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Pushing of liquid drops by marangoni force
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Acta Astronautica Vol. 51, No. 11, pp. 789–796, 2002 ? 2002 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0094-5765(02)00030-9 0094-5765/02/$ - see front matter
PUSHING OF LIQUID DROPS BY MARANGONI FORCE† R. MONTI‡, R. SAVINO and G. ALTERIO Dipartimento di Scienza e Ingegneria dello Spazio “Luigi G. Napolitano”, Universit;a degli Studi di Napoli “Federico II”, P.le V.Tecchio 80, 80125 Napoli, Italy (Received 12 May 2000; revised version received 22 February 2001)
Abstract—Following initial observations in microgravity, on ground experiments pointed out that coalescence and wetting can be prevented by Marangoni forces at liquid–gas or liquid–liquid interfaces. Pushing of liquid droplets by a solid surface due to Marangoni forces can explain a number of relevant phenomena occurring during solidiAcation, in particular the separation of immiscible alloys encountered in Material Science processings in microgravity. In this paper, numerical simulations have been performed to evaluate the Marangoni forces on a free droplet near a solid surface immersed in a transparent liquid matrix in the presence of diBerent temperature gradients and that explain a number of experiments made on ground in preparation of an experiment to be carried out in microgravity on a Sounding Rocket. Numerical results clarify questions of interest for solidiAcation processes. In particular, they may provide some explanation for the minority phase separation during solidiAcation due to the Marangoni repulsive forces that are experienced in the vicinity of the solidiAcation front and that result in pushing of the liquid droplet towards the hot site of the sample. ? 2002 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
The prevention of liquid drops coalescence of silicone oil in air originally experienced during the Mission D-2 with the Fluid Physics Module [1] has motivated extended numerical and experimental activities on ground [2–5] also in preparation of experiments to be hosted on microgravity platforms [6]. The cited references report the details of the ground experimentation that were in very good agreement with numerical results. Here we simply summarize the Andings, the numerical correlations and the extension to problems of relevant interest for Material Science and microgravity. A semi-spherical drop of a liquid is formed as a hanging drop from a circular disk. Suppose that the liquid wets the surface of a certain solid and suppose that the drop is brought in contact with this surface; as soon as the liquid contacts the surface (as expected), it spreads over it. Suppose,
†Based on paper IAF-99-J.4.05 presented at the 50th International Astronautical Congress, 4 –8 October 1999, Amsterdam, The Netherlands. ‡Corresponding author. Tel.: +39-081-768-2360; fax: +39081-593-2044. E-mail address: [email protected] (R. Monti). 789
now that the surface tension of the liquid decreases with the temperature and that the circular disk is held at a temperature TD ¿ TS (TS being the solid surface temperature), at these conditions spreading (or wetting) is prevented as long as TD is sufAciently higher than TS and the drop is not too “strongly” pushed against the surface. For instance, for a number of liquids (silicone oil, kerosene, alcohols) a NT = TH − TS ¿ 0, of ◦ the order of 3–4 C (for drops of few millimeter diameter), is suPcient to prevent wetting even if the drop is pushed against the surface [5]. This very intriguing phenomenon was explained by numerical simulations by assuming that the Marangoni Qow, generated at the drop surface, entrains a Alm of air between the (deformed) drop and the solid surface and prevents an intimate contact between the liquid and the solid surface molecules [5]. Similarly, two drops on two disks held at different temperatures will not coalesce if not pushed very hard against each other [7,8]. This behaviour is similar to the one observed when bubbles or drops coalescence is retarded by solutal Marangoni eBect induced by concentration gradients during mass transfer in liquid–liquid systems [9]. In contrast to other proposed explanations [10], the numerical simulation was able to explain this
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phenomenon also when two opposite eBects take place, i.e. the entrainment of air by the upper (hot) drop and the detrainment of air, caused by the very same Marangoni eBect, along the lower (cold) drop. As explained in [2– 4] the reason for the coalescence prevention is the diBerence between the two eBects caused by the temperature distribution: the entrainment eBect is larger than the opposite one. In [11] it was shown that, for similar reasons, if the temperature diBerence is suPciently large, a liquid drop can be completely immersed in a pool of the same liquid. What has been found for semi-spherical hanging drops was extended to full drops initially at a temperature TD larger than the solid surface temperature (TS ): the drop does not spread over the surface but tends to roll over the surface until thermalization (or when its temperature becomes almost equal to the surface temperature [6]). Similarly drops “Qoating” over a pool (of the very same liquid) have been observed [12] until thermalization occurs. All what found for drops in air was also found for drops submerged in a liquid matrix. The situation is now more complex but the explanation is basically the same. If half drop of a liquid “LD ” is formed on a circular disk immersed in a matrix of liquid “LM ” (LD and LM are immiscible) then we have the same eBect as above if a temperature gradient is established in the matrix. Again, the same eBect is being experienced by a full spherical drop in a liquid matrix that is close to a solid wall (representing a solidiAcation front) in presence of a strong temperature gradient. This speciAc case, of greater relevance for the solidiAcation of immiscible alloys, will be analysed, both numerically and experimentally, in what follows. 2. RELEVANCE IN THE FIELD OF MATERIAL SCIENCE
