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Eprints ID : 16588
To link to this article : DOI:10.1016/j.jeurceramsoc.2014.07.032
URL :
http://dx.doi.org/10.1016/j.jeurceramsoc.2014.07.032
To cite this version :
Lesseur, Julien and Bernard, Dominique and
Chung, U-Chan and Estournès, Claude and Maglione, Mario and
Elissalde, Catherine
3D mapping of anisotropic
ferroelectric/dielectric composites.
(2015) Journal of the European
Ceramic Society, vol. 35 (n° 1). pp. 337-345. ISSN 0955-2219
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3D
mapping
of
anisotropic
ferroelectric/dielectric
composites
J.
Lesseur
a,b,
D.
Bernard
a,b,
U.-C.
Chung
a,b,
C.
Estournès
c,d,
M.
Maglione
a,b,
C.
Elissalde
a,b,∗aUniv.Bordeaux,ICMCB,UPR9048,F-33600Pessac,France bCNRS,ICMCB,UPR9048,F-33600Pessac,France
cUniversitédeToulouse,UPS,INP,InstitutCarnotCirimat,118routedeNarbonne,F-31062Toulouse,cedex9,France dCNRS,InstitutCarnotCirimat,F-31062Toulouse,cedex9,France
Abstract
Macroscopicanisotropyinpolycrystallinematerialsisofkeyinterestsinceitmayhelpfillingthegapbetweenrandomlyorientedpolycrystals likeceramicsandsinglecrystals. Non-destructiveX-rayComputedMicroTomography(XCMT)isanecessarysteptowardsthefullcontrol andmodellingofsuchanisotropy,beyondthestandardscheme ofinterfaces.Toascertainthisprogress,XCMTisappliedto3Dmixturesof ferroelectricanddielectricoxidesprocessedbySparkPlasmaSintering(SPS).Insuchconditions,notonlyisthisanisotropyseenintheoverall dielectricparametersbutitalsoshowsupinferroelectricproperties.Experimentalmacroscopicparametersarelinkedtothe3Dmorphological anisotropyofindividualMgOinclusionsinducedduringSPS.
Keywords:Ceramics;Ferroelectric;Compositematerials;3Dimensionalimaging;X-raycomputedmicrotomography
1. Introduction
Three-dimensional composites are finding an increasing numbersofusessincetheyprovideimprovedfunctionalitiesas comparedtothesimpleadditionoftheirindividualconstituents. Interfacesareusuallythoughttobeinvolvedforsuch compos-iteeffectstoappearandtheiranalysisistakenasthefirststep towardsmodelling3Dmaterials.Ontheotherhand,wecannot currentlyfindouttheactualshapeofthecompositephaseswith the required resolution,withthe necessary large-scale exten-sionandin3-dimensionswithoutdestroyingthesamples.X-ray ComputedMicroTomography(XCMT)meetsallthese require-mentsprovidedthat therequiredresolutionisreached. Inthe fieldoffunctionalmaterials,XCMThasbeenmainlyusedfor fluidflowinporousmedia,1fortheinsitucontrolofsintering2 andformechanicalproperties.3Inthefirsttwocases,the result-ing3Dmapisthestartingpointfortheimplementationoffluid
∗Corresponding author at: ICMCB-CNRS, 87 avenue du Dr. Albert
Schweitzer,33608Pessac,Cedex,France.
E-mailaddress:elissald@icmcb-bordeaux.cnrs.fr(C.Elissalde).
dynamicsequations.4Inthethird,large-scaleelasticexcitations
are usually considered.5 Thisapproach isless frequent when
materialspropertiesaresensitivetomorelocalfluctuationslike dielectricsandthisisourmaingoalhere.Forthispurpose,we processed compositesmade of aBa0.6Sr0.4TiO3 (BST)
ferro-electricmatrixinwhichMgOinclusionswerespread.6–8While theformerprovidesuniquedielectricflexibility,thelatterdrives dielectric lossestowardsthe levels requiredbythe electronic industry.9,10Inthepresentstudy,wegobeyondthis accumula-tionofpropertiesandshowarealcompositeeffect.Thankstothe specificconditionsofSparkPlasmaSintering(SPS),weareable toproduceinclusionswithalargeaspectratio(ellipsoid)while preserving the chemical integrity of both phases.Asa result of such anisotropic microstructure, overall anisotropy of the dielectricandferroelectricpropertiesoftheBST/MgO compos-itesisinduced.WecheckedthatBSTaloneprocessedusingthe sameroutedidnotshowanysuchanisotropy.While piezoelec-tricceramicswithgradedporosityhavealreadybeenreported to show dielectric anisotropy,11,12 to the best of our knowl-edgenoferroelectricanisotropyhasbeenreported.Inaddition, weproposetolinkourexperimentalparametersatlowelectric fieldtothe3DmorphologicalanisotropyoftheindividualMgO
inclusionsandtothesharp inclusionedgesfacingeachother. This is the first step towards realistic mapping of a possible electricfieldfocusingthatwouldfitwiththeavailableempirical models.12–16
2. Experimental
2.1. BSTandMgOpowdermixing
Ba0.6Sr0.4TiO3 nanopowders(50nm)werepurchasedfrom
Pi-Kem(Tilley,UK)(Sections3.1and3.2)andprocessedby solid-state routeatICMCB (grainsize 200nm) (Section3.3). MgO(97%)suppliedbyMerck(Darmstadt,Germany)ismade of sphericalspray-driedgranuleswithan averagediameterin therange10–100mm.Eachgranuleiscomposedofpacked ele-mentarycrystallitesof nanometersize.Thedrypowderswere mixedbyhandinanagatemortar.Toinvestigatetheinitialstate –i.e.powderbedpriortothesinteringstep–BST-MgOpowder mixtureswereintroducedinsideanSPStoolandconsolidated usingPMMAasbinder.
