Microgravity Sci. Technol (2009) 21:59–65 DOI 10.1007/s12217-008-9048-x

Experimental Considerations of Solutocapillary Flow Initiation on Bubble/Drop Interface in the Presence of a Soluble Surfactant Konstantin G. Kostarev · Andrey L. Zuev · Antonio Viviani

Received: 15 October 2007 / Accepted: 15 April 2008 / Published online: 15 July 2008 © Springer Science + Business Media B.V. 2008

Abstract The paper deals with the experimental study of the dynamics of generation of the solutocapillary flow at the air bubble or chlorobenzene drop interfaces in inhomogeneous aqueous solutions of ethyl or isopropyl alcohols, which have low surface tension and therefore are surface-tension active with respect to water. It has been found that the onset of the solutocapillary Marangoni convection is appreciably delayed relative to the moment at which a running flux of concentrated surfactant solution reaches the surface. Critical differences in surfactant concentrations (diffusion Marangoni numbers), which are responsible for initiation of the surfactant mass transfer along the interface, were determined depending on the rate of surfactant flux and alcohols initial concentration in the solution surrounding the bubble. Keywords Solutocapillary flow · Surfactant transfer · Marangoni convection · Aqueous alcohol solutions

Introduction In the recent literature there has been enough evidence for origination of the tangential capillary stresses at K. G. Kostarev · A. L. Zuev Institute of Continuous Media Mechanics, UB Russian Academy of Sciences, Acad. Korolev Str. 1, 614013 Perm, Russia A. Viviani (B) Dipartimento di Ingegneria Aerospaziale e Meccanica, Seconda Università di Napoli, via Roma 29, 81031 Aversa, Italy e-mail: [email protected]

the interface of gas bubbles or drops of insoluble fluid placed in a liquid medium with a temperature or surfactant concentration gradient. These forces, directed towards increasing surface tension, involve in motion the near-surface zones and thus initiate in the fluid the volumetric convective flows known as the Marangoni convection. Such convection (thermocapillary or solutocapillary, respectively) plays an important role in heat/mass transfer and multi-phase media hydrodynamics and generally has a marked effect on many technological processes. However, despite much likeness between the nature of the driving forces, the solutocapillary flows can differ substantially from the thermocapillary ones. Firstly, these differences are concerned with the considerable characteristic times of admixture diffusion and, accordingly, the much greater values of the diffusion Prandtl and Marangoni numbers. As a result, the concentration inhomogeneities in the fluid exist for much longer time, whereas the intensity and continuance of capillary forces at the interface show a manifold increase. Secondly, the mechanism of the surfactant transfer (adsorption) on the interface differs from the surface temperature formation mechanism. The interface, considered as a separate phase, has inertial properties. Along the interface, both a convective transfer and surface diffusion of the surfactant may occur. This results in the appearance of solutocapillary phenomena, which have no thermocapillary analogues. Among these effects is formation of self-oscillatory modes of the solutal flows. Experimentally, the oscillatory modes of solutal convection were first found in a liquid system which was formed by a diethyl phthalate drop placed under the free water surface (Kovalchuk et al. 1999). The drop formed at the tip of a capillary was immersed into

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the water layer. Solving in water, diethyl phthalate reached its surface and being a surfactant for this liquid caused a local variation of the surface tension. In turn, the surface tension gradient, after it had reached the threshold value, caused the development of the Marangoni convection. Due to large diffusion times the arising convective motion disturbed rapidly the surfactant distribution, formed in the water and on its surface, after which it died out and was not observed for a long time (until regeneration of the critical concentration gradient of the surfactant at the free surface). The theoretical analysis has shown that solubility of the diethyl phthalate in water and its surface activity are the main characteristics determining the system behavior. Further experiments made under similar conditions but for different liquid systems supported the obtained conclusions (Kovalchuk and Vollhardt 2000, 2004, 2006). The development of the oscillatory convection was also observed around the motionless air bubbles introduced in a thin liquid horizontal layer with a vertical gradient of surfactant concentration (Kostarev et al. 2004). The tests were conducted in a cavity 2 mm thick, containing a two-layer system of mixable fluids, one of which is surface-tension active with respect to the other (water–methanol, water–acetic acid). The plane-parallel glass plates with semi-transparent reflecting coating served as the top and bottom walls of the cavity. This allowed us to apply interferometric method to visualize horizontal distribution of the surfactant concentration in the fluid in the form of the refraction index isolines pattern. The experiments revealed periodic short-time intensive disturbances of concentration round the bubble. The period of these oscillations varied in the range from a few seconds to tens of minutes depending on time, initial concentration difference, fluid properties, layer thickness and bubble diameter. With the decrease of the vertical concentration gradient due to solution stirring the oscillations occurred more and more rarely and finally abruptly ceased. It has been suggested that the observed mass transfer intensification was caused by regular ejections to the surrounding solution excess surfactant accumulated at one of the bubble poles due to solutocapillary flow over its surface. However, in the context of threedimensional problem, the experimental investigation of the disturbance structure presented serious difficulties. In order to elucidate the causes of these oscillations the dynamics of vertical distribution of the surfactant concentration and the structure of convective flows round the bubble was investigated experimentally in a thin vertical layer of a stratified fluid (Zuev and Kostarev 2006; Kostarev et al. 2007). For this purpose,

