# Hierarchical bubble size distributions in coarsening wet liquid foams.

Nicolo Galvani2,5, Marina Pasquet1, Arnab Mukherjee2, Alice Requier1, Sylvie
Cohen-Addad2,3,
Olivier Pitois5, Reinhard Höhler2,3, Emmanuelle Rio1, Anniina Salonen1,
Douglas J. Durian4,
Dominique Langevin1
1 Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405,
Orsay, France.
2 Sorbonne Université, Institut des NanoSciences de Paris, Paris, France.
3 Université Gustave Eiffel, Laboratoire Navier, Marne-la-Vallée, France.
4 Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA.
5 Lab Navier, Univ Gustave Eiffel, ENPC, CNRS, 5 Bd Descartes, Champs-sur-
Marne,
F-77454Marne-la-Vallée, France

###### Abstract

Coarsening of two-phase systems is crucial for the stability of dense particle
packings such as alloys, foams, emulsions or supersaturated solutions. Mean
field theories predict an asymptotic scaling state with a broad particle size
distribution. Aqueous foams are good model systems for investigations of
coarsening-induced structures, because the continuous liquid as well as the
dispersed gas phases are uniform and isotropic. We present coarsening
experiments on wet foams, with liquid fractions up to their unjamming point
and beyond, that are performed under microgravity to avoid gravitational
drainage. As time elapses, a self-similar regime is reached where the
normalized bubble size distribution is invariant. Unexpectedly, the
distribution features an excess of small roaming bubbles, mobile within the
network of jammed larger bubbles. These roaming bubbles are reminiscent of
rattlers in granular materials (grains not subjected to contact forces). We
identify a critical liquid fraction $\phi^{*}$, above which the bubble
assembly unjams and the two bubble populations merge into a single narrow
distribution of bubbly liquids. Unexpectedly, $\phi^{*}$ is larger than the
random close packing fraction of the foam $\phi_{rcp}$. This is because,
between $\phi_{rcp}$ and $\phi^{*}$, the large bubbles remain connected due to
a weak adhesion between bubbles. We present models that identify the physical
mechanisms explaining our observations. We propose a new comprehensive view of
the coarsening phenomenon in wet foams. Our results should be applicable to
other phase-separating systems and they may also help to control the
elaboration of solid foams with hierarchical structures.

_Keywords_ Foams $\cdot$ Coarsening $\cdot$ Ostwald Ripening

## Significancy Statement

Coarsening is a ubiquitous phenomenon in phase separations. It is widely
observed in alloys, polymers, emulsions, foams and even in biological systems.
However, coarsening of materials where the two phases have comparable volume
fractions is still poorly understood. To fill this gap, we performed
coarsening experiments on aqueous foams in microgravity – free from gravity-
driven destabilization. We discovered that coarsening naturally produces,
besides large jammed bubbles, a significant proportion of small roaming
bubbles. This hierarchical size distribution is surprising, but could be
general and exist in other coarsening systems. Foaming being a generic method
to produce solid cellular materials with many applications, making use of
these roaming bubbles opens up a new way of designing hierarchical materials.

##  1 Introduction

Liquid phase separation is a common phenomenon in material processing or
aging. In the late stages of separation, the dispersed domains grow, in order
to decrease interfacial energy. This often occurs as the dispersed phase
diffuses through the continuous phase. This process, known as coarsening or
Ostwald ripening, is sometimes referred to as ‘thermodynamic capitalism’ [58],
where big entities get bigger at the expense of small entities which
disappear.

Coarsening kinetics determines the microstructure of the materials, i.e. the
average domain size and size distribution, which is why it has been studied in
widely varying contexts. The coarsening of alloys has been extensively studied
[56], but coarsening also affects crystallization of proteins [40], separation
by chirality [42], synthesis of small particles of controlled microstructure
including quantum dots [2, 20, 47], stability of foams and emulsions [59, 6,
31] as well as other complex liquid-liquid phase separations [46]. Coarsening
has recently emerged as an important route to protein compartmentalization
within living cells [21].

The theoretical description of coarsening goes back to Lifshitz and Slyozov
[33], and to Wagner [57], and is usually referred to as ‘LSW theory’. This
theory is only valid in the limit of high volume fraction $\phi$ of the
continuous phase. Nonetheless, the LSW theory correctly predicts the
coarsening behavior in numerous systems, namely, that the time-dependent
average domain radius increases with time as $t^{1/3}$ and that the normalized
size distribution of the domains (PDF) is invariant with time. Many efforts
were subsequently made to account for the behavior at smaller $\phi$ by means
of theoretical methods and by computer simulations [3, 60]. However, the
evolution of the growth regime in $t^{1/3}$ toward the so called ’grain growth
regime’ in $t^{1/2}$ established for vanishing $\phi$ remains mostly
unexplored, both experimentally and theoretically. In view of the large
variety of phase separating systems and their existing and potential
applications, improving the knowledge on this transition is highly desirable.
We have chosen liquid foams as model systems because the domains are fluid and
isotropic.

Liquid foams are metastable dispersions of gas bubbles in a liquid matrix [59,
6, 31, 18]. They are not only interesting model systems for coarsening
studies, they have numerous practical applications. Solidifying the continuous
phase of liquid foams yields solid materials which inherit the structure of
their precursors. Solid foams are widely used for packaging, insulation or as
lightweight construction materials such as foamed cement or metallic foams.
Their solid volume fraction is frequently chosen between $20\%$ and $50\%$, to
confer sufficient mechanical strength [18]. The solid foam microstructure has
an impact on its mechanical properties, for a given density. Hierarchical foam
structures were predicted to have an order of magnitude improvement of
mechanical strength-to-density ratio with just two levels of hierarchy [28].
Therefore, such hierarchical structures self-assembled by foam coarsening, as
we report here, could be of great interest for applications.

When the liquid volume fraction $\phi$ is large, the bubbles are spherical and
isolated, the dispersions are called ‘bubbly liquids’ rather than foams.
Coarsening is expected to lead to a bubble growth proportional to $t^{1/3}$,
as in Ostwald ripening. When the liquid fraction $\phi$ is decreased below a
critical value, $\phi_{\text{rcp}}$, contacts between neighboring bubbles are
formed and their shapes progressively evolve from spheres to polyhedra in the
limit $\phi\rightarrow 0$ [6]. Equilibrium films separating neighboring
bubbles have generally thicknesses of a few tens of nanometers. They are
connected three by three to channels called Plateau borders, themselves
connected at vertices. For disordered monodisperse foams,
$\phi_{\text{rcp}}\approx 36\%$, and $\phi_{\text{rcp}}$ is expected to
decrease slightly as polydispersity increases [14]. Experiments with 3D foams
of small liquid fractions have shown that the average bubble radius grows at
long times as $t^{1/2}$ [11, 30, 23, 8], in contrast with the $t^{1/3}$
scaling observed in the case of Ostwald ripening. The value of the exponent is
related to the mechanism of gas transfer between bubbles. In dry foams, it
occurs mostly through the thin films between bubbles, whereas in bubbly
liquids, gas is transferred through bulk liquid [39]. Another important
feature of coarsening is the shape of the bubble size distribution that is
also expected to change with liquid fraction. Several experimental and
numerical works [16, 29, 35, 30, 54, 61] show that the normalized distribution
is asymmetric, of the Weibull or lognormal type, in the regime associated with
the $t^{1/2}$ growth law, whereas for the $t^{1/3}$ regime, it is more
symmetric and narrower [33].

