Late Stage Of Phase Separation Research Articles

Reaction-controlled morphology of phase-separating mixtures.

The role of externally-controlled chemical reactions in the selection of patterns in phase-separating mixtures is presented. Linearized theory and computer simulation show that the initial long-wavelength instability characteristic of spinodal decomposition is suppressed by chemical reactions, which restrict domain growth to intermediate length scales even in the late stages of phase separation. Our findings suggest that such reactions may provide a novel way to stabilize and tune the steady-state morphology of phase-separating materials. Pattern formation in reaction-diffusion systems occurs throughout nature. It is well known, for example, that spiral waves and other interesting steady-state patterns can be generated by simple chemical reactions [1]. In contrast, transient patterns are formed during phase separation by spinodal decomposition in both small molecule and polymer mixtures [2,3]. These patterns, whose characteristic length scale depends on the specificity of the components of the mixture, coarsen and disappear when macroscopic phase separation is achieved at asymptotically long times. It would be desirable to devise a mechanism by which these phase-separating morphologies could be stabilized. In this Letter, we argue that chemical reactions can be used to stabilize and tune the characteristic length scale of patterns arising in phase-separating materials. Unlike the usual scenario of spinodal decomposition, where concentration fluctuations of all length scales larger than a certain critical length scale spontaneously grow with time, we show that chemical reactions introduce two cutoff lengths, thereby

The droplet size distribution in the late stage of phase separation

We consider a collection of droplets during the late stage of phase separation in a closed system. Its coarsening is driven by surface energy and leads asymptotically to a linear growth of the mean droplet volume with time (Ostwald ripening). The droplets grow either from the supersaturated uncondensed phase (coalescence) or by collisions with subsequent fusion (coagulation). The combination of both mechanisms leads asymptotically to a self-similar evolution of the size distribution of the droplets when the coagulation kernel is homogeneous with degree zero. We calculate the scaled droplet size distribution for Brownian and constant kernel and compare the effects of coagulation with the effects of correlation and screening discussed in the literature. We compare our results for the asymptotic scaled distribution with computer simulations for the combined coalescence and coagulation processes.

Hydrodynamic interface quench effects on spinodal decomposition for symmetric binary fluid mixtures.

For nearly symmetric binary fluid mixtures, the coupling between concentration and velocity fields leads to a quick hydrodynamic coarsening: For bicontinuous phase separation, the pattern evolution is known to be governed by tube hydrodynamic instability. The interface tension is a driving force in the hydrodynamic coarsening. In the conventional theories of late-stage phase separation, local equilibrium has so far been assumed; however, this assumption might not be valid for nearly symmetric fluid mixtures even after the formation of a sharp interface. This is because the interfacial tension probably starts to play a major role in coarsening before the concentration reaches the local equilibrium concentration. In such a case, there is a possibility that the concentration diffusion cannot follow this quick geometrical coarsening. This could cause a drastic effect, which we call an interface quench effect. The interface quench effect could induce spontaneous double phase separation for bicontinous morphology, especially under a geometrical confinement.

Chemically controlled pattern formation in phase-separating materials

The role of chemical reactions in the selection of patterns in phase-separating mixtures is presented. Linearized theory and computer simulation show that the initial long-wavelength instability characteristic of spinodal decomposition is suppressed by chemical reactions, which restrict domain growth to intermediate length scales even in the late stages of phase separation. Our findings suggest that chemical reactions may provide a novel way to stabilize and tune the steady-state morphology of phase-separating materials.

Late-stage phase separation: Dynamics, spatial correlations, and structure functions.

Particle coarsening in the late stage was investigated using numerical simulations. The multiparticle diffusion problem was solved using a multipole expansion method which is valid to an arbitrary order of the expansion. The simulations were performed using both monopole and monopole plus dipole approximations. We found that the monopole approximation yields a good description of the diffusion field up to a volume fraction of approximately 0.1. Beyond this volume fraction, particle migration induced by interparticle diffusional interactions plays an important role. The simulations were performed using two different initial spatial distributions. Despite the different initial states of the system, we find that the spatial correlation functions evolve to unique scaled time independent forms. These spatial correlation functions show that depletion zones exist between small particles and that the density of small particles near large particles is less than that of a random spatial distribution. A scaled time independent structure function similar to that observed experimentally was found. The slope of the structure function in a log-log plot is close to 4 at small wave numbers and is -4 at very large wave numbers. Oscillations in the structure function, which are related to the spherical shape and size distribution of particles, are present at large wave numbers. The rate constant of the cubic growth law and the scaled particle size distribution are also determined.

