Interplay between Covalent and Non-Covalent Interactions in

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Interplay between Covalent and Non-Covalent Interactions in Electrocatalysis Jun Huang, and Shengli Chen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b07534 • Publication Date (Web): 22 Oct 2018 Downloaded from http://pubs.acs.org on October 22, 2018

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The Journal of Physical Chemistry

Interplay between Covalent and Non-Covalent Interactions in Electrocatalysis

Jun Huang,†,* Shengli Chen‡,*

† Hunan Provincial Key Laboratory of Chemical Power Sources, College of Chemistry and Chemical Engineering, Central South University, Changsha 410083, P.R. China. E-mail: [email protected] ‡ Hubei Key Laboratory of Electrochemical Power Sources, Key Laboratory of Analytical Chemistry for Biology and Medicine (Ministry of Education), Department of Chemistry, Wuhan University, Wuhan 430072, China E-mail: [email protected]

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Abstract Electrocatalysis has been advanced to a point that understanding how non-covalent interactions interplay with covalent interactions becomes essential to precision design of efficient electrocatalysts. However, a conceptual framework for facilitating such understandings is yet missing; existing

theories

of

electrocatalysis

usually

neglect

non-covalent

interactions. To fill in this gap, this study presents a refined theory of electrocatalysis by adding a non-covalent term that is self-consistently derived from a mean-field electrical double layer (EDL) model into the model Hamiltonian, in addition to the electronic interaction term described by the Anderson-Newns model and the solvent term inspired by the Marcus theory. Applying the Green function technique allows us to portray the potential energy surface, to pinpoint multiple states along the reaction pathway and further to determine the reaction free energy and activation energies. Multifaceted interplay between covalent and non-covalent interactions is revealed. On one hand, chemisorption-induced surface dipoles modify the work function and electrostatic properties of the metal, leading to non-monotonicity in the surface charging relation and then the activation energy profile. As an immediate consequence, the sum of anodic and cathodic transfer coefficient is less than unity and the cathodic transfer coefficient can even be negative. On the other hand, EDL effects modulate covalent interactions, in return, via dictating the electrochemical potential of ions. The theory brings forth a new adsorption isotherm, which provides fundamental insights into the perplexing strong indirect 2


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adsorbate-adsorbate interactions, and furnishes a theoretical approach to analyze the electrosorption valency as well.

3



Introduction Electrocatalytic reactions are governed by a multitude of processes and parameters in a multidimensional and hierarchical space. Laying at the upper end of the spectrum of scales is mass transport we feel we know fairly well under most conditions. Laying at the lower end are covalent interactions controlled by electronic factors. In this regard, first-principles calculations based on the density functional theory (DFT) have generated a great deal of understanding in past few decades, bringing into being several far-reaching ideas such as the d-band model1-3 and adsorptionenergy scaling relations.4-5 The understandings have been successfully transformed to practical approaches, viz., thermodynamic descriptorbased approaches, with proved efficacy in guiding rational design and screening of electrocatalysts.6-8 This well-developed research filed has been reviewed by several groups of authors.6,

8-13

However, understanding

towards the effect of non-covalent electrostatic interactions at the mesoscopic scale (1~100 nm) is at its infancy stage. In this direction, a seminal work from Markovic et al. unveiling the peculiar effect of the cation on the electrocatalytic activity instigates a flurry of interest.14 Whereas experimental interests are high,15-20 related theoretical endeavors are sparse. Therefore, there is an urgent need for a theoretical framework for understanding the synergy of covalent and non-covalent interactions, which is believed to be essential for “any predictive ability in tailor-making real-world catalysts”.21 As exemplified in classical studies on hydrogen reactions by Butler22, 4


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Horiuti and Polanyi,23 Parsons,24 and the more recent d-band model by Norskov and co-workders,1-2 theory has been playing an essential role in electrocatalysis. Strong covalent interactions between the metal and the redox species in solution are at the heart of electrocatalysis. Without considering it, explicitly at least, classical electron transfer theories, i.e., the

Marcus-Hush

theory25-30

and

the

Levich-Dogonadze-Kuznetzov

theory,31-34 are not suited to model electrocatalysis. In its stead, Schmickler has been making seminal contributions since 1980s, leading to a unique theory of electrocatalysis.35-40 The essence of the Schmickler theory is to extend the Anderson-Newns theory,41-42 which describes the metal-reactant covalent interactions, by incorporating the solvent effect, a crucial lesson from the Marcus theory. The Schmickler theory reveals how the electronic structure of the metal, especially the d-band center, and the strength of metal-reactant covalent interactions dictate the electrocatalytic activity, thus shedding light on the essence of electrocatalysis and providing unique guidance in the design of new materials. As put forward by the authors themselves in Ref

37,

the weakest point of

existing electrocatalysis theories is the lacking of an appropriate description of non-covalent electrostatic interactions, mainly electrical double layer (EDL) effects, i.e., how the surface free charge on the metal, the solvent environment, and the ion concentration and potential distribution in the diffuse layer affect electrocatalytic reactions. For noncatalytic outer-sphere reactions, EDL effects can be treated by the Frumkin correction,43-44 which is based on the Gouy-Chapman-Stern (GCS) model.44-46 It is our opinion that the GCS model is inadequate for 5



inner-sphere electrocatalytic reactions, as it does not consider some essential features of electrocatalytic reactions, notably that covalent interactions lead to chemisorption of ions, form surface dipole moments, and change non-covalent electrostatic properties, e.g., the work function and the surface charging relation, dramatically. These effects, central to electrocatalysis, are what we mean by the title of this article. It comes into our notice that Lin et al. recently studied EDL effects on ion transfer reactions.47 Also of note, the GCS model in the dilute limit, resulting in an exponential decay of the potential distribution, is employed therein.47 Currently, DFT-based first-principles methods cannot tackle this problem, since they are difficult to navigate the complex interplay, steered by the electrode potential, between chemisorption, surface charging, fielddependent behaviors of solvents and distributions of potential and species concentration in the diffuse layer. This self-consistency challenge has been widely discussed in numerous reviews.12,

48

Theory is still valuable,

especially for conceptual understanding, to say the least, even if DFTbased first principles methods overcome the challenge aforementioned; here we cite the authors of Ref. 37, “realistic DFT calculations or simulations for electrochemical reactions … imitate the reaction rather than explain it.” Herein, the aim is to develop a viable theoretical framework for deciphering the interplay between covalent and non-covalent interactions, and further to unveil its impact on electrocatalytic reactions. To this end, a refined theory of electrocatalysis incorporating a mean-field EDL model is developed.49,50 Note in passing that the present study focuses on 6


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conceptual understanding rather than simulation of any realistic systems. The remainder of this article is organized as follows. The theory development section starts with model specifications, followed by detailed derivation of the model Hamiltonian and the double layer model, closed with solutions of the model. The results section analyzes main results generated from the model, including the potential energy surface, activation energy profiles, and double layer effects. The discussion section dwells on the abnormal negative transfer coefficient, the origins of indirect adsorbate-adsorbate

interactions,

the

effectiveness

of

supporting

electrolyte in diminishing the EDL effects, implications of model results for electrocatalyst design, as well as limitations of this work. In the end, major findings are concluded.

Theory development Model specifications We confine our attention to the electrochemical adsorption of a proton onto a metal surface, viz. the Volmer step of hydrogen evolution reaction in acid, H + + e→Had

(1)

This prototypical electrocatalytic reaction is adiabatic on transition metals,27,

35

involves only one electron transfer and no covalent bond

breaking. Simple as it is, this reaction already captures the essence of covalent and non-covalent interactions. The reaction can be described by the charge on the reactant, z, the solvent 7



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coordinate, q, and the distance between the reactant and the metal, d. To simplify the analysis, q is normalized in the following manner.39 The solvent coordinate is by definition q when solvent molecules are in equilibrium with the reactant with a charge of – q. Therefore, the reaction can be descried by only two independent variables, q and d. In this model, we take the potential in bulk solution as the reference, namely 𝜙𝑆,𝑏𝑢𝑙𝑘 = 0. As a result, the electrode potential 𝐸 is correlated with the electrostatic potential applied to the metal, 𝜙𝑀, in a linear relation, 𝐸 = 𝜙𝑀 ― 𝜇𝑒/𝑒0 where 𝜇𝑒 is the chemical potential of electrons in the metal electrode, and 𝑒0 the elementary charge.51 The Fermi level of the metal electrode is by definition given by, 𝜖𝑓 = ― 𝑒0𝐸. Assuming that variation in 𝜙𝑀 does not alter 𝜇𝑒 leads to 𝛥𝐸 = 𝛿𝜙𝑀 and 𝛥𝜖𝑓 = ― 𝑒0𝛿𝜙𝑀 with the symbol 𝛿 representing variation.51 The framework of the present model is shown in Figure 1. We start with writing down the model Hamiltonian as a sum of three terms, 𝐻𝑠𝑦𝑠(𝑞,𝑑) = 𝐻𝑒𝑙(𝑞,𝑑) + 𝐻𝑠𝑜𝑙(𝑞,𝑑) + 𝐻𝑛𝑐(𝑞,𝑑)

(2)

with 𝐻𝑒𝑙, 𝐻𝑠𝑜𝑙 and 𝐻𝑛𝑐 being the electronic interaction term, the solvent term and the non-covalent term, respectively. Detailed consideration of these three terms is given below.

