Density functional theory excess free energy

There are many varieties of density functional theories depending on the choice of ideal systems and approximations for the excess free energy functional. In the study of non-uniform polymers, density functional theories have been more popular than integral equations for a variety of reasons. A survey of various theories can be found in the proceedings of a symposium on chemical applications of density functional methods [102]. This section reviews the basic concepts and tools in these theoretical methods including techniques for numerical implementation. 

In general, use of the ideal gas functional in terms of the molecular density requires computation. Despite the computational intensive nature of the resulting theory, this is probably the most widely used functional for polymers and is described greater detail below. As mentioned earlier, the approximations for the excess free energy functional are similar to those used for simple liquids. The exact expression for the ideal gas functional in this case is... 

The other two approaches divide the excess functional into a hard-core and an attractive part with different approximations for the two. Rosinberg and coworkers [126-129] have derived a functional from Wertheim s first-order perturbation theory of polymerization [130] in the limit of complete association. Woodward, Yethiraj, and coworkers [39,131-137] have used the weighted density approximation for the hard-core contribution to the excess free energy functional, that is,... 

Given the expression for K(T), one can construct an EOS by modeling the excess free energy density by = HS + u + ID + DI + DD + where is summed over contributions from hard-sphere (HS), ion-ion (II), ion-dipole (ID), dipole-ion (DI), and dipole-dipole interactions (DD), respectively. 4>ex also contains the contribution due to the internal partition function of the ion pair, = — p lnK(T). Pairing theories differ in the terms retained in the expression for ex. 

This equation states that the change in the free energy of the critical germ with the chemical potential per molecule of species / in the original phase (i.e., the mother liquor) equals the negative of the excess number An of molecules of type i in the nucleus over that present in the same volume of original space. The nucleation theorem is independent of the model and of the transition it holds true for classical nucleation theory, density functional theory, or cluster kinetic analysis and for gas-to-liquid or liquid-to-solid conversions. 

Over the last two decades the exploration of microscopic processes at interfaces has advanced at a rapid pace. With the active use of computer simulations and density functional theory the theory of liquid/vapor, liquid/liquid and vacuum/crystal interfaces has progressed from a simple phenomenological treatment to sophisticated ah initio calculations of their electronic, structural and dynamic properties [1], However, for the case of liquid/crystal interfaces progress has been achieved only in understanding the simplest density profiles, while the mechanism of formation of solid/liquid interfaces, emergence of interfacial excess stress and the anisotropy of interfacial free energy are not yet completely established [2],... 

A fundamental estimate of the excess free energy of the liquid film can be provided by the density functional theory, which is based directly on the microscopically specified molecular interactions. In its simplest form, the free energy functional, E, can be represented as [11]... 

For calculation of the equilibrium compositions of the liquid phase either the equilibrium constants of the dissociation and polycondensation reactions have to be known or they can be computed by methods which use the approach of minimizing Gibbs free energy [200-202]. In addition, ab initio modeling techniques such as density functional theory (DFT) in combination with reactive molecular dynamic (MD) simulations could be used. Once the liquid phase system is modeled, there are in principle two options to describe the vapor-liquid equilibrium. Either equations of state (EOS) or excess Gibbs free energy models (g -models) may be used to describe the thermodynamics of the liquid... 

Van der Waals augmented the excess free-energy density by terms that are large when the density gradient is large. In the simplest version of the theory (and retaining the same symbol for this augmented function) we have... 

In the language of the one-density van der Waals theory of Chapter 3, we have in a c-component system a density of excess free energy, - W, as a function of some density or composition variable, x, at fixed vsdues of the c +1 thermodynamic fields (of which only c - p+2 are independent if p phases are in equilibrium). The function W(x) here is like the W(p) in Fig. 3.2, except that now, to describe three-phase equilibrium, it must have three equal maxima, as in Fig. 8.5. In this figure the variable x in the bulk a, p, and y phases is shown to take the respective values x , x, and x as determined by the prescribed values of the c -1 (because p = 3) independent fields. Those are the points x at which W has its three maxima, and where W=0. The remaining c densities—those other than X—are imagined merely to follow the variations in x through the various interfaces just as they would vary with x in a bulk phase. Hiat is the essence of this one-density version of the theory, as explained in 3.3. 

The excess free energy of the hard-sphere fluid may be evaluated by means of methods following from different versions of the density functional theory. Numerous approximations can be used ... 

In the density functional theory of Tarazona, the excess hard-sphere free energy is calculated in a nonlocal manner by employing the concept of smoothed density [261-263] ... 

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