Effect of shear on nanoparticle dispersion in polymer melts:A coarse-grained molecular dynamics study

Vibha Kalra, Fernando Escobedo, and Yong Lak Jooa�

School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, New York 14853, USA

�Received 26 June 2009; accepted 4 December 2009; published online 8 January 2010�

Coarse-grained, molecular dynamics �MD� simulations have been conducted to study the effect ofshear flow on polymer nanocomposite systems. In particular, the interactions between differentcomponents have been tuned such that the nanoparticle-nanoparticle attraction is stronger thannanoparticle-polymer interaction, and therefore, the final equilibrium state for such systems is onewith clustered nanoparticles. In the current study, we focus on how shear flow affects the kineticsof particle aggregation at the very initial stages in systems with polymers of different chain lengths.The particle volume fraction and size are kept fixed at 0.1 and 1.7 MD units, respectively. Throughthis work, shear has been shown to significantly slow down nanoparticle aggregation, an effect thatwas found to be a strong function of both polymer chain length and shear rate. To understand ourfindings, a systematic study on effect of shear on particle diffusion and an analysis of relative timescales of different mechanisms causing particle aggregation have been conducted. The aggregationrate obtained from the time scale analysis is in good agreement with that determined from theaggregation time derived from the pair correlation function monitored during simulations. © 2010American Institute of Physics. �doi:10.1063/1.3277671�

I. INTRODUCTION

Polymer nanocomposites offer a new spectrum of mate-rials that combine novel functionalities such as optical, elec-trical, and magnetic, while maintaining the manufacturingand processing flexibility inherent to plastics.1 Often signifi-cant effort must be made to avoid the aggregation of nano-particles �NPs� and thereby ensure a homogeneous disper-sion of NPs in a polymeric material. Typically, one wouldmodify the surface of NPs to enhance the interaction be-tween polymer and filler, and reduce particle-particle attrac-tion. Advanced synthetic strategies have been developed tofine tune the chemical nature of the surface ligands and con-trol NP aggregation.2,3 The modified chemical surface, alongwith other secondary parameters such as ratio of NP size toradius of gyration of polymer matrix chain, aspect ratio andvolume fraction of NPs, will then define the effective inter-particle interactions and their state of dispersion in the poly-mer matrix. However, one needs to be careful before surfacemodifying NPs used to augment properties such as catalytic,magnetic, or electrical. NP surface modification can oftenlead to loss of material properties. For example, it is wellknown that chemical functionalization disrupts the extended� conjugation of carbon nanotubes and reduces the electricalconductivity of isolated nanotubes.4 A second example ofsuch NPs whose surface modification disrupts their function-alities is catalytic NPs, which for example, make the devel-opment of low cost, commercially viable fuel cells contin-gent upon prevention of catalyst NP aggregation and theirdispersion throughout the triple point surfaces. Yet, the de-

velopment of techniques for NP dispersion without sacrific-ing their inherent properties has remained a significant chal-lenge.

Balazs and co-workers5,6 developed a strategy to dis-perse sheets of clay in a polymer matrix using self-consistentfield theory and density functional theory. They added asmall volume fraction of end-functionalized polymer chainsto the polymer/clay composite such that the end group isattracted to the sheets of clay, while the remaining chain isidentical to the bulk polymer chains. Mechanical stressessuch as shear have been shown to cause the breakdown ofNP agglomerates and subsequent individualization and dis-persion of the particles. Anderson et al.7 used molecular dy-namics �MD� simulations to show how shear can break upsilicate tactoids in a polymer matrix by its sliding mecha-nism, similar to a shear-induced breakdown of carbon blacksuperstructure observed during processing of filled rubbers.8

However, no such breakdown has been observed for carbonnanotube aggregates due to strong intertube interaction. Thishas prevented the use of shear to obtain NP dispersions forother applications where particles with unmodified surfaces�and hence strong interparticle interactions� need to be used,such as catalysis. In such systems, if we can devise ways tomanipulate aggregation time scale versus the time scale ofmaterial fabrication/processing �from the solution state�, wecan develop advanced materials with improved dispersion ofNPs.

We recently obtained some exciting results on nanocom-posite nanofibers, where we used weakly modified �makingthem isoprene selective� magnetite NPs withpoly�styrene-b-isoprene� �PS-b-PI� nanofibers.9 These NPsformed aggregates for as small as 1 wt % NP in solvent castfilms and nanorods of PS-b-PI due to strong NP-NP attrac-

a�Author to whom correspondence should be addressed. Electronic mail:ylj2@cornell.edu.

