Zooming into the Inner Helmholtz Plane of Pt(111)–Aqueous Solution Interfaces: Chemisorbed Water and Partially Charged Ions

Abstract

The double layer on transition metals, i.e., platinum, features chemical metal–solvent interactions and partially charged chemisorbed ions. Chemically adsorbed solvent molecules and ions are situated closer to the metal surface than electrostatically adsorbed ions. This effect is described tersely by the concept of an inner Helmholtz plane (IHP) in classical double layer models. The IHP concept is extended here in three aspects. First, a refined statistical treatment of solvent (water) molecules considers a continuous spectrum of orientational polarizable states, rather than a few representative states, and non-electrostatic, chemical metal–solvent interactions. Second, chemisorbed ions are partially charged, rather than being electroneutral or having integral charges as in the solution bulk, with the coverage determined by a generalized, energetically distributed adsorption isotherm. The surface dipole moment induced by partially charged, chemisorbed ions is considered. Third, considering different locations and properties of chemisorbed ions and solvent molecules, the IHP is divided into two planes, namely, an AIP (adsorbed ion plane) and ASP (adsorbed solvent plane). The model is used to study how the partially charged AIP and polarizable ASP lead to intriguing double-layer capacitance curves that are different from what the conventional Gouy–Chapman–Stern model describes. The model provides an alternative interpretation for recent capacitance data of Pt(111)–aqueous solution interfaces calculated from cyclic voltammetry. This revisit brings forth questions regarding the existence of a pure double-layer region at realistic Pt(111). The implications, limitations, and possible experimental confirmation of the present model are discussed.

Introduction

There is a wave of renewed interest in the electric double layer (EDL) structure in electrocatalysis, probably due to two reasons.¹⁻⁴ On the one hand, the activity of many electrocatalytic reactions is revealed to be affected markedly by electrolyte cations⁵⁻⁹ and pH.¹⁰⁻¹³ There is growing consensus that demystifying these electrolyte effects requires fundamental knowledge of the EDL,^3,14 as discussed in several review articles.¹⁴⁻¹⁷ On the other hand, a growing body of experimental data on the EDL of single crystals of transition metals, notably platinum, has been reported recently.^4,18−23 These new data, though not fully consistent as detailed below, have already surprised us as they do not agree, even qualitatively, with classical double layer models.

A well-documented classical model is the Gouy–Chapman–Stern (GCS) model, which describes the EDL as a series combination of an inner layer and a diffuse layer.²⁴ The GCS model has many variants, with detailed treatments of the diffuse layer and the inner layer.²⁵⁻²⁷ A hallmark of GCS-like models is the appearance of a minimum in the differential double-layer capacitance (C_dl) curve, named the Gouy–Chapman minimum, at a particular electrode potential, named the potential of zero charge (pzc), in dilute electrolyte solutions. Unless otherwise noted, pzc refers to potential of zero free charge. Moving away from the pzc in cathodic and anodic directions, C_dl grows and then decreases, resulting in a camel-shaped C_dl profile with two peaks and one valley. The two peaks usually signify overcrowding of counterions near the metal surface, which are usually asymmetric due to different sizes of cations and anions. The two peaks move closer and eventually melt into a single peak as the electrolyte solution gets more and more concentrated.^28,29

The aptness of the GCS model has been corroborated on mercury-like electrodes, first by Gouy in the 1910s³⁰ and then by Grahame in the 1940s and 1950s,³¹e.g., see Parsons’s review.³² In the presence of specific adsorption of anions, Grahame discriminated an inner Helmholtz plane (IHP) and an outer Helmholtz plane (OHP) in the inner layer.³¹ More than just a refinement of the Stern model, the Grahame model sowed the seeds of subsequent developments in the EDL theory, including the consideration of field-dependent orientation of water molecules,³³⁻³⁶ discreteness of adsorbed ions,³⁷⁻⁴⁰ and quantum effects of metal electrons,⁴¹⁻⁴⁴ as discussed in a recent tutorial.²⁷ Valette and Hamelin and coworkers have demonstrated that the GCS model is valid also for silver single crystals in nonspecifically adsorbing electrolytes.⁴⁵⁻⁴⁸

The idyll reaches an end for platinum. The Gouy–Chapman minimum is absent in the study of Pajkossy and Kolb where C_dl of Pt(111) in a KClO₄ and HClO₄ mixture electrolyte with a concentration down to 1 mM was measured.^18,20,49 Instead, their data show a C_dl peak near the pzc in the anodic direction. Contrarily, Xue et al. observed Gouy–Chapman minima for Pt(111) in 0.05 M MeClO₄ electrolytes (Me = Li, Na, K, Rb, Cs), however, at potentials ∼0.2 V below the pzc.²³ More recently, Ojha et al. observed the Gouy–Chapman minimum for Pt(111) in 0.1 mM HClO₄.^21,22 In addition, they found that the Gouy–Chapman minimum turns into a maximum near the pzc when increasing the electrolyte solution above 1 mM, qualitatively consistent with previous observations of Pajkossy and Kolb.^18,20,49

One should be aware of the discrepancy in the magnitude of C_dl of Pt(111) at the pzc, denoted C_dl, pzc, in different studies. C_dl, pzc is around 20 μF cm^–2 in the study of Pajkossy and Kolb using an electrolyte concentration of 0.1 M,^18,20,49 in the range of 10–30 μF cm^–2 for 0.05 M NaOH and 0.10 M HClO₄ in the study of Schouten et al.,⁵⁰ around 30 μF cm^–2 in the study of Xue et al. with an electrolyte concentration of 0.05 M,²³ and around 100 μF cm^–2 in the study of Ojha et al. with an electrolyte concentration of 0.005 M.^21,22 Note that C_dl, pzc shall decrease with the electrolyte concentration, according to the GCS model. Surprisingly enough, recent values of C_dl, pzc in more dilute electrolyte solutions are even several times higher than previous values in more concentrated electrolyte solutions. Understanding the origin of the anomalously large C_dl, pzc is an open question on which several viewpoints already exist, which I am about to discuss, and that motivates this study.

