Comptes Rendus Mecanique...MWCNT PC

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C. R. Mecanique 343 (2015) 371396

Contents lists available at ScienceDirect

Comptes Rendus Mecanique

www.sciencedirect.com

Thermo-mechanical characterization of multi-walled carbonnanotube

reinforced

polycarbonate

composites:

A molecular

dynamics

approach

Sumit Sharma a,,1,

Rakesh Chandra b,2,

Pramod Kumar b,3,

Navin Kumar c,2

a SchoolofMechanicalEngineering,LovelyProfessionalUniversity,Phagwara,Indiab DepartmentofMechanicalEngineering,Dr.B.R.AmbedkarNationalInstituteofTechnology,Jalandhar,Indiac SchoolofMechanical,Materials&EnergyEngineering(SMMEE),IndianInstituteofTechnology,Ropar,India

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received31January2015Accepted11March2015Availableonline23April2015

Keywords:

CarbonnanotubeDampingMechanicalpropertiesMoleculardynamicsPolycarbonate

Thermal

conductivity

The

present

study

aims

at

examining

the

mechanical

properties

of

multi-walled

carbon

nanotubespolycarbonatecomposites (MWCNTPC), throughamoleculardynamics (MD)simulation.CompositesofMWCNTPCweremodeledusingMaterialsStudio 5.5software.Multiwallcarbonnanotubes (MWCNTs) compositions inpolycarbonate (PC)werevariedby

weight

from

0.5%

to

10%

and

also

by

volume

from

2%

to

16%.

Forcite

module

in

MaterialsStudiowasused forfindingmechanicalproperties.Amarked increase in theelasticmodulus(upto89%)hasbeenobserved,evenwiththeadditionofasmallquantity(upto2weight %)ofMWCNTs.Also,uponadditionofabout2volume %ofMWCNTs,theelastic

modulus

increases

by

almost

10%.

The

increase

in

mechanical

properties

is

found

to

supplement

earlier

experimental

investigations

of

these

composites

using

nano-indentation

techniques.

Better

load

transfer

property

of

MWCNTs,

larger

surface

area

and

interactionbetween reinforcementwithbasematrixare thesuggested reasons for this increase in

mechanical

properties.2015Acadmiedessciences.PublishedbyElsevierMassonSAS.All rights reserved.

1. Introduction

Carbonnanotubesareexcellentreinforcements forpolymersbecauseof theiruniquemechanicalpropertiesand largesurfaceareaperunitvolume.Experimentsandcalculationsshowthatnanotubeshaveamodulusequaltoorgreaterthanthebestgraphitefibers,andstrengthsatleastanorderofmagnitudehigherthantypicalgraphitefibers.Forexample,themeasurementofthe tensilepropertiesof individualmulti-walledcarbonnanotubes(MWCNTs)gavevaluesof1163 GPafor

the

tensile

strength

and

270950 GPa

for

Youngs

modulus,

as

obtained

by

Yu

et al. [1].

For

comparison,

the

modulus

andstrengthofgraphitefibersare300800and5 GPa,respectively. Inaddition to theiroutstandingmechanicalproper-ties, the surfaceareaperunitvolumeofnanotubes ismuch larger than thatofembeddedgraphitefibers.Forexample,30-nm-diameternanotubeshave150timesmoresurfaceareathan5-m-diameterfibers forthesamefillervolume frac-tion,suchthatthenanotube/matrixinterfacialareaismuchlargerthanthatintraditionalfiber-reinforcedcomposites.The

* Correspondingauthor.Tel.:+918146871758.E-mailaddress:[email protected](S. Sharma).

1 Assistant Professor.2 Professor.3 AssociateProfessor.

http://dx.doi.org/10.1016/j.crme.2015.03.002

1631-0721/ 2015

Acadmie

des

sciences.

Published

by

Elsevier

Masson

SAS.

All rights reserved.
http://dx.doi.org/10.1016/j.crme.2015.03.002http://www.sciencedirect.com/http://www.sciencedirect.com/mailto:[email protected]://dx.doi.org/10.1016/j.crme.2015.03.002http://crossmark.crossref.org/dialog/?doi=10.1016/j.crme.2015.03.002&domain=pdfhttp://dx.doi.org/10.1016/j.crme.2015.03.002mailto:[email protected]://www.sciencedirect.com/http://www.sciencedirect.com/http://dx.doi.org/10.1016/j.crme.2015.03.002


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372 S. Sharma et al. / C. R. Mecanique 343 (2015) 371396

unusualmechanicalstrengthofthecarbonnanotubeshasmotivatedscientiststofabricateandmodifyotherusefulmateri-alswhicharecheaplyavailableinbulkform,bycombiningthemascompositeswithcarbonnanotubes.Polycarbonate(PC)isa lightweightpolymerthat isavailable inbulk formand iswidelyused forseveralengineeringapplicationsdueto itsmoldability.Fortakingadvantageoftheusefulpropertiesofpolymersincombinationwithuniquestructuralpropertiesofcarbonnanotubes,multi-walledcarbonnanotubespolymercompositeshavebeenresearchedandfabricatedoverthepastfewyears.Inordertoexploittheusefulnessofthesecompositesforspecificmechanicalengineeringapplications,theirstaticanddynamicmechanicalpropertiesneedtobeevaluated.Amongthestaticproperties,theelasticmodulusofthespecimen

is

very

important.AlotofworkhasbeenpublishedrelatedtotensiletestingofMWCNTPCcomposites.Thesetestshaveevidencedthat

minorcompositions(upto2 wt. %)ofMWCNTinPCenhancethemodulusandtensilestrengthfrom10%toeven70%.Choiet al. [2] usedstyreneandacrylo-nitrile (SAN)graftedMWCNTswithPC insteadofpristineMWCNTsandobserved thatwhenSAN-graftedMWCNTs(1wt. %)wereusedwithPC,bothtensilestrengthandmodulus increasedbynearly5%and10%,respectively,incomparisontopristineMWCNTPCcomposites.Liuet al.[3] observedthatat3wt. %MWCNTsinaPC,thecompositesexhibitedanearly40%highertensilestrength incomparisontopurePC.However, fora5wt. %MWCNTcomposition, the strength reduced drastically. There are also contrary results obtained by Olek et al. [4], who reportednoimprovementinthepresenceofMWCNTsinthepolymerpoly-methyl-methacrylate(PMMA)foranystaticmechanicalproperty.Evenwith acompositionalchange from1% to 5%ofMWCNTs, theelasticmodulus remainedalmost the sameasthatofpurePMMA.However, iftheMWCNTswerecoatedwithsilica,thecompositeshowedremarkableresultsuponnano-indentation.Withonly4%MWCNTsilicainPMMA,themodulusmeasuredisaboutthreetimesthatofpurePMMA.Reinforcementonpoly-vinylalcohol(PVA)andPMMAwithfew-layergraphene(FG)wasalsotestedusinganano-indenter

by

Das

et al. [5].LowcompositionsofFG(0.6%)inPVAmadethemodulusincreasebyabout20%.Vivekchandet al.[6] haveexplainedthe

useofinorganicnanowires(NW)asreinforcementinPVAtobeasefficientasMWCNTs.Theelasticmodulusincreasedbyalmosttwotimesuponreinforcementby0.8%(involume)ofinorganicNW.However,MWCNTshaveaverysmoothsurface;makingthestrength impartedbyreinforcingwithMWCNTs lesserthanwithNW.Kimet al. [7]usedacompatibilizerastwopoly-g-polycapro-lactones (P3HT-g-PCLs)withbisphenol-A-PC-MWCNTcomposite.WhenaPC-MWCNTcompositewascombinedwithP3HT-g-PCL,thentherewasanincreaseofnearly22%intheYoungmodulusand30%inthetensilestrengthincomparisontopurePC.Forsmallconcentrations(0.10.5wt. %)ofMWCNTs,thisincreasewasfoundtobeconsistent.However,whentheMWCNTsconcentrationwasfurtherincreasedto1wt. %,thenbothYoungsmodulusandtensilestrengthwereconsiderablyreduced.

Eitan et al. [8] used bisphenol-A-PC with MWCNTs as composites for mechanical characterization. Tensile tests wereperformedusingaUniversalTestingMachineanditwasfoundthatforcompositeswithsurface-modifiedMWCNTs(5 wt. %),themodulusimprovedby95%incomparisontopurePC.EvenforcompositesusingpristineMWCNTs(5wt. %),themodulus

rise

was

nearly

70%

in

comparison

to

pure

PC.

Ayatollahi

et al. [9] have

used

epoxy-MWCNT

composite

under

shear

andbending loadusinga SantanUniversal TestingMachine. Theyhavealso found that there isagradual increase inelastic

modulusandtensilestrengthasMWCNTcompositionincreasedinepoxy.Compositionsof0.1%,0.5%and1.0%MWCNTsinepoxywerefabricatedand,asthecompositionofMWCNTsincreased,bothelasticmodulusandtensilestrengthincreasedby10%. Montazeri et al. [10] usedaHounsfield machineand also evaluated the viscoelastic behavior of epoxy-MWCNTcomposite.TheyreportedthatwithfurtherincreaseinMWCNTcompositionof2%,theelasticmodulusincreasedbyabout20%ascomparedtopureepoxysamples.

Computationalstudiescomplementexperimentbyprovidingeasymanipulation,analysisandinsightsintothemolecularlevel.Theextenttowhichmechanicalreinforcementcanbeachieveddependsonseveral factors, includinguniformityofdispersion,degreeofalignmentofCNTs,andthestrengthofpolymerCNTinterfacialbonding.Sinceitisdifficulttocontrolandmeasuremanyofthesepropertiesexperimentally,computationalmodelingcanprovidesomecrucialinsights.Forthisreason,theoreticalandcomputationalmethodshavebeenwidelyappliedtostudypolymer/CNTcomposites.

Due to the difficulties in the experimental characterization of nanotubes, computer simulation has been regarded as

a

powerful

tool

for

modeling

the

properties

of

nanotubes.

Among

the

available

modeling

techniques,

molecular

dynamicssimulationhasbeenusedmostextensively.First-principlesmethodsareabletogeneratereasonablyaccuratedataofstruc-

turesandenergies relevant to polymernanocomposite systems, but theycan only beused to study small systems overshorttimesduetotheircomputationalexpense. Incontrast,molecularsimulationmethodssuchasmolecularmechanics(MM)andmoleculardynamics (MD),whicharebasedonanalytic forcefields,arecomputationallycheapercompared tofirst-principlesmethods.Theycanthereforebeusedtostudylargermolecularsystemsforlongertimes.Asdescribedbelow,MDcanalsobeusedtoobtainmacroscopicpropertiessuchastheYoungmodulus.

