Cahn-Hilliard theory

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). 

In conclusion, the Cahn-Hilliard theory is a modernization of van der Waals, confirming cmd extending the latter. With these theories and their many variants and extensions the framework is basically available for computing interfacial tensions from molecular interactions. Carrying out the computations is no easy matter, especially if the molecules are not spherical and if their interactions contain contributions other than those of the Lennard-Jones type. Then the quality of the results is determined by the quality of the choice of the parameters, analytical approximations, truncations, etc. A promising alternative is to invoke computer solutions, which will be treated in the next section. 

Thus, to calculate Vj one has to select an appropriate expression for the energy density gradient and then integrate Eq 4.8 within the limit of composition in both phases, ( ) and ( )g. Huggins-Flory, and Cahn-Hilliard theories were used with good success to predict the temperature gradient, but poor as far as the effects of molecular weight were concerned. 

The basis of the Cahn-Hilliard theory [12] is the assumption that AG per unit volume of the system, AG, is given by... 

The Cahn-Hilliard theory is deterministic, ignoring the random generation of concentration fluctuations by thermal agitation. This thermal noise effect, first pointed out by Cook [14], adds a new term to the right-hand side of eq 2.4. Since the timescale of thermal agitation is much shorter than that of the slow uphill diffusive process concerned here, the added term may play a role... 

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. 

EoS models can also be used in the frame of the gradient approximation, such as the Cahn-Hilliard theory [100] of inhomogeneous systems, for the description of surface properties. In the frame of this theory, the Helmholtz s free-energy density r in a one-component inhomogeneous system can be expressed as an expansion of density p and its derivatives ... 

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. 

A conceptually different approach to the calculaticHi of interfacial tensions is the use of the generalized square-gradient approach as embodied in the work of Cahn and Hilliard [216]. The Cahn-Hilliard theory provides a means for relating a particular equation of state, based on a specific statistical mechanical model, to surface and interfacial properties. The local free energy, g, in a region of nonuniform composition will depend on the local composition as well as the composition of the immediate environment. Thus, g can be expressed in terms of an expansion in the local composition and the local composition derivatives. Use of an appropriate free energy expression derived from statistical mechanics permits calculation of the surface or interfacial tension. 

The theories discussed up to now do not hold rigorously near the critical point of demixing, and an alternative approach is, thus, required. Nose [249] studied the interfacial behavior for both polymer mixtures and polymer solutions near the critical point. The theory was based on the Cahn-Hilliard theory [216] and takes into account the dimensions of the polymer coils at the interfacial region. For a... 

Sanchez [181] used a Taylor expansion of the Flory-Huggins equation for the free energy density, and the Cahn-Hilliard theory with a constant coefficient for the gradient terms. He found the same classical mean field exponents for the temperature dependence of interfacial tension and thickness, but he predicted that, for the symmetric case, both the interfacial tension and the thickness are independent of chain length. Sanchez explained this result to be due to the fact that, in his approach, chain connectivity was only implicitly taken into crmsideration through the entropy of mixing. The theories of Nose [249] and Joanny and Leibler [246] take explicitly into account chain connectivity in various approximations. 

The PCL/SAN-27.5 blend provided an excellent opportunity to investigate phase dissolution kinetics and to add some experimental data (that are still quite rare) to this subject. First, attention was focused on phase separation above LCST (at 122 °C), and examined this qualitatively with optical microscopy (at 130 °C) and then quantitatively by light scattering at various temperatures (125-180 °C). The SD mechanism of phase separation was confirmed by a detailed data analysis based on Cahn-Hilliard theory. Second, the phase-separated structure was quenched to various temperatures below LCST (50-115 °C) to study phase dissolution. The latter was affected by the temperature of previously performed SD and, based on light-scattering data, a model of phase dissolution mechanism was created. Several possible interpretations exist to explain the shift in the peak maximum during phase dissolution, one of which is a faster decay of the high-q... 

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... 

It is worth noting that several years prior to the approach adopted Lai and Fuller [43], Pistoor and Binder [45] had already attempted to include molecular effects of shear within the Cahn-Hilliard theory. They utilised the random phase approximation which offers an alternative formulism for determining... 

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. 

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