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The coalescence of liquid drops
J O U R N A L OF COLLOID S C I E N C E
11, 391-397 (1956)
THE COALESCENCE OF LIQUID DROPS M. Linton and K. L. Sutherland Division of h~dustrial Chemistry, C.S.[.R.O., Melbourne, Australia Received April 13, 1956 ABSTRACT Liquid drops floating on a surface of the same liquid were studied using high speed p h o t o g r a p h y . T h e coalescence of a drop with the u n d e r l y i n g surface or with a n o t h e r drop on collision is delayed by the air film t r a p p e d between the opposing surfaces which deform on a p p r o a c h i n g each other. A drop remains in contact with the u n d e r l y i n g surface for a time p r o p o r t i o n a l to its size. When two floating drops collide t h e y bounce or coalesce after a time proportional to the cube of the size of the smaller drop. Drops of d i a m e t e r ~maller t h a n 0.05 cm. coalesce readily with o t h e r drops, whereas those larger t h a n 0.1 cm. in d i a m e t e r are more likely to bounce. These results have a bearing on the coalescence of colliding jets, mist, and raindrops. INTRODUCTION
The phenomenon of drops of a liquid floating on a surface of the same liquid was described in ]88t by Reynolds (1), who observed raindrops remaining on a dust-free water surface for some seconds before coalescing with the underlying water. Surface-active agents cause floating drops to last longer, and hence we suggested that they would hinder coalescence of droplets in a cloud. Telford (private cormnunication ~ tested this suggestion in a wind tunnel, but. found that small drops formed from solutions Of surface-active agents coalesced just as rapidly as drops of pure water. We sought the reason for this apparent anomaly by a study of coalescence processes. EXPERIMENTAL
The solution used to form floating drops was 0.l % w / v cationic detergent (I.C.I. "Cetav]on" containing 75% cetyl trimethyl ammonium bromide) in distilled water. Qualitatively this solution behaved like a wide variety of sprayable liquids. A fine jet which broke into drops was played at an angle of about 30 ° at a point off-center on a surface of the same solution contained in a glass vessel 3 cm. in diameter. The drops bounced off the surface and off the wall and then floated individually or in the form of a drop raft (Plate I, A). The drops were photographed from above at film speeds up to 2700 frames/see., the speed being estimated from a timing trace. 391
392
M. LIN~TOI~ AND K. L. SUTHERLAND
PLATE I. Selections from frames of film (at ca. 2000 frames/sec.) of drops of " C e t a v ]on" solution, floating on the same solution, individually, and in a p p a r e n t c o n t a c t in a drop raft. Scales are as shown. The numbers m a r k e d on the frames are the times in milliseconds from frame A. A : Overall view of vessel, jet entering from top left and h i t t i n g the surface near X . A small drop "a" (A-C) coalesces w i t h t h e surface, forming a wave p a t t e r n (C). A m e d i u m size drop "b" bounces from a larger d r o p (A and B), and t h e n from a smaller drop (A-C). Drop "b" collides w i t h a larger drop at t h e edge of the r a f t (D) and coalesces w i t h it, showing r a p i d d e v e l o p m e n t of necks (E a n d F), a small p r o t u b e r a n c e (G a n d H)~ a n d oscillation (I).
COALESCENCE OF LIQUID DROPS
393
The coalescence of a drop with the underlying surface was revealed by a wave pattern (Plate I, C). Occasionally the drop coalesced in stages (cf. (2)), a secondary drop being ejected a few millimeters into the air. Drops colliding with each other remained in contact for a period, the time of apparent contact, and then separated or bounced ("b," Plate I, A-C) or coalesced ("b," Plate I, D-I). The onset of coalescence was shown by a "neck" which widened within a few milliseconds ("b," Plate I, E-G) causing oscillations ("b," Plate I, I) the amplitude and period of which increased with the size of the drops involved. During coalescence protuberances often formed (Plate I, G and H), and occasionally one would detach as a secondary drop. The life of a drop is measured from its appearance on the surface to the appearance of the wave pattern produced by its disappearance and could usually be determined to :t:5 msec. For bounce or coalescence on collision the error in the time of contact was about 1 msec. The drop diameters were measured to 4-0.01 cm. bY comparison with the known size of the vessel. Since drop sizes ranged from 0.02 to 0.24 cm. there was a considerable error in measuring the smaller drops. RESULTS
Figure 1 shows the time of apparent contact (t in milliseconds) for coalescence between a floating drop and the free liquid surface in relation to the drop diameter (d in centimeters). The best linear fit on a double logarithmic plot is: t = 2 X 1 0 8 d °'Ts.
