# Crystalline structures in foams: guided mechanical self-assembly of bubbles in fiber arrays

Marwan Chammouma, Manon Jouanlanne, Antoine Egelé, Damien Favier, Jean Farago, Aurélie Hourlier-Fargette

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Marwan Chammouma, Manon Jouanlanne, Antoine Egelé, Damien Favier, Jean Farago, et al. Crystalline structures in foams: guided mechanical self-assembly of bubbles in fiber arrays. Journal of Physics: Materials, 2025, 8 (1), pp.015011. 10.1088/2515-7639/adaa21. hal-04989398

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# Crystalline structures in foams: guided mechanical self-assembly of bubbles in fiber arrays

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# Crystalline structures in foams: guided mechanical self-assembly of bubbles in fiber arrays

Marwan Chammouma¹ ID, Manon Jouanlanne¹ ID, Antoine Egelé ID, Damien Favier ID, Jean Farago ID and Aurélie Hourlier-Fargette* ID

Université de Strasbourg, CNRS, Institut Charles Sadron UPR22, F-67000 Strasbourg, France

¹ Equal contributions.

* Author to whom any correspondence should be addressed.

E-mail: hourlierfargette@unistra.fr

**Keywords:** self-assembly, solidifying foams, architected materials, crystalline structures

Supplementary material for this article is available [online](#)

## Abstract

Spontaneous mechanical self-assembly of monodisperse bubbles generally leads to disordered foams at low density. Producing crystalline structures requires specific care: for example, Kelvin foams—periodic assemblies of bubbles arranged on a body-centered cubic lattice—are a typical example of a structure which is challenging to obtain experimentally, despite it being a local minimum of energy. Here we show how bubbling in different arrangements of fibers enables to control foam architectures through a guided mechanical self-assembly of bubbles: for optimal ratios of bubble size to fiber spacing, Kelvin and hexagonal close packing crystalline foams are formed in square and hexagonal fiber arrays, respectively. The long-range crystalline architectures achieved in samples spanning hundreds of bubbles are then quantified through a theoretical approach analysing the orientational order in the samples. This methodology, based on the decomposition of strut orientations via spherical harmonics, is inspired by the so-called *Steinhardt's coefficients*, developed for quantifying rotational order in 3D liquids. Beyond the achievement of architecting liquid foam structures, our work demonstrates that the obtained ordering persists upon solidification of initially liquid polymeric foams, using alginate and polyurethane foams in nylon fiber arrays as model systems. The mechanically guided self-assembly of bubbles offers an attractive alternative to additive manufacturing to generate highly ordered architected polymeric materials.

## 1. Introduction

The properties of liquid and solid foams are highly dependent on their morphological features, in terms of mechanics and electrical conductivity, but also for thermal and acoustic insulation for solid foams [1, 2]. For these reasons, there is a growing interest into strategies to control the way bubbles spontaneously assemble in three dimensions. Controlling the assembly of a liquid foam that then solidifies into a cellular solid material with a specific structure is of special interest for the field of architected (meta)materials. However, packing bubbles of equal sizes into a foam generally leads to disordered structures, unless experimental strategies are specifically designed towards the obtention of crystalline architectures [3]. The self-assembly of bubbles is driven by the minimization of interfacial energy in the system, leading to the so-called Plateau's laws at low liquid contents [4]. Experimentally, structures often correspond to local minima of energy, rearrangements of the bubbles being too costly in energy to occur. As a striking example, Kelvin predicted in 1887 that packing truncated octahedrons made of 6 square faces and 8 hexagons to pave the space in 3D would minimize the surface and thus the interfacial energy [5], but such cells were elusive experimentally. In the historical work of Matzke in the 40 s [6, 7], who observed under a microscope numerous cells in a foam prepared bubble by bubble, no Kelvin cell had been observed. One isolated instance of observation occured ten years later [8], and Kelvin cells assemblies have been observed more recently, as detailed below, at the cost
of complicated strategies [9–14]. Weaire and Phelan predicted theoretically another structure composed of an arrangement of two types of bubble shapes [15], that provides a more efficient partition of space than the Kelvin cell. This structure was even harder to obtain experimentally, which was achieved only a decade ago [16]. Those two examples of experimental hunts for Kelvin and Weaire–Phelan foams illustrate the difficulty to experimentally obtain crystalline ordered foams with desired structures.

