Ind. Eng. Chem. Res. 2007, 46, 6105-6112
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Dynamics of Drop Impact on a Rectangular Slot Hariprasad J. Subramani, Talal Al-Housseiny, Alvin U. Chen, Mingfeng Li, and Osman A. Basaran* School of Chemical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907
Drop impact on substrates is of scientific importance and plays a central role in both microscale and largescale applications, e.g., ink-jet printing and spray coating. For over 100 years, researchers have studied situations where drops impact planar substrates, a beautiful free surface flow resulting in either drop deposition or splashing. By contrast, drop impact on nonplanar substrates, e.g., spheres, has become of interest only recently. Here, the impact of drops of several liquids with a slot of width comparable to the drop diameter that is dug into an otherwise planar substrate is studied experimentally as a function of impact velocity. Two different kinds of splashing arise in the new experiments: an internal splash similar to that observed on planar substrates and a new, external splash, where some of the drop liquid splashes out of the slot. Phase diagrams that delineate regimes of drop spreading and splashing are presented. Simple scaling arguments are also developed to rationalize the findings. I. Introduction The phenomenon of impact of liquid drops on solid or liquid surfaces is encountered in a variety of widely used applications such as ink-jet printing, spray painting and coating, pesticide spraying, and microfabrication of structured materials and microelectronics.1,2 Despite more than a century of research,3 the phenomenon of drop impact still fascinates researchers and remains incompletely understood. Part of the fascination is a consequence of the fact that drop impact is a beautiful free surface flow in which the liquid may fragment or splash upon colliding with the substrate.1,2 Experimental studies to date have primarily focused on the impact dynamics of drops of simple Newtonian liquids on (a) solid planar surfaces,4,5 (b) thin films of the same liquid,6,7 and (c) quiescent liquid surfaces.8-11 A number of researchers have reported that substrate roughness is an important parameter on the occurrence of splashing.12-14 While most of the previous research on drop impact has been experimental in nature, there has been an increase in activity on the theoretical and computational fronts given the recent advances in computing power.1,2 On the theoretical and computational fronts, much of the effort has focused on the prediction of crown formation due to drop impact on wet surfaces by insightful but idealized onedimensional models15-17 and direct numerical simulations by means of the volume of fluid (VOF) method.18 Recently, Pasandideh-Fard et al.,19 Bussmann et al.,20,21 and Gunjal et al.22 have utilized the VOF method to analyze the dynamics of drop impact on dry solid substrates. Motivated by emerging applications in the life sciences and the manufacture of foodstuffs, e.g., in coating of biochips, diagnostic test strips, and cereals and potato chips, among others, where drops of simple as well as complex fluids impact on nonplanar substrates, researchers have recently began to focus attention on the effects of dynamic surface tension,23,24 nonNewtonian rheology,24,25 hydrophobicity of substrates,26,27 and topology of substrates such as spheres,28 cylinders,19 pedestals,29,30 and planar confinements or obstacles.31 Other new avenues of research include drop impact on fluid spheres,32 which can be used to produce microcapsules for controlled * To whom correspondence should be addressed. Tel.: (765) 4944061. Fax: (765) 494-0805. E-mail:
[email protected].
release applications, and drop impact on elastic membranes,33 analysis of which can further our understanding of the shorttime-scale dynamics that occur when a drop collides with a surface. However, recent insightful experiments by Xu et al.5 have cast doubt on the ability of existing computational algorithms to quantitatively predict certain salient features of the outcome of experiments on drop impact. In particular, Xu et al.5 have shown that, in the case of impact of drops of Newtonian liquids on dry planar substrates, splashing, which occurs as the velocity with which a drop impacts a substrate is increased beyond a threshold, can be altogether eliminated by reducing the pressure of the gas surrounding the drop or carrying out the experiments in a vacuum. By contrast, all simulations predict that splashing results once the impact velocity exceeds the threshold velocity despite the fact that they assume that drop impact occurs in a dynamically passive ambient surroundings, i.e., vacuum.20,21 Therefore, given the existing disagreement between observation and computation, there is currently no substitute for careful experiments in particular if a new application or a novel configuration involving drop impact is being studied. Furthermore, experimental measurements on old and new applications of drop impact will be needed for the foreseeable future in benchmarking any new numerical algorithms that may be developed to model the phenomenon. Motivated by some of the aforementioned applications in the life sciences and the food industry, the major goal of this paper is to study experimentally by high-speed imaging the impact of drops of Newtonian liquids on a substrate with a small-scale feature. Here the feature is in the form of a rectangular slot or a well that has been dug into an otherwise planar, solid substrate and which in applications is designed to hold the entire volume of liquid delivered to it by the impacting drop. When the width of the slot is comparable to the drop diameter, the dynamics of impact on such a nonplanar substrate can be quite different from those on a planar substrate. Thus, another major objective of this work is to quantify the effect of various variables such as impact velocity, drop size, slot width and height, substrate roughness, and the physical properties of the drop liquid on the impact dynamics in the form of phase or operability diagrams that delineate (a) regimes in which drops simply spread and
10.1021/ie070290t CCC: $37.00 © 2007 American Chemical Society Published on Web 07/03/2007
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Figure 1. Schematic diagram of the experimental setup.
