Pattern formation in drying drops - University of Michiganrddeegan/PDF/Coffee_Rings_2_PRE.pdf ·...

PHYSICAL REVIEW E JANUARY 2000VOLUME 61, NUMBER 1

Pattern formation in drying drops

Robert D. Deegan*James Franck Institute, 5640 South Ellis Avenue, Chicago, Illinois 60637

~Received 24 November 1998!

Ring formation in an evaporating sessile drop is a hydrodynamic process in which solids dispersed in thedrop are advected to the contact line. After all the liquid evaporates, a ring-shaped deposit is left on thesubstrate that contains almost all the solute. Here I show that the drop itself can generate one of the essentialconditions for ring formation to occur: contact line pinning. Furthermore, I show that when self-inducedpinning is the only source of pinning an array of patterns—that include cellular and lamellar structures,sawtooth patterns, and Sierpinski gaskets—arises from the competition between dewetting and contact linepinning.

PACS number~s!: 47.54.1r, 05.45.2a, 47.20.Ma, 68.45.Gd

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I. INTRODUCTION

A fluid droplet on a solid surface is ostensibly so simplephysical system that one might suppose that all its behais thoroughly understood. Despite countless studies goback at least 200 years@1#, issues about the phenomena ocurring at the contact line, defined as the line beyond whthe solid is wet, continue to engage the interest of the sctific and engineering communities. One such issue is conline pinning @2,3#. Looking through a window after a rainstorm, one is struck by the fact that droplets, though overtical surface, nonetheless defy gravity. The force tholds them in place arises from a pinning of the contact lby irregularities, such as roughness or chemical heterogity, on the surface of the glass@4#. Here I explore how thecontact line can become pinned due to solute within theuid that corrupts the substrate. The capacity of solute to athe surface properties of the substrate is well known. Hoever, in ring-forming drops it achieves a new scalestrength because the solute preferentially accumulates acontact line, where it exterts the greatest influence onmotion of the contact line.

This work is an extension of an earlier study in which mcolleagues and I developed and experimentally testetheory for the formation of rings in drying drops@5,6#. Acommon manifestation of this phenomenon is the brown rleft when a drop of coffee dries on a counter top@see Fig.1~a!#. We found that contact line pinning and evaporationsufficient conditions for ring formation. Since these are comon and generic conditions, ring formation often occuwhenever a liquid with solid constituents evaporates. Tring forms because the contact line cannot move. Therefwhen evaporation removes liquid from around the contline, a flow develops to keep the substrate wet up to tpoint @see Fig. 1~b!#. The solute in the drop is dragged to thcontact line by this flow, where it accumulates to form tringlike deposit that remains after all the liquid evaporate

The substrate by itself cannot keep the contact line pinindefinitely. In Refs.@5,6# we speculated that the accumultion of solid components at the contact line perpetuates

*Electronic address: [email protected]

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pinning of the contact line. Here I will refer to this process‘‘self-pinning’’ because of the bootstrapping that would bintrinsic to ring formation: some preexisting conditions othe substrate temporarily anchors the contact line; this pmits the ring to start growing, and the additional growincreases the energy barrier the contact line must surmbefore moving. In this paper I will show that self-pinnindoes occur, and that it gives rise to pattern formation.

II. SELF-PINNING

In order to test the hypothesis of self-pinning, I isolatthe effects of solute pinning by using a smooth homogensubstrate. Freshly cleaved mica provides such a surface.substrates were prepared from a slab of mica by prying oa corner and introducing a drop of deionized water. The wter cleaves the interlayer potassium bond, and if done prerly produces an atomically flat surface. In order to minimcontamination of the surface, the amount of water used ding the cleaving process was kept as low as possible.sheets were dried in open air~this typically took less than 20s!, briefly exposed to the flame of a Bunsen burner, aplaced on a copper plate to cool. The sheets were usesoon as their temperature returned to normal.

FIG. 1. ~a! A photograph of a dried coffee drop. The dark primeter is produced by a dense accumulation of coffee particThe radius is approximately 5 cm.~b! Schematic illustration of theorigin of the advective current.~i! If the contact line were notpinned, uniform evaporation would remove the hashed layer,interface would move from the solid line to the dashed line, andcontact line would move fromA to B. However, if the contact lineis pinned then the retreat fromA to B is not possible, and there musbe a flow that replenishes the lost fluid.~ii ! Shows the actual motionof the interface and the compensating current.

