Microelectrodes are electrodes whose characteristic dimension is only a fraction of a millimeter. They represent an important tool for electroanalytical measurements. In this section, we examine some of the advantages and disadvantages of these electrodes.

Two examples of microelectrode geometries are shown in Figure 6.28, but the general concepts developed here can be applied to other geometries. The simplest type of electrode to analyze is the hemispherical microelectrode. However, as you might imagine, fabricating such an electrode is challenging. The second geometry we will consider is the disk electrode shown in Figure 6.28b. It is more complicated to analyze, but has many features in common with the hemispherical electrode. Owing to the simplicity of fabrication, the disk microelectrode is the one that is most commonly used.

Potential Drop

What is it that makes microelectrodes so useful? In order to answer this question, let’s examine the potential drop associated with a microelectrode in solution. As is typically the case, we assume that the counter electrode is large and located far away from the microelectrode. For simplicity, we will consider a hemispherical electrode and neglect concentration gradients in solution. First, let’s determine the potential drop in solution for a specified current density at the electrode surface. The problem is spherically symmetric and the relevant equation is Laplace’s equation with the following boundary conditions:

(6.49)

Solving for the potential yields

(6.50)

Because the area available for current flow increases with r² as you move further from the electrode, most of the voltage drop occurs close to the electrode. In fact, by the time you are 10 radii (10a) away from the electrode, 90% of the total voltage drop has occurred. The total potential drop associated with the microelectrode is

(6.51)

Now that we have determined the voltage drop for a given current density at a microelectrode, it is useful to look at the absolute value of the current:

(6.52)

Thus, the total current corresponding to a specific current density at the surface scales with a². With the total current and voltage, we can calculate the total ohmic resistance for the hemispherical electrode:

(6.53)

where the resistance is in ohms. With the above relations, we can now explore some of the important characteristics of microelectrodes. Because of spherical decay, the potential drop is centered at the electrode and there is no substantial potential drop far away from the electrode. The total ohmic drop scales with the electrode radius, a, and, therefore, becomes smaller as the size of the electrode is reduced. What are the practical implications? For an experiment at a given current density, i_surf,

• We can reduce the ohmic drop in solution by reducing the size of the electrode.
• Because almost all of the potential drop occurs very close to the electrode, the position of the counter electrode does not affect the experiment as long as it is greater than ∼10 radii from the electrode. For a 100 μm electrode, this means that “infinity” is only millimeters away from the electrode.
• While we commonly place the counter electrode far away, analytical experiments often try to locate the reference electrode close to the working electrode in order to minimize uncompensated iR. This is not needed for a microelectrode since the ohmic drop can be lowered by reducing the size of the electrode.
• As long as the reference electrode is greater than ∼10 radii from the working (micro)electrode, its position does not matter and the ohmic drop between the working and reference electrodes can be easily calculated from Equation 6.51.
• Because the total current required to obtain a desired current density is much less for the microelectrode, much higher current densities can be achieved with a given instrument when using a microelectrode.

Because of these characteristics, and others that we will show shortly, microelectrodes have become an important tool for electroanalytical chemistry.

The above characteristics derived for a hemispherical electrode also apply to a disk microelectrode. Analogous to the resistance presented above, the resistance for a disk electrode is

(6.54)

which is also in ohms. Besides the ease of use, there is an important difference between a hemispherical electrode and a disk electrode: The current density at the disk electrode is not uniform. This, of course, is a problem when you want to measure, for example, kinetic data, which describes the relationship between the overpotential and the current density. How does the size of the electrode affect the current distribution at the electrode? The answer is found in the Wagner number, introduced previously in Chapter 4 and shown here for Tafel kinetics.

(6.55)

As you can see, the Wagner number is inversely proportional to the disk radius, a. Therefore, as the disk gets smaller, Wa increases and the current density is more uniform. In other words, another advantage of microelectrodes is that they yield a more uniform current density.

