Cahn—Hilliard free energy approach

Non-Random Systems. As pointed out by Cahn and Hilliard(10,11), phase separation in the thermodynamically unstable region may lead to a non-random morphology via spinodal decomposition. This model is especially convenient for discussing the development of phase separating systems. In the linearized Cahn-Hilliard approach, the free energy of an inhomogeneous binary mixture is taken as ... 

The structure of the interface formed by coexisting phases is well described by the Cahn-Hilliard approach [53] (developed in a slightly different context by Landau and Lifshitz [54]) extended to incompressible binary polymer mixtures by several authors [4,49,55,56]. The central point of this approach is the free energy functional definition that describes two semi-infinite polymer phases <]), and 2 separated by a planar interface (at depth z=0) and the composition ( )(z) across this interface. The relevant functional Fb for the free energy of mixing per site volume Q (taken as equal to the average segmental volume V of both blend components) and the area A of the interface is expressed by... 

Within this continuum approach Cahn and Hilliard [481 have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy functionals with a square gradient form we illustrate it here for the important special case of the Ginzburg-Landau form. For an ideally planar interface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy functional (B3.6.11). This yields the Euler-Lagrange equation... 

Let us apply the idea of the Cahn-Hilliard approach to a diblock copolymer, where (pA and 4>b are now the reduced local densities of monomers A and B which are chemically bonded in the diblock-copolymer hnear chain molecule. As before, we shall assume that 4>a) = a—4>b i 4>) = 0) as the order parameter. It has been shown [33]-[35] that the long-range interaction of monomers in a copolymer chain can be described by an additional nonlocal term in the Ginzburg-Landau free energy functional ... 

From a theoretical standpoint, the phenomenological approach used is based on the Helmholtz free energy F that is varying with composition and composition gradient (Cahn and Hilliard, 1958 Cahn, 1961,1965, 1968 De Fontaine, 1967, 1975). 

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. 

Kammer [209] used the Cahn-Hilliard approach with the Flory-Huggins free energy of mixing and the assumption of a symmetric system to obtain ... 

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. 

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