Marina Pasquet

François Boulogne

Julien Sant-Anna

Frédéric Restagno

Emmanuelle Rio

Impact of physical-chemistry on the lm thinning in surface bubbles

In this paper, we investigate the thinning dynamics of evaporating surfactant-stabilised surface bubbles by considering the role of the physical-chemistry of solutions used in the liquid bath. We study the impact of the surfactant concentration below and above the cmc (critical micelle concentration) and the role of ambient humidity.

First, in a humidity-saturated atmosphere, we show that if the initial thickness depends on the surfactant concentration and is limited by the surface elasticity, the drainage dynamics are very well described from the capillary and gravity contributions. These dynamics are independent of the surfactant concentration. In a second part, our study reveals that the physical-chemistry impacts the thinning dynamics through evaporation. We include in the model the additional contribution due to evaporation, which shows a good description of the experimental data below the cmc. Above the cmc, although this model is unsatisfactory at short times, the dynamics at long times is correctly rendered and we establish that the increase of the surfactant concentration decreases the impact of evaporation. Finally, the addition of a hygroscopic compound, glycerol, can be also rationalized by our model. We demonstrate that glycerol decreases the bubble thinning rate at ambient humidity, thus increasing their stability.

Introduction

The study of surface bubbles have raised a large research interest due to the broad range of applications. For instance, we can cite industrial applications such as the avor of zzy drinks [START_REF] Liger-Belair | Unraveling dierent chemical ngerprints between a champagne wine and its aerosols[END_REF], or the understanding of the dispersion of pollutants above swimming pools [START_REF] Poulain | Biosurfactants change the thinning of contaminated bubbles at bacterialaden water interfaces[END_REF]. In addition, understanding the bubble bursting phenomenon is crucial for modelling the climate, as the process is involved in the quantication of the gas exchange between the ocean and the atmosphere [START_REF] Murphy | Inuence of sea-salt on aerosol radiative properties in the southern ocean marine boundary layer[END_REF][START_REF] Boucher | Clouds and aerosols[END_REF].

In presence of surfactants, the bubble lifetime is primarily xed by the thinning velocity [START_REF] Miguet | Stability of big surface bubbles: impact of evaporation and bubble size[END_REF], which in turn is due to two mechanisms, the drainage, and the liquid evaporation. Indeed, the lm surrounding surface bubbles has been proven to thin down to a few tens of nanometers, the thickness at which bursting occurs. To understand and to be able to predict the lifetime of these bubbles, it is therefore necessary to consider their thinning dynamics [START_REF] Miguet | The life of a surface bubble[END_REF].

In absence of evaporation, the drainage is driven by the competing eects of capillary suction [START_REF] Lhuissier | Bursting bubble aerosols[END_REF] and gravity [START_REF] Miguet | Stability of big surface bubbles: impact of evaporation and bubble size[END_REF]. This process is limited by the ow through a pinch developing at the bottom of the surface bubbles, in the vicinity of the meniscus [START_REF] Aradian | Marginal pinching in soap lm[END_REF]. Such a description has been veried experimentally, either in a saturated atmosphere, i.e. in absence of evaporation [START_REF] Miguet | Stability of big surface bubbles: impact of evaporation and bubble size[END_REF] or at a short time, when evaporation is negligible over the drainage [START_REF] Lhuissier | Bursting bubble aerosols[END_REF][START_REF] Poulain | Ageing and burst of surface bubbles[END_REF].

Nevertheless, evaporation must be taken into account for thicknesses typically lower than one micrometer [START_REF] Champougny | Life and death of not so bare bubbles[END_REF].

To describe the thinning rate in presence of evaporation, a constant evaporation rate can be added to the mass conservation. Such consideration provides a successful description of the data available in the literature [START_REF] Poulain | Ageing and burst of surface bubbles[END_REF][START_REF] Miguet | Stability of big surface bubbles: impact of evaporation and bubble size[END_REF].

