Macromolecule and Particle Dynamics in Confined Media

Perspective pubs.acs.org/Macromolecules

Macromolecule and Particle Dynamics in Confined Media Chia-Chun Lin, Emmabeth Parrish, and Russell J. Composto* Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6272, United States ABSTRACT: Dynamic properties play an important role in designing functional polymer nanocomposites, impacting molecular transport and phase separation kinetics. When nanoparticle (NP) size is comparable to polymer chain size, segmental relaxations may be influenced by changes in chain conformations and packing at the polymer/NP interface. Following the reptation model, these changes can perturb the longest relaxation time, in particular, the center-of-mass (COM) dynamics of polymer chains in entangled melts. This Perspective focuses on unsolved issues in polymer COM diffusion and local dynamics and segmental motions in the presence of NPs. The article introduces the effect of NP size, shape, surface modification, and enthalpic interactions on polymer diffusion and further relates dynamic studies in PNCs to macromolecular transport in bio-related systems and nanopores. Studies of local dynamics also provide insights into how entanglement density, monomeric friction, and chain conformation are influenced by NPs and how the interplay between these key parameters relates to COM dynamics to provide a unified picture across length scales. Moving forward, new studies investigating dynamics in PNCs are needed to address these unresolved problems and motivate potential applications from membranes for separations to NP carriers for drug delivery.

1. INTRODUCTION Polymer composites based on nanoparticles (NPs) have had a tremendous impact in science and technology.1 As opposed to traditional composites, the fillers in polymer nanocomposites (PNCs) have dimensions below 100 nm and therefore exhibit high surface area to volume ratios. Correspondingly, the addition of a small volume of nanofillers produces a great amount of interfacial area between polymer matrix and nanofillers, resulting in either improved or new properties without compromising attractive properties inherent to neat polymers such as toughness, processability, light weight, and optical transparency.1−3 One of the important challenges in this field is to control the NP dispersion state such that a desired property can be achieved. Whereas previous reviews focused on morphology,4,5 here we focus on dynamics of polymers in confined media, focusing on PNCs. In addition to efficiently achieving desired properties, the incorporation of nanofillers into polymers can impart superior and unique properties that traditional fillers are unable to achieve such as mechanical strength,6 flame resistance,7 and thermal,6 optical, and electrical properties.8−10 For example, tunable optical properties are observed upon adding and assembling gold nanorods in polymers,11,12 and high-efficiency polymer-based photovoltaic can be created by adding TiO2 NPs.13 Furthermore, nanofillers also change polymer viscosity,14−28 a critical parameter that influences molecular transport and flow behavior. Because polymer dynamics are typically changed upon incorporating functional NPs, understanding the dynamics of PNCs would provide insights into fundamental topics such as optimizing viscosity to improve processing conditions and manipulating assembly kinetics29 as well as applied topics, including coatings, self-healing materials, and membranes.30−32 © XXXX American Chemical Society

Although the focus of this Perspective is polymer dynamics in PNCs, an improved understanding of macromolecular dynamics in confined media can impact our understanding of biological processes and devices. The transport of macromolecules in crowded media is linked to understanding the activity of biomolecules in cellular environments as well as the functions of bio-related systems.33−37 For example, in vivo biological processes and reactions, such as protein folding, translocation, and RNA transport, take place in cells even though macromolecules and organelles occupy from 10 to 40 vol % of the medium.33,34 In many bio-related devices, macromolecule diffusion in confined media is critical to their functionality. For example, macromolecules threading through nanopores can be used to identify single-stranded DNA and RNA without amplification and labeling, making rapid biomolecular sequencing possible.38 Similarly, using gel electrophoresis in nanofabricated channels, DNA molecules are stretched which enables accurate DNA mapping;39 also, protein separation can be efficiently achieved using ultrathin porous nanocrystalline silicon membranes with nanopores from 5 to 25 nm.37 Templated nanopores provide a model system for understanding both the above confinement effects in biological systems and local dynamics of polymer chains. Hydrogels, 3D networks of crosslinked hydrophilic polymers,40,41 also provide an excellent platform for molecule separation in protein chromatography and membrane technology.42,43 Because they are biocompatible, and sensitive to changes in temperature, pH, and ionic strength, hydrogels have broad applicability, such as in biosensors44−46 Received: March 4, 2016 Revised: July 26, 2016

A

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in entangled polystyrene melts and reported N−2 dependence in agreement with the reptation model.64 A dependence of N−2.3 has also been reported62 and explained by contour length fluctuations due to random fluctuation of chain end positions in the tube. If the matrix degree of polymerization (P) is less than a characteristic value (P*) and greater than entanglement degree of polymerization (Ne), the test chain can also relax by a constraint-release mechanism65,66 because the tube relaxes before the test chain can completely escape the original tube. Because reptation and constraint release are independent mechanisms, the diffusion coefficient is the sum of two contributions

and drug delivery. 47−50 Various reviews of biomedical applications using hydrogels are given elsewhere.40,51−56 The transport of macromolecules through confined regions is important in drug delivery. For example, drug transport through hydrogels directly determines the rate of release to target cells.57−59 Current challenges in drug delivery, such as development of mucus penetrating particles and DNA transport through the cell cytoplasm, require better understanding of macromolecule interactions with local environment as well as their center-of-mass (COM) diffusion. In this Perspective, we consider key experimental studies and unresolved issues critical to advancing the field related to dynamics in confined polymer systems. We begin with a brief introduction of the seminal models of polymer diffusion. Then, we ask how crowding affects polymer COM diffusion as nanofiller concentration increases, and whether nanofiller dimension and shape influence polymer COM diffusion. As the length scale decreases to the size of a segment, we ask how chain conformation, entanglement density, and monomeric friction change adjacent to the nanofiller and how an adsorbed layer near the nanofiller influences relaxation. The goal of these sections is to begin developing a unifying picture of polymer dynamics across length and time scales. In addition to an overview relating COM and segmental dynamics in PNCs, we highlight similarities and differences between dynamics within PNCs and those within the uniform confinement of templates and the complex confinement of hydrogels and biopolymer networks. Furthermore, the dynamics of nanofillers themselves are needed to help understand polymer diffusion in crowded environments which are continually rearranging. A better understanding of dynamics in PNCs will allow us to further control their unique properties and lead to new advanced materials. Polymer Diffusion Models. The reptation model60,61 provides a framework for understanding polymer diffusion in an entangled melt or strongly cross-linked polymeric gel. To simplify the entangled matrix, Doi and Edwards developed a tube model in which a test chain is confined in a tubelike region imposed by surrounding chains, and the lateral motion of the confined test chain is restricted by the topological constraints (i.e., entanglements). Introduced by de Gennes60 in 1971, the reptation model is based on restricted chain dynamics along this tube. The time for the restricted chain to diffuse out of the original tube of average length ⟨Ltube⟩ is the reptation time, τrep. Theoretically, the reptation time scales with the degree of polymerization, N, of the diffusing chain as N3.0 whereas experimentally scaling is greater than 3, i.e., τrep ∼ N3.4. Lodge has attributed this stronger dependence to tube length fluctuations.62 The reptation model predicts that the reptation diffusion coefficient (Drep) scales as

Drep ∼

D* = Drep(N ) + Dcr (N , P) ≈

(2)

where Dcr is the contribution from constraint release and τtube is the Rouse relaxation time of the confining tube. The faster of the two mechanisms controls the diffusion of the test chain. Namely, reptation dominates when P ≫ N (i.e., since τtube > τrep), whereas constraint release dominates when P < N (i.e., τtube < τrep). Originally conceived to describe conductivity in heterogeneous media, the Maxwell model67 has been adopted to describe polymer diffusion through PNCs. In this model, the COM diffusion coefficient decreases monotonically as the concentration of filler particles increases. Namely, the model predicts that diffusion varies as 1−ϕ D = D0 1 + ϕ/2

(3)

where D0 is the diffusion coefficient in pure matrix and ϕ is the filler volume fraction. In addition to being limited to very dilute filler concentration, the model does not account for the change in chain structure near the filler surface; this impact could be more significant for nanofillers due to their size which can be comparable to the chain dimension. Furthermore, entropy and enthalpic interactions should be considered because of chain confinement and chain/nanofiller interactions, respectively. This Perspective will focus mainly on polymer dynamics in the presence of nanosized fillers, where time scales range from slow relaxation, i.e., COM diffusion, to fast segmental relaxation.

