Journal of The Electrochemical Society, 162 (7) E73-E83 (2015) E73

A Multi-Paradigm Computational Model of MaterialsElectrochemical Reactivity for Energy Conversion and StorageMatias A. Quirogaa,b and Alejandro A. Francoa,b,c,∗,z

aLaboratoire de Reactivite et Chimie des Solides (LRCS), Universite de Picardie Jules Verne and CNRS, UMR 7314,80039 Amiens Cedex, FrancebReseau sur le Stockage Electrochimique de l’Energie (RS2E), FR CNRS 3459, FrancecALISTORE European Research Institute, 80039 Amiens Cedex, France

In this paper we report a new multi-paradigm modeling approach devoted to the investigation of the electrochemical reactivity ofmaterials in electrodes for energy conversion or storage applications. The approach couples an atomistically-resolved Kinetic MonteCarlo (KMC) modeling module describing the electrochemical kinetics in an active material, with continuum modeling modulesdescribing reactants transport at the active material/electrolyte nanoscopic interface (electrochemical double layer region) and alongthe mesoscale electrode thickness. The KMC module is developed by extending the so-called Variable Step Size Method (VSSM)algorithm (called here Electrochemical-VSSM) and constitutes the first VSSM extension reported so far which allows calculatingthe electrode potential as function of the imposed current density. The KMC module can be parameterized with activation energiescalculated from Density Functional Theory (DFT), and thanks to the coupling with the transport modules, it describes the materialsreactivity in electrochemical conditions. This approach allows us to study how the surface morphology (e.g. distribution of inactivesites, size of the active material particle, etc.) impacts the performance of the electrode. As an application example, we report herea computational investigation of the Oxygen Reduction Reaction (ORR) kinetics in a Pt(111)-based Polymer Electrolyte MembraneFuel Cell (PEMFC) cathode.© The Author(s) 2015. Published by ECS. This is an open access article distributed under the terms of the Creative CommonsAttribution 4.0 License (CC BY, http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse of the work in anymedium, provided the original work is properly cited. [DOI: 10.1149/2.1011506jes] All rights reserved.

Manuscript submitted December 26, 2014; revised manuscript received February 20, 2015. Published April 1, 2015.

Because of the global warming and the fossil fuels depletion, zero-emission electrochemical devices for energy conversion and storage,such as fuel cells and secondary batteries, are called to play a signif-icant role in the sustainable development of the Humanity. The op-eration principles of these devices involve complex competitions andsynergies between electrochemical, transport and thermo-mechanicalmechanisms occurring at multiple spatial and temporal scales. Mul-tiscale modeling techniques can help on establishing correlations be-tween their materials chemical and microstructural properties, opera-tion conditions, performance and durability.1–3 Such correlations areimportant in order to suggest enhanced materials and cells designs.4–7

Several multiscale modeling approaches have been reported so far.The most popular one consists in extracting data from a lower-scaleand using it as input for the upper-scale via their parameters. Forexample, activation barriers for elementary reaction steps can be ex-tracted from Density Functional Theory (DFT) calculations8 and theninjected into Transition State Theory (TST) Eyring’s expressions toestimate the parameters which are used to solve Mean Field (MF)kinetic models simulating the evolution of the reactants, intermediatereaction species and products.9,10

Alternatively, direct multiparadigm multiscale models consist incoupling “on-the-fly” models developed in the frame of differentparadigms. For instance, continuum equations describing the transportphenomena of multiple reactants in a porous electrode may be cou-pled with atomistically-resolved simulations describing electrochem-ical reactions among these reactants. Several numerical techniquesnowadays are well established to develop such a type of models,e.g. coupling Continuum Fluid Dynamics (CFD) with Monte Carlo(MC) approaches, applied to the simulation of catalytic and electro-deposition processes.11–13 Indeed, the MC approach consists in anelegant simulation technique which allows closing the gap betweenthe atomistic and the continuum viewpoints of the mechanisms underinvestigation.

