Downloaded by guest on September 25, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1820526116 materials photonic disordered of devices. generation photonic of next scalable fabrication a and enabling produce cheap materials, to foams PBG disordered 3D poten- to self-assembled a extended have the be results maximizes to obtained tial The which systems. material, experimental dielectric in find PBG we of fraction, quantity area foam the optimal the closes filling tuning an nodes, rapidly By fourfold modes. structure defect bigger with foam of PBG formation wet by long to mainly dry PBG and the from nodes transition large threefold a A open to PBG. with advantageous con- especially organization is borders index foam Plateau slender refractive dry foam sufficient a simulated at trast, and that, experimental demonstrate Cal- in-house polarization. band structures on (TE) photonic electric based isotropic transverse culations for for be materials template can (PBG) self-assembled foams gap a 2D review for disordered as (received polydisperse 2019 used 2, slightly April that Rossky J. show Peter Member We Board Editorial by accepted and NJ, 2018) Princeton, 2, University, December Princeton Torquato, Salvatore France by Paris, Edited 75005 University, Research (PSL) Lettres et Sciences Paris CNRS, a Ricouvier Joshua band material photonic gap amorphous self-assembling a as Foam o o aevcosblwacranlmt(2.Sc suppression Such (22). limit certain fac- a below structure vectors zero exhibiting for tor (13) stealthy from point networks of derives hyperuniform One of techniques gaps. optimization band find valency) first possible to the (e.g., largest applied the class been with com- networks have same decreased disordered protocols used the with design often different Within systems is structures, 3D (21). which real cost plane), of putational (TE) properties the electric predict in threefold transverse to field in for (electric observed systems polarization network is 2D behavior gyroid air: threefold coordinated Similar or by (18–20). diamond surrounded amorphous structures material fourfold example, dielectric valency for constant of usually networks are Cham- networks features. connected PBG peculiar disordered some of share pions which gap, Anderson band the of isotropic as study such the transport, for 17). optical interesting (16, Disordered localization of are . regimes PBG photonic different a classical possessing for structures (13–15)—applications or impossible sources, pigments radiation simply color isotropic structural guides, disor- useful wave noniridescent isotropy, such to freeform directional Moreover, argued create offer (13). are to also fabricate materials PBG photonic to complete dis- dered easier large discovered potentially a Recently devices be with (10–12). potential materials crystals of sensitivity scat- ordered development photonic high Bragg the on and coherent limit based issues strongly multiple defects fabrication the sci- to However, to the (7–9). of related formation led tering usually where structures, first , is ordered and materials PBG crystals long-range com- photonic PBG consider as of for such elements to optical search community efficiency and extraction The entific 5), light (6). 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This interest.y of and conflict paper.y no J.R. the declare wrote P.Y. authors research; and The J.R. designed and P.Y. data; analyzed P.Y. and and P.T., J.R. research; J.R., contributions: Author adjacent two any between angle bor- any Plateau that four such connects 2D) in node (three Every meet ders a nodes. which as called borders, represented vertices Plateau be called at constrained can channels structure is slender foam of emulsions dry network and the laws: local foams matter Plateau The soft dry by (30). by relatively structures years of foam/emulsion for structure dry studied as and such realized scientists, been have works direct a give not techniques do method. design and fabrication proposed compatible algorithms self-assembly the mathematical of on probably most based is that are This fact resolution. the insufficient to cases, related slowness, some are or in bottlenecks photolithography and serious cost, which as for such (27–29), approach, printing 3D top-down the by inated and with hyperuniformity agreement between in correlation system, (23). PBG