doi.org/10.26434/chemrxiv.14248889.v1

Thermodynamic Cyclic Voltammograms: Peak Positions and ShapesNicolas G Hörmann, Karsten Reuter

Submitted date: 19/03/2021 • Posted date: 22/03/2021Licence: CC BY-NC-ND 4.0Citation information: Hörmann, Nicolas G; Reuter, Karsten (2021): Thermodynamic Cyclic Voltammograms:Peak Positions and Shapes. ChemRxiv. Preprint. https://doi.org/10.26434/chemrxiv.14248889.v1

Based on a mean-field description of thermodynamic cyclic voltammograms (CVs), we analyse here in fullgenerality, how CV peak positions and shapes are related to the underlying interface energetics, in particularwhen also including electrostatic double layer (DL) effects. We show in particular, how non-Nernstianbehaviour is related to capacitive DL charging, and how this relates to common adsorbate-centeredinterpretations such as a changed adsorption energetics due to dipole-field interactions and theelectrosorption valency -- the number of exchanged electrons upon electrosorption per adsorbate. UsingAg(111) in halide-containing solutions as test case, we demonstrate that DL effects can introduce peak shiftsthat are already explained by rationalizing the interaction of isolated adsorbates with the interfacial fields,while alterations of the peak shape are mainly driven by the coverage-dependence of the adsorbate dipoles. Inaddition, we analyse in detail how changing the experimental conditions such as the ion concentrations in thesolvent but also of the background electrolyte can affect the CV peaks via their impact on the potential drop inthe DL and the DL capacitance, respectively. These results suggest new routes to analyse experimental CVsand use of those for a detailed assessment of the accuracy of atomistic models of electrified interfaces e.g.with and without explicitly treated interfacial solvent and/or approximate implicit solvent models.

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Thermodynamic Cyclic Voltammograms: Peak Positionsand Shapes

Nicolas Georg HormannChair of Theoretical Chemistry and Catalysis Research Center, Technische UniversitatMunchen, 85748 Garching, GermanyFritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany

E-mail: hoermann@fhi-berlin.mpg.de

Karsten ReuterFritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany

Abstract. Based on a mean-field description of thermodynamic cyclic voltammograms(CVs), we analyse here in full generality, how CV peak positions and shapes are related tothe underlying interface energetics, in particular when also including electrostatic double layer(DL) effects. We show in particular, how non-Nernstian behaviour is related to capacitive DLcharging, and how this relates to common adsorbate-centered interpretations such as a changedadsorption energetics due to dipole-field interactions and the electrosorption valency – thenumber of exchanged electrons upon electrosorption per adsorbate. Using Ag(111) in halide-containing solutions as test case, we demonstrate that DL effects can introduce peak shifts thatare already explained by rationalizing the interaction of isolated adsorbates with the interfacialfields, while alterations of the peak shape are mainly driven by the coverage-dependence of theadsorbate dipoles. In addition, we analyse in detail how changing the experimental conditionssuch as the ion concentrations in the solvent but also of the background electrolyte can affectthe CV peaks via their impact on the potential drop in the DL and the DL capacitance,respectively. These results suggest new routes to analyse experimental CVs and use of thosefor a detailed assessment of the accuracy of atomistic models of electrified interfaces e.g. withand without explicitly treated interfacial solvent and/or approximate implicit solvent models.

2

1. Introduction

Cyclic voltammetry is a standard experimental technique for studying electrochemicalinterfaces that allows to infer surface compositions and interface reactions as a function ofthe applied electrode potential. In practice, cyclic voltammograms (CVs) are obtained byvarying the electrode potential at fixed scan rate and measuring the current response of theelectrode immersed in electrolyte solution. In general, CVs are characterized by potentialregions within the stability window of the solvent which exhibit peaks of varying shape andheight, and regions at low and high potentials, where faradaic, electrocatalytic reactions leadto exponentially increasing currents, e.g. due to the decomposition of the solvent. At high scanrates and/or in regions with faradaic reactions, CVs are governed by kinetic processes whichtypically induce a pronounced asymmetry for the forward and backward scan direction and/ora strong scan-rate dependence. Evidently, any theoretical description of such CVs necessitatesthe use of kinetic models [1], including potentially also macroscopic mass transport [2].

On the other hand, at low scan rates and in potential windows without faradaic sidereactions this asymmetry typically vanishes as does the scan-rate dependence (when currentsare appropriately normalized). In this case, CV peak positions and shapes are related directlyto the underlying thermodynamics of the electrified interface [3], and such CV experimentsprovide invaluable contributions to the understanding of the latter. In turn, such CVs canbe understood and simulated based on equilibrium thermodynamic considerations [4–12].For such thermodynamic CVs the variation of the applied potential induces changes in theequilibrium surface charges, which can be traced back to changes in adsorbate coverages,as well as changes in the double layer charge, leading in sum to the observed electriccurrent. As recently demonstrated for Ag(111) in halide containing solutions [13], doublelayer (DL) charging does not only add capacitive currents, but also affects equilibriumadsorbate coverages and the number of exchanged electrons per adsorbate – as expressedby the electrosorption valency [13–16].

In this work we therefore analyse in most general terms thermodynamic CVs withincluded DL response. The derived equations can naturally explain Non-Nernstian behaviourand introduce a sensitive dependence of CV peaks to the electrolyte, via its impact on theinterfacial capacitance [17, 18]. We hope, these results might help in the future to betterunderstand according experiments and thus also help to validate and improve theoreticalmodels. This is in particular important as at present all (atomistic) theoretical modelsof electrified interfaces necessarily introduce approximations due to computational timeconstraints, e.g in the complexity of the atomistic description (e.g. the surface with or withoutexplicit water [19–23]), the quality of the energetics (e.g. empirical potentials [24]), theapplication of the electrode potential (e.g. absent or not [25–27], with explicit ions [28, 29]or an implicit solvent model [16, 22, 30–51]) or the derivation of macroscopic quantities (e.g.sampling [6, 10, 28, 52–56], mean field models [9, 13]).

Here, we start from a fully grand canonical, ab initio thermodynamics description ofthe interface and derive in a consistent way expressions for the grand canonical mean-field energetics that includes effects of the potential up to second order. This second order

3

approximation will be referred to as the CHE+DL approximation as the terms up to linearorder in the applied potential are identical to the terms frequently used to estimate the effectsof the potential based on ab initio calculations at the potential of zero charge (PZC) andwhich is often referred to as the Computational Hydrogen Electrode [19, 57] (CHE) method.The quadratic, higher order term, on the other hand, captures capacitive DL charging withan assumed potential-independent interfacial capacitance, which was already successfullyapplied in many other works [13, 16, 37, 46, 54, 58–63].

Naturally, the derived equations can be extended in a straightforward way to includeany higher order response to the potential (e.g. of the DL capacitance), however, as wesee later, any such energetics can not be treated analytically any more. At variance, thedescribed CHE+DL level of theory is largely amenable to rigorous mathematical analysis,and yields straightforward expressions for thermodynamic CVs. These results in combinationwith additional approximations, allow us to derive proxies for the position and the shape ofpeaks in according CVs that derive from electrosorption, which we then validate against thefull theoretical description, and experiments. We demonstrate that both, peak positions andshapes, can be affected in a non-trivial way by the ion concentrations in solution as well asthe electrolyte composition, and estimate according effects for halide solutions and selectedmetallic electrodes based on theoretical density functional theory (DFT) calculations.

2. Theory

2.1. Ab initio thermodynamics of electrochemical interfaces

In this work we concentrate for clarity on the description of metallic electrodes with justa single possible adsorbate species a (generalization to multiple species is conceptuallystraightforward). In this case, the Gibbs excess energy [13, 16, 50, 64, 65] at applied electrodepotential ΦE is defined as

Gαexc = Gα

surf(Nαs ,N

αa ,N

abs,αe ) − Nα

s µs − Nαa µa + Nabs,α

e eΦE . (1)

α denotes a certain interface configuration as characterized by its surface geometry (e.g.position of adsorbates) and chemical composition of Nα

s substrate atoms, Nαa adsorbates

of type a, and Nabs,αe electrons in excess to the charge-neutral pristine electrode surface.

Expression (1) in essence determines the free energy cost of creating the interfacial system α

with total energy Gαsurf(N

αs ,N

αa ,N

abs,αe ) when its constituents are taken from the thermodynamic

reservoirs characterized by the (electro-)chemical potentials µs, µa and −eΦE for substrateatoms, adsorbates and electrons, respectively.

With the adsorbates a present in solution as dissolved ions Aqa/e(aq) of charge qa, theirelectrochemical potential µa can be determined from tabulated experimental equilibriumpotentials Φ

expa,eq of the redox reaction A Aqa/e(aq) +

qae e−(m) in combination with theoretical

calculations of the chemical potential µA in the reference phase. For hydrogen and halides,e.g., µA is conventionally determined as half the chemical potential of the biatomic moleculesat 298 K and 1 bar (µA = 1

2µA2(g)) and Φexpa,eq refers to a 1 M solution [66, 67]. In this case:

µa = µA + qaΦexpa,eq + β−1 ln (ca) , (2)

4

where ca is the molar concentration of the ions in solution and β−1 = kBT .As noted in the introduction, whenever the interfacial capacitance is independent of the

applied potential – as e.g. in many implicit solvent models for high electrolyte concentrations[13, 16, 68] (see also section 1 in the Supporting Information (SI)) – Gα

surf in eq. (1) can bedescribed with a second order polynomial in the number of electrons. In this case, the grandcanonical, charge-equilibrated excess energy (constant potential conditions) decomposes intoan excess energy term valid at the PZC – which is linear in the applied potential ΦE

and identical to the CHE expressions [16, 19, 66] – and an additional, generic DL energycontribution due to capacitive charging [13, 16, 63]. Furthermore, mean-field sampling ofconfigurations α at fixed adsorbate coverages θa = Nα

a /Nsites can provide an approximateinterface free energy landscape as a function of the coverage and the potential, from whichany thermodynamic quantity can be inferred, in principle [13].