2. MACROMOLECULE DYNAMICS IN PNCs WITH SPHERICAL NPs 2.1. Polymer Diffusion through PNCs with Hard Neutral NPs. The simplest case of geometric confinement is when NPs act as immobile neutral barriers to the COM diffusion of polymers. Therefore, enthalpic interactions between NPs and host polymer should be minimized. Because of their heterogeneous structure, semicrystalline polymers present a complex matrix in which polymer diffusion is restricted to amorphous regions whereas crystalline regions act as barriers. Using dynamic secondary ion mass spectroscopy (DSIMS), Segalman et al.68 investigated the diffusion of deuterated polystyrene (dPS) in precrystallized isotactic polystyrene (PS) matrices and reported that the tracer diffusion coefficient (D) decreases with increasing N, i.e., D ∼ N−1. This N−1 dependence of D is consistent with the scaling prediction of the entropic barrier model (EBM), proposed by Muthukumar and Baumgartner,69,70 where polymer diffusion is modeled as a self-avoiding polymer chain moving through arrays of cubic cavities connected by narrow bottlenecks.

R2kBTNe ζb2N3

R2 R2 + τrep(N ) τtube(N , P)

(1)

where R is the end-to-end distance of the chain, kB is the Boltzmann constant, T is temperature (K), Ne is the degree of polymerization of an entanglement strand, ζ is monomeric friction coefficient, and b is monomer length. A dimensionless prefactor is not shown in order to focus on the physical parameters that impact Drep. Given that ζ and Ne are constant when the tracer and matrix chairs are similar, Drep is proportional to N−2. Using forward recoil spectrometry, Mills et al.63 measured tracer diffusion coefficients of deuterated polystyrenes B

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Macromolecules The entropic barrier is the activation energy that a polymer chain has to overcome when it diffuses from a region of large volume, where the chain can access many configurations, or higher entropy, to a region of smaller volume, where the chain has fewer possible configurations. In semicrystalline polymers, the randomly distributed crystalline lamellae serve as obstacles that define bottlenecks (amorphous region) through which dPS chains diffuse, as shown in Figure 1. However, because spherulite

Figure 2. Normalized tracer diffusion coefficients plotted against the confinement parameter. Open and closed symbols correspond to systems containing NPs with d = 13 nm and d = 29 nm, respectively. ID was calculated using eq 4. Dashed curve is an empirical fit to the experimental data. Reproduced with permission from ref 72. Copyright 2012 the Royal Society of Chemistry.

radius of gyration of the tracer and ID is interparticle distance defined by72 1/3 ⎧⎛ ⎫ ⎪ 2 ⎞ ⎪ 2 ⎜ ⎟ ⎨ ID = dn ⎜ ⎟ [exp(ln ρ) ] − 1⎬ ⎪⎝ πϕNP ⎠ ⎪ ⎩ ⎭

Figure 1. Illustration of a test chain (red) diffusing through a semicrystalline polymer. Crystalline spherulites are impenetrable by polymer chains, surrounded by amorphous region where polymer chains can freely diffuse. For simplicity, this cartoon assumes an amorphous region between neighboring spherulites. The amorphous region can also be between lamellae within the spherulites. Bottlenecks are formed between crystals where chains have to overcome the entropic barrier.

(4)

where dn is the number-average NP diameter and ρ is the NP diameter polydispersity using a log-normal distribution. The collapse of the data onto a master curve suggested that the confinement parameter was able to capture the effect of NP dimension, NP size distribution, and tracer molecular weights. Moreover, the scaling behavior of the tracer diffusion coefficient was in agreement with the EBM, which accounts for the loss in chain entropy caused by bottlenecks between NPs. These results suggest that the slowdown in dPS diffusion due to confinement imposed by NPs may be explained by the EBM. 2.2. Polymer Diffusion through PNCs with Hard Attractive NPs. Attractive enthalpic interactions between NP and polymer can facilitate NP dispersion; this enthalpic contribution could also affect polymer dynamics relative to the athermal case. For example, Zheng et al.75 reported polymer diffusion as a function of distance from an attractive planar surface and found an order of magnitude decrease in diffusion due to surface−polymer interaction relative to the bulk. Hu et al.76 also reported a slowdown in the diffusion of deuterated poly(methyl methacrylate), or dPMMA, by a factor of 3 in PMMA/clay nanocomposites at clay loading of ∼5 vol %. Using molecular dynamics, simulation studies77 showed a slowdown in polymer diffusion in the presence of attractive NP−polymer interactions. Furthermore, using FRES, Lin et al.78 measured the COM diffusion of dPMMA in PMMA matrices containing hydroxyl-terminated silica NPs where attractive NP−polymer interactions were present. The diffusion of dPMMA was slowed down as a function of NP loadings and data collapse onto a master curve when the normalized diffusion coefficient (D/D0) was plotted versus the confinement parameter (ID/2Rg). Interestingly, when compared with the studies of dPS diffusion into PS matrices containing phenyl-capped silica NP,71,72 i.e., neutral NP−polymer interactions, the scaling of D/D0 vs ID/2Rg was very similar, suggesting that attractive NP−polymer interactions were insufficient to alter COM polymer diffusion. Theoretically, Meth et al.79 developed an analytical model to

size and dispersity are hard to control and the space available for diffusion is ill-defined in semicrystalline polymers, a model system is needed to rigorously test the applicability of the EBM for describing polymer diffusion through a matrix with obstacles. PNCs are an ideal platform for testing diffusion in confined systems because the size and distribution of NPs can be controlled, and thus the space available for the diffusion of the polymer chain can be well-defined. In addition, the surface of NPs can be functionalized using short ligands so that the NP− polymer interaction is nearly athermal yet maintains a sharp interface with the matrix. For these cases, “hard” NPs, with short neutral ligands, are expected to be compatible with the host polymer. Gam et al.71,72 investigated the diffusion of dPS in PS nanocomposites containing neutral NPs using forward recoil spectrometry (FRES). FRES is particularly suited to measure the depth profile of light elements such as deuterium in a hydrocarbon matrix, mainly carbon and hydrogen. Thus, the concentration of deuterated polymers can be profiled to investigate polymer diffusion as well as surface and interface segregation. Details of FRES are reviewed elsewhere.73,74 In these systems, silica NPs with particle diameters 13 and 29 nm were grafted with a coupling agent, phenyltrimethoxysilane (PhTMS). The surfaces of these phenyl-capped silica NPs were chemically identical to the host polymer, PS, which resulted in a neutral environment for polymer and NPs. NPs were welldispersed in PS up to volume fraction, ϕNP = 0.5. These studies showed that the diffusion coefficient of dPS was reduced significantly in the presence of NPs (e.g., by 80% at ϕNP = 0.5). Furthermore, as shown in Figure 2, the normalized diffusion coefficients of dPS, D/D0, collapsed onto a master curve when plotted against a confinement parameter, ID/2Rg, where Rg is the C