475 ©2000 The American Physical Society

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476 PRE 61ROBERT D. DEEGAN

Observations of the behavior of filtered deionized wadeposited on the mica attest to smoothness and homogeof the substrate.~As with all fully wetting liquids, in additionto the macroscopic contact line there is also a microscoline due to the precursor film that extends beyond the mroscopic contact line. The contact line referred to herein walways be the macroscopic contact line.! A drop with aninitial volume of 0.5ml was placed on the mica substratand its radius versus time was recorded. These data areted in Fig. 2. Initially the drop grew until reaching a maxmum radius of 2 mm, and thereafter it shrank continuouslyzero size. The smoothness of the data in Fig. 2 is a meaof the smoothness of the substrate. Repeated trials indicthat the substrates contained approximately one pinningper cm2.

The drops in the experiments that follow were made fra 2% solid volume fraction colloidal suspension of sulfaterminated polystyrene microspheres dyed yellow-grfluorescent acquired from Interfacial Dynamics, PortlaOregon. The colloid was synthesized by a surfactant-fmethod which minimizes the presence of amphiphilic mecules. I used two different microsphere diameters: 1 andmm. Volume fractions different from 2% were made by dluting the colloid with deionized water, or by sedimentinthe microspheres in a centrifuge and decanting the clearuid.

When placed on mica, a drop of colloid will grow quicklto its largest size, much like the drop of pure water shownFig. 2. Unlike with pure water, the drop of colloid remainsits maximum size for a substantial fraction of the dryitime. Nadkarni and Garoff@7# showed that a single microsphere attached to the surface can pin the contact line.same process doubtlessly occurs here but at a much strolevel; the microspheres jam into the wedge of fluid nextthe contact line, preventing it from retracting. A ringformed during this initial pinned stage. At late times, tliquid pulls away from the ring and shrinks down to zeradius. During this phase, segments of the contact line ctinually switch between a pinned state and a depinned sleaving a trail of deposits which can be highly regular. Tdepinning process begins with the nucleation of a dry salong the inner edge of the ring~see Fig. 3!. Gradually moredry spots develop until the contact between the liquid andring is completely severed. I measured the fraction of ti

FIG. 2. Plot of the radius of a water drop~mm! drying on a micasubstrate vs time.

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during which the drop remained pinned and the width ofthinnest portion of the ring.

To measure the time of the initial pinned state, the drwas dried on an analytical balance and the mass as a funof time was recorded. A magnified image of the drop wacquired using a CCD camera, and was used to visumonitor the state of the drop. The depinning timetd wasmeasured when the first hole forms. The effective total ding time of the drop,t f , was determined by a linear extrapolation of the mass versus time plot to zero, as shown in F4~a!. Since the evaporation rate depends only on the radiuthe drop,t f is equal to the total drying time that would bobtained if the drop remained pinned throughout its lifetimThe depinning time, normalized byt f , versus the initial solidvolume fractionf, is plotted in Fig. 4~b!. The data are ap-proximately linear on a log-log scale, and a best fit topower law yields an exponent of 0.2660.08. For concentra-tions below 0.1% there was insufficient contrast in the vidimage of the drop to determine the depinning time.

The self-pinning process was also characterized by msuring as a function of initial concentration the width of thring at its narrowest point. When a depinning event occthe ring at that point ceases to grow, and so the thinnportion of the ring indicates where the first depinning eveoccurred. The volume of each drop varied because of incuracies intrinsic to pipetting small volumes. Therefore,width of the ring,wd , was normalized by the drop radiusRto compensate for different size drops. Figure 5~a! shows a

FIG. 3. Photographic sequence demonstrating the formationhole. The view is from above, and the solid white band in the lowpart of the frame is the ring; the rest of the drop is above the riThe hole is created, expands, and is eventually contained byaccumulation of microspheres along its edge. The portion ofring next to the hole remains wet throughout the period coveredthe figure. The flow inside the drop remains outward directed ewhen the contact line is retreating.~The backward motion of someof the larger particles appears to contradict this statement. Howethese particles are clamped between the liquid surface and thestrate so that their motion is governed by the the interface’s motand hence by the contact line’s motion, instead of by the fluid flo!The times for frames~a!–~f! are t50, 0.23, 0.50, 0.83, 1.87, an5.90 s, respectively. The major axis of the hole is approximat150 mm.

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PRE 61 477PATTERN FORMATION IN DRYING DROPS

plot of wd /R versusf. For drops of equal volume the aveage radius maximum showed a slight concentration depdence, increasing by about 10% when the concentrachanged by a factor of 10. The different data sets plotcorrespond to 0.1- and 1-mm particle sizes. The two data secoincide and, therefore, for clarity they are offset from eaother by multiplying the 0.1-mm data by a factor of 5. I wasunable to obtain data for particle sizes outside of this rabecause larger particles sediment too quickly and smaparticles are commercially unavailable in a surfactant-fsuspension. Both data sets are well fit by a power law whfor the 0.1-mm beads gives an exponent of 0.7860.10 andwhich for the 1-mm beads gives an exponent of 0.8660.10.Within the experimental uncertainty these two exponentsequal.

