Defects in chiral columnar phases: Tilt-grain boundaries and iterated moiré maps

Biomolecules are often very long with a definite chirality. DNA, xanthan, and poly-{γ}-benzyl-glutamate (PBLG) can all form columnar crystalline phases. The chirality, however, competes with the tendency for crystalline order. For chiral polymers, there are two sorts of chirality: the first describes the usual cholestericlike twist of the local director around a pitch axis, while the second favors the rotation of the local bond-orientational order and leads to a braiding of the polymers along an average direction. In the former case, chirality can be manifested in a tilt-grain boundary phase analogous to the Renn-Lubensky phase of smectic-A liquid crystals. In the latter case, we are led to a new ``moir\'e'' state with twisted bond order. In the moir\'e state, polymers are simultaneously entangled, crystalline, and aligned, on average, in a common direction. In this case the polymer trajectories in the plane perpendicular to their average direction are described by iterated moir\'e maps of remarkable complexity, reminiscent of dynamical systems. {} 1996 The American Physical Society.