Effects of particles on spinodal decomposition: A Phase field study
EEffect of particles on spinodal decomposition:A phase field study
A Project Reportsubmitted in partial fulfilment of therequirements for the Degree of
Master of Engineering in Materials Engineering by Supriyo Ghosh under the guidance of
Prof. T. A. Abinandanan
Department of Materials EngineeringIndian Institute of ScienceBangalore – 560 012 (INDIA)JUNE 2012 a r X i v : . [ c ond - m a t . s o f t ] J a n cknowledgements I would like to thank my advisor, Prof. T. A. Abinandanan, for his guidance and encouragementthroughout my work. I cherish his motivating words and advice on my future career. I could notask for a better advisor. I thank Dr. Suryasarathi Bose for his invaluable comments, suggestionsand advise on my project. The entire faculty, staff and students in the Materials EngineeringDepartment made my learning enjoyable. I would also like to thank my family, friends, andlab mates - Naveen, Bhaskar, Chaitanya and Vinay for their support and understanding. Finally,special thanks to Rooparam, Nikhil, Santanu, Ratul, Rajdipda, Samratda, Arkoda for their usefulsuggestions. i bstract
The present work is directed towards the understanding of the interplay of phase separationand wetting which dominates the morphological evolution in multicomponent systems. For thispurpose, we have studied the phase separation pattern of a binary mixture (AB) in presenceof stationary spherical particles (C) which prefers one of the components of the binary (say,A). Binary AB is composed of critical composition(50:50) and off-critical compositions(60:40,40:60). Off-critical compositions are chosen to include two cases where either the major or minorcomponent wets the particle. Particles are fixed in position and spherical in shape. Particle sizesof 8 units and 16 units are used in all simulations. Two types of particle loading are used, 5%and 10%. Particles are well-distributed in the matrix at a certain interparticle distance followingperiodic boundary conditions.We have employed a ternary form of Cahn-Hilliard equation to model such system. Thismodel is a modification of Bhattacharya’s model to incorporate immobile fillers. Free energyof 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 isgiven by regular solution expression which is very closely related to Flory-Huggins model formonodisperse polymer mixtures. To elucidate the effect of wetting on phase separation we havedesigned three sets of χ i j and κ i j to include the effects of neutral preference, weak preferenceand strong preference of the particle for one of the binary components. We have simulated twodifferent cases where the binary matrix (A:B) is quenched critically or off-critically in presenceof stationary spherical particles. iibstract iiiIf the particles are preferentially wetted by one of the components then early stage microstruc-tures show transient concentric alternate layers of preferred and non-preferred phases around theparticles. When particles are neutral to binary components then such a ring pattern does not form.At late times neutral preference between particles and binary components yields a cocontinuousmorphology whereas preferential wetting produces isolated domains of non-preferred phasesdispersed in a continuous matrix of preferred phase. In other words lack of preference formsa nearly complete phase separate morphology for a binary of critical composition whereas anincomplete phase separation is seen if preference exists between particle and matrix components.In all the cases the binary interaction parameters are such that χ AB > | χ BC − χ AC | , which refersto a equilibrium wetting state where particles are in contact with both the components witha surplus of preferred component around it. Particles at the interface provide a resistance tointerfacial motion and thus impede domain coarsening. In addition, higher particle loading andsmaller particle size are also highly effective in reducing the kinetics of phase separation anddomain growth.For off-critical compositions we have studied two different situations where either majoror the minor component wets the network. When minor component wets the particle then abicontinuous morphology results whereas when major component wets the network a dropletmorphology is seen. In such cases early stage morphology suggests an enriched layer of preferredcomponent around the particle though it is fundamentally different than the "target" patternformed in case of critical mixture. When majority component wets the particle, a possibilityof double phase separation is reported. In such alloys phase separation starts near the particlesurface and propagates to the bulk at intermediate to late times forming spherical or nearlyspherical droplets of the minor component. ontents Acknowledgements iAbstract ii1 Introduction 12 Literature Review 4 λ ) . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.4 Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Compositional Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Mobility Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4.1 Choice of Interaction Parameters . . . . . . . . . . . . . . . . . . . . . 224.4.2 Choice of Gradient Energy parameters . . . . . . . . . . . . . . . . . . 234.5 Synopsis of simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . 244.6 Computational Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3.1 System S O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3.2 System S W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3.3 System S S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4 Off-symmetric alloys in system S S . . . . . . . . . . . . . . . . . . . . . . . . 475.4.1 A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.4.2 A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ist of Figures S O ( χ AB = 2.5, χ BC = 3.5, χ AC = 3.5)(schematic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Isothermal section of the phase diagram for system S W ( χ AB = 2.5, χ BC = 4.0, χ AC = 3.5)(schematic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.3 Isothermal section of the phase diagram for system S S ( χ AB = 2.5, χ BC = 5.0, χ AC = 3.5)(schematic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.4 Equilibrium composition profile across (a) α − β interface (b) β − γ interface(c) α − γ interface according to system S O variables . . . . . . . . . . . . . . . 315.5 Equilibrium composition profile across (a) α − β interface (b) β − γ interface(c) α − γ interface according to system S W variables . . . . . . . . . . . . . . . 325.6 Equilibrium composition profile across (a) α − β interface (b) β − γ interface(c) α − γ interface according to system S S variables . . . . . . . . . . . . . . . 335.7 Microstructures corresponding to left column is for particle radius of 8 unitsand right column is for particle radius of 16 units for the same volume fraction( ) of particles.The top picture is from some early stage (t = 1500 time steps),middle one is of intermediate stage (t = 3000 time steps) and bottom one is forlate-stage (t = 5000 timesteps). All corresponding microstructures are comparedat similar timestep and follow system S O . . . . . . . . . . . . . . . . . . . . . 365.8 Microstructures corresponding to left column is for particle radius of 8 unitsand right column is for particle radius of 16 units for the same volume fraction( ) of particles.The top picture is of some early stage (t = 1500 time steps),middle one is of intermediate stage (t = 3000 time steps) and bottom one is oflate-stage (t = 5000 timesteps). All corresponding microstructures are comparedat similar timestep and follow system S O . . . . . . . . . . . . . . . . . . . . . 37viIST OF FIGURES vii5.9 Microstructures corresponding to left column is for particle radius of 8 units andright column is for particle radius of 16 units for the same volume fraction ( )of particles.The top picture is from some early stage (t = 200 time steps), middleone is of intermediate stage (t = 500 time steps) and bottom one is for late-stage(t = 3000 timesteps). All corresponding microstructures are compared at similartimestep and follow system S W . . . . . . . . . . . . . . . . . . . . . . . . . . 395.10 Microstructures corresponding to left column is for particle radius of 8 unitsand right column is for particle radius of 16 units for the same volume fraction( ) 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 oflate-stage (t = 3000 timesteps). All corresponding microstructures are comparedat similar timestep and follow system S W . . . . . . . . . . . . . . . . . . . . . 405.11 Microstructures corresponding to left column is for particle radius of 8 units andright column is for particle radius of 16 units for the same volume fraction ( )of particles.The top picture is from some early stage (t = 50 time steps), middleone is of intermediate stage (t = 500 time steps) and bottom one is for late-stage(t = 3000 timesteps). All corresponding microstructures are compared at similartimestep and follow system S S . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.12 Microstructures corresponding to left column is for particle radius of 8 unitsand right column is for particle radius of 16 units for the same volume fraction( ) 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 oflate-stage (t = 3000 timesteps). All corresponding microstructures are comparedat similar timestep and follow system S S . . . . . . . . . . . . . . . . . . . . . 435.13 Microstructures( A B ) corresponding to left column is for particle radius of8 units and right column is for particle radius of 16 units for the same volumefraction ( ) of particles.The top picture is from some early stage (t = 100 timesteps), middle one is of intermediate stage (t = 500 time steps) and bottom oneis for late-stage (t = 3000 timesteps). All corresponding microstructures arecompared at similar timestep and follow system S S . . . . . . . . . . . . . . . . 