Optical birefringence in uniaxially compressed aerogels
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Title: Optical birefringence in uniaxially compressed aerogels
Series Title: New J. Phys. 12, 103016 (2010)
Physical Description: Journal Article
Creator: Lee, Yoonseok
Bhupathi, P.
Jaworski, L.
Hwang, J.
Tanner, D. B.
Obukov, S.
Mulders, N.
Publisher: IOP Publishing Ltd
Place of Publication: London, UK
Publication Date: October 13, 2010
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Abstract: The uniaxial strain dependence of refractive index anisotropy has been measured for a series of high-porosity aerogel samples. Uniaxial compression of compliant aerogels introduces an optical activity into the material. We report on the compression-dependent optical birefringence of samples with porosities from 95 to 99% under a uniaxial compression of up to 15% in the spectral range 320–800 nm.
Acquisition: Collected for University of Florida's Institutional Repository by the UFIR Self-Submittal tool. Submitted by Yoonseok Lee.
Publication Status: Published
Funding: Publication of this article was funded in part by the University of Florida Open-Access Publishing Fund.
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Holding Location: University of Florida
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Optical birefringence in uniaxially compressed aerogels This article has been downloaded from IOPscience. Please scroll down to see the full text article.2010 New J. Phys. 12 103016(http://iopscience.iop.org/1367-2630/12/10/103016)Download details:IP Address: 128.227.185.95The article was downloaded on 06/09/2011 at 20:15 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience

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Opticalbirefringenceinuniaxiallycompressedaerogels PBhupathi1;4,LJaworski1,JHwang1;2,DBTanner1,SObukov1,YLee1;5andNMulders3 1DepartmentofPhysics,UniversityofFlorida,Gainesville,FL32611-8440,USA2DepartmentofPhysics,PusanNationalUniversity,Busan609-735,RepublicofKorea3DepartmentofPhysicsandAstronomy,UniversityofDelaware,Newark,DE19716,USAE-mail: yoonslee@phys.u.edu NewJournalofPhysics12103016ppReceived26May2010Published13October2010Onlineat http://www.njp.org/ doi:10.1088/1367-2630/12/10/103016 Abstract.Theuniaxialstraindependenceofrefractiveindexanisotropyhasbeenmeasuredforaseriesofhigh-porosityaerogelsamples.Uniaxialcompressionofcompliantaerogelsintroducesanopticalactivityintothematerial.Wereportonthecompression-dependentopticalbirefringenceofsampleswithporositiesfrom95to99%underauniaxialcompressionofupto15%inthespectralrange320nm. 4Presentaddress:DepartmentofPhysics,SyracuseUniversity,Syracuse,NY13244,USA.5Authortowhomanycorrespondenceshouldbeaddressed. NewJournalofPhysics121030161367-2630/10/103016+20$30.00IOPPublishingLtdandDeutschePhysikalischeGesellschaft

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2 Contents 1.Introduction 2 2.Principleofmeasurement 3 3.Experimentalmethod 4 4.Resultsanddiscussion 6 4.1.Mechanicalproperties ............................... 6 4.2.Poissonratiomeasurements ............................ 6 4.3.Opticaltransmissionmeasurements ....................... 8 4.4.Effectivemediummodel ............................. 14 5.Summary 17 Acknowledgments 18 Appendix 18 References 191.IntroductionHighlyporousandtransparentsilicaaerogelswererstsynthesizedin1931[ 1 ].Manufacturedbyasolgelmethod[ 2 3 ],aerogelsarecomposedofatenuousnetworkofSiO2strandsandcanbesynthesizedinawiderangeofporosities,especiallyinthehighporositylimit,whichextendsupto99.9%only0.1%silica.Theyare,infact,thelightestsolidmaterialswiththelowestrefractiveindexevermanufactured.Owingtotheiruniquestructureandhighporosity,aerogelshavefoundawidevarietyofapplicationsinscienceandtechnology[ 4 ].SilicaaerogelshavebeenextensivelyusedinCerenkovcountersinparticlephysicssincethe1980sandhaverecentlybeenutilizedtocapturecosmicdustinouterspace[ 5 6 ].Theirlowthermalconductivityandhighsolartransmittanceofferpossibilitiesforuseassuperinsulatingllersforwindowsystemsandasasolarenergycollector[ 7 ].Onapurelyacademicside,aerogelshavealsofoundauniqueroleasimpuritiesorquencheddisorderinsystemssuchasliquidcrystals[ 8 ]andquantumuids,e.g.liquid4He[ 9 ],3He[ 10 11 ]andtheirmixtures[ 12 ].Theinuenceofquencheddisorderonunconventionalsuperuidsandsuperconductorsisofimmenseimportanceintheeldofcondensedmatterphysicsandourmotivationforthispaperspringsfromourstudiesofsuperuid3Heinaerogel.Theaerogelstrands,3nmindiameter,encloseopenvolumesorporestypically30nmwide,bringingthegeometricalmeanfreepathtoarangeof100150nmatnominally98%porosity.Thecoherencelengthinthesuperuid3HethecharacteristicsizeofaCooperpair:from16nmat0barto77nmat34bariscomparabletothemeanfreepathprovidedbytheaerogelstructure,butismuchlargerthantheaerogelstranddiameter.Therefore,liquid3Heimbibedinhigh-porositysilicaaerogelallowsasystematicinvestigationoftheeffectsofdisorderonap-wavespin-tripletsuperuid.Varioustheoreticalmodelsforsuperuid3Heinaerogelexplainsomebutnotalloftheexperimentallyobservedfeaturesofsuperuid3Heinaerogel.Inparticular,therehasbeencontinuinginterestintheconsequencesoftheinteractionbetweenanisotropicdisorderpresentedbyaerogelandtheanisotropicorderparameterofsuperuid3He[ 13 ][ 16 ].Ithasbeenproposedthatthiseffectcanbesystematicallystudiedbyintroducingcontrolledglobalanisotropythroughuniaxialdeformationofaerogel[ 16 ].Thisschemeisattractivebecause NewJournalofPhysics12103016 http://www.njp.org/