Figure 2. Schematic diagram of the substrate with a rectangular slot (not drawn to scale). The front face of the slot is cut open so that the dynamics of a drop spreading within the slot can be readily visualized.
successfully deposit in the slot and (b) others where splashing, or drop fragmentation, results. The rest of the paper is organized as follows. Section II describes the experimental setup, the materials, and the procedures used to study the impact dynamics. Section III presents the results of the experiments and discusses the findings. Section IV provides concluding remarks and outlines opportunities for future research. II. Experiments The experiments entail generating drops of incompressible, Newtonian liquids and causing them to impact a rectangular slot. High-speed imaging is then used to obtain quantitative information on the time evolution of the shape of the spreading/ splashing drops. The apparatus for doing all this is described in the first subsection, which is then followed by subsections on materials used in the experiments, experimental procedures, determination of impact velocities, and dimensionless groups. a. Description of the Apparatus. The experimental apparatus consists of a capillary tube through which the liquid is made to flow at a vanishingly small but constant flow rate by means of a syringe pump and from the tip of which a liquid drop is formed. The drop of diameter Dd that is thereby formed is then allowed to fall freely through the ambient air toward a substrate made of acrylic sheets and impact with velocity V0 within a rectangular slot, or well, dug into the substrate (cf. Figures 1 and 2). Two such slots are employed in the present study. They are both of the same length Ls ) 2.5 cm and width Ws ) 0.945 cm, but have different depths Hs ) 0.25 cm and Hs ) 0.1 cm. The acrylic sheets have been cut such that the front edge or face of the rectangular slot is open, as shown in Figure 2, in order to facilitate the ease with which images may be captured
of the drop upon impact. Other essential parts of the experimental setup include a high-speed video camera for imaging the dynamic drop shapes, the associated hardware and software for recording, storing, and analyzing the drop shape data, and a light source used in conjunction with the camera to produce silhouette images of the drop. Care is taken to shield the capillary and the drops from external air drafts that can disturb the drops during their formation and as they fall through the air. The liquid is delivered to the capillary using an Orion Sage Model M361 syringe pump which is capable of providing a wide range of flow rates from 0.0005 to 60 mL/min depending on the syringe configuration used with an accuracy of (1%. All the equipment except the syringe pump and the camera support hardware are placed on an air-floated Newport vibration isolation table (RS 4000). The capillary tube used was procured from Vici Valco Instruments Co., Inc. It is 10.16 cm in length and is made of stainless steel. The outer diameter of the tube (0.231 ( 0.003 cm) is virtually constant over its entire length, and the thickness of its wall is less than 1/20 of the diameter. The imaging system is a Kodak Motion Corder Analyzer SR Ultra that is capable of recording 30-10000 frames per second. The images are stored in digital form in the image processor with a memory capacity of 2200 frames. A Dolan-Jenner light plate, Model QV ABL, connected by a fiber-optic cable to a Dolan-Jenner Fiber Lite, Model 3100, is used to backlight the drop. Backlight intensity, along with lens aperture setting, and the camera exposure rate are adjusted to produce sharp images of the drops as they grow and subsequently detach from the capillary, and later as they impact and spread on the slot (see subsection IIc). The recorded images on the digital processor are downloaded to a personal computer (PC). The digitized
Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007 6107 Table 1. Measured Physical Properties of Various Liquids at 23 °C and Resulting Values of the Dimensionless Groups for Drops of the Indicated Diameters liquid
F (g/cm3)
µ (cP)
σ (mN/m)
Dd (cm)
Oh × 103
G
θ (deg)
water GW25 ethanol GW40 GW50 GW60 GW75 DEG
1.000 1.060 0.785 1.100 1.130 1.160 1.200 1.116
1.0 1.8 1.2 3.2 5.0 10.0 27.7 31.6
72.0 69.0 22.3 68.0 66.7 66.0 65.5 44.0
0.40 0.38 0.28 0.38 0.37 0.37 0.36 0.32
1.86 3.43 5.42 6.03 9.47 18.9 52.1 79.7
2.18 2.17 2.70 2.24 2.27 2.29 2.33 2.55
44 47 0 52 52 45 46 24
