On the theory of spinodal decomposition in solid and liquid binary mixtures

The kinetics of phase separation is discussed with emphasis on the transition between spinodal decomposition and nucleation. A reanalysis of the theory of Langer, Baron and Miller shows that it exhibits a spinodal line somewhat closer to the coexistence curve than the meanfield spinodal. There the same (as we think unphysical) critical singularities occur as in Cahn-Hilliard theory. The precise location of this spinodal line depends on the cell size of the coarse graining. For concentrations less than the spinodal one the structure factorS(k, t) converges then towards the structure factor of the metastable onephase state, implying an infinite lifetime of the latter.In order to include the effects of nucleation and growth we hence present an alternative treatment, extending our previous work on cluster dynamics. From a simple approximation for the radial concentration distribution function of clustersS(k, t) is computed numerically. Even at rather low concentrations the time evolution ofS(k, t) is then similar to what Langer et al. find at high concentrations, implying a very gradual transition from nucleation and growth to spinodal decomposition, at least for parameter values appropriate to the Ising model. This treatment, which is consistent with Lifshitz-Slyozov's coarsening law at late times, is extended to the early stages of phase separation in liquid mixtures.     

Phase separation in a chaotic flow

The phase separation between two immiscible liquids advected by a bidimensional velocity field is investigated numerically by solving the corresponding Cahn-Hilliard equation. We study how the spinodal decomposition process depends on the presence-or absence-of Lagrangian chaos. A fully chaotic flow, in particular, limits the growth of domains, and for unequal volume fractions of the liquids, a characteristic exponential distribution of droplet sizes is obtained. The limiting domain size results from a balance between chaotic mixing and spinodal decomposition, measured in terms of Lyapunov exponent and diffusivity constant, respectively.     

Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation

This paper gives theoretical results on spinodal decomposition for the stochastic Cahn–Hilliard–Cook equation, which is a Cahn–Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of a characteristic size. In more detail, our results can be summarized as follows. The Cahn–Hilliard–Cook equation depends on a small positive parameter ε which models atomic scale interaction length. We quantify the behavior of solutions as ε→ 0. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finite-dimensional subspace ?_ε is high as long as the solution stays within distance r _ε=O(ε ^R ) of the homogeneous state. The subspace ?_ε is an affine space corresponding to the highly unstable directions for the linearized deterministic equation. The exponent R depends on both the strength and the regularity of the noise. Received: 2 May 2000 / Accepted: 8 July 2001     

Stochastic description of phase separation near the spinodal curve in alloys

The earlier-developed statistical methods for nonequilibrium alloys are applied to stochastically describe phase separation near the spinodal curve. An important parameter of the theory is the size of local equilibrium regions, which is estimated using simulations for the different values of this parameter. The simulations based on this approach reveal significant changes in the type of evolution from nucleation to spinodal decomposition under variation of concentration and temperature across the spinodal curve. The scale of these changes seems to be mainly determined by the difference of the properly defined supersaturation parameters.     

Reaction-Induced Phase Separation and Structure Formation in Polymer Blends

The peculiarities of reaction-induced phase separation and the structure formation in semi- and full interpenetrating polymer networks and in the blends of linear polymers formed in situ are analyzed. It is shown that for most of these systems phase separation proceeds viathe spinodal decomposition mechanism resulting in the formation of interconnected spatially periodic structures. The possible ways for the structure regulation of the composites produced are considered.     

Experimental small-angle X-ray scattering investigation of initial stages of the spinodal decomposition in model sodium silicate glasses

The kinetics of the initial stages of the spinodal decomposition in model glasses of the Na₂O-SiO₂ system has been investigated in situ. It has been demonstrated that there is a quantitative agreement of the experimental results obtained in the framework of the Stephenson theory with the basic principles of modern theories and the data on direct determination of the viscosity, mobility, and diffusion. It has been found that the spatial-temporal evolution of the heterogeneous structure has a multistage character during spinodal decomposition. The characteristic size of the phase regions at each stage varies with time according to the power law. The sequence of stages and the values of exponents for the spinodal decomposition are as follows: 1/20, 1/4, 1/2, and 1/3.     

Structure Development via Reaction-Induced Phase Separation in Polymer Mixtures: Analysis of Early- and Late-Stage Demixing and Computer Simulations at Non-Isoquench Depths

The effects of phase separation during polymerization of a monomer/polymer mixture were studied. The kinetics of the spinodal decomposition of glycidyl phenyl ether/polystyrene (95/5) during polymerization was investigated using a light-scattering technique. This phase separation during polymerization was comparatively different from the one observed at the isoquench depth such that (i) there was significantly more time before any observable fluctuation, and (ii) the periodic distance of the resulting two-phase structure in the later stages decreased with time. To further investigate the influence of a continually increasing quench depth on phase separation, a diglycidyl aminophenol/diglycidyl ether of bisphenol A (50/50) system with a simple phase diagram was investigated using ultra-small-angle X-ray scattering analysis, and a similar phase separation behavior was observed. Furthermore, computer simulations using the dynamic self-consistent field method were performed to interpret polymer morphology during the demixing driven by chemical reactions. The simulation results suggested that the periodic distance of the resulting two-phase structure formed in the later stages decreases with time. The time dependence of the concentration fluctuations observed in these simulations provided a satisfying explanation for the morphologies formed as a result of a continuously increasing quench depth.     

