Effect of Confinement on Droplet Coalescence in Shear Flow

Oct 1, 2009 - The effect of confinement on the coalescence of Newtonian (polydimethylsiloxane) droplets in a Newtonian. (polyisobutylene) matrix is ...
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Effect of Confinement on Droplet Coalescence in Shear Flow Dongju Chen, Ruth Cardinaels, and Paula Moldenaers* Katholieke Universiteit Leuven, Department of Chemical Engineering and Leuven Materials Research Centre, Willem de Croylaan 46, B-3001 Leuven, Belgium Received May 20, 2009. Revised Manuscript Received September 14, 2009 The effect of confinement on the coalescence of Newtonian (polydimethylsiloxane) droplets in a Newtonian (polyisobutylene) matrix is investigated experimentally. A counter rotating parallel plate device, equipped with a microscopy setup, is used to visualize two interacting droplets during shear flow. The ratio of droplet-to-matrix viscosity is kept constant at 1.1. Droplet collisions are studied for a range of droplet sizes, both in bulk conditions and for gap spacings that are comparable to the droplet size. As a result, we present the first quantitative experimental data set for the coalescence of two equal-sized droplets in a pure shear flow with varying degrees of confinement. Compared to bulk conditions, for droplets smaller than roughly 0.2 times the gap spacing, a slight degree of confinement only decreases the orientation angle at which the droplets coalesce whereas the critical conditions for coalescence remain unaltered. For more confined conditions, the critical capillary number up to which coalescence can occur, increases. Therefore, confinement clearly promotes coalescence. In addition, the droplet trajectories, the time-dependent orientation angle of the droplet pair, and the droplet deformation prior to the coalescence event are systematically studied, and a comparison between the confined and the unconfined situation is provided. It is shown that the presence of two parallel walls can induce changes in the flow field around the droplet pair, which cause an increase of the interaction time between the droplets. Moreover, for sufficiently confined droplets, the additional force originating from the presence of the walls becomes comparable to the hydrodynamic force on the droplet pair, thus influencing the drainage of the matrix film between the droplet surfaces.

Introduction Coalescence of liquid droplets dispersed in an immiscible liquid matrix plays an important role in many industrial processes such as liquid-liquid extraction, emulsification, and polymer blending. Since coalescence inevitably involves droplet interactions, it is a more complex process than single droplet breakup, and hence more difficult to study both theoretically and experimentally. Coalescence is generally depicted as consisting of three consecutive steps: approach, drainage, and rupture.1 To obtain the trajectories of two non-Brownian interacting droplets, Wang et al.2 extended the creeping flow equations of Batchelor and Green3 for interacting solid particles to the case of spherical fluid droplets. In mixed linear flows, good agreement was found between the experimental trajectories and the predictions of Wang et al.4,5 However, in pure shear flow, Mousa et al.6 only obtained good agreement for small droplets. Guido and Simeone7 also studied the trajectories of interacting droplets in shear flow. Their experimental results are in agreement with the results of boundary integral simulatons,8,9 but a comparison with the trajectory theories was not provided. A great deal of work has been done on the interaction of two droplets that are in close *To whom correspondence should be addressed. E-mail: paula.moldenaers@ cit.kuleuven.be. Fax: 32(0)16 322991. (1) Chesters, A. K. Chem. Eng. Res. Des. 1991, 69, 259–270. (2) Wang, H.; Zinchenko, A. Z.; Davis, R. H. J. Fluid Mech. 1994, 265, 161–188. (3) Batchelor, G. K.; Green, J. T. J. Fluid Mech. 1972, 56, 375–400. (4) Tretheway, D. C.; Muraoka, M.; Leal, L. G. Phys. Fluids 1999, 11, 971–981. (5) Yang, H.; Park, C. C.; Hu, Y. T.; Leal, L. G. Phys. Fluids 2001, 13, 1087– 1106. (6) Mousa, H.; Agterof, W.; Mellema, J. J. Colloid Interface Sci. 2001, 240, 340– 348. (7) Guido, S.; Simeone, M. J. Fluid Mech. 1998, 357, 1–20. (8) Loewenberg, M.; Hinch, E. J. J. Fluid Mech. 1997, 338, 299–315. (9) Cristini, V.; Blawzdziewicz, J.; Loewenberg, M. J. Comp. Phys. 2001, 168, 445–463.

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contact and the drainage of the fluid film between the droplet interfaces.8,10-21 In particular, it has been shown that the drainage process depends on the droplet deformability8,10,11,13,15-17,21 and the mobility of the interfaces.1,17,18 The latter determines the boundary conditions for the flow in the thin film. Strong recirculation within the droplets can completely inhibit film drainage.9 Analytical solutions for the drainage between flat interfaces with various mobilities have been reviewed by Chesters.1 Finally, the critical factor in the coalescence process is whether there is enough time during the interaction of the droplets to drain the matrix film in between the droplet interfaces in order to allow the nonhydrodynamic attractive forces to cause coalescence. An overview of the different forces that can govern film rupture is given by Bergeron.22 Flow-induced coalescence in polymer blends and emulsions can be studied indirectly by measuring droplet sizes and droplet size distributions as a function of shearing time. A whole range of experimental techniques, including rheology, scattering (10) Davis, R. H.; Schonberg, J. A.; Rallison, J. M. Phys. Fluids A 1989, 1, 77–81. (11) Baldessari, F.; Leal, L. G. Phys. Fluids 2006, 18, 013602/1–20. (12) Zinchenko, A. Z. J. Appl. Math. Mech. 1978, 42, 1046–1051. (13) Zinchenko, A. Z. J. Appl. Math. Mech. 1981, 45, 564–567. (14) Zinchenko, A. Z.; Rother, M. A.; Davis, R. H. Phys. Fluids 1997, 9, 1493– 1511. (15) Rother, M. A.; Zinchenko, A. Z.; Davis, R. H. J. Fluid Mech. 1997, 346, 117–148. (16) Yiantsios, S. G.; Davis, R. H. J. Colloid Interface Sci. 1991, 144, 412–433. (17) Jaeger, P. T.; Janssen, J. J. M.; Groeneweg, F.; Agterof, W. G. M. Colloids Surf., A 1994, 85, 255–264. (18) Edwards, S. A.; Carnie, S. L.; Manor, O.; Chan, D. Y. C. Langmuir 2009, 25, 3352–3355. (19) Allan, R. S.; Mason, S. G. J. Colloid Sci. 1962, 17, 383–408. (20) Bartok, W.; Mason, S. G. J. Colloid Sci. 1959, 14, 13–26. (21) Rother, M. A.; Davis, R. H. Phys. Fluids 2001, 13, 1178–1190. (22) Bergeron, V. J. Phys.: Condens. Matter 1999, 11, R215–R238.

