Constant-phase-element behavior caused by inhomogeneous water uptake in anti-corrosion coatings

O

pen

A

rchive

T

OULOUSE

A

rchive

O

uverte (

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Eprints ID : 14079

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10.1016/j.electacta.2012.09.061

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http://dx.doi.org/10.1016/j.electacta.2012.09.061

To cite this version : Amand, Sylvain and Musiani, Marco and

Orazem, Mark E. and Pébère, Nadine and Tribollet, Bernard and

Vivier, Vincent Constant-phase-element behavior caused by

inhomogeneous water uptake in anti-corrosion coatings. (2013)

Electrochimica Acta, vol. 87. pp. 693-700. ISSN 0013-4686

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Constant-phase-element

behavior

caused

by

inhomogeneous

water

uptake

in

anti-corrosion

coatings

Sylvain

Amand

a

_,

_Marco

_Musiani

b,∗

_,

_Mark

_E.

_Orazem

c

_,

_Nadine

_Pébère

a

_,

Bernard

Tribollet

d

_,

_Vincent

_Vivier

d

a_{UniversitédeToulouse,CIRIMAT,UPS/INPT/CNRS,ENSIACET,4,alléeEmileMonso–BP44362,31030ToulouseCedex4,France} b_{Istitutoperl’EnergeticaeleInterfasi,ConsiglioNazionaledelleRicerche,35127Padova,Italy}

c_{DepartmentofChemicalEngineering,UniversityofFlorida,FL32611,USA}

d_{LaboratoireInterfacesetSystèmesElectrochimiques,UPR15duCNRS,UniversitéPierreetMarieCurie,75252ParisCedex05,France}

Keywords: CPE

Effectivemediumtheory Impedance

Porosity Resistivity

a

b

s

t

r

a

c

t

Theimpedanceofasubstrate/coating/electrolytesystemwascalculatedwiththeassumptions:(i)the coatinguptakeselectrolytetoanextentthatprogressivelydecreasesfromthecoating/electrolyte inter-facetothesubstrate/coatinginterfacewhereitbecomesnegligible;(ii)thevolumefractionof the electrolytevariesalongthecoatingthicknessaccordingtoapower-law;(iii)theresistivityand per-mittivityprofilesoftheelectrolyte-penetratedcoatingcanbecalculatedthroughaneffectivemedium theory(EMT)formulacorrespondingtoaparallelcombinationofthetwomedia(electrolyteand coat-ingmaterial);and(iv)someporesextendfromthecoating/electrolyteinterfacetothesubstrate/coating interface,providingalowresistancepath.Theimpedanceplotsthuscalculatedexhibitedaconstant phaseelement(CPE)behaviorinalargefrequencyrange.Someexperimentalresultsobtainedwith2024 aluminumalloy/hybridsol–gelcoatingsamplesimmersedinaNaClsolutionwereanalyzedwith refer-encetotheabovedescribedmodel.Theextensionoftherecentlydevelopedpower-lawCPEmodelto anti-corrosioncoatingsisshowntoyieldinsightintothedistributionofresistivityandassociatedwater uptake.Evaluationofmixingrulesforconductivitiesandpermittivitiesofthetwomedia(coatingand electrolyte)showedthatthelinearcombinationprovidedresultsthatwereconsistentwiththeobserved impedanceresponse;whereas,distributionsresultingfromaseriescombinationofthetwomedia,an EMTformulaproposedintheliterature,andtheMaxwellapproximationwereincompatiblewiththe observedCPEimpedanceresponse.

1. Introduction

Electrochemicalimpedancespectroscopyisapopulartechnique for the assessment of the corrosion protection performance of organiccoatings,whichhasbeen usedforsome decades[1–5]. Byfollowingtheevolutionoftheimpedanceresponse ofa sub-strate/coating/electrolytesystemoveranextendedperiodoftime, manyresearchgroupshavebeenabletomonitorthedegradationof thecoatingmaterialandtheonsetofcorrosionprocesses.The anal-ysisoftheimpedancedataisoftenbasedontheclassicalequivalent circuitproposedbyBeaunieretal.in1976[2]oritsvariants. Beau-nier’scircuitconsistsoftheuncompensatedelectrolyteresistance inserieswithaparallelconnectionof(i)thecoatingcapacitanceand (ii)abranchincludingtheresistanceoftheelectrolyte-penetrated coating,assumedtobeidenticaltotheresistanceoftheelectrolyte insideitspores,inserieswithaparallelconnectionofthedouble

∗ Correspondingauthor.