If changing the size of the drop only changes its scalenot its shape, then the width of the ring should scale withradius. That is, the volume of the wedge-shaped ring willproportional to Rw2, and this volume must contain aamount of solute proportional to volume of the dropR3 so

FIG. 4. ~a! Mass of a drying drop vs time. The early time daare suppressed because it contains transients due to the settlingof the balance. The arrows indicate the depinning time, labeledtd ,and the extrapolated drying time, labeledt f . The depinning time isdetermined by the first appearance of a hole, as in Fig. 3, neacontact line.~b! Depinning time normalized by the extrapolatedrying time vs concentration. The line running through the databest-fit to a power law, which yields an exponent of 0.2660.08.

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thatw}R. To explicitly test this assumption, I measuredwdandR for drops of different sizes but the same concentratiThe results plotted in Fig. 5~b! demonstrate that thewdscales withR over this range of radii.

It is not readily apparent that the width data of Fig. 5 aconsistent with the depinning time data of Fig. 4. A rouestimate of the width as a function of time can be obtainby assuming that the ring is an annulus with a cross secshaped like a right triangle~i.e., a wedge!. Therefore, itsvolume is

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FIG. 5. ~a! The width of the ring at depinning normalized by thdrop radius vs the concentration. The upper data set is for0.1-mm microspheres, and the lower one is for the 1-mm micro-spheres. The 0.1-mm data set has been multiplied by 5 to separthe two data sets. A best fit to a power law gives an exponen0.7860.10 for the 0.1-mm microspheres and 0.8660.10 for the1-mm microspheres. The crosses~3! correspond to the width calculated using the model given in the Appendix and using the dfrom Fig. 4~b!. ~b! The width at depinning for drops of differensizes but of equal initial concentration (1.5631023 volume frac-tion! plotted against the radius. The line running through the data fit to the functional formw5aR.

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Using the experimental resulttd;f0.2660.08 yields wd;f0.6760.05. This result is consistent within experimental ucertainty with the width data of 0.1-mm microspheres but nowith the width data of the 1-mm microspheres.

Equation~3! underestimates the growth rate of the ribecause the ring’s actual shape is different from that owedge. Consider the innermost edge of the ring to beinterface between a solid~packed microspheres! and a liquid.The speed with which the interface moves inward is demined by how fast microspheres arrive and how highly thcan be stacked. The height of the stacking depends onangle of the liquid surface relative to the substrate; a larangle can accommodate a higher stacking of microspheSince this angle must be smaller than the initial contact anand is, furthermore, decreasing in time, the ring will be widthan that calculated above. A more precise calculation forring width which explicitly includes the opening anglegiven in the Appendix. For each of the data points (f,td) inFig. 4~b!, the differential equations in the Appendix wenumerically integrated fromt50 throught5td , usingf asan input parameter. The crosses in Fig. 5~a! correspond tothe results of this integration. The agreement betweendepinning time and width data is good.

These experimental results demonstrate that self-pinndoes occur and that it has a predictable dependence oninitial concentration. It is worth emphasizing that contact lipinning due to the accumulation of microspheres is expecand has already been amply demonstrated by NadkarniGaroff @7#. What is unusual about the pinning process in tsystem is that it arises from a self-organization of the pinnsites.

The formation of a hole is the first stage of depinning.subset of possible mechanisms for its formation are dynacal instabilities. In order to determine the relevance ofnetic effects, I reduced the rate of evaporation so thatnamical quantities such as flow velocity and temperatgradients were reduced. To accomplish this, a dropplaced on a mica substrate and covered with a microscslide in which a bowl about 2 cm wide and 2 mm deep wcarved out. The drop was thus surrounded by a glasstainer which vented to the atmosphere solely throughgaps between the glass slide and the substrate. The drop20 times longer to dry than an identical drop drying in topen air. I found that the first hole appeared in the sldrying drop at the same effective time,t/t f , as its rapidlydrying counterpart. I therefore concluded that hole formatis not a dynamical effect~see Fig. 6!.

