Surface-Ascension of Discrete Liquid Drops viaExperimental Reactive Wetting and Lattice Boltzmann

Simulation

Gary C. H. Mo, Wei-yang Liu, and Daniel Y. Kwok*

Nanoscale Technology and Engineering Laboratory, Department of Mechanical Engineering,University of Alberta, Edmonton, Alberta T6G 2G8, Canada

Received November 4, 2004. In Final Form: March 14, 2005

The reactive-wetting technique is employed to move liquid against gravitational force. Experimentshave shown that the velocity of an ascending liquid drop is constant, unlike the gradual decrease intuitivelylinked to objects against gravitation. The ascending velocity decreases for increasing slope. The maximuminclination, or stopping, angle for this particular setup is >25°. Computer simulation of a reactive-wettingdrop using the lattice Boltzmann method is also performed. The results indicate that the method employedis suitable for the task, producing most experimentally observable responses. The mass flow of a liquiddrop under reactive wetting was studied through simulation results, and a general description of thereactive-wetting phenomenon was deduced.

I. Introduction

The motion of a three-phase contact line is an interestingand relevant area of current research. It is known thatthe three-phase contact line consists of a visible edge; ifthere is complete wetting, it is led by a thin precursorhaving a height of hundreds of angstroms or less.1,2 Thespreading of the precursor precedes appreciable wettingand, hence, motion in the contact line. The completewetting case proves to be more forgiving in formulationbecause a stress singularity occurs if one assumes zerofluid thickness at the contact line. The motion of the contactline under transient circumstances when the contactedsurface is under continuous modification, in other words,reactive, has also been known for a few decades. Thisexcludes the Marangoni effect, which is a phenomenon inwhich the fluid density is transient depending on tem-perature. The solid surface modification is typically amolecular adsorption, forming self-assembled monolayers.The reactive version of the moving contact line probleminvolves wetting and dewetting of the fluid.

From a lubrication-type theory, the flow within such adrop has been proposed to be a superposition of shear flowand Poiseuille flow.3 Recently, an atomistic account ofwetting/dewetting flow has also been presented, showinga rolling flow inside the liquid drop simulated by atomisticcollisions.4 As a linear theory leads to symmetric dewettingand wetting (i.e., the response of the contact line is thesame at both edges), it was shown that, for a very smalldrop size, significant evaporation at the dewetting edgeand condensation at the wetting edge occur.5 In additionto theoretical works, there has been a collection ofexperiments that induce translation of a discrete liquiddroplet via a solid surface energy gradient.6-11 These

experiments were carried out under atmospheric condi-tions on smooth substrates. The drop translates in anunpredictable manner, reflecting the intrinsic nature ofthe solid surface modification. Among many influences,the effect of gravity on droplet motion can be critical. Theinclination of a substrate may introduce fluid motion thatdeviates from one’s anticipations. We are thus interestedin the effect of gravity on the velocity of moving drops.The nature of the motion of a three-phase contact line onreactive surfaces is also of fundamental importance toour study. There have been few specific investigations inthe fluid interfacial flow of a reactive-wetting drop. Wereasoned that it would be difficult to reveal the flowinformation at the interface through an experimentalapproach, leading to an unfruitful undertaking. Thus, itwas proposed to construct a computer simulation exhibit-ing accurate behavior of the liquid drop. In this way, thesimulation may elucidate the wetting and dewettinginterfacial flows. The lattice Boltzmann computationmethod was thus adopted, which we believe to bequalitatively reliable for the understanding of the behaviorof discrete liquid drops. This paper summarizes such acombined experimental and computational study of sur-face-ascending reactive-wetting (RW) drops against gravi-tational influence.