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Macromolecules describe the slowdown of polymer diffusion in the presence of NPs by modeling polymer chains as spheres diffusing through cylindrical pores and found that the reduction in diffusion can be attributed to excluded volume. The model was able to capture the experimental results of Gam et al.71,72 and Lin et al.78 at low loadings for ID/2Rg > 2 where polymers can still maintain their Gaussian conformation. At high loadings, the model overestimated the experimental diffusion coefficients, suggesting that slowing down in highly crowded environments invokes other mechanisms, such as altered chain conformations and entanglement density as discussed later. 2.3. Polymer Diffusion through PNCs with Polymer Grafted NPs. To control NP/matrix interactions, NP surfaces can be modified with a polymer brush. The dispersion of NPs in PNCs can be enhanced by these brushes when the matrix chains wet the brush, resulting in repulsive interactions between NPs. In addition, various NP dispersion states with unique properties can be achieved by tuning brush graft densities and the ratio of degree of polymerization between the brush, Nb, and the host polymer.4,80,81 For example, optical absorption can be controlled by tuning NP−NP spacing,11,12,82 and enhanced mechanical properties can be achieved by tuning NP morphologies in the matrix.83,84 Because the grafted brush on the NP can result in a broad interface due to interpenetration between brush and matrix/tracer polymer, NPs grafted with brushes can behave as “hard” or “soft/penetrable” obstacles depending on whether the matrix chains are dry or wet, respectively. To understand how this tunable brush/matrix interface influences COM polymer diffusion, the ratio of the degree of polymerization between grafted brush and polymer matrix can be varied so that the transition from hard NP to soft NP is controlled. Choi et al.85 investigated dPS diffusion in the presence of soft NPs and compared the result with the hard NP case. Silica NPs grafted with a long PS brush (87 kg/mol) were dispersed in a PS matrix (160 kg/mol), and the diffusion of dPS (23−532 kg/mol) was measured using FRES. The dPS penetration into the brush depended on the ratio between the degree of polymerization of dPS and the brush (i.e., N/Nb). Namely, larger tracers penetrated less deeply into the brush than smaller tracers, resulting in a sharp interface. This allowed an effective particle diameter to be defined. Using self-consistent field theory (SCFT) coupled with small-angle neutron scattering (SANS), an effective ID (IDeff) could be determined using an effective particle diameter (deff) given by the hard NP diameter plus twice the dry brush thickness. For the larger dPS, the ID was reduced because dPS chains were partially excluded from the brush. Because smaller dPS chains wet the brush, IDeff was close to the hard sphere value. As shown in Figure 3, when normalized diffusion coefficients, D/D0, were plotted against the effective confinement parameters, both hard and soft NP systems showed similar scaling behaviors, suggesting that the universal scaling of polymer diffusion previously observed for hard NPs also extended to soft NPs. In addition, the bulk diffusion coefficients were recovered (D/D0 = 1) when NPs were far apart, i.e., ID/2Rg > 20, indicating the NPs had a long-range effect on polymer diffusion. A similar long-range effect was observed in a planar surface system75 where the diffusion of dPS was slower than that in the bulk up to a distance of 10Rg from the surface. This result was explained by a change in local structure in the vicinity of the surface due to an attractive force, which propagated a distance away from the surface. This long-range effect is still not well understood, and models are required to accurately explain this observation. The general principle that emerges from these studies on polymer diffusion in

Figure 3. Normalized diffusion coefficient of dPS (23, 49, 168, 532, and 1866 kg/mol) plotted as a function of IDeff/2Rg. Closed and open symbols represent dPS diffusion in PNCs having soft NP and hard NP, respectively. deff was calculated using SCFT, and IDeff was calculated using eq 4 and dn = deff. Relative to smaller dPS chains, the inset illustrates that larger dPS chains are unable to penetrate into the brush resulting in a larger effective NP size. Adapted with permission from ref 85.

PNCs is that confinement can be described by a simple parameter, ID/2Rg, which captures the slowing down of macromolecules independent of interactions between NP and polymer.

3. MACROMOLECULE DYNAMICS IN PNCs WITH ANISOTROPIC NPs 3.1. Polymer Dynamics in Polymer/Carbon Nanotube PNCs. Whereas D decreases monotonically as spherical NP concentration increases, polymer diffusion in the presence of carbon nanotubes (CNT) initially slows down, reaches a minimum, and recovers as CNT loading increases. Using FRES, Mu et al.86 investigated tracer (dPS) diffusion in PS matrices containing single-walled carbon nanotubes and reported that tracer diffusion decreased initially as SWCNT concentration increased and then recovered beyond a critical concentration, φcrit ∼ 0.4 vol %, in a PS matrix as shown in Figure 4. The critical concentration was consistent with a dynamical percolation threshold measured by rheology, indicating that the diffusion minimum correlated with the formation of a CNT network. A phenomenological trap model was proposed and simulated where polymer diffusion was anisotropic in the vicinity

Figure 4. Diffusion coefficient of dPS in SWCNT/PS (480 kg/mol) nanocomposites plotted against SWCNT volume fractions. dPS molecular weights range from 75 to 680 kg/mol. The minimum diffusion coefficient occurs near 0.4 vol % of SWCNT. Adapted with permission from ref 86. D

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short loadings (ϕNR = 0.1> ϕmin), dPS diffusion scales as 2 and 2.3 for low and high molecular weight tracers, respectively. This behavior is similar to the spherical NP case because L < 2Rg, dPS, and therefore the NRs appear as isotropic obstacles. For high molecular weight, the slightly greater exponent of 2.3 may be explained by entropic barriers imposed by stronger confinement (smaller NR mesh size) compared to the NR long systems. In contrast, for the PS/NR long system, D scales as 1.1 for dPS at high molecular weight. In this case, L > 2Rg, indicating that the tracer chains are perturbed by the anisotropic obstacles. The weaker scaling is attributed to monomer−NR friction as NRs are able to thread through polymer chains as well as to expanded chain conformation, recently observed by Tung et al.90 for PS in the presence of SWCNT. 3.3. Polymer Dynamics in the Presence of Chained NPs. Sections 2, 3.1, and 3.2 have described the COM dynamics of polymer chains in the presence of isolated individual spherical and high aspect ratio cylindrical NPs. In this section, we focus on polymer dynamics in the presence of chained NPs (cNP, shown in Figure 6a) that mimic carbon black aggregates commonly used to reinforce tires.91 Note that the cNPs have an irregular geometry whereas the model spheres71,72,78 and cylinders86,87,92 are well-defined NPs; therefore, it is important to understand whether polymer diffusion through PNCs with irregularly shaped cNPs differs from the model NP systems. Lin et al.93 probed COM tracer (dPS) diffusion in PS nanocomposites containing PS-grafted, stringlike cNPs (d = 5 nm and ∼5 individual NPs per cNP as shown in Figure 6a) using FRES. When the molecular weight of dPS was very large (1866 kg/mol), D decreased monotonically with increasing cNP core volume fraction (ϕcNP), consistent with the results for spherical NPs. In contrast, when the molecular weights of dPS were small (49, 168, and 532 kg/mol), D decreased with increasing ϕcNP, reached a minimum at ϕcNP = 0.0025, and recovered again for ϕcNP > 0.0025, consistent with the behavior for CNT and nanorod PNCs. The importance of relative size between the tracer (2Rg) and cNP (d and L) is evident in how these stringy cNPs impose polymer chains to diffuse anisotropically. In addition, the brush could affect tracer diffusion because the dPS can penetrate into the brush region when the tracer is short relative to the brush length. For a two-phase matrix, an effective diffusion coefficient (De) is defined to reflect different diffusion rates through a continuous polymer matrix (D0) and a discrete brush region around the NPs (D1), eq 5, where ϕ0 and ϕ1 are the volume fractions of the PS matrix and PS brush regions, respectively.94

