Diffusion electron hopping

In addition to the configuration, electronic stmcture and thennal stability of point defects, it is essential to know how they diffuse. A variety of mechanisms have been identified. The simplest one involves the diffusion of an impurity tlirough the interstitial sites. For example, copper in Si diffuses by hopping from one tetrahedral interstitial site to the next via a saddle point at the hexagonal interstitial site. 

An example of a layer structure mixed conductor is provided by the cathode material L CoC used in lithium batteries. In this solid the ionic conductivity component is due to the migration of Li+ ions between sheets of electronically conducting C0O2. The production of a successful mixed conductor by doping can be illustrated by the oxide Cei-jPxx02- Reduction of this solid produces oxygen vacancies and Pr3+ ions. The electronic conductivity mechanism in these oxides is believed to be by way of electron hopping between Pr4+ and Pr3+, and the ionic conductivity is essentially vacancy diffusion of O2- ions. 

Attaching the catalyst molecules to the electrode surface presents an obvious advantage for synthetic and sensor applications. Catalysis can then be viewed as a supported molecular catalysis. It is the object of the next section. A distinction is made between monolayer and multilayer coatings. In the former, only chemical catalysis may take place, whereas both types of catalysis are possible with multilayer coatings, thanks to their three-dimensional structure. Besides substrate transport in the bathing solution, the catalytic responses are then under the control of three main phenomena electron hopping conduction, substrate diffusion, and catalytic reaction. While several systems have been described in which electron transport and catalysis are carried out by the same redox centers, particularly interesting systems are those in which these two functions are completed by two different molecular systems. 

The second factor, electronic conduction through an assembly of redox centers, involves electron hopping between adjacent sites. It is analyzed so as to highlight the characteristics of this transport in terms of equivalent diffusion. Field effects will also be addressed. Although bearing some... 

Why does electron hopping between adjacent redox sites behave like a diffusional transport is the question we address now. A first answer could be found in investigations of the diffusion of redox couples in solution in which it was found that the diffusion rate is augmented by electron exchange in solution between the members of the couple.13 The resulting overall diffusion coefficient, Dap, is thus the sum of two terms ... 

A more rigorous approach consists of considering that electron hopping between fixed redox sites is fundamentally a percolation problem, each redox center being able to undergo a bounded diffusion motion.16 If these are fast enough, a mean-field behavior is reached in which (4.24) applies replacing d2 by d2 + 3 Ad2, where Adr is the mean displacement of a redox molecule out of its equilibrium position. 

What benefits and drawbacks to these problems can one expect from the use of cyclic voltammetry instead of RDEV They are related. In a general case, the application of cyclic voltammetry will be more complicated, because playing with the scan rate, one can make the diffusion layer penetrate the film or remain outside, as is the case with RDEV. We have already seen a fruitful application of the first of these possibilities in the use of cyclic voltammetry to the characterization of electron hopping transport within the redox films (Section 4.3.4). In the second situation, cyclic voltammetry may replace RDEV in a manner similar to what has been seen in Section 4.3.2 Each time a term (1 — ///a) is encountered in the analysis, it suffices to replace it by... 

Equivalent Diffusion and Migration Laws for Electron Hopping Between Fixed Sites... 

The major part of the reports discussed above provides only a qualitative description of the catalytic response, but the LbL method provides a unique opportunity to quantify this response in terms of enzyme kinetics and electron-hopping diffusion models. For example, Hodak et al. [77[ demonstrated that only a fraction of the enzymes are wired by the polymer. A study comprising films with only one GOx and one PAH-Os layer assembled in different order on cysteamine, MPS and MPS/PAH substrates [184[ has shown a maximum fraction of wired enzymes of 30% for the maximum ratio of mediator-to-enzyme, [Os[/[GOx[ fs 100, while the bimolecular FADH2 oxidation rate constant remained almost the same, about 5-8 x 10 s ... 

Above room temperature, the mobile 3 d electrons are well described by a random mixture of Fel" and FeB ions with the mobile electrons diffusing from iron to iron, some being thermally excited to FeA ions, but the motional enthalpy on the B sites is AH < kT. As the temperature is lowered through Tc, the Seebeck coefficient shows the influence of a change in mobile-electron spin degeneracy, and at room temperature the Seebeck coefficient is enhanced by correlated multielectron jumps that provide a mobile electron access to all its nearest neighbors. The electron-hopping time xi, = coi = 10" s... 

So, we have two possibilities for the case of the ion-exchange polymer film-coated electrode reduction could occur by physical diffusion of the Fe3+ through the film, or reduction could occur via electron hopping through the film. How can we know which process is operative Electrochemists have devoted a considerable amount of research effort to answering this question. The answer clearly depends on the nature of the polymer, the extent of swelling of the... 

We describe here that the redox oligomer wires fabricated with the stepwise coordination method show characteristic electron transport behavior distinct from conventional redox polymers. Redox polymers are representative electron-conducting substances in which redox species are connected to form a polymer wire.21-25 The electron transport was treated according to the concept of redox conduction, based on the dilfusional motion of collective electron transfer pathways, composed of electron hopping terms and/or physical diffusion.17,18,26-30 In the characterization of redox conduction, the Cottrell equation can be applied to the initial current—time curve after the potential step in potential step chronoamperometry (PSCA), which causes the redox reaction of the redox polymer film ... 

Taken together with the results from the photoinduced electron-transport experiments in which diffusive thermal electron hopping can occur over a few... 

Figure 3.9 Schematic illustration of the processes that can occur at a modified electrode, where P represents a reducible substance in a film on the electrode surface and A a species in solution. The processes shown are as follows (1) heterogeneous electron transfer to P to produce the reduced form Q (2) electron transfer from Q to another P in the film (electron diffusion or electron hopping in the film) (3) electron transfer from Q to A at the film/solution interface (4) penetration of A into the film (where it can also react with Q or at the substrate/film interface) (5) movement (mass transfer) of Q within the film (6) movement of A through a pinhole or channel in the film to the substrate, where it can be reduced. From A.J. Bard and L.R. Faulkner, Electrochemical Methods Fundamentals and Applications, 2nd Edition, Wiley, 2001. Reprinted by permission of John Wiley Sons, Inc...

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