^{1}, Takuma Akimoto

^{2}and Tomoshige Miyaguchi

^{3}

### Abstract

In entangled polymer systems, there are several characteristic time scales, such as the entanglement time and the disengagement time. In molecular simulations, the longest relaxation time (the disengagement time) can be determined by the mean square displacement (MSD) of a segment or by the shear relaxation modulus. Here, we propose the relative fluctuation analysis method, which is originally developed for characterizing large fluctuations, to determine the longest relaxation time from the center of mass trajectories of polymer chains (the time-averaged MSDs). Applying the method to simulation data of entangled polymers (by the slip-spring model and the simple reptationmodel), we provide a clear evidence that the longest relaxation time is estimated as the crossover time in the relative fluctuations.

This work is supported by the Core Research for the Evolution Science and Technology (CREST) of the Japan Science and Technology Agency (JST).

I. INTRODUCTION

II. SIMULATION RESULTS

A. Slip-spring model

B. Discrete reptationmodel

III. DISCUSSIONS

IV. CONCLUSIONS

### Key Topics

- Polymers
- 36.0
- Reptation
- 23.0
- Tissue ablation
- 23.0
- Trajectory models
- 19.0
- Relaxation times
- 17.0

## Figures

Ensemble-averaged MSDs of CMs in the slip-spring model for *N* = 10, 20, 40, 80, and 160. Dashed lines represent curves which are proportional to Δ^{1} and Δ^{1/2}. *b* and τ_{0} are the size and characteristic time of a segment, respectively.

Ensemble-averaged MSDs of CMs in the slip-spring model for *N* = 10, 20, 40, 80, and 160. Dashed lines represent curves which are proportional to Δ^{1} and Δ^{1/2}. *b* and τ_{0} are the size and characteristic time of a segment, respectively.

Relative fluctuations of TAMSDs of CMs in the slip-spring model for Δ/τ_{0} = 10 and *N* = 10, 20, 40, 80, and 160, where τ_{0} is the characteristic time of a segment. (a) *N* = 10 and 20, and (b) *N* = 40, 80, and 160. Dashed line represents a curve proportional to *t* ^{−1/2}. The power law exponents for the short time regions are α = 0.31, 0.24, and 0.19 for *N* = 40, 80, and 160, respectively.

Relative fluctuations of TAMSDs of CMs in the slip-spring model for Δ/τ_{0} = 10 and *N* = 10, 20, 40, 80, and 160, where τ_{0} is the characteristic time of a segment. (a) *N* = 10 and 20, and (b) *N* = 40, 80, and 160. Dashed line represents a curve proportional to *t* ^{−1/2}. The power law exponents for the short time regions are α = 0.31, 0.24, and 0.19 for *N* = 40, 80, and 160, respectively.

The longest relaxation time τ_{ d } and the crossover time τ_{ c } in the slip-spring model. τ_{ d } is determined from the shear relaxation modulus data^{22} whereas τ_{ c } is determined from the RF data of TAMSDs for Δ/τ_{0} = 10. Dashed lines represent fitting curves for large *N* (τ_{ d } ∝ *N* ^{3.48} and τ_{ c } ∝ *N* ^{3.51}).

The longest relaxation time τ_{ d } and the crossover time τ_{ c } in the slip-spring model. τ_{ d } is determined from the shear relaxation modulus data^{22} whereas τ_{ c } is determined from the RF data of TAMSDs for Δ/τ_{0} = 10. Dashed lines represent fitting curves for large *N* (τ_{ d } ∝ *N* ^{3.48} and τ_{ c } ∝ *N* ^{3.51}).

Ensemble-averaged MSDs of CMs in the discrete reptation model for *Z* = 10, 20, 40, 80, and 160. *a* is the step size of a tube segment and τ_{ l } is the characteristic time of the longitudinal motion of a segment along the tube.

Ensemble-averaged MSDs of CMs in the discrete reptation model for *Z* = 10, 20, 40, 80, and 160. *a* is the step size of a tube segment and τ_{ l } is the characteristic time of the longitudinal motion of a segment along the tube.

Relative fluctuations of TAMSDs of CMs in the discrete reptation model for Δ/τ_{ l } = 10 (see Appendix A) and *Z* = 10, 20, 40, 80, and 160. The dashed line represents a curve proportional to *t* ^{−1/2}. τ_{ l } is the characteristic time of the longitudinal motion of a segment along the tube.

Relative fluctuations of TAMSDs of CMs in the discrete reptation model for Δ/τ_{ l } = 10 (see Appendix A) and *Z* = 10, 20, 40, 80, and 160. The dashed line represents a curve proportional to *t* ^{−1/2}. τ_{ l } is the characteristic time of the longitudinal motion of a segment along the tube.

The longest relaxation time τ_{ d } and crossover time τ_{ c } in the discrete reptation model. τ_{ d } is determined from the shear relaxation modulus and τ_{ c } is determined from the RF data of TAMSDs in the same way as Fig. 3. The dashed line represents the reptation time τ_{rep}/τ_{ l } = *Z* ^{3}/π^{2}.

The longest relaxation time τ_{ d } and crossover time τ_{ c } in the discrete reptation model. τ_{ d } is determined from the shear relaxation modulus and τ_{ c } is determined from the RF data of TAMSDs in the same way as Fig. 3. The dashed line represents the reptation time τ_{rep}/τ_{ l } = *Z* ^{3}/π^{2}.

Rescaled RFs of TAMSDs of CMs in the discrete reptation model for different values of *Z*. The data are the same as Figure 5 but the observation time *t* is rescaled by the reptation time τ_{rep} = *Z* ^{3}τ_{ l }/π^{2}. All the data points collapse into one master curve except the short time region or the small *Z* data (*Z* = 10, in this case).

Rescaled RFs of TAMSDs of CMs in the discrete reptation model for different values of *Z*. The data are the same as Figure 5 but the observation time *t* is rescaled by the reptation time τ_{rep} = *Z* ^{3}τ_{ l }/π^{2}. All the data points collapse into one master curve except the short time region or the small *Z* data (*Z* = 10, in this case).

Relative fluctuations of TAMSDs of CMs in the slip-spring model for *N* = 80. The dashed line represents a curve proportional to *t* ^{−1/2}.

Relative fluctuations of TAMSDs of CMs in the slip-spring model for *N* = 80. The dashed line represents a curve proportional to *t* ^{−1/2}.

Relative fluctuations of TAMSDs of CMs in the discrete reptation model for *Z* = 40. The dashed line represents a curve proportional to *t* ^{−1/2}.

Relative fluctuations of TAMSDs of CMs in the discrete reptation model for *Z* = 40. The dashed line represents a curve proportional to *t* ^{−1/2}.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content