Deegan.PRE

Contact Line Deposits in an Evaporating Drop Robert D. Deegan, Olgica Bakajin, Todd F. Dupont, Greg Huber, Sidney R. Nagel, Thomas A. Witten James Franck Institute, 5640 South Ellis Ave, Chicago, IL 60637, USA Typeset February 3, 2001 Solids dispersed in a drying drop will migrate to the edge of the drop and form a solid ring. This phenomenon produces ring-like stains and occurs for a wide range of surfaces, solvents, and solutes.Here we show that the migration is caused by an outward °ow within the drop that is driven bythe loss of solvent by evaporation and geometrical constraint that the drop maintain an equilibriumdroplet shape with a flxed boundary. We describe a theory which predicts the °ow velocity, therate of growth of the ring, and the distribution of solute within the drop. These predictions arecompared with our experimental results. I. INTRODUCTION The residue left when cofiee dries on the counter-top or at the bottom of a mug is typically darkest, and hence most concentrated, along the perimeter of the stain giving the deposit a ring-like appearance. Given that the cofiee solutewas uniformly distributed in the liquid prior to drying, its segregation to the edge is surprising. Ring-like stains arenot particular to cofiee and are a general phenomenon of which some of the more commonplace examples are mineralrings left on washed glassware, banded deposits of salt on the sidewalk during winter, and enhanced edges in watercolor paintings (see Fig. 1). Controlling the distribution of solute during drying is vital in many industrial and scientiflc processes. For example, paint manufacturers use a variety of additives to ensure that the pigment is evenly dispersed and remains so duringdrying, and protein crystallographers are attempting to use evaporation driven convection to assemble 2-dimensionalprotein crystals [1,2]. Segregation efiects are undesirable in either of these cases. However, as in the production ofnanowires [3] or the patterning of a surface [4] the ring-efiect can be a boon. Understanding the ring formation processshould be of interest to those attempting to circumvent or harness this phenomenon. Ring-like deposits have been previously reported: Unertl et al reported on the di–culties of obtaining a uniform deposit [5], Parisse et al investigated the deformation of sessile drops due to a sol-gel transition of the solute at the boundary [6,7], Adachi et al studied the stick-slip motion of the contact line of colloidal liquids [8,9], and Co nway et almeasured the efiect of ring formation on the evaporation rate of sessile drops [10]. With this contribution we expand on an earlier paper [11] in which we reported the experimental results on ring formation and demonstrated that these could be quantitatively accounted for. In our theory, an outward °ow in adrying drop of liquid is produced when the contact line is pinned so that liquid that is removed by evaporation fromthe edge of the drop must be replenished by a °ow of liquid from the interior. This °ow is capable of transferring 100%of the solute to the contact line and thus accounts for the strong perimeter concentration of many stains. Furthermore,the theory relies only on a generic property of the substrate-solvent interaction, the presence of surface roughness orchemical heterogeneities that produce contact line pinning, and therefore it accounts for the ubiquitous occurrence ofring-like stains. This paper is arranged in flve parts; after this introduction the next part deals with preliminary experiments that show the generality of the phenomenon. The third part lays out the fundamental mechanism responsible for thisefiect and describes the expected °ow mathematically. The fourth section presents the calculated results. In the flnalsection we present the results of several experiments that test the theory’s predictions. We found that most of thosepredictions are in accord with the experiments. However, there are some discrepancies that suggest the importanceof other mechanisms as well. II. EXPLORATORY EXPERIMENTS Exploratory experiments using a variety of carrier °uids, solutes and substrates indicated that preferential deposition at the contact line is insensitive to a wide range of experimental conditions. We observed ring-like deposits wheneverthe surface was partially wet by the °uid irrespective of the chemical composition of the substrate; we tried glass,metal, polyethylene, roughened Te°on, freshly-cleaved mica, ceramic and silicon. Rings were found in big drops(15 cm) and in small drops (1 mm). They were found with aqueous and non-aqueous (acetone, methanol, toluene,and ethanol) solvents. They were found with solutes ranging in size from the molecular (sugar and dye molecules) tothe colloidal (10 „polystyrene microspheres) and with solute volume fractions ranging from 10 ¡6to 10¡1. Likewise, 1 environmental conditions such as temperature, humidity, and pressure could be extensively varied without afiecting the ring. Efiects due to solute difiusion, gravity, electrostatic flelds, and surface tension forces were considered as possible sources of contact line deposits. We rejected these on the basis of experimental evidence. For these observationswe used small drops of a dilute suspension of colloidal particles in water, deposited on a glass microscope slide andviewed through an optical microscope. We describe the experiment