Optimizing Hyperuniformity in Self-assembled Bidisperse Emulsions

Joshua Ricouvier,1 Romain Pierrat,2 Rémi Carminati,2 Patrick Tabeling,1 and Pavel Yazhgur11ESPCI Paris, PSL Research University, CNRS, IPGG,

MMN, 6 rue Jean Calvin, F-75005, Paris, France2ESPCI Paris, PSL Research University, CNRS,

Institut Langevin, 1 rue Jussieu, F-75005, Paris, France

We study long range density fluctuations (hyperuniformity) in two-dimensional jammed packingsof bidisperse droplets. Taking advantage of microfuidics, we systematically span a large range of sizeand concentration ratios of the two droplet populations. We identify various defects increasing longrange density fluctuations mainly due to organization of local particle environment. By choosingan appropriate bidispersity, we fabricate materials with a high level of hyperuniformity. Interestingtransparency properties of these optimized materials are established based on numerical simulations.

PACS numbers: 47.57.Bc, 05.65.+b, 82.70.Kj

Hyperuniform systems have been recently identified asa particular subclass of disordered systems with densityfluctuations vanishing at infinitely large length scales [1].In Fourier space, the structure factor S(q) reveals hype-runiformity by vanishing when q tends to zero. Thesesystems possess properties of crystals, such as long-rangeorder, in addition to characteristics of disordered sys-tems such as a statistically isotropic structure. Hyper-uniformity has been used as a guideline to create disor-dered photonic band gap materials (PBG) [2–4]. It pavesthe way for fabricating colloidal materials with forbid-den band gaps less sensitive to defects than those ob-served in ordered colloidal arrangements. Disorder alsobrings isotropy, useful to create free-form waveguides,laser cavities, noniridescent dyes or display devices [5–7]. It was also shown that hyperuniform materials canbe optimized to be optically dense and transparent atthe same time [8]. The obtained results have been nat-urally extended to other types of propagating waves, forexample to produce disordered phononic or electronicband gaps [9, 10]. Such active hyperuniform materialsare now mainly designed on a computer and are printedafterwards [3, 4, 11, 12]. These patterns can be opti-mized to exhibit tremendous complete photonic band gapbut suffer from severe limitations of 3D printing (timescale, defects, resolution, etc). By contrast, the bottom-up approach, potentially more interesting in terms ofthroughput and cost, has not yet led to innovative mate-rials [13, 14].

Using numerical simulations, Donev et al. pre-dicted [15] that random close packing of spheres can behyperuniform under certain conditions. Such obtainedstructures are called Maximally Random Jammed (MRJ)states and are defined as the most disordered strictlyjammed packing of solid spheres [16] which minimizesan arbitrary order parameter, e.g. a fraction of crystal-lized particles. To realize such assemblies, the systemshould avoid two kinds of defects: rattlers (particles freeto move in a confining cage) and crystalline domains.The former refers to zones less dense than the surround-

ing medium (underpacked defects), while the latter aredenser (overpacked defects) [17]. Both create long rangedensity fluctuations and are unfavorable for hyperunifor-mity.

In this Letter, we demonstrate the creation of hyper-uniform structures by self-assembling bidisperse dropletsin a Hele-Shaw configuration using a microfluidic chip.The small friction between liquid droplets, together withdeformability, allow us to close the cages around rattlingparticles and create truly jammed states, while size bidis-persity breaks the crystallisation. We take advantage ofmicrofluidics to produce a set of particle ensembles intypically a few minutes, precisely controlling the size andnumber ratios of droplets. We thus systematically varyboth ratios, unlike most studies of jammed states. Inter-estingly, we discover that hyperuniformity defects can bemuch more complex than expected. Density fluctuationsturn out to be very sensitive to the structure of clustersformed by the particles and their immediate neighbors.Thus a whole new way to describe hyperuniform jammedsystems is proposed. We believe that this can guide re-search for new materials with vanishing density fluctua-tions.

