^{1}and John D. McCoy

^{1,a)}

### Abstract

Nonlinear dynamics of a simple bead-spring glass-forming polymer were studied with molecular dynamics simulations. The energy response to sinusoidal variations in the temperature was tracked in order to evaluate the dynamic heat capacity. The amplitude dependence of the response is the focus of the current paper where pronounced nonlinear behavior is observed for large amplitudes in the temperature “driving force.” We generalize the usual linear response analysis to the nonlinear regime so that higher order terms in the Fourier series of the energy response can be compactly analyzed. This is done by grouping all Fourier terms contributing to entropy generation into a “loss” contribution and the remainder yields the “storage” term. Finally, the bead-spring system is mapped onto three simpler models. First is a potential energy inspired “trap” model consisting of interconnected potential energy meta-basins and barriers. Second is the Tool-Narayanaswamy-Moynihan (TNM) model. Third is a version of the TNM model with a temperature dependent heat capacity. Qualitatively similar nonlinear behaviors are observed in all cases.

I. INTRODUCTION

II. METHODOLOGY

III. RESULTS AND DISCUSSION

A. The MD system and the trap model

B. The TNM model

IV. CONCLUSIONS

### Key Topics

- Heat capacity
- 41.0
- Glass transitions
- 13.0
- Molecular dynamics
- 13.0
- Relaxation times
- 12.0
- Entropy
- 10.0

##### C03C3/00

## Figures

Plot of the average IS energy response of the MD (left) and trap (middle, right as labeled) models as a function of time (units of quarter period). The MD response is also smoothed from the raw data. The top three plots are for a frequency where the system is in equilibrium in the linear response regime, *ω* = 2π/20 000 ≈ 3 × 10^{−4} (inverse LJ time units), and the bottom plots are for frequency near the *α*-peak, *ω* = *ω* _{ α } = 2π/640 ≈ 1 × 10^{−2}. The different colors indicate different temperature amplitudes: Δ*T* = 0.75*T* _{0} (blue), Δ*T* = 0.5*T* _{0} (red), Δ*T* = 0.3*T* _{0} (green), Δ*T* = 0.15*T* _{0} (black), and Δ*T* = 0.05*T* _{0} (light blue, linear response).

Plot of the average IS energy response of the MD (left) and trap (middle, right as labeled) models as a function of time (units of quarter period). The MD response is also smoothed from the raw data. The top three plots are for a frequency where the system is in equilibrium in the linear response regime, *ω* = 2π/20 000 ≈ 3 × 10^{−4} (inverse LJ time units), and the bottom plots are for frequency near the *α*-peak, *ω* = *ω* _{ α } = 2π/640 ≈ 1 × 10^{−2}. The different colors indicate different temperature amplitudes: Δ*T* = 0.75*T* _{0} (blue), Δ*T* = 0.5*T* _{0} (red), Δ*T* = 0.3*T* _{0} (green), Δ*T* = 0.15*T* _{0} (black), and Δ*T* = 0.05*T* _{0} (light blue, linear response).

Plot of the unsmoothed IS energy response of a MD model at frequency *ω* = 2π/200 000 and temperature amplitude Δ*T* = 0.75*T* _{0} as a function of time with the time reduced modulo the period so that all oscillations lay on top of one another (top), and a histogram of the energies in the above plot (bottom).

Plot of the unsmoothed IS energy response of a MD model at frequency *ω* = 2π/200 000 and temperature amplitude Δ*T* = 0.75*T* _{0} as a function of time with the time reduced modulo the period so that all oscillations lay on top of one another (top), and a histogram of the energies in the above plot (bottom).

Lissajous-Bowditch-like parametric plots of the IS energy against the temperature for the different systems as labeled. The top curves (left scale) and bottom curves (right scale) correspond to the same frequencies in Fig. 1. The colors have the same meaning as in Fig. 1.

Lissajous-Bowditch-like parametric plots of the IS energy against the temperature for the different systems as labeled. The top curves (left scale) and bottom curves (right scale) correspond to the same frequencies in Fig. 1. The colors have the same meaning as in Fig. 1.

Plot of the 0th, 1st, and 2nd harmonics (labeled *E*, *a* _{1} and *b* _{1}, *a* _{2} and *b* _{2}, respectively) of the IS energy response as a function of frequency on a logarithmic scale for the three systems as labeled. The different colors indicate different temperature amplitudes: Δ*T* = 0.1*T* _{0} (blue, linear response), Δ*T* = 0.3*T* _{0} (red), and Δ*T* = 0.75*T* _{0} (green).

Plot of the 0th, 1st, and 2nd harmonics (labeled *E*, *a* _{1} and *b* _{1}, *a* _{2} and *b* _{2}, respectively) of the IS energy response as a function of frequency on a logarithmic scale for the three systems as labeled. The different colors indicate different temperature amplitudes: Δ*T* = 0.1*T* _{0} (blue, linear response), Δ*T* = 0.3*T* _{0} (red), and Δ*T* = 0.75*T* _{0} (green).

Plot of the nonlinear storage (left) and loss (right) moduli discussed in the text as a function of frequency on a logarithmic scale for the three systems as labeled. The different colors indicate different temperature amplitudes: Δ*T* = 0.75*T* _{0} (blue), Δ*T* = 0.5*T* _{0} (red), Δ*T* = 0.3*T* _{0} (green), Δ*T* = 0.15*T* _{0} (black), and Δ*T* = 0.1*T* _{0} (brown, linear response).

