hu quotes double-bowl deegan result

Evaporation of a Sessile Droplet on a Substrate Hua Hu* and Ronald G. Larson Department of Chem. Eng., Uni Versity of Michigan, Ann Arbor, Michigan 48109-2136 Recei Ved: May 14, 2001; In Final Form: September 20, 2001 The evaporation of a sessile droplet with a pinned contact line is investigated experimentally, by analytic theory and by computation using the finite element method (FEM). Because of the low value of R2/Dtf) cv(1-H)/F)1.410-5, whereRis the contact-line radius, Dis the water vapor diffusivity, cvis the saturatedwatervaporconcentration, Histherelativehumidity,and Fistheliquidwaterdensity,theevaporation canbeconsideredasaquasi-steady-stateprocess.Hence,thevaporconcentrationdistributionabovethedropletsatisfies the Laplace equation but with a time-varying droplet surface. It is found both theoretically andexperimentally that the net evaporation rate from the droplet remains almost constant with time for a smallinitial contact angle ( ı<40°), even though the evaporation flux becomes more strongly singular at the edge of the droplet as the contact angle decreases during evaporation. We also measured the critical contact angleat which the contact line starts to recede and found that it is about 2 -4°for clean water on glass. Finally, we compare the results obtained by our FEM analysis with an analytical solution and derive a very simpleapproximate evaporation rate expression mø(t))- ðRD(1-H)c v(0.27 ı2+1.30), which agrees with the theoretical results presented by Lebedev [Lebedev, N. N. Special Functions and Their Application ; Prentice Hall: Englewood Cliffs, New Jersey, 1965 and Picknett and Bexon [Picknett, R. G.; Bexon, R. J. Colloid Interface Sci. 1977, 61, 366] for any initial contact angle ıbetween 0 and ð/2 with ıin radians. The approximateexpressionisalsocomparedwithdropletevaporationdatafromtheliterature,andgoodagreementis found without any parameter fitting. 1. Introduction The evaporation of a sessile droplet is not only important in manyheattransferapplicationsbutisassociatedwithcommon,everyday phenomena, such as the annoying ring-like spots lefton dishes that are allowed to dry. Recently, important newapplicationsofthissimplephenomenonhaveemerged.Jingandco-workers 1havedevelopedahigh-throughputautomaticDNA mapping method based on drying droplet. In this technique,water evaporation is used both to induce a microscopic flowthat stretches DNA molecules and to deposit those moleculesonto a substrate where they can be subjected to a restrictiondigestion.ThelocationsofdigestionsitesalongtheDNAstrandobserved in an optical microscope then constitutes on “opticalmap” of the DNA molecule. Droplet drying is also importantin the creation of arrays of DNA spots for gene expressionanalysis. Li et al. 2found that the DNA’s stretching behavior is strongly affected by the evaporation rate of droplet. At a lowevaporationrate,theDNAmoleculesarelessstretchedandtheirmolecularconformationsarefoldedorcoiled,whereasatahighevaporation rate, DNA molecules are more stretched and theirshapes become dumbbells or half-dumbbells. Thus, there arenewmotivationsforrevisitingtheoldproblemofadryingsessiledroplet. TheevaporationofasessiledroplethasbeenstudiedbyBirdi and Winter, 3who obtained the evaporation rate by measuring the change of weight of droplets of water on a glass surfaceand concluded that for most of the time during the evaporationtheevaporationrateremainsconstant.Later,they 4reportedthat therateofevaporationofsessiledropsofwateronglass(contactangle ı)41°) andn-octane on Teflon surfaces is constant intime with the contact line pinned. Shanahan and Bourges 5,6 investigated the evaporation of droplets of water from smooth polyethylene and from both smooth and rough epoxy resinsurfaces. They measured the change of drop height, contactangle, and contact-line radius with time and observed thatevaporation occurs in several distinct stages. In the longest ofthese, the droplet flattens as it evaporates with the contact linepinned. Rowan et al. 7,8presented detailed measurements of the changeincontactangleandheightwithtimeforwateronpoly-(methyl methacrylate), PMMA, and for three alcohols restingon Teflon. Starting with a large initial contact angle ( ı80°), both the height and the contact angle decreased linearly withtime. Lebedev, 9andlaterPicknettandBexon,10independently,used the analogy between diffusive concentration fields and electro-static potential fields (they both satisfy Laplace’s equation) tothe problem of evaporation of a sessile droplet. For diffusion-controlled evaporation, the vapor concentration field is equiva-lent to the electrostatic potential field around the top half of anequiconvex lens. These theoretical results should be valid aslong as the sessile droplet remains in the shape of a sphericalcap. Bourges and Shanahan 6proposed an evaporation model for the sessile droplet by taking the concentration gradient tobe that for a hemispherical droplet of same radius as that of thesessile droplet. This approximation is not accurate for a flatdroplet because the distribution of the evaporation flux along asessile droplet surface is not uniform as it would be for ahemispherical droplet. Rowan et al. 7analyzed the problem theoreticallyusingavapor-phasediffusionmodelsuggestedbyBirdi et al. 3and derived an approximate analytic equation for the evaporation rate. Their model fits experimental results verywell for droplets with large initial contact angle. Erbil et al. 11* To whom correspondence should be addressed.1334 J. Phys. Chem. B 2002,106,1334 -1344 10.1021/jp0118322 CCC: $22.00 © 2002 American Chemical Society Published on Web 01/18/2002 modeled the droplet as an ellipsoidal cap, defined by three parameters. By adjusting these parameters, they obtained goodagreementwithexperiments.MericandErbil 12reportedanother model considering a “pseudo-spherical-cap” geometry (forwhich the droplet height is given by h 0)RRtanı/2, where R is the droplet radius and Ris an adjustable flatness parameter, with R)1 for a spherical cap), which they argued provided much better fits to experimental results. Deegan et al.13,14 presented an analytical solution for a droplet with the shape ofasphericalcap.Althoughauthorsneglectedtheevaporationfluxdistribution along the droplet surface, Deegan et al. 14used the exact analytic expression for the evaporation flux distributionderived by Lebedev. 