Liquid crystal

A liquid crystal has orientational or partial translational order while retaining some fluid mobility.

An isotropic phase has no preferred spatial direction. In a liquid crystal, its orientational nematic order parameter vanishes, although the molecules themselves may be anisotropic.

A polar liquid crystal distinguishes head from tail and is described at leading order by a vector polar order parameter $\mathbf{p}$.

A nematic liquid crystal identifies the director $\mathbf{n}$ with $-\mathbf{n}$. Its rotational order is described by a symmetric traceless tensor such as $Q_{ij}=S(n_i{n}_j-{\delta }_{ij}/3)$.

The twisted nematic field effect switches a nematic liquid crystal between a twisted optically rotating state and a largely field-aligned state. With adiabatic optical following in a twisted nematic and crossed polarizers aligned to the two planar surface directors, a quarter-turn cell passes light at low voltage and suppresses it at high voltage. A rear reflector permits reflective operation with little power expended on light generation.

When the birefringence separation of optical propagation constants $\Delta k=2\pi \Delta n/\lambda$ is large compared with the spatial rate of rotation of the optical axes, light with linear polarization in a local eigenpolarization follows the twisting nematic director. Thickness alone is insufficient if birefringence vanishes. Strong field alignment along the propagation direction suppresses the transverse optical anisotropy and hence the quarter-turn polarization rotation.

The Fréedericksz transition is a field-driven orientational instability of a confined nematic liquid crystal. Elasticity and surface anchoring resist the electric field or magnetic torque; a lowest allowed distortion mode becomes unstable at a finite threshold. Positive dielectric anisotropy of a nematic permits a normal electric field to tilt a planar anchored director.

For planar strong anchoring and a quarter-turn base profile, ${\varphi }^{\prime }=\pi /(2L)$ and $\theta =0$. Expanding the one-dimensional twisted nematic energy through quadratic order gives $\Delta F=\frac{1}{2}\int [K_1{\theta }^{\prime 2}+\{(K_3-2K_2)(\pi /(2L){)}^2-\Delta ϵE^2/(4\pi )\}{\theta }^2]dz$. The lowest zero-endpoint mode is $\sin (\pi z/L)$. Its coefficient vanishes at the displayed threshold, with $V=EL$, provided the bracket and anisotropy are positive. Halving the electric coupling doubles $V_c^2$.

The dielectric response differs along and perpendicular to the nematic director. In Gaussian units the orientational electric contribution at prescribed field is $-\Delta ϵ(\mathbf{E}\cdot \mathbf{n}{)}^2/(8\pi )$; positive anisotropy favors alignment with the electric field. The electric term does not receive the additional one-half convention used in writing the Frank elastic terms.

A nematic director is a unit vector specifying the local unoriented axis of a nematic liquid crystal. Head-tail symmetry identifies $\mathbf{n}$ with $-\mathbf{n}$, so the order-parameter space is a Real projective space rather than a sphere.

For a unit nematic director, the bulk elastic density is $[K_1(\nabla \cdot \mathbf{n}{)}^2+K_2(\mathbf{n}\cdot \nabla \times \mathbf{n}{)}^2+K_3|\mathbf{n}\times \nabla \times \mathbf{n}{|}^2]/2$. The three positive moduli penalize nematic splay, nematic twist, and nematic bend. Boundary elastic terms, chirality, or varying nematic order require additional contributions.

For a laterally uniform nematic director $(\cos \theta \cos \varphi ,\cos \theta \sin \varphi ,\sin \theta )$, its divergence is $\cos \theta {\theta }^{\prime }$ and its twist invariant is $-{\cos }^2\theta {\varphi }^{\prime }$. Thus $f=K_1{\cos }^2\theta +K_3{\sin }^2\theta$ and $g=K_2{\cos }^4\theta +K_3{\sin }^2\theta {\cos }^2\theta$. A normal electric field contributes $-c{\sin }^2\theta$. The Euler-Lagrange equations for two fields are $(g{\varphi }^{\prime }{)}^{\prime }=0$ and $f{\theta }^{\prime \prime }+f_{\theta }{\theta }^{\prime 2}/2-g_{\theta }{\varphi }^{\prime 2}/2+2c\sin \theta \cos \theta =0$.