The present study was undertaken to investigate changes in the static mechanical properties under the influence ofvaryingcompositions(perweightandpervolume)ofpristineMWCNTsinPCbyemployingtheMDtechniquewithoutanyadditionalcomponentoranysurfacemodificationinthecomposite.Theelasticmodulushasbeenstudiedtosupplementthe experimental investigationson thesecomposites made by Kumaret al. [11].They preparedcomposites of MWCNTPCbyatwo-stepmethodofsolutionblending,followedbycompressionmolding.Multiwallcarbonnanotubes(MWCNTs)compositions inpolycarbonate (PC)werevariedbyweight from0.5% to10%.Nano-indentation techniqueswereused to

evaluate

mechanical

properties

like

elastic

modulus

and

hardness.

A

marked

increase

in

the

elastic

modulus

(up

to

95%)wasobservedwiththeadditionofsmallquantities(upto2wt. %)ofMWCNTs.


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S. Sharma et al. / C. R. Mecanique 343 (2015) 371396 373

Fig. 1. (Color online.) A carbonate monomer.

Fig. 2. (Color online.) Polycarbonate chain containing ten repeat units of carbonate.

Thesepropertieswillhelpus inthemechanicalcharacterizationofPC-basedcompositesand in theevaluationof theusefulnessofMWCNTsasreinforcementfromthepointofviewofapplications.

2. Simulationstrategy

OurMDsimulationswereperformedbyMaterialsstudio5.5.WeusedCondensed-phaseOptimizedMolecularPotentialsforAtomisticSimulationStudies (COMPASS) forcefield,which is implanted inMaterials studio5.5.COMPASS is thefirst

force

field

that

has

been

parameterized

and

validated

using

condensed

phase

properties

in

addition

to

empirical

data

for

moleculesinisolation.Consequently,thisforcefieldenablesaccurateandsimultaneouspredictionofstructural,conforma-tional,vibrational,andthermo-physicalpropertiesforabroadrangeofmoleculesinisolationandincondensedphases.TheCOMPASSforcefieldconsistsoftermsforbonds(b),angles(),dihedrals(),out-of-planeangles( )aswellascross-terms,andtwonon-bonded functions,aCoulombic function forelectrostatic interactionsanda96LennardJonespotential forvanderWaalsinteractions.

Etotal= Eb+ E+ E+ E +Eb,b + Eb,+ Eb, + E, + E, + E,, + Eq+ EvdW (1)

where

Eb=energy due to bond stretching,E=energy due to bond bending,E =energy due to bond torsion,

E =energy due to out of plane inversion,Eq=electrostatic energy,EvdW=van der Waals energy andEb,b ,E, ,Eb,,Eb, ,E, ,E,, = crosstermsrepresentingtheenergyduetotheinteractionbetweenbondstretchbondstretch,bondbendbondbend,bondstretchbondbend,bondstretchbondtorsion,bondbendbondtorsionandbondbendbondbendbondtorsion,respectively.

Allthesimulationsweredoneintheconstanttemperatureandconstantvolumecanonicalensemble(NVT).Theequationsof motions were integrated using the Verlet algorithm with an integration time step of 1 fs and the temperature wascontrolledbyAndersensthermostat.Thetotalsimulationtimeforthedynamicsrunwas50 ps,withatimestepof1 fs.Afterthecompletionofdynamicsrun,thestructureobtainedwasfirstsubjectedtogeometryoptimizationusingaconjugategradientalgorithmandthenastainof0.005wasappliedtoobtaintheelasticmoduli.Periodicboundaryconditionswereusedinallsimulations.

Fig. 1and

Fig. 2show,

respectively,

the

chemical

structure

of

the

investigated

carbonate

and

PC

polymer.

Firstly,

a

repeatunitwasbuiltusingthe BuildtoolinMaterialsstudio 5.5.Then,usingthesametool,thePCstructurewasobtainedby


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Fig. 3. (Color online.) Simulation cell of pure PC containing 17,728 atoms.

Fig. 4. (Color online.) Simulation cell of MWCNT (0.75% by weight) reinforced PC containing 20,273 atoms.

Fig. 5. (Color online.) Simulation cell of MWCNT (2% by volume) reinforced PC containing 20,173 atoms.

polymerizingthecarbonaterepeatunitandtakingthechainlengthasten.Fig. 3showsasimulationcellofpurePC.The

densityofthecellwas1.5 g/cm3 andthecelldimensionsweretakenas:49.349.349.3 3

.ThemolecularmodelofMWCNT/PCcompositeswithdifferentdimensionswasbuiltwiththeuseoftheAmorphouscellmodule.Firstly,aMWCNToflength41.81 wasconstructedusingtheBuildnanostructuredialogboxinMaterialsStudio.Theconfigurationoftheinnerandouternanotubeswastakenas(3,3)and(6,6)respectively.Thistubewastheninsertedinasimulationcelloftherequireddimensions.Lastly,PCwaspackedaroundtheMWCNTusingthePackingfeatureavailableinAmorphousCell.Inthisstudy,simulationswereperformedintwoways.Inthefirstapproach,MWCNTswerepackedinPCbyweightandinthesecondapproach,packingwasdonebyvolume.Fig. 4andFig. 5show,respectively,thestructuresobtainedafterpacking

by

weight

and

by

volume.

Similar

to

Figs. 45,

a

number

of

simulation

cells

were

created

in

the

software,

and

mechanicalpropertieswereobtained.


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Fig. 6. (Color online.) Simulation cell of MWCNT (Vf=0.04, l/d=30) reinforced PC composite containing 20,798 atoms.

AnumberofsimulationswerealsoperformedforafixedMWCNTvolumefractionandbyvaryingtheaspectratio.Theaspect ratio of MWCNTs was varied from l/d=5 to l/d=100.Fig. 6 shows one of the simulation cells of an MWCNT-reinforcedPCcompositewith Vf= 0.04 and l/d=30.AsimilarapproachwasadoptedwhilemodelingzigzagandchiralMWCNT-reinforcedPCcomposites.

3.Thermalconductivity

Withthedimensionsofelectronicandmechanicaldevicesapproachingthenanometerscale,efficientheatremovalisofcrucialimportancetobothperformanceandfunction.Whileabasicunderstandingofheattransportindielectricshasal-readybeenachieved,manyimportantissuesremainunresolved.Theinterpretationofexperimentalresultsremainsdifficultbecausetypicallythecontributionsofindividualdefectscannotbedeconvoluted.MDsimulationsare ideal foraddressingsuch issuessince theycanbeused tostudy individualmicrostructuralelements, thereby identifying themost importantissues for thermal conductivity in polycrystalline materials. For example, by elucidating the correlation between grain-boundarystructureandthermal-transportproperties,onemayhope toeventuallydesignmaterialswith tailoredthermalproperties.However,priortoasystematicstudyofinterfacialeffects,itisnecessarytofirmlyestablishsuitablecomputa-tionalmethods.ThethermalconductivityrelatestheheatcurrenttothetemperaturegradientviaFourierslawas:

J=

T

x(2)

where J isacomponentofthethermalcurrent, isanelementofthethermalconductivitytensor,andT/x isthegradientofthetemperatureT.Experimentally, istypicallyobtainedbymeasuringthetemperaturegradientthatresultsfromtheapplicationofaheatcurrent.

In MD simulations, thermal conductivity can be computed either using non-equilibrium MD (NEMD) or equilibriumMD (EMD). The two most commonly applied methods for computing thermal conductivity are the direct method andtheGreenKubomethod.Thedirectmethod is anNEMDmethod that relieson imposing a temperature gradient acrossthesimulationcellandisthereforeanalogoustotheexperimentalsituation.Bycontrast,theGreenKuboapproach isanEMDmethodthatusescurrentfluctuationstocomputethethermalconductivityviathefluctuationdissipationtheorem.Schellinget al. [12] studiedandcomparedthefeaturesofeachmethod.TheypointedoutthatNEMDmightcontainnon-lineareffectsduetotheapplicationoftherequiredtemperaturegradient.TheyalsonotedthatwhilebothEMDandNEMDapproachesexhibitfinitesizeeffects,theseeffectsaremuchmoresevereinNEMDduetothepresenceofinterfacesattheheatsourceandsink.Furthermore,EMD facilitatesthermalconductivityprediction inalldirectionsusingonesimulation,

whereas

NEMD

requires

the

use

of

a

thermal

gradient

and

therefore

only

enables

the

calculation

of

thermal

conductivityinonedirection.Therefore,EMDisparticularlyusefulforgeometrieswhereperiodicboundaryconditionscanbeapplied.

However,thebasisofEMDisthefluctuationdissipationtheorem,whileNEMDsbasisinFouriersLawofconductionmakesNEMDanalogoustoexperimentalmeasurements.Furthermore,EMDhasbeenoftencomputationallymoreexpensiveandtheresultsaremoresensitivetosimulationparameters.

InEMD,thesystemissettothedesiredtemperature,andthenaconstantenergyschemeisusedwiththewell-knownGreen[13]andKubo[14]relationstocalculatethethermalconductivitytensor.MDsimulationsmaybeappliedtodifferentstatisticalensembles,namelycanonical(NVT),grand-canonical(PT),andmicro-canonical(NVE).Itisworthnotingthattheirderivationshavebeendoneindifferentensembles:theformerinmicro-canonicalandthelatteringrand-canonical.Lepriet al. [15] resolvedthisdiscrepancybynotingthatifthemicro-canonicalensembleisused,thenthethermalconductivitymight diverge non trivially unless the velocity of the center of mass of the system is set to zero or alternatively sometermsare subtracted from thecalculatedheatflux vectors.However, thecanonical ensemblecanalsobeusedwith theGreenKubo formulatopredictthethermalconductivitybyapplyingadditionalthermalforcestoalltheatoms.Dynamic

properties

such

as

thermal

conductivity

are

calculated

in

EMD

based

on

the

fluctuation

dissipation

and

linear

responsetheorem.Thismethodappliesthefactthattheheatflowinasystemofparticlesintheequilibriumstatefluctuatesaround


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zero. The heat flux vectors and their correlations are computed throughout the simulations. The time needed for HeatAuto-CorrelationFunctions(HACF)todecaytozero isthenusedthroughtheGreenKuborelationtopredictthethermalconductivity.

NEMDsimulationsprovideameanstocalculatethermalconductivityinawayanalogoustotheexperimentalmeasure-mentseitherbyimposingathermalgradientintothesystemofparticlesorbyintroducingaheatflow.

Inthereversenon-equilibriumMD(RNEMD)method,theenergyexchangeiscarriedoutbyexchangingthekineticen-ergy of two particles: the hottest particle in the cold layer and the coldest particle in the hot layer. The energyE is

therefore

variable

and

needs

averaging

over

many

exchanges.

