Abrikosov dislocation lattice in a model of the cholesteric–to–smectic-Atransition

Abstract

The nematic--to--smectic-A transition in liquid crystals is analogous to the normal to superconducting transition in metals with the Frank director n in liquid crystals playing the role of the vector potential A in metals. The liquid-crystal analog of an external magnetic field is a field, arising, for example, from molecular chirality, leading to a nonzero \ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}n in the equilibrium nematic phase. The cholesteric (twisted nematic) phase is the analog of a normal metal in an external magnetic field. In type-II superconductors in an external magnetic field, the Abrikosov flux lattice phase with partial flux penetration intervenes between the low-temperature Meissner phase and the high-temperature normal-metal phase. In this paper we study the analog in liquid crystals containing chiral molecules of the Abrikosov phase in superconductors. Using a covariant form of the de Gennes free energy, we find that in mean-field theory a state, which we call the twist-grain-boundary (TGB) state, with regularly spaced grain boundaries consisting of parallel screw dislocations, intervenes between the smectic and cholesteric phases. We calculate the liquid-crystal analogs of the upper and lower critical fields ${H}_{c2}$ and ${H}_{c1}$. The properties of the TGB phase depend on the angle 2\ensuremath{\pi}\ensuremath{\alpha} between axes of dislocations in adjacent grain boundaries. \ensuremath{\alpha} can be rational or irrational. When \ensuremath{\alpha}=p/q for mutually prime integers p and q, the TGB state has a q-fold screw axis and quasicrystalline symmetry for crystallographically forbidden q. Our calculations ignore exponentially small terms favoring lock in at rational \ensuremath{\alpha}. We calculate the x-ray scattering intensities in the cholesteric phase near the TGB phase boundary and in the TGB phase for rational and irrational \ensuremath{\alpha}. We also discuss experimental difficulties in observing the TGB state and the possible effects fluctuations not included in mean-field theory might have on its existence.