# The Life of a Surface Bubble

Jonas Miguet1 Florence Rouyer2 and Emmanuelle Rio3
1 TIPS C.P.165/67, Université Libre de Bruxelles, Av. F. Roosevelt 50, 1050
Brussels, Belgium<EMAIL_ADDRESS>
2 Lab Navier, Univ Gustave Eiffel, Ecole des Ponts, CNRS, Marne-la-Vallée,
France<EMAIL_ADDRESS>
3 Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405,
Orsay, France<EMAIL_ADDRESS>
###### Abstract

Surface bubbles are present in many industrial processes and in nature, as
well as in CO2 beverage. They have motivated many theoretical, numerical and
experimental works. This paper presents the current knowledge on the physics
of surface bubbles lifetime and shows the diversity of mechanisms at play that
depend on the properties of the bath, the interfaces and the ambient air. In
particular, we explore the role of drainage and evaporation on film thinning.
We highlight the existence of two different scenarios depending on whether the
film cap ruptures at large or small thickness compared to the thickness at
which van der Waals interaction come in to play.

##  1 Introduction

Bubbles have attracted much attention in the past for several reasons. First,
their ephemeral nature commonly awakes children’s interest and amusement.
Their visual appeal has raised interest in painting [4], in graphism [32] or
in living art. Nevertheless, these are not the only good reasons to study
these rich physical objects (see for example [18]). The more specific case of
surface bubbles has been the focus of works conducted by various scientific
communitie, due to their interaction with the environment. Indeed, a system in
which a liquid is overlain by a gas and where surface bubbles are susceptible
to appear is affected by these relatively small objects at the scale of the
entire system. For instance, earth being covered by more than 70% with oceans,
researchers[7, 19, 63] found that bubbles-mediated production of sea-salt
aerosols is a major natural source of cloud condensation nuclei. In turn, the
clouds affect the climate through their radiative properties, among other
things. Bourouiba and Bush [10] recently reviewed other examples of bubbles
found in nature. These include bubbles produced by gas exsolution in magmatic
chambers [24], local spreading of toxins through aerosolisation for instance
during a Red tide event [66, 8] or even catastrophic release of oversaturated
gas as happened in 1984 (Lake Monoun) and in 1986 (Lake Nyos) in Cameroon
[21]. Bubbles are also relevant in the glass industry where they favour
homogenization of the melt but also constitute a major source of defects [9]
and of energy loss as the formation of foam at the surface of the melt screens
the heat coming from above. As the primary building block of a foam, the study
of bubbles and their stability at the surface gets even broader, as was
recently reviewed recently by Suja et al. [58]. To take an image from the
daily life, a beer drinker may prefer a beer with some foam above it: drink a
foam means drink a beer in Belgium. A champagne consumer would probably not
trust a foamed beverage: drink bubbles means consume sparkling wine in France.
Beer drinkers and champagne consumers both love bubbles [62, 38], but not in
the same way.

Figure 1: Artistic photography by Serge Guichard of a centimetric surface
bubble during thinning. Marginal regeneration patches can be observed at the
surface of a bubble. The bubble is stabilised by TTAB at a concentration of
0.5 times the critical micelle concentration (cmc).

The entire life of a bubble covers many steps that we now shortly review to
define the scope of this review. Be it by nucleation in a supersaturated
medium (exsolution), by drop or snow impact on a surface or by gas entrapment
upon wave breaking, a bubble first needs to be born. Then, once the bubble is
entrapped in a bulk liquid, the density difference fosters a movement that
eventually brings the bubble to the surface. A typical distance for a rising
bubble to reach its terminal velocity is one centimeter [68, 3]. During this
process, the shear stresses induced by the flow may accumulate surfactants at
the bottom of the bubble and affect the bubble shape and rising velocity. Once
the bubble gets close to the surface, they start interacting because the
surface needs to be deformed and curved for the bubble to protrude. This step
can lead to rebounds as shown for example in Zawala et al. [68]. When the
bubble protrudes, a thin film is formed and it eventually adopts a metastable
shape after typically 10-100 ms. The liquid film then undergoes thinning under
the combined action of capillary and/or gravitational drainage, evaporation
and ultimately disjoining pressure. Eventually, a hole nucleates and invades
the film, leading to the bursting of the bubble. This last step can trigger
the formation of aerosols through three different processes: film drops due to
the so-called "disgregation" of the thin film during rupture [64], jet drops
due to the Worthington jet [67] and formation of daughter bubbles [6],
themselves susceptible to undergo these different processes. Film drops and
daughter bubbles formation depend upon the thickness of the thin film at burst
[36, 6], itself conditioned by the thinning rate and lifetime of the bubble,
thus the interest to study the whole process.

In these problems, both gravitational and capillary effects must a priori be
taken into account. A key dimensionless quantity is therefore the Bond number
[29] \- sometimes referred to as the Eötvös number - defined as :

$Bo=\frac{\rho_{\text{liq}}gL^{2}}{\gamma},$ (1)

where $\rho_{\text{liq}}$ [kg.m-3] is the density of the liquid bath
(considered much higher than that of the surrounding gas), g [m.s-2] the
gravitational acceleration and $\gamma$ [N.m-1] the surface tension between
the liquid and the gas. As we will see in the following, $L$ [m] is a
geometric parameter that accounts for the phenomenon under consideration. Some
authors use $L=R_{\text{B}}=\left(\frac{3V}{4\pi}\right)^{1/3}$, where V is
the volume of the bubble, in part because $V$ is an adjustable parameter in
numerical models. $R_{\text{B}}$ is then the radius of the equivalent sphere
containing the gas volume within the bubble. Another relevant length is
$R_{\text{film}}$, the radius of curvature of the emerged thin film (for large
enough bubbles). It gives the magnitude of the surface tension-induced forces
acting at the foot of the bubble cap and is easily accessible experimentally.
Quite naturally, the Bond number can be rewritten in terms of the capillary
length, $\ell_{\text{c}}=\sqrt{\frac{\rho_{\text{liq}}g}{\gamma}}$, that
appears in problems where gravity and capillarity dominate:
$Bo=\frac{L^{2}}{\ell_{\text{c}}^{2}}$.

The scope of this review will focus on the life of bubbles at the surface of a
liquid bath, i.e. the shape, thinning and lifetime of surface bubbles, meaning
we will consider bubbles with lifetimes typically of the order of 1-1000
seconds much larger than the time needed for the bubble shape to stabilise and
for the hole to nucleate. In this framework, we will consider the shape of the
bubble at the surface in section 2, the dynamic thinning of the thin film in
sections 3 and 4 and the consequences on bubbles lifetime in section 5.

