Orientation-dependent surface tension functions for surface energy minimizing calculations

Previous numerical methods that calculate equilibrium particle shape to study thermodynamic and kinetic processes depend on interfacial (surface) free energy functions γ(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\skew7\hat{n}$$\end{document}) that have cubic symmetry and thus produce Wulff shapes W of cubic symmetry. This work introduces a construction yielding the minimal surface energy density γconvex(W) that can be determined for anyW. Each γ(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\skew7\hat{n}$$\end{document}) that belongs to the equivalence class γ(W) bounded by γconvex(W) can be used in an energy-minimizing calculation that depends only on W. For practical numerical calculations, this work gives two methods taking directional distance from specified orientation minima as a parameter to produce analytic forms of γ(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\skew7\hat{n}$$\end{document}) giving W as the equilibrium shape for (an otherwise unconstrained) fixed volume. Included are several two- and three-dimensional examples that demonstrate the application and utility of the model γ(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\skew7\hat{n}$$\end{document}) functions.