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7/26/2019 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:sumit_sharma1772@yahoo.com(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:sumit_sharma1772@yahoo.comhttp://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:sumit_sharma1772@yahoo.comhttp://www.sciencedirect.com/http://www.sciencedirect.com/http://dx.doi.org/10.1016/j.crme.2015.03.0027/26/2019 Comptes Rendus Mecanique...MWCNT PC
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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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374 S. Sharma et al. / C. R. Mecanique 343 (2015) 371396
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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S. Sharma et al. / C. R. Mecanique 343 (2015) 371396 375
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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376 S. Sharma et al. / C. R. Mecanique 343 (2015) 371396
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
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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,
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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
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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.
MWCNTrange(volume %)
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-
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Fig. 13.(Color online.) Variation of the shear modulus ( G23) with the percentage of MWCNT (by volume) in PC, for different configurations of MWCNT.
Fig. 14.(Color online.) Variation of the shear modulus ( G31) with the percentage of MWCNT (by volume) in PC, for different configurations of MWCNT.
Fig. 15.(Color online.) Variation of the shear modulus ( G12) 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
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Table 7
PercentageincreaseinshearmoduliwhenincreasingtheMWCNTvolumefraction.
MWCNTrange(volume %)
Increase in moduli (in %)
Armchair Zigzag Chiral
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
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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
PercentageincreaseinG31
PercentageincreaseinG12
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
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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
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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 %)
Armchair Zigzag Chiral
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)
Increase in moduli (in %)
Armchair Zigzag Chiral
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
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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}
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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
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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 %)
Armchair Zigzag Chiral
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
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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 %)
Armchair Zigzag Chiral
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.
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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.
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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.
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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
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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.
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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;
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26/26
396 S. Sharma et al. / C. R. Mecanique 343 (2015) 371396
(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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