Equilibrium shape of a buoyant particle

Small particles on supports have been of interest for many years both scientifically and with respect to their uses in heterogeneous catalysis. One of the common and often critical questions is the shape of a small particle, i.e., spherical or faceted, since many of the properties depend upon the relative populations of different surface planes and edges. The question of the equilibrium shape of an unsupported small particle was solved many years ago first by Wulff, with later theoretical proofs in a continuum model by von Laue and Dinghas (see also Herring). The extension to a supported particle on an infinitely flat substrate was given by Winterbottom, and to twinned particles by Marks.' However, in all of these models to date the substrate has either been ignored or assumed to be infinitely flat and this latter assumption is really very dubious; in any real system diffusion can as easily change the substrate as the particle, and one should include this possibility. In some recent experimental work on small gold particles on MgO we observed that indeed the substrate can change, and there are indications that a small particle can 'sink' into a substrate; see, for instance, Fig. 1. (We feel that it is useful to draw analogies to particles floating on a liquid and in this sense we will use the idea of sinking and buoyancy. Of course, gravity is not playing any role in the physics of the process.) As part of this work we employed a very simplified model to explain the physical source of sinking as an intermediate between the particle wetting the substrate and the substrate wetting the particle. The purpose of this paper is to improve upon the theoretical model, and consider explicitly the equilibrium shape of a particle where the particle can partially sink into the substrate, i.e., a buoyant particle. In general, the problem does not appear to have a simple analytical solution, but we will show that with some assumptions about the directional dependence of the surface free energies it can be solved in a closed form by using an approach previously introduced for twinned particles by one of us.'