Azimuthal instability of the radial thermocapillary flow around a hot bead trapped at the water-air interface
G. Koleski, A. Vilquin, J.-C. Loudet, T. Bickel, B. Pouligny
AAzimuthal instability of the radial thermocapillary flow around a hot bead trapped atthe water – air interface
G. Koleski,
1, 2
A. Vilquin, J.-C. Loudet,
2, 4
T. Bickel, and B. Pouligny Univ. Bordeaux, CNRS, Laboratoire Ondes et Matière d’Aquitaine (UMR 5798), F-33400 Talence, France Univ. Bordeaux, CNRS, Centre de Recherche Paul Pascal (UMR 5031), F-33600 Pessac, France ESPCI, CNRS, Institut Pierre – Gilles de Gennes,Laboratoire Gulliver (UMR 7083), F-75005 Paris, France Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada
We investigate the radial thermocapillary flow driven by a laser-heated microbead in partialwetting at the water-air interface. Particular attention is paid to the evolution of the convectiveflow patterns surrounding the hot sphere as the latter is increasingly heated. The flow morphologyis nearly axisymmetric at low laser power ( P ). Increasing P leads to symmetry breaking with theonset of counter-rotating vortex pairs. The boundary condition at the interface, close to no-slip inthe low- P regime, turns about stress-free between the vortex pairs in the high- P regime. Theseobservations strongly support the view that surface-active impurities are inevitably adsorbed onthe water surface where they form an elastic layer. The onset of vortex pairs is the signature ofa hydrodynamic instability in the layer response to the centrifugal forced flow. Interestingly, ourstudy paves the way for the design of active colloids able to achieve high-speed self-propulsion via vortex pair generation at a liquid interface. I. INTRODUCTION
Temperature gradients arising along a fluid interfaceare responsible for local variations of its surface tension.Shear stresses ensue and set the liquid into motion. Thisphenomenon is known as the Marangoni effect [1] andthe resulting flow is said thermocapillary. Such flows areubiquitous in everyday life [2] – [4]. Over the past fewdecades, thermocapillary convection has attracted greatinterest in many research areas.As an example, Marangoni – driven instabilities havebeen recognized to severely damage the quality of crystalsgrown from the melt [5, 6] as well as that of arc welds [7].Interested readers can refer to reviews [8, 9] for a detailedpresentation of thermocapillary instabilities.Surface forces dominate over body forces at smallscales due to large surface – to – volume ratios. Surfacetension driven flows thus appear as ideal candidates forthe manipulation of microfluidic systems. For instance,many efforts have been devoted in recent years to thethermocapillary actuation of droplets sitting on the freesurface of immiscible liquid films [10, 11]. It has also beenshown that breaking the symmetry of recirculation flowswithin a microfluidic droplet allows for efficient chaoticmixing [12]. In Ref. [13], gas bubbles forming on a thinlight – absorbing liquid layer were remotely trapped andmanipulated using an optical tweezer. The authors of [14]tailored laser – induced vortical flow patterns around amicrobubble to sort polystyrene beads by size.Researchers are now going further and further in thedesign of lab – on – a – chip platforms whose processingunits enable combined and automatized operations onmicrofluidic systems such as droplet production, size –selective sorting, pumping, division and fusion, transportalong microchannels or trapping [15] – [17]. Ref. [18] is atypical example in which heaters embedded on a printed circuit board are engineered to make levitated dropletscollide and merge their reagents via thermocoalescence.A holistic view of thermocapillary – based microfluidics isprovided in the exhaustive reviews [19, 20].Light – to – work conversion has also been harnessed inthe field of active matter. Recently, Maggi et al. haveshown how to rotate asymmetrically – toothed microgearssitting on a liquid – air interface by coating them on oneside with a light – absorbing material [21]. Their work isone more example that promotes thermocapillarity as ahighly efficient mechanism for colloidal self – propulsion.The present work is motivated by recent observationsof the dynamics of a laser – heated microsphere at thewater – air (WA) interface [22, 23]. The authors of [22]report that, at sufficiently high power, the microsphereorbits the laser beam on quasicircular closed tracks. Mostinterestingly, they also observe that counter – rotatingvortex pairs escort the hot sphere along its trajectory(Fig. 1). No stable orbits but radial oscillations of theparticle arise in [23] under strong heating conditions.This echoes the situation depicted in [24], except thatcapillary — instead of optical forces — act as restoringforces. Capillarity combines with Marangoni driving tosustain oscillations, much like in Ref. [23]. One majorfeature common to both studies [22, 23] is that the light –absorbing microbead behaves as a self – propelled particleunder the action of thermocapillarity.The present study is the natural continuation of theexperiments by Girot et al. [22], with the goal to unravelthe propulsion mechanism of the hot sphere (Fig. 1).However, characterizing the multipolar flows induced bya hot particle moving at the WA interface is an arduoustask. We consider a simpler approach which consists infixing the heat source at the interface whilst monitoringthe convective flows that develop around it. Despite itsfundamental interest to interfacial hydrodynamics, thissituation has been given only little consideration so far. a r X i v : . [ c ond - m a t . s o f t ] S e p Figure 1. Vortex pair forming around an orbiting hot bead.This schematic depicts the situation shown in the SupportingInformation Video S6 of Ref. [22]. Tracers reveal a first vortexahead of the bead and a second vortex inside the orbit (redloops). The former rotates anticlockwise whereas the latterrotates clockwise, just like the hot sphere on its track (yellowcircle). The black cross marks the center of the vertical laserbeam sketched by the green halo. Experimental parameters(Ref. [22]) : beamwaist ω = 6 . µ m, laser power P = 28 mW.Scale bar: 5 µ m. Among the rare contributors to the problem, Bratukhinand Maurin derived an analytical solution of the coupledNavier – Stokes and heat transport equations in the non-linear Marangoni regime [25]. Later, Shtern and Hussainprobed the stability of axisymmetric flows in responseto azimuthal perturbations [26]. On the experimentalside, Mizev et al. [27] revisited early works pertaining tothe structure and stability of surface tension driven flows.The remainder of the paper is structured as follows.We describe the system and its operating principle inSec. II. Next comes the Materials and Methods’ Sec. III.We report our experimental observations in Secs IVand V. We first consider the base flow state which issimply axisymmetric around the heat source (Sec. IV).This type of flow is solely observed under low heating.We will see that experiments yield direct evidence for anelasticity of the interface. What happens when heatingis increased is the matter of Sec. V. We show that theradial symmetry of the low – power pattern breaks downas the flow self – organizes into counter – rotating vortexpairs. Salient characteristics of the multipolar flows arethen identified. Finally, we summarize and discuss themain outcomes in Sec. VI.