Foams evolve with time not only because of coarsening but also due to gravity
drainage [50, 6], and possibly due to rupture of liquid films separating
neighboring bubbles, called coalescence. Since gravity drainage and coarsening
are coupled, studying and modelling coarsening requires gravity drainage to be
suppressed. Pioneering foam coarsening experiments were performed with dry
horizontal 2D foams (single layers of bubbles) where drainage was not an issue
[53]. Studies of 3D foams on Earth are generally restricted to small liquid
fractions $\phi\ll 0.1$, where drainage is slow enough [5, 11].

To rule out artifacts related to gravity in 3D foams whatever the liquid
fraction, we have performed foam coarsening experiments in microgravity, on
board the International Space Station (ISS), where drainage is suppressed.
Samples with arbitrary liquid volume fractions $\phi$ can thus be studied over
long times, up to several days, as required to investigate the Scaling State
of foam containing a significant fraction of liquid.

##  2 Results and Discussion

###  2.1 Excess of small bubbles

Figure 1: Excess of small bubbles. (a) Image of foam surface ($\phi=15\%$) in
the Scaling State regime. Yellow stars have been superimposed on the image to
highlight the small bubbles corresponding to the sharp peak in the
distribution shown in (b). (b) Probability density function of normalized
bubble radius $\rho=R/\langle R\rangle$ at different foam ages as indicated,
for a foam with liquid fraction $\phi=15\%$. The curve corresponding to age
$>$ 2000 s represents the Scaling State regime, for which the normalized
distribution no longer evolves. Inset: evolution of the proportion of small
bubbles as a function of time. The number fraction f${}_{\text{small}}$ is
obtained by dividing the number of bubbles with radius $R<R_{t}$ by the total
number of bubbles in the sample (see Section 2.2 for details). A change in
$R_{t}$ by $\pm 5\%$ induces a variation of f${}_{\text{small}}$ smaller than
the point size. (c) Probability density function of normalized bubble radius
at different ages as indicated, for a sample with liquid fraction $\phi=8\%$
studied on ground.

We have investigated foam coarsening for liquid fractions between 15% and 50%
using the instrument described in [4]. Details can be found in the Materials
and Methods section. From the sample surface observations (a typical image is
shown in Fig. 1a), we measure the bubble sizes using image analysis, and
determine the bubble size distributions of the radius normalized by its
average $\rho=R/\langle R\rangle$. The initial size distributions produced by
our experimental setup are asymmetric (positive skew) with a maximum at
$\rho\approx 0.6$ (see Figure 1b for foam with $15\%$ liquid fraction as an
example). The normalized size distributions broaden with time, and a sharp
peak builds up progressively for small bubble sizes, i.e. $\rho\approx 0.3$,
until a stationary form is reached, indicating a Scaling State. This is shown
in Figure 1b for times $t>2000$ s. This evolution is typical of the
measurements we have made for foams with liquid fractions within the range
$15\%\leq\phi<\phi^{\star}$, with $\phi^{\star}\approx 39\%$. The small
bubbles corresponding to the peak in the distribution are highlighted in
Figure 1a. After an increase in the transient regime, they finally represent
about $35\%$ of the total bubble population in the scaling state (inset of
Figure 1b). We also measured the number of those small bubbles per foam vertex
to reach a maximum average value of 1.5, due to space limitation in the
vertices. As a consequence, the size distribution becomes invariant in time
(statistically self-similar) as observed.

Up to now, such an excess of small bubbles has not been reported in the
literature [35, 16, 30, 61]. In order to check if distributions with an excess
of small bubbles are also found in drier foams, we have performed coarsening
experiments using the same surfactant and a liquid fraction of $8\%$, low
enough for gravity effects to be compensated in a ground based experiment by
rotating the cell around a horizontal axis (clinostat). As shown in Figure 1c
we observed a similar excess of small bubbles. The small bubbles were thus
seemingly present, but not detected in previous studies. This is probably
because high spatial resolution together with a careful image analysis are
needed [44]. The only experimental work we have found that indirectly relates
to this is that of Feitosa and Durian [16], which reports the development of
transient bidispersity for initially monodisperse bubbles in a Steady State
column, where drainage and coarsening occur simultaneously. In their
simulations of 2D foam coarsening, Khakalo et al [26] have observed an excess
of small bubbles but the gas transfer through interstitial bulk liquid was not
taken into account. Other peculiar size distributions such as “Apollonian”
distributions were observed during the decay of beer foam [51] and with
emulsions [27]. In contrast to what happens in our systems, they arose from
coalescence events.

For $\phi>\phi^{\star}$ we have observed a different scenario: the initial
bubble size distribution shrinks until a steady state is reached where the
size distribution is notably narrow (see Fig. S1 in the SI). The latter
distribution is reminiscent of the theoretical distribution predicted for the
Ostwald regime [33, 57]. Around $\phi^{*}$, a change in the growth laws for
the average bubble size is also observed for the same foam samples[45]:

$R_{32}^{2}(t)=R_{32}^{2}(0)+\Omega_{p}\;t\quad\quad\textrm{for}\quad\phi<\phi^{*}$
(1)
$R_{32}^{3}(t)=R_{32}^{3}(0)+\Omega_{c}\;t\quad\quad\textrm{for}\quad\phi>\phi^{*}$
(2)

The Sauter mean radius $R_{32}=\langle R^{3}\rangle/\langle R^{2}\rangle$ is
defined as the ratio of third to second moments of the bubble radius
distribution.

###  2.2 Transition from jammed bubbles to roaming bubbles

Figure 2: Roaming transition: (a) Evolution of the area of individual bubbles
as a function of foam age measured as the time elapsed since the end of the
foam sample production, for $\phi=15\%$. The area $A_{t}=\pi R_{t}^{2}$
denotes the bubble area at the wall when its shrinking abruptly slows down
(see text). Each label corresponds to a different bubble. (b) The transition
to the very small shrinking rate was observed to occur when the foam bubble
has become so small that it fits inside the interstice between neighboring
larger bubbles. The corresponding geometrical transition can therefore be
described as follows: When its radius is larger than $R_{t}$, the small bubble
is a foam bubble, in the fact that it shares thin liquid films with its
neighbors. In contrast, as its radius reaches values smaller than $R_{t}$, the
bubble loses its contacts with its neighbors: it becomes a roaming bubble and
its shrinking rate is strongly decreased. (c) Coefficient $x_{n}=R_{t}/R_{32}$
as a function of $\phi$. Filled orange disks: values deduced from the tracking
of individual bubbles. Error bars show $\pm 3SD$, to highlight the observed
variability. Black stars/drawings: calculation of $x_{n}$ from the size of a
hard sphere (in red) that can be inserted into the interstice formed by three
spheres at the wall, assuming either a compact bubble cage (bottom) or slight
loosening (top) of the latter. The dotted line corresponds to equation 4 with
$\xi=2.2$.

