# Coarsening of bubble assemblies: From dry foams to dilute bubbly liquids

Nicolò Galvani, Olivier Pitois, Sylvie Cohen-Addad

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Nicolò Galvani, Olivier Pitois, Sylvie Cohen-Addad. Coarsening of bubble assemblies: From dry foams to dilute bubbly liquids. Journal of Colloid and Interface Science, 2025, 696, pp.137661. 10.1016/j.jcis.2025.137661. hal-05068078

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Regular Article

# Coarsening of bubble assemblies: From dry foams to dilute bubbly liquids

Nicolò Galvani<sup>a,b, ID</sup>, Olivier Pitois<sup>a, ID</sup>, Sylvie Cohen-Addad<sup>b,c, ID, *</sup>

<figure><img src="image_3.png" /></figure>

## GRAPHICAL ABSTRACT

<figure><img src="image_4.png" /></figure>

## ARTICLE INFO

**Keywords:**

Coarsening

Foam

Emulsion

Bubble dispersion

Ostwald ripening

Adhesion

## ABSTRACT

**Hypothesis:** Experimental data on bubble coarsening in a liquid are sparse, limited to relatively narrow ranges of liquid volume fractions, and do not encompass the full range relevant to the numerous applications of these systems. Our objective is to leverage recent data obtained aboard the International Space Station and to integrate these with dedicated measurements to produce a comprehensive coarsening rate diagram across the entire range of liquid fractions.

**Experiment:** The objective of new coarsening measurements is to eliminate gravity effects, which can make these systems highly inhomogeneous. We achieve this by using clinostat-type coarsening cells capable of maintaining the homogeneity of foams with low aqueous solution content, i.e. between 2% and 8%. For liquid fractions above 45%, we replace the aqueous solution with a concentrated emulsion of oil droplets in an aqueous solution, possessing a low yield stress to counteract gravitational effects without impacting coarsening kinetics.

**Findings:** For the first time, by combining our results with recent data obtained aboard the International Space Station using the same surfactant, we achieve coverage of the full range of liquid fractions characterizing liquid foam coarsening, from 2% to 38%, significantly expanding the range studied to date. This leads us to propose a model that describes coarsening over this entire range, accounting for both osmotic pressure effects and contact adhesion effects at higher liquid fractions. Similarly, for the first time, we have data covering the full range of liquid fractions characterizing the coarsening of bubbly liquids, from 45% to 96%, allowing us to compare existing models and identify the most robust ones. Finally, we can map the evolution of coarsening rates over a broad

<footnote><sup>a</sup> Laboratoire Navier, Université Gustave Eiffel, ENPC, Institut Polytechnique de Paris, CNRS - UMR 8205, Champs-sur-Marne, 77420, France</footnote>

<footnote><sup>b</sup> Sorbonne Université, CNRS, Institut des Nanosciences de Paris, CNRS, UMR 7588, INSP, Paris, F-75005, France</footnote>

<footnote><sup>c</sup> Université Gustave Eiffel, Champs-sur-Marne, 77420, France</footnote>

<footnote>* Corresponding author at: Sorbonne Université, CNRS, Institut des Nanosciences de Paris, CNRS, UMR 7588, INSP, Paris, F-75005, France.
E-mail address: sylvie.cohen-addad@insp.upmc.fr (S. Cohen-Addad).</footnote>
range, from 2% to 96%, thereby highlighting the overall impact of dispersed phase fraction on coarsening. This provides a valuable new tool to tackle coarsening effects in diphasic systems.

# 1. Introduction

Bubble coarsening, commonly known as *Ostwald ripening* in bubbly liquids or simply *coarsening* in foams, is a process that occurs in systems made of two immiscible phases undergoing phase separation, such as gas bubbles in a liquid. In this process, due to Laplace pressure differences, smaller bubbles gradually dissolve, while their gas diffuses towards larger bubbles, causing them to grow. This phenomenon is driven by the reduction in total system surface energy, as larger bubbles have a lower surface-to-volume ratio and therefore lower surface energy density compared to smaller bubbles.

This process is crucial in various applications and situations. Well-known examples include food products, where air is incorporated as small bubbles to provide specific textures, such as brittleness in confectionery and lightness in whipped cream. The gas dispersion may degrade in several ways, including Ostwald ripening. The microstructure of the gas phase in bread, which significantly influences its textural and sensory attributes, is also controlled by coarsening [1,2]. In other contexts, we can mention the formation of gas bubbles in blood due to decompression effects. The growth of these bubbles through coarsening appears to be related to the risk of decompression sickness (DCS) during and after diving [3]. In the production of foamed materials, like cement foam, it is necessary to counteract coarsening which alters the material's microstructure and properties [4]. Magmatic degassing also involves bubble coarsening, where bubble growth continues until they become more buoyant and mobile, potentially contributing to volcano-related seismicity [5]. Bubble coarsening occurs in soils where gas has been injected and is crucial for understanding the distribution of non-aqueous phase liquid (NAPL) pollutants in soil and aquifer systems [6], enhanced oil recovery (EOR) [7,8], and CO₂ storage in underground strata of porous rock [9,10]. In these contexts, bubbles are confined in pores and they coarsen until a stable configuration is reached [11–13].

As we have just presented, the situations in which bubbles are led to coarsen are varied and numerous. Among them, there are those where the liquid volume fraction, denoted $\phi$, is relatively low, which refer to foams, in contrast to others where the liquid volume fraction is large which correspond to bubbly liquids. The evolution of their bubble size distributions and the growth law of the average bubble size $\langle R \rangle$ upon coarsening has been the subject of dedicated studies (cf. reviews [14,15]). The diffusional gas transfer driven by surface energy leads to different coarsening regimes, depending on $\phi$. When $\phi \to 1$ (infinitely dilute limit), the diffusional gas concentration profile around each bubble extends at infinity, leading to the average bubble radius growth $\langle R \rangle \propto t^{1/3}$ on long coarsening times. In the opposite dry foam limit $\phi \to 0$, the diffusional profile sets through the films of constant thickness between neighboring bubbles, yielding to the long time growth $\langle R \rangle \propto t^{1/2}$. Conventionally, the separation between a bubbly liquid and a foam is associated to the liquid volume fraction at the onset of contacts between bubbles, the so-called jamming transition, which should also coincide with the transition between coarsening regimes. For foam structures solely driven by capillary interactions between bubbles, the transition corresponds to the random close packing fraction, $\phi_{rcp}$, and depends on structure and polydispersity [16]. However, it is not certain that the change in the coarsening regime occurs precisely at $\phi_{rcp}$. Indeed, it has been shown recently that the regime change could be observed at $\phi^*$ significantly larger than $\phi_{rcp}$ [17]. Moreover, the range of $\phi$ values for which the regime change is observed is excluded from coarsening models, which are better suited either for foams or for bubbly liquids.

In this paper, we aim to provide, for the first time, an overview of bubble coarsening across the entire range of volume fractions, from dry

foams with $\phi$ of a few percent up to dilute bubbly liquids with $\phi$ close to one. We present new experimental investigations of the coarsening of both dry foams and bubbly liquids, where bubble radius distributions and average bubble growth laws are measured as a function of ageing time. A clinostat cell is used to counteract gravity drainage. We combine our data with those previously collected in microgravity conditions with the same surfactant for an intermediate range of liquid fraction [18, 17]. This extended dataset allows us to determine the foam coarsening rate from the dry limit up to the jamming transition $\phi^*$. We propose a model inspired by granular physics which takes into account bubble adhesive forces besides repulsive capillary forces in order to predict the delayed decrease of the coarsening rate at the approach of $\phi_{rcp}$. Finally, we compare the variation of the coarsening rate with $\phi$ determined in the bubbly side, where interparticle diffusional interactions matters, to existing theoretical and numerical predictions.

# 2. Materials and methods

## 2.1. Sample preparation

Two foaming solutions are studied: i) a solution of TTAB (TetradecylTrimethylAmmonium Bromide, reference 8.40006, Sigma-Aldrich, purity 99%) dissolved in water at concentration $C = 5$ g/L, which is identical to the one studied in [17,18]; ii) an O/W emulsion of silicon oil droplets (Rhodorsil Si 47V350, Bluestar Silicones) dispersed in a water-glycerol solution with glycerol (bidistilled 99.5%, reference 24388.295 from VWR Chemicals) at concentration $G \approx 52$ wt%, and stabilized by TTAB surfactant at concentration $C = 18$ g/L. The large surfactant concentrations prevent any droplet or bubble coalescence. All the chemicals are used as received. We use ultrapure Milli-Q water. The surface tension of the TTAB solution and that of the TTAB/glycerol solution at the air-liquid interface, measured at $T = 21^\circ C$ are equal to $37 \text{ mNm}^{-1}$ and $33 \text{ mNm}^{-1}$ respectively. At this temperature the viscosity of the silicon oil is $\eta = 348$ cP.