II. DESCRIPTION OF THE SYSTEM
Water fills a cylindrical cell of radius R and height H . The z – axis coincides with the axis of revolution ofthe container. It is oriented upwards, with unit vector e z . The principle of our experiment is the following: ahot bead of radius a , which is kept fixed as described inSec. III, sits in partial wetting across the WA interface Figure 2. Schematic of the system (not to scale). W : water; A : air; P : laser power; ∇ γ : surface tension gradient. Thearrows indicate the direction of the thermocapillary flow. located at z = 0 (Fig. 2). Because of thermal gradients inthe vicinity of the hot bead, Marangoni flows appear andpoint towards the colder edges of the vessel. Since wefocus on the evolution of the convective flow surroundingthe hot bead, the main tunable parameter here is thelaser power P . The technical features of our in – houseexperimental setup are specified in the following section. III. MATERIALS AND METHODSA. Sample cell
The sample cell is a 2 R = 22 . × H = 3 mmhigh cylindrical quartz cuvette purchased from ThuetFrance. After a 24h cleaning in sulfochromic acid, thelatter is thoroughly rinsed, then filled, with MilliporeMilli-Q water. A lid drilled with a 12 mm circular holein its center is placed a few mm above the sample cell tolimit the pollution of the interface by air contaminants,prevent massive evaporation and avoid surface agitationcaused by air turbulence. B. Hot bead and power supply system
A glassy carbon bead (Alfa Aesar), whose diameterslightly varies from one experiment to another around2 a = 300 µ m, is stuck onto the extremity of an optic fiberusing an UV curing adhesive (NOA 65). To this end, thefiber has been preliminary stripped up to the cladding.Carbon is prized here for its strong capacity to absorbthe incident laser light while resisting photodegradation.A 532 nm green laser beam generated by a QuantumLaser Opus source is guided towards the glassy carbonbead by a single – mode optic fiber (F-SA-C Newport). Inour experiments, the laser power P ranges from a few mWto a few tens of mW. Because of injection losses withinthe fiber optic coupler (transmission losses through thefiber are marginal), the power P impinging on the beadis only a fraction of the power P em emitted by the lasersource. In our practical conditions, we have P = (cid:15) P em with a coupling efficiency (cid:15) ≈ ◦ .The excess temperature on the surface of the bead∆ T ˙= T ( r = a ) − T ( T = 25 ◦ C, room temperature)reads ∆ T = α ( P / πκa ) ( κ = 0 . . m − . K − , waterthermal conductivity in standard conditions) [28]. In thelatter expression, the prefactor α < P convertsinto heat within the body of the carbon particle. Whatremains of the optical power is reflected and scattered bythe fiber – bead junction, making the hot sphere radiant(photograph on top of Fig. 3). Since the value of α isnot known, we cannot calculate the temperature of thebead using the above expression. However, pushing thesystem to a maximum power P max ≈
130 mW (beyondthis value, there is a serious risk of damage) makes smallbubbles nucleate on the bead’s surface, meaning that weare close to the water boiling temperature T b . Note thatin routine conditions the setup is operated at powers farbelow this limit. Based upon bubbling, and performinga linear interpolation between T b and T , we estimatethat ∆ T ranges between a few K, in the axisymmetricflow regime, and 40 K when the azimuthal instability isfully developed. An accurate measurement of ∆ T mightbe obtained by thermography using a dedicated infraredmicroscopy hardware, however, the latter has not beenset up yet.An alternate way to estimate the temperature of thecarbon sphere is to perform a numerical simulation of thefiber – bead geometry with the bead in partial wettingconfiguration at the water – air interface. The simulationshall compute the resulting flow field, which first requiressolving the full hydrodynamic problem (including heattransport from the surface of the carbon sphere). Thisobjective, beyond the scope of the present article, will bepart of a future publication [29]. C. Bead/water contact
In order to restrict the number of factors influencingthe system, we endeavor to keep the interface as flat aspossible. To this end, the cuvette is prefilled to the brimand water is then progressively removed. In such a way,a flat interface is obtained far from the heat source bycontact line pinning on the sharp edges of the cuvette.For the axis of the fiber to be oriented perpendicularlyto the surface while passing through the protective lid,the fiber is bent using a thread of adjustable tension tiedaround its coating and stretched between the fixationpoint and the fiber mounting plate (Fig. 3, top insert).The bead is displaced with a xyz translation stage tillbeing partially immersed in water. The tension exertedby the fiber on the bead leads to the formation of a steepmeniscus which, for the same reason as above, shall beminimized as much as possible. Simply stated, we wantthe interface to remain flat around the carbon sphere.