##  2 Shape of a surface bubble

Once a bubble has reached an interface, it adopts a static shape that is
governed by the balance between the surface tension and buoyancy effects. In
this case, the relevant length scale for the forces is $R_{\text{film}}$,
defined as the radius of curvature of the spherical cap, both in the case of
relatively small [31] and large [59] bubbles. The corresponding Bond number is
therefore defined as

$Bo_{\text{film}}=\frac{\rho_{\text{liq}}gR_{\text{film}}^{2}}{\gamma}.$ (2)

Surface bubbles with varying Bond numbers are represented in Figure 2. Two
limit cases can be easily identified: if $Bo<<1$ surface tension dominates the
system, buoyancy is not strong enough to deform the interface and the bubble
remains spherical and stands below the surface. For $Bo>>1$, the emerged
spherical cap tends to a perfect hemisphere, the contact angle between the cap
and the meniscus is equal to 90° and the bottom part of the bubble, below the
free surface, is essentially flat because hydrostatic pressure then
compensates the over-pressure due to the curvature of the cap. Surface bubbles
in natural or industrial processes often have intermediary sizes and the full
resolution of the problem then requires to solve the Young-Laplace equation
for the interface. Assuming axisymmetric geometry since surfaces (or,
equivalently, free energy) must be minimum, three distinct zones have to be
considered separately : the spherical cap made of a thin liquid film with two
interfaces, above the meniscus, generally modeled as a weightless surface with
a surface tension of $2\gamma$, so that the pressure jump across the
interfaces writes $\frac{4\gamma}{R_{\text{film}}}$ as is the case in 3D; the
immersed part of the bubble up to the top of the meniscus, where the curvature
of the interface and the hydrostatic pressure have to be taken into account ;
and the outer side of the meniscus, where the curvature is balanced by the
hydrostatic pressure and that is connected to the original bath height at
infinity. No general analytical solution for the entire shape of the bubble
can be inferred but numerous authors have tackled the problem [45, 60, 52, 31,
48]. Most recently Teixeira et al. [59] derived a semi-analytical model valid
in the limit of large Bond numbers (however providing convincing results down
to approximately $Bo\sim 4$) and provided a comparison between experimental
data, Surface Evolver simulations [11] and to Howell’s 2D and 3D models[31].
The better agreement obtained with 3D models suggests that the curvature in
the horizontal plane should be taken into account, which does not contradict
the axisymmetry of the problem.

This analysis shows that the apparent contact angle $\theta$, i.e. the angle
between the horizontal direction and the tangent to the spherical cap at the
connection with the meniscus is a monotonously increasing function of the Bond
number (Figure 2). This implies that the bigger the bubbles, the larger the
area of the thin film that eventually drains and ruptures. Finally as shown by
Cohen et al.[16], the weightless film assumption significantly fails at
describing the shape of bubbles of radii beyond a typical length higher than
10 cm (corresponding to Bond numbers of order 10 000), beyond which the
emerged cap flattens and departs from its spherical shape.

Figure 2: Adapted from Nguyen et al.[44]. Pictures of surface bubbles of
varying Bond numbers.

Finally, some useful length scales can be derived to simplify the analysis of
experiments. In the case of bubbles with Bond numbers high enough to feature a
protruding thin film and yet sufficiently small ($0.1<Bo<1$), gravity does not
significantly deform the cavity that keeps a spherical shape with a radius
$R_{\text{cavity}}$. In this case, the equilibrium of the inner pressure in
the bubble requires that $R_{\text{cavity}}\sim\frac{R_{\text{film}}}{2}$
which can be conveniently used to assess the curvature of the cavity for
instance to study the subsequent formation of the Worthington jet[56]. Another
geometrical parameter, important to write mass conservation in a spherical
geometry, is the ratio between the perimeter $P$ [m] of the circle delimited
by the top of the meniscus and the surface $S$ of the spherical cap. In the
case of small bubbles Lhuissier et al. [36] have shown that this ratio can be
written $\frac{P}{S}\sim\frac{2l_{\text{c}}}{R_{\text{film}}^{2}}$. With the
use of a numerical resolution of the Young-Laplace equation for the shape of
the bubble, the authors show that this approximation works up to surprisingly
large Bond numbers ($Bo\approx 5$). Finally for larger bubbles (and as long as
the weight of the film may be neglected), this ratio may be expressed as
$\frac{P}{S}\sim\frac{1}{R_{\text{film}}}$ with an error of less than a factor
2 for $Bo\geq 10$.

##  3 Drainage

Let us now describe the state of the art concerning the liquid drainage that
is the liquid flow, mostly from the top to the bottom of the bubble, leading
to film thinning.

###  3.1 Axisymmetric bubbles

In the previous paragraph it is demonstrated that the static shape of the
bubble is axisymmetric along a vertical axis, as a consequence of
gravitational and capillary forces balance. It is thus reasonable at first
stage to assume that drainage is also axisymmetric and that the liquid flows
from top to bottom inside a film whose thickness is supposed uniform for a
given height.

The equation of motion of the liquid is governed by the Navier-Stokes equation
:

$\rho_{\text{liq}}(\frac{\partial\mathbf{v}}{\partial t}+\mathbf{v.\nabla
v})=-\mathbf{\nabla}P+\eta\Delta\mathbf{v}+\rho_{\text{liq}}\mathbf{g}.$ (3)

This differential equation of the liquid velocity $\mathbf{v}$ can be solved
numerically provided boundary conditions are defined at the liquid interfaces.
Coupled with mass conservation, this equation allows to predict the film
thickness evolution over time. For certain cases analytical solutions can be
found as will be exposed later.

The left hand terms of this equation correspond to the inertia of the fluid.
The right hand terms correspond successively to pressure, viscous and
gravitational forces. Different non dimensional numbers are used to compare
the terms of this equation with each others. As will be shown, different
scaling analysis are proposed in the literature depending on the nature of the
interfaces, i.e. boundary conditions (the presence of surfactants or not),
because it changes the characteristic length of the viscous dissipation.

The pressure scales with the Laplace pressure $\gamma/R_{film}$ and viscous
stress with $\eta v/l_{v}^{2}$ where $l_{v}$ is the characteristic length of
viscous dissipation. For practical reasons, in most numerical works, the
equation is made dimensionless using $R_{B}$ assuming $R_{\text{film}}\sim
R_{\text{B}}$.

