Heterogeneous Rouse Model Predicts Polymer Chain Translational

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Heterogeneous Rouse Model Predicts Polymer Chain Translational Normal Mode Decoupling Jui-Hsiang Hung, Jayachandra Hari Mangalara, and David S. Simmons* Department of Polymer Engineering, The University of Akron, 250 South Forge Street, Akron, Ohio 44325-0301, United States S Supporting Information *

ABSTRACT: It has been known for 50 years that polymers exhibit chain normal mode decoupling upon approach to the glass transition, with chain dynamics exhibiting a weaker temperature dependence than segmental dynamics. Inspired by Sokolov and Schweizer’s suggestion that this thermorheological complexity is a consequence of dynamic heterogeneity in the supercooled state, we generalize the Rouse model to account for a distribution of segmental mobilities. The heterogeneous Rouse model (HRM) predicts chain translational normal mode decoupling as a manifestation of diffusion/relaxation decoupling (Stokes−Einstein breakdown)a consequence of differences in how normal modes average over a distribution of segmental mobilities. Molecular dynamics simulations agree with theoretical predictions, with the HRM found to quantitatively predict deviations from Rouse scaling of the translational friction coefficient based on the observed degree of Stokes−Einstein breakdown.

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fluctuating mobilities.14,15 Most recently, Sokolov and Schweizer argued that this effect is a consequence of “spatially heterogeneous dynamics which modifies local structural relaxation but is averaged out on the chain relaxation length and time scale”.6 Dynamic heterogeneity is a ubiquitous feature of glass-forming liquids,16−22 making this an attractive explanation. Sokolov and Schweizer’s hypothesis proposes that chain normal mode decoupling and the breakdown of the Stokes− Einstein (SE) relationship between diffusion and relaxation in supercooled liquids share a common physical origin in dynamic heterogeneity. A priori, it is not clear that this should be the case: the former phenomenon involves decoupling of like relaxation functions across chain normal modes; the latter involves decoupling between potentially distinct dynamical processes. Indeed, the precise origin of SE breakdown remains a matter of debate.23 The viewpoint adopted here, consistent with several (but not all24) theoretical frameworks including the random first-order transition theory25 and the entropic barrier hopping theory,26 argues that this decoupling emerges from averaging over heterogeneous dynamics.27 Specifically, if the Einstein relation applies locally, then the friction coefficient ζ0,i of a single particle is related to its diffusion coefficient D0,i as ζ0,i ∝ 1/D0,i. The mean friction coefficient ⟨ζ0⟩ in a system of N particles is then given by

INTRODUCTION A defining characteristic of flexible and semiflexible polymers is a separation between the time scales of local segmental relaxation and whole chain relaxation. The Rouse model provides a simple understanding of this separation in terms of a summation of segmental friction coefficients at the chain level.1 More generally, this model predicts thermorheological simplicitya shared temperature dependence for segmental relaxation and all chain relaxation modes. Even in models of entangled polymers, this prediction is preserved and provides the fundamental underpinning for time−temperature superposition (TTS) methods. In contrast, it has long been observed that polymers exhibit thermorheological complexity upon approach to the glass transition temperature, Tg.2−11 Chain relaxation is commonly observed to exhibit a weaker temperature dependence than segmental relaxation, signaling a breakdown of TTS. Work probing segment/chain decoupling over a range of molecular weights demonstrates that this phenomenon spans both entangled and unentangled polymers, indicating that it requires an explanation at the level of local chain friction effects and not entanglement or gyration-radiusrelated effects.5 Moreover, the strength of chain normal mode decoupling appears to be correlated with the fragility (strength of the temperature dependence12) of a given polymer’s segmental relaxation process, again linking this effect to local segmental friction.6 Several explanations have been proposed for the origin of chain normal mode decoupling. On the basis of the coupling model, Ngai and Plazek suggested it is due to a difference in the strength of coupling of segmental and terminal processes to the environment.13 Loring developed a dynamical mean-field model for chain dynamics in which the segments possess © XXXX American Chemical Society

N

⟨ζ0⟩ ≡

1 ∑ ζ0,i N i=1

(1)

Received: January 22, 2018 Revised: March 14, 2018

A

DOI: 10.1021/acs.macromol.8b00135 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

liquids. This heterogeneity corresponds to the presence of a distribution of local mobilities. As discussed above, the distribution has been argued to account for the breakdown of the Stokes−Einstein relationship. Here we show that this “averaging effect” can account for chain normal mode decoupling in polymers. To do so, we begin by employing coarse-grained molecular dynamics simulations to test Sokolov and Schweizer’s proposition that longer strands exhibit less dynamic heterogeneity. We then present a redevelopment of the Rouse model that accounts for the presence of a distribution of segmental mobilities. Finally, the predictions of this model are shown to be in quantitative agreement with simulation, indicating that dynamic heterogeneity is the leading order origin of chain translational normal mode decoupling.

