Particle Convection in an Evaporating Colloidal Droplet - American

Using lubrication theory, this study shows that either an outward flow toward the contact line or an inward flow toward the center of the droplet can ...
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Langmuir 2002, 18, 60-67

Particle Convection in an Evaporating Colloidal Droplet Benjamin J. Fischer* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544 Received July 31, 2001. In Final Form: October 22, 2001 An emergent technique of patterning surfaces with solid particles utilizes the evaporation of colloidal droplets from a substrate. Upon complete evaporation of the liquid, the suspended particles remain adhered to the solid in a variety of patterns. Experimentally, the type of particle deposit has been correlated with the mode of liquid evaporation. It is expected that the manner in which the liquid evaporates from the droplet should significantly affect the flow of fluid inside the droplet. Therefore, the determination of the flow profiles in the droplet will aid in understanding the redistribution of particles under different experimental conditions. Using lubrication theory, this study shows that either an outward flow toward the contact line or an inward flow toward the center of the droplet can be induced, depending on the evaporative driving force. If an outward flow toward the contact line is generated inside the droplet, then a ringlike deposit remains on the substrate. Conversely, if the liquid inside the droplet flows away from the contact line, then a more uniform solute deposit will adhere to the solid.

Introduction When a colloidal droplet evaporates from a substrate, the solute particles in solution will remain attached to the solid surface. Depending on experimental conditions such as the solid volume fraction or particle size, a variety of patterns can appear on the substrate.1-8 For example, when a coffee droplet dries, a dark ring of coffee solute is left on the substrate. Initially, the deposited droplet has a uniform distribution of solute. However, as the evaporation proceeds, the solute aggregates at the edge of the droplet and forms a ring on the substrate. This ability to deposit rings on a solid could have an enormous impact on a number of surface-patterning applications. During the evaporation process, the contact line does not recede as in the case of a pure droplet evaporating on a substrate. Rather, the contact line remains pinned for a substantial portion of the drying time. This “self-pinning” is due to the colloidal particles wedging between the substrate and the contact line.7 It has been proposed that if there is self-pinning of the contact line and a nonuniform evaporation of liquid then an outward flow of liquid toward the contact line is required for mass conservation.3,8 As the droplet remains pinned, this outflow will convect the particles toward the contact line where they aggregate into a solute ring. The thickness of the developing solute ring depends on the amount of time that the droplet self-pins at a particular location. To predict the growth of the solute ring, a hydrodynamic model3,8 was applied to an axisymmetric droplet in which * Corresponding author. E-mail: [email protected]. (1) Adachi, E.; Dimitrov, A. S.; Nagayama, K. Langmuir 1995, 11, 1057. (2) Conway, J.; Korns, H.; Fisch, M. R. Langmuir 1997, 13, 426. (3) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Nature 1997, 389, 827. (4) Ohara, P. C.; Gelbart, W. M. Langmuir 1998, 14, 3418. (5) Maenisono, S.; Dushkin, C. D.; Saita, S.; Yamaguchi, Y. Langmuir 1999, 15, 957. (6) Latterini, L.; Blossey, R.; Hofkens, J.; Vanoppen, P.; DeSchryver, F. C.; Rowan, A. E.; Nolte, R. J. M. Langmuir 1999, 15, 3582. (7) Deegan, R. D. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2000, 61, 475. (8) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2000, 62, 756.

a nonuniform evaporation field always induced a heightaveraged radial flow toward the contact line. By treating the droplet as a symmetric lens of fluid about the substrate and making an analogy to electrostatics, the evaporative mass flux, Js, is given by

Js(r,t) ≈ (R - r)-λ

(1)

