Numerical computation of the Faradaic impedance of inlaid microdisk electrodes using a finite element method with anisotropic mesh adaptation

(1)

HAL Id: hal-00490666

https://hal.archives-ouvertes.fr/hal-00490666v2

Submitted on 9 Jun 2010

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Numerical computation of the Faradaic impedance of inlaid microdisk electrodes using a finite element

method with anisotropic mesh adaptation

Richard Michel, Claude Montella, Claude Verdier, Jean-Paul Diard

To cite this version:

Richard Michel, Claude Montella, Claude Verdier, Jean-Paul Diard. Numerical computation of the Faradaic impedance of inlaid microdisk electrodes using a finite element method with anisotropic mesh adaptation. Electrochimica Acta, Elsevier, 2010, 55, pp.6263-6273. �10.1016/j.electacta.2009.12.093�.

�hal-00490666v2�

(2)

Faradai impedane of

inlaid mirodisk eletrodes

using a nite element method

with anisotropi mesh adaptation

R. Mihel

1,⋆

, C. Montella

1

, C. Verdier

2

, J.-P. Diard

1

(1) Laboratoire d'Életrohimie et de Physiohimie des Matériaux et Interfaes, UMR 5631

CNRS+Grenoble-INP+UJF, Bât. PHELMA, 1130 Rue de la Pisine, B.P. 75, Domaine Uni-

versitaire, 38402 Saint Martin d'Hères, Cedex, Frane.

(2)Laboratoire de Spetrométrie Physique, UMR5588 CNRS-UJF, DomaineUniversitaire, 140

Avenue de la Physique, 38402 Saint Martin d'Hères, Frane.

⋆

orrespondingauthor.

Tel.: +33-4-76826547; fax: +33-4-76826630

e-mail: Rihard.Mihellepmi.grenoble-inp.fr

Abstrat

The Faradai impedane of a mirodisk eletrode inlaid in an insulating sur-

fae is revisited by numerial omputation using a nite element method (FEM)

with anisotropi mesh adaptation. New features of the numerial results, as om-

paredtopreviousworks,areanalyzed. Arstattrativefeatureisthatthediusion

impedanerelative to a mirodisk eletrode, evaluatedat theequilibriumpotential

of the eletrode, depends both on eletron-transfer and mass-transport kinetis, in

ontrast with the usual behaviour ofuniformly aessibleeletrodes. Next, thedo-

mainofvalidityoftheFleishmannandPonssemi-analytialformulationofdiusion

impedane is determined. Finally, the harateristi of impedane graphs, whih

arethe diusionresistane, theharateristi frequeny at theapexof theNyquist

diagramandtheimaginarypartofthediusionimpedaneatthisapex,arestudied

asfuntionsofadimensionlessparameterthatomparingthestandardrateonstant

of eletron transfer to themiroeletrode diusion onstant. Closed form approxi-

mationsareproposedfor allquantities inorder tohelptheanalysisofexperimental

data.

Keyword: Impedane, Miroeletrode, Mirodisk, Simulation, Finite element method,

Anisotropi mesh adaptation.

(3)

Despite the large literature dediated to the theory of eletrohemial impedane spe-

trosopy (EIS)[13℄ontheonehandandthetheoryofultramiroeletrodes(UMEs)[4,5℄

withappliationtosanningeletrohemialmirosopy[6℄ontheotherhand,onlyafew

papers have dealt with the theoretial derivation of the impedane of a mirodisk ele-

trode inlaidinaninsulatingsurfae. The analysispresented by FleishmannandPons [7℄

opened up the use of miroeletrodes to a impedane measurements. They alulated

the real and imaginary parts of the diusion impedane fromBessel's funtion integrals.

When ahieving this work, the numerial evaluation of Fleishmann and Pons formulae

was not an easytask. In order tomake the impedane alulationeasier, they presented

their numerial data in the form of tabulated funtions. Some additionalinformation is

available from the reent work by Navarro-Laboulais et al. [8℄. These authors derived

the theoretial formulation of the diusion resistane that is the low-frequeny limit of

the diusionimpedane. Theyevaluatednumeriallytheharateristidimensionlessfre-

queny at the apex of the Nyquist impedane graph. An algorithm for alulation of

the mirodisk impedane was outlined by these authors for implementation in omplex

non-linear least-squares tting (CNLS-Fit) programs.

