Grain Shape in a Liquid

The Wulff theorem can be derived from the equilibrium condition of an isolated system containing a single crystal, i.e. the minimum free energy condition of the system.Under equilibrium at a constant temperature, an infinitesimal change in Helmholtz free energy of the system, dF, is 

n represents the number of moles, V the volume, and c and. s the single crystal and the surrounding phase, respectively. Equation (15.21) is rewritten as 

Let hj be the distance perpendicular from the crystal centre to the plane i, = UH EiAihi) and 

To determine the equilibrium shape from a y-plot, connect all of the y points in the polar diagram to the centre of the diagram, and construct perpendicular planes (Wulff planes) on the y points. Then, the minimum volume enveloped by the Wulff planes exhibits the equilibrium shape of the crystal. This is because the surface energy of a plane is proportional to the distance from the centre to the y point, and, therefore, the total surface energy is proportional to the volume enveloped by Wulff planes. When a minimum cusp is present in a y-plot, a facet plane appears. For a point and a line of energy maximum in a y-plot, a corner and an edge appear in the equilibrium shape. 

To observe the equilibrium shape of a crystal experimentally, it is necessary to confirm that the crystal one wants to observe exhibits such a shape. Since the equilibrium shape of a crystal in a matrix is the same as the equilibrium shape of the matrix entrapped within the crystal, the equilibrium crystal shape can be determined by observing the shape of the entrapped matrix. If a number of randomly oriented grains exhibit an equilibrium shape, the shape can also be determined stereographically from the crystal plane orientations of the grains and the directions of the grain interfaces on a planar section that can be obtained by the electron backscattered diffraction technique. For metals for which the anisotropy in interfacial energy is low at their processing 

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