Two main general strategies to reach Kelvin cell assemblies have been deployed in the past: (i) using gravitational drainage [9] or osmotic effects [10] to change the liquid fraction, starting from face-centered cubic (FCC) Kepler foams at high liquid fractions to obtain body-centered cubic (BCC) Kelvin foams at a lower liquid fraction, or (ii) taking advantage of confinement and surface effects: Rosa *et al* [11] reorganized layers of bubbles into a Kelvin structure by changing the distance between a liquid interface and a glass plate that confines the bubbles. Meagher *et al* [12] used pyramidal containers to modify structures by changing the angle of the pyramid. A simple confinement of bubbles in cylindrical tubes [13] is also an efficient way to produce Kelvin cells, and geometrical constraints imposed by solid walls impose foam structures on a couple of layers against the wall [14]. However, those approaches allow only to form a rather localized crystalline structure.

## 2. Guiding bubbles with fiber arrays

Here we introduce a new strategy, considering liquid foams entering an array of rigid fibers (figure 1(a)). We show that depending on the ratio of the bubble diameter *d* to the fiber spacing *a*, arrangements of bubbles constrained by a fiber network can exhibit long-range order and even include Kelvin cells arrangements, as shown in figure 1(b). Those observations are permitted via the solidification of the samples, allowing in depth three-dimensional x-ray tomography characterizations, the structure being preserved upon solidification.

In two dimensions, the use of pinning centers and of temporal evolution of foams has been used to create well defined 2D structures [17, 18] including applications in general nanofabrication techniques [19] and optical materials [20]. In 3D, the work of Whyte *et al* [21] has highlighted modifications of structures in simple arrangements of soap films, focusing on the local deformations caused by an intruder in a simple Plateau border. However, analysis on the ordering of foams on large scales thanks to fibers, is, to our knowledge, a completely novel route.

In our experimental setup, air or nitrogen bubbles of constant size are introduced in an initially liquid mixture of alginate or polyurethane precursors, through nozzles of diameter 150, 200 and 250 µm. The obtained liquid foam is poured into a container in which a nylon fiber array has been woven (figure 1(a)), before undergoing solidification. The air fraction is tuned by controlling drainage, and the bubble diameter *d* is extracted from the tomography scans as an equivalent diameter (diameter of a sphere of volume equal to the bubble volume). Experiments were performed with two polymeric materials to highlight the generality of the approach: polyurethane, shown in the supplementary material (figure S3), leading to self-standing materials, and alginate, softer, but which has the advantage of being non toxic and is used as a model system for all results shown below (figures 2–4). The general principles of foam preparation and some information about mechanical properties after solidification are detailed in the absence of fibers in our previously

<figure><img src="image_5.png" /></figure>
<figure><img src="image_6.png" /><figcaption>**Figure 2.** (a) X-ray tomography scan of an alginate foam without fiber array with bubble diameter *d* = 4.3 ± 0.2 mm, leading to a disordered structure. (b) X-ray tomography scan of an alginate foam in a square fiber array, with bubble diameter *d* = 3.7 ± 0.2 mm and fiber spacing *a* = 4.0 mm. The same figure with red dots showing the fiber positions in figure (b) can be found in the supplementary material (figure S2).</figcaption></figure>

published work (for both alginate [22] and polyurethane [23]). Details specific to the current work are described in the *appendix*, as well as in the supplementary material. In particular, pictures of the experimental setup are shown in figure S1.

The use of such foams that are initially liquid and solidify within the fiber network allows us to (i) analyse the structures via x-ray tomography with no significant drainage as it would be the case for liquid foams [24], and (ii) demonstrate that we can create solid crystalline polymeric materials. The solidification process occurs before any significant ageing mechanism yields merging of bubbles and phase separation.

Both visual observations and x-ray tomography scans show that bubbles in a *square* array of fibers can arrange into large clusters of Kelvin cells (truncated octahedrons) in a BCC configuration. Figure 2 illustrates the difference between foams produced without (a) and with (b) a fiber array of square arrangement with appropriate *d/a* ratio. Foams without fiber arrays are disordered, while the mechanically-driven self assembly of bubbles in fiber arrays can lead to large ordered zones in the samples. In the sample of figure 2(b) for instance, we counted 350 bubbles organized in a completely crystalline region of dimensions *5d × 5d × 10d*, where *d* is the bubble diameter.