more full in Section V. Difiusion of solute inthe liquid is not the primary agent of segregation because the solute particles clearly move by advection towardsthe edge of the drop. This can be easily seen under a microscope using large (1 „to 10„) colloidal particles. Gravitational efiects appear unimportant because the ring produced when the drop is dried upside down (a pendantdrop) is indistinguishable from the one produced by a drop dried right side up (a sessile drop.) Likewise, electricflelds produced by a charged needle near the contact lines showed no perceptible efiect on the ring formation. Severaldifierent fleld conflgurations with flelds of several hundred volts per millimeter were tried without visible efiect. Finally,temperature gradients did not seem to play an essential role. When the drops was locally heated with a hot fllament,circulating °ows appeared, but the ring deposition was not fundamentally changed. While the efiects listed above appear unimportant for ring formation, two other conditions appear necessary: contact line pinning and evaporation form the edge of the drop. When either of these conditions is relaxed the flnal depositis uniform. We eliminated pinning by drying the drop on smooth Te°on. The drying drop contracted as it dried andno ring appeared. Likewise we restricted the evaporation (on a glass surface) by covering the drop with a lid thathad only a small hole over the center of the drop though which the vapor could escape. This greatly reduced theproportion of evaporation from the perimeter. The resulting deposit was uniform rather than being concentrated atthe edge. III. THEORY OF SOLUTE TRANSFER The essential physical idea behind our theory is that a pinned contact line induces an outward, radial °uid °ow when there is evaporation at the edge of the drop. This is graphically demonstrated in Fig. 2. The solid line inFig. 2(a) represents the initial position of the air-liquid interface. If the evaporation rate is spatially uniform andthe contact line is notpinned, during some interval of time the hashed region would be removed from the drop, the interface would evolve from the solid line to the dashed line, and the contact line would move from A to B. However,if the contact is pinned then there must be a °ow that replenishes the liquid that is removed from the edge. In thiscase of a pinned contact-line, the interface evolves as shown in Fig. 2(b) from the solid line to the dashed line and theelasticity of the air-liquid interface, i.e. the tendency of the interface to minimize its surface area, provides the forcedriving the outward °ow of °uid. We now develop these ideas mathematically starting with the °ow inside the drop. We consider an axisymmetric drop as shown in Fig. 3. For the moment we ignore any solute in the liquid. The conservation of °uid determines therelationship between the vertically-averaged radial °ow of the °uid, v, the position of the air-liquid interface, h, and the rate of mass loss per unit surface area per unit time from the drop by evaporation, J s. The rate of change of the amount of °uid in an inflnitesimal annular element at a radial distance rfrom the center of the drop is equal to the net °ux of liquid into the column minus the amount of mass evaporated from the surface of that element: ‰@h @t=¡‰1 r@ @r(rhv)¡Js(r;t)s 1+µ@h @r¶2 (1) wheretis time and ‰is the density of the liquid. The efiect of @h=@r -term is is nearly always small and can be neglected; we keep it here only for completeness. We can solve for vby rewriting this equation in integral form: v(r;t)=¡1 ‰rhZr 0dr r0 @Js(r;t)s 1+µ@h @r¶2 +‰@h @t1 A (2) from which it is clear that a non-zero varises when there is a mismatch between the local evaporation rate and the rate of change of the interface. To compute vwe need to specify handJs. The proflle of the drop, h(r;t), can be computed from Navier-Stokes equations, however, we will assume, as in Fig. 2, that the shape of the drop is a spherical cap so that the proflle is: h(r;t)=s•h(0;t)2+R2 2h(0;t)‚2 ¡r2¡R2¡h(0;t)2 2h(0;t)(3) 2 whereRis the radius of the drop’s base and h(0;t) is the height at the center. This corresponds physically to assuming thatRis small so that the surface tension, which tends to make drops spherical, is greater than the gravitational force, which tends to °atten drops, and that evaporation occurs slowly so that dynamic contributions to the pressure,such as viscous dissipation, are negligible. In essence, we have neglected all terms in Navier-Stokes equation, exceptthe pressure term, so that rp= 0 (i.e. we are in the hydrostatic limit). The time dependence of h(0;t) will be determined below when we compute the evaporation rate. Withhspecifled we now compute J s. The functional form Jsdepends on whether the rate-limiting step is the transfer rate across the liquid-vapor interface or is the difiusive relaxation of the saturated vapor layer immediatelyabove the drop. In the former case, J sis a constant, while in the later case, as will be shown below, Jsis strongly enhanced toward the edge of the drop. For the case where the limiting rate is the difiusion of the liquid vapor, we assume that the evaporation of the drop rapidly attains steady state so that the difiusion equation reduces to Laplace’s equation r2u=D@tu’0 (4) whereuis the mass of vapor per unit volume of air and Dis the difiusion constant for vapor in air. The