pubs.acs.org/Macromolecules Published on Web 02/01/2010 r 2010 American Chemical Society

2048 Macromolecules 2010, 43, 2048–2053

DOI: 10.1021/ma902230a

Rheology of Carbon Nanotube Networks During Gelation

D. T. N. Chen,† K. Chen,† L. A. Hough,‡ M. F. Islam,§ and A. G. Yodh*,†

†Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396,‡Complex Fluids Laboratory, Rhodia Inc., Bristol, Pennsylvania 19007, and §Department of ChemicalEngineering and Department of Materials Science and Engineering, Carnegie Mellon University,Pittsburgh, Pennsylvania 15213-3890

Received October 7, 2009; Revised Manuscript Received December 10, 2009

ABSTRACT: We reportmeasurements of gelling rigid rod networks composed of a semidilute dispersion ofsurfactant-stabilized carbon nanotubes.Microrheology is employed to follow the rheological evolution of thesuspension from a semidilute solution of unbonded tubes to a bonded gel network with finite yield stress. Therheological data at various time intervals during gelation is readily collapsed onto a single time-curesuperposition master curve. A theoretical model based on the crossing probability of rods confined to finitevolumes is developed to account for network elasticity. Model predictions compare well with computersimulation and experiments as a function of nanotube volume fraction1 and cure time.

Introduction

Filamentous networks play a crucial role in many biologicaland materials contexts. In living cells, for example, networks ofbiopolymers facilitate processes such as cell division andmotility.Understanding the macroscopic mechanical properties of suchnetworks, even in vitro, is challenging because of a complexinterplay between the flexural rigidity of constituent filamentsand interfilament interactions such as cross-linking. To date, themost intensively studied model systems are semiflexible filamentnetworks, such as those composed of F-actin wherein entropicstretching of individual filaments dominates network linear andnonlinear viscoelasticity.2,3 Rigid rod networks, by contrast, arerelatively unexplored and should differ from their semiflexiblecounterparts as a result of enthalpic effects associated withbending, compression, and interfilament bonding. Carbon na-notube networks present opportunities to explore these latterissues. In addition, interest in carbon nanotube networks hasgrown as a result of their technological potential in compositematerials.4-6 These applications often depend on network con-nectivity, thus corroborating the need for a better understandingof network formation in this system class.

In this contribution, we employ a combination of rheologicalmeasurements, analytic theory, and computer simulation to in-vestigate network formation in aqueous dispersions of single wallcarbon nanotubes (SWNTs). On the experimental side, an aqu-eous dispersion of SWNTs in surfactant is prepared, and, overtime, the SWNTs cross-link because of strong localized van derWaals interactions at contact.1 As the dispersion ages, clusters ofbonded SWNTs form and eventually percolate across the sample,driving its rheological response from that of aNewtonian fluid to agel. Microrheological measurements were made on this system atvarious time points along the sol-gel transition. Observation oftime-resolved “rigidity percolation” in this system of fixed SWNTvolume fraction suggests intertube bonding as the dominantcontributor to the elasticity. We demonstrate experimentally thatthe rheology of SWNTs can be scaled onto a single time-curesuperpositionmaster curve, consistent with other gelling systems.7

The time-resolved experiments are closely related to therheometry of fully cured SWNT gels at varying rod volumefractions φ.1 The latter work found that the low-frequency elasticmodulus (G0) exhibited rigidity percolation above a criticalvolume fraction φ* with power law form, that is, G0(φ) ≈ [(φ -φ*)/φ*]2.3. In the present article, we introduce a microscopicmodel to understand this behavior. The model accounts for thenumber of intertube contacts in a static randomly oriented rodnetwork as a function of rod volume fraction, length, anddiameter; it is based on the crossing probability of rods in finitevolumes. An assumption about the relative contributions to theshear modulus of bonds of varying degrees of connectivitypermits derivation of an analytic expression for the scaling ofshear modulus with rod volume fraction.

Finally, we extend the static model to account for time-resolved sol-gel dynamics. By incorporating bonding kineticsinto the staticmodel, we predict the variation of bonding betweenrods as a function of gelation time, t. The new model provides amarked improvement over empirical power law forms that can beand are often used to describe the data. In contrast with previoussimulations8-10 and rheological measurements1,11,12 of rigid rodnetworks, ours is the first study to relate directly the measuredelasticity of a rigid rod system to its bond connectivity. Impor-tantly, the work provides predictions about the connectivity ofrigid rod networks and, potentially, a means for tailoring themechanical, electrical, and thermal properties in materials com-posed of rigid rod networks.

