A generalized Derjaguin approximation for electrical-double-layer interactions at arbitrary separations

Abstract

Derjaguin’s approximation provides the electrical-double-layer interaction force between two arbitrary convex surfaces as the product of the corresponding one-dimensional parallel-plate interaction potential and an effective radius R (function of the radii of curvature and relative orientation of the two surfaces at minimum separation). The approximation holds when both the Debye length 1/κ and minimum separation h are small compared to R. We show here that a simple transformation, R⇒[R][K1][K2]K1K2−−−−−−√, yields an approximation uniformly valid for arbitrary separations h; here, Ki is the Gaussian curvature of particle i at minimum separation, and [ ⋅ ] is an operator which adds h/2 to all radii of curvature present in the expression on which it acts. We derive this result in two steps. First, we extend the two-dimensional ray-theory analysis of Schnitzer [Phys. Rev. E 91, 022307 (2015)], valid for κh, κR ≫ 1, to three dimensions. We thereby obtain a general closed form expression for the force by matching nonlinear diffuse-charge boundary layers with a WKBJ-type expansion describing the bulk potential, and subsequent integration via Laplace’s method of the traction over the medial surface generated by all spheres maximally inscribed between the two surfaces. Second, we exploit the existence of an overlap domain, 1 ≪ κh ≪ κR, where both the ray-theory and the Derjaguin approximations hold, to systematically form the generalized mapping. The validity of the result is demonstrated by comparison with numerical computations.

Derjaguin’s approximation provides the electrical-double-layer interaction force between two arbitrary convex surfaces as the product of the corresponding one-dimensional parallel-plate interaction potential and an effective radius R (function of the radii of curvature and relative orientation of the two surfaces at minimum separation). The approximation holds when both the Debye length 1/κ and minimum separation h are small compared to R. We show here that a simple transformation, R⇒[R][K1][K2]K1K2−−−−−−√, yields an approximation uniformly valid for arbitrary separations h; here, Ki is the Gaussian curvature of particle i at minimum separation, and [ ⋅ ] is an operator which adds h/2 to all radii of curvature present in the expression on which it acts. We derive this result in two steps. First, we extend the two-dimensional ray-theory analysis of Schnitzer [Phys. Rev. E 91, 022307 (2015)], valid for κh, κR ≫ 1, to three dimensions. We thereby obtain a general closed form expression for the force by matching nonlinear diffuse-charge boundary layers with a WKBJ-type expansion describing the bulk potential, and subsequent integration via Laplace’s method of the traction over the medial surface generated by all spheres maximally inscribed between the two surfaces. Second, we exploit the existence of an overlap domain, 1 ≪ κh ≪ κR, where both the ray-theory and the Derjaguin approximations hold, to systematically form the generalized mapping. The validity of the result is demonstrated by comparison with numerical computations.