Flexible sheet polymers or “membranes” can be regarded as two-dimensional generalizations of linear polymer chains, for which there is a vigorous theoretical and experimental literature. Flexible membranes should exhibit even more richness and complexity, for two basic reasons. The first is that important geometric concepts like intrinsic curvature, orientability and genus, which have no direct analogue in linear polymers, appear naturally in discussions of two-dimensional macromolecules. Our understanding of the interplay between these concepts and the statistical mechanics of surfaces is still in its infancy. [1] The second reason is that surfaces can exist in a variety of different phases. The possibility of a two-dimensional shear modulus in planar membranes shows that we must distinguish between solids and liquids when these objects are allowed to crumple into three dimensions. Hexatic membranes, with extended six-fold bond orientational order, provide yet another important possibility. All three phases have quite distinctive properties. [2, 3] There are no such sharp distinctions for linear polymer chains.
Distinguishing noisy crystalline structures using bond orientational order parameters