observed whole such early of the reso- the order to Mie con- local hyperuniformity can their and strong fer fluctuations of meantime, Mie long-range the superposition decreases identical networks In by of (24–26). role PBG nances the a play opening sim- elements scatters of identical degree the Such estimate ilarity. to nodes, tools individual mathematical as proposed correspond such and PBGs elements, local disordered similar best with the the networks to that to states cowork- which leads and (21), This Florescu been ers by 23). proposed has concept, (13, self-uniformity hyperuniformity publications local long-range multiple local in to uniform underlined of addition facilitate importance in to The PBG. shown topology large been of has appearance fluctuations the density long-range of owo orsodnemyb drse.Eal oharcuirwimn.ci or [email protected] Email: [email protected] addressed. be may correspondence whom To oi om oss ag stoi htncbn a and gap structures. band 3D toward photonic pathway isotropic a large discuss pho- a 2D possess that foams demonstrate tonic We self-assembly. by class can fabricated foams”—that a be materials—“photonic present top- gap we band (cost, the Here, photonic limitations of etc.). severe by resolution, its defects, dominated from timescale, suffers amorphous completely and approach such is down of materials fabrication photonic the ultraresistant wave Surprisingly, iridescent freeform or of pigments. production displays, the energy-efficient enable guides, can gap pho- band complete tonic with materials disordered discovered Recently Significance ntematm,anme fsl-sebigfufl net- fourfold self-assembling of number a meantime, the In dom- is materials PBG disordered of fabrication the far, So rer ePyiu td hmeIdsrelsd aVled Paris, de Ville la de Industrielles Chimie de et Physique de erieure ´ y NSlicense.y PNAS . y www.pnas.org/lookup/suppl/doi:10. 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APPLIED PHYSICAL SCIENCES Plateau borders is 109.47◦ in 3D (tetrahedral angle) and 120◦ To check this hypothesis, we report here the use of 2D foams in 2D (30, 31). Foams also tend to equilibrate the length of (monolayers of bubbles squeezed between two plates) as poten- Plateau borders and avoid very long and very short channels tial templates to generate self-assembled 2D material with a TE (31–33). At constant volume fraction, the pressure equilibrium (electric field in the plane) omnidirectional PBG. Foams are between bubbles also gives rise to a constant thickness of the both created experimentally and simulated numerically. Their Plateau borders (31). This gives foam a very strong local order, photonic properties are calculated by a plane wave expansion very similar to the local self-uniformity concept, which accord- method. The structural organization of relatively dry 2D foams ing to ref. 21, is crucially important to open a PBG. Moreover, (such as threefold connectivity of Plateau borders, for example) these structural properties are self-sustaining: in case of any turns out to be especially useful to open a large PBG compara- destruction event (for example, coalescence of two bubbles), the ble with the best ones described in the literature for optimized foam will immediately rearrange its structure to again fulfill the disordered materials. Our results show that the small polydis- Plateau laws. Foam has been shown to be an interesting mate- persity of sizes introduces a controlled level of disorder rial possessing a phononic band gap for acoustic waves (34). The to foam structure, which makes the band gap isotropic without described properties, such as strong similarity of foam structure strongly perturbing photonic performance. Such 2D foams are units and robustness of its local organization, as well as the over- well known to capture the main properties of real 3D foams, and all resemblance of dry foam structure to amorphous diamond therefore, it provides strong evidence that our results in 2D can allow us to speculate that, having the appropriate refractive index be directly transferred to real 3D foams. contrast between “” and “air” phases, foams could pro- We believe that this research not only can define a roadmap duce a large robust PBG for wavelengths comparable with the to a class of self-assembled photonic materials but also, sheds a bubble size. light on the fundamental questions of PBG .

