaxation de contrainte pour les ligaments articulaires a été développée par Frances et al.

[69], [70]. De plus, Bergström a proposé des équations constitutives pour les comportements des

élastomères en grande transformation sous sollicitations cycliques pour les applications sur les tissus

biologiques [129], [130], [131]. Ensuite, des mises en œuvre par la modélisation et la simulation

numérique sur l’intégration de ces lois de comportement ont été développées par Holzapfel et al.

par exemple dans [76]. Enfin, une formulation tridimensionnelle d’une structure viscoélastique non

linéaire en utilisant la méthode des éléments finis a été proposée par Ronald et al. [80], [163].

Un couplage d’une loi de comportement du type thermo-viscoélastique avec la température a été

développé par quelques auteurs [77].

En conclusion, dans le cadre de ce travail, pour modéliser la loi de comportement des tissus

mous, en particulier le comportement des hydrogels du type (HEMA-EGDMA), nous utiliserons une

loi du type thermo-visco-hyperélastique en grande transformation. Nous nous proposons d’adapter

la loi de comportement, dans un premier temps, avec des variables internes pour modéliser et tenir

compte des particularités de ce type d’hydrogel. Nous définirons ensuite une loi d’évolution pour


chaque variable interne. Ensuite, nous procédons aux caractérisations par les mesures expérimen-

tales pour identifier et valider les modèles mathématiques, nous utiliserons l’hydrogel HEMA avec

un type de réticulation EGDMA.

1.3 Couplage parabolique-hyperbolique

La modélisation thermomécanique des milieux continus ou faiblement continus fait apparaître un

système d’équation aux dérivées partielles couplées précisément l’équation d’onde et l’équation de

la chaleur avec des termes sources et des conditions aux limites dynamiques. Ces équations sont

classiquement appelées équations aux dérivées partielles parabolique et hyperbolique respective-

ment. Comme dans notre cas, ces équations sont couplées et non linéaires, il faut faire appelle aux

méthodes numériques, en particulier la méthode des éléments finis, pour résoudre le problème.

Dans cette optique, Hao et al. ont étudié le comportement asymptotique et l’existence de

solution d’un problème de couplage parabolique-hyperbolique, avec des conditions aux limites

dynamiques [162], [164], [165], [166], [174]. Ensuite, Geng et al. utilisent la méthode des éléments

finis pour résoudre un problème du type couplage parabolique-hyperbolique [167]. Par contre, Xu

et al. utilisent la méthode des différences finies pour résoudre ce type de problème [170]. Un

solveur 2D non linéaire a été développé par Xia et al. pour résoudre le problème couplé de type

parabolique-hyperbolique [171], [172], [173]. Hao et al. ont développé une étude axée sur l’existence

et l’unicité d’une solution globale et sur le comportement asymptotique de la solution du système

parabolique-hyperbolique avec des conditions aux limites dynamiques bien posées [162].

1.4 Organisation du travail de thèse

Le principal but de travail de thèse est de fournir, en combinant une approche par le biais de la

modélisation et de la caractérisation expérimentale, une meilleure compréhension des phénomènes

dissipatifs apparaissant dans des matériaux soumis à des charges cycliques. Dans ce cadre, nous

nous intéressons spécifiquement aux phénomènes physiques qui sont liés à la génération de chaleur

dans des matériaux polymériques en général, et en particulier dans les hydrogels (synthétiques

de type HEMA-EGDMA) soumis à des sollicitations cycliques (self-heating phenomena). Nous

apporterons un éclairage théorique sur ces phénomènes pour ensuite pouvoir si possible les exploiter

pour des applications nouvelles dans le domaine biomédical.

Pour la modélisation mathématiques du phénomène de self-heating dans les hydrogels nous

developperons un modèle thermo-viscoélatsique des milieux continus pour prédire l’évolution de

température dans ces matériaux sous sollicitations mécaniques dynamiques. Pour le cas d’une ac-

tivité physique (sollicitation cyclique), post-opératoire, les conditions aux limites sont dynamiques

1.5. CONTRIBUTION DE LA THÈSE 9

et varient en fonction de l’activité du patient. L’hydrogel doit répondre à toute ces contraintes.

Pour modéliser ce phénomène de self-heating dans les hydrogels, nous utiliserons la thermomé-

canique des milieux continus pour les matériaux standards généralisés, linéaire et non-linéaire.

Dans ce cadre, comme le problème est couplé (bonne propriété mécanique et production chaleur

pour assurer la diffusion des médicaments) et que l’unicité de solution n’est pas assurée donc nous

validerons d’abord les conditions aux limites dynamiques avec des phénomènes déjà validés par des

mesures expérimentales pointues. Pour traiter ce type de condition (dynamique) avec des valeurs

très petites (diffusion d’espèce thermiquement activé, cas dispositif d’administration de médica-

ments), nous introduiserons un outil numérique permettant de traiter ces conditions aux limites

dynamiques. Cet outil sera validé par des mesures précises dans le cadre de la contamination

moléculaire croisée dans l’industrie micro-électronique.

Nous avons choisi le cas de la contamination moléculaire croisée dans l’industrie microéléctron-

ique car ce phénomène est d’abord gouverné par le transfer masse thermiquement activé par des

conditions aux limites dynamiques (conditions industrielles sur l’utilisation des FOUP). Ensuite,

les outils de caractérisations au CEA nous permet d’avoir des mesures assez fines pour ce type de

phénomène. Par analogie, nous utiliserons les outils validés avec ce type de phénomène pour le cas

de self-heating et la dissipation pour la diffusion des médicaments dans les hydogels vers les tissus

vivants (diffusion contrôlé par le changement de température). Ce phénomène est un phénomène

couplé, avec une espèce diffusant de petite taille et de faible valeur ce qui est le cas du phénomène

de self-heating.

Les mesures expérimetales pointues nous permettrons de maîtriser les instabilités numériques

pour ce type de phénomène dans le but baliser ensuite celle du phénomène de self-heating. Une

fois l’outil validé, pour le cas des conditions aux limites dynamiques, nous pourrons l’utiliser dans

le cas du phénomène de self-heating dont les outil de caractérisation actuelle ne nous permet pas

de maîtriser les détails sur la réponse aux conditions de l’hydrogel aux limites dynamiques et sur

l’effet de non linéarité.

1.5 Contribution de la thèse

Compte tenu de l’organisation du travail de thèse la contribution de la thèse est structurée comme

suit:

Outils mathématique et numérique pour les conditions aux limites dynamiques. La

première partie de la contribution de la thèse se focalise sur l’introduction d’une méthode de réso-

lution d’un modèle mathématique utilisant la condition de "switch" en temps Dirichlet/Neumann


pour des conditions aux limites dynamiques. Cette partie traite d’une méthode de résolution

par la méthode des éléments finis pour le système formé par deux équations convection-diffusion

couplé avec l’effet de température donc l’équation de la chaleur. Ce modèle caractérise, dans un

premier temps, le phénomène de contamination croisé dans l’industrie micro-électronique pour

étudier la sensibilité des matériaux polymères à la contamination volatile. Il modélise, dans un

second temps, le phénomène de contamination croisé avec l’effet de la température pour la décon-

tamination à chaud et le nettoyage à chaud du FOUP dans l’industrie micro-électronique. Pour

l’application industrielle du modèle, chaque étape utilise la condition de "switch"nommé StDN.

Nous utilisons le logiciel Comsol Multiphysics pour implémenter le système. Ensuite, nous met-

tons en évidence l’effet de la température sur la décontamination. D’une part, l’augmentation de

la température pendant la décontamination favorise la diffusion des polluants dans le volume car

le coefficient de diffusion augmente. D’autre part, pour l’accumulation surfacique des contami-

nants dans l’interface, l’augmentation de température durant le nettoyage permet d’éliminer une

quantité maximale de concentration superficielle. Enfin, une étude de corrélation entre les données

de caractérisation expérimentale et le modèle mathématique avec les conditions de "switch" est

présentée pour chaque étape d’utilisation industrielle.

Nous nous focalisons sur le cas de la contamination croisée dans l’industrie micro-électronique

pour valider le modèle et les conditions de "switch" (StDN). Nous proposons une méthode pour

faire un passage en temps de la condition aux limites de Dirichlet en condition aux limites de

Neumann. Ce modèle est utilisé dans le but de maîtriser la contamination dans l’industrie micro-

électronique et de choisir le matériau optimal répondant aux critères d’utilisation. Nous utilisons

la méthode des éléments finis pour résoudre le système avec ces conditions. Ensuite, nous utilisons

le logiciel Comsol Multiphysics pour implémenter le système. Une caractérisation expérimentale

pour valider le modèle mathématique avec les conditions de "switch" est présentée pour chaque

étape. Le modèle proposé avec les conditions nommées StDN est en corrélation avec les mesures

expérimentales dans des conditions industrielles.

Mise en œuvre numériques du couplage parabolique-hyperbolique. Ensuite, nous faisons

une extension de la méthode numérique basée sur la méthode des éléments finis pour résoudre un

système d’équations couplées (cas thermomécanique), l’intégration des lois de comportement et les

autres lois d’évolution (des variables internes par exemple).

Applications du modèle de self-heating pour le cas de l’hydrogel HEMA-EGDMA sol-

licité sous un chargement cyclique. Nous faisons une application sur l’identification des lois de

comportement de l’hydrogel HEMA-EGDMA en variant quelques paramètres comme la fréquence

1.5. CONTRIBUTION DE LA THÈSE 11

de sollicitation et la densité de réticulation. Nous proposons les deux potentiels thermodynamiques

correspondants et nous identifierons par une étude de corrélation avec les données expérimentales.

Nous étudierons les réponses thermiques en agissant sur les sollicitations mécaniques ensuite nous

évaluerons les effets de la température sur le comportement mécanique de l’échantillon.

Caractérisation expérimentale du self-heating dans le cas de l’hydrogel HEMA-EGDMA,

identification. Une caractérisation expérimentale a été faite dans le but de quantifier la produc-

tion de chaleur dans un échantillon. Cette caractérisation permet de faire des études de corrélation

du modèle avec les résultats expérimentaux. Pendant les mesures, on quantifie la production de

chaleur dans l’échantillon sous sollicitation cyclique en mesurant la température. Cette première

partie nous permet aussi de valider le modèle thermomécanique.

Remark 1.1. L’échantillon est en hydrogel HEMA-EGDMA de 8 mm de diamètre et 6 mm de

hauteur. Ces échantillons ont été soumis à des sollicitations cycliques de fréquence 0.5 Hz, 1.0 Hz

et 1.5 Hz. On impose le déplacement sur la partie supérieure de l’échantillon à 20% de la hauteur

de l’échantillon.

Le chargement de l’échantillon se fait en trois parties notamment la pré-charge, la charge cy-

clique et la relaxation. Et la partie inférieure est "fixe". Nous avons choisi 30 s de pré-charge, 5[mn]

de chargement cyclique et 5 mn de relaxation. La variation de la température dans l’échantillon a

été mesurée avec un capteur dans l’échantillon. La chambre contenant l’échantillon est sous vide,

en effet, on considère qu’elle ne fait aucun échange avec le milieu extérieur (à flux nul). Nous

avons observé une variation de température de 1 à 3 oC. Cette variation dépend de la fréquence

de sollicitation et du pourcentage de réticulation (cross-linking pourcentage, 4%, 6%, 8%, 10%)

de l’échantillon. Deux échantillons ont été testés notamment HEMA-EGDMA 8% et HEMA-

EGDMA 4%. Pour une même fréquence, l’augmentation de la température pendant le chargement

et la relaxation dans l’échantillon dépend et a une influence proportionnelle avec le pourcentage de

réticulation. Plus le pourcentage de réticulation est "important" plus l’échantillon a une variation

de température importante.

Sur le calorimètre isotherme, nous avons effectué un test d’une éventuelle influence de la porosité

sur la variation de la température mesurée par le thermomètre différentielle. La supposition est

la suivante : la variation de pression dans le matériau poreux peut influencer (notable ou pas) la

variation locale de la température mesurée vu que la production de chaleur dans l’échantillon est

assez "faible". Nous avons procédé alors au test suivant: on enlève l’échantillon dans la membrane

et à la place, on met de l’eau et ensuite on procède au même test qu’avec le microcalorimètre

adiabatique (conditions de mesures). Nous avons lancé 3 tests identiques: chaque teste ne donne


pas le même résultat, on observe une perturbation non expliquée à ce jour. (Normalement on

aurait du replacer l’échantillon avec un autre matériau de même taille). Ce qu’on peut "dire"ce

qu’on a une perturbation due à la variation de pression dans le matériau poreux.

1.6 Plan de la thèse

Pour apporter un éclairage sur le phénomène de production interne de chaleur dans les polymères

et en particulier dans les hydrogels sous chargement cyclique, en positionnant par rapport aux

acquis de la littérature, l’exposé du travail suit le plan suivant :

Le premier chapitre est consacré à l’introduction générale de la thèse. Une brève étude

bibliographique a été introduite notamment sur la thermomécanique des milieux continus, les

hydrogels et ses applications dans le domaine biomédical et enfin sur la méthode numérique pour

résoudre les équations aux dérivées partielles gouvernant le phénomène de self-heating.

Le second chapitre est consacré à une présentation d’un outil mathématique et numérique

pour implémenter la condition de "switch" en temps de la condition aux limites Dirichlet/Neumann

(StDN) utilisée pour un problème comportant des conditions aux limites dynamiques. Nous

présentons la méthode de "switching" en temps, nommé StDN, de conditions aux limites de

Dirichlet vers Neumann. Ensuite, nous nous focalisons avec le cas de la contamination croisée dans

l’industrie micro-électronique pour valider le modèle et les conditions de "switch" StDN.

Le troisième chapitre introduit la méthode de résolution d’un modèle mathématique utilisant

le condition de "switch" en temps Dirichlet/Neumann (StDN) pour des conditions aux limites

dynamiques. Cette partie traite d’une méthode de résolution par la méthode des éléments finis

pour le système formé par deux équations convection-diffusion couplées avec l’effet de température

gouverné par l’équation de la chaleur. Ceci permet modéliser le phénomène de contamination croisé

avec l’effet de la température dans l’industrie micro-électronique.

Le quatrième chapitre regroupe la caractérisation et les divers résultats expérimentaux

obtenus sur la quantification de la production de chaleur dans l’hydrogel de type HEMA-EGDMA.

Nous présentons ensuite un modèle linéaire simplifié qui nous permet d’identifier les paramètres liés

à la production de température dans les échantillons. Nous traitons le cas monodimensionnel car

l’augmentation de température dans l’échantillon considéré est locale. Les résultats obtenus per-

mettent d’identifier le phénomène de self-heating dans les hydrogels. Nous terminons par l’étude

de corrélations des résultats théoriques (numériques) et expérimentaux suivi d’une proposition

d’optimisation pour identifier les paramètres influençant sur le phénomène de self-heating.

Le manuscrit se termine par des conclusions et quelques perspectives portant sur le plan ex-

périmental, sur le plan de la modélisation physique et numérique des phénomènes de dissipation

1.6. PLAN DE LA THÈSE 13

dans la matrice d’hydrogel HEMA-EGDMA voire dans les tissus biologiques.

Chapter 2

Switch conditions for coupled systemof PDEs.

1

Resumé – Ce chapitre traite une formulation d’une condition de switch en temps de la con-

dition aux limites de Dirichlet vers la condition aux limites de Neumann StDN. Ces conditions

sont souvent rencontrées dans le domaine de modélisation de phénomènes physiques couplés tel

que la contamination croisée dans l’industrie microélectronique. Une fois cette condition formulée,

nous l’appliquons à la modélisation et aux méthodes de calcul pour étudier la sensibilité de cer-

tains matériaux constituants du FOUP quand il y a risque de contamination croisée. Un modèle

mathématique couplé a été formulé pour étudier les phénomènes de la contamination croisée entre

les plaques (wafer) et le FOUP. Ensuite, des optimisations numériques et des méthodes numériques

basées sur la méthode des éléments finis pour une analyse en régime transitoire ont été établies.

Une solution analytique d’un problème monodimensionnel a été développée. Le comportement

de quelques matériaux constituants du FOUP en analyse transitoire a été déterminé. Le modèle

conserve les formes classiques de la diffusion et de la convection-diffusion avec une forme cohérente

de la loi de Fick. La cinétique d’adsorption du contaminant sur la surface (l’interface contaminant

/ polymère) a été supposée en utilisant la loi de Henry. Le processus d’adsorption et l’effet de la

rugosité de la surface ont également été modélisés sous forme de conditions aux limites en utilisant

la condition de switch de Dirichlet et Neumann (StDN) à l’interface. De nombreux tests de proces-

sus de contamination ont été effectués dans le but d’étudier la sensibilité des matériaux en fonction

des contaminants. Des résultats numériques en corrélation avec les données expérimentales sont

présentés dans ce chapitre.

1This chapter was published in International Journal of applied Mathematical Research under title: "Coupled sys-tem of PDEs to predict the sensitivity of some materials constituents of FOUP with the AMCs cross-contamination",Vol. 3 (3) pp. 233-243, 2014.

15

16 CHAPTER 2. SWITCH CONDITIONS FOR COUPLED SYSTEM OF PDES.

Abstract –This chapter is a formulation of a switch condition in time of the boundary condi-

tion of Dirichlet to Neumann boundary condition (StDN) . These conditions are often encountered

in the modeling domain of some physical coupled phenomena such as cross-contamination in the

microelectronics industry. Predictive models using modeling and computational methods are pro-

posed to investigate the sensitivity of some materials constituents of the FOUP with the airborne

molecular cross contamination. Required numerical tools, which are employed in order to study

the AMCs cross-contamination transfer phenomena between wafers and FOUPs were developed.

Numerical optimization and finite elements formulation in transient analysis were established. An-

alytical solution of one-dimensional problem was developed and the identification of the physical

constants was performed. This mode was used to study the sensitivity of some material with

the cross contamination. The behavior of the AMCs in transient analysis was determined. The

model framework preserves the classical forms of the diffusion and convection-diffusion equations

and yields to consistent form of the Fick’s law. The adsorption kinetics of the contaminant on

the surface (interface contaminant/polymer) was assumed. The adsorption process and the surface

roughness effect were also traduced as a boundary condition using the switch condition Dirichlet

to Neumann (StDN) and the interface condition. Many tests of contamination processes were

assumed in order to study the sensitivity of the materials. Optimization methods with analytical

solution were used to define physical constants for each material versus contaminant. Finite ele-

ment methods including adsorption kinetic were also used and by using Henry law on the interface

and the switch of Dirichlet to Neumann conditions. Some numerical results in correlation with

experimental measurements are presented in this chapter.

2.1 Introduction

In high-tech microelectronics engineering, more attention is required to challenge the contamination

control during the manufacture of integrated circuit (I.C)[1]. Integrated circuits are manufactured

from a monocristallin silicium plates (wafer) [1], [2]. Minutiarization of the I.C is designed with the

45, 32, 22 nm, in fact, the wafer’s surface of the wafer is very sensitive to molecular contamination.

The contamination control of the wafer is a critical subject, it can potentially cause defect on the

use and have an impact in the device performance [2], [3], [4].

In this work, we focus more attention in modeling and simulation of a molecular contamination

that can damage and induce a significant impact in manufacturing yields [2], [3], [4]. The wafer

carrier and storage play a significant role for contamination control [4]. The use of the Front

Unified Pods (FOUPs) to transport from tool to tool 25 wafers in the 300 [mm] are necessary to

protect the wafer against contamination, mainly the Airborne Molecular Contamination [3]. This

2.1. INTRODUCTION 17

container may contain a lot of contaminant including, also called AMCs or Airborne Molecular

Contaminants and can still contain a significant amount of contamination with the potential to

damage the wafer.

This enclosed mini-environment is made of porous polymers, mainly in PC, COP, PP, PEEK

and PEI [5], [6]. These materials are known with their adsorption and outgas properties. They are

also able to absorb volatile compounds present in the atmosphere coming from the connection to

equipment or from the fresh assembly of wafers just processed (post processed wafers) [4]. During

the storage, these wafers may outgas the chemicals used during the process [2], [3].

As a results, a reversible and an irreversible outgassing of contaminant previously trapped in

polymer is possible [2]. In fact, a contamined FOUP already itself may be a source of contamination

because it already adsorbs the contaminant from the wafers [4], [5]. This cross-contamination

scheme was clearly induced for volatile acids used for the manufacturization. Many works have

been published which deal with experimental measurement method to quantify and to investigate

this phenomena for each types of material e.g [2], [3], [4], [5].

Generally, when one object becomes contaminated by either direct or indirect contact with

another object which is already contaminated, we talk about cross contamination, [1]. In micro-

electronics industry, this process generally takes place at the pods which contain the wafers before

and after production, [2]. The main object sources of the contamination in microelectronic factory

are the wafer, air, FOUP by which a new wafer may be contamined before the manufacturing

processes [2].

Some manufacture processes such as dry engraving (plasma), depot and photolithography are

the source of contamination, when the wafer is already processed its surface is contamined by the

volatile acids [6], [5],[6]. After this manufacturing process, these wafers will be stored in the FOUP

[4]. Then, the wafer is exposed to the FOUPs atmosphere and an acid pollution may happen from

wafer to FOUP by the intermediate air (atmosphere). As outlined before, the FOUP’s material can

absorb the contaminant in it’s around, a adsorption phenomenon from air to the internal surface

of the FOUP followed by diffusion in volume happens.

When the wafer moved or the pods is opened, the air in the FOUP’s atmosphere changes and

desorption phenomena takes place, a cross contamination from the pods to the new wafer may

happen i.e [2], [4]. The contamination of the new wafers will be stored in the pods is obtained

indirectly with the pods already contamined by the volatile acid. Indeed, in order to successfully

ensure the miniaturization the integrated circuits manufacturing by using 300[mm] wafer new

methods that are required for facing this challenge.

To endeavor a systematic analysis and control of the underlying system, numerical simulation


should help to mimic the process behaviors [7], [8], [9]. Modeling and computational methods are

worth method to predict and to quantify physical phenomenon such as AMCs cross contamination

within FOUP.

Motivated by the above phenomena, this work describes and develops mathematical model by

using finite element formulation for AMCs cross-contamination in order to investigate the effect

of the contaminant to its close environment. The mathematical model of this phenomenon is

governed by coupled partial differential equation with dynamic boundary conditions. The industrial

condition, for the application of the model prescribed that during the simulation, the Dirichlet

boundary conditions change into Neumann boundary conditions.

Indeed, we need to switch this condition for the computation. We develop this switch condition

in this work for a coupled partial differential equation in particular the model of the AMCs cross-

contamination. We propose new strong numerical tools for AMCs cross contamination to qualify

and to quantify the residual contamination in the pods. The model validation method is based on

correlation of the observed data and the direct method together.

2.2 Problem statements and model settings

Modeling the adsorption of AMCs cross contamination between wafer and FOUPs is based on

adsorption phenomena [7]. In general two concepts of adsorption, physical and chemistry adsorp-

tion models will be considered. Sorption phenomena and degassing molecular contaminants in

the FOUPs are governed by mass transfer of the gaseous molecules the mechanisms polymers [2],

[4]. According to the results of the experimental measurement, the main molecular contaminant

of FOUPs can create defects on silicon wafers (growth TiFx crystals, silicon corrosion) at various

stages of integrated circuit fabrication processes, is the HF acid.

Figure 2.1: FOUP, F300, Entegris (left), Wafer (right).

In addition, HF and HCl may occur, in some technological steps, corrosion of metal lines [1],

2.2. PROBLEM STATEMENTS AND MODEL SETTINGS 19

[2] , [2], [4]. This chapter, in the first time, aims to determine the diffusion coefficients, solubility

and permeability of gaseous acids (HF, HCl) in the main constituent polymers of FOUPs (PC,

PEEK, PEI, COP). These results enable us first, to better understand the molecular mechanisms

of contamination FOUPs, and secondly, these coefficients can be used in numerical simulation

applied to industrial conditions. The applications are the quantification of contaminants sorbed and

degassed by the FOUP, development and optimization of conditions of FOUP cleaning methods.

Figure 2.2: Component description of the FOUP, F300, Entegris.

A detailed review of various adsorption kinetics (adsorption, desorption) models was given in

[7], [9], [6]. In this model the transport towards the surface is purely diffusive and we investigate

the concentration in the internal surface of the pods by using the thermodynamics laws [7], [9].

These thermodynamics laws are given by the Henry constant in order to connect the concentration

of the contaminant at the FOUP’s atmosphere and the concentration at the internal surface of the

pods.

The mathematical model of diffusion process in these two domains is based on Fick’s second

law of diffusion [14]. At the interface of that domain kinetics law will be established. According to

Fick’s first and second law, also known as the diffusion equation [11], [14], is defined by the first

part of the equation (2.1). The AMCs cross contamination is governed by diffusion time dependent

process in which the rate of diffusion is function of time. In this process, the contaminant moves

from a region of high concentration (wafer) to one of low concentration (internal surface of pods)

[17]. According to Fick’s first and second law, also known as the diffusion equation [14], [15] the

mathematical expression for transient contaminant transfer between the wafer and the internal

part of the FOUP is given by the equation (2.1).


Figure 2.3: Schematic illustration for the three main steps of the cross-contamination in the FOUP.The bottle represents the FOUP, the two horizontal lines in the bottle illustrate two wafers on theirsupport and the internal curved arrows models the contaminant migration (diffusion) during themain steps of contamination.

In this process, contaminant moves from a region of high concentration (wafer post processed)

to one of low concentration (internal surface of FOUP). Firstly, during the contamination phe-

nomenon, the contaminant moves from the wafers post processed to the FOUP. Secondly, during

the outgassing, phenomena the contaminant moves from FOUP to the wafers.

Hypothesis 2.1 (Source of contamination). We assume that, on the wafer’s surface, we have

the source of contamination during the contamination time. We assume that the advection and

reaction time scales are slow compared to the diffusive time scales.

2.2.1 Experimental measurement

For the experimental measurement, we refer to [1], [2] and resume the experimental measurement

as two processes: the characterization of the diffusion coefficient using a thin plate and the charac-

terization of the AMCs cross-contamination in FOUP’s scale. In order to characterize the diffusion

coefficient, we use a chemical reactor in which we insert a thin membrane of the material con-

stituent of the pods. An inlet supplies the reactor and outlet contaminant fluids with constant flow

in order to measure the adsorbed quantity of the contaminant in the polymer 2.4. To maintain a

constant concentration of acid, the polymer membrane is exposed at low flows. Waterproof reactor

at , Polymer

The polymer films used in this study were prepared and conditioned before exposure. Polymer

samples were cut into rectangular shape of dimensions 18x60 [mm] and then cleaned to remove all

traces of initial acids, for 4 successive extractions with hot deionized water (70 o C) for 8 hours.

They are then stabilized by exposure to moisture in the air of the clean room (22o C, 40% RH)

for at least a week. The reactor’s atmosphere intentionally contaminated exhibition in HF or HCl

is generated under the following conditions: Relative Humidity (RH) is equivalent to that of the

atmosphere of the clean room: 40 ± 2%; the total flow of the gas flow is of 300 ± 5 [ml/min] and


Figure 2.4: Experimental devices

the acid concentration is close to that observed in cleanrooms for microelectronics manufacturing

(hundred [ppbv]).

The fluid contaminant is obtained by mixing air with initial concentration (some [ppbv], with

three regulators (mass flow controller) RDM A, RDM B, RDM C). In general, the flow rate of the

contamined airflow is 3 [ml/min] (constant).

In order to analyze the membrane and to quantify the sorbed concentration, we dissolve the

adsorbed contaminant molecules in the polymer into the water leading to ion formation. The water

has been analyzed by ionic chromatography. This method is dedicated to volatile acids.

Figure 2.5: Experimental devices for the reactor’s atmosphere control.

It consists of six gas channels. The gas flow rate is set by RDM (Regulators mass flow rates).


Each of the channels comprises 3 RDM. The first of them is used to control the flow rate of acid

gas from a commercial bottle (Air Liquide) whose initial concentration is 1 or 10 ppmv (RDM A

2.5). RDM This adjusts the flow of acid from 0 to 500 [ml/min]. The second RDM helps regulate

the flow of dry air (RDM B), and the third to set the humidity, the air used previously dabbling

in a bottle bubbler (RDM C). Both RDM bringing dry air and moist air can be set from 0 to 2500

[mL/min].

For the characterization in the pods scale, we assume the same procedure as defined above

for the quantification. The FOUP’s atmosphere, after wafers removing, has been analyzed with

specific technics. An intentional contamination is realized in order to create the initial concentration

(this process presents the contamination from the wafer). The total amount of the volatile acids

concentration was monitored with an Ion Mobility Spectrometer (IMS). After wafers removing, the

pods was connected by specific outlet/inlet the filter ports replacing FOUP filters to a bubbling

system which is composed of two bubblers in series filled with Deionized Water (DIW).

The air in the pods was pumped through bubblers to dissolve molecules into the water leading

to ion formation. The water was then analyzed by ionic chromatography. The bubbling solution

was also analyzed by ICP-OES (Inductively Coupled Plasma-Optical Emission Spectrometry). The

amount of acids sorbed on the FOUP surface was collected by DIW leaching, and then characterized

by ionic chromatography.

Figure 2.6: APA, Adixen Pod Analyser


The total amount of the acid concentration in the FOUP’s atmosphere was measured with APA

equipment (Adixen Pod Analyzer) 2.6. The analysis principle of APA is based on Ion Mobility

Spectrometry (IMS). The APA equipment is specifically configured to measure a FOUP. IMS is a

Technical chemical gas-phase analysis [1], [2]. It consists to subject ionized molecules by β radiation

to an electric field in a gas stream. The ions move along the electric field at speeds that depend

on their interaction with the gas, that in function of their weight, their size and their shape. We

talk about separation after ion mobility. The arrival of the ions on one of the electric field causes

the plates producing an electric current that is registered. We can relate the time at which a peak

occurs with the nature of the ion having caused this peak. However, IMS-APA can not differentiate

the difference between volatile acids and gives an overall measurement: the total acids. For the

collection, the FOUP is sealingly connected with the APA by a filter FOUP 2.6. The atmosphere

of the FOUP is imposed by a pump at a flow rate of 0.7 [ml/min] for 2 [min]. The APA method

can detect the total acid in the atmosphere of the FOUP with a detection limit estimated at 0.2

[ppbv].

2.2.2 Cross-contamination model

For this purpose, let I = [0, Tf ] be a time interval and let Ωs and Ωg be a open bounded in Rd,

with d := 2 or 3 (space dimension), with sufficiently smooth boundary ∂Ωs and ∂Ωg respectively

(d − 1 dimensional surface embedded in the Rd. Ωs and Ωg denote respectively the contaminant

(wafer+air in the FOUP’s atmosphere) subdomain and the polymer subdomain (FOUP).

Figure 2.7: Subdomains and boundaries definitions

The equation for transient contaminant transfer between the wafer and the internal part of the

FOUP, for isothermal condition, is given by (2.1):


∂Cs

∂t= ∇ · (Ds∇Cs) + q in (Ωs × [0, Tf ])

B.C

Cs = h0C

g on (ΓN × [0, Tf ])

Ds∇Cs · n = 0 on ((∂Ωs − ΓN )× [0, Tf ])

I.C Cs(., t = 0) = 0 in (Ωs × 0)

∂Cg

∂t= ∇.(Dg∇Cg)−∇ · (uCg) in (Ωg × [0, Tf ])

B.C

(−Dg∇Cg + uCg) · n = Fa(C0;Cg) on (ΓD × [0, Tf ])

Cg = h0−1Cs on (ΓN × [0, Tf ])

(−Dg∇Cg + uCg) · n = 0 on (∂Ωg − ((ΓN

⋃ΓD))× [0, Tf ])

I.C Cg(., t = 0) = 0 in (Ωg × 0)

(2.1)

The initial conditions are defined: it consists to consider that at the initial time t := 0 the

FOUP and its atmosphere are not yet contamined i.e. Cs(t = 0, ·) := 0 and Cg(t = 0, ·) := 0.

Practically, these studies have done with a new FOUP for a first use. In the model, Dg ∈ R+

and Ds ∈ R+are the diffusion coefficients in gas (contaminant) and solid medium, which have no

connection with the spatial location and no variation in time, [m2/s]. u denotes the transport

advective field, and q is the volume source. ∇ and ∇· denote the gradient and the divergence

operators.

The unknown of the problem are Cs ∈ R+ and Cg ∈ R+ which are respectively the concen-

tration of the contaminant and the concentration at the internal FOUP’s surface, H(t− ε) is the

Heaviside function for any time t ∈ [0, Tf ] define the intervalle [t, Tf ], C0 ∈ R+ is the initial con-

centration in FOUP’s atmosphere when the wafers have finished to outgassing the contaminant,

h0 ∈ R+ is the Henry constant and n is the outer unit normal vector, Tf ∈ R+ is the final time

and q is the source.

Hypothesis 2.2 (Interface conditions). We suppose that, the FOUP and the membrane are ho-

mogenous and isotropic. Adsorption resolution of the contaminant is balanced at the surface

[11], [14]. The surface roughness of the internal surface of the FOUP or the membrane is ne-

glected but we take into account of this parameter when we use the Henry law on the interface

contaminant-polymer (interface between the subdomains Ωs and Ωg), Cs = h0Cg on (Γ+

N × [0, Tf ]),

Cg = h−10 Cs on (Γ−N × [0, Tf ]).

Because the concentration of the contaminant in the polymer Cs ∈ R+ depends on the con-

centration of the contaminant Cg ∈ R+, we assume the following boundary conditions: on ΓD

a prescribed inflow concentration (source of the concentration, in the post processed wafer’s sur-

face) is prescribed. We start by developing some analytical solutions for the diffusion equation

with uniform diffusivities in an unbounded domain and for very simple boundary conditions as

determined by J. Crank. Then we solve the more general equation using finite element method,

2.3. SWITCH CONDITION ON TIME STDN OF THE BOUNDARY CONDITIONS 25

a numerical technique of optimization, for any type of boundary conditions. In this approach, we

use the assumptions below.

Hypothesis 2.3 (Diffusion coefficients). We assume that, the model is under isothermal condi-

tions. Then, the diffusion coefficients are function of a given temperature (constant) during the

simulation. Indeed, we consider that the diffusion coeffcients are weakly function of the concentra-

tion, so we can write Dg 6= Dg(Cg) and Ds 6= Ds(C

s).

