# Fast Vibrational Modes and Slow Heterogeneous Dynamics in Polymers and Viscous Liquids

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## Abstract

**:**

## 1. Introduction

## 2. A model of the Slow Heterogeneous Relaxation and Transport in Terms of Vibrational Dynamics

#### 2.1. Relaxation Time

#### 2.2. Diffusion Coefficient

#### 2.3. Stokes–Einstein Product

## 3. Transport and Relaxation in Polymeric Melts

## 4. Correlation between Vibrational Fast Dynamics and Slow Relaxation

#### 4.1. Vibrational Caged Dynamics and Debye–Waller factor

#### 4.2. Debye–Waller Scaling of the Slow Relaxation

## 5. Signatures of the Heterogeneous Dynamics

#### 5.1. van Hove Function

- The self-part of the van Hove function is expressed by suitable correlation functions, see Appendix B. Then, the coincidence of ${G}_{s}(r,{\tau}_{\alpha})$ in states with equal DW factor observed in Figure 5a (the sets of states labelled as A, ⋯, E) is in harmony with Equation (3).
- Equation (3) also holds if one inspects the spatial dependence of the correlation function, e.g., the van Hove function, at ${\tau}_{\alpha}$. In particular, even in the presence of DH.
- The pattern of the D and E sets of states is not consistent with the Gaussian limit ${G}_{s}^{g}(r,{\tau}_{\alpha})$, Equation (20), predicting a progressive decay with r, i.e., the DH dynamics is not Gaussian;

#### 5.2. Non-Gaussian Parameter

## 6. Breakdown of the Stokes–Einstein (SE) Law in the Presence of Dynamical Heterogeneity

#### 6.1. SE Breakdown in Unentangled Polymers

#### 6.2. Quasi-Universal SE Breakdown of Fragile Glass-Formers

## 7. Displacement Correlation Length

## 8. Discussion

## 9. Methods

## Author Contributions

## Funding

## Acknowledgments

^{®}Italia is gratefully acknowledged.

## Conflicts of Interest

## Abbreviations

DDC | displacement–displacement correlation |

DH | dynamical heterogeneity |

DW | Debye–Waller |

ISF | Intermediate scattering function |

MD | Molecular-dynamics |

MSD | Mean square displacement |

NGP | non-Gaussian parameter |

SE | Stokes–Einstein |

## Appendix A

#### Appendix A.1. Structural Relaxation

#### Appendix A.2. Diffusion Coefficient

#### Appendix A.3. Stokes–Einstein Product

## Appendix B

## References

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**Figure 1.**Monomer arrangements at a time ${t}_{0}$ of two states of a polymer melt with (

**a**) homogeneous and (

**b**) heterogeneous dynamics. Bonds are removed for clarity reasons. Particles are coloured according to their squared displacements in the time interval $[{t}_{0}-{\tau}_{\alpha},{t}_{0}]$. Bright yellow particles have squared displacements no less than 1. Notice that the two states have comparable mean square displacement (∼0.21, homogeneous state; ∼0.28, heterogeneous state) but rather different relaxation times ${\tau}_{\alpha}$ (∼9, homogeneous state; ∼1550, heterogeneous state). Homogeneous, i.e., position-independent, dynamics of the monomers is an aspect of systems with fast relaxation. Conversely, in the presence of heterogeneous dynamics, clusters of particles with extremely high mobility coexist with nearly immobile ones, slowing down the relaxation.

**Figure 2.**Monomer dynamics in the polymer melt. (

**a**) Mean square displacement (MSD) for polymers in selected states (see below for details). For clarity reasons, MSDs are multiplied by indicated factors. Inset: corresponding MSD slope $\Delta \left(t\right)$, Equation (16); the position of the minimum at ${t}^{\star}=1.0\left(4\right)$ is signalled by the arrow in the inset and the dashed line in the main panel. (

