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Gas spreads on a heated wall wetted by liquid Y. Garrabos and C. Lecoutre-Chabot CNRS-ESEME, Institut de Chimie de la Matière Condensée de Bordeaux, Université de Bordeaux I, Avenue du Dr. Schweitzer, F-33608 Pessac Cedex, France J. Hegseth Department of Physics, University of New Orleans, New Orleans, LA 70148 ∗ V. S. Nikolayev and D. Beysens 6 ESEME, Service des Basses Températures, CEA-Grenoble, France† 1 0 2 J.-P. Delville Centre de Physique Moléculaire, Optique et Hertzienne, n CNRS, Université de Bordeaux I, Cours de la Libération, 33405 Talence Cedex, France a (Dated: January 27, 2016) J 6 This studydealswith asimplepurefluidwhosetemperatureisslightly belowitscritical temper- 2 atureanditsdensityisnearlycritical,sothatthegasandliquidphasesco-exist. Underequilibrium conditions, such a liquid completely wets the container wall and the gas phase is always separated ] fromthesolidbyawettingfilm. Wereportastrikingchangeintheshapeofthegas-liquidinterface n influenced by heating under weightlessness where the gas phase spreads over a hot solid surface y showing an apparent contact angle larger than 90◦. We show that the two-phase fluid is very sen- d sitive to the differential vapor recoil force and give an explanation that uses this non-equilibrium - u effect. We also show how these experiments help to understand the boiling crisis, an important l technological problem in high-power boiling heat exchange. f . s c I. INTRODUCTION II. EXPERIMENTAL SETUP i s y h We report results that were obtained and repeated p Singularpropertiesofasimplefluid[1,2]appearwhen using several samples of SF (T = 318.717 K, ρ = 6 c c [ itisnearitscriticaltemperature,Tc,anditscriticalden- 742 kg/m3). These samples were heated at various rates 1 sity,ρc. Whenthefluid’stemperature,T,isslightlylower in cylindrical cells of various aspect ratios on several v than Tc and the averagefluid density ρ is close to ρc the French/Russian and French/American missions on the 9 fluid exhibits perfect wetting (i.e., zero contact angle) of Mir space station using the Alice-II instrument [8]. This 6 practically any solid by the liquid phase in equilibrium. instrument is specially designed to obtain high precision 9 In this article we study a system that is slightly out of temperature control (stability of ≈15µK over 50 hours, 6 equilibrium. Our experiments, performed in weightless- repeatability of≈50µKover7 days). To place the sam- 0 ness [3–7] showed that when the system’s temperature ples near the critical point, constant mass cells are pre- . 1 T is being increased to Tc, the apparent contact angle paredwithahighprecisiondensity,to0.02%,byobserv- 0 (see Fig. 7 below for definition) becomes very large (up ing the volume fraction change of the cells as a function 6 to 110◦), and the gas appears to spread over the solid of temperature on the ground [9]. 1 surface. InsectionII wedescribeourexperimentalsetup : Afluidlayerwassandwichedbetweentwoparallelsap- v that allows the spreading gas to be observed. The gas- phirewindowsandsurroundedbyacopperalloyhousing i liquidinterfaceshapeatequilibrium,whichisconsidered X in the cylindrical optical cell, the axial section of which in section III, plays a crucial role as an initial condition is shown in Figure 1. We consider here three cells of r a for the gas spreading phenomenon. The sections IV and the same diameter D =12 mm, the other parameters of Vdealwiththeobservationsofthespreadinggas. Athe- which are shown in Table I. The liquid-gas interface was oreticalmodelthatallowsthisunusualphenomenontobe visualized throughlight transmissionnormalto the win- explained is proposed in section VI. In the section VII, dows. Since the windows were glued to the copper alloy we discuss the boiling crisis, a phenomenon that plays wall,someoftheglueissqueezedinsidethecellasshown an important role in industrial applications, and how it in Fig. 1. This glue forms a ring that blocks the light is relevant to the spreading gas. transmission in a thin layer of the fluid adjacent to the copper wall making it inaccessible for observations. Be- causeofthisgluelayer,thewindowsmayalsobe slightly tilted with respect to each other as discussed in sec. III. ∗email:[email protected] A 10 mm diameter ring was engraved on one of the †Mailing address:CEA-ESEME, Institut de Chimie de la Matière windows of each cell in order to calibrate the size of the Condensée de Bordeaux, 87, Avenue du Dr. Schweitzer, 33608 PessacCedex, France visible area of the cell images as can be seen in each im- 2 the windows (Fig. 1) due to the large aspect ratio D/H wire grid of the cell. Let us first consider an ideally cylindrical cell as op- posed to the real cell. At equilibrium, the windows and glue engraved ring CuBeCo the copper wall are wetted by the liquid phase. Because the van der Waals forces from the walls act to make the window wetting film as thick as possible, the weightless bubble liquid H gas bubble shouldbelocatedinthecell’scenter. Becausethebubble window