Constitutive modeling of nanotube–reinforced polymer composites

17 Pages·2003·0.72 MB·English

CompositesScienceandTechnology63(2003)1671–1687
www.elsevier.com/locate/compscitech
Constitutive modeling of nanotube–reinforced polymer composites
G.M. Odegarda,*, T.S. Gatesb, K.E. Wisea, C. Parka, E.J. Siochic
aNationalInstituteofAerospace,NASALangleyResearchCenter,MS390,Hampton,VA,23681,USA
bNASALangleyResearchCenter,MS188E,Hampton,VA,23681,USA
cNASALangleyResearchCenter,MS226,Hampton,VA,23681,USA
Received16August2002;receivedinrevisedform16September2002;accepted15December2002
Abstract
Inthisstudy,atechniqueispresentedfordevelopingconstitutivemodelsforpolymercompositesystemsreinforcedwithsingle-
walledcarbonnanotubes(SWNT).Becausethepolymermoleculesareonthesamesizescaleasthenanotubes,theinteractionatthe
polymer/nanotube interface is highly dependent on the local molecular structure and bonding. At these small length scales, the
latticestructuresofthenanotubeandpolymerchainscannotbeconsideredcontinuous,andthebulkmechanicalpropertiescanno
longer be determined through traditional micromechanical approaches that are formulated by using continuum mechanics. It is
proposedhereinthatthenanotube,thelocalpolymernearthenanotube,andthenanotube/polymerinterfacecanbemodeledasan
effective continuum fiber by using an equivalent-continuum modeling method. The effective fiber serves as a means for incorpor-
ating micromechanical analyses for the prediction of bulk mechanical properties of SWNT/polymer composites with various
nanotubelengths,concentrations,andorientations.Asanexample,theproposedapproachisusedfortheconstitutivemodelingof
twoSWNT/polyimidecompositesystems.
PublishedbyElsevierLtd.
Keywords:B.Mechanicalproperties;B.Modelling;C.Anisotropy;Carbonnanotubes;Nanotechnology
1. Introduction have been developed to predict the macroscopic beha-
vior of polymeric composite materials reinforced with
Inrecentyears,nano-structured,non-metallicmaterials typical reinforcement such as carbon or glass fibers.
have spurred considerable interest in the materials com- These micromechanical models assume that the fiber,
munitypartlybecauseoftheirpotentialforlargegainsin matrix, and sometimes, the interface, are continuous
mechanicaland physicalproperties ascomparedtostan- materials and the constitutive equations for the bulk
dardstructuralmaterials.Inparticular,carbonnanotube/ composite material are formulated based on assump-
polymer composites may provide order-of-magnitude tions of continuum mechanics. However, for nanos-
increasesinstrengthandstiffnesswhencomparedtotypi- tructured materials, such as nanotube-reinforced
cal carbon fiber/polymer composites. In order to facil- polymers, a typical single-walled nanotube (SWNT)
itate the development of nanotube-reinforced polymer may have a diameter of approximately 1–10(cid:1)10(cid:2)9 m,
composites, constitutive relationships must be devel- compared to the typical carbon-fiber diameter of
oped that predict the bulk mechanical properties of the 50(cid:1)10(cid:2)6 m, which leads to a breakdown of the rules
composite as a function of molecular structure of the and requirements for continuum modeling. Even
polymer,nanotube, andpolymer/nanotube interface. though a limited number of studies have attempted to
For computational simplicity, and to adequately address the applicability of continuum micromechanics
address scale-up issues, it is desirable to couple an to nanotube-reinforced polymer composites [3,4], it
equivalent-continuum model of a nanotube/polymer appears that the direct use of micromechanics for
composite with established micromechanical models to nanotube composites is inappropriate without taking
describe the mechanical behavior. As outlined by into account the effects associated with the significant
McCullough [1,2], numerous micromechanical models size difference betweenananotubeandatypical carbon
fiber, asdescribed below.
To help address this analysis deficiency, atomistic
* Correspondingauthor.Tel.:[phone].
E-mailaddress:[email protected](G.M.Odegard). simulation can be used to investigate behavior of
0266-3538/03/$-seefrontmatterPublishedbyElsevierLtd.