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techniques, and light and electron microscopy, are used for this purpose.23-27 However, in order to study the coalescence process in more detail and to gain fundamental insight, studies of the collision and coalescence of isolated droplet pairs are indispensable. Using a counter-rotating Couette device, Mason and coworkers19,20,28 studied the influence of surfactants, diffusion of a third mutually soluble component, and the presence of an electrical field on droplet coalescence in shear flow. More recently, systematic experimental studies of the flow-induced coalescence of two equal-sized droplets in two-dimensional linear flows were performed by Leal and co-workers.5,29-34 They used a miniaturized version of a four-roll mill, which enabled them to study coalescence in mixed flows with varying contents of vorticity and extension. In agreement with previous results in pure shear flow,19 they only observed coalescence at low flow intensities. Similar to droplet deformation and droplet breakup,35 the effect of the flow intensity on droplet coalescence can be described by the capillary number Ca (= ηm 3 γ_ 3 R/Γ, with ηm the matrix viscosity, γ_ the shear rate, R the droplet radius, and Γ the interfacial tension) that represents the ratio of the viscous to the interfacial stresses. Therefore, a critical capillary number, which is the upper limit for coalescence, has been determined for different systems and conditions. It was shown that this critical capillary number depends on the initial offset between the droplets in the velocity gradient direction, the viscosity ratio, the flow type, the molecular weight of the matrix fluid as well as the droplet size.5,31-34 Moreover, the presence of a block copolymer at the interface significantly inhibits droplet coalescence.31,36 In addition to the critical capillary number, the mentioned experimental studies mainly focus on the evaluation of scaling laws for the drainage time and the coalescence angle. Mousa et al.6 experimentally studied the shear-induced coalescence of two droplets with different diameters. They show that the angle at which coalescence occurs does not depend on the ratio of the droplet diameters in the range from 0.7 to 1. The literature in the broad area of simulations of merging surfaces and coalescence is quite extensive, and a complete account is beyond the scope of this work. However, it should be mentioned that three-dimensional (3D) numerical simulations of droplet coalescence in mixed linear flows are available (see, e.g., refs 21, 37, and 38). Nowadays, microfluidic devices are finding use in many scientific and industrial contexts.39-41 Multiphase flows provide several mechanisms for enhancing and extending the performance (23) Caserta, S.; Simeone, M.; Guido, S. Rheol. Acta 2004, 43, 491–501. (24) Ziegler, V. E.; Wolf, B. A. Macromolecules 2005, 38, 5826–5833. (25) Vinckier, I.; Moldenaers, P.; Terracciano, A. M.; Grizzuti, N. AIChE J. 1998, 44, 951–958. (26) Rusu, D.; Peuvrel-Disdier, E. J. Rheol. 1999, 43, 1391–1409. (27) Lyu, S.-P.; Bates, F. S.; Macosko, C. W. AIChE J. 2000, 46, 229–238. (28) Mackay, G. D. M.; Mason, S. G. Kolloid Z. 1964, 195, 138–148. (29) Borrell, M.; Yoon, Y.; Leal, L. G. Phys. Fluids 2004, 16, 3945–3954. (30) Hsu, A. S.; Roy, A.; Leal, L. G. J. Rheol. 2008, 52, 1291–1310. (31) Hu, Y. T.; Pine, D. J.; Leal, L. G. Phys. Fluids 2000, 12, 484–489. (32) Park, C. C.; Baldessari, F.; Leal, L. G. J. Rheol. 2003, 47, 911–942. (33) Yoon, Y.; Borrell, M.; Park, C. C.; Leal, L. G. J. Fluid Mech. 2005, 525, 355–379. (34) Leal, L. G. Phys. Fluids 2004, 16, 1833–1851. (35) Taylor, G. I. Proc. R. Soc. London, A 1934, 146, 501–523. (36) Ha, J. W.; Yoon, Y.; Leal, L. G. Phys. Fluids 2003, 15, 849–867. (37) Yoon, Y.; Baldessari, F.; Ceniceros, H. D.; Leal, L. G. Phys. Fluids 2007, 19, 102102/1–24. (38) Janssen, P. J. A.; Anderson, P. D.; Peters, G. W. M.; Meijer, H. E. H. J. Fluid Mech. 2006, 567, 65–90. (39) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. Rev. Fluid Mech. 2004, 36, 381–411. (40) Shui, L.; Eijkel, J. C. T.; van den Berg, A. Adv. Colloid Interface Sci. 2007, 133, 35–49. (41) Squires, T. M.; Quake, S. R. Rev. Mod. Phys. 2005, 77, 977–1026.

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of single-phase microfluidic systems.42 The design and optimization of such multiphase microfluidic applications requires a thorough scientific understanding of the relationships between the rheology of the components, the kinematic conditions, and the resulting morphology development. In order to gain fundamental insight in the underlying physics, the morphology development in “simple” confined geometries such as channel flow43 or shear flow44 is frequently studied. For the case of confined shear flows, experimental studies that focus on the effects of confinement on droplet breakup and droplet deformation have been reported.45-47 For example, for single droplets, the critical capillary number for droplet breakup is strongly affected by confinement: for low viscosity ratios, confinement suppresses breakup, whereas for high viscosity ratios, confinement promotes breakup.46 In more concentrated blends, geometrical confinement has been shown to introduce a range of particular morphological phenomena. Droplet interactions and droplet coalescence result in the formation of structures such as strings, pearl necklaces, and layered morphologies48,49 (for a recent review, see ref 44). For blends with viscosity ratios below 1, the appearance of bands with droplet-rich and depleted regions alternating along the vorticity axis, was observed. Although the ratio between droplet radius and gap size was rather low, the details of the banding phenomenon were influenced by the gap spacing.50 Despite the application potential of the aforementioned superstructures, a thorough understanding of their formation is presently lacking.44 Consequently, a good understanding of the coalescence process in confined geometries is crucial. Coalescence of isolated droplet pairs in microfluidic devices has received considerable attention during the last years. Among others, flow-induced coalescence in microfluidics has been demonstrated in expanding tapered or straight channels,51-53 in T-junctions,54 and in trifurcation junctions.53,55 However, to the authors’ knowledge, there are no experimental investigations of the coalescence of two droplets in confined shear flow. In the present paper, we report on what we believe to be the first quantitative experimental data set for the coalescence of two equal-sized droplets in a pure shear flow with varying degrees of confinement.