As mentioned before, the authors believe that the problem of wetting and coalescence prevention of drops in an external liquid matrix, up to now unexplored, is of particular interest in the Aeld of Material Science. Liquid–liquid immiscible alloys (e.g. In–Al, Al–Pb, etc.) formed by two melts at diBerent compositions can be fully exploited only if a uniform Anely dispersed minority phase can be preserved in the solid state (e.g. materials for self-lubricating bearings with a soft dispersed phase in a high mechanical strength matrix; Lead or Indium dispersions in an aluminium matrix are the most promising). In the Earth’s gravitational
Aeld the usual diBerences in density cause rapid separation of the alloy components through sedimentation or Qoatation. Material Science experiments dealing with metallic alloys processing in microgravity resulted in a number of disappointing results [13,14]. The anticipated structure of uniformly distributed intrusions of the minority phase in the external matrix after solidiAcation was not found. In many cases, a separation occurred and the minority phase accumulated in the “hot” region during the solidiAcation process. This means that the minority phase (initially uniformly dispersed in the majority phase) moved to the region that solidiAed later (one end of the cartridge, for directional solidiAcation or the internal region when cooling takes place at the external walls). Explanation of the microgravity results has been attempted by assuming that liquid drops in a liquid matrix migrate towards the hot region due to Marangoni eBect. The idea being that if the solidiAcation front advances with a velocity larger than the drop migration velocity then it engulfs the drop; vice versa there is a separation of the two materials. Recent Andings on wetting and coalescence prevention and the extended ground experiments performed in the last four years strongly suggest to study all the phenomena that occur during alloys solidiAcation (migration, dissolution, wetting and coalescence of drops). Of particular interest is the wetting prevention that could explain the drops pushing by solidiAcation fronts that may be responsible for the phases separation (as observed in the solid). 3. MARANGONI FORCE ON THE DROPLET
This paragraph reports the results of numerical simulations for the evaluation of the Marangoni forces exerted on free droplets interacting with a solid surface at diBerent temperature. The analysis is performed in terms of the forces acting on drops that are either motionless in a liquid matrix or that migrate at constant velocity (e.g. with the Young velocity in an inAnite medium). At each vertical position of the drop and for a Axed NT between the upper and lower wall, the force FM , due to Marangoni Qows, has been computed by integrating the surface forces along the drop surface: nv ds − pn ds; (1) FM = S
S
where n is the unit vector normal to the drop surface.
Marangoni force on drops
791
This force consists of two parts: (a) the integral of the viscous force, due to the viscous stress tensor: = 2 (∇V )s0 ;
(2)
where the subscript (0) and the superscript (s) denote the traceless symmetric part of the velocity gradient tensor; (b) the resultant of the normal surface forces due to the pressure along the drop, due to the motion of the liquid in the matrix and to the Archimedean eBect. The thermoQuidynamic Aelds, inside and outside of the drop, is computed for Anite dimension containers in an accurate way. A drop of liquid (LD ) is immersed in a matrix of liquid (LM ) at constant temperature. If the drop density is larger than that of the matrix then the drop falls to the bottom and if the matrix liquid wets the bottom surface more than the drop liquid then the drop will sit on the bottom and its net weight (Wg ) will be balanced by the surface reaction forces (FR ): FR = Wg = WD −
D3 L g; 6
(3)
WD being the drop weight. If a temperature gradient is established in the matrix, upwards along the vertical (cold bottom), then a Marangoni force is created due to the viscous tangential forces (S ) directed along the surface (s) and to the pressure (p), directed normal to the surface (n). The equilibrium along the vertical (FM is directed upwards, toward the hot side) now reads FR = W g − FM :
(4)
The computations made have shown an eBect similar to the “ground eBect” that appears on moving objects near the ground. Indeed when the drop is located near a solid surface the pressure forces on the drop (that are small far away from the wall) become large compared with the viscous forces (see Fig. 1). This results in a force on the drop that pushes the drop upward far from the cold wall and might explain the reason for the separation of minority phases during solidiAcation of immiscible alloys, that tends to migrate towards the hot side (that solidiAes last). Suppose now that we increase the temperature gradient (∇T ) in the liquid matrix, then the drop will start Qoating when FR ≡ 0 for a value of ∇T = (∇T )F .
Fig. 1. Computed stream-lines (a), isotherms (b) and pressure contours (c) for a droplet of Fluorinert FC43 in silicone oil 3 cS (D = 2 mm; ∇T = 20 K=cm; h = 100 m).
