Equilibrium disorder in crystals

First of all, disorder in crystals arises from thermal motion of atoms around their equilibrium positions (thermal disorder). Disorder may also originate from the presence of defects in the lattice, due, for instance, to substitution of atoms (or group of atoms) with atoms (or group of atoms) of a different chemical nature, inclusion of structural motifs in interstitial lattice positions, or when structural units may pack in the same basic crystal lattice in different orientations or in a different conformation [63,64]. The presence of defects in the crystals, in turn, may induce local deformation of the lattice (due to small displacements of atoms from their average positions), which helps relaxation of local stresses, and causes diffuse scattering. 

The reactivity of solids is brought about almost entirely as a result of the disorder in crystals. The most important lattice defects in connection with chemical reactions are point defects. In order that a chemical reaction may take place in a finite time, it must be carried out above a certain minimum temperature, where the defects which give rise to transport have a sufficiently large mobility. Therefore, in most cases it can be assumed that local defect equilibrium is attained during a reaction, as long as there are sufficient sources and sinks for point defects. 

In the following example concerning the equilibrium disorder in AgBr we shall illustrate the principles which we have been discussing, and vve shall illustrate the methods by which such problems may be treated. The methods will also apply for higher ionic crystals where only the number of components, and therefore the number of external equilibrium conditions, is increased. 

A second example will now be discussed in order to illustrate the application of the internal equilibrium condition in combination with structural constraints. Let us regard a crystal AX, such as AgBr, having Frenkel disorder in the cation sublattice (see Fig. 1-2), Structure elements which must be considered here are Aa, Xx, Va, Vj, Aj. The structural constraint reads... 

If majority point defect concentrations depend on the activities (chemical potentials) of the components, extrinsic disorder prevails. Since the components k are necessarily involved in the defect formation reactions, nonstoichiometry is the result. In crystals with electrically charged regular SE, compensating electronic defects are produced (or annihilated). As an example, consider the equilibrium between oxygen and appropriate SE s of the transition metal oxide CoO. Since all possible kinds of point defects exist in equilibrium, we may choose any convenient reaction between the component oxygen and the appropriate SE s of CoO (e.g., Eqn. (2.64))... 

Comparison between the various condis crystals shows that large variations in the amount of conformational disorder and motion is possible even in similar molecules. The tritriacontane in the condis state possesses about 3 gauche conformations per 100 carbon atoms. For cyclodocosane which is in its transition behavior similar to the tetracosane of Fig. 23, one estimates about 16 gauche conformations per 100 carbon atoms, and for the high pressure phase of polyethylene (see Sect. 5.3.2), one expects 37 gauche conformations per 100 carbon atoms 171). The concentration of gauche conformations in cyclodocosane and polyethylene condis crystals are close to the equilibrium concentration in the melt, while the linear short chain paraffin condis crystals are still far from the conformational equilibrium of the melt. 

Yet more important was the publication by Schottky and Wagner (1930) of their classical paper on the statistical thermodynamics of real crystals (41). This clarified the role of intrinsic lattice disorder as the equilibrium state of the stoichiometric crystal above 0° K. and led logically to the deduction that equilibrium between the crystal of an ordered mixed phase—i.e., a binary compound of ionic, covalent, or metallic type—and its components was statistical, not unique and determinate as is that of a molecular compound. As the consequence of a statistical thermodynamic theorem this proposition should be generally valid. The stoichiometrically ideal crystal has no special status, but the extent to which different substances may display a detectable variability of composition must depend on the energetics of each case—in particular, on the energetics of lattice disorder and of valence change. This point is taken up below, for it is fundamental to the problems that have to be considered. 

Figure 16. Model used to calculate equilibrium surface disorder in once-folded chain crystals of /7-alkanes. Left complete order, p = density (after ref 40).

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