Size Effect of Nanoparticle Diffusion in a Polymer Melt - American

Oct 9, 2014 - the ratio D/DSE was determined, where DSE is the continuum prediction from Stokes−. Einstein (SE) relation. Our results indicate gradu...
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Size Effect of Nanoparticle Diffusion in a Polymer Melt Christopher A. Grabowski and Ashis Mukhopadhyay* Department of Physics & Astronomy, Wayne State University, Detroit, Michigan 48201, United States ABSTRACT: We investigated the dynamics of gold nanoparticles (NPs) within an entangled liquid of poly(n-butyl methacrylate) (PBMA) above the glass transition temperature (Tg). The experiments were performed by using a modified version of fluctuation correlation spectroscopy (FCS), which measured the translational diffusion (D) of the isolated NPs as a function of their size (2R0 = 5−20 nm) and temperature (T). We probed the most interesting but sparsely investigated length regime where the particle size/tube diameter (dt) ratio ranges between ∼1−4. This allowed us to put into direct test recently developed theories and simulations. By measuring the bulk viscosity of the melt, the ratio D/DSE was determined, where DSE is the continuum prediction from Stokes− Einstein (SE) relation. Our results indicate gradual recovery to SE behavior and full coupling to entanglement relaxation would require 2R0 ≈ 7−10dt.

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INTRODUCTION Polymer nanocomposites (PNCs) have received a lot of attention recently because of their potential in fabricating materials with novel mechanical, electrical, and photonic properties.1 The particles in conventional composites are essentially immobile, which is in contrast to PNCs particularly above the glass transition temperature (Tg). The nanoparticle (NP) mobility can affect polymer dynamics, which in turn can change important polymer properties, such as viscosity, modulus, kinetics of the particle-cluster formation, etc.2−4 Tensile measurements have shown that below Tg conventional composites and PNCs behave similarly with respect to mechanical properties.3 But above Tg, the toughness of PNC can increase by orders of magnitude with increase of temperature. It has been suggested that the mechanism to the toughness enhancement is the mobility of the nanoparticles.4 The development of self-healing materials and coatings, where NPs migrate toward various defect sites, requires a better understanding of the NP diffusion process.5 Experimentally, there have been a couple of studies of NP diffusion within polymer melt, which were all restricted to a single particle size.6,7 Both of them measured diffusion coefficient of particles which are more than 2 orders of magnitude higher compared to continuum prediction. However, systematic measurements at the most interesting length scale, where the particle diameter (2R0) is comparable to few times of entanglement tube diameter (dt), have been lacking so far. This hinders progress as in the absence of reliable experimental data, a mechanistic understanding of the NP dynamics in PNCs remained elusive. Here, by using a series of gold nanospheres of different sizes (2R0 = 5−20 nm) but possessing identical surface chemistry, we determined the effect of particles size on diffusion in an entangled polymer melt. We observed gradual recovery to continuum hydrodynamic behavior as the particle size increases beyond few times of tube diameter and estimated the crossover © 2014 American Chemical Society

length scale. The novelty of these experiments is the use of multiphoton luminescence to probe dynamics of the individual, isolated NPs in the polymer matrix. This pristine experimental condition allowed us to compare the results with theories and simulations, which were also performed at the dilute particle limit. We identified situations where agreement and disagreement were found. This research will be of importance not only in the field of polymer science but also in fields as diverse as soft matter and biophysics, where there is complex coupling between two or more characteristic length scales that govern their dynamics. For a relatively large spherical particle in a solvent of bulk viscosity, ηb, the diffusion coefficient is given by the Stokes− Einstein (SE) relation DSE = kBT/6πηbR0, where the thermal energy is kBT (kB is the Boltzmann constant and T is the absolute temperature). The diffusion of small (∼1−2 nm) and mesoscopic probes (∼1 μm) in polymer solutions have been studied.8−13 In neat polymers, most studies have used small fluorescent molecules. The diffusivity is found to be independent of molecular weight (Mw) of the polymer, and the particles experience a local viscosity (ηc) related to the monomeric friction coefficient, η1. In recent years, more generally length-scale-dependent viscosity has received a lot of interest. Mode coupling,14 statistical dynamical,15,16 and scaling theories17,18 were used, and computer simulations19−21 have been performed. One of the most important parameters that governs the friction coefficient of a single NP immersed within an entangled polymer liquid is its radius (R0) in relation to various polymer characteristic length scales: coil radius (Rg), entanglement spacing (dt), and correlation length (ξ ≈ monomer size). The pioneering study in this field was performed by Brochard and Received: August 14, 2014 Revised: October 3, 2014 Published: October 9, 2014 7238