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tion. On the other hand, for the case of nanofibers we wereable to place 20 wt % NPs selectively to the isoprene do-main. We speculated that this is an effect of deformation andfast solvent evaporation during the electrospinning processused to fabricate our materials. We believe that the deforma-tion can have beneficial effects on kinetics of particle aggre-gation. This, combined with fast solvent evaporation that re-stricts the time scale of the process to an order of 10−3 s,helps to freeze the morphology in a state of dispersion. Al-though, such a state of dispersion is the desired one, it isbelieved to be a metastable state given the interaction poten-tials within the system.

In the current study, we investigate the effect of shearflow in controlling the kinetics of the aggregation process ofNPs that have strong interparticle interaction using coarsegrained MD. We would refer to these particles as “self-attracting.” We aim to understand what role shear plays indelaying the aggregation of self-attracting NPs. This studywill be beneficial in designing processes for fabrication ofnanocomposites with well dispersed unmodified NPs, whileretaining the inherent properties of the fillers. With this work,we hope to outline different conditions where shear would bemost beneficial. For this purpose, we study the effects ofpolymer matrix chain length, NP-NP attraction strength andshear rate on NP dispersion and discuss the mechanisms re-sponsible for such effects. In addition, to zero in on the find-ings of this work, we have addressed the diffusion of NPs inpolymer melts under shear flow.

Brochard-Wyart and de Gennes studied the viscosity atsmall length scales in entangled polymer melts.10 They re-ported that the friction experienced by small colloidal par-ticles with diameters smaller than the polymer tube diameteris much lower than the bulk polymer viscosity.10–12 The localfriction depends only on the layers of monomers that rubagainst the particle surface, making it proportional to theparticle area and independent of the molecular weight of thechains or the bulk viscosity. Such a drop in local viscositywould cause increased diffusion coefficients compared tothose predicted by Stokes–Einstein relation. Given the sizeof NPs used in this study, we expect a similar reduction inlocal viscosity. Somoza et al.13 conducted experiments tostudy the diffusion of small molecules in polymers and foundthat diffusivity of small particles is much higher as theysense a “nanoviscosity” which is much smaller than the bulkviscosity. Tuteja et al.14 recently studied the diffusion of cad-mium selenide �CdSe� NPs in a polymeric liquid under qui-escent conditions and found that the NP diffusion is muchhigher than the prediction from Stokes–Einstein relation dueto a decrease in the local friction. Similar results have alsobeen predicted by Ganesan et al.15 However, to the best ofour knowledge, most previous works on diffusion of nano-particles have been conducted under quiescent conditions.Therefore, to understand the effect of shear on flow of NPs,we study their diffusion as a function of shear rate and howthe local friction-effect changes with increasing shear flow.

A proper experimental characterization of polymer nano-composite systems under shear flow is rather complicated asthe physically relevant length scales range from macroscopicdown to molecular dimensions. Molecular dynamics is a

powerful tool that can provide insight into basic scientificphenomena at different length and time scales. Furthermore,it offers direct access to particle coordinates and helps toquantify the state of dispersion, unlike experimental tech-niques where quantification at such levels is a major chal-lenge.

Although shear flow, studied in this work, does not rep-resent dominant elongational deformation, it serves as a firststep toward understanding our results on electrospun nano-composite nanofibers.16 Furthermore, this work will openpathways to utilize shear as a dispersion strategy for self-attracting NPs and optimize processing technologies. In ad-dition, shear being an industrially important process, thisstudy will help to understand how nanocomposites with self-attracting NPs behave under standard industrial processingconditions and will help to integrate optimized conditionsinto existing processes. With this work, we hope to provide astep toward development of generalized principles for dis-persion of arbitrary particles differing in chemical nature,size, and shape.

In Sec. II we describe the manner in which we modeledthe polymer melt and the NPs. We show the pairwise poten-tials for all species, and we provide details on the computa-tional method employed. In Sec. III, we present simulationresults on the polymer nanocomposites with self-attractingNPs under shear flow. In particular, we probe the effect ofpolymer matrix chain length and shear rate on NP state ofdispersion and aggregation kinetics, and quantify our resultsusing system properties such as NP pair correlation functionand nonbonded potential energy. Finally, we suggest mecha-nisms responsible for our findings via studies such as effectof shear on NP diffusion.

