Experimental realization of Weaire–Phelan foams as photonic crystals

In recent years, open dielectric networks have gained attraction as photonic media for crystalline and amorphous designs. In contrast to packed sphere arrangements, such as opal crystals, in a network, the volume filling fraction can be continuously adjusted, even free-standing. Moreover, specific network structures possess a favorable and uniform local topology. A well-known example is the tetrahedral arrangements of dielectric rods connected to a single node in a three-dimensional diamond crystal network that is also retained in an amorphous diamond structure.¹ Already in 1971, Weaire highlighted the benefits of tetrahedrally bonded materials for opening bandgaps in his work on silicon semiconductors.² A decade ago, Florescu, Torquato, and Steinhardt proposed a scheme to transform a so-called hyperuniform point pattern in 2D or 3D into a dielectric network by triangulation and tessellation.^3,4 The idea was later used in experimental and numerical studies of amorphous photonic materials.^5,6 Using high-resolution 3D lithography, such as direct laser writing (DLW),⁷ one can fabricate dielectric networks on submicrometer length scales using computer-generated designs as templates.⁸ 

Early work on fabricating three-dimensional (3D) photonic crystals for the visible or near-infrared spectral range often concentrated either on conventional lithography or on assemblies of submicron-sized beads made of polystyrene or silica (SiO₂).^9–11 Those choices were mainly made for practical reasons. Electron-beam lithography was established when Yablonovitch first proposed photonic bandgap (PBG) materials in 1987.¹⁰ The drawback of conventional lithography is that it is cumbersome and costly, and it is almost impossible to fabricate more than a few layers of a crystal lattice.⁹ Alternatively, one can assemble colloids into photonic crystals by centrifugation or surface deposition, e.g., meniscus drying.^12,13 However, it has long been known that bead assemblies result in face-centered-cubic (fcc) crystal structures unfavorable for opening a PBG.^14,15 Moreover, well-defined spherical particles with a high refractive index contrast n/n_h, such as TiO₂ (n ≃ 2.7) or silicon (n ≃ 3.6) in air (n_h = 1), were not readily available at the time and are still challenging to make.^16–18 In addition, crystal or amorphous materials obtained by densely packing spheres lead to volume-filling fractions of 60% or more.^19–21 Therefore, researchers often inverted the opaline colloidal crystals, made of polystyrene or silica (SiO₂) beads, by infiltration with a high index material and subsequent removal of the spheres.^22–24 The higher index contrast facilitates the opening of a gap, and the inversion reduces the filling fraction to some 35%–40%, closer to values considered optimal, i.e., around twenty-five to thirty percent. Still, for inverted colloid opals (fcc-structure), a PBG opens at a refractive contrast (dielectric material-to-air) n/n_h > 2.8,^12,25 whereas diamond structures can support a PGB already for n/n_h > 1.9.^14,15,26

Direct laser writing (DLW) lithography is a newer method introduced in the early 2000s and is now widely used. For details about the technique and its implementation, we refer to the literature⁷ and the commercial instrument manufacturer (NanoScribe, Germany); see also Ref. 27. Moreover, a sketch of a typical experimental setup consisting of a light source, beam and sample motion components, beam steering and focusing optics, and computer control system is shown in Fig. 2 of Ref. 7. The technique has proven successful in producing three-dimensional photonic materials with arbitrary designs, both crystalline and amorphous.^5,26 The main limitations of DLW are the processing speed, the still limited resolution, and the small overall system sizes that can be realized. Studies typically report specimens with dimensions of (0.1 mm)³ or less,^5,26,28 and processing often takes several hours. Furthermore, obtaining band gaps or structural color in the visible range remains challenging, with some notable exceptions reported for simple ”woodpile” stacking.^29,30 We note that Direct Laser Writing (DLW) is a sequential writing procedure, and when it comes to generating volumes near the millimeter scale, it may appear slow. Nonetheless, when measuring its performance against other additive manufacturing methods, DLW stands out as the swiftest option, particularly when considering the practical metric of voxels written per second. While colloidal self-assembly is sometimes proposed as an alternative, it can be cumbersome, difficult to control, and it cannot address many structural designs.¹⁵ Another contender, DNA origami nanofabrication, offers versatile and targeted nanoscale fabrication. However, it, too, suffers from sluggishness and susceptibility to imperfections.³¹ Generally, the attainable level of flawlessness through self-assembly varies and frequently falls behind what DLW can achieve.