Experimental Section

Materials. Primary experiments were conducted on disper-sions of SWNTs made by the HiPCO process (CarbonNanotechnologies) at volume fraction φ = 0.0027. The nano-tubes were purified and suspended in filtered deionized water(Millipore) withNaDDBS surfactant (SigmaAldrich) followingthe protocol outlined in ref 13. The ratio of SWNT to NaDDBSwas 1:10 by weight. We prepared the dispersion by mechanicalagitation for 6 h in a high-frequency bath sonicator (Cole-Palmermodel 08849-00). A small amount (φ ≈ 0.0001) of fluorescentlylabeled carboxylated polystyrene spheres (Molecular ProbesFluoSpheres) of nominal diameter, 2a = 0.46 μm, was added to

*To whom correspondence should be addressed. E-mail: yodh@physics.upenn.edu.

Article Macromolecules, Vol. 43, No. 4, 2010 2049

the SWNT-NaDDBS dispersion. The samples were then loadedinto a microscope slide and hermetically sealed with optical glue(Norland 63) just prior to each run.

Methods. Particle tracking passive microrheology14,15 wasemployed to follow the rheological evolution of the network.This method is well suited for measuring viscoelastic moduli ofincipient gels because they are generally fragile under shear, andtheir moduli are often too weak to measure using conventionalrheology. The formation of the SWNT bond network wasfollowed by tracking the displacement of ∼100 tracer particlesin the field of view using digital video microscopy.16 In typicalexperiments, 1-5 min of video data were obtained every 30 minover a 4 h period spanning the gelation process. For cure timeslonger than 3 h, the displacement of the tracers was comparableto the experimental noise, and thuswe limited the data presentedherein to 3 h or less cure time.

From the tracer trajectories, we compute tracer particle meansquare displacement (MSD): ÆΔrx2(Δt)æ = ÆΔrx(t,Δt)Δrx(t,Δt)æ,where Δrx(t,Δt) = rx(tþ Δt)- rx(t) is the particle displacementin the x direction during lag time, Δt. Note that we calculatedtwo-point MSDs as well18 and obtained very similar data, buthenceforth only one pointMSD results will be shown because ofits higher statistical resolution at the longest lag times. Owing tothe difficulty of imaging through the strongly absorbing SWNTsuspension and tominimize sample heating, it was necessary forus to use a relatively long camera shutter time of σ= 1/60 s onthe video CCD camera (Hitachi KP-M1) to achieve adequatesignal-to-noise levels in imaging. This led to the introduction ofdynamic errors in the MSD, as described in ref 17. We havefollowed the procedure discussed in ref 17 to correct MSD datafor dynamic error. The details of this procedure can be found inthe Supporting Information. In theMSD results that follow, thedata exhibited are dynamic-error-corrected.

Results and Discussion

As gelation proceeds, both the magnitude and functional formof theMSDchanges. InFigure 1, we exhibit the particleMSD fordifferent waiting times during gelation. For the earliest cure time(t = 10 min), the MSD is linear over the entire measurementwindow, corresponding to a particle diffusing in a Newtonianfluid with viscosity roughly three times larger than that of water.This observation indicates that steric entanglements betweenunbonded SWNTs do not induce non-Newtonian behavior atthis volume fraction. As time progresses, the long lag timebehavior of theMSD changes markedly, becoming progressivelymore subdiffusive at t = 1 h and finally exhibiting a nearly flatplateau at t=3h.Therefore, intertubebondinghas progressed tomodify themedium’s rheological response from purely viscous tostrongly elastic.