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The apparent lack of a dynamic signature at the onsehole formation leaves open the possibility of a static cririon. The spontaneity with which holes form is reminisceof nucleation and growth phenomenon. This suggests a scriterion might be based on energetic considerations. Hever, the ingredients in the energy balance are unclear,sidering that the depinning in a ring-forming drop is unusuin several respects. First, though the free energy of a wdrop will decrease as it spreads over mica, the retracshown in Fig. 2 is what one would expect if spreading weenergetically unfavorable. In other words, though water wmica it behaves as if though it were only partially wettinSecond, the depinning of a contact line from a surface asity only involves a distortion of the contact line whereas tdepinning in a ring-forming drops involves the creation onew contact line. Third, though the hole clearly has a mroscopic contact line, the region within the hole is coatwith a thin layer of liquid, i.e., the precursor film. Any explanation of the depinning must account for these effects

Using the data of Fig. 5, I searched for a static criteribased on the geometrical proportions of the ring. I usedmodel in the Appendix to calculate the height of the ring,h,and the angle of the free surface with respect to the substQ. In Fig. 7~a!, h versusf is plotted. These data show thathis a strong function off. In Fig. 7~b!, the h versusw isplotted. The data are well described by a power law to wha best-fit yields an exponent of 0.8560.05. Sincedh/dw'Q this indicates thatQ;w20.1560.05 which together withthe dependence ofw on f givesQ;f0.1260.05. Since this isa weak function off, it appears that depinning occurs onQ reaches some threshold value.

III. PATTERN FORMATION

The focus of Sec. II was the pinning force on the contline exerted by the deposited solute. During this early st

FIG. 6. ~a! The deposit left by a drop dried in the open a~b! The deposit left by a drop, made from the same solution as~a!,dried in the chamber described in text. The drying time is 20 timlonger for ~b! than for ~a!. The images have the same scale andthe photographs indicate that~b! is approximately twice as wide a~a!. This is the result of the drop in~b! spreading for a longer timeperiod than~a! before evaporation truncated the spreading proceAlso, the images indicate that the sizes of the arches in~b! are muchlarger than those in~a!. The diameter of the deposit in~a! is ap-proximately 5 mm. The scales in~a! and ~b! are the same.

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of the drop’s life the pinning force is so large that it is soleresponsible for the contact line behavior. Uncomplicaconstraints produce simple structures: a ring. This secshifts the focus to the events that occur after the contactseparates from the ring. Dewetting@8# and pinning forces areof comparable strength during this stage. Here I will shthat the competition between these two forces leads to cplicated yet ordered behavior.

As in the self-pinning experiments, the observations tfollow are of drops of colloidal microspheres drying onfreshly cleaved mica surface.~Experiments conducted usina cleaned, optically flat, silicon substrate gave similarsults.! The additional step of briefly centrifuging the 0.1-mmsample before use was performed to remove large part~;1 mm! because otherwise the resulting deposits were iproducible. The experimental control parameters were pticle size, solute concentration, surfactant concentrationic strength, and polydispersity of the particle size. Aobservations were done with the aid of a fluorescent micscope. Therefore, the gray level in the images that followproportional to the number of particles at a given point wwhite corresponding to the largest value.

A. Patterns with 0.1-mm particles

Arranged top to bottom in order of decreasing concention, Fig. 8 shows the deposit resulting from dried dropsdifferent initial concentration of 0.1-mm microspheres. Theleft-hand column shows a complete picture of the depowhereas the right-hand column shows only magnified fr

FIG. 7. ~a! The height of the ring at depinning in units ofmmcalculated using the results of the Appendix and the data in Fig~b! The height at depinning vswd . The data are best fit by a powelaw with an exponent of 0.8560.05.

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ments. At the scale of the whole drop a bull’s-eye-like ptern is visible. In~a! there are four well defined concentrrings. In ~b! and ~c! the interior shows concentric marks bthey are qualitatively different from those of~a!. In ~d! againthe inner rings are strong though they are finer, less coplete, and less organized than those in~a!. The rings shownin ~a! and ~d! form when the contact line is pinned lonenough that a large contact line deposit can form. Thus th

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FIG. 8. Photographs of the deposit left by drops containing 0mm microspheres at various concentrations dried on mica. Thescale indicates the density of particles in a given area with wcorresponding to the highest value. The left column shows thetire deposit for initial volume fractions~a! 1%, ~b! 0.25%, ~c!,0.13%, and~d! 0.063%. The diameter of these drops is appromately 6 mm. The right column shows a closeup of a deposit mfrom the same concentration as the deposit to its immediate lefsome cases, multiple closeups for a single concentration are shThe scale bar is 500mm in ~f!–~i! and 250mm in ~j!.

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rings chronicle the moments of arrested contact line motIn contrast, direct observation of the deposition process leing to ~b! and ~c! shows that the rings in these casesformed while the contact line is moving.

