# Molecular Dynamics Simulations for Resolving Scaling Laws of Polyethylene Melts

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## Abstract

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## 1. Introduction

## 2. Methodology

## 3. Results and Discussion

#### 3.1. Static Properties

**R**is the end-to-end vector, ${\mathit{r}}_{1}$ and ${\mathit{r}}_{N}$ are the coordinates of the chain ends, and ${\mathit{r}}_{\mathrm{c}.\mathrm{m}.}$ is the center-of-mass coordinate of the chain. In Flory’s theory, $\langle {R}^{2}\rangle $ and $\langle {R}_{\mathrm{G}}^{2}\rangle $ scale as M. Figure 1 shows the results for (a) $\langle {R}^{2}\rangle $–M, (b) $\langle {R}_{\mathrm{G}}^{2}\rangle $–M scalings, and (c) the ratio $\langle {R}^{2}\rangle /\langle {R}_{\mathrm{G}}^{2}\rangle $ with respect to M. In general, universal polymer behavior is observed for $N>{N}_{\mathrm{e}}$, where ${N}_{\mathrm{e}}$ is the critical length that indicates onset of chain entanglement [1,2]. In the UA PE model, ${N}_{\mathrm{e}}\sim 80$ was estimated from the primitive pass analysis [56,57,58]. Thus, curve fitting was done for the $\langle {R}^{2}\rangle $–M and $\langle {R}_{\mathrm{G}}^{2}\rangle $–M at $M>{M}_{\mathrm{e}}$, where ${M}_{\mathrm{e}}$ corresponds to ${N}_{\mathrm{e}}$. $\langle {R}^{2}\rangle $ and $\langle {R}_{\mathrm{G}}^{2}\rangle $ scale with ${M}^{1.052}$ and ${M}^{1.121}$, respectively. These differ by 5.2% and 12%, respectively, from values expected for ideal chains. The discrepancy was observed for short-chain conditions, indicating non-Gaussian statics. The ratio $\langle {R}^{2}\rangle /\langle {R}_{\mathrm{G}}^{2}\rangle $ deviates from the behavior of an ideal chain. The slow convergence to the ideal value ($\langle {R}^{2}\rangle /\langle {R}_{\mathrm{G}}^{2}\rangle =6$) is observed with increasing M. This can be problematic when PE chains are expected to satisfy the typical polymer behavior (i.e., static universality).

