# Analysis of Elongational Viscosity of Entangled Poly (Propylene Carbonate) Melts by Primitive Chain Network Simulations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Simulations

^{3}, which is sufficiently large to accommodate the examined polymers. The equilibrium simulations were performed for a sufficiently long time, and the duration was at least 10 times longer than the longest relaxation time for each system. For statistics, $G\left(t\right)$ was averaged over 32 independent simulation runs. The obtained $G\left(t\right)$ was converted to storage and loss moduli, ${G}^{\prime}\left(\omega \right)$ and ${G}^{\u2033}\left(\omega \right)$, via the REPTATE software [66]. For uniaxial deformation, following the previous studies [15,38], we equilibrated the system filled in a flat simulation box and stretched it. The dimension of the initial flat box was 4 × 45 × 45, and that at the final elongated state was 506 × 4 × 4. This deformation attains the Hencky strain of 4.8. We acquired the uniaxial viscosity growth function, ${\eta}_{\mathrm{E}}{}^{+}\left(t\right)$, by averaging the stress growth in 8 independent simulation runs for each condition. Figure 1 exhibits a typical snapshot of the longest chain with $Z=88$ during the elongation, with the elongation rate of $\dot{\epsilon}{\tau}_{0}=4.4\times {10}^{-5}$, and the strain of 1.76.

## 3. Results and Discussion

^{−1}for both cases, followed by the steady-state at the large $\dot{\epsilon}$ limit. Although the value of ${\lambda}_{\mathrm{max}}$ is common, the viscosity for the polydisperse case is slightly suppressed by the contribution of low molecular weight components. With the friction change (solid curves), the decrease of viscosity with increasing $\dot{\epsilon}$ is similar for both cases. According to this result, we conclude that the effect of polydispersity on the friction change is not significant for this specific case due to the small polydispersity index.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Typical snapshot of a chain with $Z=88$ involved in Zw48 under elongation with the elongation rate of $\dot{\epsilon}{\tau}_{0}=4.4\times {10}^{-5}$, taken at the elongational strain of 1.76. Thin black lines are the other chains, and thick green lines are segments entangled to the test chain. The yellow frame shows the simulation box for periodic boundary conditions.

**Figure 2.**Linear viscoelasticity at $T=70\xb0\mathrm{C}$ from experiments [31] (shown by symbols and plotted against the bottom and left axes) and simulations (indicated by curves and plotted against the top and right axes) for Zw47 (PPC158k), Zw34 (PPC111k), and Zw21 (PPC69k) from left to right.

**Figure 3.**Viscosity growth curves under uniaxial elongation at $T=70\xb0\mathrm{C}$ for Zw47 (PPC158k, panel (

**a**)), Zw34 (PPC111k, panel (

**b**)), and Zw21 (PPC69k, panel (

**c**)) from top to bottom. Experimental data [32] are shown by circles. Simulation results with and without the change of friction (according to Euqation (2)) are indicated by red solid and black broken curves. Red broken curves show the linear viscoelastic envelopes. The strain rates are $2.8\times {10}^{-1},8.5\times {10}^{-2},2.8\times {10}^{-2},5.6\times {10}^{-3},1.8\times {10}^{-3},5.6\times {10}^{-4},\mathrm{and}1.8\times {10}^{-4}$ s

^{−1}for Zw47 (PPC158k), $3\times {10}^{-3},1\times {10}^{-2},3\times {10}^{-2},1\times {10}^{-1},\mathrm{and}3\times {10}^{-1}$ s

^{−1}for Zw34 (PPC111k), and $4\times {10}^{-3},6.1\times {10}^{-3},4\times {10}^{-2},6.1\times {10}^{-2},4\times {10}^{-1},\mathrm{and}6.1\times {10}^{-1}$ s

^{−1}for Zw21 (PPC69k), respectively.

**Figure 4.**Steady-state elongational viscosity (

**a**) and the friction (

**b**) at $T=70\xb0\mathrm{C}$ as a function of the strain rate for Zw47 (PPC158k, black), Zw34 (PPC111k, red), and Zw21 (PPC69k, blue). Experimental data [32] are shown by unfilled circles. In the top panel (

**a**), simulation results with and without the friction change (according to Equation (2)) are drawn by solid and broken curves, respectively.

**Figure 5.**Simulation results for a polystyrene melt with ${M}_{w}=520\mathrm{k}$ and ${M}_{w}/{M}_{n}=$ 1.3 for linear viscoelasticity (

**a**), elongational growth curve (

**b**), and steady-state elongational viscosity (

**c**) at $T=130\xb0\mathrm{C}$. Circles and curves are experimental and simulation results, respectively. Solid and broken curves in the panel (

**b**) exhibit the simulation results with the friction change and the linear viscoelastic envelope. The strain rates are $2.8\times {10}^{-4}$, $8.4\times {10}^{-4}$, and $2.8\times {10}^{-3}$ s

^{−1}, from left to right. In the panel (

**c**), solid and broken curves correspond to the simulation with and without the friction change.

**Figure 6.**Comparison between Zw47 (the polydispersity index of 1.3, black curves) and Z47 (monodisperse, red curves) for linear viscoelasticity (

**a**) and steady-state elongational viscosity plotted against strain rate (

**b**). The experimental data for PPC158 k are also shown by symbols.

Code | $\mathit{Z}$ | ${\mathit{\phi}}_{\mathit{N}}{}^{\mathrm{\#}}$ | ${\mathit{Z}}_{\mathit{w}}$ | ${\mathit{Z}}_{\mathit{w}}/{\mathit{Z}}_{\mathit{n}}$ | ${{\mathit{M}}_{\mathit{w}}}^{+}(\mathbf{kg}/\mathbf{mol})$ | ${\mathit{M}}_{\mathit{w}}/{{\mathit{M}}_{\mathit{w}}}^{+}$ |
---|---|---|---|---|---|---|

Zw47 (PPC158k *) | 11 | 0.09 | 47.3 | 1.31 | 158 | 1.30 |

22 | 0.41 | |||||

44 | 0.41 | |||||

88 | 0.09 | |||||

Zw34 (PPC111k *) | 8 | 0.09 | 33.7 | 1.30 | 111 | 1.30 |

16 | 0.41 | |||||

32 | 0.41 | |||||

62 | 0.09 | |||||

Zw21 (PPC69k *) | 4 | 0.1 | 20.6 | 1.41 | 68.8 | 1.43 |

8 | 0.4 | |||||

18 | 0.4 | |||||

38 | 0.1 | |||||

Z47 | 47 | 1 | 47 | 1 | - | - |

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**MDPI and ACS Style**

Masubuchi, Y.; Yang, L.; Uneyama, T.; Doi, Y.
Analysis of Elongational Viscosity of Entangled Poly (Propylene Carbonate) Melts by Primitive Chain Network Simulations. *Polymers* **2022**, *14*, 741.
https://doi.org/10.3390/polym14040741

**AMA Style**

Masubuchi Y, Yang L, Uneyama T, Doi Y.
Analysis of Elongational Viscosity of Entangled Poly (Propylene Carbonate) Melts by Primitive Chain Network Simulations. *Polymers*. 2022; 14(4):741.
https://doi.org/10.3390/polym14040741

**Chicago/Turabian Style**

Masubuchi, Yuichi, Lixin Yang, Takashi Uneyama, and Yuya Doi.
2022. "Analysis of Elongational Viscosity of Entangled Poly (Propylene Carbonate) Melts by Primitive Chain Network Simulations" *Polymers* 14, no. 4: 741.
https://doi.org/10.3390/polym14040741