doi:10.1016/S0266-3538(03)00063-0
1672 G.M.Odegardetal./CompositesScienceandTechnology63(2003)1671–1687
Nomenclature fiber
Molecular model Cm elasticstiffnesstensor ofmatrix
a bond number D diameter of effective fiber
DIJ well depth ofinteraction a e applied strain magnitude
a
involving atoms Iand J E Young’s modulus ofisotropic
K(cid:1) angle varianceforce constant composite
a
K(cid:2) stretching force constant E longitudinal Young’s modulusof
a L
(cid:3)m total molecular potential energy composite
(cid:1) deformed bond-anglenumbera E transverse Young’s modulusof
a T
(cid:4) undeformed bond-angle number a anisotropic composite
a
(cid:2) deformed interatomic distance of Ef longitudinal Young’s modulusof
a L
bond number a effective fiber
(cid:2)IJ Van derWaals distancefor G shear modulus ofisotropic
a
interactionainvolvingatomsIand composite
J G longitudinal shearmodulusof
L
P undeformed interatomic distance anisotropic composite
a
of bond numbera Gf longitudinal shearmodulusof
L
effective fiber
Equivalent-trussmodel Gf transverse shearmodulusof
T
Ab cross-sectional areaof rod a of effective fiber
a
truss member typeb I identitytensor
rb deformed lengthsof rod a oftruss k axisymmetric alignment factor
a
member typeb Kf transverse bulk modulusof
T
Rb undeformed lengths ofrod a of effective fiber
a
truss member typeb L length ofeffective fiber
Yb Young’s modulus ofrod a of truss n outward normal to boundary B
a j
member typeb s ,s orientation factors
1 2
(cid:3) bond-stretching interaction truss S Eshelby’s tensor
member type TðBÞ tractionsapplied onto boundary B
i
(cid:4) bond-angle variance interaction uðBÞ displacementsapplied onto
i
truss member type boundary B
(cid:3)t total strain energy of truss model v effective fiber volume fraction
f
! Van derWaals interaction truss v matrixvolume fraction
m
member type V volume of effective fiber
x coordinate systemof effectivefiber
j
Equivalent-continuum model andcomposite x000 global coordinate system
j
Af dilute mechanical strain e straintensor
kl
concentration tensor of effective (cid:7)ðy(cid:2)y(cid:7)Þ Diracdeltadistributioncenteredat
fiber y(cid:7)
c directioncosines forcoordinate (cid:8), , (cid:10) angles ofcoordinate
ij
transformation transformation
C elastic stiffnesstensor ofthe lð(cid:8); Þ orientation distribution function
composite (cid:3)f total strainenergyof effectivefiber
Cf ; Cf; Cf elastic stiffnesstensor ofeffective (cid:11) stresstensor
ijkl ij ij
materials at the nanometer length scale. Recently, Wise of the polymer/SWNT composite are accounted for in
andHinkley[5]usedmoleculardynamicssimulationsto both bulk stiffnessand strength calculations.
address the polymer/SWNT material response for a Thefirststeptowardscalculationofbulkstiffnessand
SWNT surrounded by polyethylene molecules. They strength is to establish a constitutive model that can be
predicted that the local changes in the polymer mole- used in a continuum mechanics formulation. The con-
cular structure and the non-functionalized polymer/ tinuum model must account for the fundamental
SWNT interface are on the same length scale as the assumption in continuum mechanics that the mass,
width of the nanotube. The magnitude of this localized momentum, and energy can be represented in a mathe-
effect is generally unknown and needs to be accurately matical sense by continuous functions, that is, indepen-
modeledtoensurethatthefullload-transfercapabilities dent of length scale.
G.M.Odegardetal./CompositesScienceandTechnology63(2003)1671–1687 1673
In this paper, a technique for developing constitutive property relationships. In a recent molecular dynamics
models for SWNT-reinforced polymer composite mate- studyofSWNT/polymermaterials,Franklandetal.[10]
rials is proposed which is based on the equivalent- addressed the effects of covalent bonds at the SWNT/
continuum modeling technique for nano-structured polymer interface. They have shown that for nanotube/
materials which was developed by Odegard et al. [6]. polyethylene composites there is a one to two order-of-
The modeling technique takes into account the discrete magnitude increase in the interfacial shear strength for
nature of the atomic interactions at the nanometer composites with covalent bonding between the nano-
length scale and the interfacial characteristics of the tube and adjacent polymer molecules, relative to sys-
nanotube and the surrounding polymer matrix. After tems without the covalent bonds. However, other
the constituent materials used in this paper are dis- studies [11,12] have shown that because the covalent
cussed in detail, the development of the constitutive bonding may significantly affect the properties of the
model using the presented technique is described. First, nanotube itself, it is desirable to increase the load
a model of the molecular structure of the nanotube and transfer between the nanotube and polymer by using
the adjacent polymer chains is established by using the improvednon-covalentbondingmethods.Forexample,
atomic structure that has been determined from mole- it has been shown that PmPV molecules [poly(m-phe-
cular dynamics (MD) simulations. Second, an equiva- nylenevinylene) substituted with octyloxy chains] natu-
lent-continuum model is developed in which the rallywraparoundcarbonnanotubesinahelicalpattern
mechanicalpropertiesaredeterminedbasedontheforce [12]. This wrapping allows for an improved nanotube/
constants that describe the bonded and non-bonded polymer molecule interaction through non-covalent
interactions of the atoms in the molecular model and bondedinteractions,andthusimprovedloadtransferat
reflect the local polymer and nanotube structure. the nanotube/polymer interface, compared to those
Finally, the equivalent-continuum model is used in found with most structural polymers. Because the
micromechanical analyses to determine the bulk con- PmPVpolymermoleculeswilllikelyentanglethemselves
stitutivepropertiesoftheSWNT/polymercompositewith with neighboring structural polymer molecules (such as
alignedandrandomnanotubeorientationsandwithvar- polyimides), the PmPV can be used as a highly effective
iousnanotubelengthsandvolumefractions.Inaddition, interface between the nanotube and structural polymer,
predicted values of modulus are compared with experi- and is used as the interface in the present study. The
mentaldataobtainedfrommechanicaltesting. molecular structure of a single unit of the PmPV mole-
