Frenkel equilibria

At higher oxygen partial pressures, assume that oxygen is incorporated into the structure and adds to the oxygen interstitials already present as part of the Frenkel equilibrium. The incorporation of neutral oxygen requires the abstraction of two electrons to form oxide ions, thus generating an equal number of holes ... 

The approximations to use depend upon the pressure regime and the values of the equilibrium constants. This oxide is an insulator under normal conditions, and so, in the middle region of the diagram, Frenkel equilibrium is dominant, that is, Kf > Ke and the electroneutrality equation is approximated by... 

The second term on the right hand side of Eqn. (13.5) describes the rate of recombination. In the case of diffusion controlled recombination, fc and k may be calculated in terms of defect diffusivities and steady state concentrations. Without radiation, cd = 0, and the Frenkel equilibrium, requires that cv -cA = K/k. If a steady state is attained under irradiation, the rate of radiation produced defects (cp) add to the thermal production rate, and the sum is equal to the recombination rate. Therefore,... 

Table VII lists the mole fraction of extrinsic defects at 831°C calculated from (16) at the pressures listed. The tensimetric data yield information only on the extrinsic defects. To this calculated number a value has been added which is interpreted to be the mole fraction of interstitial oxygen atoms due to the Frenkel equilibrium...

Anti-Frenkel disorder similar to Frenkel disorder except that the interstitials are anions and vacancies are therefore in the anion sublattice. In Zr02 the reaction is 0 kS + 0[ and the anti-Frenkel equilibrium constant is K p = [ko ][On- This type of thermal defect is found in lattices that have a fluorite structure (CaF2, Zr02), which means that there are many large interstitial sites where the anions can be accommodated, but not the cations because their charge is larger, and they are less well screened from each other. 

Doping with aliovalent species affects the concentrations of the defects that are formed thermally in the intrinsic equilibrium. An increase in the concentration of one defect in a pair implies a decrease in the concentration of the other defect. The Schottky or Frenkel equilibrium constants Kg or Kp, the products of the defect concentrations, are constants that depend only on temperature. This means that if one defect is added by doping, its partner in the thermal equilibrium disappears. 

The concentrations of charged atomic defects—point defects—follow the law of mass action. The considerations of thermodynamic equilibria can be applied to disorder equilibria in solid crystalline compounds, the so-called ordered mixtures. Point defects can be regarded as quasi-chemical species with which chemical reactions can be formulated. This has led to the so-called imperfection chemistry. As an example, the disorder equilibrium between vacancies and interstitial particles—the so-called Frenkel equilibrium—will be regarded. 

For all crystals there exists at equilibrium a constant product of the concentrations of interstitial particles and the corresponding vacancies, the so-called Frenkel equilibrium which depends on temperature and pressure. 

In pure and stoichiometric compounds, intrinsic defects are formed for energetic reasons. Intrinsic ionic conduction, or creation of thermal vacancies by Frenkel, ie, vacancy plus interstitial lattice defects, or by Schottky, cation and anion vacancies, mechanisms can be expressed in terms of an equilibrium constant and, therefore, as a free energy for the formation of defects, If the ion is to jump into a normally occupied lattice site, a term for... 

At the beginning of the century, nobody knew that a small proportion of atoms in a crystal are routinely missing, even less that this was not a mailer of accident but of thermodynamic equilibrium. The recognition in the 1920s that such vacancies had to exist in equilibrium was due to a school of statistical thermodynamicians such as the Russian Frenkel and the Germans Jost, Wagner and Schollky. That, moreover, as we know now, is only one kind of point defect an atom removed for whatever reason from its lattice site can be inserted into a small gap in the crystal structure, and then it becomes an interstitial . Moreover, in insulating crystals a point defect is apt to be associated with a local excess or deficiency of electrons. 

This Wilson-Frenkel law is in fact valid even more generally the essential assumption for the derivation was that the surface structure of the growing interface does not change from the equilibrium one. Every rough surface should then be able to grow on average according to this Wilson-Frenkel law ... 

