Nanorod Mobility Influences Polymer Diffusion in ... - ACS Publications

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Nanorod Mobility Influences Polymer Diffusion in Polymer Nanocomposites Chia-Chun Lin,† Matteo Cargnello,‡ Christopher B. Murray,†,‡ Nigel Clarke,§ Karen I. Winey,† Robert A. Riggleman,*,∥ and Russell J. Composto*,† †

Department of Materials Science and Engineering, ‡Department of Chemistry, and ∥Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States § Department of Physics and Astronomy, University of Sheffield, Western Bank, Sheffield S10 2TN, United Kingdom S Supporting Information *

ABSTRACT: Polymer diffusion is enhanced when nanorods (NRs) in a polymer nanocomposite are mobile relative to nanocomposites containing immobile NRs. NR mobility is tuned by varying the molecular weight of the matrix polymer and the NR concentration, and diffusion of deuterated polystyrene (dPS) tracers with varying molecular weight is studied using elastic recoil detection. When dPS diffuses faster than the NRs such that the NRs are “immobile” on the time scale of dPS, the tracer diffusion coefficient (D) monotonically decreases as the nanorod concentration increases; we interpret these results as though the nanorods provide additional topological constraints that slow diffusion of dPS. When the tracer diffusion is slow relative to NR diffusion (i.e., “mobile” NRs), diffusion is enhanced relative to the immobile NR case. This enhanced diffusion is captured by a slip-link model with two populations of topological constraints: one fixed population attributed to the PS matrix, and a second population of constraints with a finite constraint release time determined by the diffusion time of the NRs relative to dPS. These experimental and computational results provide fundamental insights into the nature of entanglements and constraint release in NR-containing polymer nanocomposites. anisotropic NPs are “immobile” on the time scale of polymer diffusion, namely, NPs diffuse by less than their size. Recent studies indicate that NP diffusion is faster than the Stokes− Einstein (SE) prediction.15−18 Thus, to understand the role of mobile NPs on the tracer diffusion of polymers, a PNC system where NP diffusion can be controlled is needed. Small mobile nanoparticles can influence polymer dynamics if the transient nature of NP location influences the topological constraints imposed on the polymer. For example, simulations19,20 showed that the entanglement degree of polymerization (Ne) increases as nanoparticles are added, although this observation may depend on how NPs are treated in the primitive path analysis.21 Viscosities of PNCs containing small spherical NPs are generally found to decrease at low NP concentrations; however, above a critical concentration, the viscosity can increase because of network formation.19,20,22,23 In addition, molecular simulations have reported16 that polymer diffusion in the presence of small NPs can increase by 40% relative to the neat polymer. In this Letter, we investigate polymer diffusion in the presence of both mobile and immobile nanorods (NR). At 190 °C, NR diffusion coefficients (DNR) decrease from 10−12 to

C

oatings and bulk materials with unique optical, electrical, mechanical, and magnetic properties can be prepared by combining nanoparticles (NPs) with polymers to form polymer nanocomposites (PNCs).1−4 In addition to adding functionality, nanoparticles can influence fundamental properties such as the diffusion of macromolecules through a PNC. Therefore, by understanding how polymer dynamics are influenced by nanofillers, one can better optimize the processing conditions of these multicomponent materials consisting of hard particles embedded in a relatively soft and flexible matrix.5,6 The diffusion of polymers is also partly responsible for determining whether the NPs in a metastable polymer/NP mixture will aggregate during processing/use or remain dispersed.7,8 Whereas diffusion in PNCs with immobile NPs has received much attention,7,9−13 polymer diffusion in PNCs containing nanoparticles that diffuse on the time scale as the polymer has not yet been reported. Recent studies have focused on polymer diffusion in PNCs containing spherical and cylindrical NPs. For spherical NPs, polymer diffusion decreases monotonically as NP concentration increases for systems with both neutral10 and weakly attractive12 interactions between the particles and the polymer matrix. In contrast, for anisotropic NPs, polymer diffusion initially slows down at low NP concentration, reaches a minimum near a concentration where the NPs percolate, and then increases.9,12−14 In these studies, the spherical and © XXXX American Chemical Society

