Letter pubs.acs.org/macroletters
Influence of the Bound Polymer Layer on Nanoparticle Diffusion in Polymer Melts Philip J. Griffin,† Vera Bocharova,‡ L. Robert Middleton,† Russell J. Composto,† Nigel Clarke,§ Kenneth S. Schweizer,∥ and Karen I. Winey*,† †
Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States § Department of Physics, University of Sheffield, Sheffield, S3 7RH, United Kingdom ∥ Department of Materials Science and Engineering, University of Illinois, Urbana, Illinois 61801, United States ‡
S Supporting Information *
ABSTRACT: We measure the center-of-mass diffusion of silica nanoparticles (NPs) in entangled poly(2-vinylpyridine) (P2VP) melts using Rutherford backscattering spectrometry. While these NPs are well within the size regime where enhanced, nonhydrodynamic NP transport is theoretically predicted and has been observed experimentally (2RNP/dtube ≈ 3, where 2RNP is the NP diameter and dtube is the tube diameter), we find that the diffusion of these NPs in P2VP is in fact well-described by the hydrodynamic Stokes−Einstein relation. The effective NP diameter 2Reff is significantly larger than 2RNP and strongly dependent on P2VP molecular weight, consistent with the presence of a bound polymer layer on the NP surface with thickness heff ≈ 1.1Rg. Our results show that the bound polymer layer significantly augments the NP hydrodynamic size in polymer melts with attractive polymer−NP interactions and effectively transitions the mechanism of NP diffusion from the nonhydrodynamic to hydrodynamic regime, particularly at high molecular weights where NP transport is expected to be notably enhanced. Furthermore, these results provide the first experimental demonstration that hydrodynamic NP transport in polymer melts requires particles of size ≳5dtube, consistent with recent theoretical predictions.
T
These theoretical and experimental studies of NP diffusion in polymer solutions and melts largely omit consideration of bound polymer layers that may form on the NP surface in many polymer nanocomposites (PNCs). While static or hydrodynamic measurements of the thickness (h) of the bound polymer layer in PNCs are relatively scarce, it is generally believed that h should be proportional to the polymer radius of gyration (Rg), similar to the case of adsorbed polymer layers on flat substrates.17−20 This viewpoint has been corroborated by the work of Anderson and Zukoski,21,22 who interpreted the results of linear viscoelastic measurements of poly(ethylene oxide)/silica PNCs as being consistent with the formation of a bound polymer layer on the NP surface with thickness ∼1.5Rg. Juoualt et al. have also found indirect evidence for a bound polymer layer in poly(2-vinylpyridine)/ silica PNCs from analysis of transmission electron micrographs.23 Certainly, the presence of ∼Rg thick bound polymer layers could significantly alter the mechanism by which NPs diffuse in polymer melts and PNCs, whether via secondary entanglements with matrix chains22 or by increasing the NP hydrodynamic size.
he complex time and length scale-dependent friction mechanisms imposed by entanglements in polymer melts are hypothesized to cause dramatic, size-dependent alterations to nanoparticle (NP) diffusion.1−5 For particles much larger than the polymer coil size, particle diffusion is controlled by the viscous drag of the continuous medium,6 with a viscosity described by reptation theory.7 Diffusion of NPs significantly smaller than the entanglement tube diameter (dtube)8 can be described in a similar manner, with the viscous drag now associated with Rouse motion of polymer segments of size on the order of the NP diameter (2RNP).1−5 Recent theoretical studies of NP diffusion in sufficiently entangled polymers have suggested an activated hopping mechanism can enhance NP mobility when 2RNP/dtube ≈ 1.9,10 Other studies predict a gradually increasing coupling of NP motion to the entanglement network with increasing NP diameter, with full coupling and the hydrodynamic Stokes− Einstein limit attained only when 2RNP/dtube > 5−10.5 The few available experimental studies generally find enhanced NP diffusion in this size regime in entangled melts11−15 and semidilute DNA solutions,16 although the crossover from nonhydrodynamic to hydrodynamic transport in melts is far from well established. For example, studies of gold NP diffusion in poly(butyl methacrylate) melts with ∼10 entanglements per chain demonstrate that NP transport is 10−100× faster than the hydrodynamic law for NPs of diameter up to ∼3.5dtube.13 © XXXX American Chemical Society