A noticeable consequence of this description lies on the role of the liquid composition, which is expected to modify the thinning process, and thus the bubble lifetime, only through the liquid viscosity, density, and surface tension. This observation conicts to the common intuition and daily observations that the choice of the soapy solution recipe is fundamental to control bubbles stability [START_REF] Frazier | How to make a giant bubble[END_REF], whereas viscosity, density, and surface tension are slightly modied.

Hence, the aim of the paper is to identify the role of physical-chemistry on surface bubble thinning, and in particular to precise separately the specic eects on drainage and on evaporation. To make this distinction, we performed experiments in a saturated atmosphere to identify the role of the surfactant concentration on thinning dynamics, without any evaporation phenomena. Then, additional experiments are presented in an atmosphere controlled in humidity to investigate the role of evaporation. C (at ±0.2 mN/m). TTAB used for the solutions at 0.5 and 1 times the cmc have been recrystallized. The denomination "TTAB/Gly" corresponds to solutions containing a mass concentration of glycerol of 20 %.

system c γ (mN/m) ρ (kg/m 3 ) η (mPa•s) TTAB 0.
2 Experimental setup

Material

In this study, we use TTAB (Tetradecyl Ammonium Bromide, purchased from Sigma-Aldrich) as well as glycerol (purity ≤ 99.5 % purchased from VWR) diluted in ultrapure water (resistivity = 18.2 MΩ•cm). The TTAB solutions are prepared for several concentrations between 0.5 and 20 times the critical micelar concentration (cmc = 3.6 mmol•L -1 ). Since impurities in surfactant solutions have a noticeable eect on surface properties below the cmc, TTAB used 0.5 and 1 times the cmc is recrystallized prior to the preparation of the solutions [START_REF] Stubenrauch | Foams and foam lms stabilized by cntab: inuence of the chain length and of impurities[END_REF].

To investigate the impact of glycerol on the stability of surface bubbles, solutions with TTAB are made by replacing ultrapure water with a mixture of ultrapure water and glycerol with an initial glycerol concentration c g0 = 20 wt%.

The surface tensions γ of our solutions are measured with a pendant drop commercial apparatus (Tracker, Teclis) and the results are reported in Table 1. The viscosity η and the density ρ of the solutions containing TTAB in water are assumed to be identical to the pure water properties. The viscosity and density of solution containing glycerol are estimated from data on waterglycerol mixtures available in the literature [START_REF] Cheng | Formula for the viscosity of a glycerol-water mixture[END_REF][START_REF] Volk | Density model for aqueous glycerol solutions[END_REF].

2.2

Bubble generation and thinning measurement

The experimental setup is schematized in Fig. 1. Bubbles are generated on the surface of a cylindrical Plexiglas tank of 4.0 cm diameter and 4.6 cm depth, lled with a solution up to the rim. To inject air in the solution, a PTFE tubing with an external diameter of 0.56 mm (from Cole-Parmer) is inserted at the base of the tank. This extremity of the tube is beveled and placed just below the surface of the solution. To circumvent the exibility of the PTFE tubing, the later is inserted in a glass capillary tube. The opposite extremity of the PTFE tubing is then connected to a exible tube of 6 mm diameter, and about 80 cm long connected to a 50 mL syringe. The syringe is placed on a syringe pump (AL-1000 from WPI), such that the airow rate Q and the injected volume Ω air are perfectly controlled. In this study, we choose to keep the injected volume Ω air = 1.50 mL constant, as it has already been varied in a previous study [START_REF] Miguet | Stability of big surface bubbles: impact of evaporation and bubble size[END_REF]. The the ow rate Q = 15 mL•min -1 is also kept constant, such that the bubble radius of curvature R = 1.03 ± 0.08 cm is also constant. The temperature is monitored and ranges between 20 and 23 • C. The whole device is enclosed in a chamber of size 40×40×50 cm 3 . The humidity is controlled through a humidity regulator [START_REF] Boulogne | Cheap and versatile humidity regulator for environmentally controlled experiments[END_REF]. For experiments conducted at 100 % humidity, a humidier (Bionaire BU1300W-I) and wet towels are also used to saturate the box with water vapor.