In

the

method

of

Jund[16],

the

energy

E is

fixed,

and

in-volvesallparticlesinthehotandcoldlayers.Bothmethodsconservetotallinearmomentumandenergyofthesystem.TheimposedfluxmethodofJund [16]alsoconservesthetotallinearmomentumofthehotandcoldlayer,i.e.nomomentumisexchanged.

In this study, the imposed flux method was used to find thermal conductivity. The number of layers in which thedirectionofthefluxisdividedwasfixedat40.Increasingthenumberoflayerscanincreasetheaccuracyofthegradients,buttoomanylayerswillleadtolargefluctuationsinthelayertemperatures.Twotypesofexchangemethodwereusedinthestudy.ThefirstisVARIABLE,whichexchangesavariableenergybetweenoneobjectinthehotlayerandoneinthecoldlayer.FIXEDexchangesaconstantenergybetweenallhotobjectsinthehotlayerandallobjectsinthecoldlayer.TheamountofenergytoexchangeineachstepwhenusingtheFIXEDexchangetypewastakenas1kcal/mol.Thefluxwasdeterminedbytheratioofexchangeenergyandnumberofsteps.Thenumberofexchangesduringtheequilibrationstagewastakenas500.Duringtheequilibrationstage,athermostat(NVT)actsonthesystem.Thenumberofexchangesduringtheproductionstagewasequalto1000.Theproductionstagewascarriedoutatconstantenergy(NVE).Atimestepof1 fs

was

used

in

the

simulation.

The

number

of

time

steps

in

between

two

exchanges

was

fixed

at

100.

Decreasing

the

numberofsteps leadstohigherfluxesandincreasesthetemperaturegradient.Toosmallvalueswereavoidedasthese introduce

nonlineareffectsandmayimpactperformance.Tofindthermalconductivity,MDsimulationswereperformedwithMWCNTvolumefractionvaryingfromVf= 00.16

and the aspect ratio was kept fixed at l/d=10. Also, simulations were performed with varying the aspect ratio (l/d)of MWCNT and the fixed volume fraction Vf= 0.04. Thermal conductivity results obtained using MD simulations werecompared with other models such as series model, parallel model, MaxwellGarnett model, LewisNielsen model, andHamiltonCrosser model. Comparison of MD results for thermal conductivity was made with Dengs model. A brief in-troductiontothesemodelswillbegiveninthenextsection.

4. Modelstocalculatethermalconductivity

Many theoreticalandsemi-theoreticalmodelsareavailable torepresent theeffective thermalconductivityofconven-

tional

polymer

composites

in

which

large-size

fillers

have

been

dispersed

in

a

polymer

matrix.

Simple

models

such

as

theseriesmodelgivethelowerbound,whereastheparallelmodel(ruleofmixture)givestheupperboundofthethermalcon-

ductivityofananocomposite.Asmightbeexpected,experimentalobservationssuggestthatrealvaluesfornanocompositesfallsomewhereinbetweenthesetwolimits.

4.1. Parallelandseriesmodels

Boththeseriesmodelandtheparallelmodelassumethateachphasecontributeindependentlytotheoverallthermalre-sistanceandconductance,respectively,andassumeaperfectinterfacebetweenanytwophasesincontact.Theseriesmodelappliesreadilytothethermalconductivityofalaminatedcompositealongthestackingdirection.However,ittypicallygivesanunderestimationofthermalconductivityduetothepresumablycompletelocalizationofthecontributionfromthefibersembeddedinthematrix,neglectingtheinteractionamongthefillers.Therefore,theseriesmodelgivesthelowerboundforthethermalconductivityofcomposites.Theseriesmodelofthermalconductivityisgivenby:

k=(km kf)/

kf (1 v f) + km v f

(3)

where,

k=thermal conductivity of compositekm=thermal conductivity of polycarbonate matrix =0.15 W/m/Kkf=thermal conductivity of armchair MWCNT =30 W/m/Kvf=MWCNT volume fraction

Incomparison,theparallelmodelpredictsthethermalconductivityofconventionalcontinuousfiber-reinforcedcompos-itesalongthefiberalignmentdirection.Theruleofmixture implicitlyassumesperfectcontactbetweenfibers.However,itgivesa largeoverestimationofthermalconductivityandgivesanupperboundforthermalconductivityofcomposites.Itisworthpointingoutthatthermalconductivitymeasurementresultsofcompositesshouldalwaysfallbetweenthepre-

dictions

of

the

series

model

(lower

bound)

and

the

parallel

model

(upper

bound),

except

for

the

cases

where

interfacialphononscatteringinnano-laminatescanyieldevenlowerthermalconductivitythanthelowerboundbytheseriesmodel.


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Fortheparallelmodel,thermalconductivityisgivenas

k=(1 v f) km+v f kf (4)

wherethesymbolshavethesamemeaningaspreviouslydescribedintheseriesmodel.

4.2. MaxwellGarnettmodel

The

problem

of

determining

the

effective

transport

properties

of

multiphase

materials

dates

back

to

Maxwell.

Based

on

thecontinuityofpotentialandelectriccurrentat the interface,and on theassumption that the interactionsamong thefibersarenegligible,whichmeansthat thedisseminatedfibersare located farenough fromeachother,Maxwellderivedananalytical formula for the effective specific resistance (K)of acompoundmediumconsistingofa substanceof spe-cificresistance K2 , inwhicharedisseminated small spheresof specificresistance K1 , the ratioof thevolumeofall thesmallspherestothatofthewhole,beingp.Whentransformedtothethermalconductivity(k)discussedhere,themodelgives:

k= km+3 v f (kf km)/(2 km+ kf v f (kf km)

(5)

Eq.(5) givessatisfactoryresultsforcompositeswith:(i) verylowvf,(ii) gooddispersion,and(iii) nointerfacialthermalresistance.ItisalsoreferredtoasMaxwellGarnett(MG)equation.

4.3.

LewisNielsen

model

Inthismodel,conductivitybecomestheanalogofstiffnessortheelasticshearmodulus,andthedisturbanceofthefluxfieldbecomesanalogous to thedisturbanceof the stressfieldby thedispersed filler.Starting from theHalpinTsai [17]equations,whicharewidelyusedinmicro-mechanics,Nielsenappliedamodifiedequation,theNielsenLewisequation,tothemodelingofthermalconductivity:

k= km(1 +A B v f)/(1 B v f)

(6)

where

A= kE 1

kE isthegeneralizedEinsteincoefficient,anddependsprimarilyupontheshapeofthefillersandhowtheyareoriented

with

respect

to

the

direction

of

the

heat

flow.

In

our

case,

A

=2 l/d

=10,

B=(kf/km) 1

/(kf/km) +A

=1 + 1.775 v f

km=thermal conductivity of polycarbonate matrix =0.15 W/m/Kkf=thermal conductivity of armchair MWCNT =30 W/m/Kvf=MWCNT volume fraction.

Although Nielsens model is a semi-empirical model, significant improvements in the model shouldbe appreciated.Theshape effect and to some extent theorientation effect are both taken into account. Reducedfiller loading () accountsfor themaximumpackingdensityof thefillerswitha specificshapeandsizedistribution,and isunique to thismodel.In comparison, most of the theoretical equations assume uniform changes of filler loading up to the point where the

dispersed

phase

makes

up

the

complete

system,

which

is

not

realistic.

The

NielsenLewis

equation

gives

a

higher

predic-tionthantheMaxwellGarnettequation,mainlybecauseofthereducedfiller loading.However,weshouldnotethatthemodelgivestoohighapredictionathighfillerloading.Inaddition,interfacialthermalresistanceisnotconsideredinthismodel.

4.4. HamiltonCrossermodel

TheHamiltonandCrossermodel [18] isanextensionofMaxwells theoryaccounting for thenon-sphericityoffillersthroughtheuseofashapefactor,n,definedasn=3/ ,with beingtheparticlesphericity.Thesphericity( = Ae/A)ofaparticleisdefinedastheratioofthesurfacearea(Ae)oftheequivalentspherehavingthesamevolumetotheactualsurfacearea(A)ofthenon-sphericalfiber.TheeffectivethermalconductivityoftheHamiltonandCrossermodelisgivenby:

k= km

kf+ 5km 5vf (km kf)kf+ 5km+v f (km kf)

(7)


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Fig. 7. (Color online.) Variation of Youngs modulus, E11 and E22, with the percentage of MWCNT (by weight) in PC.

where

km=thermal conductivity of polycarbonate matrix =0.15 W/m/K,kf=thermal conductivity of armchair MWCNT =30 W/m/K,

vf=MWCNT volume fraction.

Sinceneitherinterfacialresistancenorfiberfiberinteractionwastakenintoaccount,thefibersizewasfoundtohavenoinfluenceontheeffectivethermalconductivityofthecompositeinthismodel.

4.5. Dengmodel

Forestimatingthemechanicalandthermalpropertiesofinclusion-in-matrixcomposites,evenforthosewithhighcon-centratedinclusions,sothatthe interactionsamongtheinclusionsmustbeconsidered,somematuremethodshavebeenestablishedwithintheframeworkofmicro-mechanics.ForCNTcompositeswithlowloadingsofrandomlyorientedstraightCNTsofaverage length L and diameterd,an analytical estimate for theeffective thermal conductivities,ke , of the CNTcompositescanbegiveninthefollowingform:

ke

km =1 +

vf3

kmkf

+H (8)

where

ke=equivalent thermal conductivity of the composite,km=thermal conductivity of polycarbonate matrix =0.15 W/m/K,kf=thermal conductivity of armchair MWCNT =30 W/m/K,vf=MWCNT volume fraction,

H=1

p2 1

pp2 1

lnp+

p2 1

1

,

p= l/d.

5.

Results

and

discussion

Thissectionhasbeendividedintotwoparts.InSection5.1,wehavediscussedtheresultsobtainedforelasticmoduli.InSection5.2,wehaveexplainedthedampingresultsandinSection5.3,wehaveexaminedtheresultsofthermalconductivity.

5.1. Elasticmoduli

Fig. 7 shows thevariationofYoungsmodulus E11 and E22 ,with thepercentageofMWCNT (byweight) inPC.BothmodulishowanincreasingtrendwithanincreaseintheweightpercentageofMWCNT.However,theincreaseintransversemodulus,E22 ,istoolowincomparisontothatofE11 .ThisconfirmsthefactthatMWCNTPCcompositeswhenloadedintransversedirectionbehavepoorly.Moreover,itcanbeinferredfromTable 1andFig. 7thatwhenincreasingtheMWCNTweightpercentageby05%,theincreaseinmoduliisgreaterthanwhentheMWCNTweightpercentageincreasesby510%.TheimprovementinmechanicalpropertieswasfoundforminorcompositionsofMWCNTsinPCandtheyfurtherincreased

as

the

composition

of

MWCNTs

was

increased.