where λ ) (π - 2θc)/(2π - 2θc), R is the droplet radius, r is the radial coordinate, and θc is the contact angle. At the contact line (r ) R), the evaporative mass flux and thus the outward radial velocity diverge. This model for the evaporative flux was then used to determine how the solute ring grew in time. However, in previous work, the effect of different modes of evaporation on pattern formation was not examined with the model. Experiments have shown that three different modes of evaporation yield two different types of patterns on the substrate.8 In the first mode, a colloidal droplet evaporated normally from a substrate into an open system. Upon complete evaporation of the liquid, a solute ring was left on the substrate. A spatially uniform evaporative flux was applied across the droplet in the second mode. Again, a solute ring was left on the substrate. In the third mode, evaporation was greatest at the center of the droplet and zero at the edge. In this case, a solute ring was not deposited on the substrate. Rather, a uniform distribution of colloidal particles remained on the substrate. In this study, the relationship between the resulting deposited pattern and the mode of evaporation is examined with a hydrodynamic model. By determining the flow profiles inside the droplet for each of these three evaporation cases, the redistribution of the particles under different experimental conditions can be better understood. Problem Formulation The formation of the solute ring should be closely linked to the hydrodynamics inside the droplet. If evaporation induces an outward radial fluid flow toward the contact line, then the colloidal particles will be carried toward the edge of the droplet, creating a particle ring on the substrate. Conversely, it is expected that an inward radial

10.1021/la015518a CCC: $22.00 © 2002 American Chemical Society Published on Web 12/06/2001

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Langmuir, Vol. 18, No. 1, 2002 61

At the interface z ) h, the dimensionless pressure within the lubrication approximation is given by

p)-

1 1 ∂ ∂h r Ca r ∂r ∂r

( )

(5)

where Ca ) µ*u/c /(3σ*) ) µ*2/(3F*σ*h/o) is the capillary number and σ* is the surface tension of the liquid. For simplicity, the concept of vapor recoil13,15,16 is ignored in eq 5 because this phenomenon has a secondary effect on the flow. The temperature gradient along the interface of the droplet is assumed to be significantly weak such that Marangoni flow is not induced. Therefore, the tangential stress balance at the interface is given by Figure 1. Evaporating droplet on a solid substrate.

flow away from the contact line would result in a more homogeneous particle deposition on the substrate. Therefore, to examine the hydrodynamics of this phenomenon, a model is developed in the limit of lubrication theory. Figure 1 portrays an axisymmetric droplet on a horizontal substrate with fluid density F*, viscosity µ*, initial height h/o, and initial radius r/o. As a first approximation, it is assumed that the colloidal particles have a negligible effect on the hydrodynamics inside the droplet. Therefore, the flow of fluid is governed by the continuity and NavierStokes equations. In this study, very thin droplets will be examined. In this regime, the lubrication approximation can be applied to simplify the governing equations. This approach has been successfully applied in previous studies on the spreading of pure liquid droplets9-13 as well as on the drying of latex films.14 The radial and vertical coordinates are scaled by the droplet radius r/o and the initial droplet height h/o, respectively. Furthermore, in the thin-droplet limit,  ) h/o/r/o is less than unity. If the radial velocity u* is scaled by the characteristic viscous velocity u/c ) µ*/(F*h/o), then the continuity equation implies that the vertical velocity w* should be scaled with u/c . Upon scaling, the Reynolds number, Re ) F*u/c h/o/µ*, appears as a dimensionless parameter. In the lubrication approximation, both 2 and Re are assumed to be much less than unity. The characteristic pressure is p/c ) µ*u/c r/o/h/o2, and the characteristic time is t/c ) r/o/u/c . By introducing these scalings and applying the lubrication approximation to the continuity and Navier-Stokes equations, fluid flow is governed by

0)

∂w 1 ∂ (ru) + r ∂r ∂z

0)-

(2)

∂p ∂2u + ∂r ∂z2

(3)

∂p ∂z

(4)

0)-

where the absence of an asterisk superscript denotes a dimensionless quantity. Also, the gravitational terms have been neglected. (9) Greenspan, H. P. J. Fluid Mech. 1978, 84, 125. (10) Hocking, L. M. Q. J. Mech. Appl. Math. 1981, 34, 37. (11) Haley, P. J.; Miksis, M. J. J. Fluid Mech. 1991, 223, 57. (12) Ehrhard, P.; Davis, S. H. J. Fluid Mech. 1991, 229, 365. (13) Anderson, D. M.; Davis, S. H. Phys. Fluids 1995, 7, 248. (14) Routh, A. F.; Russel, W. B. AIChE J. 1998, 44, 2088.

