Equilibrium shape of crystal

The predicted shapes could be observed in experiments on equilibrium shapes of crystals with (110) missing-row reconstructed facets. 

The problem of the equilibrium shape of crystals was addressed 100 years ago by Wulff [74]. The problem was to find the shape that minimizes the surface energy for a given number of atoms. For an isotropic system the answer is obvious it corresponds to the shape that minimized the surface—that is, a sphere. For a real crystal, which presents an anisotropy of the surface energy, the problem is less trivial. Assuming that the equilibrium shape must be a polyhedron, Wulff showed that the minimum energy is obtained when the surface energy of a given face (n,) divided by the central distance to this face hi) is a constant ... 

Following the Wulff approach of the equilibrium shape of crystals, some authors have tried to include the presence of a substrate. The solution of this problem has been given by Kaishew [84] and Winterbottom [85]. It is known as the Wulff-Kaishew theorem. The crystal is now truncated at the interface by an amount Ahg, which is related to the adhesion energy of the crystal on the substrate (j ) and to the surface energy of the facet parallel to the interface (cr ) by the following relation ... 

Another factor, which has a strong effect on the equilibrium shape of crystals is the adsorption. Indeed, from Gibbs we know that adsorption reduces the surface energy (the number of broken bonds decreases) ... 

The Wulff plot draws a graph of y versus 0. This construction was developed to allow the equilibrium shape of crystals to be determined when the surface energy depends on crystallography. 

Experimentally we can determine the equilibrium shapes of crystals by annealing small single crystal particles at high temperature in an inert atmosphere or by annealing small voids (inverse particles) in a crystal. 

Figure 7.17 Equilibrium shape of crystal with three different surfaces (i = 1, 2, 3) with different surface free energies (7,.

The surface tensions for a certain cubic crystalline substance are 7100 = 160 ergs/cm, 7110 = 140 eigs/cm, and 7210 = 7120 = 140 ergs/cm. Make a Wulff construction and determine the equilibrium shape of the crystal in the xy plane. (If the plane of the paper is the xy plane, then all the ones given are perpendicular to the paper, and the Wulff plot reduces to a two-dimensional one. Also, 7100 = 7010. etc.)... 

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 Laplace equation (eq. 6.27) was derived for the interface between two isotropic phases. A corresponding Laplace equation for a solid-liquid or solid-gas interface can also be derived [3], Here the pressure difference over the interface is given in terms of the factor that determines the equilibrium shape of the crystal ... 

Networks of steps, seen in STM observations of vicinal surfaces on Au and Pt (110), are analyzed. A simple model is introduced for the calculation of the free energy of the networks as function of the slope parameters, valid at low step densities. It predicts that the networks are unstable, or at least metastable, against faceting and gives an equilibrium crystal shape with sharp edges either between the (110) facet and rounded regions or between two rounded regions. Experimental observations of the equilibrium shapes of Au or Pt crystals at sufficiently low temperatures, i.e. below the deconstruction temperature of the (110) facet, could check the validity of these predictions. 

Notice that the expression (8) for the free energy of the step network, irrespectively of the sign of s, is a non-convex function ofp and q. This result implies that the network is always unstable some surface orientations disappear from the equilibrium shape of the crystal and are replaced by sharp edges. 

As is well known (see, for instance, Van Beijeren and Nolden, 1987), the equilibrium crystal shape is the shape that minimizes the total surface free energy at a given fixed volume. From the minimization of the free energy calculated above we can construct the equilibrium shape of the crystal around the (110) facet. This shape depends crucially... 

Under most circumstances the equilibrium shape of silicon crystals is octahedral, ie, the slowest-growing faces are (111). However, external conditions can radically alter that shape. For example, when growth is from the vapor, concentration gradients in the gas stream may affect the shape, and when growth is from the melt, the shape is primarily determined by thermal gradients in the melt. 

Extensive theoretical work in support of both models of thermal etching was produced before 1970. The theoretical basis for the thermodynamic model was the concept of reduction in total surface energy by the preferential formation of low-energy, low-index planes. The true equilibrium shape of a crystal is the shape with the lowest surface energy, as noted by Curie (36) and Gibbs (37). The thermodynamic models provide no information regarding the process of surface rearrangement. 

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. 

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