8


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Figure 1. Model framework. The novelty of the present model is to incorporate EDL effects, derived from an EDL model accounting for ion size effect, interfacial water ordering and surface dipoles formed between metal atoms and adsorbates, into the model Hamiltonian.

The electronic interaction term 𝑯𝒆𝒍 In our case, the role of spin is neglected therefore, as only one electron is considered. We consider one orbital on the reactant, labeled as a, and a series of orbitals on the metal, labeled as k. The electronic energy is denoted by 𝜀 and the operator for the occupation number by 𝑛. The electronic part of the model Hamiltonian reads,

𝐻𝑒𝑙 = 𝜀𝑎𝑛𝑎 +

∑𝜀 𝑛 + ∑[𝑉 𝑐

∗ 𝑘 𝑘 𝑐𝑎

𝑘 𝑘

𝑘

+ 𝑉𝑘∗ 𝑐𝑎∗ 𝑐𝑘]

(3)

𝑘

where the last term accounts for electron exchange between the metal and the reactant, with 𝑐 ∗ denoting a creation and 𝑐 an annihilation operator. 𝑉𝑘 characterizes the amplitude of electronic interactions. As known from the classical work of Lang and Williams52 as well as recent DFT results by 9



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Santos et al.,37 𝜀𝑎 will shift to lower energy levels upon approaching the metal. A linear relation with an identical slope for a variety of metals, namely, 𝜀𝑎(𝑑) = 𝜀0𝑎 + 𝛽𝑎(𝑑 ― 𝑑0) with 𝛽𝑎(eV Å ―1) being the slope, is found from DFT and used here. The solvent term 𝑯𝒔𝒐𝒍 Solvent dynamics can be divided into slow and fast modes.25, 28 The slow mode is related with the arrangement of solvent nuclei, which, in an appropriate configuration, sets the stage for electron transfer. This slow solvent model is affected by the charge on the reactant. A linear relation can be used, as the first approximation, to describe the interaction. Combined, the solvent term associated with the slow mode is expressed as,37, 39 2 𝐻𝑠𝑙𝑜𝑤 𝑠𝑜𝑙 = 𝜆𝑞 + 2(1 ― 𝑛𝑎)𝜆𝑞

(4)

with the first term corresponding to the intrinsic value of the relaxation energy of the solvation molecules, described with harmonic oscillators, and the second term the solvent-reactant interaction. 𝑛𝑎is the occupation number of the orbital a, a key variable to be determined in this theory. 𝜆 is termed the slow reorganization energy. 𝜆 should vary when the reactant approaches the metal, as shown by Santos et al.37 This effect is, however, not explicitly included in the present model. Assuming a constant 𝜆 may lead to up-to-0.1 eV inaccuracy in the activation energy.37 The fast solvent mode corresponds to the electronic polarizability of solvent molecules, which shall succeed the slow solvent mode adiabatically for the reaction considered here. The solvent term in the model Hamiltonian associated with this fast mode, 𝐻𝑓𝑎𝑠𝑡 𝑠𝑜𝑙 , is actually the electrostatic energy 10


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stored in solvent molecules, which shall vary strongly as a function of 𝑑, as explained below; this effect is explicitly treated in this model. For a proton approaching the metal surface, the probability of finding it in H + and Had (ideally without net charge) state is (1 ― 𝑛𝑎) and 𝑛𝑎, respectively. As a result, 𝐻𝑓𝑎𝑠𝑡 𝑠𝑜𝑙 is written as a sum of two terms,

𝑓𝑎𝑠𝑡 𝑓𝑎𝑠𝑡 𝐻𝑓𝑎𝑠𝑡 𝑠𝑜𝑙 = (1 ― 𝑛𝑎)𝐺𝑠𝑜𝑙,𝐻 + + 𝑛𝑎𝐺𝑠𝑜𝑙,𝐻𝑎𝑑

(5)

According to the classical work of Born,53-54 we have,

𝐺𝑓𝑎𝑠𝑡,𝑏𝑢𝑙𝑘 𝑠𝑜𝑙,𝐻 +

=―

𝑒20 8𝜋𝑟𝐻 +

( ) 1 1 ― 𝜖0 𝜖

(6)

for the protons in bulk solution, and (7)

𝐺𝑓𝑎𝑠𝑡 𝑠𝑜𝑙,𝐻𝑎𝑑 = 0

for Had, where 𝑒0 is the elementary charge, 𝑟𝐻 + the effective radius of solvated protons, 𝜖0 and 𝜖 the dielectric constant of vacuum and the solution, respectively. Fletcher has given a lucid discussion on the effective radius of solvated ions in calculation of the solvent reorganization energy.55 Note that the Born equation is derived from the Debye-Huckel theory, which is restricted to an infinitely dilute solution, and based on the assumption of a spherical structure for the ion.53-54 The ion loses part of its solvent shell when approaching the metal, hence, the spherical structure symmetry is broken.36 Moreover, it is well known that 𝜖 11



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decreases near the interface compared with the bulk value due to ordering saturation of water dipoles and excluded volume effects.56 These factors render that the magnitude of 𝐺𝑓𝑎𝑠𝑡 𝑠𝑜𝑙,𝐻 + shall decrease near the metal surface. For lack of a more rigorous treatment, we tentatively assume an empirical relation, 𝑓𝑎𝑠𝑡,𝑏𝑢𝑙𝑘 𝐺𝑓𝑎𝑠𝑡 𝑠𝑜𝑙,𝐻 + (𝑑) = 𝐺𝑠𝑜𝑙,𝐻 + (1 ― 𝑘𝑓exp ( ― 𝛽𝑓(𝑑 ― 𝑑0)))

(8)

with 𝑘𝑓 being a positive coefficient, and 𝛽𝑓(Å ―1) the decaying rate. It is readily seen from Eq.(8) that 𝐺𝑓𝑎𝑠𝑡,𝑠𝑢𝑟𝑓 = 𝐺𝑓𝑎𝑠𝑡,𝑏𝑢𝑙𝑘 𝑠𝑜𝑙,𝐻 + 𝑠𝑜𝑙,𝐻 + (1 ― 𝑘𝑓) at 𝑑 = 𝑑0. The variation in the solvent energy associated with the fast mode when the reactant approaches the metal is termed potential of mean force (pmf) in the Schmickler theory39 and work terms in the Marcus theory.25, 28 The non-covalent term 𝑯𝒏𝒄 This model features the consideration of 𝐻𝑛𝑐, which is not included in most (if not all) previous theories.3,5,35,39,42,55 Non-covalent interactions include electrostatic forces, van der Waals forces and so on, in a decreasing order of significance. In spirit of the superposition principle, the non-covalent term is written as a sum of two terms,

𝑛𝑐 𝐻𝑛𝑐(𝑞,𝑑) = (1 ― 𝑛𝑎)𝐺𝑛𝑐 𝐻 + (𝑑) + 𝑛𝑎𝐺𝐻𝑎𝑑(𝑑)

(9)

𝑛𝑐 where 𝐺𝑛𝑐 𝐻 + (𝑑) and 𝐺𝐻𝑎𝑑(𝑑) could be further divided into a chemical part

and an electrostatic part,

𝑐ℎ𝑒𝑚 𝑒𝑙𝑒𝑐 𝐺𝑛𝑐 𝐻 + /𝐻𝑎𝑑(𝑑) = 𝐺𝐻 + /𝐻𝑎𝑑 + 𝐺𝐻 + /𝐻𝑎𝑑(𝑑)

12


(10)

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Regarding the chemical part, we have,

𝑐ℎ𝑒𝑚 0 𝐺𝑐ℎ𝑒𝑚 𝐻 + = 𝐺𝐻𝑎𝑑 = 𝐺𝐻 +

(11)

where 𝐺0𝐻 + is the standard value corresponding to a bare (without solvation shell) proton. Here the chemical part is identical for 𝐻 + and 𝐻𝑎𝑑 because the electronic energy and solvent effects have been included in the electronic interaction term in Eq.(3) and the solvent term in Eq.(4) and (5), respectively. Regarding the electrostatic part, we have,