II. MODEL AND METHODS

A. Potentials

The polymer chains in the current study are modeled asfully flexible bead-spring chains where the monomers arenever allowed to overlap. The excluded volume interactionsbetween different monomer beads are accounted for by thepurely repulsive, cut, and shifted, Lennard-Jones �LJ� poten-tial which is often referred to as the Weeks-Chandler-Anderson �WCA� potential,17

uREP�r� = 4����

r�12

− ��

r�6� + � = uLJ�r� + �,

r � 21/6� ,

�1�uREP�r� = 0, r � 21/6� ,

where r is the separation distance between beads, and � and� are the LJ parameters, taken to be unity for the sake ofsimplicity. Within a polymer, the neighboring monomers areconnected by a finitely extensible nonlinear elastic �FENE�potential,18

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uFENE�r� = −1

2kRmax

2 ln�1 − � r

Rmax�2� , �2�

where we set the spring constant, k, to 30, and the maximumextensibility, Rmax, to 1.5 as described by Kremer andGrest.19

NPs are modeled as spherical beads with �=1.7 MD units. The interaction between NP and monomer isalso modeled using fully repulsive WCA potential �Eq. �1��with � adjusted according to NP and monomer size. ForNP-NP interaction, we use a slightly modified LJ potentialcut and shifted at r=2.5� �Eq. �3�� to include a repulsivecore for excluded volume and an attractive tail to modelself-attracting NPs. The NP-NP attraction is adjusted usingthe k value in Eq. �2�. For k=1, the equation generates theconventional LJ attractive potential and for k=0, we obtain apurely repulsive WCA potential,

uATT�k� = 4����

r�12

− ��

r�6� + ecut, r � 21/6�

= k�4����

r�12

− ��

r�6� + uLJ�2.5�,

21/6� � r � 2.5�

= 0, r � 2.5� , �3�

where, ecut is the value by which uATT�k� needs to be shiftedto make the potential continuous at r=21/6�. NP-polymerand polymer-polymer interaction is kept fixed for this work

using Eq. �1�. For NP-NP interaction, we use k=0, 0.5, or 0.8in Eq. �3�.

Our current simulation model neglects the effect of NProtational dynamics on the rheological behavior of our sys-tems. Note that although such rotational effects are irrelevantfor thermodynamic behavior, they are important for flow be-havior. It is expected, however, that such effects will becomemore significant the higher the concentration of NPs in thesystem and the higher the shear rate. We assume that the NPsurface functionalization is such that rotation modes tend tobe suppressed and that for the relatively low NP concentra-tion used in this work �10 vol %�, such rotational effects aresmall compared to that of the translational modes. This isclearly an approximation but is expected not to greatly affectthe qualitative response of our systems under shear flow.

B. Thermostat

We used a thermostat that preserves hydrodynamic inter-actions, i.e., the dissipative particle dynamics �DPD� thermo-stat. DPD is a simulation technique that was originally con-ceived to model the interaction of mesoscopic units. Theinteractions between sites are treated as “soft” potentialsmeaning that the mesoscopic units can overlap. It was dem-onstrated by Soddemann et al.19 that DPD thermostat is ef-fective when using “hard” LJ potentials as is being used inthis work. For details on the thermostat, we request thereader to refer to our recent paper on MD simulations ofblock copolymer nanocomposites.16

TABLE I. Simulation parameters.

Parameters Symbol Value �MD units�a

Temperature kBT 1Monomer size � 1NP diameter �P 1.7Chain length N 1–400Bead density � 0.85NP fraction �P 0.1Number of chains M 100–2000MD integration time step �t 0.005–0.01NP-NP attraction parameter k 0, 0.5, 0.8Shear rate 0–0.15

aThe values in MD units are understood to be multiplied by the appropriatecombinations of three independent fundamental units of length, mass, andenergy �Ref. 23�.

TABLE II. Simulation details.

Figure No.Shear rate

Chain length

N

NP-NPattraction parameter

kNP diameter

�P

1 0 10 0.5 1.72, 3 0, 0.05, 0.1 10 0.5 1.7

4 0, 0.1, 0.15 10 0.8 1.75 0, 0.02, 0.08, 0.1, 0.15 10 0 1.7

6a, 7, 8, 9, 10 0 1–400 0 1.76b 0 60 0 1–1.7

6c, 7, 8, 9, 10 0.05 10–200 0 1.76d, 7, 8, 10 0.1 10–200 0 1.7

FIG. 1. Self-attracting NP polymer nanocomposites. �a� Snapshots for =0, =0.05, and 0.1 at t=0 �left� and t=1500 MD units �right�, and 3000MD units. Matrix chain length is N=10 and k=0.5 for NP-NP attraction.Polymer chains have been removed for clarity.

