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The Effect of Nanoparticle Shape on Polymer-Nanocomposite Rheology and Tensile StrengthScott T. Knauert

Jack F. Douglas

Francis W. StarrWesleyan University, [email protected]

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Recommended CitationKnauert, Scott T.; Douglas, Jack F.; and Starr, Francis W., "The Effect of Nanoparticle Shape on Polymer-Nanocomposite Rheologyand Tensile Strength" (2007). Division III Faculty Publications. Paper 236.http://wesscholar.wesleyan.edu/div3facpubs/236

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The Effect of Nanoparticle Shape on Polymer-NanocompositeRheology and Tensile Strength

SCOTT T. KNAUERT,1 JACK F. DOUGLAS,2 FRANCIS W. STARR1

1Department of Physics, Wesleyan University, Middletown, Connecticut 06459

2Polymers Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899

Received 2 February 2007; revised 22 February 2007; accepted 23 February 2007DOI: 10.1002/polb.21176Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: Nanoparticles can influence the properties of polymer materials by a varietyof mechanisms. With fullerene, carbon nanotube, and clay or graphene sheet nanocom-posites in mind, we investigate how particle shape influences the melt shear viscosity! and the tensile strength " , which we determine via molecular dynamics simula-tions. Our simulations of compact (icosahedral), tube or rod-like, and sheet-like modelnanoparticles, all at a volume fraction # ! 0.05, indicate an order of magnitudeincrease in the viscosity ! relative to the pure melt. This finding evidently can not beexplained by continuum hydrodynamics and we provide evidence that the ! increasein our model nanocomposites has its origin in chain bridging between the nanopar-ticles. We find that this increase is the largest for the rod-like nanoparticles andleast for the sheet-like nanoparticles. Curiously, the enhancements of ! and " exhibitopposite trends with increasing chain length N and with particle shape anisotropy.Evidently, the concept of bridging chains alone cannot account for the increase in "

and we suggest that the deformability or flexibility of the sheet nanoparticles con-tributes to nanocomposite strength and toughness by reducing the relative value ofthe Poisson ratio of the composite. The molecular dynamics simulations in the presentwork focus on the reference case where the modification of the melt structure associ-ated with glass-formation and entanglement interactions should not be an issue. Sincemany applications require good particle dispersion, we also focus on the case where thepolymer-particle interactions favor nanoparticle dispersion. Our simulations point to asubstantial contribution of nanoparticle shape to both mechanical and processing prop-erties of polymer nanocomposites. © 2007 Wiley Periodicals, Inc." J Polym Sci Part B: PolymPhys 45: 1882–1897, 2007Keywords: mechanical properties; modeling; molecular dynamics; nanocomposites;rheology

INTRODUCTION

Blends of polymers and nanoparticles, commonlycalled “polymer nanocomposites” (PNC), have

Correspondence to: F. W. Starr (E-mail: [email protected])Journal of Polymer Science: Part B: Polymer Physics, Vol. 45, 1882–1897 (2007)© 2007 Wiley Periodicals, Inc. "This article is a US Government work and, as such,is in the public domain in the United States of America.

garnered much attention due to the possibility ofdramatic improvements of polymeric propertieswith the addition of a relatively small fractionof nanoparticles.1–7 Successfully making use ofthese materials depends upon a firm understand-ing of both their mechanical and flow properties.Numerous computational and theoretical stud-ies have examined the clustering and networkformation of nanoparticles and their effect onboth the structural and rheological properties ofPNCs.8–26 The vast majority of these efforts have

1882

SHAPE EFFECTS ON NANOCOMPOSITE PROPERTIES 1883

focused on nanoparticles that are either spheri-cal, polyhedral or otherwise relatively symmetric,although there are some notable exceptions.19–21,23

In contrast, experiments have tended to empha-size highly asymmetric nanoparticles,27–38 suchas layered silicates or carbon nanotubes. It isgenerally appreciated that these highly asymmet-ric nanoparticles have the potential to be evenmore effective than spherical (or nearly spheri-cal) nanoparticles in changing the properties of thepolymer matrix to which they are added. In addi-tion to the large enhancements in viscosity andshear modulus expected from continuum hydrody-namic and elasticity theories, extended nanoparti-cles can more easily form network structures boththrough direct interaction between the nanopar-ticles, or through chain bridging between thenanoparticles,11,16,17,39 where a “bridging” chainis a chain in contact with at least two differentnanoparticles. These non-continuum mechanismsare believed to play a significant role in propertyenhancement, though the dominant mechanismdepends on the properties considered, particle-polymer and particle-particle interactions, samplepreparation, etc.

Given that the majority of previous computa-tional efforts have focused on symmetric nanopar-ticles, we wish to elucidate the role of nanoparticleshape in determining basic material properties,such as the viscosity !, and material “strength”,(i.e., breaking stress). Computer simulations arewell suited to examine the role of nanoparticleshape, since it is possible to probe the effectsof changing the shape without the alteration ofthe intermolecular interactions. As a result, thechanges due to nanoparticle shape can be iso-lated from other effects. Such a task is compli-cated experimentally, since it is difficult to modifythe shape of a nanoparticle without dramati-cally altering its intermolecular interactions. Inthis paper, we evaluate the viscosity ! and ulti-mate isotropic tensile strength " of model PNCsystems with either (i) highly symmetric icosa-hedral nanoparticles (compact particles), (ii) elon-gated rod-like nanoparticles, and (iii) sheet-likenanoparticles. These nanoparticles can be thoughtof as idealizations of common nanoparticles, suchas gold nanoparticles and fullerenes (polyhedral),nanotubes and fibers, and nanoclay and graphenesheet materials, respectively. Our results arebased on molecular dynamics (MD) computer sim-ulations, using non-equilibrium methods to eval-uate !,40,41 and exploit the “inherent structure”formalism to determine " .42,43 We find that the

rod-like nanoparticles give the largest enhance-ment to !, which we correlate with the presenceof chains that bridge between the nanoparticles.The sheet nanoparticles offer the weakest increasein !, and correspondingly have the smallest frac-tion of bridging chains. For the ultimate isotropicstrength " , we find opposite results: the sheets pro-vide the greatest reinforcement, while the rods theleast. For both of these properties, the propertychanges induced by the icosahedral nanoparticlesfall between those of the extended nanoparticles.