To rationalize the large C_dl, pzc, Doblhoff-Dier and Koper introduced weak (around several hundred meV), attractive ion–surface interaction, of which possible origins are also discussed.⁵¹ This effectively increases the ion concentration in the double layer, shortens the Debye length, and magnifies the magnitude of C_dl, pzc. Recently, Schmickler rationalized the higher C_dl, pzc as a consequence of weak specific adsorption of anions.⁵² The introduction of anion adsorption can increase the overall capacitance, and decrease the Parsons–Zobel slope. However, this model cannot explain that the Parsons–Zobel slopes are not sensitive to the identity of electrolyte anions.²² More recently, Ojha, Doblhoff-Dier, and Koper modified their previous model by further adopting a two-state submodel for the first-layer water molecules,²² which was developed earlier by Le et al.⁵³

This marked and surprising discrepancy in C_dl, pzc might be ascribed to different methods used to determine C_dl. Pajkossy and Kolb extracted C_dl from electrochemical impedance spectroscopy (EIS) data using the Frumkin–Melik–Gaikazyan (FMG) model.^18,20,49 Xue et al. extracted C_dl from EIS data using an electric circuit model involving empirical constant phase elements.²³ Ojha et al. calculated C_dl from cyclic voltammogram (CV) data.^21,22 EIS is, in principle, more accurate than CV in determining C_dl, because EIS can separate C_dl from possible pseudo-capacitances of chemisorption. In the presence of chemisorption, the FMG model expresses the total capacitance as , where C_dl is the pure double layer capacitance, R_ad is the adsorption resistance, C_ad is the adsorption capacitance, σ_ad is the coefficient of Warburg diffusional impedance, and ω is the angular frequency. Only when ω ≫ ω_c = (C_adR_ad)⁻¹ can one approximate C_total(ω) as C_dl. At low frequencies, one will find C_total(ω) ≈ C_dl + C_ad. Earlier studies have shown that ω_c is around 1 MHz for hydrogen adsorption at Pt(111) in 0.5 M perchloric acid⁵⁴ and above 10 kHz for hydroxyl adsorption at Pt(111) in 0.1 M perchloric acid.⁵⁰ A CV measurement with a scanning rate of 50 mV s^–1 over a potential range of 0.05 V consumes 1 s and, hence, has a characteristic frequency of 1 Hz, which is far lower than the typical ω_c of Pt(111) electrode processes. Ojha et al. reported the capacitance calculated from impedance at a single frequency of 18 Hz, which is also much below ω_c.²²

The fact that the CV-derived capacitance is much higher than the EIS-derived capacitance could be attributed to pseudo-capacitances of chemisorption. Based on this assumption, a theory for the large CV-derived capacitance of Pt(111) in the double layer region is developed in this paper. I am about to show that chemisorption at a small percentage of platinum sites can explain the observed large capacitance.

The remainder of this paper is organized as follows. First, I present a short review of the developments on the IHP model after the pioneering works of Grahame and Parsons.^55,56 Most of these important works that adequately reflect the depth, core, and beauty of electrochemical research are less known and only sparsely cited nowadays. This literature review also sets the baseline for the present model, which is not entirely new, but extends previous models in some aspects. Second, a detailed derivation of the model, including a water layer submodel and a chemisorption submodel, is presented. Third, the model is demystified for two cases. In the first case without chemisorbed ions, emphasis is put on understanding how metal–water interactions change the pzc and the double layer capacitance. For the second case with chemisorbed ions, emphasis is put on the consequence of chemisorption-induced surface dipole moment. The model is then used to interpret recent experimental data of Ojha et al.^21,22 in the double layer region. This revisit questions current fundamental understanding of the EDL at Pt(111). Lastly, discussion on assumptions, implications, and possible extensions of the model is presented before the conclusion section.

Models of the IHP: Grahame–Parsons Model and beyond

Above analysis has turned the spotlight on the IHP in understanding the EDL at transition metals. Figure 1 summarizes three groups of IHP models. The Grahame–Parsons model views the IHP as an ionic layer with specifically adsorbed ions rigidly lining up.^31,55,56 The potential drop from the metal surface to the IHP is composed of two parts. The first part is due to the metal surface charge, and the second part is the net charge of specifically adsorbed ions at the IHP. There is no electron transfer during specific ionic adsorption; hence, the specifically adsorbed ions retain the charge they have in the bulk solution. In the literature, e.g., ref (57), there is another Grahame–Parsons model, which describes the thermodynamics of specific ion adsorption, and determines the coverage of adsorbed ion as a function of the electrode potential. The Grahame–Parsons model of ion adsorption was later improved by the Alexeyev–Popov–Kolotyrkin model⁵⁸ and the Vorotyntsev model.⁵⁹

Figure 1.

Models of the inner Helmholtz plane (IHP). (a) Grahame–Parsons model views the IHP as an ionic layer where specifically adsorbed ions are rigidly aligned.^31,55,56 (b) Many models are focused on the field-dependent behaviors of solvent molecules, most often water molecules, at the IHP.^34,36,63 (c) Recent models consider the coexistence of solvent molecules and partially charged chemisorbed ions (A^–ξ) at the IHP.^72,73

Another group of IHP models focuses on solvent molecules at the IHP in the absence of specifically adsorbed ions, (Figure 1b); see an earlier review by Guidelli.⁶⁰ The two-state model assumes that water molecules take either the hydrogen-up or hydrogen-down configuration, depending on the local electric field.³⁴ The three-state model further considers formation of water dimers and interconversion between water dimers and monomers.³⁶ Both models are founded purely on electrostatic considerations; thus, the average dipole moment of the water layer is zero when the metal surface has a zero net charge. In addition to electrostatic interactions, chemical interactions between water molecules and the metal surface were speculated in the 1970s.⁶¹ Damaskin and Frumkin considered both electrostatic and chemical metal–water interactions at the IHP.⁶² Electrostatic interactions are treated using the approach of Bockris, Devanathan, and Miller (BDM),⁶³ which does not consider properly the configurational entropy. The fraction of chemisorbed water molecules was expressed phenomenologically as an exponential function of the electrode charge. Recently, we have seen a surge of atomistic simulations of water molecules near metal surfaces.^53,64−68 These first-principles calculation results provide unbiased, accurate atomistic details, guiding the development of phenomenological models, as demonstrated in the work by Le et al.⁵³

In most cases, the IHP is neither an ionic layer nor a solvent layer. Instead, it is a mixture of solvent molecules and chemisorbates. Chemisorbates, usually partially charged due to partial charge transfer, contribute an additional potential drop in the inner layer.⁶⁹ Therefore, a refined model of the IHP should be a binary IHP model that considers co-existing solvent molecules and partially charged ions. Carnie and Chan developed a so-called civilized model of EDLs, which considers ions as hard spheres with embedded point charge (integral charges, rather than partial charges), and solvent molecules as hard spheres with embedded dipole moment in the 1980s.⁷⁰ The Carnie–Chan model was used to calculate the dipole moment of specifically adsorbed ions at the IHP.⁷¹ More recently, the binary IHP model was further extended to consider potential-varying coverage of partially charged chemisorbates.^72,73 More than just a further complication of a model, it brings forth nontrivial implications of chemisorption-induced surface dipole moment (μ_chem). It was revealed that μ_chem of adsorbed hydroxyl leads to a nonmonotonic surface charging behavior of Pt(111)–aqueous solution interfaces. Specifically, the free charge, not the total charge nor the metal charge,⁷⁴ exhibits a negative–positive–negative transition with increasing electrode potential. This implies the occurrence of negative C_dl and more than one potential of zero free charge. Nonmonotonic surface charging behaviors of a polycrystalline Pt electrode were measured using the radioactive tracers method by Frumkin and co-workers^75,76 and later in several experiments of laser-induced potential transients.⁷⁷⁻⁷⁹ The nonmonotonic surface charging behaviors were modeled earlier by assuming that the pzc grows linearly with the coverage of adsorbed oxygen and later by a phenomenological treatment in which the potential drop in the IHP is interpolated between two limiting states.⁷⁶ The model developed in refs (72) and (73) and later modified in ref (80) provides a consistent treatment of chemisorption and surface charging behaviors as a function of electrode potential. It was also shown in ref (72) that orientational polarization of interfacial water molecules, described using the two state model, leads to a negative capacitive component to C_dl, an effect that was later elucidated in greater detail by Le et al.⁵³