Figure 3. Experimental setup (not to scale). S, sample; OF,optic fiber guiding the 532 nm green laser beam that heats thebead; CU, coupling unit to rotate the set {optic fiber + bead}on the circular guide rail, as shown by the transparent partof the schematic; LS, 514 nm green laser sheets; {BC + ZO},bottom camera + zoom objective; DR, diode ring; {SC + CL},side camera + correction lens. The top insert is a photographof the optic fiber (1) bent by a thread of adjustable tension (2)that ensures the axis of the bead is perpendicular to the WAinterface. The bottom insert shows the ring of light – emittingdiodes as imaged on the video screen. The carbon sphere isthe dark disk well visible in the middle.
The deformation of the interface is monitored using a ringencrusted with light – emitting diodes that we positionbeneath the cuvette. The light from the diodes reflectsoff the water surface yielding a luminous circle on thevideo screen (Fig. 3, bottom insert). We then finely tunethe altitude of the {fiber + bead} unit in such a way thatthe circle shrinks down and is no longer visible on theimage. The latter tuning is accurate within a few  µ m.Because of water evaporation around the hot bead, theprocedure must be repeated several times within a singleexperimental run which usually lasts for one hour. D. Flow visualization
Flow visualization is based on laser sheet tomography.This technique requires seeding the liquid with tracers.Polystyrene beads from Magsphere (density ρ PS = 1 . . µ m serve here as tracer particles.We checked with a dedicated experiment [30] that thelatter are neutral towards the investigated flows, a basicrequirement any suitable tracer shall meet. Note that avolume fraction of these beads as minute as ϕ v ∼ − is sufficient to ensure proper flow visualization. In theplane of a laser sheet, tracers emit a fluorescent lightthat is sensed by video cameras. In the end, streamlinesappear on time – lapse pictures as streaks of light left bytracers along their trajectories.A couple of laser sheets powered by a 514 nm greenlaser source (Genesis CX 514 – 2000 STM from Coherent)allows for cutting the sample cell along selected planes :a horizontal laser sheet is positioned at a short distancebeneath the interface ( z ≈ − . e ≈ µ m), the defaults ofthe optical imaging hardware (essentially astigmatism)and the size of the camera pixels. A spatial resolutionof about 10 µ m is estimated empirically from the videoimages. The latter value holds in the region { r < } not too far from the vertical z – axis where details are notsignificantly blurred by astigmatism (a correction lens isplaced between the sample and the side camera to fix thisissue as much as possible, see Fig. 3). Note that imagesare exploited only within the latter viewing area. Tracerslocated at larger distances appear motionless. The timeresolution is set by the frame rates of the cameras which,depending upon the case, are operated from 2 up to 30frames per second. E. Flow velocity measurements
Flow velocities are measured either by single particletracking or particle image velocimetry (PIV) [32] – [34].In this study we use PIV only within vertical cuts. Themethod can be applied to horizontal cuts, but not close tothe WA interface due to large variations of longitudinalvelocities over the thickness of the laser sheet [35]. Belowwe uncover correlations between the morphology of thesurface flows and the interfacial dynamics by comparingsurface with subsurface velocities. Such an analysis relieson the ability to distinguish between surface and bulktracers while tracking them on vertical cuts : we exploitthe fact that particles at a small distance below the freesurface ( z <
0) display well discernible mirror imagesresulting from their reflection off the interface, whereasparticles found at the surface ( z = 0) appear as singlebright spots. This property is very helpful for locatingthe interface and hence measuring with pixel accuracy( δz ≈ µ m) the depths at which bulk tracers travel. Figure 4. Base torus. (a) Centrifugal surface flow. (b) Flowin a vertical cross – section. The cyan line marks the positionof the interface. The hot bead (2 a ≈ µ m) is sketched by ared disk on both figures. The bright spots visible on Fig. 4(a)around the hot bead are due to clusters of tracers settled onthe cell floor. Scale bar (common to both figures) : 3 mm. IV. BASE FLOW STATE
The thermocapillary flow observed under low heating,typically for
P ∼
A. A nearly solid interface
We compare the velocity of interfacial tracers with thatof tracers situated in a shallow layer which extends downto a few tenths of a mm underwater. Measurements aremade through direct particle tracking in a cross – sectionof the base torus. In practice, the centrifugal motion offour tracers located about 2 . z = 0)while the other three lie at shallow depths ( z < | v r, max | , com-prised between 75 µ m/s and 150 µ m/s, is reached at a Figure 5. Time evolution of the radial distance r to the hotbead of surface and subsurface tracers in a cross – section ofthe base torus. The travelling depth z of each tracer is givennext to its symbol. Position uncertainty: δz = ±
10 Â µ m.The surface tracer ( z = 0) has a velocity v = 42 µ m/s, whilesubsurface tracers located at z = − , − , − µ m moveat v = 81 , , µ m/s, respectively. Velocity uncertainty: δv = ± µ m/s. P = 20 mW. depth z max such that 0 . < | z max | /H < . z max increases with increasing distance to the heatsource. Furthermore, a recirculation flow arises in thebulk owing to mass conservation, as revealed by the flowvelocity systematically changing sign below some criti-cal depth z inv (here 0 . < | z inv | /H < . B. Centrifugal motion of a surface tracer
We now aim at characterizing the centrifugal motionof surface tracers. The trajectories of tracers located onthe interface at varying distances from the heat source are‘bound together’ so as to reconstruct a full radial trajec-tory [39]. This amounts to time – shifting tracers’ posi-tions until a single representative trajectory is generatedout of the trajectories of individual particles. Note thatthis operation relies on the assumption of flow steadi-ness. Fig. 7 clearly reveals a law of motion r ∼ t / . Aswill be discussed in Sec. VI, this scaling departs from thebehavior expected for a pure thermocapillary flow.There seems to be a contradiction between the r ∼ t / behavior of surface tracers evidenced in Fig. 7 and theapparently linear behavior suggested by Fig. 5. However,the range 2 .
