Dynamics of evaporative colloidal patterning
C. Nadir Kaplan, Ning Wu, Shreyas Mandre, Joanna Aizenberg, L. Mahadevan
aa r X i v : . [ c ond - m a t . s o f t ] D ec Dynamics of evaporative colloidal patterning
C. Nadir Kaplan , Ning Wu , Shreyas Mandre , Joanna Aizenberg , , , , and L. Mahadevan , , , School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA. Department of Chemical Engineering, Colorado School of Mines, Golden, CO 80401, USA. School of Engineering, Brown University, Providence, RI 02912, USA. Wyss Institute for Biologically Inspired Engineering, Harvard University, Boston, MA 02115, USA. Kavli Institute for Bionano Science and Technology,Harvard University, Cambridge, MA 02138, USA. Department of Chemistry and Chemical Biology,Harvard University, Cambridge, MA 02138, USA Department of Physics, Harvard University, Cambridge, MA 02138, USA. (Dated: July 16, 2018)Drying suspensions often leave behind complex patterns of particulates, as might be seen in thecoffee stains on a table. Here we consider the dynamics of periodic band or uniform solid filmformation on a vertical plate suspended partially in a drying colloidal solution. Direct observationsallow us to visualize the dynamics of the band and film deposition, and the transition in betweenwhen the colloidal concentration is varied. A minimal theory of the liquid meniscus motion alongthe plate reveals the dynamics of the banding and its transition to the filming as a function ofthe ratio of deposition and evaporation rates. We also provide a complementary multiphase modelof colloids dissolved in the liquid, which couples the inhomogeneous evaporation at the evolvingmeniscus to the fluid and particulate flows and the transition from a dilute suspension to a porousplug. This allows us to determine the concentration dependence of the bandwidth and the depositionrate. Together, our findings allow for the control of drying-induced patterning as a function of thecolloidal concentration and evaporation rate.
I. INTRODUCTION
Colloidal self organization occurs in systems such asopals [1], avian skin [2], photonics [3–5], and tissue en-gineering [6]. An approach to colloidal patterning isvia evaporation-driven deposition of uniformly dispersedparticles in a volatile liquid film [7–15]. The basic mech-anism of evaporation-driven patterning involves vaporleaving the suspension more easily along the liquid-air-substrate triple line (the contact line), resulting in a sin-gular evaporative flux profile. The combination of thecontact line pinning and a singular evaporative flux theregenerates a fluid flow that carries the dissolved particlestowards the edge of the film [10, 11]. The advected col-loids then get arrested near the contact line to form pat-terns such as continuous solid films, or regular bands [16–19], that are laid down along the substrate.The evaporative patterns form via a complex dynam-ics. The propagation rate of the interface separating theliquid and the colloidal deposit is controlled by the localparticle concentration in the solution (Fig. 1(a)) and thevelocity of the viscous capillary flow transporting the col-loids. During this process, the liquid meniscus pinned tothe edge of the deposit deforms as a function of the rateof particle transport and evaporation, in turn dictatingthe formation of either a continuous film or a periodicband (Fig. 1(b)–(e) and Fig. 2(a)–(f)). As a result, theparticles self-organize into various forms of ordered anddisordered states[20] as a function of the deposition speedand the local evaporation rate [8, 21]. An additional com-plexity is that the fluid flow regime changes dramaticallyover the course of the drying process. Initially, we haveflow in a thin film that is characterized by the Stokes regime away from the deposition front where the parti-cle concentration is low. Near the deposition front, theliquid enters a porous region that is itself created by theparticulate deposits at the solid-liquid interface, leadingto a Darcy regime. Early models [16, 17, 19, 22, 23] fo-cused on understanding the singular evaporative flux andthe related particulate flux, leaving open mechanisms forthe filming-banding transition, the deposition front speedthat sets the rate of patterning, and the Stokes-Darcytransition, questions we answer here.Here we use a combination of experimental observa-tions and theoretical models of the interface growth andcolloidal patterning to understand the dynamics of pe-riodic banding, and its transition to the deposition ofa continuous film as a function of the particle concen-tration. Based on our observations, we formulate twocomplementary theories: (1) A coarse-grained two-stagemodel, which consists of a hydrostatic stage until themeniscus touches down the substrate, followed by rapidcontact line motion terminated by its equilibration. Thisminimal model allows us to explain the geometry of theperiodic bands as a function of the deposition rate. (2) Adetailed multiphase model of the drying, flowing suspen-sion allows us to account for the Stokes-Darcy transition,and couples the evaporation rate, fluid flow, the menis-cus height, the distribution of particle concentration, andthe dynamic interface velocity as a function of the initialparticle concentration. This theory leads to explicit pre-dictions for the deposition rate, and the banding-filmingtransition in good quantitative agreement with the mea-surements.The present study is organized as follows: Experi-ments are described in Sec. II. Based on experimental (cid:1) (cid:2) (cid:3)(cid:4)(cid:5)(cid:1) (cid:2)(cid:6)(cid:7) (cid:3)(cid:8)(cid:5)(cid:3)(cid:9)(cid:5) (cid:10)(cid:3)(cid:11) (cid:8) (cid:6)(cid:12)(cid:4)(cid:5) (cid:13)(cid:3)(cid:14)(cid:5)(cid:14)(cid:15)(cid:7)(cid:14)(cid:15)(cid:16)(cid:3)(cid:4)(cid:5) (cid:17)(cid:3)(cid:14)(cid:6)(cid:4)(cid:5) (cid:14) (cid:18) (cid:17) (cid:19) (cid:20)(cid:21)(cid:22) (cid:23) (cid:24)(cid:8) (cid:23) (cid:4) (cid:25) (cid:9) (cid:4) (cid:2) (cid:3)(cid:18)(cid:5) (cid:3)(cid:2)(cid:5)(cid:3)(cid:21)(cid:5) (cid:13) (cid:7) (cid:14) (cid:26)
FIG. 1.
Schematics of a drying suspension on a verti-cal substrate. (a) A schematic of the geometry and variabledefinitions for banding on a vertical substrate. (b), (c), (d),(e) The evolution of the meniscus deformation while leavingbehind a colloidal deposit, and the corresponding variable def-initions. Grey full line indicates the location of the solid-liquidinterface, while the red curves represent the local thickness ofthe solid deposit. evidence, we develop the minimal two-stage model inSec. III. This model is complemented by the multiphasemodel in Sec. IV. Concluding remarks are given in thefinal section of the paper.
