Biological tissue-inspired tunable photonic fluidXinzhi Lia, Amit Dasa, and Dapeng Bi ( )a,1

aDepartment of Physics, Northeastern University, Boston, MA 02115

Edited by Andrea J. Liu, University of Pennsylvania, Philadelphia, PA, and approved May 15, 2018 (received for review September 7, 2017)

Inspired by how cells pack in dense biological tissues, we design2D and 3D amorphous materials that possess a complete pho-tonic bandgap. A physical parameter based on how cells adherewith one another and regulate their shapes can continuouslytune the photonic bandgap size as well as the bulk mechani-cal properties of the material. The material can be tuned to gothrough a solid–fluid phase transition characterized by a vanish-ing shear modulus. Remarkably, the photonic bandgap persists inthe fluid phase, giving rise to a photonic fluid that is robust toflow and rearrangements. Experimentally this design should leadto the engineering of self-assembled nonrigid photonic structureswith photonic bandgaps that can be controlled in real time viamechanical and thermal tuning.

photonic materials | cells | tissue mechanics | metamaterials |bioinspired materials

Photonic bandgap (PBG) materials have remained an intensefocus of research since their introduction (1, 2) and have

given rise to a wide range of applications such as radiationsources (3), sensors, wave guides, solar arrays, and optical com-puter chips (4). Most studies have been devoted to the designand optimization of photonic crystals—a periodic arrangementof dielectric scattering materials that have photonic bands due tomultiple Bragg scatterings. However, periodicity is not necessaryto form PBGs, and amorphous structures with PBGs (5–8) canoffer many advantages over their crystalline counterparts (7). Forexample, amorphous photonic materials can exhibit bandgapsthat are directionally isotropic (9, 10) and are more robust todefects and errors in fabrication (7). Currently there are fewexisting protocols for designing amorphous photonic materi-als. They include structures obtained from a dense packing ofspheres (3D) or disks (2D) (5, 11–14), tailor-designed protocolsthat generate hyperuniform patterns (9, 10, 14, 15), and spinodal-decomposed structures (16, 17). While these designs yield PBGs,such structures are typically static, rigid constructions that do notallow tuning of photonic properties in real time and are unsta-ble to structural changes such as large-scale flows and positionalrearrangements.

In this work, we propose a design for amorphous 2D and 3DPBG materials that is inspired by how cells pack in dense tis-sues in biology. We generate structures that exhibit broad PBGsbased on a simple model that has been shown to describe cellshapes and tissue mechanical behavior. An advantage of thisdesign is that the photonic and mechanical properties of thematerial are closely coupled. The material can also be tunedto undergo a density-independent solid–fluid transition, and thePBG persists well into the fluid phase. With recent advancesin tunable self-assembly of nanoparticles or biomimetic emul-sion droplets, this design can be used to create a “photonicfluid.”

ResultsModel for Epithelial Cell Packing in 2D. When epithelial andendothelial cells pack densely in 2D to form a confluent mono-layer, the structure of the resulting tissue can be described by apolygonal tiling (18). A great variety of cell shape structures havebeen observed in tissue monolayers, ranging from near-regulartiling of cells that resembles a dry foam or honeycomb lattice

(19) to highly irregular tilings of elongated cells (20). To bet-ter understand how cell shapes arise from cell-level interactions,a framework called the Self-Propelled Voronoi (SPV) modelhas been developed recently (21, 22). In the SPV model, thebasic degrees of freedom are the set of 2D cell centers {ri}, andcell shapes are given by the resulting Voronoi tessellation. Thecomplex biomechanics that govern intracellular and intercellularinteractions can be coarse-grained (18, 19, 23–27) and expressedin terms of a mechanical energy functional for individual cellshapes.

