# Fractional Coupling of Primary and Johari–Goldstein Relaxations in a Model Polymer

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## Abstract

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## 1. Introduction

## 2. Model and Numerical Methods

## 3. Results

#### 3.1. Bond Correlation Function

#### 3.2. Time Temperature and Pressure Superposition of Primary Relaxation

#### 3.3. Dynamic Heterogeneity

#### 3.4. Fractional Coupling of Primary and JG Relaxations

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

^{®}Italia is gratefully acknowledged. F.P. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 754496.

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BCF | Bond correlation function |

CM | Coupling model |

DH | Dynamical heterogeneity |

JG | Johari–Goldstein |

KWW | Kohlrausch–Williams–Watts |

LJ | Lennard–Jones |

MD | Molecular dynamics |

NGP | Non-Gaussian parameter |

NPT | Constant number of monomers N, constant pressure P and constant temperature T |

NVT | Constant number of monomers N, constant volume V and constant temperature T |

TTPS | Time–temperature–pressure superposition |

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**Figure 2.**Time−temperature−pressure superposition (TTPS) of BCF for all the states in Figure 1. The curves are shifted along the horizontal axis to optimise their superposition at long times. The superimposed solid black line is the best fit with a stretched exponential proportional to ${\varphi}^{kww}\left(t\right)$, Equation (2), with ${\beta}_{\mathrm{TTPS}}=0.415$.

**Figure 3.**Stretching parameter of the secondary JG relaxation of all the investigated states according to the best fit with Equation (8). The dashed line is a guide for the eyes. Stretching increases mildly with the primary relaxation time.

**Figure 5.**Correlation plot between the JG relaxation time ${\tau}_{s}$ and the primary relaxation time ${\tau}_{p}$ of all the investigated states, as drawn by fitting BCF $C\left(t\right)$ with a weighed superposition of two stretched exponentials, Equation (8). The correlation is excellent (Pearson correlation coefficient $R=0.998$) and best fit with a power law with slope $\xi =0.71\pm 0.01$ (dashed line). The grey area is the confidence region within one standard deviation of the best-fit parameters.

T∖p | 0 | 0.5 | 1 | 1.5 | 2.5 | 5 | 7.5 | 10 |
---|---|---|---|---|---|---|---|---|

1.1 | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | |

1 | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | |||

0.95 | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | |||

0.9 | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | |||

0.85 | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ | ${}^{\circ}$ |

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**MDPI and ACS Style**

Massa, C.A.; Puosi, F.; Leporini, D.
Fractional Coupling of Primary and Johari–Goldstein Relaxations in a Model Polymer. *Polymers* **2022**, *14*, 5560.
https://doi.org/10.3390/polym14245560

**AMA Style**

Massa CA, Puosi F, Leporini D.
Fractional Coupling of Primary and Johari–Goldstein Relaxations in a Model Polymer. *Polymers*. 2022; 14(24):5560.
https://doi.org/10.3390/polym14245560

**Chicago/Turabian Style**

Massa, Carlo Andrea, Francesco Puosi, and Dino Leporini.
2022. "Fractional Coupling of Primary and Johari–Goldstein Relaxations in a Model Polymer" *Polymers* 14, no. 24: 5560.
https://doi.org/10.3390/polym14245560