The IHP model that I am about to present in this paper improves over the previous one in refs (72) and (73) in three aspects. First, the present model abandons the BDM-type two-state water model and adopts a statistical treatment of water polarization with a continuum spectrum of orientational states, following the approach developed earlier in refs (81) and (82). The configurational entropy of water molecules at the IHP is considered. In addition to electrostatic interactions, chemical interactions with the metal are considered, as inspired by ab initio molecular dynamics (AIMD) results,^53,66,67 but in a different manner compared to the Damaskin–Frumkin model. Second, chemisorption at the IHP is now energetically distributed in such a way that the equilibrium potential follows a distribution instead of taking a single value as usual. This allows treatment of chemisorption at different types of active sites, chemisorption of different species, and lateral interactions between chemisorbates in a unifying manner. Third, as suggested by recent detailed atomistic simulations, e.g., ref (83), the IHP is further divided into an adsorbed ion plane (AIP) and an adsorbed solvent plane (ASP). This refinement is necessary because the AIP and ASP have very different properties, which I am about to show in model parameterization.

Theoretical Methods

Physical Picture of the Double Layer

A schematic illustration of an electrocatalytic double layer is presented in Figure 2a, and a continuum model is depicted in Figure 2b. The metal electrode is assumed to be planarly uniform. Surface roughness is not considered here. The first-layer water molecules are usually highly polarized due to the high interfacial electric field and chemical interactions with the metal; see a recent review by Groß and Sakong.⁶⁸ The ASP denotes the outer plane of adsorbed solvent molecules. Adsorbed ions, if any, are usually located closer to the metal surface than adsorbed solvent molecules, according to atomistic simulations, e.g., by Braunwarth et al.⁸³ The central plane of adsorbed ions is denoted AIP. Different characteristic planes are used for depicting the ASP and AIP due to difference between ionic charge and dipolar charge. Nonspecifically adsorbed ions with their solvation shell reside in the diffuse layer stretching from the OHP toward the solution bulk. The regions between the metal and the AIP, the AIP and the ASP, and the ASP and the OHP are described as dielectric continua, parameterized with thicknesses δ_i and permittivities ϵ_i, with i = 1,2,3.

Figure 2.

(a) Schematic illustration of the electrocatalytic double layer with partially charged, chemisorbed ions, adsorbed solvent (water in this case) molecules, and nonspecifically adsorbed ions. (b) Continuum EDL model consists of an AIP denoting the central plane of chemisorbed ions, an ASP denoting the outer plane of adsorbed water molecules, and an OHP denoting the central plane of nonspecifically adsorbed ions. The regions between the metal and the AIP, the AIP and the ASP, and the ASP and the OHP are described as dielectric continua, parameterized with respective thicknesses δ_i and permittivities ϵ_i, with i = 1,2,3. (c) Overall potential difference between the metal bulk and the solution bulk can be divided into four components: χ_M denoting the potential drop at the metal surface due to electron distribution, Δϕ_ad the potential change due to partially charged, chemisorbed ions, Δϕ_w the potential change due to orientational polarization of adsorbed water molecules, and Δϕ_free the potential change due to excess ionic charge stored in the diffuse layer, namely, free surface charge.

Potential Differences in Different Components

As EDL is a grand canonical system with electrode potential being one of the independent variables, the overarching task of any EDL model is to calculate the electric potential distribution from the metal to the electrolyte solution. Figure 2c depicts components of the overall potential difference between metal bulk (ϕ_M) and solution bulk (ϕ_S). At the metal surface exists a large potential drop due to the metal electron tail, denoted χ_M, which is readily obtained from electronic structure calculations, e.g., in ref (84). There are, at least, three other potential differences, including one due to partially charged, chemisorbed ions, one due to adsorbed solvent (water in this work) molecules, and one due to the free surface charge density, σ_free, defined as negative of the net ionic charge in the diffuse layer.⁸⁰

The above decomposition scheme, following ref (85), is expressed as

1

Within a continuum picture, Δϕ_free is written as

2

where the first term denotes the potential change over the diffuse layer, which is derived from the Bikerman model considering finite ion size,^28,73 is the Debye length, ϵ_b is the dielectric permittivity of the bulk solution, c_b is the bulk ion concentration, R is the gas constant, F is the Faraday constant, T is the temperature, and γ = 2c_bd_h³ with d_h being the diameter of hydrated ions. Size asymmetry of cations and anions is not considered here, for the sake of an analytical expression of Δϕ_free as a function of σ_free, and effects of size asymmetry on the C_dl are known already.⁸⁶⁻⁸⁹

The other two components Δϕ_ad and Δϕ_w are determined from two submodels as introduced below.

Distributed Chemisorption Model

The amount of charge carried by partially charged, chemisorbed ions at the AIP is calculated as, , where N_ad is the number density of adsorption sites, θ_A and ξ_A are the coverage and net electron of a chemisorbed ion of type A, and e₀ is the elementary charge.⁸⁰ The parameter ξ_A is related to the partial charge transfer coefficient, λ_A, via ξ_A = z_A + λ_A, with z_A being the initial charge of chemisorbed ions. A total charge transfer process corresponds to, λ_A = – z_A. Note in passing that the net charge ξ_A and the partial charge transfer coefficient λ_A are different from the electrosorption valency that can be measured experimentally.⁸⁵

Δϕ_ad is then calculated as the potential difference across an equivalent planar capacitor with a charge amount of on each side

3

A positive-valued ξ_A means negatively charged chemisorbed ions, which introduces a positive Δϕ_ad. A chemisorption submodel is required to determine θ_A as a function of the electrode potential.

For chemisorption of hydrogen, H⁺ + e + * ↔ H_ad, the Frumkin isotherm gives^90,91

4

where θ_H is the coverage of H_ad, θ_H, max is the maximum coverage, γ_H is the lateral interaction coefficient, β = F/RT, and E_H⁰ is the equilibrium potential on the standard hydrogen electrode (SHE) scale. Recently, Hormann et al. has illustrated how to obtain E_H and γ_H from density functional theory (DFT) calculations.⁹² The existence of lateral interactions stands in the way of obtaining an analytical solution for θ_H.

The Frumkin isotherm can be transformed into a series of Langmuir-type isotherms with a distributed E_H⁰,

5

where f_H(E) denotes the probability of an equilibrium potential at E.

The distribution in equilibrium potential can be ascribed to the existence of structural defects, lateral interactions that change the electrochemical potential for later adsorbates, and a potential-dependent distribution in the electrochemical potential of reactants, among others. I will go back to this in the discussion section.

As any distribution can be expanded as a series of normal distributions, E_H⁰ is assumed to take the following general form,

6

where g_k(E; μ_k, σ_k) are normal distributions of weight p_k, expectations μ_k, and variances σ_k

7

and ∑_kp_k = 1 for the weight values.

