Fundamental Phenomena and Principles 233

behavior may be reproduced if substituting f by u0/D0 in the above equation.

3.2.6Sessile Droplet Deformation on a Surface

In addition to the phenomena discussed above, many other droplet deformation processes on a surface are driven by surface tension, gravity force, and/or droplet-surface chemical interaction, characterized by very low or zero impact velocities. The isothermal spreading of a sessile droplet on a reactive wetting substrate, for example, is driven by the liquid-substrate chemical reaction and limited by solute diffusion to the triple line.[416] Under such conditions, the spreading rate is independent of time and droplet volume, but proportional to the contact angle and linearly dependent on droplet solute concentration. Very slow droplet spreading processes can be dramatically affected by impurities, especially if they are volatile.[417] Such processes are also sensitive to any external perturbation like roughness or contamination. Inhomogeneities of concentration can give rise to surface tension gradients that may induce liquid flows or non-wetting fluid spreading, a phenomenon termed Marangoni effects.[418] Marangoni effects may be encountered in droplet deformation processes coupling with temperature/composition changes or chemical reactions. A typical example is solder droplet spreading on a metallic surface, a reactive form of the wetting problem. A metallic component may diffuse in the liquid toward the surface, where it is consumed by a reaction that forms a solid intermetallic phase. The resultant spatial variation in the composition of the droplet may cause composition gradients along the free surface of the droplet. Along with any thermal gradients, Marangoni effects may modify the transport characteristics of the spreading droplet. To account for the Marangoni effects, Braun et al.[419] extended the lubrication theory for the spreading of thin droplets in the presence of gravity and thermocapillarity to include mass transport and solutocapillarity. They used an approximate solute profile in the droplet to derive coupled evolution equations for

234 Science and Engineering of Droplets

the free surface shape and concentration field. The study showed that the reactive effects have relatively important impact on the flow patterns and spreading rates at the early stages of droplet deformation, and will phase out at the end of the spreading.

Carles et al.[417] examined the influence of surface and atmospheric contamination on the spreading dynamics of silicone oil droplets on glass surfaces. The results showed that an acceleration surface tension gradient develops under both contamination conditions. However, the spreading dynamics and profiles are different in the two cases. The atmospheric contamination gives rise to smooth surface tension gradients distributed over the whole droplet, and the profile is of a fairly general type, resembling those obtained in many cases where a flat film is made from a droplet spreading over a smooth surface. In this case, a droplet spreads and thins while losing its spherical cap shape. During the early stages, fringes are not visible due probably to too large thickness and too high slopes. Then, a bump develops near the contact line while the central part of the droplet becomes flat. This profile remains analogous to each other while it spreads and thins during the late stages of deformation. The radius increases almost linearly with time. In contrast, the surface contamination produces localized surface tension gradients near the contact line, and the profile is more comparable to those obtained when a liquid droplet spreads over a liquid layer. In this case, the profile does not retain the shape of a spherical cap, but instead shows an inflexion point near the edge, while the radius increases roughly as the square root of time.

Surface tension and gravity driven droplet spreading on a surface, has been studied with nonvolatile droplets containing sur- face-active agents at relatively low velocities.[354] At the velocities on the order of a few centimeters per second, a droplet moves spontaneously on a surface. For small droplets, this self-supported motion may be interpreted in terms of capillary theory, which provides insightful information about the deformation mechanism on the surface. For large droplets, gravity effect intervenes in. A remarkable change in the droplet profile has been observed and different spreading regimes have been identified.[354]