Materials and Methods Materials. Polydimethylsiloxane (PDMS Rhodorsil 47 V12500 from Rhodia) with a viscosity (ηd) of 11.2 Pa 3 s at the experimental temperature of 27 °C was used as the droplet phase. Polyisobutylene (PIB Glissopal V-190 from BASF) was used as the suspending fluid. Its viscosity (ηm) is 10.5 Pa 3 s at 27 °C. (42) Gunther, A.; Jensen, K. F. Lab Chip 2006, 6, 1487–1503. (43) Olbricht, W. L. Annu. Rev. Fluid Mech. 1996, 28, 187–213. (44) Van Puyvelde, P.; Vananroye, A.; Cardinaels, R.; Moldenaers, P. Polymer 2008, 49, 5363–5372. (45) Sibillo, V.; Pasquariello, G.; Simeone, M.; Cristini, V.; Guido, S. Phys. Rev. Lett. 2006, 97, 054502/1–4. (46) Vananroye, A.; Van Puyvelde, P.; Moldenaers, P. Langmuir 2006, 22, 3972– 3974. (47) Vananroye, A.; Van Puyvelde, P.; Moldenaers, P. J. Rheol. 2007, 51, 139– 153. (48) Migler, K. B. Phys. Rev. Lett. 2001, 86, 1023–1026. (49) Pathak, J. A.; Davis, M. C.; Hudson, S. D.; Migler, K. B. J. Colloid Interface Sci. 2002, 255, 391–402. (50) Caserta, S.; Simeone, M.; Guido, S. Phys. Rev. Lett. 2008, 100, 137801/1–4. (51) Bremond, N.; Thiam, A. R.; Bibette, J. Phys. Rev. Lett. 2008, 100, 024501/ 1–4. (52) Hung, L.-H.; Choi, K. M.; Tseng, W.-Y.; Tan, Y.-C.; Shea, K. J.; Lee, A. P. Lab Chip 2006, 6, 174–178. (53) Tan, Y.-C.; Fisher, J. S.; Lee, A. I.; Cristini, V.; Lee, A. P. Lab Chip 2004, 4, 292–298. (54) Christopher, G. F.; Bergstein, J.; End, N. B.; Poon, M.; Nguyen, C.; Anna, S. L. Lab Chip 2009, 9, 1102–1109. (55) Tan, Y.-C.; Ho, Y. L.; Lee, A. P. Microfluid. Nanofluid. 2007, 3, 495–499.

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Figure 1. Schematical representation of two colliding droplets in shear flow. This resulted in a viscosity ratio λ of 1.1. Both materials are Newtonian in the shear rate range of interest and have no measurable elasticity. It was shown before that PIB is slightly soluble in PDMS, but the solubility of PDMS in PIB is very limited.56 Therefore, with PDMS as the droplet phase, no droplet shrinkage was observed, and hence, the system can be considered to be completely immiscible. The interfacial tension Γ of the pair of liquids was determined by fitting the slow flow deformation data for a single droplet with the small-deformation theory of Greco,57 which resulted in a value of 1.8 mN/m. Buoyancy effects are expected to be negligible owing to the small density difference between the two liquids (0.08 g.cm-3).46 Setup. The experiments are performed with a counter rotating parallel plate shear flow device (based on a Paar Physica MCR300). It consists of two parallel quartz plates with a diameter of 50 mm. The motion of both plates can be separately controlled, and the plates rotate in opposite directions. This creates a stagnation plane, enabling the visualization of droplets with a stationary microscopy setup. The optical train consists of a stereo microscope (Wild M5A) combined with a digital camera (Basler 1394). This resulted in a resolution of 0.63 pixels/μm. The interacting droplet pairs are visualized in the velocity-velocity gradient plane. A detailed description of the device is given elsewhere.46,58 In the present work, a gap spacing of 1 mm was chosen for confined experiments, and 3 or 4 mm for unconfined experiments. The degree of confinement is expressed by means of the ratio of the droplet diameter 2R to the gap spacing H. The temperature of the sample was measured with a thermocouple and kept constant at 27 °C by controlling the temperature of the room. Terminology and Data Analysis. To illustrate the terminology and the data analysis method, a schematic of two colliding droplets is shown in Figure 1. The x-axis is in the flow direction, and the y-axis is in the velocity gradient direction. The relative trajectory of the two droplets will be expressed by ΔX = (X2 - X1) and ΔY = (Y2 - Y1). The dimensionless distance between the two droplets along the flow direction is defined as ΔX/2R. The offset is the dimensionless height difference, defined as ΔY/2R. The center-to-center distance between the two droplets is referred to as d. The orientation angle θ of the droplet pair is the angle between the line joining the droplet centers and the flow direction. Image analysis is performed with the commercial Image J Software package. Since the image obtained in the velocity-velocity gradient plane is slightly elongated along the x-axis due to the curvature of the cup around the bottom plate (see schematic of the device in the work of Vananroye et al.46), the distorted x-axis needs to be rescaled before image analysis can be performed. When the droplets come into apparent contact, their images can not be separated anymore by simple thresholding methods. In this case, the watershed algorithm was used to automatically generate (56) Guido, S.; Simeone, M.; Villone, M. Rheol. Acta 1999, 38, 287–296. (57) Guido, S.; Simeone, M.; Greco, F. J. Non-Newt. Fluid Mech. 2003, 114, 65– 82. (58) Verhulst, K.; Moldenaers, P.; Minale, M. J. Rheol. 2007, 51, 261–273.