Magnifying the deposits reveals yet more structure. Athighest concentration@frame ~e!#, the space between the inner rings is coated by multiple layers of microspheres tare deeper in the vicinity of a ring; the deepening depoheralds, and perhaps causes, the pinning episode duwhich a ring is formed. Forf50.25% @frames~b!, ~f!, and~g!#, the interior of the deposit is coated with a layermicrospheres with a thickness that varies between onetwo particles. The transition from one- to two-particle layeindicated by the a steplike increase in brightness, has a stoothed shaped pattern pointed radially outward@Fig. 8~f!#.However, the transition from a two- to a one-layer deposiabrupt, as can be seen from the termination of the outermsaw-toothed front. The saw-toothed front is caused bpiecewise transition to a double layer film. First, a few poialong the contact line begin to produce two layers. Ascontact line retreats, the segments producing double laextend along the contact line, leaving a triangular-shadeposit in their wake. Finally, the segments grow into oanother so that the entire contact line deposits double layThe piecewise growth of the double layer and the abrreturn to a monolayer suggests that there is an energy coswitching from a single layer to a double layer. If so, tsaw-toothed front is an energetically inexpensive meanachieving two-layer production because the distortion eneis localized to a few particle diameters along the contact li

At f50.25%@Fig. 8~g!# and 0.13%@Fig. 8~h!#, a gridlikepattern appears. It can exist concurrently with or separafrom other patterns at these concentrations. The producof a grid appears to be an unstable version of single-laproduction. Direct observation of the contact line shows tparts of it move steadily and that the parts between thmove in a stick-slip fashion. The steady moving segmelay down the radial lines of the grid and stick-slip segmeproduce nothing when moving and a ring when at rest;combination of the radial lines and the rings forms a grFigure 8~i! shows another type of pattern in which singllayer deposition gives way to an even slower rate of deption that does not achieve total coverage and includes loempty, radial grooves. Finally, at the lowest concentrat(f50.063%) another new mode appears in which radspokes are produced@Fig. 8~j!#.

B. Patterns with 1-mm particles

@Due to the difficulty of deducing the microscopic behaior of submicron-sized particles with light microscopy, I focused on the behavior of drops with 1-mm particles.# Thepatterns of deposit left when a drop dries are shown in Figfor different initial concentrations of 1-mm-sized particles. InFig. 9, frame~a! is the highest concentration and frame~f! isthe lowest. There are distinct features and trends in the dsition patterns. The features may be grouped by distafrom the center of the drop. Moving inward, the first zonea featureless solid packing of particles. This is the initial riformed before

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any depinning has taken place. The second zone consisarch-shaped formations, the third zone is a mixture of hformed arches and radial lines, and the central region is cposed of apparently disorganized dots. The range of thphases as a function of concentration is plotted in Fig. 10high concentration all phases are present but as the contration decreases, the mixed zone gains at the expense oothers and its composition.

A change in the velocity of the contact line might accoufor the different phases seen in the deposits. However, diobservation of the contact line shows that there is very li

FIG. 9. Deposit left by drops of 1-mm microspheres dried onmica. The initial volume fractions reading from left to right anfrom top to bottom are 2.0%, 1.0%, 0.5%, 0.25%, 0.13%, a0.063%. The diameter of each drop is approximately 6 mm.

FIG. 10. A diagram indicating the range of each phase: riarch, mixed, and disordered phases. The radial position is norized to 1 at the contact line.

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change in its speed. Another possibility is that the deposirate at the contact line changes as the drop shrinks. Thconsistent with a constant contact line velocity. MoreoverRef. @6# we showed that a gradient in the concentrationvelops and grows while the contact line is pinned. Therefothe deposition rate would be altered as the contactmoved closer to the origin.

The ring width as a function of concentration was quatified in Sec. III A. Here the focus will be on the other zonbeginning first with the arches zone. I measured the sizindividual arches from the resulting deposit of many drofor each concentration a total of approximately 1000 msurements were taken. I took the greatest width of the arca measure of its size. Closer inspection of the larger arcshowed that they were composed of multiple subarchesthese I measured the width of the subarch rather than wof the entire group. This procedure is shown in Fig. 11. Fthe first four concentrations represented in Fig. 9, the ameasurements were binned by size and plotted in F12~a!–12~d!. Since there are no arches in drops withf50.13%, instead I measured the perpendicular distancetween spokes where they ran parallel to each other. Thetribution function for these measurements is plotted in F12~e!. For the concentration represented by Fig. 9~f! nolength scale is apparent, and so no measurements were mThere are two clear trends apparent from the distributfunction: for a decreasing concentration the peak of thetribution shifts to a larger value, and the width of the distbution, except for the lowest concentration, grows. The loest concentration shows an abrupt narrowing ofdistribution; this sudden jump corresponds to switching frmeasuring arch sizes to measuring the distance betweedial lines. To extract the trends, the data in Fig. 12 were fithe functional formLl exp(2L/a), from which I extractedthe peak position and the full width at half height of thdistribution. A plot of these parameters is shown in Fig. 1The change in peak size is well fit by a linear functionconcentration.

