Wulff shape

Figure 14.8 The 7-plot for a material with a Wulff shape corresponding to a cube...

Other constructions employing the 7-plot are reviewed in Section C.3.1. These include the reciprocal 7-plot, which is also useful in treating the faceting problem above, and the Wulff construction, which is used to find the shape (Wulff shape) of a body of fixed volume that possesses minimum total surface energy. 

The expression for weighted mean curvature for any surface in local equilibrium is simplified when the Wulff shape is completely faceted [10, 12], In this case,... 

Weighted mean curvature, which is uniform on a Wulff shape, goes to zero in the limit of large body volumes. 

When the interfacial energy depends on the inclination of the interface and 7 is therefore anisotropic, Eq. 19.1 does not apply. In this case, the cluster will minimize its total interfacial energy by adopting its Wulff shape (Section C.3.1), which may be fully faceted or made up of faceted and smoothly curved regions. Several characteristic shapes are shown in Fig. 19.2. The interfacial-energy term in Eq. 19.1 can then be expressed as the sum of the interfacial energies of the various faceted or smoothly curved patches that make up the entire closed interface and... 

Solution. Cross sections of the Wulff plot and Wulff shape consistent with the symmetry of the problem are shown in Fig. 19.29. Since the a/f3 interface is isotropic, the top surface is spherical. Also, the construction is consistent with Young s equation, since from the figure, 7 = 27 cos ... 

Another topic of interest is the shape that an isolated body of constant volume with an anisotropic surface energy will adopt to minimize its total interfacial energy. This can be resolved by means of the Wulff construction shown in Fig. C.4e. Here, a line has been drawn at each point on the 7-plot which is perpendicular to the n corresponding to that point. The interior envelope of these lines is then the shape of minimum energy (i.e., the Wulff shape). The Wulff shape for the 7-plot in Fig. C.4a contains sharp edges and contains only inclinations that have been shown to be stable in Fig. C.46 and c. 

Note that when the interfacial energy is isotropic and the 7-plot is a sphere, the Wulff shape will also be a sphere. However, if the 7-plot possesses deep depressions or cusps at certain inclinations such as in Fig. C.4a, the planes normal to the radii of the plot at these inclinations will tend to dominate the inner envelope, and the Wulff shape will be faceted. In such cases, the system is able to minimize its total interfacial energy by selecting patches of interface of particularly low energy even though the total interfacial area increases. 

The weighted mean curvature is the interface divergence of the evaluated on the unit sphere. The interface divergence is defined within the interface, and if the interface is not differentiable, subgradients must be used. The convex portion of is equivalent to the the Wulff shape, so the interface divergence is operating from one interface onto another. This form can get very complicated. 

Fig. 3.7. Most common stable shapes of nanoparticles (a) icosahedron, (b) truncated octahedon (Wulff shape), (c) Marks decahedron...

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]. 

Fig. 4.41. EM images of the before (insets) and after reaction 40-rnn Pt/ceria and Pt/alumina model catalysts, (a) Pt/ceria after 5h in / = Pco/ Pco + P02) = 0.67 (stoichiometric) at 673K (b) Pt/ceria after 5h in / = 0.1 (O-rich) at 673K (c) 40-nm Pt/alumina after 5h in / = 0.66 at 673 K. It is seen that on Pt/ceria the prolonged stoichiometric CO oxidation (/ = 0.67) leads to pronounced restructuring, with a height-to-width ratio much larger than expected from thermodynamics (Wulff shape). On Pt/AbOa, the reaction induces a reshaping that approaches the thermodynamically predicted H/W-iaXio of 0.6 (from [98])...

Wulffman (NIST) is a software package that allows you to model the Wulff shapes of crystals interactively. You can specify the crystal symmetry. The source for NIST s Wulffman program is available for you to use (www.ctcms.nist.gov/wulffman). There are commercial programs that address the same problena. 

Figure 15.11 Reprinted with permission from Castell. M.R. (2003) Wulff shape of microscopic voids in UO2 crystals, Phys. Rev. B 68, 235411. Copyright 2003 by the American Physical Society.

Fig. 5.28 Molecular dynamic simulation of the Wulff shape of a small Si3N4 crystal. According to Ref. (157).

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