Substitution of the local Einstein relation into eq 1 leads to the result that ⟨ζ0⟩ ∝ ⟨1/D0⟩. In contrast, a “system level” application of SE would yield ⟨ζ0⟩SE ∝ 1/⟨D0⟩. In the presence of a distribution of relaxation timesdynamic heterogeneity these two quantities are not equal. The SE breakdown ratio rSE ≡

⟨ζ0⟩ ⟨1/D0⟩ = ⟨ζ0⟩SE 1/⟨D0⟩

(2)

is therefore greater than 1 in the presence of dynamic heterogeneity. This difference between the average of the inverse mobility and the inverse of the average mobility provides a plausible explanation for the origin of SE breakdown: a broadening of the distribution of local mobilities as dynamic heterogeneity emerges upon approach to Tg leads to a temperature-dependent value of rSE and a commensurate breakdown in a fixed SE proportionality between relaxation and diffusion. Can this same averaging effect account for chain normal mode decoupling in polymers? In answering this question, it is useful to recall the key physical assumptions of the Rouse model that lead to the prediction of coupled chain normal mode dynamics. First, the Rouse model assumes that the Einstein relation, ζ ∝ 1/D, interrelates the friction coefficient and diffusion constants at both the segmental and chain normal mode levels, such that ζ0 ∝ 1/D0 and ζn ∝ 1/Dn (where the subscript n denotes a value at the level of a subchain strand of length n). Second, Rouse assumes that segmental motions are uncorrelated, such that the chain friction coefficient can be written as the sum of segmental friction coefficients: n ζn = ∑i ζ0, i . Third, this Rouse model assumes that friction coefficients are monodisperse, such that the summation above can be rewritten as ζn,R = nζ0

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SIMULATION METHODS We employ molecular dynamics simulations of melts of bead− spring polymer chains modeled via the attractive extension28−36 of the Kremer−Grest37 model. This model is well-described in the literature and has been widely employed to study both polymer glass formation28−30,32,33,36,38 and chain dynamics.39 Here we specifically study systems composed of 12-, 24-, and 48-bead chains. In-equilibrium configurations are generated over a range of temperatures approaching Tg for each system via a quench-and-anneal procedure. Within this procedure, prior to data collection each configuration is subject to an isothermal annealing period of at least 10 times the terminal chain end-to-end reorientational relaxation time, as determined from the reorientational autocorrelation function C2(t) described below. The longest segmental relaxation times probed in this study are of order 104 LJ time units, corresponding approximately to 104 ps (based on a standard conversion of 1 LJ time unit to 1 ps). We note that longest segmental relaxation times probed decrease with increasing overall chain molecular weight due to the choice described above to anneal for a product of the whole chain relaxation time. The 12-bead chain therefore enables access to the most deeply supercooled states. The longest whole-chain relaxation times approach 106 LJ time units, corresponding to approximately 1 μs in real units. All simulations are performed at a constant pressure P = 0, employing a Nosé−Hoover thermostat and barostat with damping parameters of 2 LJ time units. Simulations are performed in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package40 using a Verlet integrator with a time step of 0.005 τLJ, with the total system momentum zeroed out every 10−100 τLJ to prevent any possible momentum buildup. Chain dynamics are quantified based upon both a translational and reorientational relaxation function. Translational chain dynamics are quantified on a strand-length-dependent basis via the self-part of the intermediate scattering function41

(3)

where the subscript “R” denotes that this is the classical Rouse prediction. Finally, this adding rule for strand friction coefficients is converted into a relaxation time prediction via the assumption of Gaussian chain statistics, leading to the result that τn ∼ R2ζn ∝ nR g 2τ0 ∝ n2τ0

(4)

Of these four physical arguments, the assumption of Gaussian chain statistics is the most commonly cited cause of breakdowns in Rouse scaling of the molecular weight dependence of the relaxation time. However, this is not a likely origin of chain normal mode decoupling, as polymer chain conformations in the melt tend to be temperature insensitive and therefore cannot be expected to account for an effect that by definition represents a temperature-dependent change in the ratio of chain to segmental dynamical rates. In order to exclude any possible contamination by the effect of non-Gaussian chain statistics, in this study we therefore focus primarily on friction coefficients ζ rather than relaxation times τ. By focusing on the breakdown of eq 3 rather than eq 4 in the glass formation range, we eliminate any assumption whatsoever regarding chain conformational statistics and focus purely on the dynamical content of the theory. Breakdowns in the Rouse prediction for ζn must therefore emerge from a breakdown in at least one of the first three assumptions described above. Notably, the third of these assumptionsthat of a monodisperse friction coefficientis directly at odds with the known presence of dynamical heterogeneity in glass-forming

N

Fs(q, t ) =

1 ∑ ⟨exp[−iq·(rj(t ) − rj(0))]⟩ N j

(5)

where q is the wavevector and rj(t) is the position of particle j at time t. In general, we average over numerous randomly chosen wavevectors q corresponding closely to a given wavenumber q to obtain Fs(q,t). We compute this relaxation function both at the level of single segments and at the level of strands of length n = N/p, B


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Macromolecules where N is the number of segments in a chain and p is the chain normal mode index. To compute Fs(q,t) for a strand of length n, we first divide each chain into p strands, each of length n. We then locate the center of mass for each of these n-strands as a function of time, and we apply eq 5 at the level of this center of mass. For example, this allows computation in 12-bead chains of a relaxation time for nonoverlapping strands with n = 1, 2, 3, 4, 6, and 12 segments (p = 12, 6, 4, 3, 2, and 1), respectively. We then define a friction coefficient ζF,n for any strand length n at a wavenumber q = 7.07 as q2 times the time at which this function relaxes to 0.2, employing a fit to a Kohlrausch− Williams−Watts stretched exponential functional form42,43 β⎤ ⎡ ⎛ t ⎞ ⎥ Fs(q , t ) = h exp⎢ −⎜ ⎟ ⎢⎣ ⎝ τKWW ⎠ ⎥⎦

(6)

for data smoothing and interpolation. Here τKWW is a time constant, h is called the nonergodicity parameter, and β is the stretching exponent. The use here of a cutoff of 0.2 to define a relaxation time or friction coefficient has been widely employed in recent simulation work.38,44 Reorientational relaxation is quantified via the reorientational autocorrelation function C2(t), given by45 C2(t ) = ⟨P2[ei(0) ·ei(t )]⟩

Figure 1. Effective friction coefficient ζF,n as determined from the selfpart of the intermediate scattering function, plotted for each strand length n as a function of inverse temperature, for chains of total length 12 (a), 24 (b), and 48 (c).

