^{1,a)}, Yuichi Masubuchi

^{2}, Michael C. Böhm

^{1}and Florian Müller-Plathe

^{1}

### Abstract

We report a multi-chain approach for dissipative particle dynamics where the uncrossability constraints of polymer chains are mimicked by temporary cross-links, so-called slip-springs. The conformational statistics of the chains are not affected by the introduction of slip-springs. Dynamical properties such as mean square displacements, diffusion coefficient, and longest relaxation time are in good agreement with the results of reptation theory. According to our analysis, the present formalism is 500 times faster and requires 7 times fewer beads than conventional generic polymer models employing Newtonian dynamics and excluded-volume potentials.

The present work has been supported by the Deutsche Forschungsgemeinschaft (DFG) within the priority program SPP1369.

I. INTRODUCTION

II. MODEL

III. SIMULATION DETAILS

IV. RESULTS

A. Chain structure

B. Dynamics

C. Computational efficiency

V. CONCLUSIONS

### Key Topics

- Monte Carlo methods
- 31.0
- Polymers
- 18.0
- Reptation
- 14.0
- Relaxation times
- 9.0
- Diffusion
- 6.0

## Figures

Measures of the chain structure: (a) Radius of gyration plotted against *N*. The line with slope +1 is a guide to the eye to show agreement with the scaling law. (b) Mean square internal distance d(s) for the chain length *N* = 100 scaled by the bead distance s. Empty symbols illustrate the system without slip-springs, i.e., classical DPD simulation. Filled symbols show the behavior of the system with slip-springs.

Measures of the chain structure: (a) Radius of gyration plotted against *N*. The line with slope +1 is a guide to the eye to show agreement with the scaling law. (b) Mean square internal distance d(s) for the chain length *N* = 100 scaled by the bead distance s. Empty symbols illustrate the system without slip-springs, i.e., classical DPD simulation. Filled symbols show the behavior of the system with slip-springs.

Mean square displacement of the central bead *g* _{1,mid }(*t*) scaled by *t* ^{1/2} for chains without (dashed curves) and with slip-springs (solid curves). The chain length *N* varies from 10 to 100. The line is a guide to the eye and shows the *t* ^{1} regime.

Mean square displacement of the central bead *g* _{1,mid }(*t*) scaled by *t* ^{1/2} for chains without (dashed curves) and with slip-springs (solid curves). The chain length *N* varies from 10 to 100. The line is a guide to the eye and shows the *t* ^{1} regime.

Mean square displacement of the central bead *g* _{1,mid }(*t*) scaled by *t* ^{1/4} for chains without (dashed curves) and with slip-springs (solid curves). The chain length *N* varies from 10 to 100. The two lines are guides to the eye and show the *t* ^{1} and *t* ^{1/2} regime, respectively.

Mean square displacement of the central bead *g* _{1,mid }(*t*) scaled by *t* ^{1/4} for chains without (dashed curves) and with slip-springs (solid curves). The chain length *N* varies from 10 to 100. The two lines are guides to the eye and show the *t* ^{1} and *t* ^{1/2} regime, respectively.

Anisotropy coefficient *A* _{ cm }(*t*) for chains without (empty symbols) and with slip-springs (filled symbols). The chain length *N* = 100 is considered either completely (circles) or without the last 20 beads at both chain ends (squares).

Anisotropy coefficient *A* _{ cm }(*t*) for chains without (empty symbols) and with slip-springs (filled symbols). The chain length *N* = 100 is considered either completely (circles) or without the last 20 beads at both chain ends (squares).

Zero shear relaxation modulus *G*(*t*) for chains without (dashed curves) and with slip-springs (solid curves). The chain length *N* ranges from 40 to 100. The straight line is a guide to the eye and demonstrates the Rouse scaling with *t* ^{−1/2}.

Zero shear relaxation modulus *G*(*t*) for chains without (dashed curves) and with slip-springs (solid curves). The chain length *N* ranges from 40 to 100. The straight line is a guide to the eye and demonstrates the Rouse scaling with *t* ^{−1/2}.

Zero shear relaxation modulus *G*(*t*) for chains without (dashed curves) and with slip-springs (solid curves) multiplied by *t* ^{1/2} to point out Rouse behavior as horizontal line. The chain length *N* ranges from 40 to 100.

