Electron hopping diffusion model

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 ... 

Since it is well-known that in a microscopic sense, diffusion can be modeled in terms of a random walk, then in three dimensions, the electron-hopping diffusion coefficient can be expressed as follows... 

FIGURE 1.10. (a) Variation of with redox site concentration according to the He-Chen model [see Eqn. 38(a)]. (b) Comparison between the He-Chen prediction and experimental data for electron-hopping diffusion coefficients for Os(bpy)f loaded in Nafion films obtained via complex impedance spectroscopy by Sharp and coworkers (Ref. 40). 

The results imply that the diffusion coefficient represents the thermally activated transport of electrons through the particle network. Indeed, these and subsequent studies have been interpreted with models that involve trapping of conduction band electrons or electron hopping between trap sites [158, 159]. An unexpected feature of the diffusion constants reported by Cao et al. is that they are dependent on the incident irradiance. The photocurrent rise times display a power law dependence on light intensity with a slope of -0.7. The data could be simulated if the diffusion constant was assumed to be second order in the electron concentration, D oc n. The molecular origin of this behavior is not well understood and continues to be an active area of study [157, 159]. 

This last equation is valid as long as the diffusion front of the diffusing species in solution phase remains within the electrode coating, a condition that applies for times shorter than 10-20 msec (Miller and Majda, 1986,1988). Dynamics of electron hopping processes have been recently modeled by Denny and Sangaranarayan (1998) using kinetic Ising model formalism. 

Finally a more comprehensive model for simulating the Ru /Fe system was solved using finite-element methods. This model takes into account mass transport due to diffusion and migration, electron transfer due to electron hopping, homogeneous chemical reaction in the membrane, heterogeneous reactions, double-layer charging, and Donnan partition equilibrium between the membrane and diffusion layer. 

FIGURE 10.2. A model for the mediated reduction of species Y by the 0/R redox couple immobilized in a polymeric film at the electrode surface. The governing process may be either substrate diffusion to the film substrate partition into and diffusion within the layer charge transport within the PME film (governed by relative rates of electron hopping, redox site diffusion, counterion transport, or polymer motions) and electrode or mediation kinetics. 

Besides electric field effects, ion association within the polymer films plays an important role in the dynamics of electron hopping within the films. (Extensive ion association might be expected due to the high ion content and the low dielectric permittivity that prevails in the interiors of many redox polymers.) According to the model that includes ion association, the sharp rise in the apparent diffusion coefiicient as the concentration of the redox couple in the film approaches saturation is an expected consequence of the shift in the ionic association equilibrium to produce larger concentrations of the oxidized form of the redox couple, which is related to rapid electron acceptance from the reduced form of the couple [176]. 

The process of intersite electron hopping has been discussed in terms of a quasi-diffusional process. We now take a more detailed view of the intersite electron transfer reaction in a fixed-site redox polymer. The approach adopted here is due to Fritsch-Faules and Faulkner. These researchers developed a microscopic model to describe the electronhopping diffusion coefficient Z>e in a rigid three-dimensional polymer network as a function of the redox site concentration c. The model takes excluded volume effects into consideration, and it is based on a consideration of probability distributions and random-walk concepts. The microscopic approach was adopted by these researchers to obtain parameters that could be readily understood in the context of the polymer s molecular architecture. A previously published related approach was given by Feldberg. ... 

The ease of preparation and the variety of electron active ions make ion-exchange polymers amenable to fundamental studies of electron transfer mechanisms. The systematic variation of the concentration (effective density) of redox sites within a coating has been useful in the construction of electron hopping models (4). These models are based on the apparent rate of electron diffusion through the film. Because the redox centers are able to diffuse within the polymer, the apparent rate is related to two parameters redox molecule diffusion and the rate of electron self-exchange. 

As the concentration of enzyme in the film is increased from 0 to 65 wt%, the diffusion constant falls by 2 orders of magnitude with a function that is close to exponential. Simplistic application of the Dahms-Ruff theory (which implicitly requires that a "mean-field" approximation holds) would predict a linear dependence on concentration. The latter approximation would require that physical diffusion of the Os sites be rapid compart to electron hopping, which is clearly not the case here. Theory based on rigorously fixed site redox molecules and extended electron transfer (ie. static percolation), would indeed predict an exponential decrease in electron hopping with concentration.(25) However, simulations by Blauch and Saveant for the case of tethered redox sites also leads to a behavior part-way between that predicted by "static percolation" and "mean-field" approximations, resulting in a functional form of close to that seen in the inset of Figure 6. (See Figure 3B of reference 15 where tjtp=0.1). It must be pointed out that the weak dependence of the film thickness, r, on enzyme content (Table I), leads to little electrochemically measurable decrease in site concentration as enzyme is increased. This of course makes a quantitative application of any of the models mentioned above rather difficult. 

Anderson s simple model to describe the electrons in a random potential shows that localization is a typical phenomenon whose nature can be understood only taking into account the degree of randomness of the system. Using a tight-binding Hamiltonian with constant hopping matrix elements V between adjacent sites and orbital energies uniformly distributed between — W/2 and W/2, Anderson studied the modifications of the electronic diffusion in the random crystal in terms of the stability of localized states with respect to the ratio W/V. 

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