𝑒𝑙𝑒𝑐 𝐺𝑒𝑙𝑒𝑐 𝐻 + (𝑑) = 𝑒0𝜙(𝑑), 𝐺𝐻𝑎𝑑 (𝑑) = 0

(12)

with 𝜙(𝑑) being the location-dependent potential. Lin et al. assumes 𝜙(𝑑) as an exponentially decaying function of 𝑑, which is valid near the potential of zero charge only, namely when the electrode surface is slightly charged.47 In the present theory, 𝜙(𝑑) is to be solved out from a selfconsistent EDL model, as detailed below. Double layer model An EDL model oriented for electrocatalysis should consider the effects of electronic interactions and the resultant chemisorption. Surface dipoles are usually formed between metal atoms and adsorbates due to the partial electron transfer. This surface dipole moment, as a function of the adsorbate coverage 𝜃, introduces an extra potential drop at the metal surface, thus dramatically changes electrostatic properties of the metal surface as well as the EDL properties. In a previous study, we have shown 13



that OHad formed from water dissociation on Pt(111) gives rise to a nonmonotonic surface charging relation and negative double-layer differential capacitances, which are in sharp contrast with the classical GCS model.4950

In addition to the adsorbate effect, the previous EDL model also

accounts for the ion size effect and the orientational ordering of surficial water molecules. More details of the EDL model can be found in the original papers.49-50 First, we should introduce the EDL structure, as depicted in Figure 2. 𝑑 = 0 denotes the metal surface, and 𝑑 = 𝑑0 locates the plane where the adsorbate and the co-planar water molecules reside. Given the radius of a hydrogen atom (0.56 Å) and the radius of a water molecules (1.37 Å), we use an approximate value of 𝑑0 = 1 Å. Outside the adsorbate plane lays the closest plane that solvated protons can approach the metal surface, namely the Helmholtz plane (HP). Herein, we give a crude estimation of the distance between the adsorbate plane and the HP, 𝛿𝐻𝑃 = 2 Å. Beyond the HP follows the diffuse layer stretching toward the solution bulk. Parameters of the EDL model are listed in Table S1 in the supporting information (SI).

14


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Figure 2. Schematic structure of the EDL model. Surface adsorbates carrying a net charge rigidly line up at 𝑑0, forming surface dipoles on the metal surface.

By adopting the previous EDL model, we can obtain the surface charge density, 𝜎𝑀, in the form below,49,50

sign(𝜎𝑀)

2𝑅𝑇 arsinh 𝐹

) ))

( ( (

1 𝜒 2 exp (𝜎𝑀) ― 1 2𝛾 2

= 𝜙𝑀 ― Δ𝜙𝑀 ―

𝜇surf 𝜖ad

+

𝑁tot𝜇w 𝜖IHP

+

(

)

𝑅𝑇𝜖 𝑀 𝑑0 𝛿HP 𝜎 + 𝜖ad 𝜖HP 𝐹𝜆𝐷

(13)

tanh (𝑋)

𝐹, 𝑅 and T have their usual meanings, and other variables are explained 15



Page 16 of 48

as follows. 𝜎𝑀 = 𝐹𝜆𝐷𝜎𝑀/𝑅𝑇𝜖 is the dimensionless surface free charge density with 𝜆D = 𝜖𝑅𝑇/2𝐹2𝑐𝑏 being the Debye length and 𝑐𝑏 the total concentration of cations (including protons and cations of the supporting electrolyte, if there exists any) in bulk solution. 𝜒 = 2𝑐𝑏𝑁𝐴𝑎3 with 𝑎 being the average diameter of ions in solution is termed the compacity factor. 𝜖ad and 𝜖HP are the dielectric constant of the space between the metal surface and the adsorbate plane and that between the adsorbate plane and the HP, respectively. Δ𝜙𝑀 is the constant potential drop at the metal surface due to electron spillover. 𝑁tot is the number density of surface atoms, 𝜇w the dipole moment of a water molecule, and 𝑋 the dimensionless total field-dependent adsorption energy of water molecules, which is found from the following relation,49

(

0.6 𝜋𝑑20

+ 𝑁tot

)

𝑁A𝜇2w 𝑑0𝜖ad𝑅𝑇

tanh (𝑋) ― 𝑋 =

𝑁A𝜇w𝜎𝑀 𝜖ad𝑅𝑇

(14)

based on a two-state water model. Here 𝑁𝐴 is the Avogadro's number. In Eq.(13), the effective surficial dipole moment 𝜇surf is approximated by a linear relation, 𝜇surf = 𝑁tot(𝜃 ∙ 𝜍𝑒0 ∙ 𝑑0)

(15)

with 𝜍 being the average charge number of the adsorbate (positive when the adsorbate carries a negative charge). The effect of 𝜇𝑠𝑢𝑟𝑓 can be equivalently viewed as chemisorption-induced modification of the 𝑀 potential of zero charge (pzc), 𝜙𝑒𝑓𝑓 𝑝𝑧𝑐 = 𝛥𝜙 + 𝜇𝑠𝑢𝑟𝑓/𝜖𝑎𝑑. As a result, 𝜙𝑝𝑧𝑐

16


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is shifted towards more negative values, viz. 𝜎𝑀 becomes more positive at a given electrode potential, when 𝜇𝑠𝑢𝑟𝑓 < 0. After obtaining 𝜎𝑀, 𝜙(𝑑) in the diffuse layer (𝑑 > 𝛿𝐻𝑃 + 𝑑0) can be determined by solving the modified Poisson-Boltzmann equation,

∂2𝜙

= ∂𝑑2

sinh (𝜙) 1 + 2𝛾sinh2

() 𝜙 2

(16)

where 𝜙 = 𝜙𝐹/𝑅𝑇 is the dimensionless potential and 𝑑 = (𝑑 ― 𝛿𝐻𝑃 ― 𝑑0 )/𝜆D is the dimensionless coordinate. Eq.(16) is closed with the boundary condition at the HP, 𝑑 = 0, ∇𝜙 = ― 𝜎𝑀

(17)

and the boundary condition in bulk solution at infinite distance, 𝑑 = ∞, (18)

∇𝜙 = 0, 𝜙 = 0

as the potential in bulk solution is taken as the potential reference. For the space in the region of 𝑑0 < 𝑑 < 𝑑0 + 𝛿𝐻𝑃, there is no net charge and therefore 𝜙(𝑑) is given by a linear extrapolation,

𝜙(𝑑) = 𝜙𝐻𝑃 ― 𝜎𝑀

𝑑 ― 𝑑0 ― 𝛿𝐻𝑃 𝜖𝐻𝑃

with 𝜙𝐻𝑃 being the potential at the HP.

17


(19)


Page 18 of 48

Self-consistency is a grand challenge for theoretical electrocatalysis. In the present theory, the mean-field EDL model bridges covalent and noncovalent interactions, and provides a viable way to solve the selfconsistency challenge. As shown in Figure 1, the present theory forms a closed-loop among dependent variables, including 𝜃, 𝜙(𝑑), the reaction free energy 𝛥𝐺 and activation energies 𝛥𝐺𝑎𝑐/𝑎 (c and a stand for the forward cathodic and backward anodic reaction, respectively), that are controlled by independent variables, namely, the electrode potential 𝐸 and the solution properties, e.g., the proton concentration in bulk solution 𝑐𝑏𝑢𝑙𝑘 𝐻+ . Model solution The total model Hamiltonian is the sum of the electronic interaction term in Eq.(3), the solvent terms in Eq.(4) and (5), and the non-covalent term in Eq.(9). Appropriate mathematical rearrangement of above terms gives,

𝑛𝑐 𝐻𝑒𝑙(𝑞,𝑑) = 𝜖′𝑎(𝑞,𝑑)𝑛𝑎 + 𝜆𝑞2 + 2𝜆𝑞 + 𝐺𝑓𝑎𝑠𝑡 𝑠𝑜𝑙,𝐻 + (𝑑) + 𝐺𝐻 + (𝑑) +

+

∑𝜀 𝑛

𝑘 𝑘

𝑘

∑[𝑉𝑘𝑐𝑘∗ 𝑐𝑎 + 𝑉𝑘∗ 𝑐𝑎∗ 𝑐𝑘]

(20)

𝑘

where 𝜖𝑎′(𝑞,𝑑) is a modified electronic energy on the hydrogen orbital defined as,

𝑓𝑎𝑠𝑡 𝜖′𝑎(𝑞,𝑑) = 𝜀𝑎(𝑑) ― 2𝜆𝑞 ― 𝑒0𝜙(𝑑) + 𝐺𝑓𝑎𝑠𝑡 𝑠𝑜𝑙,𝐻𝑎𝑑 ― 𝐺𝑠𝑜𝑙,𝐻 + (𝑑)

