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Dynamic Behavior of the Water Droplet Impact on a Textured Hydrophobic/Superhydrophobic Surface: The Effect of the Remaining Liquid Film Arising on the Pillars’ Tops on the Contact Time Xiying Li, Xuehu Ma,* and Zhong Lan Institute of Chemical Engineering, Dalian University of Technology, Dalian 116012, China Received September 23, 2009. Revised Manuscript Received December 20, 2009 We have fabricated a series of textured silicon surfaces decorated by square arrays of pillars whose radius and pitch can be adjusted independently. These surfaces possessed a hydrophobic/superhydrophobic property after silanization. The dynamic behavior of water droplets impacting these structured surfaces was examined using a high-speed camera. Experimental results validated that the remaining liquid film on the pillars’ tops gave rise to a wet surface instead of a dry surface as the water droplet began to recede away from the textured surfaces. Also, experimental results demonstrated that the difference in the contact time was subjected to the solid fraction referred to as the ratio of the actual area contacting with the liquid to its projected area on the textured surface. Because the mechanism by which the residual liquid film emerges on the pillars’ tops can essentially be ascribed to the pinch-off of the liquid threads, we further addressed the changes in the contact time in terms of the characteristic time of pinch-off of an imaginary liquid cylinder whose radius is related to the solid fraction and the maximum contact area. The match of the theoretical analysis and the experimental results substantiates the assumption aforementioned.
Introduction The wetting of a solid surface is ubiquitous and has attracted widespread attention in painting, coating, microfluidic, and mineral recovery. The contact angle of a drop residing on an ideally flat surface can be determined with the knowledge of γSV, γSL, and γ, representing different surface tensions (solid/vapor, solid/liquid, and liquid/vapor, respectively) involved in the system. The balance of these forces leads to the classical Young equation γ -γ cos θ ¼ SV SL ð1Þ γ
hydrophobic nature because of the value of r being constantly larger than 1. Unlike homogeneous wetting (the Wenzel model), heterogeneous wetting (the Cassie-Baxter model) represents the other wetting state in which a liquid droplet is resting on a composite surface. In this regime, the weighted effect of each acting phase corresponding to its own fraction is responsible for the resulting apparent contact angle. Accordingly, a liquid droplet residing on the composite surface (here especially for the composite surface of air and solid) has a resulting contact angle in the form of
where θ represents the Young contact angle or intrinsic contact angle. As a matter of fact, it is well-known that the apparent contact angle in practice is considerably influenced by both the surface chemistry and roughness. Two existing models, referred to as Wenzel3 and Cassie-Baxter4 models, can account for the apparent contact angle on a rough surface. In the case of the Wenzel scenario, the liquid conforms to the solid surface such that the actual contact area underneath the liquid drop is increased in comparison with the projected area. Thus, surface roughness r, defined as the ratio of actual area over the projected area, is introduced to compute the apparent contact angle in the Wenzel state, taking into consideration the Young contact angle. From the perspective of the surface energy, Wenzel first gave the apparent contact angle when the liquid followed all of the roughness:
cos θ ¼ -1 þ js ðcos θ þ 1Þ
1,2
cos θ ¼ r cos θ
ð2Þ
where θ* indicates the apparent contact angle. According to the Wenzel model, the surface roughness enhances the hydrophilic/ *Corresponding author. E-mail:
[email protected]. Phone: (86) 41183653402. Fax: (86) 411-83653402. (1) de Gennes, P. G. Wetting: statics and dynamics. Rev. Mod. Phys. 1985, 57, 827-863. (2) de Gennes, P. G.; Brochar-Wyart, F.; Quere, D. Capillary and Wetting Phenomena; Springer: Berlin, 2003. (3) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988–994. (4) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546–551.
