mdxv v2 2m v2 1 isBoltzmann ID: 824273
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Thetime,size,viscosity,andtemperaturedep
Thetime,size,viscosity,andtemperaturedependenceoftheBrownianmotionofpolystyrenemicrospheresDongdongJia,JonathanHamilton,LenuM.Zaman,andAnuraGoonewardeneDepartmentofGeologyandPhysics,LockHavenUniversityofPennsylvania,LockHaven,Pennsylvania17745Received22June2006;accepted6October2006AnupperdivisionundergraduatelaboratorymoduletostudyBrownianmotionwasdevelopedusingpolystyrenemicrospheressuspendedinasampleliquidcellthatwasisolatedfromitssurroundings.ThedependenceoftheBrownianmotionofthemicrospheresontheirradius,thetime,theviscosityofthesuspensionliquid,andthetemperatureweremeasuredusingaCCDcamerainterfacedwithaframegrabberandacomputer.Themotionofeachparticlewasmonitoredfor5minutesandwasfoundtobeconsistentwiththeLangevinformulaforBrownianmotion.©2007AmericanAssociationofPhysicsTeachers.DOI:10.1119/1.2386163mdxvv22m v2=1isBoltzmannÕsconstantandisthetemperature.IfwesubstituteEqs.intoEq.,weobtaind x2andobtaindtmw=2Ifwewrite,wehavedtw=2WeletandEq.dtwet+2+t=2 m =2sothatEq.isaparticularsolutionofEq..Inourcasethetimeislargeinminutesandthemassoftheissmall.TheseconditionsreducethesolutionEq.WeuseEqs.andobtain2thatis,IfweuseEqs.wehavet=2at=intwodimensions,wehaveIII.EXPERIMENTALTECHNIQUESANDAPPARATUSToobservethemotionofsmallBrownianparticles,poly-styrenemicrosphereswithradiiof0.5,1,3,and5mwerepurchasedfromBangLaboratory.ThemajorequipmentusedwasanOlympusCKX41SFinvertedopticalmicroscopeequippedwitha40objectivelens,aCCD,andadatacol-lectingsystemHitachicamerawithaframegrabbertorecordastreamofframes.Samplesuspensionsweremadebysuspendingthepoly-styrenemicrospheresintovariousliquidsataconcentrationsuchthatonlyonespherewaso
bservedintheopticalÞeld.Fortheviscositym
bservedintheopticalÞeld.Fortheviscositymeasurements1mpolystyrenemicro-sphereswereused.Liquidswithdifferentviscositieswereusedtosuspendthepolystyrenemicrospheres,forexample,water,ethylglycol,cookingoil,andwater/EGsolutioninvaryingconcentrations.Forsizedependencemeasure-ments,eachsizeofpolystyrenemicrospherewassuspendedinwater.Fortemperaturedependencemeasurements,1polystyrenemicrospheresweresuspendedinethylglycol.SampleobservationcellsweremadebyÞrstplacingtwolargedropsofcookingoilwithhighviscosityonaglassslide.Onesmalldropofsamplesuspensionwasinjectedintotheoilusinga1mlsyringe.AnotherglassslidewasthencarefullyplacedontopoftheÞrstslide.Thesuspensionandoilthenspreadoutandstabilized,allowingustousethisstructureasoursamplecell.ThecookingoilisolatesthesuspensiontoinsurethattheparticlesinthesuspensionareinBrownianmotionandnotsimplyßowingwiththeliquid.FortemperaturemeasurementsaCIR-1013/120Vcar-tridgeheaterwasusedtomaintainthesamplecellatdifferenttemperatures.Thetemperaturewasvariedbetween20¡Cand150¡C,whichisbelowtheboilingpointofethylglycol.Thesamplecellwaswrappedwithaluminumfoilforuni-formheating.Awindowwasopenedfordataacquisitionthroughthemicroscope.AKtypethermocoupleandanOmegaCNiuniversaltemperaturecontrollerwereusedtocontroltheheatingprocess.ThissetupisshowninFig.1.Thesamplecellswereplacedunderthemicroscope,ob-served,andrecordedasasequenceofframesacquiredbytheframegrabberandthecomputer.Thetimeintervalbetweentwoframeswas1s.Avideoof5minwastakenforeachmeasurementconsistingofaseriesof300frames.Theinitialpositionofthemicrospherewasselectedastheoriginwithpixelcoordinateintheframe.ThesquareofthedistanceoftheparticlefromineachframeasshowninFig.2isgivenby
Thetimeaveragevalueisgivenbyisthenumber
Thetimeaveragevalueisgivenbyisthenumberofframes.ByusingEqs.wehaveÞveparametersthatcanbemeasured.Theyarethetemperature,time,viscosity,radius,andthetimeaverageofthesquareofthedisplacementBecausetheexactdistanceinmcannotbeaccuratelyobtainedduetoampliÞcationeffectsofthemicroscope,pix-elsareusedasthecountingunit.Theminimumresolutionis0.5pixelmeasuredfromeachsphericalcenter.Fig.1.Setupofthesamplecellfortemperaturedependencemeasurements.112112Am.J.Phys.,Vol.75,No.2,February2007Jiaetal.IV.RESULTSThemeasurementofthetimedependenceofBrownianmotionwasperformedat20¡C.Figure2showsthelineardependenceofaspredictedbytheory.Duringthemea-surement,themotioninthewasobservedasadefocusingeffect.Severalmeasurementsweretakenandtheresultswithlessdefocusingeffectwereused.ToinvestigatethesizedependenceoftheBrownianmo-tion,eachsizemicrosphereisusedandthetimedependenceisplottedinFig.3.Inthiscasethetemperatureandtheviscositywereconstant,andthereforefromEq.slopeofthelinearÞtsreßectthedependenceontheradiusAlinearÞtgivesanonzerointerceptinsomecases,althoughtheinterceptsshouldallbezerotheoretically.Thenonzerointerceptcouldbeduetotwo-dimensionaldataobtainedfromthree-dimensionalmotion,uncertaintyinthedata,andthetimeintervalnotbeingsufÞcientlylong.Thesizedependenceofwouldcontainlargeerrorsifitweredeterminedbyusingdataatthesametimeforallsizemicrospheres.Instead,weusetheslopetodeterminethesizedependenceof.FromEq.theslopeofisgiven,whereisaconstant.ThevaluesoftheinverseslopewereplottedagainsttheradiusasshowninFig.3.Alinearrelationisfoundaspredicted.SeeTableI.TheviscosityisdeÞnedastheratiooftheshearingtothevelocitygradientinaßuid:TheSIunitofviscosityisPascalsecondbutisrarelyused.T
hemostcommonunitofviscosityispoiseaftert
hemostcommonunitofviscosityispoiseaftertheFrenchphysicistPoiseuille,whichisadynesecondpersquarecentimeterás/cm,thatis,10P=1Pa.s.Fig.2.TimedependenceofBrownianmotionusing1mmicrospheres.Brownianmotionofsinglemicrosphereinframe.Plotofversustheshowingalinearrelation2%slopeerrorFig.3.SizedependenceofBrownianmotion.for0.5,1,3,and5mradiipolystyrenemicrospheres.Theinverseslopeversusshowingalinearrelationerrorbaris5%TableI.Thevaluesoftheinverseslopeforeachsizemicrospherearbitraryunit0.53.616.7323.8537113113Am.J.Phys.,Vol.75,No.2,February2007Jiaetal.V.DISCUSSIONThedependenceoftheBrownianmotionofsmallparticlesonthetime,viscosity,size,andtemperatureisfoundtobeconsistentwithEq..Alargenumberofdataneedtobetakenandanalyzedbythestudents.Tosavetimewedividedthelabintothreepartsandhadstudentsdeterminethesizeandtimedependence,theviscositydependence,andthetem-peraturedependence.Wefoundthatdefocusingduetothethree-dimensionalmotionofthemicrosphereisoneofthemajorproblemsdur-ingtheexperiment.Ifsuchmotionappears,wemustneglectit.Tothisextentwemadesurethatthemotionwastwo-dimensionalandconsistentwithourassumption.Wealsofoundthatevenslightvibrationsofthemicro-scopetablecangreatlyaffectthequalityofthedata.Allhandsmustbeoffthetablewhendataistaken.ThisworkwaspartlyfundedbyaWorkForceLeadershipgrantfromthePennsylvaniaDepartmentofCommunityandEconomicDevelopmentandthemodulewassuccessfullytestedandincorporatedinupperdivisionundergraduatelaboratoriesatsevenuniversitiesinthePennsylvaniaStateSystemofHigherEducation.TheauthorswouldliketothankDonWoodhousewhohelpedproofreadthemanuscript.Se-YuenMak,ÒBrownianmotionusingalaserpointer,ÓPhys.Teach.H.GradenKirksey,ÒBrownianmotion:Aclassr
oomdemonstrationandstudentexperiment,ÓJ.
oomdemonstrationandstudentexperiment,ÓJ.Chem.Educ.,1091Ð1093PaulNakroshis,MathewAmoroso,JasonLegere,andChristianSmith,ÒMeasuringBoltzmannÕsconstantusingvideomicroscopyofBrownianmotion,ÓAm.J.Phys.,568Ð573W.EbelingandF.Schweitzer,ÒSelf-organization,activeBrownaindy-namicsandbiologicalapplications,ÓNovaActaLeopold.,169Ð188RobertBrown,TheWorldoftheAtom,editedbyH.BoorseandL.MotzBasicBooks,NewYork,1966,pp.206Ð212.AlbertEinstein,InvestigationsontheTheoryoftheBrownianMovementDover,NewYork,1956S.C.Garg,R.M.Bansal,andC.K.Ghosh,ThermalPhysicsTataMcGraw-Hill,NewDelhi,1993,pp.53Ð59.F.Reif,FundamentalsofStatisticalandThermalPhysicsNewYork,1965,pp.560Ð567.www.bangslabs.comCRCHandbookofChemistryandPhysics,59thed.,editedbyR.WeastandM.J.AstleCRC,BocaRaton,FL,1979,p.D-275.N.G.TsierkezosandI.E.Molinou,ÒThermodynamicpropertiesofwater+ethyleneglycolat283.15,293.15,303.15and313.15K,ÓJ.Chem.Eng.Data,989Ð939E.A.ChechetkinaandS.A.Dembovsky,ÒOntheformoftheviscositytemperaturedependenceforglassesandtheirmelts,ÓJ.Non-Cryst.Solids,171Ð178I.A.LavyginandS.F.Chistov,ÒTemperaturedependenceoftheviscos-ityofpolymermeltsandliquid-liquidtransitions,ÓPolym.Sci.U.S.S.R.,1478Ð1485Reference10,p.F-54.Fig.6.TemperaturedependenceofBrownianmotionusing1mmicro-=20,60,90,120,150¡C.Theslopeversus.Dataindicatedwiththesymbolshowsthatincreaseswithincreas-.Dataindicatedbythesymbolshowalinearrelationofaftertheslopesarecorrectedforthetemperaturedependenceofviscosityerrorbaris10%TableIII.Thevaluesoftheslopeattemperature10%errorTemperaturearbitraryunitViscosityCorrectedslopearbitraryunits250.042160.67600.1550.75900.342.30.781200.871.10.961501.670.61115115Am.J.Phys.,Vol.75,No.2,February2007Jiaetal