(21)

Equipped with the Green function technique, we can derive the density of states (DOS) of the hydrogen orbital as,42 18


Page 19 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


𝜌𝑎(𝑞,𝑑,𝜀) =

1 Δ(𝑑,𝜀) 𝜋(𝜀 ― (𝜖𝑎′(𝑞,𝑑) + Λ(𝑑,𝜀)))2 + Δ(𝑑,𝜀)2

(22)

where Δ(𝑑,𝜀) and Λ(𝑑,𝜀) are chemisorption functions, expressed as,

Δ(𝑑,𝜀) =

∑

1 𝜋

𝑘

Δ(𝑑,𝜀′)

∫ 𝜀 ― 𝜀′ 𝑑𝜀′

|𝑉𝑘|2𝜋𝛿(𝜀 ― 𝜀𝑘), Λ(𝑑,𝜀) = 𝒫

(23)

Δ(𝑑,𝜀) represents the width of 𝜌𝑎(𝑞,𝑑,𝜀) in a Lorentzian form, Λ(𝑑,𝜀) the shift in the center of 𝜌𝑎(𝑞,𝑑,𝜀) due to electronic interactions, and 𝒫 the principle part. The essence of electrocatalysis is embodied in the two chemisorption functions, Δ(𝑑,𝜀) and Λ(𝑑,𝜀). Moreover, Λ(𝑑,𝜀) can be derived from Δ(𝑑,𝜀). Therefore, we focus our discussion on Δ(𝑑,𝜀), particularly on how Δ(𝑑,𝜀) relates with the electronic structure of the metal. Transition metals usually have a broad sp band and a narrow d band. As a result, Δ(𝑑,𝜀) can be divided into two terms,

Δ(𝑑,𝜀) = Δd(𝑑,𝜀) + Δsp(𝑑,𝜀), which are further

expressed as,

Δd(𝑑,𝜀) = |𝑉𝑑(𝑑)|2𝜋𝜌𝑑(𝜀), Δsp(𝑑,𝜀) = Δ0sp(𝑑)

(24)

where 𝜌𝑑(𝜀) is the DOS of the metal d-band and |𝑉𝑑(𝑑)|2 characterizes the strength of electronic interactions exerted by the metal d-band. Δsp(𝑑,𝜀) is taken to be independent of 𝜀, which is a valid for a wide sp band. Note that

|𝑉𝑑(𝑑)|2 becomes larger when the reactant approaches the metal. The exact form of |𝑉𝑑(𝑑)|2 as a function of d needs to be obtained using DFT 19



Page 20 of 48

calculations. 𝜌𝑑(𝜀) is usually in a complex form, rendering analytical expressions for the occupancy of the orbital a 𝑛𝑎(𝑞,𝑑) and the electron term of the total energy 𝐸𝑒𝑙(𝑞,𝑑) intractable. Given the DOS, the expectation value of the occupancy of the orbital a is obtained by integrating the DOS up to the Fermi level. 𝜖𝑓 = ― 𝑒0𝐸,

𝑛𝑎(𝑞,𝑑) =

∫

𝜖𝑓

𝜌𝑎(𝑞,𝑑,𝜀)𝑑𝜀

(25)

―∞

0

Eq. (25) is equivalent to, 𝑛𝑎(𝑞,𝑑) = ∫ ―∞𝜌𝑎(𝑞,𝑑,𝜀)𝑑𝜀, with 𝜖𝑎′(𝑞,𝑑) in Eq.(21) replaced with an effective expression,

𝑓𝑎𝑠𝑡 𝑓𝑎𝑠𝑡 𝜀𝑒𝑓𝑓 𝑎 (𝑞,𝑑) = 𝜀𝑎 ― 2𝜆𝑞 + 𝑒0(𝐸 ― 𝜙(𝑑)) + 𝐺𝑠𝑜𝑙,𝐻𝑎𝑑 ― 𝐺𝑠𝑜𝑙,𝐻 + (𝑑)

(26)

According to Eq.(26), 𝜀𝑒𝑓𝑓 is greater at higher 𝜙𝑀. As a result, 𝑛𝑎(𝑞,𝑑) is 𝑎 smaller, viz. the reaction is shifted toward the desorption direction. For the sake of simplicity in the notation, we use 𝜀𝑒𝑓𝑓 𝑎 (𝑞,𝑑) given in Eq.(26) in the remaining part. The electron term of the total energy is expressed as,

𝐸𝑒𝑙(𝑞,𝑑) =

∫

𝜖𝑓

𝜀𝜌𝑎(𝑞,𝑑,𝜀)𝑑𝜀 + (1 ― 𝑛𝑎(𝑞,𝑑))𝜖𝑓

(27)

―∞

where the first term on the right hand side corresponds to the adsorbate and the second term to the substrate. If the metal d band is wide and structureless, that is, Δ(𝑑,𝜀) becomes independent of 𝜀, then we have, Λ(𝑑,𝜀) = 0. This limiting case is termed 20


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wide band approximation.35 Based on this approximation, widely adopted in previous studies,35,39,47 𝑛𝑎(𝑞,𝑑) and 𝐸𝑒𝑙(𝑞,𝑑) can be explicitly expressed as,

𝜀𝑒𝑓𝑓 1 1 𝑎 (𝑞,𝑑) 𝑛𝑎(𝑞,𝑑) = ― arctan 2 𝜋 Δ(𝑑)

(

) (28)

𝐸𝑒𝑙(𝑞,𝑑) = 𝜀𝑒𝑓𝑓 𝑎 (𝑞,𝑑)𝑛𝑎(𝑞,𝑑) ― 𝑒0𝐸 +

2 𝜀𝑒𝑓𝑓 𝑎 (𝑞,𝑑)

2

+ Δ(𝑑) Δ(𝑑) ln 2π (𝜀𝑒𝑓𝑓(𝑞,𝑑) ― 𝜀𝑙𝑏)2 + Δ(𝑑)2 𝑎

𝑎

where 𝜀𝑙𝑏 𝑎 is a very negative number, denoting the lower bound of the metal band. For not interested in the absolute value of the energy, we designate 𝑙𝑏 0 39 Herein, we assume 𝜀𝑒𝑓𝑓 𝑎 (𝑞,𝑑) ― 𝜀𝑎 = 𝜖𝑎, just as in the Schmickler theory.

that the electronic interaction strength decays exponentially when the reactant

moves

away

from

the

metal

surface,

viz.

Δ(𝑑) = Δ0

exp ( ― 𝛽𝑑(𝑑 ― 𝑑0)) with 𝛽𝑑(Å ―1) being the decay rate. This exponential decay is a good approximation to DFT results,37 and the same assumption is used in the work of Lin et al.47 By this point, we are ready to formulate the total Gibbs energy of the system by bringing together all above threads, 𝑛𝑐 𝐺𝑠𝑦𝑠(𝑞,𝑑) = 𝐸𝑒𝑙(𝑞,𝑑) + 𝜆𝑞2 + 2𝜆𝑞 + 𝐺𝑓𝑎𝑠𝑡 𝑠𝑜𝑙,𝐻 + (𝑑) + 𝐺𝐻 + (𝑑)

(29)

Under the wide band approximation, the total free energy of the system can be given by an analytical expression,

21



2

2 Δ(𝑑) 𝜀𝑒𝑓𝑓 𝑎 (𝑞,𝑑) + Δ(𝑑) (30 + 𝐺𝑠𝑦𝑠(𝑞,𝑑) = ln ― 𝑒0𝐸 + 𝜆𝑞2 + 2𝜆𝑞 02 2 2π 𝜖𝑎 + Δ(𝑑) ) 𝑛𝑐 + (𝑑) + 𝐺 + (𝑑) + 𝐺𝑓𝑎𝑠𝑡 𝑠𝑜𝑙,𝐻 𝐻

𝜀𝑒𝑓𝑓 𝑎 (𝑞,𝑑)𝑛𝑎(𝑞,𝑑)

The first three terms on the right hand side (rhs) correspond to the energy of the electron, the fourth to sixth terms the solvent-related energy, and the last term the electrostatic energy of a proton as explained before. Model parameters are given with notes in the model parameterization section in the SI. Using Eq.(30) obtained under the premise of the wide band approximation, the theory gives out, all at once, potential energy surfaces in Figure 3, activation energy profiles in Figure 4, as well as adsorbate formation and EDL properties in Figure 5. It should be mentioned in passing that major conclusions obtained from Figure 3-5 are not limited to the wide band approximation, as we will demonstrate in the discussion section. Also of note, we discuss in the SI that the present theory is reduced to Schmickler theory by dropping the non-covalent term, and to the Marcus-Hush theory by further neglecting the electronic interaction term.