In contrast to Einstein and his statement that “God does not playdice”, Hawking believes that the Universe is being governed by non-determinism (chance) as illustrated by his answer to Einstein “Notonly does God definitely play dice, but He sometimes confuses usby throwing them where they cannot be seen”.14 The MC approach

∗Electrochemical Society Active Member.zE-mail: alejandro.franco@u-picardie.fr

exploits this fact by extensively repeating random executions to obtainresults that can mimic those physical systems where the random char-acter is inherent. Several MC techniques have been developed so far,the most popular one being the Metropolis MC algorithm consistingon performing random swaps from a given arrangement (e.g. spatialdistribution of particles constituting a system) in order to search forthe global minimal energy configuration.15–17

The so-called Kinetic Monte Carlo (KMC) approach is a MCmethod that helps to study the evolution of the systems where theassociated transition rates are known. The KMC approach treats theevents as a simple Markov process, in other words, a series of “mem-oryless” random swaps. The probability density function Pi, that isthe probability to find the system at state i, is described by the masterequation of Markovian form18

d Pi (t)

dt= −

∑j �=i

ki j Pi (t) +∑j �=i

k ji Pj (t) [1]

where kij is the average escape rate from state i to state j. The mainKMC postulate is that during thermal vibrational motion the systemloses memory and has the same probability of finding the escape pathfrom state i to state j for any short time increment �t, leading to anexponential decay statistics.12 This is mathematically described by thePoisson distribution

Pi (t) = ki j exp(−ki j�t

)[2]

In particular, the KMC approach is relevant for investigating elec-trochemical reactions in devices for energy conversion and storagewhich show sensitivity to the microstructure of the active materials.19

In the frame of Lithium Ion Batteries (LIBs), Methekar et al.20 applieda KMC simulation approach to investigate the formation of the SolidElectrolyte Interphase (SEI) layer on an intercalation graphite anode.Van der Ven and Ceder21 used KMC simulations to study lithiumdiffusion through a divacancy mechanism in layered LixCoO2. Yuet al.22 implemented KMC simulations parameterized by molecular-dynamics-based activation energy barriers for Li+ and e− diffusion inTiO2, revealing the central role of the electrostatic coupling betweenLi+ and e− on their collective drift diffusion.

In the frame of the Polymer Electrolyte Membrane Fuel Cells(PEMFCs),23 the description of the relationship between the cata-lyst/electrolyte interfacial charge distribution and its evolution rep-resents a significant issue for the understanding of the redox kinetic

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E74 Journal of The Electrochemical Society, 162 (7) E73-E83 (2015)

Figure 1. Schematics of the classical VSSMprinciples.

processes and the prediction of the associated effective catalyst ac-tivity. The so-called first-principles methods, like the DFT approach,constitute interesting tools to study isolated redox events and can offerpowerful capabilities for predicting materials properties when coupledwith thermodynamics.24 However, the amount of competing eventsand involved species in complex reactions such as the Oxygen Reduc-tion Reaction (ORR) makes these methods unable to provide a widedescription of the redox events in realistic electrochemical operationconditions where both the electric field and the temperature are notzero. In this sense, in order to describe the electrode surface coveragedynamics in relation to processes like adsorption, desorption and reac-tions, the implementation of a KMC method arises as a highly relevanttechnique to get a proper description of surface electrocatalytic events.

Within this sense, Zhdanov23 implemented the KMC approachto study the O2 reduction on Pt(100) finding that the impact in theoverall kinetics of the lateral interaction between adsorbates is notso significant as the O2 adsorption rate. By using KMC simulations,Abramova et al.25 investigated the O2 adsorption on different metallicsurfaces with fcc(100) termination as function of the metal surfacecharge density and the molecular oxygen sticking probability.

The KMC approach was also used to study the electric field fluc-tuations in simple electrochemical reactions kinetic rates due to elec-trochemical double layer (EDL) effects.26 Furthermore, Harrington27

implemented the KMC approach to study the Pt oxide growth.Among the KMC algorithms, one of the most popular is the

so-called Variable Step Size Method (VSSM),12,28 which has beenused to describe surface reactions in catalysis (Figure 1). The VSSMalgorithm

(a) starts with a given species surface spatial distribution (called“configuration”);

(b) then, it creates a list of all possible M processes (M = 6 in theexample of Figure 1) with the corresponding transition rates, and

calculates ktot =M∑i

ki ;

(c) then it generates two random numbers ρ1 and ρ2 and selects asingle process N that fulfills

N∑i=1

ki ≥ ρ1ktot ≥N−1∑i=1

ki [3]

(d) then, it executes the selected process;(e) consequently, it updates the clock with a time step given by

�t = ln(ρ2)/ktot ;(f) if the time t < the desired total simulation time, then it backs to

(b), otherwise, it stops.

One common expression for the kinetic rate ki in equation 3 hasthe Arrhenius form

ki = κi exp

(− Ei

kB T

)[4]

where κi is a pre-exponential factor depending on the vibrationalfrequency of the transition state, Ei is the activation barrier betweenthe two states i and j while kB and T are the Boltzmann constant andthe absolute temperature, respectively.

