Surface energy orientation dependence

The surface energy, y, depends on the structure, which depends on the orientation. We have developed two approaches for considering this orientation dependence the Wulff plot and the inverse Wulff plot. 

There has been considerable elaboration of the simple Girifalco and Good relationship, Eq. XII-22. As noted in Sections IV-2A and X-6B, the surface ftee energies that appear under the square root sign may be supposed to be expressible as a sum of dispersion, polar, and so on, components. This type of approach has been developed by Dann [70] and Kaelble [71] as well as by Schonhom and co-workers (see Ref. 72). Good (see Ref. 73) has preferred to introduce polar interactions into a detailed analysis of the meaning of in Eq. IV-7. While there is no doubt that polar interactions are important, these are orientation dependent and hence structure sensitive. 

SFA has been traditionally used to measure the forces between modified mica surfaces. Before the JKR theory was developed, Israelachvili and Tabor [57] measured the force versus distance (F vs. d) profile and pull-off force (Pf) between steric acid monolayers assembled on mica surfaces. The authors calculated the surface energy of these monolayers from the Hamaker constant determined from the F versus d data. In a later paper on the measurement of forces between surfaces immersed in a variety of electrolytic solutions, Israelachvili [93] reported that the interfacial energies in aqueous electrolytes varies over a wide range (0.01-10 mJ/m-). In this work Israelachvili found that the adhesion energies depended on pH, type of cation, and the crystallographic orientation of mica. 

Crystals have spatially preferred directions relative to their internal lattice structure with consequences for orientation-dependent physico-chemical properties i.e., they are anisotropic. This anisotropy is the reason for the typical formation of flat facetted faces. For the configuration of the facets the so-called Wullf theorem [20] was formulated as in a crystal in equihbrium the distances of the facets from the centre of the crystal are proportional to their surface free energies. ... 

An essential requirement for device applications is that the orientation of the molecules at the cell boundaries be controllable. At present there are many techniques used to control liquid crystal alignment which involve either chemical or mechanical means. However the relative importance of these two is uncertain and the molecular origin of liquid crystal anchoring remains unclear. Phenomenological models invoke a surface anchoring energy which depends on the so-called surface director , fij. In the case where there exists cylindrical symmetry about a preferred direction, hp the potential is usually expressed in the form of Rapini and Popoular [48]... 

The phenomenon of the orientation dependent surface free energy of metals is theoretically and experimentally well established [1-4]. An example from the experimental work of Heyraud and Metois for Pb is shown in fig. 1 [5]. Here the relative anisotropy of y(0) is derived from the ECS of Pb particles on graphite measured by scanning electron... 

For a nematic LC, the preferred orientation is one in which the director is parallel everywhere. Other orientations have a free-energy distribution that depends on the elastic constants, K /. The orientational elastic constants K, K22 and K33 determine respectively splay, twist and bend deformations. Values of elastic constants in LCs are around 10 N so that free-energy difference between different orientations is of the order of 5 x 10 J m the same order of magnitude as surface energy. A thin layer of LC sandwiched between two aligned surfaces therefore adopts an orientation determined by the surfaces. This fact forms the basis of most electrooptical effects in LCs. Display devices based on LCs are discussed in Chapter 7. 

In general, the nuclei are not spherical. There are several reasons for this. One is that the a- 3 surface energy term, y p, depends on the orientations of the a and 3 phases. Another reason is that when nucleation occurs on oc-oc grain boundaries, the angle of contact, 0, between the a and 3 phases depends on the ratio of y p/yagb- A third reason is that the elastic strain energy is minimized if the precipitating phase is lenticular. 

The surroundings of atoms at the surface of a material are different from those in the interior. A simple way of estimating the orientation dependence of surface energy is to determine the number of missing near-neighbor bonds per area of... 

The surface energy of a crystal is anisotropic. For planes with low indices, it is possible to calculate by trigonometry the number of near-neighbor bonds missing per area and follow the procedure above to estimate the relative surface energies of different low-index planes. The orientation dependence of the free surface energy of a two-dimensional square lattice depends on the orientation of the surface and can be found as follows. 

Values of surface energy measured by this technique were found to be 1.140 J/m2 for Ag at 903 °C, 1400J/m2 for Au at 1204 °C, and 1.650 J/m2 for Cu at 1000 °C. The solid-vapor surface energy is relatively independent of the temperature, but the surface energy of a solid-vapor interface depends on its crystallographic orientation, so these measured values must reflect an average value for many orientations. 

For simplicity of the presentation we will hereafter employ the rigid-rotor approximation in which r is fixed at the equilibrium bond distance re of BC.t Note that the potential energy surface (PES) V depends only on the two distances R and r and on the orientation angle 7. 

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