eepl draft
Drop trampoline
Pierre Chantelot, Martin Coux, Christophe Clanet and
David Quere PMMH, UMR 7636 du CNRS, ESPCI Paris, 75005 Paris, France LadHyX, UMR 7646 du CNRS, Ecole polytechnique, 91128 Palaiseau, France
PACS nn.mm.xx – First pacs description
PACS nn.mm.xx – Second pacs description
PACS nn.mm.xx – Third pacs description
Abstract – Rigid superhydrophobic materials have the ability to repel millimetric water drops,in typically 10 ms. Yet, most natural water-repellent materials can be deformed by impactingdrops. To test the effect of deformability, we perform impacts of non-wetting drops onto thin ( ∼ µ m), circular PDMS membranes. The bouncing mechanism is markedly modified compared tothat on a rigid material: the liquid leaves the substrate as it is kicked upwards by the membrane.We show that the rebound is controlled by an interplay between the dynamics of the drop andthat of the soft substrate, so that we can continuously vary the contact time by playing on themembrane’s characteristics and reduce it up to 70%. Liquid droplets and soft solids can interact in many in-teresting ways. For instance, elastocapillary effects canlead to deformation of solids close to contact lines [1,2] orto bending of slender and thin structures [3, 4]. In moredynamical situations, substrate deformation can induceself-propulsion [5,6], prevent splashing [7,8] or allow liquidpenetration in soft solids [9]. It can also lead to improve-ment of the water or ice repellency of superhydrophobicmaterials [10, 11]. Such materials have the ability to re-flect impacting drops and we focus here on the way theirflexibility affects this property. The contact time τ of mil-limetric water drops on a rigid, repellent solid is on theorder of 10 ms, which can be large enough to induce freez-ing [12], significant heat transfer [13] or contamination bysurfactants [14]. Different techniques have been proposedto reduce τ . Decorating the substrate with macrotextures(such as ridges) was found to divide τ by a factor of typ-ically 2 [15], a reduction also obserbved using soft mem-branes, as recently shown by Weisensee et al. [16]. Thiseffect occurs at large impact velocity, in a regime difficultto explore due to splashing, which might explain the scat-tered nature of the results. Vasileiou et al. also stressedthe ability of soft membranes to reflect viscous drops -a point of obvious practical interest - but did not provideneither a specific study on the contact time nor a model toaccount for its reduction [11]. In order to study systemat-ically the ability of soft solids to enhance water repellency,we chose to texture liquids rather than solids, that is, touse liquid marbles as a model of non-wetting drops [17]. This allows us to show that the interplay between flexiblesubstrates and non-wetting impacts leads to the possibilityof continuously tuning the contact time, which we model.The experiment is sketched in figure 1a. Our substratesare polydimethylsiloxane (PDMS) sheets with thickness h = 20 µ m (Silex) clamped between two plexiglas rings.The clamped sheets are placed on a frame with radius a ( a = 7 . , , . ,
25 mm) and tension is adjusted byweighting the membrane with a mass m . Liquid marblesare made by coating distilled water (density ρ = 1000kg/m and surface tension γ = 72 mN/m) with ly-copodium grains (diameter ∼ µ m) treated with fluoro-decyl-trichlorosilane. These marbles have a surface ten-sion γ ’ ± ∼ ◦ )and low hysteresis ( ∼ ◦ ) typical from superhydrophobicmaterials. Water drops ( R = 1 mm and R = 1 . V can be varied be-p-1 a r X i v : . [ c ond - m a t . s o f t ] J un ierre Chantelot et al. V R a l a s e r m b ca m Fig. 1: a A liquid marble (with radius R and velocity V ) impacts a circular PDMS membrane with radius a and thickness 20 µ m put into tension by a mass m . The membrane is superhydrophobic and its deflection δ is measured by the deviation of anoblique laser sheet. b (top) A liquid marble ( R = 1 . V = 0 .
