Sign inversion in the lateral van der Waals force between an anisotropic particle and a plane with a hemispherical protuberance: an exact calculation

We investigate the lateral van der Waals (vdW) force between an anisotropic polarizable particle and a perfectly conducting plane with hemispherical protuberance radius $R$. predict, via exact calculation, sign inversion in vdW force, sense that, instead of pointing to protuberance, certain situations this points opposite direction. In literature, predictions inversions were based on perturbative solutions, valid when height is very small compared distance $z_0$ plane. Here, taking into account formulas, we how such nontrivial geometric effect depends ratio $R/z_0$, orientation anisotropy affect inversion.