assumption is good for times longer than D=R2(e.g. fo r a 1 mm-sized drop of water this time is 0.04 sec.) The boundary conditions are that (1) along the surface of the drop the air is saturated with vapor so there uis a constant us, (2) the current normal to the substrate ( J¢n=¡D@nu) is zero because the vapor cannot penetrate the substrate, and (3) uconverges to the ambient vapor concentration, u1, far from the drop. This boundary value problem is identical to that of a charged conductor if we equate uwith the electrostatic potential and Jwith the electric fleld. As in the case of an electric fleld near a sharp edge on a conductor (e.g. a lightning rod) the evaporation current diverges near the contact line of the drop (see caption for Fig. 3). Close to thecontact line J s(r;t)»(R¡r)¡‚(5) where‚=(…¡2µc)=(2…¡2µc) [12] andµcis the contact angle the liquid makes with the substrate. This edge enhancement arises from the greater probability of an evaporating molecule’s escape when leaving from the edge thanwhen leaving from the center of the drop (see Fig. 4). Furthermore, as shown in the appendix, the expression: J s(r;t)…Jof(‚)£ 1¡(r=R)2⁄‚(6) is an excellent approximation to the analytic solution to the boundary value problem. FromJs(r;t) we can determine the time-dependence of h(0;t). The change of the volume of the drop must be exactly equal to the total evaporation rate. Therefore, dM dt=‰d dtZR 0dr02…r0h(r0;t)=ZR 0dr02…r0Js(r0;t)s 1+µ@ @r0h(r0;t)¶2 (7) While we have assumed steady state, Js(r;t) is nonetheless time-dependent because of its dependence on the contact angle which changes during the course of evaporation. The velocity of the °uid can now be computed by combiningEqs. 2, 4, & 6. Given the velocity in the drop we can also compute the growth of the ring. We use our knowledge of the initial distribution of solute, namely that it is constant everywhere in the drop at t= 0, and compute the time it would take a packet of °uid to reach the contact line having started at some initial radial distance r o. Within this time all the solute that lay beyond robecomes part of the ring. Therefore, the mass of solute in the ring, mR(t) is: mR(t(ro) )=2…coZR rodr0r0h(r0;t= 0) (8) wherecois the mass of solute per unit volume of solution and t(ro) is the time it takes to go from r=roat time t=0t or=Rand can be calculated by integration of dr=dt =v(r;t) with the initial condition that r(0) =r0. Since we have used the vertically averaged velocity in this formula we have implicitly assumed that there is no verticalsegregation of the solute. Lastly, we can compute the distribution of solute. However, to do so, without resorting to a lengthier calculation of the °ow within the drop, we must assume something about the vertical distribution. For simplicity we start withthe assumption, used above, that the concentration, c(r), is a function solely of r(i.e. at a given radial distance the concentration is uniform throughout the liquid column.) Therefore, the equation of solute conservation reads: 3 @ @t(ch)+1 r@ @r(rchv)=0 ( 9 ) wherecis assumed to be independent of z, the vertical distance from the substrate, and the difiusion of solute has been neglected. The mass in the ring can also be determined from the difierence of the mass of solute in the drop at t= 0 and the mass of solute left at a later time,RR 0dr02…r0h(r0;t)c(r0;t). IV. RESULTS In the previous section we described how to compute the various quantities ( v(r;t),mR(t), andc(r;t)) that describe the state of the drop. Without simpliflcations, these computations can only be done numerically. However, thequalitative features of the theory can be understood without recourse to numerics. In this section we will highlightthese features. The velocity diverges near the edge of the drop and towards the end of the drying time. These qualitative features can be deduced directly from Eq. 2 and 5. Near the edge h’(R¡r), so that the contribution of the @ th-term to Eq. 2 is negligible in comparison with the Js-term and integration yields v»(R¡r)¡‚. What is happening physically is that the change of volume of the wedge near the edge (see Fig. 5) becomes increasingly smaller as the contact lineis approached and therefore the outgoing vapor must be matched by an equally strong in°ow of liquid. In addition,as the height of the wedge decreases the amount of liquid entering the wedge will not vary by much over the dryingtime because the amount of liquid removed from the surface of the wedge is almost constant. Therefore, hv smust remain constant so that vs»1=hand diverges as the height goes to zero at the end of the drying. The ring grows initially as a power law in time. To see this, consider a particle at a distance R¡rfrom the edge. All the particles that are at equal or lesser distance from the contact line will be swept into the ring by thetime our particle reaches the edge so that the ring will increase in mass by m R»(R¡r)2. The time this takes is t»RR rdr0=v»(R¡r)‚+1. Therefore, at early times the ring grows in time as a power law: mR»t2=(1+‚)(10) There is also a late time regime in which the growth of the ring rapidly increases and diverges. We can demonstrate this divergence by using a simplifled model in which the drop is always thin h(0;t)¿R. In this limit it follows that ‚is nearly constant because the contact angle does not vary much. For example, for a drop starting with a