Long-Range Electron Transfer through Ultrathin Polyelectrolyte Complex Films: A Hopping Model

Pinhole-free ultrathin films of polyelectrolyte complex assembled using layer-by-layer deposition were used to evaluate electron transfer from a redox species in solution to an electrode over the distance range of 1–9 nm. Over this thickness, the polyelectrolytes employed wet the surface and the polymer molecules flattened to less than their equilibrium size in three dimensions. A decay constant β for current as a function of distance of about 0.3 nm^–1 placed this system in the regime expected for multistep hopping versus a one-step tunneling event. Discreet hopping sites within the films were identified as ferrocyanide ions with an equilibrium concentration of 0.032 M and an average separation of 3.7 nm. The Butler–Volmer (BV) expression for electron transfer as a function of overpotential was modified by distributing the applied voltage evenly among the hopping sites. This modified BV expression fits both the distance dependence and the applied potential dependence well, wherein the only freely adjustable parameter was the electron transfer coefficient. The finding that β is simply the inverse of the hopping range is consistent with previous conclusions that electrons within conjugated molecule sites are delocalized, or, for nonconjugated systems, spread over more than one repeat unit by lattice distortions.