Mass-Transfer Control

There are also some very important advantages of microelectrodes with respect to mass transfer. To illustrate these, let’s return to the hemispherical electrode and examine mass transfer by diffusion for a stagnant fluid. For simplicity, we neglect the influence of bulk flow caused by the current, which is only important at high concentrations and at high rates. With no variations in the θ or ϕ directions, Equation 4.25 becomes

(6.56)

The following initial and boundary conditions apply:

(6.57)

The solution for this mass-transfer limited case for the current density is

(6.58)

Here because the hemisphere is uniformly accessible, the current density at the surface does not vary spatially and depends only on time. In contrast to large planar electrodes, we see that there are two parts to the solution. The first term on the right side represents the steady-state solution for diffusion to a sphere. The second part is transient. At short times, the transient term is much larger of the two and the solution is identical to the Cottrell equation for the planar geometry. At these short times, the concentration is depleted only in a very small region near the surface. If this region is thin enough, then the curvature of the sphere is not important. At long times, the transient term goes to zero and we are left with the steady-state solution.

As already noted, Equation 6.58 describes the transient behavior of a system that is mass-transfer controlled. A plot of i(t) versus  yields a straight line whose slope and intercept are a function of both the diffusivity and the bulk concentration, as illustrated in Figure 6.29. Consequently, values of the diffusion coefficient and the concentration can be obtained from one experiment with use of the slope and intercept.

If we want to get an idea of when each of the two terms Equation 6.58 dominates, we simply equate the two terms and solve for time. The resulting characteristic time for spherical diffusion is denoted by τ_D:

(6.59)

How does this compare with the time required to charge the double layer? As noted in Section 6.4, the time constant for charging of the double layer, τ_c, is R_ΩC. The time constant for double-layer charging of the spherical electrode is

(6.60)

where C_DL is the double-layer capacitance per unit electrode area. As the dimension of the microelectrode, a, gets smaller, the charging time decreases. Using a value of 10 S·m⁻¹ for the conductivity, 0.2 F·m⁻² for the double-layer capacitance, and 20 μm for the dimension of the electrode gives a charging time of 0.4 μs. Referring back to Figure 6.29, one doesn’t need to worry about the effect of this non-faradaic current except for very short times, and measuring a good slope is not a problem.

As mentioned previously, the spherical or hemispherical electrode is difficult to implement experimentally. The embedded disk is much preferred. The solution for the disk geometry is not as straightforward as that for the spherical geometry, but we arrive at

(6.61)

Comparison of this equation with Equation 6.58 shows that the transient behavior of the disk is very similar to that of the hemispherical electrode. In fact, the basic physics and the accompanying advantages are the same as for the hemispherical electrode. These same advantages exist for other geometries too.

Kinetic Control

Now that we have relationships for the mass-transfer limited current, how does this help our use of microelectrodes as an analytical tool? Clearly, we can use the relationships to describe behavior under mass-transfer limited conditions, as was illustrated in Figure 6.29. However, what we have learned regarding mass transfer and microelectrodes provides some important insights and advantages for experiments that are not at the limiting current. Examination of Equations 6.58 and 6.61 shows that magnitude of the steady-state mass-transfer limited current density is inversely proportional to the radius of the microelectrode. Because microelectrodes are so small, the limiting current for these electrodes is quite high, much greater than that for large electrodes. This means that microelectrodes make it possible to measure, for example, kinetic data at much higher rates without reaching the mass-transfer limit. In experiments where both rate and transport limitations are important, mass transfer reaches a steady state and does so rather quickly, facilitating both experimentation and analysis of the resulting data.

To illustrate, we’ll stick with the spherical geometry because of its simplicity. The governing differential equation remains unchanged (Equation 6.56), but the initial and boundary conditions are modified. Specifically, at the surface we replace the zero concentration boundary condition by equating the flux at the surface with the current density. The current density is then given by the Butler–Volmer equation, and the problem can be solved for a given value of the potential or, equivalently, the surface overpotential at r = a.

Assuming steady state, the boundary conditions are

(6.62)

where i is calculated for the specified potential. The solution to Equation 6.56 for these boundary conditions is shown as the solid line in Figure 6.30. The concentration at the surface of the electrode is given by

(6.63)

where a is the radius and

(6.64)

which is the steady-state solution from Equation 6.58 under mass-transfer control (surface concentration equal to zero). At low current densities (e.g., less than 10% of the limiting current), the electrode is under kinetic control; under these conditions, the concentration at the surface is close to that of the bulk. As the current density increases, the concentration at the surface decreases and the overpotential increases due to both concentration and kinetic effects; this is the mixed control region. At large current densities, a limiting current is reached; here the electrode is under mass-transfer control.

