Image analysis of cement paste: relation to diffusion transport

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transport

Vincent Tariel

To cite this version:

Vincent Tariel. Image analysis of cement paste: relation to diffusion transport. Analyse de données,

Statistiques et Probabilités [physics.data-an]. Ecole Polytechnique X, 2009. Français. �tel-00516939�

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Image analysis of cement paste:

relation to diffusion transport

TH`

ESE

présentée et soutenue publiquement le 19 février 2009

pour l’obtention du

Doctorat de l’´

ecole Polytechnique

(sp´

ecialit´

e physique)

par

Vincent Tariel

Composition du jury

Rapporteurs :

Mr. Dominique BERNARD

Directeur de recherche-CNRS/Universit´e de Bordeaux

Mr. Mark KNACKSTEDT

ARC QEII Fellow-Australian National University

Examinateurs :

Mr. Michel BORNERT

Professeur-´

Ecole Polytechnique

Mr. Dominique JEULIN

Professeur-´

Ecole des Mines

Ms. Karen SCRIVENER

Professeur-´

Ecole Polytechnique F´ed´erale de Lausanne

Codirecteur de th`

ese :

Mr Denis DAMIDOT

Professeur-´

Ecole des Mines de Douai

Directeur de th`

ese :

Mr Pierre LEVITZ

Directeur de recherche-CNRS/´

Ecole Polytechnique

Invit´

es :

Mr Emmanuel GALLUCCI

Docteur-´

Ecole Polytechnique F´ed´erale de Lausanne

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Je tiens tout d'abord à remer ier les membres du jury de ma thèse. Je remer ie Dominique Bernard

et DominiqueJeulin pourl'intérêtqu'ils ont apporté àmon manus rit. Leurs remarques onstru tives

ontpermisd'améliorer grandementsaqualité. Jeremer ie Mar kKna kstedt, quiaposédes questions

pertinentesportantsurlapartiephysique demontravailet elamalgré lafatigue desonvoyage. Ilest

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manus ritpourentirerl'information iment. Jeremer ieMi helBornertpoursa uriositéintelle tuelle.

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etpourm'avoirapportéla ulturedelarigueurs ientique. Enn,jeremer ieDenisDamidot, mon

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etleursoutientoutaulongdemathèse. Jetiensàremer ierPierreCalkaave quij'ai ollaboréausein

duprojetANRmipomodim.

Pourremer ierlesautres,voi iunepetitehistoiretypiquementgauloise! Noussommesen2000après

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d'irrédu tibles her heurs résiste en ore et toujours àl'envahisseur. Et la vie n'est pas fa ile pour les

garnisonsde te hno ratesromainsdes amps retran hésautour de e a um, evaluatum, ontrolumet

pre aratum...

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quelquegauloishabitant e village:

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Chimistrix,le druide, ueille desplantes et prépare despotions magique,telle Sol-Gelremplide nanoparti ules,

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étaitLevitz. Commeà ha undesesapprentis,ildemandait,enplusdutravail ourant,d'approfondirun

sujet. Lemienaétéde omprendrelelienentrelamorphologiedel'objetetsespropriétésphysiques. Ce

sujetétaitaussilargequeletourdetailled'Obélix. For ément,ilyaeuun ontourdusujetm'intéressant.

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plupart des autre apprentis venait des élèbres é oles de Lutè e omme Polyte hni um, Normalium...

Armé de la onan e et de la passion, je me jetai dans lamêlée du travail, par Toutatix ! Parfois, il

y avait des impasses. Par exemple, pendant une semaine, j'ai passé mon temps àessayerde marteler

le métal en vain. Il m'expliqua ensuite qu'il était très dur de travailler le métal si on ne le hauait

pas: le fameux re uit ! Par Bélénox, il aurait pu me ledire tout de suite ! Oui, mais l'apprentissage,

'estpeut-êtreça. Etanttouslesdeux deersgaulois, nosdis ussionsétaientdetantentantmus lées.

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me suisfait de nombreux amisapprentis: Jia,Laurent,Quentin, Matthieu, Julien... Ave esa olytes,

j'aieula han ed'alleré outerun/auMITauxAmériques.

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helvètes. Al'EPFL,Gallu ixetsesapprentism'onttoutexpliquétouuuuutdouuuuu ement.

Heureuse-ment ar jene omprenaispasvite. Ce queje n'ai toujourspas ompris 'estpourquoiil m'appelait le

geek. Ilsontfous eshelvètes!

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expérien esdegrainsdepollensàlasurfa edel'eaud'unré ipient. Il omptaitletempsqueprendune

parti uledePollenpourtou herlebordduré ipientpourlapremièrefois. A ha unede esexpérien es,

il riait: "aleaja taest !". Ilssontfous esrusses!

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diérentsangles. Nousétionsenphasesurbien biendes hamps.

Après des heures à marteler le métal, un gentleman venant de l'île de Bretagne nommé Lassaillix me

préparait un breuvage appelé "tea for two" qui me donnait plus de for e que la potion magique de

Panoramix. Mais bon, quand j'étais vraiment tropfatigué, il y avait juste les ervoises bues ave mes

an iensamisd'é olepourmedétendre!

