Electrostatic instability of micro-plates subjected to differential pressure

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Delft University of Technology

Electrostatic instability of micro-plates subjected to differential pressure

A semi-analytical approach

Sajadi, Banafsheh; Goosen, Hans; van Keulen, Fred

DOI

10.1016/j.ijmecsci.2018.02.007

Publication date

2018

Document Version

Final published version

Published in

International Journal of Mechanical Sciences

Citation (APA)

Sajadi, B., Goosen, H., & van Keulen, F. (2018). Electrostatic instability of micro-plates subjected to

differential pressure: A semi-analytical approach. International Journal of Mechanical Sciences, 138-139,

210-218. https://doi.org/10.1016/j.ijmecsci.2018.02.007

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ContentslistsavailableatScienceDirect

International

Journal

of

Mechanical

Sciences

journalhomepage:www.elsevier.com/locate/ijmecsci

Electrostatic

instability

of

micro-plates

subjected

to

diﬀerential

pressure:

A

semi-analytical

approach

Banafsheh

Sajadi

∗

_,

_Hans

_Goosen

_,

_Fred

_van

_Keulen

Department of Precision and Microsystem Engineering, Delft University of Technology, Delft, 2628 CD, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Keywords: Pull-in Electrostatic instability Bi-stability Micro-plate Snap-through Nonlocal elasticity

a

b

s

t

r

a

c

t

ElectrostaticinstabilityisoneofthemainfeaturesofmanyelectrostaticMEMSandNEMSdevices.Inthispaper, weinvestigatehowtheelectrostaticinstabilityofaplate-likeelectrodecanbeaffectedbyadifferentialpressure. Theresultsofthisstudyindicatethatthepresenceofdifferentialpressurecanhaveasignificantinfluenceonthe equilibriumpath,thenumberandlocationofunstablepoints,andthepost-instabilitybehavior.Asaresult,while thesystemisloadedandunloadedelectrically,theelectrostaticinstabilitymightleadtoasnappingbehavior. Thenoticedsnappingbehaviorofaflatplatemakesitveryappealingforsensingandactuatingapplications. Thisstudyisbasedonbothasemi-analyticalframeworkandfiniteelementsimulations.Theproposedanalytical solutionisshowntobeaccurateenoughtobeusedasaneffectivetoolfordesign.

1. Introduction

Electrostaticinstability(andbi-stability)isanimportantfeatureof manyelectrostaticMEMSdevices,sensorsandactuators[1–3].Asolid understandingoftheelectrostaticinstabilityisessentialtoimprovethe performancesuchMEMS/NEMSdevicesandtoobtainnewdesignsfor newapplications.

ElectrostaticMEMSdevicesessentiallyconsistofasimpleparallel platecapacitorwithatleast oneflexibleelectrode.Whenanelectric potentialisappliedtothecapacitor,anattractiveelectrostaticloadis inducedbetweenitselectrodes.Thisloaddependsontheapplied elec-tricpotential,thelocaldistancebetweentheelectrodes,andthe dielec-tricconstantofthemediumseparatingthem[4].Theelectrostaticload leadstodeformationoftheflexibleelectrode(s)tomaintainthebalance betweenelectrostaticandelasticpotentials.Thus,anysmallchangein theelectricpotentialcangeneratemechanicalmovementoftheflexible plate,whichcanbeusedforactuation[5].

InstabilityofanelectrostaticMEMSdeviceoccursmainlyduetothe nonlinearityintheelectrostaticpotential[6].Whenthesystembecomes unstable,anyperturbationcouldleadtofailureorpull-inoftheﬂexible electrode.Thisstabilityphenomenonappearsasafoldoralimitpoint intheequilibriumcurveofthesystem[7,8].

IncertainelectrostaticMEMSdevices,e.g.microphonesand pres-suresensorsoractuators,itisessentialtoavoidpull-ineﬀects,sincethe contactbetweenthetwoelectrodesinducesfailures,includingshort

cir-∗_{Corresponding}_author.

E-mailaddress:b.sajadi@tudelft.nl(B.Sajadi).

cuit,stick,wear,dielectricchanging,andbreakdown[6,9].Ontheother hand,pull-inisafeatureofMEMS/NEMSdevices[2]whichcan also provideinformationonthemechanicalandphysicalcharacteristicsof thesystem.Therefore,ithasbeenintroducedasarobustmechanismfor measuringthemechanicalpropertiesofnano-structures[10],orsensing theadsorbatestiﬀnessinnano-mechanicalresonators[11].

Inordertoavoidoremploythepull-ineffect,anin-depthknowledge ofthestabilitybehaviorandanaccuratedeterminationofthepull-in voltageof thestructure iscriticallyrequired.Inengineering applica-tions,toapproximatethecriticaldeformationoftheelectrodeandto predictthepull-involtage,asimple1-Dspring-massmodeliscommonly used(seeFig.1).Insuchamodel,theinstabilityoccurswhendueto thedeformationoftheflexibleelectrode,thegapbetweenthetwo elec-trodesbecomestwothirdsoftheinitialgap[7,12,13].Thissimplified modeliscommonlyreferredtoasthe‘1/3airgaprule’.Inpractice, how-ever,amembrane/platestructureisdifferentfromasimplespringmass model.Itisa2DstructurewhichincorporatesPoissonratioeffects,and membranestiffeningeffects.Inaddition,iftheplate-likeelectrode de-forms,theelectrostaticloadisnolongeruniform.The1/3airgapmodel inherentlydoesnotaccountforanyoftheseeffects.However,the criti-calgapbeingequaltotwo-thirdsoftheinitialgapsize,isshowntobe themostconservativecriticalgapinMEMScapacitors[12].

Thepull-inofcircularclampedplate-likeelectrodeshasbeen investi-gatedandformulatedinmanystudies[9,12,14].Thesestudiesarebased onsimplifyingassumptions,suchassmallandone-dimensional

defor-https://doi.org/10.1016/j.ijmecsci.2018.02.007

Received 12 June 2017; Received in revised form 3 January 2018; Accepted 3 February 2018 Available online 6 February 2018

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B. Sajadi et al. International Journal of Mechanical Sciences 138–139 (2018) 210–218

Fig.1. Simple1-Dmodeltypicallyusedtoapproximatethecriticaldeformation oftheelectrodeinordertopredictthepull-involtage.

mationoftheplate,oruniformityoftheelectrostaticload.Ultimately, finite-differenceandfinite-elementmethodshavebeenappliedtosolve theresultedequations[9,15,16].Differentvaluesforcritical deforma-tionwereproposedforcircularclampedelectrodes,from41.5%[9]or 41.6–45.6%[15]to72–75%[16]dependingon thethicknessof the plateandsimplifyingassumptions.However,acomprehensiveanalytical

solutionforpull-involtageofacircularclampedplate,while consider-ingthenonlinearmembraneeﬀectsandnon-uniformityofelectrostatic load,ismissingintheliterature.