isflattenedandoccupiesone-halfoftheavailablevolume, D thedistanceofsuchacenteredbubbletothecopperwall is large (Fig. 1). The lateral centering forces are then incident light much weaker than the centering forces in the direction of the cell axis. Any small externalinfluences in the real cell can displace the bubble laterally from the cell’s cen- FIG. 1: Sketch of a cross-section of the cylindrical sample ter. This displacement is illustrated in Fig. 2 that shows cell (with parallel windows). The fluid volume is contained cell 10 at room temperature. We note that there are between two sapphire windows that are glued to a CuBeCo twokindsofexternalinfluencesthatareeasilyidentified: alloy ring. The dimensions H (see Table I) and D(=12 mm) residualaccelerationsin the spacecraftandcellasymme- of the cell are indicated. Some glue is squeezed into the cell. try. The thicknessof theglue layer is exaggerated for illustration Bubble images for cells 8 and10 were recordedin four purposes. Inweightlessness,thegasbubbleshouldbelocated Mir missionsbetween1996and2000. Severalimagesare inthemiddleofsuchan’ideal’cell,seesec.IIIforthediscus- reported in [3] (Cassiopeia mission, 1996) and in Fig. 6 sion. below(GMSF2mission,1999)forcell10. Itisextremely likelythatthespacestationchangeditspositionwithre- Cell number Cell thicknessH (mm) (ρ−ρc)/ρc (%) spect to the residual gravity vector between these runs. 8 3.016 0.85 Thebubblepositionwithrespecttothecell,however,al- 10 1.664 0.25 waysremainedthesame. Thebubblelocationalsovaries fromcellto cellwithoutanydependence onthe station’s 11 4.340 0.87 orientation. Therefore, we have no reason to attribute theoff-centerpositionofthe bubbletothe residualgrav- TABLEI:Physicalparametersoftheexperimentalcells. Cell ity. 11 hasamovablepiston tochange thecell volume. However, Although the cells were manufactured with high pre- thevolume was kept constant duringthese experiments. cision, the cell windows could not be exactly parallelbe- cause of the glue layer as shown in Fig. 3. In the rest of this section we will discuss the influence of the windows’ age. An out-of-focus wire grid, designed to visualize [10] tilt on the position and on the shape of the bubble. fluidinhomogeneitiesand/orafluidflowthroughlightre- When the bubble’s surface is curved, there is a con- fraction,wasalsoused. Thegridwasoccasionallymoved stant excess pressure ∆p inside the bubble defined by out of the light path, so that it is not always present in the Laplace formula all images. ThesamplecellisplacedinsideofacopperSampleCell ∆p=σK, (1) Unit(SCU)that,inturn,isplacedinsideofathermostat. Heat is pumped into and out of the SCU using Peltier where σ is a surface tension and K is the surface curva- elementsandheaters. Thetemperatureis sampledevery ture. This excesspressureactsonallpartsofthe bubble second and is resolved to 1µK. interface. Inparticular,itactsonthepartA (wherethe sg Similar ground based experiments were done before index s standsfor “sapphire”andg for “gas”)ofthe flat these experiments using a copy of the same instrument. window surface that contacts the gas directly (or, more The gravity forces push the denser liquid phase to the accurately, through a wetting film that we assume to be bottom of the cell and completely different behavior is of homogeneous thickness). This pressure creates reac- seen, see [11]. tions forces F~(1) and F~(2) at each window, that act on s s the bubble. Each of these forces is perpendicular to the corresponding window. The absolute values of F~(1) and III. BUBBLE POSITION AT EQUILIBRIUM s UNDER WEIGHTLESSNESS F~s(2) are equalto Asg∆p. When the windowsare exactly parallel, F~(1)+F~(2) = 0 and the bubble remains at the s s The gas volume fraction φ (volume of the gas divided cell’s center. When the windows are tilted with respect bythetotalcellvolume)isdefinedbyρandthedensities to each other, the non-zero force F~(1)+F~(2) pushes the s s ofgasandliquidforthegiventemperature. Inourexper- bubble in the direction of the increasing cell thickness. iments, φ ≈ 0.5 and the gas bubble is flattened between Thismotioncontinuesuntilthe bubble touches(through 3 a b FIG. 2: The experimental image of cell 10 at room temperature (a) and the equilibrium bubble shape simulated for the tilt angle of 0.46◦ (b). When superposed, the images (a) and (b) give almost perfect match. The outer white circle in (a) (black in (b)) shows the actual location of the cell wall. The inner black circles in (a) and (b) correspond to the engraved ring that allows thesuperposition to bemade. Thedark space between two thesecircles in theimage (a) ismade bythering of glueas shown in Fig. 1. The image (b) is a frontal projection of the bubbleshown in Fig. 4. glue definedbytheknowngasvolumefractionandthecellvol- gas bubble tilt angle A ume. Onecanalsominimize thegas-liquidinterfacearea A cg sg with a bubble volume constraint. In both cases bound- aryconditionsmustbesatisfied(zerocontactangleinour case). The resulting bubble shape obviously depends on the cell geometry. It is also nearly