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ð3Þ
where φs designates the fraction of solid surface being in touch with the liquid drop. Air entrapment below the drop can remarkably increase the contact angle up to 150° or larger and simultaneously maintain very low contact angle hysteresis, namely, achieving a superhydrophobic state.5,6 In this regard, the fine, smart, textured surfaces originating from the leaves of some natural plants7,8 and some insects9,10 living on water or in desert environments bestowed the prototype for the artificial structures accessible to the superhydrophobic state. The comprehensive remarks on the superhydrophobic state can be found in the latest reviews.11-19 Although a combination of the intrinsic (5) Lafuma, A.; Quere, D. Nat. Mater. 2003, 2, 457–460. (6) Callies, M.; Quere, D. Soft Matter 2005, 1, 55–61. (7) Barthlott, W.; Neinhuis, C. Planta 1997, 202, 1–8. (8) Neinhuis, C.; Barthlott, W. Ann. Bot. 1997, 79, 667–677. (9) Gao, X. F.; Jiang, L. Nature 2004, 432, 36. (10) Parker, A. D.; Lawrence, C. R. Nature 2001, 414, 33–34. (11) Quere, D. Rep. Prog. Phys. 2005, 68, 2495–2532. (12) Quere, D. Annu. Rev. Mater. Res. 2008, 38, 71–99. (13) Quere, D.; Reyssat, M. Philos. Trans. R. Soc. A 2008, 366, 1539–1556. (14) Bush, J. W. M.; Hu, D. L.; Prakash, M. Adv. Insect Physiol. 2008, 34, 117– 192. (15) Bhushan, B. Philos. Trans. R. Soc. A 2009, 367, 1445–1486. (16) Koch, K.; Bhushan, B.; Barthlott, W. Soft Matter 2008, 4, 1943–1963. (17) Nosonovsky, M.; Bhushan, B. J. Phys.: Condens. Matter 2008, 20, 225009. (18) Xia, F.; Jiang, L. Adv. Mater. 2008, 20, 2842–2858.
Published on Web 02/12/2010
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hydrophobe and smart textures can validate the ultrahydrophobic state, only when a re-entrant structure is taken into consideration can a superolephobic surface20,21 be achieved. In addition, many investigators concentrated on the irreversible transition from the Cassie to Wenzel regime under the action of external actuation (squeezing, impacting, vibrating, and evaporating)22-30 because of its intimate relationship to the practical applications of a superhydrophobic surface. There are multiple equilibrium states corresponding to local minimum energy from the Cassie to Wenzel regime where contact angle hysteresis plays an important role.31,32 Considering the rapid development of superhydrophobic surfaces for the latest 10 years, the dynamic behavior of the interaction between liquid drops and composite surfaces came to researchers’ attention in the context of exploring the robustness of the fakir nonwetting state. Indeed, the robustness of the superhydrophobic surface plays a crucial role in its practical applications such as waterproof, self-cleaning, and antiadhesive functions, plastron respiration underwater, and reduction of friction in microfluids.33 Except for the ideas of rough wetting aforementioned, some researchers27-29,34-37 have recently argued that it is the interaction of between the liquid and solid at the three-phase contact line, instead of the contact area within the contact perimeter, that accounts for the apparent contact angle. However, more detailed information about the triple-line footprints should be taken into account.38-40 The difference of advancing contact angle θa and receding contact angle θr is referred to as contact angle hysteresis. The Young contact angle exclusively defines the angle at the equilibrium state, while contact angle hysteresis can account for the diversity of the wetting behavior especially for the dynamic (19) Zhang, X.; Shi, F.; Niu, J.; Jiang, Y. G.; Wang, Z. Q. J. Mater. Chem. 2008, 18, 621–633. (20) Tuteja, A.; Choi, W.; Ma, M. L.; Mabry, J. M.; Mazzella, S. A.; Rutledge, G. C.; Mckinley, G. H.; Cohen, R. E. Science 2007, 318, 1618–1622. (21) Tuteja, A.; Choi, W.; Mckinley, G. H.; Cohen, R. E.; Rubner, M. F. MRS Bull. 2008, 33, 752–758. (22) Reyssat, M.; Pepin, A.; Marty, F.; Chen, Y.; Quere, D. Europhys. Lett. 2006, 74, 306–312. Buguin, A.; Silberzan, P.; (23) Bartolo, D.; Bouamrirene, F.; Verneuil, E.; Moulinet, S. Europhys. Lett. 