2.3 Switch condition on time StDN of the boundary condi-

tions

The switch condition on time of the Dirichlet boundary condition and the Neumann condition

is used to switch on time, on the same bounds (ΓD × [0, Tf ]), the Dirichlet and the Neumann

conditions. Also, called StDN, by applying this conditions in the equation (2.1), during the

simulation we have to switch: during the contamination tp, the boundary condition in the model of

the contaminant is Cg = C0H(t−ε) (on the wafer surface on (ΓD×[0, Tf ]), source of contamination)

and after the contamination time to the end of the simulation we have the boundary condition

(−Dg∇Cg + uCg) · n = 0, it means that the source of contamination is stopped and the wafer

surface is neutral (on the wafer surface on (ΓD × [0, Tf ])). We have :

•if 0 ≤ t < tc, Cg = C0H(t− ε) on (ΓD× [0, Tf ]), then the model for the contamination process

(the wafer post processed contamines the air in the FOUP during tc) is given :

∂Cs

∂t= ∇ · (Ds∇Cs) + q in (Ωs × [0, Tf ])

B.C

Cs = h0C

g on (ΓN × [0, Tf ])

Ds∇Cs · n = 0 on ((∂Ωs − ΓN )× [0, Tf ])

I.C Cs(., t = 0) = 0 in (Ωs × 0)

∂Cg

∂t= ∇.(Dg∇Cg)−∇ · (uCg) in (Ωg × [0, Tf ])

B.C

Cg = C0H(t− ε) on (ΓD × [0, Tf ])

Cg = h0−1Cs on (ΓN × [0, Tf ])

(−Dg∇Cg + uCg) · n = 0 on ((∂Ωg − (ΓN

⋃ΓD))× [0, Tf ])

I.C Cg(., t = 0) = 0 in (Ωg × 0)

(2.2)

•if tc < t ≤ T, Fa(C0;Cg) = 0 on (ΓD × [0, Tf ]), then the model for the contamination process

(the wafer postprocessed stops to contamine the air in the FOUP) is given by :


∂Cs

∂t= ∇ · (Ds∇Cs) + q in (Ωs × [0, Tf ])

B.C

Cs = h0C

g on (ΓN × [0, Tf ])

Ds∇Cs · n = 0 on ((∂Ωs − ΓN )× [0, Tf ])

I.C Cs(., t = 0) = 0 in (Ωs × 0)

∂Cg

∂t= ∇.(Dg∇Cg)−∇ · (uCg) in (Ωg × [0, Tf ])

B.C

(−Dg∇Cg + uCg) · n = 0 on (ΓD × [0, Tf ])

Cg = h0−1Cs on (ΓN × [0, Tf ])

(−Dg∇Cg + uCg) · n = 0 on ((∂Ωg − (ΓN

⋃ΓD))× [0, Tf ])

I.C Cg(., t = 0) = 0 in (Ωg × 0)

(2.3)

So, during the simulation t ∈ [0, T ], we need to switch the condition Cg = C0H(t−ε) on (ΓD×

[0, Tf ]) to (−Dg∇Cg +uCg) ·n = 0 on (ΓD× [0, Tf ]). In the first approach we use a flux boundary

condition to switch on time Dirichlet to Neumann. Then, we have to switch (−Dg∇Cg+uCg) ·n =

Fa(C0;Cg) on (ΓD × [0, Tf ]) to (−Dg∇Cg + uCg) · n = 0 on (ΓD × [0, Tf ]), where,

Fa(C0;Cg) = Ng0 + kc [C0H(t− ε)− Cg] on (ΓD × [0, Tf ]) (2.4)

Also,

(−Dg∇Cg + uCg) · n = Ng0 + kc [C0H(t− ε)− Cg] on (ΓD × [0, Tf ]) (2.5)

Firstly, we assume that there is no inner flux Ng0 (no initial flux) and we assume that we

have a transient boundary conditions with a laminar gas flow on this boundary (u = 0). When is

sufficiently large, we have the Dirichlet condition i.e Cg ≡ C0H(t−ε) on (ΓD×[0, Tf ]) and if we have

the Neumann’s boundary condition (for the neutral area of the wafer), i.e. (−Dg∇Cg +uCg) ·n =

0 on (ΓD × [0, Tf ]).

Definition 2.1 (Switch condition). For a given dynamic boundary condition as defined in the

equation (2.2) and (2.3), we define the switch condition in time of the Dirichlet and Neumann

boundary conditions StDN as:

if

t < tc ⇒ Dirichlet BC

t ≥ tc ⇒ Neumann BC(2.6)

In which, tc ∈ R+ ⊂ [0, Tf ] is the critical time for the switch condition.

In order to ensure the stability, parametric study was developed. After applying the switch

StDN conditions in the model given by the equation (2.1), (2.2) and (2.3), we define the following

setting to switch as:

Definition 2.2 (Switch condition setting). We defined the switching condition as a prescribed

2.4. IDENTIFICATION OF THE PHYSICAL CONSTANTS 27

inward flux on the source of contamination i.e on (ΓD × [0, Tf ]), we have :

(−Dg∇Cg + uCg) · n = kc [C0H(t− ε)− Cg] on (ΓD × [0, Tf ]) (2.7)

Indeed, we define a constant kc ∈ R+ >> Dg, we have the following conditions to simulate the

airborne molecular cross-contamination on (ΓD × [0, Tf ]) as

if

kc 1 ⇒ Cg ' C0H(t− ε) if 0 < t ≤ (tc + tp)

kc = 0 ⇒ (−Dg∇Cg + uCg) · n = 0 if t > (tc + tp)(2.8)

2.4 Identification of the physical constants

In this section, we assume that the diffusion coefficients Ds ∈ R+ and Dg ∈ R+ are deterministic.

We have constructed analytical solution for the concentration in the polymer Cs by solving the

one dimensional model for the first approximation [10], [11], [12].

The computed quantity is obtained by the analytical formulae, such that the concentration at

the internal surface is Q∞ ∈ R+ and there is no initial distribution in the polymer membrane [13],

[14]. The diffusion coefficient Ds is the unknown; it is obtained by using the optimization method.

The Fick’s law involves the diffusion coefficient of contaminant through the polymeric material [5],

[6]. However, the literature doesn’t provide enough data concerning the molecular diffusion.

In order to find the values of the diffusion coefficients, a numerical optimization is established

by using the experiment data [14], [15]. This method is used to calculate the diffusion coefficient

for each contaminant in the polymer material constituent of the FOUP. It consists to minimize

the equation (2.13) which fits the diffusion coefficient as parameters of the model function to

experimental data of the sorbed quantity-time curves [14], [15].

Hypothesis 2.4 (Analytical solution). For the analytical solution in Ωs, we use the assumptions

that, we have no initial concentration distribution in the domain, no convective part, the maximum

quantity Q∞ of concentration is on the boundary (Γ) and finally with the hypothesis 2.3. Accord-

ing to J. Crank [14] the amount of contaminant is a parametric function of time and diffusion

coefficient, for a membrane with thickness :

Q(ti; Ds) = Q∞

[1−

∞∑n=0

8

(2n+ 1)2π2exp

(−Ds(2n+ 1)2π2ti

4L2

)](2.9)

We use the nonlinear least square method to determine the diffusion coefficient for each contam-

inant in the polymer membrane. The objective function using the experimental data and according


to Urruty et al. method, is written as

Q(X) =1

2

∥∥Qexp −Qthe(X)∥∥2

(2.10)

where Qexp and Qthe(X) are the experimental quantity of the contaminant recorded inside the

polymer membrane (FOUP) and the amount of incorporated contaminant is computed using the

analytical solution defined in the equation (2.9).

Definition 2.3 (Objective function). The objective function for a least square approximation is

defined as :

Q(Ds) =1

2

m∑i=1

[Qexpi −Qthe

i (ti; Ds)]2

=1

2

m∑i=1

r2i (Ds) (2.11)

in which Qexpi and Qthe

i (ti; Ds) are respectively the observed quantity by using chromatography

ionic method and the predicted quantity obtained with the equation (2.9). m denotes the number of

the measured quantities in time during the experimental measurement.

In this above relation, the quantity computed is first obtained by using the analytical solu-

tion, such that the concentration at the internal surface is Q∞ and such that there is no initial

distribution in the polymer membrane. The diffusion coefficient Ds is the unknown. In order to

determine Ds, we have to minimize the constrained objective function given by the equation (2.12),

the diffusion coefficient Ds must be positive, the maximum amount of the initial concentration Q∞

is positive and the computed sorbed quantity Qthei (ti; Ds) must be positive, also, we have :

infDs∈R+

Q(Ds) = infDs∈R+

1

2

∥∥∥∥∥Qexp −Q∞

(1−

∞∑n=0

8

(2n+ 1)2π2exp

(−Ds(2n+ 1)2π2t

4L2

))∥∥∥∥∥2

Qthe(t; Ds) ≥ 0

Q∞ ≥ 0

Ds > 0

(2.12)

Definition 2.4 (Constrained objective function for diffusion coefficient in membrane). We define

the objective function used to identify the diffusion coefficient as :

infDs∈R+

Q(Ds) = infDs∈R+

1

2

m∑i=1

[Qexpi −Q∞

(1−

∞∑n=0

8

(2n+ 1)2π2exp

(−Ds(2n+ 1)2π2ti

4L2

))]2

Q∞

(1−

∞∑n=0

8

(2n+ 1)2π2exp

(−Ds(2n+ 1)2π2ti

4L2

))≥ 0

Q∞ ≥ 0

Ds > 0

(2.13)

2.5. MODEL OF CONTAMINATION PROCESS 29

We attempt to maximize or minimize this function of the decision variable Ds, the values of this

variable must verify the constrains defined in the equation (2.13).

This function has one global minimum and the set the diffusion coefficient that belongs to

this minimum is defined to be optimal fitting to the experimental data. In order to evaluate the

correlation between the data and model we need to evaluate the coefficient of determination. It

measures how well the regression line represents the data.

Definition 2.5 (Correlation and determination coefficient). The correlation and determination

coefficients are defined as :

0 ≤ R2 = 1− ‖ r ‖2

‖ Qexp −QexpI ‖2≤ 1 (2.14)

where I the identity vector.

0 ≤ R2(Ds) = 1−∑m

i=1

[Qexpi −Qthe

i (ti; Ds)]2∑m

i=1

(Qexpi − 1

m

∑mi=1 Qexp

i

)2 ≤ 1 (2.15)

2.5 Model of contamination process

We start directly from the contamination event, when the wafers processes were finished, it will

be stored in the pods. The volatile acid caused by the process contaminates these wafers : the

chemical product used in wafer processing is the main sources of this AMC. In the FOUP the

wafers outgases this volatile acid during a few minutes or hours, it depends on the contamination

level. And after that, the outgassing step is completed but the contaminant continues to move

from air to the internal surface of the pods and followed by contaminant diffusion in the polymer.

To mimic the contamination process the contaminant is intentionally introduced in the FOUP.

These experimental procedures allow holding steady the concentration of contaminant close to

1000 [ppbv] in the pods. This technique mimics the process during which the wafer outgassing the

pollutant which contaminates internal the atmosphere of the pods. This is the first step of the

cross contamination. The pollutant moves from wafer to the internal surface of the pods. During

this process, the surface adsorption step takes place and the diffusion in the volume of the FOUP

polymeric materials happens. Contamination simulations consist to use the same conditions and

assumptions which have been developed in the equation (2.7) and (2.8).

Thus, we consider that the wafer is the contaminant source governed by the Heaviside function

with a delay ε. The amplitude of the contamination on the wafer is C0. The implementation of this

boundary condition is defined in the equation (2.7) and (2.8). Therefore, during tc (contamination

time), we apply on ΓD (wafer’s surface) the concentration C0 governed by the Heaviside function,


after tc = tc + to the wafer surface stops to outgas the contaminant and stays neutral. Fa(C0, Cg) = Cg ' C0H(t− ε) if 0 ≤ t ≤ tc

Fa(C0, Cg) = 0 if tc < t ≤ (tc + to)

(2.16)

The experimental measurement prescribes that the contamination time is decomposed into

characteristic times and respectively the time until the wafer outgassing is completed and the

downtime before opening the FOUP to remove the wafer The implementation of the boundary

conditions are defined in the equation (2.7) and (2.8), that was the same method validated with

the experimental measurement. When t ≤ tc, the boundary condition describes (cf.eq.(2.7) and

(2.8)) the wafer post processed outgassing the contaminant. And the second boundary condition

tc < t ≤ (tc+ to) defined the condition during which the wafer outgassing is done but it was stored

in the FOUP (i.e. we consider that the wafer surface become neutral).

2.6 Model of purging and outgassing process

This procedure takes place after opening the FOUP in the goal to remove the wafer post processed.

When the door of the FOUP is opened, airflow from the exterior atmosphere goes into and modifies

the contaminant concentration Cg ≡ 0 due to the dilution of the atmosphere by room air. An inert

gas from exterior atmosphere is supplied on their pods. Applying an inert gas purge in the pods

will be doing with an inlet and outlet ports on the pods and with the interface ports. Nevertheless,

during this process we consider only the removing atmosphere during which the front door is open.

The atmosphere of the pods is removed and the contaminant concentration during this operation

before it the front door is closed. It results a reverse flow of the contamination gradient during

which the contaminant moves from the materials (from polymeric materials to air and from air to

wafer). After the contamination, at time tc, the FOUP is opened during the opening time tp then

it is closed during the outgassing time td.

Fa(C0, Cg) = Ng

0 + kc [Cp0H(t− ε)− Cg] if tc ≤ t(tc + tp)

Fa(C0, Cg) = 0 if (tc + tp) < t ≤ (tc + tp + td)

(2.17)

where Cp0 ≡ 0 is the concentration of the contaminant during the purging. When tc < t ≤

(tc+tp) boundary condition mimics the opening door of the pods the wafer moves and the removing

atmosphere. Experimentally, the measurement during the opening door processes exhibited that

the contaminant concentration at the atmosphere is negligible and close to zero. Therefore, we this

concentration (zero) is applied during this operation and again the surface become neutral after

closing door.

2.7. MODEL OF DECONTAMINATION AND CLEANING PROCESS 31

2.7 Model of decontamination and cleaning process

In this section, we describe the application of the model to the decontamination process. This

operation consists to introduce a downward flow of inert gas in order to clean up the FOUP. Many

processes of purge are considered and are studied by theoretical analysis and experimental simu-

lation and measurement. Purging method with inert gas is one of the most popular methods, but

there are other several methods as UV or vacuum method. During purging the amount of the inert

gas is evaluated as function of temperature process in the goal to eliminate any undesirable con-

tamination. Therefore it is possible to estimate the mean value of velocity of the inert gas flow. By

using the present model various values of velocity could be applied on the process decontamination.

Subsequently outgassing, the decontamination could be done. Decontamination is the process

of removing contaminant that is accumulated inside polymeric materials of the FOUPs. Decontam-

ination is employed in order to reduce the AMCs cross contamination. In this work, we consider

two types of procedures, the cold purging and the warm purging. The goal of this section is to

study the temperature effect during cleaning processes. In general, purging method with inert gas

provides many advantages.

The decontamination process will be done during tw (the downtime) when the pods are closed

after decontamination process. The waiting time tu is important to know the cleaning efficiency.

So the final times T of the simulation is the sum of all characteristic time ti, it can be written as:

T =∑i

ti = tc + tp + td + tu + tw (2.18)

At the end, the total time is defined as t ∈ [0,∑

i ti] or t ∈[0, T ]

Fa(C0, Cg) = Ng

0 + kc[Cd0H(t− ε)− Cg

]if (tc + tp + td) < t ≤

∑i ti − tw

Fa(C0, Cg) = 0 if t >

∑i ti

(2.19)

The first boundary condition (2.19) represents the cleaning operation by purging methods with

with inert gas. Experimentally, an amount of the inert gas was introduced in the FOUP. The

present assumption gauges that the contaminant concentration at the atmosphere is negligible and

close to zero. Indeed, during this process, the equation (2.19) describes the cold cleaning operation

i.e. the cleaning operation is done with the ambient temperature (in general 19-21oC).

In order to complete the experimental measurement and to solve numerically the model, alterna-

tive approach via finite elements are used to numerically treat the AMC cross-contamination finite

element analysis. The method consists in using the standard enriched finite element approaches

with time-interpolation. It will be applied here to the transient conduction diffusion equation where


the classical Galerkin method is shown to be unstable. The proposed method consists in adding

and eliminating bubbles to the finite element space and then to interpolate the solution to the real

time step. This modification is equivalent to the addition of a stabilizing term tuned by a local

time-dependent stability parameter, which ensures an oscillating-free solution. To validate this

approach, the numerical results obtained in classical 2D problems are compared with the Galerkin

and the analytical solutions and experimental measurements.

2.8 Finite element approximations

Here we use finite element approximation to solve the problem (2.1). Standard numerical meth-

ods in order to approach the solution of a time dependent coupled diffusion on the polymer and

convection-diffusion equation in the contaminant are established. We use the finite element meth-

ods to solve numerically the AMCs cross-contamination model e.g [16], [28], [29], [19].

For this purpose, we use the Galerkin finite element formulation of the problem given by equa-

tion (2.1). It is obtained by multiplying the govering equation by an appropriate test function

respectively δCs and δCg for the concentration Cs ∈ Rd and Cg ∈ Rd, d := 2 or 3, respectively and

by integrating in over thecomputational subdomain Ωs and Ωg respectively. Throughout the sec-

tion, we use the standard notation and results on the Sobolev space and finite element formulations.

We now definite the following space :

V s :=Cs ∈ Rd, δCs ∈ [H1(Ωs)]

d; δCs = 0 on Γs

;

V g :=Cg ∈ Rd, δCg ∈ [H1(Ωg)]

d; δCg = 0 on Γg

[L2(Ωs,g)]d =

δCs,g : Ωs,g −→ R|

∫Ωs,g

|δCs,g|2 <∞

(2.20)

In this case, the weak formulation of the governing equation, a weighted residual formulation can

be obtained by multiplying the equation (2.1) by functions δCs ∈ [H1(Ωs)]d and δCg ∈ [H1(Ωg)]

d

and integrating over the respective volume

∫Ωs

∂Cs

∂tδCs dV

Ωi =

∫Ωs

∇ · (Ds∇Cs)δCs dV Ωi +

∫Ωs

q1δCs dVΩi , ∀δCs ∈ [H1(Ωs)]

d∫Ωg

∂Cg

∂tδCg dV

Ωi =

∫Ωg

∇ · (Dg∇Cg)δCg dV Ωi −∫

Ωg

∇ · (uCg)δCg dV Ωi +

∫Ωs

q2δCg dVΩi

∀δCg ∈ [H1(Ωg)]d

(2.21)

Let introduce [H1s (Ωs)]

d and [H1s (Ωg)]

d a functional space in which we are searching the solution

in accordance with its regularity H1s = δCs ∈ [H1(Ωs)]

d|δCs = s∀x ∈ Γs and H1s = δCg ∈

[H1(Ωg)]d|δCg = s∀x ∈ Γg where [H1(Ωs)]

d and [H1(Ωg)]d are Sobolev spaces.

Definition 2.6. Sobolev spaces are classicaly defined as [H1(Ωs)]d = δCs ∈ [L2(Ωs)]

d, ‖∇δCs‖ ∈

2.8. FINITE ELEMENT APPROXIMATIONS 33

[L2(Ωg)]d and [H1(Ωg)]

d = δCg ∈ [L2(Ωg)]d, ‖∇δCg‖ ∈ [L2(Ωg)]

d. The first space we need is

the space of square integrable functions.

Definition 2.7. We define [L2(Ωg)]d and [L2(Ωg)]

d respectively the Hilbert vector space of the

functions quadratically summable respectively in (Ωs) and (Ωg).

[L2(Ωs)]d =

δCs : Ωs −→ R|

∫Ωs

|δCs|2 <∞, [L2(Ωg)]

d =

δCg : Ωg −→ R|

∫Ωg

|δCg|2 <∞

The norm of these spaces is :

‖δCs‖1,Ωs =

(∫Ωs

∇δCs · ∇δCsdx) 1

2

; ‖δCg‖1,Ωg =

(∫Ωg

∇δCg · ∇δCgdx

) 12

(2.22)

According to the Green’s theorem, and selectively integration by parts of the equation (2.21)

leads to the weak formulation (2.23),

∫Ωs

∂Cs

∂tδCs dV

Ωi = −∫

Ωs

Ds∇Cs · ∇δCs dV Ωi +

∫∂Ωs

Ds∇Cs · nδCs dSΩi +

∫Ωs

q dV Ωi ,

∀δCs ∈ [H1(Ωs)]d∫

Ωg

∂Cg

∂tδCg dV

Ωi = −∫

Ωg

Dg∇Cg.∇δCg dV Ωi +

∫∂Ωg

Dg∇Cg · nδCg dSΩi

+∫

Ωg∇ · (uCg)δCg dV Ωi +

∫Ωs

q2δCg dVΩi , ∀δCg ∈ [H1(Ωg)]

d

(2.23)

in which n is the normal outward unit on dSΩs and dSΩg . By applying the Galerkin weighted

residual methods with a piecewise linear test functions which are continuous in space and in time

and after using the Green’s theorem [16], [28], [29], [19], the variational formulation corresponding

to the AMCs cross-contamination is given by the equation (2.1). The standard weak formulation

of the problem (2.1) with homogenous boundary conditions reads: find Cs ∈ [H1(Ωs)]d and Cg ∈

[H1(Ωg)]d such that,

a1

(∂Cs

∂t, δCs

)+ b1(Cs, δCs) = L1(δCs) ∀δCs ∈ [H1(Ωs)]

d

a2

(∂Cg

∂t, δCg

)+ b2(Cg, δCg) = L2(δCg) ∀δCg ∈ [H1(Ωg)]

d

(2.24)


where

L1(δCs) =

∫Ωs

q1δCs dVΩi ; L2(δCg) =

∫Ωs

q2δCg dVΩi +

∫Ωg

Ng0 δCg dS

Ωi ;

a1

(∂Cs

∂t, δCs

)=

∫Ωs

∂Cs

∂tδCs dV

Ωi ; a2

(∂Cg

∂t, δCg

)=

∫Ωg

∂Cg

∂tδCg dV

Ωi ;

b1(Cs, δCs) =

∫Ωs

Ds∇Cs · ∇δCs dV Ωi +

∫ΓN

h0CgδCs dS

Ωi

b2(Cg, δCg) =

∫Ωg

Dg∇Cg.∇δCg dV Ωi +

∫Ωg

∇ · (uCg)δCg dV Ωi +

∫ΓN

Cs

h0δCg dS

Ωi

+

∫ΓSDN

kc [C0H(t− ε)− Cg] δCg dSΩi

(2.25)

For the application of the model in microelectronic industry, we have for each main process

(contamination, purging, outgas, cleaning and decontamination) the following Dirichlet-Neumann

switch condition. These processes are linked and continue with the next order: the first and initial

process is the contamination (reference process, in application the wafer post processed is the

source of the contamination). The second process after the contamination process is the purging

and outgassing processes, in fact, the initial condition of these processes is the computed results

of the contamination process. The last processes are the cleaning and purging processes, during

which the initial conditions are the computed results of the purging and outgassing processes.

Remark 2.1 (Computational methods). The three linked simulations for each process must be

successively computed. If during the simulation the one of the processes is stopped for any reason,

in this case we must restart the simulation with the initial process (contamination).

1. Contamination process

Unknown Csc and Cgc

With the switch StDN conditions, we have:

if

kc 1 ⇒ Cgc ' C0H(t− ε) if 0 < t ≤ tc

kc = 0 ⇒ (−Dg∇Cgc + uCgc ) · n = 0 if t > tc

(2.26)

Dynamic boundary conditions (B.C) for the contaminant : Cgc ' C0H(t− ε) if 0 ≤ t ≤ tc

F aa (C0, Cgc ) = 0 if tc < t ≤ (tc + to)

(2.27)

Initial conditions (I.C) Csc (., t = 0) = 0 in (Ωs × 0) and Cgc (., t = 0) = 0 in (Ωg × 0),


we have :

Lc1(δCcs) =

∫Ωs

qc1δCcs dV

Ωi ; Lc2(δCcg) =

∫Ωs

qc2δCcg dV

Ωi +

∫Ωg

Ng0 δC

cg dS

Ωi ;

ac1

(∂Csc∂t

, δCcs

)=

∫Ωs

∂Csc∂t

δCcs dVΩi ; ac2

(∂Cgc∂t

, δCcg

)=

∫Ωg

∂Cgc∂t

δCcg dVΩi ;

bc1(Csc , δCcs) =

∫Ωs

Ds∇Csc · ∇δCcs dVΩi +

∫ΓN

h0Cgc δC

cs dS

Ωi

bc2(Cgc , δCcg) =

∫Ωg

Dg∇Cgc .∇δCcg dVΩi +

∫Ωg

∇ · (uCgc )δCcg dVΩi +

∫ΓN

Csch0δCcg dS

Ωi

+

∫ΓSDN

kc [C0H(t− ε)− Cgc ] δCcg dSΩi

(2.28)

2. Purge and outgassing processes

Unknown Csp and Cgp

With the switch StDN conditions, we have :

if

kc 1 ⇒ Cgp ' Cp0H(t− ε) if 0 < t ≤ (tc + tp)

kc = 0 ⇒ (−Dg∇Cgp + uCgp ) · n = 0 if t > (tc + tp)(2.29)

Dynamic boundary condition (B.C) for the contaminant : F pa (Cd0 , Cgp ) = Ng

0 + kc[Cp0H(t− ε)− Cgp

]if tc ≤ (tc + tp)

F pa (Cd0 , Cgp ) = 0 if (tc + tp) < t ≤ (tc + tp + td)

(2.30)

Initial conditions (I.C) Csp(., t = tc) = Csc in (Ωs×tc) and Cgp (., t = tc) = Cgc in (Ωg×tc)

We have,

Lp1(δCps ) =

∫Ωs

q1δCps dV

Ωi ; Lp2(δCpg ) =

∫Ωs

q2δCpg dV

Ωi +

∫Ωg

Ng0 δC

pg dS

Ωi ;

ap1

(∂Csp∂t

, δCps

)=

∫Ωs

∂Csp∂t

δCps dVΩi ; a2

(∂Cgp∂t

, δCpg

)=

∫Ωg

∂Cgp∂t

δCpg dVΩi ;

b1(Cs, δCps ) =

∫Ωs

Ds∇Csp · ∇δCps dVΩi +

∫ΓN

h0CgpδC

ps dS

Ωi

b2(Cgp , δCg) =

∫Ωg

Dg∇Cg.∇δCpg dVΩi +

∫Ωg

∇ · (uCgp )δCpg dVΩi +

∫ΓN

Csph0δCpg dS

Ωi

+

∫ΓSDN

kc[Cp0H(t− ε)− Cgp

]δCpg dS

Ωi

(2.31)

3. Decontamination and cleaning processes

Unknown Csd and Cgd , the cleaning time is tu =∑

i ti − tw


if

kc 1 ⇒ Cgd ' Cd0H(t− ε) if 0 < t ≤ (

∑i ti − tw)

kc = 0 ⇒ (−Dg∇Cgd + uCgd ) · n = 0 if t > (∑

i ti − tw)(2.32)


Dynamic boundary condition (B.C) for the contaminant :

F da (Cd0 , Cgd ) = Ng

0 + kc[Cd0H(t− ε)− Cgd

]if (tc + tp + td) ≤ t ≤ (

∑i ti − tw)

Fa(Cd0 , Cgd ) = 0 if t >

∑i ti

(2.33)

Initial conditions (I.C) Csd(., t = (tc + tp + td)) = Csp in (Ωs × tc + tp + td) and Cgd (., t =

tc + tp + td) = Cgp in (Ωg × tc + tp + td)

We have,

Ld1(δCds ) =

∫Ωs

q1δCds dV

Ωi ; Ld2(δCdg ) =

∫Ωs

q2δCdg dV

Ωi +

∫Ωg

Ng0 δCg dS

Ωi ;

ad1

(∂Csd∂t

, δCds

)=

∫Ωs

∂Csd∂t

δCds dVΩi ; ad2

(∂Cgd∂t

, δCdg

)=

∫Ωg

∂Cgd∂t

δCdg dVΩi ;

bd1(Csd, δCds ) =

∫Ωs

Ds∇Csd · ∇δCs dVΩi +

∫ΓN

h0CgδCds dS

Ωi

bd2(Cgd , δCdg ) =

∫Ωg

Dg∇Cgd .∇δCdg dV

Ωi +

∫Ωg

∇ · (uCgd )δCdg dVΩi +

∫ΓN

Cs

h0δCdg dS

Ωi

+

∫ΓSDN

kc[Cd0H(t− ε)− Cgd

]δCdg dS

Ωi

(2.34)

The finite element discretization and analysis of plane continua consist of the partitioning of the

structure, or the domain under consideration, into finite elements and the approximation of con-

tinuously distributed physical quantities (e.g. displacements) by discrete nodal degrees of freedom

and the assumption of their distribution over the element area. This assumption is associated with

the choice of shape functions.

In contrast to the spatial truss frame, for which a constructively discrete structure was available

already before the mathematical discretization, now a two-dimensional continuum Ωs and Ωg must

be subdivided into finite subdomains, the domains are decomposed in finite number of subdomains

Ωs := ∪eΩes and Ωg := ∪eΩeg. Inside these finite subdomains Ωes and Ωeg, or finite elements e, the

continuous field variables are approximated by means of shape functions and discrete nodal degrees

of freedom. Similarly, the boundary ∂Ωs and ∂Ωg are decomposed in ∂Ωes and ∂Ωeg. Finally the time

interval is subdivided by n subinterval. For the spatial discretization, we assume the finite element

partition T sh and T sh of Ωs and Ωg respectively into tetrahedral elements. Again for simplicity, we

will consider that the finite element partition associated to T sh and T sh are uniform, h is the size of

the element domains. Let us Csh and Cgh are the approximation solution of Cs and Cg respectively.

The source terms q is a given function assumed to be square integrable in Ωs.

The classical Galerkin approximation given by the equation (2.24) takes the form :


Find Csh ∈ V sh ⊂ [H1h(Ωg)]d and Cgh ∈ V

gh ⊂ [H1h(Ωg)]

d such that,a1

(∂Csh∂t

, δCsh

)+ b1(Csh, δCsh) = L1(δCsh) ∀δCsh ∈ [H1h(Ωs)]

d

a2

(∂Cgh∂t

, δCgh

)+ b2(Cg, δCgh) = L2(δCgh) ∀δCgh ∈ [H1h(Ωg)]

d

(2.35)

where

V sh =δCsh ∈ C0(Ωs)|δCsh/K ∈ P1(K), ∀K ∈ Fh

V gh =

δCgh ∈ C0(Ωg)|δCsh/K ∈ P1(K),∀K ∈ Fh

(2.36)

are the finite element space of continuous piecewise linear functions on Fh used to approximate

the exact solution of the model. Finally, we have a system of first order differential equation and

using the matrix notation we have,Ms 0

0 Mg

∂

∂t

Cs

Cg

+

Ds h0

h−10 Dg

Cs

Cg

=

Fs

Fg

(2.37)

in which Cs ∈ R2 and Cg ∈ R2 are the unknowns concentration vectors on nodes respectively

in the domain (Ωs) and (Ωg). Ms and Mg are the time constant matrix, Fs and Fg are the source

and external flux vectors.

The coupled system of ordinary differential equations, given by the equation (2.37) has to be

integrated in time. Let 0 = t0 < t1 < t2 < · · · < tn < · · · < tN = T be a partition of the time

interval I into steps of the length ∆t = tn − tn−1, for n = 1, 2, · · · , N . Using the finite difference

approximation and the explicit Euler scheme for Cs and Cg.

We define the time steps for the time range [0, Tf ] in the equation (2.38) and (2.39)

∆t = tn+1 − tn (2.38)

[0, Tf ] =

n⋃i=1

[ti, ti + ∆t] (2.39)

where T ∈ R+ is the range time and ∆t is the step time.

Cst+∆t =

Cst+∆t −Cs

t

∆t; Cg

t+∆t =Cgt+∆t −Cg

t

∆t; (2.40)


The system (2.37) can be written, at time t+ ∆t, as:Ms 0

0 Mg

e

Csn+1 −Cs

n

Cgn+1 −Cg

n

e

+ ∆t

Ds h0

h−10 Dg

e

Csn

Cgn

e

= ∆t

Fsn

Fgn

e

(2.41)

where

Fsn =

∫Ωe

s

qN dV Ωi ; Fgn =

∫Ωe

g

Ng0 N dV Ωi ; Ms =

∫Ωe

s

NTN dV Ωi ; Mg =

∫Ωe

g

NTN dV Ωi

Ds =

∫Ωe

s

Ds∇NT · ∇N dV Ωi +

∫ΓN

h0NTN dSΩi ; Dg =

∫Ωe

g

Dg∇NT.∇N dV Ωi

+

∫Ωe

g

∇ · (uNT)N dV Ωi +

∫ΓN

NT

h0N dSΩi +

∫ΓD

kc[C0H(t− ε)−NT

]N dSΩi

(2.42)

and N denote the linear interpolation function at each node.

And by using the assembling theory for all subdomains, we have:

m⋃e=1

Ms 0

0 Mg

e

Csn+1 −Cs

n

Cgn+1 −Cg

n

e

+

m⋃e=1

∆t

Ds h0

h−10 Dg

e

Csn

Cgn

e

=

m⋃e=1

∆t

Fsn

Fgn

e

(2.43)

where Ms,gij =

pnodes∑i,j

Ms,gij , Ds,gij =

pnodes∑i,j

Ds,gij , Fs,gj =

pnodes∑j

Fs,gj ,

Remark 2.2 (Stability). When diffusion is the only mechanism for cross-contamination transfer,

there are conditions for which the Galerkin method fails to produce smooth solutions. It is well

known that this method, based on piecewise polynomial approximations, yields poor solutions for

low thermal diffusivity materials (Ds, Dg, for the AMCs cross-contamination the diffusion of the

contaminant in the polymer is in general 1e-14 and 1e-15) and/or when the time step is small (Ds ≤

h2∆t, Dg ≤ h2∆t). Thus, one way to overcome such limitations consists in using stabilized finite

element methods. In the following, we discuss the use of enriched method on unsteady diffusion

problems.