The numerical computations have shown that, at a Axed ∇T , FM is an increasing function of the drop diameter and an inverse function of the distance between the drop and the solid surface (Fig. 2). An evaluation of the (upward) migration velocity (VM ) could be made by assuming that the Drag force is that pertaining to the viscous Stokes regime VM =
FM − Wg : 3 D
(5)
This value of the velocity should be accurate in the limit of low speed or small diameters (or Ma = (T NT= )D → 0) i.e. when the drop migration does not disturb the Qow Aeld. If h is the distance between the drop surface and the bottom wall, the total Marangoni force acting
792
R. Monti et al. Fc43 - Silicone oil 3cs 1.00E-1
In - Al
F (dyne)
1.00E-2
1.00E-3
∆ T/L=2 K/mm 1.00E-4
1.00E-5
1.00E-6 0
200
400
600
800
1000
Drop diameter (µm) 0. 12
∆ T/ L=2 K /mm
F (dyne)
D=2 mm
0. 08
Fc43 - Silicone oil 3cs
Fig. 4. Computed stream-lines (a), isotherms (b) and pressure contours (c) for a droplet in a liquid matrix between two walls at diBerent temperatures.
0. 04
In - A l
0. 00 0
40
80
120
160
200
h (µm)
Fig. 2. Computed force on the droplet versus drop diameter (D) and distance from the wall (h).
4.0
F/Fm
3.0
the limit for h=D → ∞ corresponds to the classical Young solution. Similar eBects can be found for a drop in a liquid matrix between two solid walls maintained at different temperatures. In this case the liquid around the drop is detrained, due to the Marangoni eBect, from the liquid layer between the drop and the hot wall (upper) and entrained between the drop and the lower wall. The pressure is reduced in the upper part and increased in the lower one, resulting in an overall force pushing the drop in the direction of the temperature gradient (Fig. 4).
Ma =30
2.0 Ma < 1
4. LABORATORY TESTS
Ma = 15
1.0 0.0 0.00
0.10
0.20
0.30
0.40
0.50
h/R
Fig. 3. Non-dimensional plot of the computed force (scaled by the Marangoni force FM ) versus h=R, for diBerent Marangoni numbers.
on the drop FM is a decreasing function of the distance and an increasing function of the Marangoni number (see the non-dimensional plot of Fig. 3),
A drop (D = 120 m) of Fluorinert FC 43 (1 cS, = 1700 kg=m3 )) in a matrix of Silicone Oil of 3 cS ( = 910 kg=m3 ) was conAned between two plates at diBerent controlled temperature (Tu ; Tb ). A temperature diBerence between the upper disk (Tu ) and the bottom one (Tb ) Tu − Tb = NT ¿ 0 was established by heating the upper disk: when the average temperature gradient across the cell (NT ) is above a certain value then the drop is pushed away from the wall and Ands its stable equilibrium condition at a distance h such that FM (h) = Wg =
3 D (D − L )g: 6
(6)
Marangoni force on drops
793
Table 1. Physical properties of the liquids investigated Fluid properties
Silicone oil 3 cS
Fluorinert FC43
Ratios of properties silicone oil=FC43
Density (g=cm3 ) Dynamic viscosity (centipoise) Kinematic viscosity (cm2 =s) Interface tension (dyne=cm)
0.89 2.67 0.03 18.8
1.88 5.35 0.028 16.6
Surface tension derivative T (dyne=(cm K))
0.066
0.088
0.47 0.5 1.07 Interface tension 5.35 Interface tension derivative 0.038
1:16 × 10−3
1:2 × 10−3
0.96
2:6 × 10−4 6:27 × 10−4 47.8
1:5 × 10−4 3:39 × 10−4 82.6
1.66 1.85 0.57
Thermal expansion coePcient T (K −1 ) Thermal conductivity coePcient [cal=(cm s K)] Thermal diBusivity (cm2 =s) Prandtl number Pr = =
When further increasing |∇T | (and FM ) the drop moves further up until h=D becomes large and if FM (∞) ¿ Wg , then the drops starts migrating along the temperature gradient. The liquid properties used in the numerical computations are summarized in Table 1. The surface tensions and the interface tension at the Fluorinert–silicone oil interface have been measured at diBerent temperatures using a tensiometer [15].
The numerical predictions (see Fig. 5) were indeed conArmed by experiments. The drop, initially sitting on the bottom, immediately jumped at the equilibrium position (that moves upward with increasing ∇T ). Figure 6 shows the photographs of the experiments and Fig. 7 shows the drop position and velocity versus time. What was found numerically and experimentally for high Prandtl numbers liquids (e.g. silicone oils,
Fig. 5. Computed streamlines, isotherms and pressure distributions from the numerical simulation of a drop (D = 120 m) of Fluorinert FC 43 in a matrix of Silicone Oil of 3 cS conAned between two plates at diBerent temperatures, changing the distance from the lower and upper walls.
