# 物理代写|光学代写Optics代考|ELEC-E5730

## 物理代写|光学代写Optics代考|General Stress Tensor for Nematic Liquid Crystals

The general theoretical framework for describing the hydrodynamics of liquid crystals has been developed principally by Leslie [16] and Ericksen [17]. Their approaches account for the fact that the stress tensor depends not only on the velocity gradients but also on the orientation and rotation of the director. Accordingly, the stress tensor is given by
$$\sigma_{\alpha \beta}=\alpha_1 n_\gamma n_\delta A_{\gamma \delta} n_\alpha n_\beta+\alpha_2 n_\alpha n_\beta+\alpha_3 n_\beta n_\alpha+\alpha_4 A_{\alpha \beta}+\alpha_5 n_\gamma A_{\gamma \beta}+\alpha_6 n_\beta n_\gamma A_{\gamma \alpha},$$
where the $A_{\alpha \beta}$ sare defined by
$$A_{a \beta}=\frac{1}{2}\left[\frac{\partial v_\beta}{\partial x_\alpha}+\frac{\partial v_\alpha}{\partial x_\beta}\right] .$$
Note that all the other terms on the right-hand side of Eq. (3.59) involve the director orientation, except the fourth term, $\alpha_4 A_{\alpha \beta}$. This is the same term as that for an isotropic fluid (cf. Eq. [3.57]), that is, $\alpha_4=2 \eta$.

Therefore, in this formalism, we have six so-called Leslie coefficients, $\alpha_1, \alpha_2, \ldots$, $\alpha_6$, which have the dimension of viscosity coefficients. It was shown by Parodi $[18]$ that
$$\alpha_2+\alpha_3=\alpha_6-\alpha_5$$
and so there are really five independent coefficients.
In the next few sections, we will study exemplary cases of director axis orientation and deformation, and we will show how these Leslie coefficients are related to other commonly used viscosity coefficients.

## 物理代写|光学代写Optics代考|Flows with Fixed Director Axis Orientation

Consider here the simplest case of flows in which the director axis orientation is held fixed. This may be achieved by a strong externally applied magnetic field (see Figure 3.11), where the magnetic field is along the direction $\hat{n}$. Consider the case of shear flow, where the velocity is in the z-direction, and the velocity gradient is along the $x$-direction. This process could occur, for example, in liquid crystals confined by two parallel plates in the $y$-z plane.

In terms of the orientation of the director axis, there are three distinct possibilities involving three corresponding viscosity coefficients:

1. $\eta_1: \hat{n}$ is parallel to the velocity gradient, that is, along the $x$-axis $\left(\theta=90^{\circ}, \phi=0^{\circ}\right)$.
2. $\eta_2: \hat{n}$ is parallel to the flow velocity, that is, along the $z$-axis and lies in the shear plane $x-z\left(\theta=0^{\circ}, \phi=0^{\circ}\right)$.
3. $\eta_3: \hat{n}$ is perpendicular to the shear plane, that is, along the $y$-axis $\left(\theta=0^{\circ}, \phi\right.$ $\left.=90^{\circ}\right)$.

These three configurations have been investigated by Miesowicz [19], and the $\eta \mathrm{s}$ are known as Miesowicz coefficients. In the original paper, as well as in the treatment by deGennes [3], the definitions of $\eta_1$ and $\eta_3$ are interchanged. In deGennes notation, in terms of $\eta_a . \eta_b$, and $\eta_c$. we have $\eta_a=\eta_1, \eta_b=\eta_2$, and $\eta_c=\eta_3$. The notation used here is attributed to Helfrich [6], which is now the conventional one.

