nist-借助力曲线拟合针尖几何形态提升原子力显微镜测量杨氏模量的精度

EnhancingtheAccuracyofAtomicForceMicroscopyMeasurementsofYoung’sModulusviaForce-Curve-InformedTipGeometryFitting

LoganJ.Kirsch,JasonP.Killgore,GregoryJ.Rodin,TimothyS.Weeks,FilippoMangolini*

L.J.Kirsch,F.Mangolini

WalkerDepartmentofMechanicalEngineering

TheUniversityofTexasatAustin

Austin,TX78712,USA

E-mail:

Logan.Kirsch@

J.P.Killgore,T.S.Weeks

AppliedChemicalsandMaterialsDivision

NationalInstituteofStandardsandTechnology

Boulder,CO80305,USA

E-mail:

Jason.Killgore@

G.J.Rodin

DepartmentofAerospaceEngineeringandEngineeringMechanics

TheUniversityofTexasatAustin

Austin,TX78712,USA

E-mail:

gjr@

F.Mangolini

TexasMaterialsInstitute

TheUniversityofTexasatAustin

Austin,TX78712,USA

E-mail:

Filippo.Mangolini@

(correspondingauthor)

Keywords:forcespectroscopy,indentation,nanoindentation,contactmechanics,nanomechanics

Abstract

Atomicforcemicroscopy(AFM)iswidelyusedfornanoscalemechanicaltesting.However,extractingYoung’smodulusfromtheforcevs.indentationdepthdataremainsachallenge.Inthisregard,uncertaintiesabouttheAFMtipgeometryhavebeenrecognizedasamajorsourceoferrors.HereweproposeamethodologyinwhichthegeometricapproximationoftheAFMtipisinformedbytheforcevs.indentationdepthdata.Themethodologyisbasedontwoleast-squarefits,oneinvolvingtheforcevs.indentationdepthdataandtheotherthetipprofile.Atthecoreofourmethodologyisthepropositionthatthetipgeometrymustbeproperlycharacterizedintheintervalboundedbythecontactradiuscorrespondingtothemaximumindentationdepth.Thispropositionhasasolidgeometricunderpinninganddoesnotrequireanyadditionalassumptions.Further,therearenoconceptualobstaclestoapplyingthemethodologytomulti-parametergeometricmodels,includingthosebasedonrawimagedata.Themethodologyissuccessfullyappliedtobothsyntheticandphysicaldata.

1.Introduction

Atomicforcemicroscopy(AFM)isextensivelyusedformeasuringmechanicalpropertiesatthenanoscale.OneofthefoundationalAFMtestsinvolvesquasi-staticindentingofasample.Theresultingforceversusindentationdepthcurveor‘forcecurve’isthenanalyzedusingacontactmechanicsmodeltoderivelocalmaterialpropertiesofthesample,mostcommonlyYoung’smodulus.Suchtestshavebeenproveninsightfulforapplicationsincluding,butnotlimitedto,cellsandtissues[1–8],electronicsandbatteries[9–11],andgeologicalmaterials.[12–14]Despitethewidespreaduseofthisapproach,extractingaccurateandprecisevaluesforremainsachallenge.

