物理代写|光学代写Optics代考|ELEC-E5730

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物理代写|光学代写Optics代考|Free Energy and Torques by Electric and Magnetic Fields

In this section, we consider the interactions of nematic liquid crystals with applied fields (electric or magnetic); we will limit our discussion to only dielectric and diamagnetic interactions.

For a generally applied (dc, low frequency, or optical) electric field $\vec{E}$, the displacement $\vec{D}$ may be written in the form
$$\vec{D}=\varepsilon_{\perp} \vec{E}+\left(\varepsilon_{|}-\varepsilon_{\perp}\right)(n \cdot \vec{E}) n .$$

The electric interaction energy density is therefore
$$\mu_{E}=-\int_{0}^{E} \vec{D} \cdot d \vec{E}=-\frac{1}{2} \varepsilon_{\perp}(\vec{E} \cdot \vec{E})-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^{2}$$
Note that the first term on the right-hand side of Eq. (3.24) is independent of the orientation of the director axis. It can therefore be neglected in the director axis deformation energy. Accordingly, the free-energy density term associated with the application of an electric field is given by
$$F_{E}=-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^{2}$$
in SI units (in cgs units, $F_{E}=-(\Delta \varepsilon / 8 \pi)(\hat{n} \cdot \vec{E})^{2}$ ). The molecular torque produced by the electric field is given by
$$\vec{\Gamma}_{E}=\vec{D} \times \vec{E}=\Delta \varepsilon(n \cdot \vec{E})(n \times \vec{E}) .$$

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 $\left.\varepsilon_{| \mid}\right)$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
$$\vec{P}=N \overrightarrow{\vec{\alpha}}:(\overrightarrow{\vec{K}}: \vec{E})$$
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
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}

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物理代写|光学代写Optics代考|Free Energy and Torques by Electric and Magnetic Fields

$$\vec{D}=\varepsilon_{\perp} \vec{E}+\left(\varepsilon_{1}-\varepsilon_{\perp}\right)(n \cdot \vec{E}) n .$$

$$\mu_{E}=-\int_{0}^{E} \vec{D} \cdot d \vec{E}=-\frac{1}{2} \varepsilon_{\perp}(\vec{E} \cdot \vec{E})-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^{2}$$

$$F_{E}=-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^{2}$$

$$\vec{\Gamma}_{E}=\vec{D} \times \vec{E}=\Delta \varepsilon(n \cdot \vec{E})(n \times \vec{E}) .$$

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

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

$$\Delta \varepsilon=\varepsilon_{1}-\varepsilon_{\perp} \quad=\frac{N}{\varepsilon_{0}}\left(\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\vec{K}}\rangle_{\mid}-\langle\overrightarrow{\vec{\alpha}}: \overrightarrow{\vec{K}}\rangle_{\perp}\right) .$$

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