Variants of Surface Charges and Capacitances in Electrocatalysis : Insights from Density-Potential Functional Theory Embedded with an Implicit Chemisorption Model

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Variants of Surface Charges and Capacitances in Electrocatalysis : Insights from DensityPotential Functional Theory Embedded with an Implicit Chemisorption Model © Authors 2024 Published version Huang, Jun; Domínguez-Flores, Fabiola; Melander, Marko Huang, J., Domínguez-Flores, F., & Melander, M. (2024). Variants of Surface Charges and Capacitances in Electrocatalysis : Insights from Density-Potential Functional Theory Embedded with an Implicit Chemisorption Model. PRX Energy, 3, Article 043008. https://doi.org/10.1103/prxenergy.3.043008 2024 PRX ENERGY 3, 043008 (2024) Variants of Surface Charges and Capacitances in Electrocatalysis: Insights from Density-Potential Functional Theory Embedded with an Implicit Chemisorption Model Jun Huang ,1,2,3,*Fabiola Domínguez-Flores ,4and Marko Melander 4,† 1Institute of Energy Technologies, IET-3: Theory and Computation of Energy Materials, Forschungszentrum Jülich GmbH, 52425 Jülich, Germany 2Theory of Electrocatalytic Interfaces, Faculty of Georesources and Materials Engineering, RWTH Aachen University, Aachen 52062, Germany 3Jülich Aachen Research Alliance JARA Energy & Center for Simulation and Data Science (CSD), 52425 Jülich, Germany 4Department of Chemistry, Nanoscience Center, University of Jyväskylä, P.O. Box 35 (YN), FI-40014, Jyväskylä, Finland (Received 13 May 2024; revised 28 August 2024; accepted 23 October 2024; published 15 November 2024) Prevalent electrolyte effects across a wide range of electrocatalytic reactions underscore the general importance of the local reaction conditions in the electrical double layer (EDL). Compared to traditional EDLs, the electrocatalytic siblings feature partially charged chemisorbates that could blur our long-held views of surface charge densities and differential capacitances—two interrelated quantities shaping the crucial local reaction conditions. Herein, five variants of surface charge density and three variants of differential capacitance in the presence of chemisorbates are defined and compared. A semiclassical model of electrocatalytic EDLs is developed for a quantitative analysis of the differences and interrelationships between these variants of surface charge densities and differential capacitances. It is revealed that the potentialand concentration-dependent net charge on these chemisorbates dramatically changes the surface charge densities and differential capacitances. Specifically, the free surface charge density could decrease as the electrode potential becomes more positive, implying that the corresponding differential capacitance is negative. The relationship between the free and total surface charge densities is analyzed with aid of the concept of electrosorption valency. By linking the electrosorption valency with the differential capacitance in the absence of chemisorbates, we explain the potential and concentration dependence of the former. The conceptual analysis presented in this work has important implications for experimental characterization and first-principles-based atomistic simulations of electrocatalysis and EDL effects. Particularly, we disclose a hidden yet potentially crucial disadvantage of widely employed atomistic simulation models that fix the coverage of chemisorbates. Proposing the self-consistent implicit model as an expedient remedy for this disadvantage, this work contributes to more realistic modeling of electrocatalytic EDLs. DOI: 10.1103/PRXEnergy.3.043008 I. INTRODUCTION Electrocatalytic reactions proceed via the adsorption and desorption of chemisorbed intermediates. The thermodynamic state of the chemisorbed intermediates and *Contact author: [email protected] †Contact author: marko.m.melander@jyu.fi Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. the kinetic rate of their adsorption and desorption are determined by both solid and liquid components of the electrode-electrolyte interface, as exemplified by the hydrogen evolution reaction (HER). On one hand, the exchange current density, J0,HER, can vary up to 7 orders of magnitude among different metallic electrodes [1–3]. The catalytic trends between different metals have been successfully rationalized using the Sabatier principle [4,5] and modern theories of chemisorption and electrocatalysis [6–9]. On the other hand, J0,HER at Pt(111)is around 20 times higher in 0.05MH2SO4than in 0.1MNaOH [10]. Moreover, J0,HER in alkaline solutions decreases with increasing effective cation size (Li+>Na+>K+> 2768-5608/24/3(4)/043008(21) 043008-1 Published by the American Physical Society JUN HUANG et al. PRX ENERGY 3, 043008 (2024) Cs+) at Ir, Pd, Pt electrodes, while the trend is reversed at Cu, Ag, Au electrodes [11,12]. Remarkable electrolyte effects have also been reported for oxygen reduction [13–19], oxygen evolution [20–25], and carbon dioxide reduction [26–29], as well as other electrocatalytic reactions, on various catalysts, underpinning the crucial impact of both the liquid and solid components on electrocatalytic activity and selectivity. Understanding the origins of these electrolyte effects is a long-standing and classical, yet topical, problem in electrocatalysis, as reflected by a review written by Frumkin in 1959 [30] and recent vigorous research endeavors [31–33]. Recent emphasis has been targeted towards the local reaction conditions within the nanometric electrical double layer (EDL) at the electrode-electrolyte interface [31–37]. Within the theoretical and computational electrocatalysis fields, there is a strong consensus on the need to extend the standard computational hydrogen electrode (CHE) method by incorporating the EDL effects [38–44]. However, it remains unresolved how these effects should be described in an efficient, accurate, and general way; as a result, various approaches from constant-charge to constant-potential schemes have been proposed and are under constant development. The constant-charge scheme involves complementing the canonical free energy calculated using the CHE method with various electrostatic correction terms, mostly of the form −C(U−U0)2/2, where Cis an effective capacitance, U0is the electrode potential corresponding to the canonical DFT slab model, and Uis the electrode