The present simulations are idealized mix-tures of polymers and nanoparticles in whichthe polymer-nanoparticle interactions are highlyfavorable so as to promote nanoparticle disper-sion. Moreover, we have chosen to work at rel-atively high temperature in order to avoid con-tributions to ! from the complex physics of theslowing dynamics that arise from approachingthe glass transition. Previous work9,10 has shownthat polymer-surface interaction effects in this lowtemperature range can alter, and potentially dom-inate the nanocomposite properties. We also limitthe range of chain length N studied to avoid effectsof significant polymer entanglement. These lim-itations on interaction, temperature, and chainlength are advantageous in order to develop aclear understanding of the origin of the observedchanges in properties. Such a reference calculationprovides a starting point to understand behaviorwhen these constraints are relaxed. With this inmind, caution is needed when comparing theseresults with experimental data where these com-plicating additional factors may be present – alongwith other possible effects, such as crystallizationor phase separation.

We organize this paper as follows: we firstdescribe the details of the model and method,focusing on the differences between the nanopar-ticle types used in each system. The CompositeRheology section describes our investigation ofthe rheological properties of the nanocomposites,while the Isotropic Tensile Strength section con-siders the effects of shape on " and a discussion ofthe general significance of our results is given inthe Conclusion section.

SIMULATION

To directly compare to experiments, it is desirableto use as realistic a molecular model as possi-ble. While a chemically accurate MD simulationis possible in principle, it is often more difficult to

Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb

1884 KNAUERT, DOUGLAS, AND STARR

identify basic physical trends with such models.Such attempts at chemical realism are alsodemanding in terms of the computational timesrequired, which restricts the class of problemsthat can be investigated. Coarse-grained modelsof polymeric materials provide a good compromisebetween the opposing needs of realism and com-putational feasibility. Such models can reproducequalitative experimental trends of nanocompos-ites, but precise quantitative predictions cannotbe expected.26 Building on nanocomposite modelsintroduced before,8–10,44 we study a coarse-grainedmodel for polymers45,46 and nanoparticles that

allow us to consider PNC systems over a widerange of physically interesting conditions. Herewe consider several nanoparticle shapes built byconnecting spherical force sites.

We perform MD simulations of systems consist-ing of a small fraction of model nanoparticles in adense polymer melt. For reference we also simu-late the corresponding systems of pure polymers.The polymers are modeled via the common “bead-spring” approach, where polymers are representedby chains of monomers (beads) connected by bondpotentials (springs).47 All monomers interact via amodified Lennard Jones (LJ) potential

VP#P(rij) =

!"#

"$4$P#P

%&%

rij

'12

#&

%

rij

'6(

# VLJ(rc) # dVLJ

drij

))))rc

(rij # rc) : rij $ rc

0 : rij > rc

, (1)

where rij is the distance between two monomers,where $P#P is the depth of the well of the LJpotential and % is the monomer size. The poten-tial is truncated and shifted at rc = 2.5 % , sothat the potential and force are continuous atthe cutoff. Bonded monomers in a chain interactvia a finitely extensible, non-linear elastic (FENE)spring potential45,46

VFENE(rij) = #k&

R20

2

'ln

%

1 #&

rij

R0

'2(

, (2)

where k and R0 are adjustable parameters thathave been chosen as in refs. 45, 46. Since we donot aim to study a specific polymer, we use reducedunits where % = $P#P = m = 1 (m is the monomermass). Length is defined in dimensionless unitsrelative to % , time in units of %

%m/& and temper-

ature T is expressed in units of &/kB, where kB isBoltzmann’s constant.

We use three different types of nanoparticlesfor our calculations. Figure 1 shows representa-tive images of the nanoparticles. The first type ofnanoparticle, an icosahedron, was previously stud-ied in refs. 8–10 which focused on factors control-ling nanoparticle dispersion and low temperatureeffects on transport for a similar polymer matrix.Practical realizations of polyhedral nanoparticlesinclude fullerene particles, primary carbon blackparticles, quantum dots, and metal nanoparticleadditives. The nanoparticle force sites interactwith each other via an identical VP#P given in

eq 1, with % = $P#P = m = 1. To maintain theicosahedral shape, the force site at each vertex isbonded to its 5 nearest neighbors via a harmonicspring potential

Vharm(rij) = #'r2

0

2

&rij

r0# 1

'2

(3)

where ' = 60 and r0 equals the minimum ofthe force-shifted LJ potential, approximately 21/6.To further reinforce the icosahedral geometry, acentral particle is bonded to the vertices withthe potential of eq 3 with the same value of ',but with a slightly smaller preferred bond length,r&

0 = 1/4(10 + 2%

5)1/2r0, the radius of the spherecircumscribed around an icosahedron. The result-ing nanoparticles have some flexibility, but arelargely rigid, thereby preserving their icosahedralshape.

The second type of nanoparticle is a semiflexiblerod represented by 10 LJ force sites with neigh-boring monomers bonded by the same VFENE asused for the polymers. We choose the shape torepresent nanoparticles such as carbon nanotubesor nanofibers. Carbon nanotubes typically havelengths up to several µm and a diameter of 1 nm to2 nm (single-walled tubes) or 2 nm to 25 nm (mul-tiwalled tubes can have even larger diameters).48

Unfortunately, such large rods are not feasibleto simulate, as the system size needed to avoidfinite size effects exceeds current computationalresources. As a compromise, we simulate rods with



a length-to-diameter ratio of 10. There is an addi-tional bond potential between nanoparticle forcesites

Vlin(() = 'lin(1 + cos () (4)

where ( is the angle between three consecutiveforce sites. This imparts stiffness to the rod. Wechoose 'lin = 50, so that like the icosahedra, therods have some flexibility, but are largely rigid.

