English

• Steady-state current (j)-potential (E) voltammograms provide an easy-to-follow technique for extracting the kinetic information of an electrode reaction because of the simplicity of steady-state kinetic theory. In mathematics, steady-state voltammetry avoids charging current interference [1] and possesses time-independent mass transport processes between the electrode surface and the bulk solution [2].

  Reversibility is a basic scale in the recognition of the reactivity of a reaction, and the electrode equations are often divided into reversible, irreversible and quasi-reversible types [3]. The electrode equations and their applicability are dependent on the reference potential against which the potential is quantified, e.g., standard (or formal) potential, equilibrium potential (E_eq) and half wave potential (E_1/2). The electrode equations are multifarious in terms of reversibility and reference potential. If possible, it is strongly desired to unify all the electrode equations and elaborately differentiate their applicability in steady-state voltammetry.

  "Reversible" and "irreversible" are two extreme cases; the former is only dependent on thermodynamics, while the latter is fully governed by kinetics. The variations in the kinetic features embodied in the steady-state voltammograms between the two cases have long been absent from a quantitative measure. E_1/2 is an important indicator of reaction occurrence, whereas its physical significance in the full range of reversibility has yet to be clarified.

  Recently, we found that the peak point of the derivative of a j-E curve in irreversible cases corresponded to a special kinetic state, where the peak potential was demonstrated to be E_1/2 and the peak value designated the activation feature of a reaction driven by potential [4]. Additionally, this derivative extremum analysis (DEA) can be easily demonstrated to be applicable in reversible cases; although, the peak potential and peak value have different expressions. In this work, DEA was further proven to be valid in the full range of reversibility through numerical simulations using a derived universal electrode equation, and it provided a new perspective in understanding the kinetic feature evolution in the reversibility domain. We reorganized the equations referenced with formal potential, E_eq and E_1/2 in the full reversibility scale and elucidated their applicability. Reversibility classifications were proposed according to the variation trends of DEA in the reversibility domain.

  To investigate the DEA applicability, we first derived a universal electrode equation in the full range of reversibility with three premises as follows: the uniformly accessible electrode treating the mass transport process, a large excess of supporting electrolyte to eliminate the migration effect, and a Nernst diffusion layer with stirring maintaining the bulk concentration constant. The cathodic current is defined as positive, and the anodic current is negative.

  The most basic equation describing an elementary reaction O + ne⁻=R is an original Butler-Volmer form (Eq. 1) [5]:

  (1)

  where k⁰ is the standard rate constant, E⁰ is the standard potential and can be replaced by formal potential in practice, c_O^* and c_R^* are the oxidant and reductant concentrations on the electrode surface, respectively, α and β are the electron transfer coefficients (ETCs), α + β = 1, F is the Faraday constant, and f = F/RT, where R is the ideal gas constant, and T is the temperature in Kelvin. All the symbols are defined in Table S1 (Supporting information).

  Eq. 1 is difficult to use since c_O^* and c_R^* can hardly be defined and measured. However, c_O^* and c_R^* can be replaced by introducing the mass transport relationships of reactants (O and R). In the steady state, the reaction rate equals the mass transport fluxes of O and R between the electrode surface and bulk solution (Eqs. 2) and 3:

  (2)

  (3)

  where c_O^b and c_R^b are the concentrations in bulk solution, m is the mass transport coefficient, D is the diffusion coefficient, and δ is the diffusion layer thickness. The maximum transport fluxes of O and R are expressed as the cathodic and anodic limiting current densities (j_{l, c} and j_{l, a}), which are obtained at c_O^* and c_R^* reaching zero (Eq. 4):

  (4)

  Introducing Eq. 4 into Eqs. 2 and 3 obtains (Eq. 5):

  (5)

  Finally, introducing Eq. 5 into Eq. 1 derives the following equation with the potential referenced against E⁰ (Eq. 6):

  (6)

  This equation is a universal form in consideration of the concentration polarization in the full range of reversibility, which excludes any simplification of kinetic factors. The electrode geometric size information can be introduced to the equation by using the analytical expressions of the diffusion layer thickness [6]. All other equation forms, such as reversible and irreversible, in the presence of only O or R, and with equal mass transport coefficients, can be simplified from this equation. For example, in the presence of only O, j_{l, a} = 0, Eq. 6 can be simplified to the following well-known equation proposed by Mirkin and Bard (Eq. 7) [2]:

  (7)

  Under the premise of equal mass transport coefficients (m_O = m_R), Eq. 6 can be transformed to the following form as reported by Molina et al. (Eq. 8) [6]:

  (8)

  where and .