The thickness h of the lm at the apex of the bubble cap is measured using a UV-VIS spectrometer (Ocean Optics Nanocalc 2000), whose wavelengths are between 450 and 800 nm, associated with a 200 µm diameter optical ber. The spectrometer must be positioned vertically, precisely at the apex to allow the spectrometer to be perpendicular to the lm. The tank containing the solution is thus placed on a xy-translation plate. The thickness is extracted from the spectrum of the reected light using the Oospectro [START_REF] Boulogne | Oospectro: Get thicknesses from ocean optics spectra[END_REF] library written in Python [START_REF] Miguet | Stability of big surface bubbles: impact of evaporation and bubble size[END_REF][START_REF] Miguet | Marginal regeneration-induced drainage of surface bubbles[END_REF]. The temporal evolution of the lm thickness is measured two or three times for each experimental condition. As the measurements are perfectly reproducible, the results presented will correspond systematically to two or three repetitions of the same experiment condition and for better readability, the symbols will be the same. The initial time (t = 0) corresponds to the end of the bubble generation.

3 Impact of physical-chemistry on thin lm drainage The error bars correspond to the estimated accuracy of 0.2 µm on the determination of the initial thickness. The inset shows the Gibbs-Marangoni elasticity of the lm obtained from h 0 using Eq. 1. The dashed line is a power law t: E = 0.10 c -0.20 . The initial thicknesses plotted correspond to those obtained by leaving h 0 as an adjustable parameter when we compare Eq. 2 to our data, shown in Fig. 3.

Initial thickness

In this Section, we consider the experiments performed in a saturated atmosphere, so that we expect the thinning to be due solely to drainage since no evaporation occurs. Once the bubble is inated at the surface of the surfactant solution, after the 6 seconds of generation, the thickness of the bubbles at the apex is referred to as the initial thickness and is noted h 0 . The measurements of h 0 extracted from our drainage curves are presented in Fig. 2, which shows a decrease of the initial thickness with the surfactant concentration.

To understand the variation of h 0 with the surfactant concentration, a rst approach is to adopt the description of the lm generation based on a Frankel-type mechanism [START_REF] Shinoda | Soap Films: studies of their thinning[END_REF]. In this model, the thickness is set by a balance between the capillary suction in the meniscus and the viscous dissipation in the thin lm, characterized by the capillary number Ca = ηV /γ with the entrainment velocity V . The Frankel description leads to an initial thickness h 0 ∼ ℓ c Ca 2/3 , where ℓ c = γ/(ρg) is the capillary length, with g the gravitational acceleration.

In the present study, all bubbles are generated with the same ow rate, i.e. the same entrainment velocity V .

Additionally, the surfactant concentration aects mainly the surface tension γ (table 1). The dierence of surface tension between TTAB at 0.5 and 20 cmc is of 15 mN/m. This leads to a dierence of 0.4 µm for h 0 , much lower than what is measured. As a consequence, this model predicts a nearly identical initial thicknesses for all bubbles based on Frankel's mechanism. This prediction is in contradiction with the observations presented in Fig. 2.

In previous experiments, performed at similar injection rates, the initial thickness of surface bubbles was shown to be rather independent on the ow rate [START_REF] Champougny | Life and death of not so bare bubbles[END_REF]. Therefore, we conclude that the soap lm generation is not driven by a viscous ow, but rather by elasticity [START_REF] Champougny | Surfactant-induced rigidity of interfaces: a unied approach to free and dip-coated lms[END_REF][START_REF] Seiwert | Theoretical study of the generation of soap lms: role of interfacial visco-elasticity[END_REF]. In an elasticity-dominated regime, the thickness is limited by the surface elasticity E, i.e. by the ability of the surface to sustain surface concentration gradients and thus surface tension gradients.