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de Gennes (BdG) using scaling theory.18 They argued that for R0 ≫ Rg friction is determined by bulk viscosity (ηb). As the size of the particle becomes comparable to or smaller than the entanglement tube diameter (dt), the friction force experienced by the particle exhibits a dramatic drop of about few orders of magnitude. Polymers are unentangled at length scales smaller than the tube diameter. As the particle moves, it does not have to disentangle any chain, and a simple rearrangement of chain portions comparable to particle size allows for motion. It has been argued that in this situation, i.e., R0 < dt, the local viscosity is proportional to the particle surface area (R02) and is independent of the chain’s length. A more sophisticated version of the scaling theory has been developed by Cai et al., which takes into account the effects of segmental motion and reptation dynamics on particle motion.17 They assumed that local viscosity (ηc) is length scale dependent and identified three regimes depending upon the particle size. Small particles (R 0 < ξ) are not affected by surrounding polymers, intermediate sized particles (ξ < R0 < dt) are affected by the segmental motion, and large particles (R0 > dt) are affected by entanglement. It qualitatively captures many aspects of BdG theory and is applicable both in the melt and in polymer solutions, but in their theory the dramatic change of particle mobility at 2R0 ≈ dt is broadened due to a hopping motion of the particles caused by local entanglement relaxation.22,23 The importance of this hopping motion in highly entangled melts and in cross-linked polymer networks has also been recognized in a more recent statistical dynamical theory.16 The same theory was used to determine the quantitative dependencies of the ratio D/DSE as a function of R0/dt.15 Here, D is given as a sum of hydrodynamic and nonhydrodynamic contributions. The former is given by the Stokes−Einstein (SE) relation with full polymer melt viscosity (ηb). The nonhydrodynamic force arises from direct polymer−NP forces and structural relaxation of the matrix. The theory predicts that the “crossover condition” for return to SE behavior is gradual and different for unentangled and entangled melts. For unentangled melts, it was estimated to occur at R0/Rg ≈ 3/2, in strong contrast to heavily entangled liquids, where a return to SE behavior requires 2R0/dt ≈ 5. Experimentally, to the best of our knowledge, only two studies had reported single nanosphere diffusion in polymer melts, but none of them investigated the effect of different size ratios. Tuteja et al. have studied organic ligand coated cadmium selenide (CdSe) quantum dots (QDs) of size 2R0 ≈ 10 nm in an entangled polystyrene (PS) matrix using X-ray photon correlation spectroscopy (XPCS).6 XPCS relies on the large contrast of X-ray scattering at the particle−polymer interface and requires high brilliance synchrotron radiation source. The study used about 8% volume fraction of the particles. Assuming random distribution, the interparticle surface−surface spacing (Rp) can be estimated by using the relation Rp = 2R0((ϕm/ ϕp)1/3 − 1), where ϕm = 0.638 is the maximum random packing volume fraction. So Rp ≈ 10 nm, and as root-mean-square endto-end distance of the PS used (2Rg) was about 34 nm, some polymer mediated interaction between the NPs through transient bridging cannot be ruled out.24 The addition of nanoparticles in their experiments, however, significantly reduced the blend viscosity. For their experiments 2R0/dt ≈ 1, and they found that the ratio D/DSE steadily increases from 162 to 271 as the temperature is lowered from 1.16Tg to 1.08Tg. The analysis indicated that the QDs experienced a medium with effective viscosity that is between the full

