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# Biological tissue-inspired tunable photonic fluid

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

## Significance

We design an amorphous material with a full photonic bandgap inspired by how cells pack in biological tissues. The size of the photonic bandgap can be manipulated through thermal and mechanical tuning. These directionally isotropic photonic bandgaps persist in solid and fluid phases, hence giving rise to a photonic fluid-like state that is robust with respect to fluid flow, rearrangements, and thermal fluctuations in contrast to traditional photonic crystals. This design should lead to the engineering of self-assembled nonrigid photonic structures with photonic bandgaps that can be controlled in real time via mechanical and thermal tuning.

## Abstract

Inspired by how cells pack in dense biological tissues, we design 2D and 3D amorphous materials that possess a complete photonic bandgap. A physical parameter based on how cells adhere with one another and regulate their shapes can continuously tune the photonic bandgap size as well as the bulk mechanical properties of the material. The material can be tuned to go through a solid–fluid phase transition characterized by a vanishing shear modulus. Remarkably, the photonic bandgap persists in the fluid phase, giving rise to a photonic fluid that is robust to flow and rearrangements. Experimentally this design should lead to the engineering of self-assembled nonrigid photonic structures with photonic bandgaps that can be controlled in real time via mechanical and thermal tuning.

Photonic bandgap (PBG) materials have remained an intense focus of research since their introduction (1, 2) and have given rise to a wide range of applications such as radiation sources (3), sensors, wave guides, solar arrays, and optical computer chips (4). Most studies have been devoted to the design and optimization of photonic crystals—a periodic arrangement of dielectric scattering materials that have photonic bands due to multiple Bragg scatterings. However, periodicity is not necessary to form PBGs, and amorphous structures with PBGs (5⇓⇓–8) can offer many advantages over their crystalline counterparts (7). For example, amorphous photonic materials can exhibit bandgaps that are directionally isotropic (9, 10) and are more robust to defects and errors in fabrication (7). Currently there are few existing protocols for designing amorphous photonic materials. They include structures obtained from a dense packing of spheres (3D) or disks (2D) (5, 11⇓⇓–14), tailor-designed protocols that 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 not allow tuning of photonic properties in real time and are unstable to structural changes such as large-scale flows and positional rearrangements.

In this work, we propose a design for amorphous 2D and 3D PBG materials that is inspired by how cells pack in dense tissues in biology. We generate structures that exhibit broad PBGs based on a simple model that has been shown to describe cell shapes and tissue mechanical behavior. An advantage of this design is that the photonic and mechanical properties of the material are closely coupled. The material can also be tuned to undergo a density-independent solid–fluid transition, and the PBG persists well into the fluid phase. With recent advances in tunable self-assembly of nanoparticles or biomimetic emulsion droplets, this design can be used to create a “photonic fluid.”

## Results

### Model for Epithelial Cell Packing in 2D.

When epithelial and endothelial cells pack densely in 2D to form a confluent monolayer, the structure of the resulting tissue can be described by a polygonal tiling (18). A great variety of cell shape structures have been observed in tissue monolayers, ranging from near-regular tiling of cells that resembles a dry foam or honeycomb lattice (19) to highly irregular tilings of elongated cells (20). To better understand how cell shapes arise from cell-level interactions, a framework called the Self-Propelled Voronoi (SPV) model has been developed recently (21, 22). In the SPV model, the basic degrees of freedom are the set of 2D cell centers **1**, the term *SI Appendix*, Fig. S6). **1** by *E*, using the Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm (28) starting from a random Poisson point pattern. We also test the finite temperature behavior of the SPV model by performing Brownian dynamics (21).

The ground states of Eq. **1** are amorphous tilings where the cells have approximately equal area but varying perimeters as dictated by the preferred cell perimeter *A*, we show three representative snapshots of the ground states at various values of *A*) produced by the SPV model and use them as templates to engineer photonic materials.