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3 high-porosityaerogelspossessanextremelylowYoung'smodulus0:1MPaandexcellentmechanicalcompliance.Therefore,itisimperativetohaveaquantitativecharacterizationofamountofanisotropygeneratedbycompressionorstretchingforasystematicinvestigation.Recently,Pollanenetal[ 17 ]demonstratedthepresenceofglobalanisotropyincompressedaerogelsthroughopticalbirefringence.Theymeasuredthetransmittanceofwhitelightpassingthroughcylindricalaerogelsamplesundervariousdegreesofcompressionbetweentwocrossedpolarizers.However,noquantitativeandspectroscopicinformationonbirefringencewasobtained.Inthispaper,wereportthequantitativeresultsofouropticalcharacterizationsofhigh-porosityaerogelsamplessubjectedtoauniaxialstrainofupto15%.Thisworkisanextensionofourearlierpublication[ 18 ]on98%porosityaerogels.Inthispaper,wepresentmoredetailedmeasurementsincludingthemechanicalpropertiesofaerogelswith95,97,98and99%porositiesandamodelforourresultsbasedoneffectivemediumapproximationEMA.2.PrincipleofmeasurementTheexaminationofadielectricmaterialplacedbetweentwocrossedpolarizersisastandardmethodofseparatinganisotropicmaterialsfromthosethatareisotropic.Itisbasedonthefactthatalinearlypolarizedlightpropagatingthroughauniaxiallyanisotropicmediumexperiencesdifferentindicesofrefractionfortworayswithmutuallyorthogonalpolarizations,namely,theordinaryrayORandtheextraordinaryrayER.Asaresult,thelightemergesfromthemediumwithaphasedifferencebetweenthetwopolarizationsexpressedby D2d1n ; wheredisthethicknessofthesampleandisthewavelengthoflight.1nrepresentsthebirefringenceofthesample;itisdenedas1nDne)]TJ/F37 11.955 Tf 11.318 0 Td[(no,whereneandnoare,respectively,theindicesofrefractioncorrespondingtoERandOR.Equation 1 holdstruewhentheangleofincidenceoflightisnormaltothesample,whichisthecaseforallourmeasurements.Figure 1 showsaschematicdiagramofourexperimentalsetup.Supposethattheopticaxisisalongthecompressionaxis,asindicatedingure 1 .IfabeamofintensityIoisincidentonthesample,placedbetweentwolinearpolarizers,thetransmittedintensitiesforthecrossedI?andparallelIkorientationsoftheanalyzerrelativetothepolarizeraregivenby[ 19 ] I?DIosin.2/sin2 2; IkDIosin.2/cos2 2; whereistheanglebetweenthepolarizationaxisofthepolarizerandtheopticaxis,xedat45inourexperiments.Here,wehaveneglectedtheabsorptioncoefcientsassociatedwiththeneandnodirections.Equations 2 and 3 canbeinvertedtondthephasedifference, jjDk+2tan)]TJ/F60 7.97 Tf 6.217 0 Td[(1s I? Ik;kD0;2;4;:::; jjD.k+1/)]TJ/F60 11.955 Tf 11.318 0 Td[(2tan)]TJ/F60 7.97 Tf 6.217 0 Td[(1s I? Ik;kD1;3;5;:::: NewJournalofPhysics12103016 http://www.njp.org/

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4 Figure1.Avisualizationoftheexperimentalsetup.Collimatedmonochromaticlightbeamispassedthroughapolarizerwithitspolarizingaxisxedat45tothecompressionaxisofthesample.Theoutputintensityismeasuredthroughananalyzeratanglerelativetothepolarizertransmissionaxis. BymeasuringtheintensitiesI?andIk,onecanextractthephasedifferenceandhence1nfromequation 1 .Notethattheabovesolutionsonlygivetheabsolutevalueof1nbutnotthesign.Thesignhastobeinferredbyothermeans,discussedlater.Becausewearedealingwiththeuniaxialcompressionoftheaerogel,wehavetoaccountfortheincreaseinthicknessdinthedirectionperpendiculartothecompressionbyincorporatingthePoissonratiooftheaerogel.IfdoistheoriginalthicknessandListhelengthoftheuncompressedaerogel,thenforasmalldeformation1L,thethicknessofthecompressedaerogelis dDdo1+1L L; whereisthePoissonratiooftheaerogel.Thismodiedthicknessneedstobeincorporatedintoequation 1 toextract1n.3.ExperimentalmethodWeadoptedatechniquesimilartotheonedevelopedtoinvestigatebirefringenceinliquidcrystals[ 20 ]andpolymers[ 21 ].Wehaveopticallycharacterizedthreeaerogelsampleswith98%porositybatchA,whichwillbereferredtohereafterassamples1.Anotherbatchofaerogelsampleswithporositiesrangingfrom95to99%batchBhavebeenusedtomeasuretheirPoissonratiosandopticalbirefringence.Theaerogelsusedinthisstudyweremadefollowingtheso-calledtwo-stepmethodrstdescribedbyTillotsonandHrubesh[ 2 ],usingacetonitrileasthesolventtosetthenalaerogeldensity.Thegelsaretypicallyagedfortwoweeksandthensupercriticallydriedat290C.Theresultingaerogelsarehydrophobic.Thespecicsurfacearea,asdeterminedfromheliumadsorptionisotherms,isabout1000m2g)]TJ/F60 7.97 Tf 6.217 0 Td[(1withaprimaryparticlediameterof3nm.Thedensityof98%aerogelis0:044kgm)]TJ/F60 7.97 Tf 6.217 0 Td[(3. NewJournalofPhysics12103016 http://www.njp.org/

