Article pubs.acs.org/Langmuir
Coalescence and Noncoalescence of Sessile Drops: Impact of Surface Forces Stefan Karpitschka,*,† Christoph Hanske,‡ Andreas Fery,‡ and Hans Riegler*,† †
Max Planck Institute of Colloids and Interfaces, 14424 Potsdam, Germany Physical Chemistry II Department, University of Bayreuth, 95440 Bayreuth, Germany
‡
S Supporting Information *
ABSTRACT: Due to capillarity, sessile droplets of identical liquids will instantaneously fuse when they come in contact at their three-phase lines. However, with drops of different, completely miscible liquids, instantaneous coalescence can be suppressed. Instead, the drops remain in a state of noncoalescence for some time, with the two drop bodies connected only by a thin neck. The reason for this noncoalescence is the surface tension difference, Δγ, between the liquids. If Δγ is sufficiently large, then it induces a sufficiently strong Marangoni flow, which keeps the main drop bodies temporarily separated. Studies with spreading drops have revealed that the boundary between instantaneous coalescence and noncoalescence is sharp (Karpitschka, S.; Riegler, H. J. Fluid. Mech. 2014, 743, R1). The boundary is a function of two parameters only: Δγ and Θ̅ a, the arithmetic mean of the contact angles in the moment of drop−drop contact. It appears plausible that surface forces (the disjoining pressure) could also influence the coalescence behavior. However, in experiments with spreading drops, surface forces always promote coalescence and their influence might be obscured. Therefore, we present here coalescence experiments with partially wetting liquids and compare the results to the spreading case. We adjust different equilibrium contact angles (i.e., different surface forces) with different substrate surface coatings. As for spreading drops, we observe a sharp boundary between regimes of coalescence and noncoalescence. The boundary follows the same power law relation for both partially and completely wetting cases. Therefore, we conclude that surface forces have no significant, explicit influence on the coalescence behavior of sessile drops from different miscible liquids.
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surface tensions. In essence, this surface tension difference, Δγ, has been identified as the primary origin of the temporary noncoalescence. The Δγ causes a Marangoni flow in the connecting neck region between the two drops. As a consequence, the surface tension gradient will be asymmetric with respect to the neck and localized on the side with the higher surface tension (Figure 1).19 Therefore, this flow drains the neck region and acts against the coalescence of the two drops. Under certain conditions, this can lead to a temporary situation of noncoalescence, which has been described theoretically as a quasi-stationary, traveling-wave-like solution of the thin-film equation.19 Recently a scaling analysis in the limit of lubrication flow20 has revealed that essentially two control parameters determine whether the two drops remain in a state of temporary noncoalescence or coalesce instantaneously. The two control parameters are (1) Δγ, the surface tension difference, and (2) Θ̅ a, the arithmetic mean of the two three-phase angles of the two drops in the moment of lateral drop−drop contact. Δγ characterizes the (drop-separating) Marangoni force. Θ̅ a is a measure of the negative curvature in the neck region and therefore describes the (coalescence-promoting) capillary
INTRODUCTION Understanding the interaction between droplets from miscible liquids is important. This holds for droplets with identical liquids as well as for those with different, miscible liquids. In particular, the case of sessile droplets with different, miscible liquids is ubiquitous in nature and also highly relevant to a number of technological applications such as microfluidics1−4 and inkjet printing.5−7 In the semiconductor industry, Marangoni drying is routinely used for wafer surface cleaning. It is common knowledge that capillarity promotes the coalescence of sessile drops as soon as the drops come into contact. With identical liquids, the drop−drop contact will immediately initiate coalescence. In this case, the evolution of the drop contours and the hydrodynamics developing after the drop−drop contact have been analyzed in several publications.11−15 Quite unexpectedly, for different but still completely miscible liquids the coalescence behavior is very different. In this case, sessile drops do not always coalesce instantaneously upon contact. Rather, in many cases the drop bodies remain separated in a temporary state of noncoalescence, connected only through a thin liquid bridge.16−18 In this configuration, the drops move along the substrate and slowly exchange liquid through their connecting neck. After some time (up to minutes), the drops will finally merge. Meanwhile, this unexpected coalescence behavior is understood to some degree. Different liquids usually have different © 2014 American Chemical Society
Received: February 3, 2014 Revised: May 14, 2014 Published: May 19, 2014 6826
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For the partially wetting case, the situation is very different, and this is the topic of this report. Here we probe experimentally if surface forces do have an influence on the coalescence behavior. Therefore, we study partially wetting cases with different Π(h) and compare the behavior with the case of completely wetting liquids. In contrast to the complete wetting case, for partial wetting, Π(h) favors film thinning down to an equilibrium thickness which is of molecular size. This effect could add to the other contribution acting against coalescence, namely, the Marangoni flow. Therefore, the threshold angle Θ̅ t could be larger and/or Δγ could be smaller than for the corresponding complete wetting cases. In the partial wetting case the influence of Π(h) can be measured quantitatively because the (measurable) equilibrium contact angle ΘE is directly related to the disjoining pressure Π(h) (here in the lubrication approximation):21
Figure 1. For drops with sufficiently different surface tensions, the Marangoni flow out of the connecting neck compensates for the capillary flow into the neck. Thus, the drops remain in a quasistationary twin drop configuration of temporary noncoalescence.