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the experimental cuvette having the form of Hele– Shaw cell was set vertically on its narrow face so that observation was made from the wide faces. The cuvette was filled with an aqueous solution of acetic acid or isopropyl alcohol with a stable vertical stratification of the surfactant concentration. An air bubble with a volume of 20–30 μl was inserted in the fluid. It took the form of a short horizontal cylinder, 4–6 mm in diameter with a free lateral surface. A special wire frame, exerting no effect on the development of the Marangoni convection, prevented the air bubble from floating up under the action of the buoyancy force. Using optical technique we determined that oscillations of flow near the bubble were caused by competition of two solutal (capillary and gravitational) convective mechanisms of mass transfer having different characteristic times. The analysis of time dependence of the oscillation period for different values of the average solution concentration, surfactant concentration gradient and the Marangoni and Grasshoff diffusion numbers showed that the ratio of dimensionless oscillation frequency to the Marangoni number was independent of time. The values of this ratio for different fluid solutions and even for oppositely directed concentration gradients were found to be in close agreement. The oscillations of the solutal flows described above were observed not only at the free surface of the gas bubbles. Similar oscillatory modes of convection near the phase interface of insoluble fluids were also found in experiments with chlorobenzene drops placed in the stratified aqueous solutions of isopropyl alcohol (Kostarev et al. 2006). In this case, unlike the case with the bubbles, a solutal flow affecting the period and lifetime of the oscillations developed both outside and inside the drop, the latter being caused by diffusion of the surfactant from the surrounding solution into the drop. Examination of the concentration fields revealed the main stages of surfactant absorption by the drop, development of flows inside the drop and correlation between the maximal differences in the surfactant concentration outside and inside the drop. Further investigations into this phenomenon were made for a vertical fluid layer, which was bounded above and below by the solid plates forming an extended horizontal channel of rectangular cross section (Birikh et al. 2006). An air bubble, placed in the channel, completely bridged over the channel and had free lateral surfaces. The special convenience of such a “quasi-two-dimensional” configuration of the experimental cuvette and the bubble was that it allowed us to compare the results of experiments with the results of numerical simulation. The tests were performed with aqueous solutions of methyl, ethyl and isopropyl

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alcohols at different initial distributions of the surfactant concentration. The experiments revealed an interaction between two different convective mechanisms. One of the mechanisms was related to an intensive Marangoni convection, which transported the surfactant along the free bubble surface and decayed as soon as the surface concentration of the surfactant was equalized in the process of fluid stirring near the bubble. The other mechanism was actuated by the buoyancy forces, which caused the solution with higher alcohol concentration to float up and thus to retard a relatively slow global flow in the channel. This advective flow restored the disturbed stratification of the solution near the bubble surface, after which there occurred a new outburst of intensive solutocapillary convection. These experiments allowed us to investigate time dependence of the oscillation period and vertical concentration gradient and to define the critical Marangoni and Grasshoff numbers, leading to the onset of oscillatory mode of the convective motion. The stated problem was solved numerically based on the model of convection involving diffusion transfer of the surfactant to the bubble surface (without formation of the surface phase) (Vazquez et al. 1995). Consideration was given to a rectangular horizontally elongated cavity, in which one of the vertical boundaries modeled the bubble surface impermeable to the surfactant. In the cavity, an initial horizontal gradient of the surfactant concentration was prescribed. The calculation showed that in the cavity at large values of the Schmidt number (∼103 ) the self-oscillatory convection mode developed on the background of a slow gravitational convection in the form of short-time outbursts of the Marangoni convection having an order of magnitude greater intensity. The experiment and numerical calculations were found to be in good agreement with respect to the convective flow structure and the oscillation period. The experimental investigation of the flow structure and the concentration fields near the bubble in a narrow channel (Birikh et al. 2006; Vazquez et al. 1995) revealed the time lag between the moment, at which an inhomogeneous surfactant flow contacted the free surface of the bubble, and the initiation of the Marangoni flow. Such a delay is similar to the induction period observed in liquid systems with a drop placed under the free surface (Kovalchuk et al. 1999; Kovalchuk and Vollhardt 2000, 2004, 2006) and can be explained by the development of physico-chemical processes at the interface, such as surfactant adsorption and surface diffusion. The objective of this work is to define the conditions for generation of the solutocapillary flow

