1. Noncoalescence Phenomenon
Droplets of water were released from a nozzle situated 5 mm above the water surface. The droplets grew at the tip of the nozzle to ~3 mm in diameter and then detached into free fall. Some droplets coalesced immediately with the water beneath, but the majority did not.
An example of the latter is shown in the sequence of . When the falling droplet first encountered the water surface, it broadened (0), then depressed the surface (12), and after a few minor oscillatory movements, finally came to a stop (40). The impact created a wave, which propagated away from the center point. No change was observed for the next 50 ms; the surface around the droplet remained still (92). Then, abruptly at 96 ms there was a dramatic change. A liquid connection formed between the droplet and the bulk in the shape of a cone-shaped skirt above the surface (96). The droplet seemed to expel some of its contents into the skirt beneath, while extending slightly upward. Meanwhile, the top portion of the skirt narrowed into a column (100) while the original extrusion on bulk surface turned progressively into a depression (96 to 104 ms). These movements generated a series of waves propagating away from the coalescence site.
Figure 2 Dynamics of water droplet falling onto water surface. Numbers indicate time count in milliseconds. White spots on the droplet are reflections of the illuminating light and should be ignored. Reflections of droplet on water surface and on microscope slide (more ...)
Once the liquid connection had formed (as at 96 ms, above), one of the two scenarios followed. In the first scenario, complete coalescence was achieved. In the second and more dominant scenario, as illustrated above and described in detail next, coalescence took place in a series of steps.
2. Coalescence Cascade
In the cascade scenario, the original droplet coalesced only partially. The liquid column (, 100 ms) elongated upward while thinning at the bottom to form a “bulb.” The bulb is shown again in at 135 ms. Eventually, the column broke and retracted upward into the bulb, producing a new droplet suspended in the air. The new droplet, referred to as “secondary”, was smaller than the original one, and sometimes, instead of one droplet, there were two. This happened when the water column broke in two places simultaneously along its length. The middle portion of the column then turned into a “satellite” droplet (, 104).
Figure 3 Coalescence cascade. For reference, the primary droplet is shown in the top left corner. The numbers indicate time in milliseconds after the primary droplet detached from the nozzle. For easy visualization, the trial displayed is one in which sideways (more ...)
The secondary droplet landed on the surface but did not immediately coalesce (, 200). Instead, the sequence described above in association with repeated itself. Cycles of residence and coalescence repeated several times, and after each coalescence event, a progressively smaller droplet was produced. Each such droplet jumped upward (360, 405). The smaller the droplet, the higher it jumped. The highest jumps were approximately 5 to 6 mm, close to the height from which the original droplet was released.
The sequence of coalescence steps is referred to as “coalescence cascade.” For clarity, we refer to the original droplet as the “primary”, whereas the daughter droplets are referred to as “secondary”, “tertiary”, and so on. In the majority of trials, six residence-coalescence cycles were observed.
The diameters of successive droplets in the stepwise cascade are presented in . During each step, the diameter diminished to ~0.6 of the initial diameter. The exception was the first step, when the ratio was ~0.4. The reason for this lower value may be that during the first step two daughter droplets were sometimes produced instead of one, as previously mentioned. The one that remained in place was regarded as the secondary; the other had a smaller diameter (0.3 to 0.5 mm) and frequently shot off to one side. The graph in shows up to fifth-order droplets, but sixth-order droplets were present as well, although they were too small to be accurately measured.
Average droplet diameter in coalescence cascade. Droplet order refers to primary droplet, secondary droplet, and so on. (n = 30 coalescence cascades).
The time that a droplet resides on the bulk prior to coalescence is referred to as the “residence time.” Droplet-residence times are presented in . Residence time diminished with the droplets’ size. Therefore, primary droplets were longest-lived, with residence time averaging 143 ± 9 ms (n = 30). Secondary droplets persisted for 78 ± 35 ms, and higher order droplets persisted for progressively shorter times. The residence time of each droplet was commonly 0.7 of that of the preceding droplet.
Residence times of droplets in cascade.
In the majority of trials, the pattern of primary droplet coalescence was not symmetrical; that is, secondary droplets typically jumped obliquely. Also, coalescence of primary droplets initiated waves on the bulk-water surface. Therefore, secondary droplets landed on a perturbed water surface rather than on a still surface and hence moved sideways. This affected both stability and residence time of secondary droplets, which is reflected by the high residence-time standard deviation (, bar 2). Sideways displacement occurred for all droplets in the cascade; however, droplets of higher order were smaller, and thus their coalescence produced smaller distortions of the bulk surface.
