Langmuir 2008, 24, 6395-6398
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Delayed Coalescence Behavior of Droplets with Completely Miscible Liquids Hans Riegler* and Paul Lazar Max-Planck-Institut fuer Kolloid and Grenzflaechenforschung, Am Muehlenberg, D-14476 Potsdam, Germany ReceiVed February 27, 2008. ReVised Manuscript ReceiVed May 8, 2008 Because of capillary forces, sessile droplets usually fuse instantaneously after contact. We find however a delay of the droplet fusion by many seconds if the droplets consist of different but completely miscible liquids. After the initial contact, the main bodies of the droplets remain separated, connected only through a shallow conduit with a flow from the low to the high surface tension liquid. Sporadically, this connecting film can thicken with turbulent or pulsating flows. The droplets will finally fuse when the flow has sufficiently reduced the difference in composition and surface tension. We present calculations which explain this delayed droplet fusion with the compensation of the fusion-promoting capillary pressure by a droplet-separating dynamic pressure caused by the flow between the droplets. Droplets with high contact angles fuse instantaneously. In this case, no separation-stabilizing dynamic pressure can build up because the interdroplet flow becomes turbulent.
The manipulation of tiny portions of fluid is a future key technology (“microfluidics”).1,2 An important process in microfluidics is the mixing of fluids, for example, by fusion of small droplets. Droplet fusion is driven by interfacial forces because two droplets have a larger interfacial area than a single droplet of the same volume, and because the contact region where two droplets get in touch has a curvature with a fusion-promoting capillary pressure. Therefore, one expects (and usually observes) that droplets fuse instantaneously after first contact. The early processes during the fusion of droplets of the same liquid have been investigated experimentally and theoretically by a number of research groups.3–7 The fusion of sessile droplets in particular were studied by Eggers et al.8 For droplets of different but miscible liquids, the entropy of mixing also favors droplet fusion. Hence, one also expects fast coalescence after droplet contact. However, we find that, under certain conditions, sessile droplets of different miscible liquids do not coalesce rapidly after peripheral contact. The main parts of the two droplets remain separated for quite some time (up to minutes). They are connected with each other only through a shallow conduit. The droplets even seem to push each other away (“chasing droplets”), and occasionally the connecting liquid film shows oscillations (“pulsations”). To the best of our knowledge, this delayed fusion behavior of droplets of miscible liquids has been described up to now only three times in the literature. Clerk-Maxwell mentioned it very briefly already more than a century ago.9 Hardy reports about it in 1920.10 Bangham * To whom correspondence should be addressed. E-mail: Hans.Riegler@ mpikg.mpg.de. (1) Whitesides, G. M. Nature 2006, 442(7101), 368–373. (2) de Jong, J.; Lammertink, R. G. H.; Wessling, M. Lab Chip 2006, 6(9), 1125–1139. (3) Mandre, S.; Feng, J. J. Phys. Fluids 2006, 18(5), 051705. (4) Gilet, T.; Mulleners, K.; Lecomte, J. P.; Vandewalle, N.; Dorbolo, S. Phys. ReV. E 2007, 75(3), 036303. (5) Honey, E. M.; Kavehpour, H. P. Phys. ReV. E 2006, 73(2), 051705. (6) Thoroddsen, S. T.; Etoh, T. G.; Takehara, K.; Ootsuka, N. Phys. Fluids 2005, 17(7), 071703. (7) Wu, M. M.; Cubaud, T.; Ho, C. M. Phys. Fluids 2004, 16(7), L51–L54. (8) Eggers, J.; Lister, J. R.; Stone, H. A. J. Fluid Mech. 1999, 401, 293–310. (9) Maxwell, J. C. In Encyclopedia Britannica, 9th ed.; 1875. Reprinted in The Scientific Papers of James Clerk Maxwell; Niven, W. D., Ed.; Dover: New York, 1965; Vol. II, p 541. (10) Hardy, W. B. Philos. Mag. 1920, 40(236), 201–210.