  If specifying the electrode geometric size in m_O and m_R, more detailed electrode equations hold for hemispherical and disc microelectrodes [3, 7]. Here, the electrode equations referenced with E⁰ in the full range of reversibility are summarized in Table S2 (Supporting information).

  The DEA derivation in reversible case is shown as follows. To obtain the reversible equations, Eq. 6 is transformed to the following form (Eq. 9):

  (9)

  Under reversible conditions, m_O/k⁰ → 0 and m_R/k⁰ → 0. We proved that in the first term and in the second term are negligible; the detailed derivation is shown in Supporting information. Thus, a reversible equation form can be simplified from the above equation:

  (10)

  By solving for the extremal solution of the derivative of Eq. 10, we derived the expressions of the peak potential (E_P) and peak vale (PV) in the dj/dE curve and demonstrated that E_P was identical to E_1/2. The detailed derivation process is shown in the Supporting information.

  (11)

  (12)

  The DEA derivation in irreversible case is shown here. To date, the reported irreversible equations have basically applied the forms using E_eq as the reference potential. Eq. 6 can be transformed to a E_eq-referenced form. The exchange current density (j₀) is an equilibrium current at the initial state where and and has the following expressions (Eq. 13):

  (13)

  In combination with the expressions of j₀, j_{l, c} and j_{l, a}, the cathodic and anodic terms in Eq. 6 can be rearranged as follows (Eqs. 14 and 15):

  (14)

  (15)

  Please note that the above transformations are only valid at c_O^b and c_R^b are not zero. Based on the above relationships, Eq. 6 can be transformed to the following form referenced with E_eq (Eq. 16):

  (16)

  In the irreversible case, j₀ is so small that a very high overpotential is required to drive the reaction occurrence. Thus, the term for the cathode reaction is negligible, and for the anode reaction is negligible. Eq. 16 can be simplified to the following cathodic and anodic irreversible equations (Eqs. 17 and 18):

  Cathode:

  (17)

  Anode:

  (18)

  These equations have similar forms to those derived from the multielectron and multistep irreversible process with one sole rate-limiting step [4]. The DEA applicability in irreversible cases has been demonstrated by solving for the extremal solution of the derivatives of Eqs. 17 and 18 in a previous work [4]. The expressions of E_1/2 and PV in irreversible cathodic reactions are shown as follows (Eqs. 19 and 20):

  (19)

  (20)

  In both reversible and irreversible cases, E_p is identical to E_1/2, and PV has a unit of A m⁻² V⁻¹ and can serve to characterize the kinetic slowness of a reaction driven by potential. However, E_1/2 and PV have different expressions in the two cases.

  Furthermore, the DEA simulations in the full range of reversibility using Eq. 6 are calculated to show the DEA in the transition range between reversible and irreversible cases, because mathematically solving for the derivatives of Eq. 6 is too difficult. In a given scenario, a set of j-E curves with k⁰ values varying from 0 m/s to 10⁻¹⁰ m/s are calculated first, then the dj/dE curves are calculated, and the E_p and PV values are extracted from the peak points of dj/dE curves. Finally, the plots of E_p and PV vs. lgk⁰ are obtained. For all the calculations, n = 1, α = β = 0.5, E⁰ = 0 V vs. SHE and T = 25 ℃. The other parameter values (m_O, m_R, c_O^b and c_R^b) are presented in the figure titles. The calculations were performed using MATLAB 2012a.

