Wrinkling crystallography on spherical surfacesMiha Brojana,1, Denis Terwagneb,2, Romain Lagrangec, and Pedro M. Reisa,b,3

Departments of aMechanical Engineering, bCivil and Environmental Engineering, and cMathematics, Massachusetts Institute of Technology, Cambridge,MA 02139

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

We present the results of an experimental investigation on thecrystallography of the dimpled patterns obtained through wrin-kling of a curved elastic system. Our macroscopic samples comprisea thin hemispherical shell bound to an equally curved compliantsubstrate. Under compression, a crystalline pattern of dimples self-organizes on the surface of the shell. Stresses are relaxed by bothout-of-surface buckling and the emergence of defects in the quasi-hexagonal pattern. Three-dimensional scanning is used to digitizethe topography. Regarding the dimples as point-like packing unitsproduces spherical Voronoi tessellations with cells that are poly-disperse and distorted, away from their regular shapes. We analyzethe structure of crystalline defects, as a function of system size.Disclinations are observed and, above a threshold value, dislocationsproliferate rapidly with system size. Our samples exhibit strikingsimilarities with other curved crystals of charged particles andcolloids. Differences are also found and attributed to the far-from-equilibrium nature of our patterns due to the random and initiallyfrozen material imperfections which act as nucleation points, thepresence of a physical boundarywhich represents an additional sourceof stress, and the inability of dimples to rearrange during crystalliza-tion. Even if we do not have access to the exact form of theinterdimple interaction, our experiments suggest a broader generalityof previous results of curved crystallography and their robustness onthe details of the interaction potential. Furthermore, our findings openthe door to future studies on curved crystals far from equilibrium.

mechanical instabilities | curved surfaces | pattern formation | packing |defects

The classic design of a soccer ball, with its 20 hexagonal(white) patches interspersed with 12 (black) pentagons, the

buckminsterfullerene C60 (1), virus capsules (2), colloidosomes(3), and geodesic architectural domes (4) are all examples ofcrystalline packings on spherical surfaces. In contrast with crys-tals on flat surfaces, these structures cannot be constructed froma tiling of hexagons alone. Instead, disclinations––nonhexagonalelements such as the 12 pentagons on a soccer ball––are requiredby topology (5, 6), which constrains how the crystal order mustcomply 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 local-ized stress concentrator, which can be relaxed through out-of-plane deformation with positive Gaussian curvature (7, 8). Like-wise, a heptagon (sevenfold disclination) induces a disturbancewith negative Gaussian curvature.An example of a physical realization of curved crystals is found

in experiments on colloidal emulsions, where equally chargedparticles self-organize at the curved interface of two immiscibleliquids (3, 9–11). These experiments build upon a wealth of previoustheoretical and numerical investigations, as reviewed by Bowickand Giomi (12). For small system sizes, similarly to the soccer ballabove, the “simplest” spherical crystals have exactly 12, fivefolddisclinations, located at the vertices of a regular icosahedron (13).When the number of particles is sufficiently large, additionaldefects known as dislocations (5–7 disclination dipoles, which arenot required by topology) emerge and break the translationalorder and lower the energy of the crystal more efficiently thanpentagons alone (14, 15). In spherical packings with large numberof particles, dislocations typically connect into linear chains to

form scars (16) (strings of dislocations attached to a pentagonaldisclination) and pleats (10) (strings of dislocations), which incontrast with flat space, start and terminate within the crystal (16).It is therefore organized collections of dislocations, rather thandisclinations or isolated dislocations, that predominantly screencurvature in large systems. Disclinations, dislocations, and chainsof dislocations interact not only with each other (e.g., throughelasticity of the crystal), but also with the curvature of the sub-strate by a geometric potential that depends on the particular typeof defect (17). Their total number and arrangement is primarilydictated by energetics, in addition to the topological constraintson the number of excess disclinations. The challenge in ratio-nalizing these systems is enhanced by the fact that the number ofmetastable states grows exponentially with system size (18).Crystallography on curved surfaces has also been considered

in the context of deformable elastic membranes with internalcrystalline order (8, 12). Elastic stresses in membranes, adheredto 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 rigidsubstrates. For example, out-of-plane deformations in free-standing graphene sheets have been directly linked to energyminimization 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 boundto an equally curved compliant substrate (schematic in Fig. 1A).The dimpled pattern (Fig. 1B) self-organizes from an originallysmooth surface when the sample is compressed and eventuallybuckles to relax the stress induced by depressurizing an un-dersurface cavity. Profilometry through laser scanning pro-vides access to the topography of the patterns (Fig. 1D). Fromthe positions of dimple centers, we construct spherical Voronoi tes-sellations (Fig. 1E) and find a striking agreement between the

Significance

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

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

2Present address: Faculté des Sciences, Université Libre de Bruxelles (ULB), Bruxelles 1050,Belgium.

3To whom correspondence should be addressed. Email: preis@mit.edu.

