Vacancies equilibrium number

A plot of this equation (Fig. 2.1 b) closely resembles Figure 2.1a. The minimum in the curve gives the equilibrium number of vacancies present and confirms that vacancies exist in all crystals at temperatures above 0 K. For this reason these defects cannot be removed by thermal treatment but are always present in a crystal. Such defects are thus intrinsic defects. At equilibrium, AGy will be equal to zero and the minimum in the AGy versus tiy curve is given by... 

Calculate the energy of vacancy formation in aluminum, given that the equilibrium number of vacancies at 500°C is 7.57 x 10 m. State your assumptions. 

The equilibrium number of vacancies depends exponentially on temperature ... 

Virtually all minerals contain defects. In addition to point defects (e.g., vacancies that exist in a thermodynamically determined equilibrium number, impurities etc ), macroscopic minerals contain line defects (dislocations), and planar defects such as stacking foults, antiphase boundaries and twins. Intergrown layers of different structure or composition, and polytypic disorder also may be present. 

Equation (5.21), despite its apparent simplicity, is an extremely powerful relationship because once the free energy of a system is formulated as a function of the state of equilibrium can simply be determined by differentiation of that function (i.e., locating the minimum). In Chap. 6 how this simple and yet powerful method is used to determine the equilibrium number of vacancies in a solid is described. It is important to emphasize that this condition for equilibrium is valid only when the changes are occurring at constant temperature and pressure. 

Equation (6.7) predicts that the equilibrium number of vacancies increases exponentially with temperature. To understand why. it is instructive to compare Fig. 6.2a and 6, where Eq. (6.6) is plotted, on the same scale, for two different temperatures. At higher temperatures (Fig. 6.26), the configurational entropy term becomes more important relative to the enthalpy term, which in turn shifts eq to higher values. 

On the same graph, plot Eq. (6.6) for two different values of hj for the same temperature, and compare the equilibrium number of vacancies. Which will have the higher number of defects at equilibrium Why ... 

The creep of materials can also occur solely by diffusion, i.e., without the motion of dislocations. Consider a crystal under the action of a combination of tensile and compressive stresses, as shown in Fig. 7.4. The action of these stresses will be to respectively increase and decrease the equilibrium number of vacancies in the vicinity of the boundaries. (The boundaries are acting as sources or sinks for the vacancies.) Thus, if the temperature is high enough to allow significant vacancy diffusion, vacancies will move from boundaries under tension to those under compression. There will, of course, be a counter flow of atoms. As shown in Fig. 7.4, this mass flow gives rise to a permanent strain in the crystal. For lattice diffusion, this mechanism is known as Nabarro-Herring creep. The analysis showed that the creep rate e is given by... 

Figure 11.3 is a plot of nAE, AS, and AG. From this plot you can see that introducing vacancies lowers the free energy of the crystal until an equilibrium concentration is reached adding too many vacancies increases G again. At higher temperatures the equilibrium number of vacancies increases. The implications are important. In pure crystals we expect to hnd point defects at all temperatures above OK. Since these defects are in thermodynamic equilibrium, annealing or other thermal treatments cannot remove them. 

To give you an idea of what these numbers mean calculate the equilibrium number of vacancies that would be in a single crystal of MgO the size of the earth (reanh -6400km) at room temperature it is still a pretty small number. 

In diffusional processes, such as the classic Kirkendall effect of interdiffusion in a bulk diffusion couple of A and B, the atomic flux of A is not equal to the opposite flux of B. If we assume that A diffuses into B faster than B diffuses into A, we might expect that there will be a compressive stress in B, since there are more A atoms diffusing into it than B atoms diffusing out of it. However, in Darken s analysis of interdiffusion, there is no stress generated in either A or B. But Darken has made a key assumption that vacancy concentration is in equilibrium everywhere in the sample. To achieve vacancy equilibrium, we must assume that lattice sites can be created and/or annihilated in both A and B, as needed. Hence, provided that the lattice sites in B can be added to accommodate the incoming A atoms, there is no stress. The addition of a large number of lattice sites implies an increase in lattice planes if we assume that the mechanism of vacancy creation and/or annihilation is by dislocation climb mechanism. It further implies that lattice planes can migrate. 

Calculate the equilibrium number of vacancies in a material at some specified temperature, given the relevant constants. 

The equilibrium number of vacancies A7 for a given quantity of material (usually per meter cubed) depends on and increases with temperature according to... 

Tutorial Video Computation of the Equilibrium Number of Vacancies... 

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

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... 

In thermodynamic equilibrium, the free energy has a minimum. Accordingly, F does not change with the number of introduced vacancies n. Feeding Eq. (2) into Eq. (3) results in the following equilibrium concentration of vacancies ... 

The formation of the combination of defects may be described as a chemical reaction and thermodynamic equilibrium conditions may be applied. The chemical notations of Kroger-Vink, Schottky, and defect structure elements (DSEs) are used [3, 11]. The chemical reactions have to balance the chemical species, lattice sites, and charges. An unoccupied lattice site is considered to be a chemical species (V) it is quite common that specific crystal structures are only found in the presence of a certain number of vacancies [12]. The Kroger-Vink notation makes use of the chemical element followed by the lattice site of this element as subscript and the charge relative to the ideal undisturbed lattice as superscript. An example is the formation of interstitial metal M ions and metal M ion vacancies, e.g., in silver halides ... 

For an atom in a solid, vibratory motion involves potential energy as well as kinetic ener, and both modes will contribute a term l/2kT, resulting in an average total energy of 3kT. Thus, it is the entropy of mixing that forces the creation of a certain number of vacant lattice positions above 0.0 °K. Hence, vacancies are the natural resultof thermod5mamic equilibrium md not the result of accidental growth or sample preparation. 

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... 

Because the configurational entropy of interstitial defects has the same form as that of vacancies, a population of self-interstitial atoms is also thermodynamically stable. The creation of these defects can then also be treated as a pseudochemical equilibrium, and an equation for the relationship between the number of self-interstitials and the appropriate equilibrium constant for interstitial generation, Kv is readily... 

---