Materials Geometry @ UMass Amherst

(Image credit: Zina Deretsky, National Science Foundation)

In a recent paper, we describe a method to design 3D shapes through the controlled growth of a thin polymer sheet. The method relies on a polymer film whose internal chemistry, and consequently its response to stimuli, is altered by shining ultraviolet light (UV) through a mask. The polymer film swells like a sponge in water, but will swell less where UV light has shined through the holes in the mask and more where it has not. The high swelling regions then buckle to accommodate the extra material. To make the technique useful to control the 3D shape, we borrow a technique from printing, “half-toning,” in which a pattern of low swelling dots are embedded in a high swelling background. By changing the local size of the dots, we can achieve a very fine-tuned control over the local degree of swelling.

J. Kim, J.A. Hanna, M. Byun, C.D. Santangelo, and R.C. Hayward, “Designing responsive buckled surfaces by halftone gel lithography,” Science 335, 1201-1205 (2012). [JOURNAL].

Also see the Perspective by Eran Sharon or the UMass press release.

Update: Recent press: Chemical and Engineering News, Nature, Physics Today.

We also studied a similar material system with in-plane swelling, the “bistrip”, in another article in the journal Soft Matter. There, a strip of high swelling is attached to a strip of low swelling material along their long edges. Surprisingly, the result rolls up to a radius that scales with the thickness of the strip to the 2/3 power.

J. Kim, J.A. Hanna, R.C. Hayward, and C.D. Santangelo, “Thermally responsive rolling of thin gel strips with discrete variations in swelling,” Soft Matter 8, 2375-2381 (2012). [JOURNAL].

Riemann’s minimal surface, shown here, is a singly-periodic surface of channels connecting flat layers. In a new paper, we show that it can be decomposed into two more fundamental defects, called screw dislocations. See the paper:
S.A. Matsumoto, C.D. Santangelo and R.D. Kamien, “Smectic pores and defect cores,” accepted Interfaces Focus, [ARXIV] (2012).

Abstract: Riemann’s minimal surfaces are a complete, embeddable, one-parameter family of minimal surfaces with translational symmetry along one direction. It’s infinite number of planar ends are joined together by an array of necks, closely matching the morphology of a bicontinuous, lamellar system with pores connecting alternating layers. We demonstrate explicitly that Riemann’s minimal surfaces are composed of a nonlinear sum of two oppositely-handed helicoids. This description is particularly appropriate for describing smectic liquid crystals containing two screw dislocations.

Smectic liquid crystals are composed of layers which can bend but otherwise maintain close to a uniform spacing. In such a material, the layer shapes are represented by a displacement giving their deviation from the ideal ground state configuration. The layered nature of this phase connects the bending energy of the layers to the spacing between the layers in a complex manner. A proper understanding of how the layers arrange themselves requires us to account for these connections. Similar difficulties arise in the elasticity of thin elastic sheets.

We have developed a method to explore deformations of the layers around particular configurations called edge dislocation (shown in the figure). To do so, we start with a focal texture – a possibly complex configuration of exactly equally-spaced layers. We are then able to relax the constraint of constant spacing by balancing the bending energy with the layer spacing using a technique called a BPS decomposition. It turns out the two methods are closely allied, allowing us to construct approximate layer configurations that were impossible before.

G. Alexander, R.D. Kamien and C.D. Santangelo, “Developed smectics: When exact solutions agree”, accepted to Physical Review Letters (2011). [ARXIV].

Packing problems have a long history in physics and mathematics. Often hundreds of years or more elapse between the statement of a particular packing problem and the proof that the best packing has been found. For example, the best way to pack spheres has been known, undoubtedly, for millennia. It is as a face-centered cubic lattice – much the way grocers stack oranges. The proof, however, was only in 1999. Packing problems are important in understand how molecules and particles of different shapes arrange themselves in space, and the various shapes of molecules and particles have inspired a search for the densest packings of many different shapes.

What is the densest way to pack squares into a larger square? It is a problem that is deceptively simple to state, yet subtle and difficult to solve. We have simplified this problem further by asking for the densest packing of squares into a larger square with periodic boundary conditions, in which the sides of a square are identified. Even in this packing problem, exact results are sparse. On the one hand, the addition of translation symmetry simplifies the arrangements of squares, yet numerical analysis indicates that the densest packings can be still quite unpredictable. See, for example, the densest packing of 23 squares that we found.

