Heterogeneous Segmental Dynamics during Creep and Constant

Article pubs.acs.org/Macromolecules

Heterogeneous Segmental Dynamics during Creep and Constant Strain Rate Deformations of Rod-Containing Polymer Nanocomposites Gregory N. Toepperwein,† Kenneth S. Schweizer,‡ Robert A. Riggleman,§ and Juan J. de Pablo*,∥ †

Department of Chemical and Biological Engineering, University of Wisconsin, Madison, Wisconsin 53706-1691, United States Department of Materials Science and Frederick Seitz Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801, United States § Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States ∥ Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, United States ‡

ABSTRACT: The state of a glassy material can be altered considerably through the application of stress. The origins of this effect, and its relation to molecular mobility, are not well understood. Recent experimental, theoretical, and simulation studies have examined molecular mobility under deformation for different classes of materials, ranging from colloidal suspensions to polymeric glasses. While such studies indicate that dynamic heterogeneity, one of the hallmarks of glasses, decreases with the onset of flow, less is known about the effects of stress on polymer nanocomposites, whose mechanical and rheological properties can be significantly different from those of pure polymers. In this work, molecular simulations of multistep creep deformations and constant strain rate tensile deformations are used to examine dynamic heterogeneity in entangled, rod-containing nanocomposites. It is found that polymer nanocomposites, like polymers, experience reduced dynamic heterogeneity due to deformation, but some intrinsic heterogeneity of nonglassy origin persists. A connection between the segmental relaxation time and dynamic heterogeneity, suggested on the basis of an activated barrier-hopping model that includes local density fluctuations, is found to describe pure polymer behavior over a wide range of applied stresses, material compliances, and deformation histories. Additionally, segment displacements during postyield deformation are found to agree with a simple diffusional model across 2 orders of magnitude in strain rate for both pure polymer and nanocomposite systems.

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INTRODUCTION When a liquid is cooled rapidly, relative to the rate of local structural rearrangement, it can fall out of equilibrium and form a glass. As time passes, glasses undergo structural rearrangements in a process known as “physical aging”.1 The time scale for such rearrangements depends on temperature and can range from hours or days to thousands of years. This aging history of a material, however, can be erased by reheating above the glass transition temperature (Tg) and subsequent cooling. Remarkably, the application of a large stress can cause a solid, glassy material to flow like a liquid and likewise remove the effects of physical aging in a process known as “rejuvenation”. A model originally proposed by Eyring2 qualitatively accounts for the former behavior by assuming that deformation lowers activation energy barriers, thereby permitting flow on the experimental time scale. More generally, classic phenomenological free volume and other models envision that, during aging, the amount of free volume slowly decreases with time, thereby leading to the observed loss of mobility. Physical deformation can create extra free volume and allow deformation processes to erase aging and further accelerate yielding and plastic flow. Through a careful review of the literature, however, McKenna proposed that the complete © 2012 American Chemical Society

erasure of thermal history by mechanical stimulation is overly simplistic.3 For example, large deformations well below Tg are in fact capable of increasing the apparent age of the glass. McKenna argued that structural changes arising from deformation can lead to effects similar, but not identical, to those induced by different aging times. This issue was subsequently examined in an extensive series of simulations of a binary Lennard-Jones glass by Lacks and Osborne, who analyzed the inherent structure energy after various extents of deformation.4 They found that high strains push the system to higher energy minima on the potential energy landscape, while physical aging or small strains take the system to lower energy minima. More recently, Chung and Lacks characterized the effect of both temperature and deformation on the structure of a coarse-grained model of atactic glassy polystyrene.5 Their results point toward intricate connections between deformation, aging, temperature, and mobility. Received: July 18, 2012 Revised: September 5, 2012 Published: October 3, 2012 8467

dx.doi.org/10.1021/ma301501z | Macromolecules 2012, 45, 8467−8481

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Vlasveld and co-workers performed creep experiments on polyamide-6 filled with layered silicates.21 They found a reduction in creep compliance corresponding to the increased modulus induced by the additives and that the shift rate was slightly higher for composites. Rittigstein and Torkelson monitored changes in Tg resulting from the addition of small silicon and aluminum spheres to poly(methyl methacrylate), poly(2-vinylpyridine), and polystyrene.22 They found that Tg varied depending upon whether interfaces between polymer and particle were wetted or nonwetted. Riggleman et al. simulated rough spheres dispersed in an entangled polymer undergoing creep (as well as constant strain-rate deformations).23 They found that the two modes of deformation led to different responses of the inherent structure energy and were unable to identify a state variable that was uniquely indicative of the material’s relaxation time. Two recent papers by Boucher et al. examined physical aging in PNCs. In the first,24 they found that the addition of silica particles to poly(methyl methacrylate) markedly accelerated physical aging and that this process became more pronounced for smaller particles. In a second paper,25 they obtained similar results upon addition of gold particles to polystyrene. They further noted that segmental dynamics were similar in both the pure polymer and the PNCs, suggesting a disconnect between changes in aging and changes in mobility. On the basis of those results, they proposed that aging in PNCs is aided by the diffusion of free volume holes at the particle−polymer interface. Kim and co-workers examined the dynamics of poly(ethylene glycol) silica PNC using NMR across a range of temperatures and particle concentrations.26 They found that relaxations in PNCs exhibit additional modes that are absent in the pure polymer, which they interpreted to represent distinct populations of polymer close to the fillers and at intermediate distances from the fillers, respectively. The thickness of glassy layers around the particles was found to be of order 1 nm and essentially independent of temperature. In the present work, we employ computer simulation to examine segmental relaxation and dynamic heterogeneity in rod-containing PNCs by performing creep and constant strain rate deformation. Several measures of mobility are considered. Previous examination of these systems in the equilibrium melt state and under uniaxial deformation in the glassy state revealed that the mobility of nanoscale additives is similar to that of the surrounding polymer.27 However, PNC systems tended to exhibit a broader relaxation spectrum with increasing nanorod length. Here, by extending that work to the case of creep deformation, we seek to arrive at a better understanding of how additives affect glassy heterogeneous relaxation. It is found that polymer nanocomposites, like polymers, have their dynamic heterogeneity reduced by deformation. However, their intrinsic polymeric heterogeneity of nonglassy origin remains unaffected. It is shown that the underlying relationship between heterogeneity and relaxation time across a range of multistep creep treatments can be explained based on a theory of activated barrier hopping in the presence of nanoscale domain disorder of structural origin. The distribution of segment displacements at deformations far beyond yield is likewise found to agree with a simple diffusion model across 2 orders of magnitude in strain rate.