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5 Figure2.Imagesofthesurfacesoftheaerogelinwhitelight.Cutsurfaceinaandas-grownsurfaceinb. TheopticalmeasurementswereperformedusingaZeissMPM800microspectrophoto-meterequippedwithaXelampasthelightsourceandaphotomultipliertubeasthedetector.Aplatformwasdesignedformountingtheaerogelbetweenthepolarizerandanalyzerofthemicroscope,sothatthelightpropagatesinthedirectionperpendiculartothecompressionaxis.Theaerogelsamplesweresynthesizedinglasstubesintheshapeofrightcircularcylinders.Twoplanesurfaceswerecutparalleltothecylinderaxisoneithersideofthecylinderusingahigh-speeddiamondcutter.Thisprocessprovidedtwoparallelandatsurfacesforopticalmeasure-ments,eliminatingtiltofthebeamatthesurfaceoftheaerogel.Thecutandas-grownsurfacesofanaerogelsamplewereimagedusingawhitelightandpicturesofthoseimagesareshowningures 2 aandb.Atthescaleobserved,thepicturessuggestthattheas-grownsurfaceoftheaerogelismuchsmootherthanthemachine-cutaerogel.Thiscouldbeofimportancetotrans-verseacousticimpedancemeasurementsofsuperuid3Heinaerogel,wherethecontactbetweentheaerogelandthetransducersurfacesiscrucialinobservingthesuperuidtransitionfeatures.Thepolarizedlightbeam0:50:3mm2wasfocusedonthesamplewithafocusinglensandtheoutputlightwasviewedthrougha10objectivelenslocatedbeforetheanalyzergure 1 .Theaerogelsampleswerecompressedalongthecylindricalaxisandthestrainwasdeterminedusingamicrometervisewithanon-rotatingspindle.Foreachcompressiondecompressioncycle,thewavelengthwasscannedfrom320to800nmin4nmincrementsandtheoutputintensityoflightwasmeasuredforvariousanglesof,theangleoftheanalyzerrelativetothepolarizertransmissionaxisthatwasalwaysxedat45tothecompressionaxis.Strongabsorp-tionbytheopticalcomponentslimitedtheshort-wavelengthmeasurementstoapproximately320nmandthephotomultipliercutoffdictatedthe800nmlong-wavelengthlimit. NewJournalofPhysics12103016 http://www.njp.org/

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6 Figure3.GraphshowingshrinkageofaerogelsbatchAaftereverycycleofcompressionanddecompression.Thebarsinthegraphdenotethelengthmeasuredbeforeeachcycleto15%compression.Eachcompressiondecompressioncycletypicallytookabout4h.Nosignicantrelaxationinthelengthwasobservedafterthecompletionofeachcycle. 4.Resultsanddiscussion4.1.MechanicalpropertiesBeforewediscussthebirefringence,wewouldliketobrieydescribesomemechanicaleffectsofcompressivestrainontheaerogel.Itisknownthatcompressioncanleadtoareductionintheelasticmodulusofhigh-porosityaerogels[ 22 23 ],andshrinkageordamagecanoccurinaerogelsduringthesupercriticaldryingstagesofthegel[ 17 24 25 ]orduringcapillarycondensation[ 26 ].Inourmeasurementsweobservedasubstantialamountofshrinkageinlengthalongthecompressionaxisfortheaerogelsamplesthatunderwentcyclesofcompressionanddecompression.Ingure 3 thelengthoftheaerogelbeforeeachcycleofcompressionupto15%isplottedforthreesamplesofbatchA.Ingeneral,theshrinkageisfoundtobethelargestaftertherstcompressiondecompressioncycleanddecreasesforthesubsequentcycles.Afterthecompletionoffourcycles,thetotalamountofshrinkagewas9,7.5and6.7%forsamples1,respectively.However,noshrinkagewasobservedforthesamplescycledupto5%compression.OurobservationisconsistentwiththeelasticmeasurementsofGrossetal[ 22 ],inwhichmostsamplesrecovered99.5%orbetteroftheiroriginallengthaftercompressionsofafewpercent,althoughwehavetopointoutthattheirsamplepreparationmethodwasdifferentfromours.4.2.PoissonratiomeasurementsToourknowledge,theonlyknownmeasurementsofthePoissonratioofhigh-porosityaerogelsintheliteraturearefromGrossetal[ 23 26 ]andPollanenetal[ 17 ].Grossetaldetermined NewJournalofPhysics12103016 http://www.njp.org/

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7 Figure4.Thewidthversuslengthofanaerogelsamplewith95%porosityundergoingacompressionofupto15%.Theblacksquaresredcirclesweretakenoncompressiondecompression.Thesolidlineisalinearttothedata. Table1.Poissonratiosofaerogelsamplesofdifferentporosities. Porosity%Poissonratio 950:1120:003970:0940:002980:1470:005990:1430:005 thePoissonratiofromthelongitudinalandtransversesoundvelocityintheaerogelandfound0:2,independentofporosity,whereasPollanenetaldetermined0:30:05basedontheiropticalimages.WemeasuredthePoissonratiobydirectmechanicalcompressionusingthesamemicrometerviseusedinourbirefringencemeasurement.Webelievethatthismethodismoreappropriatefordeterminingtheactualchangeindcausedbyuniaxialcompression.Thesampleswerecutasdescribedinsection 3 andwerecompressedinamicrometervisemountedonaxymicropositioningstageunderamicroscope.Thecompressioninthex-directionwasvariedfrom0to15%determinedbythemicrometerreading.Thechangeinwidthofthesampleinthey-directionwasmeasuredusingthexymicropositioningdevicewiththehelpoftheeyepiececross-hairs.Alaserbeamthreadingthroughthesampleinthey-directionwasusedtoclearlyidentifytheaerogelboundary.Atypicalplotofthewidthversusthelengthofa95%porositysampleisshowningure 4 .Theslopeofastraightlinetsolidlineingure 4 tothedatayieldsthePoissonratio.MeasurementswererepeatedondifferentsampleswiththesameporosityandtheaveragePoissonratiowascalculated.TheaveragePoissonratiosfordifferentporositysamplesaregivenintable 1 andareusedinouranalyses.Ourvaluesfalllowerthan0.2,between0.1and0.15forporositiesfrom95to99%.Thedifferencemightbecausedby NewJournalofPhysics12103016 http://www.njp.org/

