Relaxation Dynamics of Nanoparticle-Tethered Polymer Chains

Aug 17, 2015 - Thomas KinseyEmmanuel Urandu MapesaWeiyu WangKunlun .... Kim , Wentao Wang , Banghao Chen , Hedi Mattoussi , Daniel T. Hallinan...
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Relaxation Dynamics of Nanoparticle-Tethered Polymer Chains Sung A Kim, Rahul Mangal, and Lynden A. Archer* School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, New York 14853, United States S Supporting Information *

ABSTRACT: Relaxation dynamics of nanoparticle-tethered cis-1,4-polyisoprene (PI) are investigated using dielectric spectroscopy and rheometry. A model system composed of polymer chains densely grafted to spherical SiO2 nanoparticles to form self-suspended suspensions facilitates detailed studies of slow global chain and fast segmental mode dynamics under surface and geometrical confinementfrom experiments performed in bulk materials. We report that unentangled polymer molecules tethered to nanoparticles relax far more slowly than their tethered entangled counterparts. Specifically, at fixed grafting density we find, counterintuitively, that increasing the tethered polymer molecular weight up to values close to the entanglement molecular weight speeds up chain relaxation dynamics. Decreasing the polymer grafting density for a fixed molecular weight has the opposite effect: it dramatically slows down chain relaxation, increases interchain coupling, and leads to a transition in rheological response from simple fluid behavior to viscoelastic fluid behavior for tethered PI chains that are unentangled by conventional measures. Increasing the measurement temperature produces an even stronger elastic response and speeds up molecular relaxation at a rate that decreases with grafting density and molecular weight. These observations are discussed in terms of chain confinement driven by crowding between particles and by the existence of an entropic attractive force produced by the space-filling constraint on individual chains in a self-suspended material. Our results indicate that the entropic force between densely grafted polymer molecules couples motions of individual chains in an analogous manner to reversible cross-links in associating polymers.

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INTRODUCTION Understanding the behaviors of macromolecules tethered to rigid supports/fillers is a longstanding challenge in soft matter science. Movement of short, unentangled linear polymer chains tethered to a support can normally be described using models for Brownian motion of beads linked by harmonic springs, such as that proposed by Rouse,1−3 with the added constraint that the bead tethered to the support cannot move. When the molecular weights of the tethered chains become long enough to allow individual molecules to entangle, reptation and armretraction dynamics4−10 are expected to be overlaid with those associated with surface attachment.11 Model polymers with long-chain branches, such as stars, H-polymers, combs, and pompoms, exhibit aspects of these dynamics, and the concepts of arm retraction, hierarchical relaxation, and branch point diffusion play crucial roles in understanding their overall rheological properties.12−15 Dynamics of entangled polymers in a variety of branched architectures have been studied with diverse approaches, including NMR,16−18 neutron scattering,19−21 rheology,16,18−22 dielectric spectroscopy,18−21 and others. Among these techniques, dielectric spectroscopy is advantageous because a material can be interrogated in an undeformed state with minimal concern about field-induced damage during the measurement. Dielectric relaxation experiments are especially important for studying relaxation dynamics of materials that © XXXX American Chemical Society

possess a net dipole moment parallel to the polymer end-to-end vector.23−26 Polymers, such as cis-1,4-polyisoprene (PI) and poly(propylene glycol) (PPG), possess such a type A dipole, and it is possible to quantify global chain relaxation (i.e., relaxation of the chain end-to-end vector) from dielectric relaxation experiments.23−26 Dielectric relaxation studies of type A polymers in a wide range of architectures have allowed the reliability of the method to be established and have played a role in the development of strategies for analyzing dielectric spectra to extract molecular-scale information. Among these studies, dielectric relaxation of type A blocks in diblock27−35 and triblock36,37 copolymers has been particularly advantageous for understanding the effects of confinement and tethering on global chain and segmental relaxation dynamics of polymers. Alig et al.27 examined the relaxation dynamics of polyisoprene (PI) in polyisoprene−polystyrene (PI-b-PS) diblock copolymers. Both the normal mode and segmental mode relaxation of PI were reported to be slower relative to a bulk PI homopolymer of comparable molecular weight to the PI block. Stuhn28 reported a broadened dielectric loss spectrum for diblock copolymers compared to the corresponding homopolymers. Floudas and co-workers29 reported on the Received: April 16, 2015 Revised: August 4, 2015

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large changes in normal mode relaxation above the critical molecular weight for entanglements. Granick and co-worker40 showed that the normal mode relaxation slows down in PI thin films, but the segmental relaxation process remains the same, compared to bulk PI. It was surmised that the differences in the results between the three studies arise from the different thin film preparation and loading methods. Zhang and Archer11 studied relaxation dynamics of PI chains with different molecular weights confined in nanoporous glasses with pore dimensions comparable to the PI random coil size. The authors reported that the confined molecules exhibit two relaxation processes: one fast mode and one slow mode. The authors also reported significant broadening of the dielectric relaxation spectrum of the confined PI. The relaxation time for the fast process was shown to be independent of polymer molecular weight and to originate from dynamic adsorption/desorption of chain segments at the glass surface. The characteristic time τ for the slow process was found to exhibit a very strong dependence on molecular weight, τ ∼ Mw5.7, which could be eliminated by coating the CPG (controlled porous glass) with a silane that prevented adsorption. Floudas et al.44 also reported broadened normal and segmental mode spectra from dielectric loss data of unentangled PI confined in nanoporous alumina. On the basis of these observations, the authors argued that confinement of even unentangled chains produces a broader distribution of relaxation events than in their unconfined and unentangled analogues. Ding and co-workers45 showed that in filled polymers (PI with C60 nanoparticles) both the segmental and normal mode dynamics were slower. Although it is unclear whether the slower normal mode relaxation isn’t simply a straightforward consequence of the slowdown in segmental mode dynamics, the observations are significant because they show that strong interactions between confined chains and their confining substrates are not required for slower dynamics. A well-known challenge in characterizing relaxation behavior of polymers near rigid supports, or confined between surfaces, stems from the low signal-to-noise of the experimental observables from which chain dynamics must be inferred. This challenge is to some extent implicit in the nature of surface or geometric confinement, wherein the largest effects are apparent on length scales comparable to the chain dimensions. Recently, we reported that self-suspended suspensions created by densely grafting poly(ethylene glycol) chains to silica nanospheres (SiO2-PEG) provide excellent model systems for studying the effects of surface and geometric confinement on polymer structure and crystallization on a range of length scales.46,47 The approach is attractive because every polymer chain in such a self-suspended material is confined to and between the surfaces of particles, which makes it possible to employ high signal-to-noise measurements on a bulk material to examine the nanoscale physics of polymer chains under confinement. By substituting cis-1,4-polyisoprene for PEG, it is possible to create self-suspended SiO2-PI, which can be used in dielectric relaxation experiments with bulk materials to probe surface and confinement dynamics of tethered polymer chains. The first study utilizing this approach focused on systems in which the PI chains were so densely grafted that their surface dynamics were dominated by polymer crowding on the particle surface and by a form of arm retraction required by such laterally constrained chains to relax stress.48 Nonetheless, a remarkable consequence of these physics is that even short, unentangled PI chains tethered to nanosize silica particles