Glassy materials are also known to exhibit heterogeneous dynamics,6 and recent studies have sought to quantify dynamic heterogeneity under active deformation. Lee and co-workers performed photobleaching experiments on lightly cross-linked poly(methyl methacrylate) undergoing single-step and multistep creep deformations.7 They found a strong correlation between the instantaneous strain rate and mobility. These experiments were accompanied by extensive simulations that provided a direct measure of local segmental mobility on highly entangled model polymer glasses.8 Such simulations revealed that glassy heterogeneity decreases significantly following the onset of flow. Both simulation and experiments showed remarkably similar trends, even for features not captured by the Eyring model. The mechanism by which deformation imparts mobility to glassy systems remains incompletely understood.9 Riggleman et al. quantified polymer glass mobility by examining bond orientation relaxation during creep compliance across a range of stresses (positive and negative).10 They found that dynamics are strongly correlated to strain rate but show little connection to volume. They also observed higher inherent structure potential energies during periods of high strain rate. More recently, Warren and Rottler examined polymer mobility under deformation by measuring segmental hopping rates.11,12 They found that the distribution of hopping frequency of polymer segments scaled with system strain, as opposed to the strain rate or applied stress, suggesting that slower moving segments are more accelerated by deformation. Since high-mobility segments quickly rearrange during slow, prolonged deformations, their local environment relaxes before local strain becomes important. Low-mobility segments, however, are eventually forced into relaxations they may not otherwise have experienced, as their local environment inevitably feels the full impact of a large global strain. Global strain has also been identified as a useful, but not unique, variable with which to describe the mobility state of a polymer glass by Chung and Lacks.13 They observed that when single-segment van Hove correlation functions are plotted as a function of total strain (rather than time elapsed), systems subjected to varying, but low, strain rate exhibit similar distributions of site displacements. At a theoretical level, Chen and Schweizer recently extended their nonlinear Langevin equation theory of segmental relaxation in polymer glasses to describe the interplay between mechanical stimulus and physical aging.14 They predict that the nonequilibrium dynamics of the amplitude of nanometer-scale density fluctuations correlate with segmental relaxation mechanisms. The evolution of this key collective structural variable provides results for segmental relaxation time and mechanical response that are qualitatively consistent with the complicated picture of “rejuvenation” proposed by McKenna3 and the landscape-based physical scenario suggested by Chung and Lacks.13,15 The addition of nanoscale additives to a polymer matrix can drastically affect the structure and dynamics of polymer glasses.16−20 The properties of polymer nanocomposites (PNC) depend on both their physical and chemical composition as well as the technique(s) used to prepare them. Like pure polymers, PNCs are inherent glass formers, having retained the slow relaxation modes from the polymer. Despite extensive research into understanding the dynamics and structures of these composite materials, limited information is available about how additives influence their glassy nature.

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METHODS Models and Simulation. We have examined the structural and dynamic properties of rod-containing polymers in previous work.27,28 For completeness, some details of the model are

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recounted here. Fully flexible polymers are represented using a coarse-grained bead−spring model consisting of 500 beads. Nonbonded interactions are described via a Lennard-Jones potential, where σ and ε are the length and energy units, respectively. In order to smooth the nonbonded interaction, the Lennard-Jones potential is replaced with a cubic equation between the inner cutoff (2.4σ) and outer cutoff (2.5σ). The parameters of this function are chosen so that it is differentially continuous with the Lennard-Jones potential at the inner cutoff and goes to zero at the outer cutoff of 2.5σ. Polymer bonds are represented by simple harmonic springs with an equilibrium bond length also equal to σ. Nanorods, represented by a similar bead−spring model, are added to the polymer at a concentration of 10 wt %. The equilibrium nanorod bond length of the rods is (2/3)σ in order to give them a texturedcylinder-like appearance. Nanorods are subject to a strong angle potential that encourages a linear configuration but are not strictly rigid. Readers are referred to the literature for additional details.27,28 Both polymer monomers and nanorods have the same diameter; their dimensions are such that they could be viewed as short single-walled carbon nanotubes or nanowires. A relatively strong attractive interaction (3kBT at the site−site level) between polymer and particles is included to promote mixing. In a recent paper, these systems were originally prepared by growing polymer chains in a field of randomly placed nanorods;28 the equilibrated samples from that study were used as initial configurations for the work presented here (additional equilibration steps are described below). Nanorods of 1, 4, 6, 8, and 16 beads were considered in this work. For convenience, PNC systems will be referred to by the length of the rods they contain (e.g., 1mer, 4mer, etc.). Throughout this work, a constant pressure is maintained (P = 0.30 in reduced units of σ3P/ε). This protocal differs from that employed in our recent PNC work27,28 but is consistent with previous studies of the pure polymer model under creep,10 the results of which were corroborated by experimental data.7 Temperature (T) is expressed in units of kT/ε and t ̃ is used to denote Lennard-Jones time units by t ̃ = (mσ2/ε)1/2. Here m is the mass of a bead and kB is the Boltzmann constant. Six independent configurations were prepared for each PNC rod length, consisting of 184 000 polymer beads and 20 480 particle beads. Six additional configurations were prepared for the pure polymer, containing 184 000 beads. It was necessary to use larger systems here than those employed in our previous work27,28 to reduce finite-size effects and to improve the signalto-noise ratio of our results. All simulations were performed using the LAMMPS simulation package. Eighth-sized systems of 23 000 polymer beads and 2560 particle beads were prepared by performing 300 000t ̃ equilibration runs at a temperature of 1.2.28 During equilibration, rebridging or bond-swap moves were performed to relax the longest modes of the polymer.29,30 These systems were then doubled in all dimensions to their final sizes and equilibrated for another 10 000t ̃ using bond swapping to break the translational symmetry of the entangled polymer network. The equilibrated PNCs fall into two separate categories of behavior. For short-rod PNCs (1mer, 4mer, and 6mer), the rods are fully dispersed spatially and exhibit random orientations. In contrast, the 8mer and 16mer systems formed multirod clusters; such aggregation was also observed in our previous work with 16mers,28 but the 8mer remained dispersed. However, the systems considered in this work were equilibrated at a different temperature and pressure than in our previous

study. Nanocomposite aggregates consist of an array of nanorods with intercalating polymer. Structurally, the 8mer aggregates are similar to those previously reported for 16mers. These aggregates separate into a large cluster that spans the simulation box. As will be shown below, no significant difference in behavior is observed between the 8mers and 16mers. After equilibration, systems were cooled to T = 0.3 over the course of 36 000t.̃ The glass transition temperature was determined by measuring the log of molar volume as a function of temperature. These data were fit to an expression of the form y(T ) = y(Tg) + m0(T − Tg) ⎤ ⎡ ⎡ ⎛ Tg − T ⎞⎤ − 0.5α⎢δT ln⎢cosh⎜ ⎟⎥ − T + Tg ⎥ ⎥⎦ ⎢⎣ ⎝ δT ⎠⎦ ⎣

(1)

where y is the monitored quantity (log of molar volume), m0 is the slope of the linear regime above Tg, α is the difference in slope between the two linear regimes, and δT is the width of the transition region. Glass transition temperatures of 0.444, 0.610, 0.595, 0.599, 0.454, and 0.463 were found for the pure polymer, 1mer, 4mer, 6mer, 8mer, and 16mer systems, respectively. Note that the dispersed short-rod additives significantly increase the observed Tg over that of the pure polymer, but rod length among those systems has little effect. On the other hand, the longer-rod aggregated systems exhibit a Tg only slightly larger than that of the pure polymer. This suggests that the observed segmental dynamics for long rod PNCs is occurring in the polymer-rich domains with only modest influence from the filler. No second Tg corresponding to an independent transition in the composite-rich region could be detected. After cooling, the configurations from the cooling runs corresponding to T/Tg = 0.95 were extracted and aged for 10 000t ̃ at their respective temperatures. The aged systems then underwent a multistep creep deformation in the x dimension. The effective stress history applied to the system was 0.62 for 2000t,̃ 0.20 for 4000t,̃ 0.62 for 4000t,̃ and 0 for 4000t.̃ Additional creep deformations at 41.4% and 100% higher stresses across all domains were also performed. During the deformation process, the bond autocorrelation, Cb(t), was calculated according to C b(t ) = ⟨P2[b(t ) ·b(0)]⟩

