# Foam coarsening in granular packings

A. Salamé, V.-T. Nguyen, V. Langlois, O. Pitois, S. Vincent-Bonnieu

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A. Salamé, V.-T. Nguyen, V. Langlois, O. Pitois, S. Vincent-Bonnieu. Foam coarsening in granular packings. Physical Review E, 2026, 113 (1), pp.015419. 10.1103/w17t-c4tn. hal-05496916

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# Foam coarsening in granular packings.

A. Salamé, V.-T. Nguyen, V. Langlois, and O. Pitois*

Laboratoire Navier, Univ Gustave Eiffel, ENPC, CNRS

S. Vincent-Bonnieu

European Space Agency, Keplerlaan 1, 2200 AG Noordwijk, The Netherlands

(Dated: Published 27 January, 2026. DOI 10.1103/w17t-c4tn)

Foam coarsening refers to the process by which gas diffuses between bubbles, leading to an increase in their average size and a corresponding evolution of the foam's physical properties. When foam is confined within a porous medium, this process can continue, albeit in a modified manner. Anticipating the rate of coarsening under such conditions is therefore crucial. In this study, we examine the coarsening dynamics of liquid foam confined within a packing of spherical grains, systematically varying both the grain size and the foam's liquid fraction, in the range of 7% to 25%. Our key finding is that, under confinement, the coarsening rate is much less sensitive to the liquid fraction than in bulk foams. Instead, the dynamics are more strongly governed by the ratio of bubble size to grain size. This behavior is well captured by a model based on the effective liquid fraction—a measure that accounts for the substantial amount of liquid retained at grain surfaces and contacts in the form of liquid bridges. Additionally, we observe numerous small, persistent bubbles located at the periphery of the liquid bridges. These structures act as hosts, facilitating the accumulation of such bubbles, particularly at higher liquid fractions. Depending on the context, this phenomenon may be either advantageous or detrimental to the intended application.

## INTRODUCTION

It is well known that a liquid foam undergoes coarsening. This fascinating aging process, which animates the bubbles and appears to bring the foam to life, originates from gas transfers between bubbles, driven by pressure differences. Its ultimate stage is the emergence of a single large bubble that has consumed all the others, thereby minimizing the system's surface energy [1].

Bubble coarsening has been extensively studied, due to its importance in controlling bubble size distributions—relevant, for instance, in industrial processes—as well as the physical modeling challenges it poses, such as predicting the shape of the bubble size distribution once it reaches a statistically self-similar regime. For example, a self-similar growth regime in coarsening experiments of dry foams has been observed [2], building on previous work that attempts to establish this fact [3, 4]. Moreover, it was established early on that two distinct coarsening mechanisms should be distinguished. Ostwald ripening [5–7] occurs in relatively dilute bubbly systems and is characterized by a cubic growth law of the mean bubble radius with system age. In contrast, Mullins-type ripening takes place in concentrated bubbly systems, such as foams, where the coarsening follows a parabolic bubble growth law [8, 9]:

$$R_{32}^2(t) = R_{32}^2(0) + \Omega_0 t \quad (1)$$

Here, $\Omega_0$ is the coarsening rate, and the Sauter radius $R_{32} = \langle R^3 \rangle / \langle R^2 \rangle$ is often used to represent the average bubble size. Recent studies have explored the

transition between Ostwald and Mullins-type coarsening regimes [10–12].

Predicting $\Omega_0$ remains challenging, as it depends on a wide range of parameters spanning several length scales [1], such as the thickness of thin films (10–100 nm), the extent of these films on bubble surfaces (10–100 µm), the overall bubble arrangement (0.1–1 mm), and the scale of the foam (1-10 cm). Among the relative successes in modeling foam coarsening, one can cite the description of the effect of bubble concentration—or equivalently, liquid fraction $\phi_\ell$. This approach considers that gas exchange between bubbles primarily occurs through thin films [13–15]. For a given bubble size distribution, a lower liquid fraction corresponds to larger film areas, and therefore to faster gas transfer rates.

Relatively robust expressions are available to describe the evolution of the fraction $\xi$ of bubble area covered by films as a function of liquid fraction. In what follows, we will refer to the expression proposed by Höhler [16]:

$$\xi(\phi_\ell) = \frac{\tilde{\Pi}(\phi_\ell)}{\tilde{\Pi}(\phi_\ell) + 2(1 - \phi_\ell)} \quad (2)$$

where $\tilde{\Pi}(\phi_\ell)$ is the dimensionless osmotic pressure of the foam:

$$\tilde{\Pi}(\phi_\ell) = \frac{\Pi(\phi_\ell)}{\gamma/R_{32}} = k \frac{(\phi_\ell^* - \phi_\ell)^2}{\phi_\ell^{1/2}} \quad (3)$$

with $\gamma$ the surface tension of the liquid-gas interface. Values for $k$ and $\phi_\ell^*$ are respectively equal to 7.3 and 0.26 for monodisperse ordered foam, 3.2 and 0.36 for monodisperse disordered foam [16], while values corresponding to polydisperse foam in the coarsening steady state are respectively 4.5 and 0.31 [15]. $\Omega_0$ encompasses additional physicochemical parameters, including gas solubility in

<footnote>* Olivier.Pitois@univ-eiffel.fr</footnote>
the liquid and the thickness of the films. It is common to separate the contributions arising from physicochemical properties and those related to film area. This decomposition can be expressed as [1]:

$$ \Omega_0 = \frac{\gamma DH_e V_m}{h} \times \xi(\phi_l) \quad (4) $$

with surface tension $\gamma$, coefficient $D$ of molecular diffusion of the gas in the solution, Henry constant $H_e$ accounting for gas solubility, molar volume of the gas $V_m$, and film thickness $h$. Thus, for a given surfactant solution, Equations 1–4 together provide a framework for predicting the rate of bubble growth as a function of the liquid fraction.