independent of tem- → Fs(1) F→ perature as can be seen from Eq. (2), because all three → c terms of this equation are proportional to the surface F(2) s tension σ, so that this force balance remains valid even nearT ,whereσ disappears. Thereare,however,several c sources of weak temperature dependence of the bubble shape. First,thereisweakdependenceofthegasvolume fractionφ ontemperature atconstantaveragedensity ρ. FIG. 3: Sketch of a cross-section of the sample cell with the This small deviationis smallest at the criticaldensity ρ c tilted windows at equilibrium in weightlessness. The wetting andslightlygreaterinthese experimentsdue to the very film is not shown. The window tilt is possible due to the ex- small deviation (see Table I) of ρ from ρ . Second, the c istenceof aspace between thewindow’s edgeand thecopper curvatureK dependsonthethicknessofthewettingfilm wall, whichisfilledbyglue. Thisspaceandthetilt areexag- that increases near T . The wetting film remains small, geratedforillustrationpurposes. Basedonthemanufacturing c however, in comparison with the cell thickness. Both of process a maximum tilt angle of ≈ 1◦ is possible. The glue these effects are very weak. squeezed into the cell is not shown. The reaction forces that act on the gas bubble are shown with arrows. The contact The force F~ , which is directed horizontally in Fig. 3, c areas of the gas bubblewith thesolid are indicated. causes a distortion of the bubble. This distortion results in an ovalimage in Fig. 2 instead of a circle. The degree ofdistortionincreaseswiththetiltanglebecausesodoes a wetting film) the copper wall of the cell, thus forming F~ . This distortion can thus be used to estimate the tilt a contact spot of the area A , where the index c stands c cg angle. for“copper”. Thisdirectcontactwiththesolidresultsin anotherreactionforceF~ withtheabsolutevalueA ∆p, For these constant volume gas bubbles, the degree of c cg distortion should decrease with increasing cell thickness such that H forthesamewindowtilt. AlargervalueofH resultsin F~(1)+F~(2)+F~ =0 (2) a less compressed (more sphere-like) bubble shape with s s c less area in contact with the wall. This smaller bubble in equilibrium, see Fig. 3. curvature results in a smaller value for ∆p according to Therearetwoequivalentwaystofindthebubbleshape Eq. (1). Consequently, the force F~ , the area A of the c cg at equilibrium. One can solve Eq. (1) that reduces to contact with the copper wall, and the bubble distortion K =const, where the constant is obtained from the con- are smaller. This window tilt hypothesis is consistent ditionofthe givenbubble volume. The bubble volumeis withobservations: wewerenotabletodetectanydistor- 4 tion of the gas bubble in cell 8 (see Fig. 10a below that impossible to verify. corresponds to the nearly equilibrium shape) that is approximately twice as thick (Table I) as cell 10 shown in Fig. 2. There is, however, some tilt in cell 8 because the IV. CONTINUOUS HEATING EXPERIMENTS bubbletouchesthewall. Weexpectthatthetiltanglesin allofthecellsareofthesameorderofmagnitudebecause In the continuous heating experiments, the cells 8, 10 they were all manufactured using the same method. and 11 were heated nearly linearly in time t. The evo- To verify the window tilt hypothesis, we performed lution of the non-dimensional temperature τ for each of a 3D numerical simulation of the bubble surface by us- theseexperimentsisshowninFig.5. Theparameterτ is ing the Surface Evolverfinite element software [12]. The result of this calculation is shown in Fig. 4 for cell 0.01 10. The experimentally observed bubble deformation 0 c -0.01 b τ a h -0.02 0.002 g 0.001 -0.03 f 0 e -0.001 -0.04 d -0.002 1:22:00 1:24:00 1:26:00 -0.05 1:00:00 1:10:00 1:20:00 1:30:00 t (h:mm:ss) FIG. 5: Reduced temperature evolution for the image se- FIG. 4: The result of a 3D finite element calculation of the quencesshown in Fig. 6 (solid line), Fig. 8 (dottedline), and equilibrium gas-liquid interface for cell 10 with a window tilt angleof0.46◦. Theverticesofthepolygonallinesindicatethe Fig.9(dashedline). Thetemperaturevaluesthatcorrespond to each of the images (a-h) shown in these figures are indi- location of thecylindrical copperwall andthey are shown to catedbyarrowsandthecorrespondingletters. Thedefinition guide the eye. A shape of the circular cylinder was input of τ is discussed in the text. The temperature is measured to the simulation. The contact angle is zero. A part of the in the body of the SCU. The vicinity of the critical point is image marked by the square is enlarged to show the contact enlarged in theinsert. area Acg of thegas with thecopper wall (a small white rect- anglecrossed bytwosymmetrylines). ThecontactareasAsg withthewindowshavetheovalshape. Theprojection ofthis definedas(T−T )/T ,whereT isthetemperature coex c coex bubbleshape to thecell window is shown in Fig. 2b. of the coexistence curve that corresponds to the fluid’s averagedensity shown in Table I. Note that T differs coex matches the calculation performed for a tilt angle of