dynamics of PI in symmetric PI-b-PS diblock copolymers, finding the dynamics of both the homopolymer PI and the PI subchain changed in a manner qualitatively consistent with expectations from the reptation and arm retraction models for free and tethered polymer chains, respectively. Through combined dielectric and viscoelastic measurements, Watanabe’s group30 reported that diblock copolymers of polyisoprene and poly(p-tert-butylstyrene) (PtBS) exhibit thermorheological complexity, in comparison with PI/PtBS blends, because the PI block is forced to relax slower when it is anchored to the less mobile PtBS block at low temperature, whereas at high temperature the difference in the relaxation rate of the two polymer blocks is reduced. Watanabe,31 Lodge,32 and their coworkers also used dielectric relaxation and mechanical rheology to study relaxation processes in spherical micelles composed of styrene cores and isoprene corona, formed by blending PS-b-PI diblock copolymers in an unentangled PI matrix. The authors reported a slow process corresponding to Stokes−Einstein diffusion of the particle-like PS cores, an intermediate mode arising from star-like relaxation of the PI corona, and a fast relaxation due to the unentangled homopolymer matrix. Significantly, the corona dynamics were shown to be consistent with arm retraction of entangled PI stars and the slow diffusion of the cores were found to be well described by the Stokes− Einstein formula, if the star arm relaxation time is used to compute the medium viscosity. Karatasos et al.33,34 investigated the dielectric relaxation of symmetric PS-b-PI diblock copolymer melts, in both the ordered and disordered states. Far from the order-to-disorder transition (ODT), the authors reported significant broadening of the low-frequency dielectric loss spectrum and contended that the broadening reflected the effect of composition fluctuations on the normal mode relaxation. Below the ODT, they reported slow dynamics, which was argued to originate from the relaxation of conformal interfaces in the ordered lamellae. Floudas et al.35 demonstrated that global chain relaxation of PI depends on the microstructure (spheres, cylinders, lamellae), formed by ordered PS-b-PI star diblock copolymers. It was observed that the microstructures that confined the PI chains produced greatest broadening and retardation of PI chain dynamics. On this basis, it was suggested that dielectric spectroscopy could be utilized as a dynamic probe of the interface in ordered triblock copolymers.36 Watanabe’s group37 took these ideas somewhat further and showed that from comparisons of the dielectric strength of PSb-PI (SI) diblocks and PS-b-PI−PI-b-PS (SIIS) triblock copolymers with a symmetric inverted dipole in the PI block, it is possible to obtain one of the first quantitative measurements of the fraction of PI chains that adopted loop versus bridge conformations in a microphase-separated triblock. Dielectric relaxation experiments have also been used to study the combined effects of surface and geometric confinement of polymer films to delineate their impacts on relaxation dynamics. Thin films with controlled thickness deposited on rigid substrates,38−40 polymer layers confined between twodimensional nanolayers with changing interlayer spacing41−43 or into 3-dimensional nanopores with varying pore size,44,45 and polymers distributed between nanoparticle fillers in filled melts46 have all been used to produce and study confinement effects on polymer conformation and dynamics. Serghei et al.38 reported a new relaxation mode in thin films of PI induced by the immobilization of chain segments at an interface, but changes in the segmental and normal mode were not shown. Mapesa et al.39 also investigated PI thin films and observed B

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Table 1. Weight-Average Molecular Weight (Mw), Number-Average Molecular Weight (Mn), Polydispersity Index (PDI), Radius of Gyration (Rg), and Contour Length (Rmax) of Polyisoprene Synthesized with Anionic Living Polymerization in the Left Column; Silica Nanoparticle Core Volume Fraction (ϕ), Grafting Density (Σ), Stem Length (l), and Lateral Spacing (ξ) of Their Nanoparticle (10 nm Diameter)-Tethered Polyisoprene in the Right Columna free PI

a

silica nanoparticle tethered PI

Mw (g/mol)

Mn (g/mol)

PDI

Rg (nm)

Rmax (nm)

ϕ (%)

Σ (chains/nm2)

l (nm)

ξ (nm)

3200 7200

3000 6900

1.07 1.04

1.77 2.66

22.4 50.4

13000 25100

12600 23300

1.03 1.08

3.57 4.96

91.0 175.7

5.99 4.23 5.78 8.1 2.91 1.11 1.67 2.34

2.84 1.68 1.21 0.84 1.35 1.93 1.27 0.91

2.07 0.45 0 0 0 0.83 0 0

0.59 0.77 0.91 1.09 0.86 0.72 0.89 1.05

Calculated with Kuhn monomer molar mass (M0) 120 g/mol and Kuhn length (b) 0.84 nm of PI.3 Small-Angle X-ray Scattering (SAXS) Measurements. To gain insight into the degree to which the particles are correlated, SAXS measurements were performed at Sector 12-ID-B of the Advanced Photon Source with extremely small exposure time (0.2 s). Dielectric Spectroscopy. Frequency-dependent dielectric loss and dielectric constant were measured using a Novocontrol broadband dielectric spectrometer with Quatro temperature control. Isothermal frequency sweep experiments were carried out over a wide range of temperature above the glass transition temperature of PI. Mechanical Rheometry. Frequency-dependent and strain-dependent oscillatory shear measurements were performed using a stress-controlled rheometer, Anton Paar MCR 301, outfitted with cone and plate fixture of diameter 10 mm and cone angle 1°, at multiple temperatures. Samples were presheared to erase any previous history until the strain sweep data were reproducible. Strain sweep measurements were performed at a fixed shear frequency, ω = 10 rad s−1, and frequency sweep experiments were executed within 0.01 s−1 ≤ ω ≤ 100 s−1 at a fixed shear strain within the linear viscoelastic regime for each material. To construct time−temperature superposition (TTS) plots, frequency sweep rheology experiments were performed from 0.1 to 10 rad/s at each temperature. Then, moduli data at different temperatures were shifted vertically and horizontally with the reference of 30 °C moduli data for each sample. The horizontal and vertical shift factors, aT and bT respectively, are shown in Figure S3.

exhibit tube dynamics reminiscent of their much higher molecular weight, more entangled counterparts. This report considers relaxation dynamics of cis-1,4-PI in a broad range of SiO2-PI self-suspended materials deliberately created to investigate modes of polymer confinement beyond surface crowding. In particular, by systematically varying the grafting density and molecular weight of particle tethered PI, we are able to investigate how tethering to, confinement between and crowding of anchor points on nanoparticles change the dielectric and viscoelastic relaxation of the materials. By investigating SiO2-PI systems encompassing the complete molecular weight range, from unentangled to well entangled, we find that tethering restricts relaxation and broadens the relaxation spectrum of chains in all molecular weight ranges. We find that these effects are greatest for lower molecular weight molecules, where tethering also produces substantial coupling of individual chain dynamics.