2.2. Sintering
TheSparkPlasmaSinteringapparatususedwasaDrSinter SPS-2080SPSSyntexINCJapanfromthePlateformeNationale
CNRS de Frittage Flash (PNF2/CNRS) located in Toulouse
(France).Powders(withoutanysinteringaids)wereloadedonto acylindricaldieof8mminnerdiameter.Thetemperaturewas raisedto600◦Coveraperiodof3min,andwasthenmonitored
andregulatedbyanopticalpyrometerfocussedonasmallhole locatedatthesurfaceofthedie.Aheatingrateof100◦Cmin−1 wasusedtoreachthefinaltemperatureof1200◦C.AlltheSPS
cycleswere performedundervacuumandauniaxialpressure of either50MPaor100MPawasappliedimmediatelybefore and until completion of the temperature rising step. The as-obtainedSPSceramicswerere-oxidizedduringapost-annealing performedstepat800◦Cfor12hinair.
2.3. Microstructuralanddielectricstudy
The microstructure of the sintered compacts was studied using ascanningelectron microscopeJEOLJSM 6360Aand ahigh-resolutionscanningelectronmicroscopeJEOL6700F.
Dielectric measurements were performed using a Wayne–Kerr component analyser 6425 for temperatures between 200K and 500K and in the frequency range: 100Hz–100kHz. The real part of the permittivity derived fromthe capacitanceanddielectriclosses(tanδ=ε′′r/ε′r)was
directly measured. The piezoelectric resonance was recorded using a HP4194 impedance analyser and the pyroelectric currents were measured witha Keithley 6517Belectrometer. The experimentalconditions are such that the error bars are estimatedsmallerthanthesymbolsusedinthecorresponding figures.
Thetunability(A(%)=|ε(T,E)−ε(T,E=0)|/ε(T,E=0)×100, whereEistheelectricfieldandTthetemperature)wasmeasured byapplyingastaticfieldof1kV/mmtothesampleusingahigh
voltagesupply(BertanSeries225). Ahomemadedecoupling device ensured protection of the impedance meter from the bias. To avoid breakdown in air (∼1000V),the sample was placedinasiliconeoilbath.
2.4. 3Dimaging
XCMTconsistsin:firstacquiring,fordifferentangular posi-tions,alargenumberofradiographsofasamplerotatingaround afixedaxis, andsecond, inreconstructing from thisdata set a3Dmapofµ,thelinearattenuationcoefficientofthe differ-entcomponentspresentwithinthesample.17Thecoefficientµ varieswiththeenergyoftheX-raybeamand,foreach compo-nent,itdependsontheaverageatomicnumberandthedensity. Whentheµvaluesofthedifferentcomponentsaredifferent,3D µmappingcanbetransformed,usingimage-processingtools, intoa3Dimageoftheinternalmicro-geometryofthescanned sample.
DifferentX-raysourcescanbeusedforXCMT.Inthepresent work,weusedsynchrotronradiationfromID19the3DX-ray imagingbeamlineof theESRF(European Synchrotron Radi-ationFacility, Grenoble, France)and fromTOMCAT the 3D X-rayimagingbeamlineoftheSLS(SwissLightSource, Villi-gen,Switzerland).Atypicalexperimentconsistedinrecording 2000radiographs(2048×2048pixels)andabout100references duringacontinuousrotationthrough180◦.Theenergywasequal
to40keVandtheeffectivevoxelsizeofthereconstructed3D imagesequalto0.52mm(ESRF)or0.37mm(SLS).
3. Resultsanddiscussion
Different parameters can alter the effective properties of multi-materialscomposedofBSTandMgO:thevolume frac-tionofthedielectricphase,theshapeofthedielectricinclusions, andthelevelofchemicalinter-diffusionbetweenthetwophases. Wehavealreadyshownthat,inceramicsobtainedfroma mix-ture of BST andMgOpowders, increasing the MgOcontent upto10wt% withintheBST matrixefficiently decreasesthe dielectriclossesbutleadstoaloss of tunability.The optimal valueinordertodecrease the lowfrequencydielectriclosses whilekeepingsufficientlyhighpermittivitytunabilityisaround 4wt%MgO.8
3.1. Tuninginclusionshapebypressure
3.1.1. Initialmicrostructure
First,usingSEMweexaminedtheMgOgranulesbefore mix-ing.Theyappearasspheroidalparticleswithalargedispersion insize(Fig.1a).Themixingprocessismildenoughtomaintain thisshapeasshowninFig.1bwhereSEMwasusedtoscana polishedsectionthroughasampleoftheinitialmixture consol-idatedbyPMMAimpregnation.Thesectionrevealstheinternal structureoftwoMgOparticlesthatappeartobehighly hetero-geneous,beingcomposedofaseveralsphericalshells.Within theBST matrixsomeheterogeneities are alsovisiblesuch as largezonesoflowerporosity.Thecracksthatarevisiblearound theMgOparticleandthroughthematrixaremainlyduetothe
Fig.1.(a)SEMimageoftheMgOgranulesbeforemixing.Compositesampleattheinitialstate:(b)SEMimageofapolishedsection,(c)sectionofatomogram (3Dreconstructedimage)acquiredatESRF,(d)3DrenderingoftheMgOparticles.
