Phoamtonic designs yield sizeable 3D photonic band gaps

Michael A. Klatta,1, Paul J. Steinhardta,1, and Salvatore Torquatoa,b,c,d,1

aDepartment of , Princeton University, Princeton, NJ 08544; bDepartment of Chemistry, Princeton University, Princeton, NJ 08544; cPrinceton Institute for the Science and Technology of Materials, Princeton University, Princeton, NJ 08544; and dProgram in Applied and Computational , Princeton University, Princeton, NJ 08544

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved October 7, 2019 (received for review July 23, 2019)

We show that it is possible to construct foam-based heterostruc- In this paper, we consider phoamtonics, the potential applica- tures with complete photonic band gaps. Three-dimensional tion of these foam-based architectures to photonic materials. For foams are promising candidates for the self-organization of large this purpose, the essential question is whether the resulting mate- photonic networks with combinations of physical characteristics rials have complete photonic band gaps (PGBs) that prohibit that may be useful for applications. The largest band gap found is the propagation of light in all directions and for all polarizations based on 3D Weaire–Phelan foam, a structure that was originally for a substantial range of frequencies. If so, they would expand introduced as a solution to the Kelvin problem of finding the 3D the range of 3D heterostructures available for photonic appli- composed of equal-volume cells that has the least sur- cations beyond photonic (22–24), (25–27), face area. The photonic band gap has a maximal size of 16.9% and amorphous networks (28–30). (at a volume fraction of 21.6% for a dielectric contrast ε = 13) Here we demonstrate that a 3D photonic network based on and a high degree of isotropy, properties that are advantageous the Weaire–Phelan foam results in a substantial complete PBG in designing photonic waveguides and circuits. We also present (16.9%) with a high degree of isotropy, an important property results for 2 other foam-based heterostructures based on Kelvin for photonic circuits. We find substantial PBGs for the Kelvin and C15 foams that have somewhat smaller but still significant and C15 foam, but the Weaire–Phelan network has the largest band gaps. phoamtonic band gap. This provides impetus for further studies of foam-based photonic materials. Plateau’s laws for dry foam | Weaire–Phelan foam | TCP structures and There are 2 motivations for phoamtonics. First, Plateau’s laws Frank–Kasper phases | self-organization | complete photonic band gap guarantee exclusively tetrahedral vertices. The latter are gen- erally thought to be advantageous for photonic networks, as explained below. This naturally relates photonics to the Kelvin oams are both of fundamental interest (1) and important problem, that is, the search for the 3D tessellation composed Ffor a host of applications ranging from chemical filters to of equal-volume cells that has the least surface area—a prob- heat exchangers (2). They locally minimize the surface area of lem of fundamental interest in both mathematics and physics. their cells subject to volume constraints. In the limit of vanishing Its connections to famous packing and covering problems (31) fraction, the so-called dry foam is completely character- relate foams, for example, to Frank–Kasper phases (6, 14, 32–34) ized by Plateau’s laws, where each cell vertex is tetrahedral; and their dual clathrate structures (35–37). The self-assembly∗ of that is, 4 edges meet with bond angles equal to arccos(−1/3) ≈ 109◦. These laws were first observed empirically (3) and later Significance proven rigorously for area-minimizing with volume constraints (4, 5). Kelvin posed the fundamental question: What tessellation with Foams appear ubiquitously in many fields of science and cells of equal volume has the least surface area? He conjec- technology. However, applications to photonics have not tured that the solution is a relaxation of the Voronoi tessellation been possible because until now, foam-based 3D heterostruc- of