Understanding Electrochemical Reactions: An Introduction to Cyclic Voltammetry, Study notes of Law

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Cyclic Voltammetry

Denis Andrienko

January 22, 2008

Literature:

1. Allen J. Bard, Larry R. Faulkner “Electrochemical Methods: Fundamentals and Applications”
2. http://www.cheng.cam.ac.uk/research/groups/electrochem/teaching.html

4 CHAPTER 1. CYCLIC VOLTAMMETRY

1.2 Electrode Reactions

A typical electrode reaction involves the transfer of charge between an electrode and a species in solution. The electrode reaction usually referred to as electrolysis, typically involves a series of steps:

1. Reactant (O) moves to the interface: this is termed mass transport
2. Electron transfer can then occur via quantum mechanical tunnelling between the electrode and reactant close to the electrode (typical tunnelling distances are less than 2 nm)
3. The product (R) moves away from the electrode to allow fresh reactant to the surface

The ‘simplest’ example of an electrode reaction is a single electron transfer reaction, e.g. Fe3+^ + e−^ = Fe2+. Several examples are shown in Fig. 1.

Figure 1.1: Simple electrode reactions: (left) A single electron transfer reaction. Here the reactant Fe3+^ moves to the interface where it undergoes a one electron reduction to form Fe2+. The electron is supplied via the electrode which is part of a more elaborate electrical circuit. For every Fe3+^ reduced a single electron must flow. By keeping track of the number of electrons flowing (ie the current) it is possible to say exactly how many Fe3+^ molecules have been reduced. (middle) Copper deposition at a Cu electrode. In this case the electrode reaction results in the fomation of a thin film on the orginal surface. It is possible to build up multiple layers of thin metal films simply by passing current through appropriate reactant solutions. (right) Electron transfer followed by chemical reaction. In this case an organic molecule is reduced at the electrode forming the radical anion. This species however is unstable and undergoes further electrode and chemical reactions.

1.3 Electron Transfer and Energy levels

The key to driving an electrode reaction is the application of a voltage. If we consider the units of volts

V = Joule/Coulomb (1.1)

we can see that a volt is simply the energy required to move charge. Application of a voltage to an electrode therefore supplies electrical energy. Since electrons possess charge an applied voltage can alter the ’energy’ of the electrons within a metal electrode. The behaviour of electrons in a metal can be partly understood by considering the Fermi-level. Metals are comprised of closely packed atoms which have strong overlap between one another. A piece of metal therefore does not possess individual well defined electron energy levels that would be found in a single atom of the same material. Instead a continum of levels are created with the available electrons filling the states from the bottom upwards. The Fermi-level corresponds to the energy at which the ’top’ electrons sit.

1.4. KINETICS OF ELECTRON TRANSFER 5

Figure 1.2: Representation of the Fermi-Level in a metal at three different applied voltages (left). Schematic representation of the reduction of a species (O) in solution (right).

This level is not fixed and can be moved by supplying electrical energy. Electrochemists are therefore able to alter the energy of the Fermi-level by applying a voltage to an electrode. Figure 1.2 shows the Fermi-level within a metal along with the orbital energies (HOMO and LUMO) of a molecule (O) in solution. On the left hand side the Fermi-level has a lower value than the LUMO of (O). It is therefore thermodynamically unfavourable for an electron to jump from the electrode to the molecule. However on the right hand side, the Fermi-level is above the LUMO of (O), now it is thermodynamically favourable for the electron transfer to occur, ie the reduction of O. Whether the process occurs depends upon the rate (kinetics) of the electron transfer reaction and the next document describes a model which explains this behavior.

1.4 Kinetics of Electron Transfer

In this section we will develop a quantitative model for the influence of the electrode voltage on the rate of electron transfer. For simplicity we will consider a single electron transfer reaction between two species (O) and (R)

O + e−^ −k−red→ R (1.2)

R kox −−→ O + e−^ (1.3)

The current flowing in either the reductive or oxidative steps can be predicted using the following expressions

iO = F AkoxcR (1.4) iR = −F AkredcO (1.5)

For the reduction reaction the current iR is related to the electrode area A, the surface concentration of the reactant cO , the rate constant for the electron transfer kred and Faraday’s constant F. A similar expression is valid for the oxidation, now the current is labelled iO , with the surface concentration that of the species R. Similarly the rate constant for electron transfer corresponds to that of the oxidation process. Note that by definition the reductive current is negative and the oxidative positive, the difference in sign simply tells us that current flows in opposite directions across the interface depending upon whether we are studying an oxidation or reduction. To establish how the rate constants kox and kred are influenced by the applied voltage we will use transition state theory. In this theory the reaction is considered to proceed via an energy barrier, as shown in Fig 1.3. The summit of this barrier is referred to as the transition state. Using this picture the corresponding reaction rates are given by

kred,ox = Z exp

−∆Gred,ox kB T

If we plot a series of the free energy profiles as a function of voltage the free energy of R will be invariant with voltage, whereas the right handside (O + e) shows a strong dependence.