II. EXPERIMENTS
Methods.
Our experiments were performed by partlyimmersing a vertical unpatterned silicon and glasssubstrates in a dilute colloidal suspension of colloidalspheres (see Fig. 1(a) for the experimental setup).Silicon and glass substrates were used for evaporativecolloidal coating. They were first cleaned in a mixtureof sulfuric acid and hydrogen peroxide (3:1) at 80 o Cfor one hour. They were then rinsed with deionizedwater thoroughly and treated with oxygen plasma for 1minute immediately before use. Colloidal particles wereeither synthesized (375 nm PMMA spheres) by usingsurfactant-free emulsion polymerization or purchasedfrom Life Technologies (1 µ m latex spheres). Beforethe evaporative deposition, particles were centrifugedfour times and were re-dispersed in deionized water.The substrate was mounted vertically and immersedpartially in a vial containing the colloidal suspension.Water was evaporated slowly over a period of ∼
12 hr to2 d in an oven that was placed on a vibration-free table.The solvent evaporation rate was controlled by thetemperature of the oven and was measured by putting asecond vial filled with colloidal suspension but withoutthe substrate. Images of the band structures andcolloidal packing were taken by both optical microscopy(Leica DMRX) and scanning electron microscopy (Zeiss Ultra). A custom-built side view microscope (OlympusBX) was also used to image the in situ movement ofmeniscus and colloids during the band formation onglass substrates.
Results and discussion.
As evaporation proceeds,two types of patterns are observed near the contact line.When the bulk volume fraction Φ b inside the reservoir isbigger than a critical value Φ c , a continuous film of parti-cles was deposited by the receding contact line (Fig. 2(f)and Movie S1 in Supporting Information). However,when Φ b < Φ c , the contact line retreated leaving behinda periodic pattern of colloidal bands (Fig. 2(a)–(e), Fig. 3(a), (b), and Movie S2 in Supporting Information). Thebands are oriented locally parallel to the receding con-tact line; the small curvature of each band in Fig. 3(a)results from the finite size of the substrate. In Fig. 2(g),we show the values of the width ∆ d of a single band andthe spacing between adjacent bands d , against Φ b .High magnification optical images of a typical colloidalband (Fig. 3(c)) show a range of interference colors thatcorrespond to deposits with 1 (magenta), 2 (green), and3 (orange) particle layers at Φ b = 6 × − . The de-posits have a strongly asymmetric cross-sectional shape,as evidenced by scanning electron microscopy (SEM): asthe meniscus recedes, gradual deposition of wide, well or-dered colloidal layers takes place (Fig. 3(d), (e)). Whenthe band stops, a region of randomly packed particleswith a sharp deposition front terminates the colloidalband left behind the moving liquid meniscus (Fig. 3(f)).In this region that is only a few particles wide, the pack-ing is disordered (Fig. 3(f)). The transition from theordered to the disordered packings is attributed to therapidity of flow at the end of a deposition cycle, not leav-ing time for the colloids to anneal into an ordered struc-ture [8]. Once a deposition cycle is complete, the result-ing cross-sectional profile of the deposit at Φ b = 6 × − is shown in Fig. 3(g). The number of maximum layers in-crease as a function of Φ b , as demonstrated in Fig. 2(h).Light microscopy study allows us to monitor themovement of the meniscus imposed by the evaporation.Fig. 4(a)–(d), and Movie S3 in Supporting Informationshow that while the deposition front advances, the menis-cus approaches the substrate, as evidenced by the emerg-ing skewed interference rings behind the contact line(Fig. 4(e)). Once the meniscus touches the bare substrateand dewets, it breaks up and separates into two mov-ing dynamic contact lines (Fig. 4(b)). One of these re-treats towards the just formed colloidal band (Fig. 4(c)),which wicks the fluid, dries and changes its optical con-trast (Fig. 4(d)), while the other contact line slips untilits dynamic contact angle θ D re-equilibrates on the sub-strate over a time scale T D (see Table I). Colloidal par-ticles then flow towards the stabilized contact line andstart to build a new band at that location. Interferencepatterns allow us to measure the height of the fluid filmfrom the substrate, as shown in Fig. 4(f) in the frame ofthe deposit-liquid interface. Bandwidth Spacing Period θ e restoration The deposition Critical∆ d ( µ m) d ( µ m) T (s) time T D (s) speed C (m/s) concentration Φ c Experiment 70 100 500 10 < E . × − Mimimal model 100 150 200 6 0 . E –Multiphase model 55 – 200 – 0 . E . × − TABLE I. Quantitative comparison of structural variables between the experiments and theoretical models. All values areapproximate. The measurements in the experiments were obtained for the bulk colloidal volume fraction Φ b ∼ − . The bulkevaporation rate in the reservoir is E ∼ − m/s for water in atmospheric pressure and room temperature. For Φ b ≈ × − ,the deposition height is H ∼ µ m. The outputs of the minimal model correspond to ǫ = H/ℓ = 0 .
01 (Table II) and thedimensionless deposition rate β ≡ C/E = 0 . . The results of the multiphase model are given for ǫ = 0 .
01 and Φ b = 2 × − . The wavelength is given by d + ∆ d . (a) (b) (c) (d) (e) (f) (g) (h) FIG. 2.
Periodic bands and uniform films of colloidaldeposits on a silicon substrate placed vertically, as afunction of the colloidal volume fraction.
The volumefraction of colloids inside the suspension are; (a) Φ b = 8 × − , (b) Φ b = 2 × − , (c) Φ b = 6 × − , (d) Φ b =1 . × − , (e) Φ b = 2 . × − , (f) Φ b = 4 × − (uniformfilm). In (a)–(f) the scale bars are 200 µ m. (g) Band spacing d ( (cid:4) ) and bandwidth ∆ d ( ) are plotted with their error bars,as a function of the colloidal volume fraction. The wavelengthis given by d + ∆ d . (h) Layer number is plotted as a functionof the colloidal volume fraction. (cid:1) (cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3)(cid:1)(cid:5)(cid:3)(cid:1)(cid:6)(cid:3) (cid:1) (cid:7) (cid:3) (cid:1)(cid:8)(cid:3)(cid:1)(cid:9)(cid:3) FIG. 3.