In this regime, the balance between the capillary suction and the Marangoni stress due to surfactant surface concentration inhomogeneities at the interfaces leads to a thickness independent on the generation velocity [START_REF] Champougny | Surfactant-induced rigidity of interfaces: a unied approach to free and dip-coated lms[END_REF][START_REF] Seiwert | Theoretical study of the generation of soap lms: role of interfacial visco-elasticity[END_REF]:

h 0 ∼ ℓ c E γ . (1)
With this prediction, the initial thickness is expected to depend on the surfactant concentration through the surface elasticity E. Here, we propose to apply the same model to the surface bubbles, although there is likely so existing renements from the geometry and deduce the surface elasticity from the initial thickness measurements and comment the result. The accordance with the experiments performed on at lms show that this model still holds for surface bubbles. From our data h 0 (c) and Eq. 1, we plot in the inset of Fig. 2 the predicted surface elasticity. We obtain a decay of the surface elasticity with the surfactant concentration, which we describe with a power law model. Such decrease is expected since a larger surfactant concentration allows a faster repopulation of the interfaces and thus a smaller surface elasticity [START_REF] Langevin | Rheology of adsorbed surfactant monolayers at uid surfaces[END_REF]. The measurement of surface elasticity is delicate and rare in the literature. In our study, the order of magnitude of the surface elasticity is 0.1 mN/m, which is lower but in agreement with those measured by Champougny et al. [START_REF] Champougny | Surfactant-induced rigidity of interfaces: a unied approach to free and dip-coated lms[END_REF] (C 12 E 6 ), which are between 0.15 and 0.59 mN/m. This non-ionic surfactant is much less soluble that TTAB. In such a situation, the surface elasticity is expected to be slightly higher, in agreement with our experiment. In the literature, one can also nd higher values [START_REF] Couder | On the hydrodynamics of soap lms[END_REF][START_REF] Prins | Elasticity of thin liquid lms[END_REF][START_REF] Kim | Marangoni elasticity of owing soap lms[END_REF][START_REF] Prins | Composition and elasticity of thin liquid lms[END_REF] for the surface elasiticity. Nevertheless, a direct comparison can be hazardous since this parameter is expected to depend drastically on the physical-chemistry and on the characteristic time of solicitation. Our observation is that the values measured here are comparable to surface elastic moduli measured with similar systems and a similar solicitation.

Thinning dynamics

Now that the initial thickness is described, we focus our attention on the thinning dynamics. All the drainage curves obtained for the dierent TTAB concentrations are presented in Fig. 3. Remarkably, the thinning process appears to be nearly independent on the surfactant concentration, besides the initial thickness described in the previous paragraph.

In the following, we compare these thinning dynamics with the drainage model described in the introduction, in which the ow is limited by the presence of a pinch in the vicinity of the contact line [START_REF] Lhuissier | Bursting bubble aerosols[END_REF][START_REF] Poulain | Ageing and burst of surface bubbles[END_REF][START_REF] Miguet | Stability of big surface bubbles: impact of evaporation and bubble size[END_REF]. The thinning rate is therefore the combination of capillary-driven and gravity-driven ows, which reads [START_REF] Miguet | Stability of big surface bubbles: impact of evaporation and bubble size[END_REF]

∂h ∂t = - γ η h 5/2 R 5/2 + h 3 Rℓ 2 c , (2)
Indeed, the gravity-driven drainage is relevant in our situation because the Bond numbers Bo comparing the eect of gravity and surface tension (Bo = ρgR 2 /γ) corresponding to our bubbles are between 20 and 44. To solve this dierential equation we used the odeint function from the scipy module [START_REF] Jones | SciPy: Open source scientic tools for Python[END_REF] in Python. The initial thickness of the bubbles h 0 is left as an adjustable parameter for the integration of Eq. 2, via the use of the curve_t function of the scipy module [START_REF] Jones | SciPy: Open source scientic tools for Python[END_REF]. This thickness is plotted as a function of TTAB concentration in Fig. 2. We have chosen to leave it as an adjustable parameter in order to be able to describe the long time dynamics of all our systems in the presence of evaporation, as we will see later. Fig. 3 shows that the experimental data agree with this theoretical prediction (solid lines). Note that an implicit solution to the Eq. 2 can be de-termined analytically. The calculation and the result are given in S.I.

