Effects of particles on spinodal decomposition: A Phase field study

4.2 MB·

by  Supriyo Ghosh

Effect of particles on spinodal decomposition:
A phase field study
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1 A Project Report
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2 submitted in partial fulfilment of the

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a requirements for the Degree of
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] Master of Engineering
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t Materials Engineering
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Supriyo Ghosh
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1 under the guidance of
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Department of Materials Engineering
Indian Institute of Science
Bangalore – 560 012 (INDIA)
JUNE 2012
Acknowledgements
Iwouldliketothankmyadvisor,Prof. T.A.Abinandanan,forhisguidanceandencouragement
throughoutmywork. Icherishhismotivatingwordsandadviceonmyfuturecareer. Icouldnot
askforabetteradvisor. IthankDr. SuryasarathiBoseforhisinvaluablecomments,suggestions
and advise on my project. The entire faculty, staff and students in the Materials Engineering
Department made my learning enjoyable. I would also like to thank my family, friends, and
labmates-Naveen, Bhaskar, Chaitanya andVinayfortheirsupportandunderstanding. Finally,
special thanks to Rooparam, Nikhil, Santanu, Ratul, Rajdipda, Samratda, Arkoda for their useful
suggestions.
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Abstract
The present work is directed towards the understanding of the interplay of phase separation
and wetting which dominates the morphological evolution in multicomponent systems. For this
purpose, we have studied the phase separation pattern of a binary mixture (AB) in presence
of stationary spherical particles (C) which prefers one of the components of the binary (say,
A).BinaryABiscomposedofcriticalcomposition(50:50)andoff-criticalcompositions(60:40,
40:60). Off-criticalcompositionsarechosentoincludetwocaseswhereeitherthemajororminor
componentwets theparticle. Particlesare fixedin positionand sphericalinshape. Particle sizes
of 8 units and 16 units are used in all simulations. Two types of particle loading are used, 5%
and 10%. Particles arewell-distributedin the matrixat a certaininterparticle distancefollowing
periodicboundaryconditions.
We have employed a ternary form of Cahn-Hilliard equation to model such system. This
model is a modification of Bhattacharya’s model to incorporate immobile fillers. Free energy
of such an inhomogeneous system depends on both composition and composition gradients.
Composition provides homogeneous contribution to the system free energy whereas compo-
sition gradients contribute to the interfacial energies. Homogeneous form of free energy is
given by regular solution expression which is very closely related to Flory-Huggins model for
monodispersepolymermixtures. Toelucidatetheeffectofwettingonphaseseparationwehave
designedthreesetsof χ andκ toincludetheeffectsofneutralpreference,weakpreference
ij ij
andstrongpreferenceoftheparticleforoneofthebinarycomponents. Wehavesimulatedtwo
differentcases wherethe binarymatrix (A:B)is quenchedcritically oroff-criticallyin presence
ofstationarysphericalparticles.
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Abstract iii
Iftheparticlesarepreferentiallywettedbyoneofthecomponentsthenearlystagemicrostruc-
turesshowtransientconcentricalternatelayersofpreferredandnon-preferredphasesaroundthe
particles. Whenparticlesareneutraltobinarycomponentsthensucharingpatterndoesnotform.
Atlatetimesneutralpreferencebetweenparticlesandbinarycomponentsyieldsacocontinuous
morphology whereas preferential wetting produces isolated domains of non-preferred phases
dispersed in a continuous matrix of preferred phase. In other words lack of preference forms
anearly completephaseseparate morphology forabinary ofcriticalcomposition whereasan
incompletephaseseparationisseenifpreferenceexistsbetweenparticleandmatrixcomponents.
Inallthecases thebinaryinteractionparametersaresuchthat χ >|χ −χ |,whichrefers
AB BC AC
to a equilibrium wetting state where particles are in contact with both the components with
a surplus of preferred component around it. Particles at the interface provide a resistance to
interfacialmotionand thusimpededomaincoarsening. Inaddition,higher particle loadingand
smaller particle size are also highly effective in reducing the kinetics of phase separation and
domaingrowth.
For off-critical compositions we have studied two different situations where either major
or the minor component wets the network. When minor component wets the particle then a
bicontinuousmorphologyresultswhereas whenmajorcomponentwetsthenetworkadroplet
morphologyisseen. Insuchcasesearlystagemorphologysuggestsanenrichedlayerofpreferred
component around the particle though it is fundamentally different than the "target" pattern
formed in case of critical mixture. When majority component wets the particle, a possibility
of double phaseseparation is reported. In suchalloys phase separation starts near the particle
surface and propagates to the bulk at intermediate to late times forming spherical or nearly
sphericaldropletsoftheminorcomponent.