Now, θ_H is expressed analytically as

8

Similarly, the coverage of adsorbed OH is written as

9

where f_OH(E) is the distribution function of the equilibrium potential of OH adsorption, which has the same form as eq 6.

The pseudo-capacitance of chemisorption of H_ad and OH_ad is calculated as

10

with A = H (−) or OH (+).

Cautious readers may think a term (1 ± ξ_A) ( + for H_ad and – for OH_ad) should be added in eq 10 to account for the partial charge transfer. This is unnecessary because no matter how the electron is partitioned between the metal and the adsorbate, one electron is taken out from or moved into the electrode, if we separate the total surface charge into σ_free and the charge involved in specific adsorption of ions. I will explain this point later when discussing the relationship between the total capacitance, C_Aad, and C_dl in eq 24.

The Frumkin isotherm can be regarded as a Langmuir isotherm with the equilibrium potential linearly varying with the adsorbate coverage, with the linear coefficient being the lateral interaction coefficient. In this sense, the distribution function expressed in eq 6 is just a generalization of the Frumkin isotherm. In fact, the linear coverage dependence of the equilibrium potential has no fundamental reason but is just a first approximation. Nonlinear effects have been observed in many cases such as the chemisorption of OH at Pt(111) in the study of Climent et al.⁹¹ The exact form of f_A(E) corresponding to the Frumkin isotherm is unknown currently. However, I will extract f_H(E) and f_OH(E) from experimental CVs that are usually fitted using Frumkin isotherms.

Water Model with a Continuous Spectrum of Orientational States

The last term in eq 1 is the potential difference due to the average dipole moment of the first-layer water molecules,

11

where θ_w is the coverage of water molecules, μ_w is the dipole moment of a water molecule, and ⟨ cos α_w⟩ is the statistical average of cosα_w with α_w being the angle between water dipoles and the local electric field, as depicted in Figure 3. Herein, a positive electric field points toward the bulk solution, and the direction of a dipole points from its negative end to its positive end.

Figure 3.

(a) α_w is the angle between water dipoles and the local electric field. A positive electric field points toward the bulk solution, and the direction of a dipole points from its negative end to its positive end. Chemical metal–water interactions result in an energy difference between water of α_w = 0^° and that of α_w = 180^°, denoted δμ. (b) Shift in the potential of zero charge as a function of δμ̃ (the solid line: exact results, the dotted line: linear approximation). The calculation uses the following set of model parameters: N_ad = 0.15 Å^–2, θ_w = 0.5, ϵ₂ = 3.25ϵ₀, μ_w = 0.73 D, as discussed in the model parameterization section. Matlab script of this figure is provided in the Supporting Information of this paper.

In this model, α_w could take any value between 0 and 180°, instead of two values in the Watts–Tobin model and the BDM model that were widely adopted in recent work.^53,72 In this sense, the present treatment could be named an infinite-state water model.

Considering a canonical ensemble consisting of N_w water molecules at the IHP, at the mean-field level, the Hamiltonian of this canonical ensemble is written as

12

where α_i is the angle between the ith water and the local electric field and δμ is the chemical interaction energy difference between the case of α_w = 0^° and the case of α_w = 180^°; see the illustration in Figure 3a. The introduction of δμ accounts for chemical interactions between water molecules and the metal surface. Damaskin and Frumkin also considered chemical metal–water interactions.⁶² A key difference between this water model and the Damaskin–Frumkin model is that a statistical ensemble approach is adopted here.

The partition function of the water layer is written as⁹³

13

where ψ_i is the longitude of the solid angle.

The Helmholtz free energy of the water layer is calculated as

14

The statistical average of cos(α_i) is calculated as

15

Surface Charging Formula

Combining eqs 1, 2, 3, and 11 leads to a formula from which one can solve for σ_free as a function of ϕ_M,

16

At the pzc, σ_free = 0, eq 16 is thus simplified to

17

where the subscript 0 denotes the pzc condition.

If there is no chemisorbed ions at the pzc, eq 16 is further simplified to

18

where the dimensionless is used.

The potential difference ϕ_M – ϕ_S at an electrode–electrolyte interface is not measurable without involving its counterpart at a reference electrode⁹⁴ and must be differentiated from the electrode potential E in experiments.

Conversion from (ϕ_M– ϕ_S) to Electrode Potential

In potentiostatic experiments, one can control the potential of the working electrode (WE), namely, the metal in Figure 2a, with respect to a reference electrode (RE). The electrode potential is written as E = ϕ_M – ϕ_RE with ϕ_RE being the inner potential of the RE, up to an unimportant constant accounting for different work functions of WE and RE materials. The electrode potential could be rewritten as a difference between two potential differences, E = (ϕ_M – ϕ_S) – (ϕ_RE – ϕ_S). Here, it is assumed that the RE is put in the same electrolyte with the working electrode, and ϕ_RE – ϕ_S is the potential difference between the RE and the electrolyte solution. The reversible hydrogen electrode (RHE) belongs to this case. If the RHE is used as the RE, (ϕ_M – ϕ_S) becomes

19

where the subscript ‘RHE’ specifies the reference electrode. This equation can be related to the constant inner-potential DFT proposed recently by Melander et al.⁹⁵

According to the Nernst equation, the potential difference of the RHE and that of the standard hydrogen electrode (SHE) are correlated as

20

where (ϕ_RE – ϕ_S)_SHE is the value at the SHE.

Combined, eqs 19 and 20 lead to

21

given that .

Substituting eq 21 into eq 16 gives

22

where (ϕ_RE – ϕ_S)_SHE – χ_M is, in general, a function of E_SHE.

Double-Layer and Total Capacitance

In the presence of chemisorption, it is important to distinguish two types of differential capacitance. The differential double-layer capacitance, corresponding to the free surface charge, is defined as

23

The total surface charge σ_total, of which the change is measurable by counting electrons flowing through the external circuit, is the sum of σ_free, corresponding to the excess ionic charge stored in the diffuse layer, and the charge involved by specific adsorption of ions, namely,⁷⁴

with A = H (−) or OH (+). The differential total capacitance, corresponding to σ_total, is thus the sum of C_dl and pseudo-capacitances given in eq 10,

24

It is important to understand that specific adsorption of ions not only contributes to C_Aad but also influences C_dl via the resultant surface dipole moments.

Model Parameterization

The process of parameterizing a continuum EDL model is more or less a process connecting it to atomistic simulations using Kohn–Sham DFT, reactive force field molecular dynamics (MD), etc. The model contains four groups of parameters. The first group concerns the multilayer structure and dielectric properties of the EDL. The second and third group describe adsorbed water molecules and chemisorbed ions, respectively. The last group defines material properties of the metal and the electrolyte solution.

