Enhanced coarsening induced by pore confinement

A Salame, V.-T Nguyen, V Langlois, A Petit, B Soltner, O Pitois, S

Vincent-Bonnieu

► To cite this version:

A Salame, V.-T Nguyen, V Langlois, A Petit, B Soltner, et al.. Enhanced coarsening induced by pore confinement. Physical Review E, 2025, 111 (1), pp.L013401. 10.1103/PhysRevE.111.L013401. hal-04871596

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Submitted on 7 Jan 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Enhanced coarsening induced by pore confinement.

A. Salame, V.-T. Nguyen, V. Langlois, A. Petit, B. Soltner, and O. Pitois*

*Laboratoire Navier, Univ Gustave Eiffel, ENPC, CNRS*

S. Vincent-Bonnieu

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

*(Dated: January 7, 2025)*

The coarsening process is driven by the reduction in interfacial energy density, resulting in an enlargement of the size scale within two-phase systems. It plays a crucial role in the stability and the aging of densely packed arrangements of particles, which are commonly found in alloys, foams, emulsions, supersaturated solutions, ... It has been shown previously that, when properly confined, such particles undergoes the so-called *anti-coarsening* phenomenon, ultimately leading to the one particle-per-pore configuration associated with the arrest of coarsening. In this study, we quantitatively explore the coarsening dynamics of liquid foam confined within the pores of spherical bead packings, where the size of the beads has been adjusted to modify the degree of confinement. We observe that, apart from the final configuration associated with the arrest of coarsening, the more we confine this system within small spaces, the more rapidly it coarsens. Moreover, the broadening of the particle size distribution has an opposite effect with respect to that observed for unconfined systems. This unexpected behavior is shown to arise from the coupling between the liquid trapped at the surface of the grains and the effective liquid volume fraction within the core of the pores.

The growth and coarsening of spherical particles or droplets in a liquid or solid matrix, initially observed by Ostwald in 1900 [1], is prevalent in various two-phase mixtures characterized by significant diffusion. Liquid foam is a well-adapted system for studying coarsening due to its fluid and isotropic domains. It is now well known that, including very wet foams, the coarsening process reaches a self-similar regime characterized by an increase of the average bubble size as a function of age as $\langle R \rangle \sim t^{1/2}$ [2, 3]. The study of coarsening under confinement naturally arose for liquid foams, given their diverse industrial applications in soils, as an alternative to conventional flushing techniques for effectively extracting non-aqueous phase liquids from aquifers [4, 5], as a barrier reducing liquid permeability in porous substrates [6, 7], thereby aiding in containing sources of contamination, or assisting CO₂ sequestration [8]. Within confinement conditions, a pseudo-bulk-foam behaviour has been observed when the average bubble size is much smaller than the pore [9]. However, as bubbles approach the size of the pores, more significant effects lead to a decrease of the coarsening rate and eventually to the stop of coarsening [10], in a configuration where each pore contains an unique bubble [9, 11].

In this study, we investigate the coarsening dynamics of foam confined within the pores of spherical bead packings, where bead size has been adjusted to modify the degree of confinement. As we later demonstrate, the impact of confinement on coarsening is complex: before observing the decrease in coarsening rate described above, there exists an unexpected regime in which this rate increases very significantly.