Nematic bend curves the integral lines of the nematic director. For a unit field, $(\mathbf{n}\cdot \nabla )\mathbf{n}=-\mathbf{n}\times (\nabla \times \mathbf{n})$, so the bend energy measures curvature along those lines.

Nematic twist rotates the nematic director around an axis perpendicular to it. The helix $\mathbf{n}=(\cos qz,\sin qz,0)$ has zero nematic splay and nematic bend, and twist density $K_2{q}^2/2$.

Nematic splay describes local spreading or convergence of the nematic director field. Its leading elastic cost is quadratic in the divergence and invariant under the head-tail identification $\mathbf{n}\sim -\mathbf{n}$.

A nematic disclination is a topological line defect of the unoriented director field. In a two-dimensional sample its point cross-section can have half-integer winding because $\mathbf{n}$ and $-\mathbf{n}$ represent the same state.

A neutral pair of opposite nematic disclinations can annihilate under dissipative relaxation, because the total topological charge of a two-dimensional nematic disclination is zero. Its far-field elastic energy increases as $2\pi K{\lambda }_0^2{q}^2\log (R/r_0)$ with separation $R$, so the pair has an attractive elastic force proportional to $-1/R$. As opposite cores meet, the nematic order parameter may vanish locally and the two windings disappear. Integer defects may first undergo energetic fission of an integer nematic disclination, leading to encounters of elementary half-charge defects. The actual paths and annihilation time require a dynamical law; the free energy alone does not specify core mobility. Pinning, imposed boundary winding or thermal defect production can obstruct a defect-free final state.

With constant amplitude outside a core, the planar nematic divergence-elasticity identity reduces the elastic density to $K{\lambda }_0^2|\nabla \chi {|}^2/2$. A topological charge of a two-dimensional nematic disclination $q$ has $\chi =q\theta +{\chi }_0$ and $|\nabla \chi {|}^2=q^2/r^2$. Integrating over an annulus gives the displayed energy, plus finite core and boundary terms. The outer cutoff is the sample size, compensating-defect separation, or another screening length. An isolated defect in an infinite ordered plane has divergent energy. A neutral pair $q,-q$ separated by $R$ instead has $E_{\mathrm{pair}}=2\pi K{\lambda }_0^2{q}^2\log (R/r_0)$ plus core terms, since its far-field gradients cancel.

In a strictly planar nematic liquid crystal, a charge-one nematic disclination is a winding-two map into the circle ${\mathbb{RP}}^1$. Its nonzero fundamental group class prevents complete unwinding while the surrounding loop stays fixed, but permits splitting into two charge-$1/2$ defects. For separation $r_0\ll d\ll L$, the two near fields contribute $\pi K{\lambda }_0^2(1/4+1/4)\log (d/r_0)$ while their common charge-one far field contributes $\pi K{\lambda }_0^2\log (L/d)$. Compared with the unsplit defect, the change isIt is negative at sufficiently large scale separation. Thus integer-defect fission is energetically favorable despite conservation of total topological charge. This leading-log argument proves failure of global energetic stability, not absence of kinetic barriers or pinning. If escape into the third dimension is allowed, an integer winding can also become topologically trivial in ${\mathbb{RP}}^2$.

For a planar tangential charge-one nematic disclination, $\nabla \cdot Q=-({\lambda }^{\prime }/2+\lambda /r){\mathbf{e}}_r$. This follows by differentiating the tensor dyad, using $\nabla \cdot {\mathbf{e}}_{\theta }=0$ and $({\mathbf{e}}_{\theta }\cdot \nabla ){\mathbf{e}}_{\theta }=-{\mathbf{e}}_r/r$. With bulk density $a{\lambda }^2/2+b{\lambda }^4/4$ and divergence elastic density $K({\lambda }^{\prime }/2+\lambda /r{)}^2/2$, radial variation of $2\pi \int rfdr$ givesFor $a<0$, a regular melted core has $\lambda =O(r^2)$ and an ordered far field has ${\lambda }_0=\sqrt{-a/b}$. Linearizing the bulk derivative gives $\lambda /{\lambda }_0=1-K/(2|a|r^2)+O(r^{-4})$. The core scale is of order $\sqrt{K/|a|}$; the amplitude disturbance has an integrable algebraic tail, rather than exactly compact support.

