Stability of viscoelastic dynamic contact lines: An experimental studyM. A. Spaid and G. M. Homsy Citation: Physics of Fluids (1994-present) 9, 823 (1997); doi: 10.1063/1.869480 View online: http://dx.doi.org/10.1063/1.869480 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/9/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Absolute instability in viscoelastic mixing layers Phys. Fluids 26, 014103 (2014); 10.1063/1.4851295 Lubricated extensional flow of viscoelastic fluids in a convergent microchannel J. Rheol. 55, 1103 (2011); 10.1122/1.3613948 Measurement of pressure loss and observation of the flow field in viscoelastic flow through an undulating channel J. Rheol. 44, 65 (2000); 10.1122/1.551080 Numerical study of viscoelastic phase separation in dynamically asymmetric systems AIP Conf. Proc. 469, 309 (1999); 10.1063/1.58515 Anomalous Migration of a Rigid Sphere in Torsional Flow of a Viscoelastic Fluid. II: Effect of Shear Rate J. Rheol. 29, 639 (1985); 10.1122/1.549820

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Stability of viscoelastic dynamic contact lines: An experimental studyM. A. Spaida) and G. M. Homsyb)Department of Chemical Engineering, Stanford University, Stanford, California 94305

~Received 8 July 1996; accepted 17 December 1996!

An experimental study of the rivulet instability associated with spin coating a circular drop of fluidis conducted to examine the effect of elasticity on the onset and evolution of the instability. The spincoating experiments are conducted with viscoelastic drops consisting of a high molecular weightpolystyrene in tricresyl phosphate~TCP!, as well as the Newtonian solvent TCP. Results show anunequivocal delay in the onset of the instability when the appropriate Weissenberg number issufficiently large, resulting in a larger coated area and more finger arms relative to Newtonianresults. Experiments performed with the viscoelastic fluid at low Weissenberg number exhibitsimilar behavior to those performed with the Newtonian solvent as expected. Additionally, thegrowth rate of the instability is reduced for experiments in which the elastic forces are important, inagreement with the perturbation theory of Spaid and Homsy@Phys. Fluids8, 460 ~1996!#,demonstrating that elastic forces have a stabilizing influence on the contact line instability. ©1997American Institute of Physics.@S1070-6631~97!03404-1#

I. INTRODUCTION

Interfacial instabilities occur in a wide variety of coatingflows which exhibit several common features. In many coat-ing applications, some external forcing~e.g., gravity, cen-trifugal force or boundary motion!, sets the contact line inmotion and causes the fluid to spread. As the coating liquidspreads over the solid, the advancing contact line inevitablydevelops modulations which rapidly grow into rivuletsthrough which the bulk of the coating fluid travels. Althoughheterogeneities in the wetting properties of the solid cancause rivulet formation, the effects of the surface character-istics are secondary to hydrodynamic considerations. Rivuletformation in coating flows is hydrodynamic in origin, and istherefore unavoidable even for surfaces where the variationin the surface properties of the solid substrate is slight.

In all of the experimental work on rivulet instabilities,the formation of a capillary ridge near the contact line isobserved as a precursor to the instability. A theoretical analy-sis of the instability of the Newtonian capillary ridge wasperformed by Troianet al.1 This analysis considered the sta-bility of the Newtonian capillary ridge in the lubricationlimit, relieving the contact line singularity with a precursorfilm boundary condition. Stability results show that the cap-illary ridge is always unstable to a range of spanwise distur-bances and predict a most amplified wavenumber whichagrees well with available experimental results.2–4 Spaid andHomsy5 examined the stability of the Newtonian capillaryridge using a slip boundary condition at the advancing con-tact line, and obtained results nearly identical to those ob-tained with the precursor film contact line condition. By per-forming an energy analysis, they demonstrate that themechanism of the instability is insensitive to the details ofthe contact line condition. The instability of the capillaryridge arises due to local variations in the fluid thickness,whereby thicker regions of fluid advance more rapidly over

the substrate. In addition to examining the Newtonian stabil-ity problem, Spaid and Homsy examine the rivulet instabilityfor slight viscoelasticity by perturbation theory in the Weis-senberg number. Results show a stabilization of the capillaryridge relative to the Newtonian case. Analysis of the pertur-bation results indicates that the primary mechanism for thestabilization is an elastic resistance to streamwise accelera-tion of fluid.

Several studies have examined the driven contact lineinstability with either gravitational2,6,7 or centrifugalforcing,3,4 the latter are relevant here. In spin coating experi-ments, a drop of fluid is placed on the axis of rotation of aspinning disk. The drop spreads axisymmetrically for a pe-riod of time before modulations appear at the contact lineand rivulets form. Experimental studies on spin coating arein good agreement with lubrication theories for the spreadingof Newtonian fluids. Me´lo et al.4 performed experiments us-ing silicon oil on a silicon substrate, and were able to con-firm the asymptotic theoretical spreading rateR(t);t1/4. Forsmall capillary numbers, defined in the usual way as Ca5mU/s, the wavelength of the instability was found to beconstant and independent of experimental parameters such asdrop volume and rotation speed. Fryasse and Homsy3 havestudied spin coating of both Newtonian and non-Newtonianfluids on glass, with no significant difference between thetwo. They suggest that the appropriate Weissenberg number,which is a measure of the elasticity of the flow, was toosmall to alter the Newtonian results. They also show that thecontact angle plays a role in determining the critical radius atwhich the drop becomes unstable, with non-wetting fluidsexhibiting a shorter onset time for the instability. Compari-son of these experiments with the linear stability theory ofTroian et al.1 showed good quantitative agreement for boththe wavelength and growth rate of the instability as long asthe onset time and critical radius are taken from the experi-ment.