Forquantifyingexperimentalresults,oneneedstwosetsofinputs:theforcecurveandtipgeometry.Thelatterisobtainedfromeitherelectronmicroscopyimagesortheblindtipreconstruction(BTR)method.[15]Thesedifferentmethodshavetheirassociatedtrade-offs.Principally,electronmicroscopycanleadtocarbondepositionandistime-consumingasitrequirestheusertotakethetipoutoftheAFM.BTRdoesnothavetheseissues,butcanleadtotipwearandalossoflateralresolution.[16,17]Inallcases,manufacturingvariabilitynecessitatesAFMtipshapecharacterizationforaccuratenanomechanicalmeasurements.Fromtheobtainedprofile,onemustconstructageometricmodelofthetip,whichconsistsofafunctionandadomainoverwhichthisfunctionmustprovideanaccuratefittotheexperimentallydeterminedtipprofile.Uncertaintiesandinaccuraciesassociatedwiththisstephavebeenrecognizedasamajorsourceoferrorsintheextractionof.[18,19]Inpart,theseerrorsarisebecauseAFMtipsareusuallyapproximatedwithconical[4,14,20,21],parabolic[2,5,6,22,23],orpower-law[19,24–26]shapes,whichfailtoaccuratelycapturetheactualgeometry,especiallyforindentationsmuchdeeperthanthecharacteristictipsize[27],suchastheoneshowninFigure1.Thisissuehasbeeninvestigatedbyevaluatingtheaccuracyofvariousgeometricmodels[28,29]andtheirinfluenceonthepredictedforcecurves.[30,31]Furthermore,effortshavebeenmadetoimprovetheaccuracywithcorrectionfactors,[32,33]improvedgeometrycalibrationtechniques,[22,34]andintentionallymodifiedprobes.[7,13,23,35]Inmanycases,largersphericalcolloidalprobes(withradii)areutilizedbecauseoftheirwell-definedshapes.However,theselargerprobeslackthelateralresolutionofsharperAFMtips.[36,37]Whilemanyofthesestudieshaveexploredhowthegeometricmodelselectioninfluencestheresultingvalueof,thereisinsufficientrecognitionthatareliablegeometricmodelmustincludenotonlyanaccurategeometricshapebutalsothepertinentapplicabilityrange(i.e.,thespatialdomainoverwhichthetipprofileisfitwiththegeometricshapefunction).Ingeneral,thisrangeisdependentonthetipgeometryandtheforcecurve.

Inthiswork,weaimtoidentifyoptimalboundsforfittingtipprofileswithgeometricshapesanddemonstratetheirusefulnessthroughapplicationsinvolvingsyntheticandexperimentaldata.Theseboundsarechosenbasedontheportionofthetipincontactwiththesampleduringindentation.TheirapplicationlimitserrorinpredictedvaluesregardlessofthespecificAFMtipgeometryorchosenindentermodel.

Figure1.Ascanningelectronmicroscopy(SEM)imageofanAFMtip.

2.Background

2.1ModelProblem

ConsideramodelproblemwhichhelpsusbetterunderstandmethodologiesforextractingYoung’smodulusfromAFMindentationtests.Thisprobleminvolvesarigidaxisymmetricindentationofasample,modeledasanisotropiclinearelastichalf-space.Itisassumedthatthereisnofrictionoradhesionatthecontact.Theproblemisdescribedusingacylindricalcoordinatesystemwiththeupwardz-axisalignedwiththetipsymmetryaxisandtheoriginattheinitialpointofcontactbetweenthetipandthesample(Figure2).Inthiscoordinatesystem,thetipsurfaceisprescribedbytheshapefunction.Inthecontactarea,theaxialdisplacementsare

, (1)

whereistheindentationdepth.Sneddon[38]hasshownthatandtheconjugateforcecanbeexpressedparametricallyas

(2)

and

. (3)

Hereisthecontactradius(Figure2),andisthereducedelasticmodulus:

, (4)

whereisPoisson’sratioofthesample.Itisusefultocombine(2)and(3)toobtaintheforcecurvebyeliminating.Also,(2)and(3)yieldtherelationship[39]

. (5)

Figure2.Anindenterdescribedbytheshapefunctionpressedintoasamplewithaforceresultinginanindentationdepthandcontactradius.

2.2.BasicIndenterModels

Table1summarizesthefunctionsandforthreebasicmodelindenters,conical,paraboloidal,andpower-lawindentersshowninFigure3.Itisstraightforwardtoestablishthatbothandfortheparaboloidcanbederivedfromthepower-lawmodelbysettingand.Thatis,paraboloidalindentersareparticularpower-lawindenters.Also,for,thepower-lawmodeldescribesaconewiththecomplimentaryangle.Thatis,asformulated,thepower-lawmodelisnotgeneralenoughtoincludeallcones.Ifneeded,onecanintroduceanadditionalparameterintothepower-lawmodel,sothatitincludesall,butthisisnotnecessaryforourpurposes.