An alternative approah for omputing the impedane of mirodisk eletrodes is based

on nite element analysis. The pioneering work on this topi was that of Ferrigno and

Girault [9℄, whih foused onthe axisymmetrireessed mirodiskgeometry. As a limit,

the inlaid disk eletrode was reovered when the reess depth tends towards zero. A

relatively good (qualitative) agreement was observed with the semi-analytial formula-

tion of Fleishmann and Pons. However, this alulation was limited to the reessed

mirodisk geometry to avoid the presene of a singularity in the neighbourhood of the

eletrode edge where the boundary ondition hanges from the Dirihlet type (uniform

onentration perturbation) on the disk to the homogeneous Neumann type (zero ux)

on the insulator (see [10℄ for analysis of this kind of loalsingularities). The singularity

refers to the onentration perturbation eld whih is not twie dierentiable. Gabrielli

(4)

ulationoftheeletrohemialimpedane ofaninlaidmirodiskeletrode usingCOMSOL

Multiphysis (formerly FEMLAB) software. These authors investigated the inuene of

the disk radius and the total eletrode radius (eletroative disk + insulating sheath)

on the impedane diagram. Their theoretial preditions were ompared to experimen-

tal data olleted froma 10^µm^diameter^Pt ^mirodiskîmmersed ⁱⁿ â10 mM ^K3^Fê(CN)6

+ 10 mM K4^Fê(CN)6 ⁺ ^0.5 ^M ^KCl âqueous ^solution. ^The împedane ^was ^measured

at the equilibrium potential of Pt eletrode. Very good agreement was found for the

impedane diagramsimulatednumeriallyusing Fleishmannand Ponsequations,aswell

as with the simulated FEM diagram. As disussed by Gabrielli et al. [11℄, Ferrigno and

Girault[9℄usedaDirihletboundaryondition atthedisk/eletrolyteinterfae(withthe

perturbationof interfaial onentrationof eletroative speies being diretly ontrolled

by the eletrode potentialperturbation), whileFleishmannand Pons [7℄used a uniform

non-homogeneous Neumann boundary ondition (i.e. the distribution of the diusional

ux perturbation is assumed to be uniform over the disk surfae). The more rigorous

treatmentproposedby Gabriellietal.[11℄ makesuse ofthe Fourier-Robinboundaryon-

dition that is the linearized formulation with respet to the eletrode potential of the

Butler-Volmerurrent-potentialharateristi.

Theaimofthisworkisanattempttorenethenumerialsimulationoftheimpedane of

inlaid mirodisk eletrodes using a nite element method (FEM) with anisotropi mesh

adaptation. In this artile, the omputation proedure is employed for modeling the

Faradai impedane of mirodiskeletrodes atthe equilibriumpotential.

The results presented here are foused on the harateristi quantities available from

impedanegraphs,whiharethediusionresistane,theharateristifrequenyobserved

at the apex of the Nyquist diagram, and the imaginary part of the impedane measured

atthis apex. Ofourse, thewholeimpedane diagramwillbeof majorimportaneforits

implementation in CNLS-Fit programs. However, the three quantities mentioned above

are suienttoomparethe aurayofthesemi-analytial(FleishmannandPons)and

(5)

inludingboth the meshadaptation strategy and some aspets of omputer implementa-

tion, is presented in Setion 3. Next, the inuene of omputational domain size on the

auray of omputed diusion impedane is investigated in Setion 4. The validity of

Fleishmann and Pons formulae is heked in Setion 5 by omparison with adaptative

FEM omputations. Finally, in Setion 6, the harateristi elements from the diusion

impedane graph are analyzed with respet to the dimensionless number that ompares

thestandardrateonstantofeletrontransfertothediusiononstantofmiroeletrodes.

