How Deformation Enhances Mobility in a Polymer Glass

Article pubs.acs.org/Macromolecules

How Deformation Enhances Mobility in a Polymer Glass Yongchul G. Chung and Daniel J. Lacks* Department of Chemical Engineering, Case Western Reserve University, Cleveland, Ohio 44106, United States ABSTRACT: Molecular dynamics simulations and energy landscape analyses are carried out for a realistic model of glassy polystyrene under shear deformation. Deformation enhances the atomic mobility, as quantified by the van Hove function. The enhanced mobility is shown to arise from two mechanisms: First, active deformation continually reduces barriers for hopping events, and the importance of this mechanism is modulated by the rate of thermally activated transitions between adjacent energy minima. Second, deformation moves the system to higher-energy regions of the energy landscape, characterized by lower barriers. Both mechanisms enhance the dynamics during deformation, and the second mechanism is also relevant after deformation has ceased. Furthermore, neither mechanism on its own could be expected to correlate all mobility data.

I. INTRODUCTION The atomic mobility in glasses is very low, and this characteristic distinguishes glasses from liquids. However, the mobility in glasses is significantly enhanced by deformation this effect has been observed in experiments on polymer1−4 and colloidal5−7 glasses and molecular simulations of simple8,9 and polymer10 glasses. The effect can be dramatic, with the mobility increased by several orders of magnitude. The enhanced mobility can lead to changes in mechanical properties11,12 and structure − such as crystallization13−15 and shear band formation.16 The physical basis for the enhanced mobility is elusive; it cannot be attributed to free volume, which often controls dynamics, as the enhancement occurs even in compressive deformation where free volume decreases.17 We will interpret our results in terms of the energy landscape, which has become a useful framework for understanding the properties of liquids and glasses.18 The energy landscape is characterized by a huge number of local energy minima (often called “inherent structures”) with a range of depths (the energy of an inherent structure will be denoted EIS). For a liquid or glass, mobility occurs when the system hops between metabasins (groups of proximate local minima separated by low barriers),19 with these hops being relatively frequent in liquids and infrequent in glasses. As temperature is decreased, systems tend to visit deeper energy minima,20 as shown schematically in Figure 1b.i and quantitatively for our system in Figure 1b.ii; the deeper minima have larger barriers between metabasins, and the combined effect of less thermal energy and larger barriers leads to the stronger than Arrhenius slowing of dynamics upon cooling as the glass transition is approached.20 Below the glass transition temperature, the dynamics are so slow that the system cannot equilibrate, and the system slowly moves to deeper energy minima in the aging process (see result in Figure 1b.ii). © 2012 American Chemical Society

We describe here our hypothesis and then show how it is supported by our simulation results. Our idea is that the enhanced mobility arises from two mechanisms: First, active deformation continually reduces barriers for thermally activated hopping events (the “fold-catastrophe mechanism”, since this decrease of a barrier to zero is known mathematically as a fold catastrophe), as proposed originally by Eyring.21 This effect, shown schematically in Figure 1a.i, has been demonstrated in simulation studies that show that the changes in barrier heights, and curvatures of the landscape quantitatively follow the fold-catastrophe predictions.8,22,23 This deformation-induced reduction of barriers represents the physical basis for the deformation-activated events as in the soft glassy rheology,24 shear transition zone models,25 and the nonlinear Langevin equation theory.26 The distortion of the potential energy landscape comes about as follows. The potential energy (U) is a function of positions of all atoms in the simulation cell and the dimensions of the simulation cell, U = U(r;⃗ ε⃗), where r ⃗ is the vector of positions of all atoms in the simulation cell and ε⃗ is the vector of the simulation cell dimensions. The potential energy landscape corresponds to the dependence of U on r ⃗ at fixed ε⃗, and the distortion of the landscape is due to the changes in the dependence of U on r ⃗ as ε⃗ changes. Second, deformation moves the system to regions of the energy landscape with shallower energy minima (the “rejuvenation mechanism”,27 since it acts oppositely to the aging process), as shown schematically in Figure 1b.i28 and quantitatively for our system in Figure 1b.iii; the lower barriers in this region of the landscape allow more frequent hopping events. Both of these mechanisms enhance the dynamics during active deformation, and the rejuvenation mechanism is also Received: March 2, 2012 Revised: April 8, 2012 Published: May 10, 2012 4416

