Cahn-Hilliard theory spinodal decomposition

Lipson (1943, 1944), who had examined a copper-nickeMron ternary alloy. A few years ago, on an occasion in honour of Mats Hillert, Cahn (1991) mapped out in masterly fashion the history of the spinodal concept and its establishment as a widespread alternative mechanism to classical nucleation in phase transformations, specially of the solid-solid variety. An excellent, up-to-date account of the present status of the theory of spinodal decomposition and its relation to experiment and to other branches of physics is by Binder (1991). The Hillert/Cahn/Hilliard theory has also proved particularly useful to modern polymer physicists concerned with structure control in polymer blends, since that theory was first applied to these materials in 1979 (see outline by Kyu 1993). 

Snyder et al. [23] were among the earlier workers who studied the early stage of spinodal decomposition of polymer blends, with the distinct aim of testing the Cahn-Hilliard theory. They measured /total as a function of time for three PS/PVME blends and found that ln(/total ) increased linearly with time t over a certain range of early time. If R k) defined by eq 2.8 has a sharp maximum at fc = Fni> eq 2.7 may be approximated by u x,t) e.xp (//(fcni)t), so that exp (2/ (fcm)t). Therefore, the initial slope of ln(/totai ) vs. t can be equated to R(km)- Snyder et al. used this idea to analyze their data (the same idea had already been used by Nishi et al. [11]). However, we have to note that the peak of R k) given by eq 2.8 is not so sharp as to justify the approximation used. 

Hard particles with an additional soft attractive interaction have been used in a study of spinodal decomposition [6]. Such a system can be realized in a colloidal suspension and exhibits a fluid-fluid phase separation between a colloid-rich and a colloid-poor phase (see the phase diagram in Fig. 2). Quenched below the spinodal, the system undergoes spinodal decomposition which leads finally to the formation of dense and less dense clusters. The main progress, as compared to nonlinear Cahn-Hilliard theories, is the more accurate description of the short wavelength correlations, a property which will become most relevant in confined systems. 

We believe this example to be representative of a quite general principle with important implications for a large class of configurational changes in solids Often the kinetics of a phase transformation is described as continuous or homophase by means of a generalized diffusion theory, such as the Cahn-Hilliard theory for spinodal decomposition. The tacit assumption is always that a sufficiently large number of mobile diffusion-mediating defects is available in any small volume element considered in order to allow a continuum description. Whereas this requirement may always be well fulfilled in... 

More recently, DeGennes [83] and Pincus [84] have extended the Cahn-Hilliard theory of spinodal decomposition to polymer blends in the melt. Their scaling relationships show that the kinetics of the early stages of the process should depend strongly on molecular weight. This analysis has been called into question by Ronca and Russell [85] who employ a very different form of the concentration gradient term. 

We observed that the phase separation in the 107. aqueous UPC solutions is the spinodal decomposition. The linearized theory of Cahn-Hilliard predicts many of the qualitative features of the SD in the present system, but is not adequate in the quantitative comparison. The phenomenon of SD is non-linear in nature dominated by the late stage of SD. The growth mechanism has been identified to be the coarsening process driven by surface tension. The kinetics of phase separation at the 107. aqueous HPC resembles the behavior of off-critical mixtures. 

The Kinetics of Spinodal Decomposition. Cahn s kinetic theory of spinodal decomposition (2) was based on the diffuse interface theory of Cahn and Hilliard (13). By considering the local free energy a function of both composition and composition gradients, Cahn arrived at the following modified linearized diffusion equation (Equation 3) to describe the early stages of phase separation within the unstable region. In this equation, 2 is an Onsager-type... 

Kammer [1977] considered the interface between two polymers from the basic thermodynamic point of view. He derived a simple relation Vj = A 0g/S, where A 0g is the excess chemical potential of polymer B in the mixture, and S is the molar area of the interface. Near the spinodal decomposition, using Cahn-Hilliard gradient theory, he calculated ... 

We start from a brief account of the phenomenological theory of spinodal decomposition due essentially to Cahn and Hilliard [12, 13]. We consider a binary system consisting of chemically different, but mutually compatible components 1 and 2, with the volume fraction 0 assigned to component 2. For simplicity. 

Phase Separation Kinetics. There currently is considerable interest by the polymer physics community in the mechanism of phase separation in polymer blends. In an early study, Kwei, Wang and Nishi [4] employed pulsed NMR and found that the Cahn-Hilliard linearized non-statistical theory of spinodal decomposition [82] adequately described the process at short times the average composition of the two phases changed exponentially with time while their volume fractions remained constant. In addition, it was observed that no change in phase volume fraction took place at longer times when the linearized model no longer was valid. 

It has already been discussed that the phase separation in IPNs may be described in terms of the spinodal decomposition. At the initial stages the kinetics of phase separation is described by the linear theory of Cahn-Hilliard in spite of the fact that continuous changes of conversion occur in this case. This means that the system drifts along the phase diagram with a continuous change in parameter X23 and in the composition of the separating phases. Thus, one more principal feature of IPN structure formation is established. This feature consists in a continuous sequence of structures or states, which differ in composition not only at various stages of phase separation, but also... 

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