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qui aimporté de sonlointain pays la outume de jouer au moins une fois au football par semaine, la

ommunautémaghrébine,Sabrina,Amel,Linda,kamila,quim'ontouvertlesyeuxsur ertainspréjugés,

despersonnagesprovenantdusoleilLevant,Vu etXiaxin,desgensquiparlentlaplupartdutempsave

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Clotilde qui ompte plus qu'une lopinette, Carine qui est plustêtue quemoi, Ra, l'homme ritique,

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Abstra t vii

Introdu tion ix

Glossary xi

Chapter 1

Bibliography overview: Imageanalysis versus diusivetransport in ementpaste

1.1 Cementpaste . . . 1

1.2 Diusion ina onneddomain . . . 3

1.3 Linkbetweenthegeometry of ementpasteandtransportproperties . . . 8

1.3.1 Estimatingtransportpropertiesofmortarsusingimageanalysisof SEMimages. . 8

1.3.2 X-ray mi rotomographi studies of pore stru ture and permeability in Portland ement on rete . . . 10

1.4 Con lusionanddis ussion . . . 12

Chapter 2 Imagingmethods: SEM and X-ray tomography 2.1 Introdu tion. . . 13

2.1.1 Why3Dimaging? . . . 13

2.1.2 Choi eofanimagingte hnique . . . 15

2.2 S anningEle tronMi ros opy(SEM) . . . 16

2.2.1 Prin iple . . . 16

2.2.2 Stateofart . . . 17

2.2.3 Samplepreparation . . . 18

2.2.4 First ommentsonimage analysis . . . 19

2.3 SRXTM . . . 19

2.3.1 Prin iple . . . 22

2.3.2 Stateoftheartfor ementitiousmaterials . . . 26

2.3.3 Preparation . . . 30

2.3.4 First ommentsonimage analysis . . . 32

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Chapter 3

Seeded regiongrowing by pixelsaggregation/dissolution

3.1 Introdu tion. . . 33

3.1.1 Whi heldofImagePro essing? . . . 33

3.1.2 Why on eptualising? . . . 34

3.2 ExampleofSRGPAD algorithm: thewatershedtransformation . . . 36

3.2.1 Presentationofthewatershedtransformation . . . 36

3.2.2 Sele tionofapair . . . 36

3.2.3 Algorithmi implementationofthewatershedtransformation . . . 37

3.2.4 Implementation ofthewatershedtransformationusing generi lasses . . . 38

3.3 TheframeworkofSRGPAD . . . 44

3.3.1 ZI . . . 44 3.3.2 A tualisation . . . 45 3.3.3 Organisation . . . 47 3.3.4 Comparison . . . 50 3.4 SomeSRGPAD algorithms . . . 51 3.4.1 Onequeue. . . 51 3.4.2 nqueues. . . 59

3.5 Howtomanagetheregion ollisions . . . 70

3.5.1 Classi algrowingpro esses . . . 70

3.5.2 Partitionindependentoftheseededregioninitialisationorder . . . 76

3.6 Con lusion . . . 76

Chapter 4 Segmentation 4.1 Materialsandmethods . . . 81

4.1.1 Image hara teristi s . . . 81

4.1.2 Materials . . . 82

4.1.3 Computationalrequirements . . . 83

4.2 Thresholdsegmentationusing tintinformation . . . 83

4.2.1 Threshold . . . 83

4.2.2 Morphologi alltering . . . 85

4.2.3 Limitationofthethresholdsegmentation . . . 88

4.3 Watershedtransformationusingboundaryinformation . . . 89

4.3.1 Seeds- ontrolledwatershed . . . 89

4.3.2 Seedinside aphase. . . 90

4.3.3 One-stepmethod . . . 91

4.3.4 Step-by-stepmethod . . . 93

4.3.5 Appli ation . . . 93

4.3.6 Choi eofthegradientoperator . . . 97

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4.4 Artefa ts orre tion . . . 107 4.4.1 Grainsplitting . . . 107 4.4.2 Holelling . . . 109 Chapter 5 Geometri al hara terization 5.1 Introdu tion. . . 112

5.1.1 Thegeometri alorganisationis omplexatones ale . . . 112

5.1.2 Thegeometri alorganisationisspe i at ea hs ale . . . 112

5.1.3 Basi on eptsof imageanalysis . . . 113

5.1.4 Whi h lassof geometri alfeature?. . . 115

5.1.5 Whi h ementpaste? . . . 115

5.1.6 Notation. . . 115

5.2 Metri analysis . . . 116

5.2.1 Minkowskifun tionals . . . 116

5.2.2 Volumefra tion. . . 116

5.2.3 Spe i surfa earea . . . 118

5.2.4 2-pointprobabilityfun tion/Covarian e . . . 121

5.2.5 Chordlengthdistribution fun tion . . . 125

5.2.6 Volumedistribution of onne ted omponents: non-stereologi al . . . 128

5.3 Topologi al hara terisation . . . 131

5.3.1 Minkowskifun tional: Euler-Poin aré/Gaussian urvature . . . 132

5.3.2 Per olation . . . 133

5.3.3 Topologi algraph . . . 137

5.3.4 Analysisofthetopologi algraph . . . 145

5.4 De ompositioninelementarypores . . . 149

5.4.1 Materials . . . 150

5.4.2 Two onventionsforthede omposition. . . 151

5.4.3 Chara terisation . . . 155

Chapter 6 Testing 2D

→

3Dre onstru tion of multi-phaseporousmedia obtainedby SEM 6.1 Introdu tion. . . 163

