# 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

## References

- Ianniruberto, G.; Marrucci, G.; Masubuchi, Y. Melts of Linear Polymers in Fast Flows. Macromolecules
**2020**, 53, 5023–5033. [Google Scholar] [CrossRef] - Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Oxford University Press: Oxford, UK, 1986. [Google Scholar]
- Marrucci, G.; Grizzuti, N. Fast flows of concentrated polymers: Predictions of the tube model on chain stretching. Gazz. Chmica Ital.
**1988**, 118, 179–185. [Google Scholar] - Bhattacharjee, P.K.; Nguyen, D.A.; McKinley, G.H.; Sridhar, T. Extensional stress growth and stress relaxation in entangled polymer solutions. J. Rheol.
**2003**, 47, 269. [Google Scholar] [CrossRef] - Bhattacharjee, P.K.; Oberhauser, J.P.; McKinley, G.H.; Leal, L.G.; Sridhar, T. Extensional rheometry of entangled solutions. Macromolecules
**2002**, 35, 10131–10148. [Google Scholar] [CrossRef][Green Version] - Huang, Q.; Hengeller, L.; Alvarez, N.J.; Hassager, O. Bridging the Gap between Polymer Melts and Solutions in Extensional Rheology. Macromolecules
**2015**, 48, 4158–4163. [Google Scholar] [CrossRef][Green Version] - Huang, Q.; Mednova, O.; Rasmussen, H.K.; Alvarez, N.J.; Skov, A.L.; Almdal, K.; Hassager, O. Concentrated polymer solutions are different from melts: Role of entanglement molecular weight. Macromolecules
**2013**, 46, 5026–5035. [Google Scholar] [CrossRef] - Sridhar, T.; Acharya, M.; Nguyen, D.A.; Bhattacharjee, P.K. On the Extensional Rheology of Polymer Melts and Concentrated Solutions. Macromolecules
**2014**, 47, 379–386. [Google Scholar] [CrossRef] - Bach, A.; Almdal, K.; Rasmussen, H.K.; Hassager, O. Elongational viscosity of narrow molar mass distribution polystyrene. Macromolecules
**2003**, 36, 5174–5179. [Google Scholar] [CrossRef] - Nielsen, J.K.; Rasmussen, H.K.; Hassager, O.; McKinley, G.H. Elongational viscosity of monodisperse and bidisperse polystyrene melts. J. Rheol.
**2006**, 50, 453–476. [Google Scholar] [CrossRef][Green Version] - Wingstrand, S.L.; Alvarez, N.J.; Huang, Q.; Hassager, O. Linear and Nonlinear Universality in the Rheology of Polymer Melts and Solutions. Phys. Rev. Lett.
**2015**, 115, 1–5. [Google Scholar] [CrossRef] - Wagner, M.H.; Kheirandish, S.; Koyama, K.; Nishioka, A.; Minegishi, A.; Takahashi, T. Modeling strain hardening of polydisperse polystyrene melts by molecular stress function theory. Rheol. Acta
**2005**, 44, 235–243. [Google Scholar] [CrossRef] - Stephanou, P.S.; Kröger, M. From intermediate anisotropic to isotropic friction at large strain rates to account for viscosity thickening in polymer solutions. J. Chem. Phys.
**2018**, 148, 184903. [Google Scholar] [CrossRef] - Ianniruberto, G.; Brasiello, A.; Marrucci, G. Simulations of fast shear flows of PS oligomers confirm monomeric friction reduction in fast elongational flows of monodisperse PS melts as indicated by rheooptical data. Macromolecules
**2012**, 45, 8058–8066. [Google Scholar] [CrossRef] - Yaoita, T.; Isaki, T.; Masubuchi, Y.; Watanabe, H.; Ianniruberto, G.; Marrucci, G. Primitive chain network simulation of elongational flows of entangled linear chains: Stretch/orientation-induced reduction of monomeric friction. Macromolecules
**2012**, 45, 2773–2782. [Google Scholar] [CrossRef] - Masubuchi, Y.; Matsumiya, Y.; Watanabe, H. Test of Orientation/Stretch-Induced Reduction of Friction via Primitive Chain Network Simulations for Polystyrene, Polyisoprene, and Poly(n-butyl acrylate). Macromolecules
**2014**, 47, 6768–6775. [Google Scholar] [CrossRef] - Takeda, K.; Sukumaran, S.K.; Sugimoto, M.; Koyama, K.; Masubuchi, Y. Primitive chain network simulations for elongational viscosity of bidisperse polystyrene melts. Adv. Modeling Simul. Eng. Sci.