To extract the frequency-dependent (i.e.,ω-dependent) viscoe-lastic moduli,G*(ω), from theMSD, we analyze the data using ageneralized form of the Stokes-Einstein relation:G*(ω) = kBT/πaiωÆΔr2(ω)æ, whereΔr2(ω) is theFourier transformof theMSD.We approximate the transform of the MSD using the numericalapproximation schemedetailed in ref 19.Themoduli, exhibited inFigure 2, showclear rheological evidence of the sol-gel transitionin the SWNT network as a function of gelation time. Below thecritical gelation time 1< t*< 2 h, the rheology is dominated bythe loss modulus G0 0(ω). Above t*, the elastic modulus G0(ω)dominates at low frequency. For all gelation times in our data,G0(ω) exhibits a weak frequency dependence (∼ω0.3) character-istic of soft (G0 ≈ 1 Pa) physical gels and of chemical gels ofunbalanced stoichiometry.20 (Note that we expect for strong gels(G0 g 100 Pa) thatG0(f=ω/2ω=1Hz)=G0(ff 0)=G0

0 , whereG0

0 is the plateau modulus.)For times longer than t*, the moduli exhibit a point of cross-

over at which the viscous and elastic components are equal. Thisdefines a crossover modulus, Gc, and crossover frequency, ωc,both of which increase with the gelation time above t*. By scalingthe magnitude ofG byGc and the frequencyω byωc, we find thatthe network moduli exhibit a striking collapse. The main plotof Figure 3 shows the data collapse under time-cure superpo-sition.21,22 The resulting master curve reveals the viscoelasticrelaxation of the NT gel over four decades in frequency. We cancollapse the data from different NT concentrations and surfac-tant ratios onto the same master curve.23

We parametrize the extent of the gelation via the dimensionlesstime parameter ε = |t - t*|/t*. Above the gel point, a zero-frequency finite elasticmodulus appears and increases as a powerlaw with ε. Experimentally, we find Gc ≈ εz, where z ≈ 1.03(Figure 3, upper inset). For all gelation times in our data, ωc iscomparable to ω � 2πf ≈ 6.3 rad/s (Figure 3, lower inset);therefore, Gc and, indeed, the low-frequency elastic modulusG0(t, f = 1 Hz) exhibit very similar scaling with gelation extent.

In any gelling network wherein both ωc and Gc scale as powerlawswith the cure time (insets ofFigure 3), the viscoelasticmodulishould be of similar functional form and should collapse underrescaling. Intuitively, ωc is related to the mean relaxation time ofthe bonded rod clusters and Gc is related to their mean elasticmodulus, both of which scale with the size of the bonded clusters.Therefore, the effect of an increase in the number of bondscorresponds, essentially, to a rescalingof time in the curing gel.Asgelation proceeds (t > t*), the bonds percolate, producing achange in connectivity without reorganization of the network

Figure 1. Mean square displacement for 2a=0.46 μmparticles in φ=0.27 wt % SWNT, 10:1 NaDDBS/SWNT suspension for t = 10 min,1 h, 2 h, and 3 h (top to bottom). Solid line is slope= 1.0, dashed line isslope = 0.12. Figure 2. Viscoelastic moduli G0(ω) (closed symbols) and G0 0(ω) (open

symbols) derived from the MSD. t > 1.5 h are in the gel regime(G0 >G0 0) data. t<1.5 h are in the sol regime (G0 <G0 0) data.Note thatin the gel regime, there exists a crossover point [ωc, Gc] where G

0(ωc) =G0 0(ωc) = Gc indicated by the arrows.

2050 Macromolecules, Vol. 43, No. 4, 2010 Chen et al.

structure. The collapse of the viscoelastic moduli for the curingSWNT network under time-cure superposition highlights thecrucial role of bonding between rods, which we explicate furtherbelow.

Theory Section

Clearly, bonding between rods is the dominant contributor tothe elasticity in the gel because the number of rods is constant intime andG0 increases with time. Here we introduce a microscopictheory that establishes the relationship between elastic modulusG0 and number of contacts, Nc, in the system, first for static andthen dynamic networks. The first part of the theory derives, fromthe crossing probability of rods, a relation defining the number ofcontacts for a given density of randomly oriented rods. We thenderive a relation for the shear modulus given an effective numberof contacts, which is a fraction of all contacts. These results arecorroborated with computer simulations and are used to fit bothstatic1 and dynamic experimental rheology data.

Static Model. The crossing probability of randomly or-iented rods confined to a finite volume has an expectednumber of contacts, Nc, which purely depends on geometricparameters. A simple calculation yields an expression for thenumber density of contacts. (See the Supporting Informationfor full details of calculation.)