E-mailaddress:m.musiani@ieni.cnr.it(M.Musiani).

layercapacityatthebaseofthepores(coating/substrateinterface) andanimpedancerepresentingthecorrosionprocess.Inthemain variantofBeaunier’scircuit,thecoatingcapacityisreplacedbya constantphaseelement(CPE);additionalmodificationsinvolvethe replacementofthedoublelayercapacitybyasecondCPEand/orthe inclusionofadditionalelements.SubstitutingCPEsforcapacities maygreatlyimprovethequalityofthefittingbetweenthe experi-mentaldata,whichoftenshowaCPEbehavior,andtheequivalent circuit,butcreatesambiguitiesinthephysicalinterpretationofthe results,sincetheCPEmoduluscannotbesimplyidentifiedwiththe coatingcapacity(orthedoublelayercapacity)andthecalculationof theeffectivecapacityfromtheCPEparametersrequiresadetailed knowledgeonthephysicalreasonsfortheCPEbehavior[6].Thus, thepricetopayforimprovingthefittingisthelossofinformation onthecoatingcapacity,aparameterthat,followingthepioneering workofBrasherandKingsbury[7],hasbeenusedfordecadesto calculatetheelectrolyteuptakeinthecoating.

The physical origin of the CPE behavior is still incom-pletely understood, but it is generally agreed that it must be ascribed to inhomogeneous physical properties of the system

= 1 Electrolyte Coa"ng Substrate Rpore

R

pore

R

e

Z

c

CPE

(a)

(b)

Fig.1.(a)Schematicrepresentationofasubstrate/coating/electrolytesystem.(b)Equivalentcircuit;theupperboxcorrespondstoEq.(7)andthecompletecircuitistheone usedforregression.TheCPEinthelowerboxwasincludedtoaccountforanoxidelayerexistingonthealuminumalloyelectrode.

[8].With anoriginal experimentalapproach,Kittel et al. [9,10] demonstrated marked variations in the impedance of organic coatings along their thickness, and concluded that accurate models for substrate/coating/electrolyte systems should inte-grategradientsofpropertiesover thethicknessof thecoatings. However, those authors did not propose any specific resis-tivity or permittivity profiles. Inhomogeneities in the coating properties have been considered by Hinderliter et al. [11,12] who related these inhomogeneities with electrolyte uptake in the coating and, taking into account various geometries for the electrolyte-filled coating defects, developed physical mod-elsofsubstrate/coating/electrolytesystems. Rocheetal.[13,14] observed the formation of an interphase region, next to the substrate/coatinginterface,withpropertiessignificantlydifferent from those of the bulk coating material. Such an inter-phase affected the mechanical and adhesion properties of the system.

Ourgrouphasrecentlyshown thattheimpedanceofalayer exhibitsaCPEbehavioriftheresistivityofthematerialvariesalong thelayerthicknessaccordingtoa power-lawandits permittiv-ityiseitherindependentoftheposition[15]orvariesalongthe layerthicknessonlyinthelow-impedancepartsofthelayer[16]. This“power-lawmodel”hasbeenappliedtosystemsasdiverseas passiveoxidesandhumanskin[17].Inthepresentpapera possi-bleextensionofthepower-lawmodeltoanti-corrosioncoatingsis explored.Thehypothesesaremadethatinhomogeneitiesinthe coatingmorphologycauseaninhomogenouselectrolyte uptake, strongernearthecoating/electrolyteinterfaceand weakernear thesubstrate/coatinginterface,andthat thedependence ofthe electrolytevolumefractiononthepositionalongthecoating thick-nessobeysapower-law.Theprofilesofresistivityandpermittivity resultingfromtheelectrolyteuptakeprofilearecalculatedforsome effectivemediumtheory(EMT)formulas,andthen usedto cal-culatetheimpedance of thesystem.It is shown that only one oftheseformulasleadstoCPEbehavior. Finally,thetheoretical impedanceexpression is compared to experimentalimpedance dataobtainedwithhybridsol–gelcoatingsexposedtoa0.5MNaCl solution.