14–19 | PNAS | January 6, 2015 | vol. 112 | no. 1 www.pnas.org/cgi/doi/10.1073/pnas.1411559112

Voronoi construction and the network of ridges of the experi-mental pattern. We therefore regard the dimples as (point-like)packing units that repel one another, due to storage of elasticstrain energy, which themselves form a continuum elastic shell.As such, we study the nucleation process and quantify the defectstructure of our wrinkling patterns (Fig. 1F). By drawing analo-gies with the packing of particles on curved surfaces, we find thatour system relaxes stresses both by out-of-surface bucklingthrough the formation of arrays of dimples, and by simulta-neously developing topological defects in these patterns. More-over, our macroscopic system is found to mimic many of theidentifying features of other curved crystals (9, 10) (a priori farfrom obvious, given the difference of the underlying physics),despite some important differences in the morphology of defectsand their rapid growth with system size that we attribute to thefar-from-equilibrium nature of our system.

Wrinkling on Curved Surfaces: Our ExperimentsWe have recently introduced an experimental system to studywrinkling on curved surfaces, as smart morphable surfaces foraerodynamic drag reduction (23). Periodic wrinkling patternsemerge from the mechanical instability of a thin stiff film adheredto a soft foundation under compression. Whereas wrinkling offlat-plate–substrate systems is well understood (24–26), inves-tigations of the curved counterpart have been mostly limited tonumerics (26–28) and microscopic experiments (29, 30), where itis 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 werefabricated out of silicone-based elastomers using rapid proto-typing techniques (see the schematic in Fig. 1A andMaterials andMethods for the fabrication details and ranges of parameters).Using the pneumatic apparatus shown in Fig. 1C, a pressuredifference Δp= pa − pi can be set up between the outside atmo-spheric pressure, pa and pi, inside the undersurface cavity of radiusr. Above a critical load, an undulatory wrinkling pattern emergesfrom the originally smooth shell (Fig. 1B), with a characteristicwrinkling wavelength λ dictated by the combination of geometricand material properties of the film and substrate.

We have found (23) that the curvature of the substrate leavesthe wrinkling length scale unchanged, compared with that pre-dicted for flat infinite substrates (25),

λ= 2πh

"1− ν2s1− ν2f

Ef

3Es

#1=3

; [1]

where h is the thickness of the film, and Ef , νf and Es, νs are theYoung’s moduli and Poisson’s ratios of the film and substrate, re-spectively. However, curvature of the substrate (23, 29, 30) and thelevel of overstress (29, 31) can affect the pattern selection mode.For h=RK 0:01 (and low overstress) labyrinthine patterns werefound (23). On the other hand, for h=RJ 0:01 we observed dim-pled patterns that pack in a hexagonal-like crystal structure (Fig.1B). From here on, we focus exclusively on these dimpled patternsto characterize and analyze their crystallographic structure.

Three-Dimensional Scanning and Spherical VoronoiConstructionThe full 3D surface profile of the samples was digitized usinga laser scanner (Fig. 1C). In Fig. 1D, we present the resultingtopographic map of the radial surface depth d (measured fromthe outer spherical surface) for a representative fully developeddimpled pattern. Dimples (blue regions) are crater-like depres-sions, separated by ridges (red regions). Using an image pro-cessing algorithm developed in-house (Materials and Methods),we identify the spherical coordinates of the centers (local minimaof d) of all of the dimples in a sample (yellow markers super-posed in Fig. 1E). With the coordinates of the dimple locationsat hand, we then construct a spherical Voronoi tessellation(black lines superposed in Fig. 1E).It is remarkable that the skeleton provided by the Voronoi

construction accurately delineates the underlying network of ridgesof the experimental pattern (Fig. 1E), suggesting that each dimple iswell represented by the corresponding Voronoi cell. As such, thedimples can be regarded as quasi-particles with characteristic in-terparticle distance λ (given by Eq. 1). Because the system is undercompression, these quasi-particles repel one another through anelastic potential, the precise characterization of which would re-quire a detailed theoretical description that goes beyond the scopeof our experimental work. We regard our dimpled patterns as self-organized tilings of a well-defined individual unit––the dimple––thatpacks into a quasi-hexagonal arrangement constrained by the un-derlying curved surface. In Fig. 1F, we show an example of theoutput of our procedure: a Voronoi tiling, where each dimple isreplaced by the corresponding Voronoi cell. This representationwill be used extensively below, to analyze our patterns.

Crystallization of the Dimpled PatternsWe first turn to the process of nucleation and then describe thestructure of the fully developed crystalline patterns.

Nucleation. In Fig. 2 A–H we present snapshots (top views) of oneof our samples during a loading and unloading cycle, startingfrom a spherical (undeformed) configuration at Δp= 0 kPa,loading it up to a maximum of Δp= 76:4 kPa, and then unloadingto Δp= 6:5 kPa (the experimental uncertainty of all pressuremeasurements is ±0:12 kPa). This particular sample has radiusR= 20:0 mm and a characteristic dimple size of λ= 4:40± 0:60 mm(set by Ef = 2:10± 0:11 MPa, Es = 0:23± 0:01 MPa, νf = νs ≈ 0:5,and h= 0:48± 0:07 mm that is determined from Eq. 1, Materialsand Methods).A few dimples first emerge, nonuniformly (Fig. 2A, Δp= 6:5

kPa). These are small regions of the initially smooth shell thatbuckle inward and eventually act as nucleation sites from whichthe rest of the pattern progressively grows with Δp (e.g., Fig. 2 Band C, Δp= 13:1 kPa and 23.4 kPa, respectively). The front ofthe crystalline phase spreads into the undimpled portions of theshell, until full coverage of the hemisphere is attained (Fig. 2D,

