# Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity

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## Abstract

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## 1. Introduction

## 2. Characterizing Polymer Entanglement

## 3. Polymeric Materials with Weave-like Entanglements

## 4. Simulation of the Polymers

## 5. Bulk Mechanical Responses

#### 5.1. Complex Modulus

#### 5.2. Conformational Analysis

#### 5.3. Topology and Mechanical Responses

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The Gauss linking integral between two curves measures the degree to which they entangle, it is a real number that varies continuously with the chain coordinates for open chains, and it is an integer topological invariant for closed chains. Similarly, the Writhe (the Gauss linking integral over one curve) measures the degree to which a curve entangles with itself and varies continuously with chain coordinates.

**Figure 2.**We consider polymeric chains entangled with weave-like topologies. The weave0 (w0) denotes case with aligned chains, weaveI (wI) the case with smaller density of orthogonal and non-interlaced chains, weaveII (wII) the case with larger density of orthogonal and non-interlaced chains, and weaveIII (wIII) the case with alternating interlaced chains.

**Figure 3.**We show the entaglements of the polymer chains of wIII. The weave wIII has a topology with alternating interlaced chains. We see the chains in the x direction which alternatingly go over and under chains in the perpendicular y direction. We show one chain in the x-direction (orange curve) that can be seen locally to meet with three chains in the perpendicular y direction.

**Figure 4.**In Lees–Edwards boundary conditions, a monomer that exits the cell in a direction perpendicular to the shear direction will re-enter from the opposite face at position shifted by $\pm L\left(t\right)\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}{L}_{0}$.

**Figure 5.**Polymer Weave Frequency Response: Dynamic Moduli. The Elastic Storage and Viscous Loss Moduli of the infinite chain weaves are shown left and those of the open chain weaves right. The infinite weaves behave like crosslinked polymers with a primarily elastic behavior throughout the range of frequencies simulated. The exponents 1/2 and 1/4 are similar to those in the Rouse model [49]. The open weaves transition from an elastic to a viscous behavior as frequency is varied. The exponents 3/4 and 3/2 indicate the predicted scaling for semi-dilute solutions of semi-flexible chains and for the BEL model respectively [50]. The slope increases with decreasing topological complexity.

**Figure 6.**Polymer Weave Frequency Response: Loss Tangent. The infinite systems behave like crosslinked polymers with a loss tangent less than 1 at all frequencies. The open chains transition from a liquid-like behavior to that of a solid-like behavior as the frequency increases. The inset plot shows the log-log plot for open chains. These results show that the crossover frequency increases with decreasing topological complexity. Similarly, the slope of decrease increases with decreasing topological complexity.

**Figure 7.**Polymer configurations subject to oscillatory shear. Configurations at the end of simulations for the closed chain and open chain systems. In both cases, the chains tend to form bundles. By forming tubes of larger radius the chains decrease their individual deformation. For weave0, the open chains can significantly rearrange their conformations to those of random coils and the system becomes inhomogeneous, disconnected accross the periodic boundary. As the entanglement complexity and density increases, the open chains cannot fully escape their original conformations, forming tubular connected domains (weaveI), lamellar structures (weaveI) and even retain entanglement percolation in three dimensions (weaveIII).

**Figure 8.**Writhe and Loss Tangent of open chains for small frequencies of oscillation (frequencies corresponding to periods $T>6{\tau}_{D}$). We find a linear behavior of the Writhe as a function of the Loss Tangent. The data points corresponding to the different weaves form clusters. As the topological complexity of the weave decreases both the Writhe and Loss Tangent increase.

**Figure 9.**(

**Left**) Writhe and ${G}_{2}/\omega $ for open weaves at small frequencies (log-log plot). The viscosity can be obtained as the limit $\eta ={lim}_{\omega \to 0}{G}_{2}\left(\omega \right)/\omega $. We find an indication that the Writhe decreases with viscosity, while Z shows the opposite relation to viscosity [10]. (

**Right**) Periodic Linking Number as a function of ${G}_{1}$ for open chains for small frequencies (log-log plot). The equilibrium modulus can be obtained as the limit ${G}_{eq}={lim}_{\omega \to 0}{G}_{1}\left(\omega \right)$. We find an indication that ${G}_{eq}$ increases with increasing linking, similar to what was reported for rings in [32].

