Complex Geometry and Electric Double Layers
- PDF / 643,136 Bytes
- 10 Pages / 420.48 x 639 pts Page_size
- 15 Downloads / 197 Views
COMPLEX GEOMETRY AND ELECTRIC DOUBLE LAYERS
BERTRAND DUPLANTIER Service de Physique Th~oriquet de Saclay, CE-Saclay, 91191 Gif-sur-Yvette Cedex, France and The James Franck Institute, The University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637
ABSTRACT The properties of electric double layers near curved surfaces of arbitrary shape and genus are obtained exactly within the Debye-Hiickel theory by means of multiple-scattering expansion. For smooth membranes, geometric and topological feature of the electrostatic free energy then emerge through convergent expansions in inverse powers of the principal radii of curvature. Some consequences for the electrostatic stability of various membrane shapes are considered. We also study the effects of surface singularities, e.g., wedges, on the thermodynamics of electric double layers near a rough colloid. Each wedge yields an additive contribution to the free energy F that is a function of the angle. A probabilistic Brownian representation of F is given, which is entirely similar to that of vibration eigenmodes given by M. Kac long ago in "Can one hear the shape of a drum?" [Amer. Math. Monthly 73S, 1 (1966)]. The analysis yields a universal scaling law for the free energy of a rough colloid with its fractal Minkowski dimension.
INTRODUCTION When charged or held at fixed potential, surfaces immersed in an electrolyte interact via screened Coulomb interactions, which are coupled by specific boundary conditions to position or shape fluctuations. Physical systems in which this problem arises include polyelectrolytes, colloidal suspensions and crystals, and bilayer fluid membranes. The problem we want to address is the following: what is the free energy of a sMt of closed charged surfaces of arbitrary shape immersed in an electrolyte? [1,2]. Multiple-Scattering Expansions The continuum Gouy-Chapman [3,4] theory of electric double layers amounts mathematically to solving the nonlinear Poisson-Boltzmann equation, with specific boundary conditions. This nonlinearity has confined most studies to consider only highly idealized and symmetric shapes for the surfaces [1,5,6]; the same has been true for most applications of the linearized Debye-Hiickel equation, which is characterized microscopically by a single parameter, the screening length ADH [1,7,8]. This equation governs the behavior of the electrical potential 0 in an electrolyte provided that 0
Data Loading...