6.2 Metropolisalgorithmforthe3Dre onstru tion . . . 165

6.2.1 Phasespa e . . . 165

6.2.2 Obje tivefun tion . . . 166

6.2.3 Probabilityspa e . . . 166

6.2.4 Metropolisalgorithm. . . 167

6.2.5 Simulatedannealing algorithm . . . 167

6.2.6 Perturbation . . . 167

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6.2.8 Numeri ally . . . 170

6.3 Test andvalidationusing3Dtomographyimages . . . 172

6.3.1 Timeof onvergen e: temperatureandperturbation . . . 172

6.3.2 Topologi alanddiusion validation . . . 175

6.4 Con lusionanddis ussion . . . 175

Con lusion Appendi es AppendixA Annexes A.1 restri tedset andneighborhoodset . . . 184

A.1.1 neighborhood set . . . 184

A.1.2 restri tedset . . . 184

A.2 Proofofthea ualization . . . 184

A.2.1 Growthofthemyselfregion . . . 184

A.2.2 Degrowthofthemyselfregion. . . 185

A.2.3 Growthoftheotherregion . . . 186

A.2.4 Degrowthoftheotherregion . . . 187

A.3 Watershedtransformationusingmeta-programmationapproa hinmodernlanguage . . . 188

A.4 Appli ationforSEMimagesof ementpaste . . . 189

A.5 Algorithms . . . 189

A.5.1 Spe i surfa earea . . . 189

A.5.2 2-pointprobabilityfun tiousingMatlab . . . 189

A.5.3 Graphof2-pointprobabilityfun tion . . . 190

A.6 Signatureofapolydispersionofgrainsfollowingapowerlaw . . . 190

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Sin etheemergen eofimagingte hniques,MRI,CT,itisnowpossibletodire tlyprobethegeometri al

organization of systems su h as bone, ement, paper, glass, ro ks. As the physi al and me hani al

properties depend on the geometri al organization, there is as ienti and industrial interest for the

understanding of this relationship by using these imaging te hniques. In this ontext, the purpose of

thisthesis isthe developmentof atoolkit fordigitalimage analysis of thematerialgeometry,then the

appli ationofthistoolkitinthestudyoftheevolutionoftheporespa eof ementpaste.

Inpartone,afteradis ussiononthe hoi eofanimagingte hniqueadaptedtoamaterial,wepresent

thetwoimagingte hniquessele ted,s anningele tronmi ros opyandsyn hrotron tomographyforthe

analysisof ementpasteandtheexperimentalproto ol forsamplepreparation.

Inthese ondpart,weproposeageneri ,e ientandsimplemethodologyofsegmentation.

Segmen-tationisthetransformationofagrey-levelimagetoanlabeledimagewhereea hlabelrepresentsaphase

of the material. Generi means that this methodology an beused for a wide rangeof materials and

imagingte hniques. Ee tivemeans that the segmented stru ture mat hes the real stru ture. Simple

meansthatthe alibrationiseasy. Theimplementationoftheoptimizedalgorithmsasso iatedwiththis

methodologyisdonethankstothetheoreti al on eptualizationoftheregiongrowing.

Inthenalpart,wequantifythemorphologyandtopologyofthegeometryofthematerialstatisti ally.

Then,wede omposeaphaseintermofelementary omponentsalongtwoagreements:onemorphologi al

and the other topologi al. Finally, we use the stereologi al information estimated on the 2D sli e to

re onstru ta3Dmodellargerthantherepresentativeelementaryvolumeusingtheoptimizedalgorithm

of simulated annealing. The validation of the 3D re onstru tion is performed by the omparison of

propertiesofdiusivetransport.

Depuisl'émergen edeste hniquesd'imagerie,IRM,tomographie,ilestmaintenantpossibled'observer

dire tement l'organisationgéométrique de systèmes tels l'os, le iment, le papier, le verre, les ro hes.

Comme les propriétés physiques et mé aniques dépendent de l'organisation géométrique, il existe un

intérêts ientiqueet industrielde omprendreetdedénir ette relationdedépendan e àl'aidede es

te hniquesd'imagerie. S'ins rivantdans e ontexte,lebutde ettethèseestdedévelopperunensemble

d'outils numériques pour l'analyse d'image de la géométrie d'un matériau, puis d'appliquer es outils

dansl'étudedel'évolutiondelaporositédelapâtede iment.

Enpremièrepartie,aprèsunedis ussionsurle hoixd'unete hniqued'imagerieadaptéeàunmatériau,

nousprésentonslesdeuxte hniques d'imageriesséle tionnées, lami ros opieéle troniqueàbalayageet

latomographieparsyn hrotron,pourl'analysedelapâtede imentetleproto oleexpérimentalpourla

préparationdesé hantillons.

Endeuxièmepartie, nousproposonsuneméthodologiegénérique,e a eetsimpledesegmentation.

Lasegmentation est latransformationde l'imageen niveaux degrisen une imagelabellisée où haque

label représenteune phase dumatériau. Générique signie que ette méthodologieest appli able pour

unelarge lassede matériauxet dete hniques d'imageries. E a e spé ie quelastru ture segmentée

orrespondàla stru tureréelle. Simple indiquequel'étape de alibrationest fa ile. L'implémentation

de l'ensemble des algorithmes optimisés asso iés à ette méthodologie est rendue possible grâ e à la

on eptualisationthéoriquedela roissan ederégions.

