# Coincident Correlation between Vibrational Dynamics and Primary Relaxation of Polymers with Strong or Weak Johari-Goldstein Relaxation

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

_{α}/τ

_{β}to variations of pressure and temperature, keeping τ

_{α}constant. β JG, for these reasons, can be considered as the precursor to structural relaxation, having a slower dynamics due to cooperativity involving many body dynamics [6,8].

## 2. Model and Numerical Methods

_{c}= 512 linear chains made of M = 25 monomers each, resulting in a total number of monomers N = 12800. Non-bonded monomers at a distance r interact via a Lennard-Jones potential:

^{1/6}σ is the minimum of the potential, ${U}^{LJ}\left(r=\sigma \right)=-\u03f5$. The potential is truncated at r = r

_{c}= 2.5σ for computational convenience. Adjacent bonded monomers interact with each other via the harmonic potential U

^{bond}(r) = k

_{bond}(r − l

_{0})

^{2}, where the constant k

_{bond}is set to $2000\u03f5/{\sigma}^{2}$ to ensure high stiffness. We consider two distinct cases, corresponding to rest bond length l

_{0}set to either l

_{0}= 0.48σ or l

_{0}= 0.55σ. The rationale behind our choice of the two bond lengths relies on the finding that previous Molecular-Dynamics (MD) simulations of the present model [14], investigating the rotational dynamics, revealed the steep increase of the separation between the primary and the JG relaxations by decreasing the bond length below ${l}_{0}^{*}=0.5\sigma $ (corresponding to 2Å in Figure 3 of Reference [14]). Therefore, one anticipates that the JG relaxation is much more apparent if l

_{0}= 0.48σ with respect to l

_{0}= 0.55σ. A bending potential U

^{bend}(α) = k

_{bend}(cos α − cos α

_{0})

^{2}, with kbend = 2000ϵ/σ

^{2}and α

_{0}= 120°, is introduced to maintain the angle α formed by two consecutive bonds nearly constant (see Figure 1 for typical chain conformations) [17].

_{B}, where k

_{B}is the Boltzmann constant, and time in units of τ

_{MD}= (mσ

^{2}/ϵ)

^{1/}

^{2}. We set σ = 1, ϵ = 1, m = 1 and k

_{B}= 1.

_{ee}, being τ

_{ee}the relaxation time of the end-to-end vector autocorrelation function [21,22,23,24,25,26,27,28]. Production runs have been performed within the NVT ensemble (constant number of monomers N, constant volume V and constant temperature T). Additional short equilibration runs were performed when switching from NPT to NVT ensemble. No signatures of crystallization were observed in all the investigated states.

## 3. Results and Discussion

#### 3.1. Bond Reorientation

_{m,n}(t) denotes the position of the m-th monomer in the n-th chain at time t. We define the correlation function C(t):

_{p}(t) and f

_{s}(t) are two decaying functions with amplitudes A

_{p}and A

_{s}, respectively. The explicit form of f

_{i}(t) is taken as a stretched exponential:

_{i}denotes the relaxation time and β

_{i}the stretching exponent (as normally β

_{i}≤ 1). We fit the MD data concerning the correlation function C(t) with Equation (5) excluding the time window t ≤ $\widehat{t}$ where the decay of the function is controlled by the ballistic motion of monomers (0.6 ≤ $\widehat{t}$ ≤ 1 with $\widehat{t}$ decreasing slightly by increasing T).

^{MD}(t) − C

^{fit}(t) plotted in the inset of Figure 3 prove the poor performance of the single-relaxation curve to fit the MD data.

_{s}and τ

_{p}(ii) enhanced amplitude of the secondary relaxation with respect to the primary one. It is also worth noting that the primary relaxation is less stretched in the model where the secondary relaxation is more apparent (l

_{0}= 0.48) with respect to the case in which it is weak (l

_{0}= 0.55). The differences are small but significant, that is, larger than the errors on the stretching exponents. Stretched relaxation is usually associated with the presence of dynamical heterogeneities, namely the spatial distribution of mobilities, which may differ of orders of magnitude in regions only a few nanometers away. In this framework, our results suggest that the presence of a not negligible secondary relaxation process slightly decreases the degree of dynamical heterogeneity of the system. We plan to address this aspect in future works.

_{p}and f

_{s}) leading to

_{0}= 0.48, red symbols).

#### 3.2. Monomer Mobility

_{i}(t) is the position of the i-th monomer at time t. Figure 6 shows MSD curves for the two systems at all the investigated temperatures. At very short times (ballistic regime) MSD increases according to $\langle {r}^{2}\left(t\right)\rangle \cong \left(3{k}_{B}T/m\right){t}^{2}$. At later times a quasi-plateau region becomes apparent when the temperature is lowered. This signals the increased trapping of a particle in the cage of its neighbors. Once escaped from the cage, due to the presence of the chain connectivity, the monomers undergo a sub diffusive motion $\langle {r}^{2}\left(t\right)\rangle \propto {t}^{\delta}$ with δ < 1 (Rouse regime) [21]. At very long times, monomers displace in a diffusive way (δ = 1). Diffusion is hardly seen in our simulations since, due to the length of the chains, it occurs at the limit of the accessible timescales.

