Symmetry-breaking in drop bouncing on curved surfaces
Yahua Liu, Matthew Andrew, Jing Li, Julia M Yeomans, Zuankai Wang
SSymmetry-breaking in drop bouncing on curved surfaces
Yahua Liu †§ , Matthew Andrew † , Jing Li , Julia M Yeomans ∗ , Zuankai Wang , ∗ Department of Mechanical and Biomedical Engineering,City University of Hong Kong, Hong Kong 999077, China The Rudolf Peierls Centre for Theoretical Physics,1 Keble Road, Oxford, OX1 3NP, UK Shenzhen Research Institute of City University of Hong Kong, Shenzhen 518057, China
Abstract
The impact of liquid drops on solid surfaces is ubiquitous in nature, and ofpractical importance in many industrial processes. A drop hitting a flat surfaceretains a circular symmetry throughout the impact process; however a dropimpinging on
Echevaria leaves exhibits asymmetric bouncing dynamics withdistinct spreading and retraction along two perpendicular directions. This is adirect consequence of the cylindrical leaves which have a convex/concave archi-tecture of size comparable to the drop. Systematic experimental investigationson mimetic surfaces and lattice Boltzmann simulations reveal that this novelphenomenon results from an asymmetric momentum and mass distribution thatallows for preferential fluid pumping around the drop rim. The asymmetry ofthe bouncing leads to ∼
40% reduction in contact time. We expect that thecoupling of fast drop detachment on surfaces of different architectures (convex,concave or corrugated) with facile and scalable manufacturing has potential ina wide range of applications. † These authors contributed equally to this work. ∗ Correspondence should be addressed to Z.W ([email protected]) or J.M.Y. ([email protected]). § Present Address: Key Laboratory for Pre-cision & Non-traditional Machining Technology of Ministry of Education, Dalian University of Technology,Dalian 116024, China. a r X i v : . [ phy s i c s . f l u - dyn ] N ov ince Worthington’s pioneering work studying the complex dynamics of liquid dropsimpacting on solid surfaces in 1876, extensive progress has been made in understanding andcontrolling drop dynamics on various textured surfaces. Recent research in particular hasshown that the spreading and retraction dynamics of impacting drops is highly dependenton both the roughness and the wettability of the underlying substrate . Progress has beendriven by the intrinsic scientific interest and beauty of fluid impacts , together with recentadvances in the ability to fabricate micro- and nano-scale surfaces , and also becausedrop impact is central to many technological processes including DNA microarrays, digitallab-on-a-chip, water harvesting, dropwise heat removal and anti-icing .Drops hitting superhydrophobic surfaces can bounce off quickly because of the low frictionbetween drop and the substrate, either at the end of retraction or at their maximumextension in a pancake shape . Normally the drop retains a circular symmetry duringthe bouncing and the contact time is bounded below by the Rayleigh limit . However Bird et al. showed that drops impacting on surfaces where asymmetry is introduced with ridgesan order of magnitude smaller than the drop leave the surface with shortened contact time.Moreover, the contact time of drops bouncing on a superhydrophobic macrotexture can takediscrete values depending on the drop impact point relative to the texture and its impactvelocity . The left-right symmetry can be also broken by imposing a surface gradient toinduce a directional movement or by considering impacts on a moving surface . Inthese studies, the reported surfaces are still macroscopically flat, with the feature size at thescale of microns or nanometres. Inspired by the observation that many natural surfaces havemuch larger-scale convex or concave architecture, in this work we first consider the impactof drops on a natural Echevaria surface.
ResultsAsymmetric bouncing on natural surfaces.
Figures 1a-c show the optical and scanningelectron microscopic (SEM) images of the surface. The
Echevaria leaves approximate cylin-ders with a diameter of curvature ( D ) a few millimeters. The surface of the leaves is coveredby waxy nanofibers yielding an apparent contact angle over 160 ◦ . Our experimental re-sults are are very different from those conventionally reported on a flat superhydrophobicsurface . Figure 1d presents selected snapshots of a drop of diameter D = 2 . Echeveria leaf with a diameter of curvature of ∼ .
