the recoiling of liquid droplets upon collision with solid...

PHYSICS OF FLUIDS VOLUME 13, NUMBER 3 MARCH 2001

The recoiling of liquid droplets upon collision with solid surfacesH.-Y. Kima) and J.-H. ChunDepartment of Mechanical Engineering, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139

~Received 1 March 2000; accepted 5 December 2000!

Although the spreading behavior of liquid droplets impacting on solid surfaces has been extensivelystudied, the mechanism of recoiling which takes place after the droplet reaches its maximum spreaddiameter has not yet been fully understood. This paper reports the study of the recoiling behavior ofdifferent liquid droplets~water, ink, and silicone oil! on different solid surfaces~polycarbonate andsilicon oxide!. The droplet dynamics are experimentally studied using a high speed video system.Analytical methods using the variational principle, which were originated by Kendall and Rohsenow~MIT Technical Report 85694-100, 1978! and Bechtelet al. @IBM J. Res. Dev.25, 963~1981!#, aremodified to account for wetting and viscous effects. In our model, an empirically determineddissipation factor is used to estimate the viscous friction. It is shown that the model closely predictsthe experimental results obtained for the varying dynamic impact conditions and wettingcharacteristics. This study shows that droplets recoil fast and vigorously when the Ohnesorgenumber decreases or the Weber number increases. Droplets with a large equilibrium contact angleare also found to recoil faster. Here the Ohnesorge number scales the resisting force to the recoilingmotion, and is shown to play the most important role in characterizing the recoiling motion.© 2001 American Institute of Physics.@DOI: 10.1063/1.1344183#

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I. INTRODUCTION

A liquid droplet impacting with a solid surface will oscillate on the surface unless it possesses negligible iminertia, splatters, or rapidly freezes. The oscillation of tdroplet consists of the initial spreading of the droplet untireaches its maximum base diameter and subsequent ostory motions of recoiling and weak re-spreading. The dnamics of liquid droplets colliding with solid surfaces habeen extensively studied for more than a century.1,2 How-ever, the majority of this effort has been focused oninitial spreading process, i.e., from the moment of impacthe moment when the droplet reaches its maximum bdiameter.3–11 As a result, understanding of the dynamicsthe recoiling appears to be very far from complete. Thisper reports the study of the recoiling behavior of differeliquid droplets~water, ink, and silicone oil! on different solidsurfaces~polycarbonate and silicon oxide!.

Although the ‘‘mechanism’’ of recoiling has not yebeen investigated, there were some authors who reporteoscillatory motions of a droplet upon impact. Elliot anFord12 reported that the impacting drop undergoes sevstages, including spreading and retraction, before it reachsessile drop form. Cheng13 measured the dampened vibratomotion of water drops impacting onto a coal surface. Fuet al.14,15 conducted a theoretical study on the spreadingrecoiling of a liquid droplet upon collision with a solid suface. Their model was based on solving the full NavieStokes equation by utilizing deforming finite elements a

a!Author to whom correspondence should be addressed. Present adThermal/Flow Control Research Center, Korea Institute of ScienceTechnology, Seoul 130-650, Korea. Telephone: 82-2-958-6709; Fax:2-958-5689; electronic mail: [email protected]

6431070-6631/2001/13(3)/643/17/$18.00

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allowing the contact line to slip. Their prediction was inqualitative agreement with the experimental results, butinvestigation of the recoiling dynamics itself was not coducted. Pasandideh-Fardet al.16 also performed a numericastudy of droplet impact. Their model solved the full NavieStokes equation using a modified SOLA–VOF methoSOLA–VOF ~SOLution Algorithm–Volume Of Fraction! isa program for the solution of two-dimensional transient fluflow with free boundaries, based on the concept of a frtional volume of fluid. Their model accurately predictedroplet contact diameters obtained by experimental measments during the initial spreading stages. However, acrepancy was revealed during the recoiling stages. Schiafand Sonin17 discussed the oscillations of a droplet’s centline elevation after its footprint is arrested by freezing onsubcooled target of its own kind. They found that the osclation damping time of a deposited water droplet is in goagreement with the damping time of a negligibly viscouliquid drop that oscillates freely. Recently, Aziz anChandra18 photographed a vigorous recoiling of a molten tdroplet impacting on a heated stainless steel plate.

In this work, we experimentally study the spreading arecoiling of different droplets on different solid surfacesinvestigate not only the effects of dynamic impact conditiobut also the wetting effects arising from different combintions of liquids and solids. In addition, we present appromate models to predict the behavior of the droplets. Otheoretical study is based on an approach to the dynamicdroplet impact, originated by Kendall and Rohsenow19 andBechtel et al.20 The approach utilizes the variational principle rather than the Navier–Stokes equation. Its advantis that by assuming the geometry of the deforming droplesimple differential equation can be obtained to describe

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644 Phys. Fluids, Vol. 13, No. 3, March 2001 H.-Y. Kim and J.-H. Chun

droplet dynamics. Therefore, if the approximate models pvide sufficiently accurate predictions, one can save signcant computational efforts required to solve the NavieStokes equation numerically. Although there exist sevesimplified models to describe the ‘‘initial spreading’’ proceby a single equation,16,21–23simple modeling for the ‘‘recoil-ing’’ process has been sparse except those of Refs. 1920. Moreover, unlike numerical analysis based on theNavier–Stokes equation, the models presented here grfacilitate the analytical study of the recoiling mechanism athe effects of various parameters including inertia and infacial properties.

First we describe the experimental methods and tpresent the approximate models of the droplet dynamicsthe models, two different droplet shapes, i.e., a cylindera truncated sphere, are assumed and each model is bupon the method of Refs. 19 and 20, respectively. Theperimental results are compared with the predictions ofmodels. Finally, the effects of dynamic impact conditioand wetting characteristics on the recoiling behavior arecussed to elucidate the mechanism underlying the recoil

II. DESCRIPTION OF THE EXPERIMENTALAPPARATUS

The experimental apparatus is illustrated in Fig. 1.consists of a pipette which gently ejects a liquid dropletflat target on which the droplet falls, a high-speed video stem to record the shape evolution of the droplet, and a sboscope which is synchronized with the video system. Asexperimental liquids, deionized water, ink~NCR 940607!,and silicone oil~Dow Corning 704 diffusion pump oil! areused. Table I shows the physical properties of these liqu

FIG. 1. Experimental apparatus for the high speed imaging of dropletnamics.

TABLE I. Physical properties of the liquids used in the experiments.

LiquidDensity~kg/m3!

Surface tension~N/m!

Viscosity~kg/~m s!!

Water 996 0.0717 8.6731024

Ink 1052 0.055 2.631023

Silicone oil 1064 0.0373 3.6331022


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To examine various impact conditions, the velocity priorimpact is varied by changing the distance between the pipand the target. In addition, two different sizes of pipet~Gilson Pipetman P200 and P1000! are used to vary theoriginal droplet diameter.

