Wrinkles and splay conspire to give positive disclinations negative curvature

Abstract

Recently, there has been renewed interest in the coupling between geometry and topological defects in crystalline and striped systems. Standard lore dictates that positive disclinations are associated with positive Gaussian curvature, whereas negative disclinations give rise to negative curvature. Here, we present a diblock copolymer system exhibiting a striped columnar phase that preferentially forms wrinkles perpendicular to the underlying stripes. In freestanding films this wrinkling behavior induces negative Gaussian curvature to form in the vicinity of positive disclinations.disclinations and curvature | diblock copolymer | smectic | free-standing membrane | wrinkling instability N on-Euclidean geometry has been shown to be one of the most robust mechanisms used to prescribe the configuration of defects in crystalline (1, 2) or striped phases (3-5). Gaussian curvature can also stabilize more exotic defects, including scars (6), fractionalized defect charges (7), and pleats (8). Likewise, if a 2D crystal is allowed to buckle out of the plane, the elastic energy associated with isolated disclinations can be strongly reduced by screening their strain fields through curvature, trading off stretching for bending energy (9, 10). Depending on the sign of the topological charge, isolated defects can deform the membrane into cone-or saddle-shape configurations, acting as sources of Gaussian curvature (9, 11). Here we demonstrate experimentally, theoretically, and through simulations that the molecular splay distortions associated with disclinations in a free-standing smectic membrane act as sources of Gaussian curvature, resulting in a pattern of wrinkles in the membrane that form perpendicular to the underlying smectic layers. Dramatically, wrinkling changes the very nature of the curvature-defect coupling, making positive disclinations sources of negative curvature in contrast to intuition gained from geodesic domes and soccer balls. By dictating the distribution of topological defects, it should be possible to control the specific non-Euclidean geometry of the membrane.The mechanical anisotropy of striped phases can give the bulk material markedly different properties, depending on the relative orientation of the stiffer direction with respect to the underlying stripes. Although the case in which both the stiffer direction and the stripes are parallel has been previously studied (3, 4), we turn our attention to the peculiarities of the perpendicular case, in particular the preference of the perturbed system to form wrinkles orthogonal to the stripes. Defects, ubiquitous in 2D systems, also signal locations of ill-defined elasticity and thus function as sources of geometry in a free-standing membrane. We note that purely molecular splay can drive this unusual behavior in nematic elastomers due to anisotropic swelling (12-15). However, the present system relies on wrinkle formation due to anisotropic elastic moduli as a means of coupling curvature to disclinations.The two main obstacles to studying the coupling bet...

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