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## Creator

[Miao Wang](https://orcid.org/0000-0001-9483-6877), [Ken Sakaushi](https://orcid.org/0000-0003-4797-9087)

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[Bridging classical concepts and modern advances in electrode processes: a tutorial review](https://mdr.nims.go.jp/datasets/ba232625-f7ff-4633-ae1b-88cf14ebc758)

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Received 00th January 20xx,1. a. Research Center for Energy and Environmental Materials, National Institute for  Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan.Email: [SAKAUSHI.Ken@nims.go.jp](mailto:SAKAUSHI.Ken@nims.go.jp)Web: [https://www.nims.go.jp/group/eec/en/](https://www.nims.go.jp/group/eec/en/)Accepted 00th January 20xxDOI: 10.1039/x0xx00000xBridging Classical Concepts and Modern Advances in Electrode  Processes: A Tutorial ReviewMiao Wanga and Ken Sakaushi\*aElectrochemistry is becoming increasingly crucial due to its central role in energy conversion/storage and the production of  high-value chemicals via renewable energy. As these technologies advance, students and researchers, particularly those  entering the field from adjacent disciplines, are confronted with the intricate physicochemical phenomena at electrode- electrolyte interfaces, whose fundamental understandings are essential for the rational design of efficient, selective, and  robust electrocatalytic systems. In addition, several simple and practical methods have been developed in recent years to  try to address the existing stability issues in electrocatalysis. Aiming to link the multifaceted history of electrochemistry with  its promising future for diverse applications, in this Tutorial Review, which focuses on electrode processes, starting from a  brief history, we provide representative discussion on microkinetic modelling with/without assumption of rate-determining  step, demonstrate mass transport effects by reactants/products, analyse electrode material reconstruction, and explore  microkientic models for depicting degradation processes. An alternative perspective on reaction mechanisms is presented  based on graph theory. Next, we introduce a data-driven approach to facilitate the understanding of electrode reactions  including both activity description and stability issue. Finally, we propose a heuristic framework via matrix to organise  electrode states in electrode processes. We hope this Tutorial Review will provide fundamental and useful insights into  processes at the electrified electrode-electrolyte interface, inspiring the rational design of highly efficient electrochemical  reactions for green chemistry in the near future.1 Introduction 1In the pursuit of carbon neutrality and sustainable energy 2  strategies, electrocatalysis has attracted growing attention as a 3  versatile platform for driving chemical transformations.1, 2 4  Unlike to conventional thermochemical approaches  typically 5  requiring harsh conditions, for instance high temperature and 6  pressure, electrocatalytic reactions operate under 7  comparatively mild environments, offer tunable activity and 8  selectivity, and can be directly coupled with renewable 9  electricity.3 A fundamental understanding of electrode 10  processes is therefore central to advancing electrocatalysis and 11  realizing its potential in energy conversion/storage and 12  sustainable chemical production. According to the definition by 13  International Union of Pure and Applied Chemistry (IUPAC), 14  electrode process refers to the totality of changes occurring at 15  or near a single electrode during the passage of current.4 These 16  changes may be included in/on parts such as electrode 17  morphology/structure, electrode surface species, interfacial 18  structure of electrolyte, reactant distribution, and species 19  transport, with a remarkable impact on the overall electrode 20  performance. Therefore, as the fundamental building block in 21  electrochemical energy conversion and application, electrode 22  process (here for electrocatalysis) requires in-depth 23  understanding to facilitate the rational design of highly efficient 24  systems. Before outlining the long history of investigating 25  electrode processes, we first provide a brief discussion of the 26  relationships among thermocatalysis, non-catalysed 27  electrochemical reaction, and electrocatalysis. 28A process, resulting in the interconversion of chemical species, 29  is called chemical reaction.5 It is obvious that if a chemical 30  reaction occurs under an electrochemical condition (usually 31  electricity is applied to a conductive solid immersed in a 32  conducting medium with the movement of ions), the concept is 33  restricted specifically to electrochemical reaction. In addition, 34  thermocatalysis (or thermal catalysis, i.e. conventional 35  heterogeneous catalysis) refers to a chemical reaction where 36  the reaction rate is accelerated by a catalyst with heat as the 37  driven force,6 such as Haber-Bosch process for ammonia 38  synthesis and Fischer-Tropsch process for hydrocarbon 39  production.7-10 Opposite from the term catalyst, that the one 40  diminishes the rate of a chemical reaction is called inhibitor.11 41  Electrocatalysis lies at the intersection of electrochemical 42  reaction and thermocatalysis, encompassing both electric and 43  catalytic characteristics. Thus, in this sense, we can regard 44  electrocatalysis as catalyst-promoted electrochemical reactions. 45  In contrast, there are thermocatalytic reactions that are 46  electrochemically promoted, also known as electrochemical 47  promotion of catalysis (EPOC).12-14 In this review, we do not 48  discuss too much about EPOC. The readership who are 49  interested in this topic may refer to some previous papers.12, 13 50ARTICLEPlease do not adjust marginsPlease do not adjust margins[mailto:SAKAUSHI.Ken@nims.go.jp](mailto:SAKAUSHI.Ken@nims.go.jp)[https://www.nims.go.jp/group/eec/en/](https://www.nims.go.jp/group/eec/en/)According to the above discussion, thermocatalysis and 1  electrocatalysis should share the same framework of chemical 2  kinetics. Nonetheless, electrocatalysis features both bare 3  electrochemical reaction and thermocatalysis such as 1) the 4  applied potential can change the reaction free energy; 2) free 5  energy of activation is directly influenced by the applied 6  potential; 3) the catalyst changes the reaction pathway and free 7  energy of activation to tune the reaction rate/selectivity. These 8  features endow electrocatalysis with complicated processes, 9  thus a full clarification requires consideration of many factors, 10  like electrode material structure/morphology, electrified 11  electrode-electrolyte interface structure, and reactant/product 12  transfer in electrolyte. A schematic illustration with the 13  transition-state framework, for thermocatalysis, uncatalysed 14  electrochemical reaction, and electrocatalysis, is shown in Fig. 15  1. To narrow the topic, in this paper we focus on aqueous 16  electrolyte-based electrode processes. 1718Fig. 1 Schematic illustration of thermocatalysis, uncatalysed 19  electrochemical reaction, and electrolysis. 20The history of electrochemistry is over two centuries, as 21  suggested by the unit for voltage Volt, which can trace back to 22  the vital work by Alessandro Volta in 1800.15 After the 23  introduction of overpotential by Caspari in 1899,16 an important 24  work for exploring electrode processes is the well-known Tafel 25  equation by Julius Tafel in 1905 to provide a linear description 26  between overpotential and logarithm of current density.17 The 27  structure of the electric double layer (EDL) in the electrode-28  electrolyte interface was explored from 1879 by Helmholtz,18-25 29  and this EDL influences electrode processes as the research 30  initiated by Frumkin in 1930s.26 Nonetheless, for shortening the 31  length of this Tutorial Review, we do not delve into the EDL 32  effect on electrode process and the readership may refer to 33  other published papers for more details.27-44 Even though 34  nowadays it is not usually employed, the polarography method 35  initiated by Heyrovský in 1922 opens the way to modern 36  electroanalysis techniques, which enable to obtain reliable 37  electrochemical data.45-47 Later, impressive and widely used is 38  the equation to quantify the relation between current density 39  and potential, i.e. Bulter-Volmer equation (also called Erdey-40  Grúz-Volmer equation).48-50 Starting from 1940s, mass transport 41  effect for electrode reactions was well explored in a series of 42  works by rotating electrode techniques including rotating disk 43  electrode (RDE) and rotating ring-disk electrode (RRDE).51-59 44  Another breakthrough in describing electrode reactions is the 45  theoretical framework based on Marcus, Hush, and Chidsey to 46  give a microscopic understanding with quantum effects 47  included from 1956.60-64 In fact, quantum effects have been 48  discussed in electrode processes (so called quantum electrode 49  processes) from 1931 by Gurney,65 later many achievements 50  have been made, such as the Levich-Dogonadze theory in 1959 51  and 1960s-1970s,66-70 the systemized proton-coupled electron 52  transfer (PCET) theory from 1990s,71-73 and recently quantum-53  to-classical transitions by our group from 2018.74, 75 For more 54  details about the history of quantum electrode processes, the 55  readership may refer to our previous papers.76, 77 Apart from the 56  direct analysis of electrode current-potential relation, 57  impedance analysis, i.e. electrochemical impedance 58  spectroscopy (EIS), can provide informative picture for 59  electrode process kinetics, typically by theoretical model-based 60  analysis and electrical equivalent circuit-based modelling.78-81 In 61  addition, distribution of relaxation time (DRT) derived from EIS 62  can conveniently depict the time-resolved electrode processes 63  including mass transfer, which has been increasingly attractive 64  in recent years.82-85 For more about the history of EIS and 65  modelling strategies, the readership may refer to some 66  wonderful previous papers.86, 87 Apart from the above outline, 67  the interdisciplinary viewpoints on electrode processes in 68  recent years, for example, from material science with 69  consideration of electrode reconstruction kinetics and from 70  data science by using Bayesian optimisation, have provided 71  deeper insights into both electrode reaction activity and 72  stability.88-91 73Even though many achievements are listed above, we do not 74  aim to include all important contributions, and certain opinions 75  expressed may be subjective. The readership may refer to some 76  previous papers to obtain more details of the history of 77  electrode processes.92, 93 Due to the limited length of this 78  Tutorial Review, we do not attempt to discuss too much of 79  selectivity issue in (electro)kinetic process and the readership 80  may be directed to previous review papers.94-96 In addition, the 81  readership may refer to some attractive papers/books for the 82  corresponding topics to get more details if they are interested 83  in, such as active sites in electrocatalysis,97, 98 microkinetic 84  modelling in (electro)catalysis,96, 99 multi-electron transfer 85  theory,100-102 hierarchical modelling for electrode processes 86  including thermodynamics, multistep kinetics, mass transport, 87  and ELD effects,103 first-principles calculations for 88  electrochemical interfaces,104-108 first-principles-supported 89  microkinetic models,109-111 mixed quantum-classical electron 90  transfer,112 transient electrochemistry,113, 114 multi-level 91  descriptors,115, 116 and machine learning-based methods.117, 118 92  For a systematic knowledge about electrode processes, we 93  recommend the books by Bockris or Schmickler and their 94  collaborators.119-121 95Based on the above overview on the long history of electrode 96  process, on-going rapid development, and the potentials for 97  wide applications in the future, now is the best timing to 98  organise typical and basic knowledge on electrode process 99  (focusing on electrocatalysis) and to highlight recent typical 100  advances, including emerging theories and novel 101  methodologies, with our own perspective. This review takes 102ARTICLE Journal Name2 \| J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xxPlease do not adjust marginsPlease do not adjust marginselectrified electrode-electrolyte interface, comprising reacting 1  species, reaction kinetics, and electrode materials, as the 2  framework. To control the article length and to enable readers 3  to follow, we do not provide extensive discussions. First, we 4  introduce reaction kinetics based on classical considerations 5  and graph-theory viewpoint, with discussion on the rate-6  determining assumption and non-rate-determining frameworks. 7  Next, we examine the influence of reactant and product mass 8  transport on electrode reactions, with the discussion on the 9  rigorous conditions for applying the Koutecký-Levich equation 10  and corresponding experimental validation methods. 11  Technique for probing dynamic local pH during an electrode 12  reaction using RRDE is also introduced. Then, we introduce 13  recent progress in quantitative studies of electrode material 14  reconstruction, as well as the reaction stability. In the next part, 15  we provide a brief overview of fundamentals of a data-driven 16  method, i.e. data assimilation, and highlight our group’s recent 17  representative work integrating data assimilation with 18  investigations on activity and stability issues of electrode 19  processes. Finally, we propose electrode process matrix as a 20  heuristic framework to contain any state of an electrode 21  process and provide another perspective to understand 22  electrode process in the electrified electrode-electrolyte 23  interface. 