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Figure 1. Schematics of mechanisms underlying enhanced mobility. (a.i) Distortions of the energy landscape with deformation, which ultimately lead to a fold catastrophe (disappearance of an energy barrier and minimum). In quasi-static simulations, system properties change discontinuously when fold catastrophes are reached. In molecular dynamics simulations, transitions occur prior to the fold catastrophe because the system can hop over the barrier when the barrier height becomes comparable to the system’s thermal energy, kBT, which smooth out the discontinuities. (a.ii) Results for the energy as a function of shear strain, from quasi-static simulations; these results show discontinuous energy drops due to atomic rearrangements after fold catastrophes. (b.i) The system visits different regions of the energy landscape under different conditions. Aging after a temperature quench brings the system to deeper energy minima, and large strains bring the system to shallower energy minima (in general, these shallower minima are not identical to those visited during aging). (b.ii) Results for ⟨EIS⟩ as a function of temperature obtained in 30 μs simulations (black circles). Below the glass transition temperature the results are time-dependent, and the blue circle represents the result after aging for 162 μs. (b.iii) Results for ⟨EIS⟩ from MD simulations at shear rate 1.8 × 107 s−1; note that shear leads to an increase in ⟨EIS⟩.

agree very well with an atomistic model at these temperatures.31,32 We find Tg ∼ 372 K for this aPS model as the temperature corresponding to the change in the slope of the results in Figure 1b.ii. This simulation result agrees well with the experimental value Tg = 368 K;33 such close agreement is of course fortuitous, but the general agreement suggests that these potentials are valid for temperatures below Tg. The physical time in the CG model is obtained by scaling the simulation time by a factor determined in comparisons of the dynamics (e.g., mean-squared displacements34) from the CG and all-atom models.35 The factor ranges from 500 to 5000, depending on molecular weight, density, and temperature, but for simplicity we use a constant scale factor of 1000 here. Our simulations are carried out for a system of 50 aPS chains, where each chain has 192 monomer units. Periodic boundary conditions are applied in all directions. Numerical integration of the equations of motion is carried out with a 3 ps time step (after scaling), using the leapfrog method. Temperature is controlled by the Berendsen thermostat, and in constantpressure simulations the pressure is controlled by the Berendsen barostat. This thermostat and barostat are chosen because the previous tests of the CG potentials were carried out under these methods.31

relevant after deformation ceases. Note that the rejuvenation does not precisely reverse aging, in that the states reached with rejuvenation are likely different than those visited during aging. We carry out molecular dynamics (MD) simulations and energy landscape analyses on a polymer glass to test this hypothesis.

II. MODEL The coarse-grained (CG) model of atactic polystyrene (aPS) of Kremer and co-workers is used29−31 in order to allow longer time scales to be addressed. In this model, the 16 atoms of each monomer are modeled by 2 particles: A (representing the styrene backbone) and B (representing the phenyl ring group). The interaction potential parameters for the CG model were obtained by fitting to the results of atomistic simulations. In particular, the bonded interaction potentials are obtained by fitting to atomistic simulations of isolated chains, and the nonbonded interaction potentials are obtained by fitting to potentials of mean force between two short oligomers in vacuum. In principle, the potentials obtained this way will depend on temperature, but the CG model uses the potentials obtained at T = 503 K for all temperatures. To test the transferability of potentials at low temperatures, Kremer and co-workers compared the bonded and nonbonded potentials at T = 423 K with the potentials obtained at T = 503 K. They found that the deviation of potential due to the temperature is very small (a fraction of kBT). They also found that the structure properties (e.g., radial distribution function, internal distance of polymer chains, and density) of CG model