Casalongue et al.29 described the dynamics of the ORR on Pt(111)surfaces with the VSSM. Their simulations included an electrodeButler-Volmer potential as an external parameter being controlled(input parameter). A similar approach was implemented by otherauthors.19,30,31

These previous works lack of predicting electrochemical condi-tions and full polarization curves (I-V) as they do not consider theEDL effects, thus the surface charge density impact onto the electro-chemical elementary kinetic steps, neither the reactants transport atthe vicinity of the catalyst.

In this paper we present a new multiparadigm modeling approachto simulate electrochemical reactions in materials for energy conver-sion or storage. The approach combines a KMC algorithm, calledhere “Electrochemical-VSSM” (E-VSSM), with continuum trans-port models. For instance, in this way, it allows simulating electro-chemical conditions by accounting for reactants/products adsorption/desorption dynamics, reaction intermediates surface diffusion,chemical/electrochemical reactions, EDL effects and mesoscopicreactants/products transport phenomena. Thus, this approach allowscapturing the dynamics of the reactants, reaction intermediates andproducts at the atomistic/molecular level for an operating electro-chemical cell. These processes cannot be addressed with state of theart continuum kinetic models supported only on the MF approxi-mation which neglects, by construction, adspecies surface diffusionphenomena for example.32,33

This paper is organized as follows. First our model is presentedwithin the framework of PEMFCs. An illustrative application exampleis provided for the description of the ORR kinetics in a Pt(111)-basedPEMFC cathode. Then results under different simulated scenarios arediscussed. Finally, we conclude and indicate further directions for ourmodel development and application.

Our Model

For the case of PEMFCs, our model is articulated into intercon-nected modules representative of three relevant scales in a PEMFCcathode electrode (Figure 2a):

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Journal of The Electrochemical Society, 162 (7) E73-E83 (2015) E75

Figure 2. (a) Schematics of our general multiscale modeling framework; (b) schematics of the O2 mesoscopic transport model coupled with the EDL model;(c) schematics of the EDL model with some of the adspecies represented.

� mesoscopic scale describing reactants transport in gas phase ina parallel direction to the catalyst surface (Figure 2b);

� nanoscopic ionic transport scale in the electrolyte (constitutedof ionomer, water and protons) through a continuum model proposedby us in Ref. 34 (Figure 2c) and describing the external layer (EL)within the EDL region;

� the adlayer scale or inner layer (IL), where the redox reactionstake place, and described through a KMC modeling framework aiming

to rationalize the effects of the catalyst surface heterogeneity on thenet electrochemical reaction rate (Figure 3).

Mesoscopic scale.— For the simulation results reported in thispaper, as an illustrative example, O2 transport in gas phase at themesoscopic scale is described as a Darcy-like process in the paral-lel direction to the catalyst surface. The corresponding conservationequation in the discretized form, for a system of M bins along the Y

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E76 Journal of The Electrochemical Society, 162 (7) E73-E83 (2015)

Figure 3. Schematics of our E-VSSM principles.

direction (Figure 2b), is given by

�pn,O2

�t= k

J inn,O2

− J outn,O2

�y− ωkmc

n,O2[5]

where the time evolution of the O2 pressure in each bin (n) is solvedby numerical integration.

In Eq. 5 pn,O2 is the oxygen pressure in bin (n), ωkmcO2

is the oxygenconsumption rate calculated with KMC in the same bin and expressedin atm/second, k is the O2 transport coefficient at the mesoscopic scale,J out

n,O2is the oxygen flow rate going out from the bin (n) and J in

n,O2is

the oxygen flow rate going in the bin (n) defined respectively by

J inn,O2

= (pn−1,O2 − pn,O2 ) [6]

J outn,O2

= −(pn+1,O2 − pn,O2 ) [7]

where the O2 partial pressures p0 and pN are the boundaries conditionsgiven in Table II. Once the gas O2 pressure is calculated for each mesh,the corresponding absorbed O2 concentration in the electrolyte (EDLregion) is calculated from the Henry’s law cO2 = pO2/αH , where αH

is the Henry’s constant.7

As a first approximation, we neglect H2O transport in all the scalesas we suppose the EL (see below) to be fully hydrated. Furthermore,H2O2 transport is also neglected as its calculated production is verysmall for the electrocatalytic system under investigation in this paper.We underline however that the implementation of these transport mod-els in the overall framework is quite straightforward and is reservedfor a future investigation.