9However, Deegan et al. did not study the relationship between the evaporation rate or flux distributionand the contact angle. The purpose of this article is to developasimple,yetaccurate,modelforevaporationofasmalldropletwith the shape of a spherical cap and to lay the basis forcalculating the Marangoni force induced by a nonuniformdistribution of the evaporation flux and for developing anaccurate model for the complete flow field in an evaporatingdroplet, to be presented in a forthcoming paper. 15This flow model will, in future work, be used for predicting stretchingand deposition of DNA molecules in an evaporating droplet. In this paper, we first use a finite element method to solve for the outer vapor concentration and evaporation flux. Theevaporation flux distribution along the droplet surface is notuniformwhenadropletisplacedonthesurface.Thenonuniformevaporation flux eventually affects the droplet surface temper-ature distribution and therefore generates a surface tensiongradient along the droplet surface, which may be a possiblecontributing factor for the contact-line pinning during evapora-tion.Second,wedescribeaparticle-trackingmethodtomeasurethe time-dependent change in droplet shape and the rate ofdropletevaporation.Thentheexperimentalresultsarepresentedand compared with the results computed by the FEM method.Finally, our model for the rate of the evaporation is comparedtotheresultsreportedintheliterature,andasimple,yetaccurate,empirical expression for the evaporation rate as a function ofcontact angle is obtained. The empirical expression is alsocompared to the theoretical results derived by Lebedev 9and Picknett and Bexon.10 2. Theory From our experimental observations, we have found that, when the contact angle is less than 90 °, droplet evaporation generally includes two main phases. In the first phase, thecontact angle decreases while the contact line is pinned. In thesecond phase, the contact line recedes while the contact angleremains very small. Because the first phase occupies the 90 - 95% of the total drying time, we only consider this phase inwhich the contact line is pinned. In this section, we develop amathematical model and a corresponding FEM solution for thedroplet evaporation rate. 2.1.MathematicalModel. Here,weconsiderasessiledroplet having the shape of a spherical cap resting on a flat substrate.The droplet shape is controlled by the Bond number, Bo ) FgRh 0/ó, which accounts for the balance of surface tension and gravitationalforceonthedropletshape,andthecapillarynumberCa)íuj r/ó, which is the ratio of viscous to capillary forces. Here, Fis the fluid density, gis the gravitational constant, Ris the contact-line radius, h0is the initial height of the droplet, ó is the air -water surface tension, íis the liquid viscosity, and ujristheaverageradialvelocityinducedbydropletevaporation. In our experiments, with small droplets with contact-line radiiof0.8 -1.0mmandheightsofabout0.3mmandforslowflows (around 1 ím/sec), the Bond number is in a range of 0.03 - 0.04andthecapillarynumberisaround10-8,sothatthedroplet shape can be regarded as a spherical cap. In Figure 1, a small drop of water on a glass surface, whose shape is that of a spherical cap, is presented. A cylindricalcoordinate system is used with radial coordinate rand axial coordinate z.S){h(r,t)jreR}defines the surface of the droplet, where h(r,t)i s where ıis the contact angle and Ris the contact-line radius. The volume of the droplet is whereh(0,t) is the droplet height as a function of time and Ris the contact-line radius. From eq 1, when we let r)0, we have h(0,t) Because water evaporates into the ambient air, the vapor concentrationisdistributednonuniformlyabovethedroplet.Atthe interface between the liquid and the vapor, the vaporconcentration cis assumed to equal the saturation value c v. Far above the droplet, the vapor concentration approaches anambient value Hc v(whereHis the relative humidity of the ambient air). The difference in water vapor concentrationc v(1-H)drivestheevaporationofwaterintotheair,according to the diffusion equation wherecis the local water vapor mass concentration and Dis the vapor diffusivity. The boundary conditions are HereJis the vapor mass flux that is due to evaporation. The time required for the vapor-phase water concentration to adjust to changes in the droplet shape is of the order of R2/ D, whereDis the diffusivity of the vapor in air and Ris the contact-line radius. The ratio of this time to the dropletevaporationtime t fisR2/Dtfcv(1-H)/F.Inourexperiments, wecantake H)0.4,cv)2.3210-5g/cm3,and F)1g/cm3, so that we obtain R2/Dtfcv(1-H)/F)0.000 014 ,1. Hence,thewatervaporconcentrationadjustsrapidlycomparedto the time required for droplet evaporation, and the waterevaporation can be considered to be at a quasi-steady state. Wetherefore neglect the transient term in eq 4 and obtain the Figure 1. Droplet with the shape of a spherical cap rests on a flat surface. The contact angle is ı, the local height is h(r,t), and the local evaporation flux is J(r,t). h(r,t))xR2/sin2ı-r2-R/tan(ı) (1) V(t))ðh(0,t)[3R2+h2(0,t)] 6(2) h(0,t))Rtan[ı(t)/2] (3) @c @t)D¢c (4) 1.r<R,z)h(r):c)cv 2.r>R,z)0:J)0 3.r)∞,z)∞:c)Hvcv (5)Evaporation of a Sessile Droplet on a Substrate J. Phys. Chem. B, Vol. 106, No. 6, 2002 1335 Laplace equation for the vapor concentration distribution: Asthedropletevaporates,thesurfaceofthedropletdescends toward the substrate. For this moving-boundary problem, wepresent a finite element method to solve eq 6 to calculate thevapor distribution above the droplet and the evaporation fluxalong the droplet surface. At the air -liquid interface, the local evaporation flux JB(r,t) is expressed as The evaporation rate J tover the whole surface is expressed by the following equation: The above derivation assumes that the evaporation is not so rapid as to alter the droplet temperature enough to change thevalues of the parameters c vorD. 2.2. Finite Element Method. We apply the FEM method to calculatethevaporconcentrationdistributionabovethedroplet.For simplification, we describe all of the detailed derivationsof FEM model in the Appendix. The final formulations of theFEM model are expressed by eqs A-5 to A-12. Incorporatingthe boundary conditions listed in eq 5, eqs A-5 to A-12 can benumericallysolvedtoobtainthevaporconcentrationdistribution. 