Concerning networks, inverted diamond colloidal crystals are a promising possibility, but their production with the help of DNA nanotechnology is complex.¹⁵ Dry foams are an alternative and represent the closest known real-world analog to computer-designed, hyperuniform photonic networks or amorphous diamond photonic networks.^{1,3–5,32,33} Yazhgur et al.³⁴ demonstrated that two-dimensional disordered foam structures possess photonic band gaps. Klatt et al. studied the photonic properties of crystalline foams numerically in three dimensions and found large band gaps for a contrast of 3.6.³⁵ This work also provides data for a range of refractive indices and filling fractions. They show a photonic bandgap opens for materials with a refractive index contrast greater than ≈2.5.

Experimentally, monodisperse dry foams with micrometer-sized bubbles can be self-assembled in a single batch process, and the resulting fluid-filled Plateau borders form a connected network.^36–38 Nevertheless, using classical foaming methods, including microfluidics,³⁸ it is nearly impossible to make sub-micrometer sized foam bubbles small enough to obtain stop-bands for visible or near-infrared wavelengths.³⁸ Therefore, alternative ways are also being explored, such as soft colloidal templating.^39,40

Here, we investigate 3D Weaire–Phelan crystalline foams generated on the computer and fabricated in a polymer photoresist via direct laser writing lithography and subsequent shrinking by thermal processing.^41,42 We demonstrate a photonic pseudogap for wavelengths between λ ∼ 1–2 μm and study the dependence of the stop-gap position and strength on the sample height and the solid filling fraction. Our work aims to demonstrate the experimental realization of idealized foams as photonic crystals that can unravel their practical performance potential.

To create a Weaire–Phelan foam on the computer, we start with an A15 point lattice (space group $Pm3̄n$⁠) consisting of eight points arranged in a cubic cell.^42,43 Following the classical Weaire and Phelan approach, we calculate the Voronoi tessellation of lattice points to get eight bubbles. Using the Surface Evolver software,⁴⁴ we minimize the total area of the space partition while forcing all bubbles to have the same volume. We retain the plateau borders and replace them with cylindrical rods to create the desired network structure [Fig. 1(e)]. We set the lattice constant to d = 3 μm. A WP-foam has rods with different lengths, which for our choice of the lattice constant, lie in the range of $≈469−884$ nm, thus accessible by the DLW-technique. By choosing periodic boundary conditions, we replicate the structure in all directions and create larger structures on demand. We use the thus obtained digital representations as input for DLW software.

We can estimate the expected positions of the stop bands from Bragg’s law Nλ_Gap = 2 d n_eff cos(θ_mean) with N being the diffraction order, d being the lattice constant, and cos(θ_mean) ≃ 1 for backscattering. From the symmetry of the underlying A15 point pattern, only even diffraction orders N = 2, 4, 6, … should be observed, and in this case, the lowest energy stop band lies at λ_N=2 ∼ d for n_eff≅1

Klatt and coworkers investigated the band structure of Weaire–Phelan foam networks with a refractive index of silicon in air, denoted as n = 3.6, and a filling fraction of 21%.³⁵ Their research revealed the presence of the lowest energy complete bandgap when the ratio of a to the bandgap wavelength λ_Gap is ∼0.27. Here, the unit length a is defined such that the number of vertices per primitive unit cell matches its volume. In the case of the Weaire–Phelan structure, this results in 46 nodes per unit cell. Consequently, the lattice constant d can be expressed as d = 46^1/3a = 3.58a, implying that λ_Gap ≈ a/0.27 ≈ d/(3.58 ⋅ 0.27) ∼ d.