### Characterization of 2D Structure.

To better understand the ground states of the SPV model, we first probe short-range order by analyzing the pair-correlation function *Materials and Methods*) of cell centers. In Fig. 1*C*,

Next we focus on the structural order at long length scales. While the SPV ground states are aperiodic by construction, they show interesting long-range density correlations. The structure factor *Materials and Methods*) is plotted for various *B* and *D*. Strikingly, for all *SI Appendix*, Fig. S1).

It has been suggested that hyperuniform amorphous patterns can be used to design photonic materials that yield PBGs (9, 10). Florescu et al. (9) further conjectured that hyperuniformity is necessary for the creation of PBGs. However, recent work by Froufe-Pérez et al. (14) have demonstrated that short-range order rather than hyperuniformity may be more important for PBGs. The SPV model provides a unique example of a hyperuniform point pattern with a short-range order that can be turned on and off. This tunability will allow a direct test of the ideas proposed 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 of PBGs is to decorate it with a high dielectric contrast material. The simplest protocol is to place cylinders centered at each point *xy*-plane) (35). Based on this design, we first construct a material using SPV point patterns. To maximize the size of the bandgap, the cylinders are endowed with dielectric constant *SI Appendix* includes a sample script used for photonic band calculations.

The TM-optimized band structure based on a SPV ground state with *A*. Due to the aperiodic nature of the structure, the PBG is isotropic in *B*) by binning eigenfrequencies from 10 samples with the same *A* for both TM- and TM + TE-optimized structures. We find that the size of the PBG is constant in the solid phase of the SPV model (*C*. This is in agreement with the findings of Yang et al. (11) and suggests that short-range positional order is essential to obtaining a PBG that allows for collective Bragg backscattering of the dielectric material. This also shows that PBGs are absent in states that are hyperuniform but missing short-range order.

To test for finite-size dependence of the ODOS and the PBG, we carry out the photonic band calculations at *SI Appendix*, Fig. S6, while low-frequency modes may depend on the system size and bigger fluctuations exist for smaller systems, the PBG is always located between mode numbers N and

Since the model behaves as a fluid-like state above *A*), this gives rise to a photonic fluid where a PBG can exist without a static and rigid structure. To show this explicitly, we test the effects of fluid-like cell rearrangements by analyzing the photonic properties of the dynamical fluid phase at finite temperature. At a fixed value of *B*, we plot the bandgap size as a function of increasing temperature. Note that even past the melting temperature of *SI Appendix*, Fig. S5). Finally, we analyze the effect of heating on the short-range order. In Fig. 3*C*, increasing T results in another “path” in manipulating the short-range order.

### Extension to 3D Photonic Material.

To further demonstrate the viability and versatility of tissue-inspired structures as design templates for photonic materials, we extend this study to 3D. Recently, Merkel and Manning (37, 38) generalized the 2D SPV model to simulate cell shapes in 3D tissue aggregates by replacing the cell area and perimeter with the cell volume and surface area, respectively. This results in a quadratic energy functional that is a direct analog of Eq. **1**:*i*-th cell in 3D, with the preferred surface area and volume being **2**. For this purpose, we extract the cell surface areas and volumes from 3D Voronoi tessellations generated using the Voro++ library (41). N such moves constitute a Monte Carlo step, and we perform *SI Appendix*, Fig. S9). States below the transition (

We characterize the structure of the 3D cell model by calculating their pair-correlation function *C*, the short-range order behaves similar to the 2D case. At low values of *C* is an extreme example where it is possible for two cell centers to be arbitrarily near. However, the long-range order remains throughout all values of *D*. The small value of **2**. Whether they are truly hyperuniform would require sampling at higher system sizes, which is beyond the scope of this study.