A crude emulsion with an oil volume fraction $\varphi = 87.0\%$ is first prepared by hand, mixing the oil and a water-glycerol solution with $G = 52$ wt% and an initial TTAB concentration $C = 30$ g/L. The crude emulsion is refined using a Couette mixer to obtain a nearly monodisperse emulsion with average droplet radius $r = 1.6 \text{ µm}$. The resulting fine emulsion is diluted with water-glycerol solution to get the desired $C = 18$ g/L and oil fraction $\varphi = 72.5\%$ of the final emulsion employed for the coarsening study. The glycerol concentration in the emulsion is finely adjusted to match the refractive indices of the two phases. The emulsion does not ripen over the duration of the experiments.

The yield stress of the emulsion is determined using a shear strain amplitude sweep oscillatory measurement performed with a control-strain rheometer (Anton Paar MCR 501) at a frequency of 1 Hz, using a rough cone-plate geometry. After a preshear, a ramp of strain amplitude $\epsilon$ increasing from $10^{-4}$ to 10 is applied, and the complex shear modulus $G^*(\epsilon)$ is measured as a function of $\epsilon$. The yield stress $\tau_y$ is determined at the crossover between the linear regime observed at low strains and the elastoplastic regime above the yield strain $\epsilon_y$ as in [19]: $\tau_y = |G^*(\epsilon_y)|\epsilon_y$. Repeating the measurements for two cone angles (either 2° or 4°) and samples, we find: $\tau_y = (16.5 \pm 0.1)$ Pa at $T = 21^\circ C$. By measuring the variation of $\tau_y$ with $\varphi$, we deduce the surface tension at the droplet oil/water interface from the relationship $\tau_y \approx 0.5(\gamma_{o/w}/r)(\varphi - \varphi_c)^2$ knowing the average droplet size $r$ and $\varphi_c = 0.637$ [20]. We find $\gamma_{o/w} = (4.5 \pm 0.5) \text{ mNm}^{-1}$.

Foams are prepared using the double-syringe method [21]. The foaming liquid is either the TTAB/water solution for $\phi \le 8\%$ or the
O/W emulsion for $\phi \ge 45\%$. A 10 mL syringe (Fisherbrand™ Plastic PP Syringe, Luer Lock) is filled with the foaming solution and assembled using a luer-luer connector to a second identical syringe filled with air, in proportions to obtain the desired foam liquid volume fraction $\phi$, with an uncertainty ~ ±1% for large liquid fractions ($\phi \ge 40\%$). For dry foams, we weight the liquid injected in the syringe, to reduce the uncertainty to ~ ±0.1%. Foam is produced by moving back and forth the two pistons in a controlled manner so that the total volume remains constant. This operation lasts about 2 minutes until the gas is completely incorporated in the liquid phase and no air pocket larger than the average bubble is visible.

## 2.2. Clinostat cell

The foam sample is placed in a 10 mL cylindrical plastic cell (inside radius $R_{cell}$ = 15.0 mm, thickness $L$ = 10.8 mm), closed by two flat transparent windows, and hermetically sealed with o-rings. For an average bubble radius of 200 µm, the cell contains around 25 layers of bubbles, thus we consider the studied samples as 3D.

In order to counteract gravity drainage that may induce spatial heterogeneity of the liquid fraction $\phi$, the cell is placed in a slowly rotating clinostat: the cell is axially mounted in a cup attached to the shaft of a rotating step motor (Midi-Ingenierie MAC23), equipped with a planetary gearbox (Neugart PLE60-164) and aligned along the horizontal direction. The period of rotation $T_r$ is chosen equal either to 4 s for the TTAB/water foams or 10 s for the bubbly emulsions.

An external LED source (Schott KL2500 LED) is positioned in front of a lateral aperture ($\emptyset$ = 10 mm) pierced in the cup, shining light to a 45° mirror located inside, in order to transmit light though the sample at each rotation. A camera (IDS-3080CP-M-GL) is placed in front of the cell, and equipped with objectives of different magnifications in the range 2.4 – 9.5 µm/pixel to maximize the resolution while keeping a minimum of two hundreds bubbles in the field of view at the center of the cell. Images of the sample surface are recorded at instants synchronized with the instants when the aperture is facing the source of light, at logarithmically spaced time intervals to monitor the intrinsic kinetics of coarsening.

To assess the homogeneity of the liquid fraction in our samples while rotating in the clinostat, we measured the surface liquid fraction by recording images of the surface through a 45° prism, as in [22]. Our observations for the wettest TTAB/water foam (which is most prone to undergo liquid drainage) show that the surface fraction remains constant up to the end of the coarsening experiments when the bubble Sauter radius is $R_{32} \approx 600$ µm. Indeed the period of rotation $T_r = 4$ s is much smaller than the characteristic drainage time $t_d$ necessary to establish a liquid fraction gradient over a length scale $h \approx R_{32}$, that can be evaluated as follows. The liquid flow velocity through a foam of permeability $k(\phi)$ under the hydrostatic pressure gradient $\rho g$ is given by [23]: $u = k(\phi)\rho g/\eta$, where $\rho$ is the liquid density, $\eta$ the liquid viscosity, and $g$ the gravitational acceleration. In a clinostat, the gravity direction in the reference frame of the cell varies continuously upon a period of rotation $T_r$. In this situation, the liquid may flow back and forth during each rotation, and induces a liquid fraction gradient if $t_d = R_{32}/u \ll T_r/2$. This sets a criterion for the period of rotation to avoid drainage: $T_r \ll t_d = 2R_{32}\eta/\rho g k(\phi)$. For the TTAB/water foam with $\phi = 0.08$, $k(0.08) = 8 \times 10^{-5} R_{32}^2$ [23] and $R_{32} \approx 600$ µm, we should have $T_r \ll t_d \approx 30$ s, which is indeed the case with our choice of $T_r = 4$ s. Note that the maximum centrifugal acceleration evaluated at the foam edge, $R_c(2\pi/T_r)^2$, is two orders of magnitude smaller than the gravity acceleration $g$, thus negligible.

In bubbly liquids, rotation is not sufficient to avoid drainage, which motivated the use of a foaming liquid made of a concentrated O/W emulsion with a small yield stress $\tau_y$. The motion of an isolated bubble of radius $R$ is impeded if the yield stress overcomes the gravitational stress $\tau_g \ge 0.13\delta\rho g R$, where $\delta\rho$ is the density difference between the continuous and disperse phases [24]. Thus, for the yield stress value used here,

a bubble with $R \approx 1$ cm, which is one order of magnitude larger than the maximum average radius studied in the experiments, should not rise under gravity. However, since there is no prediction of bubble rise in a concentrated bubbly emulsion, we let the samples coarsen in the clinostat with a slow rotation (period $T_r = 10$ s). Image analysis of their surface confirms the absence of radial segregation up to the end of the coarsening experiments.

For experiments lasting more than one day, the sample cell is removed from the clinostat between image acquisitions, and kept on a tube roller (Cole Parmer RS-200) with the same rotation speed as that of the clinostat. This procedure makes possible the parallelization of multiple experiments over several months. Over this timescale, the liquid phase in all of the bubbly liquids remains transparent which indicates the absence of water evaporation since the water/glycerol index matching is very sensitive to the glycerol concentration.

All experiments are conducted at a constant temperature equal to 21 °C.

## 2.3. Image analysis

Images of the foam surface are first treated with a Python script, which implements algorithms from open-CV library [25] to apply denoising, contrast enhancement and edge enhancement to the images. Then an automatic image segmentation model based on YOLOv8.1 [26] is used to recognize individual bubbles and inscribe them in a 50-sided polygon. The model is trained on a dataset of images that we analysed by manual ellipse-fitting, as in [21]. The dataset contains images of more than $10^5$ bubbles analysed in images with different liquid fractions and lightning conditions. Previous work have shown that the bubble size distributions of coarsening foam are hierarchical, and span bubble radii from about $0.1 R_{10}$ to $3 R_{10}$ [18]. Thus, to avoid missing the recognition of the smallest bubbles, we combine YOLO with the Slicing Aided Hyper Inference (SAHI) algorithm [27]. The procedure is detailed in [28]. From the area $A$ of polygons, we measure the bubble radii as $R = \sqrt{A/\pi}$, with an uncertainty on the measurement on the order of the pixel size. By comparing the bubble size distribution observed on the surface of a 3D wet foam sample with that of a monolayer of bubbles obtained by diluting the same foam sample, we previously demonstrated that the surface distribution accurately represents the size distribution of the bulk foam [21].