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du matériau. Puis, nous dé omposons une phase en éléments élémentaires suivant deux onventions:

l'une morphologique, l'autre topologique. Enn, nous utilisons l'information stéréologique estimée sur

une oupe2Dpourre onstruireunmodèle3Dplusgrandquelevolumeélémentairereprésentatifàl'aide

de l'algorithme optimisé dure uit simulé. Une validation de lare onstru tion 3D est ee tuée par un

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Inmaterials ien es,aporousmediumisasolid,often alledmatrix,permeatedbyaporenetwork(voids)

lledwith auid (liquidand/orgas)su h as asponge. Manynaturalsystemssu has ro ks(e.g.

lime-stone),soils(e.g. sand),biologi altissues(e.g. bones,lung),andman madematerialssu has ements,

foamsand erami s anbe onsideredasporousmedia.

Thegeometri alorganisationoftheporous mediumae tsnumerousphysi o- hemi alphenomenasu h

asmole ular diusion, ex itation relaxation,rea tionkineti s,phase transitions,adsorptions and

apil-lary ondensation. For example, one feature of this geometri al organisation, the spe i surfa earea

measuringthesurfa eareaperunitofvolume,isrelatedtotheadsorption,heterogeneous atalysis,and

rea tions onsurfa es. In aindustrial ontext, of a tivated har oal,syntheti resins, and water

puri- ation, the main goal is to design porous media that exhibit a large spe i surfa e area in order to

maximisethepossibleintera tionwithrea tants.

In turn, for some industrial appli ations, the aim is the on eption of porous media in order to

min-imisetransportproperties. For ementitiousmaterials,thetransportpropertiesare loselyrelatedtothe

durabilitysin ethey ontroltheinvasiondynami sofvariousmole ulesinside theporousmedium. One

dire tee t ofthese potential ontaminantsisthe orrosionof thestru turalsteelby the hlorideions.

Theme hani alstrain,fromthedilationindu edbythe orrosion,isthesour eof ra ks[131,184℄.

Inporousmedia, thetwomain pro essesfortransportare diusion(dueto dieren ein on entration)

and onve tion (due to a dieren e inpressure). Mole ular diusion is usually slower. However,

on-ve tion in aporous medium, with pores smaller thanafew mi rons, is slowereddown by the vis osity

ee ts,sothatmole ulardiusionispredominantandallowsthe hemi alspe iestomoveintheporous

medium. This isthe aseof a ementpaste with apores size smallerthan afew mi rons. Theaim of

thisthesisisto understandthistransportpro ess.

In porous media, mole ular diusion is ae ted by the uid saturation, the ele tro- hemi al gradient,

the surfa e hemistry and the surfa e rea tivity. In this thesis, we assume that the pore network is

homogeneouslylled either byliquidor gas water, there isno ele tro- hemi al gradientandnally the

interfa eis onsideredasinertinordertofo usontheunderstandingoftherelationbetweenthediusive

transportandthegeometri alorganisationoftheporousmedium of ementpaste.

To a hieve this task, it is ru ial to des ribe orre tly the geometri al organisation. A rst level of

des riptionisredu edtofewnumbersthat hara terisetheglobalpropertiesofthegeometri al

organisa-tion. Themost ommonnumbersaretheporosity,

φ

,andspe i surfa earea,

S

v

. Inmostof ases,the

modelof thegeometri alorganisation isbasedonthese twonumbersasinput parameters. Thevariety

ofthe ement hemi al ompositionsimposesawiderangeofthegeometri alorganisationsoftheporous

medium. Theuseof

φ

and

S

v

doesnottakeinto a ountmanygeometri alfeatures, likethe

onne tiv-ity ofthe pore network,thestru tural orrelationand thehierar hyat dierent lengths ales. Amore

extendedanalysishasthustobeperformed.

Inthisrespe t, experimentalimagingte hniques play andwill playan importantrole in understanding

themetri andtopologyofgeometri alorganisationof porousmedium. A developmentin material

s i-en eimagingisa tuallytakingpla einvolvingforexampleX-ray,neutron,ele tron2D/3Dmi ros opies.

However,several obsta les remain. The image pro essing needsan a urate and robust segmentation.

Thisstepisoftenbadly ontrolledwithnomat hingbetweenthesegmentedphasesandtherealphases,

althoughe ienttoolsofsegmentationalreadyexist,forexamplethewatershedtransformationin

math-emati almorphology. Sin emostoftenwehaveonlya ess totheobservation of2Dse tionsthrougha

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liketheaverageporesize,themean urvature,theporeshape,thesurfa eroughnessand thestru tural

orrelation, the isotropy, the homogeneityand the lass of disorder. Using this extended metri

infor-mation,another hallengeisto re onstru tarealisti 3D ongurationfromthe2Dse tionsthat would

enablethetopologi alanddiusionpropertiesoftheoriginalmediumtoberetrieved. Finally,an

impor-tantquestionis to knowifitit possibleto simplify thegeometri aldes riptionof the3Dpore network

usingasmallnumberoffun tions thatarewellrepresentativefortheanalysisoftransportproperties.

