Electric double layers near modulated surfaces

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Abstract
The screened electrostatic potential and free energy of electric double layers near modulated boundaries are computed in terms of the height function h of the surface, as an extension of linearized Gouy-Chapman theory. Two approaches, one perturbative, the other iterative, are explored and compared, both expressing the average electrostatic potential away from the surface as a power series involving n-point correlations of h. When compared order by order in powers of the height function, the two methods are found to be equivalent, but they differ in the degree of differentiability demanded of h for a convergent expansion, the perturbative technique requiring infinite differentiability, and the iterative method needing only twice-differentiable functions. The wider applicability of the iterative method is shown to arise from the summation of certain infinite classes of terms in the perturbative expansion which remove ultraviolet divergences associated with the high-order nondifferentiability of the height function. The electrostatic free energy of interacting double layers is found to depend on height-height correlations both within and between the surfaces, a result that may also be interpreted as expressing the coefficients of capacity and induction of conductors in terms of their surface roughness. For boundaries with a well-defined modulation wavelength, the thermodynamic and electrostatic quantities are computed perturbatively in terms of the ratio of the modulation amplitude to the Debye-Hückel screening length, and the characteristic gradients of the height function. It is suggested that these long-wavelength results may find application in the study of the stability and structure of multilamellar liquid crystals composed of modulated membranes.