Linearized Poisson-Boltzmann equation function

This series arises naturally, when expressing the Coulomb potential of a charge separated by a distance s from the origin in terms of spherical coordinates. The positive powers result when r < s, while for r > s the potential is described by the negative powers. Similarly the solutions of the linearized Poisson-Boltzmann equation are generated by the analogous expansion of the shielded Coulomb potential exp[fix]/r of a non-centered point charge. Now the expansion for r > s involves the modified spherical Bessel-functions fo (x), while lor r < s the functions are the same as for the unshielded Coulomb potential,... 

Integrating the linearized Poisson-Boltzmann equation (2.123) over x, we get the ion distribution function in the diffuse layer... 

The MSA is fundamentally connected to the Debye-Hiickel (DH) theory [7, 8], in which the linearized Poisson-Boltzmann equation is solved for a central ion surrounded by a neutralizing ionic cloud. In the DH framework, the main simplifying assumption is that the ions in the cloud are point ions. These ions are supposed to be able to approach the central ion to some minimum distance, the distance of closest approach. The MSA is the solution of the same linearized Poisson-Boltzmann equation but with finite size for all ions. The mathematical solution of the proper boundary conditions of this problem is more complex than for the DH theory. However, it is tractable and the MSA leads to analytical expressions. The latter shares with the DH theory the remarkable simplicity of being a function of a single screening parameter, generally denoted by r. For an arbitrary (neutral) mixture of ions, this parameter satisfies a simple equation which can be easily solved numerically by iterations. Its expression is explicit in the case of equisized ions (restricted case) [12]. One has... 

The solution procedure of the linearized Poisson-Boltzmann equation used above is not suited to include hard core effects of the ions, the most we can do is to give a size to the central ion, but that makes the pair distribution function asymmetric. To include the hard core effects in a symmetric way, we have to change the formalism. We notice, first, that Poisson s equation (1.8) relates the potential (r) to the charge distribution We can formally integrate this equation to yield ... 

Analytical results for the Green function are obtained for a planar geometry only. For a planar surface, the linearized Poisson-Boltzmann equation yields the screened potential ... 

The linearized Poisson-Boltzmann equation usually must be solved numerically, such as via the finite difference method. The principle of this method is as follows. Consider a small cube of side length h centered at a certain point, say r (see Eigure 2). Integrating Eq. [20] over the volume occupied by the cube and applying Gauss theorem (lyV-A)dv = n do), approximating continuous functions by distinct values at indicated points inside and outside the cube, and finally approximating derivatives by the ratio of the differences, we... 

In Figure 6a, the force per unit area between surfaces with grafted polyelectrolyte brushes, plotted as a function of their separation distance 2d, calculated in the linear approximation, is compared with the numerical solution of the nonlinear Poisson—Boltzmann equations, for a system with IV = 1000, a = 1 A, ce = 0.01 M, s2 = 1000... 

The cell model from Katchalsky, Onsager and Manning, which was originally only applicable to linear PEs [26], was extended by Deshkovski in 2001 to a more general two zone model based on the mean field approximation of the nonlinear Poisson Boltzmann equation [84]. In this two zone model, the counter ions around a cylindrical rod (PE) form the area close to the PE, and a spherical volume which extends up to the distances between the PE is the area far away from the PE [84]. This model is able to explain the mean distance of the counterions from the PE chain as a function of the counterion concentration. 

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