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Article a white line between the cusps on both sides of the doublet, thus allowing separation of the images of the two droplets, as in the work by Guido and Simeone.7 Droplets were further analyzed by fitting an equivalent ellipse to the droplet contour. Experimental protocol. Droplets of PDMS were carefully injected in the PIB matrix by means of a homemade injection device. Before starting a coalescence experiment, two droplets with a similar size were obtained by breaking a single droplet into two daughter droplets. To break up the droplet, the sample was first sheared at a capillary number slightly above the critical value but not allowed to break up during shear flow. The flow was stopped when the droplet was extended in such a way that it breaks up into two daughter droplets during relaxation.59 This method resulted in two droplets with a difference in diameter of maximum a few percent. In unbounded linear flows, the critical capillary number for coalescence decreases with increasing droplet diameter,5,31,32 therefore the maximum droplet diameter resulting in coalescence within the attainable range of shear rates is 315 μm. It can be noted that the droplet sizes previously reported to result in coalescence are generally less than 200 μm in diameter.5,29,30,32-34 Since the long axis of an extended droplet in shear flow makes an angle with respect to the velocity direction, there is always a small distance between the centers of the two daughter droplets in the velocity gradient direction, which determines the initial offset ΔY/2R. By varying the Ca number and shearing time during droplet extension, the initial offset of the two droplets can be controlled. After breakup of the original droplet, shear flow was applied in such a way as to further separate the two droplets along the flow direction. Once the desired distance in the flow direction was reached, the flow was stopped, and the droplets were allowed to relax back to their spherical shape. Then, the flow direction was reversed to bring the droplets together for a coalescence experiment. To ensure that the droplets were in the middle of the gap during coalescence, the rotational velocities of the top and bottom plate were continuously monitored, while the droplet pair was kept in the stagnation plane. To systematically investigate the effect of confinement on droplet coalescence, the droplet diameter, gap spacing, and Ca number were varied, whereas the initial dimensionless distance ΔX/2R was fixed at about 1.6, and the initial dimensionless offset ΔY/2R was fixed at about 0.16; for all experiments, the maximum deviation is 0.005. The Re number Re (= F 3 γ_ 3 R2/ηm, with F the fluid density) remains below 1  10-7 in all experiments and therefore inertia is negligible. The Pe number Pe (= γ_ 3 6π 3 R3/kT, with k the Boltzmann constant and T the absolute temperature) remains above 1  107 and Brownian motion is thus negligible.

Experimental Results Droplets with diameters between 108 and 315 μm were selected to carry out the experiments. Gap spacings of 3 or 4 mm were chosen to obtain a set of results for droplets with confinement ratios 2R/H below 0.085. For these cases, the walls are not expected to influence the droplet behavior, and therefore this will be considered as the reference case of an unbounded shear flow. In addition, a set of experiments was performed for droplets with the same diameters, but in a gap with a height H of 1 mm, thus resulting in confinement ratios 2R/H between 0.108 and 0.315. Time Sequences of the Droplet Behavior. Typical examples of time sequences of the collision and coalescence of two droplets at a Ca number of 0.0036 are shown in Figure 2 for droplet diameters of 191 and 227 μm. The left columns for each droplet size demonstrate the behavior in a gap of 3 mm, the right ones represent that in a gap of 1 mm. During flow, after some time, the droplets come into apparent contact and form a rotating droplet (59) Stone, H. A.; Bentley, B. J.; Leal, L. G. J. Fluid Mech. 1986, 173, 131–158.

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Figure 2. Time sequence of the collision and coalescence of two droplets, initial offset ΔY/2R = 0.16, initial ΔX/2R = 1.6, Ca = 0.0036. Left column: 2R = 191 μm; (a-e) 2R/H = 0.064; (a0 -e0 ) 2R/H = 0.191. Right column: 2R = 227 μm; (f-j) 2R/H = 0.076, (f0 -j0 ) 2R/H = 0.227. Original images without rescaling of the x-axis.

pair. Simultaneously, the film between the droplets is being drained. Eventually, when it has become sufficiently thin, van der Waals forces cause film rupture, resulting in droplet coalescence.1 It is shown in Figure 2 that, for the droplets with a diameter of 191 μm, in both conditions, the droplets coalesce, but the angle θ at which coalescence occurs is different. The coalescence angle for the droplet pair in the 3 mm gap with 2R/H = 0.064 (65°, Figure 2c) is higher than that for the droplet pair with 2R/H = 0.191 (51°, Figure 2c0 ). Results for slightly bigger droplets with confinement ratios 2R/H of 0.076 and 0.227, respectively, are shown in the right column of Figure 2. Coalescence occurs in a gap of 1 mm (Figure 2f0 -j0 ), while the droplets in a gap of 3 mm (Figure 2f-j), under identical conditions, merely rotate over each other and separate again. Thus, for droplets with a higher confinement ratio, confinement clearly promotes coalescence. From the results of Figure 2 it can be concluded that, at a gap spacing of 1 mm, the presence of the walls alters the behavior of the droplet pairs. Therefore, confinement effects are present, and the results obtained at this gap spacing will be referred to in this work as confined experiments. Coalescence Angles. The difference in coalescence angle, as shown in Figure 2, was observed systematically in the range of droplet sizes and Ca numbers studied here. Figure 3a,b shows the coalescence angle as a function of the Ca number in confined and unconfined conditions. Since the critical Ca number is known to decrease with increasing droplet diameter,31,32 results are presented here for the two smallest droplet sizes, for which the widest range of data points is available. As droplet coalescence is an almost instantaneous process, the coalescence angle can be determined very precisely; some experiments were performed twice, and deviations up to 2° were generally found. It is clear from Figure 3 that the coalescence angle in bulk conditions increases with the Ca number, similar to the results available in literature for pure extensional flow.29,33 This can be explained by the dependence of the drainage rate on the interaction force. An increase in the hydrodynamic force, which is caused by the higher flow intensity, increases the size of the contact area 12888 DOI: 10.1021/la901807k

Figure 3. Coalescence angle as a function of capillary number Ca, initial offset ΔY/2R = 0.16: (a) 2R = 108 μm, (b) 2R = 191 μm.

between the droplets, which reduces the rate of film thinning.1 Also, for slightly confined droplets, with a confinement ratio up to 0.2, an increase of the coalescence angle with the Ca number is systematically observed. However, the coalescence angles for the confined droplets are lower than those of the droplets in an unbounded flow. This implies that either the drainage speed is increased due to confinement or the rotational speed of the doublet is reduced, providing more time for film drainage. From Figure 3a, which shows the results for the smallest droplet size, it can be seen that, at very low Ca numbers, confined and unconfined droplets show exactly the same behavior. However, with increasing Ca number, the difference between the coalescence angles in bulk and confined conditions gradually increases. Therefore, it can be concluded that, at these low confinement ratios, confinement effects are only present at sufficiently high flow intensities. This is expected since it has been shown that the effect of the walls on different aspects of the droplet behavior such as droplet deformation,47 migration of deformed droplets perpendicular to the walls49 and the additional drag force parallel to the walls on droplets in a confined shear flow60 (see also the Discussion section), all increase when higher flow intensities are applied. In addition, for the same droplet size, the portion of the gap that is blocked by the droplet pair increases when its orientation angle increases. Therefore, at the highest Ca numbers, where the highest coalescence angles are obtained, more influence of confinement is expected. For higher confinement ratios, as shown in Figure 3b for a bigger droplet size with a maximum confinement ratio of 0.191, (60) Shapira, M.; Haber, S. Int. J. Multiphase Flow 1990, 16, 305–321.

