Particles with selective wetting affect spinodal decomposition microstructures
Supriyo Ghosh, Arnab Mukherjee, T.A. Abinandanan, Suryasarathi Bose
PParticles with selective wetting affect spinodal decompositionmicrostructures
Supriyo Ghosh, ∗ Arnab Mukherjee, † T. A. Abinandanan, and Suryasarathi Bose Materials Engineering Department,Indian Institute of Science, Bangalore 560012, India (Dated: October 1, 2018)
Abstract
We have used mesoscale simulations to study the effect of immobile particles on microstructureformation during spinodal decomposition in ternary mixtures such as polymer blends. Specifically,we have explored a regime of interparticle spacings (which are a few times the characteristic spin-odal length scale) in which we might expect interesting new effects arising from interactions amongwetting, spinodal decomposition and coarsening. In this paper, we report three new effects forsystems in which the particle phase has a strong preference for being wetted by one of the compo-nents (say, A). In the presence of particles, microstructures are not bicontinuous in a symmetricmixture. An asymmetric mixture, on the other hand, first forms a non-bicontinuous microstructurewhich then evolves into a bicontinuous one at intermediate times. Moreover, while wetting of theparticle phase by the preferred component (A) creates alternating A-rich and B-rich layers aroundthe particles, curvature-driven coarsening leads to shrinking and disappearance of the first A-richlayer, leaving a layer of the non-preferred component in contact with the particle. At late simula-tion times, domains of the matrix components coarsen following the Lifshitz-Slyozov-Wagner law, R ( t ) ∼ t / . ∗ Corresponding author. Email: [email protected]. Present address: Materials Science and Engineer-ing Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA † Present address: Institute of Materials and Processes, Karlsruhe University of Applied Sciences, Moltkestr.30, 76133, Karlsruhe, Germany a r X i v : . [ c ond - m a t . m t r l - s c i ] J un . INTRODUCTION Phase separation in ternary mixtures, which may include alloys, polymers and metallicglasses, involves a complex interplay between thermodynamics and kinetics. This complexityis further amplified by a pre-existing “third phase,” which may be present in the matrix inthe form of spherical particles [1], a patterned substrate [2, 3], a network [4], a wall [5], orany arbitrary shape [6, 7]. As particulate additives, the third phase may also be mobile [8–11] or immobile [1, 12, 13], and span a wide size range from tens of nanometers to microns.The presence of such a pre-existing phase introduces several new features: its interactionbetween the matrix components may influence the shape of the phase diagram, change thephase separation temperature and extend the miscibility window or compatibility betweenthe phases.Phase separation of a binary mixture at or near a surface, referred to as surface-directedspinodal decomposition (SD), has been studied quite extensively (for a recent review see [14–16]). The most common finding is that the surface gets enriched first with the component(say, A) with the lower surface free energy. This triggers the formation of an adjacent layerrich in B, which in turn leads to the formation of a third A-rich layer, and so on. Themicrostructure finally has several such alternating A-rich and B-rich layers near the surface,co-existing with the internal region with normal spinodal microstructure. This is referredas “target pattern.” A similar pattern is found in the case of immobile grain boundariesin polycrystalline materials [17]. However, when they are mobile, they lead to a new pat-tern, termed discontinuous SD [18], in which the boundary becomes a transformation front:fast diffusion at the boundary produces alternating A-rich and B-rich lamellae perpendic-ular to, and behind the moving boundary, which keeps migrating into one of the (as yetuntransformed) grains.Experimentally, a mixture can phase separate under the effect of an external field (suchas shear flow [8], cross-linking [19], electric-field [20]) or an internal field like geometricalperturbation (such as a third-phase particle [21]). Target patterns may be produced througheither of them; for example, Tran-Cong and Harada [19] observed them when an externalinfluence (selective cross-linking reaction) triggered phase separation; Karim et al. [21] ob-served them around silica nano-particles. Interestingly, such target pattern has also beenobserved in a metallic glass in which the C-rich particle itself was formed as a result of2pinodal decomposition in a ternary blend, followed by a further phase separation of theC-poor (or, AB-rich) region around it [22].Binary mixture composition also plays a role in SD morphologies. Both