Ion hopping

Another example of alternating line width effect was found in the spectra of durosemiquinone (6),9,10 where the effect is due to alkali metal ions hopping back and forth from one oxygen atom to the other. The rates depend on the alkali metal as shown in Table 5.3. 

Ion exclusion chromatography, of ascorbic acid, 25 760 Ion hopping, 14 469 Ionic aggregates, 14 463—466 Ionically conducting polymers, 13 540 Ionic carbides, 4 647 Ionic compounds, rubidium, 21 822 Ionic conduction, ceramics, 5 587-589 Ionic crystals, 19 185. See also Silver halide crystals... 

Ion hopping is a familiar concept in the chemistry of solid-state conductors, e.g. in the semiconductor industry. In the fluoride electrode, fluoride vacancies in.side the solid LaF lattice allow for conduction of charge (see Figure 3.11), in turn registered by the electrode as a potential. The emf is zero if the internal and external solutions are the same because the same numbers of fluoride ion enter the crystal from either face. 

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 above two mechanisms may be regarded as isolated ion hops. Sometimes, especially in solid electrolytes, cooperative ion migration occurs. An example is shown in Fig. 2.1(c) for the so-called interstitialcy or knock-on mechanism. A Na" ion. A, in an interstitial site in the conduction plane of j -alumina (see later) cannot move unless it persuades one of the three surrounding Na ions, B, C or D, to move first. Ion A is shown moving in direction 1 and, at the same time, ion B hops out of its lattice site in either of the directions, 2 or 2. It is believed that interstitial Ag" ions in AgCl also migrate by an interstitialcy mechanism, rather than by a direct interstitial hop. 

This notion of occasional ion hops, apparently at random, forms the basis of random walk theory which is widely used to provide a semi-quantitative analysis or description of ionic conductivity (Goodenough, 1983 see Chapter 3 for a more detailed treatment of conduction). There is very little evidence in most solid electrolytes that the ions are instead able to move around without thermal activation in a true liquid-like motion. Nor is there much evidence of a free-ion state in which a particular ion can be activated to a state in which it is completely free to move, i.e. there appears to be no ionic equivalent of free or nearly free electron motion. 

The activation energy represents the ease of ion hopping, as already indicated above and shown in Fig. 2.5. It is related directly to the crystal structure and in particular, to the openness of the conduction pathways. Most ionic solids have densely packed crystal structures with narrow bottlenecks and without obvious well-defined conduction pathways. Consequently, the activation energies for ion hopping are large, usually 1 eV ( 96 kJ mole ) or greater and conductivity values are low. In solid electrolytes, by contrast, open conduction pathways exist and activation energies may be much lower, as low as 0.03 eV in Agl, 0.15 eV in /S-alumina and 0.90 eV in yttria-stabilised zirconia. 

In activated ion hopping processes such as occur in solid electrolytes, there is an inverse correlation between the magnitude of the activation energy and the frequency of successful ion hops this leads us to the concept of ion hopping rates. 

There are two times to consider, therefore, in order to characterise ionic conduction. One is the actual time, tj, taken to jump between sites this is of the order of 10 -10 s and is largely independent of the material. The other time is the site residence time, which is the time (on average) between successful hops. The site residence times can vary enormously, from nanoseconds in the good solid electrolytes to geological times in the ionic insulators. Ion hopping rates, cOp, are defined as the inverse of the site residence times, i.e. 

The ion hopping rate is an apparently simple parameter with a clear physical significance. It is the number of hops per second that an ion makes, on average. As an example of the use of hopping rates, measurements on Na )3-alumina indicate that many, if not all the Na" ions can move and at rates that vary enormously with temperature, from, for example, 10 jumps per second at liquid nitrogen temperatures to 10 ° jumps per second at room temperature. Mobilities of ions may be calculated from Eqn (2.1) provided the number of carriers is known, but it is not possible to measure ion mobilities directly. 

Such a mechanism is not incompatible with a Haven ratio between 0.3 and 0.6 which is usually found for mineral glasses (Haven and Verkerk, 1965 Terai and Hayami, 1975 Lim and Day, 1978). The Haven ratio, that is the ratio of the tracer diffusion coefficient D determined by radioactive tracer methods to D, the diffusion coefficient obtained from conductivity via the Nernst-Einstein relationship (defined in Chapter 3) can be measured with great accuracy. The simultaneous measurement of D and D by analysis of the diffusion profile obtained under an electrical field (Kant, Kaps and Offermann, 1988) allows the Haven ratio to be determined with an accuracy better than 5%. From random walk theory of ion hopping the conductivity diffusion coefficient D = (e /isotropic medium. Hence for an indirect interstitial mechanism, the corresponding mobility is expressed by... 

The simplest model involves ion hopping between sites on a lattice (not fixed as in a solid electrolyte such as Agl) with the ions obeying a hopping type equation... 

Using a simple electrostatic interaction-based model factored into reaction rate theory, the energy barrier for ion hopping was related to the cation hydration radius. The conductance versus water content behavior was suggested to involve (1) a change in the rate constant for the elementary ion transfer event and (2) a change in the membrane microstructure that affects conduction pathways. 

Two possible mechanisms for the movement of ions through a lattice are sketched in Figure 5.4. In Figure 5.4(a), an ion hops or jumps from its normal position on the lattice to a neighbouring equivalent but vacant site. This is called the vacancy mechanism. (It can equally well be described as the movement of a vacancy instead of the movement of the ion.) Figure 5.4(b) depicts the interstitial mechanism, where an interstitial ion jumps or hops to an adjacent equivalent site. These simple pictures of movement in an ionic lattice are known as the hopping model, and ignore more complicated cooperative motions. 

This compares well with an estimation of the minimum size of the dipolar cluster formed around nearest neighbors Nbs+ ion given by [M>5+]-1 = [0.38] 1 2.63. At elevated temperatures (T > 354 K), the Nb5+ ions hop at random between equivalent minima of their potential wells within their site in the unit cell. As the phase transition is approached, these ions form a dipolar cluster around the Cu+ ions that are randomly distributed far apart from each other. These clusters are at first of a minimum size containing only the Nbs+ ions that are closest to the Cu+ ion. As the phase transition is approached, the cluster size around the Cu+ ion grows accordingly. These represent the rearranging clusters of the A G theory. 

Nafion 117 were compared with the values in the aqueous phase at infinite dilution [53], the ratio of />Nafion ii7/f inflnite ditmion was found to be 1/41 and 1 /93 for and Al ions, respectively. Miyoshi [32] had earlier found in their work with Na, K, Ca, Cu, and ions during DMP that the drop in self-diffusion coefficient value in the membrane phase was higher for the bivalent ions than for the monovalent ions. This point can be understood from the fact that in their movement across the membrane, trivalent ions have to link themselves with three negative sites, to maintain electroneutrality, while monovalent ions need to link with only one negative site. Therefore, the efficiency with which a multivalent ion hops within an exchange membrane is significantly reduced in comparison with monovalent ions. 

Tierney, N.K. Register, R.A. Ion hopping in ethylene-methacrylic acid ionomer melts as probed by rheometry and cation diffusion measurements. Macromolecules 2002, 35, 2358. 

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