242 Electrode processes 252\.1 Electrode reaction 26Electrochemistry features a high sensitivity to the reaction and 27  the real-time electrode reaction rates under various conditions 28  can be easily recorded by monitoring the currents, in which 29  generally builds current-potential plots. An electrode reaction 30  includes the time-dependent change (e.g. transient response) 31  and time-independent process (the steady state). In this section, 32  under the steady-state approximation, we will introduce a basic 33  description of electrode process, rate-determining step, non-34  rate determining assumption-based analysis, a framework of 35  understanding of electrode reactions, and the viewpoint on 36  mechanisms based on graph theory. All the discussions are 37  based on mean-field model. As for active sites with spatial 38  inhomogeneity and transient electrode processes, the 39  readership may refer to some previous literature.122-127 For the 40  influences of environmental species like cation/anion effects, 41  the readership may refer to previous reviews.30, 37, 103, 128, 129 422\.1.1 Bulter-Volmer framework 43As the result of applied external potential at an electrified 44  electrode-electrolyte interface, electrode reaction driven by 45  electron transfer (ET) and cation transfers such as proton 46  transfer (PT) features the driven force and energy barrier that 47  are easily regulated by tuning electric properties governed by 48  potential controlling. According to Faraday's laws of 49  electrolysis,130-132 the rate of an electron reaction can be 50  quantified by the measured current (density). Potential with the 51  zero Gibbs free energy for a reaction is regarded as the 52  equilibrium potential (Eeq), and the deviation of the potential for 53  accelerating the reaction for a given current flowing the 54  electrode, is called overpotential (η).133 In a narrow sense, an 55  electrode reaction (excluding the mass transfer limitation) can 56  be described by the Butler-Volmer equation (also called Erdey-57  Grúz-Volmer equation) with an exponential relationship 58  between the current density (I) and overpotential (η). 48-50 A 59  typical anodic reaction with one-ET can be expressed as: 60大文字 I. イコール 、 大文字 I. 下付き 0 、 始めかっこ e じょう 、 分子 、   ベータ   a.   大文字 F エータ 最後 分子 , 割る 大文字 R 大文字 T 最後 上付き文字 マイナス e じょう 、 マイナス 分子 、 ベータ 下付き c 、 大文字 F エータ 最後 分子 , 割る 大文字 R 大文字 T 最後 上付き文字 、 , 終わりかっこ 𝐼 =  𝐼0 (𝑒 𝛽𝑎𝐹𝜂 𝑅𝑇− 𝑒− 𝛽𝑐𝐹𝜂 𝑅𝑇)(1) 61in which  is the exchange current density,   is the symmetry 62  factor for the forward reaction,   is the symmetry factor for 63  the backward reaction, and the sum of   and   is assumed to 64  be the unity; F is Faraday constant, R is gas constant, T is the 65  absolute temperature during the reaction. 66大文字 I. 下付き 0 𝐼0ベータ 下付き a. 𝛽𝑎ベータ 下付き c 𝛽𝑐ベータ 下付き a. 𝛽𝑎ベータ 下付き c 𝛽𝑐If the overpotential is relatively large (e.g., \> 120 mV), the 67  backward reaction is ignorable, and the above equation can be 68  simplified as: 69大文字 I. イコール 、 大文字 I. 下付き 0 、 e じょう 、 分子 、 ベータ 下付き a. 、 大文字 F エータ 最後 分子 , 割る 大文字 R 大文字 T 最後 上付き文字 𝐼 =  𝐼0𝑒 𝛽𝑎𝐹𝜂 𝑅𝑇(2) 70Here we demonstrates several typical polarization curves for an 71  anodic reaction (Fig. 2). It shows that the related parameters 72  have large influence on the electrode reaction rate, especially 73  by the symmetry factor. Given that the symmetry factor is 74  related to the potential-induced change of free energy of 75  activation, tuning this energy enables a significant improvement 76  in electrode reaction rate. Different from outer-sphere 77  electrode reaction without chemical bonding between reactant 78  and electrode, inner-sphere electrode reaction features strong 79  interaction between reactant and electrode to form chemical 80  bonding, which can readily influence the free energy of 81  activation. Therefore, the inner-sphere electrode reaction rate 82  is highly regulated by electrode surface states. These surface 83  states can be effortlessly tuned through approaches like the 84  design of electrode materials and thus favours the catalysis 85  process for the related electrochemical reaction, i.e. 86  electrocatalysis. 87It is known that the Butler-Volmer framework assumes a linear 88  dependence of the free energy of activation for ET on the 89  potential, and this can be understood by the Taylor series 90  expansion (first order) of the free energy of activation with 91  respect to the reference point (e.g., the equilibrium 92  potential).134 This causes that in principle the Butler-Volmer 93  framework should be applied at potentials close to the 94  reference point.134, 135 However, the Butler-Volmer framework 95  works well in most cases for electrode reactions in a wide 96  potential range (even though still “narrow”).135-137 This may be 97  benefited from the equal derivatives of the free energy of 98  activation due to the Brønsted-Evans-Polanyi relation.138, 139 99100Fig. 2 Simulated polarization curves for anodic reactions by eq. 101  2. (a) Various I0 (0.01, 0.03, 0.10 mA cm-2), with other 102  parameters fixed (βa = 0.5, T = 298.15 K). (b) Various βa (0.3, 0.5, 103Journal Name  ARTICLEThis journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 \| 3Please do not adjust marginsPlease do not adjust margins0\.7), with other parameters fixed (I0 = 0.01 mA cm-2, T = 298.15 1  K). 22\.1.2 Rate-determining assumption 3Many electrode reactions with high-value products are multi-4  electron transfer processes.140, 141 Although “rate-determining” 5  concept is theoretically and experimentally non-trivial,142-144 it 6  is a typical assumption that one reaction step imposes an upper 7  limit on the overall rate, much “slower” than other reaction 8  steps, and thus called rate-determining step (RDS) in a 9  reaction.145, 146 A similar concept is called potential-determining 10  step (PDS) by taking the step with the maximum thermal 11  dynamic barrier as the limiting step in the whole reaction.147, 148 12  Although there is still debate on the RDS concept and always 13  contains the risk of misleading on the interpretation of 14  observations,149 the RDS-based approximation works well in 15  describing many electrode reactions and benefits our first-step 16  understanding of the electrode kinetics.150, 151 Herein, we 17  showcase a general case of a n-ET multistep oxidation reaction 18  with single RDS and other steps are in quasi-equilibrium 19  (chemical steps are in arbitrary): 20大文字 A. 、 ⇌ 大文字 B プラス 、 e じょう マイナス 𝐴 ⇌ 𝐵 + 𝑒−(ET step 1) 21大文字 B 、 ⇌ 大文字 C プラス 、 e じょう マイナス 𝐵 ⇌ 𝐶 + 𝑒−(ET step 2) 22大文字 D 、 ⇌ 大文字 E 𝐷 ⇌ 𝐸(Chemical step 1) 2324垂直の省略記号 ⋮垂直の省略記号 、 ⋮垂直の省略記号 ⋮大文字 P 、 ⇌ 大文字 R プラス 、 e じょう マイナス 𝑃 ⇌ 𝑅 + 𝑒−(ET step γ) 25大文字 R 、 右向き矢印 大文字 S プラス 、 上付き文字 てい , r e , 最後 てい , じょう マイナス 𝑅 → 𝑆 + 𝑟𝑒−(RDS repeated ν times) 2627垂直の省略記号 ⋮垂直の省略記号 ⋮垂直の省略記号 ⋮大文字 Y 、 ⇌ 大文字 Z プラス 、 e じょう マイナス 𝑌 ⇌ 𝑍 + 𝑒−(ET step ne) 28The overall electrode current density I is: 29大文字 I. イコール 、 大文字 I. 下付き 0 、 始めかっこ e じょう 、 分子 、 アルファ 下付き a. 、 大文字 F エータ 最後 分子 , 割る 大文字 R 大文字 T 最後 上付き文字 マイナス e じょう 、 マイナス 分子 、 アルファ 下付き c 、 大文字 F エータ 最後 分子 , 割る 大文字 R 大文字 T 最後 上付き文字 、 , 終わりかっこ 𝐼 =  𝐼0 (𝑒 𝛼𝑎𝐹𝜂 𝑅𝑇− 𝑒− 𝛼𝑐𝐹𝜂 𝑅𝑇)(3) 30アルファ 下付き a. イコール ガンマ 割る ニュー プラス r ベータ 𝛼𝑎 = 𝛾 𝜈+ 𝑟𝛽(4) 31アルファ 下付き c イコール 分子 、 n マイナス ガンマ 最後 分子 , 割る ニュー マイナス r ベータ 𝛼𝑐 = 𝑛−𝛾 𝜈− 𝑟𝛽(5) 32where r is the number of electrons transferred in the RDS (r: 0 33  for chemical step, 1 for ET step),   and   are transfer 34  coefficients for the forward and backward reactions, 35  respectively. More details of the mathematical consideration 36  can be found in a previous literatures.120, 152 37アルファ 下付き a. 𝛼𝑎アルファ 下付き c 𝛼𝑐The equation above for the overall anodic current density is a 38  more general form of the Butler-Volmer equation. Apart from 39  the single RDS assumption, another prerequisite for the 40  rigorous application of the general Butler-Volmer equation is 41  that the vacant active site remains constant in the whole 42  potential range. 43In addition, the definition of transfer coefficient is 44  recommended based on a directly measured experimental 45  quantity irrespective of any possible mechanisms by IUPAC:153 46アルファ 下付き a. イコール 左小かっこ 大文字 R 大文字 T 割る 大文字 F 右小かっこ 左小かっこ 分子 、 d l n 大文字 I. 下付き a. 最後 分子 , 割る d 大文字 E 右小かっこ 𝛼𝑎 = ( 𝑅𝑇 𝐹)( 𝑑𝑙𝑛𝐼𝑎 𝑑𝐸)(6) 47アルファ 下付き c イコール マイナス 左小かっこ 大文字 R 大文字 T 割る 大文字 F 右小かっこ 左小かっこ 分子 、 d l n 縦線 大文字 I. 下付き c 、 縦線 最後 分子 , 割る d 大文字 E 右小かっこ 𝛼𝑐 = −( 𝑅𝑇 𝐹)( 𝑑𝑙𝑛\|𝐼𝑐\| 𝑑𝐸)(7) 48in which Ia and Ic are anodic and cathodic current densities, 49  respectively; E is the potential. Clearly, the mathematical form 50  of the transfer coefficients derived by the general Bulter-51  Volmer and recommended by IUPAC are the same. 52By taking the above single RDS model with the related transfer 53  coefficient (α), we can use the experimentally determined Tafel 54  slope (b) to extract some kinetic information: 55b イコール 、 、 分子 、 大文字 R 、 大文字 T 最後 分子 , 割る ぶんぼ 、 アルファ 大文字 F 、 下付き文字 てい , l o g , 最後 てい , 下付き 10 、 e 最後 ぶんぼ 𝑏 =   𝑅 𝑇 𝛼𝐹 𝑙𝑜𝑔10𝑒(8) 56where R is gas constant, T is absolute temperature, α is transfer 57  coefficient, F is Faraday constant, and e is Euler's number. For 58  example, under the framework of the single RDS model as 59  introduced above, the Tafel slope around 120 mV dec-1 60  indicates that the 1st ET step serves as the RDS; a value close to 61  40 mV dec-1 suggests the RDS for the 2nd ET step. The typical 62  Tafel slopes for a 4-ET electrode reaction are summarized in 63  Table 1. Note that reliable analysis of Tafel slope requires the 64  exclusion of mass transfer effects, especially for gas-evolving or 65  low-concentration reactions.154 In addition, as mentioned 66  above, the Tafel analysis introduced here implicitly assumes a 67  constant coverage of vacant site. The readership may refer to a 68  previous paper of Tafel analysis with considering coverage 69  change for several typical electrocatalytic reactions.155 70Table 1. Typical Tafel slopes in a 4-ET electrode reaction with a 71  single RDS. The symmetry factors (β) for all the ET steps are set 72  as 0.5 here. Note that the symmetry factor is not necessarily 0.5 73  and in the range of (0,1).112 74RDSaγνrβαTafel slope (mV dec-1)1 CT,ET0100\.50∞1 ET,CT0110\.50\.51201 ET-CT0110\.50\.51202 CT,ET1100\.51602 ET,CT1110\.51\.5402 ET-CT1110\.51\.5403 CT,ET2100\.52303 ET,CT2110\.52\.5243 ET-CT2110\.52\.5244 CT,ET3100\.53204 ET,CT3110\.53\.5174 ET-CT3110\.53\.517aCT: chemical transfer; ET: electron transfer; n CT,ET (n = 1, 2, 3, 75  or 4): RDS of CT followed by ET in the nth ET step; n ET,CT: RDS 76  of ET followed by CT in the nth ET step; n ET-CT: RDS of ET-77  coupled CT in the nth ET step. 782\.1.3 Reaction space and reaction mechanism 79In one electrode reaction with multi-electron transfer, some 80  “product” in one step, like the generated intermediate, 81  participates in the subsequent reaction step as “reactant”; or 82  reactant in the forward reaction serve as product in the 83  backward reaction. In this sense, “reactant” and “product” are 84  different roles for the same entity, i.e., they are the unity of 85  opposites. To make it clear, in this paper we define the reactant 86  and product as the initial and final species for a given reaction, 87  respectively, and use intermediate to differentiate the short-88  lifetime generated key species for a multi-step reaction. 89  Geometrically, we can grasp electrode reaction by the following 90  viewpoints: 1) reactant and product are points (zero dimension); 91  2) one pair of reactant (or intermediate with other produced 92  species) and product (or intermediate with other produced 93  species) forms the elementary step (one dimension); 3) a 94ARTICLE Journal Name4 \| J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xxPlease do not adjust marginsPlease do not adjust marginssequence of elementary steps constitutes a reaction path to 1  generate the final product from the reactant (one dimension); 2  4) multiple reaction paths build a reaction network to generate 3  (various) products from the same reactant (two dimensions); 5) 4  different reaction networks intertwine, with various reactants 5  and products, including all the possible reactions for a given 6  electrode condition, forming the reaction space (three 7  dimensions). A schematic diagram is shown in Fig. 3. 89Fig. 3 The schematic diagrams for reactant/product, elementary 10  step, reaction path, reaction network, and reaction space. R: 11  reactants, P: products, I: intermediates. For electrocatalysis, the 12  vacant site is hidden for simplification thus the electrocatalytic 13  cycle is not shown. The dash lines are possible transformations 14  for the intermediates. 15Given that a chemical reaction (e.g., as shown above) can be 16  topologically demonstrated, the graph theory for chemical 17  kinetics has been well developed especially for linear reactions 18  containing at most one intermediate on one or both sides of the 19  elementary step.156-161 As pointed out by Temkin, in a linear 20  reaction graph the vertices denote intermediates while the 21  edges represent for elementary steps (an example is shown in 22  Fig. 4a).159, 160 Based on the kinetic graphs, the rate expression 23  of a reaction network can be directly handwritten by following 24  a routine method. To control the length of this paper, we do not 25  show details here and the readership may refer to previous 26  papers such as by Temkin and by Kozuch.161, 162 An extension of 27  the graph theory for chemical kinetics is also applied to 28  electrode reaction like the oxygen evolution reaction (OER).135  29For a reaction, one of the most important is how to clarify the 30  mechanism. However, the term mechanism somewhat seems 31  “vague” as explained previously: “A detailed description of the 32  process leading from the reactants to the products of a reaction, 33  including a characterization as complete as possible of the 34  composition, structure, energy and other properties of reaction 35  intermediates, products and transition states”.5 By using the 36  graph theory for chemical kinetics, we can re-understand a 37  reaction mechanism with a clear mathematical concept as “the 38  subgraph of the network that consists of only one cycle, but still 39  includes all the intermediates, even if they are dead-ends of the 40  cycle”.161 As demonstrated in Fig. 4b, a network with eight 41  intermediates contains two catalytic cycles (i.e. 1→2→7→5→42  6→1 and 1→2→3→4→5→6→1). These two cycles with 43  pendant vertices (dead-ends) lead to five different catalytic 44  mechanisms. It is clear that for each mechanism even though 45  the pendant vertices do not participate in the reaction directly, 46  they still influence the overall reaction. This reminds us that, in 47  some cases, not only the intermediates/species lies within the 48  reaction path but also the pendant intermediates can influence 49  the overall reaction for activity, selectivity, and even stability. 