# Rotation Dynamics of Star Block Copolymers under Shear Flow

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model and Simulation Method

#### 2.1.1. Coarse-Grained Model for the Star Block Copolymer

#### 2.1.2. Multiparticle Collision Dynamics and Molecular Dynamics

#### 2.2. Rotational Dynamics

#### 2.2.1. Laboratory Frame

#### 2.2.2. Eckart Frame

#### 2.2.3. Hybrid Frame

#### 2.2.4. Geometrical Approach

## 3. Results and Discussion

#### 3.1. Global Conformation and Dynamics

#### 3.2. Reference Configuration Update

#### 3.3. Angular Momentum and Angular Frequency

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Rotation Frequencies

**Figure A1.**Values of the average magnitudes of the components of the angular velocity, $|{\omega}_{\mu}|$, and the magnitude of the whole vector, $\left|\omega \right|$, for Case 1 as a function of $Wi$ for two different values of ${t}_{\mathrm{Eckart}}$ as indicated in the legend, for Cases 1, 2 and 3.

## Appendix B. Kinetic Energy in the Eckart Frame

## Appendix C. Explicit Calculation of T_{u}

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**Figure 1.**Schematic illustration of the simulation setup, demonstrating the shear (${\hat{\mathbf{e}}}_{1}$), gradient (${\hat{\mathbf{e}}}_{2}$) and vorticity (${\hat{\mathbf{e}}}_{3}$) directions of planar, Couette flow. Yellow, red and blue spheres correspond respectively to the star core, solvophilic (A type) monomers and solvophobic (B type) monomers.

**Figure 2.**Time evolution of the (fixed) reference configuration in Eckart’s frame as seen in the laboratory frame for Case 1 and $Wi=10$.

**Figure 3.**Time evolution of the (fixed) reference configuration in Eckart’s frame as seen in the laboratory frame for Case 1 and $Wi=100$.

**Figure 4.**Time evolution of the (fixed) reference configuration in Eckart’s frame as seen in the laboratory frame for Case 1 and $Wi=400$.

**Figure 5.**Diagonal components of the (average) gyration tensor of an SBC with $f=18$, ${N}_{\mathrm{pol}}=40$, $\lambda =1.0$ (top row) and $\lambda =1.1$ (bottom row). For athermal stars, the scalings ${G}_{11}\sim W{i}^{0.4}$ and ${G}_{22}\sim W{i}^{-0.32}$ are found at high $Wi$.

**Figure 6.**Reduced orientational resistance ${m}_{G}/Wi$ for stars with $f=18$, ${N}_{\mathrm{pol}}=40$ and values of $\lambda $ and $\alpha $ as indicated on the panels. For athermal stars, we find ${m}_{G}/Wi\sim W{i}^{-0.83}$ at small $Wi$, and ${m}_{G}/Wi\sim W{i}^{-0.30}$ at large $Wi$.

**Figure 7.**Representative simulation snapshots displaying the time evolution of the star block-copolymer (SBC) in shear flow for $\{f,\alpha ,\lambda \}=\{12,0.3,1.0\}$ (Case 1), $\{15,0.5,1.1\}$ (Case 2) and $\{18,0.7,1.1\}$ (Case 3). In Case 1, individual arms of the star perform tank-treading motion, while in Case 3, the star tumbles as a whole. Case 2 presents a tank-treading-like motion, but it is performed by both individual and clustered arms. Circles and squares are guides to follow the motion of arms. In all cases, ${N}_{\mathrm{pol}}=40$ and $Wi=100$. In the panels, $\Delta t$ represents the elapsed time from the first configuration in $\tau $ units.

**Figure 8.**Comparison between the values of the kinetic energy for Case 1 evaluated in both the lab and Eckart frames at different Eckart times (Table 2).

**Figure 9.**Comparison between the values of the kinetic energy for Case 2 evaluated in both the lab and Eckart frames at different Eckart’s times (Table 2).

**Figure 10.**Comparison between the values of the kinetic energy for Case 3 evaluated in both the lab and Eckart frames at different Eckart’s times (Table 2).

**Figure 11.**The Coriolis coupling for two different values of ${t}_{\mathrm{Eckart}}$ for Case 1 as a function of $Wi$. For the meaning of the quantities ${E}_{\mathrm{C}}$, ${E}_{\mathrm{C},1}$ and ${E}_{\mathrm{C},2}$, see Equation (31).