Results Potential energy surface Potential energy surface with 𝑞 and 𝑑 as the coordinates is an informative representation of the reaction.1,

3

Taking Figure 3 (a) as an

example, one is able to identify multiple states along the reaction pathway. The initial state, corresponding to a solvated proton in bulk solution 22


Page 22 of 48

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(H +,bulk), locates at 𝑞 = ―1 and 𝑑→∞ (represented by the point at the leftupper corner in Figure 3). The intermediate state, corresponding to a surface proton (H +,surf), locates at 𝑞 = ―0.9 and 𝑑 = 3.0 Å, where 𝐺𝑠𝑦𝑠 in Eq.(30) achieves a local minimum in solution. The transition state, corresponding to the saddle point of the potential energy surface, is found at 𝑞 = ―0.5 and 𝑑 = 2.0 Å. The final state, corresponding to an adsorbed ―𝜍 hydrogen atom (Had ), is positioned at 𝑞 = ―0.05 and 𝑑 = 1.0 Å, where 𝐺𝑠𝑦𝑠

has another local minimum near the metal surface. It is directly ―𝜍 determined from Figure 3 (a) that Had bears a positive charge of 0.05 e,

i.e., ς = ―0.05. The non-zero value of ς means that the adsorbate shares its electron with the metal. As a result, the coefficient ς can be related to the electrosorption valency, see an excellent review on this topic by Schmickler and Guidelli.57 Figure 3 (c) portrays how the system state evolves when the proton approaches the metal. The whole process can be divided into two substeps: a physisorption step H +,bulk→H +,surf and an ―𝜍 electron transfer step H +,surf +(1 ― 𝜍)𝑒→Had , as already pointed out by Lin

et al.47 Below we give a theoretical analysis on the initial, final and transition state, based on Eq.(30). At the initial state, following conditions are met, 𝑛𝑎 = 0, 𝑞 = ―1, 𝑑 ≫ 𝛽𝑑―1 𝑆,𝑏𝑢𝑙𝑘 0 and Δ(𝑑) ≈ 0. In addition, 𝐺𝑛𝑐 with 𝜙𝑆,𝑏𝑢𝑙𝑘 = 0 as the 𝐻 + = 𝐺𝐻 + + 𝑒0𝜙

potential in bulk solution is taken as the potential reference. Combined, is given by, 𝐺𝑠𝑦𝑠 at the initial state 𝐺𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑠𝑦𝑠

𝐺𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝐺0𝐻 + ― 𝑒0𝐸 + (𝐺𝑓𝑎𝑠𝑡,𝑏𝑢𝑙𝑘 ― 𝜆) 𝑠𝑦𝑠 𝑠𝑜𝑙,𝐻 +

23


(31)


Page 24 of 48

where 𝐺𝑓𝑎𝑠𝑡,𝑏𝑢𝑙𝑘 is the solvation energy of H+ in bulk solution in the fast 𝑠𝑜𝑙,𝐻 + mode, and the term (𝐺𝑓𝑎𝑠𝑡,𝑏𝑢𝑙𝑘 ― 𝜆) is exactly the total solvation energy of 𝑠𝑜𝑙,𝐻 + protons in bulk solution.39 At the final state, we can write down the following conditions, 𝑞 = 𝜍, 𝑛𝑎 = 1 + 𝜍, and 𝑑 ≈ 𝑑0 (the adsorbate plane), thus, Δ(𝑑) ≈ Δ0, 𝜖𝑎 ≈ 𝜖0𝑎 as well 𝑓𝑎𝑠𝑡,𝑠𝑢𝑟𝑓 𝑛𝑐 0 as 𝐺𝑓𝑎𝑠𝑡 𝑠𝑜𝑙,𝐻 + ≈ 𝐺𝑠𝑜𝑙,𝐻 + . In addition, 𝐺𝐻 + = 𝐺𝐻 + + 𝑒0𝜙0, with 𝜙0 being the

solution phase potential at 𝑑0. Hence, 𝐺𝑠𝑦𝑠 at the final state 𝐺𝑓𝑖𝑛𝑎𝑙 is given 𝑠𝑦𝑠 by,

𝐺𝑓𝑖𝑛𝑎𝑙 𝑠𝑦𝑠 = 𝜖𝑒𝑓𝑓,𝑓𝑖𝑛𝑎𝑙 (1 + 𝜍) + 𝑎

Δ0

ln 2𝜋

2

𝜀𝑒𝑓𝑓,𝑓𝑖𝑛𝑎𝑙 + Δ20 𝑎 2 𝜖0𝑎

+ Δ20

(32 + 𝜆𝜍2 + 2𝜆𝜍 + 𝐺0𝐻 + ― 𝑒0(𝐸 ― 𝜙0) )

+ 𝐺𝑓𝑎𝑠𝑡,𝑠𝑢𝑟𝑓 𝑠𝑜𝑙,𝐻 + 𝑓𝑎𝑠𝑡,𝑠𝑢𝑟𝑓 where 𝜀𝑒𝑓𝑓,𝑓𝑖𝑛𝑎𝑙 = 𝜖0𝑎 ―2𝜍𝜆 + 𝑒0(𝐸 ― 𝜙0) + 𝐺𝑓𝑎𝑠𝑡 𝑎 𝑠𝑜𝑙,𝐻𝑎𝑑 ― 𝐺𝑠𝑜𝑙,𝐻 + . For the ideal case

of 𝜍 = 0, 𝐺𝑓𝑖𝑛𝑎𝑙 is independent of the electrode potential 𝐸. Otherwise, the 𝑠𝑦𝑠 potential dependence of 𝐺𝑓𝑖𝑛𝑎𝑙 is the rule rather than an exception. 𝑠𝑦𝑠 At the transition state (denoted as TS for brief), the modified energy level of the adsorbate orbital in Eq.(21) exactly matches the Fermi level of the metal, namely, 𝜀𝑒𝑓𝑓,𝑇𝑆 =0 𝑎

(33)

By substituting Eq.(33) into Eq.(30), we can obtain the variation in the activation energy of the forward cathodic reaction due to the effect of electronic interactions, with respect to the case without considering electronic interactions, as follows, 24


Page 25 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

𝛿Δ𝐺𝑎𝑐

Δ𝑇𝑆 Δ𝑇𝑆 = ln 2π 𝜖02 + Δ𝑇𝑆2

(34)

𝑎

which is always negative, meaning that electronic interactions between the metal and the reactant in solution brings down the activation energy. This effect is a simple illustration of the essence of electrocatalysis, which has been expounded by Schmickler.39 It is read out from the potential energy surface shown in Figure 3 (a) that = ―11.90 𝑒𝑉, 𝐺𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑠𝑦𝑠

𝐺𝑇𝑆 𝑠𝑦𝑠 = ―11.38 𝑒𝑉

and

𝐺𝑓𝑖𝑛𝑎𝑙 𝑠𝑦𝑠 = ―11.98 𝑒𝑉.

Therefore,

activation energies for this case are Δ𝐺𝑎𝑐 = 0.52 𝑒𝑉 for the cathodic reaction and Δ𝐺𝑎𝑎 = 0.60 𝑒𝑉 for the anodic reaction, see the discussion on the definition of Δ𝐺𝑎𝑐/𝑎 in the SI. Also at 0.3 V, Figure 3 (b) shows that the activation energies increase up to Δ𝐺𝑎𝑐 = 0.94 𝑒𝑉 and Δ𝐺𝑎𝑎 = 1.13 𝑒𝑉 for the case of Δ0 = 1.0 eV, in accordance with the expectation that a weaker metalreactant interaction shall impede the discharge adsorption of a single proton.

25



Page 26 of 48

Figure 3. Potential energy surface. Results are calculated based on the wide band approximation with the electronic interaction strength of Δ0 = 2 eV at the electrode potential of 𝐸 = 0.3 V. The effect of electronic interactions is examined in (b) with

Δ0 = 1 eV. The

schematic illustration at the bottom depicts multiple system states. The initial state corresponds to a solvated proton in bulk solution. The final state corresponds to an adsorbed hydrogen (with a partial charge) on the metal surface. The transition state represents the saddle point of the potential energy surface. In addition, there will be an intermediate state near the metal surface, separating the adsorption process into a physisorption step and an electron transfer step.