Foam is frequently found confined within the pore spaces formed between grains, with one of the most prominent examples being its injection into soils. Indeed, its use has emerged as a promising alternative to conventional flushing techniques for efficiently extracting non-aqueous phase liquids (NAPLs) from aquifers [17, 18]—subsurface layers of permeable rock, sediment, or soil that store groundwater. The gas trapped within foam bubbles enhances the stability of fluid displacement in porous materials, providing improved control over remediação processes in reservoirs or aquifers [19]. Moreover, foam can serve as a barrier by reducing liquid permeability in porous substrates [20, 21], helping to contain contamination sources, or assisting CO₂ sequestration [22]. In tunnel excavation through granular soils, foams are employed in Earth Pressure Balance shield machines, where they are injected at the cutting face and mixed with the excavated material to stabilize the tunnel face by balancing earth pressure and enhancing soil cohesion through capillary effects [23]. In the context of coal dust suppression, the process known as self-hardening foam-soil involves combining liquid foam with dust particles to mechanically stabilize them [24]. In all these applications, bubble size often plays a crucial role in determining the properties of the confined foam, meaning that coarsening is a phenomenon likely to significantly alter these properties. This is the case, for example, for the liquid permeability of a granular soil filled with foam [25, 26], or the apparent viscosity of a liquid foam being forced to flow through such a medium [27].

Within such confinement conditions, a pseudo-bulk-foam coarsening behavior has been observed when the average bubble size is much smaller than the pore size [28]. However, as bubbles grow and approach the size of the pores, confinement effects become increasingly significant, leading to a reduction in the coarsening rate and eventually to its arrest [29]. Notably, under appropriate confinement, a polydisperse distribution of gas bubbles can exhibit an anti-coarsening phenomenon, wherein small bubbles do not transfer their gas to larger bubbles, ultimately yielding a monodisperse distribution [28, 30–32], with each pore containing a single large bubble [28, 32]. It was recently shown that, prior to coarsening arrest, the coarsening of confined foams can be

<figure><img src="image_2.png" /><figcaption>FIG. 1. As foam fills the pore space of a granular packing, liquid is sequestered at the surface of the grains, either in the form of wall Plateau borders (in red), which are larger than their bulk counterparts (in blue), or as liquid bridges at grain contacts (in green), resulting from the steric exclusion of bubbles in these regions. As a result, the foam in the pore core has a lower liquid fraction $\phi_l^{\text{eff}}$ than the overall pore space average, $\phi_l$.</figcaption></figure>

described in a way similar to the coarsening described by Eq.1. However, confined foams were found to coarsen significantly faster than their unconfined counterparts with the same bubble size and liquid content [33]. Specifically, this rate increases with the bubble-to-grain size ratio, $r$, up to a critical threshold, $r_{\infty} \approx 0.8$, which marks the onset of the arrested state. Consequently, when coarsening occurs within the pore space of a granular packing, the coarsening velocity $\Omega$ behaves as follows: (i) $\Omega \approx \Omega_0$ for $r \lesssim 0.01$, (ii) $\Omega > \Omega_0$ for $0.01 < r < r_{\infty}$, and (iii) $\Omega \approx 0$ for $r \approx r_{\infty}$.

A physical explanation for such behavior has been proposed [33]. It is based on the idea that the foam undergoing coarsening does not contain as much liquid as the total pore space does. Indeed, it was recently shown that liquid is sequestered at the surface of the grains, either in the form of wall Plateau borders, which are larger than their bulk counterparts, or as liquid bridges at grain contacts, resulting from the steric exclusion of bubbles in these regions [34] (see Fig. 1). As a result, the foam in the pore core has a lower liquid fraction than the overall pore space average, $\phi_l$. It is therefore proposed to define an effective liquid fraction for the coarsening foam, denoted as $\phi_l^{\text{eff}}$, which is calculated based on the estimated amounts of liquid stored in wall Plateau borders and liquid bridges [34, 35]. This approach has been supported by a quantitative agreement between measured coarsening rates as a function of $r$ and predictions based on Equations 1–4, where $\phi_l$ is replaced by $\phi_l^{\text{eff}}$ [33]. However, only a single foam liquid fraction was investigated ($\phi_l = 20\%$), so it remains unproven whether this ap-
proach holds for liquid foams of arbitrary liquid fraction. The present study specifically aims to assess the validity of the proposed approach by investigating the coarsening of confined liquid foams within granular packings over the widest possible range of liquid fractions. The following sections present the effective liquid fraction model, the experimental methodology, and the resulting observations.

## EFFECTIVE LIQUID FRACTION OF THE FOAM IN THE CORE OF THE PORES

In this section, we compile the experimental and theoretical results available to us concerning the effective liquid fraction of foams confined within packings of spherical grains. Modeling the effective liquid fraction in pore foam (with bubble radius $R_b$) involves estimating the liquid volume that must be distributed at grain contacts and along grain surfaces (characterized by radius $R_g$) to ensure capillary pressure equilibrium throughout the system. This amounts to determining the mean radius of curvature of the interfaces, $\delta$, for each set of parameters $\phi_\ell$ and $r = R_b/R_g$, i.e., $\delta/R_b = f(\phi_\ell, r)$, and then deducing the corresponding liquid volumes associated with each element of the liquid network (see Fig 1). A strong consistency was observed between experimental data, geometrical models, and numerical simulations for monodisperse foam [34, 35]. Together, they provided the evolution as a function of $r$ for the effective liquid fraction in the pore foam, $\phi_\ell^{\text{eff}}$, as well as for the proportions of liquid contained in the wall Plateau borders, $\phi_\ell^{\text{wall}}$, and in the liquid bridges, $\phi_\ell^{\text{bridges}}$. It has also been shown that the ratios $\phi_\ell^{\text{eff}}/\phi_\ell$, $\phi_\ell^{\text{wall}}/\phi_\ell$ and $\phi_\ell^{\text{bridges}}/\phi_\ell$ depend only very weakly on $\phi_\ell$, so that all configurations can be described using only three functions of the parameter $r$, which are shown in Fig. 2a. It is particularly useful to approximate $\phi_\ell^{\text{eff}}/\phi_\ell$ using a polynomial fit of the experimental data presented in Langlois et al. [35] (see Figure 8a of this reference):

$$ \frac{\phi_{\ell}^{\text{eff}}}{\phi_{\ell}} = 1 + 248.48r^4 - 225.77r^3 + 78.485r^2 - 13.046r \quad (5) $$