3. Experimental Method We use a microscopic particle tracer method to measure the rate of droplet evaporation. Figure 2 is a schematic of theimagingsystemusedinourexperimentstomeasurethedropletevaporation, with Figure 2b as an enlargement of element A inFigure 2a. A Nikon inverted fluorescence microscope (Eclipse,TE200) with a motorized X -Y stage (Prior, Inc.) is employed to locate the fluorescent particles. As shown in Figure 2b, adroplet of volume about 0.5 íL is deposited on a clean glass cover slip (Dow-Corning, No.1; 22 mm) and initially sealed byacylindricalcaptoblockitsevaporation(Oncethedropletsizeand location have been precisely measured, the cap is removedand replaced by a cylinder open at the top to allow evaporationto begin while suppressing the effect of air currents onevaporation.) The glass cover slip with its holder is placed onthemotorizedX -Ystage.A40 objectiveiscoatedwithwater, and the water coating is brought into contact with the coverslip, thus eliminating the air gap between the sample and theobjective and to minimize the effect of refraction. Fluorescentparticles 0.75 ím in diameter (Polyscience, Inc.) are used as tracerstomapthedropletprofilesatdifferenttimesandtherebycalculate the residual droplet volume. Particle images aredetected by a CCD camera (PC-26C, Super Circuit Co.) with aresolution of 640 480 pixel. Then, the images are recorded onto a computer disk and analyzed by the SimplePCI imageprocessing system (Compix, Inc.). With the cylindrical cap in place to prevent evaporation, we move the X -Y stage around the perimeter of the droplet and measure the coordinates of the particles at the edge of thedroplet. The coordinates of these particles are fitted by a circleso that the center and the radius of the droplet are determined.From the experimental results for about 500 droplets, we canconclude that the contact line of the droplet is indeed a circleand the radius of the contact line is about 850 (10 microns.We also obtained the residual droplet volume by measuring the droplet surface profiles at different times by finding thecoordinatesoffluorescentparticlesonthesurfaceofthedroplet.The results are shown in Figure 3, in which the symbols arethe positions of fluorescence particles on the droplet surfaceand the line is a fit by a circular arc, representing a sphericalcap. The standard error of the fit is about 1 -2 micron. This implies the droplet shape is not affected by adding the tracerparticles. The tracer particles might enhance pinning of thecontact line, but it is hard to examine it. Because the dropletremainsintheshapeofasphericalcapatalltimesduringdropletevaporation, we need only measure the height of the droplet asa function of time in order to reach an accurate measurementof the droplet surface and volume. 4. Results 4.1. Convergence of the FEM Computation. The conver- gence of the FEM computation is tested by systematicallyrefining the mesh at the edge of the droplet according to themethod described in Appendix A.4. The mesh near the edge ofthe droplet is refined several times over so that the number of¢c)0 (6) JB(r,t))DrBc (7) Jt)s¡g(JBânb)d¡g (8) Figure 2. Experimental setup. Figure 2a is a Nikon fluorescence microscope, and Figure 2b is a blow-up of part A in Figure 2a.1336J. Phys. Chem. B, Vol. 106, No. 6, 2002 Hu and Larson elements becomes large and the size of the elements near the edge becomes smaller. Subsequently, the evaporation flux andthe total evaporation rate are calculated using eqs 7 and 8 forthe different refined meshes. Numerical calculations show thatas the number of the elements increases the total evaporationratedecreasesforsmallnumbersofelementsandlevelsoffwhenthe number of elements exceeds about 5000. The relative errordefined in eq A-13 is plotted in Figure 4, which shows that therelative error decreases linearly with the number of elements,asexpectedforlinearshapefunctions.Figure4showsthatwhenthe number of elements in the circular sector is larger than10 000 the relative error is small enough that the convergencecriterion is satisfied. We can conclude that after the mesh hasbeen locally refined 6-fold according to the method given inAppendix A.4, an acceptable FEM computational accuracy isattained. Using this protocol, the vapor concentration distribu-tion, the evaporation rate, the height of the droplet, the contactangle,andthedropletvolumesatdifferenttimesarecomputed,as described in the following. 4.2. Vapor Concentration Distribution. When using the finite element model, eqs A-5 to A-12, to solve the vaporconcentration distribution, we need to consider the boundarycondition in eq 5, which is c)Hc vatr,zf∞. We impose this condition along a boundary at ( r2+z2)1/2)KR, whereKis a constant much larger than unity. By testing different values ofKbetween 10 and 100, we find that the calculated vapor concentration at position ( r2+z2)1/2)20Ris close enough to the ambient vapor concentration c∞that the deviations are negligible; that is, ( c-c∞)/(cv-c∞)<0.002. Therefore, in our FEM analysis, we use the boundary ( r2+z2)1/2)20R to approximately represent r,zf∞. The vapor concentration distributions at different times are then computed by the FEM method and presented in Figure 5.Figure5isacontourplotofthevaporconcentrationdistributionwhen the droplet is starting to evaporate. The contour lines areconcentrated near the droplet surface, where the vapor concen-tration has a large gradient. The vapor concentration along thelinesr)0 andz)0 is plotted in Figure 6. When randzare greater than 5 R, the changes in vapor concentration are small, Figure 3. Measured droplet profiles at different times, for water evaporating from a droplet of initial radius R)0.85 mm and height h0)0.329mm.Thesymbolsshowthelocationsoffluorescentparticles on the droplet surface, and the lines are the fittings of circular arcs tothese data. The fitting errors are about 1 -2ím. Figure 4. Relative error vs the number of FEM elements. Figure 5. Contour plot of the vapor concentration distribution above a droplet of radius R)1 mm and height h0)0.364 mm. The parameters used in the FEM method are vapor diffusivity D)26.1 mm2/s, relative humidity H)0.40 (i.e., 40%), and saturated vapor concentration on the droplet surface cv)2.3210-8g/mm3, which is the value obtained form the CRC Handbook16at 25°C. These parametervaluesarealsousedinFigures6 -10.Thegraybarsrepresent the vapor concentration in g/mm3. Figure 6. Vapor concentration distribution along the directions z)0 andr)0 from FEM (solid lines) and analytic solution (dashed lines). The FEM results along each direction superimpose on the analyticresults.Evaporation of a Sessile Droplet on a Substrate J. Phys. Chem. B, Vol. 106, No. 6, 2002 1337 and the normalized vapor concentration, ( c-Hcv)/(1-H)cv, issosmall,0.02,thatthecutoff,( r2+z2)1/2)20R,isacceptable. 