and Zaweris describe in more detail in 193811 the delayed droplet fusion of various combinations of miscible liquids. However, their experiments were not particularly targeted. They ignored that delayed fusion occurs for volatile as well as nonvolatile liquids. Hence, their vague explanation, which is based on vapor transport,12 is most likely wrong. In addition, they never noticed the influence of the contact angle on the delayed fusion behavior and seemed unaware of the concept of flow driven by surface tension gradients (“Marangoni effect”) and its consequences. Figure 1 shows what happens when droplets of water and acetic acid (concentration: 25 vol %) sitting on a glass substrate with small contact angles get in contact with each other (Figure 1 is a selection of frames from the movie in the Supporting Information). The contact angles for both fluids are in the range of 6-10°13 (advancing contact angles, measured by the sessile drop method). Although acetic acid and water are miscible in all proportions, the droplets do not fuse instantaneously upon contact. Instead, after getting in contact at their perimeters, the acetic acid droplets even push away the water droplet, thereby deforming it considerably (Figure 1, t ) 1 and 2 s). Between the water and acetic acid droplets appears a gap, which is several hundreds of micrometers wide. This gap is bridged by a thin liquid film, which bears a flow of acetic acid into the water droplet. This flow can be concluded from density variations within the water droplet bulk phase indicating the mixing of acetic acid and water as well as from the movement of small dust particles. Interference colors/ fringes indicate that the bridging film has a thickness in the range of micrometers. Quite often, this film/gap between the droplets destabilizes after a few seconds suddenly into a conduit with “macroscopic” thickness (in the range of millimeters), accompanied by turbulent bulk mixing in the region where the two droplets are connected. The chaotic turbulences often organize (11) Bangham, D. H.; Saweris, Z. Trans. Faraday Soc. 1938, 34(1), 0554– 0569. (12) Their explanation is as follows: (p 565) “The phenomenon (of a droplet of one liquid chasing one of a second liquid) is unquestionably connected with the one-sided effects of the vapours on the contact angles, already discussed. The adsorbed film surrounding the drop of the first liquid (which is pursued) induces the spreading of the pursuing drop in its direction”. (13) The experiments with the delayed coalescence were performed with microscope slides as substrates. Untreated slides taken from a freshly opened package are perfect substrates with suitably small contact angles. If they are exposed to ambient air for some time, the contact angle will increase and the droplets will fuse instantaneously.
10.1021/la800630w CCC: $40.75 2008 American Chemical Society Published on Web 06/03/2008
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Figure 1. Time evolution of the interaction between sessile droplets of water and acetic acid after being brought into contact. The substrate is a glass microscope slide. The contact angles for both liquids are 6-10°. The frames are taken from the movie in the Supporting Information.
into regular vigorous pulsations (Figure 1, t ) 3, 4, and 6 s). After a few seconds of vigorous local bulk mixing, the system frequently stabilizes and separates again into two (or more) droplets (Figure 1, t ) 11s), which again are connected by the thin film. The sequence of thin film with laminar flow and thick film with turbulences (pulsating) may repeat itself several times until the system will eventually stabilize either with the droplets
being separated far enough to interrupt their interaction or with a single droplet (or several droplets) containing the final, homogeneous mixture of the two liquids. The delayed droplet fusion is observed for many combinations of liquids that mix at any proportion (e.g., water, aqueous glucose solutions, aqueous glycerol solutions). It also occurs for melts of long chain alkanes of different chain lengths. It can be concluded that under certain
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Figure 3. Geometry for the calculation of the pressure conditions in the neck between two contacting droplets. Figure 2. Schematic drawing on how the dynamic pressure induced by the flow in the neck between the two droplets balances the capillary pressure for a bridging film thickness dc. The dashed lines show the droplet shapes/menisci at the moment of first contact.
boundary conditions it is a general phenomenon for sessile droplets of miscible liquids. We find that delayed droplet fusion only occurs for droplets with low contact angles. Experiments analogous to those shown in Figure 1 that are performed with larger contact angles (e.g., on a more hydrophobic surface) result in instantaneous droplet fusion. For instance, droplets on a polysterene surface (Petri dish) with contact angles of about 60-70° (acetic acid) and 80-90° (water) fuse instantaneously (see the Supporting Information movie, with droplets on a Petri dish made from polystyrene). We found this dependence of the fusion behavior on the contact angle for all the combinations of liquids which we investigated. In particular, the coalescence behavior of droplets of melts of long chain alkanes14 supports the assumption that the contact angle is a key parameter. The contact angle of sessile droplets of long chain alkanes on silica (oxidized silicon wafer surface) increases when the temperature is lowered toward the bulk melting point.15 If two droplets, which consist of melts of alkanes with different chain lengths (e.g., C30H62 and C36H74), are brought into contact at their perimeters, they will either fuse or stay separated,16 depending on the temperature and thus on the contact angles. The time evolution of the local surface topology and of the bulk conditions where the two droplets touch each other is governed by interfacial and hydrodynamic contributions. It is complicated, and its understanding will need extensive experimental and theoretical work. In the following, we will present a first, semiquantitative explanation, which accounts for the essential features of the experimental findings. It is based on two ingredients, namely, (1) a separation-stabilizing dynamic pressure that is induced by the directed laminar flow of liquid through the film between the two droplets and (2) the collapse of this dynamic pressure stabilization due to turbulences in the case of large contact angles. Figure 2 shows how the surface and bulk flow from droplet 1 (left) to droplet 2 (right) create a dynamic pressure which balances the capillary pressure of the connecting neck. Figure 3 shows a simplified geometry of the contact region. It is assumed (14) In fact, we first observed the delayed droplet fusion and the “chasing” droplets with sessile droplets of melts of alkanes. Technically, these experiments were quite demanding. So, we decided to investigate the phenomena with the aqueous solutions which are experimentally easier to control. (15) Merkl, C.; Pfohl, T.; Riegler, H. Phys. ReV. Lett. 1997, 79(23), 4625– 4628. (16) With alkanes, we also observe occasional pulsations as well as chasing droplets.