from T only by 1−50µK because the density is very c ◦ 0.46 ,seeFig.2. Thesimulationresultedintheinterface close to ρ for all cells. A 40 min temperature equili- c curvature K = 1.389 mm−1 and in A = 0.150 mm2 bration at τ ≈ −0.033 preceded the heating. The mean cg calculated for the bubble volume Vφ = 26.675 mm3. value of dT/dt at T was ≈7.2 mK/s. c From this data, it is easy to calculate the effective ac- Figure 6 shows the time sequence of the images of celeration g that would create the equivalent buoy- the cell 10. The interface appears dark because the eff ancy force F = (ρ −ρ )Vφg = KσA . It turns liquid-gas meniscus refracts the normally incident light c L V eff cg out that g = 1.55·10−3g for T = 290 K, where g is awayfromthecellaxis. Afterthetemperaturerampwas eff thegravityaccelerationonEarth. Thisg acceleration started but still far from the critical temperature, the eff is much larger than the residual steady accelerations in bubble shape changed. The contact area A of the gas cg the Mir space station (∼10−6g) and this shows that the with the copper wall appears to increase. In other sys- observed bubble deformation is not caused by residual tems the wetting film under a growing vapor bubble is accelerations. We conclude that the window tilt hypoth- observed to evaporate [14]. In near-critical fluids, how- esis about the origin of the bubble deformation and its ever, the heat transfer processes are more complex [15]. off-center position is correct. In this system we believe that there may be a similar A similar off-center bubble position was observed un- drying process, i.e., at some time the thin wetting film der weightlessness in a cell similar to ours by Ikier et. that separates the gas from the copper wall evaporates. al. [13] and was attributed to a residual acceleration. In fact, we have observed low contrast lines that appear However, they report only one run in a single cell mak- within the A area when the heating begins. An exam- sg ing the actual cause of the bubble off-centered position ple of such a line is indicated in Figure 6a by the white 5 a b c d e f g h FIG.6: Timesequenceofimagesofthecell10duringthecontinuousheatingthroughthecriticalpoint. Thetemperaturevalues that correspond to each of the images (a-h) shown in these Figures are indicated in Fig. 5 by arrows and the corresponding letters. This run is a repeat of the run shown in Fig. 2 of [3]. The gradual increase of the apparent contact angle as the gas spreads with increasing temperature is clearly seen. The time corresponding to each image is shown to the left of the cell in themiddle. The magnified upperregions close to the contact line from theimages (e-g) are shown in Fig. 7. glue 6e 6f 6g boundary liquid-gas meniscus 8e 8f 8g 8h FIG.7: ThemagnifiedupperregionsclosetothecontactlinefromtheimagesFig.6(e-g)andfromtheimagesFig.8(e-h). The apparent contact angle can be “measured” as the angle between the tangents to the black glue boundary and the liquid-gas meniscus. Thelattercorrespondstotheboundarybetween thewidedarkand narrowbrightstripes ontheimages. Theliquid domain is to theleft from the meniscus. Onecan see that this apparent contact angle exceeds90◦ in the images 6g and 8f. arrows. The out-of-focus gridshows that these lines cor- rapidlyevolving. AtT ≈T ,thesurfacetensionvanishes, c respond to a sharp change in the wetting film thickness. thebubble’srelaxationfromsurfacetensionisnegligible, These lines are most likely triple contact lines and we so that the interface shape is defined by the variation of have actually seen them pinned by an imperfection on the localevaporationratealong the interface. The evap- the windows as they advance andretreatin other exper- orationis strongerat the parts of the interface closestto iments. Since the heat conductivity of copper is larger the copper heating wall. This effect leads to the waved than that of sapphire, the heat is supplied to the cell interface shape shown in Fig. 6g. Diffusion causes the mainly through the hotter copper wall. Therefore the disappearance of the interface at T > T as shown in c filmshouldevaporateonthecopperwallevenearlierthan Fig. 6h. on the sapphire. A more refined analysis of the contact Figure 8 shows the time sequence of the images of the line motion will be discussed elsewhere. cell 8, which is approximately twice as thick as cell 10. TheincreaseoftheAcg areaisaccompaniedbyanevi- The images in Fig. 8 were taken exactly for the same dentincreaseintheapparentcontactangle,seeFig.6d–f values of the non-dimensional temperature τ (shown in and the corresponding magnified images in Fig. 7. Near Fig. 5) as the corresponding images in Fig. 6. The force the critical temperature the apparent contact angle be- F~ pushes the bubble against the cell wall as in the case comeslarger than90◦! We willanalyzetheseeffects the- c of cell 10. As discussed above, this force is weaker than oretically in section VI. forthecell10becausethebubbleappearsalmostcircular While crossing the critical point, the vapor bubble is at equilibrium (see Fig. 6a and Fig. 8a). By comparing 6 a b c d e f g h FIG.8: Timesequenceoftheimagesfrom cell8duringcontinuousheatingthroughthecriticalpoint. Images(a-h)weretaken exactly for the same values of temperature (shown in Fig. 5) as corresponding images in Fig. 6. The magnified upper regions close to thecontact line from theimages (e-h) are shown in Fig. 7. images(e)ofbothsequences,wecanalsoseethattheva- images(a), (d) and (f) were takenfor the same values of porspreadsslowerincell8. Theincreaseoftheapparent non-dimensionaltemperatureascorrespondingimagesin contact angle is also slower. The waved shape interface Figs. 6 and 8. This cell contained three wetted thermis- appears earlier in Fig. 8f, i.e. farther from T than for tors(Fig.9i)thatconstrainthe bubble surface(Fig.9a). c the cell 10. The interface is still quite sharp in Fig. 8h, The bubble is only slightly squeezed by the windows so whileithasalreadydiffusedinthecaseofthethinnercell that the reaction forces that act on the bubble at equi- (Fig. 6h). This difference can be explained by the differ- libriumareweak. As a result,the bubble doesnottouch ence in the liquid-gas interface area, which is roughly the copper heating wallat all. Although this is notclear proportional to the cell thickness. The surface tension intheimage(a),becauseofthegluenearthecopperwall, force that tends to maintain the convex shape is not as itisclearinimage(f) wheresmallnewlyformedbubbles strongforthethickercellwherealargerfluidvolumehas separate the initial bubble from the wall. These bubbles to be moved during the same time. The diffusion time form from the local overheating of the fluid between the is larger for cell 8 because the size of the inhomogeneity large bubble and the copper wall. There is enough fluid (i.e. interface) is larger. between the large bubble and the wall so that a small Figure 9 shows the time sequence of the images from bubble may grow in it. These small bubbles push the cell11,whichisthickerthanboththecells8and10. The large bubble away from the wall before any coalescence can take place. The comparison of these three experiments clearly a d shows that in order to obtain the bubble spreading, the bubble needs to have a direct contact with the heating wall,i.e. to be pushed to the heating wallbysome force. Notethatnoneoftheimagesshowanyevidenceofsteady fluidmotionthatwouldbenecessarytomaintainthedis- torted bubble shapes in Figs. 6 and 8. We conclude that f i thisdistortionofthebubbleequilibriumshapecannotbe caused by fluid motion. Similar continuous heating experiments are reported by Ikier et. al. in [13]. However, a smaller heating rate (1.7 mK/s) and erratic accelerations of the cell did not allow gas spreading to be observed. FIG.9: Timesequenceofimagesofthecell11duringthecon- V. QUENCHING EXPERIMENTS tinuousheatingthroughthecriticalpoint. Nobubblespread- ing is seen. The bubbledoes not touch the copper wall. The images(a,d,f)weretakenexactlyforthesamevaluesoftem- Figure10showsthetimesequenceoftheimagesofcell perature (shown in Fig. 5) as corresponding images in Fig. 6 8 when it was heated by 100 mK quenches as shown in and Fig. 8. The cell 11 contains three thermistors shown in Fig.11. Whiletheheatingrateisquitelargeduringeach image (i) byarrows. This image was taken after the temper- quench, the time average of the heating rate 1.4 mK/s ature equilibration above Tc. is smaller than that during the continuous heating due 7 a b c d e f g h FIG.10: Timesequenceofimagesofcell8duringtwo100mKquenches. Thegasspreadsduringeachquenchthatlastsabout 12 s. The equilibrium position of the vapor bubble with respect to the cell is shown by the white circle in each image for comparison. 0 served during the heating of CO cells in other experi- g h 2 f ments (Pegasus BV4705, Post-Perseus F14) carried out e by our group in the Mir station. However, these experi- -0.0005 ments were not designed to study the spreading gas and we do not discuss them here. d -0.001 τ -0.0015 VI. INTERFACE EVOLUTION DURING THE HEATING b c -0.002 The above experimental data showed that the spread- a ing gas and the associated interface deformation are -0.0025 caused by an out-of-equilibrium phenomenon. This is 2:30:00 2:35:00 2:40:00 2:45:00 especially demonstrated by the analysis of the interface t (h:mm:ss) shape at equilibrium (sec. III) and by the return to the equilibrium shape after each quench in sec. V. In this section we analyze possible causes of the spreading gas. FIG.11: Temperatureevolutionduringtheseriesofquenches. Twocausesareconsidered: Marangoniconvectiondueto ThepointsthatcorrespondtoeachoftheimagesinFig.10(a- h) are indicated by arrows and corresponding letters. The thetemperaturechangeδTi alongthegas-liquidinterface temperature is measured in thebody of the SCU. and the differential vapor recoil. to the waiting time of ≈ 60 s after each quench. Dur- A. Marangoni convection ing this waiting time a partial equilibration takes place. The images (a–c) show a slight bubble spreading that IfatemperaturechangeδT exists,itwillcreateasur- i