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EXPERIMENTAL METHODS

Polymerization and Polymer End-Functionalization. Methyl and secondary amine terminated polyisoprene were synthesized using anionic living polymerization described in the literature.49 Molecular weights (Mn and Mw) and polydispersities (Mw/Mn), determined using a Waters ambient temperature gel permeation chromatography (GPC) equipped with a Waters 410 refractive index detector and 486 UV−vis detector, are summarized in Table 1. Preparation of Nanoparticle-Tethered Polymers. The method of preparing sulfonic acid functionalized silica nanoparticles was described previously.46,50 Secondary amine terminated polyisoprene was dissolved in tetrahydrofuran (THF), and then was added slowly to sulfonic acid functionalized silica nanoparticle aqueous solution. Untethered polyisoprene was removed with repeated precipitation, and finally, nanoparticle-tethered PI was dried in a vacuum oven at room temperature (see Supporting Information for details). Thermogravimetric Analysis (TGA). The total inorganic and organic content of the SiO2-PI nanoparticle−polymer hybrid materials was determined using thermal gravimetric analysis TGA Q1000 (TA Instruments). A fixed temperature scan rate of 5 °C/min was used to ramp the materials from 20 to 600 °C under nitrogen atmosphere. Using the known density of SiO2 and PI, as well as the known the molecular weight of the tethered chains, the grafting density of the SiO2-tethered PI was determined for all the samples. Differential Scanning Calorimetry (DSC). Glass transition behaviors were studied using a DSC Q2000 (TA Instruments). Samples were heated to 100 °C, then cooled to −100 °C at a fixed scan rate of 2 °C/min, and finally heated up to 100 °C at 2 °C/min under nitrogen flow. The glass transition step change was recorded from the second heating cycle.

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RESULTS AND DISCUSSION

Structural Characterization. The entanglement molecular weight (Me) of cis-1,4-PI is known to be 5000 g/mol, and the critical molecular weight (Mc) is approximately twice Me, ∼10 000 g/mol.51,52 Thus, the studied SiO2-PI materials with PI molecular weight in the range 3.2K−25.1K Da allow us to explore the effect of surface attachment on dynamics of tethered polymer chains that range from unentangled to well entangled. To understand the effect of grafting density in the different molecular weight regime on the relaxation dynamics, one PI material with molecular weight below Mc and another with molecular weight above Mc were selected to investigate the effect of surface crowding. These polymers were grafted to the nanoparticle surface at a range of grafting densities from densely tethered to sparsely tethered. The limit of the lowest grafting density of polymer chains was determined from theoretical analysis of the chain configuration near the nanoparticle surface as the extent where the polymer chains are crowded enough so that none of chain backbone segments can fold backward to adsorb on the nanoparticle surface. Figure 1 reports the scattering intensity I(q) of SiO2-PI with PI Mw = 25.1 kDa at various polymer grafting densities C

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incompressible molecular fluid in which all elements are identical. These self-suspended systems can therefore be thought of single-component materials, allowing them to be employed as model systems to examine the relaxation dynamics of nanoparticle-tethered chains. Soft Glassy Rheology and Jamming. Knowledge of the mobility of the anchor points to which polymer chains are tethered is important in studying dynamics of tethered chains. Figure 3 reports the strain-dependent storage and loss moduli

Figure 1. I(q) vs q determined from small-angle X-ray scattering experiments of silica nanoparticle-tethered PI with molecular weight of 25.1K Da and various grafting densities. Scattering curves have been displaced vertically for clarity of presentation.

obtained from small-angle X-ray scattering (SAXS) experiments. In the low wave vector (q) regime, I(q) is seen to transition to a weak plateau, indicating that the SiO2 particles are well dispersed and that there are no large scale particle aggregates. In the absence of a suspending fluid, nanoparticletethered polymers are now understood to be subject to a spacefilling constraint, which leads to extension of the chains away from the surface of particles.53,54 These physics have been shown to lead to soft glassy rheology even in the limit of moderate particle loadings (ϕ ∼ 10%)50,55,56 and to an increased degree of correlation among the particles. Results reported in Figure 2 show that the arrangement of nanoparticles in the SiO2-PI materials are quite different from those typically seen in the suspensions of hard spheres (HS) at comparable particle volume fractions (shown in dashed lines in the figures), for which a molecular solvent fills the interparticle space. The strong deviation of the measured S(q) from the structure factor values for HS suspensions in the low q region, and the enhanced correlation peaks are consistent with these expectations56 and in accord with expected results for an

Figure 3. Dynamic storage (G′) and loss (G″) moduli [Pa] versus shear strain [%] for SiO2-tethered PI with molecular weight of 7.2K Da and grafting density of 1.21 chains/nm2. Strain sweep measurements were performed at a fixed oscillatory frequency of 10 rad/s at a temperature of 30 °C.

of a representative SiO2-PI, with a polyisoprene of molecular weight 7.2K Da with the grafting density of 1.21 chains/nm2. The figure shows the telltale signatures of a soft glass, including a pronounced yielding transition at a moderate shear strain. Soft glasses, proposed by Sollich et al.,57,58 are jammed materials that exist in metastable energy states wherein individual material elements are trapped in cages formed by potential energy wells substantially greater than thermal energy.

Figure 2. Structure factor S(q) vs q for self-suspended nanoparticle suspensions: (a) SiO2 tethered PI of 7.2K Da with ϕ = 4.2%, (b) SiO2 tethered PI of 7.2K Da with ϕ = 5.8%, and (c) SiO2 tethered PI of 7.2K Da with ϕ = 8.1%; (d) SiO2 tethered PI of 25.1K Da with ϕ = 1.1%; (e) SiO2 tethered PI of 25.1K Da with ϕ = 1.7%; and (f) SiO2 tethered PI of 25.1K Da with ϕ = 2.3%. All measurements were carried out at T = 30 °C. The corresponding hard-sphere suspension S(q) are denoted by dashed lines. D