potential of interest [45–47]. In this sense, the constant-charge scheme adopts the Helmholtz model for the EDL, describing it as a capacitor with a constant capacitance in the potential range between Uand U0[48,49]. The constant-charge scheme is handy and widely used, since only a posteriori correction of EDL effects is required, without resorting to an explicit treatment of the EDL. However, these models need a good model of C. Most often, Cis treated as a constant, which cannot be considered a generally valid approximation [47]. The constant-potential scheme entails grand-canonical free energy calculations [38,40,41,43]. A fully grandcanonical scheme should allow the electron number (Ne); the coverage of chemisorbates (θi); and the numbers of cations (Nc), anions (Na), and solvent molecules (Ns)to vary as functions of the electrochemical potentials of these components [50]. Since genuine fully grand-canonical calculations are prohibitively expensive and need nontrivial methodological efforts beyond the current models, various approximations have been introduced in practical applications, including an implicit treatment of the electrolyte solution and a reduced number of grand-canonical variables [47,51]. Crucially, available implicit solvent models for DFT simulations cannot treat chemisorption or covalent bonding; instead, they are restricted to only “outerHelmholtz layer” effects dictated mainly by electrostatic interactions [51]. Therefore, current constant-potential calculations tacitly assume a fixed coverage of chemisorbates and should, hence, be referred to as electronically grand-canonical [43]orsemi-grand-canonical simulations [47]. While the constant-charge and constant-potential simulations are thermodynamically equivalent and related to each other via a Legendre transformation [47,52–54], the inability to treat chemisorption implicitly impacts them differently. Constant-potential calculations require a good computational model of the capacitance a priori.By contrast, constant-charge calculations can correct for the capacitance a posteriori using even experimental capacitance data. Therefore, provided with adequate knowledge about the differential capacitance of the EDL, it is possible to calculate the constant-potential free energy from its constant-charge counterpart [47,52–54]. While it is claimed that constant-charge calculations are more immune to inaccuracies and omissions in the computed capacitance, it should be emphasized that it is not any easier to obtain accurate differential capacitances than to self-consistently carry out constant-potential calculations directly. For instance, it has been shown in the example of the acidic Volmer step on Au(111)that the constant-charge scheme with a constant-capacitor approximation agrees quantitatively with the electronically grandcanonical scheme [54]. However, the grand-canonical calculations were performed by fixing the chemisorbate coverage in a single electronically grand-canonical calculation. Importantly, changes in coverage during the reaction, as required by a fully grand-canonical calculation of reaction kinetics [55], have not been addressed. The consequences of constraining the chemisorbate coverage have, however, barely been discussed. What consequences does the constrained chemisorbate coverage have on the grand-canonical free energies? This question is transformed to understand the influence of chemisorption and potential-dependent chemisorbate coverage on the differential capacitance that bridges canonical and grandcanonical free energies. As we show below, there are at least three capacitances in the presence of chemisorbates. It is not well agreed upon in the literature which one should be used in the conversion between the electrode charge and the electrode potential. Moreover, we show that chemisorption changes not only the magnitude of the capacitance, as tacitly assumed previously, but also the sign of the differential capacitance. Our results imply that electronically grand-canonical methods with constrained chemisorbate coverages cannot describe this essential characteristic of electrocatalytic EDLs. Likewise, the fully self-consistent implementation of constant-charge calculations should consider the impact of the chemisorptioninduced changes in the differential capacitance. As a step forward towards fully self-consistent simulations of electrocatalytic EDLs, we propose a practical way to include 043008-2 DPFT-CHEMISORPTION... PRX ENERGY 3, 043008 (2024) variable chemisorbate coverages in electronically grandcanonical simulations through an advanced semiclassical implicit electrolyte model. II. CONCEPTS As discussed in Sec. I, both constant-charge and constant-potential simulations of EDL effects require knowledge of the differential interfacial capacitance (C), which is defined as the electrode potential derivative of the surface charge density. In what follows, we begin by laying out a few basic concepts of electrocatalytic EDLs. Specifically, we define several types of differential capacitances and surface charge densities for electrocatalytic EDLs with chemisorbates. Then, we develop a semiclassical model of electrocatalytic EDLs to allow for a quantitative discussion on the differences and relationships between these variants. Afterwards, the semiclassical model is applied to understand the influence of chemisorption on the differential capacitances and surface charge densities. In the end, we relate the insights learned from the semiclassical model to experimental characterization of electrocatalytic EDLs and DFT-based models. For the latter aspect, the emphasis is put on understanding the consequence of the fixed chemisorbate coverage in electronically grand-canonical models and their future improvements towards a fully grand-canonical model of EDLs. A. Surface charge densities An atomistic simulation of an EDL yields the volumetric charge density (ρ), not the surface charge density [56]. Surface charge density is obtained by integrating the volumetric charge density along a particular line, most often the line perpendicular to the electrode surface. σ=ze zs (ρn+ρe)dz(1) where ρnis the volumetric charge density of nuclei, and ρe is that of electrons which is negative. It needs to be noted that the selection of the starting and ending points of the line, zsand ze, respectively, make the quantification of σ ambiguous. This