that the droplet shape and composition are preserved as they were before contact, except for the region between points 1 and 2. We also assume identical contact angles Θ for the two droplets. This is rather inconsistent because the two liquids as well as the two surface tensions γ1 and γ2 are assumed different. Thus, usually the contact angles will also differ. However, identical Θ values greatly simplify the calculations, and the general features of the results will be valid also for different Θ values. A spherical curvature with radius r17 with a minimum thickness d is assumed for the contacting liquid neck:
d)
r(1 - cos Θ) Θ2 ≈ r cos Θ 2
(1)
The surface tension shall change linearly18 from γ1 to γ2 along the length of the spherical segment ∆x between points 1 and 2:
∆x )
4 d Θ
(2)
The surface tension gradient ∆γ/∆x (with ∆γ ) γ1 - γ2) between points 1 and 2 induces a surface flow speed V which depends on the thickness d and the viscosity η:19
V)
∆γ d ∆γ ) Θ ∆x η 4η
(3)
According to eq 3, the surface velocity is a function of the contact angle but independent from d. With ∆γ ) 10 [mN/m], η ) 10-3 [Ns/m2], and Θ ) 0.174 (≈10°), one obtains a flow velocity typical for aqueous solutions of V ≈ 0.44 [m/s]. This flow induces a dynamic “Bernoulli” pressure pB (F is the density):
F F∆γ2 2 pB ) V2 ) Θ 2 32η2
(4)
For aqueous systems (F ) 103 [kg/m3]) and V ≈ 0.44 [m/s], the dynamic pressure is pB ≈ 3000Θ2 or pB≈ 100 [N/m2] for Θ ) 10°. This pB value may be compared to the capillary pressure pc exerted by a surface tension γ at a curvature with radius r:
pc )
2γ r
(5)
For r ) 10-3 [m] and water (γ ) 7 × 10-2 [N/m]), one obtains pc ≈ 144 [N/m2]. This is in the same range as the dynamic (17) In reality, the neck between the two droplets is not nicely spherical. For the case of two droplets of the same liquid, see ref 8. (18) In reality, the gradient will be nonlinear and increase toward the high energy droplet because of the surface flow toward this droplet. (19) Guyon, E.; Hulin, J.-P.; Petit, L.; Mitescu; C. D. Physical Hydrodynamics; Oxford University Press: Oxford, 2001.
