Crystals and Liquid Crystals Confined to Curved Geometries

Abstract

This review introduces the elasticity theory of two-dimensional crystals and nematic liquid crystals on curved surfaces, the energetics of topological defects (disclinations, dislocations and pleats) in these ordered phases, and the interaction of defects with the underlying curvature. This chapter concludes with two cases of three-dimensional nematic phases confined to spaces with curved boundaries, namely a torus and a spherical shell.

Topological vacancies in spherical crystals

“…Geometric frustration of ordered phases in twodimensional condensed matters like crystals and liquid crystals confined on curved surfaces can create a myriad of emergent static [1][2][3][4] and dynamic [5][6][7][8] defect structures. Through the Thomson problem of finding the ground state of electrically charged particles confined on a sphere [9][10][11][12][13] and the generalized versions on typical curved surfaces [4,14], several defect motifs in crystalline order like isolated disclinations, dislocations, scars and pleats have been identified in the past decades. [15][16][17][18][19] As a fundamental topological defect in crystalline order, a disclination refers to a vertex whose coordination number n is deviated from six in two-dimensional triangular lattice, and a topological charge of q = 6−n can be assigned to an n-fold disclination.…”

Phase diagram of SALR fluids on spherical surfaces

“…However, in this case one also needs to consider the additional lengthscale of the sphere radius, as well as its topology: these two factors introduce a more complex behavior in the phase diagram. One consequence is that even regular patterns include ineliminable topological defects (disclinations) 43 , which manifest as structures with a number of neighbors smaller than that expected in the plane for cluster and bubble crystals, and with pole defects in stripe patterns. The second consequence is that by changing the ratio σ /R one finds patterns with different numbers of clusters, stripes, or bubbles.…”

“…where the sum is carried over the topological charges of the defects q k , and the integral represents the Euler characteristic ξ = 2 for the 2-sphere, proportional to the integral of the local gaussian curvature G over the surface. By computing the topological charge of different kinds of disclinations 43 , one can easily predict the geometry that will emerge: for instance 12-cluster crystals must necessarily contain 12 five-fold disclinations, because their topological charge is 1 6 , while stripe patterns always contain two pole disclinations with charge 1.…”

Stress driven fractionalization of vacancies in regular packings of elastic particles

“…Topological defects are emergent structures commonly seen in various ordered condensed media [1][2][3][4][5]. As a fundamentally important crystallographic defect, vacancies are highly involved in several important physical processes in both two- [6][7][8] and three-dimensional systems, [9,10] such as in facilitating migration of atoms, [11,12] and crystallization of DNA-programmable nanoparticles, [13,14] and in resolving geometric frustrations in curved crystals [7,8,[15][16][17].…”