2022 Volume 90 Issue 10 Pages 102003
Polarization measurement is one of the major electrochemical methods used by electrochemists. The changes in current/potential with time at constant potential/current are investigated. The outcomes of these observations can be used to plot a current–potential curve. Therefore, it is important to understand the relationship between the three parameters: electrode potential, current, and time. In this paper, we described the fundamentals of the polarization, especially the current–potential curve (Butler–Volmer equation) and mass-transfer. In addition, the concept of polarization in corrosion reactions is explored.
Chronoamperometry and chronopotentiometry are important polarization measurement techniques used in electrochemistry. The term “chrono” descend from “chronos”, which means “time” in Greek. Hence, the term, chronoamperometry, describes the technique of measuring current variation as a function of time at constant potential, and chronopotentiometry describes the technique of measuring potential variation as a function of time at constant current. The charge–discharge test of batteries and capacitors, in which the cell voltage shifts with increasing time at constant current, is practically an example of a polarization reaction. Other examples include corrosion and metal finishing reactions. During the course of these reactions, the variation in the quantity of electricity, Q (= I [A] × t [s]) [C], at a constant electrode potential, is often emphasized.
Generally, the output parameters that can be detected during electrochemical measurements are current density and electrode potential. Therefore, understanding the correlation between there parameters is important for analyzing the results obtained from the polarization measurements. The details of the electrode potential are described in a related paper (see the Electrode Potential section1 in this special issue). In this paper, we first describe the origin of current in electrochemistry, and then provided an overview of the correlation between electrode potential and current density, called the polarization curve. Mass-transfer is also an important process in electrochemistry, discussed in this paper. Based on these discussions, a few cases of corrosion are briefly discussed in Section 3. The contents on electrochemical polarization are based on well-known textbooks in fundamental electrochemistry and corrosion.2–6 Cyclic voltammetry, a polarization method, has been independently described in another paper (see the Cyclic Voltammetry section7 in this special issue).
The electrochemical reduction proceeds in the following five steps: (I) arrival of reactants (Ox) near the electrode surface from the bulk electrolyte solution by diffusion, electrophoresis and convection, (II) adsorption of Ox on the electrode surface, (III) generation of the products (Red) via electron transfer reaction, (IV) desorption of Red from the electrode surface, and (V) transfer of Red from the vicinity of the electrode surface to the bulk electrolyte solution by diffusion, electrophoresis, and convection. In the oxidation reaction, the aforementioned five steps proceed in the reverse order.
First, we consider a simple electrochemical reaction and calculate the product mass of Red.
| \begin{equation} \text{Ox} + \text{$n$e$^{-}$} \to \text{Red} \end{equation} | (1) |
When the electrochemical reaction is performed at a constant current, I [A], for t [s], the product mass of Red, NRed [mol], is determined by the following equation.
| \begin{equation} N_{\text{Red}} = \frac{It}{nF} = \frac{Q}{nF} \end{equation} | (2) |
The equation is based on Faraday’s law. Here, n is the stoichiometric coefficient of electrons in the electrochemical reaction and F is the Faraday constant (96500 C mol−1). In addition, we can obtain the reaction rate, vt [mol s−1] by differentiating NRed with respect to t.
| \begin{equation} v_{\text{t}} = \frac{dN_{\text{Red}}}{dt} = \frac{I}{nF} \end{equation} | (3) |
Equation 3 can be rewritten using the current density, i [A cm−2], and the electrode square, S [cm2].
| \begin{equation} v_{\text{t}} = \frac{iS}{nF} \end{equation} | (4) |
Generally, the geometrical surface area of the electrode is used in the Eq. 4 for the sake of simplicity. However, when we employ the specific electrodes such as the high porosity electrodes or electrodes that are partially inactivated, we have to consider how their surface areas are taken into consideration in Eq. 4: real surface area or geometric surface area.
One of the fundamental aspects in electrochemistry, is that the current density and reaction rate are considered to be equivalent. For example, the amount of metal deposited on the electrode in the metal finishing process, that is, the deposition (reaction) rate can be controlled by changing the current density.