is true even for ideally polarizable EDLs, e.g., that of the Ag(111)-aqueous KPF6solution interface, because the overlap between the electron tail of the metal and the electron cloud of water molecules makes it impossible to rigorously define a surface charge density, σ. Nevertheless, the electroneutrality condition implies that σ=− (Nc−Na)|e| Asurface (2) where Ncand Naare the number of K+and PF− 6in the electrolyte solution, respectively; |e|is the elementary charge; and Asurface is the geometrical area of the metal surface. For ideally polarizable EDLs, the states at the Fermi level are fully localized on the metal side, and excess electrons are added to or removed from the metal alone, which can be quantified by the projected density of states on the metal states. In constant-charge DFT simulations of the EDL, Eq. (2) implies that one can tune σby varying the type and number of ions in the electrolyte solution [57–59]. The corresponding electrode potential is then obtained from the self-consistently computed Fermi level or the inner potential of the system. In this scheme, the potential of zero charge (PZC) usually corresponds to metal-water interfaces. Rigorously speaking, this potential is the potential of zero ions (PZI) because there are no electrolyte ions in the liquid phase. On the contrary, at the PZC an equal number of cations and anions exist in the EDL. The PZC and PZI are not necessarily identical since the presence of cations and anions in the EDL at the PZC may change the configuration of interfacial water molecules, and thereby impact both the Fermi level of the system and entropy. Though σdefined in Eq. (2) is unambiguous, it does not offer a clear connection with σfrom Eq. (1) due to the spatial overlap of metal electrons and electrons of electrolyte components. Given that electrons are indistinguishable particles, and that charge partition cannot be defined uniquely, one cannot assign the electrolyte, solvent, or electrodes charge without extrathermodynamic assumptions. It is very plausible that one cannot use Eq. (1) to unambiguously obtain the same σexpressed in Eq. (2), no matter how we choose zsand ze. Therefore, Eq. (2) is a pragmatic definition based on a modelistic picture. Compared to “simple” EDLs without chemisorption, the EDLs encountered in electrocatalysis are even more complex, as they feature chemisorbed ions and solvent molecules [31,60,61]. In a modelistic picture, these chemisorbates in the inner Helmholtz plane (IHP) bear a net charge and/or a dipole moment. The charge on chemisorbed ions is different from that of the same ions in the bulk solution, due to the formation of chemical bonds between these ions and the electrode. This has been revealed in the theory of chemisorption [6–9] and quantum chemical simulations [62]. Moreover, the chemisorbed ions do not need to carry an integer charge but are typically partially charged. This fact further complicates the definition and calculation of surface charge densities. As a result, at least five different types of surface charge densities can be defined, as summarized in Fig. 1: the total surface charge density (σtot), the free surface charge density (σfree), the metal surface charge density (σM), the total and net charge density of chemisorbates (σad,tot,σad,net). All these quantities are related but only some are measurable while others are modelistic quantities. We will explore their relationships and characteristics. σtot is a measurable quantity, as it corresponds to the total amount of electrons flowing through the external 043008-3 JUN HUANG et al. PRX ENERGY 3, 043008 (2024) FIG. 1. Surface charge densities of electrocatalytic EDLs. Total charge density, σtot, measures the total amount of electrons flowing through the external circuit when the electrode is subject to a given potential. Metal charge density, σM, is a modelistic concept, describing the surface charge density within the metal region, of which the outer boundary is indefinite in the presence of chemisorbates. Total and net charge density of partially charged chemisorbates are σad,tot and σad,net, respectively, and their difference is the amount of charge transferred between the metal and the chemisorbed ion, namely, the charge involved in the partial charge transfer process, denoted σPCT. Total charge density σtot is divided into σPCT, charging the chemisorbates, and σM, charging the metal. Amount of charge stored in the diffuse layer stretching from the outer Helmholtz plane to the bulk solution is the negative of the free charge density, σfree. Since the overall EDL is electroneutral, σfree is the sum of σMand σad,net, which establishes a rigid dichotomy between the metal and the chemisorbates. circuit upon changing the electrode potential. It also bears thermodynamic significance. The electrocapillary equation links it with the interfacial tension and electrode potential: σtot =(∂γ /∂E)˜μi, with γbeing the interfacial tension, Eis the electrode potential, and ˜μiis the electrochemical potentials of the constituents of the electrolyte solution [63,64]. σtot was often denoted as σMin earlier references, e.g., Ref. [65]. Lorenz and Salié thus termed σtot the thermodynamic charge density [66], while Frumkin and Petrii coined the name total charge density [67]. In the presence of chemisorption, σtot contains contributions from EDL charging or discharging, chemisorbate adsorption or desorption, and the nontrivial coupling of these processes. The potential-dependent changes in σtot can be understood using a simple gedanken experiment of a single anion chemisorption at constant potential. Let us first freeze all other components of the EDL and move a monovalent anion from the bulk solution to the IHP. No matter how the extra valence electron is shared between the electrode and the anion, an electron must flow out of the electrode to ensure overall electroneutrality. However, the presence of this partially charged chemisorbed anion brings up an extra potential drop between the metal surface and the IHP. To retain the electrode potential, this potential drop must be compensated for and screened by accumulation of counterions in the diffuse layer. Therefore, a certain number of cations move into the diffuse layer, rendering an equal number of electrons flowing back into the electrode. In sum, the change in σtot is less than one electron upon chemisorption of a monovalent anion. Extending from this example, we consider a general case involving multiple types of chemisorbed ions with surface coverages θi=Ni/NM, with NMbeing the areal density of metal atoms and Nithat of adsorbates i. We denote the partial charges of these chemisorbed ions as qiand their corresponding bulk charges qb i. The total charge density of chemisorbates σad,tot is σad,tot = i θiNMqb i,(3) and the net charge density of chemisorbates, σad,net,is σad,net = i θiNMqi.