To clarify the origin of the hierarchical bubble population, we have
identified bubbles that eventually disappear and tracked evolution of their
area. Figure 2a shows examples of such measurements in a foam with
$\phi=15\%$. Similar data are shown for other liquid fractions between
$\phi=20\%$ and $\phi=33\%$ in the Supplementary Information (cf. Figure S2).
Over time, the individual bubble area can either increase or decrease,
depending on the bubble’s gas exchanges with its neighbours, but most of the
observed bubbles eventually shrink (see Figure 2a). The magnitude of the
shrinking rate appears to be initially similar to that characterizing the
initial growing rate. Then, a transition occurs and the area decreases much
more slowly. Actually, the shrinking after this transition can be extremely
slow, and we think this is the underlying mechanism explaining why a peak at
smaller than average bubbles builds up in the size distribution. Remarkably,
the bubble radius at the transition, $R_{t}$, is such that its area $A_{t}=\pi
R_{t}^{2}$ increases linearly with time, which is similar to the evolution of
the squared mean radius in the Scaling State (Eq. 1). Moreover, the transition
to the very small shrinking rate appears to occur when the bubble has become
so small that it fits inside the interstice between three larger bubbles at
the surface, and possibly loses contacts with them as sketched in Fig. 2b.
(See movies S1-S3 in the SI.) They are free to move throughout the interstice
without being pressed against multiple neighbors. Such small bubbles can have
different configurations in the interstice, i.e. near the center of the
interstice or in contact with one bubble or two bubbles, but these
configurations do not last for the entire life of the bubbles because their
positions are jostled as the jammed bubbles intermittently rearrange due to
the coarsening induced dynamics [9, 6]. We call the small bubbles roaming
bubbles. Note that they are reminiscent of rattlers (grains carrying no force)
in granular media [1]. We conjecture that the bubble size at the transition,
$R_{t}$, should scale as the maximum radius of a sphere that can be trapped in
such an interstice at the wall surface.

In a coarsening foam that has reached the Scaling State, there is only one
independent length scale of the bubble packing structure. Since the bubbles
that form the interstices are bigger than the encaged roaming bubbles, we
chose to characterize their average size by the Sauter mean radius. With
respect to $\langle R\rangle$, $R_{32}$ indeed represents mainly the average
radius of the larger bubbles of the distribution and minimizes the
contribution of the small bubbles. At a time $t$, the maximum radius of a
sphere trapped in such a vertex can be written, on average:

$R_{t}(t,\phi)=x_{n}(\phi)R_{32}(t)$ (3)

where $x_{n}(\phi)$ is a dimensionless geometrical coefficient. We show in
Figure S3 of the SI the plots of $R_{t}$ versus $R_{32}$ for each liquid
fraction. The plots are reasonably described by equation 3, allowing
determination of the average coefficient $x_{n}$ for each liquid fraction (see
Figure 2c). $x_{n}(\phi)$ varies from $0.25$ to $0.55$ as $\phi$ varies from
$15\%$ to $38\%$ respectively. Using those $x_{n}$ values, the transition
radii $R_{t}$ collapse on a linear master curve when plotted versus
$x_{n}(\phi)R_{32}$ (cf. Figure S3 of the SI).

We have performed a geometrical calculation of the size of the interstice
between a plane and three perfect spheres of equal radius $R_{32}$ in contact
together and with the plane (see Figure 2c). This leads to $x_{n}=1/3$. This
value is smaller than what is measured for liquid fractions corresponding to
the bubble random close packing fraction, i.e. $\phi_{\text{rcp}}\approx 31\%$
(see section 2.4 for more details), beyond which the bubbles are spherical. As
the liquid fraction gets close to $\phi_{\text{rcp}}$, the foam osmotic
pressure, which pushes neighboring bubbles against each other at contacts,
becomes very low, and it can be inferred that the cage formed by the triplets
of bubbles of radius $R_{32}$ loosens. Note that such a geometrical loosening
effect is general and independent of friction [10]. Therefore, as a correction
to the previous calculation, a distance $\epsilon R_{32}$ is added around each
sphere (see Figure 2c). The coefficient now reads:
$x_{n}=\frac{\epsilon(2+\epsilon)+\scriptscriptstyle\frac{4}{3}(1+\epsilon)^{2}}{2(2+\epsilon)}\approx\frac{1}{3}+\epsilon$,
and it increases significantly due to the loosening effect: assuming a
moderate loosening $\epsilon\approx 0.$ gives $x_{n}\approx 0.5$ which is in
better agreement with our measurements (see Figure 2c and movie S2). It is
reasonable to assume that polydispersity may also impact the size of the
interstice. This effect can be estimated by considering two bubbles of size
$R_{32}$ and a third one with size $\beta R_{32}$. It can be shown that in
such a case, the coefficient reads $x_{n}\approx\frac{1}{3}+0.11(\beta-1)$.
Therefore, the magnitude of the polydispersity effect is much weaker than the
previous one, in addition to the fact that it can work in both directions,
depending on the value of $\beta$, which we observed to vary in the range
$0.3<\beta<1.5$ (see Figure S4 of the SI). However, it is worth noting that a
significant fraction of bubbles have a radius larger than $R_{32}$, i.e.
$1\leq\beta\leq 1.5$, and that almost half of the nodes are bounded by one
such large bubble (see Figure S4 as an example for foam with $15\%$ liquid).
These findings suggest that the effect of polydispersity is only slightly
positive, and should only slightly increase $x_{n}$ i.e. the size of the wall
interstice. We conclude that the loosening of the bubble packing is the main
effect accounting for the measured $x_{n}$ values.

To extend our prediction to any liquid fraction $\phi\leq\phi_{\text{rcp}}$,
we turn to [34], where the radius of passage of a hard sphere through the
liquid channels (so-called Plateau borders [6]) was determined as a function
of $\phi$ and bubble radius $R$ in a monodisperse foam. Due to uniformity of
the capillary pressure through the foam, which sets the radius of curvature of
the channels, and thus their cross-section, the bubble radius at the
transition $R_{t}$ should be proportional to this radius of passage. Following
the approach proposed in [34] we refer to the effective pore radius introduced
by Johnson et al. [24]:
$\Lambda\approx(8\overset{\sim}{{k}}/\overset{\sim}{{\sigma}})^{1/2}R$, where
$\overset{\sim}{{k}}$ is the dimensionless liquid Darcy’s permeability through
the foam structure, i.e. $k/R^{2}$, and $\overset{\sim}{{\sigma}}$ is the
ratio of the electrical conductivity of the foam to that of the foaming
liquid. Therefore, the expression sought for $x_{n}$ is:

$x_{n}=\xi(8\overset{\sim}{{k}}/\overset{\sim}{{\sigma}})^{\frac{1}{2}}$ (4)

where $\xi$ is a geometrical coefficient to be determined. Note that the
latter is expected to account for the loosening and polydispersity effects
discussed previously. To continue we now need expressions for
$\overset{\sim}{{k}}$ and $\overset{\sim}{{\sigma}}$. Since $\Lambda$ was
initially proposed for solid porous media, the permeability should correspond
to foam having rigid interfaces to mimic solid-like boundary conditions. As
studied by Rouyer et al. [48], its expression is given by:
$\overset{\sim}{{k}}=\phi^{2}/(312(1-2.15\phi+1.37\phi^{2})^{2})$ within the
range of liquid fractions $1\%\leq\phi\leq 40\%$. For foams and bubbly
liquids, Feitosa et al. [15] proposed an approximate analytical expression for
$\overset{\sim}{{\sigma}}$, i.e.
$\overset{\sim}{{\sigma}}=2\phi(1+12\phi)/(6+29\phi-9\phi^{2})$. Using these
expressions, we set $\xi=2.2$ in equation 4 in order to get a predicted value
of $x_{n}$ close to the measured value 0.53 for $\phi\approx\phi_{\text{rcp}}$
(see Figure 2c). Remarkably, the agreement with our experimental data is very
good over the whole range of liquid fractions, which reinforces the physical
picture that $R_{t}$ actually corresponds to the size of the interstices
formed by the jammed bubbles around the roaming bubbles. Note that in all of
the above, nothing is really specific to the fact that we are looking at the
wall. In bulk, typical interstices are formed by four bubbles in a tetrahedral
assembly. The geometrical calculation for four bubbles in contact gives
$\sqrt{\scriptscriptstyle\frac{3}{2}}-1\approx 0.225$, compared to $1/3$ at
the wall. Therefore we can estimate $x_{n}$ for bulk by using equation 4 with
coefficient $\xi=2.2\times(0.225/0.333)\approx 1.5$. Provided this value is
used, the behavior observed at the wall should be similar to the behavior
observed in the bulk of the foam.

###  2.3 Dissolution rate of the roaming bubbles

Figure 3: Roaming bubble dissolution: (a) Radius evolution of dissolving
roaming bubbles where each curve represents a single bubble. The solid lines
correspond to fits of Eq. 5. (b) Average shrinking rate of roaming bubbles
$\Omega_{r}$ as a function of liquid fraction compared to the growth rate of
average bubble size in the foam $\Omega_{p}$ (Eq. 1, data from [45]). The
lines are guides to the eye. $\Omega_{r}$ values fall within the range
(highlighted in green) predicted by the shell model (Eq. 1 in the SI),
schematically illustrated by the inside drawing. Error bars correspond to $\pm
1SD$. The growth rate $\Omega_{p}$ is strongly dependent on the liquid
fraction, at the difference of the dissolution rate $\Omega_{r}$. (c) Measured
shape parameter $\sigma_{2}$ of the jammed bubbles size distribution (Eq. 6 in
the SI) as a function of liquid fraction (blue circles). The (orange)
continuous line represents the maximum packing volume fraction predicted for a
lognormal distribution of spheres with shape parameter $\sigma$ [13, 14]. The
gray vertical area highlights the range where $\sigma$ and $\sigma_{2}$
coincide, from which we deduce $\phi_{\text{rcp}}\approx 30-32\%$. This also
corresponds to the range of liquid fractions where $\Omega_{r}$ is comparable
to $\Omega_{p}$ in b.

In this section, we focus on the dissolution rate of the roaming bubbles in
the range $\phi<\phi^{*}$. We first consider the data for times longer than
those that mark the intersection of the dissolution curve with $A(t)=\pi
R_{t}^{2}$ (Figures 2a and S2 in the SI). We follow the evolution of the
radius of roaming bubbles $R(t)$ for $R(0)\lesssim R_{t}$. For comparison, we
similarly analyze individual bubbles roaming in the bubbly liquids
($\phi>\phi^{*}$), from the instant they start to continuously shrink. Several
examples of the curves are presented in Figure 3a. We observe that the
following function fits well all the curves[12, 37]:

$R^{2}(t)=R^{2}(0)-\Omega_{r}t$ (5)

where the only fitted parameter $\Omega_{r}$ represents the dissolution rate
of the roaming bubble. Such fits were performed for all the liquid fractions
and the average values of $\Omega_{r}$ are presented in Figure 3b.
$\Omega_{r}$ is found to depend only weakly on liquid fraction:
$\Omega_{r}\approx 1-2~{}\mu$m${}^{2}/s$. We also plot on Figure 3b the growth
rate $\Omega_{p}$ that characterizes the coarsening of the foam in the Scaling
State (Eq. 1). It appears that $\Omega_{p}\gg\Omega_{r}$ for
$\phi\lesssim\phi_{\text{rcp}}\approx 31\%$, and $\Omega_{p}\approx\Omega_{r}$
for $\phi_{\text{rcp}}<\phi<\phi^{*}$. This comparison reinforces our
discussion in section 2.1: the size of the roaming bubbles, represented on the
left side of the distribution, varies more slowly than the average bubble
size. As a result, the roaming bubbles accumulate in the interstices formed by
the larger bubbles.

As the dissolution rate $\Omega_{r}$ plays a crucial role in the accumulation
mechanism of the roaming bubbles, we seek here to understand this value. The
starting point is the comparison of our data with theory for the dissolution
of isolated bubbles [12, 37], which gives the steady dissolution rate far
enough from the final instant of bubble disappearance as:
$\Omega_{r}=-dR^{2}/dt=2D_{m}V_{m}\left(c(R)-c_{\infty}\right)=2D_{m}V_{m}\text{He}P_{0}(1-\zeta)$,
where the saturation parameter $\zeta=c_{\infty}/\text{He}P_{0}$ characterizes
the gas saturation of the liquid environment, $c(R)$ and $c_{\infty}$ are
respectively the gas concentrations in the liquid at the bubble surface and at
infinity, $P_{0}$ is the gas pressure at infinity, He and $D_{m}$ are
respectively the Henry solubility and the diffusion coefficient of the air
molecules in the foaming solution. $V_{m}$ is the molar volume of the gas at
the pressure $P_{0}$. From the measured $\Omega_{r}$, we deduce an effective
value for the saturation parameter: $\zeta=0.973-0.987$, which suggests that
the bubbles dissolve faster than if they were isolated, and despite the
presence of the large neighbouring bubbles which impose at their interface a
gas concentration larger than $\text{He}P_{0}$. To explain this apparent
contradiction, it is important to understand that the gas transfer is
controlled by the concentration gradient, and not only by the concentration
difference. Due to the short distances involved between the roaming bubble
interface and the interfaces of the large neighbouring bubbles, the
concentration gradient around the roaming bubble reaches relatively high
values compared to the case of the isolated bubble. Therefore, to mimic this
situation, we consider the configuration illustrated in the inset of Figure
3b, where a roaming bubble of radius $R$ is centered in a cavity of radius
$R_{t}$, and is surrounded by a liquid shell of thickness $R_{t}-R$. The local
concentration at the outside boundary of the shell is estimated as that at the
surface of a bubble of average size $R_{32}$. From Fick’s first law, we then
predict the bubble dissolution rate $\Omega_{r}$ in that shell environment
(See more details in the SI). For the range of values of $R_{32}$ in the
scaling state in our experiments and typical ratio $R/R_{t}$, we expect
$\Omega_{r}\approx 0.75-4~{}\mu$m2/s which provides boundaries consistent with
the measured values of $\Omega_{r}$ (cf. Fig. 3b).