#### 3.3. Cage Dynamics and Correlation with Primary and Secondary Relaxations

_{0}= 0.48) or weak (l

_{0}= 0.55) secondary relaxation exhibit the same master correlation curve ${\overline{\tau}}_{p}\text{}\mathrm{vs.}\text{}\langle {u}^{2}\rangle $. Notably, this coincidence takes place even if the correlation curves ${\overline{\tau}}_{s}\text{}\mathrm{vs.}\text{}\langle {u}^{2}\rangle $ do depend on the bond length.

#### 3.4. Alternative Probe Functions of Secondary Relaxation

_{max}∼ 2π/σ, corresponding to the maximum of the static structure factor. It is seen that there is no evidence of the two-step decay observed in the bond correlation function C(t) at the same temperature, see Figure 2, left. The results presented in Figure 9 are not unexpected. Previous MD studies [13] performed by using the same model of the present work (dubbed FRC model in Reference [13]) reported that ISF needs lower temperatures to reveal a two-step process in the relaxation. This suggests that ISF has lower JG resolution.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

^{®}Italia is gratefully acknowledged.

## Conflicts of Interest

## Abbreviations

ISF | Intermediate scattering function |

JG | Johari-Goldstein |

MD | molecular-dynamics |

MSD | Mean square displacement |

DW | Debye-Waller |

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**Figure 1.**Pictorial view of the two kind of polymeric chains under consideration. L

_{0}denotes the bond length. The time axis signal, in a qualitative way, the location of the dynamical processes of interest. The size of the dot of the secondary relaxation is proportional to the strength of the relaxation. The exact definition of the symbols is given in Section 3.

**Figure 2.**Temperature dependence of the bond correlation function of the chains with different bond length. If l

_{0}= 0.48, a two-step decay—evidencing two distinct relaxations—is observed.

**Figure 3.**Illustrative example of the best-fit results via the double-relaxation function Equation (5). The best-fit with the single-relaxation function A

_{0}f

_{0}with f

_{0}in the form of Equation (6) is also shown. Inset: residuals of the best-fit with the double- and single- relaxation functions.

**Figure 4.**Plots of the temperature dependence of the best-fit parameters by using Equation (5) and Equation (6) (color codes as in Figure 1). From left to right: apparent relaxation times, relaxation strengths and stretching exponents.

**Figure 5.**Arrhenius plots of the average relaxation times of the primary (${\overline{\tau}}_{p}$, left) and secondary (${\overline{\tau}}_{s}$, right) relaxations. Color code as in Figure 1.

**Figure 6.**Monomer mean square displacement (MSD) of the two polymer melts with chains having bond lengths l

_{0}= 0.48 (left) and l

_{0}= 0.55 (right). Inset: corresponding MSD slope ∆(t), Equation (9). The vertical dashed lines mark the time t

^{*}≈ 1 where ∆(t) reaches the minimum, locating the time where caging is more effective. t

^{*}is found to independent of both the system and its physical state. The black circles indicate the values of $\langle {u}^{2}\rangle $, Equation (10).

**Figure 7.**Temperature dependence of the Debye-Waller factor $\langle {u}^{2}\rangle $. Color code as in Figure 1.

**Figure 8.**Average relaxation times ${\overline{\tau}}_{p}$ and ${\overline{\tau}}_{s}$ versus the inverse Debye-Waller factor. The correlation between the vibrational dynamics and the primary relaxation time ${\overline{\tau}}_{p}$ is unaffected by the changes of both the strength and the relaxation time of the secondary relaxation, see Figure 4. Color code as in Figure 1.

**Figure 9.**Left: comparison between the torsional autocorrelation function (TACF) and the bond-orientation correlation function, C(t) for the system with stronger secondary relaxation (l

_{0}= 0.48) at the lowest investigated temperature (T = 0.85). Notice the step observed in both functions at $t~4\xb7{10}^{3}$, signalling the secondary relaxation, see Figure 2, left. The step is much more apparent in C(t), occurring when the latter is dropped of ∼40% only. The same feature is observed in TACF when more than 90% of the decay has been completed. Right: corresponding intermediate scattering function (ISF) for different wavevectors in a range including q = q

_{max}, corresponding to the maximum of the static structure factor. No apparent two-step decay is seen.

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**MDPI and ACS Style**

Tripodo, A.; Puosi, F.; Malvaldi, M.; Capaccioli, S.; Leporini, D.
Coincident Correlation between Vibrational Dynamics and Primary Relaxation of Polymers with Strong or Weak Johari-Goldstein Relaxation. *Polymers* **2020**, *12*, 761.
https://doi.org/10.3390/polym12040761

**AMA Style**

Tripodo A, Puosi F, Malvaldi M, Capaccioli S, Leporini D.
Coincident Correlation between Vibrational Dynamics and Primary Relaxation of Polymers with Strong or Weak Johari-Goldstein Relaxation. *Polymers*. 2020; 12(4):761.
https://doi.org/10.3390/polym12040761

**Chicago/Turabian Style**

Tripodo, Antonio, Francesco Puosi, Marco Malvaldi, Simone Capaccioli, and Dino Leporini.
2020. "Coincident Correlation between Vibrational Dynamics and Primary Relaxation of Polymers with Strong or Weak Johari-Goldstein Relaxation" *Polymers* 12, no. 4: 761.
https://doi.org/10.3390/polym12040761