A

B

C

D FE

Fig. 1. Wrinkling patterns on hemispherical samples. (A) Schematic diagram ofthe setup. (B) Experimental sample: PDMS shell (radius R= 20:0 mm, thicknessh= 0:53± 0:05 mm, Young modulus Ef = 2:10 MPa, Poisson ration νf ≈ 0:5) ad-hered to a VPS substrate (Es = 0:23 MPa, νs ≈ 0:5, radius of cavity, r = 9:5 mm).Undeformed configuration (at Δp= 0) and wrinkled pattern (at Δp= 76:4 kPa)with characteristic wavelength, λ= 4:83± 0:45 mm (uncertainty is SD of alldimples). (C) Experimental apparatus: 3D scanner, pneumatic system, and pres-sure data acquisition system. (D) Surface profile of dimpled pattern. (E) Skeletonof the spherical Voronoi construction (black lines) obtained from the centers ofthe dimples (yellow circles), superposed on surface profile. (F) Voronoi tessella-tion, color coded according to the coordination number.

Brojan et al. PNAS | January 6, 2015 | vol. 112 | no. 1 | 15

APP

LIED

PHYS

ICAL

SCIENCE

S

Δp= 40:3 kPa). Beyond this point, and up to Δp= 76:4 kPa, themaximum depressurization explored (Fig. 2E), there is no rear-rangement of the dimples. The unloading path is, however, qual-itatively different. Gradually decreasing the differential pressurefrom Δp= 76:4 kPa results in patterns whose morphology remainsapproximately unchanged (Fig. 2 E–G). Back at Δp= 6:5 kPa, theconfiguration in Fig. 2H is remarkably different from that of Fig.2A, which is significant of hysteresis.In Fig. 2I, we quantify this hysteretic behavior by plotting the

average depth of dimples d as a function of Δp. For Δp< 40:3 kPa(partial coverage of the sample), there are three distinct paths in themechanical response: one for the first loading ramp and the othertwo for subsequent unloading and loading cycles. All paths convergeabove Δp≈ 40 kPa (beyond which the full sample is crystalline),and up to the maximum Δp≈ 75 kPa explored. Note that oursystem is elastic, the viscosity of the elastomers we use is negligi-ble, and there is no delamination between the film and substrate.As such, we attribute this hysteretic behavior to the series ofmultiple snap-buckling events that must occur on the initiallysmooth spherical shell for each of the individual dimples to form.We highlight that the position of each dimple remains fixed

after nucleation and throughout the evolution of the pattern (Fig.2 A–D, 0<Δp½kPa�< 40:3). Moreover, repeating the experimentswith the same sample leads to identical patterns. The loci of nu-cleation occur presumably at regions of “frozen” material imper-fections (e.g., due to small air bubbles trapped in the elastomer

during curing). The appearance of subsequent dimples propagatesfrom these nucleation sites, which we therefore refer to as anchordimples, until the full surface crystallizes. Interestingly, the depth ofthese anchor dimples does not differ significantly ðK 5%Þ from theaverage dimple depth once the pattern is fully developed. Themechanism by which anchor dimples emerge is still uncertain.However, based on the work of Paulose and Nelson (32), wespeculate that frozen imperfections may generate small soft regionson the cap, which can snap-buckle.

Structure of the Crystalized Dimpled Patterns. We now make use ofthe Voronoi representation introduced above to further analyzethe experimental patterns. In Fig. 3 A–F we show a series ofexamples of Voronoi tilings superimposed on top of the scanneddata for samples with increasing values of R=λ, a measure of therelative system size, attained by changing the shell thickness in therange 0:23< h½mm�< 0:88 (which modifies λ through Eq. 1),while keeping all other parameters fixed (R= 20:0 mm, r= 9:5mm, and Ef=Es = 9:13). Note that larger values of R=λ correspondto thinner shells because λ∼ h. Regions shaded in gray representbands of dimples which were taken into account for identifyingthe coordination number (i.e., number of neighbors) of thedimples inside the solid green line, but otherwise omitted fromfurther quantitative analysis. The coordination numbers of theseborder dimples are undetermined and they cannot be interpretedunder our framework of packing of point-like units. The domainof interest is thus reduced from a hemisphere to a spherical cap.For all samples in Fig. 3 A–F, the most prominent cells are

