Project 4: Chronoamperometry and Cyclic Voltammetry¶

Tanner Leo, University of Oregon, Electrochemistry Lab, November 28^th¶

1 Introduction¶

Kinetics: Theory and Ligand Contributions

While thermodynamics dictates the conditions for equilibrium of an electron transfer reaction, the kinetics of the reaction take precedent in governing the various aspects of the electron transfer reaction. The kinetics of electron transfer reactions describe how the system evolves with time as the conditions around the reaction change. Marcus theory presents a practical method for describing electron transfer kinetics from the microscopic level.¹

In Marcus theory, the free energy of activation describes the energy required to overcome the activation barrier for an electron transfer step. The magnitude of the activation energy barrier depends on the reorganization energy, λ. Further, the kinetic rate constants for a reaction are exponentially proportional to the reorganization energy. The reorganization energy is often split into the inner and outer reorganization energies, for the active species involved and the solvent, respectively. Equations 1 and 2 below are often used to calculate the inner and outer reorganization energies.

\begin{equation} \lambda_i = \sum_{j} \frac 1 2 k_j ( q_{O,j}-q_{R,j})^2 \tag{1} \end{equation}

Where k is a force constant and q is the displacement of the core components of the oxidized and reduced species. This equation is used to evaluate the energy difference between the vibration modes before and after the electron transfer reaction.

\begin{equation} \lambda_O = \frac{e^2}{4 \pi \epsilon_0} \left(\frac{1}{2a_1} + \frac{1}{2a_2} - \frac{1}{d}\right) \left(\frac{1}{\epsilon_{op}} - \frac{1}{\epsilon_s}\right) \tag{2} \end{equation}

Where a₁ and a₂ are the radii of the reactants including any solvated molecules, and d is the distance between the centers of the two reactants as they exist in their precursor states.¹ From these equations the following generalizations can be made. The standard rate constant, k⁰, will be larger for reactions where O and R very similar. The greater the change of distance or angle between ligands, the solvent shell, or other involved species in the oxidized and reduced states, the greater the reorganization energy will be, and therefore the smaller k⁰ will be.

An example of this can be observed with complexed iron redox couples. Three different iron redox couples are shown in Figure 1 below, where Fe is in the 2+ oxidation state. The largest redox couple with the bulkiest ligands is the Fe(phen)₃^2+/3+. This particular redox couple, when compared to the others, has shown to have the fastest kinetics.² According to Marcus theory, this is because the ligands are positioned nearly the same distance away from the Fe center before and after electron transfer. Fe(aq)^2+/3+ on the other hand, is relatively small, and experiences large changes in the Fe-ligand bond distances before and after the electron transfer reaction. Further, Fe(aq)^2+/3+ shows the slowest kinetics of the three iron redox couples.² The metal-ligand bond distances have been experimentally determined by extended X-ray fine structure analysis (EXAFS), where the predictions from Marcus theory on the kinetics of electron transfer hold true.³

Figure 1. Chemical Structures of three Fe redox couples with different ligands.

Experimental Methods

Different experimental methods are often used to determine the kinetic rates of redox couples, and gain insight into the reversibility of the reaction. One method that provides an approximate picture of the reversibility of a reaction is the analysis of cyclic voltammograms performed at various ramp rates. For a theoretically wholly reversible reaction, the difference between the potentials at the cathodic and anodic peak currents of a CV scan should be 57 mV.⁴ Further, this observation should be independent of scan rate, however no redox couple is completely reversible in practice and some increase in the difference in potentials is expected with increasing scan rate. If parameters such as concentrations, diffusion coefficients, and electrode area are known for the redox couple, the CV curves can be fit to determine the standard rate constant.

A common experimental method for determining diffusion coefficients of species are Cottrell experiments. This method utilizes chronoamperometry, where potential is held at a constant value and current is measured over time. The benefit of this method is that the current is directly proportional to the number of species being oxidized or reduced at the electrode surface, allowing for a clear picture of the concentration gradient as it evolves over time. At time increases, the diffuse layer slowly grows larger until it reaches a limit. The current has a inverse square root relationship with time, as shown in equation 3 below.

\begin{equation} i(t) = \frac{n F A D_R^{1/2}C_R^*}{\pi^{1/2}t^{1/2}} \tag{3} \end{equation}

