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# Foam as a self-assembling amorphous photonic band gap material

Edited by Salvatore Torquato, Princeton University, Princeton, NJ, and accepted by Editorial Board Member Peter J. Rossky April 2, 2019 (received for review December 2, 2018)

## Significance

Recently discovered disordered materials with complete photonic band gap can enable the production of freeform wave guides, energy-efficient displays, or iridescent ultraresistant pigments. Surprisingly, the fabrication of such amorphous photonic materials is completely dominated by the top-down approach and suffers from its severe limitations (cost, timescale, defects, resolution, etc.). Here, we present a class of photonic band gap materials—“photonic foams”—that can be fabricated by self-assembly. We demonstrate that 2D photonic foams possess a large isotropic photonic band gap and discuss a pathway toward 3D structures.

## Abstract

We show that slightly polydisperse disordered 2D foams can be used as a self-assembled template for isotropic photonic band gap (PBG) materials for transverse electric (TE) polarization. Calculations based on in-house experimental and simulated foam structures demonstrate that, at sufficient refractive index contrast, a dry foam organization with threefold nodes and long slender Plateau borders is especially advantageous to open a large PBG. A transition from dry to wet foam structure rapidly closes the PBG mainly by formation of bigger fourfold nodes, filling the PBG with defect modes. By tuning the foam area fraction, we find an optimal quantity of dielectric material, which maximizes the PBG in experimental systems. The obtained results have a potential to be extended to 3D foams to produce a next generation of self-assembled disordered PBG materials, enabling fabrication of cheap and scalable photonic devices.

Materials with a complete photonic band gap (PBG), which prevents the propagation of electromagnetic waves in all directions within a certain frequency range, have given rise to various promising industrial applications, such as lossless wave guides (1, 2), LEDs with high light extraction efficiency (3), high-Q laser cavities (4, 5), and optical elements of computers (6). The search for PBG materials first led the scientific community to consider long-range ordered structures, such as photonic crystals and quasicrystals, where formation of PBG is usually related to the multiple coherent Bragg scattering (7⇓–9). However, fabrication issues and high sensitivity to defects strongly limit the development of potential devices based on photonic crystals (10⇓–12). Recently discovered disordered materials with a large complete PBG are argued to be potentially easier to fabricate (13). Moreover, such disordered photonic materials also offer directional isotropy, useful to create freeform wave guides, isotropic radiation sources, or noniridescent structural color pigments (13⇓–15)—applications simply impossible for classical photonic crystals. Disordered structures possessing a PBG are interesting for the study of different regimes of optical transport, such as the Anderson localization (16, 17).

There exist different disordered structures possessing a wide isotropic band gap, which share some peculiar features. Champions of disordered PBG networks are usually constant valency connected networks of dielectric material surrounded by air: for example, fourfold amorphous diamond or threefold gyroid structures (18⇓–20). Similar behavior is observed in threefold coordinated 2D network systems for transverse electric (TE) polarization (electric field in the plane), which is often used to predict properties of real 3D systems with decreased computational cost (21). Within the same class (e.g., valency) of structures, different design protocols have been applied to find disordered networks with the largest possible band gaps. One of the first optimization techniques derives networks from stealthy hyperuniform point patterns (13) exhibiting zero structure factor for wave vectors below a certain limit (22). Such suppression of long-range density fluctuations has been shown to facilitate the appearance of large PBG. The importance of uniform local topology in addition to long-range hyperuniformity has been underlined in multiple publications (13, 23). This leads to the local self-uniformity concept, proposed by Florescu and coworkers (21), which states that the best disordered PBGs correspond to networks with similar local elements, such as individual nodes, and proposed mathematical tools to estimate the degree of similarity. Such identical elements play the role of identical Mie scatters opening a PBG by superposition of their Mie resonances (24⇓–26). In the meantime, strong local order of such networks decreases long-range density fluctuations and can confer hyperuniformity to the whole system, in agreement with the early observed correlation between hyperuniformity and PBG (23).

