Ray-theory approach to electrical-double-layer interactions

A novel approach is presented for analyzing the double-layer interaction force between charged particles in electrolyte solution, in the limit where the Debye length is small compared with both interparticle separation and particle size. The method, developed here for two planar convex particles of otherwise arbitrary geometry, yields a simple asymptotic approximation limited to neither small zeta potentials nor the “close-proximity” assumption underlying Derjaguin's approximation. Starting from the nonlinear Poisson-Boltzmann formulation, boundary-layer solutions describing the thin diffuse-charge layers are asymptotically matched to a WKBJ expansion valid in the bulk, where the potential is exponentially small. The latter expansion describes the bulk potential as superposed contributions conveyed by “rays” emanating normally from the boundary layers. On a special curve generated by the centers of all circles maximally inscribed between the two particles, the bulk stress—associated with the ray contributions interacting nonlinearly—decays exponentially with distance from the center of the smallest of these circles. The force is then obtained by integrating the traction along this curve using Laplace's method. We illustrate the usefulness of our theory by comparing it, alongside Derjaguin's approximation, with numerical simulations in the case of two parallel cylinders at low potentials. By combining our result and Derjaguin's approximation, the interaction force is provided at arbitrary interparticle separations. Our theory can be generalized to arbitrary three-dimensional geometries, nonideal electrolyte models, and other physical scenarios where exponentially decaying fields give rise to forces.

Figure 1

Asymptotic solution for the electric potential around a single particle.

Figure 2

The thin-double-layer solution for two particles, with $\delta \ll D$, obtained by superposing two single-particle solutions.

Figure 3

The interaction force calculated by integrating stresses on a bulk curve generated by the centers of all maximally inscribed circles.

Figure 4

Dimensionless repulsion force $F$, scaled by ${\delta }^{-3/2}{e}^{-D/\delta }$, as a function of minimum separation $D$ between two identical parallel cylinders held at a fixed low potential, $\zeta =0.3$. See Sec. 4a for details regarding the data and approximations shown.