The Wulff construction is a method to determine the equilibrium shape of a droplet or crystal of fixed volume inside a separate phase (usually its saturated solution or vapor). Energy minimization arguments are used to show that certain crystal planes are preferred over others, giving the crystal its shape.
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Theory
In 1878 Josiah Willard Gibbs proposed that a droplet or crystal will arrange itself such that its surface Gibbs free energy is minimized by assuming a shape of low surface energy. He defined the quantity
Here
In 1901 Russian scientist George Wulff stated (without proof) that the length of a vector drawn normal to a crystal face
In 1953 Herring gave a proof of the theorem and a method for determining the equilibrium shape of a crystal, consisting of two main exercises. To begin, a polar plot of surface energy as a function of orientation is made. This is known as the gamma plot and is usually denoted as
Proof
Various proofs of the theorem have been given by Hilton, Liebman, Laue, Herring, and a rather extensive treatment by Cerf. The following is after the method of R. F. Strickland-Constable. We begin with the surface energy for a crystal
which is the product of the surface energy per unit area times the area of each face, summed over all faces. This is minimized for a given volume when
We then consider a small change in shape for a constant volume
which can be written as
The second term must be zero, as it represents the change in volume [which does not make sense, as before the whole expression was supposed to be the change in volume] and we wish only to find the lowest surface energy at a constant volume (i.e., without adding or removing material). We are then given from above
and
which can be combined by a constant of proportionality as
The change in shape