The main obje tive of thisPhD work isto improve the pro essingof experimental images

in order to extra t quantitative information,thento understand the relationbetween this

quantitative informationand thediusionproperties. Theprin iple omponentsofthisworkare

asfollows:

•

In hapter 1,werapidly present ementpaste as anexampleof area tiveporousmaterial where thediusivetransport takespla e. Twopaperswill bereviewedto stresstheguiding ideasof our

work.

•

In hapter2, wedes ribethetwoexperimental te hniques: S anningEle tron Mi ros opy(SEM) and X-raytomographythat areused to imagetwo ementpastes(Portlandandalite) during the

setting.

•

In hapter3,weintrodu eatheoreti al frameworkfora lassofalgorithms, alledSeededRegion Growing by Pixels Aggregation/Dissolution (SRGPAD). This framework allows the fast

imple-mentationof advan edandoptimisedalgorithms dedi atedto segmentation,de ompositionof the

porousmediumin elementaryporesandpermutationlo alisationonthephaseboundaries.

•

In hapter 4, weapply these algorithms to get a simple, generi and robustmethod to segment experimental two- or three-dimensional images of materials obtained by X-ray tomography and

SEM.

•

In hapter 5, we hara terise the geometri al organisation at three levels. First, we present the extendedmetri analysisof2Dor3Dimages,thenwedes ribehowtoextra trelevantinformation

aboutthe onne tivityoftheporenetworkon3Dimagesandnally,wede omposeaporousmedium

in term of elementary pores using either ametri and atopologi al approa h. The guiding idea

of this de ompositionwill beto studyrestri teddiusion inside and betweenadja entpores,and

to geta oarsegrainingdes riptionofthediusivetransportinanee tivenetworkofelementary

pores.

•

In the hapter 6, we introdu e an e ient pro edure to re onstru t a representative elementary volumeofamulti-phasesmaterialusingmetri informationofabi-dimensionalimagingte hnique

su hasSEM.Thislastexperimentalsetupiswidelyusedinthe ementindustryandthepossibility

torestore3Dmodelsfrom2Dobservationshouldprovideagainofe ien yinaindustrial ontext.

(14)

Tosimplifythereading,thenotationswillalwaysbethesame.

Notationsusedin settheory

E

ave torspa e

Z

theintegersset

Ω

asubsetofE

Ω

c

the omplementof

Ω

⊕

theMinkowskiaddition,also alleddilation,operator:

A

⊕ B = {a + b : a ∈ A

and

b

∈ B}

Ax

theset

A

translatedbytheve tor

x

⊖

theerosionoperator:

A

⊖ B = {a + b : a ∈ A

c

and

b

∈ B

s

}

c

B

s

thesymmetri ofB,

B

s

₌

{x : −x ∈ B}

.

◦

theopeningoperator:

A

⊖ B = (A ⊖ B) ⊕ B

•

the losingoperator:

A

• B = (A ⊕ B) ⊖ B

⊕

k

thedilationoperatortimes

k

:

A

⊕

k

B =

timesk

z

}|

{

(.(A

_{⊕ B) . . . ⊕ B)}

+

thedisjointunionoperatorin thesettheory:

A + B =

{A ∪ B : A ∩ B = ∅}

⊎

thedisjointunionoperatorin thesettheory:

A

⊎ B = {A ∪ B : A ∩ B = ∅}

−

thein lusion restri tionoperator inthesettheory:

A

− B = {A \ B : B ⊂ A = ∅}

Ω

adomainin ludedinthespa e

E

I

the hara teristi fun tion of

Ω

:

I(x) = 1

if

x

∈ Ω, 0

otherwise

∂Ω

theboundaryofthedomain

Ω

(the losureof

Ω

withouttheinteriorof

Ω

) Notationsusedin seededregiongrowingbypixelsaggregation/dissolution

X

t

i

aregion(adomainofthespa e)attime

t

with thelabel

i

Z

t

i

azoneofinuen easso iatedtotheregion

X

t

i

:

Z

t

i

= (X

i

t

⊕ V

i)

\ (

S

j∈N

i

X

t

j

)

Ni

arestri tedset,asubsetof

N

Vi

aneighbourhood

δ(x, i)

anorderingattribute fun tion

C

x,y

theset of ontinuousappli ationfrom

[0, 1]

to

E

su h asthetwoextremitiesareequalsto

x

and

y

f

agrey-levelimage: anappli ationof

E

to

Z

∧

theoperatorandinthesymboli logi

d

Ω

_{(x, y)}

(15)

Notationsusedingeometri al hara terization

Ii

the hara teristi fun tionof thephase

i

Wi

thei-Minkowskifun tional

φi

thevolumefra tionofthephase

i

αc(t)

thedegreeofhydrationattime

t

S

i

thespe i surfa eareaofthephase

i

S2,i

the2-pointprobabilityfun tionof thephase

i

(auto- orrelationfun tion)

fi,µ

the

µ

- hordlengthdistributionfun tion ofthephase

i

li

themean hordlength

N3

theEuler-Poin aréinvariant

K

theGaussian urvature

pc

theper olationthreshold

φc

thevolumefra tionofthegivenphaseattheper olationthreshold

β0

thenumberof onne ted omponentsofthemi rostru ture

β1

thenumberofirredu ible y les

β2

thenumberofinternalsurfa esorthenumberof onne ted omponentsofthe omplementary

α0

thenumberofvertexesofthetopologi algraph

α1

thenumberofedgesof thetopologi algraph

Nc(v)

the oordinationnumberofthevertex

v

< Nc

>

themean oordinationnumber

C

theintensivetopologi alnumber

Notationsusedinannealing simulatedalgorithm

S2,i,j

the2-pointprobabilityfun tionsof thephase

i

and

j

(

i = j

auto,

i

6= j

ross)

d(

_{M, R)}

thedistan ebetweenthemodelandthereferen e

P (

_M|R)

theprobabilityof themodel

M

given

R

Q

thesele tionmatrix(the perturbation)