The nonionic surfactant Glucopon ® 225DK (BASF), was utilized at 20 g/L in water. A double-syringe setup was employed to produce the foam [12]: One syringe (50 mL) contained 3.5 mL of the solution, while the other contained 46.5 mL of air. These syringes were interconnected via a Luer adapter, facilitating foaming through pump movements, totaling 20 passes. The resulting foam possessed a liquid volume fraction of 7%, with a bubble size distribution indicating a Sauter radius $R_b$ close to 100 µm. The sample under investigation consists of a packing of glass beads with a radius $R_g \pm \Delta R_g$ (where $\Delta R_g/R_g$ is approximately 5%), ranging from 400 µm to 1500 µm, within a cylindrical glass tube with an inner diameter of 26 mm. The pore volume fraction of the bead packing was measured to be within the range of 0.38 to 0.43 depending a little bit on grain size. The roughness of the beads was estimated to be of the order of 1 µm or less. The 50 mL foam syringe was employed to fill the bead packing, assisted by a syringe pump operating at a rate of 15 mL/min. The 7% liquid volume fraction was chosen to minimize liquid drainage in the syringe during the filling step. However, to prevent bubble coalescence during the study, additional surfactant solution was co-injected with the foam, resulting in a final liquid content of $S_w = 20\%$ in the pore space of the bead packing. Therefore, following the filling process, the pore space is filled with liquid foam with bubble radius $R_b \approx 100$ µm and liquid content $S_w = 20\%$. We introduce the size ratio $r = R_b/R_g$. During the filling step and after, the sample was rotating at 80°/s (see Figure 1a). The column's rotation is programmed to halt for 1s at a specified angle before resuming its rotational motion. Two cameras positioned at +45° and -45° from the horizontal, i.e. above and below the column, capture images of the sample at the wall surface during this pause, through two right-angle prisms (15x15x21.2mm) stuck onto the

<footnote>* Olivier.Pitois@univ-eiffel.fr</footnote>
<figure><img src="image_2.png" /><figcaption>FIG. 1. Coarsening experiment: (a) A liquid foam is pushed into a packing of glass beads formed in a transparent column rotated to maintain a uniform volume fraction of liquid $S_w$. (b) Observing the samples through two prisms attached at the top and bottom walls of the column, allows the visualization in black of the liquid in contact with the wall and in white the gas (separated from the wall by a very thin liquid film). Black spots appear, which are liquid bridges formed at the wall with the grains in contact. Dotted circles indicate the beads in contact with the cylinder (red) vs the ones that do not for a bridge with the cylinder (orange). (c) Sketch of a liquid bridge formed between two grains (a similar configuration can be considered between one grain and the wall of the column). Colors are used to distinguish the liquid contained in the pore core foam (gray) from the liquid contained in the liquid bridges (blue) and in the surface liquid channels (green). (d,e,f,g) Magnified images during coarsening. Some liquid bridges (in black) can be seen, between which, the bubbles of the core pore foam grow over time, until the configuration shown in (g) is reached, where each pore contains a single bubble. With the increase of bubble size, we can also obviously observe the increase of the liquid bridges size. The width of each image is 2.5mm. (h,i) Images of the same foam ($R_b \approx 100$ µm and $S_w = 0.2$) observed for two values of the confinement degree: (h) corresponds to $r = R_b/R_g \approx 0.08$ and (i) corresponds to $r \approx 0.20$. We can clearly see how the foam appears drier between liquid bridges in (i) compared to (h). Such images were used to measure the surface liquid fraction $\phi_S^{core}(r)$ (see text). The width of the images is 5mm.</figcaption></figure>

glass wall at 180° relative from each other. The prism allows to enhance the light contrast between liquid and gas at the wall [13] (see Figure 1b). The resulting images show the liquid in black and the gas (separated from the wall by thin liquid films) in white. These images facilitate the direct measurement of the surface liquid fraction $\phi_S$. The comparison of $\phi_S$ between the top and bottom images allow us to detect possible drainage issues, i.e. the no evolution of the surface liquid fraction at the top and bottom allows us to say that the rotation speed deemed sufficient to counteract the gravity drainage. By segmentation and tessellation of the resulting images (see more details in [14]), we identify the area $A_i$ occupied by each bubble in contact with the wall (see more details in [14]), from which we calculate the corresponding bubble radius, i.e. $R_{b,i} = (A_i/\pi)^{1/2}$, and finally the Sauter radius is determined for the total bubble population observed at the surface: $R_b = \langle R_{b,i}^3 \rangle / \langle R_{b,i}^2 \rangle$. The study involves tracking the temporal evolution $R_b(t)$ for different grain sizes (see videos in [14]). The following results account data averaged over two or three experiments (for both top and bottom images).

