Mesoscale structure of wrinkle patterns and defect-proliferated liquid crystalline phases

Abstract

Thin solids often develop elastic instabilities and subsequently complex, multiscale deformation patterns. Revealing the organizing principles of this spatial complexity has ramifications for our understanding of morphogenetic processes in plant leaves and animal epithelia, and perhaps even the formation of human fingerprints. We elucidate a primary source of this morphological complexity -an incompatibility between an elastically-favored "microstructure" of uniformly spaced wrinkles and a "macro-structure" imparted through the wrinkle director and dictated by confinement forces. Our theory is borne out of experiments and simulations of floating sheets subjected to radial stretching. By analyzing patterns of grossly radial wrinkles we find two sharply distinct morphologies: defect-free patterns with a fixed number of wrinkles and non-uniform spacing, and patterns of uniformly spaced wrinkles separated by defect-rich buffer zones. We show how these morphological types reflect distinct minima of a Ginzburg-Landau functional -a coarse-grained version of the elastic energy, which penalizes nonuniform wrinkle spacing and amplitude, as well as deviations of their actual director from the axis imposed by confinement. Our results extend the effective description of wrinkle patterns as liquid crystals (H. Aharoni et al., Nat. Commun. 8:15809, 2017), and we highlight a fascinating analogy between the geometry-energy interplay that underlies the proliferation of defects in the mechanical equilibrium of confined sheets and in thermodynamic phases of superconductors and chiral liquid crystals.T hin solid bodies tend to suppress compression by developing wrinkles -elongated periodic undulations. Wrinkle patterns are ubiquitous due to the broad range of conditions that generate compression: boundary loads [1], incompatible topographical constraints [2-4], differential swelling [5], expansion on soft substrates [6,7], and growth in confined spaces [8,9], have all been recognized as potential drivers of wrinkled morphologies. A basic picture, often used to model these phenomena, is uniformly-spaced undulations along parallel lines (lower part of Fig. 1a). However, most observed patterns differ significantly from such a simplistic picture, demonstrating instead how multi-scale patterns emerge under smooth, featureless forcing.A predominant source of complexity here is a conflict between two primary features: a wavelength λ (i.e. distance between nearby peaks), and a directorn -the axis along which the sheet undulates. The former is a micro-scale object, typically determined by a local balance of bending rigidity and the stiffness of an "effective substrate" [1,10], whereas the latter is a macro-scale field that reflects the confining topography and lateral forces exerted on the body (|∇n| λ −1 ) [11][12][13]. The basic pattern of perfectly parallel wrinkles emerges when a sheet is confined uniaxially. However, deviations from this ideal picture occur when either the director or the locally-favored wavelength are non-uniform ( Fig...

Van der Waals interaction affects wrinkle formation in two-dimensional materials

“…The number of wrinkles decreases sharply with the number of layers N. b. The wrinkle number ( ) increases with radial distance r. Furthermore, a detail inspection indicates that ( ) ∝ , such that wrinkles are "nucleated" at various locations 32 , and the wavelength ℓ( ) 2 / ( ) is close to a constant value, ℓ( ) ℓ (see Supplementary Information Fig. 2).…”

“…The wrinkle number increases with radial distance r. Furthermore, a detailed inspection indicates that , such that wrinkles are “nucleated” at various locations ( 32 ), and the wavelength is close to a constant value, ( SI Appendix , Fig. S2 ).…”

“…Denoting the membrane-substrate distance by , the wrinkle pattern can be modeled as: where and are, respectively, the amplitude and number of azimuthal undulations at a distance r > R from the center of the bubble. From the AFM images we can characterize the shape and observe a few noteworthy features: The number of wrinkles decreases sharply with the number of layers N. The wrinkle number increases with radial distance r. Furthermore, a detailed inspection indicates that , such that wrinkles are “nucleated” at various locations ( 32 ), and the wavelength is close to a constant value, ( SI Appendix , Fig. S2 ).…”

“…Furthermore, a detailed inspection indicates that mðrÞ ∝ r, such that wrinkles are "nucleated" at various locations (32), and the wavelength ℓðrÞ ¼ 2πr=mðrÞ is close to a constant value, ℓðrÞ ¼ ℓ 0 (SI Appendix, Fig. S2).…”

Sculpting liquids with ultrathin shells

“…Here we focus on ultrathin (∼ 100 nm) polymer films that strongly resist in-plane stretching yet readily wrinkle, allowing them to conform to a wide range of surface topographies [12]. Such films have given a window into the rich interplay between geometry and mechanics in thin solids [5,10], including connections to pattern formation in liquid crystals [13][14][15]. Current understanding in this area has been driven primarily by studies on planar sheets [16,17].…”