2006, 74, 299–305. (24) Jung, Y. C.; Bhushan, B. Langmuir 2008, 24, 6262–6269. (25) Reyssat, M.; Yeomans, J. M.; Quere, D. EPL 2008, 81, 26006. (26) Nosonovsky, M.; Bhushan, B. Langmuir 2008, 24, 1525–1533. (27) Bormashenko, E.; Pogreb, R.; Whyman, G.; Erlich, M. Langmuir 2007, 23, 6501–6503. (28) Bormashenko, E.; Pogreb, R.; Whyman, G.; Erlich, M. Langmuir 2007, 23, 12217–12221. (29) Bormashenko, E.; Pogreb, R.; Stein, T.; Whyman, G.; Erlich, M.; Musin, A.; Machavarianib, V.; Aurbachc, D. Phys. Chem. Chem. Phys. 2008, 10, 4056– 4061. (30) Moulinet, S.; Bartolo, D. Eur. Phys. J. E: Soft Matter Biol. Phys. 2007, 24, 251–260. (31) He, B.; Patankar, N. A.; Lee, J. H. Langmuir 2003, 19, 4999–5003. (32) Ishino, C.; Okumura, K. Europhys. Lett. 2006, 76, 464–470. (33) Cottin-Bizonne, C.; Barrat, J. L.; Bocquet, L.; Charlaix, E. Nat. Mater. 2003, 2, 237–240. (34) Extrand, C. W. Langmuir 2002, 18, 7991–7999. (35) Extrand, C. W. Langmuir 2003, 19, 3793–3796. (36) Gao, L.; McCarthy, T. J. Langmuir 2007, 23, 3762–3765. (37) Gao, L.; McCarthy, T. J. Langmuir 2009, 25, 7249–7255. (38) Dorrer, C.; R€uhe, J. Langmuir 2007, 23, 3179–3183. (39) Kusumaatmaja, H.; Yeomans, J. M. Langmuir 2007, 23, 6019–6032. (40) Anantharaju, N.; Panchagnula, M. V.; Vedantam, S.; Neti, S.; Tatic-Lucic, S. Langmuir 2007, 23, 11673–11676. (41) Johnson, R. E.; Dettre, R. H. Adv. Chem. Ser. 1964, 43, 112–135. (42) Joanny, J. F.; de Gennes, P. G. J. Chem. Phys. 1984, 81, 552–562. (43) Reyssat, M.; Quere, D. J. Phys. Chem. B 2009, 113, 3906–3909. (44) Dorrer, C.; R€uhe, J. Langmuir 2006, 22, 7652–7657. (45) Nosonovsky, M.; Bhushan, B. Nano Lett. 2007, 7, 2633–2637. (46) Gao, L.; McCarthy, T. J. Langmuir 2006, 22, 6234–6237. (47) McHale, G.; Shirtcliffe, N. J.; Newton, M. I. Langmuir 2004, 20, 10146– 10149. (48) Bico, J.; Marzolin, C.; Quere, D. Europhys. Lett. 1999, 47, 220–226. (49) Patankar, N. A. Langmuir 2003, 19, 1249–1253.
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Figure 1. Representative SEM images of a patterned surface decorated by microposts. In the images above, D, H, and P denote the diameter, height of the posts, and pitch between them and have sizes of 20, 40, and 40 μm, respectively.
wetting process. Even though many investigators34,35,41-52 proposed insightful arguments about contact angle hysteresis, so far, it is impossible to clearly address the real origin or the quantitative characterization of hysteresis. In the case of the receding contact angle on the composite surface, Bico et al.48 and Patankar et al.49,50 had forever speculated that the remaining liquid film on the microstructured surface was responsible for the smaller receding contact angle in comparison to the retaining dry surface. Moreover, Roura and Fort discussed a similar phenomenon from the viewpoint of energy conservation.53 Especially for superhydrophobic surfaces, the residual liquid layer on the tops of the pillars will lead to the composite surface encompassing air and the liquid itself while a liquid drop starts to retract. According to the Cassie model, the receding contact angle has the following form:49,50
cos θ ¼ -1 þ 2js
ð4Þ
However, there is still a lack of substantial evidence to verify the prior assumption. By means of drop impact, we experimentally revealed the residual liquid film on the pillars’ tops as an impact drop began to retract and eventually detached from a patterned surface. At the same time, the receding angles derived by eq 4 were in accordance with our experimental results. To go further, on the basis of the assumption that the mechanism by which a liquid layer emerges on the tops of the pillars during the receding stage is essentially analogous to the pinch-off of liquid cylinders or liquid jets, we addressed the difference in contact time during which the liquid drop was in touch with the textured surface. The difference in the contact time can be ascribed to the influence of the pinch-off of a liquid thread with a certain radius depending on the solid fraction. The derived relationship is in good agreement with our experimental results.