1.5. MASS TRANSPORT 7

Figure 1.4: Diffusion of the reactants to the electrode

We have already seen that a typical electrolysis reaction involves the transfer of charge between an electrode and a species in solution. This whole process due to the interfacial nature of the electron transfer reactions typically involves a series of steps. In the section on electrode kinetics we saw how the electrode voltage can effect the rate of the electron transfer. This is an exponential relationship, so we would predict from the electron transfer model that as the voltage is increased the reaction rate and therefore the current will increase exponentially. This would mean that it is possible to pass unlimited quantities of current. Of course in reality this does not arise and this can be rationalized by considering the expression for the current that we encountered in the electrode kinetics section Clearly for a fixed electrode area (A) the reaction can be controlled by two factors. First the rate constant kred and second the surface concentration of the reactant (Csurf O ). If the rate constant is large, such that any reactant close to the interface is immediately converted into products then the current will be controlled by the amount of fresh reactant reaching the interface from the bulk solution above. Thus movement of reactant in and out of the interface is important in predicting the current flowing. In this section we look at the various ways in which material can move within solution - so called mass transport. There are three forms of mass transport which can influence an electrolysis reaction

• Diffusion
• Convection
• Migration

In order to predict the current flowing at any particular time in an electrolysis measurement we will need to have a quantitative model for each of these processes to complement the model for the electron transfer step(s). Diffusion occurs in all solutions and arises from local uneven concentrations of reagents. Entropic forces act to smooth out these uneven distributions of concentration and are therefore the main driving force for this process. One example of this can be seen in the animation below. Two materials are held separately in a single container separated by a barrier. When the barrier is removed the two reagents can mix and this processes on the microscopic scale is essentially random. For a large enough sample statistics can be used to predict how far material will move in a certain time - and this is often referred to as a random walk model. Diffusion is particularly significant in an electrolysis experiment since the conversion reaction only occurs at the electrode surface. Consequently there will be a lower reactant concentration at the electrode than in bulk solution. Similarly a higher concentration of product will exist near the electrode than further out into solution. The rate of movement of material by diffusion can be predicted mathematically and Fick proposed two laws to quantify the processes. The first law

JO = −DO

∂cO ∂x

relates the diffusional flux JO (ie the rate of movement of material by diffusion) to the concentration gradient and the diffusion coefficient DO. The negative sign simply signifies that material moves down a concentration gradient i. e. from regions of high to low concentration. However, in many measurements we need to know how the concentration of material varies as a function of time and this can be predicted from the first law. The result is Fick’s second law

∂cO ∂t

∂JO

∂x

8 CHAPTER 1. CYCLIC VOLTAMMETRY

Figure 1.5: Schematic of the setup.

In this case we consider diffusion normal to an electrode surface (x direction). The rate of change of the concentration cO as a function of time t can be seen to be related to the change in the concentration gradient. So the steeper the change in concentration the greater the rate of diffusion. In practice diffusion is often found to be the most significant transport process for many electrolysis reactions. Fick’s second law is an important relationship since it permits the prediction of the variation of concentration of different species as a function of time within the electrochemical cell. In order to solve these expressions analytical or computational models are usually employed.

1.6 Voltammetry

Voltammetry is one of the techniques which electrochemists employ to investigate electrolysis mechanisms. There are numerous forms of voltammetry

• Potential Step
• Linear sweep
• Cyclic Voltammetry

For each of these cases a voltage or series of voltages are applied to the electrode and the corresponding current that flows monitored. In this section we will examine potential step voltammetry, the other forms are described on separate pages For the moment we will focus on voltammetry in stagnant solution. The figure below shows a schematic of an electrolysis cell. There is a working electrode which is hooked up to an external electrical circuit. For our purposes at the moment we will not worry about the remainder of the circuit, obviously there must be more than one electrode for current to flow. But as we shall see later it is only the so called working electrode that controls the flow of current flow in the electrochemical measurement. The essential elements needed for an electolysis measurement are as follows:

• The electrode: This is usually made of an inert metal (such as Gold or Platinum)
• The solvent: This usually has a high dielectric constant (eg water or acetonitrile) to enable the electrolyte to dissolve and help aid the passage of current.
• A background electrolyte: This is an electrochemically inert salt (eg NaCl or Tetra butylammonium perchlorate, TBAP) and is usually added in high concentration (0.1 M) to allow the current to pass.
• The reactant: Typically in low concentration 10−^3 M.