(2)

where P2 is the second Legendre polynomial, b(t) is a unit vector aligned along the bond of a polymer, and the angular brackets indicate an average over all polymer bonds in the system; Cb(t) measures local segmental dynamics of polymer chains, equally weighting reorientation motions in all spatial directions. For glassy systems, one can quantify changes to the dynamics by fitting the bond autocorrelation function to a KWW stretched exponential function of the form C b(t ) = C0e−(t / τeff )

β

(3)

where C0, τeff, and β are fitting parameters referred to respectively as the pre-exponential factor, the unnormalized relaxation time, and the stretching exponent. This functional form is consistent with the view that glasses experience a range of relaxation times due to dynamical and/or structural heterogeneity. The further below unity the stretching exponent is, the more heterogeneous are the dynamics.6 Material behavior will change during different periods of a multistep 8469


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chain connectivity constraints. Under this scenario, one has for (nonaffine) displacements in the x-direction

creep deformation; it is therefore instructive to measure the dynamics within discrete “time windows”, as originally proposed by Riggleman et al.10,27 Here we use time windows of 500t ̃ except where noted otherwise. Local dynamics were also evaluated by examining the nonaffine displacements of individual particles. If all species in a system move in accordance with the macroscopic strain, the deformation is said to be affine; a nonaffine displacement ui for a given particle i is given by the deviation of the position of a particle from that corresponding to affine motion. In order to characterize the extent of heterogeneity, one can calculate the participation ratio (PR) according to PR = ⟨u 2⟩2 /⟨u 4⟩

⎛ −x 2 ⎞ Hs(x , ε) = (2Dx t )−1/2 exp⎜ ⎟ ⎝ 2Dx t ⎠

Here, Dx is an effective, stress- or strain-dependent segmental diffusion constant in the x direction; analogous expressions describe displacements in the y and z directions. Note that the values of t appearing in eq 8 are determined as a function of the target strain and applied strain rate according to eq 7. For uniaxial deformation, the behavior in two of the directions must be the same while the other would, in principle, differ, here modeled by a different diffusivity. However, experiments on glassy colloidal suspensions31 and simulations32,33 under strong flow conditions suggest the diffusion constants in all directions are quite similar and reflect an isotropically averaged hopping motion with barriers greatly reduced by deformation.34 If this physical picture applies, then the (hopping) diffusion constant is given by

(4)

where the brackets denote an average over all beads in the system. Displacements were calculated over a period of 400t.̃ PR ranges from unity, when all particles move as a concerted whole, to 1/N, when one particle out of all N particles of the system undergoes a large enough displacement to dominate the bracketed averages. A lower PR is indicative of more heterogeneous dynamics. An additional set of deformations were performed at constant strain rate using the smaller systems (23 000 polymer beads + 2560 particle beads) deformed at constant strain rate by applying tension as in our previous work.27 Deformation rates from 5 × 10−6 to 6.4 × 10−4 were considered here, in units of 1/t.̃ The resulting distribution of segment displacements can be quantified using the van Hove function: Gs(z , t ) =

1 N

Dj ≈

N

(5)

which describes the probability that a segment has moved a distance z in a set time along a given direction. Following Chung and Lacks,13 mobility was quantified through a modified van Hove function: Hs(z , ε) =

1 N

N

∑ δ(z − |uj(ε)|) j

(6)

Whereas the van Hove correlation function quantifies motion at a set time, this function measures the probability that a site has moved a distance z by the time the system has undergone a global strain of ε. In the work of Chung and Lacks,13 deformations were performed via shear and the segmental displacements were only monitored in the direction normal to deformation. This allowed for unambiguous definition of nonaffine displacement. In this work, which adopts an extensional deformation, mobility was measured in all directions with appropriate rescaling of coordinates to remove the effect of affine motion. We shall examine the nonaffine displacements in the deformation direction and its orthogonal counterparts, which are not necessarily equivalent. Theoretical Models. We are interested in the modified van Hove function at strains well beyond the nominal yield value, γ > > γyield ≈ 0.05−0.1, where strain γ is defined by γ = εṫ

Δj 2 (γ ) 2τα(γ )

(9)

where Δj(γ) is a mean diffusive jump distance in direction j that is expected to be of the order of a polymer segment size, σ, and τα(γ) is a jump time scale (proportional to the mean α relaxation) associated with a strain γ defined above. Physically, for a uniaxial deformation one expects the jumping distance in the strongly postyield regime of interest here to be modestly different along and orthogonal to the pulling direction. Based on potential energy landscape studies15,35 and also the nonlinear Langevin equation theory,36 a modest decrease of the mean jump distance with increasing strain might also be anticipated. Moreover, in the strongly postyield regime one expects that, based on simulation8,10,23 and theory,37 a nearly steady state is achieved whence the α time acquires a limiting plastic flow regime constant value at high strain. Small positive deviations of the α time are predicted in the strain hardening regime,38 but this is a modest correction that is difficult to detect in simulations.8,10,23 Combining the above relations, one has

∑ δ(z − |uj(t )|) j

(8)

1 x2 ln(Q x) − Q x 2 2 ⎛ x ⎞2 1 ≡ − ln(Q x) − Bx ⎜ ⎟ ⎝σ ⎠ 2

ln{H(x ; γ )} ∝ −

Qx ≡

ετ̇ α , ∞ 2

Δx γ

≡

Wi Δx 2 γ

(10)

(11)

where a dimensionless curvature constant, Bx, is defined; an analogous expression applies in the directions orthogonal to the pulling direction. An effective Weissenberg (Wi) number has been defined in eq 11 in terms of the applied deformation rate and the limiting steady state (isotropically averaged) segmental relaxation time. Although the above model is simple, and ignores issues such as a distribution of diffusion constants and non-Gaussian displacements, we shall demonstrate that it accurately captures our simulation data under the strongly postyield conditions studied here. This provides evidence that some aspects of dynamic heterogeneity are erased, as suggested by recent

(7)

and t is time in units of t.̃ In this postyield, plastic flow regime one might expect segmental motion to be Fickian-like, with an effective diffusion constant accelerated by the large deformation. Such a simple picture could be valid as long as displacements are not large enough to trigger dynamical 8470


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Figure 1. Full bond autocorrelation decay for T/Tg = 1.1 (left). Black denotes the pure polymer, while red, green, blue, orange, and violet denote the PNC containing 1mer, 4mer, 6mer, 8mer, and 16mer rods, respectively. Image on right shows fits of polymer autocorrelation curves to eq 3 for several temperatures with a limited fitting window (see text).

find below) might be rationalized if the theory is indeed applicable. Equation 12 was developed, and applied to polymer liquids,40 to describe changes with cooling in the equilibrated molten state. However, the central underlying idea, in its simplest form, is that activated dynamics involve a barrier that obeys a Gaussian distribution.39,40 Hence, its relevance can be explored for the systems of present interest, glassy polymers subjected to deformation where stress or strain, not temperature, modifies the relaxation time and degree of nonexponential relaxation. The usefulness of this model will only be tested for the pure polymer and mixtures with a one-site filler (1mer PNCs). For nanocomposites with extended fillers, one expects extra dynamic heterogeneity effects associated with adsorbed polymer layers surrounding the rods in addition to a population of polymers further away from nanoparticle surfaces,41,42 a thus an even more heterogeneous situation that the theory underlying eq 12 addresses. Hence, we do not apply the theoretical model to these systems.

