Electrochemical cells

Electrochemical cells are used in voltammetric experiments to drive the redox reaction of the analyte. Like other electrochemical cells, two half-cells are required, one to facilitate reduction and the other oxidation. The cell consists of an analyte solution, an ionic electrolyte, and two or three electrodes, with oxidation and reduction reactions occurring at the electrode/electrolyte interfaces.^[5] As a species is oxidized, the electrons produced pass through an external electric circuit and generate a current, acting as an electron source for reduction. The generated currents are Faradaic currents, which follow Faraday's law. As Faraday's law states that the number of moles of a substance, m, produced or consumed during an electrode process is proportional to the electric charge passed through the electrode, the faradaic currents allow analyte concentrations to be determined.^[6] Whether the analyte is reduced or oxidized depends on the analyte, but its reaction always occurs at the working/indicator electrode. Therefore, the working electrode potential varies as a function of the analyte concentration. A second auxiliary electrode completes the electric circuit. A third reference electrode provides a constant, baseline potential reading for the other two electrode potentials to be compared to.

Three electrode system

Potential as a function of time for anodic stripping voltammetry

Three-electrode setup: (1) working electrode; (2) auxiliary electrode; (3) reference electrode

Voltammetry experiments investigate the half-cell reactivity of an analyte. Voltammetry is the study of current as a function of applied potential. These curves I = f(E) are called voltammograms. The potential is varied arbitrarily, either step by step or continuously, and the resulting current value is measured as the dependent variable. The opposite, i.e., amperometry, is also possible but not common. The shape of the curves depends on the speed of potential variation, (nature of driving force) and whether the solution is stirred or quiescent (mass transfer). Most experiments control the potential (volts) of an electrode in contact with the analyte while measuring the resulting current (amperes).^[7]

To conduct such an experiment, at least two electrodes are required. The working electrode, which makes contact with the analyte, must apply the desired potential in a controlled way and facilitate the transfer of charge to and from the analyte. A second electrode acts as the other half of the cell. This second electrode must have a known potential to gauge the potential of the working electrode from; furthermore it must balance the charge added or removed by the working electrode. While this is a viable setup, it has a number of shortcomings. Most significantly, it is extremely difficult for an electrode to maintain a constant potential while passing current to counter redox events at the working electrode.^[7]

To solve this problem, the roles of supplying electrons and providing a reference potential are divided between two separate electrodes. The reference electrode is a half cell with a known reduction potential. Its only role is to act as reference for measuring and controlling the working electrode's potential and it does not pass any current. The auxiliary electrode passes the current required to balance the observed current at the working electrode. To achieve this current, the auxiliary will often swing to extreme potentials at the edges of the solvent window, where it oxidizes or reduces the solvent or supporting electrolyte. These electrodes, the working, reference, and auxiliary make up the modern three-electrode system.^[7]

There are many systems which have more electrodes, but their design principles are similar to the three-electrode system. For example, the rotating ring-disk electrode has two distinct and separate working electrodes, a disk, and a ring, which can be used to scan or hold potentials independently of each other. Both of these electrodes are balanced by a single reference and auxiliary combination for an overall four-electrode design. More complicated experiments may add working electrodes, reference, or auxiliary electrodes as required.^[7]

In practice it can be important to have a working electrode with known dimensions and surface characteristics. As a result, it is common to clean and polish working electrodes regularly. The auxiliary electrode can be almost anything as long as it doesn't react with the bulk of the analyte solution and conducts well. A common voltammetry method, polarography, uses mercury as a working electrode e.g. DME and HMDE, and as an auxiliary electrode. The reference is the most complex of the three electrodes; there are a variety of standards used. For non-aqueous work, IUPAC recommends the use of the ferrocene/ferrocenium couple as an internal standard.^[8] In most voltammetry experiments, a bulk electrolyte (also known as a supporting electrolyte) is used to minimize solution resistance. It is possible to run an experiment without a bulk electrolyte, but the added resistance greatly reduces the accuracy of the results. With room temperature ionic liquids, the solvent can act as the electrolyte.^[7]

Voltammograms

A common shape for current vs potential voltammogram measuring maximum peak current^[4]

A common shape for current vs potential voltammogram measuring limiting peak current^[4]

A common voltammogram shape showing change in current vs potential^[4]

A voltammogram is a graph that measures the current of an electrochemical cell as a function of the potential applied.^[9] This graph is used to determine the concentration and the standard potential of the analyte. To determine the concentration, values such as the limiting or peak current are read from the graph and applied to various mathematical models.^[10] After determining the concentration, the applied standard potential can be identified using the Nernst equation.^[10]

There are three main shapes for voltammograms. The first shape is dependent on the diffusion layer.^[1] If the analyte is continuously stirred, the diffusion layer will be a constant width and produce a voltammogram that reaches a constant current. The graph takes this shape as the current increases from the background residual to reach the limiting current (i_l). If the mixture is not stirred, the width of the diffusion layer eventually increases. This can be observed by the maximum peak current (i_p), and is identified by the highest point on the graph. The third common shape for a voltammogram measures the sample for change in current rather than current applied. A maximum current is still observed, but represents the maximum change in current (${\displaystyle \Delta }$i_p).^[10]