the body-centered cubic (BCC) lattice—now known as the tures with substantial complete photonic band gaps have Kelvin foam (1). More than 100 y later, Weaire and Phelan not been identified. Here we provide explicit examples based disproved the conjecture by finding a foam with a slightly smaller on the edges of dry-crystalline foams, including the famous surface area (6). Their numerical finding was later rigorously Weaire–Phelan foam. More generally, foam networks based confirmed (7). The Weaire–Phelan foam can be obtained by on Frank–Kasper phases offer new, promising approaches to relaxing a weighted Voronoi tessellation of the A15 . the rapid self-organization of large photonic networks. More- Importantly, the foam has been not only studied numerically over, foams are excellent candidates for designer materials (8) but also realized experimentally (9). Whether it is the with versatile multifunctional characteristics. globally optimal solution to Kelvin’s problem is still an open Author contributions: M.A.K., P.J.S., and S.T. designed research; M.A.K., P.J.S., and S.T. question. The Weaire–Phelan foam has 2 types of cells with dif- performed research; M.A.K. analyzed data; and M.A.K., P.J.S., and S.T. wrote the paper.y ferent pressures. So the Kelvin foam has still the least surface The authors declare no competing interest.y area among known foams with equal pressure in each cell. Kelvin cells have also been found in experiments (10–12). The A15 This article is a PNAS Direct Submission.y crystal (β-tungsten) belongs to a larger class of so-called tetrahe- Published under the PNAS license.y drally close-packed (TCP) structures (or Frank–Kasper phases) Data deposition: All data generated or analyzed for this study, including configura- tions, parameter files, raw output, and postprocessed data, are available at the Zenodo (13). This class includes, for example, the C15 crystal (cubic repository, https://zenodo.org/, as Databases 1–3 (DOI: 10.5281/zenodo.3401635). y Friauf–Laves phase). Relaxing its Voronoi diagram, subject to 1 To whom correspondence may be addressed. Email: [email protected], steinh@ the constraint that each cell has the same volume, defines the princeton.edu, or [email protected] C15 foam (14). Its surface area is larger than that of the Kelvin This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. foam. Here we study these 3 foams—Weaire–Phelan, Kelvin, 1073/pnas.1912730116/-/DCSupplemental.y and C15—as classic examples of crystalline foams. Their electri- First published November 6, 2019. cal, mechanical, and transport properties have been intensively *In this paper, we apply the term “self-assembly” to equilibrium systems, whereas “self- studied elsewhere (15–21). organization” also refers to athermal or nonequilibrium systems.

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A15, weighted 5, or (3, suitably Kelvin, laws from constraints Weaire–Phelan, Plateau’s volume obeys the to geometry construct final subject the of the tessellation minima that local a such as of defined are area that interfacial foams dry study we Here Edges Foam of Topology and Geometry circuits. photonic and 46) 45, (27, angles bending waveguides adjustable in more to efficient (44) sources and radiation isotropic is thermal highly which from ranging isotropic, applications nearly C15 for are useful and that PBGs Weaire–Phelan complete applications. demonstrate, have multipurpose networks will we certain and as for stiffness Furthermore, candidates their excellent as such may be heterostructures foams, foam-based photonic of studies (15–18), properties conductivity previous extensive fabrica- physical the rapid the to for due of Also, allow samples. may large of foams of exam- tion of For advantages self-organization materials. additional the photonic ple, the other to by compared compensated phoamtonics be may photonic crystals a into network (43). 