We stress that this expression applies to a monodisperse foam filling the pore space of a monodisperse sphere packing. However, for a polydisperse foam with a polydispersity index $p = R_{32}/\langle R^3 \rangle^{1/3} - 1 = 0.13$, using $R_b = R_{32}$, the corresponding function has been experimentally measured (see also Fig. 2b) as [27]:

$$ \frac{\phi_{\ell}^{\text{eff}}}{\phi_{\ell}} = 1 - 56.119r^4 + 59.161r^3 - 13.585r^2 - 2.7915r \quad (6) $$

The liquid bridges formed at grain contacts play a key role in determining the effective liquid fraction of the pore foam. Particular attention has been devoted to modeling their geometry and comparing it with experimental observations [34, 35]. The radial extent of the

liquid bridge, denoted $R_{\text{bridge}}$, scales proportionally with the grain radius $R_g$ and depends on the parameter $\delta$. This parameter $\delta$ effectively defines how close the bubbles—characterized by a mean curvature radius $\delta$—can approach the contact point between grains. A major geometrical difference between these liquid bridges and those typically observed in unsaturated granular media [36, 37] is the presence of bubbles at the periphery of the bridge, giving it a characteristic daisy shape [34]. It has been shown that the volume of liquid contained in the bridge depends significantly on the number of bubbles, $n_b$, that form it [35]: $n_b \approx (0.89 + 1.46\phi_\ell)(4 + r^{-0.83})$ within the range $6 < n_b \le 18$.

Since the available experimental data concern only so-called "surface" liquid bridges—i.e., those formed between grains in contact with the wall of the container holding the sample—we focus here on the geometry of such a bridge. The numerical simulation work carried out in Langlois et al. [35] was extended to accurately describe the geometry of bridges between a grain and a flat surface, and results similar to those for bridges between two grains were obtained (see Appendix A). Here, we present an approximate expression that accounts for these results:

$$ \frac{R_{\text{bridge}}}{R_g} = A(n_b) \left( \frac{\delta}{R_g} \right)^{B(n_b)} \quad (7) $$

with $\delta/R_g \lesssim 0.1$, $A(n_b) = 0.169 \times n_b + 0.942$ and $B(n_b) = 0.0113 \times n_b + 0.452$.

The values of $\delta/R_g$ can be estimated by the relation proposed in [34]:

$$ \frac{\delta}{R_g} \approx \frac{r}{\frac{2}{\sqrt{3}} - 1} \frac{0.27(\phi_{\ell}^{\text{eff}})^{0.5} + 3.17(\phi_{\ell}^{\text{eff}})^{2.75}}{1 + 0.57(\phi_{\ell}^{\text{eff}})^{0.27}} \quad (8) $$

By combining Equations 7 and 8 with Eq. 5, an expression can be obtained for the extent of the liquid bridges that are observed at the wall of the grain packing sample when filled with monodisperse foam. Curves as a function of $r$ are presented in Fig. 2c for several liquid fractions and compared to data collected for a monodisperse bubble foam [27, 34, 35] and a polydisperse bubble foam ($p = 0.13$) [33]. As can be seen, there is good quantitative agreement for all the liquid fractions studied. More surprisingly, a very satisfactory agreement is also observed for the polydisperse foam, even though the model does not account for polydispersity. It is likely that the choice of $R_b = R_{32}$ contributes significantly to this agreement. Overall, the results suggest that the model can be useful for estimating the size of liquid bridges, regardless of the type of foam filling the pore space.

The agreement observed in Fig. 2 for the size of the liquid bridges provides direct evidence that the effective liquid fraction model is suitable for describing the enhanced bubble concentration in the core of the pore space, at the expense of the grain contact zones, which instead tend to accumulate more liquid. This description of the liquid distribution will subsequently be used as a key element
<figure><img src="image_3.png" /></figure>

FIG. 2. Liquid distribution in a packing of spheres of equal size, filled with a monodisperse bubble foam. (a) The volume of liquid is distributed into three contributions (see also Fig. 1): $\phi_{\ell}^{\text{wall}}/\phi_{\ell}$: Plateau borders at the surface of the grains, $\phi_{\ell}^{\text{bridge}}/\phi_{\ell}$: Plateau borders forming liquid bridges between two contacting grains, and $\phi_{\ell}^{\text{eff}}/\phi_{\ell}$: bulk foam Plateau borders (all Plateau borders not belonging to the first two categories). The illustrations shown for these three contributions were obtained through numerical surface minimization and therefore represent the precise shape of each element. Pay particular attention to the positions of the bubbles (numbered here from 1 to 10), which surround and thereby form the liquid bridge. The curves show their evolution as a function of the parameter $r$, which represents the ratio between the bubble size and the grain size. (b) The theoretical curve $\phi_{\ell}^{\text{eff}}/\phi_{\ell}$ (solid line, eq. 5) for a monodisperse foam is compared to experimental data (filled symbols). Data for a polydisperse foam (polydispersity index $p = 0.13$) are also shown (open symbols). The polynomial function given in eq. 6 provides a fit to these data. (c) The curves $R_{\text{bridge}}/R_g$ (solid lines) as a function of $r$ for four liquid fractions $\phi_{\ell} = 7\%$ (green), 12% (blue), 20% (black), and 25% (red), are compared to experimental data for a monodisperse foam (filled symbols). Since the measurements are performed at the surface of the container, the presented theoretical values of $R_{\text{bridge}}/R_g$ correspond to the illustrated configuration, i.e., contact between a sphere and a plane. Data for a polydisperse foam (polydispersity index $p = 0.13$) are also shown (open symbols).

in modeling the observed coarsening rate. The measurement of this coarsening rate is the focus of the following section.

## COARSENING EXPERIMENTS

### Materials and Setup

The nonionic surfactant Glucopon® 225DK (BASF) is used at a concentration of 20 g/L in water. Foam generation follows the double-syringe technique described in [38], with one 50 mL syringe containing 3.5 mL of the surfactant solution and the other filled with 46.5 mL of air. The syringes are connected via a Luer-lock adapter, and foam is produced through 20 back-and-forth passes. This method yields a foam with a liquid fraction of 7% and a bubble size distribution characterized by a Sauter radius $R_b \equiv R_{32} \approx 100 \ \mu\text{m}$. The experiments are conducted in a granular medium composed of glass beads with radius $R_g \pm \Delta R_g$, where the relative polydispersity $\Delta R_g/R_g$ is about 5%. $R_g$ ranges from 0.4 mm to 2.5 mm and beads are confined within a cylindrical glass tube with an inner diameter $D_c = 26$ mm. Surface roughness of the beads is estimated to be on the order of 1 $\mu$m or less. For optimal compaction, the column is filled in thirds, each layer being compacted with a 7 mm rod.

