# Galvanostatic simulations: Fundamental electrochemistry

## UNDER CONSTRUCTION

### Fundamental electrochemistry

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

\[O + e^- \overset{k_f}{\underset{k_r}{\leftrightarrows}} R \overset{k_c}{\rightarrow} Z\]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 reverse electrochemical rate constant, $k_r$, primarily depend on the fundemental electrochemical parameters $k_0$ and $\alpha$ (Eqns 3.3.9 & 3.3.10):

\[k_f = k^0 \exp\left({-\alpha f (E - E^{0'})}\right)\] \[k_r = k^0 \exp\left({(1-\alpha) f (E - E^{0'})}\right)\]$k_0$ 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 $k_c$ 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_1 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.