Packing and Sorting Colloids at the Contact Line of a Drying Drop

4 Feb 2011 - (9) This water flux drags particles toward the contact line, which ..... The segregation effect that we report in our study in the case o...
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Packing and Sorting Colloids at the Contact Line of a Drying Drop Cecile Monteux* and Franc-ois Lequeux PPMD/SIMM, UMR 7615 CNRS-ESPCI-Universite Pierre et Marie Curie, ESPCI, 10 rue Vauquelin, 75005 Paris, France

bS Supporting Information ABSTRACT: In this article, we study the drying kinetics of a sessile droplet containing latex particles. We find that a depletion film is left at the edge of the drops, whose width is controlled by two geometric parameters, the contact angle and the diameter of the particles. We show that this effect can be used to sort colloidal mixtures because nanometric colloids always segregate at the edge of the drop, whereas micrometric colloids are blocked further away from the edge. We also provide a simple method to measure the velocity of a micrometric latex as it flows toward the contact line. We find that the particles strongly accelerate at the end of the drying process. Using Deegan’s prediction for the rate of evaporation in the vicinity of the contact line, we quantitatively explain this phenomenon by the fact that the contact angle vanishes at the end of the drying process, therefore inducing a strong increase in the flux of water and particles close to the edge. The decrease in the contact angle also controls the width of the ringlike deposit.

1. INTRODUCTION In applications such as photonics, electronics, or biotechnology, it is required to control the deposition of materials (polymers and colloids) into organized nanometer- or micrometerscaled patterns over large areas.1-4 However, in applications such as coating or inkjet printing it is required to produce a uniform deposition of material on a substrate by evaporation.5 Both applications still represent a technical challenge because of the accumulation of material at the contact line of an evaporating droplet in a ringlike deposit, known as the “coffee-stain effect”. Because of the work of Deegan,6-8 we know that the rate of evaporation of volatile droplets is not uniform and is maximized at the edge of the drop. When the contact line is pinned, this phenomenon drives a hydrodynamically driven compensation flow directed toward the outer edge of the drop to keep the contact line at a fixed position.9 This water flux drags particles toward the contact line, which accumulate in a ringlike deposit.10,11 Since this work was performed, much effort has been devoted to understanding more precisely the shape and the concentration field of the deposit, which depends on the type of solute in the drop.12-15 Indeed, as the solute concentration increases inside the edge, the viscosity may strongly increase, modifying the flow field in the vicinity of the contact line.14 Moreover, the question of whether the contact line is pinned and the mechanism for the pinning has also been recently studied.16-19 For example, for colloidal suspensions it has been shown that the shape of the forming deposit controls the pinning force,15,20 whereas for polymers the surface tension between the concentrated deposit and the drop as well as the viscosity at the contact line seems to control the motion of the contact line.15,18,21 Moreover, various parameters have been shown to influence the morphology of the deposit, such as the presence of surfactants22 as a result of the solutal Marangoni effect, the conductivity r 2011 American Chemical Society

of the substrate23 as a result of thermal Marangoni24 effects of the electrostatic force of interaction between the particles and the substrate,25 and the initial contact angle of the drops on the substrate.26,27 Although measuring the flow inside the drop seems to be crucial in helping to control the shape of the deposits, only a few articles have reported the flow field inside droplets.7,14,28 For example particle imaging velocimetry, PIV, experiments can be performed, but these probes are usually different from the colloidal particles that actually accumulate at the edge of the drop and might be characterized by different “packing” behaviors at the contact line. Moreover the validity of Deegan’s prediction of the velocity of particles approaching the edge of the drop has been proven only partially. For example, scaling with the distance from the contact line was verified by Deegan,7 but the order of magnitude of the prefactor is not discussed in the article. In the present study, we use a simple method based on optical microscopy to track individual particles as they flow toward the edge of the drop, and we measure their velocity as a function of time for a wide range of particle concentrations. We find that in the last step of the drying process there is a strong acceleration of the particles at the edge of the drop. Using Deegan’s model, we quantitatively show that this accelaration is due to the vanishing contact angle of the drop on the substrate at the end of the drying process. To our knowledge, this is the first time that this strong acceleration of the particle velocity has been reported and that Deegan’s model has been tested as a function of time. We also study the building of the colloidal deposit formed at the edge of the drop as a function of time, and we find that in all cases a particle-free thin liquid film is left at the edge of the drop, Received: October 8, 2010 Revised: January 7, 2011 Published: February 04, 2011 2917