The third type of nanoparticle is square “sheet”comprised of 100 force sites. While the sheets areobviously different from the rods, the sheets havethe same aspect ratio as the rods, where aspectratio is defined by the ratio of the largest andsmallest length scales. Aspect ratio is often con-sidered to be one of the most important propertiesfor anisotropic particles, apart from the interac-tions with the surrounding matrix. The sheetsrepresent a coarse-grained model of clay-silicatenanoparticles or graphene sheets,23 and in the lit-erature these objects are also termed “tetheredmembranes”.9,10,45,46,49 Each monomer in the 10 '10 array has non-bonded interactions described bythe same LJ forces as the polymers. The monomersare also bonded to their 4 nearest neighbors via thesame FENE bond potential of eq 4. The particlesat edges and corners of the sheet are only bondedto 3 and 2 neighboring particles, respectively. Like

the rods, the sheets are stiffened with a Vlin poten-tial to prevent the nanoparticle from folding inon itself. The sheets also include a perpendicularbonding potential

V((() = k(2

*( # )

2

+2(5)

which limits distortions from a square geometry.Without the potential V(, the sheet can deforminto a rhombus. As in ref. 23, we choose 'lin = 10and k( = 100 for the sheets.

Thus far, we have not defined a nanoparticle-monomer potential. In previous work,8–10 the sameLJ potential of eq 1 was used for nanoparticle-monomer interactions, with the well depth $P#P

replaced by $N#P. The soft 1r6 form of the attrac-

tion makes it relatively easy for monomers ofpolymers which are first and second nearest neigh-bors of a nanoparticle to exchange. As a result,the chains can “slide” relatively easily along thenanoparticle surface, thereby reducing the ben-efit of networks built by chain bridging.50 Tomake monomer exchange at the surface less favor-able, we use a “12–24” potential V12#24 ratherthan the standard “6-12” powers of VLJ. Thus,the the nanoparticle-monomer interactions are ofthe form

VN#P(rij) =

!"#

"$4$N#P

%&%

rij

'24

#&

%

rij

'12(

# V12#24(rc) # dV12#24

drij

))))rc

(rij # rc) : rij $ rc

0 : rij > rc

, (6)

where % = 1 and rc = 2.5% . This potentialhas stronger forces binding the first neighbormonomers to the nanoparticle. So that the totalenergy of the potential well, i.e. 4)

, rc%

VN#Pdr, isroughly the same as that of the LJ potential usedin refs. 9 and 10, we choose $N#P = 3. In this way,the total potential energy is comparable to thatused in refs. 9 and 10 with $N#P = 1.5, which led towell-dispersed nanoparticles in that study. Thus,we expect our systems will be well-dispersed.However, we shall see this is not entirely the casefor the systems with sheet nanoparticles. Figure 2shows snapshots of a typical configuration for eachsystem studied.

We generate initial configurations using thesame approach as in refs. 9 and 10, namelygrowing vacancies in the pure melt so that the

nanoparticles can be accommodated. However,this process generates artificial initial configu-rations, and so we generate subsequent “seed”configurations by simulating at T = 5 where reor-ganization occurs on relatively short time scales.Depending on the nanoparticle type, we extractindependent starting configurations every 105 to106 time steps. Subsequently, we make the neces-sary changes in density and chain length beforecooling and relaxing at T = 2. A possible con-cern is that at T = 2, monomers will stick to thenanoparticle surface on a time scale that is longcompared to the simulation, since $N#P = 3; how-ever, we confirmed that chain monomers exchangewith the nanoparticle surface many times over thesimulation. The equilibration time depends on thesystem type and is directly related to particle size.



Figure 1. The three different types of nanoparticles used in simulation: (a) icosahe-dron, (b) rod,and (c) sheet.The nanoparticle force sites are rendered as spheres connectedby cylinders representing FENE bonds.

Figure 2. Typical equilibrium configurations of each of the three nanocomposite sys-tems: (a) icosahedra, (b) rods, and (c) sheets. In each image all nanoparticles are shown;for clarity of the figure, only 10% of the polymers are shown.



Pure systems relax relatively quickly,needing only105 time steps. Icosahedral systems need roughly106 time steps before reaching thermodynamicequilibrium. The rods and sheet nanoparticles aremuch larger, and thus diffuse more slowly. As aresult, these systems require in excess of 5 ' 106

time steps to reach equilibrium. Given that equi-libration requires more than 106 time steps atT = 2, equilibration under conditions of entan-glement or supercooling would be computationallyprohibitive.

We integrate the equations of motion via thereversible reference system propagator algorithm(rRESPA), a multiple time step algorithm usedto improve simulation speed.40 We use a basictime step of 0.002 for a 3-cycle velocity Verletversion of rRESPA with forces divided into “fast”bonded (VFENE, Vharm, Vlin, V() and “slow” non-bonded (VP#P,VN#P) components. The temperatureis controlled using the Nose-Hoover method wherethe adjustable “mass” of the thermostat is selectedto match the intrinsic frequency obtained fromtheoretical calculations of a face-centered cubic LJsystem.

To study rheological properties, we shear equili-brated configurations using the SSLOD equationsof motion,51 integrated using the same rRESPAalgorithm used for equilibrium simulation. TheSLLOD method generates the velocity profile forCouette flow. If we choose the flow along the x-axisand the gradient of the flow along the y-axis, theshear rate dependent viscosity is given by

! = #)Pxy**̇

(7)

where )Pxy* is the average of the x-y componentsof the pressure tensor (sometimes also called thestress tensor), and *̇ is the shear rate.41 We limitour simulations to small enough shear rates toavoid potential instabilities associated with break-ing FENE bonds in the simulation.52

We choose a loading fraction # ! 0.05, definedby the ratio of the number of nanoparticle forcesites to the total number of system force sites;values of this order of magnitude are common inexperimental studies.30,34–36 Since all force siteshave the same diameter, # should also be roughlyequal to the volume fraction. Due to the fact bothpolymers and nanoparticles consist of a discretenumber of force sites, # varies slightly betweensystems. For the icosahedral systems # = 0.0494,and for the rods and sheets # = 0.0505. The sys-tem size for all three nanoparticle systems is muchlarger than either the radius of gyration of the