3D wavelengths already with has a network procedure suggested convert solidified similar silicon-coated recently a to first fact, as used In be material (39). been to dielectric al. et needs a Ricouvier by foam by coated dry then a and realize edges, experimentally thickened To of (42). such those opals exceeding synthetic geomet- or optimized like rically to crystals photonic comparable self-organizing 17%, of examples to typical up of PBGs tonic esuybt ewrsfre ytecre de fthe of edges curved the by formed networks both study We photonic diamond to compared PBG reduced somewhat The PNAS | oebr1,2019 19, November ,Kli om(Center foam Kelvin ), a TiO h ilcrccnrs is contrast dielectric The . 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PHYSICS coincide with the faces of the cells. While a Kelvin cell has 2 optimized for a maximal gap–midgap ratio ∆ω/ωm , where ∆ω squares and 4 hexagons per vertex, the TCP foams have only is the frequency width of the gap (the difference between the pentagonal and hexagonal faces (on average, 5.2 and 0.78 in the lowest-frequency eigenvalue in the lowest air band and the high- Weaire–Phelan foam and 5.3 and 0.71 in the C15 foam), which est in the HDB), and ωm is the frequency at the middle of may be related to their importance for foams (32). The A15 and the gap. The radii that maximize the gap–midgap ratio differ C15 Voronoi tessellations are the TCP structures with the most by a few percent, and so does the corresponding volume frac- or least average number of faces f¯ per cell (32). The Weaire– tion covered by the network; for more details, see SI Appendix, Phelan structure has the largest value of f¯ = 13 + 1/2, and the Table S3. C15 structure has the smallest f¯ = 13 + 1/3. For the Weaire–Phelan, Kelvin, and C15 foam networks, the In the relaxed foam, Plateau’s rules guarantee that each ver- photonic band structures are shown in Figs. 2–4. Time-averaged tex has valency 4 and that the incident bond angle between any electric-field energy of modes in the HDBs are depicted 2 edges is arccos(−1/3) ≈ 0.608π ≈ 109◦. While the first state- together with small portions of the 3D networks in Fig. 1. More ment also holds for the progenitor tessellations, the second must energy densities are presented in Movies S1–S3. be violated by a tessellation with flat faces (32). The same analysis is repeated for 4 classic photonic crys- For the A15, C15, and BCC lattice, the distributions of bond tals: the diamond, Laves, hexagonal diamond, and simple cubic lengths and angles in the weighted Voronoi diagrams with networks. Their photonic band structures are depicted in SI straight edges are plotted in SI Appendix, Fig. S1. Detailed statis- Appendix, Figs. S2–S5. tics of the bond length and angle distributions of both the foams and their progenitors are listed in SI Appendix, Table S2. Gap Sizes. The PBG sizes, measured by the gap–midgap ratio All edges of the BCC Voronoi diagram have exactly the same ∆ω/ωm , are 7.7% for the Kelvin foam, 13.0% for the C15 foam, length, in contrast to those in the TCP structures. There is less and 16.9% for the Weaire–Phelan foam. The minimization of the variation in the A15 (18.5%) than in the C15 structure (26.3%). surface area increases the gap size by 0.7, 0.3, or 1.3%, that is, The variance of bond lengths decreases for the relaxed foams comparing the progenitor tilings with straight edges and slightly (14.7 and 24.7%) with curved edges. nonuniform vertices to the relaxed foams with curved edges and The average of the bond angles in the BCC Voronoi tessella- tetrahedral vertices. The gap edges are listed in SI Appendix; for tion differs from the tetrahedral angle by about 0.5%. For the further details, see also SI Appendix, Table S3. A15 and C15 crystals, the average bond angles are an order of The TCP foam networks with effectively tetrahedral bond magnitude closer to the tetrahedral value with differences of angles have distinctly larger gap sizes than the Kelvin network. about 0.09 and 0.06%, respectively. In that sense, the weighted Between the 2 TCP structures, the Weaire–Phelan network has Voronoi diagrams of the A15 and C15 crystals have effectively the larger gap size, presumably due to the smaller variation in tetrahedral bond