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8 Figure5.Thetransmittancetheratioasdenedinsection 4.3 ofsamples1ac,respectivelyinthecrossed-polarizercongurationasafunctionofwavelength.Thesoliddashedlinesaredatatakenoncompressiondecompression. asmallnonlinearityevidentingure 4 becausethePoissonratiodeterminedfromthesoundvelocityispresumablyinthelowstrainlimit.4.3.OpticaltransmissionmeasurementsFigure 5 displaysthetransmittanceofsamples1inthecrossed-polarizercongurationD90asafunctionofwavelengthonthefourthcompressiondecompressioncycle.ThetransmittancewasdeterminedastheratioofthetransmittedintensitytoabackgroundintensitymeasuredwithoutasamplebutwiththetwopolarizersalignedparallelD0.Thetracestakenoncompressiondecompressionarerepresentedbythesoliddashedlines.Inthisconguration,anynitetransmittanceindicatesthepresenceofopticalactivityintheaerogel.Atzerocompression,thetransmittanceisverylow,becauseitisthetransmittanceofanearlyisotropicsubstancebetweencrossedpolarizers.Withincreasingcompressionthereisinitiallyanincreaseintransmittanceacrossthespectrumandthenanoscillatorybehavior,bothwithwavelengthatagivencompressionandwithcompressionatagivenwavelength.Forsample1 NewJournalofPhysics12103016 http://www.njp.org/

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9 Figure6.Transmittanceversuswavelengthofsample1,withprogressivelyincreasingcompressionshowninpanelsgoingfromlefttoright%,atdifferentanglesoftheanalyzerseethelegendontheright. ingure 5 a,thetransmittancereachesabroadmaximumby7%compression.Thepositionofthemaximummovesprogressivelytowardslongerwavelengthsforhighercompressionandthenasecondmaximumemergesfromtheshortwavelengthside.Theseoscillationsarecausedbythebirefringence.Foragivencompression,thepathlengthdandrefractiveindexanisotropy1narexed;however,thephasedifferenceincreaseswithdecreasingwavelength,andthetransmittance,inaccordwithequation 2 ,swingsbetweenthemaximumtransmittanceandzero.Oneotherfactor,scatteringoflightintheaerogel,affectsthespectra.Becausethesizesoftheaerogelstrandsaremuchsmallerthanthewavelengthoflight,Rayleighscatteringisresponsiblefortheincreasingenvelopeofthetransmittancebetween320and800nm.Similarbehaviorhaspreviouslybeenobservedinaerogelswithcomparableporosities[ 27 ].Wenotethatsample1showsnon-zerotransmittanceintheuncompressedstate.Thetrans-mittancerstdecreaseswithcompressionandthenstartstoincreaseafter2%compression.Webelievethatthisbehaviorisduetotheinherentanisotropyintheaerogelsample,probablyfromthegrowthprocessorthroughtherepeatedcompressions[ 17 ].Ourresultsalsoexhibitsubstan-tialhysteresisbetweenthecompressionanddecompression.Forexample,thetransmittancefor7%insample1ondecompressionactuallymatchesthatfor5%oncompression,asshowningure 5 a.Thisresultcanbeunderstoodbytakingintoaccounttheshrinkageinthelengthoftheaerogel2%occurringduringthefourthcycle,ascanbeinferredfromgure 3 .Thebuilt-inanisotropyisalsomanifestinsample3,ascanbeseeningure 5 c.Figures 6 8 displaythetransmittancefordifferentanglesoftheanalyzerforsamples1,respectively.Thepanelsshowdataatprogressivelyincreasingcompression,from0topleftto15%bottomright.Inanuncompressedaerogel,theeffectofopticalbirefringenceismodestduetotheweakbuilt-inanisotropymentionedabovesamples1and3.Therefore,thepolarization-angledependenceofthetransmittanceiseasilyunderstood.Whencompressed NewJournalofPhysics12103016 http://www.njp.org/

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10 Figure7.Transmittanceversuswavelengthofsample2withincreasingcompressionshowninpanelsgoingfromlefttoright%atdifferentanglesoftheanalyzer. Figure8.Transmittanceversuswavelengthofsample3withincreasingcompressionshowninpanelsfromlefttoright%.Thelegendontherightshowsthedifferentanglesoftheanalyzer. by15%,thebirefringenceproducesoscillatorybehaviorasthephasedifferenceisinverselyproportionaltothewavelength.Oneparticularfeatureofthisobservationistheappearanceofthenodesatwhichthetransmittancebecomesindependentoftheangleoftheanalyzer. NewJournalofPhysics12103016 http://www.njp.org/