To this end, we recall the equation (2.35) as: find Csh ∈ V sh ⊂ [H1h(Ωg)]d and Cgh ∈ V gh ⊂

[H1h(Ωg)]d such that,

(Cs,n+1h

∆t, δCsh

)+(Ds∇Cs,n+1

h ,∇δCsh)

= (q1, δCsh) +

(Cs,nh∆t

, δCsh

)∀δCsh ∈ V sh(

Cg,n+1h

∆t, δCgh

)+(Dg∇Cg,n+1

h ,∇δCgh)

= (q2, δCgh) +(∇ · (uCg,n+1

h ), δCgh)

+

(Cg,nh∆t

, δCgh

)∀δCgh ∈ V gh

(2.44)


We introduce the following subspaces V s∗h and V g∗h , with the inner product notation, into :

V s∗h =δCsh ∈ C0(Ωs)|δCsh/K ∈ P1(K)⊕B(K), ∀K ∈ Fh

V g∗h =

δCgh ∈ C0(Ωg)|δCsh/K ∈ P1(K)⊕B(K), ∀K ∈ Fh

(2.45)

In which B(K) is the bubble functions which satisfies φs(x), φg(x) > 0∀x ∈ K, φs(x), φg(x) =

0∀x ∈ K and φs(x), φg(x) = 1 at the barycenter of K. In fact, we decompose Csh ∈ V s∗h and

Cgh ∈ Vg∗h into its linear part Cs1 ∈ V sh and C

g1 ∈ V

gh . We have :

Csh = Cs1 +∑K∈Fh

Cs∗Kφs, Cgh = Cg1 +

∑K∈Fh

Cg∗Kφg (2.46)

where Cs∗K and Cg∗K are the unknown bubble coefficients.

(Cs,n+1h

∆t, φs

)K

+(Ds∇Cs,n+1

h ,∇φs)K

= (q1, φs)K +

(Cs,nh∆t

, φs)K(

Cg,n+1h

∆t, φg

)K

+(Dg∇Cg,n+1

h ,∇φg)K

= (q2, δCgh)K +(∇ · (uCg,n+1

h ), φg)K

+

(Cg,nh∆t

, φg)K

(2.47)

By using the decomposition (2.46) of the solution and subsituting it into the (2.47) :

(Cs,n+1

1

∆t, φs)K

+ Cs∗K

(φs,n+1

∆t, φs)K

+(Ds∇Cs,n+1

1 ,∇φs)K

+ Cs∗K(Ds∇φs,n+1,∇φs

)K

= (q1, φs)K +

(Cs,nh∆t

, φs)K(

Cg,n+11

∆t, φg)K

+ Cg∗K

(φg,n+1

∆t, φg)K

+(Dg∇Cg,n+1

1 ,∇φg)K

+ Cg∗K(Dg∇φg,n+1,∇φg

)K

= (q2, δCg1)K +(∇ · (uCg,n+1

1 ), φg)K

+

(Cg,n1

∆t, φg)K

(2.48)

We use the shape functions and vanishing the third order term. Solving the equation (2.48) for

the bubble coefficient in each element K ∈ Fh, leads to :

Cs∗K =1

1∆t ||φs||

20,K + Ds||φs||20,K

((q1, φ

s)K +

(Cs,nh∆t

, φs)K

−(Cs,n+1

1

∆t, φs)K

)

Cg∗K =1

1∆t ||φg||

20,K + Dg||φg||20,K

((q2, δCg1)K +

(∇ · (uCg,n+1

1 ), φg)K

+

(Cg,n1

∆t, φg)K

)(−C

g,n+11

∆t, φs)K

(2.49)

Where ||φs||20,K =∫KφsdΩs and ||φg||20,K =

∫KφgdΩg.

The bubbles considered here are quasi-static, i.e., that the effect of their time variation may


be neglected. Note that following the evolution of small-scales in time is an interesting method,

but for this type of equation, it could increase the computational cost without considerable gain

in accuracy. Hereafter, we need to solve equation (2.44) on the macro-scale, but not treated in

this section. The static condensation procedure will eliminate the bubbles function at the element

level and re-write the weak formulation with the stabilization coefficient. The previous method

improves stability by adding a stabilizing term obtained after condensation of the bubble function

in the original problem. But as mentioned before, this can work only in particular case when the

source term is zero. Furthermore, in order to avoid an extra diffusion effect and thus a non-realistic

result toward the steady state, a cut-off strategy is introduced. This strategy consists in modifying

the stabilization parameter making it varying with time and depending on the regularity of the

approximate solution. In practice, this diffusion correction factor can be seen as a function of the

element Péclet number often used in convection-dominated problems. In conclusion, this strategy

will at the same time ensure stability in the initial iterations and convergence toward the steady

state without extra diffusivity.

By assuming this switch condition, the matrix components in equation (2.43), can be expressed

as follows :

1. Contamination process

Unknown Csc and Cgc


if

kc 1 ⇒ Cgc ' C0H(t− ε) if 0 < t ≤ tc


(2.50)

Dynamic boundary condition (B.C) for the contaminant on (ΓD × [0, Tf ]) : Cgc ' C0H(t− ε) if 0 ≤ t ≤ tc

F ca(C0, Cgc ) = 0 if tc < t ≤ (tc + to)

(2.51)

Initial conditions (I.C) Csc (., t = 0) = 0 in (Ωs × 0) and Cgc (., t = 0) = 0 in (Ωg × 0)

We have,

m⋃e=1

(Ms)(c)

0

0(Mg)(c)

e

Csc,n+1 −Cs

c,n

Cgc,n+1 −Cg

c,n

e

c

+

m⋃e=1

∆t

(Ds)(c)

h0

h−10

(Dg)(c)

e

Csc,n

Cgc,n

e

c


=

m⋃e=1

∆t

Fsc,n

Fgc,n

e

c

(2.52)

where

(Msij

)(c)

=

pnodes∑i,j

(Msij

)(c),(Dsij)(c)

=

pnodes∑i,j

(Dsij)(c)

,(Fsj)(c)

=

pnodes∑j

(Fsj)(c)

,

(Mgij

)(c)

=

pnodes∑i,j

(Mgij

)(c),(Dgij)(c)

=

pnodes∑i,j

(Dgij)(c)

,(Fgj)(c)

=

pnodes∑j

(Fgj)(c)

, (2.53)

2. Purge and outgassing processes

Unknown Csp and Cgp


if



Dynamic boundary condition (B.C) for the contaminant on (ΓD × [0, Tf ]) : F pa (Cp0 , Cgp ) = Ng



F pa (Cp0 , Cgp ) = 0 if (tc + tp) < t ≤ (tc + tp + td)

(2.55)

Initial conditions (I.C) Csp(., t = tc) = Csc in (Ωs×tc) and Cgp (., t = tc) = Cgc in (Ωg×tc)

We have,

m⋃e=1

(Ms)(p)

0

0(Mg)(p)

e

Csp,n+1 −Cs

p,n

Cgp,n+1 −Cg

p,n

e

p

+

m⋃e=1

∆t

(Ds)(p)

h0

h−10

(Dg)(p)

e

p

Csp,n

Cgp,n

e

p

=

m⋃e=1

∆t

Fsp,n

Fgp,n

e

p

(2.56)

Where

(Msij

)(p)

=

pnodes∑i,j

(Msij

)(p),(Dsij)(p)

=

pnodes∑i,j

(Dsij)(p)

,(Fsj)(p)

=

pnodes∑j

(Fsj)(p)

,

(Mgij

)(p)

=

pnodes∑i,j

(Mgij

)(p),(Dgij)(p)

=

pnodes∑i,j

(Dgij)(p)

,(Fgj)(p)

=

pnodes∑j

(Fgj)(p)

, (2.57)

3. Decontamination and cleaning processes



i ti − tw

With the switch (StDN conditions, we have:

if

kc 1 ⇒ Cgd ' Cd0H(t− ε) if 0 < t ≤ (

∑i ti − tw)


i ti − tw)(2.58)

Dynamic boundary condition (B.C) for the contamination (ΓD × [0, Tf ]) :



]if (tc + tp + td) ≤ t ≤ (

∑i ti − tw)


∑i ti

(2.59)



We have,

m⋃e=1

(Ms)(d)

0

0(Mg)(d)

e

Csd,n+1 −Cs

d,n

Cgd,n+1 −Cg

d,n

e

d

+

m⋃e=1

∆t

(Ds)(d)

h0

h−10

(Dg)(d)

e

Csd,n

Cgd,n

e

d

=

m⋃e=1

∆t

Fsd,n

Fgd,n

e

d

(2.60)

Where

(Msij

)(d)

=

pnodes∑i,j

(Msij

)(d),(Dsij)(d)

=

pnodes∑i,j

(Dsij)(d)

,(Fsj)(d)

=

pnodes∑j

(Fsj)(d)

,

(Mgij

)(d)

=

pnodes∑i,j

(Mgij

)(d),(Dgij)(d)

=

pnodes∑i,j

(Dgij)(d)

,(Fgj)(d)

=

pnodes∑j

(Fgj)(d)

, (2.61)

We resume in the following section the computation order of the AMCs cross-contamination

under the industrial constraints. We present just the sequence and the order of the computation

but not the algorithms of computation.

2.9 Computation details

Before going to the main results of the model, let us give a key of the computation order of the

AMCs cross-contamination approximated model. For the computation we use three sub-simulations

for each main step. The time range is t ∈ [0,∑

i ti] or t ∈ [0, Tf ]. For this purpose, we separate the

2.9. COMPUTATION DETAILS 43

simulation in three sub-simulation, SUB 1 , SUB 2 and SUB 3 . Each simulation contains

respectively StDN 1, StDN 2 and StDN 3. The computation order for the cross-contamination

model is resumed by the following step.

SUB 1 for 0 ≤ t ≤ tc: COMPUTE: Contamination process

-Unknowns : Csc and Cg

c

-StDN 1, Cf. eq.(2.50)

–B.C (Contamination) on (ΓD × [0, Tf ]) Cf. eq.(2.51)

-I.C Csc (., t = 0) = 0 in (Ωs × 0) and Cgc (., t = 0) = 0 in (Ωg × 0)

COMPUTE Cf. eq.(2.52)

m⋃e=1

(Ms)(c)

0

0(Mg)(c)

e

Csc,n+1 −Cs

c,n

Cgc,n+1 −Cg

c,n

e

c

+

m⋃e=1

∆t

(Ds)(c)

h0

h−10

(Dg)(c)

e

Csc,n

Cgc,n

e

c

=

m⋃e=1

∆t

Fsc,n

Fgc,n

e

c

(2.62)

OUTPUT : Csc and Cg

c

⇓

SUB 2 tc < t ≤ (tc + tp + td) : COMPUTE : Purge and outgass

-Unknowns : Csp and Cg

p

-StDN 2 Cf. eq.(2.54)

-B.C (Purge) on (ΓD × [0, Tf ]) Cf. eq.(2.55)

-I.C Csp(., t = tc) = Csc in (Ωs × tc) and Cgp (., t = tc) = Cgc in (Ωg × tc)


m⋃e=1

(Ms)(p)

0

0(Mg)(p)

e

Csp,n+1 −Cs

p,n

Cgp,n+1 −Cg

p,n

e

d

+

m⋃e=1

∆t

(Ds)(p)

h0

h−10

(Dg)(p)

e

Csp,n

Cgp,n

e

p

=

m⋃e=1

∆t

Fsp,n

Fgp,n

e

p

(2.63)

OUTPUT : Csp and Cg

p


⇓

SUB 3 (tc + tp + td) < t ≤∑

i ti : COMPUTE : Decontamination and cleaning

processes

-Unknowns : Csd and Cg

d

-StDN 3 Cf. eq.(2.58)

-B.C (Decontamination) on (ΓD × [0, Tf ]) Cf. eq.(2.59)

-I.C Csd(., t = (tc + tp + td)) = Csp in (Ωs × tc + tp + td) and

Cgd (., t = (tc + tp + td)) = Cgp in (Ωg × tc + tp + td)


m⋃e=1

(Ms)(d)

0

0(Mg)(d)

e

Csd,n+1 −Cs

d,n

Cgd,n+1 −Cg

d,n

e

d

+

m⋃e=1

∆t

(Ds)(d)

h0

h−10

(Dg)(d)

e

Csd,n

Cgd,n

e

d

=

m⋃e=1

∆t

Fsd,n

Fgd,n

e

d

(2.64)

OUTPUT : Csd and Cg

d

This specific numerical method is implemented into a finite element code (Comsol Multiphysics)

to examine the capabilities of the chosen formulation. Comsol Multiphysics allows us to compute

easily finite element formulation based on the weak formulation and allow us to compute the sub-

simulation SUB 1 , SUB 2 and SUB 3 . Each sub-simulation is saved in the current results

and will be used as initial condition for the next sub simulation in order to link the three main

sub-simulations for the AMC cross-contamination model.

2.10 Main results, findings and discussion

This section is devoted to the presentation of the main results obtained by using numerical approx-

imation, optimization and finite element method. Two practical examples will be solved for the

validation and the correlation study of the model with the experimental measurement. The first

one comprises the test of adsorption of the contaminant in the polymer. We will solve this problem

using different material. In the second example we use the simulation of the dynamic of the AMC

cross contamination in order to study the sensitivity of the material on the contamination.

We present the validation of the model with experiment measurement. We discuss many con-

ditions corresponding to the industrial’s application (contamination process, opening and purging

2.10. MAIN RESULTS, FINDINGS AND DISCUSSION 45

process). In practice, this method is defined as the contaminant concentration increases to the

maximum value during the contamination time, and when the contamination time is finished; the

wafer area is considered as a neutral area.

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300

350

400

Time in [h]

Q in

[ng/

cm2 ]

Sorbed quantity in [ng/cm2]

Observed ±11%Computed

0 500 1000 1500 20000

5

10

15

20

25

30

35 Sorbed quantity in [ng/cm2]

Time in [h]

Q in

[ng/

cm2 ]


Figure 2.8: Sorbed quantity in the polymer in function of time for the contaminant XC1 (left): computed model in blue, experimental measurement in red. Sorbed quantity in the polymerin function of time for the contaminant XC2 (right) : computed model in blue, experimentalmeasurement in red.

0 500 1000 1500 2000 2500 30000

5

10

15

20

25

30

35

40


Time in [h]

Q in

[ng/

cm2 ]


0 200 400 600 800 10000

2

4

6

8

10

12


Time in [h]

Q in

[ng/

cm2 ]


Figure 2.9: Sorbed quantity in the polymer in function of time for the contaminant XC1 (left): computed model in blue, experimental measurement in red. Sorbed quantity in the polymerin function of time for the contaminant XC2 (right) : computed model in blue, experimentalmeasurement in red.

Figures 2.8 and 2.9, illustrate the correlation between the computed model and the experimental

measurements during the contamination process. The curves represent the sorbed quantity for two

different contaminants XC1 and XC2 in function of time. We can observe that the mathematical

model is in correlation with the experimental measurement. We use the first model governed by

the equation (2.1) to study the correlation of the model with the experimental measurements. We

measured the sorbed quantity of the contaminant in the polymer with the ionic chromatography

methods, and we obtained the diffusion coefficient with numerical optimization by using the model

given by the equation (2.1) and the equation (2.13).


0 0.5 1 1.5 2 2.5 3

x 10−5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Thickness [m]

Con

cent

ratio

n in

[mol

/m3 ]

Concentration in polymer

COPPEEKPCPEI

0 0.2 0.4 0.6 0.8 1 1.2

x 10−5

0

0.1

0.2

0.3

0.4

0.5

0.6

Thickness [m]

Con

cent

ratio

n in

[mol

/m3 ]


COPPEEKPCPEI

Figure 2.10: Contamination process, for the contaminant XC1 : after 1[h] of contamination and1[h] of waiting time just after the contamination process (for diffusion). The curve illustrates theconcentration of the contaminant in the polymer in function of the thickness. Each curve illustratesthe concentration of the contaminant for one material (polymer, PEEK, PC, COP, PEI) used forthe FOUP.

0 0.5 1 1.5 2 2.5 3

x 10−5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Thickness [m]

Con

cent

ratio

n in

[mol

/m3 ]


COPPEEKPCPEI

0 0.5 1 1.5 2

x 10−5

0

0.05

0.1

0.15

0.2

0.25

Thickness [m]

Con

cent

ratio

n in

[mol

/m3 ]


COPPEEKPCPEI

Figure 2.11: Contamination process for the contaminant XC2 : after 1[h] of contamination and1[h] of waiting time just after the contamination process (for diffusion). The curve illustrates theconcentration of the contaminant in the polymer in function of the thickness. Each curve illustratesthe concentration of the contaminant for one material (polymer, PEEK, PC, COP, PEI) used forthe FOUP.

Figures 2.10 illustrates the concentration for the contamination process for the contaminant

XC1 : after 1[h] of contamination and 1[h] of waiting time (just after the contamination process for

diffusion). The curve illustrates the concentration of the contaminant in the polymer in function

of the thickness. Each curve illustrates the concentration of the contaminant for one material

(polymer, PEEK, PC, COP, PEI) used for the FOUP. We can see that, some material absorbs great

amount of contaminant in the polymer and promotes quickly the diffusion in the volume. While

other material adsorbs loss amount of the contaminant in the polymer and the quantity diffused

is small. So we can determine directly with this figure the optimal material for the contamination

criterion.

Figures 2.11 represent the sorbed quantity in the polymer in function of time for the contam-


0 0.5 1 1.5 2 2.5 3

x 10−5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Thickness [m]

Con

cent

ratio

n in

[mol

/m3 ]


COPPEEKPCPEI

0 0.2 0.4 0.6 0.8 1 1.2

x 10−5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Thickness [m]

Con

cent

ratio

n in

[mol

/m3 ]


COPPEEKPCPEI


0 0.5 1 1.5 2 2.5 3

x 10−5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Thickness [m]

Con

cent

ratio

n in

[mol

/m3 ]


COPPEEKPCPEI

0 0.5 1 1.5 2

x 10−5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Thickness [m]

Con

cent

ratio

n in

[mol

/m3 ]


COPPEEKPCPEI


inant XC1 (left) : computed model in blue, experimental measurement in red. Sorbed quantity

in the polymer in function of time for the contaminant XC2 (right): computed model in blue, ex-

perimental measurement in red. These computed models are in correlation with the experimental

measurements.

Figures 2.12 represent for contamination process: after 2[h] of contamination and 2[h] of waiting

time just after the contamination process (for diffusion). The curve illustrates the concentration

of the contaminant in the polymer in function of the thickness. Each curve illustrates the concen-

tration of the contaminant for one material (polymer, PEEK, PC, COP, PEI) used for the FOUP.

These computed models are in correlation with the experimental measurements.

Figures 2.13 illustrate the contamination process: after 2[h] of contamination and 2[h] of waiting


Figure 2.14: Contamination process : after 1[h] of contamination and 1[h] of waiting time. Thecurve illustrates the concentration of the contaminant in the polymer in function of time (left).Outgassing : 5[mn] removing wafer, 22[h] waiting time for atmosphere concentration equilibrium.The curve illustrates, after cleaning and contamination processes, the concentration of the contam-inant in the polymer in function of thickness for the contaminant XC1 (left) and the concentrationof the contaminant in the polymer in function of thickness for the contaminant XC2 (right). Eachcurve in the figures represents the profiles for a given time. We can see the response of the modelwith the swich condition StDN for the transient analysis

Figure 2.15: Cleaning/decontamination process : after 1[h] of contamination and 1[h] of waitingtime. The curve illustrates the concentration of the contaminant in the polymer in function of thethickness. Outgassing : 5[mn] removing wafer, 22[h] waiting time for atmosphere concentrationequilibrium. The curve illustrates, after cleaning and decontamination processes, the concentrationof the contaminant in the polymer in function of thickness for the contaminant XC1 (left) and theconcentration of the contaminant in the polymer in function of thickness for the contaminant XC2(right). Each curve in the figures represents the profiles for a given time. We can see the responseof the model with the switch condition StDN for the transient analysis.

time just after the contamination process (for diffusion). The curve illustrates the concentration

of the contaminant in the polymer in function of the thickness. Each curve illustrates the concen-

tration of the contaminant for one material (polymer, PEEK, PC, COP, PEI) used for the FOUP.

These computed models are in correlation with the experimental measurements.


Figure 2.16: Cleaning/decontamination process, for the contaminant XC1 : after 1[h] of contam-ination and 1[h] of waiting time, 5[mn] of outgassing (removing wafer), 22[h] of waiting time foratmosphere concentration equilibrium. The decontamination time is 4[h] and 22[h] of waiting timeafter cleaning process. The curve illustrates the concentration of the contaminant in the FOUP’satmosphere in function of time (left). The curve illustrates the concentration of the contaminantin the polymer in function of time (right). We can see the response of the model with the switchcondition StDN for the transient analysis.

Figure 2.17: Cleaning/decontamination process, for the contaminant XC2 : after 1[h] of contam-ination and 1[h] of waiting time, 5[mn] of outgassing (removing wafer), 22[h] of waiting time foratmosphere concentration equilibrium. The decontamination time is 4[h] and 22[h] of waiting timeafter cleaning process. The curve illustrates the concentration of the contaminant in the FOUP’satmosphere in function of time (left). The curve illustrates the concentration of the contaminantin the polymer in function of time (right). We can see the response of the model with the switchcondition StDN for the transient analysis.

Figures 2.14 represent the contamination process: after 1[h] of contamination and 1[h] of waiting

time an after outgassing : 5[mn] removing wafer, 22[h] waiting time for atmosphere concentration

equilibrium. The curve illustrates the concentration of the contaminant in the polymer in function

of time (left). The curve illustrates, after cleaning and contamination processes, the concentration

of the contaminant in the polymer in function of thickness for the contaminant XC1 (left) and the


Figure 2.18: Contamination process for the contaminant XC1 (first row)/Decontamination pro-cess for the contaminant XC1 (first row) : after 1[h] of contamination and 1[h] of waiting time,outgassing : 5[mn] for removing wafer, 22[h] for waiting time for atmosphere concentration equi-librium. The isosurface illustrates the concentration of the contaminant XC1, in the polymer aftercontamination process (first row). The curve illustrates the concentration of the contaminant XC1in the polymer in function of time after decontamination and cleaning processes (second row). Wecan see the response of the model with the switch condition StDN for the transient analysis.

concentration of the contaminant in the polymer in function of thickness for the contaminant XC2

(right). Each curve in the figures represents the profiles for a given time. We can see the response

of the model with the switch condition StDN for the transient analysis.

Figures 2.15 illustrate the concentration evolution during the cleaning/decontamination process

: after 1[h] of contamination and 1[h] of waiting time. The curve illustrates the concentration of

the contaminant in the polymer in function of the thickness. Outgassing : 5[mn] removing wafer,

22[h] waiting time for atmosphere concentration equilibrium. The curve illustrates, after cleaning

and decontamination processes, the concentration of the contaminant in the polymer in function of

thickness for the contaminant XC1 (left) and the concentration of the contaminant in the polymer

in function of thickness for the contaminant XC2 (right). Each curve in the figures represents the

profiles for a given time. We can see the response of the model with the switch condition StDN for

the transient analysis. The maximum amount of concentration in the polymer stays in the polymer

after cleaning and decontamination processes. Two concentration gradients are obtained by the

switch condition and, the first gradient continues to diffuse in the polymer and the second gradient

returns back in the FOUP’s atmosphere and re-contamines the wafers (cross-contamination).

Figures 2.16 illustrate the concentration during the leaning/decontamination process, for the

contaminant XC1 : after 1[h] of contamination and 1[h] of waiting time, 5[mn] of outgassing (remov-

ing wafer), 22[h] of waiting time for atmosphere concentration equilibrium. The decontamination

time is 4[h] and 22[h] of waiting time after cleaning process. The curve illustrates the concentra-

2.11. CONCLUSION 51

tion of the contaminant in the FOUP’s atmosphere in function of time (left). The curve illustrates

the concentration of the contaminant in the polymer in function of time (right). We can see the

response of the model with the switch condition StDN for the transient analysis. The maximum

amount of concentration in the polymer stays in the polymer after cleaning and decontamination

processes. Two concentration gradients are obtained by the switch condition and, the first gradient

continues to diffuse in the polymer and the second gradient return back in the FOUP’s atmosphere

and re-contamines the wafers (cross-contamination).

Figures 2.17 represent the concentration during the cleaning/decontamination process, for the

contaminant XC2 : after 1[h] of contamination and 1[h] of waiting time, 5[mn] of outgassing (remov-

ing wafer), 22[h] of waiting time for atmosphere concentration equilibrium. The decontamination

time is 4[h] and 22[h] of waiting time after cleaning process. The curve illustrates the concentra-

tion of the contaminant in the FOUP’s atmosphere in function of time (left). The curve illustrates

the concentration of the contaminant in the polymer in function of time (right). We can see the

response of the model with the switch condition StDN for the transient analysis. The maximum

amount of concentration in the polymer stays in the polymer after cleaning and decontamination

processes. Two concentration gradients are obtained by the switch condition and, the first gradient

continues to diffuse in the polymer and the second gradient returns back in the FOUP’s atmosphere

and re-contamines the wafer (cross-contamination).

Figures 2.18 represent concentration during the contamination process for the contaminant XC1

decontamination process for the contaminant XC1 (first row) : after 1[h] of contamination and 1[h]

of waiting time, outgassing: 5[mn] for removing wafer, 22[h] for waiting time for atmosphere

concentration equilibrium. The isosurface illustrates the concentration of the contaminant XC1,

in the polymer after contamination process (first row). The curve illustrates the concentration

of the contaminant XC1 in the polymer in function of time after decontamination and cleaning

processes (second row). We can see the response of the model with the switch condition StDN for

the transient analysis. The maximum amount of concentration in the polymer stays in the polymer

after cleaning and decontamination processes. Two concentration gradients are obtained by the

switch condition and, the first gradient continues to diffuse in the polymer and the second gradient

returns back in the FOUP’s atmosphere and re-contamines the wafers (cross-contamination).

2.11 Conclusion

The AMCs cross contamination model forms a coupled partial differential equations. In this case,

it is impossible to find explicit analytical solutions. Most approaches are undertaken to the AMCs

simulation, a set of coupled partial differential equations has been solved by finite element method.


The solution has been mainly determined by the applied boundary conditions as defined in each

process. A considerable variety of boundary conditions have been implemented in Comsol multi-

physics.

Some basic mathematical properties have been analyzed to fulfill self-consistent formulations of

the boundary conditions in the device simulation. These formulations and implementations have

been analyzed from the mathematical and the numerical points of view, illustrating both correct

and inconsistent approaches with examples. An investigation of the use of the types of boundary

conditions from both the mathematical and the numerical points of view are discussed.

Consistency and convergence behavior has been illustrated with computational results and ex-

perimental measurement. The performance of the tools provides following conclusions and remarks

: during the contamination processes the concentration is adsorbed in the polymer and continues to

diffuse during the outgassing process of the wafer and after the stopping outgas. Using the Dirichlet

to Neumann’s boundary conditions performs this condition. This first step conditioned the entire

next step, it can be seen by simulation that we can obtain with this first approach a significant

value of contamination level in the FOUP compared with the experimental measurement.

The model is stable and consistent for these conditions but a mathematical development of

these mixed boundary conditions is not yet developed in this work. It will be performed in a next

work. It was demonstrated that we have exactly the same solution if we separate the part of the

Dirichlet to Neumann condition into two sub simulations i.e. we use just the Dirichlet condition

and after that we apply the Neumann condition such that the initial condition is the last computed

solution.

It can be seen that during the cleaning time the concentration of the contaminant near surface is

outgassed and purged to the internal surface with the inert gas. We remark that we have two parts

of the concentration gradient in the polymer; one part comes to the internal surface (reversible

contamination) and one part to volume (irreversible contamination). It can be seen that one part

the contaminant continues to diffuse in the volume and one part of the contaminant come through

to the surface absorbed by the purging system. Indeed the maximum of the concentration stay in

the volume of the polymer.

This residual part of the contamination can move (diffuse) in two directions : irreversible

contamination in the volume of the polymer and the other part (the reversible parts) can constitute

the AMC source which contamines the new wafer in the FOUP atmosphere after cleaning. It can be

seen that the irreversible contamination results to an accumulation with the residual concentration

in the polymer. It can be proven that the level of these parts increases and has effect in the cleaning

time.

2.11. CONCLUSION 53

The reversible part of the contamination accumulated in the polymer is the source of the

contamination of the new wafer in the FOUP. With this process it has been proven that more

the number of the cycle of the contamination increases, more the time of the cleaning time must

increase too in order to take into account of the residual contaminant added during each cycle.

We can then estimate the life time of the FOUP and the optimal time of the cleaning process in

function of the cycle number.

Another complex model not established in this work can be performed in the case we don’t

have the value of this free enthalpy. It is expected that similar analysis can be carried out in other

geometries taking into account the FOUP scale and the wafers supports. Another approach can be

undertaken for example to take into account the purging fluid flow circulation in the FOUP and

to define for each contaminant a kinetic law at the interface. The simple domain considered here

allows us to utilize a simpler conditions and methods which simplify the calculations and analysis.

It may give a best understanding of the AMCs cross contamination’s dynamics. It is natural to

suggest that when one considers other geometries, the use of some assumptions will be necessary.

The cross-contamination phenomena is thermically activated so it is natural to complete the study

with the heat effect on the decontamination process.

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inants inducing defects in integrated circuits manufacturing, Microelectronic Engineering, Vol.

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Chapter 3

Dynamic boundary conditions forcoupled system of PDEs.

1

Resumé – Dans le chapitre un, nous avons introduit un modèle isotherme de contamination

croisée avec les analyses dynamiques et les validations du modèle avec les mesures expérimentales.

Nous avons aussi mis au point dans le chapitre l’outil permettant de traiter les conditions de

switch en temps (Dirichlet / Neumann StDN). Dans ce chapitre nous étudions en plus l’effet

de température sur le transfert de masse de la contamination moléculaire croisée (AMC) entre

le Front Opening Unified Pod (FOUP) et les wafers (des substrats de silicium) au cours de la

fabrication de dispositifs électroniques tels que les circuits intégrés et les puces électroniques dans

l’industrie microélectronique. Ces phénomènes de contamination croisée conduisent à un effet

négatif sur le rendement de la production et sur la performance des dispositifs ainsi fabriqués.

Une approche prédictive en utilisant la modélisation et les méthodes de calcul est un moyen très

efficace pour comprendre l’effet de la température sur ce phénomène de contamination croisée.

D’abord, un modèle couplé, diffusion et convection-diffusion avec des effets de variation de la

température, en variant le coefficient de diffusion par la loi d’Arrhenius, est formulé pour définir

les phénomènes. Ensuite, nous généralisons le modèle en introduisant l’équation de la chaleur

pour calculer le gradient de température et ensuite on l’introduit dans le coefficient de diffusion.

Des méthodes d’optimisation utilisant la solution numérique pour définir les constantes physiques

de divers matériaux en fonction des contaminants ont été étudiées. Enfin, des caractérisations

expérimentales ont été conduites pour valider les modèles prédictifs. Nous avons utilisé la méthode

des éléments finis pour résoudre numériquement le problème proposé. Le comportement dynamique

de l’analyse AMC a été déterminé grâce à la condition de switch de Dirichlet / Neumann StDN.

1 This chapter was published in International Journal of applied Mathematical Research under title: "Dynamicboundary conditions for a coupled convection-diffusion model with heat effects : applications in cross-contaminationcontrol", Vol.4(1), pp. 58-77, 2015.

57

58CHAPTER 3. DYNAMIC BOUNDARY CONDITIONS FOR COUPLED SYSTEM OF PDES.

Le modèle mathématique préserve les formes classiques de la diffusion et diffusion de convection

de Fick. Les résultats numériques sont en corrélation avec les mesures expérimentales. Quelques

résultats numériques sont présentés dans ce travail.

Abstract –This work investigates the mass transfer with heat effect of the Airborne Molec-

ular cross-contamination (AMCs) between the Front Opening Unified Pod (FOUP) and wafer

(silicon substrates) during the microelectronics devices manufacturing using dynamic boundary

conditions. Such cross-contamination phenomena lead to detrimental impact on production yield

in microelectronic industry and a predictive approach using modeling and computational meth-

ods is a well-known way to understand and to qualify the AMCs cross-contamination processes.

The FOUP is made of polymeric materials and it is considered as a heterogeneous porous media,

it can adsorb and desorb the contaminant, thus the modeled processes are the contamination of

two-components in transient analysis. Coupled diffusion and convection-diffusion models with heat

effects are used to define the phenomena. The present methodology is, first using the optimization

methods with one dimensional analytical solution in order to define the physical constants (dif-

fusion constant) of various materials which have been studied experimentally and separately, and

the second using the finite element methods including these physical constants in the model and

relevant interface condition in order to take into account the adsorption kinetic law. Numerical

methods to solve the problem are proposed. The dynamic behavior of the AMCs analysis was

determined thanks to the switch on time of Dirichlet to Neumann boundary condition StDN. The

mathematical model preserves the classical forms of the diffusion and convection diffusion equa-

tions and yields to consistent form of the Fick’s law. The computed results are in correlation with

the experimental measurements. Some numerical results are presented in this work.

3.1 Introduction

In the semiconductor manufacturing, the particle, bacterian, metallic and molecular contamination

of the wafer are a crucial subjects. These contaminations can potentially cause defects in devices

performance. The wafer carrier and storage play an important role for contamination control [1]–

[2]. The Front Unified Pods (FOUP) is used to transport wafers from one point to another point

and to protect the wafer with contamination [2]– [3]. The use of these pods may induce new kinds

of contamination as the airborne molecular contamination [3]–[4]. Some process of the wafer uses

high temperature. It was demonstrated that the AMC- cross contamination is function of the

temperature. The wafer post processed contamines the pods with volatile acids and contamines

new wafers and vice-versa [4], [5]–[6].

The FOUP is made with porous polymers materials well-known with its adsorption and outgas

3.1. INTRODUCTION 59

properties [1]–[2], [5]–[6]. These physical phenomena are traduced by sorption phenomena as

adsorption and desorption [7], [8]. The porosity size change is in function of the temperature.

As the temperature is high, the polymer adsorbs many amount of the contaminant. Also, the

diffusion coefficient in the polymer is function of the temperature. They are also able to absorb

volatile compounds present in their atmosphere coming from the connection to an equipment or

from the release of wafers post processed [1]–[4], [9]. These phenomena result to a reversible and

an irreversible outgassing of contaminant previously trapped in polymer [9], [10], [11]. The aim of

this work is to evaluate in function of the temperature the reversible and irreversible amounts of

the contaminant during the contamination and decontamination processes.