of NPs. Namely, polymer diffusion was faster parallel to the CNT than perpendicular to the CNT, and thus beyond the critical concentration, polymer chains were able to diffuse fast along the CNT network, resulting in the diffusion recovery. A further investigation87 by the same research group using multiwalled carbon nanotube (MWCNT) suggested that this diffusion minimum was a function of relative size between the tracer polymer and the nanofiller; i.e., the diffusing chain had to be larger than the CNT diameter to observe a minimum diffusion coefficient. These initial studies motivated the need for examining tracer diffusion in model PNCs where the anisotropic NP length was shorter and relatively monodisperse (section 3.2). 3.2. Geometric Criteria for Polymer Diffusion in PNCs. Two criteria have been proposed for the observation of a minimum in the diffusion coefficient in PNCs with CNTs, namely, that (1) the size of the tracer chain (2Rg) has to be larger than the nanorod diameter and (2) the CNTs form a percolating network. However, the effect of anisotropic NP length (L) could not be investigated because the CNT’s were semi-infinite. Choi et al.88 investigated geometric factors and systematically studied anisotropic polymer diffusion in the presence of nanorods (NRs) by varying tracer molecular weight, NR diameter (dNR), and NR length (NR short: L = 43.1 nm; NR long: L = 371 nm). They observed nonmonotonic diffusive behavior when the tracer size was larger than the NR diameter (2Rg > dNR) but less than the NR length. On the contrary, monotonic diffusive behavior was observed when the tracer size was either larger or smaller than both the NR diameter and length. Figure 5 shows the dPS diffusion coefficient in NR/PS nanocomposites plotted against tracer molecular weight. At low

D1 − De D − De ϕ1 + 0 ϕ =0 D1 + 2De D0 + 2De 0

(5)

Thus, D can be normalized using D0, as done previously,71 or De, which accounts for the heterogeneity of the media. Namely, D/ D0 does not account for the effect due to the brush, whereas D/ De distinguishes between diffusion in the brush and matrix regions, incorporating the effect of polymer brush on tracer diffusion. When normalized diffusion coefficients were plotted against cNP loading, a sharp transition from a diffusion minimum to a monotonic decrease was observed for D/D0 (Figure 6b), whereas a gradual transition was observed for D/De (Figure 6c). This study demonstrates that tracer diffusion in heterogeneous matrices should take into account dynamics in both the matrix and brush regions. Additionally, this study highlights how morphology and arrangement of NPs could affect diffusion of

Figure 5. dPS diffusion coefficient in PS nanocomposites containing NR long (squares) or NR short (circles) plotted against dPS molecular weight at ϕNR = 0.04 (a) and 0.1 (b). Red and blue dashed circles show the size of tracer chains relative to NR long and NR short, respectively. Adapted with permission from ref 88.

NR loadings (ϕNR ≤ ϕmin = 0.04 where ϕmin is the NR concentration near the diffusion minimum), the scaling behavior for both short and long NRs follows D ∼ N−2, in agreement with reptation,89 suggesting that chain conformation and local friction felt by the tracer chain are not perturbed by the NRs. At high NR E

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Figure 6. (a) TEM image of neat cNP showing the morphology of the particle with scale bar = 100 nm. The cartoon illustrates that cNP is composed of individual spherical Fe3O4 NPs (black spheres) grafted with a PS brush (blue chains). Normalized diffusion coefficient plotted as a function of cNP volume fraction. Diffusion coefficients are normalized by (b) D0 and (c) De. dPS molecular weights are 49 kg/mol (black), 168 kg/mol (red), 532 kg/ mol (green), and 1866 kg/mol (blue). Solid lines are guide to the eye. Adapted with permission from ref 93.

4.1. Polymer Chain Conformation in PNCs. Upon adding NPs into a polymer matrix, the chain conformation can be altered depending on filler size and concentration. For PNCs containing spherical nanofillers, the radius of gyration, Rg, of the chain can increase when Rg is greater than the NP size, i.e., 2Rg > d; however, Rg is not significantly affected for relatively short chains, 2Rg < d. For example, using SANS, Nakatani et al.96 investigated poly(dimethylsiloxane) (PDMS) chain dimensions in the presence of trimethylsilyl-treated polysilicate and compared the result with Monte Carlo calculations. Note that SANS utilizes scattering differences between deuterated and hydrogen containing polymers to probe polymer structure and thermodynamics.97−99 They reported that 2Rg of chains larger than d ∼ 2 nm increased by 50% upon adding 20 wt % of the filler. However, for PDMS chains with similar dimensions as the filler, Rg slightly decreased as filler loading increased. In addition, a similar increase in Rg was observed in PS matrices containing intramolecular cross-linked PS NPs where the Rg increased by up to 20% when 10 vol % of NPs (d = 7 nm) was added.100 Conversely, when NPs aggregated to form larger particles, polymer chains did not expand because the relatively large aggregates were now greater than Rg.98,99,101 Simulations also revealed that Rg increased in the presence of small NPs (i.e., d < 2Rg) in various systems but was unaffected when the NP sizes were comparable to Rg.102−104 Furthermore, in PS/CNT systems, Tung et al.90 reported a 30% increase of Rg upon

polymers in PNCs. Further studies are needed to determine the mechanism of polymer diffusion through PNCs with aggregates of NPs. These studies are experimentally challenging because they require a matrix with a controllable size and distribution of aggregates that remain stable (i.e., do not coarsen) during polymer diffusion. Recent advances using numerical dynamic mean-field theory suggests that nonequilibrium morphologies can be accurately predicted in PNCs,95 and therefore the vast PNC parameter space and processing conditions can be greatly reduced for future experimental studies.

4. MACROMOLECULE LOCAL DYNAMICS IN PNCs In the reptation model,60 the motion of an entangled chain is confined by surrounding chains which form a tubelike topological constraint as described in section 1. As noted in eq 1, the COM diffusion coefficient scales as Drep ∼ R2Ne/ζ. Here R is the square root of the mean-squared end-to-end distance of a free polymer chain in the absence of confinement, Ne determines the tube diameter (dt), and ζ, the monomeric friction coefficient, derives from the viscosity (η) of the melt. Therefore, at a given thermal energy, R, Ne, and ζ govern the diffusion of entangled chains in an entangled matrix. If these parameters are perturbed by the addition of nanofillers, the diffusion coefficient is expected to change accordingly. Therefore, in this section, we examine the effect of NPs on chain conformation, segmental relaxation time, and entanglement density. F

