Modeling defect formation

Results have shown that the properties of solids can usually be modeled effectively if the interactions are expressed in terms of those between just pairs of atoms. The resulting potential expressions are termed pair potentials. The number and form of the pair potentials varies with the system chosen, and metals require a different set of potentials than semiconductors or molecules bound by van der Waals forces. To illustrate this consider the method employed with nominally ionic compounds, typically used to calculate the properties of perfect crystals and defect formation energies in these materials. 

The MEG model has been extensively used to determine lattice energies and interionic equilibrium distances in ionic solids (oxides, hydroxides, and fluorides Mackrodt and Stewart, 1979 Tossell, 1981) and defect formation energies (Mack-rodt and Stewart, 1979). Table 1.21 compares the lattice energies and cell edges of various oxides obtained by MEG treatment with experimental values. 

Defect formation and dynamics in the crystal and at the melt-crystal interface are molecular-scale events that are only adequately simulated by lattice-scale models. A discussion of lattice-scale equilibrium and calculations of molecular dynamics is beyond the scope of this chapter. 

Starting from the mixture model, the structural behavior of water in the presence of dissolved simple ions is discussed from the point of view of defect formation and lattice distortions at interfaces. The observed behavior of the ions and the water lattice is applied to a number of unsolved biological problems in an attempt to elucidate the specific interface phenomena that are characteristic of such systems. 

Ab initio methods provide elegant solutions to the problem of simulating proton diffusion and conduction with the vehicular and Grotthuss mechanism. Modeling of water and representative Nation clusters has been readily performed. Notable findings include the formation of a defect structure in the ordered liquid water cluster. The activation energy for the defect formation is similar to that for conduction of proton in Nafion membrane. Classical MD methods can only account for physical diffusion of proton but can create very realistic model... 

The lattice energy based on the Born model of a crystal is still frequently used in simulations [14]. Applications include defect formation and migration in ionic solids [44,45],phase transitions [46,47] and, in particular, crystal structure prediction whether in a systematic way [38] or from a SA or GA approach [ 1,48]. For modelling closest-packed ionic structures with interatomic force fields, typically only the total lattice energy (per unit cell) created by the two body potential,... 

Based on the ADC model, we proposed a kinetic model for defect formation in II-VI compounds. A system of rate equations describing the interaction between a II-VI crystal and chalcogen vapor was set up and solved under the assumption that only the atomic component of the vapor phase reacts with the crystal surface. In calculating the ADC equilibrium on... 

Thus, the proposed method for analyzing the kinetics of defect formation, based on the quasi-epitaxy model, clearly demonstrates the role of the surface in the ADC equilibrium and allows one to assess the effects of temperature and Jb of the equilibration of defect concentrations. Computer simulations of defect formation kinetics in II-VI crystal non-equilibrium chalcogen vapor systems indicate that steady state defect concentrations in the surface layer are reached very rapidly and, accordingly, are not... 

In concluding this section, we note that defect calculations may be used to study defect mobilities as well as defect formation and interaction processes. Assuming the validity of the hopping model of defect transport the frequency of defect jumps can then be written as ... 

Levy et al. [47] related the kinetics of decomposition of NaBr03 to the concentration of colour centres present. Rate expressions, derived from models involving defect formation followed by charge transfer, accurately described the experimental nr-time results. 

In this chapter we have seen that application of porous ceramic coatings on porous substrates for preparing membrane supports is a complex process. Every step has to be carried out successfully to obtain substrates or membranes themselves which fulfil the requirements. We have seen that models of the coating processes are useful but still far from capable of describing the processes completely. We have seen that specific aspects such as prevention of defect formation still defy quantitative and sometimes also qualitative understanding. We have seen that sccding-up is not a trivial matter and that much work has to be performed to enable successful preparation of large surface areas. 

An alternative perspective on the subject of point defects to the continuum analysis advanced above is offered by atomic-level analysis. Perhaps the simplest microscopic model of point defect formation is that of the formation energy for vacancies within a pair potential description of the total energy. This calculation is revealing in two respects first, it illustrates the conceptual basis for evaluating the vacancy formation energy, even within schemes that are energetically more accurate. Secondly, it reveals additional conceptual shortcomings associated with... 

The substitution of divalent cations for Mg2+ ions in MgO has been extensively studied within the ionic model (Mackrodt and Stewart, 1979). Recently these systems have been the subject of a series of systematic studies within the LCAO-HF formalism in which the highly symmetric nature of the system is exploited to lower computational costs and to simplify the geometrical relaxation (Freyria-Fava et al., 1993 Dovesi and Orlando, 1994 Orlando et al., 1994a). These studies contain the only published attempt to demonstrate the convergence of the supercell method for the treatment of defects within a QM formalism in ionic systems. The defect formation energy ( D) of, e.g., substitution of Ca for Mg ions in MgO may be referenced to the energies of the isolated ions ( Mg2< and ca2+)> i.e. 

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