2.2 Current-potential curveNext, we consider the following electrochemical reaction:
| \begin{equation} \text{Ox} + \text{$n$e$^{-}$}\overset{k_{\text{red}}}{\underset{k_{\text{ox}}}{\rightleftarrows}} \text{Red} \end{equation} | (5) |
Chemical reactions proceed from the initial to the final states beyond the activation energy. It is well-established that most chemical reactions follow the Arrhenius equation. In reduction reaction, the reaction rate can be represented as:
| \begin{equation} v_{\text{red}} = Ac_{\text{ox}}\exp \left(-\frac{\Delta G_{\text{red}}}{RT} \right) = k_{\text{red}}c_{\text{ox}} \end{equation} | (6) |
| \begin{equation} \Delta G_{\text{red}}{}^{*} = \Delta G_{\text{red}} + \alpha nF\Delta E \end{equation} | (7) |
Therefore, the reaction rate is rewritten by using this relation.
| \begin{equation} v_{\text{red}} = Ac_{\text{ox}}\exp \left(-\frac{\Delta G_{\text{red}}^{*}}{RT} \right) \end{equation} | (8) |
| \begin{equation} = Ac_{\text{ox}}\exp\left(-\frac{\Delta G_{\text{red}} + \alpha nF\Delta E}{RT} \right) \end{equation} | (9) |
| \begin{equation} = Ac_{\text{ox}}\exp \left(-\frac{\Delta G_{\text{red}}}{RT} \right) \cdot \exp \left(-\frac{\alpha nF\Delta E}{RT}\right) \end{equation} | (10) |
By applying Eq. 6 to Eq. 10,
| \begin{equation} = k_{\text{red}}c_{\text{ox}}\exp \left(-\frac{\alpha nF\Delta E}{RT} \right) \end{equation} | (11) |
Equation 11 can be converted to current density by multiplying vred with the Faraday constant.
| \begin{equation} i_{\text{red}} = Fv_{\text{red}} \end{equation} | (12) |
| \begin{equation} = FAc_{\text{ox}}\exp \left(-\frac{\Delta G_{\text{red}} + \alpha nF\Delta E}{RT} \right) \end{equation} | (13) |
| \begin{equation} = Fk_{\text{red}}c_{\text{ox}}\exp \left(-\frac{\alpha nF\Delta E}{RT} \right) \end{equation} | (14) |
Similarly, iox can be described as:
| \begin{equation} i_{\text{ox}} = Fv_{\text{ox}} = Fk_{\text{ox}}c_{\text{red}}\exp \left( \frac{(1 - \alpha) nF \Delta E}{RT}\right) \end{equation} | (15) |
When vred and vox are equal, it is called the equilibrium state, and the current does not flow to the external circuit (ired = iox). The current density obtained in the equilibrium state is defined as the exchange current density, whereas the electrode potential in equilibrium is called the equilibrium potential, ΔEeq. The exchange current density i0 satisfies the following relationship:
| \begin{equation} \mathit{i}_{0} = \mathit{i}_{\text{red}} = \mathit{i}_{\text{ox}} \end{equation} | (16) |
| \begin{equation} = Fk_{\text{red}}c_{\text{ox}}\exp \left(-\frac{\alpha nF\Delta E_{\text{eq}}}{RT} \right) \end{equation} | (17) |
| \begin{equation} = Fk_{\text{ox}}c_{\text{red}}\exp \left(\frac{(1 - \alpha)nF\Delta E_{\text{eq}}} {RT} \right) \end{equation} | (18) |
Once the relation between ired and iox deviates from the equilibrium state, the current flows to the external circuit (ired ≠ iox). In addition, the corresponding potential difference is called the overpotential (η), which is the difference between the electrode potential and the equilibrium potential. The current density flowing into the external circuit is expressed as:
| \begin{equation} i = i_{\text{red}} - i_{\text{ox}} \end{equation} | (19) |
| \begin{align} &= Fk_{\text{red}}c_{\text{ox}}\exp \left(- \frac{\alpha nF (\Delta E_{\text{eq}} + \eta)}{RT} \right) \notag\\ &\quad- Fk_{\text{ox}} c_{\text{red}}\exp \left(\frac{(1 - \alpha)nF (\Delta E_{\text{eq}} + \eta)}{RT} \right) \end{align} | (20) |
| \begin{equation} = i_{0}\left[\exp \left(-\frac{\alpha nF\eta}{RT} \right) - \exp \left(\frac{(1 - \alpha)nF\eta}{RT}\right) \right] \end{equation} | (21) |
This relationship between the current density and potential is known as the Butler–Volmer equation. Figure 1 shows the current–potential curves based on the Butler–Volmer equation. The iox and ired curves are indicated by blue- and red-dashed lines, respectively. In practice, it is not easy to analyze the current–potential curves using the Butler–Volmer equation. Therefore, an approximation formula is often used.