# 3. Coarsening theories

## 3.1. Bubbly liquids $\phi > \phi^*$

Fundamental work on Ostwald ripening started with the seminal works of Lifshitz & Slyozov [33], and Wagner [34] (LSW). Their mean field model predicts that diffusion-controlled coarsening in systems with vanishing volume fraction of the dispersed phase ($\phi \to 1$) reaches a statistically self-similar regime where the mean particle radius, i.e. the first moment of the radius distribution denoted $R_{10}$, increases as a function of age $t$ as:

$$R_{10}^3(t) = R_{10}^3(t_0) + K_0 (t - t_0) \quad (1)$$

The coarsening rate $K_0$ depends on the material's physicochemical properties:

$$K_0 = \frac{8}{9} \gamma H_e D v_m \quad (2)$$

where $\gamma$ is the surface tension, $H_e$ is the Henry coefficient of gas solubility in the liquid phase, $D$ is the gas diffusivity in the liquid phase, and $v_m$ its molar volume.

In the case of a finite volume fraction, particles interact via the long-range diffusional field of concentration of the gas dissolved in their surroundings. The concentration gradient near a given bubble can be significantly altered by the presence of nearby bubbles. Qualitatively,
<figure><img src="image_5.png" /><figcaption>Fig. 1. (a) Enhancement factor g predicted from the mean field theories and simulations as a function of the gas volume fraction (1 − φ): (MLSW) [29], (TM3) [30], (PS) [31] and (VG) [32]. (b) Schematic representation of a dispersion of gas bubbles in the semi-dilute regime. The dissolved gas concentration c is smaller (resp. bigger) close to the bigger (resp. smaller) bubbles than the average concentration $\bar{c}$, and its gradient extends around each bubble over the diffusional screening length $\lambda$. In the TM3 model, $\lambda$ is assumed to be equal for all bubbles and set by the average bubble radius and the gas volume fraction.</figcaption></figure>

the coarsening rate is expected to increase as $\phi$ decreases because the average interparticle distance becomes smaller, which amplifies the concentration gradients around the particles. Numerous theoretical and numerical studies have been proposed to account for these particle interactions and predict the growth law of the mean particle size, described by a cubic law of the form (see review by Baldan [35]):

$$R_{10}^3(t) = R_{10}^3(t_0) + \Omega_c (t - t_0) \quad (3)$$

The statistically averaged growth rate $\Omega_c$ is a function of the liquid volume fraction $\phi$:

$$\Omega_c(\phi) = K_0 g(\phi) \quad (4)$$

where the coarsening enhancement factor $g(\phi)$ describes the relative variation of the coarsening rate with $\phi$. By construction, $g(1) = 1$.

Among the numerous models available, we focus here on those that: (i) assume isotropic phases and a uniform composition of the dispersed phase, which is relevant for bubble dispersions (in contrast to alloys or solid precipitates); and (ii) predict an increase in $g(\phi)$ by at least a factor of approximately 5 as $\phi$ decreases from 1 to $\phi_{rcp}$, consistent with previous measurements [17]. These mean-field models are based on the same framework as the LSW theory, with an added assumption about the diffusional screening length, $\lambda$, around each bubble, where the concentration of dissolved gas matches the average value. The parameter $\lambda$ must increase with $\phi$, depending on the spatial distribution of the bubbles. In the MLSW model [29], $\lambda$ is proportional to the bubble radius, with bubbles randomly distributed within the continuous phase. In the TM3 model [30], $\lambda$ is proportional to the average bubble radius, with bubbles homogeneously dispersed in the continuous phase, as illustrated in Fig. 1b. In the PS model [31], the author introduces an ansatz, assuming $\lambda$ follows a polynomial form, with coefficients determined by the ratio of the maximum bubble size, $R_{max}$, to the critical radius for growth, $R_c$. Finally, in the VG model [32], the mean-field approximation for the diffusion process is abandoned in favour of computer simulation techniques, which provide approximated solutions to the multiparticle-diffusion problem. The predicted variations of the enhancement factor g are represented as a function of the gas volume fraction in Fig. 1a.

### 3.2. Liquid foams $\phi < \phi^*$

In foams, where bubbles are densely packed and form thin films at their contacts, coarsening kinetics are driven by Laplace pressure differences between neighboring bubbles, which induce gas exchange through the films. Assuming that, as in bubbly dispersions, the bubble assembly reaches a scaling state in which the bubble radius distribution, normalized by its average value, becomes independent of coarsening time t,

both Mullins and Lemlich predicted that the average bubble radius $\langle R \rangle$ should grow according to a parabolic law [36,37]:

$$\langle R(t) \rangle^2 = \langle R(t_0) \rangle^2 + \Omega_p (t - t_0) \quad (5)$$

where the statistically averaged growth rate $\Omega_p$ depends on the physicochemical properties of the foam and its liquid volume fraction. For finite gas volume fraction ($\phi \gtrsim 1\%$), the coarsening rate $\Omega_p$ can be decomposed in 3 factors [17]:

$$\Omega_p = C \Omega_0 f(\phi) \quad (6)$$

The constant C is a geometrical factor depending on the probed moment of the radius distribution. According to Lemlich's prediction, $\langle R \rangle = R_{21}$, i.e. the ratio of the second to the first moment of the radius distribution, which by construction corresponds to C = 1. The coarsening constant $\Omega_0$ is given by:

$$\Omega_0 = \frac{\gamma D H_e v_m}{h} \quad (7)$$

where h is the effective thickness for the gas transfer through the film. The function $f(\phi)$ represents the fraction of the bubble surface area occupied by the films. At the jamming transition when bubble contact vanishes, $f(\phi^*) = 0$. As $\phi$ decreases below $\phi^*$, bubbles are more and more compressed against each other and f increases accordingly, as well as the foam osmotic pressure $\Pi(\phi)$. If bubble interactions are repulsive and arise solely from interface deformations at their contacts, the jamming transition coincides with the random close packing fraction ($\phi^* = \phi_{rcp}$), and the osmotic pressure of a polydisperse disordered bubble packing is given by the semi empirical relation [38,39]:

$$\Pi = k \frac{\gamma}{R_{32}} \frac{(\phi - \phi_{rcp})^2}{\sqrt{\phi}} \quad (8)$$

where $R_{32}$ is the Sauter mean radius defined as the ratio of the third to the second moment of the radius distribution. Both the constant k and $\phi_{rcp}$ depend on the bubble polydispersity. In a random monodisperse packing, $\phi_{rcp} = 0.36$, $k = 3.2$ [38], whereas in a coarsened foam, $\phi_{rcp} = 0.31$ and $k = 4.5$ [17]. The average film area of a bubble, normalized by the surface area of an effective sphere with the same volume has been predicted as a function of the normalized osmotic pressure, $\tilde{\Pi} \equiv \Pi R_{32}/\gamma$, in the range of wet foams $\phi \lesssim \phi_{rcp}$ [40]:

$$f(\phi) = \frac{\tilde{\Pi}}{\tilde{\Pi} + 2(1 - \phi)} \quad (9)$$
<figure><img src="image_6.png" /><figcaption>Fig. 2. Pictures of coarsened foams and bubbly dispersions with liquid fractions $\phi$ ranging from the dry to the dilute limits, after reaching the scaling state. The samples belong to three groups of experiments: ($\phi$ ≤ 8%) dry foams studied on ground (cf. Table 1), (15% ≤ $\phi$ ≤ 50%) wet foams and bubbly liquids studied in microgravity [18], ($\phi$ ≥ 70%) bubbly emulsions studied on ground (cf. Table 1). The scale bar corresponds to 500 µm for all of the images.</figcaption></figure>

In [40], a factor $((1 - \phi)/(1 - \phi_{rcp}))^{1/3}$ was proposed to extend Eq. (9) to smaller liquid fractions. Since the latter is not appropriate to describe dry foams, we choose to stick to Eq. (9).

## 4. Results

We measure the evolution of bubble sizes driven by coarsening in two distinct regimes: dry foams with liquid fractions ($2.5 \le \phi \le 8.0\%$) and bubbly dispersions ($45.0 \le \phi \le 95.5\%$). To bridge the gap between these regimes, we include data for wet foams ($15.0 \le \phi \le 40.0\%$) with the same composition as our dry foams, previously measured under microgravity conditions [17,18]. Fig. 2 shows images of foams and bubble dispersions in the considered range of liquid fraction. Using the image processing and analysis techniques described in Section 2.3, we determine the distribution of bubble radii at successive ages $t$, where $t$ represents the time elapsed since the end of the foaming process, continuing until the average bubble radius $R_{10}$ approaches a ratio of $R_{cell}/R_{10} \approx 50$.