To maintain a uniform and stable packing, two 3 mm-thick plastic grids were positioned above and below the grain bed. Each grid has 0.3 mm circular openings spaced 0.5 mm apart, permitting foam flow while retaining the grains. Packing porosity ranges from 0.38 to 0.43, depending slightly on bead size, indicating non-negligible wall effects. Grain arrangement near the walls may thus differ from that in the bulk. This is unlikely to cause issues, as the bubbles are smaller than the wall pores; however, wall effects could become relevant when comparing results across different bead sizes. The bead packing is filled with foam using a syringe pump operating at 15 mL/min. The initial 7% liquid content is selected to limit drainage during injection, but the target value is reached thanks to co-injected additional surfactant solution, raising the final liquid fraction in the pore space to $7\% \le \phi_{\ell} \le 25\%$. Consequently, the final system consists of a foam with bubble radius $R_b \approx 100 \ \mu\text{m}$ and liquid content $\phi_{\ell}$ occupying the porous medium. We define the size ratio $r = R_b/R_g$. Throughout and following the injection phase, the sample is rotated at 80°/s (see Fig. 3). The column is programmed to pause for 1 s at regular angular positions before resuming rotation. During these pauses, two cameras located above and below the column captured images of the wall surface through right-angle prisms (15×15×21.2 mm) affixed to the glass tube 180° apart, i.e., top and bottom. These prisms enhance the
<figure><img src="image_4.png" /></figure>

FIG. 3. Experimental setup for foam injection and optical imaging in a granular medium. The porous medium consists of glass beads confined within a cylindrical glass column with an inner diameter of $D_c = 26$ mm (not at scale in the figure). Bead radii $R_g$ range from 0.4 mm to 2.5 mm. Foam with a bubble radius $R_b \approx 100$ µm and a liquid fraction in the range $7\% \le \phi_l \le 25\%$ is injected using a syringe pump at a flow rate of 15 mL/min. After the medium is filled, the injection is stopped. The column is continuously rotated at 80°/s, with 1 s pauses at regular angular intervals. During these pauses, two cameras capture images through right-angle prisms (15×15×21.2 mm) mounted on opposite sides of the column, i.e., top and bottom. These prisms enhance optical contrast between the liquid and gas phases near the wall, rendering the liquid black and the gas (separated from the wall by a thin film) white in the captured images (see Fig. 4a).

optical contrast between gas and liquid near the wall [39] (see Fig. 4), rendering the liquid black and the gas (separated from the wall by a thin film) white.

## Image processing

The step (i) of our image processing method is to detect the bubbles by preparing initially the raw image by sharpening and enhancing contrast. This allows to make the images clearer and facilitate the step (ii): segmentation. Segmentation based solely on direct gray-level thresholding to distinguish between liquid and gas phases was found to be inadequate due to parasitic reflections around the bubble-wall contact areas, likely caused by the curvature of the column and the mounting of the prism. Instead, artificial intelligence (AI)-based segmentation offered a more robust and reliable solution, as advanced algorithms can differentiate between phases with similar gray levels by leveraging additional image features such as shape, texture, and spatial context. In our methodology, we employ a dedicated deep learning model—namely the U-Net convolutional neural network—which is widely recognized for its high performance in biomedical image segmentation tasks [40]. The U-Net model is applied following an initial classification performed using a Random Forest algorithm, thus creating a hierarchical segmentation workflow.

<figure><img src="image_5.png" /></figure>

FIG. 4. Image processing of foam observed at the column wall through a prism. The width of the images is 3.5 mm. (a) The raw image shows dark regions corresponding to liquid and bright areas where bubbles are in contact with the wall via a thin liquid film. Large dark patches indicate liquid bridges formed at contact points between the grains and the wall (the grains themselves are not visible in this type of image). (b) Result of the segmentation performed using *Dragonfly* after training the model. (c) Intermediate processing step where segmented bubbles (in white) and identified liquid bridges (yellow ellipses) are overlaid. (d) Result of the Voronoi tessellation, shown superimposed on the original raw image. The liquid bridges are represented as detected from the segmentation.

This two-stage approach combines the strengths of classical machine learning and deep learning, yielding improved accuracy and robustness in phase separation (see Fig. 4b,c). All segmentation procedures are carried out using the *Dragonfly* software suite [41], which has been recently adopted for similar tasks involving granular materials [37]. The segmented bubbles are fitted with ellipses using the ImageJ software [42]. For several images within each experimental run, the fitted ellipses are overlaid on the raw images to visually assess segmentation accuracy. Additionally, *ground truth data* are generated by manually placing ellipses over the bubbles in those raw images, enabling quantitative comparison with the automated segmentation results. In the step (iii), the dark regions between bubbles—corresponding to liquid bridges—are identified. To achieve this, in images such as the one shown in Fig. 4c, we apply the so-called *top-
hat morphological filter in ImageJ to suppress the white bubbles while preserving the black regions corresponding to the liquid bridges. The filter parameters are manually adjusted to accurately capture the shapes of the bridges. The radius of each detected bridge, of area $A_{bridge,i}$, is computed using $R_{bridge,i} = \sqrt{A_{bridge,i}/\pi}$ and the arithmetic mean bridge size $R_{bridge}$ is then calculated. Each detected bridge is then fitted with an ellipse (highlighted in yellow in Fig. 4c), enabling visual inspection and identification of potential segmentation errors.