485.14 Microstructures ( A B ) corresponding to left column is for particle radius of8 units and right column is for particle radius of 16 units for the same volumefraction ( ) of particles.The top picture is of some early stage (t = 100 timesteps), middle one is of intermediate stage (t = 500 time steps) and bottom oneis of late-stage (t = 3000 timesteps). All corresponding microstructures arecompared at similar timestep and follow system S S . . . . . . . . . . . . . . . . 495.15 Microstructures( A B ) corresponding to left column is for particle radius of8 units and right column is for particle radius of 16 units for the same volumefraction ( ) of particles.The top picture is from some early stage (t = 100 timesteps), middle one is of intermediate stage (t = 500 time steps) and bottom oneis for late-stage (t = 3000 timesteps). All corresponding microstructures arecompared at similar timestep and follow system S S . . . . . . . . . . . . . . . . 51IST OF FIGURES viii5.16 Microstructures ( A B ) corresponding to left column is for particle radius of8 units and right column is for particle radius of 16 units for the same volumefraction ( ) of particles.The top picture is of some early stage (t = 100 timesteps), middle one is of intermediate stage (t = 500 time steps) and bottom oneis of late-stage (t = 3000 timesteps). All corresponding microstructures arecompared at similar timestep and follow system S S . . . . . . . . . . . . . . . . 52 ist of Tables κ . . . . . . . . . . . . . . . . . . . . . . . 234.3 Values of all simulation Variables . . . . . . . . . . . . . . . . . . . . . . . . . 245.1 Gradient energy parameters and corresponding interfcial energies . . . . . . . . 31ix hapter 1Introduction Consider a binary AB, either a polymer blend or a fluid mixture, which is homogeneous at hightemperature. If this unstable or metastable mixture is quenched below the co-existing curve,it will thrust into A-rich and B-rich domains. This phenomena is called phase separation. Inaddition, it is also possible that same AB binary can phase separate in presence of a surfacewith a preferential attraction to one of the components. This is called preferential wetting andit results into a partially or completely wetted surface by the preferred component. Thus, theinterplay of these two kinetic processes, phase separation and wetting, produce a great richnessof microstructures. This class of microstructures are of great technological, experimental andtheoretical importance.Polymer materials are hardly ever used in their pure form in applications [1].They are of-ten filled with solid additives which dramatically improve the mechanical, thermal and interfacialproperties of the material relative to the pure polymer blend [2]. For example, rubber particle in-creases toughness, carbon black/flake/tube improves conductivity and processibility, silica/glassbeads or fibers enhance modulus and strength, clay sheets modify the heat resistance of thematrix [2] etc. Moreover, wetting induced phase separation can lead us to gain a novel com-posite structure of alternating domains of polymeric and metallic materials [2]. In addition,fixed particles in a matrix induces a pinning effect and thus dictates the final domain size and1distribution (bicontinuous or isolated) [3]. All these ideas could be applied to physical de-sign of multilayer composites including polymer blends and polymer-dispersed liquid crystaldisplays [3], thin films [4], super conductors [5], shape memory alloys [5] and even nanotubes [6].While phase separation in binary systems have been studied extensively, little is known aboutphase separation in ternary systems [7, 8, 4]. When, along with phase separation another kineticprocess, preferential wetting, comes into picture then the problem becomes complex. Moreover,quantitative simulation of such situations need to consider melt condition thermodynamics (forpolymer mixture) or hydrodynamics (for fluid mixture). This complicates the problem evenfurther. However, few studies have shed some light on wetting induced phase separation wherewetting surface is provided by stationary wall [9, 10], substrate [9], spherical particle [11],network [12], pattern substrate [9] etc. Majority of research in this direction employed a Cahn-Hilliard-Cook approach to simulate phase separation of a critical mixture, via Ginzburg-Landaufunctional in conjunction with a velocity field to the surface. Phase separation of a off-criticalbinary mixture in presence of a surface also has not received much attention except only byfew [9, 13, 14, 10]. Hence, coupled wetting and phase separation of a critical/off-critical binarymixture in presence of a immobile spherical (symmetric to both components) particle can still beconsidered a new problem.Our objective is to investigate the phase separation behavior of a critical/off-critical binarymixture in presence of fixed particles of variable size and density. For this purpose, we adopt aternary spinodal model developed by Bhattacharya [15] and modified it to simulate such situation.This model is based on Cahn-Hilliard formalism [16] where bulk free energy (regular solutionmodel) is supplemented with a gradient squared term (gradient energy). We vary the pairwiseinteraction energy and gradient energy parameters to incorporate preferential attraction (wetting)between the particles and one of the components of the binary mixture. Simulations are carriedout by semi-implicit numerical integration of time-dependent Cahn-Hilliard equation on a 512 square lattice in Fourier space, subject to periodic boundary conditions in both x and y directions.The present report is organized as follows. In chapter 2, we review the available experimen-tal and theoretical literatures related to our study. Chapter 3 contains detailed description ofthe numerical model and other supplementary numerical calculations regarding ternary phaseseparation. Chapter 4 deals with the background details of our simulation including parameterestimations and the modifications adopted to incorporate immobile filler particles in a modifiedCahn-Hilliard model. In chapter 5, we present the results and discuss the possible mechanismsof the microstructural features. Finally, we conclude with our findings in chapter 6. hapter 2Literature Review In this chapter we review the experimental and numerical studies relevant to the phase separationof a binary mixture in presence of third component. Third component may be present in thematrix as filler, particle, tube, network, wall etc. and behaves as mobile or immobile. It canbe scaled as microscopic or nanoscopic. It may even induce preferential wetting to one of thecomponents of the binary mixture. Moreover, the binary mixture can be comprising of a polymerblend or a fluid mixture. All this variables leads to a multiplicity of interesting microstructures.Thus, a symbiotic interaction between experimental, theoretical and numerical studies are veryimportant for development in this class.
While phase separation in binary mixture has been studied extensively for the past two decades,available literature is few on the same in presence of solid particles. Amongst them most of thepaper reports about the phase-separation morphology of binary mixture having critical composi-tion. So, off-critical phase separation behavior has received very less attention so far.The first related experimental study was conducted by Jones et. al. [9]. They studied the spinodaldecomposition behavior of a critical binary polymer mixture of poly(ethylene-propylene) (PEP)and perdeuterated-PEP (dPEP) in a presence of surface which preferentially attracts dPEP. They4.1 Experimental Studies 5reported the origin of concentration waves from the surface. Another connected study is found inBruder et. al. [9] and Straub et. al. [9]. They have used critical mixture of deutarated polystyrene(dPS) and brominated polystyrene (PBr x S). In this situation preferential wetting resulted in ametastable CW morphology, which finally decomposed into a PW morphology.Tamai, Tran-cong and others [11] also experimented using PS/PVME with a crosslink-ableside group styrene-chloromethyl styrene random copolymer (PSCMS). The morphology exhibitsa ring pattern of alternate layers of preferred and non-preferred phases. A similar situation wasstudied by Ermi, Karim and others [11]. Karim et. al. [1] examined with PS/PVME polymerblend with immobile macroscopic silica beads with preferential interaction with PS. Duringintermediate stage, AFM images show a enriched PS composition layer about the filler. Thispattern is also ring alike, which is termed as "target pattern". Breaking of the translational phaseseparation symmetry by the filler particles is the reason reported for such circular compositionwaves. Moreover, these "target patterns" are transient in nature and disintegrate at late times.The authors also shed some light on slowing down of the phase separation process due to theinterfacial segregation of particles.To the best of our knowledge, Tanaka’s observations [17] are most closely related with ourresults. Binary fluid mixtures was comprised of oligomers of styrene (OS) and ε -caprolactone(OCL) and spherical macroscopic glass particles are used as spacer as well as immobile filler.The glass particles were sandwitched between two glass plates so that particles become essen-tially immobile due to the large friction against the glass plates. As a result during morphologyevolution the more preferable OCL-rich phase forms domains around the glass particles and theglass particles, which are close enough, are essentially bridged by it. He also mentioned thatcoarsening of droplets completely stops due to pinning of the same by the fixed glass particles.He also conducted same experiment with mobile particles. This also concluded with similarresults including bridge pattern formation and spontaneous pinning by mobile particles [3].Another related experiment was performed by Benderly