The basic physical hypothesis for this ordering are first that the need to collectively minimize the interfacial energy leads the assembly of bubbles to place itself in a rather ordered way so that all fibers are embedded in the liquid part, with a loose pinning of the bubbles to the fibers, similar to the one observed in [21]. This 'preordering' then biases the path of the foam within the space of different configurations and allows it to find the Kelvin structure among the wealth of similar-in-energy but highly disordered other metastable equilibriums. Indeed, the potential accessible states are separated by energetic barriers, which are both modest with respect to the typical energy input given during mechanical solicitations and considerably larger than the thermal fluctuations. This entails two things: on the one side, during the dynamical phase of a foam generation or deposition, the arrangement of bubbles can switch very easily between two different neighboring (in the phase space) configurations and as a result, adopt very easily a disordered configuration. On the other side, as soon as the foam has reached a quiescent state, it cannot rearrange spontaneously towards lower energy minima anymore, due to the lack of energy fluctuations with the appropriate level.

### 3. Analysis of the architectures: Steinhardt's coefficients

After the capture of the 3D structures of the solidified bubble assemblies via x-ray tomography, we perform a numerical extraction of the strut network with a in-house built Matlab code, inspired by the work of Montminy et al [25, 26]. Once the strut network is extracted, we quantify the orientational order in the sample. To do that, we adapted the methodology developed by Steinhardt, Nelson and Ronchetti to study the correlation of rotational order in 3D liquid systems [27, 28], which was an extension of the work performed on the prediction of the hexatic phase of 2D liquids [29, 30]. The Steinhardt's parameters have been only occasionally used in the context of foams [31] or cellular materials or assemblies of capsules [32].
This method consists in computing the global orientational density function of strut orientation over the unit radius 3D sphere (see figure 3(c) for an example on one of our experimental 3D reconstructions, and figure S4 of the supplementary material for schematics illustrating the process on the example of a perfect Kelvin foam):

$$ \rho(\mathbf{n}) = \frac{1}{2N_b} \sum_{\langle i,j \rangle} [\delta(\mathbf{n} - \hat{\mathbf{r}}_{ij}) + \delta(\mathbf{n} - \hat{\mathbf{r}}_{ji})] \quad (1) $$

where $\hat{\mathbf{r}}_{ij} = \mathbf{r}_{ij}/r_{ij}$ is the directional unit vector corresponding to the strut linking the vertex $i$ to its neighbour $j$. This function is defined for any 'trial' unit vector $\mathbf{n}$ and each configuration of the fluid. The sum is over the $N_b$ pairs of nearest-neighbour vertices $(i, j)$ within the foam (notice that this is different from the Steinhardt's parameters for fluids, which are defined for directional unit vectors linking nearest neighbour particles). The presence of two delta functions for each pair $\langle i, j \rangle$ reflects that the bonds are not physically oriented, a symmetry $i \leftrightarrow j$ must be preserved.

The function $\rho(\mathbf{n})$ is then represented by its expansion in terms of the spherical harmonics $Y_{\ell}^{m}(\mathbf{n}) = Y_{\ell}^{m}(\theta, \phi)$, assuming an external coordinate system at the center of the sphere

$$ \rho(\mathbf{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \rho_{\ell m} Y_{\ell}^{m}(\theta, \phi), \quad (2) $$

$$ \rho_{\ell,m} = \frac{1 + (-1)^{\ell}}{2N_b} \sum_{\langle i,j \rangle} (Y_{\ell}^{m})^{*} (\theta_{ij}, \phi_{ij}), \quad (3) $$

where $(\theta, \phi)$ are the polar and azimuthal angles of the spherical coordinates of $\mathbf{n}$, and $z^*$ is the complex conjugate of $z$. Notice that $\rho_{\ell,m} = 0$ if $\ell$ is odd, as a consequence of the $\mathbf{n} \leftrightarrow -\mathbf{n}$ symmetry.