Since the droplet motions, especially recoiling, are etremely sensitive to the cleanness of target surfaces, theget materials are selected such that the surface contamindue to dusts, oils, etc., is expected to be very low. Asconsequence, the compact disc~polycarbonate! and the ther-mally oxidized silicon wafer~silicon oxide, SiO2! are se-lected as the target surfaces. Our experiments revealed this very difficult to obtain consistent results~e.g., equilibriumcontact angle and recoiling speed! using commercial glassunless it is cleaned with extreme care. The root-mean-sq~RMS! surface roughness of polycarbonate surface was msured to be 1.3 nm with 4.8% of standard deviation by scning areas of~10 mm!2 using an atomic force microscop~Park Scientific Instruments M5!. A typical RMS roughnessof the thermal oxide (SiO2) layer is about 3 nm.24

The equilibrium contact angles between the liquids athe surfaces are measured to evaluate the wetting charaistics using the same method as employed in Schiaffinexperiment.25 The method basically measures the contangle of sessile droplets on a solid surface and the dataextrapolated to zero volume to account for the gravitatioeffect. The contact angles thus obtained are presenteTable II.

A high speed video system~Kodak Ektapro EM, Model1012! records the spreading and subsequent oscillationdroplet on a solid surface at a rate of 1000 frames per sond. An image stored in the system consists of 1923239pixels. The illumination is provided by a stroboscope, whiis synchronized with the camera, to capture very sharpages. The images stored in the digital memory are doloaded onto a video tape using an S-VHS video tapecorder, and analyzed by an image analysis software, whiccapable of measuring the dimensions of objects by the nber of pixels. An object of a known size~8 mm in diameter!is recorded by the same video setup and used for calibraIn addition, to obtain highly accurate conditions of the drolet impact, the weight of the droplet is measured for eaexperiment using a high-precision balance~Mettler Toledo,Model AB104!. The diameter deduced by the weighmeasurement method is compared with that obtained byimage calibration. The diameters measured by both methare in very close agreement in all cases, with less thandiscrepancy. Using the image analysis software, we mea

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TABLE II. Equilibrium contact angles.

Liquid Solid Contact angle

Deionized water Polycarbonate 87.4°Deionized water Silicon oxide 58.6°Ink Polycarbonate 70.9°Ink Silicon oxide 51.5°Silicone oil Polycarbonate 6.2°


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645Phys. Fluids, Vol. 13, No. 3, March 2001 The recoiling of liquid droplets upon collision with solid surfaces

the base diameter of the droplet at each frame to determthe temporal evolution of droplet shapes.

III. MODELING

As mentioned earlier, we model the spreading andcoiling of a droplet by adopting the variational principle istead of solving the Navier–Stokes equation with movboundaries. We assume two different droplet shapes inplying the variational method, i.e., a cylinder and a truncasphere. It is noted that the accuracy of a model criticadepends on how closely the assumed shape in the mresembles the real droplet. It is shown in Sec. IV thatpending on the impact conditions, one shape assumptionsults in better prediction than the other for different recoilibehavior. Kendall and Rosehnow19 were first to use thevariational principle assuming the cylindrical shape, in moeling the droplet dynamics. However, they did not consithe interaction of the liquid droplet with a solid surface athe viscous effects. Our model substantially extends thcylinder model to include the wetting properties of the liqudroplet with a solid surface and frictional dissipation.variational method using a truncated-sphere shape asstion was derived by Bechtelet al.20 In this work, we modifythe frictional dissipation term employed in their model.

A. Problem formulation using a cylinder model

We use the variational principle26 to describe the motionof a droplet colliding with a solid surface:

Et1

t2~dT* 2dV* 1dWf* !dt* 50, ~1!

whereT* denotes the kinetic coenergy of the droplet, i.the complementary state function of the kinetic energy,26 V*the potential energy, andt* the time. The frictional workdWf* is expressed as

dWf* 52F* dy* , ~2!

whereF* denotes the frictional force andy* the displace-ment of the frictional motion. In cases where the frictionforce is due to the wall shear stress, we write Eq.~1! as

Et1

t2Fd~T* 2V* !2EAb*

t* dD r* dA* Gdt* 50, ~3!

wheret* is the shear stress at the base of the droplet,D r* theradial displacement, andAb* the base area. Each term in thintegrand can be evaluated when the shape of the deformdroplet and the velocity profile are known.

The initial velocity and the diameter of the original drolet before collision areU* andD* , respectively. The droplehas the densityr, the surface tensions, and the viscositym.We chooseD* and (s/rD* )1/2 as the characteristic lengtand velocity scales, respectively. We note that most stuaddressing the ‘‘initial spreading’’ of droplets employ an iertial spreading time,D* /U* , as a characteristic time scalHowever, the time scale that best characterizes the recomotion driven by capillary action is the characteristic osclation period of a freely vibrating droplet (rD* 3/s)1/2, and


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this scale has been adopted in Refs. 17 and 20 for the stuof oscillating droplets on solid surfaces. Moreover, Wachtand Westerling27 and Foote28 showed that the time it takefor a colliding droplet to bounce off the contact surface dto vigorous vibration is close to the free oscillation perioThus we choose (r* D* 3/s)1/2 as our time scale. The following quantities are nondimensionalized based on thscales unless noted otherwise, and their forms are sumrized in Appendix A.

The cylinder model assumes the droplet on a solid sface as a cylinder whose nondimensional base diameterheight areDb and h, respectively, as shown in Fig. 2. Thvolume of the cylinder is the same as that of the droplet, t

Db2h5 2

3 ~4!

is satisfied for anyt. If eitherDb or h is specified in time, theother is known by Eq.~4!. Therefore, it is sufficient to simulate the temporal evolution of eitherDb or h to fully describethe dynamics of the oscillating cylinder.

The evaluation of the kinetic coenergy is essentiallysame as that of Kendall and Rosehnow. Assuming axismetric motion and taking the linear velocity-momentum rlation ~Ref. 26, pp. 19 and 20!, the nondimensional kineticcoenergy is written as

T5EV

~vz21v r

2!dV, ~5!

wherevz andv r denote the nondimensional axial and radvelocity of the flow inside the cylinder, respectively, andVis the volume. For the energy and volume scales, we chopsD* 2/12 andpD* 3/6. We relate the axial velocity,vz ,with h as

vz5z

hh, ~6!

where the overdot denotes the time derivative. The ravelocity, v r , is given by continuity as

v r521

2

r

hh, ~7!

where r is the nondimensional radius. The flow field giveby Eqs.~6! and ~7! is that of the potential flow which satisfies the Laplace equation. Substituting Eqs.~6! and ~7! intoEq. ~5! and performing integration over the volume of thcylinder, we obtain

T51

3h2S 11

1

16h3D . ~8!

FIG. 2. Geometry of the cylinder model.


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The potential energy of the system consists of the sface energy and the gravitational energy. The surface potial energy is a sum of the energy of the cylinder surfacecontact with gas,ps(Db* h* 1Db*

2/4), and that of the bot-tom surface touching the solid surface,pDb*

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2sSG)/4.29 HeresSL andsSG denote the interfacial tensionor free energy densities, associated with the solid–liquidsolid–gas interfaces, respectively~Ref. 30, p. 58!. Assumingthat the interfacial energy densities stay constant whildroplet moves on the surface, the difference ofsSL andsSG

can be obtained using Young’s equation, i.e.,s cosu5sSG

2sSL, whereu is the equilibrium contact angle between tliquid droplet and the solid surface. Maoet al.31 also sug-gested that the equilibrium contact angle be used in evaling the surface energy of the solid–liquid interface.