contact angle of 15o,‚would increase from 0.455 to 0.5, a 10% change, over the entire drying time. With ‚constant it follows from Eq. 7 that the height of the drop decreases linearly in time, h(0;t)=h(0;0)(1¡t=tf) whereh(0;0) is the center height att= 0 andtfis the total drying time. In this simplifled case, the integration of Eq. 8 can be done because the time and space components are separable. This would not be the case for a thick drop where ‚is time-dependent. The result for the thin drop limit is: mR=moh 1¡(1¡t=tf)(1+‚)=2i2=(1+‚) (11) wheremois the total mass of solute present initially in the drop and ‚(µc)’‚( 0 )=1=2. At early times ( t¿tf) mR’t2=(1+‚)as expected from Eq. 10. At late times mR¡mo’(tf¡t)(1+‚)=2, which implies that the growth of the ring is divergent: dmR=dt’1=(tf¡t)1=4. It is this flnal divergence that is responsible for the observed 100% transfer of solute to the edge. For a thick drop, where the contact angle varies appreciably with time as the evaporation takesplace, the late time result must be obtained numerically. However, the early time results, before the contact anglechanges become important, are the same as described above. V. EXPERIMENTAL RESULTS We studied a dilute colloidal suspension of surfactant-free, charge-stabilized, polystyrene microspheres in water. The mass versus time, shown in Fig. 6(a) for drops of various sizes, is to good approximation linear in time andthe evaporation rate rapidly stabilizes to its steady state value. Thus, the assumption that we can neglect the timedependence in the difiusion equation 4 and that h(0;t)=h(0;0)(1¡t=t f) is supported by data in Fig. 6(a). By extracting the slope of these plots (and others not shown) we can make a plot of evaporation rate versus size of the drop. Intuition might suggest that the evaporation rate should vary as the surface area of the drop ( »R2). While this is the case for drops that evaporate uniformly, it is not so for difiusion-limited evaporation. Consider the case of 4 a spherical drop evaporating. The relative humidity around the drop is u=R=r (just like a conductor charged to a potential of 1), the local evaporation rate on the surface is J=¡D@ru=DR=R2, and the total evaporation rate is J£Area = 4…Rwhich is linear in R. In general, the evaporation rate in the difiusion-limited regime is proportional toR[13]. This counter-intuitive result arises because the rate is lessened by the flnite probability that an evaporating molecule will return to the drop if it is executing a random walk. Fig. 6(b) demonstrates the rate is linear in Rand indicating that the evaporation rate is difiusion limited [14]. Given that the evaporation of our experimental drops is difiusion-limited, we can immediately surmise some general qualitative features about the edge deposition process. For an oddly shaped drop, such as the one that resultedin the deposition pattern depicted in Fig. 4(b), the evaporation rate is strongest where there is the lowest densityof neighboring evaporation sites. The heavy deposition at tips of the \clover" in Fig. 4(b) arises from the strongevaporation from that location. Also, when two drops are placed next to each other, their vapor flelds overlap andthere is less evaporation, hence less deposition, in the region of greatest proximity (see Fig. 4(c)). By photographing the drop from the side, we measured the shape of the interface at various times. These data are plotted in Fig. 7 for several equally spaced intervals in time. The solid lines running through the data are flts to Eq. 3using the same parameters Randt ffor all flts. Given the good agreement of with the data, our assumption of Eq. 3 appears reasonable. Another feature demonstrated by the data in Fig. 7, is the duration of the pinning. A drop of pure liquid (i.e. without solid components) drying on a substrate is thought to evolve in two distinct modes: an initial pinned phase,in which the base of the drop stays constant and the contact angle decreases, followed by a second unpinned phase, inwhich the base shrinks but the angle remains flxed [15,16]. Our observations of ring-forming drops indicate that thesecond phase is absent so that the addition of a solid component to the liquid which accumulates at the contact linealters the dynamics of the contact line. In this sense, we believe that the drop is self-pinning: the initial roughness orchemical heterogeneities of the substrate provides a foothold on which the contact line flrst sticks; the accumulationof material at the contact line, by the process we have been discussing, then apparently strengthens the pinning, andeventually takes over as the primary source of pinning. The difiusion-limited theory predicts a velocity divergence at the contact line that depends on the contact angle. The velocity in the °uid was measured by tracking the motion of 1 „microspheres in a drying drop, a small fraction of which were °uorescent. A region extending two-thirds of the way out from the center of the drop up to the contactline was imaged by °uorescent microscopy and recorded on videotape. At roughly 80% of the drying time the depthof the liquid became comparable to the depth-of-fleld of the lens and particles at all depths were in focus. At thisstage the contact angle was around 0.2 radians, and the corresponding exponent for the velocity was ‚=0:47§0:01. Several two-second sequences of images were digitized