Our microfluidic device shown in Fig. 1 is made ofstandard soft photolithography and replica-molding tech-niques using polydimethylsiloxane (PDMS). Two popu-lations of droplets (FC-3283 in 0.2% SDS solution) withtwo different sizes (typically less than 2% of polydisper-sity) are created in two separate T-junctions. The heightof the channels (10µm) forces the droplet and to adopta pancake shape. We can therefore consider the sys-tem as 2D disks which organization does not stronglydepend on the true thickness of the Hele-Shaw cell [18].The droplet size is in the range 20 − 40µm. The rela-tively small size of droplets and high enough surface ten-sion (22±0.5mNm−1) keep them circular due to Laplacepressure inside while being close to the jamming point.Droplet production rates, and sizes can be tuned by vary-ing pressures in four input channels. As a result, the sizeratio between droplets of different sizes SR = Rs/Rb

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Figure 1. Top. Scheme of the microfluidic chip. Bottom.Experimental images of droplet assemblies. Gray scale rep-resents the orientational order parameter ψ6. White parti-cles are crystallized in a hexagonal lattice. SR = 0.8 andNF = 0.88 (left, polycristal) orNF = 0.38(right, disordered).

(here s stands for small and b for big droplets) and thenumber fraction of big particles NF = Nb/N are ad-justable. Defined this way, both ratios vary from 0 to 1.Then, the droplets pass through the micromixer whichbrings disorder and homogeneity to the droplets by ran-domizing their entry in the observation chamber. As en-tering, they self-organize themselves into a close packingby plastic events. Down from the entrance, plastic eventsrarefy and the droplets seem to adopt their final config-uration with almost circular shapes [19]. The ceiling ofthe observation chamber is reinforced with a glass slideplaced inside the PDMS to prevent deformations. Fil-ters are placed downstream to retain droplets inside thecell without stopping fluxes. The filters do not stop thedroplets completely but let them pass slowly, allowingone to renew the packing continuously. Top view mi-croscopy images of N ' 3000 can be taken allowing toprobe wavevectors as small as qmin ∼ 1/(Dm

√N), Dm

being the mean diameter of disks. To avoid any orderingdue to the walls, deformation close to the filters at theexit, and loosely packed zones at the entrance, the pho-tographs are taken at least five layers far from the edgesof the observation chamber [19, 20]. The interval betweenimages is typically on the order of 30 seconds, which cor-responds to the duration of the filling of the observationchamber. Therefore, systems are completely refreshed,and the images are uncorrelated. For each experimentalset of parameters at least 30 images are taken. Statisti-

cal measures (e.g. structure factor, fraction of particleswith a given coordination) are calculated on each imageand then averaged over the full ensemble. In each ex-perimental set we have checked that the populations arewell mixed and no global gradients of small or big par-ticle concentrations are present. We also ensure isotropyof images by checking circular symmetry of their Fouriertransformations.

We also perform numerical simulations of 2D bidis-perse jammed assemblies of disks using freely availablecode based on the Lubachevsky-Stillinger (LS) algorithm.It is based on a collision-driven packing generation, andis shown to create highly jammed bidisperse assembly ofdisks with a low rate of rattlers [21, 22].

To explore the whole diagram of different size ratioand number fraction, we start by keeping the size ratioSR constant, at a value 0.8 which is a standard valueused in most of previous studies [23, 24], and varyingNF . We observe that for values of NF close to 0 or 1,the system forms a polycrystal with well distinguishablehexagonal crystal domains. For moderate NF values, weobserve disordered structures where crystal domains aresmall and rare, and the assembly resembles a homoge-neous mixture of small and big disks. Two experimentalpictures for both polycrystal and disordered systems areshown in Fig. 1. To be more quantitative, the experimen-tal pictures are colored using the bond orientational orderparameter ψ6, which shows how the weighted Voronoi cellfor each particle is close to a regular hexagon and givesit a score from 0 to 1, approaching 1 for particles withan ideal 6-fold symmetry. By considering a particle tobe crystallized if ψ6 > 0.9, we plot in Fig. 2a the frac-tion of crystallized particles both for experimental andsimulated samples as a function of NF . The data clearlyshow that there exists an optimum where the fractionof crystallized particles is minimized. It is close to thispoint that MRJ and the most hyperuniform systems areexpected. Fig. 2a also shows that the rate of rattlers,defined as particles having less than three neighbor con-tacts, is always rather low in our system (less than 1 %).Therefore, they should not strongly affect the structuralproperties. This also indicates that we indeed succeed inapproaching a jamming point.