When one object becomes contaminated by either direct or indirect contact with another ob-

ject which is already contaminated, in this case we talk about cross contamination [5]–[6]. In

microelectronics industry, this process takes place generally at the FOUP who contains the wafer

before and after fabrication process [2], [4]. Some non isothermal fabrication method is the source

of contamination. When the wafer is already processed its surface is contamined by the volatile

acids [1]. The wafer is exposed to the FOUP’s atmosphere and a volatile acid contamination may

happened from wafer to internal surface of the pods [2]. This source of contamination is function

of the temperature gradient in the wafer. The hot post-processed wafers outgas many amount

of contaminant as the temperature increase during the contamination time. When the wafer is

removed or the pods is opened, a cross-contamination may happened, the contaminant moves from

internal surface of the pods to new wafer [4], [5].

To ensure the integrated circuit manufacturing, especially through the utilization of 300 mm

wafer manufacturing technologies, new methods are needed for systematic characterization, the

numerical method for analysis and control of the underlying system and processes behavior [5]–[6].

Generally, we want to compare the efficiency of the cold and hot decontamination. The modeling

and computational simulation are excellent methods to predict and quantify physical phenom-

ena as AMCs cross contamination in FOUP [6]–[7]. Numerous methods are already developed to

characterize these phenomena by using experimental measurements J. Crank and al. established

semi-analytical solution of the diffusion equation and with many cases [12], [14]. Some papers deal

with kinetics adsorption and deposition the contamination process with volatile acid in the pods

[15]–[17]. A finite element simulation of the purge method and numerous method of characteriza-

tion are also developed in [3], [18], [19]. Another paper studies the kinetics of sorption and the

decontamination process characterization [20], [21], [22]. The heat effect on theses process remains

a crucial subject in microelectronic industry.

This work describes and develops a mathematical model and appropriate numerical tools using


the finite element formulation for the dynamic of the airborne molecular cross-contamination with

heat effect in microelectronic industry. We study the effect of the temperature during the indus-

trial processes mainly the contamination and decontamination processes. The model includes the

heat effects by using the Arrhenius law and the heat equation. We use the diffusion and convec-

tion/diffusion model with/without heat effect to describe the phenomena. A dynamic boundary

and interface conditions are applied to simulate the industrial conditions of the pods as in the

chapter 2. Numerical approximation is used to solve the problem with the heat effect (for the two

cases: model with Arrhenius law and heat equation). The validation of the model is based on

the correlation study between the observed data (experimental measurements) and the computed

results with the model.

3.2 Physical problem and experiment procedure

FOUP’s geometry

Thi Quynh Nguyen [1], Paola G. [2] and al. established new experimental protocol to quantify the

cross-contamination, with heat effect, for each material constituent of the FOUP in order to choose

the optimal material versus contaminant. The heat effect is generally, studied during the decon-

tamination process. The utilization of the pods in the microelectronic industry is defined in several

steps as opening, closing and cleaning (purge, outgassing, waiting) steps for the characterization

[1], [5], [6]. Let us add in this main steps another step, an intentional contamination event, in order

to understand the dynamics and to quantify the AMCs cross contamination for some contaminant

vs. FOUP’ materials types (polymers). For the simulation, we assume the following geometry

given by the figure 3.1.

3.3 Mathematical settings

The modeling of the adsorption of the airborne molecular cross-contamination between wafer and

FOUP is based on adsorption phenomena (physical adsorption, chemical adsorption) e.g [5], [6].

In general, two concepts of adsorption exist, physical and chemistry adsorption models. A detailed

review of various adsorption kinetics (adsorption, desorption) models was given e.g in [12], [15].

Generally, adsorption processes can be divided into two classes, reversible and irreversible ad-

sorption e.g [17], [21], [22]. In this work the kinetics of adsorption between the FOUP’s atmosphere

and the polymer is no yet taken into account but traduced as Henry law at the interface. The

simplest model for adsorption is diffusion based by the Fick’s law which consists to describe the

concentration with respect the space and the time as in [12].

3.3. MATHEMATICAL SETTINGS 61

Figure 3.1: Quarter of the FOUP with the wafer support and the filter port (top right), 2Delementary representative volume for the simulation. (Ωs × [0, Tf ]) denotes the polymer, (Ωg ×[0, Tf ]) denotes the FOUP’s atmosphere, (ΓD× [0, Tf ]) denotes the wafer surface and (ΓN × [0, Tf ])denotes the interface between the FOUP’s atmosphere and the polymer.

In this model, the transport towards the surface is purely diffusive and we investigate the

concentration in the internal surface of the pods by using the thermodynamics laws. This interface

law is given by the Henry constant to connect the concentration of the contaminant at the FOUP’s

atmosphere and the concentration at the internal surface of the pods.

Transport equation

For this purpose, let Ωs ⊂ Rd and Ωg ⊂ Rd with d := 2 or 3, be a open bounded domain for the

polymer s and for the contaminant g respectively. The boundary Γ is a part of ∂Ω := ∪iΓi d− 1

dimensional surface in Rd which involves smoothly in time without any self intersection and is

divided into two parts for Dirichlet boundary conditions on ΓD and Neumann boundary conditions

on ΓN , in this work ΓN := ∂Ωs∩∂Ωg (∂Ωs and ∂Ωg are defined in figure 3.2). The time is denoted

by t ∈ I := [0, Tf ]. We denote by I = [0, Tf ] the time interval. The time-range is given by [0, Tf ]

with T ∈ R+.

The FOUP geometry is very complex and it contains many singularity cf. figure 3.1. In this

work a simplified geometry will be used to solve numerically the model. We use the same REV

(Representative Elementary Volume) for the FOUP’s material (support of the wafer and the body’s

polymer). Sometimes, this simplification is useful to enhance the performance of vector rendering

or to reduce complexity of the geometry and to ensure the numerical convergence and in order to

reduce the computational subdomain. This might be especially handy for a small-scale geometry.


So, let us assume two rectangular geometries (control geometries), in 2-dimensions represented

by the domains (Ωg × [0, Tf ]) and (Ωs × [0, Tf ]) respectively the FOUP atmosphere and polymer

membrane cf. figure 3.2.

Hypothesis 3.1. For simplicity, we reduce the geometry defined in figure 3.1 to a problem on

rectangular domain by prescribed the boundary on the fictive artificial boundary (ΓD×[0, Tf ]) (wafer

surface). We will use the same notation (ΓD × [0, Tf ]) as defined on figure 3.2. (ΓN × [0, Tf ]) is

the interface between the FOUP’s atmosphere and the body’s polymer.

Figure 3.2: Simplified Geometry: the first subdomain (Ωs× [0, Tf ]) is for the contaminant and thesecond subdomain (Ωg × [0, Tf ]) is for the polymer. An exact ratio for the total surface exchangeis quantified in order to define the size of the elementary representative volume. (ΓD × [0, Tf ])respesents the wafer (source of the contaminant) and (ΓN × [0, Tf ]) is the interface between thecontaminant g and the polymer s.

The mathematical model of diffusion process in the domains (Ωg × [0, Tf ]) and (Ωs × [0, Tf ])

is based on Fick’s second law. At the interface of these domains kinetics law will be established.

According to Fick’s first and second laws as in e.g [5]–[6], [12], the flux per unit of area perpendicular

to the flux direction is given by

Ji(∇Ci, T ) = −Di(T )∇Ci in (Ωs × [0, Tf ]) (3.1)

in which i := s, g denotes the contaminant and the polymer indices.

A diffusion time dependent process in which the rate of diffusion is fraction of time governs the

AMCs cross contamination model. In this process, the contaminant moves from a region of high

concentration (wafer) to the one region low concentration (internal surface of FOUP) e.g [5]–[6],

[12]. We have respectively the following conservation equation for the species i in the polymer s

and in the contaminant g as :

∂Cs

∂t= −∇ · Js(∇Cs, T ) + qs in (Ωs × [0, Tf ]) (3.2)

∂Cg

∂t= −∇ · Jg(∇Cg, T )−∇ · uCg + qg in (Ωg × [0, Tf ]) (3.3)

where Cs in Cg are the concentrations in the polymer and in the contaminant domains (the

problem’s unknowns), (Ωg × [0, Tf ]) and (Ωg × [0, Tf ]) are the spatial computational domains, t is

the time, Di(T ), i := s, g is the diffusion coefficient in the domains (Ωg× [0, Tf ]) and (Ωg× [0, Tf ])

respectively. During the outgass phenomena the contaminant moves from FOUP to the new wafer.

3.3. MATHEMATICAL SETTINGS 63

We assume that, on (ΓD × [0, Tf ]), we have the wafer area (wafer surface), in which the source of

contamination takes place during the contamination time tc ∈ R+.

We assume that the advection parts and reaction time scale are slow compared to the diffusive

time scale. The mathematical model with heat effect for transient contaminant transfer between

the wafer and the internal part of the FOUP is given by :

∂Cs

∂t= ∇ · Js(∇Cs, T ) + q1 in (Ωs × [0, Tf ])

Cs = h0Cg on (ΓN × [0, Tf ])

Js(∇Cs, T ) · n = 0 on ((∂Ωs − ΓN )× [0, Tf ])

Cs(., t = 0) = 0 in (Ωs × 0)

∂Cg

∂t= ∇ · Jg(∇Cg, T )−∇ · (uCg) + q2 in (Ωg × [0, Tf ])

(−Jg(∇Cg, T ) + uCg) · n = Ng0 + kc [C0H(t− ε)− Cg] on (ΓD × [0, Tf ])

Cg =Cs

h0on (ΓN × [0, Tf ])

(−Jg(∇Cg, T ) + uCg) · n = 0 on ((∂Ωg − (ΓN⋃

ΓD))× [0, Tf ])

Cg(., t = 0) = 0 in (Ωg × 0),

(3.4)

Hypothesis 3.2. We suppose that, the FOUP and the thin membrane are homogenous and

isotropic. At internal surface adsorption kinetic of the contaminant is balanced by Henry law e.g

[16], [12]. The surface roughness of the internal surface of the FOUP or the membrane is neglected

but we take into account of this parameter when we use the Henry law at the surface. Then the

diffusion coefficient is weakly function of the concentration, so we can assume that Dg 6= Dg(Cg)

and Ds 6= Ds(Cs) but in function of the temperature Dg := Dg(T ) and Ds := Ds(T ).

Boundary conditions

The industrial conditions of AMCs cross contamination prescribes the use of the switch condition

in time on the boundary condition (Dirichlet-Neumann switch StDN). During the contamination

process, the wafer post processed contamines with a constant concentration C0 ∈ R+, we have

a Dirichlet condition during this process. After the contamination step, this boundary condition

changes to Neumann boundary condition because the wafer stop to outgas contaminant on the

wafer surface.

In practice, this method is defined as the contaminant concentration maximum value is C0 ∈ R+

during the contamination time tc ∈ R+, and when the contamination time is finished the wafer

surface is considered as a neutral surface. So we need to switch this two conditions during this

time-range I = [0, Tf ]. We write the condition as a flux inflow and we use the parameter kc ∈ R+

to switch the two conditions (Dirichlet/Neumann).


Definition 3.1. The boundary condition in wafer area is defined as :

(−Jg(∇Cg, T ) + uCg) · n = Ng0 + kc [C0H(t− ε)− Cg] on (ΓD × [0, Tf ]) (3.5)

We assume that a transient boundary conditions with a laminar gas flow on this boundary (u =

0), when kc is sufficiently large, we have the Dirichlet condition i.e C0H(t − ε) ' Cg on (ΓD ×

[0, Tf ]) and if kc = 0 we have the Neumann’s boundary condition (for the neutral surface of the

wafer), i.e (−Jg(∇Cg, T ) +uCg) ·n = 0 on (ΓD× [0, Tf ]). Then, we need to conditionate C0 and

kc a parametric study was made to ensure the stability.

We have,

if

kc 1 ⇒ Cg ' C0H(t− ε) if 0 < t ≤ (tc + tp)

kc = 0 ⇒ −Jg(∇Cg, T ) · n = 0 if t > (tc + tp)(3.6)

In the model, Dg(T ) ∈ R+ and Ds(T ) ∈ R+ are the gas diffusion coefficient in contaminant and

solid media (polymer), which has no connection with the spatial location and no variation in time.

H(t − ε) denotes the Heaviside function, C0 is the initial concentration in FOUP’s atmosphere

when the wafers post processed are finished to outgas the contaminant, h0 ∈ R+ is the Henry

constant and n is the outer unit normal vector on Γi, T ∈ R+ is the final time and q1 and q2 are

the source.

In order to complete the experimental measurement and to numerically solve the model, al-

ternative approach via finite elements is used to treat numerically the AMC cross-contamination

problem. It will be applied here to the transient conduction diffusion equation where the classical

Galerkin method is shown to be unstable. The proposed method consists in adding and eliminating

bubbles to the finite element space and then to interpolate the solution to the real time step. This

modification is equivalent to the addition of a stabilizing term tuned by a local time-dependent

stability parameter, which ensures an oscillating-free solution. To validate this approach, the nu-

merical results obtained in classical 2D problems are compared with the Galerkin and the analytical

solutions and experimental measurements.

3.4 Mathematical model with temperature effect

3.4.1 Introduction

In this section, we will study the temperature effect on the AMCs cross-contamination between

wafer and FOUP. For the first approximation, we use the Arrhenius law for the variation of the

diffusion coefficient. This law assumes that only the diffusion coefficient change with the temper-

ature. The diffusion coefficient is function of the temperature. The mathematical model including

this law is traduced by the equation (3.8). The Arrhenius law applied in the polymer and the

3.4. MATHEMATICAL MODEL WITH TEMPERATURE EFFECT 65

contaminant is given as :

D∗s,g(T∗) = D∗0s,0g exp

(E

RT− E

RT ∗

)in (Ωs,g × [0, Tf ]) (3.7)

where D∗s,g and D∗0s,0g denote respectively the diffusion coefficient in the polymer/contaminant

at the temperature T ∗ and the reference diffusion coefficient in the polymer/contaminant at the

reference temperature T . E denotes the activation energy of the contaminant.

3.4.2 Mathematical model using Arrhenius law

In this section, Arrhenius law gives the variation of diffusion coefficient in function of the tem-

perature. We assume that the advection parts and reaction time scale are slow compared to the

diffusive time scale. The model with heat effect for transient contaminant transfer between the

wafer and the internal part of the FOUP is given by:

∂Cs∗

∂t= ∇ · (D∗s(T ∗)∇Cs∗) + q∗1 in (Ωs × [0, Tf ])

Cs∗ = h∗0C∗g on (ΓN × [0, Tf ])

D∗s(T∗)∇Cs∗ · n = 0 on ((∂Ωs − ΓN )× [0, Tf ])

Cs∗(., t = 0) = 0 in (Ωg × 0)

D∗s(T∗) = D∗0s exp

(E

RT− E

RT ∗

)in (Ωs × [0, Tf ])

∂Cg∗

∂t= ∇.(D∗g(T ∗)∇Cg∗)−∇ · (u∗Cg∗) + q∗2 in (Ωg × [0, Tf ])

(−D∗g(T∗)∇Cg∗ + u∗Cg∗) · n = Ng

0 + kc [C0H(t− ε)− Cg∗] on (ΓD × [0, Tf ])

Cg∗ =Cs∗

h∗0on (ΓN × [0, Tf ])

(−D∗g(T∗)∇Cg + u∗Cg∗) · n = 0 on ((∂Ωg − (ΓN

⋃ΓD))× [0, Tf ])

Cg∗(., t = 0) = 0 in (Ωg × 0)

D∗g(T∗) = D∗0g exp

(E

RT− E

RT ∗

)in (Ωs × [0, Tf ])

(3.8)

In order to complete the experimental measurement and to solve numerically the model, alterna-

tive approach via finite elements is used to treat numerically the AMC cross-contamination finite

element analysis. The method consists to use the standard finite element approaches with time-

interpolation. It will be applied to compute numerically the model with the classical Galerkin

methods. The proposed method consists first in adding and second in eliminating bubbles to the

finite element space and then to interpolate the solution with the real time step. This modification

is equivalent to the addition of a stabilizing term tuned by a local time-dependent stability pa-

rameter, which ensures a free oscillating solution. To validate this approach, the numerical results


obtained in classical 2D problems are compared with the Galerkin and the analytical solutions and

with the experimental measurements.

Finite element methods

We use the Galerkin finite element formulation for numerical solution of the problem. Classically, it

is obtained by multiplying the govering equation by an appropriate test function respectively δC∗s

and δC∗g for the concentration Cs∗ ∈ Rd and Cg∗ ∈ Rd respectively, and by integrating respectively

over the computational subdomain Ωs∗ and Ωg∗.

We define the following space:

V s∗ :=Cs ∈ R, C∗s ∈ [H1(Ωs∗)]

d; δC∗s = 0 on Γs∗

;

V g∗ :=Cg ∈ R, C∗g ∈ [H1(Ωg∗)]

d; δC∗g = 0 on Γg∗

[L2(Ωs,g)]d =


∫Ωs,g

|δCs,g|2 <∞

(3.9)

Definition 3.2. Sobolev spaces are classicaly defined as [H1(Ωs)]d = δC∗s ∈ [L2(Ωs)]

d, ‖∇δC∗s ‖ ∈

[L2(Ωs)]d and [H1(Ωg)]

d = δC∗g ∈ [L2(Ωg)]d, ‖∇δC∗g‖ ∈ [L2(Ωg)]

d.

Definition 3.3. We define [L2(Ωs)]d and [L2(Ωg)]

d respectively the Hilbert vector space of the

functions quadratically summable respectively in (Ωs) and (Ωg) defined as:

[L2(Ωs)]d =

δC∗s (x)|

∫Ωs

|δC∗s (x)|2dx <∞, [L2(Ωg)]

d =

δC∗g (x)|

∫Ωg

|δC∗g (x)|2dx <∞

In this case, we have:

∫Ωs

∂Cs∗

∂tδC∗s dV

Ωi =

∫Ωs

∇ · (D∗s(T ∗)∇Cs∗)δC∗s dVΩi +

∫Ωs

q∗1δC∗s dV

Ωi , ∀δC∗s ∈ H1(Ωg)∫Ωg

∂Cg∗

∂tδC∗g dV

Ωi =

∫Ωg

∇ · (D∗g(T ∗)∇Cg∗)δC∗g dVΩi −

∫Ωg

∇ · (u∗Cg∗)δC∗g dVΩi

+

∫Ωs

q∗2δC∗g dV

Ωi , ∀δC∗g ∈ H1(Ωg)


(E

RT− E

RT ∗

)(3.10)

Let [H1s (Ωs)]

d and [H1s (Ωs)]

d a functional space in which we are searching the solution in

accordance with its regularity [H1s ]d = δC∗s ∈ [H1(Ωs)]

d|δC∗s = s∀x ∈ Γs and [H1s ]d = δC∗g ∈

[H1(Ωg)]d|δC∗g = s∀x ∈ Γg where [H1(Ωs)]

d and [H1(Ωg)]d are Sobolev spaces.

The norm of these spaces is:

‖δC∗s ‖1,Ωs=

(∫Ωs

∇δC∗s · ∇δC∗sdx) 1

2

; ‖δC∗g‖1,Ωs=

(∫Ωg

∇δC∗g · ∇δC∗gdx

) 12

(3.11)


By using the Green’s theorem, integration by parts of the (3.10) leads to,

∫Ωs

∂Cs∗

∂tδC∗s dV

Ωi = −∫

Ωs

D∗s(T∗)∇Cs∗ · ∇δC∗s dV

Ωi +

∫∂Ωs

D∗s(T∗)∇Cs · nδC∗s dS

Ωi

+

∫Ωs

q∗1δC∗s dV

Ωi , ∀δC∗s ∈ [H1(Ωg)]d∫

Ωg

∂Cg∗

∂tδC∗g dV

Ωi = −∫

Ωg

D∗g(T∗)∇Cg∗.∇δC∗g dV

Ωi +

∫∂Ωg

D∗g(T∗)∇Cg∗ · nδC∗g dS

Ωi

−∫

Ωg

∇ · (u∗Cg∗)δC∗g dVΩi +

∫Ωs

q∗2δC∗g dV

Ωi , ∀δC∗g ∈ [H1(Ωg)]d


(E

RT− E

RT ∗

)(3.12)

By applying the Galerkin weighted residual methods and the Green’s theorem, the variational

formulation corresponding the AMCs cross contamination is given by the equation (3.8). We now

introduce the weak formulation of the AMCs model:

We find Cs∗ ∈ [H1(Ωs)]d and Cg∗ ∈ [H1(Ωg)]

d such that,

a∗1

(∂Cs∗

∂t, δC∗s

)+ b∗1(Cs∗, δC∗s ) = L∗1(δC∗s ) ∀δC∗s ∈ [H1(Ωs)]

d

a∗2

(∂Cg∗

∂t, δC∗g

)+ b∗2(Cg∗, δC∗g ) = L∗2(δC∗g ) ∀δC∗g ∈ [H1(Ωg)]

d


(E

RT− E

RT ∗

) (3.13)

where

L∗1(δC∗s ) =

∫Ωs

q∗1δC∗s dV

Ωi ; L∗2(δC∗g ) =

∫Ωs

q∗2δC∗g dV

Ωi +

∫Ωg

Ng0 δC

∗g dS

Ωi ;

a∗1

(∂Cs∗

∂t, δC∗s

)=

∫Ωs

∂Cs∗

∂tδC∗s dV

Ωi ; a∗2

(∂Cg∗

∂t, δC∗g

)=

∫Ωg

∂Cg

∂tδC∗g dV

Ωi ;

b∗1(Cs∗, δC∗s ) =

∫Ωs

Ds∗(T∗)∇Cs∗ · ∇δC∗s dV

Ωi +

∫ΓN

h∗0Cg∗δC∗s dS

Ωi

b∗2(Cg∗, δC∗g ) =

∫Ωg

D∗g(T∗)∇Cg∗.∇δC∗g dV

Ωi +

∫Ωg

∇ · (u∗Cg∗)δC∗g dVΩi +

∫ΓN

Cs∗

h∗0δC∗g dS

Ωi

+

∫ΓD

kc [C0H(t− ε)− Cg∗] δC∗g dSΩi

(3.14)

The domains (Ωs× [0, Tf ]) and (Ωg× [0, Tf ]) are decomposed as a finite number of subdomains

(Ωes × [0, Tf ]) and (Ωeg × [0, Tf ]) for each element. Similarly, the boundaries ∂Ωs and ∂Ωg are

decomposed into ∂Ωes and ∂Ωeg. Finally the time is subdivised by n subinterval.

For the spatial discretization, we assume the finite element partition T sh and T gh of (Ωs× [0, Tf ])

and (Ωg × [0, Tf ]) respectively into tetrahedral elements. Again for simplicity, we will assume that

the finite element partition associated to T sh and T sh are uniform, h is the size of the element

domains. Let Cs∗h and Cg∗h be the approximations solutions of Cs∗ and Cg∗ respectively.

The Galerkin approximation given by the equations (3.13), becomes:


Find Cs∗h ∈ V sh ⊂ [H1h(Ωg)]d and Cg∗h ∈ V

gh ⊂ [H1h(Ωg)]

d such that,

a∗1

(∂Cs∗h∂t

, δC∗sh

)+ b∗1(Cs∗h , δC

∗sh) = L∗1(δC∗sh) ∀δC∗sh ∈ [H1h(Ωs)]

d

a∗2

(∂Cg∗h∂t

, δC∗gh

)+ b∗2(Cg∗, δC∗gh) = L∗2(δC∗gh) ∀δC∗gh ∈ [H1h(Ωg)]

d


(E

RT− E

RT ∗

)(3.15)

in which the spaces V sh and V gh are defined as

V sh =δC∗sh ∈ C0(Ωs)|δC∗sh/K ∈ P1(K),∀K ∈ Fh

V gh =

δC∗gh ∈ C0(Ωg)|δC∗sh/K ∈ P1(K),∀K ∈ Fh

(3.16)

Finally, we have a system of first order differential equations and using the matrix notation we

obtain,Ms∗ 0

0 Mg∗

∂

∂t

Cs∗

Cg∗

+

Ds∗(T ) h∗0

h−1∗0 Dg∗(T )

Cs∗

Cg∗

=

Fs∗

Fg∗

(3.17)

where Cs∗ ∈ Rd and Cg∗ ∈ Rd are the unknowns concentration vectors on nodes, Ms∗ and Mg

are the time constant matrix, Fs∗ and Fg∗ are the source and external flux vectors.

Using the same methods as defined in the chapter 2, the system can be written at time t+ ∆t

as:Ms∗ 0

0 Mg∗

e

Cs∗n+1 −Cs∗

n

Cg∗n+1 −Cg∗

n

e

+ ∆t

Ds∗(T ) h∗0

h−1∗0 Dg∗(T )

e

Cs∗n

Cg∗n

e

= ∆t

Fs∗n

Fg∗n

e

(3.18)

where

Fs∗n =

∫Ωe

s

q∗1N dV Ωi ; Fg∗n =

∫Ωe

s

q∗1N dV Ωi +

∫Ωe

g

Ng0 N dV Ωi ; Ms∗ =

∫Ωe

s

NTN dV Ωi ;

Mg∗ =

∫Ωe

g

NTN dV Ωi ; Ds∗ =

∫Ωe

s

D∗s(T∗)∇NT · ∇N dV Ωi +

∫ΓN

h∗0NTN dSΩi

Dg∗ =

∫Ωe

g

D∗g(T∗)∇NT.∇N dV Ωi +

∫Ωe

g

∇ · (u∗NT)N dV Ωi +

∫ΓN

NT

h∗0N dSΩi

+

∫ΓD


]N dSΩi

(3.19)

and N denote the linear interpolation function at each node.


And by using the assembly theory for all subdomains, we have:

m⋃e=1

Ms∗ 0

0 Mg∗

e

Cs∗n+1 −Cs∗

n

Cg∗n+1 −Cg∗

n

e

+

m⋃e=1

∆t

Ds∗(T ) h∗0

h−1∗0 Dg∗(T )

e

Cs∗n

Cg∗n

e

=

m⋃e=1

∆t

Fs∗n

Fg∗n

e

(3.20)

where

Ms∗ij =

pnodes∑i,j

Ms∗ij , Ds∗ij =

pnodes∑i,j

Ds∗ij , Fs∗j =

pnodes∑j

Fs∗j , Mg∗ij =

pnodes∑i,j

Mg∗ij , D

g∗ij =

pnodes∑i,j

Dg∗ij , Fg∗j =

pnodes∑j

Fg∗j ,

We use the same conditions as defined in the equation (3.5) and (3.6). These conditions are

defined to simulate the contaminant concentration C0 ∈ R+ during the contamination time tc ∈ R+,

and when the contamination time is finished the wafer surface is considered as a neutral surface.

In fact, we use the swich condition StDN (see chapter 1) to compute this process also called

contamination process. It consists to write the condition as a flux inflow and we use the parameter

kc ∈ R+ to switch the two conditions. In this case, the boundary condition in wafer surface is

given by:

(−D∗g(T∗)∇Cg∗ + u∗Cg∗) · n = Ng

0 + kc [C0H(t− ε)− Cg∗] on (ΓD × [0, Tf ]) (3.21)

We assume that Ng0 = 0 on (ΓD×[0, Tf ]) (no initial flux) with a laminar gas flow on this boundary

(u∗ = 0). When the constant kc ∈ R+ is sufficiently large, we have the Dirichlet condition i.e

C0H(t − ε) ' Cg∗ on (ΓD × [0, Tf ]) and if kc = 0 we have the Neumann’s boundary condition

(for the neutral surface of the wafer), i.e (−D∗g(T∗)∇Cg∗ + u∗Cg∗) · n = 0 on (ΓD × [0, Tf ]). So,

we need to set up C0 ∈ R+ and kc ∈ R+ as parametric study to ensure the stability of the model.

We have,

Definition 3.4. The switch condition are define as:

if

kc 1 ⇒ Cg∗ ' C0H(t− ε) if 0 < t ≤ (tc + tp)

kc = 0 ⇒ (−D∗g(T∗)∇Cg∗ + u∗Cg∗) · n = 0 if t > (tc + tp)

(3.22)

By assuming this switch conditions StDN during the contamination process, the matrix com-

ponents in equation (3.21) and the equation(3.22) can be expressed as:

Fs∗n =

∫Ωe

s

q∗1N dV Ωi ; Fg∗n =

∫Ωe

s

q∗2N dV Ωi ; Ms∗ =

∫Ωe

s

NTN dV Ωi ;

Mg∗ =

∫Ωe

g

NTN dV Ωi ; Ds∗ =

∫Ωe

D∗s(T∗)∇NT · ∇N dV Ωi +

∫ΓN

h∗0NTN dSΩi

Dg∗ =

∫Ωe

g

D∗g(T∗)∇NT.∇N dV Ωi +

∫Ωe

s∗

∇ · (u∗NT)N dV Ωi +

∫ΓN

NT

h∗0N dSΩi

(3.23)


Mathematical model with heat effects

Introduction

In the industrial conditions, the heat effect is used for decontamination processes. In order to

qualify the effect of the heat on the decontamination, we include in the model of the AMC cross-

contamination, the heat effect. In this section, we will study the temperature effect on the AMCs

cross-contamination by using the heat equation.

We assume that the diffusion coefficient is function of the temperature, and the coupling with

the concentration is small. We compute the heat equation and the numerical solution of the

temperature is used to update the diffusion coefficient in the model. Also, the diffusion coefficient

is in function of the temperature.

The Fourier’s law is given by the equation (3.24).

J iT = −κs∇T i in (Ωs × [0, Tf ]) (3.24)

And the classical heat equation is given by above the equation (3.25):

ρscs∂T i

∂t= −∇ · J iT −∇ · uT i + ρsr in (Ωs × [0, Tf ]) (3.25)

In which ρs ∈ R+ is the mass density of the material, cs ∈ R+ denotes the specific heat, κs ∈ R+ is

the conductivity coefficient, T i ∈ Rd is the temperature the problem unknown, r ∈ Rd is the heat

source and u ∈ Rd is the velocity for the convective parts. The mathematical model including this

law is expressed by the equation.

We introduce in the model already defined at the section (3), the heat conduction and in the

FOUP (polymer), at the FOUP’s atmosphere we assume that the temperature is constant. The

boundary and initial conditions are: a prescribed temperature T s = Ta on (ΓN × [0, Tf ]) this tem-

perature is the temperature of decontamination, no inner flux κs∇T s·n = 0 on ((∂Ωs − ΓN )× [0, Tf ])

and finally at the initial conditions there is no temperature distribution in the material.T s = Ta on (ΓN × [0, Tf ])

κs∇T s · n = 0 on ((∂Ωs − ΓN )× [0, Tf ])

T s(., t = 0) = 0 in (Ωs × 0), T sref = T0 in (Ωg × [0, Tf ])

(3.26)

According to the geometry simplification and the notations already defined at the section (2),

the REV the same is defined in figure (2). The model of heat transport in the polymer with the


boundary condition is given by the equation (3.27).

ρscs∂T s

∂t= ∇ · (κs∇T s)−∇ · (uT s) + ρsr in (Ωs × [0, Tf ])

T s = Ta on (ΓN × [0, Tf ])

κs∇T s · n = 0 on ((∂Ωs − ΓN )× [0, Tf ])


(3.27)

Hypothesis 3.3 (Heat equation used for AMC cross-contamination). We assume that, the tem-

perature of the fluid of decontamination is constant Ta (isothermal condition), so we use this tem-

perature as a boundary condition at the interface fluid (contaminant or purnging fluid)/polymer

(ΓD × [0, Tf ]). Thus, the heat equation is in this case especially used for the polymer.

3.4.3 Mathematical model using heat equation

We incorporate in the model of the AMC cross-contamination given by the equation (3.4) the heat

equation given by the (3.27). There is no change in the switch condition StDN for the model

in the equation (3.4). For the heat equation, the switch condition StDN is used/not used. The

mathematical model of the AMCs cross contamination with the temperature effect can be expressed

as:

∂Cs

∂t= ∇ · [Ds(T

s)∇Cs] + q1 in (Ωs × [0, Tf ])

Cs = h0Cg on (ΓN × [0, Tf ])

Ds(Ts)∇Cs · n = 0 on ((∂Ωs − ΓN )× [0, Tf ])

Cs(., t = 0) = 0 in (Ωs × 0), Ds(Tref ) = Ds0 in (Ωs × [0, Tf ])

∂Cg

∂t= ∇.(Dg(T

s)∇Cg)−∇ · (uCg) + q2 in (Ωg × [0, Tf ])

(−Dg(Ts)∇Cg + uCg) · n = Ng

0 + kc [C0H(t− ε)− Cg] on (ΓD × [0, Tf ])

Cg =Cs

h0on (ΓN × [0, Tf ])

(−Dg(T )∇Cg + uCg) · n = 0 on ((∂Ωg − (ΓN⋃

ΓD))× [0, Tf ])

Cg(., t = 0) = 0 in (Ωg × 0), Dg(Tref ) = Dg0 in (Ωg × [0, Tf ])

ρscs∂T s

∂t= ∇ · (κs∇T s)−∇ · (uT s) + ρsr in (Ωs × [0, Tf ])

T s = Ta on (ΓN × [0, Tf ])

κs∇T s · n = 0 on ((∂Ωs − ΓN )× [0, Tf ])


(3.28)

Hypothesis 3.4 (Initial conditions). The initial conditions are defined as : the initial time t := 0


the FOUP and his atmosphere are not yet contamined i.e Cs(·, t = 0) = 0 at Cs(·, t = 0) = 0.

At the initial time, we assume that the initial temperature are the ambient temperature (absolute

temperature) T∞ i.e T s(·, t = 0) = T∞ at T g(·, t = 0) = T∞.

Practically, we assume that we study with a new FOUP for a first use. In the model as defined

above, κs ∈ R+ denote the conduction coefficient, T s ∈ R+ the temperature at the FOUPs.