hexagonal (in blue) as expected from crystallinity. As R=λ is in-creased, pentagonal and heptagonal defects (yellow and red,respectively) become more prominent. The “simplest” latticestructure is found for R=λ= 2:21 (Fig. 3A), representative ofsmall system sizes, with seven hexagons and three isolated pen-tagonal disclinations. For R=λ= 3:75 (Fig. 3B), series of dislo-cation defects (5–7 disclination dipoles) appear, in addition tohexagons and isolated pentagons. This pattern comprises oneisolated pentagonal disclination, two isolated dislocations, andtwo strings of dislocations which resemble a scar and a pleat.Note, however, that a true scar and pleat would start and ter-minate in the interior of the curved crystal (33), but in our casethey often do so at the boundary. Still, the overall scenario in ourexperimental patterns is analogous to that found in other curvedcrystals (9–11). For even larger sample sizes (see Fig. 3 D–F, forR=λ> 4:71), there is a proliferation of more complex arrange-ments of defects with clusters, as well as strings of dislocationsthat are increasingly longer and branched. By contrast, branchingof linear arrays of dislocations, as well as isolated dislocations inlarge system sizes, would not occur in crystals at equilibrium.In Fig. 3G, we present the cap ratio α, the ratio between the area

of the spherical cap that is analyzed and the area of the corre-sponding full sphere 4πR2, as a function of R=λ. As expected, αincreases with R=λ; the relative size of the dimples decreases, andthe required exclusion band is increasingly smaller. The sharp dropin α for R=λJ 6:5, however, is due to a technical difficulty in our3D scanning procedure that forced us to only acquire top-viewscans for these samples with smaller dimples (instead of the fullhemispherical surface obtained from the stitching of multipleperspectives, for R=λK 6:5). The white annuli in Fig. 3 E and Frepresent the portions of the samples that were not scanned.

Polydispersity and Topology of Dimples.We proceed by quantifyingthe area of the Voronoi cells that underlies the crystalizeddimpled patterns, as well as the topology of their tilings. For this,we consider statistical ensembles of the individual cells associatedwith each dimple, whose coordination number allows for theirclassification as pentagons, hexagons, or heptagons. By way ofexample, we focus on a set of nine samples (R= 20:0 mm,Ef=Es = 9:13 and λ= ð3:07± 0:10Þ mm; the uncertainty is the SDof the mean of the set), each with ∼180 dimples.Polydispersity is measured using the ratio Ap =Ai=Ahex, where

Ai is the area of each Voronoi cell, and Ahex =ffiffiffi3

pλ2=2 is the area

A B C D

EFH G

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50 60 70 80

A

C

B

G

D

I

F

H

E

Fig. 2. Crystallization of the dimpled patterns. (A–H) Top-view snapshots ofthe surface profile during a loading–unloading cycle. (A–E) Loading withincreasingΔp= ð6:5,13:2,23:4,40:3,76:4Þ kPa. (E–H) Unloading at the same valuesof Δp. Anchor dimples in A are marked with circles. (I) Experimental measure-ments of the average dimple depth d as a function of Δp. Three loading paths(increasing Δp) and two unloading paths (decreasing Δp) are shown. Solid linesare guides to the eye and the labeled data points correspond to the patterns inA–H. The shell thickness is h= 0:48± 0:07 and other parameters identical to Fig. 1.

16 | www.pnas.org/cgi/doi/10.1073/pnas.1411559112 Brojan et al.

of a regular Euclidean hexagon with a distance λ (defined in Eq. 1)between the parallel sides. A constant value of Ap = 1 for all cellswould correspond to a perfectly monodisperse hexagonal pat-tern, which is, however, unattainable in a curved system. InFig. 4A, we present the probability density function (PDF),PðApÞ, of all Voronoi cells in the ensemble. We find that PðApÞis well described by a Gaussian distribution, PðApÞ= ½1=ðσ ffiffiffiffiffi

2πp Þ�

exp½−ðAp −ApÞ2=ð2σ2Þ�, with a mean A

p= 0:98 and SD σ = 0:12.

The near-unity value of Apis indicative of the hexagonal crys-

talline packing but the significant SD conveys that the dimplesare polydisperse. We have also classified the area of each cellaccording to its coordination number, Ap

7, Ap6, and Ap

5, for hepta-gons, hexagons, and pentagons, respectively. The correspondingrelative PDFs, normalized such that

R∞0 P7dAp

7 +R∞0 P6dAp

6 +R∞0 P5dAp

5 = 1 are also Gaussian distributed (see fits in Fig. 4A).Whereas P6ðAp

6Þ is still peaked near unity, ðAp

6; σ6Þ= ð0:99; 0:11Þ,there is a splitting for the mean areas for heptagons and pentagons,ðAp

7; σ7Þ= ð1:07; 0:12Þ and ðAp

5; σ5Þ= ð0:88; 0:10Þ, respectively. Notethat this splitting occurs nearly symmetrically from A

p. The high

coefficients of variation, σi=Api ≈ 11%, indicate a high degree of

polydispersity for all families of cells.In addition to polydispersity, we quantify the morphology of

the Voronoi cells by measuring their shape factor, ζ=C2i =ð4πAiÞ,

where Ci and Ai are the perimeter and surface area of each cell,respectively (34, 35). This quantity accounts for both the topol-ogy and the level of distortion of each Voronoi cell. The shape

factor is ζ= 1 for a circle and ζ> 1 for all other shapes. Forexample, regular pentagons, hexagons, and heptagons haveζn = n=πtanðπ=nÞ with n= 5; 6, and 7, respectively.In Fig. 4B, we plot the PDF for shape factor of all cells, PðζÞ.

