Two-dimensional matter: order, curvature and defects

Abstract Many systems in nature and the synthetic world involve ordered arrangements of units on two-dimensional surfaces. We review here the fundamental role payed by both the topology of the underlying surface and its Gaussian curvature. Topology dictates certain broad features of the defect structure of the ground state but curvature-driven energetics control the detailed structure of the ordered phases. Among the surprises are the appearance in the ground state of structures that would normally be thermal excitations and thus prohibited at zero temperature. Examples include excess dislocations in the form of grain boundary scars for spherical crystals above a minimal system size, dislocation unbinding for toroidal hexatics, interstitial fractionalization in spherical crystals and the appearance of well-separated disclinations for toroidal crystals. Much of the analysis leads to universal predictions that do not depend on the details of the microscopic interactions that lead to order in the first place. These predictions are subject to test by the many experimental soft- and hard-matter systems that lead to curved ordered structures such as colloidal particles self-assembling on droplets of one liquid in a second liquid. The defects themselves may be functionalized to create ligands with directional bonding. Thus, nano- to meso-scale superatoms may be designed with specific valency for use in building supermolecules and novel bulk materials. Parameters such as particle number, geometrical aspect ratios and anisotropy of elastic moduli permit the tuning of the precise architecture of the superatoms and associated supermolecules. Thus, the field has tremendous potential from both a fundamental and materials science/supramolecular chemistry viewpoint. Keywords: topological defectsorientational ordercurvaturedifferential geometry Acknowledgements We thank V. Vitelli, A. Hexemer, W. Irvine, P. Chaikin, A. Lucas and A. Fonseca for sharing with us some of the images and data presented in this review. We thank D. R. Nelson, H. Shin and A. Travesset for long-standing collaborations and M. Forstner for interesting discussions regarding amphiphilic membranes. Notes Note 1. The expression of the bending energy used in Equation (186b) is that originally given by Helfrich for a membrane with zero pre-existing curvature 168 Helfrich, W. 1973. Z. Naturforsch., 28C: 693[Crossref] , [Google Scholar]. Often the equivalent expression is found in the literature. In this case, however, the mean curvature H is defined as the sum of the principal curvatures rather than their average.