# Shear-Thinning in Oligomer Melts—Molecular Origins and Applications

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## Abstract

**:**

## 1. Introduction

## 2. Microscopic Model and Simulation Techniques

## 3. Shear-Thinning in Oligomer Melts—A Molecular Analysis

## 4. Hybrid Multiscale Method

`convective term`is rewritten in the conservative form and the interface integrals are approximated by means of the Lax–Friedrichs numerical flux

## 5. Two Channel Flows of a Non-Newtonian Oligomer Fluid

## 6. Summary and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Alternative Time-Scale Mapping

**Figure A1.**$\langle {R}_{ee,x}^{2}\rangle /\langle {R}_{ee}^{2}\rangle $ as a function of $\kappa $ at $\dot{\gamma}=0.001$ for melt (line with dots) and single chain simulations (only dots) with the more involved mapping described in the appendix.

## Appendix B. Snapshots

**Figure A2.**Snapshots of melts corresponding to (

**a**) $\kappa =6$ in equilibrium, (

**b**) $\kappa =6$ at $\dot{\gamma}=0.001$ (

**c**) $\kappa =10$ in equilibrium.

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**Figure 1.**(

**a**) Viscosity $\eta $ as a function of shear rate $\dot{\gamma}$ for a melt of flexible oligomer chains ($\kappa =0$) with N = 1, 2, 5, 10 and 15 beads per chain. Corresponding shear viscosities according to the Green–Kubo relation ${\eta}_{GK}$ are shown on the y-axis and displayed as a function of N in the inset. Density $\rho =0.8$ and box dimensions are $10\times 10\times 10{\sigma}^{3}$ for N = 1, 2, 5 and 10 and $15\times 15\times 15{\sigma}^{3}$ for N = 15. (

**b**) $\eta \left(\dot{\gamma}\right)$ for a melt with N = 15 and $\rho =0.8$ and varying stiffnesses. ${\eta}_{GK}$ for $\kappa =0,3,5,7$ and 10 are displayed on the y-axis. The viscosity for monomers at $\dot{\gamma}$ = 0.5 (blue triangle) is also shown for reference. If not displayed explicitly, errors are smaller than symbol sizes. All lines serve as guides to the eye.

**Figure 2.**(

**a**) $\langle {R}_{ee}^{2}\rangle $ as a function of shear rate $\dot{\gamma}$ for stiffnesses $\kappa =0$, $\kappa =5$ and $\kappa =10$ at density $\rho =0.8$. (

**b**) Ratio of the x-component $\langle {R}_{ee,x}^{2}\rangle $ and $\langle {R}_{ee}^{2}\rangle $ as a function of $\dot{\gamma}$ for $\kappa =0$, $\kappa =5$ and $\kappa =10$. The dotted line at the ratio of 1/3 marks the value for an unsheared melt. Results for a single chain in shear flow are shown as dashed lines (with points) in both figures. Values on the y-axis (in (

**a**,

**b**), colour scheme such as in Figure 1b) correspond to equilibrium simulations without shear. As there is no preferred orientation in the bulk, the value for the ratio refers to the largest component. For $\kappa \le 5$, there is no preferred orientation in the bulk. All lines serve as guides to the eye.

**Figure 3.**(

**a**) Probability distributions P(${R}_{ee}^{2}$) for stiffness $\kappa $ = 5 at shear rates $\dot{\gamma}=0,0.01$ and $0.5$. The two peaks of the distribution for $\dot{\gamma}=0.5$ correspond to U-shaped configurations and stretched configuration of individual oligomers, respectively, as indicated by typical snapshots. (

**b**) P(${R}_{ee}^{2}$) for stiffness $\kappa $ = 0 and shear rate $\dot{\gamma}=0$ and $0.5$. Results for a single chain in shear flow are shown as dashed lines in both figures.

**Figure 4.**(

**a**) Viscosity $\eta $ as a function of stiffness $\kappa $ for shear rate $\dot{\gamma}$ = 0.001. Shear viscosity at zero shear rate ${\eta}_{GK}$ are shown as green dots. (

**b**) $\langle {R}_{ee,x}^{2}\rangle /\langle {R}_{ee}^{2}\rangle $ as a function of $\kappa $ at $\dot{\gamma}=0.001$ for melt and single chain simulations (dashed lines). All lines serve as guides to the eye.

**Figure 5.**Shear viscosity of flexible oligomer chains and oligomer chains with stiffness $\kappa =5$ as obtained by non-equilibrium molecular dynamics (open symbols) and fitting by the Carreau–Yasuda rheological fluid model Equation (12) (dashed curves, compare with Figure 1b). Bold symbols on the y-axis represent viscosity values at $\dot{\gamma}=0$ obtained from equilibrium molecular dynamics simulations.

**Figure 6.**Steady-state velocity profiles of the pressure-driven channel flow of oligomer melts consisting of either flexible or semiflexible chains with stiffness $\kappa =5$. Solutions are computed by a hybrid MD-dG method (20b) for various external pressure difference parameters ${P}_{\mathrm{x}}$.

**Figure 7.**Velocity profiles of flexible and semiflexible oligomer chains in comparison with Newtonian flow profiles. The velocity profiles shown in Figure 6 were normalised by the corresponding $max\left(u\right(\xb7,y\left)\right)$.