Received: July 20, 2017 Accepted: July 25, 2017

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ACS Macro Letters nearly 10−15 cm2/s as matrix molecular weight (P) increases from 60k to 2000k.24 Compared to spheres of similar diameter, nanorod diffusion is particularly sensitive to matrix molecular weight because lateral diffusion is coupled to matrix chain dynamics. Because the tracer diffusion coefficient (D) of entangled polymer scales with tracer molecular weight (M) as D ∼ M−2,25,26 polymer and nanoparticle diffusion coefficients can be independently controlled by varying M and P, respectively, or increasing the NR concentration above the overlap concentration. In this way, NR diffusion can be varied to be either faster or slower than tracer polymer diffusion, and we show that the addition of NRs slows tracer polymer diffusion. When NRs are immobile on the time scale of tracer diffusion, the NRs add additional entanglements with the polymer. When the NRs are mobile, however, the reduction in diffusivity is smaller compared to the immobile NR system, which we model through a constraint release mechanism. Phenyl-capped TiO2 nanorods with diameter (d) = 4.5 nm, and length (L) = 31 nm were prepared using a method27,28 described in Supporting Information. The NR surface was functionalized with (chloromethyl)dimethyl phenylsilane. NRs were dispersed in polystyrene (PS; Pn = 650k and 2000k, k = kg/mol) for NR volume fractions, ϕNR = 0.004−0.08 from thermal gravimetric analysis (TGA). The PNCs were preannealed at 150 °C for 72 h. The glass transition temperatures (Tg) of the PNCs were 104 ± 1 °C, similar to pure PS, and consistent with prior studies.10 Diffusion couples were prepared by depositing a 20 nm deuterated PS film (dPS; M = 800k, 1800k and 3200k) on a thick PNC film (>2 μm). After annealing at 190 °C, dPS volume fraction profiles were measured using elastic recoil detection (ERD),29 as shown in Figure S1. The cross-sectional TEM images in Figure 1 demonstrate that NRs are well-dispersed in PS matrices having P = 650k and 2000k for ϕNR from 0.004 to 0.08. At low ϕNR, individual NRs freely diffuse. To estimate the NR overlap concentration, the

volume pervaded by a NR (Vs) is taken as a sphere with diameter = L. The critical NR overlap concentration V ϕc = 0.74 VNR ≈ 0.02 , where VNR is the NR volume and 0.74 s

is the volume fraction for hexagonal close-packed spheres. For NR diffusion within the PNC, free diffusion should be observed at ϕNR < ϕc, whereas NR diffusion should be hindered at ϕNR > ϕc. In the former case, NR diffusion is determined by the local viscosity of the matrix polymer,24 whereas NR motion is slowed/stopped due to interactions with neighboring NRs in the latter. The transition from free to hindered NP diffusion has been observed elsewhere.20,30 In addition to tuning NR mobility by varying P, the range of ϕNR in this study enables polymer tracer diffusion to be directly compared in PNCs where NRs are mobile and immobile. We note that NR diffusion in this study is the translational diffusion of the NRs, which is the weighted average of transverse and longitudinal diffusion.24 The diffusion coefficients of dPS in PNC matrices containing mobile and immobile NRs is presented in Figure 2a,b, where

Figure 2. Tracer diffusion coefficients of dPS at T = 190 °C for M = 800k (squares), 1800k (triangles), and 3200k (circle) in PS(P)/NR, P = 650k (closed symbol) and P = 2000k (open symbol), as a function of ϕNR. Stars represent NR diffusion coefficients when ϕNR → 0 in pure PS with P = 650k (closed) and P = 2000k (open). Representative error bars are shown.