  The calculation results in the presence of both O and R are shown in Fig. 1. Figs. 1a clearly presents the variation trend of the j-E curves from the reversible to irreversible case: in the condition of k⁰ > 10⁻⁴ m/s, the curves approached the calculation result of the reversible equation (Eq. 10) with k⁰ = 1 m/s; once k⁰ was small enough, namely, k⁰ = 10⁻⁹ m/s, the curves were consistent with the results of irreversible equations (Eqs. 17 and 18). In Fig. 1b, the associated dj/dE curves show that the peak point gradually shifts from one point in the reversible case to two peak points. The isolation degree between the cathode and anode reactions can be characterized by the distance between the two points and is intensified as k⁰ decreases.

  Figure 1

  

  Figure 1.  Calculation results of Eq. 6 with lgk⁰ varying from 0 to –9, c_O^b=c_R^b=10 mol/m³, and m_O = m_R = 5 × 10^–5 m/s, n = 1, E⁰ = 0 V, α = 0.5 and T = 25 ℃: (a) j-E curves, the dashed lines were calculated from the reversible equation (Eq. 10) with lgk⁰ = 0 and irreversible equations (Eqs. 17 and 18) with lgk⁰ = –9, (b) associated dj/dE curves, (c) peak potential (E_P) and potentials at j_{l, c}/2 and j_{l, a}/2 plotted vs. lgk⁰, and (d) PV plotted vs. lgk⁰, the lines were calculated from the reversible and irreversible expressions of PV.

  DownLoad: Full-Size Img PowerPoint

  To further investigate the relationship between E_p and E_1/2, the E_p and the potentials of j_{l, c}/2 and j_{l, a}/2 were plotted in Fig. 1c. E_p changed suddenly from the reversible E_1/2 feature, namely, the potential of (j_{l, c} + j_{l, a})/2, to the irreversible E_1/2 feature, namely, the potentials of j_{l, c}/2 and j_{l, a}/2. In Fig. 1d, the PV gradually shifts from the reversible case, as designated by the line calculated from Eq. 12 to the irreversible case designated by the line calculated from Eq. 20. These results confirmed that DEA is applicable in most cases in the presence of both O and R, except for a narrow transition range between reversible and irreversible cases where the exact E_1/2 can hardly be defined. In light of the variation features of E_p and PV, the exact reversible and irreversible ranges can be defined as shown by the colored regions.

  The DEA simulation results in the presence of only O are shown in Fig. 2. Figs. 2a and b demonstrate that the E_p in the dj/dE curves corresponded to the potential of j_{l, c}/2 in the j-E curves in the full range of reversibility. Thus, E_p is invariably identical to E_1/2 in this case. The E_p vs. lgk⁰ plot in Fig. 2c also presented a sudden change from the reversible E_1/2 feature to the irreversible E_1/2 feature. In Fig. 2d, PV gradually shifts from the reversible case to the irreversible case.

  Figure 2

  

  Figure 2.  Calculation results of Eq. 6 with lgk⁰ varying from 0 to –10, c_O^b = 10 mol/m³, c_R^b = 0, m_O = m_R = 5 × 10^–5 m/s, n = 1, E⁰ = 0 V, α = 0.5 and T = 25 ℃: (a) j-E curves, (b) associated dj/dE curves, (c) E_p vs. lgk⁰ and (d) PV vs. lgk⁰, the lines were calculated from the reversible and irreversible expressions of E_p and PV.

  DownLoad: Full-Size Img PowerPoint

  Finally, the above DEA simulation results confirmed that E_1/2 is a kinetically special potential corresponding to the positions of the half limiting current in the j-E curves and the peak point in the dj/dE curve. DEA is a general method to extract E_1/2 as well as PV in the full range of reversibility. The variation trends of E_1/2 and PV along the k⁰ axis can give a classification criterion of quasi-reversible and quasi-irreversible regions, namely, (ⅰ) quasi-reversible: E_1/2 is approximate to the reversible E_1/2 value; (ⅱ) quasi-irreversible: E_1/2 is approximate to the irreversible E_1/2 value.

  The above calculations indicate that E_1/2 is a good choice of reference potential because it can be steadily determined in the full range of reversibility. Here, the electrode equations referenced with E_1/2 in the full range of reversibility are summarized in Table 1. The reversible equation form (Eq. 21) can be obtained by combining Eqs. 10 and 11:

  (21)

  Table 1

  

  Table 1.  Electrode equation forms referenced with E_1/2 in the full range of reversibility.