ΘE =
pressure. Both parameters can be combined in a modified Marangoni number20 3Δγ M̃ = 2 2γ ̅ Θ̅a
2 γ
∫h
∞
Π(h) dh (2)
p
Here hP is the thickness of the precursor film in equilibrium with Π. According to eq 2, the equilibrium contact angle ΘE is a function of γ and Π(h). This means, with different ΘE, Π(h) can be different for the same γ. As a consequence, with different substrate surfaces, Π(h) can be adjusted independently from γ. When a drop does not move or change its shape, it is locally in equilibrium with its precursor according to eq 2. If the two drop bodies are now placed increasingly closer to each other, the two different precursors will interact and eventually form a thin connecting neck. This neck will most likely be asymmetric regarding its geometrical shape and composition. Although it is not straightforward how all of this affects the interaction between the main drop bodies, one could expect a decrease in M̃ t compared to the case of complete wetting. In the main part of this report, we will measure the coalescence behavior of drops for various values of ΘE and Δγ by using a wide range of different liquid/substrate combinations. From this we will derive the relation between ΘE and Δγ as well as M̃ t and compare it to the result obtained for the complete wetting case. A possible influence of surface forces will be manifest in a difference between the two cases.
(1)
where γ ̅ is the average surface tension of the two liquids. Experiments and simulations show that there is a threshold Marangoni number of M̃ t ≈ 2.0. Situations with M̃ < M̃ t lead to immediate coalescence of both drops into a single drop body, whereas M̃ > M̃ t establishes a state of temporary noncoalescence with the two drop bodies remaining separated (and moving together across the surface). Experiments as well as simulations indicate that the boundary between the two coalescence regimes is sharp with respect to M̃ . These investigations20 revealing the importance of Δγ and Θ̅ a (appropriately combined in M̃ t) focused on the coalescence behavior that emerges after the two drops already have come into lateral contact, i.e., with a connecting bridge between both drops that is thicker than molecular dimensions. Experimentally, this has been achieved by investigating the coalescence behavior of spreading drops, i.e., by using completely wetting liquids. This allowed a convenient variation of Θ̅ a without the alteration of other parameters (depositing the drops at different distances on the common substrate leads to different contact angles, Θ̅ a, in the moment of drop−drop contact resulting from the continuous spreading). According to eq 1, the occurrence of instantaneous coalescence or temporary noncoalescence depends only on Δγ and Θ̅ a. However, these might not be the only relevant parameters because initially the height of the connecting liquid bridge between the drops is on the order of the molecular size. Therefore, surface forces, i.e., the disjoining pressure Π(h) (with h being the local thickness of the liquid film) will affect the drop−drop interaction. For the hydrodynamic analysis yielding eq 1, the influence of Π(h) could be neglected because the resulting stable neck height is in the range of at least several hundred nanometers. With completely wetting liquids, the experimental results are in agreement with the purely hydrodynamic description because in this case Π(h) is repulsive, i.e., it favors the thickening of the bridging neck between the drops approaching each other laterally. So even if there was an influence of Π(h), it would not show up in the experimental results because the neck between the drops will grow until hydrodynamic conditions lead to one of the two different coalescence behaviors.