If no bond orientational order is present in the strut network of the foam, the bond vectors $\hat{\mathbf{r}}_{ij}$ are expected to uniformly populate the unit sphere, up to finite size fluctuations $\propto 1/\sqrt{N_b}$. For such an isotropic case, all coefficients but the first $\rho_{00} = \frac{1}{2\sqrt{\pi}}$ vanish. Conversely, if an orientational order develops, this isotropy is broken, and the set of mode amplitudes $\rho_{\ell,m}$ will account for this order. As they depend on the choice of the coordinate system, Steinhardt et al proposed to use only combinations of modes which are naturally rotationally invariant (consequently, they will also give identical signatures for any two ordered structures having broken the rotational isotropy in different 'directions'). The first series of these 'Steinhardt's parameters' is defined by the discrete function

$$ Q_{\ell} = \left[ \frac{4\pi}{2\ell + 1} \sum_{m=-\ell}^{\ell} |\rho_{\ell,m}|^2 \right]^{1/2}, \quad (4) $$

considered only for even integers $\ell$, since $Q_{\text{odd}} = 0$. Notice also that $Q_{\ell} = \delta_{\ell,0}$ for an isotropic foam. Another Steinhardt's coefficient is defined by the cubic combination

$$ W_{\ell} = \sum_{m_1+m_2+m_3=0} \begin{pmatrix} \ell & \ell & \ell \\ m_1 & m_2 & m_3 \end{pmatrix} \rho_{\ell,m_1} \rho_{\ell,m_2} \rho_{\ell,m_3}, \quad (5) $$

(where the matrix is the Wigner 3j symbol), or its normalized version

$$ \hat{W}_{\ell} = (W_{\ell}/Q_{\ell}^3) (4\pi/(2\ell+1))^{3/2}, \quad (6) $$

which requires that $Q_{\ell} \neq 0$. The combined inspection of $Q_{\ell}$ and $W_{\ell}$ (or $\hat{W}_{\ell}$) gives a rather univocal signature of different periodic layouts of cells and is therefore a useful tool for characterizing partially or totally ordered foams.

To compute the Steinhardt's coefficients for an ideal Kelvin structure, namely a regular space-filling network of truncated octahedrons, one notices that this polyhedron has edge unit vectors $\mathbf{u}_j \in \{(0, \pm 1, \pm 1)/\sqrt{2} \text{ and circular permutations}\}$, i.e. 12 different equal vectors, but only 6 different directions. The multiplicity of each of the 6 directions is 6 because they all play an equivalent role. As a result one can write for the Kelvin structure: $\rho_K(\mathbf{n}) = \frac{1}{12} \sum_{j=1}^{12} \delta(\mathbf{n} - \mathbf{u}_j)$. When plugged in the Steinhardt's parameter definition, it yields the theoretical values for a perfect Kelvin cells arrangement. Those are shown in hatched white bars in figure 3, and compared to the experimental values in the case of a square array of fibers.
4. Quantitative results in a square array

Figure 3 presents the results of this analysis on samples with square fiber arrays of spacing *a* = 4.0 mm, compared both to samples with no fibers, and to theoretical values for a perfect Kelvin structure. The coefficients *Q*<sub>*ℓ*</sub> (figure 3(a)) and *W*&#770;<sub>*ℓ*</sub> (figure 3(b)) are computed theoretically for a perfect Kelvin cell arrangement in white (hatched), as described in the previous paragraph, and compared to a sample with no fibers in yellow, and an ordered part of a sample with a fiber array in orange. Note that the skeleton parts that correspond to the fibers in the foam have been removed, to keep only the bubble network for the analysis. For this foam-fiber assembly, the fibers are not engulfed in struts (figure 1(b)). Results with and without removing the parts of the skeleton corresponding to the fibers are presented in the supplementary material (figure S5).

We observe that the values of Q<sub>l</sub> for l ≠ 0 with no fibers are close to zero, which is the signature of an
isotropic sample (no specific orientational order), whereas the samples exhibiting visually a Kelvin
arrangement show a signal much closer to the theoretical values. The values of Q<sub>l</sub> are rather sensitive to
noise due to the slight spreading of the strut orientation belonging to the experimental Kelvin cells and to
small global noise pervading the whole unit cell. However, we still recover a shape of signal for the Q<sub>l</sub> and W&#770;<sub>l</sub>
that is a clear signature of Kelvin cell arrangements in a BCC configuration. Figure 3(c) plots the density
function over the unit sphere for the sample with a square fiber array used for figures 3(a) and (b), with the
fibers highlighted in red. Those points highlighted in red are the ones removed for the analysis including only
the struts corresponding to the bubbles.

