Investigation of the range of validity of the pairwise summation method applied to the calculation of the surface roughness correction to the van der Waals force

It has long been recognized that stochastic surface roughness can considerably change the van der Waals (vdW) force between interacting surfaces and particles. However, few analytical expressions for the vdW force between rough surfaces have been presented in the literature. Because they have been derived using perturbative methods or the proximity force approximation the expressions are valid when the roughness correction is small and for a limited range of roughness parameters and surface separation. In this work, a nonperturbative approach, the effective density method (EDM) is proposed to circumvent some of these limitations. The method simplifies the calculations of the roughness correction based on pairwise summation (PWS), and allows us to derive simple expressions for the vdW force and energy between two semispaces covered with stochastic rough surfaces. Because the range of applicability of PWS and, therefore, of our results, are not known a priori, we compare the predictions based on the EDM with those based on the multilayer effective medium model, whose range of validity can be defined more properly and which is valid when the roughness correction is comparatively large. We conclude that the PWS can be used for roughness characterized by a correlation length of the order of its rms amplitude, when this amplitude is of the order of or smaller than a few nanometers, and only for typically insulating materials such as silicon dioxide, silicon nitride, diamond, and certain glasses, polymers and ceramics. The results are relevant for the correct modeling of systems where the vdW force can play a significant role such as micro and nanodevices, for the calculation of the tip-sample force in atomic force microscopy, and in problems involving adhesion. (C) 2016 Elsevier B.V. All rights reserved.