2. Theory

Aschematicrepresentation ofa substrate/coating/electrolyte systemispresentedinFig.1a.Thecoatingthicknessisdenotedı

andthepositionalongthecoordinateperpendiculartothe sub-strate/coating and coating/electrolyte interfaces is expressedin dimensionlesscoordinatesas

=x

ı (1)

wherexisthedistancefromthesubstrate/coatinginterface. Thecoatingwasassumedtobepenetratedbytheelectrolyte whichfillsanumberofpores,thelateraldimensionofwhichmay bedistributed and not necessarily independentof . InFig. 1a, the poresare represented as being straight, although theyare obviouslytortuousinnature.Allporeshaveamouthatthe coat-ing/electrolyteinterfaceand,mostofthemarenotasdeepasthe coating.Thus,differentplanes paralleltotheinterfaces,located withinthecoatingatdifferentpositions,crossdifferentnumbers ofpores,anddifferentelementallayersofthecoating,dthick,have differentlocalelectrolytevolumefractions().Somepores(only oneisshownintherightsideofFig.1a)mayextendthroughout thecoatingandprovidealow-resistivitypathbetweenthe elec-trolyteandthesubstrate.Thus,theelectricalequivalentcircuitof thesubstrate/coating/electrolytesystemconsistsoftheelectrolyte resistanceinserieswiththeparallelcombinationoftwo compo-nents:a resistance(Rpore), representingthethrough pores,and

animpedance(Zc),representingtheelectrolyte-modifiedcoating.

SuchacircuitisrepresentedintheupperboxshowninFig.1b. Thelowerbox,containingaCPE,accountsfortheseriesimpedance responseoftheoxidelayeronthesubstratesurface.Thescheme inFig.1a,inwhichthelocalelectrolytevolumefractionincreases asincreases,issimilartotheoneproposedbyHinderliteretal. [11]who,however,clearlyseparatedanouterelectrolyte-affected coatingregionfromaninnerpristinecoatingregionandtherefore didnotconsiderporesthatreachthesubstrate.

Variouseffectivemediumtheory(EMT)formulashavebeen pro-posedthatmaybeusedtocalculateresistivityandpermittivity profilescausedbyadistributionof().Takingintoaccountthat, ineachelementallayerparalleltotheinterfaces,eachpositionis occupiedbyeitherthecoatingmaterialortheelectrolyte,thesetwo mediamaybeassumedtobeinparalleltoeachother.Therefore,the overallconductivity(=−1_{)andpermittivity(ε)maybeobtained}

aslinearcombinationsoftheconductivitiesandpermittivitiesof thetwomedia(coatingandelectrolyte)withcoefficientsequalto therespectivevolumefractions[11],i.e.,

and

ε()=εw()+εc[1−()] (3)

wherethesubscriptswandcrefertotheelectrolyteandthecoating material,respectively.Sinceusually()≪1and−1

c ≪−1w,()−1

dependslinearlyon().

Itwasrecentlyshown[15]thataCPEbehaviorcouldbedueto avariationof()accordingtoapower-law.ForaCPEimpedance writtenas

ZCPE=

1

(jω)˛Q (4)

thepower-lawexponent andtheCPEexponent˛arelinkedby therelationship

˛= −1

(5)

Inthepresentwork,thehypothesisismadethat()changes from(0)=0at=0to(ı)at=1accordingtoapower-law,i.e., ()= (ı)

1+(ı)( ₋₁₎ (6)

The condition (0)=/ 0 was not considered because Rpore

accountsforthecontributionofthroughpores,andthereforeonly theporesnotreachingthesubstrate/coatinginterfaceneedtobe consideredforcalculatingZc.Theelectrolytevolumefraction()

ispresentedasafunctionofpositioninFig.2forsometypical cases.ThesewerecalculatedfollowingEq.(6)fordifferent(ı) val-uesbetween0.001and0.1.Anexponent =5wasassumed,which correspondsto˛=0.8,atypicalvaluefoundwithsystems exhibit-ingamarkedCPEbehavior.Assumptionofadifferentvalueof wouldproducequalitativelysimilarresults.

