Curvature-Controlled Defect Localization in Elastic Surface Crystals

Francisco López Jiménez, Norbert Stoop, Romain Lagrange, Jörn Dunkel, and Pedro M. Reis

Phys. Rev. Lett. 116, 104301 – Published 7 March 2016

Abstract

We investigate the influence of curvature and topology on crystalline dimpled patterns on the surface of generic elastic bilayers. Our numerical analysis predicts that the total number of defects created by adiabatic compression exhibits universal quadratic scaling for spherical, ellipsoidal, and toroidal surfaces over a wide range of system sizes. However, both the localization of individual defects and the orientation of defect chains depend strongly on the local Gaussian curvature and its gradients across a surface. Our results imply that curvature and topology can be utilized to pattern defects in elastic materials, thus promising improved control over hierarchical bending, buckling, or folding processes. Generally, this study suggests that bilayer systems provide an inexpensive yet valuable experimental test bed for exploring the effects of geometrically induced forces on assemblies of topological charges.

Received 20 September 2015

DOI:https://doi.org/10.1103/PhysRevLett.116.104301

Physics Subject Headings (PhySH)

Defects Elastic deformation

Crystal structures Surfaces

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Francisco López Jiménez¹, Norbert Stoop², Romain Lagrange², Jörn Dunkel^2,*, and Pedro M. Reis^1,3,†

¹Department of Civil & Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA
²Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA
³Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA

^*Corresponding author. dunkel@mit.edu
^†Corresponding author. preis@mit.edu

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Vol. 116, Iss. 10 — 11 March 2016

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Images

Figure 1
Buckling-induced crystalline surface patterns on a thin film of thickness $h$ adhered to a soft core (top), obtained by minimization of Eq. (1), and their dual hexagonal Voronoi tessellations (bottom) for different surface geometries: (a) sphere ( $R / h = 70$ ), (b) ellipsoid ( $R_{x} = 2 R_{y} = 2 R_{z} = 110 h$ ), and (c) torus ( $r / h = 16, R / h = 80$ ). The color bar represents the surface elevation. The outlined surface domain in (b) highlights a representative chain of defects. Voronoi cells are color coded by their coordination number $Z$ .
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Figure 2
(a) The total number of defects grows linearly with $\sqrt{N}$ , where $N$ is total number of lattice units, exhibiting similar slopes for all geometries (solid line: linear fit for spheres). Inset: Excess dislocations on spheres were predicted [6] to increase linearly with reduced radius $ρ = R / λ$ for colloidal crystals (solid line). For comparison, the best-fit power law $(ρ - ρ_{c})^{β}$ with $ρ_{c} = 4.5 \pm 0.4$ and $β = 0.67 \pm 0.08$ is also shown (dashed line). (b)–(d) The total defect charge grows differently with integrated Gaussian curvature $I$ for different geometries. The gray shading represents the conditional PDFs of the total charge $Q$ for a given value of $I$ . The red dashed line corresponds to linear growth $I = (π / 3) Q$ . Dark regions in the surface sketches illustrate integration domains.
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Figure 3
Curvature-induced defect localization on ellipsoids. (a) Isolated pentagonal $+ 1$ defects accumulate in high curvature regions; $p$ is the average number of single defects per ellipsoid. (b) Although defect chains form preferentially near the equator ( $| ϕ | = 0$ ) their orientation angles $α$ , measured relative to the tangent vectors $t$ , show no significant ordering. (c),(d) Voronoi tessellations for $R_{x} / h = 40$ and $R_{x} / h = 160$ . Ellipsoids with $R_{x} / h \geq 110$ show a weak alignment of the lattices along lines $ϕ = 0$ (elements with the same lattice orientation are connected by solid black lines).
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Figure 4
Defect localization and superstructures on tori. (a) At the onset of instability, $a = a_{c}$ , pure hexagonal phases are stable only for tori with $r / R < 2 / 3$ (dashed line, Eq. (2)), while striped patterns occur at angles $ϕ > ϕ_{c}$ for larger aspect ratios. Simulations for $a / a_{c} = 0.98$ agree well with the isoline estimates of the symmetry-breaking terms in the energy density, $γ (r / R, ϕ) = C$ , with $C = 0.42$ (solid line). (b) Isolated penta- and heptagonal defects segregate; $p$ and $n$ are the average numbers of positive and negative disclinations per torus. (c) Orientations of charged scars and neutral pleats correlate strongly with their transversal centroid position $| ϕ |$ . (d) Defect positions are also strongly correlated along $θ$ . (e),(f) While this positional ordering is less prominent in small systems [(e), $R / h = 40$ ], it becomes apparent for larger systems [(f) $R / h = 160$ ]. The toroidal superstructure (elements with the same lattice orientation are connected by solid black lines) follows geodesics of minimal integrated absolute Gaussian curvature (dashed green line).
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