3. Moving Droplets
Whereas sideways drift was commonly seen in secondary droplets, primary droplets occasionally drifted sideways as well. Among those that drifted noticeably, drift velocities typically ranged between 3 and 13 mm/sec. These drifting droplets had residence times of 608 ± 132 ms (n = 50). This should be compared with residence times of stationary primary droplets, whose mean value (above) was 143 ms, approximately four times longer than the persistence of stationary droplets.
Droplets with sideways drift could easily be produced by allowing water to flow freely down through the nozzle. This was achieved by disconnecting the Tygon tubing from the syringe so that a continuous water jet formed between the nozzle and the bulk surface. Occasionally, the jet broke into individual droplets, which often landed on the surface in rapid succession without coalescing (). The droplets appeared in a variety of sizes ranging between 1 and 3.5 mm, at a rate of 3–5 droplets per second. If a large droplet happened to be residing just beneath the nozzle, then successive droplets slid down the residing droplet’s surface, landing on the bulk water without coalescence and continuing to move sideways. Smaller droplets generally moved faster than the larger ones. Therefore, 1 mm droplets moved at 209 ± 67 mm/s (n = 10), whereas 3 mm droplets moved at 46 ± 20 mm/s (n = 12).
Floating droplets formed as the jet of water from the nozzle broke into individual droplets. Floating droplets quickly translated away from the region immediately beneath the tip.
Analysis of residence times versus translation speed of these droplets revealed no apparent correlation. As long as droplets moved, their residence times were enhanced relative to non-moving ones but in seemingly random fashion with regard to velocity. Size did play some role: larger droplets persisted longer than smaller ones (). The graph shows a roughly linear correlation of residence time with diameter, and, with mean residence time of 518 ± 263 ms, compared with ~140 ms with nonmoving droplets, the result confirms again that on average moving droplets persisted longer than stationary ones.
Residence times of randomly chosen moving droplets with different diameters (n = 55, from four experimental trials). Residence times were measured from the moment of droplet formation in the vicinity of the nozzle until the first coalescence step.
4. Residence-Time Consistency
During the initial experimental runs measured, residence times were inconsistent. Not only were they inconsistent within a series (, dark bars) but also different series produced different degrees of inconsistency. In some experimental series, all droplets coalesced immediately upon contact.
Comparison of trials with inconsistent (darker) and consistent (lighter) primary residence times. Where no bar is present, coalescence was instantaneous upon contact.
Reasons for inconsistencies were further explored by the addition of electrical and mechanical perturbations to the system. To test for mechanical disturbances, the entire setup was moved onto a vibration-isolation table, with no observable difference in results. To test for inconsistencies originating from minor variations in experimental procedure, the protocols prior to and during each experiment were strictly maintained; again, there was little observable change. In some experiments, the purified water was passed through a fine filter (Whatman Nylon 0.2 μm) to eliminate any microscopic particles that might have been present, but this procedure also had no apparent effect on reproducibility.
We also considered residual electric charge. The impact of electrostatic charge was confirmed by bringing a charged glass rod near the nozzle. With the charged rod nearby, the primary droplet-residence time diminished sharply to ~50 ms, and secondary drops were not seen. With electrostatic fields implicated as potentially relevant, we found that the one procedure that did make a difference in terms of consistency was grounding both the nozzle tip and the bulk water. This was done by connecting both the water-supply tube and the base of the setup to a common building ground. This procedure resulted in consistency such as that shown in (light bars).
Another factor contributing to the inconsistency was found to be water-surface purity. When some residue (dust, grease) was purposely left on the glass substrate prior to the addition of water, floating droplets often did not form at all. Bubbles of air trapped in the nozzle or water-supply line also caused inconsistencies and diminished residence times. As the droplet grew at the tip of the nozzle, the water flow partially shifted the air bubble downward to outside the nozzle. When the droplet detached, the bubble then retracted back to the nozzle. When such bubbles were present, the residence times were less consistent.
In sum, realization of consistency depended on careful control of a number of factors. Once these factors were identified and controlled, experimental results could be obtained on a consistent basis. All experiments reported above and below were carried out with these factors under control.