50  An attractive work of stability description directed by Ooka et al. 51  introduces the deactivation of the catalyst during the OER 52  cycle,163 which, in our opinion, is understandable by considering 53  pendant vertices in a reaction network based on the graph 54  theory. 5556Fig. 4 (a) Kinetic graph of a three-step catalytic reaction, with 57  the vacant site included as the intermediate 1. (b) A kinetic 58  network with eight intermediates (the vacant site is included), 59  including two catalytic cycles (1→2→7→5→6→1, 60  1→2→3→4→5→6→1). Five mechanisms were extracted (right 61  side). 622\.1.4 Non-rate-determining-step framework 63As mentioned in the above part, the RDS-based framework 64  enables simple and effective quantification of an electrode 65  process. However, as it is well-known, the precondition, i.e. the 66  existence of RDS, is not always satisfied as RDS is a very strong 67  assumption. Again, we emphasise here that the RDS assumption 68  could mislead the conclusion of our observation (See the 69  discussions provided by Enyo in 1960s-70s). In comparison to a 70  rigorous kinetic analysis (without any approximation) replete 71  with laborious mathematical treatments, another approach by 72  using a relatively “weaker” assumption than RDS,142, 164-169i.e. 73  steady-state approximation (SSA), has been widely adopted in 74  chemical kinetics.170 75The assumption of SSA is that the change rates of intermediates 76  are close to zero during the reaction period. For a reaction from 77  A to the product B via an intermediate X: 78A 、 ⟺X 、 ⟺B A ⟺ X ⟺ B79It includes the consumption of X by the backward reaction of 80  the 1st step and the forward reaction of the 2nd step, while the 81  production of X is by the forward reaction of the 1st step and the 82  backward of the 2nd step. So that the change of X concentration 83  (here denoted \[X\]) with time (t) can be expressed as following 84  based on SSA: 85分子 、 d 左大かっこ 大文字 X 右大かっこ 最後 分子 , 割る d t イコール k 下付き 1 f 、 左大かっこ 大文字 A. 右大かっこ プラス k 下付き 2 b 、 左大かっこ 大文字 B 右大かっこ マイナス k 下付き 1 b 、 左大かっこ 大文字 X 右大かっこ マイナス k 下付き 2 f 、 左大かっこ 大文字 X 右大かっこ 近似的に等しい 0 𝑑\[𝑋\] 𝑑𝑡= 𝑘1𝑓\[𝐴\] + 𝑘2𝑏\[𝐵\] − 𝑘1𝑏\[𝑋\] − 𝑘2𝑓\[𝑋\] ≈ 0(9) 86Given a serial reaction path with first-order kinetics with n 87  surface intermediates (including the vacant site) and the 88  conservation of surface coverages (θj , j = 1, 2, ..., n), a set of 89  linear equations can be obtained: 90∑ (      (10) 91和 から j イコール 1 へ n の 、 始めかっこ k 下付き j i. 、 シータ 下付き j マイナス k 下付き i. j 、 シータ 下付き i. 、 , 終わりかっこ イコール 0 最後 総和 、 、 𝑘𝑗𝑖𝜃𝑗 − 𝑘𝑖𝑗𝜃𝑖) = 0𝑛 𝑗=1∑         (11) 92和 から j イコール 1 へ n の シータ 下付き j 、 イコール 1 𝜃𝑗 𝑛 𝑗=1 = 1in which θi refers to the surface coverage of the intermediate in 93  the nearest neighbour position for the jth intermediate (note 94Journal Name  ARTICLEThis journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 \| 5Please do not adjust marginsPlease do not adjust marginsthere are two nearest neighbour positions for each 1  intermediate), kji or kij represents the reaction constant (the 2  activities of other species are assumed as unity) in the 3  elementary step from the jth or ith to the ith or jth intermediate, 4  which is related to the applied potential. Due to the SSA, the 5  reaction rate can be expressed as (if we set the direction from 6  the jth to ith intermediate as the forward): 7r イコール 、 k 下付き j i. 、 シータ 下付き j マイナス k 下付き i. j 、 シータ 下付き i. 𝑟 =  𝑘𝑗𝑖𝜃𝑗 − 𝑘𝑖𝑗𝜃𝑖(12) 8Then by combining the above equations the expression of 9  reaction rate can be obtained without any intermediate term. 10It is evident that there is no RDS assumption in the SSA analysis. 11  Thus, in principle all the kinetic parameters can be extracted 12  from the experimental data, even though, it is not easy to pick 13  the “real” ones because of the usually non-unique solution 14  space. One previous research demonstrates the deconvolution 15  of rate constants and pre-exponential factors for all the 16  elementary steps in the OER from the polarization curve.171 This 17  makes it possible to explore more microkinetic details during 18  the oxygen evolution with each forward and backward 19  elementary step, as compared to the RDS approach that the 20  information is lost after the RDS. To facilitate the reader’s 21  understanding, we give another example for the OER with the 22  widely used adsorbate evolution mechanism (AEM) in 1 M KOH 23  (4OH- + \* → 3OH- + \*OH + e- → 2OH- + \*O + H2O + 2e- → OH- + 24  \*OOH + H2O + 3e- → \* + O2 + 2H2O + 4e-). The reaction constant 25  in each step is further expanded based on transition-state 26  theory:172-174 27(13) 28k 下付き j i. イコール 、 カッパ 、 分子 、 k 下付き 大文字 B 、 大文字 T 最後 分子 , 割る h 、 e じょう 、 マイナス 分子 、 大文字 デルタ 大文字 G 下付き j i. じょう ノット イコール 最後 分子 , 割る 大文字 R 大文字 T 最後 上付き文字 𝑘𝑗𝑖 =  𝜅 𝑘𝐵𝑇 ℎ𝑒− Δ𝐺𝑗𝑖 ≠ 𝑅𝑇in which  is transmission coefficient (commonly equals unity), 29   is Boltzmann constant,  is Planck constant,   is the free 30  energy of activation (here linearly influenced by the applied 31  potential). The electrocatalytic current is finally obtained by 32  considering the four-electron-transfer process of OER, i.e. one 33  electrocatalytic cycle produces four electrons with release of 34  one oxygen molecule. The above steady-state microkinetic 35  analysis successfully modelled the polarization curve well (the 36  bottom-right insert in Fig. 5). Furthermore, the free-energy 37  diagram (here shows the potential E = 1.23 V vs. RHE) for the 38  AEM pathway with free energies of activation in each step 39  resolved (Fig. 5). For more details the readership may refer to 40  the previous paper.135 As suggested above, it should be kept in 41  mind that there may be non-unique solution problem due to 42  more parameters in the non-RDS model as compared to the 43  simplified RDS approach. One promising way to solve this 44  problem is to use different electrochemical data sources, i.e. 45  multidimensional data sources, to obtain reliable and unique 46  model parameter set.175 47カッパ 𝜅k 下付き 大文字 B 𝑘𝐵h ℎ大文字 デルタ 大文字 G 下付き j i. じょう ノット イコール Δ𝐺𝑗𝑖 ≠In recent years, an approximate framework, called energetic 48  span model, has been attractive for the new insight into the 49  “rate-determining” concept by considering the energy 50  difference between the key states (not the reaction step).176-179 51  The total reaction rate (indicated by turnover frequency, TOF) 52  can be approximated as: 53(14) 54大文字 T 大文字 O 大文字 F 近似的に等しい 、 、 分子 、 k 下付き b 、 大文字 T 最後 分子 , 割る h 、 e じょう 、 マイナス デルタ 大文字 E 割る 大文字 R 大文字 T 最後 上付き文字 𝑇𝑂𝐹 ≈   𝑘𝑏𝑇 ℎ𝑒−𝛿𝐸/𝑅𝑇where kb is Boltzmann constant, h is Planck constant, T is 55  absolution temperature, R is gas constant, and δE is energetic 56  span. A schematic diagram is shown for a four-step reaction to 57  determine δE (Fig. 5). This energetic span concept is also 58  applicable to electrocatalysis, which differs from the 59  consideration of the maximum standard free energy ( ) as 60  PDS.180 For oxygen electrolysis, including OER and oxygen 61  reduction reaction (ORR), the optimal adsorption free energies 62  of \*O ( ) are almost the same by δE and   63  considerations in large overpotentials (\> 300\~400 mV); however, 64  in the low overpotential region, the δE consideration shows 65  potential-dependent trends while   still constant. In a 66  recent work, we applied the energetic span model to simplifying 67  the OER microkinetics and proposed a degradation model to 68  quantify the OER degradation process.91 To establish this model, 69  Taylor polynomial was used to approximate the relation 70  between the energetic span δE and the potential (U) at a fixed 71  potential U0: 72増分 大文字 G 下付き m a. x じょう 0 ∆𝐺𝑚𝑎𝑥 0増分 大文字 G 下付き 、 大文字 O アスタリスク 最後 下付き文字 ∆𝐺𝑂∗増分 大文字 G 下付き m a. x じょう 0 ∆𝐺𝑚𝑎𝑥 0増分 大文字 G 下付き m a. x じょう 0 ∆𝐺𝑚𝑎𝑥 0デルタ 大文字 E 近似的に等しい デルタ 大文字 E 始めかっこ 大文字 U 下付き 0 、 , 終わりかっこ プラス 部分 デルタ 大文字 E 割る 部分 大文字 U 、 縦線 下付き 、 大文字 U 下付き 0 、 最後 下付き文字 、 左小かっこ 大文字 U マイナス 大文字 U 下付き 0 右小かっこ 𝛿𝐸 ≈ 𝛿𝐸(𝑈0) + 𝜕𝛿𝐸 𝜕𝑈\|𝑈0 (𝑈 − 𝑈0)(15) 73This resulted in analogous Butler-Volmer framework, but the 74  physicochemical meaning was obviously different especially 75  from the RDS assumption used in the general form of the Butler-76  Volmer equation (eq. 3). 7778Fig. 5 (a) Free-energy diagram (E = 1.23 V vs. RHE) for the AEM 79  pathway to release oxygen molecules, and the free energies of 80  activation as well as the reaction free energies are extracted 81  from the microkinetic modelling based on steady-state 82  approximation without RDS assumption. The bottom-right 83  insert shows the polarization curve for OER (noted as “raw 84  data”), which is fitted by the steady-state model (grey line). 85  Reproduced from Ref. 135 with permission, Copyright 2025, 86  Elsevier B.V. (b) Energy profile of a catalytic cycle. TDI: TOF-87  determining intermediate, TDTS: TOF-determining transition 88  state. The energetic span is determined by the difference 89  between TDTS and TDI. Note that TDTS and TDI may be in 90  different elementary steps. 91Apart from the energetic span model, similar considerations 92  based on microkinetic frameworks have been proposed by 93  other researchers in heterogenous catalysis, such as the 94ARTICLE Journal Name6 \| J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xxPlease do not adjust marginsPlease do not adjust marginsreaction can be controlled by the transition state with the 1  highest free energy,181 or the concept of an effective barrier to 2  model the overall catalytic reaction.182-184 A recent work by 3  Huang demonstrates another attractive viewpoint to consider 4  the “rate-determining” with a general concept: rate-5  determining term (RDT), incorporating the detailed kinetics and 6  thermodynamics in multistep electrocatalytic reactions based 7  on SSA modelling. A unified formalism for RDT with first-order 8  kinetics is proposed by decomposing the overall reaction 9  resistance -neρ/jss: 10マイナス n e ロー 割る j 下付き s s 、 近似的に等しい 和 から i. イコール 1 へ 大文字 N プラス 1 の 大文字 シータ 下付き i. 割る ぶんぼ 、 k 下付き i. じょう プラス 、 最後 ぶんぼ − 𝑛𝑒𝜌 𝑗𝑠𝑠≈ ∑ Θ𝑖 𝑘𝑖 +𝑁+1 𝑖=1(16) 11大文字 シータ 下付き i. イコール 分子 、 和 から 太字斜体 j イコール 太字 0 へ 太字斜体 大文字 N の せき から 太字斜体 s イコール 太字 1 へ 太字斜体 j の 大文字 K 下付き 、 太字斜体 大文字 P 左小かっこ i. ,プラス s 右小かっこ 最後 下付き文字 、 最後 分子 , 割る ぶんぼ 、 せき から 太字斜体 s イコール 太字 1 へ 太字斜体 大文字 N の 大文字 K 下付き 、 太字斜体 大文字 P 左小かっこ i. ,プラス s 右小かっこ 最後 下付き文字 、 最後 ぶんぼ Θ𝑖 = ∑ ∏ 𝐾𝑷(𝑖,+𝑠) 𝒋 𝒔=𝟏 𝑵 𝒋=𝟎 ∏ 𝐾𝑷(𝑖,+𝑠) 𝑵 𝒔=𝟏(17) 12where n is total number of electron transfer, e is electron charge, 13  ρ is the number of density of active sites,   is the rate constant 14  of the forward reaction in the ith elementary step, K is the 15  equilibrium constant of the ith elementary step,  is the 16  permutation operators for the electrocatalytic cycle. Usually, 17  there is one resistance term is much larger than the rest in eq. 18  16 and this unique term determines the overall reaction 19  resistance, which is defined as the RDT. For the details the 20  readership may refer to the original paper.185  21k 下付き i. じょう プラス 𝑘𝑖 +太字斜体 大文字 P 左小かっこ i. ,プラス s 右小かっこ 𝑷(𝑖,+𝑠)2\.2 Mass transfer effects 22The above discussion focuses solely on electrode reaction itself; 23  however, a complete electrocatalytic process involves multiple 24  steps, including the transport and adsorption of reactants, the 25  reaction, and the desorption/release of products. Therefore, 26  when considering a comprehensive electrode process, the 27  influence of mass transfer cannot be neglected. It is well 28  established that mass transfer encompasses convection, 29  diffusion, and migration. Since convection can be readily 30  controlled and migration can be suppressed (e.g., by employing 31  a high concentration of supporting electrolyte), this part will 32  focus exclusively on the effects of diffusion on electrode 33  reactions. We will introduce fundamental aspects and recent 34  representative advances from the perspectives of both reactant 35  transfer and product transfer. The full handling of mass transfer 36  effects can be achieved by computational fluid dynamics and 37  this is beyond our scope. 382\.2.1 Reactant transfer 39For an electrode reaction the reactant has limited concentration, 40  or the reaction rate is fast, the local concentration of the 41  reactant will form spatial and temporal distribution. In this 42  paper we focus on the spatial distribution-based diffusion 43  process, i.e., the steady-state diffusion. For an electrode 44  reaction, the reactant in the bulk (Rb) diffuses towards the 45  electrode surface (Rs) and forms the product (P) with releasing 46  n electrons (here we demonstrate anodic reaction): 47R 下付き b 、 、 右向き矢印 、 R 下付き s 、 、 右向き矢印 P プラス n e じょう マイナス 、 、 Rb  →  Rs  → P + 𝑛𝑒−48The diffusion flux Jd of the reactant from the bulk to the surface 49  is described according to Fick’s first law: 50(18) 51大文字 J 下付き d イコール マイナス 大文字 D 下付き 大文字 R 、 、 分子 、 d 大文字 C 下付き 大文字 R 最後 分子 , 割る d x 𝐽𝑑 = −𝐷𝑅 𝑑𝐶𝑅 𝑑𝑥where DR is the diffusion coefficient of the reactant, CR is the 52  concentration of the reactant,   is the concentration gradient. 53  For an ideal diffusion under steady state, each position has the 54  same diffusion flux, so that 55分子 、 d 大文字 C 下付き 大文字 R 最後 分子 , 割る d x 𝑑𝐶𝑅 𝑑𝑥大文字 J 下付き d 、 縦線 下付き 、 x 下付き 1 、 最後 下付き文字 イコール マイナス 大文字 D 下付き 大文字 R 、 、 分子 、 d 大文字 C 下付き 大文字 R 最後 分子 , 割る d x 、 縦線 下付き 、 x 下付き 1 、 最後 下付き文字 イコール 大文字 J 下付き d 、 縦線 下付き 、 x 下付き 2 、 最後 下付き文字 イコール マイナス 大文字 D 下付き 大文字 R 、 、 分子 、 d 大文字 C 下付き 大文字 R 最後 分子 , 割る d x 、 縦線 下付き 、 x 下付き 2 、 最後 下付き文字 𝐽𝑑\|𝑥1 = −𝐷𝑅 𝑑𝐶𝑅 𝑑𝑥\|𝑥1 = 𝐽𝑑\|𝑥2 = −𝐷𝑅 𝑑𝐶𝑅 𝑑𝑥\|𝑥2(19) 56This leads to a linear distribution of concentration along the 57  diffusion direction: 58大文字 J 下付き d イコール 大文字 D 下付き 大文字 R 、 、 分子 、 大文字 C 下付き 、 大文字 R 下付き b 、 最後 下付き文字 マイナス 大文字 C 下付き 、 大文字 R 下付き s 、 最後 下付き文字 最後 分子 , 割る l 𝐽𝑑 = 𝐷𝑅 𝐶𝑅𝑏 −𝐶𝑅𝑠 𝑙(20) 59where   is the reactant concentration in the bulk,   is the 60  reactant concentration on the electrode surface, l is the 61  thickness of the diffusion layer. 