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Macromolecules In Figure 3, three distinct flow regimes are noticed: (i) at low shear strains, strain independent linear viscoelastic regime, where G′ is larger than G″; (ii) at intermediate shear strains, a nonlinear viscoelastic regime, where G″ increases; and (iii) at high strains, a strain softening liquid regime where the loss modulus is larger than the storage modulus and both moduli decrease with shear strain. A distinct feature of soft glasses, different from free polymers, is the pronounced loss maximum that occurs at the onset of strain-softening in G′.55 This behavior originates from the sudden enhancement in energy dissipated during the cage breaking of a soft glass,59 and the relative size of loss maximum compared to G″ in the linear viscoelastic regime at low shear strains reflects the measure of jamming within the system. It is shown in Figure 4a that the normalized loss maximum is an increasing function of temperature, which indicates that materials are more jammed at higher temperature. As illustrated in Figure 4b, the corresponding loss tangent, G″/G′, measured in the linear regime decreases with increasing temperature, implying more solid-like or more jammed system at elevated temperature. This so-called thermal jamming behavior has been reported previously55 and is now understood to be a distinguishing feature of a single-component soft glass created by densely tethering polymers to nanoparticles to create self-suspended suspensions. It has been argued to reflect greater interpenetration of tethered ligands as they attempt to uniformly fill space between their nanoparticle anchors at elevated temperature. The jammed state can be conveniently captured using a variable termed the noise temperature (X), which signifies the energy available for hopping of particles to escape the cages and satisfies the following relationship: X = 1 + (2/π)δ, where δ is the phase lag.55 It is observed that as temperature increases, the noise temperature monotonically decreases, implying that the particles are more jammed (Figure 4c). Figure 5a shows the effect of the grafting density of 7.2K PI chains on the relative size of the loss maximum. It is apparent in Figure 5b−c that as the grafting density decreases, tan δ is lower and the noise temperature decreases, meaning that lower grafting density also leads to a more jammed or more solid-like material. This behavior can also be explained in terms of the space-filling constraint on the nanoparticle-tethered PI chains. Specifically, when low molecular weight polymer chains are sparsely attached to nanoparticles, each chain has to stretch further (Figure 6) to fill the interparticle space, compared to the densely grafted case. The stretched chains are more interpenetrated and thereby create deeper potential wells in the energy landscape the particles must negotiate. Figure 7 compares the frequency-dependent moduli for the untethered PI polymer and the same SiO2-PI materials as a function of polymer grafting density. The frequency range is extended using time−temperature superposition. It is apparent from the figure that jamming is accompanied by a dramatic increase in the width of the rubbery plateau and a noticeable slowdown in cage breaking as the grafting density of SiO2-PI is lowered from 1.68 to 1.21 chains/nm2. This behavior is analogous to what has been reported in melts of unentangled ionomer chains as the strength of the ionic interactions is increased. It implies that the interpenetration of tethered PI molecules at lower grafting density results in enhanced, entanglement-like interactions between the tethered PI chains. Figures S1 and S2 report on the effect of temperature and grafting density for SiO2-PI with a higher PI molecular weight,

Figure 4. (a) Normalized loss maximum vs strain amplitude [%]. (b) tan δ vs strain amplitude [%]. (c) Noise temperature (X) vs temperature [K] for SiO2-PI of Mw 7.2K Da with grafting density of 1.21 chains/nm2. Strain sweep measurements were performed at a fixed oscillatory frequency of 10 rad/s at different temperatures.

25.1K Da. It is apparent that these materials do not exhibit thermal jamming and that the grafting density has at most a weak effect on jamming. For nanoparticle-tethered 25.1K Da PI with grafting density of 1.93 chains/nm2, the normalized loss maximum is more or less the same, loss tangent increases, and noise temperature also increases with elevated temperature, which are the usual characteristics of viscoelastic polymer materials. E

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Figure 7. Time−temperature superposition (TTS) master curves for untethered and nanoparticle-tethered PI chains of molecular weight 7.2K Da with different grafting densities. Frequency-dependent oscillatory shear rheology measurements were performed at different temperatures for 0.1 s−1 ≤ ω ≤ 10 s−1 in the linear viscoelastic regime. T = 30 °C was used as the reference temperature for TTS.

Global Chain Relaxation Dynamics. Typical dielectric loss (ε″) spectra for untethered and tethered PI 3.2K Da shown in Figure 8 reveal several key differences. First, the loss maxima

Figure 5. (a) Normalized loss maximum vs strain amplitude [%]. (b) tan δ vs strain amplitude [%]. (c) Noise temperature (X) vs core volume fraction [%] for SiO2-PI of Mw 7.2K Da with different grafting densities. Strain sweep measurements were performed at a fixed oscillatory frequency of 10 rad/s at 30 °C.

Figure 8. (a) Dielectric loss (ε″) vs frequency [Hz] for untethered PI with Mw = 3.2K Da at different temperatures. (b) Dielectric loss (ε″) vs frequency [Hz] for nanoparticle-tethered PI with Mw = 3.2K Da at different temperatures.

Figure 6. Illustration of enhanced jamming of nanoparticle-tethered polymers by decreasing grafting density.

F

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Figure 9. (a) Relaxation time of untethered and nanoparticle-tethered polyisoprene of different molecular weights as a function of temperature obtained from H−N fits to the dielectric loss spectra. (b) Relaxation time of untethered and tethered PI of different Mw as a function of temperature calculated from the peak frequency of raw dielectric loss spectra. For each molecular weight, SiO2-PI with the highest grafting density is used for the plots.

⎤−γ /2 ⎡ ⎛ βπ ⎞ 2β β ⎜ ⎟ ε″(ω) = Δε⎢1 + 2(ωτHN) cos + (ωτHN) ⎥ ⎝ 2 ⎠ ⎦ ⎣ ⎧ βπ ⎡ ⎤⎫ sin 2 ⎪ ⎢ ⎥⎪ ⎬ × sin⎨γ arctan⎢ βπ ⎥ −β ⎪ ⎪ ωτ + ( ) cos ⎢ ⎥ HN ⎣ 2 ⎦⎭ ⎩ (1)

associate with the slow, normal mode relaxation shift dramatically toward lower frequency for the tethered PI. For the type A polymer studied here, the frequency ωp at the loss maxima is related to the end-to-end vector relaxation time, τ = 1/(2πωp).24,25 Second, the breadth of the loss maxima for the tethered PI is significantly broader than for the untethered molecules. Finally, the peak intensity and peak area are both larger as well as weaker functions of temperature for the tethered PI. It is also noted that when the molecular weight of the PI chains exceed the critical molecular weight, broadened peak widths, higher intensity, and larger peak areas continue to be observed (Figure S6), but there is little change in the frequency at which the loss maximum is seen for the tethered PI. These observations are similar to results reported by Yao et al. in their dielectric relaxation studies of microphase-separated PS−PI/PI copolymers, in which slower, coupled dielectric relaxation and substantial broadening of the dielectric loss spectrum of PI chains anchored to glassy PS domains were observed.60 A more complete understanding of the structure and relaxation of the SiO2-tethered PI chains can be obtained by fitting the dielectric loss spectra with the model function proposed in the Havriliak−Negami (H−N) analysis. Unlike the Debye model, which assumes all energy potential wells are symmetric, the H−N model considers multiple asymmetric energy potentials within the system.25 The H−N function for the frequency-dependent permittivity is given by the formula ε*(ω) = ε∞ + Δε/[(1 + (iωτHN)β)γ] = ε′(ω) − iε″(ω), where β (0 < β) describes symmetric broadening, γ (βγ ≤ 1) for asymmetric broadening, τHN for H−N relaxation time, ω for frequency, ε∞ the dielectric constant (ε′) at high frequency limit, and Δε the dielectric strength or dielectric intensity. The dielectric intensity can be calculated from the real part, ε′, using the formula Δε = εs − ε∞, where εs is the dielectric constant (ε′) in the low-frequency limit. It can also be determined from the loss, ε″, spectra by integrating the area under ε″ peaks: Δε = (π/2)∫ ε″(ω) d ln ω. In order to estimate τHN, β, and γ, experimentally measured dielectric loss spectra were fitted with the function25

