Migration Effects in Electron Hopping

Until now we have considered electron-hopping processes in isolation to other charge percolation phenomena occurring in electroactive polymer films. This of course is not valid We must also consider the 

A fundamental feature of all these analyses is that electron hopping is not only driven by a concentration gradient, it is also field-assisted. AVe therefore have a diffusional term and a migrational term. Hence the potential difference between sites is taken into account, and the electronhopping flux /x must contain a migrational contribution. In the following discussion, we adopt the analysis presented by Albery and coworkers.  

We again assume that the polymer consists of neutral sites A that can be oxidized to form positive charge carriers B . At the same time, a charge compensating counterion X enters the polymer layer from the solution as follows 

We consider three adjacent sites in the polymer located at x - 5, x, and X + 8. As before, bimolecular electron exchange occurs between these sites, and it is quantified by a second-order rate constant k. The transition states are located at x - (1 — a)S and x + ad, where a is a symmetry factor. Hence we adopt the following scheme  

Note that in Eqn. 48 we have expanded the exponential because 5 is small. We can now expand the concentrations, potential terms, and kinetic factors in terms of a Taylor expansion, as outlined in the following expressions  

---