Table1.Thefunctionsandforthreebasicmodelindenters.Thesymboldenotesthestandard-function.

Model

Shapefunction

Forcecurve

Cone[40]

Paraboloid[41]

Power-Law[40]

Figure3.Modelindenters:(a)Conicalcharacterizedbythecomplimentaryangle,(b)Paraboloidalcharacterizedbythelocalradiusofcurvature,(c)Power-Lawcharacterizedbyareferencelengthandexponent.

2.3.Data

Currentstate-of-the-artmethodologiesforprocessingAFMdatainvolvetwosetsofdiscretedata,theforcecurveandthetipprofile.Thedomainforisnaturallysetbythemaximumindentationdepth,sothat.Incontrast,thereisnoconsensusonchoosingtherelevantdomainorrangefor.Typically,onedefinestherange,whereischosenvisually,independentlyoftheforcecurve.Forthisreason,werefertosuchmethodologiesasuninformed.Inthiswork,weproposeasystematicapproachtoidentifyinganoptimaldomainfor,whichtakestheforcecurveintoaccount.Therefore,werefertotheproposedmethodologyasinformed.Theproposedmethodologyisdevelopedintwostages.First,inSection3,weexplainitusingsyntheticdata.Then,inSection4,itisappliedtoexperimentallyextractfromdataforpolydimethylsiloxane(PDMS)samples.

3.SyntheticDataAnalysis

Syntheticdataprovidesanidealsettingforassessingvariousmethodologiesforextractingthereducedelasticmodulus,asitallowsonetoisolateintrinsicerrorsofthemethodologyfromthoseassociatedwithmeasurements.Inthiswork,syntheticdataaregeneratedbychoosingthemodelindentersinFigure3andTable1.Thegoalistofitsyntheticdatawithaninappropriatemodel(i.e.adifferenttip)whilevaryingthesizeoftheprobeandthedepthofindentation.Inwhatfollows,werefertothemodelchosenforsyntheticdatagenerationastheconstructionmodel,andwedenotethecorresponding,,andas,,and,respectively.Thus,inthissection,thediscretedataandarereplacedwithcontinuousfunctionsand,respectively.Thereducedelasticmodulusoftheconstructionmodelisdenotedby.

Themodelforextractingthereducedelasticmodulusisreferredtoasthetestmodel,anditsattributesaredenotedby,,,and.Thetestmodelischosensuchthat,regardlessoftheparameters,cannotequalforallvaluesofr.Ourapproachisbasedontwoleast-square-fitminimizations,onefortheforcecurve,

(6)

andtheotherfortheindentershapefunction,

. (7)

Inthefirstfit,weminimizethedistancebetweenthetwoforcecurvesacrosstheentiredomain.Inthesecondfit,theupperlimitreferstothecontactradiusofthetestmodelcorrespondingtotheindentationdepth,anditiscalculatedusing(5)as

. (8)

Figure2demonstratesthatthefunctionmustbeanaccurateapproximationofonlyintheinterval.Thedefinitionofimpliesthatweseekthefitforthelargestcontactradius,whichcorrespondstoandapproximatedby.Further,wehypothesizethatthebestapproximatedis.Therefore,in(8),theslopeoftheforcecurveisevaluatedatandthecorrespondingcontactradiusisapproximatedby.

Theoutlinedmethodologywasappliedtotwocases.First,thepower-lawwasusedastheconstructionmodelandtheparaboloidasthetestmodel.Inthiscase,therelativeerroris

. (9)

TherelativeerrorisplottedasafunctioninFigure4for(typicalvaluesseenwithrealAFMtips).[25]Inthisplot,themaximumerrormagnitudeof29%isfor,whenthepower-lawshaperepresentsaconewiththecomplementaryangle.Ofcourse,theerroriszerofor,whenthepower-lawshaperepresentsaparaboloid.Theerroralsoapproacheszeroas,whenthepower-lawshaperepresentsaflatpunch.