(7)

where P2 is the second Legendre polynomial and ei(t) is the ith unit vector under consideration at time t. Here the vector employed is the end-to-end vector of a strand consisting of n = N/p segments. For mode p = 6 (n = 2) in a 12-bead chain, for example, we employ vectors between nearest-neighbor bonded beads. For mode 4 (n = 3) in this chain, we employ vectors between next-nearest bonded neighbors. Since single segments do not possess an internal vector within this model, we can only compute this function for n = 2, 3, 4, 6, and 12 in the 12bead chain. For consistency, reorientational relaxation times τC2(p) are defined from this relaxation function by the same criterion as aboverelaxation to a value of 0.2.

seen in Figure 2. This latter plot reveals a striking correspondence with experimental dielectric measurements of polymer chain decoupling by Sokolov and Hayashi5 (see Figure 4b in that paper) and by Urakawa et al.:47 both exhibit a nonmonotonic dependence of the normal mode ratio on

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TRANSLATIONAL DECOUPLING AND HETEROGENEITY IN SIMULATION We begin by focusing on trends in an effective mean friction coefficient ζF,n for translational relaxation of strands of length n within chains of fixed overall length 12, 24, or 48. Because we employ a fixed value of the wavenumber q = 7.07 (comparable to the segmental peak in the structure factor), ζF,n does not encode the size scales Rn of the different strands and therefore scales as ζn rather than as τn ∼ Rn2ζn. By comparison, most experimental studies of this issue focus on either rheological or dielectric response of whole chains. In contrast, prior simulation studies have sometimes employed a Rouse mode analysis that naturally encodes both the local dynamics and the spatial extent of the relaxing unit.46 As discussed in the Introduction, the present approach, based around translational friction coefficients at fixed length scale, is selected to exclude the possibility that results could be driven by these types of trends in dynamical length scale probed due to trends in Rg, facilitating “apples-to-apples” comparison of pure dynamics across modes. As shown by Figure 1, the temperature dependence of ζF,n is evidently different between modes. Pronounced decoupling between chain modes becomes particularly apparent when the data are replotted as ratios of the mode friction coefficients, as

Figure 2. Ratio of the whole-chain friction coefficient to the coefficient at each n, normalized by its peak value, and plotted versus the segmental friction coefficient, for chains of total length 12 (a), 24 (b), and 48 (c). C


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Macromolecules segmental relaxation time, with a broad power-law decay to longer times and a relatively sharp dropoff to shorter times. We note that the time scale of the peak values in these figures is comparable to the onset temperature of glass formation at which dynamics first become non-Arrhenius for this system, such that the glass formation range of temperatures is roughly found to the right of these peaks. The time scale of these phenomena is shifted to shorter times relative to dielectric experiments, which appears to be a ubiquitous feature of glass formation in bead−spring systems. The origin of this shift remains an open question and should be a major focus of ongoing work. However, as noted above, it does not appear to alter other qualitative aspects of this phenomenon. Are these data consistent with suppression of dynamic heterogeneity in longer strands? The exponent β for a stretched exponential fit to the relaxation process is often interpreted as a measure of dynamic heterogeneity. As shown by Figure 3

Figure 4. Stretching exponent β plotted for strands of length n at fixed wavenumber q = 7.07 (a), 3.79 (b), and 0.505 (c) vs inverse temperature, for chains of total length 24.


through Figure 5, β is greater for longer strands for all T and q probed, indicating a narrower translational mobility distribution. This is consistent with prior simulations suggesting more stretching in shorter strands.48 Moreover, this finding holds despite the fact that the temperature dependence of β qualitatively changes with q. Indeed, for q values comparable to the inverse segmental size (parts a and b of these figures), the stretching exponent exhibits a nonmonotonic temperature dependence. For large q approaching the diffusive limit (part c), the temperature dependence becomes simply monotonic. In all cases, β systematically increases toward unity with increasing strand length. As shown by Figures 4 and 5, these findings are insensitive to overall chain molecular weight. Moreover, the non-Gaussian parameter α2(t) = 3⟨r4⟩/5⟨r2⟩2 − 1, which quantifies deviations from a single-time Gaussian dynamic path process, is shown in Figures 6 and 7 to be consistently lower for the center-of-mass motion of longer strands. Evidently, these


chains indeed exhibit a suppression in translational dynamic heterogeneity with increasing strand-length probes.

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DERIVATION OF HETEROGENEOUS ROUSE MODEL We now generalize the Rouse model to account for a distribution of segmental mobilities. We begin the HRM D


Article

Macromolecules n

rc⃗ =

1 ∑ ri ⃗ n i=1

(8)

The displacement of the center of mass is then given by n

Δ rc⃗ (t ) ≡ rc⃗ (t ) − rc⃗ (0) =

n

1 1 = ∑ Δ ri (⃗ t ) ⃗ ∑ [ ri (⃗ t ) − ri (0)] n i=1 n i=1 (9)

For the moment we focus on displacement in one dimension and replace r in the above equations with x. ⎛1 n ⎞2 (Δxn) = ⎜⎜ ∑ Δxi⎟⎟ ⎝ n i=1 ⎠ 2

(10)

For any finite n, the value above will in general be a distributed quantity. We must now compute its mean over the entire system. To do so, we note that the quantity in parentheses on the right-hand side of eq 10 is simply the distribution of means of a sample of size n over the underlying distribution of Δx, which may in general be any distribution over the real numbers. This is an issue of statistics rather than physics: regardless of the form of this underlying distribution, the central limit theorem dictates that distribution of its sample mean is Gaussian for sufficiently large n, with a standard deviation scaling as n−1/2. This leads to the conclusion that

Figure 6. Non-Gaussian parameter plotted vs log time in reduced Lennard-Jones units for various strand lengths n in chains of total length 12 (a), 24 (b), and 48 (c), at the noted corresponding temperatures.