Plot of the nonlinear storage (left) and loss (right) moduli discussed in the text as a function of frequency on a logarithmic scale for the three systems as labeled. The different colors indicate different temperature amplitudes: Δ*T* = 0.75*T* _{0} (blue), Δ*T* = 0.5*T* _{0} (red), Δ*T* = 0.3*T* _{0} (green), Δ*T* = 0.15*T* _{0} (black), and Δ*T* = 0.1*T* _{0} (brown, linear response).

Plot of the power law exponents of the loss modulus for low frequencies (*α*, triangles, left scale), for high frequencies (*β*, inverted triangles, left scale), and the peak frequency (circles, right scale) as a function of relative temperature amplitude on a logarithmic scale for the MD system. *α* and *β* were computed from the slopes on a log-log scale of the loss modulus on the low-frequency and high-frequency sides of the peak, respectively.

Plot of the power law exponents of the loss modulus for low frequencies (*α*, triangles, left scale), for high frequencies (*β*, inverted triangles, left scale), and the peak frequency (circles, right scale) as a function of relative temperature amplitude on a logarithmic scale for the MD system. *α* and *β* were computed from the slopes on a log-log scale of the loss modulus on the low-frequency and high-frequency sides of the peak, respectively.

Plot of the average IS energy response of the MD (left) and TNM (middle, right as labeled) models as a function of time (units of quarter period). The MD response is also smoothed from the raw data. The top three plots are for a frequency where the system is in equilibrium in the linear response regime, *ω* = 2π/20 000 ≈ 3 × 10^{−4} (inverse LJ time units), and the bottom plots are for frequency near the *α*-peak, *ω* = *ω* _{ α } = 2π/640 ≈ 1 × 10^{−2}. The different colors indicate different temperature amplitudes: Δ*T* = 0.75*T* _{0} (blue), Δ*T* = 0.5*T* _{0} (red), Δ*T* = 0.3*T* _{0} (green), Δ*T* = 0.15*T* _{0} (black), and Δ*T* = 0.05*T* _{0} (light blue, linear response).

Plot of the average IS energy response of the MD (left) and TNM (middle, right as labeled) models as a function of time (units of quarter period). The MD response is also smoothed from the raw data. The top three plots are for a frequency where the system is in equilibrium in the linear response regime, *ω* = 2π/20 000 ≈ 3 × 10^{−4} (inverse LJ time units), and the bottom plots are for frequency near the *α*-peak, *ω* = *ω* _{ α } = 2π/640 ≈ 1 × 10^{−2}. The different colors indicate different temperature amplitudes: Δ*T* = 0.75*T* _{0} (blue), Δ*T* = 0.5*T* _{0} (red), Δ*T* = 0.3*T* _{0} (green), Δ*T* = 0.15*T* _{0} (black), and Δ*T* = 0.05*T* _{0} (light blue, linear response).

Lissajous-Bowditch-like parametric plots of the IS energy against the temperature for the different systems as labeled. The top curves (left scale) and bottom curves (right scale) correspond to the same frequencies in Fig. 7. The colors have the same meaning as in Fig. 7.

Lissajous-Bowditch-like parametric plots of the IS energy against the temperature for the different systems as labeled. The top curves (left scale) and bottom curves (right scale) correspond to the same frequencies in Fig. 7. The colors have the same meaning as in Fig. 7.

Plot of the 0th, 1st, and 2nd harmonics (labeled *E*, *a* _{1} and *b* _{1}, *a* _{2} and *b* _{2}, respectively) of the IS energy response as a function of frequency on a logarithmic scale for the three systems as labeled. The different colors indicate different temperature amplitudes: Δ*T* = 0.1*T* _{0} (blue, linear response), Δ*T* = 0.3*T* _{0} (red), and Δ*T* = 0.75*T* _{0} (green).

*E*, *a* _{1} and *b* _{1}, *a* _{2} and *b* _{2}, respectively) of the IS energy response as a function of frequency on a logarithmic scale for the three systems as labeled. The different colors indicate different temperature amplitudes: Δ*T* = 0.1*T* _{0} (blue, linear response), Δ*T* = 0.3*T* _{0} (red), and Δ*T* = 0.75*T* _{0} (green).

Plot of the nonlinear storage (left, linear scaling) and loss (right, logarithmic scaling) moduli discussed in the text as a function of frequency on a logarithmic scale for the three systems as labeled. The different colors indicate different temperature amplitudes: Δ*T* = 0.75*T* _{0} (blue), Δ*T* = 0.5*T* _{0} (red), Δ*T* = 0.3*T* _{0} (green), Δ*T* = 0.15*T* _{0} (black), and Δ*T* = 0.1*T* _{0} (brown, linear response).

Plot of the nonlinear storage (left, linear scaling) and loss (right, logarithmic scaling) moduli discussed in the text as a function of frequency on a logarithmic scale for the three systems as labeled. The different colors indicate different temperature amplitudes: Δ*T* = 0.75*T* _{0} (blue), Δ*T* = 0.5*T* _{0} (red), Δ*T* = 0.3*T* _{0} (green), Δ*T* = 0.15*T* _{0} (black), and Δ*T* = 0.1*T* _{0} (brown, linear response).

## Tables

Parameters for 8 level trap model (units are standard Lennard-Jones units).^{a}

Parameters for 8 level trap model (units are standard Lennard-Jones units).^{a}

Parameters for the TNM models (all units are standard Lennard-Jones units).^{a}

Parameters for the TNM models (all units are standard Lennard-Jones units).^{a}

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