Contents
Acknowledgements i
Abstract ii
1 Introduction 1
2 LiteratureReview 4
2.1 ExperimentalStudies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 TheoreticalStudies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Formulation 10
3.1 RegularSolutionModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Flory–HugginsModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 TheCahn-HilliardModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 ChemicalPotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.5 EvolutionEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.6 NumericalImplementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 SimulationDetails 18
4.1 ParticleCharacteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1.1 ParticleLoading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1.2 ParticleSizeandshape . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1.3 InterparticleDistance(λ) . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.4 Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 CompositionalDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3 MobilityMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4.1 ChoiceofInteractionParameters . . . . . . . . . . . . . . . . . . . . . 22
4.4.2 ChoiceofGradientEnergyparameters . . . . . . . . . . . . . . . . . . 23
4.5 Synopsisofsimulationparameters . . . . . . . . . . . . . . . . . . . . . . . . 24
4.6 ComputationalAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 ResultsandDiscussion 27
5.1 TernaryPhaseEquillibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 InterphaseInterfacialEnergy . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
iv
CONTENTS v
5.3 MicrostructuresinA B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
50 50
5.3.1 SystemS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
O
5.3.2 SystemS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
W
5.3.3 SystemS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
S
5.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4 Off-symmetricalloysinsystemS . . . . . . . . . . . . . . . . . . . . . . . . 47
S
5.4.1 A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
40 60
5.4.2 A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
60 40
5.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 SummaryandConclusions 55
AppendixA ThermodynamicsofTernarySystem 57
AppendixB InterfacialEnergyDetermination 60
Bibliography 61
List of Figures
2.1 Simulated(a)andexperimental(b)coreshellmorphoology . . . . . . . . . . . 8
2.2 Simulated(a)andexperimental(b)dropletmorphology . . . . . . . . . . . . . 9
2.3 Simulated(a)andexperimental(b)core/shellmorphologywithcontinuousshell 9
2.4 Simulated(a)andexperimental(b)ringpattern . . . . . . . . . . . . . . . . . 9
4.1 Periodicboundaryconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Radialcompositionprofilefromthecenteroftheparticles(schematic) . . . . . 21
4.3 GrayscalecolormapprojectedonGibbstriangle . . . . . . . . . . . . . . . . 26
5.1 Isothermal section of the phase diagram for system S (χ = 2.5, χ = 3.5,
O AB BC
χ =3.5)(schematic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
AC
5.2 Isothermal section of the phase diagram for system S (χ = 2.5, χ = 4.0,
W AB BC
χ =3.5)(schematic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
AC
5.3 Isothermal section of the phase diagram for system S (χ = 2.5, χ = 5.0,
S AB BC
χ =3.5)(schematic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
AC
5.4 Equilibrium composition profile across (a) α−β interface (b) β −γ interface
(c)α−γ interfaceaccordingtosystemS variables . . . . . . . . . . . . . . . 31
O
5.5 Equilibrium composition profile across (a) α−β interface (b) β −γ interface
(c)α−γ interfaceaccordingtosystemS variables . . . . . . . . . . . . . . . 32
W
5.6 Equilibrium composition profile across (a) α−β interface (b) β −γ interface
(c)α−γ interfaceaccordingtosystemS variables . . . . . . . . . . . . . . . 33
S
5.7 Microstructures corresponding to left column is for particle radius of 8 units
andrightcolumnisforparticleradiusof16unitsforthesamevolumefraction
(5%) of particles.Thetop picture is from someearly stage (t = 1500 time steps),
middleoneisofintermediatestage(t=3000timesteps)andbottomoneisfor
late-stage(t=5000timesteps). Allcorrespondingmicrostructuresarecompared
atsimilartimestepandfollowsystemS . . . . . . . . . . . . . . . . . . . . . 36
O