∂u )0 ∂z

(6)

As time elapses, evaporation occurs at the air-liquid interface with a spatially and time-dependent mass flux J*(r*, t*). To nondimensionalize this quantity, the method of Burelbach et al.16 is applied. The characteristic mass flux is determined from the energy-jump condition to be J/c ) k*∆T*/(L*h/o). On this scale, k* is the thermal conductivity of the liquid, L* is the heat of vaporization, and ∆T* is the temperature difference between the substrate and saturation temperatures. In the lubrication approximation, the height-evolution equation is given by

1 ∂ ∂h )(rh〈u〉) - EJ ∂t r ∂r

(7)

where E ) k*∆T*/(µ*L*) is the evaporation number. The height-averaged radial velocity 〈u〉 is given by

〈u〉 )

1 h

∫0h u dz

(8)

If eqs 3 and 4 are integrated over the boundary conditions in eqs 5 and 6 and are subject to the no-slip condition, then the radial velocity distribution is given by

u)-

∂h 1 2 1 ∂ 1 ∂ r z - hz Ca ∂r r ∂r ∂r 2

[ ( )](

)

(9)

Furthermore, using this expression, the height-averaged velocity can be substituted into eq 7 to obtain the full height-evolution equation:

[ (

1 1 ∂ ∂ ∂2h 1 ∂h ∂h rh3 )+ ∂t 3Ca r ∂r ∂r ∂r2 r ∂r

)]

- EJ (10)

This equation is the axisymmetric equivalent to the governing equation derived by Anderson and Davis13 for the spreading of a droplet on a heated substrate, neglecting Marangoni forces, vapor recoil, and slip. At the center of the droplet (r ) 0), the boundary conditions represent the symmetry and no-flux conditions given by

∂3h ∂h )0 ) ∂r ∂r3

(11)

(15) Palmer, H. J. J. Fluid Mech. 1976, 75, 487. (16) Burelbach, J. P.; Bankoff, S. G.; Davis, S. H. J. Fluid Mech. 1998, 195, 463.

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Fischer

At the contact line (r ) 1), the droplet remains pinned such that

h)0

θc ∂h )∂r 

and

(12)

where θc is the contact angle. Although the contact line will remain at a fixed location, the contact angle will change in time. To monitor the solute or particle concentration C*, it is assumed that there is a uniform vertical distribution of particles and that diffusion is negligible. Therefore, the concentration has only a spatial dependence in the radial direction and is governed by

1 ∂ ∂ (hC) ) (rh〈u〉C) ∂t r ∂r

(13)

where the concentration is scaled by the initial uniform concentration C/o. This equation is solved using the symmetry condition at the origin:

∂C )0 ∂r

(14)

Furthermore, because we are interested in the growth of the solute ring, the mass fraction of particles in the ring, χ, is defined by

χ)

Nring Ntot

(15)

where Nring and Ntot are the number of particles in the solute ring and the entire droplet, respectively. The number of particles can be calculated by integrating the concentration over the appropriate volume:

Ntot ) 2π

∫01 Chr dr

and Nring ) 2π

∫r˜1 Chr dr

pinned contact line within the lubrication approximation, the evaporative mass flux must be zero at the contact line. Physically, the contact line remains pinned because of the accumulation of colloidal particles at the edge of the droplet. Therefore, these particles must also hinder the evaporation at the contact line. For the experiment in which a colloidal droplet is allowed to evaporate freely from a surface, previous theoretical work used an electrostatic analogy to determine the evaporation given by eq 1. However, this model experiences a singularity at the contact line. An alternative method for determining the evaporative mass flux is to use heattransfer analysis. When a pure-liquid droplet evaporates from a heated substrate, the mass flux of evaporating liquid is given by13

(16)

where r˜ is the initial radial coordinate of the solute ring.