Theresistivity()anddielectricconstantε()profiles, calcu-latedaccordingtoEqs.(2)and(3)forthesamew,c,εwandεc

valuesasthoseconsideredinRef.[11],areshowninFig.3.With (ı)=0.1(correspondingtoameanelectrolytepartialvolume<2%), theresistivityvariationacrossthecoatingisca.8ordersof magni-tude.Evenwith(ı)aslowas0.001,(ı)is5ordersofmagnitude lowerthan(0).Incontrast,thepermittivityvariationsare mod-erate:for(ı)=0.1,ε(ı)islargerthanε(0)byonlyafactorofca. 2.Atlowvalues,i.e.,closetothemetal/coatinginterface,Fig.3a showsahighresistivityregioncompatiblewiththeassumptionof aninterphasewithpropertiessignificantlydifferentfromthoseof thebulkcoating[13,14].

Undertheassumptionthateachelementallayermaybe rep-resentedbyaparallel combinationofa resistance()d and a

0.0

0.2

0.4

0.6

0.8

1.0

0.00

0.02

0.04

0.06

0.08

0.10

= 0.1

= 0.05

= 0.01

= 0.001

( ) /

dimensionless

/ dimensionless

Fig.2. Wateruptakeprofileswith(ı)asaparameter,calculatedaccordingtoEq.

(6)with(0)=0and =5. 0.0 0.2 0.4 0.6 0.8 1.0 103 106 109 1012 = 0.1 = 0.05 = 0.01 = 0.001

(

)

/

cm

/ dimensionless

a

0.0 0.2 0.4 0.6 0.8 1.0 8 10 12 14 16 = 0.1 = 0.05 = 0.01 = 0.001

)

/

d

im

e

n

s

io

n

le

s

s

/ dimensionless

b

Fig. 3.Resistivity (a)and permittivity (b) profileswith (ı)as aparameter, calculatedaccording to Eqs. (2) and (3) respectively,with c=2×1011_cm, w=330cm,εc=8,εw=82,(0)=0,and =5.

capacitanceε()ε0/d,theimpedanceoftheelectrolyte-penetrated

coating(Zc)iscalculatedas Zc(ω)=ı 1

Z

0 1 ()−1+jωε()ε0 d (7)

Plotscorrespondingtofourdifferent(ı)valuesareshownin Fig.4,bothinBodecoordinates,currentlyusedinthecoating com-munity,andinNyquistcoordinates.Clearly,thecoatingimpedance becomessmallerwhen(ı)increases(andthemeanelectrolyte partialvolumeincreases).However,theshapeoftheplotsis insen-sitivetothevalueof(ı).Theplotofthephaseangleclearlyshows aCPEbehaviorinahigh-frequencyrange,approximatelybetween 100Hzand 10kHz. TheCPEbehavioris alsovisiblein thehigh frequencybranchoftheNyquistplots.

Thefrequencydependenceofthedlog|Z′′

c|/dlogfderivativeis

showninFig.5.Inafrequencyrangewherethesystemexhibits aCPEbehavior,dlog|Zc′′|/dlogf =−˛[18].Theplots

correspond-ingtofourdifferent(ı)valuescompletelyoverlapwithavalueof −0.8(correspondingto˛=0.8)inthehighfrequencyrange. Corre-spondingly,theconstant-phase-anglevalueinFig.4bisgivenby 0.8×90◦₌₇₂◦_.

To obtain the overall impedance Z of the sub-strate/coating/electrolyte system, the pore resistance Rpore in

parallel with Zc and the electrolyte resistance in series with

the parallel combination of Rpore and Zc should be considered.

10-2 10-1 100 101 102 103 104 105 103 104 105 106 107 108 = 0.1 = 0.05 = 0.01 = 0.001

|Z

c

|

/

cm 2

f

/ Hz

a

10-2 10-1 100 101 102 103 104 105 0 30 60 90 = 0.1 = 0.05 = 0.01 = 0.001

/

d

e

g

re

e

f / Hz

b

0 1x107 2x107 3x107 0 1x107 2x107 3x107

= 0.1

= 0.05

= 0.01

= 0.001

-Z

c"

/

cm

2

Zc' /

cm

2

c

1.259 Hz

Fig.4.Impedanceplotswith(ı)asaparameter,inBodecoordinates(a)and(b) andNyquistcoordinates(c),calculatedaccordingtoEq.(7)withı=20mmandthe conductivityandpermittivityprofilesshowninFig.3,i.e.,with(0)=0and =5.

