Wulff construction

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 :\Delta G_i= \sum_{j}\gamma_j O_j~ Here \gamma _j represents the surface (Gibbs free) energy per unit area of the jth crystal face and O_j is the area of said face. \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 \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 will be proportional to its surface energy \gamma_j: h_j=\lambda \gamma_j. The vector h_j is the "height" of the jth 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 1943 Laue gave a simple proof, The method was extended to include curved surfaces in 1953 by Herring with a different proof of the theorem Herring gave a method for determining the equilibrium shape of a crystal, consisting of two main exercises. ==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 :\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. This is minimized for a given volume when :\delta \left(\sum_{j}\gamma_j O_j\right)_{V_c} = \sum_{j}\gamma_j \delta (O_j)_{V_c} = 0\,\! Surface free energy, being an intensive property, does not vary with volume. We then consider a small change in shape for a constant volume. If a crystal were nucleated to a thermodynamically unstable state, then the change it would undergo afterward to approach an equilibrium shape would be under the condition of constant volume. By definition of holding a variable constant, the change must be zero, \delta (V_c)_{V_c} = 0 . Then by expanding V_c in terms of the surface areas O_j and heights h_j of the crystal faces, one obtains :\delta (V_c)_{V_c} =\frac{1}{3} \delta \left(\sum_{j} h_j O_j\right)_{V_c} = 0 , which can be written, by applying the product rule, as : \sum_{j}h_j \delta (O_j)_{V_c} + \sum_{j}O_j\delta (h_j)_{V_c}= 0 \,\!. The second term must be zero, that is, O_1 \delta (h_1)_{V_c} + O_2 \delta (h_2)_{V_c} + \ldots = 0 This is because, if the volume is to remain constant, the changes in the heights of the various faces must be such that when multiplied by their surface areas the sum is zero. If there were only two surfaces with appreciable area, as in a pancake-like crystal, then O_1/O_2 = -\delta(h_1)_{V_c}/\delta (h_2)_{V_c}. In the pancake instance, O_1 = O_2 on premise. Then by the condition, \delta(h_1)_{V_c} = - \delta(h_2)_{V_c}. This is in agreement with a simple geometric argument considering the pancake to be a cylinder with very small aspect ratio. The general result is taken here without proof. This result imposes that the remaining sum also equal 0, :\sum_{j}h_j \delta (O_j)_{V_c} = 0 \,\! Again, the surface energy minimization condition is that :\sum_{j}\gamma_j \delta (O_j)_{V_c} = 0\,\! These may be combined, employing a constant of proportionality \lambda for generality, to yield :\sum_{j}(h_i - \lambda \gamma_j) \delta (O_j)_{V_c} = 0\,\! The change in shape \delta (O_j)_{V_c} must be allowed to be arbitrary, which then requires that h_j=\lambda \gamma_j, which then proves the Gibbs-Wulff Theorem. == See also ==