The gravitational potential energy can be ignored whits change is small as compared to the change of the intecial energy. The change of the gravitational energy durdroplet motion,DVg* , is scaled asDVg* ;rV* gR* , whereR* is the radius of a droplet. The change of the interfacenergy,DVs* , can be scaled asDVs* ;psD* 2(Dmax

2 21),whereDmax is the ratio of the maximum spread diameterthe original droplet diameter. Thus the relative magnitudethe gravitational energy to the interfacial energy is scaledDVg* /DVs* ;(1/12)Bo/(Dmax

2 21), where the Bond numberBo5rgD* 2/s. In the experiments whose results are copared with the current modeling later in this paper, the valof DVg* /DVs* are commonly less than 0.1. Therefore, wneglect the gravitational effect in evaluating the potenenergy. The total potential energy of the cylinder is thus, idimensionless form,

V53Db2F4h

Db1~12cosu!G . ~9!

It is noted that our potential energy evaluation is differefrom that of Kendall and Rosehnow in that ours takesinterfacial energy between the liquid and solid surfaces iaccount. By using Eq.~4!, we express the potential energy

V52F2~6h!1/21~12cosu!

h G . ~10!

The dissipative work is estimated in a similar mannerBechtel et al.20 Since the potential flow field we obtaineabove does not afford the viscous effects, we estimateexternal viscous stress,t* , based on that of an oscillatinstagnation flow with the period (r* D* 3/s)1/2. The stresst*is written as

t* 5mv r*

dH*, ~11!

where m is the viscosity. The characteristic hydrodynamboundary layer thickness,dH* , of the oscillating flow, isscaled as32

dH* ;S m2D* 3

rs D 1/4

. ~12!

Choosing (m2s/rD* 3)1/2 as the stress scale, we obtain tfollowing nondimensional expression for the viscous stre


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whereFd is the dissipation factor, which is determined empirically through comparison between the experimental amodeling results. A detailed discussion on the dissipatfactor is given in Appendix B. Oh is the Ohnesorge numbdefined as

Oh5m

~rD* s!1/2. ~14!

The radial displacement,D r , is written as

D r5E v r dt. ~15!

Substituting Eq.~7! into Eq. ~15!, we obtain

D r52 12r ln h, ~16!

where we arbitrarily set the integration constant to zero sidD r is of our interest here.

Performing the variation of the kinetic coenergy wirespect toh and using integration by parts, we get

Et1

t2dT dt52

1

3 Et1

t2F2hS 111

16h3D23

16

h2

h4Gdh dt.

~17!

By a similar procedure, we obtain

dV52F61/2

h1/22~12cosu!

h2 Gdh ~18!

and

dWf5Fd

24Oh1/2

h

h4 dh. ~19!

On substituting Eqs.~17!, ~18!, and ~19! into Eq. ~3!, werequire that the coefficient ofdh should be zero for the variational formula to be satisfied. In consequence, we finaobtain an equation to describe the temporal evolution ofh:

h2A~h!h21B~h!h1C~h!50. ~20!

The coefficients are given by

A~h!5 332h

21~h31 116!

21, ~21!

B~h!5 23FdA~h!Oh1/2, ~22!

and

C~h!532A~h!@61/2h7/22~12cosu!h2#. ~23!

The initial conditions, i.e.,h and h at t50, should bespecified to solve Eq.~20!. Since the original droplet andcylinder have a substantial difference in shape, we seektial conditions that will ensure that the initial kinetic coeergy and potential energy of the cylinder are the samethose of the original droplet. In addition, the volume mustconserved as stated in Eq.~4!. The equality of initial poten-tial energies of the original droplet and of a cylinder yielthe following relation:


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~12cosu!

4Db~0!. ~24!

Further arrangement of Eq.~24! is made by using Eq.~4!:

~12cosu!

8Db

3~0!21

2Db~0!1

1

350. ~25!

Therefore, Eq.~25! determines the initial diameter of thcylinder and the initial height is then given by either Eq.~4!or ~24!. We note that the initial diameter and height of tcylinder depends solely on the contact angle,u, and Fig. 3shows initialDb andh versusu. The initial h is determinedby the equality of the kinetic coenergies of the original drolet and of a cylinder:

h~0!52F3 WeS 111

16h3~0! D21G1/2

, ~26!

where We is the Weber number defined as W5rU* 2D* /s.

Alternatively, the temporal evolution of the base radiuR5Db/2, can be expressed as the following, whichequivalent to Eq.~20!:

R2A~R!R21B~R!R1C~R!50. ~27!

Here the coefficients are given by

A~R!53R212 13R

23G~R!, ~28!

B~R!5 23FdG~R!Oh1/2, ~29!

and

C~R!5296R3G~R!@ 1216R

272 136~12cosu!R24#, ~30!

where

G~R!5 916R

2~ 1216R

261 116!

21. ~31!

FIG. 3. Initial diameter and height of the cylinder vs contact angle.


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,

In summary, the cylinder model presented here is baon the modeling of Kendall and Rosehnow19 on an inviscidcylindrical drop in the air. We added the effects of wettinbetween the liquid and the solid, and the frictional dissiption. To evaluate the frictional dissipation, the Bechet al.20 dissipation model was employed with modificatioOur cylinder model yields the second order nonlinear diffential equation, Eq.~20! or ~27!, which completely describesthe dynamics of a liquid cylinder impacting on a solid suface. The initial value problem can be solved numericawith the initial conditions specified by Eqs.~25! and ~26!,combined with Eq.~4!.

B. Problem formulation using a truncated-spheremodel

In this section, we present the Bechtelet al.20 truncated-sphere model with the estimation of the frictional term mofied. The geometry of a truncated sphere is shown in FigBechtelet al. assumed the same velocity profiles as in tcylinder model above. Following the same procedure asabove formulation using a cylinder model, the equationthe temporal evolution ofh is given by~for detailed deriva-tion, see Ref. 20!

2E~h!h2G~h!h21I ~h!h1J~h!50, ~32!

where the coefficients are given by

E~h!5FM 8~h!

M ~h! G2S 13

180h51

11

144h21

1

72h21D , ~33!

M ~h!5 16~2h1h4!, ~34!

FIG. 4. Geometry of the truncated-sphere model.~a! Initial shape of thesphere at impact.~b! Assumed shape of a droplet during spreading arecoiling.


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h4

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and

J~h!52@4h2h221cosu~2h1h22!#. ~37!

The parameterc is given, in the present nondimensionaliztion, by

c5b•Oh, ~38!

where b is related to the boundary layer thickness ofoscillating flow with the period (r* D* 3/s)1/2. Now wemodify the Bechtelet al. model to include the dissipatiofactor,Fd , in the estimation. Thus,b is given by

b5Fd

Oh1/2. ~39!

As mentioned above, the value ofFd is empirically deter-mined as discussed in Appendix B.

The base diameter is related toh by

Db52@ 13~h212h2!#1/2. ~40!

Since the initial droplet shape assumed by this model exarepresents the real spherical shape, the initial conditionsobtained straightforwardly. The initial height is set equalthe original diameter of the droplet and the initial velocityset equal to the velocity of the droplet prior to collisioAfter nondimensionalization, we write

h~0!51 ~41!

and

h~0!52We1/2. ~42!

However, we note that while the exact value of the initkinetic coenergy ispr* D* 3U* 2/12, its value given by thismodel is 13pr* D* 3U* 2/120, due to the assumed velociprofile. The values of the initial potential energy are eqautomatically since the assumed and real shapes aresame.

