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:

$Ox+e^-\Leftrightarrow Red,$

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 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 $c_2$ is the concentration field 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 to 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