How is this behavior affected by electrode size? In particular, how is the behavior different with use of a microelectrode? These questions are examined in Illustration 6.7.

ILLUSTRATION 6.7

Compare the limiting current and surface concentration at a current density of 10 A·m⁻² for two hemispherical electrodes, one with a radius of 1 mm and the other with a radius of 10 μm. The bulk concentration is 0.1 M, the diffusivity is 1 × 10⁻⁹ m²·s⁻¹, and the reaction is a two-electron reaction.

a (mm)	ilim (A·m−2)	% ilim	csurf
1	19.30	52	0.0482
0.01	1930	0.52	0.0995

The difference between the two electrodes is clear from these numbers. At the same current density, the microelectrode is operating under kinetic control at a very small fraction of the limiting current (less than 1%), and its surface concentration is essentially equal to the bulk concentration. In contrast, the larger electrode is under mixed control at 52% of the limiting current, with a surface concentration that is considerably lower than the bulk.

In summary, microelectrodes represent a powerful electroanalytical tool that allows us to reduce and correct for uncompensated iR, provide a more uniform current density, easily analyze behavior at the limiting current, and make measurements at much higher rates without reaching the mass-transfer limit. These advantages are explored further in homework problems. Care must be taken, however, to accurately measure the current, which can be quite small for very small electrodes, even when the current density is high. Fortunately, modern instrumentation can measure currents in the nanoampere range quite readily.

Closure

In this chapter, we have examined several different analytical methods applicable to electrochemical systems. We began with the description of an electrochemical cell and a description of a three-electrode system for electrochemical measurements. Then in the remaining part of the chapter we examined a variety of different analytical techniques at an introductory level, exploring the influence of geometry, flow condition (transport), and control of potential or current on the output from the techniques. Time-dependent behavior and the time constant associated with different physical processes represent important themes in the chapter. Our goal for this chapter was to provide a foundation upon which a more detailed understanding of electroanalytical techniques may be based.

Further Reading

1. Bard, A.J. and Faulkner, L.R. (2001) Electrochemical Methods, John Wiley & Sons, Inc., New Jersey.
2. MacDonald, J.R. and Barsoukov, E. (2005) Impedance Spectroscopy: Theory, Experiment, and Applications, Wiley Interscience.
3. Orazem, M.E. and Tribollet, B. (2008) Electrochemical Impedance Spectroscopy, John Wiley & Sons, Inc., New Jersey.
4. Wang, J. (2006) Analytical Electrochemistry, Wiley-VCH Verlag GmbH, Hoboken, NJ.

Problems

6.1. You have been asked to measure the kinetics of nickel deposition from a Watts nickel plating bath. The conductivity of the plating solution is 3.5 S·m⁻¹. The reference electrode is located 2 cm from the working electrode. The electrode area is 5 cm² (with only one side of the electrode active). You may neglect the impact of the concentration overpotential. You may also assume that the current density is nearly uniform.

1. Recommend a reference electrode for use in this system. (Hint: What is in the Watts bath?)
2. You apply a potential of 1.25 V and measure an average current density of 5 mA·cm⁻². What is the surface overpotential? Is the IR drop in solution important? (Assume a 1D uniform current density for this part.)
3. How good is the assumption of uniform current density? What type of cell geometry would satisfy this assumption? How would your results be impacted if the current density were not uniform?

6.2. One of your lab colleagues is attempting to measure the kinetics of the following reaction:

which is used in the cathode of vanadium-based redox flow batteries. To simplify things, he is making the measurement at constant current. He finds that the potential decreases slightly with time, followed by an abrupt decrease and substantial bubbling.

1. Qualitatively explain the observed behavior.
2. Given the following parameters, how long will the experiment proceed until the abrupt change in potential is observed? Assume that there is excess H⁺ in solution, and that the electrode area is 2 cm².