Periodic bands in a drying dilute suspension( Φ b = × − ). (a), (b) Optical micrographs show peri-odic bands at different magnifications ((a) scale bar: 1 mm,(b) scale bar: 200 µ m). The vertical arrow in (a) indicatesthe direction of meniscus movement during evaporation. (c)Optical image of a single band (scale bar: 40 µ m). The profileof the band is asymmetric with significant differences betweenthe advancing (gradual transition to a thicker layer and or-dered packing) and receding side (abrupt transition and ran-dom packing). (d), (e), (f) SEM images at different locationsof the colloidal deposit in (c) (scale bar: 2 µ m). (g) Cross-sectional profile of the band shown in (c). Position z is scaledwith the width of the band ∆ d . The meniscus moves fromleft to right. To explain the experimental results, we must accountfor the dynamics of meniscus deformation and break-up,the subsequent receding of the contact line, the role ofparticulate flow, and the transition from a dilute to adense suspension in the vicinity of the deposition front.Thus, we first build a minimal model of the periodicbanding and uniform filming, and then complement itwith a multiphase approach. (cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3) (cid:1)(cid:5)(cid:3) (cid:1)(cid:6)(cid:3)(cid:1)(cid:7)(cid:3) (cid:1)(cid:2)(cid:3)(cid:4) (cid:1)(cid:2)(cid:5)(cid:4) (cid:1)(cid:2)(cid:6)(cid:4) (cid:1)(cid:2)(cid:7)(cid:4)
FIG. 4.
Real-time observations of periodic banding. (a), (b), (c), (d) A sequence of snapshots shows the meniscusbreak-up and the subsequent contact line motion during theformation of a new band on a vertical glass substrate in asuspension (0.06 vol % 1 µ m latex particles in water, scalebar: 20 µ m). (e) A close-up of the red box in (a) shows theinterference fringes associated with the meniscus approach-ing the substrate (scale bar: 20 µ m). (f) The meniscus pro-file h ( z, t ) at t = 0 , , z / √ ℓ c a p meniscus ∆ dd (a) h ( z ) /H
0 0.5 1 0123 ∆ d / √ ℓ c a p β (d)0 0.5 1 0100200300 T / τ (b) β β h m a x / H (c)0 0.5 1 0.050.10.150.20.25 d / √ ℓ c a p β (e) FIG. 5.
Theoretical models of the formation of peri-odic bands and uniform films. (a) After a time period
T , a single band has formed with the width ∆ d (projected lengthof the deposit shown by the red curve), spacing d (turquoiseline), and the maximum deposition height h max (green line).The left-over tail of the receding meniscus is demonstratedby the black dashed line ( ℓ cap is the capillary length, seeTable II). The scaled dependence of (b) the period T , and(c) the maximum band height h max (green line in (a)), (d)the bandwidth ∆ d , and (e) the spacing between bands d onthe dimensionless deposition front speed β ≡ C/E , where τ ≡ H/E is the time scale (Table II). All curves in (a)–(e)are numerical solutions to the Eqs. (4)–(7) with ǫ = 0 . β = 0 . θ e, = 15 o . III. MINIMAL MODEL FOR BANDING ANDFILMING
To quantify the dynamics of the filming-banding tran-sition and the deposit height as a function of the relativemagnitude of evaporation and particle deposition rates,we must account for the kinetic effects associated withthe meniscus deformation, break-up, and the subsequentcontact line motion. Thus, we first develop a minimalhydrostatic model of the periodic band formation. Thismodel also explains the termination of each band by asharp deposition front, as observed in experiments. Al-though the meniscus break-up starts as a localized eventbelow the maximum of the forming, convex band (seeFig. 4), it then spreads laterally across the plate, result-ing in a well-defined periodic banding pattern (see Fig. 3(a), (b)). Therefore, we will limit ourselves to shapes andmotions in a two-dimensional x − z plane (Fig. 1).The liquid meniscus deforms hydrostatically when thecapillary forces, proportional to the surface tension γ ofthe liquid-air interface, dominate over the viscous hy-drodynamic forces µv f . Here µ is the dynamic viscosityof the liquid, and v f is its evaporation-induced upwardflow velocity in the vicinity of the substrate. Away fromthe far edge of the deposit on the plate, v f is in the or-der of E , the evaporation rate at the level of the bath.The condition for quasistatic meniscus evolution then be-comes Ca ≡ µE /γ ≪ , where Ca is the capillary num-ber (Table II). In this regime, the diverging evaporativeflux at the contact line alters solely v f near the singu-larity; it has no effect on the overall meniscus evolution.Then, the hydrostatic meniscus profile h ( z, t ) is deter-mined by the equilibrium condition p = p , where p is the atmospheric pressure, and the pressure p at themeniscus is given by p = − γκ − ρg [ z − L ( t )] + p . (1)Here κ ≡ ∂ (sin θ ) /∂z is the curvature of the liquid-airinterface, where the local angle θ = θ ( z ) is defined astan θ ≡ ∂h/∂z , the density of the suspension is denotedby ρ , and the gravitational acceleration by g . The dynamics of the meniscus deformation is drivenby two processes that act simultaneously. First,evaporation-induced flow results in the deposition of thesolute near the contact line with a speed C (Φ b , t ) , mov-ing the liquid-deposit wall where the meniscus is attached(Fig. 1(a)). Second, the level of fluid inside the containerdescends with a constant speed E (Fig. 1(a)). Defining L ≡ L ( t ) as the distance between the vertical level of thebath and the deposition front (Fig. 1(a)), which moverelative to each other, L ( t ) changes at a rate given by dLdt = E − C , E , C > . (2)When the bulk volume fraction is smaller than the crit-ical volume fraction (Φ b < Φ c ), the deposition happensslower than the descent of the liquid level inside the bath( C < E ) and L ( t ) increases over time ( dL/dt > L ( t ) changes the curvature of the concavemeniscus due to the gravity when p = p (see Eq. (1)),the dynamic contact angle θ e ( t ) becomes smaller thanthe equilibrium contact angle θ e, , and decreases mono-tonically in time ( dθ e /dt < , see Fig. 1(b)–(e)). Thenthe meniscus touches down on the substrate at a loca-tion z c behind the deposition front (Fig. 1(e)). Thiscauses the meniscus to break to form two contact lines,one that moves towards the deposit, and another thatrecedes rapidly until it eventually re-equilibrates at adistance where the contact angle regains its equilibriumvalue θ e, . As the process repeats, periodic bands areformed. Conversely, when Φ b > Φ c , C > E , so that dL/dt <