Experimental Section Textured surfaces with square arrays of pillars were fabricated by photolithography. By means of the lithographic technique (BP212 positive photoresist) combined with reactive ion etching, a pattern of circular shapes with a square arrangement was transferred from a photomask into an oxide layer on a 5-in. silicon wafer. Thereafter, the oxide layer acted as a masking layer in a subsequent anisotropic etching process, and then the threedimensional structure with a square array of microposts was embossed on the silicon wafer. The height of the microposts was regulated by control of the etching time. In our experiments, the height was set at the equal size of 40μm. A patterned surface topography was designed by independently adjusting the pitch and radius of the pillars. The textured surfaces are signified by TPD (50) (51) (52) (53)
He, B.; Lee, J. H.; Patankar, N. A. Colloids Surf., A 2004, 248, 101–104. Gao, L.; McCarthy, T. J. Langmuir 2006, 22, 2966–2967. Yeh, K. Y.; Chen, L. J.; Chang, J. Y. Langmuir 2008, 24, 245–251. Roura, P.; Fort, J. Langmuir 2002, 18, 566–569.
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Figure 2. Representative images of water drops residing on microstructured surfaces. The lights can transverse in the grooves, which indicates that water droplets are suspended on top of the microposts. For these images above, parts a-d respectively correspond to the 40 40 80 surfaces of T20 10, T10 , T20 , and T20 . with the subscript D designating the diameter of the pillars and superscript P indicating the pitch on the order of microns. The products including nine specimens were divided into two types according to the diameter of the pillars. The pillars for type 1 have a diameter of 10 μm and different pitches (20, 40, 60, 80, and 100 μm). Likewise, the pillars for type 2 have a diameter of 20μm and different pitches (40, 80, 120, and 150 μm). Figure 1 shows the representative SEM images of the patterned surfaces. These patterned surfaces turned out to be hydrophobic or superhydrophobic after immersion into a 1 wt % solution of octadecyltrichlorosilane (Alfa Aesar) for 5 min at room temperature. This method was addressed in detail in ref 55. Contact angles against water were measured with a goniometer (OCA20; Dataphysics Corp., Bad Vilbel, Germany) with a 1° precision. The advancing and receding contact angles were found to be θa = 105° and θr = 87° on a flat silicon wafer after silanization treatment. The equilibrium contact angle was also measured as 101°. The scenarios of drop-impinging events were recorded by a high-speed camera (CCD) (Photron, Fastcam Apx-Rs, which is equipped with a long-distance microscope, Hirox OL-35, Tokyo, Japan). CCD works at 10 000 frames/s at a resolution of 512 512 pixels.
Results and Discussion The representative images of equilibrium contact angles on different patterned surfaces are displayed in Figure 2. Also, the contact angles including equilibrium, advancing, and receding contact angles, which were respectively obtained from the measured results and theoretical models, are given in Table 1. The lights can pass through the grooves except for the T20 10 surface (likely due to its smaller pitch than the others), which confirmed that the resting droplets dwelled on composite surfaces including air and solid. The Cassie model was therefore accessible to € (54) Oner, D.; McCarthy, D. Langmuir 2000, 16, 7777–7782. (55) Quere, D.; Lafuma, A.; Bico, J. Nanotechnology 2003, 14, 1109–1112.