We note that the angle between the tangential line oftruncated sphere and the solid surface has no physical ming. As the cylinder model above always has the contangle of 90°, the contact angle of the truncated sphermerely an outcome of the intrinsic shape assumption ansuch a shape solely determined by Eq.~32!. The value ofuappearing in Eq.~37! is the equilibrium contact angle, whicis constant, as shown in Table II. Here we emphasize thacontact angle appearing in our model is not used to define


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apparent angle between the tangential lines of an assudroplet shape and a solid surface, but to evaluate the soliquid interface energy.

In summary, we introduced the Bechtelet al.20 modelwhich assumes the droplet as a truncated sphere. Howethe evaluation of the frictional dissipation was modifiedthis work to include the dissipation factor that is to be detmined empirically. The second order nonlinear differentequation, Eq.~32!, describes the dynamics of a liquid truncated sphere on a solid surface. The initial value problembe solved numerically with the initial conditions specifiedEqs.~41! and ~42!.

IV. RESULTS AND DISCUSSION

A. Water droplets

Figure 5 shows the dynamics of water droplets collidiwith polycarbonate surfaces. The water droplets, which psess relatively low viscosity and high surface tension as cpared with ink and silicone oil droplets, go through vigorooscillation due to high We and low Oh.17 Nevertheless, thetwo droplets in the figure show distinctive impact behavdepending on impact conditions as explained below. Ttemporal evolution of the oscillating droplet shapes is psented quantitatively in Fig. 6, showing the base diametetime until the droplets nearly come to rest. In the figure,vigorous primary oscillation of a droplet with higher impainertia is well observed as compared to one with lower intia. The droplet, which impacts on the surface with highspeed, and thus with greater inertia and higher Weber nber, spreads further in the initial spreading process. Figualso shows that, in the subsequent recoiling stages, the dlet with greater inertia retracts more vigorously than one wsmaller inertia, and consequently exhibits a smaller baseameter and a higher centerline elevation just prior [email protected]., in Fig. 5, 25.6 ms for~a! and 36.2 ms for~b!#. In Fig. 5 we name the first extension and retractionthe base diameter of a droplet upon impact, the primspreading, and the primary recoiling, respectively.

The droplets with greater impact inertia show relativehigher motility than those with weaker inertia in the oscilltion stages subsequent to the primary recoiling. These ssequent oscillations were experimentally observedCheng13 and Zhang and Basaran.33 However, motion in thosestages is relatively weak compared with the primary motio~see Fig. 6!. Therefore, our interest in this work mainly consists in the primary spreading and primary recoiling stag

Next, we compare the experimental data, especithose of the primary recoiling stage, with the predictionsthe models, in Fig. 7. The results of both the cylinder moand the truncated-sphere model are presented. In theimpact inertia case as shown in Fig. 7~a!, both the cylindermodel and the truncated-sphere model show qualitativgood agreement with the experimental data. It is notedthe cylinder model overestimates the strength of recoiliwhile the truncated-sphere model underestimates it. In7~b!, where impact inertia is higher than that in Fig. 7~a!, thecylinder model closely predicts the experimental data. Onother hand, the truncated-sphere model deviates significa



FIG. 5. Images of a water droplet colliding with a polycarbonate surface.~a! Original droplet diameter53.6 mm, Impact velocity50.77 m/s, We530 andOh50.0017.~b! Original droplet diameter53.5 mm, impact velocity53.47 m/s, We5582, and Oh50.0017.

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from the experimental data, failing to predict the strongcoiling. Figure 7~c!, where the impact inertia is even higheshows the closer agreement between the cylinder modelthe experimental results, while the truncated-sphere modeviates even more.

We note that the truncated-sphere model results are cto the experimental data whent is large (t52). This is be-cause the steady-state shape of the truncated-sphere mand the actual sessile droplet are identical. Here the stestate implies when the time derivatives in Eq.~32! vanish,thus the steady-state shape is given byJ(h)50. Since theradii of the droplets used in these experiments~1.6 to 1.8mm! are less than the capillary length of water,k21

5(s/rg)1/252.7 mm, the deformation of the sessile drofrom a truncated-sphere shape due to gravity, is small.34

Figure 8 shows the droplet dynamics longer than Figto better illustrate this statement. While the droplets, whbehavior is illustrated in Fig. 7, generally show weak badiameter oscillations aftert52, the droplet in Fig. 8 retainsits motility longer than those droplets owing to its highimpact inertia~see also Fig. 6!. Again, the cylinder modepredicts the primary recoiling fairly accurately, and ttruncated-sphere model’s prediction approaches the equrium state slowly as exhibiting a great discrepancy from


-

ndel

se

deldy

7e-

b--

perimental results. It is emphasized that in all the simulatioshown in Figs. 7 and 8, a single value of dissipation fachas been used for each model~Table III!.

When the impact inertia is high, i.e., We is high, fingedevelop at the rim of a spreading droplet due to tRayleigh–Taylor instability of the decelerating spreadifront.11,18,35The image at 4.2 ms of Fig. 5~b! shows the typi-cal fingers. However, the fingering is supposed to havesignificant effects on measurement and prediction of the bdiameter considering that the magnitude of the frontal unlations is less than 2% of a local spreading radius even wWe reaches 1020~see Ref. 10! which is much higher than themaximum We used in this study.

So far we have observed that the cylinder model maccurately predicts the experiments than the truncated-spmodel in the primary recoiling stages. This is mainly becauthe truncated-sphere model predicts slower recoiling thanexperiments. This may be explained by investigating theages in Fig. 5, which show that the recoiling water dropresembles a cylinder more closely than a truncated sphThe similarity of the droplet shape to a cylinder and its dcrepancy from a truncated sphere are more pronouncethe impact inertia increases and thus as the recoilingstronger. This is consistent with the observed improvem


doa.ancinucuepar

-x

wer

ul

dem

e-reee

et d

veter.tial

r-lib-ater

coan

s forcyl-

.igibly


of agreements between the experiments and the cylinmodel as inertia increases. This explains the phenomenaserved in Fig. 7~a!, where the recoiling droplet assumesshape lying between a cylinder and a truncated sphereboth Figs. 7 and 8, the cylinder model does not yieldaccurate prediction after the primary recoiling stages, sithe droplets no longer resemble a cylinder beyond that poIn addition, we note that the truncated-sphere model sceeds in predicting the primary spreading stage fairly acrately. This appears to be because a truncated sphere rsents the shape of the collapsing droplet in the primspreading stages with sufficient accuracy.

To further examine the validity of our model, we compare our modeling predictions with previously reported, eperimental and numerically simulated results. In Fig. 9,compare our predictions with the experimental and numcal results of Fukaiet al.14 on water droplets colliding withPyrex glass and wax coated plates. Their numerical simtion solved the full Navier–Stokes equation, as mentionedthe Introduction. In this paper, we used a cylinder mosince Oh and We of these experiments fall into a regiwhere a cylinder model is expected to be more accurate~seeabove and also Sec. IV E!. The figure shows that the agrement of our modeling with their experimental results agenerally good. Moreover, comparing the computationalforts to solve the full Navier–Stokes equation and to solvsingle differential equation such as Eq.~27!, the efficiency ofour model becomes apparent.

