Anisotropic free energy

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. 

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. 

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 mean-field SCFT neglects the fluctuation effects [131], which are considerably strong in the block copolymer melt near the order-disorder transition [132] (ODT). The fluctuation of the order parameter field can be included in the phase-diagram calculation as the one-loop corrections to the free-energy [37,128,133], or studied within the SCFT by analyzing stability of the ordered phases to anisotropic fluctuations [129]. The real space SCFT can also applied for a confined geometry systems [134], their dynamic development allows to study the phase-ordering kinetics [135]. 

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. 

Physical properties of liquid crystals are generally anisotropic (see, for example, du Jeu, 1980). The anisotropic physical properties that are relevant to display devices are refractive index, dielectric permittivity and orientational elasticity (Raynes, 1983). A nematic LC has two principal refractive indices, Un and measured parallel and perpendicular to the nematic director respectively. The birefringence An = ny — rij is positive, typically around 0.25. The anisotropy in the dielectric permittivity which is given by As = II — Sj is the driving force for most electrooptic effects in LCs. The electric contribution to the free energy contains a term that depends on the angle between the director n and the electric field E and is given by... 

FIGURE 7. 500 MHz H-NMR partial spectra of l-phenyl-2-propylzinc bromide in the presence of I equivalent of (- -)-45, or its racemic counterpart ( ) -45, at different temperatures. The side-product 1 -phenylpropane is represented by H. Methylene protons [c 2.75 -2.95 (2H, m, CH2) in spectrum (a)] are not shown. Assignments of the absolute configurations were made based both on free energy criteria and predicted anisotropic effects and are represented in spectram (b). Reproduced by permission of Wiley-VCH from Reference 99... 

The inclusion in the model of three optimum volumes Vy V..j2 and Vy3 for three different anisotropic bonds yi, "l 2 and y 3, introduces three new minima in the free energy when V approaches VyI Vy2 and Vy3. In Fig. 1 we have plotted the free energy as a function of the reduced density in units of l/4.8b. 

Recent studies showed that amphiphilic properties have to be taken into account for most water-soluble monomer units when their behavior in water solutions is considered. The amphiphilic properties of monomer units lead to an anisotropic shape of the polymer structures formed under appropriate conditions, which is confirmed both by computer simulation and experimental investigations. The concept of amphiphilicity applied to the monomer units leads to a new classification based on the interfacial and partitioning properties of the monomers. The classification in question opens a broad prospective for predicting properties of polymer systems with developed interfaces (i.e., micelles, polymer globules, fine dispersions of polymer aggregates). The relation between the standard free energy of adsorption and partition makes it possible to estimate semiquantitatively the distribution between the bulk and the interface of monomers and monomer units in complex polymer systems. 

Interaction energy between A and B From the zero-frequency part of the van der Waals interaction free energy per unit area between the two anisotropic semi-infinite regions A and B across a slab m of thickness / [see Eqs. (L3.237) and (L3.238)],... 

It is easy to see that these relations for anisotropic media reduce immediately to the nonretarded Lifshitz result for isotropic media [see Eq. (L2.8)] where e = ej, = e = eu P (0) = u2 + v2. As in that case, the free energy of interaction takes the form... 

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