59 mm ≤ r ≤ .
77 mm of radial distancesspanned by the surface tracer of Fig. 5 represents only alittle portion of the r = f ( t ) curve plotted in Fig. 7. Thecorresponding area is delineated by a rectangle in Fig. 7. Figure 6. Evolution of the radial velocity v r as a function ofdepth z (dimensionless unit, H = 3 mm : height of the cell).Different colors correspond to different distances to the heatsource. The solid lines are a guide to the eye. The area tothe right of the thick grey line, for which 0 . ≤ | z | /H ≤ P = 8 mW.Figure 7. Cube of the radial distance r of surface tracers tothe hot bead as a function of time t (brightly colored line).Each colored section represents the trajectory of a tracer. Asecondary ordinate axis is added to plot the corresponding r = f ( t ) curve (grey tones). The orange box marks the datarange of the surface tracer displayed in Fig. 5. P = 14 mW. One can see that the latter is located in a region where r is linear in t to a first approximation. C. Response to laser shutdowns
In the following, we probe the dynamic response of thebase torus to laser shutdowns. In practice, the laser issuddenly switched off using a beam stop. Note that thetime ∆ t on → off (∆ t off → on ) during which the laser is on(resp. off) must be chosen long enough for the flow toreach a steady state (resp. vanish). We set ∆ t >
20 s asdetermined from preliminary tests.We note the onset of a short – lived ( ≈
10 s) centripetalmotion of surface tracers at the very moment t off = 0 Figure 8. Evidence of interface elastic retraction. The verticalline marks the instant t off = 0 of the laser shutdown. Differentcolors correspond to different tracers. P = 20 mW. when the laser is turned off (see Supporting InformationVideo V1). The latter retraction phenomenon stronglysuggests that the WA interface behaves as an elasticmembrane because of the probable presence of adsorbedsurface – active species. We actually have evidence, fromongoing numerical simulations which will be the matter ofa forthcoming publication [29], that the aforementionedreversed motion of surface tracers after laser shutdownis a very likely indication of the presence of adsorbedsurfactants at the water – air interface.Elastic retraction is definitely confirmed by trackingsurface tracers along their paths. Fig. 8 shows how the ra-dial velocity of interfacial tracers sharply (angular point)reverses at t off = 0 : just after the laser extinction, trac-ers start moving in the opposite direction as revealed bya change in the sign of the slope measuring the flow ve-locity. We check that the closer the particle to the heatsource, the sharper the peak on the curve, that is thehigher its pre – and post – shutdown speed. V. QUADRUPOLAR FLOW
In a typical experiment we progressively increase thelaser power P , starting with the axisymmetric base flowpresented in Sec. IV. Beyond a critical power P ∗ , theflow looses its radial symmetry and self – organizes intovortex pairs. We observed one vortex pair (a dipole) ortwo pairs (a quadrupole), with no obvious trend that adipole shows up before a quadrupole when the power isslowly increased. In most cases, the flow was observedto transition directly from axisymmetric to quadrupolar.One strong practical limitation comes from the fact thatthe properties of the WA interface evolve significantlywithin about one hour. As will be discussed in Sec. VI,the problem is due to contamination by surface – activespecies. Even though interface pollution involves minuteamounts of contaminants it cannot be avoided, at leastunder the experimental conditions set up in the presentwork, whatever the care taken in preparing the samples.Interface contamination increases within minutes and its Figure 9. Measuring local velocities in various regions of aquadrupolar flow ( top view ). The arrows show the directionof the flow. + / − : clockwise/anticlockwise vortex rotation.The vertical laser sheets are depicted by green lines. The hotbead (2 a = 295 µ m) is sketched by a red disk. P = 70 mW.Scale bar : 1mm. amplitude definitely varies between distinct experimentalruns. Consequently, only a rough estimate of P ∗ can bemade within a single run, and the latter may stronglyvary between different runs. Typical values of P ∗ rangebetween 20 mW and 30 mW.The transition from the axisymmetric regime to themultipolar flow is qualitatively reversible, which meansthat reducing the power below some value P ∗∗ brings thesystem back to the base flow state. However, the valueof the crossover power P ∗∗ is in general lower than P ∗ .Because of the aforementioned issue, it is not possible totell whether the difference between P ∗ and P ∗∗ reflectsa truly hysteretic behavior or is simply due to growingcontamination (‘surface ageing’).In general, vortex pairs forming quadrupoles were notequal in size. We noticed that the hierarchy betweenvortex sizes could nevertheless evolve in time and evenreverse. Below we focus on an experiment where a neatquadrupole is observed and is stable enough to performa thorough analysis of its features. The investigated flowis shown in Fig. 9 : two counter – rotating vortex pairsare clearly visible with the four vortices separated bytwo ‘channels’, one centrifugal and the other centripetal,intersecting at (nearly) right angles.The flow is scrutinized in a couple of cross – sectionalplanes : a first viewing plane is positioned close to thecentrifugal channel while a second plane, parallel to thefirst one, cuts across the eddies. More precisely, local Figure 10. Time evolution of the distance x to the hot beadof surface and subsurface tracers in the cut plane AB (Fig. 9).The travelling depth z of each tracer is indicated next to itscorresponding symbol. Position uncertainty: δz = ± µ m.The surface tracer ( z = 0) has a velocity v = 666 µ m/s, whilesubsurface tracers located at z = − , − µ m move at v = 601 , µ m/s, respectively. Velocity uncertainty δv = ± µ m. P = 70 mW. flow velocities are measured in the immediate vicinityof points A to D. This study is complemented with anestimate of the velocity near point E which lies withinthe centripetal channel (Fig. 9). A. A passing interface along the channels
Tracking surface and subsurface tracer particles in thecross – section AB reveals that the flow velocity is higherat the interface than in the shallow depth region (Fig. 10).Recall that the opposite occurs in the ground flow state(Sec. IV A). The interface is here ‘passing’ in the sensethat a quasi stress – free boundary condition sets in alongthe centrifugal channel.In the present experiment, the vertical laser sheet isnot oriented parallel to the centripetal channel. It is stillpossible to estimate local velocities along the section ofthe centripetal channel close to point E (Fig. 9) based ona cut plane tangent to the interface. By doing this, wemeasure flow velocities about half those reported for thecentrifugal channel. We therefore assume a qualitativelysimilar situation for the centrifugal and the centripetalchannels, namely a ‘passing’ interface in both cases.
