Vertical vibration of liquid drops on nano-structured surfaces

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    Keywords: Irradiation; Tracks; Wetting; Drops


    ies followed with many fundamental and applied interestssuch as the measurements of both dynamic tension surfaceand contactless viscosity (by air-levitating drops method

    very high hydrophobicity. We discuss the morphologicalproperties of such surfaces and their inuence on the wet-

    2. Experimental

    We chose to study the oscillation modes of water dropsdeposited on surfaces processed by ion irradiation. For thispurpose the following steps were developed.

    * Tel.: +33 4 72 431218; fax: +33 4 72 431592.E-mail address:

    Available online at

    Nuclear Instruments and Methods in Physics RAlthough these pioneering studies have been used and ex-tended in dierent elds, the investigations concerningthe case of sessile drops on solids have been very limitedfor several decades. In the 1980s microgravity experimentshave motivated experimental [4] and theoretical [5] studiesdevoted to understanding the inuence of liquid dropvibration on crystal growth in microgravity. Dierent stud-

    ting properties. The second part is devoted to a potentialapplication of these surfaces in the study of fundamentalvibration modes of supported submillimeter-size droplets.We measure the dependence of the oscillating modes onvolume. Special care is devoted to the inuence of dissipa-tive eects on solidliquid interface in the oscillation ampli-tude of mode = 2.1. Introduction

    The vibrations of free liquid drops have been investi-gated for more than a century by Kelvin [1], Rayleigh [2]and later, Lamb [3] who established a general expressionfor the resonance mode frequencies fr of a free oscillatingdrop of volume V, surface tension c and density q in theabsence of gravity. For the mode , Lamb obtained:

    fr 1 2c


    [6,7]) and drop atomization [8,9]. In the last decade, theproblem dealing with the inuence of the wetting hysteresison the supported drop oscillations was also investigated byseveral authors [10,11]. Meanwhile, in the last few years,motivated by numerous applications in microuidics [15]and microelectronics [16], interest in this subject has beenrenewed [1214].

    The aim of this paper is twofold. In the rst part weshow that ion track etching technique constitutes an impor-tant tool for constructing nano-structured surfaces withVertical vibration of liquid dro


    Universite de Lyon, F-69000 Lyon, France and Universite de Lyon 1, La

    Available onlin


    This paper reports on both ion track etching induced nano-strution modes of submillimeter drops supported by such surfaces. A dstantially increases the mobility of the supported drop, which acquidrop volume established experimentally is confronted to predictiongravity. 2008 Elsevier B.V. All rights reserved.

    PACS: 61.80x; 68.08 Bc; 68.08p; 47.55D0168-583X/$ - see front matter 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.nimb.2008.03.089s on nano-structured surfaces

    amos *

    atoire PMCN, CNRS, UMR 5586, F69622 Villeurbanne Cedex, France

    1 March 2008

    red surfaces with high hydrophobicity and the fundamental vibra-tic reduction in the hysteresis of the contact angles (Dh 6 6) sub-a quasifree behaviour. The dependence of the oscillating modes onf classical models dealing with free oscillations of droplets without

    esearch B 266 (2008) 31433147

    NIMBBeam Interactions

    withMaterials &Atoms

  • 2.1. Surface processing and characterisation

    features can be pointed out: (i) the advancing and recedingcontact angles increase with the surface roughness whilstthe wetting hysteresis decreases and reaches a minimumvalue of 4. In agreement with previous studies [20,21], thisvery low hysteresis can be explained by the fact that themajor part of the drop base lies on the air layer formedbetween the bottom and the top of the thin walls separatingtwo adjacent holes; (ii) the driving force (per unit length)Fm for the drop mobility decreases with the surface struc-turation. It thus requires about 10 times as much force tomove a droplet on smooth rather than on processed sur-

    3144 S.M.M. Ramos /Nucl. Instr. and Meth. inAmorphous silica (a-SiO2) layers of 400 nm thicknessresulting from thermal wet oxidation of silicon substrateshave been irradiated1 with lead ions at an energy of0.63 MeV/u and a uence of 7 109 Pb cm2. The sampleswere then etched at room temperature in hydrouoric acid(HF) solution on 1% volumic concentration, according to aprocedure described in previous studies [17]. Dierent etch-ing times (given in Table 1) were used to vary the holediameter in a controlled way.

    To provide stable hydrophobic surfaces characterized bycontact angles larger than 90, tailored surfaces weregrafted with PFOTS (C8H4Cl3F13Si) molecules. Prior toPFOTS deposition, the samples were cleaned accordingto the procedure described previously [18]. The cleanedsubstrates were exposed to PFOTS vapour at low pressurefor about 10 h. Grafting of PFOTS molecules on a pristineSiO2 lm also produced a reference surface.

