Scanning Electrochemical Microscopy

Description of the process

Scanning Electrochemical Microscopy (SECM) finds many applications in current problems in the biological field. Quantitative mathematical models have been developed for different operating modes of the SECM. Except for some very specific problems, like the diffusion-controlled current on a circular electrode far away from the border, solutions can only be obtained by numerical simulation, which is based on discretization of the model in space by an appropriate method like finite differences, finite elements, or boundary elements. After discretization, a high-dimensional system of ordinary differential equations is obtained. Its high dimensionality leads to high computational cost.

We consider a cylindrical electrode in Fig.1. The computation domain under the 2D-axisymmetrical approximation includes the electrolyte under the electrode. We assume that the concentration does not depend on the rotation angle. A single chemical reaction takes place on the electrode:

$Ox+e^-\Leftrightarrow Red,$ (1)

where $Ox$ and $Red$ are two different species in the reaction. According to the theory of SECM [2], the species transport in the electrolyte is described by diffusion only. The diffusion partial differential equation is given by the second Fick's law as follows

$\frac{dc_1}{dt}=D_1\cdot \Delta ^2c_1 , \, \frac{dc_2}{dt}=D_2\cdot \Delta ^2c_2,$

where $c_1$ and $c_2$ are the concentration fields of species $Ox$ and $Red$, respectively. The initial conditions are $c_1(0)=c_{1,0}, \, c_2(0)=c_{2,0}$. Conditions at the glass and the bottom of the bath are described by the Neumann boundary conditions of zero flux $\nabla c_1\cdot \vec{n}=0, \, \nabla c_2\cdot \vec{n}=0$. Conditions at the border of the bulk are described by Dirichlet boundary conditions of constant concentration, equal to the initial conditions $c_1=c_{1,0}, \, c_2=c_{2,0}$. The boundary conditions at the electrode are described by

$\nabla c_1\cdot \vec{n}=j, \, \nabla c_2\cdot \vec{n}=-j.$

Here $j$ is related to the forward reaction rate $k_f$ and the backward reaction rate $k_b$ through the Buttler-Volmer equation,

$j=k_f \cdot c_1-k_b \cdot c_2.$

The reaction rates $k_f$ and $k_b$ are in the follow form,

$k_f=k^0\exp{(\frac{\alpha z F(v(t)-v^0)}{RT})}k_b=k^0\exp{(\frac{-(1-\alpha) z F(v(t)-v^0)}{RT})} .$

Here, $k^0$ is the heterogeneous standard rate constant, which is an empirical transmission factor for a heterogeneous reaction. $F$ is the Faraday-constant, $R$ is the gas constant, $T$ is the temperature and $z$ is the number of exchanged electrons per reaction. $u(t)=v(t)-v^0$ is the difference between the electrode potential and the reference potential. This difference, to which we refer below as voltage, is changed during the measurement of a voltammogram.

Description of the model

The control volume method has been used for the spatial discretization of (1). Together with the boundary conditions, the resulting system of ordinary differential equations is as follows,

$E\frac{d\vec{c}}{dt}+K(u(t))\vec{c}-A\vec{c}=B,\quad y(t)=C\vec{c},\quad \vec{c}(0)=\vec{c}_0 \neq 0,$

where E and $K(u(t))$ are system matrices, $K(u(t))$ is a function of voltage that in turn depends on time. The voltage appears in the system matrix due to the boundary conditions~(\ref{b3}). The vector $\vec{c} \in \mathbb{R}^n$ is the vector of unknown concentrations, which includes both the $Ox$ and $Red$ species. The vector $B$ is the load vector, which arises as a consequence of the Dirichlet boundary conditions imposed at the bulk boundary of the electrolyte. The total current is computed as an integral (sum) over the electrode surface. The matrix $K(u(t))$ has the following form,

$K(u(t))=K_1(u(t))+K_2(u(t)),$

where $K_i(u(t))=h_i D_i, \, i=1,2$, and $h_1=\exp(\beta u(t)), \, h_2=\exp(-\beta u(t))$ with $u(t)=v(t)-v^0$. The voltage $u(t)$ is a function of $\sigma$,

$u(t)=\sigma t-1, \, \text{for} t \leq \frac{2}{ \sigma}, \quad u(t)=-\sigma t+3, \, \text{for} \frac{2}{ \sigma} < t \leq \frac{4}{ \sigma},$

where $\sigma$ can take four different values, $\sigma=0.5, \, 0.05, \, 0.005, \, 0.0005$. The constant $\beta$ is computed from the parameters $\alpha, \, z, \, F, \, R$, and $T$, leading to the value $\beta=21.243036728240824$.

Although the system is a time-varying system, it can be considered as a parametrized systems with two parameters $h_1$ and $h_2$.

Data information

The data of the system matrices $E, \ D_1, \ D_2, \ A, \ B, C$ as well as the initial state $\vec{c}_0=x_0$ are in MatrixMarket format (http://math.nist.gov/MatrixMarket/), and can be downloaded here File:Matrices SECM.tgz. The interesting output of the model is the current which is computed by $I(t)=C(5,:)\vec{c}$ in MATLAB notation. The interested plot of the output is called the cyclic voltammogram, which is the plot of the current changing with the voltage $u(t)$.

Fig.1

References

[1] L. Feng, D. Koziol, E. B. Rudnyi, and J. G. Korvink, "Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms," Sensor Letters, Vol. 4, 1-10, 2006, pp.1-10.

[2] M. V. Mirkin, "Theory in scanning electrochemical microscopy," Bard, A. J. and M. V. Mirkin, Eds. (2001). New York, John Wiley & Sons. pp. 145 – 199.