### 4.1. Dry foams

In dry foams, for each liquid fraction, we observe that after a transient period, the distribution of bubble radii, normalized by their average value, becomes independent of the foam age. This marks the onset of a scaling state where bubble growth is statistically self-similar. As shown in Fig. 3(a), the probability density function (PDF) of the radii measured in the scaling state exhibits a pronounced peak corresponding to small bubbles. These bubbles, previously identified as those roaming through the interstices between larger jammed bubbles, reflect the hierarchical structure characteristic of coarsening foams [18].

We show the evolution of the Sauter mean radius $R_{32}$ as a function of foam age in Fig. 3(b). Since, in the scaling state, any $n^{th}$-moment of the radius distribution $R_n$ scales as $R_{10}^n$, we write Eq. (5) as:

$$R_{32}^2(t) = R_{32}^2(t_0) + \Omega_p (t - t_0) \qquad (10)$$

We fit this equation to the data for each liquid fraction, setting $R_{32}^2(t_0)$ at the first instant $t_0$ of measurement. As can be seen in Fig. 3(b), for each liquid fraction, the increase of $R_{32}$ with time follows such a parabolic law. The values of the fitted coarsening rates $\Omega_p$ are reported in Table 1. From the scaling state distributions, we also determine, for each liquid fraction, the moment $R_{21}$ of the distribution in order to evaluate the ratio $R_{32}/R_{21}$. Their corresponding values are given Table 1. They yield the geometrical factor $C = (R_{32}/R_{21})^2$ introduced in Eq. (6). We show in Fig. 4a the evolution of $\Omega_p/C$ as a function of the liquid fraction.

### 4.2. Bubbly dispersions

We measure the evolution of the bubble size distributions in coarsening bubbly dispersions. As shown by the cumulated density functions (CDF) in Fig. A.7, a scaling state is reached for ageing times $t \gtrsim 10^4$ s, for each studied liquid fraction. In Fig. 3(c), we present the distributions measured in the scaling state. The shape of the distribution evolves with the liquid fraction, narrowing and reducing its maximum relative radius, $R_{max}/R_{10}$, as $\phi \to 1$. This trend is qualitatively consistent with predictions from mean-field models and simulations [29–32]. The measured distributions reveal a population of bubbles significantly larger than the average, with a positive skewness resembling a lognormal function. Notably, the relative maximum radius, $R_{max}/R_{10} \approx 2$, is substantially larger than the mean-field value of 1.5 predicted in the dilute LSW limit. For $45\% \le \phi \le 58\%$, the distributions broaden and closely resemble those observed in foams in the gel-like regime, $\phi_{rcp} \le \phi \le \phi^*$, where, in contrast to our present samples, the growth laws are parabolic [18,17]. The possible mechanisms underlying these observations will be discussed in the next section.

Fig. 3(d) shows the evolution of the mean radius $R_{10}$ measured as a function of coarsening time. For each liquid fraction, we fit Eq. (3) to the data with $t_0$ as the time of the first measurement, and find a good agreement between the fits and the observed growth laws. The fitted coarsening rates $\Omega_c$ are reported in Table 1 and plotted as a function of the gas volume fraction $1 - \phi$ in Fig. 5.

For all of the liquid fractions, the average bubble size reaches a maximum value $R_{10} \approx 200$ µm, which corresponds to a capillary pressure $\Pi_c = 2\gamma/R_{10} \approx 330$ Pa. This value is much larger than the emulsion yield stress $\tau_y = 16.5$ Pa. The observation of a growth regime, characterized by the exponent $1/3$, indicates that the yield stress of the emulsion has no effect on this growth. We conclude that, in the investigated age range, the coarsening of a bubble dispersion in an emulsion with such a small yield stress exhibits a growth law similar to that the dispersion would have in a simple surfactant solution.

## 5. Discussion

### 5.1. Coarsening of adhesive foams for $\phi < \phi^*$

To gain more insight in the dependency of the coarsening rate $\Omega_p$ with the liquid fraction (Eq. (6)), we combine our present measurements with those previously made on the same system, studied in microgravity [17,18], which results are briefly recalled here: Parabolic growth laws were observed in the range ($15\% \le \phi \le \phi^* \approx 39\%$). The random close packing fraction corresponding to the bubble size distribution
<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Coarsening of dry foams: (a) Bubble size distributions in the scaling state; (b) Evolution of the Sauter mean radius $R_{32}$ as a function of foam age. The continuous lines represent growth laws obtained by fitting a parabolic law (Eq. (10)) to the data. Coarsening of bubbly dispersions: (c) Bubble size distributions in the scaling state; (d) Evolution of the mean radius $R_{10}$ as a function of age. The continuous lines represent growth laws obtained by fitting a cubic law (Eq. (3)) to the data. All samples coarsen in the clinostat cell as described in Section 2.2, with liquid fractions as indicated. Each curve represents one experiment. The fitted coarsening rates are given in Table 1.</figcaption></figure>

<caption>**Table 1**</caption>

Coarsening rates $\Omega_p$ or $\Omega_c$ deduced by fitting the growth laws, respectively Eq. (10) or Eq. (3), to the data. Foams are prepared with a TTAB aqueous solution. Bubbly dispersions are made of an O/W silicone oil droplets dispersed in an index matched TTAB aqueous/glycerol mixture with oil fraction $\varphi = 0.725$ and yield stress $\tau_v = 16.5$ Pa (cf. Section 2.1). The absolute uncertainty on liquid fractions are about 0.1% for foams, and about 1% for bubbly liquids. The Sauter radius $R_{32}$ and the ratio of second to first moment of the bubble radius $R_{31}$ are computed from the measured bubble size distributions in the scaling state. The relative uncertainty on this ratio is estimated to be around 5%. Values of $\Omega_p$ are averaged over 3 experiments and the uncertainty represents the squared sum of the standard deviations. The uncertainty on $\Omega_c$ for $\varphi = 45\%$ and $\varphi = 96\%$ is deduced from the average over two measurements, while for other $\varphi$ it represents the standard deviation for a single measurement of $\Omega_c$.

<table><thead><tr><th></th><th colspan="3">Foams</th><th colspan="9">Bubbly emulsions</th></tr></thead><tbody><tr><td>φ (%)</td><td>2.5</td><td>3.0</td><td>5.0</td><td>8.0</td><td>45</td><td>58</td><td>65</td><td>70</td><td>75</td><td>80</td><td>90</td><td>96</td></tr><tr><td>Ω<sub>p</sub> (μm<sup>2</sup>s<sup>-1</sup>)</td><td>157±4</td><td>138±4</td><td>114±4</td><td>102±1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>R<sub>32</sub>/R<sub>21</sub></td><td>1.27</td><td>1.24</td><td>1.23</td><td>1.28</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Ω<sub>c</sub> (μm<sup>3</sup>s<sup>-1</sup>)</td><td></td><td></td><td></td><td></td><td>97±4</td><td>60±2</td><td>49±2</td><td>51±2</td><td>37±3</td><td>34±3</td><td>25±1</td><td>22±2</td></tr></tbody></table>

in the steady-state coarsening regime was evaluated: $\phi_{rcp} \approx 31\%$. The coarsening rate $\Omega_p$ was determined based on the parabolic growth laws of $R_{32}$, as described by Eq. (10). The evolution of the scaled parameter $\Omega_p/C$, where the factor $C = (R_{32}/R_{21})^2$ is derived from the scaling state bubble size distributions, was analyzed as a function of the liquid fraction. The film fraction area (Eq. (9)), which is expected to be valid for $\phi \lesssim \phi_{rcp} = 31\%$, was fitted to the experimental data, as illustrated in Fig. 4a. The fitting yielded $\Omega_0 = (214 \pm 21) \ \mu m^2 s^{-1}$. Since our data extends into the dry limit, we fit the scaled coarsening rate $\Omega_p/C$ predicted by Eq. (9) with $\phi_{rcp} = 31\%$ to the entire dataset, encompassing both dry and wet foams. The fit yields $\Omega_0 = (190 \pm 10) \ \mu m^2 s^{-1}$. As shown in Fig. 4a, this prediction aligns very well with the data for dry

foams. However, it fails to capture the observed behavior of $\Omega_p/C$ as $\phi$ approaches $\phi_{rcp}$, a limitation previously highlighted in [17].