images are further analyzed using the SigmaScan Pro 4.0 image analysis software from Jandel Scientific installed on the PC. b. Materials and Their Properties. Pure diethylene glycol (DEG), ethanol, and solutions of glycerol/water are used in the experiments. Reasons for using these liquids are given below. Ethanol, glycerol, and DEG (all of which are 99% pure) were obtained from Sigma-Aldrich Chemical Co. and used as received. Millipore water (electrical resistivity of 18 MΩ‚cm) is used for making the different glycerol/water mixtures. Hereafter, the following notation is used where a mixture of x% (by weight) of glycerol in water is referred to as GWx such that the abbreviation GW40 denotes a 40% (by weight) glycerol-water mixture (cf. Table 1). All property measurements of the test liquids are performed at 23 °C. The viscosities µ of the liquids are determined using a RS-150 rheometer from Thermo Haake with a F6 temperature controller and a C25 chilled water reservoir. The viscosities change by about (1% when the shear rate is increased to as high as 6000 or 12 000 s-1, thereby justifying the Newtonian liquid assumption. The surface tensions σ of the liquids are measured using a Langmuir trough from KSV Instruments, Finland, with a roughened platinum Wilhelmy plate. Liquid densities F are measured by weighing a known volume of liquid using a Sartorius Model AC121S mass balance. Measurement errors incurred in surface tensions and densities of the liquids are about (1%. Table 1 lists the measured properties of the liquids used in the experiments. The diameters Dd of drops of various liquids are determined from drop volumes obtained from recorded images of the drops upon their formation from the capillary using the edge detection software described earlier. As expected, Table 1 shows that Dd increases as surface tension σ increases.34 The average roughness of the substrate Rs is measured to be 2.25 × 10-8 m using a Tencor Alpha Set 200 Profilometer. The equilibrium contact angles of drops of the various liquids on the substrate are obtained (with an accuracy of (1°) from the digitized images of the drops (of the same volumes as used in the impact experiments) that are gently placed on the substrate using the SigmaScan Pro 4.0 image analysis software (Table 1). c. Experimental Procedures. The liquid is first drawn into the syringe, and the syringe is fixed onto the syringe pump. The pump is then started and run at a high flow rate to cleanse the capillary and the tubing connecting the syringe to the capillary. Any bubble present in the syringe or the tubing is purged off. A vanishingly low flow rate is then set to form a drop that is allowed to fall under gravity from a desired elevation h above the substrate to impact the rectangular slot. Drops of low-viscosity liquids like water tend to oscillate and drops of all liquids may deform from spheres as they fall toward the target due to air drag. Thus, a drop of a particular liquid as it impacts the substrate at a given velocity may have a different shape if the experiment is repeated. For the conditions of the present experiments, it has been directly determined by repeating
each experiment three to four times that whether the drops are spherical or they depart slightly from spheres when they impact the substrate has a negligible effect on the results to be presented in this paper. The method for determining the velocity with which a drop impacts the substrate is described in the next subsection. The images of the dynamics of drop impact are recorded on the image processor and are subsequently downloaded onto the PC for further analysis of the dynamics using the image analysis software. The drop impact experiments are all carried out at a temperature of 23 °C. d. Determination of Impact Velocity. The velocity V0 with which the drop impacts the surface of the slot is extracted from images that capture the vertical translation of the center of mass of the drop during the millisecond prior to impact. The error incurred in measuring V0 is about (1%. The impact velocity is varied by changing the height h from which the drop falls. In the present study, V0 is restricted to a maximum value of 3 m/s. Although higher impact velocities could be attained with the current setup, values of V0 larger than 3 m/s make it difficult to ensure that the drop would land at the center of the slot after falling over such a large height. The variation of the impact velocity V0 of drops of various liquids as a function of the height h over which they have fallen is depicted in Figure 3. As expected, the experimentally measured values of V0 are lower than the values of the impact velocity predicted theoretically by assuming that the surrounding air exerts zero drag on the falling drops. e. Governing Dimensionless Groups. The dynamics of a drop of a pure Newtonian liquid that impacts a substrate with an impact velocity V0 is governed by several dimensionless groups. Foremost among these dimensionless groups are the Weber number We ) FDdV02/σ, which measures the relative importance of inertial to surface tension forces, the Ohnesorge number Oh ) µ/(FDdσ)1/2, which measures the relative importance of viscous to surface tension forces, the Bond number G ) FgDd2/σ, which measures the relative importance of gravitational to surface tension forces, and the relative roughness of
Figure 3. Experimentally measured impact velocity of drops of various liquids as a function of the height through which they fall. The solid curve labeled “No drag” gives the impact velocity of drops in the absence of air drag.