4.3. Vapor Flux above the Droplet. From the vapor concentration distribution, we calculate the vapor flux from eq7. Along the droplet surface, the flux increases as one movesfromthecentertopofthedroplettothecontactlineattheedge,where the flux is theoretically infinite, see Figure 7. The totalevaporationratecanbeevaluatedfromeq8byintegratingalongthe droplet surface. The evaporation rate for a series of dropletshapesisusedtocomputethechangeindropletvolumevstimein the next section. 4.4.DropletVolume,Height,andContactAnglevsTime. By repeating the FEM analysis for a series of droplet heights,we simulate the droplet evaporation process. The radius andthe initial height of the droplet are 1 and 0.364 mm. Theparameters used in the FEM analysis are the vapor diffusivityD)26.1 mm 2/s (from CRC Handbook of Chemistry and Physics16), the relative humidity H)0.4, and the saturated vapor concentration cv)2.3210-8g/mm3. A small time step of about 0.02 tfis used to calculate the time-dependent volume.Ateachtimestep,thelossofwaterisdeterminedfromthe product of the total evaporation rate integrated over thedroplet surface and the time step. As we described in section 2,the evaporation rate can be obtained from the vapor concentra-tion distribution by using eqs 7 and 8 on the droplet surface.The new droplet volume is thereby calculated from the loss ofthesolventandthepreviousdropletvolume,andthenewdropletsurface profile can be derived from eq 2, for a spherical cap.Thisprocedureisaccurate,becausethecapillarynumberislow(Ca10 -8; so that the droplet remains a spherical cap), the contactlineispinned,andthevaporconcentrationfieldisquasi-steady; that is, it adjusts rapidly whenever the droplet shapechanges.Thus,wecanconvertthismoving-free-surfaceprobleminto a simple series of solutions to Laplace’s equation. Theheight of the droplet and the contact angle decrease roughlylinearly with time as shown in Figure 8, suggesting a nearlyconstant evaporation rate. The total evaporation rate, shown intheinserttoFigure8,isnotquiteconstant;itdecreasesslightlyduring the droplet evaporation for an initial contact angle of40°. 4.5.ExperimentalResults. Inourexperiments,weobtained the droplet evaporation rate by measuring droplet height at aseriesoftimes.Doingsoisreliablebecause,asshowninFigure3,thedropletshaperemainsasphericalcapduringevaporation.Fromtheheightandthecontact-lineradius,thedropletvolumeat different times is calculated. The residual droplet volume vstime is plotted in Figure 9 and compared with the results of the finite element simulation using the experimentally derivedparameters. We can see that there is very good agreementbetween the experiment and the FEM calculations. The FEMresults show that the evaporation rate changes slightly duringevaporation as seen in Figure 8 and the solid line in Figure 9 isnotastraightline.Itchangesitsslopeslightlyduringtheinitialdrying process but has become almost constant near the end ofdrying. In the experiments, we also measured the critical contact angle, the angle at which the contact line starts to recede forabout 50 droplets, and obtained an average of 2 -4°. 5. Discussion 5.1. Approximate Expression for the Evaporation Rate. The excellent agreement between the FEM computation andthe experimental measurements confirms that the dropletevaporation is a quasi-steady-state process. This should be truewhenever R 2/Dtfcv(1-H)/Fcv/Fis small; that is, whenever the vapor phase has a density much smaller than thatof the liquid, which is always true except near supercriticalconditions. Thus, the quasisteady approximation for the vaporconcentration field should be valid even for very rapidlyevaporating droplets. It should be kept in mind, however, thatfor very rapidly evaporating droplets large temperature non- Figure 7. Evaporation flux along the droplet surface. The insert shows the magnitude of the evaporation flux along the droplet surface. The flux vector at the edge of the droplet does not orient along the normal direction, because when the mesh is refined several times near the edge of thedroplet, the edge of the last element at r)R, which has become very small, does not perfectly match the droplet surface profile. This artifact is, however, confined to a single element and has negligible effect on the overall flux. Figure 8. Height of the droplet and the contact angle vs time. The insert shows the evaporation rate vs time.1338J. Phys. Chem. B, Vol. 106, No. 6, 2002 Hu and Larson uniformities may develop because of latent heat and this will affect the evaporation flux. The evaporation flux on the droplet surface calculated in our FEManalysiscanbefittedbythefollowingequationsuggestedby Deegan et al.: 13,14 where ìis a fitting parameter representing the nonuniformity oftheevaporationfluxonthedropletsurfaceand r˜r/R.From fits of eq 9 to our FEM results, we obtain values of J0andì for different contact angles, see Figures 10 and 11. BecauseDeeganetal.didnotpreciselydefinetherelationshipsbetweenJ 0,ì, and the contact angle ı, we determine J0(ı) and ì(ı) versusthecontactangleempiricallybyusingthefiniteelementmethod. From Figure 10, we find that the ratio J 0(ı)/J0(ð/2) is close to unity when the contact angle ıis in a range of 70 -90°but starts to decrease linearly with ıforıless than 60 °.A t ı )0°, the ratio reaches the value of 0.6377. The error between the FEM result and the