####  3.1.1 Viscous bubble from bare to not so bare interfaces

It is common for the problem of bubbles at the surface of viscous fluids as
for example molten glass, magma or polymeric fluids to assume that the inertia
of the liquid flow is negligible. Moreover, these fluids are supposed to have
perfectly bare interfaces, i.e. no surfactants nor impurities so that the
interfaces are supposed stress free. Such hypothesis implies a plug flow of
the liquid. The flow is thus extensional, i.e. the viscous dissipation is due
to the velocity variation along the flow.

The non dimensional number, that compares inertia to viscous dissipation, is
the Reynolds number and writes $Re=\rho_{\text{liq}}^{2}gR_{B}^{3}/\eta^{2}$,
assuming a motion driven by gravity and a length scale equal to the bubble
size with a velocity scaling as $\rho gR_{B}^{2}/\eta$. Neglecting inertia
implies $Re<<1$. One can note that for air bubble in water, this criterion
sets a maximal bubble size to $10$ $\mu$m whereas for viscous fluids ($\eta=1$
Pa.s) it sets it to $1$ mm. Additionally, two viscous characteristic times can
be deduced from equation (3), one is based on bubble buoyancy
$\tau_{B}=\frac{\eta}{\rho_{\text{liq}}gR_{B}}$, and the other one is based on
capillarity at the scale of non deformed bubble: $\tau_{Cb}=\frac{\eta
R_{B}}{\gamma}$. Their ratio leads to a Bond number
$Bo_{B}=\tau_{Cb}/\tau_{B}$ and $\tau_{B}$ is assumed to be relevant for large
$Bo_{B}$ and $\tau_{Cb}$ for low $Bo_{B}$.

Figure 3: (a) View of a large bubble at the surface silicon oil, extracted
from Debregeas et al. [20], (b) Normalized film thickness $\bar{h}=h/h_{0}$ as
a function of normalized time $\bar{t}=t/\tau_{g}$ for different bubble sizes
at the surface of Ucon Oil for different $Bo_{B}$ numbers corresponding to
different buble sizes, extracted from Kocarkova et al. [34]

Since the pioneer quantitative work on the subject from G. Degregeas, P.G. De
Gennes and F. Brochard [20] for large surface bubble, it is known that the
film thickness $h$ decreases monotonously from bottom to top (Figure 3 (a))
and decays exponentially (Figure 3(b)). One can write:

$h(t)=h_{0}\exp(-at/\tau)$ (4)

where $\tau$ is a characteristic viscous time. The value of $a$ is equal to
the normalized thinning rate of the film: $a=-(dh/dt)(\tau/h_{0})$. Their
experiments with silicon oil and quite large bubble ($Bo>1$) give $a_{g}\sim
0.21$ considering $\tau=\tau_{B}$ which is consistent even though lower than
their analytical model that proposed $a_{g}=1.25$ (see [34]).

Numerical simulations, based on Stokes equation (no inertia, $Re=0$), confirm
the exponential decay of the film thickness [48] and show a decay of $a$ with
the Bond number $Bo_{B}$ which tends to $a_{g\infty}=0.34$, in the limit of
large bubbles (large $Bo$)[34, 3]. This result implies that in normalized
time, the smaller the bubble, the faster it drains. Experiments [34] with
molten glass and various viscous fluids (Ucon, Castor oil) provide $h(t)$ for
different bubble sizes (Figure 3(b)), the data are dispersed, even so the
variations of $a$ with $Bo_{B}$ are consistent with previous simulations. In a
second set of experiments, Nguyen et al [44] have measured the time $\Delta t$
between the moment at which bubble rising velocity becomes negligible and the
one at which the bubble spontaneously ruptures. Then, assuming that this time
interval corresponds to the time required for the film to drain from an
initial thickness ($h_{0}\sim 100$ $\mu$m) down to a final thickness
($h_{f}\sim 100$ nm) and assuming an exponential decay of $h(t)$, they are
able to calculate the normalized thinning rate

$a=\ln(h_{0}/h_{f})/(\Delta t/\tau).$ (5)

Their results agree quite well with previous ones, and we estimate
$a_{g\infty}=0.2$ once converting the normalization with the same parameters
than above. They also proposed a scaling with $\tau_{\text{c}}$ and find
$a_{\text{c}}=0.05$ at low $Bo$.

In the limit of small bubbles, (low Bond number) Howell [31] studied the
drainage of bubbles at the free surface using a lubrication model in 2D and
pointed out that the film thickness decays according to an algebraic function
of time.

$h(t)=h_{0}(1+ct)^{-2},\text{where
}c\propto\frac{\gamma^{2}h_{0}^{1/2}}{\eta\rho gR_{\text{film}}^{7/2}}$ (6)

Finally, recent numerical simulations of viscous bubbles ($Re$=0.1) have
modelled the thinning of the film on top of surface bubbles in presence of
surfactants [3], thus the interfaces are not bare anymore. The simulations are
able to model Marangoni stress induced by the inhomogeneity of surface tension
due to surfactants concentration variation at the interfaces. They retrieve an
exponential decay of the film thickness over time and a monotonous decay of
the film thickness from the bottom to the top, for low concentration of
insoluble surfactants or for very large bubbles ($Bo>1$). However they find an
algebric decay at low $Bo$ and large surfactants concentration which reduces
the drainage rate (long living bubbles). The presence and dynamics of
surfactants may change the equilibrium shape of the bubble, as previously
observed for a bubble and a free surface with unequal surface tensions [26].
Additionally, the presence of surface tension gradients not only rigidifies
the interfaces but also affects the usual monotonous thickness decay from top
to bottom. Indeed, for intermediate bubble size and large surfactant
concentration, a pinch (neck) appears at the base of the cap (Figure 4(a)).

####  3.1.2 Apparition of a pinch or a dimple

Figure 4: (a) Scheme adapted from reference [3]. From top to bottom, shape of
bubbles at increasing Bond numbers in presence of some surfactants. For
intermediate Bond numbers, a pinch appears, which leads to a drastic increase
of the bubble lifetime. (b) Prediction of the thickness profile of the film
during a bubble-bubble approach. $r$ is the radial coordinate with its origin
on the vertical axis of symmetry of the two approaching bubbles. A pinch
appears and thins along time. Image adapted from reference [61]. (c)
Comparison of the thickness of the film at the top of bubble with the model
proposed by Lhuissier [36] (dashed lines) complemented by a gravity driven
drainage (solid lines) for bubbles of different radius $R$. The inset
represents a sketch of the pinch in the vicinity of the meniscus, close to the
bath. Figure adapted from reference [41].