Next, to make a photonic material, we decorate the Voronoi tessellations to create a connected dielectric network. While a Voronoi tessellation is already a connected network made of vertices and edges, it possesses a large dispersion of edge lengths. This can result in vertices that are arbitrarily close to each other and could hinder the creation of PBGs (42). To overcome this, we adopt a method described in refs. 9, 15, and 43 to make the structure locally more uniform. In a 3D Voronoi tessellation, each vertex is calculated from the circumcenter of the four neighboring cell centers and edges are formed by connecting adjacent vertices. In this design protocol, the connectivity of the dielectric network is the same as the network of the vertices and edges in the Voronoi tessellation. However, the vertex positions of the dielectric network are replaced by the center-of-mass (barycenters) of the four neighboring Voronoi cell centers. The resulting structure is a tetrahedrally connected network where the edges are more uniform in length, two representative samples are shown in Fig. 5*A*. Next we decorate the network with dielectric rods of width W running along the edges. For the dielectric rods, we again use

The photonic properties of the 3D dielectric network are calculated using the MIT Photonic Bands program (36). We calculate the ODOS for structures based on different values of *SI Appendix*, Fig. S7), we generate the ODOS based on two reciprocal vectors *B*). Interestingly, the midgap frequency *C*). The appearance of the PBG here also coincides with the presence of short-range order. The PBG vanishes when there is no longer a pronounced first peak in the

## Discussion and Conclusion

We have shown that structures inspired by how cells pack in dense tissues can be used as a template for designing amorphous materials with full PBGs. The most striking feature of this material is the simultaneous tunability of mechanical and photonic properties. The structures have a short-range order that can be tuned by a single parameter, which governs the ratio between cell surface area and cell volume (or perimeter-to-area ratio in 2D). The resulting material can be tuned to transition between a solid and a fluid state, and the PBG can be varied continuously. Remarkably, the PBG persists even when the material behaves as a fluid. Furthermore, we have explored different ways of tuning the short-range order in the material including cooling/heating and changing cell–cell interactions. We propose that the results in Fig. 3*C* can be used as a guide map for building a photonic switch that is either mechanosensitive (changing

While they are seemingly devoid of long-range order (i.e., they are always amorphous and nonperiodic), these tissue-inspired structures always exhibit strong hyperuniformity. Most interestingly, there are two classes of hyperuniform states found in this work: One that has short-range order, and one that does not. While the former is similar to hyperuniform structures studied previously, the latter class is new and exotic and has not been observed before. Furthermore, we have shown that hyperuniformity alone is not sufficient for obtaining PBGs, which complements recent studies (14). Rather, the presence of short-range order is crucial for a PBG. Another recent study has suggested that the local self-uniformity (LSU) (15)—a measure of the similarity in a network’s internal structure—is crucial for bandgap formation. We believe that the states from the SPV model with short-range order also have a high degree of LSU, and it is likely that LSU is a more stringent criterion for PBG formation than the simplified measure of short-range order based on

It will be straightforward to manufacture static photonic materials based on this design using 3D printing or laser etching techniques (10, 13, 15). In addition, top–down fabrication techniques such as electron beam lithography or focused ion beam milling can also be used that will precisely control the geometry and arrangement of micro/nanostructures.

An even more exciting possibility is to adapt this design protocol to self-assemble structures. Recent advances in emulsion droplets have demonstrated feasibility to reconfigure the droplet network via tunable interfacial tensions and bulk mechanical compression (44⇓–46). Tuning interfacial interactions such as tension and adhesion coincides closely with the tuning parameter in the cell-based model

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

## Materials and Methods

### Characterization of Structure.

The pair correlation function is defined as (52)

### Photonic Band Structure Calculation.

Photonic properties are numerically calculated using the plane wave expansion method (36) implemented in the MIT Photonic Bands program. We use the supercell approximation in which a finite sample of N cells is repeated periodically. *SI Appendix* includes a sample script used for photonic band calculations in 2D.

## Acknowledgments

The authors thank M. Lisa Manning, Bulbul Chakraborty, Abe Clark, Daniel Sussman, and Ran Ni for helpful discussions. The authors acknowledge the support of the Northeastern University Discovery Cluster and The Massachusetts Green High Performance Computing Center (MGHPCC).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: d.bi{at}northeastern.edu.

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.

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

Published under the PNAS license.

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