# Nonlinear Rheology and Fracture of Disclination Network in Cholesteric Blue Phase III

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Phase Behavior

#### 2.2. Linear Rheology

#### 2.3. Nonlinear Dynamic Behavior

#### 2.4. Nonlinear Relaxation Modulus

#### 2.5. Stress Growth Behavior

## 3. Conclusions

## 4. Materials and Methods

#### 4.1. Material

#### 4.2. Methods

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Temperature dependence of the viscosity at $\dot{\gamma}$ = 1 s${}^{-1}$. Different symbols correspond to the viscosity measured during cooling and heating process. Temperature was swept at $\dot{T}$ = 0.1 ${}^{\xb0}$C/min. Arrows indicate the phase transition temperatures. Polarized microscope images at different temperatures are also shown. Scale bar indicates 100 $\mathsf{\mu}$m.

**Figure 2.**Polarized microscope image at T = 35.1 ${}^{\xb0}$C. White arrows correspond to the optic axes of polarizer and analyzer. Deviation angle of the analyzer is $\varphi $ = ±4${}^{\xb0}$.

**Figure 3.**(

**a**) Dynamic shear moduli as a function of the angular frequency $\omega $ measured at T = 35.1 ${}^{\xb0}$C. The strain amplitude of the oscillatory shear is ${\gamma}_{0}$ = 0.03, which corresponds to the linear viscoelasticity region. (

**b**) Dynamic shear moduli as a function of the strain amplitude ${\gamma}_{0}$. The angular frequency is $\omega $ = 10 s${}^{-1}$. Symbols are the same as those in (

**a**).

**Figure 4.**(

**a**) Complex shear modulus ${G}^{*}$ as a function of the angular frequency $\omega $ measured at T = 35.1 ${}^{\xb0}$C. ${G}^{*}$ obtained at various strain amplitude in the range 0.02 $\le \gamma \le $ 2.0 are compared in the same panel. (

**b**) Nonlinear shear modulus ${G}_{\mathrm{e}}^{*}$ at $\omega $ = 0.1 s${}^{-1}$ is plotted as a function of the strain amplitude ${\gamma}_{0}$. Solid curve is the best fit to the equation, ${G}_{\mathrm{e}}^{*}\simeq {G}_{\mathrm{e}}^{*}\left({\gamma}_{0}\right)/1+{\left(\xi {\gamma}_{0}\right)}^{2}$ with $\xi $ = 0.2.

**Figure 5.**(

**a**) Nonlinear relaxation modulus $G({\gamma}_{0},t)$ at T = 35.1 ${}^{\xb0}$C. Relaxation moduli obtained at various step strains in the range 0.01 $\le \gamma \le $ 2.0 are shown. Solid lines are the best fit to the stretched exponential function, ${G}_{0}$exp(−(t/$\tau $)${}^{\beta}$). (

**b**) Value of the relaxation modulus at t = 1 s is plotted as a function of ${\gamma}_{0}$. (

**c**,

**d**) Relaxation time $\tau $ and value of the stretched exponent $\beta $ obtained from the best fit to the stretched exponential function.

**Figure 6.**(

**a**) Relaxation modulus reduced by damping function $G({\gamma}_{0},t)h{\left({\gamma}_{0}\right)}^{-1}$ obtained in the range of the step strain, 0.02 $\le {\gamma}_{0}\le $ 0.1. Arrow indicates a characteristic time ${\tau}_{\mathrm{k}}$ beyond which nonlinear relaxation moduli is factorized into separate strain and time-dependent functions, i.e., the time–strain separability holds. (

**b**) Nonlinear relaxation modulus reduced by damping function $G({\gamma}_{0},t)h{\left({\gamma}_{0}\right)}^{-1}$ obtained in large step strain, 0.1 < ${\gamma}_{0}\le $ 2. (

**c**) Damping function $h\left({\gamma}_{0}\right)$. Solid line is the best fit to the equation, $h\left({\gamma}_{0}\right)=1/1+\xi {\gamma}_{0}^{a}$ with $\xi $ = 330 and a = 3.

**Figure 7.**(

**a**) Stress growth behavior as a function of time for several shear rates applied at t = 0. Shear rates are $\dot{\gamma}$ = 1, 0.1, 0.01, and 0.001 s${}^{-1}$ from left to right. The dotted lines corresponds to $\sigma \left(t\right)$ = ${G}_{0}\dot{\gamma}t$, where the shear modulus is ${G}_{0}$ = 54 Pa. The dashed lines indicate the linear viscoelastic stress response predicted by BKZ equation (Equation (3)). The solid lines are K-BKZ equation (Equation (2)). Stress reaches its maximum ${\sigma}_{\mathrm{m}}$ at a time ${t}_{\mathrm{m}}$ as shown by an arrow. (

**b**) Normalized shear stress $\sigma /{\sigma}_{\mathrm{m}}$ as a function of normalized strain $\gamma /{\gamma}_{\mathrm{m}}$.

**Figure 8.**(

**a**) The peak shear stress ${\sigma}_{\mathrm{m}}$ as a function of the strain ${\gamma}_{\mathrm{m}}$ at the overshoot point. The slope corresponds to the modulus at the yield point, ${G}_{\mathrm{y}}\simeq $ 42 Pa. (

**b**) Critical stress as a function of applied shear rate. (

**c**) Critical shear strain as a function of applied shear rate. Solid lines show the power law relations.

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**MDPI and ACS Style**

Fujii, S.; Sasaki, Y.; Orihara, H.
Nonlinear Rheology and Fracture of Disclination Network in Cholesteric Blue Phase III. *Fluids* **2018**, *3*, 34.
https://doi.org/10.3390/fluids3020034

**AMA Style**

Fujii S, Sasaki Y, Orihara H.
Nonlinear Rheology and Fracture of Disclination Network in Cholesteric Blue Phase III. *Fluids*. 2018; 3(2):34.
https://doi.org/10.3390/fluids3020034

**Chicago/Turabian Style**

Fujii, Shuji, Yuji Sasaki, and Hiroshi Orihara.
2018. "Nonlinear Rheology and Fracture of Disclination Network in Cholesteric Blue Phase III" *Fluids* 3, no. 2: 34.
https://doi.org/10.3390/fluids3020034