( )

( )

Figures S7 and S8 show that the ε″ data are fit well by eq 1 at low and high temperature; here we do not constrain β or γ. Among the four parameters deduced from the H−N model fits, the relaxation time exhibits the most pronounced and nonintuitive changes upon tethering PI chains. Figure 9a report τHN values determined using the fitting procedure for the SiO2PI and untethered PI materials over a range of polymer molecular weights and as a function of temperature. In comparing the τHN values in the figure, it is important to keep in mind that because the shape of the loss spectra itself changes significantly upon tethering PI of any molecular weight to SiO2, the τHN values deduced from these fits reflect different averages over relaxation spectra that are themselves changing. Thus, at this level of detail, we only seek to make qualitative comparisons between dynamics of the tethered and untethered chains; later we will study how the shape of the loss maximum changes upon tethering by analyzing the corresponding β and βγ values obtained from the H−N model fits. As a simple check of the reasonableness of the τHN values obtained from the fitting procedure, we also report in Figure 9b average relaxation times estimated from the frequency ωp at which the loss maxima are observed in the raw ε″ data. Comparison of the results in Figures 9a and 9b shows that the characteristic time scales deduced from τHN and (2πωp)−1 are of comparable magnitude and exhibit the same nonintuitive trends with PI molecular weight. Specifically, both sets of results show that whereas relaxation of shorter, unentangled PI chains are dramatically slowed down by tethering, those of higher molar mass, entangled polymers change very little. This finding is most succinctly presented in terms of the ratio between the average relaxation time of densely tethered to untethered PI, plotted in Figure 10. It is evident that when the molecular weight of the tethered chains is below the critical molecular weight for entanglements, this ratio is over 104. For PI of G

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are able to adopt random walk chain statistics similar to those in a bulk melt and may therefore be considered less crowded (see Figure 11a,b). Stem lengths for the range of SiO2-PI

Figure 11. Schematic illustration of tethered polymer chain configuration: (a) low-Mw chains; (b) high-Mw chains. Stem length (l) is shown with blue dashed line, and entire polymer brush height is presented with purple dashed line. The degree of restriction being imposed on chains is expressed by the intensity of the colored lines (red indicating the most confined).

Figure 10. Relaxation time ratio between densely tethered and untethered PI with different molecular weights at a fixed temperature of 293.15 K. Entanglement molecular weight, Me, and critical molecular weight, Mc, are indicated by the red lines. Open squares indicate the ratio determined from H−N fits and open triangles from the normal mode peak position in the dielectric loss spectra.

grafting densities used in the study are reported in Table 1. Although the stems are relatively short, the relative portion of a polymer chain that fits in the stem region is larger for a lower molecular weight PI, which means that the lower Mw tethered PI are more confined and likely far more stretched out on the particle surfaces than they are in a bulk melt. Reducing the PI grafting density at fixed Mw is a straightforward way to lower crowding of polymer chains on a particle surface and to test the hypothesis that crowding of chains tethered to a nanoparticle can produce highly entangledstar like relaxation even for short PI chains. Figure 12a reports relaxation times of SiO2-PI and untethered PI with Mw = 7.2K Da as a function of grafting density and temperature. It is seen that at low grafting densities the relaxation time of the sparsely tethered polymer at 60 °C is more than 100 000 times greater than that of the untethered molecule with the same molecular weight. In contrast, relaxation times for the most densely tethered polymers, at the grafting density of 1.68 chains/nm2, are only ∼2−4 times those of untethered PI, consistent with earlier findings for unentangled or weakly entangled stars.61 These results clearly show that increasing the grafting density of the tethered polymer chains has the effect of speeding up chain relaxation. For the SiO2-PI 7.2K Da materials with grafting density of 1.21 chains/nm2 and 0.84 chains/nm2, which exhibit the slowest dynamics, the lateral spacing between chains on the surface is larger than bK, and the stem length is calculated to be zero. Comparison of the observations in Figure 12a with the enhanced elasticity and jamming reported earlier, based on shear rheology measurements, as the polymer grafting density is progressively lowered (see Figures 5−7) indicates that the two effects are related but are likely not a consequence of topological constraints between polymer chains densely grafted to particles. The corresponding results for the highest molecular weight PI studied (Mw = 25.1K Da) are reported in Figure 12b. For this case it is apparent that the grafting density has little if any effect on relaxation time and that the relaxation time of the tethered chains is again only about 2−4 times that of the untethered polymer over the entire temperature range studied. Δεn (or Δε) values determined by fitting the H−N model to the experimental ε″(ω) results are reported in Figure 13 (left Yaxis) as a function of temperature, for untethered and tethered

molecular weight 3.2K Da, tethered chains relax about 70 000 times slower than untethered ones! As the molecular weight of PI chains is increased through Mc, the ratio reduces sharply and approaches a value in the range of 2−4, independent of the tethered polymer molecular weight. Boese et al.61 previously reported that the relaxation time of unentangled star polymers is around 4 times that of linear counterparts, regardless of the number of arms and the molecular weight of the arm. This result can be shown to be what one would deduce from the Rouse analysis with the appropriate boundary conditions for a chain immobilized at one end. For PS−PI/PI star-like micelles with moderately entangled arms, Sato et al. reported that the fast relaxation of the PI arms exhibits both similar mode distribution and molecular weight dependence of the relaxation time as for arm retraction of entangled star PI.31 Considering the relatively low molecular weight of the PI chains studied here and the inverse relationship between the dielectric slow-mode relaxation time on PI molecular weight, it is evident that neither the physics employed by Boese et al. nor by Sato et al. can explain the observations reported in Figures 9 and 10. Previously, we reported that crowding of polymer chains tethered to a nanoparticle surface produces topological constraints that have an analogous influence on overall polymer dynamics as those originating from chain entanglements.48 In this scenario, the “tube” is formed by neighboring chains on a single particle, which means that for the small interchain spacings (bKΣ−1/2) at the high PI grafting densities, Σ ∼ O(1), studied here, even the shortest SiO2-tethered PI chains would be highly entangled and should exhibit slow entangled-star like relaxation. A reduction in relaxation time ratio with increased PI Mw can be rationalized in this tube-conf inement by crowding framework in terms of the inevitably lower grafting density of tethered chains produced by steric crowding of tethered PI to a fixed number of grafting sites on the SiO2 particles.62 To illustrate this point, we compute the distance from the surface, l = R(bK∑1/2 − 1), or stem length, at which the interchain spacing between tethered chains just exceeds the Kuhn step length bK,46 where R is the radius of nanoparticle and Σ is the grafting density. On length scales larger than l, tethered chain segments H