We then investigate the range of parameters in which such Kelvin arrangements can be observed. The
first characterization, shown in figure 3(d), consists in varying the ratio of the bubble size over the fiber
spacing *d/a*. Note that *d* corresponds to the diameter of a sphere of same volume as the bubble. We observe

<figure><img src="image_7.png" /></figure>

**Figure 3.** (a) Coefficients *Q*<sub>*ℓ*</sub> extracted in yellow for a sample with no fibers (*d* = 4.3 ± 0.2 mm), and in orange for a reduced zone of size 4*d* × 2*d* × 5*d* (approximately 80 bubbles) in the sample with a square fiber array from figure 2(b) (*d* = 3.7 ± 0.2 mm, only the skeleton corresponding to the bubbles is kept, struts corresponding to the fibers are removed). Those experimental data are compared to the theoretical values for a BCC arrangement of perfect Kelvin cells (in white, hatched). (b) Coefficients *W*&#770;<sub>*ℓ*</sub> calculated for the same set of systems. Notice that *W*&#770;<sub>2</sub> is not defined in this case, as the theoretical value of *Q*<sub>2</sub> is zero for the structures studied in this work. (c) Unit sphere showing the bond orientational density for the ordered sample analysed in (a) and (b), highlighting in red the points corresponding to parts of the skeleton actually being fibers. (d) Evolution of the obtention of Kelvin cells as a function of the ratio of bubble size to fiber spacing *d/a*, using the ratio *Q*<sub>6</sub>/*Q*<sub>6th</sub> of the value of *Q*<sub>6</sub> obtained experimentally for a cubic central zone of 15 mm side in the sample, composed of approximatively 100 bubbles, divided by the theoretical value *Q*<sub>6th</sub> for a perfect Kelvin cell arrangement. *d* corresponds to the average bubble size over the whole sample. (e) Phase diagram highlighting the zones in which clusters of Kelvin cells are obtained, when varying the structure thickness and the polydispersity of bubble sizes. The gray zone on the left corresponds to a zone where stability of foams is not sufficient to allow x-ray tomography scans.
that Kelvin cells are mostly observed near an optimal value of $d/a$, which is close to the calculated optimum $d/a = 0.985$ for an infinitely dry foam (see supplementary material for the calculation and tomographic images (figure S6)). Note that the values of $d$ correspond to an average over the whole foam, and that the polydispersity is comprised between 3 and 21% depending on the sample. Figure 3(d) is thus showing that, among all the experimental points gathered for this study, being close to the optimal theoretical value of $d/a$ is a necessary condition to have points with high $Q_6/Q_{6th}$ values, corresponding to structures close to perfect Kelvin foams.

We set the bubble size near this optimum by choosing a nozzle inner diameter of 250 µm and construct a phase diagram (figure 3(e)) showing the role of the structure thickness (defined as the average thickness of elementary constituents of the material network (mainly struts), measured by the x-ray tomography analysis software VG Studio MAX [33]) and of the sample polydispersity. Details are provided in table S1 of the supplementary material. Several zones can be observed in figure 3(e): (i) thin structures (highlighted in gray) are not stable enough for x-ray tomography with the current protocol, (ii) polydisperse foams (yellow) are disordered, and (iii) only sufficiently monodisperse foams (orange) exhibit a significant amount of Kelvin cells. Note that we manage to obtain Kelvin cells even for thick structures: the highest material fraction measured by the x-ray tomography software is 7.2%, corresponding to a liquid content for which BCC Kelvin foams without fiber are higher in energy than FCC foams: Theoretically, the BBC structure has a lower energy compare to the FCC structure, up to a liquid fraction of 6% only [10].

The obtention of crystalline structures—with cells paving the space with periodic patterns—can be of specific interest at different scales in terms of wave propagation, potentially transforming foams into elastic, acoustic or photonic metamaterials [34, 35]. Beyond theoretical predictions and simulations, additive manufacturing allows to explore properties of crystalline cellular materials: the present work aims at offering a new route towards mechanical self-assembling strategy of such architected materials.

