Sublattice mobile

Every ionic crystal can formally be regarded as a mutually interconnected composite of two distinct structures cationic sublattice and anionic sublattice, which may or may not have identical symmetry. Silver iodide exhibits two structures thermodynamically stable below 146°C sphalerite (below 137°C) and wurtzite (137-146°C), with a plane-centred I- sublattice. This changes into a body-centred one at 146°C, and it persists up to the melting point of Agl (555°C). On the other hand, the Ag+ sub-lattice is much less stable it collapses at the phase transition temperature (146°C) into a highly disordered, liquid-like system, in which the Ag+ ions are easily mobile over all the 42 theoretically available interstitial sites in the I-sub-lattice. This system shows an Ag+ conductivity of 1.31 S/cm at 146°C (the regular wurtzite modification of Agl has an ionic conductivity of about 10-3 S/cm at this temperature). 

In densely packed solids without obvious open channels, the transport number depends upon the defects present, a feature well illustrated by the mostly ionic halides. Lithium halides are characterized by small mobile Li+ ions that usually migrate via vacancies due to Schottky defects and have tc for Li+ close to 1. Similarly, silver halides with Frenkel defects on the cation sublattice have lc for Ag+ close to 1. Barium and lead halides, with very large cations and that contain... 

Chemical diffusion has been treated phenomenologically in this section. Later, we shall discuss how chemical diffusion coefficients are related to the atomic mobilities of crystal components. However, by introducing the crystal lattice, we already abandon the strict thermodynamic basis of a formal treatment. This can be seen as follows. In the interdiffusion zone of a binary (A, B) crystal having a single sublattice, chemical diffusion proceeds via vacancies, V. The local site conservation condition requires that /a+/b+7v = 0- From the definition of the fluxes in the lattice (L), we have... 

Conceptually it is often convenient to formulate transport only in terms of point defect fluxes since point defects are the primary mobile species. Regular SE s in ionic crystals are then rendered mobile by point defect jumps. We assume (in accordance with many systems of practical importance) that the X anions are (almost) immobile and refer the fluxes to the X sublattice. At sufficiently low concentrations of point defects, their individual elementary jumps are independent. Thus... 

In a single sublattice crystal (A, B) with a fixed number of lattice sites and a negligible fraction of vacancies, the sum of the fluxes of A and B has to vanish if the number of sites is to be conserved. We just noted that if we formulate the A and B fluxes in the binary system as usual, they will not be equal in opposite directions because of the differing mobilities (bA 4= bB). However, if we have a local production (annihilation) of lattice sites which operates in such a way as to compensate for any differences in the two fluxes by the local lattice shift velocity, vL, we then obtain... 

However, a shift of the AO crystal does not always occur in gradients. If, for example, in the oxygen potential gradient, cations are immobile and anions are the mobile species (e.g., in U02), the cation sublattice is a closed subsystem and thus cannot be shifted. Therefore, if oxygen is transported via anionic (plus electronic) defects across the AO slab, the whole crystal is stationary. Likewise, if the solid solution (A,B)0 is exposed to an oxygen potential gradient and transport is by way of anionic point defects, there is again no crystal shift. 

In many cases, p is rather insensitive to the composition (NAO) because both A21 and B2+ are rendered mobile by the same vacancies in the same sublattice. In deriving Eqn. (8.11), we have assumed that (A, B)0 is an ideal quasi-binary solid solution. Analogous to Eqn. (8.6), Eqn. (8.11) has to be integrated under the restricting condition of the conservation of cation species A and B. There is no analytical solution to this problem, but a numerical solution has been presented in [H. Schmalzried, et al. (1979)]. 

AX at the AY/AX boundary. The main feature of this interface reaction (i.e., the transport of building elements across b) is the injection of mobile point defects into available vacant sites and the subsequent local relaxation towards equilibrium distributions. According to Figure 10-9 b, two different modes of cation injection can take place in the relaxation zone R. 1) Cations are injected into the sublattice of predominant ionic transference in AX by the applied field. In this case, no further defect reaction is necessary for the continuation of cation transport. 2) Cations are injected into the wrong sublattice which does not contribute noticeably to the cation transport in AX. Defect reactions (relaxation) will occur subsequently to ensure continuous charge transport. This is the situation depicted in Figure 10-9b, and, in view of its model character, we briefly outline the transport formalism. 