the diffusion coefficients DdPS of dPS decreases monotonically as ϕNR increases from 0.004 to 0.08. We attribute the decrease in DdPS to the ability of the NRs to provide an independent set of entanglements with the tracer polymers, consistent with the findings from previous simulations.31 This behavior is also consistent with previous studies which showed a monotonic decrease in D when polymer size (2Rg) is greater than d and L of anisotropic nanoparticles.9 As expected,7,9,10 at fixed ϕNR, dPS (800k) exhibits the fastest diffusion, though these tracers are still slowed by the NRs. We calculate the probability of a binary contact between a tracer and any NR by calculating the

Figure 1. Cross-sectional TEM images of PS(P)/NR matrices containing TiO2 with dimensions 4 nm × 31 nm. (a) P = 2000k, ϕNR = 0.004; (b) P = 2000k, ϕNR = 0.06; (c) P = 650k, ϕNR = 0.0065; (d) P = 650k, ϕNR = 0.08. Scale bars are 100 nm. 870

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ACS Macro Letters dimensionless concentration of pervaded spheres with radius n L R rperv ≈ 2NR + 2E , Pcontact = ⎛ VNR . At the smallest ϕNR and sys ⎞ ⎜

⎟

⎝ Vperv ⎠

lowest M, we calculate Pcontact ≈ 0.12, confirming that it is reasonable to expect a fraction of the tracers to contact the NRs and, thus, for the NRs to affect tracer diffusion. A significant finding is that the matrix molecular weight also affects the measured diffusivites. For ϕNR < 0.02, dPS (1800k and 3200k) diffuses faster in the composite matrix with the lower P (650k) compared to the higher P (2000k) system. For example, dPS (3200k) diffusion is ∼25% greater at ϕNR = 0.01. However, dPS (800k) diffusion is the same for P = 650k and 2000k. This result is consistent with the suppression of constraint release by matrix chains because P is relatively high (≥650k).32 The changes in the matrix molecular weight do not affect the diffusion of the tracer chains because the diffusion time scales of the tracers and the matrix are not sufficiently separated;33−35 this expectation is explicitly demonstrated for the dPS (3200k) system at ϕNR = 0 for P = 650k and 200k in Figure 2b, which shows that the diffusivites are essentially unchanged. Figure 2a,b also show the NR diffusion coefficients, DNR, in PS matrixes (P = 650k and 2000k). As ϕNR → 0, DdPS for the 800k tracer is about 3× and 10× greater than DNR for P = 650k and 2000k, respectively. In addition, all dPS (800k) data points collapse onto one curve, signifying that NR diffusion does not influence dPS (800k) diffusion, and these NRs are immobile on the time scale of dPS (800k) diffusion. In contrast, for the two higher molecular weight tracers, dPS diffuses faster than NRs in a matrix with P = 2000k, but slower than NRs for P = 650k. When NRs diffuse faster than the tracer dPS, enhanced tracer diffusion is observed. For example, at ϕNR = 0.01, dPS (3200k) diffusion is ∼25% faster in the lower P matrix relative to the higher P matrix. This suggests that mobile NRs do not slow tracer diffusion as much as immobile NRs. To further examine this interplay, we need to carefully define NR mobility. At P = 650k, NR diffusivity at ϕNR = 0 is ∼4 times faster than dPS (3200k) diffusion. Namely, when dPS diffuses by about Rg, NRs diffuse about 2Rg using the characteristic diffusion distance (i.e., x ∼ Dt ), suggesting that NRs can migrate away from a dPS tracer. Similar arguments hold for the dPS (1800k) system. However, when NR diffusion is slightly slower than dPS diffusion, that is, NRs move ∼0.8 Rg as dPS (3200k) moves by Rg, dPS diffusion is dominated by nominally immobile NRs. Accordingly, in this study, when DNR > 2.5DdPS, enhanced dPS diffusion relative to the fixed NR case is observed. Although dPS diffusion is enhanced at dilute concentrations, diffusion in PNCs at concentrations above ϕNR = 0.02 is independent of P. Figure 3 shows the reduced diffusion coefficient, D/D0, where D0 is the diffusion coefficient in a pure PS matrix versus ϕNR. We find that the overlap concentration ϕNR = 0.02 approximately separates free and hindered NR diffusion. Note that dPS (800k) diffusion in both regimes is independent of P, because NRs are immobile on the time scale of polymer diffusion for ϕNR < 0.02. On the other hand, for dPS (1800k and 3200k) and ϕNR ≲ 0.02, NRs are mobile on the time scale of dPS diffusion when P = 650k, and dPS diffusion is enhanced relative to the fixed NR case (P = 2000k). This enhancement in dPS diffusion is absent above ϕNR = 0.02, because NR motion is slowed or quenched due to network formation of NRs, as shown in Figure 1b,d. Moreover, Figure 3 shows that the enhancement in D/D0 increases as the tracer