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and to ascertain the critical difference in the surfactant concentration provoking its initiation.

Experimental Technique The experimental setup is represented schematically in Fig. 1. The working fluid (water or homogeneous aqueous solution of ethyl or isopropyl alcohol) filled the cavity in the form of extended horizontal channel of rectangular cross-section of height h = 2 mm and thickness 1.2 mm. The channel was confined between two vertical interference glass walls coated with reflecting semi-transparent material (see Fig. 1a) purposely for making observations. Due to a small thickness of the layer the arising flows and distribution of concentration (averaged across the layer) were two-dimensional. Air bubbles or drops of immiscible fluid (chlorobenzene) were injected by means of a syringe into a channel cavity from one of its ends in such a way that the bubble blocked the channel passage. Then, from the other end, the cavity was gradually filled with water– alcohol solution with initial mass concentration C0 of alcohol ranging from 1% to 40%, while excess of water was pumped out. Since alcohol is lighter than water, this led to formation of a rather slow large-scale advective gravitational flow in the channel, during which a narrow “tongue” of a more concentrated surfactant solution flowed along the upper channel boundary toward the bubble /drop surface forming near the surface a region

a) h

3

2

b)

1 6 2

5 4 7

7

Fig. 1 Schematics of experimental setup: 1 — interferometric semi-transparent glasses; 2 — channel with fluid; 3 — air bubble; 4 — objective lens; 5 — semi-transparent mirror; 6 — laser; 7 — video camera

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with upward-directed concentration gradient. Varying the velocity and concentration of the inflowing surfactant flux and the initial concentration of the solution in the channel we could thus change the value of the vertical concentration gradient near the surface and the rate of its formation. The surfactant distribution in the fluid mixture was visualized with the aid of the Fizeau interferometer (Fig. 1b) as a pattern of isolines of the refraction index, which under isothermal conditions depends on the solution concentration. Bearing this in mind, a transition from one monochrome fringe to another in the interferogram may be considered to be correspondent to variation of the surfactant concentration by 0.34% for all isopropyl alcohol solutions. In ethyl alcohol solutions, a new interference fringe appeared when concentration changed by 0.53% at C0 = 5%, decreasing gradually to 0.38% at C0 = 40%. The interference patterns were recorded in transmitted and reflected light by video camera (25 s−1 , resolution of 640 × 480). The maximum error of concentration measurements did not exceed 0.1%. All measurement were carried out at constant ambient temperature 20 ± 1◦ C.

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a)

b)

c)

d)

e)

f) Results and Discussion Figures 2 and 3 show respectively the interference patterns of the concentration filed near the lateral boundary of the air bubble and the drop of chlorobenzene in the isopropyl alcohol solution. In distinction from the thermocapillary flow known to develop in a nonthreshold manner at arbitrary small values of the interface temperature gradient, the solutocapillary flow was initiated not at the time when the surfactant “tongue” reached the bubble surface, but with some delay. Thus, in experiments with air bubbles (Fig. 2a, b) a time-lag t between the arrival of the surfactant flow at the bubble surface, and initiation of the convective vortex (Fig. 2c) was 28 s. At the same time the difference of the surfactant concentration C∗ caused by a continued motion of the “tongue” reached 2.2% (C∗ = C2 − C1 where C2 and C1 were the alcohol concentrations at the upper and lower bubble poles, respectively. Note that from here on, we will describe the interface processes in terms of the volumetric characteristics of the medium near the interface assuming that variations of the surface characteristics (surface concentration, surface diffusion, etc.) is proportional to variation of the volumetric characteristics. Then the equilibrium was abruptly disturbed and very quickly (in a matter of about 0.2 s) a rather intensive Marangoni flow was initiated in the form of a vortex cell, in which the

g)