Depending on the material’s (effective) refractive index and filling fraction, the corresponding vacuum wavelength will vary, but these examples generally support our estimate that the (lowest energy) stop band wavelength should be roughly equal to the lattice constant d. In our experiments, we do not anticipate observing a full bandgap. Therefore, the precise position of the stop gap will also be influenced by the crystal’s orientation and the (oblique) angle of incidence when utilizing the Cassegrain objectives in our infrared microscope setup.²⁸ 

We use direct laser writing (DLW) based on the commercial platform Photonics Professional (Nanoscribe GmbH, Germany). As a photoresist, we use IP-Dip (Nanoscribe GmbH, Germany), refractive index n_IP-Dip = 1.53, and a 63× oil immersion objective from Zeiss with 1.4 NA. The instrument’s light source is a femtosecond pulsed laser with a wavelength of 780 nm and a laser power of 28 mW. The device uses an acoustic-optical modulator to adjust the laser power. We print the structures from the bottom to the top over a silica glass slide. Two different writing modes are used: piezo-mode and galvo-mode. In the piezo mode, we write rod by rod using a piezo scanning stage with a positioning accuracy of better than 10 nm and a maximum writing speed of 300 μm/s. The piezo mode provides a high lithographic resolution, and the fs-laser beam can be moved in any direction [Figs. 1(a) and 1(c)]. In the galvo-mode, by scanning the laser beam, we write layer-by-layer, as most standard 3D printers do, and the x–y writing is controlled by a galvanic mirror scanner, which provides a much higher writing speed of up to 40 000 μm/s. We continue using the piezo stage for displacements in the z-direction.

After printing, the unexposed resist is removed by immersing the specimen twice for 10 minutes in a bath containing the developer solution [propylene glycol monomethyl ether acetate (PGMEA)]. The sample is washed in a low-surface tension liquid such as ethanol or isopropanol. To prevent structural deformations caused by capillary forces, the fabrication process concludes with the critical point drying (CPD) technique. This final step is crucial to ensuring the integrity of the structure. We use a commercial critical point dryer (Leica EM CPD 300), where ethanol is exchanged several times with carbon dioxide at 10 °C and a typical pressure of 50 atm. Later, after most of the ethanol has been removed, the carbon dioxide is heated to a temperature of 40 °C, and the pressure reaches values above 70 atm, passing the critical point of carbon dioxide (73 atm, 31 °C). Finally, the pressure is released to return the carbon dioxide to its gaseous state. Figures 1(c) and 1(d) show scanning electron images of specimens fabricated in piezo-mode and galvo-mode.

The printed structure slightly differs from the digital template due to the asymmetry of the laser pen and intrinsic shrinkage. Due to the point spread function of the microscope objective, rods are printed with an ellipsoidal cross-section with an aspect ratio of up to three, as discussed in detail previously.^5,46 The aspect ratio depends on the rods’ angle with the x–y plane. Rods that lie perpendicular to the x–y plane have an aspect ratio of about one, while parallel rods have the highest aspect ratio. In addition, the structures shrink slightly during development. When directly attached to the fused silica substrate, shrinkage occurs mainly in the z direction, while x–y shrinkage is more pronounced at greater distances from the substrate. In this study, we have employed an approach that achieves isotropic shrinkage by introducing a support structure, as discussed previously in detail in Refs. 45 and 47 [see also Fig. 1(f)].

We use a temperature-controlled heating stage (Linkam Scientific Instruments, Ltd., FTIR600) to shrink the samples with a controlled high-temperature treatment. First, we increase the temperature at a constant rate of 10° C/min until 450° C is reached; then, we keep the temperature constant for about 10–20 min. Finally, we decrease the temperature to room temperature at 10° C/min. During the process, the chamber is flooded with nitrogen to prevent oxidation. Thermogravimetric analysis (TGA) shows that heat treatment reduces mass.⁴⁵ We obtain nearly isotropic shrinkage of the polymer structures using a spider-leg support structure that holds the specimen freestanding above the substrate.^28,30 We attach the legs in a concentric circle, which helps to release stress more efficiently. Using a similar procedure, Liu et al.³⁰ report an increase in the refractive index n_IP-Dip for wavelengths in the 400–900 nm range. Moreover, they reported a reduction of the aspect ratio of the elliptical rods by up to 35%. To characterize the amount of shrinkage we obtain, we define the ratio x = L_H/L_o, where L_o is the edge size of the digital foam template and L_H is the edge size in the x–y plane after the heat-induced shrinkage. We checked that the shrinkage is isotropic, and all edge lengths are reduced by the same factor x.