⟨(Δxn)2 ⟩ =

∞

∫−∞ Δx 2

⎛ −nx 2 ⎞ n ⎟ exp⎜ 2 2π σ0 ⎝ 2σ0 ⎠

(11)

Performing this integral gives ⟨(Δxn)2 ⟩ =

σ0 2 n

(12)

We note that, in general, this form is not guaranteed to apply in the limit of small n. However, the leading-order description of the underlying distribution of Δx is itself Gaussian,41 such that we can expect this expression to provide a reasonable leadingorder approximation at modest values of n. In this special case of an approximately Gaussian distribution of Δx, σ0 is simply the standard deviation of this underlying distribution. For a Gaussian distribution of displacements, the mean-square displacement is simply the variance σ02 of the distribution, such that the above equation can be rewritten as ⟨(Δxn)2 ⟩ =

⟨Δx02⟩ n

(13)

Because the product of Gaussian random walks in three spatial dimensions is itself a Gaussian random walk,1 this can be trivially extended to three dimensions as ⟨(Δrn)2 ⟩ =

Figure 7. Peak value from Figure 6 and other similar data of the nonGaussian parameter for each mode, plotted vs inverse temperature for various strand lengths n in chains of total length 12 (a), 24 (b), and 48 (c).

⟨Δr0 2⟩ n

(14)

Under the definition D ≡ D(t) ≡ ⟨r (t)⟩/6t, this becomes 2

⟨Dn⟩ =

⟨D0⟩ n

(15)

We note that this equation holds exactly only in the limit of a Gaussian distribution of displacements; in the presence of deviations from this distribution, this equation will provide a leading order description of the diffusion coefficient, with higher order terms quantifying non-Gaussian effects. As reported above in Figures 6 and 7, this system indeed exhibits

derivation by obtaining a generalized statement of the core Rouse expectation of friction coefficient additivity, ⟨ζn⟩R = nζ0. We begin with the equation of the centroid rc⃗ of this group of particles, given by E


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The HRM thus indicates that Rouse overpredicts ⟨ζn⟩ by a factor equal to the Stokes−Einstein breakdown ratio for the strand over that of the segment. Within this model, breakdown of the Rouse model thus occurs as a direct result of the attenuation of Stokes−Einstein breakdown for long strands. We can gain further insight into the expected n-dependence of this effect by introducing a functional form for the distribution of D0. We employ a log-normal distribution, which is equivalent within a simple activated model of relaxation D0 ∝ exp[−Ea/kT] to assuming a Gaussian distribution of activation energies Ea. This choice is also consistent with results from simulations that have indicated that distributions of local free volume, mobility, and stiffness are well described by log-normal distributions,49,50 indicating that this may provide a common form for properties associated with local mobility in glass-forming liquids. This is unsurprising, as many systems obey an alternate form of the central limit theorem leading to log-normal distributions of randomly distributed variables.51,52 To make use of this distribution, we first rewrite eq 17 as

deviations from a Gaussian distribution in the glass-formation range. However, as shown below, the leading-order Gaussian statistics are sufficient to predict the decoupling behavior of the systems simulated here at a quantitative level. Considering now a single strand j of length n, the center-ofmass motion of this strand is simply the average segmental diffusion coefficient reduced by a factor n: n

Dn , j =

1 1 ⟨D0⟩n , j = 2 ∑ D0, i , j n n i=1

(16)

where D0,i,j denotes the segmental diffusion coefficient of a single segment i within strand j, and ⟨D0⟩n,j denotes an average diffusion coefficient over all n segments in strand j. Unlike the classical Rouse model, eq 16 does not encode an assumption that the system is characterized by a monodisperse diffusion coefficient. We now define an effective friction coefficient as ζ ≡ 4π2τ/R2 where τ is the relaxation time and R2 = 4π2/q2 is the square characteristic length scale of the relaxing object. These definitions of D and ζ obey the Einstein relationship when the distribution of particle displacements is Gaussian and relaxation is exponential (see Supporting Information). Both of these relations are thus expected to hold locally. Substitution of the local Stokes−Einstein relationship into eq 16 then yields ζn , j =

n 1 n 1 ∑ n i = 1 ζ0, i , j

n n = ∝ ⟨1/ζ0⟩n , j ⟨D0⟩n , j

⟨ζn⟩ 1 ∝ n⟨D0⟩ ⟨ζ0⟩SE ⟨D0⟩n

We treat D0 as being log-normally distributed, with parameters μ and σ such that its mean m and variance s02 are given by ⎡ σ2 ⎤ m = exp⎣μ + 2 ⎦ = ⟨D0⟩ and s02 = ⟨D0⟩2[exp(σ2) − 1]. ⟨D0⟩n will then also be given by a log-normal distribution, with the same mean m but with variance given by sn2 = s02/n. In general, the first inverse moment m−1 of a log-normal distribution can be given in terms of its mean m and variance s2 as

