Mean sphere approximation

The interfacial solution layer contains h3 ated ions and dipoles of water molecules. According to the hard sphere model or the mean sphere approximation of aqueous solution, the plane of the center of mass of the excess ionic charge, o,(x), is given at the distance x. from the jellium metal edge in Eqn. 5-31 ... 

Carnie and Chan and Blum and Henderson have calculated the capacitance for an idealized model of an electrified interface using the mean spherical approximation (MSA). The interface is considered to consist of a solution of charged hard spheres in a solvent of hard spheres with embedded point dipoles, while the electrode is considered to be a uniformly charged hard wall whose dielectric constant is equal to that of the electrolyte (so that image forces need not be considered). 

This was averaged over the total distribution of ionic and dipolar spheres in the solution phase. Parameters in the calculations were chosen to simulate the Hg/DMSO and Ga/DMSO interfaces, since the mean-spherical approximation, used for the charge and dipole distributions in the solution, is not suited to describe hydrogen-bonded solvents. Some parameters still had to be chosen arbitrarily. It was found that the calculated capacitance depended crucially on d, the metal-solution distance. However, the capacitance was always greater for Ga than for Hg, partly because of the different electron densities on the two metals and partly because d depends on the crystallographic radius. The importance of d is specific to these models, because the solution is supposed (perhaps incorrectly see above) to begin at some distance away from the jellium edge. 

In a gas or a mixture of two gases 1 and 2, the mean free path must be obtained as an average over the velocity distribution. We quote here a simple formula based on the (crude) hard sphere approximation,... 

Frequency of collisions. The mean frequency of collisions is similarly expressed in the hard spheres approximation as... 

For rough estimates, the collisional cross section may be assumed to be velocity-independent, Qn(vn) = Qo = constant (hard-sphere approximation), so that the mean time between collisions becomes... 

In all of the discussion above, comparisons have been made between various types of approximations, with the nonlinear Poisson-Boltzmann equation providing the standard with which to judge their validity. However, as already noted, the nonlinear Poisson-Boltzmann equation itself entails numerous approximations. In the language of liquid state theory, the Poisson-Boltzmann equation is a mean-field approximation in which all correlation between point ions in solution is neglected, and indeed the Poisson-Boltzmann results for sphere-sphere [48] and plate-plate [8,49] interactions have been derived as limiting cases of more rigorous approaches. For many years, researchers have examined the accuracy of the Poisson-Boltzmann theory using statistical mechanical methods, and it is... 

The value of DFT is evidently dependent on the accessibility and accuracy of the grand potential functional, Si [p(r)]. The usual practice is to treat the molecules as hard spheres and divide the fluid-fluid potential into attractive and repulsive parts. A mean field approximation is used to simplify the former by the elimination of correlation effects. The hard sphere term is further divided into an ideal gas component and an excess component (Lastoskie etal., 1993). The ideal component is considered to be exactly local, since this part of the Helmholtz free energy per molecule depends only on the density at a particular value of r. 

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. 

The latest models propose to represent electrolyte solutions as a collections of hard spheres of equal size, ions, immersed in a dielectric continuum, the solvent. For such a system, what is called the Mean Spherical Approximation, MSA, has been successful in estimating osmotic and mean activity coefficients for aqueous 1 1 electrolyte solutions, and has provided a reasonable fit to experimental data for dilute solutions of concentrations up to -0.3 mol dm". The advantage in this approach is that only one... 

This equation was first used by Born, Huggins, and Meyer and therefore bears their names. The first two terms represent, respectively, the attractive and repulsive potentials. The last two terms represent dipole-dipole and dipole-quadrupole potentials, respectively. In spite of allowing for the dipole interactions, the calculation is still a hard-sphere one, a mean spherical approximation, because the forces are not allowed to change the shape and the position of the particles. Later on, Saboungi et al. 

Blum, L., Kalyuzhnyi, Yu.V., Bernard, O., and Herrera-Pacheco, J.N. Sticky charged spheres in the mean-spherical approximation A model for colloids and polyelectrolytes. Journal of Physics - Condensed Matter, 1996, 8, No. 25A, p. A143-A167. 

Kalyuzhnyi, Yu.V., and Cummings, P.T. Multicomponent mixture of charged hard-sphere chain molecules in the polymer mean-spherical approximation. Journal of Chemical Physics, 2001, 115, p. 540-551. 

This approximation is known as the mean spherical approximation (MSA). For the case of a hard-sphere fluid for which u r) = 0, the MSA is equivalent to the PY approximation. For the case that the hard spheres have embedded point charges, the function u(r) is simply Coulomb s law. Although the MSA provides the least detailed expression for c(r), it is popular because the OZ equation can often be solved using this approximation to yield an analytical expression for g(r). The equation for g(r) within a hard sphere is... 

From the density, it is now possible to construct the potential for the next iteration, and also to calculate total energies. During the self-consistency loop, we make a spherical approximation to the potential. This is because the computational cost for using the full potential is very high, and that the density converges rather badly in the corners of the unit cell. In order not to confuse matters with the ASA (Atomic Sphere Approximation) this is named the Spherical Cell Approximation (SC A) [65]. If we denote the volume of the Wigner-Seitz cell centered at R by QR, we have fiR = QWR — (4 /3), where w is the Wigner-Seitz radius. This means that the whole space is covered by spheres, just as in the ASA. We will soon see why this is practical, when we try to create the potential. 

The van der Waals approximation discussed in Section LA applies the mean field approximation in the solid phase in the same way as in the fluid phase. Baus and co-workers [150,151] have recently presented an alternative formulation in which the localization of the molecules in the solid phase is taken into account. They have applied this to the understanding of trends in the phase diagrams of systems of hard spheres with attractive tails as the range of attractions is changed. For the mean field term in the solid phase they use the static lattice energy for the given interaction potential and crystal lattice. A similar approach was used earlier to incorporate quad-rupole-quadrupole interactions into a van der Waals theory calculation of the phase diagram of carbon dioxide [152]. 

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