[1]
t = 4 × 103 d
[2]
However, a simpler relation:
is consistent with the results. The times of apparent contact for mutual bounce and coalescence between two drops were found to depend on the diameter of the smaller drop as shown in Fig. 2. A relation which represents the results is: t = 1.8 × 104d 3.
[3]
These results, as shown by the two curves in Fig. 2, strikingly bear out the visual observations that drops less than 0.1 cm. in diameter coalesce with one another much more rapidly than with a free liquid surface. In qualitative accord with relations [1]-[3] are the visual observations (1, 3, 4) that single drops a few millimeters in diameter float for about 2 sec. Some of the larger drops shown in Plate I were present throughout the 2.5 sec. duration of the film. The scatter of the experimental results indicates that the errors of
394
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Fie. 2. Bounce and coalescence between colliding drops floating on the surface.
Variation of apparent contact time with the size of the smaller drop involved. measurement are outweighed by uncontrolled factors such as a variation in velocity of impact and oscillations of the surface or of the drops. Figure 3 shows that in mutual collision drops smaller than 0.05 cm. in diameter are more likely to coalesce than are drops larger than 0.10 cm. The larger drops often bounce many times. The increased probability of coalescence with decrease in drop size seems to be related to the decrease in contact time as expressed by relation [3]. The above observations agree with the high percentage of coalescences observed by Telford et al. (5)
COALESCENCE 10"
~
OF '
l BouNcE
LIQUID
DROPS
395
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n
ALESCENCE 8 I n II! ,,!,,,,,! 0
0-05 0"10 0'15 DROP DIAMETER,cm.
0.20
0-25
F~G. 3. The dependence of the frequency of bounce and coalescence on the size of the smaller of two colliding drops.
for a mist of 0.015 cm. water drops, and also for drops of "Cetavlon" solution (private communication). DISCUSSION
The Mechanism of Floating Drops N e w t o n ' s rings can b e seen b e t w e e n t h e floating d r o p s a n d t h e u n d e r l y i n g liquid surface (6). T h e s e i n d i c a t e a n air g a p of 0.1-1.0 ~ in thickness.
That an air film can be formed is shown by the observation of Riedel (7) that drops of soap solution falling from a sufficient height can penetrate a plane liquid surface and remain completely surrounded by an air film of thickness 6 # for a drop 0.3 cm. diameter. We observed this same phenomenon with "Cetavlon" solution. Small air bubbles are also trapped in the underlying liquid or the remaining drop after coalescence. By replacing air by other media, including viscous oils, Mahajan found that the life of floating drops increased more or less linearly with the viscosity of the surrounding medium (8). By means of interference rings, Deryaguin and Prokhorov (9) studied an air gap between two drops of a volatile liquid pressed against each other. A stream of unsaturated air passing around the drops caused the diffusion of air and vapor through the gap sufficient to prevent coalescence indefinitely. In some recent work they showed that the vapor saturation also played an appreciable although less important part in the coalescence of moving water drops and drop mists (9). We considered the application, to this problem, of two simple models used by Elton (i0) in his studies of the approach of a gas bubble in a liquid to a solid surface. In the first model a sphere approaches a plane in avis-
396
M. L I N T O N A N D K . L. S U T H E R L A N D
cous medium and this gives a calculated rate of approach much greater than the observed rate. When floating drops are obtained the underlying liquid is depressed and the drops are flattened. Thus the approach of the two opposing surfaces might resemble that of two curved parallel discs. If we take two plane parallel discs as equivalent for our second model, the time t to approach from an initial separation hi cm. to a final separation h2 is (10): t -
:3TroD4( 1
1)
64rag ~ -
~
[4]
'
where D is the disc diameter (in our ease the diameter of the equivalent deformed area), v the viscosity of the air between the discs (1.8 X 10-~ poise), mg is the force on the discs (in our case the weight of the drop). If h2 << hi, the 1/h~ term can be neglected. If D is proportional to d, relation [4] agrees in form with relation [2]. With the assumption of plausible values of D and h, the calculated times can be made to agree with the experimental. Hence the disc model is at least plausible. There is experimental evidence that the flow of entrapped air between the two liquid surfaces does not alone control the stability of the floating drops. We have confirmed that dust and electric charges hinder or completely prevent the formation of floating drops. The increased stability of floating drops with decrease in surface tension is ascribed to the increased deformation of the drops and underlying surface and consequently to the larger area of the air gap. A monolayer of insoluble surface-active material also increasesdro p stability, but an excess has the same effect as dust.