Once both the bubbles and the liquid bridges are identified, we proceed to the next stage (iv): a Voronoi tessellation of the image (see Fig. 4d). This geometric partitioning aims to estimate the effective area of contact between each bubble and the wall. This step is necessary because relying solely on the actual contact areas between the bubbles and the wall would introduce a bias due to the decrease in these contact areas as the liquid fraction increases. The tessellation divides the image into regions around each bubble, where the boundaries are defined by points that are equidistant from the centers of the two nearest neighboring bubbles. From these areas, individual bubble radii are computed using $R_{b,i} = \sqrt{A_i/\pi}$, and the Sauter mean radius is then derived over the entire population as $R_b = \langle R_{b,i}^3 \rangle / \langle R_{b,i}^2 \rangle$. The error between our automated image processing pipeline—comprising AI-based segmentation and subsequent manual analysis—was consistently below 5%, and in many cases, below 2%. We determine also the polydispersity index as $p = R_b / \langle R_{b,i}^3 \rangle^{1/3} - 1$.

The evolution of $R_b(t)$ is tracked over time for various grain sizes and liquid volume fractions.

# RESULTS

## Reproducibility and Drainage Issues

Figure 5a presents the evolution of bubble size, expressed as the ratio $r = R_b/R_g$ measured at the top and bottom of the sample in several experiments performed with a grain size $R_g = 1$ mm and a liquid fraction of 25%. First, the results demonstrate good reproducibility. We also observe that, in this specific case, there is a significant difference between the curves obtained at the top and those at the bottom: on average, the measurements at the top exceed those at the bottom, indicating faster coarsening in the upper region. In other words, we observe the effects of gravitational drainage, which, in this example, is not effectively counteracted. Notably, the final bubble size—whether at the top or bottom—is constrained by the size of the pores between the grains. As a result, both curves tend to converge toward the same value over comparable time scales, so the observed difference remains limited. It is therefore justified to average the top and bottom curves in order to obtain a representative curve describing the increase in bubble size at constant liquid fraction. In many cases, the curves are

so close that they become indistinguishable, indicating that bubble size measurements are, in those instances, unaffected by drainage.

To gain a more general understanding of drainage effects, we attempted to determine the value of the ratio $r$ for which it becomes clear that the top curve exceeds the bottom one. This threshold will be denoted as $r_d$ (see Fig. 5b). As previously explained, if drainage induces a difference in coarsening between the top and bottom, this difference is typically only observable for $r_d \lesssim 0.3$, as the curves tend to converge naturally for higher values. Figure 5b presents the obtained values, highlighting that, as generally expected, samples with higher liquid content and larger grains tend to exhibit drainage effects earlier, while those with lower liquid content and smaller grains show no noticeable signs of drainage. A more detailed analysis of these differences is provided in the discussion.

## Temporal evolution of liquid bridges

Our measurements of the sizes of liquid bridges $R_{bridge}$ at the column wall are presented in Fig. 6. The bridge sizes are normalized by the grain size, with which a direct dependence is expected. We can start by noting that all samples exhibit liquid bridges, even those with the lowest liquid fractions. We observe that the scaling with $R_g$ is well respected: for a given liquid fraction, all the curves converge very closely, even though the grain size varies by a factor of approximately 1 to 6. The vertical positioning of the curves is also consistent with expectations: the size of the liquid bridges increases with the liquid fraction. Overall, the size of these bridges increases with bubble size — or, equivalently, with $r$ — consistently with Fig. 2c. In our experiments, however, the parameter $r$ evolves during a single run as a result of coarsening.

## Temporal evolution of the bubble sizes

Results for $R_b$ as a function of foam age are presented in Fig. 7(a-d) for the four investigated $\phi_\ell$ values. The curves result from averaging the raw curves obtained for top and bottom views (see Fig. 5) over several experimental runs (from 2 to 5). Several observations can be made: (i) Bubble growth correlates with grain size, with the average bubble size increasing more rapidly as the grains decrease in size. (ii) The form of the growth curve likewise varies with grain size. (iii) The curves ultimately reach saturation sooner when the grains are smaller.

The polydispersity index, defined as $p = R_{32} / \langle R^3 \rangle^{1/3} - 1$, is shown in Fig. 7(e-h) as a function of $r$. For all curves, $p$ initially increases from approximately 0.13 to a peak value around 0.6 at $r \approx 0.5$, before decreasing—either gradually or abruptly depending on the case.
<figure><img src="image_6.png" /><figcaption>FIG. 5. (a) Results from multiple measurements of the time evolution of the average bubble size, normalized by the grain size to yield the ratio r, for sample characterized by R_g = 1 mm and φ_ℓ = 25%. The purple curves correspond to measurements taken at the top of the sample, while the pink curves represent those taken at the bottom (see Fig. 3). The specific value r_d indicates the point at which it becomes evident that the average of purple curves surpass the pink ones. The shaded area represents the range of values corresponding to the uncertainty margin we apply in determining r_d. (b) Values of r_d as a function of grain radius for several liquid fractions. (c) Liquid permeability of the samples determined according to eq. 12 as a function of r_d. Horizontal dashed line represents the average value. Symbols used for liquid volume fraction φ_ℓ: ○ (circle) 20%, □ (square) 25%. Colors for grain size: R_g = 2.5 mm (black), 1.5 mm (blue), 1 mm (ochre), 0.75 mm (grey).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIG. 6. Radius of the liquid bridges R<sub>bridge</sub> observed at the sample wall normalized by the grain radius R<sub>g</sub> as a function of the ratio r = R<sub>b</sub>/R<sub>g</sub>, which accounts for the increase in bubble size over time. The inset in (a) shows a drawing of a liquid bridge between a grain and the wall of the container. The results are presented for four liquid volume fractions: (a) 7%, (b) 12%, (c) 20%, and (d) 25%. Colors for grain sizes are indicated. The red curves show the numerical results (eqs. 7 and 8), although they do not extend over the full range of measured r values</figcaption></figure>

The inset image in (d) shows a liquid bridge at the early stages of the coarsening experiment with R_g = 1.5 mm diameter grains and 25% liquid fraction: numerous small bubbles can be seen very close to the bridge. These bubbles contribute to a reduction in the size of the bridge.

