# 物理代写|光学代写Optics代考|EGR 558

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• Statistical Computing 统计计算
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## 物理代写|光学代写Optics代考|Elastic Constants, Free Energies, and Molecular Fields

Upon application of an external perturbation field, a nematic liquid crystal will undergo deformation just as any solid. There is, however, an important difference. A good example is shown in Figure 3.1a, which depicts a “solid” subjected to torsion, with one end fixed. In ordinary solids, this would create very large stress, arising from the fact that the molecules are translationally displaced by the torsional stress. On the other hand, such twist deformations in liquid crystals, owing to the fluidity of the molecules, simply involve a rotation of the molecules in the direction of the torque; there is no translational displacement of the center of gravity of the molecules, and thus, the elastic energy involved is quite small. Similarly, other types of deformations such as splay and bend deformations, as shown in Figure $3.1 \mathrm{~b}$ and $\mathrm{c}$, respectively, involving mainly changes in the director axis $\hat{n}(\vec{r})$, will incur much less elastic energy change than the corresponding ones in ordinary solids. It is evident from Figure $3.1 \mathrm{a}-\mathrm{c}$ that the splay and bend deformations necessarily involve flow of the liquid crystal, whereas the twist deformation does not. We will return to these couplings between flow and director axis deformation in Section 3.5.

Twist, splay, and bend are the three principal distinct director axis deformations in nematic liquid crystals. Since they correspond to spatial changes in $\hat{n}(\vec{r})$, the basic parameters involved in the déformation ennergiês are various spatial derivatives (i.é. curvatures of $\hat{n}(\vec{r})$, such as $\nabla \times \hat{n}(\vec{r})$ and $\nabla \cdot \hat{n}(\vec{r})$, etc.). Following the theoretical formalism first developed by Frank [1], the free-energy densities (in units of energy per volume) associated with these deformations are given by
splay : $f_{1}=\frac{1}{2} K_{1}(\nabla \cdot n)^{2}$,
twist : $f_{2}=\frac{1}{2} K_{2}(n \cdot \nabla \times n)^{2}$,
bend : $f_{3}=\frac{1}{2} K_{3}(n \times \nabla \times n)^{2}$,
where $K_{1}, K_{2}$, and $K_{3}$ are the respective Frank elastic constants.

## 物理代写|光学代写Optics代考|DC and Low-frequency Dielectric Permittivity

Typical values of $\varepsilon_{|}$and $\varepsilon_{\perp}$ are on the order of $5 \varepsilon_{0}$, where $\varepsilon_{0}$ is the permitlivity of free space. Similarly, the electric conductivities $\sigma_{|}$and $\sigma_{\perp}$ of nematics are defined by
$$J_{|}=\sigma_{|} E_{|}$$
and
$$J_{\perp}=\sigma_{\perp} E_{\perp},$$
where $J_{|}$and $J_{\perp}$ are the currents flowing along and perpendicularly to the director axis, respectively. In conjunction with an applied dc electric field, the conductivity anisotropy could give rise to space charge accumulation and create strong director axis reorientation in a nematic film, giving rise to an orientational photorefractive [6] effect (see Chapter 8 ).

Most nematics (e.g. E7, pentyl cyanobiphenyl [5CB], etc.) are said to possess positive (dielectric) anisotropy $\left(\varepsilon_{|}>\varepsilon_{\perp}\right)$. On the other hand, some nematics, such as MBBA, possess negative anisotropy (i.e. $\varepsilon_{| \mid}<\varepsilon_{\perp}$ ). The controlling factors are the molecular constituents and structures.

In general, $\varepsilon_{|}$and $\varepsilon_{\perp}$ have different dispersion regions, as shown in Figure $3.4$ for 4-methoxy-4′-n-butylazoxy-benzene [7], which possess negative dielectric anisotropy $(\Delta \varepsilon<0)$. Also plotted in Figure $3.4$ is the dispersion of $\varepsilon_{\text {iso }}$, the dielectric constant for the isotropic case. Notice that for frequencies of $10^{9} \mathrm{~Hz}$ or less, $\varepsilon_{\perp}>\varepsilon_{| 1}$. At higher frequencies and in the optical regime, $\varepsilon_{|}>\varepsilon_{\perp}$ (i.e. the dielectric anisotropy changes sign).

For some nematic liquid crystals, this changeover in the sign of $\Delta \varepsilon=\varepsilon_{|}-\varepsilon_{\perp}$ occurs at a much lower frequency (cf. Figure $3.5$ for phenylbenzoates [8]). This changeover frequency $f_{\mathrm{co}}$ is lower because of the long three-ringed molecular structure, which is highly resistant to the rotation of molecules around the short axes.

# 光学代考

## 物理代写|光学代写Optics代考|Elastic Constants, Free Energies, and Molecular Fields

splay 给出: $f_{1}=\frac{1}{2} K_{1}(\nabla \cdot n)^{2}$,

## 物理代写|光学代写Optics代考|DC and Low-frequency Dielectric Permittivity

$$J_{\mid}=\sigma_{\mid} E$$

$$J_{\perp}=\sigma_{\perp} E_{\perp},$$

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