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Langmuir which dries out only at the very end of the evaporation process. The size of this depleted film depends on the contact angle of the drop on the substrate and on the size of the particles. In the case of bidisperse mixtures, we show that this depletion phenomenon can be used to sort mixtures of colloids because the small particles can enter further away into the edge.

2. EXPERIMENTAL SECTION Colloidal suspensions are made from deionized water and latex particles (IDC Corporation) with diameters of 40 and 100 nm and 1, 3, and 5 μm as well as fluorescent 100 nm particles. The weight fraction ranges between 0.16 and 1.6%. The solutions are deposited onto a glass plate, which is first polished with a monodisperse powder of cerium oxide (CeroxGG, Rhodia, 2 μm diameter) and then rinsed with ethanol and distilled water. Using this protocol, we obtain reproducible values of the contact angle of water on glass slides of around 10-15. The evaporation process is studied using an optical microscope in transmission. The size of the ring can be easily measured as a function of time, and the velocity of the particles can be obtained by varying the exposition time (between 500 ms and 3 s) and measuring the length of the trace left by the displacement of the particles. The glass slide is fixed on a balance to measure the loss of mass of the drop by evaporation. We use these mass measurements to deduce the contact angle, which is needed to test Deegan’s model. Assuming that the drop initially takes the shape of a spherical cap, the volume V remaining at any time, the cap radius R, and the drop height h are linked by the following relation V(t) = πh(t)(3R2(t) þ h2(t))/6, leading to V(t) ≈ [πR3(t) θ(t)]/2 for small contact angles θ (because h , R, h can be approximated as h(t) = θ(t) R(t)), which is generally the case for a water drop sitting on a sufficiently hydrophilic substrate. During the drying process, provided that the contact line is pinned, which is the case for our suspensions, the contact angle can be deduced from the mass m(t) of the

Figure 1. (a) Images of the early stage of the drying experiment showing 5 μm particles as they reach the edge of the drop and are stoped a few micrometers from the edge. The time between the two images is 110 s. (b) Magnified image of 5 μm particles observed in the later stage of the evaporation process. The particles leave a thin liquid film of pure water of width λ.

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drop by θ(t) ≈ 2 m(t)/πFR3 (in radians), with F being the density of the liquid.

3. RESULTS AND DISCUSSION In this part, we study the building of the ring composed of latex particles during the evaporation of the drop. We first discuss the early stage of the drying process where individual particles are tracked using optical microscopy as they reach the edge of the drop. In the second part, we investigate the late stage in the drying kinetics where the velocity of the particles is measured. 3.1. Early Step in the Drying Process: Depletion of Particles at the Edge of the Drop. Using optical microscopy, we