experimentsand simulaions have shown that a critical 50:50 mixture phase separates into bicontinuousmorphologies, while droplet morphologies are prevalent in off-critical mixtures [4, 5, 10, 14,21, 23–25]. Two different situations were observed in off-critical mixtures based on whetherthe majority or the minority component is attracted to the surface. The Bulk was charac-terized by a bicontinuous morphology in the former and droplet in the latter. In a recentexperiment by Jiang et al. [26], selective interaction between the bulk components resultedin percolation which further effected the transition of bicontinuous morphology to dropletin a symmetric PMMA/SAN blend, while asymmetric blends retained their bicontinuity.The experiments by Tanaka et al. [27] showed that when particles are mobile, phase separa-tion could induce them to localize inside the preferred phase. The experiments by Morkvedand co-workers [28, 29] demonstrate an interesting application in which gold is made toself-assemble through selective aggregation of a phase separated mixture. In a symmetric(PS-PMMA) blend, gold particles self-assemble into PMMA phase [28], whereas in an asym-metric (PS-b-PVP) blend, gold particles self-assemble on PVP [29]. The experimental workof Herzig et al. [30] and Sanz et al. [31] on the particle effects on phase separating liquidsystems is particularly interesting in that it points to a novel possibility of a highly porousmaterial produced by getting particles confined to the A-B interfaces, and draining the liq-uids. These studies provided interesting yet significant insights into the influence of additivesurface on phase separating mixtures. However, the dependence of the morphologies on sizeand fraction of these particles has received little attention.Simulations of particle effects have taken several approaches. The most common one,adopted by Lee et al [1], Chakrabarti [4], Millett [6, 32], Oono and Puri[33], and Balazsand coworkers[8, 12], uses a Cahn-Hilliard-Cook (CHC) model for phase separation in bi-nary blends, along with a surface interaction term at the particle-matrix interfaces. Suppa et al. [13] employed a lattice Boltzmann approach, in which the particles are small com-pared to the characteristic length scale of SD. Other approaches include Langevin particledynamics [34], fluid particle dynamics [35], dissipative particle dynamics [9], cell dynam-ics [11, 33, 36], and molecular dynamics [10, 37].In this paper, we address the role of immobile particles during spinodal decomposition3ith a view to elucidating interesting new microstructural features. We use a ternary phasefield model that allows us to treat the particle as a C-rich phase that co-exists with theinitial binary mixture AB. The role of these particles, then, is determined primarily throughhow their interfaces interact with the mixture during the early stages of phase separation.This paper extends this framework in which of the role of particles on phase separation isexamined through a study of the effect of the interface between the matrix and particlephases. While the matrix-particle interface could act very much like a free surface, thereare two key differences: (a) the matrix-particle interface possesses a curvature, and there-fore curvature-dependent phenomena such as domain coarsening (Ostwald ripening) becomepossible, and (b) the presence of particles at finite volume fractions introduces a new lengthscale: interparticle spacing, λ . In particular, when λ is of the same order of magnitude asthe spinodal wavelength, we may expect a richer variety of spinodal morphologies that arisefrom an interplay of phase separation, wetting and curvature-driven coarsening.In the present study, we have used a Cahn-Hilliard formulation of a ternary system. Inthis system, the particle phase ( γ ) is rich in C, with a C-poor (or, A-B rich) matrix phaseseparates to produce A-rich ( α ) and B-rich ( β ) phases. The free energies of α - β , β - γ and α - γ interfaces can then be tailored easily through an appropriate choice of interaction energies,and the gradient energy coefficients. The rationale behind this work is to explore the regimewhere the interparticle spacing λ is of the same order of magnitude as (but larger than) thespinodal length scale. We compare the effect of particle in two systems: one in which γ hasa strong preference for one of the phases, and the other in which they have no preferencefor either of the phases.Following a description of the ternary Cahn-Hilliard model in Sec. II, we present ourresults on particle effects on SD microstructures in the neutral system and in the stronglyinteracting system in Sec. III. We discuss our results in Sec. IV and summarize the mainconclusions in Sec. V. II. MODEL