For a continuous director angle around a closed counterclockwise circuit, the enclosed topological charge is $q=\Delta \theta /(2\pi )$. Since $\theta$ is defined modulo $\pi$, elementary nematic disclinations can have $q=\pm 1/2$.

A two-dimensional director winding can sometimes unwind when the director may tilt out of the plane. For a three-dimensional nematic the fundamental group of the order-parameter space is ${\pi }_1({\mathbb{RP}}^2)=\mathbb{Z}/2\mathbb{Z}$, so two disclination lines are topologically equivalent to none.

In the one-elastic-constant approximation, slow director distortions have a quadratic gradient energy with one modulus. For a planar angle field it reduces to $F=(K/2)\int |\nabla \theta {|}^2$ up to the convention-dependent effective modulus.

For a planar nematic director $\mathbf{n}=(\cos \chi ,\sin \chi )$ and constant amplitude ${\lambda }_0$, the nematic order parameter isDifferentiation gives $\nabla \cdot Q={\lambda }_0(-\sin 2\chi {\chi }_x+\cos 2\chi {\chi }_y,\cos 2\chi {\chi }_x+\sin 2\chi {\chi }_y)$. The coefficient matrix is orthogonal, proving the identity. Hence $K|\nabla \cdot Q{|}^2/2$ is the angle-gradient energy with effective modulus $K{\lambda }_0^2$. The full tensor-gradient norm instead obeys ${\sum }_{ijk}({\partial }_k{Q}_{ij}{)}^2=2{\lambda }_0^2|\nabla \chi {|}^2$ in this normalization; the two local densities must not be confused.

A uniaxial nematic order parameter has one distinguished nematic director and equal eigenvalues in its perpendicular plane. In the displayed normalization the eigenvalues are $\lambda ,-\lambda /2,-\lambda /2$, giving $\mathrm{Tr}{Q}^2=3{\lambda }^2/2$ and $\mathrm{Tr}{Q}^3=3{\lambda }^3/4$. Positive and negative $\lambda$ describe prolate and oblate orientational anisotropy, respectively; reversing the nematic director leaves the tensor unchanged.

Under rotation by a rotation matrix, a symmetric second-rank tensor transforms by conjugation, so its traces of powers are invariant. For a three-dimensional traceless second-rank tensor, the Cayley-Hamilton theorem implies $\mathrm{Tr}{Q}^4=(\mathrm{Tr}{Q}^2{)}^2/2$. In two dimensions its eigenvalues are $s,-s$, so $\mathrm{Tr}{Q}^3=0$. These identities organize the Landau-de Gennes free energy.

The Landau-de Gennes free energy is a rotational symmetry preserving Landau free energy of the nematic order parameter. One three-dimensional normalization through fourth order isThe cubic invariant generically produces a first-order phase transition. Its absence in two dimensions is an algebraic identity, not a guarantee that a mean-field approximation captures all fluctuations.

The leading scalar coupling of an electric field to the nematic order parameter is quadratic in the field and linear in the tensor. For uniaxial nematic order, it is $-3\chi \lambda E^2({\cos }^2\vartheta -1/3)/2$. When $\chi \lambda >0$, minimization aligns the nematic director with the field axis and adds $-\chi E^2\lambda$ to the scalar Landau free energy.

For a scalar Landau-de Gennes free energy with $\bar{b}>0$, $\bar{c}<0$, and coupling $-\chi E^2\lambda$ with $\chi >0$, the critical endpoint of a quartic Landau free energy is reached at the displayed field strength and ${\bar{a}}_c=3{\bar{c}}^2/(8\bar{b})$, ${\lambda }_c=-\bar{c}/(4\bar{b})$. It terminates the field-biased first-order phase transition when both field and quadratic coefficient can be tuned. This is a quartic small-field mean-field approximation; higher terms may alter its location.

An unoriented local axis: $\mathbf{n}$ and $-\mathbf{n}$ describe the same orientation.