Electric fields in an electrolyte solution near a strip of fixed potential

Electrostatic fields produced by flat electrodes are often used to manipulate particles in solution. To study the field produced by such an electrode, we consider the problem of an infinite strip of width 2a with imposed constant potential immersed in an electrolyte solution. Sufficiently close to the edge of the strip, the solution is determined by classical electrostatics and results in a field singularity. We examine two limiting cases, (a) when strip width a<<1k, the Debye screening length, and (b) when strip width is much greater than the Debye screening length, a>>1k. We present exact results for the two cases in the limit of small potentials where the Poisson-Boltzmann equation can be linearized. By drawing on an analogy with antiplane shear deformations of solids, and by employing the path-independent J integral of solid mechanics, we present a new method for determining the strength of the edge singularity. The strength of the singularity defines an exact near-field solution. In the far field the solution goes to that of a line of charge. The accuracy of the solution is demonstrated by comparison with the numerical solutions of the Poisson-Boltzmann equation using the finite element method.