0. Then, θ e > θ e, and dθ e /dt >
0, so that themeniscus moves away the plate everywhere, and a con-tinuous deposition film will be laid out by the depositionfront in this regime. The transition between the bandingand filming happens at Φ b = Φ c where the evaporationand deposition speeds are matched, i.e. E = C (Φ c ). Inthis case the size of the domain will always be equal toits equilibrium value L = L . In the frame comoving with the deposit-liquid wall ata dimensionless deposition speed β ≡ C/E , the wall isalways at z = 0 (Fig. 1(a)), where θ = θ e ( t ) (Fig. 1(c)).In order to calculate the height of the fluid film h ( z, t ) , Eq. (1) needs two boundary conditions: (i) ∂h/∂z →∞ at z = L ( t ) , (ii) h ( z = 0 , t ) = h i ( t ) in the movingframe, where h i ( t ) is the time dependent height of thesolid-liquid interface (Fig. 1(d)). In the rest frame, h i satisfies ∂h i /∂t = 0 , since the deposit is assumed to beincompressible. In dimensional units, this condition isrewritten in the moving frame as ∂h∂t = C ∂h∂z (cid:12)(cid:12)(cid:12)(cid:12) z =0 . (3)Eq. (3) satisfies the local conservation of the solute masswhen the particle concentration at the interface is aslowly varying function in space and time.We describe our model in dimensionless units for con-venience, indicated using tildes, by z ≡ ℓ ˜ z , h ≡ H ˜ h , L ≡ ℓ ˜ L , and t ≡ τ ˜ t (see Table II for the length and timescales). Using the boundary condition ∂ ˜ h/∂ ˜ z → ∞ at˜ z = ˜ L (˜ t ) (equivalent to θ → π/ p = p yields the first-order equationsin θ = 1 − ( z − L ) , where θ = tan − (cid:18) ǫ ∂h∂z (cid:19) . (4)Similarly, Eq. (2) in dimensionless units may be writtenas dLdt = ǫ (1 − β ) , (5)where ǫ ≡ H/ℓ (Table II), and the dimensionless depo-sition speed is β ≡ C/E . From Eq. (4), the distancebetween the deposition front at z = 0 and the liquid level inside the bath is obtained as L ( t ) = p − sin θ e ( t ) . The time evolution of L ( t ) is determined by imposing itsequilibrium size as the initial condition, which is given by L ≡ L (0) = p − sin θ e, [24]. In dimensionless units,the boundary condition given in Eq. (6) is rewritten inthe moving frame as ∂h∂t = ǫβ ∂h∂z (cid:12)(cid:12)(cid:12)(cid:12) z =0 . (6)In the banding regime, Eqs. (4)–(6) determine alto-gether the meniscus touch-down location z c ≡ L ( t c ) − t c is the instant of touch-down), which is followed bythe formation of two contact lines with vanishing con-tact angles. Unless the substrate is perfectly wetting(when θ e, = 0), these contact lines are out of equilib-rium and so start to move with a velocity U ; one runsinto the porous deposit, while the other moves with adynamic contact angle θ D until it is restored to its equi-librium value θ D = θ e, . For a dynamic contact line with θ e, ≪
1, the velocity is given by [24, 25] U = U ∗ ξ θ D ( θ e, − θ D ) , (7)where U ∗ ≡ γ/µ ( U ∗ ∼ ξ ≡ log ( ℓ/a ) ∼ a ≡ b/θ a (logarithmic cutoff). Here b is the slip length with amolecular size, and θ a is the apparent contact angle,measured at x = a [24, 25] ( a ∼ − m , b ∼ − m , θ a ∼ − ). The travel time of the dynamic contact line T D can be roughly estimated as follows: When θ e, ≈ o and θ D ≪ θ e, , assuming that the distance d D traveledby the contact line is nearly equal to the spacing betweenadjacent bands, T D ≈ d D /U ≈ µm/ ( U ∗ θ D θ e, / ξ ) ∼ . θ D ∼ o . Eq. (7) is evalu-ated more precisely by assuming a wedge-shaped liquidborder such that h = xθ D ( i.e. κ = 0) [24]. θ D = θ a isextracted from Eq. (7) by replacing U with E , as the ve-locity of the contact line at the meniscus touchdown willbe equal to the rate of decrease of the liquid level insidethe container. Then the distance d and time T D at whichthe contact line travels from the instant of break-up tothe moment of re-equilibration at θ D = θ e, are given by d D = R t ( θ e, ) t ( θ a ) U dt and T D = R t ( θ e, ) t ( θ a ) dt , respectively. Lo-cal mass conservation at x = a yields a dθ D dt − θ D U = 0 inthe frame of the contact line. Defining d D ≡ ℓ ˜ d D , a ≡ ℓ ˜ a , T D ≡ τ ˜ T D , and dropping the tildes leads to the followingdimensionless travel distance and time of the contact line d D = a Z θ e, θ a dθ D θ D , and T D = aAǫ Z θ e, θ a dθ D U θ D , (8)where A ≡ ξE /U ∗ ∼ × − . Then the spacing be-tween two adjacent bands is given by d = d S + d D . Thelength d S of the static left-over tail of the meniscus formswhen the meniscus touches down and leaves behind asmall fluid tail (shown by the dashed curve in Fig. 5(a)).This tail evaporates completely over time and its lengthcontributes to the band spacing d . Substituting the nu-merical values into Eq. (8) and switching back to realunits, the travel distance and time of the contact line arefound as d D ∼ µm and T D ∼ s , in good agreementwith experiments (Table I). The difference in the precisevalue of T D with the rough estimation of T D ∼ .