appears during a quench that is farther from the criti- face tension change δσ = (dσ/dT)δT that will drive a i cal point than the quench shown in images (d-h). After thermo-capillary (Marangoni) flow in the bulk of both each quench as soon as the heating stops, the bubble in- fluids[16–18]. Theimagesobtainedinourexperimentare terface begins to return to its initial form (Fig. 10c,d). capable of visualizing convective flows from the shadow- This shows that the spreading vapor is caused by a non- grapheffect. Wehavenotseenanyevidenceofthesteady equilibrium effect. The second quench that precedes the convectionthatisrequiredtocreateandmaintaintheob- crossing of the critical point (Fig. 10d–h) shows very servedbubbleshapecontinuouslyduringtheheating. We rapid interface motion accompanied by fluid flows. conclude that the Marangoni convection is absent. While the interface returns to its initial state during This conclusion is an apparent contradiction with the waiting time of the first quench (Fig. 10c), it does many works that study the Marangoni effect caused by not return in the second quench (Fig. 10h). This occurs evaporation (see e.g. [19]). The main difference between because the characteristic equilibration time grows dra- these works and ours is in the conditions of evaporation. matically near Tc. Theseworksconsiderthe evaporationintoanopen space The same phenomenon of spreading gas was also ob- where the vapor pressure is very small. The interface 8 temperature thus follows the temperature in the bulk of According to the quasi-staticargument[23], the inter- the liquid and a very large evaporation rate is possible, face shape canbe determined from the modified Laplace limitedonlybytheaveragevelocityofthefluidmolecules. equation In our case, the gas phase is almost at saturation pres- σK =∆p+P (~x). (4) sure. This means that the total evaporation (over the r whole gas liquid interface) is small and limited by the The 3D curvature K is equal to the sum of the 2D cur- amountofthesuppliedheatconsumedbythelatentheat. vature c in the image plane and the 2D curvature in the Therefore, any variation δT is rapidly dampened by the i perpendicular plane shown in Fig. 1. For the small cell correspondingchangeintheevaporationrate,stabilizing thickness H, this latter curvature can be accurately ap- the interface against Marangoni convection, see [15] for proximated by the constant value 2/H. This is possible an extended discussion. This conclusion is confirmed by because the relatively small heat flow through the less theexperiments[20],inwhichMarangoniconvectionwas conductivesapphirewindowsimpliesasmallP nearthe carefullystudiedinaclosed cellwithverycleanwaterin r contact line on the windows, as compared to the large contactwithitsvapor. No surface-tension-drivenconvec- value of ∆p at this small H. The interface shape can tionwasregisteredinspite ofa largeMarangoninumber thus be obtained from the 2D equation that was much greaterthan its critical value obtained in the classical Marangoni-Benard experiments with non- σc=∆p′+P (l), (5) r volatile liquids [16]. It was argued in [16] that the con- ′ vection was absent due to a hypothetical interface con- where∆p isaconstanttobedeterminedfromtheknown tamination present in spite of many careful preventive bubblevolumeandlisacoordinatethatvariesalongthe measures. According to our reasoning, a variation δT bubble contour in the image plane. i would have been strongly dampened in [16] because of In order to find the distribution n(~x) at the interface the saturationconditionsinthe sealedcell. We alsonote it is necessary to solve the entire heat transfer problem. that even in evaporative driven Marangoni convection Thisproblemiscomplicatedbyseveralimportantfactors. far from saturation, the convection cells may also tend First,wedealwithaproblemthatcontainsafreebound- tostabilizetheinterfaceresultinginintermittentcellular ary (gas-liquid interface) the position of which should formation as was observed in [17]. It was also observed be determined. Second, this interface contains lines of in [17] that the velocity of convection and frequency of singularities (gas-liquid-solid contact lines) where vari- intermittent cell formation decreases as the external gas ous divergences are possible. Third, the adiabatic heat becomes more saturated. transfer[15,21,22](“the pistoneffect”) shouldbe taken into account for near-critical fluids. The first two com- plications were addressed in [23–25] for plane geometry, B. Differential vapor recoil i.e. for the gas bubble growing on a plane. We have shown that n(~x) can exhibit a divergence at the contact line and that it decreases exponentially far away from We now analyze another possible source of bubble de- it. Because the bulk temperature varies sharply in the formingstressthatdoesnotrequireatemperaturegradi- boundary layer adjacent to the walls of the cell [21] and ent alongthe interface. The bubble may be deformedby theinterfacetemperatureisconstant,thelargestportion the normal stress exerted on the