The generation of RW drop movement relies on theconversion of a high-energy surface to a low-energy surfacethat induces a change in wettability sufficient to providethe energetic contribution for drop movement. For ourbase surface, the adsorption of a mercaptoalkanoic acid[HS(CH2)nCO2H] onto gold generates a densely packedmonolayer film that exposes carboxylic acid groups at itssurface. These high-energy surfaces are (partially) wetby most liquids including water. Previously, Lee et al.8reported that these surfaces could be modified by contactwith a nonpolar solution (decahydronaphthalene, DHN)containing an n-alkylamine. In this process, the amino

* Author to whom correspondence should be addressed[telephone (780) 492-2791; fax (780) 492-2200; e-maildaniel.y.kwok@ualberta.ca].

(1) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827.(2) Leizerson, I.; Lipson, S. G.; Lyushnin, A. V. Nature 2003, 422,

395.(3) Brochard, F. Langmuir 1989, 5, 432.(4) Freund, J. B. Phys. Fluids 2003, 15, L33.(5) Leizerson, I. I.; Lipson, S. G. Langmuir 2004, 20, 8423.(6) Chaudhury, M. K.; Whitesides, G. M. Nature (London) 1992, 256,

1539.

(7) Santos, F. D. D.; Ondarcuhu, T. Phys. Rev. Lett. 1995, 75, 2972.(8) Lee, S. W.; Kwok, D. Y.; Laibinis, P. E. Phys. Rev. E 2002, 65,

051602.(9) Ichimura, K.; Oh, S.-K.; Nakagawa, M. Science 2000, 288, 1624.(10) Ross, D.; Rutledge, J. E.; Taborek, P. Science 1997, 278, 664.(11) Daniel, S.; Chaudhury, M. K.; Chen, J. C. Science 2001, 291,

633.

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10.1021/la0472854 CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 05/19/2005

groups adhere non-covalently to the surface carboxylicacid groups and generate another oriented alkylaminemonolayer at the surface. The resulting bilayer exposesthe methyl groups (CH3) of the alkylamine at its surface,as evidenced by its wetting properties by water and varioushydrocarbon liquids. Exposure of the bilayer to a polarsolvent removes the amine layer and regenerates the high-energy carboxylic acid surface. The resulting surfaces areless wettable (i.e., exhibit higher values of advancing andreceding angles) when alkylamines of longer chain lengthsor solutions of higher amine concentrations are employed.The adsorption process to generate the amine adlayer caneffect a RW drop movement on the carboxylic acid (high-energy) surface. To experiment with a controlled movingdrop, the RW experiment is complemented by regions ofhydrophobic restriction as two-dimensional (2D) barrierson the solid-vapor interface prior to an experiment. Thishydrophobic region does not accept further surfacemodification and can direct the movement of liquid drops.

II. ProceduresA. Experimental Methods and Materials.The experiments

were performed on smooth, flat gold substrates prepared ac-cording to the following procedures. A test grade silicon waferof orientation ⟨100⟩ is treated with resistively evaporated metalflux under an ambiance of 10-6 Torr, resulting in a thin-film (100nm) coverage of gold. The silicon wafer has a spot height variationof <10 nm after this process and has been confirmed with atomicforce microscopy. The smooth surface ensures low contact anglehysteresis, so that the contact line may translate with the leastpossible resistance. Airborne particulates on the surface wereremoved using pressurized, filtered nitrogen. A microcontactprinting technique12 was then used to specify the hydrophobicregions; self-assembly of 1-octadecanethiol is employed for thispurpose. The remaining area on the gold surface is functionalizedwith 16-mercaptohexadecanoic acid to allow wetting by thedroplets. A 5 mM solution of dodecylamine in DHN was used asthe droplet, and its movement was recorded using an alignedCCD camera at 30 fps. In each experiment, a 2.0 µL drop of theDHN solution is deposited on a hydrophilic track under atmo-spheric conditions. The substrate is placed such that an acclivityis present against the direction of drop motion (see the schematicdiagram of Figure 2). A general, captured image of a RW dropon zero (0°) slope is presented in Figure 1 to convey a betterunderstanding of the physical situation. The surface-ascensionof RW drops was performed on a setup similar in function to theschematics of Figure 2. The substrate plane, or the incline of thesolid surface, was measured by an inclination angle, R. Betweenexperiments, R was varied in increments of 5°. The trials wereperformed on an vibration-isolated air-table. A CCD cameracaptured the experiments at 30 fps in gray scale for imageanalysis. The camera was carefully aligned with the substrateto minimize distortion.