We model a binary mixture with embedded particles using a ternary system containingcomponents i = A, B, and C (assumed to be of similar molecular sizes). If local volumefractions of A ( c A ) and B ( c B ) are considered independent, then c C = 1 − c A − c B becomes a4ependent variable. The present work uses the formulation of Bhattacharyya’s [38] ternaryCahn-Hilliard model where total free energy F couples bulk free energy f with a gradientsquared term of conserved parameter c as [39] FK B T = N V (cid:90) V f ( c A , c B , c C ) + (cid:88) i = A,B,C κ i ( ∇ c i ) dV, (1)where N V is number of molecules per unit volume and κ i are the bare gradient energy coef-ficients associated with gradients in composition of the components i . Bulk or homogeneousfree energy is given by regular solution expression:1 K B T f ( c A , c B , c C ) = 12 (cid:88) i (cid:54) = j χ ij c i c j + (cid:88) i c i ln c i , (2)where χ ij is the pair-wise ( i and j ) interaction parameter, K B is Boltzmann’s constant and T is absolute temperature. Note that χ ij is inversely proportional to T . If a homogeneousternary blend is quenched thermally or compositionally, it will thrust into A-rich, B-richand C-rich domains. To track the temporal evolution of respective composition fields, thecontinuity equation is used: ∂c i ∂t = −∇ · (cid:126)J i , (3) (cid:126)J i is net flux of component i . It is formulated [40] by combining results of Kramer [41], Gibbs-Duhem equations and Onsager relations. Thus we obtain the following kinetic equations formicrostructural evolution: ∂c A ∂t = M AA (cid:104) ∇ g A − κ A + κ C ) ∇ c A − κ C ∇ c B (cid:105) − M AB (cid:104) ∇ g B − κ B + κ C ) ∇ c B − κ C ∇ c A (cid:105) , (4) ∂c B ∂t = M BB (cid:104) ∇ g B − κ B + κ C ) ∇ c B − κ C ∇ c A (cid:105) − M AB (cid:104) ∇ g A − κ A + κ C ) ∇ c A − κ C ∇ c B (cid:105) . (5)Here g A = ( ∂f /∂c A ) and g B = ( ∂f /∂c B ). M AA , M BB and M AB are the effective mobiltieswhich are given by M AA = (1 − c A ) M A + c A ( M B + M C ) ,M BB = (1 − c B ) M B + c B ( M A + M C ) ,M AB = (1 − c A ) c B M A + c A M B (1 − c B ) − c A c B M C . (6)5ubstituting M C = 0 and adjusting the matrix composition (A:B) accordingly, we obtainthe effective mobilities. In all simulations, the scaled mobilities used are M AA = M BB = 1 . M AB = 0 .
98. We start with a system containing immobile C-rich spherical particlesand then allow the homogeneous matrix to phase separate following the kinetic Eqs. 4, 5.Simulations are carried out by semi-implicit numerical integration [42] of the non-linearequations on a 512 ×
512 lattice, subject to periodic boundary conditions in both x and y directions. A. Simulation Details
The particle effect on phase separation is primarily through the strength of the interactionbetween the particle and product phases, and the interparticle spacing λ . While we havestudied systematically the role of both these parameters, we focus our attention on twokinds of systems: the first one, system S o , is neutral in terms of preference for either ofthe product phases (i.e., σ αγ = σ βγ ), and the second, system S s , in which the particle hasa strong preference for the A-rich α phase (i.e., σ αγ < σ βγ ). We refer to this selectivepreference of A about the C particles as wetting. The wetting in the present scenario issolely due to relative interfacial energies between phases. In the system S o , all the threeinterfaces have the same interfacial energy. However, in the strongly interacting system S s ,sum of the α - γ and α - β interfacial energies is still lower than the β - γ interfacial energy(refer to Table 1). Thus, the α phase truly wets the β - γ interface. In our ternary phasefield model, the values of the three interfacial energies (in Tab. I) are determined by theinteraction parameters ( χ AB , χ BC and χ AC ), and gradient energy coefficients ( κ A , κ B and κ C ) in Eq. 1. A short description for calculation of σ is given in Appendix.TABLE I: Inferfacial energy of corresponding interfaces system σ αβ σ βγ σ αγ S o S s Similarly, our results in the following section will be specifically for two values of particlespacings: large λ systems have a smaller volume fraction ( V = 5%) and larger particles6 R = 16), while the low λ systems have a higher volume fraction ( V = 10%) and smallerparticles ( R = 8). These two conditions correspond to interparticle spacings of λ (cid:39)
126 and45, respectively.In our simulations, particles start with a composition given by that of the γ phase inequilibrium with α and β phases in the ternary phase diagram, as shown in Fig. 1a. Particlesare also rendered immobile by making the mobility of component C nearly zero; i.e., M C =0 in Eq. 6. After randomly placing the particles in a two-dimensional simulation box with512 ×
512 grid points (with grid spacing ∆ x = ∆ y = 1 . ± .