However,

beyond

a

certain

limit

of

composition,

this

pattern

was

not

fol-lowed.Ithasbeenarguedthatproperdispersionatlowercompositions,stronger interactionbetweenMWCNTsand


9/26


Table 1

PercentageincreaseinmoduliwhenincreasingtheweightpercentofMWCNT.

MWCNT (weight %) Percentageincreaseinlongitudinalmodulus(E11)

Percentageincreaseintransversemodulus(E22)

05 89.65 54.13510 21.81 3.87

Table 2

PercentagedifferencebetweenKumaret al. [11] andpresentstudyforthelongitudinalmodulus(E11).

MWCNT (weight %) Percentage difference

0 2.030.50 5.400.75 5.752 5.205 4.18

10 4.14

Fig. 8. (Color online.) Variation of the shear modulus with the percentage of MWCNT (by weight) in PC.

Table 3

PercentageincreaseinshearmoduliwhenincreasingtheMWCNTweightpercentage.

MWCNTrange(weight %)

Increase in shear moduli (in %)

G23 G31 G12

02 110 83.33 63.33210 42.85 36.36 35.45

basematrix,combinationoflargeaspectratioandhighsurface-to-volumeratioofMWCNTsandimprovedloadtransferca-pabilityofMWCNTsassistedintheimprovementoftheproperties,butforhighercompositions,aggregatesofMWCNTswereformedandthecurvyandslipperynatureofMWCNTsdidnotassistinfurtherimprovementofthemechanicalproperties.

From

Table 2and

Fig. 7,

it

can

be

inferred

that

the

results

of

our

study

supplement

the

experimental

study

conductedbyKumaretal.forthesamematerial.AscanbeobservedfromTable 2,MDsimulationresultsfromthisstudyareclose

totheexperimentalresults.Fig. 8showsthevariationofshearmoduli,G23 ,G31 andG12 withMWCNTweightpercentage.ComparingFig. 7andFig. 8,itcanbeconcludedthatMWCNT-reinforcedPCcompositesareweakwhenloadedinshearincomparisontowhenloadedintension.ThoughtheshearmoduliincreasewiththeMWCNTweightpercentage,theriseismuch lesserwhencomparedwiththerisein longitudinalmodulus E11 .Table 3showsthatinitially,whenincreasingtheMWCNTweightpercentfrom02,theriseinalltheshearmoduliisgreaterincomparisontotheincreaseinshearmoduliwhenMWCNTweightpercentincreasesby2to10.ThisisduetothecurvyandslipperynatureofMWCNTs.

Fig. 9showsthevariationofYoungsmoduli,E11 andE22 ,withthepercentageofMWCNT(byvolume)inPC.Itshowsthatthe longitudinalmodulus (E11) increaseswhen increasingtheMWCNTvolume fraction.Thepercent increase in E11when increasing theMWCNTvolume fraction isshown inTable 4.FromFig. 9 andTable 4, itcanbeobserved that E11increasessharplytilltheMWCNTvolumefractionis2%.Thereafter,theriseisminimalduetotheaggregationofMWCNTsandthecurvyandslipperynatureofMWCNTsdoesnotassistinfurtherimprovementofthemechanicalproperties.Fig. 9

also

shows

that

the

increase

in

the

transverse

modulus

(E22)

is

negligible

in

comparison

to

the

rise

in

E11 .

This

is

due

tothefactthatthefibersarealignedinthelongitudinaldirection.AsthepercentageofMWCNT(byvolume)inPCincreases,


10/26


Fig. 9. (Color online.) Variation of Youngs modulus, E11 and E22, with the percentage of armchair MWCNT (by volume) in PC.

Table 4

Percentagevariationinlongitudinalmodulus(E11)whenincreasingtheMWCNTvolumefraction.

MWCNT

(volume %)

Percentagedifferenceinlongitudinal

modulus

(E11)

w.r.t.

previous

value

Percentage increase in E11

0 from Vf=0 to Vf=6% from Vf=6 to Vf=16%2 925.86

1362.07 88.63

4 20.506 18.278 15.09

10 14.1612 13.5114 12.6916 12.22

Fig. 10. (Color online.) Variation of the shear modulus with the percentage of armchair MWCNT (by volume) in PC.

thevolumeofthefibersisincreasing.SinceMWCNTshaveahighmodulusinthelongitudinaldirection,thecompositehasimprovedpropertiesinthesaiddirection.Inthetransversedirection,thepropertiesofthecompositearematrixdominatedandhencelowervaluesofE22 areobserved.

Fig. 10showsthevariationofshearmoduliwhenincreasingtheMWCNTvolumefraction.WhenincreasingtheMWCNTvolume fraction by 00.06%, the percent increase in transverse shear modulus G23 is approximately 183%, whereas thepercentincreasereducestomerely10%whentheMWCNTvolumefractionincreasesby0.060.16.Also,theriseinalltheshearmoduli isnegligible incomparison to the rise in E11 .This isagaindue to the fact that inevery shearmode, thepropertiesofthecompositearematrixdominatedandhencelowervaluesofallshearmoduliareobserved.Table 5showsthepercentageincreaseinshearmoduliwhenincreasingtheMWCNTvolumefraction.

Fig. 11 showsthevariationofYoungsmodulus(E11)withthepercentageofMWCNT(byvolume)inPC, fordifferentconfigurationsofMWCNT.AllthesimulationswereperformedwithMWCNTaspectratio(l/d=10).Thecarbonatomsin

a

CNT

are

in

sp2 configurations

and

connected

to

one

another

by

three

strong

bonds.

Due

to

the

geometric

orientationofthecarboncarbonbondsrelativetothenanotubeaxis,armchairMWCNT-reinforcedPCcompositesexhibithightensile


11/26


Table 5

PercentageincreaseinshearmoduliwhenincreasingtheMWCNTvolumefraction.

MWCNTrange(volume %)

Increase in shear moduli (in %)

G23 G31 G12

06 183.33 175 166.66616 9.73 9.69 9.15

Fig. 11. (Color online.) Variation of Youngs modulus ( E11) with the percentage of MWCNT (by volume) in PC, for different configurations of MWCNT.

Fig. 12. (Color online.) Variation of Youngs modulus ( E22) with the percentage of MWCNT (by volume) in PC, for different configurations of MWCNT.

Table 6

PercentageincreaseinelasticmoduliwhenincreasingMWCNTvolumefraction.


Increase in moduli (in %)

Armchair Zigzag Chiral

E11 E22 E11 E22 E11 E22

06 1362.07 13.10 1234.82 10.35 1137.58 8.27616 98.63 37.20 95.17 35.00 95.16 32.16

strengthandYoungsmodulivaluescomparedtothezigzagMWCNT-reinforcedPCcomposites.ChiralMWCNT-reinforcedPCcompositeshavelowerelasticmodulibecauseofthecurvednatureofallC=Csp2 bonds.

Fig. 12 showsthevariationofYoungsmodulus(E22)withthepercentageofMWCNT(byvolume) inPC,fordifferentconfigurationsoftheMWCNT.ComparingFig. 11andFig. 12,itcanbeobservedthattheincreaseinE22 issmallerthanthatofE11 .Table 6showsthepercentageincreaseinelasticmoduliuponincreasingtheMWCNTvolumefraction.FromFig. 11andTable 6,itcanbeinferredthatinitially,whenincreasingtheMWCNTvolumefractionby06%,theriseintheelasticmodulusE11 isfasterwhencomparedwiththeriseinmoduluslateron.FromFigs. 1112andTable 6,itcanbeconcludedthatthepercentageriseinmoduliisgreatestforarmchairMWCNT-reinforcedPCcomposites.

Fig. 13shows

the

variation

of

the

shear

modulus

(G23)

with

the

percentage

of

MWCNT

(by

volume)

in

PC,

for

differentconfigurations of MWCNT. It can be observed that the increase in shear modulus is the greatest for armchair MWCNT-


12/26


Fig. 13.(Color online.) Variation of the shear modulus ( G23) with the percentage of MWCNT (by volume) in PC, for different configurations of MWCNT.



reinforcedPCcomposites.Also,thevaluesareconsiderablysmallerincomparisontothevaluesofthelongitudinalelasticmodulus.

Fig. 14showsthevariationoftheshearmodulus(G31)withthepercentageofMWCNT(byvolume)inPC,fordifferentconfigurationsofMWCNT.Thevaluesof G31 aresmaller incomparison tothevaluesofG23 .Fig. 15 showsthevariationoftheshearmodulus(G12)withthepercentageofMWCNT(byvolume)inPC,fordifferentconfigurationsofMWCNT.ThevariationissimilartothatobservedforG31.

Table 7showsthepercentageincreaseinshearmoduliwithanincreaseintheMWCNTvolumefraction.Itshowsthat

the

percentage

increase

in

modulus

is

the

greatest

for

armchair

MWCNT-reinforced

PC

composites.

Initially,

the

increasein all the shear moduli is faster, when the MWCNT volume fraction increases by 06%, in comparison to the increase


13/26


Table 7

PercentageincreaseinshearmoduliwhenincreasingtheMWCNTvolumefraction.




G23 G31 G12 G23 G31 G12 G23 G31 G12

06 183.33 175.00 166.66 143.33 141.66 133.33 132.45 130.00 80.00616 28.23 19.70 13.87 20.00 19.31 13.57 15.83 13.40 13.14

Fig. 16.(Coloronline.)VariationofYoungsmoduliE11 andE22,withtheaspectratio(l/d)ofarmchairMWCNT-reinforcedPCcompositewithfixedvolumefraction(Vf=0.04).

Fig. 17. (Coloronline.)Variationoftheshearmoduliwiththeaspectratio(l/d)ofarmchairMWCNT-reinforcedPCcompositewithfixedvolumefraction(Vf=0.04).

inmodulus lateronwhen theMWCNTvolume fraction increasesby616 %.ThecurvyandslipperynatureofMWCNTscausestheshearmodulustodecreasewhenincreasingtheMWCNTvolumefraction.Moreover,Table 7alsoshowsthatthepercentageincreaseinshearmoduliofthezigzagMWCNT-reinforcedPCcompositeisgreaterthanthepercentageincrease

in

the

shear

moduli

of

chiral

MWCNT-reinforced

PC

composites.

Comparing

Table 6and

Table 7,

it

can

be

concluded

thatthepercentageincreaseinshearmoduliissmallerincomparisontothepercentageincreaseinlongitudinalelasticmodulus.