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For a typical case of In–Al alloys the value of FM (D; ∇T; h) was computed that show the very same trend as for large Prandtl number Quids. In Fig. 8 the numerical results obtained for a drop of Quorinert FC43 in a matrix of silicone oil 3 cS are compared to those corresponding to the case In–Al. It is interesting to note that the behaviour at large and small Prandtl numbers are diBerent for relatively large drops (diameter of the order of 1 mm), whereas for suPciently small drops (diameter of the order of 100 m) the Qow and temperature behaviour are dominated by diBusion eBects, so that the two systems exhibit the same behaviour. This consideration allows to experiment in rather high Prandtl number, transparent matrices and to extend the results to cases of greater applicative interest (immiscible liquid metals).
5. CONCLUSIONS
Fig. 6. Photographs of the laboratory experiment (a, b, c correspond to an increasing temperature gradient; d shows the time history of the temperatures).
Fluorinert, ethyl alcohol, etc.) can be extended to the cases of metal melts (very low Prandtl numbers) in the presence of strong temperature gradient at solidiAcation fronts. Experimentally it is diPcult to visualize the drop position in melts of real interest in Material Science and only numerical runs have so far be performed.
Numerical simulations and the preliminary laboratory experiments performed in this work show that Marangoni eBects are responsible for a pushing (or attracting) force on a droplet interacting with a solid wall in the presence of a temperature gradient. If the temperature of the wall is smaller than the drop temperature (as for the case of a solidiAcation front) repulsive eBects arise that prevent the wetting. Similarly, during solidiAcation processes, Marangoni eBects play a role in the solidiAcation front=drops interaction by favouring the pushing (or, equivalently, by preventing engulfment=entrapment of liquid drops). Laboratory tests with a drop of Fluorinert in a matrix of silicone oil, conAned between two plates at diBerent controlled temperatures, have conArmed the numerical predictions. When the average temperature gradient across the cell is above a certain value the drop, which is heavier than the surrounding liquid matrix, is pushed away from the wall and Ands its stable equilibrium condition at a distance such that the pushing force, due to the Marangoni eBect, balances the net weight of the drop. Further experimental and numerical studies are in progress along this line. Acknowledgements—This work has been partially supported by the European Space Agency (ESA) and by the Italian Space Agency (ASI)
Marangoni force on drops
795
Fig. 7. Experimental results for a drop of FC43 in Silicone Oil 3 cS; D = 120 mm.
Fig. 8. Comparison of computed streamlines and isotherms (grey contains) for the two cases: (1) drop of FC43 in silicone oil; (2) drop of In in Al, for two diBerent drop diameters. REFERENCES
1. Monti, R. and Dell’Aversana, P., Microgravity experimentation in non coalescing systems. Microgravity Quarterly, 1994, 4, 123. 2. Monti, R. and Savino, R., Numerical model of non coalescence liquid drops and correlation with experimental results. Microgravity Quarterly, 1996, 6, 102. 3. Monti, R. and Savino, R., Correlation between experimental results and numerical solutions of the Navier–Stokes problem for two non coalescing liquid drops with Marangoni eBect. Physics of Fluids, 1997, 9, 260. 4. Savino R. and Monti R., Modelling of non coalescing liquid drops in the presence of thermocapillary convection. Meccanica, 1997, 32, 115. 5. Monti R., Savino R. and Tempesta S., Wetting prevention by thermal Marangoni eBect. European. Journal of Mechanics B=Fluids, 1998, 17, 51.
6. Monti R., Savino R., Lappa M. and Fortezza R., Behavior of droplets on solid surfaces in presence of Marangoni eBect. ESA-SP-433, 1999, p. 225. 7. Dell’Aversana P., Banavar J.R. and Koplik J., Suppression of coalescence by shear and temperature gradients. Physics of Fluids, 1996, 8, 15. 8. Dell’Aversana P., Tondodonato V., Carotenuto L., Suppression of coalescence and wetting: the shape of interstitial Alm. Physics of Fluids, 1997, 9, 2475. 9. Groothuis H., Zuiderweg F.J., InQuence of mass transfer on coalescence of drops. Chemical Engineering Science, 1960, 12, 288. 10. Bratukin, Yu. K. and Gershuni, G. Z., On condition of coalescence of drops in the presence of thermocapillary convection. Microgravity Quarterly, 1994, 4, 193. 11. Monti, R. and Savino, R., Lappa M. and Tempesta, S., Behavior of drops in contact with pool surfaces
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of diBerent liquids. Physics of Fluids, 1999, 10, 2786. 12. Monti, R. and Savino, R., Coalescence and wetting prevention by Marangoni eBect. In Foams and Films, ed. D. Weaire and J. Banhart, Verlag, Bremen 1999, p. 25. 13. Andrews, J. B., SolidiAcation of Immiscible Alloys. In Immiscible Liquid Metals and Organics, ed.