1s is dueto the fact that the contact line spends a lot of time toincrease its speed when θ D ∼ θ a , as there is a singularityat θ D = 0 in Eq. (7) [24, 25].Given the ratio of the deposition front speed to theevaporation rate β = C/E , the height of the fluid film h ( z, t ) until touch-down is calculated by solving Eq. (4)subject to the boundary condition in Eq. (6), while thedynamics of L is obtained by solving Eq. (5). Once themeniscus touches the substrate, a single band with awidth ∆ d has formed (Fig. 1(e)). From that instant on,the dynamics of the contact line is governed by Eq. (7)until θ D = θ e, , resulting in a spacing d between bands.When contact line motion ceases at θ D = θ e, , one cy-cle is complete. The resulting shape of the band, theassociated structural quantities, and the instantaneousmeniscus profile are shown in Fig. 5(a). Our model alsopredicts the termination of the band by a sharp frontas observed in experiments, since the deposition growthvanishes when the meniscus breaks up.Our minimal theory of banding is in quantitative agree-ment with experiments for e.g. β ∼ . T , h max , and ∆ d in Fig. 5(b)–(d) for the range of β between 0 and 1, whichdiverge when β → . To investigate the scaling behav-ior of these quantities when β → , we first integrateEq. (5), which yields L ( t ) as L ( t ) = L + ǫ (1 − β ) t . (9)At t = t c ≡ T − T D , the critical time of meniscus touch-down, L ( t ) should always stay finite as a function of β . This leads to the scaling form of t c lim β → t c ∼ (1 − β ) − , (10)such that when β → , L ( t c ) > L and is finite. Thus, T also diverges with (1 − β ) − . Similarly, the bandwidthis given by ∆ d = ǫβt c , which, when β →
1, leads tolim β → ∆ d ∼ ǫβ (1 − β ) − . (11)In the limit β → h max as well. While a single bandis forming around h max , ∂h/∂z ≪
1. Then evaluatingEq. (4) in this limit, and substituting the result in Eq. (6)at z = 0 in the moving frame, we obtain the interfacecondition ∂h∂t (cid:12)(cid:12)(cid:12)(cid:12) h = h max = βθ e ( t ) , where θ e ( t ) = 1 − L ( t ) . (12)Integrating Eq. (12) over time, and in the limit β → h max ∼ (1 − β ) − . (13) Parameter Definition Magnitude ℓ ≡ √ ℓ cap length scale ∼ − m ℓ cap ≡ p γ/ρg capillary length ∼ − m τ ≡ H/E time scale 1 −
10 s ǫ ≡ H/ℓ aspect ratio 10 − Ca ≡ µE /γ capillary number ∼ − P e ≡ E ℓ/D s P´eclet number 10 ν ≡ H/ √ kµ scaled deposit thickness 1 α ≡ ǫ /ν Ca dimensionless no. 10 γ surface tension 0.1 N/m ρ density of water 10 kg/m g gravitational constant 10 m/s H deposit thickness 10 − − − m E evaporation rate 10 − m/s µ dynamic viscosity 10 − Pa · sof water D s diffusion constant 10 − m /s k permeability 10 − − − m /Pa · sTABLE II. List of parameters.
Length scales, time scales,and dimensionless numbers (above), and auxiliary physicalparameters (below).
That is, as β →
1, the time instant at which h max formsshould scale again with t ∼ (1 − β ) − .Eqs. (10), (11), and (13) manifest a continuous transi-tion between the formation of uniform solid deposits andperiodic bands. However, the spacing between bands d , shown in Fig. 5(e), depends only weakly on β , a conse-quence of the fact that this is controlled solely by theleft-over fluid tail (the dashed line in Fig. 5(a)) when themeniscus touches down.This minimal model captures the essential features ofbanding and filming, as well as relevant time and lengthscales. However this can only be done in terms of thedimensionless deposition rate β = E /C , which is a freeparameter. Therefore, a more sophisticated model is re-quired to determine the dependence of the front propa-gation speed C = C (Φ b , t ) on the bulk volume fractionΦ b and thence β . This will further allow the determina-tion of the bandwidth ∆ d as a function of Φ b , as well asthe experimentally observed concentration Φ c associatedwith the transition between the two patterns. IV. MULTIPHASE FLOW MODEL FORBANDING AND FILMING
To determine β = C (Φ b , t ) /E and the bandwidth ∆ d in terms of the bulk volume fraction Φ b , as well as thecritical concentration Φ c at the banding–filming transi-tion, we develop a multiphase flow model of colloids dis-solved in a container of liquid. This approach couplesthe inhomogeneous evaporation at the meniscus and theheight of the deposit to the dynamics inside the suspen-sion, i.e. the fluid flow and the particle advection. Anessential component of this dynamics is the change inthe particulate flow from the Stokes regime at low col- (cid:1) (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:2)(cid:6)(cid:7)(cid:7)(cid:7) (cid:5)(cid:8)(cid:6) FIG. 6.
Multiphase model for deposition on a verticalplate in a suspension. (a) Schematic of the meniscus on thesubstrate (dark gray). The stagnation point (black dot) liesat the coordinate (
R, h ( R )). The stagnation line (red dashedcurve) divides the domain into two flow regimes: (I) capillary-driven viscous shear flow, (II) recirculation flow. The lightgray arrows indicate the evaporation profile. (b), (c) Themeniscus height h ( z, t ) in the moving frame of the depositioninterface for small and large initial colloidal concentrations, inthe domain z ∈ [0 , R ] , as shown by the red rectangle in (a).(d), (e) The evolution of the depth-averaged colloidal con-centration Φ( z, t ) corresponding to (b) and (c), respectively.These results follow from Eqs. (4), (16), (17), (20), subjectto the boundary conditions given by Eqs. (21)–(23) at z = R and Eqs. (25)–(27) at z = 0. In (b)–(e), the grayscale changesfrom dark to light with increasing time. (f) shows the meandimensionless deposition speed ¯ β as a function of the bulkvolume fraction Φ b , where the time average is calculated overan interval t/τ ∈ [4 , . The time when the interface velocityreaches a quasi-steady state is given by t ∼ τ . The pointsin (f) correspond to Φ b ∈ (cid:8) × − ∪ (cid:2) − , . (cid:3)(cid:9) increas-ing in increments of 10 − from bottom to top. (g) shows thebandwidth ∆ d as a function of Φ b . The points in (g) cor-respond to Φ b ∈ (cid:8) × − ∪ (cid:2) − , × − (cid:3)(cid:9) increasing inincrements of 10 − from bottom to top. The dashed linesat Φ c = 6 . × − in (f) and (g) denote the phase bound-ary between banding and filming, namely when ¯ β = 1 . In(g), the red ∞ sign represents the unbounded growth of thebandwidth ∆ d in the filming regime ( ¯ β > loidal concentrations away from the deposition front, toa porous flow characterized by the Darcy regime in thevicinity of the deposition front. The porous region is it-self created by the particle advection towards the contactline as the suspension turns first to a slurry and eventu-ally a porous plug over the course of drying.
4 6 8 10 12 14012 β ≡ C / E (a) t/ τ t/ τ L / √ ℓ c a p (b) FIG. 7.
Multiphase model for deposition on a verti-cal plate in a suspension (continued).
For ǫ = 0 .