B. Ink and silicone oil droplets

Ink and silicone oil, the liquids whose behavior will bdiscussed in this section, possess physical properties tha

FIG. 6. The temporal evolution of the base diameter of water dropletsliding with a polycarbonate surface. The impact conditions for squarestriangles are the same as those in Figs. 5~a! and 5~b!, respectively. Theimpact conditions for circles: Original droplet diameter53.7 mm, impactvelocity51.63 m/s, We5137, and Oh50.0017.


erb-

Innet.c--re-y

-ei-

a-inle

f-a

if-

fer significantly from those of water. For example, they haa greater viscosity and a smaller surface tension than waIt is shown here that these differences make substanchanges in the droplet impact behavior. Figure 10~a! showsimages of an ink droplet colliding with a polycarbonate suface. Although the ink droplet has comparable size, equirium contact angle, and impact speed to those of the w

l-d

FIG. 7. Predictions of the models and the experimental measurementwater droplets on a polycarbonate surface. Modeling results using theinder model~CY! and the truncated-sphere model~TS! are both presentedBeyond the time range shown here, the base diameter change is neglsmall. ~a! We530 and Oh50.0017,~b! We5137 and Oh50.0017,~c! We5207 and Oh50.0018.


gallritioelgercfareh

n-

eritthnlurre

osiuledrieaca

ro-

ob-gett theentsofabertact

rtiaitialtactthehlythat

l,

ourtion


droplet in Fig. 5~b!, their behaviors, especially their recoilinmotions, are very different. The ink droplet does not recoilvigorously as the water droplet but rather, recoils graduauntil it reaches its equilibrium shape. Due to this weak pmary recoiling, the subsequent re-spreading and oscillaare absent in this case. At this point, we may qualitativremark that the high viscosity of the ink induces a larresistance to a recoiling flow and also that the restoring fois weak due to a small surface tension. These combinedtors contribute to the relatively weak recoiling motion. Modiscussion on this is postponed to the following sections. Tmotion of a silicone oil droplet impacting with a polycarboate surface is shown in Fig. 10~b!. Silicone oil has a viscositytwo orders higher than that of water and it also has a vlow equilibrium contact angle on polycarbonate. Due tohigh viscosity, its droplet spreads less than those of the oliquids in the primary spreading stage, and it shows oslight recoiling. Afterwards, since the droplet wets the sface almost perfectly in its equilibrium state, the dropletspreads slowly until it comes to rest as a thin film.

After examining both Figs. 10~a! and 10~b!, which showthe shapes of the recoiling droplets, a truncated-sphere mseems a clear choice to predict the dynamics of ink andcone oil droplets. Figure 11 shows the experimental resand model predictions. The predictions of the truncatsphere model agree with the experimental results surpingly well, considering the approximate nature of the modThe figure shows the base diameter of an ink droplet imping on a silicone oxide surface as well as on a polycarbon

FIG. 8. Behavior of a water droplet on a polycarbonate surface untilt55.CY and TS denote the cylinder model and the truncated-sphere modespectively. We5582 and Oh50.0017.

TABLE III. Dissipation factors used in this work.

Liquid Assumed shape in model Fd

Water Cylinder 15Water Truncated sphere 5.3Ink Truncated sphere 3.2Silicone oil Truncated sphere 3.2


sy-n

y

ec-

e

ysery--

delli-ts-s-l.t-te

surface. The effect of the target surface on the recoiling pcess will be discussed in Sec. IV D.

C. Effects of Weber and Ohnesorge numbers

So far we have presented the experimental resultstained by varying such conditions as droplet liquids, tarsurfaces, and impact inertia. It has also been shown thaapproximate models predict the experimental measuremfairly well. The models we employ state that the motionthe droplet, more specifically the temporal evolution ofnondimensional base diameter, is governed by the Wenumber, the Ohnesorge number, and the equilibrium conangle. This is consistent with Schiaffino and Sonin’s17 resultof a similarity analysis for the liquid deposition process.

The Weber number, a measure of the impact inecompared to the surface energy, appears only in the inconditions, while the Ohnesorge number and the conangle exclusively appear in the equations governingdroplet dynamics. Consequently, both of the models rougstate that We determines the initial spreading stage and

re-

FIG. 9. Comparison of Fukaiet al. ~Ref. 14! results and the predictions bya cylinder model. Circles denote experimental results by Fukaiet al. In thefigure, Ref. 14 and CY denote the computation results of Ref. 14 and ofcylinder model, respectively. In the cylinder model, the same dissipafactor as droplets on polycarbonate, i.e.,Fd515, was used.~a! Water drop-let on a Pyrex glass plate. We5115, Oh50.001 68, andu554°. ~b! Waterdroplet on a wax coated plate. We5128.2, Oh50.001 68, andu576°.



FIG. 10. ~a! Images of an ink droplet colliding with a polycarbonate surface. Original droplet diameter53.2 mm, impact velocity51.75 m/s, We5190, andOh50.0060.~b! Images of a silicone oil droplet colliding with a polycarbonate surface. Original droplet diameter52.8 mm, impact velocity51.44 m/s,We5166, and Oh50.109.

e

ane

ren

ne

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aril

esth

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the following droplet motion is, under the influence of thprevious motion history, ruled by Oh andu. The foregoingstatement is consistent with the observation of SchiaffinoSonin17 that the early stage of spreading is mainly governby the inertial effect~We! and that the later stage, whedroplet motion is slow, is appropriately described by Oh au. This section discusses the role of We and Oh and thesection will address the effect of the contact angle.

The Ohnesorge number scales the resisting force inrecoiling process. As shown above, whether an impacdroplet will assume a shape close to a cylinder or a truncasphere depends on the strength of recoiling. An ink drophas a relatively high viscosity and a low surface tension, ta high Oh, which implies that the resistance is dominandue to viscosity. Therefore, the flow induced by the capillity force is entirely dissipated by the viscosity and the recoing is relatively slow. As a result, the recoiling droplet donot deviate much from the truncated sphere. On the ohand, a water droplet has a relatively low viscosity andhigh surface tension, thus low Oh. This implies that thesistance is caused by the inertia as well as the viscoTherefore, the potential energy stored in an excessivspread state is partially converted into kinetic energy ass


dd

dxt

egdts

y--

era-y.lyi-

ated with the recoiling motion, despite the viscous dissition. Hence, the water droplets recoil faster and a cylindethe more appropriate shape to model the vigorously recoidroplet.

The foregoing argument is verified by Fig. 12. The fiure shows the relative magnitudes of resisting force termsrecoiling ~inertia and viscosity! with respect to the drivingforce term~potential energy! for both the water and ink droplets. For water droplets, the kinetic~inertia! and viscousterms are comparable in the recoiling stage, and both pimportant roles to balance with the potential term. For idroplets, however, the kinetic term is negligible, and thonly the viscous term is balanced with the potential term

From the fact that inertia terms become negligibly smin the ink droplet, it follows that the recoiling motion can bestimated using only viscous and potential terms. TherefEq. ~32! reduces to

F~h!h1G~h!50, ~43!

which is the first order equation, thus requiring only oinitial condition for h. The initial condition should be satisfied at the time when the recoiling starts, that is, when


th-

s

aal

mr-n-it

en

one

owof

tersur-

ntter-inggre

ngnd

cyl-h.al

ith

thes-.


base diameter~height! reaches its maximum~minimum!. It ispossible to further simplify Eq.~43! in the limit h!1, i.e.,when the droplet is very flat, which is usually the case inbeginning of primary recoiling. Then we obtain the following equation:

h2kh250, ~44!

where k52(12cosu)/c, k being always positive unlesu50. With the initial condition,h5hmin , wherehmin signi-fying the minimum height, att5t i , the solution of Eq.~44!is given by

h5F 1

hmin2k~ t2t i !G21

. ~45!

