Cahn theory

J.W. Cahn. Theory of crystal interface motion in crystalline materials. Acta Metall., 8(8) 554—562, 1960. 

Phase dissolution in polymer blends. The reverse process of phase separation is phase dissolution. Without loss of general validity, one may assume again that blends display LCST behavior. The primary objective is to study the kinetics of isothermal phase dissolution of phase-separated structures after a rapid temperature-jump from the two-phase region into the one-phase region below the lower critical solution temperature. Hence, phase-separated structures are dissolved by a continuous descent of the thermodynamic driving force responsible for the phase separation. The theory of phase separation may also be used to discuss the dynamics of phase dissolution. However, unlike the case of phase separation, the linearized theory now describes the late stage of phase dissolution where concentration gradients are sufficiently small. In the context of the Cahn theory, it follows for the decay rate R(q) of Eq. (29) [74]... 

Rg is the polymer radius of gyration, Xs is the value of the x parameter (see Section 2.3.1) at the spinodal point, and D is the mutual diffusion coefficient of the two polymer components. Bates and Wiltzius (1989) have confirmed the predictions of Eqs. (9-4) and (9-5) for early-time SD of binary blends of perdeuterated and protonated 1,4-polybutadiene. Neutron-scattering studies of SD on a similar system by Jiimai et al. (1993a, 1993b) also confirm the Cahn theory at early times, but the spinodal growth rates deviate somewhat from Eq. (9-5). 

Figure 9-5 presents a simplified picture of SD, with three regimes (1) the linear regime described by the Cahn theory, in which is independent of time t (2) the slow diffusive coarsening regime in which oc and (3) the fast hydrodynamic regime in which... 

Mechanisms of liquid-liquid phase separation were studied in the binary styrene-acrylonitrile copolymer/poly(methyl methacrylate) system. Evidence is presented which suggests that spinodal decomposition occurs in this system. The Cahn theory provides an interpretation of key experimental results. Both a dispersed, two-phase structure and a highly interconnected, two-phase structure can be formed. These two structures coarsen at significantly different rates. The dispersed-phase structure coarsens by Ostwald ripening, an extremely slow process in the polymer-polymer system. The interconnected structure coarsens more rapidly. Data suggest that the mechanism of coarsening is viscous flow driven by interfacial tension. 

The Cahn theory provides a selfconsistent interpretation of the experimental results obtained in this study. It is not the only possible interpretation, but it does provide logical explanations for all observations. Hence, it is invoked in interpreting the results. Complete verification will require further experimentation. 

Figures 7 and 8 were obtained with a 25% SAN composition near the critical composition. At the critical composition, roughly equal phase volumes are expected. The SAN-rich phase constitutes roughly 35% of the phase volume in the micrographs of Figure 8. However, the key to the mechanism lies not in the phase volume ratio rather, it lies in the observed kinetics of the phase separation process. Figure 7 shows clearly that phase separation proceeds by a gradual change in composition over fairly well-defined regions in space. The scale of the phase separation (500-1000 A) is quite consistent with the data in Table II for spinodal decomposition in polymer-polymer systems. These observations, along with the observed high level of phase interconnectivity, are all consistent with theoretical predictions based on the Cahn theory, and they confirm semiquantitatively that spinodal decomposition is indeed the mechanism of phase separation.

The Cahn theory of spinodal decomposition explains the essential features of phase transition in the unstable region. The scale of phase separation was predicted and exceeded by an order of magnitude the scale in low molecular weight systems. A comparison of the diffusion coefficient estimated by the rate of spinodal decomposition agrees with the value predicted by the Bueche equation. The high level of phase interconnectivity predicted by the Cahn theory was also observed experimentally. 

Kinetics of demixing of fluid and solid mixtures in the bulk has been intensively studied for decades [2, 17]. A general conclusion of these studies is that the linearized Cahn theory of spinodal decomposition is not quantitatively valid (with the exception of symmetrical... 

Figure 4.34. The behaviour of the interface width w and the average domain size R at various stages of spinodal decomposition (top) the early stage - well described by the linear (Cahn) theory of spinodal decomposition, (centre) the intermediate stage and (bottom) the late stage - djmamic scaling should hold.

A homogeneous mixture quenched from a stable phase into a thermodynamically unstable state within a phase diagram develops into an inhomogeneous system. Spinodal decomposition (SD) is induced by the instability of an order parameter, which is usually concentration [101]. In the early stages, the SD is interpreted within the framework of the Cahn theory for an isotropic SD [102,103]. In contrast, in the late stages, the SD is governed by diffusion or hydrodynamic processes and exhibits slow coarsening [104]. There are two types of order parameters that describe a system. One is a conserved order parameter such as concentration of binary mixtures. The other is a non-conserved order parameter such as the polarization of a ferroelectric... 

We first consider the behavior oftheSD with lo = OinEq. (2.77). When lo = 0,we obtain b(q) = d(q) = 0 and then the kinetic equations (2.82) and (2.83) have no cross term between the gradients of the concentration and orientation order parameters. Such behaviors of SD in PDLC have been discussed theoretically (73, 119-122]. The time evolution of the structure factor for the concentration fluctuations falls into the Cahn-Hilliard classical category [102,103] and Eq. (2.84) results in the Cahn theory of SD for isotropic solutions. The structure factor for concentration in Eq. (2.92) is given by... 

To summarize, in the Cahn theory there are three main aspects which characterize the early stages of spinodal decomposition. 

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