**b**) corresponding ISF curves. The figure shows a manifestation of Equation (3), see Section 5.1 for details, i.e., if states have equal DW factor $\langle {u}^{2}\rangle $, both the MSD and ISF curves coincide at least in the time window $[{t}^{\star},{\tau}_{\alpha}]$ (${\tau}_{\alpha}$ is marked with dots on each curve). The physical states are labelled by the string (M, $\rho $, T, q, p) where M is the number of monomers per chain, $\rho $ the number density, T the temperature and the pair $(q,p)$ refers to the characteristic parameters of the non-bonding potential, Equation (25). The six sets of states are as follows. Set A: (2,1.086,0.7,7,6), (3,1.086,0.7,7,6), (10,1.086,0.7,7,6), (10,1.033,0.7,8,6). Set B: (2,1.033,0.7,10,6), (3,1.039,0.7,11,6), (3,1.041,0.7,11,6). Set C: (2,1.033,0.5,10,6), (3,1.056,0.7,12,6), (5,1.033,0.6,12,6), (10,1.056,0.7,12,6). Set D: (3,1.086,0.7,12,6), (5,1.086,0.7,12,6), (10,1.086,0.7,12,6). Set E: (2,1.0,0.7,12,11), (3,1.1,1.1,15,7). Data from [48].

**Figure 3.**Correlation function of the end-to-end vector joining the two ends of a polymer chain. Each group of curves corresponds the physical states A, ..., E with identical DW factor detailed in Figure 2. Polymer states contributing to one cluster of scaled curves have not necessarily equal chain length. However, the scaled time removes the chain length dependence. Dots mark the time $4{\tau}_{ee}/{M}^{2}$. The results prove that Equation (3) holds also at times ${\tau}_{ee}$ much longer than ${\tau}_{\alpha}$. Data from [48].

**Figure 4.**The structural relaxation time ${\tau}_{\alpha}$ and the scaled reorientation time ${\tau}_{ee}$ of the polymer chains vs. the DW factor $\langle {u}^{2}\rangle $. Empty circles highlight the cases plotted in Figure 2. The other states are detailed in Ref. [48]. The dashed line across the ${\tau}_{\alpha}$ curve is Equation (A1). The dashed curve across the chain reorientation time curve is a guide for the eyes. Data from [48].

**Figure 5.**(

**a**) Self part of the van Hove function ${G}_{s}(r,t)$ of the states of Figure 2 at the structural relaxation time $t={\tau}_{\alpha}$. The curves are multiplied by indicated factors. The sets of clustered curves A–E show that, if states have equal DW factor, they have coincident van Hove functions too. As ${G}_{s}(r,t)$ may be expressed in terms of correlation functions, the coincidence reflects Equation (3). Data from [49]. (

**b**) The ratio ${N}_{s}(r,{\tau}_{\alpha})$, Equation (21), of the states of Figure 2. On increasing the structural relaxation time from A states to E states, the system tends to increase the fractions of monomers with either much lower or much higher mobility with respect to the fraction predicted by the Gaussian approximation. Data from [53].

**Figure 6.**Non-Gaussian parameters (NGPs) of states with different relaxation times ${\tau}_{\alpha}$ (marked with grey dots). The physical states A, ⋯, E are the states with identical DW factor detailed in Figure 2. Note that they have coinciding NGPs in the time window $[{t}^{\star},{\tau}_{\alpha}]$ at least, in agreement with Equation (3). The curve labelled as F is the state (M, $\rho $, T, q, p) = (3, 1.1, 0.65, 12, 6) with ${\tau}_{\alpha}\simeq 2\xb7{10}^{3}$, see Figure 4. Inset: the NGP maximum ${\alpha}_{2}^{max}$ vs. the ratio R, Equation (9). The dot with the largest ${\alpha}_{2}^{max}$ value corresponds to the state with the longest structural relaxation time ${\tau}_{\alpha}$ in Figure 4 with parameters (M, $\rho $, T, q, p) = (3, 1.2, 0.95, 6,12). Data from [29].