experiments on glassy colloidal suspensions31 and polymer glasses.6 Recent theoretical work for polymer glasses37 predicts the effective Wi number in the flow regime is significantly less than unity and increases only weakly with deformation rate. The question of how deformation modifies segmental heterogeneous dynamics is also of interest. Here we explore the applicability of an existing simple model to organize and interpret our simulation results. Specifically, an activated barrier-hopping theory of glassy dynamics based on the assumed existence of nanometer-scale domains characterized by different local densities determined by small-scale thermodynamics, and hence different local barriers and relaxation times, will be employed.39 This model has been successfully applied to provide insights into the decoupling of segmental and chain scale relaxation in deeply supercooled polymer liquids.40 Its relevant aspect here is the prediction of a stretched exponential relaxation of Cb(t) with a KWW exponent directly relating the mean relaxation time for temperatures below the dynamic crossover temperature, Tc (typically ∼1.3Tg or so37), where collective barriers emerge:39 β

−2

A=

q≡

⎛ τ ⎞ − 1 ≅ A ln⎜ α ⎟ ⎝ τα(Tc) ⎠

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RESULTS AND DISCUSSION Melt State. Before examining the polymer and PNC systems under glassy deformation, it is necessary to establish a baseline for comparison by briefly characterizing the dynamics of the liquid state. The left side of Figure 1 shows the segmental bond relaxation function Cb(t) at a temperature T/Tg = 1.1. Additional simulations were performed at T/Tg = 1.3 and T/Tg = 1.8 (results are not shown). Results for the pure polymer (black curve) show two relaxation regimes. The fast (“β” process) regime accounts for only about 0.05 of the bond relaxation, while the slow (“α” process) regime accounts for the majority of the observed behavior; at very long times and low amplitudes, deviations from the KWW form occur due to dynamical constraints of chain connectivity. The slow regime has a decreasing relaxation time with increasing temperature, while the fast regime is relatively insensitive to temperature. At the highest temperature considered here (T/Tg = 1.8), this causes the two regimes to become conflated. To better separate these distinct modes of relaxation, polymer curves for all three temperatures were fit to the sum of two KWW functions. While such an approach is likely an overfit of the data, it provides a rough estimate of what time scales are relevant to each set of processes. It is found that the contribution from the fast modes is nearly zero by the time t ̃ = 1 regardless of temperature. As such, all further fits of Cb(t) are limited to t ̃ ≥ 1, allowing us to focus on the slow modes. The results of fitting only to data where t ̃ ≥ 11 and Cb(t) ≥ 0.5 (explained below) are shown in the last image of Figure 1.

(12)

8q (1 + q)3

(13)

σF 2 2FB̅

(14)

By construction, Debye (single exponential) relaxation is recovered above Tc. Note that this is not generally true for real polymers, where chain connectivity alone, even in the absence of glassy physics, leads to stretching. The prefactor A is determined by the ratio of the variance in barrier fluctuation to the mean dynamic free energy barrier, q. It has been estimated a priori for both glassy hard sphere colloidal suspensions and supercooled polymer melts to be essentially constant (independent of temperature and/or volume fraction), and ≈0.06−0.15 based on the idea that the dynamical domain diameters are 3−4 times the elementary length scale (colloid particle diameter or polymer segment diameter).39 The recent application to understand decoupling phenomena of a wide range of chemically distinct polymer melts led to a chemistryindependent value of q ∼ 0.13,40 which implies A ∼ 0.72. One does not expect the same value to apply to the computersimulated bead−spring model, but a value of order unity (as we 8471


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Figure 2. Results of bond-relaxation fits to KWW stretched exponential function in the liquid state. The left panel shows the relaxation time, while the right panel presents stretching exponents. Black denotes the pure polymer, while red, green, blue, orange, and violet denote the PNC containing 1mer, 4mer, 6mer, 8mer, and 16mer rods, respectively.

Figure 3. Strain as a function of time for multistep creep deformation. From left to right, applied stress was 0.62, 0.20, 0.62, and 0. Black denotes the pure polymer, while red, green, blue, orange, and violet denote the PNC containing 1mer, 4mer, 6mer, 8mer, and 16mer rods, respectively. The figure on the right shows the corresponding change in inherent structure energy during this process.

methods of fitting give identical results for the pure polymer, but differences arise for the PNC. The value of τeff, a mean segmental relaxation time, varies depending on the choice of fitting equation. Since the available data for fitting during creep deformations are limited to a range of Cb(t) that is inadequate for extrapolating C∞, values for the fit to eq 3 are reported in Figure 2. As can be expected, τeff decreases strongly with increasing temperature, especially at temperatures closer to the glass transition. Aggregated PNCs exhibit the highest τeff across the temperature range. Results using eq 15 gives τeff values for the PNCs that are always within a factor of 2 of the pure polymer and in most cases deviate by less than 30%. This suggests that a complete description of Cb(t) may separate pure polymer relaxation modes from the influence of nanocomposite modes in the PNC. However, we currently lack an adequate theoretical model to distinguish between these overlapping time scales. Note that these results for Cb(t) describe only the motion of the polymer component of these systems. Values for the stretching exponent β for the two fitting procedures agree with each other (within the uncertainty of the fits). However, the uncertainty in the fit is much lower using eq 15, allowing us to more easily discern trends. Results are shown in Figure 2. Well above Tg, the pure polymer has the highest β, whereas the PNCs exhibit varying levels of heterogeneity, even in the melt state. Across all systems, β changes little with temperature between T/Tg = 1.8 and T/Tg = 1.3, suggesting that such differences result primarily from “high temperature” polymeric, chain-connectivity effects. However, the stretching exponent decreases as the systems approach and go below Tg. This is consistent with the view that heterogeneity due to glass physics emerges as the deeply supercooled state is entered, and within the limited number of temperatures investigated here, T/Tg = 1.3 is a reasonable estimate of a dynamic crossover

Aside from the fast modes, the only deviation of the fit from the source data is a small error in the slowest part of the relaxation. Figure 1 also reveals the onset of additional slower modes of relaxation in the PNC. These are especially notable in the aggregated 8mer and 16mer systems, suggesting a separation of modes for polymer-rich and particle-rich domains. This seems qualitatively consistent with Kim et al.26 whose larger particles create distinct populations of differing mobility. Additionally, our previous examination of dynamic structure functions for high temperature melts also showed that polymer near aggregated particles have slower dynamics.27 Fits to the complete Cb(t) for PNC using the sum of two KWW functions for t̃ > 1 were not capable of fully describing the curve despite the large number of fitting parameters this entails. This suggests that additional insight is required to describe the PNC bond relaxation for these slower modes. This difficulty occurs well above Tg, suggesting the issue is fundamental to PNC; they have inherent heterogeneity unrelated to the bulk glassy state. However, the focus of the current work is to describe heterogeneity during glassy creep. These long-time modes cannot be observed in the limited frame of rapid deformation and are beyond the scope of the current work. The reader is referred to the work of Kim et al.26 and additional references cited therein for further discussion of these slower modes of behavior. Instead, we limit fitting to Cb(t) > 0.5 to separate these slower modes. To ensure this method provides an adequate description of the data, fits were performed with both eq 3 and the following modified form: β

C b(t ) = (C0 − C∞)e−(t / τeff ) + C∞

(15)

where C∞ accounts for the contribution of the slower modes just as C0 is historically used to separate out fast modes. Both 8472