The Mechanism of Mutual Bounce and Coalescence of Floating Drops Large drops floating in the form of a drop raft show flattening between drops (Plate I). Again the considerable times for which large drops rest against each other and the rapidity of coalescence after a neck forms suggest that it is an air film between the deformed surfaces which controls coalescence. Thus, qualitatively, the factors discussed above which influence the stability of floating drops will similarly influence mutual coalescence. However, the contact time depends on the forces between the drops and hence on the slope of the surface surrounding the drops. The slope probably depends on the experimental arrangement, the number and size of drops constituting the raft as well as the surface tension and density of the liquid.
Bounce and Coalescence of Colliding Jets From the above results we would expect, as Lord Rayleigh found (11), that small drops from two jets which collide in the air would rapidly
COALESCENCE OF LIQUID DROPS
397
coalesce and that large drops would bounce. Some factors affecting floating drops influence colliding jets similarly. Thus soluble surface-active agents facilitate bounce but dust and electric charges prevent bounce (ll). CONCLUSIONS
The coalescence of a drop with a free surface or with another drop on collision is delayed by the air film trapped between the opposing surfaces which deform on approaching each other. The delay increases with the size of the drop because of the increased volume of air trapped. The delay is such that on mutual collision floating drops of 0.1 cm. diameter readily bounce but drops of 0.05 cm. are more likely to coalesce. Drops colliding in jets behave similarly. The results should have a bearing on the coalescence of rain drops and mists of the size range studied, i.e., > 0.01 cm. ACKNOWLEDGMENT We gratefully acknowledge the assistance of F. D. Lugton with the high-speed photography and preparation of Plate I. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
REYNOLDS~0., Chem. News 44, 211 (1881). WARK, I. W., AND COX, A. B., Nature 136, 182 (1935). MAH•JAN, L. D., Kolloid-Z. 69, 16 (1934). :KATALINIC,M., Z. Physik 38, 511 (1926); Nature 19.7, 627 (1931); ibid. 136, 916 (1935). TELFORD, J. W., THORNDIKE, N. S., AND BOWEN, E. G., Quart. J. Roy. Meterol. Soc. 81, 241 (1955). MAHAJAN,L. D., Z. Physik 84, 676 (1933). RIEDEL, W., KolIoid-Z. 83, 31 (1938). MAHAJAN,L. D., Phil. Mag. [7], 10, 383 (1930). P~ox~o~ov, P., Discussions Faraday Soc. 18, 41 (1954). ELTON, G. A. H., Proc. Roy. Soc. (London) A194, 259 (1948). LORO RAYLEIGH, Proc. Roy. Soc. (London) 27, 406 (1879); ibid. 29, 71 (1879); ibid. 34, 130 (1882).