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Figure 4. Time evolution of the shape of a water drop impacting and spreading on a rectangular slot of width Ws ) 0.945 cm and height Hs ) 0.25 cm at a low impact velocity of V0 ) 1.575 m/s. Here the drop diameter Dd ) 0.4 cm. In this figure and ones that follow, the time interval between successive frames is 1 × 10-3 s.
the substrate Rr ) Rs/Dd, which is the substrate roughness made dimensionless with the drop diameter. The values of Oh and G in the experiments are given in Table 1. Thus, the choice of liquids used allows the attainment of low and high values of the Ohnesorge number (cf. Table 1) while slightly affecting the Bond number (cf. Table 1) and the relative roughness of the substrate. Other dimensionless groups arise from the dimensions of the rectangular slot, namely, the dimensionless width W ) Ws/Dd and depth H ) Hs/Dd of the slot. Care is taken to ensure that the drop impacts at a location that is virtually at the center of the slot with respect to its width, i.e., a distance Ws/2 from the two side walls, and sufficiently far away from the slot’s back wall (i) to minimize and/or eliminate the effect of lateral offcentering and (ii) to eliminate the effect of the back wall on the impact dynamics. III. Results and Discussion a. Impact Dynamics of Water Drops. This subsection reports results on the dynamics of impact of water drops on the slot of Hs ) 0.25 cm as the impact velocity is varied between 0.5 and 3 m/s. As shown in Table 1, the diameter Dd of all the water drops, which are formed from a capillary tube of outer diameter Dt ) 0.231 cm, is about 0.4 cm. At low impact velocities, the water drops gently spread across the bottom surface of the slot (not shown). Such drops eventually come to rest as sessile drops that sit on the bottom surface of the slot, in accord with intuition and earlier studies of drop impact on planar substrates. Figure 4 shows the evolution in time of the shape of a water drop shortly before and after it impacts the slot at a moderate impact velocity of V0 ) 1.575 m/s. In Figure 4 and in what follows, the time interval between successive frames shown is 1 ms. Following impact, the drop rapidly spreads and thins, and the contact line quickly reaches the side walls of the slot. Figure 4d depicts a snapshot of the dynamics at about this instant in time when the spreading drop resembles a circular thin film with a small bump at its center. When the liquid in the expanding drop comes in contact with the side walls, the pressure within the thin film in that locale rises as a consequence of the arrest
Figure 5. Time evolution of the shape of a water drop impacting and splashing on a rectangular slot of width Ws ) 0.945 cm and height Hs ) 0.25 cm at a high impact velocity of V0 ) 2.475 m/s (We > Wex). Here the drop diameter Dd ) 0.4 cm.
of any further outward expansion of the drop in the horizontal direction. Thereafter, this rise in the pressure in the liquid can have two consequences. On the one hand, the rise in pressure at the wall tries to drive liquid from the wall toward the center of the slot. However, if the inertia of the liquid in the spreading drop is sufficiently high, liquid near the wall cannot instantaneously change its direction of motion and continues to move toward the wall. On the other hand, the increase in pressure of the liquid just underneath the free surface can only be balanced by the surface tension generated capillary pressure if the curvature of the drop’s surface in the vicinity of the wall increases. Thus, the portion of the drop’s surface adjacent to the two side walls starts to deform in the vertical direction and the liquid begins to climb the side walls of the slot, as shown in Figure 4e. However, Figure 4f-l shows that, in the present situation, while the film may climb all the way up the side walls to the slot’s edge, the impact velocity is small enough that none of the liquid spills out of the slot. As the impact velocity is further increased, there exists a critical impact velocity or, equivalently, a critical Weber number Wex such that for We g Wex, the impacting drop has sufficient inertia that the film climbing the side walls of the slot continues to shoot upward past the edges of the slot as a free film and a portion of the liquid splashes out of the slot. The aforementioned type of splashing is quite distinct from the type of splashing that occurs when a drop impacts on a planar surface and is henceforward referred to as “external splashing.” Figure 5 depicts the evolution in time of the dynamics that arises when a water drop impacts on the slot at a higher impact velocity (V0 ) 2.475 m/s) than that in Figure 4, resulting in an external splash. Figure 5i-l clearly shows the ejection of several small drops from the edge of the free film. For the situation depicted in Figure 5, the impact velocity V0 has a value that makes the corresponding Weber number just exceed that required by water drops to exhibit external splashing, Wex. b. Impact Dynamics of Drops of Other Liquids. To gain insights into the effects of surface tension, which is a physical property of key importance in many free surface flows, results are next reported from a set of experiments in which the drop liquid is changed from water to ethanol while the slot of Hs ) 0.25 cm used in the previously reported experiments on water is retained. Once again, the impact velocity is varied between
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Figure 6. Time evolution of the shape of an ethanol drop impacting and internally splashing on a rectangular slot of width Ws ) 0.945 cm and height Hs ) 0.25 cm at an impact velocity of V0 ) 2.3 m/s. Panel j, which is a blowup of panel b, highlights the occurrence of internal splashing. Here the drop diameter Dd ) 0.28 cm.