01 and θ e, = 15 o the length of the domain L( t c ) between the inter-face and the level of the container at the time of the meniscusbreak-up t c . These results follow from Eqs. (4), (16), (17),(20), subject to the boundary conditions given by Eqs. (21)–(23) at z = R and Eqs. (25)–(27) at z = 0 . The timewhen the interface velocity reaches a quasi-steady state isgiven by t ∼ τ . The lines in (a) and (b) correspond toΦ b ∈ (cid:8) × − ∪ (cid:2) − , . (cid:3)(cid:9) increasing in increments of10 − (a) from bottom to top, (b) from top to bottom. In a meniscus that forms on a vertical plate dipped ina suspension or liquid, there exist two qualitatively dif-ferent flow regimes when evaporation is present [26, 27].Near the contact line, the thickness of the liquid filmis much smaller than its lengthwise dimension. In thisregime (I in Fig. 6(a)), there is a capillary-driven viscousshear flow which extends partially into the bulk solution,and compensates for the liquid lost by evaporation. Inthe bulk of the liquid, the meniscus thickness tends toinfinity towards the level of the bath. In this region (IIin Fig. 6(a)) there is a recirculation flow where the trans-verse component of the flow is dominant [26, 27]. Thesetwo regions are separated by a stagnation curve whichterminates at a surface stagnation point h ( R ) , R < L on the liquid air-interface. For small capillary numbers Ca ≪ R and L , the distortion of the fluid-air interfacedue to the recirculating fluid flow is negligible, resultingin a local hydrostatic profile. This may be justified us-ing the method of matched expansions for Ca ≪ h ( R ) of the meniscus height h ( z, t ) isdetermined by hydrostatics at z = R (see Eq. (4)).For h ( R ) ≪ ℓ cap , the viscous forces in the z − directionare balanced by the pressure gradients along the slen-der film in the domain z ∈ [0 , R ] (region I), and theforces in the transverse direction are negligible. Thissimplification of the Navier-Stokes equations is knownas the lubrication approximation [29]. In this limit, theproblem becomes one dimensional by averaging the lo-cal particle volume fraction φ ( x, z, t ), local solute ve-locity v s ( x, z, t ), and local solvent velocity v f ( x, z, t )over the meniscus height h ( z, t ) . Then, the depth-averaged solute volume fraction is given by Φ( z, t ) ≡ h − R h φ ( x, z, t ) dx , the depth-averaged solvent volumefraction is 1 − Φ( z, t ) , the depth-averaged solute velocity V s ≡ h − R h v s ( x, z, t ) dx , and the depth-averaged liquidvelocity V f ≡ h − R h v f ( x, z, t ) dx . Furthermore, we as-sume that the deposition front speed C ( t ) only variestemporally, whereas the evaporation rate is a functionof space alone, i.e. E = E ( z ). In a frame comovingwith the deposition front, the depth-averaged equationsof local mass conservation for the fluid and solvent are ∂∂t [(1 − Φ) h ]+ ∂∂z [(1 − Φ) h ( V f − C )] = − E ( z ) p ∂ z h ) , (14) ∂∂t [Φ h ] + ∂∂z [Φ h ( V s − C )] = ∂∂z (cid:20) D s h ∂ Φ ∂z (cid:21) , (15) where D s is the diffusion constant of the solute (Table II).In Eq. (14), we note the presence of an evaporative sinkon the right-hand side, with E ( z ) = E / p z/ℓ + ∆ d ( t )being the singular functional form of the local evapo-ration rate along the meniscus [10, 11]. Here ∆ d ( t ) isthe distance of the wall from the edge of the deposit ata given instant, so that the position of the far edge isgiven by z d = − ∆ d ( t ) in the moving frame. In Eq. (15),we note the right side associated with the diffusion ofparticles (that prevents the formation of an infinitelysharp deposition front); experiments suggest that dif-fusion is dominated by advection [15, 30], so that theP´eclet number P e ≡ E ℓ/D s ≫ α ≡ ǫ /ν Ca and ν ≡ H/ √ kµ , where k is the permeability of the porousplug (Table II). The dimensionless evaporation rate isgiven by ˜ E ( z ) ≡ E ( z ) /E . Then, dropping the tildes,Eqs. (14) and (15) in dimensionless form are rewritten as ∂∂t [(1 − Φ) h ]+ ∂∂z [(1 − Φ) h ( αV f − ǫβ )] = − E ( z ) q ǫ∂ z h ) , (16) ∂∂t [Φ h ] + ∂∂z [Φ h ( αV s − ǫβ )] = ∂∂z (cid:20) P e − ǫh ∂ Φ ∂z (cid:21) . (17) To complete the formulation of the problem, we need todetermine the fluid and particle velocities. In the bulkof the fluid where Φ = Φ b ≪
1, the solute and solventvelocities should match ( V s ≈ V f ), as the particles areadvected by the fluid. Beyond the deposition front, thesolute velocity V s must vanish as with Φ → Φ , the maxi-mum packing fraction corresponding to the close packingof particles (Φ ≈ .
