A Worked Example using Graphics Primitives: The Wulff Construction

This next example shows a clever way to perform a famous thermodynamic graphical construction called  the  Wulff construction.  For surface tensions that are a function of their orientation, the Wulff construction gives the enclosing surface of lowest total energy.  The following code creates the graphics objects corresponding to the Wulff construction; it will be combined with an orientation-dependent surface tension to produce a prediction of the energy-minimizing shape

These last two plots have to do with the orientation dependence of the surface energy of a 2-D crystal (GammaPlot)and the equilibrium shape that a single crystal having this surface energy will adopt to minimize its total surface energy.

The inner envelope of the tangent lines to the GammaPlot in the figure above illustrates the energy-minimizing shape of the crystal. For this particular GammaPlot, the shape is approximately a square but the corners are rounded.

Created by Wolfram Mathematica 6.0  (28 August 2007)