Analytical model for the deformation of a fluid-fluid interface beneath an AFM probe

We present an analytical solution for the shape of a fluid-fluid interface near a nanoscale solid sphere, which is a configuration motivated by common measurements with an atomic force microscope. The forces considered are surface tension, gravity, and the van der Waals attraction. The nonlinear governing equation has been solved previously using the method of matched asymptotic expansions, and this requires that the surface tension forces far exceed those of gravity, i.e., the Bond number is much less than one. We first present this method using a physically relevant scaling of the equations, then offer a new analytical solution valid for all Bond numbers. We show that one configuration with a large effective Bond number, and thus one requiring our new solution, is a nanothick liquid film spread over a solid substrate. The scaling implications of both analytical methods are considered, and both are compared with numerical solutions of the full equation.