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11 Atthesenodestheemerginglightiscircularlypolarized,withthephasedifferencebetweenORandERbeinganoddintegermultipleof=2.Forexample,the15%compressedaerogelsample1behavesasaquarter-waveplateatthewavelengthsofthenodesnear375,500,and800nmseegure 6 ,whichisperiodicinfrequency,inaccordancewiththebirefringenceinthecompressedaerogel.Itisworthmentioningthatcompressedaerogelshavethepotentialtoserveastunablewaveplates,withsomeadvantagesoverconventionaltunablewaveplatessuchasBabinetcompensators,includingsmallerreectance,absenceofgradienteffectandnolimitationonthebeamsize.Inordertounderstandfullyourresults,weneedtoconsidertheeffectsofbirefringenceassociatedwithanisotropy,Rayleighscatteringanddispersion.Therefore,theexpressionfortransmittanceinthecrossed-polarizercongurationshouldbemodiedfromequation 2 to TDAexp)]TJ/F37 11.955 Tf 10.234 0 Td[(B 4sin.2/sin2d1n : TherstexponentialtermaccountsfortheRayleighscattering.Thedispersioneffectisincorporatedin1nbyasimpliedformoftheSellmeierdispersionequationcalledtheCauchyformula[ 19 ]: 1nDC+D 2: TheSellmeierformhasbeenpreviouslyusedintheliteraturetotthebirefringencedispersioninglassesandliquidcrystallinematerials[ 21 28 29 ].Thelightpathlengthisalsocorrectedbyequation 6 ,usingthereportedvalueofthePoissonratiosD0:2[ 23 26 ]thePoissonratiosofthesesampleswerenotmeasuredinthiswork.Theabovetwoequationsareusedtotourtransmittanceforallsamplesat15%compressionbyanonlinearleastsquarettingmethod,withA,B,CandDasourttingparameters[ 18 ].Theresultsareshowningure 9 a[ 18 ]assolidlines.Ingure 9 bweplotthe1n./directlyobtainedfromourmeasurementsofI?=Ikusingequations 1 4 and 5 .Forexample,thethreeredtracesrepresentthebirefringenceforsample1,assumingthreedifferentbranchesofkinequations 4 and 5 .Ineachtraceonlythenearlyhorizontalportionsaremeaningfulandtheconjoinedtracesofthehorizontalsectionscorrectlyreectthebirefringenceofthesample.Alsodisplayedintheplotis1n./,calculatedusingequation 8 andCandDobtainedfromthet.Theyareinexcellentagreementwitheachother.Adoptingthesamettingprocedure,wendverygoodagreementwiththemeasured1n./atallcompressions.Theseresultsareshownforsample3ingure 10 forstrainsbetween2and13%overthewholewavelengthrange.Figure 11 ashowsthestraindependenceoftheopticalbirefringenceforthethreesampleswith98%porosity.Allsamplesstudiedfollowaquasi-lineardependencewithaweaknonlinearity.Thebuilt-inanisotropyismanifestinthisplotforsamples1and3.Wecanonlydeterminetheabsolutevalueofbirefringencethroughourmeasurements.Therefore,itisreasonabletothinkthatthebirefringenceofsample1actuallychangesitssignaroundthe3%strainpoint,suggestingthatthesmalluniaxialcompressioncompensates,ineffect,thebuilt-inanisotropyinthesample.Incontrast,thebuilt-inanisotropyinsample3isinthesamedirectioncausedbyuniaxialcompression.1nforallcompressioncyclesofsample3isshowningure 11 b.Ineachcyclethestrainwasdeterminedbasedonthesamplelengthatthebeginningofeachcycle.1nshiftsupwiththenumberofcyclesandseemstosaturateatthefourthcycleofcompression.However,theoverallchangein1nforagivenstrainrangeremainsalmostthesame. NewJournalofPhysics12103016 http://www.njp.org/

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12 Figure9.Transmittanceandbirefringencedatainthecross-polarizedsetupforsamples1at15%compression.Thesolidlinesinbothpanelsaretheresultsofthetasexplainedinthetext. Figure10.Birefringencedispersioninaerogelsample3forvariousstrainsseethelegendrangingfrom2to13%.Theblacklinesarethe1ncalculatedbyttingthewavelength-dependenttransmittancedatausingequations 7 and 8 NewJournalofPhysics12103016 http://www.njp.org/

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13 Figure11.a1nversusstrainat632nmforsamples1ontheirfourthcycleofcompression.b1nversusstrainat632nmforsample3atdifferentcyclesofcompressionfromitsvirginconditioncycle1. Figure12.aTotaltransmittancehT?iinthecross-polarizedcongurationintegratedforallwavelengthsnmasafunctionofstrainforallsamples.bThetransmittanceT?ofsample2betweenthecrossedpolarizersforseveralwavelengths. Figure 12 aillustratesthetotaltransmittancehT?iofthethreesamplesintegratedoverthewholewavelengthrangeasafunctionofappliedstrain.Thetransmittanceforsamples2and3risesinitiallyandstartstodropbeyond10%.Sample1behavesalittledifferently, NewJournalofPhysics12103016 http://www.njp.org/

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14 Figure13.StraindependenceofbirefringenceforaerogelsamplesbatchBwithvariousporosities. withamorepronouncedoscillatorybehavior.Asexplainedpreviouslyinthediscussionofgures 6 8 ,thetransmittanceisexpectedtooscillatesincetheaerogelplacedbetweenthecrossedpolarizersactsjustlikeawaveplatetunablewiththecompressivestrain.OurresultsareincontrasttotherecentobservationbyPollanenetal[ 17 ].Theyperformedasimilarmeasurementwitha98%cylindricalaerogelsample8mmdiameterbetweentwocrossedpolarizers.Theirtransmittanceusingwhitelightshowedalinearstraindependencetotheirhighestuniaxialcompressionat20%andwasuniformthroughoutthesample.Inthiscase,oneeffectivelymeasurestheaverageintensityoflighttraversingdifferentpathlengthsthroughoutthecylindricalsample.Figure 12 bplotsthewavelength-dependenttransmittanceT?ofsample2,whichconformstothewaveplatecharacterofthecompressedaerogel,wherethemaximuminthetransmittanceoccursatdifferentcompressionsinducedatdifferentwavelengths.Asimilarbehaviorwasobservedforsampleswithporositiesotherthan98%.Figure 13 depictstheopticalbirefringenceasafunctionofstrainonsampleswithporositiesof95,97,98and99%batchB.InevaluatingthebirefringenceourmeasuredPoissonratiovalues,asdeterminedintheprevioussection,wereused.Thedatashowningure 13 wereobtainedduringthesecondcompression-decompressioncycle.Asseenfromthegure,thebirefringenceforthevariousporositiesisofcomparablemagnitudeandthe99%aerogelsamplealsoshowsbuilt-inanisotropyalmostidenticaltothatofsample1.4.4.EffectivemediummodelInthissection,wedevelopasimplemodelforourresultsinanattempttocorrelatethebirefringencewiththestructuralchanges.Theopticalpropertiesoftheaerogelcanbemodeledbythewell-knownEMAs[ 30 31 ],theBruggemannEMAortheMaxwellGarnetttheory NewJournalofPhysics12103016 http://www.njp.org/