11, 391-397 (1956)
THE COALESCENCE OF LIQUID DROPS M. Linton and K. L. Sutherland Division of h~dustrial Chemistry, C.S.[.R.O., Melbourne, Australia Received April 13, 1956 ABSTRACT Liquid drops floating on a surface of the same liquid were studied using high speed p h o t o g r a p h y . T h e coalescence of a drop with the u n d e r l y i n g surface or with a n o t h e r drop on collision is delayed by the air film t r a p p e d between the opposing surfaces which deform on a p p r o a c h i n g each other. A drop remains in contact with the u n d e r l y i n g surface for a time p r o p o r t i o n a l to its size. When two floating drops collide t h e y bounce or coalesce after a time proportional to the cube of the size of the smaller drop. Drops of d i a m e t e r ~maller t h a n 0.05 cm. coalesce readily with o t h e r drops, whereas those larger t h a n 0.1 cm. in d i a m e t e r are more likely to bounce. These results have a bearing on the coalescence of colliding jets, mist, and raindrops. INTRODUCTION
The phenomenon of drops of a liquid floating on a surface of the same liquid was described in ]88t by Reynolds (1), who observed raindrops remaining on a dust-free water surface for some seconds before coalescing with the underlying water. Surface-active agents cause floating drops to last longer, and hence we suggested that they would hinder coalescence of droplets in a cloud. Telford (private cormnunication ~ tested this suggestion in a wind tunnel, but. found that small drops formed from solutions Of surface-active agents coalesced just as rapidly as drops of pure water. We sought the reason for this apparent anomaly by a study of coalescence processes. EXPERIMENTAL
The solution used to form floating drops was 0.l % w / v cationic detergent (I.C.I. "Cetav]on" containing 75% cetyl trimethyl ammonium bromide) in distilled water. Qualitatively this solution behaved like a wide variety of sprayable liquids. A fine jet which broke into drops was played at an angle of about 30 ° at a point off-center on a surface of the same solution contained in a glass vessel 3 cm. in diameter. The drops bounced off the surface and off the wall and then floated individually or in the form of a drop raft (Plate I, A). The drops were photographed from above at film speeds up to 2700 frames/see., the speed being estimated from a timing trace. 391
392
M. LIN~TOI~ AND K. L. SUTHERLAND
PLATE I. Selections from frames of film (at ca. 2000 frames/sec.) of drops of " C e t a v ]on" solution, floating on the same solution, individually, and in a p p a r e n t c o n t a c t in a drop raft. Scales are as shown. The numbers m a r k e d on the frames are the times in milliseconds from frame A. A : Overall view of vessel, jet entering from top left and h i t t i n g the surface near X . A small drop "a" (A-C) coalesces w i t h t h e surface, forming a wave p a t t e r n (C). A m e d i u m size drop "b" bounces from a larger d r o p (A and B), and t h e n from a smaller drop (A-C). Drop "b" collides w i t h a larger drop at t h e edge of the r a f t (D) and coalesces w i t h it, showing r a p i d d e v e l o p m e n t of necks (E a n d F), a small p r o t u b e r a n c e (G a n d H)~ a n d oscillation (I).
COALESCENCE OF LIQUID DROPS
393
The coalescence of a drop with the underlying surface was revealed by a wave pattern (Plate I, C). Occasionally the drop coalesced in stages (cf. (2)), a secondary drop being ejected a few millimeters into the air. Drops colliding with each other remained in contact for a period, the time of apparent contact, and then separated or bounced ("b," Plate I, A-C) or coalesced ("b," Plate I, D-I). The onset of coalescence was shown by a "neck" which widened within a few milliseconds ("b," Plate I, E-G) causing oscillations ("b," Plate I, I) the amplitude and period of which increased with the size of the drops involved. During coalescence protuberances often formed (Plate I, G and H), and occasionally one would detach as a secondary drop. The life of a drop is measured from its appearance on the surface to the appearance of the wave pattern produced by its disappearance and could usually be determined to :t:5 msec. For bounce or coalescence on collision the error in the time of contact was about 1 msec. The drop diameters were measured to 4-0.01 cm. bY comparison with the known size of the vessel. Since drop sizes ranged from 0.02 to 0.24 cm. there was a considerable error in measuring the smaller drops. RESULTS
Figure 1 shows the time of apparent contact (t in milliseconds) for coalescence between a floating drop and the free liquid surface in relation to the drop diameter (d in centimeters). The best linear fit on a double logarithmic plot is: t = 2 X 1 0 8 d °'Ts.
