Christian D. Santangelo

Designing Responsive Buckled Surfaces by Halftone Gel Lithography

Smooth Operator When thin sheets are compressed they can buckle and wrinkle, such as when the edges of a sheet of paper, or two areas of skin, are pushed together. Variations in local thickness and stiffness will alter the buckling patterns, but controlling this in a simple and predictable way is difficult. Kim et al. (p. 1201; see the Perspective by Sharon) used halftone lithography with two photomasks to create highly cross-linked dots embedded in a lightly cross-linked matrix of a swellable polymer. This material could generate “smooth” swelling profiles on thin sheets with arbitrary two-dimensional geometries so that complex three-dimensional structures could be produced. Halftone lithography can pattern two-dimensional swellable gels to produce complex three-dimensional shapes. Self-actuating materials capable of transforming between three-dimensional shapes have applications in areas as diverse as biomedicine, robotics, and tunable micro-optics. We introduce a method of photopatterning polymer films that yields temperature-responsive gel sheets that can transform between a flat state and a prescribed three-dimensional shape. Our approach is based on poly(N-isopropylacrylamide) copolymers containing pendent benzophenone units that allow cross-linking to be tuned by irradiation dose. We describe a simple method of halftone gel lithography using only two photomasks, wherein highly cross-linked dots embedded in a lightly cross-linked matrix provide access to nearly continuous, and fully two-dimensional, patterns of swelling. This method is used to fabricate surfaces with constant Gaussian curvature (spherical caps, saddles, and cones) or zero mean curvature (Enneper’s surfaces), as well as more complex and nearly closed shapes.

Using origami design principles to fold reprogrammable mechanical metamaterials

Folding robots and metamaterials The same principles used to make origami art can make self-assembling robots and tunable metamaterials—artificial materials engineered to have properties that may not be found in nature (see the Perspective by You). Felton et al. made complex self-folding robots from flat templates. Such robots could potentially be sent through a collapsed building or tunnels and then assemble themselves autonomously into their final functional form. Silverberg et al. created a mechanical metamaterial that was folded into a tessellated pattern of unit cells. These cells reversibly switched between soft and stiff states, causing large, controllable changes to the way the material responded to being squashed. Science, this issue p. 644, p. 647; see also p. 623 Origami folded sheets can be structurally altered by adding defects to control the mechanical properties. [Also see Perspective by You] Although broadly admired for its aesthetic qualities, the art of origami is now being recognized also as a framework for mechanical metamaterial design. Working with the Miura-ori tessellation, we find that each unit cell of this crease pattern is mechanically bistable, and by switching between states, the compressive modulus of the overall structure can be rationally and reversibly tuned. By virtue of their interactions, these mechanically stable lattice defects also lead to emergent crystallographic structures such as vacancies, dislocations, and grain boundaries. Each of these structures comes from an arrangement of reversible folds, highlighting a connection between mechanical metamaterials and programmable matter. Given origami’s scale-free geometric character, this framework for metamaterial design can be directly transferred to milli-, micro-, and nanometer-size systems.

Programming Reversibly Self‐Folding Origami with Micropatterned Photo‐Crosslinkable Polymer Trilayers

Self-folding microscale origami patterns are demonstrated in polymer films with control over mountain/valley assignments and fold angles using trilayers of photo-crosslinkable copolymers with a temperature-sensitive hydrogel as the middle layer. The characteristic size scale of the folds W = 30 μm and figure of merit A/ W (2) ≈ 5000, demonstrated here represent substantial advances in the fabrication of self-folding origami.

Origami structures with a critical transition to bistability arising from hidden degrees of freedom

Origami is used beyond purely aesthetic pursuits to design responsive and customizable mechanical metamaterials. However, a generalized physical understanding of origami remains elusive, owing to the challenge of determining whether local kinematic constraints are globally compatible and to an incomplete understanding of how the folded sheets material properties contribute to the overall mechanical response. Here, we show that the traditional square twist, whose crease pattern has zero degrees of freedom (DOF) and therefore should not be foldable, can nevertheless be folded by accessing bending deformations that are not explicit in the crease pattern. These hidden bending DOF are separated from the crease DOF by an energy gap that gives rise to a geometrically driven critical bifurcation between mono- and bistability. Noting its potential utility for fabricating mechanical switches, we use a temperature-responsive polymer-gel version of the square twist to demonstrate hysteretic folding dynamics at the sub-millimetre scale.