#### 3.2. Dynamic Properties

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Oxford University Press: Oxford, UK, 1988; Volume 73. [Google Scholar]
- Ferry, J.D. Viscoelastic Properties of Polymers; John Wiley & Sons: Hoboken, NJ, USA, 1980. [Google Scholar]
- Masubuchi, Y. Simulating the flow of entangled polymers. Annu. Rev. Chem. Biomol. Eng.
**2014**, 5, 11–33. [Google Scholar] [CrossRef] [PubMed] - Baaden, M.; Marrink, S.J. Coarse-grain modelling of protein–protein interactions. Curr. Opin. Struct. Biol.
**2013**, 23, 878–886. [Google Scholar] [CrossRef] [PubMed] - Brini, E.; Algaer, E.A.; Ganguly, P.; Li, C.; Rodríguez-Ropero, F.; van der Vegt, N.F. Systematic coarse-graining methods for soft matter simulations—A review. Soft Matter
**2013**, 9, 2108–2119. [Google Scholar] [CrossRef] - Everaers, R.; Sukumaran, S.K.; Grest, G.S.; Svaneborg, C.; Sivasubramanian, A.; Kremer, K. Rheology and microscopic topology of entangled polymeric liquids. Science
**2004**, 303, 823–826. [Google Scholar] [CrossRef] [PubMed] - Gay, J.; Berne, B. Modification of the overlap potential to mimic a linear site–site potential. J. Chem. Phys.
**1981**, 74, 3316–3319. [Google Scholar] [CrossRef] - Groot, R.D.; Warren, P.B. Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation. J. Chem. Phys.
**1997**, 107, 4423. [Google Scholar] [CrossRef] - Jury, S.; Bladon, P.; Cates, M.; Krishna, S.; Hagen, M.; Ruddock, N.; Warren, P. Simulation of amphiphilic mesophases using dissipative particle dynamics. Phys. Chem. Chem. Phys.
**1999**, 1, 2051–2056. [Google Scholar] [CrossRef] - Karimi-Varzaneh, H.A.; van der Vegt, N.F.; Müller-Plathe, F.; Carbone, P. How good are coarse-grained polymer models? A comparison for atactic polystyrene. ChemPhysChem
**2012**, 13, 3428–3439. [Google Scholar] [CrossRef] [PubMed] - Kremer, K.; Grest, G.S. Dynamics of entangled linear polymer melts: A molecular-dynamics simulation. J. Chem. Phys.
**1990**, 92, 5057. [Google Scholar] [CrossRef] - Riniker, S.; Allison, J.R.; van Gunsteren, W.F. On developing coarse-grained models for biomolecular simulation: A review. Phys. Chem. Chem. Phys.
**2012**, 14, 12423–12430. [Google Scholar] [CrossRef] [PubMed] - Barducci, A.; Bonomi, M.; Parrinello, M. Metadynamics. Wiley Interdiscip. Rev.
**2011**, 1, 826–843. [Google Scholar] [CrossRef] - Baumgärtner, A.; Binder, K.; Hansen, J.P.; Kalos, M.; Kehr, K.; Landau, D.; Levesque, D.; Müller-Krumbhaar, H.; Rebbi, C.; Saito, Y.; et al. Applications of the Monte Carlo Method in Statistical Physics; Springer Science & Business Media: Berlin, Germany, 2013; Volume 36. [Google Scholar]
- Binder, K. Monte Carlo and Molecular Dynamics Simulations in Polymer Science; Oxford University Press: Oxford, UK, 1995. [Google Scholar]
- Rapaport, D.C. The Art of Molecular Dynamics Simulation; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Baig, C.; Mavrantzas, V.G.; Kröger, M. Flow effects on melt structure and entanglement network of linear polymers: Results from a nonequilibrium molecular dynamics simulation study of a polyethylene melt in steady shear. Macromolecules
**2010**, 43, 6886–6902. [Google Scholar] - Barrat, J.L.; Baschnagel, J.; Lyulin, A. Molecular dynamics simulations of glassy polymers. Soft Matter
**2010**, 6, 3430–3446. [Google Scholar] [CrossRef] - Chung, H.S.; Piana-Agostinetti, S.; Shaw, D.E.; Eaton, W.A. Structural origin of slow diffusion in protein folding. Science
**2015**, 349, 1504–1510. [Google Scholar] [CrossRef] [PubMed] - Do, C.; Lunkenheimer, P.; Diddens, D.; Götz, M.; Weiß, M.; Loidl, A.; Sun, X.G.; Allgaier, J.; Ohl, M. Li+ transport in poly (ethylene oxide) based electrolytes: Neutron scattering, dielectric spectroscopy, and molecular dynamics simulations. Phys. Rev. Lett.
**2013**, 111, 018301. [Google Scholar] [CrossRef] [PubMed] - Hossain, D.; Tschopp, M.; Ward, D.; Bouvard, J.; Wang, P.; Horstemeyer, M. Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene. Polymer
**2010**, 51, 6071–6083. [Google Scholar] [CrossRef] - Hur, K.; Jeong, C.; Winkler, R.G.; Lacevic, N.; Gee, R.H.; Yoon, D.Y. Chain dynamics of ring and linear polyethylene melts from molecular dynamics simulations. Macromolecules
**2011**, 44, 2311–2315. [Google Scholar] [CrossRef] - Mitchell, J.S.; Harris, S.A. Thermodynamics of writhe in DNA minicircles from molecular dynamics simulations. Phys. Rev. Lett.
**2013**, 110, 148105. [Google Scholar] [CrossRef] - Ndoro, T.V.; Voyiatzis, E.; Ghanbari, A.; Theodorou, D.N.; Böhm, M.C.; Müller-Plathe, F. Interface of grafted and ungrafted silica nanoparticles with a polystyrene matrix: Atomistic molecular dynamics simulations. Macromolecules
**2011**, 44, 2316–2327. [Google Scholar] [CrossRef] - Stephanou, P.S.; Baig, C.; Tsolou, G.; Mavrantzas, V.G.; Kröger, M. Quantifying chain reptation in entangled polymer melts: Topological and dynamical mapping of atomistic simulation results onto the tube model. J. Chem. Phys.
**2010**, 132, 124904. [Google Scholar] [CrossRef] [PubMed] - Harmandaris, V.A.; Kremer, K. Dynamics of polystyrene melts through hierarchical multiscale simulations. Macromolecules
**2009**, 42, 791–802. [Google Scholar] [CrossRef] - Pant, P.K.; Han, J.; Smith, G.D.; Boyd, R.H. A molecular dynamics simulation of polyethylene. J. Chem. Phys.
**1993**, 99, 597–604. [Google Scholar] [CrossRef] - Jorgensen, W.L.; Madura, J.D.; Swenson, C.J. Optimized intermolecular potential functions for liquid hydrocarbons. J. Am. Chem. Soc.
**1984**, 106, 6638–6646. [Google Scholar] [CrossRef] - Martin, M.G.; Siepmann, J.I. Transferable potentials for phase equilibria. 1. United-atom description of n-alkanes. J. Phys. Chem. B
**1998**, 102, 2569–2577. [Google Scholar] [CrossRef] - Boyd, R.H.; Gee, R.H.; Han, J.; Jin, Y. Conformational dynamics in bulk polyethylene: A molecular dynamics simulation study. J. Chem. Phys.
**1994**, 101, 788–797. [Google Scholar] [CrossRef] - Harmandaris, V.A.; Mavrantzas, V.G.; Theodorou, D.N. Atomistic molecular dynamics simulation of polydisperse linear polyethylene melts. Macromolecules
**1998**, 31, 7934–7943. [Google Scholar] [CrossRef] - Jin, Y.; Boyd, R.H. Subglass chain dynamics and relaxation in polyethylene: A molecular dynamics simulation study. J. Chem. Phys.
**1998**, 108, 9912–9923. [Google Scholar] [CrossRef] - Kavassalis, T.; Sundararajan, P. A molecular-dynamics study of polyethylene crystallization. Macromolecules
**1993**, 26, 4144–4150. [Google Scholar] [CrossRef] - Moore, J.; Cui, S.; Cochran, H.; Cummings, P. A molecular dynamics study of a short-chain polyethylene melt.: I. steady-state shear. J. Non-Newton. Fluid Mech.
**2000**, 93, 83–99. [Google Scholar] [CrossRef] - Paul, W.; Smith, G.; Yoon, D.Y.; Farago, B.; Rathgeber, S.; Zirkel, A.; Willner, L.; Richter, D. Chain motion in an unentangled polyethylene melt: A critical test of the rouse model by molecular dynamics simulations and neutron spin echo spectroscopy. Phys. Rev. Lett.
**1998**, 80, 2346. [Google Scholar] [CrossRef] - Ramos, J.; Vega, J.F.; Theodorou, D.N.; Martinez-Salazar, J. Entanglement relaxation time in polyethylene: Simulation versus experimental data. Macromolecules
**2008**, 41, 2959–2962. [Google Scholar] [CrossRef] - Rissanou, A.N.; Power, A.J.; Harmandaris, V. Structural and dynamical properties of polyethylene/graphene nanocomposites through molecular dynamics simulations. Polymers
**2015**, 7, 390–417. [Google Scholar] [CrossRef] - Zhang, Y.; Zhuang, X.; Muthu, J.; Mabrouki, T.; Fontaine, M.; Gong, Y.; Rabczuk, T. Load transfer of graphene/carbon nanotube/polyethylene hybrid nanocomposite by molecular dynamics simulation. Compos. Part B Eng.