005 is superimposed at each grid point.There is no noise in the particles.Tab. II lists all the parameters used in our simulations.TABLE II: Binary interaction ( χ ) and gradient energy ( κ ) parameters system χ AB χ BC χ AC κ A κ B κ C S o S s The local concentrations in ternary microstructures are represented using a gray scalemap in Fig. 1b; with this map, α , β and γ phases appear, respectively, white, light gray anddark gray, and interfaces acquire a black edge. III. RESULTSA. Microstructure Evolution in System S o We begin with a description of spinodal decomposition in a neutral system ( S o ) in whichthe particle phase has no preference for either component. In a symmetric blend in thissystem, phase separation leads to the well known bicontinuous microstructures shown inFig. 2a. In the presence of a single particle, a similar microstructure is obtained.In a system with multiple particles (at both high and low interparticle spacing λ ), theSD microstructures (Fig. 3) show a nearly bicontinuous pattern. We can easily discernthe underlying bicontinuous pattern if we imagine replacing the particles randomly with7 + β + α + β γα + γ β + γγ βα C BA (a) (b)
FIG. 1: (a) Isothermal section of the ternary phase diagram representing S system isdepicted (schematic). The A-rich, B-rich and C-rich phases are labelled as α , β and γ ,respectively. Simulations begin with particles having equilibrium composition of γ ( c A , c B , c C = 0.04, 0.04, 0.92) and matrix having the composition in square ( c A , c B , c C = 0.45,0.45, 0.1). This matrix composition eventually phase separates in α and β phases in thegiven directions (schematic).(b) Gray scale color map projected on Gibbs triangle (i.e. concentration triangle).Comparing the projection of (a) on (b), C-particles are represented as dark gray and initialmatrix as black which spinodally decomposes to white α and light gray β .either of the phases in the microstructure. The main difference between the high λ andlow λ conditions is in the length scale. For example, thickness of α or β regions of themicrostructure in the former is larger than that in the latter. B. System S s In system S s , the particle phase has a strong preference for component A. This is purelydue to σ αγ < σ βγ , which depends both on the interaction parameter χ and the gradientenergy parameter κ in Tab. II. Thus, the microstructure around a single particle (in Fig. 4b)is significantly altered from that in a system with no particle (Fig. 4a). Specifically, wefind a pattern of concentric, alternating rings of α and β phases around the particle inFig. 4b. This ring pattern also referred to as a “target pattern”, and its developments arerationalized as follows: species of A segregates preferentially to the particle-matrix interface,8 a) t = 3000 (b) t = 3000 FIG. 2: A B : Typical microstructures in system S o (a) without particles (b) with asingle particle (a) t = 3000 (b) t = 3000 FIG. 3: A B : Typical microstructures for (a) R = 16 and V = 5%, (b) R = 8 and V =10% using system S o parametersand forms an α layer around the particle. The region around this layer gets enriched withB, leading to the formation of a layer of β . This process sets up a composition wave thatpropagates outward from the particle [1]. The propagation is arrested when the outer-mostring meets the interior that has phase separated to a significant extent; therefore, the ringpattern around the particle coexists with the normal SD microstructure in the interior [43].As the microstructure evolves, we also find another interesting feature in the ring patternitself: since the rings have a curvature, they undergo coarsening due to the Gibbs-Thomsoneffect [44]. This effect causes solute concentration adjacent to a curved surface to increase asthe radius of curvature of the surface decreases. A concentration gradient therefore results,allowing the solute to diffuse in the direction of small curvature from the large, so thatlarge curvature shrink and eventually dissolve while small curvature grow. The inner-most9ing of α phase has the largest curvature, and therefore, shrinks the fastest; when this ringdisappears, the particle finds itself surrounded by the (non-preferred) β phase.Surface-directed SD [14] would also lead to the presence