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Figure 4. Coalescence angle as a function of droplet diameter for gap spacings of 1 mm and 3 mm, initial offset ΔY/2R = 0.16: (a) Ca = 0.0022, (b) Ca = 0.0036. Lines are drawn to guide the eye.

the effect of confinement is already significant at the lowest Ca. Under the same flow conditions (same Ca number), the effects of the walls thus become more pronounced at higher confinement ratios, as can be seen in a comparison of panels a and b of Figure 3. This is not surprising since the distance between the droplet interface and the walls decreases for increasing values of the confinement ratio. Similar results were obtained for droplet breakup46 and droplet deformation.45,47 In addition, the results in Figure 3b show that, for these bigger droplets, the effect of the walls also increases when the flow intensity is increased. The coalescence angle as a function of the droplet diameter is shown in Figure 4 for gap spacings of 1 and 3 mm. Figure 4a,b shows the results at Ca numbers of 0.0022 and 0.0036, respectively, values that were chosen to provide results for the maximum possible range of droplet diameters. For droplets in an unbounded shear flow at both Ca numbers, starting from the same initial offset ΔY/2R, an increase in the droplet size postpones droplet coalescence to higher orientation angles of the droplet pair. Loewenberg and Hinch8 have theoretically shown that the extent of the near-contact region between two droplets increases with the droplet size, which decreases the rate of film drainage.1 Therefore, an increase of the droplet size is expected to postpone coalescence. The results obtained at gap spacings of 1 mm and 3 mm coincide for the smallest droplet size, showing that bulk behavior still prevails under these conditions. When the droplet size is increased, however, it is clear that droplets in a confined environment have lower coalescence angles than their bulk counterparts. This effect was already shown in Figure 3 for Langmuir 2009, 25(22), 12885–12893

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different Ca numbers. For small droplets, an increase of the coalescence angle with increasing droplet size is still present, while for bigger droplets, the coalescence angle in confined conditions decreases with increasing droplet size. Under these conditions, wall effects thus either enhance drainage or retard rotation of the doublet that much that this effect overcomes the extra difficulties in film drainage caused by increasing the droplet size. Therefore, under certain conditions, bigger droplets are more advantageous to obtain coalescence than smaller ones. This might be an important factor in the formation of strings in concentrated blends in a confined environment.48 Droplet coalescence increases the confinement ratio due to an increase in the droplet size, which then enhances coalescence, leading eventually to very big droplets or strings. Similar to the results of Figure 3, these confinement effects set in at a lower confinement ratio when the flow intensity is higher, as can be deduced from a comparison between panels a and b of Figure 4. For the Ca numbers and droplet sizes studied in the present work, coalescence was always observed at orientation angles of the droplet pair that are below 90°. Therefore, coalescence in the extensional quadrant of the flow, which was present for systems with a similar viscosity ratio and offset, but in pure extensional flow,29,33,51 was never observed here. Critical Ca Number. Figure 2 clearly shows that confinement can promote coalescence. Therefore, the critical Ca number for coalescence was systematically determined for droplets with different diameters. Figure 5 shows Cacrit versus the droplet diameter 2R for both unconfined and confined droplets. The closed symbols represent the highest Ca number at which coalescence occurred, the open symbols the lowest Ca number at which no coalescence was present. For unconfined droplets, Cacrit decreases with increasing droplet size; the smaller the droplets, the broader the range of Ca numbers in which coalescence can occur. These results are in qualitative agreement with literature results that were obtained for mixed linear flows.31,32,34 When the droplet diameter is above 315 μm, Cacrit is close to 0, which does not allow for accurate experiments with our setup. For droplets in a smaller gap of 1 mm, the trend is clearly different. For small droplets (2R < 200 μm, 2R/H < 0.2), the results coincide exactly with those for the droplets in bulk shear flow. Although a decrease of the coalescence angle with respect to bulk conditions was observed (see Figure 3), for confinement ratios up to a value of 0.2, the critical capillary number is the same in both confined and unconfined conditions. For bigger droplets (2R > 200 μm, 2R/H > 0.2), however, the Cacrit increases with increasing droplet size and confinement ratio. This trend is similar to the results for the coalescence angle, that are shown in Figure 4. Clearly, coalescence enhancement due to confinement can overrule the coalescence suppression due to the increase in droplet size. The previous results thus show that confinement promotes coalescence, as big droplets that would not be able to coalesce in unconfined conditions may coalesce when confined. At very high confinement ratios and high flow intensities, wall migration will gain importance.49 As a consequence, droplets will migrate toward the center line during their approach, and their offset will become very low. In the latter case, the time for the droplets to move toward each other is very long, and they may reach the center line before coalescence can occur. Therefore, the curve for the critical Ca number of confined droplets in Figure 5 is not expected to maintain its increasing trend up to a confinement ratio of 1. However, perfectly density-matched systems are needed to perform accurate experiments at those high confinement ratios. On the other hand, coalescence in highly confined geometries is observed in more concentrated blends, where f.e. droplets in a DOI: 10.1021/la901807k

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Figure 5. Critical capillary number for coalescence Cacrit as a function of droplet diameter 2R. Filled symbols represent the highest Ca where coalescence occurred, open symbols indicate the lowest Ca number where no coalescence was observed.