The equilibrium concentration of defects is obtained by applying the law of mass action to Eq. (7) or (8). This leads in the case of Frenkel disorder to... 

CeHot. From thermodsmamic measurements, it was found that the intrinsic defects were Anti-Frenkel in nature, i.e.- (H + Vh). An equilibrium constant was calculated as ... 

Given ksh = 3 x 10-3 for CaQ2, calculate the number of intrinsic defects present in this crystal. If CaCl2 is face-centered cubic, use the same equilibrium constant to calculate the intrinsic Frenkel, Anti-Frenkel and Interstial defects expected in this crystal. 

The Boltzmann constant is represented by kB. It is more difficult to use Monte Carlo methods to investigate dynamic events as there is no intrinsic concept of time but an ensemble average over the generated states of the system should give the same equilibrium thermodynamic properties as the MD methods. A good review of both MD and the Monte Carlo methods can be found in the book by Frenkel and Smit [40]. 

Fig. 17 B/E-p dependence of the critical temperatures of liquid-liquid demixing (dashed line) and the equilibrium melting temperatures of polymer crystals (solid line) for 512-mers at the critical concentrations, predicted by the mean-field lattice theory of polymer solutions. The triangles denote Tcol and the circles denote T cry both are obtained from the onset of phase transitions in the simulations of the dynamic cooling processes of a single 512-mer. The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

Fig. 18 The equilibrium temperatures (circles) and the heights of the free-energy barrier at these temperatures (triangles) for a single 512-mer as a function of B/Ep. The dashed line shows the demarcation for the occurrence of a prior collapse transition (Hu and Frenkel, unpublished results)...

A number of textbooks and review articles are available which provide background and more-general simulation techniques for fluids, beyond the calculations of the present chapter. In particular, the book by Frenkel and Smit [1] has comprehensive coverage of molecular simulation methods for fluids, with some emphasis on algorithms for phase-equilibrium calculations. General review articles on simulation methods and their applications - e.g., [2-6] - are also available. Sections 10.2 and 10.3 of the present chapter were adapted from [6]. The present chapter also reviews examples of the recently developed flat-histogram approaches described in Chap. 3 when applied to phase equilibria. 

The estimation of the number of Frenkel defects in a crystal can proceed along lines parallel to those for Schottky defects by estimating the configurational entropy (Supplementary Material S4). This approach confirms that Frenkel defects are thermodynamically stable intrinsic defects that cannot be removed by thermal treatment. Because of this, the defect population can be treated as a chemical equilibrium. For a crystal of composition MX, the appropriate chemical equilibrium for Frenkel defects on the cation sublattice is... 

At all temperatures above 0°K Schottky, Frenkel, and antisite point defects are present in thermodynamic equilibrium, and it will not be possible to remove them by annealing or other thermal treatments. Unfortunately, it is not possible to predict, from knowledge of crystal structure alone, which defect type will be present in any crystal. However, it is possible to say that rather close-packed compounds, such as those with the NaCl structure, tend to contain Schottky defects. The important exceptions are the silver halides. More open structures, on the other hand, will be more receptive to the presence of Frenkel defects. Semiconductor crystals are more amenable to antisite defects. 

An intrinsic defect is one that is in thermodynamic equilibrium in the crystal. This means that a population of these defects cannot be removed by any forms of physical or chemical processing. Schottky, Frenkel, and antisite defects are the best characterized intrinsic defects. A totally defect-free crystal, if warmed to a temperature that allows a certain degree of atom movement, will adjust to allow for the generation of intrinsic defects. The type of intrinsic defects that form will depend upon the relative formation energies of all of the possibilities. The defect with the lowest formation energy will be present in the greatest numbers. This can change with temperature. 

It is important that the complete diagram displays prominently information about the assumptions made. Thus, the assumption that Schottky defect formation was preferred to the formation of electronic defects is explicitly stated in the form Ks > Ke (Fig. 7.9e). As Frenkel defect formation has been ignored altogether, it is also possible to write Ks > Ke > > Kt , where A p represents the equilibrium constant for the formation of Frenkel defects in MX. 

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