Figure4.Therelativeerrorasafunctionoffortheinformedmodel.Theconstructiongeometryispower-law,andthetestgeometryisparaboloid.Todemonstratethattheerrortendstozeroforlargen,theinsetshowsthesamefunctionintheinterval.

Inthesecondcase,theparaboloidwasusedastheconstructionmodelandtheconeasthetestmodel.Therelativeerroris

foranyparaboloid.Itisremarkablethatbothresultsaresimpleandforce-curveindependent.

Letusnowconsiderchoicesfortheupperboundofuninformedbytheforcecurve,consistentwithtraditionalmethods[32,42].Inpractice,thosearespecifiedbychoosinganupperboundonratherthan.Thus,forachosenupperbound,thecorrespondingupperboundonis.Becauseistypicallychosenarbitrarily,weexamineawiderangeofvalues.Specifically,isvariedfromonetenthtotentimesthecharacteristiclengthoftheconstructiongeometry.Accordingly,forthefirstcase,wheretheconstructionmodelisthepower-law,weconsiderthreechoices,,,and.TherelativeerrorcorrespondingtothesechoicesisplottedinFigure5asafunctionoffor.Thismethodologyleadsnotonlytoanerrorthatisdependenton,butalsotolargeerrorswhichvarysignificantlyamongthethreechosenupperbounds.Forreference,for,theinformedmethodologypredictsanerrorof-1%.

Figure5.Therelativeerrorasafunctionofforthreeupperbounds:isshowninorange,isshowninblue,andisshowningreen.Theconstructiongeometryisapower-lawwithandthetestgeometryisaparaboloid.

Anotheroptionistoset,whichwecallthesemi-informedmethodology.Thisapproachyieldsresultsindependentof.However,acloserlookatFigure2revealsthatitincludesthedeformedsurfaceregionnotincontactwiththeindenter,increasingerror.Inthiscase,weobtainanerrorof14%,whichisasignificantlyworsethantheerrorobtainedwiththeinformedmethodology,lessthan1%.

Similarly,forthesecondcase,wheretheconstructionmodelistheparaboloid,weconsiderthreechoices,,,and.TherelativeerrorcorrespondingtothesechoicesisplottedinFigure6asafunctionof.Thismethodologyissimplyunacceptableformostpracticalpurposes.If,weobtainanerrorof23%;forreference,theerrorfortheinformedmethodologyis-7%.

Overall,thesyntheticdataresultsindicatethatthestandarduninformedapproachofarbitrarilyselectingucanleadtoextremeerrors.Conversely,theinformedmethodologyproducedrelativelysmallerrorsevenincaseswithextremedifferencesbetweenand.Therefore,theinformedmethodologydeservesfurtherconsiderationwithdiscreteexperimentaldata.

Figure6.Therelativeerrorasafunctionofforthreeupperbounds:isshowninorange,isshowninblue,andisshowningreen.Theconstructiongeometryisaparaboloid,andthetestgeometryisacone.

4.ExperimentalDataAnalysis

Inthissection,themethodologiesdescribedinSection3areappliedtoAFMtestdatageneratedbyindentationtestsonaPDMSsample.Twosetsofmeasurementswerecollected,eachwithadifferentsiliconAFMtipandconsistingofhundredsofforcecurves.AtypicalforcecurveobtainedfromasingleindentationtestisshowninFigure7a.Sincetherearemanysimilarcurves,thedataallowsustogenerateastatisticaldescriptionofineachcase.Further,forvalidationpurposes,wasalsodeterminedfromstandarduniaxialtensiletests;fordetails,seeSupportingInformation.

Figure7.Forceversusindentationdepthdata:(a)AtypicalforcecurveusedforgeneratingdataforthePDMSsample,(b)Aheatmapofoverlappingforcecurvesshowingtheapproachsectionsforthe248indentationsperformedinthefirstsetofmeasurements.Thecolorbardisplaysthepercentageoftheforcescurvesoverlappingateachpoint.

