Wulff constructions

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.

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. 

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.)... 

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... 

Figure 4.15. Atomically resolved TEM images of a Cu/ZnO model catalyst in various gas environments together with the corresponding Wulff construction of the Cu particle (a,b) Cu nanocrystal faceted by (100), (110) and (111) surfaces the TEM image was recorded at 1.5 mbar of H2 at 220 °C with the electron beam parallel to the [Oil] zone-axis of copper. The insert shows EELS data at the Cu L2,3-edge...

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...

The construction relies on Wulff s assumption that the distance from the surface of a specific plane to the center of the crystallite is proportional to the surface energy i.e. hi <=<= Yi. Thus, if we have a surface plane of small surface energy, its distance from the center of the crystallite will be small and this plane will then cut of all others and dominate the polyhedron. 

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. 

Figure 2. Morphological evolution of unstressed thin-films under the zero-torque condition, in which all the interfaces are allowed to pucker freely. The corresponding Wulff constructions are also pictured for comparison.

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 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. 

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

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. 

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 ... 

---