Finally, our rst conclusion is that, in absence of evaporation, the concentration in TTAB does not impact the drainage dynamics. This is conrmed by the results of Poulain et al [START_REF] Poulain | Ageing and burst of surface bubbles[END_REF] for bubbles of radius R = 4.8 ± 0.1 mm with another surfactant, SDS (Sodium Dodecyl Sulfate). The only impact of the physical-chemistry is a shorttime eect. The liquid composition changes the initial thickness through the variation of the surface elasticity. After a few tens of seconds, which is short compared to the bubble draining time, all the dynamics becomes independent of the TTAB concentration. We thus expect a negligible impact on the bubble lifetime. [START_REF] Boucher | Clouds and aerosols[END_REF] Impact of physical-chemistry on thin lm evaporation To explore the impact of physical-chemistry on evaporation, we now measure the thinning dynamics at relative humidity values RH equal to 39 and 55 %. As expected, the dynamics is much faster in presence of evaporation so that the bubble lifetime is more than 5 times shorter. Additionally, in conditions for which evaporation proceeds, the thinning dynamics for dierent surfactant concentrations do not collapse in a single curve as shown in Fig. 4. This observation contrasts with the results in a saturated atmosphere obtained in the previous Section. Thus, although the surfactant concentrations studied in this article have a little impact on the drainage dynamics, this concentration aect signicantly the thinning of the bubbles when evaporation proceeds.

To describe the thinning dynamics, we add an evaporation rate [START_REF] Poulain | Ageing and burst of surface bubbles[END_REF][START_REF] Miguet | Stability of big surface bubbles: impact of evaporation and bubble size[END_REF] to the former drainage Eq. 2. For this, we can consider a diusive evaporation rate [START_REF] Fuchs | Evaporation and droplet growth in gaseous media[END_REF], which is proportional to the dierence in pressure between the saturation vapour pressure at the interface p sat and in the surrounding environment p ∞ . We write the contribution of the evaporation on the lm thinning as k e (1 -RH),

where k e has the dimension of a velocity. This coecient contains the geometry of the system, kept constant in this study, the diusion coecient of water in air, and the saturated vapor pressure of the chemical solution. This leads to the thinning rate

∂h ∂t = - γ η h 5/2 R 5/2 + h 3 Rℓ 2 c -k e (1 -RH). ( 3
)
The model describes very well the data for TTAB concentration smaller or equal to the cmc for both RH = 39 and 55 %. For the solutions containing TTAB above the cmc, the proposed model does not allow to account Figure 4: Film thinning at the bubble apex in a environment of controlled humidity of (a) RH = 55 ± 4 % and (b) RH = 39 ± 1 %, for bubbles stabilized by dierent concentrations of TTAB ranging in 0.5-20 cmc. The continuous lines correspond to Eq. 3 using k e (given in (c)) as a tting parameter. For c > cmc, the t was made from t = 18 s: the extrapolation to the rst instants is indicated by a dotted line.

for the beginning of the thinning curves. We observe that, in presence of evaporation, the thinning dynamic is delayed at the beginning for these concentrations, especially for the highest concentration (20 times the cmc) where a plateau, marked by a black arrow, can be observed. Thus, a t for these "high" concentrations is proposed only at times greater than 18 seconds (Fig. 4 (a) and (b)). In this case, there is a good agreement between our model and the experimental data. It can be noted that for concentrations above cmc, the higher the TTAB concentration, the slower the evaporative thinning dynamics, as indicated by the measured k e constant values given in the Fig. 4 (c).

