Crystals equilibrium shape

We noted in Section VII-2B that, given the set of surface tension values for various crystal planes, the Wulff theorem allowed the construction of fhe equilibrium or minimum firee energy shape. This concept may be applied in reverse small crystals will gradually take on their equilibrium shape upon annealing near their melting point and likewise, small air pockets in a crystal will form equilibrium-shaped voids. The latter phenomenon offers the possible advantage that adventitious contamination of the solid-air interface is less likely. 

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.)... 

An enlarged view of a crystal is shown in Fig. VII-11 assume for simplicity that the crystal is two-dimensional. Assuming equilibrium shape, calculate 711 if 710 is 275 dyn/cm. Crystal habit may be changed by selective adsorption. What percentage of reduction in the value of 710 must be effected (by, say, dye adsorption selective to the face) in order that the equilibrium crystal exhibit only (10) faces Show your calculation. 

In the case of supported metalhc particles, the construction is modified by introducing the adhesion energy (Wulff-Kaishew construction) [Henry, 1998]. The equilibrium shape is a Wulff polyhedron, which is truncated at the interface by an amount Ahs, according to the relation Ahs/hj = /3/(t where /3 is the adhesion energy of the crystal on the substrate. 

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 ... 

Equilibrium Shape ofSmall Crystal, Surface Science, 221 233 (1989). 

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... 

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

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. 

Fig. 2. A schematic Wulff construction for an equilibrium crystal shape using the polar y plot of the surface tension, (a) The equilibrium shape is that found from the inner envelope of tangents to the y plot, (b) An ECS with (001) facets produced by cusps in the y plot (39). (Reprinted from Prog. Surf. Sci., Volume 39, E. H. Conrad, Page 65, Copyright (1992), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 IGB, UK.)...

A major complication in the analysis of convection and segregation in melt crystal growth is the need for simultaneous calculation of the melt-crystal interface shape with the temperature, velocity, and pressure fields. For low growth rates, for which the assumption of local thermal equilibrium is valid, the shape of the solidification interface dDbI is given by the shape of the liquidus curve Tm(c) for the binary phase diagram ... 

---