Lattice conservative reactions

Fig. 31, Schematic of physicochemical processes that cwcur within a passive film according to the point defect model m = metal atom Mm = metal cation in cation site Oo = oxygen ion in anion site VjjJ = cation vacancy Vq = anion vaccancy Vm = vacancy in metal phase. During film growth, cation vacancies are produced at the film/solution interface, but are consumed at the metal/film interface. Likewise, anion vacancies are formed at the metal/film interface, but are consumed at the film/solution interface. Consequently, the fluxes of cation vacancies and anion vacancies are in the directions indicated. Note that reactions (i), (iii), and (iv) are lattice-conservative processes, whereas reactions (ii) and (v) are not. Reproduced from J. Electrochem, Sec. 139, 3434 (1992) by permission of the Electrochemical Society.

The second-generation point defect model (PDM-II) [39] addressed the deficiencies of the previous model by incorporating a bilayer structure of the film consisting of a defective oxide layer on the metal surface and an outer layer that is formed by precipitation of products firom the reaction of transmitted cations firom the underlying metal with species in the environment. PDM-II assumed that the barrier layer controls the passive current and recognized the barrier layer dissolution and the need to distinguish whether the reactions are lattice conservative or nonconservative. The model also introduced the metal interstitials to the suite of defects. The model is in agreement with experimental results. Model PDM-III extends the apphcation of the PDM model and addresses the formation of multiple passive layers at the outer layer [40]. 

If all fluxes vanish and the number of lattice sites is conserved, only two types of homogeneous reactions between SE s are possible... 

In Section 4.7, we discussed the relaxation process of SE s in a closed system where the number of lattice sites is conserved (see Eqn. (4.137)). A set of coupled differential equations was established, the kinetic parameters (v(x,iq,x )) of which describe the rate at which particles (iq) change from sublattice x to x. We will discuss rate parameters in closed systems in Section 5.3.3 where we deal with diffusion controlled homogeneous point defect reactions, a type of reaction which is well known in chemical kinetics. 

Fortunately, the enthalpy of this reaction has been experimentally measured to be —423 kJ inol . Adding in the value —500 + 20 kJ mol for the lattice energy provides an estimate of the beat of the reaction in Eq. 4.25 that is essentially zero. This 1 somewhat discoursing, since if Eq. 425 is not exothermic, entropy will drive the reaction to the left because all of those spedes are gases, and dioxygenyl letra-fluoroborate would oot be expected to be stable. Recall, however, (hat our estimates were on the conservative side. We would therefore expect that dioxygenyl tetra-fluoroborate is either energetically unfavorable or may form with a i tively low stability. It certainly is worth an attempt at synthesis. 

If one is interested in properties that vary on very long distance and time scales it is possible that a drastic simplification of the molecular dynamics will still provide a faithful representation of these properties. Hydrodynamic flows are a good example. As long as the dynamics preserves the basic conservation laws of mass, momentum and energy, on sufficiently long scales the system will be described by the Navier-Stokes equations. This observation is the basis for the construction of a variety of particle-based methods for simulating hydrodynamic flows and reaction-diffusion dynamics. (There are other phase space methods that are widely used to simulate hydrodynamic flows which are not particle-based, e.g. the lattice Boltzmann method [125], which fall outside the scope of this account of MD simulation.)... 

An even more drastic simplification of the dynamics is made in lattice-gas automaton models for fluid flow [127,128]. Here particles are placed on a suitable regular lattice so that particle positions are discrete variables. Particle velocities are also made discrete. Simple rules move particles from site to site and change discrete velocities in a manner that satisfies the basic conservation laws. Because the lattice geometry destroys isotropy, artifacts appear in the hydrodynamics equations that have limited the utility of this method. Lattice-gas automaton models have been extended to treat reaction-diffusion systems [129]. 

In writing equations for defect reactions using this notation, there must be conservation of charge, mass and ratio of lattice sites. 

In compound crystals, balanced-defect reactions must conserve mass, charge neutrality, and the ratio of the regular lattice sites. In pure compounds, the point defects that form can be classified as either stoichiometric or nonstoichiometric. By definition, stoichiometric defects do not result in a change in chemistry of the crystal. Examples are Schottky (simultaneous formation of vacancies on the cation and anion sublattices) and Frenkel (vacancy-interstitial pair). 

As indicated by Equation (1), intercalation reactions are usually reversible, and they may also be characterized as topochemical processes, since the structural integrity of the host lattice is formally conserved in the course of the forward and reverse reactions. Typically, these reactions occur near room temperature, but this is in sharp contrast with most conventional solid-state synthetic procedures which often require temperatures in excess of 600 °C, the term Chemie Douce has been coined to describe this type of low-temperature reaction. Remarkably, a wide range of host lattices has been found to undergo these low-temperature reactions, including framework (3D), layer (2D), and linear chain (ID) lattices. 

For instance, if MgO is used to dope AI2O3, because the ionic radii of Mg " and Al with coordination number of six are very close, the Mg ions can enter the lattice of AI2O3 to form solid solution as substitutional defects. AI2O3 has the corundum structure, in which one-third of the octahedral sites formed by the close-packed O ions are vacant, so that it is also highly possible for the Mg ions to sit on the interstitial sites. The defects with lower energy are more favorable. In AI2O3, the cation sites and anion sites have a number ratio of 2 3. If substitutional defects are formed, every two Mg atoms on cation sites will replace two A1 sites and two O sites are involved. In this case, the third O site should be a vacancy for site conservation. Therefore, on the basis of mass and site balance, the defect reaction is given by ... 

Defect exarriDles V Iron vacancy in e.g. FejO Vp Oxygen vacancy in a metal oxide Zn" Zinc interstitial in e.g. ZnO Al( Al substitutional dopant in e.g. SrTIOj Defect reaction reouirements 1. Conservation of mass 2. Conservation of lattice site stoichiometry 3. Conservation of charge... 

Note that this is an irreversible reaction, since spontaneous demixing will not occur. This means that one cannot define an equilibrium constant for a dissolution reaction. Closer inspection shows that reaction (2.13) indeed fulfills the required conservation of mass, charge, and lattice site stoichiometry. 

One may try to get rid of the oxygen vacancy in reaction (2.13) by performing the s5mthesis in an oxygen-rich atmosphere. In this case, both mass and lattice site stoichiometry are conserved when adding the oxygen gas, which means that the charge has to be balanced by the addition of a hole ... 

Site ratio conservation The ratio of the number of regular cation sites to the number of regular anion sites in the crystal remains constant. For example, in the compound MO, the ratio of the regular M and O sites must remain in the ratio of 1 1. Sites may be created or destroyed in the defect reaction, but they must occur in such a way that the site ratio in the regular lattice is maintained. 

Defect reactions can be written in a similar manner as ordinary chemical reactions. They must confer with the requirements for conservation of mass, charge, and ratios of lattice sites, but are allowed to increase or decrease the total number of lattice units. 

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