Our second result is thus that the variation of the TTAB concentration aects the thinning dynamics of surface bubbles through the evaporation rate. A simple model, in which a constant evaporation ux is added to the drainage dynamics allows describing the data in the rst order. This is actually a simple model, which cannot capture the entire thinning dynamics. Indeed, at high concentration, the variation of h with time is poorly described by our model. Our strong result is that the physical chemistry aects the thinning dynamics mainly through evaporation and not through drainage. Similar behaviour has already been observed by Poulain et al. [START_REF] Poulain | Biosurfactants change the thinning of contaminated bubbles at bacterialaden water interfaces[END_REF]. Nevertheless, the exact mechanism behind this observation will deserve more investigation in the future: capturing the evaporation mechanisms in a single parameter, which is the evaporation rate is actually a simplied model. Indeed evaporation can also lead to temperature gradients, surfactant bulk and surface concentration gradients, which can lead to Marangoni stresses at the interface and aect the thinning dynamics.

Control of evaporation via the addition of an hygroscopic component

In the following, the surface bubbles studied are generated at RH = 55 ± 5 % in water-glycerol mixtures for two concentrations of the TTAB solution (equal to 0.5 and 20 times the cmc).

The thinning curves obtained for bubbles stabilized by TTAB/glycerol/water solutions, with an initial glycerol concentration c g0 = 20 wt.%, are plotted in Fig. 5. Comparing these results to those obtained with TTAB/water only, we can see that the addition of glycerol stabilizes the bubbles by decreasing their thinning rate at long times (in particular for the concentration corresponding to 20 times the cmc). We also note that the bursting thickness is still high, around 1.5 µm in Fig. 5 (a). It has been demonstrated that the lifetime can have a wide distribution since the mechanism behind is partially stochastic [START_REF] Lhuissier | Bursting bubble aerosols[END_REF]. Thus, this is not necessarily a robust feature that the bubbles burst at such high thickness but the thinning dynamics holds in any case.

To rationalize this observation, we propose to take into 4. For the data in the presence of glycerol, the black dotted line corresponds to the Eq. 5 with a constant concentration c g = c g0 . The black solid line corresponds to this equation taking into account the variation of c g with time, by combining Eq. 6, 7 and 5. The inset shows the variation of c g and V e for these two cases.

account the addition of glycerol in the thinning dynamics. Glycerol is a hygroscopic component, known to affect the solution activity [START_REF] Bouchaudy | Steady microuidic measurements of mutual diusion coecients of liquid binary mixtures[END_REF] and thus, its evaporation rate [START_REF] Producers | Physical properties of glycerine and its solutions[END_REF][START_REF] Chen | Hydrogen bonding analysis of glycerol aqueous solutions: a molecular dynamics simulation study[END_REF]. The saturation vapour pressure p sat (c g ) at the evaporating interface is modied by the presence of glycerol in the solution and can be written as [START_REF] Producers | Physical properties of glycerine and its solutions[END_REF]:

p sat (c g ) = p sat (c g = 0) 1 -c g 1 + c g (a -1) (4)
with a = 0.248 [START_REF] Roux | Everlasting bubbles and liquid lms resisting drainage, evaporation, and nuclei-induced bursting[END_REF][START_REF] Producers | Physical properties of glycerine and its solutions[END_REF]. As in Eq. 2, the drainage caused by the capillary suction and gravity remains unchanged. Only the evaporation term is modied by taking into account the change in saturation vapour pressure at the interface given by Eq. 4, which leads to writing the thinning rate such as

∂h ∂t = - γ η h 5/2 R 5/2 + h 3 Rℓ 2 c -k e 1 -c g 1 + c g (a -1)
-RH .

(5) The thinning rate corresponding to evaporation, the second term on the right hand side of this equation, can therefore be positive or negative, which means that in the presence of glycerol, the water can either evaporate (positive term) or condense (negative term). This depends on both the glycerol concentration and the relative humidity.