6.3. An estimate of the diffusivity can be obtained by stepping the potential so that the reaction is mass-transfer limited as described in the chapter. From the following data for V²⁺ in acidic solution, please estimate the diffusivity. The reaction is as follows:

t (s)	I (mA)
0.5	6.2
1.0	4.1
5.0	1.7
10.0	1.28
25.0	0.86
60.0	0.58
600	0.17
6,000	0.052
10,000	0.043

The bulk concentration of V²⁺ is 0.01 M, and the area of the electrode is 1 cm².

6.4. In Section 6.4 we examined the time constant associated with charging of the double layer. In doing so, we assumed that the physical situation could be represented by a resistor (ohmic resistance of the solution) and a capacitor (the double layer) in series. However, the actual situation is a bit more complex since there is a faradaic resistance in parallel with the double-layer capacitance as shown in Figures 6.6 and 6.25. This problem explores the impact of the faradaic resistance on double-layer charging. Our objectives are twofold: (1) Determine the time constant for double-layer charging in the presence of the faradaic resistance, and (2) determine an expression for the charge across the capacitor as a function of time.

1. Initially, there is no applied voltage, no current, and the capacitor is not charged.
2. At time zero, a voltage V is applied.
3. Your task is to derive an expression for the charge across the double layer as a function of time, and report the appropriate time constant. Use the symbols shown in Figure 6.17 for the circuit components.

Approach: The general approach is identical to that used in the chapter with the simpler model. In this case, you will need to write a voltage balance for each of the two legs, noting that the voltage drop must be the same. Remember that  where “1” is the capacitor leg and “2” is the faradaic leg. Once you have written the required balances, you can combine them into a single ODE and solve that equation for the desired relationship and time constant.

Finally, please explain physically how the characteristic time that you derived can be smaller than that determined for the simpler situation explored in the chapter.

6.5. GITT (galvanostatic intermittent titration technique) uses short current pulses to determine the diffusivity of solid-phase species in, for example, battery electrodes where the rate of reaction is limited by diffusion in the solid phase. This situation occurs for several electrodes of commercial importance. The concept behind the method is to insert a known amount of material into the surface of the electrode (hence the short time), and then monitor the potential as it relaxes with time due to diffusion of the inserted species into the electrode. In order for the method to be accurate, the amount of material inserted into the solid must be known. For this reason, the method uses a galvanostatic pulse for a specified time, which permits determination of the amount of material with use of Faraday’s law assuming that all of the current is faradaic (due to the reaction).

1. While it is sometimes desirable to use very short current pulses, what factor limits accuracy for short pulses?
2. Assuming that you have a battery cathode, how does the voltage change during a current pulse?
3. For a current of 1 mA, what is the shortest pulse width (s) that you would recommend? Assume that you have a small battery cathode at open circuit, and that the drop in voltage associated with the pulse is 0.15 V. The voltage during the pulse can be assumed to be constant. The error associated with the pulse width should be no greater than 1%.

6.6. Assume that you have 50 mM of A²⁺ in solution, which can be reduced to form the soluble species A⁺. Assume that the reaction is reversible with a standard potential of 0.2 V. There is essentially no A⁺ in the starting solution. Please qualitatively sketch the following:

1. The IV curve that results from scanning the potential from a high value (0.5 V above the standard potential of the reaction) to a low value (0.5 V below the standard potential of the reaction).
2. The IV curve that results from scanning the potential from a low value (0.5 V below the standard potential of the reaction) to a high value (0.5 V above the standard potential of the reaction).
3. Why are the curves in (a) and (b) different?
4. Assuming that you started from the open-circuit potential, in which direction would you recommend scanning first? Why?

6.7. The following CV data were taken relative to a Ag/AgCl reference electrode located 1 cm from the working electrode. You suspect that the results may be impacted by IR losses in solution. The conductivity of the solution is 10 S·m⁻¹.

1. Determine whether or not IR losses are important and, if needed, correct the data to account for IR losses.
2. Is it possible to determine n for the reaction from the data? If so, please report the value. If not, please explain why not.