exact solution shown by the dashedlineinFigure10islessthan1.5%.Surprisingly,Figure11showsthatìdecreases linearly with ıover the range of 0 -90°, and soìcan be expressed by a simple linear function, ì(ı))0.5 -ı/ð, which is different from a result cited by Deegan et al., ì(ı))(ð-2ı)/(2ð-2ı). However, this latter formula for ì(ı) applies to the equation J(r)(1-r˜) -ìand not to the equation used here (and by Deegan et al.), namely, J(r)(1- r˜2)-ì. Obviously, the formula for the exponent ì(ı) will differ for these two expressions. By using the formula ì(ı))0.5- ı/ð, the largest relative error between the prediction from eq 9 and the results from the FEM analysis over the range of 0 -90° is less than 6%, and this value decreases toward zero as thecontactangleapproacheseither0 °or90°.FromFigures10and 11, we conclude that the evaporation rate is a function of thecontact angle ı. Using eq 9, the evaporation rate becomes whereSistheareaofintegrationand @h(r,t)/@risthederivative ofh(r,t) with respect to r, given for a spherical cap by We find that the term [( @h(r,t)/@r) 2+1]1/2in the integration kernel in eq 10 can be approximated by where ä(ı)isanempiricalfunctionofcontactangle ı.Inserting eq 12 into eq 10, we have where ⁄(ı))ì(ı)+ä(ı) is a combination of two factors, one is a parameter reflecting the nonuniformity of the evapora-tion flux and the other reflects the droplet surface area per unitarea of substrate at each value of r˜. When the contact angle is Figure 9. Symbols giving the residual droplet volume vs time calculated from the height of the droplet at different times and thecontact-line radius. The thin solid line is the result of the FEMcalculations, using the experimental conditions as parameters, i.e., R )0.95mm, h 0)0.364mm,vapordiffusivity D)26.1mm2/s,relative humidity H)0.38 (i.e., 38%), and saturated vapor concentration on the droplet surface cv)2.3210-8g/mm3at temperature 25 °C. The dashedlineiscalculatedbytheapproximateevaporationrateexpressioneq 21 using the same values of parameters as the FEM method. Thethin dashed line is calculated by Picknett and Bexon’s model 9using the same value of parameters as the FEM method. Figure 10. RatioJ0(ı)/J0(ð/2) versus the contact angle. The solid line is calculated by FEM analysis, whereas the dashed line is from theanalytical solution. (JBânb))J0(1-r˜2)-ì(9) Figure 11. Evaporation rate exponent ìas a function of the contact angle, obtained by fitting eq 9 to the FEM results. -mø(t))sS(JB(r,t)ânb)dS) s012ðr˜âR2âJB0(ı)(1-r˜2)-ì(ı)x(@h(r,t) @r)2 +1dr˜ (10) @h(r,t) @r)-r˜ x1/sin2ı-r˜2(11) x(@h(r,t) @r)2 +1)(1-sin2ır˜2)-0.5=(1-r˜2)-ä(ı)(12) -mø(t))s012ðr˜âR2âJ0(ı)(1-r˜2)-⁄(ı)dr˜ (13)Evaporation of a Sessile Droplet on a Substrate J. Phys. Chem. B, Vol. 106, No. 6, 2002 1339 90°, because ì(ð/2) is 0 and ä(ð/2) is 0.5, ⁄(ð/2) equals 0.5. When the contact angle is 0 °, because ì(0) is 0.5 and ä(0) is 0, ⁄(0) also equals 0.5. Using the results of our FEM analysis, we fit the term (1 -r˜2)-ì(ı)[(@h(r,t)/@r)2+1]1/2in eq 10 with the equation (1 -r˜2)-⁄(ı)at a given contact angle to obtain ⁄(ı) and plot this in Figure 12 as solid diamonds. In the insert to Figure 12, we plot the term ln[ K(r,ı)])ln{(1-r˜2)-ì(ı)- [(@h(r,ı)/@r)2+1]1/2}against -ln(1 -r˜2), and the slopes of the lines give ⁄(ı). The points in Figure 12 can be fitted well by a parabola, namely, where ıis given in radians. Substituting eq 14 into eq 13, and integrating, gives Equation 15 is a general formula for the droplet evaporation rate for any contact angle (0 °<ı<90°). In eq 15, we note thatJ0(ı), the evaporation flux at the center of the droplet, is also a function of the contact angle ı, as shown in Figure 10. To simplify eq 15, we calculated the ratio of J0(ı)/J0(ð/2)(1 - ⁄(ı)) using the FEM results and plot this in Figure 13, which isgivenbysolidsquares.InFigure13,thelineisobtainedfromthe fitting function where ıis given in radians. The results predicted by eq 16 are compared to the exact solution for contact angle ranging from0t o9 0°, and the largest error is less than 6%. Substituting eq 16 into eq 15, we derive an approximate expression for the droplet evaporation rate at any contact angle(0°<ı<90°), which is whereJ 0(ð/2) is the evaporation flux for the contact angle 90 °.Forı)90°, the evaporation flux is uniform everywhere along the droplet surface. The solution of eq 6 for a contact angle of90°is, by symmetry, the same as the solution for a droplet suspended in the air without the presence of a substrate. For asuspended droplet, the boundary conditions are ( r 2+z2)1/2) R:c)cvand (r2+z2)1/2)∞:c)HcV, and we derive the evaporation flux J0(ð/2) from eq 7, which is Combining eqs 17 and 18 gives From eq 19, we can see that at a given contact angle the evaporationrateisproportionaltothecontact-lineradius R,the vapor concentration difference (1 -H)cV, and the diffusivity Danddependsweaklyonthecontactangle ı.Figure13shows that when the contact angle is less than 40 °(0.7 rad) the dependence on ıalmost becomes flat, and therefore, the evaporation rate is almost constant. Equation 19 agrees well with the theoretical results obtained by Picknett and Bexon.10 We calculate the evaporation rate as a function of the contactangle according to eq 19 and compare the result to that derivedby Picknett and Bexon. The averaged relative error betweenthetwopredictionsislessthan1%.However,eq19isdifferentfrom the prediction of the model presented by Bourges andShanahan. 