Revisiting electrocatalytic oxygen evolution reaction microkinetics from a mathematical viewpoint: implicit rate expression, ambiguous rate constant, and confusing overpotentials

Oxygen evolution reaction (OER) is attractive for many sustainable energy storage and conversion devices, and microkinetic analysis is critical to gain vital reaction details for clarifying the underlying reaction mechanisms. Although many microkinetic studies have been conducted for OER and remarkable achievements have been obtained in both theory and experiment, several “clouds” over reaction microkinetics still need to be swept: (1) the implicit and complex rate expression by conventional equation sets; (2) the ambiguous exponential relationship between the rate constant and the applied potential; (3) the inconsistently used overpotentials for the microkinetic analysis. In this article, we clarify the above points by introducing graph theory for chemical kinetics to illustrate the OER microkinetic process, by which we straightforwardly obtain the steady-state expression of OER kinetic current. Taylor’s theorem and transition state theory are further applied to precisely describe the relationship among rate constant, free energy of activation, and applied potential. Through this, the Butler-Volmer equation can be deduced from the 1st-order Taylor polynomial, and analogous Marcus equation is accessible by the 2nd-order Taylor polynomial. Finally, we clarify two overpotentials (nominal and elementary overpotentials) commonly used in microkinetic OER and find that they are equally reliable for steady-state rate analysis. This mathematical discussion will be conducive to understanding fundamental electrochemical processes and designing highly-efficient electrocatalysts.


Background and Scope
Microkinetic analysis enables to access fundamental electrochemical processes and extract critical information of reaction parameters, thus providing a powerful tool to understand and design superior electrocatalysts for high-value chemical production. 1,2 As a typical electrochemical process, oxygen evolution reaction (OER) is pivotal to many electrochemical energy applications, including but not limited to water electrolysis for green hydrogen production, CO2 reduction for valuable chemical generation, and rechargeable metal-air batteries for electricity storage. 3 OER involves complex electron-transfer processes in elementary steps, elucidating the microkinetics of which is crucial to confirming the potential rate-determining step, clarifying the catalytic mechanism, and further boosting the performance. 4,5 With the rapid development of modern electrochemistry, OER microkinetic process has been extensively investigated, including diverse reaction pathways (e.g., adsorbate evolution mechanism, lattice-oxygen-mediated mechanism), kinetic modeling (e.g., quasi-equilibrium analysis, energetic span model), reactive intermediate detection (e.g., in situ/operando electrochemical spectroscopy, methanol-probe method), theoretical calculation (e.g., computational hydrogen electrode), reaction environment effects (e.g., local pH effects, external physical fields), etc. [6][7][8][9][10][11][12][13][14][15] Despite the above-mentioned achievements for OER, efficient and reliable analysis of its microkinetics still suffers from several confusions in the present theory treatments, which prevents an accurate and precise understanding of the reaction process: (1) One main task in microkinetics is to obtain the rate expression.
However, conventionally deriving the rate expression is implicit with the complex kinetic equation sets, which requires extra calculation tricks and hinders the straightforward understanding of the electrocatalytic process. 16 How to explore the OER process and derive the rate expression through an explicit approach remains challenging; (2) To describe the specific relationship between 4 reaction rate and applied potential, the theory in Butler-Volmer equation (also called Erdey-Grúz-Volmer equation) is usually used. However, due to the essentially empirical expression under the Butler-Volmer framework, the plausible exponential relation between the rate constant and the applied potential is ambiguous and should be justified; 17 (3) There are inconsistent overpotentials usually used in OER microkinetics: one is determined by the difference between the applied potential (E) and the overall reaction equilibrium potential (Eeq,0) (here termed nominal overpotential, η0), 18 while another is calculated by the difference between E and the respective standard potential (Eeq,ij) in each electron-transfer step (here termed elementary overpotential, ηi). 19 This plausible inconsistency of using overpotentials in kinetic equations may hamper the agreement of the basic reaction process toward OER. Thus, the reliability of adopting η0 or ηi for the analysis of OER microkinetics should be clarified. In a word, efforts are required to answer the above questions.
In this article, we will apply a graph theory-based method to interpret the electrocatalytic microkinetic OER process for the first time, demonstrate a rational understanding of the relationship between the rate constant and the applied potential with Taylor's theorem, and discriminate the overpotentials in electrode reaction steps. We hope that this revisiting of the microkinetic OER process from a mathematical view may help clean the "grey zone" with ambiguous knowledge on principles of the electrochemical process, and contribute to the rational design of electrocatalysts for energy devices.