The first group of parameters includes ϵ_i and δ_i. Recent years have witnessed detailed DFT and MD studies of Pt(111) electrodes with chemisorbed H and OH. Le et al. showed that the density of H_ad peaks at ∼1 Å above the metal surface.⁶⁷ Braunwarth et al. gave a bond length of 1.05 Å for adsorbed O on Pt(111).⁸³ Therefore, I use δ₁ = 1Å for the AIP. In the presence of an electron tail in the space between the metal and the AIP, ϵ₁ should be higher than the vacuum permittivity ϵ₀; otherwise, C_dl is bounded by C_AIP = 8.85 μF cm^–2. Herein, I assume ϵ₁ = 10ϵ₀ to account for the effects of electrons spilled from Pt(111). These studies also indicated that the adsorbed, first-layer water molecules are around 2.1 Å above the metal surface. Therefore, δ₂, the thickness of the gap between the AIP and ASP, is approximated as δ₂ = 1.1 Å. Since orientational (inertial, slow, low-frequency) polarization of water dipoles is explicitly considered in Δϕ_w, ϵ₂ is the electronic (noninertial, fast, high-frequency) permittivity, which is known to be ϵ₂ = 3.25 ϵ₀ at room temperature.⁹⁶ Le et al.⁶⁷ used ϵ₂ = 4ϵ₀. An earlier estimate by Bockris et al. reads ϵ₂ = 6ϵ₀.⁶³ For the region between the ASP and OHP, I used δ₃ = 3.5 Å, a typical radius of hydrated ions, and ϵ₃ = 78.5 ϵ₀, the permittivity of bulk solution.

As for adsorbed water molecules, Le et al. determined a coverage of ∼0.5.⁵³ Hence, I use θ_w = 0.5. Schnur and Groß calculated the adsorption energy of the H-down and H-up water at several solids using an ice-like water bilayer model.⁹⁷ The H-down and H-up configurations correspond to α_w = 180^° and α_w = 0^°, respectively. Their data show that δμ̃ ≈ 0 at Ru(0001) and δμ̃ ≈ 1 at Au(111), Ag(111), and Pt(111). It is understood that δμ̃ has the unit of k_BT. In more recent AIMD simulations, a more detailed picture of interfacial water molecules has emerged.^53,66,98 At the Pt(111)–water interface, a fraction of water molecules are bound strongly through their oxygen atom with the metal so that the hydrogen atoms of the water molecule are above the oxygen atom. This is called the O-down configuration, corresponding to α_w = 0^° in this model. A bit further away from the metal surface resides another type of water molecules, which can be further divided into the H-down (α_w = 180^°) and H-up (α_w = 0^°) configurations. These recent AIMD results suggest δμ̃ < 0 for the present model. The intrinsic dipole moment of a water molecule is 1.85 D. However, in order to reproduce the permittivity of bulk water using the dipolar Poisson–Boltzmann theory, Abrashkin et al. used an effective value of 4.86 D.⁹⁹ Herein, I determine μ_w from the interfacial potential change due to water orientation, which was determined to be −0.2 V for a clean, uncharged Pt(111) surface by Le et al.⁶⁵ From eq 22, the present model determines the orientation-induced potential change as . Assuming δμ̃ = – 1, I thus determine . If δμ̃ = – 2, μ_w = 0.43 D. The lower bound of μ_w is 0.23 D. A lower effective value of μ_w accounts for the unevenness of the first-layer water molecules as well as offset effects by second-layer water molecules.

Malek and Eikerling calculated the work function change due to chemisorbed oxygen at Pt(111), from which they estimated the partial charge residing on each adsorbed oxygen.¹⁰⁰ Using a vacuum permittivity, they determined that each adsorbed oxygen on Pt(111) carries a partial charge of ∼0.012e. Since ϵ₁ = 10ϵ₀ is used here, I should use ξ_OH = 0.12 to reproduce the same magnitude of work function change induced by chemisorbed ions. Chemisorbed H is assumed to be electroneutral, namely, ξ_H = 0. The present model does not need a single-valued equilibrium potential and lateral interaction coefficient. Instead, this model only needs f_H(E) and f_OH(E), which are determined from experimental CVs.

With Pt(111) as the metal, I have N_ad = 0.15 Å^–2 based on a lattice constant of 3.92 Å, and the potential of zero charge is E_pzc = 0.29 V_SHE.¹⁰¹ Given that water dipole orientation lowers the pzc by 0.2 eV, E_pzc⁰ without considering this water dipole effect should be 0.49 V_SHE. In accordance with experiments by Ojha et al.,^21,22 I consider an electrolyte solution of 0.1 mM HClO₄ + x mM LiClO₄. The diameter of hydrated cations and anions is taken uniformly as d_h = 7 Å.

The Case without Chemisorbed Ions

The central formula expressed in eq 22 contains the interplay between the diffuse layer, water molecules at the ASP, and partially charged chemisorbed ions at the AIP. To simplify the matter, I consider in this section a simple case without chemisorbed ions, namely, θ_A = 0 in eq 22. The focus is put on how chemical interactions between adsorbed water molecules and the metal influence the pzc and furthermore how field-dependent reorientation of water molecules changes the C_dl profile.

Influence of Metal–Water Interactions on the pzc

In the absence of chemisorbed ions, the pzc is obtained from eq 22 as

25

where the first term is the potential drop at the metal surface determined by the electron density profile at the interface, the second term is a constant of the SHE, and the third term represents the influence of water molecules at the ASP.

E_pzc, SHE⁰ is defined as the value for the case of δμ̃ = 0, namely, the case in which metal–water interactions are the same for water molecules possessing α_w = 0^° and α_w = 180^°. This artificial case is hardly met. In the general case of δμ̃ ≠ 0, I obtain

26

where the second term on the right-hand side represents the pzc shift due to orientational polarization of adsorbed water molecules. It is important to emphasize that the pzc shift due to electronic polarization of adsorbed water molecules is included in E_pzc, SHE⁰.

When |δμ̃| ≤ 1, E_pzc, SHE is asymptotic to

27

Equation 27 implies that E_pzc, SHE is greater (smaller) than E_pzc, SHE⁰ when δμ̃ > 0 (<0), as shown in Figure 3b. For the case of δμ̃ < 0, the first-layer water molecules prefer the α_w = 0^° (H-up or O-down) configuration. For this case, the resulting potential rise due to the water dipole moment compensates partly for the potential drop at the metal surface, leading to a smaller E_pzc, SHE. In the opposite case, the first-layer water molecules prefer the α_w = 180^° (H-down) configuration, resulting in an additional potential drop and thereby a higher E_pzc, SHE.

Influence of Water Orientational Polarization on C_dl

In this section, I analyze how water orientational polarization affects the differential double-layer capacitance C_dl. In addition to the full model, two simplified models are introduced to differentiate multiple coupled effects. The simplified model without ion size effects is described by a variant of eq 22 in the limit of γ = 0,

28

In eq 28, I have neglected the chemisorption-induced dipole moment for this case without chemisorbed ions. The simplified model without water effects is expressed as a variant of eq 22 in the limit of θ_w = 0,

29

In Figure 4, I compare the three models for the case of δμ̃ = 0 with the base set of model parameters. The model without water effects has a camel-shaped C_dl curve with two very broad peaks corresponding to overcrowding of counterions. The C_dl curve is symmetric around the pzc as I have neglected size asymmetry. Otherwise, the peak corresponding to smaller counterions will be higher, see, e.g., ref (27).

