# 物理代写|光学代写Optics代考|EGR558

## 物理代写|光学代写Optics代考|Equilibrium Temperature and Order Parameter Dependences

The two principal refractive indices $n_{\perp}$ and $n_{|}$of a uniaxial liquid crystal and the anisotropy $n_{|}-n_{\perp}$ have been the subject of intensive studies for their fundamental importance in the understanding of liquid crystal physics and for their vital roles in applied electro-optic devices. Since the dielectric constants $\left(\varepsilon_{\perp}\right.$ and $\left.\varepsilon_{|}\right)$enter directly and linearly into the constitutive equations (Eqs. (3.30a)-(3.30c)), it is theoretically more convenient to discuss the fundamentals of these temperature dependences in terms of the dielectric constants.

From Eq. (3.34) for the local field $\vec{E}^{\text {loc }}$ and Eq. (3.31) for the induced dipole moments, we can express the polarization $\vec{p} \equiv N \vec{d}$ by
$$\vec{P}=N \overrightarrow{\bar{\alpha}}:(\overrightarrow{\bar{K}}: \vec{E})$$
where $\overrightarrow{\bar{\alpha}}$ is the polarizability tensor of the molecule, $N$ is the number of molecules per unit volume, and the parentheses denote averaging over the orientations of all molecules.
The dielectric constant $\overrightarrow{\vec{\varepsilon}}$ (in units of $\varepsilon_0$ ) is therefore given by
$$\overrightarrow{\bar{\varepsilon}}=1+\frac{N}{\varepsilon_0} \overrightarrow{\vec{\alpha}}: \overrightarrow{\vec{K}}$$
and
\begin{aligned} \Delta \varepsilon &=\varepsilon_{|}-\varepsilon_{\perp} \ &=\frac{N}{\varepsilon_0}\left(\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\bar{K}}\rangle_{|}-\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\bar{K}}\rangle_{\perp}\right) . \end{aligned}
From these considerations and from observations by deJeu and Bordewijk [13] that
$$\Delta \varepsilon \propto \rho S$$ and
$$\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\vec{K}}\rangle_{|}-\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\bar{K}}\rangle_{\perp} \propto S,$$
we can write $\varepsilon_{|}$and $\varepsilon_{\perp}$ as
$$\varepsilon_{|}=n_{|}^2=1+\left(\frac{N}{3 \varepsilon_0}\right)\left[\alpha_l K_l(2 S+1)+\alpha_t K_t(2-2 S)\right]$$
and
$$\varepsilon_{\perp}=n_{\perp}^2=1+\left(\frac{N}{3 \varepsilon_0}\right)\left[\alpha_l K_l(1-S)+\alpha_t K_t(2+S)\right]$$
respectively, where $K_l$ and $K_t$ are the values of $\overrightarrow{\bar{K}}$ along the principal axis and $S$ is the order parameter.

## 物理代写|光学代写Optics代考|FLOWS AND HYDRODYNAMICS

One of the most striking properties of liquid crystals is their ability to flow freely while exhibiting various anisotropic and crystalline properties. This dual nature of liquid crystals makes them very interesting materials to study; it also makes theoretical formalism very complex.

The main feature that distinguishes liquid crystals in their ordered mesophases (e.g. the nematic phase) from ordinary fluids is that their physical properties are dependent on the orientation of the director axis $\vec{n}(\vec{r})$; these orientation flow processes are necessarily coupled, except in very unusual cases (e.g. pure twisted deformation). Therefore, studies of the hydrodynamics of liquid crystals will involve a great deal more (anisotropic) parameters than studies of the hydrodynamics of ordinary liquids.

We begin our discussion by reviewing first the hydrodynamics of an ordinary fluid. This is followed by a discussion of the general hydrodynamics of liquid crystals. Specific cases involving a variety of flow-orientational couplings are then treated.

Consider an elementary volume $d V=d x d y d z$ of a fluid moving in space as shown in Figure 3.9. The following parameters are needed to describe its dynamics:
position vector: $\vec{r}$,
velocity: $\vec{v}(\vec{r}, t)$,
density: $\rho(\vec{r}, t)$,
pressure: $p(\vec{r}, t)$, and
forces in general: $\vec{f}(\vec{r}, t)$.
In later chapters where we study laser-induced acoustic (sound, density) waves in liquid crystals, or generally, when one deals with acoustic waves, it is necessary to assume that the density $\rho(\vec{r}, t)$ is a spatially and temporally varying function. In this chapter, however, we “decouple” such density wave excitation from all the processes under consideration and basically limit our attention to the flow and orientational effects of an incompressible fluid. In that case, we have
$$\rho(\vec{r}, t)=\text { constant }$$
For all liquids, in fact for all gas particles or charges in motion, the equation of continuity also holds
\nabla \cdot(\hat{\rho} \vec{v})=-\begin{aligned} &\partial \rho \ &\partial t \end{aligned}

## 物理代写|光学代写Optics代考|Equilibrium Temperature and Order Parameter Dependences

$$\vec{P}=N \overrightarrow{\vec{\alpha}}:(\overrightarrow{\bar{K}}: \vec{E})$$

$$\overrightarrow{\vec{\varepsilon}}=1+\frac{N}{\varepsilon_0} \overrightarrow{\vec{\alpha}}: \overrightarrow{\vec{K}}$$

$$\Delta \varepsilon=\varepsilon_1-\varepsilon_{\perp} \quad=\frac{N}{\varepsilon_0}\left(\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\vec{K}}\rangle_{\mid}-\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\vec{K}}\rangle_{\perp}\right)$$

$$\Delta \varepsilon \propto \rho S$$

$$\langle\vec{\alpha}: \overrightarrow{\vec{K}}\rangle_1-\langle\vec{\alpha}: \overrightarrow{\vec{\alpha}}\rangle_{\perp} \propto S$$

$$\varepsilon_{\mid}=n_{\mid}^2=1+\left(\frac{N}{3 \varepsilon_0}\right)\left[\alpha_l K_l(2 S+1)+\alpha_t K_t(2-2 S)\right]$$

$$\varepsilon_{\perp}=n_{\perp}^2=1+\left(\frac{N}{3 \varepsilon_0}\right)\left[\alpha_l K_l(1-S)+\alpha_t K_t(2+S)\right]$$

## 物理代写|光学代写Optics代考|FLOWS AND HYDRODYNAMICS

$$\nabla \cdot(\hat{\rho} \vec{v})=-\partial \rho \quad \partial t$$

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