Spherical solid model system: Exact evaluation of the van der Waals interaction between a microscopic or submacroscopic spherical solid and a deformable fluid interface

In many previous research work associated with studying the deformation of the fluid interface interacting with a solid, the theoretical calculation of the surface energy density on the deformed fluid interface (or its interaction surface pressure) is often approximately obtained by using the expression for the interaction energy per unit area (or pressure) between two parallel macroscopic plates, e.g. σ(D) = − A / 12πD2 or π(D) = − A / 6πD3 for the van der Waals (vdW) interaction, through invoking the Derjaguin approximation (DA). This approximation however would result in over- or even inaccurate-prediction of the interaction force and the corresponding deformation of the fluid interface due to the invalidation of Derjaguin approximation in cases of microscopic or submacroscopic solids. To circumvent the above limitations existing in the previous DA-based theoretical work, a more accurate and quantitative theoretical model, available for exactly calculating the vdW-induced deformation of a planar fluid interface interacting with a sphere, and the interaction forces taking into account its change, is presented in this paper. The validity and advantage of the new mathematical and physical technique is rigorously verified by comparison with the numerical results on basis of the previous Paraboloid solid (PS) model and the Hamakerʹs sphere-flat expression (viz. F = − 2Aa3 / (3D2(D + 2a)2)), as well as its well-known DA-based general form of F / a = − A / 6zp02.