Shape and fluctuations of positively curved ribbons

Abstract

We study the shape and shape fluctuations of incompatible, positively curved ribbons, with a flat reference metric and a spherelike reference curvature. Such incompatible geometry is likely to occur in many self-assembled materials and other experimental systems. Ribbons of this geometry exhibit a sharp transition between a rigid ring and an anomalously soft spring as a function of their width. As a result, the temperature dependence of these ribbons' shape is unique, exhibiting a nonmonotonic dependence of the persistence and Kuhn lengths on the temperature and width. We map the possible configuration phase space and show the existence of three phases: At high temperatures it is the ideal chain phase, where the ribbon is well described by classical models (e.g., wormlike chain model). The second phase, for cold and narrow ribbons, is the plane ergodic phase; a ribbon in this phase might be thought of as made out of segments that gyrate within an oblate spheroid with extreme aspect ratio. The third phase, for cold, wide ribbons, is a direct result of the residual stress caused by the incompatibility, called the random structured phase. A ribbon in this phase behaves on large scales as an ideal chain. However, the segments of this chain are not straight; rather they may have different shapes, mainly helices (both left and right handed) of various pitches.

Shaping by Internal Material Frustration: Shifting to Architectural Scale

“…The reference curvature in such systems stems from variations in the inelastic strain, 𝜖, across the thickness t. The typical magnitude of the reference principal curvatures is |k| ≈ 𝜀 t . Incompatible shells with Euclidean reference metric and isotropic, [13,27] or saddle-like [29][30][31] reference curvature, in either disc or ribbon geometries, were studied. Such systems demonstrate a sharp shape transition, between the curvature-dominated regime and the metric-dominated regime.…”

“…Beyond its pure scientific importance and its relevance to biological [ 10 , 11 , 12 ] and nanoscale [ 13 , 14 , 15 ] systems, such ability to morph thin plates into complex shapes is appealing to various domains. Specifically, it can have a high impact in architecture and industrial design, where the manufacturing of complex curved surfaces with the investment of minimal energy/effort, is a major challenge.…”

On the Packing of Stiff Rods on Ellipsoids Part II -- Entropy and Self Avoidance

“…exhibit residual stresses, in the absence of any external forcing, that arise directly from the filaments being limited onto a confining manifold [16][17][18]. As a result of this, a complex configuration space is expected to arise, affecting the statistics and mechanics of such system [19,20].…”

Cumulative geometric frustration and superextensive energy scaling in a nonlinear classical XY-spin model

“…Instead, the ground state is obtained via a compromise that best resolves the conflict, which might be non-local. In recent years, geometric frustration was shown to underlie the mechanism that leads to a variety of non-trivial morphological responses, including size limitation [1], filamentation [2] and other non-trivial response properties [3][4][5][6]. Such unique characteristics have been observed in several different settings including a colloidal crystal confined to a spherical interface [2], filamentation of irregular elastic hexagons [7], twisted molecular crystals [8] and bundles of twisted filaments [9].…”