62大文字 C 下付き 、 大文字 R 下付き b 、 最後 下付き文字 𝐶𝑅𝑏大文字 C 下付き 、 大文字 R 下付き s 、 最後 下付き文字 𝐶𝑅𝑠However, in a real system part of the diffusion region overlaps 63  (forced) convection region. As a result, the distribution is not 64  always linear from diffusion to convection region. To this end, 65  an effective thickness (δ) of the diffusion layer (Fig. 6a) is used 66  and the diffusion flux can be simply expressed as: 67大文字 J 下付き d イコール 大文字 D 下付き 大文字 R 、 、 分子 、 大文字 C 下付き 、 大文字 R 下付き b 、 最後 下付き文字 マイナス 大文字 C 下付き 、 大文字 R 下付き s 、 最後 下付き文字 最後 分子 , 割る デルタ 𝐽𝑑 = 𝐷𝑅 𝐶𝑅𝑏 −𝐶𝑅𝑠 𝛿(21) 68Considering the electrode reaction with current density Je and 69  the steady state, we have: 70大文字 J 下付き e イコール n 大文字 F k 下付き e 、 大文字 N 大文字 C 下付き 、 大文字 R 下付き s 、 最後 下付き文字 イコール n 大文字 F 大文字 J 下付き d 𝐽𝑒 = 𝑛𝐹𝑘𝑒𝑁𝐶𝑅𝑠 = 𝑛𝐹𝐽𝑑(22) 71where ke is the reaction rate constant tuned by potential, N is 72  the active site density. Combine eq. 21 and eq. 22, we obtain 73  the Koutecký-Levich (K-L) equation:52, 53 741 割る 大文字 J 下付き e 、 イコール 1 割る 大文字 J 下付き k 、 プラス 1 割る 大文字 J 下付き l 、 1 𝐽𝑒= 1 𝐽𝑘+ 1 𝐽𝑙(23) 75大文字 J 下付き k イコール n 大文字 F k 下付き e 、 大文字 N 大文字 C 下付き 、 大文字 R 下付き b 、 最後 下付き文字 𝐽𝑘 = 𝑛𝐹𝑘𝑒𝑁𝐶𝑅𝑏(24) 76大文字 J 下付き l イコール 下付き文字 てい , n 大文字 F 大文字 D , 最後 てい , 下付き 大文字 R 、 、 大文字 C 下付き 、 大文字 R 下付き b 、 最後 下付き文字 割る デルタ 𝐽𝑙 = 𝑛𝐹𝐷𝑅 𝐶𝑅𝑏 𝛿(25) 77where Jk is the kinetic current density, Jl is the limiting current 78  density by diffusion control. In a RDE (i.e. rotating disk electrode) 79  system, for the reactant species with low diffusivity the 80  effective thickness is formulated: 81デルタ イコール 上付き文字 てい , 0.620 , 最後 てい , じょう 、 マイナス 1 最後 上付き文字 、 大文字 D 下付き 大文字 R じょう 左小かっこ 1 割る 3 右小かっこ 、 ニュー じょう 、 1 割る 6 最後 上付き文字 、 オメガ じょう 、 マイナス 1 割る 2 最後 上付き文字 𝛿 = 0.620−1𝐷𝑅 1/3 𝜈1/6𝜔−1/2(26)  82where  is kinematic viscosity,  is angular rotation rate of the 83  electrode. 84ユプシロン 𝜐オメガ 𝜔The K-L equation has been widely used in the field of mass 85  transfer-influenced electrocatalytic reactions, such as the ORR 86  with oxygen dissolved in the electrolyte leading to “plateau” 87  current densities under high overpotentials due to the limited 88  supply of oxygen from the electrolyte to the electrode 89  surface.186 According to the above equations, it is clear that the 90  K-L equation-based method assumes the reaction of order as 91  unity.56, 187, 188 Thus, it is recommended to check the reaction 92  order (m) for rigorous use of K-L equation. One effective 93  method is based on the RDE system, as the equations shown 94  below: 95大文字 J 下付き e イコール n 大文字 F k 下付き e 、 大文字 N 大文字 C 下付き 左小かっこ 大文字 R 下付き s 右小かっこ じょう m イコール 大文字 B 下付き 1 、 大文字 C 下付き 左小かっこ 大文字 R 下付き s 右小かっこ じょう m 𝐽𝑒 = 𝑛𝐹𝑘𝑒𝑁𝐶𝑅𝑠 𝑚 = 𝐵1𝐶𝑅𝑠 𝑚(27) 96大文字 J 下付き e イコール n 大文字 F 大文字 D 下付き 大文字 R 、 、 分子 、 大文字 C 下付き 、 大文字 R 下付き b 、 最後 下付き文字 マイナス 大文字 C 下付き 、 大文字 R 下付き s 、 最後 下付き文字 最後 分子 , 割る ぶんぼ 、 上付き文字 てい , 0.620 , 最後 てい , じょう 、 マイナス 1 最後 上付き文字 、 大文字 D 下付き 大文字 R じょう 左小かっこ 1 割る 3 右小かっこ 、 ニュー じょう 、 1 割る 6 最後 上付き文字 、 オメガ じょう 、 マイナス 1 割る 2 最後 上付き文字 、 最後 ぶんぼ , イコール 大文字 B 下付き 2 、 左小かっこ 大文字 C 下付き 、 大文字 R 下付き b 、 最後 下付き文字 マイナス 大文字 C 下付き 、 大文字 R 下付き s 、 最後 下付き文字 右小かっこ オメガ じょう 、 1 割る 2 最後 上付き文字 𝐽𝑒 = 𝑛𝐹𝐷𝑅 𝐶𝑅𝑏 −𝐶𝑅𝑠 0.620−1𝐷𝑅 1/3 𝜈1/6𝜔−1/2= 𝐵2(𝐶𝑅𝑏 − 𝐶𝑅𝑠 )𝜔1/2(28) 97Combine the above two equations, we have: 98ln 、 大文字 J 下付き e 、 、 イコール m ln 2 番目の始めかっこ 大文字 C 下付き 、 大文字 R 下付き b 、 最後 下付き文字 マイナス 大文字 J 下付き e 割る ぶんぼ 、 大文字 B 下付き 2 、 オメガ じょう 、 1 ぶんぼ 2 最後 上付き文字 、 最後 ぶんぼ , , 2 番目のかっこ閉じ プラス l n 大文字 B 下付き 1 ln 𝐽𝑒 = 𝑚 ln (𝐶𝑅𝑏 − 𝐽𝑒 𝐵2𝜔 1 2) + 𝑙𝑛𝐵1(29) 99It is easy to calculate the reaction order (m) by deriving the 100  slope from the   plots under various 101  potentials. The readership may refer to previous literature for 102  more discussion on the reaction order issue in a RDE system.187 103ln 、 大文字 J 下付き e 、 、 \~ln 2 番目の始めかっこ 大文字 C 下付き 、 大文字 R 下付き b 、 最後 下付き文字 マイナス 大文字 J 下付き e 割る ぶんぼ 、 大文字 B 下付き 2 、 オメガ じょう 、 1 ぶんぼ 2 最後 上付き文字 、 最後 ぶんぼ , , 2 番目のかっこ閉じ ln 𝐽𝑒 \~ ln (𝐶𝑅𝑏 − 𝐽𝑒 𝐵2𝜔 1 2)In an electrode system, the same target product may be 104  generated by different reactants. On account of various 105  concentrations, reaction activities, and diffusion capabilities of 106  the reactants, the local concentrations on electrode surface will 107  decrease due to the continuous consumption during the 108Journal Name  ARTICLEThis journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 \| 7Please do not adjust marginsPlease do not adjust marginsreactions. This will cause potential-dependant reactant 1  evolution. One typical clue is the “stair” current in the middle of 2  the whole polarization curve because of the diffusion limitation, 3  such as the two reactants (proton and water molecule) in the 4  hydrogen evolution reaction (HER).189-191 Similar observation 5  was also discussed in OER. Shinagawa et al. found that in mild 6  pH conditions, there was “reactant switching” between H2O and 7  OH-, owing to the diffusion limitation of OH- as suggested by the 8  “plateau” regions under various rotation speeds by RDE 9  measurements (Fig. 6b).192 Therefore, to clarify a multi-10  reactant-involved electrode reaction with an obvious diffusion 11  effect, a rotating electrode (like RDE or RRDE) is useful by virtue 12  of homogeneously distributed and thickness-controllable 13  diffusion layer, which we believe that it is exactly the role of an 14  RDE/RRDE system in fundamental electrochemistry from the 15  past to future.51, 56, 193 1617Fig. 6 (a) The schematic for the effective thickness of diffusion 18  layer. (b) Reactant switching from OH- to H2O in the OER with 19  increasing the potential. 202\.2.2 Product transfer 21The generated products may cause “feedback” on the electrode 22  reaction. For example, the produced gaseous oxygen/hydrogen 23  through water electrolysis will block the active sites if without 24  in-time release from the electrode surface, hindering the overall 25  reaction performance.194 In addition, the accumulated products 26  will influence the reaction thermodynamics as indicated by 27  Nernstian equation for a reaction between oxidant (Ox) and 28  reductant (Red) with transferring n electrons (Ox + ne-  29  Red):195, 196  30両方向矢印 ↔大文字 E 下付き e q イコール 大文字 E じょう 0 プラス 大文字 R 大文字 T 割る n 大文字 F 、 l n 、 a. 下付き o x 割る a. 下付き r e d 、 𝐸𝑒𝑞 = 𝐸0 + 𝑅𝑇 𝑛𝐹𝑙𝑛 𝑎𝑜𝑥 𝑎𝑟𝑒𝑑(30) 31where Eeq is equilibrium potential, E0 is the standard potential, 32  aox and ared are the activities of the oxidant and reductant, 33  respectively. Thus, the extra products lead to the increase in 34  actual thermodynamic potential, resulting in consumption of 35  additional energy to drive the reaction. This resultant extra 36  potential is called concentration overpotential in the field of 37  electrochemistry,197 and it is increasingly attractive in modern 38  electrochemical applications.198-200 For example, Yokoyama et al. 39  discussed the expanded potential window by local pH change 40  caused by OER/HER during water electrolysis because of the 41  production of protons/hydroxides (Fig. 7a).201 42Given the fact that protons participate in many electrochemical 43  reactions,73 here we discuss more of the local pH topic. To 44  elucidate the local pH effects (thermodynamics) caused by the 45  reaction as mentioned in Fig. 7a, a prerequisite task is to 46  measure the local pH during the electrode reaction. There have 47  been several methods to detect the local pH,202 including 48  scanning probe techniques and laser fluorescence 49  microscopy,203-205 RRDE voltammetry,206 infrared spectroscopy-50  based method,207 Raman spectroscopy-based method,208 and 51  RRDE-based open-circuit potential (RRDE-OCP) method.209-212 52  Here we introduce the RRDE-OCP method, which can be 53  conveniently conducted in a laboratory. 54In the RRDE-OCP method for non-buffer electrolyte, usually a 55  pH sensor (like IrOx) is loaded on the ring. There is a relation 56  among protons near the ring/disk, hydroxides near the ring/disk, 57  and protons/hydroxides in the bulk, as formulated below: 58c 下付き 、 r ,大文字 H じょう プラス 、 最後 下付き文字 マイナス 、 c 下付き 、 r ,上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 イコール 、 大文字 N 下付き 大文字 D 、 始めかっこ   c   d ,  大文字 H   プラス    マイナス 、 c 下付き 、 d ,上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 、 , 終わりかっこ プラス 左小かっこ 1 マイナス 大文字 N 下付き 大文字 D 右小かっこ 左小かっこ c 下付き 、 b ,大文字 H じょう プラス 、 最後 下付き文字 マイナス 、 c 下付き 、 b ,上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 右小かっこ 𝑐𝑟,𝐻+ − 𝑐𝑟,𝑂𝐻− = 𝑁𝐷(𝑐𝑑,𝐻+ − 𝑐𝑑,𝑂𝐻−) + (1 − 𝑁𝐷)(𝑐𝑏,𝐻+ −59         (31) 60c 下付き 、 r ,大文字 H じょう プラス 、 最後 下付き文字 マイナス 、 c 下付き 、 r ,上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 イコール 、 大文字 N 下付き 大文字 D 、 始めかっこ   c   d ,  大文字 H   プラス    マイナス 、 c 下付き 、 d ,上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 、 , 終わりかっこ プラス 左小かっこ 1 マイナス 大文字 N 下付き 大文字 D 右小かっこ 左小かっこ c 下付き 、 b ,大文字 H じょう プラス 、 最後 下付き文字 マイナス 、 c 下付き 、 b ,上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 右小かっこ 𝑐𝑏,𝑂𝐻−)大文字 N 下付き 大文字 D イコール 1 マイナス 1 6 番目 大文字 F 左大かっこ 2 番目の始めかっこ r 下付き 2 割る r 下付き 1 、 , 2 番目のかっこ閉じ 3 じょう マイナス 1 右大かっこ マイナス 2 ぶんぼ 3 大文字 F 左大かっこ 2 番目の始めかっこ 分子 、 r 下付き 2 プラス r 下付き 3 最後 分子 , 割る ぶんぼ 、 2 r 下付き 1 、 最後 ぶんぼ , , 2 番目のかっこ閉じ 3 じょう マイナス 1 右大かっこ マイナス 1 6 番目 大文字 F 左大かっこ 2 番目の始めかっこ r 下付き 3 割る r 下付き 1 、 , 2 番目のかっこ閉じ 3 じょう マイナス 1 右大かっこ 𝑁𝐷 = 1 − 1 6𝐹 \[( 𝑟2 𝑟1) 3 − 1\] − 2 3𝐹 \[( 𝑟2+𝑟3 2𝑟1) 3 − 1\] − 1 6𝐹 \[( 𝑟3 𝑟1) 3 − 1\]61    (32) 62大文字 F 左大かっこ シータ 右大かっこ イコール 、 、 3 じょう 、 1 ぶんぼ 2 最後 上付き文字 割る 4 パイ 、 l n 左大かっこ 始めかっこ 1 プラス シータ じょう 、 1 3 番目 最後 上付き文字 、 , 終わりかっこ 3 じょう 割る ぶんぼ 、 1 プラス シータ 最後 ぶんぼ , 右大かっこ プラス 3 割る 2 パイ 、 a. r c t a. n 2 番目の始めかっこ 分子 、 2 シータ じょう 、 1 3 番目 最後 上付き文字 マイナス 1 最後 分子 , 割る 3 じょう 、 1 ぶんぼ 2 最後 上付き文字 、 , 2 番目のかっこ閉じ プラス 1 4 番目 𝐹\[𝜃\] =   3 1 24𝜋 𝑙𝑛 \[ (1+𝜃 1 3) 3 1+𝜃\] + 3 2𝜋𝑎𝑟𝑐𝑡𝑎𝑛 ( 2𝜃 1 3−1 3 1 2) + 1 4(33) 63where  and  are the concentrations of protons and 64  hydroxides near the ring, respectively,  and   are the 65  concentrations of protons and hydroxides near the disk, 66  respectively,  and   are the concentrations of 67  protons and hydroxides in bulk electrolyte, respectively.  is 68  the detection efficiency. Note that   is different from 69  collection efficiency usually used in RRDE experiments. r1 is the 70  disk radius, r2 and r3 are inner and outer radii of the ring, 71  respectively. 72c 下付き 、 r ,大文字 H じょう プラス 、 最後 下付き文字 𝑐𝑟,𝐻+c 下付き 、 r ,上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 𝑐𝑟,𝑂𝐻−c 下付き 、 d ,大文字 H じょう プラス 、 最後 下付き文字 𝑐𝑑,𝐻+c 下付き 、 d ,上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 𝑐𝑑,𝑂𝐻−c 下付き 、 b ,大文字 H じょう プラス 、 最後 下付き文字 𝑐𝑏,𝐻+c 下付き 、 b ,上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 、 、 𝑐𝑏,𝑂𝐻−大文字 N 下付き 大文字 D 𝑁𝐷大文字 N 下付き 大文字 D 𝑁𝐷The proton concentration near the ring during the electrode 73  reaction can be obtained by monitoring the OCP of the ring 74  followed by the transformation from OCP to pH based on a pre-75  determined standard curve for pH and OCP at the ring. Next, the 76  concentration of protons near the disk can be calculated 77  through the above equations by combining the autoprotolysis 78  of water for proton concentration ( ) and hydroxide 79  concentration () with the autoprotolysis constant (): 80c 下付き 、 大文字 H じょう プラス 、 最後 下付き文字 𝑐𝐻+c 下付き 、 上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 𝑐𝑂𝐻−大文字 K 下付き w 𝐾𝑤大文字 K 下付き w イコール c 下付き 、 大文字 H じょう プラス 、 最後 下付き文字 、 c 下付き 、 上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 𝐾𝑤 = 𝑐𝐻+𝑐𝑂𝐻−(34) 81Finally, the local pH near the disk ( ) during the 82  electrode reaction is obtained according to the calculated : 83下付き文字 てい , p 大文字 H , 最後 てい , 下付き 、 d ,、 l o c a. l 最後 下付き文字 𝑝𝐻𝑑,𝑙𝑜𝑐𝑎𝑙c 下付き 、 d ,大文字 H じょう プラス 、 最後 下付き文字 𝑐𝑑,𝐻+下付き文字 てい , p 大文字 H , 最後 てい , 下付き 、 d ,、 l o c a. l 最後 下付き文字 イコール 、 マイナス log 左小かっこ c 下付き 、 d ,大文字 H じょう プラス 、 最後 下付き文字 右小かっこ 𝑝𝐻𝑑,𝑙𝑜𝑐𝑎𝑙 = −log (𝑐𝑑,𝐻+)(35) 84A typical process for obtaining the local pH near the 85  electrocatalyst is shown in Fig. 7b, with the OER via water 86  molecule oxidation as an example: the electrocatalyst on the 87  disk catalyses water under the applied potential and the 88  produced protons transfer to the ring electrode with the OCP 89  recorded by the potentiostat, then the pH near the ring 90  electrode is converted from the OCP according to the a pre-91  determined pH-OCP standard curve, finally the local pH near the 92  disk electrode is calculated through the above equation. 93This RRDE-OCP method is low-cost and operatable in an 94  electrochemistry laboratory. However, it is noticeable that the 95  “steep” regions of the curves may lead to large disk errors for 96  proton release or consumption electrode reactions (Fig. 7c). For 97  example, the proton-released/hydroxide-consumed OER 98  causes a lower pH nearby disk as compared to that nearby ring. 99  Even a small error of the pH at the ring electrode can result in a 100  significantly large error of the calculated pH at the disk 101  electrode with bulk pH \> 7; in contrast, the error of the disk pH 102  is much decreased with bulk pH \< 7 due to the mild change of 103  the curves (as marked by dash lines in Fig. 7c). An error analysis 104ARTICLE Journal Name8 \| J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xxPlease do not adjust marginsPlease do not adjust marginswas conducted by our group (Fig. 7d),213 which suggests that for 1  the OER the bulk pHs \< 7 ensure small pH errors at the disk 2  electrode. In contrast, for proton consumption/hydroxide 3  release (e.g. the HER), bulk pHs \> 7 benefit smaller disk pH 4  errors as compared to those \< 7. Therefore, for practical 5  application of this RRDE-based approach, it is recommended to 6  consider the errors for different bulk pHs and reaction types 7  (like proton release or consumption). 8Note that the above method is effective in electrolytes without 9  buffers. Koper group modified this equation in a buffered 10  system (e.g. phosphates), with corrected concentration for OH- 11  ( ): 12c 下付き 左小かっこ 上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 右小かっこ じょう プライム 𝑐𝑂𝐻− ′c 下付き 左小かっこ 上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 右小かっこ じょう プライム イコール c 下付き 、 上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 プラス 左大かっこ 大文字 H 下付き 2 、 大文字 P 大文字 O 下付き 4 じょう マイナス 、 右大かっこ プラス 2 左大かっこ 大文字 H 大文字 P 大文字 O 下付き 4 じょう 左小かっこ 2 マイナス 右小かっこ 、 右大かっこ プラス 3 左大かっこ 大文字 P 大文字 O 下付き 4 じょう 左小かっこ 3 マイナス 右小かっこ 、 右大かっこ 𝑐𝑂𝐻− ′ = 𝑐𝑂𝐻− + \[𝐻2𝑃𝑂4 −\] + 2\[𝐻𝑃𝑂4 2−\] + 3\[𝑃𝑂4 3−\](36) 13大文字 N 下付き 大文字 D イコール 分子 、 c 下付き 左小かっこ 上付き文字 てい , r ,大文字 O 大文字 H , 最後 てい , じょう マイナス 右小かっこ じょう プライム 、 、 マイナス 、 c 下付き 無限大 じょう プライム 最後 分子 , 割る ぶんぼ 、 c 下付き 左小かっこ 上付き文字 てい , d ,大文字 O 大文字 H , 最後 てい , じょう マイナス 右小かっこ じょう プライム 、 、 マイナス 、 c 下付き 無限大 じょう プライム 、 最後 ぶんぼ 𝑁𝐷 = 𝑐𝑟,𝑂𝐻− ′  − 𝑐∞ ′ 𝑐𝑑,𝑂𝐻− ′  − 𝑐∞ ′(37) 14in which   is the concentration before the correction, 15  \[ , \[ , and \[  are the concentrations of 16  ,  , and  , respectively.   