Mathematical Description
In this section, we will discuss the electrocatalytic OER microkinetic process under a steady state by exploring the relationship among OER kinetic current, rate constant, free energy of activation, and applied potential, on the basis of graph theory for chemical kinetics, Taylor's theorem, and transition state theory. Through this mathematical investigation, we hope to provide an elegant and rigorous theoretical framework to promote the understanding of steady-state OER microkinetics. All the discussions are based on standard free energy/condition and mean-field model.

(1) Steady-State Rate Expression of Oxygen Evolution Reaction via Graph Theory
The reaction mechanism for OER is complex and there are various proton-electron transfer paths proposed, such as the conventional adsorbate evolution mechanism (AEM) and latticeoxygen-mediated mechanism (LOM). 8 Herein, we use the typical AEM for discussion, in which four concerted proton-electron transfer (CPET) steps occur subsequently on one active site to finally release one oxygen molecule (Figure 1a). To grasp the steady-state process straightforwardly, we introduce the graph theory for chemical kinetics to obtain the rate expression toward OER microkinetics. [20][21][22][23][24] It shall be noticed that some inspiring pioneer works have been well conducted, including the species-reaction graph for reaction invariant analysis and the schematic diagram for general multi-electron electrocatalytic reaction with a unified formalism. 25,26 To control the length of this article, the readership may refer to the comprehensive theoretical work by Kozuch for the detail term definitions and discussions in graph theory for chemical kinetics. 24 According to the graph theory for chemical kinetics, the rate for oxygen molecule production per site (r) can be directly written as: includes activity terms of other reactants (we will discuss more details in the following part). kf reflects the forward electrocatalytic cycle for OER, while kb reflects the backward electrocatalytic cycle for OER. As for the denominator item |A| in eqn (1), it can be written as the sum of |Ai|: Each |Ai| above belongs to the directed spanning trees corresponding to each vertex (Figure 1b), which is easy to depict just by hand.
Given that one oxygen molecule contributes to four electrons, the OER kinetic current (I) can be written as: In eqn (9), F is the Faraday constant, Csite is the number of active sites for OER. It is clear that the OER kinetic current expression under a steady state is directly related to the rate constants in all reaction steps (including forward and backward steps). Prior to discussing the relationship between rate constant and applied potential for electrocatalytic OER, in the next part, we will turn into the free energy of activation, which is the key parameter for reaction microkinetics.