6Whenthecontactangle ıiscloseto0 °,theirmodel predicts an evaporation rate of 0, which is incorrect. This isbecause that Bourges and Shanahan assume that the vaporconcentrationgradientisuniformalongthesurfaceofthesessiledroplet and is set by the radius of curvature of the dropletsurface,whichbecomesinfinitewhenthedropletbecomesflat,leadingtozeroevaporationflux.Actually,fromourFEMresults,Figure7,wecanseetheconcentrationgradientalongthesurfaceof a sessile droplet is not uniform compared to the uniformconcentration gradient on the surface of a spherical droplet. Equation 19 gives the exact solution for two limiting cases, for a contact angle of 0 °and a contact angle of 90 °. When the contact angle is close to 90 °, eq 19 gives Figure 12. ⁄(ı) versus the contact angle ı. The solid symbols are the slopes of the curves in the insert obtained by fitting ln[ K(r,ı)]) ln{(1-r˜2)-ì(ı)[(@h(r,ı)@r)2+1]1/2}with -⁄(ı)ln(1 -r˜2). The line is a parabola, eq 14, fitted to the solid points. ⁄(ı))0.2239( ı-ð/4)2+0.3619 (14) -mø(t))ðR2J0(ı) 1-⁄(ı))ðR2J0(ı) 1-(0.2239( ı-ð/4)2+0.3619) (15) J0(ı) 1-⁄(ı))J0(ð/2)(0.27 ı2+1.30) (16) -mø(t))ðR2J0(ð/2)(0.27 ı2+1.30) (17) Figure 13. RatioJ0(ı)/J0(ð/2)(1 -⁄(ı)) versus the contact angle, calculated from Figures 10 and 12. J0(ð/2))D(1-H)cv R(18) -mø(t))ðRD(1-H)cv(0.27 ı2+1.30) (19) -mø(t))2ðD(1-H)cvR (20)1340J. Phys. Chem. B, Vol. 106, No. 6, 2002 Hu and Larson whereas when the contact angle ıis close to 0 °, eq 19 gives Both eqs 20 and 21 are consistent with the results of Picknett and Bexon10for the two cases of the contact angles, 90 and 0 °, respectively. We use eq 21 to calculate the droplet volume vs time and compare this with the results from the FEM method and theexperimentalmeasurementasshowninFigure9.Figure9showsthat the results calculated by eq 21 are consistent with thosefrom the FEM method, Picknett and Bexon’s model, and theexperiments. The relative error between eq 21 and the experi-ments is 3.6%. This implies that when the initial contact angleis less than 40 °the evaporation rate can be approximated by eq 21 and the evaporation rate can be regarded as a constant. Theprediction from the FEM analysis, which is the thin solid line,is very close to the prediction of Picknett and Bexon’s model,which is the thin dashed line. The relative error in the dryingtimes predicted by FEM and the theoretical model of Picknettand Bexon is less than 1.3%. 5.2. Comparison with the Exact Solution. Deegan et al. 14 reported an analytical solution for the Laplace equation, which wasderivedbyLebedev.9Lebedevconsideredachargedsurface formed by the union of two spherical domains, and derived theelectrostatic potential distribution in the space exterior to thissurface by using toroidal coordinates. In toroidal coordinates,the potential distribution is given by where Randâare the toroidal coordinates, Vis the potential on the surface, and â 1andâ2are two angles of ð-ıandð+ ı,P-(1/2) +iô(cosh R) is the Legendre function of the first kind and is expressed by Although eq 22 has a different form from Picknett and Bexon’s solution, they are identical because they are derivedfrom the same model. Thetoroidalcoordinates Randâarerelatedtothecylindrical coordinates randzby whereRis the contact-line radius. The electrostatic potential around a lens-shaped object with uniform surface potential and the vapor concentration distribu-tionsaboveanevaporatingdropletareequivalent,becausetheyare both solutions of the Laplace equation. By symmetry, thesolutiontoLaplace’sequationinthehalf-planeaboveasphericalcap is identical to that of a lens-like shape formed from twoback-to-back spherical caps. Thus, we can apply the solutionof the electrostatic potential in toroidal coordinates for a lens-like shape to the vapor concentration distribution above thedroplet. We consider the electrostatic potential uas a dimen- sionless vapor concentration, u)(c-c ∞)/(cv-c∞), wherecv andc∞are the vapor concentration on the droplet surface and intheambient,respectively.Then u)1onthedropletsurface. For the lens-like geometry, we let â1)ð-ıandâ2)ð+ ı,where ıisthecontactangle.Fromeq25,weobtainthevapor concentration distribution Once the vapor concentration is known, from eq 7, we can obtaintheevaporationfluxdistributionalongthedropletsurface,which in toroidal coordinates is Inserting eq 26 into eq 27, the evaporation flux is which corrects eq A2 of Deegan et al. 14 Eqs 6 and 28 are the expressions for the vapor concentration distribution above the droplet and the evaporation flux alongthedropletsurface,respectively.Bothofthemhavecomplicatedintegrations, so closed forms are not available. A numericalmethod is therefore used to solve these two formulas. Now we evaluate eqs 26 and 28 for two special cases, ı)0 and 90°. For the case of ı)90°, eq 26 becomes wherer′(r 2+z2)1/2 Equation 29 becomes Equations 9 and 30 are consistent with the results obtained by solvingtheLaplaceequationincylindricalcoordinates.Insertingeq 30 into eq 10, the evaporation rate obtained is identical toeq 20. For the case of ı)0°, eqs 26 and 28 can be solved: In cylindrical coordinates, eq 32 becomes-mø(t))4D(1-H)cvR (21) u)Vx2 cosh R-2 cos âs0∞ cosh[( ð-â1)ô] sinh[( â-â2)ô]+cosh[( ð-â2)ô] sinh[(2 ð+â1-â)ô] cosh( ðô) sinh[(2 ð+â1-â2)ô] P-(1/2) +iô(cosh R)dô(22) P-(1/2) +iô(cosh R))2 ðcoth(ðô)sR∞sin(ôt) x2 cosht-2 cosh Rdt (23) r)Rsinh R cosh R-cosâ(24) z)Rsinâ cosh R-cosâ(25)c-c∞ cv-c∞)x2 cosh R-2 cos âs0∞ cosh( ıô) cosh( ðô)cosh[(2 ð-â)ô] cosh[( ð-ı)ô]P-(1/2) +iô(cosh R)dô(26) (JBânb))-D(cosh R-cosâ) R@u @âjâ)3ð-ı(27) (JBânb))-D(cv-c∞) R[sinı 2+(cosh R+ cosı)(3/2)s0∞cosh( ıô) cosh( ðô)tanh[( ð-ı)ô]ôP-(1/2) +iô(cosh R)dô] (28) c-c∞ cv-c∞)xcosh R-cosâ cosh R+cosâ)R r′(29) (JBânb))D(cv-c∞) R(30) c-c∞ cv-c∞)3 2+1 ðarctan[cosâ-sinh2(R 2) cos(â 2)x2 cosh R-2 cos â](31) (JBânb))D(cv-c∞) R2 ðcosh(R 2)(32) (JBânb))D(cv-c∞) R2 ð(1-r˜2)-0.5(33)Evaporation of a Sessile Droplet on a Substrate J. Phys. Chem. B, Vol. 106, No. 6, 2002 1341 Equation 33 implies that the fitting parameter ìin eq 9 becomes 0.5 when the contact angle reaches the limiting valueof 0°. From eqs 30 and 33, we find the ratio J 0(0)/J0(ð/2)) [(2/ð)(D(cv-c∞)/R)]/(D(cv-c∞)/R))2/ð. The ratio from Figure 10 computed by the FEM method for ı)0 is 0.6377, which is indeed very close to the value 2/ ð)0.6366. Substituting eq 33 into eq 10, we also rederive the same evaporation rate for ı)0°given by eq 21. We can now compare the FEM results with the analytical resultsobtainedbysolvingeqs26and27.InFigure6,thevaporconcentrationdistributionsalongthe r)0andz)0directions calculated by FEM are indistinguishable from the resultsobtained from eq 26. Figure 10 shows that the ratio J 0(ı)/J0- (ð/2) computed by FEM and from the analytical solution are nearly identical. We note that the analytic solution is only available for the specialcaseofasphericalcap,whichisvalidforsmalldroplets(Re1 mm). For larger droplets, the bond number exceeds 0.1, and gravitational sag becomes important. The analytic solution also fails when the evaporation rate is fast