On the van der Waals force between two spheres or a sphere and a wall

Abstract

Exact, explicit, analytical expressions are derived for the van der Waals free energy of interaction between two spheres or between a sphere and a wall for arbitrary radii, dielectric constants and separation. The analysis utilizes van Kampen's formulation and solves the boundary value problem for the dispersion relation in the system of bispherical coordinates, where solutions of the Laplace equation are non-orthogonal. This non-orthogonality leads to sets of difference equations which are solved in a manner analogous to the solution of differential equations in terms of Greens functions. Comparison is made with the Hamaker results for small dielectric differences, and with the asymptotic results of Mitchell and Ninham for both large and small separations. The asymptotic expressions of Langbein for large separations and for large dielectric differences are also compared with the exact result. Criteria are given for the validity of the asymptotic forms.