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Table 1. Comparison of the Theoretical Prediction and Experimental Results sample T20 10 T40 10 T60 10 T80 10 T100 10
from the Cassie model
experimental results
θe = 147° θa = 149° θr = 142° θe = 164° θa = 165° θr = 162° θe = 169° θa = 170° θr = 168° θe = 172° θa = 172° θr = 171° θe = 173.5° θa = 173.7° θr = 172.6°
θe = 148° θa = 165° θr = 130° θe = 157° θa = 165° θr = 150° θe = 161° θa = 164° θr = 158° θe = 164° θa = 165° θr = 163° θe = 166° θa = 166° θr = 164°
cos θr = -1 þ 2φs θr = 127° θr = 154° θr = 163° θr = 167° θr = 170°
evaluate the contact angles on these surfaces. As shown in Table 1, the advancing contact angles lie within a very narrow band around 165° and remain independent of the surface topography. € Similar phenomena can be found in the work of Oner and 54 44 McCarthy and discussed by Dorrer and R€uhe and Gao and McCarthy.51 These contact angles on type 2 are similar to those on type 1 because their solid fractions are consistent in sequence. Thus, the data of the contact angles on type 2 are not given here. Referring to the contact angles shown in Table 1, this confirms that eq 4, taking into consideration the remaining liquid film giving rise to a wetted surface in its receding phase, has priority over the theoretical results established for a dry surface in determining the receding contact angles on textured hydrophobic surfaces. Additionally, Zhang et al.19 and Dorrer and R€uhe44 argued that once the contact angles are above 156°, the contact DOI: 10.1021/la903603z
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angle cannot represent the real wettability of the solid because of the confinement of the computing software used by the goniometer. Accordingly, the tendency of the contact angles appears to be more useful. As to drop impact, the dynamic behavior of the interaction between liquid drops and patterned hydrophobic/superhydrophobic surfaces was experimentally investigated. In our experiments, a high-speed camera was placed with an inclined angle 3-5° relative to horizontal direction such that the delicate vestiges emerging on the textured surface could be observed by inspection of the recorded images. By carefully adjusting the luminance of the backlight source, we could obtain the optimum contrast of the landscape before and after collision on a patterned surface, thus making the slight vestiges of the remaining liquid film on the pillars’ tops clearly discernible. As shown in Figures 3 and 4 (detailed information demonstrated in the Supporting Information including the related videos), an additional dark stripe appeared after the water drop departed from the solid surfaces in comparison with the images before contact, which was indicative of the residual liquid film emerging on the posts’ tops. Moreover, the residual liquid film only existed on the pillars’ tops rather than in the interstices because liquid impregnation in the grooves would inevitably lead to considerable contact angle hysteresis and, hence, a sticky state,5,55 which is completely different from the present bouncing water. As a matter of fact, referring to ref 56, a similar phenomenon can also be found in its Figure 6 except for the subsequent pinch-off of the rising water droplet thanks to the transition from the Cassie to Wenzel state at the center of the contact area, but the article had no remarks on this point. The reason for the remaining liquid film emerging on pillars’ tops may be lies in the synergistic action of hysteresis force (from the difference between the dynamic contact angle and the equilibrium) and the interfacial instability of liquid threads known as Plateau-Rayleigh instability, which is triggered by surface tension and tends to happen for the large space between two adjacent pillars. The liquid thread suspended by two adjacent pillars becomes thinner and thinner in the receding phase and then is subjected to the interfacial instability known as PlateauRayleigh instability. As a result, the liquid thread ultimately breaks off and gives rise to the remaining liquid on the pillars’ tops. Likewise, Hamamoto-Kurosaki theoretically discussed the instability of the receding film on the textured surface. In our 80 surface was not as clear as experiments, the dark stripe on the T20 40 that on the T20 surface because of the solid fraction (φs = 0.196) on the latter being much larger than that (φs = 0.0491) on the former. However, the dark stripe will exist less than 1 s, allowing for the large Laplace pressure in relation to the small radius of curvature on the same order of the pillars’ radii. This phenomenon is essentially similar to transient evaporation on the eyes of a mosquito, which can still remain in a nonwetting state even after exposure to moist environments.57,58 In fact, Patankar et al.49,50 and Bico et al.48 previously argued that the existence of a liquid film on the pillars’ tops was responsible for the smaller contact angle than that of the counterpart on a nonwetted surface after triple-line retraction. For drop impact on textured hydrophobic/superhydrophobic surfaces, we also examined the differences in the contact time (56) Tsai, P.; Pacheco, S.; Pirat, C.; Lefferts, L.; Lohse, D. Langmuir 2009, 25, 12293–12298. (57) Hamamoto-Kurosaki, M.; Okumura, K. Eur. Phys. J. E: Soft Matter Biol. Phys. 2009, 30, 283–290. (58) Gao, X. F.; Yan, X.; Yao, X.; Xu, L.; Zhang, K.; Zhang, J. H.; Yang, B.; Jiang, L. Adv. Mater. 2007, 19, 2213–2215.