samplepreparation,impregnationonlyprovidesalow mechan-icalstrengthtothecomposite.Theobservationsonthepolished sectionscannedbySEMwereconfirmedbyXCMT(ESRF).The sectionthroughthe3DimagepresentedinFig.1cpresentsthe samefeaturesasinFig.1b.Thenumberofcracksissmaller, par-ticularlyaroundMgOparticles,becausesamplepreparationfor XCMTislessdamagingthanforSEM.Ofcourse,havinga3D imageoftheinteriorofthesampleenablesmuchmorecomplete characterizationsthanwithsuperficialdataonly. Forinstance, the number,the shape, thevolume, and severalother param-eters canbedetermined for the MgOparticlesafter asimple segmentationofthe3Dimage(Fig.1d).
3.1.2. Finalmicrostructure
Theuseof SPSnot onlyallowsdensitieshigherthan95% tobereachedbutitalsopreventschemicalinter-diffusionand
cracks at interfaces between the two phases. In addition the pressure applied during SPS is used to flatten the dielectric inclusions. Herewefocusoneffectofthe shapeof the inclu-sionsonthedielectricproperties.Anincreaseofpressurefrom 50 to 100MPa for the composite BST – 4%MgO, strongly depresses permittivityinparticularinthevicinityofthe tran-sitionleadingtonearlyconstantvaluesbetween200and300K (Fig.2).Notethat suchapressureincreasehadnoimpacton thedensitythatwasstillhigherthan95%.Thetunability mea-sured at100kHzwithanelectricfieldof1kV/mmdecreased from25%to18%uponincreasingthepressurefrom50MPato 100MPa.Globally,thedoublingofthepressuredecreased per-mittivityby40%whereasthedecreaseintunabilitywasonlyof 7%(Table1).Wecheckedwhetherthismodulationof dielec-tric propertiescould be related toachange in the composite microstructureandinterfaces.Theimpactofhigherpressureon
Fig.2.ThermalvariationsofdielectricpermittivityofBST+4%MgOsintered bySPSatdifferentpressure.Inset:SEMimageoffracturesurfaceoftheceramic sinteredbySPSaunderuniaxialpressureof100MPa.
microstructure resultsinanarrowergrain sizedistributionof closestBSTtotheMgOinclusions(200–500nm)andinaslight grainsizedecreaseoftheMgOitself(200–300nm).Inaddition, thinnerandmoreelongatedMgOinclusionswereobservedin thecomposite sinteredunder100MPapressure(insetFig.2). Theinterfaceswerestill verycleanandwithoutanytracesof delaminationor cracks.Thesoft plasticbehaviour underSPS oftheMgOnanoparticlesconstitutingtheinclusions,andgrain rearrangementbyslidingcouldbeattheoriginofsuch accom-modation that allowstuning of the microstructure anisotropy withoutdegradationoftheceramic.18–20 The3Drenderingof thesample(Fig.3)obtainedbyXCMT(SLS)clearlyrevealsthat theinclusionsappearingasellipsesontheSEMimagepresented inFig.2aredisc-shaped.Thisisverynaturalsincetheonly fac-torthatbreakstheoverallcompositesymmetryistheuniaxial pressurethatisappliedduringtheSPSsintering.Inaddition,the 3Dimages confirmthattheinterfacesbetweeninclusionsand matrixarewelldefined.Asmallsub-volumecanbeextracted fromthe completesample(black cubeinFig.3)defining 3D computationdomainsdirectlyfrom the3Dgeometry.Wecan thusexcludedensity,grainsizeandinterfacesasasourceofthe changeinthedielectricparameters.Itfollowsthattheincrease inthe3Dmicrostructureanisotropy,adjustablebytheincrease inpressureduringtheSPSprocessingisanefficientwayto sig-nificantlyreducetheeffectivepermittivitywhilemaintaininga hightunability. Theseresultsare ingoodagreement withthe theoreticalapproachproposedbyShermanetal.13and demon-stratethatthedielectricpropertiesarestronglyaffectedbythe
Table1
ComparisonofdielectricpermitivityatTcandroomtemperatureelectricfield
tunabilityofthepermittivityat100kHzofBST+4%MgOsinteredbySPSat differentpressures. SPSBST−4% MgO Permittivity (Tc-100KHz) Tunability (RT-1kV/mm) 1200◦C–3′– P=50MPa 2500 25% 1200◦C–3′– P=100MPa 1500 18%
Fig.3.3Drenderingofasampleinthefinalstate:theexternalsurfacesof thesampleandofthedielectricinclusionsarevisualized.Theblackcube corre-spondstoasub-volumesimilartotheoneusedforthepermittivitycomputations.
shapeandthe distribution of the“passive component” inthe ferroelectricmatrix that affectthe electric fieldredistribution betweenthedifferentcomponents.Beingabletoaffectthe over-alldielectricproperties,inclusionanisotropycouldalsoinduce anisotropyintheferroelectricpropertiesofthematrix.Thiswas checkedinthenextpart.
