Outline DavidNecasetal 227 Outline IntroductionQuanticationHeightdistributionpropertiesSpatialpropertiesAutocorrelationPowerspectrumdensityNonuniformsurfacesConclusion Roughnesscharacterisation I ID: 609808
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Roughnesscharacterisation Outline DavidNecasetal. 2/27 Outline IntroductionQuanticationHeightdistributionpropertiesSpatialpropertiesAutocorrelationPowerspectrumdensityNon-uniformsurfacesConclusion Roughnesscharacterisation Introduction DavidNecasetal. 4/27 Atomicforcemicroscopydata AFMprovidesheighteldsoflimitedIsizeIresolution(oftenduetotipconcolution)Imaximumslope(duetotipconcolution)Toone,toocoarseortoosteepfeaturesarenotadequatelycaptured.Thedatararelycanbedirectlyanalysed.Usually,correctionsarenecessary:Ilevelling(plane,polynomial)Iline-by-line-matchIlocaldefectcorrectionIFourier-domainlteringAllrepresentseriousdatamanipulation. Atomicforcemicroscopescheme Roughnesscharacterisation Introduction DavidNecasetal. 5/27 VisualassessmentAwidespreadpracticeistojustpublishAFMimages.Because,ofcourse,everyonecaneasilytellfromatopographicalimagehowroughorsmooththesurfaceis.Forinstance,theleftsurfaceisobviouslysmoothwhiletherightoneisrough: Oops!Weforgotthescales(alsocommon).Butwait,nowitisnolongerclearwhichiswhich. Roughnesscharacterisation Introduction DavidNecasetal. 6/27 MoredifcultiesItcangetevenmorecomplicated.Largetopographicfeaturescanbemixedwithroughnessinvariousways. MicrochipsurfaceTiO2lmatacertaingrowthstageMaybewehavetoresorttomathematicsafterall.Andconsiderthesimplecaserst:Surfacescorrespondingtowide-sensestationarystochaticrandomprocesses.Inotherwords,theyarethesameeverywherefromthestatisticalpointofview. Roughnesscharacterisation Quantication DavidNecasetal. 7/27 QuanticationTogoalistocharacterisetheroughness,i.e.torepresenttheinformationfromtheimageusingjustasmallnumberofquantities.Balanceisneededbetweennotbeingabletocomparebecausewehavetoomuchdataandbecausewehavereducedthedatatoomuch.Iscalar–asinglenumber(orsetofthem)Ione-dimensionalcharacteristic–distributions,correlationfunctionsItwo-dimensional–rare,usuallyanintermediatestepApproaches:IUseanorm:ASMEB46.1-1995,ASMEB46.1-1985,ISO4287-1997,ISO4287/1-1997,...Thisisthesafepathtocomparableresults–sortof.Butwemaynothaveanyideahowistheheightofthethirdhighestpeakfromthethirdlowestvalleypersamplinglengthrelatedtoanythingwearetryingstudy.Mostnormshaveorigininprolometry–i.e.forproles,dening1DparameterscalledRsomething.2DarecalledSsomething.ITrytocalculatequantitiesthatarerelevantforothermeasurementsorprocesses–forinstancepowerspectrumdensityforoptics.Thismaybeisdifcultandevenmoredifculttocompare.Severalbasicconceptsandquantitiesarecommontobothapproaches. Roughnesscharacterisation Heightdistributionproperties DavidNecasetal. 10/27 BeyondSaandSq TheseprolesweretakenfromsurfaceswithquitesimilarbothSaandSq.Buttheroughnessisobviouslydifferent.Andincertainapplicationsthesurfaceswouldbehavequitedifferently.Anumberofotherheightdistributionparametersareaimedatcapturingthis. Peakandvalleyparameters–orminimumandmaximuminplaintermsSp=maxi;jzi;j;Sv=mini;jzi;jUsefulforindicationofunusualsharpspikesorcracks. SkewnessSsk,kurtosisSku–thirdandfourthordermomentsof%(z).One-dimensionalcharacteristics:bearingratiocurve(BRC).Inplainterms,thecummulativeheightdistribution. Roughnesscharacterisation Spatialproperties DavidNecasetal. 