From observing the processes transpiring at the conline a partial picture of the pattern formation proceemerges. The outermost structure, the ring, was explabriefly in Sec. I and in more detail elsewhere@5,6,9–12#. Thearches are formed by the process shown in Fig. 3: a honucleated, the contact line begins to recede, but its growarrested by the accumulation of particles. Hole nucleationnanometer-thick films has been previously observed@13,14#,

FIG. 11. A closeup view of a large arch shows that it is in facomposed of multiple subarches. The size of these smaller arwas measured as shown.

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and mechanisms for their formation were proposed in R@13# and @15#. However, these mechanisms are inapplicato the micron-thick films present in my experiments. Tgrowth of arch size with decreasing concentration mayplausibly explained by the fact that there are fewer particwith which to pin the contact line, and this allows the archto grow larger. By frame~e! of Fig. 9 there is an insufficiennumber of particles to stop the contact line from moving athe arches grow without bound; this accounts for the abseof arches in frames~e! and~f!. Ohara and Gelbart previousl

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FIG. 12. Plot of the unnormalized distribution function of tharch sizes. Plots~a!–~e! correspond to initial concentrations o2.0%, 1.0%, 0.5%, 0.25%, and 0.13%, respectively.

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proposed the same mechanism for pinning of the contactby nanoparticles. They predicted thatL}1/f for a holespreading in a film of uniform thickness and concentratiThe result is not consistent with my data but it is not epected to hold given that a drop is not uniformly thick, ththere are additional flows in a ring-forming drops, and ththe concentration is not uniform.

The mixed zone begins when the number of holes timtheir average size becomes comparable to the circumferof the drop. Unlike the arch creation period in which tmajority of the contact line is pinned, large portions of tcontact line become free to move. At high concentratiothis motion is heavily constrained by the accumulationparticles, and is hence erratic. The half-formed archesvelop when a small portion of the contact line is temporarpinned. However, for lower concentrations the contact lorganizes itself into a series of cusps that emit particles. Tdeposition appears to be a release of particles controlledthat there is no net accumulation at the contact line. Durthis type of motion, radial lines are formed. The ability of tcontact line to do this is thwarted at higher concentratiobecause the influx of particles is overwhelming. As the ccentration is lowered and fewer particles reach the conline per unit time the ejection process can handle allparticles that arrive; this is the situation in Fig. 9~e!. At even

FIG. 13. ~a! The most likely size of an arch vs concentratiotaken from the peak position in the curve of Fig. 12.~b! The fullwidth at half height vs concentration of those curves.

e

.-tt

sce

sfe-

eissog

s-cte

lower concentrations@Fig. 9~f!# the output from the cuspscan exceed the influx of particles and the emission procbecomes discontinuous, leaving disjointed trails of particwhich are nonetheless radially oriented.

In other pattern forming systems, such as solidifyingloys, the wavelength selection criterion depends on thelocity of interface. In order to determine if an analogorelation exists in drying drops, I measured the velocity of tminima between two cusps and the cusp-to-cusp distaduring the stage when the radial spokes in Fig. 9~e! weregenerated. The data are plotted in Fig. 14, and they shonly a very weak relation between the velocity and the cuto-cusp distance.

I also measured the shape of the interface to compawith the shape expected from contact line elasticity@16# inthe low velocity limit. For measurement purposes, I definthe contact line as the outline of the particles caught inwedge of fluid next to the contact line. Using a fluorescemicroscope and fluorescent dyed particles, I captured imaof the contact line and extracted the outline of the macscopic contact line. Since there was no predefined zerdefined it as the line connecting the lowest point on each sof the cusp. An elaboration of this is shown in Fig. 15. Theight of the cusp above the zero line is plotted againsttance in Fig. 16, where the zero of the abscissa was choseachieve the maximum symmetry between the two sides.data are well fit by an exponential function. This fit is bettthan the logarithmic fit expected from contact line elasticiThis discrepancy may be accounted for by viscous disstion in the fluid, the interaction of the particles through caillary force @17#, or deviations from the logarithm due tdefect size.

