The Wulff construction

The surface energy is different for different facets. For a crystallite consisting of a given number of atoms, the equilibrium shape is the shape which minimizes the surface (Gibbs) energy. 

If we know the surface energies, the equilibrium shape can easily be determined from the Wulff construction  

in 2 dimesions In a polar coordinate system, draw a vector parallel to the normal of the surface and with length proportional to the energy of the surface. At the endpoint of the vector, draw a tangent line. Repeat for all surfaces. The equilibrium shape is the area limited by the tangent lines. 

It can easily happen that a particular surface has such a high energy, that is will not be present at all in the final construction. 

The construction in 3 dimensions is in principle the same. You use a sperical (not polar) coordinate system, you draw the vectors and then the tangent plane (not line). 

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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.

By putting the right-hand side in Eqs. (52) and (53) equal to zero, one receives the equilibrium value of local radius of curvature R (or, Rt), which is nothing but the Wulff construction. For an anisotropic step tension 7(0), there is a local critical radius defined as... 

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... 

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 figure below shows the Wulff construction for a typical FCC metal. The object is a cubo-octahedron. The larger hexagonal facets have a (111) orientation and the smaller, square facets have a (100) orientation. 

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. 

C(n) is related to v(h) in the same way that the capillarity vector, , is related to 7(ri) and is constructed in the same way. The Wulff construction applied to v(n) produces the asymptotic growth shape. This and other relations between the Wulff construction and the common-tangent construction for phase equilibria are discussed by Cahn and Carter [16]. 

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. 

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. 

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. 

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 dimension-less surface area AfW of the different facets as well as the total area calculated from the Wulff construction assuming that the (110) plane of the particle is attached to the substrate. A The surface area is shown as a function of the contact surface free energy y/yo between particle and substrate. B The surface area is shown as a function of the reduction potential of the gas phase [21]. The relation between contact surface free energy and reduction potential is given by equation (16). 

In the case of the SiNWs with the axes along the <001> directions, the Wulff construction predicts the 001 and 011 facets [8] to form their shape as that is shown in Fig. 1. 

To summarize, large NPs with low surface to volume ratio have bulk-like crystalline structures and are terminated by low energy surfaces (the Wulff construction). 

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)... 

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