Thepull-involtageandcriticaldeflection ofcantilever-,beam-or plate-likeelectrodesdependonthestiffnessoftheflexiblecomponent, aswellastheinitialdistancebetweentheelectrodes.Amechanicalload appliedtothecomponentcandirectlyaffectbothparameters,and con-sequently,influencestheelectrostaticinstabilityofthesystem.The sen-sitivityofelectrostaticinstabilitytoamechanicalloadistheconcept behindusingpull-ininstabilityasamechanismforsensing,for exam-ple,temperature[17],surface-stress[4],orresidualstressesinclamped structures[7].Furthermore,MEMSsensorsandactuatorsarefrequently designedtooperateunderamechanicalload,suchasincapacitive pres-suresensors.Sometimes,theadditionalmechanicalloadinthese de-vicesisundesired,butalsoinevitable,suchasthermalloadsorresidual stressesinclampedstructures.Therefore,anin-depthknowledgeabout theeffectofthesemechanicalloadsonthestabilityofthemicro me-chanicalcomponentisparamount.

Thedependencyofthepull-involtageofMEMSdevicestoexternal mechanicalloads,suchasauniformtransverse,or,in-planeload,has beeninvestigatedintheliterature[7,13,18–22].Particularly,theeffect ofauniformdifferentialpressureontheelectrostaticinstabilityofa cir-cularmicroplatehasbeenstudiedaswell[23].Thelater,usinga numer-icalcontinuationscheme,calculatedthecombinationofpressureand voltagewhichcanleadtotheinstabilityofthesystem.Thisstudy consid-ersthedifferentialandtheelectrostaticpressuretobeinthesame direc-tionanditshowsthatthedifferentialpressurealwayscausesthesystem tobemorepronetoinstability.Itshouldbenoticedthattheproposed numericalmethod,evenifhighlyaccurate,isanexpensivetools,and,for eachnewsetofdesignparameters(radiusorthickness)thesimulation hastobe repeated.Anaccurateanalyticalmodel–ifavailable– could provideaclosedformsolutionforapproximatingthepull-involtageand thecriticaldeformationalmostwithoutanytimecost.Inaddition,it pro-videsmoreinsighttotheproblemwhichisfavoredfordesignpurposes. Inthispaper,weproposeananalyticalapproximatemethodology tostudythestabilityandpull-inbehaviorofacircularflexible elec-trode,while,loadedwithelectrostaticanddifferentialpressure.Inthis analysis,thedirectionofthepressureis notpredefined.Instead,itis consideredtobeadifferentialpressurepositivewhenopposingthe elec-trostaticload,andnegativeotherwise.Thenon-linearstretchingofthe thinplateandthenon-uniformityofelectrostaticloadduetodeflection oftheflexibleplate,areincorporatedinthissolution.Theaccuracyof theproposedanalyticalapproximationisevaluatedwithacomparison tothefiniteelementsimulations[24].

Usingtheproposedsolutions,first,theinstabilityandpull-in behav-ioroftheelectrodeanditsdependencyontheelectrode’sthicknessand radiusarediscussed.Next,weexplorehowadifferentialpressureand itsdirectionwouldaffectequilibrium,stability,andthecriticalvoltages anddeflections.Inaddition,thepost-instabilitybehaviorofthesystem

Fig.2. Schematicofthecrosssectionofthecapacitorwithoneflexibleelectrode ina)undeformedconfiguration,and,b)deformedundercombinedelectrostatic anddifferentialpressure.

andpossiblesnappingbehaviorwillbeaddressed.Wewillshowthat thepresenceofadiﬀerentialpressurecantriggerbi-stabilityinthe sys-tem.Therequiredcriteriatoattaintheadditionalstablesolutionand thesnappingbehavior,whichcanbeofgreatinterestforsensingand actuationpurposes,willbethoroughlydiscussed.

Finally,weshallremindthattheeffectiveelasticpropertiesof struc-tures at nano andsometimeseven at microscalesareknown tobe stronglysize-dependent[25–29].Theclassicalcontinuumtheoryis in-herentlysizein-dependentandhence,itcannotprovideagood predic-tionwhenthethicknessoftheplateisverysmall.Forsmalllengthscales, size-dependentcontinuumtheoriesthataccountforthesescaleeffects should be utilized[30–36]. These theories,embed amaterial length scale(e₀a) whichmakes itpossible toqualifythesizeof astructure aslarge” orsmall” relativetoitsmateriallengthscale.Ifthesizeofthe structureisrelativelylarge,thenthenonlocalorhigher-ordergradient theoriesconvergetoclassicalelasticitytheoryandtherefore,employing theclassicaltheorywillleadtosimilarresults.Onceplasticityplaysa role,anotherlengthscaleshouldbeconsidered[37].Inthispaper,using astrongnonlocalelasticitytheory[35,38],thepossibilityofcapturing thescalingeffectsintheproposedformulationisbrieflyinvestigated, andthesize-dependenceofstabilityofamicro-platewhilesubjectedto electrostaticanddifferentialpressureisaddressed.

2. Analyticalformulation

Theanalyticalmodelproposedhereisbasedonaparallelplate ca-pacitorwithathin,circular,fullyclampedplateasoneelectrode,while theotherisﬁxedandrigid.Theshapeofthecapacitorischosentobe cir-cular,sincetheMEMSdeviceswithacircularplategenerallyyieldbetter structuralﬂexibilityascomparedtorectangularplates.Inaddition,they havenocornersorsharpedgeswhichmayinducehighresidualstresses duringfabricationprocess[15].Theschematicoftheassumedmodelis showninFig.2.

