A statistical-mechanical derivation of the Derjaguin approximation in a molecular solvent

The Derjaguin approximation (DA) relates the force between curved surfaces to the interaction free energy between parallel planes. It is typically derived by considering the direct interaction between the bodies inolved, thus treating the effect of an intervening solvent implicitly by a rescaling of the corresponding Hamaker constant. Here, we provide a generalization of DA to the case of a molecular medium between the bodies, as is the case in most applications. The derivation is based on an explicit statistical-mechanical treatment of the contribution to the interaction force from a molecular solvent using a general expression for intermolecular and molecule-surface interactions. Starting from an exact expression for the force, DA is arrived at by a series of well-defined approximations. Our results show that DA remains valid in a molecular solvent as long as (i) the surface-molecule interactions are of much shorter range than the radius $R$ of the sphere and (ii) the density correlation length in the solvent is smaller than $R$. We furthermore show that DA remains valid for forces due to capillary induced phase transitions between the interacting bodies, in spite of the discontinuous density profiles involved.