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Macromolecules adding 10 wt % of SWCNT (dSWCNT/2Rg ∼ 0.4), whereas a slight decrease in chain dimension was observed when 2 wt % of MWCNT was added (dSWCNT/2Rg ∼ 1). On the contrary, Crawford et al.105 reported that PS chain dimensions were not influenced by adding Si NP (d = 13 nm) for loadings up to ϕNP = 32.7 vol %, regardless of chain size relative to NP diameter. Using Monte Carlo calculations and ϕNP = 20 vol %, Vacatello106 also reported no change in chain dimensions when 2Rg was ∼2.5× greater than the filler diameter. While critical length scales that influence chain conformation still remain unclear, other dimensions should be considered in future studies such as particle size relative to dt. Furthermore, chain conformation could depend on an additional parameter, such as particle anisotropy as in the PS/CNT composite studied by Tung et al.,90 and particle−polymer interactions.100 However, a comparison between the filler and polymer sizes provides a simple approach to predict the length scale at which confinement is expected to produce a change in polymer dynamics. 4.2. Segmental Relaxation in PNCs. The monomeric friction coefficient (ζ) is directly proportional to the shortest Rouse relaxation time (τ0) that is experimentally measurable. This segmental relaxation time is typically on the order of a nanosecond at T − Tg = 100 K, and thus neutron scattering and NMR spectroscopy techniques107 are useful methods to observe chain motion. For systems with weak or neutral interactions between polymer and NPs, experimental studies98,108−110 showed that segmental relaxations were unaffected. For example, Schneider et al.108 used neutron spin echo spectroscopy (NSE) to investigate segmental dynamics in poly(ethylene−propylene) (PEP) matrices containing silica NPs (d = 17 nm) coated with short hydrocarbons. They reported that the Rouse relaxation rate remained unaffected as NP loading increased. Conversely, simulation studies111,112 suggested enhanced segmental dynamics when NPs were added to a polymer matrix. Using molecular dynamics simulations, Kalathi et al.113 pointed out the importance of NP size relative to entanglement mesh size. Namely, in a PNC where NPs were smaller than the mesh size, NPs reduced the monomer friction, enhancing segmental relaxation, whereas for NPs larger than the mesh size, segmental relaxation times were not affected significantly even at high NP loadings. This indicates that NP size is an important parameter in determining segmental dynamics in athermal systems and possibly explains the divergent experimental results. For systems with favorable polymer−NP interactions, bound layers are reported in the vicinity of the NPs, suggesting that attractive interactions can suppress the translational diffusion of chains near the NP surface. Whereas a review of polymer−filler interactions in intercalated nanocomposites is given elsewhere,114 this Perspective will provide a more general discussion of how these interactions impact segmental relaxation. On a shorter length scale, different segmental behavior of adsorbed chains has been reported including (i) a glassy shell, (ii) slower segmental mobility, and (iii) no change in segmental mobility. The glassy shell is a layer of highly immobilized segments attached onto the surface of NPs due to attractive interactions.115−120 Berriot et al.115,116 reported an immobilized polymer layer on colloidal silica (d = 60 nm) in a poly(ethyl acrylate) PNC. The thickness of this immobilized layer decreased with increasing temperature. In addition, studies suggest that the thickness of the immobilized layer increases as NP size increases or curvature decreases. For example, Harton et al.117 reported a 1 nm thick bound layer around 15 nm diameter silica NPs compared to a 5 nm layer on a flat silica surface.

Estimations of the thickness of the glassy shell are based on the assumption that polymers in the shell are completely immobilized compared to bulk polymers, whereas simulation and experimental studies of local dynamics show that chain mobility in the vicinity of the NPs exhibits slower dynamics relative to bulk polymer.113,121−126 For example, using broadband dielectric spectroscopy (BDS) and SAXS, Holt et al.122 investigated segmental dynamics near the NP surface in poly(2vinylpyridine)/silica nanocomposites and showed that the interfacial region (4−6 nm) exhibited a gradient in segmental mobility, i.e., slower segmental relaxation near the NP surface that recovered to its bulk value near the interfacial region. Using rheology, Archer and co-workers observed bulk slowing of segmental dynamics of PMMA containing silica NPs grafted with poly(ethylene glycol) (PEG).127 In contrast, using dielectric relaxation measurements and neutron spin-echo spectroscopy, Bogoslovov et al.128 and Glomann et al.109 reported no change in segmental relaxations. These divergent results raise the question of how to interpret data measured at different time scales and length scales probed by various techniques.129 Moreover, whether the relative size between the polymer entanglement and NP is an important parameter affecting local dynamics in PNCs remains an open question. Future studies are needed to address these issues. 4.3. Entanglement Density in PNCs. The entanglement degree of polymerization, Ne, is a microscopic quantity that is usually determined from the measured plateau modulus of a pure melt. However, in PNCs, traditional viscoelastic measurements are confounded by the simultaneous polymer and NP relaxation. Neutron scattering and NMR techniques can directly measure local dynamics using deuterium/proton labeling. Specifically, in a weakly interacting system, using neutron spin-echo spectroscopy (NSE), Schneider et al.108 reported that PEP chains underwent significant disentanglement (increase in Ne) in the presence of short hydrocarbon-coated silica NP (d = 17 nm) at high loadings. Simulation studies also showed that Ne increases at high NP loadings111,130,131 and that the NP size relative to tube diameter influences Ne. For example, Kalathi et al.113 investigated segmental dynamics using molecular dynamics simulation and reported that an increase in Ne was more pronounced in the presence of NPs smaller than the dt compared to that of NPs larger than the dt. Namely, small NPs acted akin to plasticizers that increased Ne by 40% more than the value obtained with the addition of larger NPs, leading to enhanced segmental relaxation. In contrast, a decrease in Ne was reported in the presence of a neutral SWCNT132 and NRs.133 This was explained by direct contact between fillers and polymers that increased entanglements when the NR diameter was less than the dt. Furthermore, Ne decreases due to favorable polymer−particle interactions134 because NPs are able to induce chain entanglements at their surfaces.

5. MACROMOLECULE DYNAMICS IN CYLINDRICAL NANOPORES Whereas random immobile NPs present a tortuous path of confined regions, templates with parallel walls similarly confine macromolecules, but in a uniform manner. Polymer-filled porous media are of fundamental and technological interest because of their use in lubrication,135−137 filtration membranes,138 drug delivery,139,140 and nanoscaffolds for tissue engineering.141−143 When polymer chains are confined within these nanoscopic pores, polymer dynamics are perturbed, and as a result chain relaxation and physical aging of the polymer change. Therefore, a G

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the polymer chain, low compressibility of polymer melts, and impenetrability of the pore walls that limited the lateral motion of polymer chains.161−163 Tube diameter can also be influenced when chains are confined in nanopores. When N < Ne, unentangled chains show a crossover from Rouse dynamics to cooperative reptation motion as the capacity of free volume is reduced as confinements increases, indicating a reduction of the dt or a decrease in Ne.163−166 For example, using NMR, Fatkullin et al.165 reported a transition from Rouse dynamics in bulk to slower dynamics for perfluoropolyether (PFPE) chains (Rg = 10 nm) confined in SiO2 porous glass (dpore = 4 nm), where the confined PFPE chains showed an effective dt of 2 nm. In contrast, the nanoporous confinement can dilute the entanglement network (i.e., increase Ne).156−159 Specifically, Martin et al.159 used NSE to show that the dt of PEO in an AAO template increased by 15% (5.3 to 6 nm) consistent with strongly confined polymer chains (2Rg/d ∼ 2), in agreement with simulations by Sussman et al.157 As shown in the reptation model (eq 1), chain diffusivity can be influenced by chain conformation, segmental relaxation, and entanglement density. 5.2. Relation of Local Dynamics to Bulk Diffusion. In this section, we will discuss COM polymer diffusion in cylindrical confinement and its relationship to changes in segmental dynamics. Using proton pulsed-gradient NMR, Lange et al.167 investigated COM diffusion of entangled poly(butadiene) in cylindrically confined AAO template. In weak confinement (dpore/2Rg ∼ 10), chain diffusion was moderately reduced (∼20%) with respect to the pure polymer case, whereas at stronger confinement (dpore/2Rg ∼ 4), chain diffusivity was reduced by 50%. In both cases, the decrease in chain diffusivity was independent of Rg, suggesting that pore diameter played an important role in slowing down chain diffusion. This slowing down was explained by a short-range interaction on the length scale of the molecular size at the polymer−wall interface where molecular friction could increase by up to 10-fold. A unified picture of the effect of cylindrical confinement on the local dynamics and polymer COM diffusion has been proposed. Using FRES coupled with molecular dynamics simulations, Tung et al.158 investigated how changes in end-to-end distance (R), Rouse relaxation time, and entanglement density influenced polymer COM diffusion when polymer (PS) chains were confined in cylindrical AAO templates. As confinement increases (dpore/R), Figure 8 shows that experimentally measured chain diffusivity (red circles) also increases. This trend is in qualitative agreement with MD simulation results where chain diffusivities are determined from mean-squared displacement, MSD, of the chain along the cylindrical axis (blue squares) and the calculated chain diffusivity from R, τ0, and Ne using the reptation model (green squares). Although faster diffusion with increasing confinement was observed, the slower diffusion in experimental studies may be attributed to favorable polymer−wall interactions which were not considered in the simulations or model. Despite this difference, this study qualitatively relates local dynamics and long-range diffusion and suggests that polymer chains are disentangled within strongly confined pores, leading to faster COM diffusion compared to the bulk polymer case.