2 Theory

2.1 Geometry

We onsider a mirodisk eletrode inlaid in an insulating surfae. The mirodisk radius

and the total eletrode radius (eletroative disk + insulating sheath) are respetively

denoted by re ând rmax^. ^The ^geometry ôf ^the ^devie îs axisymmetri, so the alulation

domainΩîs^redued ^toâ^2-D ^meridian^setionôf^the ^domainôupied ^by ^theêletrolyti

solution. This domain is skethed in Fig. 1 where (r, z) ^denote ^the ^usual ^ylindrial

oordinates. Its boundary Γ ^deomposes înto Γs^, ^the ^symmetry âxis; Γe^, ^the êletrode

surfae;Γi^,^the^insulator^surfae;^andΓb^,^the^bulkeletrolyte. Thesepartsoftheboundary

Γ ^are ^dened ^as^follows:

Γs = {(r, z) ; r= 0 ând 0< z < zmax} ^(1a) Γe = {(r, z) ; 0 < r < re ând z = 0} ^(1b) Γi = {(r, z) ; re< r < rmax ând z = 0} ⁽¹⁾ Γb = {(r, z) ; (r=rmax ând 0< z < zmax) ôr( 0≤r ≤rmax ând z =zmax)}^(1d)

Γb ^represents ^the ^part ôf ^the êletrolyte^whih îs^loatedâtâ^suient ^distane ^from^the

mirodisk so that the inuene of the eletrohemial reation (2) ourring at the disk

surfaeanbenegletedonΓb^. ^Thisâssumptionâllows^toîmpose^the ^boundaryôndition

(6)

ofsemi-innitediusiononΓb^while,ⁱⁿâ^rigorous^way^, ît^should^beîmposedâtânînnite

distane fromΓe^. ^The ^validityôf ^this âssumption^mainly^depends ôn^the ^ratiormax/re âs

will bedisussed inSetion 4.

2.2 Phenomenology: eletron-transfer and mass-transport pro-

esses

A one-step eletrohemialreation ours at the disk surfae and involves a n^-eletron-

transfer proess (usuallyn = 1⁾^between ^two^soluble ^speies⁽O^and R⁾^at^the ^metal^(e.g.

Pt)/eletrolyte interfae Γe^:

O + ne −→←− R ⁽²⁾

The reationrate is desribed by Butler-Volmer kinetis [3,13℄and, atthe eletrode po-

tentialE^,ît îsêxpressed ⁱⁿ ^the ûsual ^way ^by:

v =k⁰ (exp (−αrξ) cO − exp (αoξ)cR ) ^with ξ= (n F/RT)

E−E⁰^′

on Γe ⁽³⁾

wherecX ^denotes ^theonentrationof speiesX = O,R^,^and^the eletron-transferproess isharaterizedbyitsstandardrateonstantk⁰^,^the^standard^(formal)^potentialE⁰^′^,^and

the symmetry oeients αo ^and αr^, ^with αo +αr = 1^, ^the ^other ^symbols ^having ^their

usual meaning.

Duetothe quiesentsolutionandthepreseneofasuitablesupportingeletrolyte, migra-

tion and onvetion eets onmass-transportproesses an be negleted 1

, thusresulting

in apure diusion proess suh that the ux-vetor J_X of speies X = O,R ^is^given ^by:

J_X =−DX∇cX ⁱⁿ Ω ⁽⁴⁾

where the diusion oeient DX îs â ônstant ⁱⁿ^the ^presene ôf ^supporting eletrolyte, 1

Stritlyspeaking,mass-transportofeletroativespeiesbyunforedonvetionshouldbetakeninto

onsiderationusing themodel presentedby Amatoreet al.[14,15℄. Nevertheless, theseauthors showed

that theontributionofunfored onvetionanbe negletedforaneletroderadius re <25^µm^. ^This

(7)

and ∇cX îs ^the onentration gradient whose omponents in ylindrial oordinates are the partialderivatives, ∂rcX ând ∂zcX^,ôf cX ^with^respet ^tor ând z respetively.