On the theory of late-stage phase separation in off-critically quenched binary systems

The late stage of phase separation in off-critically quenched binary systems is studied from an unifying viewpoint by numerically solving not only the kinetic equation for the single droplet distribution function ƒ( R,t) but also the linear equation with the source term for the structure function S( k , t), both of which were recently derived by the present authors to study the dynamics of phase separation. The scaling behaviors, ƒ(R,t) ≈ 〈R〉(t) −4F L ( R 〈R〉(t) ) and S(k,t) ≈ k M (t) −3Ψ L ( k k M (t) ) , are thus recovered, where 〈R〉( t) is the average droplet radius, and k M( t) the peak position of S( k , t) as a function of k . The temporal power behavior of 〈R〉 and k M are also shown to agree with those expected on the basis of the Lifshitz-Slyozov-Wagner theory. The universal features of the time-independent scaling functions F L(ρ) and Ψ L( x) are thus investigated explicitly, including their volume fraction dependence, and the x 4 dependence of Ψ L( x) for small x. The results are in good agreement with experiments for both scaling functions.

Sans Study of Phase Separation in Solid 3He-4He

ABSTRACTSmall angle neutron scattering has been used to study phase separation in a quantum alloy, solid 3Hex-4He1-x. The onset of phase separation is marked by a dramatic increase in the measured scattering. A simple interpretation of the results suggests that the late-stage phase separation kinetics are dominated by an increase in the concentration of 3He atoms in preexisting precipitate regions.

Phase separation morphology in the critical to off-critical crossover region

The crossover behavior from a cocontinuous structure to a droplet structure during the late stage of phase separation in the polystyrene/poly(2-chlorostyrene) blend was investigated by observing domain structure in real space

Late-stage phase separation and sedimentation in a binary liquid mixture.

Visualization studies of the phase separation of the critical binary liquid mixture, aniline-cyclohexane, in the late stage of the gravity-dominated regime are reported. The sedimentation behavior, following the formation of the bulk interface, shows that the residual domains grow in size according to a(t)=${\mathit{At}}^{\mathrm{\ensuremath{\varphi}}}$, with \ensuremath{\varphi}=0.27\ifmmode\pm\else\textpm\fi{}0.06 and A=(7\ifmmode\pm\else\textpm\fi{}5)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}7}$ m ${\mathrm{s}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varphi}}}$. Domain growth in the earlier stages of spinodal decomposition and nucleation mechanisms also yield similar growth exponents and prefactors. The differences are probably due to some hydrodynamic interactions during the sedimentation.

Role of Interfaces in Coarsening and Grain Growth

AbstractTheories of late-stage phase separation are discussed from the perspective of statistical mean-field approaches, whereby the material interfaces are assumed to interact with the appropriate average microstructural environment. For coarsening of two- and three-dimensional phases, recent progress is presented for selecting the most appropriate averages that characterize the microstructural environment surrounding a domain. The effective mean field, as well as the average interaction distance over which transport occurs, may be determined self-consistently by imposing global constraints that reflect the microstructural phase fractions and by using explicit spatial and ensemble averages of the matrix transport fields. Comparison of new theoretical coarsening rates with liquid-phase sintering experiments show good agreement over a wide range of phase fractions. Extension of this approach to grain growth, which involves more complex topological interactions, is also discussed.

Late stages of delta ' precipitation in an Al-Li alloy by small-angle neutron scattering.

The late stage of phase separation in an Al--Li(3 wt. %) alloy aged at 463 K has been studied by small-angle neutron scattering. The time dependence of the distribution of \ensuremath{\delta}' particle sizes has been obtained from an analysis of the cross-section curves measured in absolute units. The coarsening of the \ensuremath{\delta}'-${\mathrm{Al}}_{3}$Li precipitates obeys Ostwald ripening kinetics but diffusion interaction effects are needed to account for the width and shape of the size distribution functions. The center-to-center interparticle distances have been estimated from an analysis of the interparticle structure factor describing the interference effects due to the spatial correlation between the precipitates.

Late-stage two-dimensional coarsening of circular clusters.

The size distribution of circular clusters resulting from a computer simulation of the late stages of phase separation in two dimensions was recently reported by Rogers and Desai [Phys. Rev. B 39, 11 956 (1989)]. The distribution that evolved was considerably broader than that predicted by an analytical version of the Lifshitz-Slyozov theory of late-stage coarsening in two dimensions. The results of the numerical analysis are compared with an older version of the two-dimensional theory which includes the effect of diffusive correlations among nearest-neighbor clusters; the correlations are related to the volume (area) fraction of clusters. The size distribution predicted by the older theory is in much better agreement with the numerical data. The implications of this agreement for the late-stage coarsening of three-dimensional clusters are discussed.

A light‐scattering study of phase separation kinetics in polystyrene/poly(2‐chlorostyrene) blends

AbstractThe angular dependence of the intensity of scattered light from polystyrene/poly(2‐chlorostyrene) blends was measured as a function of time in early and late stages of phase separation. The results were discussed in terms of Cahn's theory and scaling law for the late stage. Some distinctive behaviors which had not been found in other polymer blends were observed: the intensity of low‐angle scattering increased significantly during phase separation, possibly because of inhomogeneities in density. In the power laws km α tϕ and Im α tθ for the time (t) evolution of the peak position km and peak heigh Im of the structure function in the late stage; the value of ϕ was around −1, almost independent of composition and temperature; the ratio |θ/ϕ| was less than the theoretical value 3. Possible explanations for these behaviors are discussed in relation to the glass transition.

Numerical study of multitime scaling in a solid system undergoing phase separation.