0.5 and 3 m/s. As shown in Table 1, while ethanol has a viscosity that is similar to that of water (1.2 cP versus 1 cP), it has a low surface tension (22.3 mN/m) which is about onethird that of water (72.0 mN/m). Consequently, owing to its low surface tension and as also shown in Table 1, the diameter Dd of all the ethanol drops which are formed from a capillary tube of outer diameter Dt ) 0.231 cm is about 0.28 cm. At low impact velocities, the ethanol drops, like the water drops of the previous subsection, gently spread across the bottom surface of the slot (not shown). However, in contrast to all the other liquids used in this work, ethanol perfectly wets the substrate in that the contact angles made by ethanol drops on the substrate are virtually zero. The impact dynamics of ethanol drops start to differ starkly from that of water drops as impact velocity increases. An example is depicted in Figure 6, where an ethanol drop impacts the substrate with a velocity of V0 ) 2.3 m/s. Figure 6b, and its blowup in Figure 6j, shows that the ethanol drop splashes virtually immediately upon impact with the substrate, when the drop’s contact line is still quite far from the side walls of the slot. This type of splashing, which is henceforward referred to as “internal splashing,” occurs when the impact velocity, and the corresponding Weber number, exceeds a threshold. The critical value of the Weber number for internal splashing is hereafter denoted by Wei. Such an internal splashing phenomenon is reminiscent of the commonly observed “crown formation” that occurs when a drop impacts a flat substrate, which is quite different from the external splashing of water drops described earlier wherein the walls of the slot play a major role in causing the liquid to splash out of the slot. Because of the relatively large value of the impact velocity (V0 ) 2.3 m/s), after exhibiting internal splashing, the ethanol drop of Figure 6 continues to spread toward the side walls of the slot and climbs them all the way up to the brim of the slot, as shown in Figure 6. When the impact velocity is increased beyond that of Figure 6, the ethanol drop not only splashes internally, but also splashes out of the slot. Thus, ethanol drops exhibit both types of splashing when the impact velocity or the Weber number reaches sufficiently high values. It is next instructive examine how the impact dynamics changes as the viscosity of the drop liquid is
systematically increased while retaining the slot used in the experiments of Figures 4-6. Just as ethanol drops having a sufficiently large impact velocity can exhibit both types of splashing, evidently such behavior is also exhibited by drops of liquids having somewhat larger viscosities than ethanol. An example is provided by Figure 7, which shows a GW40 drop, which has a viscosity of 3.2 cP (cf. Table 1), impacting a slot with a velocity of V0 ) 2.717 m/s. Figure 7b, and its blowup in Figure 7m, shows internal splashing, which is then followed, as shown by Figure 7e and its blowup in Figure 7n, by external splashing. Figures 8 and 9 show, however, that once the viscosity of the drop liquid becomes sufficiently large, drops may exhibit neither internal splashing nor external splashing, as shown for a drop of GW75 in Figure 8 and a drop of DEG in Figure 9. The viscous resistance to flow in these high-viscosity liquids is so large that they fail to exhibit either type of splashing over the range of impact velocities (0.5 m/s e V0 e 3.0 m/s) investigated in this paper. c. Phase Diagram Summarizing Fate of Drops Impacting a Rectangular Slot. The responses exhibited by drops of liquids listed in Table 1 upon their impact on the rectangular slot of Hs ) 0.25 cm can be conveniently summarized in the form of a phase diagram in the (We,Oh) parameter space, as shown in Figure 10. Each data point in the phase diagram corresponds to an experiment in which a drop of a specific liquid and diameter, or equivalently a specific value of Oh, is made to impact the substrate at a certain impact velocity, or equivalently a certain value of We. Thus, for each value of the Ohnesorge number Oh, the Weber number We is varied from a low to a high value to probe whether the outcome is spreading or splashing. The phase diagram also identifies the threshold Weber numbers above which internal splashing (Wei) and external splashing (Wex) occur. The curves drawn through the critical values of We, which are hereafter referred to as the Wei and the Wex curves, delineate the (We,Oh) parameter space into operating regimes where drops (a) simply spread within the slot, (b) exhibit internal splashing only, (c) exhibit external splashing only, and (d) undergo both internal and external splashing upon impact (cf. Figure 10).
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Figure 7. Time evolution of the shape of a GW40 drop impacting and both internally and externally splashing on a rectangular slot of width Ws ) 0.945 cm and height Hs ) 0.25 cm at an impact velocity of V0 ) 2.717 m/s. Panels m and n, which are blowups of panels b and e, highlight the occurrence of internal and external splashing, respectively. Here the drop diameter Dd ) 0.38 cm.
Figure 8. Time evolution of the shape of a GW75 drop impacting and spreading on a rectangular slot of width Ws ) 0.945 cm and height Hs ) 0.25 cm at an impact velocity of V0 ) 2.313 m/s. Here the drop diameter Dd ) 0.36 cm.
Figure 9. Time evolution of the shape of a DEG drop impacting and spreading on a rectangular slot of width Ws ) 0.945 cm and height Hs ) 0.25 cm at an impact velocity of V0 ) 2.338 m/s. Here the drop diameter Dd ) 0.32 cm.