angles. Moreover, the standard deviations of bond lengths. The Weaire–Phelan foam has not only the cur- their bond angles are smaller than those of the BCC lattice (SI rently smallest known interfacial area; its corresponding network Appendix, Table S2). also has the largest PBG. Perhaps due to the variations in bond lengths, the gap sizes of Phoamtonic Networks our foam-based photonic networks are as expected smaller than Next, we construct a photonic network based on the edges of dry the largest known gaps, which are found in networks based on foam. Its volume fraction φ is the fraction of volume covered by the diamond (31.6%) and Laves (28.3%) graph; for more details the high dielectric material. Typical values of φ that maximize and other classic networks, see Classic Photonic Crystals and, e.g., the PBG are about 20% (49). The Weaire–Phelan foam becomes refs. 24, 30, 40, and 49. unstable at about 15% volume fraction (8). So we do not con- sider wet foam, where the Plateau borders have tricuspoid cross sections. Instead, we start from dry foam structures and homoge- neously thicken each edge. Each edge is finally the medial axis of a spherocylinder, where the radius is a tuning parameter. Such a foam-based network is not literally a foam, but for simplicity we will henceforth refer to heterostructures derived from foams in this way as “foams” or “foam networks.” Fig. 1 illustrates these 3D networks. For an accurate prediction of the photonic band structure, we solve the frequency-domain eigenproblem with the plane- expansion method implemented in the Massachusetts Institute of Technology (MIT) Photonic Bands (MPB) soft- ware package (50). To achieve high numerical precision, we choose a high resolution of the discretization, a low toler- ance of the eigensolver, and a high sampling of the k-points; for more details, see Creating and Analyzing Phoamtonic Net- works. Using primitive unit cells and the standard notation for high- k-points of the Brillouin zone, we choose paths along the boundary of the irreducible Brillouin zones similar to those in ref. 51. The dielectric contrast ε is the ratio of the dielectric constants of the high- to low-dielectric material (24). Unless otherwise stated, we assume a dielectric contrast of 13 to 1, which is com- monly used in the literature as a standard when comparing the performance of different photonic material designs (28, 30, 40). To facilitate experimental realizations, our map of gap sizes Fig. 2. Photonic band structure for the Weaire–Phelan foam in a cubic below shows the gap–midgap ratios at lower dielectric contrast. primitive unit cell at a dielectric contrast ε = 13 and a volume fraction For each network, the radius of the edges (spherocylinders) is φ = 21.7%. For a 3D sample, see Fig. 1 (Left).

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Gap contrast dielectric a at fraction crystal cubic body-centered a of 3. Fig. lt tal. et Klatt the from deviations require generally properties photonic physical ferent excellent Sizes. Gap its of Maps for commonly reasons is zone the Brillouin of the the properties. of one of isotropy as boundary relative understood the the that at dia- hexagonal Note diamond and networks. (8.8%) the (9.7%) cubic comparably and simple mond still (3.4%) the are to network possibi- networks relative Laves isotropic These the (4.2%). the to network remarkably similar diamond provide is with which (3.5%), foams networks TCP Weaire–Phelan photonic the the self-organizing PBGs. isotropic by of So followed lity (1.2%). closely very foam (1.0%), index See S4. 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PHYSICS Fig. 5. Maps of gap sizes: gap–midgap ratio ∆ω/ωm as a function of (Left) the dielectric contrast, where the radii are optimized at each value of ε to maximize ∆ω/ωm [for precise values, see Database S1 (53)], and (Right) the volume fraction at a dielectric contrast ε = 13. The vertical lines indicate the maximal gap sizes shown in Figs. 2–4. For more details on the other photonic crystals, see SI Appendix. For 3D maps of gap sizes, see Fig. S6.