External layer.— The EL model is a continuum model whichcalculates the proton molar concentration at the reaction plane, i.e.x = L, for each KMC time step. This EL model is based on the trans-port model in EDLs we have published in Ref. 34. According to thismodel, the molar concentration ci in the electrolyte of a specie i isgiven by the solution of

∂ci

∂t= Bi RT ∇2ci + Bi �∇ ·

(ci �∇

∑i

ai j c j

)

+ Bi zi F �∇ · (ci �∇ϕ

) − NA Bi �∇ · (ci �∇(�pi · �∇ϕ

))+ f1 (βi , ci ) − f2 (βi , ci ) [8]

In the above expression f1 and f2 are terms capturing the finite sizeand relative finite size entropic diffusion and their mathematical for-mulations are given in Ref. 34. In Eq. 8 Bi is the i specie diffusioncoefficient over RT where R is the ideal gas constant, NA is the Avo-gadro number, ϕ is the electrostatic potential, zi is the i specie charge,�pi is the i specie dipolar moment, aij is the quantum interaction energybetween the i and the j species, βi is the i specie relative size relatedto the smallest specie.

For illustrative purposes, we consider here the steady-state solutionof the Eq. 8 ( ∂ci

∂t = 0) with a single species size (f1 and f2 = 0) for H+,for the water molecule and for counter ions concentration (SO−

3) ofthe ionomer (e.g. Nafion), then Eq. 8 becomes for protons,

0 = BH+ RT ∇2cH+ + BH+ zH+ F �∇ · (cH+ �∇ϕ

)[9]

for water

0 = −NA BH2 O �∇ · (cH2 O �∇(�pH2 O · �∇ϕ

))[10]

and for SO−3

0 = BSO−3

RT ∇2cSO−3

+ BSO−3

zSO−3

F �∇ · (cSO−

3�∇ϕ

)[11]

Because the SO−3 and water concentrations are assumed to be uniform

along the EL thickness (L), expressions [10] and [11] arise to therelation for the electrostatic potential

ϕ = a1x + a2 [12]

where a1 and a2 are constants related to the SO−3 and water concen-

trations and calculated from the boundary conditions.In one dimension Eq. 9 becomes

JH+ = −BH+ RT∂cH+

∂x− BH+ zH+ FcH+

∂ϕ

∂x[13]

where JH+ is an integration constant related to the proton flux at thecatalyst surface.

We can then rewrite Eq. 13 as

JH+ = BH+ RT exp

(− Fϕ

RT

)∂

∂x

[cH+ exp

(Fϕ

RT

)][14]

so

JH+ exp

(Fϕ

RT

)= BH+ RT

∂

∂x

[cH+ exp

(Fϕ

RT

)][15]

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Journal of The Electrochemical Society, 162 (7) E73-E83 (2015) E77

As we assume steady-state conditions (JH+ uniform along the coor-dinate X), by integrating in the EL thickness L we obtain

JH+ = −BH+ RTcH+ (L) exp

[ Fϕ(L)RT

] − cH+ (0) exp[ Fϕ(0)

RT

]L∫

0

exp[ Fϕ(x)

RT

]dx

[16]

In Eq. 12 we have an expression for the electrostatic potential profileand by fixing the boundary conditions as ϕ(0) = 0 and dϕ

dx (L) = E(where E is the electric field that is obtained at the IL boundary fromthe surface charge density σ in the same way as it has been donein Ref. 34) we can explicitly solve Eq. 16 and calculate the protonconcentration at x = L as,

cH+ (L)=

⎛⎜⎜⎜⎝

JH+L∫0

exp[Fϕ(x)]dx

BH+ RT+ cH+ (0) exp[Fϕ(0)]

⎞⎟⎟⎟⎠ exp[−Fϕ(L)]

[17]

Because of the small value of L (typically few nanometers), O2 con-centration in the EL is assumed to be uniform.

Since H2O transport is neglected an uniform and constant electricpermittivity is assumed for the EL. A description of the dependenceof this electric permittivity on the space can be included by followingour model in Ref. 34, something that we plan to do in close future.