negligible, and henceforth will be disregarded. Thus Zmay be simplycalculatedas

Z(ω)=



Zc(ω)−1+R−1pore



−1

(8) Fig.6 showsimpedanceplotscalculated accordingtoEq. (8) for variousRpore values and for theresistivity and permittivity

10-1 101 103 105 -1.0 -0.5 0.0 0.5 1.0 = 0.1 = 0.05 = 0.01 = 0.001

d

lo

g

|Z

c

"|

/d

lo

g

(f

)

f / Hz

= 0.8

Fig.5.Frequencydependenceofdlog|Zc"|/dlogf,calculatedwiththesame param-etersasforFig.4.

profilescorrespondingto(0)=0,(ı)=0.1and =5.Aslongas Rpore>Zc(0),whereZc(0)=ı

1

Z

0

()disthezerofrequencylimitof Zc,ithasonlyaslighteffectontheoverallimpedance.WhenRpore

becomesprogressivelysmaller,theimpedancemodulusdecreases andtendstoapproachRpore,asshown inFig.6a;thetransition

fromaCPEtoaresistivebehaviorisshiftedtohigherfrequency, asshowninFig.6b;andtheshapeoftheNyquistplotsbecomes moresymmetricalwithrespecttothemaximumofZ”,asshownin Fig.6c.

Hinderliteretal.[11]consideredothereffectivemedium theo-ries,inadditiontoEqs.(2)and(3).Assuminganelectrolyteuptake describedbyEq.(6),(0)=0, (ı)=0.1and =5,resistivityand permittivityprofileswerecalculatedcorrespondingto

(i) aseriescombinationofthetwomedia[11],i.e.,

()=wϕ()+c[1−ϕ()] (9) ε()=



1−() εc + () εw



−1 (10) (ii)anEMTformulaproposedbyBrasherandKingsbury[7],i.e.,

log[()−1]=() log[−1

w]+[1−()] log[c−1] (11)

log ε()=ϕ()logεw+[1−ϕ()] logεc (12)

and

(iii)theMaxwellapproximation[19],i.e.,

()−1=−1c −1 w +2−1c +2ϕ()(w−1−−1c ) w−1+2c−1−ϕ()(−1w −c−1) (13) ε()=εcεw +2εc+2ϕ()(εw−εc) εw+2εc−ϕ()(εw−εc) (14) TheresultsareshowninFig.7.Eqs.(9),(11)and(13)giveriseto muchsmootherresistivityprofilesthandoesEq.(3).The(0)/(ı) ratioiscloseto10fortheBrasherandKingsburyformula,andbelow 2forboththeseriescombinationandtheMaxwellapproximation, inagreementwithHinderliteretal.[11].Thepermittivityprofiles computedaccordingtoEqs.(10),(12)and(14)arealsosignificantly smootherthanthatcalculatedaccordingtoEq.(3).Theprofiles cal-culatedaccordingtoEqs.(9)–(14)wereusedtoobtaintheNyquist plotsshowninFig.8a,togetherwithaplotcorrespondingtoa sim-pleRCparallelcombinationwithR=cı,andC=εcε0/ı. Allplots

haveanalmostperfectsemicircularshape,i.e.,thereisnoevidence ofaCPEbehavior.Thisresultisconfirmedbythelogarithmicplots

10-2 10-1 100 101 102 103 104 105 102 103 104 105 106 107 108 Infinite Rpore Rpore = 108 cm2 Rpore = 107 cm2 Rpore = 106 cm2

|

Z

| /

cm 2

f / Hz

a

10-2 10-1 100 101 102 103 104 105 0 30 60 90 Infinite Rpore Rpore = 108 cm2 Rpore = 107 cm2 Rpore = 106 cm2

/ degree

f / Hz

b

0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00

Infinite Rpore

Rpore = 108 cm2 Rpore = 107 cm2 Rpore = 106 cm2

-Z

"/

R

p

/

dimensionless

Z'/Rp / dimensionless

c

25.12 Hz 3.162 Hz 1.585 Hz 1.259 Hz

Fig.6.ImpedanceplotswithRporeasaparameter,inBodecoordinates(a)and(b) andNyquistcoordinates(c),calculatedaccordingtoEq.(8)with(0)=0,(ı)=0.1, and =5.Tofacilitatethecomparisonoftheshapeofthediagrams,theimpedance isnormalizedbythepolarizationresistanceRp.