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Figure 4. Effective relaxation time during multistep creep deformation, nondimensionalized by the stress-free value (left panel), and the corresponding stretching exponents (right panel). Black denotes the pure polymer, while red, green, blue, orange, and violet denote the PNCs containing 1mer, 4mer, 6mer, 8mer, and 16mer rods, respectively.

that the application of stress raises the inherent structure energy of both the pure polymer and PNC systems. The subsequent removal of stress allows for relaxation to lower basins in the potential energy landscape, but not to the original undeformed state. Despite the limited strain response of the PNCs, their inherent structure energy experiences changes of comparable magnitude to those of the pure polymer. There is no clear trend with rod length; the 4mer and 6mer behave similarly, as do the 8mer and 16mer. The only qualitative difference consists of an increase for the pure polymer between 1000t ̃ and 2000t ̃ as well as between 7000t ̃ and 8000t.̃ These times correspond to periods of local plastic flow. In order to explore dynamic heterogeneity during creep, it is of interest to measure the local polymer relaxation. Figure 4 shows the stretching exponent (β) and the characteristic relaxation time (τeff) extracted from KWW fits to results for Cb(t). These parameters refer to bond-orientation rotational relaxation. Fits only included data for t ̃ ≥ 10 (“α” process), beyond the regime of fast modes (“β” process). Note that Cb(t) is greater than 0.7 in all windows examined. Values of τeff in Figure 4 are normalized by the relaxation time for an undeformed sample. We begin our analysis with the pure polymer, shown by the black line. A sharp, essentially instantaneous, decrease of roughly 1 order of magnitude in relaxation time is observed with application of stress, as anticipated from the Eyring model and polymer NLE theory.14 An additional order of magnitude of decrease is then seen within the first stress window, consistent with the view that flow enhances mobility. Upon the reduction of stress entering the second window, the relaxation time increases drastically as the material approaches a more “glassy” state. Application of stress in the third step again reduces the relaxation time; however, this process reaches a minimum near 8000t.̃ This return to a slowing of the dynamics while still under deformation coincides with the onset of strain hardening in Figure 3, where strain rate decreases. In the fourth window, the bond relaxation dynamics again slow down when the stress is removed. PNC relaxation times decrease in the first and third domains and recover to their undeformed values in the second and fourth domains. However, the magnitudes of the changes in τeff are much smaller for the PNC than for the pure polymer, suggesting that the observed change in inherent energy is not a sufficient basis to predict enhanced mobility. The limited enhancements observed here do not follow a clear trend with rod length. We have previously shown that τeff is proportional to an instantaneous strain rate in pure polymers.10 The PNCs, which exhibit little change during the applied stress domains,

temperature to strongly activated dynamics. The pure polymer and 1mer PNC show the largest changes across Tg, while other PNC show their largest drops in β just above Tg. For the remainder of the text all systems are examined at T/ Tg = 0.95. Creep Deformation. Application of creep deformations allow for examination of both dynamic strain at constant stress as well as local segmental relaxation. The effective mechanical history consists of applying a stress of 0.62 for 2000t,̃ 0.20 for 4000t,̃ 0.62 for 4000t,̃ and 0 for 4000t.̃ Figure 3 shows the dynamic strain response resulting from that applied multistep creep sequence. The response of the pure polymer is given by the black line. The four distinct regions of behavior correspond to the different creep steps. The initial strong applied stress produces a small strain jump of 0.02, followed by an approximately exponential increase (indicative of plastic flow or yielding) in strain up to a value of 0.23, when the stress is released. In the second step, a small, positive stress is applied. The abrupt decrease in stress leads to a drop in the strain of nearly 0.02. The deformation is then resumed in the direction of the applied stress, albeit at a near-zero rate. In the third domain, a large stress is again applied, causing the system to jump back to the strain it had at the end of the first domain. Thereafter, the polymer deforms rapidly until it reaches 8000t,̃ the onset of strain hardening. At this inflection point, the still rapid deformation begins to slow down. Finally, in the fourth domain, the external stress is released and the polymer rapidly relaxes toward a steady, final deformed state characterized by a large residual strain. The strain response of the PNCs is much more muted than that of the pure polymer, consistent with the increased mechanical strength that the nanorod additives confer to the polymer.27 Importantly, the PNC systems do not yield (no increase in strain rate during deformation). There is a trend of increasing strain response with decreasing filler length. The longer, aggregated rods show the smallest strain response, consistent with the view that the large nanorod clusters provide additional reinforcement. Recall that the aggregates created by the 8mer and 16mer include trapped polymer, creating a networked state that may be responsible for this additional reinforcement. Also recall that these dispersed PNCs exhibit a higher Tg than the pure polymer, resulting in their runs being performed at higher absolute temperatures. The inherent structure energies were determined by minimizing the energy of configurations obtained during the multistep creep. Note that points on the cusp of a change in applied stress in Figure 3 correspond to configurations immediately before the change. Examination of Figure 3 reveals 8473


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Figure 5. Strain as a function of time for multistep creep deformation. For the left image, the applied stress sequence was 0.877, 0.283, 0.877, and 0 (41.4% higher across all domains than Figure 3). For the right image, the applied stress sequence was 1.24, 0.400, 1.24, and 0 (100% higher across all domains than Figure 3). Black denotes the pure polymer, while red, green, blue, orange, and violet denote PNCs containing 1mer, 4mer, 6mer, 8mer, and 16mer rods, respectively.

Figure 6. Effective segmental bond relaxation times deduced from the KWW fits during multistep creep deformation at 41.4% higher applied stress across all domains than Figure 3 (left panel). The right panel shows the corresponding stretching exponents. Black denotes the pure polymer, while red, green, blue, orange, and violet denote the PNC containing 1mer, 4mer, 6mer, 8mer, and 16mer rods, respectively.

show behavior consistent with the view that changes in τeff require changes in strain rate. One sees from Figure 4 that the stretching exponent also changes with the deformation for the pure polymer. If left undeformed, the pure polymer has β ∼ 0.66. During periods of active deformation, β jumps to higher values, indicative of a more homogeneous distribution of bond relaxation times. Once stress is removed, the stretching exponent relaxes back to lower values, more typical of a heterogeneous glassy material. For PNCs, most shifts in β are not statistically significant. The undeformed PNCs have β values between 0.47 and 0.64, indicating they are more heterogeneous than the undeformed pure polymer. Stretching exponents of the PNC systems increase in the order 6mer, 4mer, 8mer, 16mer, and 1mer. This ordering matches that seen in the liquid state (Figure 2), suggesting that the observed differences are inherited from the liquid. Since the PNCs are mechanically stronger than the pure polymer,27 for a given stress they exhibit a reduced strain response. As both the strain and strain rate are possible contributors to the degree or rejuvenation, the trends discussed above may be a byproduct of the lower strains, rather than a qualitative difference in polymer and PNC behavior. To explore this possibility, we also considered larger stresses. Specifically, those stresses were 41.4% and 100% higher than reported above across all windows. Results are shown in Figure 5 for applied stress sequences of 0.877, 0.283, 0.877, and 0 for the former and 1.24, 0.400, 1.24, and 0 for the latter. The pure polymer under the original stress deformed to a strain of 0.224 in the first window. At both higher applied stresses, the pure polymer undergoes a pronounced initial strain response wherein the system almost immediately enters the strain hardening regime. Despite this, the latter domains of behavior are qualitatively similar across all applied stresses.