interface by the recoil of mass transfer acrossthe interface takes place near the from departing vapor [19]. Let n(~x) be the evaporating triple contact line. Thus n(~x) is large in the vicinity mass per unit time per unit interface area atthe point~x of the contact line. In this work, we present first the onthe interface. The evaporatinggasmovesnormallyto scaling arguments and then an approximate calculation theinterface,andexertsaforceperunitarea(a“thrust”) of the bubble shape to illustrate our explanation of the on the liquid of spreading gas in the cylindrical geometry. We assume that n(~x) has the following form: Pr(~x)=n2(~x)(1/ρG−1/ρL), (3) n(~x)=g(~x)(Tc−T)a (6) where ρ denotes mass density and the subscripts L and as T → T , i.e., it has the same local behavior with re- c G refer to liquid and gas respectively. spect to temperature as the critical temperature is ap- Theinterfaceshapecanbeobtainedfromaquasi-static proached. The integral rate of change of mass M of the argumentwhentheexperimentallyobservedinterfaceve- gas bubble is defined as locity v is smallerthanthe characteristichydrodynamic i velocity σ/η, where η is the shear viscosity. A numerical dM/dt= n(~x)d~x∼(T −T)a, (7) Z c estimateshowsthatthequasi-staticapproximationholds for the images (a-f) in Fig. 6, in which the spreading is where the integration is performed over the total gas- observed. The quasi-static approximation does not ap- liquid interface area. On the other hand, peartoholdforthe quenchexperiments (Fig.10), where the interface moves rapidly. dM/dt=d/dt(Vφρ ), (8) G 9 whereV isthecellvolume,andφ=0.5isassumed. Near Eq.(10)canbedeterminedfromEq.(11), wherethe up- the critical point, the co-existence curve has the form per integration limit can be replaced by infinity without ρ =ρ −∆ρ/2,where∆ρ∼(T −T)β withtheuniversal any loss of accuracy. Although the expression Eq. (10) G c c exponentβ =0.325,so thatdM/dt∼(T −T)β−1dT/dt for the vapor recoil pressure is not rigorous, it contains c as T → T according to Eq. (8). Thus Eq. (7) results themainphysicalfeaturesofthesolutionoftheheatcon- c in a =β−1 and the curvature change due to the vapor duction problem: a weak divergence at the contact line recoil scales as and a rapid decay away from it. It is shown in [25] that the rigorous numerical solutions obtained far from the Pr/σ ∼(Tc−T)3β−2−2ν, (9) critical point follow this behavior. TheresultofthiscalculationisshowninFig.12. Since whereEq.(3)andthescalingrelationshipσ ∼(T −T)2ν c (ν =0.63)wereemployed. Becausethiscriticalexponent N=0.01 N=0.5 (3β −2−2ν ≈ −2.3) is very large, it should manifest itself even far from the critical point in agreement with the experiments. In summary, as T → T , the vapor c mass growthfollows the growthof its density (the vapor volume remains constant), so that the diverging vapor productionnearthecriticalpointdrivesadivergingrecoil ¹¹¹``‰˚(cid:28)fiÛ´````¿L_`,˜``````,`(`Û+&```^•T`z•T`à•T`¹¹ñ•T`¹¹¹¹¹¹¹¹``````````(`»`Û´òj_`¿L_`````Wˆ(cid:31)˜`¿````````````````˚`´`˚`````l•T`````Ł```````````````````````````````````X`h`Û´òj_`¿L_`````````````−`L`Û´òj_`¿L_`````````````Ÿ`–`−`Û+Ł````````````````` force. This curvature change has a striking effect on the N=1.0 N=1.5 bubble shape because it is not homogeneously dis- tributed along the bubble interface. Since the evaporation is strongest near the copper heating wall where the strongest temperature gradients form, both P and c in- r creasestronglynearthis wall,i.e. nearthe triple contact line. Note that c is proportionalto the secondderivative of the bubble shape function, i.e. to the first derivative of the bubble slope. If c is large, then the slope of the FIG. 12: Calculated bubble shape for different values of bubble contour changes sharply when moving along the the non-dimensional strength of vapor recoil N that goes to bubble contour towards the contact line, see [5, 23, 24] infinity at the critical point. Note that the actual contact for more details. Because the interface slope changes so angle is zero for all thecurves. abruptly near the contact line, the apparent contact angle should be much larger than its actual value. Eq. (9) implies Because c is proportional to the second derivative of the bubble shape function, Eq. (5) is a differential equa- N ∼(T −T)−2.3 →∞ (12) c tion with the boundary condition given by the actual as T → T , the N increase mimics the approach to contact angle [23]. This actual contact angle defines the c the criticalpoint andqualitatively explains the observed first derivative (the slope) of the bubble shape function shape of the vapor bubble (see Fig. 6). The increase of at the solid wall. It is also specified by the interfacial the apparent contact angle and of the gas-solid contact tension balance andmust be zero near the criticalpoint. area A can be seen in Fig. 12. Note that such a cal- Thisconditionofthezerocontactanglegivesaboundary cg culation is not able to predict the wavy interface shapes conditionforEq.