A BC

D E

Fig. 1. (A) Binary image of a simulated dry-like foam generated from a random close pack configuration followed by Voronoi and annealed with Surface Evolver software. (Inset) Close-up view of the Fourier transform of the image. (B) Photograph of an experimental 2D foam generated in a Hele–Shaw cell. Typical bubble diameter, 2 mm. (Inset) Close-up view of the Fourier transform of the image. (C) Disordered disk assembly representing very wet foam. (Inset) Close-up view of the Fourier transform of the image. For all three systems, the ratio of bubble sizes is about 0.75, while the fraction of big bubbles is about 0.55. The area fraction of the continuous phase is about 0.40. (D) Corresponding band diagrams. We set the refractive index contrast to p 3.4. The frequencies are normalized with characteristic length defined as a = L/ Nbubbles, with L being the dimension of the image and Nbubbles being the number of bubbles inside the image. (E) Normalized density of optical states for each system. Band gap widths are 28 and 22% for the simulated dry-like foam and the experimental 2D foam, respectively. Only a pseudogap can be observed for disordered disk assemblies. More details on PBG calculations can be found in SI Appendix—notably a convergence test in SI Appendix, Fig. S1.

2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1820526116 Ricouvier et al. Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 iove tal. et Ricouvier calculations our in silicon with phase (n liquid the vir- replacing By domains. tually circular extended spatially A peaks without foam Bragg borders). isotropic of absence Plateau 1B, and of transform (Fig. Fourier length foam the the of of as crystal- self-similarity avoid the (such to perturbing elements chosen strongly about is is without polyspersity areas, lization low cell’s relatively Voronoi as This corresponding defined 0.75. bubbles, of small root to square big of the ratio size The cell. Hele–Shaw 1B Fig. Discussion and Results oa rdcdNO)sat ogl ttesm rqec for pseu- frequency same or the 0) at = roughly starts (NDOS NDOS) PBG (reduced dogap the slightly ones), and disordered crystallized polydisperse monodisperse (both systems considered incompatible completely is dielectric one appropriate PBG. wet the with the at while PBG fraction, large We material a 0.8. and gives as nodes borders high well-defined with as Plateau structure fractions foam dry area that for struc- conclude area persists can crystal any gap and foam band for dry-like the gap disordered simulated tures, of band with case least sizeable simulations the any at In Our fraction. show limit, view. never wet of higher assemblies the point for disk approach PBG observed to be the starts can from Foam PBG fractions. no and area closes, abruptly Surprisingly, foam foams. dry-like simulated about for at close calculated quantitatively 2). also one is (Fig. the system fraction to experimental same the for the PBG at cor- The approximately both the maximum crystals, in disk a observed triangular having one or network the honeycomb to responding similar maxi- qualitatively a is through behavior passes It fraction. at area mum the with PBG increases the fraction foam of area width the material material, dielectric tric high the 2 Fig. φ in plot we limit, 1D). sign (Fig. no dimin- PBG but strongly true (NDOS) with a states time, of pseudogap of density same a photonic only the normalized has At ished band system large one. foam a experimental wet has the the foam with dry-like comparable simulated gap the mate- fraction, dielectric material same area the dielectric for rial that, the see the from can study One separately influence. to fraction structure us foam allow systems of Con- foams. simulated effect physical two real these represent of not siderations do they dielectric of if later). content even low material, discussed with frac- assemblies area disk is high disordered with and transition foams tion dry-like consider (the can when still limit increases we wet However, fraction the to area dry the the from transition found a be experiences can foam information More dry limit. wet mimic in very rep- to a and assemblies in disk foams laws bidisperse resent disordered Plateau obtained use also The fulfill We software. foams. always Evolver structures Surface the network and with assemblies disk annealed pro- disordered are are of tessellation foams Voronoi dry-like by Simulated duced cases. extreme two the represent PBG of disordered one self-assembled foam 2D of makes (30% materials. examples This literature promising (13). the 2D) most in in polarization described TE systems disordered almost best size are width direction gap gap gap 25% the band band band the the of on that to ratio depend find close not We modes flat. does the which propagation: of PBG, isotropic an eedneo B width PBG of dependence lsrlo ttePGfeunisrvasta,i all in that, reveals frequencies PBG the at look closer A wet to dry from evolves response PBG foam real how see To 1 Fig. aeil n Methods and Materials 3 = o xeietl2 om hc scmaal ihthe with comparable is which foam, 2D experimental for ,w bev htsc iodrdfasdemonstrate foams disordered such that observe we .4), hw htgaho iipre2 ompoue na in produced foam 2D bidisperse of photograph a shows φ A φ 0 = rv htw aaet raeadisordered a create to manage we that prove Inset) and 0 = .3 .45 C − hw w eeec iuae ytm,which systems, simulated reference two shows 0.4 − h adgpo h xeietl2D experimental the of gap band the 0.55, n trst erae pt hspit the point, this to Up decrease. to starts and ∆ and ω ∆ h xeietl2D experimental The Appendix. SI omdl frequency middle to ω/ω 0 ∆ o o uniyo dielec- of quantity low For . ω/ω ∆ 0 ω/ω o xeietl2D experimental for 0 enda the as defined , ω 0 reaches , for aso h orsodn ytm r niae in indicated are systems corresponding the of foams. gaps 2D experimental for and (Left PBG circles of purple frequencies foams; (C upper dry-like upper and simulated and and lower for lower lower squares, line, PBG orange line, of disks; of green frequencies lattice frequencies crystal; upper triangular upper the network for and PBG honeycomb lower of line, the frequencies Blue for PBG. PBG and TE Lower of foam the (B) dry-like squares). for (purple simulated frequencies foam line), upper 2D (green experimental and disks circles), crystal of (orange network lattice honeycomb triangular structures: line), different (blue for fraction area material 2. Fig. NDOS where frequencies, of assemblies, range disk narrow disordered very the a for have even elec- (13). which valid the structures stays for network similar result bubbles for This the shown previously around as material field tric dielectric field the magnetic the in for bubbles and individual in localized strongly are various and between width systems PBG various of edge for systems. change upper different the be the for responsible to meantime, is out the thus, turns In gap band 2). the (Fig. of fraction area given a C B A htgah fteeprmna Dfa o ifrn rafractions: area different for foam 2D experimental the of Photographs ) ssoni i.3 h ilcrcbnsjs eo h PBG the below just bands dielectric the 3, Fig. in shown As 6 (Center 26, ) B it o Eplrzto safnto ftedielectrical the of function a as polarization TE for width PBG (A) 26 % 1 n ( and 41, ) Right 4,tpclbbl imtr2m.Band mm. 2 diameter bubble typical 54%, )