Received: August 22, 2016 Accepted: September 21, 2016
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ACS Macro Letters We address this problem by measuring the diffusion of silica NPs (2RNP = 26 nm) in poly(2-vinylpyridine) (P2VP) using Rutherford backscattering spectrometry (RBS). P2VP of varying molecular weight (28, 49, 90, 122, and 301 kDa, PDI < 1.1) was purchased from Polymer Source or Scientific Polymer Products and used as received. The P2VP weightaveraged molecular weight was verified via static light scattering. These polymers exhibit light-to-moderate degrees of entanglement, with M/Me ∼ 1−15 (P2VP entanglement molecular weight Me ≈ 18 kDa estimated from viscosity measurements, see Supporting Information). Thick (∼10 μm) P2VP matrix films were prepared as follows: P2VP/MeOH solutions of varying concentration were cast onto silicon wafer substrates, dried for several hours in ambient conditions, and then annealed at 180 °C (Tg + 85 °C) in a vacuum oven (∼1 mbar) for 24 h. Thin (100−200 nm, measured via reflectometry) polymer nanocomposite (PNC) tracer films were prepared by mixing monodisperse silica nanoparticles (log-normal geometric mean diameter 2RNP = 26.1 nm and standard deviation eσ = 1.2 determined by transmission electron microscopy, Figure SI.1) suspended in EtOH (cNP ≈ 15 mg/mL) with P2VP of varying molecular weight dissolved in MeOH. Silica NPs were synthesized following the modified Stöber method.24,25 The necessary amount of NP solution to achieve a NP volume fraction φNP ≈ 0.1 in the PNC tracer film was added dropwise to the P2VP solution under constant stirring and was stirred further for at least 12 h to promote homogeneous NP dispersion. PNC tracer films were prepared by spin coating these solutions onto a silicon wafer, which had been previously coated with a thin (30 nm) 200 kDa polystyrene film. This underlayer serves to readily delaminate the PNC thin film from the substrate and prevents potential NP aggregation at the silicon surface. The PNC thin films exhibited homogeneous NP dispersion as determined via transmission electron microscopy (Figure SI.1). As-prepared PNC tracer layers were transferred onto the corresponding P2VP matrix films (the matrix and tracer layer P2VP molecular weights are equal) and then annealed in a vacuum oven at Tanneal = 180 °C for variable annealing times such that the characteristic diffusion lengths were ∼500 nm (∼20 × 2RNP). Concentration profiles of silica NPs in the annealed diffusion couples were measured using RBS, as previously described.14,26 Representative profiles of silica NPs infiltrated into 49 kDa P2VP after annealing for 0, 5, and 30 min are depicted in Figure 1 (all measured profiles are shown in Figure SI.2). To extract the NP diffusion coefficients, we fit these concentration profiles to the solution to Fick’s second law for a finite source diffusing into a semi-infinite medium, convoluted with a Gaussian resolution profile (half width = 60 nm).27 It should be noted that the NP volume fraction in all concentration profiles used to extract diffusion coefficients is less than the initial tracer layer concentration, with φNP varying nominally between 0−0.05 around an average value φNP ≈ 0.025 (representative of the dilute limit), which we employ below as the average NP concentration in these PNCs. The as-determined diffusion coefficients are independent of annealing time for all diffusion couples, varying at most by ∼10−15%. If NPs were aggregating during the annealing process or undergoing polymer bridginginduced flocculation, we would expect a dramatic reduction in diffusion coefficient with increasing annealing time. We see no evidence for such behavior in any of these diffusion couples, verifying that the particles remain well dispersed during the course of these experiments.
Figure 1. Concentration profiles of silica NPs diffused into 49 kDa P2VP after annealing at 180 °C for 0, 5, and 30 min. Symbols depict measured data, and the solid lines depict fits used to determine the NP diffusion coefficient. The inset shows a schematic representation of the RBS geometry.
