Helmholtz free energy theory

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... 

The problem at hand is the evaluation of the activity coefficient defined in Eq. (76). It will be assumed that only pairwise interactions between the defects need be considered at the low defect concentrations we have in mind. (The theory can be extended to include non-pairwise forces.23) Then the cluster function R(n) previously defined in Eq. (78) is the sum of all multiply connected diagrams, in which each bond represents an /-function, which can be drawn among the set of n vertices, the /-function being defined by Eqs. (66), (56), and (43). The Helmholtz free energy of interaction of two defects appearing in this definition can be written as... 

Reciprocal molten salt systems are those containing at least two cations and two anions. We shall deal with the simplest member of this class, that containing the ions A+, B+, X-, and Y-. The four constituents of the solution, AX, BX, AY, and BY, will be designated by 1, 2, 3, and 4 respectively. There are four ions in the system and one restriction of electroneutrality. Consequently, of the four constituents, there are only three which are independent components. In order to calculate the Helmholtz free energy of mixing conveniently, we must (arbitrarily) choose the three components. Here we choose BX, AY, and BY. This choice requires that in order to make mixtures of some compositions a negative quantity of BY must be used. This presents no difficulty in the theory and is thermodynamically self-consistent. One mole of some arbitrary composition (XA, XB, Yx, XY) can be made by mixing Arx moles of BX (component 2), XA moles of AY (component 3), and (XY — XA) moles of BY (component 4). ... 

In the MSF theory, the function,/, in addition to simple reptation, is also related to both the elastic effects of tube diameter reduction, through the Helmholtz free energy, and to dissipative, convective molecular-constraint mechanisms. Wagner et al. arrive at two differential equations for the molecular stress function/ one for linear polymers and one for branched. Both require only two trial-and-error determined parameters. 

The free energy of the two surface system is calculated from the partition function expressed in terms of Flory-Huggins solution theory. The Helmholtz free energy per site, relative to pure solvent, and at constant [Pg.175]

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. 

To further speed up this approach, one can replace the expensive explicit-solvent simulations with implicit ones. Statistical mechanical theory gives the Helmholtz free energy A, apart from the scaling constant of the classical partition function that cancels out in binding energy calculations, as... 

Tarazona and Navascues have proposed a perturbation theory based upon the division of the pair potential given in Eq. (3.5.1). In addition, they make a further division of the reference potential into attractive and repulsive contributions in the manner of the WCA theory. The resulting perturbation theory for the interfacial properties of the reference system is constructed through adaptation of a method developed by Toxvaerd in his extension of the BH perturbation theory to the vapor-liquid interface. The Tarazona-Navascues theory generates results for the Helmholtz free energy and surface tension in addition to the density profile. Chacon et al. have shown how the perturbation theories based upon Eq. (3.5.1) may be developed by a series of approximations within the context of a general density-functional treatment. 

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