L. Ratke. Obernrsel Germany, DGM, Information gesellschaft-Verlag, 1993, p. 199. 14. Ratke, L., Korekt, G. and Drees, S., SolidiAcation of immiscible alloys. ESA-SP-385, 1996, p. 247. 15. Alterio G., Ph.D. thesis in Aerospace Engineering, 1999.
www.elsevier.com/locate/actaastro
Acta Astronautica Vol. 51, No. 11, pp. 789–796, 2002 ? 2002 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0094-5765(02)00030-9 0094-5765/02/$ - see front matter
PUSHING OF LIQUID DROPS BY MARANGONI FORCE† R. MONTI‡, R. SAVINO and G. ALTERIO Dipartimento di Scienza e Ingegneria dello Spazio “Luigi G. Napolitano”, Universit;a degli Studi di Napoli “Federico II”, P.le V.Tecchio 80, 80125 Napoli, Italy (Received 12 May 2000; revised version received 22 February 2001)
Abstract—Following initial observations in microgravity, on ground experiments pointed out that coalescence and wetting can be prevented by Marangoni forces at liquid–gas or liquid–liquid interfaces. Pushing of liquid droplets by a solid surface due to Marangoni forces can explain a number of relevant phenomena occurring during solidiAcation, in particular the separation of immiscible alloys encountered in Material Science processings in microgravity. In this paper, numerical simulations have been performed to evaluate the Marangoni forces on a free droplet near a solid surface immersed in a transparent liquid matrix in the presence of diBerent temperature gradients and that explain a number of experiments made on ground in preparation of an experiment to be carried out in microgravity on a Sounding Rocket. Numerical results clarify questions of interest for solidiAcation processes. In particular, they may provide some explanation for the minority phase separation during solidiAcation due to the Marangoni repulsive forces that are experienced in the vicinity of the solidiAcation front and that result in pushing of the liquid droplet towards the hot site of the sample. ? 2002 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
The prevention of liquid drops coalescence of silicone oil in air originally experienced during the Mission D-2 with the Fluid Physics Module [1] has motivated extended numerical and experimental activities on ground [2–5] also in preparation of experiments to be hosted on microgravity platforms [6]. The cited references report the details of the ground experimentation that were in very good agreement with numerical results. Here we simply summarize the Andings, the numerical correlations and the extension to problems of relevant interest for Material Science and microgravity. A semi-spherical drop of a liquid is formed as a hanging drop from a circular disk. Suppose that the liquid wets the surface of a certain solid and suppose that the drop is brought in contact with this surface; as soon as the liquid contacts the surface (as expected), it spreads over it. Suppose,
†Based on paper IAF-99-J.4.05 presented at the 50th International Astronautical Congress, 4 –8 October 1999, Amsterdam, The Netherlands. ‡Corresponding author. Tel.: +39-081-768-2360; fax: +39081-593-2044. E-mail address: [email protected] (R. Monti). 789
now that the surface tension of the liquid decreases with the temperature and that the circular disk is held at a temperature TD ¿ TS (TS being the solid surface temperature), at these conditions spreading (or wetting) is prevented as long as TD is sufAciently higher than TS and the drop is not too “strongly” pushed against the surface. For instance, for a number of liquids (silicone oil, kerosene, alcohols) a NT = TH − TS ¿ 0, of ◦ the order of 3–4 C (for drops of few millimeter diameter), is suPcient to prevent wetting even if the drop is pushed against the surface [5]. This very intriguing phenomenon was explained by numerical simulations by assuming that the Marangoni Qow, generated at the drop surface, entrains a Alm of air between the (deformed) drop and the solid surface and prevents an intimate contact between the liquid and the solid surface molecules [5]. Similarly, two drops on two disks held at different temperatures will not coalesce if not pushed very hard against each other [7,8]. This behaviour is similar to the one observed when bubbles or drops coalescence is retarded by solutal Marangoni eBect induced by concentration gradients during mass transfer in liquid–liquid systems [9]. In contrast to other proposed explanations [10], the numerical simulation was able to explain this