entangled melt viscosity and the Rouse viscosity of an equivalent unentangled melt. In the other work, the authors of this paper studied gold NPs of radii 2.5 nm in both unentangled and entangled poly(butyl methacrylate) (PBMA) melts.7 In unentangled PBMA (Mw = 2.5 kg/mol, R0/Rg ≈ 2.5) a slight increase of the ratio D/DSE was observed with lowering of temperature and the averaging over temperature yielded D/ DSE about unity. In entangled PBMA (Mw = 180 kg/mol), the NPs diffuse more than 2 orders of magnitude faster compared to SE prediction. In the following, we performed systematic measurements of NP diffusion using particles of different sizes. We determined the ratio D/DSE by measuring the bulk viscosity of the melt and demonstrated the gradual recovery of SE behavior as the particle size is increased beyond a few times of the tube diameter.

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MATERIALS AND METHODS

The experiments were performed by using monodisperse (Mw/Mn = 1.12) poly(butyl methacrylate) (Polymer Source, Inc.) of Mw = 180 kg/mol and gold nanospheres (Corpuscular, Inc.). The typical size variation of the particles was about 20% as determined by TEM measurements (Figure 1, inset). The scattering signal from small

Figure 1. Inset (a): transmission electron micrograph of gold colloids deposited on carbon film magnified 800000×. A JEOL FasTEM 2010 TEM with a LaB6 filament working at 200 kV was employed to capture the image. Inset (b): a histogram obtained from measuring the diameters of gold colloids. The average diameter measured is 4.7 ± 1.1 nm. Main figure: intensity decay curve for R0 = 5 nm particle in 180 kg/mol PBMA at ∼75 °C. The fitting is with stretched exponential function with weight-average decay time, τD ≈ 150 s. metallic nanospheres is typically low, but they have high luminescence efficiency upon multiphoton excitation.23 Gold NPs are well suited for our experiments because they are photostable for several hours of continuous excitation and do not blink like semiconductor quantum dots, and their size can be tuned across the most important length scales in polymer melts while keeping the surface chemistry identical. For the preparation of the sample, the gold NP solution is evaporated in a controlled manner and the particles are redispersed in toluene. The mixture is sonicated for 30 min. The absence of aggregation of the particles in the organic solvent is verified by performing a fluctuation correlation spectroscopy (FCS) measurement to determine their hydrodynamic radius and verifying that it agrees with the estimate using SE relation. To prepare the NP-embedded polymer composites, the polymer sample of suitable amount is mixed with the particle 7239

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least 30 °C above the glass transition temperature of PBMA (Tg ≈ 23 °C). As the particle size increases, the measurements were performed at even higher temperatures due to the increasingly sluggish dynamics at lower temperatures. The viscoelastic measurements of PBMA were performed by using an ARES rheometer (TA Instruments, Inc.). The elastic modulus as a function of angular frequency is shown in the Figure 3 inset. We determined the entanglement plateau

solution. This was followed by sonication for another few minutes, and the solution is then spun-cast onto a glass slide. To remove the residual solvent, the films were annealed at an elevated temperature for approximately 24 h Because of very slow dynamics of the particles in the melt, we used a modified version of FCS which has been outlined in ref 7. Briefly, each experimental run measured the photon counts, I(t), as a function of time from a single NP as it diffused out of the laser focus created by a high NA objective (NA = 1.25) and a femtosecond near-infrared laser (Figure 1). The dynamics were quantified by fitting each intensity decay curve with a stretched exponential function, I(t) ≈ exp[−(t/ τ)β], and from the fitting parameters τ and β we calculated a weighted average decay time (τD). To get the statistical distribution, the whole process is repeated for many particles. Note that Figure 1 is the intensity decay curve for an individual particle and not the intensity− intensity correlation function or the dynamic structure factor as obtained in XPCS, dynamics light scattering (DLS), or FCS measurements. In a crowded medium such as in concentrated polymer or protein solutions it had been observed that β < 1,25 and the exponent was found to decrease with increasing polymer concentration. But the interaction between the probe and the matrix as well as the size of the probe relative to various characteristic length scales could also influence the magnitude of β. The subdiffusive behavior also can be transient, which returns to normal diffusion at long times. In contrast, a compresses exponential (β > 1) had been observed for a wide range of complex fluids, such as gels, polymer composites, and jammed systems. XPCS experiments had shown that the magnitude of β may be affected by several factors, such as temperature, the viscoelasticity of the medium, local internal stress, and interparticle interaction.26,27 These experiments, however, were done at high particle loading, and in the probed wave-vector range, the XPCS measured the collective dynamics of the particles. In our experiments, we did not observe any clear dependence of β on the particle size. The intensity decay rate is determined by the diffusion time of the particle though the laser focus, which can be approximated as a Gaussian cylinder with half-width ω0 ≈ 0.22 μm and half-height z0 ≈ 1 μm. As the laser is more tightly focused in the lateral direction compared to the axial direction, τD is determined primarily by the lateral diffusion of the particles. The diffusion coefficient of the particle was thus calculated by using the relation D = ω02/8τD.

