Model ionic hopping

Point defects in solids make it possible for ions to move through the structure. Ionic conductivity represents ion transport under the influence of an external electric field. The movement of ions through a lattice can be explained by two possible mechanisms. Figure 25.3 shows their schematic representation. The first, called the vacancy mechanism, represents an ion that hops or jumps from its normal position on the lattice to a neighboring equivalent but vacant site or the movement of a vacancy in the opposite direction. The second one is an interstitial mechanism where an interstitial ion jumps or hops to an adjacent equivalent site. These simple pictures of movement in an ionic lattice, known as the hopping model, ignore more complicated cooperative motions. 

Equation (5.7) is a general equation defining conductivity in all conducting materials. To understand why some ionic solids conduct better than others it is useful to look at the definition more closely in terms of the hopping model that we have... 

When sodium lignosulfonate or sulfur lignin are compounded, for instance, with iodine or bromine, complexes supposedly form (16-17). These systems are conductors with mixed ionic and electronic nature. Presumably they are charge transfer complexes, since the electronic conductivity predominates (18-19). These compounded materials form charge transfer structures (20). Water is supposed to introduce ionic conductivity to the system. Impurities affect conductivity, too (21). In any case, the main models of conductivity are probably based on the band model and/or the hopping model. 

The second necessary condition for crystalline or vitreous solid to have high ionic conductivity is that the mobile ions have a high diffusion coefficient, i.e. it is indeed a fast ion conductor . Much attention has been given to developing models of ionic motion. The simple hopping models applied successfully in the case of defect transport are not appropriate because of the high density of mobile ions in solid electrolytes, and... 

The discussion here is a brief summary of some of the considerations in a simple model for lithium conduction in an ionic solid. In practice, the movement of a lithium ion may be strongly correlated the vibrational modes of the lattice and the movement of other lithium ions and is likely to involve some degree of local relaxation of the lattice around sites which switch from containing vacancies to accommodating a lithium ion. In some remarkable cases, such as lithium sulfate, the lithium ion conductivity is associated with partial melting of the anion sublattice within the crystalline material. Despite the complexities of individual systems the basic hopping model provides a useful framework to discuss ion conductivity in the solid state and it is supported by considerable experimental evidence. 

The first half of this chapter concentrates on the mechanisms of ion conduction. A basic model of ion transport is presented which contains the essential features necessary to describe conduction in the different classes of solid electrolyte. The model is based on the isolated hopping of the mobile ions in addition, brief mention is made of the influence of ion interactions between both the mobile ions and the immobile ions of the solid lattice (ion hopping) and between different mobile ions. The latter leads to either ion ordering or the formation of a more dynamic structure, the ion atmosphere. It is likely that in solid electrolytes, such ion interactions and cooperative ion movements are important and must be taken into account if a quantitative description of ionic conductivity is to be attempted. In this chapter, the emphasis is on presenting the basic elements of ion transport and comparing ionic conductivity in different classes of solid electrolyte which possess different gross structural features. Refinements of the basic model presented here are then described in Chapter 3. 

The model was constructed by decomposing the electrolyte into layers, which contain a certain number of cations, anions, and vacancies, as shown in Figure 1. During the simulations, the cations remain fixed, while the anions and vacancies can hop from layer to layer (while preserving the overall charge neutrality of the system). An important consideration in this work is the treatment of the electric field, and its influence on the diffusive ionic motion within the electrolyte. For instance, a hopping event (i.e., diffusion of an ion from one layer to a neighboring... 

Elliott (1987, 1988 and 1989) approached the relaxation problem differently. In his diffusion controlled relaxation (DCR) model, Elliott, like Charles (1961) considers ionic motion to occur by an interstitialcy mechanism. There is a local motion of cations (for example Li ion in a silicate glass) among equivalent positions located around a NBO ion. Motions of cations among these positions causes the primary relaxational event and it occurs with a characteristic microscopic relaxation time t. The process gives rise to a polarization current. However, when another Li ion hops into one of the nearby equivalent positions with a probability P(/), a double occupancy results around the anion and this makes the relaxation instantaneous. Since the latter process involves the diffusion of a Li ion, the process as a whole involves both polarization and diffusion currents. Thus the relaxation function can be written as [l-P(/)]exp(-t/r). [1-P(0] is a function of the jump distance and the diffusion constant. Making use of the Glarum-Bordewijk relation (Glarum, 1960 Bordewijk, 1975) for [1-/ (/)] Elliott (1987) has shown that... 

Gileadi et al. [22], in their study of the conductivity characteristics of certain salts, namely AlBr3-LiBr and AlBr3-KBr, in toluene, have observed a behavior similar to that found by Fuoss and Kraus. The model proposed by them is based on a hopping mechanism of ionic species from one cluster to another. 

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