Using focused ion beam milling (FIB), we cut a cross-section of the polymer foams, followed by scanning electron microscopy (SEM) of the sample’s interior. FIB-SEM allows us to study the structure’s integrity in bulk and determine the elliptical polymer rods’ minor and principal axes [Fig. 1(b)]. Using the values obtained, we can estimate the volume occupied by the polymer and then calculate the filling fraction ϕ.

We use a Fourier transform infrared (FTIR) spectrometer coupled to a microscope (Hyperion 2000, Bruker, Germany) to measure the transmittance T(λ) and R(λ), as described earlier.^5,28 Using Cassegrain mirror objectives, the instrument collects and detects light along a hollow cone with an acceptance angle between θ = 15° and θ = 30° as measured perpendicular to the sample [see Fig. 2(a)]. The surface of our structure lies parallel to the unit cubic cell. The surface normal corresponds to the Γ-X direction within reciprocal space. As a result, during the experiment, we investigates a cone with an aperture of 15° around an oblique angle centered at 22.5°. For an in-depth examination of a similar experiment conducted on a cubic crystal, please refer to Ref. 28 and its supplementary information. We focus the incident light over an area of typically 30 × 30 μm² or slightly less, and average of 100 measurements. For calibration, we measure the spectra of the bare silica glass for transmission and a gold-coated mirror for the reflectance spectra. For the transmission measurements, we consider a wavelength range from 1 to 2.5 μm over which the light absorption in the polymer is negligible.²⁸ For more details, we would like to refer to the supplementary material in Ref. 28, where clear experimental evidence is provided. We note that despite the absence of absorption, the sum of reflection and transmission (R + T) does not reach the the value of 1 expected for a crystalline structure (without diffraction), as shown in Fig. 2(d). This is a common observation made in many similar studies²⁶ and can be attributed to scattering by imperfections,⁴⁸ surface roughness, and challenges in calibrating the FTIR spectrometer due to reflection and refraction at the material-to-air or substrate interfaces.^4,41

We first produced samples with a footprint of 40 × 40 μm² and heights of 25 μm in piezoelectric mode. One typical sample is shown in Figs. 1(a) and 1(b). Despite the complex digital template, we succeeded in fabricating high-quality photonic crystals. As for the footprint of the polymer foam, we find no measurable difference from the digital template in the SEM image. However, we observe a reduction in height of ∼20%. To study the role of the polymer filling fraction ϕ, we fabricate foams with DLW using different nominal laser powers of $3.010.38,3.270.43,3.460.46$⁠, and $3.640.53$ mW. The filling fraction increases with the laser power, and the estimated values are given in brackets. The results are shown in Fig. 2(b). Increasing the filling fraction leads to a red shift of the N=4 stop band and a plateauing of the peak height. For higher filling fractions, we expect the peak height to drop again, as shown in earlier work,⁵ but in the present study, we did not explore this regime. Without thermal shrinkage, the gap wavelength is located in the shortwave infrared. We note that in the digital representation, the average rod length is 700 nm, while the smallest rod length is 470 nm. Due to the finite resolution of standard DLW, we cannot fabricate shorter rods and, therefore, cannot reach stop-gap wavelengths much smaller than 2 μm.