Figure 3. Normalized tracer diffusion coefficients of dPS for M = (a) 800k, (b) 1800k, and (c) 3200k, in PS(P)/NR where P = 650k (closed symbol) and P = 2000k (open symbol), as a function of NR volume D 1 fraction. The solid lines are fits using D = 1 + αϕ to model systems 0

NR

with immobile nanorods (P = 2000k or ϕNR > ϕc). The dashed line (offset for clarity) represents ϕc ≈ 0.02.

molecular weight increases, because dPS is further slowed down relative to the NRs. Namely, the enhancement is more significant for dPS (3200k) than dPS (1800k), which supports the concept that the relative difference between polymer and NR diffusion is responsible for enhanced polymer diffusion at ϕNR < 0.02. This study presents the first experimental evidence that polymer diffusion is enhanced when NRs in the matrix are mobile relative to the immobile NRs. Simulation studies of PNCs 36−38 provide some insight into the underlying mechanism for mobile-nanoparticle-assisted polymer diffusion by investigating how entanglements between polymers and nanorods reinforce mechanical properties. Using dissipative 871

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ACS Macro Letters particle dynamics (DPD), Karatrantos et al.37 reported that NRs with a radius smaller than Rg increases the entanglement density because NR anisotropy facilitates polymer-NR entanglements.20 When analyzing entanglements, Toepperwein et al.36 showed that removing NRs without modifying the polymer configurations produces an entanglement density similar to the neat polymer, whereas polymer-NR entanglements increased the entanglement density. This result motivated us to treat polymer topological constraints as independent of constraints from NRs, which we take as proportional to ϕNR. In other words, the number of entanglements per chain is Z(ϕNR) = Z0(1 + αϕNR), where Z0 is the average number of entanglements per tracer molecule without NRs and α is a constant of proportionality without NRs and α is a constant of proportionality that depends on the tracer molecular weight and likely the NR geometry. The number of monomers between entanglements is NE(ϕNR ) =

N Z(ϕNR )

=

NE(0) . 1 + αϕNR

Figure 4. Normalized diffusivity plotted against number of polymernanorod entanglements per chain relative to the pure polymer at constraint release rates k = 0, 3.9, 10.9, and 109. The lines are fit to model data (see Figure S2), and filled symbols are experimental results for dPS (1800k, k ≈ 3.9; red diamonds) and dPS (3200k, k ≈ 10.9; blue squares). The mapping of the experiments to Z(ϕNR)/Z(0) is described in the text.