**2015**, 2, 11. [Google Scholar] [CrossRef] - Masubuchi, Y.; Matsumiya, Y.; Watanabe, H.; Marrucci, G.; Ianniruberto, G. Primitive chain network simulations for Pom-Pom polymers in uniaxial elongational flows. Macromolecules
**2014**, 47, 3511–3519. [Google Scholar] [CrossRef] - Masubuchi, Y.; Ianniruberto, G.; Marrucci, G. Primitive Chain Network Simulations of Entangled Melts of Symmetric and Asymmetric Star Polymers in Uniaxial Elongational Flows. J. Soc. Rheol. Jpn.
**2021**, 3, 171–178. [Google Scholar] [CrossRef] - Yaoita, T.; Masubuchi, Y.; Watanabe, H. Concept of Stretch/Orientation-Induced Friction Reduction Tested with a Simple Molecular Constitutive Equation. Nihon Reoroji Gakkaishi
**2014**, 42, 207–213. [Google Scholar] [CrossRef][Green Version] - Desai, P.S.; Larson, R.G. Constitutive model that shows extension thickening for entangled solutions and extension thinning for melts. J. Rheol.
**2014**, 58, 255–279. [Google Scholar] [CrossRef] - Ianniruberto, G. Extensional Flows of Solutions of Entangled Polymers Confirm Reduction of Friction Coefficient. Macromolecules
**2015**, 48, 6306–6312. [Google Scholar] [CrossRef] - Costanzo, S.; Huang, Q.; Ianniruberto, G.; Marrucci, G.; Hassager, O.; Vlassopoulos, D. Shear and Extensional Rheology of Polystyrene Melts and Solutions with the Same Number of Entanglements. Macromolecules
**2016**, 49, 3925–3935. [Google Scholar] [CrossRef][Green Version] - Park, G.W.; Ianniruberto, G. Flow-Induced Nematic Interaction and Friction Reduction Successfully Describe PS Melt and Solution Data in Extension Startup and Relaxation. Macromolecules
**2017**, 50, 4787–4796. [Google Scholar] [CrossRef] - Matsumiya, Y.; Watanabe, H.; Masubuchi, Y.; Huang, Q.; Hassager, O. Nonlinear Elongational Rheology of Unentangled Polystyrene and Poly(p-tert-butylstyrene) Melts. Macromolecules
**2018**, 51, 9710–9729. [Google Scholar] [CrossRef][Green Version] - Morelly, S.L.; Palmese, L.; Watanabe, H.; Alvarez, N.J. Effect of Finite Extensibility on Nonlinear Extensional Rheology of Polymer Melts. Macromolecules
**2019**, 52, 915–922. [Google Scholar] [CrossRef] - Masubuchi, Y.; Yaoita, T.; Matsumiya, Y.; Watanabe, H.; Ianniruberto, G.; Marrucci, G. Stretch/orientation Induced Acceleration in Stress Relaxation in Coarse-grained Molecular Dynamics Simulations. Nihon Reoroji Gakkaishi
**2013**, 41, 35–37. [Google Scholar] [CrossRef][Green Version] - Ianniruberto, G.; Marrucci, G. Molecular Dynamics Reveals a Dramatic Drop of the Friction Coefficient in Fast Flows of Polymer Melts. Macromolecules
**2020**, 53, 2627–2633. [Google Scholar] [CrossRef] - O’connor, T.C.; Hopkins, A.; Robbins, M.O. Stress Relaxation in Highly Oriented Melts of Entangled Polymers. Macromolecules
**2019**, 52, 8540–8550. [Google Scholar] [CrossRef] - Kida, T.; Doi, Y.; Tanaka, R.; Uneyama, T.; Shiono, T.; Masubuchi, Y. Rheological properties of linear and short-chain branched polyethylene with nearly monodispersed molecular weight distribution. Rheol. Acta
**2021**, 60, 511–519. [Google Scholar] [CrossRef] - Yang, L.; Uneyama, T.; Masubuchi, Y.; Doi, Y. Linear Rheological Properties of Poly (Propylene Carbonate) with Different Molecular Weights. Nihon Reoroji Gakkaishi
**2021**, 49, 267–274. [Google Scholar] [CrossRef] - Yang, L.; Uneyama, T.; Masubuchi, Y.; Doi, Y. Nonlinear Shear and Elongational Rheology of Poly(Propylene Carbonate). Nihon Reoroji Gakkaishi
**2022**, 50, 127–135. [Google Scholar] [CrossRef] - Masubuchi, Y.; Takimoto, J.-I.; Koyama, K.; Ianniruberto, G.; Marrucci, G.; Greco, F. Brownian simulations of a network of reptating primitive chains. J. Chem. Phys.
**2001**, 115, 4387–4394. [Google Scholar] [CrossRef] - Masubuchi, Y.; Ianniruberto, G.; Greco, F.; Marrucci, G. Entanglement molecular weight and frequency response of sliplink networks. J. Chem. Phys.