et. al. [18].The system comprised binary.2 Theoretical Studies 6mixture (off-critical) of polypropylene (PP) and polyamide-6 (PA -6) along with macroscopicglass beads/fibers as filler with preferential attraction to PP. SEM images show an encapsulatedmorphology when PP is the minor phase and there was no encapsulation reported when PPis major phase. Benderly et. al. [18] have done another set of experiment with PP/PC/glassblend. They predicted an encapsulated morphology about PC. But, they observed a differentmorphology (separately dispersed) and they blamed the kinetic factors like viscosity etc., for thekinetic hindrance to encapsulation. Majority of the related theoretical modelling literature deals with the phase separation behaviorof a critical binary mixture with presence of solid additives with preferential attraction to one ofthe components of that mixture. However, available literature (known to us) regarding off-criticalphase separation is very few [14, 10, 12, 19]. Filler surfaces are introduced in the modelsas stationary wall [9, 10], substrate [9], sphere [11], network [12] etc. Various groups havesimulated with patterned substrates like checkerboard or strip pattern so that different regionshave different interaction with the third component [9].The experiment of Karim et. al. [1] was motivated by theoretical study of Lee et. al. [11].Scope of our simulation seems to be most closely related with the same. They adapted a Cahn-Hilliard-Cook (CHC) model and the free energy functional considered was Ginzburg-Landau(GL) form subject to an initial condition of random thermal noise (white noise). The put a isolatedspherical immobile particle in the center of 128 matrix, and allowed the A/B critical binary tophase separate with the constraints of variable polymer-filler interaction (controlled by parameterh). In case of strong interaction the early stage microstructures suggest a concentric ring patternof alternate layers of wetting phase and non-wetting phase, whereas neutral interaction betweenpolymer-filler does not produce any "target pattern". Moreover, the authors reported that suchfiller induced composition wave is transient and breaks up when the background spinodal patterncoarsens to a scale larger than filler particle. They also accounted an off-critical composition and.2 Theoretical Studies 7claimed that if minor phase is the preferred phase then an encapsulated layer forms around fillerparticle though it can not be considered as "target pattern".A recent simulation study by Hore et. al. [20] claimed that nanoparticles segregate at theinterfaces if the mutual pairwise interaction parameters are such that χ AB > | χ BC − χ AC | . Theirsimulation was motivated by an experiment of Chung et. al. [Nano Lett. 5, 1878 (2005)]. Themodel was based on Dissipative particle dynamics (DPD) and a rigid body dynamics was em-ployed to impart the velocity field to the spherical nanoparticles. Moreover, they reported thatdomain growth of a A/B binary decreases if the volume fraction of the particles increases andradius of the same decreases.A series of simulations by Balazs et. al. [21] shed light on this situation [22, 23]. Theyfocus on both issues of fixed particle [22, 23] and mobile particle [21] which was subject toselective interaction to one of the components in a critical binary mixture. The authors usedCahn-Hilliard type approach, where free energy functional was considered as GL form alongwith a coupling contribution to incorporate the interaction between particles and the order pa-rameter field. Navier-Stokes equation or Langevin dynamics were employed to incorporatehydrodynamic effects or particle mobility. All these simulations reported late stage morphologywhen particles sit at the interface and act as an obstacle to interface motion resulting a distributionof non-wettable phase as isolated islands in a continuous sea of wettable phase.Chakrabarti [12], Brown [10, 24] have also used similar models to probe the surface directedspinodal decomposition [24] and surface directed nucleation [12, 10]. They modeled off-criticalbulk phase separation by a CHC approach via GL functional in conjuction with a surface potentialterm or long range interaction term. They used fumed silica network [12] or block coplymerof strip pattern [10] to study the phase separation. Following them, if major component wetsthe surface then minor droplets nucleates near the surface before they nucleates in bulk. Theyreported a late time arrested growth of the wetting layer in case of minor component wets thenetwork. Another similar numerical study is due to puri and Binder [14]. They simulated bulk.2 Theoretical Studies 8off-critical A/B phase separation with preferential attraction to a stationary surface. Followingthem, if minor component wets the surface, droplets of wetting components form whereas incase of major component wets the network non-wetting component form the droplets. They alsoreported a enriched layer of preferred component at the surface, in case of minor componentwets the surface.Finally, we will conclude this chapter with a beautiful review paper of Nauman and He [4]. Thisreview discusses their contribution to this stimulating field in pedagogical framework. Here,the authors talk about the non-ideal diffusion in small and large binary/ternary systems. Theyconsidered enthalpic and entropic contribution to the gradient energy parameter, and supplied apre-gradient mole fraction in flux equation. All these result a modified Cahn-Hilliard equationwhich leads to a better agreement with experiments. According to them, white noise (initialcondition of random noise) can be considered as reasonable estimate to thermal noise and magni-tude of such noise does not have any significant effect in middle to late stage phase separation.We will reproduce some of the related microstructures from this review to establish qualitativeagreement with our results.Figure 2.1: Simulated (a) and experimental (b) core shell morphoology (Reproduced from [4]).2 Theoretical Studies 9Figure 2.2: Simulated (a) and experimental (b) droplet morphology (Reproduced from [4,12])(surface directed nucleation)Figure 2.3: Simulated (a) and experimental (b) core/shell morphology with continuous shell(Reproduced from [4])Figure 2.4: Simulated (a) and experimental (b) ring pattern (reproduced from [11])(surfacedirected spinodal decomposition) hapter 3Formulation In this chapter we deal with the formulation for the dynamics of the spinodal decomposition internary alloys. At first we derive a free energy functional by the local free energy expressionfrom regular solution model and the inhomogeneous free energy expression from Cahn-Hilliardmodel. Then sequentially with the help of continuity equation, Fourier transform and finitedifference approach we obtain the kinetic equations which governs the temporal evolution ofcomposition field. We adopted the similar numerical model practiced by Bhattacharya [15].
We consider a ternary alloy system consisting of three different species A,B and C.Let c i ( r , t ) for i = A, B, C represent the mole fraction of the ’i’th component as a function of position r andtime t . Since c i is the mole fraction we have the following conditions: ∑ i = A , B , C c i ( r , t ) = f ( c A , c B , c C ) is given by f ( c A , c B , c C ) = ∑ i (cid:54) = j χ i j c i c j + ∑ i c i lnc i (3.2)10.2 Flory–Huggins Model 11where i,j = A,B,C and χ i j = χ ji is the effective interaction energy between components i and j.Here we considered only pair–wise interactions or nearest neighbour approximation between theatoms. χ AB = Z [ E AB − E AA − E BB ] K B T (3.3)where E AB , E AA , E BB are the bond energies between A/B,A/A and B/B bonds respectively, Z isthe number of bonds per atom, K B is the Boltzmann constant and T is the absolute temperature.If χ AB > χ AB > The Flory–Huggins model for polymer solutions is a close relative of the regular solution model.According to this model in case of an incompressible A - B -C polymer blends ( φ A + φ B + φ C = 1) : fKT = φ A ln φ A N A + φ B ln φ B N B + φ C ln φ C N C + χ AB φ A φ B + χ BC φ B φ C + χ AC φ A φ C (3.4)f is the free energy of mixing of the ternary solution. φ A , φ B , φ C are volume fractions and N A , N B , N C are degree of polymerization of components A, B, C respectively. Degree of polymerizationdepends on polymer chain length and the number of chain segments ( mer content). If it isassumed that N A = N B = N C =
1, then this model resembles to regular solution model. Similarto regular solution model χ AB , χ BC , χ AC are the effective interaction energy between A–B, B–C,A–C binaries. The Cahn-Hilliard model adds a correction to the homogeneous free energy function to accountfor spatial inhomogeneity.This correction comes from a Taylor expansion of f ( c a , c b , c c ) inpowers of ∇ c combined with symmetry considerations. While composition in a homogeneoussystem is scalar, composition becomes a field for an inhomogeneous system.Thus the total free.4 Chemical Potential 12energy becomes a functional of the compositional field[[16] given by: F = N v (cid:90) V (cid:34) f ( c A , c B , c C ) + ∑ i = A , B , C κ i ( ∇ c i ) (cid:35) dV (3.5)where f is the homogeneous free energy,and N V is the number of sites per unit volume. κ iscalled the gradient energy co-efficient. κ i ( ∇ c i ) is the gradient energy which, is the first ordercorrection for inhomogeneity, introduces a penalty for sharp gradients and make the interface adiffuse one. Volume fraction also can be treated similarly, instead of composition, as a conservedphase field variables to simulate a polymer solution. According to Onsager relations, the flux of the i element, J i , is proportional to the gradient of thechemical potential J i ( x , t ) = − M i ∇ µ i ( x , t ) (3.6)where M i is the onsager coefficient (mobility of i th component) and is always positive. Sincemobility is isotropic for cubic materials it may be replaced by a scalar instead of