III. COMPUTATIONAL METHODS Shear deformation is carried out in the xy-plane by imposing Lees−Edwards boundary condition, which shifts the simulation cell element at a constant rate. Energy minimizations are carried 4417

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out using l-bfgs method until the maximum force in the system becomes less than 10−7 N. The GROMACS 4 package is used for all simulations.36 The study is carried out at room temperature (T = 300 K), which is far below the glass transition temperature, Tg. Systems at T = 300 K are prepared by the following sequence of simulations: (1) 30 μs NPT simulation at T = 503 K (where the liquid readily equilibrates); (2) 30 μs NPT simulation at T = 363 K (slightly below Tg, but with significant atomic mobility); (3) 102 μs NVT simulation at T = 363 K and (4) 30 μs NPT simulation at T = 300 K. Next, MD simulations with shear are then carried out, at shear rates, ε̇, spanning 7 orders of magnitude (ε̇ = 1.8 × 103−1.8 × 109 s−1); these simulations are carried out until the shear strain is ∼220% (thus the simulation time is different for each shear rate). For each shear rate, simulations are carried out for 100 independent starting configurations, and the results are averaged to reduce statistical noise (for the slowest shear rate, simulations are carried out for only 10 configurations due to the computational intensity). Note that our lowest shear rates overlap with the experimentally accessible deformation rate. For example, experiments on glassy polycarbonate and PMMA have been carried out at shear rates up to 6 × 103 s−1.37 The energy landscape analyses are carried out two ways: First, the depths of energy minima visited during the MD simulations are found using energy minimizations that start from configurations obtained from the MD trajectoriesthe average inherent structure energy, ⟨EIS⟩, at a particular time is obtained by averaging results for the EIS from configurations at this time in independent simulation runs. Second, quasi-static simulations are carried out; this procedure begins with an energy minimum obtained as described above and repeatedly alternates between imposing a small shear deformation (0.17%) and carrying out an energy minimization. The quasi-static simulations correspond to shear in the low temperature-low strain rate limit, where thermally activated transitions are excluded.

Figure 2. Results for van Hove functions, at various strain rates (coded by color), and compared to the unsheared result in part a and the quasi-static result in parts b and c. (a) Gs(z,t), with t = 6 μs; (b) Hs(z,ε), with ε = 109%; (c) Hs(z,ε), with ε = 109%, for a wider range of shear rates. N

Hs(z , ε) = 1/N ∑ δ(z − |zi(ε0 + ε) − zi(ε0)|) i=1

IV. RESULTS The atomic mobility is quantified by the van Hove function, Gs(z,t), which is the probability that a particle moves the distance z in the time interval t

As shown in Figure 2b, the results for Hs at these shear rates nearly collapse onto a single curve; i.e., in these cases the dynamics appears to be a function of the strain increment ε. This result concurs with recent simulation9,38 and experimental7 studies that show that dynamics correlate with the strain increment. The Hs results obtained from the quasi-static simulations nearly collapse with the MD results at low shear rate (Figure 2b); this collapse demonstrates that these dynamics are mainly controlled by the fold-catastrophe mechanism shown schematically in Figure 1a.i. Recall that the quasi-static simulations exclude thermal energy, so that the only means for mobility in such simulations is via fold catastrophes that cause barriers to disappear. The existence of fold catastrophes and the subsequent atomic rearrangement events is demonstrated by the results for the energy in our quasi-static simulations (Figure 1a.ii)the discontinuous drops in energy correspond to the atomic rearrangements that follow fold catastrophes. These atomic rearrangements are randomly directed in the dimensions perpendicular to shear and lead to a random walk (diffusive motion) along these dimensions. The chance of a fold catastrophe occurring is proportional to the strain increment, which is why the results for Hs collapse. In the molecular dynamics simulations, which include thermal motion, the

