The difference between the moduli of the glassy PS cylinders and the rubbery PEP matrix gives rise to a strong contrast in atomic force microscopy (AFM) images. ![]() This polymer was chosen in part for its thermal properties, as room temperature is below the T g of the polystyrene (PS) block and above the T g of the poly(ethylene-alt-propylene) (PEP) block. In particular, we consider the cylinder-forming block copolymer polystyrene-block-poly(ethylene-alt-propylene) (PS-PEP). ![]() At a constant temperature above the glass transition temperature ( T g) of both blocks, the average distance between disclinations measured by the orientational correlation length ξ 2 scales with time as ξ 2 ∼ t 1 / 4 ( 16). Both challenges can be overcome by studying thin films of cylinder-forming block copolymers a monolayer of cylinders behaves as a 2D smectic, wherein the density of defects can be controlled quite easily through coarsening. The two main obstacles to studying the coupling between disclinations and elastic deformations are accurately controlling the distribution of disclinations and simultaneously measuring both the height and the smectic phase of a free-standing membrane. By dictating the distribution of topological defects, it should be possible to control the specific non-Euclidean geometry of the membrane. 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. 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. 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). 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). Gaussian curvature can also stabilize more exotic defects, including scars ( 6), fractionalized defect charges ( 7), and pleats ( 8). Non-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).
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