Wulff construction

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. 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. 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 γ j {displaystyle gamma _{j}} represents the surface (Gibbs free) energy per unit area of the j {displaystyle j} th crystal face and O j {displaystyle O_{j}} is the area of said face. Δ G i {displaystyle Delta G_{i}} represents the difference in energy between a real crystal composed of i {displaystyle i} molecules with a surface and a similar configuration of i {displaystyle 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 Δ G i {displaystyle Delta G_{i}} . In 1901 Russian scientist George Wulff stated (without proof) that the length of a vector drawn normal to a crystal face h j {displaystyle h_{j}} will be proportional to its surface energy γ j {displaystyle gamma _{j}} : h j = λ γ j {displaystyle h_{j}=lambda gamma _{j}} . The vector h j {displaystyle h_{j}} is the 'height' of the j {displaystyle 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 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 γ ( n ^ ) {displaystyle gamma ({hat {n}})} , where n ^ {displaystyle {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 n ^ {displaystyle {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. 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