is the detection 17  efficiency,  ,  , and   are corrected concentrations 18  of OH- on the ring electrode, on disk electrode, and in the bulk, 19  respectively. For more details, the readership please refers to 20  the original papers.206, 214 21c 下付き 、 上付き文字 てい , 大文字 O 大文字 H , 最後 てい , じょう マイナス 、 最後 下付き文字 𝑐𝑂𝐻−左大かっこ 大文字 H 下付き 2 、 大文字 P 大文字 O 下付き 4 じょう マイナス 、 右大かっこ 𝐻2𝑃𝑂4 −\]左大かっこ 大文字 H 大文字 P 大文字 O 下付き 4 じょう 左小かっこ 2 マイナス 右小かっこ 、 右大かっこ 𝐻𝑃𝑂4 2−\]左大かっこ 大文字 P 大文字 O 下付き 4 じょう 左小かっこ 3 マイナス 右小かっこ 、 右大かっこ 𝑃𝑂4 3−\]大文字 H 下付き 2 、 大文字 P 大文字 O 下付き 4 じょう マイナス 𝐻2𝑃𝑂4 −大文字 H 大文字 P 大文字 O 下付き 4 じょう 左小かっこ 2 マイナス 右小かっこ 𝐻𝑃𝑂4 2−大文字 P 大文字 O 下付き 4 じょう 左小かっこ 3 マイナス 右小かっこ 𝑃𝑂4 3−大文字 N 下付き 大文字 D 𝑁𝐷c 下付き 左小かっこ 上付き文字 てい , r ,大文字 O 大文字 H , 最後 てい , じょう マイナス 右小かっこ じょう プライム 𝑐𝑟,𝑂𝐻− ′c 下付き 左小かっこ 上付き文字 てい , d ,大文字 O 大文字 H , 最後 てい , じょう マイナス 右小かっこ じょう プライム 𝑐𝑑,𝑂𝐻− ′c 下付き 無限大 じょう プライム 𝑐∞ ′In our previous work, we used the RRDE-based OCP method to 22  investigate the local pH effect on OER behaviours in electrolytes 23  with different bulk pHs.213 We found that the difference of 24  between local and bulk pHs increased with the OER current 25  density increasing, resulting in the dynamic shift of the 26  equilibrium potential for OER during the reaction (Fig. 7e). We 27  made calibration for the reversible hydrogen electrode (RHE) 28  with the dynamic local pH during the OER by 29大文字 E 下付き 、 大文字 R 大文字 H 大文字 E ,l o c a. l 、 p 大文字 H 最後 下付き文字 イコール 大文字 E 下付き 、 大文字 R 大文字 H 大文字 E ,b u l k 、 p 大文字 H 最後 下付き文字 プラス 2.303 大文字 R 大文字 T 割る 大文字 F 左小かっこ 下付き文字 てい , p 大文字 H , 最後 てい , 下付き l o c a. l マイナス 下付き文字 てい , p 大文字 H , 最後 てい , 下付き b u l k 右小かっこ 𝐸𝑅𝐻𝐸,𝑙𝑜𝑐𝑎𝑙 𝑝𝐻 = 𝐸𝑅𝐻𝐸,𝑏𝑢𝑙𝑘 𝑝𝐻 + 2.303𝑅𝑇 𝐹(𝑝𝐻𝑙𝑜𝑐𝑎𝑙 − 𝑝𝐻𝑏𝑢𝑙𝑘)30   (38) 31where   is the RHE calibrated by local pH, 32   is the RHE calibrated by bulk pH,   and 33   are the local and bulk pHs, respectively. With the local-34  pH calibrated RHE, it revealed that the various OER 35  performances in different bulk pHs were almost identical, 36  indicating that the local pH effect dominated the performance 37  differences by influencing the thermodynamic equilibrium 38  potential for OER. This work demonstrates a “feedback” effect 39  from the products for an electrode reaction, highlighting the 40  dynamic electrode processes. 41大文字 E 下付き 、 大文字 R 大文字 H 大文字 E ,l o c a. l 、 p 大文字 H 最後 下付き文字 𝐸𝑅𝐻𝐸,𝑙𝑜𝑐𝑎𝑙 𝑝𝐻大文字 E 下付き 、 大文字 R 大文字 H 大文字 E ,b u l k 、 p 大文字 H 最後 下付き文字 𝐸𝑅𝐻𝐸,𝑏𝑢𝑙𝑘 𝑝𝐻下付き文字 てい , p 大文字 H , 最後 てい , 下付き l o c a. l 𝑝𝐻𝑙𝑜𝑐𝑎𝑙下付き文字 てい , p 大文字 H , 最後 てい , 下付き b u l k 𝑝𝐻𝑏𝑢𝑙𝑘42Fig. 7 (a) The local pH changes by OER or HER. Reproduced from 43  Ref. 201 with permission. Copyright 2018, IOP. (b) The RRDE-OCP 44  method for measuring the ring pH. Reproduced from Ref. 212 45  with permission. Copyright 2020, Oxford University Press. (c) 46  Theoretical relationship between disk surface pH and ring 47  surface pH.  means the pH in the bulk. Reproduced from 48  Ref. 211 with permission. Copyright 2019, John Wiley \& Sons, Inc. 49  (d) The derivative of pH@Disk with respect to pH@Ring and (e) 50  the pH changes between bulk and local electrolyte with the OER 51  current density increasing, as well as the dynamic drift of the 52  OER equilibrium potential. Reproduced from Ref. 213 with 53  permission. Copyright 2025, John Wiley \& Sons, Inc. 54下付き文字 てい , p 大文字 H , 最後 てい , 下付き 無限大 𝑝𝐻∞2\.3 Electrode material reconstruction 55At the surface of a solid metal electrode (e.g., single-crystal 56  gold),215-218 the surface atoms may rearrange with structure 57  significantly changed and this process is regarded as surface 58  reconstruction.217, 219, 220 In addition, an applied potential or 59  electrolyte species can induce surface reconstruction, indicating 60  dynamic electrode surface structures during an electrode 61  process.217 As triggered by the promising non-noble metal-62  based materials for the OER,221-231 a large amount of materials 63  have been explored in recent years, such as metal oxides,232-235 64  phosphides,236, 237 and nitrides.238, 239 Similar to the single-crystal 65  system with surface reconstruction, those metal-based 66  materials usually experience reconstruction to form  active 67  species under the electrochemical conditions, and thus these 68  “precursors” to the real active electrocatalytic reactions are 69  called precatalyst (Fig. 8a).240 In this paper, reconstruction refers 70  to the changes of electrode materials, such as structures, 71  morphologies, and components, triggered by electrochemical 72  processes and the formed species usually serve as active species 73  for the corresponding electrode reactions. This understanding 74  is similar to “surface restructuring” in heterogenous catalysis.241  75In recent years, reconstruction phenomena have been 76  increasingly noticed and utilized to regulate the electrode 77  reaction performance.88, 242-247 Quantification of the 78  reconstruction is believed to benefit the in-depth 79  understanding for guiding the rational design of the electrode 80  reaction systems. In 2023, an equation was proposed to 81  calculate the surface reconstruction degree (Dr) for LiCo2O4 82  during electrochemical water oxidation:248 83大文字 D 下付き r イコール 大文字 V 下付き r e c o n s t r u c t i. o n 割る 大文字 V 下付き l o a. d i. n g 、 イコール 分子 、 分子 、 c 下付き 大文字 L i. 、 下付き文字 てい , 、 大文字 N , 最後 てい , 下付き 大文字 A. 最後 分子 , 割る ぶんぼ 、 6.94 、 x 最後 ぶんぼ , 、 下付き文字 てい , 、 大文字 V , 最後 てい , 下付き m o l 最後 分子 , 割る ぶんぼ 、 4 ぶんぼ 3 パイ 2 番目の始めかっこ へいほうこん の 、 分子 、 m 下付き l o a. d i. n g 、 、 大文字 B 大文字 E 大文字 T 最後 分子 , 割る 4 パイ 最後 へいほうこん , , 2 番目のかっこ閉じ 3 じょう 、 最後 ぶんぼ 𝐷𝑟 = 𝑉𝑟𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑜𝑛 𝑉𝑙𝑜𝑎𝑑𝑖𝑛𝑔= 𝑐𝐿𝑖 𝑁𝐴 6.94 𝑥𝑉𝑚𝑜𝑙 4 3𝜋(√ 𝑚𝑙𝑜𝑎𝑑𝑖𝑛𝑔 𝐵𝐸𝑇 4𝜋) 3(39) 84where   and   are the volumes of the 85  reconstructed portion and the totally loaded one, respectively. 86   is the content of Li in the cycled electrolyte,   is the 87  Avogadro constant,   is the molecular volume of spinel 88  LixCo3-xO4, x is the stochiometric number in spinel LixCo3-xO4, 89   is the mass loading, and BET is the surface area by BET 90  method of the oxide. By using the above equation, quantitative 91  discussion on the reconstruction with the Li stochiometric 92  number in the oxide was carried out. For more detail the 93  readership may refer to the original paper.248 94大文字 V 下付き r e c o n s t r u c t i. o n 𝑉𝑟𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑜𝑛大文字 V 下付き l o a. d i. n g 𝑉𝑙𝑜𝑎𝑑𝑖𝑛𝑔c 下付き 大文字 L i. 𝑐𝐿𝑖大文字 N 下付き 大文字 A. 𝑁𝐴大文字 V 下付き m o l 𝑉𝑚𝑜𝑙m 下付き l o a. d i. n g 𝑚𝑙𝑜𝑎𝑑𝑖𝑛𝑔Given the contact between electrode material and electrolyte 95  in an electrochemical system, the reconstruction process is 96  usually companied with mass transfer towards/outwards the 97  electrode material. It reveals that layered β-Co(OH)2 98  reconstructs to β-CoOOH, driven by the continuously applied 99  potential with hydroxide intercalation from the electrolyte and 100Journal Name  ARTICLEThis journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 \| 9Please do not adjust marginsPlease do not adjust marginsthe subsequent extraction of water molecules/protons (Fig. 1  8b).249 Wang et al. investigated the reconstruction process of 2  layered α-Co(OH)2 by tuning the intercalated ions.90 It unveiled 3  that the electrochemical reconstruction process could be 4  regarded as intercalated species-coupled electron transfer 5  process, with the change of Co-O local configuration. They 6  quantified the reconstruction by modelling the Co redox peak 7  and the time-dependant Co oxidation rate curve based on 8  simplified reaction models. For the redox peak modelling: 9大文字 I. イコール 大文字 S n 大文字 F k 下付き 0 、 シータ じょう x 、 e じょう 、 分子 、 アルファ 左小かっこ 大文字 E マイナス 大文字 E 下付き 0 右小かっこ 最後 分子 , 割る 大文字 R 大文字 T 最後 上付き文字 𝐼 = 𝑆𝑛𝐹𝑘0𝜃 𝑥𝑒 𝛼(𝐸−𝐸0) 𝑅𝑇(40) 10分子 、 マイナス d シータ 最後 分子 , 割る d t イコール 大文字 I. 割る ぶんぼ 、 積分 の 大文字 I. d t 最後 ぶんぼ , イコール c 大文字 I. −𝑑𝜃 𝑑𝑡= 𝐼 ∫ 𝐼𝑑𝑡= 𝑐𝐼(41) 11d 大文字 E 割る d t イコール ニュー 𝑑𝐸 𝑑𝑡= 𝜈(42) 12where I is redox current, S is the site density, n is the electron 13  number, F is Faraday constant, k0 is the reaction coefficient, θ is 14  the reaction of the remaining Co(OH)2, x is the apparent 15  reaction order, α is the apparent transfer coefficient, E is the 16  applied potential, E0 is the (apparent) equilibrium potential of 17  the related reconstruction reaction, R gives the gas constant, T 18  is the temperature, t is the time, c represents for the reciprocal 19  of the integration of I in the whole reaction region,   is the 20  potential scan rate. 21ニュー 𝜈For the time-dependant oxidation rate analysis, the reaction 22  current i is expressed as: 23i. イコール 大文字 C 、 e じょう 、 マイナス t 割る タウ 最後 上付き文字 𝑖 = 𝐶 𝑒− 𝑡 𝜏(43) 24where C is a constant for a certain reaction system, τ is the time 25  constant (the reciprocal gives the reaction frequency). 26The two parameters, i.e. E0 and τ, were further used to quantify 27  the ability to reconstruct (reconstruction activity) by taking as -28  E0 and τ-1 as the thermodynamic and the kinetic descriptors of 29  reconstruction activities, respectively. A volcano-type plot for 30  the OER performance and the reconstruction activity were 31  finally established, indicating that a moderate reconstruction 32  activity favoured a high OER rate (Fig. 8c). This case study shows 33  that not only the reaction based on the electrolyte, but also the 34  reaction occurred in the electrode materials should be carefully 35  examined for a comprehensive understanding of the electrode 36  process. 3738Fig. 8 (a) Metal-based materials may serve as precatalysts to 39  form the active species for electrode reactions. (b) 40  Reconstruction of β-Co(OH)2 during the OER. (c) The volcano-41  type relationship between the OER performance and the 42  reconstruction activity of the electrode materials. 432\.4 Stability quantification 44Apart from the electrode reaction activity, i.e. the current of a 45  Faradaic process, the capability to keep the activity (i.e. high 46  stability) has been increasingly attractive nowadays for large-47  scale applications of electrochemistry-based technologies.2 48  According to previous literature, a trade-off between activity 49  and stability can be found in an electrode reaction (like the 50  OER).250 Therefore, to achieve both high activities and stabilities, 51  it is crucial to understand the stability issue and thus 52  quantitative investigation into the electrode stability process is 53  strongly required. In 2018, Geiger et al. proposed a concept for 54  describing the OER stability for various iridium-based oxides, 55  called stability number (S-number, Fig. 9a, noted as S) and 56  defined as the ratio between the amounts of produced oxygen 57  () and dissolved iridium ():251  58n 下付き o x y g e n 𝑛𝑜𝑥𝑦𝑔𝑒𝑛n 下付き i. r i. d i. u m 𝑛𝑖𝑟𝑖𝑑𝑖𝑢𝑚大文字 S イコール n 下付き o x y g e n 割る n 下付き i. r i. d i. u m 、 𝑆 = 𝑛𝑜𝑥𝑦𝑔𝑒𝑛 𝑛𝑖𝑟𝑖𝑑𝑖𝑢𝑚(44) 59The electrode lifetime can be related to the S-number: 60l i. f e t i. m e イコール 分子 、 大文字 S ドット z 、 ドット 大文字 F 、 ドット m 最後 分子 , 割る ぶんぼ 、 j ドット 大文字 M 最後 ぶんぼ 𝑙𝑖𝑓𝑒𝑡𝑖𝑚𝑒 = 𝑆⋅𝑧 ⋅𝐹 ⋅𝑚 𝑗⋅𝑀(45) 61where z is electrons per transferred oxygen molecule, F is 62  Faraday constant, m is the loaded mass of iridium, j is the 63  applied current density, M is the molar mass of iridium. 64As reported in previous literature, the S-numbers are nearly 65  unchanged under various current densities from 0.01 to 1 A mg 66  Ir-1, suggesting a direct relationship between the 67  electrocatalytic OER and dissolution for the electrode 68  materials.251 69In 2020, a microkinetic network was built to detail the stability 70  of OER electrocatalysts by considering dissolution of 71  intermediate species with the RDS assumption, and the analytic 72  expression of stability number was given.252 This hints the 73  inclusion of both electrocatalytic reaction and electrode 74  material loss in one electrocatalytic cycle. In 2024, Ooka et al. 75  proposed a theoretical framework to formulate the reaction 76  intermediate dissolution-based catalytic cycle (Fig. 9b) dictated 77  by the following ordinary differential equation:163 78d x 割る d t イコール 始めかっこ 太字 大文字 C マイナス 太字 大文字 D , 終わりかっこ 、 太字 x 𝑑𝑥 𝑑𝑡= (𝐂 − 𝐃)𝐱(46) 79太字 x イコール 始めかっこ x 下付き 1,ドット ドット ドット ,x 下付き n 、 , 終わりかっこ じょう 大文字 T 𝐱 = (𝑥1, … , 𝑥𝑛)𝑇(47) 80太字 大文字 C イコール 左大かっこ 5 基準 5 行列、 マイナス k 下付き 1 最後 列 , 0 、 水平中央の省略記号 、 0 、 k 下付き n 、次の行、 k 下付き 1 最後 列 , マイナス k 下付き 2 最後 列 , 水平中央の省略記号 最後 列 , 0 、 0 、次の行、 垂直の省略記号 最後 列 , 垂直の省略記号 最後 列 , 右下がりの省略記号 最後 列 , 垂直の省略記号 最後 列 , 垂直の省略記号 、次の行、 0 、 0 、 水平中央の省略記号 、 マイナス k 下付き 、 n マイナス 1 最後 下付き文字 、 0 、次の行、 0 、 0 、 水平中央の省略記号 、 k 下付き 、 n マイナス 1 最後 下付き文字 、 マイナス k 下付き n 、 最後 行列 右大かっこ 𝐂 = \[         −𝑘1 0 ⋯ 0 𝑘𝑛 𝑘1 −𝑘2 ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ −𝑘𝑛−1 0 0 0 ⋯ 𝑘𝑛−1 −𝑘𝑛\](48) 81太字 大文字 D イコール 左大かっこ 5 基準 5 行列、 d 下付き 1 最後 列 , 0 、 水平中央の省略記号 、 0 、 0 、次の行、 0 、 d 下付き 2 、 水平中央の省略記号 、 0 、 0 、次の行、 垂直の省略記号 最後 列 , 垂直の省略記号 最後 列 , 右下がりの省略記号 最後 列 , 垂直の省略記号 最後 列 , 垂直の省略記号 、次の行、 0 、 0 、 水平中央の省略記号 、 d 下付き 、 n マイナス 1 最後 下付き文字 、 0 、次の行、 0 、 0 、 水平中央の省略記号 、 0 、 d 下付き n 、 最後 行列 右大かっこ 𝐃 = \[         𝑑1 0 ⋯ 0 0 0 𝑑2 ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ 𝑑𝑛−1 0 0 0 ⋯ 0 𝑑𝑛\](49) 82in which t refers to the time, xi (i = 1, 2, ..., n) refers to the 83  amount of reaction intermediate, ki (i = 1, 2, ..., n) is the related 84  rate constant for catalytic reaction, di (i = 1, 2, ..., n) is the 85  related rate constant for dissolution. By assuming that the 86  deactivation is much slower than the catalytic reaction and 87  using the steady-state approximation, the time-dependent total 88  amount of intermediates (N) can be solved, as well as the 89  expression for lifetime defined as the time for N delaying to 1/e 90  of the original value. For more details the readership may refer 91  to the original paper.163 As mentioned above, this integration of 92  reaction and deactivation (generally, the deactivation is also 93ARTICLE Journal Name10 \| J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xxPlease do not adjust marginsPlease do not adjust marginsregarded as “reaction”) can be understood in a reaction 1  network by considering the deactivation as the pendant vertices 2  (Fig. 4b). 3To directly describe the degradation process of the OER, 4  recently our group proposed a simple and practical model based 5  on catalyst dissolution and substrate passivation (as illustrated 6  in Fig. 9c).91 The relation among the applied potential (Eapp), OER 7  current (I), and time (t) by chronopotentiometry is: 8大文字 E 下付き a. p p イコール 1 割る アルファ 、 l n 、 分子 、 大文字 I. マイナス c 最後 分子 , 割る ぶんぼ 、 k 下付き 0 、 大文字 S 左小かっこ t 右小かっこ 最後 ぶんぼ , プラス 大文字 I. 