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15 MGT.Theseapproachesareapplicableherebecausethewavelengthofthelightusedismuchlongerthantheinter-stranddistance.Theeffectivedielectricfunctionofthecompositemediumiscalculatedbyaveragingthedielectricpermittivitiesofitsconstituents,namelyairandSiO2grainswiththeirrespectivevolumefractions.Inthecaseofcompressedaerogels,thedielectricfunctionisatensorduetotheanisotropygeneratedandthedepolarizationeffectsbecomeimportant.Notethattherearenoabsorptionbandsintheaerogelforthemeasuredwavelengthrange,sothattherefractiveindexcanbesimplycalculatedbysettingnDp ".IntheBruggemanmodel,consideringtheaerogelasacollectionofuniformlydistributedneedle-shapedSiO2pieces,thedielectricconstant"Bisformulatedas f"SiO2)]TJ/F132 11.955 Tf 11.317 0 Td[("B g"SiO2+.1)]TJ/F37 11.955 Tf 11.581 0 Td[(g/"B+.1)]TJ/F37 11.955 Tf 13.936 0 Td[(f/"air)]TJ/F132 11.955 Tf 11.318 0 Td[("B g"air+.1)]TJ/F37 11.955 Tf 11.581 0 Td[(g/"BD0; wherefisthellingfractionofSiO2,"SiO2D2:34and"airD1:00.gisthedepolarizationfactor,relatedtothetopologyoftheinclusionmaterialandtheorientationoftheappliedelectriceld.Forthecylindricalaxisparallelandperpendiculartotheelectriceld,gkD0andg?D1=2,respectively.WithfD0:02%porosity,theindicesofrefractionfortwoperfectlyalignedcasesareevaluatedasnoD1:0133fortheordinarycomponentgkD0andneD1:0081fortheextraordinarycomponentg?D1=2.Thissetstheupperboundofbirefringenceforthismodel1nDne)]TJ/F37 11.955 Tf 11.317 0 Td[(no)]TJ/F60 11.955 Tf 21.306 0 Td[(510)]TJ/F60 7.97 Tf 6.217 0 Td[(3,areasonableresultconsideringthatourbirefringenceat15%is510)]TJ/F60 7.97 Tf 6.217 0 Td[(5.TheindexofrefractionforanisotropicaerogelcanbeestimatedtobenD.1=3/no+.2=3/ne1:010,ingoodagreementwiththepreviouslyreportedvalues[ 32 ].IntheMGT,wemodelthesystemasacollectionofSiO2ellipsoidsembeddedinairwiththerespectivedielectricpermittivities.Theprocedurediscussedherehasbeensuccessfullyappliedbeforeinunderstandingtheformbirefringenceofporoussemiconductorsanddielectrics[ 33 34 ].TheMGTdielectricfunctionfororientedellipsoids[ 31 ]isgivenby "G?;kD"air+"airf."SiO2)]TJ/F132 11.955 Tf 11.317 0 Td[("air/ g?;k.1)]TJ/F37 11.955 Tf 13.936 0 Td[(f/."SiO2)]TJ/F132 11.955 Tf 11.318 0 Td[("air/+"air: ThespectraldispersionoftherefractiveindexofSiO2intheaboveequationisgivenbyathree-termSellmeierformgivenbyMalitson[ 28 ]: n2SiO2./D1+XiAi2 2)]TJ/F132 11.955 Tf 11.317 0 Td[(2i; whereforiD1;2;3,AiandiaretheconstanttparametervaluesdeterminedfromtheexperimentallymeasuredvaluesoftherefractiveindexofSiO2see[ 28 ].Theconstantsgkandg?appearinginequation 10 arethegeometricaldepolarizationfactorssimilarlydenedasintheBruggemannmodel[ 35 ],correspondingtotheelectriceldbeingparallalorperpendiculartotheprincipalaxesoftheellipsoid.Theirvaluesdependontheratiosofthepolartotheequatorialaxesofanellipsoid.Tabulatedvaluesofgcanbefoundintheliterature[ 36 ].Forexample,aplanehasgkD1;g?D0,aninnitecylinderhasgkD0;g?D0:5andaspherehasgkDg?D0:333.Iftheellipsoidisaspheroidwheretwooftheprincipalvaluesofgareequal,wehavetheconditiongk+2g?D1.Usinggkasthettingparameter,thebirefringencene)]TJ/F37 11.955 Tf 11.318 0 Td[(noDp "k)]TJ 11.317 7.774 Td[(p "?iscalculatedusingequation 10 ,whichalsogivesanegativesigntoourbirefringencevalues.Thisiscomparedagainstourmeasuredvaluesofbirefringencein NewJournalofPhysics12103016 http://www.njp.org/

