Nucleation versus spinodal decomposition in confined binary solutions

Basic features of spinodal decomposition, on one side, and nucleation, on the other side, and the transition between both mechanisms are analyzed within the framework of a generalized thermodynamic cluster model based on the generalized Gibbs approach. Hereby the clusters, representing the density or composition variations in the system, may change with time both in size and in their intensive state parameters (density and composition, for example). In the first part of the analysis, we consider phase separation processes in dependence on the initial state of the system for the case when changes of the state parameters of the ambient system due to the evolution of the clusters can be neglected as this is the case for cluster formation in an infinite system. As a next step, the effect of changes of the state parameters on cluster evolution is analyzed. Such depletion effects are of importance both for the analysis of phase formation in confined systems and for the understanding of the evolution of ensembles of clusters in large (in the limit infinite) systems. The results of the thermodynamic analysis are employed in both cases to exhibit the effect of thermodynamic constraints on the dynamics of phase separation processes.     

The phase dynamics and wetting layer formation mechanisms in two-step surface-directed spinodal decomposition

The two-step quench process of surface-directed spinodal decomposition is numerically investigated by coupling the Flory-Huggins-de Gennes equation with the Cahn-Hilliard-Cook equation. The phase dynamics and formation mechanisms of the wetting layer in two-step surface-directed spinodal decomposition have been concerned in detail. The results demonstrate that a parallel strip structure forms near the wetting layer and propagates into the bulk, when the first quench depth is very shallow and the bulk does not undergo phase separation, and the second quench depths are various points with deeper quench depths. In this case, the wetting layer turns to be unchangeable at the intermediate and later stages of the second quench process, compared to the growth with a time exponent 1/2 during the first quench process. When the first quench depth is deeper and phase separation occurs in the bulk during the first quench process, it is found that a deeper second quench depth can stimulate a more obvious secondary domain structure, and the formation mechanism of the wetting layer changes from logarithmic growth law to Lifshitz-Slyozov growth law.     

Computational analysis of spinodal decomposition dynamics in polymer solutions

In this work a model, composed of the nonlinear Cahn-Hilliard and Flory-Huggins theories, is used to numerically simulate the phase separation and pattern formation phenomena of oligomer and polymer solutions when quenched into the unstable region of their binary phase diagrams. This model takes into account the initial thermal concentration fluctuations. In addition, zero mass flux and natural nonperiodic boundary conditions are enforced to better reflect experimental conditions. The model output is used to characterize the evolution and morphology of the phase separation process. The sensitivity of the time and length scales to processing conditions (initial condition c) and properties (dimensionless diffusion coefficient D) is elucidated. The results replicate frequently reported experimental observations on the morphology of spinodal decomposition (SD) in binary solutions: (1) critical quenches yield interconnected structures, and (2) off-critical quenches yield a droplet-type morphology. As D increases, the dominant dimensionless wave number k increases as well, but the dimensionless transition time t from the early stage to the intermediate stage decreases. In addition, t is shortest when c is at the critical concentration, but increases to infinity when c is at one of the two spinodal concentrations. These results are found when the solute degree of polymerization N₂ is in the range 1 ≤ N₂ ≤ 100. When N₂ > 100, however, a problem of numerical nonconvergence due to diverging relaxation rates occurs because of the very unsymmetric nature of the phase diagram. A novel scaling procedure is introduced to explain the phase separation phenomena due to SD for any value of N₂ during the time range explored in this study.     

Homogeneous crystal nucleation triggered by spinodal decomposition in polymer solutions

We report dynamic Monte Carlo simulations of polymer crystal nucleation initiated by prior spinodal decomposition in polymer solutions. We observed that the kinetic phase diagrams of homogeneous crystal nucleation appear horizontal in the concentration region below their crossovers with the theoretical liquid-liquid spinodal. When the solution was quenched into the temperature beneath this horizontal boundary, the time evolution of structure factors demonstrated the spinodal decomposition at the early stage of crystal nucleation. In comparison with the case without a prior liquid-liquid demixing, we found that the prior spinodal decomposition can regulate the nanoscale small polymer crystallites toward a larger population, more uniform sizes, and a better spatial homogeneity, whereas chain folding in the crystallites seems little affected.     

