Free energy functionals density functional theory

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

In order to accurately describe such oscillations, which have been the center of attention of modern liquid state theory, two major requirements need be fulfilled. The first has already been discussed above, i.e., the need to accurately resolve the nonlocal interactions, in particular the repulsive interactions. The second is the need to accurately resolve the mechanisms of the equation of state of the bulk fluid. Thus we need a mechanistically accurate bulk equation of state in order to create a free energy functional which can accurately resolve nonuniform fluid phenomena related to the nonlocality of interactions. So far we have only discussed the original van der Waals form of equation of state and its slight modification by choosing a high-density estimate for the excluded volume, vq = for a fluid with effective hard sphere diameter a, instead of the low-density estimate vq = suggested by van der Waals. These two estimates really suggest... 

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, different approximations are invoked for the hard-core contribution and the attractive contribution to the free energy functional. For the hardcore contribution, two accurate approximations can be obtained from the fundamental measure theory [108] and the weighted density approximation... 

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,... 

Density Functional Theory does not require specific modifications, in relation to the solvation terms [9], with respect to the Hartree-Fock formalism presented in the previous section. DFT also absorbs all the properties of the HF approach concerning the analytical derivatives of the free energy functional (see also the contribution by Cossi and Rega), and as a matter of fact continuum solvation methods coupled to DFT are becoming the routine approach for studies of solvated systems. 

By contrast, the alternative PCM-LR approach [15-17] determines in a single step calculation the excitation energies for a whole manifold of excited states. This general theory may be combined with the Time-Dependent Density Functional Theory (TDDFT) as QM level for the solute. Within the PCM-TDDFT formalism, the excitation energies are obtained by proper diagonalization of the free energy functional Hessian. 

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. 

With the chemical potential and pressure obtained in the form of the closed expressions (4.A.9) and (4.A.11) in Chapter 4, the phase coexistence envelope can be localized directly by solving the mechanical and chemical equilibrium conditions (1.134) and (1.135) for the vapor and liquid phase densities, Pvap and puq, whether or not the solution exists for all intermediate densities. Provided the isotherm is continuous across all the region of vapor-liquid phase coexistence, Eqs.(1.134) and (1.135) are exactly equivalent to the Maxwell construction on either pressure or chemical potential isotherm. This stems from the fact that the RISM/KH theory yields an exact differential for the free energy function (4.A. 10) in Chapter 4, which thus does not depend on a path of thermodynamic integration. 

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]... 

Field theoretical simulations [74,75,80] avoid any saddle point approximation and provide a formally exact solution of the standard model of the self-consistent field theory. To this end one has to deal with a complex free energy functional as a fimction of the composition and density. This significantly increases the computational complexity. Moreover, for certain parameter regions, it is very difficult to obtain reliable results due to the sign problem that a complex weight imparts onto thermodynamical averages [80]. We have illustrated that for a dense binary blend the results of the field theoretical simulations and the EP theory agree quantitatively, i.e., density and composi-... 

Fredrickson and Helfand developed a weak segregation theory with fluctuations [77], starting by mapping the Leibler free energy functional onto one treated earlier by Brazovskii [78]. They used the UCA and treated the three classical phases. This has since been extended to the G phase [79,80] and to triblocks [81], and modified to include a multihannonic approximation for the density profiles, although not for the G phase [82]. 

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