Smooth-density sphere model

When the smoothed or nonlocal density approximation (or NL-DFT model) is used, the weighting function is chosen so that the hard-sphere direct pair-correlation function is well described for the uniform fluid over a wide range of densities. One example of such a weighting function is the model proposed by Tarazona [69], which uses the Percus-Yevick theory for approximating the correlation function over a wide range of density. In this case, the weighting function is expanded as a power series of the smoothed density. The use of a smoothed density in NL-DFT provides an oscillating density profile expected of a fluid adjacent to a sohd surface, the existence of which is corroborated by molecular simulation results [17,18]. 

The form of the function efr ( ) is different in different versions of the smoothed-density approximation proposed by Somo-za and Tarazona [71, 72] and by Poniwier-ski and Sluckin [69, 73]. The density functional model of Somoza and Tarazona is based on the reference system of parallel hard ellipsoids that can be mapped into hard spheres. In the Poniwierski and Sluckin theory the effective weight function is determined by the Maier function for hard sphe-rocylinders and the expression for Ayr (p) is obtained from the Carnahan-Starling ex-... 

We start with the simplest model of the interface, which consists of a smooth charged hard wall near a ionic solution that is represented by a collection of charged hard spheres, all embedded in a continuum of dielectric constant c. This system is fairly well understood when the density and coupling parameters are low. Then we replace the continuum solvent by a molecular model of the solvent. The simplest of these is the hard sphere with a point dipole[32], which can be treated analytically in some simple cases. More elaborate models of the solvent introduce complications in the numerical discussions. A recently proposed model of ionic solutions uses a solvent model with tetrahedrally coordinated sticky sites. This model is still analytically solvable. More realistic models of the solvent, typically water, can be studied by computer simulations, which however is very difficult for charged interfaces. The full quantum mechanical treatment of the metal surface does not seem feasible at present. The jellium model is a simple alternative for the discussion of the thermodynamic and also kinetic properties of the smooth interface [33, 34, 35, 36, 37, 38, 39, 40]. 

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