Last updated: May 10, 2018

This cyclic voltammetry simulation couples a one-electron electrochemical reduction with a subsequent chemical reaction of the reduced species, as below:

$O + e^- \overset{k_f}{\underset{k_b}{\rightleftarrows}} R \overset{k_c}{\rightarrow} Z$

This simulation assumes an aqueous electrochemical system. A metal electrode in solution supplies and extracts electrons from aqueous species O and R, such as in the case of ferro-/ferricyanide.

In this simulation, we model three mechanistic processes: the electrochemical reaction (charge transfer), the chemical reaction, and diffusion. This type of coupled electrochemical-chemical process is often called an $EC$ reaction.

#### Electrochemical reaction (charge transfer)

In this model, the charge-tranfer kinetics follow the Bulter-Volmer formulation. The forward electrochemical rate constant, $k_f$, and backwards electrochemical rate constant, $k_b$, primarily depend on the fundemental electrochemical parameters $k^0$ and $\alpha$ (Eqns 3.3.9 & 3.3.10):

$k_f = k^0 \exp\left({-\alpha f (E - E^{0'})}\right)$ $k_b = k^0 \exp\left({(1-\alpha) f (E - E^{0'})}\right)$

$k^0$ is the standard electrochemical rate constant, defined as the “kinetic facility” of a redox couple - in other words, it measures the ease of electron transfer in the redox reaction (see section 3.3.3). $k^0$, $k_f$, and $k_b$ all have units of $\text{cm/s}$.

$\alpha$ is the transfer coefficient, which is a measure of the “symmetry of the energy barrier” between the forward and reverse reactions (section 3.3.4). $\alpha$ is dimensionless.

Three more terms:

• $f$ is the normalized Faraday’s constant, $\text{38.92 } V^{-1}$ at room temperature
• $E$ is the applied potential, in $V$
• $E^{0’}$ is the formal potential, in $V$. Thus, $E - E^{0’}$ is the overpotential.

The surface concentrations of O and R ultimately control the measured current (3.3.11):

$i = F A k^0 \left[C_O(0,t)\exp\left({-\alpha f (E - E^{0'})}\right) - C_R(0,t)\exp\left({(1-\alpha) f (E - E^{0'})}\right)\right]$

Here, $C_O(0,t)$ and $C_R(0,t)$ are the concentrations of O and R at the electrode surface.

#### Chemical reaction

The reduction product can also chemically react in a simple unimolecular reaction ($R \overset{k_c}{\rightarrow} Z$). Standard first-order reaction kinetics describe this unimolecular reaction:

$\frac{\partial C_R(x,t)}{\partial t} = -k_c C_R$

The units of $k_c$ are $\text{s}^{-1}$, as expected for a first-order reaction rate constant.

#### Diffusion

Fick’s laws model diffusion in this system. Fick’s first law of diffusion is:

$J_i(x,t) = -D_i\frac{\partial C_i(x,t)}{\partial x}$

Here, $J_i(x,t)$ is the flux ($\text{mol/cm}^2 \text{s}$) of species $i$, and $D_i$ ($\text{cm}^2 \text{s}$) is the diffusion coefficient of species $i$. A common assumption in aqueous electrochemistry is that the diffusion coefficients of $O$ and $R$ are equal, or $D_O = D_R$. This assumption is reasonable if $O$ and $R$ are molecularly similar, like the ferro/ferricyanide system.

Fick’s second law is:

$-\frac{\partial C_i(x,t)}{\partial t} = \frac{\partial J_i(x,t)}{\partial x}$

The combined first and second law is:

$\frac{\partial C_i(x,t)}{\partial t} = D_i\frac{\partial C_i^2(x,t)}{\partial x^2}$

#### Summary

The interplay between the electrochemical reaction at the surface, the chemical reaction of the reduced product $R$, and the diffusion of both $O$ and $R$ yield the cyclic voltammetry result. Specifically, two dimensionless numbers—$\Lambda$, an indicator for charge transfer vs mass transfer rates, and $k_c t_k$, an indicator for chemical reaction rate vs experiment time—define the system’s electrochemical reversibility and chemical reversibility, respectively. I discuss these in greater detail here.