In this section, we will not consider any adjustable parameter to integrate the Eq. 5: the initial thickness h 0 corresponds to the experimentally measured value and the evaporation constant k e to the value determined in the previous paragraph, in the absence of glycerol. The prediction of Eq. 5, for a constant glycerol concentration c g (t) = c g0 , is shown in black dotted lines in Fig. 5.

However, the mass concentration c g varies over time and we can attempt to quantify this eect. To integrate Eq. 5, the time evolution of the glycerol concentration c g (t) must be precised from the exchanges with the bath and the atmosphere. We consider that the bubble has a spherical cap of volume Ω(t) = 2/3π(3R 2 h(t) + 3Rh(t)+h(t) 3 ), for which the liquid composition is homogeneous. The weight of glycerol in the bubble is dened as m g (t) = c g (t)ρΩ(t). From the denition of the glycerol concentration c g (t) = m g (t)/(m g (t) + m w (t)), we have m w (t) = (1 -c g (t)) ρΩ(t). The time variation of the glycerol weight in the spherical cap is due to drainage of velocity

V d [h(t)] = γ/η h 5/2 /R 5/2 + h 3 /(Rℓ 2 c ) , which gives dm g dt = -ρ g V d [h(t)]SΦ g (t), (6)
where S = 2πR 2 is the surface area of the hemispherical cap and Φ g (t) = (1 + ρ g /ρ w (1/c g (t) -1)) -1 is the volume concentration of glycerol. Similarly, for the water content, the variations is due to the drainage velocity V d but also the evaporation velocity

V e [c g (t)] = k e ((1 -c g )/(1 + c g (a -1)) -RH), such that dm w dt = -ρ w V d [h(t)]SΦ g (t) -ρ w V e [c g (t)]S. (7)
We solve equations 5, 6 and 7 for the following initial conditions, using an Euler-type method: c g0 = 0.2, m w0 = (1 -c g0 )ρΩ 0 and m g0 = c g0 ρΩ 0 with Ω 0 = 2/3π(3R 2 h 0 + 3Rh 0 + h 3 0 ) the volume of the liquid in the bubble cap at t = 0. Note that the measured surface tension of water/glycerol mixtures in presence of TTAB remains almost constant up to c g = 0.4, so we used the value given in the Tab. 1.

The results of this integration are shown in Fig. 5 with black solid lines, describing relatively well the temporal evolution of the measured thickness for the two concentrations. They show a weak eect, to rst order, of taking into account the variation of c g with time. The inset shows the evolution of the glycerol concentration and the evaporation rate during the thinning of the bubbles. As time goes by, the glycerol concentration increases, which leads to a decrease in the evaporation rate and therefore the bubble thinning rate. We attribute the enhanced stability of these bubbles at longer times to the reduction of the thinning rate.

Conclusion

In absence of evaporation, physical-chemistry has no impact on the thinning dynamics of surface bubbles. It aects only the initial thickness, which can be explained by a surface elasticity depending on the surfactant concentration. In presence of evaporation, the thinning dynamics is aected by the TTAB concentration. This effect can be explained at the rst order by an evaporation ux depending on the TTAB concentration. Nevertheless, a more precise description of this dynamic probably requires a better understanding of the role of surfactants on evaporation. This role can be due to a barrier exerted by the surfactants to evaporation [START_REF] Victor | Retardation of evaporation by monolayers: transport processes[END_REF], which could explain the variation of the evaporation rate with the surfactant concentration. Evaporation could also generate additional solutal or thermal surface tension gradients, which could add a stabilizing or destabilizing liquid ow [START_REF] Poulain | Ageing and burst of surface bubbles[END_REF][START_REF] Li | Eect of humidity on dynamic foam stability[END_REF]. Additionally, the water evaporation could lead to a variation of the local concentration in surfactants, which can also aect the ow and/or evaporation rate. This work thus opens new questions, which shall be answered in a future work and will benet from the exploration of a wider range of chemical systems. Furthermore, we observed that the addition of glycerol slows down the thinning dynamics. Even if we know that numerous potential mechanisms at stake due to evaporation in presence of surfactants or surfactants/glycerol solutions can occur [START_REF] Li | Gravitational eect in evaporating binary microdroplets[END_REF][START_REF] Shin | Bénard-Marangoni instability driven by moisture absorption[END_REF][START_REF] De Suduiraut | Hydrodynamique des lms liquides hétérogènes[END_REF], the simple model presented in this article allows a good description of the data. The eect of the addition of glycerol can be described by considering both the modication of the saturation vapor pressure in the presence of glycerol and the increasing glycerol concentration in time due to water evaporation.