Nc

V¼ C

L2σN2rod

V2≈ φ2 ð1Þ

Here φ= (πNrodLσ2)/4V is the volume fraction of rods, and

rod diameter, σ, is assumed to be constant. Equation 1predicts the number of contacts in a randomly orientednetwork of rods as a function of the number of rods, Nrod,rod length, L, and rod diameter, σ. (Note that similar L2σscaling is found in the excluded volume analysis of percola-tion at large rod aspect ratios.24) The volume of the samplespace is V.

Because we cannot directly observe the bonding betweennanoscale rods in solution, computer simulations are em-ployed to test the predictions of this theory. We constructstatic networks of monodisperse rods of length, L = 10,diameter, σ=0.05, and aspect ratio,L/σ=200, chosen to becomparable to the SWNTs in our experiments. (Note thatthe SWNTs used in the experiments are polydisperse in

length. We also carried out simulations for rods with lengthsdrawn from a Gaussian distribution of comparable poly-dispersity to the SWNTs used in the experiments; a signifi-cant deviation of Nc/V from the results for monodisperserods was not found.) The results that follow are fromsimulations of monodisperse rods.

Simulations.Rods are deposited randomly (off-lattice) in a3D periodic cube with linear dimension l ¼ 20-40. Then,we determine whether the randomly deposited rod centralaxes approach one another within a prescribed distance.Physically, we choose this distance to be the rod diameter.A contact is said to form between two rods when the distancebetween their points of closest separation is less than or equalto the rod diameter. Note that this definition for contactpermits rods to interpenetrate (i.e., soft core). A snapshottaken from a simulation is given in Figure 4, where darkspheres mark the points of intersection between rods locatedby the algorithm.

To test the scaling prediction of eq 1, we varied the samplevolume (V ranged from 203 to 403) and rod length (L rangedfrom 5 to 10) while keeping rod diameter constant (σ=0.05)in the simulations. In Figure 5A, we plot the number ofcontacts, Nc, versus the total number of rods, Nrod, in oursimulations. Rescaling the Nc by V/L

2 in accordance with eq 1yields a collapse of the data, as shown in Figure 5B. Thiscollapse validates the first piece of our theoretical model forthe crossing probability of rigid rods. We next extend themodel in two successive steps: first, we derive the macro-scopic shear modulus from consideration of only elasticallyeffective bonds, and, second, we derive the temporal evolu-tion of elasticity assuming first-order bonding kinetics.

Some bonds do not contribute to the shear modulus of thenetwork; for example, some rods will have only a singlebond, and these noncontributing bonds need to be excludedwhen the shear modulus is computed. Physically, thesebonds are akin to “dangling” strands in polymer melts.25

In Figure 6B, we illustrate two types of bonds that occur in acluster of rods. The bonds denoted by circles belong to a pairof rods that are both connected to other rods, that is,multiply connected bonds. The bonds denoted by starsbelong to a pair of rods for which one of the rods is notconnected to any other rods, that is, singly connected bonds.Physically, we expect only the multiply connected bonds torespond elastically under shear and thus to contribute to themeasured shear modulus in a rheology measurement, asdepicted in Figure 6D. We define an exclusion probabilityPexc = Ns/Nc, where Ns is the number of noncontributing

Figure 3. Collapsed rheologicalmaster curve obtained by scalingG0(ω)(closed symbols) and G0 0(ω) (open symbols) by their respective cross-over frequency ωc and modulus Gc. Upper inset: Gc versus gelationextent, scaling asGc≈ [(t- t*)/t*]1.03 with t*= 6777 s. Lower inset:ωc

versus gelation extent, scaling as ωc ≈ [(t- t*)/t*]0.66 with t*= 6957 s.

Figure 4. Snapshot of simulation for N = 100 rods of aspect ratioL/σ = 200 confined to a volume V = 103. Spheres indicate contactsbetween rods.