better understanding of polymer dynamics in nanoconfined media is required to precisely control the properties of PNCs. Templates with cylindrical channels, such as the anodic aluminum oxide (AAO) membranes shown in Figure 7A, have

Figure 7. (A) SEM images showing top surfaces of AAO templates with pore diameter of 30 ± 2 nm and a center-to-center distance of 68 ± 3 nm. (B) Schematic of the random confinement imposed by NPs in a PNC. (C) Schematic of the uniform confinement imposed by cylindrical nanopores.

received growing interest in recent years. Cylindrical pores provide 2-D channels where polymer chains are able to relax perpendicular and parallel to the pore wall. Thus, relative to dispersed NPs in a polymer matrix, cylindrical pores provide a model system for understanding how nanoconfinement influences polymer dynamics, shown schematically in Figure 7B,C. Furthermore, because pore diameter (dpore) and interpore distance are well-controlled, geometric parameters can be systematically tuned. 5.1. Local Segmental Dynamics in 2-D Confinement. The segmental dynamics of polymers confined to 2D channels have been probed using different techniques,129,144 such as inelastic neutron scattering,145−148 nuclear magnetic resonance,149,150 dielectric spectroscopy, and calorimetry151−154 across various time and length scales. Here we focus on the effect of nanoconfinement on chain conformation, segmental relaxation time, and entanglement density because these three parameters can influence the reptation of polymers as noted in eq 1. The conformation of polymer chains confined inside nanoscopic pores has been investigated using SANS. Noirez et al.155 reported that the conformation of PS did not change when confined in AAO templates having pore sizes comparable to the Rg of PS. Furthermore, when the pore size (15 nm) was less than the PS chain dimension (2Rg = 45 nm), Shin et al.156 found that such strong confinement did not significantly influence chain conformation along the pore axis, consistent with theoretical and simulation studies.157,158 Whereas the local segmental dynamics at short times (t < 1 ns) are relatively unaffected by confinement, at intermediate times (Rouse), dynamics are slowed down when polymers are confined in nanopores.146,149,159 For example, using NSE, Martin et al.159 investigated dynamics of PEO chains confined in AAO templates and revealed a slowdown in Rouse dynamics, consistent with NMR studies.149 These studies also suggested that the slowdown is not due to an increase in monomeric friction but is attributed to the adsorption of PEO chains onto the AAO surface.160 Using field-cycling NMR relaxometry, studies suggested rotational polymer segment dynamics can become slower for confinements ranging from nano- to micrometers due to excluded volume of

6. MACROMOLECULE DYNAMICS IN BIOLOGICAL MEDIA Whereas a complete review of macromolecule diffusion in biological systems is beyond the scope of this Perspective, the translatability of findings in PNCs to biopolymer systems can be H

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Figure 9. Extrapolated mobility, μ0, as a function of number of base pairs, n. Open diamonds represent DNA in PEG solutions where only Rouse diffusion is observed. Open circles, open squares, and open triangles represent PEG networks with Ns of 113, 227, and 454, respectively. Filled inverted triangles represent data where μ0 was independent of Ns. Scaling of μ0 with n is −0.17 and −0.81 at low and high regions of n, respectively. Inset depicts DNA chains entering a homogeneous gel. Adapted with permission from ref 177.

Figure 8. Normalized diffusion coefficients along the direction of cylindrical pores. The green, blue, and red symbols represent the reptation model with parameters (R, Ne, and τ0) predicted from MD simulations, the diffusivity extracted from the MSD of a chain diffusing along a pore, and experimental results from ion beam measurements. Adapted with permission from ref 158.

described. For example, one of the earlier successes of reptation theory was the explanation of gel electrophoresis, where DNA strands of different length separate due to differences in diffusion.168 Hydrogels are of interest for studying polymer diffusion in confinement because their network size can be tuned by varying the cross-link density.169,170 Models have been developed to describe the complex arrangement of gel systems as a random arrangement of fibers with specific fiber radii.171,172 Combining these models and the scaling relationship between chain size and diffusion coefficient, the mesh size of the hydrogel networks can be estimated or the mesh size measured by scattering can be corroborated.173−175 Pluen et al. used fluorescence recovery after photobleaching to study DNA diffusion within agarose gels.173 For DNA ranging in number of base pairs (n) from 125 to 23 000, D scaled as n−0.52 and n−1.55 in the low and high n regimes, in agreement with Rouse and reptation dynamics, respectively. Using fluorescent microscopy, Carlsson, Larsson, and co-workers directly visualized DNA chains extending and contracting during electrophoresis, as expected for chains undergoing reptation.168,176 In a more recent study177 on the electrophoretic mobility, μ, of double-stranded DNA, a transition from Rouse to reptation diffusion mechanisms was observed in homogeneous cross-linked PEG gels when n was approximately equal to the persistence length of DNA, n ∼ 150. This transition was found to be independent of the degree of polymerization between strands, Ns. As shown in Figure 9, rigid chains, shorter than the persistence length, diffused in a Rouse manner, while semiflexible chains, longer than the persistence length, migrated by reptation. In the Rouse and reptation regimes, μ is predicted to scale with n as 0 and −0.8 for real chains, respectively.177 μ can be related to D by eq 6, where Q is the charge of the DNA molecule.178

μ=

DQ k bT

with Rouse and reptation scaling, respectively. Because only Rouse diffusion was observed in PEG solutions (open diamonds in Figure 9), these studies concluded that the confinement due to the cross-links in the gel was responsible for DNA reptation.177 Importantly, these studies demonstrate that theory can aid our understanding of biopolymer diffusion. As discussed for polymer diffusion in PNCs (section 2.2), enthalpic interactions could influence chain dynamics in hydrogels. Liu et al.169 investigated the diffusion of fluorescently labeled dextran and cationic avidin protein in a hydroxyethyl methacrylate (HEMA)/methacrylate acid (MAA) anionic hydrogel. For the nonadsorbing solutes, dextran, diffusion strongly decreased as mesh size decreased. The Ogston172 model, which considers the gel to be a network of fibers with free volume between fibers, was used to describe solute size, hydrodynamic drag, and the distribution of mesh sizes. Like the interparticle spacing for PNCs, this model used polymer fraction and characteristic length scales of the system to define confinement. The model did not quantitatively capture all the experimental data because it overestimated the volume available to the solutes. Nevertheless, the decrease of dextran diffusion with increasing confinement in hydrogels mirrors the behavior of tracer diffusion in PNCs.71,72 Conversely, cationic avidin became strongly bound to the negatively charged carboxylate groups of the hydrogel matrix, which greatly reduced diffusion relative to dextran in the same hydrogel. This significant slowing due to strongly attractive interactions was not observed in the dPMMA/ PMMA:silica system78 discussed previously, perhaps due to bound layers of PMMA on the NPs that screen dPMMA−NP interactions. The confinement in PNCs and hydrogels is echoed in complex biological and food biopolymer networks. A recent review by de Kork highlighted how probe molecules, both polymer chains and NPs, can be used to characterize food biopolymers’ nanoscale organization.179 This organization affects properties such as taste and moisture retention.179 Polymer chain diffusion models have been used to understand the nanoscale structure of cheese,180 gelatin,181 milk gels,182 and whey protein gels.183 Understanding the confined dynamics of macromolecules is critical to characterizing the structure of tissues and cells. Zador et al.184