左小かっこ 大文字 R 下付き 0 プラス r 下付き 、 大文字 R 大文字 W 大文字 E ,1 最後 下付き文字 、 t 右小かっこ 𝐸𝑎𝑝𝑝 = 1 𝛼𝑙𝑛 𝐼−𝑐 𝑘0𝑆(𝑡)+ 𝐼(𝑅0 + 𝑟𝑅𝑊𝐸,1𝑡)(50) 9in which  is the coefficient for the influence of the potential on 10  the intrinsic OER activity, c is the substrate current, k0 is the pre-11  exponential factor for OER, S(t) is the active site number 12  changing with time t, R0 is the resistance in the beginning and 13   is rate of resistance change during the degradation. 14アルファ 𝛼r 下付き 、 大文字 R 大文字 W 大文字 E ,1 最後 下付き文字 𝑟𝑅𝑊𝐸,1To describe the time-dependent active site number, we used a 15  semi-empirical description analogous to the law of mass action: 16r イコール k じょう d i. s 、 上付き文字 てい , 大文字 S の t , , 最後 てい , じょう x 𝑟 = 𝑘𝑑𝑖𝑠𝑆(𝑡)𝑥(51) 17k じょう d i. s イコール k 下付き 0 じょう d i. s 、 e じょう 、 アルファ じょう d i. s 、 2 番目の始めかっこ 大文字 E 下付き a. p p マイナス 大文字 I. 大文字 R 下付き u 、 , 2 番目のかっこ閉じ 、 最後 上付き文字 𝑘𝑑𝑖𝑠 = 𝑘0 𝑑𝑖𝑠𝑒𝛼𝑑𝑖𝑠(𝐸𝑎𝑝𝑝−𝐼𝑅𝑢)(52) 18d 大文字 S の t , 割る d t イコール マイナス r 、 w i. t h 、 大文字 S の 0 , イコール 大文字 S 下付き m a. x 𝑑𝑆(𝑡) 𝑑𝑡= −𝑟 𝑤𝑖𝑡ℎ 𝑆(0) = 𝑆𝑚𝑎𝑥(53) 19where r is the rate of dissolution,   is the dissolution 20  coefficient, x is the reaction order for the dissolution,   is the 21  pre-exponential factor for dissolution,   refers to the 22  coefficient by the influence of potential on dissolution,  is the 23  resistance. By combining the above equations, the final 24  equation for S(t) is obtained: 25k じょう d i. s 𝑘𝑑𝑖𝑠k 下付き 0 じょう d i. s 𝑘0 𝑑𝑖𝑠アルファ じょう d i. s 𝛼𝑑𝑖𝑠大文字 R 下付き u 𝑅𝑢大文字 S の t , イコール 始めかっこ 3 番目の始めかっこ k プライム 、 t プラス 上付き文字 てい , 大文字 S 下付き m a. x , 最後 てい , じょう 、 1 マイナス x プライム 、 最後 上付き文字 割る ぶんぼ 、 マイナス 1 プラス x プライム 、 最後 ぶんぼ , , 3 番目のかっこ閉じ 3 番目の始めかっこ マイナス 1 プラス x プライム 、 , 3 番目のかっこ閉じ , 終わりかっこ じょう 、 1 割る ぶんぼ 、 1 マイナス x プライム 、 最後 ぶんぼ , 最後 上付き文字 𝑆(𝑡) = ((𝑘′𝑡 + 𝑆𝑚𝑎𝑥 1−𝑥′ −1+𝑥′) (−1 + 𝑥′)) 1 1−𝑥′(54) 26k プライム イコール k 下付き 0 じょう d i. s 、   始めかっこ 分子 、 大文字 I. マイナス c 最後 分子 , 割る k 下付き 0 、 , 終わりかっこ じょう 、 アルファ じょう d i. s 割る アルファ 最後 上付き文字 𝑘′ = 𝑘0 𝑑𝑖𝑠  ( 𝐼−𝑐 𝑘0) 𝛼𝑑𝑖𝑠 𝛼(55) 27x プライム イコール x マイナス アルファ じょう d i. s 割る アルファ 𝑥′ = 𝑥 − 𝛼𝑑𝑖𝑠 𝛼(56) 28Eq. 50 and eq. 54 serve as the mathematical degradation model 29  for quantifying the OER degradation process. This model is 30  simple and sufficient to handle the degradation, with a good 31  fitting result for long-term testing. We hope the research 32  progress introduced in this part for OER stability issue will 33  trigger further research on microkinetic analysis-based 34  frameworks for promoting comprehensive understandings of 35  the deactivation issue of electrode materials, not limited to 36  dissolution mechanisms but other factors such as 37  agglomeration, bubble effect, and catalyst detachment. 3839Fig. 9 (a) Stability number for the OER by considering iridium 40  dissolution. (b) Catalytic cycle including catalytic (the blue 41  arrows) and deactivation (the orange arrows) reactions. (c) A 42  degradation model for quantifying the OER deactivation 43  process.  443 Data assimilation and electrode processes 45Benefitted from the effective combination of experimental 46  data/observed data and as-built model, data assimilation (DA) 47  has achieved great success in geophysics, autonomous, and 48  assisted navigation,253, 254 and recently it showed prospective 49  applications in other fields like materials science and 50  electrochemistry.89, 91, 255-261 In this section, we will introduce 51  basic knowledge of DA and two typical approaches, i.e. Kalman 52  filter and four-dimensional variational assimilation. Then we 53  will offer our recent advances by using DA to contribute to the 54  understanding and description of electrode processes with the 55  ORR and OER as model reactions. 563\.1 Introduction of data assimilation 57DA features a “bridge” between observation (experimental data) 58  and simulation (theoretical model). The target of DA mainly 59  locates on obtaining reliable initial states/parameters and 60  future forecasting based on established models. Before 61  introducing typical DA methods, a basic method for linear 62  minimum variance estimation is shown first to give perceptual 63  cognition for the readership.262 Given the truth state xt, and two 64  estimates x1 and x2 with errors  and , respectively: 65小文字イプシロン 下付き 1 𝜀1小文字イプシロン 下付き 2 𝜀2x 下付き 1 イコール x 下付き t プラス 小文字イプシロン 下付き 1 𝑥1 = 𝑥𝑡 + 𝜀1(57) 66x 下付き 2 イコール x 下付き t プラス 小文字イプシロン 下付き 2 𝑥2 = 𝑥𝑡 + 𝜀2(58) 67We assume that x1 and x2 are unbiased estimates and their 68  errors are not correlated, then for the mathematical 69  expectations: 70小なり 小文字イプシロン 下付き 1 大なり 、 イコール 、 小なり 小文字イプシロン 下付き 2 大なり 、 イコール 0 \< 𝜀1 \> = \< 𝜀2 \> = 0(59) 71小なり 小文字イプシロン 下付き 1 、 小文字イプシロン 下付き 2 大なり 、 イコール 0 \< 𝜀1𝜀2 \> = 0(60) 72Accordingly, the estimate xa is formulated as the linear 73  combination of x1 and x2 with weights  and , respectively: 74アルファ 下付き 1 𝛼1アルファ 下付き 2 𝛼2x 下付き a. イコール アルファ 下付き 1 、 x 下付き 1 プラス アルファ 下付き 2 、 x 下付き 1 𝑥𝑎 = 𝛼1𝑥1 + 𝛼2𝑥1(61) 75The two weights for giving the minimum error variance   for 76  the optimal estimate xa can be solved as: 77シグマ 下付き a. 2 じょう 𝜎𝑎 2アルファ 下付き 1 イコール 分子 、 シグマ 下付き 2 2 じょう 最後 分子 , 割る ぶんぼ 、 シグマ 下付き 1 2 じょう プラス シグマ 下付き 2 2 じょう 、 最後 ぶんぼ 𝛼1 = 𝜎2 2 𝜎1 2+𝜎2 2(62) 78アルファ 下付き 2 イコール 分子 、 シグマ 下付き 1 2 じょう 最後 分子 , 割る ぶんぼ 、 シグマ 下付き 1 2 じょう プラス シグマ 下付き 2 2 じょう 、 最後 ぶんぼ 𝛼2 = 𝜎1 2 𝜎1 2+𝜎2 2(63) 79シグマ 下付き a. 2 じょう イコール 分子 、 シグマ 下付き 1 2 じょう 、 シグマ 下付き 2 2 じょう 最後 分子 , 割る ぶんぼ 、 シグマ 下付き 1 2 じょう プラス シグマ 下付き 2 2 じょう 、 最後 ぶんぼ 𝜎𝑎 2 = 𝜎1 2𝜎2 2 𝜎1 2+𝜎2 2(64) 80It is easy to know that   is smaller than   or   by 81  transforming the above equation to 82シグマ 下付き a. 2 じょう 𝜎𝑎 2シグマ 下付き 1 2 じょう 𝜎1 2シグマ 下付き 2 2 じょう 𝜎2 21 割る ぶんぼ 、 シグマ 下付き a. 2 じょう 、 最後 ぶんぼ , イコール 1 割る ぶんぼ 、 シグマ 下付き 1 2 じょう 、 最後 ぶんぼ , プラス 1 割る ぶんぼ 、 シグマ 下付き 2 2 じょう 、 最後 ぶんぼ 1 𝜎𝑎 2= 1 𝜎1 2+ 1 𝜎2 2(65) 83Therefore, the optimal state xa is the linear combination of x1 84  and x2 with a reduced error variance, i.e. improved precision. 85Following the above understanding, we exemplify one system 86  (such as a dynamic or reaction system) for demonstrating DA, 87  including a state by modelling/simulation, i.e. the forecast xf, 88  another state obtained by experiments/observations, i.e. the 89  observation xo, and the optimal estimate (also called “analysis”) 90  xa with the weight α:259 91x 下付き a. イコール 始めかっこ 1 マイナス アルファ , 終わりかっこ 、 x 下付き f プラス アルファ x 下付き o 𝑥𝑎 = (1 − 𝛼)𝑥𝑓 + 𝛼𝑥𝑜(66) 92which is usually expressed as: 93x 下付き a. イコール x 下付き f プラス アルファ 左小かっこ x 下付き o マイナス x 下付き f 右小かっこ 𝑥𝑎 = 𝑥𝑓 + 𝛼(𝑥𝑜 − 𝑥𝑓)(67) 94Journal Name  ARTICLEThis journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 \| 11Please do not adjust marginsPlease do not adjust marginsAssume that the expectations of the error variances for xf and 1  xo follow 2小なり 小文字イプシロン 下付き f 大なり 、 イコール 、 小なり 小文字イプシロン 下付き o 大なり 、 イコール 0 \< 𝜀𝑓 \> = \< 𝜀𝑜 \> = 0(68) 3小なり 小文字イプシロン 下付き f 、 小文字イプシロン 下付き o 大なり 、 イコール 0 \< 𝜀𝑓𝜀𝑜 \> = 0(69) 4Then we can calculate the error variance of the optimal 5  estimate as following: 6シグマ 下付き a. 2 じょう イコール 上付き文字 てい , 左小かっこ 1 マイナス アルファ 右小かっこ , 最後 てい , 2 じょう 、 シグマ 下付き f 2 じょう プラス アルファ 2 じょう 、 シグマ 下付き o 2 じょう 𝜎𝑎 2 = (1 − 𝛼)2𝜎𝑓 2 + 𝛼2𝜎𝑜 2(70) 7with the weight α calculated as: 8アルファ イコール 分子 、 シグマ 下付き f 2 じょう 最後 分子 , 割る ぶんぼ 、 シグマ 下付き o 2 じょう プラス シグマ 下付き f 2 じょう 、 最後 ぶんぼ 𝛼 = 𝜎𝑓 2 𝜎𝑜 2+𝜎𝑓 2(71) 9for letting the minimum  and it is thus expressed as: 10シグマ 下付き a. 2 じょう 𝜎𝑎 2シグマ 下付き a. 2 じょう イコール 分子 、 シグマ 下付き o 2 じょう 、 シグマ 下付き f 2 じょう 最後 分子 , 割る ぶんぼ 、 シグマ 下付き o 2 じょう プラス シグマ 下付き f 2 じょう 、 最後 ぶんぼ 𝜎𝑎 2 = 𝜎𝑜 2𝜎𝑓 2 𝜎𝑜 2+𝜎𝑓 2(72) 11This result is like eq. 64 and it shows a lower error variance of 12  the optimal estimate xa after combining the information of xo 13  and xf, as illustrated in Fig. 10a, demonstrating the power of DA. 14  The situation for multiple states can be extended by considering 15  the state vector x: 16太字 x イコール 上付き文字 てい , 左小かっこ x 下付き 1,x 下付き 2,ドット ドット ドット ,x 下付き n 右小かっこ , 最後 てい , じょう 大文字 T 𝐱 = (𝑥1, 𝑥2, … , 𝑥𝑛)𝑇(73) 17太字 x 下付き 太字 a. イコール 太字 x 下付き 太字 f プラス 太字 大文字 K 左小かっこ 太字 z マイナス 太字 大文字 H 太字 x 下付き 太字 f 右小かっこ 𝐱𝐚 = 𝐱𝐟 + 𝐊(𝐳 − 𝐇𝐱𝐟)(74) 18in which xa is the optimal estimate for the state vector, xf is the 19  predict for the state vector, z is the observation for the state 20  vector, H is the observation function, K is the weight matrix. 21Next, we introduce one typical DA approach, called Kalman 22  filter, which was proposed by Kalman in 1960,263 for estimating 23  the state of a system over time based on measurements and the 24  model. Consider a time-dependent controlled system governed 25  by the linear stochastic difference equation with state vector x 26  at the nth time step: 27太字 x 下付き 左小かっこ 太字斜体 n プラス 太字 1 右小かっこ じょう 太字 t 太字 r 太字 u 太字 e イコール 太字 大文字 F 太字 x 下付き 太字斜体 n じょう 太字 t 太字 r 太字 u 太字 e プラス 太字 大文字 G 太字 u 下付き 太字斜体 n プラス 太字 w 下付き 太字斜体 n 𝐱𝒏+𝟏 𝐭𝐫𝐮𝐞 = 𝐅𝐱𝒏 𝐭𝐫𝐮𝐞 + 𝐆𝐮𝒏 + 𝐰𝒏(75) 28with a measurement z at the nth time step: 29太字 z 下付き 太字斜体 n イコール 太字 大文字 H 太字 x 下付き 太字斜体 n じょう 太字 t 太字 r 太字 u 太字 e プラス 太字 v 下付き 太字斜体 n 𝐳𝒏 = 𝐇𝐱𝒏 𝐭𝐫𝐮𝐞 + 𝐯𝒏(76) 30in which   and   are the true state vectors at the (n+1)th 31  and nth time steps, respectively; F is the state transition matrix, 32  G is the control input matrix, un is the control vector, wn is the 33  process noise, zn is the observation vector of the true state xn, 34  H is the observation matrix, vn is the observation noise (or 35  measurement noise). wn and vn follow zero mean normal 36  distribution with the respective covariance matrices Qn and Rn: 37太字 x 下付き 左小かっこ 太字斜体 n プラス 太字 1 右小かっこ じょう 太字 t 太字 r 太字 u 太字 e 𝐱𝒏+𝟏 𝐭𝐫𝐮𝐞太字 x 下付き 太字斜体 n じょう 太字 t 太字 r 太字 u 太字 e 𝐱𝒏 𝐭𝐫𝐮𝐞太字 w 下付き 太字斜体 n 、 、 \~ 、 大文字 N 左小かっこ 太字 0 ,、 太字 大文字 Q 下付き 太字斜体 n 右小かっこ 𝐰𝒏 \~ 𝑁(𝟎,𝐐𝒏)(77) 38太字 v 下付き 太字斜体 n 、 、 \~ 、 大文字 N 左小かっこ 太字 0 ,、 太字 大文字 R 下付き 太字斜体 n 右小かっこ 𝐯𝒏 \~ 𝑁(𝟎, 𝐑𝒏)(78) 39The Kalma filter algorithm consists of predict-update cycle for 40  each time step. For the predict part, the predicted state 41  estimate () and predicted estimate covariance ( 42  are calculated: 43太字 x ハット 下付き 、 太字斜体 n ,太字斜体 n マイナス 太字 1 最後 下付き文字 𝐱̂𝒏,𝒏−𝟏太字 大文字 P 下付き 、 太字斜体 n ,太字斜体 n マイナス 太字 1 最後 下付き文字 右小かっこ 𝐏𝒏,𝒏−𝟏)太字 x ハット 下付き 、 太字斜体 n ,太字斜体 n マイナス 太字 1 最後 下付き文字 イコール 太字 大文字 F 太字 x ハット 下付き 、 太字斜体 n マイナス 太字 1 ,太字斜体 n マイナス 太字 1 最後 下付き文字 プラス 太字 大文字 G 太字 u 下付き 、 太字斜体 n マイナス 太字 1 最後 下付き文字 𝐱̂𝒏,𝒏−𝟏 = 𝐅𝐱̂𝒏−𝟏,𝒏−𝟏 + 𝐆𝐮𝒏−𝟏(79) 44太字 大文字 P 下付き 、 太字斜体 n ,太字斜体 n マイナス 太字 1 最後 下付き文字 イコール 太字 大文字 F 太字 大文字 P 下付き 、 太字斜体 n マイナス 太字 1 ,太字斜体 n マイナス 太字 1 最後 下付き文字 、 太字 大文字 F じょう 大文字 T プラス 太字 大文字 Q 下付き 、 太字斜体 n マイナス 太字 1 最後 下付き文字 𝐏𝒏,𝒏−𝟏 = 𝐅𝐏𝒏−𝟏,𝒏−𝟏𝐅 𝑇 + 𝐐𝒏−𝟏(80) 45For the update part: 46太字 大文字 K 下付き 太字斜体 n イコール 太字 大文字 P 下付き 、 太字斜体 n ,太字斜体 n マイナス 太字 1 最後 下付き文字 、 太字 大文字 H じょう 大文字 T 、 始めかっこ 太字 大文字 H 太字 大文字 P 下付き 、 太字斜体 n ,太字斜体 n マイナス 太字 1 最後 下付き文字 、 太字 大文字 H じょう 大文字 T プラス 太字 大文字 R 下付き 太字斜体 n 、 , 終わりかっこ じょう 、 マイナス 1 最後 上付き文字 𝐊𝒏 = 𝐏𝒏,𝒏−𝟏𝐇 𝑇(𝐇𝐏𝒏,𝒏−𝟏𝐇 𝑇 + 𝐑𝒏) −1(81) 47太字 x ハット 下付き 、 太字斜体 n ,太字斜体 n 最後 下付き文字 イコール 太字 x ハット 下付き 、 太字斜体 n ,太字斜体 n マイナス 太字 1 最後 下付き文字 プラス 太字 大文字 K 下付き 太字斜体 n 、 左小かっこ 太字 z 下付き 太字斜体 n マイナス 太字 大文字 H 太字 x ハット 下付き 、 太字斜体 n ,太字斜体 n マイナス 太字 1 最後 下付き文字 右小かっこ 𝐱̂𝒏,𝒏 = 𝐱̂𝒏,𝒏−𝟏 + 𝐊𝒏(𝐳𝒏 − 𝐇𝐱̂𝒏,𝒏−𝟏)(82) 48太字 大文字 P 下付き 、 太字斜体 n ,太字斜体 n 最後 下付き文字 イコール 左小かっこ 太字 大文字 I. 、 マイナス 、 太字 大文字 K 下付き 太字斜体 n 、 太字 大文字 H 右小かっこ 太字 大文字 P 下付き 、 太字斜体 n ,太字斜体 n マイナス 太字 1 最後 下付き文字 𝐏𝒏,𝒏 = (𝐈 −  𝐊𝒏𝐇)𝐏𝒏,𝒏−𝟏(83) 49in which Kn is the Kalman gain,   is the updated state 50  estimate,  is the updated estimate covariance. By running 51  the above predict-update cycle, the state at each time step can 52  be well estimated by using the information from both the model 53  and the observations/measurements. An information flow in 54  the Kalman filter cycle is shown in Fig. 10b. 55太字 x ハット 下付き 、 太字斜体 n ,太字斜体 n 最後 下付き文字 𝐱̂𝒏,𝒏太字 大文字 P 下付き 、 太字斜体 n ,太字斜体 n 最後 下付き文字 𝐏𝒏,𝒏Different from the sequential data processing by Kalman filter, 56  another typical DA approach, i.e. four-dimensional variational 57  assimilation (4D-Var), features non-sequential and has been 58  widely applied in weather forecasting.264 The target of 4D-Var is 59  to find the best estimate of the initial state of a dynamic system 60  by minimizing the misfit between modelled/simulated results 61  and the measurements/observations over a time window 62  (called assimilation window). To this end, the cost function J is 63  built by the combination of a background term Jb and an 64  observation term Jo in the assimilation window (t0, tn): 65大文字 J 始めかっこ 太字 x 下付き 太字 0 、 , 終わりかっこ イコール 大文字 J 下付き b プラス 大文字 J 下付き o 𝐽(𝐱𝟎) = 𝐽𝑏 + 𝐽𝑜(84) 66大文字 J 下付き 太字斜体 b イコール 1 ぶんぼ 2 始めかっこ 太字 x 下付き 太字 0 マイナス 太字 x 下付き 太字 0 じょう 太字斜体 b 、 , 終わりかっこ じょう 大文字 T 、 太字 大文字 B じょう 、 マイナス 太字 1 最後 上付き文字 、 左小かっこ 太字 x 下付き 太字 0 マイナス 太字 x 下付き 太字 0 じょう 太字斜体 b 右小かっこ 𝐽𝒃 = 1 2(𝐱𝟎 − 𝐱𝟎 𝒃) 𝑇 𝐁−𝟏(𝐱𝟎 − 𝐱𝟎 𝒃)(85) 67大文字 J 下付き o イコール 和 から 太字斜体 t イコール 太字斜体 t 下付き 太字 0 へ 太字斜体 t 下付き 太字斜体 n の 、 1 ぶんぼ 2 始めかっこ 太字 大文字 H 太字 x 下付き 太字斜体 t マイナス 太字 z 下付き 太字斜体 t 、 , 終わりかっこ じょう 大文字 T 、 太字 大文字 R 下付き 太字斜体 t じょう 左小かっこ マイナス 1 右小かっこ 、 左小かっこ 太字 大文字 H 太字 x 下付き 太字斜体 t マイナス 太字 z 下付き 太字斜体 t 右小かっこ 最後 総和 𝐽𝑜 = ∑ 1 2(𝐇𝐱𝒕 − 𝐳𝒕) 𝑇𝐑𝒕 −1(𝐇𝐱𝒕 − 𝐳𝒕) 𝒕𝒏 𝒕=𝒕𝟎(86) 68in which x0 is the state vector at the initial time,   is the 69  background state at the initial time, B is the background error 70  covariance matrix. Here H is the observation operator, which 71  converts from model space to observation space; zt is the vector 72  of observations at time t. xt can be calculated through the model 73  integration from t0 to ti: 74太字 x 下付き 太字 0 じょう 太字斜体 b 𝐱𝟎 𝒃太字 x 下付き 、 太字斜体 t 下付き 太字斜体 i. 、 最後 下付き文字 イコール 大文字 M 下付き 、 t 下付き 0 右向き矢印 t 下付き i. 、 最後 下付き文字 、 左小かっこ 太字 x 下付き 太字 0 右小かっこ 𝐱𝒕𝒊 = 𝑀𝑡0→𝑡𝑖 (𝐱𝟎)(87) 75イコール 大文字 M 下付き 、 t 下付き 0 、 最後 下付き文字 、 大文字 M 下付き 、 t 下付き 1 、 最後 下付き文字 水平中央の省略記号 大文字 M 下付き 、 t 下付き i. 、 最後 下付き文字 𝑀𝑡0→𝑡𝑖 = 𝑀𝑡0 𝑀𝑡1 ⋯𝑀𝑡𝑖(88) 76where   is the model operator for state vector evolution in 77  each time step. Through the variational approach the initial 78  state vector for the minimum J can be found, and the forecast 79  with higher reliability for the future data is thus available, as a 80  schematic shown in Fig. 10c. Apart from the above two typical 81  approaches, there are other methods like extended Kalman 82  filter, ensemble Kalman filter/smoother, and particle filter, 83  which can be found in previous literature for more details.265, 266 84大文字 M 下付き t 𝑀𝑡85Fig. 10 (a) The probability density distribution of the analysis is 86  improved after combining simulation and observation. (b) The 87  predict-update cycle of Kalman filter. (c) Schematic diagram of 88  the 4D-Var method to minimize the misfit between the 89  observation and model simulation for better forecasting. 