Liquid Crystals

Instructor: Yao Zhenwei
中文版：液晶
Back to: Soft Matter Physics Course Notes

Core Picture

Liquid crystals sit between liquids and crystals. A nematic phase has orientational order but no positional order. The director n represents average orientation, with n equivalent to -n.

Scalar and Tensor Order Parameters

Orientation can be described by f(theta, phi). With cylindrical symmetry around the director, it depends only on theta.

scalar S=$S=\left\langle\frac12(3\cos^2\theta-1)\right\rangle$

Legendre=$S=\langle P_2(\cos\theta)\rangle$

tensor=$Q_{\alpha\beta}=S(n_\alpha n_\beta-\delta_{\alpha\beta}/3)$

trace=$\mathrm{Tr}\,Q=0$

S=1 means perfect alignment. S=0 means isotropic orientation. The Q tensor removes the isotropic part and keeps anisotropy.

Macroscopic Response Tensor

Nematic anisotropy can be measured through response to an external field.

response=$M_\alpha=\chi_{\alpha\beta}H_\beta$

isotropic=$\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$

nematic=$\chi_{\alpha\beta}=\chi_\perp\delta_{\alpha\beta}+\chi_a n_\alpha n_\beta$

anisotropy=$\chi_a=\chi_\parallel-\chi_\perp$

Frank Free Energy

Spatial variation of director costs elastic energy. The three basic deformations are splay, twist, and bend.

Frank=$F=\frac12\int[K_1(\nabla\cdot\mathbf n)^2+K_2(\mathbf n\cdot\nabla\times\mathbf n)^2+K_3|\mathbf n\times\nabla\times\mathbf n|^2]dV$

splay=$\nabla\cdot\mathbf n$

twist=$\mathbf n\cdot\nabla\times\mathbf n$

bend=$\mathbf n\times\nabla\times\mathbf n$

In two dimensions and the one-constant approximation, the director is represented by an angle field theta.

2D n=$\mathbf n=(\cos\theta,\sin\theta)$

one K=$F=\frac{K}{2}\int|\nabla\theta|^2d^2r$

Topological Defects

The total rotation of director around a defect defines its topological charge. Because n is equivalent to -n, half-integer defects are allowed.

charge=$q=\frac{1}{2\pi}\oint d\theta$

defect=$\theta=q\varphi+\theta_0$

energy=$F_{defect}\approx\pi Kq^2\ln(R/a)$

sphere=$\sum_i q_i=2$

A nematic on a sphere must carry total topological charge 2, often realized by four +1/2 defects.

Exam Cue

For liquid crystals, first state n is equivalent to -n, then write S or Q. If director varies, use Frank free energy. If defects appear, compute winding number and remember the spherical charge constraint.