63 m s − , corresponding to We = 7 . Oh =0.0028. Here,2 e = ρv r /γ is the Weber number, where r is the drop radius, ρ is the liquid density and γ is the liquid-vapor surface tension, and Oh = µ/ √ ργr is the Ohnesorge number, with µ the liquid viscosity. The impacting drop initially spreads isotropically, but the drop spread-ing becomes increasingly anisotropic as the drop starts to retract (Supplementary Movie 1).Interestingly, when the liquid in the axial (straight) direction has started to retract at ∼ . (cid:112) ρr /γ ), the drop leaves the surface maintaining an elongated shapealong the azimuthal direction, indicating that the asymmetric bouncing can be driven bypreferential retraction along just one axis. The contact time ( t ) is ∼
30% faster than thaton the equivalent flat substrate (which is compared in Supplementary Movie 2, rightand Supplementary Fig. 1) and sphere (Supplementary Fig. 2). A reduction in contact timedue to bouncing asymmetry was first reported by Bird et al. who considered drop breakupon surfaces with ridges of size ∼
100 microns.
Symmetry-breaking in droplet bouncing on synthetic surfaces.
Inspired by this un-expected result we hypothesize that new physics comes into play when symmetry-breakingmechanisms are introduced by the convexity of the surface . To explore these, we fab-ricated curved surfaces with varying diameters of curvature D between 4 mm and 20 mm(Supplementary Fig. 3). The surfaces are coated with hydrophobic rosettes of diameter ∼ µ m to give an intrinsic contact angle of ∼ ◦ .Drop impact on the fabricated surfaces reveals similar bouncing dynamics to that on thenatural surface. Figure 2a shows the time-evolution of the spreading diameters in the axialand azimuthal directions on the curved surface with D = 8 mm at We = 7 . of ∼ We . To quantify the spreading asymmetry we define k as the ratio of the maximum values of the drop spreading diameters in the azimuthal andaxial directions. Figure 2b plots the variation of k as a function of the diameter of curvaturenormalized by the initial drop diameter ( D/D ). It is apparent that an increase in the struc-tural anisotropy gives rise to a larger k . Note that for the flat or spherical superhydrophobicsurfaces, the k is equivalent to unity, suggesting that the asymmetric spreading is modulatedby the structural anisotropy. Figure 2c plots the variation of the contact time as a functionof normalized diameter of curvature for different We . The contact time is also significantly3ffected by the anisotropy: at a constant We , the symmetry-breaking surfaces with smallerdiameters of curvature corresponds to smaller contact times. To better elucidate the depen-dence of the contact time on the surface structure, we decompose the contact time into thespreading time t and retraction time t along the axial direction in Fig. 2d. It is apparentthat the spreading time is almost independent of surface curvature, partially due to the factthat the spreading is mainly dominated by the inertia. However the retraction time showsa strong decrease with decreasing D : for D = 6 mm the total decrease in contact timecompared to a flat substrate is ∼
40% for We ∼
15. These results, in conjunction with thespreading dynamics shown in Fig. 2a and b, indicate that the asymmetric bouncing is indeedmodulated by the asymmetric curvature whose size is comparable to that of the impactingdrop. This argument is also confirmed by our control experiment on the spherical surfacewhere the bouncing is symmetric and the contact time is the same as that on a flat surface(Supplementary Fig. 2).
Mechanisms for symmetry-breaking and contact time reduction.