Thus,MWCNT-reinforcedPCcompositesareweakerinshearthaninothermodesofloading.Fig. 16showsthevariationofYoungsmoduli E11 and E22 withtheaspectratio (l/d)ofarmchairMWCNT-reinforced

PC composite with volume fraction (Vf=0.04). When increasing the MWCNT aspect ratio (l/d) till l/d=50, both thelongitudinalandtransverseelasticmoduliincreaserapidly.Thereafter,theincreaseissmaller.WhenthelengthoftheCNTsincreases,theefficiencyofloadtransferisincreaseddramatically.Butforlargerlengths,theincreaseinvanderWaalsforcescausesthemodulustoincreaseataslowerrate,ascanbeobservedfromFig. 16.Fig. 17showsthevariationoftheshearmoduluswiththeaspectratio(l/d)ofarmchairMWCNT-reinforcedPCcompositeswithafixedvolumefraction(Vf=0.04).Itcanbe inferred thatthe increase inshearmoduli issignificantlysmaller incomparisontothe increase in longitudinalandtransverseelasticmoduli.Alltheshearmoduliincreasetilll/d=50.Afterthat,theriseisminimal,whichmaybeduetotheincreaseinthevanderWaalsforceforhigheraspectratios.Fig. 17alsoshowsthattheincreaseinG31 andG12 issmallerthantheincreaseinG23 .

Table 8 shows

the

percentage

increase

in

elastic

moduli

when

increasing

the

MWCNT

aspect

ratio

(l/d)

for

a

fixedvolumefraction(Vf= 0.04).FromFig. 16andTable 8,itcanbeconcludedthatE11 andE22 increasesharplyfroml/d=5


14/26


Table 8

PercentageincreaseinelasticmoduluswhenincreasingtheMWCNTaspectratio(l/d).

MWCNT aspect ratio (l/d) Percentage increase in E11 Percentage increase in E22

550 418.07 381.5650100 6.43 46.41

Table 9

Percentage

increase

in

shear

moduli

when

increasing

the

MWCNT

aspect

ratio

(l/d).

MWCNTaspectratio(l/d)

PercentageincreaseinG23



550 220.34 169.04 132.8950100 19.42 25.00 26.83

Fig. 18. (Coloronline.)Variationofthelongitudinalmodulus(E11)withtheaspectratio(l/d)fordifferentconfigurationsoftheMWCNT-reinforcedPCcompositewithfixedvolumefraction(Vf=0.04).

to l/d=50. A possible reason for this is the presence of shear stress concentration at numerous fiber ends in a short

fiber

composite.

However,

for

l/d

>50,

both

moduli

increase

slowly.

This

is

because,

after

l/d

=50,

higher

fiber

axial

stressoccurs,whichcausesthemodulitoincreaseataslowerrate.

Table 9 shows thepercentage increase inshearmoduliwhen increasing theMWCNTaspectratio (l/d). Itshows thatthepercentage increase inall the shearmoduli isgreaterwhen theaspectratiovaries from l/d=5 to l/d=50.This isprobablyduetothefactthatat lowfiberaspectratio,thenumberoffibersinthecompositewillbemoreforthesamevolumefraction.Thehigherthenumberofdiscontinuousfibersis,thehigherthenumberoffiber-endsavailableforstressconcentrationis.

Thus,atlowvaluesofl/d,thestressconcentrationatnumerousfiberendsleadstoariseinthevaluesofalltheshearmoduli.Asthefiberaspectratioincreases,shearmodulibecomeconstant,asshowninFig. 17.

Fig. 18 shows the variation of the longitudinal modulus (E11) with the aspect ratio (l/d) for different configurationsof MWCNT-reinforced PC composites with fixedvolume fraction (Vf= 0.04). It can be observed that armchair MWCNT-reinforced PC composites have a higher elastic modulus E11 , in comparison to zigzag and chiral MWCNT-reinforced PCcomposites. E11 increasesrapidlytilll/d=50,afterwhichtheincreasetakesplaceataslowerrate.Duetothegeometric

orientation

of

the

carboncarbon

bonds

relative

to

the

nanotube

axis,

armchair

MWCNT-reinforced

PC

composite

exhibithigherYoungsmodulivaluescomparedtothezigzagMWCNT-reinforcedPCcomposites.ChiralMWCNT-reinforcedPCcom-

positeshave lowerelasticmodulibecauseof thecurvednatureofallC=C sp2 bonds.At lowervaluesof l/d, thestressconcentrationatnumerousfiberendsleadstoariseinthevaluesofalltheshearmoduli.

Fig. 19 shows the variation of transverse modulus (E22) with the aspect ratio (l/d) for different configurations ofMWCNT-reinforced PC composites with a fixed volume fraction (Vf=0.04). It can be observed here also that armchairMWCNT-reinforcedPCcompositeshavehigherelasticmodulus E22 incomparisontozigzagandchiralMWCNT-reinforcedPCcomposites.ItisinferredbycomparingFigs. 18 and 19thatthevaluesof E22 aresignificantlysmallerthanthecorre-spondingvaluesofE11 .ThisisbecauseMWCNTsarealignedinthedirectionoftheappliedstrain.

Fig. 20showsthevariationoftheshearmodulus(G23)withtheaspectratio(l/d)fordifferentconfigurationsofMWCNT-reinforcedPCcompositeswithfixedvolumefraction(Vf= 0.04).ArmchairMWCNT-reinforcedPCcompositeshavehighershearmoduliincomparisontozigzagandchiralMWCNT-reinforcedPCcomposites.G23 increasesrapidlytilll/d=50,afterwhichtheincreaseoccursataslowerrate.

Fig. 21shows

the

Variation

of

the

shear

modulus

(G31),

with

the

aspect

ratio

(l/d)

for

different

configurations

of

MWCNT-reinforcedPCcompositeswithfixedvolumefraction(Vf= 0.04).G31 increasesrapidlytilll/d=50.Afterthat,theriseis


15/26


Fig. 19. (Coloronline.)Variationofthetransversemodulus(E22)withtheaspectratio(l/d)fordifferentconfigurationsoftheMWCNT-reinforced PCcompositewithfixedvolumefraction(Vf=0.04).

Fig. 20.(Color

online.)

Variation

of

the

shear

modulus

(G23)

with

the

aspect

ratio

(l/d)

for

different

configurations

of

the

MWCNT-reinforced

PC

composite

withfixedvolumefraction(Vf=0.04).

Fig. 21. (Color

online.)

Variation

of

the

shear

modulus

(G31)withtheaspectratio(l/d)fordifferentconfigurationsoftheMWCNT-reinforcedPCcompositewith

fixed

volume

fraction

(Vf=0.04).

minimal,whichmaybedue to the increase invanderWaals force forhigheraspect ratios.Fig. 22 shows thevariationshearmodulus (G12) with theaspect ratio (l/d) fordifferentconfigurations of MWCNT-reinforced PCcomposites withafixedvolumefraction(Vf=0.04).ThetrendissimilartothatofG23 .

Table 10showsthepercentageincreaseinelasticmoduliwhenincreasingtheMWCNTaspectratio(l/d)forfixedvolume

fraction

(Vf= 0.04).

From

Figs. 1819and

Table 10,

it

can

be

observed

that

the

percentage

increase

in

elastic

moduli

isgreater for MWCNT aspect ratio (l/d=5 to l/d=50) than the percentage increase in elastic moduli when l/d=50 to


16/26


Fig. 22.(Coloronline.)Variationoftheshearmodulus(G12)withtheaspectratio(l/d)fordifferentconfigurationsofMWCNT-reinforcedPCcompositewithfixedvolumefraction(Vf=0.04).

Table 10

PercentageincreaseinelasticmoduliwhenincreasingtheMWCNTaspectratio(l/d).

MWCNT

aspect

ratio(l/d)Increase in moduli (in %)


E11 E22 E11 E22 E11 E22

550 418.07 381.56 413.94 371.78 411.85 369.5450100 6.43 46.41 12.84 58.55 14.87 59.56

Table 11

PercentageincreaseinshearmoduliwhenincreasingtheMWCNTaspectratio(l/d).

MWCNTaspectratio(l/d)



G23 G31 G12 G23 G31 G12 G23 G31 G12

550 220.34 169.04 132.89 203.57 151.25 124.48 180.24 132.25 121.4250100 19.42 25.00 45.48 27.05 36.81 47.87 36.96 48.88 53.54

l/d=100.Table 11showsthepercentageincreaseinshearmoduliwhenincreasingtheMWCNTaspectratio(l/d).Itshowsthatthepercentageincreaseinalltheshearmoduliisgreaterwhentheaspectratiovariesfroml/d=5 tol/d=50.Thisisprobablyduetothefactthatatlowfiberaspectratios,thenumberoffibersinthecompositewillbemoreforthesamevolumefraction.Thehigherthenumberofdiscontinuousfibersis,thehigherthenumberoffiber-endsavailableforstressconcentrationis.Thus,atlowvaluesofl/d,thestressconcentrationatnumerousfiberendsleadstoariseinthevaluesofalltheshearmoduli.Asthefiberaspectratioincreases,theshearmodulibecomeconstant,asshowninFigs. 2022.

5.2. Damping

Whilemostoftheresearchesonnanocompositeswithcarbonnanotubesfocusonelasticproperties,relativelylittleef-

fort

to

date

has

been

put

in

the

studies

of

their

damping

characteristics.

In

fact,

there

exists

a

great

potential

in

developingnanocomposites with high damping capacity using carbon nanotubes, since the interfacial slips between nanotubes and

polymerresinandbetweennanotubesthemselvesarebelievedtobesignificant.Thisisduetothenanoscaledimensionsand thehighaspectratioofnanotubes,whichresults ina large interfacialcontactareaandhigh frictionenergydissipa-tionduringtheslidingofnanotubesurfaceswithinthecomposite.Tomodelananocomposite,moleculardynamics(MD)methodsareoftenused.

BuldumandLu[19] investigatedtheinterfacialslidingandrollingofcarbonnanotubesusingMDmethods.Itwasfoundthatananotubefirststickandthenslipssuddenlywhentheforceexertedonitissufficientlylarge.Suhret al. [20] inves-tigatedthedampingcharacteristicsofpolymersreinforcedwithmulti-wallednanotubesandconcludedthatthestick-slipmotionbetweenthenanotubesthemselvesisbelievedtobethemajorcontributiontotheoveralldampingofthenanocom-posite material. Because of the small size of nanotubes, the surface area to mass ratio (specific surface area) of carbonnanotubearrays isextremely large.Therefore, incompositeswithCNTfillers, it isanticipated thathighdampingcanbeachievedbytakingadvantageoftheweakbondingandinterfacialfrictionbetweenindividualCNTsandresin.

In

this

study,

damping

properties

of

carbon

MWCNT/PC

composites

have

been

expressed

analytically

from

the

theory

ofshortfibercomposites.Theelasticviscoelastic correspondenceprinciplealinearelastostaticanalysiscanbeconvertedto


17/26


Table 12

DynamicpropertiesofMWCNTsandPC.