AC/ is known as the overpotential in the electrode kinetics of electrochemistry. Let us summarize the essence of this modeling. If we know the applied driving forces, the mobilities of the SE s in the various sublattices, and the defect relaxation times, we can derive the fluxes of the building elements across the interfaces. We see that the interface resistivity Rb - AC//(F-y0) stems, in essence, from the relaxation processes of the SE s (point defects). Rb depends on the relaxation time rR of the (chemical) processes that occur when building elements are driven across the boundary. In accordance with Eqn. (10.33), the flux j0 can be understood as the integral of the relaxation (recombination, production) rate /)(/)), taken over the width fR. 

We summarize what is special with these prototype fast ion conductors with respect to transport and application. With their quasi-molten, partially filled cation sublattice, they can function similar to ion membranes in that they filter the mobile component ions in an applied electric field. In combination with an electron source (electrode), they can serve as component reservoirs. Considering the accuracy with which one can determine the electrical charge (10 s-10 6 A = 10 7 C 10-12mol (Zj = 1)), fast ionic conductors (solid electrolytes) can serve as very precise analytical tools. Solid state electrochemistry can be performed near room temperature, which is a great experimental advantage (e.g., for the study of the Hall-effect [J. Sohege, K. Funke (1984)] or the electrochemical Knudsen cell [N. Birks, H. Rickert (1963)]). The early volumes of the journal Solid State Ionics offer many pertinent applications. 

In view of the long-time operation we have to rely on thermodynamically stable structures and compounds, or on pronouncedly metastable situations. Under such conditions, given the nature of the constituents, the relevant control parameters are temperature T, component potentials or partial pressures (Pj, and doping content (C). For given operation conditions, Tand Pare fixed leaving the nature of the major chemical elements and the concentrations of dopants (Cl) as the only variable parameters. (In multinary oxides usually not all sublattices are mobile, with the consequence of having the additional freedom to varying the fine composition... 

At low temperatures the hydrogen atoms in a hydrogen-palladium system would be expected to form a Debye sublattice, but at higher temperatures when the well known, but little understood, diffusion processes set in, heat capacities characteristic of hindered translation for the hydrogen atoms might be expected. Rapid cooling of a mobile system of hydrogen atoms would be expected to produce nonequilibrium conditions. Experimentally the system does behave somewhat as expected, but some unusual consequences of this situation became evident only after the experimental observations. 

Under usual conditions at least one sublattice is very rigid and—in the case of interest (in particular when dealing with solid ion conductors)—one sublattice exhibits a significant atomic mobility. The selectivity of the conductivity (cf. also the selective solubility of foreign species) is indeed a characteristic feature of solids. 

Figure 42 shows the basic elementary ion migration processes in a low defective isotropic ion conductor with a mobility in the A-sublattice. The vacancy mechanism (Fig. 42 top) can be described by a transport process (Zv= effective charge of the A-vacancy) such as... 

The procedure is best illustrated by an example. Suppose that a nonstoichiometric phase of composition MA can have an existence range, which spans both sides of the stoichiometric composition, MX, oo. Assume that in this phase only vacancies are of importance, so that the stoichiometric composition will occur when the number of vacancies on the cation sublattice is exactly equal to the number of vacancies on the anion sublattice, which is, therefore, due to a population of Schottky defects. At other compositions, electrical neutrality is adjusted via mobile electrons or holes, leading to n-type or p-type semiconductivity. Thus there are four defects to consider, electrons, e, holes, h, vacancies on metal sites, Vm, and vacancies on anion sites, Vx. Finally, assume that the most important gaseous component is X2 as is the case in most oxides, halides, and sulphides. 

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