5. Abrupt Residence-Time Change
In some trials, residence time suddenly switched from one stable value to another. An example is shown in . Residence times of the initial group of droplets were ~50 ms. After the 15th droplet, residence time jumped to 160 ms and stayed at approximately that value for some time. Then, it jumped back to 50 ms, only to return to the higher value after the 26th droplet.
Residence time versus droplet number, with examples of abrupt residence-time changes.
Such bimodal behavior was observed in ~10% of all recorded series. From the recorded videos, we could find no visual indications of any differences in the setup, the droplet size, or the way droplets formed during these abrupt residence-time shifts. In , the two residence times were 53 ± 8 and 164 ± 14 ms; they were separated by eight SDs. Of the five such records examined, the average separation was 5.8 standard deviations.
6. Effect of Nozzle Height
Nozzle height was increased from 5 to 10 mm in 1 mm increments and then from 10 to 18 mm in 2 mm increments. At each height, 20 droplets per trial were observed in three separate trials. For droplets released from heights 5 to 10 mm, primary residence time did not vary appreciably from the ~140 ms reported above. At heights of 12–16 mm, most droplets coalesced shortly after impacting the bulk surface, with residence times between 0 and 30 ms. However, droplets that were able to survive the initial impact usually persisted on the surface for ~120 ms. At 18 mm and above, coalescence occurred instantly.
When a very thin (1 to 0.5 mm) layer of bulk water was used on the receiving surface instead of the standard-thickness layer, the behavior of droplets was different. The layer was created by first dispensing 10 mL of deionized water on the glass slide and then removing most of the water with a needle attached to the house vacuum. Only a thin wetting layer remained on the glass. Droplets were then released from a 10 mm height. After the impact, droplets flattened as though they were hitting a solid surface and then recoiled back up to a height ~4 mm above the surface (). The droplets bounced two to three times, the height diminishing with each cycle. Sometimes, a water bridge was visible between the droplet and the bulk during the initial portion of the bounce (, 60 ms inset), which disappeared within one or two frames (5–10 ms). Between the bounces, no reduction in droplet size was noticed. When the droplet came to rest, it coalesced after ~150 ms.
Water droplet bouncing on a thin layer of water. Inset shows apparent water bridge between droplet and surface. The impact flattened the droplet and produced surface waves visible on the 50 ms panel.
7. Effect of Reduced Air Pressure
To check the effect of air pressure on residence time, the pressure in the experimental chamber was reduced in 10 kPa increments from atmospheric pressure (~100 kPa) down to 10 kPa. Two sets of experiments were performed at each pressure. In one set, water was taken directly from the purification unit and used immediately in the experiment; in the other set, water was kept at the respective air pressure while being stirred for 1 h prior to use to produce “equilibrated” water. The need for this latter procedure arose particularly at relatively low air pressures when air bubbles began to appear in the water-supply line. With bubbles present, droplet production became sporadic and unpredictable. This problem could be averted through the use of the equilibrated water.
Residence time diminished linearly with diminishing air pressure (). For the nonequilibrated water, 20 and 10 kPa data points are missing because bubbles upset the experiment. Residence times were slightly higher for equilibrated water than nonequilibrated water at all pressures, the difference being slightly more substantial at lower pressures.
Influence of air pressure on residence time.
The effect of pressure on the number of coalescence steps is shown in . Median values are shown. The median remained constant at four in the higher-pressure range and then diminished as pressure was reduced. At the lower pressures, secondary droplets were much smaller than those formed at atmospheric pressure and had residence times of 5–10 ms.
Effect of air pressure on the number of steps in coalescence cascade. Median values are shown at each pressure.
A noteworthy effect of the air-pressure change was that each time the pressure was reduced, the first few droplets coalesced instantly; then, longer residence times would gradually appear. Even when the pressure reduction was preceded by a pressure increase, the same effect appeared. Somehow, reducing chamber pressure temporarily upset the ability of water to form floating droplets.
Even at reduced air pressures, there were occasional droplets with surprisingly long residence times. On one occasion at 60 kPa, three consecutive droplets were observed with residence times of ~250 ms. Because there were only a small number of such droplets and their residence times were vastly different from the majority, they were considered to be exceptional and were not included into the mean residence time.