The delay observed in the transition between the foam and bubbly liquid coarsening regimes, i.e. $\phi_{rcp} < \phi^*$ was previously attributed to the attractive interactions between bubbles, which cause them to adhere to one another through their contact films. This adhesion effect is quantified by the contact angle $\theta$, formed when two bubbles come into contact, as illustrated in Fig. 4b. In a foam, the contact angle is observed at the junction between a film and its neighboring liquid channels, known as Plateau borders in dry foams [16]. For our foaming liquid, the contact angle value was measured to be equal to $\theta = (4 \pm 1)°$ [17].
<figure><img src="image_8.png" /><figcaption>Fig. 4. (a) Foam coarsening rate $\Omega_p$ scaled by the geometrical factor C (Eq. (6)) as a function of the liquid fraction: (●) present data obtained with a clinostated cell using the values of $C = (R_{32}/R_{21})^2$ reported in Table 1, (■) previous data obtained in microgravity with $C = 1.69$ for $\phi \le 28\%$ and $C = 1.44$ for $\phi \ge 31\%$ (data from [17]). For the present data, the average over three repetitions is shown, with vertical bars highlighting the minimum and maximum values. The lines represent fits to the data with $\phi_{rcp} = 31\%$: (—) Eq. (9) in the range $15.0\% \le \phi \le \phi_{rcp}$ with $\Omega_0 = 214 \mu m^2 s^{-1}$ (from [17]); (...) Eq. (9) in the range $2.5\% \le \phi \le \phi_{rcp}$ with fitted $\Omega_0 = 190 \mu m^2 s^{-1}$; (■) Eq. (14) in the range $2.5 \le \phi \le \phi^* \approx 39\%$ with fitted parameters $\Omega_0 = 173 \mu m^2 s^{-1}$, $\theta = 5.8^\circ$, and $a = 40$ (see text). (Inset) Average coordination number $z(\phi)$ given by Eq. (15) with $\phi_{rcp} = 31\%$ and $a = 40$. The green area represents a $\pm 1\sigma$ variation of $a$. (b) Scheme of two bubbles sharing an adhesive contact film, with contact angle $\theta$. The area of the film is set by the compressive force, $F_\Pi$, and the adhesive force, $F_\theta$. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)</figcaption></figure>

The coarsening rates measured near $\phi_{rcp}$ are substantially higher than anticipated. This behavior is attributed to adhesive forces, which significantly expand the film area as the liquid fraction approaches the random close packing value — where, in the absence of adhesion, the film area would be zero. Conversely, adhesive forces are expected to have a negligible impact on the dry side. In the following, we propose a model to predict the fraction of bubble surface area occupied by the films taking into account adhesive forces, characterized by the contact angle $\theta$.

In foams with a finite film contact angle, bubbles experience adhesive contact forces, denoted $F_\theta$, which press them against one another, in addition to the effect of osmotic pressure that we find in purely repulsive foams. It must be acknowledged that, in terms of modeling the effect of adhesive forces on the area of contact films between bubbles, we are faced with an unexplored field. For now, we draw inspiration from existing results in the domain of unsaturated granular media, such as those found in [41,42]. In the case of a granular material where liquid menisci exert tensile forces at each contact point, the mechanical response of the material is determined by the effective stress, which accounts for the externally applied stress augmented by an internal stress associated with the forces exerted by the menisci on the grains. Applying this approach to our case amounts to introducing an effective stress, $\sigma^*(\phi, \theta)$, arising from two contributions: one of osmotic origin, as in a foam with $\theta = 0$, denoted $\Pi(\phi)$ and introduced in Eq. (8), and a second one of adhesive origin, denoted $\sigma_\theta(\phi)$:

$$ \sigma^*(\phi, \theta) \approx \Pi(\phi) + \sigma_\theta(\phi) \qquad (11) $$

We are aware of the differences between adhesive bubbles and adhesive solid grains, but adhesive forces are expected to have a significant impact when $\sigma_\theta(\phi) \gg \Pi(\phi)$, i.e., when the bubbles are reasonably spherical, and the contact areas are relatively small, making them, in this case, closer to solid grains. Conversely, when $\sigma_\theta(\phi) \ll \Pi(\phi)$, i.e., at sufficiently low liquid fractions, where bubbles are primarily flattened by osmotic pressure, modeling the adhesive forces and their effect on the contact areas becomes less critical. Thus, to account for these adhesive forces across the entire range of liquid fractions, we evaluate the forces $F_\theta$ near $\phi_{rcp}$ and then consider them as constant.

Near $\phi_{rcp}$, the pressure $\sigma_\theta$ can be related to $F_\theta$, like the osmotic pressure $\Pi$ to compressive forces, as in [40]. Assuming that the pressure environment around bubbles in a polydisperse foam is homogeneous [43], we take the Sauter radius as the effective bubble radius, and write:

$$ \sigma_{\theta}(\phi) = \frac{1 - \phi}{4\pi R_{32}^2} z(\phi) F_{\theta} \qquad (12) $$

where $z(\phi)$ is the average number of contacts per bubble. The adhesion force between two bubbles in a doublet is independent of the liquid fraction and is assumed to scale as [44,45]: $F_\theta = 2\pi R_{32}\gamma \sin^2\theta$. By replacing it in Eq. (12) and normalizing the pressures by $\gamma/R_{32}$, Eq. (11) yields the dimensionless effective pressure:

$$ \bar{\sigma}^*(\phi, \theta) \approx \bar{\Pi}(\phi) + \frac{1}{2}(1 - \phi) z(\phi) \sin^2\theta \qquad (13) $$

As a consequence of the adhesion, for a given liquid fraction, the average area of the contact film increases with respect to its value for $\theta = 0$. This has been modeled in 2D adhesive foams [43], and observed in simulated 3D adhesive foams [46]. We deduce the average area fraction of a bubble surface occupied by the films, $f(\phi, \theta)$, from the effective pressure using Eq. (9):

$$ f(\phi, \theta) \approx \frac{\bar{\sigma}^*(\phi, \theta)}{\bar{\sigma}^*(\phi, \theta) + 2(1 - \phi)} \qquad (14) $$

Note that for $\phi = \phi_{rcp}$, $\bar{\Pi}(\phi_{rcp}) = 0$, thus $f(\phi_{rcp}, \theta) \approx (1/4)z\sin^2\theta$, which is consistent with [17]. This approach is also consistent with the effective pressure predicted by numerical simulations for 3D adhesive foams, which show that, for a given $\phi$, the effective osmotic pressure decreases as the contact angle increases [46].

Furthermore, the effective number of bubble neighbors must be considered. Numerical simulations of 2D adhesive foams predict a decrease of the mean number of neighbors $z(\phi)$ with $\phi$, which does not vanish at $\phi_{rcp}$ but instead smoothly decreases for values of $\phi$ well above $\phi_{rcp}$. The rate of decrease becomes slower as the contact angle increases [43]. To describe the evolution of $z(\phi)$ in our 3D foams, we empirically choose a sigmoid function:

$$ z(\phi) = \frac{z_0}{1 + \exp[a(\phi - \phi_{rcp})]} \qquad (15) $$

and fix $z_0 = 14$ as measured in coarsening dry foams [47]. The dimensionless parameter $a$ should be large enough so that at the jamming transition, where contacts become rare, $z(\phi^*) \approx 0-2$. Indeed, visual inspection of images of foams at $\phi^*$ reveals the persistency of a few bubble doublets.

We fit $\Omega_p/C$ predicted by Eqs. (6), (14) and (15) to our data in the range $2.5\% \le \phi \le \phi^*$ with fixed $\phi_{rcp} = 31\%$, while $a, \theta$ and $\Omega_0$ are considered as fitting parameters. As shown in Fig. 4, we get a very good agreement between the fit and the data, with $\Omega_0 \approx (173 \pm 2) \mu m^2 s^{-1}$, $a = 40 \pm 30$, and $\theta = (5.8 \pm 0.6)^\circ$. Note that the value of $\Omega_0$ is about 17% smaller than that deduced from the set of wet foams data only [17]. In the range where repulsive bubble interactions are dominant, i.e. up to $\phi \le 0.10$, the difference between both predictions of $\Omega_p/C$ using either $f(\phi)$ (Eq. (9)) or $f(\phi, \theta)$ (Eq. (14)), is negligible, as shown in Fig. 4. For larger $\phi$, the effect of adhesion on the coarsening rate is well captured by our model (Eq. (14)) with the sigmoid function shown in the inset of
<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Coarsening rate of bubbly dispersions $\Omega_c$ as a function of the gas fraction $1 - \phi$. The error bars represent the standard deviation for a single measurement of $\Omega_c$, or the squared sum of the standard deviations from the average over two measurements for the extreme gas fractions (4%, 55%). The lines show Eq. (4) using the enhancement factors $g(\phi)$ predicted by the models: (—) MLSW [29]; (·–) TM3 [30]; (·–) VG [32]; (...) PS [31]. The corresponding fitted coarsening constants $K_0^{O/W}$ are given in Table 2.</figcaption></figure>

<caption>**Table 2**</caption>

Coarsening rate in the dilute limit $K_0$ and corresponding standard deviation deduced by fitting the coarsening enhancement factor $g$ to the measured values of the coarsening rate $\Omega_c$ (data of Table 1 and Fig. 5), for 4 models: (TM3) [30], (VG) [32], (PS) [31], (MLSW) [29]. The corresponding adjusted coefficients of determination $\bar{r}^2$ are given for each.