Let us now concentrate on this pinch, which can appear close to the meniscus.
Equivalently, this pinch can be seen as a thicker zone or dimple at the top of
the bubble. As mentioned in the previous paragraph, the pinch indeed leads to
a slowing down of the liquid flow, and to long living bubbles [3]. As we will
see in the following, this pinch can destabilise in small patches leading to a
loss of axisymmetry. Here we concentrate on the situations, in which an
axisymmetric description holds.

This pinch was already mentioned and observed by Mysels in the 60’ [43] in
vertical films. It is attributed to the capillary suction in the meniscus and
Aradian et al provided a theoretical description of the pinching dynamics [2].
They proposed scaling laws predicting the velocity at which the pinch thins
and enlarges along time. Although this pinch has been observed in many
different experimental studies (see for example [15, 25, 5]), its dynamics has
never been compared to the theoretical prediction by Aradian. This is due to a
fast destabilisation in many experimental studies, which will be discussed in
the next section and to difficulties to measure the thickness with high enough
accuracy.

The presence of a surface tension gradient seems necessary for this pinch to
appear. Indeed, Howell and Stone [30] showed theoretically that no pinch can
develop in absence of surface tension gradients. This has been confirmed more
recently by Atasi et al [3] with numerical simulations.

Chan and collaborators [15] also described a pinching at the periphery or,
equivalenty, a dimple at the center, when two bubbles approach at controlled
velocity in an Atomic Force Microscope (AFM). They link the apparition of the
dimple to a balance between the Laplace pressure, which tend to keep a round
bubble on one side and the hydrodynamical and disjoining pressure, which tend
to flatten the film on the other side. The result leads to the evolution of
the bubble profile along time plotted in Figure 4(b). In this figure, the
axisymmetric profile exhibits a pinch, whose thickness decreases along time
until the bubbles coalescence. They obtain a very nice description of the
experimental bubbles profiles obtained by interferometry. As noticed in the
review on bubbles and drops in AFM written by Wang[65], a direct consequence
of these results is that a dimple is less expected to appear in small bubbles
because they are less deformable. This in agreement with Atasi et al [3],
whose numerical solutions exhibit a range of Bond numbers, typically, between
0.3 and 3, in which a pinch can be observed (see typical shapes in Figure 4
(a)).

Lhuissier et al [36] proposed a scaling law to take into account the presence
of the pinch in the drainage dynamics. The principal ingredient of their model
is that the pinch limits the liquid flow driven by capillary pressure toward
the bottom of the surface bubble. The principal steps of the calculation are
described in the following. For small bubbles, the mass conservation in a
spherical cap writes

$\frac{\partial h}{\partial t}+\frac{\ell_{c}}{R_{\text{Film}}^{2}}Vh=0.$ (7)

where $V$ is the average fluid velocity, set by the capillary suction in the
meniscus and can be extracted by scaling the Stokes equation

$\eta\frac{V}{\delta^{2}}\sim\frac{\gamma}{R_{\text{film}}\ell},$ (8)

in which $\delta$ and $\ell$ are respectively the thickness and the width of
the pinch (insert figure 4(c). The thickness $\delta$ is expected to scale
with $h$ while $\ell$ is set by a curvature matching between the curvature
$\frac{h}{\ell^{2}}$ in the pinch and the curvature
$\frac{1}{R_{\text{film}}}$ of the bubble. Altogether, this leads to a
prediction for the thinning law,

$h(t)\sim\left(\frac{\eta}{\ell_{c}\gamma
t}\right)^{2/3}R_{\text{film}}^{7/3}.$ (9)

Note that the fact that $\delta$ scales with $h$ comes from the fact that the
pinch thinning is interrupted by the pinch destabilisation (marginal
regeneration), which we will describe in the next paragraph. The model is thus
axisymmetric even though it necessitates the existence of this
destabilisation.

As we will show later (section 4), this prediction only holds if evaporation
can be neglected, i.e. for reasonably thick films or in saturated atmosphere.
In these conditions, the scaling has been verified experimentally using
various methods to measure the bubble thickness. Miguet et al [41] used a
spectrometer to benefit from the interferences and extract the thickness at
the top of the bubble cap. Another classical measurement [51, 50, 36, 42]
consists in measuring the opening velocity of the cap when the bubble bursts.
The Culick law [17] then allows to link the film thickness to the opening
velocity $v=\sqrt{\frac{2\gamma}{\rho h}}$. For example, in Figure 4.c, the
thickness measured by interferometry at the top of the bubble is plotted as a
function of time for different bubble sizes in a saturated environment. The
dashed lines represent the integration of Eq. 7 using the capillary driven
drainage to calculate the velocity. The solid lines represent a similar
integration taking into account not only the capillary driven drainage but
also the gravity driven one. It is not surprising to see that the gravity must
be taken into account to describe accurately the drainage of the bigger
bubbles [41].

###  3.2 Loss of axisymmetry

Figure 5: (a) Top view of a bubble thinning along time. The bubbles are
stabilised by DPPC. The experiment allows to control accurately the surface
pressure, which is set to 5 mN.m-1 (top) or to 25 mN.m-1. The former exhibits
some loss of axisymmetry. Figure adapted from reference [5] (b) View of the
marginal regeneration patches in the meniscus of a surface bubble. Figure
adapted from reference [36] (c) Observation of marginal regeneration patches
rising at the surface of vertical foam film during its extension at a
controlled velocity. The white arrow indicates the position of the frame used
to extend the foam film. Figure adapted from reference [57].

The pinch described in the preceding section happens to be unstable in certain
conditions. The axisymmetric thin zone at the bottom of the bubble can indeed
destabilise in thin patches. Their diameter is around a few hundreds of
micrometers. These thinner and therefore lighter zones tend to rise along the
interface due to buoyancy because of their smaller apparent density [1] and
are reminiscent of the dynamic colored zones, which can be observed in Figure
5 (b) and (c). Mysels chose to call this phenomenon marginal regeneration
because new thin zones enter the foam film and replace the aspiration of
thicker zones inside the meniscus. The regeneration mentioned by Mysels thus
refers to a regeneration of the surface with thinner and more fragile zones.
This finally contributes to the global film thinning. As noticed by Aradian
[2], Atasi et al [3] as well as by Chan et al [15], one of the future
challenge to understand fully bubble drainage is to better describe this loss
of axi-symmetry and its impact on drainage.

