^{1}, Hong Liu

^{1,a)}and Zhong-Yuan Lu

^{1,b)}

### Abstract

We introduce a highly coarse-grained model to simulate the entangled polymer melts. In this model, a polymer chain is taken as a single coarse-grained particle, and the creation and annihilation of entanglements are regarded as stochastic events in proper time intervals according to certain rules and possibilities. We build the relationship between the probability of appearance of an entanglement between any pair of neighboring chains at a given time interval and the rate of variation of entanglements which describes the concurrence of birth and death of entanglements. The probability of disappearance of entanglements is tuned to keep the total entanglement number around the target value. This useful model can reflect many characteristics of entanglements and macroscopic properties of polymer melts. As an illustration, we apply this model to simulate the polyethylene melt of C_{1000}H_{2002} at 450 K and further validate this model by comparing to experimental data and other simulation results.

This work is subsidized by the National Basic Research Program of China (973 Program, 2012CB821500), and supported by National Science Foundation of China (Grant Nos. 50930001, 21025416, and 20974040).

I. INTRODUCTION

II. MODEL AND METHOD

A. The way of entanglement appearance

B. The way of entanglement disappearance

C. Simulation method

III. RESULTS AND DISCUSSION

A. Simulation setup

B. Entanglement characteristics

C. Dynamics

IV. CONCLUSIONS

### Key Topics

- Polymers
- 29.0
- Diffusion
- 15.0
- Shear rate dependent viscosity
- 15.0
- Polymer melts
- 14.0
- Molecular dynamics
- 10.0

## Figures

The topological constrains (entanglements) are created when the centers of mass of two adjacent chains move apart or together. The entanglement interactions slow the relative movement of center of mass of the chains and even draw them back.

The topological constrains (entanglements) are created when the centers of mass of two adjacent chains move apart or together. The entanglement interactions slow the relative movement of center of mass of the chains and even draw them back.

The radial distribution function *g*(*r*) for both systems with and without entanglements. The circles are for the entangled system and the solid line is for the system without entanglements.

The radial distribution function *g*(*r*) for both systems with and without entanglements. The circles are for the entangled system and the solid line is for the system without entanglements.

Distribution of the number of entanglements *Z* per chain computed by our model in equilibrium state. Symbols: simulation data; line: fitted curve.

Distribution of the number of entanglements *Z* per chain computed by our model in equilibrium state. Symbols: simulation data; line: fitted curve.

The *P* _{ t } of the entanglements as a function of *t* _{ age }.

The *P* _{ t } of the entanglements as a function of *t* _{ age }.

The distribution of the deviation of the distance of centers of mass of two entangled chains from the initial states with different strengths of entanglement interactions. The *k** is the harmonic force constant in reduced unit.

The distribution of the deviation of the distance of centers of mass of two entangled chains from the initial states with different strengths of entanglement interactions. The *k** is the harmonic force constant in reduced unit.

The mean square displacement *g*(*t*) of the system with (scatted cycles) and without (full line) entanglements.

The mean square displacement *g*(*t*) of the system with (scatted cycles) and without (full line) entanglements.

The stress auto-correlation *G*(*t*) of entangled PE system.

The stress auto-correlation *G*(*t*) of entangled PE system.

Shear viscosity as a function of shear rate. Linear fitting the data and extrapolating to zero shear rate gives the zero shear viscosity.

Shear viscosity as a function of shear rate. Linear fitting the data and extrapolating to zero shear rate gives the zero shear viscosity.

## Tables

The input parameters for the model of *C* _{1000} *H* _{2002} at 450 K. The mass density ρ_{ M } is taken from Ref. 36, the isothermal compressibility κ_{ T } from Ref. 38, and the radius of gyration *R* _{ G } from Ref. 29. *r* _{ c } is the cutoff radius. Δ*t* and Δ*t** are the integration time step in real units and in reduced units, respectively. ρ* is the reduced number density, and *k** is the entanglement interaction force constant in reduced units. *Pc* ^{0} and *Pa* ^{0} are the probabilities of creation and annihilation of entanglements, respectively. Parameters *A* and *B* are calculated from Eq. (12) for MDPD.

The input parameters for the model of *C* _{1000} *H* _{2002} at 450 K. The mass density ρ_{ M } is taken from Ref. 36, the isothermal compressibility κ_{ T } from Ref. 38, and the radius of gyration *R* _{ G } from Ref. 29. *r* _{ c } is the cutoff radius. Δ*t* and Δ*t** are the integration time step in real units and in reduced units, respectively. ρ* is the reduced number density, and *k** is the entanglement interaction force constant in reduced units. *Pc* ^{0} and *Pa* ^{0} are the probabilities of creation and annihilation of entanglements, respectively. Parameters *A* and *B* are calculated from Eq. (12) for MDPD.

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