Figure 3. D vs size ratio in semilog plot at two different temperatures much above the glass transition temperature. Also plotted are the simulation results from ref 21, which is for shorter chains in the relevant length scales. The solid lines in the figure have the slope of −1. Inset: the master curve showing the elastic modulus, G′(ω), of PBMA; Mw = 180 kg/mol at 45 °C.

modulus, G0N ≈ 0.2 MPa, from which we obtained the molecular weight between entanglement, Me ≈ 14 kg/mol. Therefore, our experiments (Mw/Me ≈ 12) allow us to study the effect of moderate entanglement on particle dynamics. The tube diameter (dt) is related to Me, and using well-known relation in polymer physics, we have estimated that dt ≈ 6 nm.28 Figure 3 shows measured D as a function of particle size/ tube diameter at two different temperatures, which are at same temperature above Tg. Within the probed size regime, 2R0/dt ≈ 0.8−3.5, our data did not indicate a clear scaling relation as a function of particle size. The predicted sharp drop of diffusion at 2R0 ≈ dt as in BdG theory was not observed in our experiments. Kalathi et al. has recently developed a molecular dynamics simulation for NP diffusivity in composites.21 Their simulation, which is consistent with the microscopic force based theory by Yamamoto et al.,15 observed a gradual slowdown of diffusion as the particle size is increased beyond the tube diameter. The computer simulation extended to shorter chains (N ≈ 400) compared to our system (N ≈ 1300), and the particle size is also limited to the ratio 2R0/dt = 1.5. In Figure 3 we compare the simulation predictions with our data in the relevant length regime. We have assumed that the simulation was performed at much higher temperature than the glass transition temperature. The simulation predicted a stronger dependence of diffusion with respect to the particle size. Using scaling theory, Cai et al. had argued for a hopping like motion of NPs that are slightly larger compared to tube diameter.17 In Figure 3, we show that data in a semilog plot with the straight lines indicating the exponential

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RESULTS AND DISCUSSION In Figure 2 we plot D as a function of temperature for four different particle sizes. All measurements were carried out at

Figure 2. Measured D as a function of temperature for different sized nanoparticles. The indicted sizes are the radii of the particles. The error bars are standard deviation from ≈25−30 particles. The glass transition temperature of PBMA was ∼23 °C. 7240