With knowledge of other parameters such as concentration, the diffusion coefficient can be extracted from the slope of the Cottrell plot, or inverse root time vs current. Using the correct region of the Cottrell plot for determining the diffusion coefficient is important for accurate results. At early times just after the potential is stepped to some value above the equilibrium potential, a significant portion of the current will be contributing towards charging the double layer. Only after about 5R_uC_d, where R_u is the uncompensated resistance and C_d is the double layer capacitance, will the current be purely faradaic. At long times, typically longer than 20 s, the temperature fluctuations in the solution cause convection and disruption of the uniformity of the diffuse layer, which leads to greater-than-expected currents.¹

2 Experimental¶

Three solutions were characterized using cyclic voltammetry and potential step chronoamperometry experiments. A glassy carbon (GC) working electrode was polished using progressively larger grit polishing paper. The first solution contained 5 mM of K₄Fe(CN)₆, and 20 mL of 200 mM KCl as the supporting electrolyte. A three-neck round-bottom flask was used to hold the solution and the electrodes. The solution was purged with N₂ gas for 10 minutes before any experiments were conducted. When positioning the working electrode in the cell, care was taken to ensure that the working electrode was at least 2 cm away from a cell wall. An Ag/AgCl reference electrode and a graphite rod counter electrode were used for all the experiments.

Chronoamperometry experiments were conducted at 100 mV increments from the open circuit potential, towards more anodic potentials. Each potential step collected data for 10 s, followed by a 15 second reductive potential followed by a short open circuit step to allow the diffuse layer to equilibrate. The current range was manually specified for these experiments, and the number of data points collected was increased to near the maximum that the potentiostat allowed for. Cyclic voltammograms were collected using scan rates between 10 mV/s and 1600 mV/s, in a potential window of -0.4 to 0.8 V vs Ag/AgCl.

The second solution contained 5 mM of FeSO₄, and 20 mL of 200 mM K₂SO₄, and was purged prior to experimentation similar to the first solution. Potential step experiments were conducted in an identical manner. The CV experiments were collected at scan rates between 10 mV/s and 400 mV/s, across a potential range of -0.5 to 1.2 V vs Ag/AgCl.

The third solution contained 20 mM of 1,10-phenanthroline and 4.5 mM FeSO₄, in 20 mL of 200 mM K₂SO₄, and was prepared by adding to the second solution. The solution went from having no color to turning a dark red color upon adding 1,10-phenanthroline, as shown in Figure 2. This solution was purged for 10 minutes with inert gas similar to the first two solutions. Potential step experiments were collected in a similar manner as the first two experiments, although at 50 mV increments starting at 100 mV. CV experiments were conducted at scan rates between 10 and 400 mV/s, in a potential range of -0.3 to 1.2 V vs Ag/AgCl.

Figure 2. Image of electrochemical cell after the addition of 1,10-phenanthroline, showing the color change from translucent to dark red.

Lastly, the collected CV data was used to fit simulated CV curves using the EC-Lab CV simulator analysis tool. The nominal values for the active species concentration, transfer coefficient, scan rate, potential range, and electrode area were entered into the simulation parameters, while the measured values for the diffusion coefficient and the standard potential were used. The value of standard rate constant was altered in the simulation parameters until the simulated results fit the measured data. If needed, the other parameters were altered until the simulated results fit the measured data.

3 Results¶

3.1 Potential Step Experiments¶

Potential step experiments were conducted on the Fe(CN)₆^4- containing solution, at 100, 200, 300, and 400 mV overpotential. E_eq was measured at -135 mV vs Ag/AgCl. Figure 3 below shows the resulting Cottrell slopes extracted from the various potential steps and the Cottrell slope that was extracted from the 400 mV potential step. The slopes appear linear between 8 and 17 s^-1/2, where at lower t^-1/2, the slope begins to flatten out, and the slopes of the different potential steps appear more similar. Using the Cottrell equation, given in equation 1, the diffusion coefficient was extracted from the slope, and found to be 3.17 x 10^-6 cm²/s.

Figure 3. Left: Cottrell plot for the Fe(CN)₆^4- CA experiments at four different potential steps. Right: Cottrell plot of the 400 mV potential step and the linear fit used to determine the diffusion coefficient.
The results of the FeSO₄ potential step experiments and the Cottrell slope was extracted from the 400 mV potential step are shown in Figure 4 below. Here, the slopes appear steep at small t^-1/2, and flatten out at larger t^-1/2. The slopes at larger t^-1/2 also appear more similar across the different potentials steps. The diffusion coefficient from this potential step was found to be 3.68 x 10^-6 cm²/s.