C. Patterns from mixtures

The parameter space of the deposition patterns wasther probed by combining colloidal microspheres with sfactant, salt, and microspheres of a different size. A sampof the results is given below. It is clear that there are macontrol parameters and that there are many different deption patterns.

FIG. 14. The velocity of the contact line between two cuspsthe distance between the two cusps. The initial concentration ofdrops wasf56.2531024. The fit is to a linear equation whichgivesv5(50.710.16)d in units of mm/s.

ame

or

ig.ed

at-im-

eenate isic

ition

areca.

llyif

le-ted

atedde-ndd

linendat-

tio

sg

gh

rolinve

lthe

p to


The anionic surfactant sodium dodecyl sulphate~SDS!was added to the 0.1-mm colloid. This was motivated bydifferences observed in the patterns left by drops of the sconcentration but containing colloidal particles synthesizby different manufacturers. By the addition of surfactant tsurfactant-free sample, I was able to produce similar patteto those formed by other colloid brands.~The inexact matchmay be due to differences in the ionic strength.! In addition,

FIG. 15. Schematics for demonstrating the contact line posimeasurement. The position of the contact line~the dashed line inthe closeup! was inferred from the position of the microsphereThe zero height position was chosen as the segment connectintwo minima between a pair of cusps, and the zerox position waschosen by finding the horizontal value at which the left and rihand parts of the curve are most symmetric.

FIG. 16. ~c! The position of the contact line relative to the zepoint vs distance plotted on a log-linear scale. The straight solidrunning through the data is a fit to an exponential, and the cursolid line is a fit to a logarithm.

edans

I found a number of new patterns which are shown in F17. These patterns where made from a colloid with a fixvolume fraction of microspheres (f50.0013) but variableconcentrations of SDS. During the production of these pterns the contact line sweeps from top to bottom in theages of Fig. 17.

Since the strength and range of the interaction betwmicrospheres and between a microsphere and the substrdetermined by screening ions, I tried altering the ionstrength of the solvent and observed the resulting depospattern. Starting with a 2% solid volume-fraction of 1-mmmicrospheres, I diluted the sample down tof50.25% with aNaCl and water solution. The images shown in Fig. 18the patterns left when a drop of this mixture is dried on miThe molarity of the solution used to make~a! was 0M , ~b!was 0.01 mM , ~c! was 0.1 mM , and ~d! was 1 mM . Fromthese images it is clear that the addition of salt radicaalters the resulting deposit. It would be interesting to knowthis is due to a change in the particle-particle or particsubstrate interaction, or to the pinning effects of deposisalt.

Last I investigated the effect of combining 0.1- and 1-mmmicrospheres. Two of the most interesting deposits generby this binary combination are shown in Fig. 19. Theseposits were formed from a one to one mixture of 0.1- a1-mm colloids with an initial volume fraction of 0.125% an0.0625%.

IV. CONCLUSION

The major results presented here are that the contactcan self-pin, that the competition between this pinning adewetting give rise to pattern formation, and that these p

n

.the

t

ed

FIG. 17. Deposit left by 0.1-mm microspheres of 0.5% initiavolume fraction when a SDS surfactant is added. The center ofdrop lies below the image so that the contact line moved from tobottom. In each image the scale bar corresponds to 50mm. Inframes~a!–~d! the concentration of surfactant is 8.131024M , 4.331024M , 1.431024M , and 4.831025M , respectively.

nger—ioereahie

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ofa

heneeldie

tact

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tionhere

ternof

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in

tot

20

reeis aionsthe


terns exhibit evidence of wavelength selection. A dryidrop is a new, rich, and unexplored example of a pattforming system. It can be controlled in numerous wayssolute concentration, particle size, surfactant concentrationic strength, and particle mixtures were partly treated hAll the tools that have been developed to treat nonlinsystems far from equilibrium can be brought to bear on tproblem. It will be interesting to know if this system fits thstandard paradigm of pattern formation.

Investigating pattern formation in drying drops is appeing because a much is already known about contact linehavior and so it ought to be theoretically tractable, and it irelatively simple experimental system. However, some fdamental experimental and theoretical obstacles must firstackled. On the experimental side, the circular geometrydrop is too complicated. The phases shown in Fig. 10most likely a manifestation of this problem. In addition, tfree drying conditions do not provide sufficient control othe kinetic processes in the system. It is clear that a linanalog of the drop is needed. The result of a simple, wcontrolled experiment ought to provide the stimulus thatrectional solidification and Hele-Shaw experiments providfor moving interface problems@18,19#. Furthermore, the

FIG. 18. Deposit left by a 0.25% solid volume-fraction of 1-mmmicrospheres in a solvent with different ionic strengths~see text!.The diameter of the deposits is approximately 6 mm.