    A Nanoscope III Digital Instruments atomic forcemicroscope (AFM) operating in tapping mode was used.The surfaces were probed under ambient conditions withsilicon tips having a nominal curvature radius of 10 nm.

    The sessile drop method was used to characterize thewettability properties of the processed surfaces. Steady-state advancing contact angles, hA and receding contactangles hR, were measured, with an accuracy of 3, using1ll drops of deionised water (with a surface tensionc = 72 mN m1). The following two parameters were deter-mined from contact angle measurements: (i) the contactangle hysteresis, dened as Dh = hA hR; (ii) the drivingforce per unit length of triple line dened from [19] asFm = c (coshR coshA). These two parameters describethe dissipative eects present at liquidsolid interface.

    2.2. Oscillating system

    A loudspeaker is used to vibrate the structured samplevertically. It is linked to a power amplier connected to adigital function generator. An ultra-pure water drop isdeposited on the substrate that is bound up with the mov-ing part of the loudspeaker. Although frequency andamplitude of excitation are adjustable in a wide range, wechose to work with a sinusoidal excitation of relativelylow amplitude. This choice is based on the fact that in suchconditions only the contact angle is oscillating. In order tocompare our results with theoretical predictions for a freedrop (without gravity inuence) the volume of water dropswas limited to a range of 0.21.8 ll. The radius of suchdroplets is at least three times lower than the capillarylength j1 (j 1 = 2.7 mm). In this case, gravity is a smallperturbation compared to surface tension and the dropremains essentially spherical at equilibrium.1 The irradiations were performed at GANIL (Grand AccelerateurNational dIons Lourds) Caen (France).Considering that the contact angle of deposited dropranges between [hA, hR] [19], particular care to the deposi-tion of drops before each measurement was required. Towork in well-dened conditions we imposed a verticalvibration to the drop with a frequency of 55 Hz duringabout 1 min at variable amplitude, thus inducing a periodicmotion of the contact line. By reducing the excitationamplitude we begin the measurement with a contact anglevalue at rest equal to the mean equilibrium angle hE. Thevalues measured are summarized in Table 1.

    3. Results and discussion

    Fig. 1 shows a typical AFM image along with the corre-sponding surface prole recorded on the processed (etchingtime: 16 min) a-SiO2 lm. It reveals a random distributionof etch pits characterized by an average basal diameterD = 101 20 nm and a mean depth d = 69 15 nm. It isworth noting that the depth value is certainly underesti-mated due to the fact that the bottom of the holes wasnot probed by the AFM tip. In this paper, the parameteradopted to characterize the surface roughness is the maxi-mum peak to valley height, Rt. For the sample presented inFig. 1, a value of Rt = 214 nm was measured on a scannedzone of 25 lm2. It is about 30 times the Rt measured on aat surface (see Table1).

    Fig. 2 shows a photograph of a droplet being advancedon a processed surface when water is added with a syringe.The advancing and receding contact angles are hA = 161and hR = 157 whilst the at surface exhibits water contactangles hA = 113 and hR = 101. The contact angle valuesmeasured for the dierent surfaces are summarized inTable 1. From the analysis of the whole result, two main

    Table 1Etching time tHF, peak to valley surface roughness Rt, measuredequilibrium hE, advancing hA and receding hR contact angles, contactangle hysteresis Dh and restoring force per length unity Fm

    tHF (min) Rt (nm) hE () hA () hR () Dh () Fm (mN/m)

    0 6.6 106 113 101 12 14.414 107.0 123 154 142 12 8.016 214.0 136 157 151 6 3.317 208.0 136 161 155 6 2.818 184.0 135 161 157 4 1.8

    Phys. Res. B 266 (2008) 31433147faces. This dierence is a direct consequence of the wettinghysteresis. On the at surface (Dh = 14) the contact line is

  • 0. inZ D







    S.M.M. Ramos /Nucl. Instr. and Methmore strongly pinned than on structured surfaces (Dh 66). Such surfaces are usually named superhydophobic.

    A scanning in frequency allows us to measure the reso-nance frequency fr. We have limited our investigations tothe three rst modes ( = 2, 3 and 4). The drop prole char-acteristic of each one of these modes is presented in Fig. 3for a sample etched during 16 mn. As indicated in this g-ure, is the half number of nodes along the drop prole.Note that in static regime ( = 0), the drop deposited onthe surface exhibits a quasi-spherical shape. This occursfor two reasons: the rst one is the small volume of thedrops investigated, the second one is due to the high hydro-phobicity of our processed surfaces. In this congurationthe contact between solidliquid interfaces is reduced to a

    Fig. 1. AFM top view and surface prole of a SiO2 lm irradiated with a uesilanized. The areal concentration of the hole structures agrees well with the i

    5 m5 m

    100 nm

    A = 161

    = 4

    Fig. 2. Photograph of advancing contact angle on a superhydrophobicsurface (a). 3D view of silanized SiO2 lm pre-irradiated with a uence of7 109 ions cm2 and etched during 18 mn (b).Distance (m) 1 2 3 4 5

    nce of 7 109 Pb cm2, then etched in 1% HF during 16 min and nallyrradiated uence.