of alternating layers of α and β phases, much like the rings in Fig. 4b. However, since such layers are not curved, thephase inversion observed in the ring microstructure is not found in surface-directed SD. In (a) t = 3000 (b) t = 3000 FIG. 4: A B : Typical microstructures in system S s (a) without particles (b) with singleparticlesystems with multiple particles (at finite V ), we expect the propagation of the compositionwave emanating from each particle to be stopped by those from neighboring particles. Thisimplies that ring pattern around each particle would have a smaller number of rings thanin the single particle case; this number is decided by the interparticle distance λ . Thus, in asystem with a large separation (large particles R = 16 at small V = 5%), in Fig. 5c), we findtwo rings around each particle; remnants of the third ring from each particle have met upto form a continuous background. On the other hand, in the system with a small separation(see Fig. 6a for R = 8 and V = 10%), the first ring of α phase itself comes in contact withthat from neighboring particles, with the β phase being confined to interparticle regions.Thus, at intermediate stages of phase separation, the bicontinuity is broken with the β islands embedded in a continuous α matrix. However, the microstructures are quite differentin systems with high λ and low λ . With large interparticle separation, in Fig. 5d, we findparticles surrounded by just one ring of the non-preferred β phase, and β islands embeddedinside the α matrix. With small λ , in Fig. 6b, the α matrix has both γ particles andelongated β phase islands embedded inside it.In Fig. 7, we show the microstructural pathways in an asymmetric blend with β being themajority phase. With large interparticle spacing λ , each particle initially has a thin α and a10 a) t = 100 (b) t = 300(c) t = 500 (d) t = 3000 FIG. 5: A B : Typical microstructures from various simulation times are shown usingsystem S s parameters. Simulations begin with particles of R = 16 and V = 5% distributedrandomly in the homogeneous liquid. Compostion of the particles and initial liquid matrixare given in Fig. 1a. Particles are represented as dark gray and initial matrix as black, asdescribed in Fig. 1b, which spinodally decomposes to white α and light gray β . (a, b)Concentric rings of α and β initally forms around the particles. (c, d) α rings meet witheach other on third rings, which eventually form the continuous background of α . For thesake of brevity, we do not show the initial snapshots henceforth.thicker β ring around it, with the remaining α forming a thin, meandering, river-like featurein the interparticle regions. When λ is small, however, the ring patterns around particlesimpinge at the first or the second ring; when they impinge at the first ring (e.g., aroundclosely spaced particles), we find chains of γ particles inside the α phase. At other places,we find the β as a continuous phase. What is noteworthy, however, is that, at intermediatetimes, the α phase regions (with the particles inside them) become interconnected (e.g., bypiercing the continuous β phase at its thinnest parts), and develop bicontinuity of β and α phases in Fig. 7d. 11 a) t = 500 (b) t = 3000 FIG. 6: A B : Typical microstructures for (a, b) R = 16 and V = 5%, (c,d) R = 8 and V = 10% using system S s parameters (a) t = 500 (b) t = 3000(c) t = 500 (d) t = 3000 FIG. 7: A B : Typical microstructures for (a, b) R = 16 and V = 5%, (c,d) R = 8 and V = 10% using system S s parametersFor the sake of completeness, we present the microstructural pathways in blends with α phase as the majority phase (Fig. 8), which is also preferred at the particle matrix interfacein system S s . The microstructural development promotes the formation of islands of theminority β phase embedded in a continuous α phase. At t = 3000, the microstructures in12he large λ system exhibits β phase islands surrounding the particles, clearly as a result oftheir origin as a second ring. In the small λ system, however, small β islands are formedright at the beginning at the interparticle regions. (a) t = 500 (b) t = 3000(c) t = 500 (d) t = 3000 FIG. 8: A B : Typical microstructures for (a, b) R = 16 and V = 5%, (c,d) R = 8 and V = 10% using system S s parameters