Internal layer.— Once again we follow the strategy reported inour Ref. 34 where a correction in the activation energy is introducedin Eq. 4 for all those kinetic parameters k where a charge transfer isoccurring, thus

kl = cH+ (L) κkB T

hexp

(−Eact,l ∓ f (σ)

RT

)[18]

where cH+ (L) is the proton concentration at the boundary between ILand EL (x = L); Eact,l is the activation energy of an elementary reactionl without influence of the interfacial electric field (extracted from DFTcalculations), h is the Planck constant. f(σ) is a function of the chargedensity σ, the so-called surface potential, a quantity different from zeroonly for electrochemical steps. f(σ) is also equal to the electrostaticpotential drop through the IL between the catalyst surface and theelectrolyte.34 We note that in the exponential argument Eact,l ± f(σ) isan effective activation energy (the sign depends on whether reductionor oxidation is occurring).

The charge density σ is calculated in terms of the imposed elec-tronic current density J (input for our model) through

J − JF AR = −∂σ

∂t[19]

where JFar is the faradaic current density which is equal to FJH+ andcalculated here as the total charge δQ transferred by the electrochem-ical reactions per unit of time δt and per unit of area A in the catalystsurface:

JF AR = 1

A

∂ Q

∂t[20]

Note that JFAR impacts the time evolution of σ which affects in turnthe kinetic rates in Eq. 18. The expression for f(σ) is given by34

f (σ) = F

NAε0(σ − �) H [21]

where F is the Faraday constant, ε0 the electric permittivity of vacuoand � is the adlayer dipolar charge density calculated by adding thedipolar moment of all the adspecies divided by the catalyst surfacearea (as a first approximation, only the dipolar moments of H2O andOH have been considered in this paper). The adlayer dipolar densityscreens the effective charge density, and H is the IL thickness (thedistance between the catalyst surface and the assumed location of thereaction plane).

Table I. List of activation energies used in the KMC module.

Activationenergy [kJ.mol−1]

Reaction Forward Backward

O2ads + s ←→k2/k−2

2 Oads 30 149

H+ + e− + O2ads ←→k3/k−3

HO2ads 38 43

2OHads ←→k4/k−4

H2Oads + Oads 1 158

H+ + e− + Oads ←→k5/k−5

OHads 88 94

H+ + e− + OHads ←→k6/k−6

H2Oads 19 80

H+ + e− + HO2 ads ←→k7/k−7

H2O2 ads 24 47

O2ads diffusion 30Oads diffusion 40

OHads diffusion 45H2Oads diffusion 20H2O2ads diffusion 20O2Hads diffusion 20

The algorithm behind the E-VSSM proposed in this paper is struc-turally similar to the VSSM but with some major differences:

(a′) first, the system starts with a given surface charge density (zeroin the present work) and initial configuration (free metal surfacein the present work);

(b′) then the total kinetic rate is calculated according to Eq. 18and 21;

(c′) this step remains the same as in the classical VSSM (step (c));(d′) when the algorithm executes the process it also calculates the

total charge transferred;(e′) once the time step is obtained in the same way as in the classical

VSSM (step (e)), JFAR is calculated from Eq. 20, and for the givenimposed current density J, the new charge density is calculatedby numerical integration of Eq. 19;

(f′) this step remains the same as in the classical VSSM (step (f)).

The O2 and the H2O adsorptions are modeled with the samemethodology that has been used in Ref. 32 and 35. The activationbarriers used in the KMC module for the considered ORR elementarysteps, are presented in Table I and the parameters values used in thecontinuum modules are listed in Table II. All the reaction activationbarriers were taken from Refs. 35 and 36 and the diffusion activationbarriers were assumed. The initial configuration was set as a perfectlyclean Pt(111) surface but non-zero initial coverage can also be as-sumed for O2 and H2O. Because the simulation starts with a cleansurface, the only possible steps at the beginning are the competitive

Table II. Parameter values implemented in the continuummodules.

Parameter Value

κ 7.29 × 10−6 s−1

O2 sticking coefficient 0.045a,b

Proton diffusion coefficient 1.0 × 10−9 m2s−1

H2O dipolar moment 5.6 × 10−10 C.mOH dipolar moment 6.7 × 10−10 C.m

EL thickness 5 × 10−9 mIL thickness 2 × 10−10 m

ε0 8.854 × 10−12 F.m−1

Temperature 350 KInlet pressure p0 1.2 atm

Outlet pressure pN 1.0 atm

aReference 37.bReference 38.

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E78 Journal of The Electrochemical Society, 162 (7) E73-E83 (2015)

adsorption of H2O and O2. The H2O concentration was fixed constant(55000 mol.m−3).

