Liquid drop shapes on hexagonal substrates: corner dewetting in the context of vapor-liquid-solid growth of nanowires

We consider the equilibrium shape of a liquid drop on a hexagonal substrate as motivated by vapor-liquid growth of nanowires. We numerically determine the energy-minimizing liquid drop shape on a hexagonal base using the software Surface Evolver in conjunction with an efficient regridding algorithm and convergence monitoring. The drop shape depends on two nondimensional parameters, the drop volume, and the equilibrium contact angle. We show that sufficiently large drops are well approximated away from the base by a spherical cap drop with geometric parameters determined by the area of the hexagonal base. Notably, however, the drop/base contact region does not extend to the corners of the hexagonal base, even in the limit of large volume V. In particular, there is a self-similar structure to the dry corner region with a length scale proportional to V-3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V<^>{-3/2}$$\end{document}. Since steady-state growth of faceted hexagonal nanowires by vapor-liquid-solid growth requires the liquid drop to be commensurate with the underlying wire cross-section, our findings mean that steady-state growth of hexagonal wires is not strictly compatible with an equilibrium liquid drop acting as a catalyst.