# The Connection between Biaxial Orientation and Shear Thinning for Quasi-Ideal Rods

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## Abstract

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## 1. Introduction

## 2. Theory

#### 2.1. Smoluchowski Theory

#### 2.2. Doi–Edwards–Kuzuu (DEK) Theory

#### 2.3. Scaling

## 3. Experiments, Materials and Methods

#### 3.1. Measurements and Materials

#### 3.2. Obtaining the Full Orientation Tensor and the Biaxiality

## 4. Results

## 5. Discussion

#### 5.1. Shear Thinning

#### 5.2. Biaxiality

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

ODF | orientational distribution function |

DEK | Doi–Edwards–Kuzuu |

SANS | small angle neutron scattering |

## References

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**Figure 1.**(

**a**) Definition of angles in our scattering geometry as defined by the flow field, dashed lines are guides to the eye; (

**b**) scattering planes visualized on behalf of the Couette cell, red circles indicate moving parts, red arrows show the direction of motion.

**Figure 2.**Porod plot for $c=11$ mg/mL without shear flow. The solid line is a ${q}^{-1}$ fit and the dashed line a ${q}^{-2}$ fit to the data. The q-ranges I and II of analysis are parted by the vertical lines. Inset: comparison of the azimuthal intensity profiles for q-ranges I and II.

**Figure 3.**Corrected viscosity versus the orientational order parameter $\langle {P}_{2}\left(\psi \right)\rangle $. The full lines are calculated by Smoluchowski theory for the indicated concentrations, while the dashed line is based on Doi–Edwards–Kuzuu (DEK) theory for the smallest given concentration, and the dotted thick line is an empirical fit.

**Figure 4.**Zero-shear viscosity scaled by the solvent viscosity as a function of the volume fraction from data shifting. The dotted line is a least squares log-linear fit to the data points, while the other lines represent the different theories.

**Figure 5.**Shear alignment given by the orientational order parameter $\langle {P}_{2}\rangle $, as well as shear thinning given by ${\eta}_{\text{corr}}:=\eta /{\eta}_{0}$ versus the effective Peclet number for the given concentrations. Theoretical curves are given as solid lines, and the dashed line is a least squares fit of the measurement.

**Figure 6.**Projected order parameter as a function of the (effective thickness-) corrected Peclet number for the measured concentrations; the numbers j–k indicate the unit vectors in the measurement plane.

**Figure 7.**(

**a**) Measured largest eigenvalue of S as a function of the corrected Peclet number compared to the Smoluchowski theory, evaluated for two effective thickness values ${d}_{\text{eff}}^{(1-3)}=8.6$ (dotted line) and ${d}_{\text{eff}}^{(1-2)}=12$ (dashed line) at a concentration of 11 mg/mL. Inset: largest eigenvalue compared to Smoluchowski theory for two different rotational diffusion coefficients. Two regimes are marked, separated at ${\tau}_{r}\dot{\gamma}=1$; (

**b**) Measured azimuthal tilt angle θ [t] and largest projected order tensor eigenvalue ${\lambda}_{1}\left(\theta \right)$ [l] as a function of the corrected Peclet number, for two concentrations, compared to the Smoluchowski theory. The solid and dashed lines are theoretical curves for the two given concentrations; (

**c**) Measured biaxiality parameter T (squares) versus corrected Peclet number compared to Smoluchowski theory for the two effective thicknesses ${d}_{\text{eff}}^{(1-3)}=8.6$ (solid line) and ${d}_{\text{eff}}^{(1-2)}=12$ (dashed line).

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**MDPI and ACS Style**

Lang, C.; Kohlbrecher, J.; Porcar, L.; Lettinga, M.P. The Connection between Biaxial Orientation and Shear Thinning for Quasi-Ideal Rods. *Polymers* **2016**, *8*, 291.
https://doi.org/10.3390/polym8080291

**AMA Style**

Lang C, Kohlbrecher J, Porcar L, Lettinga MP. The Connection between Biaxial Orientation and Shear Thinning for Quasi-Ideal Rods. *Polymers*. 2016; 8(8):291.
https://doi.org/10.3390/polym8080291

**Chicago/Turabian Style**

Lang, Christian, Joachim Kohlbrecher, Lionel Porcar, and Minne Paul Lettinga. 2016. "The Connection between Biaxial Orientation and Shear Thinning for Quasi-Ideal Rods" *Polymers* 8, no. 8: 291.
https://doi.org/10.3390/polym8080291