Computational Details

The KMC part of the model has been implemented within thein-house LRCS software MESSI (Monte Carlo ElectrochemistrySimulation Software for Innovation), fully coded in Python program-ming language. The continuum parts of the model (EL and mesoscaletransport modules) are implemented within the MS LIBER-T frame-work which is also coded in Python (www.modeling-electrochemistry.com). All the simulations were carried out in our University calcula-tion platform Plateforme Modelisation et Calcul Scientifique.39 Typi-cal calculation time for one single simulation on one single processoris comprised between ten hours for a single constant current simula-tion and three days for the calculation of a full polarization curve. Theelectrode potential of the cathode is obtained by the addition of theelectrostatic potential jump across the IL, equal to f (σ), and the elec-trostatic potential at the reaction plane calculated from the equation12 in the EL.7,34

The Pt(111) surface was modelled with a supercell with 30 × 30atoms unit cells (with a cell parameter a = 2.81 Å) with periodicboundary conditions in order to guarantee simulation reproducibilityin a reasonable computational time. For all the results presented inthe following, each simulation was run at least five times, in order tocheck that no significant deviation is occurring due to the probabilisticdistribution.

The EL thickness was fixed at 5 nm. The mesoscopic O2 transportmodule was numerically solved with a single bin in most of the cases,except for the polarization curve simulation where it was discretizedin three bins. The solved mesoscale transport length is 0.09 m. Thetemperature for the simulations was fixed at 350 K unless otherwisestated.

Open circuit condition.— In this section we report results withzero imposed current density (J = 0 mA.cm−2 in Eq. 19) in order tosimulate the Open Circuit Voltage (OCV) condition.

The initial condition of the surface charge density, for all thesimulated potentials, is set to be zero. It has been demonstratedthat for uncharged surfaces adsorbed water molecules adopt a “flat”configuration40 (their dipolar moments prefer being parallel to thesurface). Once the surface becomes charged through the protonsreduction the water molecules reorient and their dipolar momentbecome perpendicular to the surface. As our model does not describethe spatial configuration changes of each adspecie, it implementsan algorithm in order to hold at zero the polarization field of theadsorbed water molecules at zero charge density until the first protonreduction event occurs.

From these simulations we observe that, when some of the electro-chemical steps start to occur, the surface becomes positively chargedand the argument of the exponential in Eq. 21 becomes negative;in other words, electrons involved in the reduction reactions are at-tracted by the catalyst surface which increases the effective activationbarrier. Once the exponential argument becomes sufficiently negativethe electrochemical steps become inhibited and no more charge trans-fer occurs, leading to an equilibrium regime. The calculated speciescoverage evolutions and the corresponding final configuration are pre-sented in Figure 4.

As illustrated in Figure 5 for a simulation representative of OCVconditions, our model is able to track the IL electric permittivityevolution. Notice that for previous IL models (e.g. Ref. 32) treatthe electric permittivity as a constant value fitted from macroscopicexperimental data.

As one can see in Figure 5, the IL electric permittivity is notconstant throughout the time. Indeed, the electric permittivity dependson the dipolar moment of the adsorbed species, their coverage and thesurface charge density, thus on the applied external current density.

The calculated OCV is reported in Figure 6.

b

a

Figure 4. a) Calculated coverage evolution of the ORR species under theOCV condition; b) final obtained configuration.

In Figure 7, we investigate the impact of temperature onto theOCV and the ORR intermediates surface coverage evolution (zero ap-plied current density). One can see that the calculated OCV increaseswith the temperature (Figure 7a) due to the increase of the effectivekinetic rates (cf. Eq. 18). Additionally we observe that the calculatedwater coverage increases with temperature which leads to a slight O2

coverage decrease.Because of its atomistic nature, the KMC approach allows inves-

tigating the impact of surface defects on the effective electrochemicalactivity. This is due to the fact that it explicitly accounts for thestructural correlations between the adspecies locations and the spatialconfiguration of active sites. This is in contrast to the MF approachwhere it is assumed that all the adspecies are perfectly mixed (no in-termediates surface diffusion limitations). We report here simulationresults at OCV conditions for a Pt(111) catalyst with surface inactivesites randomly distributed to mimic the properties of a polycrystallineor degraded surface (e.g. due to electrochemical dissolution), noted inthe following Pt∗.

In Figure 8 we can see that the presence of inactive sites does notonly impacts the coverage evolution but also the final coverage of theadspecies. The calculated final water coverage is 0.13 ML for Pt(111)surface and 0.18 ML for Pt∗. This result is due to the presence of theinactive sites making less favorable the adspecies surface diffusionand consequent reactions.