Theradiusof theﬂexible electrodeisRandits thicknessis h.It ismodeled withalinearelastic,homogeneousandisotropicmaterial model.TheYoungsmodulusandPoissonratiooftheplatearedenotedE

and𝜈,respectively.Theplateissuspendedoverthegroundedelectrode withsimilarradiusandtheinitialgapbetweentheelectrodesisd.The plateisloadedwithadiﬀerentialpressureP,andanelectricpotentialV

isappliedtotheelectrodes.

TheplateismodeledwithvonKárḿansplatetheory,whichaccounts for finitedeflection butmoderaterotations andis adequatefor thin plates[39].Theloadsareconservative,whichimpliesthatfirst,to es-timatethedeflectionintheequilibriumstate,approximationsbasedon minimizingthetotalpotentialenergycanbeapplied,andsecond,no dy-namicconsiderationisrequiredtoassessstabilityofequilibriums. Mini-mizingthetotalpotentialenergyisavariationalproblemanditssolution can beestimatedusingRitz’smethod.Inthismethodaparametrized displacementfieldsatisfyingtheclampingboundaryconditionis con-sidered,whereastheunknownparametersarecalculatedbyrequiring thetotalpotentialenergytobestationary.

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Duetotheaxisymmetricconditionintheproblemathand,theonly appearingdisplacementcomponentsaretheradial(u)andtransverse (w)components.Although,thenonlinearitymightcausethesymmetry tobreakup,weconsiderthesymmetrytomaintainduringdeformation. Thisassumptionhasbeenverifiedusingafiniteelementmodelwhich willbedescribedinthenextsection.Theclampingboundarycondition forcesthedisplacementcomponentsandalsothefirstderivativeofthe transversedisplacementwithrespecttotheradialcoordinatetobeequal tozeroattheboundaries.WeadoptTimoshenko’ssimpleapproximate displacementfieldforuniformlyloadedcircularplates[40],to approx-imatetheradial(u)andtransverse(w)displacements:

𝑤=𝐶1𝑑(1−𝜌2)2,

𝑢=𝑅𝜌(1−𝜌)(𝐶2+𝐶3𝜌), (1)

where𝜌 =_𝑅𝑟 isthenon-dimensionalradialcoordinateand,Ci,(i=1–3,)

aretheparameterstobecalculated.Next,theassociatedtotalpotential energyisevaluated.Thetotalpotentialenergyconsistsof fourterms namely,theelectrostaticpotential(Ue),thepotentialsassociatedwith

elasticdeformationduetothebending(Ub)andthestretching(Us)of

theplate,andthepotentialassociatedwiththemechanicalpressure(W):

𝑈=𝑈_𝑒+𝑈_𝑏+𝑈_𝑠−𝑊. (2)

Assumingtheparallel-platecapacitortheory,theelectrostaticpotential followsas[2,41]: 𝑈𝑒=−𝜋𝜖𝑉2𝑅2 ∫ 1 0 𝜌𝑑𝜌 𝑑+𝑤, (3)

where𝜖 istheelectricpermittivityofthedielectricbetweenthe elec-trodes.Noticethat thelocaldistance betweentheelectrodes (𝑑+𝑤) is employed to calculate the electrostatic potential. Thus, the non-uniformityoftheelectrostaticloadduetothedeﬂectionoftheﬂexible electrodeisincorporated.

Provided that themicro-plate is isotropic andhomogeneous, the bending-extensioncouplingstiﬀnessequalstozero.Therefore,the po-tentialsassociated withelasticdeformationcan be decoupled tothe bendingenergy(Ub)andthestretchingenergy(Us)[39]:

𝑈𝑏 = 𝜋𝐷_𝑅2 ∫ 1 0 (₍ 𝜕2_𝑤 𝜕𝜌2 )2 + ( 1 𝜌𝜕𝑤𝜕𝜌 )2 + ( 2𝜈 𝜌 𝜕𝑤𝜕𝜌𝜕 2_𝑤 𝜕𝜌2 )) 𝜌𝑑𝜌, (4a) 𝑈𝑠 = ₍₁𝜋𝐸ℎ₋_𝜈2₎∫ 1 0 (₍ 𝑢 𝜌 )2 + ( 𝜕𝑢 𝜕𝜌+ 1 2𝑅 ( 𝜕𝑤 𝜕𝜌 )2)2 +2𝜈𝑢 𝜌 ( 𝜕𝑢 𝜕𝜌+ 1 2𝑅 ( 𝜕𝑤 𝜕𝜌 )2)) 𝜌𝑑𝜌, (4b) where𝐷= 𝐸ℎ3

12(1−𝜈2₎isthebendingstiffnessoftheflexibleplate.Notice thatnonlinearmembraneeffectshavebeenincorporatedintheelastic potential.

Astherotationsintheplateduetomechanicalandelectrostaticloads aresmall,thepressureisassumedtobealwaysperpendiculartothe un-deformedsurface.Therefore,thepotentialassociatedwiththepressure canbecalculatedas:

𝑊 =2𝜋𝑃𝑅2 ∫

1

0 𝑤𝜌𝑑𝜌.

(5) BysubstitutingEq.(1)intoEqs.(3)–(5),anapproximationforthetotal potentialenergycanbederivedanalytically.Sincetheanalytical expres-sionoftheintegralinEq.(3),dependsonthesignoftheparameterC1, weshallcalculatethetotalpotentialenergyandsolvetheproblemfor

C₁<0andC₁>0,separately: 𝑈 =−𝜖𝑉2𝜋𝑅 2 2𝑑 ( ϝ(𝐶1) ) −𝑃𝜋𝑅 2_𝑑 3 𝐶1 + 32𝜋𝑑 2 3𝑅2 𝐷𝐶 2 1+𝐸ℎ𝑅 2_𝜋 (1−𝜈2₎ ( 𝛼1𝐶22 +𝛼2𝐶32+𝛼3𝐶2𝐶3−𝛼4𝐶12𝐶2𝑑 2 𝑅2 +𝛼5𝐶3𝐶12 𝑑2 𝑅2+𝛼6𝐶 4 1 𝑑4 𝑅4 ) , (6)

where, 𝛼i is introduced for compactness, with 𝛼1=0.250, 𝛼2=0.117 𝛼3=0.300,𝛼4=0.068,𝛼5=0.055,𝛼6=0.305.Theseparametersare de-terminedbytheselectedbasis-functions,andrepresentthelinearand nonlinearstretchingstiﬀnesscomponentsinthestrainenergy. More-over, ϝ(𝐶1)= atanh√−𝐶1 √ −𝐶1 if𝐶1<0, (7a) ϝ(𝐶1)=1 if𝐶1=0, (7b) ϝ(𝐶1)= atan√𝐶1 √ 𝐶1 if𝐶1>0. (7c)