To obtain the relations between $\eta_{1,2,3}$ and the Leslie coefficients $\alpha_{1,2, \ldots, 6}$, one could evaluate the stress tensor $\sigma_{\alpha \beta}$ and the shear rate $A_{\alpha \beta}$ for various director orientations and flow and velocity gradient directions. From these considerations, the following relationships are obtained [3]:

\begin{aligned} &\eta_1=\frac{1}{2}\left(\alpha_4+\alpha_5-\alpha_2\right) \ &\eta_2=\frac{1}{2}\left(\alpha_3+\alpha_4+\alpha_6\right) \ &\eta_3=\frac{1}{2} \alpha_4 \end{aligned}
In the shear plane $x-z$, the general effective viscosity coefficient is actually more correctly expressed in the form [20]
$$\eta_{\mathrm{eff}}=\eta_1+\eta_2 \cos ^2 \theta+\eta_2$$
in order to account for angular velocity gradients. The coefficient $\eta_{1,2}$ is related to the Leslie coefficient $\alpha_1$ by
$$\eta_{1,2}=\alpha_1 .$$

## 物理代写|光学代写Optics代考|General Stress Tensor for Nematic Liquid Crystals

$$\sigma_{\alpha \beta}=\alpha_1 n_\gamma n_\delta A_{\gamma \delta} n_\alpha n_\beta+\alpha_2 n_\alpha n_\beta+\alpha_3 n_\beta n_\alpha+\alpha_4 A_{\alpha \beta}+\alpha_5 n_\gamma A_{\gamma \beta}+\alpha_6 n_\beta n_\gamma A_{\gamma \alpha},$$

$$A_{a \beta}=\frac{1}{2}\left[\frac{\partial v_\beta}{\partial x_\alpha}+\frac{\partial v_\alpha}{\partial x_\beta}\right] .$$

$$\alpha_2+\alpha_3=\alpha_6-\alpha_5$$

## 物理代写|光学代写Optics代考|Flows with Fixed Director Axis Orientation

3.11），其中磁场沿方向 $\hat{n}$. 考虑萠切流的情况，其中速度在 z 方向，速度梯度沿 $x$-方向。例如，这个过程 可能发生在由两个平行板限制的液晶中。 $y-z$ 平面。

1. $\eta_1: \hat{n}$ 平行于速度梯度，即沿 $x$-轴 $\left(\theta=90^{\circ}, \phi=0^{\circ}\right)$.
2. $\eta_2: \hat{n}$ 平行于流速，即沿 $z$-轴并且位于剪切平面内 $x-z\left(\theta=0^{\circ}, \phi=0^{\circ}\right)$.
3. $\eta_3: \hat{n}$ 垂直于剪切面，即沿 $y$-轴 $\left(\theta=0^{\circ}, \phi=90^{\circ}\right)$.
Miesowicz [19] 研究了这三种配置，并且 $\eta$ s称为 Miesowicz 系数。在原始论文以及 deGennes [3] 的处理 中，定义 $\eta_1$ 和 $\eta_3$ 被互换。在 deGennes 表示法中，根据 $\eta_a . \eta_b$ ，和 $\eta_c$. 我们有 $\eta_a=\eta_1, \eta_b=\eta_2$ ，和 $\eta_c=\eta_3$. 这里使用的符号归功于 Helfrich [6]，它现在是传统的符号。
获得之间的关系 $\eta_{1,2,3}$ 和莱斯利系数 $\alpha_{1,2, \ldots, 6}$, 可以评估应力张量 $\sigma_{\alpha \beta}$ 和剪切速率 $A_{\alpha \beta}$ 适用于各种导向器方向 以及流动和速度梯度方向。从这些考虑，得到以下关系 [3]:
$$\eta_1=\frac{1}{2}\left(\alpha_4+\alpha_5-\alpha_2\right) \quad \eta_2=\frac{1}{2}\left(\alpha_3+\alpha_4+\alpha_6\right) \eta_3=\frac{1}{2} \alpha_4$$
在盼切平面 $x-z$ ，一般有效粘度系数实际上更正确地表示为 [20]
$$\eta_{\text {eff }}=\eta_1+\eta_2 \cos ^2 \theta+\eta_2$$
为了考虑角速度梯度。系数 $\eta_{1,2}$ 与莱斯利系数有关 $\alpha_1$ 经过
$$\eta_{1,2}=\alpha_1 .$$

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