Sincewenolongerdealwithcontinuousdata,expressions(6)through(8)mustbemodified.Tothisend,onecanreplacewiththesum

, (10)

andwiththesum

. (11)

Here,isthenumberofdatapointssatisfyingin.Likewise,isthenumberofdatapointssatisfyingin.Further,since,inthissection,weusedthepower-lawtestmodel,wasminimizedwithrespecttobothand,while,asbefore,wasminimizedonlywithrespectto.

Applyingtheinformedmethodologytothediscretedatarequiresfurthermodification.Here,isminimizedrepeatedleyasapartoftheminimization,asillustratedinFigure8.Theiterativeminimizationschemeresultsinahighercomputationalcostcomparedtotheuninformedmethodologies.However,theexampleMATLABimplementation(seenintheSupportingInformation)wasabletoprocess248experimentalforcecurvesinthreeminutesonastandarddesktopcomputer.Aninitialorder-of-magnitudeguessforbasedonaprioriknowledgeofthesamplecanspeedupcomputationbutisnotrequired.Subsequentguessesforwilldependonthechosenoptimizationalgorithm.

Figure8.Theprocessforapplyingtheinformedmethodologytodiscretedataassumingapower-lawindentermodel.

Forthefirstsetofmeasurements,248forcecurveswerecollected.AnSEMimageofthesilicontipusedforthistestingisshowninFigure9a.ThetipgeometrywasextractedfromthisSEMimageandfittedwiththethreebasicmodelslistedinTable1usingvaryingfittingupperboundsfor.AsimilartipprofilecouldhavebeenextractedusingBTR.Thefitispoorfortheconicalmodel.Incontrast,bothpower-lawandparaboloidalmodelsprovidedexcellentfitsforthefittingupperboundsabove2nm.ThisisevidencedbyFigure9b,wherethecoefficientofdeterminationversustheupperboundisplottedforallthreemodels.Thetipisthereforetreatedasatrueparaboloidwith,themeanfittedvaluefromtheSEMtipprofile(thefittedparametersforallthemodelsandfittingregionscanbefoundintheSupportingInformation).Thisvalueisusedfornormalizationoftheuninformedboundsbutisnotrequiredfortheinformedmethodology.

ThefieldofviewfortheprovidedSEMimagelimitstheuseoftheforcecurveto.Forthesamereason,usingisnotpossible.Therefore,theanalysiswaslimitedtotwouninformedchoices,and,thesemi-informedchoice,andtheinformedchoice,withtheunderstandingthatthevalueofisdeterminedaspartofthesolutionstrategy.Forallfourchoices,oncehadbeendeterminedusing(10)and(11),wascalculatedfromageneralizationof(4)whichaccountsforthetipcompliance:

. (12)

Here,weused[43–45]andthetipstiffnessfor<100>Siis.[46]

Figure9.TheAFMtipusedintheindentationtests:(a)SEMimageandfittedgeometries;(b)Coefficientofdeterminationvs.thefittingupperboundforthebasicindentermodels.

Figure10presentsthepredictionsforobtainedusingfourdifferentchoicesfortheupperbound.Thesepredictionsarecomparedwiththereferencevalueof,measuredusingstandarduniaxialtensiletestsconductedonthesamematerial.Alongwiththeabsolutemagnitudespredictedfor,Figure10reportstherelativeerrorcomparedtothemeanreferencevalue.UponexaminingtheresultsinFigure10weofferthefollowingconclusions:

Asexpected,foreverychoiceoftheupperbound,thepredictionsfortheparaboloidalandpower-lawindentersaresuperiortotheircounterpartsfortheconicalindenter.

Withoneexception,theinformedmethodology,characterizedbytheupperbound,outperformsitscompetition.Theexceptionisthepower-lawindenter,forwhichtheinformedmethodologyinvolvesa2%error,whereastheuninformedmethodologycharacterizedbytheupperboundinvolvesa1%error.Ofcourse,thesetwopredictionsareaccurate,butthelatteroneissomewhatfortuitousastheupperboundresultsina15%error.