Noticethatϝ(x)isacontinuousandsmoothfunctionaroundzero. Next,thestationarypointsoftotalpotentialenergy(U)canbefoundby equatingitsderivativetotheunknownparameters(Ci)tozero,

𝜕𝑈 𝜕𝐶1 = 𝜕𝑈 𝜕𝐶2 = 𝜕𝑈 𝜕𝐶3 =0. (8)

SolvingEq.(8)forparametersC2andC3,leadstoarelationbetween thestretchingoftheelectrodeanditstransversedeﬂection,independent of theappliedloads,VandP.Asamatteroffact,C₂andC₃ canbe calculatedasafunctionofC1andsubstitutedintoEq.(6).Hence,the degreesoffreedomcanbereducedtoC1only,whileincorporatingthe in-planedeformation,aswell.Then,equilibriumrequires:

𝜕𝑈 𝜕𝐶1

=0, (9)

whichleadsto: −𝜖𝑉2𝜋𝑅 2 2𝑑 ( 1 2𝐶1(1+𝐶1) −ϝ(𝐶1) 2𝐶1 ) +64𝜋 3 ( 𝑑 𝑅) 2 𝐷 ( 𝐶1+0.488(𝑑_ℎ) 2 𝐶3 1 ) −𝑃𝜋𝑅 3 3 𝑑 𝑅=0. (10)

ItisworthtonotethatinEq.(10),twosourcesofnonlinearityare in-corporated:(1)thecubictermduetothegeometricalnonlinearityand, (2)thenonlinearityofelectrostaticload.Duetothepresenceof non-linearity,multipleequilibriumstatesmightbefoundforoneloadcase (PandV).Therefore,theequilibriumpathmightexhibitoneoreven morebifurcations,atwhichsolutionbranchesmeet.Thestabilityofthe solutioncanbedeﬁnedbythesignofthesecondderivativeofthetotal potentialenergywithrespecttotheonlydegreeoffreedomleft(C1). Infact,thesystemisstable,whenthesecondderivativeispositive,and unstable,ifitisnegative.

Thecriticalpoint(s)canbecalculatedbyequatingthesecond deriva-tiveofthetotalpotentialenergywithrespecttotheonlydegreeof free-domtozero.This,fromaphysicspointofview,meansthatthesystem wouldhavenostiﬀnessinthedirectionofthesubjecteddegreeof free-dom.Therefore,thesecondderivativeofthetotalpotentialenergyat thecriticalpointscanbecalculatedas:

𝜕2_𝑈 𝜕𝐶12 =−𝜖𝑉2𝜋𝑅 2 2𝑑 ( − 5𝐶1+3 4(𝐶1+1)2𝐶12 +3ϝ(𝐶1) 4𝐶12 ) +64𝜋𝑑 2 3𝑅2 𝐷 ( 1+𝛼7( 𝑑 ℎ )2 𝐶12 ) =0. (11)

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where𝛼7=1.464.

Recallthatatthecriticalpoints,thesystemisstillinequilibrium. Thus,Eqs.(10)and(11)shouldbesolvedsimultaneouslyinorderto calculatethecriticaldeflection(s)andvoltage(s).Thecriticalvalueof voltageanddeflectionaredenotedwithsuperscribecr.Withsuchan analyticalsolution,onecanaccuratelyapproximatethevoltagelevel(s) atwhichinstabilityoccursasafunctionofthematerialproperties,the geometricalparameters,andtheapplieddifferentialpressure.Itisworth notingthatsinceEqs.(10)and(11)arebothhighlynonlinearinC1, solvingtheseequationsnumericallyisrelativelydifficult.Therefore,as analternative,onecansimplysolvetheseequationsforPandVfora feasiblerangeofcritical𝐶𝑐𝑟

1 (e.g.-0.99to+1).Thisapproachwillresult inobtaining𝐶𝑐𝑟

1 asanumericfunctionofPandV.

3. Finiteelementanalysis

Toverifytheresultsoftheanalyticalestimation,a3Dcircular elec-trodewasmodeledusingfiniteelements(COMSOLMultiphysics[24]). Inthemodel,theelectrodeisconsideredtobeflexible,clampedonthe edgeanditwasdiscretizedwithsolidelementsusingfreetetrahedral meshing.Thematerialpropertiesandspecificationsofthemodel,that areusedfor thetestcasefor thissolution, are:𝐸=80GPa,𝜈 =0.2,

𝜖 =8.854× 10−12_m−3_kg−1_s4_A2_and_𝑑₌₂_μm._To_study_the_eﬀect_of_the dimensionsoftheelectrode,diﬀerentcombinationsofthicknessand ra-diushavebeenstudied.

The electrostatic anddiﬀerential pressures have been applied as boundaryloadstotheplate,asPand𝜖 −𝑉2

2(𝑑+𝑤)2,wherewisthetransverse displacementfieldofthemicro-plateandVisavariablerepresentingthe voltage.Inthesolidmechanicsmodule,aglobalequationisintroduced todefinethevoltage(V)asafunctionoftheaveragedeflectionofthe plate.Hence,therequiredelectricpotentialtomaintaintheequilibrium oftheplateforaspecifiedaveragedeflection(w0)canbecalculated.

Thiscalculationisrepeatedoverarangeofaveragedeflectionsand asaresult,theequilibriumpathofthesystemisachieved.Itshouldbe noticedthatinthefiniteelementmodel,thesymmetryofthe displace-mentfieldisnotimposedtothesystem.However,thedisplacementfield appearstobeaxi-symmetricforboththeresultingstableandunstable solutionbranches.Theresults fromthismodelis comparedwiththe proposedanalyticalsolutioninthe“Resultsanddiscussion”.

4. Scalingeﬀects

InordertousetheformulationproposedinSection2fordesign pur-poses,oneshouldconsiderathicknessrangeatwhichtheelastic coeﬃ-cientsforbulkmaterialscanstillbeemployed.Otherwise,apropersize dependenttheoryshallbeemployedtocapturethescaleeﬀectsinthe formulation.