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pressure. The surface tension γ in the contact region may be taken as the mean between the two surface tensions γ1 and γ2 (γ ≈ 0.5(γ1 + γ2)). Thus, eqs 1, 4, and 5 yield the conditions at which (contact angles, Θ, and thicknesses, dc) both pressures balance each other:
dc )
32η2γ F∆γ2
(6)
For aqueous solutions with ∆γ ) 10 [mN/m], one obtains dc ≈ 22.5 × 10-6 [m]. This roughly agrees with the experimentally observed thickness of the bridging film. It is quite remarkable that this simplistic theory predicts a stabile configuration/balance between dynamic and capillary pressure. Still, we are fully aware that a more detailed fluid dynamic analysis of the problem is necessary to provide more reliable criteria of the stability of the film. The result of eq 6 can be understood because according to eq 4 the flow induces a dynamic pressure that is independent from the thickness of the bridging film, whereas the capillary pressure is proportional to the reciprocal film thickness (eqs 1 and 5). For films thinner than dc, the dynamic pressure is smaller than the capillary pressure. Hence, the film thickens and the capillary pressure decreases until it is balanced by the dynamic pressure. The separated droplets remain connected by a film of thickness dc bearing a flow of liquid from the droplet with the low surface tension to the one with the high tension. Only after the liquids have mixed sufficiently will the flow and thus the dynamic pressure break down. The droplets will fuse. On the time scale of the instantaneous fusion of droplets of the same liquid, the droplet fusion of different liquids is delayed by many orders of magnitude. According to eq 6, dc and thus the delayed droplet fusion are independent from the contact angle. This may indeed be the case as long as there exists a connecting film between the two droplets with a laminar flow causing the separation of the two droplets due to the balance between the capillary and dynamic pressures. Future experimental investigations will show to what extent the simple model and eq 6 are in quantitative agreement with the real system if there is a stabilizing film. For large contact angles, we observe no delay of the droplet fusion. The droplets fuse rapidly, and their volumes mix in a turbulent volume flow process. There is no film connecting two separated droplets. Our simple model seems to fail. However, this is not the case. Only one crucial assumption becomes invalid shortly after the droplet contact. We can assume that, even for high contact angles, right after the first contact of the droplet perimeters, still a connecting film between the droplets will build up with a Marangoni flow and a resulting dynamic pressure. However, because according to eq 3 the flow speed is proportional to the contact angle, at high contact angles it may get so high that the flow becomes turbulent. Thus, eq 4 and a stabilizing film thickness according to eq 6 are obsolete. The droplets fuse in a turbulent volume process as observed. The transition from the delayed to the rapid droplet fusion as a function of the contact angle due to the transition between the laminar and turbulent flow can be estimated within the framework of our simple model via the Reynolds number Re. For the system shown in Figure 3, it is
Re )
FV 32γ d) Θ η ∆γ
(7)
Here, for simplicity, the stabilizing film thickness dc according to eq 6 is taken as the “size” of the system (most likely this is a lower limit for Re). The Reynolds number indeed increases with Θ. With values typical for an aqueous system, eq 7 yields Re ≈ 200Θ. This agrees roughly with the observed rapid fusion for contact angles in the range of tens of degrees. In conclusion, we present experiments which show that sessile droplets of different liquids that are miscible at all proportions may not fuse instantaneously after peripheral contact. This is in remarkable contrast to the rapid coalescence of sessile droplets of the same liquid. Experimental results on delayed droplet fusion were reported in the literature only twice (and a long time ago). However, these previous experiments were not very targeted and the vague theoretical explanation is most likely wrong. In particular, there are no reports about the influence of the contact angle on the droplet coalescence behavior. We present new experiments and a new explanation based on the observation of (1) a flow of liquid through the neck between the two connected but segregated droplets and (2) the breakdown of the droplet separation at high contact angles. We assume that the different composition of the droplets and especially their different surface tensions induce a directed flow from the low to the high surface tension droplet. The resulting dynamic pressure compensates the capillary pressure that promotes coalescence. The droplets are in contact but remain separated until the mixing flow breaks down and the main bodies of the droplets finally fuse into a single droplet. At high contact angle, the flow is turbulent and no repulsive dynamic pressure can sustain. The droplets fuse rapidly after the first contact. Calculations indeed reveal a stabile solution for the balance between dynamic and capillary pressure at low contact angles and a breakdown of the laminar flow at high contact angles. The calculations are roughly in quantitative agreement with the experimental data. We are fully aware that the spatio-temporal evolution of such a system of two contacting sessile droplets of different miscible liquids is complicated. More experimental data and detailed calculations than those presented here are necessary to understand the experimental findings in detail. In particular, the neck pulsations are not explained with our simple calculations (although they are typical for the predicted transition from laminar to turbulent flow). Nevertheless, we believe that we have presented interesting new experimental observations and the essence of their explanation. Acknowledgment. We would like to thank Helmuth Moehwald for his support and Len Pismen for elucidating discussions. Supporting Information Available: Movie showing in real time the delayed (rapid) coalescence of droplets of acetic acid (25%) and water on a microscope slide (Petri dish made from polysterene). The experiments were performed on the image plane of an overhead projector. The movie shows the images projected to the wall. Thus, the geometrical contours of the droplets and density gradients within them are visible. The experiments are neither caused nor significantly affected by the slightly elevated temperature on the overhead glass plate. This material is available free of charge via the Internet at http://pubs.acs.org. LA800630W