Thermally responsive rolling of thin gel strips with discrete variations in swelling

The process by which spatial variations in growth transform two-dimensional elastic membranes into three-dimensional shapes is both a fundamentally interesting mechanism of shape selection and a powerful tool for the preparation of responsive materials. From the perspective of lithographic patterning of thin gel sheets, it is most straightforward to prepare materials consisting of discrete regions with different degrees of swelling. However, the sharp variations in swelling at the boundaries between such regions make it impossible for the sheet to adopt a configuration that is free of in-plane stresses everywhere. Thus, the deformation of such materials is not well understood. Here, we consider the geometrically simple case of a photo-crosslinkable poly(N-isopropylacrylamide) copolymer patterned into thin rectangular strips divided into one high- and one low-swelling region. When swelled in an aqueous medium at 22 °C, the sheet rolls into a three-dimensional shape consisting of two nearly cylindrical regions connected by a transitional neck. Heating to 50 °C leads to fully reversible de-swelling back to a flat configuration. We propose a scaling argument based on a balance between stretching and bending energies that relates the curvature of the 3D shape to the width and thickness of the strip, find good agreement with experimental data and numerical simulations, and further demonstrate how this simple geometry provides a powerful route for the fabrication of self-folding stimuli-responsive micro-devices.

Soft spheres make more mesophases

We use both mean-field methods and numerical simulation to study the phase diagram of classical particles interacting with a hard-core and repulsive, soft shoulder. Despite the purely repulsive interaction, this system displays a remarkable array of aggregate phases arising from the competition between the hard-core and shoulder length scales. In the limit of large shoulder width to core size, we argue that this phase diagram has a number of universal features, and classify the set of repulsive shoulders that lead to aggregation at high density. Surprisingly, the phase sequence and aggregate size adjusts so as to keep almost constant inter-aggregate separation.

Computing Counterion Densities at Intermediate Coupling

By decomposing the Coulomb interaction into a long-distance component appropriate for mean-field theory, and a non-mean-field short distance component, we compute the counterion density near a charged surface for all values of the counterion coupling parameter. A modified strong-coupling expansion that is manifestly finite at all coupling strengths is used to treat the short-distance component. We find a nonperturbative correction related to the lateral counterion correlations that modifies the density at intermediate coupling.

Swelling-driven rolling and anisotropic expansion of striped gel sheets

Swelling of spatially patterned gel films promises many new opportunities for understanding and exploiting differential growth as a means to define three-dimensional (3D) configurations of thin sheets. Here, we consider the swelling-driven deformation of sheets patterned with alternating parallel strips of material with high and low equilibrium extents of swelling, and the dependence of this process on film thickness, strip width, and swelling contrast. Temperature-responsive, photo-crosslinkable hydrogels are prepared from poly(N-isopropyl acrylamide) (PNIPAm) copolymers with pendent benzophenone groups that allow the degree of crosslinking to be programmed by the dose of UV light. Photo-patterned multi-strips deform by rolling around the axis perpendicular to the interface between high- and low-swelling regions, due to a balance between stretching energy in the transition region and bending of the entire sheet. However, when the strip width falls below a critical size proportional to the film thickness, the patterned sheets instead remain flat, undergoing greater expansion along the direction normal to the interfaces than in the direction parallel. This configuration suggests possibilities for patterning anisotropic growth, and also provides fruitful information on the contrast in modulus between the regions.

Distribution of counterions near discretely charged planes and rods

Realistic charged macromolecules are characterized by discrete (rather than homogeneous) charge distributions. We investigate the effects of surface charge discretization on the counterion distribution at the level of mean-field theory using a two-state model. Both planar and cylindrical geometries are considered; for the latter case, we compare our results to numerical solutions of the full Poisson-Boltzmann equation. We find that the discretization of the surface charge can cause enhanced localization of the counterions near the surface; for charged cylinders, counterion condensation can exceed Oosawa-Manning condensation.

Topological Mechanics of Origami and Kirigami.

Origami and kirigami have emerged as potential tools for the design of mechanical metamaterials whose properties such as curvature, Poisson ratio, and existence of metastable states can be tuned using purely geometric criteria. A major obstacle to exploiting this property is the scarcity of tools to identify and program the flexibility of fold patterns. We exploit a recent connection between spring networks and quantum topological states to design origami with localized folding motions at boundaries and study them both experimentally and theoretically. These folding motions exist due to an underlying topological invariant rather than a local imbalance between constraints and degrees of freedom. We give a simple example of a quasi-1D folding pattern that realizes such topological states. We also demonstrate how to generalize these topological design principles to two dimensions. A striking consequence is that a domain wall between two topologically distinct, mechanically rigid structures is deformable even when constraints locally match the degrees of freedom.