Here,webrieflydiscussthescalingeffectsontheobtained formu-lationusingastrongnonlocalplateformulation[38,42,43].Assuming thattheradiusoftheplateismuchlargerthanitsthickness,andthe stressderivativesinradialdirectionaresmall,wecanneglectthe non-localeffectsinin-planedirection.Therefore,thescalingmodification factorprovidedbyRef.[35]can beadoptedforimposingthescaling effectsonthebendingandstretchingrigiditiesoftheplateasafunction ofitsthickness: 𝜆 = 1 𝜂√𝜋 ( exp(−𝜂2)−1)+erf(𝜂), (12) and, 𝛽 =erf(𝜂)−_√1 𝜋 ( 2 𝜂exp(−𝜂2)+(3𝜂−1−2𝜂−3)(1−exp(−𝜂2)) ) . (13) whereerfistheerrorfunction,𝜂 = _𝑒ℎ

0𝑎ase0aisthemateriallengthscale, andtheobtained𝜆 and𝛽 arethemodiﬁcationfactorsforstretchingand bendingrigidities,respectively.Thesefactorsareobtainedbyusinga threedimensionalstrongnonlocalformulationandaGaussiannonlocal

kernelforaplateofwhichtheradiusismuchlargerthanthethickness. MoredetailsofthederivationofthesefactorscanbefoundinRef.[35]. As a consequenceof employing these factors, thestretching and bendingenergytermsinEqs.(4b)and(4a)canbemodiﬁed. Follow-ingtheprocedureasdiscussedinSection2resultsinequilibriumand instabilityconditions: 𝜕𝑈 𝜕𝐶1 =−𝜖𝑉2𝜋𝑅 2 2𝑑 ( 1 2𝐶1(1+𝐶1) −ϝ(𝐶1) 2𝐶1 ) + 64𝜋 3 ( 𝑑 𝑅 )2 𝐷 ( 𝜆𝐶1+0.488𝛽( 𝑑_ℎ )2 𝐶3 1 ) −𝑃𝜋𝑅 3 3 𝑑 𝑅=0. (14) 𝜕2_𝑈 𝜕𝐶12 =−𝜖𝑉2𝜋𝑅 2 2𝑑 ( − 5𝐶1+3 4(𝐶1+1)2𝐶12 +3ϝ(𝐶1) 4𝐶12 ) +64𝜋𝑑 2 3𝑅2 𝐷 ( 𝜆 +𝛼7𝛽( 𝑑 ℎ )2 𝐶12 ) =0. (15)

In orderto obtainthesizedependent criticaldeflection(s) and volt-age(s),Eqs.(14)and(15)shouldbesolvedsimultaneously.Inthenext section,wewillbrieflydiscusstheeffectsofusingtheproposed formu-lation,andcapturingthescalingeffects,onthestabilityassessmentofa micro-platewhilesubjectedtodifferentialandelectrostaticpressures. 5. Resultsanddiscussion

Inthissection,theinfluenceofauniformpressureonthecritical de-flectionandvoltageofaparallelplatecapacitorwithacircularflexible electrode,willbestudied.Theresultsoftheproposedanalytical approx-imationwillbediscussedandcomparedwithfiniteelementsimulations. Forthispurpose,normalizedloadparametersareintroducedas:

normalizedvoltage:𝑉′₌_𝑉 √ 12𝜖𝑅4₍₁₋_𝜈2₎ 𝑑6_𝐸 , normalizedpressure:𝑃′₌_𝑃1−𝜈2 𝐸 . (16)

Inaddition,themaximumdeﬂectionoftheplateisnormalizedwiththe initialgapsized.

First,considerthecasewithnopressure(𝑃′₌₀_)._The_{corresponding} deﬂectiondeformationismodeledasexpressedbyEq.(1).The maxi-mumdeﬂectionoccursatthemidpoint(𝜌 =0),andisequaltoC1d.

Fig.3showsthechangeofthemidpointdeflectionasafunctionof theappliedvoltage.Thepresentedcurvesaredeterminedanalytically fordifferentthicknessesoftheflexibleelectrode.Theresultsofthe fi-niteelementsimulationsarealsoshowninthisfigure,andascanbe observed,theyconfirmtheaccuracyoftheapproximateanalytical so-lution.Theerrorbetweenthesetwosolutionsinworstcase(ℎ∕𝑑=0.1) occursatthelimitpointandislessthan8%.Infact,theaccuracyofthe analyticalsolutionisbetterforthickerelectrodes.

AsFig.3indicates,thedeﬂectionofthemidpointoftheﬂexible elec-trodeincreasesmonotonicallywiththeappliedvoltageuntilthesystem reachesalimitpointorsaddle-nodebifurcation.Atthiscriticalpoint, thesystembecomesunstable,andifthevoltageisincreased,itleadsto pull-in.

Itcan beobservedfromFig.3thatthecriticaldefectiondepends onthethicknessofthestructure.Infact,solvingEqs.(10)and(11)for

𝑃 =0,resultsinacriticaldeﬂection(𝑤𝑐𝑟

𝑑 =𝐶𝑐𝑟1)whichisonlyafunction ofh/d.ThisfunctionisshowninFig.4.Thecriticaldeﬂectioncalculated withtheproposedmethodvariesbetween51–71%oftheinitialgap be-tweentheelectrodesandisalwayshigherthan1/3oftheinitialgap whichiscalculatedwithasimple1Dspringmodel.This,asmentioned before,is becausemodelingtheelasticrestoringforceswith alinear springdoesnotaccountforthenon-uniformelectrostaticforceonthe 213

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Fig.3.Theequilibriumpathofthemidpointofthecircularflexibleelectrode fordifferentthicknesses,andradius𝑅=100μm.—— stableequilibrium,--- -unstableequilibrium,and finiteelementsimulations(COMSOL Multi-physics).

Fig.4. Thenormalizedcriticaldeﬂectionatthemidpointofacircularplate withradius𝑅=100μm,asafunctionofitsnormalizedthicknessh/d.

plateafterdeflection,and,thenonlinearstiffeningeffectof the flexi-bleelectrode.Thelattereffectismorepronouncedforthinnerplates, causing𝑤𝑐𝑟

𝑑 tobelarger.