(17)

Since this finding holds only for a single strand, we must average over all strands in a system containing a total of N segments (N/n strands of length n). Normalizing the result by the Rouse prediction of ⟨ζn⟩R = nζ0 gives N /n

⟨ζn⟩ 1 = ∑ N j=1 ⟨ζn⟩R

n 1 n ⟨ζ0⟩ ∑ n i = 1 ζ0, i , j

=

1/⟨ζ0⟩ ⟨1/ζ0⟩n

= N /n

⟨1/⟨D0⟩n ⟩N / n

m−1 =

⟨1/D0⟩

Here the quantities ⟨1/ζ0⟩n and ⟨D0⟩n denote averages of 1/ζ0 and D0, respectively, over n segments in a single strand. If the system is dynamically heterogeneous, for finite n these averages yield distributions rather than single values. The outer bracket subscripted with N/n then denotes an average of this distributed quantity over N/n strands in the system. Equation 18 bears a striking similarity to the SE breakdown ratio rSE (eqs 2). We thus generalize the definition of rSE, which is defined for segments, to quantify the internal Stokes− Einstein decoupling ratio rn of any strand of length n: rn ≡ ⟨ζn⟩/ ⟨ζn⟩SE. Employing eq 18 together with the consequence of eq 16 that ⟨Dn⟩ = ⟨D0⟩/n leads to the result

m−1 =

nm3

+

=

s02/n,

1 m

this can be

(23)

⎡ ⎤ ⟨ζn⟩ ⎛ 1⎞ = n⎢1 − ⎜1 − ⎟(1 − exp(−σ0 2))⎥ ⎝ ⎠ ⎣ ⎦ ⟨ζ0⟩ n

(24)

We note that a similar derivation applies for the Stokes− Einstein breakdown ratio given by eq 2 in the text, which is simply the ratio of the inverse moment m−1 to the inverse of the first moment 1/m1. If the D0 is distributed log-normally, then this is given by

(19)

This definition is consistent with the segmental SE ratio, such that r1 = rSE. For large n eq 19 predicts rn → 1, indicating that SE is restored in the long-strand limit due to “preaveraging” over dynamic heterogeneity. With eqs 2 and 19, eq 18 can be written in terms of SE breakdown ratios as ⟨ζn⟩ r = n ⟨ζn⟩R rSE

s02

(22)

sn2

With m = ⟨D0⟩, substituting this into eq 21 leads to a prediction for the n dependence of ⟨ζn⟩ in a dynamically heterogeneous system:

⟨1/⟨D0⟩n ⟩N / n 1/⟨D0⟩

s2 1 + 3 m m

For the sample distribution with rewritten as

(18)

rn =

(21)

N /n

rSE =

⟨1/D0⟩ = exp[σ 2] 1/⟨D0⟩

(25)

Notably, this is exactly the same functional form for the Stokes−Einstein breakdown ratio previously obtained via a distinct approach by Schweizer and Saltzman.26 In that work the authors began with a nonlinear Langevin model for relaxation and, assuming a Gaussian distribution of barrier heights for relaxation, arrived at the conclusion that rSE was

(20) F


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Macromolecules exponential in the variance of barrier heightsthe same functional form as above. Evidently both a simple activation model and a theory built on a nonlinear Langevin model yield results consistent with a log-normal distribution of diffusivities under the assumption of a Gaussian distribution of barrier heights. As noted above, this is consistent with the observation that the local relaxation rate is expected to be Arrhenius in the activation barrier, D0 ∝ exp[−Ea/kT]. Equation 25 also provides a means of rewriting eq 24 in terms of the segmental, Stokes−Einstein breakdown ratio as ⎡ ⟨ζn⟩ ⎛ 1 ⎞⎛ 1 ⎞⎤ = n⎢1 − ⎜1 − ⎟⎜1 − ⎟⎥ ⎝ ⎢⎣ ⟨ζ0⟩ n ⎠⎝ rSE ⎠⎥⎦

(26)

As illustrated in Figure 8a, eq 26 predicts that the n dependence of the strand to segment friction coefficient ratio is modified

Figure 9. A log−log plot of the ratio of strand to segmental friction coefficient as a function of n, at several tempreatures shown in the legend for 12-bead (a), 24-bead (b), and 48-bead (c) chains. The black line is the Rouse prediction, and dashed lines are a fit of equation to each of these data sets. Figure 8. Strand friction coefficient ⟨ζn⟩ normalized by (a) the segmental friction coefficient ⟨ζ0⟩ (eq 26) or (b) the Stokes−Einstein prediction of the segmental friction coefficient ⟨ζ0⟩SE (eq 27), plotted vs strand length n. The black line in (a) is the Rouse prediction employing the true segmental ⟨ζ0⟩; the black line in (b) is the Rouse prediction employing the Stokes−Einstein prediction ⟨ζ0⟩SE. Each curve corresponds to a different value of the Stokes−Einstein breakdown ratio rSE, as shown in the legends.

considered. The fit values rfit SE of rSE are shown in parts a of Figure 10, Figure 11, and Figure 12. Because we employ a fixedlength scale ζ rather than τ, deviations from Gaussian chain statistics48 cannot account for these deviations from Rouse behavior (nor can they account for the temperature dependence of these effects given the temperature insensitivity of chain

from its Rouse form in the presence of dynamic heterogeneity, as quantified by rSE. This modification results from anomalous behavior of segments rather than of long strands: if we rewrite eq 26 in terms of the Stokes−Einstein prediction ⟨ζ0⟩SE for the segmental friction coefficient ⟨ζn⟩ ⎡1 ⎤ = n⎢ (rSE − 1) + 1⎥ ⎣n ⎦ ⟨ζ0⟩SE