The objective of this experimental study is to determinethe effect of elastic forces on the capillary ridge instability.As discussed by several authors, the instability in question is

a!Current address: National Institute of Standards and Technology, PolymersDivision, Gaithersburg, Maryland 20899.

b!Corresponding author.

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both hydrodynamic and local in its nature. It is found that thewavelength and growth rates are only weakly dependent onthe contact angle, with the dominant mechanism being thechange in resistance accompanying a perturbation in theheight of fluid near the capillary ridge.5 Furthermore, sincethe mechanism is a local one associated with the capillaryridge, quasistatic stability theories can be successfully ap-plied to flows produced by a variety of driving forces. Herewe choose to study the instability in the context of spin coat-ing, but our results are expected to apply in more generality.As discussed below, spin coating is a particularly attractiveexperimental setting to study the effect of elasticity on theinstability since, as we will see, the relevant Weissenbergnumber may be systematically varied over a significant rangeby adjusting the rotational frequency.

II. EXPERIMENTAL APPARATUS AND PROCEDURE

The experimental apparatus is described in detail in aprevious paper,3 therefore we only describe the main featureshere. Our spin coating apparatus has been designed to pro-vide a range of rotational frequencies and to allow continu-ous observation of the entire drop. A sketch of the apparatusis represented in Fig. 1. A circular glass plate of 14.6 cmdiameter is used as a substrate, which is supported by a cy-lindrical aluminum annular ring which rests on a set of twoball bearings and is coupled via a belt to a pulley mounted onthe shaft of a variable speed motor. A circumferential beltdrive leaves the central area of the set-up clear of mechanicalparts, which is essential for measuring film thickness throughthe use of dyed liquids and absorption measurements. ACCD video camera with a high speed electronic shutter isabove the set-up and focused on the glass plate. The camerais connected to a video-tape recorder and a laboratory micro-computer with a video digitizing board.

A positive-displacement microliter dispenser is used todeposit drops in a range of volumes from 90 to 100ml. Inorder to center the drop, we use a holder for our syringewhich has been specially machined to fit the cylindrical ro-tating part of the apparatus on the one hand and the syringebody on the other hand, with a total play less than 0.1 mm.To ensure reproducible surface characteristics for the glasssubstrate, the cleaning protocol of Fryasse and Homsy3 wasused.

The experimental procedure is as follows. The rotationspeed of the motor is first preselected. Using the microliterpipette and its holder, a drop having the chosen volume isdeposited on the motionless glass plate and it is allowed torest for at least 15 s before the motor is switched on in orderto let the fluid completely relax. This is particularly impor-tant for experiments with viscoelastic liquids. Spin-up is be-gun and each experiment is video-taped until the liquid hasalmost reached the edge of the substrate. Images are thendigitized from the tape, and analyzed on a Macintosh Quadra840AV.

III. FLUID PREPARATION AND CHARACTERIZATION

We studied spin coating of both viscoelastic and New-tonian fluids. The viscoelastic fluid is a polymer solutionconsisting of 1% by weight polystyrene~PS! ~Mw5203106

g/mol! dissolved in tricresyl phosphate~TCP!. The 1%TCP/PS polymer solution was prepared by first dissolvingthe PS in a known quantity of methylene chloride using amagnetic stirrer at low speed for approximately two days.TCP was then added to the solution, and the methylene chlo-ride, which is highly volatile at room temperature, was com-pletely evaporated by heating the solution slightly aboveroom temperature while continuously stirring at low speedfor 3 days. The Newtonian fluid is simply the solvent of thepolymer solution, TCP, although some validation experi-ments with the Newtonian fluid polydimethylsiloxane~PDMS! were performed to verify the reproducibility of pre-vious work.3,4

Comparison of the theoretical prediction of elasticstabilization5 would ideally involve spin coating of fluidwhich is well described by the Oldroyd-B constitutive equa-tion. In previous experimental work with Boger fluids,3 noapparent difference was observed for the viscoelastic fluidand its Newtonian solvent. However, the parameter govern-ing the viscoelastic correction,3,5,8 We(3Ca)1/3, was quitesmall ~,0.2!, suggesting deviations from Newtonian behav-ior would also be quite small. In these previous experiments,it was also determined that the contact angle plays a signifi-cant role in the fingering instability, with fluids of largercontact angle exhibiting an earlier transition to the instabil-ity, with fewer rivulets around the circumference of thespreading droplet. Therefore, it is desirable to examine aNewtonian/viscoelastic fluid pair with approximately equalcontact angles, which is the case for the TCP and TCP/PSsystem. The major drawback of the 1% TCP/PS fluid is thatit is not well modeled by the Oldroyd-B constitutive equa-tion, exhibiting both shear and normal stress thinning. How-ever, as we will see, the shear rate range of the experimentsis relatively small, leading to approximately constant shearviscosities over the course of an experiment. We also antici-pate that the primary mechanism of elastic stabilization,5 i.e.the decrease in streamwise acceleration of fluid, does notsignificantly depend on the particular constitutive model.