Let us first raise that this destabilisation necessitates $mobile$ enough
interfaces. The extreme opposite case is the presence of a solid surface.
Indeed, if the thin film is deposited on a solid surface, the capillary
suction by the meniscus is the same so that the pinching should occur and its
dynamics should be described by Aradian et al [2]. Nevertheless, the presence
of a solid happens to kill the instability [27]. This is actually the only
situation, in which quantitative measurements have been performed. Hack et al
[27] measured the film profile around drops by Digital Holography Microscopy
(DHM). They enlighten the presence of the pinch and measure widening of the
pinch position scaling with $t^{1/4}$ as predicted by Aradian et al [2].
Nevertheless, they observe no destabilisation. This is confirmed by the
experiments performed by Bhamla and collaborators, who observed this
destabilisation for liquid/air interfaces (Figure 5(a)) and not for
solid/liquid ones [5].

The mechanical resistance of the interfaces can thus prevent the appearance of
marginal regeneration. This is reminiscent of the observations by Mysels [43]
showing that the marginal regeneration is observed when the films are
stabilised by Sodium Dodecyl Sulfate (a standard surfactant) but disappears if
a co-surfactant, such as dodecanol is added to the solution. This co-
surfactant is indeed known to allow the surface to sustain large surface
tension gradients and thus to exhibit a large surface Gibbs elasticity. Mysels
chose to call such interfaces rigid because of the absence of motion due to
marginal regeneration and we can only guess that this rigidity has an effect
similar to the solid surface mentioned above. Bahmla et al [5] have also
observed that large surface elasticity can prevent the development of the
instability. They use surfaces covered by a phospholipid, the DPPC
(dipalmitoylphosphatidylcholine) and they control the surface tension by using
a Langmuir trough. Then, they create a bubble at the liquid/air interface and
visualize the drop profile thanks to the colors characteristic of the
interferences. Figure 5 (a) shows such bubbles created at smaller (top) and
higher (bottom) surface concentrations in DPPC. The axial symmetry is broken
for small concentrations, which also corresponds to small surface elasticity,
showing that the surface rigidity can prevent the pinch destabilisation.

Finally, in vertical soap films as well as in horizontal ones or in surface
bubbles, the pinch instability and the subsequent marginal regeneration is
expected for large enough characteristic sizes, in presence of standard
surfactants [57, 25, 40, 36] leading to low enough surface elasticity. Mysels
already proposed that the dynamics of these thin films is crucial to
understand the foam film thinning since thin patches tend to rise and replace
thicker ones. It has recently been shown that this sliding puzzle dynamics, as
Gros and collaborators have called this phenomenon, can indeed fully explain
drainage in a film [55] or in a surface bubble [40].

Surprisingly we now have two mechanisms to describe the drainage of a thin
film in presence of marginal regeneration: the model proposed by Lhuissier et
al [36], in which the flow is determined by the friction in the pinch and the
sliding puzzle dynamics, in which the thinning is due to the reorganization of
the thin and thick zones. Both are susceptible to describe very well the
drainage in presence of marginal regeneration [41, 50, 55]. We are currently
investigating the sliding puzzle dynamics on surface bubbles. Both phenomenon
are actually coupled. The pinch thins until a critical thickness at which the
destabilisation occurs. This thickness fixes the viscous dissipation in the
film. It also fixes the patch thickness and thus its efficiency to contribute
to film drainage. The film must find a working point in which both thinning
velocities are compatible.

##  4 Evaporation

Because it is not relevant in a number of thin film stability problems (foam
or emulsion coalescence for instance), evaporation is a rather overlooked
subject in literature. As was pointed out by Li et al.[37], the relevance of
this parameter is however accessible in the everyday life. If the reckless
pouring of a carbonated beverage like a beer can lead, at least in a first
time, to high foam to liquid ratio, it does not always end up in overflow. The
rise of the foam indeed often ends just at the surface of the glass which is
not only due to luck, but also because the destruction rate of the foam is
higher there, because evaporation is higher than within the glass column. The
experiment of these authors is represented in figure 6(a). A cylindrical
column at the bottom of which a soapy liquid is constantly bubbled and at the
top of which the relative humidity is controlled is used to measure the height
of the foam as a function of time. The results show that, in the case of low
humidity, a constant height of equilibrium is reached while the foam
continuously rises to the top of the column when the humidity is saturated.
The explanation is that the rate of destruction of the produced foam is high
enough to balance the rate of production when evaporation proceeds. On the
contrary, when humidity is close to saturation and therefore evaporation
eliminated, the foam destruction rate is much lower and the foam height can
reach the limit of detection of the set-up. This illustrates the leading order
role of evaporation in surfactant-stabilised films.

Going one step further, Champougny et al.[12] measured the thickness evolution
of planar thin films during the continuous extraction of a frame from a soapy
solution in a humidity controlled environment. The data are presented in
figure 6(b). The main conclusion is that the evaporation rate does not
significantly influence the thinning rate of the film but determines to
leading order the time (or, equivalently, the length) at which it bursts. A
satisfactory explanation for such observations is then presented by Poulain et
al. [50] on millimetric-sized surface bubbles. As can be seen when derivating
equation 9, the thinning rate due to drainage $\frac{dh}{dt}$ is a
monotonically decreasing function of time (in absolute value) so that the
thinning due to drainage is expected to become very slow along time. As a
first approach, the tinning rate due to evaporation $J$ [m.$s^{-1}$] is
constant and can be estimated from diffusive models written for drops [22],
giving typical values or 1 to 10 nm.s-1 respectively for centimetric and
milimetric bubbles. By equalizing the thinning rate due to drainage with the
one due to evaporation, one can find an expression for a critical thickness,
above which the thinning is dominated by drainage and below which evaporation
becomes more important. This thickness is approximately 1 $\mu$m for
millimetric bubbles. To go one step further, mass conservation needs to be
rewritten to take this contribution into account and reads :

$\frac{\partial h}{\partial t}+\frac{\ell_{c}}{R_{\text{film}}^{2}}Vh+J=0,$
(10)

This law is verified on figure 6(c). These results demonstrate the importance
of evaporation on the thinning dynamics. Additional results concerning the
impact of evaporation on the thinning of surface bubbles were obtained by
Miguet et al.[41] and confirm the approach of Poulain et al.[50]. In
particular, they show the importance of gravitational convection due to water
vapor being lighter than dry air. This convection is shown to determine the
water vapor concentration field and therefore the evaporation rate above the
bubbles for a circular bath with a radius of 2 cm. A typical value for the
evaporation rate in this case is a few tenths of nanometers per second,
independently of the bubble size. This is expected to be true as soon as the
characteristic size of the system is above a few millimeters. This remark thus
probably holds for carbonated beverages and more generally for most natural
and artificial systems where surface bubbles appear.