In the following, we will use expressions that are normalized to the number of surfacesites (denoted by lower case letters), such that the mean-field CHE excess energy (per surfacesite) reads

gθa,CHEexc,0 = gclean

exc,0 + θaGθaads,0 − θaβ

−1 ln (ca) + θaqa(ΦE − Φexpa,eq) , (3)

(4)

where gcleanexc,0 is the excess energy of the clean surface and Gθa

ads,0 the average adsorption energyper adsorbate, at coverage θa. The subscript 0 denotes values determined at the PZC. Bothterms can be approximated e.g. from (charge-neutral) DFT calculations via

gcleanexc,0 ≈

1Nsites

[Eclean,DFTsurf,0 − NsEDFT

s,bulk] (5)

Gθaads,0 ≈

⟨1

Nαa[Eα,DFT

surf,0 + ∆Fα,corrsurf,vib − Eclean,DFT

surf,0 − Nαa µ

DFTA ]

⟩MFT

. (6)

Eα,DFTsurf,0 and Eclean,DFT

surf,0 are the DFT 0 K energies of the adsorbate-covered system consistingof Nα

a adsorbates and Ns substrate atoms and of the pristine substrate system (Ns substrateatoms). EDFT

s,bulk is the DFT 0 K bulk energy per atom of the substrate material. Here and inthe following we assume all DFT surface energetics to be obtained using implicit solvationmodels. The corresponding terms could also be evaluated with explicit interfacial water tocapture specific solvation effects. This can evidently affect the numerical values of bothquantities, but would not affect the general thermodynamic description within the presentwork. The MF average 〈·〉MFT at fixed coverage θa = Nα

a /Nsites can be efficiently approximatedby random sampling or via the use of special quasi random structures [69–71]; for theconsidered showcase of halides on Ag(111) using only maximally isotropic structures atdifferent coverages yielded already a good agreement between predicted and experimental CVpeak shapes [13]. As none of the present discussion depends on the way these MF averagesare determined (they might as well be determined by a fit to experimental data) we will referto these averages at given coverage θa simply by a corresponding superscript. In the case ofa first-principles parametrization (e.g. based on eqs. (5), (6)) inaccuracies due to the use of 0K energy differences are largely corrected by addition of the term ∆Fα,corr

surf,vib – the vibrationalfree energy contributions of the adsorbates only. As noted previously, within the CHE+DLapproximation [13,16,63] the DL energy contributions per site due to capacitive charging are

5

simply given by a plate-capacitor-like term

gθa,DLexc = −1

2 AsiteCθa0 (ΦE − Φ

θa0 )2 , (7)

with the area per site Asite = A/Nsites. Φθa0 is the PZC (aka work function of the interface in

solution) and Cθa0 the area-normalized DL capacitance at the PZC for the coverages θa. Both

intensive quantities can be determined again from ab initio calculations, e.g. with an implicitsolvent model. In the context of the discussion here, they are to be seen as system-specificparameters which can vary for different adsorbates and experimental conditions. Finally, theMF free energy landscape per site as a function of θa and ΦE is given by

g θa,MFTexc ≈ gθa,CHE

exc,0 + gθa,DLexc − T sθa

conf , (8)

with sθaconf corresponding to the MF configurational entropy term

sθaconf = − θmax

a kB

[(θaθmax

a

)ln

(θaθmax

a

)+

(1 − θa

θmaxa

)ln

(1 − θa

θmaxa

)], (9)

where θmaxa is the maximum achievable coverage (per site) for the adsorbates a [13].

Above, and in the following, all DL-charging-related terms are underlined to make theimpact of the latter more clear.

2.2. Equilibrium interface compositions and thermodynamic CVs

The equilibrium surface coverage θa(ΦE) at given applied electrode potential ΦE minimizesg θa,MFT

exc (ΦE) (eq. (8)) with respect to θa. Knowledge of θa(ΦE) then directly yields the expectednumber of electrons nabs

e per site as a function of the potential via

nabs,θae = θa

qae −

1e AsiteC

θa0 (ΦE − Φ

θa0 ) , (10)

from which thermodynamic CV currents j can be derived [13]. At a fixed scan ratev = d

dt ΦE, this current can be written as j = vCpseudo, where the pseudocapacitance Cpseudo isscan-rate-independent. Within the MF model, CMFT

pseudo = −e ddΦE

nabs,θae and thus given by

CMFTpseudo = AsiteC

θa0 + CMFT,sorp

pseudo

= AsiteCθa0 − elMFT

ad

dΦEθa . (11)

lMFTa is the electrosorption valency [13–16] – the number of exchanged electrons per adsorbate

– which is given by

lMFTa = 1

e

[qa + AsiteC

θa0

ddθa

Φθa0 − Asite

ddθa

C θa0 (ΦE − Φ

θa0 )

]. (12)

Note that expression (11) includes a double-layer charging, baseline contribution AsiteCθa0

plus a contribution due to electrosorption which we refer to as CMFT,sorppseudo . The latter is naturally

proportional to the change in surface coverage ( ddΦE

θa) times the electrosorption valency lMFTa .

6

3. Results

Until now we have put much attention to clarifying how the respective mean-field equationsderive from an ab initio thermodynamics-based ansatz. However, as this mean-fielddescription is completely generic, these equations and the parameters therein form a generaltheory of thermodynamic CVs, where all mean-field parameters are to be seen as fundamentalsystem-inherent descriptors. All CHE terms are thereby terms associated with the energeticsat the PZC, and all underlined DL terms arise from the capacitive charging of the interface,aka the second order response to the application of a potential while leaving surface coveragesfixed.

3.1. CV Peaks

θa minimizes g θa,MFTexc and is thus given by

ddθa

g θa,MFTexc = 0 = ε θa + β−1 ln

(θaθmax

a

/ (1 − θa

θmaxa

)), (13)

which can be rewritten into the typical form of an adsorption isotherm as

θa = θmaxa

[1 + exp

(βε θa

)]−1. (14)

While eq. (14) defines θa(ΦE) in general only implicitly, it is possible to derive ananalytic expression for the inverse function Φ

θaE B ΦE(θa), which is given as the solution

to the quadratic equation (in ΦE) defined by eq. (13) and with

ε θa = ddθa

(gθa,CHE

exc,0 + gθa,DLexc

)= a(ΦE − Φ

θa0 )2 + b(ΦE − Φ

θa0 ) + E (15)

a =[−1

2 Asited

dθaC θa

0

](16)

b = qa +[AsiteC

θa0

ddθa

Φθa0

]= qa(1 + δa) (17)

E = Gθaads,0 + θa

ddθa

Gθaads,0 − β

−1 ln (ca) + qa(Φθa0 − Φ

expa,eq) , (18)

The general solution to this second order equation in ΦE (eq. (13), (15)) howeverhampers meaningful further analysis and we restrict ourselves in the following to the casewhere a = 0 in eq. (15). This approximation implicitly assumes d

dθaC θa

0 = 0 and thus refers to

systems, where the interfacial capacitance is independent on the surface coverage (C θa0 = C0)

or where this is a good approximation. In this case

ΦθaE = − 1

qa

[G + qa∆a

]= Φ

θa,CHEE − ∆a (19)

G = E − qaΦθa0 + β−1 ln

(θaθmax

a

/ (1 − θa

θmaxa

))(20)

∆a = δa∆ΦE with ∆ΦE = ΦθaE − Φ

θa0 , (21)

where δa is a unitless measure for DL effects as defined in eq. (17). Note, that the CHE resultis given by the first term in eq. (19) with Φ

θa,CHEE = − Gqa

, whereG is the differential, adsorption-energy-like per adsorbate quantity defined in eq. (20). Inclusion of the (underlined) DL terms

7

shifts Φθa,CHEE thus by an amount of −∆a = −δa∆ΦE at given coverages θa with ∆ΦE = Φ

θaE −Φ

θa0

the difference between the applied potential and the PZC Φθa0 . Note that δa = DaC0 is

the product of an adsorbate-dipole-related quantity Da =Asiteqa

ddθa

Φθa0 and the DL capacitance

C0 (assumed independent on coverage already before). Thus the DL-induced shifts in theelectrosorption potential can be rationalized by a dipole field interaction with the field in thedouble layer ∝ C0∆ΦE. This term corresponds to the interface charge at applied potentialthat is used as relevant variable in other works [18]. Furthermore, qaδa relates directly tothe electrosorption valency via elMFT

a = qa(1 + δa) (cf. eq. (12) for ddθa

C θa0 = 0 and eq.

(17)). The close connection between electrosorption valency and dipole has been clarifiedpreviously [15, 16]. Although adsorbate dipoles and interfacial fields are more commondescriptors when discussing adsorbates at electrochemical interfaces, we use here mostly theunitless quantity δa, as it represents a direct measure of the size of dipole-field interactions (atgiven DL capacitance) relative to the work by the electron transfer process upon adsorption,and it relates in a straightforward way to lMFT

a .Evidently, Φ

θaE (eq. (19)) can be approximated as a perturbation to the CHE result Φ

θa,CHEE

via

ΦθaE ≈ Φ

θa,CHEE − δa∆ΦCHE

E , (22)

where ∆ΦCHEE = Φ

θa,CHEE − Φ

θa0 is the potential drop in the double layer, as estimated from the

CHE electrosorption potentials Φθa,CHEE . With Cl@Ag(111) having electrosorption valencies

of ∼ −0.5 [16] we can directly derive |δa| ≈ 50% and thus estimate the DL-charging-relatedshift of the CHE peak position as ≈ 0.5|∆ΦCHE

E |, which is easily in the range of hundreds ofmeV, and thus far from negligible.‡

3.1.1. Peak position As discussed above, CVs are typically dominated by the electrosorptionpeak CMFT,sorp

pseudo (eq. (11)). Whenever lMFTa depends only weakly on the coverage, the position

and width of the CV peak are hardly affected by its neglect, as it enters only as a multiplicativeconstant in eq. (11)§. Thus, we focus in the following on the quantity C = σ d

dΦEθa , with

σ =−qa|qa |

. This expression yields positive electrosorption peaks, independent on the sign of theadsorbate charge qa for physically sound, i.e. convex free energy landscapes‖.

Thus, CV peak position and shape can be evaluated from the moments of C via theintegrals

〈ΦnE〉 = 1

N

∫ ∞

−∞

C · ΦnEdΦE (23)

and the normalization N =∫ ∞−∞C · dΦE. These can be rewritten as integrals over the coverage

‡ As the peak position is directly related to adsorption energies as e.g. used in kinetic theories, this error has adramatic impact on the predictive quality of those.§ The appropriate inclusion of the electrosorption valency is performed in the SI in section 3. The resultingequations do however not allow further insights as gained by the presented analysis.‖ Systems where adsorption energies decrease with increasing coverage can lead to non-convex free energylandscapes in mean-field theory, which we do not consider here.