pearl necklace form strings, or several strings coalesce into a ribbon.48 For these systems, the situation is far more complex. Interactions between multiple droplets do not only cause changes in the droplet trajectories, but also in the local flow intensity, thereby possibly stretching and rotating droplets that are in close contact, which might enable them to touch. Migler48 attributed the coalescence of strings into a ribbon to a hydrodynamic attraction between the strings, which is caused by the presence of the walls. Time-Dependent Behavior of the Droplet Pairs. To clarify the observed differences between the coalescence angles of unconfined and confined droplets, a methodology to study the complete time-dependent behavior of the droplet pairs, as proposed by Guido and Simeone,7 is applied. For this purpose, the droplet trajectories, approach velocities, and the droplet deformation are investigated in detail for the cases depicted in Figure 2. The results are given in Figures 6-9. Figure 6 shows the offset ΔY/2R of two droplets with a diameter of 227 μm as a function of ΔX/2R during approach and collision, up to the moment of coalescence or separation. For the unconfined droplet, it can be seen that the offset ΔY/2R starts to increase at a dimensionless distance ΔX/2R = 1.2 (end of region 1). The droplets thus start to interact hydrodynamically when they are still visibly separated. After the two droplets come into apparent contact (ΔX/2R = 0.92, d/2R = 1, end of region 2), the droplet pair starts to rotate, resulting in a swift increase of the offset (region 3). Subsequently, the offset goes through a maximum, and finally, after separation of the droplets, reaches a new steady-state value. The final value of the offset is clearly higher than the value before collision. This result is in agreement with the experiments of Guido and Simeone7 and the numerical simulations of Loewenberg and Hinch.8 Allan and Mason19 showed that this is only the case for fluid droplets, while hard spheres show a symmetrical behavior around ΔX/2R = 0. A comparison between the droplet trajectories for an unbounded and a bounded droplet with a diameter of 191 μm is included in Figure S1 in the Supporting Information. This figure shows that, up to a confinement ratio of 0.2, the relative trajectories followed by the centers of both droplets are completely similar in bulk and confined conditions. The droplet pair in the unconfined situation however, rotates further before coalescence occurs. When comparing the results for the confined droplet in Figure 6 with the results for the unbounded droplet, it becomes clear that 12890 DOI: 10.1021/la901807k

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Figure 6. Offset ΔY/2R as a function of ΔX/2R corresponding to Figure 2f-j and f0 -j0 for droplets with 2R = 227 μm.

there is a difference between the behavior of droplets with a confinement ratio 2R/H below and above 0.2. It can be seen that, for the more confined droplets, the offset decreases very slightly up to the point where the droplet interfaces almost touch (ΔX/2R ≈ 1), a feature that is not present for unconfined droplet pairs. Most probably this decrease is caused by differences in the flow field around the droplet pair, as for highly confined droplets, and at high flow intensities numerical simulations have shown that large recirculation zones are present at the front and rear of the droplets.61 Therefore, a second droplet that is positioned on such a stream line will not easily flow around the first droplet but remains trapped in the dead zone in front of this droplet for a long time. The exact velocity and pressure profiles for the modestly confined droplets at lower flow intensities that are studied in the present work, are unknown however. Once the droplet doublet has started to rotate, the difference between the trajectory for the unconfined and the confined droplets vanishes. The trajectory followed by the centers of the two droplets of the rotating droplet pair, depends on the extent of overall droplet deformation.5,7,11,34 At high Ca numbers, the external flow might deform the initially spherical droplets into ellipsoids. In pure extensional flow, both Baldessari and Leal11 and Yang et al.5 observed center-to-center distances that were significantly below 2R at Ca numbers as low as 0.01. Therefore, the trajectory ΔY/2R - ΔX/2R that represents the rotation of a pair of hard spheres is added to Figure 6. In that case, the distance between the droplet centers remains constant throughout the interaction process, and the trajectory thus obeys the following equation: 1¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y 2

ð1Þ

It is clear from the figure that, under the present conditions, the droplets behave as a rotating pair of spheres. In order to study the global droplet deformation due to the external flow in more detail, the droplet deformation parameter D is presented as a function of the dimensionless distance ΔX/2R in Figure 7 for droplets with a diameter of 191 μm. Here, the Taylor35 definition for the droplet deformation parameter is used: D = (L - B)/(L þ B), with L and B being , respectively, the major and minor axis of the deformed droplet in the velocity-velocity gradient plane. The average of the deformation parameter for the two droplets in each doublet is shown. However, the droplet deformation parameter D of a spherical droplet obtained with the (61) Renardy, Y. Rheol. Acta 2007, 46, 521–529.

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Figure 7. Droplet deformation parameter D as a function of ΔX/2R corresponding to Figure 2a-e and a0 -e0 for droplets with 2R = 191 μm.

used image analysis technique can rise up to 0.002. Therefore, this can be seen as the uncertainty of the data. Since the contact area between two droplets in a doublet is limited at these low Ca numbers, the accuracy of determining the deformation parameter for an isolated droplet and a droplet in a doublet is expected to be similar. Before the droplets start to interact, the deformation parameter is approximately zero due to the very low Ca (= 0.0036). The value of the bulk deformation parameter D that is expected from the Taylor small deformation theory35 at this Ca number is 0.004. This corresponds to an almost spherical droplet, which is confirmed by the experimental data. It has been shown that the droplet deformation parameter for isolated, confined droplets is well described by the Shapira-Haber theory.45,47,60 This theory states that the droplet deformation in confined conditions can be obtained by multiplying the Taylor bulk deformation parameter with a Shapira-Haber factor that accounts for the effects of the walls. For the conditions of the present experiments, this factor is 1.009. Therefore, no difference in deformation parameter in confined and unconfined conditions is expected to be detectable with our experimental setup, as is confirmed in Figure 7. A slight increase of the deformation parameter arises when the two droplets come into apparent contact (ΔX/2R = 0.92, region 3). Within experimental error, the results for the droplets in bulk shear flow, and that in bounded shear flow remain, however, indistinguishable. The results for the deformation parameter of droplets with different diameters or Ca numbers are similar to those shown in Figure 7 and are omitted for the sake of brevity. Figure 8 shows the velocity in the flow direction between the centers of mass of two droplets with a diameter of 191 μm. The velocity is calculated as d(ΔX)/dΔt, which is plotted as a function of ΔX/2R. At a dimensionless distance ΔX/2R ≈ 1.2, where the droplets start to feel each other’s presence, the relative velocity starts to decrease, and it reaches a minimum when the two droplets come into apparent contact (ΔX/2R = 0.92, d/2R = 1). Thereafter, when the droplets start sliding over each other (region 3), the velocity increases rapidly up to a maximum where coalescence occurs. The curves for the confined and unconfined droplets coincide during the complete approach and collision process. However, the droplets in bulk shear flow rotate further before coalescing. A prediction for the approach velocity of two droplets that move with the applied velocity Vx in the flow direction is included in Figure 8. The latter is calculated as Vx= _ γΔy, where γ_ is the shear rate, and Δy is the experimental value Langmuir 2009, 25(22), 12885–12893

Figure 8. Velocity as a function of ΔX/2R corresponding to Figure 2a-e and a0 -e0 for droplets with 2R = 191 μm.

