Hard-sphere model density functional theory

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

Integral equation methods provide another approach, but their use is limited to potential models that are usually too simple for engineering use and are moreover numerically difficult to solve. They are useful in providing equations of state for certain simple reference fluids (e.g., hard spheres, dipolar hard spheres, charged hard spheres) that can then be used in the perturbation theories or density functional theories. 

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. 

It is also possible to employ pure computational methods (sometimes called in silico methods) to predict shape, or at least isomer preferences, for complexes. The simplest approach to modelling employs molecular mechanics this relies on a classical model that treats atoms as hard spheres with the bonds as springs. This is introduced in Chapter 8.3. More sophisticated approaches, such as density functional theory (DFT) are growing in popularity and capacity. Molecular modelling promises to provide excellent predictive capacity in the future without the need for laboratory synthesis, at least in the initial stages. However, laboratory-free chemistry is still far off, and synthesis and product identification remains the essence of chemistry. 

Although most of the studies of this model have focused on the fluid phase in connection with the theory of electrolyte solutions, its solid-fluid phase behavior has been the subject of two recent computer simulation studies in addition to theoretical studies. Smit et al. [272] and Vega et al. [142] have made MC simulation studies to determine the solid-fluid and solid-solid equilibria in this model. Two solid phases are encountered. At low temperature the substitutionally ordered CsCl structure is stable due to the influence of the coulombic interactions under these conditions. At high temperatures where packing of equal-sized hard spheres determines the stability a substitutionally disordered fee structure is stable. There is a triple point where the fluid and two solid phases coexist in addition to a vapor-liquid-solid triple point. This behavior can be qualitatively described by using the cell theory for the solid phase and perturbation theory for the fluid phase [142]. Predictions from density functional theory [273] are less accurate for this system. 

Rosenfeld Y Free-energy model for the inhomogeneous hard-sphere fluid mixture and density functional theory of freezing, Phys Rev Lett 63(9) 980—983, 1989. 

Semiconductor cluster polarizabilities have been the subject of some very important experimental studies via beam-deflection techniques (Backer 1997 Schlecht et al. 1995 Schnell et al. 2003 Schafer et al. 1996 Kim et al. 2005) while they have been extensively studied using quantum chemical and density functional theory. In this research realm, one of the areas intensively discussed is the evolution of the cluster s polarizabilities per atom (PPA) with the cluster size. The PPA is obtained by dividing the mean polarizability of a given system by the number of its atoms. Such property offers a straightforward tool to compare the microscopic polarizability of a given cluster with the polarizability of the bulk (see O Fig. 20-16) as the latter is obtained by the hard sphere model with the bulk dielectric constant via the Clausius-Mossotti relation ... 

The results of the simple DHH theory outlined here are shown compared with DH results and corresponding Monte Carlo results in Figs. 10-12. Clearly, the major error of the DH theory has been accounted for. The OCP model is greatly idealized but the same hole correction method can be applied to more realistic electrolyte models. In a series of articles the DHH theory has been applied to a one-component plasma composed of charged hard spheres [23], to local correlation correction of the screening of macroions by counterions [24], and to the generation of correlated free energy density functionals for electrolyte solutions [25,26]. The extensive results obtained bear out the hopeful view of the DHH approximation provided by the OCP results shown here. It is noteworthy that in... 

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. 

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