J)

∂〈u〉 ∂h ∂h ) -h - 〈u〉 - h〈u〉 - EJ ∂t ∂r ∂r

J)

(17)

At the contact line, the height is zero and does not change in time. Furthermore, because the contact line does not move, the fluid velocity, 〈u〉, must be zero at r ) 1. According to the lubrication approximation and the no-slip requirement, eq 10 shows that the fluid velocity will be zero if the second-order and third-order spatial derivatives of the height do not diverge at the contact line. Because only physically realistic solutions are of interest, the fluid velocity must be zero at the contact line. In this limit, eq 17 reduces to EJ(r ) 1) ) 0. Therefore, to maintain a

(18)

where K is a dimensionless nonequilibrium parameter. At the edge of the droplet, where h ) 0, the evaporative flux is maximized. Therefore, this form cannot be applied to a droplet in which the contact line is pinned. However, far from the contact line, this model is a very good approximation of the evaporation from a colloidal droplet because the particles will have a negligible effect in this region. Consequently, the manner in which the evaporative flux decays to zero at the contact line needs to be determined. In this study, the interest lies in the overall behavior of the macroscopic droplet. The specific nature of the flux decay in this small region of flow is not the most important parameter of the model. Therefore, as a first approximation it is assumed that the flux will decrease exponentially near the contact line. Although this is a qualitative hypothesis, it is reasonable to believe that the effect of the colloidal particles will be strongest near the contact line but will quickly decrease toward the center of the droplet. Future work should closely examine the behavior near the contact line because the presence of particles affects not only the evaporation but also the hydrodynamics in this region. To that end, the evaporative flux for this first case is given by

Evaporation Models As previously mentioned, the focus of this work is to understand why the three different evaporative conditions give rise to two different colloidal patterns on the substrate. Therefore, three functional forms for the evaporative mass flux, J(r,t), are examined in this study. However, before introducing the three evaporation models, a requirement on the evaporative flux at the contact line needs to be presented. To understand the behavior of the droplet at the contact line, eq 7 is expanded into the form

1 K+h

2 1 [1 - e-A(r-1) ] K+h

(19)

where A is related to the length over which the colloidal particles affect the evaporation. As previously mentioned, Deegan et al8 experimentally altered the manner in which evaporation occurred from the droplet. To mimic these experiments, the following qualitative evaporative flux functions are also examined:

J)

1 [1 - tanh A(r - ro)] 4h(r ) 0)

(20)

2 2 e-Ar h(r ) 0)

(21)

J)

where A is an adjustable constant. The function in eq 20 is proposed to approximate the experiment in which evaporation is held constant across the droplet. The hyperbolic tangent function approximates a step function in the evaporative flux. The third model uses a Gaussian distribution centered at r ) 0 to mimic evaporation that is concentrated at the center of the droplet. The height of the droplet will decrease in time. Therefore, the mass flux values given in eqs 19-21 will increase in time. This behavior is consistent with that

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Langmuir, Vol. 18, No. 1, 2002 63

proposed by the model in eq 18. However, the assumption that the flux will increase in time is valid only when the time interval is short. At low solute mass fractions, a colloidal droplet is expected to behave similarly to a pureliquid droplet. Thus, the evaporative mass flux should increase in time. However, as the liquid evaporates and causes the solute mass fraction to increase, evaporation will be affected by the presence of the solute; therefore, the models for the evaporative mass flux need to be altered for long-time behavior. This observation is verified by previous experiments2,7 that show that the rate of change of the droplet mass slows at long times. Therefore, these evaporative flux models are valid only for short evaporation times. Results and Discussion There are two important parameters in this system: the capillary number, Ca, and the evaporation number, E. Using physical constants that are typical for water, Ca can range from ∼0.01 to ∼10. Similarly, representative values for E will lie between ∼0.01 ∼ 1. Therefore, the two regimes of interest are (1) Ca-1 . E and (2) Ca-1 ≈ E. Numerical Procedure. The two governing eqs 10 and 13 were solved by the method of lines,17 which implements second-order centered differences for the spatial derivatives and a fully implicit Gear’s method for the time integration.18 The number of grid points used in the computations was 101. At the start of each simulation, the dimensionless parameters Ca and E were specified. More importantly, the type of evaporative flux was chosen from eqs 19-21. At t ) 0, the droplet was assumed to be a spherical cap given by

h(r,0) ) 1 - r2

(22)

while the solute was assumed to be uniformly distributed throughout the droplet such that

C(r,0) ) 1.0

(23)