second rankproperty tensor as in the previous equation.In constructing the kinetic equation of a substitutional alloy undergoing diffusion we adoptthe results of Kramer et al. [25], who proposed that there must be a net vacancy flux operatingduring the diffusion process with the constraint of local thermal equilibrium of vacancies every-where. Thus the net flux of component i , ¯J i , across a fixed lattice plane (not with respect to inertmobile markers) is the sum of the diffusion flux of A plus the A transported by the vacancy flux. ¯J i = J i + c i J V J V = − ( J A + J B + J C ) (3.7).4 Chemical Potential 13where J V is the vacancy flux. Combining equations 3.7 we get : ¯J i = J i − c i ∑ i = A , B , C ( J i ) (3.8)from eqn. 3.8 and eqn. 3.1, it is clear that ∑ i = A , B , C (cid:0) ¯J i (cid:1) = ¯J A = − ( − c A ) M A ∇ µ A + c A M B ∇ µ B + c A M C ∇ µ C ¯J B = − ( − c B ) M B ∇ µ B + c B M A ∇ µ A + c A M C ∇ µ C ¯J C = − ( − c C ) M C ∇ µ C + c C M A ∇ µ A + c A M B ∇ µ B (3.10)We need to solve for the only two compositional variables, say c A and c B as other compositioncan be directly computed by substraction of c A and c B from unity. Now we need to derive anexpression for ¯J i by Gibbs-Duhem equation: c A ∇ µ A + c B ∇ µ B + c C ∇ µ C = ∇ µ A = ( − c A ) ∇ µ e f fA − c B ∇ µ e f fB ∇ µ B = ( − c B ) ∇ µ e f fB − c A ∇ µ e f fA ∇ µ C = − c A ∇ µ e f fA − c B ∇ µ e f fB (3.12)w here ∇ µ e f fA = ∇ µ A − ∇ µ c & ∇ µ e f fB = ∇ µ B − ∇ µ C .4 Chemical Potential 14using the expressions for ∇ µ A , ∇ µ B , ∇ µ C in eqn. 3.10 and arraging we can write: ¯J A = − (cid:104) ( − c A ) + c A ( M B + M C ) (cid:105) ∇ µ e f fA + [ c B M A ( − c A ) + c A M B ( − c B ) − c A c B M C ] ∇ µ e f fB (3.13) ¯J B = − (cid:104) ( − c B ) + c B ( M A + M C ) (cid:105) ∇ µ e f fB + [ c A M B ( − c B ) + c B M A ( − c A ) − c A c B M C ] ∇ µ e f fB (3.14)Now let us define the effective mobilities: M AA = ( − c A ) M A + c A ( M B + M C ) M BB = ( − c B ) M B + c B ( M A + M C ) M AB = M BA = ( − c A ) c B M A + c A M B ( − c B ) − c A c B M C (3.15)Using the relations in eqn. 3.15, we can rewrite the flux equations 3.13 and 3.14 in a morecompact form : ¯J A = − M AA ∇ µ e f fA + M AB ∇ µ e f fB ¯J B = − M BB ∇ µ e f fB + M AB ∇ µ e f fA (3.16)Chemical potential in homogeneous system is proportional to the partial derivative of bulkfree energy: µ e f fA = ∂ f ( c A , c B ) ∂ c A (3.17)Chemical potential in a inhomogeneous system is away from global equilibrium. So if we assumelocal equilibrium, we can define this potential field by employing the calculus of variations: µ e f fi = δ F δ c i where , i = A , B (3.18).5 Evolution Equations 15The variational derivative may be found by applying the Euler-Lagrange equation: δ F δ c i = ∂ F ∂ c i − ∂∂ x ∂ F ∂ ∇ c i (3.19)So, we obtain the following equations: µ e f fA = ∂ f ∂ c A − ( κ A + κ C ) ∇ c A − κ C ∇ c B µ e f fB = ∂ f ∂ c B − ( κ B + κ C ) ∇ c B − κ C ∇ C A (3.20)where, ∂ f ∂ c A = ln c A − ln c C + ( χ AB − χ BC ) c B + χ AC ( c C − c A ) ∂ f ∂ c B = ln c B − ln c C + ( χ AB − χ AC ) c A + χ BC ( c C − c B ) (3.21) Since c is a conserved quantity, it obeys a conservative (continuity) law, which can be used toobtain the expressions for the temporal evolution of composition field: ∂ c i ∂ t = − ∇ · ¯J i (3.22)using equations 3.16 and 3.20 in equation 3.22 we get the following two independent kineticequations: ∂ c A ∂ t = M AA (cid:20) ∇ (cid:18) ∂ f ∂ c A (cid:19) − ( κ A + κ C ) ∇ c A − κ C ∇ c B (cid:21) − M AB (cid:20) ∇ (cid:18) ∂ f ∂ c B (cid:19) − ( κ B + κ C ) ∇ c B − κ C ∇ c A (cid:21) (3.23).6 Numerical Implementation 16 ∂ c B ∂ t = M BB (cid:20) ∇ (cid:18) ∂ f ∂ c B (cid:19) − ( κ B + κ C ) ∇ c B − κ C ∇ c A (cid:21) − M AB (cid:20) ∇ (cid:18) ∂ f ∂ c A (cid:19) − ( κ A + κ C ) ∇ c A − κ C ∇ c B (cid:21) (3.24)where, for the sake of convenience we denoted κ AA = κ A + κ C , κ BB = κ B + κ C & κ AB = κ BA = κ C and mobility is considered independent of composition field. We extended the 2-D semi-implicit Fourier spectral method to the ternary systems and obtaina sequence of time dependent ordinary differential equation in Fourier space. If we consider afunction g A = ∂ f ∂ c A and g B = ∂ f ∂ c B then the expression becomes: ∂ ˜ c A ( k , t ) ∂ t = M AA (cid:20) − k (cid:18) ∂ f ∂ c A (cid:19) k − κ AA k ˜ c A − κ AB k ˜ c B (cid:21) − M AB (cid:20) − k (cid:18) ∂ f ∂ c B (cid:19) k − κ AB k ˜ c A − κ BB k ˜ c B (cid:21) (3.25) ∂ ˜ c B ( k , t ) ∂ t = M BB (cid:20) − k (cid:18) ∂ f ∂ c B (cid:19) k − κ BB k ˜ c B − κ AB k ˜ c A (cid:21) − M AB (cid:20) − k (cid:18) ∂ f ∂ c A (cid:19) k − κ AB k ˜ c B − κ AA k ˜ c A (cid:21) (3.26)where k = ( k x , k y ) is the reciprocal lattice vector: k = | k | ; ˜ c A ( k , t ) and ˜ c B ( k , t ) are the Fourier trans-forms of the respective compositions in the real space. Using finite difference method for ∂ c A ∂ t and ∂ c B ∂ t we get the following equations: ∂ c i ∂ t = ˜ c i ( k , t + ∆ t ) − ˜ c i ( k , t ) ∆ t (3.27).6 Numerical Implementation 17where i = A,B. We treated the linear terms , ˜ c A and ˜ c B implicitly and the non-linear terms, ˜ g A ˜ g B are treated explicitly i.e. ˜ g i ( k , t + ∆ t ) = ˜ g i ( k , t ) . We solve the following equations iteratively toobtain the microstructures.˜ c A ( k , t + ∆ t ) = ˜ c A ( k , t ) − k ∆ t [ M AA ˜ g A ( k , t ) − M AB ˜ g B ( k , t )] − k ∆ t ˜ c B ( k , t ) [ M AA κ AB − M AB κ BB ] + ∆ tk ( M AA κ AA − M AB κ AB ) (3.28)˜ c B ( k , t + ∆ t ) = ˜ c B ( k , t ) − k ∆ t [ M BB ˜ g B ( k , t ) − M AB ˜ g A ( k , t )] − k ∆ t ˜ c A ( k , t ) [ M BB κ AB − M AB κ AA ] + ∆ tk ( M BB κ BB − M AB κ AB ) (3.29) hapter 4Simulation Details In this chapter we deal with the different variables or factors which influence the microstructureobtained by phase field simulations. First we discuss about loading of the particles, their sizedistribution, interparticle distance and distribution of particles within the matrix. Then, we definethree systems ( S O , S W , S S ) which includes the parameters like pair–wise interaction coefficient( χ ) and gradient energy coefficient ( κ ). Finally, how the composition of the phases are distributedin the matrix and what are the constant mobility values assigned to the components are described. Simulations are performed with two types of particle loading : 5% and 10%.
Shape of the Particles are circular in 2D and spherical in 3D. In 2D, we have considered circlesof two different radius : 8 unit and 16 unit. According to the volume fraction ( ≈ area fraction) ofthe particles, we calculated the numbers of circles of each radius (8 or 16) that can be arrangedin 512 ×
512 matrix. 18.1 Particle Characteristics 19Area fraction Radius (R) No. of particles5% 8 6516 1610% 8 13016 33Table 4.1: Particle size and number of particles λ ) To obtain a well dispersed spatial arrangement of the particles we assigned an interparticledistance of greater than 5 × R in cases of 5% loading and a distance of greater than 4 × R incases of 10% loading. In other words, in case of 5% loading domain size of the particle(diameter) is greater than interparticle distance and in case of 10% loading domain size ofparticle approximately resembles to the interparticle distance. particles are positioned at a interparticle distance in the matrix following periodic boundaryconditions. This trick helps us to minimize the surface effects and also to simulate the propertiesof a system more closely. In periodic boundary conditions the cubical simulation box is replicatedin all directions to form a infinite lattice. In the course of the simulation, when a molecule movesin the central box, its periodic image in every one of the other boxes moves with exactly the sameorientation in exactly the same way. Thus, as a molecule leaves the central box, one of its imageswill enter through the opposite face. There are no walls at the boundary of the central box, andthe system has no surface. The central box simply forms a convenient coordinate system formeasuring locations of the N molecules [26].Fig 4.1 is a 2D version of such a periodic system. As a particle moves through a boundary, all itscorresponding images move across their corresponding boundaries..2 Compositional Distribution 20Figure 4.1: Periodic boundary conditions
It seems to be computationally more expensive when we assign the compositions to the particlesof fixed radius (R), and also to matrix and simulate the system to evolve. So we used thefollowing trick to distributed the composition throughout the system. c p is the composition ofthe particles with is confined by the region of (R - dR), c m is composition of the matrix whichoccupies space at a distance greater than R + dR and compositions within the region in betweenthem follows as a straight line relationship (refer 4.1, 4.2). This approach gives us a c–rich phase(particle) of desired mean radius. All the simulations are performed keeping dR constant, 4 unit. c i ( r ) = c p r < R − dRstraight line R − dR < r < R + dRc m r > R + dRc m − c p dR = c − c p r − ( R − dR ) (4.1) c = c p + [ r − ( R − dR )] c m − c p dR (4.2).3 Mobility Matrix 21Figure 4.2: Radial composition profile from the center of the particles (schematic) In all our simulations particles are kept nearly immobile. Our objective is to show how theseimmobile particles affect the phase separation behaviour of the binary matrix phase. So,toachieve that we assign M c = M AA = ( − c A ) M A + c A M B M BB = ( − c B ) M B + c B M A M AB = c B ( − c A ) M A + c A ( − c B ) M B (4.3)we judiciously selected the values of c A and c B with the following constraints : • determinant of mobility matrix must be positive definite • it represents the matrix composition of an A–B binary mixture. • scaled mobilities of components A and B, M AA and M BB respectively, equals to one. • every particle in the matrix survives..4 Systems 22Thus, the mobility matrix becomes the following and used throughout all simulations. M = M AA M AB M AB M BB = . . .