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C. Simple shear flow

In our MD code, we implemented shear flow with theSLLOD algorithm using the following equations ofmotion:20

dri,

dt=

pi,

m

+ y�i, �4�

dpi,

dt= Fi, − py,�i, �5�

where ri,, pi,, and m are the position vector, peculiar mo-mentum, and mass of the th particle/bead, respectively. isthe shear rate, and �i is the unit vector in x direction. By thismethod we can impose a linear velocity profile in the x di-rection with a constant gradient in the y direction �with z asthe neutral direction�. These equations of motion are imple-mented with compatible Lee–Edwards periodic boundaryconditions,21 wherein two opposing periodic images are in-crementally moved in opposite directions consistent with theapplied shear rate. Plots of shear viscosity versus shear ratefor polymeric fluids have been widely published in theliterature,22 and they clearly demonstrate these fluids exhibita Newtonian regime at low shear rates and shear thinning athigher shear rates. The pressure �total stress� tensor can becalculated from simulations using23

Pij =1

V�

nMpi,pj,

m

+ ���

nM

ri,�Fj,�, �6�

where ri,� and Fj,� are the position and force of the thparticle/bead relative to the �th one �i.e., ri,�=ri,−ri,� andFi,�=Fi,−Fi,��.

From this, we can determine the shear viscosity by thefollowing relationship,

= −Pxy

. �7�

D. Computational details

The particle volume fraction was fixed at 0.1 for all sys-tems. The site density, �, was kept fixed at 0.85 and thetemperature, kBT, was kept at 1.0. The velocity verlet algo-rithm was used to integrate the equations of motion with atime step, �t=0.01 for lower shear rates and 0.005 for highshear rates. To make the code efficient, we used a cell listalgorithm.23 The system was initialized such that the chainswere well equilibrated and particles were well dispersed inthe polymer melt as seen from the NP pair correlation func-

FIG. 2. Self-attracting NP polymer nanocomposites. Snapshots of system at=0, =0.05, and 0.1 at different elapse MD units, and 3000 MD units. Forall figure parts, matrix chain length is N=10 and k=0.5 for NP-NP attrac-tion. Polymer chains have been removed for clarity.

FIG. 3. NP pair correlation function at =0.0 �a�, 0.05 �b�, and 0.1 �c� atdifferent elapsed times. For all figure parts, matrix chain length N=10 andk=0.5 for NP-NP attraction. The curves have been normalized with themaximum peak value at =0.0 �a�.

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tion �g�r��.18,23 To calculate the diffusion of NPs during shearflow, we calculated the unconvected mean squared displace-ment of NPs in y and z direction such that

�r2 = �ry2 + rz

2 = 4DNPt , �8�

where DNP is the diffusivity of NPs and t is the time. Alldiffusion coefficients in this work are calculated for a systemwith k=0 �in Eq. �3��, i.e., NP-NP interaction is same asNP-polymer. This, we believe is a valid approximation as wemonitor time scales and dynamics of different systems beforethe actual onset of aggregation. So we do not expect anyeffect of NP-NP attraction on NP diffusion in any system. Wediscuss this in more detail in Sec. III.

NP size used for the current work is 1.7 MD units, whichhas been shown to have an approximate real diameter of1.18 nm by Kairn et al.24 Although quite small, the particlesdo lie in the NP regime. We vary the polymer chain size from1 to 400 and shear rate from 0 to 0.15 MD units. To keep NPvolume fraction at 0.1, we use two NPs �size=1.7� for every90 monomer beads �size=1.0�. To quantify the state of dis-persion of NPs, we use a fixed peak value of NP pair corre-lation function as a reference point, which is identified as theonset of aggregation using simulation snapshots. The aim ofthis project is to devise methods that can help in increasingthe time scale of self-attracting NP aggregation beyond thefabrication time scale itself in order to obtain well dispersedsystems of self-attracting NPs. Therefore, we concentrate onnanocomposite properties such as NP diffusion mostly at theinitial stages before the onset of aggregation and tagg is de-fined as the time elapsed before the onset of aggregation. Theinitial condition used in all runs is a well dispersed system.The idea is to be able to use shear flow in solution basedmaterial fabrication where the NPs are initially well dis-persed �either due to stabilization by solvent in case ofweakly modified particles or due to techniques such as ultra-

sonication used for pure NP solution� and later start to ag-gregate as the solvent evaporates. However, in the currentstudy we use a simplified system by using a polymer meltand not taking into account any solvent evaporation. Table Isummarizes some of the parameters used in the current study.

