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# Wrinkling crystallography on spherical surfaces

Edited by Howard A. Stone, Princeton University, Princeton, NJ, and approved November 24, 2014 (received for review June 19, 2014)

## Significance

Curved crystals cannot comprise hexagons alone; additional defects are required by both topology and energetics that depend on the system size. These constraints are present in systems as diverse as virus capsules, soccer balls, and geodesic domes. In this paper, we study the structure of defects of the crystalline dimpled patterns that self-organize through curved wrinkling on a thin elastic shell bound to a compliant substrate. The dimples are treated as point-like packing units, even if the shell is a continuum. Our results provide quantitative evidence that our macroscopic wrinkling system can be mapped into and described within the framework of curved crystallography, albeit with some important differences attributed to the far-from-equilibrium nature of our patterns.

## Abstract

We present the results of an experimental investigation on the crystallography of the dimpled patterns obtained through wrinkling of a curved elastic system. Our macroscopic samples comprise a thin hemispherical shell bound to an equally curved compliant substrate. Under compression, a crystalline pattern of dimples self-organizes on the surface of the shell. Stresses are relaxed by both out-of-surface buckling and the emergence of defects in the quasi-hexagonal pattern. Three-dimensional scanning is used to digitize the topography. Regarding the dimples as point-like packing units produces spherical Voronoi tessellations with cells that are polydisperse and distorted, away from their regular shapes. We analyze the structure of crystalline defects, as a function of system size. Disclinations are observed and, above a threshold value, dislocations proliferate rapidly with system size. Our samples exhibit striking similarities with other curved crystals of charged particles and colloids. Differences are also found and attributed to the far-from-equilibrium nature of our patterns due to the random and initially frozen material imperfections which act as nucleation points, the presence of a physical boundary which represents an additional source of stress, and the inability of dimples to rearrange during crystallization. Even if we do not have access to the exact form of the interdimple interaction, our experiments suggest a broader generality of previous results of curved crystallography and their robustness on the details of the interaction potential. Furthermore, our findings open the door to future studies on curved crystals far from equilibrium.

The classic design of a soccer ball, with its 20 hexagonal (white) patches interspersed with 12 (black) pentagons, the buckminsterfullerene C_{60} (1), virus capsules (2), colloidosomes (3), and geodesic architectural domes (4) are all examples of crystalline packings on spherical surfaces. In contrast with crystals on flat surfaces, these structures cannot be constructed from a tiling of hexagons alone. Instead, disclinations––nonhexagonal elements such as the 12 pentagons on a soccer ball––are required by topology (5, 6), which constrains how the crystal order must comply with the geometry of the underlying surface. For example, seeding a hexagonal crystal with a pentagon (fivefold disclination) disrupts the perfect hexagonal symmetry and introduces a localized stress concentrator, which can be relaxed through out-of-plane deformation with positive Gaussian curvature (7, 8). Likewise, a heptagon (sevenfold disclination) induces a disturbance with negative Gaussian curvature.

An example of a physical realization of curved crystals is found in experiments on colloidal emulsions, where equally charged particles self-organize at the curved interface of two immiscible liquids (3, 9⇓–11). These experiments build upon a wealth of previous theoretical and numerical investigations, as reviewed by Bowick and Giomi (12). For small system sizes, similarly to the soccer ball above, the “simplest” spherical crystals have exactly 12, fivefold disclinations, located at the vertices of a regular icosahedron (13). When the number of particles is sufficiently large, additional defects known as dislocations (5–7 disclination dipoles, which are not required by topology) emerge and break the translational order and lower the energy of the crystal more efficiently than pentagons alone (14, 15). In spherical packings with large number of particles, dislocations typically connect into linear chains to form scars (16) (strings of dislocations attached to a pentagonal disclination) and pleats (10) (strings of dislocations), which in contrast with flat space, start and terminate within the crystal (16). It is therefore organized collections of dislocations, rather than disclinations or isolated dislocations, that predominantly screen curvature in large systems. Disclinations, dislocations, and chains of dislocations interact not only with each other (e.g., through elasticity of the crystal), but also with the curvature of the substrate by a geometric potential that depends on the particular type of defect (17). Their total number and arrangement is primarily dictated by energetics, in addition to the topological constraints on the number of excess disclinations. The challenge in rationalizing these systems is enhanced by the fact that the number of metastable states grows exponentially with system size (18).