98 1 . (4.4) Wettability and phase separation behavior between components depends on mutual interactionenergy( χ ) and interfacial energy ( κ ). we have used three distinct systems of different combinationof χ and κ . Interaction energy is introduced in the system in terms of pairwise interactionparameters of regular solution model and gradient energy is integrated in terms of gradientenergy coefficients of CH model. All this different variables will lead to three different ternaryisothermal phase diagram from where we utilized the equilibrium compositions of three phases α , β and γ . Following equation 3.3, regular solution interaction parameter (per mole) can be written as : χ = Ω / RT where, Ω = Z (cid:2) E i j − E ii − E j j (cid:3) χ is inversely proportional to critical temperature. So, instead of calculating a critical temperaturefor polymer solutions, we calculated χ crit , the critical value of interaction parameter at the onsetof miscibility gap. χ > χ crit = Ω / RT crit (4.6)It can be shown that the critical temperature (inflection point) in regular solution model is T crit = Ω / R (4.7)Combining equations 4.6 and 4.7 we can conclude that at a value of χ greater than 2.0 i/j binarywill phase separate..4 Systems 23Our objective was to preferentially wet the C rich particles by component A. So we intention-ally attributed high interaction parameter values to B/C interface to make the system reluctant toform B/C interface in order to minimize its Gibbs free energy. χ system- S O system- S W system- S S χ AB χ BC χ AC Our system is composed of A–rich and B–rich phases as a matrix and C-rich phases (particles)are dispersed in between the A/B binary phase. Our focus was to preferentially wet the particlesby one of the components (A). That’s why we we assign higher interfacial energy to B/C interfacein terms of larger gradient energy coefficient which eventually punishes the B–C interface moreand creates a more diffuse interface between them.From table 4.2, below, we can calculate three independent values of κ A , κ B , κ C for eachtype of system. Then we combine the corresponding values to form the computationally usedparameters – κ AB , κ BC , κ AC . κ system- S O system- S W system- S S κ AB = κ A + κ B κ BC = κ B + κ C κ AC = κ A + κ C κ .5 Synopsis of simulation parameters 24 Simulation parameters system- S O system- S W system- S S ∆ x ∆ y ∆ t ± . ± . ± . M AA M BB M AB κ A κ B κ C κ AA = κ A + κ C κ BB = κ B + κ C κ AB = κ BA = κ C χ AB χ BC χ AC c mA ) 0.45 0.45 0.45 c mB c mC c PA ) 0.04 0.037 0.035 c PB c PC (i) Put requisite number of circles of fixed radius (8 and 16 units) at certain inter-particledistance using periodic boundary conditions in a 512 matrix so that the area fractionoccupied by the circles equals to the intended volume fraction of particles (5% and 10%)..6 Computational Algorithm 25(ii) The mobility of the particles (circles in 2D) is attributed to nearly zero and mobility ofother two components are scaled accordingly.(iii) Initial composition profile within the particles is taken as the equilibrium composition ofC-rich phase from the isothermal section of ternary phase diagram and the composition ofA/B components is taken accordingly (refer section 4.3, 4.5). Now provide a compositionalfluctuation of 0.5% to the composition at each grid point of a 512 matrix.(iv) Compute g i in real space which is a function of c i & χ i j where i , j = A & B (v) Perform forward Fourier transform (using FFTW library [27]) to convert the real space c i ( r , t ) and g i ( r , t ) values to Fourier space values ˜ c i ( k , t ) and ˜ g i ( k , t ) (vi) Get the modified value of ˜ c i ( k , t + ∆ t ) and ˜ g i ( k , t + δ t ) after calculation of evolutionequations for c A and c B respectively in 2-D Fourier space.(vii) Return to real space after performing backward Fourier transform and scale the c i ( k , t ) values.(viii) Iterate steps 4 to 7 again and again upto desired timestep achieved and make sure that the˜ c i ( k , t ) and ˜ g i ( k , t ) values get modified after each timestep.(ix) After a certain time interval (timestep × stepsize) save the modified values of ˜ c i ( k , t ) and˜ g i ( k , t ) in a data file.(x) For ternary plotting, we used grey scale representation of gibbs triangle, which is shown inthe next page, is a four distinct region, center region darkest, top region darker, bottomright region dark and bottom left region white. Darker area signifies C-rich region, darkarea denotes B-rich region and white area constitutes of A-rich region. Using this approachwe are able to distinguish three distinct regions after plotting the modified compositionprofiles in GNUPLOT..6 Computational Algorithm 26Figure 4.3: Gray scale color map projected on Gibbs triangle hapter 5Results and Discussion In this chapter we present our results obtained after carrying out simulations. The present workis directed at understanding of : • The dynamics of phase separation of a binary mixture ( critical and off-critical composi-tions) in presence of immobile solid particles. • Influence of preferential wetting on phase evolution. • Morphology and domain growth characteristics of such systems. • Role of effective interaction energy and relative interfacial energy on the morphology. • How particle size, shape and density affect the phase behavior of underlying binarypolymeric pattern.We have studied the above mentioned aspects following three systems ( S O , S W , S S ) whichdiffers in terms of effective binary interaction parameter ( χ ), gradient energy parameter ( κ )etc. Each system is simulated for different particle radius and volume fractions. Compositionfields of the matrix and particles have considered from the ternary equilibrium phase diagram,described in section 5.1, corresponding to each system. In section 5.2, we show the relativeinferfacial energy between the existing phases and produce the equilibrium composition profiles.Microstructures are illustrated in section 5.3, 5.4.27.1 Ternary Phase Equillibria 28 We used three distinct systems (section 4.5) for better understanding of our objectives. Threesystems of different χ give rise to three distinct equilibrium phase diagrams. The phase diagramsconsist of single phase region (A–rich α , B–rich β , C–rich γ ), two phase region ( α + β , β + γ , α + γ ) and three phase region ( α + β + γ ). Equilibrium composition of α , β and γ inthe three systems are depicted below. In our simulation, composition field inside the parti-cle is taken as equilibrium composition of γ phase and the matrix composition is consideredas the composition of the point denoted as ’+’ in the following ternary isothermal phase diagrams.In case of system– S O ( χ AB = 2.5, χ BC = 3.5, χ AC = 3.5),( c α A , c α B , c α C ) = ( 0.767, 0.174, 0.059 )( c β A , c β B , c β C ) = (0.174, 0.767, 0.059)( c γ A , c γ B , c γ C ) = (0.04, 0.041, 0.919)In case of system– S W ( χ AB = 2.5, χ BC = 4.0, χ AC = 3.5),( c α A , c α B , c α C ) = ( 0.779, 0.168, 0.053)( c β A , c β B , c β C ) = ( 0.159, 0.807, 0.034)( c γ A , c γ B , c γ C ) = (0.037, 0.023, 0.94)In case of system– S S ( χ AB = 2.5, χ BC = 5.0, χ AC = 3.5),( c α A , c α B , c α C ) = ( 0.803, 0.154, 0.043)( c β A , c β B , c β C ) = ( 0.145, 0.843, 0.012)( c γ A , c γ B , c γ C ) = ( 0.035, 0.008, 0.957).1 Ternary Phase Equillibria 29Figure 5.1: Isothermal section of the phase diagram for system S O ( χ AB = 2.5, χ BC = 3.5, χ AC =3.5)(schematic)Figure 5.2: Isothermal section of the phase diagram for system S W ( χ AB = 2.5, χ BC = 4.0, χ AC =3.5)(schematic).2 Interphase Interfacial Energy 30Figure 5.3: Isothermal section of the phase diagram for system S S ( χ AB = 2.5, χ BC = 5.0, χ AC =3.5)(schematic) One of our objectives was to study the effect of relative interfacial energies between co-existingphases to the microstructural evolution. For that purpose we have used three different sets of κ A , κ B , κ C values ( system S O , system - S W , system - S S respectively). These combinations result inthree sets of interfacial energies corresponding to α − β , β − γ , α − γ interfaces.To calculate interfacial energy of an interface at equilibrium, for example α − β interface,a 1-D simulation is set up with one half of the system with α and other half with β . Then theCahn-Hilliard equations (eqns. 3.23, 3.24) are soved to steady state to get the equilibriumcomposition profiles which are shown in Fig. [5.4,5.5,5.6]. Relative interfcial energies betweenco-existing phases, calculated using Eqns. B.1 and B.2, are listed in Table 5.1..2 Interphase Interfacial Energy 31system κ A κ B κ C σ αβ σ β γ σ αγ S O S W S S α − β interface (b) β − γ interface (c) α − γ interface according to system S O variables.2 Interphase Interfacial Energy 32(a)(b)(c)Figure 5.5: Equilibrium composition profile across (a) α − β interface (b) β − γ interface (c) α − γ interface according to system S W variables.2 Interphase Interfacial Energy 33(a)(b)(c)Figure 5.6: Equilibrium composition profile across (a) α − β interface (b) β − γ interface (c) α − γ interface according to system S S variables.3 Microstructures in A B A B S O , S W , S S ), tabulated in section 4.5,which are designed judiciously to incorporate the effects of variable χ , κ etc. More precisely,systems are designed in such a way that incorporate no preference ( S O ), weak preference ( S W )and strong preference ( S S ) for the particles by one of the components (say, A). The ternarymicrostructures can be interpreted following the gray scale color map [Fig. 4.3], described insection 4.6. S O This subsection refers to, when system S O is attributed to the morphological evolution, for phaseseparation of a critical binary mixture, in presence of spherical particles. For the case volumefraction of particles is 5%, the corresponding figure is 5.7. Figure 5.8 is the case when volumefraction of the particles is 10%. We show three snapshots of each case, one from early stage, onefrom the intermediate stage and one from last stage. All other necessary details are provided atthe caption of each figure.According to parameters corresponding to system S O , components do not have any preferencefor the particles or in other words particles interacting symmetrically to both components.Microstructures corresponding to Fig. [ 5.7, 5.8] show that at early times both phases startappearing simultaneously which eventually leads to nearly complete phase separation at latetimes and a bi-continuous pattern is observed irrespective of particle size or volume fraction.Interfacial energy driven coarsening takes place after phase separation. Late times morphologydemonstrates a surplus of A at some places and surfeit of B at some other places around the.3 Microstructures in A B A B ) of particles.Thetop picture