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16 Figure14.Measuredbirefringencevaluesofsample1at15%compressioncirclesonthelefty-axisand2%compressiontrianglesontherighty-axis.SolidblacklinesarefromtheMGTequation 10 ,withgkD0:340for15%compressedaerogelandgkD0:333for2%compressedaerogel. gure 14 .Thebottomtraceofthegureshowsthebirefringenceofsample1at2%compressionandthetoptraceat15%compression.ThebesttstoourexperimentaldatayieldvaluesofgkD0:340for15%compressedaerogeland0.333for2%compressedaerogel,indicatingthatat2%compression,theaerogelparticlesmaintainasphericalshape,andwhencompressedby15%theydeviatetowardsaspheroidbyabout3%inlength.Bothmodelspredict1n/f,whichisinqualitativeagreementwithourresultsseegure 13 .Whentheneedle-shapedaerogelstrandsarenotperfectlyalignedthebirefringencecanbeexpectedtobelowerthantheperfectlyalignedcase.ThedegreeofmisalignmentcanbedescribedbytheHermansorientationfunction./[ 37 ],suchthat 1nD1nmax./; where./isgivenby1 2.3hcos2i)]TJ/F60 11.955 Tf 17.697 0 Td[(1/and1nmaxisthemaximumbirefringenceforthecaseofperfectalignment.hcos2iistheaverageoverpolaranglebetweentheopticaxisandtheneedledirection.Thisquantityis1=3foranisotropicconguration.TherearevarioustheoreticalmodelsinpolymerphysicsthatdescribethemolecularorientationthroughuniaxialdeformationandonecanderivetheappropriateformoftheHermansfunctionforthesystemunderstudy.Inourcase,assumingalineardeformationoftheaerogelneedleaboutthenetworkofjoints,thehcos2icanbegivenas hcos2iD1 31)]TJ/F132 11.955 Tf 12.513 8.38 Td[(1L L2: Thisrelationisderivedinthe appendix .Usingequations 12 and 13 andthevalueof1nmaxDne)]TJ/F37 11.955 Tf 11.317 0 Td[(no)]TJ/F60 11.955 Tf 21.307 0 Td[(510)]TJ/F60 7.97 Tf 6.217 0 Td[(3fromourpreviousconsideration,j1njisplottedingure 15 asafunctionofstrainalongwithourexperimentallymeasuredvaluesforsamples1.Asseeninthisgure,aswellasingures 11 and 13 ,themeasured1nshowasmallcurvature,indicating NewJournalofPhysics12103016 http://www.njp.org/

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17 Figure15.Straindependenceofbirefringenceforaerogelsamples1andcalculatedvaluesusingtheHermansorientationfunction. thepresenceofasmallquadraticstraindependencewithanegativecoefcient.Thisbehaviorisconsistentwithequation 13 1n<0.However,themagnitudeofopticalbirefringenceisabout1=3ofthevaluepredictedbythismodel.Here,weshouldnotethatthemagnitudeof1nmaxisestimatedinamodelofidealneedle-likecylindersmadeofSiO2.Itisalsoassumedthatalltheneedleswouldrotateinresponsetouniaxialcompression.Thisidealizationisfarfromtheactualstructureoftheaerogelstrand,whichisformedbyrandomclusterclusteraggregationofnanometer-scaleaerogelparticles[ 38 39 ].Inarealaerogel,therearemanydead-endclustersattachedtothebackbonestructureatoneendandwiththeotherenddangling[ 39 ].Thesedead-endclustersneithercontributetothemechanicalstrengthofthematerialnorgeneraterotationtotherstorderinresponsetouniaxialstrain[ 40 ].Amoresophisticatedmodelshouldalsoincorporatelocalvariationsofclusterorientationbnlocinastrandwhoseorientationisrepresentedbybnstseegure 16 .Forarandomstructure,bnstandbnlocarecorrelatedandarecharacterizedbythecorrelationfunctionD3 2.bnstbnloc/2)]TJ/F60 7.97 Tf 12.513 4.83 Td[(1 2.Therefore,thiscorrelationeffectshouldbeincludedasaprefactorinequation 13 .Thevalueofdependsonthedetailedstructureoftheaerogelbutshouldbe0<<1.Basedonourmeasurements,wendthat1 3assumingnodeadclusters,whichlimitsthevariationofbnlocwithin60withrespecttobnst.5.SummaryWehavemeasuredthebirefringenceofuniaxiallycompressedhigh-porosityaerogelsinthevisiblerangeofthespectrum.Amechanicalcompressionbeyond5%ontheaerogelshowshysteresisintransmittancedata,whichcanbecorrelatedtothenon-recoveryoftheaerogeltoitsuncompressedlength.Thebirefringenceexhibitsquasi-lineardependenceonthecompressionandislargeenoughtoobservethewaveplatephenomenon.Thebirefringence,andits NewJournalofPhysics12103016 http://www.njp.org/

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18 Figure16.Acartoonoftheaerogelstructuredepictinglocalvariationsofclusterorientationofastrand.Thethickredarrowrepresentsbnstandthesmallblackarrowsindicatethedirectionoflocalclusterorientation,bnloc. dispersion,isinexcellentagreementwiththeSellmeieroscillatormodel.Theeffectivemediumapproximationmodelprovidesasatisfactorypictureoftheobservedphenomena.AcknowledgmentsWeacknowledgesupportfromtheNSFgrantsDMR-0239483andDMR-0803516toYLandtheDOEgrantDE-FG02-02ER45984toDBT.AppendixHere,wederiveequation 13 relatingtheHermansanisotropyfunctiontomechanicalstrain.LetLbetheinitiallengthoftheuncompressedaerogeland1Lbethechangeinlengthaftercompression.Denethecompressionorstretchratioas DlnL Lmin; A.1 whereLminDL)]TJ/F132 11.955 Tf 11.318 0 Td[(1L,sothatintheuncompressedstateD0and>0whencompressed.Weconsideranaerogelstrandoflength`initiallymakingananglewithrespecttotheOz-axis.Afterasmalldeformationbyanangle,thechangeinlengthalongtheOz-axisis zD`fcos.+/)]TJ/F60 11.955 Tf 11.318 0 Td[(cos./g; A.2 which,forsmall,reducesto zD)]TJ/F132 11.955 Tf 21.306 0 Td[(`sin: A.3 NewJournalofPhysics12103016 http://www.njp.org/

