Effect of concurrent geometry and roughness in interacting surfaces

We study the interaction energy between two surfaces, one of them flat, the other describable as the composition of a small-amplitude roughness and a slightly curved, smooth surface. The roughness, represented by a spatially random variable, involves Fourier wavelengths shorter than the (local) curvature radii of the smooth component of the surface. After averaging the interaction energy over the roughness distribution, we obtain an expression which only depends on the smooth component. We then approximate that functional by means of a derivative expansion, calculating explicitly the leading- and next-to-leading-order terms in that approximation scheme. We analyze the resulting interplay between shape and roughness corrections for some specific roughness models in the cases of electrostatic and Casimir interactions.

Figure 1

Ratio between the electrostatic force at second order in the roughness and the corresponding force at zeroth order $\left({\chi }_{\mathrm{P}}=\frac{{\mathcal{F}}_{P,2}}{{\mathcal{F}}_{P,0}}\right)$, as a function of the distance $\delta =d/R$, for the sphere-plane geometry. Parameters are $p_{\max }R=10000,p_{\min }R=100$, and $\epsilon /R=0.001$.

Figure 2

Ratio between the Dirichlet Casimir energy at second order in the roughness and zeroth order as a function of the distance $\delta =d/R$, for the sphere-plane geometry. Parameters are $p_{\max }R=10000,p_{\min }R=100$, and $\epsilon /R=0.001$.

Figure 3

Ratio between the Neumann Casimir energy at second and zeroth order in the roughness as a function of the distance $\delta =d/R$, for the sphere-plane geometry. Parameters are $p_{\max }R=10000,p_{\min }R=100$, and $\epsilon /R=0.001$.