Table 1. The Table Details All System VariantsSimulatedSystem Type N NC # n nN

Pure 10 100 0 N/A N/A20 100 0 N/A N/A40 100 0 N/A N/A

Icosahedra 10 400 0.0494 13 1620 200 0.0494 13 1640 100 0.0494 13 16

Rods 10 944 0.0505 10 5020 472 0.0505 10 5040 236 0.0505 10 50

Sheets 10 944 0.0505 100 520 472 0.0505 100 540 236 0.0505 100 5

Listing: Chainlength N, Number of Chains NC, Loading Frac-tion #, Number of Force Sites per Nanoparticle n, and Numberof Nanoparticles nN.

polymers, or the largest nanoparticle dimension,to avoid finite size effects. The number of chainsNC ranges from NC = 10 in the smallest system upto NC = 994 for the largest systems. For each sys-tem we examine chain lengths N = 10, 20, and 40.For this polymer model, the entanglement length! 30 to 40.53 Hence, only the longest chains stud-ied may exhibit any effects of entanglement, andthe effects should be small in the present work. Wereiterate that effects of dynamics due to proximityto the glass transition should not play a signifi-cant role at the relative high T of our systems. Asummary of the system parameters can be foundin Table 1.

COMPOSITE RHEOLOGY

Shear Viscosity !

In this section, we evaluate the role of nanopar-ticle geometry on ! and clarify the influence ofchain bridging on ! for a range of shear rates (*̇ =0.005, 0.007, 0.01, and 0.02) and chain lengths. Inthe nanocomposite simulations of ref. 8, ! becamenearly constant at *̇ = 0.005 – in other words, atthe smallest *̇ , we are approaching the Newton-ian limit where ! is independent of *̇ ; however,our data are still clearly in the shear thinningregime. Ideally, we would estimate ! in the *̇ = 0limit directly—for example, by using an Einsteinor Green-Kubo relation—but the accurate eval-uation of ! using these methods is difficult.13,44

We note that if *̇ is considered in physical units,these rates would be quite large by experimentalstandards.



Figure 3. The viscosity ! as a function of shear rate *̇

for chain length N = 20 at + = 1.00. The rods showthe largest !, followed by the icosahedra, and lastlythe square sheets. We show the statistical uncertaintyof ! for each system at *̇ = 0.005, where the fluc-tuations are largest, and hence represents an upperbound for the relative uncertainty for all ! calculations.The uncertainty is the result of “block averaging”.41

The inset includes the bulk polymer and shows thatthe pure system has much lower viscosity than anyof the nanocomposites by roughly an order of magni-tude. The lines in this and subsequent figures are drawnonly as guide for the reader’s eyes. [Color figure canbe viewed in the online issue, which in available atwww.interscience.wiley.com]

Figure 3 shows ! for the three nanocompositesand pure polymer as a function of *̇ . These datashow a clear change in ! between the systems forall *̇ simulated, with the rods having the largest !,followed by the icosahedra, sheets, and finally thebulk polymer. The decrease of ! with increasing *̇

is indicative of shear thinning. We note that ! isnot constant at the lowest *̇ , indicating we havenot yet reached the Newtonian regime. Nonethe-less, it is clear that the pure polymer has a viscos-ity approximately an order of magnitude smallerthan any of the composites. Comparable differ-ences in ! have been observed in experiments.29,31

This demonstrates an enhancement of ! throughthe addition of nanoparticles, which is often adesired goal of adding nanoparticles to a polymermelt. While the figure only shows ! for the N = 20and + = 1.00, other chain lengths and densitiesindicate the same trend. Figure 4(a) shows theeffect of varying N on ! at + = 1.00 and *̇ = 0.007.An increase in ! with increasing N is expectedfrom basic polymer physics since the chain frictioncoefficient increases with N.47 Interchain inter-actions and “entanglement” interactions enhance

this rate of increase since the friction coefficient ofeach chain increases linearly with N.

While it is clear that addition of nanoparticlesincreases ! of the resulting composite, the reasonsfor this are not obvious. Since the pure polymerhas a relatively low ! it is reasonable to assumethat the rigid nanoparticles themselves inherentlyraise ! as in any suspension of particles in fluidmatrix. To separate out this continuum hydrody-namic effect of the nanoparticles from the effectsspecifically due to polymer-nanoparticle interac-tions, we calculate intrinsic viscosity [!] for theindividual nanoparticles. Specifically, continuumhydrodynamics provides an estimate of the incre-mental change of the reduced viscosity !r(#) =!(#)/! (# = 0) through the addition of particlesto a fluid,

!r(#) = 1 + [!]# + O(#2) (8)

where the velocity of the fluid is taken to be zeroon the surface of the particle (“stick” boundaryconditions). Such expansions are only quantita-tively reliable for # ! 0.01 for nearly sphericalparticles; for larger #, we expect eq 8 to recover

Figure 4. The (a) viscosity ! and (b) reduced viscos-ity !r as a function of chain length N. All values arefrom systems at + = 1.00 and *̇ = 0.007. Chain lengthappears to have no effect on the ordering of ! amongstthe systems. [Color figure can be viewed in the onlineissue, which in available at www.interscience.wiley.com]



only qualitative variations in !r. For reference,the “intrinsic” viscosity [!] = 5/2 for spheres inthree dimensions.54 It is also known that [!] isgenerally larger for isotropically oriented spher-ical shapes.55 Refs. 19–21 and 56 have reviewedthe modeling of polymer composites containingextended particles based on continuum mechan-ics. This work also considers the effect of particleclustering on ! which is important under poordispersion conditions.

To compare the predictions of hydrodynamictheory to our simulation, we use the Zeno packageto calculate [!] for our particles.57 The compu-tational method involves enclosing an arbitrary-shaped object within a sphere and launching ran-dom walks from the border or launch surface. Theprobing trajectories either hit the object or returnto the launch surface. From these path integra-tion calculations, [!] for the rigid particles canbe estimated for objects of essentially arbitraryshape.58 We determine [!] of the nanoparticles asan average over 100 different conformations foundin the equilibrium system as they adopt a range ofshapes due to bond flexibility.