75 m/s. Thedrop leaves the substrate after 22.5 ms. (bottom) Same liquid marble impacting a flexible membrane ( a = 10 mm, m = 12 . V = 1 m/s. The drop is kicked off after 7 ms, a reduction of 70 % compared to the rigid case, with a pancake shape. c Deflection δ of the center of the membrane for the impact shown in b, from which we obtain the maximal deflection δ max andits time τ d . Later, the free oscillations of the membrane give access to its fundamental period τ m . tween 0.5 and 1.5 m/s by adjusting the height from whichthese marbles fall. We record side views of the impact andmonitor the membrane deflection δ through the observa-tion of a laser sheet in oblique incidence using two fastvideo cameras (Phantom V7) working at typically 10000frames per second. The vertical position of the membraneis directly proportionnal to the displacement of the lasersheet.Marbles bounce off flexible substrates differently fromwhat they do on rigid materials (figure 1b). In thelatter case (top sequence), they spread, recoil and takeoff with an elongated shape, here after 22.5 ms. Thecontact time τ of a marble with radius R = 1 . V andplateaus at τ = 22 . ± . τ in figure 1b is 7 ms, that is, about onethird of τ , a reduction even larger than that reportedin [11, 16]. Recoiling takes place later, while the dropis in the air. Figure 1c shows the time evolution of thedeflection δ of the center of the membrane. Firstly, themembrane sinks down to its maximal deflection δ max at time τ d (see supplementary figure SI2). Then, thesubstrate moves back and goes above the horizontal,which kicks the marble and makes it take off (at time τ )at the membrane’s uppermost position. Later, it freelyoscillates, allowing us to measure its natural period,here τ m = 3 . ± .
10 ms. These free oscillations arefaster than the first oscillation forced by impact, showingthat our system has a characteristic time τ intermediatebetween τ m and τ , the respective response times of the membrane and of the drop.The time τ m (and corresponding frequency f m = 1 /τ m )can be varied by tuning the membrane geometry (throughthe radius a ) and tension (through the mass m , seesupplementary figure SI3). We show in figure 2a how thecontact time τ varies as a function of the impact velocity V for various frequencies f m . For each value of f m , thecontact time is roughly independant of the impact velocity V , apart from a weak increase at low V also observedfor drops on rigid substrates [19]. More importantly, weconfirm our main observation: the contact time τ on softmembranes is reduced compared to τ , the plateau valueon a rigid substrate indicated in figure 2a with a dashedline. Specifically, τ decreases as we increase the frequencyof the membrane, showing the influence of the responseof the substrate on the timescale at which the liquid isrepelled. The marble size R also influences the contacttime, small drops being shed faster than large ones, asshown in supplementary figure SI4 where it is observedthat τ varies slower than R / , the usual inertio-capillarybehavior [20]. The response of the substrate can becharacterized spatially, and we plot in figure 2b themaximal deflection δ max as a function of V . δ max varieslinearly with V , and its value is typically millimetric.When R is fixed (filled circles, R = 1 . δ max and f m . At fixed f m ( f m = 290 Hz, green circles and triangles), δ max increaseswith R , a logical consequence of the change in liquid mass.Our aim is to understand how the liquid and the mem-brane cooperate in an original bouncing mechanism. Ouranalysis holds for τ m < τ , the only regime where we ex-pect contact time reduction. We model the solid/liquidsystem as coupled oscillators (figure 3a and 3b). On theone hand, the marble can be represented as a spring ofp-2rop trampoline ab Fig. 2: a Contact time τ of liquid marbles with radius R = 1 . f m asa function of impact velocity V . The dashed line representsthe contact time τ on a rigid substrate. b Maximal deflection δ max at the center of the membrane as a function of V . Thedotted lines are linear fits. stiffness k d and a mass m d , a system with an oscillatingfrequency f d = π q k d m d . Lord Rayleigh [21] calculatedthe frequency of freely oscillating drops, and showed thatit writes: f d = q γ πm d , which provides the stiffness k d ofthe spring: k d = π γ . On the other hand, the membranecan be modelled as a spring of stiffness k , mass m m andfundamental frequency f m = π q km m . We assume thatthe droplet-membrane system behaves during contact asoscillators in series, as sketched in figure 3b. Then theposition z of the membrane obeys a 4th-order differentialequation: d zdt + k d m m (1 + m m m d + kk d ) d zdt + kk d m d m m z = 0 (1)Equation 1 has two natural limits. (1) On a rigidsubstrate ( k → ∞ ), it reduces to: ¨ z + 4 π f d z = 0.The contact time on rigid repellent materials is simplyproportional to the Rayleigh period 1 /f d [20]. (2) A rigidbead ( k d → ∞ ) hitting a flexible membrane is describedby the equation ¨ z + 4 π f b z = 0, with f b = π q km d + m m ,that is, the frequency of a membrane of stiffness k and mass m d + m m . We performed experiments withpolypropylene beads ( R b = 1 mm and R b = 1 .