is from some early stage (t = 1500 time steps), middle one is of intermediate stage(t = 3000 time steps) and bottom one is for late-stage (t = 5000 timesteps). All correspondingmicrostructures are compared at similar timestep and follow system S O ..3 Microstructures in A B ) of particles.Thetop picture is of some early stage (t = 1500 time steps), middle one is of intermediate stage(t = 3000 time steps) and bottom one is of late-stage (t = 5000 timesteps). All correspondingmicrostructures are compared at similar timestep and follow system S O ..3 Microstructures in A B S W This subsection refers to, when system S W is attributed to the morphological evolution, for phaseseparation of a critical binary mixture, in presence of spherical particles. For the case volumefraction of particles is 5%, the corresponding figure is 5.9. Figure 5.10 is the case when volumefraction of the particles is 10%. We show three snapshots of each case, one from early stage, onefrom the intermediate stage and one from last stage. All other necessary details are provided atthe caption of each figure.According to the parameters corresponding to system S W , component A is weakly preferred tothe particle surface. Early stage to intermediate stage morphology illustrates a ring pattern whereparticles are completely wetted by A components. This can be referred as core-shell morphologywith particle as core and A (preferred component) as shell. As larger size particles are greaterdistance apart, it seems a thick layer of B forms at the surface of A components. However, forsmaller size particles (no. of particles are more to keep the volume fraction same) the process ofengulfing of A around the particles breaks the nucleating domains of B. Hence, for such case theconcentric circular layer of B is not apparent. Moreover, Fig. [ 5.9, 5.10] clearly demonstratethat phase separation is not complete at late times, and bi-continuous morphology is also notobserved. The late stage morphology consists of isolated B domains in a continuous sea of A.Coarsening prevails to these domains to reduce the interfacial energy. It is obvious from the latestage behavior, that the phase evolution and growth in A/B binary happens in such a fashion thatparticles appear at the interfaces. It is worthwhile to mention that for all the cases early stagecore-shell morphology breaks towards late stage, although in few cases (5.9, R = 16 units) suchmorphology remains but with a different shell, i.e., B, around the core particles..3 Microstructures in A B ) of particles.Thetop picture is from some early stage (t = 200 time steps), middle one is of intermediate stage(t = 500 time steps) and bottom one is for late-stage (t = 3000 timesteps). All correspondingmicrostructures are compared at similar timestep and follow system S W ..3 Microstructures in A B ) of particles.Thetop 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 = 3000 timesteps). All correspondingmicrostructures are compared at similar timestep and follow system S W ..3 Microstructures in A B S S This subsection refers to, when system S S is attributed to the morphological evolution, for phaseseparation of a critical binary mixture, in presence of spherical particles. For the case volumefraction of particles is 5%, the corresponding figure is 5.11. Figure 5.12 is the case whenvolume fraction of the particles is 10%. We show three snapshots of each case, one from earlystage, one from the intermediate stage and one from last stage. All other necessary details areprovided at the caption of each figure.According to the parameters corresponding to system S S , component A is strongly preferredto the particle surface. At early times, microstructure consists of concentric rings of A and Baround the particles. Such core-shell morphology survives longer due to higher interactionsbetween A and particles. Late times, interfacial energy driven coarsening dominates and particlestend to arrange at the interfaces between co-existing phases. Furthermore, the morphology atlate times almost resembles to that of system S W , where B components are distributed as isolatedislands in a continuous A phase, and characteristic domain size of B is smaller in case of systemswith higher particle density. In addition, here also early stage core-shell morphology with shellA breaks towards late stage, forming a different shell, i.e., B, around the core particles..3 Microstructures in A B ) of particles.Thetop picture is from some early stage (t = 50 time steps), middle one is of intermediate stage(t = 500 time steps) and bottom one is for late-stage (t = 3000 timesteps). All correspondingmicrostructures are compared at similar timestep and follow system S S ..3 Microstructures in A B ) of particles.Thetop 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 = 3000 timesteps). All correspondingmicrostructures are compared at similar timestep and follow system S S ..3 Microstructures in A B This subsection is an account to address the microstructures obtained when a binary mixture isquenched critically ( A B ). Here, we systematically study the effect of spherical solid particleson the phase evolution due to background spinodal decomposition of a critical binary mixture.When the particles are neutral to A and B (system S O ) then ring pattern does not form at earlytimes. Whereas, if the particles are preferably wetted by one of the components (system S W / S S ),ring pattern appears in the microstructure. The ring or alternate layers of A and B forms dueto the propagation of composition waves originated from the particle surface [11]. Preferentialinteraction between component A and particle develops a higher concentration of A near theparticle surface than in the bulk. So, there must be depletion of A somewhere, which is immediatenext to the A enriched layer away from particle surface. This explains the formation of alternatelayers of preferred and non-preferred components around the particle. In case of system S O , thesurface directed composition waves are composed of equally likely phases A/B. Hence, ringmorphology does not form and A/B domains appear irregularly in the matrix.The early to intermediate stage morphology in systems S W and S O can be referred as core-shell morphology. As component A is thermodynamically preferable to the particles, A formsthe shell with particles at core. At late times few core-shell structure still remains, but nowB (non-preferred component) serves as shell around particles. Such shell structure transitioncan be due to Gibbs Thomson effect [28]. Large radius of curvature of the A layer around theparticles causes a higher concentration gradient of A in the shell relative to the matrix. Such con-centration gradient expels A from the particle surface, allowing the B to form shell about particles.During intermediate to late times the particle density at the interface increases. Considerthe two case: (a) a particle is completely enveloped by a preferred component (say, A) and (b)it resides at the interface between co-existing phases (A/B). Second case is more concerned interms of lowering the interfacial energy of the system. Roughly speaking, when a particle staysin interface than in bulk, the system reduces its energy by an amount Π γ AB R N , where R N is theparticle radius and γ AB is the A/B interfacial energy [20]. At early stages the particles may be.3 Microstructures in A B χ ) between the components [18]. In case of system S W / S S , the values of energeticparameters ( χ and κ ) confirmed γ BC > γ AC and hence A is preferentially attracted to the C(particle) interface. However, at late times particles tend to sit at the interface, though there isa surplus of preferred component A at the interface. This mimics a situation like morphologytransition from complete wetting (CW) to partial wetting (PW). Now, to achieve such a partiallywet morphology, one of the following conditions has to be satisfied. γ AB > | γ BC − γ AC | (5.1) χ AB > | χ BC − χ AC | (5.2)If free energy can be described by a regular solution model, gradient energy coefficient is directlyrelated to pairwise interaction parameter [16] by the relation κ = χ a /
2, where a is the interac-tion distance. So, gradient energy and interaction energy both affects the wetting behavior of theparticles.Therefore, if both conditions [ 5.1, 5.2] are satisfied by imposed parameters, completewetting prevails.Another explanation related to the interfacial segregation of particles is found in literarure [18].Spreading co-efficient (S) is defined as S = γ BC − γ AC − γ AB . S > 0 yields CW morphology and S< 0 results into PW morphology. This parameter stems from the young equation 5.1 and equallyuseful to predict the thermodynamically preferred morphology. However, kinetic factors canprevent us to achieve that morphology. Viscosity, quench depth ( ( χ − χ c ) / χ c = ( T − T c ) / T c ) etc.are well-known kinetic factors. In our case for system S W / S S , the thermodynamically favoredmorphology is CW. However, we obtain a PW morphology. Higher viscosity and deeper quench.3 Microstructures in A B S W / S S ) also seems to develop this type of interlocking and encapsulation is impeded atlate times.In case of system S W / S S , the late time morphology demonstrates a single infinite domainof the wetting phase (A) and the B components are trapped as isolated islands in this domain [11].The process of engulfing of the preferred component around the particle and subsequent domaincoarsening effectively breaks the continuity of domain B. Further coarsening of B domains isinhibited as the particles behave as obstacles to the motion of interfaces. It is obvious from latestage microstructures that bi-continuous morphology does not form and hence, phase separationis also incomplete. To elucidate the effect of wetting on the slowing down of domain growth, weperformed one set of simulation where components have no preference for the filler (system S O ).At late times, this results into complete phase separation , assured by a bi-continuous pattern.Thus, effect of wetting on the slowing down of domain growth is confirmed.Particle size,shape and density also affects the dynamics of phase separation of a critical binarymixture. Comparing the microstructures obtained, it is qualitatively true that higher volumefraction and smaller radius of particles are more effective for slowing down the kinetics ofphase separation and domain growth. Homogeneous distribution of particles leads to effectiveincrease in the viscosity. As smaller particles are distributed more homogeneously comparedto bigger particles, viscosity increment is pronounced in case of smaller particles. Moreover,volume fraction is also directly related with the viscosity [20]. There are two competing factorsat