Thermal Marangoni bubbles

Saurabh Nath, Guillaume Ricard, Panlin Jin, Ambre Bouillant, David Quéré

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Saurabh Nath, Guillaume Ricard, Panlin Jin, Ambre Bouillant, David Quéré. Thermal Marangoni bubbles. Soft Matter, 2022, 18 (38), pp.7422 - 7426. 10.1039/d2sm00961g. hal-05347427

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Thermal Marangoni bubbles†

Saurabh Nath, Guillaume Ricard, Panlin Jin, Ambre Bouillant and David Quéré

When air reaches the surface of a pool (or bath) of pure liquid, it does not form long-lasting bubbles, as opposed to when the bath contains surfactants. Here we describe what happens when the pool is pure (consisting of oil), yet hot. The bubbles dwelling at the surface can then live for minutes or even longer, which we interpret as a consequence of the gradients of temperature generated in this experiment. Indeed, oil is observed to be constantly drawn to the apex of the bubble, which opposes its gravitational drainage. Since their existence relies on ascending Marangoni flows, thermal bubbles are found to be dynamical in essence, which endows the oil film with remarkable stability and persistence.

Air injected into soapy water produces lasting bubbles at the surface of the bath,¹⁻³ as is also often seen when raindrops impact a pool to create a crown that engulfs air.⁴ The persistence of the bubbles has been discussed, and it is generally assumed that surfactants make the water/air interface solid-like, which hinders the gravitational drainage in water films.⁵,⁶ Conversely, bubbles at the surface of a bath of pure liquid are ephemeral, except for ultra-viscous liquids where the drainage is delayed by viscosity.⁷,⁸ Volatile liquids can behave differently: bubbles made of pure alcohol were observed to be unexpectedly stable, which was interpreted as a consequence of the appari-tion of evaporation-driven thermal gradient. The resulting Marangoni flows can compensate for the evaporative and gravitational effects depleting the film.⁹ Volatility can also favour the vertical transport of a mixture, as seen for wine in a glass, where solutal Marangoni effects drive the upward motion, accumulation of liquid, and production of tears.¹⁰⁻¹²

In order to eliminate possible evaporation-induced effects as alluded to in ref. 9 we consider here a non-volatile liquid and look for a purely thermal Marangoni effect.¹³,¹⁴ More specifi-cally, we wonder if differential heating in itself is sufficient to stabilize large air bubbles sitting on the surface of the liquid. Our experiment thus complements the classical study by Young et al.¹⁵ where air bubbles in a bath were observed to be possibly stabilized by a vertical temperature gradient $T' = dT/dz$, with $T$ denoting the temperature and $z$ the vertical coordinate. If the liquid is hotter at the bottom and because surface tension $\gamma$ decreases with temperature (at a rate $\alpha = d\gamma/dT$ of typically $-10^{-4} \text{ N m}^{-1} \text{ K}^{-1}$), a differential tension $\Delta\gamma$ scaling as $\alpha T'R$

appears along a bubble with radius $R$. The total (upward) Marangoni force $\Delta\gamma R$ acting on the bubble can thus exceed the Archimedes thrust $\rho g R^3$ if the bubble size is smaller than $\alpha T'/\rho g$, where $\rho$ is the liquid density and $g$ the acceleration of gravity. This Marangoni capillary length varies between $10 \ \mu\text{m}$ and $100 \ \mu\text{m}$ when the temperature gradient increases from $10 \ ^{\circ}\text{C cm}^{-1}$ to $10 \ ^{\circ}\text{C mm}^{-1}$, which corresponds to the size of the arrested bubbles reported in ref. 15. Only tiny bubbles can be stopped by an adverse thermal Marangoni effect, and we wonder here whether there is an equivalent effect for a floating bubble.

To that end, we consider a silicone oil with moderate viscosity ($\eta = 96 \ \text{mPa s}$ at $T = 25 \ ^{\circ}\text{C}$), for which we expect neither surface-active contaminants, nor strong viscous effects, nor significant evaporation. The oil, supplied by Sigma Aldrich, has a surface tension $\gamma = 20 \ \text{mN m}^{-1}$ and a density $\rho = 960 \ \text{kg m}^{-3}$ at ambient temperature. All parameters $\eta$, $\gamma$ and $\rho$ decrease when $T$ increases from $25 \to 100 \ ^{\circ}\text{C}$, by a relative amount of 30% and 10% for $\gamma$ and $\rho$, and by a factor 4 for the viscosity (Fig. S1 in the ESI†). The experiment consists of heating silicone oil on a hot plate to a temperature of typically $100 \ ^{\circ}\text{C}$. The bath has a depth of $5 \ \text{cm}$ sufficient to eliminate the thermal cells generated by heat on a shallow bath, a case where surface bubbles would break at the boundaries of the cells. Air at room temperature is injected inside the bath with a syringe, producing clusters of millimetric bubbles that rise until they reach the surface of the oil where they eventually form a hemispherical bubble having a radius $R$ of about $1 \ \text{cm}$ (Fig. 1a and b). If the floating bubbles are free (Fig. S2, ESI†), they are drifted by the slow convective currents generated by heat at the bath surface. Hence, it appears convenient to place a copper wire that holds the bubbles in place, as seen in Fig. 1b. We will show further that the wire also serves as a probe to quantify the dynamics of oil in our experiment.