Using Zeno, we find that the icosahedralnanoparticles have [!] = 3.93, the rods have [!] =16.3, and the sheets have [!] = 10.8. The intrinsicviscosity of a regular icosahedron has been esti-mated to be 2.47.58 The discrepancy between thisvalue and our icosahedral nanoparticles arisesfrom the fact that our nanoparticles are not regu-lar solids, but rather consist of a group of sphericalforce sites with icosahedral symmetry. We can useeq 8 to predict !r for the MD simulations, and find(for the N = 20 systems) !r = 1.20 for the icosa-hedra, !r = 1.82 for the rods, and !r = 1.54 forthe sheets. Such a prediction underestimates theN = 20 MD results by 92% for the icosahedra,90% for the rods, and 89% for the sheets. Clearlythis hydrodynamic estimate of !r for our nanocom-posites is not adequate, even accounting for theorder-of-magnitude nature of the estimate basedon eq 8. Consistent with the full MD simulations,the rods have the largest value, but in contrastto the MD simulation, the order of the sheets andicosahedra are reversed. While the larger [!] valueof the rods are generally consistent with a largervalue of !, the reversal of the sheet and icosahe-dra [!] values indicates that nanoparticle and/orpolymer interactions play a role in the overall vis-cosity. Moreover, the continuum theory indicatesthat !r should be independent of N, while we seefrom Figure 4(b) that !r from the MD simulationsis strongly dependent on N. Hence we conclude

that the continuum model of ! does not provide anadequate explanation of the changes we see in oursimulations.

Evidently, we must also examine other contri-butions to the nanoparticle viscosity. One possi-bility that has recently been discussed21 is thatslip rather than stick boundary conditions mightprovide a better description of nanoparticle bound-ary conditions. However, this would result in adecrease of the predicted value of !r

22,44 relativeto the stick case, which is in the wrong directionfor explaining our results. At a molecular level,the boundary conditions are clearly neither sticknor slip, as particles have finite residence timesat the surface, although it is difficult to deter-mine the appropriate hydrodynamic boundaryconditions directly from molecular considerations.Nanoparticle-polymer interactions and nanopar-ticle clustering are evidently factors that mightcause the property enhancements we observe, andin the next section we focus on this possibility.

Relationship of Viscosity to Fluid Structure

We next turn to the structural results obtainedfrom MD simulation to better quantify the rela-tionship of polymer and nanoparticle structureto !. We first calculate the fraction of nanopar-ticles in contact with other nanoparticles, fN#N.Nanoparticles are said to be “in contact” if oneor more comprising force sites of a nanoparti-cle are within the nearest neighbor distance of aforce site belonging to another nanoparticle. Thenearest-neighbor distance is defined by the firstminimum in the radial particle distribution func-tion g(r); the first minima is at r ! 1.5 for allsystems. Figure 5 shows fN#N as a function of N for+ = 1.00 and for both *̇ = 0 and *̇ = 0.007. Whileit is not readily apparent from Figure 2(c), thepolymers “intercalate” between the sheets to somedegree, and hence, the nanoparticles are not actu-ally in direct contract with each other, resulting ina very low value of fN#N for the sheets. With theexception of the N = 40 equilibrium system, thesheets have the fewest nanoparticle-nanoparticlecontacts. For the systems under steady shear, fN#N

evidently stays fairly constant as N varies. Whilethe order of the systems for fN#N is consistent withthe trend for !, the similarity between the rods andicosahedra suggests that the fraction of nanopar-ticles alone is not responsible for determiningcomposite viscosity. Of note, is the large fractionof nanoparticle contacts for the N = 40 sheet sys-tem at equilibrium. We have conducted further



Figure 5. The fraction of nanoparticles in contact withother nanoparticles fN#N as a function of N for each ofthe composite systems at (a) *̇ = 0 and (b) *̇ = 0.007.[Color figure can be viewed in the online issue, which inavailable at www.interscience.wiley.com]

simulations at various + (not shown) that suggestthere is a tendency for the sheets to favor a trulyclustered state at lower density, that is, chains nolonger intercalate between the sheets for N = 40.The lack of intercalating chains may also be a sam-pling issue, since the timescale on which chainsenter or exit between the sheets is a significantfraction of the total simulation time. Thus, we maynot sample the equilibrium fraction of contactsbetween the sheets well, and our average maybe dominated by either an intercalated or non-intercalated state. Nonetheless, it is reasonableto think that the tendency of the sheets to stackis a + dependent phenomena, possibly caused byentropic interactions that result from polymer-depletion induced attractions between the sheets.Such depletion interactions could arise due tophysically adsorbed polymers on the sheet surface,much like the depletion interactions of polymercoated colloids.59 This interaction should be mostpronounced for the sheets, presumably due to thelarge and relatively flat surface of a sheet, whichmore effectively reduces the entropy of chains atthe surface.

Given that nanoparticle–nanoparticle contactsdo not appear to be the origin of the relative

ordering, we next consider the role of polymer-nanoparticle interactions. To gauge the possiblerole of such interactions we calculate the fractionof polymer chains in direct contact with a nano-particle, fN#P. Using the same nearest-neighborcriteria as used to determine nanoparticle-nanoparticle contacts, we compute fN#P as a func-tion of N for + = 1.00 at *̇ = 0 and *̇ = 0.007in Figure 6. We find that fN#P is approximatelythe same for both the quiescent and sheared sys-tems. There are also clear trends: fN#P increaseswith increasing N, and for every value of N thesystem containing the rods has the largest valueof fN#P, followed by the system containing theicosahedra, and finally the system containing thesheets. These same trends in ! appear in Figure 4,suggesting that the nanoparticle-polymer contactsindeed have a significant effect on relative magni-tude of !.