(5). In orderto illustrate a possibleso- like those in Fig. 6g or Fig. 8f-h, because these images lution of Eq. (5), we solved it using the same expression correspondeither to T >T (images g and h of the both for P (l) as in [23] c r figures) or to the close vicinity of T where σ < v η, see c i P (l)∝−N log(l/L) exp{(−[l/(0.1L)]2}, (10) the discussion of the validity of the quasi-static approxi- r mation earlier in this section. where l ∈ [0,L], L being a length of the bubble half- contour with l = 0 at the solid wall. We use a non- VII. SPREADING GAS AND THE BOILING dimensionalparameterN tomeasuretheinfluenceofthe CRISIS vapor recoil force relative to the surface tension. It is defined as Averysimilarbubblespreadingwasobservedfarfrom L T during boiling at large heat flux [26, 27]. When the 1 c N = P (l)dl. (11) heating to a surface is increased past a Critical Heat σ Z r Flux (CHF) there is a sudden transition to “film” boil- 0 ing,wherethe heaterbecomescoveredwithgasandmay wheretheintegrationisperformedoverthedropcontour burnout [14]. This “boiling crisis” is an important prac- intheimageplane. Thenumericalcoefficient(see[23])in tical problem in many industries where large heat fluxes 10 arefrequentlyused. Weinterpret[23,25]the boilingcri- apparent contact angle that appeared despite the zero sis to be similar to the gas spreading shown here. The actual contact angle with the solid. The spreading gas main difference is that the large value of N is made by is a phenomenon that can occur in a sealed heated fluid a large vapor production that can be achieved during cell only when the bubble is pressed against the heating strong overheating rather than by the critical effects. wall. The 3D numerical calculation of the equilibrium It is well-documented from experiments [14] that the bubble shape showed that the slightly tilted windows of CHF decreases rapidly when the fluid pressure p ap- theexperimentalcellpressedthebubbleagainstthecop- proaches the critical pressure p , i.e., when T → T in per side-wall. Weightless conditions are needed in the c c our constant volume system. Previously, this tendency near-critical region in order to observe this phenomenon hasnotbeenwellunderstood. Thedivergenceofthefac- whenthesurfacetensionissmallandabubble-likeshape tor N, discussed above, helps to understand it. We first persists. Thesamephenomenoncanbeobservedfarfrom note that the evaporation rate n scales as the applied thecriticalpointduringboilingathighheatfluxeswhere heat flux q and N ∼ q2, where Eqs. (3) and (11) are itisknownasthe “boilingcrisis”. While the gasspreads used. By assuming that the boiling crisis (q = q ) veryquicklyduringthe boilingcrisisfarfromthe critical CHF begins when N attains its critical value N ∼ 1 (see point, the near-critical region allows a very slow spread- CHF [23]), one finds that ing gas to be observed in great detail. q ∼(T −T)1+ν−3β/2 ∼(T −T)1.1 (13) We explain this phenomenon as induced by the vapor CHF c c recoil force that changes the shape of the vapor-liquid interface near the triple contact line. Our preliminary from Eq. (12). The same exponent is also valid for the calculations of the gas-liquid interface shape are quali- pressure scaling, tatively consistent with the experimental images. The q ∼(p −p)1.1. (14) scaling analysis gives the critical exponent for the criti- CHF c calheatfluxdecreasenearthecriticalpointandexplains Eq. (14) explains the observed tendency q → 0 as the increase of the vapor recoil effect near the critical CHF p→p . point. Webelievethatthereismuchtobelearnedabout c Although the strict requirements on temperature sta- theboilingcrisisinthenear-criticalregionandhopethat bility and the necessity of weightlessness lead to experi- these experiments inspire more investigations. mental difficulties to study the boiling crisis in the near- critical region, they also present some important advan- tages. Only a very small heating rate (heat flux) is needed to reach the boiling crisis because q is very CHF small. At such low heat fluxes, the bubble growth is ex- Acknowledgments tremelyslowdue tothe criticalslowing-down. Inourex- periments wewereable toobservethe spreadinggas(i.e. the dry-out that leads to the boiling crisis, see Fig. 6) This work was supported by CNES and NASA Grant during 45min! Such experiments notonly permit anex- NAG3-1906. A part of this work was made during the cellent time resolution, but also allow the complicating stayofV.N.atthe UNO andhe wouldliketo thank the effects of rapid fluid motion to be avoided. Department of Physics of the UNO for their hospitality. WethankalloftheAliceIIteamandeveryoneinvolvedin theMirmissions. WeespeciallythankJ.F.Zwilling,and VIII. CONCLUSIONS the french cosmonauts Claudie André-Deshays, Léopold Eyharts, and Jean-Pierre Haigneré. We thank Kenneth In our experiments we observed a gas bubble spread- Brakkefor creating the Surface Evolversoftware and for ◦ ing over a solid wall and a large value (> 90 ) of the making it available for the scientific community. [1] M. Schick, in: Liquids at Interfaces, Eds. J. 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