Mean-field Boltzmann approximation

In addition to the nearest-neighbor interaction, each ion experiences the electrostatic potential generated by the other ions. In the literature this has generally been equated with the macroscopic potential 0 calculated from the Poisson-Boltzmann equation. This corresponds to a mean-field approximation (vide infra), in which correlations between the ions are neglected. This approximation should be the better the low the concentrations of the ions. 

The electric field within each cell is determined in the mean field approximation from the Poisson Boltzmann equation (2.3.1), written for the prototypical case of a symmetric low molecular electrolyte of valency z added to a polyelectrolyte with a single type of proper counterion of va-... 

As known from nuclear physics, a direct extension of quantal mean-field is delicate [36]. Fortunately enough, in the high excitation regime, where semi-classical approximations are likely to become acceptable, Boltzmann-like kinetic equations, offer an efficient alternative. They thus have been extensively used in nuclear physics for describing heavy-ion collisions (VUU,... 

Most of the water-mediated interactions between surfaces are described in terms of the DLVO theory [1,2]. When a surface is immersed in water containing an electrolyte, a cloud of ions can be formed around it, and if two such surfaces approach each other, the overlap of the ionic clouds generates repulsive interactions. In the traditional Poisson-Boltzmann approach, the ions are assumed to obey Boltzmannian distributions in a mean field potential. In spite of these rather drastic approximations, the Poisson-Boltzmann theory of the double layer interaction, coupled with the van der Waals attractions (the DLVO theory), could explain in most cases, at least qualitatively, and often quantitatively, the colloidal interactions [1,2]. 

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 name, DLYO, originates from the first letter in the surname of the four authors (Derjaguin, Landau, Verwey and Overbeek) from two different groups, which originally published these ideas. The theory is based on the competition between two contributions, a repulsive electric double layer and an attractive van der Waals force [4,5]. The interaction in the electric double layer was originally obtained from mean field calculations via the Poisson-Boltzmann equation [Eq. (4)]. However, the interaction can also be determined by MC simulations (Sec. II. B) and by approximate integral equations like HNC (Sec. II. C). This chapter will focus on the first two possibilities. 

In Chapter 2, we saw that the configuration integral is the key quantity to be calculated if one seeks to compute thermal properties of classical (confined) fluids. However, it is immediately apparent that this is a formidable task because it reejuires a calculation of Z, which turns out to involve a 3N-dimensional integration of a horrendously complex integrand, namely the Boltzmann factor exp [-C7 (r ) /k T] [ see Eq. (2.112)]. To evaluate Z we either need additional simplifjfing assumptions (such as, for example, mean-field approximations to be introduced in Chapter 4) or numerical approaches [such as, for instance, Monte Carlo computer simulations (see Chapters 5 and 6), or integral-equation techniques (see Chapter 7)]. 

If one accepts the continuum, linear response dielectric approximation for the solvent, then the Poisson equation of classical electrostatics provides an exact formalism for computing the electrostatic potential (r) produced by a molecular charge distribution p(r). The screening effects of salt can be added at this level via an approximate mean-field treatment, resulting in the so-called Poisson-Boltzmann (PB) equation [13]. In general, this is a second order non-linear partial differential equation, but its simpler linearized form is often used in biomolecular applications ... 

A number of alternatives to the GB, both below and above it on the "approximations tree" have been tested in molecular dynamics simulations. Approaches that make fewer fundamental approximations to reality, such as those based directly on the Poisson-Boltzmann treatment of solvation or ones that even go beyond the mean-field level, are particularly attractive from the accuracy point of view. More testing is needed to better characterize the overall performance of these models in practical MD simulations. 

Here the gradient square term describes the extra free energy cost due to concentration inhomogeneities. Boltzmann s constant is denoted as and the parameter r then has dimensions of length (in microscopic models, e.g., lattice models of binary mixtures treated in mean-field approximation, r has the meaning of the range of pairwise interactions among the particles). 

Figure 2.16. The surface tension of a lattice gas as a function of the temperature, assuming that the surface is the (111) face of a FCC lattice, with units chosen so that the lattice spacing is imity. Boltzmann s constant is unity and the interaction energy e = — 1. In these units the critical temperature is 3. The solid line is the prediction of square gradient theory, whereas the points are the predictions of an analogous mean-field theory in which no small-gradient approximation is made.

A second nonlinear problem to consider is the Poisson-Boltzmann (PB) mean-field theory of ionic solution electrostatics. " " This approximate theory has found wide application in biophysics involving problems such as protein-protein and protein-membrane interactions. At a qualitative level, the PB equation arises from replacement of the exact mobile-ion charge distribution by its average, assuming that the average is given by the Boltzmann factor for the ions interacting with the mean electrostatic potential ... 

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