Frustrated Order on Extrinsic Geometries

Abstract

We study, analytically and theoretically, defects in a nematically-ordered surface that couple to the extrinsic geometry of a surface. Though the intrinsic geometry tends to confine topological defects to regions of large Gaussian curvature, extrinsic couplings tend to orient the nematic in the local direction of maximum or minimum bending. This additional frustration is unavoidable and most important on surfaces of negative Gaussian curvature, where it leads to a complex ground state thermodynamics. We show, in contradistinction to the well-known effects of intrinsic geometry, that extrinsic curvature expels disclinations from the region of maximum curvature above a critical coupling threshold. On catenoids lacking an "inside-outside" symmetry, defects are expelled altogether.PACS numbers: 61.30. Dk, 61.30.Jf,61.30.Hn The intrinsic curvature of a surface frustrates the order of materials living upon it, making defects necessary even in the ground state [1]. This frustration derives from the incompatibility of straight and parallel directions on surfaces with Gaussian curvature. A nematic texture on a sphere, for example, must have a net topological charge of +2 [2-4], as do, for example, lines of latitude on the globe. On more complex surfaces with non-uniform intrinsic curvature, the ground-state organization of defects is well-described theoretically by competition between defect-defect interactions, which favor separation of like-charged defects, and curvature-defect interactions, which favor the localization of defects in regions of relatively high Gaussian curvature [5,6]. This picture is far from complete, however, as real surfaces are endowed with an extrinsic geometry that depends in detail on how they sit in space [7][8][9]. Stripes on a curved surface, for example, tend to orient along"flat" directions to minimize their bending energy, an effect observed in block copolymer films [10].In this Letter, we demonstrate that the extrinsic geometry of a surface generates a strong and previously unstudied source of frustration which controls the ground state of anisotropically-ordered surfaces. While the intimate relationship between topological defects and intrinsic curvature is well-established, we show that the coupling between the spatial organization of defects and extrinsic geometry is as important, and in some cases more important, than intrinsic geometry for determining the ground state of nematic order on anisotropic surfaces. Away from umbilics, extrinsic geometry gives rise to symmetry-breaking fields in regions of high curvature that favor uniform order most strongly in precisely those regions where the intrinsic geometry favors the locally "isotropic" textures generated by defects. The frustration between textures favorable to intrinsic and extrinsic energies becomes particularly pronounced on surfaces of negative Gaussian curvature. As prototypical example this frustration, we study the ground state of nematic order on the surface of catenoid (see Fig. 1). While the intrinsic curvature favors...

Perspective: Geometrically frustrated assemblies

“…20 To clarify, a surface (a membrane or a substrate) is intrinsically curved when is it has non-zero Gaussian curvature, K g = κ 1 κ 2 0. While extrinsic curvature of the surface, as described by, say, the mean curvature H = (κ 1 + κ 2 )/2, may impose costs on assemblies due to shape-dependent strains on the assembly, 45 the requirement of non-zero extrinsic curvature does not, itself, introduce geometric frustration. For example, in a model where a membrane assembly has preferred mean curvature H = H 0 , assemblies with K g = 0 (i.e., cylinders) achieve this geometry without introducing gradients in the long-range ordering of sub-units.…”

Curvature-Controlled Topological Defects

“…We consider two qualitatively different geometries. First, we study possible defect structures on a catenoid's [21] surface (see Figure 1) defined by…”

“…In our modelling, we considered only the so-called intrinsic free energy curvature terms. However, as illustrated in [3,21,29], in general, extrinsic curvature terms are also present. The intrinsic terms penalize in-plane distortions of ordering which are independent of embedding of a 2D manifold in 3D.…”

“…The ETCC mechanism is useful for the geometries that possess surface patches exhibiting significantly different characteristic Gaussian curvatures. We illustrated its efficiency on an open catenoid [21] geometry, where topology does not require the presence of TDs. Our simulations reveal that if a catenoid's neck is squeezed enough, so that the local curvature is strong enough to nucleate the formation of pairs {defect, antidefect}, four m = −1/2 TDs appear at the neck in order to compensate the local negative Gaussian curvature.…”

“…Consequently, on increasing the extrinsic term strength TDs would be expelled from the neck region. This was demonstrated in [21] where a simple nematic director field description was employed. The competition of these contradicting ordering mechanisms is the focus of our future paper.…”