ofZ”asafunctionoffrequency,showninFig.8b.Owingtothe mod-eratevariationofresistivityacrossthecoating,theimpedancesare comparabletothatofacoatingthathasuptakennoelectrolyte, andaremorethan10timeslargerthanthatshowninFig.4for thesameıand(ı)value.Thus,thedistributionsresultingfrom aseriescombinationofthetwomedia[11],describedbyEqs.(9) and(10);theEMTformulaproposedbyBrasherandKingsbury[7], describedbyEqs.(11)and(12);andtheMaxwellapproximation [19],described byEqs.(13)and(14)areincompatible withthe observedCPEimpedanceresponse.

In the next two sections a preliminary test of the theory describedaboveispresented.

0.0 0.2 0.4 0.6 0.8 1.0 103 105 107 109 1011

/

cm

/ dimensionless

Eq. 2, parallel Eq. 9, series Eq. 11, Brasher-Kingsbury Eq. 13, Maxwell

a

0.0 0.2 0.4 0.6 0.8 1.0 8 10 12 14 16

b

Eq. 3, parallel Eq. 10, series Eq. 12, Brasher-Kingsbury Eq. 14, Maxwell

/ dimensionless

/ dimensionless

Fig.7.Resistivity(a)andpermittivity(b)profilescalculatedfor(0)=0,(ı)=0.1, and =5,accordingtotheequationsindicatedonthefigure.

3. Experimental

Thesol–gelcoatingwaspreparedbyusingapolyaminoamide (PAA), 3-glycidoxypropylethoxysilane (GLYEO), tetraethylortho-silicate (TEOS), butanol and an additional epoxy resin. The compoundswereintroducedwithaGLYEO/TEOS/PAA/Epoxyratio of 1:1:10:8. The complete chemical structures of the various components cannot be reported herebecause they are partof proprietaryformulations.Thesol–gelfilmdidnotcontainany pig-mentsorfillers.

Thecoatingsweredepositedontoa2024T3aluminum alloy currentlyusedintheaerospaceindustry.Thespecimensconsisted of125mm×80mm×1.6mmplatesmachinedfromrolledplate. Beforepainting,thesamplesweredegreasedinanalkalinebath at 60◦_{C (pH=9) for 15min, rinsed twice with distilled water,}

then etched in an acid bath at 52◦_{C for 10min, and rinsed}

again with distilled water. The liquid paints were applied by air spraying and cured at 100◦_{C for 1h. The coating}

thick-ness was measured at several locations from the observation of the cross-sections of the samples by SEM and found to be 20±1mm.

Athree-electrodeelectrochemicalcell,filledwith0.5MNaCl, wasusedinelectrochemicalimpedancemeasurements.Acoated specimenwasusedasworkingelectrode.AcylindricalPlexiglas tubewasassembledontopofthecoatedsample(exposedsurface areaof24cm2_{).Thiscellconfiguration(recessedsystem)allowed}

eliminationofthefrequencydispersionduetocurrentand poten-tialdistributionsatthediskelectrode[20].Alargeplatinumsheet andasaturatedcalomelelectrodewereusedascounterand ref-erenceelectrodes,respectively.Theelectrochemicalcellwaskept atroomtemperatureandopentoair.Electrochemicalimpedance measurementswerecarriedoutusingaBiologic VSPapparatus. Theimpedancediagramswereobtainedunderpotentiostatic con-ditionsatthecorrosionpotentialoverafrequencyrangeof60kHz

0

1x108 2x108 3x108 4x108

0

1x108 2x108 3x108 4x108

-Z

"

/

cm

2

Z' /

cm

2

Eq. 9 and 10, series

Eq. 11 and 12, Brasher-Kingsbury Eq. 13 and 14, Maxwell

No electrolyte uptake

a

1.259 Hz 10-2 10-1 100 101 102 103 104 105 103 104 105 106 107 108

Eq. 9 and 10, series

Eq. 11 and 12, Brasher-Kingsbury

Eq. 13 and 14, Maxwell

No electrolyte uptake

-Z

"

/

cm 2

f / Hz

b

Fig.8.Impedanceplots,inNyquist(a)andlogZ”vs.logf(b)coordinates,calculated accordingtoEq.(7),fortheresistivityandpermittivityprofilesshowninFig.7.The solidlineisthecoatingimpedanceintheabsenceofelectrolyteuptake.

to10mHzwith6pointsperdecade,usinga20mVpeak-to-peak sinusoidalpotential.