E =

N∑i=1

[KA(Ai −A0)2 +KP (Pi −P0)2

]. [1]

The SPV energy functional is quadratic in both cell areas ({Ai})with modulus KA and cell perimeters ({Pi}) with modulus KP .The parameters A0 and P0 set the preferred values for area andperimeter, respectively. Changes to cell perimeters are directlyrelated to the deformation of the acto-myosin cortex concen-trated near the cell membrane. After expanding Eq. 1, the termKPP

2i corresponds to the elastic energy associated with deform-

ing the cortex. The linear term in cell perimeter, −2KPP0Pi ,represents the effective line tension in the cortex and gives riseto a “preferred perimeter” P0. The value of P0 can be decreasedby up-regulating the contractile tension in the cortex (19, 24, 27),and it can be increased by up-regulating cell–cell adhesion. Wesimulate tissues containing N cells under periodic boundary con-ditions; the value of N has been varied from N = 64 to 1600 tocheck for finite-size effects (SI Appendix, Fig. S6). A0 is set to beequal to the average area per cell, and

√A0 is used as the unit

of length. After nondimensionalizing Eq. 1 by KAA20 as the unit

energy scale, we choose KP/(KAA0) = 1 such that the perime-ter and area terms contribute equally to the cell shapes. Thechoice of KP does not affect the results presented. The preferredcell perimeter is rescaled p0 =P0/

√A0 and varied between 3.7

Significance

We design an amorphous material with a full photonicbandgap inspired by how cells pack in biological tissues. Thesize of the photonic bandgap can be manipulated throughthermal and mechanical tuning. These directionally isotropicphotonic bandgaps persist in solid and fluid phases, hence giv-ing rise to a photonic fluid-like state that is robust with respectto fluid flow, rearrangements, and thermal fluctuations in con-trast to traditional photonic crystals. This design should leadto the engineering of self-assembled nonrigid photonic struc-tures with photonic bandgaps that can be controlled in realtime via mechanical and thermal tuning.

Author contributions: D.B. designed research; X.L., A.D., and D.B. performed research;X.L., A.D., and D.B. analyzed data; and X.L., A.D., and D.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1 To whom correspondence should be addressed. Email: d.bi@northeastern.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1715810115/-/DCSupplemental.

Published online June 11, 2018.

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(corresponding to the perimeter of a regular hexagon with unitarea) and 4.6 (corresponding to the perimeter of an equilateraltriangle with unit area) (27). We obtain disordered ground statesof the SPV model by minimizing E, using the Limited-memoryBroyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm (28)starting from a random Poisson point pattern. We also test thefinite temperature behavior of the SPV model by performingBrownian dynamics (21).

The ground states of Eq. 1 are amorphous tilings wherethe cells have approximately equal area but varying perime-ters as dictated by the preferred cell perimeter p0. It has beenshown that at a critical value of p∗0 ≈ 3.81, the tissue collec-tively undergoes a solid–fluid transition (27). When p0< p∗0 , cellsmust overcome finite energy barriers to rearrange and the tissuebehaves as a solid, while above p∗0 , the tissue becomes a fluid witha vanishing shear modulus as well as vanishing energy barriersfor rearrangements (27). Coupled to these mechanical changes,there is a clear signature in cell shapes at the transition (21,29, 30): The shape-based order parameter calculated by averag-ing the observed cell perimeter-to-area ratio s = 〈P/

√A〉 grows

linearly with p0 when p0> p∗0 in the fluid phase but remainsat a constant (s ≈ p∗0 ) in the solid phase. In Fig. 1A, we showthree representative snapshots of the ground states at variousvalues of p0. We take advantage of the diversity and tunabilityof the point patterns and cell structures (Fig. 1A) produced bythe SPV model and use them as templates to engineer photonicmaterials.