**Figure 10.**Periodic Linking Number and Loss Tangent of open systems for small frequencies of oscillation (corresponding to periods $T>6{\tau}_{D}$). We find that the responses corresponding to the different weaves form clusters. The Periodic Linking Number increases with weave complexity and decreases with loss tangent.

**Figure 11.**The ratio of the Periodic Linking Number over the Writhe as a function of the loss tangent for open systems at low frequencies. We see that the ratio $\langle |L{K}_{P}|\rangle /\langle \left|Wr\right|\rangle $ decreases with the loss tangent.

**Table 1.**Densities associated with the polymeric weaves shown in Figure 2.

Weave | Topology | Density | MW (open) |
---|---|---|---|

W0 | parallel, non-interlaced | 0.0625 (15 amu/nm${}^{3}$) | 20 ${m}_{0}$ |

WI | orthogonal (non interl.) | 0.1875 (45 amu/nm${}^{3}$) | 20 ${m}_{0}$ |

WII | orthogonal (non interl.) | 0.33 (80 amu/nm${}^{3}$) | 15 ${m}_{0}$ |

WIII | alternating interlaced | 0.35 (84 amu/nm${}^{3}$) | 21–17 ${m}_{0}$ |

**Table 2.**Parameterization for the polymer weave models. These parametrizations (and Table 3), are chosen for relatively stiff polymers exhibiting worm-like chain responses for polymeric solutions and melts.

Parameter | Description | Value |
---|---|---|

$\sigma $ | monomer radius | 1.0 nm |

$\u03f5$ | energy scale | 2.5 amu·nm${}^{2}$/ps${}^{2}$ |

${m}_{0}$ | reference mass | 1 amu |

${w}_{c}$ | energy potential width | $2.5\sigma $ |

m | monomer mass | 240 ${m}_{0}$ |

$\tau $ | LJ-time-scale | $\sigma \sqrt{{m}_{0}/\u03f5}$ = 0.6 ps |

${k}_{B}T$ | thermal energy | 1.0 $\u03f5$ |

$\rho $ | solvent mass density | 39 ${m}_{0}/{\sigma}^{3}$ |

$\mu $ | solvent viscosity | 25 ${m}_{0}/\tau \sigma $ |

$\mathsf{{\rm Y}}$ | drag coefficient | 476 ${m}_{0}/\tau $ |

Parameter | Description | Value |
---|---|---|

${E}_{b}$ | harmonic bonds potential constant | 619.5 amu/ps${}^{2}$ |

b | harmonic bonds rest length | 1.0 nm |

${E}_{\theta}$ | harmonic angle potential constant | 19.8 amu·nm${}^{2}$/ps${}^{2}$ |

${\theta}_{0}$ | harmonic angles’ rest length | 180${}^{\circ}$ |

Parameter | Description | value |
---|---|---|

${\tau}_{A}$ advection time | propagation in fluid | 0.0013 ps |

${\tau}_{D}$ diffusion time | monomer moves a dist. $\sigma $ | 302 ps |

${\tau}_{R}$ Rouse time | ideal chain $N=20$ | 6937 ps |

${\tau}_{0}$ critical time | cross-over reference time | 598 ps |

T | period of oscillation | 6 ps < T < 3600 ps |

t simulation time | longest simulation time | 150 ps < t < 90,000 ps |

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**MDPI and ACS Style**

Panagiotou, E.; Millett, K.C.; Atzberger, P.J.
Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity. *Polymers* **2019**, *11*, 437.
https://doi.org/10.3390/polym11030437

**AMA Style**

Panagiotou E, Millett KC, Atzberger PJ.
Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity. *Polymers*. 2019; 11(3):437.
https://doi.org/10.3390/polym11030437

**Chicago/Turabian Style**

Panagiotou, Eleni, Kenneth C. Millett, and Paul J. Atzberger.
2019. "Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity" *Polymers* 11, no. 3: 437.
https://doi.org/10.3390/polym11030437