Liquid-liquid phase separation in multicomponent polymer solutions

Summary In conformity with a prediction byTompa, polymer solutions were found which, in a limited range of concentrations and temperatures, separate into three liquid phases. This was demonstrated with solutions in diphenylether of two polyethylene fractions with narrow molecular weight distributions, which represented the closest possible approximations of ternary systems. The phenomenon takes place exactly under the conditions indicated byTompa on the basis of theFlory-Huggins free enthalpy of mixing function.

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Zusammenfassung In Übereinstimmung mit einer Voraussage vonTompa ließen sich polymere Lösungen finden, die sich in einen begrenzten Bereich von Konzentration und Temperature in drei flüssige Phasen trennen. Dies wurde an Lösungen von zwei Polyäthylenfraktionen mit schmaler M-G-Verteilung in Diphenyläther gezeigt, die die engste mögliche Annäherung an ternäre Systeme repräsentieren. Das Phänomen tritt exakt unter den Bedingungen auf, dieTompa auf Grundlage derFlory-Hugginsschen Freien Mischungsenthalpie angegeben hat.

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With 9 figures in 26 details and 3 tables     

Kinetics of phase separation by spinodal decomposition in a liquid-crystalline polymer solution

Abstract

Time-resolved light scattering studies have been undertaken for elucidating the dynamics of phase separation in aqueous HPC (hydroxypropyl cellulose) liquid-crystalline solutions. The HPC/water system phase separates during heating and returns to a single phase upon cooling. The phase diagram of thermally induced phase separation was subsequently established on the basis of cloud point measurements. For kinetic studies, T (temperature) jump experiments of 10 per cent aqueous HPC solutions were undertaken. Phase separation occurs in accordance with the spinodal decomposition mechanism. At low T jumps or in reverse quenched experiments, the scattering maximum remains invariant as predicted by the linearized Cahn-Hilliard theory. However, at large T jumps, the SD is dominated by non-linear behaviour in which scattering peaks move to low scattering angles. The latter process has been identified to be a coarsening mechanism associated with the coalescence of phase separated domains driven by a surface tension. A reduced plot has been established with dimensionless variables Q and t. It was found that the scaling law is not valid over the entire spinodal process. The time evolution of the scattering profiles of 10 per cent HPC solutions, following a Tjump to 49°C, is tested with the scaling law of Furukawa. It seems that the kinetics of phase separation at 10 per cent solution resemble the behaviour of off-critical mixture.     

Kinetics of phase separation by spinodal decomposition in a liquid-crystalline polymer solution

Time-resolved light scattering studies have been undertaken for elucidating the dynamics of phase separation in aqueous HPC (hydroxypropyl cellulose) liquid-crystalline solutions. The HPC/water system phase separates during heating and returns to a single phase upon cooling. The phase diagram of thermally induced phase separation was subsequently established on the basis of cloud point measurements. For kinetic studies, T (temperature) jump experiments of 10 per cent aqueous HPC solutions were undertaken. Phase separation occurs in accordance with the spinodal decomposition mechanism. At low T jumps or in reverse quenched experiments, the scattering maximum remains invariant as predicted by the linearized Cahn-Hilliard theory. However, at large T jumps, the SD is dominated by non-linear behaviour in which scattering peaks move to low scattering angles. The latter process has been identified to be a coarsening mechanism associated with the coalescence of phase separated domains driven by a surface tension. A reduced plot has been established with dimensionless variables Q and t. It was found that the scaling law is not valid over the entire spinodal process. The time evolution of the scattering profiles of 10 per cent HPC solutions, following a Tjump to 49°C, is tested with the scaling law of Furukawa. It seems that the kinetics of phase separation at 10 per cent solution resemble the behaviour of off-critical mixture.     