Inallcases,evenfortheconicalindenter,theinformedmethodologypredictswithanaccuracybetterthan10%.Theresultshowstheinformedmethodologycanyieldacceptableaccuracyfor,regardlessofthechosenmodel.

Figure10.Young’smodulusforthePDMSsamplefromAFMindentationtestsperformedwiththeAFMtipinFigure8ausingthebasicindentermodelsandfourchoicesfortheupperboundin(10).Thestandarddeviationsareshownaserrorbars.Thereferencevaluewasfoundusingstandarduniaxialtensiontestsonthesamematerial.Thelistedpercentagesaretheerrorrelativetothismeanreferencevalue(seetheSupportingInformationforplotsoftherelativeerror).

Inthesecondsetofmeasurements,209indentationswereperformedonthesamePDMSsample.ThesemeasurementsusedtheAFMtipshowninFigure1.ThefitsoftheSEMimageforthistipwerepoorforboththeconeandtheparaboloidshapefunctions(SeetheSupportingInformationformoredetails).Thetipwasthereforetreatedasatruepower-lawwith,themeanfittedvaluefromtheSEMtipprofile(thefittedparametersforallthemodelsandfittingregionscanbefoundintheSupportingInformation).Similartothepreviousmeasurements,thisvaluewasusedtonormalizetheuninformedbounds.

Alsosimilartothepreviousmeasurements,theforcecurveswerefitupto.Theanalysiswasperformedfortwouninformedbounds,and,thesemi-informedbound,andtheinformedbound.

Figure11presentsthepredictionsforobtainedusingfourdifferentchoicesfortheupperboundwiththesecondsetofmeasurements.Thesepredictionsareagaincomparedwiththeuniaxialtensionbenchmarkvalue.Alongwiththeabsolutemagnitudespredictedfor,Figure11reportstherelativeerrorcomparedtothemeanreferencevalue.UponexaminingtheresultsinFigure11weofferthefollowingconclusions:

ThemeasurementsshowedsignificantlymorevariationcomparedtothevaluesreportedinFigure10,asevidencedbythedepictederrorbars.Thisvariabilityisbelievedtobecausedbygreaterinstrumentalnoise,butadditionalstudiesareneeded.Nevertheless,themeanvaluespointtosimilarconclusionsasthepreviousmeasurements.

Foreverychoiceoftheupperbound,thepredictionsforthepower-lawindenteraresuperiortotheircounterpartsfortheparaboloidalandconicalindenters.

Withoneexception,theinformedmethodology,characterizedbytheupperbound,outperformsitscompetition.Theexceptionistheparaboloidalindenter,forwhichtheinformedmethodologyinvolvesa19%error,whereastheuninformedmethodologycharacterizedbytheupperboundinvolvesa16%error.Thisresultagainappearstobehappenstance,astheupperboundresultsina30%error,andtheupperboundresultsina51%errorfortheconicalindenter.

Inallcases,theinformedmethodologypredictswithanerror.Again,theseresultsindicatethat,regardlessoftheindentermodel,theinformedmethodologyleadstoasystematicimprovementinthepredictionof.

Figure11.Young’smodulusforthePDMSsamplefromAFMindentationtestsperformedwiththeAFMtipinFigure1usingthebasicindentermodelsandfourchoicesfortheupperboundin(10).Thestandarddeviationsareshownaserrorbars.Thereferencevaluewasfoundusingstandarduniaxialtensiontestsonthesamematerial.Thelistedpercentagesaretheerrorrelativetothismeanreferencevalue(seetheSupportingInformationforplotsoftherelativeerror).