So far, the fabrication of disordered PBG materials is dominated by the top-down approach, such as photolithography or 3D printing (27⇓–29), for which serious bottlenecks are slowness, cost, and in some cases, insufficient resolution. This is probably related to the fact that most of the proposed design techniques are based on mathematical algorithms and do not give a direct self-assembly compatible fabrication method.

In the meantime, a number of self-assembling fourfold networks have been realized and studied for years by soft matter scientists, such as dry foam/emulsion structures (30). The local structure of relatively dry foams and emulsions is constrained by Plateau laws: the dry foam structure can be represented as a network of slender channels called Plateau borders, which meet at vertices called nodes. Every node connects four Plateau borders (three in 2D) such that any angle between any two adjacent Plateau borders is

To check this hypothesis, we report here the use of 2D foams (monolayers of bubbles squeezed between two plates) as potential templates to generate self-assembled 2D material with a TE (electric field in the plane) omnidirectional PBG. Foams are both created experimentally and simulated numerically. Their photonic properties are calculated by a plane wave expansion method. The structural organization of relatively dry 2D foams (such as threefold connectivity of Plateau borders, for example) turns out to be especially useful to open a large PBG comparable with the best ones described in the literature for optimized disordered materials. Our results show that the small polydispersity of bubble sizes introduces a controlled level of disorder to foam structure, which makes the band gap isotropic without strongly perturbing photonic performance. Such 2D foams are well known to capture the main properties of real 3D foams, and therefore, it provides strong evidence that our results in 2D can be directly transferred to real 3D foams.

We believe that this research not only can define a roadmap to a class of self-assembled photonic materials but also, sheds a light on the fundamental questions of PBG physics.

## Results and Discussion

Fig. 1*B* shows a photograph of bidisperse 2D foam produced in a Hele–Shaw cell. The size ratio of big to small bubbles, defined as the square root of corresponding Voronoi cell’s areas, is about 0.75. This relatively low polyspersity is chosen to avoid crystallization without strongly perturbing the self-similarity of foam elements (such as the length of Plateau borders). A circular symmetry of the Fourier transform and absence of Bragg peaks (Fig. 1*B*, *Inset*) prove that we manage to create a disordered isotropic foam without spatially extended crystal domains. By virtually replacing the liquid phase with silicon in our calculations (

Fig. 1 *A* and *C* shows two reference simulated systems, which represent two extreme cases. Simulated dry-like foams are produced by Voronoi tessellation of disordered disk assemblies and are annealed with the Surface Evolver software. The obtained network structures always fulfill Plateau laws and mimic dry foams. We also use disordered bidisperse disk assemblies to represent foams in a very wet limit. More information can be found in *Materials and Methods* and *SI Appendix*. The experimental 2D foam experiences a transition from the dry to the wet limit when the area fraction increases (the transition is discussed later). However, we still can consider dry-like foams with high area fraction and disordered disk assemblies with low content of dielectric material, even if they do not represent real physical foams. Considerations of these two simulated systems allow us to study the effect of foam structure separately from the dielectric material fraction influence. One can see that, for the same dielectric material area fraction, the simulated dry-like foam has a large band gap comparable with the experimental one. At the same time, the wet foam system has only a pseudogap with strongly diminished normalized photonic density of states (NDOS) but no sign of a true PBG (Fig. 1*D*).

To see how real foam PBG response evolves from dry to wet limit, we plot in Fig. 2 the high dielectric material area fraction ϕ dependence of PBG width

A closer look at the PBG frequencies reveals that, in all considered systems (both monodisperse crystallized and slightly polydisperse disordered ones), the PBG (NDOS = 0) or pseudogap (reduced NDOS) starts roughly at the same frequency for a given area fraction (Fig. 2). In the meantime, the upper edge of the band gap turns out to be different for various systems and thus, is responsible for the change of PBG width between various systems.