enough to changethe temperature of the droplet enough to affect the saturatedvapor concentration c v. In the case of gravitational sag, eq 1 for the droplet surface profile must be replaced by an ordinarydifferential equation for the static droplet shape under theinfluence of gravity and surface tension, which must be solvedeach time step. If temperature variations become large, thetemperature field inside the droplet must be obtained each timestep by solving a quasisteady heat equation (as is done in aforthcomingpaper 15).Inbothcases,however,themostimportant simplifications of our approach remain valid: the vaporconcentration field is at quasi-steady-state (because R 2/Dtf) cv(1-H)/Fis small), and the droplet shape is at static equilibrium (because Ca )íujr/óis small). Thus, the simple quasi-steady-stateFEMmethodpresentedhereremainsaccuratefor a very wide range of conditions, including conditions forwhich eq 26, the electrostatic solution, does not apply, such aswhen the droplet is not a spherical cap (because of gravity) orthe temperature is not uniform within the droplet because ofrapid evaporation. 5.3. Comparison of Model Predictions with Other Ex- perimental Results. We now compare the results calculated from the simple empirical formula eq 19 (which fits the FEMand analytic results almost perfectly) with the experimentalresults reported by Birdi et al. 3and Rowan et al.7Birdi et al.3 studiedtheevaporationofdropletsofwateronglassbyweighingthe residual droplet mass. In general, we should use eq 19 tocompare with their experiments, but this equation is wellapproximated by eq 21 when the initial contact angle ıis les than 40°, as is the case in these experiments. Their paper does not give the parameters D,H, andc v. Therefore, we obtain the experimental term 4 D(1-H)cvfrom one of their experiments and then apply this constant to calculate the results for otherdropletswithdifferentcontactlineradii.Theresultsareplottedin Figure 14, which shows that if we arbitrarily fit the set ofexperimentaldataforthedropletwithcontact-lineradiusof2.01mm by eq 21 we obtain the fitting constant 4 D(1-H)c v)6.5 10-5gs-1mm-1with correlation coefficient R)0.9999. With this constant, the residual masses of the droplets of radii2.53 and 2.93 mm are calculated by the integration of eq 21with time. The results are plotted in Figure 14, in which thesolid lines are the calculated results, which are very close tothe experimental results. The average relative errors betweenthe theoretical predictions and the experiments are 3.1% and3.5% for droplet radii 2.53 and 2.93 mm, respectively.We also compare our model’s predictions with Rowan’s 7 results in Figure 15. Rowan et al. studied the height and the contact angle as functions of time for a water droplet on aPMMAsubstratebyusingaKru ¨sscontact-anglemeter.Intheir experiments, the droplets have an initial contact angle of about80°.Therefore,weemployeq19tocalculatetheresidualdroplet volume versus time. Here, the parameters in eq 19 are directlyobtained form Rowan’s paper. They reported that the watervapor diffusivity is D)17 mm 2/s (which is a fitting value by using their model to their experimental results), the relativehumidity is H)0.55 (i.e. 55%), and the saturated vapor concentration on the droplet surface is c v)1.910-8g/mm3. Thecontactlineradiiforsixdifferentdropletsare0.585,0.491, Figure 14. Comparison of the time-dependent weight from the model predictions(eq21)andtheresultspublishedbyBirdietal.3fordroplets of water of radii 2.01, 2.53, and 2.93 mm on glass at T)22°C. The theory for R)2.01 mm was fit to the data by adjusting 4 D(1-H)cv to the best-fit value of 0.000 065 g mm-1s-1, and this was held fixed for the other experiments. The symbols are experimental results andthe lines are the model predictions. Figure 15. Comparison between the model prediction (eq 19) and the results published by Rowan et al.7for water droplets with radii R) 0.585, 0.491, 0.451, 0.381, 0.324, and 0.293 mm on PMMA substratesatT)21.5°C. All parameters in eq 19 are obtained from Rowan’s paper,andtheyarevapor-phasewaterdiffusivity D)17mm 2/s,relative humidity H)0.55 (i.e., 55%), and saturated vapor concentration on the droplet surface cv)1.910-8g/mm3. The symbols are the experimental results, and the lines are the model predictions.1342J. Phys. Chem. B, Vol. 106, No. 6, 2002 Hu and Larson 0.451, 0.381, 0.324, and 0.293 mm. From Figure 15, we can see that, when droplet is bigger ( R)0.585 and 0.491 mm), our model predicts the experimental results fairly well, and theaverage relative errors between the predictions and the experi-ments are 7.4% and 9.7%, respectively. As the droplet sizebecomes smaller, the relative errors between the model predic-tion and the experiments become larger and are in the range of10-25%. It is nearly impossible that evaporation cooling produces such big errors because in the forthcoming paper 15 we have calculated that the temperature drops in the dropletare only about 0.02 °C, which has hardly any effects on c vand D. Possible reasons for the errors are that when droplet size is smaller (below 0.451 mm) the contact angle of the droplet ishardtodeterminepreciselybythemethoddescribedinthepaperor that the humidity used here is not the true value. In all of these cases, the overall evaporation rate remains nearlyconstantasthedropletdriesout,andthus,thesingularityin evaporation flux at the droplet edge is of little consequencefor the overall rate of droplet drying. However, this singularityinevaporationfluxcreatesadropletflowfieldwithasingularityat the contact line. In addition, small temperature variationsalong the droplet surface, which have negligible effects on thedryingrate,induceMarangonistresses(Scriven 17),whichaffect qualitatively the flow inside the droplet. In our forthcomingpaper, 15this flow field will be examined both theoretically and experimentally. 