Current–potential curves based on Butler–Volmer equation.
(a) In case of |η| < 5 mV
We can obtain the following approximation formula using a Taylor expansion in the Butler–Volmer equation.
| \begin{equation} \eta = -\frac{RT}{i_{0}nF}i \end{equation} | (22) |
| \begin{equation} -\frac{\eta}{i} = \frac{RT}{i_{0}nF} \end{equation} | (23) |
| \begin{equation} = R_{\text{ct}}\ (\text{$R_{\text{ct}}$: charge transfer resistance}) \end{equation} | (24) |
Thus, the current–potential curve can be linearly approximated near the equilibrium potential, which is similar to Ohm’s law.
(b) In case of |η| > 70 mV
(i) When large overpotential is applied to the electrode (η < −70 mV), the term iox is negligible.
| \begin{equation} i = i_{0} \exp \left(-\frac{\alpha nF\eta}{RT} \right) \end{equation} | (25) |
| \begin{equation} \eta = \frac{RT \ln i_{0}}{\alpha nF} - \frac{RT \ln i}{\alpha nF} \end{equation} | (26) |
| \begin{equation} = \frac{2.303RT \log i_{0}}{\alpha nF} - \frac{2.303RT \log i}{\alpha nF} \end{equation} | (27) |
(ii) In case of η > +70 mV, i can be described as:
| \begin{equation} i = - i_{0}\exp \left(\frac{(1 - \alpha) nF\eta}{RT}\right) = |i_{0}| \exp \left(\frac{(1 - \alpha) nF\eta}{RT}\right) \end{equation} | (28) |
| \begin{equation} \eta = -\frac{RT \ln i_{0}}{(1 - \alpha)nF} + \frac{RT \ln i}{(1 - \alpha)nF} \end{equation} | (29) |
| \begin{equation} = -\frac{2.303RT \log i_{0}}{(1 - \alpha)nF} + \frac{2.303RT \log i}{(1 - \alpha)nF} \end{equation} | (30) |
These equations correspond to the Tafel equation expressed in the following equation:
| \begin{equation} \eta = a \pm b \log i. \end{equation} | (31) |
Relationship between η and log|i| (Tafel plot).
To proceed with the electrochemical reaction on the electrode surface continuously, reactants should be seamlessly provided from the bulk electrolyte solution to the electrode surface. Products must simultaneously course to the bulk electrolyte; otherwise, the concentration of the product near the electrode reaches a saturated concentration. This causes the deposition of the product on the electrode surface, suppressing the electrochemical reaction. Therefore, mass-transfer process is an important concern in electrochemistry.
As mentioned at the beginning of this section, mass-transfer in the electrolyte solution occurs due to diffusion, electrophoresis, and convection. Among these, we can remove the influence of convection on the measurement by shortening the measurement time. In addition, the effect of electrophoresis can be cancelled by adding a supporting electrolyte, which leads to a dramatic increase in the conductivity of the electrolyte solution. Therefore, the mass-transfer near the electrode can be governed by diffusion.
Again, we consider the reaction (1).
When the reaction rate on the electrode increases with increasing overpotential, the reactant near the electrode is drastically consumed. For instance, the concentration of reactants near the electrode is lower than that in the bulk electrolyte solution because of the supply delay of the reactants, which can be described using Fick’s first law:
| \begin{equation} J = -D\frac{dc_{0}}{dx} \end{equation} | (32) |
The concentration gradient near the electrode can be approximated linearly in Fig. 3, and the region in which the concentration of the reactant changes with distance from the electrode is called the Nernst diffusion layer, δ.