<table><thead><tr><th>Model</th><th>TM3</th><th>VG</th><th>PS</th><th>MLSW</th></tr></thead><tbody><tr><td>$K_0^{O/W}$ (µm³s⁻¹)</td><td>12.1 ± 0.3</td><td>15.1 ± 0.5</td><td>16.2 ± 0.6</td><td>4.2 ± 0.2</td></tr><tr><td>$\bar{r}^2$</td><td>0.97</td><td>0.94</td><td>0.92</td><td>0.89</td></tr></tbody></table>

**Fig. 4.** We notice that the agreement between the fit and the data is not sensitive to a ±75% relative variation of the parameter $a$, which means that the choice of how the number of contacts varies with $\phi$ is not critical. Note that since $f(\phi, \theta)$ is rather insensitive to $z(\phi)$ in the dry limit, it cannot be expected to quantitatively describe the average number of neighbors in this limit. The fitted contact angle is very close to the value previously measured $\theta = (4 \pm 1)°$. Given the agreement between the fit and the data, as well as the values of the fitted parameters, we conclude that the function $f(\phi, \theta)$ (Eq. (14)) provides a good prediction of the film area fraction. This, in turn, accurately describes the evolution of the foam coarsening rate $\Omega_p$ with respect to $\phi$ in adhesive foams, from the dry limit up to the jamming transition at $\phi^*$.

## 5.2. Coarsening of bubbly dispersions for $\phi^* < \phi < 1$

We discuss the evolution of the coarsening rate $\Omega_c$ with the gas fraction $1 - \phi$, and compare it to the predictions mentioned in Section 3.1. We compute the enhancement factor $g(1 - \phi)$, either using the analytical expressions given in [31] or by fitting a cubic spline interpolation to the data provided in [29], in [30] or in [32]. Then, for each prediction, we perform a linear regression on the data set constituted by the 8 points ($g, \Omega_c$). This yields values of the fitted parameter $K_0^{O/W}$, reported in Table 2. The corresponding predicted coarsening rates $\Omega_c$ are compared to the data in Fig. 5. We see that the MLSW model predicts an almost linear scaling within the studied range of gas fractions, which is inconsistent with our observations. Conversely, the other three predictions fit well to the data, exhibiting an inflection point around

$1 - \phi = 0.20$ and an increasing slope beyond this point. To provide a quantitative goodness-of-fit, we compute for each of them the adjusted coefficient of determination, $\bar{r}^2$, corresponding to 8 samples and 1 variable $g(1 - \phi)$. They are reported in Table 2. Among the four tested models, the TM3 model achieves the best correlation with the data, with $\bar{r}^2 = 0.97$. Qualitatively, the primary distinction between TM3 and the two closest models, VG and PS, lies in its prediction of a faster evolution of $\Omega_c$ in the range $1 - \phi > 0.4$, which agrees with the data. Consequently, we now consider the TM3 model to provide the most accurate prediction of the coarsening rate in the dilute limit and retain the coarsening constant value as $K_0^{O/W} = (12.1 \pm 0.3) \ \mu\text{m}^3\text{s}^{-1}$.

Now, we fit Eq. (4) to the coarsening rates of bubbly liquids previously measured in [17], using the factor $g$ given by the TM3 model. We get $K_0^{W} = (29 \pm 5) \ \mu\text{m}^3\text{s}^{-1}$, with a large uncertainty due to the limited range of liquid fractions $40\% \le \phi \le 50\%$ of those data. For this TTAB aqueous foaming liquid and air, we have $\gamma = 37 \ \text{mNm}^{-1}$, $H_e = 7.6 \cdot 10^{-6} \ \text{mol Pa}^{-1}\text{m}^{-3}$, $D = 2 \cdot 10^{-9} \ \text{m}^2\text{s}^{-1}$ [17], thus $K_0^{W}$ is ~2.4 times larger than the predicted mean-field LSW value $K_0 = 12 \ \mu\text{m}^3\text{s}^{-1}$. The same increased ratio ($K_0^{W} / K_0 \approx 2.5$) was previously reported in Ostwald ripening of O/W emulsions with a continuous phase volume fraction equal to 90% [48]. This quantitative discrepancy could be related to the actual shape of the bubble size distributions [18], compared to the predicted one.

It is more difficult to compare the value of $K_0^{O/W}$ measured in our bubbly emulsions since the gas (air) can diffuse in both the aqueous phase (TTAB/glycerol/solution) and the oil phase (silicon oil). If we first assume that air diffuses solely through the glycerol aqueous phase, we have: $D = 3.6 \cdot 10^{-10} \ \text{m}^2\text{s}^{-1}$, $H_e = 3 \cdot 10^{-6} \ \text{mol Pa}^{-1}\text{m}^{-3}$ (see details in Appendix B) and Eq. (2) yields: $K_0 = 0.8 \ \mu\text{m}^3\text{s}^{-1}$. Note that fitting the data using MLSW model (cf. Table 2) gives the smallest estimation of $K_0^{O/W} \approx 4.2 \ \mu\text{m}^3\text{s}^{-1}$, which is still an order of magnitude larger than the dilute limit prediction. Taking into account the experimentally measured ratio $K_0^{W} / K_0 = 2.4$, we see that a factor 2-6 remains between the $K_0^{O/W}$ values and the dilute limit prediction Eq. (2). This enhanced coarsening constant could be due to the solubility of air in oil which is 26 times larger than that in the aqueous phase (cf. Appendix B). However the diffusivity of air could be damped by the tenuosity induced by the presence of the oil droplets. Treating the problem as a parallel diffusion in the two liquid phases, we estimate (see Appendix B): $D = 0.011 \cdot 10^{-9} \ \text{m}^2\text{s}^{-1}$, $H_e = 57 \cdot 10^{-6} \ \text{mol Pa}^{-1}\text{m}^{-3}$, and accordingly the predicted mean-field LSW value $K_0 = 0.45 \ \mu\text{m}^3\text{s}^{-1}$. We conclude that the air solubility in oil cannot account for the observed large coarsening constant $K_0^{O/W}$.

By analogy with diffusion through a porous medium, the coarsening constant in an emulsion could be enhanced by surface diffusion [49]. Adsorption/desorption cycles of the solute, effectively “jumping” along the interface, could induce an effective diffusion along the pore surfaces and give rise to an anomalous diffusion through the medium. Moreover, since oil droplets exhibit a gas solubility constant, $H_e$, that is 26 times higher than that of the glycerol solution, but with a diffusivity, $D$, that is 60 times lower, they could exhibit a high capacity to accumulate and supply the glycerol/aqueous phase with solute molecules. This behavior could perturb the steady equilibrium of the dissolved solute concentration around the bubbles. However, these effect would currently require substantial theoretical work to achieve a quantitative prediction.

From these considerations, we propose an interpretation for the shape of the bubble size distributions for $45\% \le \phi \le 60\%$ in Fig. 3. At these liquid fractions, we observe frequent contacts between bubbles, as in the gel-like adhesive foams with $\phi_{rcp} \le \phi \le \phi^*$ studied in [18], where the contacts are promoted by the attractive energy due to the adsorption: $U_a = 2\gamma(1 - \cos\theta)A$, with $\theta$ the contact angle, and $A = \pi R_{10}^2 \sin^2\theta$ the contact area between two equal-sized bubbles of radius $R_{10}$. In a foamed emulsion, the attraction between bubbles can be enhanced by the depletion of the oil droplets. The attractive energy due to depletion between two spheres can be evaluated as [50]: $U_d = V_{ex} \Pi_d$, where
$V_{ex} \sim 4\pi r^2 R_{10}$ is the excluded volume for oil droplets of radius $r$ between the bubbles, and $\Pi_d$ is the osmotic pressure of the emulsion. Concentrated emulsions with oil volume fraction $\varphi > 0.637$ exhibit an osmotic pressure analogous to that of foams, described by Eq. (8). For $R_{10} = 300$ µm, $\theta = 4^\circ$, $r = 1.6$ µm, $\varphi = 0.725$, and surface tension at the oil/water interface $\gamma_{o/w} = 4.5$ mNm$^{-1}$, we get attractive energies of the same order of magnitude: $U_a \approx 4 \cdot 10^{-12}$ J and $U_d \approx 2 \cdot 10^{-12}$ J. Thus the bubble attraction could be enhanced by the depletion of the oil droplets around the bubbles, and enable the persistence of a few contact films for liquid fractions $\varphi > \varphi^* \approx 39\%$. This would in turn maintain a small population of jammed bubbles, which is compatible with the observed distributions, as in [18].