Crystallography on curved surfaces has also been considered in the context of deformable elastic membranes with internal crystalline order (8, 12). Elastic stresses in membranes, adhered to curved substrates (19⇓–21), can be relaxed either by (*i*) out-of-surface buckling through wrinkling for compliant substrates, or (*ii*) the in-surface proliferation of topological defects for rigid substrates. For example, out-of-plane deformations in freestanding graphene sheets have been directly linked to energy minimization in the neighborhood of topological defects (22).

Here, we study a macroscopic model system in which a curved crystal arises from the wrinkling of a hemispherical shell bound to an equally curved compliant substrate (schematic in Fig. 1*A*). The dimpled pattern (Fig. 1*B*) self-organizes from an originally smooth surface when the sample is compressed and eventually buckles to relax the stress induced by depressurizing an undersurface cavity. Profilometry through laser scanning provides access to the topography of the patterns (Fig. 1*D*). From the positions of dimple centers, we construct spherical Voronoi tessellations (Fig. 1*E*) and find a striking agreement between the Voronoi construction and the network of ridges of the experimental pattern. We therefore regard the dimples as (point-like) packing units that repel one another, due to storage of elastic strain energy, which themselves form a continuum elastic shell. As such, we study the nucleation process and quantify the defect structure of our wrinkling patterns (Fig. 1*F*). By drawing analogies with the packing of particles on curved surfaces, we find that our system relaxes stresses both by out-of-surface buckling through the formation of arrays of dimples, and by simultaneously developing topological defects in these patterns. Moreover, our macroscopic system is found to mimic many of the identifying features of other curved crystals (9, 10) (a priori far from obvious, given the difference of the underlying physics), despite some important differences in the morphology of defects and their rapid growth with system size that we attribute to the far-from-equilibrium nature of our system.

## Wrinkling on Curved Surfaces: Our Experiments

We have recently introduced an experimental system to study wrinkling on curved surfaces, as smart morphable surfaces for aerodynamic drag reduction (23). Periodic wrinkling patterns emerge from the mechanical instability of a thin stiff film adhered to a soft foundation under compression. Whereas wrinkling of flat-plate–substrate systems is well understood (24⇓–26), investigations of the curved counterpart have been mostly limited to numerics (26⇓–28) and microscopic experiments (29, 30), where it is challenging to independently tune the control parameters.

Our centimetric hemispherical samples (radius *R*) comprise a thin–stiff shell adhered to a soft–thick substrate and were fabricated out of silicone-based elastomers using rapid prototyping techniques (see the schematic in Fig. 1*A* and *Materials and Methods* for the fabrication details and ranges of parameters). Using the pneumatic apparatus shown in Fig. 1*C*, a pressure difference *r*. Above a critical load, an undulatory wrinkling pattern emerges from the originally smooth shell (Fig. 1*B*), with a characteristic wrinkling wavelength λ dictated by the combination of geometric and material properties of the film and substrate.

We have found (23) that the curvature of the substrate leaves the wrinkling length scale unchanged, compared with that predicted for flat infinite substrates (25),*h* is the thickness of the film, and *B*). From here on, we focus exclusively on these dimpled patterns to characterize and analyze their crystallographic structure.

## Three-Dimensional Scanning and Spherical Voronoi Construction

The full 3D surface profile of the samples was digitized using a laser scanner (Fig. 1*C*). In Fig. 1*D*, we present the resulting topographic map of the radial surface depth *d* (measured from the outer spherical surface) for a representative fully developed dimpled pattern. Dimples (blue regions) are crater-like depressions, separated by ridges (red regions). Using an image processing algorithm developed in-house (*Materials and Methods*), we identify the spherical coordinates of the centers (local minima of *d*) of all of the dimples in a sample (yellow markers superposed in Fig. 1*E*). With the coordinates of the dimple locations at hand, we then construct a spherical Voronoi tessellation (black lines superposed in Fig. 1*E*).

It is remarkable that the skeleton provided by the Voronoi construction accurately delineates the underlying network of ridges of the experimental pattern (Fig. 1*E*), suggesting that each dimple is well represented by the corresponding Voronoi cell. As such, the dimples can be regarded as quasi-particles with characteristic interparticle distance λ (given by Eq. **1**). Because the system is under compression, these quasi-particles repel one another through an elastic potential, the precise characterization of which would require a detailed theoretical description that goes beyond the scope of our experimental work. We regard our dimpled patterns as self-organized tilings of a well-defined individual unit––the dimple––that packs into a quasi-hexagonal arrangement constrained by the underlying curved surface. In Fig. 1*F*, we show an example of the output of our procedure: a Voronoi tiling, where each dimple is replaced by the corresponding Voronoi cell. This representation will be used extensively below, to analyze our patterns.