study individual latex particles flowing toward the edge of the sessile water droplet in the initial stage of the drying process. In the two photographs presented in Figure 1 (a movie is provided as Supporting Information), it can be observed that a few seconds after the deposition of the drop a radial flow drags particles toward the edge. The outward flow causes the accumulation of these particles during the drying process, which gives rise to the well-known “coffee-stain effect” (i.e., the ringlike deposit composed of particles, whose growth is discussed in the next section). Tracking the first particles flowing toward the edge of the drop shows that the particles suddenly stop a few micrometers away from the edge. The depleted thin liquid film dries out only at the very end of the evaporation process because the outward flow continuously replenishes the edge of the drop with water. As evaporation continues, other particles flow toward the edge and form a monolayer over several rows as shown in Figure 1. The force, which prevents the particles from going deeper into the edge, is probably of a capillary nature because the interface would necessarily have to deform if the particles entered closer to the edge. The deformation of the interface by the dewetting of water on the particle described in ref 29 is unlikely to occur because the contact angle of water with such charged latex particles is usually low.30 Figure 2 is a schematic drawing representing the profile of the droplet and introducing the symbols used in the following. By varying the diameter Dp of the particles and the contact angle θ of water drops on the substrate by appropriate surface treatments, we find that the width of the depleted zone, referred to as λ in the following text, is nearly equal to Dp/tan θ (Figure 2), which supports the fact that the interface is not deformed by the particles. For sufficiently high contact angles, for example, when drops are deposited on a plastic Petri dish, λ falls to zero, meaning that there is no depletion of the colloidal particles. Because the depletion length λ depends on the size of the particles, it is interesting to study whether this depletion effect

Figure 2. Schematic diagrams of the droplet (left side) and the edge (right side) introducing the notations used in the calculations. λ is the width of the zone close to the edge of the drop that is depleted by the particles. L is the width of the ringlike deposit. d is the distance from the inner edge of the deposit at which the velocity U of the particles is measured, and it is fixed equal to 50 μm in the whole study. U is the velocity of the particles flowing to the contact line. ξ(t) is defined as ξ(t) = L(t) þ d. θ is the contact angle, Dp is the diameter of the particles, and R is the radius of the droplet. 2918

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Langmuir can be used to separate mixtures of particles because we expect small particles to penetrate further into the edge of the drop. Using optical microscopy, scanning electron microscopy, and fluorescence microscopy, we find that for all bidisperse colloidal suspensions of various particles sizes or concentration ratios the smallest particles always segregate at the edge of the drop. Figure 4a,b provides evidence of such segregation. These two images were obtained simultaneously using fluorescent detection and optical microscopy for a mixture of 100 nm fluorescent and 5 μm white particles. The images show that the nanometric particles segregate at the edge of the drop. The combination of fluorescence detection and optical microscopy used here could provide a simple method to study the deposition of mixtures of particles but is beyond the scope of this article. In Figure 5 are presented the photographs obtained by SEM and optical microscopy of the deposits left after the evaporation of various mixtures of particles: 1 μm/5 μm as well as 100 nm/1 μm and 100 nm/5 μm particles. The length λ of the segregated ring composed of small particles is reported in Figure 3 (open symbols) and falls on the λ = f(Dp/tan θ) curve obtained for monodisperse colloidal suspensions (closed symbols). These results show that this segregation phenomenon can be used to sort particles in a controlled yet simple manner. Moreover, one

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should be particularly aware of such a segregation effect when studying the drying dynamics of polymer solutions, which are never perfectly monodisperse or when using micrometric tracers to study the field of velocity in drops of colloidal suspensions of nanometric size, which may have a different packing behavior from the nanometric particles. 3.2. Later Stage of the Evaporation Process: Strong Acceleration of the Particles toward the Contact Line. In this section, we study the growth of the ring deposit formed at the edge of the drop as well as the radial velocity of the particles accumulating in the ring as a function of time. In Figure 6, we report the time variation of the radial velocity U of the particles measured at a distance ξ from the contact line. Here we define ξ as ξ(t) = (L(t) þ d) from the contact line, with d = 50 μm for a suspension of 1 μm particles with weight fractions from 0.16 to 1.6%. (See Figure 2 for a schematic drawing of the drop, including the symbols.) L is the width of the ringlike deposit formed at the edge of the drop. First, let us point out that we observe no reversal of flow of the particles in any experiment, which is probably due to the fact that the contact angle is very low.24 Although the radial flow directed toward the edge of the drop is characterized by a velocity of around 1 μm/s at early times, the particles accelerate strongly in the last stage of the drying process, where the velocity is on the order of 20 μm/s. In Figure 7, we report the length L of the ringlike deposit formed by the particles at the edge of the drop as a function of time. Figure 7 shows that the growth of the ring also tends to accelerate at the end of the drying process. In what follows, we use Deegan’s model as well as a simple mass balance approach to explain the spectacular increase in the particle velocity. On the basis of Deegan’s prediction for the local rate of evaporation as a function of the distance x from the contact line, we write the flux per unit length as jðxÞ 