To interpret themechanism behind the asymmetric bouncing observed in our experiments, we first consid-ered the effect of surface topography on initial drop momentum. Two factors will lead tomore momentum being transferred in the azimuthal direction than in the axial direction.First, distinct from the flat surface, the impact area on the curved surface is approximatelyelliptical. As a result of such an asymmetric footprint, more momentum will be transferredperpendicular to the long axis of the ellipse, i.e., along the azimuthal direction. Second,fluid landing on the curved sides of the asymmetric surface has a tangential component ofmomentum which will continue unperturbed.To test this interpretation, we modelled the drop impact, using a lattice Boltzmann algo-rithm to solve the continuum equations of motion of the drop . Details of the equationsand the numerical algorithm are given in the Supplementary Information. Figure 3a showssnapshots of the time evolution of a drop impacting on the asymmetric surface obtainedfrom the numerics. Comparison of the simulation results with those obtained in the experi-ments (Fig. 1d and Supplementary Movie 1) shows that the evolution of the drop shape isqualitatively the same during the rebound, with the axial direction starting to retract first.The colour shading represents the relative heights of the fluid at each time, red high to bluelow. Note that in particular (the 3 rd image) a large rim develops in the azimuthal direction.The arrows in the figure show the local fluid velocity field. Consistent with our experimental4bservation, during the initial spreading stage, the fluid exhibits a radial outwards flow. Asspreading progresses, there is a preferential flow to the azimuthal direction that drives theformation of the larger liquid rims. The liquid pumped around the rim to the azimuthaldirection acts to amplify the contrast in the drop retraction between the two directions,leading to a positive feedback that enhances the asymmetry of the bouncing. This scenariois in striking contrast to that on the flat surface.The simulations allow us to understand how the asymmetric surface topography affectsthe drop bouncing. Figure 3b displays the variation of the momentum in the horizontaldirection relative to the initial impact momentum as a function of time during the impacton the surface with D/D = 1 .
2. In the figure, a positive momentum corresponds to dropspreading whilst a negative momentum corresponds to drop retraction. From the graph,it can be clearly seen that the momentum in the azimuthal direction is always larger thanthat in the axial direction. When the momentum in the axial direction starts to reverseits direction, the azimuthal momentum remains positive and indeed increases slightly. Thisis consistent with our experimental observations that the drop sustains a spreading statewithout retracting in the azimuthal direction, demonstrating the positive feedback from theaxial direction that enables this to occur. As a comparison, we also plotted the variation ofmomentum on a flat surface (blue curve in Fig. 3b), which shows that the momentum alongany given direction for a flat substrate lies between the two curves for the asymmetric surface.In order to quantify how the momentum anisotropy is dictated by the surface topography,we calculated the ratio between the maximal momentum in the azimuthal direction andthat in the axial direction. As shown in Fig. 3c, the momentum anisotropy decreases withincreasing diameter of curvature.To further validate that the momentum asymmetry is responsible for the asymmetricbouncing, we simulated drop impact on a flat surface by introducing a momentum asym-metry into the simulation manually immediately after the initial collision. The momentumin the azimuthal direction was increased by a factor of 2 while the momentum in the axialdirection was reduced by a factor of 2. Indeed, as shown in Supplementary Fig. 4, the dropshows qualitatively the same bouncing pathway as that in Fig. 3a. In particular, the dropretraction in the axial direction is much faster than that in the azimuthal direction, whichis consistent with our experimental results. Moreover, to validate that a momentum asym-metry can be induced by an elliptical drop footprint, we also simulated an initially elliptical5rop impacting a flat substrate which, again, led to a very similar asymmetric bouncing(Supplementary Fig. 5).For surface obstacles much smaller than the drop, such as those used in Bird et al. , theinitial momentum asymmetry is largely suppressed. Notably, the plot of the variation of thecontact time relative to that on a flat surface as a function of D/D (Fig. 3d) displays aminimum at D/D ∼
1. This is expected since, as the obstacle size decreases the surfacebecomes more comparable to a flat surface (
D/D ∼
0) and the momentum anisotropy startsto gets smaller and has less effect. However, in this regime, the critical Weber number fordrop splitting (
