scaling laws标度律

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1、ProgramofNano1(Nanoscience)aswasgivenin2012ScalinglawsScalingingeometry.Scalinginrigidbodydynamics.TheTrimmerforcescalingvector–scalinginelectrostaticforces,electromagneticforces,scalinginelectricityandfluidicdynamics,scalinginheatconductingandheatconvection.Forcescalingvector:F=[lF]=[l1l2l3l4]Sur

2、face-to-volumeratioscalesasA/V→L2/L3=L-1;Mass(weight)m=rV~L3MomentofinertiaI=mR2~L5Torquet=(ma)R~L4DiffusionD=kT/6πηr,η=viscosityr=isthehydrodynamicradiusFrequency:f=1/2p(k/m)½Naturalfrequencywn=an(EI/mL4)½EquivalentRigidity:Keq=EI/L3whereE:ElasticModulus;I:Theareamomentofinertia;m:Massperunitleng

3、th;L:Lengthofthebeam;an:anappropriatecoefficientforboundaryconditionsPiezolectriceffect:Hook’sLawD=eEwhereDistheelectricdisplacement,eispermittivityandEiselectricfieldstrength.ConstitutiveequationfordielectricsS=sTwhereSisstrain,siscompliance,Tisstress.CoupledEquationsS=sET+dtE;D=dT+eTEElectricpot

4、entialinparallelplates:E=½CV2=ee0WidthLengthV2/distanceScalingofclassicalmechanicalsystemsandelectricalsystems:’displacementxÛchargeqforceFÛvoltageVvelocitydx/dtÛcurrentI(=dq/dt)massmÛinductanceLspringKÛcapacitance1/CdampingbÛresistanceRProblemexample:frequencyscalingderivetheequationonthedependen

5、ceoffrequencyandequivalentrigidityoncantileverdensity.Problemexample:ScalinginelectrostaticsConsiderchargingupthecapacitorplatesthroughalargeresistorRNowmovetheplatesasmallamountdd.Forabiasofafewvoltsandaspacingof100nm,andknowingthatitiseasytomeasuremicrovolts,calculatethedetectabledisplacementsof

6、theplates.ForcesinthenanoworldIntermolecularforces(Debye,KeesomandLondoninteractions;Lennard-Jonespotential,vanderWaalsforces).Dipole-dipoleinteractionsUd=-2m12m22/3kTr6WheremisthedipolemomentTheinductioneffectUi=–[a2m12+a1m22]/r6whereaisthepolarisabilityLondondispersionforces:betweentwoatomswhere

7、andarethedipolepolarizabilitiesoftherespectiveatoms.IAandIBandarethefirstionizationpotentialsoftheatomsandRistheintermoleculardistance.Repulsion,shortdistances(Fµr12,Eµr11)CasimirforceFC=-p2ħc/240d4Cohesiveenergy