Thecriticalvoltageofthesystem,dependsonthematerial proper-tiesandthedimensionsofthecapacitor.SolvingEqs.(10)and(11)for

𝑃=0showsthatthepull-involtageisproportionalto1/R2_,_which_is inagreementwiththeexperimentalresultspresentedby[44].The nor-malizedcriticalvoltageasdefinedusingEq.(16),onlydependsonthe normalizedthickness,seeFig.5.Forcomparison,thefiniteelement re-sultsandtheresultsofasimplesolutionbasedon1/3-air-gaptheory withuniformelectrostaticload(asexplainedin[12]),arealsoshown. Itisworthtonotethattheresultspresentedinthisgraphareclosely sim-ilar(5%different)totheclassicallimitprovidedbyAnsarietal.[20]. Inthelatter,theauthorshaveemployedcouplestressandstrain gradi-entelasticitytheorytoobtainthesizedependentpull-incharacteristics foramicro-platewithℎ∕𝑑=0.83.However,sincethegeometric nonlin-earityisnotconsideredinthementionedarticle,theobtainedcritical deflectionsaresignificantlydifferent.

Forthinnerplateswherethenonlinearstiffeningeffectismore sig-nificant,thesimple1Dlinearspringmodel(1/3-air-gaprule)predicts asignificantlylowercriticalvoltageascomparedtothefiniteelement solution;while,theapproximateanalyticalsolutionpresentedherecan

Fig.5. Thenormalizedpull-involtageofacircularplatewithradius𝑅=100 μm,asafunctionofitsnormalizedthicknessh/dcalculatedwithdiﬀerent meth-ods.

Fig.6. Themidpointdeﬂectionofthecircularﬂexibleelectrodewiththickness

ℎ=0.2μmandradius𝑅=100μmasafunctionofappliedvoltage,for diﬀer-entialpressuresindiﬀerentdirections.—— stableequilibrium,----unstable equilibrium.

predictverypreciseresults.However,althoughthecriticalvoltage cal-culatedwiththe1Dlinearspringmodelisinaccurateforthecircular membranes,itprovidesamoreconservativeapproximationforthe crit-icaldeﬂection.

Next,considerthecasewhereadifferentialpressure,positivein op-posingdirectionoftheelectrostaticload,isapplied(P′≠ 0).Thistime, twoloadparameters,i.e.pressureandelectrostaticloadareinvolved inthestabilityanalysis.Inordertocalculatethelimitvoltage,we pre-servedthepressureandconsiderthevoltageasthevaryingload param-eter.Themidpointdeflectionoftheelectrodeasafunctionofapplied voltageisshownforthreedifferentdifferentialpressuresinFig.6.

Infact,pressurizingtheflexibleelectrodecansignificantlyaffectthe shapeoftheequilibriumpath:firstofall,amechanicalpressureleads toaninitialdeflectionintheplatewhen𝑉=0.Thisinitialdeflection dependsontheamountanddirectionoftheappliedpressure.Second, adifferentialpressuremightinfluencethepositionand/ornumberof limitpoints.

AsFig.6shows,whenanegative(downwardinFig.2)pressureis applied,thepull-involtagedropsandthecriticaldeﬂectionslightly in-creases.Thisisbecauseanegativediﬀerentialpressuredecreasesthe av-erageinitialdistancebetweentheelectrodes.Though,theoverallshape oftheequilibriumpathremainsthesame.

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Fig.7.Themidpointdeﬂectionofthecircularﬂexibleelectrodewiththickness

ℎ=0.2μmandradius𝑅=100μmasafunctionofappliedvoltage,when𝑃′₌

2× 10−9_.

Forpositivepressures,however,theshapeoftheequilibriumpath mightdiffersignificantly(see𝑃′₌₂_{× 10}−9_in_Fig.₆_)._In_such_a_case,_the systemexhibitsoneorthreesaddle-nodebifurcationsinitsequilibrium path[39].Onelimitpoint(PointCinFig.6)isclosetothelimitpointin anunloadedsystem,i.e.𝑃′₌₀_._Only,_due_to_the_initial_deflection_of_the plateandtheassociatedadditionalgeometricalstiffness,thislimitpoint occursataslightly differentvoltageanddeflection.Werefertothis criticalpointasthe“ultimate” limitpoint.Anotherlimitpointoccurs earlierwhenthedeflectionoftheplateisstillinthepositivedirection (PointAinFig.6).Weshallrefertothispointasthe“primary” limit point.Theotherlimitpointisalocalminimumintheappliedvoltage (PointB).Ifwerampupthevoltageontheupperstablebrancharound PointA,or,rampdownthevoltageonthelowerstablebrancharound PointB,thesystemmightjump fromonestableconfigurationtothe other.

Similarbi-stabilitybehaviorhasbeenobservedforshallowarched structures[45].Thesestructuresmayexhibittwodifferentstable con-figurationsunderthesameappliedelectrostaticloadandtheycansnap fromonetotheother.Fortheproblemathand,thepressureiscausing theinitiallyflatflexibleelectrodetobehavelikeanarchedstructure.

Inordertoverifytheanalyticalapproximate,theequilibriumpath calculatedbytheﬁniteelementmodelisprovidedinFig.7.Theresults ofthenumericalsolutionconﬁrmtheaccuracyoftheapproximate an-alyticalsolution.Theerrorbetweenthesetwosolutionsappearstobe themostattheultimatelimitpoint(approximately4%).Similartothe caseofnopressure,theaccuracyoftheanalyticalsolutionisbetterfor thickerelectrodes.

Afterthesystempassestheprimarylimitpoint,thepost-instability behaviorstronglydependsontheappliedpressure.Fig.8showsthe mid-pointdeﬂectionasafunctionofappliedvoltage,fordiﬀerentpositive pressures.Itcanbeobservedthattheprimarylimitpointcanonlybe noticedifthepressureishigherthanacertainthreshold.Iftheapplied pressureistoosmall(see𝑃′₌₀_.₆_{× 10}−9_in_Fig.₈_),_then,_the_shape_of theequilibriumpathchangesslightly,andtheprimaryinstabilityisnot observed.Forhigherpressure,though,theprimarylimitpointexists.

Formoderatepressures,theprimarylimitvoltageislowerthanthe ultimatelimitvoltage.Therefore,theinstabilityleadstothesnapping behaviordiscussedbefore(see𝑃′₌₂_.₄_{× 10}−9_in_Fig.₈_)._For_larger pres-sures,theprimarycriticalvoltageexceedstheultimatepull-involtage andthus,asmallperturbationmayleadtopull-inoftheﬂexible elec-trode(see𝑃′₌₄_.₈_{× 10}−9_)._For_larger_pressures,_the_so-called_secondary andultimatelimitpointstotallyvanish.