Using the relationship between

diffusion coefficient and number of entanglements per chain from the tube model, the normalized diffusion coefficient is given by

D(ϕNR ) D(0)

=

1 1 + αϕNR

coefficient versus the normalized number of entanglements per chain for constraint release rates of k = 0 (immobile), 3.9, and 10.9, which correspond to the diffusion of dPS (1800k) and dPS (3200k) in PS (P = 650), and a higher rate of k = 109. For dPS (800k), k is ∼0.4, which is nearly equivalent to the immobile case. Due to the additional constraints imposed by NRs, the normalized D decreases monotonically as the entanglements per chain increases for all values of k. However, as k increases, the monotonic decrease becomes weaker because constraints imposed by the NRs are released. Similarly, at a fixed entanglement value, the diffusivity increases as k increases from 0 to 109. This increase in normalized D becomes slightly stronger at higher values of Z(ϕNR)/Z0 because the number of releasable constraints increases. Thus, the relative enhancement of D is stronger for dPS (3200k; red line) compared to dPS (1800k; blue line). To compare the SS model to experiments for dPS (1800k) and dPS (3200k) in PS (P = 650k), α from experiments with immobile NRs (i.e., Figure 4) are used to

when NRs are immobile and do not

release their constraints on the time scale of tracer diffusion. The solid lines in Figure 3 are fits to all diffusion data where we expect NRs to be immobile (for P = 2000k and ϕNR > ϕc for P = 650k), and the experimental data are well-described by this model. When NRs are mobile on the time scale of tracer diffusion, tracer polymer/NR entanglements can be released at a rate faster than the tracer polymer/matrix polymer entanglements. Although it may be possible to use an analytic model33,35 to capture constraint release, the diffusion times of the NRs and the tracers are not well separated and therefore we choose a variation of a slip-spring (SS) model to describe our data. Our model, detailed in the Supporting Information, is a combination of prior models.39,40 Briefly, our model consists of a single chain with Z(ϕNR) = Z0 + ZNR(ϕNR) total slip-links per chain, where Z0 are constraints from entanglements with the matrix and is assumed constant. ZNR additional constraints are added to model the overlap with NRs, and these additional constraints may be released at a finite rate. We define the constraint release rate k by the ratio of the diffusion time for a NR relative to the τpoly tracer polymer, k = τ , and the diffusion times are

determine

τNR =

τpoly =

R g2 Dpoly

Z0

= 1 + αϕNR . The magnitude of the enhance-

ment from the model is comparable to but smaller than that observed experimentally. For example, if k is increased to 109 (i.e., order of magnitude faster diffusion of NRs), the model (green line) better captures the enhanced diffusivity of dPS (3200k). These discrepancies could be due to the assumptions in the model. For example, we implicitly assume that constraint release is linear in Z(ϕNR)/Z0 and that the friction by the coarse-grained monomers is unmodified by nanorods. Because the value of Z0 is relatively small and not allowed to fluctuate the effect of constraint release could increase in a more highly entangled polymer. Nevertheless, the good qualitative agreement between the models supports the insight that mobile nanorods can release any constraints they provide to the tracer polymers. In this Letter, the effect of NR mobility on tracer polymer diffusion was measured in NR-containing polymer nanocomposites. By either varying the molecular weight of both the matrix and the tracer polymers or increasing the NR concentration above the overlap concentration, we varied the relative diffusion coefficients of the NRs to the tracers. The NRs behave as a source of additional entanglements under all conditions; when the NRs are immobile on the time scale of

NR

L2 and D NR

Z(ϕNR )

. When k = 0, SSs are not released

unless the end of the tracer chain passes through the SS. To span the relevant range of Z(ϕNR)/Z0, our SS model uses Z0 = 6, a relatively small number of entanglements per chain. Figure S2 shows the monomer mean-squared displacement used to determine diffusion coefficients. The simulated diffusion coefficient data with k = 0 is well-described by −1 ⎛ D(ϕNR ) ⎡ Z(ϕNR ) ⎤⎟⎞ ⎜1 + α 1 = − ⎢ ⎥ ⎣ Z0 ⎦⎠ , which is equivalent to the D0 ⎝ functional form to describe experiments in Figure 3 if we assume Z(ϕNR) = Z0(1 + αϕNR). To estimate the influence of the relative mobility of the tracer to the NRs, we allow the number of nanorod constraints ZNR to fluctuate as a function of time. At each time step, SSs can be created or destroyed with probability proportional to τpoly/k, where k is taken from experiments and τpoly is from simulations using Z(ϕNR = 0) = Z0. Figure 4 shows the normalized diffusion 872

