Defect crystals, Gibbs energies

If vacancies exist in the crystal under equilibrium, the Gibbs energy must be less than it is in a perfect crystal. However, if the introduction of more and more defects causes the Gibbs energy to continually fall, then ultimately no crystal will remain. 

Figure 2.1 Change in Gibbs energy, AG, of a crystal as a function of the number of point defects present (a) Variation of AG with number of point defects (schematic) at equilibrium, neq defects are present in the crystal (b) calculated variation of AG for hy = 0.6 eV, kT = 0.1, N = 1000 the equilibrium number of point defects is 2.5 per 1000.

In the knowledge that the Gibbs energy of a crystal containing a small number of intrinsic defects is lower than that of a perfect crystal, the defect population can be treated as a chemical equilibrium. In the case of vacancies we can write... 

Only one sort of defect is supposed to be found in a crystal. Broadly speaking, this assumption is based upon the fact that the number of defects, nd, is related to the Gibbs energy of formation, AG, by an equation of the form ... 

The introduction of Schottky defects causes the Gibbs energy of the crystal to change by an amount AGs ... 

The calculation of the number of Frenkel defects in a crystal proceeds along lines parallel to those above. The introduction of Frenkel defects causes the Gibbs energy of the crystal to change by an amount AGp ... 

Although the statistical approach to the derivation of thermodynamic functions is fairly general, we shall restrict ourselves to a) crystals with isolated defects that do not interact (which normally means that defect concentrations are sufficiently small) and b) crystals with more complex but still isolated defects (i.e., defect pairs, associates, clusters). We shall also restrict ourselves to systems at some given (P T), so that the appropriate thermodynamic energy function is the Gibbs energy, G, which is then constructed as... 

Here, n0 is the number of lattice sites per unit crystal volume and Ag the formation Gibbs energy of defects without the applied stress. 

Next, let us consider the fact that a given solid of known crystal structure has at least two additional degrees of freedom which may change its behavior. The presence of lattice defects, such as dislocations, and any alteration of particle size or specific surface will change its Gibbs energy. Since our present knowledge of the influence of lattice defects on solubility is rather limited, we shall restrict ourselves to a discussion of the particle size effect only. 

Let us consider an elemental crystal first (with defect d). If Nd identical defects are formed in such a crystal of N identical elements, a local free enthalpy of Agd° is required to form a single defect, and if interactions can be neglected, the Gibbs energy of the defective crystal (GP refers to the perfect crystal) is... 

Figure 3.12 The Gibbs energy of a crystal as a function of the number of point defects present. At equilibrium, defects are present in the crystal...

Under equilibrium conditions, the Gibbs energy of a crystal, G, is lower if it contains a small population of Schottky defects, similar to the situation shown in Figure 3.12. This means that Schottky defects will always be present in crystals at temperatures above 0 K, and hence Schottky defects are intrinsic defects. The approximate number of Schottky defects, ns, in crystal with a formula MX, at equilibrium, is given by ... 

The presence of a small number of Frenkel defects reduces the Gibbs energy of a crystal and so Frenkel defects are intrinsic defects. The formula for the equilibrium concentration of Frenkel defects in a crystal is similar to that for Schottky defects. There is one small difference compared with the Schottky defect equations the number of interstitial positions that are available to a displaced ion, N, need not be the same as the number of normally occupied positions, N, from which the ion moves. The number of Frenkel defects, np. present in a crystal of formula MX at equilibrium is given by ... 

Point defects, both vacancies and interstitials, are thermodynamically stable since they lower the Gibbs energy of the crystal. The equilibrium concentrations of point defects rapidly increase with temperature. In metals, vacancies are the predominant point defects in the equilibrium state and their concentrations at high temperatures are much larger than those of interstitials. 

An ideal single crystal has no defects. However, since the Gibbs energy of crystal formation AG = AH-TAS) is a balance between the energetic A/f term (or the tendency to have the most perfect and well-packed structure) and the entropic TAS term (or the tendency for disorder), the minimum AG for a real crystal in the equilibrium state at r 54 0 K could be attained only if a certain non-zero concentration of equilibrium defects is present. Thus, defects are a natural and thermodynamically permitted feature of any existing crystal. 

The creation of single, unassociated point defects in an elemental, crystalline solid increases the internal energy of the system and the enthalpy of the defect formation is positive. But the configurational entropy of the system also increases, and the equilibrium concentration of the defects will be reached when the Gibbs energy of the system is at minimum. Thermodynamically, point defects will thus always be present in a crystal above 0 K. 

The papers of Wagner and Schottky contained the first statistical treatment of defect-containing crystals. The point defects were assumed to form an ideal solution in the sense that they are supposed not to interact with each other. The equilibrium number of intrinsic point defects was found by minimizing the Gibbs free energy with respect to the numbers of defects at constant pressure, temperature, and chemical composition. The equilibrium between the crystal of a binary compound and its components was recognized to be a statistical one instead of being uniquely fixed. 

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