(2) Free Energy of Activation -Applied Potential Relationship by Taylor's Theorem
Applying potential enables to conveniently tune the free energy of activation (△ ≠ ) in each electron-transfer step (from the i th intermediate to the j th intermediate) for OER, which is one of the most attractive merits of electrocatalysis as compared with traditional catalysis. We hypothesize that the free energy of activation is the function of applied potential (E), i.e., △ , ≠ .
By Taylor's theorem, we describe the relationship between △ , ≠ and E in a mathematical view: is the free energy of activation at applied potential E0, △ , 0 ≠ ′ and △ , 0 ≠ ′′ are the 1 storder and 2 nd -order derivatives of free energy of activation at E0, respectively, Rn(E) is the remainder term. 8 Considering the 1 st -order Taylor polynomial, we have the free energy of activation in the forward step (from the i th to the j th intermediate): Similarly, the free energy of activation in backward step (from the j th to the i th intermediate): It is accepted that the free energies of activation in forward and backward steps hold the below relationship: 27 in which △ , is the reaction free energy in the forward step, △ , 0 is the reaction free energy in the forward step at applied potential E0. Combing eqn (11)-(13), we have: From the identical relation of eqn (14), we can derive the expression of △ , ≠ and △ , ≠ : where βij holds below relationship with △ , 0 ≠ ′ and △ , 0 ≠ ′ : It is evident that eqn (15) and eqn (16) hold the same formation as the free energy of activation related to applied potential introduced in a standard textbook for electrochemistry. 28 In addition, according to the widely existing Brønsted-Evans-Polanyi (BEP) relation in catalytic processes, [29][30][31] it is plausible that in general more positively applied potential will lead to smaller/larger free energy of activation in forward/backward step for OER. To this point, we set △ , 0 ≠ ′ negative and △ , 0 ≠ ′ positive. Therefore, according to eqn (17) and eqn (18), we have: In this situation, it is natural to notice that βij can be termed as symmetry factor, which is commonly known in electrochemistry. 28 When it comes to the 2 nd -order Taylor polynomial, the free energy of activation can be written as: We set E0 as the potential where the reaction free energy equals zero, then: Combing eqn (20)- (22), and we hypothesize that the free energy of activation is not negative, so that: Note that eqn (23) and eqn (24) can be merged as: Interestingly, eqn (26) is analogous to the expression of free energy of activation in Marcus theory for electron transfer. [32][33][34] This is not surprising that a quadratic relation between the free energy of activation and the applied potential is restrained via eqn (20) and eqn (21).
By the above discussion, the description of free energy of activation via Taylor's theorem from a mathematical view is to some extent reliable, in which the 1 st -order Taylor polynomial leads to the commonly used linear relation between free energy of activation and applied potential regulated by symmetry factor, and the 2 nd -order Taylor polynomial enables to derive analogous Marcus equation. 35

(3) Rate Constant -Applied Potential Relationship with Transition State Theory
In the above parts we obtain the OER kinetic current expression by rate constants (kij), and describe the relation between free energy of activation (△ ≠ ) and applied potential (E) through Taylor's theorem. In fact, when it comes to the concept of free energy of activation, transition state theory is implied. 36 In this part, we further apply transition state theory to establish a complete relation among kij, △ ≠ , and E.
According to transition state theory, we have: kTS,ij is the rate constant expressed in transition state theory. As mentioned previously, kij contains activity terms of other reactants, here we write the full formalism: According to the relation between △ , ≠ and E in eqn (10), kij can be fully described as: When we adopt the 1 st -order Taylor Polynomial and combine eqn (17) and eqn (18), the expressions of kij (specifically for forward step) and kji (specifically for backward step) are simplified to: in which, From the exponential relation between rate constant and applied potential described by eqn (31) and eqn (32), we can readily obtain the widely-used Butler-Volmer equation. we will not repeat the deduction here and the readership may refer to some standard textbooks for the details. 28,37 In short, through this systematic and delicate analysis of rate constant and applied potential by Taylor's theorem (Figure 2), we provide a comprehensive theoretical framework to understand the empirical Butler-Volmer theory by 1 st -order approximation of free energy of activation, which is also consistent with the idea briefly mentioned by Chen and Liu. 38

Overpotentials in Microkinetic Modeling for Oxygen Evolution Reaction
After establishing the mathematical framework for OER microkinetics under a steady state, next we come into the overpotential issue, i.e., the two overpotentials (nominal and elementary overpotentials) used in the microkinetic analysis for OER. In this section, we will clarify the essence of these two overpotentials and more importantly discuss their reliability applied in OER microkinetics, especially for the widely used Butler-Volmer framework.