75 mm , ρ b = 900 kg/m ) with mass m b , and our data plotted infigure 3c show that the reduced contact time τ f b collapseson a single curve, confirming that f b is the frequency ofthe bead-membrane system. However this added massargument does not capture the contact time reduction ob-served for drops as shown in the supplementary figure SI5.Coming back to the impact of a water drop on a flexiblesubstrate, we can notice that equation 1 provides two nat-ural frequencies, that is, f ∗ = π ( k d m m (1+ m m m d + kk d )) / and f = π ( kk d m d m m ) / . As shown in the SI, we have f ∗ > f whatever the values of the physical parameters, which sug-gests that the dynamics of the system is set by 1 /f , thelonger timescale. When we rescale the contact time τ bythe frequency f and plot it as a function of the impactvelocity V (figure 3d), data for various f m (such as in fig-ure 2a) and various R indeed collapse. Apart from theincrease observed at low V [19], contact time is found toplateau at a value τ ∼ . f . The frequency f turns outto be the geometric mean of that of the drop and of themembrane, f = √ f m f d , a formula capturing how the twoobjects conspire to generate fast bouncing. Interestingly,the frequency f scales as R − / , a behavior very differentfrom that on a rigid substrate (where it varies as R − / ),in agreement with our results in the supplementary figureSI4. Knowing the frequency f yields a simple predictionfor δ max . Before impact, the membrane is immobile andthe droplet with mass m d moves at speed V ; during thefirst oscillation, drop and membrane both oscillate at thefrequency f . Conserving the momentum provides the fol-lowing scaling: m d V ∼ ( m m + m d ) δ max f . Figure 3e and3f represent δ max f b (1+ m m m b ) and δ max f (1+ m m m d ) as a func-tion of V . For both solid and liquid marbles, we observethe predicted linear relationship, with respective slopes 0.4 ± ± . τ (a very lowvalue) and τ , by playing on the membrane characteristicsand the drop radius. The opposite case ( f m < f d ) woulddeserve a separate study. On the one hand, we classicallyexpect in this limit a Rayleigh bouncing time scaling as1 /f d . On the other hand, the soft nature of the substrateimplies that the membrane should be deformed at impactand takeoff, making it usable as a force sensor for dropimpacts.p-3ierre Chantelot et al. c de f zm d m m kk d f d f m ab Fig. 3: a The drop and membrane are modelled as oscillators of respective frequencies f d and f m . b We assume that this systembehaves as two oscillators in series during contact. c Normalized contact time τf b for solid beads impacting a flexible membraneat velocity V . f b = q km m + m d is the frequency of a membrane with mas m d + m m . d Normalized contact time τf of liquidmarbles impacting a flexible membrane at velocity V where f = √ f d f m is extracted from equation 1. e Maximum deflection δ max of the membrane as a function of the impact velocity V for various membrane frequencies and bead radii. Collapse of datais obtained by multiplying δ max by the quantity f b (1 + m m m b ) as suggested in the text. f Same plot for marble impact. δ max isnow multiplied by f (1 + m m m d ). In both cases we observe a linear behavior as predicted in the text. Dotted lines are linear fitswith slopes 0.4 ± ± ∗ ∗ ∗ We thank Direction G´en´erale de l’Armement (DGA)for contributing to the financial support. We also thankHadrien Bense, Lucie Domino and Benoˆıt Roman for in-sightful comments.
