Particle−Surface Capillary Forces

Particle−surface interactions due to capillary effects between spherical, paraboloidal, and conical particles and a flat substrate are considered. A recent approach given in a previous issue of Langmuir (Marmur, A. Langmuir 1993, 9, 1922) to obtain analytical estimates of the resulting capillary forces is discussed. Because Marmur's analytic approximation includes unnecessary simplifications, an improved formula for the approximation is given. Furthermore, the contribution of the surface tension forces acting along the contact line to the total adhesion force, which was deliberately omitted in the above cited work, is shown to be, in some cases, of similar order of magnitude as the capillary forces or even to exceed the latter. The range of applicability of the improved approach is verified by comparison with numerical solutions of the capillary equation for the liquid thread joining particle and substrate. The improved analytic approach is shown to be within ±5% of the exact solution for spherical and paraboloidal probes over a wide range of contact angles and contact line positions of the liquid bridge. However, for extreme configurations such as comparatively long and thin liquid bridges or when the absolute slopes of the interface at the contact lines on probe and surface differ greatly, large deviations might occur between approximation and the numerical solution. In such cases the numerical solution should be preferred to any approximation.