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40 Figure 3. Dark stripe present on the T20 surface with a contact angle of 146° after water drop detachment, which indicated the liquid layer left behind on the microposts. The impact velocity was set at 0.54 m/s, and the radius of the drop was 1.35 mm. The water drop is shown (a) before impact, (b) at maximum spreading, and (c) after departing from the solid surface.
during which the impinging drops keep in touch with the solid surfaces. Taking into account the initial conditions of our experiments, the experimental dimensional parameters, mainly including Re, We, and Oh numbers, which are respectively 729, 5.47, and 0.0032, determine that the impact dynamics are governed by a competition between inertia and capillarity for both spreading and retraction in light of the phase diagram that Bartolo et al gave.59 Richard et al.60 and Okumura et al.61 had investigated the contact time in the context of a water droplet colliding with the superhydrophobic surface. As a result, the relationship derived was proposed, taking into consideration a balance of the initial and capillary effects: τ ¼ ð2:6 ( 0:1ÞðFR0 3 =γÞ1=2
ð5Þ
(59) Bartolo, D.; Josserand, C.; Bonn, D. J. Fluid Mech. 2005, 545, 329–338. (60) Richard, D.; Clanet, C.; Quere, D. Nature 2002, 417, 811. (61) Okumura, K.; Chevy, F.; Richard, D.; Quere, D.; Clanet, C. Europhys. Lett. 2003, 62, 237–243.
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80 Figure 4. Dark stripe present on the T20 surface with a contact angle of 157° while the water drop was receding, which indicated the liquid layer left behind on the microposts. The impact velocity was set at 0.88 m/s, and the radius of the drop was 1.35 mm. The water drop is shown (a) before impact, (b) in the intermediate stage of retraction, and (c) after departing from the solid surface.
where τ represents the contact time and R0 denotes the initial radius of the impact drop. Also, Watchters et al.62 gave a similar relationship, τ = π(FR03/2γ)1/2, for the Leidenfrost drops, namely, the first-order vibration period of a freely oscillating drop. However, it is noteworthy that eq 5 was achieved under the circumstances of very low contact angle hysteresis (θa - θr e 5°).63 Actually, a wide range of contact angle hysteresis was exhibited on the superhydrophobic surfaces, as manifested by € and McCarthy.54 Dorrer and R€uhe,44 Bico et al.,48 and Oner Moreover, the introduction of an accommodation coefficient in eq 5 was unable to essentially resolve the differences in the contact time for various superhydrophobic surfaces. In our experiments, the differences in the contact time on various patterned hydrophobic/superhydrophobic surfaces are shown in Figures 5 and 6. In order to keep the captured images as sharp as possible, here we placed the camera horizontally such that the traces of the residual liquid layer did not emerge in Figure 5. Also, because the scenarios of impacting events on the surfaces of type 2 are almost identical with the ones of type 1 thanks to their matching solid (62) Wachters, L. H. J.; Westerling, N. A. J. Chem. Eng. Sci. 1966, 21, 1047– 1056. (63) Richard, D.; Quere, D. Europhys. Lett. 2000, 50, 769–775.
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fractions, we did not exhibit these images. As shown in Figure 5, the images on each patterned surface were very similar in 40 morphology except for the counterparts on the T20 10 and T20 surfaces whose static contact angles were about 146°. Also, the differences in the contact time were further manifested in Figure 6. 40 In the cases of drop impact on the T20 10 and T20 surfaces, the motion of the receding triple line remained quiescent for a moment because of the transition from the Cassie to Wenzel state at the center of the impact area.22-24 Thereafter, the receding motion should be relevant to the pinch-off of the liquid column. So, its contact time was apparently longer than the ones on the superhydrophobic surfaces. Furthermore, there was a definite slip with the length scale of less than 2(P - D) on the T150 20 surface, referring to the results in Figure 6. Although few articles recently have been concerned with the special issue of the contact time with respect to liquid droplet impingements on superhydrophobic surfaces, we can access relevant information by referring to the latest literature.24,56,64 According to eq 5 with the numerical prefactor 2.6 and taking into account a drop radius with a size of 1.35 mm in our experiments, (64) Zorba, V.; Stratakis, E.; Barberoglou, M.; Spanakis, E.; Tzanetakis, P.; Anastasiadis, S. H.; Fotakis, C. Adv. Mater. 2008, 20, 4049–4054.