903\.2 Data assimilation in decoding electrode processes 91Given the complexity of electrode processes, the microkinetic 92  model may use some assumptions, such RDS approximation, 93  which may cause the loss of some kinetic information hidden in 94  the experimental data. Encouraged by the booming data 95  science integrated with electrochemistry field in recent years, 96ARTICLE Journal Name12 \| J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xxPlease do not adjust marginsPlease do not adjust marginsour group used DA (ensemble Kalman filter and smoother) to 1  contribute to the understanding of the ORR electrode process 2  by using single-crystal Pt as the model electrocatalyst.89 We 3  used a four-electron ORR model via two reaction paths without 4  the RDS assumption, as shown as R1→R3→R4 and R2→R4 (Fig. 5  11a). The free energies of activation for each step at the zero 6  overpotential are defined as  ,  ,  ,  , and the 7  adsorption energy of reaction intermediate \*O and \*OH at zero-8  overpotential   and  . One example of the free energy 9  diagram is shown in Fig. 11b. The state vector xt including ORR 10  current density it was written as: 11増分 大文字 G 下付き 1 じょう 0 ∆𝐺1 0増分 大文字 G 下付き 2 じょう 0 ∆𝐺2 0増分 大文字 G 下付き 3 じょう 0 ∆𝐺3 0増分 大文字 G 下付き 4 じょう 0 ∆𝐺4 0増分 大文字 G 下付き 大文字 O じょう 0 ∆𝐺𝑂 0増分 大文字 G 下付き 大文字 O 大文字 H じょう 0 ∆𝐺𝑂𝐻 0太字 x 下付き 太字斜体 t イコール 左大かっこ 7 基準 1 行列、 i. 下付き t 、次の行、 増分 大文字 G 下付き 1 じょう 0 、次の行、 増分 大文字 G 下付き 2 じょう 0 、次の行、 増分 大文字 G 下付き 3 じょう 0 、次の行、 増分 大文字 G 下付き 4 じょう 0 、次の行、 増分 大文字 G 下付き 大文字 O じょう 0 、次の行、 増分 大文字 G 下付き 大文字 O 大文字 H じょう 0 、 最後 行列 右大かっこ 𝐱𝒕 = \[               𝑖𝑡 ∆𝐺1 0 ∆𝐺2 0 ∆𝐺3 0 ∆𝐺4 0 ∆𝐺𝑂 0 ∆𝐺𝑂𝐻 0 \](89) 12Then the parameters of the free energies were estimated with 13  the potential changes. For the free energies of activation,   14  featured dynamic change with potential while others showed 15  nearly constant. This potential-dependent free energy of 16  activation for the first step (R1 step) is different from the 17  constant assumption in the microkinetic model (Fig. 11c), 18  providing an awareness of microscopic ORR mechanism 19  evolution under various electrochemical driven forces. The 20  above work demonstrates prospective applications of DA into 21  clarifying electrode processes by extracting the hidden 22  information in the experimental data that is usually lost in the 23  knowledge-based microkinetic model. For more details the 24  readership may refer to the original paper.89 25増分 大文字 G 下付き 1 じょう 0 ∆𝐺1 03\.3 Data assimilation in electrode lifetime forecasting 26For practical applications of electrolysers for reaction like water 27  splitting, the stability evaluation usually costs long time, such as 28  thousands of hours or even more.2 Thus, it is necessary to 29  explore effective methods to forecast the electrode lifetime in 30  a short time frame. To this end, based on a simple degradation 31  model (as shown in Fig. 9c) and with the support of the 32  collaborator (Ishii), our group used a nonsequential data 33  assimilation method minimizing the cost function using tree-34  structured Parzen estimator (DMC-TPE) proposed by Ishii et 35  al.,260 to build an assimilation-forecasting-diagnosis cycle as 36  shown in Fig. 11d.91 The state vector xt with time t (  37  ) was built as: 380 小なり t 、 しょうなり イコール 、 t 下付き e n d 0 \< 𝑡 ≤0 小なり t 、 しょうなり イコール 、 t 下付き e n d 𝑡𝑒𝑛𝑑太字 x 下付き 太字斜体 t イコール 、 左大かっこ 2 基準 1 行列、 大文字 E 下付き a. p p 、 縦線 下付き t 、次の行、 分子 、 d 大文字 E 下付き a. p p 最後 分子 , 割る d t 、 縦線 下付き t 、 最後 行列 右大かっこ 𝐱𝒕 = \[ 𝐸𝑎𝑝𝑝\|𝑡 𝑑𝐸𝑎𝑝𝑝 𝑑𝑡\|𝑡 \](90) 39太字 x 下付き 太字斜体 t イコール 大文字 M 左小かっこ 太字 x 下付き 太字 0 ,t 右小かっこ 𝐱𝒕 = 𝑀(𝐱𝟎, 𝑡)(91) 40where   refers to the length of the assimilation window, 41   is the applied potential for the OER, M indicates the 42  degradation model, x0 is the vector consisting of the parameters 43  of the model to be estimated. For the best estimate, a cost 44  function J(x0) was made to evaluate the mismatch between xt 45  and the observation vector zt (the recorded applied potential 46  during the chronopotentiometry testing and its first-order 47  derivative with respect to time): 48t 下付き e n d 𝑡𝑒𝑛𝑑大文字 E 下付き a. p p 𝐸𝑎𝑝𝑝大文字 J 始めかっこ 太字 x 下付き 太字 0 、 , 終わりかっこ イコール 、 、 1 ぶんぼ 2 始めかっこ 太字 x 下付き 太字 0 マイナス 太字 x 下付き 太字 0 じょう 太字斜体 b 、 , 終わりかっこ じょう T 、 太字 大文字 B じょう 、 マイナス 1 最後 上付き文字 、 始めかっこ 太字 x 下付き 太字 0 マイナス 太字 x 下付き 太字 0 じょう 太字斜体 b 、 , 終わりかっこ プラス 和 から t へ t 下付き e n d の 、 1 ぶんぼ 2 始めかっこ 大文字 M 3 番目の始めかっこ 太字 x 下付き 太字 0 ,t , 3 番目のかっこ閉じ マイナス 太字 z 下付き 太字斜体 t 、 , 終わりかっこ じょう T 、 上付き文字 てい , 太字 大文字 R 下付き 太字斜体 t , 最後 てい , じょう 、 マイナス 1 最後 上付き文字 、 左小かっこ 大文字 M 始めかっこ 太字 x 下付き 太字 0 ,t , 終わりかっこ マイナス 太字 z 下付き 太字斜体 t 右小かっこ 最後 総和 𝐽(𝐱𝟎) =   1 2(𝐱𝟎 − 𝐱𝟎 𝒃) T 𝐁−1(𝐱𝟎 − 𝐱𝟎 𝒃) + ∑ 1 2(𝑀(𝐱𝟎, 𝑡) − 𝑡𝑒𝑛𝑑 𝑡49   (92) 50大文字 J 始めかっこ 太字 x 下付き 太字 0 、 , 終わりかっこ イコール 、 、 1 ぶんぼ 2 始めかっこ 太字 x 下付き 太字 0 マイナス 太字 x 下付き 太字 0 じょう 太字斜体 b 、 , 終わりかっこ じょう T 、 太字 大文字 B じょう 、 マイナス 1 最後 上付き文字 、 始めかっこ 太字 x 下付き 太字 0 マイナス 太字 x 下付き 太字 0 じょう 太字斜体 b 、 , 終わりかっこ プラス 和 から t へ t 下付き e n d の 、 1 ぶんぼ 2 始めかっこ 大文字 M 3 番目の始めかっこ 太字 x 下付き 太字 0 ,t , 3 番目のかっこ閉じ マイナス 太字 z 下付き 太字斜体 t 、 , 終わりかっこ じょう T 、 上付き文字 てい , 太字 大文字 R 下付き 太字斜体 t , 最後 てい , じょう 、 マイナス 1 最後 上付き文字 、 左小かっこ 大文字 M 始めかっこ 太字 x 下付き 太字 0 ,t , 終わりかっこ マイナス 太字 z 下付き 太字斜体 t 右小かっこ 最後 総和 𝐳𝒕) T𝐑𝒕 −1(𝑀(𝐱𝟎, 𝑡) − 𝐳𝒕)where   is the background vector, B is the covariance matrix 51  of the background error, Rt is the observation error matrix. By 52  the DA method with using 34% of the whole lifetime data, we 53  realized a lifetime forecast with the error lower than 10% for a 54  \~900-h long-term stability testing, and it showed much better 55  forecasting capability than the method by a simple fitting (Fig. 56  11e). Further diagnosis showed that the dissolution factor 57  (noted k’ herein) changed remarkably with the increase in the 58  assimilation window (Fig. 11f), indicating that k’ was particularly 59  important in the degradation mode for accurate forecast. This 60  DA-assistant degradation model suggested meaningful 61  interpretation for the dissolution process towards various 62  materials and conditions: NiOx preferred dissolution followed 63  by CoOx and FeOx (Fig. 11g); protons might accelerate the 64  electrode dissolution during OER (Fig. 11h). For further 65  improved capability of electrode lifetime forecasting, the 66  present degradation model needs to be updated, as well as 67  more reliable error matrices. 68太字 x 下付き 太字 0 じょう 太字斜体 b 𝐱𝟎 𝒃69Fig. 11 (a) The ORR paths, (b) the free energy diagram of one 70  ORR path R1→R3→R4 at zero-overpotential, (c) the parameter 71   obtained by ensemble Kalman filter/smoother (the solid 72  origin line with the area of solid orange as the standard 73  deviations) and by fitting the kinetic model (the green solid line). 74  Reproduced from Ref. 89 with permission. Copyright 2022, 75  American Chemical Society. (d) The assimilation-forecasting-76  diagnosis cycle for the OER anode lifetime forecasting, (e) the 77  forecasting degradation curve by using 34% of the whole 78  experimental data via DA, (f) the dissolution factor k’ in the 79  degradation model by dynamic DAs with various assimilation 80  windows, and k’ obtained from the DA method for different (g) 81  materials and (h) conditions. Reproduced from Ref. 91 with 82  permission. Copyright 2025, American Chemical Society. 83増分 大文字 G 下付き 1 じょう 0 ∆𝐺1 04 A general framework for unifying electrode 84  processes 85An electrode process encompasses various phenomena, such as 86  interfacial reaction, mass transfer, electrode material 87  reconstruction, dissolution, and surface charging. This makes 88  the electrode process much more complicated and 89Journal Name  ARTICLEThis journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 \| 13Please do not adjust marginsPlease do not adjust marginscomprehensive than just considering the “traditional” current-1  potential relationship. In this section, we propose a unified 2  framework for coding and re-understanding electrode 3  processes, which may favour the computer language-based 4  descriptions. We hope that this heuristic framework can 5  promote the overall understanding of electrode processes 6  including both temporal and steady states, benefitting the 7  combination of electrochemical kinetics and data science for 8  accelerating the development of sustainable energy-based 9  technologies. 104\.1 Electrode process matrix 11Matrix is capable of organising information effectively and 12  provide clear formulas for mathematical calculations, especially 13  for the present information age, and they have showed 14  widespread applications in many fields.267-269 Given the 15  electrode processes, there are usually many species under 16  various conditions, the power of matrix is obvious and it has 17  been used to investigate multistep reactions with a matrix 18  impedance theory developed by Harrington.270 Herein, we start 19  with considering a potential-driven electrode process under 20  steady-state with all kinds of states like current ( ), surface 21  coverage of the kth species ( ), the lth bond length of the 22  surface active species (), charge at position x (), and free 23  energy of reaction/activation for the tth step (), and all the 24  above states are quantified by related 25  functions/methods/mechanisms: 26大文字 I. 𝐼シータ 下付き k 𝜃𝑘r 下付き l 𝑟𝑙q 下付き x 𝑞𝑥大文字 G 下付き t 𝐺𝑡\[                 (93) 27左大かっこ 6 基準 3 行列、 大文字 I. 、 イコール 、 f 左小かっこ 大文字 E 右小かっこ 、次の行、 シータ 下付き k 最後 列 , イコール 最後 列 , g 左小かっこ 大文字 E 右小かっこ 、次の行、 r 下付き l 最後 列 , イコール 最後 列 , h 左小かっこ 大文字 E 右小かっこ 、次の行、 q 下付き x 最後 列 , イコール 最後 列 , p 左小かっこ 大文字 E 右小かっこ 、次の行、 大文字 G 下付き t 最後 列 , イコール 最後 列 , 大文字 G 左小かっこ 大文字 E 右小かっこ 、次の行、 垂直の省略記号 最後 列 , 垂直の省略記号 最後 列 , 垂直の省略記号 最後 行列 右大かっこ 𝐼 = 𝑓(𝐸) 𝜃𝑘 = 𝑔(𝐸) 𝑟𝑙 = ℎ(𝐸) 𝑞𝑥 = 𝑝(𝐸) 𝐺𝑡 = 𝐺(𝐸) ⋮ ⋮ ⋮ \]where E is the potential, f, g, h, p, and G are the descriptions 28  between E and the related states, e.g., Bulter-Volmer equation, 29  steady-state approximation-based surface coverage equation, 30  reconstruction effect, electric double layer effect, and 31  transition-state theory, respectively. We generalize the above 32  equation including n states sji (j = 1, 2, ..., n) under the ith applied 33  potential Ei: 34\[      (94) 35左大かっこ 3 基準 1 行列、 s 下付き 1 i. 、次の行、 垂直の省略記号 、次の行、 s 下付き n i. 、 最後 行列 右大かっこ イコール   左大かっこ 3 基準 1 行列、 f 下付き 1 i. 、 左小かっこ 大文字 E 下付き i. 右小かっこ 、次の行、 垂直の省略記号 、次の行、 f 下付き n i. 、 左小かっこ 大文字 E 下付き i. 右小かっこ 最後 行列 右大かっこ 𝑠1𝑖 ⋮ 𝑠𝑛𝑖 \] =  \[ 𝑓1𝑖(𝐸𝑖) ⋮ 𝑓𝑛𝑖(𝐸𝑖) \]where fji (j = 1, 2, ..., n) is the description function bridging Ei and 36  sji. For the left side we use a state vector Si to include all the 37  electrode states at Ei, the right side we use Fi(Ei) for simplifying 38  the expression. Consider all the applied potentials, we use the 39  set of Si, i.e. , to note the related state vectors and marked 40  as S. Similarly, we use the set of Fi, i.e. , to label all the 41  related description functions and marked as F. For all the 42  applied potentials, , they are labelled as E. Then we have: 43左中かっこ 太字 大文字 S 下付き 太字 i. 右中かっこ {𝐒𝐢}左中かっこ 太字 大文字 F 下付き 太字 i. 右中かっこ {𝐅𝐢}左中かっこ 太字 大文字 E 下付き 太字 i. 右中かっこ {𝐄𝐢}太字 大文字 S イコール 太字 大文字 F 左小かっこ 太字 大文字 E 右小かっこ 𝐒 = 𝐅(𝐄)(95) 44where we call S, F, and E as state space, transition function 45  space, and potential space, respectively. The above equation is 46  extended as a   matrix with m potentials and each 47  potential n states as following: 48n 倍 m 𝑛 × 𝑚\[  (96) 49左大かっこ 3 基準 3 行列、 s 下付き 11 最後 列 , 水平中央の省略記号 最後 列 , s 下付き 1 m 、次の行、 垂直の省略記号 最後 列 , 右下がりの省略記号 最後 列 , 垂直の省略記号 、次の行、 s 下付き n 1 最後 列 , 水平中央の省略記号 最後 列 , s 下付き n m 、 最後 行列 右大かっこ イコール 左大かっこ 3 基準 3 行列、 f 下付き 11 、 左小かっこ 大文字 E 下付き 1 右小かっこ 最後 列 , 水平中央の省略記号 最後 列 , f 下付き 1 m 、 左小かっこ 大文字 E 下付き m 右小かっこ 、次の行、 垂直の省略記号 最後 列 , 右下がりの省略記号 最後 列 , 垂直の省略記号 、次の行、 f 下付き n 1 、 左小かっこ 大文字 E 下付き 1 右小かっこ 最後 列 , 水平中央の省略記号 最後 列 , f 下付き n m 、 左小かっこ 大文字 E 下付き m 右小かっこ 最後 行列 右大かっこ 𝑠11 ⋯ 𝑠1𝑚 ⋮ ⋱ ⋮ 𝑠𝑛1 ⋯ 𝑠𝑛𝑚 \] = \[ 𝑓11(𝐸1) ⋯ 𝑓1𝑚(𝐸𝑚) ⋮ ⋱ ⋮ 𝑓𝑛1(𝐸1) ⋯ 𝑓𝑛𝑚(𝐸𝑚) \]On the left side, each column refers to the electrode states 50  under a potential; on the right side, the row direction displays 51  the applied potentials from E1 to Em. The above matrix formula 52  includes all states during an electrode process under applied 53  potentials and we call this matrix equation as electrode process 54  matrix. 55We further consider the dynamic response, so that the time-56  related (t) electrode process matrix is written as: 57太字 大文字 S 左小かっこ t 右小かっこ イコール 太字 大文字 F 左小かっこ 太字 大文字 E 左小かっこ t 右小かっこ ,t 右小かっこ 𝐒(𝑡) = 𝐅(𝐄(𝑡), 𝑡)(97) 58\[  59   (98) 60左大かっこ 3 基準 3 行列、 s 下付き 、 11,t 最後 下付き文字 最後 列 , 水平中央の省略記号 最後 列 , s 下付き 、 1 m ,t 最後 下付き文字 、次の行、 垂直の省略記号 最後 列 , 右下がりの省略記号 最後 列 , 垂直の省略記号 、次の行、 s 下付き 、 n 1,t 最後 下付き文字 最後 列 , 水平中央の省略記号 最後 列 , s 下付き 、 n m ,t 最後 下付き文字 、 最後 行列 右大かっこ イコール 左大かっこ 3 基準 3 行列、 f 下付き 11 、 左小かっこ 大文字 E 下付き 1 、 左小かっこ t 右小かっこ ,t 右小かっこ 最後 列 , 水平中央の省略記号 最後 列 , f 下付き 1 m 、 左小かっこ 大文字 E 下付き m 、 左小かっこ t 右小かっこ ,t 右小かっこ 、次の行、 垂直の省略記号 最後 列 , 右下がりの省略記号 最後 列 , 垂直の省略記号 、次の行、 f 下付き n 1 、 左小かっこ 大文字 E 下付き 1 、 左小かっこ t 右小かっこ ,t 右小かっこ 最後 列 , 水平中央の省略記号 最後 列 , f 下付き n m 、 左小かっこ 大文字 E 下付き m 、 左小かっこ t 右小かっこ ,t 右小かっこ 最後 行列 右大かっこ 𝑠11,𝑡 ⋯ 𝑠1𝑚,𝑡 ⋮ ⋱ ⋮ 𝑠𝑛1,𝑡 ⋯ 𝑠𝑛𝑚,𝑡 \] = \[ 𝑓11(𝐸1(𝑡), 𝑡) ⋯ 𝑓1𝑚(𝐸𝑚(𝑡), 𝑡) ⋮ ⋱ ⋮ 𝑓𝑛1(𝐸1(𝑡), 𝑡) ⋯ 𝑓𝑛𝑚(𝐸𝑚(𝑡), 𝑡) \]Under some circumstances, we use current rather than 61  potential to conduct the electrode process, then we have a 62  similar formula of the electrode process matrix: 63太字 大文字 S プライム 、   左小かっこ t 右小かっこ イコール 太字 大文字 G 左小かっこ 太字 大文字 I. 左小かっこ t 右小かっこ ,t 右小かっこ 𝐒′ (𝑡) = 𝐆(𝐈(𝑡), 𝑡)(99) 64where   is the state space under the current control, G is 65  the related transition function space, and I is current space. We 66  then give a general equation: 67太字 大文字 S プライム 、   左小かっこ t 右小かっこ 𝐒′ (𝑡)太字 大文字 S 左小かっこ t 右小かっこ イコール 太字 大文字 F 左小かっこ 太字 大文字 U 左小かっこ t 右小かっこ ,t 右小かっこ 𝐒(𝑡) = 𝐅(𝐔(𝑡), 𝑡)(100) 68where S(t) here refers to the general time-dependent state 69  space, F here refers to the general transition function space, U 70  here refers to electrical quantity space. From this general 71  electrode process matrix, we can know that an electrode 72  process includes all the states (such as reaction, transport, and 73  species distribution) vary with electricity (such as potential and 74  current) and time under a given electrode-electrolyte system. 75  Herein, “given” means conditions, like temperature, pressure, 76  concentration, and convection, are set in the beginning. 774\.2 Re-understanding electrode processes 78Based on the above electrode process matrix, if we assume a 79  constant Ei(t): 80分子 、 d 大文字 E 下付き i. 