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Pierre Chantelot, Martin Coux, Christophe Clanet and
David Quere PMMH, UMR 7636 du CNRS, ESPCI Paris, 75005 Paris, France LadHyX, UMR 7646 du CNRS, Ecole polytechnique, 91128 Palaiseau, France
PACS nn.mm.xx – First pacs description
PACS nn.mm.xx – Second pacs description
PACS nn.mm.xx – Third pacs description
Abstract – Rigid superhydrophobic materials have the ability to repel millimetric water drops, intypically 10 ms. Yet, most natural water-repellent materials can be deformed by impacting drops.To test the effect of deformability, we perform impacts of non-wetting drops onto thin ( ∼ µ m),circular PDMS membranes. The bouncing mechanism is markedly modified compared to that on arigid material: the liquid leaves the substrate as it is kicked upwards by the membrane. We showthat the rebound is controlled by an interplay between the dynamics of the drop and that of thesoft substrate, so that we can continuously vary the contact time by playing on the membrane’scharacteristics and reduce it up to 70%. Contact time of liquid marbles. –
We measure the contact time of liquid marblesimpacting a rigid superhydrophobic substrate (made water repellent by spray coating ofUltra Ever Dry) as a function of the impact velocity V . Figure 1 shows the contact time τ of water droplets and marbles (black circles and dots respectively) as a function of V .Above 0.5 m/s, τ does not depend on V for marbles and droplets and takes roughly thesame values. We measure the plateau value τ = 22 . ± . Fig. 1: Contact time τ of liquid marbles (black dots) and water droplets (black circles) with radius R = 1 . V . Above 0.5 m/s for marbles, τ is independant of V and we measure τ = 22 . ± p-1 a r X i v : . [ c ond - m a t . s o f t ] J un ierre Chantelot et al. Maximal deflection time τ d . – Our setup allows us to measure the deformation ofthe membrane and to monitor the time τ d at which maximal deflection δ = δ max occurs.Figure 2a shows the variation of τ d as a function of V . τ d is roughly constant althoughit slightly increases at small V as observed for τ . The variation of τ d with f m is of moreinterest, τ d decreases as f m increases. Our coupled oscillator model enables us to plot therescaled deflection time τ d f as a function of impact velocity (figure 2b). All the data collapseon a single curve meaning that the timescale of the first oscillation is proportional to thatof the full rebound. We find that τ is roughly equal to twice τ d . a b Fig. 2: a Deflection time τ d as a function of impact velocity V for membranes with differentfrequencies f m (dots) and marbles with different radius R . b Normalized deflection time τ d f as afunction of V . The data collapse on a single curve. The legend is the same as in a. Membrane modelling. –
We model the PDMS sheets as two dimensional ( a ∼ cmand h = 20 µ m) elastic ( E = 1 MPa, ρ = 965 kg/m ) membranes under tension. Themembranes are deposited on frames of radius a and put into tension by adding a mass m tothe clamps maintaining the edge of the PDMS sheets. Table 1 summarises the characteristicsof the membranes studied.The tension T applied to the membrane is T = mg πa . The fundamental frequency of a circularmembrane under tension T is: f thm = χ πa s Tµ where χ is the first 0 of the Bessel function of the first kind and µ is the area density ofthe PDMS sheet. a (mm) m (g) f m (Hz) f thm (Hz)7.5 3.41 302 3097.5 12.23 444 58610 3.41 210 20110 12.23 290 37917.5 13.5 128 17225 4.68 60 5925 13.5 75 10125 29.2 96.8 148 Table 1: Characteristics of the eight membranes used in the accompagnying paper: radius a , mass m , measured and predicted frequencies f m and f thm respectively. Figure 3 compares the frequencies measured during free oscillations of the membrane tothe above prediction. For low imposed masses m (3.41 and 4.68 g), we have a quantitativep-2rop trampoline Supplementary information Fig. 3: Predicted oscillating frequency f thm of the membrane compared to the fundamental frequency f m measured during free oscillations. The grey dotted line has a slope 1 and shows quantitativeagreement between measurement and prediction for the low m data. The light grey line has a slope1.3, it highlights that while we overpredict the frequency a linear relationship between measurementand prediction is kept. agreement between the predicted and measured frequencies. For larger m ( m > .