(27)

we find that the Rouse prediction is recovered at large n if one employs ⟨ζ0⟩SE rather than ⟨ζ0⟩ in the Rouse model. In essence, long strand behavior follows a combined Einstein−Rouse model, as shown in Figure 8b, but segmental and short-strand behavior deviates from both Einstein and Rouse in the presence of dynamic heterogeneity. Because glass-forming liquids exhibit T-dependent dynamic heterogeneity upon approach to Tg,16−22 rSE and ⟨ζn⟩/⟨ζ0⟩ are T-dependent. On the other hand, when the system is dynamically homogeneous, rSE = 1 and classical Rouse scaling is recovered.

Figure 10. For 12-bead chains: (a) Plot vs segmental friction coefficient of the fit values of rSE for fits shown in Figure 9 to data at all simulated temperature (left axis, black) and of the measured ratio of segmental friction coefficient to inverse mobility (right axis, red). (b) Plot of the two y-axis values in figure (a) against one another, with a fit proportionality relation shown by the red dashed line.

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TEST OF HETEROGENEOUS ROUSE MODEL As shown by Figure 9, eq 26 describes ⟨ζn⟩/⟨ζ0⟩ data over the entire T-range simulated for all chain molecular weights G


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Indeed, as shown by Figures 10−12, the equality rfit SE = aζF,n=1/q2t* holds quantitatively over the entire temperature range probed with values of a in the range of 20, indicating that the fit value of rfit SE genuinely reflects the segmental SE breakdown. This agreement suggests that it should be possible to predict the temperature and n dependence of ⟨ζn⟩ simply by employing the quantity aζF,n=1/q2t* as rSE in eq 26, using the values of a determined in Figures 9−11. Accordingly, we rearrange eq 26 employing the substitution rSE = aζF,n=1/q2t*: (1 − ⟨ζn⟩/n⟨ζ0⟩) (1 − q2t */aζF, n = 1)

= 1 − 1/n (28)

We test this equation against simulation data for all temperatures and n probed in these simulations, which, as shown by Figure 13a−c, exhibit a range of large negative deviations from the classical Rouse prediction. As shown by Figure 13d, a plot of the left-hand side vs the right-hand side of eq 28 with a = 19.3 yields a collapse (R2 = 0.98) of the data in Figure 13a−c for all molecular weights considered to a single master curve of slope 1, indicating that the HRM is



conformation in the melt). Moreover, because our simulations probe the dynamics of subchain strands at fixed chain molecular weight, these deviations cannot be ascribed to trends in segmental activation behavior with molecular weight.53 The question is now whether these fit values truly reflect the segmental SE breakdown ratio or are arbitrary fit parameters. To test this, we compute an inverse mobility t* as the time at which the segmental mean-square displacement is equal to one (the segment size in reduced Lennard-Jones units). Consistent with eq 2, rSE is then proportional to the ratio of friction coefficient to inverse mobility, ζF,n=1/q2t*, where q = 7.07 is the wavenumber at which ζF,.n was determined and with a constant unknown prefactor reflecting the arbitrarily chosen length scale employed in determining t*.

Figure 13. (a−c) Plot of the ratio of strand to segmental friction coefficient as a function of n, for all temperatures simulated, for 12bead (a), 24-bead (b), and 48-bead (c) chains. Black lines are the Rouse prediction. (d) Plot of the left-hand side vs the right-hand side of equation in the text for these combined data at all molecular weights, with the heterogeneous Rouse model prediction given by the black line. H


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Macromolecules quantitatively predictive. Weak deviations from the master curve are consistent with the approximations and assumptions employed in arriving at this result: Gaussian or near-Gaussian relaxation, a log-normal distribution of local mobilities, and the neglect of explicit spatial correlations in the local mobility.

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COMPARISON TO REORIENTATIONAL RELAXATION A central objection to the hypothesis of simple averaging over dynamic heterogeneity as an origin of various types of decoupling has centered around the behavior of the stretching exponent β as probed by dielectric spectroscopy or dynamic light scattering.23 In particular, the interpretation of the reorientational β as reflecting the distribution of local mobilities implies a connection between the reorientational β and the breakdown ratio that is apparently absent in some systems. In order to probe this issue, we additional quantify decoupling and relaxational stretching based on the C2 relaxation function. This reorientational relaxation function is commonly probed by light scattering and is closely related to the C1 reorientational relaxation function probed by dielectric spectroscopy. As shown by Figure 14, relaxation times τC2 extracted from this relaxation function exhibit behavior similar to the friction Figure 15. Reorientational stretching exponent βC2 vs inverse temperature for strands of length n, for chains of total length 12 (a), 24 (b), and 48 (c).