790
R. Monti et al.
phenomenon also when two opposite eBects take place, i.e. the entrainment of air by the upper (hot) drop and the detrainment of air, caused by the very same Marangoni eBect, along the lower (cold) drop. As explained in [2– 4] the reason for the coalescence prevention is the diBerence between the two eBects caused by the temperature distribution: the entrainment eBect is larger than the opposite one. In [11] it was shown that, for similar reasons, if the temperature diBerence is suPciently large, a liquid drop can be completely immersed in a pool of the same liquid. What has been found for semi-spherical hanging drops was extended to full drops initially at a temperature TD larger than the solid surface temperature (TS ): the drop does not spread over the surface but tends to roll over the surface until thermalization (or when its temperature becomes almost equal to the surface temperature [6]). Similarly drops “Qoating” over a pool (of the very same liquid) have been observed [12] until thermalization occurs. All what found for drops in air was also found for drops submerged in a liquid matrix. The situation is now more complex but the explanation is basically the same. If half drop of a liquid “LD ” is formed on a circular disk immersed in a matrix of liquid “LM ” (LD and LM are immiscible) then we have the same eBect as above if a temperature gradient is established in the matrix. Again, the same eBect is being experienced by a full spherical drop in a liquid matrix that is close to a solid wall (representing a solidiAcation front) in presence of a strong temperature gradient. This speciAc case, of greater relevance for the solidiAcation of immiscible alloys, will be analysed, both numerically and experimentally, in what follows. 2. RELEVANCE IN THE FIELD OF MATERIAL SCIENCE
As mentioned before, the authors believe that the problem of wetting and coalescence prevention of drops in an external liquid matrix, up to now unexplored, is of particular interest in the Aeld of Material Science. Liquid–liquid immiscible alloys (e.g. In–Al, Al–Pb, etc.) formed by two melts at diBerent compositions can be fully exploited only if a uniform Anely dispersed minority phase can be preserved in the solid state (e.g. materials for self-lubricating bearings with a soft dispersed phase in a high mechanical strength matrix; Lead or Indium dispersions in an aluminium matrix are the most promising). In the Earth’s gravitational
Aeld the usual diBerences in density cause rapid separation of the alloy components through sedimentation or Qoatation. Material Science experiments dealing with metallic alloys processing in microgravity resulted in a number of disappointing results [13,14]. The anticipated structure of uniformly distributed intrusions of the minority phase in the external matrix after solidiAcation was not found. In many cases, a separation occurred and the minority phase accumulated in the “hot” region during the solidiAcation process. This means that the minority phase (initially uniformly dispersed in the majority phase) moved to the region that solidiAed later (one end of the cartridge, for directional solidiAcation or the internal region when cooling takes place at the external walls). Explanation of the microgravity results has been attempted by assuming that liquid drops in a liquid matrix migrate towards the hot region due to Marangoni eBect. The idea being that if the solidiAcation front advances with a velocity larger than the drop migration velocity then it engulfs the drop; vice versa there is a separation of the two materials. Recent Andings on wetting and coalescence prevention and the extended ground experiments performed in the last four years strongly suggest to study all the phenomena that occur during alloys solidiAcation (migration, dissolution, wetting and coalescence of drops). Of particular interest is the wetting prevention that could explain the drops pushing by solidiAcation fronts that may be responsible for the phases separation (as observed in the solid). 3. MARANGONI FORCE ON THE DROPLET
This paragraph reports the results of numerical simulations for the evaluation of the Marangoni forces exerted on free droplets interacting with a solid surface at diBerent temperature. The analysis is performed in terms of the forces acting on drops that are either motionless in a liquid matrix or that migrate at constant velocity (e.g. with the Young velocity in an inAnite medium). At each vertical position of the drop and for a Axed NT between the upper and lower wall, the force FM , due to Marangoni Qows, has been computed by integrating the surface forces along the drop surface: nv ds − pn ds; (1) FM = S
S
where n is the unit vector normal to the drop surface.
Marangoni force on drops
791
This force consists of two parts: (a) the integral of the viscous force, due to the viscous stress tensor: = 2 (∇V )s0 ;
(2)
where the subscript (0) and the superscript (s) denote the traceless symmetric part of the velocity gradient tensor; (b) the resultant of the normal surface forces due to the pressure along the drop, due to the motion of the liquid in the matrix and to the Archimedean eBect. The thermoQuidynamic Aelds, inside and outside of the drop, is computed for Anite dimension containers in an accurate way. A drop of liquid (LD ) is immersed in a matrix of liquid (LM ) at constant temperature. If the drop density is larger than that of the matrix then the drop falls to the bottom and if the matrix liquid wets the bottom surface more than the drop liquid then the drop will sit on the bottom and its net weight (Wg ) will be balanced by the surface reaction forces (FR ): FR = Wg = WD −
D3 L g; 6
(3)
WD being the drop weight. If a temperature gradient is established in the matrix, upwards along the vertical (cold bottom), then a Marangoni force is created due to the viscous tangential forces (S ) directed along the surface (s) and to the pressure (p), directed normal to the surface (n). The equilibrium along the vertical (FM is directed upwards, toward the hot side) now reads FR = W g − FM :
(4)
The computations made have shown an eBect similar to the “ground eBect” that appears on moving objects near the ground. Indeed when the drop is located near a solid surface the pressure forces on the drop (that are small far away from the wall) become large compared with the viscous forces (see Fig. 1). This results in a force on the drop that pushes the drop upward far from the cold wall and might explain the reason for the separation of minority phases during solidiAcation of immiscible alloys, that tends to migrate towards the hot side (that solidiAes last). Suppose now that we increase the temperature gradient (∇T ) in the liquid matrix, then the drop will start Qoating when FR ≡ 0 for a value of ∇T = (∇T )F .
Fig. 1. Computed stream-lines (a), isotherms (b) and pressure contours (c) for a droplet of Fluorinert FC43 in silicone oil 3 cS (D = 2 mm; ∇T = 20 K=cm; h = 100 m).