(4) We note that |σad,net|≤|σad,tot|. One can also use Frumkin’s definition of free surface charge, σfree [67], where “free” means that the addition (removal) of these ions does not involve the formation (cleavage) of chemical bonds. Put in an equivalent way, the addition (removal) of these ions is purely of electrostatic origin. Since the outer Helmholtz plane (OHP) is the closest plane that nonspecifically adsorbed ions can approach the electrode surface, a pragmatic definition of σfree is [56] σfree =−z∞ zOHP  i niqb idz,(5) where zOHP denotes the OHP, z∞is the bulk solution, and niis the number density of ions. As in the example of anion chemisorption, the total surface charge density is then a sum of the negative of the total adsorbate charge density and the surface free charge density: σtot =−σad,tot +σfree,(6) where the negative sign before σad,tot means that the electrode charge tends to counterbalance the charge carried 043008-4 DPFT-CHEMISORPTION... PRX ENERGY 3, 043008 (2024) by the chemisorbates. In some cases, the negative sign is omitted when σad,tot is defined with respect to the electrode charge. It is also important to note that σfree and σad,tot are coupled. The immediate influence of this coupling is that the chemisorption of a monovalent anion results in a fraction of an elementary charge flowing in the external circuit. This leads to the concept of electrosorption valency, l [64,66], which is formally defined in Eq. (9). Because σtot is measurable, Eq. (6) indicates that σfree is known if θican be measured, e.g., using optical spectroscopy approaches [68–70]. However, as σfree is much smaller than both σtot and σad,tot, the accuracy of calculating the difference between two large numbers is a perplexing issue. It is important to note that the relationship σM=σfree does not hold for the case with chemisorption due to the presence of nonzero adsorbate charge, σad,net. Instead, we have σM+σad,net =σfree,(7) as illustrated in Fig. 1. Substituting Eq. (7) into Eq. (6) leads to σtot =−σPCT +σM,(8) where σPCT =(σad,tot −σad,net)is the partial charge transferred from chemisorbates to the electrode. We wish to emphasize that these formal definitions of surface charge variants are not solely driven by academic curiosity; rather, they play a crucial role in delineating modelistic quantities from measurable quantities, thus fundamentally shaping the modeling and understanding of capacitance and, consequently, EDL effects in electrocatalysis. B. Electrosorption valency The electrosorption valency, li, is a key quantity in understanding the influence of chemisorption on the differential capacitances, as it measures how the total charge density depends on the surface coverage of a given species [64,66]: li=−1 NM∂σtot ∂θiE,θj=i .(9) The electrosorption valency quantifies the negative of the total charge flowing through the external circuit upon chemisorption of an ion iat constant electrode potential and coverages of other chemisorbates. liis different from qb isince σfree varies during the chemisorption process, as seen in the gedanken experiment. For the usual case with only one type of chemisorbed ion, the electrosorption valency can be reformulated as li=−qb i∂σtot ∂σad,tot E . (10) Combining Eqs. (6) and (10) leads to li=qb i−qb i∂σfree ∂σad,tot E . (11) In summary, lireflects the strength of the coupling between σfree and σad,tot at constant potential. If σfree and σad,tot are decoupled, we retrieve the trivial result, li=qb i. C. Differential capacitances The charge density variants lead to corresponding differential capacitance variants defined as C(E)=∂σ ∂E˜μi , (12) where we highlight that the capacitance depends on the electrode potential, E, and that the electrochemical potentials, ˜μi, are kept constant. According to Eq. (6), the relationship between the total capacitance, Ctot =∂σtot/∂E; the double layer capacitance, CDL =∂σfree/∂E; and the pseudocapacitance of chemisorption coined by Conway and Gileadi [71], Cad =−(∂σad,tot/∂E), is obtained as Ctot =Cad +CDL. (13) Inherited from the magnitude difference between σad,tot and σfree,Cad is usually much larger than CDL. Therefore, Ctot is usually dominated by Cad in the presence of chemisorption, see examples by Refs. [61,72]. Note that Cad and CDL are not independent; instead, CDL is influenced by the adsorbates. Usually, the charging and discharging of the diffuse layer is much faster than the adsorption and desorption of chemisorbates, which allows the separation of Cad and CDL based on the difference of their time constants using, e.g., electrochemical impedance spectroscopy (EIS) [73–77]. In this scenario, the differential capacitance measured at a sufficiently high frequency is CDL,HF, with the subscript “HF” denoting high frequency, while that measured at a sufficiently low frequency is the sum of CDL,LF, with the subscript “LF” denoting low frequency, and Cad. It is important to note that CDL,HF and CDL,LF are characteristically different due to the influence exerted by Cad;we come back to this in Sec. V. III. THEORETICAL METHODS The key question to answer is how chemisorption changes the EDL properties. Specifically, we are interested 043008-5 JUN HUANG et al. PRX ENERGY 3, 043008 (2024) FIG. 2. Schematic diagram of electrocatalytic EDLs with partially charged chemisorbates (intermediates, denoted A) located at the IHP. This work develops a grand-canonical, semiclassical model for electrocatalytic EDLs with an implicit treatment of chemisorption. Specifically, the metal electrons are described using an orbital-free DFT, the electrolyte solution is described using a classical statistical field theory, and metal-solution short-range interactions are described using Morse potentials. Chemisorption of anions is treated using the Anderson-Newns model Hamiltonian approach. The semiclassical model itself does not involve classical modelistic concepts like the IHP