Article Macromolecules, Vol. 43, No. 4, 2010 2051

bonds.When the volume fraction is low, almost all bonds arenoncontributing bonds. The exclusion probability decreases

as the packing fraction increases because it is progressivelymore difficult for a rod or a cluster of rods to be isolated fromthe rest of the sample. We extract the volume fractiondependence of the Pexc from a simulated network with V =203,L=10, and σ=0.05. The results, exhibited in Figure 6A,show that the exclusion probability is well approximatedby an exponential function: Pexc = e-B(φ-φ*)/φ*, where φ* isthe volume fraction at which the sample starts to develop ashear modulus and B is a dimensionless parameter charac-terizing the rate of decrease in noncontributing bonds withincreasing φ. Note that this theoretical form is one of severalpossible functions; here we chose a natural form with aminimum number of free parameters.

Therefore, the density of bonds that contribute to thesample shear modulus is

Nc0

V¼ Nc

Vð1-PexcÞ ¼ Aφ2ð1-e-Bðφ-φ

�=φ

�ÞÞ ð2Þ

where A is a constant of proportionality. The number ofelastically effective contacts (Nc

0 ) is given by the total numberof contactsminus the number of single bonds (i.e.,Nc

0 =Nc-Ns). From the simulation data of Figure 6A, we obtain thenumber of effective contacts and plot it versus the volumefraction of rods. The results, exhibited in Figure 6C, showthat Nc

0 is well fit by eq 2.

Comparison with Rheology Experiments

Our previous rheologicalmeasurements yielded a scaling of thelow frequency elastic modulus G0(φ,f = 1 Hz) with rod volumefraction that was well described by the critical power law formA[(φ- φ*)/φ*]βwithφ*=0.0027( 0.0002 and β=2.3( 0.1.1 Itis worth noting that the simple power law Aφ2 does not fit therheology data at all, confirming that the data are in a regime(indicated in the shaded areas of Figure 6A,C), where we expect arelatively high fraction of single bonds to have a significant effecton themeasured shearmodulus. It follows that ifG0 ≈Nc

0 , then eq2 should also fit the volume-fraction-dependent G0 rheologicaldata with only a different constant of proportionality. Indeed, asFigure 7 attests, we find comparable fit quality when comparingeq 2 againstA[(φ- φ*)/φ*]β for the rheological data of ref 1.Notethat both expressions have three free parameters. Whereas thecritical power law form ismore commonly used to fit scaling data

Figure 5. (A) Number of contacts versus number of rods in thesimulation box. Data shown are for different box volumes and rodlengths. (B) Data collapse under rescaling of contact number by V/L2.Solid line is slope 2.0.

Figure 6. (A) Ratio of number of single bonds to number of totalcontacts (Ns/Nc) versus volume fraction from simulation. Solid line is fitto e-B(φ-φ*)/φ* with φ* = (1.0 ( 0.1) � 10-3 and B = 0.345 ( 0.036.Shaded region corresponds to concentration regime of rheology data inFigure 7. (B) Cartoon of rod network showing multiply connectedbonds (circles) and single bonds (stars). Multiply connected rods areblack. (C) Number of contacts with single bonds removed (Nc

0 = Nc -Ns) versus (φ - φ*)/φ* from simulation. Solid line is fit to eq 2 (see thetext) withA=(781.5( 2.2)� 105, φ*= (8.35( 1.46)� 10-4, andB=0.253 ( 0.063. (D) Cartoon illustrating that only the nonsingle bondscontribute to an elastic response under shear.

Figure 7. Low-frequency elastic modulus G0(f = 1 Hz) versus volumefraction from rheology. Data is taken from ref 1. Dashed line is fit toeq2withφ*=0.0028( 0.0001andB=0.053( 0.007. Solid line is fit tocritical power law A[(φ- φ*)/φ*]β with φ*= 0.0027( 0.0002 and β=2.3 ( 0.1.

2052 Macromolecules, Vol. 43, No. 4, 2010 Chen et al.

for gelation, it is largely empirical. Equation 2, on the other hand,has been derived from the crossing probability of rods in aconfined geometry, augmented with minimal assumptions aboutthe relative contribution to the shear modulus from bonds withdiffering degrees of connectivity. The discrepancy between thevalues of φ andB obtained from fitting eq 2 to simulations (φ*=8.35( 1.46)� 10-4, B=0.253( 0.068) and experiments (φ*=0.0028( 0.0001, B=0.053( 0.007) is likely due to the fact thatour model does not exclude the bonds in higher-order structuressuch as nonspanning clusters and dangling closed loops. In a realnetwork, these structures will not contribute to elasticity, result-ing in a higher value for φ*, consistent with our fitted values forφ*. Instead, we have focused on excluding the simplest structures(single bonds), which, while sidestepping complicated considera-tions such as finite-size effects, may have come at the expense ofexact quantitative agreement between φ* and B between thesimulations and experiments.

