Solids equilibrium crystal shapes

For solids, the surface tension is anisotropic — it is different for different crystal faces, defined by their normal n y = y h). The equilibrium crystal shape is not a sphere, but is determined by the Wulff construction (for a proof see Ref. 2)... 

In this chapter the consequences of the orientation-dependent surface energies for crystalline sohds have been described. The effects on equilibrium crystal shape and the thermodynamics of grain-boundary behavior and faceting have been used as examples. Two common techniques for measuring y for solid surfaces, namely zero creep and multiphase equilibrium, have been described. 

It is recommended that concentration measurements for this type of modeling work are based on analytical standards of mole or mass fraction, to avoid the conversion error caused by density effects. The excess solid phase should always be characterized by a suitable analytical technique, before and after the equilibrium solubility measurements, to confirm that the polymorphic form is unchanged. It should be noted that the crystal shape (habit) does not always change significantly between different polymorphic forms, and visual assessments can be misleading. 

Near the transition temperature, SMAs also exhibit the curious effect of pseudoelasticity, in which the metal recovers (apparently in the usual manner) from an isothermal bending deformation when the stress is removed. However, the elasticity is not due to the usual elastic modulus of a fixed crystalline form, but instead results from strain-induced solid-solid phase transition to a more deformable crystalline structure, which yields to the stress, then spontaneously returns to the original equilibrium crystal structure (restoring the original macroscopic shape) when the stress is removed. 

We noted in Section VII-2B that, given the set of surface tension values for various crystal planes, the Wulff theorem allowed the construction of fhe equilibrium or minimum firee energy shape. This concept may be applied in reverse small crystals will gradually take on their equilibrium shape upon annealing near their melting point and likewise, small air pockets in a crystal will form equilibrium-shaped voids. The latter phenomenon offers the possible advantage that adventitious contamination of the solid-air interface is less likely. 

Of the major solids formed from melts, many, but not all, at equilibrium, the overwhelming influence is of cooperative interaction between ionic units of similar shape and size as we see in crystals. Trace elements apart from forming isolated minerals are fractioned in bulk oxides, for example, in particular orders as the melt solidifies, and this reduces the relative availability of some elements such as Cr and Ni (see Williams, and Williams and Frausto da Silva (1999) in Further Reading). Again the interaction of selective molten minerals and water creates extremely reactive environments and such environments still exist, especially in the deep sea black smokers (hydrothermal vents), around which particular mixed minerals form, which could also have been involved in prebiotic chemistry and are still involved in the peculiarities of life in these smokers . In Figure 1.6 we summarise... 

The Laplace equation (eq. 6.27) was derived for the interface between two isotropic phases. A corresponding Laplace equation for a solid-liquid or solid-gas interface can also be derived [3], Here the pressure difference over the interface is given in terms of the factor that determines the equilibrium shape of the crystal ... 

Three-Phase Transformations in Binary Systems. Although this chapter focuses on the equilibrium between phases in binary component systems, we have already seen that in the case of a entectic point, phase transformations that occur over minute temperature fluctuations can be represented on phase diagrams as well. These transformations are known as three-phase transformations, becanse they involve three distinct phases that coexist at the transformation temperature. Then-characteristic shapes as they occnr in binary component phase diagrams are summarized in Table 2.3. Here, the Greek letters a, f), y, and so on, designate solid phases, and L designates the liquid phase. Subscripts differentiate between immiscible phases of different compositions. For example, Lj and Ljj are immiscible liquids, and a and a are allotropic solid phases (different crystal structures). 

From equation (8) it can be seen that sohds and liquids will form equilibrium shapes in an effort to minimize their surface area and thus the free energy of the system. Indeed, crystal faces with the closest packing of surface atoms have the lowest surface area and tend to be the most stable. When one considers a two-component system, with one material on top of the other, the interaction between the two will be defined by the surface tensions. The surface tensions of some selected solids and hquids are listed in Table 2. From these values, it can be easily predicted which materials will be capable of wetting another. In general, most liquids have lower surface tensions than clean solids and will therefore spread to cover them. 

A) in hydrocarbon solutions, NMR spectra indicated a monomer dimer equilibrium with the energy of dissociation = 12.8kcalmol (equation 109). From an electron diffraction study, [(Mes Si)2CH]2 Sn in the gas phase has a V-shaped monomeric stmcture, with C Sn = 97(2)°. The related stannylene (77) is monomeric in the solid state, with the shortest Sn- - -Sn distance of 7.4 A. The dark-red air- and moistnre-sensitive componnd (77) exists in a dimer monomer eqnUibrium in solntion. Two crystal modifications of [2,4,6-(F3C)3C6H2]2Sn have been reported one is monomeric and the other is a weakly associated dimer with a Sn-Sn distance of 3.639(1) A. There are Sn F interactions (eqna-tion 110). 

To truly control crystallization to give the desired crystalline microstructure requires an advanced knowledge of both the equilibrium phase behavior and the kinetics of nucleation and growth. The phase behavior of the particular mixture of TAG in a lipid system controls both the driving force for crystallization and the ultimate phase volume (solid fat content) of the solidified fat. The crystallization kinetics determines the number, size, polymorph, and shape of crystals that are formed as well as the network interactions among the various crystalline elements. There are numerous factors that influence both the phase behavior and the crystallization kinetics, and the effects of these parameters must be understood to control lipid crystallization. 

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