Surface Energy Forms

In order to solve equations (10.20) and (10.21) we must know the explicit angular dependence of functions F and El. Their simplest form is the so-called Rapini energy [16]  

8cp = cp — cpI is an angle of director deflection from the equilibrium angle cpI and W is usually referred to as anchoring energy. 

When both angles 9 and cp are changed, two Rapini energies should be introduced Azimuthal (for fixed 9 )  

The zenithal anchoring is often called polar, but this word is misleading because polar anchoring is related to polar director L as given by Eq. (10.13). Thus, the Rapini form corresponds to the sine-squared shape potential well for any director deviation (3 (89 or 8cp ) from the easy direction (9, cPq)  

The Rapini term is the first one (j = 1) in an expansion of E in the Legendre polynomial series in terms of sin p  

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Often, and as a good approximation, and the surface energy are not distinguished, so Eq. III-6 can be seen in the form... 

The calculation of the surface energy of metals has been along two rather different lines. The first has been that of Skapski, outlined in Section III-IB. In its simplest form, the procedure involves simply prorating the surface energy to the energy of vaporization on the basis of the ratio of the number of nearest neighbors for a surface atom to that for an interior atom. The effect is to bypass the theoretical question of the exact calculation of the cohesional forces of a metal and, of course, to ignore the matter of surface distortion. 

Metals A and B form an alloy or solid solution. To take a hypothetical case, suppose that the structure is simple cubic, so that each interior atom has six nearest neighbors and each surface atom has five. A particular alloy has a bulk mole fraction XA = 0.50, the side of the unit cell is 4.0 A, and the energies of vaporization Ea and Eb are 30 and 35 kcal/mol for the respective pure metals. The A—A bond energy is aa and the B—B bond energy is bb assume that ab = j( aa + bb)- Calculate the surface energy as a function of surface composition. What should the surface composition be at 0 K In what direction should it change on heaf)pg, and why ... 

The resistance to nucleation is associated with the surface energy of forming small clusters. Once beyond a critical size, the growth proceeds with the considerable driving force due to the supersaturation or subcooling. It is the definition of this critical nucleus size that has consumed much theoretical and experimental research. We present a brief description of the classic nucleation theory along with some examples of crystal nucleation and growth studies. 

If the contact angle is zero, as in Fig. XIII-8e, there should be no tendency to adhere to a flat surface. Leja and Poling [63] point out, however, that, as shown in Fig. XIII-8/, if the surface is formed in a hemispherical cup of the same radius as the bubble, then for step la, the free energy change of attachment is... 

Permanent Set. When an elastomer is stretched and then allowed to relax, it will not completely recover its original dimensions. This divergence from its original form is called its permanent set. It is principally affected by the affinity of the elastomer for the filler surface and is, therefore, primarily a function of the surface energy or wetting of the filler. 

The quantitative relationship between the degree of adsorption at a solution iaterface (7), G—L or L—L, and the lowering of the free-surface energy can be deduced by usiag an approximate form of the Gibbs adsorption isotherm (eq. 9), which is appHcable to dilute biaary solutions where the activity coefficient is unity and the radius of curvature of the surface is not too great ... 

Zirconia prepared by the thermal decomposition of zirconium salts is often metastable tetragonal, or metastable cubic, and reverts to the stable monoclinic form upon heating to 800°C. These metastable forms apparently occur because of the presence of other ions during the hydrolysis of the zirconium their stabiUty has been ascribed both to crystaUite size and surface energy (152—153) as well as strain energy and the formation of domains (154). 

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