Results for $R_b$ as a function of $t$ are presented in Figure 2. 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. Even by

<figure><img src="image_3.png" /><figcaption>FIG. 2. Average bubble radius as a function of foam age for five values of the grain radius $R_g = 1500, 1000, 750, 550$ and $400$ µm. Symbols: experimental data, each circle represents a treated image. Solid lines: values provided by the following equation: $R_b(t) = (R_b^{3/2}(t_0) + \Omega_{2/3}(t-t_0))^{2/3}$. Dotted line: values provided by the following equation: $R_b(t) = (R_b^2(t_0) + \Omega_{1/2}(t-t_0))^{1/2}$. The green arrows indicate the times corresponding to the four images (d,e,f,g) shown in Figure 1 for $R_g = 1000$ µm.</figcaption></figure>

excluding the data corresponding to the approach of saturation, it is not possible to describe these curves with the classical parabolic law (i.e., with an exponent of 1/2), as is usually the case for bulk foam. If we really aim to describe these curves with a power law, then
the 2/3 exponent proves to be much more appropriate: $R_b(t) = (R_b^{3/2}(t_0) + \Omega_{2/3}(t-t_0))^{2/3}$. The observed increase in the exponent suggests that bubble growth is accelerated compared to bulk foam.

In order to progress into our analysis, we perform the derivative of the curves presented in Figure 2 and we smooth the values obtained using the Savitzky-Golay filter. We determine the coarsening rate as:

$$ \Omega(r) = 2R_b \frac{dR_b}{dt}, \qquad (1) $$

which implies that the underlying mechanism for coarsening is the gas transfer through the films, usually associated with the parabolic law already presented [2]. We consider here that this law is not observed because the coarsening rate is not constant. The values obtained for $\Omega(r)$ are plotted in Fig. 3a. We recall that for bulk foam, within the scaling state, one would obtain a constant value equal to $\Omega(0) = \Omega_0 \times \xi(\phi_l)$. The constant $\Omega_0 = \gamma DH_e V_m / h$ accounts for the effects of surface tension ($\gamma$), molecular diffusion of the gas in the solution (D), gas solubility through Henry constant ($H_e$), molar volume of the gas ($V_m$) and effective film thickness (h). The function $\xi(\phi_l) = \tilde{\Pi}(\phi_l)/(\tilde{\Pi}(\phi_l) + 2(1-\phi_l))$ accounts for the evolution of the surface area covered by the films on the bubbles as a function of liquid volume fraction [2, 15], where $\tilde{\Pi}(\phi_l) = \Pi(\phi_l)/(\gamma/R_b) = k(\phi_l^* - \phi_l)^2/\phi_l^{1/2}$ is the reduced osmotic pressure of the bulk foam. Typical values for $k$ and $\phi_l^*$ are 3.2 and 0.36 respectively [2]. In Figure 3a, $\Omega(r)$ increases significantly before reaching $r \approx 0.7$ which initiates the arrest of coarsening, where the single bubble-per-pore configuration begins to appear [9, 11, 16, 17]. The microstructure of the porous medium studied here involves pores of different sizes at wall, i.e., single pores, formed by 3 grains, or larger pores, formed by 4 or 5 grains. Thus, when the coarsening of single pores stops, larger pores may still be evolving, and the coarsening rate does not completely vanish.

Since we do not obtain a master curve for $\Omega(r)$ in Figure 3a, it can be concluded that at least one other parameter plays a significant role in the coarsening rate. The analysis of bubble size distributions shows that the polydispersity index, $p = R_{32}/\langle R^3 \rangle^{1/3} - 1$, increases during coarsening, from its initial value ($p_0 \approx 0.13$) to a value that depends on the initial size ratio, i.e., $r_0$. The polydispersity increases up to reach a maximum value ($\approx 0.30$) for a size ratio $r$ close to 0.40 (not shown here), which can be approximately described by the following empirical relation $p/p_0 = 10(r - r_0) + 1$. Note that for larger $r$ values the system tends to the one-bubble-per-pore configuration, which is associated with the decrease of the polydispersity and ultimately the arrest of coarsening. As the parameter $p$ is known for each point of the $\Omega$ curves, points having a chosen $p$ value (within a range of 5%) are linked together to construct constant-$p$ curves. Those curves are then superimposed to $\Omega(r)$ curves in Fig. 3a: It becomes very clear that as the polydispersity index increases, the coarsening rate decreases.