Figure 4. Left: Cottrell plot for the FeSO₄ CA experiments at four different potential steps. Right: Cottrell plot of the 400 mV potential step and the linear fit used to determine the diffusion coefficient.

The results of the 1,10-phenanthroline Fe solution (Fe(phen)₃) potential step experiments are shown in Figure 5 below. All of the potential steps from this experiment yielded relatively linear curves across the entire t^-1/2 range. The 450 mV potential step was used for Cottrell analysis and the diffusion coefficient was found to be 3.31 x 10^-6 cm²/s.

Figure 5. Left: Cottrell plot for the Fe(phen)₃ experiments at eight different potential steps. Right: Cottrell plot of the 450 mV potential step and the linear fit used to determine the diffusion coefficient.

3.2 Cyclic Voltammetry Experiments¶

CV experiments were performed on the three different solutions across a range of scan rates. The CV curves collected for Fe(CN)₆^4- are shown in Figure 6 below. The difference between the potentials of peak anodic and cathodic current were calculated as a function of scan rate. At the lowest scan rate of 10 mV/s, the peak difference was 72.9 mV, while at 400 mV/s it was found to be 115.4 mV. E^1/2 was measured to be 0.246 V vs Ag/AgCl.

Figure 6. Cyclic voltammograms of Fe(CN)₆^4- at scan rates between 10 mV/s and 1600 mV/s. The red dots on the CV curve correspond to the measured anodic wave peaks and the blue dots correspond to the measured cathodic wave peaks.

The CV experiments performed on the FeSO₄ solution are shown in Figure 7 below. The peaks of the cathodic and anodic waves are clearly separated by a significant degree. The separation in peak potentials at 10 mV/s was found to be 388.6 mV, and at 400 mV/s the separation was found to be 700.5 mV. E^1/2 was measured to be 0.441 V vs Ag/AgCl.

Figure 7. Cyclic voltammograms of FeSO₄ at scan rates between 10 mV/s and 400 mV/s. The red dots on the CV curve correspond to the measured anodic wave peaks and the blue dots correspond to the measured cathodic wave peaks.

The CV experiments performed on the Fe(phen)₃ solution are shown in Figure 8 below. The peaks current potentials of the anodic and cathodic sweeps appear close to one another at all scan rates. At 10 mV/s, the separation in peaks was 59.5 mV, while at 400 mV/s it was 72.7 mV. E^1/2 was measured to be 0.895 V vs Ag/AgCl.

Figure 8. Cyclic voltammograms of Fe(phen)₃ at scan rates between 10 mV/s and 400 mV/s. The red dots on the CV curve correspond to the measured anodic wave peaks and the blue dots correspond to the measured cathodic wave peaks.

3.3 Simulated Results¶

The measured CV curves were used to determine parameters such as the standard rate constant, k⁰. Due to the limitations of the simulation software, the minimum value for the diffusion coefficients was 1 x 10^-6 cm²/s, and the minimum value for k⁰ was 1 x 10^-6 cm/s. In some cases, a better fit may have been obtained by adjusting these parameters outside of the specified range. The radius of the electrode was set to the 1.5 mm for all simulation runs. The measured and simulated CV curves for the Fe(CN)₆^4- solution are shown in Figure 9 below. The parameters for the simulation are shown in Table 2 below, and compared to the nominal values. All three simulated CV curves performed at 10, 50, and 400 mV/s were well-fitted to the data. The bulk concentration was set to 8 mM, instead of the nominal 5 mM because a portion of the electrolyte was spilled from the flask before the active species was added to the solution. It is expected that the bulk concentration would be larger than the nominal concentration.

Figure 9.
Table 2. Nominal and Simulation Parameters for CV Curve Fitting of Fe(CN)₆^4-/3-

	E⁰ (V vs Ag/AgCl)	k⁰ (cm/s)	α	C_R^* (mM)	D_R/D_O (cm²/s)	C_dl (μF)	R_Ω (ohms)
Simulated	0.246	5 x 10^-3	0.5	8	3.17 x 10^-6	60	0
Nominal	0.246	N/A	0.5	5	3.17 x 10^-6	N/A	N/A

The measured and simulated CV curves for the FeSO₄ solution are shown in Figure 10 below. The parameters for the simulation are shown in Table 3 below, and compared to the nominal values. The simulated CV curve shows a reasonable fit to the data, although the width of the peak potentials are more narrow than the measured results. This becomes apparent when the scan rate is changed, as the potentials of peak currents do not align with the slower scan rates. The relative magnitude of the peak currents are in agreement with the measured data.