26


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The potential energy surface is modulated by the electrode potential 𝐸. According to Eq.(31), increasing 𝐸 from 0.3 V to 0.6 V should decrease by an amount of 0.3 eV from ―11.90 𝑒𝑉 to ―12.20 𝑒𝑉, as shown in 𝐺𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑠𝑦𝑠 Figure S1. Moreover, 𝐺𝑓𝑖𝑛𝑎𝑙 is also decreased by 0.01 eV to ―11.99 𝑒𝑉 and 𝑠𝑦𝑠 𝑎 𝑎 𝐺𝑇𝑆 𝑠𝑦𝑠 to ―11.45 𝑒𝑉. Therefore, we have Δ𝐺𝑐 = 0.75 𝑒𝑉 and Δ𝐺𝑎 = 0.54 𝑒𝑉. The

transfer coefficient of the forward cathodic reaction, which is by definition the ratio of the change of the activation barrier with respect to the change of 𝑒0𝐸, is therefore given by, αc = 0.77. Moreover, the transfer coefficient of the backward anodic reaction is αa = 0.20. As a result, we arrive at a surprising conclusion that the sum of αc + αa deviates from unity (the deviation is more significant at other potentials, as to be discussed in the next section). This abnormality is an immediate consequence of the fact ―𝜍 that Had bears a positive charge, hence, 𝐺𝑓𝑖𝑛𝑎𝑙 too changes as a function 𝑠𝑦𝑠

of the electrode potential 𝐸. Without considering EDL effects or for outersphere electron transfer reactions, the relation, αc + αa = 1, will be retained. As a result, EDL effects have a marked impact on the transfer coefficient for electrocatalytic reactions, an issue we will return to in the discussion section. Activation energy profiles Activation energies as a function of the electrode potential 𝐸 varying between 0 and 0.6 V are calculated for three cases in Figure 4. The 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 abscissa axis is the Gibbs free energy of the reaction, 𝛥𝐺 = 𝐺𝑓𝑖𝑛𝑎𝑙 𝑠𝑦𝑠 ― 𝐺𝑠𝑦𝑠 .

The base case considers EDL effects with ς = ―0.05 and a potential of zero charge, 𝜙𝑝𝑧𝑐 = 0.3 𝑉. Two additional cases are included for comparison, including a case without considering surface dipoles (ς = 0), and a case 27



Page 28 of 48

without considering EDL effects realized by fixing 𝜙(𝑑) = 𝜙𝑆,𝑏𝑢𝑙𝑘 (or 𝜎𝑀 = 0 equavilently). In Figure 4 (b), the effect of 𝜙𝑝𝑧𝑐 is examined by comparing the base case and the case with 𝜙𝑝𝑧𝑐 = 0.5 𝑉. Note that ς can be self-consistently calculated from the model, rather than given as an input. However, in the potential range of 0 < 𝐸 < 0.6 V, it is found that ς is nearly constant around -0.05. Consequently, to a first approximation, we label the curves in Figure 4 and Figure 5 with ς = ―0.05. Below is the theoretical analysis on ς. At the final state, 𝐺𝑠𝑦𝑠(𝑞,𝑑) has its minimum value, that is, ∂𝐺𝑠𝑦𝑠(𝑞,𝑑)/∂𝑞 = ∂𝐺𝑠𝑦𝑠(𝑞,𝑑)/∂𝑑 = 0. Assuming that the final state is achieved at the adsorbate plane (𝑑 = 𝑑0), the minimum value is achieved at 𝑞𝑓𝑖𝑛𝑎𝑙, given by, Δ0 ∙ tan (𝑞𝑓𝑖𝑛𝑎𝑙 + 1)𝜋 ― 2𝑞𝑓𝑖𝑛𝑎𝑙𝜆 + 𝐴 = 0

(35)

𝑓𝑎𝑠𝑡,𝑠𝑢𝑟𝑓 0 𝑓𝑎𝑠𝑡,𝑠𝑢𝑟𝑓 𝑓𝑎𝑠𝑡 with 𝐴 = 𝜖0𝑎 + 𝑒0(1 ― 𝛾)𝐸 + 𝐺𝑓𝑎𝑠𝑡 = ―8.7 𝑠𝑜𝑙,𝐻𝑎𝑑 ― 𝐺𝑠𝑜𝑙,𝐻 + . As 𝜖𝑎 + 𝐺𝑠𝑜𝑙,𝐻𝑎𝑑 ― 𝐺𝑠𝑜𝑙,𝐻 +

eV, 𝐴 varies from ―8.7 eV to ―8.4 eV, when 𝐸 varies from 0 to 0.6 V, assuming 𝛾 = 0.5. As a result, 𝑞𝑓𝑖𝑛𝑎𝑙 is around -0.05, with a variance less than 3 × 10 ―3, in the examined potential range. This analysis also indicates that 𝑞𝑓𝑖𝑛𝑎𝑙 is more negative when Δ0 is higher, which is in line with the expectation that the adsorbate shares more electrons with the metal when the electronic interaction is stronger. This simple analysis demonstrates that the presented model enables a theoretical interrogation of the electrosorption valency as well.

28


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Figure 4. Activation energies. (a) Model results are calculated with the wide band approximation using the basic parameter set in the potential range of [0, 0.6 V]. The case without considering EDL effects is realized by fixing 𝜙(𝑑) = 𝜙𝑆,𝑏𝑢𝑙𝑘 in the present model. The abscissa 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 axis is the Gibbs free energy of the reaction, 𝛥𝐺 = 𝐺𝑓𝑖𝑛𝑎𝑙 𝑠𝑦𝑠 ― 𝐺𝑠𝑦𝑠 . In

panel (b), the effect of the potential of zero charge is examined.

In Figure 4 (a), without considering EDL effects, the relations between activation

energies

ΔG𝑎𝑐/𝑎

and

the

reaction

free

energy

𝛥𝐺

are

approximately linear and symmetrical for the forward and backward reaction, namely αa ≈ 𝛼𝑐 ≈ 0.5. However, for the case of ς = 0 (viz., 𝜇𝑠𝑢𝑟𝑓 = 0), namely in presence of EDL effects but in absence of surface dipoles, the symmetry between the forward and backward reaction is broken. Specifically, the relation of Δ𝐺𝑎𝑐 becomes more inclined while that of Δ𝐺𝑎𝑏 less, viz., 𝛼𝑐 > 0.5 and αa < 0.5. Moreover, both curves are down shifted by EDL effects when 𝐸 < 𝜙𝑝𝑧𝑐, namely, 𝜎𝑀 < 0. This down shift in activation energies can be understood as follows. As the reactant at the transition 29



𝑀 state carries a positive charge, 𝐺𝑇𝑆 goes more 𝑠𝑦𝑠 shall decrease when 𝜎

negative, resulting in a smaller activation barrier. The same reasoning can be applied to explain why both curves are up shifted by EDL effects when 𝐸 > 𝜙𝑝𝑧𝑐, namely, σM > 0. For the case of 𝜍 = ―0.05, namely in presence of both EDL effects and surface dipoles, anomalies manifest in the lower potential region involving adsorption. It is revealed that the decrease in Δ𝐺𝑎𝑐 upon lowering 𝐸 is suppressed, while the increase in Δ𝐺𝑎𝑎 is, instead, enhanced. In addition, a positive shift in 𝜙𝑝𝑧𝑐, bringing about a negative shift in σM, lowers both Δ𝐺𝑎𝑐 and Δ𝐺𝑎𝑎, see Figure 4 (b). In order to rationalize the above anomalies, we need to examine EDL properties as shown in Figure 5. Double layer properties Figure 5 (a) shows 𝜎𝑀 and 𝜃 as functions of 𝐸. For comparison, we also include the case of 𝜍 = 0, namely the case without chemisorption-induced surface dipoles, for comparison. For the case of 𝜍 = 0, 𝜎𝑀 is an approximately linear function of 𝐸, and becomes negative below 𝜙𝑝𝑧𝑐 = 0.3 V. Moreover, 𝜙0, the potential at the HP, is a monotonic function of 𝐸. The slope of the 𝜙0~𝐸 relation is calculated to be 𝛾 = ∂𝜙0/∂𝐸 ≈ 0.6. In contrast, for the case of considering surface dipoles (the line marked with 𝜍 = ―0.05), non-monotonic phenomena are seen in the relations of 𝜎𝑀~𝐸 and 𝜙0~𝐸, resulting in negative slopes, 𝛾 < 0, in the potential region involving adsorption. Chemisorption-induced surface dipoles, pointing from the metal to the adsorbate, shift 𝜎𝑀 and 𝜙0 to more positive values upon decreasing 𝐸. This effect can be understood in an equivalent way 30


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that 𝜙𝑝𝑧𝑐 is lowered, namely, the work function is increased, caused by the chemisorption-induced surface dipoles 𝜇𝑠𝑢𝑟𝑓, as clearly expressed in Eq. (13). In agreement with the present work, a DFT study on Pt(111) by Bonnet and Marzari reported that the work function decreases from 5.69 eV at a zero coverage 𝜃 = 0 to 5.21 eV at a full coverage 𝜃 = 1.58 DFT calculations of Anderson et al. also indicated 𝜙𝑝𝑧𝑐 of Pt(111) increases at higher coverages of OHad, implying 𝜎𝑀 < 0 in the potential region with OHad formation (0.6-0.9 VRHE).59 The non-monotonic surface charging relation has been modeled by Huang et al.49-50 and corroborated by a recent experiment of Feliu et al.60

Figure 5. EDL properties. (a) 𝜃 and 𝜎𝑀, (b) 𝜙0 as functions of 𝐸 for two cases: with (marked with

ς = ―0.05) and without (ς = 0)

chemisorption-induced surface dipoles. As far as the adsorbate formation is considered, upon decreasing 𝐸, 𝜃 31



Page 32 of 48

climbs up below 0.2 V and nearly saturates at 0 V. Note that the theory per se does not incorporate any direct adsorbate-adsorbate interactions. The fact that the 𝜃~𝐸 relation gradually rises implies surprisingly strong adsorbate-adsorbate interactions. These interactions must therefore have an indirect nature. We will revisit this issue in the discussion section.