The numerical computations have shown that, at a Axed ∇T , FM is an increasing function of the drop diameter and an inverse function of the distance between the drop and the solid surface (Fig. 2). An evaluation of the (upward) migration velocity (VM ) could be made by assuming that the Drag force is that pertaining to the viscous Stokes regime VM =
FM − Wg : 3 D
(5)
This value of the velocity should be accurate in the limit of low speed or small diameters (or Ma = (T NT= )D → 0) i.e. when the drop migration does not disturb the Qow Aeld. If h is the distance between the drop surface and the bottom wall, the total Marangoni force acting
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R. Monti et al. Fc43 - Silicone oil 3cs 1.00E-1
In - Al
F (dyne)
1.00E-2
1.00E-3
∆ T/L=2 K/mm 1.00E-4
1.00E-5
1.00E-6 0
200
400
600
800
1000
Drop diameter (µm) 0. 12
∆ T/ L=2 K /mm
F (dyne)
D=2 mm
0. 08
Fc43 - Silicone oil 3cs
Fig. 4. Computed stream-lines (a), isotherms (b) and pressure contours (c) for a droplet in a liquid matrix between two walls at diBerent temperatures.
0. 04
In - A l
0. 00 0
40
80
120
160
200
h (µm)
Fig. 2. Computed force on the droplet versus drop diameter (D) and distance from the wall (h).
4.0
F/Fm
3.0
the limit for h=D → ∞ corresponds to the classical Young solution. Similar eBects can be found for a drop in a liquid matrix between two solid walls maintained at different temperatures. In this case the liquid around the drop is detrained, due to the Marangoni eBect, from the liquid layer between the drop and the hot wall (upper) and entrained between the drop and the lower wall. The pressure is reduced in the upper part and increased in the lower one, resulting in an overall force pushing the drop in the direction of the temperature gradient (Fig. 4).
Ma =30
2.0 Ma < 1
4. LABORATORY TESTS
Ma = 15
1.0 0.0 0.00
0.10
0.20
0.30
0.40
0.50
h/R
Fig. 3. Non-dimensional plot of the computed force (scaled by the Marangoni force FM ) versus h=R, for diBerent Marangoni numbers.
on the drop FM is a decreasing function of the distance and an increasing function of the Marangoni number (see the non-dimensional plot of Fig. 3),
A drop (D = 120 m) of Fluorinert FC 43 (1 cS, = 1700 kg=m3 )) in a matrix of Silicone Oil of 3 cS ( = 910 kg=m3 ) was conAned between two plates at diBerent controlled temperature (Tu ; Tb ). A temperature diBerence between the upper disk (Tu ) and the bottom one (Tb ) Tu − Tb = NT ¿ 0 was established by heating the upper disk: when the average temperature gradient across the cell (NT ) is above a certain value then the drop is pushed away from the wall and Ands its stable equilibrium condition at a distance h such that FM (h) = Wg =
3 D (D − L )g: 6
(6)
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Table 1. Physical properties of the liquids investigated Fluid properties
Silicone oil 3 cS
Fluorinert FC43
Ratios of properties silicone oil=FC43
Density (g=cm3 ) Dynamic viscosity (centipoise) Kinematic viscosity (cm2 =s) Interface tension (dyne=cm)
0.89 2.67 0.03 18.8
1.88 5.35 0.028 16.6
Surface tension derivative T (dyne=(cm K))
0.066
0.088
0.47 0.5 1.07 Interface tension 5.35 Interface tension derivative 0.038
1:16 × 10−3
1:2 × 10−3
0.96
2:6 × 10−4 6:27 × 10−4 47.8
1:5 × 10−4 3:39 × 10−4 82.6
1.66 1.85 0.57
Thermal expansion coePcient T (K −1 ) Thermal conductivity coePcient [cal=(cm s K)] Thermal diBusivity (cm2 =s) Prandtl number Pr = =
When further increasing |∇T | (and FM ) the drop moves further up until h=D becomes large and if FM (∞) ¿ Wg , then the drops starts migrating along the temperature gradient. The liquid properties used in the numerical computations are summarized in Table 1. The surface tensions and the interface tension at the Fluorinert–silicone oil interface have been measured at diBerent temperatures using a tensiometer [15].
The numerical predictions (see Fig. 5) were indeed conArmed by experiments. The drop, initially sitting on the bottom, immediately jumped at the equilibrium position (that moves upward with increasing ∇T ). Figure 6 shows the photographs of the experiments and Fig. 7 shows the drop position and velocity versus time. What was found numerically and experimentally for high Prandtl numbers liquids (e.g. silicone oils,
Fig. 5. Computed streamlines, isotherms and pressure distributions from the numerical simulation of a drop (D = 120 m) of Fluorinert FC 43 in a matrix of Silicone Oil of 3 cS conAned between two plates at diBerent temperatures, changing the distance from the lower and upper walls.
