Crystal Wulff construction

Since the crystal shape, or habit, can be determined by kinetic and other nonequilibrium effects, an actud crystal may have faces that differ from those of the Wulff construction. For example, if a (100) plane is a stable or singular plane but by processing one produces a plane at a small angle to this, describable as an (xOO) plane, where x is a large number, the surface may decompose into a set of (100) steps and (010) risers [39]. 

Fig. Vn-2. Conformation for a hypothetical two-dimensional crystal, (a) (lO)-type planes only. For a crystal of 1 cm area, the total surface firee energy is 4 x lx 250 = 1000 eigs. (b) (ll)-type planes only. For a crystal of 1-cm area, the total surface free eneigy is 4 x 1 x 225 = 900 ergs, (c) For the shape given by the Wulff construction, the total surface free energy of a 1-cm crystal is (4 x 0.32 x 250) + (4 x 0.59 x 225) = 851 ergs, (d) Wulff construction considering only (10)- and (ll)-type planes.

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.)... 

How does a support affect the morphology of a particle on top of it Which surface planes does the metal single crystal expose The thermodynamically most stable configuration of such small crystallites is determined by the free energy of the surface facets and the interface with the support, and can be derived by the so-called Wulff construction, which we demonstrate for a cross section through a particle-support assembly in two dimensions (Fig. 5.13). 

Figure 5.13. Wulff construction for a two-dimensional crystal where the surface energy has the following order...

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.)...

Figure 8.10 Wulff construction for determining the equilibrium configuration of a crystal.

As might be expected, this is easiest to visualize in two dimensions, but the principles apply directly to three dimensions as well. For sodium chloride, at low temperatures, a plot qualitatively similar to Figure 2.5a is obtained. In this two-dimensional section [taken through the (100) plane], the center square corresponds to the Wulff construction described above and it is the absolute minimum energy. The 100 facets are lower in energy than the 110 or 111 facets, the latter not shown in this 2D section. Accordingly, in three dimensions, a cube represents the lowest-energy crystal form for sodium chloride at low temperatures. 

With the monatomic face-centered-cubic (FCC) metals, a qualitative two-dimensional Wulff construction taken through the (111) plane and a truncated octahedron equilibrium crystal shape similar to those shown in Figure 2.5b are expected. The symmetry of a cubic crystal tells us that we will have six (100) vectors and eight (111) vectors. This 2D section is therefore sufficient to allow... 

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

Figure 5. Wulff construction. Left definition and general form of construction. Right Wuhf construction showing the equilibrium form of a forsterite crystal. After Zoltai and Stout (1984).

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,... 

The relative surface energies of different crystal planes determine the equilibrium shape of a metal crystal, and this can be predicted from knowledge of the energy terms by use of the Wulff construction.In this procedure the surface energies are drawn as vectors normal to the planes described by the indices, and these planes set at the tops of the vectors define the crystal shape. 

For solids, the surface tension is anisotropic — it is different for different crystal faces, defined by their normal n y = y h). The equilibrium crystal shape is not a sphere, but is determined by the Wulff construction (for a proof see Ref. 2)... 

Figure 15.7. Polar plot of surface energy y and Wulff construction for an equilibrium crystal...

FIGURE 7.6. A Wulff construction for a hypothetical, two-dimensional crystal with surface energies X and IX, from which the ideal geometric shape of the crystal can be predicted. The arrows emanating from the common point are proportional to the surface free energy of the intersecting crystal faces. 

For the shape of a crystal, the equilibrium and growth form have to be distinguished. The equilibrium form follows from the Wulff construction, while the growth form is determined by the relative growth rates of the different faces. 

This procedure would yield the square shape for the two-dimensional example of Figure 4.2. The Wulff construction is an elegant concept firom the standpoint of understanding crystal shapes hut is not of significant practical value since the entire Wulff plot is rarely available. 

Fig. 1.9. Representative polar plot of surface free energy in a symmetry plane of a crystal the free energy of a surface at a certain orientation is the radial distance from the origin to the lobed curve in the direction of the surface normal. The Wulff construction is illustrated by the family of lighter weight lines, with one member of the family and its corresponding perpendicular radial line shown in a heavier weight. The constructed equilibrium shape of the crystal is shown by the dashed line.

Calculations also examine the ordered structures of icosahedra, decahedra, and the close-packed fee truncated octahedra that are favored from the energetic point of view. Figure 19.4d-f show the close-packed structures with a total number of Nioi. Ni47, and N201 atoms, respectively. Icosahedra and decahedra are noncrystalline structures that cannot form in bulk crystals because of the fivefold symmetry. Icosahedra are quasi-spherical, where atoms are arranged in the concentric shells. Marks-truncated decahedra have reentrant (111) facets that are introduced via a modified Wulff construction, fee truncated octahedra possess a crystalline structure and have the open (100) facets. 

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