Coarsening rates

We perform the derivative of the bubble growth curves and we smooth the values obtained using the Savitzky-Golay filter. We define the coarsening rate as:

$$ \Omega(r) = 2R_b \frac{dR_b}{dt}, \qquad (9) $$

assuming a coarsening mechanism dominated by the gas transfer through the films (border blocking approximation) and associated with the parabolic law already presented in Eq. 1. While for bulk coarsening, Ω is constant, i.e., Ω = Ω₀, in our experiment the coarsening rate is evolving due to the increasing confinement effects in the pore space. The values obtained for Ω are plotted as a function of r in Fig. 7(i-k), where it is shown that Ω(r) increases significantly before reaching r ≈ 0.7 which initiates the arrest of coarsening. A comparison of the values

for different liquid fractions reveals that, for a fixed r, Ω increases as φ_ℓ decreases.

DISCUSSION

In this section we analyze the results presented in the previous section, focusing on four issues: (i) liquid drainage revealed by comparing top and bottom views, (ii) liquid bridge sizes, (iii) final bubble size, and (iv) coarsening rate.

Analysis of liquid drainage

To evaluate the effect of drainage within the sample of Darcy permeability k(φ_ℓ), it is necessary to compare the sample's rotation period, T_R ≈ 5s, with the characteristic
<figure><img src="image_8.png" /><figcaption>FIG. 7. Color code for the grain size is the following: $R_g = 2.5$ mm (black), 1.5 mm (blue), 1 mm (ochre), 0.75 mm (grey), 0.55 mm (green), and 0.4 mm (burgundy). (a-d) Average bubble radius $R_b$ as a function of foam age for the four investigated liquid volume fractions (as indicated). In (d), $R_{b,\infty}$ refers to the final bubble size. (e-h) Polydispersity index $p$ as a function of the bubble-to-grain size ratio $r$. (i-l) Coarsening rate $\Omega$ as a function of the bubble-to-grain size ratio $r$.</figcaption></figure>

drainage time, $\tau_d$. The latter can be estimated based on the gravity-driven drainage velocity [1],

$$u = k(\phi_{\ell}) \frac{\rho g}{\mu} \qquad (10)$$

and a characteristic length scale, $L$: $\tau_d = L/u$. To eliminate the effects of drainage, it is in principle necessary

that $T_R \ll \tau_d$, i.e., $\tau_d > xT_R$ with $x \gg 1$. As a result, the criterion for the absence of observable drainage can be written as:

$$C \frac{\mu}{\rho g} > k(\phi_{\ell}) \qquad (11)$$
where $C = L/xT_R$ is a numerical constant to be determined by comparison with observations. The crucial point in Eq. 11 is the estimation of the liquid permeability of the grain packing filled with foam. This has been investigated recently, and the associated permeability can be expressed as follows [25, 26]:

$$k(\phi_{\ell}) = g(r) \tilde{k}_{f0}(\phi_{\ell}) R_{b}^{2} \quad (12)$$

where $\tilde{k}_{f0}(\phi_{\ell})$ is the dimensionless permeability of bulk foam with liquid volume fraction $\phi_{\ell}$ and the function $g(r)$ accounts for the effects of the grains:

$$\tilde{k}_{f0}(\phi_{\ell}) = \frac{1}{312.5} \frac{\phi_{\ell}^{3/2}}{(1 - 2.7\phi_{\ell} + 2.2\phi_{\ell}^{2})^{2}} \quad (13)$$

$$g(r) = 0.92 \times \exp(-\frac{r}{0.045}) + 0.08 \times \exp(-\frac{r}{0.28}) \quad (14)$$

This provides a clearer understanding of the key factors driving the onset of drainage in our samples: in addition to the expected effect of the liquid fraction, drainage is more likely to occur when the grain size is large and when the bubble size increases (for instance, over time for a given grain size). Therefore, for each sample for which we determined a $r_d$ value, we determined the corresponding permeability using Eq. 12 and we plot it as a function of $r_d$ in Fig. 5c. It is reassuring to observe that the permeability values are very close to a common value, indicating that for our experimental setup, drainage effects on coarsening are negligible as long as the permeability remains below the identified threshold, i.e., $5 \times 10^{-12} \text{ m}^2$. It follows that the value of the constant C introduced in Eq. 11 is therefore $5 \times 10^{-5} \text{ m/s}$.

## Analysis of liquid bridges

To analyze the liquid bridge sizes, we compare our measurements with numerical results (eq. 7), bearing in mind that these simulations pertain to fully monodisperse systems [35]. There is overall good agreement, as evidenced by the trend of bridge size increasing with $r$ and its growth with increasing $\phi_{\ell}$. Both trends can be attributed to the increased ability to displace bubbles toward the core of the pores when they are larger and the foam is wetter, i.e., when the foam’s osmotic pressure is lower (see eq. 3).

We note that for small $r$, the curves tend to lie below the numerical curves, starting from the very first measurement point. By examining the images (see the image in Fig. 6d), we find that many small bubbles tend to accumulate around the periphery of the bridges, which automatically reduces the minimum distance at which these bubbles can approach the center of contact of the corresponding grain, thereby decreasing the size of the bridge. These bubbles are present from the start, likely deposited in these regions due to segregation effects induced by the foam flow during the filing process, but they

<figure><img src="image_9.png" /><figcaption>FIG. 8. Shrinking of a small bubble at the periphery of a liquid bridge (sample: $R_g = 1.5$ mm, $\phi_{\ell} = 25\%$). Bubble area as a function of time. The shaded region indicates the transition between two distinct regimes: on the left, the bubble remains embedded within the foam, maintaining contact with neighboring bubbles; on the right, the bubble has lost these contacts and is now immersed within the liquid bridge. The shrinking rates measured in each regime are indicated.</figcaption></figure>

eventually disappear. At that point, the bridge sizes become more quantitatively consistent with the numerical simulations.