Discussion In this section, we are to delve into the EDL effects on the transfer coefficient and to search for the origins of indirect adsorbate-adsorbate interactions. The discussion is also extended to the effectiveness of supporting electrolyte in diminishing the EDL effects, implications of model results for electrocatalyst design, as well as limitations of this work and the corresponding remedy. Negative transfer coefficient? Transfer

coefficient

is

a

fundamentally

important

concept

in

electrochemistry. Definition of the transfer coefficient may still be a topic for controversy regarding multi-electron reactions,61 but it is welldocumented and accepted without question as long as an elementary oneelectron reaction is concerned. In the influential textbook Electrochemical Methods by Bard and Faulkner,62 𝛼𝑎 (𝛼𝑐) is defined as the fraction of the total energy change of an electrochemical reaction that changes the activation energy of the anodic (cathodic) reaction, upon a shift in the electrode potential from the original value E1 to a new value E2. The subtlety here is the definition of the “total energy change”, which is taken 32


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to be 𝑒0(𝐸2 ― 𝐸1). This working definition is adopted in many textbooks and monographs.63,64 Under this premise, two properties of the transfer coefficient are readily available: (1) 0 < αc,αa < 1, (2) αc + αa = 1. Strikingly, the present model casts doubt into the above two properties. Analysis upon Figure 3 shows that αc + αa < 1 and surface dipoles may lead to αc ≤ 0. Therefore, there is a call for an overhaul of the definition of transfer coefficients for electrocatalytic reactions. To expound this point in a more transparent manner (the merit of analytical theories!), we present a highly simplified theoretical analysis below. According to Eq.(31), variation in Δ𝐺𝑖𝑛𝑖𝑡𝑖𝑎𝑙 at the initial state upon a 𝑠𝑦𝑠 potential shift from 𝐸1 to 𝐸2 is given by 𝛿Δ𝐺𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = ― 𝑒0(𝐸2 ― 𝐸1). At the 𝑠𝑦𝑠 ―𝜍 final state, Had carries a net charge of –ς, therefore, variation in Δ𝐺𝑓𝑖𝑛𝑎𝑙 is 𝑠𝑦𝑠

given by 𝛿Δ𝐺𝑓𝑖𝑛𝑎𝑙 𝑠𝑦𝑠 = ς(1 ― γ)𝑒0(𝐸2 ― 𝐸1) after neglecting variation in the second term on the rhs of Eq.(32). Given that ς = ―0.05 and γ = 0.6, a rough estimate gives 𝛿Δ𝐺𝑓𝑖𝑛𝑎𝑙 𝑠𝑦𝑠 = ―0.02 eV upon shifting E from 0.3 V to 0.6 V, which is close to the “true” value (calculated using the model without further simplifications) of ―0.01 eV. At the transition state, variation in 𝑇𝑆 𝑇𝑆 𝑇𝑆 is the slope of Δ𝐺𝑇𝑆 𝑠𝑦𝑠 is written as, 𝛿Δ𝐺𝑠𝑦𝑠 = ― (1 ― 𝛾 )𝑒0(𝐸2 ― 𝐸1), where 𝛾

the relation between 𝜙𝑇𝑆, the potential at the transition state point in the solution phase, and the electrode potential 𝐸, 𝛾𝑇𝑆 = ∂𝜙𝑇𝑆/∂𝐸. The total energy change is

𝛿Δ𝐺 = (1 + ς(1 ― γ))𝑒0(𝐸2 ― 𝐸1). Therefore,

according to the working definition of transfer coefficient, we have αa + 𝛼𝑐 = 1 + ς(1 ― γ) < 1 when ς < 0 and γ < 1. This deviation of αa + 𝛼𝑐 from unity is originated from the partial charge transfer (ς ≠ 0) with a covalent nature, and influenced by electrostatic interactions (γ is an electrostatic 33



parameter of the EDL). Moreover, it is readily seen that, 𝛼𝑐 = γTS and 𝛼𝑎 = 1 ― γTS + ς(1 ― γ). As we have shown in Figure 5 (b), surface dipoles may render γ < 0. Consequently, it is possible that 𝛾𝑇𝑆 could also be negative, namely 𝛼𝑐 < 0, in presence of sufficiently strong surface dipoles, viz. more negative ς. For the case of ς = ―0.1 (an artificial number) given in Figure 6 (a), it is found that Δ𝐺𝑎𝑐 decreases with increasing Δ𝐺 in the range of ―0.3 eV < Δ𝐺 < ―0.25 eV, namely, 𝛼𝑐 < 0.

Figure 6. (a) Negative αc is observed in the activation energy relation for the case of ς = ―0.1 while the adsorbate coverage relation remains unaltered. (b) Comparison between the case with a ultra-concentrated supporting electrolyte in regime (9 M), the base case (0.1 M), and the case without EDL effects, in terms of activation energies as a function 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎 of ΔG = 𝐺𝑓𝑖𝑛𝑎𝑙 𝑠𝑦𝑠 ― 𝐺𝑠𝑦𝑠 . The upper bundle of lines: 𝛥𝐺𝑎 and the lower

bundle: 𝛥𝐺𝑎𝑐.

Indirect adsorbate-adsorbate interactions 34


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Under equilibrium, the net rate of the reaction, expressed in Eq.(S3) in the SI, should be zero, leading to an expression relating the adsorbate coverage 𝜃 to solution properties (𝑐𝑏𝑢𝑙𝑘 𝐻 + ) and the electrode potential 𝐸. This equilibrium adsorption isotherm can be written in the familiar form of the Langmuir isotherm, (36)

𝑒0 𝜃 (𝐸 ― 𝐸𝑒𝑞) ln ― ln 𝑐𝑏𝑢𝑙𝑘 𝐻+ = ― 1―𝜃 𝑘𝑏𝑇

with the equilibrium potential, 𝐸𝑒𝑞, given by,

𝑒𝑞

𝐸 =

(𝐺𝑓𝑎𝑠𝑡,𝑏𝑢𝑙𝑘 ― λ) ― 𝜖0𝑎 𝑠𝑜𝑙,𝐻 + 𝑒0

―

Δ0

ln 2𝜋𝑒0

2

𝜀𝑒𝑓𝑓,𝑓𝑖𝑛𝑎𝑙 + Δ20 𝑎 2

𝜖0𝑎 + Δ20

―

𝜍(𝜖𝑒𝑓𝑓,𝑓𝑖𝑛𝑎𝑙 + 𝜍𝜆) 𝑎

(37)

𝑒0

where the significance of variables can be found in the theory development section. The first term on the rhs of Eq.(37), denoted as 𝐸𝑒𝑞,0, represents the standard equilibrium potential of a complete one-electron transfer reaction, namely 𝜍 = 0, and without electronic interactions, namely, Δ0 = 0. The second term corresponds to the effect of electronic interactions, and the third term that of the net charge remaining on the adsorbate. Without considering these two effects, 𝐸𝑒𝑞 is a constant, which is the case for the Langmuir isotherm. The fundamental insight of the Frumkin isotherm is to realize that 𝐸𝑒𝑞 varies linearly as a function of the adsorbate coverage 𝜃 due to adsorbate-adsorbate interactions, that is, 𝐸𝑒𝑞 = 𝐸𝑒𝑞,0 ―𝜉𝜃/𝑒0. The lateral interaction coefficient 𝜉 has been employed as a versatile yet 35