The impedance data analysis was performed using a non-commercialsoftware developed at the LISECNRS, Paris, which allowedregressionofamodelconsistingofacombinationof pas-sivecircuitelementsandanalyticalexpressions.Thissoftwaredid notprovideconfidenceintervals.

4. Resultsanddiscussion

Theimpedance of a 2024T3 aluminum alloy/hybridsol–gel coating/0.5MNaClsolutionsystemwasmeasuredat the corro-sionpotentialasafunctionoftheimmersiontime.Asanexample, theimpedancerecordedafter72hispresentedinFig.9(points)in BodeandNyquistformats.Thehigh-frequencyloopwasattributed tothecoating,andthelow-frequencyquasi-verticalstraightline wasattributedtoanoxidefilmpresentatthesurfaceofthealloy [21,22].Theexperimentaldatawereanalyzedwithreferencetoa modelconsistingofaseriescombinationofanimpedancedescribed byEq.(8),correspondingtothecoating,andaCPE,corresponding totheoxidefilm.Anequivalentcircuitforsuchamodelisshown inFig.1b(allcircuitelements).Theoxideresistancewasassumed tobeverylarge,inagreementwiththequasi-capacitiveblocking behaviorobservedatlowfrequencyinallexperiments,therefore, noresistancewasconsideredinparallelwiththeCPEassociated withtheoxidelayershowninFig.1b.Suchabehaviorsuggests that,fortheimmersiontimesexploredinthepresentwork,the

10-2 10-1 100 101 102 103 104 104 105 106

Experiment

Fit

|Z

|

/

Ω

cm 2

f / Hz

a

10-2 10-1 100 101 102 103 104 0 30 60 90

Experiment

Fit

θ

/

d

e

g

re

e

f / Hz

b

0 _1x105 2x105 3x105 0 1x105 2x105 3x105

Experiment

Fit

Z

"

/

cm 2

Z'

/

cm2

c

260.4 Hz 5.208 Hz 0.1045 Hz

Fig.9.Impedanceplotsrecordedwith2024aluminumalloy/hybridsol–gelcoating, aftera72-himmersionin0.5MNaClsolution.Theexperimentaldata(points)are comparedwiththeregressionresults(solidline).

coating-oxideensembleeffectivelyprotectstheunderlyingmetal fromcorrosion,andcircuitelementsaccountingforcorrosion reac-tionsneednotbeconsidered.Duetotheresistivityoftheelectrolyte (22cm)andtheelectrodesize(24cm2_{),theelectrolyteresistance}

couldbeneglected.

Intheregression procedure, , (ı)andRporeweretheonly

adjustableparametersforthecoating;whereas,w,c,εwandεc

weregivenfixedvalues(w=2.22×101_cm,

Table1

Dependenceof(ı), andRporeonthedurationoftheimmersionofthe2024aluminumalloy/hybridsol–gelcoatingsamplein0.5MNaCl.

Immersiontimein0.5MNaCl(h) (ı)(dimensionless) (dimensionless) Rpore(cm2)

24 1.46×10−5 _{3.5 1.04×10}6 48 2.24×10−5 _{3.4 3.12×10}5 72 3.21×10−5 _{3.1 2.11×10}5 168 1.16×10−4 _{3.0 5.15×10}4 0.0 0.2 0.4 0.6 0.8 1.0 105 106 107 108 109 1010 1011 1012 24 h 72 h 168 h

/

cm / dimensionless

Fig.10.Resistivityprofilesinanelectrolyte-penetratedhybridsol–gelcoating, cal-culatedaccordingtoEqs.(2)and(3),withthebestfitted and()valuesobtained forthethreeimmersiontimesreportedonthefigure.