Calculated response to an applied current density step.— In thissection we study the influence of applied current density steps on thedynamical response of the potential and the associated intermediatecoverage evolution. These current density steps are applied once theOCV state regime is reached.

The calculated apparent reactivity results from the interplay be-tween kinetic processes and surface transport of adspecies. Calcu-lated ORR adspecies coverage evolutions for a 0 to 1 mA/cm2 current

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Journal of The Electrochemical Society, 162 (7) E73-E83 (2015) E79

Figure 5. a) Calculated relative electric permittivity evolution at zero appliedcurrent density (OCV condition); b) calculated surface charge density anddipolar screening charge density evolutions.

density step applied at 10−2 seconds (once the stationary OCV isreached) are presented in Figure 9. After the application of the currentdensity step, most of the ORR adspecies become significant in thesurface, in contrast to the OCV case (Figure 4), where only O, O2 andH2O are significantly present in the surface.

Figure 6. Calculated potential evolution at zero applied current density (OCVcondition).

Figure 7. a) Calculated potential evolution and b) associated O, O2 and H2Ocoverage evolution, at 290 and 350 K at zero applied current density (OCVcondition).

Figure 8. (a) coverage evolution along the time in OCV condition; (b) calcu-lated adlayer final configurations for Pt(111) and for Pt∗.

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E80 Journal of The Electrochemical Society, 162 (7) E73-E83 (2015)

Figure 9. a) Calculated coverage evolution of the ORR adspecies with a 0 to1 mA/cm2 current density step applied at 10−2 seconds; b) calculated finalconfiguration.

The application of the current density step leads to the increaseof the H2O, O2H and OH production and the O2 and O coveragedecrease vs. the OCV case. This is expected, since, H2O, O2H andOH productions need a non-zero current density to form through H+

reduction. Another significant result is that the system shows a delayin its response following the applied current density step, as it can beseen for the coverage evolution. Indeed, when the current density stepis applied, the potential does not change instantaneously. Instead itchanges once the surface charge density is sufficiently high to inducea sufficient number of electrochemical steps.

In Figure 10, we report the calculated potential evolution for twocurrent steps: from 0 to 1 and from 0 to 5 mA/cm2, both applied at10−2 seconds. The noisy aspect observed for the calculated potentialis due to the stochastic character of the faradaic current and the chargedensity dynamics.

In Figure 11, we show the calculated ORR species coverage evolu-tion for the case of a current density step from 0 to 5 mA/cm2 appliedat 10−2 seconds. Once again, other species become significant as O2Hand OH and the H2O production increase depleting the O2 and Ocoverage.

In the following we investigate the system response to a currentstep from 0 to 5 mA/cm2 applied at 10−2 seconds for a surface with in-active sites randomly distributed (called Pt∗) and mimicking partiallydegraded Pt, and we compare the results with the ones obtained withPt(111) at the same conditions.

The calculated ORR species coverage evolution is illustrated inFigure 12 in logarithmic scale. One can see that the inactive sitesimpact significantly into the arising coverage especially for O, wherewe find 0.17 ML coverage for Pt(111) vs. 0.06 ML coverage for Pt∗.Indeed the presence of the inactive sites makes more difficult the O2

Figure 10. Calculated potential evolutions following the current steps from0 to 1 and 5 mA/cm2 applied at 10−2 seconds.

splitting into two O atoms as less free sites to receive the O atomsare available. Other adspecies coverage show also lower coverage forPt∗, H2O (0.76 vs. 0.64), OH (0.17 vs. 0.13) and O2H (0.11 vs. 0.08).Indeed, the inactive sites make more difficult the adspecies surfacediffusion reducing the chances to have lateral reactions.

b

Figure 11. Calculated coverage evolution of the ORR species following anapplied current step from 0 to 5 mA/cm2 at 10−2 seconds; b) calculated adlayerfinal configuration.

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Figure 12. Calculated coverage evolution in logarithmic scale of the ORRadspecies following a current step from 0 to 5 mA/cm2 applied at 10−2 secondsfor the Pt∗ surface (b) and comparison with the Pt(111) results (a).

Polarization curve.— In the present section the mesoscopic trans-port model has been discretized in three identical bins. The external O2

input pressure was set in 1.2 atm and the external O2 output pressurewas set in 1 atm.