3.2. Dielectricandferroelectricanisotropy
Weinvestigatedthedielectric,pyroelectricandpiezoelectric propertiesoftheBST/MgOcompositesintwodirectionsrelative tothelongaxisoftheMgOinclusions.Inthefollowing,wewill usethetermparalleltorefertotheconfigurationweretheelectric fieldisappliedalongtheinclusionslongaxisandperpendicular whenitisalongthethinnestinclusiondimensions.Thisnotation wasusedforboththesmallacelectricfield,usedindielectric experimentsandforthepolingelectricfieldusedinpiezoelectric andpyroelectricexperiments.Startingfromthepelletsthatwere SPSsintered, parallelepipedswere cut withtheirmajor faces paralleleithertothepelletsurfaceortothepelletedges.Samples ofBSTfreefromMgOinclusionswereprocessedinthesame waytoprovidereferencesforallexperiments.
Theparallel andperpendicular dielectricpermittivities are displayed in Fig. 4. The dielectric permittivity is clearly anisotropic,theparallelpermittivitybeingtwicethe perpendic-ularoneoverthewholetemperaturerange.Weshouldfirstpoint outthatBSTsamplesaloneprocessedinthesamewayasthe BST/MgO composites did not display such anisotropy when usingexactlythesamedielectricsetuponsamplesof similar shape.SincetherearenoinclusionsinpureBSTceramics,the parallelandperpendiculardirectionsweretakenrelativetothe axisof theapplieduniaxialpressureduringSPS (Fig.5).We showbelowthatnotonlythedielectricpropertiesbutalsothe
Fig.4.PermittivityversustemperatureofBST+4%MgOSPSceramicswith electrodesparallelandperpendiculartotheinclusionlongaxis.
ferroelectricfeaturesareanisotropicintheBST/MgO compos-ites.
To this end, we performed pyroelectric and piezoelectric investigationsversustemperature.Forthepyroelectric experi-ments,bothperpendicularandparallelsampleswerepoledunder thesameelectricfieldof4.7kVcm−1oncoolingfrom350Kto 100K.Thepyroelectriccurrentwasthenrecordedunderheating ataconstantrateof5K/min.Integrationofthisdepolarization currentandnormalizationtothesamplesurfacethenledtothe curvesofpolarizationversustemperature(Fig.6).Thesteady stateparallelpolarizationwas about4-fold(7.8mCcm−2)the
perpendicularone(1.5mCcm−2).Thisisstrongevidenceofa realcompositeeffectsincenosuperimpositionofthedielectric propertiesoftheferroelectricBSTmatrixonthoseofMgO inclu-sionaltertheoverallferroelectricpropertiesofthematrix.This featurewasnotobservedinceramicsthatincludedvoid inclu-sionsofanisotropicshape.11,12Anotherprobeforferroelectric anisotropyisthepiezoelectricresonancethatislesssensitiveto allspuriousfreechargesthat couldalwaysbeaccumulatedat theferroelectric/dielectricinterfacesandthatcouldcontribute tothedepolarizationcurrents.
Fig.5.PermittivityversustemperatureofBSTSPSceramics,thenotationsare thesameasforFig.4fortherelativeorientationoftheelectricfieldandthe pressureappliedduringSPS.
Fig.6.PolarizationversustemperatureofBST+4%MgOSPSceramicswith electrodesparallelandperpendiculartotheinclusionlongaxis.
Forthe piezoelectric experiments,the parallelsample was abar5.59×1.48×0.82mm3 andthe perpendicularsamplea Plate2.28×1.33×0.53mm3.Inbothcases,anelectricfieldof 7.3kVcm−1wasappliedoncoolingandthepiezoelectric reso-nancewasrecordedatfixedtemperaturewhileheating.Wefocus onthefundamentalpiezoelectricresonancemodethatisa trans-verse oneusing the d31 piezoelectriccoefficient, 3 being the polingdirectionand1theresonatingone. Followingstandard analysis,itisthelongestdimensionperpendiculartotheelectric fieldaxisthatleadstothed31resonancemode.21 Fig.7gives
theresultingpiezoelectricresonanceseeninconductanceinboth casesatabout−35◦C.Theresonancefrequencybeingexactly
inverselyproportionaltothelengthoftheresonatingdimension, onecanseeinFig.6thattheratiobetweentheresonance fre-quencies ofperpendiculartoparallelsamples,f(⊥)/f(q),is2.6
Fig.7.Conductanceversusfrequencyofceramicswithelectrodesparalleland perpendiculartotheinclusionlongaxis.Thepiezoelectricresonanceinthe transversemodeismuchstrongerfortheparallelorientation.Inset:temperature dependenceofthepiezoelectricresonance(parallelconfiguration).