Characterization of 2D Structure. To better understand the groundstates of the SPV model, we first probe short-range order byanalyzing the pair-correlation function g(r) (see details in Mate-rials and Methods) of cell centers. In Fig. 1C, g(r) for thesolid phase (p0< 3.81) shows mutual exclusion between near-est neighbors and becomes constant at large distances. Thesefeatures are similar to those observed in other amorphousmaterials with short-range repulsion and a lack of long-rangepositional order, such as jammed granular packings (31) anddense colloidal arrays (32). When p0 is increased, the tissueenters into the fluid phase at p0> 3.81. To satisfy the higherpreferred perimeters, cells must become more elongated. Andwhen two neighboring elongated cells align locally, their cen-

ters can be near each other, whereas cells not aligned will havecenters that are further apart. As a result, the first peak ofg(r) broadens. Hence, as p0 is increased, short-range orderis reduced. Deeper in the fluid regime, when p0> 4.2, g(r)starts to develop a peak close to r = 0, which means that cellcenters can come arbitrarily close. Interestingly, the loss of short-range order does not coincide with the solid–fluid transition,which yields an intermediate fluid state that retains short-rangeorder.

Next we focus on the structural order at long length scales.While the SPV ground states are aperiodic by construction, theyshow interesting long-range density correlations. The structurefactor S(q) (see details in Materials and Methods) is plotted forvarious p0 values in Fig. 1 B and D. Strikingly, for all p0 valuestested, the structure factor vanishes as q→ 0, corresponding toa persistent density correlation at long distances. This type of“hidden” long-range order is characteristic of patterns that areknown as hyperuniformity (33, 34). In real space, hyperunifor-mity is equivalent to when the variance of the number of pointsσ2R in an observation window of radius R grows as function of the

surface area of the window—that is, σ2R ∝Rd−1, where d is the

space dimension (33). This is in contrast to the σ2R ∝Rd scaling

that holds for uncorrelated random patterns. Indeed, real spacemeasurements in the SPV model also confirm that the distribu-tion of cell centers is strongly hyperuniform for all values of p0and system sizes tested that include both solid and fluid states (SIAppendix, Fig. S1).

It has been suggested that hyperuniform amorphous patternscan be used to design photonic materials that yield PBGs (9,10). Florescu et al. (9) further conjectured that hyperuniformityis necessary for the creation of PBGs. However, recent workby Froufe-Perez et al. (14) have demonstrated that short-rangeorder rather than hyperuniformity may be more important forPBGs. The SPV model provides a unique example of a hyperuni-form point pattern with a short-range order that can be turnedon and off. This tunability will allow a direct test of the ideasproposed in refs. 9, 10, and 14.

2D Photonic Material Design and Properties. For any point pattern(crystalline or amorphous), the first step in the engineering ofPBGs is to decorate it with a high dielectric contrast material.

A

B

C D

Fig. 1. Tissue structure in the SPV model. (A) Simulation snapshots at three different values of the preferred cell perimeters p0. Cell centers are indicatedby points, and cell shapes are given by their Voronoi tessellation (red outlines). (B) Contour plot of the structure factor S(qx , qy ) corresponding to the statesshown in A. Scale bar has length 2π/D in reciprocal space, where D is the average spacing between cell centers. (C) Pair-correlation function g(r) at differentvalues of p0. (D) Structure factor S(q) at different values of p0.

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The simplest protocol is to place cylinders centered at each point{ri = (xi , yi)} that are infinitely tall in the z−direction. Suchdesign typically yields bandgaps in the transverse magnetic (TM)polarization (the magnetic field is parallel to the xy-plane) (35).Based on this design, we first construct a material using SPVpoint patterns. To maximize the size of the bandgap, the cylin-ders are endowed with dielectric constant ε= 11.56 and radiusr/D = 0.189 (14). We will refer to this construction as “TM-optimized.” We also use a second decoration method (9, 14)that has been shown to yield complete PBGs—that is, gaps inboth TM and Transverse Electric (TE) polarizations. We usea design based on the Delaunay triangulation of a point pat-tern. Cylinders with ε= 11.56 and radius r/D = 0.18 are placedat the nodes of the Delaunay triangulation while walls withε= 11.56 and thickness w/D = 0.05 are placed on the bonds ofthis trivalent network. We refer to this construction as “TM +TE-optimized.” Photonic properties are numerically calculatedusing the plane wave expansion method (36) implemented in theMIT Photonic Bands program. We use the supercell approxima-tion in which a finite sample of N cells is repeated periodically.The photonic band structure is calculated by following the pathof kq = (0, 0)→ ( 1

2, 12)→ (− 1

3, 13)→ (0, 0) in reciprocal space.