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(8) Cui, D.; Xu, J.; Zhu, T.; Paradee, G.; Ashok, S.; Gerhold, M. Harvest of near Infrared Light in PbSe Nanocrystal-Polymer Hybrid Photovoltaic Cells. Appl. Phys. Lett. 2006, 88 (18), 183111. (9) Choi, J.; Clarke, N.; Winey, K. I.; Composto, R. J. Fast Polymer Diffusion through Nanocomposites with Anisotropic Particles. ACS Macro Lett. 2014, 3, 886−891. (10) Gam, S.; Meth, J. S.; Zane, S. G.; Chi, C.; Wood, B. a.; Seitz, M. E.; Winey, K. I.; Clarke, N.; Composto, R. J. Macromolecular Diffusion in a Crowded Polymer Nanocomposite. Macromolecules 2011, 44 (9), 3494−3501. (11) Lin, C. C.; Ohno, K.; Clarke, N.; Winey, K. I.; Composto, R. J. Macromolecular Diffusion through a Polymer Matrix with PolymerGrafted Chained Nanoparticles. Macromolecules 2014, 47 (15), 5357− 5364. (12) Lin, C. C.; Gam, S.; Meth, J. S.; Clarke, N.; Winey, K. I.; Composto, R. J. Do Attractive Polymer-Nanoparticle Interactions Retard Polymer Diffusion in Nanocomposites? Macromolecules 2013, 46 (11), 4502−4509. (13) Mu, M.; Composto, R. J.; Clarke, N.; Winey, K. I. Minimum in Diffusion Coefficient with Increasing MWCNT Concentration Requires Tracer Molecules to Be Larger than Nanotubes. Macromolecules 2009, 42 (21), 8365−8369. (14) Mu, M.; Clarke, N.; Composto, R. J.; Winey, K. I. Polymer Diffusion Exhibits a Minimum with Increasing Single-Walled Carbon Nanotube Concentration. Macromolecules 2009, 42 (18), 7091−7097. (15) Grabowski, C. A.; Mukhopadhyay, A. Size Effect of Nanoparticle Diffusion in a Polymer Melt. Macromolecules 2014, 47 (20), 7238− 7242. (16) Kalathi, J. T.; Yamamoto, U.; Schweizer, K. S.; Grest, G. S.; Kumar, S. K. Nanoparticle Diffusion in Polymer Nanocomposites. Phys. Rev. Lett. 2014, 112 (10), 1−5. (17) Tuteja, A.; Mackay, M. E.; Narayanan, S.; Asokan, S.; Wong, M. S. Breakdown of the Continuum Stokes-Einstein Relation for Nanoparticle Diffusion. Nano Lett. 2007, 7 (5), 1276−1281. (18) Yamamoto, U.; Schweizer, K. S. Microscopic Theory of the Long-Time Diffusivity and Intermediate-Time Anomalous Transport of a Nanoparticle in Polymer Melts. Macromolecules 2015, 48 (1), 152−163. (19) Kalathi, J. T.; Grest, G. S.; Kumar, S. K. Universal Viscosity Behavior of Polymer Nanocomposites. Phys. Rev. Lett. 2012, 109 (19), 198301. (20) Li, Y.; Kröger, M.; Liu, W. K. Nanoparticle Effect on the Dynamics of Polymer Chains and Their Entanglement Network. Phys. Rev. Lett. 2012, 109 (11), 118001. (21) Riggleman, R. A.; Toepperwein, G.; Papakonstantopoulos, G. J.; Barrat, J.-L.; de Pablo, J. J. Entanglement Network in Nanoparticle Reinforced Polymers. J. Chem. Phys. 2009, 130 (24), 244903. (22) Mackay, M. E.; Dao, T. T.; Tuteja, A.; Ho, D. L.; van Horn, B.; Kim, H.-C.; Hawker, C. J. Nanoscale Effects Leading to Non-Einsteinlike Decrease in Viscosity. Nat. Mater. 2003, 2 (11), 762−766. (23) Nusser, K.; Schneider, G. J.; Pyckhout-Hintzen, W.; Richter, D. Viscosity Decrease and Reinforcement in Polymer-Silsesquioxane Composites. Macromolecules 2011, 44 (19), 7820−7830. (24) Choi, J.; Cargnello, M.; Murray, C. B.; Clarke, N.; Winey, K. I.; Composto, R. J. Fast Nanorod Diffusion through Entangled Polymer Melts. ACS Macro Lett. 2015, 4 (9), 952−956. (25) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986. (26) Rubinstein, M., Colby, R. H. Polymer Physics; Oxford, 2003. (27) Buonsanti, R.; Grillo, V.; Carlino, E.; Giannini, C.; Kipp, T.; Cingolani, R.; Cozzoli, P. D. Nonhydrolytic Synthesis of High-Quality Anisotropically Shaped Brookite TiO2 Nanocrystals. J. Am. Chem. Soc. 2008, 130 (33), 11223−11233. (28) Gordon, T. R.; Cargnello, M.; Paik, T.; Mangolini, F.; Weber, R. T.; Fornasiero, P.; Murray, C. B. Nonaqueous Synthesis of TiO 2 Nanocrystals Using TiF 4 to Engineer Morphology, Oxygen Vacancy Concentration, and Photocatalytic Activity. J. Am. Chem. Soc. 2012, 134 (15), 6751−6761.