Figure 4.

Differential double layer capacitance as a function of the electrode potential for the case of δμ̃ = 0 for two effective values of water dipole moment: (a) μ_w = 0.73 D, (b) μ_w = 0.90 D. E_pzc, SHE⁰ represents the potential of zero charge without water dipole effects, which is used as the electrode potential reference here. Three models, including the full model, a simplified model without ion size effects, and a simplified model without water effects. The electrolyte solution is 0.1 mM HClO₄. δμ̃ = 0 and other model parameters have their base values. Matlab script of this figure is provided in the Supporting Information of this paper.

Upon an increment in the magnitude of σ_free, adsorbed water molecules at the ASP could reduce the potential drop across the EDL via rearranging their orientations. Consequently, C_dl is increased due to water polarization effects, as seen in the comparison between the full model and the model without water effects. Equivalently, water molecules at the ASP contribute a negative capacitance in series with the diffuse layer capacitance, as pointed out earlier by Huang et al.⁷² and Le et al.⁵³

The two peaks of the C_dl curve become sharper and move closer to the pzc, compared to their counterparts of the model without water effects. The two peaks signify saturation of water polarization effect, rather than overcrowding of counterions in previous modified Poisson–Boltzmann theories.^28,29 Mathematically, it corresponds to ⟨ cos α_w⟩ = 1 or – 1 depending on the sign of σ_free. These two peaks are significantly elevated when μ_w increases from the base value of 0.73 to 0.9 D, as shown in Figure 4b.

Since the electrolyte solution is ultradilute, c₀ = 0.1 mM, the ion size effect is unimportant near the pzc as the C_dl curves of the full model and the model without ion size effects overlap in the region of |E_SHE – E_pzc, SHE⁰| < 0.2 V. As σ_free grows in magnitude and the EDL becomes more crowded with counterions, the ion size effect cannot be neglected any longer. Without considering the ion size effect, the two peaks are higher.

Figure 5a examines the influence of δμ̃ on the C_dl curve. As already discussed, the pzc, corresponding to the valley of the C_dl curve, shifts to more negative electrode potentials as δμ̃ goes negative. The C_dl curve becomes highly asymmetric with the left peak being much higher than the right one for δμ̃ = – 1. A negative δμ̃ means that in the absence of electrostatic interactions, water molecules prefer the configuration of α_w = 0^° (the O-down or H-up configurations). Therefore, these water molecules have a higher capacity to screen negative σ_free, which intends to rotate water molecules toward α_w = 180^°, than positive σ_free. Consequently, a higher C_dl peak is observed in the negatively charged region. Interestingly, the peak at the negatively charged surface almost disappears for δμ̃ = – 6. For the case of such a very negative δμ̃, the orientation of water molecules is dominated by the strong chemical interactions and is no longer tunable by the local electric field. Therefore, the C_dl curve resembles that of the model without water effects in Figure 4, except a horizontal shift due to δμ̃.

Figure 5.

Differential double layer capacitance as a function of electrode potential for (a) different values of δμ̃ normalized to thermal energy k_BT when the electrolyte concentration is 0.1 mM and (b) different electrolyte concentrations when δμ̃ = – 1. E_pzc, SHE⁰ represents the potential of zero charge without water dipole effects, which is used as the electrode potential reference here. Other model parameters have their base values. Matlab script of this figure is provided in the Supporting Information of this paper.

Figure 5b examines the concentration dependence of the C_dl curve while δμ̃ = – 1 is fixed. C_dl near the pzc is higher because the Debye length becomes smaller as the electrolyte concentration increases. In addition, the two peaks come closer in more concentrated electrolyte solution. These peaks are ascribed to saturation of water polarization. Let us denote as σ_free^c the critical magnitude of σ_free leading to saturated water polarization. At higher concentrations, C_dl is higher and σ_free can be achieved at a smaller potential difference away from the pzc.

The Case with Chemisorbed Ions

Effect of Chemisorption-Induced Dipole Moment

Now chemisorption is brought into the play, and its influence on the C_dl curve is examined. Let us consider the Pt(111)–0.1 M HClO₄ interface. The CV of this interface is usually divided into a hydrogen adsorption region up to ∼0.4 V_RHE, a hydroxyl adsorption region above ∼0.55 V_RHE, and a double layer region in between. Climent et al. have conducted a comprehensive analysis of CVs of this interface at different temperatures using Frumkin isotherms.⁹¹ Herein, a revisit from a different view assuming distributed equilibrium potentials of H_ad and OH_ad is presented, with the focus put on the effect of chemisorption-induced dipole moment on C_dl.

Figure 6a shows a model fitting of the CV within the potential range of [0.15, 0.75]V_RHE and a model-based decomposition of the total capacitance C_tot into C_dl and adsorption capacitances (C_Had and C_OHad). The extracted distribution functions f_H and f_OH, defined in eq 6, are shown in Figure 6b. f_H is composed of four Gaussian functions with preset mean values of 0, 0.1, 0.2, and 0.3 V_RHE, and f_OH is composed of two Gaussian functions with preset mean values of 0.7 and 0.8 V_RHE. Twelve free parameters are therefore the variances σ_k and weights p_k of these six distributions. Other EDL parameters have their base values. The basic idea of this approach resembles that of the distribution of relaxation times (DRT) proposed by von Schweidler in 1907; see a recent review on DRT.¹⁰² The present approach could be termed distribution of chemisorption energies (DCE). A better fitting of C_tot can be obtained if a greater number of Gaussian functions are included, which is not the purpose of this work. The mere purpose of this fitting is to confirm that a Frumkin isotherm with repulsive lateral interactions can be mapped into a series of Langmuir isotherms with a DCE.

Figure 6.

(a) Differential double-layer capacitance (C_dl), adsorption capacitance of H_ad and OH_ad (C_Had, C_OHad), and their sum (C_tot) for the double layer of Pt(111)–0.1 M HClO₄ interface. Experimental data, shown in circles, are taken from ref (22). (b) Model-based distributions of equilibrium potential of H and OH adsorption. (c) Comparison of differential double-layer capacitance for three values of the net electron number of OH_ad. The case of ξ_OH = 0 corresponds to the case without chemisorption-induced dipole moment, μ_OH. Matlab script of this figure is provided in the Supporting Information of this paper.

Figure 6a indicates that C_dl is a small percentage of C_tot. Figure 6c shows C_dl for the cases with and without Δϕ_ad caused by chemisorption. Three values of the net electron number of OH_ad, ξ_OH, are compared, including the base value of 0.12, a zero value corresponding to the case of a complete charge transfer during chemisorption and hence a zero Δϕ_ad, and also a higher value of 0.20 to increase the magnitude of Δϕ_ad. When Δϕ_ad = 0, the C_dl curve contains a higher peak at ∼0.2 V_RHE, and a much depressed peak at potentials more positive than E_pzc = 0.29 V_SHE (0.346 V_RHE for this case), similar to the results shown in Figure 5b. In comparison, C_dl drops above 0.5 V_RHE due to a positive Δϕ_ad when ξ_OH = 0.12. Particularly, negative C_dl is found when ξ_OH = 0.20.