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APPLIED PHYSICAL SCIENCES approaches zero. Clearly, the lower edge frequency depends create defect states that start to fill the PBG of dry foam from mainly on the mean bubble size and area fraction, also influ- the upper edge and gradually diminish it with an increase of encing the separation between the bubbles. It does not strongly the area fraction. To highlight this strong correlation between depend on the exact bubble shape (polygons or disks) or on PBG width and percentage of fourfold nodes, we show for exper- the structural organization of the system (crystallized or not). imental 2D foams in Fig. 4A, Insets the number of defect modes Passing through the band gap, both fields change their local- as a function of fourfold nodes number. The number of modes ization, such as it is well known for both crystalline and dis- inside the PBG clearly scales linearly with the number of fourfold ordered PBG materials (8, 13). The air bands at the upper nodes throughout all of our data. This gives a straightforward edge of the band gap are rather localized around individual way to estimate the PBG performance of the foams simply foam nodes. The magnetic field is localized inside the nodes, by considering their topology. This linear scaling also means while the electric field is concentrated in bubbles adjacent to that, if we removed the states corresponding to these fourfold the nodes. nodes, the PBG of experimental foams would follow the PBG of We should naturally expect that the change of the PBG upper simulated dry-like foams in the whole range of studied area frac- edge for different systems, such as a rapid closing for wet foams tions. This, for example, reveals that geometric factors, such as in comparison with the dry ones, should be related to the change local curvature of nodes or circularity of bubbles (SI Appendix, of the node organization. Indeed, we observe that, for relatively Fig. S2 shows circularity of experimental and simulated dry-like wet 2D foams (area fraction above 0.4), where the PBG width foams), are not particularly important for the PBG in TE polar- ∆ω/ω0 starts to diminish, the first modes after the PBG corre- ization. This phenomenon can also be seen for the crystalline spond to the fourfold nodes as shown in Fig. 3. Such fourfold structures, where both honeycomb networks and triangular lat- nodes clearly do not respect Plateau laws and can appear only tice of disks have almost identical PBG for TE (Fig. 2 and SI when the foam goes to the wet limit. To highlight that such Appendix, Fig. S3). nodes are indeed responsible for the PBG suppression for wet The analogy between the PBG of crystallized materials and foams, we plot a fraction of such nodes as a function of area our systems stays valid for TM (transverse magnetic) polariza- fraction as shown in Fig. 4. One can see that these defect nodes tion. No sizeable TM PBG is found in honeycomb network lattice appear for area fraction higher than 0.4. The number of fourfold as well as in our systems (SI Appendix, Fig. S4). Similar net- nodes rapidly explodes with increasing dielectric material quan- works of walls described in the literature possess a large PBG tity in the same range of area fractions where the PBG abruptly for TE polarization but no band gap for TM (23). However, closes down. We recall that our simulated dry-like foams obey this 2D network has given rise to 3D PBG structures, such as the Plateau laws even for relatively high area fractions and do amorphous diamond, underlining a particular importance of TE not have such fourfold nodes by construction. The rate of four- polarization. fold nodes can also serve to define the limit (about 35–40% of Summing up, we can conclude that the lower frequency of the area fraction) where experimental foam cannot be represented TE PBG of the studied systems is mainly related to the fraction by simulated dry-like foam. of the dielectric material and the size of the bubbles no matter If we compare experimental 2D foams and simulated dry- how they are organized (crystallized or not) or the shapes that like foams with the same area fraction, electromagnetic modes the bubbles adopt (disks or polygons). The upper frequency is related to these fourfold nodes can be considered defect modes related to several phenomena; the most important one is the in the PBG of idealized dry foams. The fourfold nodes have a diversity of node shapes: the more diverse the nodes are, the lower characteristic frequency than threefold nodes related to smaller the PBG is. This approach also explains why disordered their bigger characteristic size. Therefore, the fourfold nodes disk assemblies, representing very wet foams at the jamming