The correlation of ! with fN#P is consistent withthe idea that bridging between nanoparticles playa major role in determining the rheological prop-erties of polymer nanocomposites.11,16,17,39 Thus,we extend our analysis to test for a correlation

Figure 6. The fraction of polymer chains in directcontact with a nanoparticle fN#P shown as function ofchain length N for each of the three nanoparticle sys-tems at (a) *̇ = 0 and (b) *̇ = 0.007. [Color figurecan be viewed in the online issue, which in availableat www.interscience.wiley.com]



Figure 7. The effect of chainlength N on the fractionof bridging chains fB in each of the three nanoparti-cle systems at + = 1.00 for (a) *̇ = 0 and (b) *̇ =0.007. The order of fB with respect to nanoparticle typematches the trend in Fig. 3 for !. [Color figure canbe viewed in the online issue, which in available atwww.interscience.wiley.com]

between ! and chain bridging. We define a “bridg-ing chain” as a chain that is simultaneously incontact with two or more nanoparticles. Figure 7shows fB as a function of N for + = 1.00. The trendin fB is consistent with both fN#P and !, showing anincrease in fB with increasing N as well as a clearordering between systems for every chain length.Although not shown, the results seen in Figure 7occur for every value of + that we have simulated.Hence, the fraction of bridging chains seems to bea useful “order parameter” for characterizing thepolymer-nanoparticle interactions. If the polymer-nanoparticle interactions were sufficiently small,the bridging would not be expected to play sig-nificant role, and it is likely that the continuumhydrodynamic approach would be applicable.

Similar to the notion of bridging is the ideathat the nanoparticles can act as transient cross-linkers that can lead to effects equivalent with theformation of higher molecular weight chains.11,39

To test this idea, we define an “effective chain”as the collection of chains that are connected bynanoparticles. We then define an effective chain

length Neff as the mean mass of chains connectedby the nanoparticles. Figure 8 shows that Neff

is almost an order of magnitude larger than N.Hence, we expect that the largest contribution tothe ! increase comes from this effect. Even inthe simplest Rouse theory, an order of magnitudeincrease in N would lead to an order of magnitudeincrease in !. The formation of longer effectivechains is expected to lead to entanglement interac-tions that would further amplify the ! increase, asemphasize by ref. 60. However, this entanglementcontribution is hard to quantitatively interpret inthe present context.

ISOTROPIC TENSILE STRENGTH "

Calculation of "

The ultimate isotropic tensile strength " of a mate-rial is defined as the maximum tension a homoge-neously stretched material can sustain before frac-ture. While this definition can be directly probedexperimentally, the situation is less straightfor-ward in an MD simulation. Reference 43 hasdeveloped an approach to estimate " based onpotential energy-landscape (PEL) formalism that

Figure 8. The effective chainlength Neff as a func-tion of N in each of the three composite systems at+ = 1.00 for (a) *̇ = 0 and (b) *̇ = 0.007. [Color figurecan be viewed in the online issue, which in available atwww.interscience.wiley.com]



is accessible by MD simulation. Although the PELis complex and multidimensional, one can envi-sion it as a series of energy minima connectedby higher energy transition pathways. By defin-ition, the minima, or inherent structures (IS) aremechanically stable, that is, there is no net force atthe minimum.The method of ref. 43 relates " to themaximum tension the IS can sustain, determinedby an appropriate mapping from equilibrium con-figurations to energy minimized configurations. Inother words, " is the upper bound on the tensionat the breaking point of an ideal glass state in theT + 0 limit. Here we evaluate " for the variouspossible nanoparticle geometries. As a caution-ary note, we point out that this method has notbeen directly validated by comparison with exper-iments; thus conclusions drawn from these resultsshould be considered tentative. Reference 61 dis-cusses an alternate approach using the PEL toevaluate the elastic constants.

To determine " , we must first generate theenergy minimized configurations from the equi-librium configurations. For a system of N atomsin the canonical ensemble, the most commonlyused mapping from equilibrium configurations tothe IS takes each force site along the steepestdescent path of the energy of the system. Thisconfigurational mapping procedure corresponds tothe physical process of instantaneous cooling tothe T + 0 limit to obtain an ideal glass with nokinetic energy. Therefore, by sampling thermallyequilibrated configurations and minimizing theirenergies we can estimate the average inherentstructure pressure, PIS, which will be negativewhen the system is under tension. The maximumtension then defines " . The value of PIS has beenfound to be weakly dependent on the T at whichthe sampling is performed. For example, whilecyclopentane has a strongly temperature depen-dent structure, ref. 62 has shown that equilibriumT only weakly effects PIS. Since the structures inour systems do not display such dependence, weexpect our results also to be independent of thestarting equilibrium T. Thus, we can use the samesystems at T = 2 that we used to determine ! andstill obtain reliable results for " – even thoughthe starting configuration is a highly fluid state.This calculation of " should be an upper boundfor the tensile strength that would be obtained forany finite T system, and hence is referred to theas the ultimate tensile strength. The energy min-imization process eliminates the high frequencyeffects that would normally be associated withsuch high T states and thus the method refers

Figure 9. The inherent structure pressure PIS as afunction of + for the pure polymer and the nanocompos-ite systems at N = 20. The pure, icosahedron, and rodsystems all have a Sastry density +S ! 0.99, while forthe sheets +S ! 0.93. The uncertainty intervals (“errorbars”) represent the statistical uncertainty in our aver-age for PIS from block averaging.41 [Color figure canbe viewed in the online issue, which in available atwww.interscience.wiley.com]

to the strength of an ideal glass material havingessentially no configurational entropy.

Figure 9 shows the PIS curves generated forthe N = 20 variant of each system. We find thatPIS decreases at large +, until a minimum valueis reached. The density of the minimum, referredto as the “Sastry density” +S,42 characterizes thedensity where the system fractures, and voids firststart to appear in the minimized configurations.As + decreases below this point, PIS begins to rise.Since this is the greatest tension achievable, wehave PMAX

IS = #" . For pure, icosahedral, and rodsystems we find +S ! 0.99, while for the sheets it ismuch lower +S ! 0.93. Similarly, " is considerablylarger for the sheets than for the other systems.Wefind that the increase in " for the sheets is not assizable as observed experimentally.29 This may berelated to the fact that (experimentally) the chainsoften form stronger associations with layered sil-icates than in our simulations; additionally, thechain lengths we examine are small compared toexperiments, and we will see that " increases withN. We note that at this chain length, the icosahe-dra and rods actually reduce the strength of thenanocomposite as compared to the pure melt.