Negative C_dl means that σ_free decreases with increasing electrode potential, resulting in a nonmonotonic surface charging behavior. Chemisorption-induced nonmonotonic surface charging behaviors were revealed in a previous work⁷² and elucidated in a, hopefully, more accessible manner recently.⁸⁰ The basic idea, as expressed in eq 1, is that negatively charged chemisorbed OH introduces a potential drop Δϕ_ad, which is tantamount to increasing the pzc. As a result, σ_free increases at a lower rate with the electrode potential, namely, a smaller C_dl, or decreases with the electrode potential, namely, a negative C_dl, depending on the magnitude of μ_OH.

Double Layer Region

Let us proceed to analyzing the recent experimental data by Ojha, Doblhoff-Dier, and Koper, focusing on the double layer region between 0.40 and 0.65 V_RHE.²² In order to distance the pzc as much as possible from the hydrogen and hydroxyl adsorption/desorption regions, they delicately used 0.1 mM HClO₄ as the base electrolyte solution with a pH of 4, where the pzc is 0.53 V_RHE. They repeated the measurements on the Pt(111)–0.1 mM HClO₄ interface three times, allowing us to examine how experimental uncertainties affect model results in Figure 7. Afterward, experimental data in the presence of different concentrations of LiClO₄ are analyzed in Figure 8.

Figure 7.

(a–c) Model–experiment comparison for the electric double layer at Pt(111)–aqueous interfaces. The experiment on a Pt(111) electrode in 0.1 mM HClO₄ was repeated three times.²² Model results are marked as solid lines and experimental data as circles. The total capacitance (C_total) is decomposed into the capacitance due to H adsorption, C_Had, that is due to OH adsorption, C_OHad, and the pure double-layer capacitance, C_dl. (d–f) Model-based distributions of the equilibrium potential of H and OH adsorption. (g–i) Coverages of adsorbed H and OH. Matlab script of this figure is provided in the Supporting Information of this paper.

Figure 8.

(a) Model–experiment comparison in the presence of different concentrations of LiClO₄ for the electric double layer at Pt(111) in 0.1 mM HClO₄ + x mM LiClO₄.²² Model results are marked as solid lines and experimental data as circles. For the sake of clarity, a vertical shift of 50 between different lines is used in panel (a). Coverages of adsorbed (b) H_ad and OH_ad, (c) the pure double-layer capacitance, C_dl, and (d) adsorption capacitances C_Had and C_OHad. Matlab script of this figure is provided in the Supporting Information of this paper.

Figure 7a–c shows the model–experiment comparison for three pieces of capacitance data in 0.1 mM HClO₄. The EDL parameters are fixed at their base values, and the free fitting parameters are the weight p_k, expectations μ_k, and variances σ_k of distribution functions of H_ad and OH_ad. One Gaussian distribution for H_ad and two for OH_ad are sufficient in this narrow potential region. The fitted distributions are displayed in Figure 7d–f. The coverages of H_ad and OH_ad are shown in Figure 7g–i.

In Figure 7a–c, solid lines represent model results, and circles represent experimental data. The total capacitance (C_total) is further decomposed, with the aid of the model, into the capacitance due to H adsorption, C_Had, that is due to OH adsorption, C_OHad, and the pure double-layer capacitance, C_dl. The measured capacitance C_total is contributed majorly by pseudo-capacitances C_Had and C_OHad. Three measurements show a moderate difference in between: the terrace in the potential range of 0.55–0.6 V_RHE, attributed to water adsorption in ref (22), almost disappears in the third measurement.

The present model attributes different profiles of C_total to variations in the distributed equilibrium potential of OH_ad, f_OH, as shown in Figure 7d–f. The OH adsorption has a bimodal normal distribution and the one at lower potentials with p₁ < 6% is responsible for the terrace of C_total in the potential range of 0.55–0.6 V_RHE. The terrace appears when the two normal distributions are well resolved, as in Figure 7d,e, and disappears when they come closer and become connected, as in Figure 7f. The variations in f_OH imply that the assumed distribution of equilibrium potential is likely affected by stochastic factors.

The coverages of adsorbed H_ad and OH_ad, denoted θ_H and θ_OH, are less than 5% in the double-layer region. However, such a small amount of adsorption results in adsorption capacitance several times higher than C_dl, which demonstrates the importance of and also difficulty in separating C_dl from adsorption capacitances for electrocatalytic double layers.

Experimental data in the presence of different concentrations of LiClO₄ are analyzed in Figure 8. The experiments were conducted on Pt(111) in 0.1 mM HClO₄ + x mM LiClO₄ (x = 0.1,0.3, 0.6, 1.0).²² Experimental capacitance curves are shifted vertically in Figure 8a for visual clarity. The model results in solid lines neatly agree with experimental data in circles in the examined potential region. The fitted coverages of H_ad and OH_ad, the pure double layer capacitance, and pseudo-capacitances of chemisorption are shown in Figure 8b–d, respectively. The correspondence between line colors and ionic concentrations indicated in Figure 8a is retained in all subfigures. As for the pure double-layer capacitance C_dl in Figure 8c, a Gouy–Chapman minimum is found at 0.53 V_RHE. The asymmetry around the pzc is caused by water adsorption. As shown in Figure 5a, a negative value of δμ̃, namely, the O-down configuration is more favorable, results in a more tilted increase in C_dl when the electrode potential moves away from the pzc in the more negative direction. The valley and the terrace of the measured C_total shifts to lower potentials with increasing concentration of LiClO₄. In the present model, the shift in the capacitance minimum and terrace is associated with adsorption of OH, as shown in Figure 8d. Possible reasons for this are discussed in the next section.

Discussion

Is There a Pure Double Layer Region at Pt(111)?

In contrast with previous works,^22,51,52 the present model maintains that voltammogram-derived capacitance of Pt(111) is majorly contributed by C_Had and C_OHad from a small percentage of active sites, Figure 8. One of the implications of this claim is that there is no pure double-layer region for the Pt(111) electrode.

The occurrence of chemisorbed OH_ad in the so-called double layer potential region could have two causes. On the one hand, there may be structural defects occupying a small percentage of the solid surface.^103,104 Adsorption of OH (H) occurs at lower (higher) potentials at these structural defects. In a study by Góez-Marín and Feliu,¹⁰⁵ they experimentally investigated the relation between oxygen reduction activity and step density and extrapolated this relation to obtain the oxygen reduction activity of an ideally defect-free Pt(111). They found that the extrapolated activity of this ideal Pt(111) is lower than that of a real Pt(111). This comparison leads them to conclude that “This is true even for those electrodes that do not apparently show any step contribution signal in the blank voltammogram, pointing out that a real electrode has always surface defects.” Recent works on stepped Pt electrodes, such as Pt(553)¹⁰⁶ and Pt(311),¹⁰⁷ have shown low-potential water dissociation in the hydrogen adsorption/desorption region, and the onset potential moves toward the “double layer” region at higher pHs.

