Wulff Construction - Theory

Theory

In 1878 Josiah Willard Gibbs proposed that a droplet or crystal will arrange itself such that its Gibbs free energy is minimized by assuming a configuration of low surface energy. He defined the quantity

where represents the surface energy per unit area of the jth crystal face, and is the area of said face. 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

In 1901, Georg Wulff stated -without proving- that the length of a vector drawn normal to a crystal face will be proportional to its surface energy : . The vector is the "height" of the th face, and is 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, which consists 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 where 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 is drawn at each point where it intersects the gamma plot. The inner envelope of these planes forms the equilibrium shape of the crystal.