Evolution of the structure factor in a hyperbolic model of spinodal decomposition

We consider the modification of the Cahn-Hilliard equation when a time delay process through a memory function is taken into account. The memory effects are seen to affect the dynamics of phase transition at short times. The process of fast spinodal decomposition associated with a conserved order parameter - concentration is studied numerically. Details of a semi-implicit numerical scheme used to simulate the kinetics of spinodal decomposition and evolution of the structure factor are discussed. Analysis of the modeled structure factor predicted by a hyperbolic model of spinodal decomposition is presented in comparison with the parabolic model of Cahn and Hilliard. It is shown that during initial periods of decomposition the structure factor exhibits wave behavior. Analytical treatments explain such behavior by existence of damped oscillations in structure factor at earliest stages of phase separation and at large values of the wave-number. These oscillations disappear gradually in time and the hyperbolic evolution approaches the pure dissipative parabolic evolution of spinodal decomposition.     

Spinodal method of vacancy diffusion study in ionic mixed crystals at room temperature

Some aspects of the spinodal method of deducing diffusion coefficients are considered. The decomposition kinetics yield the interdiffusion coefficient which is, however, not an intrinsic property of ionic crystals at low temperatures since it depends on the nonequilibrium vacancy concentration. Comparing, though, the spinodal kinetics in crystals doped with aliovalent impurity and undoped crystals enables one to obtain the vacancy diffusion coefficient which is an intrinsic property. The spinodal decomposition has been studied in nominally pure and Ca²⁺-doped mixed crystals of NaCl-KCl by the thermal gradient method and the cation vacancy diffusion coefficient D_v = 2 × 10⁻¹²cm²s⁻¹ at room temperature.     

Details of Structure Formation During the Induction Period of Spinodal-Type Polymer Crystallization

An unexpected type of primary crystal nucleation is described, involving spinodal decomposition (SD) type microphase separation due to the orientation fluctuations of rigid segments prior to crystal nucleation. This type of mechanism was found by the present authors about 10 years ago, and recently, it was theoretically revealed by Olmsted et al. to be one of three types of primary crystal nucleation: the well-known homogeneous crystal nucleation directly from the liquid–crystal coexistence domain, which occurs at higher temperatures above the binodal temperature T _b , crystal nucleation after binodal microphase separation between T _b and spinodal temperature T _s , and that after SD below T _s . The detailed experimental results for the spinodal-type crystal nucleation, especially the temperature dependence of characteristic wavelength in SD, are explained as well.     

The Effect of Preparation Temperature on Phase Transitions of N-Isopropylacrylamide Gel

The spinodal decomposition of N-isopropylacrylamide (NIPA) gels prepared at various onset temperatures, T _on was studied by photon transmission. It was observed that the increase in turbidity is much faster in a gel prepared at higher T _on than at lower T _on values, which indicated that the NIPA-water system reaches the spinodal decomposition much faster for a gel prepared at high T _on. It is understood that a NIPA gel prepared at high T _on possesses more heterogeneities which are gained during gelation and has a low spinodal temperature, T _s. However, NIPA gels prepared at lower T _on values go to spinodal decomposition at higher T _s values.     

STRUCTURE-PROPERTY RELATIONS IN REACTION-INDUCED,PHASE-SEPARATED POLY(2-PHENOXYETHYL ACRYLATE)/POLYSTYRENE BLENDS

Structure-property relations were studied in reaction-induced, phase-separated polymer blends. An amorphous-amorphous system consisted of polystyrene (PS) dissolved in the monomer 2-phenoxyethyl acrylate (POA). When the POA was polymerized to poly(2-phenoxyethyl acrylate) (PPOA), phase separation and phase inversion were induced, and a polymer blend was formed. The reaction kinetics were measured by monitoring the reduction in the intensity of the C╤C stretching vibration band in the Raman spectrum of POA. The phase separation kinetics were determined using light transmission experiments and were combined with the reaction kinetics so that a ternary phase diagram could be defined for the reactive system. Structure development was monitored using small-angle laser light scattering (SALLS) and optical microscopy, which showed that spinodal decomposition was the mechanism of liquid-liquid phase separation. Plots of the relative invariant with time showed an increase in the degree of phase separation. The Fourier transforms of the microscopy images had peaks in the radial intensity distributions, again implying that spinodal decomposition was the phase separation mechanism. Tensile testing showed that PPOA was soft and rubbery at 20°C. Both PS and PPOA had comparable toughness when tested to failure; however, the blend containing 17 wt% PS had a toughness more than 10 times that of either PS or PPOA in isolation. Both modulus and tensile strength increased with PS content, while the ultimate strain decreased. The Nielsen model best described the tensile modulus data, providing further evidence for co-continuous phase structure.     