1.6.1 Potential Step Voltammetry

In the potential step measurement the applied voltage is instantaneously jumped from one value V 1 to another V 2 The resulting current is then measured as a function time. If we consider the reaction

Fe3+(s) + e−^ kred −−→ Fe2+(s) (1.13)

10 CHAPTER 1. CYCLIC VOLTAMMETRY

Figure 1.8: Linear increase of the potential vs time.

• The rate of the electron transfer reaction(s)
• The chemical reactivity of the electroactive species
• The voltage scan rate

In LSV measurements the current response is plotted as a function of voltage rather than time, unlike potential step measurements. The scan begins from the left hand side of the current/voltage plot where no current flows. As the voltage is swept further to the right (to more reductive values) a current begins to flow and eventually reaches a peak before dropping. To rationalise this behaviour we need to consider the influence of voltage on the equilibrium established at the electrode surface. Here the rate of electron transfer is fast in comparsion to the voltage sweep rate. Therefore at the electrode surface an equilibrum is established identical to that predicted by thermodynamics. You may recall from equilibrium electrochemistry that the Nernst equation The exact form of the voltammogram can be rationalised by considering the voltage and mass transport effects. As the voltage is initially swept from V 1 the equilibrium at the surface begins to alter and the current begins to flow. The current rises as the voltage is swept further from its initial value as the equilibrium position is shifted further to the right hand side, thus converting more reactant. The peak occurs, since at some point the diffusion layer has grown sufficiently above the electrode so that the flux of reactant to the electrode is not fast enough to satisfy that required by the Nernst equation. In this situation the curent begins to drop just as it did in the potential step measurements. In fact the drop in current follows the same behaviour as that predicted by the Cottrell equation. The above voltammogram was recorded at a single scan rate. If the scan rate is altered the current response also changes. The figure 1.8 shows a series of linear sweep voltammograms recorded at different scan rates. Each curve has the same form but it is apparent that the total current increases with increasing scan rate. This again can be rationalised by considering the size of the diffusion layer and the time taken to record the scan. Clearly the linear sweep voltammogram will take longer to record as the scan rate is decreased. Therefore the size of the diffusion layer above the electrode surface will be different depending upon the voltage scan rate used. In a slow voltage scan the diffusion layer will grow much further from the electrode in comparison to a fast scan. Consequently the flux to the electrode surface is considerably smaller at slow scan rates than it is at faster rates. As the current is proportional to the flux towards the electrode the magnitude of the current will be lower at slow scan rates and higher at high rates. This highlights an important point when examining LSV (and cyclic voltammograms), although there is no time axis on the graph the voltage scan rate (and therefore the time taken to record the voltammogram) do strongly effect the behaviour seen. A final point to note from the figure is the position of the current maximum, it is clear that the peak occurs at the same voltage and this is a characteristic of electrode reactions which have rapid electron transfer kinetics. These rapid processes are often referred to as reversible electron transfer reactions. This leaves the question as to what would happen if the electron transfer processes were ‘slow’ (relative to the voltage scan rate). For these cases the reactions are referred to as quasi-reversible or irreversible

1.7. LINEAR SWEEP VOLTAMMETRY 11

Figure 1.9: Change of the rate constant.

Figure 1.10: Voltage as a function of time and current as a function of voltage for CV

electron transfer reactions. The figure below shows a series of voltammograms recorded at a single voltage sweep rate for different values of the reduction rate constant kred In this situation the voltage applied will not result in the generation of the concentrations at the electrode surface predicted by the Nernst equation. This happens because the kinetics of the reaction are ’slow’ and thus the equilibria are not established rapidly (in comparison to the voltage scan rate). In this situation the overall form of the voltammogram recorded is similar to that above, but unlike the reversible reaction now the position of the current maximum shifts depending upon the reduction rate constant (and also the voltage scan rate). This occurs because the current takes more time to respond to the the applied voltage than the reversible case.

1.7.1 Cyclic Voltammetry

Cyclic voltammetry (CV) is very similar to LSV. In this case the voltage is swept between two values (see below) at a fixed rate, however now when the voltage reaches V 2 the scan is reversed and the voltage is swept back to V 1 A typical cyclic voltammogram recorded for a reversible single electrode transfer reaction is shown in below. Again the solution contains only a single electrochemical reactant The forward sweep produces an identical response to that seen for the LSV experiment. When the scan is reversed we simply move back through the equilibrium positions gradually converting electrolysis product (Fe2+^ back to reactant (Fe3+. The current flow is now from the solution species back to the electrode and so occurs in the opposite sense to the forward seep but otherwise the behaviour can be explained in an identical manner. For a reversible electrochemical reaction the CV recorded has certain well defined characteristics.

1. The voltage separation between the current peaks is
2. The positions of peak voltage do not alter as a function of voltage scan rate