apply to amorphous foams in 3D. To date, complete band gaps photonic networks. Applications at visible wavelengths are more in amorphous networks have only been demonstrated for rel- challenging with current technology. atively small systems and are not guaranteed to persist as the system is made larger (30). Amorphous foams are interesting Multifunctionality of Foams. Foams provide an excellent class of candidates because they would have the advantages of perfect multifunctional materials that provide a good trade-off between isotropy and tetrahedral bond angles. In addition, they may have a number of different physical properties. Phoamtonics thus less constraining conditions for manufacture compared to crys- offers versatile solutions for multipurpose technologies that talline foams. On the other hand, the bond length distribution require further optimizations besides a maximal PBG. Foams is much broader than for crystalline foams. At present, the only are particularly promising lightweight materials because other examples of amorphous networks in 3D (including foams) have favorable properties have already been explored, identified, and either no complete PBGs or PBGs that may not persist in the utilized for a broad spectrum of technologies (2). Besides elastic limit of large system sizes; see, for example, ref. 30. We are cur- moduli (17) and conductivities (18, 62), these include, for exam- rently using the successful experience with Weaire–Phelan foams ple, phononic band structures (19), heat transfer (21), diffusive to guide us in constraining the constructing of amorphous foam transport of photons (20), or the mechanical strength of metallic heterostructures that will have robust complete PBGs. foams (63–65). The elastic properties of 3D networks based on foams, in par- Utilizing Phoamtonics. Exciting prospects of phoamtonics for ticular the Weaire–Phelan foam, have been intensively studied applications are 1) the self-organization of photonic networks as exemplary cellular structures; e.g., see ref. 16. At low volume and 2) the flexible trade-off between physical properties. fractions, the mechanical properties of the Kelvin foam, its bulk Importantly, the gap size of the Weaire–Phelan network exceeds and shear moduli, are almost isotropic (within a few percent) in that of geometrically optimized synthetic opals and other typi- the small strain range (15, 17, 66). They are exactly isotropic for cal examples of manmade† self-organizing large-scale photonic circular cross sections and zero Poisson’s ratio (bulk material) crystals (<15% at ε = 13) as reported in ref. 42. (15). At finite volume fractions, the shear moduli of the Kelvin Self-assembly of crystal structures that correspond to the foam are more anisotropic than those of the Weaire–Phelan and Kelvin or Weaire–Phelan foam as well as to other Frank–Kasper C15 foams (16). The average shear moduli are comparable, but phases have recently been reported in a growing list of soft- the TCP foams are perceptibly stiffer (16). The Young’s moduli matter systems; see ref. 38 and references therein. Phoamtonics have been reported to be more isotropic for the Kelvin than for might thus help with the self-organization at various length the Weaire–Phelan network (67). scales and offer new opportunities to the recent progress and Further intriguing insights were gained for the effective foam remaining challenges of the self-assembly of photonic nanomate- conductivity, both in the dry limit (2, 18, 62) and at higher vol- rials (56, 57). Particularly interesting is the connection between ume fractions (68, 69). Starting from the Lemlich formula (62), foams and self-assembling supramolecular micellar materials or Durand et al. (18) found sufficient criteria to maximize the con- block-copolymers (38, 58). An alternative approach could use ductivity of a network for a given topology: straight edges and the self-assembly of emulsions (37), e.g., with suitably patchy a balancing of the incident directions at each vertex. The latter droplets and emulsion templating for the fabrication of open-cell condition is guaranteed for dry foam by Plateau’s laws. Devi- foams. ations from a maximal conductivity are due to curved edges We envisage that phoamtonics may be an effective way of fab- only. Computing the orientational averaged conductivities of the ricating large photonic network for applications involving Kelvin, C15, and Weaire–Phelan foam, Durand et al. (18) found longer wavelengths, such as THz (59, 60) and X band radia- them to be nearly optimal, as expected. tion (61). These bands require cell sizes in the millimeter or submillimeter regime, where standard techniques of solid open- Materials and Methods cell foams might be highly useful for the fabrication of extended Classic Photonic Crystals. We compared the photonic foam networks to 4 classic photonic crystals: diamond, hexagonal diamond, Laves, and simple cubic networks. Hexagonal diamond is also known as Lonsdaleite and closely related to the Wurtzite crystal. The Laves graph (70) is also known as K4 †Despite recent insights into the formation of biological photonic nanostructures (54, graph (41) or SRS net (71). Its symmetric embedding in Euclidean space is 55), their technological utilization remains a challenge. the medial axis of the gyroid, which is a triply periodic (71).