W e c ) is low (Fig. 3e) and drops in the experiments tend to break up givingthe mechanism for contact time reduction described by Bird et al. The drop retracts fasteralong the ridge than perpendicular to it. As a result it tends to fragment and the newlyformed inner rims retract away from the obstacle resulting in the contact time reduction.These two regimes for contact time reduction serve to emphasise the richness of the physicsunderlying bouncing on curved and irregular surfaces.We performed a simple hydrodynamic analysis to explain the contact time reductionassociated with the asymmetric bouncing. Since the drop spreading is mainly governed bythe inertia, we consider the drop retraction process here. The drop retraction is primarilydriven by the decrease in surface energy of the thinner central film which leads to a forcepulling the rim of the drop inwards. For conventional bouncing, the drop retraction issymmetric and the surface energy of a central film of radius r is E s ≈ πr γ (1 − cos θ ), where θ is the apparent contact angle, giving a retraction force , F s = ∂E s ∂r ≈ πrγ (1 − cos θ ). Fastdrop detachment requires not only a large driving force in the central film, but also a smallinertia of the rim. However, due to the symmetric retraction and mass conservation, thesetwo processes are mutually exclusive, since a reduction in the central film radius r leads toan increase in the mass of the liquid rim as is apparent in Supplementary Fig. 1.Interestingly, for the symmetry-breaking surface this conflict is resolved by the preferentialfluid flows around the drop rim (Figs. 3a and 4a). To demonstrate this, in Fig. 4c weshowed selected plan-view images of drop retraction on the asymmetric surface. The panelsjust above (Fig. 4b) and below (Fig. 4d) this figure correspond to the side view of dropretraction in the axial direction and azimuthal direction, respectively. It is clear that owingto the preferential liquid pumping to the azimuthal direction (blue arrows), the size of rim inthe axial direction is almost unchanged throughout the retraction stage, whereas that in the6zimuthal direction shows a significant increase. This is confirmed by the simulation resultsin Fig. 4e, which plots the time evolution of the rim heights in the two directions and theheight of the central film. Notably, the mass per unit length of the rim along the azimuthaldirection is more than twice that in the axial direction when the rims from opposite sidesof the drop meet just prior to bouncing, confirming the apparent symmetry-breaking in themass distribution. Moreover, the desirous reduction in the mass of rim in the axial directionis achieved without compromising the retraction force. Due to the preferential spreading onthe curved surface, the central film can be approximated by an ellipse with a major axis b (inthe azimuthal direction) and minor axis a (in the axial direction). Thus, the surface energyof the central film is E a ≈ πabγ (1 − cos θ ). As the asymmetric retraction proceeds the lengthof the major axis b remains constant while there is a continuous reduction in a . Hence theretraction force is now F a ≈ πbγ (1 − cos θ ), and the ratio of the force acting on the rim onthe curved surface to that on the flat surface is b/ r . Experimentally, the drop diameter2 r on the symmetric surface continually decreases whilst the azimuthal diameter b on thecurved surface remains unchanged. Thus, the synergy of the enhanced retraction force andreduced mass of rim rendered by the symmetry-breaking structure results in a remarkablyefficient pathway for fast drop retraction. The convergence of liquid in the axial directiontranslates into motion perpendicular to the surface and drives the drop upwards. Moreover,as shown in Fig. 4d, as the drop retracts on the curved surface, the surface tension energyconverts to kinetic energy with a velocity component in the vertical direction (red arrows)which will aid the bouncing. By contrast, on the flat surface, the drop transition from theoblate shape (7.6 ms) to a prolate one (16.1 ms) prior to its jumping takes a longer time(Supplementary Fig. 1). Discussion
In a broad perspective, we expect that rapid bouncing driven by an asymmetric momentumtransfer occurs on many surfaces that have asymmetric structure on the order of the dropsize. The most obvious extension is to a surface which is concave in one direction and flatin the perpendicular direction. Fig. 5 shows snapshots of a water drop hitting a concavesurface with a diameter of curvature D = − W e = 7 .
9. By contrast to the im-pact on the convex surface, there is a preferential fluid flow from the azimuthal directionto the axial direction. After the drop reaches its maximum spreading in the azimuthal di-rection at ∼ . ∼ . . (cid:112) ρr /γ )(Supplementary Movie 3). Indeed the contact time reduction is even more pronounced thanthat on the convex surface, suggesting that corrugated surfaces may be excellent candidatesfor enhanced water repellency and other applications. For example, many pathogens anddiseases are transmitted through drops , and thus the fast drop detachment from naturalplant and our synthetic surfaces might significantly decrease the likelihood of virus and bac-teria deposition. Additionally, the presence of large-scale curved topography on corrugatedsurfaces could offer promise for enhanced heat transfer performances and anti-icing .Moreover, these corrugated surfaces with such millimeter-scale features are scalable in man-ufacturing. Thus we envision that the asymmetric bouncing discovered on curved surfacesnot only extends our fundamental understanding of classical wetting phenomenon, but alsooffers potential for a wide range of applications . MethodsPreparation of the asymmetric surfaces.