and OHP; instead, it describes the whole EDL with continuous density and potential profiles. in the influence of chemisorption on σfree and CDL, the coupling between σad,tot and σfree, and the coupling between Cad and CDL. To address these questions, we need a method capable of modeling the EDL with chemisorption under constant electrode and electrochemical potential conditions. For this purpose, we extend recent density-potential functional theory (DPFT) from previous models developed for ideally polarizable EDLs to EDLs with chemisorption [78–81]. As illustrated in Fig. 2, the DPFT method combines orbital-free DFT of the metal electrons, thermodynamic statistical classical DFT theory of the liquid solution, and parameterized potential relationships for short-range interactions between the metal and the electrolyte solution. In the absence of chemisorption, the DPFT approach is an orbital-free version of the recently developed constant inner-potential DFT (CIP-DFT) approach [82]. DPFT expresses the grand potential of the EDL as a hybrid explicit functional of the electron density and the electrostatic potential. The variational analysis of the grand potential functional is carried out using the Euler-Lagrange equation, resulting in two coupled partial differential equations in terms of the electron density (ne)and the electrostatic potential (φ). These two controlling equations are solved under constant electrochemical potentials of electrons, electrolyte ions, and solvent conditions, namely, under typical experimental conditions of constant electrode potential and electrolyte activity. As a result, one obtains self-consistent and continuous profiles of electron density, electrostatic potential, ion density, and solvent density. As the model describes the entire interphase [83] from the metal phase to the electrolyte or solution phase, the electrostatic potential and all the densities are calculated for the entire interphase. Properly parameterized short-range interactions between the metal and the electrolyte solution separate the two phases, guaranteeing that ions and solvent molecules do not penetrate the metal. Currently, the orbital-free DFT description cannot reliably treat chemisorption on metal surfaces [84,85]. Here, we adopt a different strategy and treat chemisorption in an implicit modelistic way. The present implicit chemisorption model can also be combined with common KohnSham DFT implementations in electrocatalysis as an extension of available implicit solvent models. Specifically, we employ the Anderson-Newns model Hamiltonian to describe the hybridization of the ion’s valence electron orbital with the metal orbitals (bands) [6–9]. The key result from the implicit chemisorption model is the partial charge of the ion, qi, which depends on the electrode potential and ion position. Since the resultant qienters the Euler-Lagrange equation for the electrostatic equation, the electronic structure, adsorbate charge, and electrostatic 043008-6 DPFT-CHEMISORPTION... PRX ENERGY 3, 043008 (2024) potential, as well as the electrolyte densities, are solved self-consistently and are naturally coupled to each other. A. DPFT-Chem model Here, we present a brief review of the key equations of the DPFT and refer interested readers to a detailed derivation in a previous work [79]. To extend the DPFT model to electrosorption problems, we reformulate the DPFT equations for the specific case of an Ag(111)electrode immersed in an electrolyte. Motivated by an experimental study [65], we consider the Ag(111)electrode in c(1−f)MKPF6+cfM KCl solution, where cis the total molar concentration of anions and 0 ≤f≤1 is the fraction of the chemisorbing Cl−anion of the total anion concentration. We denote the Ag cationic cores with the abbreviation “CC,” metal electrons with “e,” solvent with “s,” electrolyte cations with “c,” nonspecifically adsorbed PF− 6anion with “NA,” and chemically adsorbing Cl− anion with “CA.” The metal electrode is modeled as a jellium. The controlling equation for the electron density described with orbital-free DFT at the semilocal level is [75] ¯ ∇¯ ∇¯ne=20 3¯neω θTω−θXC ∂tTF ∂¯ne+∂u0 X ∂¯ne+∂u0 C ∂¯ne−(|e|φ+˜μe) ea.u.  +θTω−4 3θXC 2¯ne(θTω−θXC)(¯ ∇¯ne)2, (14) where ω=(2/5)π5/331/3(¯ne)1/3is a density-dependent coefficient; ¯ne=ne(a0)3, using the Bohr radius, a0,as the reference length is the dimensionless electron density; ¯ ∇=a0∇is the dimensionless gradient operator; θTand θXC are the coefficients of the “semilocal” gradient terms in kinetic energy and exchange-correlation energy functional, respectively; tTF is the Thomas-Fermi kinetic energy functional, u0 Xand u0 Care the exchange and correlation energy in the Perdew-Burke-Ernzerhof (PBE) functional, respectively; |e|is the elementary charge; ea.u. =27.2 eV is the reference energy in atomic units; and ˜μeis the electrochemical potential of electrons, which corresponds to the electrode potential and is the key control variable in constant-potential experiments. Equation (14) is general and can be applied to both atomic and continuum descriptions of the electrode. Herein, we adopt a jellium description of the electrode. The controlling equation for the electrostatic potential is [79] −¯ ∇(¯OP ¯ ∇¯ φ+¯ns¯psκLs) =κ((¯nCC −¯ne)+¯nc+¯nCAqCA −¯nNA), (15) where ¯ φ=|e|φ/kBTis the dimensionless electrostatic potential with the Boltzmann constant, kB, and temperature, T;¯OP is the dimensionless optical permittivity referenced to the permittivity of vacuum, 0;¯nl=nl(a0)3,l∈ {s, CC, CA, NA}are the dimensionless number densities; ¯ps=ps/|e|ais the dimensionless dipole moment of the solvent; κ=e2 0/kBT0a0is a unitless number originating from the nondimensionalization process; and Ls= coth(ps¯ E)−(ps¯ E)−1is the Langevin function with the dimensionless electric field, ¯ E=¯ ∇¯ φ. Recently, Shibata et al. extended the DPFT model with an improved description of the electrolyte solution [86]. The number densities are obtained from variational analysis of the grand potential, and are given by ¯nl=¯nb l l χv+χss+γCAχCACA +γNAχNANA +γcχcc,l∈{s, c,CA, NA}, (16) where ¯nb lis the dimensionless bulk density of species l, χl=¯nb l/l¯nb l,γlis the relative size of species lreferenced to that of the solvent molecule, and χvis the volume fraction of unoccupied space in the bulk solution. The Boltzmann factors, l,are l=exp(−(ql¯ φ+wl)),l∈{c, CA, NA}, s=exp−−lnsinh(ps¯ E) ps¯ E+ws,(17) where qlis 1 for cations (l=c),−1 for nonspecifically adsorbing anions (l=NA), and fractional for chemisorbed anions (l=CA). The short-range interaction terms, wl,l∈ {s,c, CA,NA}, are described using Morse potentials [78, 87]: wl(r)=Dl(exp(−2βl(d(r)−bl)) −2exp(−βl(d(r)−bl))), (18) with Dlbeing the well depth, related to the binding strength; βlis a coefficient controlling the well width; d(r)is the distance from a point with coordinate rreferenced to the metal surface; and blthe distance to the well bottom. Throughout this paper, we use rfor coordinates of a point in the space, zfor the coordinate in the 043008-7 JUN HUANG et al. PRX ENERGY 3, 043008 (2024) direction normal to the surface, and d(r)for the distance between a point and the metal surface. When ris within the metal, d(r)is negative and wl(r)increases exponentially, resulting in a negligible probability of species lin the bulk metal. As the electrode potential varies, the extent of the electron spillover of the metal changes, which ultimately affects the binding strength of the anion on the metal surface. Specifically, the potential-dependent covalent binding strength of the chemisorbed anion, Cl−in the present work, grows at more positive electrode potential [62,88]. This is approximated through a linear relationship: DCA =1 kBT(D0 CA +αCA ˜μe), (19) where the coefficient αCA <0, and D0 CA denotes the intercept at ˜μe=0 eV. Note that Eq. (19) does not include the electrostatic interactions, which are described separately through Eq. (17). The charge of the chemisorbed anion, Cl−in the present case, is derived from an Anderson-Newns model Hamiltonian treatment detailed in the Supplemental Material [89] (including Refs. [50,90–101]): qCA(d)=− 1 2+1 πarctanCA , (20) where =0exp(−κ(d−dcut)θ(d−dcut)) is the strength of the electronic metal-anion interaction, which decreases exponentially with the anion moving away from the metal surface; κis the coefficient determining the exponential decay of ;dis the distance of the anion away from the metal surface; dcut is the cutoff distance; and θ(x)is the Heaviside function. CA = −˜μe+(0 CA −(1−exp(−κ(d−dcut)θ(d−dcut)))) is the energy of the valence orbital of the anion referenced to the Fermi level of the metal. CA also changes with the distance d, characterized with a magnitude of >0anda coefficient of κ.0 CA is the reference value at d=dc.The increase in CA with decreasing dfacilitates transfer of the electron from the anion to the metal. In summary, the current model has three coupled components, which are to be solved self-consistently. The first component describes the metal phase using orbital-free DFT theory in Eq. (14). Different from standard KohnSham DFT calculations, the electrostatic potential and the electron density are not dual variables here; instead, they have the same status. The second component describes the electrolyte phase using an extended Poisson-Boltzmann equation in Eq. (15). The extension is threefold: solvent orientational polarization of solvent molecules is treated as in dipolar Poisson-Boltzmann theory [102,103]; a unified description of the metal and electrolyte phases removes problems associated with the metal-solution boundary, particularly the image charge problem [104,105]; and the ion charge is not constant but varies spatially and as a function of the electrode potential due to chemisorption. The partial variable charge of chemisorbed anions is calculated from an Anderson-Newns model Hamiltonian, which is the third component of the current model. B. Model parameters The developed model has four groups of parameters. The first group of parameters is related to orbital-free DFT of metal electrons, including θTcalibrated using the experimental PZC and θXC with recommended values in the PBE functional. The second group of parameters describes the short-range effects between the solvent, electrolyte, and the metal through the Morse potentials, as shown in Eq. (18). The third group of parameters is the structural and electrical properties of electrolyte ions and solvent molecules. The fourth group of parameters is in the Anderson-Newns Hamiltonian describing the chemisorption of anions. The first three groups of parameters are inherited without changes from our previous study on the EDL at the Ag(111)-KPF6aqueous solution interface [79]. These parameters were calibrated by benchmarking the model results with experimental differential double layer capacitance profiles for a series of electrolyte concentrations [106]. Additionally, the short-range interaction between Ag(111)and the chloride anion is studied using grand-canonical DFT (GC-DFT) calculations [93]. The details of implementing the GC-DFT calculations are provided in the Supplemental Material [89]. In the previous work [79], the density profiles of electrons, ions, and solvent molecules from the metal phase to the electrolyte phase were obtained for a range of electrochemical potentials. Herein, we disclose the underlying factors determining the density profiles, as these are instrumental to understand the interplay among different components of the model. Equation (16) indicates that ¯nl is determined by the competition between four Boltzmann factors, l, which are further determined by wl,¯ φ,and ln(sinh(ps¯ E)/ps¯ E), according to Eq. (17). These terms are displayed in Fig. 3. Nondimensional short-range interactions between the metal and nonspecifically adsorbing anions (wNA), cations (wc), solvent molecules (ws), and chemisorbed anions (wCA)are illustrated in Fig. 3(a). As expressed in Eq. (18), all wiprofiles transit from zero in the bulk solution to negative values near the metal surface and then to positive values at smaller d. The dip with a value of −Dlis obtained at bl. Solvent molecules and chemisorbed anions approach the metal surface closer than other species, namely, the values of blare smaller for solvent and chemisorbed anions. In addition, the well depth, −Dl, is greater for solvent molecules than nonspecifically adsorbing anions and cations. 