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Emulsions, and Foams eculaires ´ he,Y .C ag .Lrneu .ChnAdd soi rsueand pressure Osmotic Cohen-Addad, S. Lorenceau, E. Sang, C. Y. Y. ohler, ¨ – (2012). 1–6 92, 314(2000). 93–104 37, ttqeExp Statique h evnProblem Kelvin The .Ap.Mech. Appl. J. hls a.Lett., Mag. Philos. Guhe-ilr,Prs 1873). Paris, (Gauthier-Villars, u.Py.J E J. Phys. Eur. 0–1 (1994). 107–110 69, Math., Ann. h evnPolm omSrcue fMnmlSraeArea, Surface Minimal of Structures Foam Problem: Kelvin The x.Math. Exp. maximize rmnaee Th et erimentale ´ h hsc fFoams of Physics The 8–9 (1997). 787–794 64, 2–4 (2002). 129–140 7, eflo h o omnpoeuet start to procedure common now the follow We 8–3 (1976). 489–539 103, 9–0 (1994). 297–303 69, Tyo rni,Lno,1997). London, Francis, & (Taylor 8–9 (1995). 181–192 4, ae nBetsagrtmo h LIBCTL the of algorithm Brent’s on based and ) oiu e iudsSui u elsForces Seules aux Soumis Liquides des eorique ´ CmrdeUiest rs,Cambridge, Press, University (Cambridge nalother all In S6. Fig. Appendix, SI Langmuir CaednPes xod 1999). Oxford, Press, (Clarendon h eouinof resolution The S2–S5: Figs. rc .Sc odn e.A Ser. London, Soc. R. Proc. 1–2 (2008). 418–425 24, oeaut h photonic the evaluate To olcinAe-aly Mono- Alea-Saclay: Collection olisSr.A Surf. ‡ r eitdtogether depicted are ) IAppendix, SI 117–124 309, c.Am. Sci. hls Mag. Philos. n.J Solids J. Int. −7 and , 460, 235, 6 .OKef,“rsa tutrsa eidcfasadvc es”in versa” vice and foams periodic in as foams” structures and “Crystal bubbles O’Keeffe, M. of 36. geometry “The Sullivan foams. three-dimensional M. J. Quasicrystalline Sadoc, J.-F. 35. Mosseri, R. Graner, F. Cox, J. S. 34. photonic in self-uniformity tetrahedra. with Local covering and Florescu, tiling, Packing, M. Torquato, S. Sahba, Conway, H. S. J. Man, 31. W. Sellers, R. S. large, with 30. materials disordered Designer Steinhardt, J. P. a Torquato, with S. structure Florescu, diamond M. amorphous Photonic 29. Notomi, two-dimensional M. in Kanoko, gaps S. band Edagawa, Complete K. Steinhardt, 28. . J P. Optimized Torquato, Steinhardt, S. J. P. Florescu, Torquato, M. S. Chaikin, 27. M. P. Jeong, H.-C. Rechtsman, C. M. 26. foam. Kelvin a in light of transport Diffusive Stark, H. Madadi, E. Miri, M. 20. H R. Spadoni, A. 19. 3 .F ao,R osr,QaieidcFakKse hssdrvdfo h square– the from derived phases Frank–Kasper Quasiperiodic Mosseri, R. Sadoc, J.-F. 33. in structures” close-packed tetrahedrally “New Sullivan, M. J. 32. the of measurement Experimental Chaikin, M. P. Steinhardt, J. P. Megens, M. Man, Meade, W. D. R. 25. Winn, N. J. Johnson, G. S. superlattices. Joannopoulos, dielectric D. disordered J. certain 24. in photons of localization Strong John, S. electronics. 23. and Physics solid-state in emission spontaneous Inhibited Yablonovitch, E. 22. Cunsolo S. 21. noainFn o e da nteNtrlSciences. Natural the University in Princeton Ideas the Sci- New by resources Computational for supported Fund for was computational Innovation work Institute This on Princeton Engineering. the performed and ence by substantially supported presented were and simulations managed The article the software. insightful this MPB for for the Florescu in scripts and Marian useful photonics thank We on his samples. discussions acknowledge foam and gratefully Evolver we Surface particular, In foams. ACKNOWLEDGMENTS. photonic the of data. in parameters postprocessed all available and contains are output, (53) raw networks, calculations, S3 photonic Database crystalline (53). classic relaxed S2 the the Database and as diagrams well Voronoi the as unrelaxed