**Figure 7.**(

**a**) The product $D\phantom{\rule{0.166667em}{0ex}}M\phantom{\rule{0.166667em}{0ex}}{\tau}_{\alpha}$ vs. the ratio R, Equation (9) (

**b**) the same product vs. ${\alpha}_{2}^{max}$, the maximum of the non-Gaussian parameter, Equation (22). The onset of the Stokes–Einstein (SE) violation for ${\alpha}_{2}^{max}>{\alpha}_{2,c}^{max}$ and $R>{R}_{c}$, respectively, is indicated with the full vertical lines (uncertainty marked by dashed lines). The thick line in the panel (

**a**) is the master curve between $log{\tau}_{\alpha}$ and the DW factor, Equation (A1), recast in terms of R and the thin line, is the corresponding linear approximation for small R values. Note that the SE violation is apparent where the linear approximation is poor. Data from [29].

**Figure 8.**Stokes–Einstein product ${K}_{SE}$, normalised by its high temperature value ${K}_{0}$ (${\tau}_{\alpha}\simeq 1$ ps), as a function of the reduced DW factor $\langle {u}^{2}\rangle /{u}_{g}^{2}$, ${u}_{g}^{2}$ being the DW factor at the glass transition. In addition to unentangled polymers, the plot also considers MD data concerning atomic binary mixtures (atoms labelled as A and B) and metallic alloys made by Cu and Zr atoms, as well as experimental data for ortho-terphenyl (OTP) [74,75]. Two predictions of the master curve are presented in terms of the quantity ${\widehat{K}}_{SE}$ and ${\tilde{K}}_{SE}$, Equations (A8) and (A10), respectively. Both quantities have no adjustable parameters. ${\widehat{K}}_{0}$ and ${\tilde{K}}_{0}$ are suitable constants to ensure the unit limit value at large $\langle {u}^{2}\rangle /{u}_{g}^{2}$. A third master curve, drawn from the fractional SE law ${\tau}_{\alpha}^{1-\kappa}$ with $\kappa =0.85$ (orange curve), is superimposed to the other curves. For numerical data, ${u}_{g}^{2}$ is obtained according to the procedure outlined in [47]. Data from [30].

**Figure 9.**Radial dependence of the correlation of the direction (

**a**) and the mobility (

**b**) displacements occurring in a time range as wide as the structural relaxation time ${\tau}_{\alpha}$. For comparison, the radial distribution function $g\left(r\right)$ (dashed line) of the state with $\{M=2,\rho =1.086,T=0.7,q=7,p=6\}$ is plotted. Note that $g\left(r\right)$ is virtually state-independent. The insets are semi-log plots of the corresponding main panels. Note the approximate exponential decay of the peak amplitudes with slopes ${\xi}_{\overrightarrow{u}}\left({\tau}_{\alpha}\right)$ and ${\xi}_{\delta u}\left({\tau}_{\alpha}\right)$, respectively. Data from [53].

**Figure 10.**The direction ${\xi}_{\overrightarrow{u}}\left({\tau}_{\alpha}\right)$ (full symbols) and modulus ${\xi}_{\delta u}\left({\tau}_{\alpha}\right)$ (open symbols) correlation lengths vs. the structural relaxation time ${\tau}_{\alpha}$ of selected set of states of Figure 2. Dashed lines are guides for the eyes. States with equal DW factor, i.e., belonging to the same set B, ⋯, E exhibit equal directional and mobility correlation lengths. Data from [53].

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Puosi, F.; Tripodo, A.; Leporini, D.
Fast Vibrational Modes and Slow Heterogeneous Dynamics in Polymers and Viscous Liquids. *Int. J. Mol. Sci.* **2019**, *20*, 5708.
https://doi.org/10.3390/ijms20225708

**AMA Style**

Puosi F, Tripodo A, Leporini D.
Fast Vibrational Modes and Slow Heterogeneous Dynamics in Polymers and Viscous Liquids. *International Journal of Molecular Sciences*. 2019; 20(22):5708.
https://doi.org/10.3390/ijms20225708

**Chicago/Turabian Style**

Puosi, Francesco, Antonio Tripodo, and Dino Leporini.
2019. "Fast Vibrational Modes and Slow Heterogeneous Dynamics in Polymers and Viscous Liquids" *International Journal of Molecular Sciences* 20, no. 22: 5708.
https://doi.org/10.3390/ijms20225708