A drawback of this shell-like model is that the roaming bubble is assumed to
remain at the center of the interstice, which is not always the case. Indeed,
we often noticed transient apparent contacts between the roaming bubble and
either one of the bubbles delimiting the interstice or two larger bubbles
forming a corner. These transient contacts can result from adhesive forces. We
have indeed observed that under microgravity conditions, persistent aggregates
form spontaneously in dilute bubble dispersions. In complementary ground-based
experiments, we have observed a contact angle close to $3-4^{o}$ [45]. The
underlying configuration may be an adhesive contact with the formation of a
liquid film that slightly flattens the bubbles or it can be a near-contact
with a small separation distance so that the roaming bubble is spherical.
Since it was not possible to distinguish between these two types of contact,
we estimated the dissolution rate for both cases (See details of the
calculation in the SI). In the range of average bubble sizes $R_{32}$ of our
experiments, assuming a film thickness effective for the transport of gas of
the order of 40-60 nm [45], we found that the expected rates fall within the
range of values measured for $\Omega_{r}$. This remains broadly true if the
bubble is in a corner, where the corresponding dissolution rate is twice
larger. Therefore, whatever the configuration considered for the roaming
bubble in the interstice, we find values for its dissolution rate that are
compatible with our measurements, which gives robustness to the proposed
mechanism based on the accumulation of long-lasting roaming bubbles in the
foam interstices.

###  2.4 Bubble size distributions and random close packing fraction in the
Scaling State

Let us analyze now the role of liquid fraction on the distribution shape.
Details on the analysis are given in the SI. Figure 4 shows the normalized
bubble size distributions observed in the Scaling State for each sample liquid
fraction.

The PDF for $\phi=15\%$ is the same as that of Figure 1 in the Scaling State.
It exhibits a prominent narrow peak, that we identified to the roaming bubble
population in Section 2.1, followed by a broad peak for the foam bubble
population. These features qualitatively persist up to $\phi<38\%$ but the
narrow peak progressively shifts towards larger $\rho$ while its height
decreases. For $\phi\geq 40\%$, PDFs exhibit a single peak, which is
consistent with the fact that all bubbles should be roaming bubbles. PDFs
become narrower as $\phi$ increases and their peak height increases. This
qualitative change is also captured by the abrupt variation of statistical
quantities like polydispersity and standard deviation (cf. Fig. S5 of the SI).
None of the existing theories predict such distributions [3]. These findings
indicate a cross-over between qualitatively different PDFs occurring for a
liquid fraction $\phi^{\star}\approx 39\%$. This transition coincides with the
observed change of growth laws Eq. 1 and Eq. 2 and it is attributed to the
onset of the formation of a foam gel due to weak attraction between bubbles as
evidenced by finite contact angle at films junctions [45].

The expected jamming liquid fraction for randomly close-packed monodisperse
hard spheres is $\phi_{\text{rcp}}=36\%$. However, polydispersity will reduce
this value since smaller bubbles can fit into the interstices between larger
ones. This effect has been predicted by numerical simulations of polydisperse
close packings of spherical particles with lognormal PDF, as a function of the
shape parameter $\sigma$ [13, 14]. In our foams, the close packing concerns
the population of jammed bubbles, which are connected to each other via films.
Therefore, we compare the measured shape parameter of the foam bubble
distribution $\sigma_{2}$ to the predicted ones (cf. Fig. 3c). We find them to
coincide within the range $\phi=30\%$ and $\phi=32\%$: we expect the close
packing fraction $\phi_{\text{rcp}}$ of our foams to lay inside the range
between these 2 values.

To provide an independent result of the close packing fraction of frictionless
spheres with the polydispersity observed in our samples in the Scaling State,
we have performed molecular dynamics simulations. Since here we are only
interested in the geometrical sphere packing problem at the jamming point
where the confinement pressure and interaction forces drop to zero with
increasing $\phi$, we expect the nature of the interaction law used in the
simulations to have only a minor impact. Using Hertzian interactions, in the
framework of the molecular dynamics code LAMMPS (see Materials and Methods),
we obtained $\phi_{\text{rcp}}=$31.0$\pm 0.5\%$, in remarkable agreement with
our analysis based on the work of Farr and Groot [14]. Note that strictly
speaking, our simulations only provide an upper bound for the optimal random
close packing fraction of such polydisperse spheres, which may be obtained by
more sophisticated simulation procedures described in the literature [25].
However, in the context of our experiments, the truly relevant packing
fraction is the one of a coarsening foam. In this case we expect a local
packing which is not exactly the most compact possible one. A jammed foam
regularly undergoes rearrangements, helping it to settle into new minimal
energy configurations. This implies that in between rearrangements, the
packing is not always optimally close-packed. Simulations of this where we
also replace the Hertzian interaction by the more realistic Morse Witten law
[38, 22] are the subject of ongoing work.

###  2.5 Potential consequences on foam properties

The roaming bubbles represent a significant proportion of the total number of
bubbles. As a result, exclusion of the roaming bubbles from the determination
of the average radius leads to an up to 3-fold overestimation of coarsening
rates, depending on the liquid fraction [45].

In terms of volume fraction with respect to the liquid volume, the roaming
bubbles represent up to ten percent depending on $\phi$. It can therefore be
expected that their impact is important for certain properties. Roaming
bubbles can also modify foam drainage, where they slow down the flow of the
liquid. A study with solid spheres, located in the nodes of the liquid network
of the foam, showed that such an amount of particles in the liquid could
reduce the permeability of the foam by $40\%$ [49]. This shows the bias of
systematically ignoring their presence. Let us mention that to date, this
effect has never been taken into account in permeability modelling.

The presence of the roaming bubbles can be highly detrimental for applications
of foams where the microstructure is an important parameter, all the more so
if they migrate and accumulate in large proportions in certain places.
Moreover, for a number of these applications, the interstitial liquid is a
complex fluid, possibly with yield stress properties that will prevent gravity
from evacuating the roaming bubbles: one expects to find relatively high
volume fractions of roaming bubbles in such systems.

Figure 4: Bubble size distributions of normalized radius $\rho=R/\langle
R\rangle$ for each liquid fraction as labelled. The data are represented by
black continuous lines. The green dashed lines represent the bi-lognormal PDFs
(see Eq. 6 in the SI) fitted to the data. The red (resp. blue) shaded area
corresponds to the roaming bubble PDF $w\;\mathcal{L}(r;m_{1},\sigma_{1})$
(resp. to the foam bubble PDF $(1-w)\;\mathcal{L}(\rho;m_{2},\sigma_{2})$)
with the parameters given in Fig. S5 of the SI. In the plots for $\phi$ up to
$38\%$ the width of the roaming bubble distributions is characterized by
$\rho_{t}$, defined in Eq. 8 in the SI. For $\phi=15\%$, the dotted line is
the PDF predicted for wet foams by Markworth [36] based on Lemlich’s model
[32] for that $\phi$. As a comparison, for $\phi=50\%$, the dotted line is the
LSW prediction [3] ($\phi=1$).

By examining some recent papers on foamed construction materials, we
recognised traces of the presence of such roaming bubbles. For example in
[17], very small pores can be seen in the bulk nodes of cement foam solidified
after coarsening, as revealed by the micro-tomography image. There is every
reason to believe that literature is full of such examples.

On the other hand, one can make use of these roaming bubbles. Note that the
stakes are high in terms of producing solid foam structures with hierarchical
porosity, including both macro- and micro-scaled pores. Such structures have
recently been produced by 3D printing [7] and they were found to present
enhanced energy absorption properties and enhanced mechanical resistance to
cyclic loading.