The main issue in designing an experiment to measurethe effect of elasticity on the stability of a spreading of fluidis to determine a combination of operating conditions andfluid properties in which one would expect elasticity to beimportant. With the TCP/PS fluid, there exists such a com-

FIG. 1. Sketch of the experimental apparatus.

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bination, which was the primary motivation for choosing thisparticular polymeric fluid. We estimate the effect of elastic-ity on the inner region by determining the magnitude of therelevant parameter governing first order elastic effects. Aspreviously discussed,3,5,8 this parameter is a local Weissen-berg number given by

x5lU0N / l , ~1!

whereU0N is the velocity of the advancing contact line,l isthe characteristic relaxation time of the fluid, andl is theinner region length scale. Note that with this choice of innerlength scale,x is identical to We~3Ca!1/3. The parameterx isexpected to be the relevant measure of elasticity in the vicin-ity of the contact line for any viscoelastic fluid, as it corre-sponds to the definition of the Weissenberg number, We5Ul/L, using the local inner length and velocity scalesappropriate to the hydrodynamics near the capillary ridge.Thus in any experimental study, the numerical value ofx isthe dimensionless measure of the importance of elasticity. Itis possible to expressx in terms of the experimental param-eters and fluid properties. The length scale observable in theexperiments is the radius of the dropR(t). Assuming thedrop approximates a spreading flat disk of fluid, we mayinfer the height of the drop through volume conservationaccording to

pR~ t !2h~ t !5V, ~2!

whereV is the volume of the drop. Using the volume con-servation condition, the viscoelastic parameterx may by re-lated to the measurable experimental parameters as follows:

x5lV5/3r4/3v8/3

3p5/3hs1/3R2 . ~3!

Note thatx is strongly dependent on the rotation speedv,indicating that elasticity becomes increasingly important asthe rotation speed increases. It should be noted thatx variesover the course of an experiment, decreasing like the squareof the drop radius. Typical values ofx are reported at the endof this section, after a discussion of the fluid properties. Tworheological quantities are necessary to estimatex, namely atypical relaxation timel of the fluid, and the shear viscosityh. We have studied the rheology of the TCP/PS fluid in orderto determine these fluid properties.

The shear viscosity of the TCP/PS fluid was character-ized on a Rheometrics Dynamic Stress Rheometer~DSR!using a parallel disk geometry at room temperature~20.5 °C60.1 °C!, and is shown in Fig. 2. Over three decades inshear rate~1022–10 s21!, the viscosity varies by a factor of5. However, as will be discussed, typical experimental con-ditions occur over a very limited shear rate range, resultingin only minor variations in viscosity. At very low shear rates~less than 0.05 s21! the viscosity is approximately constant atthe zero shear rate valueh0510 Pa s.

The relaxation time of the fluid was estimated using twodifferent techniques. The first technique was performed on aRheometrics Dynamic Analyzer II rheometer, using a coneand plate geometry. In this technique, the relaxation time isrelated to the decay time in the normal force in the cone andplate device upon cessation of steady shear~step shear ex-

periments!. Figure 3 shows the relaxation of the normalstress after achieving a steady state at a shear rateg515 s21.In Fig. 4 we plot the data on semi-log coordinates to extractinformation about the modes of relaxation. As viscoelasticfluids exhibit a spectrum of relaxation times, it is not surpris-ing that the curve in Fig. 4 is not perfectly linear. The dataindicate that there is a fast relaxation time occurring imme-diately after the cessation of steady shearing, followed by alonger relaxation time which characterizes the subsequentdecay in normal stress. We have fit the normal stress decaywith a double exponential of the form

y5a1be2t/l11be2t/l2. ~4!

As shown by the solid line in Fig. 3, the double exponentialcaptures the normal stress decay quite well with the longrelaxation timel153.3 s, and the short relaxation timel250.6 s. The long relaxation time measured by this methodwas found to be independent of the magnitude of the step inshear for steps ranging from 10–25 s21. As the characteristictime for rivulet formation for the TCP/PS experiments dis-cussed in this paper is long compared to the relaxation timeof the fluid, we adopt the long relaxation timel153.3 s asthe relevant relaxation time for the experiments. Unfortu-nately, the rheometer is not sensitive enough to performsimilar normal stress relaxation experiments for lower shearrates relevant to our experiments.

FIG. 2. Shear viscosityh and stresst as a function of the shear rateg.

FIG. 3. Decay in the normal stress after cessation of steady shear.

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We also measured low shear relaxation time of theTCP/PS fluid from dynamic measurements with the Rheo-metrics DSR. Using small amplitude oscillatory shear, it ispossible to determine the elastic and loss modulus,G8 andG9, respectively, as a function of the modulation frequencyv, as shown in Fig. 5. In the limit of low frequencies, thenormal stress coefficientc1 is related to the elastic modulusaccording to9,10

c152G8/v252lh0 . ~5!