2.3 Struture of the Faradai impedane

2.3.1 Steady-state problem

The rst step to analyse the Faradai impedane is to alulatethe steady-state regime

orresponding toastati value ES ôf ^the ^potentialîmposed ^to^the êletrode. ^This ^regime

isharaterizedbythestationentrationscS,X^,^for^speiesX = O,R^,^whih^are^solutions

of the following boundary value problem where the subsript 'S^' ^stands ^for steady-state

onditions:

Given ES^, ^nd, ^for X = O,R^,cS,X : (r, z)∈Ω→cS,X(r, z)∈R suhthat 2

:

−DX∆cS,X= 0 ⁱⁿ Ω ^(5a) DX∇cS,X·n=ǫXvS(ES, cS,O, cS,R) ^on Γe ^(5b)

DX∇cS,X·n= 0 ^on Γi ⁽⁵⁾

cS,X =c^bX ^on Γb ^(5d)

whereǫO =−1^,ǫR= +1^,c^bX ^denotes^the ^bulkonentrationofspeiesX^,nistheoutward unit vetor normal to Γ^, ând ∆ îs ^the ^Laplae ôperator ^whose ^ylindrial êxpression îs

given by ∆cS,X =∂_rr² cS,X+¹_r∂rcS,X+∂²_zzcS,X^. ^Thesteady-state reationrate vS ⁱⁿ ^Eq. ^(5b)

omes from Eq. (3) where E ând c^X âre respetively replaed by E^S ând c^S,X^. Ône ^the

problem(5)hasbeensolved,theinterfaialonentrationsareknownandthesteady-state

Faradai urrent isobtained by integration along the radial diretion(r⁾ ^as ^follows:

I^S,f =−2π n F Z

Γe

v^Sds =−2π n F Z re

0

v^S(E^S, c^S,O, c^S,R) r dr ⁽⁶⁾

2

The boundary ondition on Γs ^takes ^on ^the formulation, r D^X∇c^S,X·n = 0^, ^whih ^results ^from

the weak (variational) formulation of the boundaryvalueproblem used in the FEM framework. This

is straightforwardlysatisedbeauseofr = 0 ônΓs^. ^Hene, îtîs ^not^neessary^to ^write ^this^boundary

onditioninEq. (5). Thesameremark appliesto theharmoniboundaryvalueprobleminEq. (9).

(8)

diusion boundary ondition cS,X(r, z)−→c^bX ^when r ^orz −→ ∞^.

2.3.2 Harmoni problem

In the seond step, a smallharmoni perturbation of the eletrode potential isimposed:

E(t) =ES+EH exp(jωt) ⁽⁷⁾

where j = √

−1^, ω = 2πf îs ^the ângular ^frequeny, f îs ^the ^frequeny ând EH îs ^the

amplitude of perturbation. Under the above onditions, the permanent regime resulting

from the potentialperturbation leads tothe following onentrationelds:

cX(r, z, t;ω) =cS,X(r, z) +cH,X(r, z;ω) exp(jωt) ⁽⁸⁾

The spatial part of the onentration elds perturbations cH,X(r, z;ω) ^are ^solutions ^of

the following boundary value problem where the subsript 'H^' ^stands ^for ^the ^permanent

harmoni regime and where the notation ';ω^' îs ûsed ^for highlighting the role of ω âs â

parameter:

Given ES^, EH ^and ω^, ^nd, ^for X = O,R^,cH,X : (r, z)∈Ω→cH,X(r, z)∈C suhthat

j ω c^H,X−D^X∆c^H,X = 0 ⁱⁿ Ω ^(9a) DX∇cH,X·n=ǫXvH(ES, cS,O, cS,R, EH, cH,O, cH,R) ^on Γe ^(9b)

DX∇cH,X·n= 0 ^on Γi ⁽⁹⁾

cH,X = 0 ^on Γb ^(9d)

The harmoni perturbation of the reation rate vH ^results ^from linearization of the expression in Eq. (3) aroundthe stati polarizationpoint (ES, cS,O, cS,R)^. ^Using ^Eq. ⁽⁷⁾ ^for