## Conclusions

Studies of foam samples where the liquid fraction remains constant over
periods of several days, without any confounding effects of gravitational
drainage, reveal that, as demonstrated earlier for dry foams, wet foams evolve
towards a Scaling State. In this state, the bubble size distributions show a
well-defined peak towards smaller than average sizes, i.e. an excess of
bubbles for sizes close to $0.3\langle R\rangle$. This feature is not
predicted by existing theories. During coarsening, as shrinking bubbles become
smaller than the size of interstices between the larger bubbles, they can move
independently from the jammed bubble network to become roaming bubbles.
Surprisingly, although we have been able to reproduce this effect on Earth, no
previous experimental study mentions the presence of these small bubbles,
except for a study of draining foams [16] but where roaming bubbles have not
been identified as such.This suggests that, although their study has been
underestimated, hierarchical bubble size distributions can build up on Earth
if drainage is not too fast compared to dissolution and coarsening.

The dissolution rate of these roaming bubbles is approximately constant,
whatever the liquid fraction of the samples. The dissolution rate is
consistent with calculations based on the gas transfer through the liquid
shell that surrounds the roaming bubble, or through the “contact“ between one
roaming bubble and larger jammed bubbles surrounding them. The key point in
the accumulation of the small bubbles in the interstices formed by the larger
bubbles, is the fact that the rate of disappearance of these bubbles is much
smaller than the average growth rate of the jammed bubbles. This behaviour is
observed for foams with liquid fractions smaller than the random close packing
fraction $\phi_{\text{rcp}}$. For $\phi$ between $\phi_{\text{rcp}}$ and
$\phi^{\star}$, where the bubble assembly approaches the regime of bubbly
liquids, the rate of dissolution of the roaming bubbles reaches progressively
the growth rate of the jammed bubbles, which suppresses the accumulation
mechanism. As a consequence, the peak initially observed for liquid fractions
$\phi<\phi_{\text{rcp}}$ shifts towards $\langle R\rangle$ and a distribution
almost centered on $\langle R\rangle$, characteristic of bubbly liquids, is
eventually observed. For $\phi$ above $\phi*$, none of the bubbles are
confined.

In closing, we have shown the existence of naturally developed hierarchical
bubble size distributions in coarsening foams. We present a comprehensive view
of coarsening of wet foams, completely different from expectations, with a
persistent co-existence of jammed bubbles with small roaming bubbles, and the
existence of a range between $\phi_{\text{rcp}}$ and $\phi*$ where foam
bubbles are still jammed although not close-packed. These findings challenge
our current understanding of foam coarsening and has potential implications in
the design and performance of foamy materials. This view should not be
restricted to foams but also be applicable to other two-phase systems driven
by interfacial effects, such as emulsions, alloys and binary fluid/polymer
mixtures. It should be mentioned that recent studies of alloys with small
volume fraction of the continuous phase suggest that the Ostwald ripening
regime persists when $\phi$ is smaller than $\phi_{\text{rcp}}$ [60]. The
difference between coarsening of foams and alloys remains to be clarified.

## Materials and Methods

The foams were made with aqueous solutions of a ionic surfactant, tetradecyl-
trimethyl-ammonium bromide (TTAB), with purity $\geq 99$ % and used as
received from Sigma-Aldrich. It was dissolved at 5 g/L in ultrapure water
(resistivity 18.2 M$\Omega\cdot$cm). This concentration is 4 times larger than
the critical micellar concentration and large enough to prevent coalescence.
The surface tension of the TTAB solution measured at room temperature is:
$\gamma=37.1$ mN/m. The Henry solubility coefficient of the air molecules in
the foaming solution is [45] $He=7.4~{}10^{-6}$ mol m-3 Pa-1 [45] and their
diffusion coefficient in the foaming solution is [45] $D_{m}=2.0~{}10^{-9}$
m2s-1.

The majority of the experiments were performed on board the International
Space Station using the experiment container described in [4]. In this
environment, the residual gravity acceleration fluctuations are reported to be
on the order of or less than a µg, for frequencies below 0.01 Hz [41]. Each
foam cell was filled on Earth with a given volume of foaming solution
(measured by weight at controlled temperature) and air, then hermetically
sealed. The liquid volume fraction $\phi$ contained in each cell was deduced
from the liquid volume and the total cell volume. After the completion of the
experiments, the cells were send back to Earth and we checked that their
weight had varied by less than 1%. All the experiments were repeated three
times and found reproducible, even a few months apart.

In addition, we made a Ground experiment with a foaming liquid of composition
identical to that of the ISS experiments. The foams were produced with the
double syringe method [44] filled with air and a volume of the foaming
solution in order to set the liquid fraction to 7.8%$\pm 0.2\%$. Note that the
initial bubble size distribution with this foam production is close to that of
the Scaling State. The sample was placed in a cylindrical cell (diameter 30
mm, thickness 12.8 mm) with transparent flat faces. The cell was kept with its
symmetry axis aligned in the horizontal direction and rotated about this axis
with a speed of rotation equal to 15 rpm.

Foam age is counted from the instant when the foaming process stops. Bubbles
at the surface of the sample are recorded using a video camera. Every image
(such as the one shown in Fig. 1a) was analysed as described in [44]. We
checked that the radial profile of liquid fraction remained constant
throughout the measurement duration, indicating that the effect of gravity
drainage was indeed counteracted and that the rotation did not induce radial
drainage either.The bubble area $A$ deduced was from the area inside the
contour of the bubbles measured using the ellipses method [44]. Finally, the
bubble radius is calculated as $R=\sqrt{A/\pi}$. In the ISS experiments,
simultaneously to the video recording, the intensity of light transmitted
through the sample was recorded, which provided the average bubble size in the
bulk of the sample as explained in [44]. Our results showed that the evolution
of the average bubble radius measured either at the surface or in the bulk are
similar.

We also performed numerical simulations to evaluate the random close packing
liquid fraction of the bubbles. In the framework of the molecular dynamics
code LAMMPS [55], a cubic simulation box was filled by spheres with repulsive,
Hertzian interactions with radii randomly chosen from a distribution
corresponding to the one we observe experimentally for $\phi=33\%$ in the
Scaling State (see Figure 4). The number of spheres was of the order of 2000,
similar to our foam coarsening experiments at the largest investigated foam
ages. To fill the simulation cell, we started with an initial cell volume so
large that the sphere dispersion was highly diluted. Using the pressostat
provided by LAMMPS, we then shrunk the cubic cell and compacted these
structures until a very small osmotic pressure appeared. We then turned off
the pressostat and equilibrated the sample for imposed simulation box volumes,
varied by small steps around the previous value. The close packing fraction
was estimated by plotting confinement pressure versus packing fraction, and by
detecting the $\phi$ value where zero pressure is reached within numerical
accuracy. We did this for 5 different initial random seeds, and found
$\phi_{\text{rcp}}=$31.0$\pm 0.5\%$. The way you compact a packing has a large
impact on the final close packing fraction in frictional granular materials,
and to a lesser extent also in frictionless systems. To investigate this
effect, we applied simulated gravity to dilute sphere dispersions as an
alternative to the initial pressostat procedure. Kinetic energy was dissipated
by introducing viscous friction in the contact law. This procedure mimics
foams that form when a bubbly liquid is subjected to buoyancy, as it is common
on earth. Once equilibrium was reached, we switched off gravity, and simulated
pressure versus packing fraction as previously. The final values of
$\phi_{\text{rcp}}$ are within statistical errors the same as the those
obtained with the pressostat.