An important remark concerns the general shape of the curves depicting
thickness evolution. It turns out that exponential laws have been found to
match experimental data very well in the case of TTAB (Tetradecyl Trimethyl
Ammonium Bromide) bubbles far from saturation conditions[14]. The results
shown in [51, 41] however show that the power law behaviour performs clearly
better in saturated conditions. The addition of the evaporation rate to a
power law drainage regime describes very well the data, even so they look like
exponential decays.

Figure 6: Representative experiments of thin films stability with controlled
humidity. (a)From Li el al.[37]. Foam height as a function of time in a
cylindrical column which height is 55 cm between the liquid interface in the
absence of foam and a humidity sensor. The bottom of the column is immersed in
a 2.92 g.L-1 SDS aqueous solution, where a constant bubbling produces the
foam. The relative humidity at the top of the column is controlled. Cross
symbols represent repeat of experiments. (b) From Champougny et al. [13].
Thickness evolution 1 mm below the superior frontier of a thin film extracted
at 0.5 mm.s-1 from a 5 cmc TTAB solution. The h=0 nm points correspond to the
instant when the film rupture is detected. (c) From Poulain et al.[51]. Film
thickness evolution of surface bubbles with $R$=5.2 $\pm$ 0.4 mm in solutions
of TTAB of concentration 0.05, 0.5 and 10 times de cmc. The unsaturated
experiments are performed in ambient conditions ie RH=26% while the saturated
ones are about RH=96%. The continuous and dotted lines are the solutions of
the integration of equation 10, respectively with RH=26% and RH=96% The
thickness measurements are operated through Taylor-Culick velocity
measurements at burst. Filled symbols represent spontaneous burst events while
open ones are obtained through manual bursts with a needle.

It should also be noted that Poulain et al. [51] reported results for surface
bubbles made of "dirty" or salt water with poor stability (generally less than
20 s) for which the temperature of the liquid bath was systematically
controlled thus tuning the evaporation rate. They make the counter-intuitive
observation that bubbles submitted to higher evaporation rates feature thicker
films. The results are rationalised by invoking thermal Marangoni backflow
from the hot bath to the cold film, where surface tension is then decreased. A
similar behaviour was reported for flat films made of melted glass [47].

The stabilisation effect of evaporation is mentioned in the review by Suja et
al. [58] and supported by Lorenceau et al. [39] notably with the remark that
bubbles formed in an ethanol/water binary solution are completely unstable if
the ambient atmosphere is saturated in ethanol while they live generally tens
of seconds when evaporation is allowed to proceed.

The condition(s) that determine the path from the stabilising effect of
evaporation at the beginning to its destabilising effect at late stage
remain(s) unknown. Nevertheless, evaporation is crucial for the determination
of the bubbles lifetime as will be shown in the following.

##  5 Bubble lifetime

Who has never observed at the surface of a puddle under the rain one bubble
that bursts instantly and another one that stays for more than 10 seconds?
Same can be said for bubbles at a surface of sodas or beverage with alcohol,
and fruit juice.

At first sight, these liquids have no stabilising molecules in a sense that no
surfactants are present in the solution to generate a positive disjoining
pressure which balances van der Waals forces and thus prevents the interfaces
of a thin films from merging [33]. Moreover, we highlight that such
stabilising molecules would act at very short distance between both interfaces
(typically 100-10 nm), whereas it is unlikely, based on previous paragraphs,
that the liquid film on top of a millimetric bubble has reached such low
thickness value in less than a second.

Moreover, it appears that even so the bubbles look similar, their lifetime is
not determined uniquely but follows some stochastic law. Studying the lifetime
of bubbles quantitatively then implies to lead a statistical study[28], all
the more complex that many parameters affect the time evolution of the liquid
film thickness on top of a bubble. Indeed drainage (section 3) and evaporation
(section 4) do not only depend on the chemistry of the solutions but also on
the bubble generation and the humidity of the room. Thus, studying lifetime of
surface bubbles and comparing one system to another requires careful
attention. Taking all precautions (control of humidity, and of bubble
generation), Miguet et al. [41] have shown two distinct behaviors of bubble
lifetime distribution (figure 7.a). One set of bubbles generated in
sucrose/water solutions (red curves) has its lifetime spreading from zero to
about 30 seconds with a maximum around 8 seconds. A second set of bubbles
obtained in sweeteners/water solutions (green and blue curves) features a much
more pronounced maximum around 23 seconds while random bursting events at
earlier times, so presumably for thicker films, are still measured. As the
bubbles size, viscosity, density and surface tension of the solutions are
quite similar for all baths, one would expect the same drainage laws. Thus the
first set of bubble appears to be "fragile" and to burst during film drainage
for thickness much larger than 100 nm, whereas the second set of bubble
presents a better resilience with respect to perturbations. For instance,
films as thick as 20 $\mu$m have been measured at the rupture in Mezcal
solutions [53].

In the following, we will describe the state of the art for both families of
bubbles.

Figure 7: (a) Probability density function of lifetimes of bubbles at the
surface of aqueous solutions of sucrose and sweeteners, extracted from Miguet
et al. [40] (b) Normalized probability density function of lifetimes of
bubbles at the surface of water baths of different quality, extracted from
Poulain et al. [51] (c) Average lifetime of bubbles as a functions of $\tau
Sc(R/l_{c})^{1/2}$ at the surface of ethanol-water mixture (each $\times$
corresponds to a different $\%$ of ethanol in water) extracted from Lorenceau
et al. [39] and at the surface of sea water (each $\diamond$ corresponds to a
different bubble size) extracted from Zheng et al. [69]

.,

###  5.1 In absence of stabilising molecules

In this part, we assume that no stabilising molecules are in the bath solution
and thus that the bursting of the bubble occurs during film drainage and for
thicknesses that might be larger than $100$ nm. In 1983, Zheng et al. [69]
measure the lifetime of about one hundred bubbles at the surface of different
water baths (tap, river, ocean), they found small variation of the average
bubble lifetime (from 2.2 to 3.9 s) but the statistical features seem
independent of the bath and seem to follow a Rayleigh distribution.