74 for hexagonal packing in three di-mensions). Given that the two limits of Stokes flow andDarcy flow are both linear, to correctly account for bothlimits and calculate V f and V s , first we need to deter-mine the depth-dependent velocities v f and v s . As theparticles accumulate near the contact line, the result-ing deposit will serve as a porous medium for the fluid.Then, in the lubrication limit, for slender geometries, thetransition from the Stokes regime for dilute suspensions(Φ ≪
1) to Darcy flow through porous medium (Φ ≃ Φ )is governed by the Darcy-Brinkman equation [31] ∂p∂z = µ ∂ v f ∂x − ( v f − v s ) /k , (18) where the pressure p is given by Eq. (1). When the dragterm vanishes ( v s = v f ), Eq. (18) reduces to the usuallubrication balance between pressure gradient and thedepth-wise shear gradients. In the limit when the particlevelocity vanishes and the fluid velocity gradients are dom-inated by shear against the particles that form a porousplug, we recover the Darcy limit. Since the particle veloc-ity will become vanishingly small as their packing fractionapproaches the close-packing limit, this suggests a simpleclosure of Eqs. (16) and (17) v s = h − (Φ / Φ ) Γ i v f [32],which is also valid for the depth-averaged velocities V f and V s . Here, the exponent Γ controls the slope of thecrossover between the two regimes. In combination withthe functional relation between v f and v s , Eq. (18) can besolved analytically, subject to the stress-free and no-slipboundary conditions ∂v/∂x | x = h = 0 and v ( x = 0 , t ) = 0.Then the depth-averaged speeds are obtained as V f = 1 a µh ∂p∂z (tanh ah − ah ) , V s = (cid:0) − a µk (cid:1) V f , (19)where a ≡ ( µk ) − (Φ / Φ ) Γ , with 1 /a being the effec-tive pore size. When a → →
0) Eq. (19)reduces to the Stokes expression for the depth-averagedvelocity, while when a ≫ → Φ we recover theDarcy limit. Defining V s,f ≡ ( ǫ γ/µν ) ˜ V s,f and droppingthe tildes from the dimensionless velocities ˜ V s,f , Eq. (18)becomes ν ∂p∂z = ∂ V f ∂x − (cid:18) ΦΦ ∗ (cid:19) Γ V f , Φ ∗ ≡ Φ (cid:18) kµH (cid:19) / Γ , (20)in dimensionless units. Here Φ ∗ is a characteristic scaledvolume fraction of the colloids, at which the Stokes-Darcytransition occurs; we note that this occurs before Φ = Φ ,the close packing fraction, i.e. when the colloids are stillmobile [32].The coupled sixth order system of Eq. (1) in dimen-sionless form, Eqs. (16), (17), and (20) constitutes aboundary-value problem and requires the specificationof seven boundary conditions in order to find the heightof the free surface h ( z, t ) , the particle (and fluid) vol-ume fraction Φ( z, t ) , and the deposition rate β . Thefirst boundary condition is given by Eq. (4) evaluated at z = R , sin θ (cid:12)(cid:12) z = R = 1 − ( R − L ) . (21)The second and third boundary conditions at z = R aregiven by p = p , (22)(see Eq. (1)), and Φ = Φ b . (23)Eq. (21) and (22) are the consequence of the liquid-air interface deformations beyond R being hydrostatic.Eq. (23) follows from the fact that Φ must converge tothe bulk volume fraction Φ b sufficiently away from thesolid-liquid wall.At z = 0 , the solute flux should satisfy the continuitycondition across the deposition front, which is given byΦ h ( αV s − ǫβ ) − P e − ǫh ∂ Φ ∂z = − ǫβ Φ h , (24)in the frame moving with speed β . The left-hand side ofEq. (24) is the flux of colloids at the liquid side of theinterface. As the colloids are arrested inside the deposit,the solute flux vanishes as given by the right-hand side.Eq. (24) then yields the deposition rate β ( t ) = 1 ǫ (Φ − Φ ) (cid:18) α Φ V s − P e − ǫ ∂ Φ ∂z (cid:19) . (25)The solvent flowing into the deposit at the interface mustreplenish the liquid lost due to the evaporation over thesolid. In the moving frame this condition in the differen-tial form becomes1 ǫβ ∂∂t [(1 − Φ) hαV f ] = − E ( z ) q ǫ∂ z h ) . (26)The remaining boundary conditions at z = 0 are givenby h = 1 , Φ = Φ i ≡ Φ − × − where Φ = 0 . . (27)At the deposit-liquid interface ( z = 0), we set the filmthickness constant ( h (0 , t ) = 1) in dimensionless units.Note that this boundary condition in Eq. (27) is assumedfor simplicity, and a fixed deposit thickness as a functionof Φ b is observed in experiments [7]. The deviation ofΦ i from Φ ensures the asymptotic determination of β from Eq. (25). Finally, we specify the initial condition ofthe meniscus height as a hydrostatic profile (see Eq. (4))between z ∈ [0 , R ], and assume the initial particle dis-tribution Φ( z,
0) = (Φ i − Φ b ) exp − z/z +Φ b underneaththe meniscus, where z ≪ . The divergence of the evap-oration rate E ( z ) = 1 / p z + ∆ d ( t ) which is present at t = 0 is resolved by assuming the initial wall distance∆ d ( t = 0) = 10 − . The numerical values of the dimensionless quantitiesare given in Table II. We choose ν to be unity ( ν = 1)since the effect of bigger ν on the dynamics is unimpor-tant when α ≫ R = ℓ/ , which models a narrow crossover regime betweenthe Stokes and Darcy flow regimes as a function of Φ.Using the COMSOL finite element package [33], we nu-merically solve Eqs. (4), (16), (17), (20), subject to theboundary conditions given by Eqs. (21)–(23) at z = R and Eqs. (25)–(27) at z = 0, for the height of the freesurface h ( r, t ) , the particle volume fraction Φ( r, t ), andthe deposition front velocity β ( t ). In Fig. 6(b) we showthe time evolution of the meniscus, which corresponds to the formation of a single band with Φ b = 3 × − .Here, since Φ b < Φ c , the interface velocity satisfies β < L > h ( R ) decreases monotonically, resulting in an overall de-crease in the height of the fluid film along z < R .Hence, the film surface will approach the substrate overtime, followed by the meniscus break-up as exemplifiedin Fig. 6(b). For Φ b = 3 × − the break-up location isat z c = 0 . ≈ L ( t c ) −
1, in agreement with the globalminimum of a quasi-hydrostatic profile. In Fig. 6(c) weshow the formation of a continuous deposit for a muchlarger bulk concentration with Φ b = 0 . > Φ c . Here β > L < z -dependence of the colloidalvolume fraction Φ is demonstrated inside the meniscusfor Φ b = 3 × − and Φ b = 0 .
01. On both sides ofthe phase boundary (namely when β = 1), at Φ b = Φ c ,Φ changes rapidly near the solid-liquid interface, whichis a natural result of the high P´eclet number P e . Thisbehavior shows qualitative agreement with experiments,where near the interface Φ of the particles is observed tobe much lower than Φ .The dependence of the mean dimensionless interfacespeed ¯ β ≡ t − c R t c βdt on Φ b is shown in Fig. 6(f), where t c is the time of meniscus touch-down in Fig. 1(e). Thebulk volume fraction Φ b at which ¯ β = 1 corresponds to Φ c (Table I). When h i is constant for all deposition speeds,the bandwidth ∆ d (Fig. 6(g)) depends linearly on Φ b when ¯ β < , and becomes infinite in the filming regime¯ β ≥ . This behavior implies an abrupt transition interms of Φ b , preempting the continuous transition ac-companied by the diverging behavior suggested by theminimal model. V. CONCLUSION
Our direct observations of the dynamics of the menis-cus, contact line, and the shape of the colloidal depositsupon evaporation of dilute colloidal suspensions lead toa simple picture of how deposition patterns arise in thesesystems. At low Φ b , meniscus pinning, deformation,touch-down and depinning leads to periodic bands whosespacing is determined by the relative motion of the inter-face and the evaporation rate, as well as the dynamics ofthe receding contact line. Meniscus touch-down does notoccur at large Φ b , leading to a continuous colloidal film.A minimal and a detailed quantitative theory capture thetransition between banding and filming, the correspond-ing critical volume fraction, the deposit growth speed,as well as the salient length and time scales, consistentwith our observations. Thus, our work reveals the condi-tions and the dynamics of the concentration-dependentevaporative patterning which has various practical appli-cations [6, 7, 14].0 ACKNOWLEDGMENTS
This research was supported by the Air Force Office ofScientific Research (AFOSR) under Award FA9550-09-1-0669-DOD35CAP and the Kavli Institute for BionanoScience and Technology at Harvard University.