## 5. Extending the range of accessible structures: hexagonal array

When changing the way fibers are arranged from a *square* array to a *hexagonal* array, we observe a hexagonal close packing (HCP) array of trapezorhombic dodecahedron bubbles—again for a narrow range of bubble diameter to spacing ratio $d/a$ (see supplementary material for results with different values of $d$, figure S7). The ordering obtained with the optimal $d/a$ is highlighted on tomographic images in figure 4(a). The Steinhardt's coefficients $Q_\ell$ and $\tilde{W}_\ell$ are shown for the ordered zone of a sample in a hexagonal fiber array in blue, and compared with the theoretical values for a perfect HCP arrangement of trapezorhombic dodecahedron bubbles in hatched blue and a perfect FCC arrangement of rhombic dodecahedron bubbles in hatched green. The Steinhardt's parameters for the rhombic dodecahedron have been computed as for the Kelvin structure, the vertex coordinates of the trapezorhombic dodecahedron being directly accessible from the Mathematica software [36].

The discrimination between the two structural arrangements is clear: the obtained structure is a HCP arrangement. It is interesting to note the crucial role played here by $\tilde{W}_4$ which alone allows a clear-cut disambiguation between the two types of structures. Although the energies of HCP and FCC are equal in the limit of low liquid content, and slightly lower for the HCP structure for wet foams [37, 38], FCC seems to be generally prefered in experimental systems [31, 39]. We show here a strategy to induce preferentially a HCP ordering.

## 6. Conclusion and perspectives

This work demonstrates that taking advantage of the confinement induced by a fiber array can lead to crystalline foam structures usually challenging to observe experimentally. The final structure (Kelvin foam (BCC), or HCP foam) is determined mainly by the fiber pattern (square or hexagonal) and by ratio of the bubble diameter to fiber spacing. For each type of structure, the presence of fibers allows to reach an architecture that is usually not preferred in classical foams, by guiding the assembly. The use of polymeric materials able to solidify not only permits a three dimensional characterization of the structures by x-ray tomography, but also paves the road towards novel fabrication techniques of highly ordered solid foams: The present desktop scale experiments could be suitable for miniaturisation, with potential applications in acoustic, elastic or photonic metamaterials [34, 35, 40]. In addition, multiple fiber configurations could be investigated to extend the zoology of foam structures reached through similar processes.
<figure><img src="image_8.png" /><figcaption>**Figure 4.** (a) Obtention of a HCP crystalline structure of trapezorhombic dodecahedra with a hexagonal fiber array: x-ray tomographic scan of the structure, together with single cell views and the corresponding schematics. (b) Coefficients $Q_ℓ$ calculated in light blue for the structured zone of a sample in a hexagonal fiber array of size $3d × 4d × 3d$, with fiber spacing $a = 4$ mm and bubble diameter $d = 4.3 ± 0.5$ mm, compared to the theoretical values obtained for a perfect HCP arrangement (hatched light blue) and for a perfect FCC arrangement (hatched green). Insets: schematics of the HCP and FCC cells. (c) Coefficients $\tilde{W}_ℓ$ calculated for the same set of systems.</figcaption></figure>

## Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

The supplementary material contains details on the materials and methods, results on polyurethane foams, and details on the Steinhardt's analysis, with 7 figures and 1 table. All the datasets describing figures 2(a), (b) and 4(a) with their description are available as supplementary data. The full x-ray tomography 3D reconstructions are available upon requests to the authors only, due to size constraints, but an overview of the main samples is given with three short supplementary movies, corresponding to the samples of figures 2(a), (b) and 4(a).

## Acknowledgments

We are grateful to Imene Ben-Djemaa and Leandro Jacomine for their help with the alginate foams formulation and to Wiebke Drenckhan, François Schosseler, Simon Cox, Christian Gauthier, Patrick Kékicheff and Reinhard Höhler for fruitful discussions. We thank the MiNaMec platform of ICS for x-ray tomography scans. This work of the Interdisciplinary Institute HiFunMat, as part of the ITI 2021-2028 program of the University of Strasbourg, CNRS and Inserm, was supported by IdEx Unistra (ANR-10-IDEX-0002) and SFRI (STRAT'US Project, ANR-20-SFRI-0012) under the framework of the French Investments for the Future Program. We also acknowledge funding from the IdEx Unistra framework (A Hourlier-Fargette) and from the ANR (FOAMINT Project, ANR-23-CE06-0014-01).