Intuition would dictate that, as Oh increases, viscous drag exerted by the solid becomes so effective in opposing the spreading of the drop that the splashing limits should rise as Oh rises. As shown by the Wex curve in Figure 10, the Weber number required for a drop to exhibit external splashing does indeed increase monotonically with the Ohnesorge number. The variation of the internal splashing limit Wei with Oh, however, is more complex. Indeed, Figure 10 shows that the internal splashing limit Wei first decreases with Oh as the Ohnesorge number is increased from a low to a moderate value but then increases with Oh once the Ohnesorge number exceeds about 10-2. The form of the Wei curve for Oh g 10-2, that is, that the Weber number for drops to exhibit internal splashing increases with Ohnesorge number, accords with intuition. The form of the Wei curve for Oh e 10-2, however, requires further scrutiny. The experimentally observed decrease in Wei as Oh increases for small Ohnesorge numbers may be attributed to differences in (i) the equilibrium contact angles of the various liquid drops
on the acrylic substrate, (ii) the relative roughness of the substrate Rr ) Rs/Dd for different drops, and (iii) the thickness of expanding films for different liquids.5 However, a closer inspection of Figure 10 reveals that even though Wei for drops of GW50 is much smaller than that for drops of GW40 drops, the equilibrium contact angles of these two drops are about the same (cf. Table 1). Hence, differences in equilibrium contact angles of the various liquid drops cannot account for the decrease in Wei with Oh. As the Ohnesorge number Oh is increased from a low to a moderate value (by increasing the viscosity and decreasing the surface tension of the liquids), the diameters of the formed drops decrease as surface tension decreases. The decrease in drop diameter results in an increase in the relative roughness of the substrate (Rr ) Rs/Dd). Previous studies have shown that surface roughness has a strong influence on the splashing limits of drops impacting on planar substrates.1,12,13 Thus, that the gradual increase in Rr results in a decrease in the Weber number Wei required for internal splashing as Oh is increased from a low to a moderate value
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Figure 10. Phase diagram in (We,Oh) parameter space that delineates regions of the parameter space where drops spread and others where they splash upon impacting a slot of width Ws ) 0.945 cm and height Hs ) 0.25 cm. The relative roughness of the substrate measured relative to the drop diameter of various liquids varies moderately between 5.63 × 10-6 and 8.03 × 10-6. In this figure and the next, “Ext.” and “Int.” are abbreviations for “external” and “internal”.
Figure 11. Phase diagram in (We,Oh) parameter space that delineates regions of the parameter space where drops spread and others where they splash upon impacting a slot of width Ws ) 0.945 cm and height Hs ) 0.1 cm. The relative roughness of the substrate measured relative to the drop diameter of various liquids varies moderately between 5.63 × 10-6 and 8.03 × 10-6.
(cf. Figure 10) accords with these previous results. Alternatively, Xu et al.5 have shown that splashing becomes easier, or equivalently that Wei required for internal splashing decreases, as drop viscosity, or equivalently Oh, increases in experiments involving low-viscosity liquids such as ethanol, methanol, and 2-propanol. These authors argued that this is because the thickness of the expanding liquid film increases as viscosity increases, thereby making the expanding film easier to destabilize for more-viscous liquids. It should be noted from Figure 10 that water drops do not splash internally and GW60 drops do not splash externally for the range of impact velocities studied (V0 e 3.0 m/s). Figure 10 further shows that GW75 drops and DEG drops splash neither internally nor externally for the range of impact velocities considered. d. Effect of Slot Height on the Impact Dynamics. A set of drop impact experiments have also been conducted with the second rectangular slot having a smaller height of Hs ) 0.1 cm. The results of these experiments are summarized by means of the phase diagram shown in Figure 11. Clearly, the measured variation of the internal splash limit Wei as a function of Oh parallels that reported earlier for the slot of height Hs ) 0.25 cm as both the substrate material and the relative roughness
Figure 12. Cartoon depicting a drop before it impacts a slot and after it has spread into a thin film of nearly uniform thickness hf when it is on the verge of overshooting the brim of the slot (not drawn to scale): definition of variables used in the scaling analysis.
