Exact Results for Classical Casimir Interactions: Dirichlet and Drude Model in the Sphere-Sphere and Sphere-Plane Geometry

Analytic expressions that describe Casimir interactions over the entire range of separations have been limited to planar surfaces. Here we derive analytic expressions for the classical or high-temperature limit of Casimir interactions between two spheres (interior and exterior configurations), including the sphere-plane geometry as a special case, using bispherical coordinates. We consider both Dirichlet boundary conditions and metallic boundary conditions described by the Drude model. At short distances, closed-form expansions are derived from the exact result, displaying an intricate structure of deviations from the commonly employed proximity force approximation.

Figure 1

Geometry of two spheres and a sphere plate. Shown are the centers ($C$, $C^{\prime }$) of the spheres and their foci ($F$, $F^{\prime }$). The dashed curves correspond to curves of constant bispherical coordinate $\mu$ with $\mu =\mp \pi$, $\mp \pi /2$, $\mp \pi /10$, $\mp \pi /20$ with the upper (lower) sign for $z<0$ ($z>0$), beginning with the smallest sphere.

Figure 2

Deviations from proximity force approximation in the sphere-plate geometry, measured by the function $\beta (\ell )$ defined by $E^{(\mathrm{D})/(\mathrm{Dr})}=-\zeta (3)k_{B}T/8[1/\ell +{\beta }^{(\mathrm{D})/(\mathrm{Dr})}(\ell )]$ with $\ell =L/R$. Shown are the exact results for Dirichlet conditions [Eq. (3)] and the Drude model [Eq. (8)] as solid lines and the short-distance expansions of Eqs. (9, 10), as dashed lines as a function of the logarithm (with base 10) of $\ell$. Note that ${\beta }^{(\mathrm{D})}>0$ and ${\beta }^{(\mathrm{Dr})}<0$ in the shown range. The inset shows the logarithmic relative difference between the exact result ${\beta }_{\mathrm{exact}}$ from Eqs. (3, 8), and the $\beta$ obtained from the short-distance expansions of Eqs. (9, 10).