B. Swirling flow region
Examining the flow in the subsurface region of the cutplane CD (Fig. 9), we find velocities that are markedlylower than those measured in either the centrifugal orthe centripetal channel (Table I). Most importantly, inthe swirling flow region along the cross – section CD,subsurface velocities are larger than interfacial ones,contrary to what occurs along the centrifugal channel z ( µ m) v ( µ m/s)0 6.6 −
75 31 −
94 45 −
206 73 −
280 122Table I. Centripetal velocity of surface and subsurface tracersas a function of depth z in the cross – section CD (Fig. 9).Measurements (uncertainty δv = ± µ m/s) are made in theleft half of the laser sheet where the local velocity is positive. where an intense surface flow is observed. The interfaceis here ‘nearly solid’ like in the toroidal state (Sec. IV A).To sum up, tracers move along the centrifugal channelat very high velocities up to v ≈ / s. The relativeincrease in the velocity is more pronounced at the surface( ×
30) than in the bulk ( × × VI. DISCUSSION
The present study yields cogent evidence of the highsensitivity to azimuthal perturbations of the divergentflow driven by a heat source at the water – air interface.We show how, from a base torus under low heating, theaxisymmetry of the flow breaks down into multipoles atsufficiently high powers [27]. The instability takes theshape of a corolla of counter – rotating vortex pairs thatself – organize periodically all around the source of heat.In this paper, both the toroidal base flow (Sec. IV) andthe quadrupolar mode of the instability (Sec. V) wereinvestigated.Most importantly, we observed the elastic response ofthe interface to laser shutdowns (Sec. IV C) as evidencedby the reversed motion of surface tracers, from centrifugalto centripetal (Fig. 8). We assume that the elasticity ofthe water – air interface stems from the presence of anadsorbed surfactant layer. This is a natural hypothesissince water, due to its high surface tension relative tomany common liquids ( γ water = 72 . . m − at 20 ◦ C),acts as a receptacle for most surface – active impuritiesunavoidably present in the environment.Water contamination is a long – standing problem ininterface science [40, 41]. For instance, the pivotal roleof surfactants in retarding the motion of rising bubbleshas been recognized in [42] – [44]. Surface contaminationis also suspected to affect the shape of ‘coffee rings’ left byevaporating droplets [45] – [47]. Interfacial effects becomeincreasingly dominant while downsizing the system owingto magnified surface – to – volume ratios. As an example,microfluidic experiments highlighted that a tiny amountof surfactants can severely undermine the drag reductionpotential of superhydrophobic surfaces [48]. Impuritiesalso have the ability to alter the viscoelastic responseof a water – air interface, as brought to light by AFMmeasurements [49, 50]. Other experiments suggest thatsurface – active contaminants can promote the rupture of µ m-thick free liquid films [51]. Strikingly, the influenceof surfactants manifests itself even at the nanoscale : thestability of interfacial nanobubbles has been attributedto impurities [52, 53], while nanomolar concentrationsof charged contaminants have been invoked to explainanomalous surface tension variations (Jones – Ray effect)reported for electrolyte solutions [54].Besides that, we focused our efforts on determiningthe interfacial boundary condition associated with thebase toroidal flow state. We evidenced that the latterdisplays a ‘nearly solid’ interface which can be describedwith a ‘quasi no – slip’ boundary condition for the fluidvelocity. A liquid – gas interface is classically modeled asa free surface, namely a no stress boundary. Conversely, asolid wall imposes such a strong constraint that the fluidvelocity is zero everywhere on its surface. Here, however,we are in an intermediate situation as we assume that thewater – air interface is partially covered with surfactants.Before proceeding further, let us precise what being‘partially covered’ means in our experimental conditions.According to a recent theoretical study [55], a naturalrelaxation time for the elastic retraction of the surfactantfilm (Sec. IV C) is given by τ = ηr/E ( η ∼ − Pa . s,water dynamic viscosity under standard conditions; E ,Gibbs elasticity; r , radial distance from the heat source).Note that the above expression is based on dimensionalanalysis. Fig. 8 yields a relaxation time τ ≈ r = 2 mm. Using then the relation E = Γ k B T between the equilibrium Gibbs elasticity E and surfactant concentration Γ , one ends up withthe extremely small value Γ ∼
100 molecules /µ m [56].The latter concentration is consistent with the very lowsurfactant contamination invoked either in [55] to explainthe stiffening of the interface or in [57] to account for thesuppression of Marangoni flows in evaporating droplets.Here the take – home message is that a surfactantconcentration as minute as Γ ∼
100 molecules /µ m isyet sufficient to provide the water surface with finiteelasticity, thereby transforming it into a ‘nearly solid’interface that imposes a ‘quasi no – slip’ boundarycondition for the fluid velocity.This fact is supported by the following observations:• As a first experimental proof, surface velocities arelower than subsurface ones in the toroidal regime (see Figs 5 and 6). Such a ‘reversed stratification’of the velocities is at odds with the image of afree surface solely subject to a thermally – drivenMarangoni flow, for which we naturally expect thevelocity to be highest at the locus of flow onset,namely at the interface ( z = 0).• The velocity scale of the pure thermocapillary flowgiven by [28, 58] U = γ T ∆ T / η ≈ γ T P / πκηa ,where by definition γ T ˙= | ∂ γ/ ∂ T | , is on the orderof U ∼ . s − (taking γ T ≈ .
