Peak, valley, and intermediate regimes in the lateral van der Waals force

We study the van der Waals (vdW) interaction between a polarizable particle and a grounded conducting corrugated surface. For sinusoidal corrugations, one knows that, under the action of the lateral vdW force, an isotropic particle is always attracted to the nearest corrugation peak, with such behavior called in the present paper as peak regime. Here, considering an anisotropic polarizable particle, and making analytical calculations valid beyond the proximity force approximation (PFA), we show that the attraction is not only toward the peaks, but, for certain particle orientations and distances from the surface, the lateral force attracts the particle to the nearest corrugation valley (valley regime), or even to an intermediate point between a peak and a valley (intermediate regime). We also show that in the configurations of transition between the peak and valley regimes the lateral vdW force vanishes, even in the presence of a corrugated surface. In addition, we find that these regimes occur in general, for periodic and nonperiodic corrugated surfaces. Moreover, we demonstrate that similar regimes arise in the classical interaction between a neutral polarized particle and a rough surface. The description of these valley and intermediate regimes, which are out of reach of the predictions based on the PFA, may be relevant for a better understanding of the interaction between anisotropic particles and corrugated surfaces in classical and quantum physics, with experimental verifications feasible in both domains.