^{1,a)}and Jörg Rottler

^{2}

### Abstract

Molecular dynamics simulations are used to investigate the effects of deformation on the segmental dynamics in an agingpolymer glass. Individual particle trajectories are decomposed into a series of discontinuous hops, from which we obtain the full distribution of relaxation times and displacements under three deformation protocols: step stress (creep), step strain, and constant strain rate deformation. As in experiments, the dynamics can be accelerated by several orders of magnitude during deformation, and the history dependence is entirely erased during yield (mechanical rejuvenation). Aging can be explained as a result of the long tails in the relaxation time distribution of the glass, and similarly, mechanical rejuvenation is understood through the observed narrowing of this distribution during yield. Although the relaxation time distributions under deformation are highly protocol specific, in each case they may be described by a universal acceleration factor that depends only on the strain.

We thank M. D. Ediger for many helpful discussions. We acknowledge the Natural Sciences and Engineering Council of Canada (NSERC) for financial support. Computing time was provided by WestGrid. Simulations were performed with the LAMMPS MD package.^{50}

I. INTRODUCTION

II. SIMULATIONS

III. HOPPING DYNAMICS

IV. RESULTS

A. Aging at zero pressure

B. Step stress

C. Constant strain rate

D. Step strain

E. Acceleration factor

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Glasses
- 72.0
- Relaxation times
- 36.0
- Polymers
- 32.0
- Stress relaxation
- 20.0
- Creep
- 19.0

## Figures

Nonaffine displacements , , and (black/blue/red) for a monomer undergoing creep at . The vertical dotted lines indicate relaxations identified by our hop detection algorithm. The total strain during this time is approximately 0.8.

Nonaffine displacements , , and (black/blue/red) for a monomer undergoing creep at . The vertical dotted lines indicate relaxations identified by our hop detection algorithm. The total strain during this time is approximately 0.8.

Mean-squared displacement as a function of the number of hops that a particle has experienced for the polymer model (○) and the BMLJ model (◻) (simulation details of the BMLJ results can be found in Ref. 46). Lines have slopes of 1/2 (blue) and 1 (red). Inset shows the corresponding mean-squared displacement for both models with time.

Mean-squared displacement as a function of the number of hops that a particle has experienced for the polymer model (○) and the BMLJ model (◻) (simulation details of the BMLJ results can be found in Ref. 46). Lines have slopes of 1/2 (blue) and 1 (red). Inset shows the corresponding mean-squared displacement for both models with time.

Relaxation dynamics during the creep experiment. (a) Creep compliance for (dashed) and 0.5 (dotted). Squares show the curve for chains of length 100. Distributions of (b) first hop times, (c) persistence times, and (d) displacements for the undeformed sample (solid lines), (dashed), and (dotted). In all plots (black), 7500 (red), and 75 000 (green). Circles in (c) show for (see text) and . Straight lines indicate power law with the given slopes.

Relaxation dynamics during the creep experiment. (a) Creep compliance for (dashed) and 0.5 (dotted). Squares show the curve for chains of length 100. Distributions of (b) first hop times, (c) persistence times, and (d) displacements for the undeformed sample (solid lines), (dashed), and (dotted). In all plots (black), 7500 (red), and 75 000 (green). Circles in (c) show for (see text) and . Straight lines indicate power law with the given slopes.

Strain rate vs hop rate for (○), 0.4 (◻), 0.3 (△), 0.2 (◇), and (black), 7500 (red), and 75 000 (green). Solid symbols show results for chains of length 100. Inset shows the mean hop displacement vs the strain rate for the same data.

Strain rate vs hop rate for (○), 0.4 (◻), 0.3 (△), 0.2 (◇), and (black), 7500 (red), and 75 000 (green). Solid symbols show results for chains of length 100. Inset shows the mean hop displacement vs the strain rate for the same data.

Relaxation dynamics during the constant strain rate experiment. (a) Stress vs time for (dashed) and (dotted). Inset shows stress vs strain for the same data. Distributions of (b) first hop times, (c) persistence times, and (d) displacements for the undeformed sample (solid lines) and (dashed) and (dotted). In all plots (black), 7500 (red), and 75 000 (green). Straight lines indicate power law with the given slopes.

Relaxation dynamics during the constant strain rate experiment. (a) Stress vs time for (dashed) and (dotted). Inset shows stress vs strain for the same data. Distributions of (b) first hop times, (c) persistence times, and (d) displacements for the undeformed sample (solid lines) and (dashed) and (dotted). In all plots (black), 7500 (red), and 75 000 (green). Straight lines indicate power law with the given slopes.

