Maximum bubble pressure method

The Wulff construction is a method for determining 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.

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

$\mathrm{\Delta }{G}_i=\sum _j{\gamma }_j{O}_j$

where ${\gamma }_j$ represents the surface energy per unit area of the $j$th crystal face and $O_j$ is the area of said face. $\mathrm{\Delta }{G}_i$ represents the difference in energy between a real crystal composed of i molecules with a surface, and a similar configuration of i molecules located inside an infinitely large crystal. This quantity is therefore the energy associated with the surface. The equilibrium shape of the crystal will then be that which minimizes the value of $\mathrm{\Delta }{G}_i$

In 1901 Georg Wulff stated (without proof) that the length of a vector drawn normal to a crystal face $h_j$ will be proportional to its surface energy ${\gamma }_j$: $h_j=\lambda {\gamma }_j$. The vector $h_j$ is the "height" of the $j$th face, drawn from the center of the crystal to the face; for a spherical crystal this is simply the radius. This is known as the Gibbs-Wulff theorem.

In 1953 Conyers 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 $\gamma (\hat{n})$ where $\hat{n}$ denotes the surface normal, e.g., a particular crystal face. The second part is the Wulff construction itself in which the gamma plot is used to determine graphically which crystal faces will be present. It can be determined graphically by drawing lines from the origin to every point on the gamma plot. A plane perpendicular to the normal $\hat{n}$ is drawn at each point where it intersects the gamma plot. The inner envelope of these planes forms the equilibrium shape of the crystal.

Proof

Various proofs of the theorem have been given by Hilton, Liebman, von 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

$\mathrm{\Delta }{G}_i=\sum _j{\gamma }_j{O}_j$

which is the product of the surface energy per unit area times the area of each face, summed over all faces, which is minimized for a given volume when

$\delta \sum _j{\gamma }_j{O}_j=\sum _j{\gamma }_j\delta O_j=0$

We then consider a small change in shape for a constant volume

$\delta V_c=\frac{1}{3}\delta \sum _j{h}_j{O}_j=0$

which can be written as

$\sum _j{h}_j\delta O_j+\sum _j{O}_j\delta h_j=0$

the second term of which must be zero, as it represents 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

$\sum _j{h}_j\delta O_j=0$

and

$\sum _j{\gamma }_j\delta O_j=0$

which can be combined by a constant of proportionality as

$\sum _j(h_i-\lambda {\gamma }_j)\delta O_j=0$

The change in shape $(\delta O_j)$ must be allowed to be arbitrary, which then requires that $h_j=\lambda {\gamma }_j$ which then proves Gibbs-Wulff Theorem.