Actually, as the rupture of the film is a random failure event, the statistic
law of lifetime which is a random positive variable $t_{b}$ can be described
using a Weibull distribution, in which the probability $p$ to observe a time
$t_{b}$ is given by

$p(t_{b})=\beta/(t_{b}/t_{b0})^{\beta-1}\exp(-(t_{b}/t_{b0})^{\beta}),$ (11)

where $\beta>0$ is the shape parameter and $t_{b0}=<t_{b}>/\Gamma(1+3/4)$ is
the scale parameter with $\Gamma$ the Gamma function and $<t_{b}>$ the average
value of lifetimes. If the rupture of the film is a rare event, possibly
triggered by an external parameter, that occurs at any time independently of
the film thickness and age, the lifetime of a bubble could take any value
between zero and infinity and is expected to follow a Weibull distribution
with an exponent $\beta=1$. Note that $\beta=2$ corresponds to the Rayleigh
distribution previously mentioned. These different situations have been
reviewed recently by Suja et al. [58] .

Recently, experimental studies with better statistics, concerning more than
thousand bubbles [36, 51, 39, 40, 23] have shown that the lifetime of bubbles
at the surface of water (hot, deionized, river … ) or mixture of water and
alcohol is well described by Weibull distributions with $\beta=4/3$.
Surprisingly this result is quite robust (Figure 7(b)). The rather small
number of bubbles observed by Zheng et al [69] may explain why their data are
well described by a Rayleigh distribution.

The exponent $\beta=4/3$ is supported by a model [36] that associates film
lifetime to the rupture of the thin cells of the marginal regeneration due to
the destabilization of the pinch, as described in section 3.2. Perfectly bare
interfaces indeed never exist and even a small amount of impurities can create
surface tension variations so that the pinch at the foot of the bubble can
form. The authors introduce an ad hoc parameter, the "efficiency of rupture"
to calculate the average lifetime $<t_{b}>$. Poulain et al. [51] completed
this model and introduced the side-to- side diffusion of impurities from one
interface of the film to another. Considering that rupture occurs if the
diffusion time is smaller than the residence time of the contaminant in the
cap, they propose

$<t_{b}>=\alpha\tau_{v}Sc(R_{\text{film}}/\ell_{c})^{1/2},$ (12)

where $\tau_{v}=\eta\ell_{c}/\gamma$ and $Sc=\frac{\eta}{\rho D}$, with $D$
the diffusion coefficient of the impurities in the aqueous solution, is the
Schmidt parameter and $\alpha$ is a parameter that accounts for thinning or
thickening of the film via Marangoni effects. Note that $D\propto 1/\eta$
leads to $<t_{b}>\propto\eta^{3}$ which makes a very strong impact of the
viscosity on the bubble lifetime. In the case of Mezcal and water-ethanol
mixture, Rage et al. [53] show a dependence of the lifetime with alcohol
concentration, with a maximum around 50 $\%$ that they correlate to a maximum
of viscosity. Lorenceau and Rouyer [39] have tested this scaling for water-
ethanol mixture and found a relatively good agreement for the impact of
ethanol concentration varying from 0 to 12 $\%$ in mass on the bubble lifetime
(Figure 7-c). This work has shown that the average lifetime of bubbles at the
surface of an ethanol-water bath increases considerably with the concentration
of ethanol. The ethanol concentration heterogeneities induced by evaporation,
which in turn create upward Marangoni stresses induce a large value of
$\alpha$ around 350 that is twice larger than the coefficient for sea water
bubbles [69]. As previously exposed in section 4 evaporation can be
responsible for longer bubble lifetime.

###  5.2 In presence of stabilising molecules

In the presence of surfactants to stabilise the interfaces, different studies
propose that the thinning time gives a good prediction for the bubble
lifetime. This means that the stabilisation of the interfaces is efficient
enough to avoid any rupture of thick film and to allow the film to thin until
the end, i.e. until its thickness reaches a few tens of nanometers. In that
case, the lifetime is expected to be fixed by the thinning rate and by the
initial and final bubble cap thicknesses. The initial bubble cap thickness
does not impact dramatically the final lifetime since the drainage of a thick
film is quite fast. The exact value of the final thickness is still an open
question [54]. Most of the time, it is proposed to be around 100 nm because
van der Waals interactions become important at such scales. Nevertheless, it
is well known that Newton black film, of thickness around a few nanometers,
can exist and even be stable. Thus, thin films do not necessarily rupture as
soon as van der Waals forces become important and the whole story is not yet
available. Nevertheless, it has been shown [41] that the lifetime is not
really sensitive to the final thickness value if it is chosen under $100$ nm.
Indeed, the evaporation leads to a thinning rate of a few tens of nanometers
per second (Section 4). An incertitude on the final thickness of a few tens of
nanometers thus leads to errors around a few seconds on the lifetime, which is
small compared to the lifetime of a soap bubble. This mechanism leads to a
distribution of the bubble lifetime very different from what is observed in
absence of surfactant. The distribution is actually much narrower and a
typical shape is shown in Figure 7(a) in water in presence of sweeteners.
Unfortunately, no theoretical description leading to such a distribution is
available in the literature. In the following, we will concentrate on the
prediction of the average bubble lifetime based on the thinning and on the
comparison with the available experimental data.

Let us first explore the influence of the bubble size. As detailed in section
3 and 4, the thinning rate drastically depends on the bubble size. Thus, this
parameter is also expected to influence the bubble lifetime. In fizzy
beverages, the bubbles are mostly quite sub-millimetric. For a radius of $500$
$\mu$m, a density of 1000 kg/m3 and a typical surface tension of $50$ mN/m,
the Bond number defined in Eq. 1 is equal to 0.05. In this case, the buoyancy
is too small for the bubble to protrude and the surface area of the thin film
is almost zero so that the probability for a hole to appear is very small. In
such situations, a large stability of the bubbles can be observed, with static
bubbles lasting hundreds of seconds [46] (Figure 8 (a)). Their lifetime is
predicted by the hydrodynamic description of the bubble approaching the
interface. Politova et al propose that this velocity is given by an
interpolation between the Stokes velocity, at which a sphere rises in an
infinite bath and the Taylor velocity at which a sphere approaches a solid
surface [49]. This model allows to predict the scaling of the lifetime of
surface droplets with their radius up to 10 $\mu$m, which has every reasons to
apply also for small surface bubbles. Nevertheless, as soon as the bubble
impacts the surface with some non-zero velocity, the precise dynamics of the
bubble-surface approach is crucial to predict the bubble lifetime [61, 15]. In
this dynamic situation, a Stokes-Reynolds-Young-Laplace model taking into
account the hydrodynamic-Laplace-disjoining pressure equilibrium describes the
data very well [15, 61] (see Figure 8 (b)).