To check hyperuniformity of the obtained samples, wecalculate the spectral density χ(q), which is an extensionof the classical structure factor for bidisperse and moregenerally polydisperse particle assemblies. In our calcu-lations we use the spectral density definition which treatseach particle as a point object with weight equal to theparticle area [25, 26]:

χ(q) =1

N 〈s2i 〉|φ(q)|2, (1)

where N is the number of particles and si are their areas.In this expression, φ(q) is the Fourier transform of the

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Experiment NF=0,88 Simulation NF=0,88 Experiment NF=0,38 Simulation NF=0,38

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min Simulation min Experiment

min

NFc)

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0,02

Optical thickness L/

L/

Figure 2. (a) Fraction of crystalline particles and rattlers for simulated and experimental binary mixtures of solid particlesfor different fraction of big ones NF . (b) Spectral density of a binary disk mixture for different fractions of big particles NF .Linear dependence S ∼ q characteristic of MRJ systems is shown as a guide for eyes. Inset: linear fit of experimental data forNF = 0.38, y-intercept (2± 3)× 10−4. (c) Spectral density at minimum experimentally available wavevector (qDm/π ≈ 0.1)and optical thickness of the simulated 2D jammed assemblies L/` as a function of NF . The refractive index is 1 for the hostmedium and 1.3 for the droplets. A strong correlation between the fraction of crystal particles, the minimum spectral densityχmin and optical thickness L/` is clearly observed. All figures correspond to SR ≈ 0.80.

surface fraction defined as

φ(r) =∑i

siδ(r− ri), (2)

where ri are the positions of individual particle centers.Fig. 2b shows two spectral densities χ(q) in the poly-

crystal and disordered cases (SR ≈ 0.8). The spectraldensity of a polycrystalline system is not hyperuniform.Indeed, χ(q) does not monotonously decrease while ap-proaching zero, but rather passes through a local max-imum. The latter can be related to the size and dis-tance between the crystal domains. On the contrary, fordisordered systems close to the minimum of crystallizedparticle fraction, the spectral density approaches a lin-ear behavior as predicted for MRJ (see Fig. 2b) [15]. Alinear fit shown in the inset reveals a very low spectraldensity limit χ(q = 0) = (2± 3)× 10−4. This confirmshyperuniformity of our optimized systems [26, 27].

Strictly speaking, hyperuniformity is defined only atinfinite lengthscales, which are not relevant for real ma-terials, nor accessible experimentally [14, 27]. To es-timate the degree of hyperuniformity in the finite sys-tem, we choose the spectral density χmin at the mini-mum wavevector (qDm/2π ≈ 0.1) available from our ex-perimental data. It probes density fluctuations at largelengthscale about ten particle diameters which are suf-ficient for the sought optical properties [28]. Also com-pared to y-intercept, χmin does not assume any linearbehavior which is, for instance, disputed in [29]. Bothexperiments and simulations reveal that χmin has a sin-gle minimum at NF ≈ 0.3 − 0.4 (Fig. 2c) similarly tothe previously described ψ6. This result proves the dom-inating role of crystalline domains in the behavior of thespectral density, and guides us towards the most hyper-uniform systems available experimentally for a given SR.

We also use numerical simulations of light scattering todemonstrate that the optimized bidisperse emulsions are

good candidates for the production of high-density andtransparent disordered materials. In order to explore arange of scattering wavevectors minimizing the structurefactor, we have chosen λ = 500µm which corresponds tok0Dm/(2π) = 0.12, so that |q| ∼ k0 lies in the regionwhere the spectral density takes very small values. Lightscattering from the numerically generated 2D jammed as-semblies (TE polarization, electric field perpendicular tothe plane) is simulated by solving Maxwell’s equations us-ing the coupled-dipoles method [30]. We use the Mie the-ory to compute the scattering cross sections σs of a sin-gle disk, and replace each one by an electric point dipolewith an effective polarizability giving the same scatter-ing cross-section as the real droplet. From the calcula-tion of the average field inside the medium, we deducethe real scattering optical thickness L/` of the structures(see SM for details on the method). In Fig. 2c we observethat scattering is strongly suppressed for samples exhibit-ing the highest degree of hyperuniformity. This analysissupports that the microfluidic fabrication technique de-scribed in this Letter is an important step towards theassembly of hyperuniform materials for photonics.