Higher stresses induce a PNC creep that is qualitatively similar to that of the pure polymer. At 41.4% higher stress, the 1mer undergoes a strain of 0.185 in the first window, and the overall curve is qualitatively identical to that for the low-stress deformation of the pure polymer. The 4mer and 6mer still exhibit a limited response but do exhibit flow in the first and third windows. The aggregated 8mer and 16mer systems behave as they did under the original, lower stress protocol. Interestingly, only the dispersed PNCs undergo flow. Previous examination of PNCs under constant strain−rate tension and compression found yield stress to be similar for all of the PNCs examined there,27 which would suggest these systems would have similar creep compliances. However, it should be noted that both the pressure and temperature used to prepare the samples in the current study differ from those of that previous work. At 100% higher stress, the 1mer, 4mer, and 6mer systems undergo strains 0.705, 0.603, and 0.513, respectively, in the first window. This is larger than that of the pure polymer under the original stress. All systems, including the aggregated PNC, exhibit similar response curves at this stress. Exploring a range of applied stresses allows for a comparison of the polymer and the PNCs at similar strains. Examination of the inherent structure energy during higher applied stresses (not shown) provides similar trends to those shown in Figure 3, but with larger magnitudes of change. As before, the bond vector relaxation function is used to extract τeff and β. Results for the 41.4% higher stress results are shown in Figure 6. The τeff values are lower than under the original applied stresses across all systems, consistent with larger deformations having a greater impact on mobility. The pure polymer τeff response in the first window exhibits a large initial drop (nearly 3 orders of magnitude), followed by an increase. The observed increase in τeff in the first window is contrary to the τeff decrease seen in Figure 4 for polymer 8474


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Figure 7. Results of bond-relaxation fits to KWW function during multistep creep deformation at a stress 100% higher than that applied in Figure 3. The left figure shows the relaxation times and the right figure shows the stretching exponents. Black denotes the pure polymer, while red, green, blue, orange, and violet correspond to PNCs containing 1mer, 4mer, 6mer, 8mer, and 16mer rods, respectively.

experiencing lower strains. This is attributed to the strain undergoing an inflection point in the creep curve and crossing over to the strain-hardening regime almost immediately. Behavior beyond the first window mirrors the results at lower stress. The 1mer exhibits a τeff comparable to that found for the pure polymer in Figure 4. This suggests that the presence or absence of flow, regardless of applied stress, determines the τeff response. Other PNCs show a clearer response than that exhibited in Figure 4, but the magnitude of the effect remains limited. Examination of the β exponent values tells a similar story. Despite nearly a factor of 2 increase in strain response, the heterogeneity of the pure polymer exhibits the same behavior under this multistep creep. The PNCs become more homogeneous with increased stress but are still heterogeneous as compared to the pure polymer. The 1mer again mimics the behavior of the pure polymer. Relaxation was also examined for the 100% higher stress system in Figure 7. Pure polymer results are virtually unchanged from the 41.4% higher tests for both β and τeff, suggesting that an effectively fully rejuvenated state has already been reached in which collective activation barriers have been destroyed. The 1mer system at this highest applied stress also reproduces the behavior of the pure polymer. Other PNC curves are shifted notably toward the response of the pure polymer for both β and τeff, suggesting that mobility enhancement for PNCs mirrors that of polymer once a sufficiently large stress is applied. However, some residual heterogeneity can be seen in the PNC, which again may reflect a baseline heterogeneity intrinsic to the liquid state. Using the values of τeff and the exponent β collected across the range of multistep deformations performed, one can test the activated barrier-hopping dynamic heterogeneity model discussed in the Theoretical Models section. The effective relaxation time is normalized by the high temperature, quiescent liquid dynamic crossover value, here identified as T/Tg = 1.3. Examining only data from the pure polymer and 1mer, Figure 8 plots the effective relaxation time τeff and stretching exponent β in the format suggested by eq 12. There is a good collapse of the data regardless of applied stress, material compliance, or what window of the multistep process the system is in. This supports the idea of a direct connection between the effective mean α time and the degree of relaxation function stretching. The level of collapse is not significantly sensitive to the choice of the dynamic crossover temperature in the range of T/Tg between 1.1 and 1.8 (not shown). Two regimes can be seen in Figure 8. For large τeff corresponding to the glassy state, the data collapse onto a line with a slope of approximately unity, the value expected

Figure 8. β versus τeff for pure polymer (black symbols) and 1mer PNC (red symbols) in the format suggested by eq 12. Circles denote results from Figure 4, squares denote results from Figure 6, and diamonds denote results from Figure 7. Symbol size corresponds to the error bars for the point having highest uncertainty.

from the theoretical model as discussed in the Theoretical Models section. The physical implication is that as the deformation reduces the α time, the dynamics become more homogeneous. This linear behavior persists until β ≈ 0.8 (β−2 − 1 ≈ 0.56). This seems consistent with the theoretical picture that the heterogeneity associated with deeply supercooled dynamics and barrier hopping is largely erased above this temperature. At this point, heterogeneity is still present but is independent of τeff, suggesting a simpler, largely polymer connectivity origin. Note that this value of β at which the transition in Figure 8 occurs (0.80) is above that found in the liquid state (0.73). Other PNC systems (not shown) exhibit similar curves but demonstrate a larger degree of scatter and fail to collapse upon each other or the pure polymer curve, as expected based on the discussion in the Theoretical Models section. Returning to the original set of deformations (Figure 3), the participation ratio PR is examined as an alternative means to describe the heterogeneity of these systems under multistep creep deformation, as shown in Figure 9. Note that only the nonaffine contribution to displacement is included in the calculation of PR. Whereas β describes segmental motion on the length scale of bonds, PR describes translational motion over a time scale of 400t.̃ For clarity, results for the polymer component and the nanorods of each system are shown separately. For the polymer component, starting with the first window, there is an increase in the PR for the pure polymer. In the second window, the pure polymer exhibits a drop in PR that is indicative of increased heterogeneity in the system. The third and fourth steps exhibit behavior similar to that seen in the first and second steps. As reported in the literature,23 deformation homogenizes pure polymer motion with increasing strain rate. 8475


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Figure 9. Participation ratio during multistep creep deformation. Left figure shows results for polymer segments. Right figure shows results for the filler sites. Black denotes the pure polymer, while red, green, blue, orange, and violet symbols correspond to the PNCs containing 1mer, 4mer, 6mer, 8mer, and 16mer rods, respectively.

Figure 10. Participation ratio during multistep creep deformation for polymer sites at 41.4% higher stresses (left) and 100% higher stresses(right) than those reported in Figure 9. Black denotes the pure polymer, while red, green, blue, orange, and violet denote the PNCs containing 1mer, 4mer, 6mer, 8mer, and 16mer rods, respectively.

changes in the applied stress, PR only slowly moves to a more homogeneous state as τeff drops. In the second, third, and fourth windows, dispersed PNCs exhibit the same qualitative behavior as the pure polymer (PR results shown in Figure 9). The small peak between the third and fourth domains of 8mers and 16mers seen in Figure 9 is much more pronounced in Figure 10 and is clearly visible in other systems. Upon application of yet higher stress, 100% higher stress across all windows, the PR for the polymer component of the PNCs begins to behave like that of the pure polymer. This is consistent with the previous examination of β showing PNC heterogeneity approaching that of the pure polymer. For the particle components (not shown), as the deformation becomes more severe, the PR of the particles begins to more closely resemble that of the corresponding polymer components. It is also possible to directly visualize dynamic heterogeneity by examining images of the displacement fields. Figure 11 shows projections of the nonaffine displacement vectors perpendicular to the axis of deformation for both pure polymers and PNCs. For each system, the corresponding arrows of length proportional to their magnitude are shown for segments with the highest 10% displacements in a randomly chosen slice whose thickness is 3σ. Two snapshots of each slice are shown in the figure, corresponding to times of high PR (t ̃ = 1500−2000) and low PR (t ̃ = 13 500−14 000). An overlay of the positions of nanorods is included for all PNCs. Recall that larger configurations were produced by doubling small, well-equilibrated configurations in each dimension and then allowing for further relaxation. While polymer segments and short PNCs can become decorrelated during this final relaxation, residual correlations between the copied images of slow-moving aggregates are visible. By examining independent configurations, it is possible to average