cule is shown in Fig. 1. The subscripts on the atomic
labels in Fig. 1 correspond to the subscripts shown in
2. Constituent materials Tables 1–3,which arediscussed below.
The constitutive models developed in this study are 2.3. Polymer matrix
for SWNT/polyimide composites with a PmPV inter-
face. The properties of the constituent materials are The nanotubes used in this study are assumed to be
described first. well dispersed inside a bulk polymer matrix. Two poly-
mers were considered in the present study. The first
2.1. Carbon nanotube polymer considered is LaRC-SI, a thermoplastic poly-
imide that has been shown to have good mechanical
In1991Iijima[7]obtainedtransmissionelectronmicro- properties for various processing and testing conditions
graphsofelongated,nano-sizedcarbonparticlesthatcon- [13–16]. The properties of LaRC-SI used in this study
sisted of cylindrical graphitic layers, known today as have been taken from Whitley et al. [16] for the system
carbonnanotubes.Becauseoftheirhighinter-atomicbond with a 3% stoichiometric imbalance at room tempera-
strength and perfectlattice structure, a Young’s modulus ture. The Young’s modulus and Poisson’s ratio of this
as high as 1 TPa and a tensile strength approaching 100 material are3.8 GPa and 0.4, respectively.
GPahavebeenmeasuredforsingle-walled carbon nano- The second polymer considered in this study is the
tubes (SWNT) [8].Theseproperties, inaddition totheir colorless polyimide LaRC-CP2 [17]. This optically
relatively low density, make SWNT an ideal candidate transparent polyimide is resistant to radiation and may
as a reinforcing constituent. For this study, a (6,6) (see be used to make thin polymer films for building large
Ref. [9] for an explanation of this notation), straight, space structures. The mechanical properties of this
finite-length,single-walled carbon nanotube ismodeled. polymer are not known a priori, and have been experi-
mentally determined asdescribed in Section 7.2.
2.2. Nanotube/polymer interface In order to synthesize the nanotube/LaRC-CP2 com-
posite, a batch of purified laser-ablated SWNTs was
Accurate representation of the nanotube-to-polymer obtained from Rice University, and transmission elec-
interface is essential for modeling nano-scale structure– tron microscope (TEM) and atomic force microscope
1674 G.M.Odegardetal./CompositesScienceandTechnology63(2003)1671–1687
Fig.1. PmPVmolecularstructure.
(AFM) observations revealed an average SWNT radius addition, a sample of purified SWNTs was additionally
and length of approximately 0.7 nm and 3000 nm, treated with an acid mixture in order to study the effect
respectively. The diamine and dianhydride used to syn- of additional SWNT treatment on the dispersion of
thesize the LaRC-CP2 were 1,3-bis(3-aminophenoxy) nanotubes. The acid mixture was prepared in a round-
benzene (APB) and 2,2-bis(3,4-anhydrodicarbox- bottomflaskwithconcentratedH SO andHNO inthe
2 4 3
yphenyl) hexafluoropropane (6FDA), respectively. As- ratio of 3 to 1. The purified SWNT was added into the
received anhydrous dimethyl formamide (DMF) was acidmixtureandrefluxedat70(cid:6)Cfor30mintomakea
used as a solvent. The details of the synthesis are dark brown solution. The dark solution was diluted
described elsewhere [18]. with distilled water and allowed to settle overnight. A
A dilute SWNT solution, typically around 0.05%
weight fraction in DMF, was prepared by homogeniz- Table2
ing for 10 min (750 rpm with a 6 mm diameter rotor Bond-anglevariationparameters
homogenizer) and sonicating for an hour at 47 kHz. In Bond-anglevariation (cid:4)((cid:6)) Ky(kcal/mol/rad2)
C–C–C 112.7 58.4
Table1 t t t
C–C–H 110.7 37.5
Bondstretchingparameters t t t
C–C–O 107.5 59.7
t t
Bondstretching P(A˚ ) Kr(kcal/mol/A˚ 2) Ht–Ct–Ht 107.8 33.0
H–C–O 108.9 59.0
t t
C–C 1.529 268.0 C–C–H 120.0 40.0
t t v v v
C–H 1.090 340.0 C–C–C 120.0 50.0
t t v v a
C–O 1.415 201.4 H–C–H 120.0 40.0
t v v v
C –O 1.355 431.6 H–C–C 120.0 40.0
a v v a
C –C 1.400 469.0 C–C–C 120.0 50.0
a a v a a
C –H 1.080 367.0 C–C–C 120.0 63.0
a a a a a
C –C 1.320 520.0 C–C–H 120.0 35.0
a v a a a
C –C 1.320 520.0 C–C–O 121.9 43.2
v v a a
C –H 1.080 367.0 C–O–C 108.9 49.6
v v t a
G.M.Odegardetal./CompositesScienceandTechnology63(2003)1671–1687 1675
Table3 molecular model (Fig. 2). The individual energy con-
VanderWaalsinteractionparameters tributions are summed over the total number of corre-
VanderWaalsinteraction DI(kcal/mol) (cid:2)I(A˚ ) sponding interactions in the molecular model. Various
functional forms may be used for these energy terms
Ct 0.066 3.50 depending on the particular material and loading con-
H 0.030 2.50
t ditionsconsidered [19].