The measured NP diffusion coefficients decrease strongly with increasing P2VP molecular weight, qualitatively consistent with entanglement controlled friction (Figure 2).28 To gain a better understanding of the NP diffusion mechanism in these P2VP/silica systems, we compare the measured NP diffusion coefficients to the values predicted by the Stokes−Einstein (SE) relation assuming stick boundary conditions6,29 DSE =
kBT 6πηpoly RNP
(1)
where kB is Boltzmann’s constant and ηpoly is the zero-shear viscosity of bulk P2VP (measured at T = 180 °C, Figure SI.3). For NPs much larger than the tube diameter (dtube ≈ 8 nm estimated from Me and assuming a Kuhn length of 1.8 nm30), the polymer can be treated as a continuous hydrodynamic medium through which the NP executes Brownian diffusion. This type of NP motion has been observed for large (∼200− 1000 nm) particles diffusing in dilute, unentangled polymer solutions.31,32 Alternatively, several recent experimental studies, in which the size of the NP was comparable to or smaller than dtube, have demonstrated that NP diffusion can be several orders of magnitude faster than what is predicted according to the SE relation when the polymer−NP interaction is nominally athermal or weakly attractive.11−15 Theory predicts the crossover to SE-controlled diffusion for hard sphere, nonadsorbing NPs when 2RNP/dtube ≈ 5−10 in the light-to-
Figure 2. Measured (D, solid symbols) and Stokes−Einstein predicted (DSE, open symbols, eq 1) nanoparticle diffusion coefficients (at T = 180 °C) are shown plotted as a function of P2VP molecular weight. Error bars on D are determined from measurements of duplicate samples, and DSE is calculated using the measured bulk viscosity (Figure SI.3) and nanoparticle-core diameter (2RNP = 26 nm). 1142
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ACS Macro Letters moderately entangled melts used in this study.3,5 This prediction is consistent with recent experimental studies of semidilute entangled DNA solutions.16 Interestingly, we observe neither of these behaviors for 26 nm silica NPs (2RNP/dtube ≈ 3) diffusing in P2VP. Instead, Figure 2 shows that the NP diffusion coefficients in P2VP are consistently lower than the SE-predicted values. Our results appear to be qualitatively in line with previous experiments of Shull et al., who observed slower than SE diffusion of 15 nm diameter gold NPs (2RNP/dtube ≈ 2) in P2VP of only two different molecular weights (Mw = 88 and 152 kDa).33 Plotting the measured NP diffusion coefficient normalized by the SE-predicted value (Figure 3) reveals the dependence of
ηPNC = ηpoly (1 + 2.5φeff + 6.2φeff 2)
where φeff = φNP
3
( ) R eff RNP
(3)
is the effective NP volume fraction.
This approach was recently used to accurately describe the zero-shear viscosity of poly(ethylene oxide)/silica nanocomposites over a wide range of nanoparticle concentrations and polymer molecular weights.21,22 Combining eqs 1−3, we write the normalized diffusion coefficient −1 ⎡ ⎛ RNP·ηpoly ⎛ R eff ⎞3 ⎛ ⎞6⎞⎤ R eff ⎜ D 2 R eff ⎢ ⎥ ⎟ = = 1 + 2.5φNP⎜ ⎟ + 6.2φNP ⎜ ⎟ DSE R eff ·ηPNC ⎢⎣ RNP ⎜⎝ ⎝ RNP ⎠ ⎝ RNP ⎠ ⎟⎠⎥⎦
(4)
Recall that the NP-core volume fraction varies from 0−0.05 for all concentration profiles used to extract diffusion coefficients, with an average φNP ≈ 0.025 (see Figure 1 and Figure SI.2). As such, we solve eq 4 to determine Reff at φNP = 0.025 for all measured, molecular weight-dependent ratios D/ DSE. Because this choice of φNP is to some extent arbitrary, we also present the solution for Reff with φNP = 0.01 and 0.04 as error bars in Figure 4.
Figure 3. Measured nanoparticle diffusion coefficient normalized by the Stokes−Einstein predicted values plotted as a function of P2VP molecular weight. Error bars are determined from multiple RBS measurements of duplicate samples. The solid line depicts D/DSE calculated via eq 4 using nanoparticle-core volume fraction φNP = 0.025 and the bound layer-modified nanoparticle size 2Reff = 2RNP + 0.061 nm × Mw1/2.