B. Computer Simulation. The computer simulation of RWdrop motion is performed using a two-component lattice Boltz-

mann method (LBM), which considers direct external rather thanmean-field forces, specifically those proposed by Kang et al.13

Recently, there has been some interest in the literature regardingother LBM investigations.14-18 We believe that continuous flowsimulation techniques such as the finite-element method (FEM)are unsuitable for our purpose. This is primarily due to abottleneck in the translation of physical surface effects intoboundary conditions sufficient for FEM. In addition, LBM relieson a density distribution instead of a sharp contact edge todescribe the interface. Thus, problems of the contact-linesingularity and its derivatives, which plague many contact lineinvestigations, can be avoided entirely in LBM simulations. Webelieve the statistical nature of a reactive-wetting system is bettersimulated with an evolutionary dense-gas-automaton than witha set of deterministic fluid equations. Modification was made inthe LBM implementation to affect a transient wall density,simulating the first-order adsorption of hydrophobic surfactants.The lattice cell under study is 50(X) by 20(Y) by 20(Z) units insize, with 19 lattice velocities (D3Q19). The cell was subjectedto periodic boundary conditions in the X and Y directions; bounce-back boundaries were implemented at Z ) 1 and Z ) 20. Theplane of Z ) 20 was treated so that surface interactions wereomitted.

Briefly, the LB method employed utilizes the followingequations:

f represents the number density distribution of the componentsin the cell, where the designations a () 1, 2, or w) represents theidentity (fluid 1, 2, or wall, respectively) of the components. ei(i ) 1, ..., 19) represents the 19 lattice directions. The solid wallis attributed a density for the purpose of calculating theinteraction strength. Equation 1 utilizes the BGK single-relaxation collision term, which simplifies the known Boltzmanncollision expression; τ is the relaxation time. Expression for theequilibrium density distribution f eq

a is lattice dependent and canbe found in the literature.19 Equation 2 shows the application ofthe direct force, where na () ∑if i

a) is simply the number densityof fluid a, and u′ is a common velocity. This second equationgives the equilibrium velocity ua

eq, which is used to compute thelattice-dependent equilibrium distribution as f eq

a ) f eqa (ei, ua

eq).A typical simulation is initialized using a previously equili-

brated hemispherical drop of ∼14 lattice units in diameter anda contact angle of ∼57°. The initial drop was brought to anequilibrium without solid surface interactions so as not to

(12) Xia, Y.; Whitesides, G. M. Angew. Chem., Int. Ed. 1998, 37, 550.

(13) Kang, Q.; Zhang, D.; Chen, S. Phys. Fluids 2002, 14, 3203.(14) Iwahara, D.; Shinto, H.; Miyahara, M.; Higashitani, K.Langmuir

2003, 19, 9086.(15) Briant, A. J.; Papatzacos, P.; Yeomans, J. M. Philos. Trans.:

Math., Phys. Eng. Sci. 2002, 360, 485.(16) Zhang, J.; Kwok, D. Y. Langmuir 2004, 20, 8137.(17) Zhang, J.; Kwok, D. Y. J. Colloid Interface Sci. 2004, 282, 434.(18) Melchionna, S.; Succi, S. J. Chem. Phys. 2004, 120, 4492.(19) Wolf-Gladrow, D. A. Lattice Gas Cellular Automaton and Lattice

Boltzmann Models: An Introduction; Lecture Notes in Mathematics;Springer: Berlin, Germany, 2000.

Figure 1. Captured image of (a) top view of the front andwake of a moving liquid volume under sufficient, rectangularwetting restriction and (b) macroscopic observation of the sameliquid drop from the side; note that the reflection of the profileis also visible. The line in (a) represents the length of 10 mm;(b) is scaled differently.