at 6 Hz and processed using image analysis algorithms fortracking particles [17]. The velocities were binned on a logarithmic scale and plotted versus log( R¡r) in Fig 8. We measured an exponent of 0 :54§0:06 (the uncertainty arises from §5„uncertainty in the position of the contact line) which is within error of the predicted value of 0 :47§0:01. Several morphological features of the deposition from a drying drop can be appreciated from Eq. 10. Three identical drops were prepared. The flrst was allowed to evaporate normally. The second was mounted on a pedestaland surrounded by a bath of water such that the level of the bath coincided with the base of the drop. In this case,the evaporation rate is to good approximation spatially uniform. The third drop was surrounded by a chamber witha small hole above the center of the drop so that evaporation primarily occurred at the center. The results of theexperiment are shown in Fig. 9. The flrst and second drop produced ring-like deposits, whereas the third produced auniform deposit. As is shown schematically to the right of each picture in Fig 9, the evaporation rates of drops 1 and2 difier from the proflle of the drop and, therefore according to Eq. 2, give rise to a non-zero velocity that is able torearrange the mass within the drop. However, the third drop had an evaporation proflle that was qualitatively similarto the proflle of the drop. This produced little redistribution of the solute. The difierence in the thickness of the rings of drop 1 and 2 can also be accounted for qualitatively by our theory. The theory predicts that the flrst ring will grow at early time as t 2=(1+‚)and that the second ring will grow as t2(because Js(r;t)=Jois constant and therefore ‚= 0). Since the two curves must tend towards the same value at the end of the drying (i.e. all the mass must be transferred) the initial growth rate must be larger for the singular evaporationcase. The inner edge of the ring can be considered as a solid-liquid interface that moves inward as material arrives.The propagation velocity of the interface is inversely proportional to the height of the liquid surface there since thedeposited material piles up to height h at that point (i.e. the solute piles up to the maximum height allowed by theliquid surface). The height is frozen-in by the solute accumulated at earlier times; a faster initial growth rate leadsto a freezing in of a larger height. Therefore the interface velocity is smaller and thinner rings are grown. Using the same experimental setup as used we to measure the velocity, we counted the total number of microspheres arriving at the contact line at early times. This data is plotted in Fig. 10a. The solid curve overlaying the data isa one-parameter flt to Eq. (10) with the exponent determined from the measured contact angle. The prefactor toEq. (10) is: 5 ‘`Ntanµc 2µ(dV=dt )(1 +‚) 2‚…R2¡‚tanµc¶2=(1+‚) (12) where‘is the length of the contact line observed, `Nis the number of microspheres per unit volume, and dV=dt is the evaporation rate of the drop in units of volume per unit time, which can be approximated by V=tfwhereV is the initial mass of the drop and tfis the total drying time. Despite the excellent flt to the expected power law there is a discrepancy in the value of the prefactor: given the parameters of our experiment, the predicted value ofthe prefactor is 3.5 but the flt in Fig. 10a gives a value of 1.7. Therefore, at early times only about 50% of expectedsolute is reaching the edge. Figure 10(b) shows data for the growth of the ring throughout the drying time. Superimposed on the data is a one-parameter flt of the data to a curve obtained by numerical integration of Eq. 11. The theoretical curve is asexpected from the above analysis: a steadily increasing growth rate at early times (power-law behavior) followed by alate-time surge in the growth rate. The flt accounts well for the early time behavior, as it did in Fig. 10(a), however,the theory fails to predict a late-time surge as great as the one seen experimentally. If we force the theory curveto terminate at the same point in ( N;t)-plane as the data, then the theory curve will be ofiset above the data by a multiplicative factor, which for this data set is about 2. This is in agreement with the 50% shortfall we found above.To summarize, the experimental data shows and the theory predicts a complete transfer of the solute to the contactline, however, the theory predicts that the material arrives at the contact line earlier than it actually does. The mass distribution of solute was measured throughout the drying time. A one microliter drop of 0 :1„°uorescent microspheres at 10 ¡4volume fraction was placed on a microscope coverslip, isolated from ambient light sources and air currents, and imaged using °uorescent video microscopy. The intensity of °uorescent light, I(r;t), was calibrated and found to be proportional to the number of particles present. A representative set of results for I(r,t) is shown inFig. 