Expressing Eq.~45! in terms ofDb using Eq.~40! in the limith!1, we obtain a closed form solution forDb :

Db5@Dmax2 2 4

3k~ t2t i !#1/2, ~46!

whereDmax is the maximum base diameter att5t i . Figure13~a! compares the experimental data during recoiling ofink droplet with modeling results using the origintruncated-sphere model, Eq.~32!, and the simplified rela-tions, Eqs.~43! and~46!. It is as expected that Eqs.~32! and~43! result in almost identical predictions since inertia terare negligibly small. However, it is remarkable that the futher simplified relation, Eq.~46!, generates a fairly close result to that of the original model, notwithstanding the icrease of discrepancy asDb gets smaller, that is, as the limh!1 deviates from the real shape.

Different results are obtained for water droplets whthe inertia terms are neglected in the modeling~it is recalledthat the inertia terms are not negligible in reality!. Using thecylinder model in terms ofR, we get the following equationwith the inertia terms ignored:

B~R!R1C~R!50. ~47!

FIG. 11. Spreading and recoiling of an ink droplet~circles, We5190 andOh50.0060! and a silicone oil droplet~triangles, We5166 and Oh50.109!on polycarbonate and an ink droplet on silicon oxide~squares, We5170 andOh50.0061!. The lines are the modeling results and they agree well weach experimental result.


e

n

s-

This equation is of the first order and we again need onlyinitial condition, namelyR5Rmax at t5t i . Figure 13~b!compares the solutions of Eq.~47! with those of the originalmodel and the experimental results. As predicted, they shconsiderable difference, indicating the significant effectskinetic terms even in the recoiling stage.

The recoiling speeds of water droplets, whose diamerange from 3 to 3.7 mm, impacting on the same target sface~polycarbonate! with different velocities are collected inFig. 14. Although Oh does not change very much~0.001 68to 0.001 87!, varying the Weber number causes a significadifference in the recoiling speed. This is because We demines the maximum base diameter in the primary spreadstage, which becomes the initial condition for the followinrecoiling motion. Highly spread droplets tend to recoil movigorously, or faster, owing to the great driving force arisifrom the large difference between its equilibrium state athe initial ~maximum! spread area.

Figure 15 simulates droplet base diameter using theinder model to evaluate the individual effects of We and OFigure 15~a! shows that the Weber number plays a critic

FIG. 12. Relative magnitudes of resisting force term with respect todriving force term. VIS~INR! denotes the relative magnitude of the viscoity ~inertia! term to the potential term.~a! Water droplet on polycarbonateVIS5uB(R)Ru/uC(R)u and INR5uR2A(R)R2u/uC(R)u @see Eq.~27!#. We5137 and Oh50.0017. ~b! Ink droplet on polycarbonate. VIS5uI (h)hu/uJ(h)u and INR5u2E(h)h2G(h)h2u/uJ(h)u @see Eq.~32!#. We5190 and Oh50.0060.


h

n

thi

tes

thtefalelamuc

w

y

fromocitydi-


role in determining the initial spreading process, while trecoiling speed~the downslope in the recoiling stage! is notmuch affected owing to the constant Oh. On the other hawhen the Weber number is kept constant@Fig. 15~b!#, theinitial spreading processes are almost identical. However,recoiling is fairly dependent on the Ohnesorge numbersuch a way that the droplet with smaller Oh recoils fasThese tendencies are consistent with our experimental obvations as discussed above.

D. Effects of the contact angle

This section examines the effects of target solids onrecoiling of water and ink droplets. Our study investigathe dependence of the droplet motion upon target surmaterials through measuring the equilibrium contact angThis is based on the assumption that the recoiling takes pmainly because the base area of the droplet spreadsthan its equilibrium state, which is determined by the eqlibrium contact angle. Therefore, the equilibrium contaangle,u, determines the strength of recoiling, and thus

FIG. 13. Simplified modeling results for recoiling.~a! Ink droplet on poly-carbonate. We5190 and Oh50.0060. LinesA and B are obtained by theoriginal model, Eqs.~32! and ~43!, respectively, and they are practicallundistinguishable. LineC is from Eq. ~46!. Lines B andC both start fromwhere the experimental base diameter is the maximum~Db53.312 at t50.26!. ~b! Water droplet on polycarbonate. We5137 and Oh50.0017.Line A is from the original model and lineB is from Eq. ~47!. Line B hasbeen shifted to the right byDt50.2 to be better compared with lineA.


e

d,

enr.er-

esces.ceorei-te

FIG. 14. Nondimensional recoil speed. The Ohnesorge number ranges0.0017 to 0.0019. The recoil speed is the average nondimensional velof the base diameter from its maximum until it passes the equilibriumameter during the primary recoiling.

FIG. 15. Individual effect of We and Oh.~a! Oh is fixed at 0.0017 and Wevaries.~b! We is fixed at 150 and Oh varies. Along the arrow, Oh50.0005,0.001, 0.0015, and 0.002.


ctteal

tata

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intia-

o-

, tyhrer

ee-

ob

leons.ilibe

itsen-

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ere

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be-thether-

edel

et

rily

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xidula-


assume thatu is the major parameter to describe the effeof wetting. The models presented in this paper are consiswith this argument as the surface potential energy is evated usingu, as shown in Eq.~9!, for example. In addition,Ref. 17 suggested that considering the equilibrium conangle be sufficient to define the phenomena at the conline at least to the lowest order.

Figure 16 compares the base diameters of water dropon different target surfaces, polycarbonate and silicon oxIt is noted that water wets the silicon oxide surface bethan the polycarbonate surface~see Table II!. Under similarimpact conditions, the base diameters in the initial spreadstage are essentially alike since the flow is inerdominated.14 However, in the recoiling stage, the droplet impacting on the polycarbonate~poor wetting surface! recoilsfaster than the droplet on the silicon oxide~good wettingsurface!. This shows the strong dependence of recoiling upthe wetting between liquid and solid. As the wetting improves, i.e., as the equilibrium contact angle decreasesrecoiling gets slower. We present the predictions of the cinder model together with the experimental results. Tmodel employing different ‘‘equilibrium’’ contact angles fodifferent target surfaces closely predicts the dependencthe recoiling behavior on the wetting. The same behavioobserved in the ink droplet~Fig. 11!. As with the water drop-let, the ink droplet recoils faster on the poor wetting~poly-carbonate! surface than on the good wetting~silicon oxide!surface. The truncated-sphere model employing differ‘‘equilibrium’’ contact angles predicts the experimental rsults surprisingly well for both the target surfaces.