FIG. 19. Deposit left by a 50-50 mixture of 0.1- and 1-mmmicrospheres diluted down to a volume fraction of 0.125% for~a!and 0.0625% for~b!. The bright parts of the deposit corresponddeposits of 1-mm microspheres, and the gray parts corresponddeposits of 0.1-mm microspheres. The scale bar corresponds tomm.

n

n,e.rs

-e-a-bea

re

arl--d

physical mechanism that drives the retraction of the conline has yet to be determined.

On the theoretical side, the most obvious unknowns~1! the flow within a spreading drop of volatile fluid, and~2!the wavelength selection mechanism. The spreading of vtile liquids is a research topic in its own right. However,appears to be an essential prerequisite for understandingpatterns in drying drops because the flow in the drop demines the distribution of solvent. Though there has besome progress on this spreading problem@20–22#, there is asof yet no widely held consensus. The wavelength selecproblem is completely open because the result presentedare the first of its kind.

Overall the prospects are good for new ideas about patformation emerging from this system. There are a wealthexperimental and theoretical avenues to pursue.

ACKNOWLEDGMENTS

I would like to thank my advisor Sid Nagel for suggestinthis problem and for the stream of useful ideas, and TWitten and Todd Dupont for many fruitful discussions. Thwork was supported by the NSF under Grant No. DM9410478 and the MRSEC Program of the NSF under GrNo. NSF-9808595.

APPENDIX

To model the shape of the deposit at the contact line,in particular to obtain its width versus time, the drop is seprated into three regions~region I, region II, and the ring! asshown in Fig. 20. Regions I and II are composed of solutiand the ring is a solid formed from the solute with packifractionp. The total volume of liquid at any given moment

VL5VI1VII 1~12p!Vr.pR2h1pR3Q

4, ~A1!

whereVr is neglected because it is small compared toVI andVII , Q is assumed to be small, and the shape of regionassumed to be a thin spherical cap. Furthermore, the volof the drop is assumed to decreases linearly, so that

VL~ t !5pR3uc

4~12t/t f !, ~A2!

whereuc is the initial contact angle. From Fig. 20 we obtathe geometrical relation

o0

FIG. 20. Diagram depicting the separation of a drop into thregions. Region I is shaped like a spherical cap, and region IIcylinder. The shape of ring deposit is determined by the equatin the text. At any given time the interface between the ring andsolution is located atw from the contact line and ish high. Theaspect ratio is greatly exaggerated.

-

al

-

the.


dh

dw5tanQ.Q. ~A3!

The ring grows inward at the ratedw/dt. Therefore, thevolume of the ring changes as solute is added and

dVr

dt52pRh

dw

dt. ~A4!

Using Eq.~2! gives

dVr

dt5p21

pR3u0

4f„12~12t/t f !

3/4…

1/3

~12t/t f !1/4 . ~A5!

Putting together Eqs.~A1!–~A5! and introducing the nondimensionalized variablesx[4w/R, y[4h/(u0R), and t

ry,

R

R

ale

-

t[t/t f we arrive at the pair of coupled ordinary differentiequations that govern the shape of the ring:

2f

p

„12~12t!3/4…

1/3

~12t!1/4 5ydx

dt, ~A6!

~12t2y!dx

dt5

dy

dt. ~A7!

It is clear from the absence ofR from nondimensional formof the equations thatw will scale with R. For calculationpurposes, I assume thatp50.656, and that the initial conditions are thatw(t50)50 and h(t50)50. The equationswere integrated numerically to achieve the conversion oftime data in Fig. 4 into the width data displayed in Fig. 5

ys.

-

ett.

e

@1# T. Young, Philos. Trans. R. Soc. London95, 65 ~1805!.@2# P. G. de Gennes, Rev. Mod. Phys.57, 827 ~1985!.@3# E. L. Decker and S. Garoff, Langmuir13, 6321~1997!.@4# R. Johnson and R. Dettre, inContact Angle, Wettability and

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the as-of-yet undetermined forces that drive the contact linretreat.

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tion in Waterborne Coatingsedited by T. Provder, M. A. Win-nik, and M. W. Urban~American Chemical Society, Washing

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lto

ton, DC, 1996!, p. 419.@12# E. Adachi, A. S. Dimitro, and K. Nagayama, Langmuir11,

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312 ~1958!.@20# L. M. Hocking, Phys. Fluids7, 2950~1995!.@21# M. Elbaum, S. G. Lipson, and J. S. Wettlaufer, Europhys. L

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