ρ

thea eptan ematrix

N

i,i

I

t

(d)

thenumberofauto- orrelationofthephase

i

,atthedistan e

d

,forthepartition

I

t

N

I

t

i,i

(d)

thenumberof hordsofthephase

i

whi hsizeis

d

,forthepartition

I

t

p

~

e

g

(y)

thephaselabelatleftofthevoxel,

y

,onthedire tion

~e

p

~

e

d

(y)

thephaselabelatrightofthevoxel,

y

,onthedire tion

~e

l

~

e

g

(y)

thelengthofthe hordatleftofthevoxel,

y

,onthedire tion

~e

l

~

e

(16)

Bibliography overview: Image analysis

versus diusive transport in ement

paste

Contents

1.1 Cementpaste . . . 1

1.2 Diusionin a onneddomain . . . 3

1.3 Linkbetweenthegeometryof ementpaste and transport properties. . . 8

1.3.1 EstimatingtransportpropertiesofmortarsusingimageanalysisofSEMimages 8

1.3.2 X-raymi rotomographi studiesofporestru tureandpermeabilityinPortland

ement on rete . . . 10

1.4 Con lusionand dis ussion . . . 12

1.1 Cement paste

Cementismadebyheatinglimestonewithsmallquantitiesofothermaterials(su has lay)to1450

o

Cin

akiln. Theresultinghardsubstan e, alled' linker',isthengroundwithasmallamountofgypsuminto

apowderto make'OrdinaryPortlandCement',themost ommonlyusedtypeof ement(oftenreferred

toas OPC). Cementrefersto adry powdersubstan e. Upon addition ofwater, the ementmixture is

referredto as ementpaste, withtheadditionofwaterandofsand,itis referredto amortar, andwith

theadditionofofwater,ofsandandofaggregate(generallya oarseaggregatesu hasgravel,limestone,

or granite)is referredto as on rete. The ementsset and hardenbe auseof hemi al rea tions when

itis mixed with water. A ategoryof ement, alled hydrauli ements, retains strength and stability

even under water. The key requirement for this strength and stability is that the hydrates formed by

immediate rea tionwith water be essentially insoluble in water. Most onstru tion ementstoday are

hydrauli ,andmostofthesearebasedonPortland ement.

Thetypi al onstituentsofPortland linkerare

name hemistnotation ement hemistnotation Mass

Tri al iumsili ate (CaO)

3

(SiO

2

) C3S 45-75%

Di al iumsili ate (CaO)

2

(SiO

2

) C2S 7-32%

Tri al iumaluminate (CaO)

3

(Al

2

O

3

) C3A 0-13%

Tetra al iumaluminoferrite (CaO)

4

(Al

2

O

3

)(Fe

2

O

3

) C4AF 0-18%

Gypsum (CaSO

4)

(H

2

O)

2

2-10%

Upon the addition of water, the hydration rea tions of the anhydrous phases begin. These pro esses

(17)

on rete sets (i.e. be omesrigid) in about 6hours, and developsa ompressivestrength of 8

∼

MPain

24 hours. The strength rises to 15

∼

MPa at 3 days, 23

∼

MPa at one week, 35

∼

MPa at 4 weeks, and

41

∼

MPaatthreemonths. Inprin iple,thestrength ontinuestoriseslowlyaslongaswaterisavailable

for ontinuedhydration,but ementpasteisusuallyallowedtodryoutafterafewweeks,andthein rease

in strength isstopped. Hydrationprodu ts formedin hardened ement pastes(HCP) are ompli ated,

be ausemanyoftheseprodu tshavenearlythesameformula(thehyphensinC-S-Hindi ateaphaseof

variable omposition).

name hemistnotation ementnotation

Cal iumSili ateHydrate (CaO)

x

(SiO

2

)

y

(H

2

O)

z

C-S-H

Cal iumhydroxide Ca(OH)

2

CH

Ettringite [Ca

3

Al(OH)

6

.12H

2

O℄

2

.2H

2

O C

3

A.3CaSO

4

.32H

2

O

AluminateFerritetrisulfate ontainsthreeanhydritemole ules: C

3

A.3CaSO

4

.32H

2

O AFt

AluminateFerritemonusulfate ontainsone anhydritemole ule: C

3

A.CaSO

4

.A2H

2

AFm

Thehydrationrea tion an bede omposedinto twosteps(seegure1.1):

1. dissolutionof theionsin thesolutionfromthe anhydrousphases. For example,thedissolutionof

the al iumandsili atefromC3S:

Ca3SiO5(s)

+ 3H2O(l)

⇄ 3Ca

2+

(l)

+ H2SiO

2−

4(l)

+ 4OH

−

(l)

where thesubs ript

(s)

meansasolidphase,

(l)

meansaliquidphase.

