The thermodynamics of nonuniform solutions usually employs a gradient energy term in the expression for the local free energy. This arises from retaining the first nonvanishing terms of a MacLaurin expansion of the free energy in the composition and its derivatives. The nature of this expansion is discussed, and the formulation is extended to include derivatives higher than the second. General, a priori arguments suggest that a reasonable next approximation is f*=f(c) +κ1∇2C+κ2(∇c) 2+κ3∇4c+κ4(∇ 2c)2+κ5∇c·∇ 3c+κ6(∇c)2∇ 2c+κ7(∇c)4, for an isotropic solution. The results for a cubic system are also presented. The significance of the coefficients are discussed using a simple mean field pair interaction analysis. This interpretation provides physical insight into the nature of the terms and suggests that in the absence of specific knowledge about a particular situation, the derivative terms may be ordered in the following sequence of decreasing importance: κ1∇2c, κ3∇4c, [k2(∇c)2, κ5∇c·∇3c], [κ 4(∇2c)2, κ6(∇c) 2∇2c, κ7(∇c)4]. The effect on the profile and interfacial tension of a flat diffuse interface of the K3V*c and the -4(Vc)*terms are determined. The pair interaction analysis indicates that the κ3∇4c term sharpens the interface profile and decreases the interface tension. When K? is positive, the κ7(∇c)4 term increases the interfacial tension and makes the interface more diffuse.
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Physical and Theoretical Chemistry