We have enlightened that the description of surface bubbles thinning dynamics requires a better understanding of the eect of physical-chemistry on evaporation. This eect can come from dierent mechanisms (i) the concentration of non volatile components along time [START_REF] Roux | Everlasting bubbles and liquid lms resisting drainage, evaporation, and nuclei-induced bursting[END_REF] (ii) the existence of a barrier to evaporation due to the presence of surfactants at the interface, which can aect the evaporation ux [START_REF] Victor | Retardation of evaporation by monolayers: transport processes[END_REF] or the apparition of solutal or thermal surface tension gradient [START_REF] Poulain | Ageing and burst of surface bubbles[END_REF][START_REF] Li | Eect of humidity on dynamic foam stability[END_REF], which can lead to additional liquid velocities.

Figure 1 :

 1 Figure 1: A container with the solution of interest is set in a humidity-controlled chamber. To generate the bubble, air is injected by a syringe pump. The lm thickness at the apex of the surface bubble is measured by white light spectrometry.

Figure 2 :

 2 Figure 2: Variation of the initial thickness of surfactant solutions as a function of the TTAB concentration.The error bars correspond to the estimated accuracy of 0.2 µm on the determination of the initial thickness. The inset shows the Gibbs-Marangoni elasticity of the lm obtained from h 0 using Eq. 1. The dashed line is a power law t: E = 0.10 c -0.20 . The initial thicknesses plotted correspond to those obtained by leaving h 0 as an adjustable parameter when we compare Eq. 2 to our data, shown in Fig.3.

Figure 3 :

 3 Figure 3: Film drainage in a saturated environment for bubbles stabilized by dierent concentrations of TTAB ranging in 0.5-20 cmc. The continuous lines correspond to Eq. 2. The same graph is represented in inset in logarithmic scale.

4 . 1

 41 Control of evaporation via the atmospheric humidity

Figure 5 :

 5 Figure 5: Time evolution of the lm thickness h at the bubble apex in a environment of controlled humidity (55 ± 5 %) for bubbles stabilised with or without an initial glycerol concentration c g0 = 20 wt.%, for xed TTAB concentrations: (a) 0.5 cmc and (b) 20 cmc. The data in the absence of glycerol and the corresponding colour adjustment correspond to those shown in Fig.4. For the data in the presence of glycerol, the black dotted line corresponds to the Eq. 5 with a constant concentration c g = c g0 . The black solid line corresponds to this equation taking into account the variation of c g with time, by combining Eq. 6, 7 and 5. The inset shows the variation of c g and V e for these two cases.

Table 1 :

 1 Surface tension γ measured for TTAB solutions at dierent concentrations noted c in the table, at

		5 cmc	50.2	998	1.00
	TTAB	1 cmc	38.2	998	1.00
	TTAB	5 cmc	35.9	998	1.00
	TTAB	10 cmc	35.7	998	1.00
	TTAB	20 cmc	35.2	998	1.00
	TTAB/Gly 0.5 cmc	42.3	1047	1.74
	TTAB/Gly 20 cmc	23.4	1047	1.74

Acknowledgments

The authors are grateful to the referees for their careful reading of the manuscript and their comments that helped to improve it. Funding supports from ESA (MAP Soft Matter Dynamics) and CNES (through the GDR MFA) are acknowledged.