The zero shear rate viscosity may be obtained from limitingvalues of the dynamic viscosityh8 according to

h05 limv→0

h85 limv→0

G8/v. ~6!

Using the results of the oscillatory shear experiments andequations~5! and ~6!, the relaxation time of the fluid wasfound to be approximatelyl51.5 s. Based on these twomeasurements, we estimate the longest relaxation time of theTCP/PS fluid to lie between 1.5 and 3.3 s. It should be notedthat only an approximate value of the relaxation time is nec-essary to interpret the experimental results.

Fluid properties relevant to the spin coating experimentsare summarized in table I for TCP and the polymer solutionTCP/PS. The surface tensions of the fluids was measuredusing the pendant drop technique in its simplestimplementation.11 We have also measured the macroscopicstatic contact angle of the fluids on glass by directly imagingthe profile of sessile drops. We found the static contact angleof the polymer solution TCP/PS to be approximately 5°larger than that of the Newtonian solvent TCP. Since onlysmall changes in both the surface tension and the contactangle are observed with the addition of the polymer, we con-clude that polystyrene is not surface-active in TCP.

We now consider the operating conditions for the experi-ments considered in this study, and calculate the viscoelasticparameterx for those performed with the TCP/PS fluid.Table II indicates the drop volumeV, rotation speedv andinitial radiusR0 for each of the experiments together with theestimates of the viscoelastic parameterx. The range ofxreported corresponds to the two estimates of the relaxationtime. As mentioned previously,x depends on time throughthe radius of the dropR(t). Accordingly we report the valueof x at two different times,t0 and tc , corresponding to theinitial radius of the dropR0 and the critical radiusRc atwhich the drop goes unstable, with a decrease inx by afactor of 3–4 betweent0 and tc . In calculatingx, the shear

FIG. 4. Semi-log plot of the normal stress decay.

FIG. 5. Elastic and loss modulus as a function of the modulation frequencyv.

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viscosity h was estimated based on the average shear rateoccurring during the experiment.

From these values, one expects elasticity to be importantfor the 550 rpm experiments~TCP/PS 1 and TCP/PS 2!,wherex is anO~1! quantity. Typical values ofx for the 350rpm experiment~TCP/PS 3! are 5–6 times smaller than forthe 550 rpm experiment, suggesting that near Newtonian be-havior is likely. Note that experiments TCP 2 and TCP/PS 2are repeat experiments which are reported to demonstrate thereproducibility of the results.

IV. EXPERIMENTAL RESULTS

A. Qualitative results

The features observed are similar to those of Meloet al.4

and Fryasse and Homsy.3 After placing the drop on the sta-tionary glass plate, the drop edge appears circular andspreads to an initial radius which depends on the drop vol-ume and static contact angle of the fluid. Starting the motorimpulsively brings the disk to a constant angular velocity,and the circular drop spreads under the action of the centrifu-gal force, increasing the radius of the drop. The fluid thick-ness is visualized with the help of the image analysis soft-ware, as the intensity of light absorbed through the drop isrelated to the local thickness of the free surface. Figure 6shows the evolution of a PDMS drop, where the gray level ofthe pixels comprising the image has been used to create asurface plot. This figure clearly shows the evolution of thefree surface from a spherical cap into a flat profile with acapillary ridge near the contact line. The drop spreads in aconcentric and axisymmetric manner for a time prior to theonset of the instability. From the figure, it is evident thatthere is an onset time for the instability, as the drop remainscircular for some time even after the capillary ridge forms.At this onset timetc , the contact line becomes modulatedand the free surface appears nearly polygonal. It is at thisstage in the evolution of the drop where the linear stabilitytheory1,5 is applicable, and exponential growth of the modu-lations is expected. The modulations are evenly and axisym-metrically distributed, and eventually grow into rivulets. Thenumber of fingers observed is closely tied to the onset timetcof the instability, corresponding to a critical radiusRc whenmodulations first appear, as drops with larger critical radiican accommodate more fingers of a given wavelength. Thusthe onset timetc and critical radiusRc are important experi-mental parameters which cannot be determined a priori bythe existing quasi-static linear stability theories. The finalstage in the evolution of the drop is the non-linear growth ofthe long and thin finger arms which eventually extend to theedge of the rotating disk.

It is useful to make some qualitative observations for theNewtonian fluid TCP and the non-Newtonian fluid TCP/PS.Previous work has established that the onset time and num-ber of fingers is independent of the drop volume and rotationspeed for Newtonian fluids. In light of these observations, wehave examined drops of approximately constant volume~93–98ml! in these experiments. For the Newtonian fluidTCP, we present results at a representative rotation speedv5400 rpm, while for the viscoelastic fluid TCP/PS, wepresent results atv5350 rpm andv5550 rpm. The tworotation speeds for the viscoelastic fluid were selected to in-vestigate the importance of elasticity, as the viscoelastic pa-rameterx increases with rotation speed, as previously dis-cussed.