Comparison with Microrheology Experiments

To compare with the dynamic results from the present micro-rheology experiments, we extend our theoretical model for staticrod networks to account for the time evolution of rod bonding inan adequately dispersed sample. At any given time, Nf free rodsare not bonded to any other rods, and Nb rods are bonded Nf þNb=Ntot. At t=0, we takeNf =Ntot andNb= 0. Conversely,at t= ¥, we take Nb = Ntot and Nf = 0. The rate of bonding isproportional to the number of free rods that are actively seekingbonds and to the total number of rods that are candidates foradditional bonding. Accordingly, the time dependence of bondformation is given by the rate equation dNf/dt = -γNfNtot,where γ is the bonding rate. Integrating the rate equation andapplying boundary conditions yields the number of bonded rodsas a function of time

Nb ¼ Ntotð1-e-γNtottÞ ð3ÞSubstituting Nb for Nrod in the static analysis of eqs 1 and 2

yields the time evolution of the low-frequency elastic modulus

G0 ¼ Aφ2ð1-e-γφtÞ2½1-e-Bðφð1-e-γφtÞ-φ�=φ

�Þ� ð4Þ

Equation 4 suggests that G0 will eventually saturate (i.e., G0 fAφ2 as tf ¥) when all possible bonding rods are exhausted. Theelastic modulus G0(t,f= 1Hz) for different cure times, shown in

Figure 8 can be fit by a power law form A[(t - t*)/t*]z with z =1.3 ( 0.2. This is not surprising because the sample is ratherdilute, and the time it takes for G0 to saturate lies outside ourexperimental window. Physically, however, G0 must saturate onapproach to its fully cured value, corresponding to the modulusat which all available rods are bonded. Clearly, this saturationbehavior is not captured in the power law, which grows indefi-nitely (G0 fA(t/t*)z as tf¥). Therefore, the power law is at bestan empirical local approximation to a saturating functional form.We can fit the microrheology data equally well to either eq 4 orthe power lawA[(t- t*)/t*]z, as shown inFigure 8, because of thelimited dynamic range of the data. In fitting eq 4, we have fixedφ* = 0.0028 and B = 0.053, the values extracted from therheology data fitting of Figure 7. As a result, both functionalforms have three parameters. In principle, we could havefurther constrained φ in eq 4. However, to account for modulusvariations between the two data sets due to sample preparation, itwas necessary to let φ vary. Nonetheless, the nearly indistinguish-able fitting over the dynamic range of our data suggests thatits time dependence is well captured by ourmodel.Measurementsfor longer cure times are clearly needed to test conclusivelyeq 4.

Conclusions

We have performed microrheological measurements of thegelation of a semidilute suspension of single-wall carbon nano-tubes. The results implicate intertube bonding as the dominantcontributor to elasticity in the system. To elucidate the quanti-tative dependence of the number of bonds on geometric para-meters characterizing the rods, we have derived an expression,based on the crossing probability of rods confined to a finitevolume, that yields the dependence of the number of contacts onthe density, length, and diameter of the constituent rods. Therelation is shown to be in agreement with the scaling of thenumber of contacts for simulated rigid rod networks. To makeconnection with the shear modulus measured in rheology experi-ments,wehave assumed that only the fraction of bondsbelongingtomultiply connected rods contributes to the network’s elasticity.With this assumption, we derived a relation that fits the staticmacro- and dynamic microrheological data with a goodness-of-fit comparable to empirically derived critical power laws. Futurerheological measurements or detailed finite element simulationswith larger dynamic range are needed to test decisively themodels.

Acknowledgment. We acknowledge stimulating discussionswith Mateusz Bryning, Andy Lau, Tom Lubensky, and BrianDiDonna. This work has been partially supported by the NSFMRSEC (DMR-0505048 and DMR 05-20020) and NASA(NAG-2939).

Supporting Information Available: Detailed discussion ofdynamic error correction procedure for theMSDdata. Detailedderivation of eq 1 in the main text. This material is available freeof charge via the Internet at http://pubs.acs.org.