ℎ=0.2μmandradius𝑅=100μmasafunctionofappliedvoltage,fordiﬀerent positivepressures.—— stableequilibrium,----unstableequilibrium.

Fig.9. Thecriticalvoltage(s)foratestcasewiththicknessℎ=0.2μmandradius

𝑅=100μm,asafunctionoftheappliedmechanicalpressure.

Ifduringtheelectrostaticloading,asnap-troughoccursfromthe up-perstablebranchtothelowerbranch(e.g.for𝑃′₌₂_.₄_{× 10}−9_in_Fig.₈_), theunloadingofthesystemcanalsoleadtoasnap-backfromthelower stablebranchtotheupperone.However,thesnapbackoccursatalower voltageatthesecondarylimitpoint.Thislimitpointisonlyobserved forthepressurerangethatbothprimaryandultimatelimitpointsare present.

Clearly,thecriticalvoltage(s)andlimitdeﬂection(s)dependonthe appliedmechanicalpressure.Thevariationofthelimitvoltage(s)versus theappliedmechanicalpressureisshowninFig.9.Ascanbeseen,the resultsofanalyticalandﬁniteelementsimulationsareingood agree-ment,whichagaindemonstratestheaccuracyoftheapproximate solu-tion.

InFig.9,fornegativepressures,only onelimitpointisobserved whichisassociatedwiththeultimatelimitpointorthepull-inofthe ﬂexibleelectrode.Inthisregion(P′<0),thereisanear-linearrelation betweenthepull-involtageandtheappliedpressure.Thepull-involtage monotonicallydecreaseswithincreasingtheamplitudeofthepressure innegative(downward)direction.

Forpositivepressures,threediﬀerentregionscanbeobserved.First, forverysmallpressures,onlytheultimatelimitpointisobserved.This is associatedwiththelimitpointfor 𝑃′₌₀_.₆_{× 10}−9 _in_Fig.₈_. _Then, thereisaregioninwhichthesystemexhibitsallthreelimitpoints.The 215

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Fig.10. Thecriticaldeﬂection(s)ofthetestcasewiththicknessℎ=0.2μmand radius𝑅=100μm,asafunctionoftheappliedmechanicalpressure.

examplesof𝑃′₌₂_.₄_{× 10}−9_,₄_.₈_{× 10}−9_and₇_.₂_{× 10}−9_in_Fig.₈_belong_to thisregion.Dependingonthevalueoftheappliedpressure,theprimary limitvoltagemightbelessormorethantheultimatelimitvoltage.This deﬁnesthepost-instabilitybehaviorofthesystem.Theﬁnalregionin

Fig.9isthepressurerangeatwhichagainthesystemexhibitsonlyone limitpoint,whichisassociatedwiththeso-calledprimarylimitpoint. Theexampleof𝑃′₌₉_.₆_{× 10}−9_in_Fig.₈_belongs_to_this_region.

Fig.10showshowthecriticaldeflectionsvarywiththeapplied me-chanicalpressure.Itcanbeobservedthattheprimarycriticaldeflection variesbetween0–50%oftheinitialgapsizeinthepositivedirection.At theultimatepull-inpoint,thedeflectionoftheplateis65–73%ofthe initialgapsize.

Thesnappingoftheflatflexibleelectrode,whensweepingthe ap-pliedvoltageupanddown,isaninterestingphenomenonthatcouldbe usedinelectrostaticallydrivenswitches,sensorsandactuators. How-ever,asexplained,onlyacertainrangeofpressureallowsforexistence ofthisbehavior.Therangeofpressuresallowingforsnappingmainly dependsonthemechanicalpropertiesoftheflexibleelectrodeandits dimensions(thicknessandradius).

Itshould be noticed thatsnap-through is adynamic processand whentheﬂexibleplateissnappingfromanunstabletoastablestate,it hasnonzerovelocity.However,sincetheloadsystemisconservative,no dynamicconsiderationisrequiredtoassessstability.Instead,thetotal potentialenergyisagoodcriteriatoensurethatthedynamicprocess doesordoesnotleadtofailure:Ifthetotalpotentialenergyattheprimary limitpointexceedsthepotentialattheultimatelimit point,theexceeding energyappearsaskineticenergycausinganovershoottooccur.

Fig.11illustratestherequiredcombinationofdiﬀerentialpressure andthicknessoftheplate,inordertoobservethesnap-through phenom-ena.ThisgraphisdeterminedusingbothanalyticalandFEMsolutions forthetestcaseat hand.Thegoodagreementbetweenthesolutions againdemonstratestheaccuracyoftheanalyticalapproximate.

AsFig.11shows,ifthepressureistoolow,theprimarylimitpoint isnotobserved;andifthepressureistoohigh,thentheprimarylimit voltageexceedstheultimatelimit voltageandthesystemwouldfail afterreachingtheﬁrstinstability.

Ifthe pressure is highenough, the ultimate andsecondary limit pointsvanishandsnap-backbehaviorwillnotbeobservedeither. How-ever,onecanconcludefromFigs.8and9thatthepressurerangefor havingsnap-throughinloadingisasubsetoftherangeforhavingthe snap-backinunloading.Infact,ifthesnappinginloadingisobserved, theoccurrenceofsnap-backinunloadingiscertain.

Fig.12showstheadmissiblecombinationsoftheappliedpressure andthicknessforexistenceofsnapping,fordiﬀerentradiiofthe

elec-Fig.11. Thepressurerangeinordertotriggertheprimarylimitpointandthe snap-through,asafunctionofthicknessoftheﬂexibleelectrode,with𝑅=100 μm.

Fig.12. Thepressurerangeinordertotriggertheprimarylimitpointandthe snap-through,asafunctionofthicknessoftheﬂexibleelectrode,fordiﬀerent radii.

trode.Ascanbeobservedforsmallerradiioftheelectrode(forexample,

𝑅∕𝑑=40inFig.12) awiderangeof pressuresmightresultin snap-throughbehavior.However,forlargerelectrodestherangeof admissi-blepressuresdrops.Itisinterestingthattherequiredthickness,resulting insnap-through,isalwayslessthan33%ofthegapsize.Forathicker electrode,theprimaryinstability,ifobserved,leadstodirectpull-in.