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chains in the diblock, Δεn is measurably smaller in magnitude relative to untethered linear PI of comparable molecular weight.60 The dielectric intensity for a type A polymer of molecular weight M has been reported previously to be related to the mean squared end-to-end distance ⟨r2⟩24,25,63 Δεn =

4πNAμp 2 FOnsager 3kBTM

⟨r 2⟩

(2)

Here NA is the Avogadro number, μp is the dipole moment, FOnsager is the so-called Onsager correction factor, kB is the Boltzmann constant, and T is the absolute temperature. Equation 2 can be shown to be rigorously valid for uncorrelated type A chains at equilibrium and in weak fields, where the endto-end vector distribution is symmetric and ⟨r4⟩/⟨r2⟩ = κ⟨r2⟩2; here κ is a numerical constant.24 Using Δεn at 293.15 K for an untethered linear PI as a reference, we employ eq 2 to estimate the ratio ⟨r2⟩1/2/2Rg for all of the studied materials, to within a constant factor α. The results are reported on the right Y-axis of Figure 13, where we note that that ⟨r2⟩1/2/2Rg is a constant, √6/2 ≈ 1.22, for an ideal melt of linear chains. Comparison of the size of the values for the tethered and untethered chains may be used to qualitatively assess the degree of stretching of the tethered PI chains. Analysis of the results in Figure 13 indicates that tethering PI with Mw = 3.2K Da causes the chains to stretch by a factor of about 7 compared to the free polymer. This stretching by tethering is less effective as the molecular weight increases, and for 25.1K Da, stretching of the most densely tethered chains is about 1.5 times. It is also seen that the degree of stretching for the tethered PI is a weak, increasing function of temperature while it is a weak, decreasing function for the untethered polymer. The latter can be understood in terms of the decreases in the spring constant with temperature. For the nanoparticle-tethered chains, the weak increase is consistent with observations of thermal jamming discussed earlier and with the explanation that the phenomenon is a consequence of greater stretching and more interpenetration of tethered chains at higher temperature. Results reported in Figure 13b,d lend additional support to this argument. In particular, it is noticed that when the grafting density for the PI 7.2K Da polymer is reduced, the tethered chains appear to stretch more. The effect is significant as the most sparsely tethered 7.2K Da PI chains are on average about 4−5 times more stretched than the most densely grafted ones. Again, this

Figure 12. (a) Relaxation time of untethered and nanoparticletethered PI 7.2K Da at different grafting densities as a function of temperature. (b) Normal mode relaxation time of untethered and tethered PI 25.1K Da with different grafting densities as a function of temperature.

PI with various molecular weights and over a range of grafting densities. It is apparent from the figure that Δεn for tethered PI is generally larger than for an untethered polymer of comparable molecular weight. It is also seen that the enhancement in Δεn is largest when the tethered polymer molecular weight and grafting desity are lowest. These results are different from what has been observed in PS−PI block copolymers, where after correction for the concentration of PI

Figure 13. Dielectric strength or dielectric intensity, Δε, and characteristic dimensionless length of silica nanoparticle-tethered PI chains with molecular weights of (a) 3.2K, (b) 7.2K, (c) 13K, and (d) 25.1K with various grafting densities. Black color for Δε (left axis) and red color for characteristic length (right axis) are used. Open circle (○) indicates untethered, and closed circle (●) presents densely tethered chains. For sparsely tethered PI with Mw 7.2K and 25.1K in (b) and (d), respectively, lower grafting density PI chains are presented with closed triangle (▲) and closed square (■). Grafting densities are in the unit of chains/nm2. I

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cross-links between the tethered PI chains. The existence of such cross-links provides a straightforward explanation for the observed rubbery plateau, particularly in the SiO2-PI systems at lower polymer grafting density (Figure 7). It would also explain the slowdown in tethered chain relaxation and coupled chain relaxation dynamics that likely lead to the failure of eq 2. Fitting the temperature-dependent relaxation time to the Vogel− Fulcher−Tamman (VFT) expression, log τ = A + [B/(T − C)],48,65 yields the high temperature limit of the relaxation time (A), the apparent activation energy (B), and the Vogel temperature (C). Figure 14 reports the activation energy of

result is consistent with the enhanced jamming and greater dissipation at the yielding transition observed as the grafting density of PI 7.2K Da is reduced. A similar trend of greater chain stretching with reduced grafting density is seen for the highest molecular weight polymer (Mw = 25.1K Da), but the differences are much smaller. The ratio of the fully stretched out length to the equilibrium root-mean-square end-to-end distance of a flexible linear polymer is R/⟨r2⟩1/2 ≈ NK1/2, where NK is the number of Kuhn segments per chain. Thus, for a PI chain of molecular weight 3.2K Da, a chain can extend by about 5 times its equilibrium size, whereas for PI of molecular weight 7.2K Da the maximum stretch ratio is around 7 and for a polymer with Mw = 25.1K Da it is approximately 14. Analysis of the ⟨r2⟩1/2/ 2Rg results in Figure 13 clearly shows that the stretch ratios for the two lowest molecular weight polymers computed using eq 2 exceed their maximum values, implying that the expression relating Δεn to ⟨r2⟩ is incorrect. There are at least three physical mechanisms in our self-suspended SiO2-PI materials that can produce the high dielectric intensities observed and can invalidate application of this formula. (i) Strong coupling of polymer chains densely grafted to particles. This effect should be especially strong in cases where the stem length of the tethered chain approaches the chain contour length. (ii) Confinement between closely spaced particles can produce chain extension by a combination of the first two effects. (iii) The existence of a strong field on each chain produced by the requirement that the tethered chains uniformly fill interparticle space in the absence of a solvent. The existence of such a field has been previously shown by theory53 and experiment54 to be critical for explaining low q deviations in the structure factor S(q) for a self-suspended suspension and a suspension of hard spheres at comparable particle volume fraction (see Figure 2). Results reported in Figure 13, which show that Δεn is highest and the stretch ratios computed using eq 2 largest for tethered chains at the lowest grafting density, imply that mechanism (i) alone cannot explain our observations. The space-filling constraint can be thought to produce an effective entropic attraction force between chains tethered to adjacent particles.47,53 Mechanism (iii) could therefore be the source of a strong field between tethered chains that may lead to correlated motions of multiple PI molecules, which would invalidate eq 2 and at the same time explain the large increases in Δεn. Coupled relaxation of PI molecules can also explain the broadened dielectric loss spectra observed upon tethering short PI chains to nanoparticles. Finally, the greater difficulty of filling space in a self-suspended material composed of shorter and/or more sparsely tethered chains would lead to stronger constraints on each chain, which is in qualitative accord with the observed trends in dielectric relaxation as the PI molecular weight is reduced and the grafting density lowered. It is nonetheless understood that confinement of PI chains between particles could produce qualitatively similar behaviors, meaning that additional information is needed to evaluate the relative importance of mechanisms (ii) and (iii) in explaining our observations. The activation energy for chain relaxation should be sensitive to the presence of an effective attractive force between tethered ligands. By analogy to the role associative, electrostatic forces play in providing temporary cross-links that delay relaxation and produce an extended rubbery plateaus in unentangled ionomer chains,64 entropic attraction force between tethered chains in a self-suspended material can produce long-lived