Similarly to PðApÞ above, we also superpose the relative PDFsfor the families of heptagons, hexagons, and pentagons, nor-malized such that

R∞1 P7dζ7 +

R∞1 P6dζ6 +

R∞1 P5dζ5 = 1. The pro-

bability of finding a particular polygon peaks sharply after thevalue of shape factor that corresponds to its regular shape:ζ≈ 1:073; 1:103, and 1.156 for regular heptagons, hexagons, andpentagons, respectively. After these peaks, the corresponding pro-babilities decrease but remain finite for an extended range of ζ.Moreover, the PDF for all cells PðζÞ is nonzero in the broad in-terval of shape factors 1:073< ζ< 1:299, meaning that the tilings ofour dimpled patterns consist of irregular Voronoi cells. To illustratethis spread, we show in Fig. 4C representative examples of cellsobtained by sampling the PDFs at specific values of ζ, for each ofthe polygon families. For example, at ζ= 5=πtanðπ=5Þ, the value fora regular pentagon, distorted hexagons, and heptagons are alsofound. Likewise, for a specific family, increasing ζ corresponds toincreasingly more distorted polygons.

Quantification of the Defect StructureThus far, we have learned that the tilings of our dimpledpatterns consist of polydisperse Voronoi cells, with a distribu-tion of distorted polygons, away from their regular shapes.Whereas previous studies focused on more monodispersesystems (9, 10, 16), Euler’s packing theorem is applicable togeneral tilings. As such, we follow an approach similar to thatof refs. 9, 10 and quantify the defect structure versus systemsize R=λ for 32 samples in the range 2:09< λ½mm�< 8:01(0:23< h½mm�< 0:88, while fixing R= 20 mm and Ef=Es = 9:13).

Net Defect Charge. The topological charge q= πs=3 is commonlyused to quantify defects of curved crystals (12), where the dis-clination charge,

A B C

D E F

G

Fig. 3. Voronoi tessellations and structure of defects. (A–F) Top views of tilingssuperimposed on the scanned data with increasing values of dimensionless sys-tem size, R=λ= ð2:21,3:75,4:71,6:20,7:56,9:06Þ, respectively. Pentagonal, hexag-onal, and heptagonal cells are colored in yellow, blue, and red, respectively.Physical edges (equators) of the shell samples are encircled black. Only theregions inside the pseudoboundary represented by the thick-solid green enve-lope were analyzed. Dimples outside this green boundary were not considered.The surface regions in the white annulus were not scanned. (G) Ratio betweenthe area of a spherical cap and the area of the corresponding full sphere α asa function of R=λ. The samples from A–F are marked with red diamonds.

0.5 1.51.0

10-1

100

101

10-2 1.20 1.25

5

10

15

20

25

A B

R

R

RC

1.050

1.151.10

Fig. 4. Polydispersity and distortion of Voronoi cells. (A) Probability densityfunction of dimensionless area of Voronoi cells A*. Area of each cell isnormalized by the area of regular hexagons, Ahex =

ffiffiffi3

pλ2=2, for a character-

istic dimple size λ. Gaussian distribution fits are represented with solid lines.(B) PDF of shape factor ζ. Vertical lines located at the corresponding valuesfor regular pentagons, hexagons, and heptagons, ζn =n=πtanðπ=nÞ, withn= 5, 6, and 7, respectively. (C) Representative Voronoi cells for specificvalues of ζ. Regular polygons are marked with “R.”

Brojan et al. PNAS | January 6, 2015 | vol. 112 | no. 1 | 17

APP

LIED

PHYS

ICAL

SCIENCE

S

s= 6−Z; [2]

is the deviation of the coordination number Z from that ofa perfect hexagonal packing; s=+1 for a pentagon and s=−1 fora heptagon, i.e., q=+π=3 and q=−π=3, respectively. From theGauss–Bonnet and Euler theorems (5, 6), it follows that, for anytriangulation over a spherical surface, there exists a fixed topo-logical constraint on the sum of these discrete charges (+12 ona sphere). We define the net topological charge for an ensembleof Nα lattice units on a spherical cap with area 4πR2α (the arearatio α was quantified in Fig. 3G for our samples) as

Qα =Qα=XNα

i

qi = 4πα: [3]

We now analyze our data following a procedure recently used forcurved colloidal crystals (10). For a given sample, we measure thenet defect charge Qβ of spherical “patches” of variable area (de-fined by the ratio β of their area with that of a sphere; Fig. 5A,Inset) up to the maximum possible cap allowed for that sample(Fig. 3G), such that 0 < β ≤ α. Such a measurement includes boththe effects of isolated disclinations and the polarization charge dueto the nonuniform distribution of disclination dipoles [by analogywith electrostatics (10)]. In this procedure, the contribution to thepolarization charge is accounted for by the cumulative countingwhen the boundary of the analyzed patches with increasing sizesdissects a pleat or a dislocation and adds toward the total topo-logical charge, which would not occur for a full sample. In Fig. 5A,we plot this net defect charge as a function of the integratedGaussian curvature,

RGdA= 4πβ, of the patches (G= 1=R2 in our

spherical case) for multiple samples with different values of R=λ.The data are consistent with a linear relation between defectcharge and integrated Gaussian curvature, with unit slope, that waslikewise previously observed on curved crystals, albeit with a sig-nificant level of scatter that is also consistent with the experimentsin ref. 10. It is striking that, despite the differences in the un-derlying physics of the two curved systems, our macroscopicwrinkling patterns and colloidal packings can be analyzed andinterpreted similarly.