Fig. 2 Interferograms of concentration fields for an air bubble in isopropyl alcohol solution. t = 0 s (a), 28.0 s (b), 28.1 s (c), 28.2 s (d), 29.2 s (e), 34.2 s (f), 34.3 s (g)

surfactant under the action of solutocapillary forces was carried along the bubble surface toward the lower pole (Fig. 2d). Owing to fluid continuity the arising flow accelerated the flux of the concentrated surfactant solution along the upper boundary of the channel towards the bubble surface, adding thereby intensity to the existing convective vortex. However, the originated vortex cell, entrapping more and more portions of highly concentrated surfactant solution, became increasingly light. Rising up, it eventually cut off the arriving jet of the alcohol from the top of the bubble. As a result, the vortex flow ceased abruptly and the bubble surface turned out to be surrounded by a thin layer of the surfactant solution having the uniform concentration (Fig. 2e). After that the vertical stratification of the solution began to slowly restore under the action of the buoyancy force.

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63

30

1

b)

Ma*, 10

6

a)

10

c)

e)

f)

g)

Fig. 3 Interferograms of concentration fields for a chlorobenzene drop in isopropyl alcohol solution. t = 0 s (a), 60 s (b), 61 s (c), 66 s (d), 70 s (e), 76 s (f), 77 s (g)

However, decay of the capillary flows did not imply cessation of the fluid motion in the channel because equalizing of the generated horizontal gradient of the surfactant concentration again made the advective flow draw a concentrated surfactant solution to the upper pole of the bubble. After the flux of the surfactant touched the bubble surface, the solutocapillary vortex recurred (Fig. 2f, g). The cycle repeated iteratively, with the difference that the oscillation period increased with time whereas intensity of the vortex flow decreased due to a gradual decrease of the vertical concentration gradient as a result of convective stirring of the fluid. The Marangoni convection entirely ceased at the time when concentration of the solution was equalized throughout the whole length of the channel. Characteristic feature of this process is that during subsequent cycles the capillary flow was initiated

0

0

5

10 N

15

20

Fig. 4 Critical Marangoni numbers at the beginning of each cycle of intensive convection around the bubble. 1 — Ethanol, 2 — isopropanol

at progressively less concentration differences at the bubble surface (see Fig. 2f, the onset of the second cycle, C∗ ∼ 0.6%). Figure 4 presents critical values ∂σ of the diffusion Marangoni number Ma∗ = ρvhD ∂C C∗ , at the moments of beginning of intensive convection cycles (here σ — coefficient of the surface tension, ρ — density, v — kinematic viscosity, D — bulk diffusion coefficient of the solution). At the same time the average concentration of the surfactant at the bubble surface gradually increased with each cycle of the vortex convection (Fig. 5). Divergence of the corresponding curves for ethanol and isopropanol in Fig. 4 and Fig. 5 is evidently caused (as in the case of alcohols with middle chain length; Kovalchuk and Vollhardt 2004),

9

1 2

6

,%

d)

2

20

3

0 0

5

10

15

20

N Fig. 5 Average surfactant concentration at the bubble surface at the onset of intensive convection cycles. 1 — Ethanol, 2 — isopropanol

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by differences in their solubility in the water and their surface activities. A similar situation was also observed at the interface of the drops of chlorobenzene in the isopropanol solution (Fig. 3). It can be seen from interferograms that the alcohol reaches the surface of the drop and even diffuses through it, generating in chlorobenzene a gradient of surfactant concentration well before the beginning of convective motion in the channel. Initially, this process proceeds quite uniformly over the drop surface, and thus, the concentration gradient at the interface is absent. In the solution, an intensive Marangoni convection develops some time (t ∼ 1 min) after the “tongue” of the surfactant has touched the drop surface and C∗ ∼ 4%. Further, a more detailed consideration is given to the case of a bubble which allows us to ignore surfactant diffusion through the interface. In experiments with the air bubbles the time interval t can vary from a few seconds to minutes depending on the velocity of the surfactant “tongue” motion. Such time scattering is evidently caused by the fact that the interval t is composed of two time moments. The former is the time, during which the surfactant molecules overcome in the diffusion mode several molecule layers of water near the free surface, and the latter is the time taken to create an initial surfactant distribution required for generation of the surface tension gradient. Since the time of molecule emergence at the interface is defined by bulk diffusion, at large flow velocities and, accordingly, at the intensive surfactant flow the time interval before the outburst of the capillary convection proves to be quite sufficient for generation of a large concentration gradient in immediate proximity to the surface (Fig. 6).