This argument leaves an open question: why do we measure cubic growth laws, even if the bubbles still present sparse contacts? In such concentrated bubbly liquids, gas transfer between two “kissing bubbles” occurs through the bulk surrounding liquid and through their contact film. For foamed emulsions, the bulk flow is characterized by the coarsening constant $K_0^{O/W}$, whereas the flow across the film is characterized by the diffusivity $D$ and the solubility of the gas $H_e$ in the glycerol aqueous solution since the oil droplets are excluded from the film. The ratio of both flow rates, for a doublet of bubbles of radius $R_{10}$ and film thickness $h$, is given by [17]:

$$ \frac{Q_{film}}{Q_{bulk}} \approx \frac{\gamma H_e D v_m}{K_0^{O/W}} \frac{R_{10} \sin^2 \theta}{h \ln(R_{10}/h)} \quad (16) $$

Here we have (cf. Appendix B): $K_0^{O/W} = 12.1 \ \mu\text{m}^3\text{s}^{-1}$, $\gamma = 33 \ \text{mNm}^{-1}$, $H_e = 3 \cdot 10^{-6} \ \text{molPa}^{-1}\text{m}^{-3}$, $D = 3.6 \cdot 10^{-10} \ \text{m}^2\text{s}^{-1}$, $v_m = 0.024 \ \text{m}^3\text{mol}^{-1}$. Thus, we expect the bulk flow to dominate over the gas transfer through films ($Q_{film}/Q_{bulk} \ll 1$) for $R_{10}/h \ll 3 \cdot 10^4$, with a contact angle $\theta \sim 4^\circ$. For a film thickness $h \approx 40$ nm [17], we get $R_{10} \ll 1.2$ mm, a value much larger than the maximum average size here studied $R_{10} \approx 300$ µm. We conclude that the gas should mainly be transferred through the bulk liquid, leading to a cubic growth law as observed, even in the presence of a few persistent contact films.

### 5.3. Wrapping up coarsening from dry foams to dilute bubble dispersions

We summarize our findings in Fig. 6, where we show the coarsening rate of both foams and bubbly liquids. We normalize them by their respective physico-chemical coarsening constants, either $C\Omega_0$ or $K_0$, and plot them as a function of the liquid fraction for foams or the gas fraction for bubbly liquids. The geometrical factor $C$ (given in Table 1) is deduced from the measured bubble size distributions. The coarsening constant $\Omega_0$ is deduced by fitting to the data the rate predicted by Eq. (6) where the average fraction of bubble surface area covered by the film is given by our prediction for adhesive foams $f(\phi, \theta)$ Eq. (14). The assumed sigmoid function $z(\phi)$ given by Eq. (15) yields a phenomenological prediction of the average number of bubble contacts persisting besides the random close packing fraction $\phi_{rcp}$. Note the large underestimation of $\Omega_p/C\Omega_0$ when the effect of adhesive forces is neglected with $\theta = 0^\circ$. Above the jamming transition $\phi^*$ where the average number of contacts become smaller than $\approx 2$, the coarsening constant $K_0$ is obtained by fitting the coarsening enhancement factor $g(1 - \phi)$ predicted by the TM3 model [30]. This result, evaluated for a dilute bubble dispersion in a continuous phase made of an emulsion with a low yield stress, is representative of the coarsening dynamics of bubble dispersions in a simple surfactant solution.

A quantitative comparison to previously published data about coarsening rates requires a precise knowledge of the physico-chemical properties of the system which tune the coarsening constants $K_0$ and $\Omega_0$: solution viscosity, gas solubility in the solution, and for foams the effective film thickness for gas transfer. The knowledge of the contact angle $\theta$ which characterizes the magnitude of the adhesive forces and has an impact on the value of $\phi^*$ at the jamming transition is also needed. Moreover, it is necessary to compare the same average of the bubble

<figure><img src="image_10.png" /><figcaption>Fig. 6. Coarsening diagram, showing the normalized coarsening rate from dry foams to dilute bubbly liquids. (top left corner) $\Omega_p/C\Omega_0$ as a function of the liquid volume fraction $\phi$ of the foam: (●,■) same data as in Fig. 4 with $\Omega_0 = 173 \ \mu\text{m}^2\text{s}^{-1}$. The line (—) represents the average fraction of bubble surface area covered by the films $f(\phi, \theta)$, predicted by Eqs. (9) and (15) with parameters $\phi_{rcp} = 31\%$, $\theta = 5.8^\circ$ and $a = 40$; the dotted line (...) represents $f(\phi, \theta = 0)$ with $\phi_{rcp} = 31\%$ (Eq. (9)). (bottom right corner) $\Omega_c/K_0$ as a function of the gas volume fraction $1 - \phi$ of the bubbly liquid: (□) same data as in Fig. 5 with $K_0 = 12.1 \ \mu\text{m}^3\text{s}^{-1}$; (■) data from [17] with $K_0 = 29 \ \mu\text{m}^3\text{s}^{-1}$. The line (—) represents the coarsening enhancement factor $g(1 - \phi)$ predicted by [30] (see main text). The global relative uncertainty is estimated to be around 20% for the ratios $\Omega_p/C\Omega_0$ and $\Omega_c/K_0$. The absolute uncertainty on the liquid fraction is $\pm 1\%$ for $\phi > 40\%$, $\pm 0.5\%$ for data from [17], and $\pm 0.1\%$ for $\phi < 10\%$. The two x-axes are aligned on the common value $\phi = 1 - \phi = 0.5$. The two y-axes are aligned on the values at the jamming transition $\phi^* \approx 0.4$: $f(\phi^*, 5.8^\circ) = 4 \cdot 10^{-3}$ and $g(1 - \phi^*) = 7.5$.</figcaption></figure>

radius for the growth law. The lack in the literature of any set of data specifying these informations (except in [17]) preclude a quantitative comparison with our master curve shown in Fig. 6. This plot highlights the overall influence of the dispersed phase fraction on coarsening, offering a valuable new tool for addressing coarsening effects in biphasic systems.

## 6. Conclusions

We investigate the coarsening of liquid foams as a function of liquid volume fraction within the 2–8% range using a clinostat cell. The average bubble radius exhibits a parabolic growth, consistent with previous studies (see for example [15] for a review) and with recent observations [17] on wetter foams (15–38%) using the same surfactant (TTAB 5 g/L). By combining the latter dataset with our data, we establish the evolution of the coarsening rate over an unprecedented liquid fraction range, spanning 2–38%.

As discussed in [17], the observation of the parabolic growth at liquid fractions as high as 38% can be explained by the presence of weak adhesion forces between bubbles. These forces help maintain small contact areas between bubbles, even when osmotic pressure alone no longer drives it. However, no existing model can describe the coarsening rate across this newly explored range of liquid fractions, highlighting the need for further theoretical developments. Drawing inspiration from studies on cohesive granular media, which utilize the concept of effective stress, we propose an expression (Eq. (14)) of the average fraction of bubble surface area covered by films. The latter combines the classical osmotic pressure [38] with an additional pressure accounting for internal adhesive forces. This approach successfully captures the dependence of the coarsening rate on liquid fraction, provided the evolution of the average number of contacts per bubble is modeled using a sig-
<figure><img src="image_11.png" /><figcaption>**Fig. A.7.** Coarsening of bubbly emulsions: Cumulative bubble radius distributions as a function of the normalized bubble radius for liquid fractions $\phi$ as indicated. The colors indicate the duration of coarsening since the end of the sample production. For all cases, the average bubble capillary pressure is much larger than the emulsion yield stress $2\gamma/R_{10} \gg \tau_y$.</figcaption></figure>

moid function with one fitting parameter. The agreement is achieved without requiring extensive fine-tuning of parameters and is consistent with the physical principles of sphere packings.

To investigate the coarsening of bubbly liquids, in the extended range of liquid fractions from 45% to 96%, which has not been previously explored in long-term coarsening experiments, we use the same clinostat cell but replace the aqueous surfactant solution with a concentrated emulsion of oil droplets in a TTAB/glycerol/water solution. This emulsion has a low yield stress, preventing drainage - which would otherwise make the system inhomogeneous - without affecting the coarsening kinetics. We observe that the average bubble radius grows according to a cubic law, as expected in the dilute regime. The coarsening rate decreases with the liquid fraction, and we compare its evolution with the prediction given by several models (cf. Fig. 5). We find that the best prediction is given by the TM3 mean-field coarsening model [30]. By comparing our bubbly liquid data to a recent dataset [17], we observe discrepancies in the values of the coarsening constant $K_0$ (Eq. (2)), which reflects the effects of the physicochemical properties of the solutions used.