The mechanism of a droplet recoiling faster on a powetting solid surface than on a good wetting surface caneasily understood as the following. In equilibrium, a dropon a poor wetting surface does not spread as much asgood wetting surface, that is, the poor wetting surface haweaker affinity for the liquid than the good wetting surface30

Therefore, when the droplet spreads wider than its equrium state due to its initial impact inertia, the liquid on th

FIG. 16. Effect of wetting on recoiling behavior. The triangles and the soline represent water droplet on polycarbonate~We5150 and Oh50.0017!.The circles and the dotted line represent water droplet on silicon o~We5166 and Oh50.0019!.


snt

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tse.r

g

n

hel-e

ofis

nt

re

ta

a

-

poor wetting surface has a stronger tendency to reducecontact area than one on the good wetting surface. As mtioned above, the models reflect this effect by incorporatthe contact angle in the surface energy term. In the cylinmodel, asu increases, the magnitude ofC(R) increases@itshould be noted thatC(R) is positive so as to be balanceroughly, with the negativeB(R)R in the recoiling stagewhere R,0#. The same applies to the truncated-sphmodel where the magnitude ofI (h) increases asuincreases—in this case,I (h) is negative sinceh.0 in therecoiling stage. Since the potential term corresponds todriving force of the recoiling motion, higheru induces fasterrecoiling. Figure 17 illustrates this effect ofu on the recoil-ing behavior using the cylinder model.

E. Discussion of the regime for each model

Now we discuss in which conditions the cylinder or thtruncated-sphere model would be appropriate. We hshown above that this is equivalent to searching for regimwhere the recoiling is resisted entirely by viscos~truncated-sphere model! and where inertia plays a significant role~cylinder model! in resisting the motion as well aviscosity. In our study, the primary spreading of both twater and ink droplets belongs to the inviscid, impact-drivflow regime17 since We@1 and Oh!We1/2. However, fromthe final stages of primary spreading, the viscous effectscome significant and the motion gets slow. Consequently,regime that characterizes the recoiling alters to that ofcapillarity-driven flow. In this regime, it is Oh that detemines whether the resistance is dominantly viscosity~highOh! or inertia ~low Oh!. Therefore, Oh is considered to bthe most important parameter to determine which mowould be appropriate.

Schiaffino and Sonin17 suggested that when a dropl‘‘spreads’’ in a capillarity-driven flow regime~We!1!, thetransition from the region where the resistance is prima

eFIG. 17. Effect of the contact angle on the droplet dynamics. The simtions are based on the cylinder model. We5150, Oh50.0017, andFd515.


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due to inertia, to the region where the resistance is duviscosity, occur in the broad range of 0.007,Oh,0.7, in thepresent definition of Oh. However, in the ‘‘recoiling’’ process studied here, the ink droplets, whose Ohnesorge nbers are approximately 0.006, exhibit highly viscoucapillarity-driven dynamics. On the other hand, the wadroplets, whose Ohnesorge numbers are less than 0.002hibit such dynamics that the flow is resisted by both ineand viscosity. Therefore, our experimental results suggthat in the capillarity-driven recoiling flow, Oh50.006 ~We,200, see below for the discussion on We! falls into anentirely viscosity resisted regime. The transition from anertia resisted regime to a viscosity resisted regime certaoccurs below Oh50.002. It is important to realize that thcylinder model represents a regime where both inertiaviscosity have significant effects in resisting the flow, athus this regime corresponds to the aforementioned transregion. It is not possible here to precisely predict the rangthe transition region, but our study shows that the rangeOh, corresponding to the transition region for the recoiliflow, is located much lower than that for the spreading mtion.

It is worth noting that We also affects the recoiling floalthough its effects are minor compared to those of Oh.have shown that We determines the maximum spread deter that is the initial condition of the recoiling motion. Iaddition, its effects on the recoiling speed have been shin Fig. 14. The lower We gets, the more significant thesistance due to viscosity becomes. Nonetheless, as mtioned above, the effects of We in determining the recoilflow regime are much weaker than Oh. Especially, whenranges where the corresponding regime completely fallseither the inertia-resisted region or the viscosity-resisted oWe plays a negligible role in determining the flow regimFor example, when We of an ink droplet was increasedapproximately 600 while keeping Oh at a constant, thecoiling pattern was the same as that of the low We ink drlets shown here. However, the quantitative analysis ofmotion is not presented since at that high We, the ink drosplattered upon collision with tiny droplets around the priphery splashing off the main body that continued to recThis suggests that when Oh is high enough~0.006!, the flowremain in the viscosity-resisted regime in spite of an increof We. As a droplet with a sufficiently low Oh, a mercudroplet can be considered. The experiments of Ref.showed that in the very low We range~We'2!, the smallmercury droplets with Oh'0.0007 still vigorously recoilupon impact, even disengaging from the surface. Therefwhen Oh is low enough, the recoiling flow belongs to tinertia-resisted regime even when We is very low.

On the other hand, the water droplets with 0.0016,Oh,0.0019 considered in this study exhibit the recoiling dnamics belonging to the transition region. In this regionappears that We plays a relatively important role in demining the flow characteristics. As discussed above, at hWeber numbers, the droplets resemble a cylinder, whilelow Weber numbers the shape of droplets tends to approa truncated sphere~Fig. 5!.

It is also noted thatu affects the flow characteristic


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since it is related to the driving force of the recoiling flowEspecially, in cases whereu is an extreme value, either verclose to 0 or 180°, it may alter the flow characteristics. Hoever, when the value ofu lies in a moderate range~0!u!180°!, its effects are not so critical as to determine tflow regime.

In summary, our ‘‘experimental’’ results suggest thwhen Oh is higher than 0.006, the truncated-sphere moshould be appropriate in modeling the droplet recoiling.the range of 0.0016,Oh,0.0019, the cylinder model closelestimates the recoiling when We is high~We.30!. As Ohdecreases to even lower values, the cylinder model ispected to predict the droplet dynamics more accurately wless dependence on We. The determination of the exact tsition region calls for more study, but our discussion sugests that Oh play a major role in characterizing the flowhile We has relatively minor effects.

V. CONCLUDING REMARKS

In this paper, we have investigated the recoiling behior of water, ink, and silicone oil droplets upon collision witdifferent solid surfaces. In the experiments, images ofimpacting droplets were captured sequentially by a hspeed video system. The images were analyzed to obtaintemporal evolution of the base diameter of the droplet.understand the physics underlying the recoiling, we devoped a model based on the variational principle, assumthe droplet shape to be cylindrical, and modified ttruncated-sphere model of Bechtelet al. to accommodate thedissipation factor. The models and the experimental reswere in good agreement for various dynamic and wettconditions. It was shown that when Oh is low~Oh,0.002,say!, the cylinder model appropriately predicts the droprecoiling dynamics and that when Oh is high~Oh50.006!,the truncated-sphere model is appropriate.