8

θa (see section 2 in the SI for more details), as

〈ΦnE〉 = 1

θmaxa

∫ θmaxa

0

(ΦθaE

)ndθa . (24)

The peak position P on an absolute potential scale is given by the first moment

P = 〈ΦθaE 〉 ≈ 〈Φ

θa,CHEE 〉 − 〈∆a〉 , (25)

which can be readily computed from the coverage-dependent adsorption energetics (cf.eqs. (19), (24)). This result clarifies the above mentioned influence of the dipole-fieldinteraction ∆a on the expected peak position relative to the CHE estimate. Furthermore, itis straightforward to derive peak shifts induced by a variation of the concentration ca via theidentities

dd log(ca)〈Φ

nE〉 = 〈nΦn−1

E∂Φ

θaE

∂ log(ca)〉 , (26)

∂ΦθaE

∂ log(ca) ≈β−1 ln(10)

qa(1 − δa) . (27)

The latter approximation is valid for small δa (cf. eq. (22) and section 2 in the SI). Evidently,the first term in eq. (27) describes the perfect Nernstian behaviour of the CHE energetics,whereas the DL terms induce Non-Nernstian shifts according to

dd log(ca) P = d

d log(ca)〈ΦθaE 〉 ≈

β−1 ln(10)qa

(1 − 〈δa〉) , (28)

which can be directly evaluated from the coverage dependence of the PZC and knowledge ofthe DL capacitance C0 (cf. eq. (17)). Therefore, non-Nernstian peak shifts are directly relatedto non-ideal electrosorption valencies elMFT

a = qa + qaδa , qa, which depend on the adsorbatedipole and the interfacial capacitance, not however on the magnitude or direction of the fieldin the double layer (within the present approximations).

3.1.2. Proxies for peak position and peak shape In analogy to the previous analysis,peak widths can be analyzed via the second central moment, as done in the SI (sections2,3), however, the intricate averages over the coverage hamper further insightful, analyticalanalysis. Therefore, we follow a simpler route to derive proxies for the position and peakshape of CVs by estimating above averages over the coverage 〈·〉 by simple evaluation of thecoverage-dependent terms at an intermediate coverage, e.g. the half maximum coverage. Thisprocedure allows to derive in full rigour general and more intelligible proxies for CV peakproperties, as elaborated in the SI. We refer the interested reader to the respective section 4 inthe SI, also to understand better the conditions under which the various approximations madeare expected to remain valid. All derived expressions are reported in a self-contained wayfor CV peak position and shape (table 1) as well as for their variation when changing the ionconcentration ca and the double layer capacitance C0 (table 2).

Indeed, the respective equations show that changing the ion concentration ca can affectCV peak positions and shapes, and these changes can be understood from the changes indipole-field-like interactions which rationalize the respective descriptors (see more detaileddiscussion later on).

9

Table 1. Approximate expressions for the CV peak position P, the potential drop in the DL∆ΦE and the CV peak shape as defined by its width W and height H. While all other variablesare as introduced in the main text, the variables σ = −

qa|qa |

= ±1 and K > 0 = O(1) canbe understood from the detailed derivations in section 4 of the SI. For halides on Ag(111)K ≈ 0.27. All coverage-dependent terms are to be evaluated at half maximum coverage(θa = 1

2θmaxa ).

property proxy

CHE peakposition

PCHE = − Gqa(29)

G = Gθaads,0 + θa

ddθa

Gθaads,0 − qaΦ

expa,eq

− β−1 ln (ca) + β−1 ln(

θaθmax

a

/ (1 − θa

θmaxa

))CHE+DLpeakposition

P = PCHE − δa∆ΦE (30)

DL poten-tial drop ∆ΦE = Φ

θaE − Φ

θa0 = P − Φ

θa0 ≈ ∆ΦCHE

E = PCHE − Φθa0 (31)

CHE+DLwidth

W = WCHE + WDL (32)

WCHE = σK( ddθa

Φθa,CHEE ) (33)

WDL = − σK · ddθa

∆a

= −WCHE · δa −|qa |K

AsiteC0· δ2

a − σK ddθaδa · ∆ΦE (34)

CHE+DLheight H = |qa(1 + δa)|

KW (35)

In the same way, a change of the DL capacitance C0 can introduce changes in peakposition and shape. As C0 changes with electrolyte concentrations but also with electrolyteion type [17] due to variations in ionic solvation shell sizes [18], the derived dependencieson C0 are highly important to rationalize non-trivial changes under different experimentalconditions.

We will provide a more detailed discussion and evaluation of these proxies in table 1 and2 later on.

10

Table 2. Proxies for the concentration dependence dd log(ca) and the capacitance dependence d

dC0

for the CV peak position P, the width W and the height H.property proxy

Change with ion concentration log(ca)CHE+DLpeakposition

dd log(ca) P =

β−1 ln(10)qa

(1 − δa) (36)

CHE+DLwidth

dd log(ca)W = K · β

−1 ln(10)|qa |

· ddθaδa (37)

CHE+DLheight

dd log(ca) H = − H

W ·β−1 ln(10)|qa |

· ddθaδa (38)

Change with DL capacitance C0

(C0 = C θa0 , coverage-independent; δa

C0= Da, independent on C0)

CHE+DLpeakposition

ddC0

P = − ∆ΦEδaC0

(39)

CHE+DLwidth

ddC0

W =

−WCHE −2|qa |KδaAsiteC0

− σKd

dθaδa

δa∆ΦE

· δaC0

(40)

CHE+DLheight

ddC0

H =|qa |

|qa(1+δa)|H · M ·δaC0

(41)

M = 2 +(1 + 2H

AsiteC0

)δa + σ

ddθa

δa

δa

H∆ΦE|qa |

(42)

11

I

BrCl

Experiment MFT&CHE+DL

MFT&CHE

Cps

eudo

sorp

/Figure 1. Experimental [72] and theoretical [13] CV peaks for Ag(111) in halide containingsolutions at ion concentrations of ca = 0.5 mM. The theoretical results are based on the presentmean-field description at the CHE+DL and CHE level of theory as parametrized in Ref. [13]based on DFT calculations in an implicit solvent model. All curves are without capacitivebaseline currents and plot the scan-rate-independent property Csorp

pseudo (area-normalized values,see section 6 in the SI for more details). Only the theoretical CHE+DL results can replicate theexperimental variations in peak shapes for Cl, Br, and I, both qualitatively and quantitatively,which supports the importance of capacitive DL charging for these systems.

I

Cl

Br

5·10-7cI = 5·10-4

5·10-1010-13

10-1 50

cBr = 5·10-4

5·10-7

104

505·10-2

cCl = 5·10-4

Figure 2. Theoretical CV peaks (area-normalized values of CMFT,sorppseudo ) for Ag(111) in halide

containing solutions as a function of the concentrations ca (MFT&CHE+DL parametrizationfrom Ref. [13]). The CVs corresponding to the central panel of Fig. 1 at concentrationsca = 5 · 10−4 M are colored and bold. All additional grey peaks are simulated at theindicated molar concentrations (values partly unphysical). The plot demonstrates firstly astrong concentration-dependence of all halide CVs and and secondly that all halide peaks areexpected in essence identical, whenever their position P on the potential axis is the same,in agreement with the importance of the potential drop in the double layer as the shape-determining factor [13].

3.2. Accuracy validation for the proxies: halide electrosorption on Ag(111)

DL effects were found relevant for understanding thermodynamic CVs of Ag(111) in halidecontaining solutions [13], as evidenced by comparing experimental results of Foresti etal. [72] with theoretical predictions of the present mean-field description at the CHE and

12

Exp CHE+DLCHE CHE+

a)

b) PZC@Ag(111)

0.5 mM0.5·102 0.5·10-2

CHECHE+DLCHE+

Figure 3. a) Experimental and theoretical peak positions P and widths W for ca = 0.5 mM,computed via first and second central moment of Csorp

pseudo(ΦE) (see text). CHE and CHE+DLresults are plotted as crosses and squares. The DL-induced CHE perturbations are indicated asarrows (horizontal arrow: eq. (25); vertical dashed, dotted and solid arrows: first, second andthird term in eq. (34)). Adding those to the CHE data points (CHE+↑, filled circles) explainapproximately the CHE+DL results. The grey regions are a guide to the eye to highlight theunderestimated correlation between positions P and widths W in the CHE model. b) P andW for the three theoretical methods in a concentration range of ca = 0.5 · 10±2 mM. The greyregion corresponds to the PZC of half-covered Ag(111) surfaces.

CHE+DL level of theory, parametrized from DFT calculations in an implicit solvent model(see Fig. 1 and Ref. [13]). The inability of the pure CHE description to replicate theexperimental variations in the peak shapes, hints at the importance of capacitive DL chargingfor these systems.

This becomes even more clear, when studying the changes in the peak shapes withchanging the ion concentrations ca for the according CHE+DL mean field models, as reportedin Fig. 2¶. Results at the experimental conditions (ca = 5 · 10−4 M [72], as in Fig.1) are colored and bold, while results for selected alternative concentrations are in grey(molar concentrations as indicated, partly unphysical). The plot demonstrates the strongconcentration-dependence of the according CVs and clarifies that all halide peak shapes areexpected in essence identical, given that their position P on the potential axis is the same. Thiscan be easily understood from the previous discussions and the fact that all adsorbate-relateddescriptors but the adsorption energy are essentially the same for Cl, Br and I (see Ref. [13]and Fig. SF1 in the SI).

In order to assess the accuracy of the derived proxies, we use these results for the CV

¶ Details on the parameters in Ref. [13] and the SI, section 6

13

peaks for Ag(111) in Cl, Br and I solutions, to test in how far the differences between the CHEand the CHE+DL results (Fig. 1) and the relevant changes with the concentration (Fig. 2)can be described and understood from the derived proxies. Unfortunately, the non-negligibledegree of asymmetry in the peak shapes (see Figs. 1, 2) hampers an unambiguous definitionof position P, width W and height H. Therefore, we restrict ourselves to an analysis of P andW, for which the above moments-based definition can provide unambiguous and well-definedresults. Verification of the proxies for P and W will also validate the proxies for H as it is theformer two that enter the derivation of the peak-height-related proxies (see section 4 in theSI).

Figure 3 a reports peak positions P and peak widths W for the experiments (encircledstars), as well as for the ab initio parametrized mean-field models (CHE: crosses, CHE+DL:squares) with quadratic coverage dependencies (see Ref. [13] and section 6 in the SI), at anion concentration of ca = 0.5 mM. P and W are computed from the mean and 2nd centralmoment of the experimental and theoretical curves of Csorp

pseudo(ΦE) according to

P =

∫Csorp

pseudo(ΦE)·ΦEdΦE∫Csorp

pseudo(ΦE)dΦE(43)

W = 2

√ ∫Csorp

pseudo(ΦE)·(ΦE−P)2dΦE∫Csorp

pseudo(ΦE)dΦE. (44)

We then evaluate the accuracy of the proxies by adding to each CHE data point(PCHE,WCHE) the approximate expressions for the DL-induced peak shift (eq. (25), horizontalvectors in Fig. 3 a) and the width change (eq. (34), vertical vectors). The individualcontributions to the DL-induced width change (first, second and third term in eq. (34)) areillustrated as dashed, dotted and solid vertical arrows, respectively. All estimates are evaluatedat half coverage (0.5θmax

a ) and with K = 0.27 in eq. (34) (see section 4 in the SI for thedetailed derivation of this value). All approximate results that derive from adding to eachCHE data point the respective, perturbative shifts are plotted as filled circles and denoted as”CHE+↑”. Evidently, these estimates fall close to the CHE+DL solutions (squares) whichprovides confidence in the accuracy of the derived proxies.

