# Simulation of Individual Polymer Chains and Polymer Solutions with Smoothed Dissipative Particle Dynamics

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## Abstract

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## 1. Introduction

## 2. Particle Model of the Polymer Suspension

#### 2.1. Mesoscopic Modelling of the Solvent

#### 2.2. Mechanical Modelling of the Polymer Chain

## 3. Simulation of a Single Polymer

#### 3.1. Static Properties

**Figure 1.**Scaling of the radius of gyration $\u2329{R}_{G}^{2}\u232a$ for chain lengths corresponding to $N\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}5,10,20,40,60,80$ beads. The solid line represents the best fit of the static exponent.

**Figure 2.**Equilibrium static structure factor vs. wave vector magnitude k for polymers with length $N=20,40,60,80$. All the curves collapse on a master line within the region of ${R}_{G}^{-1}<k<{a}_{0}^{-1}$.

**Table 1.**Estimated static exponent from the fit of the linear part of $S\left(k\right)$ within the region of ${R}_{G}^{-1}\phantom{\rule{3.33333pt}{0ex}}<\phantom{\rule{3.33333pt}{0ex}}k\phantom{\rule{3.33333pt}{0ex}}<\phantom{\rule{3.33333pt}{0ex}}{a}_{0}^{-1}$.

N | 20 | 40 | 60 | 80 |
---|---|---|---|---|

ν | 0.59 | 0.57 | 0.62 | 0.61 |

#### 3.2. Diffusion Coefficient

**Figure 3.**Log-log plot of the diffusion coefficient D as a function of the chain length N. The solid line represents the best fit with the theoretical power law.

#### 3.3. Longest Polymer Relaxation Time

**Figure 4.**Log-log plot of the longest relaxation time, computed from the autocorrelation of the radius of gyration, as a function of the chain length $N=10,20,40,60,80$. The solid line is the best fit to determine the static exponent according to the Zimm model.

#### 3.4. Rouse Modes

**Figure 5.**Log-log plot of the relaxation time of Rouse modes ${\tau}_{p}$ vs $N/p$ for $N=5,10,20,40,60,80,100$. The solid line is the best fit to determine the scaling exponent.

**Figure 6.**Log-log plot of the relaxation time of Rouse modes ${\tau}_{p}$ vs ${\left[p\langle {\mathbf{R}}_{p}^{2}\rangle \right]}^{1/2}$ for $N\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}5,10,20,40,60,80,100$. The solid line is the best fit to determine the scaling exponent.

## 4. Polymer Melt and Solution

**Table 2.**Parameters used for the simulation of polymer melts. Here $\Delta x$ is the initial smoothed dissipative particle dynamics method (SDPD) particle spacing and ${c}_{s}$ is sound speed.

$\Delta x$ | ${k}_{B}T$ | ρ | ${c}_{s}$ | η | k | ${R}_{0}$ | f |
---|---|---|---|---|---|---|---|

$2.5\times {10}^{-5}$ | $1.38\times {10}^{-14}$ | 1000 | 0.3 | 0.015 | $5.3\times {10}^{-4}$ | $1.4\Delta x$ | 6.12 |

**Figure 7.**Snapshot of the polymer melt simulation: x is a direction of the body force and y is the direction of the velocity gradient. The visualization was done with the Visual Molecular Dynamics (VMD) program [47].

**Figure 8.**Velocity profiles for solvents and three melts with chains characterized by $N=2,5,25$. Power-law indices p are $0.99,0.96,0.87,0.77$, respectively.

**Figure 10.**The calculated and imposed normalized shear-stress distribution for the melt composed by 25-bead chains.

**Figure 13.**Velocity profile for the melt of N = 25 bead chain for bulk concentration 50%. Power-law index p is $0.90$.

**Figure 14.**Normalized bead density profile for the melt of N = 25 bead chain for bulk concentration 50%.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Litvinov, S.; Xie, Q.; Hu, X.; Adams, N.; Ellero, M.
Simulation of Individual Polymer Chains and Polymer Solutions with Smoothed Dissipative Particle Dynamics. *Fluids* **2016**, *1*, 7.
https://doi.org/10.3390/fluids1010007

**AMA Style**

Litvinov S, Xie Q, Hu X, Adams N, Ellero M.
Simulation of Individual Polymer Chains and Polymer Solutions with Smoothed Dissipative Particle Dynamics. *Fluids*. 2016; 1(1):7.
https://doi.org/10.3390/fluids1010007

**Chicago/Turabian Style**

Litvinov, Sergey, Qingguang Xie, Xiangyu Hu, Nikolaus Adams, and Marco Ellero.
2016. "Simulation of Individual Polymer Chains and Polymer Solutions with Smoothed Dissipative Particle Dynamics" *Fluids* 1, no. 1: 7.
https://doi.org/10.3390/fluids1010007