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In this article, we derive an approximate asymptotic analytical expression for the long-time chronoamperometric current response at an inlaid microband (or laminar) electrode. The expression is applicable when the length of the microband is much greater than the width, so that the diffusion of the electrochemical species can be regarded as two-dimensional. We extend the previously known result for the diffusion-limited current response (

Chronoamperometry is a widely used voltammetric technique [_{f}_{b}^{−1}):
^{2}/_{O}_{R}

The majority of the theoretical modelling for the current response at an inlaid microband electrode has involved numerical simulations, using a variety of different techniques [

All of the expressions derived previously are only valid for the diffusion-limited regime, either due to extreme polarization of the electrode, or when the diffusion coefficients of the oxidant and the reductant are equal, _{O}_{R}^{2}/^{2}/^{−1}) is detailed in ^{−2} ≤ ^{3} in the

The theoretical problem is depicted in _{f}_{b}_{O}_{R}^{−3}), satisfy the diffusion equation in _{O}_{R}^{2}·s^{−1}). Hence

Initially the bulk concentrations of each species are constant everywhere:
^{−1}), is given by
^{−1}) and we recall that

To non-dimensionalise the problem, we choose the following scalings, where

Here _{O}_{R}

Then, in terms of the non-dimensional variables, the problem becomes:

To solve this problem, we take the Laplace transform in time of the expressions detailed in

The long-time behaviour, ^{2}/_{O,R}_{O,R}

We use the method of matched asymptotic expansions to solve the problem given by ^{−}^{1/2}), such that

In this outer region, the terms on both sides of the governing equations in

In the inner region, we denote the approximate solutions by _{O,R}_{O,R}_{O,R}_{O,R}

The solutions to the inner problem given by _{0} and _{0} are constants, and the solutions for _{i}_{i}_{i}_{i}^{2} = ^{2} + ^{2}):
_{i}_{i}_{0}(·) is a modified Bessel function of the second kind [_{i}_{i}_{i}_{i}

We match the inner and outer solutions using the intermediate variable technique [^{2} = ^{2} + ^{2} and 0 < _{i}_{i}

In the intermediate region, therefore, the inner expansion can be written:

At
_{i}_{i}_{i}

Eliminating _{i}_{i}_{i}_{i}_{1} = −1 from _{O}_{i}

In dimensional terms, the expression in ^{−1}) through the microband electrode becomes:
^{−2} ≤ ^{3} as the file “

We remark that the Faradaic current per unit axial length through a laminar electrode in free space is simply double the expression in

It is useful for the experimentalist to have a working curve to evaluate the function

For a particular value of _{i}_{i}_{i}_{i}_{j}_{j}^{−6} at _{j}

Asymptotic expressions can also be derived for small and large

We include a log-log plot of

In the previous section, we have derived the asymptotic solution detailed in

The diffusion-limited reduction current per unit axial length due to extreme polarization corresponds to _{f}_{b}

Similarly, the diffusion-limited oxidation current per unit axial length due to extreme polarization corresponds to _{f}_{b}

For reversible reactions, _{f}_{b}_{f}_{b}

To calculate the current numerically, we employed the fully implicit finite-difference method (FIFD) devised by Gavaghan [_{max} = 201, 0 ≤ _{max} = 200 for 0 < _{max} = 100. Considerations of symmetry allowed us to simulate only half the domain (_{O}_{R}_{max} = 201 and _{max} = 200, were chosen to satisfy the criterion that _{max},
_{O}_{R}_{max} and _{max}, instead of infinity, does not distort the current through the electrode. Gavaghan's method uses a spatial grid that expands exponentially away from the edge of the electrode in order to resolve accurately the large flux in the neighbourhood of the edge. Here we used the same grid parameters as suggested in [_{last} = 8 × 10^{−5}, and the expansion factor is given by ^{−5}, and was increased by a factor of 10 every thousand steps.

We plot _{f}_{b}_{f}_{b}_{f}_{b}^{−1} in the non-dimensionalisation of the current in _{O}_{R}_{O}_{R}_{O}_{R}

So far we have not assumed any model for the rate constants of the redox reaction, _{f}_{b}_{0} (m·s^{−1}) is the standard kinetic rate constant, 0 < _{0} (V) is the formal oxidation potential. The remaining parameters are Faraday's constant, ^{−1}), the universal gas constant, ^{−1}·mol^{−1}), and the temperature, _{f}_{b}_{0}, and the electron transfer coefficient, _{0} and _{0} and _{0} and

We have derived the approximate asymptotic expression given in

To derive the large-

As described in the main body of the article, the function _{∞}, satisfies:
_{R}_{∞} ∼ log _{∞}, and this breaks down as a singular perturbation in an ^{−1}) region of ^{−1}, and using symmetry about ^{−1}_{∞}, so that for

_{3})

_{6}]

^{3+}and [Ru (NH

_{3})

_{6}]

^{2+}in aqueous solution using microelectrode double potential step chronoamperometry

Cartoon of an inlaid microband electrode of width ^{2}/

Schematic of the theoretical two-dimensional diffusion problem to be solved for the Faradaic current through an inlaid microband electrode. Two redox species with concentrations _{O}_{R}^{−3}) diffuse with unequal diffusion coefficients _{O}_{R}^{2}·s^{−1}) above the electrode. Initially the bulk concentrations are uniform everywhere and equal to and; we assume these remain undisturbed as ^{2} + ^{2} → ∞. On the surface of the electrode, |^{−1}), through the electrode is proportional to the integral of the diffusive flux at the electrode surface.

Log-log plot of the function

Plots of _{f}_{b}_{f}_{b}_{f}_{b}_{O}_{R}_{O}_{R}_{O}_{R}

Log-linear plots of the time-varying percentage difference between the analytical expression given by _{f}_{b}_{f}_{b}_{f}_{b}_{O}_{R}_{O}_{R}_{O}_{R}

Q_beta_working_curve.csv (CSV, 53 KB)