The simulations were run until about half of the original liquid volume had been evaporated. Near this point, it was shown experimentally that the contact line will begin to depin.7 Therefore, the model is not applicable beyond this time. Small Capillary Number. The capillary number reflects the ratio of viscous/ surface-tension forces. When the capillary number is small, surface-tension forces dominate viscous forces. For Ca ) 0.01 and E ) 0.1, the solute deposition process was examined for the three flux models given in eqs 19-21. Edge-Enhanced Evaporation. If a colloidal droplet is placed on a substrate and allowed to evaporate freely, then the evaporative flux is expected to be similar to that given in eq 18 but with the constraint of no evaporative flux at the contact line. To model this situation, eq 19 is used. Far from the contact line toward the center of the droplet, the evaporative mass flux will be identical to that given in eq 18. However, as the contact line is approached, J will quickly decrease to zero. Figure 2 shows the evolution of the droplet profile and the evaporative mass flux when K ) 1 and A ) 250. In time, the height of the droplet decreases while the mass flux increases. At the contact line, the droplet remains (17) Schiesser, W. E. The Numerical Method of Lines; Academic Press: San Diego, CA, 1991. (18) Hindmarsh, A. C. In Scientific Computing; Stepleman, R. S., Ed.; North-Holland: Amsterdam, 1983; p 55.

Figure 2. Time evolution of (a) the droplet height profile and (b) the evaporative mass flux for the evaporation model given by eq 19, with Ca ) 0.01, E ) 0.1, K ) 1, and A ) 250. Profiles are shown for t ) 0, 1, 2, 3, 4, 5, and 6.

pinned and the droplet height is constant while the evaporative flux disappears. In Figure 3, the volume of the droplet and the contact angle are shown to decrease in time. Initially, the volume of liquid is lost at a constant rate. However, as the evaporative flux increases in time, the rate at which the droplet evaporates increases. Experimentally, it has been shown that the mass of the colloidal droplet decreases in a nearly linear fashion at early times.2,7,8 A linear fit to the early-time volume data shown in Figure 3a has a nondimensional slope of R ) 0.031. If this rate of change is dimensionalized using parameters for liquid water and r/o ) 0.15 cm, then the mass of the droplet decreases at a rate of R* ) - 4.6 µg/s. It was reported8 that the mass of a colloidal droplet of the same radius will decrease at a rate of R* ≈ -2.3 µg/s. Although the value predicted by the model is twice this experimental value, the orders of magnitude are comparable. This result is promising because exact values of the parameters E, Ca, and K were not used. If E ) 0.05, which is within the experimental range, then the model predicts R* ) -2.3 µg/s. Therefore, this model achieves good agreement with experimental data. The streamlines and a representative set of velocity vectors at t ) 5 are shown in Figure 4. When the evaporative flux increases toward the edge of the droplet, flow is induced toward the contact line. Because fluid evaporates faster near the edge of the droplet than in the center of the droplet, capillary forces will create a flow toward the contact line to replenish the fluid at the edge. As a result of this flow, the colloidal particles will be carried

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Figure 3. (a) Volume of the droplet and (b) contact angle as functions of time for the evaporation model given by eq 19, with Ca ) 0.01, E ) 0.1, K ) 1, and A ) 250.

Fischer

Figure 5. (a) Evolution of the particle distribution function for t ) 0, 1, 2, 3, 4, and 5. (b) Fraction of particles that are located in the solute ring at r˜ ) 0.8. Both plots are for the evaporative mass flux given by eq 19, with Ca ) 0.01, E ) 0.1, K ) 1, and A ) 250.

as P ) CVdrop where Vdrop is the total volume of the liquid. Figure 5 shows the evolution of the particle distribution function and χ when r˜ ) 0.8. The initially flat profile of P develops a spike near the contact line but decreases near the center of the droplet. This spike is representative of the solute ring. As time evolves, more solute is convected to the edge of the droplet where it accumulates in the ring. Because of the influx of particles to the edge, dramatic growth of the solute ring occurs. The previous theoretical model predicts that at early times the solute ring will grow in time as a power law:8

mr ≈ t2/(1+λ)

Figure 4. Representative streamlines and velocity vectors at t ) 5 for a pinned droplet in which the evaporative mass flux is given by eq 19, with Ca ) 0.01, E ) 0.1, K ) 1, and A ) 250. Fluid flows from the center of the droplet toward the contact line.