Fundamental Phenomena and Principles 235

3.2.7Spreading and Splashing of Droplets into Shallow and Deep Pools

In addition to the phenomena discussed above, other issues of practical importance include the phenomena of droplet impingement into shallow and deep pools. Splash and breakup of liquid almost always occur when a droplet at high velocity impinges onto a pool of the same fluid, regardless of pool depth if surface tension and/ or viscosity are negligibly small.[397] However, the splashing behavior for finite ratios of droplet diameter to pool depth is distinctly different from that for very large, or infinite ratios of droplet diameter to pool depth. Even a shallow pool with a very thin film of liquid is sufficient to interact appreciably with the lateral jet of a flattening droplet to produce an upward motion, i.e., splash. For a shallow pool, the region of impact tends to empty and a liquid sheet is ejected from the periphery of the region of impact. In addition, secondary droplets tend to form at the edge of the splash sheet. For a deep pool, a crater forms in the region of impact, followed by the ejection of liquid from the center of the crater if impact kinetic energy is high enough. A splashing droplet may also rupture into two parts during penetration into a deep pool at high impact kinetic energy. The droplet material is deposited onto the bottom and the sides of a crater, the bulk of the material lying near the bottom. As the cavity sides collapse, the bottom simultaneously tends to rise back towards the original surface level. The two processes compete. For a low droplet impact velocity, the crater is shallow and the droplet rebounds to the surface before the sides collapse. For a high droplet impact velocity, the crater is much deeper and the sides collapse earlier, trapping much of the original droplet material well below the surface. The material deposited on the sides of the crater is carried up in the rebound splash, isolating two different parts of the original droplet fluid from each other. If the pool fluid is less dense and immiscible with droplet fluid, then each part of the droplet may coalesce and drift slowly downwards. As the droplet impact velocity approaches the order of 5000 m/s, the effects of compressibility become progressively more

236 Science and Engineering of Droplets

important. Upon impact, there is almost no splash. Shocks are generated in both the droplet and the pool, but their lateral motion in the pool is relatively slow so that very little expansion occurs. Once the shock reaches the top of the droplet, there is a strong reexpansion that violently expulses the material back out in the direction from which it enters. The outward moving velocity of the droplet top is very close to the initial impact velocity. The droplet material vaporizes and subsequently behaves much like a gas.

For a liquid droplet falling at low velocities onto a liquid pool of a different fluid, Marangoni flows induced by the surface tension difference between the droplet and the pool may dominate the interaction, and the Marangoni effects may control both the direction and intensity of the pool circulation.[420] The surface tension force may generate the strongest flow as compared to buoyancy force and stirring force induced by the falling droplet. The relative difference in the surface tension between the droplet and the pool determines the flow pattern and direction. If the droplet has a larger surface tension than the pool, the surface flow is directed inward, i.e., the Marangoni forces cause the pool surface to be drawn toward the droplet, and a deep flow loop may be created while the droplet is injected deep into the pool. If the droplet has a smaller surface tension, the Marangoni forces cause the pool surface to be drawn away from the droplet while the droplet spreads out onto the pool surface. If the droplet has the same surface tension as the pool, the droplet may penetrate into the pool.

Al-Roub et al.[421] identified three basic modes of liquid breakup during droplet impingement onto a liquid film: (1) rim breakup, (2) cluster breakup, and (3) column breakup. The rim breakup mode involves the breakup and ejection of one or a few small droplets at the outer edge of the film, while the cluster breakup mode involves the breakup of liquid into clusters of many small droplets at the outer edge of the film. In the column breakup mode, liquid breaks up into one or a few droplets from a column of liquid at the center of the spreading droplet as a result of the surface waves reflecting back to their source. The diameter and number of the

Fundamental Phenomena and Principles 237

secondary droplets have been formulated as a function of impact conditions (such as the Ohnesorge number and Reynolds number) and surface conditions.[356][422]

To determine if a droplet experiences spreading or splashing when it impinges onto a liquid film on a solid surface, the correlation between the Weber number and Ohnesorge number derived by Walzel[398] may be used:

A We-Oh map can also be plotted on the basis of this correlation, similarly to that for the droplet spreading and splashing on a dry, solid surface (Fig. 3.21), but with smaller slope. The WeOh relation delimits the regimes of droplet spreading and splashing onto a liquid film on a solid surface in the We-Oh map. If the fluid dynamic conditions of a droplet are within the spreading regime, i.e., below the threshold curve in the We-Oh map, liquid splashing and ejection from the impact surface can be avoided. The corresponding threshold velocity can be formulated as:

Comparing Eq. (51) to Eq. (43) or Eq. (52) to Eq. (44a), it is clear that for the same liquid properties and droplet diameter at impact, splashing takes place at lower impact velocities on a liquid film than on a dry surface.