In order to study the effect of temperature, we assume that the diffusion coefficient is in function

of the temperature Ds = Ds(T ) and T a ∈ R+ denotes the temperature of cleaner fluid prescribed

at the internal surface of the FOUP. We use the same boundary condition in the model. This

concentration conditions will be defined on (ΓD× [0, Tf ]) and we use the same boundary condition

on the other boundary. And for the temperature condition in (ΓN × [0, Tf ]) we have a prescribed

emperature (the temperature of the cleaning fluid) Ta ∈ R+ at the interface between (Ωs× [0, Tf ])

and (Ωg × [0, Tf ]). The industrial conditions are characterized by many step. For each step the

difference is the time characteristic and the boundary condition in (ΓD × [0, Tf ]).

Finite element methods

In order to solve the coupled model given by the equation (3.28), with the boundary condition

and under the switch condition StDN, we use the numerical methods based on finite element

approximation. For this purpose, we use the Galerkin finite element formulation for numerical

solution of the problem given by equation (3.28). It is obtained by multiplying the equilibrium

equation by an appropriate test function respectively δCs, δCg and δT for the concentration Cs ∈

Rd, Cg ∈ Rd and the temperature T s respectively, and by integrating over the computational

domain. For this purpose, we define the following space V s, V g and V Ts

respectively for the

concentration in the polymer, the concentration in the contaminant and for the temperature:

V s :=Cs ∈ Rd, δCks ∈ [H1(Ωs)]

d; δCks = 0 on Γs

;

V g :=Cg ∈ Rd, δCkg ∈ [H1(Ωg)]

d; δCkg = 0 on Γg

;

V T :=T ∈ Rd, δT ∈ [H1(Ωs)]

d; δT = 0 on Γs

[L2(Ωs,g)]d =


∫Ωs,g

|δCs,g|2 <∞

(3.29)

Definition 3.5. The Sobolev spaces are classicaly defined as [H1(Ωs)]d = δCks ∈ [L2(Ωs)]

d, ‖∇δCks ‖ ∈

L2(Ωs), [H1(Ωg)]d = δCkg ∈ [L2(Ωg)]

d, ‖∇δCkg ‖ ∈ [L2(Ωg)]d and [H1(Ωs)]

d = δT ∈ [L2(Ωs)]d, ‖∇δT k‖ ∈

[L2(Ωs)]d.

Where [L2(Ωs)]d and [L2(Ωg)]

d are the Hilbert vector spaces of the functions quadratically

summable respectively in (Ωs) and (Ωg) defined in the equation. We use the classical finite element


formulation for the numerical solution of the problem given by equation (3.28) which can be

written:

∫Ωs

∂Cs

∂tδCks dV

Ωi =

∫Ωs

∇ · (Ds(T )∇Cs)δCks dVΩi +

∫Ωs

q1δCks dV

Ωi , ∀δCks ∈ [H1(Ωg)]d∫

Ωg

∂Cg

∂tδCkg dV

Ωi =

∫Ωg

∇ · (Dg∇Cg)δCkg dVΩi −

∫Ωg

∇ · (uCg)δCkg dVΩi

∫Ωg

q2δCkg dV

Ωi

∀δCkg ∈ [H1(Ωg)]d∫

Ωs

ρscs∂T s

∂tδT k dV Ωi =

∫Ωs

∇ · (κs∇T s)δT k dV Ωi −∫

Ωg

∇ · (uT s)δT k dV Ωi

∫Ωs

ρsrδT k dV Ωi

∀δT k ∈ [H1(Ωs)]d

(3.30)

Let [H1s (Ωs)]

d and [H1s (Ωs)]

d a functional space in which we are searching the solution in

accordance with its regularity H1s = δCks ∈ [H1(Ωs)]

d|δCks = s∀x ∈ Γs, [H1s ]d = δCkg ∈

[H1(Ωg)]d|δCkg = s∀x ∈ Γg and [H1

s ]d = δT k ∈ [H1(Ωg)]d|δT k = s∀x ∈ Γs where [H1(Ωs)]

d

and [H1(Ωg)]d are a Sobolev space.

The norm of these spaces are :

‖δCks ‖1,Ωs=

(∫Ωs

∇δCks · ∇δCks dx) 1

2

; ‖δCkg ‖1,Ωs=

(∫Ωg

∇δCkg · ∇δCkg dx

) 12

;

‖δT k‖1,Ωs=

(∫Ωs

∇δT k · ∇δT kdx) 1

2

By using the Green’s theorem, integration by parts leads to,

∫Ωs

∂Cs

∂tδCks dV

Ωi = −∫

Ωs

Ds∇Cs · ∇δCks dVΩi +

∫∂Ωs

Ds∇Cs · nδCks dSΩi

+

∫Ωs

q1δCks dV

Ωi , ∀δCks ∈ [H1(Ωg)]d∫

Ωg

∂Cg

∂tδCkg dV

Ωi = −∫

Ωg

Dg∇Cg.∇δCkg dVΩi +

∫∂Ωg

Dg∇Cg · nδCkg dSΩi

−∫

Ωg

∇ · (uCg)δCkg dVΩi +

∫Ωs

q2δCkg dV

Ωi , ∀δCks ∈ [H1(Ωg)]d

ρscs∫

Ωs

∂T s

∂tδT k dV Ωi = −

∫Ωs

κs∇T s · ∇δT k dV Ωi +

∫∂Ωs

κs∇T s · nδT k dSΩi

−∫

Ωg

∇ · (uT s)δT k dV Ωi +

∫Ωs

ρsrδT k dV Ωi , ∀δCks ∈ [H1(Ωg)]d

(3.31)

By applying the Galerkin weighted residual methods and the Green’s theorem, the variational

formulation corresponding the AMCs cross contamination is given by the equation (3.28). We now

introduce the weak formulation of the AMCs model : We find Cs ∈ [H1(Ωs)]d and Cg ∈ [H1(Ωg)]

d


such that,

ak1

(∂Cs

∂t, δCks

)+ bk1(Cs, δCks ) = Lk1(δCks ) ∀δCks ∈ [H1(Ωs)]

d

ak2

(∂Cg

∂t, δCkg

)+ bk2(Cg, δCkg ) = Lk2(δCkg ) ∀δCkg ∈ [H1(Ωg)]

d

ak3

(∂T s

∂t, δT k

)+ bk3(T s, δT k) = Lk3(δT k) ∀δT k ∈ [H1(Ωs)]

d

(3.32)

where

Lk1(δCks ) =

∫Ωs

q1δCks dV

Ωi ; Lk2(δCkg ) =

∫Ωg

q2δCkg dV

Ωi +

∫Ωg

Ng0 δC

kg dS

Ωi ;

Lk3(δT k) =

∫Ωs

ρsrδT k dV Ωi ; ak1

(∂Cs

∂t, δCks

)=

∫Ωs

∂Cs

∂tδCks dV

Ωi ;

ak2

(∂Cg

∂t, δCkg

)=

∫Ωg

∂Cg

∂tδCkg dV

Ωi ; ak3

(∂T s

∂t, δT k

)=

∫Ωs

ρscs∂T s

∂tδT k dV Ωi

bk1(Cs, δCks ) =

∫Ωs

Ds∇Cs · ∇δCks dVΩi +

∫ΓN

h0CgδCks dS

Ωi

bk2(Cg, δCkg ) =

∫Ωg

Dg∇Cg.∇δCkg dVΩi +

∫Ωg

∇ · (uCg)δCkg dVΩi +

∫ΓN

Cs

h0δCkg dS

Ωi

+

∫ΓD

kc [C0H(t− ε)− Cg] δCkg dSΩi

bk3(T s, δT k) =

∫Ωg

κs∇T s.∇δT k dV Ωi +

∫Ωs

∇ · (uT s)δT k dV Ωi (3.33)

The domains Ωs and Ωg are decomposed into a finite number of subdomains Ωes and Ωeg for each

elements. Similarly, the boundary ∂Ωs and ∂Ωg are decomposed into ∂Ωes and ∂Ωeg. Finally the time

interval is subdivised by n subinterval. For the spatial discretization, we assume the finite element

partition T sh and T gh of Ωs and Ωg respectively into tetrahedral elements. Again for simplicity, we

will assume that the finite element partition associated to T sh and T gh are uniform, h is the size of

the element domains. Let us Csh and Cgh the approximation solution of Csh and Cgh respectively.

The Galerkin approximation above became : find Csh ∈ V sh ⊂ [H1h(Ωg)]d, Cgh ∈ V

gh ⊂ [H1h(Ωg)]

d

and T kh ∈ V sh ⊂ [H1h(Ωs)]d such that,

ak1

(∂Csh∂t

, δCksh

)+ bk1(Csh, δC

ks ) = Lk1(δCksh) ∀δCksh ∈ H1h(Ωs)

ak2

(∂Cgh∂t

, δCkgh

)+ bk2(Cg, δCkgh) = Lk2(δCkgh) ∀δCksh ∈ H1h(Ωg)

ak3

(∂T sh∂t

, δT kh

)+ bk3(T s, δCkgh) = Lk3(δT kh ) ∀δT kh ∈ H1h(Ωs)

(3.34)

Where

V sh =δCksh ∈ C0(Ωs)|δCksh/K ∈ P1(K),∀K ∈ Fh

V gh =

δCkgh ∈ C0(Ωg)|δCksh/K ∈ P1(K),∀K ∈ Fh

V Th =

δT k ∈ C0(Ωg)|δT kh/K ∈ P1(K),∀K ∈ Fh

(3.35)


One way to overcome such limitations consists in using stabilized finite element methods. In

the following, we discuss the use of enriched method on unsteady diffusion problems. To this

end, we recall the equation (3.32) as: find Csh ∈ V sh ⊂ [H1h(Ωg)]d, Cgh ∈ V gh ⊂ [H1h(Ωg)]

d and

T kh ∈ V Th ⊂ [H1h(Ωs)]d such that,(

Cs,n+1h

∆t, δCksh

)+(Ds∇Cs,n+1

h ,∇δCsh)

=(q1, δC

ksh

)+

(Cs,nh∆t

, δCksh

)∀δCksh ∈ V sh(

Cg,n+1h

∆t, δCkgh

)+(Dg∇Cg,n+1

h ,∇δCkgh)

=(q2, δC

kgh

)+(∇ · (uCg,n+1

h ), δCkgh)

+

(Cg,nh∆t

, δCkgh

)∀δCkgh ∈ V

gh(

ρscsT g,n+1h

∆t, δT kh

)+(κs∇T k,n+1

h ,∇δT kh)

=(ρsr, δT kh

)+(∇ · (uT g,n+1

h ), δT kh)

+

(ρscs

T k,nh

∆t, δT kh

)∀δCkgh ∈ V

gh(3.36)

We introduce the following subspaces V s∗h , V g∗h and V T∗h , with the inner product notation, into :

V s∗h =δCksh ∈ C0(Ωs)|δCksh/K ∈ P1(K)⊕B(K), ∀K ∈ Fh

V g∗h =

δCkgh ∈ C0(Ωg)|δCksh/K ∈ P1(K)⊕B(K),∀K ∈ Fh

V T∗h =

δT kh ∈ C0(Ωg)|δT kh/K ∈ P1(K)⊕B(K),∀K ∈ Fh

(3.37)

In which B(K) is the bubble functions which satisfies φs(x), φg(x), φT (x) > 0∀x ∈ K, φs(x), φg(x),

φT (x) = 0∀x ∈ K and φs(x), φg(x), φT (x) = 1 at the barycenter of K. In fact, we decompose

Csh ∈ V s∗h , Cgh ∈ Vg∗h and T kh ∈ V T∗h into its linear part Cs1 ∈ V sh , T s1 ∈ V Th and Cg1 ∈ V

gh . We have:

Csh = Cs1 +∑

K∈FhCs∗Kφ

s, Cgh = Cg1 +∑

K∈FhCg∗Kφ

g and T sh = T s1 +∑

K∈FhTg∗Kφ

T . Where

Cs∗K , Cs∗K and Tk∗K are the unknown bubble coefficients.

(Cs,n+1h

∆t, φs

)K

+(Ds∇Cs,n+1

h ,∇φs)K

= (q1, φs)K +

(Cs,nh∆t

, φs)K(

Cg,n+1h

∆t, φg

)K

+(Dg∇Cg,n+1

h ,∇φg)K

= (q2, δCgh)K +(∇ · (uCg,n+1

h ), φg)K

+

(Cg,nh∆t

, φg)K(

ρscsT k,n+1h

∆t, φT

)K

+(κs∇T k,n+1

h ,∇φT)K

=(ρr, δT kh

)K

+(∇ · (uT k,n+1

h ), φT)K

+

(ρscs

T k,nh

∆t, φT

)K

(3.38)


By using the decomposition of the solution and subtsituting it into the (3.36) :

(Cs,n+1

1

∆t, φs)K

+ Cs∗K

(φs,n+1

∆t, φs)K

+(Ds∇Cs,n+1

1 ,∇φs)K

+ Cs∗K(Ds∇φs,n+1,∇φs

)K

= (q1, φs)K +

(Cs,nh∆t

, φs)K(

Cg,n+11

∆t, φg)K

+ Cg∗K

(φg,n+1

∆t, φg)K

+(Dg∇Cg,n+1

1 ,∇φg)K

+ Cg∗K(Dg∇φg,n+1,∇φg

)K

= (q2, δCg1)K +(∇ · (uCg,n+1

1 ), φg)K

+

(Cg,n1

∆t, φg)K(

ρscsT k,n+1

1

∆t, φT

)K

+ Tk∗K

(ρscs

φT,n+1

∆t, φT

)K

+(κs∇T g,n+1

1 ,∇φT)K

+ TT∗K(κs∇φT,n+1,∇φT

)K

= (ρsr, δT s1 )K +(∇ · (uT k,n+1

1 ), φT)K

+

(ρscs

T k,n1

∆t, φT

)K

(3.39)

We use the shape functions and vanishing the third order term. Solving the equation (3.39) for

the bubble coefficient in each element K ∈ Fh, leads to:

Cs∗K =1

1∆t ||φs||

20,K + Ds||φs||20,K

((q1, φ

s)K +

(Cs,nh∆t

, φs)K

−(Cs,n+1

1

∆t, φs)K

)Cg∗K =

11

∆t ||φg||20,K + Dg||φg||20,K

((q2, δCg1)K +

(∇ · (uCg,n+1

1 ), φg)K

+

(Cg,n1

∆t, φg)K

)(−C

g,n+11

∆t, φs)K

Tk∗K =1

ρscs

∆t ||φT ||20,K + κs||φT ||20,K

((ρsr, δT s1 )K +

(∇ · (uT k,n+1

1 ), φT)K

+

(ρscs

T k,n1

∆t, φT

)K

)(−ρscsT

k,n+11

∆t, φT

)K

(3.40)

Where ||φs||20,K =∫KφsdΩs, ||φg||20,K =

∫KφgdΩg and φT ||20,K =

∫KφT dΩT .

The bubbles considered here are quasi-static, i.e., that the effect of their time variation may

be neglected. Note that following the evolution of small-scales in time is an interesting method,

but for this type of equation, it could increase the computational cost without considerable gain

in accuracy. Hereafter, we need to solve equation (3.36) on the macro-scale, but not treated in

this section. The static condensation procedure will eliminate the bubbles function at the element

level. And re-write the weak formulation with the stabilization coefficient.

When diffusion is the only mechanism for cross-contamination transfer, there are conditions for

which the Galerkin method fails to produce smooth solutions. It is well known that this method,

based on piecewise polynomial approximations, yields poor solutions for low thermal diffusivity

materials (Ds, Dg, for the AMCs cross-contamination the diffusion of the contaminant in the

polymer is in general 1e − 14, 15) and/or when the time step is small (Ds ≤ h2∆t, Dg ≤ h2∆t).

Thus, one way to overcome such limitations consists in using stabilized finite element methods. In


the following, we discuss the use of enriched method on unsteady diffusion problems.

Finally, we have a system of first order differential equations and using the matricial notation

we have, where the appropriate test function respectively δCs, δCg and δT for the concentration

Cs ∈ R2, Cg ∈ R2 and the temperature T s ∈ R2 respectively. The discretization is similar into

the discretization defined in section (3).

By using the time discretization, as defined at section (3) of this work, we have to solve :

m⋃e=1

Msk 0 0

0 Mgk 0

0 0 MsTk

e

k

Csn+1 −Cs

n

Cgn+1 −Cg

n

Tsn+1 −Ts

n

e

k

+

m⋃e=1

∆t

Dsk(Ts

n) hk0 0

h−1k0 Dgk 0

0 0 Ksk

e

k

Csn

Cgn

Tsn

e

k

=

m⋃e=1

∆t

FsnFgnFsTn

e

k

(3.41)

Where

Fsn =

∫Ωe

s

q1N dV Ωi ; Fgn =

∫Ωe

s

q1N dV Ωi +

∫Ωe

g

Ng0 N dV Ωi ; FsTn =

∫Ωe

s

ρsrN dV Ωi ;

Msk =

∫Ωe

s

NTN dV Ωi ; Mgk =

∫Ωe

g

NTN dV Ωi ; MsTk =

∫Ωe

g

ρscsNTN dV Ωi ;

Dsk =

∫Ωe

s

Ds(T )∇NT · ∇N dV Ωi +

∫ΓN

h0NTN dSΩi ;

Dgk =

∫Ωe

g

Dg(T )∇NT.∇N dV Ωi +

∫Ωe

g

∇ · (uNT)N dV Ωi +

∫ΓN

NT

h0N dSΩi

+

∫ΓD


]N dSΩi ;Ks

k =

∫Ωe

g

κs∇NT.∇N dV Ωi +

∫Ωe

g

∇ · (uNT)N dV Ωi

and N denotes the linear interpolation function at each node.

And by using the assembly theory for all subdomains, we have :

m⋃e=1

Msk 0 0

0 Mgk 0

0 0 MsTk

e

k

Csn+1 −Cs

n

Cgn+1 −Cg

n

Tsn+1 −Ts

n

e

k

+

m⋃e=1

∆t

Dsk(Ts

n) hk0 0

h−1k0 Dgk 0

0 0 Ksk

e

k

Csn

Cgn

Tsn

e

k

=

m⋃e=1

∆t

FsnFgnFsTn

e

k

(3.42)


Where

(Ms,gk

)ij

=

pnodes∑i,j

(Ms,gk

)ij

;(Ds,gk

)ij

=

pnodes∑i,j

(Ds,gk

)ij

;(Fs,gk

)j

=

pnodes∑j

(Fs,gk

)j

;

(MsTk

)ij

=

pnodes∑i,j

(MsTk

)ij

;(Ksk

)ij

=

pnodes∑i,j

(Ksk)ij ;

(FsTk

)j

=

pnodes∑j

(FsTk

)j

;

(3.43)

3.5 Applications of the model in industrial processes

3.5.1 Heat effect on contamination process

This phenomenon illustrates the process during which the wafer post processed outgas the contam-

inant and contamines the internal surface FOUP. This is the first step of the cross contamination.

The contamination moves from wafer to FOUPs. During this process, the surface adsorption step

takes place and the diffusion in the volume of the FOUP happened. Contamination process consists

using the same condition in section (3) of this work, in which we consider that the wafer is the

contaminant source governed by the Heaviside function with a delay ε. The implementation of this

boundary condition is defined in equation (3.4). In fact, during tc ∈ R+ (contamination time), we

apply on (ΓD × [0, Tf ]), the condition can expressed as:

Unknown Csc and Cgc

With the switch StDN conditions, we have:

if

kc 1 ⇒ Cgc ' C0H(t− ε) if 0 < t ≤ tc


(3.44)

Dynamic boundary condition (B.C) on (ΓD × [0, Tf ]) for the contaminant holds : Cgc ' C0H(t− ε) if 0 ≤ t ≤ tc

F ca(C0, Cgc ) = 0 if tc < t ≤ (tc + to)

(3.45)

Initial conditions (I.C) Csc (., t = 0) = 0 in (Ωs × 0) and Cgc (., t = 0) = 0 in (Ωg × 0) Then,

we have,

m⋃e=1

Msk 0 0

0 Mgk 0

0 0 MsTk

e

k

Csc,n+1 −Cs

c,n

Cgc,n+1 −Cg

c,n

Tsc,n+1 −Ts

c,n

e

k

+

m⋃e=1

∆t

Dsk(Ts

n) hk0 0

h−1k0 Dgk 0

0 0 Ksk

e

k

Csc,n

Cgc,n

Tsc,n

e

k

3.5. APPLICATIONS OF THE MODEL IN INDUSTRIAL PROCESSES 79

=

m⋃e=1

∆t

Fsc,nFgc,nFsTc,n

e

k

(3.46)

The equation eq.3.43 becomes,

(Msk

)(c)

ij=

pnodes∑i,j

(Msk)

(c)ij ;

(Dsk)(c)

ij=

pnodes∑i,j

(Dsk)(c)ij ;

(Fsk)(c)

j=

pnodes∑j

(Fsk)(c)j ;

(Mgk

)(c)

ij=

pnodes∑i,j

(Mgk

)(c)ij

;(Dgk)(c)

ij=

pnodes∑i,j

(Dgk)(c)ij

;(Fgk)(c)

j=

pnodes∑j

(Fgk)(c)j

;

(MsTk

)(c)

ij=

pnodes∑i,j

(MsTk

)(c)ij

;(Ksk

)(c)

ij=

pnodes∑i,j

(Ksk)

(c)ij ;

(FsTk

)(c)

j=

pnodes∑j

(FsTk

)(c)j

;(3.47)

The experimental process prescribes that the contamination time tc ∈ R+ is decomposed into two

characteristic time tc ∈ R+ and to ∈ R+ respectively the time until the wafer outgas is finished

and the waiting time before opening the FOUP to remove the wafer.

For the implementation of the boundary conditions also defined in the equation (3.28) and

(3.45), we use the same method as with the switch condition StDN given by the equation (3.44)

using the inflow concentration flux. We run this test with many kind of material and with different

level of initial concentrations C0.

3.5.2 Heat effect on outgassing process

Outgassing process takes place after opening the FOUP in the goal to remove the wafer. The

FOUP’s atmosphere change and the contaminant concentration Cg ' 0 during the operation, after

tp ∈ R+ the front door will be close. This step results into a reverse flow of the contamination

gradient during which the contaminant moves from FOUP to wafer. After the intentional contam-

ination of the FOUP during the contamination tc ∈ R+, just after tc ∈ R+ the FOUP has been

opened during the opening time tp ∈ R+ then the FOUP has been closed during the outgassing

time td ∈ R+.

Unknown Csp and Cgp


if



Dynamic boundary condition (B.C) on (ΓD × [0, Tf ]) for the contaminant : F pa (Cp0 , Cgp ) = Ng



F pa (Cp0 , Cgp ) = 0 if (tc + tp) < t ≤ (tc + tp + td)

(3.49)


Initial conditions (I.C) Csp(., t = tc) = Csc in (Ωs×tc) and Cgp (., t = tc) = Cgc in (Ωg ×tc)

We have,

m⋃e=1

Msk 0 0

0 Mgk 0

0 0 MsTk

e

k

Csp,n+1 −Cs

p,n

Cgp,n+1 −Cg

p,n

Tsp,n+1 −Ts

p,n

e

k

+

m⋃e=1

∆t

Dsk(Ts

n) hk0 0

h−1k0 Dgk 0

0 0 Ksk

e

k

Csp,n

Cgp,n

Tsp,n

e

k

=

m⋃e=1

∆t

Fsp,nFgp,nFsTp,n

e

k

(3.50)

The equation eq.3.43 becomes,

(Msk

)(p)

ij=

pnodes∑i,j

(Msk)

(p)ij ;

(Dsk)(p)

ij=

pnodes∑i,j

(Dsk)(p)ij ;

(Fsk)(p)

j=

pnodes∑j

(Fsk)(p)j ;

(Mgk

)(p)

ij=

pnodes∑i,j

(Mgk

)(p)ij

;(Dgk)(p)

ij=

pnodes∑i,j

(Dgk)(p)ij

;(Fgk)(p)

j=

pnodes∑j

(Fgk)(p)j

;

(MsTk

)(p)

ij=

pnodes∑i,j

(MsTk

)(p)ij

;(Ksk

)(p)

ij=

pnodes∑i,j

(Ksk)

(p)ij ;

(FsTk

)(p)

j=

pnodes∑j

(FsTk

)(p)j

;(3.51)

3.5.3 Heat effect on decontamination process

In this section, we describe the application of the model in decontamination process of the pods.

After the outgassing process, the decontamination begins. Decontamination is used to remove the

contaminant already accumulated at the internal surface and diffused in the FOUP’s material.

Decontamination is an operation used of reducing the AMCs cross contamination risk by purging

methods. In this work, we assume two types of methods, the cold purging and the hot purging. In

fact, we denote these two types of decontamination by cold decontamination and hot decontamina-

tion. In this section, many processes of purge are assumed and are studied by theoretical analysis

and are validated with the experimental measurement.

The goal of this section is to study the temperature effect during the FOUP’s cleaning. Purging

the pods with inert gas is the one of the most popular method, but there are many several methods

as UV or vacuum methods. In general purging the pods with inert gas provides the many advan-

tages. The decontamination process will be done during tu ∈ R+ the decontamination duration

and we denote tw ∈ R+ the waiting time when the pods is closed after decontamination process.

This waiting process is important to known the cleaning efficiently. So the final time T ∈ R+ of

the simulation is the sum of all characteristic time ti ∈ R+, it can be written :

T =∑i

ti = tc + tp + td + tu + tw (3.52)

3.5. APPLICATIONS OF THE MODEL IN INDUSTRIAL PROCESSES 81

Ended, the total time is defined as t ∈ [0,∑

i ti] or t ∈ [0, Tf ]

Hot decontamination

We compute in this case the boundary conditions using the switch condition. The diffusion coef-

ficient is function of the temperature. We assume, in this section that the initial condition is the

step phase before (after, removing wafer, outgassing for equilibrium atmosphere).


i ti − tw


if

kc 1 ⇒ Cgd ' Cd0H(t− ε) if 0 < t ≤ (

∑i ti − tw)


i ti − tw)(3.53)

Dynamic boundary condition (B.C) on (ΓD × [0, Tf ]) for the contaminant holds :



]if (tc + tp + td) ≤ t ≤ (

∑i ti − tw)


∑i ti

(3.54)



m⋃e=1

Msk 0 0

0 Mgk 0

0 0 MsTk

e

k

Csd,n+1 −Cs

d,n

Cgd,n+1 −Cg

d,n

Tsd,n+1 −Ts

d,n

e

k

+

m⋃e=1

∆t

Dsk(Ts

n) hk0 0

h−1k0 Dgk 0

0 0 Ksk

e

k

Csd,n

Cgd,n

Tsd,n

e

k

=

m⋃e=1

∆t

Fsd,nFgd,nFsTd,n

e

k

(3.55)

where

(Msk

)(d)

ij=

pnodes∑i,j

(Msk)

(d)ij ;

(Dsk)(d)

ij=

pnodes∑i,j

(Dsk)(d)ij ;

(Fsk)(d)

j=

pnodes∑j

(Fsk)(d)j ;

(Mgk

)(d)

ij=

pnodes∑i,j

(Mgk

)(d)

ij;

(Dgk)(d)

ij=

pnodes∑i,j

(Dgk)(d)

ij;

(Fgk)(d)

j=

pnodes∑j

(Fgk)(d)

j;

(MsTk

)(d)

ij=

pnodes∑i,j

(MsTk

)(d)

ij;

(Ksk

)(d)

ij=

pnodes∑i,j

(Ksk)

(d)ij ;

(FsTk

)(d)

j=

pnodes∑j

(FsTk

)(d)

j;(3.56)

where tc = tc + ta, in which tc ∈ R+ is the contamination time (outgassing time from wafer to

atmosphere and FOUP), ta ∈ R+ is the waiting time before opening the FOUP, in this time the

post processed wafers contamination steps is already finished (the post processed wafers stopped


to outgas the source of contamination).

3.6 Computation order

By using the same methods, let us give a key of the computation order of the AMCs cross-

contamination approximated model for the hot decontamination. For the computation we use

three sub-simulations for each main step. The time range is t ∈ [0,∑

i ti] or t ∈ [0, Tf ]. For this

purpose, we separate the simulation in three sub-simulations, SUB∗ 1 , SUB∗ 2 and SUB∗ 3 .

Each simulation contains respectively StDN 1, StDN 2 and StDN 3. For a given temperature,

the diffusion coefficient is updated. We compute the firstly the heat equation in order to have the

temperature distribution in the polymer after; we have the diffusion coefficient variation in the

polymer, because the connection of the diffusion coefficient and the concentration is small. We

have to compute with the following order the approximated AMC cross-contamination model with

the heat effect,

SUB∗ 1 : for 0 ≤ t ≤ tc: COMPUTE : Contamination process

-Unknowns : Csc and Cg

c

-StDN 1 Cf. eq.(3.44)

-B.C (Contamination) on (ΓD × [0, Tf ]) Cf. eq. (3.45)

-I.C Csc (., t = 0) = 0 in (Ωs × 0) and Cgc (., t = 0) = 0 in (Ωg × 0)


m⋃e=1

Msk 0 0

0 Mgk 0

0 0 MsTk

e

k

Csp,n+1 −Cs

p,n

Cgp,n+1 −Cg

p,n

Tsp,n+1 −Ts

p,n

e

k

+

m⋃e=1

∆t

Dsk(Ts

n) hk0 0

h−1k0 Dgk 0

0 0 Ksk

e

k

Csp,n

Cgp,n

Tsp,n

e

k

=

m⋃e=1

∆t

Fsp,nFgp,nFsTp,n

e

k

(3.57)

OUTPUT : Csp, Cg

p, Ts and Ds(Ts)

⇓

3.6. COMPUTATION ORDER 83

SUB∗ 2 tc < t ≤ (tc + tp + td) : COMPUTE : Purge and outgass

-Unknowns : Csp and Cg

p

-StDN 2 Cf. eq.(3.48)

-B.C (Purge) on (ΓD × [0, Tf ]) Cf. eq.(3.49)

-I.C Csp(., t = tc) = Csc in (Ωs × tc) and Cgp (., t = tc) = Cgc in (Ωg × tc)


m⋃e=1

Msk 0 0

0 Mgk 0

0 0 MsTk

e

k

Csp,n+1 −Cs

p,n

Cgp,n+1 −Cg

p,n

Tsp,n+1 −Ts

p,n

e

k

+

m⋃e=1

∆t

Dsk(Ts

n) hk0 0

h−1k0 Dgk 0

0 0 Ksk

e

k

Csp,n

Cgp,n

Tsp,n

e

k

=

m⋃e=1

∆t

Fsp,nFgp,nFsTp,n

e

k

(3.58)

OUTPUT : Csp, Cg

p, Ts and Ds(Ts)

⇓

SUB∗ 3 (tc + tp + td) < t ≤∑

i ti : COMPUTE : Hot decontamination and cleaning

processes

-Unknowns : Csd and Cg

d

-StDN 3 Cf. eq.(3.53)

-B.C (Decontamination) on (ΓD × [0, Tf ]) Cf. eq.(3.54)

-I.C Csd(., t = (tc + tp + td)) = Csp in (Ωs × tc + tp + td) and

Cgd (., t = (tc + tp + td)) = Cgp in (Ωg × tc + tp + td)


m⋃e=1

Msk 0 0

0 Mgk 0

0 0 MsTk

e

k

Csp,n+1 −Cs

d,n

Cgd,n+1 −Cg

d,n

Tsd,n+1 −Ts

d,n

e

k

+

m⋃e=1

∆t

Dsk(Ts

n) hk0 0

h−1k0 Dgk 0

0 0 Ksk

e

k

Csd,n

Cgd,n

Tsd,n

e

k

=

m⋃e=1

∆t

Fsd,nFgd,nFsTd,n

e

k

(3.59)

OUTPUT : Csd, Cg

d, Ts and Ds(Ts)


3.7 Results and discussion

This section provides discussion with illustrations of some results for the computed model with

and without temperature effect with its correspondence to industrial applications. A correlation

study between the experimental measurement and the computed model is discussed. Finally, some

concluding remarks about for the effect of temperature change on the decontamination and cleaning

process are presented.

0 500 1000 1500 2000 2500 3000 35000

100

200

300

400

500


Time in [h]

Q in

[ng/

cm2 ]


05

1015

x 106

0

0.5

1

1.5

x 10−4

0

0.2

0.4

0.6

0.8

1

Time in [s]


Thickness in [m]

C(x

,t)/C

0 in [n

g/cm

2 ]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.3: Sorbed quantity in the polymer in function of time for the contaminant XC1 : Com-puted model in blue, experimental measurement in red (left). Sorbed quantity in the polymer infunction of time and space for the contaminant XC1 : Computed model (right). The experimentalmeasurements are obtained by ionic chromatography methods.

0 500 1000 1500 20000

5

10

15

20

25

30


Time in [h]

Q in

[ng/

cm2 ]


0

5

10

x 106

0

2

4

6

x 10−5

0

0.2

0.4

0.6

0.8

1

Time in [s]


Thickness in [m]

C(x

,t)/C

0 in [n

g/cm

2 ]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.4: Sorbed quantity in the polymer in function of time for the contaminant XC2 : Com-puted model in blue, experimental measurement in red (left). Sorbed quantity in the polymer infunction of time and space for the contaminant XC2 : Computed model (right). The experimentalmeasurements are obtained by ionic chromatography methods.

Figures 3.4, fig.3.3 and fig.3.5, illustrate the correlation between the computed model and the

experimental measurement during the contamination process. We can see that the mathematical

model is in correlation with the experimental measurement for the two tested contaminants (XC1

and XC2) versus polymer (PC, PEEK, PEI, COP). The curves illustrate the sorbed quantity in

3.7. RESULTS AND DISCUSSION 85

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300

350

400

Time in [h]

Q in

[ng/

cm2 ]



05

1015

x 106

0

0.5

1

1.5

x 10−4

0

0.2

0.4

0.6

0.8

1


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.5: Sorbed quantity in the polymer in function of time for the contaminant XC2: Com-puted model in blue, experimental measurement in red (left). Sorbed quantity in the polymer infunction of time and space for the contaminant XC2 : Computed model (right). The experimentalmeasurements are obtained by ionic chromatography methods.

the polymer in function of time and in function of time-space for the contaminants XC1 and XC2

: the computed model is in blue, and the experimental measurements in red. We measured the

sorbed quantity of the contaminant in the polymer with the ionic chromatography method, and

we obtained the diffusion coefficient with numerical optimization by using the model.