[1]
t = 4 × 103 d
[2]
However, a simpler relation:
is consistent with the results. The times of apparent contact for mutual bounce and coalescence between two drops were found to depend on the diameter of the smaller drop as shown in Fig. 2. A relation which represents the results is: t = 1.8 × 104d 3.
[3]
These results, as shown by the two curves in Fig. 2, strikingly bear out the visual observations that drops less than 0.1 cm. in diameter coalesce with one another much more rapidly than with a free liquid surface. In qualitative accord with relations [1]-[3] are the visual observations (1, 3, 4) that single drops a few millimeters in diameter float for about 2 sec. Some of the larger drops shown in Plate I were present throughout the 2.5 sec. duration of the film. The scatter of the experimental results indicates that the errors of
394
M. LINTON AND K. L. SUTHERL&ND 0"20 -:~
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0"01
,,,
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I
I
,
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I
i
w
i
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I00 200 500 CONTACT TIME, t MILLISEC. (LOGARITHMIC),
I000
FIG. 1. Coalescence with the underlying surface. Variation of apparent contact time with drop size. I '00
o BOUNCE. E
~ 104&_~."~'~ ~
. COALESCENCE. 18
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100 CONTACT TIME, t MILLISEC. (LOGARITHMIC).
n
I
1000
Fie. 2. Bounce and coalescence between colliding drops floating on the surface.
Variation of apparent contact time with the size of the smaller drop involved. measurement are outweighed by uncontrolled factors such as a variation in velocity of impact and oscillations of the surface or of the drops. Figure 3 shows that in mutual collision drops smaller than 0.05 cm. in diameter are more likely to coalesce than are drops larger than 0.10 cm. The larger drops often bounce many times. The increased probability of coalescence with decrease in drop size seems to be related to the decrease in contact time as expressed by relation [3]. The above observations agree with the high percentage of coalescences observed by Telford et al. (5)
COALESCENCE 10"
~
OF '
l BouNcE
LIQUID
DROPS
395
'
n
ALESCENCE 8 I n II! ,,!,,,,,! 0
0-05 0"10 0'15 DROP DIAMETER,cm.
0.20
0-25
F~G. 3. The dependence of the frequency of bounce and coalescence on the size of the smaller of two colliding drops.
for a mist of 0.015 cm. water drops, and also for drops of "Cetavlon" solution (private communication). DISCUSSION
The Mechanism of Floating Drops N e w t o n ' s rings can b e seen b e t w e e n t h e floating d r o p s a n d t h e u n d e r l y i n g liquid surface (6). T h e s e i n d i c a t e a n air g a p of 0.1-1.0 ~ in thickness.