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Figure 5. Sequential images indicative of temporal evolution of the impact drop on different surfaces. The impact velocity was set at 0.54 m/s. 40 40 Here, parts a-e respectively represent drop events on the T20 ,80 and T100 10, T10, T20,T20 20 textured surfaces. The scale bar is 2 mm. The same magnification applies to parts a-d and is larger than that for part e.
the contact time can be calculated as 15.2 ms, which is in good 80 and T150 agreement with our experimental results on the T10 20 surfaces where there is very little contact angle hysteresis of about 2°. However, the rest demonstrated evident discrepancies. According to the preceding discussion, we argue that the underlying mechanism by which the remaining liquid on the top of the pillars emerges in the receding stage lies in the pinch-off of liquid threads subjected to interfacial instability or the liquid jets, and thus the differences in the contact time can be ascribed to the influence of the pinch-off of liquid threads on the receding motion of the triple line. If the fluid viscosity is negligible, the equilibrium of inertia (65) Eggers, J.; Villermaux, E. Rep. Prog. Phys. 2008, 71, 036601.
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and capillary terms yields a characteristic time scale65 of the breakup of the liquid thread with the following form of τ ¼ ðFh0 3 =γÞ1=2
ð6Þ
where h0 indicates the radius of the liquid thread. So, we can introduce an imaginary liquid thread whose cross-sectional area is the same as that occupied by all of the pillars contacting the liquid while the spreading drop approaches its maximum area. As a consequence, eq 6 was therefore recast as τ ¼ ½FðAm js =πÞ3=2 =γ1=2
ð7Þ
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Figure 6. Differences of the contact time between impact droplets and various textured surfaces. The height of the microposts is 40 μm. Table 2. Comparison of the Experimental and Derivative Results sample
solid fraction
contact time, t (ms)
time of pinch-off, τ (ms) from eq 8
experimental Δt (ms)a
derivative Δτ (ms)a
T40 0.0491 16.3 ( 0.2 3.60 10 0.0218 15.6 ( 0.2 2.94 Δt1 = 0.7 Δτ1 = 0.66 T60 10 0.0120 15.2 ( 0.2 2.54 Δt2 = 1.1 Δτ2 = 1.06 T80 10 0.00785 14.9 ( 0.2 2.28 Δt3 = 1.4 Δτ3 = 1.32 T100 10 0.0491 16.3 ( 0.2 3.60 T80 20 120 0.0218 15.7 ( 0.2 2.94 Δt4 = 0.6 Δτ4 = 0.66 T20 0.0140 15.3 ( 0.2 2.63 Δt5 = 1.0 Δτ5 = 0.97 T150 20 40 60 40 80 40 a Here Δt respectively denotes the differences in the contact time as follows: Δt1 = t(T10 ) - t(T10 ), Δt2 =t(T10 ) - t(T10 ), Δt3 = t(T10 ) - t(T100 20 ), Δt4 = 80 80 150 ) - t(T120 ), and Δt = t(T ) t(T ), and so does Δτ. t(T20 20 5 20 20
where Am is the maximum spreading area of the impact droplet. The maximum deformation is independent of the surface wettability, as shown in Figure 6, which is consistent with the argument proposed by Clanet et al.66 However, each of the dilute pillars can be regarded as a strong defect43 the way the triple line cannot synchronously, axisymmetrically recede off the edges of the pillars’ tops. In this sense, the simple analogue of a single liquid thread based on the approximation of an identical crosssectional area remarkably underestimates the influence of the pinch-off of liquid threads on the contact time. Considering the fact that the inconsistency of the movement of the triple line inevitably reinforced the effect of the pinch-off of the liquid thread on the resulting contact time, a numerical prefactor P/D (the reciprocal of the line fraction of solid contacting with liquid) is therefore incorporated as a multiplier inserted into eq 7, which is reminiscent of the argument proposed by Gao and McCarthy36 that contact angle behavior (advancing, receding, and hysteresis) is determined by interactions between the liquid and the solid at the three-phase contact line rather than the interfacial area within the contact perimeter. Certainly, a more rigorous derivation based on theoretical analysis and numerical simulation in the three-dimensional structure is needed in the future. As a conse(66) Clanet, C.; Beguin, C.; Richard, D.; Quere, D. J. Fluid Mech. 2004, 517, 199–208.