Dynamic properties MWCNT Polycarbonate

Storage modulus (E), GPa 1000 2.9(Esawi and Farag[26]) (Kumar et al. [11])

Loss modulus (E), GPa 1.5 1.0Loss factor () 0.0015 0.345MWCNT volume fraction (Vf) 0.02

avibratorylinearvisco-elasticanalysisbyreplacingtheelasticmoduliwiththecorrespondingcomplexmodulihasbeenusedtodefinethedynamicproperties.Thecomplexmodulushasarealandanimaginarypart.Thedampinglossfactorcanbeexpressedastheratiooftheimaginarytotherealpartofthecomplexmodulus.

=C

C (9)

where

C =C + iC =complex modulus of the compositeC =storage modulus of the composite

C

=loss modulus of the composite=damping loss factor of the composite.

Bycombining the self-consistentapproachofHill [21] with the solutionsofHermans [22] andmakinga fewadditionalassumptions,HalpinandKardos[17]provideasimpleranalyticalformforpredictingmaterialpropertiesoffibercomposites.HalpinTsaiequationsneedonlyoneequationtofindallthecompositemoduliandthelongitudinalPoissonsratioissimplyfoundfromtheruleofmixtures.

ThedynamicpropertiesofMWCNTsandpolycarbonatematrixasgivenbyKhatuaet al.[23]areusedasinputandshowninTable 12.

TandonandWeng[24] derivedexplicitexpressionsfortheelasticconstantsofashortfibercompositeusingtheMoriTanakaapproach.Their formulae fortheplane-strainbulkmodulusk23 andthemajorPoissonratio12 arecoupled,andmust be solved iteratively. Using Tandon and Weng [24] equations, Eshelbys tensor E is evaluated and further used inEq. (10) to find the strain concentration tensor A. The strain concentration tensor is then used in Eq. (11) to find the

stiffness

and

damping

of

the

composite.

AEshelby =

I+ ESm

CfCm1

(10)

C= Cm +Vf

Cf CmAEshelby (11)

where

Vf=fiber volume fractionCm =stiffness matrix of PC matrixCf =stiffness matrix of MWCNTAEshelby =strain concentration tensor

BywritingaprograminMatlab,wecansolvetheequationsandcanstudytheeffectofvariousparameterssuchasaspect

ratio

and

fiber

volume

fraction

on

loss

modulus,

storage

modulus

and

damping

loss

factor.

Forcite

mechanical

propertiescalculationsusetheConstantstrainapproach.Theprocessstartsbyremovingsymmetryfromthesystem,followedbyan

optionalre-optimizationofthestructure,wherethecellparameterscanbevaried.Optimizationatthisstageisalwaysad-vised,asincorrectresultscanbeobtainedifthestructureisfarfromitslowestenergyconfiguration.Foreachconfiguration,anumberofstrainsareapplied,resultinginastrainedstructure.Theresultingstructureisthenoptimized,keepingthecellparametersfixed.Forexample:

numberofstepsforeachstrain= 4max.strainamplitude= 0.003strainpatterns100000,010000

Thisdefinesarangeofvalues{0.003,0.001,0.001,0.003},whichareappliedtoeachstrainpattern:

strain

pattern

100000

gives

e

= {0.003,

0,

0,

0,

0,

0},

{0.001,

0,

0,

0,

0,

0},

{0.001,

0,

0,

0,

0,

0},

{0.003,

0,

0,

0,

0,

0}strainpattern010000givese= {0,0.003,0,0,0,0},{0,0.001,0,0,0,0},{0,0.001,0,0,0,0},{0,0.003,0,0,0,0}


18/26


Fig. 23. (Color online.) Variation of loss factor (11) with the percentage of armchair MWCNT (by volume) in PC.

Table 13

Percentagevariationinlossfactor(11)whenincreasingtheMWCNTvolumefraction.

MWCNT

(volume %)

Percentagedifferenceinlossfactor(11)

w.r.t.

previous

value

using

MD

Percentagedifferencebetween

MoriTanaka

model

and

MD0 2 5.57 3.914 2.58 5.146 2.39 6.818 2.03 9.05

10 1.86 11.8112 1.76 12.9614 1.19 13.6216 0.35 13.64

Each strain pattern represents the strainmatrix in Voigt notation. It is converted into the strain matrix E, such thatE(0,0)=e(0),E(1,1)=e(1),E(2,2)=e(2),E(2,1)=E(1,2)=0.5 e(3). . .

ThesearethenusedtogeneratethemetrictensorG:

G =H0[2E + I]H0

where H0 is formed from the lattice vectors; I is the identity matrix and H0 is the transpose of H0. The new latticeparameterscanbederived fromG;theseare thenusedtotransformthecellparameters(fractionalcoordinatesareheldfixed).Followingthesesteps,thestructureisoptimizedandthestressiscalculated.Astiffnessmatrixisbuiltupbyfromalinearfitbetweentheappliedstrainandtheresultingstresspatterns.Inthecaseofatrajectory,thisisaveragedoverallframes.

InMaterialsStudio5.5,theenergydissipatedcanbeobtainedaftercompletionofthemechanicalpropertycalculationtask. From the Forcite mechanical properties task, we get two values of energy. Energy in frame 1 gives that duringloading,whereasenergyinframe2givestheenergyduringunloadingcycle.Areabetween loadingandunloadingcurvesgivesus theenergydissipated.Dividing thisdissipatedenergyby theenergyobtainedbefore theunloadingcyclebeginsgives themeasureofdamping ().Resultsof MDhavebeencomparedwith the results obtained from theMoriTanakamodel(Eq. (11)).

Fig. 23 shows

the

variation

of

the

loss

factor

(11)

with

the

percentage

of

armchair

MWCNT

(by

volume)

in

PC.

Itshowsthatthelossfactor(11)decreaseswhenincreasingtheMWCNTvolumefraction.Thepercentdecreasein11 when

increasing the MWCNT volume fraction is shown in Table 13. From Fig. 23 and Table 13, it can be observed that 11decreasessharplytilltheMWCNTvolumefractionis2%.Thereafter,thefallsteadiesduetotheaggregationofMWCNTsandthecurvyandslipperynatureofMWCNTsdonotassistinafurtherdeclineofdampingproperties.ItcanalsobeobservedthatthedifferencebetweentheMoriTanakamodelandtheMDmodelgoesonincreasingasthevolumefractionincreases.

Fig. 24 shows the variation of the loss factor (11) with the percentage of MWCNT (by volume) in PC, for differentconfigurationsofMWCNT.AllthesimulationswereperformedwiththeMWCNTaspectratio(l/d=10).ThecarbonatomsinaCNTareinsp2 configurationsandconnectedtooneanotherbythreestrong bonds.Duetothegeometricorienta-tionofthecarboncarbonbondsrelativetothenanotubeaxis,armchairMWCNT-reinforcedPCcompositesexhibithigherYoungsmodulivaluescomparedtozigzagMWCNT-reinforcedPCcomposites.SincethelossfactorisinverselyproportionaltoYoungsmodulus,thereforearmchairMWCNT-reinforcedPCcompositesexhibit lowestdampingvalues.ChiralMWCNT-reinforcedPCcompositeshavehighestlossfactorbecauseofthecurvednatureofallC=Csp2 bonds.

Table 14shows

the

percentage

decrease

in

loss

factor

when

increasing

the

MWCNT

volume

fraction.

From

Fig. 24andTable 14, itcanbe inferred that initially,when increasing theMWCNTvolume fractionby06%, the fall in11 is faster


19/26


Fig. 24. (Color online.) Variation of loss factor (11) with the percentage of MWCNT (by volume) in PC, for different configurations of MWCNT.

Table 14

PercentagedecreaseinlossfactorwhenincreasingtheMWCNTvolumefraction.

MWCNTrange

(volume %)

Decrease in loss factor (in %)


06 11.37 10.15 8.93616 8.37 8.64 8.91

Fig. 25. (Coloronline.)Variationoflossfactor(11)withtheaspectratio(l/d)ofarmchairMWCNT-reinforcedPCcompositeswithfixedvolumefraction(Vf=0.04).

Table 15

Percentagedecreaseinlossfactor(11)whenincreasingtheMWCNTaspectratio(l/d).

MWCNTaspect

ratio

(l/d)

Percentagedecrease

in

11

Percentage difference between MoriTanaka model and MD model at

l/d=5 l/d=60 l/d=100

560 61.75 0.28 25.54 41.3060100 50.26

whencomparedwiththefallinmoduluslateron.Fig. 25showsthevariationofthelossfactor(11)withtheaspectratio(l/d)ofthearmchairMWCNT-reinforcedPCcompositewithfixedvolumefraction(Vf=0.04).WhenincreasingtheMWCNTaspectratio(l/d)tilll/d=60,thelongitudinal lossfactordecreasesrapidly.Thereafter,thedecreaseissmaller.WhenthelengthoftheCNTsincreases,theefficiencyofloadtransferisincreaseddramatically.Butforlargerlengths,theincreaseinvanderWaalsforcecausesthelossfactortodecreaseataslowerrate.TheMoriTanakamodelgiveshighervaluesofthelossfactor(11)thanMD.

Table 15showsthepercentagedecreasein11 whenincreasingtheMWCNTaspectratio(l/d)forafixedvolumefraction(Vf=0.04).FromFig. 25andTable 15,itcanbeconcludedthat11 decreasessharplyfroml/d=5 tol/d=60.Apossible

reason

for

this

is

the

presence

of

shear

stress

concentration

at

numerous

fiber

ends

in

a

short

fiber

composite.

However,

forl/d>60,11 decreasesslowly.Thisisbecause,afterl/d=60,higherfiberaxialstressoccurs,whichcausesthelossfactor


20/26


Fig. 26. (Coloronline.)Variationoflossfactor(11)withtheaspectratio(l/d)fordifferentconfigurationsofMWCNT-reinforcedPCcompositeswithfixedvolumefraction(Vf=0.04).

Table 16

Percentagedecreaseinlossfactor(11)whenincreasingtheMWCNTaspectratio(l/d).

MWCNTaspect

ratio

(l/d)

Decrease in 11 (in %)


560 61.75 50.26 44.4660100 50.26 47.05 42.10

todecreaseataslowerrate.Itcanalsobeobservedthatatlowaspectratios,theMoriTanakamodelandMDgivealmostsimilarresults.Atl/d=5,thepercentagedifferencebetweenthemisonly0.28%.

Fig. 26 shows thevariationofthe loss factor(11)withtheaspectratio (l/d) fordifferentconfigurationsofMWCNT-reinforcedPCcompositeswithfixedvolume fraction(Vf= 0.04). ItcanbeobservedthatarmchairMWCNT-reinforcedPCcompositesexhibitlowervaluesoflossfactor(11),incomparisontozigzagandchiralMWCNT-reinforcedPCcomposites.11 decreasesrapidlytilll/d=60,afterwhichthedecreasetakesplaceataslowerrate.Duetothegeometricorientationof thecarboncarbonbondsrelative tothenanotubeaxis,thearmchairMWCNT-reinforcedPCcompositesexhibithigherYoungsmodulivaluescomparedtozigzagMWCNT-reinforcedPCcomposites.ChiralMWCNT-reinforcedPCcompositeshave

lower

elastic

moduli

because

of

the

curved

nature

of

all

C=C

sp2 bonds.

