Contact topology and the structure and dynamics of cholesterics

Abstract

Using tools and concepts from contact topology we show that non-vanishing twist implies conservation of the layer structure in cholesteric liquid crystals. This leads to a number of additional topological invariants for cholesteric configurations, such as layer numbers, that are not captured by traditional descriptions, characterises the nature and size of the chiral energy barriers between metastable configurations, and gives a geometric characterisation of cholesteric dynamics in any context, including active systems, those in confined geometries or under the influence of an external field.

Layering transitions and metastable structures of cholesteric liquid crystals in cylindrical confinement

“…Additionally, the director configurations of local minima coexist as metastable domains with point-like defects between them. We demonstrate that a chiral layer number invariant distinguishes these configurations, protects the distinct layer configurations (45), and explains the existence of the topological defect where the invariant changes.…”

“…An alternative perspective, chiral topology, can give a deeper understanding of these structures (45)(46)(47). In this approach, the key role is played by the twist density, given in terms of the director field n by n • ∇ × n. This choice is not arbitrary; the Frank free energy prefers n • ∇ × n ≈ −q0 = 2π/p0 with a helical pitch p0, and, from a geometric perspective, n • ∇ × n = 0 defines a contact structure (48).…”

“…In this approach, the key role is played by the twist density, given in terms of the director field n by n • ∇ × n. This choice is not arbitrary; the Frank free energy prefers n • ∇ × n ≈ −q0 = 2π/p0 with a helical pitch p0, and, from a geometric perspective, n • ∇ × n = 0 defines a contact structure (48). This allows a number of new integer-valued invariants of chiral textures to be defined (45). A configuration with a single sign of twist is chiral, and two configurations which cannot be connected by a path of chiral configurations are chirally distinct, and hence separated by a chiral energy barrier.…”

“…A configuration with a single sign of twist is chiral, and two configurations which cannot be connected by a path of chiral configurations are chirally distinct, and hence separated by a chiral energy barrier. Within each chiral class of configuration, additional topological invariants may be defined using methods of contact topology (45)(46)(47)(48), such as layer numbers. Changing these chiral topological invariants requires passing through a nonchiral configuration.…”

The topology of knots and links in nematics

“…For example, in the case of the smectic there are subtle issues to do with whether the knot is fibred, in cholesterics one expects a far greater number of metastable states due to the tendency of the cholesteric to twist. We have proposed contact topology as an appropriate tool with which to study such structures [49], but it remains far from a complete picture.…”