On the other hand, interfacial water molecules have an energy distribution, as seen in AIMD simulations,¹⁰⁸ and higher-energy water molecules are decomposed and then adsorbed as OH_ad at lower potentials. Such a distribution, intrinsically stochastic to a certain extent, echoes the experimental uncertainties found in three repeated measurements on the Pt(111)–0.1 mM HClO₄ interface. Model results in Figure 8 further indicate that a higher concentration of LiClO₄ promotes earlier adsorption of OH. Berna et al. have adopted a similar idea to interpret chemisorption of OH at Pt(111).¹⁰⁹ Specifically, they attributed the broad peak in the low potential region and the sharp peak in the high potential region to the dissociation of two kinds of water molecules with different electrochemical potentials. Their treatment could be regarded as a special case of this model where the distribution function is actually composed of two delta functions.

Currently, it is hard to detect adsorbed intermediates at such a low coverage. Nevertheless, Kondo et al. deduced from their in situ surface X-ray scattering data that a small amount of oxygen-containing species (unspecified) and/or ClO₄^– is adsorbed on Pt(111) immersed in 0.1 M HClO₄ at 0.57 V_RHE within the double-layer region.¹¹⁰ It has also been shown that the ClO₄ concentration impacts the CV of Pt(111) in the double layer region.^110,111 Though I have assumed H_ad and OH_ad as the chemisorbates when explaining the experimental data of Ojha et al., the existence of other chemisorbates such as ClO₄^– and organic impurities cannot be excluded for this moment.^110,111

Limitations and Possible Extensions of the Model

The present double layer model belongs to GCS-type models. An essential feature of these models is the decomposition of the double layer into several subregions, which are then described as dielectric continua. I am aware of the harsh criticisms on such GCS-type models regarding drastic simplifications of the double layer structure and its parameterization procedure that arbitrariness may slip in.¹¹² Nevertheless, it is undeniable that GCS-type models are simple yet effective in describing physicochemical properties of the double layer. Fortunately, accurate atomic details provided by careful first-principles calculations can help to avoid unphysical assumptions, omission of key factors, overparameterizaton and error cancellation, etc., in phenomenological modeling. Having said this, I stress that first-principles approaches have its own purpose, rather than just serving to elaborate phenomenological models.

Though the model results are obtained for a Pt(111) electrode in HClO₄-based solution, I do not see principal difficulties in applying it to the alkaline regime, to other electrolyte solution with specifically adsorbing anions such as sulfate and halide ions and to stepped Pt electrodes. In these new scenarios, the model needs to be extended by including new specific adsorption and/or chemisorption processes and to be re-parameterized on a case-by-case basis.

The model can be extended in the following directions in the future. First, the locations of chemisorbed ions and the first water layer, which are fixed in the present work, can be allowed to change as a function of electrode potential. Similar considerations of relaxation of the metal–solution gap have been shown to be able to bring forth negative C_dl.¹¹³ Second, the model may also be extended to consider how electronic effects of chemisorbed ions and water molecules change χ_M.^65,114 Third, a constant δμ̃ has been used so far, and it is meaningful to consider a potential-dependent change of δμ̃, caused by interactions between chemisorbed ions and water molecules on the one hand and orientational dependency of water binding energy on the other.¹¹⁵ Lastly, noncovalent interactions between chemisorbed ions and electrolyte species, key to understanding cation effects as shown in a recent study,¹¹⁶ are also important factors to be treated. Lastly, the model can also be extended to study other chemisorption processes, including those of electrolyte anions and organic impurities, at different types of active sites.

Conclusions

A modified double layer model has been formulated for metal–aqueous interfaces in electrocatalytic systems. The present model divides the inner Helmholtz plane (IHP) further into an AIP (adsorbed ion plane) and ASP (adsorbed solvent plane), accounting for different locations of chemisorbed ions and solvent molecules. The present model employs a statistical treatment of the first-layer water molecules with chemical interactions imposed by the metal and a distributed treatment of chemisorption of electrolyte species.

The model shows that the potential of zero charge, pzc for short, depends on metal–water chemical interactions. I introduce δμ to characterize the chemical interaction energy difference between water dipoles of α_w = 0^° and that of α_w = 180^°, where α_w is the angle between water dipole direction and the local electric field. It is found that the pzc becomes higher (lower) when δμ > 0 (<0). In addition to a shift in the pzc, the first-layer water molecules also dramatically elevate the double-layer capacitance C_dl. Equivalently, it means that the first-layer water molecules contribute a negative capacitance component in series with other components, which is a known effect. However, the formula derived in this work to describe this effect, eq 26, is, to my knowledge, a new contribution. The model also shows that chemisorption-induced dipole moment could lead to a new valley of the capacitance curve and the capacitance in the valley could take negative values.

The model is then employed to interpret recent CV-based capacitance data of Pt(111)–aqueous solution interfaces. The explanation is based on the ansatz that the experimental data are not the pure double-layer capacitance but include pseudo-capacitances of chemisorption of electrolyte species. This ansatz resolves the discrepancy that the CV-based capacitance values are several times higher than the values obtained using electrochemical impedance spectroscopy. In addition, it constitutes an alternative angle of understanding stochastic uncertainties found in these experiments. The ansatz and the resulting explanation, though being appealing, must be taken with care, as it implies that there is no pure double layer region even for Pt(111), an aspect that is yet to be confirmed.

Before closing, three experiments to confirm this ansatz are sketched here. One can examine the frequency dependence of interfacial capacitance in the double layer region, especially, at the pzc. If it is a pure double-layer capacitance, there should not be any frequency dispersion below the characteristic frequency of EDL capacitance, which is expressed as f_dl ∝ D/λ_D² with D being the diffusion coefficient of electrolyte ions and λ_D being the Debye length, according to an exact analytical solution developed in ref (117). With typical values of D = 10^–9 m²s^–1 and λ_D = 1 – 10 nm, f_dl has a range of 10⁷ – 10⁹ Hz. It is thus safe to claim that a pure double-layer capacitance should not depend on frequency below 1 MHz. In other words, shall one observe that the interfacial capacitance decreases with increasing frequency, it is very likely that the interfacial capacitance contains pseudo-capacitance(s). If it is difficult to obtain reliable high-frequency impedance data, a time-domain method that is easier to implement is to measure the current response to a potential step from the pzc by a small amount, say 20 mV. If it is pure double layer charging, the time constant, expressed as t_dl ∝ 10Lλ_D/D with L being the thickness of the diffusion layer,¹¹⁸ has a range of 1 – 10 ms using typical values of D = 10^–9 m²s^–1, L = 100 μm, and λ_D = 1 – 10 nm. In other words, shall one observe a current ramping with a time constant longer than 10 ms, it is very likely that other slower process(es) is (are) involved. The third experiment involves ultrafast CV measurements, which have been demonstrated on the order of megavolt per second,¹¹⁹ and then check the scan-rate dependency of the obtained capacitance profiles. This would require careful ohmic drop compensation and noise control.

Data Availability Statement

No new experimental data. The scripts used to calculate the model results are available from the author upon request.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacsau.2c00650.

• Matlab scripts for reproducing results reported in this paper (PDF)

The author declares no competing financial interest.