Relaxation dynamics during the strain step experiment. (a) Strain modulus vs time for (dashed) and 0.04 (dotted). Distributions of (b) first hop times, (c) persistence times, and (d) displacements for the undeformed sample (solid lines) and (dashed) and 0.04 (dotted). In all plots (black), 7500 (red), and 75 000 (green). Straight lines indicate power law with the given slopes.

Relaxation dynamics during the strain step experiment. (a) Strain modulus vs time for (dashed) and 0.04 (dotted). Distributions of (b) first hop times, (c) persistence times, and (d) displacements for the undeformed sample (solid lines) and (dashed) and 0.04 (dotted). In all plots (black), 7500 (red), and 75 000 (green). Straight lines indicate power law with the given slopes.

Cumulative probability distribution of the first relaxation event for the undeformed glass (solid line) and under deformation with a step stress (○), a step strain (△), and a constant strain rate (◻) for . Dashed line shows the cumulative for the undeformed glass.

Cumulative probability distribution of the first relaxation event for the undeformed glass (solid line) and under deformation with a step stress (○), a step strain (△), and a constant strain rate (◻) for . Dashed line shows the cumulative for the undeformed glass.

Acceleration factor as a function of global strain for three different deformation protocols. Stress step: (○), 0.4 (◻), 0.5 (△); constant strain rate: (◇), (▽); strain step (◅). For each: (black), 22 500 (red). Also shown is the acceleration factor for the persistence times for constant strain rate deformation at (blue ). Solid symbols show results for chains of length 100.

Acceleration factor as a function of global strain for three different deformation protocols. Stress step: (○), 0.4 (◻), 0.5 (△); constant strain rate: (◇), (▽); strain step (◅). For each: (black), 22 500 (red). Also shown is the acceleration factor for the persistence times for constant strain rate deformation at (blue ). Solid symbols show results for chains of length 100.

Incoherent scattering function with averaged over the non-deformed ( and ) directions for the undeformed glass (solid line) and under deformation with a step stress (○), a step strain (△), and a constant strain rate (◻) for . Dashed line shows for the undeformed glass.

Incoherent scattering function with averaged over the non-deformed ( and ) directions for the undeformed glass (solid line) and under deformation with a step stress (○), a step strain (△), and a constant strain rate (◻) for . Dashed line shows for the undeformed glass.

Acceleration factor calculated from the incoherent scattering function as a function of global strain for three different deformation protocols. Stress step: (○), 0.4 (◻), 0.5 (△); constant strain rate: (◇), (▽); strain step (◅). For each: (black), 7500 (red).

Acceleration factor calculated from the incoherent scattering function as a function of global strain for three different deformation protocols. Stress step: (○), 0.4 (◻), 0.5 (△); constant strain rate: (◇), (▽); strain step (◅). For each: (black), 7500 (red).

(a) A typical particle trajectory in the glass. Red curve shows raw position data in the -direction, and the black curve shows the running average. (b) Standard deviation in the three dimensional particle position over the averaging window. Hops are identified by a threshold in the standard deviation, shown here as a horizontal dashed line, and marked in both frames as vertical dashed lines. In panel (a) the first hop time, persistence times, and particle displacements are also labeled.

(a) A typical particle trajectory in the glass. Red curve shows raw position data in the -direction, and the black curve shows the running average. (b) Standard deviation in the three dimensional particle position over the averaging window. Hops are identified by a threshold in the standard deviation, shown here as a horizontal dashed line, and marked in both frames as vertical dashed lines. In panel (a) the first hop time, persistence times, and particle displacements are also labeled.

Probability distribution of the standard deviation in the particle position over an averaging time window of 400 for all particle trajectories. Dashed line shows the threshold for detecting a hop.

Probability distribution of the standard deviation in the particle position over an averaging time window of 400 for all particle trajectories. Dashed line shows the threshold for detecting a hop.

The distributions (a) , (b) , and (c) for (circle), 37.5 (square) and 375 (triangle).

The distributions (a) , (b) , and (c) for (circle), 37.5 (square) and 375 (triangle).

The distributions (a) , (b) , and (c) for (square), 0.25 (circle), and 0.35 (triangle). Solid lines in (c) indicate power laws with slope of −1.5 and −1.1.

The distributions (a) , (b) , and (c) for (square), 0.25 (circle), and 0.35 (triangle). Solid lines in (c) indicate power laws with slope of −1.5 and −1.1.

(a) The distribution of hop correlations, (b) , and (c) computed with all of the hops detected (circles), and with only uncorrelated hops (squares).

(a) The distribution of hop correlations, (b) , and (c) computed with all of the hops detected (circles), and with only uncorrelated hops (squares).

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