Vacancy enthalpy

Typical values of the energy to form vacancies are for silver, lOSkJmol and for aluminium, 65.5kJmol These values should be compared with the values for the activation enthalpy for diffusion which are given in Table 6.2. It can also be seen from the Table 6.2 that die activation enthalpy for selfdiffusion which is related to the energy to break metal-metal bonds and form a vacant site is related semi-quantitatively to the energy of sublimation of the metal, in which process all of the metal atom bonds are broken. 

For small enough temperature steps (< lOK) during small step annealing the vacancy concentration practically remains constant and corresponds to the instantaneous aimealing temperature. This allows for an easy analysis of SRO-kinetics yielding SRO-relaxation times and SRO-activation enthalpies, which by usual interpretation correspond to H +Hf. 

M. Migschitz and W Pfeiler, Vacancy activation enthalpies of Au-Fe alloys determined from SRO-induced resistivity changes, A/a/. Sci. Eng. A206 I55 (1996)... 

The enthalpy change involved, AHy, is not explicitly calculated. It is assumed that the enthalpy to form one vacancy, Ahy, is constant over the temperature range of interest, so that the total enthalpy change, Ally, is given by... 

Pure potassium bromide, KBr, which adopts the sodium chloride structure, has the fraction of empty cation sites due to Schottky defects, ncv/Nc, equal to 9.159xl0-21 at 20°C. (a) Estimate the enthalpy of formation of a Schottky defect, Ahs. (b) Calculate the number of anion vacancies per cubic meter of KBr at 730°C (just below the melting point of KBr). The unit cell of KBr is cubic with edge length a = 0.6600 nm and contains four formula units of KBr. 

The favored defect type in strontium fluoride, which adopts the fluorite structure, are Frenkel defects on the anion sublattice. The enthalpy of formation of an anion Frenkel defect is estimated to be 167.88 kJ mol-1. Calculate the number of F- interstitials and vacancies due to anion Frenkel defects per cubic meter in SrF2 at 1000°C. The unit cell is cubic, with a cell edge of 0.57996 nm and contains four formula units of SrF2. It is reasonable to assume that the number of suitable interstitial sites is half that of the number of anion sites. 

TABLE 5.3 Approximate Enthalpy Values for Formation and Movement of Vacancies in Alkali Halide Crystals... 

To obtain a solid with a high conductivity, it is clearly important that a large concentration, c, of mobile ions is present in the crystal [Eq. (6.1)]. This entails that a large number of empty sites are available, so that an ion jump is always possible. In addition, a low enthalpy of migration is required, which is to say that there is a low-energy barrier between sites and ions do not have to squeeze through bottlenecks. Hence the structure should ideally have open channels and a high population of vacancy defects. 

Considerable attention has also been paid to modelling the thermodynamics of defects. This includes, for example, studies of the enthalpies of formation of vacancies or interstitial atoms and the association energies associated with the clustering of such defects. It is usually crucial to allow for the relaxation of the... 

Here, ct is the shear stress in the transition liquid layer, y oi is a molecular volume, h is Planck s constant, AG is the free energy change of the movement of a molecule into a vacancy, and A/Zyac is the enthalpy of formation of a vacancy. The rate of deformation of a hquid is the strain rate, y [see Eq. (2)], so the right-hand side of Eq. (34) can be used to estimate the viscosity of the transition layer. 

The activation energy for oxide ion conduction in the various zirconia-, thoria- and ceria-based materials is usually at least 0.8 eV. A significant fraction of this is due to the association of oxide vacancies and aliovalent dopants (ion trapping effects). Calculations have shown that the association enthalpy can be reduced and hence the conductivity optimised, when the ionic radius of the aliovalent substituting ion matches that of the host ion. A good example of this effect is seen in Gd-doped ceria in which Gd is the optimum size to substitute for Ce these materials are amongst the best oxide ion conductors. Fig. 2.11. 

Vm = migration enthalpy of cationic vacancy = migration enthalpy of anionic vacancy /7m = migration enthalpy of cationic interstitial //xj = migration enthalpy of anionic interstitial... 

N is here the number of lattice defects (vacancies or interstitials) which are responsible for non-stoichiometry. AHfon is the variation of lattice enthalpy when one noninteracting lattice defect is introduced in the perfect lattice. Since two types of point-defects are always present (lattice defect and altervalent cations (electronic disorder)), the AHform takes into account not only the enthalpy change due to the process of introduction of the lattice defect in the lattice, but also that occurring in the Redox reaction creating the electronic disorder. 

Here we present thermodynamic discussions and developments based on recent in situ ETEM studies. These are important in predicting the enthalpy of formation of vacancies in oxide catalysts, the probability for CS planes to form and catalyst performance. They are also important in the design of new or improved oxide catalysts. 

The important parts of Eq. (4.76) are the exponential terms. The first exponential, which contains the entropies associated with potassium ion movement and vacancy formation, respectively, form the temperature-independent contributions to Do, as discussed in the previous section. The second exponential, which contains the enthalpies of the two processes and the temperature dependence, form the activation energy, Ea, and temperature dependence of Eq. (4.71). 

As noted in Sections 5.3-5.5, vacancies in the anion or cation array can exist in equilibrium with the vapor of the depleted element. If the enthalpy of formation of a vacancy in the anion array is markedly greater than that of one in the cation sublattice—for example, in ZnTe, where it is about 1 electron volt (1 eV = 96.5 kJ mol-1) higher—the heated solid will tend to develop an excess of anions and so will become a p-type semiconductor. The enthalpies of vacancy formation correlate with the anion-cation radius ratios thus, very large anions such as Te2- matched with relatively small cations such as Zn2+ favor doping with vapor of the anionic element for... 

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