C. Coarsening kinetics of the wetting phase α Domain growth in the above microstructures is characterized by a time-dependent struc-ture function, S ( k , t ) [45, 46]. In case of ternary systems, there exist three linearly indepen-dent structure functions [47, 48] and assuming evolution process to be isotropic the circularlyaveraged structure factor in the xy -plane with N lattice points is given by S ii ( k, t ) = 1 N (cid:104) c ∗ i ( k , t ) c i ( k , t ) (cid:105) , (7)where k is the magnitude of the wave vector k . The k value corresponding to the maximumof S ii ( k, t ) is a measure of domain size. Precise location of this maximum however is difficultto extract due to the discretization involved in the phase-field simulations. Domain size is13herefore monitored through some moment of the structure function, usually the first or thesecond. Here we use the first moment of S ii to represent the average size of the A-rich α domains (or B-rich β domains) which is given by R ( t ) = 1 k ( t ) = (cid:80) S ii ( k, t ) (cid:80) kS ii ( k, t ) . (8)At intermediate to late stages of spinodal decomposition, bulk domains supposed to growfollowing the Lifshitz-Slyozov-Wagner (LSW) law, which gives R ( t ) ∼ t / in case of binarymixtures due to the diffusion [49, 50].In the above reference, R ( t ) of the α domains is presented in Fig. 9 for the symmetriccase. Several interesting points can be deduced from this. First, in the absence of particles,coarsening kinetics are consistent with the LSW law. Though the domain size as well asthe coarsening rates are markedly affected by the presence of particles indicated by differentslopes of the curves, the coarsening law in itself is not significantly altered in both S o and S s systems.
400 600 800 1000 1200 1400 1600 1800 2500 3000 3500 4000 4500 5000 R ( t ) t No particleLow λ (R = 8, V = 10%)High λ (R = 16, V = 5%) (a) R ( t ) t No particleLow λ (R = 8, V = 10%)High λ (R = 16, V = 5%) (b) FIG. 9: A B : Average size of the α domains R ( t ), given by Eq. 8, is plotted againsttime t in (a) S o and (b) S s systems. Note that domain growth follows the LSW t / law.Note that timescales are different in the above figures.Second, in the S o system, presence of particles suppress the coarsening rates, the effectbeing more prominent with low λ , leading to a smaller R ( t ). Spinodal decompositionproceeds via amplification of the composition fluctuations with maximal growing wavelengthgiven by λ SD = (cid:114) π ( κ i + κ j ) − ∂ f/∂c , after Cahn [51]. Since the spinodally decomposed matrix isessentially binary αβ , the above expression yields λ SD ≈
36. Note that λ SD is comparableto the small λ used in the phase-field simulations, whereas large λ used is ≈ . λ SD .14s the bulk domains grow beyond λ SD , the subsequent coarsening is dictated in the scaleof λ , which is represented by the density of particles and thus acts as a constraint. Whilegrowth is hindered from early stages with low λ , the domains are able to coarsen at a rapidrate with high λ , before the particles “see” the phase separation.Third, the above arguments do not hold in the S s system, where, interestingly, coarseningin the large λ system overwhelms that of neat blends. This indicates that in addition to λ , wetting also plays a significant role in the kinetics of coarsening. Wetting-induced phaseseparation proceeds via formation of concentric rings of preferred α and non-preferred β phases around the γ particles. The length scale in such a network is governed by the rapidgrowth of the wetting rings and then merging of these rings about the adjacent particleswhich subsequently extends into the background spinodal pattern. Referring to Figs. 5c and6a, with high λ the third rings meet, as compared to the first rings with small λ , resultingin a coarser length scale in the background. Moreover, wetting promotes the continuity ofthe α domain from an earlier time as opposed to that of neat blends. These factors mayhave an influence on enhancing the coarsening kinetics of α . The trends for the asymmetricblends also follow a similar course as shown in Fig. 10 where α , which is always continuous,shows an enhanced coarsening rate. The effect is more prominent when α also happens tobe the majority phase, i.e., A B .
