Mean field electrostatics spherical distribution

Given a molecular or supra-molecular system embedded in a solvent charge distribution, the solute-solvent interaction can be modelled using a mean field (MF) approach [21-24], which treats the solvent as a continuum fully defined by a dielectric constant (s) and by a shape function, uniquely identifying the space regions where the solute and the solvent are placed. The boundary between the two domains is a compact cavity S, which in Fig. 17.2 has been represented as a spherical boundary surface including the explicit molecules. Then, the major issue related to such a scheme is how to model the interactions between the continuum and the explicit molecules placed inside the cavity. In a static picture, such interactions are of two types, electrostatic and non-electrostatic, whereas when such a model is used in MD simulations, an additional potential is included, in order to ensure that all the molecules remain confined inside the cavity during the simulation with a correct density up to the boundary [21]. 

To determine the distribution of electric potential, as modified by surface conductance, we again take the electric field to be spherically symmetric and to satisfy the Laplace equation. A thin spherical double layer shell is considered to surround the particle, and the conductivity of this shell is taken to have the mean value cr(. In reality the conductivity in the thin double layer varies continuously. Outside of the double layer shell the bulk conductivity is that of the electrolyte. This electrostatics problem is a straightforward one in which, from the Laplace equation, the solution for the potential is... 

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