2. EXPERIMENTAL SECTION The experimental procedures are sketched in Figure 2. In contrast to our previous study,20 the drops were not brought into contact with nonequilibrium adjoining three-phase angles Θa resulting from the continuous spreading on a shared support. Instead, first a single drop 1 (with liquid 1) was placed onto the substrate with a syringe from the
Figure 2. Side-view sketch of the experiments probing the coalescence behavior of partially wetting drops. First, drop 1 is placed on the substrate from the top with a syringe, and then very slowly drop 2 is created/enlarged from beneath through a hole in the substrate (1). During the process, the contact angles are continuously monitored. At some point, both drops are in contact with contact angles ΘE,1 and ΘE,2 (2). This will lead to either rapid coalescence (3a) or quasistationary noncoalescence (3b). 6827
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top. This drop quickly established a constant shape with an equilibrium contact angle of ΘE,1 (the contact angle hysteresis is very small and can be neglected, see below). Then, drop 2 was created by slowly pumping liquid 2 through a small hole in the substrate. Great care was taken to fill drop 2 very slowly to ensure that it is very close to its equilibrium shape all the time, i.e., it was ensured that the threephase angle of drop 2 was virtually its equilibrium contact angle ΘE,2 in the moment when both drops came into lateral contact. Due to the delay between placing drop 1 and initial contact, we can assume that drop 1 is locally in equilibrium with its precursor film. Drop 2, in contrast, has to move in order to establish contact, so its precursor cannot be in equilibrium. The liquid supply to drop 2 was stopped as soon as both drops came into contact. The moment of contact is rather clearly observed and well-defined because the drops either merge rapidly or start to move together in a twin-drop configuration for the case of noncoalescence. The liquids were (1) linear n-alkanes (CnH2n+2 with n = 13 to 16), (2) two branched alkanes, 2,6,10,14-tetramethylpenta-decane (C19H40, pristane) and 2,6,10,15,19,23-hexamethyltetracosane (C30H62, squalane), and (3) a branched alkene 2,6,10,15,19,23-hexamethyl2,6,10,14,18,22-tetracosahexaene (C30H50, squalene). The liquids were used as supplied (alkanes: purity >99.5%, Alfa Aesar; pristane: purity ≥95%, Sigma). The measured surface tensions (tensiometer K1, Krüss) agreed with the literature values.22 See Table 1 for a summary of the liquid properties.
Figure 3. Contact angles of the liquid/substrate combinations. The pF-DDMCS fraction gives the volume fraction of the coating agents in the coating vessel. Numerical values are listed in the Supporting Information.
Table 1. Liquid Properties22−24a
a
liquid
γ (mN/m)
η (mPa·s)
tetradecane pentadecane hexadecane squalane squalene
26.68 27.10 27.55 28.24 31.50
2.32 2.85 3.45 28.3 12.5
Figure 4. Coalescence experiments on partially wetted, silanized silica surfaces (drop 1, squalene; drop 2, tetradecane). The substrate coating for (A) was slightly less fluorinated than for (B). The slightly different equilibrium contact angles lead to qualitatively very different behaviors of noncoalescence (A) and immediate coalescence (B), respectively.
Crosscheck measurements agreed within ±5 to 10%.