**Figure 13.**Left panel: Rotational energy ($\mathrm{LF}\to \frac{1}{2}\mathit{\omega}\xb7\mathbf{J}\xb7\mathit{\omega}$, $\mathrm{EF}\to \frac{1}{2}\mathsf{\Omega}\xb7\hat{\mathbf{J}}\xb7\mathsf{\Omega}$, $\mathrm{HF}\to \frac{1}{2}\mathit{W}\xb7\hat{\mathbf{J}}\xb7\mathit{W}$) for the different frames as a function of the Weissenberg number $Wi$. Right panel: reduced angular frequency for Case 1 ($\mathrm{LF}\to \mathit{\omega}/\dot{\gamma}$, $\mathrm{EF}\to \mathsf{\Omega}/\dot{\gamma}$, $\mathrm{HF}\to \mathit{W}/\dot{\gamma}$, $\mathrm{GA}\to {\mathit{\omega}}_{G}/\dot{\gamma}$) as a function of $Wi$ for different Eckart times.

**Figure 14.**Same as Figure 13, but for Case 2.

**Figure 15.**Same as Figure 13, but for Case 3.

**Figure 16.**The reduced angular frequencies for Case 1 (

**left**) and Case 3 (

**right**) evaluated at the LF, the GA and the EF with the longest Eckart time employed, ${t}_{\mathrm{Eckart}}=8000\phantom{\rule{0.166667em}{0ex}}\tau $.

Abbreviation | Meaning |
---|---|

Case 1 | $\{f,\alpha ,\lambda \}=\{12,0.3,1.0\}$ |

Case 2 | $\{f,\alpha ,\lambda \}=\{15,0.5,1.1\}$ |

Case 3 | $\{f,\alpha ,\lambda \}=\{18,0.7,1.1\}$ |

LF | Laboratory frame |

EF | Eckart frame |

HF | Hybrid frame |

GA | Geometric approximation |

**Table 2.**The various contributions to the total kinetic energy in the laboratory, the Eckart frames and hybrid frame.

Rotational | Vibrational without Angular Momentum | ${\mathit{T}}_{\mathit{u}}$ = Vibrational with Angular Momentum + Coriolis Coupling | |
---|---|---|---|

Laboratory Frame | $\frac{1}{2}\mathit{\omega}\xb7\mathbf{J}\xb7\mathit{\omega}$ | $\frac{M}{2}{\sum}_{k}{\tilde{\mathbf{v}}}_{k}\xb7{\tilde{\mathbf{v}}}_{k}$ | – |

Eckart frame | $\frac{1}{2}\mathsf{\Omega}\xb7\hat{\mathbf{J}}\xb7\mathsf{\Omega}$ | $\frac{M}{2}{\sum}_{k}{\tilde{\mathbf{v}}}_{k}\xb7{\tilde{\mathbf{v}}}_{k}$ | $\frac{M}{2}{\sum}_{k}{\mathbf{u}}_{k}\xb7{\mathbf{u}}_{k}+M{\sum}_{k}(\mathsf{\Omega}\times {\mathbf{c}}_{k})\xb7{\mathbf{u}}_{k}$ |

Hybrid frame | $\frac{1}{2}\mathit{W}\xb7\hat{\mathbf{J}}\xb7\mathit{W}$ | $\frac{M}{2}{\sum}_{k}{\tilde{\mathbf{v}}}_{k}\xb7{\tilde{\mathbf{v}}}_{k}$ | – |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Jaramillo-Cano, D.; Likos, C.N.; Camargo, M. Rotation Dynamics of Star Block Copolymers under Shear Flow. *Polymers* **2018**, *10*, 860.
https://doi.org/10.3390/polym10080860

**AMA Style**

Jaramillo-Cano D, Likos CN, Camargo M. Rotation Dynamics of Star Block Copolymers under Shear Flow. *Polymers*. 2018; 10(8):860.
https://doi.org/10.3390/polym10080860

**Chicago/Turabian Style**

Jaramillo-Cano, Diego, Christos N. Likos, and Manuel Camargo. 2018. "Rotation Dynamics of Star Block Copolymers under Shear Flow" *Polymers* 10, no. 8: 860.
https://doi.org/10.3390/polym10080860