2. pre ipitationofhydratephasesfromthesolution. Forexample,thepre ipitationofC-S-HandCH

from watersolutionof al iumandsili ateions:

1.5Ca

2+

_(l)

+ H2SiO

2−

_4(l)

+ OH

_(l)

−

+ H2O(l)

⇄ ((CaO)1.5

(SiO2)(H2O)2.5

)(s)

Ca

2+

_(l)

+ 2OH

_(l)

−

⇄ Ca(OH)2(s)

Thedissolution-pre ipitationrea tionandthepatternformationoftheporousmedium(spa edistribution

ofthesolidandliquidphases)depend onmanyparameters:

1. the onstituentsofthedrypowdersubstan eof ement,

2. theratioofwaterto ement(W/C),

3. thetemperature[110℄,

4. therelativehumidity[157℄,

Theporousmediumexhibitsageometri alorganisationinawiderangeoflengths alesfromnanometres

to millimetres. Powers and Brownyard[136℄ distinguished twokinds of pores: gel waterpores (under

theinuen eofadsorbingfor es)fromthenanometreto0.1mi rometerand apillarywaterpores(free

water)form0.1mi rometertomillimetre. Thegeometri alfeatures hara terisingtheporousmediumare

relatedto awiderangeof dierentme hani al,physi aland hemi alproperties. Oneimportantissue,

onne tedto the on retedurability, istheunderstanding ofthetransport properties. This isa ru ial

pointinorderto ontroltheinvasiondynami sofvariousmole ulesinsidetheporousmedium. Oneee t

ofthese possible ontaminantsis the orrosionofthestru turalsteelindu edbythesteeldepassivation

when pH de reased (for example the hemi al rea tion of portlandite, Ca(OH)

2

, and al ium sili ate

hydrate, C-S-H, in the ementmatrix with arbondioxidegas leadingto al ite CaCO

3

for example).

Sin eironoxideformationisa ompaniedbyanin reaseinvolume,itshould leadtomi ro- ra kingin

thesurrounding ementpaste[184,131℄(seegure1.2). Thetwomainoriginsformole ulartransportare

diusion(duetoadieren ea on entration)and onve tion(duetoadieren einpressure). Mole ular

diusion is usually slower. However, for a porous medium, having pores smaller than a few mi rons,

sin e onve tion is slowed down by the vis osity ee ts, mole ular diusion is predominantand allows

the hemi alspe iestomoveintheporousmedium. Asthestudiedmaterialis ementpastewithapore

(18)

Anhydrous

water

phase

(a)

Anhydrous

Hydrates

Water

phase

(b)

100 microns

Anhydrous

phase

Hydrate (portlandite)

Hydrates

Capillary pores

( )

Figure1.1: Dissolution-pre ipitation rea tion. (a) attimeof hydration =0,(b)atthebeginningofthe

hydration,( )Portland ementimageobtainedbySEMattimeofhydration=1day.

1.2 Diusion in a onned domain

Letus onsiderabath at uniformtemperaturewithoutuid motion. Atinitial state,abla kink drop

falls in the bath. Propagation of the olour o urs due to the mole ular diusion. There is a ux,

~j

,

ofmole ules (pigments) from regions of high on entration(where thedrop is fallen) to regionsof low

on entration. In 1845, Fi k introdu ed a ma ros opi law of diusion, whi h governs the mole ular

diusion:

~j = −D

0 ∇c

~

with

∇

~

thegradientoperator

.

(1.1) where

c

isthemole ular on entrationand

D0

isthefreediusion oe ientdependingonthe

temper-atureandtheintera tionsbetweenthespe iesandthesolution.

Usingthemass onservationlaw,

∂c

∂t

+

∇ · ~j = 0

with

∇·

thedivergen eoperator

,

(1.2) wegetthese ondFi k'slaw:

∂c

∂t

= D0

∇

2 _c

with

∇

2

theLapla eoperator. (1.3)

Bydimensionalanalysis,wendthatthediusion hara teristi lengthisproportionalto

√

t

(forawater mole uleat 20

o

C, thislengthis in theorderof 100

µm

for

t = 1s

). Thisslowdynami sisobservedin

thediusionofbla kinkdropinthebath. Thediusioninsideaporousmediumof ementpasteisalso

slow(seegure1.3). Ina onneddomain(the porousmedium of ementpaste),agoodunderstanding

(19)

PH

Corrosion−>Oxydes formation with bigger size−>Pression

CO

2 Ca(OH)

2 + CO

2 → CaCO

3 + H

2 O

(a)

(b) ( )

Figure 1.2: Carbonation. (a)Transport of the arbondioxide gasinside the porous medium, (b) front

ofPMCLaboratory( ) zoomonthetopoffrontofPMCLaboratory- ra ksduetothe orrosionofthe

stru turalsteel.

•

thewatersaturationinsidetheporousmedium sin ethewaterdiusion oe ientsinliquidphase and in gas phaseare dierent. Inmost materialsmade of ement, therelativewaterhumidityis

above30

%

,largeenoughtohavebothwatergasinsidethe apillaryporosityandwaterliquidinside

thegelporosity[13℄.