Shaded regions enclosing experimental and theoretical data points across the halideseries support a strong overall dependence of peak widths W on the peak position P for theexperiments, which is only fully captured by theory when including the DL energetics. Note,that the inclusion of only the first two terms of eq. (34) (dashed and dotted vertical arrows)leads to identical widths across the halide series. Thus only the latter term, which varies withthe potential drop in the double layer (∆ΦE), seems responsible for the variation of the halidepeak shapes, in line with the results in Fig. 2 and the conclusions of Ref. [13].

Figure 3 b plots the theoretical values for P and W when changing the ion concentrationby two orders of magnitude around the experimental value of ca = 0.5 mM (ca = 0.5 ·10±2 mM). While the CHE method only yields trivial Nernstian shifts in the peak position,the CHE+DL theory and their approximation (CHE+↑) lead to non-trivial position and widthchanges (cf. Fig. 2). As analysed subsequently, the former is related to the fact that halides onAg(111) exhibit a normal work function change (see Fig. SF1 in the SI). Furthermore, as thework function change per adsorbate increases with the coverage for these systems (see Fig.

14

SF1 in the SI) eq. (37) results in a decreasing peak width with increasing ca as ddθaδa<0, which

explains the behaviour observed in Fig. 3 b.Figure 3 b includes as well the PZC of Ag(111) at half maximum coverage. At these

potentials most DL-induced alterations to the CHE results are expected to vanish as there arenearly no interfacial fields present. Indeed, the concentration-dependencies of the CHE+DLand CHE+↑ methods extrapolate nicely to the CHE results in this potential regime, whichunderlines again that the CHE results are intrinsically only valid at the PZC (cf. Figs. 1, 2).

Halides on Ag(111) exhibit significant coverage dependencies (e.g. for C θa0 and δa, see

Fig. SF1 in the SI), however, their approximate treatment by evaluation at half maximumcoverage and their complete neglect (a = 0; C θa

0 = C0 in eq. (15)) does not prevent the proxiesfrom reproducing the CHE+DL results even in a semi-quantitative way. This is even moreso surprising, as |δa| ∈ 0.2 . . . 0.5 for halides on Ag(111) and thus not small compared to 1,which is assumed at certain steps of the derivations (see SI).

Therefore, we expect these proxies to remain valid also for other systems, where theapproximations are typically less problematic than for the here discussed showcase system.

3.3. Analysis of the proxies

An insightful and accessible interpretation of the physical meaning and origin of the derivedproxies in tables 1 and 2 can be achieved by analysing a Frumkin model, where the adsorptionenergy Gθa

ads,0 is assumed to vary only linearly with the coverage, and where adsorbate dipolesare assumed as coverage-independent. Such systems show no shape dependence on log(ca),but exhibit Non-Nernstian peak shifts. The according analysis is not only interesting due toits straightforward interpretability, but also as it can be treated fully analytically. We refer thereader to section 5 in the SI for details.

3.3.1. The quantity δa As already stated the quantity δa is a unitless measure of DL effects,and at the same time it relates trivially to the electrosorption valency via elMFT

a = qa(1 + δa).Its meaning is thus very accessible and becomes even more clear when we reformulate e.g.the traditional chemical reaction equation for the electrosorption of Cl−

Cl− + ∗ → Cl ∗ + e− , (45)

into a more general electrosorption equation, namely

Cl− + ∗ → ClδCl ∗ + (1 + δCl)e− (46)

with δCl being identical to the according quantity used throughout this work and withδCl ≈ −0.5 [16]. Reaction equation (46) thus clarifies that some negative charge remainson the adsorbed Cl and thus only (1 + δCl) ≈ 0.5 electrons are actually exchanged upon theelectrosorption process. Note, however, that this is only a simplified picture, as above charge-conserving reaction equations are problematic in view of a fully grand canonical picture wherecharge is not a conserved quantity.

15

In general, δa discriminates between systems with normal and anomalous work functionchange: δa is negative for adsorbates with a normal work function change ( d

dθaΦθa0 < 0 for

qa > 0 and ddθa

Φθa0 > 0 for qa < 0). On the other hand δa > 0 for adsorbates with anomalous

work function change (cf. eq. (17)). Thus it makes a significant difference, whether adsorbatesbehave normally or anormally in this respect.

3.3.2. Peak positions P Indeed the work function changes due to the adsorption of halideson metallic surfaces can be surprisingly complex [73–75], and it is thus certainly interestingto analyse what these results mean in terms of their electrosorption behaviour.

Based on the approach outlined in section 7 of the SI, we use published theoreticaldata [74] for Gθa

ads,0 and ddθa

Φθa0 in combination with our own results for halides on Ag(111), in

order to estimate the values of δa and ∆ΦE at interfacial capacitances C0 ≈ 40µF/cm2 for Cl,Br and I and for a wide range of substrate materials (see section 7 in the SI for more details).This allows us directly to estimate their Non-Nernstian peak shifts as well as the accuraciesof the CHE peak positions (eq. (30)), which depend only on the sign and order of magnitudeof δa and the peak position relative to the PZC.

Figure 4 plots the derived numerical values for δa and lMFTa = −(1 + δa) as a function of

the vacuum work function of the substrate material. Evidently, for many adsorbate-substratecombinations, |δa| takes values that are significantly different from zero. In addition, negativeand positive values are observed, in line with the fact that halides can induce normal andanomalous work function changes [73–75].

As a result of eqs. (28),(36), adsorbates with normal work function change exhibit largerpeak shifts with log(ca) than expected from the ideal Nernst behaviour. The reverse is truefor adsorbates with an anomalous work function shift. This is perfectly in line with theinterpretations gained from the Frumkin model analysed in the SI, and the above intuitivechemical reaction equation (46): With δa being directly related to the electrosorption valencylMFTa =

qae (1 + δa), non-zero δa values are directly linked to a rescaled driving force for the

adsorption process. Thus, systems with normal(anomalous) work function change exchangeless(more) electrons upon adsorption, and thus necessitate larger(smaller) applied potentialsto drive the adsorption process. While these results hold in general, e.g. independent on thecharge of the adsorbate, we exemplify it in Fig. 4 for the negatively charged halides by havingincluded also the shift with log(ca) on the right y-axis (eq. (36)), which evidently exhibits thedescribed dependence and demonstrates the expected magnitude of Non-Nernstian behaviourfor the considered systems (Nernstian: d

d log(ca) P = 60mV/decade).

The combination of the δa values with the reported [74] adsorption energies Gθaads,0 allows

to assess directly the expected CHE errors in the peak positions P− PCHE (eq. (30)) as well asthe peak shift with DL capacitance d

dC0P (eq. (39)). Note that both proxies are trivially related

via the DL capacitance C0. The DL potential drop is approximated via ∆ΦE ≈ PCHE − Φθa0

and with PCHE from the theoretically reported values of Gθaads,0 (eq. (29)) and the PZC Φ

θa0

from experimentally measured values, which are available for the major part of the studiedsubstrates [76] (see SI, section 7). The results are plotted in Fig. 5 for an ion concentration in

16

normal

anomalouswork functionchange

(eV)

(-1.8)

(-1.4)

(-1.0)

(-0.6)

(-0.2)

an

d (

)

Figure 4. Numerical values for δa and lMFTa = −(1 + δa) for Cl, Br, and I on (111)

surfaces of fcc metals, as derived from published theoretical data [74] (see section 7 in the SI).Positive and negative values of δa classify systems according to their work function change(anomalous/normal) and are directly related to peak shifts d

d log(ca) P (eq. (36)) which we reporton the right y-axis (Nernstian shift: d

d log(ca) P = 60mV/decade).

solution of ca = 1 M. Evidently, the CHE error P − PCHE can easily vary in a range ±0.2 V+,which is far from negligible and on the same scale as inherent DFT inaccuracies. Furthermore,while the latter is often transferable (e.g. between substrates), Fig. 5 does not suggest thisfor the DL-induced shifts. However, a certain degree of correlation in the data points exists,which might enable a more accessible understanding in the future.

Note as well, that the magnitude and sign of peak shifts ddC0

P (eq. (39)) follow the samebehaviour of the CHE error and thus depend on the product of δa and ∆ΦE. As a result, themagnitude and direction depends on how far the electrosorption peak is positioned left or rightof the PZC, and whether the adsorbates induce a normal(anomalous) work function change

(δa

(>)<0, cf. Fig. 5).

3.3.3. Peak shapes, H and W Above considerations with respect to the rescaled driving forcedue to non-zero δa indicates that δa should as well have an impact on the peak width. Indeed,the analysis of the Frumkin model suggests (see section 5 in the SI), that this behaviour iscaptured by the first term in eq. (34) that is proportional to the product of CHE width WCHE

and δa. This term induces a larger(smaller) peak width than expected from the CHE resultsfor systems with normal(anomalous) work function change in line with the reduced(increased)driving force considerations.

The second term in eq. (34) is always negative, representing thus a peak contraction.This is tentatively related to the fact that in a grand canonical setup, the DL potential dropadapts to compensate for adsorbate dipoles. The dipole-field interaction with the so-inducedcounter-oriented field is always ”attractive” and always reduces the total energy irrespectiveof the sign of charge, dipole or DL field, thus explaining in sign and magnitude the second

+ Extreme values as 0.8 V for I on Pt are at very low potentials, and outside the stability window of water alsoin alkaline conditions.

17

Pb

Ag

Cu

Ni

Au

Pd

Pt

ClBrI

40

Figure 5. CHE peak accuracy P − PCHE and peak shift as a change of the DL capacitanced

dC0P. The contour lines are in logarithmic spacing with the indicated values in V for P−PCHE

and in V/40µFcm2 for d

dC0P (Note that both proxies are identical up to a multiplicative factor, cf.

eqs. (30), (39)). Positive regions in the plot are colored in light red. Data points are obtainedby combining experimental [76] and theoretical results [74] (see section 7 in the SI).

term in eq. (34) (see also SI). Note that these first two terms in eq. (34) are independent on∆ΦE and thus represent a renormalization of the CHE results, as e.g. plotted in Fig. 3 a.

Finally the third term in eq. (34) takes the change of the adsorbate dipole with thecoverage into account via the term d

dθaδa, which is directly related to the curvature of Φ

θa0 . It

is only this third term in eq. (34) that varies with ∆ΦE and thus the only one that inducesNon-Nernstian shape changes. This clarifies that alterations of the peak shape with changingexperimental conditions, e.g. via ca, are mainly driven by the coverage-dependence of theadsorbate dipoles.

Similarly as for δa, the sign of ddθaδa – which also occurs in the other shape-related

proxies in tables 1 and 2 – depends on whether systems show a normal or an anomalous workfunction change and whether the magnitude of the adsorbate dipole increases or decreaseswith coverage. For systems with a normal work function change, d

dθaδa<0 when the magnitude

of the adsorbate dipole increases with coverage as e.g. for halides on Ag(111). Respectivealternative statements can be made for the 3 other possible scenarios thus allowing to explaina variety of non-trivial peak shape changes with changing experimental conditions.

