Vacancy chemical potential

Let us now discuss some details of practical relevance. From the Gibbs phase rule, it is evident that crystals consisting of only one component (A) become nonvariant by the predetermination of two thermodynamic variables, which for practical reasons are chosen to be Pand T. In these one-component systems, it is easy to recognize the (isobanc) concentration dependence of the point defects on temperature. From the definition of the vacancy chemical potential for sufficiently small vacancy mole fractions Nv, namely //v = /A (P, T) + RT- In Vv, together with the condition of equilibrium with the crystal s inerL surroundings (gas, vacuum), one directly finds... 

Equation (2.57) can be compared with Eqn. (2.50). Noting that the standard value of the vacancy chemical potential of a crystal is only slightly dependent on T, Nv is in essence exponentially dependent on 0/T). This Arrhenius-type of temperature dependence is also found for the interstitials, since in view of the site-preserving formation reaction Aa+V) = Af +VA, the equilibrium condition for this defect formation reaction shows that... 

We wish to determine under isothermal and isobaric conditions the concentration of defects as a function of the solid solution composition (e.g NB in alloy (A, B)). Consider a vacancy, the formation Gibbs energy of which is now a function of NB. In ideal (A, B) solutions, we may safely assume that the local composition in the vicinity of the vacancy does not differ much from Ns and /VA in the undisturbed bulk. Therefore, we may write the vacancy formation Gibbs energy Gy(NE) (see Eqn. (2.50)) as a series expansion G%(NE) = Gv(0) + A Gv Ab+ higher order terms, so that AGv = Gv(Nb = l)-Gv(AfB = 0). It is still true (as was shown in Section 2.3) that the vacancy chemical potential /Uy in the homogeneous equilibrium alloy is zero. Thus, we have (see Eqn. (2.57))... 

Chemical potentials entering these equations are relative to the vacancy chemical potential, h ai = Hai - fiv and = hzt - t v- We use to express diffusion fluxes the Onsager reciprocity condition, Laizt = LztM-... 

Several authors " have suggested that in some systems voids, far from acting as diffusion barriers, may actually assist transport by permitting a dissociation-recombination mechanism. The presence of elements which could give rise to carrier molecules, e.g. carbon or hydrogen , and thus to the behaviour illustrated in Fig. 1.87, would particularly favour this mechanism. The oxidant side of the pore functions as a sink for vacancies diffusing from the oxide/gas interface by a reaction which yields gas of sufficiently high chemical potential to oxidise the metal side of the pore. The vacancies created by this reaction then travel to the metal/oxide interface where they are accommodated by plastic flow, or they may form additional voids by the mechanisms already discussed. The reaction sequence at the various interfaces (Fig. 1.87b) for the oxidation of iron (prior to the formation of Fe Oj) would be... 

Figure 7.4 shows the vacancy concentrations computed in this way for three different O2 pressures. Because of the dependence of the oxygen chemical potential on pressure [see Eq. (7.6)], increasing the oxygen pressure dramatically reduces the vacancy concentration. The results in Fig. 7.4 justify two approximations that were made in our calculations above. First, the vacancy concentrations under physically relevant conditions are always very small, so determining the properties of a vacancy in a DFT calculation representing... 

With the introduction of the lattice structure and electroneutrality condition, one has to define two elementary SE units which do not refer to chemical species. These elementary units are l) the empty lattice site (vacancy) and 2) the elementary electrical charge. Both are definite (statistical) entities of their own in the lattice reference system and have to be taken into account in constructing the partition function of the crystal. Structure elements do not exist outside the crystal and thus do not have real chemical potentials. For example, vacancies do not possess a vapor pressure. Nevertheless, vacancies and other SE s of a crystal can, in principle, be seen , for example, as color centers through spectroscopic observations or otherwise. The electrical charges can be detected by electrical conductivity. 

While Eqn. (2.24) justifies the introduction of virtual chemical potentials of SE s including vacancies, it also assumes that the lattice molecule M, according to Eqn. (2.23), is in equilibrium with all the vacancies V, x = 1,..., K. 

The results of the discussion on the phenomenological thermodynamics of crystals can be summarized as follows. One can define chemical potentials, /jk, for components k (Eqn. (2.4)), for building units (Eqn. (2.11)), and for structure elements (Eqn, (2.31)). The lattice construction requires the introduction of structural units , which are the vacancies V,. Electroneutrality in a crystal composed of charged SE s requires the introduction of the electrical unit, e. The composition of an n component crystal is fixed by n- 1) independent mole fractions, Nk, of chemical components. (n-1) is also the number of conditions for the definition of the component potentials juk, as seen from Eqn. (2.4). For building units, we have (n — 1) independent composition variables and n-(K- 1) equilibria between sublattices x, so that the number of conditions is n-K-1, as required by the definition of the building element potential uk(Xy For structure elements, the actual number of constraints is larger than the number of constraints required by Eqn. (2.18), which defines nk(x.y This circumstance is responsible for the introduction of the concept of virtual chemical potentials of SE s. 

By evaluating 8Q and Sy when SNy vacancies are destroyed, an expression for can be obtained. The quantity SQ is just —/lySNy, where the chemical potential of the vacancies, fiy, is given by Eq. 3.66. If a climbing edge dislocation destroys SNy vacancies per unit length, the climb distance will be Sy = (12/6) SNy. The osmotic force is therefore... 

Hint The chemical potential of excess vacancies is given by Eq. 3.66. 

If growth of the AB layer is due mainly to diffusion of the A atoms, a row (a plane) of vacancies of component A moves in its bulk. The driving force for this movement is the difference in values of the chemical potential of component A in initial phases A and B, which clearly remains constant in spite of their interaction. 

Erhart and Albe also calculated zinc diffusion in ZnO [130]. The results are displayed in Fig. 1.18 together with a comparison to experimental data. Depending on chemical potential and Fermi level position either zinc vacancy or zinc interstitial diffusion can dominate. In the case of n-type material, where the Fermi level is close to the conduction band, zinc diffusion is mostly accomplished via the vacancy mechanism. 

Fig. 1.17. Oxygen diffusion in ZnO [129]. Top Dependence of diffusivity on chemical potential and Fermi level at a temperature of 1 300 K illustrating the competition between vacancy and interstitialcy mechanisms. The dark grey areas indicate the experimental data range around 1 300 K. Bottom Comparison between calculation and experiment. Experimental data from Moore and Williams [131], Hofmann and Lauder [132], Robin et al. [133], Tomlins et al. [134], Haneda et al. [135], and Sabioni et al. [136]. Solid and dashed lines correspond to regions I (interstitialcy mechanism dominant) and II (vacancy mechanism dominant) in the top graph, respectively. Copyright (2006) by the American Physical Society...

The doped semiconductor materials can often be considered as well-characterized, diluted solid solutions. Here, the solutes are referred to as point defects, for instance, oxygen vacancies in TiC - phase, denoted as Vq, or boron atoms in silicon, substituting Si at Si sites, Bj etc. See also -> defects in solids, -+ Kroger-Vink notation of defects. The atoms present at interstitial positions are also point defects. Under stable (or metastable) thermodynamic equilibrium in a diluted state, - chemical potentials of point defects can be defined as follows ... 

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