Large‐Scale Fluctuation Behavior Prior to the Crystallization of Poly(ethylene terephthalate) Glass: A Small‐Angle Light Scattering Study

The crystallization processes of amorphous, glassy‐state poly(ethylene terephthalate) (PET) at two temperatures, a low temperature near T _g where PET has a slow crystallization speed and a middle temperature (about 55°C above T _g ) where PET crystallization is rapid, were monitored in situ by a time‐resolved small‐angle light scattering (SALS) device. It was found that large‐scale fluctuations happened prior to the crystallization at both temperatures, but the kind of fluctuation had a temperature dependence: at the middle temperature, pure density fluctuation took place during the induction period, whereas at low temperature, both density fluctuation and orientation fluctuation occurred, but the latter was the dominant factor. Analyses of the kinetics of these two kinds of fluctuation processes demonstrated that the spinodal decomposition (SD) type of phase‐separation character was undistinguishable in the SALS scale, while the nucleation‐growth (NG) type of phase behavior could describe the scattering results as well.     

Reaction‐Induced Phase Separation in Polyoxyethylene/Polystyrene Blends. I. Ternary Phase Diagram

Abstract

Reaction‐induced, phase separation has been studied in polymer blends. A model crystalline‐amorphous system consisted of semicrystalline polyoxyethylene (POE) dissolved in the monomer styrene, which was used as a reactive solvent to ease processing. When the styrene was polymerized to polystyrene (PS) in the mold, phase separation and phase inversion are induced, and a polymer blend was formed. Polyoxyethylene was selected with a molar mass, M _n  = 8578 g mol^?1 and a polydispersity of 1.19, as determined by using gel permeation chromatography. The polymerization of styrene was initiated by using 1 wt% benzoin methyl ether and 0.2 wt% 2,2′‐azobisisobutyronitrile under ultraviolet light. The polymerization kinetics were determined by monitoring the reduction in the intensity of the C?C stretching vibration band at 1631 cm^?1 in the Raman spectrum of styrene. The onset times for the liquid–solid (L–S) phase separation and crystallization of POE from styrene/PS were observed by using simultaneous small‐angle x‐ray scattering (SAXS) and wide‐angle x‐ray scattering. Onset times for L–S phase separation determined from the SAXS data were combined with the styrene polymerization kinetics to plot the L–S phase separation data onto a ternary phase diagram for the reactive system POE/styrene/PS at 45°C and 50°C.     

Kinetics of phase separation in polymer mixtures

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.     

Thermal analysis study for the phase determination and instable to metastable transformation of the Co–13Cu alloy

The differential thermal analysis (DTA) is utilized to determine the phase boundary and phase equilibria of Co–Cu alloy having a miscibility gap. Regions for various phase i.e., spinodal and coherent binodal, peritectic, magnetic and the transformation from the two phases to α phase transformations are distinctly determined for the Co–13 at%Cu alloy. The obtained results might indicate the continuity of the decomposition kinetics at the boundary between instable and the metastable regions but an activation barrier is needed. The activation energy for the magnetic transformation is determined to be 182.2 ± 2.1 kJ mol^?1.     

Investigation of Phase Separation in a Partially Miscible Polymer Blend by Rheology

The phase boundary and phase‐separation dynamics of a binary poly(styrene‐co‐maleic anhydride)/poly(methyl methacrylate) blend have been investigated by rheological measurements and light scattering. The phase diagram was experimentally established by rheology, in which the binodal line was obtained by dynamic temperature ramps and the spinodal temperatures were quantitatively estimated on the basis of the theory developed by Ajji et al. The phase‐separation dynamics from rheological viewpoints have been further investigated on the basis of the obtained phase diagram. Rheological measurements can sensitively detect the rather early stages of phase separation compared with light scattering techniques. It was found that the dynamic storage modulus initially increases over time and subsequently decreases during nucleation and growth; on the other hand, it always decreased over time during spinodal decomposition. Compared with light scattering techniques, rheological measurements were found to be relatively reliable for probing phase‐separation mechanisms.     

Phase diagrams and kinetics of phase transitions in protein solutions

The phase behavior of proteins is of interest for fundamental and practical reasons. The nucleation of new phases is one of the last major unresolved problems of nature. The formation of protein condensed phases (crystals, polymers, and other solid aggregates, as well as dense liquids and gels) underlies pathological conditions, plays a crucial role in the biological function of the respective protein, or is an essential part of laboratory and industrial processes. In this review, we focus on phase transitions of proteins in their properly folded state. We first summarize the recently acquired understanding of physical processes underlying the phase diagrams of the protein solutions and the thermodynamics of protein phase transitions. Then we review recent findings on the kinetics of nucleation of dense liquid droplets and crystals. We explore the transition from nucleation to spinodal decomposition for liquid-liquid separation and introduce the new concept of solution-to-crystal spinodal. We review the two-step mechanism of protein crystal nucleation, in which mesoscopic metastable protein clusters serve as precursors to the ordered crystal nuclei. The concepts and mechanisms reviewed here provide powerful tools for control of the nucleation process by varying the solution thermodynamic parameters.