144 mN . m − . K − , P ∼
10 mW, a ∼ µ m, η ∼ − Pa . s and κ ≈ . . m − . K − ). The latter theoretical order ofmagnitude exceeds by far the surface flow velocities(a few tens of µ m/s) measured experimentally inthe toroidal flow state. Note that some very recentstudies (e.g. [38]) also report a significant velocitydrop at the interface relative to what is expectedfor a pure thermocapillary flow, with surface flowvelocities that are one to two orders of magnitudelower in the case of a surfactant – laden interface.• The scaling law r ∼ t / found out while trackingsurface tracers (Fig. 7) departs from the theoreticalprediction for a pure thermocapillary flow. Thethermal Péclet number defined as Pe th ˙= U a/D (bead radius a ∼ µ m; experimental Marangonispeed U exp ∼ µ m/s; coefficient of thermal diffu-sivity D ∼ − m . s − ) is on the order of 10 − inour operating conditions, meaning that heat trans-port is diffusion – dominated. Solving the heat dif-fusion equation in the steady regime yields the ra-dial velocity [28] v r ( r, z = 0) = U (cid:16) ar (cid:17) . (1)The latter expression is inconsistent with r ∼ t / ,the empirical scaling which corresponds to a fasterpower decay v r ˙= d r/ d t ∼ /r . This discrepancycomes as one more hint supporting the existenceof a surfactant elastic layer adsorbed at the water –air interface and responsible for the damping of thesurface dynamics.The situation gets more involved when the forced flowis strong enough to overcome the elastic resistance of thesurfactant – laden interface and create multipolar flows,in which case marked differences in the magnitude of thesurface velocity are measured along the interface : intenseflows are observed within the ‘channels’, in contrast towhat occurs in the regions close to vortex centers wherethe flow almost vanishes (compare Fig. 10 and Table I).Most importantly, the channels are in a ‘passing state’,i.e. characterized by velocities larger at the surface thanin the bulk. The reverse trend, reminiscent of the ‘nearlysolid’ interface reported for the base torus (Sec. IV A), isobserved in the swirling flow regions.From a theoretical perspective, the main difficulty ofthe problem lies in its highly nonlinear nature. Indeed,the flow velocity, the temperature and the surfactantconcentration fields are tightly coupled to one anotherthrough advection. In its highest level of generality, thisproblem calls for the joint solving of the Navier – Stokesequation together with the heat and mass advection –diffusion equations, a situation far too intricate to beaddressed analytically.Still, inertia is irrelevant in the present case since weare working in the creeping flow regime : the Reynoldsnumber defined as Re ˙= U a/ν , with U and a the sametypical velocity and length scales as above and ν thekinematic viscosity of water ( ν ∼ − m . s − understandard conditions), is on the order of 10 − in usualexperiments. Note that this goes against previous workswhich attribute the origin of the azimuthal instabilityto inertia. For instance, the authors of [26] do predictthe onset of the azimuthal instability but for a criticalReynolds number [59] Re c = 115 (see [26], Fig. 8, limit ofinfinite Prandtl Pr number) which is much higher thanthe one encountered in our experiments. Based on thisargument, we believe that the Shtern – Hussain scenarioof the azimuthal instability presented in [26] can be ruledout in our practical conditions.In the present situation, further simplification stemsfrom the fact that the solutal Péclet number Pe s is muchlarger than the thermal Péclet number Pe th , owing to amass diffusivity D S ∼ − − − m . s − that is twoto three orders of magnitude smaller than the coefficientof thermal diffusivity D T ∼ − m . s − . The physics ofthe system is hence dominated by the advective transportof surfactant molecules along the interface.The present study leads us to postulate a scenario forthe onset of the azimuthal instability. Surfactants areswept by the dilatational flow towards the edges of thecontainer, which induces a solutocapillary counterflow inresponse to the inhomogeneous distribution of surface –active impurities along the interface [38]. We conjecturethat the instability results from the elastic deformationof the depletion front, as illustrated in Fig. 11.In fact, such an instability mechanism has been alreadyproposed by Couder et al. to account for multipolar flowpatterns arising on the surface of soap films blown by avertical air jet [60]. The scenario depicted here to explainthe onset of the instability does not depend intrinsicallyon the nature — thermal, chemical, mechanical — of thesource. The elasticity imparted to the interface by thesurfactant film appears as a necessary condition for theonset of the instability. By the way, the fact that surface –active species destabilise the system goes against formerstudies in which they are shown to exert a stabilizinginfluence over convective instabilities [61].To finish, let us come back to our starting point. Thereported experiments, with a fixed hot bead, tell us aboutthe source mechanism for the orbital motion of the free laser – heated particle (Fig. 1). The phenomenon clearlyoriginates from the kind of azimuthal instability