Nonlinear kinetics of spinodal decomposition,and dissolution of inhomogeneities formed by spinodal decomposition in polymer blends

Nonlinear kinetics of both spinodal decomposition at early stages, and the dissolution of homogeneities formed during spinodal decomposition, is studied. Variation of the scattering intensity during a complete cycle consisting of a step temperature change from T₁ in the one-phase region to T₂ in the two-phase region, a period of spinodal decomposition followed by a temperature drop from T₂ back to T₁, and the subsequent relaxation to the original equilibrium state, is investigated at various wavenumbers. Step temperature changes within one-phase region are also investigated.     

Hydrodynamic fluctuations and spinodal decomposition in ternary systems

Time correlation functions of concentration fluctuations are obtained from hydrodynamic fluctuation theory extended to ternary systems. The coupling between the concentrations of the different species is accounted for and leads naturally to symmetric correlation functions unlike those previously reported. The time correlation functions of concentration fluctuations incorporating the effect of thermal noise are used to obtain the initial scattering intensity of ternary polymer-polymer-solvent systems undergoing spinodal decomposition. It is shown that the scattering function is represented by a sum of three exponentials. Single exponential growth is predicted under certain conditions which appear to have been met in the limited experimental data available. © 1994 John Wiley & Sons, Inc.     

Manifestation of spinodal decomposition in oscillatory shear measurements

The linear dynamic moduli of a PαMSAN/PMMA blend have been measured during spinodal decomposition and the subsequent coarsening of the co‐continuous morphology. The feasibility of probing this morphology development by rheological measurements has been investigated. During phase separation, the storage modulus shows a power law behaviour at low frequency and its value decreases as time proceeds. However, effects of the dynamic measurement on the morphology development have been observed, even for strain amplitudes as low as 0.01. This effect of oscillatory shear on the coarsening of a co‐continuous structure is consistent with the predictions of the Doi‐Ohta theory.     

Kinetics of phase separation at the early stage of spinodal decomposition in epoxy resin modified with PEI blends

The behavior at the early stage of spinodal decomposition (SD) for polyetherimide (PEI)/epoxy blends was investigated. It was found that the phase separation of PEI/epoxy blends took place by SD mechanism. The development of molar mass in the epoxy resin was gradual and then the three blends could still be considered as concentrated solutions of thermoplastic. The kinetics at the early stage of phase separation for these blends could be described by the Cahn–Hilliard–Cook linearized theory.     

Adsorption behaviour of sugars versus their activity in single and multicomponent liquid solutions

In this work the activity of three carbohydrates (sucrose, glucose and fructose) in highly concentrated aqueous solutions was studied along with its effect on the adsorption behaviour of the investigated compounds. Activities of individual sugars in aqueous solutions of single solute as well as in binary mixtures were quantified on the basis of solubility properties. Solid–liquid equilibria of sugars were correlated with the NRTL (nonrandom, two liquid) model of activity coefficient formulation. Activities of individual sugars were incorporated into the single component adsorption isotherm model, which reproduced accurately the course of the adsorption equilibria of sugars in aqueous solutions obtained experimentally in previous work using an ion-exchange resin. Activities of sugars determined in binary solute systems along with the single component isotherms were used to predict competitive adsorption equilibria. To calculate adsorbed phase concentrations of individual sugars in binary mixtures the adsorbed solution theory was adopted. The isotherm shapes calculated were compared to the data of competitive adsorption from the former study and found to be able to describe these experimental results.     