Figure 7 shows the evolution of the contact line for atypical experiment with TCP at a rotation rate of 400 rpm,while Fig. 8 shows the evolution of the contact line forTCP/PS at 550 rpm. The most striking observation is the

FIG. 6. Free surface evolution of a PDMS drop. Drop volume: 98ml.Rotation rate: 300 rpm.

TABLE I. Properties of the fluids used in the spin coating experiments.

TCP TCP/PS

Viscositym ~Pa s! 0.085 see Fig. 2Densityr ~g cm23! 1.16 1.14Surface tensions ~dynes cm21! 4161 4061Static contact angleu 20–23° 25–27°

TABLE II. Experimental conditions and the Weissenberg numberx for thespin coating experiments.

Exp.number

Drop volumeml

Rotation speedrpm

Initial radiusmm

xt5t0

xt5tc

TCP 1 98 400 7.1 0 0TCP 2 97 400 7.2 0 0TCP/PS 1 93 550 6.1 1.7–3.7 .5–1.1TCP/PS 2 94 550 6.6 1.4–3.1 .4–.9TCP/PS 3 95 350 6.2 0.3–0.7 .1–.2

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difference in the number of fingers for the two cases. Threedistinct fingers develop for the TCP experiment, while 7–8major fingers are visible in the TCP/PS experiment, withsmaller fingers developing well after the onset of the insta-bility. It is also apparent that the critical radius of the insta-bility, i.e. the radius at which the contact line loses its circu-lar shape, or equivalently the onset time, is larger for theTCP/PS experiments. Thus, the TCP/PS drop coats a largerarea before the onset of the instability relative to the New-tonian solvent TCP.

We now consider an additional experiment for TCP/PS,at a lower rotation speed~350 rpm!. Figure 9 shows thecontact line evolution of the drop, which eventually develops5 fingers, with the central part of the drop appearing nearlypentagonal. We observed that a significant reduction in therotation speed with TCP/PS resulted in a reduction in thenumber of fingers. Typical experiments with TCP/PS at 350rpm resulted in 4–5 fingers, as in Fig. 9, while results at 550rpm yielded 6–8 fingers. Comparison of the TCP/PS experi-ments at 550 rpm with the TCP/PS experiment at 350 rpmalso indicates a change in the critical radius of the instability,with the onset occurring at a smaller radius for the 350 rpmexperiment. It should be noted that such dependencies re-garding the number of fingers and critical radius on the ro-tation speed have not been observed in previous experimen-tal studies3,4 for Newtonian fluids, further indicating theobservations are influenced by elasticity.

The non-linear evolution of the fingers appears to bequite similar for all of the experiments. We observe littlemodification in the shape and thickness of the fingers, indi-cating that although elasticity delays the onset time of theinstability, the ultimate finger thickness and shape are notsignificantly altered by elastic forces.

B. Quantitative results

In this section, we outline the analysis of a typical ex-periment, which was developed and described in the paperby Fryasse and Homsy.3 Using standard edge detection soft-ware, we are able to determine the position of the contactline with a precision of a few tenths of millimeters. Analysisof a typical experiment involves locating the contact line fora sequence of images at successive instants in time as shownin Figs. 7–9. Prior to the instability, we measure the averageradius of the drop as a function of time, yielding the spread-

FIG. 7. Contact line evolution of a drop of TCP~drop volume 98ml, rota-tional rate 400 rpm!, experiment TCP 1.

FIG. 8. Contact line evolution of a drop of 1% TCP/PS~drop volume 94ml,rotational rate 550 rpm!, experiment TCP/PS 2.

FIG. 9. Contact line evolution of a drop of TCP/PS~drop volume 95ml,rotational rate 350 rpm!, experiment TCP/PS 3.

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ing rate. Analysis of the drop after the onset of the instabilityinvolves the Fourier transform of the deformation of the con-tact line relative to a circle centered on the axis of rotationcovering an equal surface area as the drop, as follows:

Dk5 (n50

N21

DS 2pn

N,t De2p ik~n/N!, ~7!

wherek is the azimuthal wavenumber andN is the numberof data points comprising the deformation curve.

Figure 10 is a typical power spectrum plot correspondingto the contact line evolution data of Fig. 7. The most ampli-fied mode for this experiment isk53, verifying what is quiteobvious from the contact line data where three fingers areobserved, with harmonics and less amplified modes appear-ing at later times.

The onset of the instability is characterized by a criticaltime tc which corresponds to a critical radiusRc , derivedfrom the experimental data. We impose a criteria for thecritical radius and critical time at those values at which thedeformation of the drop exceeds 10% of the average radiusof the drop, which in turn is at least five times the pixelresolution of the drop edge, indicating that pixel noise is notimportant in determining the critical time. Figure 11 showsthe time evolution of the amplitude of several modes for the

TCP experiment corresponding to Figs. 7 and 10. In thisparticular experiment, the critical time was found to be 2.5 s.

The dimensional exponential growth rate of the instabil-ity is calculated by plotting the mode amplitude versus timeon a semi-log scale. Figure 12 is a semi-log plot of the modeamplitude results shown in Fig. 11, where we have indicatedthe best linear fit for modek53 for the time interval 2.5–3.5s, where the data are approximately linear. The slope ofthe line gives the dimensional exponential growth rateb53.46 s21.