O 0.140 2.90
In the present study, the total molecular potential
C 0.070 3.55
a
H 0.030 2.42 energy of themolecular model is takento be:
a
Cv 0.076 3.55 (cid:3)m ¼XK(cid:2)ð(cid:2) (cid:2)P Þ2þXK(cid:1)ð(cid:1) (cid:2)(cid:4) Þ2
Hv 0.030 2.42 a a a a a a
a a
X "1(cid:2)(cid:2)IJ(cid:3)12 (cid:2)(cid:2)IJ(cid:3)6# ð1Þ
þ DIJ a (cid:2) a
clear amber supernatant was decanted and the remain- a a 2 (cid:2)a (cid:2)a
ing dark solution containing the sediment was filtered
through a sintered glass filter and washed thoroughly where the terms P and (cid:4) refer to the undeformed
a a
with distilled water and methanol under vacuum. A interatomic distance of bond number a and the unde-
dried SWNT paper was peeled off from the filter and formed bond-angle number a, respectively. The quan-
dried in a vacuum oven at 60 (cid:6)C overnight. The acid- tities (cid:2) and (cid:1) are the distance and bond-angle after
a a
treated SWNT solution was also prepared as a dilute stretchingandanglevariance,respectively.Thesymbols
solution in DMF in the same manner described above. K(cid:2) andK(cid:1) representtheforceconstantsassociatedwith
a a
ThesonicatedSWNTsolutionwasusedasasolventfor the stretching and angle variance of bond and bond-
the poly(amic acid) synthesis with the diamine and dia- angle number a, respectively. The well depth and nat-
nhydride. The entire reaction was carried out while ural van der Waals distance for interaction a are given
stirringinanitrogen-purgedflaskimmersedina40kHz by, respectively [19]:
ultrasonicbathuntilthesolutionviscosityincreasedand
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
stabilized. Sonication was ceased and stirring was con- DIJ ¼ DI(cid:9)DJ
a a a
tinued for several hours to form a SWNT-poly(amic
qffiffiffiffiffiffiffiffiffiffiffi
acid) solution. Acetic anhydride and pyridine were (cid:2)IJ ¼ (cid:2)I(cid:9)(cid:2)J ð2Þ
a a a
added as catalysts with stirring to imidize the SWNT-
poly(amic acid) chemically. where the superscripts I and J denote the two atoms
A series of SWNT/LaRC-CP2 nanocomposite films involved in an individual van der Waals interaction.
were prepared with a SWNT concentration of up to Onlythebondstretching,bond-anglevariation,andvan
1.0% nanotube weight fraction for both the as-received der Waals parameters were considered in Eq. (1)
(which is the baseline system) and acid-treated systems. because other energy terms were found to have a negli-
The SWNT-poly(amic acid) solution prepared was cast gible contribution to the total molecular potential
onto a glass plate and dried in a dry air-flowing cham- energy. The values of the force constants, well depths,
ber. Subsequently, the dried tack-free film was cured in natural van der Waals distances, equilibrium bond
a nitrogen-circulating oven to obtain solvent-free, free- lengths, and equilibrium bond angles associated with
standing SWNT–polyimide film. Examination of the the Carbon, Hydrogen, and Oxygen atoms shown in
films with Transmission Electron Microscopy and an Fig. 1are listed inTables1–3.
optical microscope revealed that thin SWNT bundles
weredisperseduniformlythroughoutthewholepolymer
matrix [18].
3. Molecular potential energy
The bonded and non-bonded interactions of the
atoms in a molecular structure can be quantitatively
describedbyusingmolecularmechanics.Theforcesthat
exist for each bond, as a result of the relative atomic
positions,aredescribedbytheforcefieldsuchthatthese
forcescontributetothetotalmolecularpotentialenergy
of a molecular system. The molecular potential energy
foranano-structuredmaterialissubsequentlydescribed
bythesumoftheindividualenergy contributionsinthe Fig.2. Molecularmechanicsmodeling.
1676 G.M.Odegardetal./CompositesScienceandTechnology63(2003)1671–1687
4. Molecular dynamics simulation ether linkage were adapted from the MM3 force field
[26–28]. All simulations were carried out with the TIN-
Molecular dynamics (MD) simulation has become an KER1 3.8 [29] molecular modeling package and were
effective tool for studying the physics of condensed performed in the constant NPT ensemble, using the
mattersystemsinwhichtheforcesactingonparticlesin Berendsen weak-coupling method to maintain the tem-
adefinedcellarecalculatedandtheclassicalNewtonian perature and pressure near their specified values [30]. A
equations of motion are integrated numerically [20–22]. modifiedBeemanintegrationalgorithm,asimplemented
In general, each particle is allowed to interact with all in the Tinker 3.8 package, was used to integrate the
theotherparticles in thesimulation. equationsofmotion[29].Theresultingmolecularmodel
In the present study, a MD simulation was used to isshownon the left side of Fig. 3.