of the foams, configurations both The in edges contrast. of dielectric networks or fractions, volume radii, rod and 5 Fig. in sizes the gap at available are data, postprocessed (53). and repository Zenodo output, raw projects. files, the parameter of websites the on available publicly are (47), Evolver Surface Data. and Code to Access the in density sampling finite a Appendix and (SI eigenfrequencies error the systematic in absolute errors the index of anisotropy the estimate of conservative eigensolver Our the of negligible. errors are statistical The edges ordered). curved (decreasingly of segments approximation discretization via the radii, and eigensolver, the the of of optimization tolerance effects, of resolution precision a are contributions chose main we see of 5, data, number and Fig. values The precise Data. in length). and For Code required. sizes unit as per gap crystals voxels the of between of maps (number 24 the of For plots. these ture between path cubic high-symmetry chosen simple all each the at for evaluated 40 were and k eigenvalues network, frequency Laves and The diamond network. the for 60 about pit fteirdcbeBilunzn n t8itreit onsfor points intermediate 8 at and zone Brillouin irreducible the of -points aaaeS 5)cnan l a–igprto hw ntemp of maps the in shown ratios gap–midgap all contains (53) S1 Database configurations, including study, this for analyzed or generated data All The 1%. than smaller is errors systematic the of estimate conservative Our .F ao,N iir d.(pigr odeh,TeNtelns 1999), Netherlands, The Dordrecht, (Springer, Eds. Rivier, 403–422. N. pp. Sadoc, F. 379–402. J. pp. Emulsions, 1999), Netherlands, The Dordrecht, (Springer, Eds. Rivier, N. Sadoc, F. J. Matter Condens. Phys. J. networks. gaps. band (2009). photonic complete gap. band photonic 3d quasicrystals. photonic quasicrystals. photonic for structures (2014). metamaterials. resonant Self-assembling, foams: rageddcgnltiling. dodecagonal triangle Technology, of University (Delft Eds. 2000). Verbist, Delft, G. Banhart, J. Zitha, P. Applications, Their U.S.A. Sci. Acad. quasicrystals. icosahedral of properties photonic Light of Flow the Lett. Rev. Phys. Lett. Rev. Phys. drop. pressure (2005). 031111 72, a.Commun. Nat. ). odKli n eiePea ommdl:Ha rnfrand transfer Heat models: foam Weaire–Phelan and Kelvin Lord al., et .Ha Transf. Heat J. 4628 (1987). 2486–2489 58, (1987). 2059–2062 58, PNAS 01–01 (2006). 10612–10617 103, PictnUiest rs,Pictn d ,2008). 2, ed. Princeton, Press, University (Princeton he,S oe-da,V oonty,Coe-elcrystalline Closed-cell Dorodnitsyn, V. Cohen-Addad, S. ohler, ¨ oehrwt h corresponding the with together S6 , Fig. Appendix, SI hs e.B Rev. Phys. hs e.Lett., Rev. Phys. etakAd ryi o epu icsin on discussions helpful for Kraynik Andy thank We | A 101(2017). 114001 29, 43 (2017). 14439 8, h otaeue nti td,MB(0 n the and (50) MPB study, this in used software The oebr1,2019 19, November is tut Chem. Struct. 261(2015). 022601 138, O ,wihi anycue ysystematic by caused mainly is which (0.1%), rc al cd c.U.S.A. Sci. Acad. Natl. Proc. 512(2009). 155112 80, hs e.Lett. Rev. Phys. k pit sidctdi h adstruc- band the in indicated as -points 191(2008). 013901 100, 37 (2017). 63–73 28, Nature .Aos.Sc Am. Soc. Acoust. J. | 792(2008). 073902 101, o.116 vol. 9–9 (2005). 993–996 436, htncCytl:Molding Crystals: Photonic om n Emulsions, and Foams om,Eusosand Emulsions Foams, | 20658–20663 106, o 47 no. k pit varied -points 1692–1699 135, hs e.E Rev. Phys. om and Foams cesto Access rc Natl. Proc. | k -space 23485

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