<footnote>Physique et Mécanique des Milieux Hétérogènes, UMR 7636 du CNRS, PSL Research University, ESPCI, 75005 Paris, France
† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2sm00961g</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 1** Thermal bubbles. (a) Sketch of the experiment: a bath of oil is heated on a hot plate (in red), and air bubbles are injected into the bath so as to form a hemispherical bubble with radius *R* at the surface. Air is at an ambient temperature so that temperature gradients naturally appear along the bubble, which creates ascensional Marangoni flows and stabilizes the bubble. (b) Bubble obtained after injecting air in a bath of hot silicone oil whose surface temperature *T*<sub>s</sub> is 73 °C. The bubble is held with a copper wire contacting its North pole. At the bottom, we see its reflection at the bath surface. (c) Bubble lifetime τ<sub>L</sub> as a function of *T*<sub>s</sub>, for more than 250 bubbles with radii between 4 mm and 7 mm. We highlight the average lifetimes (per interval of temperature) with full symbols and data dispersion with bars. All times are far above the freefall time (R/g)<sup>1/2</sup> shown as a strip that takes into account the dispersity in the bubble size *R*. For all experiments, we used a silicone oil of viscosity η = 96 mPa s at 25 °C.</figcaption></figure>

Without temperature gradient and for moderate liquid viscosity, the lifetime τL of bubbles is expected to be small since the oil film then just falls in the field of gravity, in a time scaling (R/g)1/2 ≈ 30 ms. As seen in Fig. 1c, bubbles in our experiment live much longer, minutes or more, far above the free-fall time indicated with a shaded line in the figure. We interpret this longevity as a consequence of the temperature difference between the hot bath surface and the cooler bubble apex, exposed to air whose temperature far from the bath is 25 °C. The measured τL are spread out, a property shared with soap bubbles²,¹⁶ that reflects the fragility of these objects. However, the trend is clear: both the maximum value of τL and its average (full symbols) markedly increase with heating,

by about two orders of magnitude in the explored range of temperatures.

These bubbles are essentially dynamical objects. As seen in Fig. 2a and in Movie (ESI†), oil accumulates at the apex of the bubble until its mass makes it fall as tears or drips into the bulk bath. This phenomenon reveals the existence of upward flows drawing oil from the bath to the bubble top. If the flows are strong enough to compensate gravity, the lifetime of the bubbles gets extended. This effect is also seen in the absence of the pinning wire (Fig. S2, ESI†), showing that the wire alone does not explain the stability of the bubbles. Their thermal nature is further evidenced in the thermographic images of Fig. 2a, where we note a temperature difference of about 25 °C

<figure><img src="image_4.png" /><figcaption>**Fig. 2** Infrared imaging of thermal bubbles. (a) Successive infrared images of a thermal bubble (R ≈ 5 mm) sitting on a hot bath of silicone oil (T_s = 73 °C) and viewed from the top. Time is given in seconds (right bottom corner), and the bar indicates 1 cm. The bubble has a distinctly colder top – a region where oil accumulates and eventually drips after about 0.7 s, only to restart the process again. The snapshots are extracted from Movie (ESI†). Images are taken on a thermal camera (FLIR A600 series) using a calibration range from –40 °C to 150 °C, where the effective emissivity of silicone oil increases from 0.9 to 0.95 as the temperature difference ΔT is increased from 10 °C to 50 °C.¹⁷ (b and c) An axis x is drawn across the bubble (dashed line), along which we measure the temperature distribution T(x) and thus the temperature difference ΔT between the base and the apex. (d) When plotted as a function of the bath temperature T_s, ΔT increases linearly from 0 to 50 °C with a slope of 0.5.</figcaption></figure>
between the bath surface and the bubble top. This generates a gradient of surface tension and thus upward Marangoni flows. Oil climbs and gradually accumulates in the cold region, until the resulting “tears” fall when their size exceeds a few millimetres. The process of liquid accumulation followed by an inevitable drip appears to be periodic, with a period $τ ≈ 0.7$ s in the example of Fig. 2a. Drips are cold, and thermal bubbles are observed to burst systematically at the drip/bubble boundary, a region where the vertical gradients of temperature can be erased. This may weaken the bubble, which qualitatively explains the dispersity of times reported in Fig. 1c.

