Exact expressions for colloidal plane–particle interaction forces and energies with applications to atomic force microscopy

Abstract

We begin by deriving a general useful theoretical relationship between the plane–particle interaction forces in solution, and the corresponding plane–plane interaction energies. This is the main result of the paper. It provides a simple tool to obtain closed-form particle–plane forces from knowledge of plane–plane interaction energies. To illustrate the simplicity of use of this general formalism, we apply it to find particle–plane interactions within the Derjaguin–Landau–Verwey–Overbeek (DLVO) framework. Specifically, we obtain analytical expressions for forces and interaction energies in the van der Waals and the electrical double layer cases. The van der Waals expression is calculated here for benchmarking purposes and is compared with well-established expressions from Hamaker theory. The interactions for the electric double layer situation are computed in two cases: the linear superposition approximation and the constant surface potential. In both cases, our closed-form expressions were compared with existent numerical results. We also use the main result of this paper to generate an analytical force-separation expression based on atomic force microscope experiments for a tip and surface immersed in an aqueous solution, and compare it with the corresponding numerical results. Finally, based on our main result, we generalize the Derjaguin approximation by calculating the next order of approximation, thus obtaining a formula valuable for colloidal interaction estimations.