In contrast to the results shown for the pure polymer, the PNCs fail to exhibit a meaningful change in the participation ratio. The aggregated 8mer and 16mer systems both produce an interesting response at the transition from the third to fourth steps: a sharp rise and fall in PR. This sudden change at the transition from applied stress to recovery is generally found, but it becomes smaller with increasing PR. As seen in Figure 3, there is a sharp change in the shape of the response during this rapid transition. This sudden motion forces numerous local rearrangements. For both the PR (Figure 9) and β (Figure 4), these rearrangements appear to be more homogeneous than the normal glassy motion. For the 8mer and 16mer, whose glassy PR is especially heterogeneous, these rearrangements produce a clear bump on the curves. For the nanoparticle sites, all PNCs show only minimal changes in PR upon deformation or recovery. In general, the rod sites have a higher PR than their polymer counterparts, especially for the aggregated 8mer and 16mer systems. This is unsurprising as aggregates can, in general, be expected to move as a single object. This additionally explains the increased noise in the measurement in these systems, as clustered movement reduces the number of independent sites, hence resulting in poorer statistics. The participation ratio was also calculated for multistep creep with applied stresses 41.4% and 100% higher, as shown in Figure 10. For the applied stress sequence 0.877, 0.283, 0.877, and 0 (41.4% higher across all domains than Figure 3), the PR of the pure polymer and 1mer systems jump to high values upon the initial application of stress. For the pure polymer, the values slowly relax to lower, more heterogeneous values, in the first window. For the dispersed PNCs (1mer, 4mer, and 6mer), an increase in PR is observed. Whereas β responded instantly to 8476


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Figure 11. Cross section of site displacement corresponding to slice thickness of 3σ. Images on left correspond to regime with high participation ratio (t ̃ = 1500−2000) and images on right correspond to regime having low participation ratio (t ̃ = 13 500−14 000). Top, middle, and bottom images correspond to pure polymer, 4mer, and 16mer systems, respectively. Blue arrows show the nonaffine displacement of individual sites during the time interval, and black bars show the location of the rods.

Figure 12. Fits to displacement distributions for the pure polymer at 0.4 strain along the axis of deformation using different fitting windows. Lines correspond to fits of the data, and points correspond to the displacement distribution. Each color corresponds to a different deformation rate in units of 1/t;̃ values given in legend. The figure on the left provides the basis for the following figures.

motion where the movement of one site allows another to fill the vacancy left behind, creating a new vacancy and so on. For the high PR plot (e.g., the system with a more homogeneous distribution of vectors magnitudes) mobile regions mainly contain randomly oriented vectors. For the low PR plot,

over different samples of cluster orientations, but individual systems do retain few independent cluster orientations. For the pure polymer, there is a clear contrast between the two frames, consistent with previous work.8 Both images show regions of higher and lower mobility as well as “strings” of 8477


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Figure 13. Fits to displacement distribution for pure polymer at 0.8 strain for displacements along axis of deformation (left) and perpendicular to axis of deformation (right). Lines correspond to fits and points correspond to the displacement distribution. Each color denotes a different deformation rate, in units of 1/t;̃ values given in legend.

Figure 14. Fits to displacement distribution for PNC at 0.8 strain for 4mer (left) and 16mer (right). Lines correspond to fits and points correspond to the displacement distribution. Each color denotes a different deformation rate in units of 1/t;̃ values given in legend.

virtually all mobile sites are part of the many “strings” in the system. That is, in the more homogeneous system, the most mobile sites have uncorrelated behavior aside from relative proximity. In the heterogeneous system, mobile sites only exist as part of a population undergoing cooperative rearrangements. In contrast, the PNCs show less qualitative change between the two images, consistent with the small change in PR observed. Regions of higher mobility do tend to occur farther from the additives. The relationships between rod location, mobility, mechanical strength, and failure in these systems have been examined at length in previous work.43 Modified van Hove Correlation Function. We now study the nonaffine component of the polymer segmental displacement distributions at fixed global strain, i.e., the modified van Hove function of eq 6. Results are analyzed using the simple Fickian diffusion model discussed in the Theoretical Models section in the strongly postyield (large strain) regime. Although we have touched upon the role of strain by examining multiple stress levels, it still remains to be seen if recent results from Chung and Lacks13 describing strain as an important indicator of mobility are applicable to PNCs. To explore this issue more directly, an alternative set of tensile deformations were performed across a range of constant strain rates. A representative subset of modified van Hove distribution data is shown in Figures 12−14. For all systems, we find a near collapse of the distributions across different strain rates and the expected roughly parabolic form. It is important to emphasize that the time spent deforming these samples differs by a factor of 2 between successive curves, yet the displacement of sites is very similar. The tails of the distributions (e.g., the most mobile handful of sites) do show differences with strain rate. These results are consistent with those of Chung and Lacks for the transition regime observed in their systems around strain rates of 107 s−1.13 Going from lowest to highest strain rate, a weak

trend is observed of higher strain rates yielding slightly less displacement. This trend is valid at all strains considered here. For the Fickian diffusion model under consideration, it is necessary to fit the tails of the distributions of polymer segment displacements by the functional form implied by eq 10 so that values for the curvature parameter (B) can be extracted. However, these tails exhibit the largest uncertainty as the inclusion or absence of a small number of sites can significantly affect probabilities on the order of 10−4. To help address this issue, displacement data were binned in x2 space such that wider bins were used for higher displacements to reduce noise. This allows trading off the number of points for fitting to obtain higher precision for the rarer displacements. Figure 12 shows the results of fitting in the x2 space overlaid on the histogram in normal x space. We are currently interested in segments of enhanced mobility which have displacements describable by eq 10, and as such only values of x greater than unity are used in the fits. This avoids conflating diffusive Fickian behavior with more local caging effects, which indeed may be non-Gaussian. The left plot corresponds to inclusion of data of probability greater than 10−4 while the right figure only includes data with probability greater than 10−3.5. In the latter case, some of the curves are able to roughly predict the low-distance part of the histogram (2σ). To ensure a reasonable description of the tail, the former data window will be used. Figure 13 compares the distributions associated with displacements in different directions at a strain of 0.8. At this higher strain, fitting only the tail portion is sufficient to describe the entire distribution along the axis of deformation (x). However, this does not work in the orthogonal directions, where the tail distribution remains distinct from the local distributions. Hence, we draw the interesting conclusion that 8478


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Figure 15. Relaxation time during constant strain rate deformations (left) and the corresponding Weissenberg numbers (right) at an applied strain of 0.4.

