An exact solution governing a nanofilm beneath an AFM probe

In Atomic Force Microscopy (AFM), tiny forces on a probe are used to map the fine-scale structure of a surface. We re-derived the forces on AFM probes for the special case where a nanofilm of liquid rests on the surface. We demonstrate how our solution offers a new way to measure a material’s Hamaker constant (a property governing van der Waals forces). (This work is from PI Quinn’s research prior to the SFS Lab and is archived here for reference.)


Authors: Daniel Quinn, Jie Feng, Howard Stone

Abstract: 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.

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