**2014**, 63, 27–33. [Google Scholar] [CrossRef] - Harmandaris, V.A.; Daoulas, K.C.; Mavrantzas, V.G. Molecular dynamics simulation of a polymer melt/solid interface: Local dynamics and chain mobility in a thin film of polyethylene melt adsorbed on graphite. Macromolecules
**2005**, 38, 5796–5809. [Google Scholar] [CrossRef] - Hu, M.; Keblinski, P.; Schelling, P.K. Kapitza conductance of silicon–amorphous polyethylene interfaces by molecular dynamics simulations. Phys. Rev. B
**2009**, 79, 104305. [Google Scholar] [CrossRef] - Taylor, D.; Strawhecker, K.; Shanholtz, E.; Sorescu, D.; Sausa, R. Investigations of the intermolecular forces between RDX and polyethylene by force—Distance spectroscopy and molecular dynamics simulations. J. Phys. Chem. A
**2014**, 118, 5083–5097. [Google Scholar] [CrossRef] [PubMed] - Hur, K.; Winkler, R.G.; Yoon, D.Y. Comparison of ring and linear polyethylene from molecular dynamics simulations. Macromolecules
**2006**, 39, 3975–3977. [Google Scholar] [CrossRef] - Yi, P.; Locker, C.R.; Rutledge, G.C. Molecular dynamics simulation of homogeneous crystal nucleation in polyethylene. Macromolecules
**2013**, 46, 4723–4733. [Google Scholar] [CrossRef] - Henry, A.; Chen, G. High thermal conductivity of single polyethylene chains using molecular dynamics simulations. Phys. Rev. Lett.
**2008**, 101, 235502. [Google Scholar] [CrossRef] [PubMed] - Henry, A.; Chen, G. Anomalous heat conduction in polyethylene chains: Theory and molecular dynamics simulations. Phys. Rev. B
**2009**, 79, 144305. [Google Scholar] [CrossRef] - Kim, J.M.; Locker, R.; Rutledge, G.C. Plastic deformation of semicrystalline polyethylene under extension, compression, and shear using molecular dynamics simulation. Macromolecules
**2014**, 47, 2515–2528. [Google Scholar] [CrossRef] - Lavine, M.S.; Waheed, N.; Rutledge, G.C. Molecular dynamics simulation of orientation and crystallization of polyethylene during uniaxial extension. Polymer
**2003**, 44, 1771–1779. [Google Scholar] [CrossRef] - Yeh, I.C.; Andzelm, J.W.; Rutledge, G.C. Mechanical and structural characterization of semicrystalline polyethylene under tensile deformation by molecular dynamics simulations. Macromolecules
**2015**, 48, 4228–4239. [Google Scholar] [CrossRef] - Vu-Bac, N.; Lahmer, T.; Keitel, H.; Zhao, J.; Zhuang, X.; Rabczuk, T. Stochastic predictions of bulk properties of amorphous polyethylene based on molecular dynamics simulations. Mech. Mater.
**2014**, 68, 70–84. [Google Scholar] [CrossRef] - Hess, B.; Bekker, H.; Berendsen, H.J.; Fraaije, J.G. LINCS: A linear constraint solver for molecular simulations. J. Comput. Chem.
**1997**, 18, 1463–1472. [Google Scholar] [CrossRef] - Pronk, S.; Páll, S.; Schulz, R.; Larsson, P.; Bjelkmar, P.; Apostolov, R.; Shirts, M.R.; Smith, J.C.; Kasson, P.M.; van der Spoel, D.; et al. GROMACS 4.5: A high-throughput and highly parallel open source molecular simulation toolkit. Bioinformatics
**2013**, 29, 845–854. [Google Scholar] [CrossRef] [PubMed] - Hoover, W.G. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A
**1985**, 31, 1695–1697. [Google Scholar] [CrossRef] - Nosé, S. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys.
**1984**, 81, 511–519. [Google Scholar] [CrossRef] - Hockney, R.W. The potential calculation and some applications. Methods Comput. Phys.
**1970**, 9, 135–211. [Google Scholar] - Flory, P.J. The configuration of real polymer chains. J. Chem. Phys.
**1949**, 17, 303–310. [Google Scholar] [CrossRef] - Hoy, R.S.; Foteinopoulou, K.; Kröger, M. Topological analysis of polymeric melts: Chain-length effects and fast-converging estimators for entanglement length. Phys. Rev. E
**2009**, 80, 031803. [Google Scholar] [CrossRef] [PubMed] - Kröger, M. Shortest multiple disconnected path for the analysis of entanglements in two-and three-dimensional polymeric systems. Comput. Phys. Commun.
**2005**, 168, 209–232. [Google Scholar] [CrossRef] - Shanbhag, S.; Kröger, M. Primitive path networks generated by annealing and geometrical methods: Insights into differences. Macromolecules
**2007**, 40, 2897–2903. [Google Scholar] [CrossRef] - Rouse, P.E., Jr. A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys.
**1953**, 21, 1272–1280. [Google Scholar] [CrossRef] - Likhtman, A.E.; Sukumaran, S.K.; Ramirez, J. Linear viscoelasticity from molecular dynamics simulation of entangled polymers. Macromolecules
**2007**, 40, 6748–6757. [Google Scholar] [CrossRef]