**Figure 8.**Steady-state viscosity distributions across the channel flow of flexible and semiflexible oligomer melts at various external pressure drops ${P}_{\mathrm{x}}$.

**Figure 9.**(

**a**) Structured quadrilateral mesh used in the Armaly experiment: a cut–out [0…2] × [−1…1] of the full domain [0…10] × [−1…1] is shown. Mesh resolution increases smoothly at $x\approx 1$ and at $y\in \{-1,-0.5,0,0.5,1\}$. Mesh step varies between ${h}_{x}=1/20$ and $1/100$ for the x–direction and between ${h}_{y}=1/20$ and $1/1000$ for the y–direction. (

**b**) Velocity and streamlines of the non-Newtonian flows of flexible, semiflexible chain molecules, and of the Newtonian flow, ${U}_{\mathrm{inlet}}=10$, $\kappa =0$ (

**top**), $\kappa =5$ (

**middle**), $\eta =1$ (

**bottom**).

**Figure 10.**Velocity profiles at $x\in \{1.2,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}6\}$ in Newtonian (blue curves) and non–Newtonian flows of flexible $\kappa =0$ (black circles) and semiflexible $\kappa =5$ (red squares) chain molecules in the Armaly experiment with ${U}_{\mathrm{inlet}}\in \{1,10\}$. Insets: zooming into the region of the secondary flow.

**Figure 11.**Viscosity profiles at $x\in \{1.2,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}6\}$ in non–Newtonian flows of flexible (black circles) and semiflexible (red squares) chain molecules in the Armaly experiment with ${U}_{\mathrm{inlet}}\in \{1,10\}$. Newtonian flow viscosity $\eta =1$ is shown with blue lines.

$\mathit{\kappa}=0$ | ||||

${\mathit{\eta}}_{\mathbf{0}}=\mathbf{7.76}$ | ${\mathit{\eta}}_{\mathbf{\infty}}=\mathbf{1.084}$ | $\overline{\mathit{\eta}}=\mathbf{4.422}$ | ||

${\mathit{P}}_{\mathrm{x}}$ | $\mathit{U}$ | $\mathit{R}{\mathit{e}}_{\mathbf{0}}$ | $\mathit{R}{\mathit{e}}_{\mathbf{\infty}}$ | $\mathit{R}\mathit{e}$ |

0.02 | 3.25 × 10${}^{-4}$ | 4.18 × 10${}^{-5}$ | 2.99 × 10${}^{-4}$ | 7.34 × 10${}^{-5}$ |

0.2 | 3.70 × 10${}^{-3}$ | 4.76 × 10${}^{-4}$ | 3.41 × 10${}^{-3}$ | 8.36 × 10${}^{-4}$ |

1 | 3.29 × 10${}^{-2}$ | 4.24 × 10${}^{-3}$ | 3.04 × 10${}^{-2}$ | 7.45 × 10${}^{-3}$ |

$\mathit{\kappa}\mathbf{=}\mathbf{5}$ | ||||

${\mathit{\eta}}_{\mathbf{0}}\mathbf{=}\mathbf{36.05}$ | ${\mathit{\eta}}_{\infty}\mathbf{=}\mathbf{1.093}$ | $\overline{\mathit{\eta}}\mathbf{=}\mathbf{18.57}$ | ||

${\mathit{P}}_{\mathrm{x}}$ | $\mathit{U}$ | $\mathit{R}{\mathit{e}}_{\mathbf{0}}$ | $\mathit{R}{\mathit{e}}_{\mathbf{\infty}}$ | $\mathit{R}\mathit{e}$ |

0.02 | 7.08 × 10${}^{-5}$ | 1.96 × 10${}^{-6}$ | 6.48 × 10${}^{-5}$ | 3.81 × 10${}^{-6}$ |

0.2 | 3.69 × 10${}^{-3}$ | 1.02 × 10${}^{-4}$ | 3.38 × 10${}^{-3}$ | 1.99 × 10${}^{-4}$ |

1 | 5.49 × 10${}^{-2}$ | 1.52 × 10${}^{-3}$ | 5.03 × 10${}^{-2}$ | 2.96 × 10${}^{-3}$ |

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Datta, R.; Yelash, L.; Schmid, F.; Kummer, F.; Oberlack, M.; Lukáčová-Medvid’ová, M.; Virnau, P.
Shear-Thinning in Oligomer Melts—Molecular Origins and Applications. *Polymers* **2021**, *13*, 2806.
https://doi.org/10.3390/polym13162806

**AMA Style**

Datta R, Yelash L, Schmid F, Kummer F, Oberlack M, Lukáčová-Medvid’ová M, Virnau P.
Shear-Thinning in Oligomer Melts—Molecular Origins and Applications. *Polymers*. 2021; 13(16):2806.
https://doi.org/10.3390/polym13162806

**Chicago/Turabian Style**

Datta, Ranajay, Leonid Yelash, Friederike Schmid, Florian Kummer, Martin Oberlack, Mária Lukáčová-Medvid’ová, and Peter Virnau.
2021. "Shear-Thinning in Oligomer Melts—Molecular Origins and Applications" *Polymers* 13, no. 16: 2806.
https://doi.org/10.3390/polym13162806