Distance dependence of the reactant concentration under diffusion-limited condition.
By using δ, Eq. 32 can be rewritten as follows:
| \begin{equation} J = -\frac{D(c_{\text{bulk}} - c_{0})}{\delta} \end{equation} | (33) |
The diffusion flux (J) is the mole number that passes through a unit area of the cross-section in a unit time; thus, the current density, i, can be exhibited as follows:
| \begin{equation} i = -\frac{nFD (c_{\text{bulk}} - c_{0})}{\delta} \end{equation} | (34) |
When the concentration of the reactant near the electrode becomes zero (c0 = 0), the current density reaches a maximum, which is called diffusion-limiting current density.
| \begin{equation} i = -\frac{nFDc_{\text{bulk}}}{\delta} \end{equation} | (35) |
| \begin{equation} \delta = \sqrt{\pi Dt} \end{equation} | (36) |
Corrosion is a spontaneous reaction in which the Gibbs free energy change of the reaction is negative, and therefore, proceeds without any external bias. Even if the reaction may not appear to proceed at first glance, it proceeds steadily. Corrosion may proceed on a yearly basis, indicating that it is in a steady process. As a result, it is easily to envisage that stationary polarization can be utilized to analyze and evaluate the corrosion behavior. In corrosion, metals are converted into ions via an oxidation reaction or are transformed into corrosion products that are deposited on metal surfaces. A pair of reduction reactions, mostly, the reduction of hydrogen ions or dissolved oxygen occurs on metal surfaces to proceed with the aforementioned oxidation reactions. The following section provides a brief review of the qualitative description of oxidation and reduction reactions based on the concept of stationary polarization.
3.2 Stationary polarizationThe mixed potential theory is explained here to quantitatively discuss the corrosion behavior. For this purpose, the coupling of metal dissolution and the reduction of hydrogen ions was considered. The metal dissolution and reduction reactions are expressed by the following half-cell reactions:
| \begin{equation} \text{M$^{2+}$} + \text{2e$^{-}$} \rightleftarrows \text{M} \end{equation} | (37) |
| \begin{equation} \text{2H$^{+}$} + \text{2e$^{-}$} \rightleftarrows \text{H$_{2}$} \end{equation} | (38) |
(a) Corrosion potential, Ecorr, and corrosion current density, icorr, estimated from two stationary polarization curves. (b) Overpotential changes due to IR-drop in solution.
In the previous section, stationary polarization has been discussed, assuming that the reduction of hydrogen ions is predominantly a counter reaction to the dissolution of metal species. However, in the actual corrosion process of metallic materials, the reduction of dissolved oxygen is often the counter reaction. Therefore, the process where oxygen reduction becomes a counter reaction to the dissolution of metal species is considered in the following. It is commonly known that aqueous solutions near room temperature dissolve approximately 8 ppm of oxygen molecules, that is, the concentration of dissolved oxygen in an aqueous solution is relatively low. Therefore, once the reduction rate of dissolved oxygen increases, even slightly, the oxygen molecules in the vicinity of the surface of metallic materials are depleted, immediately resulting in the diffusion control of dissolved oxygen (diffusion-limiting process). The electrochemistry of this diffusion phenomenon is explained assuming a one-dimensional diffusion as the simplest description. Under diffusion-limiting conditions, the concentration of dissolved oxygen on the metal surface is close to zero, and the flux of dissolved oxygen, J, expressed in Eq. 39, becomes maximum if the thickness of the diffusion layer is assumed to be constant.
| \begin{equation} |J| = \left|-D_{\text{O${_{2}}$}} \frac{dc_{\text{O${_{2}}$}}}{dx}\right| \end{equation} | (39) |
| \begin{equation} |i_{\text{diff}}| = \left|-nFD_{\text{O}_{2}}\frac{dc_{\text{O}_{2}}}{dx} \right| = nFD_{\text{O}_{2}} \frac{c_{\text{O}_{2},\text{bulk}}}{\delta} \end{equation} | (40) |
Corrosion potential, Ecorr, and corrosion current density, icorr, estimated in the case the reduction reaction is diffusion-controlled.