Figure 10.
Table 3. Nominal and Simulation Parameters for CV Curve Fitting of Fe(aq)^2+/3+

	E⁰ (V vs Ag/AgCl)	k⁰ (cm/s)	α	C_R^* (mM)	D_R/D_O (cm²/s)	C_dl (μF)	R_Ω (ohms)
Simulated	0.441	1.2 x 10^-5	0.5	5	3.68 x 10^-6	20	0
Nominal	0.441	N/A	0.5	5	3.68 x 10^-6	N/A	N/A

The measured and simulated CV curves for the Fe(phen)₃ solution are shown in Figure 11 below. The parameters for the simulation are shown in Table 4 below, and compared to the nominal values. The simulated CV curves fit the measured data well. The peaks were well aligned in width, potential, and magnitude.

Figure 11.
Table 4. Nominal and Simulation Parameters for CV Curve Fitting of Fe(phen)₃^2+/3+

	E⁰ (V vs Ag/AgCl)	k⁰ (cm/s)	α	C_R^* (mM)	D_R/D_O (cm²/s)	C_dl (μF)	R_Ω (ohms)
Simulated	0.895	5 x 10^-2	0.5	4.5	3.31 x 10^-6	20	0
Nominal	0.895	N/A	0.5	4.5	3.31 x 10^-6	N/A	N/A

Discussion¶

Cottrell Experiments

The Cottrell experiments for the three different redox couples yielded curves with different early-time current behavior. For example, the ferri/ferrocyanide redox couple showed a slope which was steeper at early times, while the aqueous Fe redox couple showed slopes which were less steep at early times. Additionally, the current passed with the ferri/ferrocyanide redox couple was much greater than the aqueous Fe redox couple. It is expected that at early times that the results will not be accurate due to the charging of the double layer. For this reason the Cottrell slope was fit to the region roughly between 0 and 5 s^-1/2. Since the experiment was only conducted for 10 seconds, there was no concern of fitting the slope to a region where convection played a significant role in the mass transport of species. The ferri/ferrocyanide Cottrell experiment was performed under slightly different conditions than the other two redox couples, which may be the reason the current response is different. More specifically, the ferri/ferrocyanide experiments omitted an open circuit step before the potential step; the oxidative potential step was immediately proceeded by a reductive potential step used to reduce the Fe(CN)₆^3- that was produced in the previous oxidative step. Omitting the open circuit step led to data being collected from a time when the concentration gradients were not at steady state, and the double layer was charged. The larger current experienced at early times may be a result of discharging and later recharging the double layer. Mass transfer does not play a limiting role in the discharging of the double layer; the adsorbed ions should immediately be ejected from the electrode surface back into the solution, resulting in an initial current spike above what would be expected.

Another explanation would be due to the formal charges on the ferri/ferrocyanide species being negative. Since a positive potential is applied to the electrode surface, the Fe(CN)₆^4- will be initially driven to the electrode surface from migration effects before diffusion can take precedent. It is also the case that before the double layer fully develop, specifically past the outer Helmholtz plane and into the diffuse layer, a concentration gradient of Fe(CN)₆^4- would first need to develop. Therefore, the early-time migration of Fe(CN)₆^4- would be unopposed by the charged species in the electrolyte. This argument would also explain why the aqueous iron redox couple shows lower currents at early times. Since Fe(aq)²⁺ is positively charged, migration will initially drive species away from the electrode surface until diffusion can become the dominant factor in the mass transport of species.