studiedin the present work. As described by Girot et al. in [22],the particle is initially centered on the laser beam axis Figure 11. Conjectural onset mechanism of the instability.Surfactant molecules (orange balls) are swept away beyond aradius r d from the heat source (red disk) and accumulate nearthe cell walls (not drawn) where they lower the surface tension( γ low ). The competition between the centrifugal forced flow(red arrows) and the solutocapillary counterflow (blue arrows)deforms periodically the depletion elastic front (four – lobedline), eventually resulting in the azimuthal instability. (black cross in Fig. 1), but escapes to a finite distanceand starts orbiting around the axis. The configurationof a free hot sphere within a dipolar or quadrupolar flowpattern is presumably unstable. It is not surprising thenthat the hot particle moves out of the optical trap. Theconfiguration sketched in Fig. 1 cannot be static, whichmeans that the pair of counter – rotating vortices acts asa propulsor, thereby transforming the hot particle into amicroswimmer. What is still not understood is why theparticle remains at a given distance from the laser beamaxis. The latter property is far from evident. Hopefully,it could be elucidated based on a theory of the azimuthalinstability that has not been developed yet.Surface flow velocities on the order of 1 mm . s − arehere attained with just a few tens of mW of laser power.So if the hot bead (radius a ∼ µ m) were freed, itwould be entrained by the self – induced flow at a speedabout ten times its size per second. Thermocapillaritytherefore appears as an efficient mechanism to achievehigh – speed self – propulsion of active colloids [62] – [65].In that respect, our study follows the line of recenttheoretical works such as [28]. On top of that, thegeneration of vortex pairs highlighted in the presentpaper offers an alternative to the autonomous motion ofJanus colloids confined at a water – air interface, whosepropulsion is fueled by an asymmetric catalytic reactiontaking place on one side of the particle [66, 67]. Despiteits apparent simplicity the system investigated here maystill harbor a plethora of interesting phenomena.0 SUPPLEMENTARY MATERIAL
See the supplementary material for a video showingthe elastic retraction of the interface at laser shutdown.
ACKNOWLEDGMENTS
LOMA and CRPP acknowledge financial support fromIDEX-Bordeaux under “ Propulsion de micro-nageurspar effet Marangoni ” PEPS program. We are grateful to B. Gorin, A. Mombereau, A. Girot and N. Danné forfruitful collaboration, and to F. Nadal for stimulatingdiscussions. We thank H. Kellay and C. Pradère for thekind loan of a FLIR infrared camera (see note [30]). Theauthors declare no conflict of interest.
DATA AVAILABILITY
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest. [1] L. E. Scriven and C. V. Sternling. The Marangoni Effects.
Nature
186 – 188 (1960).[2] J.-B. Fournier and A.-M. Cazabat. Tears of Wine.
Euro-phys. Lett.
20 (6),
517 – 522 (1992).[3] F. Wodlei, J. Sebilleau, J. Magnaudet, and V. Pimienta.Marangoni – driven flower – like patterning of an evapo-rating drop spreading on a liquid substrate.
Nat. Com-mun.
820 – 831 (2018).[4] L. Keiser, H. Bense, P. Colinet, J. Bico, and E. Reyssat.Marangoni Bursting : Evaporation – Induced Emulsifica-tion of Binary Mixtures on a Liquid Layer.
Phys. Rev.Lett.
Chapter from the book Superhard Ma-terials, Convection, and Optical Devices , pp. 75 – 112,Springer – Verlag (1988).[6] Y. Kamotani, S. Ostrach, and J. Masud. Oscillatory ther-mocapillary flows in open cylindrical containers inducedby CO laser heating. International Journal of Heat andMass Transfer
555 – 564 (1999).[7] G. M. Oreper, T. W. Eagar, and J. Szekely. Convectionin Arc Weld Pools.
Welding Journal
62 (11),
307 – 312(1983).[8] S. H. Davis. Thermocapillary Instabilities.
Ann. Rev.Fluid Mech.
403 – 435 (1987).[9] M. F. Schatz and G. P. Neitzel. Experiments on Thermo-capillary Instabilities.
Annu. Rev. Fluid Mech.
93 –127 (2001).[10] E. Yakhsi-Tafti, H. J. Cho, and R. Kumar. Droplet ac-tuation on a liquid layer due to thermocapillary motion :Shape effect.
Appl. Phys. Lett.
Phys.Rev. E
New Journal of Physics
Phys. Chem. Chem. Phys.
Applied Physics Letters
Lab on a Chip
Ad-vances in Microfluidics, Intech Ed. (2012).[17] A. S. Basu, S. Y. Yee, and Y. B. Gianchandani. Virtualcomponents for droplet control using Marangoni flows :size – selective filters, traps, channels, and pumps.
Pro-ceedings of the IEEE 20th International Conference onMicro Electro Mechanical Systems (2007).[18] A. Davanlou and R. Kumar. Thermally induced collisionof droplets in an immiscible outer fluid.
Nature ScientificReports Micromachines Annu.Rev. Fluid Mech.
425 – 455 (2005).[21] C. Maggi, F. Saglimbeni, M. Dipalo, F. De Angelis, andR. Di Leonardo. Micromotors with asymmetric shapethat efficiently convert light into work by thermocapillaryeffects.
Nature Communications Langmuir
Optics Express
25 (3),
Phys. Rev. Lett.
J. Appl. Math.Mech.
577 – 580 (1967).[26] V. Shtern and F. Hussain. Azimuthal instability of diver-gent flows.
J. Fluid Mech.
535 – 560 (1993).[27] A. Mizev. Influence of an adsorption layer on the struc-ture and stability of surface tension driven flows.