Notably, the measurable polarization curve is obtained by the superimposition of the anodic and cathodic branches of each half-cell reaction.
3.4 Stationary polarization in galvanic coupleBased on previous discussions, galvanic corrosion is discussed as a complex example from the standpoint of stationary polarization in this section. Galvanic corrosion is a type of localized corrosion that occurs when different metals come into contact both electrically and through the transport of ions in solution. Metals M and M′ are assumed to be in contact with each other in an aqueous solution. Under the situation, these metals are assumed to exhibit half-cell reactions (37) and (41).
| \begin{equation} \text{M$'{}^{2+}$} + \text{2e$^{-}$} \rightleftarrows \text{M$'$} \end{equation} | (41) |
| \begin{equation} E_{\text{M}'^{2+}/\text{M}'} < E_{\text{M}^{2+}/\text{M}} < E_{\text{H}^{+}/\text{H}_{2}} \end{equation} | (42) |
Corrosion potential, Egal/corr, estimated in the case the metal M and the metal M′ are in contact. Noted that the illustration is presented under the assumption that the equilibrium potential of each half-cell reaction satisfies the relationship of $E_{\text{M}^{\prime 2 + }/\text{M}' } < E_{\text{M}^{2 + }/\text{M}} < E_{\text{H}^{ + }/\text{H}_{2}}$ and the area of anode site and that of cathode side are in equivalent and unchanged.
In this paper, we described the fundamentals of polarization and stationary polarization used in corrosion research. The definition of current in electrochemistry is briefly discussed, and then the relationship between the current and the electrode potential, that is, Butler–Volmer equation is explained along with its applications to assess the charge transfer resistance and the exchange current density for a single-electrode reaction. In addition, the current flow under diffusion-controlled condition is also described. These theoretical backgrounds of stationary polarization have already been established, so that various kinds of analyses, such as the estimation of i0 and δ, are possible etc. As the cases introduced for corrosion reaction are limited; some general basis on the stationary polarization and its application to galvanic corrosion are described, brief descriptions on other type of corrosion are added. Current oscillations are occasionally observed in the passive region of the anodic polarization of metals and alloys that form passive films on their surfaces. A detailed analysis of the data obtained in the stationary polarization can help in understanding the pitting behavior of metals and alloys. In contrast, in a system where the diffusion of substances and migration significantly affect the corrosion behavior, such as crevice corrosion, the stationary polarization behavior significantly differ from that usually observed on an opened smooth surface, making the interpretation of data difficult. However, if the oxidation current is increased, the change in the potential where the metals and alloys are passivated, and the current in the passive region are thoroughly analyzed, the crevice corrosion behavior can be predicted to some extent. The concept of stationary polarization plays a crucial role in the field of corrosion research. We hope that this manuscript helps in recognizing the experimental data.
The data that support the findings of this study are openly available under the terms of the designated Creative Commons License in J-STAGE Data at https://doi.org/10.50892/data.electrochemistry.20616198. The authors' profiles of this paper can be found on the preface.8
Kentaro Kuratani: Writing – original draft (Lead)
Kazuhiro Fukami: Writing – original draft (Lead)
Hiroaki Tsuchiya: Writing – original draft (Lead)
Hiroyuki Usui: Writing – review & editing (Equal)
Masanobu Chiku: Writing – review & editing (Equal)
Shin-ichi Yamazaki: Writing – review & editing (Equal)
The authors declare no conflict of interest in the manuscript.
This paper constitutes a collection of papers edited as the proceedings of the 51st Electrochemistry Workshop organized by the Kansai Branch of the Electrochemical Society of Japan.
K. Kuratani, K. Fukami, and H. Tsuchiya: These authors contributed equally to this work.
K. Kuratani, K. Fukami, H. Tsuchiya, H. Usui, M. Chiku, and S.-i. Yamazaki: ECSJ Active Members