## Acknowledgments

We acknowledge funding by ESA and CNES (via the projects “Hydrodynamics of Wet
Foams”) focused on the Soft Matter Dynamics instrument and the space mission
Foam-C, as well as NASA via grant number 80NSSC21K0898. Marina Pasquet, Nicolo
Galvani and Alice Requier benefited from CNES and ESA PhD grants. The authors
are grateful to the BUSOC team for their invaluable help during the ISS
experiments. We also want to warmly thank Marco Braibanti and Sébastien
Vincent-Bonnieu from ESA, Christophe Delaroche from CNES and Olaf Schoele-
Schulz from Airbus for their continuing support.

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Supplementary Information

## Dissolution rate models

We consider the configuration illustrated in the inset of Figure 3b, where a
roaming bubble of radius $R$ is centered in a cavity of radius $R_{t}$ and
surrounded by a liquid shell of thickness $R_{t}-R$. Denoting $r$ the radial
distance from the bubble center, the gas concentrations at $r=R$ and $r=R_{t}$
are imposed by the bubble Laplace pressures, i.e. $c(R)=He\;(P_{0}+2\gamma/R)$
and $c(R_{t})=He\;(P_{0}+2\gamma/R_{32})$, where the local concentration at
the outside boundary of the shell is estimated as that at the surface of a
bubble of average size $R_{32}$. With these boundary conditions, the dissolved
gas concentration profile is only a function of the radial coordinate $r$. It
is a solution of the steady diffusion equation and is given by
$\frac{c(r)-c(R)}{c(R_{t})-c(R)}=\frac{R_{t}}{R_{t}-R}(1-R/r)$. From Fick’s
first law, the molar rate of gas transfer outwards from the roaming bubble is
equal to $-4\pi R^{2}D_{m}(dc/dr)_{r=R}$, and the resulting dissolution rate
is:

$\Omega_{r}=\frac{4\gamma
D_{m}V_{m}He}{R}\frac{(\frac{1}{R}-\frac{1}{R_{32}})}{(\frac{1}{R}-\frac{1}{R_{t}})}$
(S1)

Values provided by equation S1 depend on estimates of the different radii that
have been introduced. We choose $R_{32}$ within the range of values
corresponding to our experiment in the Scaling State, i.e. $300~{}\mu
m\lesssim R_{32}\lesssim 500~{}\mu m$ [45], and we limit the ratio $R/R_{t}$
within the range $0.2-0.8$ (for $R/R_{t}\lesssim 0.2$ the size of roaming
bubbles is measured with less precision, and for $R/R_{t}\gtrsim 0.8$ it is
difficult to be sure that the tracked bubble is still a roaming bubble). Thus,
we get values for $\Omega_{r}$ within the range $0.75-4~{}\mu$m2/s, as
represented in Figure 3b.

As explained in the Main text, we noticed transient apparent contacts between
the roaming bubble and either one of the bubbles delimiting the interstice or
two larger bubbles forming a corner. The underlying configuration may be a
real contact, i.e. with the formation of a liquid film that slightly flattens
the bubbles, or it can be a near-contact with a small separation distance, in
which case the roaming bubble remains spherical. Since it is not possible to
distinguish between these two types of contact, we estimate the dissolution
rate for both cases. First we consider the case of the near-contact. We refer
to the work of Schimming $\&$ Durian [52], who considered the so-called
kissing bubbles configuration, where the distance $h$ between the two
spherical bubbles is such that $h/R\ll 1$. We assume two bubbles of radii
$R\approx R_{t}/2\approx R_{32}/6$ and $R_{32}$ (radius of surrounding jammed
bubbles). The dissolution rate of the small bubble then writes:

$\Omega_{r}\approx 2\gamma
D_{m}V_{m}He\left(\frac{1}{R}-\frac{1}{R_{32}}\right)\times\ln\left(0.8+0.6\frac{R_{32}}{3\;h}\right)\;\approx\frac{10\gamma
D_{m}V_{m}He}{R_{32}}\ln\left(0.8+0.6\frac{R_{32}}{3\;h}\right)$ (S2)

Because of the logarithm values provided by equation S2 are weakly dependent
on $h$. A typical value for $\Omega_{r}$ is $2~{}\mu$m2/s.

Now we consider the case of a small bubble of radius $R$ and a big bubble of
radius $R_{32}$ sharing a film, of thickness $h$ and area $A$, which meets the
free surface of the bubbles with a contact angle $\theta$ [19] (cf. Fig. S7).
The dissolution rate of the small bubble is set by the Laplace pressure
difference $\Delta P$ between both bubbles, which drives the gas transfer
through the contact film and sets its curvature. It writes:

$\Omega_{r}\approx\frac{D_{m}V_{m}He}{2\pi\;R\;h}\Delta P\;A$ (S3)

The film is a spherical cap surface with its radius of curvature $R^{\star}$
(cf. Fig. S7) given by:

$\Delta
P=\frac{4\gamma}{R^{\star}}=2\gamma\left(\frac{1}{R}-\frac{1}{R_{32}}\right)$
(S4)

The surface area $A=2\pi R^{\star 2}\left(1-cos(\psi/2)\right)$. Taking the
same range of bubble radii as above $R\approx R_{t}/2\approx R_{32}/6$ and for
the small contact angles considered here, geometrical considerations shows
that at leading order $\psi/2\approx 5\theta/7$, thus
$A\approx\frac{\pi}{2}R^{\star 2}\theta^{2}$. Then the dissolution rate(Eq.
S5) writes:

$\Omega_{r}\approx\frac{12\gamma D_{m}V_{m}He}{5\;h}\;\theta^{2}$ (S5)

Values provided by Eq. S5 depend on the film thickness and contact angle. For
our foams, we have $\theta\approx 4^{\circ}$ [45]. We remark that an effective
film thickness in the range $h\approx 40-60$ nm gives a dissolution rate close
to the rate calculated for the kissing bubbles configuration, i.e. $\approx
2.6-4~{}\mu$m2/s. Note that the effective thickness accounts for both the
aqueous core film thickness and the effective length related to gas transfer
resistance from the two surfactant monolayers. Therefore, $h\approx 40-60$ nm
may well correspond to either a Newton Black Film (NBF) with significant
monolayer effect, or to a Common Black Film (CBF) with limited monolayer
effect. It seems difficult to go further in this quantitative analysis but it
is in good agreement with the effective thickness obtained from the foam
coarsening rate with the same solution [45].

To estimate the dissolution rate when the bubble is in a corner, we can, as a
first approximation, multiply the previous values by a factor of two. For the
kissing bubbles configuration we get a dissolution rate equal to the upper
range of values provided by the shell-model, i.e. $4~{}\mu$m2/s. Values for
roaming bubbles sharing two liquid films with neighboring larger bubbles
exceed a little bit that range, i.e. $5-8~{}\mu$m2/s. As we did not measure
such high values, it suggests that such a configuration, if it really exists,
is rather rare.