Figure 3.6: Contamination process: after 1[h] of contamination and 1 [h] of waiting time (storagetime of the wafers in the FOUP) and after outgassing process: 5 [mn] removing wafer, 22 [h] waitingtime for atmosphere concentration equilibrium. The curve illustrates the computed concentration ofthe contaminant in the polymer in function of time (left) and the concentration of the contaminantin the FOUP’s atmosphere in function of time, waiting time after removing the wafers (right). Thecomputed results is obtained by using the model without heat effect (right).

Figure 3.6 illustrates the computed concentration in the polymer after 1 [h] of contamination

and 1 [h] of waiting time (storage time of the wafers in the FOUP) and after outgassing process: 5

[mn] removing wafer, 22 [h] waiting time for atmosphere concentration equilibrium. These results

are obtained with the model without heat effect. This model and the results are in correlation


Figure 3.7: Outgassing process: 5[mn] removing wafer, 22 [h] waiting time for atmosphere concen-tration equilibrium. The curve illustrates the computed concentration of the contaminant in thepolymer in function of time (left) after contamination process. Decontamination/cleaning process:the curve illustrates the concentration in the polymer in function of the space after decontamina-tion and cleaning processes (Cf. switch StDN). The computed results are obtained by using themodel without heat effect (right).

with the experimental measurement, during the contamination process. We can see that we have

two part of the concentration gradient: the first parts is during the contamination process and

the second part is during the waiting time. We use the first model without the temperature effect

given by the equation (3) to study this process.

Figure 3.7 represents the concentration in the FOUP’s atmosphere and the polymer after Out-

gassing process : 5 [mn] removing wafer, 22 [h] waiting time for atmosphere concentration equi-

librium. The curve illustrates the computed concentration of the contaminant in the polymer in

function of time (left) after contamination process. Decontamination/cleaning process: the curve

illustrates the concentration in the polymer in function of the space after decontamination and

cleaning processes (Cf. switch StDN). We can see the effect of the switch conditions in the model.

The computed results are obtained by using the model without heat effect (right). The step before

is the contamination process gived by the figure fig.3.6. We can see that the concentration profile

includes two concentration gradient, and the maximum amount of concentration after cleaning

process stay in the polymer, in fact, the first gradient continues to diffuse in the volume and the

second gradient returns back to the FOUP’s atmosphere and contamines the wafers.

Figures 3.8, 3.9, and 3.10 illustrate the computed concentration in the polymer after hot de-

contamination and cleaning processes of the FOUP. This step is governed by the model given the

equation and under the switch condition (Cf. switch StDN). Curve in these figures represent the

computed concentration in polymer function of the thickness. We observe two parts of the con-

centration gradient during the decontamination, the first part continues to diffuse in the polymer

3.7. RESULTS AND DISCUSSION 87

Figure 3.8: Contamination/hot decontamination and cleaning processes for the contaminant XC1: 4 [h] of cleaning time and 22 [h] of waiting time (wafers storage time). The curves illustratethe computed concentration in the polymer in function of the space after decontamination andcleaning processes. The computed results are obtained by using the model without heat effect(left). Contamination : the curve illustrates the concentration in the polymer in function of thethickness after decontamination. The computed results are obtained by using the model withoutheat effect (right) and under the switch condition (Cf. switch StDN). Each curve represents onecharacteristic time.

Figure 3.9: Hot decontamination and cleaning processes for the contaminant XC2 : after 4 [h]of cleaning time and 22 [h] of waiting time (wafers storage time). The curves illustrate the com-puted concentration in the polymer in function of the thickness after decontamination and cleaningprocesses. The computed results are obtained by using the model with heat effect (left). Decon-tamination : the curve illustrates the concentration in the polymer in function of the thickness afterdecontamination. The computed results are obtained by using the model with heat effect (right)and under the swicth condition (Cf. switch StDN). Each curve represents one characteristic time

and the second part returned back to the FOUP’s atmosphere and contamines the new wafer. The

dynamic of the AMCs cross contamination is determined by this phenomenon (adsorption and

desorption properties of this material). The new wafer is contamined by the contaminant already

adsorbed in the polymer but with the effect of decontamination illustrated in the figure 3.10 the


Figure 3.10: Hot decontamination and cleaning process for the contaminant XC1 : after 4 [h] ofcleaning time and 22 [h] of waiting time (wafers storage time). The curve represents the computedconcentration in the FOUP’s atmosphere in function of time (left). (Cleaning process : 4 [h] ofcleaning time and 22 [h] of waiting time, wafers storage time). The curve represent the computedconcentration in the polymer in function of time (right) and under the swicth condtion (Cf. switchStDN).

contamination returns back to the FOUP’s atmosphere. We can see that the hot decontamination

and the cleaning processes have a benefit advantage for the decontamination; the heat effect allows

avoiding a maximum amount of the contaminant in the FOUP. However, the hot decontamination

and the cleaning processes have a disadvantage because the diffusion coefficient is proportionally

in function of the temperature change, indeed, the temperature induces a quick diffusion in the

polymer.

0

5

10

x 105

0

2

4

6

x 10−5

0

0.2

0.4

0.6

0.8

1

Time in [s]

Concentration in polymer [mol/m3]

Thickness in [m]

C(x

,t)/C

0 in [m

ol/m

3 ]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

05

1015

x 105

0

0.5

1

1.5

x 10−4

0

0.2

0.4

0.6

0.8

1

Time in [s]


Thickness in [m]

C(x

,t)/C

0 in [m

ol/m

3 ]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.11: Desorption (ideal desorption) process for two contaminants just after contaminationprocess. We illustrate here the computed concentration in the polymer for ideal desorption. Thecurve represent the computed concentration in function of the thickness and the time in the polymerfor the contaminant XC1 (left) and the computed concentration in function of the thickness andthe time in the polymer (right).

Figures 3.11 and 3.12 illustrate respectively the concentration in the polymer and in the FOUP’s

atmosphere during the FOUP’s cleaning or decontamination for the contaminant XC1 and XC2.

3.8. CONCLUDING REMARKS 89

05

1015

x 106

0

0.5

1

1.5

x 10−4

0

0.2

0.4

0.6

0.8

1

Time in [s]


Thickness in [m]

C(x

,t)/C

0 in [m

ol/m

3 ]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

02

46

x 105

0

1

2

3

x 10−5

0

0.2

0.4

0.6

0.8

1

Time in [s]


Thickness in [m]

C(x

,t)/C

0 in [m

ol/m

3 ]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.12: Desorption (ideal desorption) process for two contaminants just after contaminationprocess. We illustrate here the computed concentration in the polymer for ideal desorption. Thecurve represent the computed concentration in function of the thickness and the time in the polymer(left) for the contaminant XC1 and the computed concentration in function of the thickness andthe time in the polymer for the contaminant XC2 (right).

Figure fig.3.12 illustrates the concentration in function of time and thickness during the ideal

desorption. During the ideal desorption we assume that there is no residual contaminant adsorbed

in the polymer after decontamination process ; all of the adsorbed contaminant will be desorbed (all

amount of adsorbed contaminant in the polymer will be decontaminated, non physical assumptions

for some material but possible for other material).

3.8 Concluding remarks

The aim of this section is to understand and to quantify the effect of the temperature on the

decontamination of the FOUP, already contamined by the AMC cross-contamination, by using the

modeling and simulation method. Mathematical model with the temperature effect on the AMC

cross-contamination is developed. Then, we predict with the modeling and simulation the behavior

of the contaminant in each material constituent of the FOUP during the decontamination.

This method is used to study the sensitivity of each material constituent of the FOUP with

a given contaminant, in order to optimize the specific methods of the decontamination for each

material. After the study of the behavior, one of the objectives of this work is to study the decon-

tamination process with/without heat (temperature) effect. Mathematical model and numerical

methods are established with the switch condition Dirichlet to Neumann. It is developed to predict

the transient reversible and irreversible diffusion in the FOUP’s polymer constituents of the FOUP.

We can also observe the temperature effect from the computed results that we have a benefit

effects during decontamination process with the use of the hot decontamination.

The hot purging is better in term of efficiency of the contaminant removal during the cleaning

process or decontamination of the FOUP. We have found through the simulation that the hot


purging is more efficient than the cold purging. One problem is that this method induces long

diffusion profile in the polymer i.e. the contaminant diffuses with maximum rate in the polymer

because the phenomena is thermically activated. However, we need to determine for each contam-

inant the diffusion coefficient with temperature change. In fact, by using the approach based by

the Arrhenius law, the value of the free enthalpy of each contaminant is needed.

The use of the hot decontamination cleans maximum amount of contaminant in the polymer

(support of the wafer and the body). One of the disadvantages of the hot decontamination is

that it promotes the diffusion of the contaminant in the polymer (body) because the diffusion

increases proportionally with the temperature. The used of the model with heat effect using the

Arrhenius’s law is the most benefit than using the model with the heat equation. The results

are so similar but the last model has many calculations number for the computation. The second

application of the simulation focused on the study of two cleaning methods: the purge 4 [h] at

room temperature and purge 4 [h] at 70 oC. The obtained results clearly show that a purge may

reduce very significantly the contaminant near the polymer surface and reduce contamination in

the FOUP. For a given industrial conditions, the following parameters were studied to evaluate and

optimize the cleaning efficiency: waiting time before FOUP’s cleaning (after removal of plates): 1

[h] or 24 [h]; the temperature of the cleaning step (purge at 22 oC or 70 oC); storage time of the

plates and contaminant source (short or long contamination). We have shown that the waiting time

between the plates removal and cleaning steps does not influence significantly in the short term

(22 [h]) the concentration of the contaminant in the FOUP’s atmosphere. By against the residual

amount of contaminant in the polymer is greater after 24 [h] waiting time for the contaminant

diffuse deeper (and therefore, this results in a outgassing of the residual impact long term probably

negative). It is therefore recommended to clean the FOUP as soon as possible after removing the

plates. We have also shown that increasing the temperature has a positive effect on cleaning are

some contamination scenarios. Purge at 70oCmore significantly reduces the amount of contaminant

present in the air and the FOUP’s polymer with respect to a purge at 22 oC. However, after 4 [h] of

purging it remains a contaminant in the materials of the FOUP. These results are explained by the

fact that the diffusion coefficient increases with temperature according to the Arrhenius law. This

therefore promotes the desorption of contaminant during the cleaning step and also promotes the

distribution by volume of polymer. Purge at 22 oC essentially acts near the polymer surface while

a purge at 70 oC is more effective over a greater depth of the polymeric material. These results

"computed" are in perfect qualitative and quantitative agreement with the results obtained in the

experimental study of different FOUP’s cleaning techniques. The effectiveness of decontamination

by purging the FOUP (cold or hot) is higher on contaminated FOUP for a short time compared

to those contaminated during a long time. Indeed, the decontamination acts primarily on the

3.8. CONCLUDING REMARKS 91

near surface and is more effective that the contaminant had little time to diffuse deeply into the

material. This behavior is consistent with the results experimental results. This study allowed us to

understand the behavior of the contaminant in the atmosphere and in the polymer wall of FOUP

at the various events that are contaminated, withdrawal plates, a waiting and decontamination

phase. To reduce contamination of molecular FOUPs in production, so we propose to minimize the

storage time sheets after steps "critical" source of contamination, and waiting time FOUP before

cleaning. The waiting time before cleaning depends on the availability of the decontamination

equipment. The results showed that a simple opening of the FOUP can evacuate the contaminant

contained in the atmosphere leading to a slight desorption of contaminant of FOUP’s materials

and a gradient reversal. A storage with the opened door in a specific room during this waiting

phase could be very beneficial.

The recontamination process is necessary to evaluate and to quantify the AMCs cross contam-

ination cycle after cleaning in the FOUP already used. During each cycle, contamination, purge,

cleaning processes and another amount of contaminant may be added in the residual contaminant

already adsorbed in the FOUP. A theoretical of this additional problem is required for example

including the adsorption kinetics and the deposition kinetics. The present analysis has been for-

mulated with all of the processes used in industry applications. The model, the behavior and the

computed results are in correlation with the experimental measurements. This methodology using

the model and the switch condition is relevant for the industrial applications in cross-contamination

control, cleaning control of the FOUP, we are focused on the dynamics of cross-contamination

mainly on the evolution of the concentration level and the profile through the polymeric material,

and finally on the effect of the material.

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[2] P.Gonzàlez, H.Fontaine, C.Beitia and al. A comparative study of the HF sorption and outgassingability of different Entegris FOUP platforms and materials, Microelectronic Engineering, Vol. 150,(2013), pp. 113–118.

[3] S.Hu, T.Wu, and al. Design and evaluation of a nitrogen purge system for the front opening unifedpod, Applied Thermal Engineering, Vol. 27, (2007), pp. 1386–1393.

[4] H.Fontaine, H.Feldis and al. Impact of the volatile Acid Contaminant on Copper Interconnects, Elec-trical Perform, Vol. 25, No: 5, (2009), pp. 78–86.

[5] N. Santatriniaina, J.Deseure, T.Q.Nguyen, H.Fontaine, C. Beitia, L.Rakotomanana. Mathematicalmodeling of the AMCs cross-contamination removal in the FOUPs: Finite element formulation andapplication in FOUP’s decontamination, Inter. Journ. of Math., comput. sci. engrg., Vol. 8, No: 4,(2014), pp. 409–414.

[6] N. Santatriniaina, J.Deseure, T.Q.Nguyen, H.Fontaine, C. Beitia, L.Rakotomanana. Coupled systemof PDEs to predict the sensitivity of the some material consituents of the FOUP with the AMCscross-contamination, International Journal of Applied Mathematical Research, Vol. 3, No: 3, (2014),pp. 233–243.

[7] Alemayeuhu Ambaw, Randolph Beaudry, Inge Bulens, Mulugeta Admasu Delele, Q.Tri Ho, AnnSchenk, Bart M. Nicolai, Pieter Verboven, Modelling the diffusion adsorption kinetics of 1-methylcyclopropene (1-MCP) in apple fruit and nontarget materials in storage rooms, Journal ofFood Engineering, Vol. 102, (2011), pp. 257–265.

[8] J.A.Boscoboinik, S.J. Manzi, V.D.Pereyra Adsorption-desorption kinetics of monomer-dimer mixture,Physics A, Vol. 389, (2010), pp. 1317–1328.

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[12] J. Crank, The mathematics of diffusion, second edition, 1975 Clarendon Press, Oxford.

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[14] K.J.Kuijlaars, C.R.Kleijin, H.E.A. van den Akker, Multi-component diffusion phenomena in multiple-wafer chemical vapour deposition reactors, The chemical Engineering Journal, Vol. 57, (2009), pp.127–136.

[15] Koichi Aoki, Diffusion-controlled current with memory, Journal of electroanalytical Chemistry, Vol.592, (2006), pp. 31–36.

[16] Shengping Ding, William T. Petuskey, Solutions to Ficks second law of diffusion with a sinusoidalexcitation, Solide State Ionics, Vol. 109, (1998), pp. 101–110.

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[23] Takashi Kako and Kentarou Touda, Numerical Approximation of Dirichlet –to Neumann Mapping andits Application to Voice Generation Problem, The University of Electro-Communications, Departmentof Computer Science.

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Chapter 4

Experimental identification ofself-heating in HEMA-EGDMA.

1

Resumé – Ce chapitre se concentre sur la quantification de la production de chaleur dans

l’hydrogel de type HEMA-EGDMA sous chargement dynamique. Ce chapitre commence par la

mesure de la production de chaleur par microcalorimétrie à déformation. On compare les ré-

sultats théoriques (numériques) avec les résultats expérimentaux combinés avec une proposition

d’optimisation pour identifier les paramètres influençant le phénomène de self-heating. Dans un

premier temps, nous présentons un modèle simplifié qui nous permet d’identifier les différents

paramètres liés à la production de chaleur dans les échantillons. Nous traitons le cas monodi-

mensionnel car l’augmentation de température dans l’échantillon considéré est locale. Les mesures

expérimentales montrent que la production de chaleur dans les hydrogels est liée fortement à la

densité de réticulation EGDMA et aussi à la fréquence de sollicitation.

Abstract –This chapter is dedicated to quantifying the heat production in the hydrogel

HEMA-EGDMA under dynamic loading. After, we compare the theoretical results (computed)

with experimental results combined with an optimization proposal for identifying the parameters

dependency on the self-heating phenomenon. First, we present a simplified model that allows

us to identify the various parameters related to the heat production in the samples. We treat

the one-dimensional case because the heat production in the sample is assumed local (there is no

temperature gradient in sample). Experimental measurements show that the production of heat

in the hydrogels is strongly related to the cross-linking density EGDMA and also to the frequency

1 This chapter was presented as oral presentation at the 9th European Solid Mechanics Conference, ESMC, July6h-10th, 2015, Madrid, Spain. under title: " Nonlinear thermomechanics of heat production in high dissipativehydrogel HEMA-EDGMA for biomedical applications"

95

96CHAPTER 4. EXPERIMENTAL IDENTIFICATION OF SELF-HEATING IN HEMA-EGDMA.

of loading.

4.1 Introduction

Hydrogels have been widely employed in biomedical areas [24], [27], [29], [30]. The thermomechan-

ical response of these materials depends strongly on temperature, cross-link density or frequency if

the hydrogel is under cyclic loading [7]. Particular hydrogel possessing high dissipation properties

may induce heat production under cyclic loading [8]. Due to the heat production, a local tem-

perature increase can be observed in the material, a phenomenon called self-heating. In turn, the

temperature increase has an effect on its properties and on the thermomechanical behavior [8], [16].

Modeling and simulation methods are one of the strong characterization methods of the physical

phenomena in this kind of material. When the sample is subjected to mechanical and heat loads,

we need to develop a material coupled formulation to investigate these quantities. The goal of this

work is to develop a constitutive law based on generalized standard materials. Numerical methods

for a coupled partial differential equation with dynamic boundary conditions are developed with

the conservation laws as in the two previous sections [95], [96]. Nonlinear constitutive law for

viscoelastic material without heat effect has been established by Pioletti, Rakotomanana et al.

for biological tissues in large deformation [12]. The present work extends this model to nonlinear

constitutive law for thermo-viscoelastic model with heat effect in the particular case of matrix

HEMA-EGDMA hydrogel. In this work, a general continuum thermomechanical framework de-

scribing the effect is adapted to the description of the self-heating phenomenon. Numerical studies

are then carried out to examine the capability of the model to predict the heat production and the

nature of the coupling as well as to evaluate the influence of the main parameters such as cross-link

density and frequency of loading. In parallel, microcalorimetric experimental measurements are

performed to quantify the heat production in the HEMA-EGDMA hydrogel sample.

4.2 Microcalorimetric test

In order to characterize the heat production in the hydrogel samples, an adiabatic deformation

microcalorimeter is used (figure 4.1). The system consists of a test chamber (1) with two layer

walls (2), where vacuum can be created in the gap between the two walls (3) to have a highly

insulated system. A thermistor in the center of the chamber monitors the temperature during the

test (4). Mechanical loading is applied directly on the sample in the test chamber with a piston

(5) passing through a diaphragm (6). The initial temperature of the test chamber can be set by

thermostated water circulation around the chamber. In the chamber (1) hydrogel is placed. The

hydrogel sample consists of cylindrical samples. Cylindrical hydrogel samples 5 mm of diameter

4.2. MICROCALORIMETRIC TEST 97

and 8 mm of height are subjected to cyclic mechanical load at various frequencies f = 0.5, ..., 1Hz.

For the mechanical boundary conditions, on the top of the cylinder we apply the cyclic load, while

the bottom is fixed. For the thermal boundary condition, we have an adiabatic condition (non

inward and outward flux). The initial conditions are : initial stress null and initial temperature

θ0. The heat production is measured with a specific sensor inserted within the sample and the

data acquisition is directly obtained with a computer. For a more detailed description, the reader

is reffered to [7] .

Figure 4.1: Adiabatic microcalorimeter

The displacement is prescibed on the top of the sample to 20% of the sample height. The

sample loading is done in three parts including preload, cyclic loading and relaxation. And the

bottom of the sample is "fixed". We chose 30 s of preload, 5 mn cyclic loading and 5 mn relaxation.

Tables 4.1 to 4.3 summarize the different experimental conditions.

Samples Composition Diameter [mm] Height [mm] Water [%] Crosslink density [%]d h w φ

Sample 1 HEMA-EGDMA 8.93 5.33 40 6Sample 2 HEMA-EGDMA 8.91 5.50 40 8

Table 4.1: Characteristics and composition of the sample. The sample composition is given by:HEMA+40%w+φ% EGDMA.

Samples Preload/t Cyclic load/t Relax./t[%]/[mn] [%]/[mn] [%]/[mn]

Sample 1 15/0.5 5/5 0/5Sample 1* 15/0.5 5/10 0/5Sample 2 15/0.5 5/5 0/5Sample 2* 15/0.5 5/10 0/5

Table 4.2: Characteristics of the tests for each sample. The notation (*) denotes the same samplebut the cyclic load is during 10 min.


Samples Freq./cycl. Freq./cycl. Freq./cycl.[Hz]/[-] [Hz]/[-] [Hz]/[-]

Sample 1 0.5/150 1/300 1.5/450Sample 1* 0.5/150 1/300 1.5/450Sample 2 0.5/150 1/300 1.5/450Sample 2* 0.5/150 1/300 1.5/450

Table 4.3: Characteristics of the tests for each samples, preload percentage (ratio with the heightof the sample), cyclic load and relaxation time, t denotes the time.

4.3 Mathematical model

The self-heating phenomena are governed by a nonlinear-coupled partial differential equation sys-

tem deduced from two conservation equations of classical continuum thermomechanics. We assume

the postulate of the existence of two thermodynamic potentials the strain energy function and the

dissipation potential defined per unit of the reference volume. The model is obtained by construct-

ing with the free energy method, new non-negative convex energy functions given by the equation

(4.1). For physical and mathematical considerations, convexity/polyconvexity of the strain energy

and dissipation functions are an essential point since the common methods in computer simulation

depend on gradient methods.

ψ(E, θ) =λ

2tr2E + µtrE2 − (3λ+ 2µ)αtrE(θ − θ0)− cv

2θ0(θ − θ0)2

χ(E,∇θ) =λ′

2tr2E + µ′trE2 +

κ

2||∇θ||2 (4.1)

where λ, µ, α, cv, λ′, µ′ and κ are respectively the Lamé constants, the thermal expansion coef-

ficient, specific heat capacity coefficient, viscosity coefficient and heat conduction coefficient. The

reference temperature is denoted by θ0. Parameters α, cv and κ are considered as constants.

Hypothesis 4.1. For the thermodynamic potentials given by the relations (4.1), the Lamé’s con-

stants λ, µ are known for the hydrogel HEMA-EGDMA, the specific heat capacity coefficient is

estimated by microcalorimetric test. The remaining constant are unknowns (α[1/K], λ′[MPa.s],

µ′[MPa.s] and κ[W/(m.K)]). We assume the following mechanical properties for the sample:

Samples E[MPa] ν λ[MPa] µ[MPa] cv[J/(kg.K)]Sample 1 10-30 0.45 3.10-9.3 0.34-1.02 2900-3200Sample 2 20-50 0.40 2.86-7.15 0.71-1.78 2900-3200

The balance of linear momentum and the energy conservation allow us to express the governing

4.3. MATHEMATICAL MODEL 99

equations of the hydrogel sample and can be formulated as: DivFSe + DivFSv + ρB = ρ∂2u

∂t2in (B × [0, T ])

ρe = (Se + Sv) : E−DivQ + ρr in (B × [0, T ])(4.2)

where Se(E, θ) = ρ∂ψ

∂E(E, θ) and Sv(E,∇θ) =

∂χ

∂E(E,∇θ) are respectively the elastic and viscous

parts of the second Piola-Kirchhoff stress tensor;Q

θ= − ∂χ

∂∇θ(E,∇θ) is the heat flux, e = ψ(E, θ)+

sθ the internal energy, s = −∂ψ∂θ

(E, θ) the entropy density and E = ∇u + ∇Tu + ∇u∇Tu/2 is

the Green-Lagrange strain tensor. Div means the Lagrangian divergence operator with respect

X ∈ B, B is the Lagrangian body force vector per unit of mass of B and R is the Lagrangian heat

source per unit of mass of B. ρ denotes, in this work, the Lagrangian mass density.

Equations of the three-dimensional continuum, developed avove, define the initial boundary

value problem of thermomechanics. In detail, these were the description of deformation in the

context of kinematics, the formulation of the force equilibrium based on kinetic considerations,

the constitutive equation as well as the initial and boundary conditions. We assume the following

mechanical boundary conditions which include three parts, preloading, cyclic loading and relaxation

(StDN).

u · n = −

up

(t

τ

)if t < tp

up

(tpτ

)+ u0 cos(2πft) if tp ≤ t ≤ tc

on (Γt × [0, T ])

P · n = 0 if t > tc on (Γt × [0, T ])

P · n = 0 on (Γl × [0, T ])

u · n = u0 on (Γu × [0, T ])

P = F(Se + Sv) in (B × [0, T ]), I.C u(t = 0, ·) := 0, P(t = 0, ·) := 0 in (B × 0)(4.3)

where τ ∈ R+ is a time constant. up ∈ R denotes the prescribed displacement during the preloading

and the relaxation. u0 ∈ R denotes the prescribed displacement during the cyclic loading. We

consider two time characteristics tp ∈ R+ the preloading time and tc ∈ R+ the time during which

the cyclic load is applied. Experimentally, we apply the preload as a ramp form during the preload

time tp. Then we apply the mechanical cyclic loading during the load time tc. Finally, after

tc + tp, the discharge and relaxation time are beginning for a new tp. For the heat boundary

condition, we use the same continuous media B ∈ Rd with the V B the volume. The boundary

of B is ∂B = Γq ∪ Γl ∪ Γc with the surface SB. For each time t ∈ R+ this volume is under heat

production density ρr, a heat flux q0 on one parts of the boundary of B and with a prescribed


tM (n,M)

Γt

Γu

ΓlΓl

Γq

Γc

ΓlΓl(B) (B)

Figure 4.2: Boundary conditions: mechanical boundary conditions (left), heat transfer boundaryconditions (right).

temperature θ0 on other parts of the boundary of B. The heat boundary can written as:

Q · n = q0 on (Γq × [0, T ])

Q · n = 0 on (Γl × [0, T ])

Q · n = kc(θ − θ∞) on (Γc × [0, T ])

I.C θ(t = 0, ·) := θref in (B × 0)

(4.4)

in which, q0 is the prescribed heat flux on (Γq × [0, T ]), kc denotes the convection coefficient and

θ0 is the prescribed temperature, θref is the initial local temperature of the sample and θ0 is the

thermodynamic temperature.

By using the definition of the potential ψ and χ in the equation (4.1), the elastic and viscous

parts of the second Piola-Kirchhoff hold:

Se = λtr(E)I + 2µE− (3λ+ 2µ)α(θ − θ0)I; Sv = λ′tr(E)I + 2µ′E (4.5)

In order to identify the numerical parameters of the self-heating model with the experimental

measurements and for the correlation study, we compute the Cauchy stress tensor in the current

configuration. For this purpose, we use the classical formulation with the deformation gradient.

Then, the elastic part and the viscous part of the Cauchy stress tensor are given successively by:

σe =λ

Jtr(E)FIFT + 2

µ

JFEFT − (3λ+ 2µ)

α

J(θ − θ0)FIFT

σv =λ′

Jtr(E)FIFT + 2

µ′

JFEFT (4.6)

By introducing the equation (4.5) in the governing equation (4.2), the self-heating governing

4.3. MATHEMATICAL MODEL 101

equation can be expressed as:

Div [ρ(λtr(E)FI + 2µFE)− (3λ+ 2µ)α(θ − θ0)FI] + Div[λ′tr(E)FI + 2µ′FE

]+ ρB = ρ

∂2u

∂t2

in (B × [0, T ])

ρcvθ0

∂θ

∂tθ =

[(3λ+ 2µ)αθI + λ′trEI + 2µ′E

]: E + Div(κθ∇θ) + ρr in (B × [0, T ])

B.C and I.C (Cf. eq.(4.3) and (4.4))(4.7)

Starting from the expression of the heat flux Q = −κθ∇θ in (B × [0, T ]) , by using the divergence

theorem and rearranging the terms in the heat equation, the governing equation (4.7) can be

written as:

Div [ρ(λtr(E)FI + 2µFE)− (3λ+ 2µ)α(θ − θ0)FI] + Div[λ′tr(E)FI + 2µ′FE

]+ ρB = ρ

∂2u

∂t2

in (B × [0, T ])

ρcvθ0

∂θ

∂tθ = (3λ+ 2µ)αθtrE + λ′tr2E + 2µ′trE2 − κθ∆θ + κ||∇θ||2+ρr in (B × [0, T ])

B.C and I.C (Cf. eq.(4.3) and (4.4))(4.8)

We assume two cases:

• Case 1: Local self-heating model κ ≡ 0, Q ≡ 0 For the hydrogel HEMA-EGDMA, the

heat conductivity coefficient is very small (κ ≡ 0), then the heat flux by conduction in the

sample is neglected (Q ≡ 0). Analogously, the change in internal energy caused by the

sources of heat is local vanishes and there is no heat diffusion in the media.

Hypothesis 4.2 (Local self-heating model). We assume for this case that we have a local heat

production. The internal heat production is not function of the space but just function of time

θ := θ(t). In this case, the quantity Div [(3λ+ 2µ)α(θ − θ0)FI] ≡ 0 (effect of the temperature

change on stress) in the governing equation (4.8). In fact, we have the effect of the velocity on the

internal heat production.

For the second approximation we assume that, for the hydrogel HEMA-EGDMA, the heat con-

ductivity coefficient of the sample is significant (κ 6= 0), then the heat flux by conduction in the

sample is also significant (Q 6= 0). Indeed, the change in internal energy is caused by the sources

of heat and the deformation.

• Case 2: κ 6= 0,

Hypothesis 4.3 (Total self-heating model). In this case, we assume that the total heat is

function of the space, the gradient of temperature and displacement. In fact, the heat conductivity

is not neglected, then, the internal heat production is function of the space and time θ := θ(x, t).


In this case, the quantity Div [(3λ+ 2µ)α(θ − θ0)FI] 6= 0 (effect of the temperature on stress) in

the governing equation (4.8). In fact, we have the two coupling terms: the effect of the velocity on

the internal heat production and the effect of the temperature change on the stress.

The character of the initial boundary value problem of structural mechanics depends on the types

of structure and loading that have to be described, which, on the other hand, decisively affect the

modeling of the load-carrying behavior. In the previous sections, the essential modeling aspects

were already discussed on geometrical and material levels. In summary, the modeling can be

categorized, in essence, according to the aspects of geometrical linearity or non-linearity, material

linearity or non-linearity, and time-dependence or time-independence. The various approximation

levels differ significantly in the complexity of the numerical solution of the underlying physical

problem. The correlation between the simplification of the physical problem and the complexity of

the numerical solution is illustrated in this work. Furthermore, the dynamic or static formulation

of the problem is decisive for the effort expanded on the numerical solution.

We assume linearity of the temperature and the displacement. For physical consideration, the

sample dimension is small for the hydrogel HEMA-EGDMA, we therefore assume that the heat

production in the sample is local.

Hypothesis 4.4 (Linearity in temperature). We assume small variation of the temperature

distribution in the sample the prescribed cyclic displacement. The temperature θ ∈ R+ is expessed

as a reference temperature θ0 ∈ R+ plus the perturbation δθ ∈ R+. We have:

θ = θ0 + δθ θ = δθ (4.9)

By using the linearity in temperature, the equation (4.8) becomes:

Div [ρ(λtr(E)FI + 2µFE)− (3λ+ 2µ)αδθFI] + Div[λ′tr(E)FI + 2µ′FE

]+ ρB = ρ

∂2u

∂t2

in (B × [0, T ])

ρcv∂δθ

∂t= (3λ+ 2µ)α(θ0 + δθ)trE + λ′tr2E + 2µ′trE2 − κθ0∆δθ + ρr

in (B × [0, T ])

B.C and I.C (Cf. eq.(4.3) and (4.4))(4.10)

• Case 1: Local self-heating model κ ≡ 0, Q ≡ 0, Cf. hypothesis 4.2

4.4. 2D AND 1D APPROACHES 103

The governing equation can be written as follows:

Div [ρ(λtr(E)FI + 2µFE)− (3λ+ 2µ)αδθFI] + Div[λ′trEFI + 2µ′FE

]+ ρB = ρ

∂2u

∂t2

in (B × [0, T ])

ρcv∂δθ

∂t= (3λ+ 2µ)α(θ0 + δθ)trE + λ′tr2E + 2µ′trE2 + ρr

in (B × [0, T ])

B.C and I.C (Cf. eq.(4.3) and (4.4))(4.11)

• Case 2: Total self-heating model κ 6= 0, Q 6= 0, Cf. hypothesis 4.3.



]+ ρB = ρ

∂2u

∂t2

in (B × [0, T ])

ρcv∂δθ

∂t= (3λ+ 2µ)α(θ0 + δθ)trE + λ′tr2E + 2µ′trE2 − κθ0∆δθ + ρR

in (B × [0, T ])

B.C and I.C (Cf. eq.(4.3) and (4.4))(4.12)

In order to show the solution of the problem with the applicability of the thermoviscoelastic

model as defined in the equation (4.12), we firstly assume one and two dimensional problem.

4.4 2D and 1D approaches

As preliminary steps, it is important to recall the two and monodimensional formulation. The

thermomechanical formulation will help us to understand each term appearing in the equation

(4.12). We assume one-dimensional compression. For the deformation analysis of two-dimensional

continua, the plane stress and the plane strain states are of interest. The plane strain state is

mostly used in cases where the dimension in one direction is very large with the loading in this

direction remaining unchanged. The derivation of these equations can be found in the following

sections.