ΘE is varied systematically by changing the substrate surface coatings. As substrates, we used silicon wafers (see ref 17 and Supporting Information for details). The bare wafer surfaces are completely wetted by water and the hydrocarbons. To achieve different nonzero ΘE values, the wafer surfaces were coated with layers of different mixtures of perfluorinated (pF-DDMCS) and nonfluorinated (DDMCS) decyl-dimethyl-chlorosilanes via gas-phase silanization (see Supporting Information for details on the coating procedure). With appropriate ratios of fluorinated and nonfluorinated silanes, the contact angles ΘE can be adjusted quite systematically, ranging from ∼0° (tetradecane on a pure DDMCS coating) to ∼50° (squalene on a pure pF-DDMCS coating). Figure 3 shows the variation of the experimentally determined equilibrium contact angles ΘE as a function of the amount of pF-DDMCS in the coating chamber for different hydrocarbons. Both the surface tension γ of the liquid and the relative amount of pF-DDMCS increase the contact angle. In all cases, the contact angle hysteresis was rather small (typically ≲2°). An ellipsometric characterization of the surfaces and additional contact angle data can be found in the Supporting Information. During the experiments, the contact angles were monitored continuously. ΘE,1 and ΘE,2 used for evaluation are the values immediately before contact. For each combination liquid 1/liquid 2/ substrate, at least three measurements were performed. The reproducibility regarding the contact angles and the same coalescence behavior was ∼±1°. This was also the range of the contact angle hysteresis. Two experiments, representative of noncoalescence and immediate coalescence, are depicted in Figure 4A,B, respectively. The drop sizes (h ≲ 700 μm) were below the capillary length, and gravitation can be neglected. The relevant contact angles are below 30°, and the flow velocities are comparable to those in ref 19. In these limits, typical wetting phenomena are well described by the lubrication approximation. The experiments were performed at T = 20.0 ± 0.5 °C under
a dry nitrogen atmosphere (the vapor pressures of the liquids are rather low and evaporation effects can be neglected). The coalescence behavior was observed and analyzed by video imaging from the top and from the side with the setup described earlier.17,19
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RESULTS Figure 4 presents two experiments with the same combination of liquids (tetradecane vs squalene, Δγ ≈ 4.82 mN/m) on differently coated substrates (case A, 0.1% pF-DDMCS; case B, 0.2% pF-DDMCS). In all experiments, we used only hydrocarbons as liquids and always marked the liquid with the higher surface tension with the index 1. Therefore, the geometry at the moment of drop−drop contact was always asymmetric with ΘE,1 > ΘE,2. In ref 20, we characterize the geometry of the neck region in the moment of mutual drop−drop contact by the arithmetic mean of the individual dynamic contact angles of both drops. In the case of partial wetting that we study here, we will use the arithmetic mean of the two equilibrium contact angles, Θ̅ E = (ΘE,1 + ΘE,2)/2. As shown in Figure 4, the slightly different Θ̅ E values lead to qualitatively very different coalescence behaviors. In case (A), we observe noncoalescence. Even 6.5 s after initial contact, the neck between the drops did not grow significantly. In contrast, for case B, after 0.6 s the neck has nearly reached the apex height of drop 2. The results of a series of experiments with different liquid combinations and differently coated substrates are presented in Figure 5. The liquid combinations and their relevant properties 6828
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This is the square-root scaling behavior, which we already found for the case of spreading drops and which we now also find for the partially wetting case depicted in Figure 5. We also determine the same M̃ t by the fit of eq 5 to the data in Figure 5:
M̃ t = 2.0 ± 0.2
The coalescence experiments for the partial wetting cases were performed with the same liquid combinations as in ref 20 (where the substrates were prepared such that the liquids wet them completely). Here, the substrate surfaces were modified to achieve partial wetting with nonzero equilibrium contact angles ΘE. Therefore, the liquid properties are identical for both studies, but the equilibrium thickness of the precursor film, hP, and the disjoining pressure, Π(h), are different (eq 2). By and large, within the errors, we find quantitative agreement between the partially wetting and the completely wetting case, and we can explain the findings without explicitly taking into account any surface force or disjoining pressure contributions. We can conclude that the disjoining pressure is not relevant, or at least of minor influence, for the threshold behavior, i.e., the selection of the system for either noncoalescence or rapid coalescence. This also holds for the influence of the viscosity. The liquids cover a viscosity range of approximately one decade (Table 1). Nevertheless, an influence of the viscosity on the boundary between immediate coalescence and noncoalescence could not be observed. This can be understood by the scaling arguments for the threshold three-phase angle: the threshold results from a competition between capillary flow (coalescence-promoting) and Marangoni flow (separation-promoting). Their relative strengths are given by M̃ from eq 4. The viscosity determines the strength of both fluxes in the same way. It will set the overall time scale of the flow dynamics, but it does not change the competition between the two fluxes.