(9)

vH(ES, cS,O, cS,R, EH, cH,O, cH,R) = −k⁰ n F

R T gSEH − exp (−αrξS) cH,O + exp (αoξS) cH,R

(10)

where

gS =αr exp (−αrξS) cS,O + αo exp (αoξS) cS,R ^and ξS=n F

ES−E⁰^′

/(R T) ⁽¹¹⁾

2.3.3 Harmoni perturbation of Faradai urrent and Faradai impedane

Onetheharmoniproblem(9)issolved,theharmoniperturbationsofinterfaialonen-

trationsare known and the permanentperturbationI^H,f ôf ^the ^Fâradaiûrrent^resulting

from Eqs. (7) and (8) an be derived by integration along the radial diretion (r⁾ ^as

follows:

IH,f =−2π n F Z

Γe

vHds =−2π n F Z re

0

vH(ES, cS,O, cS,R, EH, cH,O, cH,R) r dr ⁽¹²⁾

This expression isevaluatedfor eahvalue of the angularfrequeny ω^. ^The^ratio I^H,f/E^H

denes the Faradaiadmittane Yf(ω) âsâ ^funtionôf ω^. Îts formulationresults diretly from Eq. (10). It an be derived as:

Yf(ω) = Gct+YdO(ω) +YdR(ω) ^(13a) Gct = k⁰ n²F²

R T 2π Z re

0

g^S(r,0)r dr ^(13b)

YdO(ω) = −k⁰ n F EH

exp(−αrξS) 2π Z re

0

cH,O(r,0;ω)r dr ⁽¹³⁾ YdR(ω) = k⁰ n F

E^H exp(αoξS) 2π Z re

0

cH,R(r,0;ω)r dr ^(13d)

whereGct ^is^theeletron-transfer ondutane, andYdX ^denotes^the onentrationadmit-

tane relative to the speies X = O,R^. Ôf ôurse, ^the ^Fâradai împedane îs ôbtained

as

Zf(ω) = 1/Yf(ω) ⁽¹⁴⁾

(10)

respondene EH −→ IH,f^. ^So ît îs ^the âdmittane ^whih ômes ^naturally ^from ^the âbove

equations and not the impedane.

3 Numerial resolution

3.1 Dimensionless formulation and limiting onditions

3.1.1 Dimensionless numbers

It is easy to show that the solutions cS,O ^and cS,R ^of ^the steady-state boundary value problem (5)satisfy the relation:

DOcS,O(r, z) +DRcS,R(r, z) =DOc^bO+DRc^bR ^for (r, z)∈Ω ⁽¹⁵⁾

This property makes it possible to redue the problem (5) to the determination of one

onentration eld only. Arbitrarily,wehoose this onentration as cS,R ^and ^we ^use ^the

diusionoeientand bulkonentrationof speiesR^as^the^referene^quantities^for^the

denition of the dimensionlessvariablesr^⋆ =r/re^, z^⋆ =z/re^,t^⋆ =t DR/r²_e^, c^⋆_S,X =cS,X/c^bR

and c^⋆H,X = cH,X/c^bR ^where ^supersript ^'⋆^' ^indiates ^a dimensionless variable or operator.

The dimensionless numbers resultingfromthis salingare:

Λ = k⁰re

DR

and u= ω r²_e DR

(16)

3.1.2 Inuene of Λ^: ^stati ^onditions

Λ ômpares ^the ^standard ^rate ônstant ôf êletron ^transfer ^to ^the ^diusion ônstant ôf

miroeletrodes. It omes from the dimensionless form of the stati boundary ondition

(5b) writtenforthespeiesX = Rând ômbined^together ^withÊq. ⁽¹⁵⁾ ⁱⁿôrder^toêlim-

inate cS,O^. ^The ^resultingêquation îs êxpressed ^by ^the ^following ^Fourier-Robin boundary

(11)

∇^⋆c^⋆_S,R·n^⋆ = Λα(ξS)

c^⋆NS,^R(ξS)−c^⋆_S,R

on Γ^⋆_e ^(17a)

with:

c^⋆NS,^R(ξS) =

DOc^bO

DRc^bR

+ 1

1

1 + ^D_D^O_R exp(ξS)

and α(ξS) = exp(αoξS) + DR

DO

exp(−αrξS)

(17b)

where c^⋆NS,^R(ξS) ^stands ^for^the dimensionless stati interfaialonentration for Nernstian systems.

