Spatially limited diffusion coupled with ohmic potential drop and/or slow interfacial exchange: a new method to determine the diffusion time constant and external resistance from potential step (PITT) experiments

Abstract

We have analyzed chronoamperometric curves, I( t), after small-amplitude potential steps Δ E (PITT technique) for the model of linear diffusion of a species inside an electroactive film, taking into account ohmic effects in the external media (solution and electrode) as well as a finite rate of the interfacial exchange. For its short-time interval, t≪ τ d ( τ d is the diffusion time constant, corresponding to unlimited diffusion from the interface), three approximate analytical expressions have been proposed. One of these represents an interpolation formula between the value of the current at the start of the diffusion process, I(0)=Δ E/ R ext (after the end of the EDL charging), and the Cottrell equation: I≅I(0)/(1+Λ( π t/τ d ) 1/2) , Eq. (9) where Λ= R d/ R ext is the ratio of diffusion and external (solution, electrode, etc.) resistances. Its comparison with the exact analytical solution derived recently by [Montella, J. Electroanal. Chem. 518 (2002) 61] shows the ability of this simple approximation to reproduce qualitatively the current-time dependence within the short-time interval for a wide range of Λ including the case where the external resistance is dominant. Another similar formula, Eq. (10), but with fractional exponents results in even better agreement with the exact result. Both analytical expressions enable one to evaluate the parameters, τ d and Λ, of a real system by linear fitting of its experimental data in the corresponding coordinates, e.g., in coordinates, [ I( t) t 1/2] −1 vs. t −1/2 for the above analytical expression. The treatment of experimental data in these coordinates also allows one to determine the upper limit of the “short-time range” in which the analytical approximation is applicable so that only the points for this time interval are used for the fitting procedure. We have also derived an analytical expression (11) for the same short-time interval, which reproduces the exact solution with a much higher precision. Since the use of this formula does not allow one to extract parameters of the process from experimental data by the simple linear fitting, we have proposed an original procedure of the data treatment to determine τ d and Λ without complicated calculation or optimization schemes.