Magnetic self-assembly of three-dimensional surfaces from planar sheets

Abstract

This report describes the spontaneous folding of flat elastomeric sheets, patterned with magnetic dipoles, into free-standing, 3D objects that are the topological equivalents of spherical shells. The path of the self-assembly is determined by a competition between mechanical and magnetic interactions. The potential of this strategy for the fabrication of 3D electronic devices is demonstrated by generating a simple electrical circuit surrounding a spherical cavity.folding ͉ microfabrication ͉ 3D structure ͉ soft lithography ͉ soft electronics T he strategies used to form 3D micro-and nanostructures in cells and by humans differ. Proteins, RNAs, and their aggregates, the most complex, 3D molecular structures in nature, form by the spontaneous folding of linear precursors (1). The ubiquity of this strategy reflects the efficiency with which the cell synthesizes linear precursors by sequential formation of covalent bonds. Microelectronic devices, the most complex 3D structures generated by humans, are fabricated by stacking and connecting planar layers (2). This strategy is dictated by the availability of highly developed methods for parallel microfabrication in 2D and the absence of effective, general methods for fabrication in 3D (3).Folding of connected, 2D plates [using robotics (4) or spontaneous folding (5-9)] can yield 3D microelectromechanical systems (MEMS) and microelectronic devices. We (10, 11) and others (4, 12) have explored a number of routes to small 3D shapes based on self-assembly. These strategies are still early in their development.Here, we explore a new strategy for formation of 3D objects that combines the advantages of planar microfabrication with those of 3D self-assembly. Our approach comprises four steps (Fig. 1a): (i) cutting the 3D surface of interest into connected sections that ''almost'' unfold into a plane (unpeeling a sphere as one unpeels an orange is an example); (ii) flattening this surface and projecting it onto a plane; (iii) fabricating the planar projection in the form of an elastomeric membrane patterned with magnetic dipoles; and (iv) allowing this patterned membrane to fold into an ''almost-correct'' 3D shape by self-assembly. This strategy offers the potential to transform easily patterned, functionalized planar sheets into 3D structures and devices. It also raises the problem of designing and generating stable 3D structures by decomposing and projecting these structures into 2D shapes and then balancing the shapes of 2D cuts, the placement of magnetic dipoles, and the mechanical characteristics of the membrane.Converting sheets into 3D objects by folding and creasing is a very old, remarkably interesting, and still incompletely resolved problem in applied mathematics (13-16). The inverse problemmapping the surface of a 3D shape (specifically, the surface of the Earth) onto a flat sheet-has been at the core of cartography since the times of Frisius (1508Frisius ( -1555 and Mercator (1512Mercator ( -1594. Although it is known that a flat, inextensible surface cannot fold i...

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