Page 36 of 48

phenomenological parameter, enabling a better fit to experimental data.6567

However, fundamental understanding of

𝜉

is missing and the

unexpected large value of 𝜉 for Had on Pt(111), ca. 0.25 eV (25 kJ mol-1), remains hitherto unexplained. Equating the derivative of 𝐸𝑒𝑞 with respect to 𝐸 obtained from Eq.(37) and that from the Frumkin isothermal relation leads to,

( )

∂𝜃 𝜉 = 𝑒0(1 ― 𝛾) ∂𝐸

(

―1

)

(38)

Δ0 1 𝜀𝑒𝑓𝑓,𝑓𝑖𝑛𝑎𝑙 𝑎 𝜍+ 𝜋𝜀𝑒𝑓𝑓,𝑓𝑖𝑛𝑎𝑙2 + Δ2 𝑎

0

where 𝛾 = ∂𝜙0/∂𝐸, an electrostatic property of the EDL, is usually in the range of 0 < 𝛾 < 1, as shown in the dotted line in Figure 5 (b). However, chemisorption-induced surface dipoles may lead to 𝛾 < 0, as shown in the solid line in Figure 5 (b). Bonnet and Marzari reported 𝜍 ≈ ―0.09 for Pt(111).58 We know from the properties of inequalities that

2

+ Δ20) ≥ ―0.5 Δ0/(𝜀𝑒𝑓𝑓,𝑓𝑖𝑛𝑎𝑙 𝜀𝑒𝑓𝑓,𝑓𝑖𝑛𝑎𝑙 𝑎 𝑎

when

is negative and Δ0 is positive. Combined, a quick estimate gives 𝜀𝑒𝑓𝑓,𝑓𝑖𝑛𝑎𝑙 𝑎 𝜉 ≈ 0.1 eV (10 𝑘𝐽 𝑚𝑜𝑙 ―1), given that 𝛾 = 0, (∂𝜃/∂𝐸) ―1 = ―1. If the net charge on Had is more positive, 𝜉 will be greater for two synergetic effects: On one hand, 𝜍 is more negative; one the other hand, chemisorption-induced surface dipoles shift 𝛾 to more negative values. Note in passing that 𝛾 embodies the effect of non-covalent electrostatic interactions on the covalent chemisorption. The estimate is smaller than the typical value of 𝜉 ≈ 0.25 𝑒𝑉 at Pt(111) reported in several experimental studies. This discrepancy has several 36


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origins, the most obvious one being that Eq.(38) is obtained under the premise of the wide band approximation, which is obviously inaccurate for Pt(111). At any rate, above analysis do partially account for the strong adsorbate-adsorbate repulsion for Had on Pt(111). In a more general sense, it is revealed that indirect interactions that one adsorbate changes electronic and electrostatic properties of the metal and thereby alters the adsorption energy of another adsorbate play a key role in adsorbateadsorbate interactions. Effectiveness of supporting electrolyte in electrocatalysis Above analysis advocates the important role of EDL effects. A question follows naturally: are EDL effects still important in presence of supporting electrolyte? We address this question by concentrating on two states, the transition state and the final state, for 𝐺𝑖𝑛𝑖𝑡𝑖𝑎𝑙 remains unaltered with 𝑠𝑦𝑠 respect to variations in the electrolyte at a given electrode potential. As long as 𝐺𝑇𝑠 𝑠𝑦𝑠 is concerned, EDL effects can be neglected only when 𝜆𝐷 < (𝑑𝑇𝑆 ― 𝑑0) with 𝑑𝑇𝑆 being the distance from the metal surface to the transition state site in solution. From Figure 3, a rough approximation gives, 𝑑𝑇𝑆 ― 𝑑0 ≈ 1 Å. Consequently, the total concentration of cations (anions) in bulk solution, 𝑐𝑡𝑜𝑡, should be higher than 9 M to satisfy the condition, λD < 1 Å. As a result, for most cases with 𝑐𝑡𝑜𝑡 < 1 M, EDL effects 𝑓𝑖𝑛𝑎𝑙 on 𝐺𝑇𝑠 𝑠𝑦𝑠 are not negligible. As to 𝐺𝑠𝑦𝑠 , the final state is always near the

metal surface, viz. 𝑑𝑇𝑆 ― 𝑑0 ≈ 0, hence, EDL effects are always present unless the orbital of the adsorbate is fully occupied, that is, there is no net charge remaining on the adsorbate. 37



Figure 6 (b) compares the case with an ultra-concentrated supporting electrolyte (9 M), the base case (no supporting electrolyte, 0.1 M), and the case without EDL effects. It is observed that the activation energy relations are closer to those without considering EDL effects and less curved in presence of a concentrated supporting electrolyte. However, EDL effects are still remarkable. In summary, EDL effects play an important role in electrocatalysis, and introducing supporting electrolyte, even with a high concentration, is unlikely to eliminate EDL effects altogether. Implications for electrocatalyst design The d-band theory developed by Norskov and co-workers pinpoints a clear pathway toward more efficient electrocatalysts, that is, by fine tuning the binding energy of a certain reaction intermediate, which is closely related with the d-band center.1-3 The Schmickler theory gives similar suggestions that a good electrocatalyst should have a d-band spanning the Fermi level and strong electronic interactions between the metal and redox species in solution.37-38 Nevertheless, these two theories concentrate on covalent interactions, while the role of non-covalent interactions in electrocatalyst design is not discussed. The present work suggests that non-covalent interactions are important and activation energies of a reaction can be modulated by changing electrostatic properties of the metal, such as the potential of zero charge. As a concrete example, activation energies of the hydrogen adsorption reaction become smaller when the metal surface is negatively charged. It has to be emphasized that covalent and noncovalent interactions are interwoven, thus, cannot be separated in precision design and screening of electrocatalysts. 38


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Limitations and the remedy To this point, the most obvious limitation of the present work is the wide band approximation adopted in above analysis, of which the value lays in permitting analytical solutions and transparent interrogations of the theory, although the theory developed is generally flexible to any band structure. To check the generality of conclusions obtained under the premise of the wide band approximation, we reexamine above analysis by assuming a semielliptical distribution for the d-band,42,

68

as detailed in

the SI. We conclude that above analysis still holds well, see discussion on Figure S2 in the SI. Moreover, such practice allows us to study the effect of the d-band center and width. We reproduce the conclusion made by the Schmickler theory that a good electrocatalyst should have a d-band crossing the Fermi level, as schematically illustrated in Figure S3. The second limitation concerns about model parameters. The parameter set used in above analysis does not attempt to describe any specific electrocatalyst; they, being within a reasonable range, merely sever the goal of achieving a qualitative understanding of the theory. Extensive DFT calculations are required to tailor the parameter set for a specific electrocatalyst, as demonstrated in the works from the Schmickler group.37-39

Conclusions We have developed a refined theory of electrocatalysis that incorporates EDL effects in a self-consistent manner, furnishing the first conceptual framework for discussing the interplay between covalent and non-covalent 39



interactions in electrocatalysis. Taking the hydrogen adsorption on metals as a prototypical electrocatalytic reaction, the theory reveals that EDL effects strongly modulate the reaction, specifically, smaller (higher) activation energies are seen when the metal surface charge is more negative (positive). Chemisorption-induced surface dipoles give rise to a non-monotonic surface charging relation, thus, exerting dramatic influence on the activation energy profile, and resulting in anomalies of the transfer coefficient. The theory predicts that αa + 𝛼𝑐 < 1 and even a negative αc under certain circumstance. Moreover, the theory elucidates how non-covalent electrostatic interactions affect the adsorption isotherm, and provides fundamental insights into the strong indirect adsorbateadsorbate repulsion of Had. A parametric analysis indicates that EDL effects can hardly be neglected by introducing supporting electrolyte for electrocatalytic reactions. The present study suggests a new direction of designing more efficient electrocatalysts by fine tuning electrostatic properties of the electrode so as to create an enabling interfacial environment for electrocatalytic reactions. We argue that a new paradigm of electrocatalysis may be born from understanding and harnessing the synergy between covalent and non-covalent interactions.

Supporting information Double layer parameters; Expressions of activation barriers and reaction rates; Comparison with other theories; Model parameterization; Potential energy surface at 0.6 V; Beyond the wide band approximation.

Acknowledgement 40


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Dedicated to Professor Bingliang Wu ( 吴秉亮 ) on occasion of his 80th birthday. J Huang appreciates financial support from National Natural Science Foundation of China (no. 21802170), and the starting fund for new faculty members at Central South University (no.502045001). S. Chen appreciates financial support from Natural Science Foundation of China (no. 21832004). The authors appreciate Yu Gao at Wuhan University, Dr. Junxiang Chen at FJIRSM and Professor Yanxia Chen at University of Science and Technology of China for useful discussion and comments.

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