When Λα(ξS) îs ^very ^large, ^the ^Fourier-Robin boundary ondition (17a) leads to the Dirihlet boundary ondition c^⋆_S,R = c^⋆NS,^R(ξS) ôver ^the ^disk ^surfae Γe^. În ôntrast, ât

verylowvaluesof Λα(ξS)^, ^theFourier-Robinboundaryondition leadstoauniformnon- homogeneous Neumann boundary ondition, i.e. assuming a uniform perturbation ux

over the disk surfae, exept ina smallneighbourhoodof the eletrode edge.

3.1.3 Inuene of Λ^: ^harmoni ^onditions

WhenDO =DR^,^theperturbationsofonentrationseldssatisfycH,O(r, z;ω)+cH,R(r, z;ω) = 0 ^for (r, z)∈Ω^, ^whih ^leads ^to ^the ^simpliedformulation of Eqs. (9b) and (10):

∇^⋆c^⋆H,R·n^⋆ =−Λα(ξ^S)

c^⋆NH,^R+c^⋆H,R

on Γ^⋆_e ^(18a)

with:

c^⋆NH,^R = ξH

exp (−αrξS) + exp (αoξS) gS

c^bR

(18b)

where c^⋆NH,^R ^stands ^for ^the dimensionless harmoni interfaial onentration perturbation

for Nernstian systems and ξH = ^{n F E}_{R T}^H. ^Note^that gS^, ôbtained ^from Êq. ^(11), îs ônstant

over the eletrode surfae when the eletrode impedane isalulated at the equilibrium

potential. The situation would be muh more intriate for impedane alulations per-

formed away from the equilibriumpotential.

(12)

So the system kinetis is still governed by Λα(ξS) ^under ^harmoni ^onditions. ^T^wo ^lim-

iting situations an be predited. When Λα(ξS) ^is ^large, ^the ^harmoni perturbations of onentration elds tend to satisfy Dirihlet onditions at the disk/eletrolyte interfae

likeinthe workof Ferrignoand Girault[9℄. When Λα(ξ^S) ^is^small, ^the ^harmoni^pertur-

bations of diusional uxes present approximately uniform values over the disk surfae,

and then, like in the work by Fleishmann and Pons [7℄, the Fourier-Robin boundary

ondition (18a) an be replaed by a uniform Neumann ondition on Γe^. ^Both ^limiting

onditions willbenumerially veriedin Setion6.2.

At the opposite, when DO 6= DR^, ^no simpliation of harmoni equations is possible, so FEM omputationsshould be performedwith two onentration perturbationelds.

In the present work, we use the same approah as Gabrielli et al. [12℄, i.e. the Fourier-

Robin boundary ondition dened by Eq. (9b) without any approximation, exept that

the steady-state potential of the eletrode is equal to its equilibrium potential, so ξ^S ^is

given by the Nernst equation: ξS = ln(c^bO/c^bR)^,^therefore ξS= 0 ^whenc^bO =c^bR^.

3.1.4 Inuene of u

The dimensionlessangularfrequeny u^omes^diretly^from^the dimensionlessformofEq.

(9a):

j u c^⋆H,X− DX

DR

∆^⋆c^⋆H,X = 0 ⁱⁿ Ω^⋆ ⁽¹⁹⁾

Itompares theangularfrequenyω ^with^the^reiproal^of^diusion^time^onstantr_e²/D^R^.

When u îs ^very ^large, ît ^follows ^from ^the ^partial ^dierential êquation ⁽¹⁹⁾ ^that ^the ôn-

entration perturbations are vanishing inthe eletrolyte, exept in the immediate neigh-

bourhood ofthe disksurfae (due tothe boundaryondition (18a)),so aboundary layer

develops near Γ^⋆_e ât ^high frequenies. Conversely, at low frequenies, the onentration elds perturbations extendfrom the interfae Γ^⋆_e înto^the êletrolyteûntil ^they ^vanish ôn Γ^⋆_b^.