**2003**, 119, 6925–6930. [Google Scholar] [CrossRef] - Masubuchi, Y.; Ianniruberto, G.; Greco, F.; Marrucci, G. Molecular simulations of the long-time behaviour of entangled polymeric liquids by the primitive chain network model. Model. Simul. Mater. Sci. Eng.
**2004**, 12, S91–S100. [Google Scholar] [CrossRef] - Masubuchi, Y.; Ianniruberto, G.; Greco, F.; Marrucci, G. Quantitative comparison of primitive chain network simulations with literature data of linear viscoelasticity for polymer melts. J. Non-Newton. Fluid Mech.
**2008**, 149, 87–92. [Google Scholar] [CrossRef] - Masubuchi, Y. PASTA and NAPLES: Rheology Simulator. In Computer Simulation of Polymeric Materials; Springer: Singapore, 2016; pp. 101–127. ISBN 9789811008153. [Google Scholar]
- Yaoita, T.; Isaki, T.; Masubuchi, Y.; Watanabe, H.; Ianniruberto, G.; Marrucci, G. Primitive chain network simulation of elongational flows of entangled linear chains: Role of finite chain extensibility. Macromolecules
**2011**, 44, 9675–9682. [Google Scholar] [CrossRef] - Takeda, K.; Sukumaran, S.K.S.K.; Sugimoto, M.; Koyama, K.; Masubuchi, Y. Test of the Stretch/Orientation-Induced Reduction of Friction for Biaxial Elongational Flow via Primitive Chain Network Simulation. Nihon Reoroji Gakkaishi
**2015**, 43, 63–69. [Google Scholar] [CrossRef][Green Version] - Bhattacharjee, P.K.; Nguyen, D.A.; Masubuchi, Y.; Sridhar, T. Extensional Step Strain Rate Experiments on an Entangled Polymer Solution. Macromolecules
**2017**, 50, 386–395. [Google Scholar] [CrossRef] - Masubuchi, Y. Contraction of Entangled Polymers After Large Step Shear Deformations in Slip-Link Simulations. Polymers
**2019**, 11, 370. [Google Scholar] [CrossRef][Green Version] - Takeda, K.; Masubuchi, Y.; Sugimoto, M.; Koyama, K.; Sukumaran, S.K. Simulations of Startup Planar Elongation of an Entangled Polymer Melt. Nihon Reoroji Gakkaishi
**2020**, 48, 43–48. [Google Scholar] [CrossRef][Green Version] - Harmandaris, V.A.; Mavrantzas, V.G.; Theodorou, D.N. Atomistic molecular dynamics simulation of stress relaxation upon cessation of steady-state uniaxial elongational flow. Macromolecules
**2000**, 33, 8062–8076. [Google Scholar] [CrossRef] - Stephanou, P.S.; Mavrantzas, V.G. Quantitative predictions of the linear viscoelastic properties of entangled polyethylene and polybutadiene melts via modified versions of modern tube models on the basis of atomistic simulation data. J. Non-Newton. Fluid Mech.
**2013**, 200, 111–130. [Google Scholar] [CrossRef] - Behbahani, A.F.; Schneider, L.; Rissanou, A.; Chazirakis, A.; Bačová, P.; Jana, P.K.; Li, W.; Doxastakis, M.; Polińska, P.; Burkhart, C.; et al. Dynamics and Rheology of Polymer Melts via Hierarchical Atomistic, Coarse-Grained, and Slip-Spring Simulations. Macromolecules
**2021**, 54, 2740–2762. [Google Scholar] [CrossRef] - Spyriouni, T.; Tzoumanekas, C.; Theodorou, D.; Müller-Plathe, F.; Milano, G. Coarse-Grained and Reverse-Mapped United-Atom Simulations of Long-Chain Atactic Polystyrene Melts: Structure, Thermodynamic Properties, Chain Conformation, and Entanglements. Macromolecules
**2007**, 40, 3876–3885. [Google Scholar] [CrossRef] - Masubuchi, Y.; Uneyama, T.; Watanabe, H.; Ianniruberto, G.; Greco, F.; Marrucci, G. Structure of entangled polymer network from primitive chain network simulations. J. Chem. Phys.
**2010**, 132, 134902. [Google Scholar] [CrossRef][Green Version] - Masubuchi, Y.; Kida, T.; Doi, Y.; Uneyama, T. Radial Distribution Functions of Entanglements in Primitive Chain Network Simulations. Nihon Reoroji Gakkaishi
**2021**, 49, 337–345. [Google Scholar] [CrossRef] - Masubuchi, Y. Simulating the Flow of Entangled Polymers. Annu. Rev. Chem. Biomol. Eng.