SI Appendix includes a sample script used for photonic bandcalculations.

The TM-optimized band structure based on a SPV groundstate with N = 64 cells and p0 = 3.85 is shown in Fig. 2A. Due tothe aperiodic nature of the structure, the PBG is isotropic in kq.We also calculate the ODOS (Fig. 2B) by binning eigenfrequen-cies from 10 samples with the same p0 but different initial seeds.The relative size of the PBG can be characterized by the gap–midgap ratio ∆ω/ω0, which is plotted as a function of p0 in Fig.3A for both TM- and TM + TE-optimized structures. We findthat the size of the PBG is constant in the solid phase of the SPVmodel (p0< p∗0 ) with width ∆ω/ω0≈ 0.36 for the TM-optimizedstructure and ∆ω/ω0≈ 0.1 in the TM + TE-optimized structure.In the fluid phase, ∆ω/ω0 decreases as p0 increases yet staysfinite in the range of 3.81< p0 . 4.0. At even larger p0 values, thePBG vanishes. The location at which the PBG vanishes appearsto coincide with the loss of short-range order in the structure. Toquantify this, we plot ∆ω/ω0 versus the first peak height of thepair-correlation g1 in Fig. 3C. This is in agreement with the find-

A B

Fig. 2. Characterizing photonic properties in 2D. (A) Photonic band struc-ture of Transverse Magnetic (TM) for the material constructed by placingdielectric cylinders at the cell centers exhibits a PBG. Design is based on aground state of the SPV at p0 = 3.85. kq is the in-plane wave vector. (B)The optical density of states (ODOS) of TM at different values of p0. Thewidth of the bandgap ∆ω has a strong dependence on p0, while the midgapfrequency ω0 remains constant (SI Appendix, Fig. S4).

ings of Yang et al. (11) and suggests that short-range positionalorder is essential to obtaining a PBG that allows for collectiveBragg backscattering of the dielectric material. This also showsthat PBGs are absent in states that are hyperuniform but missingshort-range order.

To test for finite-size dependence of the ODOS and the PBG,we carry out the photonic band calculations at p0 = 3.7 for var-ious system sizes ranging from N = 64 to N = 1600. At eachsystem size, the ODOS was generated by tabulating TM fre-quencies along the kq = (0, 0) and kq = (0.5, 0.5) directions inreciprocal space for 10 randomly generated states. As shown inSI Appendix, Fig. S6, while low-frequency modes may dependon the system size and bigger fluctuations exist for smaller sys-tems, the PBG is always located between mode numbers Nand N + 1 and has a width that does not change as a func-tion of N .

Since the model behaves as a fluid-like state above p∗0 ≈ 3.81(21, 27) and the PBG does not vanish in the fluid-like regime(Fig. 3A), this gives rise to a photonic fluid where a PBG canexist without a static and rigid structure. To show this explicitly,we test the effects of fluid-like cell rearrangements by analyz-ing the photonic properties of the dynamical fluid phase at finitetemperature. At a fixed value of p0 = 3.7, we simulate Brown-ian dynamics in the SPV model at different temperatures T .At each T , we take 10 steady-state samples and construct theTM-optimized structures to calculate their ODOS. In Fig. 3B,we plot the bandgap size as a function of increasing tempera-ture. Note that even past the melting temperature of T ≈ 0.25(21), the PBG does not vanish, again giving rise to a robustphotonic fluid phase. Inside the fluid phase, we also find thatthe PBG is not affected by the positional changes due to cellrearrangements (SI Appendix, Fig. S5). Finally, we analyze theeffect of heating on the short-range order. In Fig. 3C, increas-ing T results in another “path” in manipulating the short-rangeorder.