## Appendix: Materials and methods

### Foaming solutions

For alginate foams, our formulation is inspired from [22]: 0.02 M of calcium phosphate dibasic (CaHPO₄—purchased from Sigma-Aldrich) is dispersed into a 1 wt% solution of alginic acid sodium salt low viscosity alginate (LVA—purchased from Alfa Aesar). We then add to this initial solution 2 g of the surfactant Disponil APG 425 (from BASF) per 100 g of solution under gentle stirring to prevent the formation of bubbles. Mixing is done for 5 min using Ultra Turrax tube drive at ~1700 rpm (speed 3). 3 g of D-(+)-Gluconic acid δ-lactone (GDL—from Sigma Aldrich) per 100 g of solution are added just before
foaming to initiate the gelation by acidification. For polyurethane foams, the formulation is given in the supplementary material.

## Fiber arrays preparation

Nylon fishing fibers of 120 µm diameter (Flashmer Magix Fluoro)—for all alginate experiments—and of 500 µm diameter (Savage gear Regenerator)—for the polyurethane sample shown in the supplementary material—are stretched and ordered between two walls with regularly spaced holes, made with a laser cutter or with a drill press. Those walls are screwed to a 3D-printed base and the fibers are inserted into the holes located 15 mm above the base, on a square surface of 30 mm on each side. To ensure that the fibers do not move, they are taped to the outside of the container.

## Foaming process

To obtain alginate foams, the foam is first generated in a circular perspex tube (ID 25 mm), closed at the bottom. The plug closing the bottom of the tube is made of silicone elastomer (Zhermack Elite Double 22) to ensure a waterproof seal and to allow two, three or four nozzles (ID 150 µm) to pass through. Air is blown through the nozzles into 50 g of the gelling LVA solution at a constant pressure (600 mbar) via an Elveflow pressure controller (OB1 MK3+). The foam then flows from the circular tube to a 50 × 50 × 80 mm container via a connecting ramp made of a PVC sheet. The container is then filled to the brim with foam and covered with parafilm in contact with the foam. The gelled foam is stable for a sufficient amount of time (almost 2 h) to be characterized via x-ray micro-tomography. To obtain polyurethane foams, the differences with the alginate foam protocol are the following: (i) nitrogen is used instead of air, with a pressure set to 100 mbar, and it enters the tube via five nozzles instead of two, three or four, and (ii) the protocol requires the production of two superimposed foams, the bottom one without catalyst, and the top one with catalyst that drains into the bottom one, as described in [23].

## X-ray tomography acquisitions

Characterization of 3D structures have been performed with the x-ray microtomograph EasyTom 150/160 from RX Solutions, and stacks of images are imported into Xact software to reconstruct 3D structures, that are then imported into VG Studio MAX 2023.1 software for visualization and analysis. All tomographies have been performed at a resolution of 25 µm (average scanning time of about 2 h for polyurethane foams and 20 min for alginate foams, to avoid evaporation and ageing of the samples). For alginate foams, the whole container is covered with parafilm and placed into the tomograph, while for polyurethane foams, a cubic sample with sides of 35 ± 2 mm is cut with a thermocut (Proxxon), as described in [22, 23] for foams without fiber arrays.

## ORCID iDs

Marwan Chammouma https://orcid.org/0009-0003-8256-0668

Manon Jouanlanne https://orcid.org/0000-0002-8269-0281

Antoine Egelé https://orcid.org/0000-0003-0504-8356

Damien Favier https://orcid.org/0000-0001-7588-8065

Jean Farago https://orcid.org/0000-0001-9858-0522

Aurélie Hourlier-Fargette https://orcid.org/0000-0002-0148-4145

## References

[1] Drenckhan W and Hutzler S 2015 *Adv. Colloid Interface Sci.* **224** 1–16

[2] Andrieux S, Quell A, Stubenrauch C and Drenckhan W 2018 *Adv. Colloid Interface Sci.* **256** 276–90

[3] Drenckhan W and Langevin D 2010 *Curr. Opin. Colloid Interface Sci.* **15** 341–58

[4] Plateau J A F 1873 *Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires* (Gauthier-Villars)

[5] Sir Thomson W 1887 *Acta Math.* **11** 121–34

[6] Matzke E B 1945 *Proc. Natl Acad. Sci.* **31** 281–9

[7] Matzke E B 1946 *Am. J. Bot.* **33** 58–80

[8] Dodd J D 1955 *Am. J. Bot.* **42** 566–9

[9] van der Net A, Delaney G W, Drenckhan W, Weaire D and Hutzler S 2007 *Colloids Surf. A* **309** 117–24