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behavior with slope −1. This would indicate that hopping through the entanglement mesh will make diffusion coefficient to decrease exponentially with the size ratio 2R0/dt. The hopping motion as a mechanism for particle diffusion has been observed in a more recent statistical dynamical theory for heavily entangled liquid.16 But the difference is that its contribution has a much stronger dependence on the size ratio and is operative for a vanishingly short window of 2R0/dt ≈ 1.5−2. The recent molecular dynamics simulation, however, did not find any evidence of hoplike motion at least for shorter chains.21 Certainly, more experiments and theoretical efforts are needed in this area. There are few key differences between our experiment and simulation conditions. The simulation was performed with slightly attractive NP−polymer mixture to ensure miscibility. In our control experiments with NPs in PBMA−organic solvent mixture, we did not find any evidence of polymer adsorption. This indicated that the interaction between nanoparticles and PBMA is not very strong. Some studies have shown that the local conformation and segmental density of the chain can be affected by monomer−particle interaction.29 These may alter the entanglement mechanism that occurs in the bulk. The effect can be different for different particle sizes, depending upon the ratio R0/Rg. The near surface chain dynamics and hence the local Tg can also be affected by the interaction,30 which in turn can influence the NP dynamics. This can cause curvaturedependent changes in diffusion, which could explain the observed differences in particle mobility compared to simulation. We note that the studies of chain conformation and dynamics near particle surfaces for PNCs are still a matter of active research and debate.31 The volume fraction of particles used in simulation (ϕNP ≈ 0.1) was also higher compared to our experiments. In our experiments, the particle concentration was kept deliberately very low, ϕNP ≈ 10−9, so that the interparticle separation is at least a few micrometers. This is needed to avoid overlap of the particle trajectories within the laser focus, which can complicate the interpretation of the data. The low concentration of the particle also eliminates the chance of particle−particle interaction and possible aggregation in spite of the weakly interacting mixture. Our experimental conditions also ensured that the glass transition temperature (Tg) or the effective macroscopic viscosity (ηb) of the composite is minimally affected. Let us next compare the measured diffusion (D) with the prediction from SE relation. Unlike microrheology experiments, which measure the local probe displacement to determine the viscoelastic response of the medium,32 our measurements were performed at the long-time scale, where the viscous response should dominate. Zero-shear viscosity of PBMA is shown in Figure 4a for the relevant temperature range, and in Figure 4b, we plotted the ratio D/DSE as a function of temperature. The ratio gradually decreases as the size of the particle increases and in the measured narrow temperature range there is no systematic variation of the ratio. This is contrast with the result of Tuteja et al., where they found that the ratio D/DSE increases systematically with lowering of temperature for 10 nm sized particles.6 In Figure 4 we plot the temperature average D/ DSE as a function of 2R0/dt. At the highest 2R0/dt ≈ 3.4, we determined D/DSE ≈ 10, which is on consistent with the finding in recent microrheology experiments with entangled DNA solutions that a return to continuum SE behavior requires 2R0/dt ≈ 6.33 Theoretical calculation by Yamamoto et al. predicted for heavily entangled polymer liquids (N/Ne ≫ 1),

Figure 4. (a) Bulk viscosity of PBMA as a function of temperature. (b) Ratio D/DSE is plotted for different sized particles as a function of temperature. (c) Temperature averaged D/DSE as a function of size ratio. The dotted line is a prediction from ref 15 using M/Me ≈ 12.

the crossover to SE friction would occur at 2R0/dt ≈ 6/√S0, where S0 is material dependent dimensionless compressibility, which depends weakly upon temperature.15 For PBMA, the temperature averaged S0 is ≈0.17, giving a theoretical estimate of crossover at 2R0/dt ≈ 14. In our experiments, the full return to SE behavior was not observed because the dynamics becomes extremely slow for particles larger than 20 nm. But from Figure 4 we can conclude that a ratio of 2R0/dt ≈ 7−10 would be required so that the probe motion is fully coupled with the chain relaxation. Apart from SE crossover behavior, the theory by Yamamoto et al. provided quantitative prediction for D/DSE for all size ratios which is plotted in Figure 4 as comparison.23 As shown, the theory predicted a lower D/DSE compared to experiments. The current theory only considered the role of constraint release but ignored the possible role of NP mobility, which can be significant for smaller particles. A more sophisticated version of this theory has been developed recently by treating them in self-consistent manner, and its overall effect is to increase the ratio D/DSE.34 As mentioned earlier, our control experiments suggested that the interaction between AuNPs and PBMA is weak. In this situation, a depletion layer can form around the particles, where the segmental density increases from zero at the particle surface to the bulk value. Assuming that viscosity is proportional to the density, the viscosity gradient can generate slip as the particle moves through the medium. The effect of low viscosity depletion layer on sphere motion was had been treated theoretically in solution,35 but the overall effect had been found to be small. It can increase the particle mobility at most by a factor of 2. Another possible mechanism is the thermally induced activated hopping of the particle, the role of which is still a matter of debate.16,17,21 It is generally agreed that its contribution is strongest near 2R0 ≈ dt and decays as the size of the particle increases. But there are disagreements on the exact functional dependence of diffusion coefficient on 2R0/dt and experimentally verifiable numerical prefactors. We note that if NP mobility on structural relaxation and hopping motion are properly taken into account, the agreement with experimental results would be closer. 7241