are kept the same. Figure 11 also shows that the Wex curve in this case is lowered considerably compared to that shown in Figure 10 as decreasing Hs enables drops to more easily overcome the lowered drag force exerted by the shorter side walls of this slot compared to the one considered earlier. e. Scaling Arguments for the External Splashing Limit. In this subsection, simple scaling arguments are developed to rationalize the variation of Wex with Oh in Figures 10 and 11. The limits of low and high Ohnesorge numbers will be considered separately as they involve balances between different pairs of forces. When Oh is small, the inertia of the impacting drop must overcome the surface tension force that holds the drop intact. Thus, balancing the initial kinetic energy of the drop, i.e., the drop’s kinetic energy as it is impacting the slot (cf. Figure 4c), with the surface energy when the drop is a flattened-out thin film of nearly uniform thickness hf and lateral extent Ds ≈ Ws + 2Hs and is about to splash out of the slot, as shown by the experimental image in Figure 4f and the cartoon in Figure 12, it is found that FV02Dd3 ∼ σDs2. The former expression can be rearranged as FV02Dd/σ ∼ (Ds/Dd)2. Therefore, Wex should virtually be independent of Oh but should increase with Hs, in accord with the experimental results shown in Figures 10 and 11. When Oh is large, the inertia of the impacting drop must overcome the viscous drag exerted on the spreading drop. The viscous shear stress exerted by a solid substrate on a thin film of thickness hf scales as µV0/hf. Thus, the work done against the viscous resistance of the solid as the drop spreads to the state depicted in Figure 12 can be estimated as (µV0/hf)(Ds2)(Ds). This expression of course overestimates the viscous resistance because (i) the instantaneous thickness of the spreading drop is larger than hf except when the drop is on the verge of splashing and (ii) the radius of the spreading drop in contact with the solid is smaller than Ds except in the very last stages of the spreading process. Equating the latter estimate with the initial kinetic energy of the drop gives FV02Dd3 ∼ (µV0/hf)Ds3. However, Ds2hf is roughly the volume of the spread-out thin film and, since the liquid is incompressible, can be replaced by Dd3 in this scaling analysis. Canceling the common factors of V0 from both sides, squaring the resulting expression, and rearranging yields We ∼ Oh2(Ds/Dd)10. Therefore, this simple scaling analysis predicts that the We for external splashing should vary as the square of the Oh and that Wex should increase with Hs, in accord with the experimental results shown in Figures 10 and 11.
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IV. Conclusions In this paper, the findings from an experimental study are reported in which drops of several liquids are made to impact at various velocities a solid substrate with a small-scale feature that is in the form of a rectangular slot or a well. According to the foregoing results, the impact dynamics on such nonplanar substrates differ substantially from their counterparts on planar substrates. Specifically, two different kinds of splashing are observed when a drop impacts a rectangular slot: one is an internal splash that is reminiscent of a crown splash on a planar substrate and the other is an external splash where some of the drop liquid splashes out of the slot. The experimental observations have been summarized in the form of phase diagrams in the (We,Oh) parameter space that delineate regimes of (a) spreading, (b) internal splashing alone, (c) external splashing alone, and (d) both internal and external splashing. For small values of Oh, the internal splashing limit Wei decreases with Oh due to the combination of (i) the increase in the relative roughness of the substrate and (ii) that in the thickness of the expanding liquid film as Oh is increased from a low to a moderate value, both of which conspire to destabilize the expanding liquid film. However, as Oh is increased further, viscous resistance opposing the spreading of the drop rises sufficiently fast so that Wei increases as Oh increases. By contrast, experiments show that the external splashing limit Wex increases as Oh increases over the entire range of Ohnesorge numbers considered in this work. Simple scaling arguments are also developed to rationalize the variation of Wex with Oh. The findings reported here should be useful in emerging applications in the life sciences and food production that entail depositing liquids on or coating substrates incorporating a multitude of slots, wells, or other features. Ongoing work in our laboratory includes a systematic experimental investigation of the role of surface-active species and complex fluid effects in the impact dynamics of drops on nonplanar substrates. Acknowledgment The authors thank the BES Program of the U.S. DOE, NIH, and NSF ERC for Structured Organic Particulate Systems for financial support. The authors also thank the Roche Diagnostics Corporation for an unrestricted grant in support of this research. T.A. participated in this research as part of the SURF Program at Purdue University and carried out some of the experiments reported here as part of his Undergraduate Honors B.S. thesis research in the School of Chemical Engineering. Literature Cited (1) Yarin, A. L. Drop impact dynamics: splashing, spreading, receding, bouncing, .... Annu. ReV. Fluid Mech. 2006, 38, 159-192. (2) Rein, M. Phenomena of liquid drop impact on solid and liquid surfaces. Fluid Dyn. Res. 1993, 12, 61-93. (3) Worthington, A. M. A study of splashes; Longman, Green, and Co.: London, 1908. (4) Chandra, S.; Avedisian, C. T. On the collision of a droplet with a solid surface. Proc. R. Soc. London, Ser. A 1991, 432, 13-41. (5) Xu, L.; Zhang, W. W.; Nagel, S. R. Drop splashing on a dry smooth surface. Phys. ReV. Lett. 2005, 94, 184505. (6) Cossali, G. E.; Coghe, A.; Marengo, M. The impact of a single drop on a wetted solid surface. Exp. Fluids 1997, 22, 463-472. (7) Wang, A. B.; Chen, C. C. Splashing impact of a single drop onto very thin liquid films. Phys. Fluids 2000, 12, 2155-2158.
