Geometry effects on the van der Waals force in atomic force microscopy

We calculate the attractive van der Waals force in a model system consisting of a sphere and a semi-infinite slab representing the tip and sample in an atomic force microscopy experiment. The objective of the calculation is to establish estimates of the errors involved when using Hartmann's approximation scheme [Phys. Rev. B 42, 1541 (1990)], which accounts for tip and sample geometry by treating them as a set of piecewise parallel planes. Neglecting retardation effects, we can evaluate the electromagnetic stress tensor and the van der Waals force exactly in the model system. Comparing the results with those obtained within the Hartmann scheme we find that the difference is not larger than 5-10% in practice in an atomic force microscopy experiment, and 25% in a worst-case situation. The accuracy of Hartmann's scheme is, however, to some extent the result of canceling errors, since the maximum stress is overestimated while the tip-sample interaction area is underestimated.