100 mV·s−1Potential [V]	Current density [mA·cm−2]	10 mV·s−1Potential [V]	Current density [mA·cm−2]
0.815	11.49	0.755	5.48
0.928	18.78	0.813	7.28
0.952	17.27	0.841	6.07
0.964	14.44	0.868	4.83
1.008	10.87	0.900	4.04
1.069	9.04	0.972	3.17
1.137	7.91	1.047	2.70
1.173	7.48	1.124	2.39
1.246	6.81	1.201	2.17
1.248	6.54	1.201	2.08
1.164	6.08	1.120	1.94
1.081	5.71	1.039	1.82
0.998	5.41	0.958	1.72
0.915	5.09	0.876	1.60
0.821	3.68	0.790	0.91
0.742	−0.23	0.730	−1.02
0.611	−9.44	0.652	−4.82
0.504	−16.08	0.597	−6.32
0.477	−14.84	0.569	−5.11
0.467	−11.87	0.542	−3.87
0.450	−9.61	0.510	−3.08
0.396	−6.98	0.438	−2.22
0.331	−5.54	0.363	−1.75
0.261	−4.61	0.286	−1.46
0.188	−3.95	0.208	−1.25
0.176	−3.68	0.208	−1.16
0.260	−3.25	0.289	−1.02
0.343	−2.90	0.370	−0.92
0.425	−2.62	0.451	−0.82
0.507	−2.34	0.532	−0.72
0.599	−1.16	0.619	−0.05
0.672	2.20	0.677	1.85
0.815	11.49	0.755	5.48

6.8. For hydrogen adsorption on polycrystalline platinum, the accepted loading is 2.1 C·m⁻². Using the (100) face, calculate the amount of H adsorbed on this FCC surface assuming one H per Pt atom. Then, convert this number to the corresponding amount of charge per area. Assume a pure platinum surface with an FCC lattice parameter of 0.392 nm, and compare your results with the polycrystalline number. Provide a possible explanation for any differences between the calculated and accepted values.

6.9. The behavior of an inductor is described by the following differential equation:

where L is the inductance. Use this equation and the procedure illustrated in Section 6.7 to derive an expression for the complex impedance, Z. Compare your answer to that found in Table 6.4.

6.10. Create a Nyquist plot for the following system considering only electrolyte resistance, kinetic resistance for a single-electron reaction, and double-layer capacitance:

1. C_DL = 10 μF·cm⁻².
2.  for disk electrode with counter electrode located far away.
3. Counter electrode is large and located more than 10 radii from the disk electrode.
4. i₀ = 0.001 A·cm⁻²
5. α_a = α_c = 0.5.
6. r₀ = 1 mm.
7. κ = 10 S·m⁻¹

6.11. Please examine your response to the Problem 6.10 and address the following:

1. How does the magnitude of the kinetic and ohmic resistances compare to those calculated in Illustration 6.5? Please rationalize the differences and/or similarities.
2. How is it possible to use just the formula for the disk electrode to estimate the ohmic resistance? Do you expect this to be accurate? Why or why not?
3. In what ways does a large counter electrode influence the impedance results?

6.12. Please include the influence of the semi-infinite Warburg impedance on the system above (Problem 6.10) for which you generated the Nyquist plot. Provide a Nyquist plot and a Bode plot of the results.

6.13. EIS data were taken for a system at the open-circuit potential. Given the Nyquist diagram below:

1. Estimate the ohmic resistance.
2. Estimate the kinetic resistance.
3. Is it likely that the experimental system included convection? Why or why not?

6.14. When measured about the open-circuit potential, the kinetic resistance is frequently larger than the ohmic resistance. However, for systems where mass transfer is not limiting, the ohmic drop inevitably controls at high current densities.

1. Given that the relative magnitude of the ohmic and kinetic resistance at high current densities has changed, is this because the ohmic resistance has increased or because the kinetic resistance has decreased? Please justify your response.
2. For the resistance that changed (kinetic or ohmic), please derive a relationship that describes how that resistance depends on the value of the current density.

6.15. The following data were taken with a RDE operating at the limiting current for a range of rotation speeds. The radius of the disk is 1 mm, and the reaction is a two-electron reaction. Assume a kinematic viscosity of . The concentration of the limiting reactant is 25 mol m⁻³. Please use a Levich plot to determine the diffusivity from the data given. Make sure that all quantities are in consistent units.