**Figure 1.**Results for (

**a**) $\langle {R}^{2}\rangle $–M; (

**b**) $\langle {R}_{\mathrm{G}}^{2}\rangle $–M scalings; and (

**c**) the ratio $\langle {R}^{2}\rangle /\langle {R}_{\mathrm{G}}^{2}\rangle $ with respect to M. Fitting curves for data at $M>{M}_{\mathrm{e}}$ are also plotted. $\langle {R}^{2}\rangle $ and $\langle {R}_{\mathrm{G}}^{2}\rangle $ scale with ${M}^{1.052}$ and ${M}^{1.121}$, respectively. These differ by 5.2% and 12%, respectively, from values expected for ideal chains. The ratio $\langle {R}^{2}\rangle /\langle {R}_{\mathrm{G}}^{2}\rangle $ deviates from ideal chain behavior. The slow convergence to the ideal value ($\langle {R}^{2}\rangle /\langle {R}_{\mathrm{G}}^{2}\rangle =6$) is observed with increasing M.

**Figure 2.**Results for $S\left(q\right)$. The unique $S\left(q\right)$ shape is observed for $q>2.0\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}/\mathrm{nm}$; however, the fractal scattering is clearly different from that expected for an ideal chain. The fractal scattering of $S\left(q\right)$ at $2.0\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}/\mathrm{nm}<q<10\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}/\mathrm{nm}$ was estimated to be ${q}^{-1.342}$ ($\nu =0.7452$), which differs by 33% from the expected value.

**Figure 4.**Semi-log plot of $C\left(t\right)$. For the range $0.1<C\left(t\right)<1/e$, $C\left(t\right)$ have linear slopes, irrespective of chain length.

**Figure 5.**${\tau}_{\mathrm{R}}$–M scaling law. For $M<{M}_{\mathrm{e}}$, ${\tau}_{\mathrm{R}}$ scales with ${M}^{2.1}$, and is 5% different from the scaling exponent predicted from the Rouse theory. For $M<{M}_{\mathrm{e}}$, ${\tau}_{\mathrm{R}}$ scales with ${M}^{2.7}$ and is 10% different from the scaling exponent predicted from the reptation theory.

**Figure 6.**Results for ${g}_{1}\left(t\right)$. The expected shape of ${g}_{1}\left(t\right)$ from Equation (10) for $M=2807\phantom{\rule{0.166667em}{0ex}}\mathrm{g}/\mathrm{mol}$ is also plotted. The results of ${g}_{1}\left(t\right)$ at $M=2807\phantom{\rule{0.166667em}{0ex}}\mathrm{g}/\mathrm{mol}$ roughly have the same scaling-law, as expected. However, the threshold values for Equation (10) are unclear from the mean-square displacement results.

**Figure 7.**Results for the D–M scaling law. For $M<{M}_{\mathrm{e}}$, ${\tau}_{\mathrm{R}}$ scales with ${M}^{-1.4}$, while the expected value is ∼${M}^{-1}$. For $M<{M}_{\mathrm{e}}$, ${\tau}_{\mathrm{R}}$ scales with ${M}^{-2.1}$, which is 5% different from the scaling exponent predicted from the reptation theory.

**Figure 8.**(

**a**) log-log plot of $G\left(t\right)$; and (

**b**) semi-log plot of $G\left(t\right){t}^{1/2}$. For $M=2106$ and $2807\phantom{\rule{0.166667em}{0ex}}\mathrm{g}/\mathrm{mol}$, peaks that indicate entanglement were observed at long times. Deviation from the Rouse theory that indicates onset of entanglement was observed at $10\phantom{\rule{0.166667em}{0ex}}\mathrm{ps}$. The Rouse behavior can only be seen over the short-time range (5–$10\phantom{\rule{0.166667em}{0ex}}\mathrm{ps}$). For $M=1405\phantom{\rule{0.166667em}{0ex}}\mathrm{g}/\mathrm{mol}$, the tendency is roughly the same; however, the entanglement is weak (i.e., the peak is small). For $M=703.4$ and $983.9\phantom{\rule{0.166667em}{0ex}}\mathrm{g}/\mathrm{mol}$, Rouse behavior is observed at 2–$100\phantom{\rule{0.166667em}{0ex}}\mathrm{ps}$.

**Table 1.**United-atom polyethylene systems studied in the present work ($\rho =0.65\phantom{\rule{0.166667em}{0ex}}\mathrm{g}/\mathrm{mol}$ and T = 500 $\mathrm{K}$).

M (g/mol) | No. of chains | Simulation time (ns) | No. of initial structures |
---|---|---|---|

422.8 | 1000 | 100 | 3 |

703.4 | 600 | 100 | 3 |

983.9 | 428 | 100 | 3 |

1405 | 300 | 500 | 6 |

2106 | 200 | 500 | 6 |

2807 | 150 | 800 | 6 |

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**MDPI and ACS Style**

Takahashi, K.Z.; Nishimura, R.; Yasuoka, K.; Masubuchi, Y.
Molecular Dynamics Simulations for Resolving Scaling Laws of Polyethylene Melts. *Polymers* **2017**, *9*, 24.
https://doi.org/10.3390/polym9010024

**AMA Style**

Takahashi KZ, Nishimura R, Yasuoka K, Masubuchi Y.
Molecular Dynamics Simulations for Resolving Scaling Laws of Polyethylene Melts. *Polymers*. 2017; 9(1):24.
https://doi.org/10.3390/polym9010024

**Chicago/Turabian Style**

Takahashi, Kazuaki Z., Ryuto Nishimura, Kenji Yasuoka, and Yuichi Masubuchi.
2017. "Molecular Dynamics Simulations for Resolving Scaling Laws of Polyethylene Melts" *Polymers* 9, no. 1: 24.
https://doi.org/10.3390/polym9010024