Figure 5. Arithmetic mean of the equilibrium contact angles of the two drops (Θ̅ E) vs reduced surface tension difference (Δγ/γ). ̅ Open symbols: immediate coalescence was observed. Full symbols: temporary noncoalescence was observed. Black solid line: power-law fit to eq 5. Gray dashed line: data from ref 20 for spreading drops.
are listed in Table 2. Figure 5 shows the average equilibrium contact angle Θ̅ E as a function of the reduced surface tension Table 2. Liquid Combinations for Coalescence Experiments drop 1
drop 2
γ̅ (mN/m)
Δγ (mN/m)
Δγ/γ̅
pentadecane tetradecane tetradecane tetradecane
hexadecane hexadecane squalane squalene
27.32 27.12 27.46 29.09
0.44 0.87 1.56 4.82
0.016 0.032 0.058 0.17
difference Δγ/γ ̅ . Immediate coalescence is depicted by open symbols. Full symbols represent (temporary) noncoalescence. There are two well-defined, separate regimes for the two coalescence behaviors. The boundary between the regimes can be fitted with a power law relation between the threshold average three-phase angle and the reduced surface tension difference: Θ̅ E, t ≈ (0.86 ± 0.09)(Δγ /γ ̅ )0.45 ± 0.04
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SUMMARY AND CONCLUSION We present experimental results on the coalescence behavior of sessile drops from partially wetting drops with completely miscible, different liquids as a function of their contact angles in the moment of mutual lateral contact at the three-phase line. A wide range of contact angles for different liquid/substrate combinations was achieved by coating the substrates with different mixtures of perfluorinated (pF-DDMCS) and nonfluorinated (DDMCS) decyl-dimethyl-chlorosilanes via gasphase silanization. In addition to the contact angles, the viscosities were also varied over approximately one order of magnitude by using different liquids. In particular, the experiments aimed at the question of an influence of surface forces or disjoining pressures on the coalescence behavior. Theoretical deliberations suggest that such an influence cannot be excluded in the case of partially wetting drops (in contrast to the case of completely wetting liquids, where stabilizing effects from surface forces can be excluded). Different equilibrium contact angles of the same liquid on substrates with different surface coatings are the result of different surface forces and disjoining pressures. Thus, a series of drop coalescence experiments with different contact angles directly reveals the influence of surface forces. The observations reveal a sharp transition between the regimes of noncoalescence and immediate coalescence. The boundary between the two regimes follows a power law relationship between the threshold contact angle Θ̅ t and the surface tension difference Δγ with Θ̅ t ∝ (Δγ/γ)1/2. The viscosity has no explicit impact. Within the experimental accuracy, this power law is identical to the case of
(3)
This threshold is visualized by the black line in Figure 5. It can be compared to the threshold obtained by the dynamic contact angle experiments as presented earlier.20 The dynamic contact angle experiments yield the gray dashed line when the average equilibrium contact angle, Θ̅ E, is replaced by the (nonequilibrium) average adjoining contact angle, Θ̅ a. The difference between the dynamic and the static case is not very pronounced. Both cases show approximately square root behavior.
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DISCUSSION In ref 20, we show that in the lubrication equation25 the relative strengths of the Marangoni flux ΦM and the capillary flux ΦC are determined by the modified Marangoni number: ΦM 3Δγ ∝ M̃ = ΦC 2γ ̅ Θ2 (4) Thus, the threshold three-phase angle is related to the surface tension difference and the threshold Marangoni number M̃ t:
Θ̅ t =
3Δγ 2M̃ t γ ̅
(6)
(5) 6829
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spreading drops reported earlier.20 We explained this power law in ref 20 by a hydrody.namic model. The negligible impact of viscosity variations on the coalescence behavior can also be understood within this model. Thus, we can conclude that variations in surface forces and disjoining pressures significantly influence neither the very early, transient evolution of the drop−drop coalescence behavior nor the later stages of rapid coalescence or quasi-stationary noncoalescence.