ΔY corresponding to each ΔX/2R. Before collision (region 1), the offset and thus the relative velocity Vx is constant, and the calculated curves coincide with the experimental data. As a result of the rotation of the droplet pair, the offset ΔY and thus the relative velocity Vx increase. However, the calculated increase in relative velocity Vx significantly exceeds the experimental one, since one drop retards the other during interaction (regions 2 and 3), an aspect that is not taken into account in the calculations. However, no influence of confinement can be noticed, neither for the calculated nor for the experimental curves. The orientation angle of the droplet pairs as a function of time is shown in Figure 9 for droplets with a diameter of 227 μm, enabling a comparison of the rotation speed of the doublets in bulk and confined conditions. The starting point of the x-axis for each curve is the time at which ΔX/2R = 1.2, the point where the trajectories of unconfined and confined droplets start to differ (see Figure 6). In bulk conditions, the second droplet immediately moves around the first one, resulting in an increase of the orientation angle. However, for the confined droplet, the difference in the trajectories (as shown in Figure 6) causes a much slower increase of the orientation angle. A comparison of the curves for the orientation angle as a function of time for smaller droplets is given in Figure S2 of the Supporting Information. It is clear from Figure S2 that, up to a confinement ratio of 0.2, the rotation speed of the droplet pair is not influenced by the presence of the walls. Therefore, the difference in rotation speed for doublets with confinement ratios above 0.2 is clearly related to the change in the relative trajectory of the droplets. Drainage Times. By taking the instant at which d/2R = 1 as the onset of film drainage, as in Yang et al.,5 the drainage times could be determined for different Ca-numbers and droplet sizes. Although it was pointed out by Hsu et al.30 that this method suffers from experimental limitations at low Ca numbers, consistent results were obtained here for all cases. For droplets with a confinement ratio below 0.2, as shown in Figure S2 in the Supporting Information, it is clear that the drainage time is shorter in confined than in bulk conditions. Because of the presence of the walls, minor differences in the film thickness and shape at the instant that is defined as the starting point of the drainage process can, however, not be excluded. Therefore, although the initial time instant of drainage is exactly the same DOI: 10.1021/la901807k

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Confinement between two parallel walls can influence different aspects of the coalescence process. First, it was shown in Figures 6 and 9 that, for moderately confined droplets, confinement causes changes in the flow field around the droplet pair that postpone the rotation of the first droplet around the second one. This increases the interaction time of the droplets at low orientation angles and thus directly alters the time history of the contact force, which is an important factor in the drainage process.29 In addition to changes in the flow field, the presence of two parallel walls also gives rise to wall forces that are superimposed on the forces exerted on a droplet pair in bulk shear flow. A general force balance on a droplet pair in pure shear flow results in the following equation: Fh þ Fvdw þ Fwall ¼ Fl Figure 9. Orientation angle of the droplet pair as a function of

time corresponding to Figure 2f-j and f0 -j0 for droplets with 2R = 227 μm; (A) and (B) indicate the instants where d/2R = 1.

when considering the external flow and the position, velocity, and global deformation of the droplets, the initial conditions with respect to the film drainage process might be profoundly different due to variations in the initial state of the draining film. For droplets with a confinement ratio above 0.2, the different trajectories experienced by unconfined and confined droplets (as shown in Figure 6) also have implications on the contact times between the droplets (see Figure 9). In this case, the rotational speed of the droplet pairs depends on the confinement ratio, which makes a comparison less straightforward. In Figure 9, the onset of film drainage is indicated by points A and B, respectively, for the bulk and the confined case. Since in a shear flow, droplets are being pulled apart at orientation angles above 90°, this is the end point of film drainage for the unconfined droplet pair. It is clear from Figure 9 that the interaction time in bulk shear flow exceeds that in confined shear flow, but yet the unconfined droplets do not coalesce. A longer drainage time is thus required in bulk conditions. Therefore, the experimental data indicate that the interaction time is not the dominant factor, which suggests that the drainage process itself is altered due to confinement. From a comparison between the coalescence process in a glancing collision and a head-on collision with a timedependent force that mimics the force history in the glancing collision, Borrell et al.29 concluded that the time history of the force along the line of centers is the dominating factor that determines the outcome of a collision event. Since confinement induces changes in the flow and pressure fields around the droplets, both confined and unconfined droplets are exposed to different force histories that might lead to differences in the thin film drainage process.

Discussion Flow-induced coalescence of two Newtonian droplets in a Newtonian matrix is generally described by considering an “inner” and an “outer” problem.1 The outer problem consists of two colliding droplets in an external flow field and determines the duration of the collision and the time dependence of the contact force. The inner problem consists of the thinning of the film in between the droplet interfaces, which will eventually lead to film rupture and thus coalescence if sufficient time is available. In general, the outer problem sets the boundary conditions for the inner problem, but due to the complexity of the coupling of both problems, the limiting cases of a constant contact force or a constant approach velocity are often used in modeling.38 12892 DOI: 10.1021/la901807k

ð2Þ

where Fl is the total lubrication force in the gap between the droplets that counteracts the sum of the hydrodynamic force along the line of centers Fh, the van der Waals attraction force between the interfaces Fvdw, and the additional force from the confining walls Fwall. The time history of the total lubrication force determines the details of the thin film thinning. If the pressure in the film exceeds the Laplace pressure, deformation of the interface takes place. The hydrodynamic force Fh along the line of centers can be described by the following expression:17   2=3 þ λ π 3 ηm 3 γ_ 3 R2 3 sinð2ð90 - θÞÞ Fh ¼ 4:34 1þλ

ð3Þ

The van der Waals attraction force Fvdw only becomes important at small film thicknesses and depends on the radius of the deformed film and the distance between the interfaces.17 The presence of two parallel walls generates forces perpendicular and parallel to those walls.62 Shapira and Haber60 theoretically derived the drag force parallel to the walls for nearly spherical droplets in a confined shear flow between two parallel plates: Fwall ¼ 4 3 π 3 ηm 3 γ_ 3 R