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For a typical case of In–Al alloys the value of FM (D; ∇T; h) was computed that show the very same trend as for large Prandtl number Quids. In Fig. 8 the numerical results obtained for a drop of Quorinert FC43 in a matrix of silicone oil 3 cS are compared to those corresponding to the case In–Al. It is interesting to note that the behaviour at large and small Prandtl numbers are diBerent for relatively large drops (diameter of the order of 1 mm), whereas for suPciently small drops (diameter of the order of 100 m) the Qow and temperature behaviour are dominated by diBusion eBects, so that the two systems exhibit the same behaviour. This consideration allows to experiment in rather high Prandtl number, transparent matrices and to extend the results to cases of greater applicative interest (immiscible liquid metals).
5. CONCLUSIONS
Fig. 6. Photographs of the laboratory experiment (a, b, c correspond to an increasing temperature gradient; d shows the time history of the temperatures).
Fluorinert, ethyl alcohol, etc.) can be extended to the cases of metal melts (very low Prandtl numbers) in the presence of strong temperature gradient at solidiAcation fronts. Experimentally it is diPcult to visualize the drop position in melts of real interest in Material Science and only numerical runs have so far be performed.
Numerical simulations and the preliminary laboratory experiments performed in this work show that Marangoni eBects are responsible for a pushing (or attracting) force on a droplet interacting with a solid wall in the presence of a temperature gradient. If the temperature of the wall is smaller than the drop temperature (as for the case of a solidiAcation front) repulsive eBects arise that prevent the wetting. Similarly, during solidiAcation processes, Marangoni eBects play a role in the solidiAcation front=drops interaction by favouring the pushing (or, equivalently, by preventing engulfment=entrapment of liquid drops). Laboratory tests with a drop of Fluorinert in a matrix of silicone oil, conAned between two plates at diBerent controlled temperatures, have conArmed the numerical predictions. When the average temperature gradient across the cell is above a certain value the drop, which is heavier than the surrounding liquid matrix, is pushed away from the wall and Ands its stable equilibrium condition at a distance such that the pushing force, due to the Marangoni eBect, balances the net weight of the drop. Further experimental and numerical studies are in progress along this line. Acknowledgements—This work has been partially supported by the European Space Agency (ESA) and by the Italian Space Agency (ASI)
Marangoni force on drops
795
Fig. 7. Experimental results for a drop of FC43 in Silicone Oil 3 cS; D = 120 mm.
Fig. 8. Comparison of computed streamlines and isotherms (grey contains) for the two cases: (1) drop of FC43 in silicone oil; (2) drop of In in Al, for two diBerent drop diameters. REFERENCES
1. Monti, R. and Dell’Aversana, P., Microgravity experimentation in non coalescing systems. Microgravity Quarterly, 1994, 4, 123. 2. Monti, R. and Savino, R., Numerical model of non coalescence liquid drops and correlation with experimental results. Microgravity Quarterly, 1996, 6, 102. 3. Monti, R. and Savino, R., Correlation between experimental results and numerical solutions of the Navier–Stokes problem for two non coalescing liquid drops with Marangoni eBect. Physics of Fluids, 1997, 9, 260. 4. Savino R. and Monti R., Modelling of non coalescing liquid drops in the presence of thermocapillary convection. Meccanica, 1997, 32, 115. 5. Monti R., Savino R. and Tempesta S., Wetting prevention by thermal Marangoni eBect. European. Journal of Mechanics B=Fluids, 1998, 17, 51.
6. Monti R., Savino R., Lappa M. and Fortezza R., Behavior of droplets on solid surfaces in presence of Marangoni eBect. ESA-SP-433, 1999, p. 225. 7. Dell’Aversana P., Banavar J.R. and Koplik J., Suppression of coalescence by shear and temperature gradients. Physics of Fluids, 1996, 8, 15. 8. Dell’Aversana P., Tondodonato V., Carotenuto L., Suppression of coalescence and wetting: the shape of interstitial Alm. Physics of Fluids, 1997, 9, 2475. 9. Groothuis H., Zuiderweg F.J., InQuence of mass transfer on coalescence of drops. Chemical Engineering Science, 1960, 12, 288. 10. Bratukin, Yu. K. and Gershuni, G. Z., On condition of coalescence of drops in the presence of thermocapillary convection. Microgravity Quarterly, 1994, 4, 193. 11. Monti, R. and Savino, R., Lappa M. and Tempesta, S., Behavior of drops in contact with pool surfaces
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of diBerent liquids. Physics of Fluids, 1999, 10, 2786. 12. Monti, R. and Savino, R., Coalescence and wetting prevention by Marangoni eBect. In Foams and Films, ed. D. Weaire and J. Banhart, Verlag, Bremen 1999, p. 25. 13. Andrews, J. B., SolidiAcation of Immiscible Alloys. In Immiscible Liquid Metals and Organics, ed.
L. Ratke. Obernrsel Germany, DGM, Information gesellschaft-Verlag, 1993, p. 199. 14. Ratke, L., Korekt, G. and Drees, S., SolidiAcation of immiscible alloys. ESA-SP-385, 1996, p. 247. 15. Alterio G., Ph.D. thesis in Aerospace Engineering, 1999.