Each experiment was analyzed according to the preced-ing method. Table III summarizes the results of the analysisfor each of the experiments, where the critical time, criticalradius, number of fingers and the growth rate of the instabil-ity are reported.

V. ANALYSIS AND DISCUSSION

In this section, we analyze the quantitative results inorder to draw conclusions as to how viscoelasticity affectsthe spin coating process. As shown previously, the TCP/PSpolymer solution is shear-thinning, and it is necessary to es-timate the degree of viscosity variation over the course of anexperiment. The shear rate based on the thickness of thespreading drop may be determined from the location of theradius of the drop as a function of time, by assuming thedrop profile is approximately a flat disk, whereh(t)5V/pR(t)2. The velocityU of the advancing contactline may be calculated by dividing the radial distance trav-eled for a given time interval. The variation of the character-istic shear rateU/h may then be used to determine the varia-tion in the shear viscosity over the course of an experiment.

FIG. 10. Power spectrum for experiment TCP 1.

FIG. 11. Mode amplitude versus time for experiment TCP 1.

FIG. 12. Semi-log plot of the mode amplitude for experiment TCP 1.

TABLE III. Critical time, critical radius, number of fingers and growth rateof the instability for each of the experiments.

Exp. numberCritical time

tc ~s!Critical radiusRc ~mm!

Number offingersNf

Growth rateb ~s21!

TCP 1 2.5 9.3 3 3.46TCP 2 2.7 8.8 3 2.7TCP/PS 1 40 11.5 6 .14TCP/PS 2 25 12.1 7 .17TCP/PS 3 95 9.5 4 .044

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As an example, consider the experiment TCP/PS 2~see Fig.8! for which the rotation speed is 550 rpm. Figure 13 showsa plot of the shear rateU/h as a function of time for 10s,t,30 s. This time interval describes the spreading of thedrop from the initial development of the inner and outerregions through the critical time of the instability. Over thistime interval, the shear rate is approximately constant at avalue of 0.8 s21, indicating that the shear viscosity also re-mains approximately constant. From the rheometric data, weestimate the maximum variation in viscosity over the courseof the experiment to be 10%. We conclude that the effect ofshear-thinning rheology for the TCP/PS experiments issmall, and any deviation from Newtonian spin coating resultswill most likely arise as a result of fluid elasticity.

Spreading of the drops is governed by the balance be-tween viscous and centrifugal forces in the outer region. It ispossible to compare the evolution of the drop radius as afunction of time for the experiments by choosing character-istic scalings appropriate for the outer region. The radiusR(t) of the drop is the appropriate length scale for the outerregion, and we non-dimensionalize the radius data with theinitial radius R0. A characteristic time scale for spreadingmay be obtained from the characteristic initial velocityU0and the initial radiusR0:

3

T0'R0

U0'

p2hR04

rv2V2 . ~8!

In equation~8!, we have once again assumed that the dropprofile approximates a planar disk with volumeV. In Fig. 14we show the dimensionless radiusR* as a function of thedimensionless timet* for each of the experiments, alongwith the lubrication theory solution which assumes a flatprofile for the drop and predictsR* (t)5(11 4

3 t* )1/4. The

dimensionless time has been uniformly shifted by an arbi-trary amount for each experiment in order to form a mastercurve. Uniform shifts in the data are justified since changesin the initial conditions for each experiment can lead to dis-placements along the abscissa from one experiment to thenext. The asymptotic spreading rate may be attained at dif-ferent dimensionless timest* for each of the experiments,depending on the initial configuration.

The data in Fig. 14 appear to be approaching theasymptotic spreading rate, but never reach it due to the onsetof the instability, corresponding to the termination of eachcurve. The discrepancy between the TCP and TCP/PS ex-periments most likely results from variation in initial con-figurations due to the difference in contact angle: however inlight of the large difference in viscosity between the twofluids, the agreement is relatively good. We do not observeany effect of elasticity on the spreading of the TCP/PS drops,to the accuracy of the experiments. These results are inagreement with the perturbation theory of Spaid andHomsy,5 where viscoelastic corrections to the asymptoticspreading rate of drops become vanishingly small after a fewcharacteristic relaxation times of the fluid. As seen fromtable III, the critical times for the TCP/PS experiments are onthe order of~10–30!l.