最後 分子 , 割る d t イコール 0 𝑑𝐸𝑖 𝑑𝑡= 0(101) 81we set: 82の極限 t   右向き矢印 s h o r t の 、 太字 大文字 S 左小かっこ t 右小かっこ 、 イコール 太字 大文字 S 下付き 太字 t 太字 s 、   lim 𝑡 →𝑠ℎ𝑜𝑟𝑡 𝐒(𝑡) = 𝐒𝐭𝐬(102) 83の極限 t   右向き矢印 p r o p e r の 、 太字 大文字 S 左小かっこ t 右小かっこ 、 イコール 太字 大文字 S 下付き 太字 s 太字 s lim 𝑡 →𝑝𝑟𝑜𝑝𝑒𝑟 𝐒(𝑡) = 𝐒𝐬𝐬(103) 84の極限 t   右向き矢印 l o n g 、 t e r m の 、 太字 大文字 S 左小かっこ t 右小かっこ 、 イコール 太字 大文字 S 下付き 太字 s 太字 b lim 𝑡 →𝑙𝑜𝑛𝑔 𝑡𝑒𝑟𝑚 𝐒(𝑡) = 𝐒𝐬𝐛(104) 85So that   with short time refers to the transient electrode 86  process,  with proper time framework refers to the steady-87  state process, and   with a long-term period reflects the 88  stability testing. From the above viewpoint, the activity and 89  stability of an electrocatalyst are essentially the same electrode 90  state in short and long terms, respectively. This suggests that 91  we should consider both electrode activity and stability issues 92  in one framework, which exactly fits the work discussed in Fig. 93  9b and Fig. 9c. Selectivity can be understood by comparing the 94  relevant states, such as the distinctive reaction currents 95  corresponding to different products. According to eq. 100, to 96  some extent, our task on electrode process research is to fill the 97  electrode process matrix as possible: the number of states may 98  be infinite and the more we can fill the better we understand 99  the electrode process, from both macroscopic and microscopic 100  points of view. We may integrate various approaches, such as 101  microkinetic analysis, electrical double layer modelling, in-102太字 大文字 S 下付き 太字 t 太字 s 𝐒𝐭𝐬太字 大文字 S 下付き 太字 s 太字 s 𝐒𝐬𝐬太字 大文字 S 下付き 太字 s 太字 b 𝐒𝐬𝐛ARTICLE Journal Name14 \| J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xxPlease do not adjust marginsPlease do not adjust marginssitu/operando electrochemical spectroscopy, electrochemical 1  probe-based techniques, and in silico calculations. 2A further discussion comes to decomposition of the matrix. We 3  take the steady-state electrode process matrix as an example, 4  i.e. , which can be decomposed as the product of 5   () and   (Ed) matrices: 6太字 大文字 S イコール 太字 大文字 F 左小かっこ 太字 大文字 E 右小かっこ 𝐒 = 𝐅(𝐄)n 倍 p 𝑛 × 𝑝太字 大文字 M 下付き 太字 大文字 F 、 始めかっこ 太字 大文字 E , 終わりかっこ 𝐌𝐅(𝐄)p 倍 m 𝑝 × 𝑚太字 大文字 S イコール 太字 大文字 F 始めかっこ 太字 大文字 E , 終わりかっこ イコール 左大かっこ 3 基準 3 行列、 s 下付き 11 最後 列 , 水平中央の省略記号 最後 列 , s 下付き 1 m 、次の行、 垂直の省略記号 最後 列 , 右下がりの省略記号 最後 列 , 垂直の省略記号 、次の行、 s 下付き n 1 最後 列 , 水平中央の省略記号 最後 列 , s 下付き n m 、 最後 行列 右大かっこ イコール 左大かっこ 3 基準 3 行列、 f 下付き 11 、 2 番目の始めかっこ 大文字 E 下付き 1 、 , 2 番目のかっこ閉じ 最後 列 , 水平中央の省略記号 最後 列 , f 下付き 1 m 、 2 番目の始めかっこ 大文字 E 下付き m 、 , 2 番目のかっこ閉じ 、次の行、 垂直の省略記号 最後 列 , 右下がりの省略記号 最後 列 , 垂直の省略記号 、次の行、 f 下付き n 1 、 2 番目の始めかっこ 大文字 E 下付き 1 、 , 2 番目のかっこ閉じ 最後 列 , 水平中央の省略記号 最後 列 , f 下付き n m 、 2 番目の始めかっこ 大文字 E 下付き m 、 , 2 番目のかっこ閉じ 、 最後 行列 右大かっこ 𝐒 = 𝐅(𝐄) = \[ 𝑠11 ⋯ 𝑠1𝑚 ⋮ ⋱ ⋮ 𝑠𝑛1 ⋯ 𝑠𝑛𝑚 \] = \[ 𝑓11(𝐸1) ⋯ 𝑓1𝑚(𝐸𝑚) ⋮ ⋱ ⋮ 𝑓𝑛1(𝐸1) ⋯ 𝑓𝑛𝑚(𝐸𝑚) \] 7イコール 、 、 太字 大文字 M 下付き 太字 大文字 F 、 始めかっこ 太字 大文字 E , 終わりかっこ 、 、 太字 大文字 E 下付き 太字 d =  𝐌𝐅(𝐄) 𝐄𝐝(105) 8太字 大文字 M 下付き 太字 大文字 F 、 始めかっこ 太字 大文字 E , 終わりかっこ 、   イコール 、 左大かっこ 3 基準 3 行列、 m 下付き f 11 最後 列 , 水平中央の省略記号 最後 列 , m 下付き f 1 p 、次の行、 垂直の省略記号 最後 列 , 右下がりの省略記号 最後 列 , 垂直の省略記号 、次の行、 m 下付き f n 1 最後 列 , 水平中央の省略記号 最後 列 , m 下付き f n p 、 最後 行列 右大かっこ 𝐌𝐅(𝐄) = \[ 𝑚𝑓11 ⋯ 𝑚𝑓1𝑝 ⋮ ⋱ ⋮ 𝑚𝑓𝑛1 ⋯ 𝑚𝑓𝑛𝑝 \](106) 9太字 大文字 E 下付き 太字 d イコール 、 左大かっこ 3 基準 3 行列、 大文字 E 下付き 11 最後 列 , 水平中央の省略記号 最後 列 , 大文字 E 下付き 1 m 、次の行、 垂直の省略記号 最後 列 , 右下がりの省略記号 最後 列 , 垂直の省略記号 、次の行、 大文字 E 下付き p 1 最後 列 , 水平中央の省略記号 最後 列 , 大文字 E 下付き p m 、 最後 行列 右大かっこ 𝐄𝐝 = \[ 𝐸11 ⋯ 𝐸1𝑚 ⋮ ⋱ ⋮ 𝐸𝑝1 ⋯ 𝐸𝑝𝑚 \](107) 10where   is the transition matrix related to E,   is the 11  decomposed potential matrix. It suggests that  contains 12  intrinsic information of the electrified electrode-electrolyte 13  interface as well as the possible bulk influence. Each column of 14   implies the redistribution of the driven force for the 15  evolution of electrode state vector with applying potential. For 16  a clear explanation, we consider an electrode reaction under 17  steady state with sufficient convection and potential 18  distribution in the electrified interface as schematically 19  represented in Fig. 12, with three different regions for potential 20  change: E1i, E2i, and E3i, respectively. Briefly, E1i is caused by 21  electron spillover from the electrode to solution due to 22  quantum mechanical effects.22, 25, 219, 271-273 As for E2i and E3i, 23  they can be understood by the ions in the compact layer and 24  thermal motion of ions with electrostatic force in the diffuse 25  layer, respectively.19-21, 23 The understanding of E3i can be 26  further detailed by density functional theory with more 27  accurate description especially in high-concentration 28  electrolyte.274 We take one column of S and note as Si (n rows); 29  similarly, one column of  and note as . According to the 30  above matrix decomposition, we have: 31太字 大文字 M 下付き 太字 大文字 F 、 始めかっこ 太字 大文字 E , 終わりかっこ 𝐌𝐅(𝐄)太字 大文字 E 下付き 太字 d 𝐄𝐝太字 大文字 M 下付き 太字 大文字 F 、 始めかっこ 太字 大文字 E , 終わりかっこ 𝐌𝐅(𝐄)太字 大文字 E 下付き 太字 d 𝐄𝐝太字 大文字 E 下付き 太字 d 𝐄𝐝太字 大文字 E 下付き 、 太字 d ,太字斜体 i. 最後 下付き文字 𝐄𝐝,𝒊太字 大文字 S 下付き 太字斜体 i. イコール 太字 大文字 M 下付き 太字 大文字 F 、 始めかっこ 太字 大文字 E 下付き 、 太字斜体 d ,太字斜体 i. 最後 下付き文字 、 , 終わりかっこ 、 、 太字 大文字 E 下付き 、 太字斜体 d ,太字斜体 i. 最後 下付き文字 𝐒𝒊 = 𝐌𝐅(𝐄𝒅,𝒊) 𝐄𝒅,𝒊(108) 32太字 大文字 S 下付き 太字斜体 i. イコール 、 左大かっこ 3 基準 1 行列、 s 下付き 1 i. 、次の行、 垂直の省略記号 、次の行、 s 下付き n i. 、 最後 行列 右大かっこ 𝐒𝒊 = \[ 𝑠1𝑖 ⋮ 𝑠𝑛𝑖 \](109) 33太字 大文字 M 下付き 太字 大文字 F 、 始めかっこ 太字 大文字 E 下付き 、 太字斜体 d ,太字斜体 i. 最後 下付き文字 、 , 終わりかっこ イコール 、 左大かっこ 3 基準 3 行列、 m 下付き f 11 最後 列 , m 下付き f 12 最後 列 , m 下付き f 13 、次の行、 垂直の省略記号 最後 列 , 右下がりの省略記号 最後 列 , 垂直の省略記号 、次の行、 m 下付き f n 1 最後 列 , m 下付き f n 2 最後 列 , m 下付き f n 3 、 最後 行列 右大かっこ 𝐌𝐅(𝐄𝒅,𝒊) =  \[ 𝑚𝑓11 𝑚𝑓12 𝑚𝑓13 ⋮ ⋱ ⋮ 𝑚𝑓𝑛1 𝑚𝑓𝑛2 𝑚𝑓𝑛3 \](110) 34太字 大文字 E 下付き 、 太字斜体 d ,太字斜体 i. 最後 下付き文字 イコール 、 左大かっこ 3 基準 1 行列、 大文字 E 下付き 1 i. 、次の行、 大文字 E 下付き 2 i. 、次の行、 大文字 E 下付き 3 i. 、 最後 行列 右大かっこ 𝐄𝒅,𝒊 = \[ 𝐸1𝑖 𝐸2𝑖 𝐸3𝑖 \](111) 35It is obvious that the potential Ei (vs. bulk potential) is the sum 36  of the decomposed potentials: 37大文字 E 下付き i. イコール   和 から j イコール 1 へ 3 の 大文字 E 下付き j i. 、 𝐸𝑖 =  ∑ 𝐸𝑗𝑖 3 𝑗=1(112) 38It is thus noted that each row in the transition matrix  39  contains the influence of electrified interface structure on the 40  corresponding state. For example, if s1i represents for the 41  current, then mf11, mf12, and mf13 serve as the process 42  “admittances” for electrode-Helmholtz layer, Helmholtz layer, 43  and diffuse layer. Despite being in its infancy as a 44  theoretical/conceptual framework, we hope that the 45  decomposition of electrode process matrix can provide 46  heuristic insights into complex electrode processes. 47太字 大文字 M 下付き 太字 大文字 F 、 始めかっこ 太字 大文字 E 下付き 、 太字斜体 d ,太字斜体 i. 最後 下付き文字 、 , 終わりかっこ 𝐌𝐅(𝐄𝒅,𝒊)48   49Fig. 12 Schematic illustration of an electrode-electrolyte 50  interface. Four regions are simply distinguished: electrode-51  Helmholtz layer interface, Helmholtz layer, diffuse layer, and 52  bulk. “Conc.” in the figure means concentration. 53Conclusions 54In this Tutorial Review, we consider the electrified electrode-55  electrolyte interface, comprising reactive species, electrode 56  reactions, and electrode materials, as a platform to discuss 57  some typical fundamental aspects of electrode processes, 58  including reaction kinetics, mass transport, and material 59  reconstruction. Furthermore, in this Review, we employ only a 60  few typical reactions, such as HER, however of course the 61  concept provided here can be applied to a wide spectrum of 62  electrode processes, for instance nitrogen reduction reaction or 63  CO2 reduction reaction, CO2 capturing reaction, and more.275-281 64  While the concept of a RDS remains effective for many reaction 65  systems, it can be difficult to define in complex cases, 66  highlighting the need for non-RDS-based microkinetic analysis. 67  From a practical perspective, simplifying these complex 68  frameworks leads to more accessible models, such as the 69  energetic span model introduced in this review. Beyond the 70  well-recognized impact of reactant diffusion on electrode 71  performance, as the well-known diffusion-controlled region in 72  the ORR, the thermodynamic effects arising from product 73  accumulation also warrant careful consideration. The activity 74  and stability of electrodes can be understood within a unified 75  framework grounded in fundamental kinetics, offering deeper 76  insights into deactivation phenomena for electrode processes. 77  DA serves as an effective strategy to integrate experimental 78  data and reaction models, promoting the understanding of 79  electrode processes. Recognising that an electrode process 80  includes all electrode states, such as current, surface coverage, 81  charge distribution, and electrode structure, thus, we propose 82  a matrix-based framework to mathematically describe and 83  deconstruct electrode behaviour. This perspective suggests that 84  electrode activity and stability are descriptions at different 85  timescales of the same electrode state. Selectivity can be 86  understood by comparing the relevant states. Further matrix 87  decomposition may yield more detailed structural information 88  related to the electrode-electrolyte interface. 89Journal Name  ARTICLEThis journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 \| 15Please do not adjust marginsPlease do not adjust marginsHowever, in view of dynamic electrode processes, especially the 1  reconstruction of electrode materials under strong 2  oxidative/reductive electrochemical conditions, the possibly 3  resultant changes of reaction mechanisms may lead to different 4  electrode performance; and the performance may in turn 5  influence the electrode reconstruction behaviour. Building such 6  a reconstruction-mechanism-performance feedback loop and 7  clarifying its evolution with various electrode states is 8  challenging. Another concern comes to the steady state 9  approximation, which is widely used in electrocatalysis. Even 10  though this approximation shows greater tolerant than the RDS 11  assumption for microkinetic analysis, it is still based on an 12  assumption that the reaction intermediates are independent 13  with reaction time. This causes incomplete description and 14  insufficient understanding of the electrode process. As for the 15  microkinetic analysis, the more complicated (usually more 16  parameters and even atomic scale) of the model provides more 17  precise description of the target process; however, it requires 18  greater efforts for practical applications. Similar to density 19  function theory that balances the in-depth description and low-20  cost practicability, it may be borne in mind that a proper 21  simplification of a comprehensive electrode process model is 22  necessary to connect the microscale and macroscale smoothly 23  and solve practical issues. As demonstrated in this review, the 24  DA owns promising capability to refine the theoretical model-25  based description by leveraging the experimental/observed 26  data. One of the keys to improving the generalizability of the 27  model by DA is using reliable (also objective) model errors 28  (matrices), which contributes to further combination of DA and 29  electrochemistry. 30Under the circumstances of sustainable energy-based modern 31  electrochemistry industries for high-value chemicals and energy 32  conversion/storage, the deep integration of theoretical 33  fundamentals (e.g., microscopic electrode process analysis, 34  double-layer interface investigation), in-situ/operando 35  electrochemical techniques (e.g., in-situ/operando 36  electrochemical Raman/infrared/X-ray absorption 37  spectroscopy), in-silico calculations (e.g., molecular dynamic 38  simulation, density function theory-based calculation, 39  computational fluid dynamics), and data science (e.g., machine 40  learning and data assimilation) is becoming increasingly 41  attractive for contributing to the resolutions of present 42  challenges in electrode processes. This indicates that apart from 43  highly developed devices for benefitting data collection, the 44  knowledge and ability to employ mathematics as well as 45  programming are of growing importance to grasp the key points 46  in electrode processes, especially for the fact that we are now 47  in the artificial intelligence (AI) era.282 283-288 In our opinion, this 48  requires higher standards for our human beings: know “why” 49  and “how” and let the machine do “what”. To this end, is it 50  necessary for us to master every detail of electrode processes? 51  Herein, we do not give an explicit answer “yes” or “no”. 52  Nonetheless, after the golden age of electrochemistry in the 53  mid-to-late 20th century, we believe that another boom of this 54  field is coming in the trend of re-realizing the importance of 55  mathematics in advancing modern electrocatalysis and 56  electrochemical energy conversion/storage triggered by the AI 57  wave. What to do for treating electrode processes may follow a 58  previous comment for interfacial electrochemistry: “In the 59  future, the role of interfacial electrochemistry in many fields of 60  science and technology may be similar to some extent to that of 61  mathematics: One does not have to be a mathematician to be a 62  first class chemist or biologist, but it is necessary to know and be 63  able to apply some aspects of mathematics.”289 64Author contributions 65MW performed formal analysis, investigation, visualization, 66  methodology, writing – original draft. KS conducted 67  conceptualization, supervision, funding acquisition, project 68  administration, writing – review \& editing. 69Conflicts of interest 70There are no conflicts to declare. 71Data availability 72No primary research results, software or code have been 73  included, and no new data were generated or analysed as part 74  of this review. 75Acknowledgements 76The authors are indebted to National Institute for Materials 77  Science. This work was supported by the JSPS KAKENHI Grant 78  Number 23H01928 and the MEXT Program: Data Creation and 79  Utilization-Type Material Research and Development Project 80  Grant Number JPMXP1122712807. 81Notes and references 821\. A. J. Shih, M. C. O. Monteiro, F. Dattila, D. Pavesi, M. Philips, 83  A. H. M. da Silva, R. E. Vos, K. Ojha, S. Park, O. van der 84  Heijden, G. Marcandalli, A. Goyal, M. Villalba, X. Chen, G. T. 85  K. K. Gunasooriya, I. McCrum, R. Mom, N. López and M. T. 86  M. 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