2 ðηm

þ 3=2ηd Þðηm þ 5=2ηd Þ ðηm þ ηd Þ2

 2 R 3 H 3 CD ð4Þ

In this equation, CD is the drag coefficient that depends on the distance between the droplet center and the nearest wall. Values for CD can be found in the original paper.60 Expressions for the migration velocities perpendicular to the walls can be found in several works (e.g., Chan and Leal63). However, if the matrix fluid is Newtonian, migration perpendicular to the walls does not occur for spherical droplets.60,62 Since it is shown in Figure 7 that the droplet deformation remains negligible during a collision event, and the effect of confinement on the deformation parameter is immeasurably small, the wall force parallel to the walls is expected to be dominant under the present conditions. Figure 10 provides a comparison between the projection on the x-axis of the hydrodynamic force on the droplet doublet and the wall force along the x-axis on a droplet. The calculations have been performed for droplets in a gap of 1 mm both for a very small droplet where no effect of confinement was recorded (2R = 108 μm, Ca = 0.0022 and 2R/H = 0.108) and for a droplet size where effects of confinement were observed (2R = 315 μm, Ca = 0.0072 (62) Uijttewaal, W. S. J.; Nijhof, E.-J.; Heethaar, R. M. Phys. Fluids A 1993, 5, 819–825. (63) Chan, P. C. H.; Leal, L. G. J. Fluid Mech. 1979, 92, 131–170.

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Figure 10. Hydrodynamic force (Fh) and wall force (Fwall) as a function of the orientation angle of the droplet pair for a small droplet with 2R = 108 μm, Ca = 0.0022 and 2R/H = 0.108 and a big droplet with 2R = 315 μm, Ca = 0.0072 and 2R/H = 0.315.

and 2R/H = 0.315). It is clear from eq 3 that the hydrodynamic force shows a maximum value at an orientation angle of 45° and changes sign after 90° where the droplets are pulled apart. The wall force increases throughout the rotation process up to 90° since the distance between the droplet and the adjacent wall decreases during this process. Therefore, Figure 10 shows that, for both droplet sizes, the relative importance of the wall force increases when the orientation angle of the droplet pair increases. More effect of confinement is thus expected for systems with a higher coalescence angle. This agrees with the experimental results, where it was shown that confinement effects are most obvious at high Ca numbers and for big droplets (see Figures 3, 4, and 5). In addition, it is shown in Figure 10 that, whereas for the small droplet the wall force only amounts to approximately 1% of the hydrodynamic force at the initial stage of the coalescence process; for the biggest droplet the wall force starts at 10 % of the hydrodynamic force. Therefore, it is clear that this wall force can influence the dynamics of the film drainage. It needs to be mentioned, however, that the calculations performed here only serve to provide an indication of the relative order of magnitude of the different forces since the drag force was derived for a single droplet confined between two parallel walls, while, in the present experiments, a droplet pair in a confined shear flow is studied. In addition, the asymmetry of the doublet in the gap will produce an additional wall force perpendicular to the wall. Nevertheless, it is clear that wall forces are expected to play a role in the observed phenomena. Numerical simulations will be needed to resolve the details of the pressure and velocity fields around a confined droplet pair and to elucidate the effects of the walls on the thinning of the matrix film in more detail. Finally, it should be mentioned that the rate of film drainage depends on the extent of the near-contact region between two droplets. It has been shown both numerically and experimentally that this region expands due to droplet deformation and therefore global droplet deformation reduces the rate of film drainage.5,8,21

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Baldessari and Leal11 numerically studied the effect of the overall droplet deformation on the film drainage during a head-on collision. They show that, at an inter droplet distance d = 2R, where the interfaces of spherical droplets would touch, the film thickness due to global droplet deformation for deformed droplets is O(Ca 3 R). At the Ca number of the present experiments, this would give a value of 340 nm. On the basis of the Hamaker constant (A) for our system, which is 1  10-20 J,64 the order of magnitude of the critical film thickness at which film rupture due to the van der Waals forces occurs, hc = (A 3 R/8πΓ)1/3, is roughly estimated to be 27 nm.1 Thus, even very small differences in the global droplet deformation could have a significant effect on the thin film properties. In our experiments, the deformation parameters are very low, therefore, in both confined and unconfined conditions, the droplet pair behaves essentially as a rotating pair of two contacting spheres that glide over each other. Slight differences in droplet deformation might, however, have an influence on the film thickness or shape. The dimensions and details of this near-contact region can not, however, be resolved precisely enough from the present experiments to detect any differences between the confined and the unconfined situations.

Conclusions The effect of confinement on droplet coalescence in simple shear flow is investigated systematically. In bulk conditions, the critical capillary number for coalescence decreases with increasing droplet size. At low degrees of confinement, the coalescence angle of droplets is decreased with respect to bulk conditions, but the critical capillary number remains unchanged. The droplet trajectories and time-dependent behavior of a rotating droplet pair are also completely identical in bulk and slightly confined conditions. For moderately confined droplets on the other hand, both the coalescence angle and the critical Ca number are influenced by the presence of the walls. Above a confinement ratio 2R/H ∼ 0.2, a substantially higher Cacrit is obtained for confined droplets. At these higher confinement ratios, changes in the flow field postpone the rotation of the second droplet around the first one. We showed that, in addition to changes in the flow field, additional forces originating from the presence of the walls also have an influence on the coalescence process and are in line with our observations. Numerical simulations are, however, needed to resolve the details of the interaction between the droplet coalescence process and the confining walls. Acknowledgment. R.C. is indebted to the Research Foundation-Flanders (FWO) for a Ph.D. Fellowship. This work is partially funded by Onderzoeksfonds K.U.Leuven (GOA09/002). Supporting Information Available: Offset ΔY/2R as a function of ΔX/2R corresponding to Figure 2a-e and a0 -e0 for droplets with 2R = 191 μm. Orientation angle as a function of time corresponding to Figure 2a-e and a0 -e0 for droplets with 2R = 191 μm. This material is available free of charge via the Internet at http://pubs.acs.org. (64) Peters, G. W. M.; Hansen, S.; Meijer, H. E. H. J. Rheol. 2001, 45, 659–689.

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