toward the contact line where they will accumulate in a ring pattern. The formation of the solute ring is examined by monitoring the particle concentration C. Because the volume of liquid decreases as a result of evaporation while the number of colloidal particles remains constant, the particle concentration will also increase. Therefore, to avoid confusion, a particle distribution function is defined

(24)

Furthermore, at early times, the solute ring of a thin droplet is expected to grow as mr ≈ t 4/3 because λ ≈ 1/2. This relation was shown to represent the experimental data at early times, but as time elapsed, the model significantly underpredicted the growth of the solute ring. To compare this previous work with the current results, χ is fitted to a power law of the form

χ ) χo + at b

(25)

As shown in Figure 5b, a best-fit analysis of the earlytime behavior of χ predicts values of a ) 0.028 and b ) 1.24. Also, a plot of eq 25 when a ) 0.025 and b ) 1.33 is shown in Figure 5b to show the prediction of the earlier

Particle Convection in an Evaporating Droplet

Figure 6. Time evolution of (a) the droplet height profile and (b) the evaporative mass flux for the evaporation model given by eq 20, with A ) 100, ro ) 0.95, Ca ) 0.01, and E ) 0.1. Profiles are shown for t ) 0, 1, 2, 3, 4, and 5.

Figure 7. Representative streamlines and velocity vectors at t ) 5 for a pinned droplet in which the evaporative mass flux is given by eq 20, with Ca ) 0.01 and E ) 0.1. Fluid flows from the center of the droplet toward the contact line.

model. As the Figure shows, the two models predict similar growth rates at early times. However, at later times, χ significantly deviates from the power-law behavior and exhibits rapid growth. This behavior is characteristic of the experimental results8 but was not represented by the earlier model. Constant Evaporation. To examine the effect of the evaporation process on the deposition pattern, Deegan et

Langmuir, Vol. 18, No. 1, 2002 65

Figure 8. Time evolution of (a) the droplet height profile and (b) the evaporative mass flux for the evaporation model given by eq 21, with A ) 10, Ca ) 0.01, and E ) 0.1. Profiles are shown for t ) 0, 1, 2, 3, 5, and 10.

al.8 controlled the experimental environment such that evaporation was relatively constant across the droplet. Under this condition, a solute ring still formed on the substrate. To mimic this experiment, the mass flux given by eq 20 is used in eq 10. For A ) 100 and ro ) 0.95, the evolution of the droplet height profile and evaporative mass flux are shown in Figure 6. Again, as time evolves, the droplet height decreases while the contact line remains pinned at r ) 1. Similarly, the evaporative flux increases in time but always disappears at the contact line. Although not shown, the volume of the droplet and the contact angle decrease in a fashion similar to that shown in Figure 3. At early times, these quantities linearly decrease in time. However, near the end of evaporation, the curves deviate from linearity because of the increasing rate of evaporative flux. Figure 7 shows the streamlines and velocity field for this evaporation process at t ) 5. The flow that develops under these conditions is very similar to that shown in Figure 4. Again, a radially outward flow of fluid toward the contact line is induced by the evaporation and capillary forces. Although not shown, the particle distribution function and χ exhibit similar behavior to those variables included in Figure 5. The outward flow convects the particles toward the contact line, causing substantial growth of the solute ring. Center-Enhanced Evaporation. In another experiment,8 a colloidal droplet was placed in an environment in which evaporation primarily occurred at the center of

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Fischer

Figure 9. Representative streamlines and velocity vectors at t ) 10 for a pinned droplet in which the evaporative mass flux is given by eq 21, with Ca ) 0.01 and E ) 0.1. Fluid flows from the edge of the droplet toward the center.

Figure 11. Evaporation of a pinned droplet when Ca ) 10 and E ) 0.1. (a) When the evaporation is modeled with eq 19 with K ) 1 and A ) 250, the droplet flattens near the contact line. The plots shown are for t ) 0, 1, 2, 2.5, 3, and 3.5. (b) Conversely, the droplet flattens at its center when the evaporation is given by eq 21 with A ) 10. These profiles are for t ) 0, 1, 2, 5, and 8.