(1) Nominal Overpotential vs. Elementary Overpotential
Defined by the International Union of Pure and Applied Chemistry (IUPAC), overpotential refers to the deviation of the potential of an electrode from its equilibrium value required to cause a given current to flow through the electrode. 39 This closely connects the name overpotential with electrochemical kinetics. Here, we talk about two kinds of overpotentials that usually appear in OER microkinetics, i.e., nominal overpotential (η0) and elementary overpotential (ηij). It should be noted that the name "nominal overpotential" is occasionally used by other researchers, such as Scott et al. 40 They consider "nominal overpotential" as the difference between the applied potential and standard equilibrium potential of OER (i.e., 1.23 V vs. RHE), specifically contrasted with 13 another overpotential ("Nernst overpotential") as the difference between the applied potential and the real equilibrium potential. In our situation, "nominal overpotential" covers the overall reaction steps, specifically contrasted with the distinct overpotential (i.e., the elementary overpotential) in each electron-transfer step. Thus, there is nuance when stating "nominal overpotential" in this article and the work by Scott et al.
According to the OER kinetic current expression (eqn (1)- (9)) and the relation between rate constant and applied potential (eqn (30)), the OER kinetic current can be related to the applied potential as: We use kij here to refer to all the rate constants and represents the relation between I and kij. If we set E0 as Eeq,0, the OER kinetic current becomes: If we set E0 as Eeq,ij, the OER kinetic current becomes: It is obvious that both eqn (36) and eqn (37) quantitatively describe the same OER kinetic current with the same free energy of activation term by different overpotentials (η0 or ηij). The difference between eqn (36) and eqn (37) essentially resides in the selected points for Taylor's formula of the free energy of activation, i.e., one is Eeq,0, another is Eeq,ij. Thus, using nominal overpotential or elementary overpotential for OER microkinetics is intrinsically identical.

Taylor Polynomial
Since Butler-Volmer equation, which can be deduced by the 1 st -order Taylor polynomial term in our theoretical framework, is widely applied in microkinetic analysis for electrocatalytic OER, it is necessary to discuss the equivalent condition by using nominal overpotential or elementary overpotential for microkinetic study with the 1 st -order approximation of free energy of activation, in order to avoid inconsistent results due to the "subjective" use of overpotentials. For the free energy of activation functioned by nominal overpotential (η0): By elementary overpotential (ηij): We further apply the 1 st -order approximation to △ , , ≠ at ,0 : Combing eqn (38)

Summary and Outlook
Through the article, we convey a graph theory-based method to interpret the OER microkinetics and straightforwardly obtain the reaction rate expression, re-explore the relation between the rate constant and the applied potential with a comprehensive theoretical framework by Taylor's theorem and transition state theory, and clarify the two types of overpotentials (nominal overpotential and elementary overpotential) applied in the microkinetic analysis of OER.
Important points are summarized below: (1) The expression of OER kinetic current in a steady state is readily accessible by graph theory with a visual reaction process, unnecessary to solve the complex equation sets.
(2) The free energy of activation -applied potential relationship can be mathematically (4) Provided that the 1 st -order derivatives of free energies of activation at corresponding equilibrium/standard potential are equal, using either nominal or elementary overpotential to analyze OER microkinetics under the Butler-Volmer framework can be approximately regarded as equivalent.
We anticipate that our mathematics-based discussion will be conducive to understanding the basic electrochemical process and will motivate more researchers to join the booming field of electrocatalysis. In addition to the explicit graph-theory method, further development of microkinetic analysis might resort to computer science-assisted approaches, such as kinetic Monte Carlo simulations, first-principle calculation, molecular dynamics simulation, artificial intelligence (e.g., machine learning), and high-throughput methods, which provide microscopic and fundamental understandings toward electrocatalytic processes (both steady and transient states) and accelerate the development of new electrode materials, especially when coupled with in situ/operando electrochemical techniques that enable to extract real-time information. 42,43 In addition, proton-coupled electron transfer theory presents an informative perspective to seize the key reaction processes and offers a quantum-based understanding of OER. 44,45 The quantum-toclassical transition phenomenon (or vice versa) in the microscopic electrode process shall be also paid attention to. 46 We believe future research in electrocatalytic microkinetics (not limited to OER) will be strongly supported by the fast development of in silico technologies, advanced in situ/operando equipment, and emerging theories for microscopic electrode process.

Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Funding Sources
The authors acknowledge the support from the City University of Hong Kong through project 9610537 and the Department of Science and Technology of Guangdong Province through project 2022A1515010212.

Notes
There are no conflicts to declare.