N

Gs(z , t ) = 1/N ∑ δ(z − |zi(t0 + t ) − zi(t0)|) i=1

(2)

(1)

where zi is the z-coordinate of particle i (we examine dynamics only in the one dimension perpendicular to the plane in which shear is applied), N is the number particles (of one type), and t0 is the time when the measurement starts. The van Hove function is independent of t0 in equilibrium systems but can depend on t0 in nonequilibrium systems. The van Hove function is shown in Figure 2a for the three slowest shear rates (ε̇ = 103, 104, and 105 s−1) in the postyield regime and compared to the result in the absence of shear. These results indicate that shear enhances the mobility (although the enhancement appears small for the slowest shear rate)the atoms move further in a given time interval at higher shear rates. While the van Hove function depends strongly on shear rate, more insight is gained by examining a variant of the van Hove function, Hs(z,ε), which is the probability that a particle moves the distance z during the strain increment ε 4418

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system can hop over a barrier via thermal motion when the barrier has become sufficiently small (before the fold catastrophe occurs)for this reason, the mobility is slightly greater for the MD simulations than in the quasi-static simulations, and the mobility in the MD simulations increases slightly with decreasing shear rate (because the system has more chances to hop over the barrier when the shear rate is lower). Also for this reason, when the strain rate is decreased while the temperature remains constant, the system has more chances to hop over an energy barrier during a strain interval, which moves the behavior away from the quasi-static limit (as seen in Figure 2). The collapse of the Hs results breaks down as the shear rate becomes very high, as shown in Figure 2c. The probability of finding atoms with large displacements diminishes with increasing shear rate because the system moves along the energy landscape in the direction of shear fast enough to reduce the number of attempts (within a strain window) to hop over barriers to lower energy states in the transverse directions. This effect can be quantified by the Peclet number, Pe0 = ε̇τ0, where τ0 characterizes the time scale in which an atoms initially reaches its cage of neighbors. The breakdown of the collapse of the Hs results can be predicted to occur when Pe0 > 1. Figure 3

Figure 4. Results for the probability that an atom moves more than 5 Å in a time interval (time intervals are denoted by the symbol color; times given in μs). These results can be also considered in terms of strain intervals; solid black lines connect results at a particular strain interval.

disrupt the collapse. This effect can be quantified by the Peclet number, PeD = ε̇τD, where τD is the time scale that characterizes diffusive motion; τD can be taken as τD = d2/D, where d is the diameter of an atom and D is the diffusivity in one dimension (note our definition, which uses the long-time diffusivity, is distinct from the Peclet number used in colloidal studies based on short-time diffusivity). In this case the breakdown of the collapse of the Hs results is expected to occur when PeD < 1. From the mean-squared displacement results (Figure 3), an upper bound for D is obtained as D < 3 × 10−17 m2/s (obtained as D = (1/2)[r2(tmax)/tmax], where tmax is the longest time examined; this is an upper bound because the diffusive limit has not been yet reached at tmax); thus, a lower bound to τD is τD ≈ 0.01. Computational limitations prevent us from carrying out simulations for ε̇ < 103 s−1 (these simulations would take too long), and so all our simulation results are for the regime PeD > 1. However, the fold catastrophes are not the only factor affecting the mobility enhancement. Figure 5 shows that the

Figure 3. Results for the mean-squared displacement (in one dimension).