The diffusion coefficients obtained from the Cottrell experiments were compared to literature values, shown in Table 5 below. The diffusion coefficients were determined using the geometric surface area of the electrode, instead of the microscopic area. This is because at long times when a relatively larger diffuse layer is able to develop, the flux of species to the electrode surface occurs over the geometric area.¹ In order for the assumptions of the Cottrell equation to be true, the potential needs to be sufficiently larger than the equilibrium potential as so the concentration of the species of interest at the electrode surface is 0 mol/cm³. This assumption was satisfied in most cases by using a potential 400 mV above the equilibrium potential. The Cottrell equation is also valid for all potentials above the "threshold" potential, given there are no other faradaic currents confounding the results. The only other faradaic current that could invalidate the results would be the oxidation of water on the GC electrode, although this is only expected to happen at 1.2 V vs Ag/AgCl. None of the potential steps approached this potential.

All of the experimentally determined diffusion coefficients (Table 5) were smaller than the literature-found values, although the relative magnitude of the diffusion coefficients are mostly in agreement with literature. The Fe(aq)^2+/3+ yielded the largest diffusion coefficient, followed by Fe(phen)₃^2+/3+ and lastly Fe(CN)₆^4-/3-. The literature values suggested a similar diffusion coefficient for Fe(CN)₆^4-/3- and Fe(aq)^2+/3+, and a measurably smaller diffusion coefficient for Fe(phen)₃^2+/3+. However, Fe(phen)₃^2+/3+ yielded a diffusion coefficient which was larger than Fe(CN)₆^4-/3- which was unexpected. It is possible that Fe(aq)^2+/3+ did not completely dissociate and bond with the 1,10-phenanthroline to form Fe(phen)₃^2+/3+, which would lead to a diffusion coefficient more similar to Fe(aq)^2+/3+. Extending the length of the experiment may have produced slightly larger diffusion coefficients for all species, as the range of data used for analysis was somewhat limited.

Table 5. Diffusion Coefficients for Iron Complexes

Species	D_R (cm²/s x 10^-6) (Meas.)	D_R (cm²/s x 10^-6) (Lit.)
Fe(CN)₆^4-/3-	3.17	7.35⁵
Fe(aq)^2+/3+	3.68	7.19-7.9² ^,5
Fe(phen)₃^2+/3+	3.31	5.56⁶

Cyclic Voltammetry Experiments

The three redox couples were subjected to CV experiments performed over a range of different scan rates. The peak separation from these experiments were measured to determine reversibility. Since all three redox couples showed both an anodic and cathodic peak on the CV scans, they may all be considered chemically reversible. The following discussion will be considered the electrochemical reversibility of the redox couples. The Fe(phen)₃^2+/3+ redox couple had the smallest peak separation at 10 mV/s scan rate of 59.5 mV, which is only slightly larger than the ideal 57 mV for a reversible redox couple. Thus, this redox couple may be considered to be reversible. The Fe(CN)₆^4-/3- showed the second smallest peak separation at 10 mV/s scan rate of 72.9 mV. This redox couple may be considered as quasi-reversible. Lastly, the Fe(aq)^2+/3+ couple had the largest peak separation of 388.6 mV at the slowest scan rate, and may be considered as irreversible. The E^1/2 for all three redox couples were measured, and used to approximate the E^0' for the redox couples. Fe(phen)₃^2+/3+ displayed the largest E^0' of 0.895 V vs Ag/AgCl. Fe(aq)^2+/3+ has the second largest E^0' of 0.441 V vs Ag/AgCl. Fe(CN)₆^4-/3- displayed the smallest E^0' of 0.246 V vs Ag/AgCl.

Simulated Results

The results of the CV experiments for the three redox couples were used to fit simulated data, and estimate the standard rate constants. All three redox couples were simulated using the nominal bulk concentration and diffusion coefficients, with the exception of the ferri/ferrocyanide redox couple for reasons mentioned in the results. The Fe(CN)₆^4- redox couple was successfully simulated using a standard rate constant of 5 x 10^-3 cm/s. This rate is in the quasi-reversible range. The Fe(aq)^2+/3+ redox couple yielded a smaller standard rate constant of 1.2 x 10^-5 cm/s, which falls in the range of slow kinetics and electrochemical irreversibility. The Fe(phen)₃ redox couple was fit using the largest standard rate constant of 5 x 10^-2 cm/s, which confirms that the kinetics are fast and electrochemically reversible. The Fe(aq)^2+/3+ simulated CV curves did not fit the data as well as the other redox couples, especially when considered over the range of scan rates used. The reason for the poor fit may be from limitations of the software, or some factor which was not accounted for properly in the model. Despite this, the results provided standard rate constants which can be taken as confident estimates of the true standard rate constants.