Hypothesis 4.5 (Small strain assumption). As a first approximation the essential components of

the description are small, linear elastic deformations

The governing equation for one dimension self-heating obtained from the conservation law are


obtained for 1D plane strain problem.ρ(λ+ 2µ)

(∂2u

∂x2

)+ (3λ+ 2µ)α

(δθ

∂x

)+ (λ′ + 2µ′)

(∂2u

∂x2

)+ ρB = ρ

∂2u

∂t2in (B × [0, T ])

ρcv∂δθ

∂t= (3λ+ 2µ)α(θ0 + δθ)

(∂u

∂x

)+ (λ′ + 2µ′)

(∂2u

∂x2

)+ κθ0

∂2δθ

∂x2+ ρR in (B × [0, T ])

(4.13)

Nondimensional equations

In order to analytically solve the one-dimensional problem (4.13), we used the dimensionless form

of the govering equation. For this purpose, we introduce new variables as defined in the equation

(4.14):

x =x

`; u =

u

u0; ˆu =

u

u0; t =

t

t0; δθ =

δθ

θ0. (4.14)

The governing equation (4.13), With the initial and the boundary conditions, and keeping the

notation u but not u can be written in the following form:

A

C

(∂2u

∂x2

)+G

C

(∂δθ

∂x

)+B

C

(∂2u

∂x2

)+ ρB =

∂2u

∂t2in (B × [0, T ])

∂δθ

∂t=D

F(θ0 + δθ)

(∂u

∂x

)+E

F

(∂2u

∂x2

)+H

F

∂2θθ

∂x2+ ρR in (B × [0, T ])

δθ(x, 0) = θref ;

(−κ∂δθ

∂x

)x=0

= 0;

(−κ∂δθ

∂x

)x=`

= 0; u(0, t) = 0

u(`, t) = u` sin(ωt); u(x, 0) = 0; u(x, 0) = 0

(4.15)

In which,

A = ρ(λ+ 2µ)

`2u2

0; B =(λ′ + 2µ′)

`2u2

0; C = ρu0

t20; F = ρc

θ0

t0; (4.16)

D =(3λ+ 2µ)αθ0

`u0; E =

(λ′ + 2µ′)

`2u2

0; G =(3λ+ 2µ)ρα

`θ0; H =

κθ20

`2. (4.17)

• Case 1: Local self-heating model, κ ≡ 0, GC∂θ∂x ≡ 0, HF

∂2θ∂x2 ≡ 0, Cf. hypothesis 4.3.

According to the equation (4.15) and including the initial and the boundary conditions, the gov-

erning equations for the local self-heating, with the hypothesis of linearity in temperature, take

the following forms given by the equations

A

C

(∂2u

∂x2

)+B

C

(∂2u

∂x2

)+ ρB =

∂2u

∂t2in (B × [0, T ])

∂δθ

∂t=D

F(θ0 + δθ)

(∂u

∂x

)+E

F

(∂2u

∂x2

)+ ρR in (B × [0, T ])

θ(x, 0) = θref ; u(0, t) = 0

u(`, t) = u` sin(ωt); u(x, 0) = 0; u(x, 0) = 0

(4.18)


For the first approximation, we assume that the heat source ρR = 0 and the body force ρB = 0,

then, we introduce K1 := AC , K2 := B

C , K3 := DF , K4 := E

F , the system can be written as:K1

(∂2u

∂x2

)+K2

(∂2u

∂x2

)=∂2u

∂t2in (B × [0, T ])

K3(θ0 + δθ)

(∂u

∂x

)+K4

(∂2u

∂x2

)=∂δθ

∂tin (B × [0, T ])

θ(x, 0) = θref ; u(0, t) = 0; u(`, t) = u` sin(ωt); u(x, 0) = 0; u(x, 0) = 0

(4.19)

For the first equation, we use the variable (space-time) separation u(x, t) = φ(x)T (t) in the first

equation, for a physic solution we have:

φ′′(x)

φ(x)

(K1T (t) +K2T (t)

)= T (t) in (B × [0, T ])

K3(θ0 + δθ)

(∂u

∂x

)+K4

(∂2u

∂x2

)=∂δθ

∂tin (B × [0, T ])

θ(x, 0) = θref ; u(0, t) = 0; u(`, t) = u` sin(ωt); u(x, 0) = 0; u(x, 0) = 0

(4.20)

We can see that the heat equation takes the form of:

∂δθ(t)

∂t+ B(x, t)δθ(t) + C(x, t) = 0 in (B × [0, T ]) (4.21)

The solution taking into account the initial condition, of the temperature can be expressed as:

δθ(t) =

[∫ t

0

C(x, τ) exp

(∫ t

0

−B(x, τ)dτ

)dτ + θ0

]exp

(∫ t

0

B(x, τ)dτ

)(4.22)

Then, we have:

δθ(t) =

[∫ t

0

(K3θ0

∂u

∂x+K4

∂2u

∂x2

)exp

(∫ t

0

−K3∂u

∂x(x, τ)dτ

)dτ + θ0

]exp

(∫ t

0

K3∂u

∂x(x, τ)dτ

)(4.23)

For a physical relevant solution (see [12]), we have:

φ′′(x)

φ(x)= −k2, φ(x) = a sin(kx) + b cos(kx) (4.24)

In this case, the first equation can be written as:

T (t) +K2k2T (t) +K1k

2T (t) = 0 in (B × [0, T ]) (4.25)

The characteristic equation is given by r2 + K2k2r + K1k

2 = 0, the discriminant is ∆ = K22k

4 −

4K1k2. We define a critical damping for ∆ = 0, Kc

2 = 2√K1

k , the damping coefficient is defined as

ζ := K1

Kc2

= K2k2√K1

. We denote by Ω0 = K1k2, the equation (4.25) can be written as:

T (t) + 2ζkΩ0T (t) + Ω20T (t) = 0 in (B × [0, T ]) (4.26)


The characteristics equation is given by s2+2ζΩ0s+Ω20 = 0, the discriminant is ∆s = 4Ω2

0

(ζ2 − 1

).

For the solution, we assume that T (0) = T0, T (0) = 0 and consider following three cases:

1. Critical damping ζ = 1, ∆s = 0

s = −Ω0 (4.27)

The solution is T (t) = aest = ae−Ω0t, the expression bte−Ω0t also satisfies the differential

equation. We have T (t) = (a+ bt)est, in which a = T0 and b = T0ω0, In this case we have

T (t) = T0(1 + Ω0t)e−Ω0t

u(x, t) = T0

+∞∑n=1

u`sin(ωt)

sin(k`)sin(kx)(1 + Ω0t)e

−Ω0t

δθ(t;x) =

+∞∑n=1

K4

K3ktan(kx)

+

+∞∑n=1

exp

(−e−Ω0tkcos(kx)sin(ωt)K3T0u` (1 + Ωt0)

sin(k`)

)(θref −

K4

K3ktan(kx)

)S33 =

+∞∑n=1

T0u`k`

sin(k`)e−Ω0tcos(kx)

[ωcos(ωt)K2 (1 + Ω0t) + sin(ωt)

(−K2Ω2

0t+K1 (1 + Ω0t))]

2. Sub-critical damping ζ < 1, ∆s < 0

s1 = −Ω0

(ζ + j

√1− ζ2

), s2 = −Ω0

(ζ − j

√1− ζ2

), j2 = −1 (4.28)

We denote by Ω = Ω0

√1− ζ2 the solution can be written as:

T (t) =T0

2

[(1 +

jζΩ0

Ω

)e−(Ω0ζ+jΩ)t +

(1− jζΩ0

Ω

)e−(Ω0ζ−jΩ)t

]T (t) =

T0

2e−Ω0ζt

[(1 +

jζΩ0

Ω

)e−jΩt +

(1− jζΩ0

Ω

)ejΩt

](4.29)

Using the transformation of e−jΩt and ejΩt, we have

T (t) = T0e−Ω0ζt

[cos(Ωt) +

Ω0ζ

Ωsin(Ωt)

]u(x, t) = T0

+∞∑n=1

u`sin(ωt)

sin(k`)sin(kx)e−Ω0ζt

[cos(Ωt) +

Ω0ζ

Ωsin(Ωt)

]δθ(t;x) =

+∞∑n=1

K4

K3ktan(kx)

+

+∞∑n=1

exp

(−e−ζΩ0tkcos(kx)sin(ωt)K3T0u` (Ωcos(Ωt) + ζsin(Ωt)Ω0)

sin(k`)Ω

)(θref −

K4

K3ktan(kx)

)S33 =

+∞∑n=1

T0u`k`

sin(kx)Ωe−ζΩ0tcos(kx) [ωcos(ωt)K2 (Ωcos(Ωt) + ζsin(Ωt)Ω0)]


+

+∞∑n=1

T0u`k`

sin(kx)Ωe−ζΩ0tcos(kx)

[sin(ωt)

(K1 (Ωcos(Ωt) + ζsin(Ωt)Ω0)− sin(Ωt)K2

(Ω2 + ζ2Ω2

0

))]3. Super-critical damping ζ > 1, ∆s > 0

s1 = −Ω0

(ζ +

√ζ2 − 1

), s2 = −Ω0

(ζ −

√ζ2 − 1

)(4.30)

The solution is

T (t) =T0

2e−ζΩ0t

[(1− Y ) e−Ω0

√ζ2−1t + (1 + Y ) eΩ0

√ζ2−1t

]u(x, t) =

T0

2

+∞∑n=1

u`sin(ωt)

sin(k`)sin(kx)e−ζΩ0t

[(1− Y ) e−Ω0

√ζ2−1t + (1 + Y ) eΩ0

√ζ2−1t

]δθ(t;x) =

+∞∑n=1

K4

K3ktan(kx)

+

+∞∑n=1

exp

[−2e−ζΩ0tkcos(kx)sin(ωt)ζsinh (YsΩ0t)K3T0u`

sin(k`)√ζ2 − 1

](θref −

K4

K3ktan(kx)

)

+

+∞∑n=1

exp

−2e−ζΩ0tkcos(kx)sin(ωt)√ζ2 − 1cosh

(√ζ2 − 1Ω0t

)K3T0u`

sin(k`)√ζ2 − 1

(θref −

K4

K3ktan(kx)

)S33 =

+∞∑n=1

T0u`k`cos(kx)

sin(kl)√ζ2 − 1

e−(ζ+Ys)Ω0t[(−1 + e2

√ζ2−1Ω0t

)ζ +

(1 + e2

√ζ2−1Ω0t

)Ys

]sin(ωt)K1 +K2

[(−1 + e2

√ζ2−1Ω0t

)ζ +

(1 + e2

√ζ2−1Ω0t

)Ys

]ωcos(ωt)

−K2

(−1 + e2

√ζ2−1Ω0t

)sin(ωt)Ω0

In which Y = ζ√

ζ2−1and Ys =

√ζ2 − 1.

• Case 2: Total self-heating model, κ 6= 0, GC(∂θ∂x

)6= 0, HF θ

∂2θ∂x2 6= 0, Cf. hypothesis 4.3.


A

C

(∂2u

∂x2

)+G

C

(∂δθ

∂x

)+B

C

(∂2u

∂x2

)+ ρB =

∂2u

∂t2in (B × [0, T ])

∂δθ

∂t=D

F(θ0 + δθ)

(∂u

∂x

)+E

F

(∂2u

∂x2

)+H

F

∂2δθ

∂x2+ ρR in (B × [0, T ])

δθ(x, 0) = θref ;

(−κ∂δθ

∂x

)x=0

= 0;

(−κ∂δθ

∂x

)x=`

= 0; u(0, t) = 0

u(`, t) = u` sin(ωt); u(x, 0) = 0; u(x, 0) = 0

(4.31)

Remark 4.1. The local behavior of a thermoviscoelastic body for one dimensional problem was

totally described in the previous section by means of the initial boundary value problem. Generally,

the solution of this differential equation is not analytically explicit. Therefore, approximation meth-

ods, in particular the Finite Element Method, are used in order to find an approximate solution.


This method does not solve the strong form of the differential equation. It merely solves its integral

over the domain, the so-called weak form of the differential equation. This weak formulation forms

the basic prerequisite for the application of approximation methods.

4.5 Identification of the model parameters

For a given thermodynamic potential, the main problem after the formulation is to calculate or

measure the physical constants in the model. If the physical constants can be identified with

the experimental measurement, it is appropriate to determine these constants by using classical

identification procedures. In the opposite case, we need to identify these constants by using ana-

lytical/numerical approaches. For that, we use the one dimension analytical description in order

to identify the physical constant in the model.

4.5.1 Cost functions

According to the classical method of optimization, the identification method of physical constant

in the model of self-heating (thermoviscoelasticity) can be expressed using complex parameters.

The parameters to be identified are α, λ′, µ′ and κ

Definition 4.1 (Cost functions). The cost function is defined as f(σcomp (α, λ′, µ′)− σobse

)for the

displacement and the cost function for the temperature is g(δθcomp (α, λ′, µ′, κ)− δθobse

). Then,

for the self-heating model, we have to minimize the following coupled cost function:

[α, λ′, µ′, κ] = infα∈R+

infλ′∈R+

infµ′∈R+

infκ∈R+

f(σcomp (α, λ′, µ′)− σobse

)g(δθcomp (α, λ′, µ′, κ)− δθobse

) (4.32)

Then, for the large deformation, this self-heating cost function can be written as:

[α, λ′, µ′, κ] = infα∈R+

infλ′∈R+

infµ′∈R+

infκ∈R+

f

((2

JFSFT

)comp(α, λ′, µ′)− σobse

)g(δθcomp (α, λ′, µ′, κ)− δθobse

) (4.33)

Where f and g are the functions used to measure the difference between the computed and observed

quantity, in general we use the square function f, g := 12 ‖ · ‖

2.

Remark 4.2. In this section the notation (· · · )comp denotes the analytical or numerical quantity.

Firstly we use analytical solution to optimize the constants and secondly we use the numerical

solution. The notation (.)obse denotes the observed quantity.

S = λtr(E)I + 2µE− (3λ+ 2µ)α(θ − θ0)I + λ′tr(E)I + 2µ′E (4.34)

4.5. IDENTIFICATION OF THE MODEL PARAMETERS 109

Definition 4.2 (Least square cost functions). For the first approximation, we define least square

cost functions to identify the physical parameters of the model:

α, λ′, µ′, κ = infα∈R+

infλ′∈R+

infµ′∈R+

infκ∈R+

1

2

∥∥∥∥∥(

2

JFSFT

)comp/33

(α, λ′, µ′)− σobse∥∥∥∥∥

2

∥∥δθcomp (α, λ′, µ′, κ)−(δθobse + 273.15

)∥∥2

(4.35)

For the computation, we have

σobse =

0 0 0

0 0 0

0 0F obs(t)

SB

; E =

ET 0 0

0 ET 0

0 0 1−uobsp (t)

h

(4.36)

In which SB and h denote respectevely the surface and the top of the sample.

up(t) = −

up

(t

τ

)if t < tp

up

(tpτ

)+ u0 cos(2πft)if tp ≤ t ≤ tc

(4.37)

4.5.2 Computation, splitting

We present in this section the computation setting using splitting methods. The main step is

summarized by the following scheme.

1) Define: Initialization [α0, λ′0, µ′0, κ0]; δθ0 = θ0 + 273.15, ν

2) Minimize Self-heating model:

•LOOP (k = 0 · · ·n)

a) Minimize wave equation: (input [αk, λ′k, µ′k, κk])

[αk, λ′k, µ′k, κk] = inf

αk∈R+

infλ′k∈R+

infµ′k∈R+

infκk∈R+

1

2

∥∥∥∥∥(

2

JFSFT

)comp/33

(αk, λ′k, µ′k)− F obs(t)

SB

∥∥∥∥∥2

ifλ′k

2(λ′k + µ′k)≥ ν (physical condition)

LOOP wave equation (k ←− k + 1)

else

End (output [αk, λ′k, µ′k, κk])

b) Minimize heat equation: (input [αk, λ′k, µ′k, κk])

[αk, λ′k, µ′k, κk] = inf

αk∈R+

infλ′k∈R+

infµ′k∈R+

infκk∈R+

1

2

∥∥δθcomp (αk, λ′k, µ′k, κk)−

(δθobse + 273.15

)∥∥2


if |αk − αk+1, λ′k − λ′k+1, µ

′k − µ′k+1, κk − κk+1| ≥ ε

•LOOP (k ←− k + 1)

else

End (output [αk, λ′k, µ′k, κk])

Hypothesis 4.6 (Cost functions for one dimensional model). For the one dimensional model, the

constant K1 is known via λ, µ. The unknowns are K2,K3,K4. We have to minimize the following

cost function.

K2,K3,K4 = infK2∈R+

infK3∈R+

infK4∈R+

1

2

∥∥∥∥(2S)

comp/33 (K1,K2)− F obs(t)

SB

∥∥∥∥2

,

‖ δθcomp (K1,K2,K3,K4)−(δθobse + 273.15

)‖2

(4.38)

4.6 Numerical approximations

In this section, we propose a finite element method for a 2D stess elasticity problem. The equations

established in the previous section are solved using a finite elements discretization in space. In

time, an implicit Euler scheme is applied for the time integration. In fact, we consider finite element

approximations of the pure dynamic displacement traction/compression boundary value in three-

dimensional nonlinear thermomechanical viscoelasticity associated with a homogenous viscoelastic

material. We use the weak form of the governing equation; firstly, we multiply each equation by a

test function that is compatible with the geometric boundary conditions. Secondly, the equation

is integrated on the volume of the media.

4.6.1 Governing equation

For the numerical approximation we use directly the second case for the self-heating governing

equation given by the equation (4.40) which takes into account the fully coupled problem.

We find a displacement field u : (B × [0, T ]) −→ Rd, a velocity field u : (B × [0, T ]) −→ Rd,

a stress field S : (B × [0, T ]) −→ Sd, and a temperature θ : (B × [0, T ]) −→ R. Similarly, we can

consider the same boundary and initial conditions as defined in the equation cf. eq. (4.3) and (4.4)

in (B × [0, T ]) and in (∂B × [0, T ]). Also, we recall the govering equation and boundary conditions

involve the stress, body force, surface loads, heat flux, heat source, prescribed displacement and

4.6. NUMERICAL APPROXIMATIONS 111

temperature


]+ ρB = ρ

∂v

∂t

in (B × [0, T ])

ρcv∂δθ

∂t= (3λ+ 2µ)α(θ0 + δθ)trE + λ′tr2E + 2µ′trE2 − κθ0∆δθ + ρr

in (B × [0, T ])

v =∂u

∂tin (B × [0, T ])

B.C and I.C (Cf. eq.(4.3) and (4.4)) in (B × [0, T ])

(4.39)

4.6.2 Variational principle

To derive the principle of virtual work, the strong form of the differential equation, which corre-

sponds with the local balance of momentum, as well as the static boundary condition are multiplied

by a vector-valued test function and integrated over the volume of the body under consideration.

As test function the virtual displacements and temperature δu and δθ∗ are chosen. This special

test function has the following properties. To formulate the finite element methods for (4.40), we

introduce the following finite element space, vector and scalar-valued δu and δθ∗ respectively as

V u :=u ∈ Rd, δu ∈ [H1(B)]d; u = 0 on Γ

; V δθ :=

δθ ∈ Rd, δθ∗ ∈ [H1(B)]d; δθ∗ = 0 on Γ

[L2(B)]d =

δu : B −→ R|

∫B|δu|2 <∞

; [L2(B)]d =

δθ∗ : B −→ R|

∫B|δθ∗|2 <∞

Let [H1

s (B)]d be a functionnal space in which we are searching the solution in accordance with

its regularity [H1s (B)]d = δu ∈ [H1(B)]d|δu = s∀x ∈ Γ and [H1

s (B)]d = δθ∗ ∈ [H1(B)]d|δθ∗ =

s∀x ∈ Γ where and [H1(B)]d is a Sobolev spaces.

Definition 4.3. We define the Sobolev spaces as [H1(B)]d = δu ∈ [L2(B)]d, ‖∇δu‖ ∈ L2(B) and

[H1s (B)]d = δθ∗ ∈ [L2(B)]d, ‖∇δθ∗‖ ∈ [L2(B)]d.

where [L2(B)]d is the Hilbert vector space of the functions quadratically summable respectively

in (B).

The corresponding weak formulation in space-time is obtained by multiplying by the test func-

tions: firstly, for the balance of momentum, by the scalar product with a vector-valued test func-

tion δu which has to be compatible with the geometric boundary conditions. Then, this equation

is integrated over the volume of the sample. Secondly, the balance of the energy is multiplied

with a scalar test function δθ∗ and also integrated over the volume. In a weak sense, we find


δΦ = δu, δu, δθ∗ ∈W , such that:

∫B

Div [ρ(λtr(E)FI + 2µFE)− (3λ+ 2µ)αδθFI] δu dV B +

∫B

Div[λ′tr(E)FI + 2µ′FE

]δu dV B

+

∫BρBδu dV B =

∫Bρ∂v

∂tδu dV B ∀δu ∈ [H1(B)]d∫

Bρcv

∂δθ

∂tδθ∗ dV B =

∫B

(3λ+ 2µ)α(θ0 + δθ)trEδθ∗ dV B +

∫Bλ′tr2Eδθ∗ dV B +

∫B

2µ′trE2δθ∗ dV B

−∫Bκθ0∆δθδθ∗ dV B +

∫Bρrδθ∗ dV B ∀δθ∗ ∈ [H1(B)]d

v =∂u

∂tin (B × [0, T ])

(4.40)

For all [δΦ] = (δu, δθ∗). In which, dV B and dSB are respectively the volume and surface ele-

ment. Using the divergence theorem and taking into account the boundary conditions, the final

representation of the weak form of the coupled self-heating model reads as follows:

−∫Bρ(λtr(E)I + 2µE) : sym[∇δu] dV B −

∫B

(3λ+ 2µ)ραδθI : sym[∇δu] dV B

+

∫B

(λ′tr(E)I + 2µ′E

): sym[∇δu] dV B +

∫BρBδu dV B =

∫Bρ∂v


Bρcv

∂δθ

∂tδθ∗ dV B =

∫B



∫B


+

∫Bκθ0∇δθ.∇δθ∗ dV B −

∫∂Bκθ∇δθ.n.δθ∗ dSB +


v =∂u

∂tin (B × [0, T ])

(4.41)

in which, sym[∇δu] := ∇δuT(∇u + I). n is the Langrangian unit outer normal. For simplicity, we

only consider the case where B is a bounded convex domain throughout this paper. In this case,

we have:

−∫Bρ(λtr(E)I + 2µE) : ∇δuT(∇u + I) dV B −

∫B

(3λ+ 2µ)ραδθI : ∇δuT(∇u + I) dV B

+

∫B

(λ′tr(E)I + 2µ′E

): ∇δuT(∇u + I) dV B +

∫BρBδu dV B =

∫Bρ∂v


Bρcv

∂δθ

∂tδθ∗ dV B =

∫B



∫B


+

∫Bκθ0∇δθ.∇δθ∗ dV B −

∫∂Bκθ∇δθ.n.δθ∗ dSB +


v =∂u

∂tin (B × [0, T ])

(4.42)

We denote by Fδu(u, δθ, δu) the first equation and by Fδθ∗(u, δθ, δθ∗) the second equation in the

system (4.41). For the numerical solution of the weak form (4.40) with the finite element methods,

the linear form of the equation (4.41) using Newton-Raphson methods can be expressed as:

L(Φ)∆Φ = ∆u,∆δθ (4.43)


Then we have, DuFδu(u, δθ, δu) DθFδu(u, δθ, δu)

DuFδθ∗(u, δθ, δθ∗) DθFδθ∗(u, δθ, δθ∗)

∆u

∆δθ

=

Fδu(u, δθ, δu) [δu]

Fδθ∗(u, δθ, δθ∗) [δθ∗]

(4.44)

We denote by: Luu Lδθu

Luδθ Lδθδθ

∆u

∆δθ

=

Fδu(u, δθ, δu) [δu]

Fδθ∗(u, δθ, δθ∗) [δθ]

(4.45)

where Lij := DiFδj(δu, δθ) for i, j := u, δθ. The right hand side is linear in argument, Di is

the directional derivative with respect the indicated arguments. The finite element discretization

allows us to write the equation: Luu Luδθ

Lδθu Lδθδθ

∆u

∆δθ

=

Fδu

Fδθ∗

(4.46)

4.6.3 Computations

For the computation we use Comsol Multiphysics to compute the model by using general form

of PDE. This tool allows us to solve systems of time-dependent or stationary partial differential

equations in one, two, and three dimensions with complex geometry. There are two forms of the

partial differential equations available, the general form and the coefficient form. They read

ea∂2u

∂t2+ da

∂u

∂t+∇ · Γ = F in (B × [0, T ])

−n · Γ = G+

(∂R

∂u

)Tµ; 0 = R on (∂B × [0, T ])

ea∂2u

∂t2+ da

∂u

∂t+∇ · (−c∇u− au + γ) + au + β · ∇u = f in (B × [0, T ])

−n(−c∇u− au + γ) + qu = g − hTµ;hu = R on (∂B × [0, T ]) (4.47)

respectively. The second kind of equation (coefficient form) can only be used for mildly nonlinear

problems. For most nonlinear problems, the general form needs to be used.

Remark 4.3. The coefficients of the coefficient form may depend both on x, t, and u. Observe

that a dependence on u is not recommended. The flux vector Γ and the scalar coefficient F , G and

R can be function of the spatial coordinates the solution u and the space and time derivatives of u.

The variable µ is the Lagrange multiplier, and T denotes the transpose. q and g are respectively

the boundary absorption coefficient and the boundary source term.

The second method, to solve numerically the non-linear mechanics in this software is to define

directly the thermodynamic potential in the software. The thermodynamic conditions as convexity


must be verified before introducing the thermodynamic potential.

∇ · (σe + σv) + ρb = ρ∂2u

∂t2; σe = J−1FSeFT σv = J−1FSvFT in (B × [0, T ])

F = ∇u + I; J = detF; E = (C− I)/2; C = FTF = I +∇u +∇u +∇Tu∇u/2

Se = 2ρ∂ψ

∂C; Sv = 2

∂χ

∂C(4.48)

In which, F is the deformation gradient, I is the identity matrix, E and C denote respectively the

Green-Lagrange and the Cauchy-Green strain tensors. To solve numerically the self-heating model

we assume: for the first approximation, we use the general form of PDE given by the equation

(4.47) (first equation) for the wave and the heat equations. In a second approximation, we use the

second method (4.48), it consists to introduce directly the thermodynamic potential for the wave

equation and the general form of PDE for the heat equation. In this work, we use these methods

to compare the numerical solution of the self-heating model. For the first approximation, we use

the following notation, for the self-heating model with the boundary conditions using the notation

of the software. eua 0

0 eθa

∂2

∂t2

u

δθ

+

dua 0

0 dθa

∂

∂t

u

δθ

+∇ ·

Γu

Γθ

+∇ ·

Γu

Γθ

=

Fu

Fθ

−n · (Γu + Γu) = 0, G = 0, on (Γ` × [0, T ])

R = −u on (Γu × [0, T ])

R = −u− u0 on (Γt × [0, T ])

−n · (Γθ + Γθ) = 0, G = 0, on (∂B − Γc × [0, T ])

−n · Γθ = h(δθ − δθref ), G = 0, on (Γc × [0, T ])(4.49)

Then, ρ 0

0 0

∂2

∂t2

u

δθ

+

0 0

0 ρcv

∂

∂t

u

δθ

+∇ ·

FSe

κ∇θ

+∇ ·

FSv

0

= ρ

B

r

(4.50)Implementation in Comsol Multiphysics software is based on the equation 4.50.

4.6.4 Numerical approximations for local self-heating

Using the hypothesis for local self-heating in the sample, (Cf. hypothesis 4.2). The equation (4.50)

becomes: ρ 0

0 0

∂2

∂t2

u

δθ

+

0 0

0 ρcv

∂

∂t

u

δθ

+∇ ·

FSe

0

+∇ ·

FSv

0

= ρ

B

r

(4.51)


In which

B = 0

r = (3λ+ 2µ)α(θ0 + δθ)trE + λ′tr2E + 2µ′trE2

Se = λtr(E)I + 2µE− (3λ+ 2µ)α(θ − θ0)I

Sv = λ′tr(E)I + 2µ′E (4.52)

We compute two-dimensionnal plane strain, in fact we have two components u and v in displace-

ment, with the components written as:eua 0 0

0 eva 0

0 0 eθa

∂2

∂t2

u

v

δθ

+

dua 0 0

0 dva 0

0 0 dθa

∂

∂t

u

v

δθ

+∇ ·

(FSe)11 (FSe)12 0

(FSe)21 (FSe)22 0

0 0 0

1

1

1

+∇ ·

(FSv)11 (FSv)12 0

(FSv)21 (FSv)22 0

0 0 0

1

1

1

=

Bu

Bv

rθ

−n · Γu = 0, G = 0, on (Γ` × [0, T ])

R =

−u

−v

on (Γu × [0, T ])

R =

0

−v − v0

on (Γt × [0, T ])

−n · Γθ = 0, G = 0, on (∂B − Γc × [0, T ])


In which, the pure displacement boundary value traducing the dynamic load for two dimensional

homegenous isotropic hydrogel is given by:

v0(t) = −

vp

(t

τ

)if t < tp

vp

(tpτ

)+ u0 cos(2πft)if tp ≤ t ≤ tc

0 if t > tc

(4.54)

4.6.5 Numerical approximations for non-local self-heating

Cf. hypothesis 4.3. The equation (4.49) becomes: ρ 0

0 0

∂2

∂t2

u

δθ

+

0 0

0 ρcv

∂

∂t

u

δθ

+∇ ·

FSe

κ∇δθ

+∇ ·

FSv

0

= ρ

B

r

(4.55)


in which

B = 0

r = (3λ+ 2µ)α(θ0 + δθ)trE + λ′tr2E + 2µ′trE2

Se = λtr(E)I + 2µE

Sv = λ′tr(E)I + 2µ′E (4.56)

We compute two-dimensionnal plane strain, with the components can be written as:eua 0 0

0 eva 0

0 0 eθa

∂2

∂t2

u

v

δθ

+

dua 0 0

0 dva 0

0 0 dθa

∂

∂t

u

v

δθ

+∇ ·

(FSe)11 (FSe)12 0

(FSe)21 (FSe)22 0

0 0 κ∇δθ

1

1

1

+∇ ·

(FSv)11 (FSv)12 0

(FSv)21 (FSv)22 0

0 0 0

1

1

1

=

Bu

Bv

rθ

−n · Γu = 0, G = 0, on (Γ` × [0, T ])

R =

−u

−v

on (Γu × [0, T ])

R =

0

−v − v0

on (Γt × [0, T ])

−n · Γθ = 0, G = 0, on (∂B − Γc × [0, T ])


Implementation in comsol multiphysics software is based on the equation (4.57).

4.7 Experimental and numerical results

As a first result, we want to verify that the experimental measurement of the temperature in

the sample is not biaised by the friction between the hydrogel and the temperature sensor in the

microcalorimeter during the deformation. It can be observed in figure 4.3 that the measured tem-

perature does not change during the preloading. We can then conclude that there is no temperature

increase due to the friction and, then, eventual temperature increase will be due to self-heating

phenomenon of the tested sample.

The effect of the self-heating and corresponding temperature increase in the hydrogel is pre-

sented in figure 4.4. A clear temperature increase is obtained over time for the three different

frequencies and two different cross-linkers concentration. The temperature increases between the

initial and last cycles read 2.5oC. There is clear dependency of the temperature increase to the

4.7. EXPERIMENTAL AND NUMERICAL RESULTS 117

0 5 10 15 20 25 3021.5

21.55

21.6

21.65

21.7

21.75

21.8

21.85

21.9

Time in [s]

Tem

pera

ture

in [°

C]

Temperature in HEMA−EGDMA

Test 1Test 2Test 3

Figure 4.3: Temperature [oC] vs. time [s], during the preloading.

50 100 150 200 250 30021

21.5

22

22.5

23

23.5

Time in [s]

Tem

pera

ture

in [°

C]

Observed temperature in HEMA−EGDMA

Observed 1.5 HzObserved 1 HzObserved 0.5 Hz

50 100 150 200 250 30021

21.5

22

22.5

23

23.5

24

24.5

Time in [s]

Tem

pera

ture

in [°

C]

Observed temperature in HEMA−EGDMA

Observed 1.5 HzObserved 1 HzObserved 0.5 Hz

Figure 4.4: Observed temperature in the sample of HEMA-EGDMA vs. time for φ = 6% (left)and φ = 8% (right), f = 0.5 [Hz], f = 1 [Hz] and f = 1.5 [Hz].

applied frequency. The higher the frequency is, the higher the temperature increases. These ex-

perimental temperature evolution were used to identify the parameters present in the analytical

1D model (see equation (4.18)). A good correlation is obtained between the experimental data and

the model as shown in figure 4.5.

Based on the these correlations, the obtained identified parameters of the model are reported

in Table 4.4.

Samples λ′[MPa.s] µ′[MPa.s] α[1/K]Sample 1 357.93 39.77 1.9e-4Sample 2 393.646 51.701 2.1e-4

Table 4.4: Optimized constants of the samples after equation (4.38)

Finally the parameters reported on table 4.4 were injected in the FEM model (see equation

(4.40)) and the computed temperature evolutions were then plotted in figure 4.6 It can be obtained

that the obtained curves closely match the experimental measurement of the hydrogel self-heating,


50 100 150 200 250 30021

21.5

22

22.5

23

23.5

Time in [s]

Tem

pera

ture

in [°

C]

Computed−observed temperature in HEMA−EGDMA

Observed 1.5 HzComputed 1.5 HzObserved 1 HzComputed 1 HzObserved 0.5 HzComputed 0.5 Hz

50 100 150 200 250 30021

21.5

22

22.5

23

23.5

24

24.5

Time in [s]

Tem

pera

ture

in [°

C]

Computed−observed temperature in HEMA−EGDMA

Observed 1.5 HzComputed 1.5 HzObserved 1 HzComputed 1 HzObserved 0.5 HzComputed 0.5 Hz

Figure 4.5: Correlation between computed (analytical solution) and observed temperature in thesample of HEMA-EGDMA vs. time. for φ = 6% (left) and φ = 8% (right), f = 0.5 [Hz], f = 1[Hz] and f = 1.5 [Hz].

50 100 150 200 250 30021

21.5

22

22.5

23

23.5

Time in [s]

Tem

pera

ture

in [°

C]

Computed temperature in HEMA−EGDMA

Computed 1.5 HzComputed 1 HzComputed 0.5 Hz

50 100 150 200 250 30021

21.5

22

22.5

23

23.5

24

24.5

Time in [s]

Tem

pera

ture

in [°

C]


Computed 1.5 HzComputed 1 HzComputed 0.5 Hz

Figure 4.6: Computed (numerical model) temperature in the sample of HEMA-EGDMA vs. timefor φ = 6% (left) and φ = 8% (right), f = 0.5 [Hz], f = 1 [Hz] and f = 1.5 [Hz].

not only the frequency dependence, but also the cross-linkers dependence could be caught by the

developed model.