As shown in Fig. 3, the dielectric bands just below the PBG are strongly localized in individual bubbles for the magnetic field and in the dielectric material around the bubbles for the electric field as previously shown for similar network structures (13). This result stays valid even for the disordered disk assemblies, which have a very narrow range of frequencies, where NDOS approaches zero. Clearly, the lower edge frequency depends mainly on the mean bubble size and area fraction, also influencing the separation between the bubbles. It does not strongly depend on the exact bubble shape (polygons or disks) or on the structural organization of the system (crystallized or not). Passing through the band gap, both fields change their localization, such as it is well known for both crystalline and disordered PBG materials (8, 13). The air bands at the upper edge of the band gap are rather localized around individual foam nodes. The magnetic field is localized inside the nodes, while the electric field is concentrated in bubbles adjacent to the nodes.

We should naturally expect that the change of the PBG upper edge for different systems, such as a rapid closing for wet foams in comparison with the dry ones, should be related to the change of the node organization. Indeed, we observe that, for relatively wet 2D foams (area fraction above 0.4), where the PBG width

If we compare experimental 2D foams and simulated dry-like foams with the same area fraction, electromagnetic modes related to these fourfold nodes can be considered defect modes in the PBG of idealized dry foams. The fourfold nodes have a lower characteristic frequency than threefold nodes related to their bigger characteristic size. Therefore, the fourfold nodes create defect states that start to fill the PBG of dry foam from the upper edge and gradually diminish it with an increase of the area fraction. To highlight this strong correlation between PBG width and percentage of fourfold nodes, we show for experimental 2D foams in Fig. 4*A*, *Insets* the number of defect modes as a function of fourfold nodes number. The number of modes inside the PBG clearly scales linearly with the number of fourfold nodes throughout all of our data. This gives a straightforward way to estimate the PBG performance of the foams simply by considering their topology. This linear scaling also means that, if we removed the states corresponding to these fourfold nodes, the PBG of experimental foams would follow the PBG of simulated dry-like foams in the whole range of studied area fractions. This, for example, reveals that geometric factors, such as local curvature of nodes or circularity of bubbles (*SI Appendix*, Fig. S2 shows circularity of experimental and simulated dry-like foams), are not particularly important for the PBG in TE polarization. This phenomenon can also be seen for the crystalline structures, where both honeycomb networks and triangular lattice of disks have almost identical PBG for TE (Fig. 2 and *SI Appendix*, Fig. S3).

The analogy between the PBG of crystallized materials and our systems stays valid for TM (transverse magnetic) polarization. No sizeable TM PBG is found in honeycomb network lattice as well as in our systems (*SI Appendix*, Fig. S4). Similar networks of walls described in the literature possess a large PBG for TE polarization but no band gap for TM (23). However, this 2D network has given rise to 3D PBG structures, such as amorphous diamond, underlining a particular importance of TE polarization.

Summing up, we can conclude that the lower frequency of the TE PBG of the studied systems is mainly related to the fraction of the dielectric material and the size of the bubbles no matter how they are organized (crystallized or not) or the shapes that the bubbles adopt (disks or polygons). The upper frequency is related to several phenomena; the most important one is the diversity of node shapes: the more diverse the nodes are, the smaller the PBG is. This approach also explains why disordered disk assemblies, representing very wet foams at the jamming point, never show a sizeable PBG. Nodes in such system can have various shapes, even more complicated than the fourfold nodes presented in Fig. 3, and they create defect states that completely fill the PBG.

The appearance of a band gap is often related to the degree of hyperuniformity in the system, and therefore, it is also interesting to evaluate the hyperuniformity of our foams. Fig. 4 shows the spectral density of the foam, which is an extension of the structure factor for bidisperse and generally, polydisperse systems (35, 36). In Fig. 4*B*, *Inset*, we also plot the spectral density at the minimum wave vector