(8) Prosperetti, A.; Oguz, H. N. The impact of drops on liquid surfaces and the underwater noise of rain. Annu. ReV. Fluid Mech. 1993, 25, 577602. (9) Thoroddsen, S. T. The ejecta sheet generated by the impact of a drop. J. Fluid Mech. 2002, 451, 373-381. (10) Mohamed-Kassim, Z.; Longmire, E. K. Drop impact on liquidliquid interface. Phys. Fluids 2003, 15, 3263-3273. (11) Fedorchenko, A. I.; Wang, A. B. On some common features of drop impact on liquid surfaces. Phys. Fluids 2004, 16, 1349-1365. (12) Stow, C. D.; Hadfield, M. G. An experimental investigation of fluid flow resulting from the impact of a water drop with an unyielding dry surface. Proc. R. Soc. London, Ser. A 1981, 373, 419-441. (13) Range, K.; Feuillebois, F. Influence of surface roughness on liquid drop impact. J. Colloid Interface Sci. 1998, 203, 16-30. (14) Stow, C. D.; Stainer, R. D. The physical products of a splashing water drop. J. Meteorol. Soc. Jpn. 1977, 55, 518-532. (15) Yarin, A. L.; Weiss, D. A. Impact of drops on solid surfaces: selfsimilar capillary waves, and splashing as a new type of kinematic discontinuity. J. Fluid Mech. 1995, 283, 141-173. (16) Trujillo, M. F.; Lee, C. F. Modeling crown formation due to the splashing of a droplet. Phys. Fluids 2001, 13, 2503-2516. (17) Roisman, I. V.; Tropea, C. Impact of a drop onto a wetted wall: description of crown formation and propagation. J. Fluid Mech. 2002, 472, 373-397. (18) Rieber, M.; Frohn, A. A numerical study of the mechanism of splashing. Int. J. Heat Fluid Flow 1999, 20, 455-461. (19) Pasandideh-Fard, M.; Bussmann, M.; Chandra, S.; Mostaghimi, J. Simulating droplet impact on a substrate of arbitrary shape. Atomization Sprays 2001, 11, 397-414. (20) Bussmann, M.; Chandra, S.; Mostaghimi, J. Modeling the splash of a droplet impacting a solid surface. Phys. Fluids 2000, 12, 3121-3132. (21) Bussmann, M.; Mostaghimi, J.; Chandra, S. On a three-dimensional volume tracking model of drop impact. Phys. Fluids 1999, 11, 1406-1417. (22) Gunjal, P. R.; Ranade, V. V.; Chaudhari, R. V. Dynamics of drop impact on solid surface: Experiments and VOF simulations. AIChE J. 2005, 51, 59-78. (23) Zhang, X.; Basaran, O. A. Dynamic surface tension effects in impact of a drop with a solid surface. J. Colloid Interface Sci. 1997, 187, 166178. (24) Crooks, R.; Cooper-Whitez, J.; Boger, D. V. The role of dynamic surface tension and elasticity on the dynamics of drop impact. Chem. Eng. Sci. 1997, 56, 5575-5592. (25) Cooper-White, J. J.; Crooks, R. C.; Boger, D. V. A drop impact study of worm-like viscoelastic surfactant solutions. Colloids Surf., A 2002, 210, 105-123. (26) Richard, D.; Clanet, C.; Quere, D. Contact time of a bouncing drop. Nature 2002, 417, 811. (27) Michel, T.; Mock, U.; Roisman, I. V.; Ruhe, J.; Tropea, C. The hydrodynamics of drop impact onto chemically structured surfaces. J. Phys.: Condens. Matter 2005, 17, S607-S622. (28) Hardalupas, Y.; Taylor, A. M. K. P.; Wilkins, J. H. Experimental investigation of sub-millimeter droplet impingement on to spherical surface. Int. J. Heat Fluid Flow 1999, 20, 477-485. (29) Roshkov, A.; Prunet-Foch, B.; Vignes-Alder, M. Impact of water drops on small targets. Phys. Fluids 2002, 14, 3485-3501. (30) Roshkov, A.; Prunet-Foch, B.; Vignes-Alder, M. Impact of drops of polymer solutions on small targets. Phys. Fluids 2003, 15, 2006-2019. (31) Josserand, C.; Lemoyne, L.; Troeger, R.; Zaleski, S. Droplet impact on a dry surface: triggering the splash with a small obstacle. J. Fluid Mech. 2005, 524, 47-56. (32) Yeo, Y.; Chen, A. U.; Basaran, O. A.; Park, K. Solvent Exchange Method: A novel microencapsulation technique using dual microdispensers. Pharm. Res. 2004, 21, 1419-1427. (33) Courbin, L.; Bird, J. C.; Stone, H. A. Splash and anti-splash: observation and design. Chaos 2006, 16, 041102. (34) Yildirim, O. E.; Xu, Q.; Basaran, O. A. Analysis of drop weight method. Phys. Fluids 2005, 17, 062107.
ReceiVed for reView February 26, 2007 ReVised manuscript receiVed May 7, 2007 Accepted May 11, 2007 IE070290T