4.7.1 Influence of the cross-link density on the self-heating

In order to have a closer look to the influence of cross-link density on the self-heating, we report

on the same graph the temperature evolution of the hydrogels for the two different cross-linker

density (6% and 8%). It can be observed on figure 4.7 that the decrease in the cross-linker density

caused a significant change in the heat production and consequently a more limited temperature

increase during cyclic loading. The effect of the cross-link density is implicitly taken into account

in the model through the dependency of the cross-link density in the model parameters (see table

4.4).

4.7. EXPERIMENTAL AND NUMERICAL RESULTS 119

0 100 200 300 400 500 600 70021

21.5

22

22.5

23

23.5

Time in [s]

Tem

pera

ture

in [°

C]

Temperature in HEMA−EGDMA

Cross−link density 6%Cross−link density 8%

Figure 4.7: Temperature (in [oC]) vs. time (in [s]) in the HEMA-EGDMA samples. The curvesshow the effect of the cross-link density φ on the temperature during test (preloading, cyclic loadingand relaxation). f = 1 [Hz] for the cyclic loading.

4.7.2 Dissipation in function of frequency and cross-link density

In this subsection, we present the experimental results for the dissipation in the hydrogel obtained

from the force-displacement hysteresis curves. We evaluate the effect of the temperature increase

on the dissipation during the different phase of the test (preloading, cyclic loading and relaxation).

We also illustrate the variation of the hydrogel dissipation in function of the cross-link density

and the frequency. Without surprise, it can be seen in figures 4.8 and 4.9 that the dissipation is

function of the cross-link density and the frequency of loading as for the temperature evolution.

−19.4 −19.2 −19 −18.8 −18.6 −18.4 −18.2−140

−120

−100

−80

−60

−40

−20

0

20

Total displacement [mm]

For

ce [N

]

Force−displacement during preloading, cyclic loading and relaxation

PreloadingFirst cycleIntermediate cycleLast cycleRelaxation

−19.4 −19.2 −19 −18.8 −18.6 −18.4 −18.2−160

−140

−120

−100

−80

−60

−40

−20

0

20


For

ce [N

]



Figure 4.8: Hysteresis cycle. The curves represent the response of the sample, force in function ofthe total displacement (during the test, preloading, cyclic loading 5 [mn] and relaxation). φ = 6%,f = 0.5 [Hz] (left) and f = 1 [Hz] (right).

More interestingly, we can also observe from this figure that the shape of the hysteresis curves

depends on the number of loading cycles. For the same sample under the same loading condition,

the shape of the hysteresis curves is completely different if we consider the first, the intermediate

or the last cycles. As there is a direct correspondence between the number of cycles and the


−19.6 −19.4 −19.2 −19 −18.8 −18.6 −18.4−400

−350

−300

−250

−200

−150

−100

−50

0

50


For

ce [N

]



−19.6 −19.4 −19.2 −19 −18.8 −18.6 −18.4−500

−400

−300

−200

−100

0

100


For

ce [N

]



Figure 4.9: Hysteresis cycle. The curves represent the response of the sample, force in function ofthe total displacement (during the test, preloading, cyclic loading 5 [mn] and relaxation). φ = 8%,for f = 0.5 [Hz] (left) and f = 1 [Hz] (right).

corresponding temperature in the sample (through the temperature evolution presented in figure

4.4 (for example), we can deduce that the dissipation is then also function of the temperature.

Indeed, a closer look to the Figures 4.8 and 4.9 highlights that the behavior of the hydrogel

presents a shift between elastic, viscoelastic and again elastic behaviors at two critical temperatures.

This unexpected (and to the best of our knowledge not reported before) behavior was observed for

all tested samples. The values of the critical temperatures are reported in tables 4.5 and 4.6.

Time in [s] T eb · · · T vb · · · T va · · · T ea(elastic) (viscoelastic) (elastic)

Temp. in [oC] 21.49 · · · 21.65 · · · 21.85 · · · 22.20for f = 0.5 [Hz]Temp. in [oC] 21.32 · · · 21.55 · · · 21.93 · · · 22.63for f = 1 [Hz]Temp. in [oC] 21.18 · · · 21.39 · · · 21.70 · · · 22.88for f = 1.5 [Hz]

Table 4.5: Critical temperatures in the sample of HEMA-EGDMA vs. time for φ = 6%, f = 0.5[Hz], f = 1 [Hz] and f = 1.5 [Hz].

Time in [s] T eb · · · T vb · · · T va · · · T ea(elastic) (viscoelastic) (elastic)

Temp. in [oC] 22.34 · · · 22.54 · · · 22.85 · · · 23.54for f = 0.5 [Hz]Temp. in [oC] 21.89 · · · 22.00 · · · 22.66 · · · 23.39for f = 1 [Hz]Temp. in [oC] 21.53 · · · 21.79 · · · 23.58 · · · 24.13for f = 1.5 [Hz]

Table 4.6: Critical temperatures in the sample of HEMA-EGDMA vs. time. for φ = 8%, f = 0.5[Hz], f = 1 [Hz] and f = 1.5 [Hz].

Remark 4.4. Between the temperatures T eb and T vb the hydrogel behavior is elastic, between T vb

4.8. DISCUSSION 121

and T va its the behavior is viscoelatsic and then, between T va and T ea its behavior becomes again

elatsic.

4.8 Discussion

In this chapter a combined analytical-numerical-experimental approach was developed to evaluate

the self-heating phenomenon in a specific hydrogel. The proposed methods are general enough to

be used to characterize other types of materials.

We demonstrate in this study that the developed model could adequately describe the self-

heating behavior of the hydrogel. The influence of two main parameters (cross-link density and

loading frequency) on the temperature evolution could also be taken into account in the model.

We have to mention that the ranges of the frequency in this work were limited to 0.1-2 Hz for the

numerical approaches and to 0.5-1.5 Hz for the experimental measurements. The cross-link density

of the hydrogel was limited to 6% and 8% and the percentage in water is prescribed to 40%. The

model prediction has then to be considered initially in these ranges but could be further extended

with new experimental data including wider ranges of parameters. In addition, the obtained results

are valid only for linearly thermo-viscoelastic materials. An extension to non-linear behavior is

presented at the end of this chapter.

As presented, the identification process followed an indirect path, using first an analytical so-

lution for a 1D problem in order to determine the model parameters. The numerical model with

the identified 1D parameters was then used to verify that the simulated temperature evolution

matches the experimental data. While this approach could not formally be considered as a valida-

tion process, it allows us to have more confidence on the developed finite element model. A formal

validation would be obtained if the model could predict the temperature evolution of a hydrogel

presenting a different cross-link density and being subjected to new loading frequencies. For this

however, an explicit relationship between the model parameters and the cross-link density should

be established.

From the experimental data, it has been observed that the hysteresis characterizing the dis-

sipation through the loop force-displacement during the harmonic loading changes its shape in

function of the cycle numbers. Two phenomena could be taken into account to explain this ob-

servation. First, we can consider that during the loading, the internal structure of the hydrogel

changes adapting its structure to the loading. This kind of behavior has been observed for the

initial loading cycles and is known under the name of the Mullin effect. However, this effect is

usually happening only during the initial load cycles and then vanishes. While we could not exclude

this kind of phenomenon, it seems anyway unlikely that the structure of the hydrogel could con-


stantly adapt to its external loading over a high number of cycles. The second phenomenon, which

could explain the change of the hysteresis curve over time, is the change in temperature of the

self-heating hydrogel. As mentioned in the result section, as the number of cycles increases so do

the hydrogel temperature. It can then be considered that the increase of temperature changes the

mechanical parameters of the hydrogel. For example, in the situation where the elastic parameters

would increase with the temperature, as the same displacement was experimentally imposed on

the hydrogel, an increase mechanical energy will then be transmitted to the hydrogel. The shape

of the hysteresis curve could then be different because of this situation.

More surprisingly is the observation of the transition between elastic, viscoelastic and again

elastic behaviors at increasing temperatures. This behavior seems not to have been previously

reported in the literature. Some recent works published more investigation of the self-heating phe-

nomenon viscoelastic materials subjected to cyclic loading, we observe that there is no ivestigation

on the behavior change during the cyclic loading and then the critical temperatures e.g [176], [177],

[178], [179].

We need to verify if this behavior is intrinsic to the used hydrogel or if it could be more

general. Nevertheless, for the present study, we may also explain this behavior by the change of

the mechanical properties with respect to the temperature. At some critical temperatures, due to

the coupling between thermal and mechanical behaviors, change in elastic or viscous parameters

could significantly change the general behavior of the material as what is observed in thermal failure

for example. However, more theoretical and experimental investigations are necessary before a clear

explanation of this phenomenon can be proposed.

In general, the developed model could be useful in the phase of design of the hydrogel for a

particular application. For example, with the idea of using this kind of dissipative hydrogel for

the controlled delivery of a drug through the temperature increase [7], a link has to be established

between the number of cycles and the targeted temperature increase. The developed model would

then be useful in this situation to determine the cross-link density needed and/or the mechanical

loading regime that the hydrogel should be exposed to. In another application, it has been shown

that the toughness of the hydrogel could be increased by increasing its dissipative properties [11].

Again in this situation, the developed model could be used to design the most dissipative hydrogel

under known mechanical conditions.

4.9. NONLINEAR EXTENSION 123

4.9 Nonlinear extension

4.9.1 Thermodynamic potentials

Constitutive laws are based on Helmholtz’ free energy ψ and dissipation potential χ depending on

Cauchy-Green strain C, its rate-time and temperature gradient∇θ. As an example we may consider

the thermo-viscoelastic behavior in the sample in case of large-applied perturbations near the

thermodynamic equilibrium. We propose the following thermodynamic potentials for the hydrogel

HEMA-EGDMA:

ψ(Ii(i=1...3)(C), θ) =λ

2exp [µ(I1 − 3)]− λµ

4(I2 − 3)− (λ+ 2µ)(I3 − 1)

−(3λ+ 2µ)α(θ − θ0)(I3 − 1)− c

2θ0(θ − θ0)

2

χ(Jj(j=1...3)(C), Jk(k=4...7)(C,C),∇θ; C, θ) =η

2J2(I1 − 3) +

1

2κ∇θ · ∇θ

≡ η

2J2(I1 − 3) +

1

2κ‖∇θ‖2

(4.58)

The second Piola-Kirchhoff stress tensor has two parts the elastic Se and viscous parts Sv, then

S = Se + Sv such that,Se(Ii(i=1...3)(C), θ) = 2ρ

[λµ

2exp[µ(I1 − 3)]I− λµ

4(I1C−C)− [(3λ+ 2µ)α(θ − θ0) + (λ+ 2µ)] I3C

−1

]Sv(χ(Jj(j=1...3)(C), Jk(k=4...7)(C,C),∇θ; C, θ) = η(I1 − 3)C

(4.59)

Definition 4.4 (Cost function).

α, κ, η = infα∈R+

infκ∈R+

infη∈R+

1

2

∥∥∥∥∥(

2

JFSFT

)comp/33

(α, η)−(F (t)

SB

)obse∥∥∥∥∥2

‖ δθcomp (α, η, κ)−(δθobse + 273.15

)‖2

(4.60)

In which

S = 2ρ

[λµ

2exp[µ(I1 − 3)]I− λµ

4(I1C−C)− (3λ+ 2µ)κ(θ − θ0)I3C

−1

]ρ(λ+ 2µ)I3C

−1 + η(I1 − 3)C (4.61)

4.9.2 Numerical results

Another numerical experiment is performed to study the self-heating phenomena in the hydrogel

HEMA-EGDMA in order to separate the reversible and the irreversible heat production. We iden-

tify the given model with the experimental measurement. This section incorporates numerical

results and experimental measurements, we present successively the results obtained by using nu-

merical approach and its correlation with the experimental measurements. The numerical method

is obtained by using the finite element methods.


Samples η[MPa.s] α[1/K]Sample 1 219.12 0.8e-4Sample 2 316.4 1.02e-4

Table 4.7: Optimized physical constants of the sample for local self heating after equation (4.60).The constants λ and µ are not optimized but given by the table 4.1.

Samples λ[MPa] µ[MPa] η[MPa.s] α[1/K]Sample 1 2.76 0.77 212.0 2.8e-4Sample 2 1.36 1.3 309.179 3.6e-4

Table 4.8: Optimized physical constants of the sample for local self heating after the (4.60). Theconstants λ and µ are optimized.

We observe exponential increasing of temperature during cycling loading, and decreasing when

unloading of relaxation test. As the frequency and the cross-link density in the material increases,

the internal heat production in the material increases. The curves report the temperature in

the sample. More accurate experimental measurements are required for the analysis of nonlinear

50 100 150 200 250 30021

21.5

22

22.5

23

23.5

Time in [s]

Tem

pera

ture

in [°

C]


1.5 Hz1 Hz0.5 Hz

50 100 150 200 250 30021

21.5

22

22.5

23

23.5

24

24.5

Time in [s]

Tem

pera

ture

in [°

C]


1.5 Hz1 Hz0.5 Hz

Figure 4.10: Temperature in the sample of HEMA-EGDMA vs. time for φ = 6% (left) and φ = 8%(right), f = 0.5 [Hz], f = 1 [Hz] and f = 1.5 [Hz].

thermodynamics of hydrogel.

Acknowledgment

Financial supports by the International Doctoral College (CDI) of the Brittany European Univer-

sity (UEB), the Brittany Region Council (France) and the Laboratory of Biomechanical Orthope-

dics (Lausanne, Switzerland) are greatly appreciated.

Chapter 5

Conclusion générale

Les phénomènes de self-heating dans les polymères, dans les tissus biologiques et dans les hydrogels

peuvent engendrer une augmentation locale de température dans ces derniers. Cette augmenta-

tion de température peut influencer le comportement mécanique des matériaux, l’activité d’un

médicament par exemple.

Le principal objectif de la thèse est, dans un premier temps de proposer un modèle numérique

pour la qualification et la quantification des phénomènes de self-heating. Ensuite, l’idée est de pro-

poser une méthode d’identification et d’interprétation des paramètres influençant ces phénomènes

dans le but d’augmenter ou diminuer ces effets thermiques. Enfin, il est nécessaire de mesure

la production de chaleur en utilisant la méthode micro-calorimétrique à déformation en prenant

comme échantillon l’hydrogel de type HEMA-EGDMA.

Comme le problème inclut des conditions aux limites dynamiques, la première partie de la

contribution de la thèse se focalise sur le développement d’une méthode de résolution numérique

d’un modèle mathématique utilisant la condition de "switch" en temps Dirichlet/Neumann (StDN)

pour des conditions aux limites dynamiques. Dans la première partie, nous avons traitée une

méthode de résolution par la méthode des éléments finis d’un système formé par deux équations

convection-diffusion couplées avec des conditions aux limites dynamiques nécessitant le switch

StDN.

Ce modèle gouverne le phénomène de contamination croisée dans l’industrie micro-électronique

qui permet étudier la sensibilité des matériaux polymères à la contamination volatile. Il a pour

but de trouver le matériau optimal qui répond aux exigences sur la condition de contamination et

sur les autres critères comme conditions mécaniques et de résistance à la température. Nous avons

mis en évidence les différents profils de contamination et sensibilité à la contamination de quelques

matériaux testés. Il est maintenant possible de combiner ces informations avec les autres critères.

Ces résultats sont en corrélation avec les données expérimentales.

125

126 CHAPTER 5. CONCLUSION GÉNÉRALE

Dans la deuxième partie, nous avons développé une méthode de résolution par la méthode des

éléments finis pour le système formé par deux équations convection-diffusion couplés avec l’effet de

température donc l’équation de la chaleur. Ce modèle gouverne le phénomène de contamination

croisée avec l’effet de la température dans l’industrie micro-électronique. Pour l’application indus-

trielle du modèle, chaque étape utilise la condition de "switch"nommé StDN. Nous avons utilisé le

logiciel Comsol Multiphysics pour implémenter les équations. Ensuite, nous avons mis en évidence

l’effet de la température sur la décontamination (avantages et inconvénients).

Nous avons montré d’une part que, l’augmentation de la température pendant la décontami-

nation favorise la diffusion des polluants dans le volume car le coefficient de diffusion augmente.

D’autre part, pour l’accumulation surfacique des contaminants dans l’interface, l’augmentation

de température durant le nettoyage permet d’éliminer un maximum de concentration superficielle.

Enfin, la corrélation entre les données de caractérisation expérimentale et le modèle mathématiques

avec les conditions de "switch" est vérifiée pour chaque étape du processus industrielle.

Ensuite, le travail présenté dans ce manuscrit porte sur la modélisation thermomécanique des

phénomènes de self-heating et sur les lois de comportement des matériaux de type, hydrogels sous

sollicitations cycliques afin de quantifier la production interne de chaleur. Nous avons contribué à

la résolution numérique du système d’équation thermomécanique couplé, système d’équations aux

dérivées partielles paraboliques-hyperboliques. Nous avons utilisé un microcalorimètre à déforma-

tion pour la quantification de cette production de chaleur.

Au regard de leurs applications potentielles, l’objectif de ce travail s’est naturellement orienté

vers la qualification du phénomène self-heating et caractérisation de la production de chaleur dans

ces hydrogels, en particulier le type HEMA-EGDMA. L’exposition de l’hydrogel à une contrainte

mécanique cyclique pendant une période suffisante conduit à un accroissement local de la tempéra-

ture dans l’hydrogel.

Afin d’analyser l’influence des phénomènes mis en jeu lors de la caractérisation de l’auto-

échauffement dans l’hydrogel sous sollicitation cyclique, nous avons développé différents modèles

théoriques thermomécaniques. Dans le cadre de ce travail, nous avons utilisé la théorie des matéri-

aux standards généralisés pour un milieu continu.

Pour la partie numérique, nous avons utilisé la méthode d’approximation numérique basée sur

la méthode des éléments finis pour résoudre le problème couplé. Une approche semi-analytique

monodimensionnelle a été utilisée durant ce travail. Avec cette approche nous avons admis une

linéarité en température. Nous avons utilisé deux différents potentiels thermodynamiques pour

identifier le comportement de l’échantillon avec les données expérimentales. Pour chaque loi de

comportement, une solution analytique a été proposée pour le cas monodimensionnel. Nous avons

127

remarqué que l’augmentation locale de la température est de forme exponentielle: une accumulation

de température irréversible en exponentielle avec une production réversible due à la sollicitation

mécanique dynamique sinusoïdale.

Partant de l’observation que le phénomène de self-heating est dépendant de la fréquence de

sollicitation et de la densité de réticulation, nous nous sommes, dans un premier temps, intéressé à

l’effet de fréquence sur la production de chaleur pour une densité de réticulation fixe. Nous avons

montré que plus la fréquence augmente plus la quantité de chaleur produite est importante. Ensuite,

nous nous sommes, dans un second temps, intéressés à l’effet de la variation de la production de

chaleur en fonction de la densité de réticulation pour une fréquence de sollicitation fixe. Nous

avons observé que la quantité de chaleur produite est fortement liée à la densité de réticulation.

Cette observation nous amené à conclure que, dans le cas de l’hydrogel HEMA-EGDMA, plus la

densité de réticulation EGDMA est importante plus la production de température est importante.

Ces dépendances de la fréquence et de la densité de réticulation sur la production de chaleur sont

présentées dans les modèles numériques.

Enfin, on a pu constater l’existence de deux domaines élastiques de l’échantillon d’hydrogel

HEMA-EGDMA quelle que soit la densité de réticulation et la fréquence de sollicitation. Un

comportement viscoélastique se trouve entre ces deux domaines élastiques. Le passage de com-

portement entre les domaines élastiques et viscoélastiques est lié à la variation de la température.

D’une part, avec une température initiale au début de l’essai, l’échantillon se comporte comme un

matériau élastique jusqu’à une certaine température. D’autre part, de cette valeur de température,

l’échantillon se comporte comme un matériau viscoélastique jusqu’à un autre niveau de tempéra-

ture. Par ailleurs, une fois la température critique atteinte, l’échantillon se comporte de nouveau

comme un matériau élastique. Ce comportement est observé avec tous les échantillons utilisés.

Ces deux températures critiques et la durée de chaque domaine de changement de comportement

dépendent de la densité de réticulation et de la fréquence de sollicitation.

Les originalités de la modélisation et de cette observation sont notamment, premièrement

de permettre d’isoler le comportement de l’hydrogel en fonction de la gamme de production de

chaleur voulue. Pour une application ciblée dans le domaine médicale, le paramètre de "design" de

l’hydrogel sera la densité de réticulation pour une température donnée. Ensuite, on peut atteindre

l’objectif d’optimiser ou de supprimer cet effet de production de chaleur en agissant mutuellement

sur le stimulateur mécanique de l’hydrogel et sa composition.

On observe que les modèles proposés sont en corrélation avec les mesures expérimentales de

production de chaleur pour le cas de l’hydrogel du type HEMA-EGDMA. Plusieurs simulations

ont été effectuées afin d’identifier les modèles de comportement avec les données expérimentales.

128 CHAPTER 5. CONCLUSION GÉNÉRALE

La corrélation qualitative entre les résultats numériques et données expérimentales nous a amené

à conclure que le modèle pourrait être utilisé comme un outil prédictif de la production de chaleur

dans ces matériaux et leur éventuel changement de propriété en fonction de la température.

Les applications possibles de ce travail concernent d’abord les applications biomédicales des

phénomènes de self-heating au travers des outils numériques prédictifs. En particulier, l’application

concerne le relargage d’un médicament à partir d’un hydrogel sollicité mécaniquement. Ces résul-

tats obtenus peuvent aussi s’appliquer aux comportements des matériaux polymères en général.

Ces résultats sur le self-heating de l’hydrogel incitent à bien modéliser la genèse de cette produc-

tion de chaleur parmi les différents phénomènes mis en jeu. Pour améliorer les modèles prédictifs,

quelques perspectives sont avérées nécessaires pour bien maîtriser ces phénomènes.

Perspectives

Les résultats (modèle, méthodologie, loi de comportement, etc...) obtenus dans cette thèse on été

confrontés à des données expérimentales ayant pour objectifs des applications dans le domaine

biomédical. Les résultats sur le self-heating de l’hydrogel suggèrent la nécessité de bien modéliser

la genèse de production de chaleur parmi les différents phénomènes mis en jeu. Pour améliorer les

modèles développés, quelques perspectives semblent se profiler.

Comme l’hydrogel est un matériau fibreux et réticulé avec une certaine densité de cross-linking

(réseau maillé), il apparait primordial de tenir compte de ces structures et leurs propriétés intrin-

sèques (porosité, densité de réticulation, etc...). Une formulation non locale de milieu continu à

gradient est une piste sérieuse afin de prendre en compte les effets non locaux ainsi que la structure

physique de l’hydrogel.

On pourrait ainsi utiliser la thermomécanique des milieux faiblement continus pour une meilleure

compréhension de ce phénomène de self-heating.

Quelques aspects doivent faire l’objet d’un développement plus poussé pour accroître une bonne

fiabilité sur la compréhension du phénomène de self-heating dans les hydrogels en particulier et sur

la méthodologie aussi. Nous proposons les pistes suivantes:

Propositions

Chaque extension du modèle comportera une partie formulation mathématique, une méthode

numérique, validation avec les mesures et les applications. Avec la courbure de Ricci comme

variable supplémentaire pour tenir compte et pour modéliser les cross-link, on peut proposer alors

les potentiels thermodynamiques suivants:

• Cas 1: L’énergie libre d’Helmholtz sera fonction d’une métrique de la température et de la

courbure. Le potentiel de dissipation sera fonction de la dérivée temporelle de la métrique du

gradient de température et ensuite implicitement fonction de la courbure, de la température

et de la métrique.

129

130 PERSPECTIVES

• Case 2: L’énergie libre d’Helmholtz sera fonction d’une métrique de la température et la

courbure. Le potentiel de dissipation sera fonction de la dérivée temporelle de la métrique,

du gradient de température, de la dérivée temporelle de la courbure et ensuite implicitement

fonction de la courbure, de la température et de la métrique. L’introduction de la courbure

a un intérêt car elle pourrait être corrélé à la densité de réticulation.

Méthode numérique

Concernant les méthodes numériques, quelques aspects doivent encore faire l’objet d’efforts de

développement afin d’accroître la pertinence et la fiabilité de la méthode. Il nous faut une nouvelle

technique pour traiter le problème de couplage avec le nouveau formalisme (milieu à gradient

d’ordre supérieur). Il faudrait une astuce pour pouvoir implémenter la courbure de Ricci et sa

dérivation temporelle dans les équations de la thermomécanique comme celle traitée dans cette

thèse. La forme générale du système d’équations thermomécanique final dépend des potentiels

thermodynamiques choisis.

Méthodes de caractérisations et études de corrélation

Dans le but de continuer la caractérisation du phénomène de self-heating on pourrait proposer

quelques méthodes de caractérisation qui s’appuiera sur les nouveaux systèmes d’essais mécaniques

statiques et dynamiques. Ces méthodes vont nous permettre, dans un premier temps, de réaliser

des essais de fatigue mecano-thermique statique et dynamique dans les hydrogels. Ensuite, elles

vont nous permettre, dans un deuxième temps, de caractériser la production d’entropie réversible

et irréversible dans les échantillons. Cette identification pourrait être couplée avec des simulations

numériques.

Ces essais seront combinés à d’autres techniques de caractérisation microscopique optique, élec-

tronique et diffraction X pour relier les performances mécaniques aux caractéristiques moléculaires

et microscopiques.

Appendix AContinuum thermomechanics

A-1 Strain and stress

Let X ∈ B be a material point in the reference configuration, x ∈ S denotes its position in the

current configuration after transformation ϕ.

ϕ :

B × [0, T ] −→ S

(X, t) −→ ϕ(X, t)(1)

The corresponding tangent and dual spaces are represented by TXB, T ∗XB and TxS, T ∗xS respec-

tively. The time is noted by t ∈ [0,T ] with t ∈ T and the time range is T ⊂ R+. The finite

transformation ϕ is an homomeorphism, ϕ ∈ C1 et ϕ−1 ∈ C1. The classical theory of continuous

medium requires that during the transformation ϕ two neighboring points X and X + dX in the

reference configuration B still remain neighbors in the current configuration S after transformation.

Assuming that ϕ is sufficiently regular, we can write:

(B) (Bf )(Bf )

(Bp)

(Γi)(Γi) (Γo)

(B) (Bf )(Bf )

(Bp)

(Γi)(Γi) (Γo)

(∂B − Γh ∪ Γb)

(Γb)

(Γh)(Γp)

(Γt)(Γi)(Γo)

(B) (S)

X x

ϕ

ϕ−1

(B) (S)X x

ϕ

ϕ−1

ϕ∗ ϕ∗ϕ−1Figure 1: Transformation from the reference configuration B to the current configuration S. Themapping ϕ is a bijective, holonomic and diffeomorphic application.

As part of the classical mechanics for continuous media, each material particle occupies a

material point in an Euclidean spaceRd. The transition between these two states (B,S) is obtained

with a transformation map ϕ, which describes the mouvement of the media B and can be holonomic

(see. fig. 1) or non-holonomic (see . fig. 2). These transformations are defined as follows:

ϕ(X + dX, t) = ϕ(X, t) +∇ϕ(X, t)[dX] +O(‖dX‖2) (2)

131

132 APPENDIX A CONTINUUM THERMOMECHANICS

(B) (Bf )(Bf )

(Bp)

(Γi)(Γi) (Γo)

(B) (Bf )(Bf )

(Bp)

(Γi)(Γi) (Γo)

(∂B − Γh ∪ Γb)

(Γb)

(Γh)(Γp)

(Γt)(Γi)(Γo)

(B) (S)X x

ϕ

ϕ−1

(B) (S)X x

ϕ

ϕ−1

ϕ∗ ϕ∗ ϕ−1

(S∗)

s

Figure 2: Kinematics of continuous media with and without microstructural effect

We define the deformation gradient as F := ∇ϕ(X, t) : TXB −→ TxS with detF > 0 in which, ∇

denotes the gradient in space with respect to the reference configuration. The material vector dX

becomes a vector dx by the intermediate of the deformation gradient. The displacement vector

and the velocity fields are defined respectively as u(X, t) = x−X and v = ∂u∂t (X, t).

For large deformation, we measure the deformation of B, by means of the right Cauchy-Green

strain tensor C : TXB −→ T ∗XB

C(X, t) = FTF (3)

C(X, t) = I +∇u +∇Tu +∇Tu∇u (4)

The Piola-Boussinesq stress tensor P : T ∗XB −→ TxS or the nominal stress (or the first Piola-

Kirchhoff stress tensor) is defined as follows:

P = JσF−T = FS (5)

where σ : T ∗xS −→ TxS is the Cauchy stress tensor. J = det(F), which describes the local change

of the volume from the reference configuration. The Piola-Kirchhoff stress tensor S : T ∗XB −→ TxB

(or second Piola-Kirchhoff stress tensor) is a symmetric tensor (S = ST) defined as:

S = JF−1σF−T = F−1P (6)

A-2 Conservation laws

Assuming the following transformation:

ϕ :

B × [0, T ] −→ S

(X, t) −→ ϕ(X, t)(7)

we write the conservation laws in the Eulerian and the Lagrangian configurations.

A-2. CONSERVATION LAWS 133

Conservation of mass

In the reference configuration B, a continuum body has the mass density ρ0, while in the current

configuration, S the mass density is ρ. The conservation of mass in the Eulerian and Langangian

takes the form of respectively

d

dt

(∫Sρdv

)= 0; ∂tρ+ div(ρv) = 0 (8)

d

dt

(∫BρdV

)= 0; ρ0 = ρdetF (9)

Balance of linear and angular of momentum

The variation of the linear momentum is equal to the sum of the external forces. For the Eulerian

formalism, we denote by the b(x, t) the body force on V S , and the surfacic force on dSS given by

the Cauchy stress tensor σ(x, t)nS(x, t), we have,

d

dt

(∫Sρ(x, t)v(x, t)

)=

∫∂Sσ(x, t)nS(x, t)dSS +

∫Sρ(x, t)b(x, t)dVS (10)

By integrating by parts and using the mass conservation, we have

ρvt + ρ(v.∇)v = divσ + ρb (11)

For the Lagrangian formalism, we have to write the conservation law in the reference configuration.

ρ0(X)∂2u

∂t2(X, t) = Div(P(X, t)) + ρ0b(X, t) (12)

Evaluation of the angular momentum shows that the second Piola-Kirchhoff stress tensor S defined

by S = F−1P is symmetric. Therefore, PFT = FPT

Conservation of energy

The first of thermodynamics postulates the conservation of energy.

ρde

dt= σ : D− divq + ρr; D :=

1

2(∇v +∇Tv) (13)

ρ0∂e

∂t= S : E−DivQ + ρ0r (14)

In which e is the internal energy, q is the Eulerian heat flux, r is the heat source, Q is the Eulerian

heat flux.


A-3 Constitutive laws

Constitutive laws should be admissible (compatible) with the thermodynamics laws.

Second principle of thermodynamics

Combining the equation (14), with the following equation

ρ0∂e

∂t=∂ψ

∂t+∂θ

∂ts+ θ

∂s

∂t(15)

The entropy inequality or Clausius-Duhem inequality can be expressed as

S : E− ρ0∂ψ

∂t− ρ0

∂θ

∂ts− Q

θ· ∇θ ≥ 0 (16)

Constitutive law of soft tissues

To define the constitutive law of B during an holonomic transformation ϕ, we use theory of gen-

eralized standard materials. This theory are based on the existence and the definition of two

thermodynamic potentials: the Helmholtz free energy ψ which is derived the laws status reflecting

properties of equilibrium states, and potential dissipation χ which is derived the laws evolution

and specify the dissipative nature of mechanisms of evolution.

States variables

The thermodynamic behavior of the system is described by introducing the observable variable

and the state variables (internal variable). The observables are the total strain (here represented

by the strain tensor Green − Lagrange via displacement) and the thermodynamic temperature

field θ. The internal variables are used to describe phenomena not observable in the B (dammage,

dislocation, etc ...). We denote by ξi the n internal variables in B.

Generalized standard material

We define the potentials based on continuum classical theory and the existence of these thermo-

dynamic potential. We will use in space varying deformation and Helmholtz free energy ψ and

the dissipation potential χ as potential status. This theory will make this energy balance, and

take into account the dissipative effects and coupling mechanisms. The potentials ψ and χ are in

function of history of deformation, temperature and n internal variables. These internal variables

A-3. CONSTITUTIVE LAWS 135

can be scalar or tensor. Then we have: ψ=ψ(C, θ, ξi) and χ=χ(C,∇θ, ξi; C, θ, ξi). ψ = ψ(C, θ, ξi) ∀ C ∈Md

χ = χ(C,∇θ,∇ξi; C, θ, ξi) ∀ C, C ∈Md(17)

Where C and C denote respectively the right Cauchy-Green strain tensor and its rate time, θ the

temperature, ξi the internal variable, ∇θ and ∇ξi denotes the gradient of the temperature and the

internal variable. Md denotes a set of three order square matrix in Rd with positive determinant

(det(Md) > 0). For an arbirtrary set of tensor R under full orthogonal group O, we have:

ψ(RCRT, · · · ) = ψ(C, θ, ξi) ∀ C ∈Md, ∀ R,RT ∈ g ⊂ O (18)

Such that the Green-Lagrange strain tensor is given by E = 1/2(C − I). The continuous media

B ∈ Rd (in hydrogel HEMA-EGDMA) undergoes in large deformation. In fact, we use the right

Cauchy-Green strain tensor. In this work, the thermodynamic potentials are function of the strain

tensor for the elastic and viscous part.(S− ρ∂ψ

∂E

): E− Q

θ· ∇θ ≥ 0 (19)

hence,

S− ρ∂ψ∂E

=∂ψ

∂E; −Q

θ=

∂χ

∂∇θ(20)

Thermodynamic admissibility

The dissipation potential χ(C,∇θ,∇ξi; C, θ, ξi) must be convex in C and ∇θ, positive and null for

C = 0 and ∇θ = 0. The free energy satisfies the usual convexity of elasticity e.g [157], [159], [160].

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VU: VU:

Le Directeur de Thèse Le Résponsable de l’Ecole Doctorale(Nom et Prénom)

VU pour autorisation de soutenance

Rennes, le

Le Président de l’Université de Rennes 1

Guy CATHELINEAU

Vu après soutenance pour autorisation de publication:

Le Président de Jury,(Nom et Prénom)

nirina j.t. santatriniaina

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