Figure 10. (a) Evolution of the particle distribution function for t ) 0, 1, 2, 3, 5, and 10. The value of P does not diverge at r ) 0 for t ) 10 but attains a maximum value of P ) 39.3. (b) Fraction of particles that are located in the solute ring at r˜ ) 0.8. Both plots are for the evaporative mass flux given by eq 21 with, A ) 10, Ca ) 0.01, and E ) 0.1.

the droplet. In this instance, a ring did not form; rather, a uniform deposit of particles remained on the substrate. Using eq 21 as a model for evaporation, the evolution of the droplet and J are shown in Figure 8 for A ) 10. Except at r ) 1, where the droplet remains pinned, and J ) 0, the height profile decreases, and the evaporative flux increases.

Contrary to the two previous cases, this type of evaporation does not induce a flow of fluid toward the contact line. Figure 9 shows the streamlines and velocity field at t ) 10 for a droplet in which evaporation is enhanced at the center. As fluid is lost from the center of the droplet, an inward flow develops to replenish the evaporated fluid. This behavior shows that an outward flow is not always induced in a pinned droplet. Because the fluid flows toward the center of the droplet, the particles will not accumulate at the contact line as shown in Figure 10. Rather, P decreases near the contact line but increases at the center of the droplet. Additionally, P does not diverge at r ) 0 for t ) 10 but attains a maximum value of P ) 39.3. The scale of Figure 10 is reduced to display the features of P. Because the solute is convected to the center of the droplet, a solute ring does not develop under these conditions. Again, these results are qualitatively consistent with experimental observations. Large Capillary Number. If the capillary number is large, then evaporation forces will dominate surface tension forces. If Ca ) 10 and E ) 0.1, then the development of the deposited solute pattern with evaporative flux is qualitatively similar to that in the small capillary-number limit. If the evaporative flux is given by either eq 19 or 20, then flow toward the contact line is induced inside the droplet. This behavior leads to the accumulation of particles at the edge of the droplet.

Particle Convection in an Evaporating Droplet

Similarly, for evaporative flux that is dominant at the center of the droplet, flow toward the center is observed. When evaporation is the dominant mechanism, droplet profiles evolve in a different manner than when surfacetension forces dominate. When the evaporative flux is given by eq 19 or 20 with the same values of K, A, and ro previously used, the flow of fluid toward the contact line cannot replenish the volume of fluid lost from evaporation. Therefore, the droplet flattens near the contact line. This behavior, shown in Figure 11, is indicative of the contact line trying to depin; therefore, the late-time behavior of this system cannot be described by this model because the droplet will not remain pinned for an extended period of time. However, the early-time analysis is still valid and shows that a solute ring will develop for these two flux functions in the large capillary-number limit. If evaporation primarily occurs from the center of the droplet, as described by eq 21 for A ) 10, then the droplet will flatten at its center when evaporation forces are dominant. Again, the flow induced toward the center of the droplet by capillary forces cannot replenish the fluid faster than the rate at which it is lost from evaporation. Conclusions Using lubrication theory, the deposition of colloidal particles from an evaporating droplet onto a substrate is examined. It is shown for the first time that under different evaporation conditions fluid can either flow toward or away from the contact line. When the evaporative flux of liquid

Langmuir, Vol. 18, No. 1, 2002 67

is relatively constant across the droplet or moderately enhanced at the edge of the droplet, an outward flow develops in the liquid. This outward flow convects the solute toward the contact line, leading to the development of a solute ring on the substrate. Conversely, if evaporation primarily occurs at the center of the droplet, then a solute ring does not form because the liquid flows toward the center of the droplet. Therefore, the key to understanding the pattern formation is the flow profile that is induced inside the evaporating droplet. If the contact line is to remain pinned, the evaporative mass flux must disappear at the edge of the droplet within the lubrication approximation. The manner in which the flux decays to zero in this small region of flow will depend on the colloidal particles. The objective of this study, however, was to understand the macroscopic flow inside the evaporating droplet; therefore, simple decays were chosen as first approximations. While these models are good approximations for understanding macroscopic flow, future work should attempt to develop models of the evaporative flux and hydrodynamics near the contact line. In this region of the droplet, the colloidal particles will have a significant effect on both the evaporation and flow of liquid. Therefore, these effects should be included in future models. Acknowledgment. I thank Jeff Davis for some helpful discussions. LA015518A