For reasons of length, we will only take a closer look at eq. (41) in table 2, as itdescribes how peak heights can change with changing C0. This is not only important for abetter understanding of experiments, but also as it allows to estimate potential errors made intheoretical calculations with inaccurate DL capacitance C0, e.g. in implicit solvation models.

Note that M in eq. (41) is a unitless scaling factor where the numerical value of allcontributing terms (eq. (42)) can be estimated in a straightforward way (e.g. the ratio betweenpeak height H and the baseline DL capacitance C0). Furthermore, it is gratifying that alldependencies on K have vanished, which makes us confident that the derived equation canbe used to discuss any localized peak, without knowledge e.g. of the amount of coverage

18

change that it encloses. All quantities have then to be interpreted as quantities at the respectiveinterface composition. Note also, that eq. (41) suggests that the height of a CV peak changesexponentially with the capacitance, which indicates possible dramatic shape changes inducedby moderate changes in C0. Furthermore, the highest sensitivity to changes in C0 is observedfor high and narrow peaks. Equations (41) and (42) can be analysed for certain limiting cases:

(i) Case: Da =δaC0→ 0; d

dθaDa → 0

(small adsorbate dipoles, non-zero C0)Then d

dC0H → 0, consistent with the absence of capacitive charging effects.

(ii) Case: C0 → 0; Da 6→ 0(small C0, non-zero adsorbate dipoles)Then d

dC0H 6→ 0 as M 6→ 0. Note that in this limit M is largely independent on C0, with

the central term in eq. (42) vanishing.

(iii) Case: Da → 0; ddθa

Da 6→ 0 (small overall dipole magnitude, however dipole magnitudestill varies with coverage)Then

ddC0

H ≈ σH2∆ΦE|qa |

ddθa

Da. (47)

Note that very narrow and high peaks (large H) studied away from the PZC (large |∆ΦE |)can thus be extremely sensitive to small changes in the DL capacitance C0. In addition,the height/shape change with C0 is expected to switch signs, whenever the system isstudied below or above the PZC (sign of ∆ΦE). These results clarify, that even in thecase of small adsorbate dipoles (Da → 0), and thus naively expected small double layereffects, a non-vanishing coverage dependence of the adsorbate dipoles ( d

dθaDa , 0) can

lead to significant peak shape changes.

(iv) Case: Halides on Ag(111)Here, all terms in eq. 41 are of similar magnitude (theoretically), and thus the(theoretical) behaviour as a function of C0 is extremely complicated and non-trivial,which we will not further elaborate here, but potentially come back to in subsequentwork.

4. Conclusions

In this work we showed how all properties of thermodynamic CV peaks can in principlebe mapped back to the coverage and potential dependence of the underlying fully grandcanonical energetics. Note that eq. (15) represents in essence the first two terms in apossibly more complex Taylor expansion around the PZC of the interface energetics, withthe difference between applied potential and the PZC (ΦE − Φ

θa0 ) as the relevant expansion

variable. As a result, any possible further potential dependence (e.g. of the adsorbate dipoles,or the interfacial capacitance) would naturally induce additional and potentially higher ordercontributions, which can not be analysed analytically considering eq. (15). However, alreadyat the present level of theory (quadratic equation in (ΦE − Φ

θa0 )) the results are not any more

19

amenable to insightful analysis. On the other hand, when restricting ourselves to the casewhere the quadratic term is neglected (a = 0 in eq. (15)) a straightforward analytical analysisis possible, where only knowledge of the coverage-dependence of quantities evaluatedat the PZC is necessary. The relevant quantities exhibit a straightforward interpretatione.g. via adsorbate dipoles and can be accessed directly e.g. from ab initio calculations.This approximation corresponds to systems where C θa

0 is largely coverage and potentialindependent. Note, that all mathematical terms that we referred to with ”adsorbate dipole” arerelated fundamentally to coverage-induced work function changes and might also arise fromother physical realities, e.g. a work function change due to the alteration of the interfacialwater structure, which only changes the physical cause of these terms in the equations but notthe derived results.

Using this and sensible additional approximations, as detailed in the SI, we were ableto derive a list of proxies, that allow an insightful analysis and understanding of how ionconcentrations and other electrolyte properties can lead to non-trivial changes in the observedCV peak positions and shapes. The successful test of these proxies on Ag(111) in halidecontaining solutions, where some of the assumptions made along the derivations are expectedproblematic, convinces us that they do indeed hold as well for other systems and providevaluable new information for an appropriate analysis of experimental results.

Among other things a markedly different behaviour for adsorbates with normal andanomalous work function is predicted. Furthermore, we have shown, that a variation in thecapacitance C0 can easily induce complex peak shifts and shape changes (cf. table 2). Asa result, experimental CV measurement without controlled and known capacitance value C0,which is often not reported, can be hardly analysed and understood in full completeness.In addition, the provided proxies allow us to pinpoint the main challenges of accuratelypredicting thermodynamic CVs via ab initio methods:

These are first and foremost the intrinsic inaccuracies of the adsorption energetics, asthese values not only directly affect the predicted peak position but also directly impact theaccuracy of the DL potential drop ∆ΦE, and thereby any related quantity. Interestingly,however, such inaccuracies do not affect the dependencies on log(ca) (cf. table 2).The other relevant quantities are the PZC as well as its change with coverage and theinterfacial capacitance. Evidently, also here, ab initio methods struggle, either due to overlysimplified atomistic models (e.g. neglect of explicit interfacial water), and/or due to inherentinaccuracies of implicit solvent models.

While fully acknowledging these existing problems of theoretical modelling and notproviding solutions to these, the present work provides important insight in how the describederrors transfer to CHE+DL (fully grand canonical) results. In addition, the proxies allowerror estimates for the more simple, but widely used, CHE energetics and thus clarifies caseswhere this method is problematic (cf. Fig. 5). In addition, we hope this work will help in thefuture to understand better the intricate details of thermodynamic CVs, e.g. their change withexperimental conditions, and allow the reconstruction of atomistic properties of the interfacethat can be resolved and studied in a straightforward way via theoretical simulations.

20

Data availability statement

All relevant data of this work is either compiled in the SI or available from the citedReferences.

Acknowledgements

The authors acknowledge financial support through the EuroTech Postdoc Programmewhich is co-funded by the European Commission under its framework programme Horizon2020 and Grant Agreement number 754462 as well as support from the DeutscheForschungsgemeinschaft (DFG, German Research Foundation) under Germany’s ExcellenceStrategy - EXC 2089/1-390776260. This work was supported by a grant from the GaussCentre for Supercomputing e.V. (www.gauss-centre.eu) by providing computing time throughthe John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUWELS[77] at Juelich Supercomputing Centre (JSC).

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SUPPORTING INFORMATION:Thermodynamic Cyclic Voltammograms: Peak Positionsand Shapes

Nicolas Georg HormannChair of Theoretical Chemistry and Catalysis Research Center, Technische UniversitatMunchen, 85748 Garching, GermanyFritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany

E-mail: hoermann@fhi-berlin.mpg.de

Karsten ReuterFritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany

1. The interfacial capacitance

Evidently, implicit solvent models behave similarly as classical interface models such as theGuy-Chapman theory and are thus, in general, only approximately correct [1,2]. Furthermoresmall variations between different implementations exist. However, for high electrolyteconcentrations most implicit solvent models behave very similar, namely that the interfacialcapacitance is dictated by the relative spatial position of quantum-mechanical region and theonset of the implicit solvent model (dielectric). This can be easily rationalized by the factthat the major potential drop between electrode and bulk solvent occurs within this ”vacuumgap” that mirrors the plane of closest approach of solvent molecules to the surface (Helmholtzplane) in real systems. In the high concentration regime, electrolyte counter charges residevery close to the electrode surface and the Helmholtz plane and thus – considering the highdielectric constant of water – induce only a minor additional potential drop inside the solventregion. As a result the total DL capacitance typically only weakly depends on e.g. differentelectrolyte models, in the high electrolyte concentration regime. In addition, the potentialdependence of the (combined) DL capacitance is rather small compared to its absolutemagnitude as the variation with potential is mainly due to electrolyte effects e.g. from amodified Poisson-Boltzmann model [3, 4]. In selected real systems, under extremely cleanand controlled conditions, some more complex behaviour was observed which is most likelyrelated to the intricate response of interfacial water to the application of a potential [5,6]. Suchdetails will be ignored in this work, as they go beyond the capabilities of common, implicitsolvent models, and would also hamper the analytic treatment, that is central to this work.

2

2. Electrosorption peak position P and width W

2.1. Analytic expressions for P and W (neglect of non-ideal electrosorption valencies)

As clarified in the main text, CVs are typically dominated by the pseudocapacitance peakCMFT,sorp

pseudo = −elMFTa

ddΦE

θa due to electrosorption. As the electrosorption valency enters onlyas a multiplicative prefactor, we focus first on the quantity C = σ d

dΦEθa , with σ =

−qa|qa |

. Theintroduction of σ ensures that C is positive (for physically sound, i.e. convex free energylandscapes), independent on adsorbate charge qa.

The position and shape of C can be computed in a straight forward way from its moments.The nth moment is defined as

〈ΦnE〉 = 1

N

∫ ∞

−∞

[σ d

dΦEθa(ΦE)

]Φn

EdΦE (1)

with the normalization N =∫ ∞−∞

[σ d

dΦEθa(ΦE)

]dΦE. By virtue of the equilibrium condition

ΦθaE = ΦE(θa) ↔ θa(ΦE), all such integrals can be rewritten as integrals over the coverage θa,

as:

σ

∫ ∞

−∞

ΦnE

[d

dΦEθa(ΦE)

]dΦE (2)

= σ

∫ θa(ΦE=∞)

θa(ΦE=−∞)

(ΦθaE

)ndθa = (3)

=

∫ θmaxa

0

(ΦθaE

)ndθa (4)

and therefore

〈ΦnE〉 = 1

θmaxa

∫ θmaxa

0

(ΦθaE

)ndθa (5)

For the first moment, and thus the peak position P, we obtain

P = 〈ΦE〉 ≈ 〈ΦCHEE 〉 − 〈∆a〉, (6)

with ΦCHEE and ∆a as defined in the main text.

Thus the peak position P = 〈ΦE〉 is shifted from the CHE result 〈ΦCHEE 〉 by the per-

adsorbate dipole-field interaction averaged over the coverage 〈∆a〉. The squared width W2 canbe estimated as

W2 ∝ 〈(ΦE − 〈ΦE〉)2〉 (7)

= 〈(ΦCHEE − 〈ΦCHE

E 〉)2〉 − 〈2ΦCHEE

(∆a − 〈∆a〉

)〉 + 〈(∆a − 〈∆a〉)

2〉 (8)

If the dipole-field interaction ∆a is independent on the surface coverage θa, then (∆a −

〈∆a〉) = 0∀θa, and therefore the latter terms vanish. In this case the expected peak widthcorresponds identically to the CHE result, with no impact of the DL energy terms.