  DownLoad: CSV

  The E_1/2-referenced equations of irreversible cathode and anode reactions can be derived by introducing the irreversible expression of E_1/2 (Eq. 19) into Eqs. 17 and 18, respectively (Eqs. 22 and 23):

  Cathode:

  (22)

  Anode:

  (23)

  In irreversible cases, it is not necessary to distinguish the equations in the presence of both O and R and in the presence of only O because the anode and cathode reactions are evidently separated from each other.

  In quasi-reversible/irreversible cases, E_1/2-referenced equations are not available because the exact expression of E_1/2 and PV can hardly be derived. Nonetheless, DEA is still applicable.

  Furthermore, the electrode equations referenced with E_eq in the full range of reversibility are summarized in Table 2. The reversible equation is derived as follows. Simultaneously, dividing by j₀ in the numerator and denominator of Eq. 16, it becomes (Eq. 24):

  (24)

  Table 2

  

  Table 2.  Electrode equation forms referenced with E_eq in the full range of reversibility.

  DownLoad: CSV

  In the reversible case, j₀ is so large that 1/j₀ is negligible. The above equation can be transformed to the following reversible form (Eq. 25):

  (25)

  Although irreversible E_eq-referenced equations are derived and are as given in Table 2, users should be highly cautious about their applications. The applications of E_eq and j₀ are introduced in the following part. The E_eq used as a reference potential in kinetics is the open circuit potential of an electrode reaction, which satisfies the Nernst relationship (Eq. 26):

  (26)

  At this potential, the cathodic and anodic rates reach balance and are expressed as exchange current density (j₀) (Eq. 27):

  (27)

  It is obvious that E_eq and j₀ are relevant only in the presence of both O and R. Once c_O^b or c_R^b approaches zero, E_eq and j₀ in Eqs. 26 and 27 become an infinite problem without explicit value. Moreover, E_eq is experimentally indeterminable in irreversible cases because the zero current is not a point but a wide gap between the cathode and anode reactions (Fig. 1). Thus, E_eq and j₀ are applicable for the kinetically fast reactions in the presence of both O and R.

  Despite this, E_eq and j₀ are still frequently applied in irreversible cases. Here, we demonstrate that E_eq and j₀ are an interdependent pair in irreversible cases. Choosing any potential in the zero current stage as a formal equilibrium potential (E_eq^*), the deviation of E_eq^* from the theoretical E_eq can be described by E_eq = E_eq^* + ΔE. Introducing it to the irreversible equation (Eq. 17) obtains Eq. 28:

  (28)

  It revealed that the application of E_eq^* only results in the variation of j₀. In irreversible cases, E_eq is indeterminable and is usually determined as the onset potential, the lowest potential where the faradaic current is observed [8]. Therefore, the j₀ value has little physical significance in irreversible cases. If selecting the same E_eq value, j₀ is still useful to compare the activity of a reaction between different conditions or catalysts.

  In addition, the activation overpotential (η_act) in the irreversible case is also meaningless because it is dependent on the j₀ and E_eq values. Nonetheless, the overpotential analysis of the mass transport effect is feasible, viz., the concentration overpotential assigned to reactant mass transport [9] and the pH overpotential assigned to H⁺/OH⁻ transport [10], because the overpotential part E − E_eq − η_act has excluded the E_eq value.

  Currently, the E_eq-referenced equations are typically used to fit the j-E curves for parameter determinations, such as the Tafel fitting method and a recently reported nonlinear fitting method [11]. The Tafel method either directly ignores the mass transport effect by fitting a narrow j-E curve range [12, 13] or corrects the mass transport current part to improve the accuracy of parameter values [14-16]. The nonlinear fitting method is performed based on Eqs. 17 or 18 to fit the entire j-E curve range, which overcomes the disadvantages of the Tafel method to obtain reliable ETC, j₀ and j_l values.