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(17) Cai, L.-H.; Panyukov, S.; Rubinstein, M. Macromolecules 2011, 44, 7853−7863. (18) Brochard Wyart, F.; de Gennes, P. G. Eur. Phys. J. E 2000, 1, 93−97. (19) Ganesan, V.; Pryamitsyn, V.; Surve, M.; Narayanan, B. J. Chem. Phys. 2006, 124, 221102. (20) Liu, J.; Cao, D.; Zhang, L. J. Phys. Chem. C 2008, 112, 6653− 6661. (21) Kalathi, J. T.; Yamamoto, U.; Schweizer, K. S.; Grest, G. S.; Kumar, S. K. Phys. Rev. Lett. 2014, 112, 108301. (22) Guo, H.; Bourret, G.; Lennox, B.; Sutton, M.; Harden, J. L.; Leheny, R. Phys. Rev. Lett. 2012, 109, 055901. (23) Kohli, I.; Mukhopadhyay, A. Macromolecules 2012, 45, 6143− 6149. (24) Cole, D. H.; Shull, K. R.; Baldo, P.; Rehn, L. Macromolecules 1999, 32, 771−779. (25) Ernst, D.; Hellmann, M.; Kohler, J.; Weiss, M. Soft Matter 2012, 8, 4886−4889. (26) Narayanan, S.; Thiyagarajan, P.; Lewis, S.; Bansal, A.; Schadler, L. S.; Luruo, L. B. Phys. Rev. Lett. 2006, 97, 075505. (27) Narayanan, S.; Lee, D. R.; Hagman, A.; Li, X.; Wang, J. Phys. Rev. Lett. 2007, 98, 185506. (28) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: New York, 2003. (29) Zhang, Q.; Archer, L. A. J. Chem. Phys. 2004, 121, 10814. (30) Starr, F. W.; Schroder, T. B.; Glotzer, S. C. Macromolecules 2002, 35, 4481−4492. (31) Crawford, M. K.; Smalley, R. J.; Cohen, G.; Hogan, B.; Wood, B.; Kumar, S. K.; Melnichenko, Y. B.; He, L.; Guise, W.; Hammouda, B. Phys. Rev. Lett. 2013, 110, 196001. (32) Mason, T. G.; Weitz, D. A. Phys. Rev. Lett. 1995, 74, 1250− 1253. (33) Chapman, C. D.; Lee, K.; Henze, D.; Smith, D. E.; RobertsonAnderson, R. M. Macromolecules 2014, 47, 1181−1186. (34) Yamamoto, U.; Schweizer, K. S. Submitted 2014. (35) Fan, T. H.; Dhont, J. K. G.; Tuinier, R. Phys. Rev. E 2007, 75, 011803.

CONCLUSION We have determined the frictional force experienced by nanoparticles in a polymer melt in situations where their sizes are comparable to the entanglement tube diameter. The particles diffuse faster compared to prediction from the Stokes−Einstein relation using the bulk polymer viscosity and their size needs to be greater than a few times the tube diameter to follow SE prediction. The novelty of this study is the direct measurements of diffusion coefficients by using a series of nanospheres with different sizes but identical surface chemistry to vary the size ratio 2R0/dt. This allowed us to determine the functional dependence of D and D/DSE on the size ratio and make comparison with recent theories and simulation, which were not possible in previous studies with single particle size. The experimental methodology that we apply here can be extended to investigate other complex fluids and rotational diffusion for particles possessing different shapes and geometries.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Tel 313-577-2775; Fax 313-5773932 (A.M.). Present Address

C.A.G.: Air Force Research Laboratory, Wright-Patterson AFB, OH. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. K. Koynov for performing the rheology measurements and Prof. K. S. Schweizer for valuable comments. Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research fund (PRF # 51694-ND10) for support of this research.

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REFERENCES

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dx.doi.org/10.1021/ma501670u | Macromolecules 2014, 47, 7238−7242