In the meantime, the spectral density or the minimum spectral density

With the results obtained for real and simulated dry 2D foam structures, we can expect that dry 3D foam structures should exhibit PBG as well. We can give a rough estimate of potential PBG width for 3D foams from the results published for structurally similar amorphous diamond networks of dielectric rods. Amorphous diamonds have been numerically optimized to possess a large isotropic PBG of around 20% for a volume fraction of dielectric material of around 20–30% and the refractive index contrast 3.4 (18). This volume fraction of dielectric material is beyond the accepted limit of dry foams in 3D, which is rather about 5–10% (31). This would mean that, to produce a good candidate for complete PBG 3D photonic foam by self-assembly, first a dry solid foam from high refractive index material should be fabricated and then, the quantity of dielectric material should be additionally increased by coating the Plateau borders. A high reliability to disorder of 2D dry foams also suggests that the PBG of proposed 3D photonic foams should be much more robust than the PBG of widely used inverse opal crystals: inverse opals are structurally close to wet foams; any disorder creates a large variety of nodes and rapidly closes the PBG. This makes 3D photonic foams a promising candidate for the next generation of self-assembled complete isotropic PBG materials.

We have demonstrated that 2D foams can be used as a self-assembling template for disordered materials with a large isotropic PBG for TE polarization. We show that the size and position of the band gap can be changed by adjusting the fraction of dielectric material, opening a way for tunable band gap applications. The dry foam structure turns out to be especially profitable to open the PBG—high level of hyperuniformity at the relevant length scale and significant similarity of dry foam nodes (for example, the threefold coordination in 2D) guarantee a large and isotropic PBG to photonic foams. However, the photonic performance strongly diminishes in the wet limit due to the appearance of defect states in large fourfold nodes. Disordered disk assemblies, mimicking the structure of bubbly liquids, never show a sizable PBG. This can be correlated both with the increase of long-range density fluctuations and the loss of a uniform local topology. The described 2D photonic foams can already be used as a platform to fabricate planar optical circuits. Moreover, we also expect that our results stay valid for real 3D foams since they are tetravalent connected structures. Our research suggests that photonic foams are potentially an excellent candidate for materials with a large complete isotropic PBG. Photonic foams are fully scalable to a wide range of wavelengths from visible range to microwaves and potentially, can be produced in a large quantity by self-assembly.

## Materials and Methods

### Experimental 2D Foam.

Experimental 2D foams are produced using an in-house Hele–Shaw cell consisting of two vertical glass plates separated by 1.5 mm. Two populations of bubbles are generated by blowing air through two orifices into a 12 g/L SDS solution. To ensure the constant liquid fraction profile, the experiments are performed in the forced drainage regime: foaming liquid is added from the top at a controlled flow rate (31).

### Disordered Disk Assembly.

We generate bidisperse 2D jammed disk assemblies with periodic boundary conditions using freely available code based on the Lubachevsky–Stillinger algorithm (40, 41). To vary the area fraction, we subtract a constant value from the radius of each disk.

### Simulated Dry-Like Foam.

Weighted Voronoi tessellation is performed over obtained disordered disk assemblies. We anneal the structure with Surface Evolver (42) to better approach experimental foams. By increasing the thickness of the walls, we get periodic networks with a controlled, variable quantity of dielectric material.

### PBG Simulation.

The band gap structures are calculated using periodic supercell approximation implemented in MIT PBG software (43). We check that the periodicity of the supercell does not influence our results (*SI Appendix*, Fig. S1).

More details can be found in *SI Appendix*.

## Acknowledgments

We thank K. Morozov, A. Leshansky, N. Stern, L. Froufe-Pérez, F. Scheffold, and A. Salonen for fruitful discussions and suggestions made along the work. This work has been supported by Ecole Supérieure de Physique et de Chimie Industrielles de la Ville de Paris, Paris Sciences et Lettres (PSL) Research University and Institut Pierre-Gilles de Gennes. Microflusa receives funding from the European Commission Horizon 2020 Future and Emerging Technologies Program Grant 664823.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: joshua.ricouvier{at}weizmann.ac.il or pavel.yazhgur{at}unifr.ch.

Author contributions: J.R., P.T., and P.Y. designed research; J.R. and P.Y. performed research; J.R. and P.Y. analyzed data; and J.R. and P.Y. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. S.T. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1820526116/-/DCSupplemental.

Published under the PNAS license.

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