this ratio on polymer molecular weight, which monotonically decreases from a factor of ∼1.5 slower in 28 kDa P2VP to a factor of ∼5 slower in 301 kDa P2VP. This large reduction in NP diffusion cannot be explained by an enhancement of viscosity due solely to the presence of 26 nm silica NPs, which at these low NP volume fractions (φNP ≈ 0.025) would produce hydrodynamic viscosity increases of 2RNP, and (2) the Stokes−Einstein relation in fact holds for these effective NPs for all studied P2VP molecular weights. Correspondingly, we define the measured diffusion coefficient
D=
kBT 6πηPNCR eff
Figure 4. (a) Effective nanoparticle diameter in the melt (2Reff, black symbols) and in dilute MeOH solution (2RDLS, green symbols). (b) Effective thickness of the P2VP bound layer on SiO2 nanoparticles in the melt (heff, black symbols) and in dilute MeOH solution (hDLS, green symbols). The black symbols depict (a) 2Reff and (b) heff calculated from RBS-measured diffusion coefficients with nanoparticlecore volume fraction φNP = 0.025, while the upper and lower error bars represent the calculated values with φNP = 0.01 and 0.04, respectively. The solid black (∼Mw1/2) and dashed green lines (∼Mw5/6) are power law fits to the corresponding data.
(2)
Figure 4a depicts the calculated values of 2Reff, which increases from 35 to 58 nm with increasing P2VP matrix molecular weight. The effective NP diameter is consistently larger than the silica core diameter 2RNP, increasing to an effective size ∼2.25× larger than 2RNP in 301 kDa P2VP. To gain a clearer understanding of how this effective size (in excess of 2RNP) depends on molecular weight, we subtract the known NP core radius from the calculated values of Reff (Figure 4b). This excess size heff = Reff − RNP follows a power law
which includes the new effective NP diameter, 2Reff, and the zero-shear viscosity of the PNC, ηPNC. To determine ηPNC, we assume that the effective nanoparticles interact as hard spheres, hydrodynamically enhancing the viscosity of the polymer melt in a manner identical to relatively dilute colloidal suspensions.34 In this case, the relevant viscosity experienced by a NP as it diffuses through the polymer matrix at these low NP volume fractions is 1143
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ACS Macro Letters dependence, scaling as Mw1/2 across the measured range of P2VP molecular weights. Specifically, we find that heff = 1.1Rg for the P2VP matrix molecular weights studied, that is, the excess NP hydrodynamic radius is approximately equal to the equilibrium polymer radius of gyration (Rg ≈ 5−15 nm, estimated using a Kuhn length of 1.8 nm) when diffusing in a P2VP melt (calculations of heff at φNP = 0.01 and 0.04 yield values of 1.4Rg and 0.8Rg, respectively). Moreover, our measured values of 2Reff/dtube ≈ 4.5−7 are consistent with the theoretical value predicted to be required for SE-controlled NP diffusion in modestly entangled melts,3,5 thereby supporting the idea of an extended crossover for the coupling of NP diffusion to the entanglement network. As stated above, the molecular weight dependence of 2Reff (or heff) is well described as a power law 2Reff = 2RNP + a × Mw1/2 (where the constant a = 0.061 nm) over the measured range of P2VP molecular weights (Figure 4, solid black line). In a self-consistent manner, we insert this power law function for Reff into eq 4, where it is seen in Figure 3 to accurately account for the molecular weight-dependent reduction of NP diffusion relative to the SE relation (eq 1). This good agreement corroborates our original postulate that the diffusion of 26 nm silica NPs in P2VP is well described by the hydrodynamic SE relation in entangled P2VP melts. The intriguing connection between the excess NP hydrodynamic size and the polymer radius of gyration clearly points to the underlying mechanism for the observed slow nanoparticle diffusion in silica/P2VP nanocomposites. As a result of the favorable interactions between the native hydroxyl moieties on the NP surface and the amine functionality of the polymer, P2VP physically adsorbs to the surface of the NP and substantially augments the NP size. From our diffusion measurements, it is clear that this bound layer thickness in the P2VP melt extends ∼Rg from the surface and apparently adopts an equilibrium, bulk-like chain