In order to simulate the polarization curve I-V, we scan the inputcurrent density between 0 and 5 mA/cm2 with incremental currentdensity steps of 0.5 mA/cm2. First we allow the potential to stabilize at0 mA/cm2 for 0.1 seconds, and then the first current density incrementis applied. Then the system is allowed to relax again for another0.1 seconds. Once the steady state is reached, a new current densityincrement is applied. The procedure is repeated until the whole rangeof current densities is explored. As shown in Figure 13, our calculatedpolarization curve shows good agreement with the experimental dataobtained by Markovic et al.41 on Pt(111).

In Figure 13, the potential drop close to 4 mA/cm2 is due to watersaturation at the surface. In other words, since the reduction of H+ isprevented because of the lack of O2 and O at the surface, the Faradaiccurrent no longer compensates the imposed current density J (Eq. 19),and the surface charge density changes significantly.

The corresponding coverage of ORR species evolution in time allalong the applied current density steps for each mesh is presented inFigure 14 in logarithmic scale.

The calculated steady state O2 pressure for each bin is 1.16 atm(mesh #1), 1.05 atm (mesh #2) and 1.02 atm (mesh #3). In Figure 14 we

Figure 13. calculated (in black) vs. experimental (in blue) I-V curve (experi-mental curve extracted from Ref. 41).

can see that the resulting O2 concentration distribution only impactssignificantly in the OH production, but not in the water production. Wealso notice that the variation in the O2 pressure through the mesoscaledoes not impact significantly into the ORR intermediates coveragedynamics. The impact of larger pressure gradients and water transportonto this dynamics merit further investigation.

Conclusions

In this paper we presented a new multi-paradigm simulation frame-work of electrochemical systems coupling an atomistically resolvedmodel of electrochemical reactions with continuum models describ-ing transport of charges and reactants in the electrolyte. To the bestof our knowledge, a model with these characteristics has not beenpreviously reported.

The atomistically resolved model is based on a new KMC algo-rithm developed by us (called MESSI), extending the VSSM methodby accounting explicitly of the electrochemical conditions. This arisesin particular through the coupling of the KMC algorithm describing thereactions with a non-equilibrium electrolyte/active material interfaceelectrochemical double layer model based on statistical mechanicsand reported by us in Ref. 34. The continuum models are supportedon a set of coupled differential equations solved in the pre-existingMS LIBER-T simulation package available at LRCS.

This simulation framework allows us to investigate how the macro-scopic electrode performance responds to nanoscopic and atomisticscale mechanisms by capturing details such as the surface morphol-ogy, inactive sites distribution, nanoparticle facets and sizes. In otherwords, this approach allows capturing the dynamics of the reactants,reaction intermediates and products at the atomistic/molecular levelfor an operating electrochemical cell. This phenomena cannot be ad-dressed with state of the art continuum kinetic models supported onlyon the MF approximation which neglects, by construction, adspeciessurface diffusion phenomena for example. As an application exam-ple, a Pt(111)-based PEMFC cathode was studied. The model allowspredicting simultaneously electrochemical observables (potential vs.time, polarization curves. . . ) and the associated surface intermedi-ates reaction coverage. We have found that O, H2O and OH are thedominant adsorbed species in good agreement with the literature.29,42

Note that in comparison to previous works, in the present work moreelementary steps are considered (Table I) which allow the productionof other species like O2H and H2O2.

This approach provides new capabilities for investigating theinterplaying between the EDL structure and electrochemical re-actions since our simulations are carried out at galvanostatic

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E82 Journal of The Electrochemical Society, 162 (7) E73-E83 (2015)

b

Figure 14. Calculated coverage evolution in logarithmic scale for the three bins (a) located at 2, 5 and 8 cm; (b) three meshes model scheme.

or galvanodynamic conditions, i.e. we do not impose cell potentialand constant capacities. Furthermore, the electric permittivity of theIL is calculated on the fly from the coverage of the adsorbing andforming species on the surface.

Because of the generality and the flexibility of the computationalmodeling algorithm proposed in this paper, it is readily applicablefor other reactions where material heterogeneity plays an importantrole. Indeed, we believe that the approach can offer interesting ca-pabilities for the investigation of the ORR mechanism in lithium airbatteries or the electrochemical processes in lithium sulfur batterycathodes.2,43,44

Acknowledgments

The authors deeply acknowledge the funding by the EuropeanProject PUMA MIND, under the contract 303419. Prof. DominiqueLarcher (LRCS, Amiens, France), Dr. David Loffreda, Dr. PhilippeSautet and Dr. Federico Calle Vallejo (ENS, Lyon, France) are alsoacknowledged for fruitful discussions.

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