To quantify the chain length dependence of " ,we plot the minimum of each PIS curve as a func-tion of N in Figure 10. For the pure polymers,we find that " decreases with increasing chain



Figure 10. Tensile strength " as a function of chainlength N for the pure polymer and the nanocompositesystems. We find that " increases as N increases. Thisis in contrast to the behavior of the pure polymer where" decreases with increasing N. Only for systems withthe sheet nanoparticles—or the system with icosahedralnanoparticles at N , 40—does the addition of nanopar-ticles produce a net benefit relative to the bulk polymer.[Color figure can be viewed in the online issue, which inavailable at www.interscience.wiley.com]

length. A similar decrease of " was observed insimulations of n-alkanes43 and experimentally ithas been found that ultimate tensile strength ofcertain polymeric systems, such as amorphouspolyethylene, decreases with increasing molecularweight.63 Such a decrease in " is non-trivial,since one expects longer chains to exhibit morechain “entanglement”. Such a view neglects theimportance of packing, however, which is cruciallyimportant in the low temperature glass regime.

For n-alkanes, it was argued that these packingeffects play an important role for the behavior of" .43 As chains grow longer, it becomes more diffi-cult to pack effectively due to the constraints ofchain connectivity. The poorer chain packing inturn can lead to an increase in the“fragility”, in thecontext of fragile and strong glass formers.64 Poorchain packing in fragile glass formers is knownto give rise to decreases in the tensile stress atbreakage, and so it is reasonable to expect that "

would decrease with increasing N as the fragilityof glass formation increases. This interpretation ofthe trend in " being related to the fragility of theglass-formation requires further investigation, ofcourse, and we mention this as a tentative expla-nation of the trend in " that we and others haveobserved. Reference 43 also finds a maximum "

for N = 3, and a tendency for +IS to saturatenear N = 8. Since the smallest chain we simu-late is N = 10, we are above the regime wherethis feature occurs. Consistent with these facts, "

decreases with N and +S is roughly independentof N. More importantly, Figure 10 shows that theaddition of the nanoparticles reverses the N depen-dence of " when compared to the pure system.Thus, while the presence of icosahedral or rod-like nanoparticles decreases the material strengthfor most chain lengths studied, " increases withincreasing N, and if this continues, " will sur-pass the pure melt for all nanoparticle shapesat large enough N. Indeed, " for the icosahedrananocomposite already exceeds that of the melt atN = 40.

Relationship of Structure to Tensile Strength

To better understand the predicted N dependenceof " , we perform a parallel analysis to the struc-tural analysis of shear runs discussed above. Theanalysis of Figures 5–7 focuses on + = 1.00. Werelate the structure to " by examining the struc-ture at +S. Evidently, +S for the sheets differs fromthe other systems studied, and it is possible that +

dependent changes in connectivity properties areresponsible for the difference of " . Thus, we cal-culate the quantities fB, fN#N, and fN#P at +S foreach system using the equilibrium and IS config-urations at the Sastry density +S of the system.The results for the equilibrium and IS configura-tions show the same qualitative trends, so we willpresent only data for the IS.

We first focus on fB because our investiga-tions into the rheological properties suggestedthat bridging chains played a large role in deter-mining !. However, when we plot fB in Figure 11for each system at +S, we see that the N depen-dence is the reverse of that seen for " , even thoughthis quantity follows !. The ordering of fB amongthe systems is the same as for fB in the case ofthe ! calculations. The sheet composites have thesmallest fB, followed by those with icosahedra, andthose with rods. While bridging chains do increasewith increasing N, fB does not seem to be a majorfactor in the relative value of " between the com-posites. For example, the N = 10 nanocompositeswith icosahedra and sheets have almost the samevalue of fB, yet they have significantly differentvalues for " .

The fewer bridging chains in the systems withsquare sheets suggests that the sheets may beclustered, and we find that this is indeed the case



Figure 11. The effect of chainlength N on the fractionof bridging chains fB in each of the three composite sys-tems calculated using the inherent structure configura-tions at + = +S. [Color figure can be viewed in the onlineissue, which in available at www.interscience.wiley.com]

for N = 40 [Fig. 5(a)]. Thus, we plot fN#N inFigure 12 and find indeed that fN#N ! 1 for thesheets. The sheets prefer a stacked state at low +

due to entropic interactions with polymers whichleads to a reduced explicit energetic interactionwith the surrounding polymers. In particular, therelative ordering of fN#N matches that of " . Naïvely,this would seem to suggest a potential correlation

Figure 12. The fraction of nanoparticles in contactwith other nanoparticles fN#N as a function of N for eachof the composite systems calculated using the inher-ent structure configurations at + = +S. [Color figurecan be viewed in the online issue, which in availableat www.interscience.wiley.com]

between fN#N and " . However, this apparent corre-lation is problematic. Firstly, the N dependence of" and fN#N are different, namely " monotonicallyincreases,while fN#N is roughly constant. Secondly,if the nanoparticles interactions are the origin ofthe increased strength, then one might expect tofind the largest effect when the nanoparticles’ arewell dispersed.8 This is not the case, since theclustered sheets give the largest effect.

To confirm that the nanoparticles actuallyimpart additional strength, we visually examinedthe location of fracture in our systems at +S. Tofind empty spaces that are at least the size ofa particle, we discretize the system into a cubiclattice of overlapping spheres, each with a radiusrV and a nearest neighbor separation d. We thenidentify spheres that do not contain any systemforce site. By adjusting the parameters rV and dwe can ensure that the voids we find are at leastlarge enough for a single particle. Since the forcesites of both the nanoparticles and polymers havediameter 1, we must choose rV > 0.5 to have phys-ical relevance. We find that values of rV = 0.6 andd = 0.3 work best for visualizing voids.

Figure 13 shows that the voids in the systemoccur in regions of pure polymer for the icosahe-dral system. The same is also true for the rodsand square sheets. Quantitative analysis of theforce sites within nearest-neighbor distance r =1.5 of the voids confirmed this suggestion, with

Figure 13. Typical fractures for the IS of the icosa-hedral nanocomposite. The large contiguous blobs arethe fractures. The surrounding spheres represent forcesites in contact with the rupture. Note that nearlyall sites in contact are polymers. [Color figure canbe viewed in the online issue, which in available atwww.interscience.wiley.com]



Figure 14. The fraction of polymer chains in directcontact with a nanoparticle fN#P as function of chain-length N for each of the three nanoparticle systemscalculated using the inherent structure configurationsat + = +S. [Color figure can be viewed in the online issue,which in available at www.interscience.wiley.com]

over 99% of the resulting force sites belongingto polymer chains. Thus, while the nanoparticlesimpart an increased strength to the nanocom-posite, the correlation to fN#N is not apparentlycausal. Evidently, there is something more subtlecontrolling the magnitude of " in the sheet fillednanocomposite that we have not yet identified.