Using these models, a significant computational efforsaved by solving a single second-order, nonlinear differenequation instead of solving the full Navier–Stokes equatwith the moving boundary. As mentioned in the Introdution, a simplified description of the recoiling motion habeen very sparse. Especially, the closed form solution ofrecoiling diameter of a high Oh droplet, Eq.~46!, is the firstin its kind to the authors’ knowledge. In addition, by makinuse of the approximate models, an analytical study ofrecoiling mechanism could be performed with great ease

It is noteworthy that Pasandideh-Fardet al.16 developeda composite model which assumes the spreading droshape to be a cylindrical disk under a truncated sphere~Fig.11 therein!. Our study shows that both the cylinder modand the truncated-sphere model have their own regimwhere they are adequate, and this suggests that the compmodel may turn out to be more generally applicable. Hoever, it is difficult to come up with a simple parameter thdefines the geometry of the shape as contrary to our curshape assumptions. Although a single parameter~eitherheight or base diameter! is only necessary to define the shaof either a cylinder or a truncated sphere, the compomodel requires more than one parameter including the b


lye

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ely

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thkeing

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diameter and the height of a cylindrical disk, to completedefine the shape. Therefore, a single equation to describdroplet motion such as Eq.~27! or ~32! is not readily ob-tained in case of the composite model. Nevertheless, wepect that if a simple geometric parameter that can definecomposite shape is derived, which is beyond the scopthis work, the composite model could yield more broadapplicable predictions.

It has been reported that when a droplet with a hsurface tension collides with a relatively nonwettable sosurface, it tends to disengage from the surface upon hitthe surface. This total rebounding can be predicted bytruncated-sphere model and the corresponding resultsshown in Bechtelet al.20 However, the cylinder model is noadequate to accommodate total rebounding since as thediameter gets very small, its height becomes so large asatisfy the volume conservation. The resultant shape wobe unrealistic to simulate the real droplet. Here we notemany computational programs solving the Navier–Stoequation also have difficulties in modeling total reboundsince it involves a singular process of the droplet~computa-tional domain! disengaging from the surface.

Our study shows that the recoiling behavior of water aink droplets is determined by We, Oh, and the equilibriucontact angle. The Weber number determines the maximspread diameter, which becomes the initial condition forrecoiling. The recoiling flow is driven by the capillarity forcarising from the difference between the equilibrium state athe excessively spread shape. The flow is resisted by inand viscosity, and the resisting force is scaled by Oh. Inrecoiling of the water droplets, whose Ohnesorge numbare relatively small, both the inertia and the viscosity psignificant roles in resisting the flow. Thus, the recoilingvigorous and the cylinder model predicts the experimenresults well. On the other hand, the recoiling of the ink drolets, whose Ohnesorge numbers are relatively large, issisted dominantly by the viscosity. The ink droplets recrelatively slowly and resemble a truncated sphere. Thus,predictions of the truncated-sphere model agree with theperimental results fairly well. It has been shown that trecoiling is greatly affected by the wetting between the liqudroplet and the solid surface. Good wetting weakensslows down the recoiling process, while poor wetting pmotes the recoiling. The models incorporating the equirium contact angle in evaluating the potential energy tesuccessfully predicted the dependence of the recoilingcess on wetting.

ACKNOWLEDGMENTS

The authors are grateful to the Aluminum CompanyAmerica and to the National Science Foundation for the sport of this work under Grant No. DMI-9634931.

APPENDIX A: NONDIMENSIONAL QUANTITIES

The following gives the definitions of the nondimesional quantities. Note that all the asterisked symbols den


the

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of

h

gere

asetoldats

d

me

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d--

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te

dimensional quantities, whereas nonasterisked symbolsthe corresponding nondimensional quantities except theterial properties:

Db5Db*

D*, h5

h*

D*, t5

t*

~rD* 3/s!1/2,

T5T*

~psD* 2/12!, V5

V*

~psD* 2/12!,

vz5vz*

~s/rD* !1/2, v r5v r*

~s/rD* !1/2, z5z*

D*,

D r5D r*

D*, t5

t*

~m2s/rD* 3!1/2, V5V*

~pD* 3/6!.

APPENDIX B: DETERMINATION OF THEDISSIPATION FACTOR AND THE SENSITIVITY OFMODELING RESULTS TO ITS VALUES

The dissipation factor,Fd , has been introduced as aempirical constant in scaling the viscous stresses in the mels above@Eqs. ~13! and ~39!#. Since our modeling of theviscous dissipation relies on the scaling analysis, the appriate coefficients must be established by experimentcomputation.17 This is a common limitation of the order-ofmagnitude analyses. Therefore, several studies to anacally model the viscous dissipation during droplet spreadhave either assumed or empirically determined the coecients. For example, Chandra and Avedisian7 assumed thecoefficients in their models of the viscous dissipation tounity as seen in their Eqs.~13!–~15!. Pasandideh-Fardet al.16 also implicitly assumed the proportional constanttheir modeling to be unity in their Eq.~9!, which modelingturned out to be a fairly close approximation when compawith experimental results of the maximum spread factorsaddition, Bechtelet al.20 included an adjustable proportionaconstant in modeling the shear stress at the droplet bottomshown in their Eq.~17!. Mao et al.31 used a least-squarmethod to deduce several constants appearing in their dpation model@their Eq. ~16!# based on their experimentaresults.

TABLE IV. Comparison of selected maximum spread factors of water drlets obtained by experiments (Dmax,exp) and by model predictions~Dmax,cyl

by the cylinder model!. PW denotes present work.

We Oh u~°! Dmax,exp Dmax,cyl Reference

30 0.0017 87.4 2.27 2.59 PW93.6 0.0020 67 3.09 3.38 Ref. 31

130 0.0020 37 3.70 3.79 Ref. 31130 0.0020 67 3.67 3.58 Ref. 31130 0.0020 97 3.60 3.42 Ref. 31137 0.0017 87.4 3.16 3.58 PW166 0.0019 58.6 3.71 3.84 PW207 0.0018 87.4 3.91 3.85 PW288 0.0020 37 4.50 4.31 Ref. 31288 0.0020 67 4.42 4.17 Ref. 31288 0.0020 97 4.32 4.02 Ref. 31519 0.0020 97 4.78 4.52 Ref. 31582 0.0017 87.4 5.13 4.75 PW


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We determine the dissipation factors on an empiricalsis, specifically by matching the maximum spread factobtained by experiments conducted in this work. Moreovfor water droplets, we also used experimental results~maxi-mum spread diameters! by Mao et al.31 who reported theequilibrium contact angles with other impact conditions suas We and Oh. Based on the experimental results, the vaof Fd , which best matchDmax experimentally measured, araveraged. Table III shows theFd values thus obtained, anTable IV shows selected experimental results for the mamum spread diameter used in this procedure along wmodel predictions, for water droplets.

Now we discuss the sensitivity of the modeling resultsFd . As Fd increases, both the maximum base diameterthe recoil speed decrease. The simulation results obtainevaryingFd in modeling a water droplet on polycarbonate aan ink droplet on polycarbonate are shown in Fig. 18. Inwater droplet, whose dynamic conditions are typical one

FIG. 18. Sensitivity of simulation results to the dissipation factor,Fd . ~a!Water droplet on polycarbonate. We5150 and Oh50.0017. Along the ar-row, Fd511, 13, 15, 17, and 19.~b! Ink droplet on polycarbonate. We5150and Oh50.006. Along the arrow,Fd52.4, 2.8, 3.2, 3.6, and 4.0.


-sr,

hes

i-th

dby

ein

the experiments in this work, asFd changes about627%,the maximum base diameter and the recoil speed changeproximately68% and618%, respectively. In the ink droplet, asFd changes about625%, the maximum base diametechanges approximately29% to 115%, and the recoil speechanges approximately24% to 118%. Nevertheless, thegeneral trend is invariant.

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the recoiling of liquid droplets upon collision with solid...

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