
Overview
Wulffman is a program for
interactively examining the Wulff shapes of crystals with specified
symmetries. The Wulff shape is the shape that possesses the
lowest surface energy for a fixed volume, and hence represents the
ideal shape that the crystal would take in the absence of other
constraints. For a periodic crystal, i.e., one that can be generated by
periodic repetition of a simple unit cell , the Wulff shape
must be consistent with the crystallographic point group symmetry of the
underlying crystal. The point group is simply the set of all
point isometries (rotations, rotoinversions, and reflections)
that leave the environment around a point unchanged. In the most
general case, the Wulff shape will be a convex polyhedron whose faces
(facets) correspond to crystal planes that are low in energy.
In some cases, Wulff shapes appear in more complex circumstances. For
example, a precipitate at an interphase grain boundary in a material
will in many cases have a shape that corresponds to a combination of
the Wulff shapes for the pure materials on either side of the
boundary. The Wulff shape in these cases is the intersection of two
Wulff shapes that may or may not have displaced origins and rotated
reference frames (coordinate systems). Wulffman possesses the
capability to construct intersections of an arbitrary number of Wulff
shapes, each of which can have a different symmetry, origin, and
orientation of coordinates.
Wulffman requires the point group symmetry of a crystal (or
crystals) of interest, a set of crystal planes (facets) for each
symmetry, and their respective surface energies. From this
information, Wulffman constructs the Wulff shape and sends its output
to Geomview , a 3D visualization tool. Surface energies can
be changed at will, and Wulffman reconstructs the lowest energy
polyhedron. In this manner, the user can directly and easily
visualize the effects of surface energy anisotropy on the equilibrium
forms of crystals.
A simple example of Wulff shape construction and intersection is
illustrated in Figure 1. Cubic symmetry is chosen for two Wulff shapes
with slightly displaced origins. The [100] facet direction is
selected for each Wulff shape, and the coordinate system of shape 2 is
rotated around the 113 axis of shape 1 by 33 degrees.
The individual Wulff shapes are, of course, cubes (Figures 1a and 1b)
that are rotated with respect to one another. If we simply overlap
these cubes, the resulting shape is the (nonconvex) polyhedron in
Figure 1c. This does not, however, correspond to the intersected
Wulff shape, since a convex polyhedron with lower surface energy can
be constructed. The final intersected Wulff shape is the convex shape
shown in Figure 1d.
Figure 1: The process of Wulff shape
intersection for two cubic Wulff shapes with displaced origins and
rotated coordinate systems. Each individual shape has cubic symmetry
m3m and [100] facets.


(a) Wulff Shape I 
(b) Wulff Shape II 


(c) Union of I and II 
(d) Intersection of I and II 
Center for Theoretical and Computational Materials Science, NIST
Questions or comments:
wulffman@ctcms.nist.gov