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(15) Borcia, R.; Borcia, I.; Bestehorn, M. Nonlinear dynamics of thin liquid films consisting of two miscible components. Phys. Rev. E 2012, 86, 056319. (16) Riegler, H.; Lazar, P. Delayed coalescence behavior of droplets with completely miscible liquids. Langmuir 2008, 24, 6395−6398. (17) Karpitschka, S.; Riegler, H. Quantitative Experimental Study on the Transition between Fast and Delayed Coalescence of Sessile Droplets with Different but Completely Miscible Liquids. Langmuir 2010, 26, 11823−11829. (18) Borcia, R.; Bestehorn, M. Partial Coalescence of Sessile Drops with Different Miscible Liquids. Langmuir 2013, 29, 4426−4429. (19) Karpitschka, S.; Riegler, H. Non-coalescence of sessile drops from different but miscible liquids: Hydrodynamic analysis of the twin drop contour as self stabilizing, traveling wave. Phys. Rev. Lett. 2012, 109, 066103. (20) Karpitschka, S.; Riegler, H. Sharp transition between coalescence and noncoalescence of sessile drops. J. Fluid Mech. 2014, 743, R1. (21) Frumkin, A. Zh. Fis. Khim. 1938, 12, 337. (22) Wohlfarth, C.; Wohlfahrt, B. Landolt Börnstein IV Physical Chemistry; Springer: Berlin, 1997; Vol. 16. (23) Korosi, G.; Kovats, E. Density and surface tension of 83 organic liquids. J. Chem. Eng. Data 1981, 26, 323−332. (24) Shrestha, L.; Aramaki, K.; Kato, H.; Takase, Y.; Kunieda, H. Foaming Properties of Monoglycerol Fatty Acid Esters in Nonpolar Oil Systems. Langmuir 2006, 22, 8337−8345. (25) Oron, A.; Davis, S.; Bankoff, S. Long-scale evolution of thin liquid films. Rev. Mod. Phys. 1997, 69, 931−980.
ASSOCIATED CONTENT
S Supporting Information *
Details of silicon wafer cleaning, the coating process, additional contact angle measurements, and an ellipsometric characterization of the surfaces. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank H. Möhwald for scientific advice and general support. S.K. was supported by the DFG (RI529/16-1) and LAM Research AG, Austria. C.H. and A.F. acknowledge the DFG for funding within the SFB840.
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REFERENCES
(1) Sellier, M.; Nock, V.; Verdier, C. Self-propelling, coalescing droplets. Int. J. Multiphase Flow 2011, 37, 462−468. (2) Sellier, M.; Nock, V.; Gaubert, C.; Verdier, C. Droplet actuation induced by coalescence: Experimental evidences and phenomenological modeling. Eur. Phys. J.: Spec. Top. 2013, 219, 131−141. (3) Christopher, G.; Bergstein, J.; Poon, M.; Nguyen, C.; Anna, S. Coalescence and splitting of confined droplets at microfluidic junctions. Lab Chip 2009, 9, 1102−1109. (4) Li, Z. G.; Ando, K.; Yu, J. Q.; Liu, A.; Zhang, J.; Ohl, C. Fast ondemand droplet fusion using transient cavitation bubbles. Lab Chip 2011, 11, 1879−1885. (5) Ihnen, A.; Petrock, A.; Chou, T.; Fuchs, B.; Lee, W. Organic Nanocomposite Structure Tailored by Controlling Droplet Coalescence during Inkjet Printing. ACS Appl. Mater. Interfaces 2012, 4, 4691−4699. (6) Hanyak, M.; Darhuber, A.; Ren, M. Surfactant-induced delay of leveling of inkjet-printed patterns. J. Appl. Phys. 2011, 109, 074905. (7) Stringer, J.; Derby, B. Formation and Stability of Lines Produced by Inkjet Printing. Langmuir 2010, 26, 10365−10372. (8) Leenaars, A.; Huethorst, J.; van Oekel, J. Marangoni drying: a new extremely clean drying process. Langmuir 1990, 6, 1701−1703. (9) Marra, J.; Huethorst, J. Physical principles of Marangoni drying. Langmuir 1991, 7, 2748−2755. (10) Matar, O.; Craster, R. Models for Marangoni drying. Phys. Fluids 2001, 13, 1869−1883. (11) Eddi, A.; Winkels, K.; Snoeijer, J. Influence of droplet geometry on the coalescence of low viscosity drops. Phys. Rev. Lett. 2013, 111, 144502. (12) Castrejón-Pita, J.; Kubiak, K.; Castrejón-Pita, A.; Wilson, M.; Hutchings, I. Mixing and internal dynamics of droplets impacting on a solid surface. Phys. Rev. E 2013, 88, 023023. (13) Ristenpart, W.; McCalla, P.; Roy, R.; Stone, H. Coalescence of Spreading Droplets on a Wettable Substrate. Phys. Rev. Lett. 2006, 97, 064501. (14) Hernandez-Sanchez, J.; Lubbers, L.; Eddi, A.; Snoeijer, J. H. Symmetric and Asymmetric Coalescence of Drops on a Substrate. Phys. Rev. Lett. 2012, 109, 184502. 6830
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