(6)

To compare μ for systems of different polymer concentrations, μ0, the extrapolated value of μ at zero polymer volume fraction, was determined by an exponential fit of the dependence of μ on polymer volume fraction. The experimentally determined scaling of μ0 with n in PEG gels was −0.17 and −0.81, in good agreement I

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s196−198and how mucus is structured on the nanoscale199,200 is of current relevance. Chetri et al.199 used gold NRs to probe the organization of two biopolymer networks: extracellular matrix (ECM) models and human bronchial−epithelial mucus. The authors found that NR diffusivity was sensitive to physiologically relevant changes in the structure of these biomaterials. For example, diffusion decreased as mucus solid concentrations increased from a healthy level to one characteristic of airway diseases. Also, increasing collagen concentration in the ECM decreased particle diffusion, indicating that confinement increased. Models developed for NP diffusion in polymer systems can aid in the quantification of these biopolymer structures. The Stokes−Einstein (SE) model201 has been used to describe diffusion of a spherical particle in various media. This model is only applicable when particles are large compared to the medium molecules and the medium is a continuum fluid. The diffusion coefficient of a particle described by the SE relation, DSE, is proportional to the thermal energy at a given temperature, i.e., the Boltzmann constant kB times temperature, T, and inversely proportional to the pure solvent viscosity, η, and the particle diameter, d. Phillies found that SE relation was applicable for colloidal PS spheres in water−glycerol and water−sorbitol solutions;202 however, in polymer−water solutions, where the solvent and solute are comparable in size, the SE relationship failed.203,204 Similar to colloids in polymer solutions, the SE relation could also underestimate the diffusion of NPs for NPs smaller than the characteristic length, i.e., dt in an entangled polymer melt. Brochard Wyart and de Gennes205 argued that the bulk viscosity did not capture the behavior of surrounding flows near NPs, and thus NP diffusion was decoupled from the SE relation. Namely, when NPs were smaller than the mesh size, the friction felt by the particle depended on contact with monomers, resulting in a length-scale-dependent friction that was less than the bulk value. Such breakdown in SE relation when NP size is comparable to the characteristic length is reported in various studies for polymer solution206−211 and melts.191,212−219 For example, Grabowski et al.213 showed that the diffusion coefficient of gold NPs in a poly(butyl methacrylate) melt was approximately 200× faster than the SE prediction when the NP diameter was comparable to the mesh size, 6 nm. Archer et al.220 also reported interesting behavior when NPs became comparable in size to the entanglement mesh. In PMMA melts, NPs grafted with PEG exhibited a transition from diffusive to hyperdiffusive when Mw became greater than the entanglement molecular weight. Brochard Wyart and de Gennes205 also argued that as NP size approached the mesh size, there was a sharp crossover in friction “felt” by NPs. Thus, for NPs of comparable size to the mesh, the viscosity experienced by the NP was the bulk value and SE behavior was recovered. In contrast, theoretical studies219 showed a continuous transition from NP diffusion determined by local viscosity to bulk SE behavior, which was recovered when d was ∼10× greater than the dt, as seen in Figure 10. The transition region suggests that other mechanisms affect NP dynamics. To account for this behavior, Yamamoto and Schweizer219 proposed a constraint-release mechanism for NP diffusion. Namely, NPs were trapped by a polymer network as NP motions became gradually coupled to the entanglement network, and a slight density fluctuation resulted in the escape of NPs, resulting in faster NP diffusion. On the other hand, Cai et al.190,210 proposed a NP hopping mechanism to explain NP diffusion, which is depicted in Figure 11. NPs (diameter = d)

characterized the structure of brain extracellular space (ECS) by studying the diffusion of macromolecules. Using fluorescently labeled dextran, diffusion was found to vary in different regions of the brain. For example, dextran diffusion was reduced 4-fold in the cortex ECS and 7.4-fold in the thalamus ECS in comparison to water. The differences in diffusion can be attributed to increased heterogeneity in the confinement of the thalamus ECS.184 Xiao et al.185 studied the diffusion of dextran to characterize the anisotropy of brain ECS, which contained aligned fibers, and found that diffusion was faster parallel to fiber alignment than the perpendicular direction. Moreover, this anisotropy increased as molecular weight increased and then reached a plateau for the two highest molecular weights, indicating that dextran had to elongate to diffuse parallel to the fibers.185 The faster diffusion of macromolecules along anisotropic obstacles, as opposed to perpendicular to them, observed in biological systems with anisotropic organization of fibers185,186 is similar to anisotropic diffusion observed in polymer CNT composites and NR PNCs described in section 3.86−88,93

7. FUTURE DIRECTIONS Whereas sections 1 to 6 focus on macromolecular dynamics in crowded environments with immobile NPs, the dynamics of nanofillers also need to be understood to completely characterize nanocomposites when both macromolecules and fillers are mobile. For example, by understanding the controlling length scales and mechanisms of particle diffusion in polymer melts, new insight into the structure of more complex networked systems, such as thermoresponsive187,188 and heterogeneous gels,189 can be gained. A recent study by Jee and Granick,189 using PS beads and fluorescent dye molecules in methylcellulose, demonstrated that NP diffusion behavior could be used to interrogate the structural environment. Using fluorescent correlation spectroscopy, dye molecule diffusion was described by two diffusion coefficients, Dslow and Dfast. Whereas Dslow was independent of polymer concentration, Dfast decreased as polymer concentration increased, suggesting that diffusion was dictated by two confinement environments, namely, a smaller fiber spacing and larger mesh size, respectively. The diffusion of the larger PS beads decreased with increasing polymer concentration, indicating that PS beads diffused only through the larger mesh. While further studies are needed to test the proposed mechanisms of NP diffusion190,191 in gels, this study shows that nanoscale probes can provide information about the structure of the media in which they are diffusing. Future studies should seek to understand how NP diffusion depends on network heterogeneity and transient network structures using gels with reversible cross-links. Using probe molecules on the size scale of the intrinsic confinement of biological systems similarly provides insight into the structure of their confined environment. Many current areas of biological research, such as drug delivery and disease pathogenesis, require knowledge of how small entities move within complex environments and how those environments are structured on the nanoscale. From an environmental and health standpoint, characterizing NP diffusion in biofilms, where bacterial cells surround themselves with a matrix of extracellular polymeric substances, is critical.192−194 Additionally, the study of the structure of the local environment of cellular organelles will be aided by understanding NP diffusion in crowded media.195 Mucus is a barrier to both drug delivery and infectious agents; therefore, understanding how NPs diffuse within mucuJ

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understand how chain dynamics alone are influenced. Moving forward, more studies should be done to obtain a comprehensive understanding of the effect of mobile NPs on polymer dynamics. Additionally, superresolution techniques222 that probe individual polymer chains should be considered in order to decouple the NP dynamics from polymer dynamics.