**2014**, 5, 11–33. [Google Scholar] [CrossRef] - Masubuchi, Y. Molecular modeling for polymer rheology. In Reference Module in Materials Science and Materials Engineering; Elsevier: Amsterdam, The Netherlands, 2016; pp. 1–7. ISBN 9780128035818. [Google Scholar]
- Masubuchi, Y.; Doi, Y.; Uneyama, T. Entanglement Molecular Weight. Nihon Reoroji Gakkaishi
**2020**, 48, 177–183. [Google Scholar] [CrossRef] - Uneyama, T.; Masubuchi, Y. Detailed balance condition and effective free energy in the primitive chain network model. J. Chem. Phys.
**2011**, 135, 184904. [Google Scholar] [CrossRef] - Uneyama, T.; Masubuchi, Y. Plateau Moduli of Several Single-Chain Slip-Link and Slip-Spring Models. Macromolecules
**2021**, 54, 1338–1353. [Google Scholar] [CrossRef] - Masubuchi, Y.; Uneyama, T. Comparison among multi-chain models for entangled polymer dynamics. Soft Matter
**2018**, 14, 5986–5994. [Google Scholar] [CrossRef] - Masubuchi, Y.; Watanabe, H.; Ianniruberto, G.; Greco, F.; Marrucci, G. Comparison among Slip-Link Simulations of Bidisperse Linear Polymer Melts. Macromolecules
**2008**, 41, 8275–8280. [Google Scholar] [CrossRef] - Masubuchi, Y.; Yaoita, T.; Matsumiya, Y.; Watanabe, H. Primitive chain network simulations for asymmetric star polymers. J. Chem. Phys.
**2011**, 134, 194905. [Google Scholar] [CrossRef][Green Version] - Masubuchi, Y.; Matsumiya, Y.; Watanabe, H.; Shiromoto, S.; Tsutsubuchi, M.; Togawa, Y. Primitive chain network simulations for comb-branched polymer under step shear deformations. Rheol. Acta
**2012**, 51, 1–8. [Google Scholar] [CrossRef][Green Version] - Masubuchi, Y. Multichain Slip-Spring Simulations for Branch Polymers. Macromolecules
**2018**, 51, 10184–10193. [Google Scholar] [CrossRef] - Masubuchi, Y.; Ianniruberto, G.; Marrucci, G. Primitive chain network simulations for H-polymers under fast shear. Soft Matter
**2020**, 16, 1056–1065. [Google Scholar] [CrossRef] - Furuichi, K.; Nonomura, C.; Masubuchi, Y.; Watanabe, H.; Ianniruberto, G.; Greco, F.; Marrucci, G. Entangled polymer orientation and stretch under large step shear deformations in primitive chain network simulations. Rheol. Acta
**2008**, 47, 591–599. [Google Scholar] [CrossRef] - Furuichi, K.; Nonomura, C.; Masubuchi, Y.; Ianniruberto, G.; Greco, F.; Marrucci, G. Primitive Chain Network Simulations of Damping Functions for Shear, Uniaxial, Biaxial and Planar Deformations. Nihon Reoroji Gakkaishi
**2007**, 35, 73–77. [Google Scholar] [CrossRef][Green Version] - Furuichi, K.; Nonomura, C.; Masubuchi, Y.; Watanabe, H. Chain contraction and nonlinear stress damping in primitive chain network simulations. J. Chem. Phys.
**2010**, 133, 174902. [Google Scholar] [CrossRef][Green Version] - Masubuchi, Y.; Watanabe, H. Origin of stress overshoot under start-up shear in primitive chain network simulation. ACS Macro Lett.
**2014**, 3, 1183–1186. [Google Scholar] [CrossRef] - Masubuchi, Y.; Ianniruberto, G.; Marrucci, G. Stress Undershoot of Entangled Polymers under Fast Startup Shear Flows in Primitive Chain Network Simulations. Nihon Reoroji Gakkaishi
**2018**, 46, 23–28. [Google Scholar] [CrossRef][Green Version] - Peterlin, A. Gradient Dependence of Intrinsic Viscosity of Freely Flexible Linear Macromolecules. J. Chem. Phys.
**1960**, 33, 1799. [Google Scholar] [CrossRef] - Boudara, V.A.H.; Read, D.J.; Ramírez, J. Reptate rheology software: Toolkit for the analysis of theories and experiments. J. Rheol.
**2020**, 64, 709–722. [Google Scholar] [CrossRef][Green Version] - Yang, L. Linear and Nonlinear Rheological Properties of Poly(Propylene Carbonate); Nagoya University: Nagoya, Japan, 2022. [Google Scholar]
- Wagner, M.H.; Rolon-Garrido, V.H. Nonlinear rheology of linear polymer melts: Modeling chain stretch by interchain tube pressure and Rouse time. Korea-Aust. Rheol. J.
**2009**, 21, 203–211. [Google Scholar]

**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