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19 UsingD)]TJ/F132 11.955 Tf 21.307 0 Td[(z=z,wehave D tan: A.4 Therefore, hcos2iD)]TJ/F60 11.955 Tf 28.35 0 Td[(2hcos2i; A.5 whichcanberewritteninthefollowingform: @lnhcos2i @D)]TJ/F60 11.955 Tf 21.307 0 Td[(2: A.6 Integratingequation A.6 ,weobtain hcos2iD1 3e)]TJ/F60 7.97 Tf 6.216 0 Td[(2; A.7 wherethefactor1/3istheintegrationconstantsothatatzerocompression,whenD0,hcos2iD1=3intheisotropiccase.Substitutingequation A.1 forandLminDL)]TJ/F132 11.955 Tf 11.318 0 Td[(1Linequation A.7 leadstoequation 13 : hcos2iD1 31)]TJ/F132 11.955 Tf 12.513 8.381 Td[(1L L2: A.8 References [1] KistlerSS1931Nature 127741 [2] TillotsonTMandHrubeshLW1992J.Non-Cryst.Solids 14544 [3] ZimmermannA,GrossJandFrickeJ1995J.Non-Cryst.Solids 186238 [4] AkimovYuK2003Instrum.Exp.Tech. 46287 [5] BurchellMJ,GrahamGandKearsleyA2006Annu.Rev.EarthPlanet.Sci. 34385 [6] PoelzGandRietmuellerR2006Nucl.Instrum.Methods34491 [7] ReimM,KornerW,ManaraJ,KorderS,Arduini-SchusterM,EbertHPandFrickeJ2005Sol.Energy 79131 [8] BelliniT,ClarkNAandSchaeferDW1995Phys.Rev.Lett. 742704 [9] ChanMHW,BlumKI,MurphySQ,WongGKSandReppyJD1988Phys.Rev.Lett. 611950 [10] PortoJVandParpiaJM1998Phys.Rev.B 5914583 [11] HalperinWP,ChoiH,DavisJPandPollanenJ2009J.Phys.Soc.Japan77111002 [12] KimSB,MaJandChanMHW1993Phys.Rev.Lett. 712268 [13] VolovikGE1996J.Exp.Theor.Phys.Lett. 63301 [14] FominI2004J.Exp.Theor.Phys. 98974 [15] AoyamaKandIkedaR2005Phys.Rev.B 72012515 [16] VicenteCL,ChoiHC,XiaJS,HalperinWP,MuldersNandLeeY2005Phys.Rev.B 72094519 [17] PollanenJ,ShirerKR,BlinsteinS,DavisJP,ChoiH,LippmanTM,HalperinWPandLurioLB2008J.Non-Cryst.Solids 3544668 [18] BhupathiP,HwangJ,MartinRM,BlanksteinJ,JaworskiL,MuldersN,TannerDBandLeeY2009Opt.Express 1710599 [19] BornMandWolfE1999PrinciplesofOptics7thednCambridge:CambridgeUniversityPress [20] WuST,EfronUandHessLD1984Appl.Phys.Lett. 441033 [21] EscutiMJ,CairnsDRandCrawfordGP2004J.Appl.Phys. 952386 [22] GrossJ,FrickeJ,PekalaRWandHrubeshLW1992Phys.Rev.B 4512774 [23] GrossJ,ReichenauerGandFrickeJ1988J.Phys.D:Appl.Phys. 211447 NewJournalofPhysics12103016 http://www.njp.org/

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20 [24] SchererGW,SmithDM,QiuXandAndersonJM1995J.Non-Cryst.Solids 186316 [25] SmithDM,SchererGWandAndersonJM1995J.Non-Cryst.Solids 188191 [26] HermanT,DayJandBeamishJ2006Phys.Rev.B 73094127 [27] VenkateswaraRaoA1998J.Mater.Synth.Process. 637 [28] MalitsonIH1965J.Opt.Soc.Am. 551205 [29] SuttonLEandStavroudisON1961J.Opt.Soc.Am. 51901 [30] SimanekE1977Phys.Rev.Lett. 381161 [31] CarrGL,PekowitzSandTannerDB1985InfraredandMillimeterWavesvol13,VedKJButtonOrlando,FL:Academicpp171 [32] BuzykaevAR,DanilyukAF,GanzhurSF,KravchenkoEAandOnuchinAP1999Nucl.Instrum.Methods 433396 [33] KunznerN,DienerJ,GrossE,KovalevD,TimoshenkoVYuandFujiiAM2005Phys.Rev.B 71195304 [34] GolovanLA,KashkarovPKandTimoshenkoVYu2007Crystall.Rep. 52672 [35] StrattonJA1941ElectromagneticTheoryNewYork:McGraw-Hill [36] OsbornJA1945Phys.Rev. 67351 [37] WardIM1985StructureandPropertiesofOrientedPolymersLondon:AppliedSciencePublishers [38] SchaeferDWandKeeferKD1986Phys.Rev.Lett. 192199 [39] EmmerlingAandFrickeJ1997J.SolGelSci.Technol.8781 [40] MaH-S,PrvostJ-HandSchererGW2002Int.J.SolidsStruct. 394605 NewJournalofPhysics12103016 http://www.njp.org/


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