11 for the drop at several stages of evaporation. The solid lines are the proflles for the same reduced time, t=t f, calculated by numerical integration of Eq. 9. Comparison of the calculated curves with the data suggests the presence of extra solute around r’0:8R. The efiect is more pronounced in drops with larger-sized microspheres where we observe directly a layer of excess materialas shown in Fig. 12(a). The layer responds to mechanical probing with a needle as though it were a solid fllm; itresists shear and it can be fracture. Identifying the surface layer with the excess density in Fig. 11, we estimate fromthe integrated difierence between the observed and theoretical proflle that the surface layer accounts for 10% of thetotal solute mass in the drop. We observe that the surface cap grows at the expense of the ring as solute is reroutedback to the center by a inward surface °ow. At late times the surface cap breaks-up and the particles in it are draggedto the contact line. VI. DISCUSSION Additional efiects not accounted for in the theory can modify the deposition signiflcantly. Among these are vertical mixing, viscous stresses, efiects of solute interaction, difiusion, sedimentation, and Marangoni efiects. A homogeneousvertical distribution of solute was assumed for Eq. 9 which is equivalent to assuming that vertical mixing is vigorous.It is likely that this is not realized, so that the theory would need to be modifled to include stratiflcation efiects.Viscous stresses may in principle perturb the drop away from the spherical-cap shape we assumed. As the velocity diverges near the edge so does the viscous term in Navier-Stokes equation ( r 2v»@2 zv»v=h2) so that at some flnite distance from the edge the surface of the drop will be distorted. For low viscosity liquids, such as water, this lengthscale is insigniflcant. However, for higher viscosity liquids it may become a signiflcant fraction of the radius. Formore concentrated solution the viscosity of the °uid is altered and the physical dimensions of the ring may be large.Both of these efiects may distort the shape of the drop. If the size of the solute particles is small, difiusive currentsbecome comparable to the advective currents and Eq. 9 would need to modifled. Finally, if the size of the soluteparticles becomes too large, sedimentation of the solute will afiect the amount of material arriving at the edge. Thismay account for some of the shortfall discussed earlier. These efiects provide additional avenues to be explored insurface deposition processes. We believe that Marangoni °ows are a signiflcant factor in our experiments. For a water drop drying on a glass substrate the apex of the drop ought to be coolest because it is farthest from the heat source provided by the substrate.Surface tension decreases with rising temperature, and therefore the surface tension would be greatest at the pole ofthe drop. This gradient would drive a radially inward surface °ow. However, it is di–cult to obtain a quantitativeprediction of the °ow speed because there are a number of additional efiects of unknown strength{such as surfacepressure due to adsorbed surfactant or solute, and surface viscosity in the presence of adsorbed molecules{that tendto counteract Marangoni °ows. 6 Nonetheless, the phenomena we observe in the drop is qualitatively in agreement with what is expected given the temperature distribution. We expect and observe an inward °ow. We also expect that there will be a stagnation pointbeyond which the surface °ow will be outward. This reversal of direction would occur at the radial point at whichthe drag on the surface from underlying liquid, which is diverging with the velocity as the contact line is approached,overcomes the oppositely directed surface tension gradient (see Fig. 12(b)). We observe such a stagnation point.Furthermore, we expect that this point will move towards the center of the drop because the overall scale of the bulk°ow velocity, and hence the drag on the surface, increases as the height of the drop decreases. The ultimate dispersalof the surface cap observed experimentally is consistent with this scenario. Given the 50% shortfall measured for the ring growth, it is appropriate to question the success of our theory. The pragmatic answer is that the theory is an increasingly useful predictive tool in the limit of small solute particles. Thering growth measurement was done with 1 „particles and the theory could account for 50% of the observed behavior. For the concentration measurement 0.1 „particles were used and accuracy of the theory improved signiflcantly. For solute of molecular size there is no discernible stratiflcation as is seen in drops of 1 „-sized microspheres (Fig. 12). Another answer is that our theory captures the essential mechanism for ring formation and that the understandingachieved through it is not diminished by the presence of other types of °ows that ultimately do not alter the fact thatthe 100% of the solute is transferred to the edge. VII. CONCLUSION Despite its shortcomings, our theory is a useful tool for understanding and predicting the ring formation process. It accounts for the widespread occurrence of solute rings, because few ingredients are required{a weakly pinningsubstrate and evaporation{and these ingredients occur commonly; it accounts in a natural way for nearly completetransfer of the solute to the periphery; and it indicates how manipulating the vapor fleld around the drop provides ameans of controlling the deposition process. The potential for controlling deposition on a surface using this contact line deposition remains virtually unexplored. Two aspects seem particularly promising. On the applied side, this mechanism provides an means of concentratinga deposit into flne lines