Structure . In the cholesteric phase, the director rotates layer by layer, ultimately creating a twisted supramolecular structure. Due to their properties intermediate between pure
liquids and
crystalline solids, liquid crystals are known as
mesogens, a name deriving from
Greek for
mésos, or "intermediate". The property underpinning all liquid crystals is
anisotropy (directional nonuniformity, typically manifested by an elongated rodlike shape), which under appropriate conditions (ex. high
temperatures and
concentrations) allows for local order around a preferred axis, named the director. There exist two classes of liquid crystals based on the conditions under which they form:
thermotropic and
lyotropic. Thermotropic liquid crystals undergo phase transitions based on temperature, whereas lyotropic liquid crystalline phases transition based on concentration within a
solvent, most commonly
water. For example,
5CB — a classic example of an achiral nematic thermotropic LC — undergoes an isotropic-nematic transition at 308K and a nematic-crystalline transition at 252K. Similarly, poly(n-hexyl isocyanate), a lyotropic liquid crystal, undergoes the analogous isotropic-nematic transition at weight fractions ranging from 0.225 to 0.438 in
toluene, depending on molecular weight of the polymer. Cholesteric liquid crystals comprise both classes. Both small molecules and polymers can form cholesteric liquid crystals. In nature, examples include
DNA,
chitin,
cellulose, and
collagen, among others.S_{\alpha \beta} (\mathbf{r}) = \frac{1}{N} \sum_{i} \left( v_{\alpha}^{(i)} \otimes v_{\beta}^{(i)} - \frac{1}{3} \delta_{\alpha \beta} \right)Here, for an anisotropic molecule, the nematic order tensor is a function of number of molecules N, the
outer product of the
unit vector along the long axis v(i), and a
traceless correction term, the
Kronecker delta δ. The orientation of the nematic director at a certain distance along the director twist axis (usually defined as the z-axis in
Cartesian coordinates) is:n_x = \cos(qz)n_y = \sin(qz)n_z = 0Here, q is defined as the
helicity of a ChLC, \frac{2\pi}{p}. The helicity is positive for a right-handed cholesteric helix, and negative for left-handed helices. The origin of the helical pitch can be described with Frank-Oseen elastic free energy density:f_{FO} = \frac{1}{2} K_1 (\text{div} \, \mathbf{n})^2 + \frac{1}{2} K_2 (\mathbf{n} \cdot \text{curl} \, \mathbf{n} + q)^2 + \frac{1}{2} K_3 (\mathbf{n} \times \text{curl} \, \mathbf{n})^2Where div is the
divergence for a
vector field n (representing the individual molecular long-axis vectors) and curl is the
curl of the same vector field. In 3D space, with unit vectors i, j, and k along each coordinate axis. The constants K are known as Frank elastic constants, and are empirical. becomes cholesteric when exposed to a small amount of chiral dopant. The mechanism by which this transition occurs is via the slight displacement of the
racemic mixture to a small
enantiomeric excess, which then drives the formation of cholesteric helices. Different chiral dopants may be quantitatively compared using their empirical helical twisting power:\beta_M=\frac{1}{PC}Where C is the
mole fraction of the dopant, corrected for enantiomeric purity. Dopants also induce chirality on small molecules by biasing a specific chiral spatial configuration, which has an amplifying effect that ultimately leads to the formation of a chiral phase from a small enantiomeric biasing. An example of inherently chiral ChLCs is poly-γ-benzyl-l-glutamate (PBLG), a lyotropic liquid crystal that forms cholesteric phases without dopant. Examples of small-molecule ChLCs include cholesterol-doped
5CB and 1,2-dipalmitoyl-sn-glycero-3-phosphocholine. The pitch of thermotropic ChLCs is temperature-dependent.
Optical Textures Due to their anisotropy, liquid crystals are
birefringent. Formally, this means that the
index of refraction is directionally dependent, with characteristic indices defined along perpendicular optical axes. Upon incident light, these different indices break up the waves into multiple with different wavelengths. Among the most common textures is the oily streak texture, which was the first texture experimentally observed in
cholesteryl benzoate. Another texture is the classical fingerprint texture, where the director helix axis is perpendicular to incident light. Here, the cholesteric helix can be easily observed and measured, as the pitch is calculated as the distance between two dark fringes. This information can be used to measure helical twisting power of the liquid crystal or monitor changes to the physical structure of the cholesteric helix in applications such as
optical sensing. Particularly long pitches arranged this way give rise to the focal-conic texture. The textures can be tuned with external stimuli. Pijper and coworkers invented a ChLCs whose pitch can be dynamically controlled via light irradiation. A chiral,
photoswitchable chromophore was functionalized onto the ends of PHIC polymers, whose
enantiomeric excess could be tuned with irradiation time. Upon irradiation with characteristic wavelengths of light, the texture changed from fingerprint to nematic to the opposite-handed fingerprint. == Characterization ==