Wulff theorem

Since an actual crystal will be polyhedral in shape and may well expose faces of different surface tension, the question is what value of y and of r should be used. As noted in connection with Fig. VII-2, the Wulff theorem states that 7,/r,- is invariant for all faces of an equilibrium crystal. In Fig. VII-2, rio is the... 

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. 

Making use of Eqs. (2) and (3) the condition for the equilibrium form dO = 0, dV = 0 leads to Wulff s rule (Gibbs-Curie-Wulff theorem [iii-v]) generalized by - Kaischew [i] to account for the crystal-substrate interaction ... 

This is the Gibbs-Wulff theorem as generalized by Kaischew for a crystal in contact with a substrate, i.e., for the case of heterogeneous nucleation and growth. Relation (4.10) allows the construction of the equilibrium form of a crystal ... 

For the equilibrium form, the Gibbs-Wulff-Kaischew theorem can be applied. It states that the normal distances hi of all faces from the Wulff point, including the top face and the contact face /, are proportional to the specific surface energies (cf. eq. (4.10)). Hence, the specific surface energies in eq. (4.21b) can be replaced by the respective distances, hi/A = ov, so that... 

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 ... 

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 ... 

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 ... 

In this case, the Wulff shape is truncated at the interface by an amount Ah, which is proportional to the adhesion energy [3. The latter represents the work to separate the supported crystal from the substrate at an infinite distance, hg and 7s are the central distance and the surface energy of the facet parallel to the interface, respectively. In particular, this theorem shows that the stronger the particle-substrate interaction (given by / ) is the flatter is the supported particle. Equation (3.7) offers a simple way for determining the adhesion energy of a supported crystal from TEM pictures of supported particles observed in a profile view [47]. 

Wulff s theorem [25] states that cr/r is invariant for all faces. Therefore, the result obtained from the Kelvin s equation must be independent of the choice of a face. 

If the growth of an ice crystal from the vapour were a simple near-equilibrium process, then the resulting crystal habit could be determined by Wulff s theorem (Wulif, 1901) which states that, in equilibrium, the distance of any face from the centre of the crystal... 

In practice, the latter term appears to be proportional to the surface itself, the proportionality factor being related to be arrangement of the surface elements, viz. to the nature of the face under consideration. The theory of crystal shapes rests on the same principle as that of the surface tension of liquids. For a given number of elements, the form to be adopted will tend to minimize the surface energy. The minimization condition is simplified by the fact that the faces are planar It reduces to the Wulff theorem °. ... 

The application of the procedure just outlined to lithium clusters leads to satisfactory values of the surface energy per atom b, in agreement with theoretical results obtained by more complicated methods and with experimental values for the liquid metal and the vapor pressure near the melting point (Tables 19 and 20). The values obtained for b satisfy the theorem of Wulff. For instance, for a cfc cuboctahedron bounded by (111) and (110) faces, one finds... 

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... 

The use of surface energies instead of reticular densities was based on the fact that the former was roughly inversely proportional to the latter Equation (12) is commonly called the Wulfftheorem, and the analytical description of the Wulff theorem was later given by Laue (30] as ... 

Figure 11.4 (a) Graphical interpretation of Wulff theorem. Note that the equilibrium shape is... 

The value of a for a crystal face relative to the other crystal faces can be found using Wulff s theorem, which states that there exists an interior point such that the surface free energy of a crystal face is proportional to the perpendicular distance from that point to the crystal face (Wolff and Gualtieri, 1962). 

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