Kinetics of the droplet formation at the early and intermediate stages of the spinodal decomposition in homopolymer blends

The kinetics of the droplet formation during the spinodal decomposition (SD) of the homopolymer blends has been studied by numerical integration of the Cahn‐Hilliard‐Cook equation. We have found that the droplet formation and growth occurs when the minority phase volume fraction, f_m , approaches the percolation threshold value, f_thr = 0.3 ± 0.01. The time for the formation of the disperse droplet morphology (coarsening time) depends only on the equilibrium minority phase volume fraction, f_m . f_m approaches its equilibrium value logarithmically at the late SD stages, and, therefore, the coarsening time decreases exponentially as the average volume fraction or the quench depth decrease. Since the temporal evolution of the total interfacial area does not depend on the quench conditions and blend morphology, the average droplet size and the droplet number density is determined by the coarsening time. Within the time scale studied, the droplet number density decreases with time as t ^{–0.63±0.03}; the average mean curvature decreases as t ^{–0.35±0.05}; the average Gaussian curvature decreases as t ^{–0.42±0.03}, and the average droplet compactness ˜V/S^3/2 where S is the surface area and V is the volume) approaches a spherical limit logarithmically with time. The droplets with larger area have lower compactness and in the low compactness limit their area is a parabolic function of compactness. The size and shape distribution functions have been also investigated.     

Kinetically stable structures in the nonlinear theory of spinodal decomposition

A nonlinear diffusion equation is used to study the spinodal decomposition of binary polymer mixtures. The structure of steady-state solutions is analyzed. The free energy of the system for the solutions is shown to be a nonincreasing time function. The solution with the minimum free energy is the only stable solution and corresponds to complete phase separation in the system. A numerical analysis shows that transition to complete equilibrium takes place through a succession of fast concentration structure changes interspersed by periods in which the process is abruptly showed down. The free energy of the system is almost constant in the kinetically stable periods. All these facts indicate a relaxation process of an absolutely new type. Slow variables are introduced to describe the spinodal decomposition in macroscopic regions. These variables govern the spatial transformation of the forms and sizes of microphases. The perturbation theory is used to derive the equations for slow variables. The final relations of the simplest type give an exponential time dependence of the average size of a microphase. The problem of a contact of two pure components that are not compatible in the entire composition range is set.     

Hydrodynamic effects on spinodal decomposition kinetics in planar lipid bilayer membranes

The formation and dynamics of spatially extended compositional domains in multicomponent lipid membranes lie at the heart of many important biological and biophysical phenomena. While the thermodynamic basis for domain formation has been explored extensively in the past, domain growth in the presence of hydrodynamic interactions both within the (effectively) two-dimensional membrane and in the three-dimensional solvent in which the membrane is immersed has received little attention. In this work, we explore the role of hydrodynamic effects on spinodal decomposition kinetics via continuum simulations of a convective Cahn-Hilliard equation for membrane composition coupled to the Stokes equation. Our approach explicitly includes hydrodynamics both within the planar membrane and in the three-dimensional solvent in the viscously dominated flow regime. Numerical simulations reveal that dynamical scaling breaks down for critical lipid mixtures due to distinct coarsening mechanisms for elongated versus more isotropic compositional lipid domains. The breakdown in scaling should be readily observable in experiments on model membrane systems.     

Percolation-to-droplets transition during spinodal decomposition in polymer blends, morphology analysis

Phase separation kinetics of the off-critical mixture of polystyrene and poly(methylphenylsiloxane) is studied by the time-resolved light scattering and optical microscopy. The results from the light scattering experiments are correlated with the images obtained by the optical microscopic observation in order to find characteristic features of the scattering intensity during the percolation-to-droplets morphology transition. At the beginning of the spinodal decomposition process only a bicontinuous network is present in the system and the light scattering intensity has only one peak. The network coarsens and at the same time small droplets appear in the system resulting in a growth of the scattering intensity at very small wave vectors. When the large network starts to break up into disjoint elongated domains a second peak in the scattering intensity appears. Finally, both peaks merge into a single peak at zero wave vector, indicating a complete transformation of elongated domains into spherical droplets of variable sizes. The comparison of the direct microscopic observations with the light scattering spectra shows that the process of breaking up of the bicontinuous network starts when the growth of the first peak, corresponding to the bicontinuous pattern, becomes very slow (essentially pinned down).