late times: a) temporal decrease in interfacial tension and b) increase in viscosity. For smallerparticles second effect is more striking than the first one and slower kinetics is more pronounced.In case of bigger particles factor (a) dominates the other and hence, kinetics of phase separationand domain growth is faster compared to smaller particles..4 Off-symmetric alloys in system S S S S This section is meant for presenting the microstructures obtained when a binary alloy of off-critical composition undergoes phase separation in presence of spherical particles. Particlecharacteristics are similar to the same used in case of binary critical mixture. Our objective wasto probe the microstructures in the following two cases: • minor component wets the particle surface • major component wets the particle surfaceThat’s why we simulated microstructures for two alloys A B and A B . As system S S is bestsuited to fulfill our purpose, we only run the simulation for the case of system S S parametersenforced on the morphological evolution of off-critical binary alloys. A B In this case minor component (A) wets the particle. For the case volume fraction of particlesis 5%, the corresponding figure is 5.13. Figure 5.14 is the case when volume fraction of theparticles is 10%. We show three snapshots of each case, one from early stage, one from theintermediate stage and one from last stage. All other necessary details are provided at the captionof each figure.Selective wetting interaction between A and the particles causes the preferred phase to forma dense enriched/encapsulated layer around the particles. This can be referred as wetting inducedprimary phase separation which happens at early limes. This causes a composition partitioningin the matrix and a depleted domain of preferred phase results in the particle free region. Arich phase appears in that region at intermediate times due to secondary phase separation. Atlate times, most of particles are bridged by a interconnected narrow pathways of A rich phase.However, system with higher particle radius (16) does not show this bridge structure. Instead forthe case of smaller volume fraction, a strictured interconnected network of A rich phase formssurrounding all the particles in a particular particle dense region. For higher volume fraction, thelate times morphology appears to be as irregular domains of preferred phase dispersed in thematrix and there also a thin wetting layer of minor phase survives about the particle..4 Off-symmetric alloys in system S S A B ) corresponding to left column is for particle radius of 8units and right column is for particle radius of 16 units for the same volume fraction ( )of particles.The top picture is from some early stage (t = 100 time steps), middle one is ofintermediate stage (t = 500 time steps) and bottom one is for late-stage (t = 3000 timesteps). Allcorresponding microstructures are compared at similar timestep and follow system S S ..4 Off-symmetric alloys in system S S A B ) corresponding to left column is for particle radius of 8units and right column is for particle radius of 16 units for the same volume fraction ( ) ofparticles.The top picture is of some early stage (t = 100 time steps), middle one is of intermediatestage (t = 500 time steps) and bottom one is of late-stage (t = 3000 timesteps). All correspondingmicrostructures are compared at similar timestep and follow system S S ..4 Off-symmetric alloys in system S S A B In this case major component (A) wets the particle. For the case volume fraction of particlesis 5%, the corresponding figure is 5.15. Figure 5.16 is the case when volume fraction of theparticles is 10%. We show three snapshots of each case, one from early stage, one from theintermediate stage and one from last stage. All other necessary details are provided at the captionof each figure.There is no encapsulation layer found about the preferred phase (A). Composition partitioningin the bulk causes a moderately depleted region of A around the wetting layer of particles. In thisregion B rich (minor) phase nucleates. Sometimes this is referred as "surface induced nucleation".This nucleation further propagates into bulk and B rich phase nucleates in there at still latertimes. At intermediate to late times the minor phase appears as nearly circular droplets, whichare predominantly distributed near to the particle. Smaller radius and higher loading of particlesseems to freeze the domains faster at late times..4 Off-symmetric alloys in system S S A B ) corresponding to left column is for particle radius of 8units and right column is for particle radius of 16 units for the same volume fraction ( )of particles.The top picture is from some early stage (t = 100 time steps), middle one is ofintermediate stage (t = 500 time steps) and bottom one is for late-stage (t = 3000 timesteps). Allcorresponding microstructures are compared at similar timestep and follow system S S ..4 Off-symmetric alloys in system S S A B ) corresponding to left column is for particle radius of 8units and right column is for particle radius of 16 units for the same volume fraction ( ) ofparticles.The top picture is of some early stage (t = 100 time steps), middle one is of intermediatestage (t = 500 time steps) and bottom one is of late-stage (t = 3000 timesteps). All correspondingmicrostructures are compared at similar timestep and follow system S S ..4 Off-symmetric alloys in system S S This subsection is an account to address the microstructures obtained when a binary mixture isquenched off-critically ( A B & A B ). More wettable phase is minor phase [Fig. 5.13, 5.14]: system S S parameters induce aselective interaction of component A and the particles. This eventually leads to a primary phaseseparation resulting a formation of a uniform A-enriched layer around the particle. Growthof the layer occurs due to the flow of component A from the bulk to the wetting layer. Thisexplains the formation of A-depleted region in the bulk. This is the region where secondary phaseseparation takes place and a percolating interconnected network of the preferred phase evolves.This morphology is completely different from the case of critically quench microstructures wherea ring pattern of alternate layers of A and B forms around the particles or in other words incase of off-critically quench microstructures there is no transient concentric composition wavesaround the particle [11].At intermediate to late times, the wetting layer seems to feed the growing A-rich domain inthe bulk. At late times, the domain size seems to be comparable to the smaller particle (radius= 8) and majority of the particles are located within the interconnected narrow pathways ofA-rich phase. Thus, at late times morphology appears to be a bridge pattern [17]. The immersedparticles in the bridge act as an obstacle to interface motion and stabilize the percolated A- richdomain size. In case of higher density of smaller particles, the bridge accommodates all theparticles within it and accordingly domain size is scaled by the immersed particles. In case ofbigger particles (radius = 16), the final domain size is not sufficient thick to hold the particleswithin it. So, A-rich phase become strictured in shape and envelopes a whole region of moderateparticle density at every possible location. Thus, at late times more A-B contacts form and systemenergy is minimized. In case of higher density of larger particles (10%), particles effectivelycut the strictured envelope of A-rich phase, yielding disperse domains of A-rich phase. Here,at late times A-B and A-C contacts are predominant and hence energetically favorable. It iswell known that off-critical mixture is supposed to produce a droplet morphology of the minorphase. However, we find a interconnected morphology of the minor phase which is preferredto the particle surface. This phenomena is sometimes referred as inverse percolation to droplet.4 Off-symmetric alloys in system S S More wettable phase is major phase [Fig. 5.15, 5.16]:
Droplet morphology evolution canbe explained as follows. There seems to be double phase separation in this case also. Primaryphase separation occurs near to the particle surface at early times and secondary phase separationoccurs in the bulk at late stages.Interphase boundary is a potent site for heterogeneous nucleation [28]. Preferential wettingof component A by the particles causes a lowering of contact angle. The key to reduction ofthe nucleation barrier is the smaller contact angle. Thus, there is a reduction in local nucleationbarrier near the wetted particles. This causes nucleation of minor droplets. This can be referredas "surface induced nucleation" [13] and it is the primary one. An interesting explanation ofthis phenomenon is reported in the paper of Brown and Chakrabarti [10]. The surface inducedselective interaction allows the A rich phase to wet around it and expels the B-rich phase to ashorter distance from the wall. This mass transport decreases the nucleation barrier of the Bphase because now it is easier to form minor nucleate on a low energy A-B interface.The optimum shape of the droplets are "two abutted spherical caps" which minimizes thetotal interfacial energy [28]. As the already formed B droplets grow, composition partitioningoccurs in the bulk. Moreover, driving force for nucleation in the bulk, which is far away fromwetted particle surface, is very less. That’s why kinetics of phase separation in bulk is smallerand the system takes intermediate to late times to form minor droplets in there. At late timesthe droplets undergo "diffusive coalescence" and attain a nearly spherical shape to reduce theinterfacial energy. It is worthy to mention that higher density of smaller particles produce a finestdroplet morphology. At late times particles act as an obstacle to interface motion and restricts thegrowth of domain and droplets. The effect of particle density, particle loading, interfacial effectsetc. on the morphology is already described in case of critically quenched microstructres[5.3.4]and same explanation can be applied to off-critical case also. hapter 6Summary and Conclusions
We have computationally investigated the surface induced phase separation of a binary A:Bmixture of critical and off-critical compositions. In our case, surface is provided by sphericalparticles fixed to the substrate and it exhibits a preferential attraction to one of the componentsof the binary A:B. To probe this systematically, simulations were carried out following threecases with zero selective interaction, weak selective interaction and strong selective interactionbetween the particle and preferred component. Off-critical phase separation behavior is studiedwith strongest selective interaction case only.