  The E_1/2-referenced equations are more significant for quantifying the activity of irreversible reactions. The information of E_eq and j₀ is involved in E_1/2 (see Eq. 19) to reveal the priority of reaction occurrence. The nonlinear fitting method can also be performed using the irreversible E_1/2-referenced equations and obtain the values of ETC, E_1/2 and j_l. In Fig. S1 (Supporting information), the nonlinear fitting results of the two types of equations (Eqs. 17 and 22) showed that the fitting lines were totally identical and could well fit the entire j-E curve. All the parameter values (Table S3 in Supporting information) determined from Eq. 22 have a narrow 95% confidence interval, and the intervals of ETC, E_1/2 and j_{l, a} are less than 1.10%, 0.06% and 0.25%, respectively. This confirms that the nonlinear fits based on the irreversible E_1/2-referenced equations are a reliable parameter determination method of irreversible reactions.

  Moreover, the irreversible E_1/2-referenced equations are also applicable in conditions deviating from the irreversible case, wherein the ETC reveals reversibility variations. The reversible E_1/2-referenced equations are the special form of irreversible E_1/2-referenced equations with ETC equal to 1. With the aid of micro- and nanoscale electrode techniques to measure steady-state voltammograms [17], the practical application scope of irreversible E_1/2-referenced equations can be greatly extended.

  Importantly, the variation trends of E_1/2 and PV in DEA in the reversibility domain provide classification criteria of reversible, irreversible, quasi-reversible and quasi-irreversible cases. Here, the influences of transport coefficients (m_O and m_R) and concentrations (c_O^b and c_R^b) on the reversibility classifications are investigated. In the presence of only O, the effect of m_O with a k⁰ scale is shown in Figs. 3a–c. The reversible and irreversible ranges presented by the colored regions gradually shift toward the decreasing direction of lgk⁰ as m_O decreases from 10⁻⁴ m/s to 10⁻⁶ m/s. The transition region can be divided into quasi-reversible and quasi-irreversible parts bounded by lgm_O as indicated by the dashed lines in Figs. 3a–c.

  Figure 3

  

  Figure 3.  E_p and PV vs. lgk⁰ plots in the presence of only O calculated from Eq. 6 with c_O^b=10 mol/m³, n = 1, α = 0.5, E⁰ = 0 V and T = 25 ℃: (a) lgm_O = –4, (b) lgm_O = –5, and (c) lgm_O = –6; (d) plots of E_p in Fig. (a–c) vs. lg(k⁰/m_O).

  DownLoad: Full-Size Img PowerPoint

  The E_p vs. lgk⁰ plots in Figs. 3a–c were redrawn using a logarithmic k⁰/m_O scale as shown in Fig. 3d, and the results at different m_O conditions were the same. It revealed that k⁰/m_O was a dimensionless parameter and the most basic scale for reversibility classification. Hence, reversibility was a relative measure between reaction kinetics and mass transport of electroactive species. Roughly,

  In theory, the full "reversible" and "irreversible" are two extreme cases that can hardly be reached, and all the reactions may be regarded as quasi-states [3]. Here, a detailed classification of the effectively quasi-reversible and quasi-irreversible cases was proposed, as listed in Table 3, namely,

  Table 3

  

  Table 3.  Reversibility classifications in terms of k⁰/m_O and k⁰ in the presence of only O.

  DownLoad: CSV

  100 > k⁰/m_O > 1: quasi-reversible, where E_1/2 approaches the reversible E_1/2 value

  0.01 < k⁰/m_O < 1: quasi-irreversible, where E_1/2 approaches the irreversible E_1/2 value

  The k⁰ scale for the reversibility classification is dependent on the m_O and/or m_R values. In practice, the diffusion coefficient or mass transport coefficient values of most solutes rarely differ by more than one order of magnitude. The diffusion coefficient usually has a magnitude of 10⁻⁹ m²/s [18, 19], and the mass transport coefficient has a magnitude of 10⁻⁵ m/s [20]. The reversibility classification in terms of the k⁰ scale with m_O = 10⁻⁵ m/s (see Fig. 3b) is listed in Table 3.