That an air film can be formed is shown by the observation of Riedel (7) that drops of soap solution falling from a sufficient height can penetrate a plane liquid surface and remain completely surrounded by an air film of thickness 6 # for a drop 0.3 cm. diameter. We observed this same phenomenon with "Cetavlon" solution. Small air bubbles are also trapped in the underlying liquid or the remaining drop after coalescence. By replacing air by other media, including viscous oils, Mahajan found that the life of floating drops increased more or less linearly with the viscosity of the surrounding medium (8). By means of interference rings, Deryaguin and Prokhorov (9) studied an air gap between two drops of a volatile liquid pressed against each other. A stream of unsaturated air passing around the drops caused the diffusion of air and vapor through the gap sufficient to prevent coalescence indefinitely. In some recent work they showed that the vapor saturation also played an appreciable although less important part in the coalescence of moving water drops and drop mists (9). We considered the application, to this problem, of two simple models used by Elton (i0) in his studies of the approach of a gas bubble in a liquid to a solid surface. In the first model a sphere approaches a plane in avis-
396
M. L I N T O N A N D K . L. S U T H E R L A N D
cous medium and this gives a calculated rate of approach much greater than the observed rate. When floating drops are obtained the underlying liquid is depressed and the drops are flattened. Thus the approach of the two opposing surfaces might resemble that of two curved parallel discs. If we take two plane parallel discs as equivalent for our second model, the time t to approach from an initial separation hi cm. to a final separation h2 is (10): t -
:3TroD4( 1
1)
64rag ~ -
~
[4]
'
where D is the disc diameter (in our ease the diameter of the equivalent deformed area), v the viscosity of the air between the discs (1.8 X 10-~ poise), mg is the force on the discs (in our case the weight of the drop). If h2 << hi, the 1/h~ term can be neglected. If D is proportional to d, relation [4] agrees in form with relation [2]. With the assumption of plausible values of D and h, the calculated times can be made to agree with the experimental. Hence the disc model is at least plausible. There is experimental evidence that the flow of entrapped air between the two liquid surfaces does not alone control the stability of the floating drops. We have confirmed that dust and electric charges hinder or completely prevent the formation of floating drops. The increased stability of floating drops with decrease in surface tension is ascribed to the increased deformation of the drops and underlying surface and consequently to the larger area of the air gap. A monolayer of insoluble surface-active material also increasesdro p stability, but an excess has the same effect as dust.
The Mechanism of Mutual Bounce and Coalescence of Floating Drops Large drops floating in the form of a drop raft show flattening between drops (Plate I). Again the considerable times for which large drops rest against each other and the rapidity of coalescence after a neck forms suggest that it is an air film between the deformed surfaces which controls coalescence. Thus, qualitatively, the factors discussed above which influence the stability of floating drops will similarly influence mutual coalescence. However, the contact time depends on the forces between the drops and hence on the slope of the surface surrounding the drops. The slope probably depends on the experimental arrangement, the number and size of drops constituting the raft as well as the surface tension and density of the liquid.
Bounce and Coalescence of Colliding Jets From the above results we would expect, as Lord Rayleigh found (11), that small drops from two jets which collide in the air would rapidly
COALESCENCE OF LIQUID DROPS
397
coalesce and that large drops would bounce. Some factors affecting floating drops influence colliding jets similarly. Thus soluble surface-active agents facilitate bounce but dust and electric charges prevent bounce (ll). CONCLUSIONS
The coalescence of a drop with a free surface or with another drop on collision is delayed by the air film trapped between the opposing surfaces which deform on approaching each other. The delay increases with the size of the drop because of the increased volume of air trapped. The delay is such that on mutual collision floating drops of 0.1 cm. diameter readily bounce but drops of 0.05 cm. are more likely to coalesce. Drops colliding in jets behave similarly. The results should have a bearing on the coalescence of rain drops and mists of the size range studied, i.e., > 0.01 cm. ACKNOWLEDGMENT We gratefully acknowledge the assistance of F. D. Lugton with the high-speed photography and preparation of Plate I. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
REYNOLDS~0., Chem. News 44, 211 (1881). WARK, I. W., AND COX, A. B., Nature 136, 182 (1935). MAH•JAN, L. D., Kolloid-Z. 69, 16 (1934). :KATALINIC,M., Z. Physik 38, 511 (1926); Nature 19.7, 627 (1931); ibid. 136, 916 (1935). TELFORD, J. W., THORNDIKE, N. S., AND BOWEN, E. G., Quart. J. Roy. Meterol. Soc. 81, 241 (1955). MAHAJAN,L. D., Z. Physik 84, 676 (1933). RIEDEL, W., KolIoid-Z. 83, 31 (1938). MAHAJAN,L. D., Phil. Mag. [7], 10, 383 (1930). P~ox~o~ov, P., Discussions Faraday Soc. 18, 41 (1954). ELTON, G. A. H., Proc. Roy. Soc. (London) A194, 259 (1948). LORO RAYLEIGH, Proc. Roy. Soc. (London) 27, 406 (1879); ibid. 29, 71 (1879); ibid. 34, 130 (1882).