Therefore, whatever the configuration considered for the roaming bubble in the
interstices, we find values for its dissolution rate that are compatible with
our measurements, which gives robustness to the proposed mechanism based on
the accumulation of long-lasting roaming bubbles in the foam interstices.

## Analysis of bubble size distributions

We first transformed the measured discrete histograms into continuous PDFs
using a Gaussian Kernel estimation (with bandwidth equal to 2.5 the image
pixel size $\approx 14~{}\mu$m). We then described the two bubble populations
(either roaming bubble or jammed bubble) in the domain $\phi<\phi^{\star}$
with a bimodal lognormal PDF defined as:

$\mathcal{F}=w\cdot\mathcal{L}(\rho;m_{1},\sigma_{1})+(1-w)\cdot\mathcal{L}(\rho;m_{2},\sigma_{2})$
(S6)

where $w$ is the proportion of roaming bubbles in the foam, and $\mathcal{L}$
is a lognormal distribution parameterized as:

$\mathcal{L}(\rho;m,\sigma)=\frac{1}{\rho\sigma\sqrt{2\pi}}\exp{\left[-\frac{\log^{2}{(\rho/m)}}{2\sigma^{2}}\right]}.$
(S7)

where the shape parameter $\sigma$ is the log-scale standard deviation and $m$
is the linear-scale median. We fitted the bimodal function to the measured
PDFs, and observed that the parameter $m_{2}=1.66\pm 0.03(SEM)$ is almost
independent of the liquid fraction. In the following, we fix $m_{2}=1.66$ and
fit the other parameters. Their variations with liquid fraction are given in
Fig. S5. The results of the Kolmogorov-Smirnov statistical test [43] applied
to the bimodal lognormal fits provide quantitative evidence for fit quality:
no fit could be rejected according to the $5\%$ rule, and many are above
40$\%$ (with the exceptions of $p(\phi=15\%)=21\%$ and $p(\phi=25\%)=14\%$).

Finally, since $R_{t}$ should represent the maximum size of the roaming
bubbles, we conjecture that it should be correlated to the width of the
roaming bubble PDF. To test this, we measure the width at the foot of the PDF
$\mathcal{L}(\rho;m_{1},\sigma_{1})$ by the normalized radius $\rho_{t}$
estimated such that number of roaming bubbles with $\rho>\rho_{t}$ equals the
number of jammed bubbles with $\rho<\rho_{t}$:

$(1-w)\int_{0}^{\rho_{t}}\mathcal{L}(\rho;m_{2},\sigma_{2})=w\int_{\rho_{t}}^{\infty}\mathcal{L}(\rho;m_{1},\sigma_{1}).$
(S8)

We have estimated $\rho_{t}(\phi)$ up to $\phi=\phi^{*}$, and in Figure S6 we
compare it to $x_{n}(\phi)$. Since we employed $R_{32}$ as the reference
radius in the estimation of $x_{n}$, we must rescale $\rho_{t}$ by $R_{32}$
for a comparison with $x_{n}$. In the range of liquid fractions up to
$\phi=\phi_{\text{rcp}}$, we find $\rho_{t}\;\langle R\rangle/R_{32}\approx
x_{n}$, which is consistent.

## Additional Figures

The additional figures support discussions in the Main text. We did not
include text there, the captions being self-explanatory.

Figure S1: Probability density function of the normalized bubble radius
$\rho=R/\langle R\rangle$ at different foam ages as indicated, for a foam with
liquid fraction $\phi=50\%$. The curve corresponding to age $>$ 20000 s
represents the Scaling State regime (observed up to 300000s - end of the
experiment), for which the normalized distribution no longer evolves. This
distribution is an example of a concentrated bubbly liquid, with a single peak
and a narrow distribution with bubble sizes strictly smaller than $\rho=2$.
Figure S2: Evolution of the area of individual bubbles as a function of foam
age versus the time elapsed since the end of the foam sample production, for a
series of liquid fractions. The area $A_{t}=\pi R_{t}^{2}$ denotes the bubble
area when its shrinking abruptly slows down (see text). Each color corresponds
to the evolution of a different bubble. For samples with $\phi=30\%$ or
$38\%$, data were acquired in parallel with other samples. As a consequence,
there are some blanks in the image in between the vertical lines.

Figure S3: (left) Bubble radius $R_{t}$ as a function of the Sauter mean
radius $R_{32}$, measured at the instant of the transition where the shrinkage
rate slows down and the bubble starts to roam. A linear relation (dashed line)
is fitted to the data for each liquid fraction $\phi$, to determine the
coefficient x${}_{n}(\phi)$ defined in Eq. 3. (right) Master curve of the
bubble radius $R_{t}$ plotted as a function of $x_{n}(\phi)\;R_{32}$. The
dotted line has a slope equal to unity.

Figure S4: Image of a sample with $\phi=15\%$ in the Scaling State (foam age
$t=9645s$). The maximum radius of a roaming bubble, $R_{t}$, at this foam age
is predicted from the value of $R_{32}$ at this age using Eq. 3. Each bubble
is then classified as a roaming (resp. foam) bubble if its radius is smaller
(resp. bigger) than $R_{t}$. In the figure, roaming (resp foam) bubbles are
identified by red (resp. cyan) spots at their centers. We used the jammed
bubbles’ center positions to analyze the foam structure by triangulation
(black lines joining the centers in overlay). We estimated the number of
surface nodes, identified by green spots, and localized at the barycenter of
the triangles. On average, we counted 1.5 roaming bubbles per node, and 1.2
nodes per jammed bubble. (right) CDF of the number fraction of nodes around
bubbles as a function of their normalized radius $R/R_{32}$, evaluated for the
sample shown in the left. We see that the median is very close to $R=R_{32}$
which means that nearly half of the nodes are delimited by a jammed bubble
larger than $R_{32}$. This is consistent with the choice of $R_{32}$ as the
characteristic radius of jammed bubbles constituting the nodes. Similar
findings are found for liquid fractions up to $\phi_{rcp}\approx 31\%$. Figure
S5: Variation of the fitted PDF parameters, as defined in the text (Eq. 6 and
7), with liquid fraction. a) Natural-scale median $m_{1}$ of the roaming
bubbles (the median $m_{2}=1.66$ is fixed). b) Logarithmic-scale standard
deviation : $\sigma_{1}$ for the roaming bubble PDFs (red disks), $\sigma_{2}$
for the jammed bubble PDFs (blue diamonds). c) Relative weight $w$ of the
roaming bubble distribution. d) Polydispersity (squares), defined as
${R_{32}}/{\langle R^{3}\rangle^{1/3}}-1$, and standard deviation (triangles)
evaluated from raw data. Error bars are of the size or smaller than the symbol
size. Figure S6: Characteristic width of the roaming bubble PDFs $\rho_{t}$
(defined by Eq. 8) rescaled by $\langle R\rangle/R_{32}$ as a function of the
liquid fraction. The dotted line represents Eq. 4 with $\xi=2.2$ as in fig 2c.
Figure S7: Bubbles of radii $R$ and $R_{32}$ sharing a film (in blue) due to
adhesion forces accounted for by the contact angle $\theta$. $R^{\star}$ is
the radius of curvature of the shared film.


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