•

theele tro- hemi algradientsin e,ifthediusing spe ies areioni ,theirmotiondependsalso on the ele tro- hemi algradient(

j

e

₌

−uc∇ψ

where

j

e

is theowdue to ele tro- hemi al gradient,

c

is the on entration,

u

is the ioni mobility,

∇ψ

is the gradient of lo al potential) [145℄. The water inside the ement paste is an ioni solution (pH=13): it is a ele trolyte. By imposing an

ele tro- hemi al gradient (see gure 1.4), the measurement of the diusivity of ions spe ies in

water-saturatedhardened ementpastes[130,100℄ anbedoneusingtheNernst-Einsteinrelation:

u =

DzF

RT

wherethe

z

thevalen eoftheioni parti le,

F = N e

istheFaraday's onstant,

T

isthetemperature

and

R

isthegas onstant.

•

thesurfa e hemistryandthesurfa erea tivity,forexample,thediusingspe ies anbe hemi ally transformedintootherspe iesafterhittingthesurfa e[136℄. In ementpaste,thereisapossibility

ofastrong arbonation,the hemi alrea tionoftheinterfa e(portlandite,Ca(OH)

2

,and al ium

sili ate hydrate, C-S-H, in the ement matrix) with diusing spe ies su h as arbon dioxide gas

(20)

C0

C1

S

l

Cement paste

Figure1.3: Experimental measurementofthediusion oe ient. Insteadystate,the onstantowis

givenby

jx

=

∆Q

S∆t

where

∆Q

is thediused quantity in thetime interval

∆t

throughthe se tion

S

of thespe imen. Theee tivediusion oe ientis al ulatedusingtherstFi k'slawwiththerelation:

De

=

j

x

|c

1 −c

0 |

l

where

c1

and

c0

arethe on entrationsinthetwo hambers. Fora on retewithathi kness

equalto3 m,wherea onstant on entrationgradientisappliedbetweenthetwofa es,thesteadystate

owisobtainedafterone year[45℄( orrespondingto thelinearpartofthe urveatlongtime).

Figure1.4: Migrationoftheioni parti lesunder aele troni eld. Theadvantageofthisexperimentis

thatthesteadystateisrapidlyrea hed,allowingthediusion oe ienttobemeasuredinareasonable

(21)

•

thegeometryofthe onnedporenetwork.

Inthisthesis,wefo usonthelinkbetweenthediusiontransportandthegeometryoftheporousmedium

of ementpaste. Weassumethattheporenetworkishomogeneouslylledeitherbyliquidorgaswater,

thereisnoele tro- hemi algradientandnallytheinterfa eis onsidered asinert.

Inahomogeneousisotropi porousmedium,theee tiveFi klawis:

∂ce

∂t

= φDe

∇

2 _ce.

(1.4) where

De

(m

2

/s) is the ee tivediusion oe ientand

ce

is the ee tivemole ular on entration (

φ

is the volume fra tion of pore phase). Experimentally [130, 100℄, we measure the ee tive diusion

oe ient(seegure1.5).

De

mayberelatedtoitsfreediusivity,

D0

(m

2

/s),bythefollowingequation:

000000000000

111111111111

000000000000

111111111111

00

11

0

1

1 00000000000000000000

11111111111111111111

000

111

0

1

00

11

11 00000

00000

11111

000

111

111 0000

0000

1111

000000000

111111111

0000000000

1111111111

0000000

1111111

00

11

11 00000000

00000000

11111111

0000000000

1111111111

000000

111111

000000

111111

00000000

11111111

000

111

0

1

1 00000

00000

11111

L

S

0

1

0

1

0

1

1 l_c

0

1

1 c

1,macro

= φc

1,micro

c

_1,micro

c

2,micro

c

_2,macro

= φc

2,micro

c

_2,micro

= C

2 c

_1,micro

= C

1

Figure 1.5: Experimentalmeasurementofthe diusion oe ient. Insteady state,the onstantowis

givenby

jx

=

∆Q

φS∆t

where

∆Q

isthediused quantityin thetimeinterval

∆t

throughthese tion

S

of thespe imen. The mi ros opi diusion oe ientis

De

=

j

x

|c

1,e

−c

2,e

|

L =

j

x

|C

1 −C

2 |

L

sin etheboundary

onditionsimpose

Ci

= ci,e

.

De

=

1 τ

D0

(1.5)

where

τ

isdimensionlessparameter(

τ

≥ 1

) alledthetortuosityoftheporousmedium. Infa t,theabove

relationshipattemptstos ale

D0

to

De

byin ludingafa torthatisthein reasedtransportpathlength

due to thegeometri alorganisation oftheporous mediumof ementpaste (PMCP) (

τ

). This physi al

parameteris dierent to thegeometri al tortuosity. The dire t or indire testimation of thetortuosity

anbedoneby:

•

theutilisationoftheAr hie'sempiri allaw[4℄where wehave:

τ

_∼

1 φ

α

with

1

2 < α <

3

2

(1.6)

Although this approa h is ommonly used in reservoir engineering, it is useless to predi t the

tortuosityforanunknownmaterial.

•

the proposition of atoys model of the porous medium basedon experimental onstraints [105℄ or the oarsegraining des riptionin using aperturbation method [156℄ or self onsistentmethod

[17,43℄. The ommoninputparametersofthesemethodsaretheporosity,thespe i surfa earea,