6. Conclusion AnFEMmodelisdevelopedtosolvethevaporconcentration distributionandtheevaporationfluxaboveadropletthatissmallenoughthatitsshapeisnotinfluencedbygravityandisthereforea spherical cap. The vapor phase water concentration fieldadjustsrapidlytochangesindropletheightandcanberegardedas quasisteady. The evaporation flux along the droplet surfaceis not uniform and becomes singular at the edge of the dropletand can be fitted by the expression J 0(ı)(1-(r/R)2)-ì(ı) suggested by Deegan and co-workers, where ris the radial position along the droplet, Ris the droplet radius, and J0(ı) andì(ı) are empirical functions of the contact angle ı, which give the precise predictions comparing to the results obtainedby Deegan and co-workers. 14Neglecting the nonuniformity of theevaporationfluxalongdropletsurfacewillleadaninaccuratetheoretical result. FEM results also show that the overallevaporation rate is almost constant over the whole evaporationperiod when the initial contact angle is less than 40 °. The FEM resultsagreewellwiththeexperimentalmeasurements.Finally,for contact angles between 0 and 90 °, an accurate approximate expression for the evaporation rate is presented: -mø(t)) -ðRD(1-H)c v(0.27 ı2+1.30), where Dis the gas-phase diffusivity, His the relative humidity, ıis the contact angle in radians, and m(t) is the time-dependent droplet mass. This equation is confirmed by Picknett and Bexon’s results. For ı <40°, this expression can be reduced to just -mø(t))4RD(1 -H)cv. The results predicted by this model compare very well with literature data without parameter fitting. The FEM resultsarealsoconfirmedbyananalyticalformuladerivedbyLebedev 9 for the electrostatic potential produced by a charged lens andby the theoretical solution of Picknett and Bexon. 10 Acknowledgment. WethankNASAMicrogravityResearch Division for supporting this study through Grant NAG3-2134. Appendix We here derive the FEM element model that is used to calculate the vapor concentration distribution.A.1. Weak Formulation. From a variational analysis, the weak form of eq 6 in the cylindrical coordinate system is where ¿is the domain of the vapor phase, which is the semiinfinite half plane above the droplet excluding the dropletvolume, and wis the weighting function. Integrating eq A-1 by parts gives ThesecondtermineqA-2isanintegrationovertheboundary ¡, where ¡)¡ g+¡h, and ¡g\¡h)L, andnbis the outward- pointing unit vector along the surface. ¡gis the droplet free surface, and ¡his the dry substrate surface excluding the area onwhichthedropletrests.Boundarycondition1ineq5appliesalong ¡ g, and boundary condition 2 applies along ¡h. The properties of the weighting function ware as follows: along ¡)¡g,w)0; along ¡)¡h,w)1, which we have used to derive eq A-2. A.2. Galerkin Method. To complete the finite element model, we apply the Galerkin method to approximate theconcentration cand the weighting function w: d AandxAare the coefficients on node A,nis the total number of nodes, and NAis shape function for node A. WheneqsA-2,A-3,andA-4arecombined,thefiniteelement model discretization of eq 6 is obtained as follow: In this equation, K,F, andDare A.3.ShapeFunction. Weemploylineartriangularelements inthefiniteelementmodel,eqsA-6toA-8.Theshapefunctionsare L 1,L2,andL3arethelocalvariablesineachtriangularelement, and only two of them are independent variables, as specifiedby eq A-12. A.4.ConvergenceCriteriaandMeshRefinement. Because twokindsofboundaryconditionsmeetattheedgeofthedroplet,the evaporation flux becomes singular there. To overcome thiss¿w(rBâ(DrrBc)) d¿)0 (A-1) s¿rBwâ(DrrBc)d¿-s¡h(rDrBc)ânbd¡)0 (A-2) c)∑ A)1n dANA (A-3) w)∑ A)1n xANA (A-4) KD)F (A-5) K)[kAB])[s¿rBNAârBNBrd¿] (A-6) F)[fA])[s¡rBNAânbrd¡] (A-7) D)[dA])[d1,d2,d3,âââ,dA]T(A-8) N1)L1 (A-9) N2)L2 (A-10) N3)L3 (A-11) L1+L2+L3)1 (A-12)Evaporation of a Sessile Droplet on a Substrate J. Phys. Chem. B, Vol. 106, No. 6, 2002 1343 difficulty, we refine the mesh near the edge of the droplet. For convenience, we use a commercial software package, Ansys5.6, to generate the mesh for our FEM analysis. Initially, themesh is generated coarsely, and the FEM code is then used tocalculatethevaporconcentrationandtheevaporationrate.Next,we choose a circular sector with a radius of r 1)0.5Rcentered at the edge of the droplet in the initial mesh and refine all ofthe elements in this area. The vapor concentration and theevaporation rate are calculated again for the new mesh. Wecomparethecurrenttotalevaporationrate J t,ionmeshrefinement itoJt,i-1,whichisforthelessrefinedmesh i-1,toseewhether they satisfy the following criteria: whereJtis expressed by eq 8 To obtain an accurate vapor concentration distribution, the mesh near the edge of the droplet is refined continuously untilthe criterion in eq A-13 is satisfied. However, the new, ith, refinement is concentrated in a circular sector with radius r i) 0.68ri-1, whereri-1is the radius of the circular sector used intheri-1th refinement. The final refined mesh, after six iterated refinements, has 11071 elements in the circular sector. References and Notes (1) Jing, J. P.; Reed, J.; Huang, J.; Hu, X.; Clarke, V.; Edington, J.; Housman, D.; Anantharaman, T. S.; Huff, E. J.; Mishra, B.; Porter, B.;Shenkeer, A.; Wolfson, E.; Hiort, C.; Kantor, R.; Aston, C.; Schwartz, D.C.Proc. Natl. Acad. Sci .U.S.A.1998,95, 8046. (2) Li, L.; Hu, H.; Larson, R. G. To be submitted, 2001.(3) Birdi, K. S.; Vu, D. T.; Winter, A. J. Phys. Chem. 1989,93, 3702. (4) Birdi, K. S.; Vu, D. T. J. Adhes. Sci. Technol. 1993,7, 485. (5) Shanahan, M. E. R.; Bourges, C. Int. J. Adhes. Adhes .1994,14, 201. (6) Bourges, C.; Shanahan, M. E. R. Langmuir 1995,11, 2820. (7) Rowan, S. M.; McHale, G.; Newton, M. I. J. Phys. Chem. B 1995, 99, 13268. (8) Rowan,S.M.;McHale,G.;Newton,M.I.;Toorneman,M. J.Phys. Chem. B 1997,101, 1265. (9) Lebedev,N.N. SpecialFunctionsandTheirApplication ;Prentice- Hall: Englewood Cliffs, New Jersey, 1965. (10) Picknett, R. G.; Bexon, R. J. Colloid Interface Sci .1977,61, 336. (11) Erbil, H. Y.; Meric, R. A. J. Phys. Chem. B 1997,101, 6867. (12) Meric, R. A.; Erbil, H. Y. Langmuir 1998,14, 1915. (13) Deegan,R.D.;Bakajin,O.;Dupont,T.F.;etal. Nature1997,389, 827. (14) Deegan,R.D.;Bakajin,O.;Dupont,T.F.;etal. Phys.Re V.E2000, 62, 756. (15) Hu, H.; Larson, R. G. Phys. Fluids 2001, to be submitted. (16) Weast,R.C.;Astle,M.J. CRCHandbookofChemistryandPhysics , 62nd ed.; CRC Press: Boca Raton, FL, 1981 -1982. (17) Scriven, L. E. Chem. Eng. Sci. 1960,12,9 8-108.)jJt,i-Jt,i-1j Jt,i<0.005 (A-13) Jt)s¡g(JBânb)d¡g (A-14)1344J. Phys. Chem. B, Vol. 106, No. 6, 2002 Hu and Larson