12/27 PeakcountingNormsusuallyemploypeakcountingandrequiretochoosecertainthresholds. Peakmustextendabovetheupperthresholdandthenfallbelowthelowerthresholdtobecounted. Highspotissimpler–denedbycrossingonlyonethreshold.PeakcountPc–numberofpeaksperunitlength.HighspotcountHSC–numberofhighspotsperunitlength.MeanspacingSm–averagedistancebetweenzerocrossings(inonedirection).Similarlyforvalleys. Roughnesscharacterisation Autocorrelation DavidNecasetal. 15/27 Modelling GaussianGx( x)=2exp � 2x T2!CommonmodelsinceGaussianhasmanynicepropertiesthatsimplifycalculations. ExponentialGx( x)=2exp� x TArguablymorelikelytobetheresultofcertainphysicalproceses. HowdoweactuallygettherootmeansquareroughnessandautocorrelationlengthT? Usingtting:IcalculatediscretisedGkfromexperimentaldata(anefcientalgorithmisFFT-based)IselectamodelforGx( x)–usuallyoneofthetwoaboveItthemodelonthediscretisedACF ThisworkswellifGkconvergestoGx( x)asthemeasurementareagetslarger.Itdoes(pointwise)butwithsomequirks. Roughnesscharacterisation Powerspectrumdensity DavidNecasetal. 18/27 PSDFversusACF Roughnesscharacterisation Powerspectrumdensity DavidNecasetal. 19/27 Modelling Gaussian(correspondstoGaussianACF)Wx(kx)=2T 2p exp �k2xT2 4! Exponential(correspondstoexponentialACF)Wx(kx)=2T 1 1+k2xT2 FittingofthesemodelfunctionsisagainusedtoobtainTand.DiscretisedPSDFiscalculatedusingdiscreteFouriertransformofthedata:W/jZj2;whereZ=1 p MM�1Xj=0zje�2ijWeonlywrite/becausenogenerallyagreed-uponconventionexists:IcontinuousFTexponentcanbeeitherikxxor2ikxx–circularvs.straightfrequency,IdirectFTfollowedbyinverseFTmustgivetheoriginalfunctionbutthisstillpermitsanarbitrarysplitofmultiplicativefactorsbetweenthedirectandinverseFT,IwecannotprescribeaspecicnormalisationofboththeFouriercoefcientsandspectrumdensity,Idiscretisationofnormalisationforniteenergyvs.niteaveragepowerfunctions.Theresult:amess.PSDFfromdifferentsoftwareisdifculttocompare. Roughnesscharacterisation Powerspectrumdensity DavidNecasetal. 20/27 EnergytheoremAresomePSDFconventionsbetterthanothers?Weimpose:Plancherel/Parsevaltheorem–sum/integralofpowerspectrumisthesameassum/integralofsquareddatavalues.Thereisnoshortageofthem: DiscreteFouriertransform M�1Xj=0z2j=M�1X=0W ClassicFourierseriesofaperiodicfunction 1 2Z�z(x)2dx=1Xn=�1janj2 ContinuousFouriertransform Z1�1jz(x)j2dx=Z1�1W(kx)dk ContinuousFouriertransformofniteaveragepowerfunctions limL!11 2LZL�Ljz(x)j2dx=Z1�1Wx(kx)dk Butthencomesthesampling-independencerequirement(recallit?):PSDFshouldbeindependentofhowlargepartofthesurfacewemeasureandthesamplingstep.ThenthediscretePSDFmustbejustsampledcontinousWx(kx).Noadditionalfactors. Roughnesscharacterisation Powerspectrumdensity DavidNecasetal. 21/27 Sampling-independence DiscretisedPSDFcorrectlysamplingthecontinuousone(circularfrequency)isthus:W=x 2jZj2;Z=1 p MM�1Xj=0zje�2ijUnitsofWare[dimension][height2].UnitscanbeusedtodistinguishFouriertransformconventions:I[height2]–plainDFTI[dimension2][height2]–nite-energycontinousFTDatasamplingchangeeffect:Icoarserdata–smallermax.frequencyIsmallerdata–coarserPSDFsamplingChangeofvariables,e.g.circularversusnon-circular.UsethatPSDFisdensity:Wx(a)da=Wx(b)db. Roughnesscharacterisation Non-uniformsurfaces DavidNecasetal. 25/27 Overalltopography Texturecansplittoroughnessandwaviness(overallshape)usingsomekindofhigh-passandlow-passlters:IsimplesplittinginfrequencydomainIGaussianltersIbidirectionalandmedianlters... Butthisisnotreallymeanstoseparatearbitraryoveralltopography...