  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 involved, 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 then extend our
analysis to the case where a phase transition occurs between the surfaces,
which cannot easily be covered using a statistical-mechanical formalism due to
the discontinuous change in the density of the medium. Instead using a
continuum thermodynamic description, we show that this phase transformation
induces an attractive force between the bodies, and that the force between
curved surfaces can be related to the free energy in the corresponding planar
case, in accordance with DA.