conformation. If the conformation of the bound layer were indeed different from the bulk Rg, for example, possessing a more extended chain conformation, the excess NP size heff would be observed to scale closer to ∼Mw1.0,7 which is clearly not the case. We have also characterized the hydrodynamic size of the silica NPs in dilute P2VP/MeOH solution (2RDLS). They are substantially larger than the NP core size (Figure 4a and Figure SI.4) for all studied P2VP molecular weights, indicating the presence of an adsorbed or effectively bound polymer layer in solution as well. The conformations of bound polymer chains in MeOH solution are substantially different than in the bulk melt, where the thickness of the adsorbed layer hDLS = RDLS − RNP (Figure 4b) is observed to scale as ∼Mw5/6. Interestingly, this scaling was predicted by Guiselin for an adsorbed brush-like polymer layer formed on a flat surface in a good solvent.36 The above conclusions and physical picture concerning the adsorbed polymer layer are consistent with prior static SANS studies of concentrated solutions of silica and PEO where hydrogen bonding between the NP surface and monomer is also present as in our system.37,38 Physically, such a strong segment-NP surface attraction, along with the ∼√N contacts made by a random coil polymer with a (nearly) flat surface, suggests equilibration of the adsorbed layer is a very slow, activated process, which is rendered even slower if the chains are entangled as here. Moreover, glassy (or at least slower than bulk-relaxing) polymer layers with thickness on the order of a few segments39−41 appear to be present in P2VP−SiO2
nanocomposites, which even further frustrates equilibration/ chain desorption on the time scale of NP diffusion. In summary, we have measured the diffusion coefficient of monodisperse silica nanoparticles in poly(2-vinylpyridine) melts over a range of polymer molecular weights at a temperature well above Tg using Rutherford backscattering spectrometry. While bare NPs of this size (2RNP ≈ 3dtube) are expected to exhibit nonhydrodynamic diffusion, we find that diffusion of these NPs in P2VP is well described by the Stokes− Einstein relation with a diameter 2Reff that is significantly larger than the NP diameter and strongly dependent on P2VP molecular weight. We demonstrate that this molecular weightdependent hydrodynamic size is consistent with the presence of a bound polymer layer on the NP surface, with thickness heff that increases with increasing P2VP molecular weight, heff = 1.1Rg. This bound layer substantially augments the hydrodynamic size of NPs in polymer melts with attractive polymerNP interactions, effectively transitioning the mechanism of NP diffusion from the nonhydrodynamic to hydrodynamic regime, particularly at higher polymer molecular weights. We anticipate that the presence of bound polymer layers will lead to similar alterations of the NP diffusion mechanism in other enthalpically attractive polymer−NP systems, and we speculate that similar behavior may occur for grafted NPs diffusing in attractive polymer melts, particularly when the grafting density is low and the grafted chains are well-wetted by the matrix chains. Finally, our results provide the first experimental evidence that the crossover to hydrodynamic NP transport in polymer melts requires particles of size ≳5dtube, consistent with recent theoretical predictions.3,5
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.6b00649. Supporting figures and table (PDF).
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The University of Pennsylvania team acknowledges funding of this work from the NSF Division of Materials Research through Grant Nos. DMR-1120901, DMR-1210379 (K.I.W.), and DMR-1507713 (R.J.C.). R.J.C. acknowledges funding from the American Chemical Society PRF Grant No. 54028-ND7 and DuPont CR&D. V.B. and K.S.S. acknowledge financial support from the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. We thank J. S. Meth (Dupont) for assistance with molecular weight characterization and C.-C. Lin for transmission electron microscopy.
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REFERENCES
(1) Wyart, F. B.; De Gennes, P. Eur. Phys. J. E: Soft Matter Biol. Phys. 2000, 1, 93−97. (2) Cai, L.-H.; Panyukov, S.; Rubinstein, M. Macromolecules 2011, 44, 7853−7863.