displays the mean-squared displacement as a function of time, which shows that τ0 ≈ 10−8 s. Thus, the breakdown of the collapse of the Hs results observed for ε̇ > 108 s−1 (Figure 2c) concurs with the prediction that the breakdown will occur at Pe0 > 1. The mobility enhancement due to this fold-catastrophe mechanism is modulated by thermally activated transitions to adjacent energy minima. The enhanced mobility as a function of strain rate is compared to results for the unsheared system in Figure 4, in terms of the probability that an atom moves more than 5 Å during the time interval (i.e., the integral of Gs from z = 5 Å to z = ∞). The finite shear rate results approach the unsheared system results (for a given time interval) as the shear rate decreases because thermally activated transitions begin to dominate at low shear rates. These results can also be viewed in terms of Hs, where results in Figure 4 for a particular ε are joined by solid lines. It can be seen here as well that the collapse of Hs with strain rate occurs only for lower shear rates and where the strain increment is large enough for the foldcatastrophe mechanism to dominate. This analysis also shows the collapse is never exactas the shear rate becomes very low, the thermally activated events will eventually dominate and

Figure 5. Results for the probability that an atom moves more than 5 Å in the strain interval 5%, for a simulation at a shear rate of 1.8 × 104 s−1 (filled black circles), and ⟨EIS⟩ at each strain increment (open red squares).

displacement for a given value of strain increment increases as the total extent of shear increases and appears to plateau at strains over ∼100% strain (all results presented above were from this plateau region). The mobility enhancement is smaller for preyield deformations than postyield, which concurs with previous simulation results.39 Also shown in Figure 5 are the results for ⟨EIS⟩ during these simulationswhen the system is 4419

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significant in the preyield regime (see Figure 5). Both experiments and simulations show that in multistep creep tests the relaxation times sometimes (but not always) correlate with strain rate43this result can be understood in that the correlation applies only when the enhancement is due to the fold-catastrophe mechanism, and we suspect that the deviations from this correlation (e.g., Figure 9 of ref 45) would themselves correlate with EIS. Also, previous simulations showed that in some cases,44 but not others,45 enhanced mobility correlates with shallower energy minima visitedwe show here the correlation is only expected to hold when the enhancement from fold-catastrophe mechanism is negligible. Finally, we note that the fold-catastrophe mechanism will be important in the regime where Pe0 < 1 but PeD > 1, and the rejuvenation mechanism will be important in systems where aging effects are significant.

sheared, the system moves to a region of the energy landscape with shallower energy minima.28 Since shallower energy minima have smaller barriers,20 this strain-induced change in position on the landscape leads to faster dynamics. Figure 5 shows that the mobility enhancement does in fact correlate with ⟨EIS⟩, demonstrating the applicability of this mechanism. The rejuvenation mechanism remains in effect after strain has ceased. In this regard, we apply a cycle of shear that returns the system to zero strain and then run a long MD simulation (1000 μs) to examine the changes in both mobility enhancement and ⟨EIS⟩ after deformation has ceased. The results are shown in Figure 6the mobility is significantly greater after the shear

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant DMR-0705191. We thank the Ohio Supercomputing Center for computational resources used in this study and Kurt Kremer and Dominik Fritz for their help with the CG model.

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REFERENCES

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Figure 6. Results for van Hove functions after a cycle of shear deformation: (a) van Hove functions with t = 300 μs, at different waiting times (t0) after a cycle of shear has ceased, in comparison with the result for the unsheared system; (b) the probability that an atom moves more than 5 Å (filled black circles) in the time interval t = 300 μs, and ⟨EIS⟩ (open red squares), as a function of t0.

cycle than in the unsheared case, and with time the mobility slowly ages back to the behavior of the unsheared system. In regard to ⟨EIS⟩, these values decrease with time as the system ages to a region of the energy landscape with deeper energy minima. Again, as shown in Figure 6b, the mobility enhancement correlates with ⟨EIS⟩, which demonstrates the applicability of the rejuvenation mechanism.

V. DISCUSSION AND CONCLUSION This picture of two mechanisms for enhanced mobility clarifies previous experimental and simulation results. For example, both experiments7 and simulations9 show cases where the mobility correlates with strain increment, but this is only expected to hold in the appropriate time scale and when the effects of the rejuvenation mechanism becomes negligible. The experimental result that stress in the preyield regime may enhance dynamics27 but not alter the extent of aging40−42 can be understood in that only the fold-catastrophe mechanism is 4420

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