5.8 Microstructures corresponding to left column is for particle radius of 8 units
andrightcolumnisforparticleradiusof16unitsforthesamevolumefraction
(10%)ofparticles.Thetoppictureis ofsomeearlystage(t=1500timesteps),
middleoneisofintermediatestage(t=3000timesteps)andbottomoneisof
late-stage(t=5000timesteps). Allcorrespondingmicrostructuresarecompared
atsimilartimestepandfollowsystemS . . . . . . . . . . . . . . . . . . . . . 37
O
vi
LISTOFFIGURES vii
5.9 Microstructurescorrespondingtoleftcolumnisforparticleradiusof8unitsand
rightcolumnisforparticleradiusof16unitsforthesamevolumefraction(5%)
ofparticles.Thetoppictureisfromsomeearlystage(t=200timesteps),middle
oneis ofintermediatestage (t=500time steps)andbottom oneisfor late-stage
(t = 3000 timesteps). All corresponding microstructures arecompared at similar
timestepandfollowsystemS . . . . . . . . . . . . . . . . . . . . . . . . . . 39
W
5.10 Microstructures corresponding to left column is for particle radius of 8 units
andrightcolumnisforparticleradiusof16unitsforthesamevolumefraction
(10%) of particles.The top picture is of some early stage (t = 200 time steps),
middle one is of intermediate stage (t = 500 time steps) and bottom one is of
late-stage(t=3000timesteps). Allcorrespondingmicrostructuresarecompared
atsimilartimestepandfollowsystemS . . . . . . . . . . . . . . . . . . . . . 40
W
5.11 Microstructurescorrespondingtoleftcolumnisforparticleradiusof8unitsand
rightcolumnisforparticleradiusof16unitsforthesamevolumefraction(5%)
ofparticles.Thetoppictureisfromsomeearlystage(t=50timesteps),middle
oneis ofintermediatestage (t=500time steps)andbottom oneisfor late-stage
(t = 3000 timesteps). All corresponding microstructures arecompared at similar
timestepandfollowsystemS . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
S
5.12 Microstructures corresponding to left column is for particle radius of 8 units
andrightcolumnisforparticleradiusof16unitsforthesamevolumefraction
(10%) of particles.The top picture is of some early stage (t = 50 time steps),
middle one is of intermediate stage (t = 500 time steps) and bottom one is of
late-stage(t=3000timesteps). Allcorrespondingmicrostructuresarecompared
atsimilartimestepandfollowsystemS . . . . . . . . . . . . . . . . . . . . . 43
S
5.13 Microstructures(A B ) corresponding to left column is for particle radius of
40 60
8 units and right column is for particle radius of 16 units for the same volume
fraction(5%)ofparticles.Thetoppictureisfromsomeearlystage(t=100time
steps),middleoneisofintermediatestage(t=500timesteps)andbottomone
is for late-stage (t = 3000 timesteps). All corresponding microstructures are
comparedatsimilartimestepandfollowsystemS . . . . . . . . . . . . . . . . 48
S
5.14 Microstructures(A B )corresponding toleftcolumn isforparticle radiusof
40 60
8 units and right column is for particle radius of 16 units for the same volume
fraction (10%) ofparticles.The toppicture isof some earlystage (t= 100 time
steps),middleoneisofintermediatestage(t=500timesteps)andbottomone
is of late-stage (t = 3000 timesteps). All corresponding microstructures are
comparedatsimilartimestepandfollowsystemS . . . . . . . . . . . . . . . . 49
S
5.15 Microstructures(A B ) corresponding to left column is for particle radius of
60 40
8 units and right column is for particle radius of 16 units for the same volume
fraction(5%)ofparticles.Thetoppictureisfromsomeearlystage(t=100time
steps),middleoneisofintermediatestage(t=500timesteps)andbottomone
is for late-stage (t = 3000 timesteps). All corresponding microstructures are
comparedatsimilartimestepandfollowsystemS . . . . . . . . . . . . . . . . 51
S
LISTOFFIGURES viii
5.16 Microstructures(A B )corresponding toleftcolumn isforparticle radiusof
60 40
8 units and right column is for particle radius of 16 units for the same volume
fraction (10%) ofparticles.The toppicture isof some earlystage (t= 100 time
steps),middleoneisofintermediatestage(t=500timesteps)andbottomone
is of late-stage (t = 3000 timesteps). All corresponding microstructures are
comparedatsimilartimestepandfollowsystemS . . . . . . . . . . . . . . . . 52
S
List of Tables
4.1 Particlesizeandnumberofparticles . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Theoreticallyconsideredvaluesofκ . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 ValuesofallsimulationVariables . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.1 Gradientenergyparametersandcorrespondinginterfcialenergies . . . . . . . . 31
ix