Extension to 3D Photonic Material. To further demonstrate theviability and versatility of tissue-inspired structures as designtemplates for photonic materials, we extend this study to 3D.Recently, Merkel and Manning (37, 38) generalized the 2D SPVmodel to simulate cell shapes in 3D tissue aggregates by replac-ing the cell area and perimeter with the cell volume and surfacearea, respectively. This results in a quadratic energy functionalthat is a direct analog of Eq. 1:

E =

N∑i=1

[KS (Si −S0)2 +KV (Vi −V0)2

], [2]

where Si and Vi are the surface area and volume of the i-th cell in3D, with the preferred surface area and volume being S0 and V0,respectively. Similar to the 2D version of the model, we have twomoduli—KS for the surface area and KV for volume. Follow-ing ref. 37, we make our model dimensionless by setting V

1/30

as the unit of length and KSV4/30 as the unit of energy. This

gives us a dimensionless energy and dimensionless shape fac-tor s0 =S0/V

2/30 as the single tunable parameter in 3D, which

is the anaolog of the parameter p0 in the 2D scenario. For the3D model, we perform Monte Carlo simulations (39) with thecell centers at a scaled temperature T = 1, expressed in the unitKSV

4/30 /kB where kB is the Boltzmann constant. During the

simulation, each randomly chosen cell center is given by a dis-placement following the Metropolis algorithm (40), where theBoltzmann factor is calculated using the energy function in Eq.2. For this purpose, we extract the cell surface areas and volumesfrom 3D Voronoi tessellations generated using the Voro++library (41). N such moves constitute a Monte Carlo step, and

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A B C

Fig. 3. Tunability of photonic properties in 2D. (A) The structure of the PBG as a function of p0. The TM-optimized structure corresponds to the size of theTM bandgap for a material constructed by placing dielectric cylinders at cell centers. The TM + TE-Optimized structure corresponds to complete PBGs for amaterial constructed using a trivalent network design. Phases are colored according to the mechanical property of the material. At p0≈ 3.81, the materialundergoes a solid–fluid transition where the shear modulus vanishes. The dependence of ω0 on p0 is weak as shown in SI Appendix, Fig. S4A. (B) Effectof heating on the PBG in the TM-optimized structure. At fixed p0 = 3.7, temperature T is gradually increased. At T ≈ 0.25, the material begins to fluidizethrough melting, while the bandgap still persists into the mechanical fluid phase. (C) The relationship between the bandgap size in the TM-optimizedstructure and the short-range order of the system. The explicit dependence of short-range order on p0 and T is shown in SI Appendix, Figs. S2 and S3.

we perform 105 such steps to ensure that the cells have reacheda steady state. Then we run for another 105 steps to computethe g(r) and S(q). The average volume per cell is held constantfor this procedure. Periodic boundary conditions were appliedin all directions. For all of the simulations, we keep the scaledmodulus KVV

2/30 /KS = 1. It was previously found (37) that the

T = 0 ground states undergo a similar solid-to-fluid transitionat s0∼ 5.4—that is, solid for s0< 5.4 and fluid when s0≥ 5.4.Our Monte Carlo simulations also recover this transition nears0 = 5.4 when we plot an effective diffusivity, extracted from themean-square displacement (MSD) of the cells, at different s0 val-

ues (SI Appendix, Fig. S9). States below the transition (s0 = 5)and above the transition (s0 = 5.82) are shown in Fig. 4 for asystem of N = 100 cells.

We characterize the structure of the 3D cell model by cal-culating their pair-correlation function g(r). In Fig. 4C, theshort-range order behaves similar to the 2D case. At low val-ues of s0 that correspond to rigid solids, the first peak in g(r) ispronounced, suggesting an effective repulsion between nearestneighbors. As s0 is increased, this short-range order gets erodedas a preference for larger surface area-to-volume ratio allowsnearest neighbors to be close. The s0 = 6.1 state in Fig. 4C is

A C D

B

Fig. 4. Structure characterization in the 3D Voronoi cell model. (A) Cell centers and their corresponding Voronoi tessellation as well as the decorated 3Dphotonic structure are shown for a state at s0 = 5 and N = 100 cells. The cell centers are drawn with a finite size and different colors only to aid visualization.(B) Cell centers and their corresponding Voronoi tessellation as well as the decorated 3D photonic structure are shown for a state at s0 = 5.82 and N = 100cells. (C) Pair-correlation function g(r) at different values of s0. (D) Structure factor S(q) at different values of s0.

