Anisotropic surface free energy

Therefore, we first look at the question of how a crystal looks in thermodynamical equilibrium. Macroscopically, this is controlled by its anisotropic surface (free) energy and the shape can be calculated via the Wullf construction. 

Shuttleworth [26] (see also Ref. 27) gives a relation between surface free energy and stretching tension as follows. For an anisotropic solid, if the area is increased in two directions by dAi and dA2, as illustrated in Fig. VII-1, then the total increase in free energy is given by the reversible work against the surface stresses, that is. 

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. ... 

The partial wetting of a melt on a low-index plane of its solid has already been predicted - and observed on Cd", Ga , Ge, NaCT and KCl . However, as mentioned earlier, indications of anisotropic wetting have been limited to Ge and icel In the case of Pb, the wetting angle is shown to increase as the surface free energy, or the atomic density of the surface plane, decreases. Germanium was quoted as having a similar behaviouf. 

However, G) is not a true thermodynamic quantity, because it does not express the equilibrium instead it depends on the history of the solid surface formation, as we see above. So far, we have considered only the isotropic solids. If, however, the solid is anisotropic, such crystals can respond differently in different directions when increasing the surface area. This property increases the number of required equations twofold, and the matter is rather complex because it is unclear which shape the anisotropic crystal would prefer for a given volume, if its total surface free energy is minimized this subject is beyond the scope of this book. 

Fig. 4. Scanning electron microscopy image of the anisotropic faceted morphology of a lead crystallite. This equihhrium shape has been analysed to derive the orientation dependence of the surface free energy at various temperatures, see chapter 4.4.7.3.1 [83Hey],...

The reason for this behavior is the angular dependence of the free energy of the interface between the ordered sohd and its vapor, ysv( ) (see Figure 8.1.9b). It is useful to define the anisotropic excess surface free energy... 

There have been several theoretical investigations analysing the nematic distortions of the LC surrounding anisotropic particles for different anchoring conditions at the LC-particle interface. In modelling the surface-anchoring free-energy density fa it is usual to adopt the Rapini-Papular expressions, which can be written as... 

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