043008-8 DPFT-CHEMISORPTION... PRX ENERGY 3, 043008 (2024) (a) (b) FIG. 8. (a) Electrosorption valency of chloride anion at Ag(111)immersed in an electrolyte solution of c(1−f)MKPF6+ cfM KCl. Two ion concentrations (c=0.1 and 0.01M) are compared. Electrosorption valency is obtained from a linear fitting of the σtot −σad,tot relationships at five values of f=0, 10−3,10 −2,10 −1, and 0.5; see Fig. S1 within the Supplemental Material [89]. (b) Differential double layer capacitance curves of the Ag(111)-aqueous KPF6solution interface at two concentrations of 0.1 and 0.01M without chloride anions. However, this approach suffers from low accuracy, as σfree is much smaller than σtot and σad,tot. Another possible method is to calculate σfree from CDL using σfree =σref free +E Eref Ceq DL(E)dE. (25) It is important to note that Ceq DL(E)in this context is an equilibrium quantity [114]. In general, we can use EIS to separate CDL(E)from the pseudocapacitance due to chemisorption, Cad, using a proper model, such as the FrumkinMelik-Gaikazyan (FMG) model, as demonstrated in a series of works from Pajkossy et al. [73–77]. In extracting CDL(E)from experimental data using the FMG model, one needs to extrapolate to sufficiently high frequencies, where the chemisorption process does not occur due to timescale differences. Therefore, CDL(E)determined from EIS measurements this way is not Ceq DL, as stressed recently [114]. Instead, it is the high-frequency component, CDL,HF, which is characteristically different from Ceq DL.CDL,HF is always positive while Ceq DL could be negative due to the net negative charge on the chemisorbates. Negative Ceq DL was proposed in an earlier Gouy-Chapman-Stern type model of electrocatalytic EDLs [115,116], and later confirmed by atomistic simulations [117,118]. The nonmonotonic surface charging behaviors of Pt(111)-aqueous solution interfaces due to anion chemisorption were also obtained by Zhang et al. using a thermodynamic analysis [113]. Wippermann et al. found nonmonotonic surface charging of Pt electrodes immersed in ionic liquids with a trace amount of water [119]. Importantly, a negative Ceq DL does not violate the stability requirement of the EDL, which is ensured by the positivity of total different capacitance, Ctot, in the presence of chemisorption. Interested readers are referred to a discussion on the stability of electrochemical interfaces by Partenskii and Jordan [120]. The above analysis on the difference between Ceq DL and CDL,HF implies that one should not calculate σfree using Eq. (25) with CDL(E)determined from EIS. The first method based on σfree =σtot +σad,tot seems to be a better option. However, it is very important to gauge the errors involved in thermodynamic estimation of σad,tot.For instance, Zhang et al. [113] and Garcia-Araez et al. [110] observed nonmonotonic behaviors in σfree at Pt(111)with chemisorption of sulfate anions or OH from water dissociation. While Eq. (21) indicates that the free charge can be determined from either simulated or measured electrosorption valences, such data remains scarce. Foresti et al. measured the electrosorption valency of Cl−,Br −, I−,andS 2−on Ag(111)in a total concentration of 0.1M [65]. Despite the fact that the Foresti study is more than three-decades old, it constitutes the most important piece of original experimental data in recent ab initio thermodynamic modeling of electrosorption by Hörmann et al. [39, 88,108,109]. Their data show that both lBr and lIexhibit a“V” shape with minima obtained at around −0.8 and −1.0 VSCE, respectively, which is qualitatively consistent with our results in Fig. 8. However, their work examined only a single electrolyte concentration and the concentration dependence of the licurve was not addressed. Hence, our observation that the licurve follows the CDL 043008-15 JUN HUANG et al. PRX ENERGY 3, 043008 (2024) curve without chemisorption at more dilute concentrations remains, to the best of our knowledge, a model-based prediction. B. Implications for electrochemical DFT simulations The semiclassical model developed and used herein identifies an important ingredient missing from atomistic (DFT) simulation methods used for studying electrochemical interfaces: the potential-dependent chemisorbate coverage. As our results show that the varying chemisorbate coverage has a significant impact on the potentialdependent capacitance and surface charge, we argue that the common practice of fixing the surface coverage has farreaching implications for both canonical constant-charge and semi-grand-canonical constant-potential methods. In particular, the differential capacitance used in current DFT simulations within an implicit electrolyte model is actually CDL,HF, not Ceq DL, since the relaxation of the coverage of chemisorbates is forbidden. Since CDL,HF and Ceq DL are characteristically different in the presence of chemisorption, see, e.g., Fig. 7, the conversion between the charge and electrode potential in current implicit electrolyte models is not fully correct. The influence of chemisorption on Ceq DL is expected to be more significant for more charged adsorbates, like the chemisorbed CO2ion with a net charge of around one electron on gold electrodes [29]. This is a notable deficiency because the accuracy of constant-charge and semi-grand-canonical constantpotential methods hinges on the quality of the differential capacitance used for converting between the total charge and electrode potential. As the semi-grand-canonical calculations are by design fully self-consistent, they need an accurate description of the differential capacitance apriori to correctly predict the total charge needed to maintain a constant electrode potential. While it is, in principle, possible to use an experimentally measured capacitance to map between the charge and electrode potential, this will impair the self-consistent nature of GC-DFT calculations. In practice, the capacitance is often calculated from an implicit electrolyte model. Therefore, if the implicit electrolyte model cannot describe chemisorption and its influence on the capacitance, the charge and number of electrons in the system at a given electrode potential will be inaccurate. For instance, a recent critical analysis on the timescales of electrochemical interfaces suggests that the EDL is in equilibrium during an electron transfer process if the activation barrier is higher than 0.3 eV [55]. This means that the local reaction conditions are determined by Ceq DL, not CDL,HF. Therefore, semi-grand-canonical simulations may wrongly describe the electrode potential and local reaction conditions in the EDL with considerably charged chemisorbates. A similar limitation is present in canonical constantcharge