<figure><img src="image_4.png" /><figcaption>FIG. 3. (a) Coarsening rate $\Omega(r) = 2R_b dR_b / dt$ as a function of the confinement parameter $r = R_b / R_g$ for all studied samples characterized by the grain size $R_g$. The dotted vertical line shows the $r$ value for which the coarsening is entering the regime characterized by the one-bubble-per-pore configuration. Solid lines connect the points of $\Omega$ having constant values of the polydispersity index $p$ (see main text). As indicated, black is for $p=0.13$, red is for $p=0.30$ and in between in dark gray for $p=0.15, 0.20$ and $0.25$. (b) $\Omega(r)$ divided by $\Omega(0)$, i.e. the coarsening rate of the corresponding bulk foam (same $R_b$ and same $S_w$), as a function of $r$. Small circles correspond to data shown in (a). Large black circles correspond to the initial points of each curve in (a) where $p = 0.13$. Large red circles correspond to the same points in (a) where $p = 0.30$. The black solid line ($p = 0.13$) describes the initial coarsening rate (right after filling) and represents eq. 3 with $\phi_l^{\text{core}}$ given by the blue curve shown in Figure 4. It can be described by: $\Omega(r)/\Omega(0) = 1 + 26r^{3/2}$. The red curve was fitted to the large red circles and is described by: $\Omega(r)/\Omega(0) = 1 + 57r^{5/2}$.</figcaption></figure>

This behavior is opposite to the coarsening observed in bulk foams [18].

We will now seek to explain why this acceleration in coarsening velocity occurs as confinement becomes more significant. Our starting point is to observe that the surface liquid fraction of the foam at wall appears to decrease as the parameter $r$ increases. This effect is striking when referring to the images of Figure 1(h,i), where the same foam ($R_b$ and $S_w$) is confined between grains of two different sizes. From now, we will refer to the so-called pore core foam, which is obtained by removing the half-layer of bubbles in direct contact with the grain walls and liquid bridges (see Figure 1c). Because we removed liquid mobilized at the surface of the grains, the effective liquid fraction of the pore core foam, $\phi_l^{\text{core}}$, is expected to be smaller than $S_w$. The corresponding surface liquid
<figure><img src="image_5.png" /><figcaption>FIG. 4. Surface liquid fraction $\varphi_S^{core}$ and associated volume liquid fraction $\phi_l^{core}$ of the foam as a function of $r$. Symbols: values measured immediately after filling the porous space ($\varphi_S^{core}$) and values of $\phi_l^{core}$ deduced from $\varphi_S^{core}$ using equation 2. Lines: values assumed for $\phi_l^{core}$ (and therefore $\varphi_S^{core}$) in equation 3 to adjust the initial coarsening rate (right after filling) represented by the black solid line in Figure 3b. Inset: Variation of the surface liquid fraction $\varphi_S^{core}$ measured for different values of the polydispersity index, $p$, for $r = 0.2$.</figcaption></figure>

fraction, $\varphi_S^{core}(r)$, was measured at the wall immediately after filling the porous space of grain packings with the foam. The results are shown in Figure 4. The associated decrease in volume liquid fraction $\phi_l^{core}$ can be deduced from $\varphi_S^{core}(r)$ using the relationship proposed by Höhler et al. [15]:

$$ \varphi_S = 1 - \frac{\tilde{\Pi}}{2 + \tilde{\Pi}/(1 - \phi_l)} \quad (2) $$

and we plot it in Figure 4. $\phi_l^{core}$ is shown to decrease significantly as a function of $r$. This decrease can be understood by the fact that the liquid amount mobilized at the surface of the grains increases a function of $r$. In particular, see the size increase of liquid bridges in Figure 1(d,e,f,g). A recent study [19] focused on these liquid bridges and, in doing so, proposed an estimate for the evolution of the liquid volume fraction $\phi_l^{core}$ for monodisperse foam as a function of $r$. This estimate is quite compatible with the results we produce here. Therefore, we now have in mind the mechanism responsible for the observed acceleration in coarsening velocity: as the bubble size increases, the ratio $r$ increases, which results in decreasing the effective volume liquid fraction of the foam, thus, in turn, increasing both the area of the films covering the bubbles and coarsening velocity. This effect is fully described by the previously introduced function $\xi$. Therefore, the coarsening rate of the confined foams writes:

$$ \Omega(r) = \Omega(0) \frac{\xi(\phi_l^{core}(r))}{\xi(S_w)} = \frac{\gamma D H_e V_m}{h} \times \xi(\phi_l^{core}(r)) \quad (3) $$

In Figure 3b, we aimed to adjust $\phi_l^{core}$ from eq. 3 to best describe the initial velocities, i.e., those corresponding to

$p \approx 0.13$, and we plot the corresponding curve in Figure 4 alongside the measured values for $\phi_l^{core}$. As can be seen in Figure 3b, using eq. 3 with values of $\phi_l^{core}$ that are very close to our measurements perfectly accounts for the measured coarsening velocities, which seems to validate the mechanism we propose. We also plot in Figure 3b the curve describing the $p = 0.3$ velocities. Such a value for the polydispersity index is rather close to $\approx 0.4$ which characterizes the scaling state in bulk foam [2]. Thus, the full curve at $p = 0.3$ gives a very good idea of the effect of confinement compared to a bulk foam in the scaling state: for $r \approx 0.3$, the coarsening velocity has been increased by nearly 4 times. We can compare the value of $\Omega(0)$ we used in Figure 3b, i.e. $20 \ \mu m^2/s$, with the value published recently for bulk foam with such liquid content: $\Omega(\phi_l = 0.2) \approx 30 \ \mu m^2/s$ [2], which falls within the same range of values and lends consistency to eq. 3.

Finally, we focus on the effect of the polydispersity index. We measured the surface liquid fraction for our samples at different values of $p$, either at the beginning or during the coarsening process, but with constant value of $r$, i.e. 0.2. The measured variation is presented in the inset of Figure 4 and shows that, with constant $r$, $\varphi_S^{core}$ increases as $p$ increases. Thus, the mechanism responsible for the effect of $p$ on the coarsening velocity is of the same nature as that previously described for $r$: the increase in $p$ leads to an increase in $\varphi_S^{core}$, which in turn slows down the coarsening velocity. This effect, which is related to how bubbles of different sizes distribute themselves within the porous space, would need to be studied in detail in the future.

As a conclusion: Through experiments conducted with liquid foam in the pore space of grain packings, we have measured the effect of geometrical confinement on coarsening. While we expected to find a slowing down of the coarsening process until it completely stopped, we instead measured a significant acceleration, with a coarsening rate more than three times higher than that characteristic of the reference case (bulk foam with same liquid content). Based on our observation, the mechanism we propose to explain this behavior is the following: Liquid is extracted from the foam to form liquid bridges at contact between grains or wall liquid channels at their surface. This is necessarily accompanied by an increase in osmotic pressure of the foam, associated with the decrease of its liquid fraction, resulting in the increase in the film areas between all the bubbles and therefore the increase of the gas fluxes involved in coarsening. This mechanism turns out to be driven primarily by the ratio between the sizes of the bubbles and the grains, and to a lesser extent, by the polydispersity index of the bubble assembly. This unexpected behavior seems to represent a new factor to consider in foam applications in porous media, such as in soils. However, some aspects deserve further investigation, such as the effect of the liquid volume fraction within the porosity, the shape and roughness of the grains, and the understanding of the impact of bubble
polydispersity on the effective liquid fraction of the foam.

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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