[10] Höhler R, Yip Cheung Sang Y, Lorenceau E and Cohen-Addad S 2008 *Langmuir* **24** 418–25

[11] Rosa M E, Fortes M A and Vaz M F 2002 *Eur. Phys. J. E* **7** 129–40

[12] Meagher A, McAteer D, Hutzler S and Weaire D 2013 *Phil. Mag.* **93** 4138–50

[13] Saadatfar M, Barry J, Weaire D and Hutzler S 2008 *Phil. Mag. Lett.* **88** 661–8

[14] Skrypnik A, Knüpfer L, Trtik P, Tholan V, Parkes S and Heitkam S 2023 *Soft Matter* **19** 8552–60

[15] Weaire D and Phelan R 1994 *Phil. Mag. Lett.* **69** 107–10

[16] Gabbrielli R, Meagher A J, Weaire D, Brakke K A and Hutzler S 2012 *Phil. Mag. Lett.* **92** 1–6

[17] Krichevsky O and Stavans J 1992 *Phys. Rev. B* **46** 10579–82
[18] Huang Z, Su M, Yang Q, Li Z, Chen S, Li Y, Zhou X, Li F and Song Y 2017 Nat. Commun. 8 14110

[19] Bae J, Lee K, Seo S, Park J G, Zhou Q and Kim T 2019 Nat. Commun. 10 3209

[20] Li Z, Huang Z, Yang Q, Su M, Zhou X, Li H, Li L, Li F and Song Y 2017 Adv. Opt. Mater. 5 1700751

[21] Whyte D, Haffner B, Tanaka A, Hjelt T and Hutzler S 2017 Colloids Surf. A 534 112–9

[22] Jouanlanne M, Ben-Djemaa I, Egelé A, Jacomine L, Farago J, Drenckhan W and Hourlier-Fargette A 2024 Adv. Mater. Interfaces 11 2400337

[23] Jouanlanne M, Egelé A, Drenckhan W, Farago J and Hourlier-Fargette A 2024 Macromol. Rapid Commun. 45 2400254

[24] Cantat I, Cohen-Addad S, Elias F, Graner F, Höhler R, Pitois O, Rouyer F and Saint-Jalmes A 2013 Foams: Structure and Dynamics (Oxford University Press)

[25] Montminy M D, Tannenbaum A R and Macosko C W 2001 J. Cell. Plast. 37 501–15

[26] Montminy M D, Tannenbaum A R and Macosko C W 2004 J. Colloid Interface Sci. 280 202–11

[27] Steinhardt P J, Nelson D R and Ronchetti M 1981 Phys. Rev. Lett. 47 1297–300

[28] Steinhardt P J, Nelson D R and Ronchetti M 1983 Phys. Rev. B 28 784–805

[29] Kosterlitz J M and Thouless D J 1973 J. Phys. C: Solid State Phys. 6 1181

[30] Nelson D R and Halperin B I 1979 Phys. Rev. B 19 2457–84

[31] Meagher A J, Whyte D, Banhart J, Hutzler S, Weaire D and García-Moreno F 2015 Soft Matter 11 4710–6

[32] Jose J, Blab G A, van Blaaderen A and Imhof A 2015 Soft Matter 11 1800–13

[33] VGStudio 2022 Foam/powder analysis module (available at: www volumegraphics.com/en/products/vgsm/foam-analysis.html)

[34] Spadoni A, Höhler R, Cohen-Addad S and Dorodnitsyn V 2014 J. Acoust. Soc. Am. 135 1692–9

[35] Klatt M A, Steinhardt P J and Torquato S 2019 Proc. Natl Acad. Sci. 116 23480–6

[36] Wolfram Research, Inc. 2024 Mathematica, Version 14.1 (Wolfram Research, Inc.) (available at: www.wolfram.com/mathematica)

[37] Whyte D, Weaire D, Drenckhan W and Hutzler S 2015 Phil. Mag. Lett. 95 319–23

[38] Hutzler S, Dunne F F, Kraynik A M and Weaire D 2020 Soft Matter 16 8262–71

[39] Heitkam S, Drenckhan W and Fröhlich J 2012 Phys. Rev. Lett. 108 148302

[40] Huang Z *et al* 2020 ACS Appl. Mater. Interfaces 12 1757–64