Mathematical models

To determine analyte concentrations, mathematical models are required to link the applied potential and current measured over time. The Nernst equation relates electrochemical cell potential to the concentration ratio of the reduced and oxidized species in a logarithmic relationship.^[6] The Nernst equation is as follows:

${\displaystyle E=E^{0}-{\frac {RT}{zF}}\ln Q}$

Where:

• ${\displaystyle E}$: Reduction potential
• ${\displaystyle E^{0}}$: Standard potential
• ${\displaystyle R}$: Universal gas constant
• ${\displaystyle T}$: Temperature in kelvin
• ${\displaystyle z}$: Ion charge (moles of electrons)
• ${\displaystyle F}$: Faraday constant
• ${\displaystyle Q}$: Reaction quotient

This equation describes how the changes in applied potential will alter the concentration ratio. However, the Nernst equation is limited, as it is modeled without a time component and voltammetric experiments vary applied potential as a function of time. Other mathematical models, primarily the Butler-Volmer equation, the Tafel equation, and Fick's law address the time dependence.

The Butler–Volmer equation relates concentration, potential, and current as a function of time.^[5] It describes the non-linear relationship between the electrode and electrolyte voltage difference and the electrical current. It helps make predictions about how the forward and backward redox reactions affect potential and influence the reactivity of the cell.^[11] This function includes a rate constant which accounts for the kinetics of the reaction. A compact version of the Butler-Volmer equation is as follows:

${\displaystyle {\displaystyle j=j_{0}\cdot \left\{\exp \left[{\frac {\alpha _{\rm {a}}zF\eta }{RT}}\right]-\exp \left[-{\frac {\alpha _{\rm {c}}zF\eta }{RT}}\right]\right\}}}$

Where:

• ${\displaystyle j}$: electrode current density, A/m² (defined as j = I/S)
• ${\displaystyle j_{0}}$: exchange current density, A/m²
• ${\displaystyle E}$: electrode potential, V
• ${\displaystyle {\displaystyle E_{\rm {eq}}}}$: equilibrium potential, V
• ${\displaystyle T}$: absolute temperature, K
• ${\displaystyle z}$: number of electrons involved in the electrode reaction
• ${\displaystyle F}$: Faraday constant
• ${\displaystyle R}$: universal gas constant
• ${\displaystyle \alpha _{c}}$: so-called cathodic charge transfer coefficient, dimensionless
• ${\displaystyle \alpha _{a}}$: so-called anodic charge transfer coefficient, dimensionless
• ${\displaystyle \eta }$: activation overpotential (defined as ${\displaystyle {\displaystyle \eta =E-E_{\rm {eq}}}}$).

At high overpotentials, the Butler–Volmer equation simplifies to the Tafel equation. The Tafel equation relates the electrochemical currents to the overpotential exponentially, and is used to calculate the reaction rate.^[11] The overpotential is calculated at each electrode separately, and related to the voltammogram data to determine reaction rates. The Tafel equation for a single electrode is:

${\displaystyle {\displaystyle \eta =\pm A\cdot \log _{10}\left({\frac {i}{i_{0}}}\right)}}$

Where:

• the plus sign under the exponent refers to an anodic reaction, and a minus sign to a cathodic reaction
• ${\displaystyle \eta }$: overpotential, V
• A: "Tafel slope", V
• ${\displaystyle i}$: current density, A/m²
• ${\displaystyle i_{0}}$: "exchange current density", A/m².

As the redox species are oxidized and reduced at the electrodes, material accumulates at the electrode/electrolyte interface.^[5] Material accumulation creates a concentration gradient between the interface and the bulk solution. Fick's laws of diffusion is used to relate the diffusion of oxidized and reduced species to the faradaic current used to describe redox processes. Fick's law is most commonly written in terms of moles, and is as follows:

${\displaystyle {\displaystyle J=-D{\frac {d\varphi }{dx}}}}$

Where:

• J: diffusion flux (in amount of substance per unit area per unit time)
• D: diffusion coefficient or diffusivity. (in area per unit time)
• φ: concentration (in amount of substance per unit volume)
• x: position (in length)

Voltammetric sensors

A number of voltammetric systems are produced commercially for the determination of species that are of interest in industry and research. These devices are sometimes called electrodes but are actually complete voltammetric cells, which are better referred to as sensors. These sensors can be employed for the analysis of organic and inorganic analytes in various matrices.^[28]

The oxygen electrode

The determination of dissolved oxygen in a variety of aqueous environments, such as sea water, blood, sewage, effluents from chemical plants, and soils is of tremendous importance to industry, biomedical and environmental research, and clinical medicine. One of the most common and convenient methods for making such measurements is with the Clark oxygen sensor, which was patented by L.C. Clark, Jr. in 1956.