Hz

AB C D

E Photonic Bandgap EF G H

Fig. 3. (A–D) Azimuthal magnetic field distribution in an experimental 2D foam for TE polarization: the fraction of dielectric material is 44%.(A) Extended mode before the PBG. (B) Localized mode before the PBG. (C) Localized mode after the PBG. (D) Extended mode after the PBG. Notice that the localized modes present four- and fivefold “symmetry.” (E–H) Electric field magnitude for TE polarization for the same modes and the same system. Please note that the electric field belongs to the plane for TE polarization, and therefore, the magnetic field is along the z axis.

4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1820526116 Ricouvier et al. Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 iove tal. hyperunifor- et to Ricouvier due PBG only robust not and is foams. large foams experimental the photonic wet that our to in more degradation dry observed once the of supports as transition result well the This at as PBG assemblies the disk of for systems: PBG studied of the all between assess absence the difference to the of tool explains concentration defects sensitive the local above, more discussed a As is is performance. the nodes, This PBG by fourfold example, bars. the for error represented, of order, the rate local within that believe similar we always why are simulated foams and 2D dry-like experimental of spectral Minimal density 0.4. can- spectral density minimum spectral We the the nodes. for for fourfold either or transition of limit special appearance wet any find the the not to revealing dry as the well from as going foams experimental for PBG density previ- the hyperuniformity with and agreement PBG in 23). between (13, is correlation result with observed correlated This ously be opening. to seems PBG the assemblies the fraction, disk area with comparison fixed in at (lower Thus, hyperuniformity PBG. of of decrease large a level Our possessing higher foams while hyperuniform. (area even 2D experimental be point an and to jamming dry-like provide simulated the accepted that at show are results assemblies works 16%) disk previous around to such fraction According 39), PBG. 38, no foams (36, have 2D which experimental assemblies, both (lower fewer see fluctuations have fraction, density systematically can long-range structures area One foam dry-like systems. given simulated considered and a all for for density fraction that, spectral area minimum the The with experimen- (37). of hyperuniformity systems the tal estimate to way convenient a ttemnmmwv vector the wave of sys- minimum 4B, extension the polydisperse Fig. at an generally, In is 36). and (35, which shows bidisperse tems foam, 4 for Fig. the factor foams. of structure our density of interest- spectral hyperuniformity also the is the it evaluate therefore, to and ing system, the in hyperuniformity completely that states defect PBG. create the nodes they fill fourfold and 3, the Fig. than in complicated have presented can more system even such in shapes, Nodes various PBG. sizeable a show (B) never mm. point, 2 diameter, bubble squares). Typical (black shown. foam is 2D node experimental fourfold and typical line), (Inset a (black fractions. of foams area photograph dry-like several a simulated for foams; line), 2D foams (red experimental assembly 2D in experimental nodes of fourfold density of Spectral number the vs. PBG the 4. Fig. AB ntematm,teseta est rtemnmmspectral minimum the or density spectral the meantime, the In h perneo adgpi fe eae otedge of degree the to related often is gap band a of appearance The ubro eetmdsin modes defect of Number (Insets) foams. 2D experimental for material dielectric of fraction area the along plotted nodes fourfold of Rate (A) χ min χ sntsniieeog ocpuetecoigo the of closing the capture to enough sensitive not is min 1 o iuae r-ieadeprmna foams experimental and dry-like simulated for 1 eas ltteseta density spectral the plot also we Inset, χ min hc a enpoe obe to proven been has which , χ min χ min bu h rafraction area the about hndsree disk disordered than ) χ min increases χ χ(q min ) ) pcrldniya iia aevco o ifrn ytm:dsree disk disordered systems: different for vector wave minimal at density Spectral ) el3 om ic hyaettaaetcnetdstructures. for connected valid tetravalent cir- stay are results optical they our planar since that fabricate foams expect to 3D also real platform we can a Moreover, foams as with photonic cuits. used a both 2D of be described correlated loss The already the be topology. and can fluctuations local density This uniform long-range PBG. of increase sizable the Disor- a , nodes. bubbly show of fourfold structure never large the mimicking in assemblies, states disk dered defect to the of due However, limit appearance wet foams. the the photonic in to diminishes guaran- strongly PBG performance 2D) foam isotropic photonic in and dry coordination large of a threefold similarity tee the at significant example, hyperuniformity and (for of scale nodes especially level length be PBG—high relevant to the gap the out open band turns tunable frac- to structure for the profitable foam way adjusting dry a by The and opening changed applications. size material, be the dielectric can that of gap tion show band We the large polarization. of a position TE with for materials PBG disordered isotropic for template self-assembling PBG isotropic complete self-assembled for the candidate of materials. promising closes generation a rapidly next foams any and photonic the foams; nodes 3D opal wet of makes inverse This variety to used PBG. large close widely a structurally be creates of are should disorder PBG opals foams the inverse photonic than 3D crystals: sug- robust proposed also more of foams dry PBG much 2D the of that disorder to gests Plateau reliability the high coating A by index borders. increased dielectric refractive of additionally quantity be high the should then, from material and foam by fabricated be foam should photonic dry material 3D 3D, a PBG in pro- complete first to foams for that, self-assembly, candidate dry mean would good of This a (31). vol- limit duce 5–10% accepted about a rather the is for the beyond which dielec- of and 20% is fraction 20–30% material volume around This around tric (18). of of 3.4 contrast material PBG index dielectric refractive isotropic of optimized large fraction for numerically ume a published been possess results have poten- dielectric to the diamonds of of from networks Amorphous estimate foams diamond rods. rough 3D amorphous a should similar for give structures structurally can width foam We PBG 3D well. tial dry as that PBG expect exhibit can we structures, such coordination. system, node the of example, organization for local as, the to related also, but mity ehv eosrtdta Dfascnb sda a as used be can foams 2D that demonstrated have We foam 2D dry simulated and real for obtained results the With NSLts Articles Latest PNAS | f6 of 5