Average Coordination Number. Given the scatter in Qβ, we turn tothe average coordination number Z= ð1=NÞPN

i=1Zi for a tilingwith N dimples. However, before quantifying Z, we step back andconsider the number of dimples Nα in our samples, as a functionof R=λ. For large systems, where R is large compared with λ andthe hexagonal cells far outnumber disclinations, we assume thatthe area of each dimple is Ai ≈ ð ffiffiffi

3p

=2Þλ2. This is supported by theabove finding that A

p ≈ 1 (Fig. 4A). In turn, the total numberof dimples on a spherical cap is Nα ≈ 4πR2α=Ai, which yields

Nα ≈8παffiffiffi3

p�Rλ

�2

: [4]

In Fig. 5B, we plot the experimental measurements for Nα=α(extrapolated to a sphere), which are in excellent agreement withEq. 4 (solid line).Toward determining Z, the average net topological charge per

lattice unit is Qα=Nα = ð1=NαÞPNα

i=1qi = 4πα=Nα, through Eq. 3.Combining this result with the definition of q and making useof Eqs. 2 and 4 gives

Z≈ 6

"1−

ffiffiffi3

p

4π

�Rλ

�−2#: [5]

A more general version of Eq. 5 was provided by Nelson (7) but,for completeness, we have reproduced the argument appliedspecifically to our system.In Fig. 5C, we plot Z measured directly from our samples, as

a function of system size, finding that the data are in very goodagreement with Eq. 5. A perfect hexagonal packing (e.g., ona plane or a cylinder) would have Z= 6, but the presence of

defects forces Z< 6. Moreover, the deviations from the planarresult (horizontal dashed line in Fig. 5C, at Z= 6) become morepronounced for smaller systems, as the relative importance ofpentagonal disclination increases.

Number of Dislocations. Further, we quantify the total number ofdislocations Nd

α for our dimpled patterns. In Fig. 5D, we plot Ndα=α

(experimental value for the spherical cap extrapolated to a sphere)as a function of R=λ. We find that the number of dislocations growslinearly with system size, Nd

α=α∼ ðR=λÞ, with a slopem= 28:3± 2:8and an intercept with the horizontal axis, ðR=λÞc = 3:2± 0:7. Thisfinite value of ðR=λÞc is significant of a threshold system size for theonset of dislocations, below which only isolated, topologically re-quired disclinations are found. Through Eq. 4, this translates intoa sample with a threshold number of dimples, Nc ≈ 150 (after ex-trapolating to a sphere using α).Remarkably, this scenario is qualitatively identical to that of

Bausch et al. (9), who found ðR=λÞc ≈ 5, for a system of colloidalparticles on the surface of spherical oil droplets which had pre-viously been predicted by theory (16, 33), and a critical system sizeof Nc ≈ 360 particles. These similarities are despite the fact thatthe system of Bausch et al. (9) was microscopic, whereas ours ismacroscopic, and in the measurement procedure for the numberof dislocations, they took the average number of dislocations perchain, detached from the boundary, whereas we used all chainsbecause they regularly emanate from the boundary.

Discussion and ConclusionWe have introduced a macroscopic experimental model systemwhere a curved crystalline pattern of dimples self-organizes fromthe wrinkling of an originally smooth thin elastic shell. The system

A B

C D

-2.0 10.06.04.0 8.0

1.0

1.0

Fig. 5. Quantification of defects. (A) Net defect charge (sum of disclination andpolarization charges) on spherical patches of variable size (within a given sam-ple, with a patch-to-sphere area ratio in the range 0 < β ≤ α), as a function ofintegrated Gaussian curvature 4πβ. The adjacent color bar represents system sizeR=λ. Different markers are used to better identify different data points. (Inset)Purple lines represent the edges of the analyzed spherical patches, with in-creasing values of β. (B) Number of counted dimples Nα=α (extrapolated to a fullsphere) vs. R=λ. Solid line is prediction from Eq. 4. (C) Average coordinationnumber Z as a function of R=λ. Solid line is prediction from Eq. 5, dashed line isZ =6. (D) Number of dislocations Nd

α=α (extrapolated to a full sphere) asa function of R=λ. The solid line is a linear fit with slope 28.3 (±2:8 SD) and in-tercept with horizontal axis at ðR=λÞc = 3:2 (±0:7 SD.). See Materials and Meth-ods for details on the geometric and material properties of the 32 samples used.