generate the equilibrium structure of the composite
system, which consisted of a (6,6) single-walled
nanotube and five PmPV oligomers, each ten repeat- 5. Equivalent-continuum modeling
ing units in length. The initial structure was con-
structed by placing the nanotube at the center of the The equivalent-continuum model of the composite
MD cell, and by inserting the PmPV molecules at material can be developed based on the equilibrium
random, non-overlapping positions within the MD molecular structure obtained with the MD simulation
cell. This sample was equilibrated for approximately by using the methods of Odegard et al. [6]. This
500 ps at 800 K and 500 atm of hydrostatic pressure to approach relies on an equivalent-continuum modeling
relax the initial configuration and compress the system technique that is used to predict the bulk mechanical
to an appropriate density. This initial procedure was behavior of nano-structured materials. The method
followed by an additional 500ps of simulation at 300 K consists of two major steps. First,a suitable representa-
and 1 atm of pressure. By the end of the final equili- tive volume element (RVE) of the nano-structured
bration run,the total energyand density had stabilized. material is chosen. The RVE of a typical nano-struc-
No constraints were placed on the periodic MD cell tured material is on the nanometer length scale, there-
shape or size. fore, the material of the RVE is not continuous, but is
The parameters used in the MD simulation are listed an assemblage of many atoms. Interaction of these
inTables1–3,withtheatomlabelsdefinedinFig.1.All atoms is described in terms of molecular mechanics
parameters,otherthanthoseinvolvingtheoxygenatom force constants, which are known for most atomic
were taken from the OPLS-AA force field developed by structures [19]. In the second step, an equivalent-con-
Jorgensen and coworkers [23–25]. Parameters for the tinuum model of the RVE is developed in which the
Fig.3. Equivalent-continuummodelingofeffectivefiber.
G.M.Odegardetal./CompositesScienceandTechnology63(2003)1671–1687 1677
total strain energies in the molecular and equivalent- 2K(cid:2)R(cid:3)
E(cid:3) ¼ a a ð4Þ
continuum models, under identical loading conditions, a A(cid:3)
a
aresetequal.Theeffectivemechanicalproperties,orthe
effective geometry, of the equivalent-continuum is then
determined from equatingstrain energies. 32K(cid:1) (cid:2) (cid:4) (cid:3)2
For the most general approach, an equivalent-truss E(cid:4) ¼ a sin a ð5Þ
a A(cid:4)R(cid:4) 2
modeloftheRVEmaybedevelopedasanintermediate a a
step to link the molecular and equivalent-continuum where K(cid:2), K(cid:1), and (cid:4) are the same parameters asso-
a a a
models. Each atom in the molecular model is repre- ciatedwithEq.(1),andthesuperscripts(cid:3)and(cid:4)indicate
sented by a pin-joint, and each truss element represents primary bonding and bond-angle variance interactions,
an atomic bonded or non-bonded interaction. The respectively.
moduliofthetrusselementsarebasedonthemolecular- Upon examination of Eq. (1), it is clear that the
mechanics force constants. Therefore, the total mole- energy associated with van der Waals interactions is
cular potential energy of the molecular model and the highly non-linear with respect to interatomic distance.
strain energy of the equivalent-truss are equal for the The determination of the Young’s moduli of truss ele-
same loading conditions. ments that representvan der Waals interactions is com-
plicated by accounting for this non-linearity and the
5.1. Trussmodel large range of values for the interatomic distance of the
interacting atoms in an equilibrium configuration.
Intraditionalmolecularmodels,theatomiclatticehas Therefore, linear relationships for the Young’s mod-
been viewed as an assemblage of discrete masses that ulus, such as those given by Eqs. (4) and (5), are not
areheldinplacewithatomicforcesthatresembleelastic realisticfor thevanderWaals interactions.
springs [31]. The mechanical analogy of this model is a To address this problem, the energy associated with
pin-jointed truss model in which each truss member the van der Waals interaction given in Eq. (1) and the
represents either a bonded or non-bonded interaction strain energy of a truss element given by Eq. (3) were
between atoms. Therefore, the truss model allows the equated. The Young’s modulus that represents the
mechanical behavior of the nano-structured system to mechanical stiffness of a van der Waals interaction is
be accurately modeled in terms of displacements of the given by:
atoms.Thedeformationofeachbondedornon-bonded " #
2(cid:2)IJDIJ 1(cid:2)(cid:2)IJ(cid:3)12 (cid:2)(cid:2)IJ(cid:3)6
interaction corresponds to the axial deformation of the E! ¼ a a a (cid:2) a ð6Þ
corresponding truss element. a Aa(cid:7)(cid:8)(cid:2)a(cid:2)(cid:2)IaJ(cid:9)2 2 (cid:2)a (cid:2)a
The total mechanical strain energy, (cid:3)t, of the truss
model is: where the superscript ! indicates van der Waals bond-
ing. Clearly, the Young’s modulus is highly dependent
(cid:3)t ¼XXAabEab(cid:8)rb(cid:2)Rb(cid:9)2 ð3Þ on the interatomic spacing. However, because of the
2Rb a a
b a a difficulty of assigning an individual Young’s modulus
value for every van der Waals interaction in a nano-
where Ab and Eb are the cross-sectional area and structured material, discretevalues ofYoung’smodulus
a a
Young’s modulus of rod a of truss member type b, may be approximated for ranges of interatomic spacing
respectively. The term rb(cid:2)Rb is the stretching of rod a for each combination of atoms based on Eq. (6). The
a a
of truss member type b, where Rb and rb are the unde- process forestablishing these ranges is discussedbelow.