Figure 2. Schematic of the experimental setup. Note the CCDcamera is aligned parallel to the substrate plane.

f ia(x + eiδt, t + δt) - f i

a(x, t) ) -f i

a(x, t) - f eqa (x, t)

τa(1)

nauaeq ) nau′ + τaFa (2)

5778 Langmuir, Vol. 21, No. 13, 2005 Mo et al.

introduce bias into its density distribution. The density is takenafter ∼1000 steps and then employed for all subsequent simula-tions of surface-ascending drops. In the computer study, thesubstrates were also deviated from the horizontal position in 5°increments. To effect reactive wetting, the wall node densities(nw) in simulation are modified by the following equation:

H is the surfactant-dependent hydrophobic limit, and k is theparameter determining the rate of adsorption. The densityincreases with simulation time in which the node is exposed tosufficiently dense liquid, giving a first-order response. Thestrengths of interaction, whether solid-liquid or liquid-liquid,are adjusted through multiplicative parameters gab:

where a and b indicate the identity of the fluid component orwall node. The effect of gravity is implemented in a similar fashionusing the parameter g, affecting only fluid components but notthe wall nodes. In this study, the parameters were as follows:τ ) 1; g ) 7.5 × 10-5; g12 ) 0.02; g1w ) g2w ) g11 ) g22 ) 0; H) 0.35; and k ) 0.01. The average number densities of fluids 1and 2 were of the order of 1.

III. ResultsA. Experimental Results. The ascension of a liquid

drop against gravity is shown in Figure 3. We note thatthe general motion of such a drop is similar to that of adrop moving on a horizontal plane. A wetting (front) withapproximately constant geometry precedes a dewetting(wake) edge, the contact line of which is transient. Thedynamic contact angles differ, with the advancing anglebeing slightly smaller than the receding angle. As expectedfor a drop under body force, the empirical, longitudinaldrop profile is asymmetric. Figure 4 provides the sche-matics and the light-enhanced images of an ascendingdrop supporting the schematic diagram. It is evident fromthese figures that the profile of a drop under gravitationin the non-normal direction is altered. This has unforeseenconsequences for the velocity of the drops. Complementaryto change in drop profile, the phenomenological length ofa drop in the hydrophobic track also varies from thehorizontal case. In our experiments, the cases when the

substrate acclivities were 10° and 15° are worth notingand will be discussed following an account of the velocityof the drop.

It is perhaps surprising to note that the velocity withwhich a RW drop climbs is constant. This is a clear resultfrom the time-sequence photography and not a conclusiondeduced from observation with the naked eye. Thehysteresis of contact angle in these experiments isnegligible and cannot effect the gradual slowing normallyobserved in a descending raindrop on, for example, awindshield. We attribute the constant velocity to that factthat the RW phenomenon is a dynamic equilibrium state.The phenomenological velocity is the balanced result ofthe effects from liquid dewetting, surfactant adsorption,and gravity. Intuitively, a significant decrease in speedis produced from an increase in the slope of the substrateplane, and vice versa (see Figure 5). Dewetting of the three-phase contact line is inhibited through surface tension bythe bulk of the drop, which is in turn affected by gravitationon a slanted plane. The maximum, or stopping, angle atwhich a 2 µL drop may climb spontaneously is >25° asimplied in Figure 5; at higher slope the drop is unable toascend.

To gain a better understanding of the velocity trend, westart by assuming that the volumetric flow rate Q of anascending drop is

Figure 3. Captured images of a surface-ascending liquid dropin two proof of concept experiments.

nw ) H{1 - exp[ln(1 -nw

H ] - k)} (3)

Fa(x) ) na∑x′

gabnb(x′)(x′ - x) (4)

Figure 4. Schematic and captured image of asymmetriclongitudinal profile; a horizontal case is provided for comparison.It is clear that the front of the asymmetric drop is more thinnedthan its own wake.

Figure 5. Experimental velocity of the surface-ascendingdrops. Linear least-squares regression gives v ) 5.49 - 10.462sin R, with an R2 value of 0.95. The constants carry the unitof velocity.