18 | www.pnas.org/cgi/doi/10.1073/pnas.1411559112 Brojan et al.

relaxes stresses both by out-of-surface buckling through the forma-tion of arrays of dimples, and by simultaneously developing defectsas nonhexagonal dimples in the otherwise hexagonal patterns. Di-rect parallels were established between the structure of defects inour system and previous studies on the packing of charged particleson curved surfaces (14–16) and curved colloidal crystals (9, 10),despite the differences in the underlying physics. These similaritiesinclude the ability to treat the dimples as point-like units, the use ofVoronoi tessellation to characterize their packing, as well as thepresence of disclinations and, above a threshold system size, theprominent growth of the number of dislocations to screen the un-derlying curvature. There are however few important distinctions.We observe dislocations that form branched arrays and clusters andproliferate more rapidly than in curved colloidal crystals (9, 16), andwith a different threshold value of system size. We speculate thatthese differences may be attributed to the fact that we havea different repulsive potential and our system is far from equi-librium due to the random and initially frozen material imper-fections that nucleate the pattern, as well as the inability for thedimples to rearrange during crystallization. Consequently, theseconstraints prevent our system from exploring phase space andlead to additional frustration that increases disorder.The interaction potential between neighboring dimples in our

system is still unknown. However, Bowick et al. (33) find thatpotentials of the form 1=rγ , with 0< γ < 2, lead to similar defectstructures. This provides a possible explanation as to why,despite the different nature of the interdimple elastic potential inour wrinkling system, we still find many of the general features ofother 2D curved crystals. We hope that our experimental resultswill instigate further studies of curved crystallography in morecomplex geometries (e.g., on a torus) and in instances of far fromequilibrium (e.g., with anchoring imperfections), which remainlargely unexplored.

Materials and MethodsFabrication of Samples. We manufactured 32 hemispherical samples, made ofsilicone-based elastomers, polydimethylsiloxane (PDMS) and vinylpolysiloxane(VPS), for the film and the substrate, respectively, using a protocol that wasdescribed previously (23). First, a thin outer shell was made by coatinga previously vacuum-formed polystyrene mold with the desired radius. The

coating process included wetting the surface of the mold and then draining theexcess polymer by gravity. A balance between gravity, viscosity, surface tension,and polymerization rate yielded shells of constant thickness (to within ∼ 10%variation). This process could be repeated multiple times to obtain thicker shells.Next, pouring the VPS into the mold (now containing the fully polymerized thinshell) produced the soft foundation. Immediately after the elastomer was cast,the mold was covered with an acrylic plate containing a spherical 3D printedpart, which produced the undersurface cavity. Upon curing and demolding, thesamples were stored in a ventilated area for 1 wk to fully cure before the ex-perimental tests were performed.

Material and Geometric Properties. The mechanical properties of PDMS and VPSwere measured on cylindrical specimens subjected to uniaxial compression usinga material testing machine (Zwick). We found a linear stress–strain response ofthese materials within the levels of compression relevant to our experiments. Theratio between the Young moduli of the film and substrate could be controlledwithin the range 1:0≤ Ef=Es ≤ 162:0 (for differentmixtures of the base and curingagents), whereas the Poisson ratios of the film and the substrate was νf = νs ≈ 0:5.The thickness of the shell was varied in the range 0:23<h½mm�< 0:88, whilefixing the radii of shell, R= 20:0 mm, and the cavity, r = 9:5 mm.

Three-Dimensional Scanning and Image Analysis. The surface topography of thedimpled patterns on the hemispherical samples was digitized using a 3D laserscanner (NextEngine) and the resulting cloud of points was postprocessed usingMATLAB. The Cartesian coordinates ðx,y,zÞ of each point were converted intospherical coordinates ðϕ,θ,ρÞ, where ϕ and θ are the azimuth and polar angles,and ρ is the radial distance from each point to the centroid of the sphere thatbest fits the outer surface of the sample. In our analysis we used a stitchedcombination of ðϕ,θ,ρÞ and ðx,y,ρÞ coordinate representation to address thedistortion of the ρðϕ,θÞ map at small polar angles. A gray-scale image usingρ as the field variable was thresholded into black and white binaries, corre-sponding to valleys of the dimples and the ridges, respectively. Determiningthe centroids of all black blobs yielded the coordinates of the centers of alldimples, from which the final Voronoi tessellation was constructed.

ACKNOWLEDGMENTS. The authors thank A. Bresson for help with preliminaryexperiments. P.M.R. acknowledges support from MIT’s Charles E. Reed facultyinitiative fund, the National Science Foundation, CMMI-1351449 Faculty EarlyCareer Development (CAREER) Program, and is grateful to U. Deusto for hos-pitality. M.B. thanks the Fulbright Program. D.T. thanks the Belgian AmericanEducation Foundation (BAEF), the Fulbright Program, and the Wallonie-Brux-elles International Excellence Grant WBI World.

1. Kroto HW, Heath JR, O’Brien SC, Curl RF, Smalley RE (1985) C60: Buckminsterfullerene.Nature 318:162–163.

2. Caspar DLD, Klug A (1962) Physical principles in the construction of regular viruses.Cold Spring Harbor Symposium 27:1–24.

3. Dinsmore AD, et al. (2002) Colloidosomes: Selectively permeable capsules composedof colloidal particles. Science 298(5595):1006–1009.