a a
formed and deformed lengths of the truss elements, Toimplementtheresultantequivalent-trussstructure,
respectively. afiniteelementmodelwasdevelopedbyusingANSYS1
In order to represent the mechanical behavior of the 6 [32] (Fig. 3). Each element (LINK8) was a three-
molecular lattice model with the truss model, Eq. (3) dimensional pin-jointed truss element with six degrees
mustbeequatedwithEq.(1)inaphysicallymeaningful of freedom (three displacement components on each
manner. Both equations are the sum of energies for end) that represented a single atomic interaction. Each
particulardegreesoffreedom.Themaindifficultyinthe node corresponded to an atom in the equilibrium struc-
substitution is specifying Eq. (3), which has stretching ture of the molecular model. A total of 14,501 elements
terms only, for Eq. (1), which also has bond-angle var- and 1818 nodeswereusedin the model.
iance and van derWaals terms.
It was shown by Odegard et al. [6] that for small 5.2. Continuummodel
deformations, the Young’s moduli of the rods repre-
senting primary bonds and the bond-angle variance With the equivalent-truss structure in place, the con-
interactions may be determined as a function of the tinuum model could be constructed. The geometry of
force constants: the linear-elastic, homogeneous, equivalent-continuum
1678 G.M.Odegardetal./CompositesScienceandTechnology63(2003)1671–1687
RVE was assumed to be cylindrical, similar to that of 1(cid:10) (cid:11)
Gf ¼ Cf (cid:2)Cf
the molecular and truss models (Fig. 3). With this T 2 22 23
approach, the mechanical properties of the solid cylin- 1(cid:10) (cid:11)
der were determined by equating the total strain ener- KTf ¼2 C2f2þC2f3
gies of the equivalent-truss and equivalent-continuum
Gf ¼Cf
modelsunderidenticalloadingconditions.Examination L 44
of the molecular model revealed that it was accurately 2Cf2
Ef ¼Cf (cid:2) 12 ð9Þ
described as having transversely isotropic symmetry, L 11 Cf þCf
22 23
withtheplaneofisotropyperpendiculartothelongaxis
of the nanotube. There are five independent material
parametersrequiredtodeterminetheentiresetofelastic Conversely, the elastic stiffness components can be
constants for a transversely isotropic material. Each of described interms ofthe fourelasticparameters:
the five independent parameters may be determined
Cf ¼Gf
from a single boundary condition applied to both 44 L
equivalent-truss and equivalent-continuum models. Cf ¼Gf þKf
22 T T
Once the mechanical properties of the equivalent-con- Cf ¼Kf (cid:2)Gf
tinuum RVE are determined, then the model may be 23 T T
h (cid:10) (cid:11)i1
used in subsequent micromechanical analyses as an Cf ¼ Kf Cf (cid:2)Ef 2 ð10Þ
12 T 11 L
effective fiber. The method employed in this study was
adapted from the approach used by Hashin and Rosen
[33] to determine elastic properties for fiber reinforced At this point, both the elastic parameters and the
compositematerials. elastic stiffness components are unknown. These values
are determined by applying five identical sets of bound-
5.2.1. Effective-fiber constitutive model ary conditions to the equivalent-truss model and the
The constitutive relationship of the transversely iso- effective fiber, and by subsequently equating the strain
tropic equivalent-continuum RVE (which is referred to energies by adjusting the five independent elastic prop-
as the effective fiber throughout the remainder of the erties. Boundary conditions must be chosen to yield
paper) is: uniquevalues forthe independentelasticproperties.
(cid:11) ¼Cf " ð7Þ
ij ijkl kl 5.2.2. Boundaryconditions
Five sets of boundary conditions were chosen to
where (cid:11) and " are the stress and strain components, determineeachofthefiveindependentelasticproperties
ij kl
respectively (i,j=1,2,3), and Cf are the elastic stiffness such that a single property could be independently
ijkl
components of the effective fiber (denoted by super- determined for each boundary condition. The displace-
scriptf).Alternatively,Eq.(7)canbesimplifiedbyusing ments and tractions applied at the boundaries of the
the usual contracted notation for the elastic stiffness RVE are generalized, respectively, by:
components and transversely-isotropic symmetry:
uðBÞ¼" x ð11Þ
i ij j
(cid:11) ¼Cf " þCf " þCf "
11 11 11 12 22 12 33 TðBÞ¼(cid:11) n ð12Þ
i ij j
(cid:11) ¼Cf " þCf " þCf "
22 12 11 22 22 23 33
whereBistheboundingsurface,x isdefinedinFig.3,and
(cid:11) ¼Cf " þCf " þCf " j
33 12 11 23 22 22 33 njarethecomponentsoftheoutwardnormalvectortoB.