whiletheinverseratiooflengths,L(q)/L(⊥),is2.45.Takinginto accountthedifferencesintheshapeofthetwosamples,thisis agoodagreement.Whatismorestrikingisthattheamplitude oftheresonancesisatleastoneorderofmagnitudesmallerin theperpendicularsamplethanintheparallelone(Fig.7).Even takingintoaccountthedifferentshapesofthetwosamples,such alargedifferencemustbeduetoanintrinsiceffectonthe piezo-electriccoupling coefficientsandpiezoelectrictensorelement d31.Thisisinlinewiththesmallerdielectricpermittivityand polarizationintheformerascomparedtothelatter.Also consis-tentwiththedielectricandpyroelectricdata,thepiezoelectric resonancedisappearsatabout5◦C,thetransitiontemperature
totheparaelectricnon-piezoelectricstate.
We canthus summarize this part by stating that not only the overall dielectric propertiesof the BST/MgO composites are anisotropic, butthe pyroelectricandpiezoelectric proper-tiesoftheBSTmatrixalsofollowthesametrend.Itshouldbe noted that theinfluence of theheterogeneity of the matrixin terms of grainsize isnot tobe discarded.At first sight,it is temptingtoascribesuchanisotropytotheanisotropicshapeof theMgOinclusionsasevidenced bytheSEMpicturesshown inFigs.2and3.However,the2Dprojectionisinsufficientto revealallthefeaturesthatcouldleadtoferroelectricanisotropy. Inferroelectric/dielectriccompositesanumberofmodelshave beendevelopedtoaverageoutthecontributionsofthetwo com-ponents of thecomposites.22–24 Shermanetal.13 haveshown theoreticallythatdielectricinclusionswithsharpedgesmaylead to large dielectric non-linearities. However, the spatial orga-nization ofthe phasesisthree-dimensionalandit isof prime importancetotakeitintoaccountasshownsubsequently.
3.3. 3Dproperties
Therelationbetweentheferroelectricpropertiesandthe spa-tialdistributionoftheMgOinclusionswithintheBSTmatrixis studiedusinga3Dnumericalmodelabletohandledirectlythe realistic3DimagesprovidedbyXCMT.
TheequationtobesolvedisPoisson’sequation:
∇·[ε(X)∇V]=0 (1)
whereVisthepotential,Xthe3Dlocalpositionandεthelocal permittivity. A permittivity value is attributed to each of the regionsidentifiedinthe3Dimage.
AtthemacroscopicscaleanexternalvoltageVLisappliedto twoparallelfaces(distance=L)ofacompositesample impos-inganaverageappliedelectricfield(Eapp=VL/L).Assumingthat thehigh-resolution3DimageprovidedbyXCMTis representa-tiveofthesample(i.e.alltheaveragedpropertiesofthesample canbecomputedusingthis3Dimage),theeffective permittiv-ity(εeff)canbecomputedresolvingEq.(1)atthemicroscopic
scale(i.e.thescaleof theinclusions).Forthattheproblemis reformulatedwithanewunknown,w,definedbythefollowing relation:V=Eapp(w+x),wherexisthecoordinateparallelto
theaveragedappliedfield.Eappwisthedifferencebetweenthe
localpotentialandtheaverageone(Eappx).Theproblem
allow-ingthecomputationofwisdevelopedfromEq.(1)(seeAnnex).
Usingafinitevolumeschemeforthespatialdiscretization,the effectivepermittivityinthedifferentdirectionscanbecomputed foranysamplefromarepresentative3Dimage(Fig.3).
To evaluate the relevance of this modelling approach, the experimental results presented in Fig. 8 are com-pared with numerical results. From the 3D volume of 1800×1800×1800voxels reconstructed from the data acquired at SLS with a pixel size of 0.37mm, a sub vol-ume of 300×300×300voxels (111×111×111mm3) is extractedafter checking its representativeness (same volume fractionsthatthecomplete3Dimage).Themicro-geometryis described with the initial resolution (0.37mm/pixel) and the resultingnumericalproblemismuchsmaller, butnonetheless representative.
Forthissample theBSTmatrixisveryhomogeneouswith micrometricgrainsize(Fig.8a)anditspermittivityisconstant atagiventemperature.TheMgOpermittivitybeingconstantin thetemperaturerangeconsideredhere,theeffectivepermittivity ofthecompositeonlydependsupontwoparameters:thelocal geometry(invariantwithtemperature)andthematrix permittiv-ity(varyingwithtemperature).Thecomparisonisperformedat differenttemperaturesfollowingthesubsequentprotocol: 1. The numericaleffective parallelpermittivity is fitted with
the experimentalvalueby varying the matrixpermittivity. Withthisstep thevariability of theBST permittivity with processingconditionsistakenintoaccount.
2. Thenumericaleffectiveperpendicularpermittivity is com-putedusingthematrixpermittivitydeterminedatstep1. Agoodagreementbetweentheexperimentalandnumerical effectiveperpendicularpermittivityvaluesdemonstratesthatthe numericalmodeliscorrectlytakingintoaccounttheeffectsof theelectricalfielddistortionsinducedbythedielectricinclusions (Fig.8bandc).AsshowninFig.8,itiseffectivelythecasefor thefourtemperaturestested.