To circumvent the DLW resolution problem, we apply a thermal shrinkage procedure proposed earlier³⁰ and confirmed by us recently using a slightly modified procedure.⁴⁵ Operating the DLW in “galvo-mode,” we write a cubic foam structure with an edge length of 60 μm. First, we report the supporting structure with a writing speed of 4000 μm/s and a nominal laser power of 14 mW. Subsequently, we fabricate the foam networks with a fixed writing speed of 10 000 μm/s and a laser power ranging from 12.5 to 14.7 mW. We apply heat-induced shrinkage at 450° C for 10–20 min depending on the targetted extent of shrinkage. Figure 1(f) shows a typical sample after shrinkage. The SEM image shows that the sample and the collapsed supporting legs have not suffered any visible damage or deformation. The foams shrink differently depending on the laser power we used during fabrication. The smaller the laser power, the more the samples shrink. In Fig. 2(c) and Fig. 2(d), we plot the transmittance and reflectance of foams with different x values. As shown in Fig. 2(c) inset, the stop-gap wavelength decreases as the value of x decreases, which aligns with our expectations. It is important to note that the relationship between the gap wavelength (λ_N=2) and x is not linear across the entire range. For x = 1 we estimate the filling ratio to be about ϕ ≈ 0.2. For smaller values of x, the increase in filling fraction leads to a higher n_eff, which counteracts the reduction in lattice constant (d ∝ x). As shown in Fig. 2(d), we were successful in shifting the gap wavelength to telecom wavelengths (1.3–1.5 μm) in the near-infrared (NIR) spectral range. For values of x ≥ 0.38, all the spectra look similar and show comparably high reflection peaks. For x ≤ 0.35, the peak height decreases. Currently, we cannot tell whether the decrease in the reflection peak is to a gradual deterioration of the sample or simply due to the increased filling fraction. As shown in the inset of Fig. 2(c), there are two different regimes λ_N=2 ∼ x with different slopes. As stated earlier, we can estimate the positions of the pseudogaps from Bragg’s law Nλ_N = 2 dxn_eff cos(θ_mean) with N = 2, 4, 6, … being diffraction order, d being the lattice constant, n_eff being the effective refractive index, and θ_mean = 22.5° being the average angle between the plane and the surface normal. For example, for x = 0.45 and n_eff = 1.4 (ϕ≈0.37), the second and fourth order stop bands are predicted at approximately λ_N=4 ≃ 1 μm and λ_N=2 ≃ 2 μm in good agreement with reflection spectra maxima.

The inspiration for this experimental investigation comes from two computer-based studies conducted in 2019, one focusing on photonic 2D foam structures³⁴ and one studying 3D foam structures and their bandgaps³⁵ (see also Ref. 38). The 3D study is particularly relevant for our work as it highlighted the potential of the Weaire–Phelan foam as a promising photonic structure. Our present work validates this theoretical forecast and shows that the suggested material is practically realizable. We could experimentally show that mono-disperse Weaire–Phelan foams possess a photonic stop-band or pseudo-gap. The structure of the foams is relatively complicated, but we nonetheless succeeded in producing them using DLW. However, the complex architecture of the foams sets narrow limits concerning the experimental resolution in the fabrication and the directly related question of the position of the bandgap. To shift the wavelength of maximum reflection into the NIR range, we have applied a recent technique with which structures can be isotropically shrunk. We were able to show that the polymer structures retain their shape, and the optical spectra change little except for the shift in wavelength, at least for moderate shrinkage x > 0.35. Moreover, the reflection spectra provide compelling evidence that galvo-mode fabrication can attain lower filling fractions, enabling the fabrication process to achieve smaller values of x. Our experiments lead to a new type of photonic crystal that has only been studied numerically so far.³⁵ The bandgap of WP foams is very pronounced, which could make Weaire–Phelan foams an exciting material for photonic applications.

The Swiss National Science Foundation financially supported this project via Project No. 188494. This work also benefited from support from the Swiss National Science Foundation through the National Center of Competence in Research Bio-Inspired Materials (Grant No. 205603). We acknowledge Luis S. Froufe-Pérez, Geoffroy Aubry, and Sofia Magkiriadou for insightful discussions. Additionally, we would like to acknowledge Stefan Aeby for his assistance in preparing Fig. 1(a) and for fabricating some preliminary foam structures not included in the final study.

The authors have no conflicts to disclose.

A. Aguilar Uribe: Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (supporting); Software (equal); Validation (lead); Visualization (lead); Writing – review & editing (supporting). P. Yazhgur: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (supporting); Software (equal); Supervision (supporting); Validation (equal); Writing – review & editing (supporting). F. Scheffold: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.8318863.