Figure 14. Relaxation time τC2 obtained from the C2 reorientational end-to-end vector autocorrelation function for strands of length noted in the figure, plotted vs inverse temperature, for chains of total length 12 (a), 24 (b), and 48 (c). Figure 16. Ratio of the whole-chain τC2 to τC2 at each n, normalized by its peak value, and plotted versus τC2 for chains of total length 12 (a), 24 (b), and 48 (c).

coefficient ζF extracted from the self-intermediate scattering function. Unlike ζF, however, τC2 scales to leading order as n2 since the end-to-end vector incorporates the explicit increase in strand length scale with n. As shown by Figure 16, the normal mode decoupling observed in τC2 is likewise similar to that observed in ζF, albeit with a somewhat delayed and muted onset. This delayed onset (lower temperature of onset) of pronounced decoupling, following a plateau of normal mode coupling spanning several decades at high temperature, may

explain in part why experimental data, which commonly employ metrologies that probe reorientation, often exhibit a lower onset temperature of decoupling than simulation, which more frequently employs relaxation functions based on density (such as Fs(q,t)). I


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Macromolecules Finally, as shown in Figure 15, the stretching exponent βC2 for stretched exponential fits to the C2 relaxation function is nearly constant with respect to both temperature and mode. This is in marked contrast to the translational stretching exponents shown in Figures 3−5, which exhibited a pronounced reduction in stretching with increasing strand length. We note that the translational relaxation function Fs(q,t) has a much simpler and more direct relationship to particle displacement than does C2 (through the Gaussian approximation and its higher order correction terms), such that stretching in Fs(q,t) is a very strong indicator of a genuine distribution of segmental mobilities. These data therefore suggest that reorientational relaxation functions, or at least the C2 function, may be less sensitive to dynamic heterogeneity than Fs(q,t). In particular, caution would appear to be warranted in interpreting the stretching exponent of reorientational relaxation functions as being a direct measure of dynamic heterogeneity. This finding may account for observations in some systems of poor correlations between decoupling and relaxational stretching as measured by dielectric spectroscopy or dynamic light scattering.

HRM predicts this decoupling from measured Stokes−Einstein breakdown ratios to at least a reasonable approximation, suggesting that some type of heterogeneity is likely involved. However, the types of dynamical heterogeneity that have been the major focus of this literature are generally understood to grow on cooling, and they are not expected to play a significant role at high temperature. One possible explanation for this apparent contradiction can be inferred from recent work of White and Lipson.55 In that paper, they demonstrated that kinetic prefactor effects (i.e., the T1/2 scaling of particle velocities with temperature) play an important role in dynamics at high temperature. This observation naturally raises the question of why dynamics ever exhibit an Arrhenius scaling regime at high temperature (which is certainly not a new question), suggesting that this regime may result in part from a cancellation of this factor with equation of state effects. If so, it is possible that the “heterogeneity” present at high temperature in the Arrhenius scaling regime is related to high-frequency compressibility-related thermal density fluctuations, which at these very high temperatures may involve time scales comparable to the segmental relaxation time. This proposition is consistent with the observation that the onset of this hightemperature decoupling regime is observed in our simulations to be similar to the onset temperature of TA of Arrheniusness on heating. This issue clearly demands additional investigation and should be a focus of future work. Finally, a natural question is whether this fundamental reformulation of the Rouse model, which we have shown to explain translational normal mode decoupling in a model polymer, applies equally well to decoupling of other relaxation functions commonly probed in experiment, such as dielectric or viscous measures of relaxation. Data above indicate that reorientational chain-normal-mode decoupling behaves in a similar manner to that seen in translational dynamics (albeit with a delay in the onset of pronounced decoupling), suggesting the likelihood of a shared mechanism. At the same time, trends between normal modes in the reorientational stretching exponent β are found to be strongly suppressed relative to those reported above for translational relaxation. Combined with some experimental studies reporting poor correlations between β and chain normal mode decoupling,23,56 these results suggest the possibility that stretching of translational relaxation functions may report more directly on the distribution of local mobilities than does the stretching of reorientational relaxation functions. This would suggest the presence of some additional physical mechanism driving reorientational stretching. A better understanding of these differences between translational and reorientational behavior, as well as extension of the HRM to deforming systems, would have the potential to address other aspects of thermorheological complexity such as in creep compliance measurements. Similarly, a framework formally connecting these decoupling phenomena to non-Gaussian relaxation would be of considerable value. Ultimately, an extension of this framework to incorporate spatial correlations would have the potential to provide insights into chain dynamics in thin films and nanostructured polymers possessing standing dynamic gradients.57

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CONCLUSIONS The heterogeneous Rouse model (HRM) quantitatively predicts both deviations from Rouse scaling of the friction coefficient and translational normal mode decoupling in this model bead−spring polymer, over a range of simulated molecular weights, based on the observed breakdown of the Stokes−Einstein relationship between translational relaxation and diffusion. Simulations are consistent with the prediction that this occurs as a consequence of a progressive averaging out of temperature-dependent translational dynamic heterogeneity in longer strands. In essence, results indicate that chain-normalmode decoupling shares precisely the same physical origin as Stokes−Einstein breakdown. We emphasize that the HRM demonstrates that this occurs due to issues of statistical sampling over a distribution of mobilities even in the absence of spatial correlations; caution is therefore warranted in attempting to infer length scales of correlations based on the molecularweight dependence of strand dynamic heterogeneity. These effects emerge in the limit of shorter chains, explaining the empirical observation that chain-normal-mode decoupling does not require especially high molecular weight.5 Finally, as noted above, some evidence points toward a correlation between fragility of glass formation and chain-normal-mode decoupling;6 if this correlation proves to be universal, the present results would then be consistent with a proposed54 correlation between fragility and the breadth of the distribution of segmental mobilities. Additional tests of this model within a broader range of more chemically diverse models would have the potential to provide further insight into this issue and additional confirmation of the HRM. Similarly, extension of this test to time scales more deeply supercooled than the ∼100 ns time scale probed here would be of value in validating the applicability of this physical scenario to temperatures closer to the experimental glass transition. Such tests must necessarily be experimental in nature given present limitations on times accessible to molecular simulation. One significant remaining question is the nature of the hightemperature inverse decoupling regime observed here and in experiments,5,47 where the temperature dependence of the longer-strand dynamics is evidently stronger than that of shorter-strand dynamics. As indicated by Figures 10−12, the J