We now consider the extent of spreading for each of theexperiments, as measured by the ratio of the initial radius tothe critical radius,Rc/R0 . This ratio provides quantitativeinformation regarding the extent of spreading prior to theonset of the instability. Fryasse and Homsy3 found this ratioto be independent of all experimental parameters except thecontact angle. For non-wetting fluids, the ratioRc/R0 wasmuch smaller than for wetting fluids, suggesting that increas-ing the contact angle of the fluid is destabilizing. As men-tioned previously, the static contact angle of the polymersolution TCP/PS is approximately 5° larger than that of theNewtonian solvent TCP. Based on the contact angle aloneand in the absence of visco-elasticity, one would expect theratio Rc/R0 to be larger for the TCP experiments relative tothe TCP/PS experiments. In Table IV we show the ratioRc/R0 for each of the experiments. The results indicate thatsignificantly more spreading occurs prior to the onset of theinstability for the TCP/PS fluid at the high rotation speed~550 rpm!, in contradiction to what one would expect from aconsideration of the effect of contact angle variation. Theratio of Rc/R0 for the 350 rpm experiment~TCP/PS 3! wasfound to lie between the TCP results and the TCP/PS resultsat 550 rpm. Both results are consistent with a stabilizingeffect of elasticity. For the 550 rpm experiments with

FIG. 13. Shear rate of experiment TCP/PS 2 as a function of time. FIG. 14. Dimensionless time evolution of the drop radius for the experi-ments. The curves have been uniformly shifted in time by an arbitraryamount, in the attempt to form a master curve.

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TCP/PS where the viscoelastic parameterx is the largest, wefind the largest value ofRc/R0 . The visco-elastic parameterx is substantially smaller for the 350 rpm experiment, con-sistent with the reduction ofRc/R0 relative to the 550 rpmexperiments. The smallest values ofRc/R0 occur for theNewtonian solvent wherex50.

The number of fingers observed in a particular experi-ment also provides information regarding stability. The num-ber of fingers is linked closely to the critical radius, as dropswith larger critical radii can accommodate more fingers of agiven wavelength. Linear stability theory1,5 predicts the mostamplified wavelength as

l;14l , ~9!

where l is the length scale appropriate for the inner region.As discussed above, the capillary ridge in the experiments istime dependent, and therefore can only be compared to thequasi-static theory when the critical time for the onset of theinstability is known. We therefore compute the capillarylength scale based on the measured critical time as

l ~ tc!5hN~ tc!@3Ca~ tc!#21/35

shN~ tc!

rv2RN~ tc!5

sV

prv2Rc3 .

~10!

We can relate the number of fingersNf predicted by thetheory to the most amplified wavelength as follows:

Nf52pRc

l5

p

7Rc2S prv2

sV D 1/3. ~11!

Table IV compares the theoretical number of fingers pre-dicted by equation~11!, obtained from a knowledge of thecritical radiusRc together with the linear stability theory, tothe experimental number of fingers inferred from the spectralanalysis described in section IV B. We found reasonableagreement between the theory and experiment, indicatingthat the linear stability theory accurately predicts the mostamplified wavenumber when the critical radius is inferredfrom the experiment. In the experiments, the selected wave-number must produce an integer number of fingers. Thetheory is obviously not subject to such a restriction, and anywavenumber is possible regardless of whether an integernumber of fingers is selected. Therefore, it is not surprisingthat the experiment and theory may differ by 1–2 fingers.Fryasse and Homsy3 also found good agreement for the the-oretical and experimental most amplified wavelength forboth PDMS and Boger fluids. It does not appear that elastic-

ity alters the wavelength of the instability, at least to theaccuracy of the experiments, consistent with our theoreticalresults for smallx.5

We have also compared the growth rates of the instabil-ity for each of the experiments. The dimensional growthratesb~s21! vary from 0.044 s21 to 3.46 s21, reflecting thewide range of fluid properties and rotation speeds consideredin this study. Before drawing conclusions, it is necessary tomake the growth rates dimensionless. We have compared thedimensionless growth rates using two different time scalest0andt1 defined as

t05hN~ tc!/U~ tc!, t15 l ~ tc!/U~ tc!. ~12!

The first time scalet0 is the the time it takes the advancingfront to travel the average thickness of the spreading drop,while the second time scalet1 is a characteristic time for thefront to travel the length of the capillary ridge. We havecompared the dimensional growth rates using both of thesetime scales.t1 is the time scale appropriate for quantitativecomparison with the linear stability theory, whilet0 is a timescale which does not require estimation of fluid properties.The two time scales are related byt05~Ca!1/3t1, and sinceCa!1 it is evident thatt0,t1. The characteristic timet0 iscompletely determined from the experimental data where thevelocity of the contact line at the critical time is computedfrom the data, and thus no fluid parameters such as viscosity,density or surface tension are required in computing it. Al-thought0 is not the time scale used in the stability analysis,it is a convenient time scale to employ, as it is independentof the fluid properties. Computing the time scalet1 requirestwo fluid parameters, namely the viscosity and surface ten-sion. For the viscoelastic experiments, we estimate the vis-cosity based on the shear rate at the critical time.

Table IV summarizes the results for the dimensionlessgrowth ratesbt0 andbt1. We observe a significant reductionin the growth rates for the TCP/PS experiments at 550 rpmrelative to the Newtonian results and the TCP/PS result at350 rpm. It is interesting to note that the growth rates madedimensionless witht0 where the relevant length scale is thethickness of the spreading film agree closely with the linearstability theory, in which dimensionless growth rates are ap-proximately 0.5, while those based on the capillary lengthscale are 4–6 times larger than expected by the theory. Thisdiscrepancy is largely due to an overprediction of the char-acteristic spreading velocityU(tc) from what is observedexperimentally, as the drops go unstable before theasymptotic spreading theory is accurate. Nevertheless, scal-ing the data with either time scale allows the conclusion ofsmaller growth rates for the 550 rpm experiments in whichviscoelastic effects are important. The results show no appar-ent difference between the TCP/PS results at 350 rpm andthe Newtonian results for TCP. As the viscoelastic parameterx is quite small~0.1–0.2 att5tc! for the TCP/PS experimentat 350 rpm, no deviation from Newtonian behavior is ex-pected.

