The structure of TGB(C) phases near the upper critical twist k(c2)

The analogy between superconductors and smectic liquid crystals leads to the prediction of the intriguing twist-grain-boundary (TGB) phase, the liquid crystal analog of the Abrikosov vortex lattice phase in type-II superconductors. Motivated by the idea proposed by de Gennes, Renn and Lubensky constructed a detailed description of the TGB phase in smectic-A systems and the theory for the transition to it from the cholesteric phase. They determined, in mean field theory, that the TGB state, which appears between the disordered cholesteric phase and the ordered smectic phase, consists of uniformly spaced grain boundaries composed of equally spaced parallel screw dislocations. The analogous critical fields were calculated and the most favourable lattice structure for the TGBA phase was also determined following the Abrikosov theory for the vortex lattice system. In the present work, we investigate the transition from the cholesteric to TGB C phases near the upper critical twist field h c2. Analyzing the chiral version of a generalized phenomenological De Gennes free energy used in the Chen-Lubensky model, we find the differential equations for the smectic order parameter near the proposed transition. From the stability analysis of the cholesteric phase within the harmonic theory, two different configurations are found to be stable. In one the layer normal to the smectic slabs is on the plane perpendicular to the pitch-axis (the Renn-Lubensky phase), and in the other the layers tilt toward the pitch-axis and their normals rotate on a cone coaxial to it (the Bordeaux phase). We analyze the order paramater equations using both variational and numerical methods, and we introduce an analytical interpolation formula that closely tracks the numerical solution and that can be used to calculate nonlinear terms. We calculate the upper critical field hc2 for both phases and analyze the stability for a given set of parameters in the phenomenological free energy. We minimize the free energy with respect to the dislocation lattice parameters and determine the ratio of the distance between the grain boundaries ( lb) to the distance between the dislocations within a grain boundary (ld) and their dependence on cholesteric pitch P and smectic layer spacing d. We find that both the order parameter profile and the grain boundary structure for these phases are similar to that of the TGBA phase. We have found no sign of a melted grain boundary phase suggested by Dozov in which the smectic order collapses not only at the dislocations but actually throughout the plane of the grain boundary. In contrast, we find that the smectic order parameter, having a robust value at the grain boundaries, is nearly constant along the pitch axis