Althoughthesnap-through hasbeen illustratedforconstant pres-suresandavaryingvoltage,asimilarbehaviorwillbeobservedifthe voltageispreservedandthepressureisvaried.Themidpointdeflection oftheelectrodeasafunctionoftheappliedpressure,fordifferent volt-ages,isprovidedinFig.13.ItcanbeobservedfromFig.13thatforany voltagelargerthanzero,atleastonelimitpointexistsinthe equilib-riumpath(e.g.PointAforV’=0.45).However,forlargervoltages,two otherlimitpointsmightappear.Forexample,inFig.13,inthecurve correspondingtoV’=0.45,ifwevarythepressurearoundPointBorC, thesystemsnapsfromapositivetoanegativedeflection,orviceversa. Forverylargevoltages,ontheotherhand,varyingthepressureoverthe limitpointsleadstopull-inofthesystem.

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ℎ=0.2μmandradius𝑅=100μmasafunctionoftheappliedpressure.—— stableequilibrium,----unstableequilibrium.

Fig.14. Themidpointdeflectionofthecircularflexibleelectrodeobtainedby nonlocalcontinuumtheorywithdifferentmateriallengthscalesasafunctionof appliedvoltage,forℎ=0.2μm,𝑅=100μm,and𝑃′₌₂_{× 10}−9_.

Itisworthtopointoutthatthecomplianceofthesystemtoa differ-entialpressureisminimumincasenovoltageisappliedtothecapacitor. Withavoltageincreasethestiffnessofthesystemdrops,andfinallyat acriticalvoltage,thesystemallowsforsnap-throughbehavior.When snap-throughoccurs,thesystemhaszerostiffness.Thesnap-through andbi-stablebehaviornoticedforpressurizedclampedelectrodes,can beemployedinsensingandactuationapplications.Thisphenomenon canbenefitfromhighsensitivityduetolowcompliance,androbustness andsimplicityofpull-involtagemeasurements.

Finally,itshouldbenotedthatthehypotheticalpropertiesthatwere utilizedforillustratingtheresultsareclosetothoseofgoldoraluminum thinfilms.Forthesetwomaterials,scaleeffectsarenotsignificantatthe thicknessesusedinthepresentstudy[26–29].Therefore,providedthat werestrictthematerialtoaluminumandgoldorothersimilarly be-havingmaterials,theclassicalcontinuumtheorycan beemployedat thediscussedlengthscales.However,forsmallerlengthscales,a size-dependentcontinuumtheoryshallbeutilized.Here,webrieflyshowthe resultsofusinganonlocalplatetheorytogetherwiththeproposed for-mulationtocapturetheeffectsofscalingonassessmentofthestability ofthemicro-plate.Fig.14showshowthepredictedsizedependent de-flectionvarieswiththeappliedvoltage.Forcomparisontheresultsfor differentmateriallengthscalesfromclassicallimit(𝑒0𝑎∕ℎ=0)tothe

limitof applicationof thenonlocaltheory(𝑒0𝑎∕ℎ=1)areillustrated inthisgraph.Itcanbeobservedthatthethreelimitpoints(primary, secondaryandultimatelimitpoints)stillexistintheequilibriumpath ofthemicro-plateobtainedbynon-localtheory.Thisfigureshowsthat whenthethicknessofthemicro-plategetscomparabletothematerial lengthscale,(i)theinitialdeflectionincreases,(ii)thecriticalvoltages decrease,and(iii)thecriticaldeflectionsslightlyincrease.Itis note-worthythattheseresultsarequalitativelyinagreementwiththeresults providedbyRef.[30]whichinvestigatesthesize-dependentdynamic pull-inanalysisofmicro-platesusingmodifiedcouplestresstheory. 6. Conclusions

Inthispaper,ananalyticalmodelwasproposedforacircularflexible electrodeinaparallelplatecapacitor,whileitisloadedwitha differen-tialpressure.Usingthisapproximatesolution,astabilityanalysiswas performedontheeffectofpressureonthecriticalvoltageand deflec-tion.Intheproposedmodel,thegeometricalnon-linearityoftheflexible electrodewastakenintoaccount.

Theresultssuggestthatapressurecantriggeradditionallimitpoints andanunstablesolutionbranchtooccur.Thepost-instabilitybehavior afterreachingtheﬁrstlimitpoint,dependsonloadparameters, thick-nessandradiusoftheelectrodeandtheairgap.Aftertheprimarylimit point,thesystemmightsnaptoanewstableconﬁguration,or,exhibit pull-in.

Itisworthtomentionherethatwhensnap-throughoccurs,the sys-temhasverysmallstiﬀness,andismechanicallyverycompliant.This conditionmakes thesystemverysuitable forsensingapplications. In particular,thesensitivityofthelimitvoltages tothepressurecanbe employedtomeasurethepressure.However,westressherethateven withoutobservationofinstability,thecombinationofpositivepressure andelectrostaticloadontheﬂexibleelectroderesultsinahigh compli-anceofthesystem,whichmaybeveryappealingforsensing applica-tions.

Moreover,tocapturethesizedependencyofstabilityofamicro-plate whileloadedwithelectrostaticanddifferentialpressures,aformulation inframeworkofnonlocalcontinuumtheoryhasbeensuggested.This formulationincludesthescalingeffectsofthethicknessofthe micro-plateandhence,itissuitableforbeingusedforsmallerlengthscales wheretheapplicationofclassicalcontinuumtheoryislimited.The re-sultsofthesize-dependentmodelexhibitsimilaraspectsofthe mechan-icalbehavioroftheplatesuchastheadditionallimitpointsand snap-throughbehavior.Consequently,thedemonstratedbehaviorcouldserve asabasisfornovelmicroaswellasnanoelectromechanicalsystems. Usingtheprimaryinstabilityofpressurizedelectrodesinsensingor actu-ationcanbenefitfromtherobustnessandsimplicityofpull-in measure-ments,andinaddition,itcanbenefitfromthesnap-throughbehavior whichpreventsthesystemfromfailure.

Acknowledgment

ThisworkissupportedbyNanoNextNL,amicroandnanotechnology consortiumoftheGovernmentoftheNetherlandsand130partners. References

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