Zero shear relaxation modulus *G*(*t*) for chains without (dashed curves) and with slip-springs (solid curves) multiplied by *t* ^{1/2} to point out Rouse behavior as horizontal line. The chain length *N* ranges from 40 to 100.

Diffusion coefficient of the center of mass *D* _{ com } scaled by 6*N* as a function of chain length for chains without (empty symbols) and with slip-springs (filled symbols). The two lines are guides to the eye and demonstrate the diffusion power law for reptation dynamics (−2) and the experimentally observed scaling behavior (−2.3).

Diffusion coefficient of the center of mass *D* _{ com } scaled by 6*N* as a function of chain length for chains without (empty symbols) and with slip-springs (filled symbols). The two lines are guides to the eye and demonstrate the diffusion power law for reptation dynamics (−2) and the experimentally observed scaling behavior (−2.3).

Rotational relaxation time *τ* _{ rot } scaled by *N* ^{−2} against the chain length *N* for chains without (filled symbols) and with slip-springs (empty symbols). The two lines are guides to the eye and show the power 3 scaling law for reptation dynamics and power 3.4 scaling behavior observed from experiments.

Rotational relaxation time *τ* _{ rot } scaled by *N* ^{−2} against the chain length *N* for chains without (filled symbols) and with slip-springs (empty symbols). The two lines are guides to the eye and show the power 3 scaling law for reptation dynamics and power 3.4 scaling behavior observed from experiments.

Relaxation modulus *G*(*t*) in comparison with KG simulations to evaluate computational efficiency of the DPD slip-spring model. The KG chains are shown by symbols and their length *N* = 50, 100 and 200 from left to right. The DPD slip-spring chains are shown by solid curves with the length *N* = 8, 15, and 30 from left to right.

Relaxation modulus *G*(*t*) in comparison with KG simulations to evaluate computational efficiency of the DPD slip-spring model. The KG chains are shown by symbols and their length *N* = 50, 100 and 200 from left to right. The DPD slip-spring chains are shown by solid curves with the length *N* = 8, 15, and 30 from left to right.

Mean square displacement of the central bead *g* _{1,mid }(*t*) scaled by *t* ^{1/4} for systems with different MC sequence lengths. The DPD sequence length is same for all systems, i.e., n_{DPD} = 500. Hereby, n_{MC} = 500 is the system on which the dynamical analysis from the Results section was carried out.

Mean square displacement of the central bead *g* _{1,mid }(*t*) scaled by *t* ^{1/4} for systems with different MC sequence lengths. The DPD sequence length is same for all systems, i.e., n_{DPD} = 500. Hereby, n_{MC} = 500 is the system on which the dynamical analysis from the Results section was carried out.

Mean square displacement of the central bead *g* _{1,mid }(*t*) scaled by *t* ^{1/4} for systems with different DPD and MC sequence lengths. The ratio n_{DPD}/n_{MC} is unity for all systems. Hereby, n_{DPD} = n_{MC} = 500 refers to the system on which the dynamical analysis from the Results section was carried out.

Mean square displacement of the central bead *g* _{1,mid }(*t*) scaled by *t* ^{1/4} for systems with different DPD and MC sequence lengths. The ratio n_{DPD}/n_{MC} is unity for all systems. Hereby, n_{DPD} = n_{MC} = 500 refers to the system on which the dynamical analysis from the Results section was carried out.

Mean square displacement of the central bead *g* _{1,mid }(*t*) scaled by *t* ^{1/2} to amplify the onset of the disengagement time *τ* _{ d } for systems with different DPD and MC sequence lengths. The ratio n_{DPD}/n_{MC} is unity for all systems. Hereby, n_{DPD} = n_{MC} = 500 refers to the system on which the dynamical analysis from the Results section was carried out.

Mean square displacement of the central bead *g* _{1,mid }(*t*) scaled by *t* ^{1/2} to amplify the onset of the disengagement time *τ* _{ d } for systems with different DPD and MC sequence lengths. The ratio n_{DPD}/n_{MC} is unity for all systems. Hereby, n_{DPD} = n_{MC} = 500 refers to the system on which the dynamical analysis from the Results section was carried out.

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