The coexistence of cholesteric and 2-dimensional orders

Abstract

Some solutions of rigid polymers like DNA or PBLG display at the same time cholesteric order and local hexagonal order. The coexistence of two such orders is impossible on long distances. Two cylindrical geometries of molecular arrangements, extending transversally on a distance of the order of the pitch, and achieving a good compromise between these two incompatible tendencies, are studied. Both of them are such that the molecules are parallel to two families of surfaces, i.e. they display 2-d ordering, but still suffer double twist One of the geometries, called below geometry a, seems to apply to recent observations of the chromosome of dinoflagellate [9] ; it is shown that it is favoured when K3 > K1 and that its nucleation is indeed encouraged by the tendency to local 2-d ordering. The other one, b, is the already well-known double twisted structure used as an element of construction of the blue phase. It is favoured when K1 < K3, and it is shown that it nucleates easily when the saddle-splay constant K24 is positive and large enough. These results suggest a new conception of the blue phase as a compromise between cholesteric and nematic order, and shed a new light on the question of BP III, the amorphous blue fog. Finally a curved space description is given of an ordered medium which is at the same time cholesteric and 2-d ordered. This curved space « crystal » generalizes the curved space description of the blue phase and suggests very new and fundamental questions about the nature of the frustration in molten polymers