Our results highlight the challenge of accurately accounting for the physico-chemical properties of the continuous phase either in foams or in bubbly liquids. It should be noted that, unlike previous studies, we aim to describe data across ranges extending over both liquid foams and bubbly liquids, while certain physicochemical parameters are common to both corresponding models. This leaves no flexibility to adjust these parameters, as would have been possible when working with only one dataset. Our findings show the need for further quantitative work on the bubbly liquid coarsening constant $K_0$ (Eq. (2)), in relation to the foam coarsening constant $\Omega_0$ (Eq. (7)), knowing that each one is related to different moments of the bubble size distribution. Experiments conducted using foams with different chemical compositions and varying contact angles $\theta$ would be very useful. Additionally, simulations of 3D packings of adhesive bubbles, such as those recently performed for 2D foams [43], which explicitly consider effective contact areas, could refine the expression for the evolution of the mean number of contacts with $\phi$ (Eq. (15)) and offer a physical interpretation for the sigmoid function.

Finally, predicting coarsening rates as a function of the volume fraction of the continuous phase should be a significant contribution towards

coarsening control in dilute and concentrated bubbly systems in various fields of applications such as food science, materials engineering and geophysics, as examples.

### CRediT authorship contribution statement

**Nicolò Galvani:** Writing – original draft, Software, Investigation, Formal analysis, Conceptualization. **Olivier Pitois:** Writing – original draft, Supervision, Software, Funding acquisition, Formal analysis, Conceptualization. **Sylvie Cohen-Addad:** Writing – original draft, Supervision, Software, Funding acquisition, Formal analysis, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

This work was supported by CNES (Centre National d'Études Spatiales). N. G. benefited of PhD grants from CNES and from ESA (the European Space Agency), in the frame of the MAP "Hydrodynamics of Wet Foams". O. P. thanks J.-N. Roux and X. Chateau for fruitful discussions on effective stress in cohesive granular media, and R. Höhler for constructive discussions on adhesive forces in foam. S. C.-A. thanks S. Cox for fruitful discussions on coarsening with gas mixtures.

### Appendix A. Scaling state in bubbly emulsions

We present in Fig. A.7 the cumulative bubble size distributions of the studied bubbly emulsions corresponding to the coarsening ages reported in Fig. 3(d). For all of the liquid fractions, we determine that the scaling state is reached for coarsening times $t \ge 10^4$ s.

### Appendix B. Gas diffusion in an emulsion

The diffusion of air through an emulsion involves two gas species ($N_2$, $O_2$) and two immiscible liquid phases, with an oil phase dispersed
<caption>**Table B.3**</caption>

Chemical properties relevant to gas diffusion, in the studied liquids, at a temperature $T = 25°C$. The superscript in the left column labels the corresponding values of $H_e$, $D$, $K_0$. Values gathered from literature are in plain text, those estimated in the text are in *italic*. Data for silicon oil come from [56]. The dynamic viscosity $\eta$ of the water glycerol solution is interpolated from [53] using Eq. (B.1). The gas solubility $H_e$ is determined for three gases (di-nitrogen, di-oxygen, air) as explained in the text; for glycerol solutions, we use values from [53] for $N_2$ and [54] for $O_2$, and estimate the missing values at $G = 52$ wt% as explained in the text. $D$ represents the diffusion coefficient of air in the liquid and $\gamma$ is the surface tension of the aqueous phase / air interface. $K_0$ is the coarsening rate predicted by LSW in the dilute limit by Eq. (2) for air as the diffusive species.

<table><thead><tr><th>Liquid phase</th><th>η</th><th>H<sub>e,N2</sub></th><th>H<sub>e,O2</sub></th><th>H<sub>e,Air</sub></th><th>D</th><th>γ</th><th>K<sub>0</sub></th></tr><tr><td></td><td>mPa s</td><td>10<sup>-6</sup> mol</td><td>Pa<sup>-1</sup> m<sup>-3</sup></td><td></td><td>10<sup>-9</sup> m<sup>2</sup>s<sup>-1</sup></td><td>mN m<sup>-1</sup></td><td>μm<sup>3</sup>s<sup>-1</sup></td></tr></thead><tbody><tr><td>Pure water [W]</td><td>1</td><td>6.5</td><td>12.7</td><td>7.6</td><td>2</td><td>37</td><td>12</td></tr><tr><td>Aqueous phase (G = 0.52) [W<sub>G</sub>]</td><td>5.6</td><td>2</td><td>6</td><td>3</td><td>0.36</td><td>33</td><td>0.8</td></tr><tr><td>Silicon oil [O]</td><td>348</td><td>69</td><td>110</td><td>77</td><td>0.006</td><td>-</td><td>0.3</td></tr><tr><td>Emulsion φ = 0.725 [O/W]</td><td>-</td><td>-</td><td>-</td><td>57</td><td>0.011</td><td>33</td><td>0.45</td></tr></tbody></table>

inside an aqueous one, forming a porous-like medium. In our study, the oil phase is constituted of silicon oil, and the aqueous phase is a water/TTAB/glycerol solution with glycerol concentration $G = 52$ wt%.

At 25°C, the dynamic viscosity $\eta$ of an aqueous glycerol solution with concentration $G$ is given by [51]:

$$ \eta(G) = 0.91e^{-6.9(1-G+1.37G(1-G)/0.66G+2.1(1-G))}. \quad (B.1) $$

We deduce the diffusion coefficient of air in the aqueous glycerol solution $D^{W_G}$ with $G = 52$ wt% using Eq. (B.1) and $2 \cdot 10^{-9} \text{ m}^2\text{s}^{-1}$ as the diffusion coefficient in pure water, and report it in Table B.3, together with the other physico-chemical properties estimated for $T = 25°C$.

The solubility of a gas in a mixture of two liquids, like in an aqueous-glycerol solution, scales with the molar ratio $x$ [52]:

$$ H_e^{W_G}(x) = (H_e^G)^x (H_e^W)^{1-x} e^{\beta \cdot x \cdot (1-x)} \quad (B.2) $$

where $H_e^G$ (resp. $H_e^W$) is the solubility in pure glycerol (resp. water), and $\beta$ is the interaction parameter between the two liquids. We plot the measured solubilities $H_{e,N2}$ of $N_2$ [53] and $He, O_2$ of $O_2$ [54] in aqueous glycerol solutions as a function of the glycerol concentration in Fig. B.8(a), and fit Eq. (B.2) to them. We deduce $He, O_2$ and $H_{e,N2}$ at $G = 52$ wt%, and we determine $H_{e,Air}^{W_G}$ as the sum of the individual solubilities, weighted by the respective molar fractions: $H_{e,Air}^{W_G} = (0.78H_{e,N2}^{W_G} + 0.21H_{e,O_2}^{W_G})/0.99$. Thus, for a glycerol/water solution at $G = 52$ wt%, we estimate the coarsening constant for air: $K_0^{W_G} \approx 0.8 \text{ µm}^3\text{s}^{-1}$.

The estimated values of diffusivity and solubility for air in silicon oil, multiplied together, are in the same order of magnitude of what we get for the glycerol/water solution. Therefore, the oil droplets could contribute to the gas diffusion process as well. To evaluate the corresponding coarsening constant, we consider diffusion in emulsion as a parallel diffusion in a biphasic system, as in [55]. The effective solubility in the emulsion is the sum of the solubilities of the two phases, weighted by their volume fractions:

$$ H_e^{O/W} = H_e^O \varphi + H_e^{W_G}(1 - \varphi). \quad (B.3) $$

Similarly, the effective diffusion coefficient of the emulsion is the sum of the diffusion coefficients of the two phases, weighted by their solubilities and volume fractions:

$$ D^{O/W} = \frac{D^O H_e^O \varphi + D^{W_G} H_e^{W_G}(1-\varphi)}{H_e^{O/W}}. \quad (B.4) $$

For $\varphi = 0.725$, using Eqs. (2), (B.3) and (B.4), we get: $H_e^{O/W} \approx 57 \cdot 10^{-6} \text{ mol Pa}^{-1} \text{m}^{-3}$, $D^{O/W} \approx 0.011 \cdot 10^{-9} \text{ m}^2\text{s}^{-1}$, and the coarsening constant for air: $K_0^{O/W} \approx 0.45 \text{ µm}^3\text{s}^{-1}$.

<figure><img src="image_12.png" /><figcaption>**Fig. B.8.** Solubility $H_e$ of water/glycerol solutions, as a function of the glycerol weight fraction $G$, for: $N_2$ [53], $O_2$ [54]. The dashed lines show Eq. (B.2) fitted to the data, with: fixed parameters the solubility in pure water (units $10^{-6} \text{ mol Pa}^{-1} \text{m}^{-3}$) $H_{e,N2}^W = 6.5$, $H_{e,O_2}^W = 12.7$; fitted interaction parameter $\beta = -1.6$, and fitted solubilities in pure glycerol (units $10^{-6} \text{ mol Pa}^{-1} \text{m}^{-3}$) $H_{e,N2}^G = 1.1$, $H_{e,O_2}^G = 5.6$.</figcaption></figure>

## Data availability

Data will be made available on request.

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