(cid:11) ¼2Cf " Thegeneralizedtotalstrainenergyoftheeffectivefiberis:
12 44 12
(cid:11) ¼2Cf " V (cid:13)D2L
13 44 13 (cid:3)f ¼ (cid:11) " ¼ (cid:11) " ð13Þ
(cid:10) (cid:11) 2 ij ij 8 ij ij
(cid:11) ¼ Cf (cid:2)Cf " ð8Þ
23 22 23 23
whereV,D,andLarethevolume,diameter,andlength
oftheeffectivefiber,respectively(Fig.3).Theboundary
Five independent elastic properties may be chosen to conditions and strain energies for each of the five inde-
describe the complete set of elastic stiffness components, pendentelasticproperties are described below.
namely, the elastic stiffness component, Cf , and four
11
elastic parameters: transverse shear modulus, Gf, trans- 5.2.2.1. Transverseshearmodulus.Forapuretransverse
T
versebulkmodulus(also knownastheplane-strain bulk shear strain, e, applied to the boundary of the equiva-
modulus),Kf,longitudinalshearmodulus,Gf,andlong- lent-truss RVE and the effective fiber, " ¼e=2, with
T L 23
itudinal Young’s modulus, Ef. The four elastic para- the remaining strain components equal to zero. From
L
metersarerelatedtotheelasticstiffnesscomponentsby: Eq. (11), the boundarydisplacementsare:
G.M.Odegardetal./CompositesScienceandTechnology63(2003)1671–1687 1679
5.2.2.4. LongitudinalYoung’smodulus.Thelongitudinal
u ðBÞ¼0
1
Young’s modulus was determined by prescribing a
e
u2ðBÞ¼2x3 strain along the x1 axis, "11 ¼e, with all of the shear
strain components set to zero. Since a transverse Pois-
e ð14Þ son contraction is allowed in this case, the transverse
u ðBÞ¼ x
3 2
2 normalstressesaresettozero,(cid:11) ¼(cid:11) ¼0.FromEqs.
22 33
(11) and (12),the boundary conditionsare:
From Eq. (13), the total strain energy of the effective
u ðBÞ¼ex
fiber is: 1 1
1 T ðBÞ¼0
(cid:3)f ¼ (cid:13)D2LGfe2 ð15Þ 2
8 T ð20Þ
T ðBÞ¼0
3
where D and L are the diameter and length of the
effective fiber, respectively (Fig. 3). The effective fiber The strain energy is:
strain energy, (cid:3)f, is equated to the strain energy of the
1
truss subjected to the boundary conditions in Eq. (14). (cid:3)f ¼ (cid:13)D2LEfe2 ð21Þ
8 L
Since D and L are known, and e is arbitrarily chosen in
determining (cid:3)f (in the range of small deformations, i.e. The longitudinal Young’s modulus was evaluated
e(cid:10)1), then the transverse shear modulus of the effec- using Eq. (21).
tive fiber isdirectlyevaluated from Eq. (15).
5.2.2.5. Elastic stiffness component Cf . The elastic
11
5.2.2.2. Transverse bulk modulus. The transverse bulk stiffness tensor component, Cf , may be determined by
11
modulus was obtained in a similar manner by prescrib- applying a strain parallel to the x axis, while con-
1
ing transverse strains " ¼" ¼e to the RVE bound- straining the strains along the x and x axes, therefore
22 33 2 3
ary with all remaining strain components set to zero. preventing a Poisson contraction. For a prescribed
From Eq. (11), thedisplacementsare: strain, " ¼e, with the remaining strains held at zero,
11
the displacementsfrom Eq. (11) are:
u ðBÞ¼0
1
u2ðBÞ¼ex2 u1ðBÞ¼ex1
u ðBÞ¼ex ð16Þ
3 3 u ðBÞ¼0
2
ð22Þ
The strainenergyof theeffective fiberfor this caseis: u ðBÞ¼0
3
1
(cid:3)f ¼ (cid:13)D2LKfe2 ð17Þ From Eq. (13), the strain energy for these displace-
2 T
ments is:
The transverse bulk modulus was determined from
1
Eq. (17). (cid:3)f ¼ (cid:13)D2LCf e2 ð23Þ
8 11
5.2.2.3. Longitudinal shear modulus. The longitudinal Therefore, the elastic stiffness component Cf can be
11
shear modulus was determined by prescribing a pure used as one of the five independent parameters that
shear strain in the x (cid:2)x plane, " ¼e=2, with the describe the overall properties of the transversely iso-
1 2 12
remaining strain components set equal to zero. There- tropiceffective fiber.
fore, the applied boundary displacementsare:
e 5.2.3. Boundaryregion
u ðBÞ¼ x
1 2 The displacements and tractions specified above were
2
applied to each node in the boundary region of the
e
u ðBÞ¼ x equivalent-trussmodel(indicatedinFig.3),andthecorre-
2 2 1
sponding strain energies were calculated by summing the
ð18Þ
u ðBÞ¼0 strainenergiesofeachindividualtrussmemberintheRVE.
3
To determine the size of the boundary region, it was
The resulting strainenergyof theeffective fiberis: assumedthattherangeoftheboundaryregionisrelated
to the interatomic distance between the minimum non-
1
(cid:3)f ¼ (cid:13)D2LGfe2 ð19Þ bonded spacing found in the equilibrium structure and
8 L
the maximum distance for which a positive-definite
The longitudinal shear modulus was evaluated by relationship exists between the force and displacement.