calculations, even though some studies indicate that they are more robust against inaccuracies in capacitance [51,52]. In practice, however, these canonical constantcharge calculations require mapping the surface charge to the electrode potential via the differential capacitance. This also raises two issues. If experimentally measured capacitances are used, the DFT calculations are no longer self-consistent or ab initio. Furthermore, accurate experimental capacitances are only available for a handful of surfaces, which limits the use of such constant-charge calculations. The above discussion delineates that the accuracy and applicability of canonical and semi-grand-canonical simulations hinge on the description of the differential capacitance. Despite the promising performance of recent advances in implicit electrolyte models in DFT codes, such as the implementation of the dipolar Poisson-Boltzmann theory [121], a self-consistent and grand-canonical treatment of potential-dependent chemisorbate coverage is missing in currently available implicit continuum electrolyte models. We propose to mitigate this deficiency by augmenting the implicit solvent models with an implicit treatment of chemisorption through an Anderson-Newns model Hamiltonian. This approach achieves a good and transparent description of chemisorption-induced changes in the surface charge densities and differential capacitances. The chemisorption model can also be implemented in implicit electrolyte models available in different KohnSham DFT codes with only modest efforts. VI. CONCLUSION We have extended the use of DPFT of EDLs to the case with chemisorption. Specifically, chemisorption is treated in an implicit manner based on the AndersonNewns model Hamiltonian. This implicit chemisorption model has been compared with a semi-grand-canonical DFT model. Decent agreement between both models in terms of the potential and distance dependences of the partial charge number on the chloride anion adsorbed on Ag(111)has been obtained. The extended DPFT model constitutes a viable approach to model and understand the interplay between electrode charging, chemisorption, and diffuse layer charging under realistic experimental conditions. For instance, it can describe the replacement of water molecules with chemisorbed anions in the inner layer of the EDL at very positive electrode potentials. Armed with the extended DPFT model, we have presented a detailed conceptual analysis on five variants of surface charge density and three variants of differential capacitance of electrocatalytic EDLs. We have focused on the influence of chemisorbed chloride on the free surface charge density, σfree, which is not directly measurable, but which plays a crucial role in tuning the local reaction conditions within the EDL. Specifically, we have revealed that partially charged chemisorbed Cl−possesses 043008-16 DPFT-CHEMISORPTION... PRX ENERGY 3, 043008 (2024) a potential-dependent negative net charge density, resulting in a peculiar nonmonotonic behavior of σfree:σfree first increases and then decreases, as the electrode potential becomes more positive and anion chemisorption proceeds. This implies the existence of a negative differential double layer capacitance at highly positive potentials with respect to the potential of zero charge. The influence of chemisorption on the free surface charge density has also been examined from the perspective of the electrosorption valency. A relationship between the electrosorption valency, the differential capacitances, and the partial charge number is derived. This simple relationship explains a large part of the potential and concentration dependence of the electrosorption valency. The model predicts that a dip in the electrosorption valency would appear at the PZC in more dilute concentrations. We have also discussed the implications of our results for the experimental characterization and theoretical modeling of electrocatalytic EDLs. Regarding experimental measurements of σfree, we reveal a caveat in employing EIS to determine CDL and using it in the integration of CDL over a range of electrode potentials to evaluate σfree. The problem lies in the mismatch of frequency between CDL obtained from EIS measurements and σfree: while σfree is an equilibrium quantity, CDL extracted from EIS measurements, with the aid of a proper physical model to disentangle CDL and Cad, is a high-frequency nonequilibrium quantity. Therefore, we recommend determining σfree from the difference between σtot and σad,tot. However, the error involved in calculating the difference of two large surface charge densities should be carefully considered. With regard to theoretical modeling, fixing the chemisorbate coverage in current DFT models misses out an essential feature of electrocatalytic EDLs, namely, the influence of potential-varying coverage of chemisorbates on σfree and CDL. Because both constant-potential and constant-charge DFT models need an accurate description of the differential capacitance, the missing contribution from the varying surface coverage negatively impacts the accuracy and generality of current DFT models of electrochemical systems. For instance, as an immediate consequence, CDL obtained from such canonical or grandcanonical DFT models is always positive, and the nonmonotonic surface charging behavior is totally missing. This leads to errors in the description of the local reaction conditions and the EDL. The influence of this neglect on the thermodynamics and kinetics of chemisorption and electrocatalytic reactions remains to be studied in the future. To remedy this, we propose the implicit chemisorption model as a practical and feasible approach to improve the quality of self-consistent DFT simulations of electrochemical interfaces and electrocatalytic reactions. The data that support the findings of this study are available from the corresponding author upon reasonable request. ACKNOWLEDGMENT This work is supported by the Initiative and Networking Fund of the Helmholtz Association (Grant No. VH-NG1709) and the Academy of Finland through Project No. 338228. Computational resources were provided by the CSC-IT Center for Science, Espoo, Finland. [1] S. Trasatti, Work function, electronegativity, and electrochemical behaviour of metals: III. Electrolytic hydrogen evolution in acid solutions, J. 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