  The van der Waals force established between two surfaces plays a central role
in many phenomena, such as adhesion or friction. However, the dependence of
this forces on the distance of separation between plates is very complex. Two
widely different non-retarded and retarded regimes are well known, but these
have been traditionally studied separately. Much less is known about the
important experimentally accesible cross-over regime. In this study, we provide
analytical approximations for the van der Waals forces between two plates that
interpolates exactly between the short distance and long distance behavior, and
provides new insight into the crossover from London to Casimir forces at finite
temperature. At short distance, where the behavior is dominated by non-retarded
interactions, we work out a very accurate simplified approximation for the
Hamaker constant which adopts analytical form for both the Drude and Lorentz
models of dielectric response. We apply our analytical expressions for the
study of forces between metallic plates, and observe very good agreement with
exact results from numerical calculations. Our results show that contributions
of interband transitions remain important in the experimentally accessible
regime of decades nm for several metals, including gold.