APPLIED PHYSICAL SCIENCES Our research suggests that photonic foams are potentially an Simulated Dry-Like Foam. Weighted Voronoi tessellation is performed over excellent candidate for materials with a large complete isotropic obtained disordered disk assemblies. We anneal the structure with Surface PBG. Photonic foams are fully scalable to a wide range of wave- Evolver (42) to better approach experimental foams. By increasing the thick- lengths from visible range to microwaves and potentially, can be ness of the walls, we get periodic networks with a controlled, variable produced in a large quantity by self-assembly. quantity of dielectric material. Materials and Methods PBG Simulation. The band gap structures are calculated using periodic supercell approximation implemented in MIT PBG software (43). We check Experimental 2D Foam. Experimental 2D foams are produced using an in- that the periodicity of the supercell does not influence our results ( house Hele–Shaw cell consisting of two vertical glass plates separated by SI , Fig. S1). 1.5 mm. Two populations of bubbles are generated by blowing air through Appendix More details can be found in . two orifices into a 12 g/L SDS solution. To ensure the constant liquid frac- SI Appendix tion profile, the experiments are performed in the forced drainage regime: foaming liquid is added from the top at a controlled flow rate (31). ACKNOWLEDGMENTS. We thank K. Morozov, A. Leshansky, N. Stern, L. Froufe-Perez,´ F. Scheffold, and A. Salonen for fruitful discussions and suggestions made along the work. This work has been supported by Ecole Disordered Disk Assembly. We generate bidisperse 2D jammed disk assem- Superieure´ de Physique et de Chimie Industrielles de la Ville de Paris, blies with periodic boundary conditions using freely available code based Paris Sciences et Lettres (PSL) Research University and Institut Pierre-Gilles on the Lubachevsky–Stillinger algorithm (40, 41). To vary the area fraction, de Gennes. Microflusa receives funding from the European Commission we subtract a constant value from the radius of each disk. Horizon 2020 Future and Emerging Technologies Program Grant 664823.

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