III. RESULTS AND DISCUSSION

As mentioned above, this work has been initiated as astep toward understanding the effect of deformation in con-trolling �self-attracting� NP location in electrospunnanofibers.9 In addition to providing a reasonable mechanismresponsible for our experimental findings, this work will pro-vide a systematic study on the effect of deformation on poly-mer nanocomposites and establish the possibility of benefi-cial use of flow to fabricate well dispersed NP-basedsystems. Table II gives the overall summary of the simula-tions performed in this study with the figure numbers wherethe results are shown. Please refer to the figure captions andtext for more details.

Figure 1 shows snapshots of polymer nanocompositemelt under quiescent conditions �=0� for N=10 and k=0.5 �NP-NP interaction�. The polymer chains are blankedout for clarity and only NPs are shown in the Figure. TheFigure shows how the particles aggregate into clusters in1500 MD time units. Figure 2 shows similar snapshots forsystems under different shear rates. At t=1500 MD units, notrace of aggregation is seen for both =0.05 and =0.1.Slight onset of aggregation is seen for =0.05 at t=3000,while for =0.1, NPs are still well dispersed. These resultsare quantified in Fig. 3 using pair correlation function fordifferent shear rates at different times. A strong second orderpeak for =0 �at t=1500� shows the formation of NPaggregates.25 The absence of a second order peak clearlyshows that the particles are well dispersed for =0.1 even at

FIG. 4. �a� NP pair correlation function at = �left�0.0, �center� 0.1, and �right� 0.15. �b� Nonbonded potential energy between NP pairs at different shear ratesas a function of time. Matrix chain length N=10 and k=0.8 for NP-NP attraction. The curves have been normalized with the maximum peak value at =0.0 �a�.

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t=3000, while a small second peak for =0.05 at t=3000shows the onset of aggregation. These results demonstratethat shear can significantly slow down the aggregation pro-cess, and the higher the shear rate, the slower the aggregationis.

We note that under shear flow the pair correlation func-tion peaks always grow with time and stabilize once all par-ticles cluster into a single aggregate. This shows that noshear induced rupture or breakage of particle clusters takesplace. This effect, we believe, is a strong function of NP-NPattraction and shear rate. For the interparticle attraction �rela-tive to shear rate� used in this work, adhesive forces betweenany two particles overpower the shear forces, resisting anybreakage of clusters. The shear effect, in such a system, isthe most prominent in the initial stages of the aggregationprocess where the NPs are relatively far away from eachother and the process is diffusion limited. Such a system isexperimentally very relevant, since the surface energies ofmost inorganic fillers are many orders of magnitude higher

than polymer surface energies, making the interparticle at-tractions very strong and resistant to any breakage once clus-ters start to form.

Similar studies were conducted for stronger NP-NP at-traction �k=0.8�. Figure 4 shows the pair correlation functionfor different shear rates at various times for k=0.8. The ma-trix chain length is kept fixed at N=10. For =0.1, the sheareffect is not as drastic as for the k=0.5 system due to stron-ger NP-NP attraction. A significant shear induced dispersionof particles can be achieved for =0.15. The results can befurther quantified by tracking the interaction energy betweenNPs. As we see in Fig. 4�b�, the nonbonded potential energybetween NP pairs decrease with time as NPs start to mini-mize the distance between them. The potential energy curveof NPs also presents the energetic reason favoring NP aggre-gation and shows that shear changes the kinetics of this en-ergetically favorable process. Since the simulations are run ata constant temperature, the drop in potential energy is aneffect of clustering only.

We speculate two main effects of shear, first being de-formation leading to increase in instantaneous interparticledistance and second being the effect of shear on NP diffusioncoefficients. One or both of these effects would play a roledepending on the time scale of shear versus the time scale ofNP diffusion in a particular system.