41 g), wepredict higher frequencies than the measurements but interestingly the linear relationshipbetween measurement and prediction is still observed when the mass is kept constant (lightgrey dotted line). That overprediction of the oscillating frequency may come from frictionbetween the membrane and the frame, although talc powder is placed on the frame to reduceit, and/or from stretching of the membrane that leads to a reduced effective tension.
Contact time of marbles of different R . – We measure the contact time of marbleswith two different radii, R = 1 . R = 1 mm, for two distinct membrane frequencies atwhich contact time reduction occurs ( τ m < τ ). We plot in figure 4 the contact time τ as afunction of impact velocity V . τ is roughly constant with V and it decreases with increasing f m and with decreasing R . For the two frequencies f m , the contact time is increased by afactor 1 . ± . R by a factor 1 .
8. This increase is in good agreement with ourscaling analysis that predicts a change of τ as R / , that is, a factor 1 .
55 when R varies from1 to 1 . . R / ). Fig. 4: Contact time τ of marbles as a function of impact velocity V . τ depends on both f m and R . p-3ierre Chantelot et al. Normalized contact time τ f b of liquid marbles. – The contact time of a rigid beadimpacting a flexible membrane is determined by the timescale 1 /f b , where f b = π q km m + m d is the frequency of a membrane with stiffness k and mass m m + m d . Figure 5 represents thecontact time τ of drops rescaled by f b as a function of impact velocity V . Data does notcollapse on a single curve suggesting that an added mass argument is not enough to modelthe system and that the deformability of the liquid has to be taken into account. Fig. 5: Normalized contact time τf b of liquid marbles impacting a flexible membrane as a functionof impact velocity V where f b is the frequency of a membrane with stiffness k and mass m d + m m .Data does not collapse suggesting that the deformability of the liquid has to be modelled. Natural frequencies of the drop-membrane system. –
In our model, the position z of the membrane obeys the equation: d zdt + k d m m (1 + m m m d + kk d ) d zdt + kk d m d m m z = 0 (1)From (1) we deduce two natural frequencies of the system, that is f ∗ = π ( k d m m (1 + m m m d + kk d )) / and f = π ( kk d m d m m ) / . Let us show that we always have f ∗ > f . Assuming thisinequality, we get k d m m (1 + m m m d + kk d ) > km d . Introducing α = m m /m d and β = k/k d , thiscan be rewritten (1 + α + β ) > αβ , which indeed is obeyed for positive α and β . Movies. – Movie S1 : Impact of a liquid marble with radius R = 1 . V = 0 .
75 m/s. We measure τ = 22 . Movie S2 : Impact of a liquid marble with radius R = 1 . a = 10 mm, m = 12 .
23 g) at velocity V = 1 m/s. The contact time is τ = 7ms, reduced by 70% compared to τ . The lycopodium grains dispersed in air during impactallow us to visualize the airflow after takeoff. Movie S3 : Impact of a rigid bead with radius R b = 1 .
75 mm on a flexible superhydrophobicsubstrate ( a = 10 mm, m = 3 .
41 g) at velocity V = 1 .
25 m/s. We measure τ = 6 ..