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an extreme example where it is possible for two cell centers to bearbitrarily near. However, the long-range order remains through-out all values of s0 tested as shown by the S(q) in Fig. 4D. Thesmall value of S(q→ 0) is a consequence of suppressed den-sity fluctuations across the system when cells all prefer the samevolume dictated by Eq. 2. Whether they are truly hyperuniformwould require sampling at higher system sizes, which is beyondthe scope of this study.

Next, to make a photonic material, we decorate the Voronoitessellations to create a connected dielectric network. While aVoronoi tessellation is already a connected network made of ver-tices and edges, it possesses a large dispersion of edge lengths.This can result in vertices that are arbitrarily close to each otherand could hinder the creation of PBGs (42). To overcome this,we adopt a method described in refs. 9, 15, and 43 to makethe structure locally more uniform. In a 3D Voronoi tessella-tion, each vertex is calculated from the circumcenter of the fourneighboring cell centers and edges are formed by connectingadjacent vertices. In this design protocol, the connectivity of thedielectric network is the same as the network of the verticesand edges in the Voronoi tessellation. However, the vertex posi-tions of the dielectric network are replaced by the center-of-mass(barycenters) of the four neighboring Voronoi cell centers. Theresulting structure is a tetrahedrally connected network wherethe edges are more uniform in length, two representative sam-ples are shown in Fig. 5A. Next we decorate the network withdielectric rods of width W running along the edges. For thedielectric rods, we again use ε= 11.56, and their width W ischosen such that the volume filling fraction of the network isVrod/Vbox = 20% (42).

The photonic properties of the 3D dielectric network arecalculated using the MIT Photonic Bands program (36). We cal-culate the ODOS for structures based on different values ofs0. Here we have chosen structures containing N = 100 cells.Due to the isotropic nature of the photonic band structure (SIAppendix, Fig. S7), we generate the ODOS based on two recip-rocal vectors k = (0, 0, 0) & k = (0.5, 0.5, 0). At each value of s0,we average over 10 different random samples. We find the firstcomplete PBG between mode number nV and nV + 1, wherenV is the number of vertices. In the solid phase of the model(s0< 5.4), we find an average gap–midgap ratio of ∼ 6%; this

A B

C

Fig. 5. Characterizing photonic properties in the 3D design. (A) The opticaldensity of states is shown at various values of s0. Here frequency is in unitsof 2πc/L, where L is the average edge length in the photonic network. (B)The size of the gap–midgap ratio ∆ω/ω0 as a function of s0. (C) Midgapfrequency ω0 as a function of s0.

decreases as s0 is increased and becomes vanishingly small whenthe model is in its fluid phase (s0> 5.4) (Fig. 5B). Interestingly,the midgap frequency ω0 also shifts slightly toward lower val-ues with increasing s0 (Fig. 5C). The appearance of the PBGhere also coincides with the presence of short-range order. ThePBG vanishes when there is no longer a pronounced first peakin the g(r).

Discussion and ConclusionWe have shown that structures inspired by how cells pack indense tissues can be used as a template for designing amor-phous materials with full PBGs. The most striking feature ofthis material is the simultaneous tunability of mechanical andphotonic properties. The structures have a short-range orderthat can be tuned by a single parameter, which governs theratio between cell surface area and cell volume (or perimeter-to-area ratio in 2D). The resulting material can be tuned totransition between a solid and a fluid state, and the PBG canbe varied continuously. Remarkably, the PBG persists evenwhen the material behaves as a fluid. Furthermore, we haveexplored different ways of tuning the short-range order inthe material including cooling/heating and changing cell–cellinteractions. We propose that the results in Fig. 3C can beused as a guide map for building a photonic switch that iseither mechanosensitive (changing p0 or s0) or thermosensitive(cooling/heating).