    Phys. Res. B 266 (2008) 31433147 3145small area delimited by a radius rc, given by rc = R sinhEwhere R is the drop radius.

    The loglog plot of the resonance frequencies as a func-tion of the drop volume is shown in Fig. 4(a). As can beseen the frequencies increase with the mode number anddecrease with the drop volume. The exponent of the powert varies between 0.485 and 0.505, which is in satisfac-tory agreement with theoretical predictions (fr / V0.5).However, a discrepancy is found between the absolute val-ues measured for fr and the theoretical predictions. Such adierence can be easily understood with a more quantita-tive analysis of mode = 2, which can be described as asuperposition of two ellipsoids having their maximum totalcurvature either along the equator (oblate ellipsoid) or con-centrated at two poles (prolate ellipsoid). Investigationsbased in simulated microgravity [4] and experiments ofaerodynamic levitation [7] have proposed dierent empiri-

    = 3 = 4



    = 2



    Fig. 3. Photographs of the drop proles (lying on surface presented inFig. 1) for both the as deposited drop and the modes = 2 (4 nodes), = 3(6 nodes) and = 4 (8 nodes). The nodes are indicated by the white spots.

  • . in2

    4f r (Hz


    V (l)100


    0.1 10

    f r (Hz






    3146 S.M.M. Ramos /Nucl. Instr. and Methcal relations that describe well the evolution of the reso-nance frequencies versus the volume of attened drops.On the same basis we propose a simpler empirical relation-ship based in the following features: the superhydrophobiccharacter of our surfaces (high hA values and low Dh) andthe quasi-spherical shape of the drops lying on such sur-faces. It is given by

    fr 8


    s b

    sin hE


    where the rst factor is the Lambs expression for = 2 andthe second is a geometrical parameter taking into accountthe deviation of the drop from a spherical shape. This cor-rective factor is controlled by both the equilibrium contactangle and by the ratio between the maximal elongation ofprolate Lp and oblate Lo ellipsoids (see Fig. 3), named hereb. The resulting t presented in Fig. 4(b) illustrates quite agood agreement between the measured and calculated data.Fig. 5 plots b as function of the driving force Fm. As couldbe expected b varies conversely with Fm. For the lowest Fmvalues, the drop undergoes a larger deformation and thevalues of maximal elongations along polar and equatorialdirections are very close (b = 0.95) to each other. It evi-

    Experiments Relation (1) Relation (2)

    V (l)


    0.1 1 10

    Fig. 4. (a) Resonance frequencies, fr, as a function of the drop volume V.(b) Experimental data for = 2, theoretical fr versus V from Eq. (1) andempirical t from Eq. (2).dences a quasisymmetry of the drop oscillation amplitudeindicating thus that on our processed surfaces the drop be-haves as quasifree. This feature is original and of greatinterest for microuidic devices where a very low dissipa-tive eect on the ow motion is required. The same param-eter b is plotted versus the etching time tHF in the inset. Theclose correlation between b and tHF evidences that thetrack etching technique provides an excellent tool for con-trolling the wetting properties of surfaces.

    4. Conclusion

    In this paper we have discussed the fundamental impor-tance of irradiation eects on the a-SiO2 layers for both the

    Fm (mN/m)















    tHF (mn) 0










    Fig. 5. Ratio between the maximal elongation of prolate and oblateellipsoids b, characteristics of mode = 2, as a function of driving force.Inset: b as a function of the etching time, tHF. The dotted line is drawnonly to guide the eye.

    Phys. Res. B 266 (2008) 31433147construction of surfaces with a controlled high hydropho-bicity and for the study of the fundamental vibration modeof supported small drops. The fact that two properties ashigh hydrophobicity and small contact angle hysteresisare combined on the same surface results in a high dropmobility. The drops behave as quasifree. The evolution ofthe resonance frequencies as a function of the drop volumecan thus be compared to theoretical predictions concerninga completely free drop. A simple empirical relationship,taking into account the drop deviation from sphericalshape, allows us to bring into agreement the measuredand calculated frequency values.


    It is a pleasure to thank A. Benyagoub, I. Monet and B.Canut for precious help in the surface processingexperiments.


    [1] Lord Kelvin, Math. Phys. Pap. 3 (1890) 384.[2] Lord Raylegh, The Theory of Sound, Macmillan, 1894.

  • [3] H. Lamb, Hydrodynamics, Cambridge University P...


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