Allthecomputationshavebeenperformedin3D.The effec-tiveperpendicularandparallelpermittivitiesareclearlydistinct, butparallelpermittivityisalmostindependentfromthe orienta-tion.Consequentlyitisreasonabletothinkthat2Dcomputation canbesufficienttorevealtheeffectsofthemicrogeometryonthe permittivityvalues.Thissuppositionhasbeentestedperforming 2Dcomputationsonasectionofthe3Dvolumeperpendicularto theinclusionshavinganMgOsurfacefractionequaltotheMgO volumefractionof the 3Dimage. Parallel permittivityvalues areinaccordancewiththe3Dresults(Fig.8),butperpendicular permittivityvaluesaresystematicallylowerthanthe3Dresults, givinganoticeableoverestimationofanisotropy.3D computa-tionsarethenpreferableto2Dcomputationsfromsectionsif anisotropyofelectricalpropertiesisaddressed.
EffectiveMediumTheories(EMT)arewidelyusedto repre-senttheeffectsonpermittivityof inclusionsinamatrix.Two very popular laws deduced from these theories are those of Maxwell–Garnett:
εeff =ε1
ε1φ1(1−A)+ε2(φ2+φ1A)
ε1+Aφ1(ε2−ε1)
Fig.8.PermittivityversustemperatureofBST(ICMCBpowder)+4%MgOSPSceramicswithelectrodesparallelandperpendiculartotheinclusionlongaxis.(a) SEMimageofafractureshowingthehomogeneityofthematrix,(b)and(c)3Drenderingoftheinclusions(ingrey)andofthecomputedelectricalfield.
andBruggeman: εeff −ε2 ε1−ε2 ε 1 εeff −A =φ1 (3)
whereεeffistheeffectivepermittivity,ε1thepermittivityofthe
inclusionswithvolumefraction φ1,ε2 thepermittivity ofthe
matrixwithvolumefractionφ2,andAthedepolarization
fac-tordependingontheshapeoftheinclusions.24Toevaluatethe applicabilityofEqs.(2)and(3),thevaluesofAareestimated fromtheexperimentalandnumericalresultsobtainedpreviously (Table2).Bothapproachesgiveastrongcontrastbetween paral-lelandperpendiculardepolarizationfactorsindicatingthatAis effectivelycharacterizingtheanisotropyoftheinclusions. Fur-thermoretheobtainedvaluesforoneequationandonedirection (⊥orq)areremarkablyconsistent(±15%aroundtheaverage). PredictabilityhasbeenevaluatedusingtheAvaluesat268Kto computethe effectivepermittivities atthe othertemperatures. Mostoftheobtainedvaluesarewithin1%oftheexperimental values(8over12),theothershavingdifferencesalwayssmaller than10%.
In spiteofthe significantqualitiesoftheEMTapproaches shownabove,3Dnumericalmodellingcansurpasssomeoftheir weaknesses:
1. Simultaneous estimates of ε2 andA are not possiblewith
Eqs. (2) and(3) while 3D numerical modellingallows it. Thisisimportantwhen,likeinourcase,thepermittivityof thematrixisnotpreciselyknown.
2. Most of the relations produced using EMT are limited to two materials. 3D numerical modellingcan potentially handle any number of materials. The composite mate-rial,whichcharacterizations are presentedin Figs.2–7,is heterogeneous with a BST matrix comprising zones with large and small grains. As BST permittivity is consider-ably varying withgrainsize,14,25 the attemptsmade to fit experimentalandnumericalresultsconsideringa homoge-neousmatrixtotallyfailed.Encouragingpreliminaryresults havebeenobtainedintroducinganewregionwitha differ-entpermittivityaroundtheinclusions,butas XCMTisnot sensitivetograinsizewhenthechemicalcompositionis con-stant, an adapted methodhas to be developedin orderto
Table2
EstimatesofthedepolarizationfactorAappearinginMaxwell–Garnett(subscriptMG,Eq.(2))andBruggeman(subscriptB,Eq.(3))intheperpendicular(⊥)or parallel(q)directionsforthefourtestedtemperatures.
Temperature(K) 250 262 268 276
AMG⊥ 4.36×10−3 5.37×10−3 6.11×10−3 6.25×10−3
AMGq 5.40×10−4 5.65×10−4 5.82×10−4 7.11×10−4
AB⊥ 1.67×10−1 2.00×10−1 2.18×10−1 2.15×10−1
Fig.A1.Diagramoftheproblemtobesolvedatthemacroscopicscale(left)andthemicroscopicscale.
objectivelydelineatetheboundarybetweenlargeandsmall grainsregions.
3. Extensionofthe3Dnumericalapproachtonon-linear ferro-electricmaterialsiseasierthanforEMT.Suchanextension hasbeentestedandcomparisonswithexperimentaltunability dataareinprogress.Comparisonsarecomplexbecauseone ofthecriticalkeypointsappearstobethenon-linearrelation used forthepermittivityevolutionwiththelocalelectrical field.