Since

the

loss

factor

is

inversely

proportional

toYoungsmodulus,thereforearmchairMWCNT-reinforcedPCcompositesexhibitlowestdampingvalues.

Table 16showsthepercentagedecreasein11 whenincreasingtheMWCNTaspectratio(l/d)fora fixedvolumefraction(Vf= 0.04).FromFig. 26andTable 16,itcanbeobservedthatthepercentagedecreasein11 isgreaterfortheMWCNTaspect ratio (l/d=5 to l/d=60) than the percentage decrease in 11 when l/d=60 to l/d=100. Because armchairMWCNT-reinforcedPCcompositesexhibitthehighestelasticmoduli,thereforetheyhavethelowestdampingvalues.

TherearetwopossiblemechanismsthatcouldberesponsiblefordampinginMWCNT-reinforcedpolymercompositesatthemolecularlevel:(a)energydissipationcausedbyinterfacialslidingatthenanotubepolymerinterface,and(ii)energydissipation caused by interfacial stick-slip sliding at the nanotubenanotube interface. When a normal tensile stress isappliedtoacomposite,itstartselongating.Asaresultoftheappliedstress,theresinstartsapplyingashearstressonthenanotube,thuscausingtheloadtobetransferredtonanotubes.Consequently,normalstrainstartsappearinginnanotubesandtheystartelongatingaccordingly.Whentheappliedstressissmall,thenanotuberemainsbondedtothematrix(stickingphase).Boththeresinandthenanotubemovetogetherduringthisphaseandthestrainsareequalinbothepoxyresinand

nanotube.

As

the

applied

stress

is

increased,

the

shear

stress

on

CNT

increases.

At

a

certain

value

of

shear

stress,

called

thecriticalshearstress,thenanotubedebondsfromtheresin.Whentheshearstressonthenanotube increasesbeyondthis

value(asaresultofincreasedappliedstress),theepoxystartsflowingoverthesurfaceofthenanotube.The strain in the nanotube remains constant at its maximum level while the strain in the epoxy increases (slipping

phase). In this phase, there is no transfer load between CNT and matrix, and because of thisenergydissipationdue toslippageoccurs,whichresultsinstructuraldamping.

5.3. Thermalconductivity

ThermalconductivityresultsobtainedusingMDsimulationswerecomparedwithothermodelssuchasseriesmodel,parallelmodel,MaxwellGarnettmodel,LewisNielsenmodel,andHamiltonCrossermodel.ComparisonofMDresultsforthevariationofthermalconductivitywithMWCNTaspectratiowasmadewithDengsmodel.Fig. 27showsthetemperatureprofilegeneratedwhilecalculatingthethermalconductivityforMWCNTPCcompositewithVf=0.10 andl/d=10.Fig. 28

shows

the

variation

of

thermal

conductivity

with

time

for

MWCNTPC

composite

with

Vf=10%.

It

shows

that

the

value

ofthermalconductivitystabilizesafterapproximately60 ps.


21/26


Fig. 27. (Color online.) Temperature profile generated while calculating thermal conductivity for MWCNTPC composites with Vf=10%.

Fig. 28. (Color online.) Variation of the thermal conductivity with time for MWCNTPC composites with Vf=10%.

Fig. 29. (Color online.) Variation of the energy flux with time for MWCNTPC composites with Vf=10%.

Fig. 29showsthevariationofenergyfluxwithtimeforMWCNTPCcompositewithVf=10%.Itshowsthatthevalueofenergyfluxstabilizesafterapproximately60 ps.ThesizeoftheMWCNTsandtheirshapeplayanimportantroleintheheattransferbetweenpolymermatrixandtheincorporatedfiller.FillerswithahigherthermalconductivitythanPCimprovetheheattransferofcompositesonecanconsiderthatPC isathermalbarrier forheatpropagationwhilethefillermaterialtransmitstheheatmuchfaster.Thethermalconductivityofnanocompositesmighthaveacompletelydifferentmechanism

in

contrast

to

micro-composites.

In

the

case

of

micro-composites,

the

heat

is

transported

by

micro-fibers

much

faster

thaninPC.


22/26


Fig. 30. (Color online.) Variation of the thermal conductivity with the volume fraction for different types of MWCNTs with fixed aspect ratio ( l/d=10).

Phonons,whichareresponsible forheatconduction indielectricmaterials,arescatteredattheinterfacebetweendis-similarmaterials.Theheatdissipatesonthesurfaceofnano-fiberstoahigherdegreethanonthesurfaceofmicro-fibers.Inthecaseofnanocompositesystemswitha surface-modifiedfiller,heattransportiscontrolledbytheinterfaceprovidedbyacouplingagentthatconnectsinorganicparticlesononesideandthepolymerhostontheotherside.Whennanosized

fillers

are

used,

the

relative

surface

area

of

the

interface,

and

thus

the

volume

of

the

interfacial

zone,

is

significant.

Hence,theinterfacialzonewilldeterminethethermalconductivityofthesystem,sinceitcanconductheatmuchbetterthanthe

constituentsthemselves.Thismeansthatultimatelythethermalconductivityisaffectedmorebytheinterfacialzonethanbythepolymerandnano-fibers.

WhenassemblingCNTsintolarge-scalecomposites,itisdifficulttomakefulluseoftheexcellentthermalconductivityof individualCNTs (3000 W/mK).Weak tubetubecoupling,dangling tubeends,misalignmentand structuraldefectsallcontributetoquenchingofphononmodesandthusdecreasethethermalconductivityofthecomposites.LongerandthickerCNT would amount to a more efficient heat conduction path, which also allows the transport of phonons with longerwavelengths.Inaddition,larger-diameterMWCNTsaremorelikelytoformarigidandcompactstructureandthusreducethethermalcontactresistanceattheintertubeandtube-polymerinterfaces.

Forshortnanotube-reinforcedcomposites,thephononmobilityisrestrictedattheinterfaceasaresultofthethermallyinsulating nature of the polymer. In such case, the overall thermal conductivity is mechanistically limited by the highCNT/polymerinterfacialthermalresistance.TheprerequisiteforinvestigatingCNTlengtheffectonthermalconductivitiesof

composites

is

a

near-ideal

structure

of

the

composites.

Factors

including

CNT

length,

volume

fraction,

alignment

and

contactareaallcontributetotheefficiencyofphonontransportinthecomposites.

Sincethemechanismofheatconductionbyphononsorelectronsdependsprofoundlyonthebandgapsofmaterials,theheattransfermechanismofCNTsisfoundtodependstronglyonthechirality,whichdeterminesthesizeoftheirbandgapsandelectronicproperties.The largestbandgap(on theorderof1.5 eV) is found innanotubeswith (n,m) chiral indicesdefiningthechiralvectorsatisfyingthecondition: |nm|=3p,wherep isaninteger.Forothertypesofnanotubes,thebandgapisconsiderablysmallerinthecaseofarmchairnanotubes(n=m).Thus,theelectroniccontributiontothethermalconductivitywillbesignificantinmetallicCNTswithasmallbandgap.Ontheotherhand,thermalconductivityofchiralCNTismainlygovernedbythephononcomponent.

Fig. 30showsthevariationofthermalconductivitywiththevolumefractionofdifferenttypesofMWCNTswithfixedaspect ratio (l/d=10). It can be observed that armchair MWCNT-reinforced PC composites exhibit the highest thermalconductivityandthechiralMWCNT-reinforcedPCcompositeshavethelowestthermalconductivityamongstalltheconfig-urationsofMWCNT.BecauseofthelargebandgapinchiralMWCNTs,thecompositesreinforcedwiththesetubesdisplay

poor

thermal

conductivity,

as

can

be

observed

from

Fig. 30.

The

thermal

conductivity

increases

when

increasing

the

volumeofMWCNTs.ItisbecauseoftheexcellentthermalconductivityofMWCNTsthattheconductivityofMWCNT-reinforcedPC

compositesincreasessignificantly.Table 17showsthepercentageincreaseinthermalconductivitywithVf forarmchairMWCNTPCcompositeusingMD.

ItshowsthatthethermalconductivityofarmchairMWCNT-reinforcedPCcompositeincreasesapproximatelyby380%whenincreasingtheMWCNTvolume fraction from0to16%.Table 18showsthepercentagedifference inthermalconductivityfordifferentconfigurationsofMWCNT-reinforcedPCcomposites. Itshowsthatthedifference inthermalconductivitybe-tweenthearmchairandothertypeofMWCNTsincreasesasVfincreases.ThisisduetothelargebandgapforzigzagandchiralMWCNTs in comparison witharmchair MWCNTs. Moreover, the percentagedifference betweenarmchairMWCNT-reinforced and zigzag MWCNT-reinforced PC composites is smaller in comparison to the percentage difference betweenarmchairMWCNT-reinforcedandchiralMWCNT-reinforcedPCcomposite.

Fig. 31showsthecomparisonofMDresultsofthermalconductivityforarmchairMWCNT-reinforcedPCcompositewithothermodelswithfixedaspectratio(l/d=10).ThermalconductivityshowsanincreasingtrendwhenincreasingMWCNT

volume

fraction.

The

series

model

and

the

parallel

model

both

assume

that

each

phase

contribute

independently

to

theoverallthermalresistanceandconductance,respectively,andassumeaperfectinterfacebetweenanytwophasesincontact.


23/26


Table 17

PercentageincreaseinthermalconductivitywithVf foranarmchairMWCNTPCcompositeusingMD.

MWCNTVf(in %)

Thermalconductivity(W/m/K)

Percentageincreasew.r.t.previousvalue

Percentageincreasew.r.t.Vf=0

0 0.15 2 0.19 26.66 26.664 0.24 26.31 60.006 0.30 25.00 100.008 0.37 23.33 146.66

10 0.44 18.91 193.3312 0.52 18.18 246.6614 0.61 18.07 309.3316 0.72 17.26 380.00

Table 18

PercentagedifferenceinthermalconductivityfordifferentconfigurationsofMWCNT-reinforcedPCcomposites.

MWCNTVf(in %)

Percentage difference in thermal conductivity

BetweenarmchairMWCNT-reinforcedandzigzagMWCNT-reinforcedPCcomposites

BetweenarmchairMWCNT-reinforcedandchiralMWCNT-reinforced PCcomposites

0 0 02 5.26 10.52

4 8.33 16.666 13.33 23.338 16.21 27.0210 18.18 27.2712 19.23 28.8414 20.19 31.5916 22.22 33.33

Fig. 31. (Coloronline.)ComparisonofMDresultsofthermalconductivityforarmchairMWCNT-reinforcedPCcompositeswithothermodelswithfixedaspectratio(l/d= 10).