200 400 600 800 1000 1200 1400 1600 1600 1800 2000 2200 2400 2600 2800 3000 R ( t ) t No particleLow λ (R = 8, V = 10%)High λ (R = 16, V = 5%) (a) R ( t ) t No particleLow λ (R = 8, V = 10%)High λ (R = 16, V = 5%) (b) FIG. 10: Average size of the α domains R ( t ), given by Eq. 8, is plotted against time t in(a) A B and (b) A B S s systems. Note that domain growth follows the LSW t / law.15 . Coarsening kinetics of the non-wetting phase β In a symmetric mixture with neutral interactions, the coarsening kinetics of β are similarto that of α . As a result, in a spinodally decomposed matrix, the respective domains areof the same size (compare Figs. 9a and 11a). In the strongly interacting system S s , onthe other hand, coarsening rates of β are enhanced at early stages of growth (in Fig. 11b)as it is rapidly expelled from the particle surfaces in order to accommodate α . However,with progress in time, β no longer remains contiguous and forms isolated chains, and thecoarsening rates of β are overwhelmed by that of neat blends. Referring to Figs. 5d and 6b,note that in large λ system, the chains of dispersed β phase are thinner when compared tothe same with low λ , where a higher density of particles leads to thicker β chains along withthinner continuity of the α matrix. These factors are reflected in the coarsening rates of β in Fig. 11b.
400 600 800 1000 1200 1400 1600 1800 2500 3000 3500 4000 4500 5000 R ( t ) t No particleLow λ (R = 8, V = 10%)High λ (R = 16, V = 5%) (a)
200 400 600 800 1000 1200 500 1000 1500 2000 2500 3000 R ( t ) t No particleLow λ (R = 8, V = 10%)High λ (R = 16, V = 5%) (b) FIG. 11: A B : Average size of the β domains R ( t ), given by Eq. 8, is plotted againsttime t in (a) S o and (b) S s systems. Note that domain growth follows the LSW t / law.Note that timescales are different in the above figures.In asymmetric mixtures, β exhibits enhanced coarsening rates as compared to that ofneat blends in both low and high λ systems (see Fig. 12). With low λ conditions in A B mixture (in Fig. 7d), majority β forms a thicker continuous network along with continuousyet thinner minority α . α domains, in this particular case, accommodate the γ particlesinside it, thereby, limiting the coarsening rates of α in a way similar to particle pinning,leading the coarsening rates of α that are lower than β (compare Figs. 10a and 12a). Inthe large λ system (in Fig. 7b), however, sizes of the α domains are not large enough to16ccommodate the particles, leading to entrapment of the γ particles within the surrounding β within an α network; such arrangement of β leads to lower coarsening rates than that of α , as evident in Figs. 10a and 12a.In A B systems, minority β is characterized by isolated droplets in a continuous matrixof α (in Fig. 8). In low λ conditions, high density of the particles breaks the β phase intomany small globular domains, which remain trapped within the interparticle regions, leadingto lower coarsening rates (in Fig. 12b) as opposed to that of in large λ conditions, where β domains coarsen relatively unhindered partially engulfing the particles.
300 500 700 900 1100 1600 1800 2000 2200 2400 2600 2800 3000 R ( t ) t No particleLow λ (R = 8, V = 10%)High λ (R = 16, V = 5%) (a)
400 600 800 1000 1200 1600 1800 2000 2200 2400 2600 2800 3000 R ( t ) t No particleLow λ (R = 8, V = 10%)High λ (R = 16, V = 5%) (b) FIG. 12: Average size of the β domains R ( t ), given by Eq. 8, is plotted against time t in(a) A B and (b) A B S s systems. Note that domain growth follows the LSW t / law. IV. DISCUSSION
We begin this section by highlighting four key conclusions from this work:(a) Importance of interparticle spacing: In strongly interacting system S s , the numberof rings formed around particles is smaller when λ is smaller. Thus, in Fig. 5c, we findtwo or three rings around each particle, while in Fig. 6a, we find just one or two. Thiscauses the bicontinuity to be broken, even in this symmetric blend, early in the processof microstructure formation. While the role of interparticle spacing is implicit in previousstudies, its importance is revealed in our study quite sharply.(b) Bicontinuity may emerge from a non-bicontinuous microstructures, and in asymmetricblends. Even though the early microstructure in an A B blend (in Fig. 7c) has isolated17 filaments (with γ particles inside them) embedded inside a continuous background of β phase, α filaments connect up by pinching off the β phase at its thinnest regions (Fig. 7d).(c) Importance of curvature effects: While the early microstructures in system S s mimicthose expected from surface-directed SD, curvature leads to the shrinkage and eventualdisappearance of the inner-most ring of α phase, thereby bringing the (non-preferred) β phase in contact with the particle.