In this figure, τ = h2 /v is convective time. At the same time, at sufficiently small flow velocities (for t longer than 1 min) the process of generation of a certain surfactant concentration gradient at the surface plays a dominate role, and therefore the critical Marangoni numbers Ma∗ prove to be nearly constant and rather close for different surfactants. We also investigated the dependence of the critical Marangoni numbers, corresponding to the time of formation of the first vortex, on the initial concentration of the surfactant in the solution surrounding the bubble. To this end, the channel was filled not with pure water but with homogeneous aqueous solutions of the alcohols with C0 ranging from 1% to 8%. With appearance of the surfactant in the surrounding liquid its surface tension, as well as its density, change. Variation of the density may cause an essential change of the critical Marangoni number. Since we consider the disturbance of the mechanical equilibrium of the fluid near vertical (to first approximation) interface, we should also take into account the buoyancy forces, because surfactant density differs from that of the basic fluid. The surfactants used in our experiments are lighter than water. Therefore the growth of their concentration in the incoming flow causes an increase of both the capillary forces directed downward along the interface and the upward-directed buoyancy forces, acting on the near-interface fluid layer. Variation in concentration of the surrounding fluid leads to the opposite effect. To allow for the ratio of gravitational to capillary forces, we introduce the Bond number Bo = ρgh2 /σ , where ρ is the difference in density between the surfactant solution and water, g is the gravity acceleration.

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45

1

2 6

40

Ma*, 10

Ma*, 10

6

1

20

0

0

50

100 Δt/τ

150

200

Fig. 6 Dependence of the critical Marangoni numbers on the time interval between the moment when the surfactant touches the bubble surface and the moment of vortex generation. 1 — Ethanol, 2 — isopropanol

30

2

15

0 0,0

2,5

5,0

Bo, 10

7,5

10,0

–3

Fig. 7 Critical Marangoni numbers versus Bond numbers for bubbles in surfactant solutions with different initial concentration. 1 — Ethanol, 2 — isopropanol

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65

30

Ma*, 10

6

1 20

2

10

0

0

3

6 –3 Bo, 10

9

12

Fig. 8 Critical Marangoni numbers versus Bond numbers for bubbles at the onset of intensive convection cycles. 1 — Ethanol, 2 — isopropanol

In Fig. 7 we can observes an explicit correlation between the critical Marangoni numbers and the values of the Bond number, despite the essential dominance of the capillary forces (Bo ∼ 10−3 ). The obtained estimations suggest that such values of the Bond number can be obtained in some technological experiments conducted in microgravity conditions. Therefore refinement of our understanding of solutocapillary convection phenomenon will be valuable for implementation of this task. A monotonic decrease of Ma∗ with the growth of C0 may be one of the explanations of an essential decrease in the concentration difference at the bubble/drop surface at the beginning of each next cycles of the convective vortex flow. The cumulative diagram in Fig. 8 presents critical Marangoni numbers at the beginning of different cycles of the vortex flow (from the plot in Fig. 4) as the functions of the Bond numbers Bo, which were calculated using the values of the surface tension corresponding to the average surfactant concentration Ca = (C1 + C2 )/2 at the air bubble surfaces at the time of motion intensification (from the plot in Fig. 5). On the same diagram we also present (solid line) the results of measurements of Ma∗ as a function of Bo, obtained for solutions with different initial alcohol concentration (from the plot in Fig. 7). It is seen that qualitatively all curves for Ma∗ obtained in different tests and under various conditions match well.

Conclusion The experimental investigations discussed in this paper have shown that in contrast to thermocapillary

Marangoni convection, in the case of solutocapillary flow the formation of the surfactant gradient at the bubble/drop interface, contacting the surfactant solution with nonuniformly distributed concentration, occurs with noticeable delay. This delay, lasting sometimes for rather long periods of time (up to minutes), is related to the fact that both the diffusion transfer of the surfactant molecules to the interface and their surface diffusion proceed much slower (compared to the diffusion of heat). On its own, the development of the motion under the action of tangential surface forces occurs in a threshold manner as soon as the gradient of the surfactant concentration at the interface reaches some critical value. The threshold value depends on the initial alcohol concentration around inclusion, the rate of formation of its concentration gradient at the interface, its solubility in the water and surface activity. Acknowledgements The work was supported by Russian Foundation for Basic Research (Grant No. 06-01-00221).

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