Interestingly, we can also remark that while the con-centration of crystallized particles is similar for experi-mental and simulated systems (Fig. 2a), considering χmin

there exists a small discrepancy for polycrystalline sam-ples (Fig. 2c). The latter is affected not only by thefraction of crystallized particles but also by the size andshape of crystal domains, which are rather sensitive tothe generation processe. In the meantime, productionof hyperuniform systems is much more robust, makingthem even more attractive for potential applications.

Surprisingly, we cannot simply extend this correlationbetween χmin and ψ6 for systems with various size ratios.Actually, by varying the size ratio for a fixed numberfraction NF = 0.5, we reveal the existence of a few lo-

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Simulation 3 4 5 6Experiment 3 4 5 6

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Simulation Experiment 6 6 7 7 8 8 9 9 10 10

c) 0,0 0,2 0,4 0,6 0,8 1,00,02

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cl

SR

Experiment Simulation

d)

Figure 3. (a) Spectral density χmin at minimum experimen-tally available wavevector (qDm/π ≈ 0.1). Peaks and valleysare signatures of complex internal organization of differentdefects. Two examples of overpacked defects: a bidispersecrystal domain with four fold symmetry SR = 0.40 (left), acluster of a big particle surrounded by seven small particlesSR = 0.70 (right). (b) Fraction of small particles having agiven coordination number as a function of size ratio SR. (c)Fraction of big particle having a given coordination numberas a function of size ratio SR. (d) Variance for cluster localfraction. All figures correspond to NF ≈ 0.50.

cal minima for χmin, pointing out a much more complexbehavior (see Fig. 3a). Several size ratios, ranging from0.3 to 0.8 seem to have a low rate of density fluctuations.The crystallisation can still account for the general be-havior of the system, namely for the increase of densityfluctuations for SR close to 0 and 1. But, peaks and val-leys for χmin for medium range of SR (0.3-0.8) cannot beexplained solely with the hexagonal crystal domains, andthus second order defects should be introduced. A simi-lar nonmonotonic behavior has been previously describedfor the surface fraction of simulated bidisperse jammedassemblies, although no explanation has thus far beenprovided [31, 32].

The initial reduction of density fluctuations with de-creasing SR from 1 is explained by the destruction ofhexagonal crystal zones by analogy with NF varying sys-tems. The minimum spectral density χmin reaches itsextremum about SR ≈ 0.8. As shown in Fig. 3b and3c, for size ratio below 0.8, small particles with coordina-tion five (number of edges of their weighted Voronoi cell)and big particles with coordination seven can be iden-tified [32]. The particles under consideration, togetherwith their closest neighbors, can arrange in compact clus-ters at appropriate size ratios. Actually, the first type ofsuch clusters is a big particle surrounded by seven smallones (see Fig. 3a). From simple geometrical considera-tions, we can predict the size ratio SR where these clus-

ters appear by calculating the ideal cluster conformation:sin(π/7)/[1− sin(π/7)] ≈ 0.76. To give a quantitative es-timate of the cluster density, we define, for each particle,the cluster fraction ϕi

cl as the ratio of the occupied surfacein the polygon, whose vertices are the centers of the sur-rounding particles, over its area Ai (Fig. 4). For instance,in the considered example, we find a cluster fraction onthe order of 0.909, slightly higher than that correspond-ing to hexagonal crystals (0.906) and much higher thanthe average surface fraction (about 0.84). Such clustersare geometrical defects that locally increase the densityand create additional fluctuations.

Each type of cluster is compact only at a particularsize ratio SR, and exists mainly in the vicinity of thisSR. Typically, below and above the characteristic size,the clusters are loosely packed and have a density compa-rable to the surrounding medium, while close to the char-acteristic SR they form overpacked defects (see Fig. 4).This gives a kind of geometric resonance for the spectraldensity. In the range of SR about 0.5-0.7, a rich morphol-ogy of clusters exists with coordination five and seven forthe central particle. For example, we can identify smallparticles surrounded by two big and three small parti-cles, or by three big and two small particles, etc. In thiswork, we do not intend to precisely describe the wholevariety, although a tentative classification has been pro-posed [33]. Resonances for different cluster types overlapgiving a plateau of χmin. In comparison to hexagonalcrystal domains, these overpacked defects do not havelarge spatial extensions and are often limited to one clus-ter, thus creating only second order fluctuations.

