Understanding cyclic voltammetry: Electron Transfer Mechanism

Cyclic voltammetry (CV) is a widely used electrochemical technique for analyzing the charge transfer of a redox-active specie during a linearly cycled potential sweep. It provides valuable information about interfacial processes, redox thermodynamics, diffusion coefficients, electrode surface properties, and charge-transfer kinetics. However, despite its popularity, CV is often misunderstood due to the simultaneous occurrence of multiple processes, complicating the interpretation of voltammograms.

Let's consider the following redox reaction:

$$F{e}^{2+}\underset{k_b}{\stackrel{k_f}{\rightleftharpoons }}F{e}^{3+}$$

The rate of this reaction can be described phenomenologically as:

$$r=k_f[F{e}^{2+}]-k_b[F{e}^{3+}]$$

In the classical chemical kinetics, the kinetic constant depends exponentially with temperature according to the Arrhenius equation:

$$k=A\exp (-\frac{Ea}{RT})$$

This equation tells us that the rate constant increases exponentially with temperature, emphasizing the importance of thermal energy in overcoming the activation barrier. In electrochemistry the situation is slightly different. The rate constant for an electrochemical reaction depends on the applied potential (voltage), which provides the energy needed for the charge transfer reaction to occur at the electrode surface. This relationship is given by the Butler-Volmer equation, which describes the kinetics as a function of  the difference between the applied potential and the equilibrium potential.

$$k_f=k_0\exp \left(\frac{\alpha nF(E-E_0)}{RT}\right)$$

$$k_b=k_0\exp (-\frac{(1-\alpha )nF(E-E_0)}{RT})$$

In these equations, k_f represents the rate of oxidation, while k_b represent the rate of reduction. Here, k₀ is the global charge transfer constant, F is the faraday constant, n denotes the number of charges being transferred, and α is the dimensionless transfer coefficient. E is the applied potential, E₀ is the equilibrium potential, which is the potential at which the rates of oxidation and reduction are equal. 

The figure below illustrates how the rates of oxidation and reduction change exponentially with the applied potential. When the polarization is more negative than E₀, reduction is favoured. Conversely, when the polarization is more potive than E₀, oxidation is favoured.  However, it's important to note that both processes occur at all potentials. Changing the polarization favours one process over the other, but it doesn't mean that the other process ceases to exist. 

Another key aspect of these redox processes is that the charge transfer process only occurrs at the interface between the electrode and the electrolyte solution. When the applied potential is far from the equilibrium potential (E₀), the rate of oxidation is very high, rapidly consuming all Fe²⁺ ions near the interface. As a result, a concentration gradient is generated, causing Fe²⁺ ions to diffuse from the bulk solution to the interface, leading to the formation of a diffusion layer. This diffusion layer has a concentration gradient that differs from the bulk solution. Tipically, the length of this layer is about 400 micrometers. The following animation illustrates this behaviour.

As seen in the animation, the peak current is not an intrinsic property of the system but instead indicates the potential at which the rate of oxidation decreases due to a signification reduction in the concentration of Fe²⁺ near the interface. At this point, most of the Fe²⁺ ions available come from the diffusion from the bulk to the interface. However, because diffusion  is a slow process, the current decays until it becomes null or reaches a steady state. It is often said, that at this stage, the current is controlled by mass transport.

According to Butler-Volmer equation, the reaction rate is proportional to the current, given by:

$$i=FA(k_f[F{e}^{2+}{]}_0-k_b[F{e}^{3+}{]}_0)$$

where A is the area of the electrode, and subscript 0 denotes the concentration of species near the electrode|electrolyte interface. However, as previously discussed, mass transport plays a crucial role in this process. The current can be defined as the flux of Fe²⁺ in the proximity of the electrode:

$$i=FA{D}_{F{e}^{2+}}{\left(\frac{\partial [F{e}^{2+}]}{\partial x}\right)}_{x=0}$$ 

where D_Fe²⁺ is the diffusion coefficient of Fe²⁺. This equation emphasizes the importance of the mass transport in determining the current at the electrode surface.

Finally, it is very common  to determine the half-wave potential (E_1/2) which is the midpoint between the cathodic (E_c) and the anodic (E_a) potentials. This potential is usually associated to E₀ 

$$E_{1/2}=E_c-E_a$$

The following app simulates a cyclic voltammogram and the corresponding diffusion layer, allowing the user to explore how different parameters affect the output. The parameters you can adjust include:

• Concentration of  Fe²⁺ and Fe³⁺
• Diffusion coefficient of Fe²⁺ and Fe³⁺ (D_Fe(II) and D_Fe(III), respectively): Observe how diffusion affects the symmetry of the CV. Note that slider values are on a logarithm scale.
• The global transfer charge constant (k₀): Determine when the redox process is reversible or irreversible. The slider values for k₀ are also on a logarithm scale.
• The length of the bulk solution (x) in mm: Explore the changes in the CV as the length of the bulk becomes shorter than the diffusion layer.
• The charge transfer coefficient (α): The influence of alpha is significant at low values of k₀ but becomes neglegible at high values of k₀.
• The number of charges being transferred (n): This value can be fractional as it relates to the moles of charges transferred.

By adjusting these parameters, you can gain a deeper understanding of the factors influencing cyclic voltammetry and diffusion layers in electrochemical systems. In addition, the plot can be exported as csv files.

References

1. Stephens, L.I., Mauzeroll, J.; Desmytifying Mathematical Modeling of Electrochemical Systems. J. Chem. Educ. 2019, 96, 10, 2217-2224. https://doi.org/10.1021/acs.jchemed.9b00542.
2. Compton, R.G, Laborda, E. Ward, K.R. (2014) Understanding Voltammetry: Simulation of Electrode Processes. Imperial College Press., 2014.  https://doi.org/10.1142/p910 
3. Bard, A.J., Faulkner, L.R., White, H.S. (2022) Electrochemical Methods: Fundamentals and Applications. (3rd ed). John Wiley & Sons. https://doi.org/10.1023/A:1021637209564
4. Elgrishi, N., Dempsey, J.L., et al.; A practical Beginner's Guide to Cyclic Voltammetry. J. Chem. Educ. 2018, 95, 2, 197-206. https://doi.org/10.1021/acs.jchemed.7b00361

If you want to cite this blog post use: Robayo, I. (2024). Understanding cyclic voltametry: EC Mechanism [online] Principles of Electrochemistry Available at: https://electrochemeisbasics.blogspot.com/2024/07/understanding-cyclic-voltammetry-ec.html [Accessed Date Accessed].