Defect entropy

Because a defect entropy (Ai j), a defect enthalpy and a defect vol-... 

Defect thermodynamics, as outlined in this chapter, is to a large extent thermodynamics of dilute solutions. In this situation, the theoretical calculation of individual defect energies and defect entropies can be helpful. Numerical methods for their calculation are available, see [A. R. Allnatt, A. B. Lidiard (1993)]. If point defects interact, idealized models are necessary in order to find the relations between defect concentrations and thermodynamic variables, in particular the component potentials. We have briefly discussed the ideal pair (cluster) approach and its phenomenological extension by a series expansion formalism, which corresponds to the virial coefficient expansion for gases. 

In addition to energies, defect entropies may also be calculated by related, although distinct, techniques. As with the case of the perfect lattice, defect entropies have two terms - configurational and vibrational and the former can be calculated using simple combinatorial expressions. The latter arises from the perturbation of the vibrational frequency of the surrounding lattice atoms by the defect. The vibrational defect entropy 5 vib is a function of the ratio of the perturbed to the unperturbed frequencies, that is,... 

Work of Harding and coworkers has clearly established the value of defect entropy calculations using these methods. Moreover, we note that comparisons have been made between entropies and energies calculated using supercell and embedded crystallite techniques. It is reassuring that the techniques yield the same defect parameters for large sizes of the supercell and of the crystallite. 

Ong (1976) of 2.71 eV and 5.5k where k is Boltzmann s constant. (The constant volume numbers are 2.71 eV for the defect energy and —1.65k for the defect entropy, underlining the points made above about constant pressure and constant volume calculations). The agreement here is exceptionally close the interionic potentials for calcium fluoride are well established. However, there is no difficulty in principle in carrying out the calculations for more complex systems. The problem is often the lack of reliable experimental data for comparison. 

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

Brunauer (see Refs. 136-138) defended these defects as deliberate approximations needed to obtain a practical two-constant equation. The assumption of a constant heat of adsorption in the first layer represents a balance between the effects of surface heterogeneity and of lateral interaction, and the assumption of a constant instead of a decreasing heat of adsorption for the succeeding layers balances the overestimate of the entropy of adsorption. These comments do help to explain why the model works as well as it does. However, since these approximations are inherent in the treatment, one can see why the BET model does not lend itself readily to any detailed insight into the real physical nature of multilayers. In summary, the BET equation will undoubtedly maintain its usefulness in surface area determinations, and it does provide some physical information about the nature of the adsorbed film, but only at the level of approximation inherent in the model. Mainly, the c value provides an estimate of the first layer heat of adsorption, averaged over the region of fit. 

The most direct effect of defects on tire properties of a material usually derive from altered ionic conductivity and diffusion properties. So-called superionic conductors materials which have an ionic conductivity comparable to that of molten salts. This h conductivity is due to the presence of defects, which can be introduced thermally or the presence of impurities. Diffusion affects important processes such as corrosion z catalysis. The specific heat capacity is also affected near the melting temperature the h capacity of a defective material is higher than for the equivalent ideal crystal. This refle the fact that the creation of defects is enthalpically unfavourable but is more than comp sated for by the increase in entropy, so leading to an overall decrease in the free energy... 

The critical size of the stable nucleus at any degree of under cooling can be calculated widr an equation derived similarly to that obtained earlier for the concentration of defects in a solid. The configurational entropy of a mixture of nuclei containing n atoms widr o atoms of the liquid per unit volume, is given by the Boltzmann equation... 

Several points are worth noting about these formulae. Firstly, the concentrations follow an Arrhenius law except for the constitutional def t, however in no case is the activation energy a single point defect formation energy. Secondly, in a quantitative calculation the activation energy should include a temperature dependence of the formation energies and their formation entropies. The latter will appear as a preexponential factor, for example, the first equation becomes... 

Approximate formulae for the point defect concentrations close (but not too close) to the stoichimetric composition in AB alloys have been derived. They show that the prefactors in the Arrhenius formulae are sensitive functions of the stoichiometry, besides representing the usual formation entropy term. 

Point defects are always present in every material in thermodynamic equilibrium. Considering the formation of n vacancies, the increase in configuration entropy is determined by the number of different possible ways of taking n atoms out of the crystal comprising N atoms in total. This number, c1, is given by... 

We have already stated that some defects are related to the entropy of the solid, and that a perfeet solid would violate the second law of thermodynamics. The 2nd law states that zero entropy is only possible at absolute zero temperature. However, most solids exist at temperatures far above absolute zero. Thus, most of the solids that we eneounter are defeet-solids. The defects are usually "point defeets", which are atomlstie... 

All of these point defects are intrinsic to the heterogeneous solid, and cirise due to the presence of both cation and anion sub-lattices. The factors responsible for their formation are entropy effects (stacking faults) and impurity effects. At the present time, the highest-purity materials available stiU contain about 0.1 part per billion of various impurities, yet are 99.9999999 % pure. Such a solid will still contain about IQi impurity atoms per mole. So it is safe to say that all solids contain impurity atoms, and that it is unlikely that we shall ever be able to obtain a solid which is completdy pure and does not contain defects. 

Equation 3.6.10. given above shows that intrinsic defect concentrations will increase with increasing temperature and that they will be low for high Enthalpy of defect formation. This arises because the entropy effect is a positive exponential while the enthalpy effect is a negative exponential. Consider the following examples of various types of compounds and the natural defects which may occur (depending upon how the compounds were originally formed) ... 

The following gives the standard Enthalpy and Entropy of these defect reactions, according to Kroeger (1965) ... 

Here, AEy and ASy as the stemdard internal energy and standard entropy of the defects, respectively. This gives us a final result of ... 

Grain boundaries form junctions between grains within the particle, due to vacancy and line-defect formation. This situation arises because of the 2nd Law of Thermodjmamics (Entropy). Thus, if crystallites are formed by precipitation from solution, the product will be a powder consisting of many small particles. Their actual size will depend upon the methods used to form them. Note that each crystallite can be a single-crystal but, of necessity, will be limited in size. 

Scheffler M, Dabrowski J. 1988. Parameter-free calculations of total energies, interatomic forces and vibrational entropies of defects in semiconductors. Phil Mag A 58 107-121. 

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