The Equilibrium Shape of a Crystal

Since the crystal shape, or habit, can be determined by kinetic and other nonequilibrium effects, an actud crystal may have faces that differ from those of the Wulff construction. For example, if a (100) plane is a stable or singular plane but by processing one produces a plane at a small angle to this, describable as an (xOO) plane, where x is a large number, the surface may decompose into a set of (100) steps and (010) risers [39]. 

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Here /, is the surface energy of the crystal surface i. The equilibrium shape of a crystal is thus a polyhedron where the area of the crystal facets is inversely proportional to their surface energy. Hence the largest facets are those with the lowest surface energy. 

The means to determine the minimum-energy shape for a crystal of fixed volume was developed by Wulff (38), who showed that the equilibrium shape can be determined if the surface tension, y, at all crystallographic orientations is known. As illustrated in Fig. 2, on a polar y plot of the surface tension as a function of orientation, the inner envelope of the planes drawn perpendicular to and at the ends of the radius vectors gives the equilibrium shape of a crystal of constant volume. Faceting in the equilibrium crystal shape is due to cusps in the polar y plot. 

WulfF construction — a way to obtain the equilibrium shape of a crystal introduced by Wulff in 1901. 

It is known that adsorption decreases the surface energy [78]. Because the adsorption energy often depends on the type of facet, one expects a change of the equilibrium shape. Shi and Masel [106,107] have calculated the equilibrium shape of a crystal at 0 K in a presence of gases. The calculations show drastic changes of the equilibrium shape already at coverages around 0.1 ML. Then during a catalytic reaction the particle shape can evolve, which can affect the reaction kinetics, as shown recently by Monte Carlo simulations [108]. 

The equilibrium shape of a crystal is that of its minimum energy. This is called the Wulff condition and indicates that the area of faces present will be such as to minimize the Gibbs free energy of the crystal. Unfortunately, the observed habit of crystals grown from solutions is often quite different from the prediction by the Wulff condition. 

The growth rate, characterized by the partial current density is the factor determining the equilibrium shape of a crystal. The faces with the smallest partial current densities... 

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... 

Constructing the equilibrium shape of a crystal begins with checking the minimum segment that can represent the entire crystal by a symmetry operation. This... 

Shape control of particles is particularly important in pharmaceutical industry and in nanotechnology. The equilibrium shape of a crystal is given by WulfPs construction which is based on the minimization of the overall surface energy as the sum over all indexed surfaces of a crystal. Shape control in the Hquid phase is achieved by either choosing appropriate solvents in which the crystals are grown to the desired equilibrium shape or by shape controUing surface active compounds. The latter adsorb on specific surfaces and inhibit their growth (Mann and Ozin, 1996 Mersmarm, 2001). 

Fig. 5.27 The Wulff construction [154] for the generation of the equilibrium shape of a crystal (see text). The dotted lines represent schematically the direction-dependent surface tension, the thick lines the normals (normal planes) to the direction vector the magnitude of which is determined by the surface tension of low-energy surfaces. The inner envelopes (not dashed peirts) represents the equilibrimn shape.

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