Surface Ascension of Discrete Liquid Drops Langmuir, Vol. 21, No. 13, 2005 5779

Q0 and Qg are, respectively, the volumetric flow rates dueto surface and gravitational effects. Because the drop isconstrained by rectangular hydrophobic tracks and itsprofile is approximately constant through its ascent, wewrite

By definition, v0 is the velocity of the drop on the trackdue to dewetting flow and hence the velocity exhibitedwhen the incline is 0°. vg is a velocity due to gravity. Asgravity is a body force acting on the entire drop, it isreasonable to argue that

The coefficient C, however, cannot be sought exactly dueto the randomness in surface wetting characteristic andthe transience in the drop profile. Using the assumedrelationship of v ) v0 - C‚g sin R, we are able to generatea good fit to our experiment as shown in Figure 5; in thiscase C ) 1.07 s. The stopping angle, as predicted by thisrelationship, is 31.63° (>25°). This is in accordance withthe experimental result.

There is now an opportunity to discuss the 10° and 15°cases. Previously, we have reported that the antagonisticshear flow at the surface may increase when the longi-tudinal length of the drop is large, causing longer dropsto move more slowly.20,21 Specifically, an optimal length(usually predicted at the capillary length) can be foundif the surface is positioned horizontally. For our specificsystem of dodecylamine in DHN on carboxylic acidmonolayer, we found that this empirical optimal lengthis ∼3.5 mm. The average length of an ascending dropagainst a 10° incline is 3.08 mm; that against a 15° inclineis 3.84 mm. We argue that the positive deviation seen inthe 10° case (see Figure 5) is due to a smaller shear flow,which in turn is the result of the short drop length. Onecannot expect to employ the optimal length exactly in themore complex situation with gravity under consideration.However, the 10° and 15° cases obviously are situationswhen the reduction in shear flow is significant. Thereduction in length is a simple effect of the surfactantadsorption. It is assumed that an amine monolayer ofapproximately homogeneous wetting characteristic is leftbehind the trail of a reactive-wetting drop. Because thevelocity of an ascending drop is smaller than that athorizontal plane, the amine monolayer coverage would begreater and, hence, the contacting surface has lowerenergy. Less wetting of the wake (dewetting) edge is thusobserved from shorter length of the drop. Thus, theobserved trend is that an increase in substrate planecauses a decrease in drop length.

B. Computational Results. Before simulation of anysurface-ascending drops, it is prudent to confirm that asimulated liquid drop does not display longitudinal motion,save perhaps spreading when a simulated adsorption isomitted from the implementation. This is shown in Figure6. In these two simulations, the substrate plane is 0°, andthe bottom solid surface is set to a given surface energy.In terms of wall node density, nw, the values used in partsa and b of Figure 6 are 0.35 and 0, respectively. We observespreading, complete wetting, and coalescence in Figure6b and dewetting and a high contact angle, signifying a

low surface energyy,in Figure 6a. In both simulations thedrop does not display appreciable motion from its initialposition. Thus, we can be confident that any translationof the drop in our studies is due to gravity, simulatedsurfactant adsorption, or a combination of these factors.In many of the simulation results presented, a roll-shapeddrop is visible on the top surface of the simulation cell(see, for example, Figure 6b). This is simply a condenseddrop due to the bounce-back boundary condition on theparticular surface. It is reasoned that due to its appreciabledistance, this condensate does not significantly affect theliquid drop under study. This will become clear once it isshown in a later figure that the mass flow in the volumebetween the liquid drop and the condensate, generally >7lattice units in height, is nearly zero. The simulation ofa drop against a 10° incline is employed as an exemplarycase in most of the discussion that follow, for the sake ofconsistency.

In Figure 7 we show an example of a moving liquid dropvia LBM simulation. As is evident from the figure, thevelocity of the drop is constant, similar to the RW dropsin our experiments. As it is difficult to translate theparameters used in a LBM simulation to exact physicalquantities, the gravitational force is adjusted through theparameter g until it is comparable to the effect ofhydrophobic surface forces. The resulting parameter (g )7.5 × 10-5) is 3 magnitudes smaller than the fluidintermolecular interactions (g12 ) 0.02), which would beelectrostatic in nature. This is physically correct asgravitational force is in reality magnitudes smaller than

(20) Mo, G. C. H.; Kwok, D. Y. Colloids Surf. A 2004, 232, 169.(21) Thiele, U.; John, K.; Bar, M. Phys. Rev. Lett. 2004, 93, 027802.