4. Conclusions
ThankstothespecificconditionsofSparkPlasmaSintering (SPS),anisotropiccompositesmadeofdielectricinclusionswith alargeaspectratiowithinaferroelectricmatrixwereperformed. X-rayComputedMicroTomographyallowed3Drenderingof the sample in both the initial mixed powders and the final ceramic.Theinitialsoft MgOgranulesaredeformedinto flat discshapedinclusionswithsharpedgesfacingeachother.The 3D anisotropy enhanced by the increaseof the SPS pressure affectstheelectricfieldredistributionbetweenthetwophases andappearstobeanefficientwaytoreducethedielectric per-mittivity whilemaintaining electricfieldtunability.Dielectric measurements performedwithelectrodesparalleland perpen-diculartotheinclusionlongaxisshowedoverallanisotropyof thedielectricpropertiesoftheBST/MgOcomposites.In addi-tionweshowedthatinclusionanisotropyalsoinducesanisotropy inthepiezoelectricandpyroelectricpropertiesofthematrix.To evaluate theeffectof themicrogeometry onthe ferroelectric propertiesweuseda3Dnumericalmodelabletohandlethe real-istic3DimagesprovidedbyXCMTdirectly.XCMTappearsto bearelevanttooltolinkthemacroscopicexperimentaldatato 3Dmorphologyinfunctionalmulti-materials.
Acknowledgements
M.PatéandJ.P.GannefromThalesResearch&Technology, rd.128,91767PalaiseauCedex,Franceareacknowledgedfor tunabilitymeasurements.ThisworkwassupportedbytheANR
ARCHIFUN project, grantANR-12-BS08-009 of the French NationalResearchAgency.
Some of these micro-tomography experiments were per-formed on the ID19 beamline at the European Synchrotron RadiationFacility(ESRF), Grenoble,France. Weare grateful toE.BolleratESRFforprovidingassistanceinusingbeamline ID19.
Some of these micro-tomography experiments were per-formedonthe TOMCATbeamlineattheSwissLight Source (SLS), Paul Scherrer Institut, Villigen, Switzerland. We are gratefultoR.MoksoatSwissLightSourcewhoseoutstanding effortsmadetheseexperimentspossible.
AppendixA. Annex
The objective is to evaluate the effective permittivity of a composite sample simulating the macroscopic experience schematizedinFig.A1.AdifferenceofpotentialVLisimposed betweentwoparallelopposingfacesofasample.Thesample thicknessLisverylargecomparedtothesizeoftheinclusions andadirectnumericalmodellingrequires ameshwithsmall enoughelementsinordertocapturethemicrostructurerevealed by3DXCMTimages.Handlingsuchalargediscreteproblemis difficultandgenerallyunnecessary.Abetterapproachconsists inreformulating the problemat the scale of a representative sub-volumesmallcomparedtoL,butlargeenoughtoprecisely representthecompositematerial(Fig.A1).
At thislocal scale Eq. (1) is still valid, but the boundary conditionsaredifferent.ThepotentialVisdecomposedintoa sumoftwoterms:
V =Eapp(x+w) (A1)
whereEapp=VL/L is the average applied electric field, x the coordinateperpendiculartothefacesatconstantpotential,and
wthenewunknown.Eappxrepresentstheaveragepotentialand
Eappwthelocaldeviationofthepotentialfromtheaverageone. Eq.(1)canbereformulatedforthenewunknownw:
whereexistheunitvectorofdirectionX.Attheinternal bound-aries betweentwo different regions (notedi and jhere) w is continuousandthefollowingboundaryconditionapplies: (εi∇wi)·nij=(εj∇wj)·nij+(εj−εi)ex·nij (A3)
wherenijisthevectornormaltotheinterfacebetweenregions
iandjdirectedfromitoj.
Toclosethesystemboundaryconditionsmustbespecified attheexternallimits.Asthe computationdomainisa Repre-sentativeElementaryVolume(REV,i.e.avolumewhichsizeis smallerthanLandlargerthanthe characteristiclength ofthe inclusions,andavolumecontainingenoughinclusionsto repre-senttheaveragebehaviourofthematerial),byanalogywiththe volumeaveragingapproach26wcanbeconsideredasperiodic. Periodicityisaweakboundaryconditiongivingthebestresults whenthegeometrywithinthecomputationdomainisnotstrictly periodic.27wiscontinuousandtherelation(A3)applieswhen
inclusionsarecutbytheexternalboundary.
ThevolumesourcetermspresentinEq.(A2)andthesurface sourceterms presentinEq.(A3) arefunctionsof the permit-tivity contrasts and of the local geometry. The system (A2) and(A3)isdiscretizedusingaclassicalfinitevolumescheme andthe resultingsparse linear systemsolvedusing aparallel solver.Theeffectivepermittivityiscomputedfromthe hetero-geneouswfieldthroughthefollowingrelationthatexpressesthe conservationoftheelectrostaticenergyofthesystem:
εeff =E2appω=
Z Z Z
ε(X)E2dω (A4)
whereωisthesizeofthelocalcomputationdomainandEthe localelectricfield.
References
1.BernardD,NielsenØ,SalvoL,CloetensP.Permeabilityassessmentby3D interdendriticflowsimulationsonmicrotomographymappingsofAl-Cu alloys.MaterSciEngA2005;392:112–20.
2.BernardD,GendronD,HeintzJ-M,BordèreS,EtourneauJ.Firstdirect3D visualisationofmicrostructuralevolutionsduringsinteringthroughX-ray computedmicrotomography.ActaMater2005;53:121–8.
