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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|光学代写Optics代考|Free Energies

Since the equilibrium configuration of a CLC is a helical structure with a pitch wave vector $q_0\left(q_0=2 \pi / p_0\right)$, its elastic free energy will necessarily reflect the presence of $q_0$. The evolution of a cholesteric to a nematic liquid crystal may be viewed as the “untwisting” of the helical structure (i.e. the twist deformation energy is involved). This is indeed rigorously demonstrated in the Frank elastic theory [1]. where consideration of the absence of mirror symmetry in cholesterics results in the addition of another factor to the twist deformation energy:
$$\begin{array}{cc} K_2(\hat{n} \cdot \nabla \times \hat{n})^2 & \rightarrow K_2\left(\hat{n} \cdot \nabla \times \hat{n}+q_0\right)^2 \ \text { nematic } & \text { cholesteric } \end{array}$$
In Figure 4.1, the director axis is described by $\hat{n}=\left(n_x, n_y, n_z\right)$, where $n_x=\cos \theta(z)$, $n_y=\sin \theta(z)$, and $n_z=0$. Note that this configuration corresponds to a state of minimum free energy $$F_2=\frac{1}{2} K_2(\hat{n} \cdot \nabla \times \hat{n})^2=\frac{1}{2} K_2\left(-\frac{\partial \theta}{\partial z}+q_0\right)^2=0 .$$
Equation (4.2) gives $q_0=-\partial \theta / \partial z$, and so we have:
$$\theta=q_0 z .$$
For a general distortion, therefore, the free energy of a CLC is given by $F_d=\frac{1}{2} K_1(\nabla \cdot \hat{n})^2+\frac{1}{2} K_2(\hat{n} \cdot \nabla \times \hat{n})^2+\frac{1}{2} K_3(\hat{n} \times \nabla \times \hat{n})^2 .$

## 物理代写|光学代写Optics代考|Field-induced Effects and Dynamics

In the purely dielectric interaction picture (i.e. no current flow), the realignment or alignment of a CLC in an applied electric or magnetic field results from the system’s tendency to minimize its total free energy. Clearly, the equilibrium configuration of the director axis depends on its initial orientation and the direction of the applied fields, as well as the signs of $\Delta \chi^m$ and $\Delta \varepsilon$. We will not delve into the various possible cases as they all involve the same basic mechanism; that is, the director axis tends to align parallel to the field for positive dielectric anisotropies, and normal to the field for negative dielectric anisotropies.

In the case of positive dielectric anisotropies, the field-induced reorientation process is analogous to that discussed for nematics (cf. Section 3.6). In cholesterics, however, the realignment of the director axis in the direction of the applied field will naturally affect the helical structure.

Figure $4.2 \mathrm{a}$ shows the unperturbed director axis configuration in the bulk of an ideal CLC. Upon the application of a magnetic field, some molecules situated in the bulk regions $A, A^{\prime}$, and so on are preferentially aligned along the direction of the field; others, situated in regions $B, B^{\prime}$, and so forth are not and would tend to reorient themselves along the field direction. As a result, the pitch of the helical structure will be increased; the helix is no longer of the ideal sinusoidal form. Finally, when the field is sufficiently high, this untwisting effect is complete as the pitch approaches infinity; that is, the CLC is said to have undergone a transition to the nematic phase [2].
This process can be described by the free-energy minimization process. The total free energy of the system is given by
$$F_{\text {total }}=\frac{1}{2} \int d z\left[K_2\left(\frac{\partial \phi}{\partial z}-q_0\right)^2-\Delta \chi^m H^2 \sin ^2 \phi\right]$$
This, upon minimization, yields the Euler equation:
$$K_2 \frac{d^2 \phi}{d z^2}+\Delta \chi^m H^2 \sin \phi \cos \phi=0$$
which is analogous to Eq. (3.75b). One can define a coherence length $\xi_H$ by
$$\xi_H=\frac{K_2}{\Delta \chi^m H^2}$$

# 光学代考

## 物理代写|光学代写Optics代考|Free Energies

$$K_2(\hat{n} \cdot \nabla \times \hat{n})^2 \rightarrow K_2\left(\hat{n} \cdot \nabla \times \hat{n}+q_0\right)^2 \text { nematic cholesteric }$$

$$F_2=\frac{1}{2} K_2(\hat{n} \cdot \nabla \times \hat{n})^2=\frac{1}{2} K_2\left(-\frac{\partial \theta}{\partial z}+q_0\right)^2=0 .$$

$$\theta=q_0 z$$

## 物理代写|光学代写Optics代考|Field-induced Effects and Dynamics

$$F_{\text {total }}=\frac{1}{2} \int d z\left[K_2\left(\frac{\partial \phi}{\partial z}-q_0\right)^2-\Delta \chi^m H^2 \sin ^2 \phi\right]$$

$$K_2 \frac{d^2 \phi}{d z^2}+\Delta \chi^m H^2 \sin \phi \cos \phi=0$$

$$\xi_H=\frac{K_2}{\Delta \chi^m H^2}$$

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## MATLAB代写

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assignmentutor™您的专属作业导师
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