The drips can be exploited to access the magnitude of the Marangoni effect. The copper wire holding the bubbles retains the oil flowing to the bubble apex, and it thus enables us to assess the volume $Ω$ of the drips. Since both $Ω$ and the drip periodicity $τ$ are accessible, we can eventually determine the flow rate $Ω/τ$ of oil rising from the bath – a quantity of utmost interest in our problem. Simultaneously, infra-red images can be analysed frame by frame to map the temperature distribution (Fig. 2b). Oil being opaque to infra-red, the camera measures the surface temperature of the oil as long as the film is thicker than the submicrometric penetration length of light wavelength in oil. Being at a maximum distance from the bath and exposed to an ambient atmosphere, the North pole of the bubble is cooler than its equator, by a quantity $ΔT$ on the order of 25 °C in Fig. 2c. We find that $ΔT$ increases linearly with $T_s$, the bath surface temperature determined independently using a thermocouple. This provides a calibration curve shown in Fig. 2d. Heating the bath from the lab temperature of 25 °C to 130 °C allows us to vary the control parameter $ΔT$ from 0 to 50 °C.

When examined as a function of time as in Fig. 3a, the spatially averaged bubble temperature exhibits oscillations that can be ascribed to the periodic dripping. The drip period $τ$ deduced from such curves is found to strongly depend on the oil temperature, falling by a factor 6 as the bath temperature is elevated from 41 °C to 126 °C. When plotting the drip period as a function of the temperature differential $ΔT$ across the bubble

(Fig. 3b), we report that drips are about 20 times more frequent when $ΔT$ increases from 10 °C to 50 °C. However, this frequency alone does not provide the flow rate for which we also need to measure the drip volume $Ω$. This is achieved by supporting the top infrared camera with a synchronized sideview camera (recording at 24 fps). By comparing the volume of oil adhering to the wire before and after each drip, we eventually deduce the volume $Ω$.

A drip can correspond to the maximum quantity of oil retained by the wire. If we denote as $r$ the characteristic drip size, oil is held as long as the drop weight $ρgr^4/R$ (where we introduced the mean slope $r/R$ experienced by the drop) remains smaller than the capillary suction $γd$ exerted by the wire. Altogether, the drip volume $Ω$ is found to scale as $(dRκ^{-2})^{3/4}$, where $κ^{-1} = (γ/ρg)^{1/2}$ is the capillary length. This volume is on the order of 10 µL, the typical value at low $ΔT$ in Fig. 3c. It weakly depends on temperature since both the density $ρ$ and surface tension $γ$ decrease with $T$, which narrows the variation of capillary length (Fig. S1, ESI†). This regime of minimum drip volume is found in Fig. 3c to be followed by a linear increase in $ΔT$. Indeed, the drip fall is also resisted by a Marangoni force acting toward the cold region and scaling as $Δγr$, that is, $αΔTr$. We expect the Marangoni force to exceed the capillary force for $ΔT > γd/αr$, namely a few tens of degrees. Then, the drip volume is given by the balance $αΔTr ∼ ρgr^4/R$, which yields $Ω ∼ αRΔT/ρg$. This quantity increases with temperature, as indeed seen in Fig. 3c, where the increase of volume at large $T$ is about 0.3 µL per degree, in fair agreement with the value of $αR/ρg$ given by our scaling arguments, which is typically 0.1 µL per degree.

Since heating the bath shortens $τ$ and increases $Ω$, the upward flow rate $\dot{Ω} ∼ Ω/τ$ responsible for the steadiness of the bubble is expected to be highly sensitive to temperature. Fig. 4a confirms this strong dependency, with a gain in rate by two orders of magnitude when the bath temperature is brought from 30 °C to 150 °C. The data can be made dimensionless by normalizing the surface tension differential $Δγ$ by $γ$ and the rate $\dot{Ω}$ by a visco-capillary flux $γκ^{-2}/η$, where all values of the