However,

it

typically

gives

an

underestimation

for

a

particulate

composite

due

to

the

presumably

complete

localization

ofthecontributionfromtheparticlesembeddedinthematrix,thatis,neglectingtheinteractionamongthefillers.Therefore,

theseriesmodelgivesthelowerboundforthermalconductivityofcomposites.Incomparison,theparallelmodelpredictsthe thermal conductivity of conventional continuous fiber-reinforced composites along the fiber-alignment direction. Forcomposites with fibrous inclusions, the rule of mixture implicitly assumes perfect contact between particles in a fullypercolatingnetwork.

However,itgivesalargeoverestimationofthermalconductivityforothertypesofcomposites,andgivesanupperboundfor the thermal conductivity of composites. It is important to point out that thermal conductivity measurement resultsofcompositesshouldalwaysfallbetweenthepredictionsoftheseriesmodel(lowerbound)andtheparallelmodel(upperbound)exceptforthecaseswhereinterfacialphononscatteringinnano-laminatescanyieldevenlowerthermalconductivitythanthelowerboundbytheseriesmodel.

Basedonthecontinuityofpotentialandelectriccurrentatthe interface,andontheassumptionthatthe interactionsamongthesphericalfillersarenegligible,whichmeansthatthedisseminatedsmallspheresare located farenough from

each

other,

Maxwell

derived

an

analytical

formula

for

the

effective

specific

resistance

(K)

of

a

compound

medium

consistingof a substance of specific resistance K2 , in which are disseminated small spheres of specific resistance K1 , the ratio of


24/26


Table 19

ComparisonofMDresultsforthermalconductivityofarmchairMWCNT-reinforcedPCcompositewithothermodels.

MWCNTVf(in %)

Percentage difference in thermal conductivity between

MaxwellGarnett model and MD MD and LewisNielsen model

0 0 02 13.75 4.214 12.66 10.00

6 10.52 16.338 10.00 21.8910 8.63 25.2212 6.15 28.4614 2.60 31.9216 1.25 34.86

Fig.32. (Coloronline.)Variationofthethermalconductivitywiththeaspectratio(l/d)fordifferenttypesofMWCNTswithfixedvolumefraction(Vf=0.04).

the volume of all the small spheres to that of the whole beingp. This model gives satisfactory results for compositeswith:(i) sphericalinclusions,(ii) verylowVf,(iii) gooddispersion,and(iv)nointerfacialthermalresistance.SinceneitherinterfacialresistancenorparticleparticleinteractionwastakenintoaccountintheHamiltonCrossermodel,thefibersize

was

found

to

have

no

influence

on

the

effective

thermal

conductivity

of

the

composite

in

this

model.Although Nielsens model isa semi-empirical model, at least three important improvements to the model shouldbe

appreciated.First,theshapeeffectandtosomeextenttheorientationeffectarebothtakenintoaccount.Second,reducedfiller loadingaccountsforthemaximumpackingdensityofthefillerswithaspecificshapeandsizedistribution,andisuniquetothismodel.Incomparison,mostofthetheoreticalequationsassumeuniformchangesoffillerloadinguptothepoint where the dispersed phase makes up the complete system, which is not realistic. Third, the earliest definition ofeffectiveunitisreflectedinthediscussiononaggregatesofspheres.

Table 19 shows a comparison of MD results for thermal conductivity of armchair MWCNT-reinforced PC compositeswithothermodels.ItcanbeinferredthatatlowMWCNTvolumefractions,MDresultsagreewellwiththeLewisNielsenmodel.But at higher volume fractions, resultsofMDare inagreement with the MaxwellGarnett model. Fig. 32 showsthevariationofthermalconductivitywiththeaspectratio(l/d)fordifferenttypesofMWCNTswithfixedvolumefraction(Vf= 0.04).ArmchairMWCNT-reinforcedPCcompositesexhibitthehighestvaluesofthermalconductivityforvaryingl/dand fixed Vf. Heat transport in MWCNT/PC composites is carried out by phonons of various wavelengths. CNT lengths

are

much

longer

than

the

mean

free

path

of

phonons.

The

effect

of

short-wavelength

phonons

probably

reaches

a

stablelevelwhilelong-wavelengthphononscontinuetocontributetotheheattransportprocess.Asaresult,alongerCNTwould

amounttoamoreefficientheatconductionpath,whichalsoallowsthetransportofphononswithlongerwavelengths.Thisresultsinanincreaseofthermalconductivitywhenincreasingl/d.

Fig. 33 shows the variation of thermal conductivity of armchair MWCNT-reinforced PC composite with other modelswith fixed Vf=0.04 and varying l/d. Longer CNTs with larger tubular diameter and more walls led to more efficientlong-distance phonon and electron conductions, resulting in higher thermal conductivities in the composites. It can beinferredfromFig. 33thatatlowMWCNTaspectratios,MDresultsareinagreementwiththeresultsgivenbyDengmodel.However,afterl/d=50,theresultsofMDstartdeviatingfromthoseobtainedbyDengmodel.

AlargevolumeofliteraturereportsthecharacterizationofthePC-basedmaterialsbyusingauniversaltestingmachine.However,Kumaret al. [11] usedanano-indenter thathasalsobeenused for thecharacterizationofvariousother typesof materials. However, there is no study available at a molecular dynamic level for MWCNTPC composites, which cansupplementtheexperimentalfindings.Thecompositematerialsusedinthisstudycomprisenano-sizedfillersintheform

of

MWCNTs.

Hence

it

becomes

essential

to

study

the

properties

of

nano- or

micro-scale

owing

to

the

size

of

the

filler

underobservationandthepropertiesatthislevelarecomplementarytothepropertiesatthemacrolevel.


25/26


Fig. 33. (Coloronline.)VariationofthethermalconductivityofarmchairMWCNT-reinforcedPCcompositeswithothermodelswithfixedvolumefractionVf=0.04.

InanearlierworkbyJindalet al.[25],dynamicloadtestsonsimilarsamplesusingSplitHopkinsonPressureBar(SHPB)

were

reported,

but

the

findings

were

somewhat

different.

It

was

found

that

for

low

concentrations

of

0.5%

MWCNTs,

theincreaseindynamicstrengthwasmostsignificantandremainedalmostthesameupto2%ofMWCNTs,whileforahigher

concentrationof5%MWCNTs,thisincreasewasnegligible.ThemethodusedtodeterminethedatabasedonSHPBwasnotdependentonasingle-pointmeasurement,butratheronatotaleffectonthewholecrosssection,andtheextentoftheappliedloadwasmuchhigher,whichcausedcrushingofthespecimen.

From the results of this paper, theelasticmodulus,which isa measureof the stiffness of anelasticmaterial and isaquantityusedtocharacterizematerials,increaseswhenincreasingtheweightandthevolumeofMWCNTs.Consideringacombinedeffectofbothweightandvolume studies, it issafe tostate that5%MWCNTcomposition inpurePC is themostsuitableonetoenhancethemechanicalpropertiestoasufficientextentforanymechanicalloadandpressure-relatedapplication.Plasticdeformation inamorphouspolymersoccursdue tonucleationandpropagationofshearbands. Intheunreinforcedpolymermatrix,shearbandspropagateuncheckedastherearenobarriersfortheirmovement.Ontheotherhand,thepresenceofMWCNTsinthecompositescouldofferresistanceforthepropagationofshearbands.ThereasonsfortheenhancementofelasticmodulusinpolymerMWCNTnanocompositesaregoodmechanicalinterlockingandthepresence

of

obstacles

to

the

motion

of

shear

bands.

6. Conclusions

ThestudypresentsresultsofMDsimulationforelasticmodulusobtainedbyusingMaterialsStudio5.5anditsupple-mentsanearlierexperimental studyconductedbyusinganano-indenteronaMWCNT-basedPCcomposite.The resultsindicateunambiguously thatstaticmechanicalpropertiesofpurePCaregreatlyenhancedbycomposing thesewith lessthan10%ofMWCNT.OnthebasisofthefindingsofthispaperontheelasticpropertiesofMWCNTPCcomposites,itissafetoconcludethataconcentrationaround2%ofMWCNTisagoodcompromisetoformcompositessuitable forenhancingbothstaticaswellasdynamicproperties.Themainfindingsofthestudyaresummarizedasbelow:

(a) whentheMWCNTweightpercentincreasesfrom05%,thepercentageincreasein E11 and E22 isgreaterthanwhen

the

weight

percent

increases

by

510%;(b) our study supplements an earlier experimental study conducted by Kumar et al. [11] using a nano-indenter on aMWCNTbasedPCcomposite;

(c) withanincreaseinweightpercentofMWCNTinPC,allthemodulishowanincreasingtrend,buttheincreaseinshearmoduliissmall;

(d) withonly2%additionbyvolumeofMWCNTinPC, E11 increasesby925%.Thereafter,theincreaseoccursata lowerrate.TheincreaseinE22 ismuchlessthantheincreaseinE11;

(e) armchairMWCNT-reinforcedPC compositesexhibit thehighestvalues of moduli in comparison to zigzagand chiralMWCNT-reinforcedPCcomposites;

(f) tilll/d=50,bothlongitudinalandtransverseelasticmoduliincreaserapidly.Thereafter,theincreaseissmaller;(g) the loss factor (11)decreaseswhen increasing theMWCNTvolume fraction.11 decreases sharply till theMWCNT

volumefractionis2%.Thereafter,thefallsteadiesduetotheaggregationofMWCNTsandthecurvyandtheslipperynatureofMWCNTsdonotassistinfurtherdeclineofdamping;

(h) chiral

MWCNT-reinforced

PC

composites

exhibit

the

highest

values

of

damping

in

comparison

to

zigzag

and

armchairMWCNT-reinforcedPCcomposites;


26/26


(i) whenincreasingtheMWCNTaspectratio(l/d)tilll/d=60,thelongitudinallossfactor(11)decreasesrapidly.There-after,thedecreaseissmaller.Forlargerlengths,theincreaseinvanderWaalsforcecausesthelossfactortodecreaseataslowerrate;

(j) because of the large band gap in chiral MWCNTs, the composites reinforced with these tubes display poor thermalconductivity.ThethermalconductivityofcompositesincreaseswhenincreasingthevolumeforalltypesofMWCNTs.Thethermalconductivityincreasesapproximatelyby27%uponadditionofonly2%byvolumeofMWCNTs;

(k) theresultsofthermalconductivityobtainedfromMDareinagreementwiththeMaxwellGarnettandLewisNielsen

models;(l) longerCNTswith largertubediameterandmorewallsleadtomoreefficient long-distancephononandelectroncon-

ductions,resultinginhigherthermalconductivitiesinthecomposites.

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