tracer diffusion, these additional entanglements are static, while when the NRs are mobile on the time scale of tracer diffusion, these NR constraints are released at a rate that is approximately given by the relative diffusion times of the NRs to the tracer polymer. This study establishes criteria for how polymer diffusion is enhanced by mobile nanorods relative to immobile nanorods, and it also has practical importance for controlling the morphology of PNCs where particle diffusion can limit the size of aggregates upon phase separation.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.7b00533.

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Experimental details and supporting figures (PDF).

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the National Science Foundation NSF/EPSRC Materials World Network DMR1210379 (R.J.C, K.I.W) the EPSRC EP/5065373/1 (N.C). Support was also provided by the NSF/MRSEC-DMR 1120901 (K.I.W, R.J.C., R.A.R.) and Polymer Programs DMR1507713 (R.J.C), as well as ACS/PRF 54028-ND7 (R.J.C). C.B.M. is grateful for the support of the Richard Perry University Professorship. Computational resources were provided on the National Institute for Computational Sciences at the University of Tennessee through an XSEDE allocation, TG-DMR150034. Discussions with Prof. Kenneth Schweizer (UICU), Prof. Michael Rubinstein (NC), Dr. Umi Yamamoto, Dr. Jeffrey Meth, and Dr. Argyrios Karatrantos are gratefully acknowledged.