Q ) Q0 - Qg (5)

v ) v0 - vg (6)

vg ∝ C‚g sin R (7)

Figure 6. LBM simulations of statics drops with each frameseparated by 100 simulation steps: (a) low-energy surface; (b)high-energy surface. The substrate plane is horizontal in bothcases.

5780 Langmuir, Vol. 21, No. 13, 2005 Mo et al.

forces of an electrostatic origin. Because the LBM simula-tions generate drops with an average interfacial thicknessof 1-2 lattice units, the volume of the simulated dropsare in principle very small. This is also reflected by thedynamic contact angles exhibited by the LBM-simulateddrop, which does not compare to experiments such asFigure 3. Digressing briefly, in Figure 8 we plot thesimulated surface energy (hence, wall node density) ofthe solid surface supporting the drop. One can immediatelyconfirm from Figure 8 the assumption that a surface whichhas supported a RW drop has a homogeneous wettingcharacteristic. There are very small differences betweenthe average values of wetted wall node density for varioussubstrate acclivities. We attribute the lack of lengthvariation in the simulated surface-ascending drops to thisfact. This situation can be remedied by adjusting the rateof simulated adsorption through the parameter k. Analo-

gous to actual experiments, we have also used therelationship v ) v0 - C‚g sin R in obtaining a regressionof the simulated ascending velocities; this is shown inFigure 9. The simulated results are in good agreementwith the regression with a coefficient (C‚g) of 0.0441simulation time unit. It is interesting to note that thisrelationship predicts that there is no stopping angle forthe particular drop size. We infer from this that a smalldrop may climb vertical substrates. This has in fact beenobserved in experiments. Small droplets with a volume ofthe order of nL can ascend substrates positioned vertically.We conclude that the general behavior of the LBMsimulations mirrored those observed in experiments andthat the predictive employment of LBM on reactive-wetting systems is qualitatively reliable.

We have shown that it is possible to produce the generaltranslational behavior of a surface-ascending RW dropvia LBM. To gain further insights in the fluid transportof reactive wetting, we also report the mass-flow field ofsuch a drop from the simulation. In Figure 10 we plot thestreamline of an XZ plane-section within the simulationcell. Figure 11 shows a collection of cross-sectional velocityplots of the mass-flow field in the simulation cell. Theincline is again 10° in both figures. We first note that

Figure 7. LBM simulation of an ascending drop with eachframe separated by 100 simulation steps. The incline is 10°.Each frame of the two-dimensional view has been rotated bythe incline angle to convey better understanding. A three-dimensional view is provided for comparison.

Figure 8. Hydrophobicity of the solid surface (XY plane at Z) 1) supporting a surface-ascending reactive-wetting drop;substrate plane is 10°.

Figure 9. Climbing velocity from LBM simulations. Least-squares regression is also plotted (v ) 0.0462 - 0.0441 sin R;constants carry the unit of simulated velocity), with an R2 valueof 0.99.

Figure 10. Streamline of a surface-ascending drop via LBMsimulation; this XZ plane is given at Y ) 10 and intersects withthe center of the drop.