3

2.2. Analytic expressions for the concentration dependence of P and W (neglect of non-idealelectrosorption valencies)

Furthermore, it is straight forward to study the non-Nernstian peak shifts due to DL-relatedenergy contributions via the identities

dd log(ca)〈Φ

nE〉 = 〈nΦn−1

E∂ΦE

∂ log(ca)〉 (9)∂ΦE

∂ log(ca) =β−1 ln(10)qa(1+δa) ≈

β−1 ln(10)qa

(1 − δa) , (10)

where β−1 = kBT . The latter approximation is valid up to linear orders in δa. Evidently,the first term in (10) describes the perfect Nernstian behaviour of the CHE energetics. Theexpected peak shift is thus

dd log(ca) P = d

d log(ca)〈ΦE〉 =β−1 ln(10)

qa(1 − 〈δa〉) , (11)

with the brackets signifying the above defined averages over coverage. In perfect analogy, weobtain an influence of log(ca) on the peak width W of

dd log(ca)W = 1

2Wd

d log(ca)W2 = 1

W 〈(ΦE − 〈ΦE〉)( ∂ΦE∂ log(ca) −

∂〈ΦE〉

∂ log(ca) )〉 (12)

= −β−1 ln(10)

qaW 〈(ΦE − 〈ΦE〉)(δa − 〈δa〉

)〉 . (13)

(ΦE − 〈ΦE〉) is typically a monotonously decreasing(increasing) function of θa for qa > 0(qa < 0), with the slope proportional to the peak width W. Considering the prefactor, thissuggests that peak broadening or narrowing is only given by the behaviour of δa, where thewidth is expected to increase(decrease) with log(ca), whenever (δa−〈δa〉) increases (decreases)monotonously. This is in perfect analogy to the formulas derived below based on a simpleransatz which finds the width to change with ∝ d

dθaδa, as reported in the proxies in the main text

in table II.

3. Analytic expressions for P and W (with non-ideal electrosorption valencies)

While above formulas only analyse the electrosorption peak C = σ ddΦE

θa, we have yet toinclude in the analysis the electrosorption valency, which yields finally the experimentallymeasurable CV pseudo capacitance peak as CMFT,sorp

pseudo = −elMFTa

ddΦE

θa. It turns out that CVpeaks that include this non-trivial prefactor can be evaluated in perfect analogy with theapproach above, with

〈ΦnE〉CV = 1

N

∫ ∞

−∞

CMFT,sorppseudo (ΦE)Φn

EdΦE (14)

with the normalization N =∫ ∞−∞

CMFT,sorppseudo (ΦE)dΦE. Rewriting this as integrals over the

coverage θa, as above, yields:

4

∫ ∞

−∞

ΦnECMFT,sorp

pseudo (ΦE)dΦE (15)

= −

∫ θa(ΦE=∞)

θa(ΦE=−∞)Φn

E(θa)elMFTa dθa = (16)

= − σ

∫ θmaxa

0Φn

E(θa)elMFTa dθa (17)

and therefore

〈ΦnE〉CV =

∫ θmaxa

0Φn

E(θa)ω(θa)dθa (18)

ω(θa) =lMFTa∫ θmax

a0 lMFT

a dθa, (19)

with the only difference to above, that a normalized weighting function ω(θa) isintroduced in the integrals. Whenever the electrosorption valency is independent on coverageand potential (i.e. coverage independent adsorbate dipoles) ω is a constant and identicalresults to above will be obtained for all moments, in particular peak position and width. Onthe other hand the overall peak height is affected due to the altered number of exchangedelectrons. In addition, whenever the electrosorption valency is independent on the potentialthe introduced weight ω is independent on the bulk ion concentration ca, which allows a directanalysis of how the peak position P = 〈ΦE〉CV and its second moment (=width) changes as afunction of ca. With

dd log(ca)〈Φ

nE〉CV = 〈nΦn−1

E∂ΦE

∂ log(ca)〉CV (20)

and elMFTa = qa(1 + δa) as well as eq. (10), the peak shift is

dd log(ca)〈ΦE〉CV =

kBT ln(10)qa

[ ∫ θmaxa

0 (1+δa)dθa

θmaxa

]−1

≈β−1 ln(10)

qa(1 − 〈δa〉) , (21)

identical to the previous result in eq. (11). In analogy to above the width change isd

d log(ca)W = 1W 〈(ΦE − 〈ΦE〉CV)( ∂ΦE

∂ log(ca) −∂〈ΦE〉CV∂ log(ca) )〉CV (22)

= −β−1 ln(10)

qaW 〈(ΦE − 〈ΦE〉CV)(δa − 〈δa〉

)〉CV , (23)

which leads to identical conclusions as before (note however the weights ω whenperforming the averages 〈·〉CV).

Note, that these general conclusions hold, for potential and coverage independentinterfacial capacitance (as assumed), and of course only if the interfacial capacitance is leftunchanged by the change in the bulk ion concentration ca.

5

4. Derivation of approximate expressions for P and W

As above derivations show, it is difficult to gain insightful proxies for understanding better theeffect of DL energy contributions on CVs keeping a fully rigorous description. Thereforewe fall back to a simpler ansatz that circumvents the intricate coverage dependencies inabove integrals. We proceed with the formulas given in the main text, keeping track of thecoverage dependencies in the derivations. Instead of, however, relying on averages w.r.t.coverage as done in the previous 2 sections, these averages are approximated by evaluatingthe respective coverage-dependent expressions at some intermediate coverage, e.g. the halfmaximum coverage. As will be shown, above derived general conclusions prevail also in thissimpler ansatz. As we had already made the approximation that the double layer capacitanceC θa

0 should not depend on θa, we will simply refer to it by the term C0 in the following.

4.1. Peak position P

The CHE peak position PCHE = Φθa,CHEE is thus written as (see also main text),

PCHE = − Gqa(24)

G = Gθaads,0 + θa

ddθa

Gθaads,0 − qaΦ

expa,eq (25)

− β−1 ln (ca) + β−1 ln(

θaθmax

a

/ (1 − θa

θmaxa

)), (26)

and we implicitly assume that G is evaluated at e.g. θa = 12θ

maxa . Indeed, for a fully symmetric

peak (e.g. within the Frumkin approximation, as elaborated below), this result is correct andequivalent to explicit treatment via the integral formulas of above.

As a result in this simplified approach, we obtain for the peak position P:

P ≈ PCHE − δa∆ΦE ,

where the potential drop in the double layer ∆ΦE can be expressed as

∆ΦE = ΦθaE − Φ

θa0 = P − Φ

θa0

≈ ∆ΦCHEE = PCHE − Φ

θa0 . (27)

Evicently, ∆ΦE is the potential difference between peak position P and the PZC of thesurface (Φθa

0 , as e.g. approximated from the PZC of the half covered surface or simply thePZC of the clean surface, which is often a close enough guess). Non-Nernstian changes of thepeak position with log(ca) are thus trivially introduced via the change of the potential drop inthe double layer ∆ΦE with log(ca),

dd log(ca)∆ΦE ≈

d PCHE

d log(ca) =β−1 ln(10)

qa, (28)

which leads to

dd log(ca) P =

β−1 ln(10)qa

(1 − δa) (29)

6

4.2. Peak width W

As shown above in all generality, the peak width is not affected by the inclusion (or not) ofthe electrosorption valency term in the expression for CMFT,sorp

pseudo (if lMFTa does not vary strongly

with coverage) from which we would naturally derive the peak width. We therefore derive thepeak width from C = σ d

dΦEθa as one over the derivative of the inverse function, namely

C =σ

ddθa

ΦθaE

≈σ

ddθa

Φθa,CHEE − d

dθa∆a

. (30)

As C must integrate to a constant (θmaxa ), we can approximate the peak width W via

W = K/C where K is a constant and C is the peak height. Under these approximations,the peak width can be written approximately as a sum of the CHE peak width and a widthcontribution due to the DL energetics with

W ≈ WCHE + WDL (31)

WCHE ≈ σK( ddθa

Φθa,CHEE ) (32)

WDL ≈ − σK · ddθa

∆a

≈ −WCHE · δa −|qa |K

AsiteC0· δ2

a − σK ddθaδa · ∆ΦE (33)

Note, how eq. (32) is in line with the idea that the peak width is related to the secondderivative of the free energy landscape in θa (and the peak height thus with its inverse).

The sign and magnitude of the last term in eq. (33) depends on the field in the doublelayer, and is therefore the only one that contributes to the change in peak width with log(ca).With ∆ΦE ≈ ∆ΦCHE

E this change is thusd

d log(ca)W ≈ K · β−1 ln(10)|qa |

· ddθaδa . (34)

The numerical value of K depends on the exact definition of W and C. As shown below,when we estimate the peak width from the second central moment and use the peak height Cat half maximum coverage, then K ≈ 0.8 ·θmax

a (≈ 0.27 for the case of halides on Ag(111) withθmax

a = 0.33 [7]). With this value for K and in combination with the given proxies evaluatedat half maximum coverage, the peak behavior of halides on Ag(111) can indeed be replicated(see main text).

Similarly as for the peak position, changes with log(ca) are only related to quantitiesrelated to the adsorbate dipole (here d

dθaδa), and are not affected by the magnitude and direction

of interfacial fields. Non-vanishing ddθaδa values are obtained for adsorbates with coverage-

dependent dipoles, aka a non-vanishing curvature of the PZC as a function of the coverage.

4.3. Peak height

Along the lines of the previous section, CV peak heights H = Csorp,MFTpseudo are obtained in a

straight forward way by multiplication of C with |elMFTa | via

H ≈ |elMFTa | KW (35)

7

With eq. (34), the change in peak height H with log(ca) is thus:d

d log(ca) H ≈ − |elMFTa | K

W2dW

d log(ca)

≈ − HW ·

β−1 ln(10)|qa |

· ddθaδa . (36)

4.4. Analysis of electrolyte effects: The impact of the DL capacitance

In order to study the the effect of varying the DL capacitance C0, we have to recall thatδa is related to C0 via δa = DaC0, the product of the adsorbate-dipole-related quantityDa =

Asiteqa

ddθa

Φθa0 and the DL capacitance C0. Therefore e.g.

ddC0δa = Da =

δaC0, (37)

which is independent on C0 within the given approximations. Thus, it is straight forward toderive from above formulas

ddC0

P ≈ − ∆ΦEδaC0, (38)

when neglecting higher order terms in δa.As only the DL contribution to the peak width can be affected by a change in C0 we can

directly derive

ddC0

W ≈−WCHE −

2|qa |KδaAsiteC0

− σKd

dθaδa

δa∆ΦE

· δaC0, (39)

for small δa. Note, that this proxy is independent on C0 as the dependence on C0 cancelsin all individual terms. Note also, that whenever the magnitude of the central term is notimportant this simplifies to

ddC0

W ≈ 1C0

WDL , (40)

which clarifies how experiments at different C0 can give direct insights into theimportance of DL effects, and allow a direct estimate of the respective quantities.