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quence, the total time influenced by the pinch-off of the liquid thread can be given considering eq 7: τ ¼ 0:5π -1=4 ðF=γÞ1=2 Am 3=4 js 1=4
ð8Þ
It is evident that the solid fraction of the textured surface can account for the differences in the characteristic time scale of the pinch-off of a liquid thread and thus the total contact time. As shown in Table 2, the results derived from eq 8 are in good agreement with experimental data. In addition, we argue whether the action of the pinch-off of the liquid thread or hysteresis was “eliminated”; for example, for the Leidenfrost droplet, the contact time should be unequivocally equal and correspond to the first-order vibration period of a freely oscillating drop62 τ = π(FR03/2γ)1/2 similar to the asymptotical value τ = 2.3(FR03/γ)1/2 that Okumura et al.61 gave. Actually, the time computed via the equation τ = π(FR03/2γ)1/2 is 12.9 ms and conforms to the experimental value 12.7 ms, which can be achieved by deducting the pinch-off time from the total contact time. In this regard, the preceding argument also supports the validity of ascribing the differences in the contact time to the pinch-off of the liquid cylinder. In the end, we must stress that eq 8 cannot figure out the effects of the contact area (changing the height between the released liquid drop and the solid surface) unless the influence of the DOI: 10.1021/la903603z
4837
Article
Li et al.
receding motion of the triple line on the pinch-off of the liquid threads iss considered. For the water droplet impact on the superhydrophobic surfaces, the retraction dynamic conforms to inertial dewetting, where the dynamic is determined by a competition between the capillary tension from the thin film and the inertia of the liquid rim at the periphery of the receding droplet. In this regime, the retraction velocity has the following form:59 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vret ∼ Rmax ðFR0 3 =γÞ -1=2 1 -cos θr
ð9Þ
where Rmax is the maximum spreading radius of the water droplet. It is evident that the receding velocity of the triple line is proportional to the maximum spreading diameter. According to more experimental results associated with the changes in the impact velocity (not given here), it appeared that increasing the receding velocity would weaken the pinch-off effect to some degree. However, up to now, we cannot elucidate the way the movement of the receding triple line will essentially influence the pinch-off of the liquid threads on textured hydrophobic/ superhydrophobic surfaces.
association with composite surfaces. Also, we shed light on the difference in the contact time in terms of the characteristic time scale for the pinch-off of the liquid threads. For cases of similar phenomena involved in rolling water or bouncing drop, the argument above is a candidate to account for the energy lost in relation to the irreversible dissipative energy of break-off of the liquid cylinder and the surface energy stored in the liquid/gas interface [with the form of ΔE = γ(1 - cos θ) per unit area] thanks to the remaining liquid layer, which will ultimately influence the descent velocity or bouncing height of the liquid droplet. Moreover, the argument that the pinch-off of the liquid threads does influence the resulting contact time as well as the receding motion of the triple line will inspire more insightful thoughts in the context of the interaction between the liquid and textured surface. Acknowledgment. It is a real pleasure to thank Bai-tai Qian for heuristic suggestions and precious discussions in preparing this manuscript. We are thankful for the financial support provided by the National Natural Science Foundation of China (Contract 50776012).
Conclusion In conclusion, we revealed the remaining thin liquid film on the pillars’ tops after the liquid drop receded away from the textured hydrophobic/superhydrophobic surfaces, which has an important bearing on the receding angle and contact angle hysteresis in
4838 DOI: 10.1021/la903603z
Supporting Information Available: Videos associated with the drop events shown in Figures 3 and 4. This material is available free of charge via the Internet at http://pubs. acs.org.
Langmuir 2010, 26(7), 4831–4838