The first effect can be viewed as being analogous toshear induced breakdown observed during processing offilled rubbers, except here this rupture is taking place whenparticles are still not part of any cluster. For the particularparameters in this work, as we mentioned above, a real rup-ture is not seen once a cluster is formed. This rupturelikeeffect of shear is anticipated as positive in all conditionsleading to slower aggregation. This effect is expected to in-crease with increasing shear rate and be independent of thematrix chain length. However, for the second effect, a de-tailed study would be very interesting and beneficial. To un-derstand the effect of shear on NP diffusion, we first plottedmean squared displacement of NPs in a polymer matrix withN=10 for shear rate=0 to 0.15 in Fig. 5�a�. Figure 5�b�shows the calculated diffusion coefficient values for NPs asfunction of shear rate using the slope of the curves from Fig.5�a� and Eq. �8�. We find that the diffusivity increases withshear rate. Diffusion constants in liquids have been shown tosignificantly increase under shear flow.26–28 This effect hasalso been seen for systems of particles suspended in liquidsboth using experiments29–31 and simulations.32,33 Malandro etal.34 conducted nonequilibrium MD simulations to study theself-diffusion of liquids. They showed that local minima ofpotential energy disappears under shear strains, leading tomechanical instabilities forcing the system toward alternatelocal minima causing a shear induced enhancement in diffu-sion. Due to the small size of the NPs,13 their diffusion canbe compared to that of liquids and an enhancement of diffu-sion under shear for our system is reasonable. In the currentstudy, we focus on the interplay between matrix chain lengthand shear rate in the NP diffusion. This is particularly impor-tant as most industrially relevant polymers have extremelyhigh degree of polymerization and are way beyond the en-tanglement length �for bead spring model the entanglement

FIG. 5. Diffusion measurements for system with N=10 and k=0. �a� Un-convected NP mean squared displacement vs time for different shear rates.�b� Diffusion calculated using Eq. �8� as a function of shear rate.

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length is N�35�.18 Once we conclusively establish the effectof shear on NP diffusion in various matrix chain lengths, theimplication of enhanced NP diffusion on aggregation kineticsis investigated. To understand the effect of various param-eters on kinetics of NP aggregation, we study the diffusion ofNPs in polymer matrices with N varying from 1 to 400.

We would again like to point out here that diffusion co-efficients in this work �as shown below� are calculated for asystem with k=0 �in Eq. �3��, i.e., NP-NP interaction is sameas NP-polymer. This, we believe, is a valid approximation aswe monitor time scales and dynamics of different systemsbefore the actual onset of aggregation. In addition, for allsystems, the time taken by NPs to even start aggregating ismuch larger than the diffusion time scales; therefore, we as-sume that at time scales relevant to diffusion, the NPs do notfeel attracted to each other.

Figure 6�a� shows a mean squared displacement of NPsin various matrices for =0. As shown in the Figure, thediffusion rate decreases with increasing chain length initiallyup to N=20. This is expected due to a rise in viscosity withincreasing chain length. A curve of diffusivity versus inverseof viscosity �not shown� for N=1–20 falls on a straight line,in accordance with Stokes–Einstein relation. However, theinteresting observation in this Figure is that beyond N=20,

FIG. 6. Unconvected NP mean squared displacement for different matrix chain lengths as a function of time �k=0� �a� =0, �c� =0.05, and �d� =0.1. Part�b� shows the diffusion coefficient values for different NP diameters at =0 and N=60.

FIG. 7. Diffusion coefficient as a function of matrix chain length �N� fordifferent shear rates.

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all curves collapse on the same line, in spite of significantincrease in the bulk polymer viscosity. Previous theoreticaland experimental works10–12,14 have reported that for par-ticles smaller than the polymer “tube diameter,” the localfriction is dependent only on a layer of monomers rubbingnext to the particle surface, and is therefore proportional tothe NP area and is independent of the full chain length. A

remarkable prediction of this theory is that the diffusion rateof such a small particle will be independent of the polymerchain length or bulk viscosity. Based on previous work, webelieve that the diffusion of NPs in a polymer melt shouldbecome constant beyond a certain chain length N=Ncrit �suchthat the area spanned by a NP of diameter d=cross-sectional area of the polymer chain with N=Ncrit