5.Conclusion

Inthispaper,weproposedasimplemethodologyforextractingthereducedelasticmodulusfromAFMindentationdata,undertheassumptionofaxialsymmetry.Werefertoitasinformedbecauseitsgeometricaspectisinformedbytheforcecurve.Themethodologyisbasedontwoleast-squarefits,andatitscoreisthepropositionthatthetipgeometrymustbeproperlycharacterizedintheinterval,whereisthecontactradiuscorrespondingtothemaximumindentationdepth.Thispropositionhasasolidgeometricfoundation(Figure2)anddoesnotrequireanyadditionalassumptions.Further,thepropositionisapplicabletoanyaxisymmetrictips.InSection3,wherewereliedonsyntheticdata,weconsideredone-parametertestmodelsonly,theconeandparaboloid.InSection4,whereweusedphysicaldata,wealsoconsideredthetwo-parameterpower-lawmodel.Therearenoconceptualobstaclestoapplyingthemethodologytomulti-parametergeometricmodels,includingthosebasedonrawdigitaldata.

InSection4,weconsideredatipwhosegeometryiswellcharacterizedbyparaboloidalandpower-lawbutnotconicalmodels.Nevertheless,thelattermodelresultedinwhichwasonly8.6%belowthereferencevalue.Encouragedbythisresult,weattemptedtodetermineusingthetipshowninFigure1.Withthismorecomplexgeometry,onlythetwo-parameterpower-lawmodelhadthefittingflexibilitytopredictwithanerroroflessthan10%.However,theinformedmethodologyagainproducedacceptableestimatesforallthreeindentermodels.Additionalexperimentswithatipwhosegeometryiswellcharacterizedbyaconicalmodelwouldhavemadetheexperimentalevaluationmorecomprehensive.However,constraintsoncommerciallyavailabletipgeometriesdonotenablesuchexperiments.

Theproposedmethodologyislimitedtoaxisymmetrictips.However,pyramidalAFMtipsincludeanaxisymmetricregionneartheapex,suchasthetipinFigure1.Assuch,themethodologyisstillvalidforindentationsbelowthetransitionpointtonon-axisymmetry(forthetipinFigure1).Moreover,theexperimentalresultsforthisparticulartipshowtheinformedmethodologycontinuestooutperformtheuninformedmethodologybeyondthistransitionpoint.Therefore,despiteitslimitations,theinformedmethodologywillgenerallyimproveaccuracy.

Theproposedmethodologyisalsolimitedtoindentationsonhomogenous,thick,fullyelasticsampleswithoutadhesion.However,themethodcouldbeadaptedusingotherforcecurveanalysistechniquesintheliteraturetoaccountfornon-idealeffects.Specifically,varioustechniquesincorporateheterogeneity,[47]viscoelasticity,[31,48]depth-dependence,[49]plasticity,[31,50]adhesion,[51,52]andfinite-sizeeffects.[31,53]Becauseallthesemethodsrequireknowledgeofthetipgeometry,theycouldbefurtherimprovedbyimplementingtheinformedmethodologyfortipprofilefitting.

Topracticallyimplementtheproposedapproach,onebeginswithaforcecurveandatipprofileobtainedfromSEM,BTR,orsomeothertipshapecharacterizationmethod.Fromthisdataandatestvalue,onewill:(i)calculateatestcontactradiusfromtheOliver-Pharrrelation;(ii)fitthetipprofileuptothetestcontactradiusforachosenindentermodel;and(iii)calculatethepredictedforcesbasedonthetipprofilefit.Thesestepsarerepeatedwithiterativevaluesforuntilthepredictedforcesbestmatchtheexperimentalforcecurve,asshowninFigure8.

Overall,weexploredmethodsforfittingtipgeometriesthatlimiterrorinpredictedvalues,atopicnotpreviouslydiscussedintheliterature.Theproposedmethodologydefinesthefittingregionbasedontheportionoftheindenterincontactwiththesampleduringindentation,boundedbythemaximumpredictedcontactradius.Wealsoprovidedaniterativeschemeforapplyingthemethodologytoexperimentaldata(Figure8),alongwithanexampleMATLABimplementationintheSupportingInformation.Thisschemeyieldsasystematicimprovementintheaccuracyofpredictedvalues.ThemethodcanbeappliedtotipgeometriesobtainedfromSEM,BTR,oranysimilartechnique.ItcanalsobeusedindependentlyorinparallelwithotherAFMforcecurveanalysistechniques.Broadly,thisworkprovi