using Eq. (19). Itwasalsoassumedthatthecontributionoftheenergies
1680 G.M.Odegardetal./CompositesScienceandTechnology63(2003)1671–1687
associated with van der Waals forces between atoms ness properties of the effective fiber/polymer composite
with a separation distance larger than this maximum material. This method has been successfully applied to
were relativelysmallandcould beneglected. transversely-isotropic inclusions by Qui and Weng [35]
TherecentMDsimulationofaSWNTsurroundedby and for this current method, the complete elastic stiff-
polyethylene molecules performed byWise and Hinkley nesstensorforthecompositeisgivenbyBenveniste[36]
[5] predicted that the local changes in the polymer
C¼Cmþv (cid:14)(cid:8)Cf (cid:2)Cm(cid:9)Af(cid:15)(cid:8)v Iþv (cid:14)Af(cid:15)(cid:9)(cid:2)1 ð24Þ
molecular structure and the non-functionalized SWNT/ f m f
polymer interface are on the same length scale as the
width of the nanotube. This recent study and the afore- where v and v are the fiber and matrix volume frac-
f m
mentioned assumptions led to the selection of a cylind- tions, respectively, I is the identity tensor, Cm is the
rical boundary region that extends from a radius of 0.9 stiffness tensor of the matrix material, Cf is the stiffness
nm measured from the center of the nanotube to the tensorofthefiber,andAfisthedilutemechanicalstrain
outer edge of the molecular model (Fig. 3). Within the concentration tensor for the fiber
0.9 nm radius, the RVE includes the nanotube, nano- Af ¼hIþSðCmÞ(cid:2)1(cid:8)Cf (cid:2)Cm(cid:9)i(cid:2)1 ð25Þ
tube/polymer interface, and polymer molecules imme-
diately adjacent to the interface.
The tensor S is Eshelby’s tensor, as given by Eshelby
5.2.4. Material propertysummary [37] and Mura [38]. The terms enclosed with angle-
The five independent parameters and the resulting brackets in Eq. (24) represent the average value of the
stiffness tensor components, from Eq. (10), were deter- term over all orientations defined by transformation
mined for an effective fiber diameter, D, of 1.8 nm, from the local fiber coordinates ðx ; x ; x Þ to the
1 2 3
length, L, of 3.2 nm, and applied strain, e, of 0.1%. globalcoordinates(cid:8)x000; x000; x000(cid:9)(Fig.4).Forexample,
1 2 3
These values arelisted in Table 4. the transformed dilute mechanical strain concentration
tensor for the fiber with respect to the global coordi-
nates is
6. Micromechanical analysis
A(cid:7)f ¼c c c c Af ð26Þ
ijkl ip jq kr ls pqrs
The effective fiber accurately accounts for the struc-
ture–property relationships at the nanoscale and pro- wherec arethedirectioncosinesforthetransformation
ij
vides a bridge to the continuum model. With this indicated inFig. 4;that is,
process firmly established, constitutive models of the
c ¼cos (cid:8) cos (cid:2)sin (cid:8) cos (cid:10) sin
effective fiber/polymer composite may be developed 11
c ¼sin (cid:8) cos þcos (cid:8) cos (cid:10) sin
with a micromechanical analysis by using the mechan- 12
ical properties of the effective fiber and the bulk poly- c ¼sin sin (cid:10)
13
mer matrix material. For the composite considered in c ¼(cid:2)cos (cid:8) sin (cid:2)sin (cid:8) cos (cid:10) cos
21
this study, the PmPV molecules that were near the
c ¼(cid:2)sin (cid:8) sin þcos (cid:8) cos (cid:10) cos
polymer/nanotube interface were included in the effec- 22
c ¼sin (cid:10) cos
tive fiber, and it was assumed that the matrix polymer 23
surrounding the effective fiber had mechanical proper- c ¼sin (cid:8) sin (cid:10)
31
ties equaltothoseofthebulkLaRC-SIandLaRC-CP2 c ¼(cid:2)cos (cid:8) sin (cid:10)
32
resin.Becausethebulkpolymermoleculesandthepoly-
c ¼cos (cid:10) ð27Þ
mer molecules included in the effective fiber are physi- 33
cally entangled, perfect bonding between the effective
fiberandthesurroundingpolymermatrixwasassumed. In general, the orientation average of the dilute
Toaddressscale-up,themicromechanics-basedMori– mechanical strain concentration tensor is [39]
Tanakamethod[34]wasusedtopredicttheelasticstiff-
Ð(cid:13) Ð(cid:13)Ð(cid:13)=2A(cid:7)fð(cid:8);(cid:10); Þ lð(cid:8); Þ sinð(cid:10)Þd(cid:8)d(cid:10)d
Table4 (cid:14)Af(cid:15)¼ (cid:2)(cid:13) 0Ð(cid:13)0 Ð(cid:13)Ð(cid:13)=2 lð(cid:8); Þ sinð(cid:10)Þd(cid:8)d(cid:10)d ð28Þ
Effective-fiber independent parameters and elastic stiffness compo- (cid:2)(cid:13) 0 0
nents(GPa)
Gf ¼4:4 Cf ¼457:6 wherelð(cid:8); Þis theorientation distributionfunction
T 11
Kf ¼9:9 Cf ¼8:4
GTf ¼27:0 C1f2¼14:3 lð(cid:8); Þ¼exp(cid:17)(cid:2)s1(cid:8)2(cid:18)exp(cid:17)(cid:2)s2 2(cid:18) ð29Þ
L 22
Ef ¼450:4 Cf ¼5:5
L 23 and where s and s are factors that control the orien-
Cf ¼27:0 1 2
44 tation.Three cases considered inthis paper are