Spatially varying dielectric constant

Lowered and spatially varying dielectric constants are readily applied in Poisson theory through a suitable definition of e(r) in Eq. (1). However, it is less obvious how to modify Eq. (2) fo implemenf such models within the GB formalism. In fact, there are two separate issues that have to be addressed ... 

A similar approach to the boundary condition for the potential at the metal-solution interface has been applied by Biesheuvel et al., in consideration of diffuse charge effects in galvanic cells, desalination by porous electrodes, and transient response of electrochemical cells (Biesheuvel and Bazant, 2010 Biesheuvel et al., 2009 van Soestbergen et al., 2010). However, their treatment neglected the explicit effect of In principle, the PNP model could be modified to incorporate size-dependent and spatially varying dielectric constants in nanopores, as well as ion saturation effects at the interface. However, in a heuristic fashion, such variations could be accounted for in the Helmholtz capacitance of the Stern double layer model. 

Think in terms of a capacitor. With a pure, nonconducting dielectric material there is a constant electric field between plates (see Fig. L3.20). But across a salt solution between nonreactive, nonconducting, ideally bad electrodes (no chemical reactions at interfaces), there is a spatially varying electrostatic double-layer field set up by the electrode walls (see Fig. L3.21). 

First, as is apparent from the previous section, each nanostmctured material is associated with a specific novel property or a significant improvement in a specific property resulting from the nanoscale structuring. As a consequence, the type of nanostmcturing used must be based on a spatial dependence of some parameter related to the property under consideration. This parameter could be, for example, the material density, transport parameters, or the dielectric constant. Another consequence is that the upper size limit of the stmctural features varies depending on the property considered. 

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