Equivalently, we obtain for the peak height change

ddC0

H = ( ddC0|elMFT

a |) KW + |elMFT

a |( ddC0

KW ) (41)

≈|qa |

|elMFTa |

δaC0|elMFT

a | KW − |elMFTa |2 K2

W2 ·

ddC0

W

|elMFTa |K (42)

≈|qa |

|elMFTa |

H δa

C0− H

ddC0

W

|qa |K

. (43)

With eq. (39) and the assumption W ≈ WCHE the latter term inside the brackets can besimplified to

−Hd

dC0W

|qa |K≈

|elMFTa |

|qa |+

2HδaAsiteC0

+ σ H|qa |

ddθa

δa

δa∆ΦE

· δaC0, (44)

8

leading thus to the final resultd

dC0H =

|qa |

|elMFTa |

H · M · δaC0

(45)

M = 2 +(1 + 2H

AsiteC0

)δa + σ

ddθa

δa

δa

H∆ΦE|qa |

(46)

Note that we have already grouped all variables in eq. (46) in a way that the respectiveterms are unitless and can be estimated in a straight forward way (e.g. the ratio between peakheight H and the baseline DL capacitance C0).

4.5. Determination of K: The relation between peak width and height

For a normalized gaussian peak C = 1√πω

exp(−

(xω

)2), with a peak height of C = 1

√πω

, weobtain for the second central moment

〈(x2)〉 = 1√πω

∫ ∞

−∞

x2 exp(−

(xω

)2)

dx = ω2

2 . (47)

Note that there are different choices for the Gaussian peak width:As an example, one might choose W = 2

√〈(x2)〉 =

√2ω, or equally the FWHM =

2√

2 ln 2ω ≈ 2.35ω. Note, that both choices make the product between peak height C andwidth independent on the shape parameter ω, consistent with the shape-independent areabelow a normalized Gaussian.

Identically, the electrosorption peak C, as defined above, integrates to θmaxa , independent

on its shape, and thus we assume an equivalent condition to hold for electrosorption peaks:namely that the product between peak height and width is a constant K with the value K∗θmax

a ,where K∗ is the respective value for a normalized peak, that integrates to 1 instead of θmax

a . Forthe considered Gaussian peak and a width definition based on the second moment (eq. (47))with W = 2

√〈(x2)〉 =

√2ω we obtain the relation

W · C =√

2√π≈ 0.8 = K∗ . (48)

As the considered halide peaks on Ag(111) integrate to θmaxa = 1

3 , we obtain approximately

K = K∗ · 13 ≈ 0.27 . (49)

As demonstrated in the main text, this parameter indeed leads to very good estimatesfor explaining the differences between CHE and CHE+DL calculations for these systems,thus convincing us that the drastic approximations (e.g. Gaussian shape to estimate K∗) aresufficiently accurate, and can be used in a straight forward way to analyse other systems.

5. The Frumkin Model

An insightful interpretation of the derived formulas for the changes in peak position andwidth can be obtained by analysis of the Frumkin isotherm, which is often used to describeexperimental adsorption peaks. A Frumkin-isotherm-CV peak arises within the presentframework, when the adsorption energy Gθa

ads,0 as well as the PZC Φθa0 are assumed to vary only

9

linearly with the coverage according to Φθa0 = Φ0

0 + Φ′

0θa and Gθaads,0 = G0

ads,0 + G′

ads,0θa. Thiscorresponds to a common first order approximation for adsorbates with mutual interactions,and interfacial capacitances and adsorbate dipoles that are independent on the surfacecoverage. Then a = 0 ↔ C θa

0 = C00 = const. in the defining eq. (15) in the main text for

ε θa and qaδa = const. = AsiteC00Φ

′

0, yielding the expression

ε θa =

modified differentialadsorption energy︷︸︸︷

E

driving force:electrode potential︷ ︸︸ ︷

+qa[1 + δa](ΦE − Φ0

0

)(50)

E = G0ads,0 + qa

(Φ0

0 − Φexpa,eq

)− β−1 ln (ca)︸ ︷︷ ︸

E0

+

(2G

′

ads,0 −q2

a

AsiteCθa0

· δ2a

)︸ ︷︷ ︸

E′

θa (51)

The full Frumkin isotherm with DL interactions is then

θa = θmaxa

[1 + exp

(βE0

)exp

(βE

′

θa

)exp

(βelMFT

a

(ΦE − Φ0

0

))]−1, (52)

with the three variables E0, E′

, and elMFTa = qa(1 + δa) independent on coverage and potential.

It allows a meaningful understanding of non-Nernstian peak shifts and the former two termsin the width-determining eq. (33):

(i) As said already a non-vanishing term δa is equivalent to an electrosorption valency lMFTa

different from the ideal integer value, which thus rescales the total work performed atelectrode potential ΦE by a factor of (1 + δa) (latter term in eq. (50)), leading ultimatelyto the Non-Nernstian peak shifts and a rescaling of the peak width (first term in eq. (33),see also analytical discussion of the Frumkin model below).

(ii) The term E′

(eq. (51)) is traditionally related to adsorbate-adsorbate interactions thatlead to a coverage-dependent adsorption energy. Repulsive interactions are associatedwith positive values and attractive interactions with negative values, leading to peakbroadening and narrowing, respectively. The latter, DL-related term in E

′

correspondsto the central term in eq. (33), which leads always to a peak contraction, which can beinterpreted as an effective, attractive adsorbate-adsorbate interaction of size − q2

a

AsiteCθa0

· δ2a .

Our tentative explanation is as follows: In the given grand canonical framework anadditional adsorbate yields a work function shift according to its dipole ∝ δa. In agrand canonical setup such as the CHE+DL approach, this work function shift needsto be compensated by an according change in the DL potential drop. The associatedcompensating field is opposing the dipole and thus always reduces the total energy.With the compensating field magnitude proportional to the total work function shift,this counter-acting field (per adsorbate) scales linearly with coverage (for coverageindependent dipoles) which yields the ”attractive” DL interaction ∝ −δ2

a , independenton the sign of charge, dipole or DL field.

Note, that although the Frumkin isotherm is already more realistic than the very commonLangmuir-isotherms (no concentration dependence of the adsorption energy), it still suffers

10

from several limitations: As evident from the analytic solution below for CV peak position,height and width, any Frumkin-like electrosorption peak is by construction fully symmetricat variance with many experimental peaks. In addition, while non-Nernstian peak shifts arerecovered, no change in the peak widths with log(ca) can be observed aka explained withinthe Frumkin approximation, due to the assumed independence of adsorbate dipoles on thecoverage and thus eq. (34) identical 0.

5.1. Analytical solution

The Frumkin isotherm given by eq. (52) can be rewritten as:

ΦθaE − Φ0

0 = −1

qa(1 + δa)

[E0 + E

′

θa + +β−1 ln(θaθmax

a

/ (1 − θa

θmaxa

)]. (53)

The CV pseudocapacitance peak is given as the product of the electrosorption valencyand the inverse of the derivative of Φ

θaE w.r.t. θa. Note that as lMFT

a ∝ (1 + δa) is coverage-independent here, its effect is a trivial rescaling of the adsorption peak height while the peakposition and width are not affected. Using θ∗a = θa/θmax for the normalized coverage, we obtainfor the pseudocapacitance peak

Csorp,MFTpseudo = − e lMFT

a

[d

dθaΦθaE

]−1(54)

=[qa(1 + δa)

]2 [E′

+ [βθmaxθ∗a(1 − θ∗a)]−1

]−1(55)

The Frumkin isotherm and the connected Csorp,MFTpseudo are by construction symmetric around

θ∗a = 0.5, such that the peak height H is given by the maximum of Csorp,MFTpseudo with

H =[qa(1 + δa)

]2 [E′

+ 4[βθmax]−1]−1

, (56)

with

E′

+ 4[βθmax]−1 > 0 (57)

for meaningful results. This condition is equivalent to demanding the second derivative(in θa) of the potential energy landscape g θa,MFT

exc (eq. (8) in the main text) being positive. Inperfect analogy to basic thermodynamics discussions of free energy landscapes, any non-convex parts in a free energy are unstable and thus lead to divergencies in the responsefunctions, which is – in the case of electrochemistry – the pseudocapacitance. Therefore,within in the quadratic model which yields an explicit expression for ΦE(θa) = Φ

θaE , it

is straight forward to analyze when and whether phase-transition-like divergencies in thepseudocapacitance occur, namely whenever d

dθaΦθaE becomes zero.

For the symmetric Frumkin model, the maximum of the pseudocapacitance is obtainedat potentials P with

11

P = −1

qa(1 + δa)

(E0 + 0.5E

′

θmaxa

), (58)

which directly rationalizes the Non-Nernstian shifts for δa , 0 as E0 depends on log(ca).The peak width W can be computed explicitly from the full width of half maximum (FWHM)which can be computed from the coverage-dependent expression for Csorp,MFT

pseudo (eq. (55)) andthe peak height H (eq. (56)):

Csorp,MFTpseudo /H = 1

2 (59)

E′

+ 4[βθmax]−1 = 12 (E

′

+ [βθmaxθ∗a(1 − θ∗a)]−1) (60)

E′

+ 8[βθmax]−1 = [βθmaxθ∗a(1 − θ∗a)]−1 (61)

θ∗,±a = 12 ±

√14 −

[βθmax

(E′+ 8[βθmax]−1)]−1 (62)

W = − 2 qa|qa |

(Φθ∗,+aE − Φ

θ∗a=0.5E

)(63)

= 2E′

|qa |(1+δa)

√14 −

[βθmax

(E′+ 8[βθmax]−1)]−1 (64)

+ 2|qa |(1+δa)β

−1 ln

12 +

√14−[βθmax(E′+8[βθmax]−1)]−1

12−

√14−[βθmax(E′+8[βθmax]−1)]−1

(65)

This explicit expressions for W is perfectly in line with the more abstract analysis ofbefore, highlighting that peak heights and widths within the given Frumkin approximation areindependent on log(ca). This is easily visible as only E0 depends on log(ca), which is not anymore present in the expressions for W. On the other hand a shift of the peak position (eq.(58)) w.r.t. log(ca) is obtained via the concentration dependence of E0 as:

dlog(ca) P = 1

qa(1+δa) ≈β−1 ln(10)

qa(1 − δa) . (66)

Note, however, that both δa =AsiteC0

0qa

Φ′

0 and E′

can vary with the electrolyte properties, asboth are influenced by the interfacial capacitance C0

0.