The asymmetric convex surfaces were fab-ricated on copper plate by combined mechanical wire-cutting and chemical etching. Aseries of convex surfaces were first cut with arc diameters ranging from 6 mm to 20 mm.Then the as-fabricated surfaces were coated with hydrophobic rosettes of average diameter ∼ . µ m to render them superhydrophobic. More specifically, after ultrasonic cleaning inethanol and deionized water for 10 min, respectively, the surfaces were washed by dilutedhydrochloric acid (1 M) and deionized water, followed by drying in nitrogen stream. Theywere then immersed in a freshly mixed aqueous solution of 2 . − sodium hydroxideand 0 . − ammonium persulphate at room temperature for ∼
60 min, after which theywere fully rinsed with deionized water and dried again in nitrogen stream. After the oxi-dation process, the surfaces were uniformly coated by CuO flowers of diameter ∼ . µ m.All the surfaces were modified by silanization by immersion in a 1 mM n-hexane solutionof trichloro-(1H,1H,2H,2H)-perfluorooctylsilane for ∼
60 min, followed by heat treatment at ∼ ◦ C in air for 1 h to render superhydrophobic.
Characterization of the surfaces . The optical images of
Echeveria were recorded by aNikon digital camera (Digital SLR Camera D5200 equipped with a Micro-Nikkor 105 mmf/2.8G lens). The micro/nano structures of protuberances and nanofibers were characterizedby a field-emission scanning electron microscope (Quanta
T M
250 FEG). Due to the extremely8ow conductivity of the
Echeveria surface, a thin layer of carbon was coated on the surfacebefore the SEM measurement.
Contact angle measurements.
Owing to the asymmetric structure, it is difficult tomeasure the contact angle of the as-fabricated asymmetric surfaces. For the flat surfacecoated with rosettes (subject to the same treatment), the apparent, advancing ( θ a ) andreceding contact angles ( θ r ) are 163 . ◦ ± . ◦ , 165 . ◦ ± . ◦ and 161 . ◦ ± . ◦ , respectively.These values are the average of five measurements. Impact experiments.
Impact experiments were performed in ambient environment, atroom temperature with 60% relative humidity. Briefly, the Milli-Q water drop of ∼ µ L(with drop diameter ∼ . Acknowledgments
We acknowledge support from the Hong Kong General Research Fund (No. 11213414),National Natural Science Foundation of China (No. 51475401) to Z.W., ERC AdvancedGrant, MiCE, to J.M.Y.
Author contributions
Y.L. and Z.W. conceived the research. Z.W. and J.M.Y. supervised the research. Y.L. andJ.L. designed and carried out the experiments. Y.L. and M.A. analysed the data. M.A. andJ.M.Y. ran the simulations. Z.W., J.M.Y., Y.L. and M.A. wrote the manuscript. Y.L. andM.A. contributed equally to this work.
Additional information ompeting financial interests
The authors declare no competing financial interests.
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100 µm ba D A x i a l Azimuthal
Figure 1 | . The surface morphology of
Echeveria and drop impact dynamics. a , Opticalimage of
Echeveria showing the curvature of individual leaves. b , Low-resolution scanning electronmicroscope (SEM) image of an Echeveria surface showing protuberances on the scale of 100 microns. c , Magnified SEM image of a single protuberance consisting of countless nanofibers. d , Selectedsnapshots showing a drop ( D = 2 . Echeveria leaf at We = 7 .