(d) Importance of wetting effects: Due to attractive interaction, α wetting layers areformed rapidly (or β layers are expelled rapidly) on the particles which further speed up thedynamics of the α (or β ) domains in the spinodally decomposed matrix, resulting in moreprominent coarsening rates than that of neat blends.The ring (or target) pattern we have seen in Figs. 5c and 6a has been observed in sim-ulations by Lee et al. [1]; there are at least two key differences between the their studyand ours. First, Lee et al used a Cahn-Hilliard-Cook model of a binary alloy and added alocal surface interaction energy for inducing preferential segregation. Thus, while the A-Binterfaces are treated in a phase field formalism, the particle and the matrix phases share asharp, Gibbsian interface. Our study, in contrast, uses a Cahn-Hilliard model in a ternarysetting which allows us to treat all the three interfaces within the same phase field formalism(which has the advantage that it could be extended easily to studying the role of particleswith irregular or ramified shapes). Second, Lee et al studied the effect due to a single par-ticle in a symmetric mixture; the present study examines multi-particle effects (in terms ofparticle size and volume fraction) in both symmetric and asymmetric mixtures.The picture that emerges from our results is consistent with previous studies. Targetpatterns in symmetric mixtures lead to a normal bicontinuous SD pattern at intermediatetimes. Asymmetric mixtures become bicontinuous or droplet depending on whether themajority or the minority component is preferred to the particles. The microstructures inFigs. 7c and 7d for a A B blend are particularly interesting in that particles are localizedinside the preferred phase. Such clustered arrangements of particles in the minority phasein a copolymer-particle mixture can be used to design composite architectures [52]. Theexperiments by Tanaka et al. [27] and Jiang et al. [26] and simulations by Ma [10, 36] andGinzburg et al. [11] also produced such clustered microstructures; however, the particle sizein their work was far smaller than the length scale of spinodal decomposition. Our simu-lations indicate that the bulk domains in critical and off-critical mixtures undergo diffusive18oarsening as ∼ t / , which is seen in most of the experiments and simulations.Finally, in the present work we do not consider the hydrodynamic interactions, interpar-ticle interactions, or processing conditions (i.e., shear) on the morphological evolution, allof which have a significant influence on phase separation dynamics [8, 12, 13]. In addition,we restricted our simulations for spherical particles, though our phase field method is notrestricted to such geometrically simple shapes. Our current simulations, with their emphasison early and intermediate stages of microstructure formation, do not allow us to examineparticle effects on late stage evolution due to computational limitations. Our ongoing workfocuses on the influence of various particle configurations and particle aggregates such asfractal surfaces on the formation and stability of interference patterns due to interactionof the composition waves about the particles; by engineering the locations and structuresof the particle phases, phase separating morphology of the mixtures can be designed andcontrolled [43]. V. SUMMARY AND CONCLUSIONS
1. We have studied the effect of immobile particles on phase separation microstructures internary mixtures through computer simulations based on a ternary phase-field modelin which the particles which are C-rich are embedded in an A-B blend.2. We have explored a regime of interparticle separation distances (a few times the char-acteristic length scale of spinodal decomposition) in which interesting new effects maybe expected.3. Our study shows four new effects in systems in which the particle phase has a strongpreference for one of the components.4. Microstructures in a symmetric blends are not bicontinuous in the presence of stronglyinteracting particles.5. Initially non-bicontinuous microstructures in asymmetric blends may evolve to becomebicontinuous at intermediate times.6. Even though the particle phase may be wetted by the preferred component, this wet-ting layer may dissolve due to curvature-driven coarsening leaving the particle in con-19act with the non-preferred component.7. α and β domains in the bulk patterns coarsen at intermediate times and scale as R ( t ) ∼ t / . While continuous wetting phase α exhibits enhanced coarsening ratesdue to its preference to the γ particles, dispersed non-wetting phase β within theinterparticle regions exhibits lower coarsening rates. APPENDIX: CALCULATION OF INTERFACIAL ENERGIES
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