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ACS Macro Letters (3) Yamamoto, U.; Schweizer, K. S. J. Chem. Phys. 2011, 135, 224902. (4) Kalathi, J. T.; Yamamoto, U.; Schweizer, K. S.; Grest, G. S.; Kumar, S. K. Phys. Rev. Lett. 2014, 112, 108301. (5) Yamamoto, U.; Schweizer, K. S. Macromolecules 2015, 48, 152− 163. (6) Hansen, J.-P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1986. (7) De Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (8) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986. (9) Dell, Z. E.; Schweizer, K. S. Macromolecules 2014, 47, 405−414. (10) Cai, L.-H.; Panyukov, S.; Rubinstein, M. Macromolecules 2015, 48, 847−862. (11) Tuteja, A.; Mackay, M. E.; Narayanan, S.; Asokan, S.; Wong, M. S. Nano Lett. 2007, 7, 1276−1281. (12) Grabowski, C. A.; Adhikary, B.; Mukhopadhyay, A. Appl. Phys. Lett. 2009, 94, 021903. (13) Grabowski, C. A.; Mukhopadhyay, A. Macromolecules 2014, 47, 7238−7242. (14) Choi, J.; Cargnello, M.; Murray, C. B.; Clarke, N.; Winey, K. I.; Composto, R. J. ACS Macro Lett. 2015, 4, 952−956. (15) Mangal, R.; Srivastava, S.; Narayanan, S.; Archer, L. A. Langmuir 2016, 32, 596−603. (16) Chapman, C. D.; Lee, K.; Henze, D.; Smith, D. E.; RobertsonAnderson, R. M. Macromolecules 2014, 47, 1181−1186. (17) Scheutjens, J.; Fleer, G. J. Phys. Chem. 1980, 84, 178−190. (18) De Gennes, P. Adv. Colloid Interface Sci. 1987, 27, 189−209. (19) Granick, S.; Hu, H.-W. Langmuir 1994, 10, 3857−3866. (20) Hu, H.-W.; Granick, S.; Schweizer, K. S. J. Non-Cryst. Solids 1994, 172, 721−728. (21) Anderson, B. J.; Zukoski, C. F. Macromolecules 2008, 41, 9326− 9334. (22) Anderson, B. J.; Zukoski, C. F. Macromolecules 2009, 42, 8370− 8384. (23) Jouault, N.; Moll, J. F.; Meng, D.; Windsor, K.; Ramcharan, S.; Kearney, C.; Kumar, S. K. ACS Macro Lett. 2013, 2, 371−374. (24) Kamiya, H.; Suzuki, H.; Kato, D.; Jimbo, G. J. Am. Ceram. Soc. 1993, 76, 54−64. (25) Iijima, M.; Kamiya, H. Langmuir 2010, 26, 17943−17948. (26) Composto, R. J.; Walters, R. M.; Genzer, J. Mater. Sci. Eng., R 2002, 38, 107−180. (27) Crank, J. The Mathematics of Diffusion; Oxford University Press: New York, 1979. (28) Rubinstein, M.; Colby, R. H. Polymers Physics; Oxford University Press: New York, 2003. (29) Einstein, A. Ann. Phys. 1905, 322, 549. (30) Holt, A. P.; Bocharova, V.; Cheng, S.; Kisliuk, A. M.; White, B. T.; Saito, T.; Uhrig, D.; Mahalik, J. P.; Kumar, R.; Imel, A. E.; Etampawala, T.; Martin, H.; Sikes, N.; Sumpter, B. G.; Dadmun, M. D.; Sokolov, A. P. ACS Nano 2016, 10, 6843−6852. (31) Lu, Q.; Solomon, M. J. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2002, 66, 061504. (32) Poling-Skutvik, R.; Krishnamoorti, R.; Conrad, J. C. ACS Macro Lett. 2015, 4, 1169−1173. (33) Shull, K. R.; Kellock, A. J. J. Polym. Sci., Part B: Polym. Phys. 1995, 33, 1417−1422. (34) Batchelor, G. J. Fluid Mech. 1977, 83, 97−117. (35) Hao, T.; Riman, R. E. J. Colloid Interface Sci. 2006, 297, 374− 377. (36) Guiselin, O. Europhys. Lett. 1992, 17, 225. (37) Kim, S. Y.; Schweizer, K. S.; Zukoski, C. F. Phys. Rev. Lett. 2011, 107, 225504. (38) Kim, S. Y.; Zukoski, C. F. Soft Matter 2012, 8, 1801−1810. (39) Holt, A. P.; Griffin, P. J.; Bocharova, V.; Agapov, A. L.; Imel, A. E.; Dadmun, M. D.; Sangoro, J. R.; Sokolov, A. P. Macromolecules 2014, 47, 1837−1843. (40) Gong, S.; Chen, Q.; Moll, J. F.; Kumar, S. K.; Colby, R. H. ACS Macro Lett. 2014, 3, 773−777.
(41) Cheng, S.; Mirigian, S.; Carrillo, J.-M. Y.; Bocharova, V.; Sumpter, B. G.; Schweizer, K. S.; Sokolov, A. P. J. Chem. Phys. 2015, 143, 194704.
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