ASSOCIATED CONTENTSupporting InformationMovie 1: Continuous movement of meniscus
The movie shows that the movement of meniscus andparticle deposition are continuous when particle concen-tration is high. Here, one micron latex particles are dis-persed in deionized water with a volume concentration of0.1%. The field of view is 470 ×
350 microns. And themovie duration in real time is 2150 sec.
Movie 2: Periodic formation of colloidal bandduring solvent evaporation
A silicon substrate was vertically immersed in a dilutecolloidal suspension (PMMA with negatively charged sul-fate end groups, 0.002 wt%, diameter ∼
375 nm). Theobjective and camera were facing the meniscus and theplane of substrate. As water evaporated naturally, themeniscus moved downwardly in a non-smooth and peri-odic fashion, leaving periodic bands of colloidal films be-hind. The whole deposition process shown in the movietook place over ∼ ∼ Movie 3: Meniscus touch-down and break-up
The formation of a new band on a vertical glass sub-strate in a suspension via optical microscopy (0.06 vol %1 µ m latex particles in water). The movie shows themeniscus break-up and the subsequent contact line mo-tion. Over time, the interference rings occur, which areassociated with the meniscus approaching the substrate.The meniscus break-up starts as a localized event belowthe maximum of the forming, convex band. [1] Bones JB, Sanders JV, Segnit ER, Hulliger F (1964)Structure of opal. Nature 204:990-991.[2] Prum RO, Torres R (2003) Structural coloration of avianskin: convergent evolution of coherently scattering der-mal collagen arrays. J. Exp. Biol. 206:2409-2429.[3] Blanco A, et al. (2000) Large-scale synthesis of a sili-con photonic crystal with a complete three-dimensionalbandgap near 1.5 micrometres. Nature 405:437440.[4] Rinnie SA, Garcia-Santamaria F, Braun PV (2008) Em-bedded cavities and waveguides in three-dimensional sil-icon photonic crystals. Nat Photonics 2:5256.[5] Eun Sik K, Wonmok L, Nam-Gyu P, Junkyung K, Hyun-jung L (2009) Compact inverse-opal electrode using non-aggregated TiO2 nanoparticles for dye-sensitized solarcells. Adv Funct Mater 19(7):10931099.[6] Sung-Wook C, Jingwei X, Younan X (2009) Chitosan-based inverse opals: Three-dimensional scaffolds withuniform pore structures for cell culture. Adv Mater21:29973001.[7] Hatton B, Mishchenko L, Davis S, Sandhage KH, Aizen-berg J (2010) Assembly of large-area, highly ordered,crack-free inverse opal films. PNAS 107:10354-10359.[8] Mar´ın ´AG, Gelderblom H, Lohse D, Snoeijer JH (2011)Order-to-disorder transition in ring-shaped colloidalstains. Phys Rev Lett 107:085502.[9] Bigioni TP, et al. (2006) Kinetically driven self assemblyof highly ordered nanoparticle monolayers. Nature Mat5:265-270.[10] Deegan RD, et al. (1997) Capillary flow as the cause ofring stains from dried liquid drops. Nature 389:827-829.[11] Deegan RD, et al. (2000) Contact line deposits in anevaporating drop. Phys Rev E 62:756-765.[12] Deegan RD, et al. (2000) Contact line deposits in anevaporating drop. Phys Rev E 61:475-485.[13] Popov YO (2005) Evaporative deposition patterns: Spa-tial dimensions of the deposit. Phys Rev E 71:036313. [14] Kim HS, Lee CH, Sudeep PK, Emrick T, Crosby AJ(2010) Nanoparticle stripes, grids, and ribbons producedby flow coating. Adv Mater 22:4600-4604.[15] Bodiguel H, Leng J (2010) Imaging the drying of a col-loidal suspension. Soft Matter 6:5451-5460.[16] Adachi E, Dimitrov AS, Nagayama K (1995) Stripe pat-terns formed on a glass surface during droplet evapora-tion. Langmuir 11:1057-1060.[17] Shmuylovich L, Shen AQ, Stone HA (2002) Surface mor-phology of drying latex films:multiple ring formation.Langmuir 18:3441-3445.[18] Maheshwari S, Zhang L, Zhu Y, Chang HC (2008) Cou-pling between precipitation and contact-line dynamics:multiring stains and stick-slip motion. Phys Rev Lett100:044503.[19] Abkarian M, Nunes J, Stone HA (2004) Colloidal crys-tallization and banding in a cylindrical geometry. J AmChem Soc 126:5978-5979.[20] Yunker PJ, Still T, Lohr MA, Yodh AG (2011) Suppres-sion of the coffee-ring effect by shape-dependent capillaryinteractions. Nature 308:308-311.[21] Li J, Cabane B, Sztucki M, Gummel J, Goehring L (2011)drying dip-coated colloidal films. Langmuir 28:200-208.[22] Witten TA (2009) Robust fadeout profile of an evapora-tion stain. EPL 86:64002.[23] Kaya D, Belyi VA, Muthukumar M (2010) Pattern for-mation in drying droplets of polyelectrolyte and salt. JChem Phys 133:114905.[24] de Gennes PG., Brochard-Wyart F., Qu´er´e D Capillar-ity and Wetting Phenomena (Springer Science+BusinessMedia, New York, NY, USA).[25] de Gennes PG (1985) Wetting: statics and dynamics.Rev Mod Phys 57:827-863.[26] Scheid B, et al. (2010) The role of surface rheology inliquid film formation. Europhys Lett 90:24002.[27] Colosqui CE, Morris JF, Stone HA (2013) Hydrodynam-1