While they are seemingly devoid of long-range order (i.e., theyare always amorphous and nonperiodic), these tissue-inspiredstructures always exhibit strong hyperuniformity. Most interest-ingly, there are two classes of hyperuniform states found in thiswork: One that has short-range order, and one that does not.While the former is similar to hyperuniform structures stud-ied previously, the latter class is new and exotic and has notbeen observed before. Furthermore, we have shown that hype-runiformity alone is not sufficient for obtaining PBGs, whichcomplements recent studies (14). Rather, the presence of short-range order is crucial for a PBG. Another recent study hassuggested that the local self-uniformity (LSU) (15)—a measureof the similarity in a network’s internal structure—is crucialfor bandgap formation. We believe that the states from theSPV model with short-range order also have a high degree ofLSU, and it is likely that LSU is a more stringent criterionfor PBG formation than the simplified measure of short-rangeorder based on g(r). This will be a promising avenue forfuture work.

It will be straightforward to manufacture static photonic mate-rials based on this design using 3D printing or laser etchingtechniques (10, 13, 15). In addition, top–down fabrication tech-niques such as electron beam lithography or focused ion beammilling can also be used that will precisely control the geometryand arrangement of micro/nanostructures.

An even more exciting possibility is to adapt this design pro-tocol to self-assemble structures. Recent advances in emulsiondroplets have demonstrated feasibility to reconfigure the dropletnetwork via tunable interfacial tensions and bulk mechanicalcompression (44–46). Tuning interfacial interactions such as ten-sion and adhesion coincides closely with the tuning parameter inthe cell-based model p0 (2D) and s0 (3D), which are inspired bythe interplay of intracellular tension and cell–cell adhesion in abiological tissue. Experimentally this may lead to construction ofa tunable range of artificial cellular materials that vary in surfacearea-to-volume ratios. Further the dynamically reconfigurablecomplex emulsions via tunable interfacial tensions (45) couldlead to switching of these properties in real time. Another possi-bility is the creation of inverted structures or “inverse opals” byusing the emulsion droplet network as a template (47, 48), whichis often able to enhance the photonic properties over the orig-inal template. Alternatively, nanoparticles grafted with polymer

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brushes (49, 50) could be designed to tune the effective surfacetension of individual particles as well as the adhesion strengthsbetween particles. This could allow the mimicking of the inter-action between cells and give rise to a controllable preferred cellperimeter in 2D.

Two recent open-source software packages, cellGPU (22) andAVM (51), have made it convenient to simulate very large systemsizes in 2D. This would allow in-depth analysis of the system-sizedependence of long- and short-range order in this model, whichis an exciting possibility for future research.

Materials and MethodsCharacterization of Structure. The pair correlation function is defined as (52)

g(r) =1

Nρ

⟨∑i

∑i 6=j

δ(~r +~ri −~rj

)⟩.

~ri is the position of the ith particle, N is the number of particles, and ρ isthe average particle density of the isotropic system. 〈·〉 denotes n ensemble

average over different random configurations at the same parameter set.The structure factor is given by (53)

S(~q) =1

N

⟨∑i,j

ei~q·(~ri−~rj

)⟩.

〈·〉 denotes n ensemble average over different random configurations at thesame parameter set. S(q) is obtained by further averaging over a sphere ofradius q in reciprocal space.

Photonic Band Structure Calculation. Photonic properties are numerically cal-culated using the plane wave expansion method (36) implemented in theMIT Photonic Bands program. We use the supercell approximation in whicha finite sample of N cells is repeated periodically. SI Appendix includes asample script used for photonic band calculations in 2D.

ACKNOWLEDGMENTS. The authors thank M. Lisa Manning, BulbulChakraborty, Abe Clark, Daniel Sussman, and Ran Ni for helpful discussions.The authors acknowledge the support of the Northeastern University Dis-covery Cluster and The Massachusetts Green High Performance ComputingCenter (MGHPCC).

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