4. Fuller R (1954) Building construction. US Patent 2,682,235.5. Sadoc JF, Mosseri R (1999)Geometrical Frustration (Cambridge Univ Press, Cambridge, UK).6. Richeson DS (2008) Euler’s Gem: The Polyhedron Formula and the Birth of Topology

(Princeton Univ Press, Princeton, NJ).7. Nelson DR (1983) Order, frustration, and defects in liquids and glasses. Phys Rev B

28(10):5515–5535.8. Seung HS, Nelson DR (1988) Defects in flexible membranes with crystalline order. Phys

Rev A 38(2):1005–1018.9. Bausch AR, et al. (2003) Grain boundary scars and spherical crystallography. Science

299(5613):1716–1718.10. Irvine WTM, Vitelli V, Chaikin PM (2010) Pleats in crystals on curved surfaces. Nature

468(7326):947–951.11. Irvine WTM, Bowick MJ, Chaikin PM (2012) Fractionalization of interstitials in curved

colloidal crystals. Nat Mater 11(11):948–951.12. Bowick MJ, Giomi L (2009) Two-dimensional matter: Order, curvature and defects.

Adv Phys 58(5):449–563.13. Dodgson MJW (1996) Investigation on the ground states of a model thin film su-

perconductor on a sphere. J Phys Math Gen 29(10):2499–2508.14. Dodgson MJW, Moore MA (1997) Vortices in a thin-film superconductor with a

spherical geometry. Phys Rev B 55(6):3816–3831.15. Perez-Garrido A, Dodgson JW, Moore MA (1997) Influence of dislocations in

Thomson’s Problem. Phys Rev B 56(7):3640–3643.16. Bowick MJ, Nelson DR, Travesset A (2000) Interacting topological defects on frozen

topographies. Phys Rev B 62(13):8738–8751.17. Vitelli V, Lucks JB, Nelson DR (2006) Crystallography on curved surfaces. Proc Natl

Acad Sci USA 103(33):12323–12328.18. Erber T, Hockney GM (1991) Equilibrium configurations of N equal charges on a

sphere. J Phys Math Gen 24(23):1369–1377.

19. Grason GM, Davidovitch B (2013) Universal collapse of stress and wrinkle-to-scar transitionin spherically confined crystalline sheets. Proc Natl Acad Sci USA 110(32):12893–12898.

20. Funkhouser CM, Sknepnek R, de la Cruz MO (2013) Topological defects in the buck-ling of elastic membranes. Soft Matter 9:60–68.

21. Azadi A, Grason GM (2014) Emergent structure of multidislocation ground states incurved crystals. Phys Rev Lett 112(22):225502.

22. Zhang T, Li X, Ga H (2014) Defects controlled wrinkling and topological design ingraphene. J Mech Phys Solids 67:2–13.

23. Terwagne D, Brojan M, Reis PM (2014) Smart morphable surfaces for aerodynamicdrag control. Adv Mater 26(38):6608–6611.

24. Bowden N, Brittain S, Evans AG, Hutchinson JW (1998) Spontaneous formation of orderedstructures in thin films ofmetals supported on an elastomeric polymer.Nature 393:146–149.

25. Chen X, Hutchinson JW (2004) Herringbone buckling patterns of compressed thinfilms on compliant substrates. J Appl Mech 71:597–603.

26. Li B, Cao YP, Feng XQ, Gao H (2012) Mechanics of morphological instabilities andsurface wrinkling in soft materials: A review. Soft Matter 8:5728–5745.

27. Yin J, Cao Z, Li C, Sheinman I, Chen X (2008) Stress-driven buckling patterns inspheroidal core/shell structures. Proc Natl Acad Sci USA 105(49):19132–19135.

28. Li B, Jia F, Cao YP, Feng XQ, Gao H (2011) Surface wrinkling patterns on a core-shellsoft sphere. Phys Rev Lett 106(23):234301.

29. Cai S, Breid D, Crosby AJ, Suo Z, Hutchinson JW (2011) Periodic patterns and energystates of buckled films on compliant substrates. J Mech Phys Solids 59:1094–1114.

30. Breid D, Crosby AJ (2013) Curvature-controlled wrinkle morphologies. Soft Matter9:3624–3630.

31. Audoly B, Boudaoud A (2008) Buckling of a stiff film bound to a compliant substrate -Part I: Formulation, linear stability of cylindrical patterns, secondary bifurcations.J Mech Phys Solids 56:2401–2421.

32. Paulose J, Nelson DR (2013) Buckling pathways in spherical shells with soft spots. SoftMatter 9:8227–8245.

33. Bowick MJ, Cacciuto A, Nelson DR, Travesset A (2006) Crystalline particle packings ona sphere with long-range power-law potentials. Phys Rev B 73:024115.

34. Reis PM, Ingale RA, Shattuck MD (2006) Crystallization of a quasi-two-dimensionalgranular fluid. Phys Rev Lett 96(25):258001.

35. Moucka F, Nezbeda I (2005) Detection and characterization of structural changes inthe hard-disk fluid under freezing and melting conditions. Phys Rev Lett 94(4):040601.

Brojan et al. PNAS | January 6, 2015 | vol. 112 | no. 1 | 19

APP

LIED

PHYS

ICAL

SCIENCE

S