Symmetric alloy : A B Off-symmetric alloys : A B , A B ppendix AThermodynamics of Ternary System In case of a heterogeneous ternary system (multicomponent and multiphase) there are threecomponents say A, B, C and A–rich, B–rich and C– rich phase constitutes the α , β and γ phases respectively. Ternary phase equilibrium is represented as isothermal sections at constantpressure. At equilibrium chemical potential of the components in the existing phases becomeequal. So, three phase equilibrium in a ternary isothermal phase diagram can be calculated fromthe following relationships [29] : µ α A = µ β A = µ γ A µ α B = µ β B = µ γ B µ α C = µ β C = µ γ C (A.1)the above relationships give six individual equations which are as follows : µ α A = µ β A (A.2) µ α A = µ γ A (A.3) µ α B = µ β B (A.4) µ α B = µ γ B (A.5)578 µ α C = µ β C (A.6) µ α C = µ γ C (A.7)Now, in a system comprising three components the chemical potential of each components canbe formulated as follows [29] : µ A = f − c B ∂ f ∂ c B − c C ∂ f ∂ c C µ B = f + ( − c B ) ∂ f ∂ c B − c C ∂ f ∂ c C µ C = f − c B ∂ f ∂ c B + ( − c C ) ∂ f ∂ c C (A.8)Using the above formalism we can calculate µ α A and µ β A — µ α A = f α − c α B ∂ f α ∂ c α B − c α C ∂ f α ∂ c α C µ β A = f β − c β B ∂ f β ∂ c β B − c β C ∂ f β ∂ c β C (A.9)With the help of A.9, equation A.2 can be expanded as :ln c α A − ln c β A + (cid:16) c α B c α C − c β B c β C (cid:17) ( χ AB − χ BC + χ AC ) + (cid:16) c α B − c β B (cid:17) χ AB + (cid:16) c α C − c β C (cid:17) χ AC = µ γ A , µ α B , µ β B , µ γ B , µ α C , µ β C , µ γ C and equations A.3, A.4, A.5, A.6, A.7becomesln c α A − ln c γ A + (cid:0) c α B c α C − c γ B c γ C (cid:1) ( χ AB − χ BC + χ AC ) + (cid:16) c α B − c γ B (cid:17) χ AB + (cid:16) c α C − c γ C (cid:17) χ AC = c α B − ln c β B + (cid:16) c α B c α C − c β B c β C (cid:17) ( χ AB − χ BC + χ AC ) + χ AB (cid:104)(cid:16) c α A − c β A (cid:17) − (cid:16) c α B − c β B (cid:17) + (cid:16) c α B − c β B (cid:17)(cid:105) + (cid:16) c α C − c β C (cid:17) ( χ BC − χ AC ) + (cid:16) c α C − c β C (cid:17) χ AC = c α B − ln c γ B + (cid:0) c α B c α C − c γ B c γ C (cid:1) ( χ AB − χ BC + χ AC ) + χ AB (cid:104)(cid:0) c α A − c γ A (cid:1) − (cid:0) c α B − c γ B (cid:1) + (cid:16) c α B − c γ B (cid:17)(cid:105) + (cid:0) c α C − c γ C (cid:1) ( χ BC − χ AC ) + (cid:16) c α C − c γ C (cid:17) χ AC = c α C − ln c β C + (cid:16) c α B c α C − c β B c β C (cid:17) ( χ AB − χ BC + χ AC ) + χ AC (cid:104)(cid:16) c α A − c β A (cid:17) − (cid:16) c α C − c β C (cid:17) + (cid:16) c α C − c β C (cid:17)(cid:105) + (cid:16) c α B − c β B (cid:17) ( χ BC − χ AB ) + (cid:16) c α B − c β B (cid:17) χ AB = c α C − ln c γ C + (cid:0) c α B c α C − c γ B c γ C (cid:1) ( χ AB − χ BC + χ AC ) + χ AB (cid:104)(cid:0) c α A − c γ A (cid:1) − (cid:0) c α C − c γ C (cid:1) + (cid:16) c α C − c γ C (cid:17)(cid:105) + (cid:0) c α B − c γ B (cid:1) ( χ BC − χ AB ) + (cid:16) c α B − c γ B (cid:17) χ AB = c α A + c α B + c α C = c β A + c β B + c β C = c γ A + c γ B + c γ C = ppendix BInterfacial Energy Determination Interfacial energy between two co-existing phases, for example α – β , can be calculated followingCahn-Hilliard (CH) formalism [16]. If a flat one-dimensional system is considered, derivativesof second or higher order are neglected and the cross-sectional area is considered as unity thenthe equation 3.5 reduces to the specific interfacial energy ( σ ) which is given by the followingequation : σ αβ = N v (cid:90) ∞ − ∞ (cid:34) ∆ f ( c i ) + ∑ i = A , B , C κ i ( ∇ c i ) (cid:35) dx (B.1)where ∆ f ( c ) = f ( c i ) − ∑ i µ α / β i i = A , B , C (B.2) µ αβ i is the chemical potential of the species ’i’ in phase α or β as at equilibrium chemicalpotential of a given component is the same in both phases. For a system comprising three phases α , β , γ – σ β γ , σ αγ also can be calculated by similar fashion described above.60 ibliography [1] A. Karim, J. F. Douglas, G. Nisato, D. Liu, and E. J. Amis. Transient target patterns inphase separating filled polymer blends. Macromolecules , 32:5917–5924, 1999.[2] A. C. Balazs. Interactions of nanoscopic particles with phase separating polymeric mixtures.
Current Opinion in Colloid and Interface Science , 4:443–448, 2000.[3] H. Tanaka. pattern evolution caused by dyanamic coupling between wetting and phaseseparation in binary liquid mixture containing glass particles.
Phys. Rev. Letters. , 72:2581–2584, 1994.[4] E.B. Nauman and D.Q. He. Nonlinear diffusion and phase separation.
Chemical Engineer-ing Science , 56:1999–2018, 2001.[5] The kirkendall effect. .[6] H. J. Fan, M. Knez, R. Scholz, K. Nielsch, E. Pippel, D. Hesse, M. Zacarias, and U. Gosele.Monocrystalline spinel nanotube fabrication based on the kirkendall effect.
Nature Materi-als , 5:627–631, 2006.[7] L. Q. Chen. Computer simulation of spinodal decomposition in ternary systems.
Actametall. mater. , 42:3503, 1994.[8] D. J. Eyre. Systems of cahn-hilliard equations.
SIAM J. Appl. Math. , 53:1686–1712, 1993.[9] S. Puri. Topical review: Surface-directed spinodal decompoaition.
J. Phys.: Condens.Matter , 17:101–142, 2005. 61IBLIOGRAPHY 62[10] G. Brown and A. Chakrabarti. Surface-induced ordering in block copolymer melts.
J.Chem. Phys. , 101:3310, 1994.[11] B. P. Lee, J. F. Douglas, and S. C. Glotzer. Filler induced composition waves in phase-separating polymer blends.
Physical Rev. E. , 60:5812, 1999.[12] A. Chakrabarti. Effects of fumed silica network on kinetics of phase separation in polymerblends.
J. Chem. Phys. , 98:2451, 1993.[13] S. Puri and H. L. Frisch. Surface-directed spinodal decompoaition: modelling and numericalsimulations.
J. Phys.: Condens. Matter , 9:2109–2133, 1997.[14] S. Puri and K. Binder. Msurface directed phase separation with off-critical composition:Analytical and numerical results.
Physical Review E , 66:(061602)1–10, 2002.[15] S. Bhattacharya. Phase separation in ternary alloys: A diffuse-interface approach.M.Sc.(Engg.) Thesis, Department of Metallurgy, Indian Institute of science, Bangalore,2002.[16] J. W. Cahn and J. E. Hilliard. Free energy of a non-uniform system.i.interfacial free energy.
J. Chem. Phys. , 28:258, 1958.[17] H. Tanaka. Interplay between wetting and phase separation in binary fluid mixtures: rolesof hydrodynamics.
J. Phys.: Condens. Matter , 13:4637–4674, 2001.[18] D. Benderly, A. Siegmann, and M. Narkis. Structure and behaviour of multicomponentimmiscible polymer blends.
J. Polymer Engg , 17:461–489, 1997.[19] Y. Tang and Y. Ma. Controlling structural organization of binary phase-separting fluidsthrough mobile particles.
J. Chem. Phys. , 116:7719, 2002.[20] M. J. A. Hore and M. Laradji. Microphase separation induced by interfacial segregation ofisotropic, spherical nanoparticles.
J. Phys.: Condens. Matter , 126:244903, 2007.[21] A. C. Balazs. Multi-scale model for binary mixtures containing nanoscopic particles.
J.Chem. Phys. , 104:3411–3422, 2000.IBLIOGRAPHY 63[22] D. Suppa, O. Kuksenok, A. C. Balazs, and J. M. Yeomans. Phase separation of a binaryfluid in the presence of immobile particles: A lattice boltzman approach.
J. Chem. Phys. ,116:6305–6310, 2002.[23] F. Qiu, G. Peng, A. C. Balazs, V. V. Ginzburg, H. Y. Chen, and D. Jansow. Spinodaldecomposition of a binary fluid with fixed impurities.
J. Chem. Phys. , 115:3779–3784,2001.[24] G. Brown and A. Chakrabarti. Phase separation dynamics in off-critical polymer blends.
J.Chem. Phys. , 101:3310, 1994.[25] E. J. Kramer, P. Green, and C. J. Palstrom. Interdiffusion and marker movements inconcentrated polymer-polymer diffusion couples.
Polymer , 25:473–480, 1984.[26] Molecular dynamics simulation. http://physicomp.technion.ac.il/~talimu/md2.html .[27] M. Frigo and S. G. Johnson. FFTW: An Adaptive Software Architecture For The FFT,ICASSP 98. , 1998.[28] D. A. Porter and K. E. Easterling.
Phase Transformations in Metals and alloys . CRC press,third edition, 2008.[29] C. H. P. Lupis.