We now address possible comparisons between the ex-perimental results and our theory.5 Our predictions of theeffect of viscoelasticity on the rivulet instability are for aspecific constitutive relation—the OldroydB equation—and

TABLE IV. Ratio of the critical radius to the initial radiusRc/R0 , and acomparison of the number of fingers and dimensionless growth rates foreach of the experiments.

Exp. Number Rc/R0

Nf

ExperimentalNf

Theoretical bt0 bt1

TCP 1 1.3 3 5 0.46 2.6TCP 2 1.2 4 4 0.50 3.0TCP/PS 1 1.9 8 6 0.25 0.97TCP/PS 2 1.8 9 7 0.24 0.97TCP/PS 3 1.5 4 4 0.46 2.6

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are perturbation theory results valid for smallx. Quantitativecomparison between theory and experiment must address~i!the degree to which the theoretical predictions are indepen-dent of the details of the constitutive equation, and~ii ! theapplication of a perturbation theory results to experimentsdone at finitex. With regard to the first issue, as we show inRef. 5, the primary mechanism of stabilization is due to theviscoelastic resistance to stream-wise acceleration of fluid,which we expect to be robust to changes in the constitutivebehavior. Any rivulet formation involves this stream-wiseacceleration, which is kinematic in origin, and any viscoelas-tic fluid is expected to have a resistance to it whose originlies in the finite relaxation time. With this perspective, wecan reasonably expect any constitutive model to give predic-tions that are quantitative, at least in order of magnitude.Accordingly, our predictions5 show that the dimensionlessgrowth rate,b, normalized by its Newtonian value, is ap-proximately

b

bN51.02cx1O~x2!, ~13!

wherec is anO~1! quantity dependent on the model used torelieve the contact line singularity. Our experimental resultsare in qualitative agreement with these predictions, indicat-ing a 50% drop in the growth rate whenx is near unity@TCP/PS 1–2#, and a negligible drop whenx is 0.1–0.2@TCP/PS 3#. Since the fluid used is not well-modeled by theOldroyd-B equation, and the values ofx in the experimentexceed the range where the perturbation analysis is likely tobe quantitatively accurate, nothing more than qualitativeagreement in order of magnitude can be concluded.

VI. SUMMARY

We have performed a number of spin coating experi-ments in order to determine what effect elasticity has on therivulet instability associated with forced contact line motion.We performed the spin coating experiments with a polymericfluid consisting of 1% by weight polystyrene dissolved intricreysl phosphate, for which it is possible to generate ex-perimental conditions in which elasticity is important. As agauge for determining whether elasticity is actually affectingthe spin coating process, we performed spin coating experi-ments with the Newtonian solvent TCP. Additionally, it ispossible to examine the effect of elasticity by performingexperiments at different rotation speeds with the TCP/PSfluid.

The experimental evidence suggests that elastic forcesstabilize the capillary ridge. Qualitatively, it is quite obviousthat the experiments performed with the polymeric fluidTCP/PS at the high rotation speed~550 rpm! coat a largerarea of the disk before becoming unstable relative to experi-ments performed with either the Newtonian solvent TCP, orthe polymeric solution TCP/PS at the lower rotation speed~350 rpm!. Although elastic forces stabilized the drop by

increasing the radius at which modulation first appears, thenon-linear evolution of the fingers appears qualitatively simi-lar to Newtonian results.

In our quantitative analysis, we showed that the criticalradius is substantially increased when elastic forces are im-portant. The ratio of the critical radius to the initial radiusRc/R0 provides a measure of the extent spreading prior to theonset of the instability. We found that the ratioRc/R0 is thelargest for the TCP/PS experiments performed at 550 rpm inwhich elastic forces are important. For the TCP/PS experi-ments performed at 350 rpm, the ratioRc/R0 is significantlysmaller, indicating that when elastic forces are not as impor-tant, Newtonian behavior is recovered.

We compared the experimentally determined number offingers to that predicted by the linear stability theory. All ofthe experiments were in agreement with the theory when thecritical radius is obtained from the experiment itself. Al-though elasticity stabilizes the TCP drops by increasing thecritical radius at which the drops become unstable, it doesnot appear that elastic forces alter the wavelength of the in-stability.

The dimensionless growth rates of the instability alsopoint to a stabilization due to elastic effects. Two time scaleswere chosen to make the growth rates dimensionless, bothindicating a decrease in the growth rate in the instability forthe TCP/PS experiments performed at 550 rpm. The reduc-tion of the growth rate of the instability is in qualitativeagreement with the corrections to the eigenvalues due to firstorder effects of elasticity.

ACKNOWLEDGMENT

This work was supported by the U.S. Department of En-ergy, Office of Basic Energy Sciences through Grant No.DE-FG03-87ER13673.

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