These effects are more pronounced at 25% presumably because the larger amount of liquid provides greater freedom of movement for the small bubbles to undergo segregation. Moreover, the lifetime of these bubbles is longer at such liquid fractions, which reinforces the observed effect on the curves. In order to clarify that issue, we have tracked small bubbles located at the periphery of the liquid bridges and measured their area as a function of time. A typical example, obtained in a sample characterized by $R_g = 1.5$ mm and $\phi_{\ell} = 25\%$, is presented in Fig. 8. Being smaller than the average, the bubble shrinks over time. Its shrinking rate is approximately $50 \text{ µm}^2/\text{s}$, which is of the same order of magnitude as $\Omega$ for this sample (see Fig. 7l). This suggests that the bubble is part of the foam network — it maintains contact with neighboring bubbles and exchanges gas with them efficiently. As a result, it disappears at the rate by which the average bubble size increases, allowing the overall bubble size distribution to shift toward larger sizes. At a certain point, however, the shrinking rate drops significantly — by more than one order of magnitude. This transition can be attributed to the loss of mechanical contact with neighboring bubbles: the bubble is now partly immersed in the liquid bridge, without being constrained by surrounding bubbles. This behavior closely resembles that of the so-called *roaming bubbles*, recently identified in wet foams through experiments conducted aboard the International Space Station [43]. The key difference here is that these bubbles are hosted by the liquid bridges, whereas in bulk wet foams they reside in the nodes of
<figure><img src="image_10.png" /><figcaption>FIG. 9. Final bubble radius $R_{b,\infty}$ normalized by the radius of the grains $R_g$ as a function of $R_g$. The symbols represent the measured values for all the studied liquid fractions $\phi_\ell$: ◇ (diamond) 7%, △ (triangle) 12%, ○ (circle) 20%, □ (square) 25%. The dotted lines labeled $\alpha = 0.4$ and $\alpha = 0.6$ correspond to eq. 15 evaluated for the respective values of $\alpha$, and represent the maximum size of a bubble that can be accommodated within a triangular wall pore (see the inset for a schematic illustration, where the bubble is shown in orange). The dotted line labeled square pore represents the maximum bubble size that can fit inside a square pore structure. The image shows examples of triangular (labeled "3") and square (labeled "4") pores, as well as even larger pores, for a sample characterized by $R_g = 0.4$ mm and $\phi_\ell = 25\%$.</figcaption></figure>

the liquid network.

## Analysis of the final bubble size

The curves in Fig. 7(a–d) show that bubble growth eventually saturates at long times, defining a final bubble size denoted by $R_{b,\infty}$ (see Fig. 7d). This maximum size occurs because a bubble filling a pore cavity must pass through a narrower inter-pore region to continue growing. In these narrower zones, they are forced to adopt curvatures similar to those of smaller, less confined bubbles, thereby balancing their respective pressures. This average final bubble size could be estimated from the size distribution of the pores at the wall. Here we consider pores formed by a triangular arrangement of the grains

(see inset in Fig. 9), each pore being occupied by a single bubble of approximately triangular shape (as observed at the wall), with the corners truncated by the surrounding liquid bridges. The radius of these bridges is given by $R_{bridge} = \alpha R_g$, where the proportionality factor $\alpha$ ranges from 0.4 to 0.6 depending on the liquid volume fraction (see Fig. 6 at $r \approx 0.6$). As a result, the final bubble size can be expressed as:

$$R_{b,\infty} = \frac{1}{\sqrt{\pi}} \left( \sqrt{3} - \frac{\pi \alpha^2}{2} \right)^{1/2} R_g \quad (15)$$

We plot Eq. 15 in Fig. 9 for both $\alpha = 0.4$ and $\alpha = 0.6$, i.e. $R_{b,\infty}/R_g = 0.69$ and 0.61 respectively. The agreement with experimental data is very good for 1.5 mm grains. However, Eq. 15 underestimates $R_{b,\infty}$ for smaller grain sizes, particularly at $R_g = 0.4$ mm. Image analysis reveals that many bubbles occupy pores formed by irregular grain arrangements—such as larger-than-average pores—or pores resulting from missing grain contacts (see the image in Fig. 9). For example, numerous grain arrangements form nearly square configuration (in green in Fig. 9), for which calculations similar to the previous estimate yield $R_{b,\infty}/R_g \approx 1$. The significant proportion of such pores—along with even larger ones—helps explain the observed large values of $R_{b,\infty}$.

## Analysis of coarsening rates

Results obtained for all liquid fractions show a similar trend: $\Omega$ increases with $r$ up to a maximum, then decreases and eventually vanishes. The approach to arrest is not as sharply marked as one might expect, because when the smallest pores are filled, larger pores still contain bubbles that can continue to grow. The evolution of the polydispersity index $p$ follows the same pattern across all liquid fractions: $p$ increases with $r$ up to approximately 0.5, then decreases to a minimum value ($p \lesssim 0.1$) at a maximum $r$, corresponding to $r_\infty = R_{b,\infty}/R_g$. At this point, the bubble size is constrained by the geometry of the pore space. Importantly, for a fixed value of $r$ within the range $r \lesssim 0.4$, $\Omega$ increases as $\phi_\ell$ decreases. This behavior suggests that gas exchange through the thin foam films—whose area increases as $\phi_\ell$ decreases—remains the dominant mechanism driving the observed coarsening.

In order to obtain a global view of the effect of liquid fraction on the coarsening rate, we plot $\Omega$ at $p \approx 0.13$ as a function of $\phi_\ell$ for all the investigated samples (see Fig. 10a). The resulting data points are highly scattered, highlighting the strong influence of confinement through the parameter $r$. The most striking result is that the effect of liquid fraction does not necessarily dominate over that of confinement. To illustrate this, consider the coarsening rates $\Omega$ measured for grains of size $R_g = 1.5$ mm at liquid fractions $\phi_\ell = 25\%$ and $\phi_\ell = 7\%$, which are approximately 16 and 89 $\mu m^2/s$, respectively. Now compare this to the case of $R_g = 1.5$ mm and $R_g = 0.4$ mm,
<figure><img src="image_11.png" /><figcaption>FIG. 10. (a) Measured coarsening rate $\Omega$ as a function of the liquid volume fraction $\phi_{\ell}$ for all studied grain sizes. Color code: $R_g = 2.5$ mm (black), 1.5 mm (blue), 1 mm (ochre), 0.75 mm (grey), 0.55 mm (green), and 0.4 mm (burgundy). (b) Measured coarsening rate $\Omega$ as a function of the effective liquid volume fraction $\phi_{\ell}^{\text{eff}}$ for the same grain sizes (symbols and color codes identical to panel a). The dotted line corresponds to Eq. 16, together with Equations 2 and 3 with parameters $k = 4.5$ and $\phi_{\ell}^{*} = 0.31$, fitted using the parameter group $\gamma DH_e V_m / h = 185 \ \mu\text{m}^2/\text{s}$, with $\phi_{\ell}^{\text{eff}}$ given by Eq. 6.</figcaption></figure>

both at $\phi_{\ell} = 25\%$, for which $\Omega$ is approximately 16 and 85 $\mu\text{m}^2/\text{s}$, respectively. In both scenarios, reducing either the liquid fraction or the grain size by roughly a factor of 4 leads to an increase in the coarsening rate by a factor of about 5. Noteworthy, for a bulk foam, referring to Eqs.4 and 2, one would expect a coarsening rate at $\phi_{\ell} = 7\%$ nearly 15 times larger than that at $\phi_{\ell} = 25\%$[15]. This highlights an expected weaker effect of liquid fraction on coarsening of confined foam compared to bulk foam.