Rotation speed (rpm)	I (μA)
100	104
500	230
1000	325
1500	404
2000	470
2500	520
3000	565
3500	607
4000	660

6.16. Illustration 6.6 is a Koutecký–Levich for oxygen reduction in water, where the bulk concentration is the solubility of oxygen in water as given in the problem. These data represent oxygen reduction in acid media, and the potential values given are relative to SHE. The equilibrium potential of oxygen is 1.23 V versus SHE under the conditions of interest.

1. Using the data from the illustration, calculate the rate of reaction for oxygen at the bulk concentration at each value of the overpotential given in the illustration.
2. Determine the exchange-current density and Tafel slope assuming Tafel kinetics.
3. What assumption was made regarding the concentration dependence of i₀ in the analysis above? Is the assumption accurate for oxygen reduction?

6.17. Suppose that you have a disk-shaped microelectrode that is 100 μm in diameter. At what value of time would the electrode be within 1% of its steady-state current density? At what value of time would the electrode be within 10% of its steady-state current density? What is the value of the limiting current at steady state in amperes? Assume a two-electron reaction with a diffusivity of 1 × 10⁻⁹ m² s⁻¹ and a bulk concentration of 25 mM.

6.18. You have been asked to design a disk-shaped microelectrode for use in kinetic measurements. You need to make measurements up to a maximum current density of 15 mA cm⁻². The concentration of the limiting reactant in the bulk is 50 mol m⁻³, and its diffusivity is 1.2 × 10⁻⁹ m² s⁻¹. The conductivity of the solution is 10 S·m⁻¹. Assume a single-electron reaction.

1. What size of microelectrode would you recommend? Please consider the impact of the limiting current and the uniformity of the current distribution.
2. What would the measured current be at the maximum current density for the recommended electrode?

Hint: Can you do kinetic measurements at the mass-transfer limit? How does this affect your response to this problem?

6.19. Derive an expression for the ratio of the iR drop associated with a microelectrode to that associated with a large electrode. Each of these two working electrodes (the microelectrode and the large electrode) is tested in a cell with the same current density at the electrode surface, and with the same reference electrode and counter electrode. Assume that any concentration effects can be neglected and that the current distribution is one-dimensional for the large electrode. Also assume that the distance L from the working electrode to the reference electrode is the same in both cases, and that L is large enough to be considered at infinity relative to the microelectrode.

6.20. Qualitatively sketch the current response of a microelectrode to a slow voltage scan in the positive direction from the open-circuit potential. Assume that the solution contains an equal concentration of the reduced and oxidized species in solution. How does this response differ from that of a typically sized electrode? Please explain. Hint: What is the steady-state behavior of a microelectrode and how might this impact the shape of the CV curve?

6.21. You need to measure the reduction kinetics of a reaction where the reactant is a soluble species. The reaction is a single-electron reaction. The diffusivity is not known. As you answer the following, please include the equations that you would use and consider the implications of both mass-transfer and the current distribution.

1. Can a rotating disk electrode be effectively used to make the desired measurements? If so, how would you proceed? If not, why not?
2. Is it possible to use a microelectrode to measure the quantities needed to determine the reduction kinetics? If so, how would you proceed? If not, why not?
3. What are the advantages and disadvantages of the two methods? Which would you recommend? Please justify your response.
4. What role, if any, does a supporting electrolyte play in the above experiments?

6.22. A CV experiment is performed using a microelectrode with a diameter of 100 μm at room temperature. The potential is swept anodically at ν = 10 mV·s⁻¹. The double-layer capacitance is 0.2 F·m⁻². Recall that the charging current is . The diffusivity of the electroactive species is 3 × 10⁻⁹ m²·s⁻¹. Assume that the fluid is stagnant. The concentration of the redox species is 100 mol·m⁻³ and the solution conductivity is 10 S·m⁻¹. From the data for a sweep in the positive direction, determine the exchange-current density and the anodic-transfer coefficient. The potentials are measured relative to a SCE reference electrode located far away from the microelectrode. The equilibrium potential of the reaction relative to SHE is 0.75 V.