εw=82 and εc=8). The adjustable parameters for the oxide, Q

and ˛, are not discussed here as theyare not relevant tothe modelpresentedaboveforthecoating.Theregressionresultsare shown in Fig.9 as solid lines.The agreementwiththe experi-mentaldata wasgoodover theentirefrequency range,both in NyquistandBodecoordinates.The(ı) andRporevalues

mea-suredfor fourimmersiontimes arereportedin Table1.Asthe coating degradation proceeded, the electrolyte uptake became progressivelystronger,althoughit remainedatlow levels((ı) justexceeded1×10−4_{after168h),andR}

porebecamesmaller.In

Fig.9,thelow-frequencylimitofthecoatingimpedanceisclose toRpore,i.e.,thepseudo-diameterofthehigh-frequencyloopmay

beidentifiedwiththeporeresistanceasfor theclassical Beau-nier’scircuit[2],and,therefore,thehigh-frequencyloopisalmost symmetrical with respect to the Z” maximum (compare with Fig.6c).

The()–dependencecalculatedwiththebestfitted and (ı)valuesisshowninFig.10forthreedifferentimmersiontimes. Sharp resistivityprofiles wereobtained, even for thesmall fit-tedvalues of (ı).The ε()–dependence wasalso calculated; however, the permittivity underwent only negligible variation (between 8 and 8.009, for 168h) and, therefore, no plot is shown.

Themodel didnotfit aswell forsomeotherhybrid sol–gel coatingswith different formulations, tested in electrolytes less aggressivethan0.5MNaCl.Presumably,amoresophisticated inter-pretationisneededforthosesystems,andworkisunderwayto refinethemodel.Theaboveresults,however,providea prelimi-naryvalidationofthemodelproposedinthepresentmanuscript. Theresultsconfirmtheassumptionthatthevalueofεinthe power-lawmodelcanbeconsideredasaconstantparameter.Theuseofthe power-lawmodelshowsthatthewateruptakethroughthecoating inducesaCPEbehaviorwithoutsignificantvariationofthe per-mittivity.EISdatarelevanttoothersubstrate/coating/electrolyte systemsarebeingexaminedandwillbedescribedinafollowing paper.

5. Conclusions

Observation of a CPE behavior in the impedance of a substrate/coating/electrolytesystemmaybeexplainedbyan inho-mogeneouselectrolyteuptakegivingrisetoapower-law()– profile,ifthelocalresistivityandpermittivityvaluesateach posi-tionalongthecoatingthickness,()andε(),arecalculatedbyan EMTformulacorrespondingtotheparallelcombinationofthe coat-ingmaterialandtheelectrolyte.Sincetheindividualresistivitiesof thetwomedia,candw,typicallydifferbyseveralordersof

mag-nitude,theelectrolyteuptakecausesmarkedresistivityvariations acrossthecoating.Incontrast,thepermittivityundergoesonlya mildvariationsincetheindividualpermittivities,εcandεw,differ

byafactorofabout10.Theseresultsallowtheextensionofa pre-viouslyderivedmodel[15,16],appliedsofartopassiveoxidesand humanskin[17],toanti-corrosioncoatings.

Whenexperimental EIS datarecorded with2024 aluminum alloycoatedwithhybridsol–gelcoatings,immersedin aqueous NaClsolution,wereanalyzedwithreferencetothemodeldescribed in thepresent work,goodqualityfitting wasobtainedusingas adjustableparametersforthecoatingonly ,(ı)andRpore.The

fitted(ı)valueswerequitelow(oftheorderof10−4 _orlower),

butsufficienttocauseresistivityvariationsoveratleast5orders ofmagnitude(andvirtuallynovariationofpermittivity).The coat-ingdegradationwascausedbythecombinedeffectofincreasing (ı)andincreasingthenumberand/orcross-sectionofthethrough pores,witnessedbythedecreasingRporevalues.

Acknowledgments

Theexperimentaldatawereobtainedintheframeworkofthe SMILEproject(SurfaceMonoInnovativeLayerforEnvironment), withthefinancial supportof theDGCIS(Direction Généralede laCompétitivité,del’IndustrieetdesServices).S.AmandandN. Pébère thanks the partners of the project: Mapaero(Pamiers), Rescoll(Bordeaux),Airbus(Toulouse),Dassault(Paris),EADSIW (Suresnes),l’Electrolyse(Bordeaux),andthelaboratoriesLCPOand US2B(UniversitédeBordeaux).M.Musianiacknowledgesfinancial supportfromtheCNRShortTermMobilityprogram.M.E.Orazem acknowledgesfinancialsupportfromtheCharlesA.Stokes Profes-sorship.

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