which may rival or exceed what is currently possible using lithographic or etching processes.Such lines can be useful for exploring electronic quantum conflnement or for writing information at high density.Achieving narrow deposits requires increasing dilution of the depositing solute. Thus we are led to explore contact-line deposition when the solute becomes more and more dilute. On the fundamental side, our mechanism seems torely on a mutual reinforcement between contact line pinning and deposition. This reinforcement appears crucial tothe ring-forming process, but has yet to be explored. This reinforcement must weaken, and ultimately fail, as thesolute is diluted. The mechanism of this failure would seem to have widespread implication of its own. Here too, theregime of interest is the regime of highly dilute solutes. Our studies of this regime are in progress. ACKNOWLEDGEMENTS We thank H. Li, X. Shi, and M. Baildon for their early contributions to this project; J. Crocker, D. Grier, and A. Marcus for sharing their expertise, their image analysis code, and their facilities; and S. Garofi, L. Mahadevan, S.Esipov, R. Leheny, D. Mueth, E. Ehrichs, J. Knight, S. Blanton, N. Menon, J. Cina, L. Kadanofi, T. Halsey, andH. Zhou for valuable discussions and suggestions. This work was supported by NSF grant DMR-9410478 and theMRSEC Program of the NSF under Grant No. NSF DMR-9808595. APPENDIX A: THE SOLUTION TO THE BOUNDARY VALUE PROBLEM OF A LENS SHAPED CONDUCTOR. The problem a spherical cap on an impermeable substrate can be replaced by the problem of a lens (as shown in Fig. 3) in free space. The general solution for the electrostatic potential of a charged conductor with a shapedeflned by two intersecting spheres is given in reference [18]. This solution for our speciflc problem (translated intothe language of difiusion) is: u(fi;fl)=u 1+(us¡u1)p 2(coshfi¡cosfl) £Z1 0d¿P¡1=2+{¿(coshfi)coshµc¿cosh(2…¡fl)¿ cosh…¿cosh(…¡µc)¿(13) 7 whereP¡1=2+{¿(x) are Legendre functions of the flrst kind, fiandflare toroidal coordinates in which toroidal surfaces are labeled with fiand spherical surfaces with fl, and the two surfaces are at fl1=3…¡µcandfl2=…+µc. Since we are interested in the evaporation at the surface we calculate J=¡Drunormal to the surface: Js=D(coshfi+ cosfl)@flujfl=3…¡µc=¡D(us¡u1) £• sin(µc)=2+p 2(x+ cosµc)3=2Z1 0d¿P¡1=2+{¿(x)¿coshµc¿tanh(…¡µc)¿ cosh…¿‚ (14) wherer=Rp 1¡x2=(x+ cosµc) whereRis the radius of the drop and xis a parameter ranging from 0 to 1. The value of Eq. 14 was computed numerically and compared to the approximate form given in Eq. 6. In Fig. 13 bothJexact andJapproximate are plotted. The inset is a plot of the percent difierence between the two Js; as can be seen the error is no greater than 10%. [1] N. D. Denkov, O. D. Velev, P. A. Kralchevsky, I. B. Ivanov, H. Yoshimura and K. Nagayama, Langmuir 8, 3183 (1992). [2] A. S. Dimitrov, C. D. Dushkin, H. Yoshimura, and K. Nagayama, Langmuir 10, 432 (1994). [3] T. Ondar» cuhu and C. Joachim, Europhysics Letters ,42, 215 (1998). to be published. [4] J. Boneberg, F. Burmeister, C. Sh˜ a°e, P. Leiderer, D. Reim, A. Fery, and S. Herminghaus, Langmuir 13, 7080 (1997). [5] A. B. El Bediwi, W. J. Kulnis, Y. Luo, D. Woodland and W. N. Unertl, Mat. Res. Soc. Symp. Proc. 372, 277 (1995). [6] F.Parisse and C. Allain, J. Phys II 6, 1111 (1996). [7] F. Parisse and C. Allain, Langmuir 13, 3598 (1996) [8] E. Adachi, A. S. Dimitro, and K. Nagayama, p.419 in Film Formation in Waterborne Coatings Edited by T.Provder, M. A. Winnik, and M. W. Urban (American Chemical Society, Washington, DC, 1996). [9] E. Adachi, A. S. Dimitro, and K. Nagayama, Langmuir 11, 1057 (1995). [10] J. Conway, H. Korns, and M. R. Fisch, Langmuir ,13, 426 (1997) [11] R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, and T. A. Witten, Nature 389, 827 (1997). [12] J. D. Jackson, Classical Electrodynamics , (J. Wiley, New York, 1962) (1st edition). [13] J. T. Davies & E. K. Rideal, Interfacial Phenomenon , (Academic Press, New York, 1963). [14] This is a well known result. For example, see K.Histek et al, Journal of Applied Physics ,73, 7395, (1993). [15] C. Bourgµ es-Monnier and M. E. R. Shanahan, Langmuir 11, 2820 (1995). [16] M. E. R. Shanahan, Langmuir 11, 1041 (1995). [17] J. C. Crocker and D. G. Grier J. Colloid and Interface Science 179, 298 (1996). [18] N. N. Lebedev, Special Functions and Their Applications , (Prentice-Hall, New Jersey, 1965). CAPTIONS: Fig.1: (a) cofiee stain; (b) mineral deposit on washed glassware; (c) salt deposits on sidewalk; (d) water color painting Fig.2: Schematic illustration of the origin of the advective current. (a) When the contact line is not pinned, uniform evaporation removes the hashed layer, the interface moves from the solid line to the dashed line, and the contact linewill move from A to B. However, if the contact line is pinned then the motion from A to B must be prevented by anout°ow to replenish the liquid removed from the edge. (b) Shows the actual motion of the interface. Fig.3: Schematic of relevant parameters for the theory. The thick solid line represents the air-liquid interface of half a drop viewed from the side. The contact line is deflned as the intersection of the interface with the dashed linedenoted \Plane of Symmetry". In an experiment the substrate would deflne the plane of symmetry. The drop isaxisymmetric so that the vertical dashed