(modest) anisotropy in the segmental displacement distributions remains even well beyond the yield strain. Figure 14 shows the x-direction (the pulling direction) modified van Hove distribution for the PNCs. While undergoing the same strain as the pure polymer, PNCs tend to exhibit larger nonaffine displacements. The PNC segmental displacement modified van Hove distribution functions are remarkably similar to those of the pure polymer. PNCs demonstrate the same near-collapse across various strain rates, better collapse for lower displacements and a trend of better collapse at higher strains. Unlike the pure polymer, a simple fit to eq 10 does not describe the entire distribution of displacements for the PNC systems, although it remains possible to describe the tail of the distribution. Fitting the same functional form to PNCs at a strain of 0.4 provides results (not shown) similar to those of the pure polymer. In order to relate the values of the curvature parameter, B, extracted from these fits back to the theoretical model provided by eq 8 it is also necessary to determine the Weissenberg numbers (Wi) during these deformations. Figure 15 shows the relaxation times at a strain of 0.4 extracted by fitting Cb(t) using the time-window approach. At the high strains examined here, one is in the (nearly) steady-state plastic flow regime, where the effective relaxation time and stress are nearly constant, though weakly dependent on strain rate.37 Figure 15 shows that the relaxation times exhibit strong deformation rate “thinning”, as expected theoretically.37 Nondimensionalization of these data by strain rate provides Wi. Both the pure polymer and 4mer show a trend of weakly increasing Wi (of order 0.1) with increased deformation rate, consistent with theory,37 with both curves having the same general shape. The 16mer shows a greater increase in Wi with strain rate, going from the lowest values of Wi of any system for slow deformation rates, to the highest at faster deformation rates. Results obtained at applied stress of 0.8 are not shown. Finally, we can plot the values for Wi/γ against the B extracted from fits of log(Hs(x,ε)), as shown in Figure 16. According to the theoretical description provided by eq 10, a linear behavior should be obtained. The slope of the lines shown in the figure corresponds to Δ−2, the effective inverse square hopping or jumping distance of the model. From Figure 16 and equivalent plots for the other systems (not shown), one can extract the hopping distance. We find jump lengths of 0.827 (0.4 strain) or 0.716 (0.8 strain) for the pure polymer, 1.134 (0.4 strain) or 0.979 (0.8 strain) for the 4mer, and 1.402 (0.4 strain) or 1.478 (0.8 strain) for the 16mer. One sees that the length scale extracted decreases with increasing strain, qualitatively consistent with a prior theoretical analysis for activated hopping in glassy collodial suspensions

Figure 16. Sample plot (4mer) relating fitted values from displacement distributions to the linear functional form suggested by in eq 10. Black points from data obtained at a strain of 0.4, and red points from data obtained at a strain of 0.8. Lines denote linear fits to the data set of the same color with an intercept of zero. Green line is a fit to all data.

under strong deformation conditions.36 Interestingly, the jump length scale grows as fillers become longer, perhaps suggestive of growing additional heterogeneity. The mean jump distance associated with displacements along the axis of deformation versus those perpendicular has also been examined. Averaging over both strains, we find 0.871 versus 0.733 for the pure polymer and 1.209 versus 0.989 for the 4mer. This suggests that larger nonaffine displacements underlie a diffusive step in the tensile direction. No clear trend was found for the aggregated systems.

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CONCLUSION The results presented here extend previous studies of polymer glassy heterogeneous segmental dynamics into the realm of PNCs. The cooling of a material through the glass transition is associated with an increase in relaxation time. In the present work, the resulting change in molar volume was used to identify Tg. Dispersed additives significantly increased the observed Tg as compared to the pure polymer, while aggregated fillers phase separated from the polymer, allowing the bulk matrix Tg to be recovered. This provided two distinct classes of nanocomposites to evaluate throughout this work. Despite this range of behaviors, trends of increasing τeff and decreasing β upon cooling occurred for all PNCs in a fashion similar to that of the pure polymer. The values of β observed above Tg were much lower for the PNCs than the pure polymer, suggesting that the presence of additional heterogeneity in these systems is not directly related to that induced by the formation of a glass and could have other origins, including the physical adsorption of polymer molecules onto nanorod inclusions. It was additionally possible to identify an approximate dynamic crossover temperature, Tc ∼ 1.3Tg, above which β became 8479


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process once a site has escaped its local cage via activated hopping. Using this idea, we proposed a simple diffusion model and found it to be capable of describing the tails of the distributions across 2 orders of magnitude in strain rate for both pure polymer and PNCs. Additionally, this model provides an estimate of the hopping (diffusive step) distance associated with these distributions, which was found to be roughly a polymer segment size. The hopping distance exhibits systematic variations as a function of displacement direction and system type.

nearly temperature independent for both pure polymer and PNCs. These systems were cooled to a temperature of T/Tg = 0.95, briefly aged, and then subjected to various deformations in order to explore the question of how stress and strain modify segmental heterogeneous dynamics. One simple model for the origin of heterogeneity is a Gaussian distribution of barriers due to small scale fluctuations of density and the presence of distinct nanodomains.14,39 This model predicts a direct connection between the stretching exponent and the mean segmental relaxation time normalized by its value at the dynamic crossover temperature. This correlation was tested by cross-plotting the effective relaxation time and the stretching exponent deduced from multistep creep experiments on the pure polymer and 1mer PNC across a broad range of stress conditions in the format predicted by the theory. It was found that the connection between τeff and β prescribed by this model (eq 12) is well obeyed and allows diverse data to be coherently organized regardless of applied stress, material compliance, or what domain of the multistep process the system was in. Extending multistep creep to the PNCs revealed new insights into the behavior of this complicated subclass of glasses. First, at low stresses the inherent energy of both pure polymer and PNCs experienced changes of comparable magnitude, despite the limited strain response from the PNCs. We monitored β, which measures segmental dynamic heterogeneity, PR, which describes translational dynamic heterogeneity, and the effective relaxation time of the material, τeff. These three quantities revealed little change in the dynamics of the PNCs, suggesting that the inherent energy is not sufficiently correlated with PNC dynamics to provide a general correlation in this context. Second, once stresses were applied capable of producing comparable levels of strain response in the PNCs as in the pure polymer, it was found that the segmental dynamics of the PNCs began to approach the behavior of the pure polymer. As we had shown previously,8 the pure polymer exhibits a distinct connection between instantaneous strain rate with τeff and with various measures of heterogeneity. This behavior was also found in the PNCs considered here. However, β revealed that some level of residual heterogeneity persisted in the PNCs, even at high stress. This heterogeneity is likely connected to the intrinsic dynamics in the undeformed liquid state. Third, both β and PR revealed increased homogeneity during deformation and a subsequent return to heterogeneity during relaxation. Despite β measuring rotational segmental dynamics and PR measuring translation segmental dynamics, both metrics display similar results, suggesting that they provide equivalent descriptions of heterogeneity. The only difference found was that β responded almost immediately to changes in deformation state, while PR had a slightly more gradual response. These systems were further explored through constant strain rate deformations during which the nonaffine displacements were tracked. Chung and Lacks recently demonstrated13 that the distributions of these displacements for pure polymers are nearly independent of strain rate (within a certain range) when samples of the same global strain are compared. The PNC systems considered in this work were found to follow the same relation. Furthermore, we examined these displacements both along and perpendicular to the axis of deformation. It was found that a weak anisotropy existed, with segments being slightly more mobile along the axis of deformation. One might predict that such nonaffine displacements are the result of a simple (although direction dependent) Fickian-like diffusion

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the UW NSEC, DMR-0832760, by the Semiconductor Research Corporation, SRC, and by the Office of Naval Research through MURI Award N00014-11-10690. We thank the Grid Laboratory of Wisconsin (GLOW) for use of their computational resources.

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