Inhomogeneous system, Helmholtz free

Let us consider a multicomponent two-phase system with a plane interface of area A in complete equilibrium, and let us focus on the inhomogeneous interfacial region. Our approach is a point-thermodynamic approach [92-96], and our key assumption is that in an inhomogeneous system, it is possible to define, at least consistently, local values of the thermodynamic fields of pressure P, temperature T, chemical potential p, number density p, and Helmholtz free-energy density xg. At planar fluid-fluid interfaces, which are the interfaces of our interest here, the aforementioned fields and densities are functions only of the height z across the interface. 

Let us first consider the case where density gradient terms do not enter explicitly into the expression for the Helmholtz free energy P. For the free energy T of the entire inhomogeneous system of volume V, entropy S, and of A number of molecules of component i (i= 1, 2,. .., t), we may write [92-96,107]... 

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 ... 

Ideas that go back to van der Waals [217, 218] and Lord Rayleigh [219] on inhomogeneous systems were applied by Cahn and Hilliard [216] to the interface problem. In inhomogeneous fluids, the Helmholtz free energy is a functional of the component density distributions. Although exact formal expressions for this functional have been derived [220,221] from statistical mechanics, they are impractical without approximation [222]. In the gradient approximation, this functional has been expressed as the sum of two contributions one is a function of the local composition and the other is a function of the local composition derivatives [216, 223, 224]. The free energy for a binary system is postulated to have the form ... 

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