line denoted the \Axis of Symmetry" is the axis of rotation. The dashedcurve is the re°ection of the interface through the plane of symmetry and shows how the boundary value problem ofa drop, detailed in the text, can be replaced by the boundary-value problem of a lens with surfaces deflned by thethick solid line and dashed curve line. The diagram also indicates the origin of the divergent evaporation current atthe contact line: the discontinuous change in the curvature that occurs as one passes from the upper surface (solid)to the lower surface (dashed). Fig. 4: (a) Shows how the probability of escape of an evaporating molecule is afiected by its point of departure. A random walk initiated at the center of the drop results in the molecule being reabsorbed so that the flnal step is 8 not completed (indicated by the dashed line). However, the same random walk initiated from the edge allows the evaporating molecule to escape. This demonstrates why the evaporation rate is larger at the edge. (b) A clovershaped drop of dried cofiee. The extremeties of the drop have the darkest deposits because the probability of escapefrom these regions is greatest. (c) Two drops dried simultaneously side by side. Region of greatest proximity to eachother shows the weakest deposit because, again, the evaporation rate is lowest where the vapor flelds of the two dropsoverlap. Fig. 5: Illustrates why v(R¡r) must scale like J s(R¡r) Near the contact line, the compensation for liquid leaving by evaporation through the interface must be increasingly shouldered by the in°ow of liquid because the decrease involume due to lowering the surface becomes increasingly negligible near the contact line. Fig. 6: (a) The mass of the drop, M, versus time, t. (b) The rate of mass loss by evaporation, dM=dt , normalized to 0% relative humidity and 20 oC versus the radius of the drop. The data is well flt by the form fiRindicating that the evaporation rate is difiusion limited. The fltting parameter fi=1 6:1§0:4 g/cm-s is in excellent agreement with the value of 16.5 g/cm-s, calculated from Eq. 14. Fig. 7: Plot of the position of the interface above the substrate versus distance from center in units of pixels The data are taken at equal intervals of time. The solid lines running through the data are flts to a circular proflle withthe same value of Randt fused for all flts. Fig. 8: The log of the velocity of solute, v, versus the log of the distance from the contact line, R¡r. The solid line is the best flt to a power law and yields an exponent of 0.54. Fig. 9: Photographs of the resulting deposit left by three identical drops dried under difierent conditions are shown in the left column. The flrst was allowed to evaporate normally. The second was mounted on a pedestal andsurrounded by a bath of water such that the level of the bath coincided with the base of the drop. In this case, theevaporation rate is to good approximation spatially uniform. The third drop was surrounded by a chamber with asmall hole above the center of the drop so that evaporation primarily occurred at the center. The evaporation rate is plotted schematically in the right column. As discussed in the text, the flrst drop has a singular proflle, the second drop has a uniform proflle, and the third has a proflle that is largest at the center andgoes to zero at the edge. The second curve (dashed line) in the right column represents the change of the interfaceheight. When the change of the interface height is difierent from the evaporation proflle, a ring is produced as in theflrst and second drop. However, if proflles are similar then a uniform deposit is formed. Also note the difierence inthe ring’s thickness in (a) and (b). Fig. 10: (a) The number of microspheres ( N) arriving at a section of the contact line of length ‘, versus time, t. For this particular run t f=326 s,`N=5:4£106mm¡3,‘=2 1„,R=2:08 mm, and µc=0:26 radians. The solid line i safl tt oN=Np(t=tf)1:38which yields a value of 1 :7 forNp. As elaborated in [11], the Noandtoare adjustments to compensate for the transients as the evaporation rate decays to its steady-state value. For this plot, Nowas set to 2 andt0was set to 2. (b) Nvstmeasured throughout the drying time. The inset shows a blow up of the late time behavior (note change to linear scale). Fig.11: Intensity of light, I(r;t), for various times from a drop containing °ourescent microspheres versus the scaled radius,r=R. The intensity of light is proportional to h(r;t)c(r;t) and the factor of proportionality is determined by comparing I(r;0) withh(r;0)c(r;0). The solid lines are curves numerically calculated from Eq. 9 and converted into intensity curves using the proportionality factor. The systematic bulge around ~ r’0:8 corresponds to an unanticipated congregation of solute on the surface of the drop. The growth of the ring can be seen in the inset. Fig.12: (a) The surface excess layer of solute revealed by dragging a wire through the drop. (b) Circulation pattern in drop due to temperature driven surface °ow. Fig.13: The value of Jscalculated with the full analytic form compared to the approximate form. The two curves are approximately equal. As is shown in the inset, the upper bound on the difierence is 10%. 9 Figure 1 10 Figure 2 11 Figure 3 12 Figure 4 13 Figure 5 14 Figure 6 15 Figure 7 16 Figure 8 17 Figure 9 18 Figure 10 19 Figure 11 20 Figure 12 21 Figure 13 22