��Rgyr�2�. This explains the constant diffusion rate for higherN values after an initial drop in diffusivity with chain size.We plotted the diffusivity of NPs with different diameters forN=60��Ncrit� at =0 and found that diffusivity is inverselyproportional to the NP area ��d2� as expected �Fig. 6�b��.Next, we studied the effect of chain size on NP diffusionunder shear flow. Figures 6�c� and 6�d� show the meansquared displacement for NPs with diameter=1.7 at =0.05 and 0.1, respectively, for different N. The diffusionmonotonically decreases with increasing N up to N=100 andbecomes constant beyond that for =0.05, while for =0.1,diffusion monotonically decreases for the entire range of Nstudied. The result in Figs. 6�a�, 6�c�, and 6�d� is summarizedin Fig. 7 by plotting diffusivity as a function of N for differ-ent shear rates. From the results of Fig. 7, it appears thatdiffusion rate versus N under shear flow follows the sametrend as for quiescent system, except Ncrit shifts to a largervalue, indicating that the area spanned by the NPs is a lotlarger under shear flow than under quiescent conditions.Therefore, for example, for the case of =0.05, the totalmatrix or bulk viscosity due to long chained polymers makesa significant contribution to the NP diffusion rate up toNcrit=100.

For a closer look at the effect of shear on NP diffusionrates, we revisited the study we did in Fig. 5, except this timeit is for N=150�N�Ncrit�. Figures 8�a� and 8�b� show the NPdiffusivity measurements in a polymer melt with N=150 fora range of =0–0.15. The diffusivity initially decreases upto =0.05 and then starts to increase. From the results ofFigs. 5, 7, and 8, we speculate that there are two main factorsthat play a role in defining NP diffusivity in polymer meltsunder shear flow; first is shear enhanced diffusion as we sawin Fig. 5 and second is the effect of local matrix viscosity,which increases with shear rate as the area spanned by theNPs increases, and therefore, this effect works to reduce thediffusion rate as a function of shear. The latter effect can beseen only for large chain polymers. Figure 6 shows that forN=150, the second force is dominant for lower shear rates,while for higher shear rates, the first effect takes over. In Fig.8�c�, we plot the diffusion versus shear rate for differentmatrix chain lengths. Again we find that for short chains thediffusion monotonically increases with shear rate due toshear enhanced diffusion, while for longer chains we see aninitial decrease in diffusion due to the increase in matrixviscosity, followed by shear enhanced diffusion.

Finally, to zero in on the effect of diffusion and defor-mation during shear flow on the kinetics of NP aggregation,we do a time scale analysis as a function of both chain lengthand shear rate for a fixed NP-NP attractive potential �k=0.5�. We define three time scales in this study, tdiff, tshear,and tagg. tdiff is the time scale of diffusion and is �R2 /DNP,where R and DNP are the radius and diffusivity of NPs. tshear

FIG. 8. �a� Unconvected NP mean squared displacement for a polymernanocomposites system with N=150 and k=0 at different shear rates as afunction of time. �b� Diffusion coefficient calculated using slope of part �a�using Eq. �8� as a function of shear rate. �c� Diffusion coefficient vs shearrate for different matrix chain lengths.

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is the time scale of deformation and is �1 / . tagg is the timeelapsed before the onset of aggregation for a system with k=0.5. This is determined using a reference value of the paircorrelation function �fixed using snapshots of various nano-composite systems�. We find that for the systems studied inthis work, tagg is at least 1–2 orders of magnitude higher thantdiff or tshear. Hence, at the time scales where diffusion andshear deformation play a role, system does not see any effectof NP-NP attraction, thereby behaving like a homogeneoussystem with equal interaction between different components.This leads us to the assumption that for the time period be-fore any real clusters are seen, the main forces responsiblefor the state of the system or for tagg are solely diffusion andshear deformation. Based on this assumption, we predict theaggregation rates of various systems taking into account thetime scales of NP diffusion and deformation only and com-pare these predicted values to the actual relative aggregationrates calculated using tagg. Again, what we mean by aggre-gation rate here is the rate at which the onset of aggregationtakes place.

In Fig. 9�a�, we plot the tdiff and tshear as a function ofshear rate for N=10. For N=10, at low shear rates tdiff

� tshear. Both tdiff and tshear decrease with shear rate as ex-

pected; however, tshear decreases at a much faster rate and for�0.03, tshear� tdiff. As we mentioned above, at the timescales of diffusion and shear, the system appears to be ho-mogeneous with no effect of NP-NP attractions. Therefore,we expect diffusion to be a homogenizing force, helping theparticles to eliminate any concentration differences and dis-perse uniformly. For a similar reason, shear/deformation isexpected to be a rupture-like positive force and homogenizethe system at time scales dependent on the shear rate. How-ever, it still needs to be established which force �diffusion orshear� plays a dominant role in different scenarios. For thiswe define, another time scale= thomogenizing defined as thedominant force responsible for homogenizing the system. Asseen from Fig. 9�a� �left�, thomogenizing can be taken= tdiff, for�0.03, while for �0.03, thomogenizing= tshear. Due to thehomogenizing character of both these forces, we expect thetime scale of aggregation to be proportional to 1 / thomogenizing

and the aggregation rate to be proportional to thomogenizing. InFig. 9�a� �right�, we plot the nondimensional “predicted” ag-gregation rate as thomogenizing�� / thomogenizing�=0�, wherethomogenizing�=0� is the normalizing factor. On the samecurve, we also plot the nondimensional “calculated” aggre-