Surface Ascension of Discrete Liquid Drops Langmuir, Vol. 21, No. 13, 2005 5781

a large portion of the simulation cell has very small massflow. From Figure 10 it is also evident that a plane existsoutside the condensate which carries the mass flowdownstream; in other words, the condensate captures orreleases fluid mass only by evaporation and condensation.These two facts support our earlier claim that thecondensate drop on the top surface of the cell does notinterfere with the proposed investigation through flow orpressure. In each cross section in Figure 11, mass flow inthe direction normal to the section is indicated by coloras shown on a color bar. Density plots of the fluid areprovided for positional comparison. It is important to notethat this flow information is difficult to obtain experi-mentally even with a fluorescent agent, and it reflectsone possible route that a fluid undertakes to result in anobservable reactive wetting. Similar flow patterns wereobtained from the LBM simulation of RW drops onhorizontal substrates. Figure 11a,c shows that the solid-liquid interfacial mass flow can be described as follows.The solid surface under the drop is under continualsurfactant adsorption that modifies its hydrophobicity.The more hydrophobic surface under the drop contributesto an upward flow away from the bulk of the moving drop,in this case, toward the back of the drop. The morehydrophilic surface, having less surfactant coverage, alsoinduces an upward (albeit smaller) flow away from thebulk, toward the wetting edge of the drop. To extractfurther information, the pressure within the lattice cellin simulation is also plotted in Figure 12. The negativenumber signifies expansion/evaporation, and positivevalues represent compression/condensation. It is obviousfrom Figure 12 that condensation occurs at the solid-liquid interface, more so at the front than at the wake ofthe drop. Evaporation occurs in the rear portion of thedrop, which we term the “cap”. Employing these figures(Figures 10-12), it is convenient to summarize the natureof a reactive-wetting drop from a LBM simulation. Thecap evaporates and moves toward the rear of the drop.Captured by surface tension, this mass flow drives themajority of the drop mass forward in the X direction, whichis the main contribution to the visible translation of thedrop. As gravity is a body force, it contributes to a flowof the entire cell in the antagonistic direction, thus slowing

the drop. Contrary to predictions of the lubrication theoryor that from an atomistic evaluation, no rolling flow isfound within the RW drop in a LBM simulation. In additionto velocities in the longitudinal direction, there is also asignificant lateralmovement in thedrop, theeffect ofwhichis to widen the span of the drop (see Figure 11b). This isin fact an experimentally observable effect and furtherattests to the accuracy of the LBM simulation. Overall,the motion of a RW drop is due to the transport of fluidin the interfacial region, which in turn drives a bulk flowthat pedestals on the interface. This is markedly differentthan that previously proposed by some authors on similarsubjects (see, for example, refs 3 and 22). We believe thatthe RW scheme is noncontinuous in the sense that it isnot strictly momentum-driven. Thus, treatment of theproblem using Newtonian dynamics may not yield anaccurate description.

IV. Summary

These results lead us to the conclusion that the substrateplane for reactive wetting discrete drops should becarefully designed to account for acclivity or declivity. Adiscrete, RW drop behaves differently when subjected togravitational forces in the direction of its motion. Theeffect of gravity can be obviously made negligible througha reduction in drop volume. The nature of fluid flow insuch a moving drop is also demonstrated using latticeBoltzmann simulation of the ascending drops. We an-ticipate that a large portion of the drop is driven by thesmall interfacial transport, which accounts for the sig-nificant effect of a volumetric force such as gravity on theRW system.

Acknowledgment. We gratefully acknowledge finan-cial support from the Canada Research Chair Program,a Research Tools and Instrument Grant, and a DiscoveryGrant from the Natural Sciences and Engineering Re-search Council of Canada (NSERC) in support of thisresearch. G.C.H.M. acknowledges financial support fromthe Alberta Ingenuity through a studentship grant.

LA0472854

(22) Braun, R. J.; Murray, B. T.; Boettinger, W. J.; McFadden, G. B.Phys. Fluids 1995, 7, 1797.

Figure 11. Slices of the mass-flow field of an ascending dropagainst an incline of 10°: (a) cross sections with constant Xvalues; (b) cross sections with constant Y values; (c) crosssections with constant Z values. Each slice shows only the flownormal to its plane, and the direction is indicated in color: bluerepresents the positive and red the negative direction. Whiterepresents stagnant flow. Density plots similar to Figure 7 areprovided for the purpose of position comparison.

Figure 12. Pressure in a surface-ascending liquid drop throughLBM simulation, obtained through material derivative DF/Dt.

5782 Langmuir, Vol. 21, No. 13, 2005 Mo et al.