Physicsof Fluids
J. FluidMech.
589 – 601 (2014).[29] Loudet et al. , in preparation . [30] We observed infrared images of the WA interface using athermal camera (FLIR SC7000). IR images do not showthe flow patterns but the temperature ( T ) map on theinterface. As the T – distribution is connected to flowpatterns through heat advection, symmetry breaking isreadily revealed by IR images. Main features of T – mapsturned out similar with and without polystyrene beads.[31] Strictly speaking, we do not observe the flow at the WAinterface but in a subsurface layer extending down to afew tenths of a millimeter below the surface. Indeed, the(low) deformation of the meniscus strongly impairs anyproper observation of the interface based on laser sheettomography. The vertical laser sheet is kept tangent tothe bead as the obstruction by the optic fiber located onthe cell centerline prevents us from performing diametralsections of the flows.[32] M. Raffel, C. E. Willert, and J. Kompenhans. Particleimage velocimetry : a practical guide. Springer Science& Business Media (2007).[33] E. J. Stamhuis. Basics and principles of particle imagevelocimetry (PIV) for mapping biogenic and biologicallyrelevant flows.
Aquatic Ecology
40 (4),
463 – 479 (2006).[34] W. Thielicke and E. J. Stamhuis. PIVlab: Towards User –fiendly, Affordable and Accurate Digital Particle Im-age Velocimetry in MATLAB.
Journal of Open ResearchSoftware (2014).[35] In principle, it is possible to get interfacial velocity mapsby running PIV on horizontal cuts. In practice, however,large errors may be committed below the onset of theinstability due to surface and subsurface tracers movingat markedly different speeds over the whole thickness ofthe laser sheet (Sec. IV). The latter issue disappears incase of perfect slip at the interface for which surface andsubsurface tracers move at comparable speeds (Sec. V).[36] As can be noticed in the upper right corner of the topfigure, the axisymmetry of the base flow is not perfect :the streamlines are not strictly radial but slightly curvedalong some preferred direction. The latter varies from oneexperiment to another leaving the impression that it isdue to parasitic temperature gradients and/or geometricflaws of the system (e.g., the rough surface of the bead).[37] E. Favre and L. Blumenfeld. Instabilities of a liquid layerlocally heated on its free surface.
Phys. Fluids
J. Fluid Mech.
495 – 533 (2019).[39] In practice, there are at least two reasons why surfacetracers can be tracked only over short distances relativeto the cell width : the vertical laser sheet is not perfectlyaligned with the radial trajectories of the tracers [31] and,to a lesser extent, thermal agitation tends to push tracersout of the viewing plane.[40] H. Kim, K. Muller, O. Shardt, S. Afkhami, and H. A.Stone. Solutal Marangoni flows of miscible liquids drivetransport without surface contamination.
Nature Phys.
Curr. Opin. Electrochem.
166 – 173 (2019).[42] C. Ybert and J.-M. di Meglio. Ascending air bubbles inprotein solutions.
Eur. Phys. J. B
313 – 319 (1998). [43] R. Palaparthi, D. T. Papageorgiou, and C. Maldarelli.Theory and experiments on the stagnant cap regimein the motion of spherical surfactant – laden bubbles.
J.Fluid Mech.
Annu. Rev. Fluid Mech.
615 – 636 (2011).[45] R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R.Nagel, and T. A. Witten. Capillary flow as the cause ofring stains from dried liquid drops.
Nature
827 – 829(1997).[46] H. Hu and R. G. Larson. Marangoni Effect Reverses Cof-fee – Ring Depositions.
J. Phys. Chem. B
Phys. Rev. Lett.
PNAS
114 (28),
Phys. Rev.Lett.
Phys. Rev. Lett.
J. Fluid Mech.
192 – 221 (2018).[52] W. A. Ducker. Contact Angle and Stability of InterfacialNanobubbles.
Langmuir
25 (16),
Phys. Rev. E
J. Phys. Chem. Lett.
189 –193 (2018).[55] T. Bickel, J.-C. Loudet, G. Koleski, and B. Pouligny.Hydrodynamic response of a surfactant – laden interfaceto a radial flow.
Phys. Rev. Fluids E = Γ k B T only holds in the dilute regimein which the surfactant elastic layer behaves as a 2D idealgas (see Eq. (5) in [55] for further details). The very lowconcentration Γ ∼
100 molecules / Â µ m estimated herelegitimates a posteriori the working assumption we made.[57] H. Hu and R. G. Larson. Analysis of the Effects ofMarangoni Stresses on the Microflow in an EvaporatingSessile Droplet. Langmuir
Eur. Phys. J. E
42 (10),
131 – 139(2019).[59] In [26], the authors introduce the particular form of theReynolds number Re = r < v r > c /ν ( < v r > c , averageradial velocity with respect to the azimuthal angle; r ,radial distance to the heat source; ν , kinematic viscosity)which they explain is convenient for comparison of thestability of the various flows they consider. [60] Y. Couder, J.-M. Chomaz, and M. Rabaud. On the hy-drodynamics of soap films. Physica D
384 – 405(1989).[61] J. C. Berg and A. Acrivos. The effect of surface – activeagents on convection cells induced by surface tension.
Chem. Eng. Sci.
20 (8),
737 – 745 (1965).[62] V. Frumkin, K. Gommed, and M. Bercovici. Dipolar ther-mocapillary motor and swimmer.
Phys. Rev. Fluids Phys. Rev. E (99),
Eur. Phys. J. E
41 (11),
137 –146 (2018).[65] A. Zöttl and H. Stark. Emergent behavior in active col-loids.
J. Phys.: Condens. Matter
28 (25),
Langmuir
33 (48),
Soft Matter11,