Diffusion and Migration of Redox Active Molecules in Nano-Electrochemical Systems
In a system containing no added inert electrolyte, both diffusion and migration contribute to the total flux of the substance transported to the electrode. The flux for each of the three solutes (i.e. S, P, and Y- the substrate, the product, and the counterion of the substrate (if it is charged), respectively) can be represented by the Nernst-Planck equation.
If the solution containing a redox species is placed between two closely-spaced plane electrodes (separated by a distance l) oriented parallel to each other, and both electrodes are independently polarized, then the mass transport will take place in the limited linear space and the steady-state conditions will be easily attainable. In such the electrochemical cell the product of the electrooxidation at one electrode is transported to- and re-electroreduced at the other electrode (communicating electrodes).
The obtained analytical solutions of the transport equations  allowed derivation of the analytical expressions for the limiting electrostatic potential and the limiting faradaic current recorded in a thin layer dual electrode system . Interestingly, the expression for the limiting electrostatic potential at the interface is identical to that derived for the semi-infinite diffusional-migrational transport to a hemispherical microelectrode in the absence of supporting electrolyte. The corresponding expression for the limiting current, iL, normalized with respect to the purely diffusional response, idL, is identical to its equivalent derived for the semi-infinite diffusional-migrational transport to a hemispherical microelectrode when either θ (DP / DS) is equal to 1 (equal diffusion coefficients of electroactive species, any zS, zP, zS ≠ zP) or zS is zero (uncharged substrate, any θ and zP, and zS ≠ zP) or zP is zero (uncharged product, any θ and zS, and zP ≠ zS). If both electroactive species bear charge a specific relation between iL / idL and θ is established for various zS, zP, zY values.
The theoretical modeling revealed high sensitivity of the thin-layer electrode system response to the diffusion coefficients diversity. It is particularly pronounced for the charge reversal processes: the θ - dependence of the limiting current “blows up” at θ = 1 where the limiting conditions can not be reached and the transport is purely migrational.
The theory allowed one to examine quantitatively the limitation of the derived expressions due to the application of the electroneutrality condition at any point in the system. The electroneutrality approximation in the mass transport modeling associated with the electron exchange implies that the electric field resulting from the flow of current through a solution is not big enough to induce the charge accumulation at any point in the solution. This assumption is generally valid if the electrical double layer occupies an insignificant fraction of the transport depletion layer. This may become questionable when the distance between the electrodes tends to nanometer range. The electrode system modeled in this work gave one the opportunity to test the validity of the electroneutrality condition. It was found that the departure from the electroneutrality diminishes with the increase in the distance between the electrodes. For a 100-nm separation of electrodes and comparable diffusion coefficients of the redox species, the error (expressed by the local net charge density in the solution, ρexc) does not exceed 1% of the initial substrate concentration (CS0). For the electrode processes with charge sign retention (for which the limiting conditions can be reached) the error ρexc does not exceed 10% of CS0 down to 60-nm gap between the electrodes for the entire θ range examined (0.2 – 5.0). The calculations indicated also a strong dependence of ρexcwith respect to θ. However, the direction of this dependence is a function of the type of the electrode process.
The theoretical predictions derived for ultra-thin layer dual electrode system provide quantitative characteristics of the redox cycling process affected by the migrational contribution. The results may be of great significance for the extension of applicability of many devices utilizing redox cycling, including the electrochemical scanning microscopy in low-supported media. Additionally, by decreasing the distance between the electrodes to the nanometer range it may be possible to perform electrochemical detection of individual redox-active molecules affected by migration .
 W. Hyk, Z. Stojek Anal. Chem.74 (2002) 4805.
 W. Hyk, Z. Stojek Electrochemistry Communications34 (2013) 192.
 M.A.G. Zevenbergen, P.S. Singh, E.D. Goluch, B.L. Wolfrum, S.G. Lemay, Nano Lett. 11 (2011) 2881.