  In addition, j_0, in comparison to the transport limiting current (j_{l, c} or j_{l, a}), is also frequently applied to the quantification of reversibility in electrochemistry [21]. The ratio between j₀ and j_{l, c} can be derived from Eqs. 4 and 27 as follows (Eq. 29):

  (29)

  This indicates that j₀/j_{l, c} is an extended form of k⁰/m_O that involves an extra concentration term. Obviously, j₀/j_{l, c} is applicable in the presence of both O and R and is identical to k⁰/m_O in the condition of c_O^b = c_R^b. The DEA calculation result under the condition of c_O^b = c_R^b is shown in Fig. S2 (Supporting information), confirming that the reversibility scale is independent of the concentration extent. Under the condition of c_O^b ≠ c_R^b (Fig. S3 in Supporting information), the reversible and irreversible regions can be well identified but the quasi-state regions for anode and cathode reactions become very complicated.

  In this work, numerical simulations of DEA using a derived universal electrode equation successfully verified the validity of DEA in the full range of reversibility and gained panoramic insight into electrochemical kinetics. The equations referenced with E⁰, E_1/2 and E_eq in conjunction with their applicability were reorganized in three tables in terms of the reversibility scale. Reversibility classifications in terms of the variation features of DEA in the reversibility domain were proposed.

  The outcomes of this work are especially important for reactions suffering from mass transport limitations. One can apply the DEA method to extract E_1/2 and PV from the steady state j-E curves. E_1/2 is a potential feature parameter pronouncing the priority of reaction occurrence on a potential scale. PV is an activation feature parameter revealing the kinetic slowness driven by potential. Alternatively, the nonlinear fitting method of j-E curves using the irreversible E_1/2-referenced equations can obtain reliable ETC, E_1/2 and j_l values. In summary, the E_1/2-referenced equations have wide application scope in the presence of only O, while the E_eq-referenced equations are primarily useful for the kinetically fast reactions in the presence of both O and R. Users should be highly cautious about the applications of E_eq and j₀ in irreversible cases because they are an interdependent pair and have little physical significance.

  Declaration of competing interest

  The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

  Acknowledgment

  This work was financially supported by the National Natural Science Foundation of China (Nos. 52131003, 52170059, 51808526, 51727812).

  Supplementary materials

  Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.cclet.2022.01.078.

  
  
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• 

  Figure 1  Calculation results of Eq. 6 with lgk⁰ varying from 0 to –9, c_O^b=c_R^b=10 mol/m³, and m_O = m_R = 5 × 10^–5 m/s, n = 1, E⁰ = 0 V, α = 0.5 and T = 25 ℃: (a) j-E curves, the dashed lines were calculated from the reversible equation (Eq. 10) with lgk⁰ = 0 and irreversible equations (Eqs. 17 and 18) with lgk⁰ = –9, (b) associated dj/dE curves, (c) peak potential (E_P) and potentials at j_{l, c}/2 and j_{l, a}/2 plotted vs. lgk⁰, and (d) PV plotted vs. lgk⁰, the lines were calculated from the reversible and irreversible expressions of PV.

  下载: 全尺寸图片 幻灯片
  

  Figure 2  Calculation results of Eq. 6 with lgk⁰ varying from 0 to –10, c_O^b = 10 mol/m³, c_R^b = 0, m_O = m_R = 5 × 10^–5 m/s, n = 1, E⁰ = 0 V, α = 0.5 and T = 25 ℃: (a) j-E curves, (b) associated dj/dE curves, (c) E_p vs. lgk⁰ and (d) PV vs. lgk⁰, the lines were calculated from the reversible and irreversible expressions of E_p and PV.

  下载: 全尺寸图片 幻灯片
  

  Figure 3  E_p and PV vs. lgk⁰ plots in the presence of only O calculated from Eq. 6 with c_O^b=10 mol/m³, n = 1, α = 0.5, E⁰ = 0 V and T = 25 ℃: (a) lgm_O = –4, (b) lgm_O = –5, and (c) lgm_O = –6; (d) plots of E_p in Fig. (a–c) vs. lg(k⁰/m_O).

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  Table 1.  Electrode equation forms referenced with E_1/2 in the full range of reversibility.

  下载: 导出CSV

  Table 2.  Electrode equation forms referenced with E_eq in the full range of reversibility.

  下载: 导出CSV

  Table 3.  Reversibility classifications in terms of k⁰/m_O and k⁰ in the presence of only O.

  下载: 导出CSV
•