Let us now compare these results with the effective liquid fraction model. The basic idea is to assume that, for a foam characterized by $R_b$ (and also by $p$) and $\phi_{\ell}$, the only difference between $\Omega_0$ and $\Omega(r)$ lies in the fact that the former is governed by the actual liquid fraction $\phi_{\ell}$, whereas the latter is controlled by an effective liquid fraction $\phi_{\ell}^{\text{eff}}$, all other parameters being equal. Under this assumption, and according to Eq. 4, one can deduce that $\Omega(r)$ is given by:

$$ \Omega(r) = \frac{\gamma DH_e V_m}{h} \times \xi(\phi_{\ell}^{\text{eff}}) \quad (16) $$

Eq. 16 can be tested by plotting, in Fig. 10b, the values of $\Omega$ (at $p \approx 0.13$) as a function of the effective liquid fraction $\phi_{\ell}^{\text{eff}}$, as defined by Eq. 6. This time, we obtain a highly consistent representation, in which all data points follow the same trend: $\Omega$ decreases monotonically with increasing $\phi_{\ell}^{\text{eff}}$. This global behavior is in excellent agreement with the model expressed by Eq. 16, which is also plotted in Fig. 10b by globally fitting the parameter group $\gamma DH_e V_m / h = 185 \ \mu\text{m}^2/\text{s}$, which fall within the expected range [15]. The resulting agreement is convincing, particularly for liquid fractions above 7%. For the lowest liquid fraction ($\phi_{\ell} = 7\%$), however, Eq. 16 slightly underestimates (by 25%) the experimental values (except for grains of size $R_g = 2.5$ mm). This suggests that our current analysis may overlook some relevant parameters of the problem.

## CONCLUSION

We performed liquid foam coarsening experiments in granular packings to investigate the effect of pore confinement on bubble growth. Our primary objective was to assess the ability of a recent model based on the effective liquid fraction [33] to account for new experimental data obtained over an unprecedentedly wide liquid fraction range, from $\phi_{\ell} = 7\%$ to 25%.

Our results reveal several important general features of the foam coarsening process occurring within the pore space of grain packings: (i) All bubble growth curves exhibit the same behavior: the coarsening rate increases with the bubble-to-grain size ratio $r$, reaches a maximum at $r \approx 0.5$, then decreases and ultimately vanishes at $r_{\infty} \approx 0.7-1$. (ii) Simultaneously, the polydispersity index increases to a maximum and then decreases back to very low values as $r$ approaches $r_{\infty}$. (iii) The liquid fraction has only a marginal effect on $r_{\infty}$, which is primarily determined by the geometry and size of the cavities available to the bubbles within the pore space. (iv) The influence of the liquid fraction on coarsening is significantly weaker than in bulk foams. For instance, starting with a sample of grains of size $R_g = 1.5$ mm at $\phi_{\ell} = 25\%$, the observed increase in coarsening rate is comparable whether $\phi_{\ell}$ is reduced to 7% or the grain size is decreased to 0.4 mm. In contrast, for bulk foams, such a change in liquid fraction is expected to lead to a coarsening rate variation nearly three times greater. (v) The effective liquid fraction model, which is based on the idea that the only difference between bulk and confined foams lies in the fact that the former is governed by the actual liquid fraction, whereas the latter is controlled by an effective liquid fraction, all other parameters being equal, provides a reasonable description of our coarsening rate data, further supporting its relevance for modeling liquid foams in confined porous environments. That said, some deviations in coarsening rates observed at the lowest liquid fractions and highest size ratios suggest that additional
effects—unaccounted for in the present model—may play
a secondary role. Moreover, the effective liquid fraction
is expected to depend on the bubble polydispersity in-
dex, as suggested by [33]. Unfortunately, in the course
of our study, we were not able to measure this quantity
with sufficient accuracy throughout the coarsening pro-
cess, particularly as the system explored varying degrees
of polydispersity. It would be valuable to seek improved
experimental conditions in order to establish a clearer re-
lationship between polydispersity and the effective liquid
fraction. Another limitation of our work is that mea-
surements were conducted exclusively at the sample sur-
face. Observing coarsening in such systems using ultra-
fast X-ray microtomography would provide particularly
valuable insights. (vi) We have highlighted the presence
of numerous small bubbles at the periphery of the liquid
bridges. Measurement of their shrinking rates indicates
that they resemble the so-called *roaming bubbles* recently
identified in wet foams [43]. The main difference here is
that these bubbles are hosted within the liquid bridges
themselves, enabling a high number of roaming bubbles
to accumulate, particularly at high liquid fractions. This
phenomenon may be either sought after or avoided, de-
pending on its impact on the targeted application.

Appendix A: Radial extension of a liquid bridge
formed by bubbles between a spherical grain and a
flat surface

We present here some additional results complement-
ing the work of Langlois et al. [35] concerning the ge-
ometry of liquid bridges formed between grains and a
flat surface. This work was carried out using the Sur-
face Evolver surface-minimization code, and all technical
details regarding these calculations can be found in ref-
erence [35]. Figure A1 shows the results for the radial
extension of these bridges as a function of the mean cur-
vature radius of the interfaces of the bubbles surround-
ing the bridge, for several configurations that differ in
the number of bubbles involved. The numerical data are
reasonably well described by the analytical relation given
by Equation 7.

ACKNOWLEDGMENTS

The authors acknowledge the ANR-FRANCE (French
National Research Agency) for its financial support of
the BONDINGFOAM Project No. ANR-23-CE51-0030

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