## Crystallization of the Dimpled Patterns

We first turn to the process of nucleation and then describe the structure of the fully developed crystalline patterns.

### Nucleation.

In Fig. 2 *A–H* we present snapshots (top views) of one of our samples during a loading and unloading cycle, starting from a spherical (undeformed) configuration at **1**, *Materials and Methods*).

A few dimples first emerge, nonuniformly (Fig. 2*A*, *B* and *C*, *D*, *E*), there is no rearrangement of the dimples. The unloading path is, however, qualitatively different. Gradually decreasing the differential pressure from *E–G*). Back at *H* is remarkably different from that of Fig. 2*A*, which is significant of hysteresis.

In Fig. 2*I*, we quantify this hysteretic behavior by plotting the average depth of dimples *d* as a function of

We highlight that the position of each dimple remains fixed after nucleation and throughout the evolution of the pattern (Fig. 2 *A–D*,

### Structure of the Crystalized Dimpled Patterns.

We now make use of the Voronoi representation introduced above to further analyze the experimental patterns. In Fig. 3 *A–F* we show a series of examples of Voronoi tilings superimposed on top of the scanned data for samples with increasing values of **1**), while keeping all other parameters fixed (

For all samples in Fig. 3 *A–F*, the most prominent cells are hexagonal (in blue) as expected from crystallinity. As *A*), representative of small system sizes, with seven hexagons and three isolated pentagonal disclinations. For *B*), series of dislocation defects (5–7 disclination dipoles) appear, in addition to hexagons and isolated pentagons. This pattern comprises one isolated pentagonal disclination, two isolated dislocations, and two strings of dislocations which resemble a scar and a pleat. Note, however, that a true scar and pleat would start and terminate in the interior of the curved crystal (33), but in our case they often do so at the boundary. Still, the overall scenario in our experimental patterns is analogous to that found in other curved crystals (9⇓–11). For even larger sample sizes (see Fig. 3 *D*–*F*, for

In Fig. 3*G*, we present the cap ratio *α*, the ratio between the area of the spherical cap that is analyzed and the area of the corresponding full sphere *α* increases with *α* for *E* and *F* represent the portions of the samples that were not scanned.

### Polydispersity and Topology of Dimples.

We proceed by quantifying the area of the Voronoi cells that underlies the crystalized dimpled patterns, as well as the topology of their tilings. For this, we consider statistical ensembles of the individual cells associated with each dimple, whose coordination number allows for their classification as pentagons, hexagons, or heptagons. By way of example, we focus on a set of nine samples (

Polydispersity is measured using the ratio **1**) between the parallel sides. A constant value of *A*, we present the probability density function (PDF), *A*). Whereas

In addition to polydispersity, we quantify the morphology of the Voronoi cells by measuring their shape factor,

In Fig. 4*B*, we plot the PDF for shape factor of all cells, *C* representative examples of cells obtained by sampling the PDFs at specific values of ζ, for each of the polygon families. For example, at

## Quantification of the Defect Structure

Thus far, we have learned that the tilings of our dimpled patterns consist of polydisperse Voronoi cells, with a distribution of distorted polygons, away from their regular shapes. Whereas previous studies focused on more monodisperse systems (9, 10, 16), Euler’s packing theorem is applicable to general tilings. As such, we follow an approach similar to that of refs. 9, 10 and quantify the defect structure versus system size

### Net Defect Charge.

The topological charge

is the deviation of the coordination number *Z* from that of a perfect hexagonal packing; *α* was quantified in Fig. 3*G* for our samples) as

We now analyze our data following a procedure recently used for curved colloidal crystals (10). For a given sample, we measure the net defect charge *β* of their area with that of a sphere; Fig. 5*A*, *Inset*) up to the maximum possible cap allowed for that sample (Fig. 3*G*), such that 0 < *β* ≤ *α*. Such a measurement includes both the effects of isolated disclinations and the polarization charge due to the nonuniform distribution of disclination dipoles [by analogy with electrostatics (10)]. In this procedure, the contribution to the polarization charge is accounted for by the cumulative counting when the boundary of the analyzed patches with increasing sizes dissects a pleat or a dislocation and adds toward the total topological charge, which would not occur for a full sample. In Fig. 5*A*, we plot this net defect charge as a function of the integrated Gaussian curvature,

### Average Coordination Number.

Given the scatter in *N* dimples. However, before quantifying *R* is large compared with λ and the hexagonal cells far outnumber disclinations, we assume that the area of each dimple is *A*). In turn, the total number of dimples on a spherical cap is

In Fig. 5*B*, we plot the experimental measurements for **4** (solid line).