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REFERENCES

(1) Balazs, A. C.; Emrick, T.; Russell, T. P. Nanoparticle Polymer Composites: Where Two Small Worlds Meet. Science 2006, 314 (5802), 1107−1110. (2) Talley, C. E.; Jackson, J. B.; Oubre, C.; Grady, N. K.; Hollars, C. W.; Lane, S. M.; Huser, T. R.; Nordlander, P.; Halas, N. J. SurfaceEnhanced Raman Scattering from Individual Au Nanoparticles and Nanoparticle Dimer Substrates. Nano Lett. 2005, 5 (8), 1569−1574. (3) Zeng, H.; Li, J.; Liu, J. P.; Wang, Z. L.; Sun, S. Exchange-Coupled Nanocomposite Magnets by Nanoparticle Self-Assembly. Nature 2002, 420 (6914), 395−398. (4) Winey, K. I.; Vaia, R. A. Polymer Nanocomposites. MRS Bull. 2007, 32 (April), 314−358. (5) Schneider, G. J.; Nusser, K.; Willner, L.; Falus, P.; Richter, D. Dynamics of Entangled Chains in Polymer Nanocomposites. Macromolecules 2011, 44 (15), 5857−5860. (6) Tuteja, A.; Duxbury, P. M.; Mackay, M. E. Multifunctional Nanocomposites with Reduced Viscosity. Macromolecules 2007, 40 (26), 9427−9434. (7) Choi, J.; Hore, M. J. A.; Meth, J. S.; Clarke, N.; Winey, K. I.; Composto, R. J. Universal Scaling of Polymer Diffusion in Nanocomposites. ACS Macro Lett. 2013, 2 (6), 485−490. 873

DOI: 10.1021/acsmacrolett.7b00533 ACS Macro Lett. 2017, 6, 869−874

Letter

ACS Macro Letters (29) Composto, R. J.; Walters, R. M.; Genzer, J. Application of Ion Scattering Techniques to Characterize Polymer Surfaces and Interfaces. Mater. Sci. Eng., R 2002, 38 (3−4), 107−180. (30) Yamamoto, U.; Schweizer, K. S. Theory of Entanglements and Tube Con Fi Nement in Rod−Sphere Nanocomposites. ACS Macro Lett. 2013, 2, 955−959. (31) Toepperwein, G. N.; Karayiannis, N. C.; Riggleman, R. A.; Kroger, M.; De Pablo, J. J. Influence of Nanorod Inclusions on Structure and Primitive Path Network of Polymer Nanocomposites at Equilibrium and under Deformation. Macromolecules 2011, 44, 1034− 1045. (32) Green, P. F.; Kramer, E. J. Matrix Effects on the Diffusion of Long Polymer Chains. Macromolecules 1986, 19 (4), 1108−1114. (33) Watanabe, H. Viscoelasticity and Dynamics of Entangled Polymers. Prog. Polym. Sci. 1999, 24 (9), 1253−1403. (34) Frischknecht, A. L.; Milner, S. T. Linear Rheology of Binary Melts from a Phenomenological Tube Model of Entangled Polymers. J. Rheol. (Melville, NY, U. S.) 2002, 46 (3), 671−684. (35) Milner, S. T. Relating the Shear-Thinning Curve to the Molecular Weight Distribution in Linear Polymer Melts. J. Rheol. (Melville, NY, U. S.) 1996, 40 (2), 303. (36) Karatrantos, A.; Composto, R. J.; Winey, K. I.; Clarke, N. Structure and Conformations of Polymer/SWCNT Nanocomposites. Macromolecules 2011, 44, 9830−9838. (37) Karatrantos, A.; Clarke, N.; Composto, R. J.; Winey, K. I. Topological Entanglement Length in Polymer Melts and Nanocomposites by a DPD Polymer Model. Soft Matter 2013, 9 (14), 3877−3884. (38) Toepperwein, G. N.; Karayiannis, N. C.; Riggleman, R. A.; Kröger, M.; de Pablo, J. J. Influence of Nanorod Inclusions on Structure and Primitive Path Network of Polymer Nanocomposites at Equilibrium and Under Deformation. Macromolecules 2011, 44, 1034− 1045. (39) Likhtman, A. E. Single-Chain Slip-Link Model of Entangled Polymers: Simultaneous Description of Neutron Spin-Echo, Rheology, and Diffusion. Macromolecules 2005, 38 (14), 6128−6139. (40) Ramírez-Hernández, A.; Peters, B. L.; Andreev, M.; Schieber, J. D.; de Pablo, J. J. A Multichain Polymer Slip-Spring Model with Fluctuating Number of Entanglements for Linear and Nonlinear Rheology. J. Chem. Phys. 2015, 143 (24), 243147.

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