References and Notes

(1) Hough, L. A.; Islam,M. F.; Janmey, P. A.; Yodh, A. G.Phys. Rev.Lett. 2004, 93, 168102.

(2) Gardel, M. L.; Shin, J. H.; MacKintosh, F. C.; Mahadevan, L.;Matsudaira, P.; Weitz, D. A. Science 2004, 304, 1301–1305.

(3) Storm, C.; Pastore, J. J.; MacKintosh, F. C.; Lubensky, T. C.;Janmey, P. A. Nature 2005, 435, 191–194.

(4) Geng, H. Z.; Rosen, R.; Zheng, B.; Shimoda, H.; Fleming, L.; Liu,J.; Zhou, O. Adv. Mater. 2002, 14, 1387–1390.

(5) Huang, Y. Y.; Ahir, S. V.; Terentjev, E. M. Phys. Rev. B 2006, 73,125422.

Figure 8. Low-frequency elastic modulusG0(f=1Hz) versus cure timefrom microrheology. Dashed line is fit to eq 4 with φ= 0.006 ( 0.001,γ = 0.0175 ( 0.007, φ* = 0.0028, and B = 0.053. Solid line is fit topower law A((t - t*)/t*)z with t* = 5793 ( 479 s and z = 1.3 ( 0.2.

Article Macromolecules, Vol. 43, No. 4, 2010 2053

(6) Moniruzzaman, M.; Winey, K. I.Macromolecules 2006, 39, 5194–5205.

(7) Larsen, T. H.; Furst, E. M. Phys. Rev. Lett. 2008, 100, 146001.(8) Wilhelm, J.; Frey, E. Phys. Rev. Lett. 2003, 91, 108103.(9) DiDonna, B. A.; Morse, D. C. arXiv: cond-mat/0703418v1.(10) Foygel, M.; Morris, R. D.; Anez, D.; French, S.; Sobolev, V. L.

Phys. Rev. B 2005, 71, 104201.(11) Hobbie, E. K.; Fry, D. J. J. Chem. Phys. 2007, 126, 124907.(12) Lin, Y. C.; Koenderink, G. H.; MacKintosh, F. C.; Weitz, D. A.

Macromolecules 2007, 40, 7714–7720.(13) Islam,M.F.;Rojas, E.; Bergey,D.M.; Johnson,A.T.;Yodh,A.G.

Nano Lett. 2003, 3, 269–273.(14) Mason, T. G.; Ganesan, K.; van Zanten, J. H.; Wirtz, D.; Kuo,

S. C. Phys. Rev. Lett. 1997, 79, 3282–3285.(15) Chen, D. T.;Weeks, E. R.; Crocker, J. C.; Islam,M. F.; Verma, R.;

Gruber, J.; Levine, A. J.; Lubensky, T. C.; Yodh, A. G. Phys. Rev.Lett. 2003, 90, 108301.

(16) Crocker, J. C.; Grier, D. G. J. Colloid Interface Sci. 1996, 179, 298–310.

(17) Savin, T.; Doyle, P. S. Biophys. J. 2005, 88, 623–638.(18) Crocker, J. C.; Valentine, M. T.; Weeks, E. R.; Gisler, T.;

Kaplan, P. D.; Yodh, A. G.; Weitz, D. A. Phys. Rev. Lett. 2000,85, 888–891.

(19) Dasgupta, B. R.; Tee, S. Y.; Crocker, J. C.; Frisken, B. J.; Weitz,D. A. Phys. Rev. E 2002, 65, 051505.

(20) Curro, J. G.; Pincus, P. Macromolecules 1983, 16, 559–562.(21) Adolf, D.; Martin, J. E. Macromolecules 1990, 23, 3700–3704.(22) Winter, H. H.; Chambon, F. J. Rheol. 1986, 30, 367–382.(23) Chen, D. T. N. Ph.D. Thesis, University of Pennsylvania, Phila-

delphia, PA, 2010.(24) Balberg, I.; Anderson, C. H.; Alexander, S.;Wagner, N.Phys. Rev.

B. 1984, 30, 3933–3943.(25) deGennes, P. G. Scaling Concepts in Polymer Physics; Cornell

University Press: Ithaca, NY, 1990.