Figure 14. Activation energy, derived from fitting relaxation time versus temperature with VFT equation, for untethered and tethered polymer chains versus core volume fraction (%).

untethered and tethered PI over the entire range of polymer molecular weights and grafting densities studied; the corresponding values of A and C are reported in Figure S9a,b of the Supporting Information. The results are reported in terms of the SiO2 volume fraction to allow for straightforward comparisons between tethered and untethered PI and polymers with dissimilar molecular weights. Figure 14 shows that the activation energy for chain relaxation is in general higher for particle-tethered polymer chains. It is also apparent that the increase is a strong function of SiO2 nanoparticle volume fraction (i.e., polymer grafting density) with the largest increases in activation energy, relative to values for untethered PI, being observed for the most sparsely grafted particles, irrespective of the polymer molecular weight. This finding is qualitatively consistent with what we would expect from mechanism (iii). It, however, cannot rule out mechanism (ii) because at the highest SiO2 nanoparticle volume fractions, the interparticle space is lowest and, for a fixed polymer molecular weight, confinement effects would also have their greatest effect. As seen from Figure S9a,b the two other VFT parameters A and C also change with polymer grafting density and molecular weight. In particular, it is seen that while A generally decreases as the polymer grafting density is reduced, there is no clear trend with tethered polymer molecular weight. Additionally, with the exception of the highest molecular weight polymer studied (PI 25.1K), which exhibits a decrease in Vogel temperature with decreased polymer grafting density, C is generally insensitive to PI molecular weight and grafting density and remains fixed at a J

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Macromolecules average value around 165 K, which is about 40 K below the glass transition temperature for tethered PI (see Figure 17). Table 2 reports the interparticle distance, the distance from the center of one particle to the center of another, assuming a Table 2. Particle Distances of Silica Nanoparticle-Tethered PI 7.2K Da with Different Grafting Densities (Σ)a theoretical calculation

empirical values from S(q)

Σ (chains/nm )

λ (nm)

ds‑s (nm)

λ (nm)

ds‑s (nm)

1.68 1.21 0.84

24.7 22.3 19.9

14.7 12.3 9.9

22.42 11.25 11.25

12.42 1.25 1.25

2

a

Interparticle distance (λ) is the distance between the centers of nanoparticles, and the surface-to-surface distance (ds‑s) is calculated by subtracting the nanoparticle diameter 10 nm from λ. The theoretical values are obtained assuming random close packing of the SiO2 cores, and the empirical values are deduced from the first maximum in S(q).

random distribution of the SiO2 particles,66 λ = d(0.63/ϕ)1/3. Here d and ϕ are respectively the particle diameter and volume fraction. The distance from surface of one nanoparticle to surface of another nanoparticle is calculated by subtracting silica nanoparticle diameter, d, from interparticle distance. In addition, the experimental interparticle distance is determined from S(q) with λ = 2π/q1, where q1 is the location of the first S(q) peak. Rg for the 7.2K PI is 2.66 nm, and the Rmax is 50.4 nm. At a grafting density of 1.68 chains/nm2, the ⟨r2⟩1/2 value or polymer brush height is approximately 10 nm, while the empirical surface-to-surface distance between the SiO2 particles is around 12.42 nm. When the grafting density is 1.21 chains/ nm2, the experimental surface-to-surface distance is approximately 1.25 nm and the end-to-end distance is around 40 nm. This means that polymer chains are stretched over 3 times the size of a single hairy nanoparticle (brush + core + brush: 1.25 nm + 10 nm + 1.25 nm). At the lowest PI grafting density studied, 0.84 chains/nm2, the end-to-end distance is close to the fully stretched out value ∼50 nm, but the average nanoparticle surface-to-surface distance is just 1.25 nm, indicating that as the PI surface coverage is reduced, the tethered chains become more stretched and progressively more geometrically confined in the interparticle space. This can be contrasted with the analogous results for PI 25.1K Da where the chain end-to-end distance at the lowest grafting density is estimated to be 16.8 nm and the surface-to-surface distance between particles computed to be ∼15 nm, indicating that these chains are rarely confined. This means that mechanism (iii) is likely the dominant source of the behaviors observed. We close this section by pointing out two additional consequences on dynamics of tethering polymer chains. Inspection of Figures 8 and Figure S6 clearly shows that the dielectric loss peaks are substantially broader and moderately less symmetric when PI chains are tethered to nanoparticles. These effects are both reflective of underline coupled chain dynamics and can be quantified using the two remaining parameters β and γ recovered by fitting the experimental data to the H−N expression. β describes the symmetric broadening of relaxation spectrum, whereas γ presents the asymmetric broadening above the peak frequency in dielectric loss spectra (Figure 15a). Results for untethered and tethered PI 7.2K and 25.1K are summarized in Figure 15b,c. It is clear from these results that both β and βγ are generally lower for tethered polymer chains. More remarkable is the fact that β for the

Figure 15. (a) Havriliak−Negami (HN) fitting parameter β for symmetric broadening and γ for asymmetric broadening above the peak frequency of the dielectric loss spectra. (b) β and βγ values of untethered PI 7.2K Da and tethered PI 7.2K Da with grafting densities of 1.68 and 0.84 chains/nm2. (c) β and βγ values of untethered PI 25.1K Da and tethered PI 25.1K Da with grafting density of 0.91 chains/nm2.

tethered materials is a decreasing function of temperature, indicating that relaxation of the tethered chains becomes more corporative at temperature rises. This behavior is consistent with the enhanced jamming of the SiO2-PI materials evident from bulk rheology experiments and is in qualitative accord with expectations based on greater interpenetration of the PI corona at higher temperature. Segmental Mode Relaxation Dynamics and Glass Transition. At high frequencies, dielectric relaxation measurements can be used to gain insights into segmental relaxation of immobilized polymers.67−70 Kirst et al.67 reported on the segmental mode relaxation of poly(dimethylsiloxane) (PDMS) adsorbed on hydrophobic and hydrophilic fumed silica. The K