D. Blair, C.D. Santangelo and J. Machta, “Packing squares in a torus,” accepted to J. Stat, [ARXIV] (2011).

Want to try your hand at packing regular polygons? Visit http://donblair.org/grains6a/

There are a great many proteins that localize to and collectively generate curvature in biological fluid membranes. We study changes in the topology of fluid membranes due to the presence of highly anisotropic, curvature-inducing proteins. Generically, we find a surprisingly rich phase diagram with phases of both positive and negative Gaussian curvature. As a concrete example modeled on experiments, we find that a lamellar phase in a negative Gaussian curvature regime exhibits a propensity to form screw dislocations of definite burgers scalar but of both chirality. The induced curvature depends strongly on the membrane rigidity, suggesting membrane composition can be a factor regulating membrane sculpting to to curvature-inducing proteins.

Updated reference:
K. Akabori and C.D. Santangelo, “Membrane morphology induced by anisotropic proteins,” Phys. Rev. E 84, 061909 (2011). [ARXIV]

We study, analytically and theoretically, defects in a nematically-ordered surface that couple to the extrinsic geometry of a surface. Though the intrinsic geometry tends to confine topological defects to regions of large Gaussian curvature, extrinsic couplings tend to orient the nematic in the local direction of maximum or minimum bending. This additional frustration is unavoidable and most important on surfaces of negative Gaussian curvature, where it leads to a complex ground state thermodynamics. We show, in contradistinction to the well-known effects of intrinsic geometry, that extrinsic curvature expels disclinations from the region of maximum curvature above a critical coupling threshold. On catenoids lacking an “inside-outside” symmetry, defects are expelled altogether.

Badel Mbanga, Greg M. Grason and C.D. Santangelo, accepted to Physical Review Letters (2011). [ARXIV]

Recent experiments have imposed controlled swelling patterns on thin polymer films, which subsequently buckle into three-dimensional shapes. We develop a solution to the design problem suggested by such systems, namely if and how one can generate particular three-dimensional shapes from thin elastic sheets by mere imposition of a two-dimensional pattern of locally isotropic growth. Not every shape is possible. Several types of obstruction can arise, some of which depend on the sheet thickness. We provide some examples using the axisymmetric form of the problem, which is analytically tractable. In particular, we find that drum-like shapes are swellable in principle, though perhaps currently impractical.

M.A. Dias, J.A. Hanna and C.D. Santangelo, “Programmed buckling by controlled lateral swelling in a thin elastic sheet”, to appear in Physical Review E (2011). [arXiv]

Differential growth processes play a prominent role in shaping leaves and biological tissues. Using both analytical and numerical calculations, we consider the shapes of closed, elastic strips which have been subjected to an inhomogeneous pattern of swelling. The stretching and bending energies of a closed strip are frustrated by compatibility constraints between the curvatures and metric of the strip. To analyze this frustration, we study the class of “conical” closed strips with a prescribed metric tensor on their center line. The resulting strip shapes can be classified according to their number of wrinkles and the prescribed pattern of swelling. We use this class of strips as a variational ansatz to obtain the minimal energy shapes of closed strips and find excellent agreement with the results of a numerical bead-spring model. Within this class of strips, we derive a condition under which a strip can have vanishing mean curvature along the center line.

B.G. Chen and C.D. Santangelo, “Minimal resonances in non-Euclidean strips,” to appear in Physical Review E (2010); arXiv:1007.2862

UPDATE: Figure from this paper selected as PRE Kaleidoscope image.

The mechanism by which a patterned state accommodates the breaking of translational symmetry by a phase boundary or a sample wall has been addressed in the context of Landau branching in type-I superconductors, refinement of magnetic domains, and compressed elastic sheets. We explore this issue by studying an ultrathin polymer sheet floating on the surface of a fluid, decorated with a pattern of parallel wrinkles. At the edge of the sheet, this corrugated profile meets the fluid meniscus. Rather than branching of wrinkles into generations of ever-smaller sharp folds, we discover a smooth cascade in which the coarse pattern in the bulk is matched to fine structure at the edge by the continuous introduction of discrete, higher wavenumber Fourier modes.
The observed multiscale morphology is controlled by a dimensionless parameter that quantifies the relative strength of the edge forces and the rigidity of the bulk pattern.

Paper

J. Huang, B. Davidovitch, C.D. Santangelo, T.P. Russell and N. Menon, “A smooth cascade of wrinkles at the edge of a floating elastic film,” to appear in Physical Review Letters (2010).
[Arxiv]