FIG. 9. Time scale analysis of various forces and events during aggregation of NPs under shear flow as a function of shear rate, �a� N=10 and �b� N=150.In both figure parts, the left curve shows the time scales of shear/deformation and diffusion and the right curve shows the “predicted” and “calculated”normalized nondimensional aggregation rates of NPs. Please see text for explanation of various terms.

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gation rate. This value is obtained from tagg at different shearrates, normalized using tagg at =0. Here, the nondimen-sional “calculated” aggregation rate is computed as tagg�=0� / tagg�� since the aggregation rate is inversely propor-tional to the time elapsed. We use the term “calculated” astagg is obtained from the actual simulation snapshots. As seenin Fig. 9�a� �right�, curves of both aggregation rates qualita-tively follow a similar trend. However, there are deviationsin aggregation rate values, particularly at smaller shear rates.

This result suggests that at lower shear rates, where tagg isrelatively smaller, the difference between tagg and tdiff / tshear

reduces. In such a scenario, the assumption that NP-NP at-traction does not play a role might be an oversimplification.

Figure 9�b� shows a similar time scale analysis for N=150. Here we find that tshear is the homogenizing force forthe entire range of shear rates studied and we again see that“predicted” and “calculated” aggregation rates follow thesame trend. Figure 10 shows the time scale analysis as a

FIG. 10. Time scale analysis of various forces and events during aggregation of NPs under shear flow as a function of matrix chain length �a� =0.02, �b�=0.05, and �c� =0.1. In all figure parts, the left curve shows the time scales of shear/deformation and diffusion and the right curve shows the “predicted”and “calculated” normalized nondimensional aggregation rates of NPs. Please see text for explanation of various terms.

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function of chain length N at three different shear rates �=0.02,0.05,0.1�. Here, tshear remains constant with chainlength, while tdiff follows the trend, as already seen in Fig. 7.We note that the dimensionless aggregation rate for =0.02,0.05,0.1 remains almost constant as the chain lengthN increases and the constant value of dimensionless aggre-gation rate decreases with increasing shear rate, indicatingthat shear delays aggregation kinetics at the initial stage.Again, we see qualitatively similar trends for “predicted” and“calculated” aggregation rates. Deviations in quantitativecomparisons again indicate the importance of including theeffect of NP-NP attraction in the time scale analysis.

IV. CONCLUSIONS

In this work, we have conducted a study on how shearflow affects the dispersion state of a polymer nanocompositesystem, in which NP-NP attraction exceeds NP-polymer in-teraction. For the systems studied, the final state is always anaggregated particle cluster as it minimizes the energy of thesystem. However, as seen in this work, shear flow can greatlyalter the kinetics of NP aggregation and can increase the timescale of aggregation by up to two orders of magnitude. Wefind that the effect of shear is two-fold: first is effect of shearon NP diffusion coefficient and second is the rupture-likedeformation caused due to the shear. To zero in on these twomechanisms, we varied different parameters such as shearrate and polymer chain length. We find that effect of shearflow on NP diffusion is a strong function of polymer chainlength and therefore can affect the aggregation time scale. Atime scale analysis was done, comparing the time scales ofdifferent mechanisms responsible for aggregation/homogenization. This study provides a first step toward un-derstanding how shear flow can be used to an advantage toobtain well dispersed states even in cases where NP-NP at-tractions are strong. A more detailed study is needed answer-ing questions, such as the following; �1� how does NP diffu-sion vary with changing NP-NP attraction potential?, �2� canfaster diffusion lead to faster aggregation in certain sce-narios?, �3� can shear cause the particles to remain dispersedeven at steady state in a certain scenario?, �4� in what cases,does shear lead to faster aggregation?.

Further studies, particularly on effect of changingNP-NP attraction and NP volume fraction �or NP-NP dis-tance�, are underway.

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