Toward determining **3**. Combining this result with the definition of *q* and making use of Eqs. **2** and **4** gives

A more general version of Eq. **5** was provided by Nelson (7) but, for completeness, we have reproduced the argument applied specifically to our system.

In Fig. 5*C*, we plot **5**. A perfect hexagonal packing (e.g., on a plane or a cylinder) would have *C*, at

### Number of Dislocations.

Further, we quantify the total number of dislocations *D*, we plot **4**, this translates into a sample with a threshold number of dimples, *α*).

Remarkably, this scenario is qualitatively identical to that of Bausch et al. (9), who found

## Discussion and Conclusion

We have introduced a macroscopic experimental model system where a curved crystalline pattern of dimples self-organizes from the wrinkling of an originally smooth thin elastic shell. The system relaxes stresses both by out-of-surface buckling through the formation of arrays of dimples, and by simultaneously developing defects as nonhexagonal dimples in the otherwise hexagonal patterns. Direct parallels were established between the structure of defects in our system and previous studies on the packing of charged particles on curved surfaces (14⇓–16) and curved colloidal crystals (9, 10), despite the differences in the underlying physics. These similarities include the ability to treat the dimples as point-like units, the use of Voronoi tessellation to characterize their packing, as well as the presence of disclinations and, above a threshold system size, the prominent growth of the number of dislocations to screen the underlying curvature. There are however few important distinctions. We observe dislocations that form branched arrays and clusters and proliferate more rapidly than in curved colloidal crystals (9, 16), and with a different threshold value of system size. We speculate that these differences may be attributed to the fact that we have a different repulsive potential and our system is far from equilibrium due to the random and initially frozen material imperfections that nucleate the pattern, as well as the inability for the dimples to rearrange during crystallization. Consequently, these constraints prevent our system from exploring phase space and lead to additional frustration that increases disorder.

The interaction potential between neighboring dimples in our system is still unknown. However, Bowick et al. (33) find that potentials of the form

## Materials and Methods

### Fabrication of Samples.

We manufactured 32 hemispherical samples, made of silicone-based elastomers, polydimethylsiloxane (PDMS) and vinylpolysiloxane (VPS), for the film and the substrate, respectively, using a protocol that was described previously (23). First, a thin outer shell was made by coating a previously vacuum-formed polystyrene mold with the desired radius. The coating process included wetting the surface of the mold and then draining the excess polymer by gravity. A balance between gravity, viscosity, surface tension, and polymerization rate yielded shells of constant thickness (to within

### Material and Geometric Properties.

The mechanical properties of PDMS and VPS were measured on cylindrical specimens subjected to uniaxial compression using a material testing machine (Zwick). We found a linear stress–strain response of these materials within the levels of compression relevant to our experiments. The ratio between the Young moduli of the film and substrate could be controlled within the range

### Three-Dimensional Scanning and Image Analysis.

The surface topography of the dimpled patterns on the hemispherical samples was digitized using a 3D laser scanner (NextEngine) and the resulting cloud of points was postprocessed using MATLAB. The Cartesian coordinates *ϕ* and *θ* are the azimuth and polar angles, and *ρ* is the radial distance from each point to the centroid of the sphere that best fits the outer surface of the sample. In our analysis we used a stitched combination of *ρ* as the field variable was thresholded into black and white binaries, corresponding to valleys of the dimples and the ridges, respectively. Determining the centroids of all black blobs yielded the coordinates of the centers of all dimples, from which the final Voronoi tessellation was constructed.

## Acknowledgments

The authors thank A. Bresson for help with preliminary experiments. P.M.R. acknowledges support from MIT’s Charles E. Reed faculty initiative fund, the National Science Foundation, CMMI-1351449 Faculty Early Career Development (CAREER) Program, and is grateful to U. Deusto for hospitality. M.B. thanks the Fulbright Program. D.T. thanks the Belgian American Education Foundation (BAEF), the Fulbright Program, and the Wallonie-Bruxelles International Excellence Grant WBI World.

## Footnotes

↵

^{1}Present address: Faculty of Mechanical Engineering, University of Ljubljana, SI-1000 Ljubljana, Slovenia.↵

^{2}Present address: Faculté des Sciences, Université Libre de Bruxelles (ULB), Bruxelles 1050, Belgium.- ↵
^{3}To whom correspondence should be addressed. Email: preis{at}mit.edu.

Author contributions: M.B. and P.M.R. designed research; M.B., D.T., R.L., and P.M.R. performed research; M.B. and R.L. contributed new reagents/analytic tools; M.B., D.T., and P.M.R. analyzed data; and M.B. and P.M.R. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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