V (SCE)	I (nA)
0.600	0.50
0.650	1.35
0.700	3.30
0.750	9.10
0.800	25.0
0.850	62.5
0.900	168
0.950	425
1.001	1200
1.052	3150
1.104	8000

6.23. Given an elementary single-electron reaction described by the following kinetic expression:

where the bulk concentration of each of the two reactants is 50 mM. You are to use a rotating disk electrode to measure the current density as a function of V for two different disk sizes, one with a 10 mm diameter and a second with a diameter of 1 mm. V is measured against a SCE reference electrode located more than 5 cm from the disk, and the standard potential of the reaction is 0.1 V SCE. Plot the i versus V curve for each of the two electrodes for a range of current densities from −150 to 150 A·m⁻² at a rotation speed of 500 rpm. Comment on any similarities and differences between the two curves. How does the size of the disk impact the mass transfer and the ohmic losses? You should account for the difference between the surface and bulk concentrations, including its impact on the equilibrium potential. Hint: It is easier to start with the current than it is with the voltage.

6.24. Our goal in this problem is to numerically generate a portion of the CV curve for a reversible system, where there are no kinetic limitations. Under these conditions, the potential at the surface remains essentially at equilibrium due to the very small overpotential needed to drive the reversible reaction. The surface potential does not remain constant, however, and the ratio of concentrations at the surface changes so that the equilibrium potential at the surface matches the applied potential. At the concentrations considered, the rate of reaction and hence the current depends completely on mass transfer. We will consider a generic single-electron reaction where both the oxidized and reduced species are in solution and there is excess supporting electrolyte (migration is not important). Given this, the following relations apply:

These expressions for the current can be combined with the concentration ratio above to yield the required boundary conditions as we will see below. Your task is to solve numerically the diffusion equation (Fick’s second law) for the concentration and the current as a function of time (and, therefore, potential). The required time can be determined from the starting and ending potentials and the scan rate. We will only do a single scan, although you should code the scan so that it can be run in either the positive or negative direction.

The discrete form of the diffusion equation needed for numerical solution is

where explicit time integration has been assumed. This equation can be written for both “O” and “R” to solve explicitly for the concentrations at the internal node points at each time step in terms of the other values at the previous time step. In the discrete equation,  represents the concentration of i at node [n] at the previous time step, where n = 0 at the surface. We recommend using about 200 spatial grid points uniformly spaced by Δx, the distance between grid points. Assume that x = 0 at the surface, and that x increases as you move away from the surface. Explicit time integration simplifies the problem by eliminating iteration, but is stable only when

This equation can be used to determine the time-step size that is needed for stability. Far away from the electrode surface (e.g., 1 cm), the concentrations remain at their initial values. Use a simple (approximate) boundary condition at the surface where the surface gradient is approximated by

Using this approximation for the gradients of both “O” and “R,” it can be shown that

where ɛ is the ratio of concentrations calculated from the applied potential by the equation listed previously. Your program will need to specify your x grid points, determine the time-step size needed for stability, define your parameters and the number of time steps required, and define and initialize the required arrays. In your time loop, the concentration values at the internal nodes can be updated at each time step from the values at the previous time step (no iteration required). Subsequently, the equations above can be used to calculate the surface concentrations of c_ox and c_red at the new time step, where the ratio is determined from the value of the applied potential at that time step. The current at the new time step can then be calculated. Please use the following parameters:

1. Please provide the current–voltage (CV) plot for a voltage sweep from 0.4 to 1.1 V at 5 mV·s⁻¹. Assume initial concentrations of  and . Also provide the concentration profiles at the ending voltage (1.1 V).
2. Repeat (a) for scan rates of 10 and 20 mV·s⁻¹. How does the peak current change? Why?
3. Why is the voltage profile flat at the beginning of the scan in (a)? Hint: Look at concentration.
4. Repeat (a) for initial concentrations of  and . Before running the simulation, predict and record what you think will happen. How accurate were your predictions?
5. Provide the CV plot for a voltage sweep from 0.7 to 1.1 V with an initial concentration of  and . Why is the initial current equal to zero?
6. Provide the CV plot for a voltage sweep in the negative direction from 0.7 to 0.4 V with the same initial conditions as (e). How does the curve generated compare with that from (e)?
7. Change the value of D_red equal to that of D_ox. How does this change the curve generated in (f)?

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