In Fig. 3a, the peak around SR ≈ 0.4 is particular as itis due to the appearance of four fold symmetry bidispersecrystal domains. The footprints of this crystal are foundin the increased fraction of small particles with coordi-nation four, and of big particles with coordination eight(see Fig. 3b and 3c). A peculiarity of these clusters isthat they are often organized in small crystallized zones(Fig. 3a). As hexagonal crystal zones, such domains havehigher density and therefore increase the density fluctu-ations. They are overpacked defects that often extend tomore than one cluster. Actually, in a first approximation(low rate of defects), we can estimate that the influence ofany defect on χmin scales with its concentration and thesurface area square. This power law suggests that evensmall crystal zones influence the spectral density muchmore than the individual clusters. Also other types ofbidisperse crystal domains are expected to appear (seeRefs. [33, 34] for some examples).

If SR decreases further, the four-fold crystals start tobe mechanically unstable, and below SR = 0.15 they arereplaced by Appolonian packing (small particles in the in-terstices between three big ones). This packing is clearlyobserved by the appearance of a coordination three forsmall particles and larger than nine for big particles. Thebig particles are organized in hexagonal crystal domains

5

0.62 0.67 0.78

0.85 0.91 0.85clφSR

Figure 4. Geometrical resonance for a cluster of one small andfive big particles. Three size ratios are shown: slightly before,close and slightly after the resonance size ratio SR ≈ 0.7corresponding to a compact cluster. The compact cluster hasa density ϕcl well above the loosely packed ones.

which then cause large increase of χmin. Some additionaldensity fluctuations may arise due to the fact that smallparticles may occupy only a part of interstices. Thesesmall particles in the interstices also show an example ofrattlers (particles with less than three neighboring con-tacts) which can be overpacked defects or even have acluster fraction close to the mean value. Hence we pre-fer to describe the system in terms of overpacked andunderpacked defects.

In order to properly monitor the quantity of variousclusters, we can calculate for the set of obtained clusterlocal fractions ϕi

cl the standard deviation σcl weighted bythe cluster area Ai/

∑Ai. The evolution of this quantity

versus the size ratio is presented in Fig. 3d for NF = 0.5,showing that we are able to recover all peaks and valleysgiven by χmin. The standard deviation σcl reports localphenomenon by measuring the disparity of every clustertowards the mean surface fraction of the system, and,χmin reports density fluctuations at large length scale.Therefore for jammed particle assemblies we can corre-late hyperuniformity at large lengthscales and the localenvironment of particles. It makes σcl also a useful toolto estimate density fluctuations with finite size images.The fact that hyperuniformity is related to local particleenvironment also gives expectations to make a statisticaldescription of the problem using a kind of granocentricmodel such as in Ref. [35, 36].

To conclude, we have successfully produced large setsof two dimensional bidisperse jammed assemblies withwell controlled size and number ratios. In an optimizedrange of parameters, structures with reduced large scaledensity fluctuations, or equivalently a high level of hy-peruniformity, emerge. Thus this work confirms the rel-evance of microfluidics for the rapid production of largescale self-assembling hyperuniform materials [13]. Thegeometrical features of the fabricated structures are sup-ported by numerical simulations, in very good agreementwith the experiments. Our results establish links be-tween the overall local parameters such as coordinationand cluster fraction, and long range density fluctuations.This can potentially help to suggest hyperuniformity for

a much wider range of systems incompatible with longrange analysis. The proposed experimental approach alsohas the potential to be extended to study 3D packings[35, 36]. and could lead to the fabrication of opticallydense and transparent materials, as suggested by lightscattering simulations.

JR and PY contributed equally to this work. We aremuch grateful to C. Cejas, J. McGraw, P. Chaikin, S.Torquato, O. Dauchot and L. Berthier for fruitful dis-cussions and suggestions made along the work. TheMicroflusa project receives funding from the EuropeanUnion Horizon 2020 research and innovation programmeunder Grant Agreement No. 664823. This work was alsosupported by LABEX and EQUIPEX IPGG, LABEXWIFI (Laboratory of Excellence within the FrenchProgram “Investments for the Future”) under refer-ences ANR-10-IDEX-0001-02,ANR-10-LABX-31, ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*.

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