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Macromolecules

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(17) Matharoo, G. S.; Razul, M. S. G.; Poole, P. H. Structural and Dynamical Heterogeneity in a Glass-Forming Liquid. Phys. Rev. E 2006, 74, 050502. (18) Starr, F. W.; Douglas, J. F.; Sastry, S. The Relationship of Dynamical Heterogeneity to the Adam-Gibbs and Random FirstOrder Transition Theories of Glass Formation. J. Chem. Phys. 2013, 138, 12A541−12A541−18. (19) Zetterling, F. H. M.; Dzugutov, M.; Simdyankin, S. I. Formation of Large-Scale Icosahedral Clusters in a Simple Liquid Approaching the Glass Transition. J. Non-Cryst. Solids 2001, 293-295, 39−44. (20) Pedersen, U. R.; Schrøder, T. B.; Dyre, J. C.; Harrowell, P. Geometry of Slow Structural Fluctuations in a Supercooled Binary Alloy. Phys. Rev. Lett. 2010, 104, 105701. (21) Tanaka, H.; Kawasaki, T.; Shintani, H.; Watanabe, K. Criticallike Behavior of Glass-Forming Liquids. Nat. Mater. 2010, 9, 324−331. (22) Debenedetti, P. G.; Stillinger, F. H. Supercooled Liquids and the Glass Transition. Nature 2001, 410, 259−267. (23) Ngai, K. L.; Plazek, D. J.; Roland, C. M. Comment on “Resolving the Mystery of the Chain Friction Mechanism in Polymer Liquids. Phys. Rev. Lett. 2009, 103, 159801. (24) Berthier, L.; Chandler, D.; Garrahan, J. P. Length Scale for the Onset of Fickian Diffusion in Supercooled Liquids. EPL Europhys. Lett. 2005, 69, 320. (25) Xia, X.; Wolynes, P. G. Diffusion and the Mesoscopic Hydrodynamics of Supercooled Liquids. J. Phys. Chem. B 2001, 105, 6570−6573. (26) Schweizer, K. S.; Saltzman, E. J. Activated Hopping, Barrier Fluctuations, and Heterogeneity in Glassy Suspensions and Liquids. J. Phys. Chem. B 2004, 108, 19729−19741. (27) Cicerone, M. T.; Wagner, P. A.; Ediger, M. D. Translational Diffusion on Heterogenous Lattices: A Model for Dynamics in Glass Forming Materials. J. Phys. Chem. B 1997, 101, 8727−8734. (28) Varnik, F.; Baschnagel, J.; Binder, K. Reduction of the Glass Transition Temperature in Polymer Films: A Molecular-Dynamics Study. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2002, 65, 021507. (29) Buchholz, J.; Paul, W.; Varnik, F.; Binder, K. Cooling Rate Dependence of the Glass Transition Temperature of Polymer Melts: Molecular Dynamics Study. J. Chem. Phys. 2002, 117, 7364−7372. (30) Starr, F. W.; Douglas, J. F. Modifying Fragility and Collective Motion in Polymer Melts with Nanoparticles. Phys. Rev. Lett. 2011, 106, 115702. (31) Hanakata, P. Z.; Douglas, J. F.; Starr, F. W. Local Variation of Fragility and Glass Transition Temperature of Ultra-Thin Supported Polymer Films. J. Chem. Phys. 2012, 137, 244901−244907. (32) Baschnagel, J.; Varnik, F. Computer Simulations of Supercooled Polymer Melts in the Bulk and in Confined Geometry. J. Phys.: Condens. Matter 2005, 17, R851−R953. (33) Barrat, J.-L.; Baschnagel, J.; Lyulin, A. Molecular Dynamics Simulations of Glassy Polymers. Soft Matter 2010, 6, 3430−3446. (34) Simmons, D. S.; Douglas, J. F. Nature and Interrelations of Fast Dynamic Properties in a Coarse-Grained Glass-Forming Polymer Melt. Soft Matter 2011, 7, 11010−11020. (35) Ruan, D.; Simmons, D. S. Glass Formation near Covalently Grafted Interfaces: Ionomers as a Model Case. Macromolecules 2015, 48, 2313−2323. (36) Mangalara, J. H.; Simmons, D. S. Tuning Polymer Glass Formation Behavior and Mechanical Properties with Oligomeric Diluents of Varying Stiffness. ACS Macro Lett. 2015, 4, 1134−1138. (37) Kremer, K.; Grest, G. S. Dynamics of Entangled Linear Polymer Melts: A Molecular-dynamics Simulation. J. Chem. Phys. 1990, 92, 5057−5086. (38) Hanakata, P. Z.; Douglas, J. F.; Starr, F. W. Local Variation of Fragility and Glass Transition Temperature of Ultra-Thin Supported Polymer Films. J. Chem. Phys. 2012, 137, 244901. (39) Hou, J.-X.; Svaneborg, C.; Everaers, R.; Grest, G. S. Stress Relaxation in Entangled Polymer Melts. Phys. Rev. Lett. 2010, 105, 068301.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00135. Additional supporting derivations (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (D.S.S.). ORCID

David S. Simmons: 0000-0002-1436-9269 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge generous support from the W. M. Keck Foundation. The authors thank T. Patra, K. Schweizer, A. Dobrynin, and J. Lipson for helpful discussions.

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Heterogeneous Rouse Model Predicts Polymer Chain Translational

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