<figure><img src="image_5.png" /><figcaption>**Fig. 3** Dripping characteristics. (a) Temporal variation of the mean oil temperature $T$ measured across a bubble meridian as shown in Fig. 2c and d. The temperature $T$ fluctuates, due to the drips emitted at the bubble North pole, from which we deduce the drip period $τ$. In the top plot, the bath is heated at $T_s = 126$ °C, to which correspond a temperature drop $ΔT = 53$ °C and a drip period $τ = 0.5 ± 0.1$ s; in the bottom plot, we have $T_s = 41$ °C, $ΔT = 10$ °C and $τ = 3.7 ± 0.3$ s. (b) Plotting $τ$ as a function of $ΔT$ confirms that the stronger the temperature differential, the more frequent the drips. (c) Dripping volume $Ω$ as a function of $ΔT$. Dashed lines show the two drip regimes predicted in the text.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4 Feeding rate of a thermal Marangoni bubble. (a) Measured upward flow rate of oil as a function of the bath temperature, showing the high sensitivity of this flux to the temperature. The dashed line is the fit provided by the model (see the text). (b) Volume flow rate in a normalized by the capillary flux $\gamma\kappa^{-2}/\eta$, as a function of the relative difference of surface tension $\Delta\gamma/\gamma$ between the bath and the bubble apex. The dashed line shows the slope 2 obtained when assuming that the Marangoni flow sustaining the bubble is balanced by the film weight.</figcaption></figure>

parameters are taken at the corresponding temperature. In Fig. 4b, we observe a power-law behaviour with an exponent close to 2, the value indicated by the dashed line.

These results guide us for modelling the phenomenon. A temperature differential $\Delta T$ between the bath and the bubble apex generates a surface tension differential $\Delta\gamma = \alpha\Delta T$ and thus a gradient in the surface tension of order $\Delta\gamma/R$, since temperature gradients are observed at the bubble scale (Fig. 2b and c). For films climbing along solids$^{18,19}$ or even free yet viscous,$^{20}$ it has been proposed to balance the Marangoni force $\Delta\gamma/R$ by the viscous friction $\eta V/h$ arising from velocity gradients across the film thickness $h$. This yields $V \sim h\Delta\gamma/\eta R$, where $h$ is unknown. If we assume that the thickness is selected by a Landau-Levich mechanism, $h \sim \kappa^{-1}(\eta V/\gamma)^{2/3}$, we deduce that $h$ scales as $(\kappa^{-3}/R^2)(\Delta\gamma/\gamma)^2$, that is micrometric in magnitude and thus too small to account for frequent drips of millimetric size. In addition, the flow rate $hVR$ then varies as $\Delta\gamma^5$, much sharper than that seen in Fig. 4b.

We rather propose to simply consider a strong slip at the free interfaces, a natural assumption for a film of oil without surface-active contaminants. Then, we just balance the Marangoni force $\Delta\gamma/R$ by the oil weight $\rho g h$, where both forces are written per unit area. This equation yields the (maximum) quantity of liquid $h \sim \Delta\gamma/\rho g R$ that can be drawn by the surface tension differential. We finally obtain the characteristic oil velocity $V$ by equating the Marangoni stress $\Delta\gamma/R$ and the viscous friction, which we consider purely extensional in the presence of interfacial slip. Then it scales as $\eta V/R$, which yields $V \sim \Delta\gamma/\eta$ after its balance with the driving force. Putting together these two laws, we get an ascending flow rate $\dot{\Omega} \sim \Delta\gamma^2/\rho g \eta$.

The oil flow rate can be rewritten as $\dot{\Omega} \sim (\Delta\gamma/\gamma)^2(\gamma/\eta)\kappa^{-2}$, a quantity expected to be around $10 \, \mu\text{L s}^{-1}$, close to the values in Fig. 4a. Moreover, the normalized rate $\eta\dot{\Omega}/\gamma\kappa^{-2}$ is convincingly fitted by a law quadratic in $\Delta\gamma/\gamma$ (dashed line in Fig. 4b) in the whole interval explored in the experiments, about one decade in $\Delta\gamma/\gamma$. The best fit provides a numerical factor of 0.7, of order unity, suggesting that our arguments provide not only the scaling but also consistent magnitudes. We can finally evaluate

$\dot{\Omega} = 0.7\Delta\gamma^2/\rho g \eta$ as a function of the bath temperature $T_s$, since all functions $\Delta\gamma(T_s)$, $\rho(T_s)$ and $\eta(T_s)$ are known (Fig. 2d and Fig. S1, ESI†). As seen in Fig. 4a, the agreement with the data is quite satisfactory, supporting our main assumptions, namely that the force resisting the driving Marangoni thrust is nothing but gravity, and that oil slips at the film interfaces.

