Mean-field theory Helmholtz free energy

Theoretically, several aspects of the Thommes-Findenegg experiment can be analyzed at the mean-field level [157]. A key quantity of a mean-field theory of confined fluids is the (Helmholtz) free energy, given by... 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

Before setting out on the exact mean field theory solution to the one-dimensional colloid problem, I wish to emphasize that the existence of the reversible phase transition in the n-butylammonium vermiculite system provides decisive evidence in favor of our model. The calculations presented in this chapter are deeply rooted in their agreement with the experimental facts on the best-studied system of plate macroions, the n-butylammonium vermiculite system [3], We now proceed to construct the exact mean field theory solution to the problem in terms of adiabatic pah-potentials of both the Helmholtz and Gibbs free energies. It is the one-dimensional nature of the problem that renders the exact solution possible. 

Several flaws in the Flory—Huggins mean-field theory have been discussed in the previous section. The theory does not explicitly show any volume dependence of the free energy. The so-called equation-of-state theories have in common that they are founded on the expression A = f T, V) where A is the Helmholtz free energy. An equation of state, p = f T, V), can be formulated because p = dA/dV)T. 

On the basis of statistical thermodynamic arguments and the mean field approximation, the lattice fluid theory provides the following expression for the equilibrium Helmholtz free energy per polymer mass in the limit of infinite coordination number (12) ... 

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