9. The firstrow is a cross section parallel to the azimuthal direction and the second is a plan view from abovethe drop. After spreading to its maximum extension in the axial direction at 3.8 ms, the dropcontinues its spreading in the azimuthal direction and reaches its maximum spreading at 5.0 ms,while retracting in the axial direction. The drop bounces off the surface at 11.8 ms with a muchshortened contact time compared to that on the flat surface (16.1 ms, Supplementary Fig. 1). C on t a c t li ne l eng t h ( mm ) Time (ms) Axial direction Azimuthal direction t ( m s ) t ( m s ) D / D We = 7.9 We = 11.9 We = 15.8 We = 19.5 We = 23.6 Infinity82 3 4 5 6 7 96 1000.981.051.121.191.261.33 Infinity k D / D We = 7.9 We = 11.9 We = 15.8 We = 19.5 We = 23.6
82 3 4 5 6 7 96 1001011121314151617 Infinity C on t a c t t i m e ( m s ) D / D We = 7.9 We = 11.9 We = 15.8 We = 19.5 We = 23.6 d b c a Figure 2 | . Asymmetric bouncing on bioinspired asymmetric surface. a , The variations ofcontact line length in the axial and azimuthal directions as a function of time. The drop continuesspreading in the azimuthal direction after it reaches its maximum extension in the axial directionat 3.8 ms. While retracting in the axial direction, the lateral extension in the azimuthal directionremains almost constant. b , c and d , The variation of the k (defined as the ratio of the maximumspreading diameters in the azimuthal and axial directions), the contact time, the axial spreadingtime t (left) and axial retraction time t (right) as a function of surface curvature D (normalisedby drop radius D ) under different We . t / t D / D M o m en t u m r a t i o / c Azimuthal direction Axial direction Symmetric surface M o m a z i m u t ha l / M o m a x i a l D / D Infinity t / t = 0 0.15 0.31 0.46 0.77 1 ab cd e W e c D / D Regime of this study Regime of this study
Figure 3 | . Asymmetric bouncing verified by simulation. a , Selected snapshots obtainedusing the lattice Boltzmann simulation showing the time evolution of a drop bouncing on anasymmetric surface for We =10.6 and Oh = 0.0028. The top panel corresponds to the cross-sectionview parallel to the azimuthal direction, and the bottom panel is the plan view from above thedrop. The colours in the plan view are indicative of the relative height of the liquid at each time(see Fig. 4e for quantitative data) and arrows indicate the velocity flux. b , The time evolutionof the momentum normalised by the total initial momentum along the axial (red) and azimuthal(black) directions on the curved surface ( D/D =1.2), and for any direction on a flat surface (blue), We =10.6 and Oh = 0.0028. The positive values correspond to the spreadingstage whilst the negative values correspond to drop retraction. c , The ratio between themaximum momentum transferred into the azimuthal direction and the axial direction as afunction of normalized surface curvature D/D . d , Simulations of the variation of the contacttime relative to that for a flat superhydrophobic surface showing a different dependence on D/D in different regimes. e , Experimental results for the variation of the critical W e fordrop break-up (
W e c ) with D/D . Error bars denote the range of the measurements.16 r b a Symmetric surface Asymmetric surface a h / D t / t Central film Axial direction Azimuthal direction e b Azimuthal directionAxial direction A x i a l Azimuthal cd Figure 4 | . Drop retraction and bouncing dynamics. a , Schematic drawings of the rim andcentral film on the symmetric and asymmetric surfaces, respectively. On the symmetric surface, therim retracts uniformly inwards towards the central film. On the asymmetric surface, the centralfilm is an ellipse with major axis b (in the azimuthal direction) and minor axis a (in the axialdirection) and retraction is primarily along the axial direction. b , Selected side-view images ofdrop retracting on the asymmetric surface in the axial direction. The drop rim is drawn inwardsby the central film and the size of rim remains almost unchanged in the majority of retractionprocess. c , Selected plan-view images of drop retracting on the asymmetric surface. d , Selectedside-view images of drop retracting in the azimuthal direction. Due to preferential liquid pumping e ,Comparison of the time evolution of the normalized rim heights in the axial and azimuthaldirections based on the simulation. The height is scaled by the drop diameter D and timeby the contact time t . During the retraction stage the axial rim height stays roughly con-stant whilst the azimuthal rim height increases greatly due to the preferential flow and masstransfer. The reduced mass of the axial rim rendered by the symmetry-breaking flows resultsin a remarkably efficient pathway for fast drop retraction. a b Figure 5 | . Selected snapshots showing a drop ( D = 2 . D = 8 . We = 7.9 both from the side view ( a ) and planview ( b ). After spreading to its maximum extension in the azimuthal direction at 3.0 ms, the dropcontinues its spreading in the axial direction and reaches its maximum spreading at 6.6 ms. Itfinally bounces off the surface after a contact time of 10.3 ms, reduced by ∼
40% compared to thaton a symmetric surface. More details are shown in Supplementary Movie 3.40% compared to thaton a symmetric surface. More details are shown in Supplementary Movie 3.