A thermal gradient thus enables bubbles made of pure liquid to defy their ephemeral character. Their lifespan is then extended from tens of milliseconds to hundreds of seconds, owing to the tendency of thin films to flow towards the cold. Contrasting with quasi-static soap bubbles, thermal bubbles are highly dynamical objects where liquid is constantly transported by surface tension gradients (present on both sides of the film) so as to compensate its gravitational fall. This scenario fixes the film thickness $h$, expected to scale as $\Delta\gamma/\rho g R$, a variation hyperbolic in $R$ from which we anticipate that it becomes challenging to form large thermal bubbles, unlike soap films. Plugging numbers into the scaling, we find a thickness of a few tens of micrometers, much thicker than the micrometric thickness of soap bubbles and wetting films climbing a wall subject to a thermal gradient, two cases where viscous effects tend to thin the films.$^{19}$ This evaluation allows us to compare the global Marangoni force to the buoyancy that also drives oil upwards. The ratio between these two forces can be written as $M = \Delta\gamma/\Delta\rho g h R$. The density differential $\Delta\rho$ being of order $100 \, \text{kg m}^{-3}$ (Fig. S1, ESI†), the number $M$ is typically 10 in our experiments, justifying why we could neglect the action of thermal buoyancy.

Marangoni forces are known to generate fascinating and exotic effects.$^{9–12,15,20–23}$ Here we reported how their ability to pull thick films of a pure liquid against gravity can be exploited to build large, persistent bubbles at the surface of a pool. Interestingly, the stability of these objects can be switched on and off by temperature, an original property in the context of free-standing films. A more detailed description of the flow within the film and of the accumulating drips would be natural extensions of our work. Moreover, the fragility of the bubbles would require a devoted study in order to better understand the dispersity of their lifetime. We could also look at their
interactions – how proximal bubbles merge, and whether it is
possible to build a thermal foam. Finally, these ideas could be
extended to water: can a bubble of pure water be made this
way? And how do these findings modify our views on soap
bubbles subjected to thermal gradients?

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 722497. We also thank Caroline Cohen for her help with the infra-red camera.

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Thermal Marangoni bubbles

Saurabh Nath, Guillaume Ricard, Panlin Jin, Ambre Bouillant & David Quéré

*Physique et Mécanique des Milieux Hétérogènes, UMR 7636 du CNRS, PSL Research University, ESPCI, 75005 Paris, France.*

*Supplementary Information*

**Influence of the temperature on the oil properties**

<figure><img src="image_7.png" /></figure>

**Figure S1.** a), b), c) Viscosity η, surface tension γ and density ρ of the silicone oil used in our experiments, as a function of the oil temperature T. d) We also display the temperature variation of the capillary length κ⁻¹ = (γ/ρg)¹/² deduced from the plots in b and c. In the explored range of temperature (between 50°C and 150°C), κ⁻¹ decreases by 20%, while the viscosity falls by a factor ~7.

**Influence of the wire**
Herein, we question the influence of the pinning wire used to hold the bubbles in place (figure 1b). The experiment described in the main text is reproduced with free bubbles. As shown in figure S2, the bubble, viewed from above with an infra-red camera, dwells on the oil bath for at least a few seconds, a lifetime much longer than expected for bubbles floating on a pure liquid bath without temperature gradients. The thermographs also reveal the existence of cooler tears of oil falling along the bubble from its North pole, as observed with pinned bubbles. The wire retains slightly more oil that here; more interestingly, it allows us to quantify the volume accumulated in each drip, without altering the essence of the thermal bubbles.

<figure><img src="image_8.png" /><figcaption>**Figure S2.** Snapshots of a free bubble (base diameter of 5.4 mm) sitting on a hot bath of silicone oil and viewed from the top. The bubble is obtained after injecting air in a bath whose surface temperature is 71°C. Observations are similar to that made when the bubble is pinned by a wire (figures 1 and 2 in the accompanying paper). Bubbles have a long lifetime, their North pole is cooler than their equator, and oil regularly drips along them. However, the volume of the drips is harder to measure than with the device in figure 1, making this experiment more qualitative than in the accompanying paper.</figcaption></figure>

**Movie caption**

Infra-red video of a thermal Marangoni bubble with a radius $R = 7$ mm. The movie is played in real time and it shows the bubble from the top. The bubble is obtained after injecting air inside a heated bath of silicone oil whose surface temperature is 125°C; as the injected, millimetric bubbles reaches the bath surface, they gather into a hemispherical, centimetric bubble that lives for 20 seconds. The movie also evidences a temperature differential between the North pole of the bubble and its equator, whose mean value is $\Delta T = 47^\circ C$.