3.KnackstedtM,ArnsCH,SaadatfarM,SendenT-J,LimayeA,Sakellariou A,etal.Elasticandtransportpropertiesofcellularsolidsderivedfrom three-dimensionaltomographicimages.ProcRSocA2006;462:2833–62.
4.FlukigerF,BernardD.Anewnumericalmodelforporescaledissolutionof calciteduetoCO2saturatedwaterflowin3Drealisticgeometry:principles
andfirstresults.ChemGeol2009;265:171–80.
5.LimodinN,RéthoréJ,BuffièreJ-Y,GravouilA,HildF,RouxS.Crack closureandstressintensityfactormeasurementsinnodulargraphitecastiron usingthree-dimensionalcorrelationoflaboratoryX-raymicrotomography images.ActaMater2009;57:4090–101.
6.ElissaldeC,EstournèsC,MaglioneM.Tailoringdielectricpropertiesof multilayer composites usingspark plasma sintering.J AmCeram Soc
2007;90:973–6.
7.ChungU-C,ElissaldeC,EstournèsC,PatéM,GanneJ-P,MaglioneM. Lowlosses,highlytunableBa0.6Sr0.4TiO3/MgOcomposite.ApplPhysLett
2008;92:0429021–429023.
8.ElissaldeC,ChungU-C,ArtemenkoA,EstournèsC,CostesR,PatéM,etal. StoichiometryandgrainboundariescontrolbySparkPlasmaSinteringin Ba0.6Sr0.4TiO3:Mn/MgOcomposites.JAmCeramSoc2012;95:3239–45.
9.SenguptaL.Brevet.U.S.5635434;1997.CeramicFerroelectricComposite materialBSTO-MagnesiumbasedCompound.
10.ChangW,SenguptaL.MgO-mixedBa0.6Sr0.4TiO3bulkceramicsandthin
filmsfortunableapplications.JApplPhys2002;92:3941–6.
11.PiazzaD,CapianiC,GalassiC.Piezoceramicmaterialwithanisotropic gradedporosity.JEurCeramSoc2005;25:3075–8.
12.OlariuCS,PadurariuL,StanculescuRE,BaldisserriC,GalassiC,Mitoseriu L.Investigation oflowfield dielectricproperties ofanisotropicporous Pb(Zr,Ti)O3 ceramics:experimentandmodeling.JApplPhys2013;114,
214101/1-9.
13.ShermanVO,TagantsevAK,SetterN.Modelofalow-permittivityand high-tunabilityferroelectricbasedcomposite.ApplPhysLett2007;90, 162901/1-3.
14.CurecheriuL,BuscagliaM-T,BuscagliaV,ZhaoZ,MitoseriuL.Grainsize effectonthenon-lineardielectricpropertiesofbariumtitanateceramics.
ApplPhysLett2010;97,242909/1-3.
15.PadurariuL,CurecheriuL,GalassiC,MitoseriuL.Tailoringnon-linear dielectricpropertiesbylocalfieldengineeringinanisotropicporous ferro-electricstructures.ApplPhysLett2012;100,252905/1-5.
16.PadurariuL,CurecheriuL,BuscagliaV,MitoseriuL.Field-dependent per-mittivityinnanostructuredBaTiO3ceramics:modelingandexperimental
verification.PhysRevB2012;85,224111/1-9.
17.BuzugT-M.Computedtomography.Berlin,Germany:Springer;2008.
18.ChaimR,MarderR,EstournèsC,ShenZ.Densificationandpreservation ofceramicnanocrystallinecharacterbysparkplasmasintering.AdvAppl Ceram2012;111:280–5.
19.CramponJ.Creepmicrostructureofultrafine-grainedMgOpolycristals.
ActaMetall1980;28:123–8.
20.ConradH,YangD.Influenceofanelectricfieldontheplastic deforma-tionoffine-grainedMgOathighhomologoustemperatures.ActaMater
2000;48:4045–52.
21.MasonW-P.Piezoelectriccrystalsandtheirapplicationtoultra-sonics. NewYork:VanNostrand;1950.
22.TuncerE,SerdyukYV,GubanskiSM.Dielectricmixtures:electrical prop-ertiesandmodeling.IEEETransDielectElectInsul2002;9:808–28.
23.SareniB,KrähenbühlL,BeroualA,BrosseauC.Effectivedielectricconstant ofperiodiccompositematerials.JApplPhys1996;80:1688–96.
24.MyroshnychenkoV,BrosseauC.Finite-elementmethodforcalculationof theeffectivepermittivityofrandominhomogeneousmedia.PhysRevE
2005;71,016701/1-16.
25.FreyM-H,PayneDA.Grain-sizeeffectonstructureandphase transforma-tionsforbariumtitanate.PhysRevB1996;54:3158–68.
26.WhitakerS.Themethodofvolumeaveraging.Dordrecht:KluwerAcademic Publication;1999.
27.KantiT,ForestS,GallientI,MounouryV,JeulinD.Determinationofthe sizeoftherepresentativevolumeelementforrandomcomposites:statistical andnumericalapproach.IntJSolidsStruct2003;40:3647–79.
