Surface energy anisotropy

The wetting experiments to be described were performed on Pb single-crystals. Interpretation of the results requires a knowledge of the surface energy anisotropy of solid Pb, as well as the atomic scale physical state of Pb surfaces of different orientations. These features of Pb surfaces have been studied by several authors in great detail and are described in the following seetion. 

In the present context, where our interest is in controlling surface character, it is important to note that surface energy anisotropy frequently leads to missing crystallographic orientations. For example, consider the two-... 

For an FCC structure one gets a truncated octahedron (see Fig. 1) that we generally call a Wulff polyhedron. It contains only (111) and (100) faces. At OK and for a simple broken-bond model, the ratio between the surface energies <7(100)/<7(111) is 2/v, which fixes the relative extensions of these facets. However, when the temperature increases, the surface-energy anisotropy decreases, disappearing at the melting point. 

Another way of representing the trends in surface energies is to examine the anisotropy in passing from one low-index surface to another. The surface energy anisotropy between the (100) and (111) surfaces is characterized by the ratio yioo/yiii- Fig- 9.6 reveals the surface energy anisotropies for the 4d transition metals where it is seen that there are systematic, but rather small, variations in the surface energy anisotropies. 

Fig. 9.6. Surface energy anisotropies for 4d transition metals (adapted from Vitos et al. (1998)).

E. Preuss, N. Freyer, and H.P. Bonzel. Surface Self-Diffusion on Pt(IIO) Directional Dependence and Influence of Surface-Energy Anisotropy. Appl. Phys. /141 137 (1986). 

Abstract Some aspects of self-assembly of quantum dots in thin solid films are considered. Nonlinear evolution equations describing the dynamics of the fihn instability that results in various surface nanostructures are analyzed. Two instability mechanisms are considered the one associated with the epitaxial stress and the other caused by the surface-energy anisotropy. It is shown that wetting interactions between the film and the substrate transform the instability spectrum from the long- to the short-wave type, thus yielding the possibility of the formation of spatiaUy-regular, stable arrays of quantum dots that do not coarsen in time. Pattern formation is analyzed by means of ampbtude equations near the insta-bibty threshold and by numerical solution of the strongly nonlinear evolution equations in the small-slope approximation. 

Here the coefficients a and b characterize the surface-energy anisotropy and can be computed from the surface-energy dependence on the surface orientation. Naturally, the nonhnear operator Too i is invariant with respect to rotations by 7t/2, as well as any of the transformations x —s- —x, y — —y, x y, while Finis invariant with respect to rotations by 27t/3 as well as the transformation y —s- —y, b —b. The functions Wo 2,zih) are determined by the type of a wetting interaction model and can also differ for different orientations of the film surface. 

The analysis of the conditions (56) and (57) shows that the short-wave instability of the film surface that can lead to pattern formation can occur only if the film thickness is above a threshold value determined only by the surface-energy anisotropy and the wetting length, namely, for... 

The situation is different for [111] orientation when the firee-energy functional has anisotropic cubic terms leading to anisotropic quadratic terms in the evolution equation for the surface shape and the complex quadratic resonant interaction coefficient. In this case, the imaginary part of the resonant interaction coefficient depends on the surface-energy anisotropy coefficient, q, and the pattern orientation within the [111] plane (angle po). As one can see from... 

Figure 15.12 Reprinted from Santala, M.K. and Glaeser, A.M. (2006) Surface-energy-anisotropy-induced orientation effects on Rayleigh instabilities in sapphire, Surf. Sci., 600, 782. Copyright 2006, with permission from Elsevier.

In this section, situations are considered for which the surface of a stressed solid is initially flat, or nearly so, and for which the slope of the evolving surface is everywhere small in magnitude throughout the evolution process. Chemical potential for a one-dimensional sinusoidal surface shape was developed in Section 8.4.1, for a two-dimensional sinusoidal shape in Section 8.5.3, and for a general small amplitude surface profile in Section 8.5.2. These results are used to examine surface evolution by either the mechanism of surface diffusion or condensation, as described in Section 9.1. In all cases considered in this section, surface energy is assumed to have the constant value 70, independent of surface orientation and surface strain. Implications of surface energy anisotropy and strain dependence are examined subsequently. 

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