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Wulff 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. [Pg.280]

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.)... [Pg.285]

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. [Pg.512]

In search for true equilibrium shapes, a smaller system with a periodic length equal to 100 fl is used to study isolated islands. In Fig. 5, such isolated islands are examined as a function ofmisfit strain, e. All other input conditions are similar to those ofFig. 3. Whene = 0, the island becomes a semi-circle in the infinite-torque condition, whereas a lense shape with a dihedral angle of 120° should be established in the zero-torque condition. Albeit some facetting due to the discrete nature, the shapes are consistent with the Wulff construction. For 8 0, the shapes of e = 0 are used as the initial configurations. As e increases, the... [Pg.128]

Figure 7 Equilibrium shape of the (110) facet obtained from the Wulff construction, for the same values of 35x and p5j, as shown in Fig. 6. Notice the cusps, due to instabilities of steps perpendicular to the missing rows. Figure 7 Equilibrium shape of the (110) facet obtained from the Wulff construction, for the same values of 35x and p5j, as shown in Fig. 6. Notice the cusps, due to instabilities of steps perpendicular to the missing rows.
The equilibrium shape of the (110) facet can be found by applying a one-dimensional Wulff construction to the step free energy as function of orientation (Van Beijeren and Nolden, 1987). The result of this construction, for a representative choice of step energies and temperature, is shown in Fig. 7. Steps with orientations close to ( ) = Jt/2 are unstable and would phase separate into combinations of two steps of orientations ( )o and-( )o the shape of the (110) facet resembles that of an almond, with cusps along the [110] direction. [Pg.222]

If we know the surface energies, the equilibrium shape can easily be determined from the Wulff construction ... [Pg.68]

We wish to prove by means of the Wulff construction (Section C.3.1) that the equilibrium shape of the grain boundary nucleus in Fig. 19.12 is indeed composed of two spherical-cap-shaped interfaces. [Pg.497]

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. [Pg.368]

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.)... 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.)...
The cusp points on the Wulff plot generally correspond to low-index planes. This is because the surface tension of a solid is primarily determined by the strength of the bonding of the individual surface atoms. Atoms in the low-index planes form the greatest number of bonds and hence have less energy than atoms in less densely packed planes. Thus, the equilibrium shape generally consists almost entirely of low-index planes. [Pg.369]

Gibbs realized that the equilibrium shape of solid is one that minimizes the total surface energy, Us = fy dA. Wulff showed that the minimum energy configuration can be found by constructing planes (Wulff planes) perpendicular to... [Pg.123]

The equilibrium shape for any crystal can be obtained by an ingenious construction due to Wulff (1901) for a discussion,. see Herring (195.3, p. 5). Wulff noted that if ones makes a polar plot of surface energy and constructs planes perpendicular to the polar vector at each point on the surface, the volume con-... [Pg.231]

WulfF construction — a way to obtain the equilibrium shape of a crystal introduced by Wulff in 1901. [Pg.595]

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 ... [Pg.269]

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 ... [Pg.271]

The equilibrium shape of a macroscopic crystal is an old problem first addressed by Wulff [37], who showed the equilibrium shape at OK to be a polyhedron. At the equilibrium, the surface energy is given by the famous Wulff s theorem ... [Pg.251]

If the crystal is lying on a support, the equilibrium shape is modified by the interaction with the substrate. This problem has been solved independently by Kaichew [45] and by Winterbottom [46]. The equilibrium shape is expressed by the Wulff-Kaichew theorem represented by the following equation ... [Pg.252]

The coalescence of crystallites in contact is much more complex than the coalescence of a Newtonian liquid. For coalescing liquids, the equilibrium shape i.s a sphere. For crystallites in contact, the equilibrium form is presumably determined by a Wulff construction exact calculations for such complex configurations have not been made. To estimate Vf for crystallites, it is customary to assume a simple geometry— for example, two spheres in contact as in the case of liquid droplets in contact. The particle properties are assumed to be isotropic. [Pg.341]

The factors that determine the crystal structure of particles formed in aerosol reactors have not been studied systematically. In this section, we identify key theoretical concepts and review relevant experimental observations. Consideration is limited to single-component systems. Panicle crystal structure depends on a combination of thermodynamic (equilibrium) factors and rate processes. The equilibrium shape of a particle is detennined by the surface energies of its crystal face.s according to the Wulff construction (Chapter 8). Another factor that inay enter into the process is the excess pressure inside small particles according to the Laplace formula (Chapter 9). Thus the equilibrium form may vary with panicle size depending on the phase diagram,... [Pg.355]

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. [Pg.42]


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