8. SUMMARY Considerable progress has been made toward understanding polymer COM diffusion and local segmental dynamics. The studies included in this Perspective provide guidance for developing new models that capture polymer dynamics in the presence of impenetrable and immobile NPs and biomacromolecular dynamics in biological and hydrogel systems. However, there are still unresolved questions that require future investigation. Developing a unified picture connecting local and bulk dynamics should be the focus of future studies. This will include understanding the effect NP mobility has on polymer COM diffusion and local dynamics. Experimental and theoretical studies on polymer systems should highlight how their findings can relate to dynamics in biological systems. We believe that these important issues will be solved with the effort from many scientists and researchers, and understanding polymer dynamics in the presence of NPs will allow us to further control the properties of PNC materials as well as to develop and discover new applications such as self-healing materials, separation techniques, and sensing devices.

Figure 10. Diffusion coefficient of a sphere predicted by self-consistent generalized Langevin equation (SCGLE) normalized by DSE versus the d-to-dt ratio with P/Ne = 4 (black curve), 8 (red curve) and 16 (blue curve). Insets illustrate the correlation between NP and entanglement where NPs are free of entanglement (bottom left) and are entanglement relaxation controlled. Adapted with permission from ref 219.

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Figure 11. Illustration of a particle larger than the mesh size hopping from initial position to a neighboring cage with a mesh loop (red) slipping around the particle by overcoming an energy barrier. Adapted with permission from ref 190.

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (R.J.C.).

moderately larger than dt are trapped in cages of the entanglement network. To move, NPs must overcome a hopping free energy barrier. The entanglement strands slip around the NPs, resulting in localization of the NP in a neighboring cage. Future experimental studies should focus not only on testing current theories and mechanism but also on examining system parameters lacking in these theories, such as the effect of NP shape and NP-polymer interactions. The next question of interest is how these highly mobile NPs impact polymer dynamics. As PNC-based devices develop, small NPs (d < 101 nm) are incorporated into polymeric materials to add optical, magnetic, electrical, and other properties. Polymer dynamics determine the dispersion of NPs, which controls the efficiency of devices. Furthermore, understanding the effect of mobile NPs on polymer dynamics addresses unresolved issues related to altered COM and segmental dynamics. The addition of these small NPs decreases nanocomposite viscosity,15−17,221 in contrast to typical observations of mechanical reinforcement of polymer composites in the presence of large NPs, i.e., d ≫ mesh size.222−224 For example, Mackay et al.16 reported a decrease in viscosity of PS filled with 50 wt % cross-linked PS NPs due to an increase in free volume induced by NPs. Other studies suggested that NPs acted like plasticizers that diluted entanglements17,225−227 or that polymer−particle solubility played an important role.15 However, mechanical testing measures the combined effect of NP and polymer, and thus it is difficult to

Notes

The authors declare no competing financial interest. Biographies

Chia-Chun Lin received his B.S. in Chemical Engineering from National Tsing Hua University in 2009 and obtained his Ph.D. from the Department of Materials Science and Engineering at the University of Pennsylvania in 2015 under the direction of Dr. Russell Composto. His research is focused on polymer and nanoparticle diffusion in polymer nanocomposites. K

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dynamics in nanocomposites. We also acknowledge editing contributions from Dr. P. Griffin and SEM images of AAO nanopore membranes from James F. Pressly. Finally, we dedicate this Perspective to Prof. E. J. Kramer, who made long-standing contributions to our understanding of polymer dynamics.

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NOTATION b monomer length d nanoparticle diameter deff effective nanoparticle diameter dn number-average nanoparticle diameter dNR nanorod diameter dpore pore diameter of a cylindrical confinement dSWCNT diameter of a single-walled carbon nanotube dt tube diameter of entangled chains D tracer diffusion coefficient D* diffusion coefficient of a test chain D0 bulk diffusion coefficient of the tracer D1 diffusion coefficient of tracer in brush Dcr constraint-release diffusion coefficient De effective tracer diffusion coefficient Drep reptation diffusion coefficient DSE Stokes−Einstein diffusion coefficient ID interparticle distance IDeff effective interparticle distance kB Boltzmann constant L chained nanoparticle/nanorod length Ltube tube length in reptation model n number of base pairs N degree of polymerization Nb degree of polymerization of a brush Ne degree of polymerization of an entanglement strand Ns degree of polymerization between cross-links P degree of polymerization of the matrix chain Q charge of DNA strand R end-to-end distance of the chain Rg radius of gyration T temperature Tg glass transition temperature ζ monomeric friction coefficient η viscosity μ electrophoretic mobility μ0 mobility at zero polymer fraction ν scaling exponent ρ nanoparticle diameter polydispersity σ brush grafting density τ0 shortest Rouse relaxation time τtube Rouse relaxation time of the confining tube τrep reptation time for the diffusing chain to leave the tube ϕ filler volume fraction ϕcNP core volume fraction of chained nanoparticle ϕmin nanorod concentration near the diffusion minimum ϕNP nanoparticle volume fraction ϕNR nanorod volume fraction

Emmabeth Parrish received her B.S. in Material Science and Engineering from the University of Tennessee in 2012. She is a recipient of the NSF Graduate Research Fellowship. She is currently a PhD candidate at the University of Pennsylvania studying nanoparticle diffusion in polymer gels and in the cytoplasm of individual cells.

Russell J. Composto received his Ph.D. in Materials Science and Engineering from Cornell University in 1987, working with Professor Edward J. Kramer. He has been the faculty of the University of Pennsylvania since 1990 and currently the Associate Dean of Undergraduate Education. He is also a professor of Materials Science & Engineering, with secondary appointments in the Departments of Chemical and Biomolecular Engineering and Bioengineering. He is actively investigating dynamics and assembly in polymer nanocomposites, blends, and copolymer films. In addition, his research extends to adhesion, adsorption, and diffusion at nano−bio interfaces. He is currently the Director of the Scanning Probe Facility, and the National Science Foundation, Partnerships for International Research and Education (PIRE) program at the University of Pennsylvania.

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ACKNOWLEDGMENTS Support is acknowledged from NSF/Polymer DMR-1507713 (R.J.C., C.L.), NSF/MRSEC DMR-1120901 (R.J.C., C.L.), and NSF/MWN DMR-1210379 (R.J.C., E.P.) grants. Support was also provided by the ACS/PRF program 54028-ND7 (R.J.C., E.P.) and E. I. DuPont De Nemours and Company. R.J.C. extends a special acknowledgement to the Division of Materials Research at the National Science Foundation for a Special Creativity Award. R.J.C. also thanks the financial support and hospitality of the MRL at UCSB (NSF DMR 1121053) during his sabbatical. The authors also thank Professors K. I. Winey, N. Clarke, R. A. Riggleman, and J. Choi as well as Drs. W. Tung, S. Gam, M. Mu, A. Karatrantos, M. Caporizzo, M. Grady, and others who contributed to our study of polymer and nanoparticle

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REFERENCES

(1) Winey, K. I.; Vaia, R. A. Polymer nanocomposites. MRS Bull. 2007, 32 (4), 314−319. (2) Jancar, J.; Douglas, J. F.; Starr, F. W.; Kumar, S. K.; Cassagnau, P.; Lesser, A. J.; Sternstein, S. S.; Buehler, M. J. Current issues in research on structure-property relationships in polymer nanocomposites. Polymer 2010, 51 (15), 3321−3343.

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DOI: 10.1021/acs.macromol.6b00471 Macromolecules XXXX, XXX, XXX−XXX