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becomes larger, meaning that there are stronger intermolecular interactions leading to enhanced correlation between dipole moments (see Figure S10a). In contrast, Δεseg changes relatively little for the 25.1K Da PI in Figure S10b. Analogous to the normal mode relaxation, the grafting density for the 25.1K Da PI has little effect on the rate of segmental mode relaxation. To understand the origins for the slowdown in segmental relaxation, particularly for the lower molecular weight PI, the glass transition temperatures of all of the SiO2-PI materials used in the study as well as one other system (SiO2-PI 5K Da) reported in our earlier work48 were characterized. Data for untethered and tethered PI of varying molecular weights and grafting densities are again presented as a function of the SiO2 nanoparticle core volume fraction to facilitate comparisons across the full material set. Figure 17 shows that, with the

authors observed that the PDMS chain segments in the adsorption layer were more movable in the hydrophobic fumed silica compared to the hydrophilic one. Roland and coworkers68 investigated the segmental dynamics in the nanocomposites of poly(vinyl acetate) (PVA) and silica nanoparticles. It was shown that the segmental mode relaxation speed in the nanocomposites remained the same in comparison with that of bulk PVA, whereas the dielectric intensities of segmental mode relaxation in dielectric loss spectra reduced in the nanocomposites. Kumar, Mijovic, Gidley, and co-workers69 immobilized poly(2-vinylpyridine) (P2VP) on silica nanoparticles and found that the nanoparticle fillers marginally slowed down the segmental dynamics. Schonhals’s group70 studied PVA adsorbed on silica nanoparticles, and two different segmental mode relaxations were detected: one related to chain motions near the particle surface and the other associated with the motion of bulk-like chains. They also found that glass transition temperature increased because of reduced chain mobility. It was conjectured that the contradictory results in the similar material system of polymers adsorbed on nanoparticles originate from differences in the sample preparation methods, including the annealing conditions. Figure 16 reports the segmental mode relaxation times of untethered and SiO2 tethered PI, Mw 3.2K Da and 25.1K Da,

Figure 17. Glass transition temperature of untethered and tethered PI chains of different molecular weights vs silica nanoparticle core volume fraction. Points at core volume fraction of 0 are glass transition temperature of untethered PI. As grafting density decreases, the core volume fraction increases.

exception of the highest molecular weight PI, Tg for the tethered chains is generally higher than for the untethered PI by a modest 5 °C. The figure also shows that for a given PI molecular weight the Tg values are largely insensitive to grafting density. Viewed in the context of the much larger effects tethering has on the segmental and normal mode relaxation times as well as on rheology, we conclude that it is unlikely that the observed effects originate from surface- or confinementinduced changes on the glass transition.

Figure 16. Segmental mode relaxation time of untethered and tethered PI Mw 3200 and 25 100 g/mol. Segmental mode relaxation times of different grafting densities of 25.1K are shown. Grafting densities are in the unit of chains/nm2

deduced from dielectric relaxation. Results for the higher molecular weight polymer are presented for several grafting densities. It is apparent that while the segmental relaxation time for the 25.1K Da polymer changes little upon tethering to nanoparticles, for the 3.2K Da material it is around 2500 times slower for the tethered PI relative to its untethered analogue. As remarkable as these levels of dynamic slow down are for PI 3.2 k Da, it is noted that the effect of tethering on polymer segmental mode relaxation is relatively small in comparison to the effect on global chain relaxation. Kirkwood and Frohlich proposed an expression for the dielectric intensity of the segmental modes, Δε seg = g(4πFNρμ2/3kBT),45,71 where Δεseg is dielectric intensity of segmental mode relaxation peak in ε″, F is local field correction, and g is Kirkwood−Frohlich correlation factor measuring the correlation between dipole moments in neighboring units. Tethering enhances Δεseg for 3.2K Da, which means that g

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CONCLUSIONS Dielectric relaxation, linear viscoelasticity, and small-angle X-ray scattering experiments were used to characterize chain configuration, structure, and relaxation dynamics of tethered polymer chains. Using a model system of solventless SiO2 nanoparticles densely grafted with cis-1,4-polyisoprene (PI) chains, we find that tethering to the nanoparticle surface profoundly impacts polymer relaxation behavior on time scales ranging from those associated with fast segmental relaxation to slow terminal chain relaxation. In particular, we find that the normal mode relaxations of PI chains grafted to a surface are generally many orders of magnitude slower than for untethered polymer of the same molecular weight and chemistry. The effect is a strong decreasing function of tethered polymer L

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the Argonne National Laboratory, was supported by the US DOE under Contract No. DE−AC02−06CH11357.

molecular weight and grafting density, such that tethering of PI chains to surfaces slows their dynamics most when the polymer grafting density is lower and the polymer molecular weight is below the critical molecular weight Mc ≈ 2Me for the onset of entanglement effects. Through analysis of the dielectric strength of tethered and untethered PI, we further show that these changes in dynamics are accompanied by significant changes in coupling of the tethered molecules, which leads to large increases in the dielectric strength of the normal mode and significant broadening of the dielectric loss spectrum. At the lower polymer molecular weights and grafting densities, where the greatest dynamic slowing down is observed, analysis of the dielectric strength using a conventional formula that ignores interchain correlations leads to unphysically large chain extension ratios. This assessment is consistent with results from bulk rheology experiments, which show that even at modest particle volume fractions and for unentangled PI chains, the SiO2-PI materials that exhibit slowest relaxation exhibit long, rubbery plateaus and soft glassy rheology. Our findings contest previous studies, which show that crowding of polymer chains on nanoparticles is the principal source of chain stretching and slower dynamics. We instead suggest that an entropic attraction force produced by the space-filling constraint on tethered PI chains in the self-suspended materials produce linkages between individual chains that couples their relaxation dynamics. Finally, we conclude that the dynamics of tethered PI are more corporative than untethered PI and show that this leads to substantial broadening of the dielectric loss spectrum for short tethered chains. These effects lead to a clear, but less dramatic, slowdown in segmental relaxation time of polymer chains in the most confined materials. However, the glass transition temperature Tg is found to be only modestly affected by tethering and is independent of the polymer grafting density.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b00791. Preparation of nanoparticle-tethered polymers; structure factor calculation; Figures S1−S10 (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (L.A.A.). Notes

The authors declare the following competing financial interest(s): Professor Archer is the co-founder and holds a financial interest in NOHMs Technologies, a technology concern focused on commercialization of electrodes and nanoparticle-based electrolytes for secondary batteries.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation Award No. DMR-1006323 and by Award No. KUS-C1-018-02 made by King Abdullah University of Science and Technology (KAUST). Facilities available though the Cornell Center for Materials Research (CCMR) were used for this study (DMR1120296). Use of the Advanced Photon Source, operated by M

DOI: 10.1021/acs.macromol.5b00791 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.5b00791 Macromolecules XXXX, XXX, XXX−XXX