Figure 8: (a) Lifetime of microbubbles as a function of their diameter. An
outstanding stability is observed for bubbles whose diameter is smaller than
200 $\mu$m. The figure is extracted from reference [46] (b) Comparison between
measured coalescence time between two bubbles and the Stokes-Reynolds-Young-
Laplace model proposed by the authors of reference [61], where the figure
comes from. (c) Comparison between the measured lifetime of centimetric
bubbles obtained for different atmospheric humidities and bubble size and the
predicted lifetime obtained taking into account the film drainage and
evaporation. Figure extracted from reference [41].

Pagureva et al. [46] also predict a slower increase of the bubble lifetime
with the bubble size for higher bubble radius, which is reminiscent of the
results obtained by different authors [41, 36, 50]. This is actually due to
the fact that, although gravity forces are greater for big bubbles, the
decrease of the capillary suction leads to longer lifetimes. Nevertheless, for
big bubbles, it is mandatory to take into account the influence of evaporation
on the thinning to predict the right lifetime. This is true as soon as the
Bond number is high enough for a thin film to appear. Indeed, as explained in
section 4, evaporation becomes important only in very thin films. For surface
bubbles submitted to evaporation and to the thinning through the marginal
pinch described in section 3.1.2, Poulain et al [50] proposed that the bubble
lifetime is given by

$\tau=1.23R\left(\frac{\eta}{\gamma}\right)^{2/5}\left(\frac{\rho}{J}\right)^{3/5}.$
(13)

This equation is obtained by integrating Eq. 7 using the flow velocity given
by Eq. 9. For big bubbles, this lifetime actually overestimates the actual
lifetime by up to 10 % since an additional gravity driven drainage needs to be
taken into account [41]. Doing so, the experimental data are very well
predicted by the thinning rate (Figure 8(c)).

Another parameter which is expected to influence drastically the bubble
lifetime is the surfactants used to stabilise the interfaces [14, 46].
Unfortunately, their influence is really mostly an open question. The models
mentioned in the previous paragraph include the physical-chemistry only
through the density, the viscosity and the evaporation rate. Whereas both
first parameters are not expected to vary a lot in presence of surfactants,
the effect of surfactants on the third one is mostly unknown. Nevertheless,
the evaporation rate is expected to vary with the chemical potential and thus,
with the surfactant concentration, which can vary along time due to the
evaporation of water [35]. Additionally, as mentioned in section 3.2, the
surface elasticity can affect the marginal regeneration and, thus the
drainage. These are first leads to take into account the influence of the
surfactants used to stabilise the bubbles.

##  6 Conclusion and open questions

To sum-up, we have presented the state of the art concerning the prediction of
the lifetime of surface bubbles. In general, the bubble unstability is linked
to two facts: (i) the bubble cap is constituted by a thin film, whose
thickness decreases along time due to both drainage and evaporation and (ii)
this thin film is unstable and eventually bursts. We have shown that the
current understanding is that two different behaviors exist depending on
whether the film thins until its thickness reaches a few hundreds of
nanometers or bursts at higher thicknesses. In the first case, determinist
models that describe the thinning of the film down to a rupture thickness of
the order of tenth to hundreds of nanometers perform correctely to calculate
the bubbles lifetime. In the second case, the presence of a fatal impurity
within the film and its propension to break it being a more random process,
lifetime distributions are much more spread and only stochastic models may
capture the physical mechanism(s) at play. The scenario depends on whether or
not surfactants are present to stabilise the thick film.

In absence of surfactants, the distribution of lifetimes is given by a Weibull
distribution. The bursting mechanism available in the literature involves the
diffusion of impurities in the film, which cause the film rupture. Film
thinning due to evaporation is likely to be rather negligible in such
experiments since its impact is small on thick films.

In presence of surfactants, the film is expected to thin until its thickness
reaches a few tens of nanometers. The prediction of the bubble lifetime thus
depends on our ability to predict the thinning rate of the film. It is fixed
by the evaporation and the drainage. For tiny bubbles, no stable thin film
appears and the evaporation is negligible. The lifetime is fixed by the
approach velocity of the bubble to the bath. For bigger bubbles, evaporation
and drainage must be taken into account. The evaporation is a constant rate,
which depends on external conditions such as atmospheric humidity, on the
diffusion/convection ratio and on the chemical potential of the solution. It
has been shown that an accurate description of the evaporation rates
necessitates to take into account the natural convection.

The drainage mechanism depends on the viscosity of the solution, on the bubble
size and on the surfactants used to stabilise the interfaces. We have
identified three main mechanisms. For viscous bubbles, the cap is axisymmetric
and the thickness decreases continuously from the bottom to the top of the
bubble. The drainage is then expected to be exponential with time. In presence
of surfactants, a pinch is expected to appear in the vicinity of the meniscus,
which slows down the drainage. The destabilization of this pinch may lead to
marginal regeneration, that in turn can affect the drainage.

Several references show that the drainage and evaporation rates are sufficient
to predict the average lifetime of the surfaces bubbles in these different
cases [49, 50, 41, 61].

Many questions remain open and deserve to be addressed in a near future and we
try to list some of them below.

  * •

The mechanism at the origin of the eventual bursting of the film, whether they
are thick (micrometers) or thin (tens of nanometers) is mostly unknown.

  * •

The marginal regeneration phenomenon, the dynamics of the pinch, the origin of
its destabilisation and its contribution to drainage are under current
investigation.

  * •

The impact of the chosen surfactants on bubble drainage and evaporation is
crucial but remains an open question.

  * •

There is still a lack of data concerning the distributions observed.
Additionally, there is no theoretical prediction of the distribution in the
presence of surfactants stabilising the interface.

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