12

1/12 ML 2/12 ML 3/12 ML 4/12 ML

a,

aa(ML)

a

a(ML) (ML)

a

2nd-order fitsb) c) d)

atomistic interface structures

a)

Figure SF1. a) Atomistic adsorption configurations for halides on Ag(111) in a√

12 ×√

12surface supercells, as used in periodic DFT calculations in an implicit solvent model to obtainto obtain the relevant quantities: b-d) Average adsorption energies per adsorbate Gθa

ads,0, thePZC Φ

θa0 and the interfacial capacitances Cθa

0 for Cl (green circles), Br (red triangles), and I(brown squares) at the different coverages, including 2nd-order polynomial fits (dotted lines).The data points plotted in b)-c) derive from density functional theory calculations in animplicit solvent model (PBE [8] exchange-correlation functional, Quantum ESPRESSO &ENVIRON [9, 10], SCCS implicit solvation model [11, 12], see Ref. [7] for details).

Experiment Theory: MFT & CHE+DL Theory: MFT & CHE

I

Br

Cl

Gads,0:2 Gads,0:1a) b) c) d) e) f)

Cps

eudo

sorp

Gads,0:2:2

C0 :20θa

θa

θa Gads,0:2:2

C0 :00θa

θa

θa θa θaGads,0:1:1

C0 :00θa

θa

θa

Figure SF2. a) Experimental pseudocapacitances Cexp,sorppseudo (per area) for Ag(111) in halide

containing solutions (ca = 0.5 mM) derived from cyclic voltammograms by Foresti et al. [13](see text for details on the performed normalization procedure). b-f) Theoretical values forCMFT,sorp

pseudo for the b-d) CHE+DL and e-f) pure CHE approach. The polynomial orders used to

interpolate the coverage-dependencies for the adsorption energetics (Gθaads,0), the work function

(Φθa0 ) and the DL capacitance (C θa

0 ) that enter the mean field theory expressions are remarkedin the plots. Results for Cl, Br and I are in orange and light and dark blue, respectively.

6. Halide electrosorption on Ag(111): Experiments vs Theory

As shown in Ref. [7], the discussed mean-field model within the CHE+DL approximation andparametrized based on DFT calculations in an implicit solvent model succeeds in reproducingthe experimental cyclic voltammograms for Ag(111) in halide-containing solutions. Fig. SF1plots the relevant quantities of the discussed mean-field model as determined by accordingDFT calculations as reported in Ref. [7] (PBE [8] exchange-correlation functional, Quantum

13

ESPRESSO & ENVIRON [9, 10], SCCS implicit solvation model [11, 12], see Ref. [7] formore details). Fig. SF2 plots the electrosorption related CV peak contribution Csorp

pseudo fromthe experimental results by Foresti et al. [13], and from the MFT models at the CHE+DLand CHE level of theory, parametrized based on the data plotted in Fig. SF1. ExperimentalCV currents were digitized in the relevant halide-electrosorption region from Ref. [13] andnormalized to the currents before the onset of electrosorption at -0.75 V (Cl, Br) and -0.95V (I) vs. SHE. Using the experimentally measured DL capacitances Cref of the pristinesurface, it is thus possible to derive the according scan-rate-independent pseudocapacitancevalues Cexp

pseudo by appropriate rescaling (see Ref. [7] for more details). Fig. SF2 a reportsCexp,sorp

pseudo (conveniently normalized to the area) – the pseudocapacitance related only to specificelectrosorption processes – which is obtained by the additional substraction of the baselinevalue Cref. Panels b-f) report the theoretical results for CMFT,sorp

pseudo (equally area-normalized),with the CHE and CHE+DL mean-field models parametrized using the indicated polynomialorders for the coverage dependence of the adsorption energy Gθa

ads,0, the work function Φθa0 and

the interfacial capacitance C θa0 . The polynomials are determined based in a least-squares fit to

the ab-initio results reported in Fig. SF1 b-c (see also Ref. [7]).Evidently, only the CHE+DL method with a quadratic coverage dependence for Gθa

ads,0

and Φθa0 can reproduce peak heights and shapes across the halide series appropriately. The

description of C θa0 is of smaller importance (cf. panels b and c), justifying thus a-posteriori

the approximations made in the derivations within this work (a = 0 in eq. (15) in the maintext). However, the proxy validation of the main text does compare with respect to modelswith quadratic coverage dependence for all relevant quantities (Fig. SF2 b)

Note that, in agreement with the statements above, the Frumkin-like approximations(linear coverage dependencies) can not explain the observed peak asymmetry, and variationof the peak shapes across the halide series (cf. panel a and panels d,f), which supports, at leastwithin the theoretical model, that the shape(width) differences between the halides deriveprimarily from the differences in the electrosorption potentials and thus ∆ΦE in combinationswith the non-vanishing values of d

dθaδa, where the magnitude of the latter essentially equivalent

for all halides (see Fig. SF1 and Fig. 1, 2 of the main text).

7. Analysis and treatment of literature data

In Ref. [14] Roman et al. report adsorption energies as well as work function changes forhalide adsorbates on the (111) surface of fcc metals. These results were obtained by periodicdensity functional theory calculations, in vacuum and in the low coverage regime. As theauthors report as well the respective studied coverages it is straight forward to derive therelevant variables for the present work. As no values for higher coverages are reported, weuse these low-coverage values in our proxies, which should be sufficiently accurate at leastfor the desired purpose to estimate signs and orders of magnitude of the relevant descriptors.

Table ST1 compiles all relevant adsorbate properties for Cl, Br and I, namely θa – thestudied coverage, Asite – as derived from the reported lattice constants, Gads,0 – the averageadsorption energy at the studied coverages, ∆Φ0 – the induced work function change at those

14

coverages, and the derived estimates for δa. Any not tabulated data in the correspondingreference was obtained by digitizing the relevant figure in Ref. [14]. The estimated values forδa are derived as follows:

δa = DaC0 =AsiteC0

qa

ddθa

Φθa0 is estimated by approximating d

dθaΦθa by the finite difference

∆Φ0/θa. A comparison to our data on halides on Ag(111) indicated that these vacuumestimates for the work function change are larger in magnitude by approximately a factor2. Own calculations showed, indeed that the implicit solvent environment can lead first toan altered work function change also at the SCF level and second to significant structuralrelaxations, which also affect the work function. In particular, both effects seem to impactthe coverage-dependence of the work function change. As we are still analyzing suchobservations in more detail, we will not further dwell on these here. However, as we seethat the implicit solvent calculations for halides on Ag(111) seem to show the correct ordersof magnitude e.g. for the electrosorption valencies, we introduce a scaling factor of 0.5 to”downscale” the vacuum work function changes reported in Ref. [14], in order to estimaterelevant d

dθaΦθa values in the solvent. In addition, we assume an interfacial capacitance of

C0 = 40µF, which finally allows to estimate realistic values for δa, as reported in table ST1(Note the similarity of the so derived values for δa with the ones reported in Ref. [7]).

The estimate of the potential drop in the double layer which allows estimating e.g. theinacccuracy of the CHE method, is obtained in a straight forward way by computing PCHE

and assuming ∆ΦE ≈ PCHE − Φ0, where for Φ0 we take the experimental potentials of zerocharge of the respective clean electrode surfaces, which were collected from Ref. [15] andare reported in table ST2. PCHE can be readily accessed from eq. (24) with the adsorptionenergies as tabulated in table ST1 and the literature reference potentials Φ

expa,eq as tabulated in

table ST3.

15

Table ST1. Relevant adsorbate and substrate properties for halides on (111) surfaces of fccmaterials as reported by Roman et al. [14]. If not tabulated therein directly, the relevant datawas obtained by digitizing the relevant figures in Ref. [14]. ∆Φ0 denotes the adsorbate-inducedwork function change at the studied coverage θa. The estimated values for δa are derived asdescribed in the text.

Relevant properties of Cl adsorbatessubstrate θa Asite (Å2) Gads,0 (eV) ∆Φ0 (V) est. δa

Sr 0.25 15.74 -3.898 0.151 -0.12Ca 0.25 13.25 -3.841 0.076 -0.05Pb 0.25 11.00 -1.543 1.357 -0.75Al 0.11 7.11 -2.018 0.233 -0.19Ag 0.11 7.51 -1.610 0.559 -0.47Cu 0.11 5.72 -1.858 0.108 -0.07Ni 0.11 5.38 -2.027 -0.028 0.02Au 0.11 7.54 -0.974 0.387 -0.33Pd 0.11 6.80 -1.665 0.108 -0.08Pt 0.11 6.85 -1.211 -0.221 0.17

Relevant properties of Br adsorbatessubstrate θa Asite (Å2) Gads,0 (eV) ∆Φ0 (V) est. δa

Sr 0.25 15.74 -3.560 0.273 -0.21Ca 0.25 13.25 -3.458 0.176 -0.12Pb 0.25 11.00 -1.397 1.350 -0.74Al 0.11 7.11 -1.702 0.298 -0.24Ag 0.11 7.51 -1.543 0.48 -0.41Cu 0.11 5.72 -1.746 -0.003 0.0Ni 0.11 5.38 -1.915 -0.182 0.11Au 0.11 7.54 -0.998 0.216 -0.18Pd 0.11 6.80 -1.711 -0.056 0.04Pt 0.11 6.85 -1.326 -0.439 0.34

Relevant properties of I adsorbatessubstrate θa Asite (Å2) Gads,0 (eV) ∆Φ0 (V) est. δa

Sr 0.25 15.74 -3.018 0.423 -0.33Ca 0.25 13.25 -2.939 0.291 -0.19Pb 0.25 11.00 -1.115 1.250 -0.69Al 0.11 7.11 -1.273 0.319 -0.25Ag 0.11 7.51 -1.408 0.319 -0.27Cu 0.11 5.72 -1.600 -0.207 0.13Ni 0.11 5.38 -1.826 -0.439 0.27Au 0.11 7.54 -0.988 -0.06 0.05Pd 0.11 6.80 -1.859 -0.328 0.25Pt 0.11 6.85 -1.576 -0.786 0.61

16

Table ST2. Experimental PZC Φ0 from Ref. [15] on an absolute potential scale in V forthe respective metallic surfaces (data for (111) surfaces if available, experimental accuracytypically ±0.2 V).

Ni Pd Pt Cu Ag Au Pb4.04 4.70 4.78 4.24 3.99 4.96 3.82

Table ST3. Literature [16–18] reference potentials Φexpa,eq for halogens, and electrode potential

of the standard hydrogen electrode (SHE) on an absolute scale.

Reference and electrode potentials in VΦ

expCl,eq Φ

expBr,eq Φ

expI,eq Φ

expSHE

5.80 5.53 4.98 4.44

17

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