The multitime-scaling rule for time correlation functions is shown in systems undergoing phase separation. With the use of multitime scaling, the asymptotic form of the structural function, S\ifmmode \tilde{}\else \~{}\fi{}(x)\ensuremath{\propto}${x}^{4}$, at small scaled wave numbers x is shown in a late state of phase separation in the diffusional system. A numerical simulation is performed to examine the multitime scaling. We use a discretized Cahn-Hilliard equation with the Oono-Puri free energy. The chemical potential is not conserved at all and may satisfy multitime scaling in the late stage of phase separation. Hence the suggested ${x}^{4}$ law of the structure function is justified.

Late stage spinodal decomposition in an oligomeric blend of polystyrene/polybutadiene: A test of the scaling law for the structure function

The late stages of phase separation by spinodal decomposition have been studied by light scattering in an oligomeric blend of polystyrene (PS) with polybutadiene (PB) as a function of both quench depth and composition. The scattering function I(q,t) follows a dynamical scaling law of the form S(q,t)=qm(t)−3S̃[q/qm(t)] where the wave vector qm corresponds to the maximum in I(q,t). The profiles of the normalized scaled structure functions are compared with that predicted by Furukawa for critical mixtures (percolation regime) which is given by S̃(x)=4x2/(3+x8)[S̃(1)=1]. Apart from one exception, Furukawa’s function adequately describes the form of the normalized structure functions for a 60% PS mixture. However there are problems with some of the comparisons with the 50% and 70% PS blends. These disparities may in part be attributable to the fact that some of the data may lie outside the ‘‘true’’ late stages.

Asymptotic Behavior of Structure Function at Small Wave Numbers in Late Stage of Phase Separation

The asymptotic behavior of the structure function at small wave numbers in systems undergoing phase separation is investigated. Brief reviews of the multi-time scaling and the k4-asymptotic behavior of the structure function for the Cahn-Hilliard equation are presented. Then the validity of the k 4-behavior for other systems is investigated. Numerical simulations for the spin exchange kinetic Ising model, for the Cahn-Hilliard model with the surface diffusivity only and for the simple liquid mixture are performed. It is found that the k4-behavior is consistent with these simulations.

Dynamical theory of polymer melt morphology

A general kinetic equation of the monomer density variables for polymer blends and block copolymer melts is obtained which describes slow morphology variations. The general theory is applied to a polymer blend adopting the biased reptation model of a polymer chain under mean field. We obtain an equation of motion of interfaces in a phase-separated polymer blend, which contains an interface reaction term for length scales shorter than l c ≡ R 2 G/ξ, where R G is the gyration radius of a polymer chain and ξ the interfacial width. We also discuss some problems associated with the incompressibility requirement for phase separation kinetics of binary systems not limited to polymers. For length scales greater than l c the interface dynamics involves diffusion in bulk pure phases even in the strong segregation limit in a way different from that for the usual time-dependent Ginzburg-Landau equation for the conserved order parameter. Implications of the existence of the new term on the late stage phase separation kinetics of polymer blend are discussed. A phenomenological model to study morphology dynamics not relying on the reptation model is also proposed.

Dynamics of fluctuations in ostwald ripening: A new equation of motion for the structure function

A new method of finding an equation of motion for the structure function S k ( t) from the kinetic equations recently derived for the late stage of phase separation is presented. S k ( t) is shown to be scaled by two kinds of characteristics lengths, the average droplet radius R (t) and the screening length l( t), as S k(t)∝ R(t) 3S(x, q; Q) 1 2 ) with x=k R(t) and q= kl( t) where Q is the volume fraction of the minority phase, and R(t) and l(t) satisfy the power laws (t)∼Q −1 2 R (t) and R(t)∼t 1 3 . The scaled structure function S(x, q; Q 1 2 ) is found to obey a first-order differential equation with two source terms which originate from two kinds of fluctuations; initial thermal fluctuations and nonthermal fluctuations generated by soft collisions among droplets. The soft-collision process is shown to produce a new type of narrowing of S(x, q; Q 1 2 ) which is different from the usual one that occurs as Q increases. The results are compared with other theories.

Late-stage phase separation and hydrodynamic flow in a binary liquid mixture.

Light-scattering measurements were made in a binary mixture in the late stage of spinodal decomposition, when gravity has become important. We observe macroscopic flow in the mixture, following which the scattering I(k,t) falls as ${k}^{\mathrm{\ensuremath{-}}4}$. This implies that the scattering is mainly from surfaces between growing domains. The measurements reveal an unexpectedly large rate of disappearance of the surface area per unit volume in the phase-separating system. The effect is probably related to a gravity-driven instability which gives rise to the flow and which was also investigated.

Kinetics of phase separation in polymer mixtures

Abstract Characteristic features of the liquid-liquid phase separation process in polymer systems are presented, followed by a concise review of existing studies on phase-separation kinetics in polymer solutions and mixtures. Brief discussions have been made on two selective topics in this field, i.e., applicability of Cahn's linearized theory for spinodal decomposition to polymer mixtures and the scaling law in the late stage of phase separation.