To complete the structural analysis we calcu-late fN#P for the IS. Figure 14 shows that thesheet nanoparticles are not in contact with manypolymers, consistent with the fact that fN#N ! 1.In fact, after minimization the difference in fN#P

between the rods and the sheets is even largerthan observed in the equilibrium configurationsshown in Figure 6. Figure 14 shows that the rela-tive ordering of fN#P is opposite to that of " , hencethe number of nanoparticle-polymer contactsalone also does not provide a good indicator of " .

A potential clue to the increase of " of the sheetnanocomposite is the fact that +S is significantlysmaller than for the other systems. This allowsthe sheet composites to undergo a greater defor-mation before fracture. This large increase in boththe strength and toughness of the polymer matrixwith incorporation of nanoparticles is reminiscentof the changes in the properties of natural and syn-thetic rubbers with the inclusion of carbon blackand nanofiller additives.39,65–67 Extraordinarilylarge increases in both the strength and tough-ness of materials have recently been observedin the case of exfoliated clay sheets dispersed

in the polymer polyvinylidene fluoride (PVDF).68

Although some of this change is associated withthe modification of the crystallization morphologyby the clay nanoparticles, this does not explainin itself the observed toughening mechanism. Itis known that nanofiller particles can behave astemporary cross-linking agents that impart vis-coelastic characteristics to the fluid. Gersappe39

further emphasizes the role of this transient net-work formation in impeding cavitation events thatinitiate material rupture and the potential sig-nificance of this phenomenon in understandingthe nature of biological adhesives (abalone), fibers(spider silk), and shell material (nacre).69,70 Ger-sappe further suggests that the relative mobilityof nanoparticles has a role on the toughening ofmaterials. However, such reasoning is inconsis-tent with the fact that the sheet nanoparticleslead to the strongest material while being the leastmobile of our additives.

Recent work indicates that flexible sheet-likestructures, such as studied here, are character-ized by a negative Poisson ratio ,,71–73 that is, thematerial expands normal the direction of stretch-ing, as opposed to normal materials which expandin the same direction as the stretching. The for-mation of a composite material with negativePoisson ratio (“auxetic” materials) is expected tolead to a reduction of the Poisson ratio of thecomposite as a whole.74,75 A large reduction in ,

can lead to materials that are strong and frac-ture resistant.76 Related to this, we observe inour simulations that the voids that form in thesheet-filled nanocomposites tend to be fewer andsmaller than those for the other nanoparticlesat +S. As we further reduce + for the sheet sys-tem, the voids grow more numerous as opposedto growing larger as in the other nanocomposites.Microvoid formation has also been established asa mechanism for toughening polymer materials.77

Previous work devoted to understanding the highimpact strength of polycarbonate and other glassypolymers has likewise emphasized the importanceof large “free volume” within the polymer materialas a necessary condition for large toughness.78,79

Based on our simulation results and previousobservations, we tentatively suggest that the addi-tion of these “springy” sheet materials reducesthe effective Poisson ratio of our nanocomposites,and that the microvoid formation process that weobserve is a manifestation of the non-uniformitiesin the elastic constants within the nanocomposite.In principle, the elastic constants can be deter-mined by the PEL approach,61 but this analysis



will be deferred to a future work since it is ratherinvolved. Our tentative interpretation of the pre-dicted " variation in the sheet nanocompositesalso suggests the need for better characterizationof the geometric rigidity properties of the sheetnanoparticles, since these variables may be impor-tant for understating how such particles modifythe stiffness and toughness of nanocomposites.

CONCLUSION

In this work we have focused on how nanopar-ticle shape influences the viscosity of polymer-nanoparticle melt mixtures at high temperatureand the ultimate tensile strength (estimated fromthe PEL formalism) of polymer nanocompositesin the ideal glass state. Our results suggest thatchain bridging between the nanoparticles canhave a large effect on ! of the mixture whenthe polymer-particle interactions are attractive, sothat the nanoparticles disperse readily. In addi-tion, there is a relatively weak increase in ! forsheet nanoparticle composites, which tend to clus-ter in our simulations, supporting the expectationthat nanoparticle clustering diminishes the vis-cosity enhancement. However, the formation of“open” or percolating fractal clusters may havethe opposite effect on the viscosity. Curiously, thetensile strength of the sheet nanocomposite isgreatest, in spite of the sheet stacking. One ofthe most intriguing effects of the nanoparticles isthat, regardless of shape, the dependence of thetensile strength on chain length for the nanocom-posites is opposite to that for the pure polymermelt. We reiterate that our results for " rely onthe PEL approach, and this approach should becarefully compared with experimental measure-ments to test the expected relationship between" from simulations and tensile strength foundexperimentally.

The trends observed for " are more difficult tounderstand than those for !. There is evidentlyno clear-cut correlation of " with the formationof bridging chains or with the attractive particle-particle interactions. In the absence of such a cor-relation, we suggest that the nanoparticles effectthe mechanical properties by modifying the ratioof the bulk and shear moduli in the low tempera-ture nanocomposite state (i.e., the nanocompositePoisson ratio). Recent work has noted that mole-cularly thin sheets have a negative Poisson ratiowhen they are flexible enough to crumple by ther-mal fluctuations. Such additives could reduce the

Poisson ratio of the material as a whole andprovide a potential rationale for interpreting theincreases in both the strength and toughnessobserved in our simulation of sheets dispersed in apolymer matrix, as well as in experiments on claynanocomposites.

The authors thank V. Ganesan, S. Kumar, andK. Schweizer for helpful discussions. We thank the NSFfor support under grant number DMR-0427239.

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