Figure 3. Width λ of the depleted film as a function of the geometric parameters of the edge: Dp, the particle diameter and θ, the contact angle. Solid circles represent the case of monodisperse solutions, and open circles represent the case of bidisperse particles, with the length, λ, being the width of the zone where the small particles are segregated.

Dvap ðCsat - C¥ Þ Fliq xR R 1 - R

ð1Þ

with R = (π/2 - θ)/(π - θ), where θ is the contact angle of the drop on the substrate, Dvap is the diffusion coefficient of vapor in air, Csat is the concentration of water in air at saturation, C¥ is the concentration of water in air a large distance from the drop, Fliq is the density of water, and R the radius of the drop. To get the total flux of water, J(ξ), per unit length lost by evaporation across the

Figure 4. Photographic images of the colloidal deposit obtained for a mixture of 100 nm fluorescent and 5 μm white particles. (a) Photographic image of the deposit obtained using optical microscopy. (b) Photographic image of the same deposit obtained by fluorescence microscopy. It can be seen that the zone of width λ is depleted of micrometer-sized particles but is filled with nanoparticles. 2919

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Figure 5. Examples of the deposits obtained for binary mixtures of colloids using optical microscopy as well as SEM. (a) Mixture of 5 and 1 μm particles (optical microscopy). (b) Mixture of 100 nm and 1 μm particles. The top image is obtained by optical microscopy, and the bottom one is obtained by scanning electronic microscopy. (c) Mixture of 5 μm/100 nm particles obtained by SEM.

the total flux of water flowing inside the drop has to be equal to J(ξ), which is written as Z ξ jðxÞ dx ð2Þ Uθξ ¼ 0

with U being the velocity of the liquid and assuming that the height of the water-air interface is h(x) ≈ θx. From eqs 1 and 2, it follows that the fluid velocity is UðξÞ ¼

Figure 6. Velocity U(t) of the particles at a distance of ξ(t) = (L(t) þ d) from the contact line, where d = 50 μm. U is plotted as a function of time for various weight fractions: 1.6% ((), 0.8% (þ), 0.27% (4), and 0.16% (0).

Figure 7. Width L(t) of the ringlike deposit as a function of time for various weight fractions: 1.6% ((), 0.8% (þ), 0.27% (4), and 0.16% (0).

interface between 0 and ξ, we have to integrate j(x) with respect to x. To keep the contact line pinned as observed experimentally,

Dvap ðCsat - C¥ Þ Fliq ξR R 1 - R θ

ð3Þ

It can be seen from eq 3 that the velocity of the liquid and hence the velocity of the particles is expected to rise for vanishing contact angles. We deduce the contact angle from the measurement of the weight of the drop as a function of time (Experimental Section), assuming that the drop takes the shape of a spherical cap whose volume is V ≈ πR2h ≈ πR3θ. This calculation holds for low volumes, low initial contact angles, and for a pinned drop whose radius R does not vary with time, which the case of the drops that we study. In Figure 8, we plot the particle velocity as a function of 1/θξRR1 - R for colloidal suspensions with varying weight fractions. All curves collapse on a single master curve and follow a power law with an exponent equal to 1. This first point shows that our results are consistent with the scaling derived from Deegan’s model. Moreover, from Figure 8 we find that the prefactor of the curve is equal to 5  10-11 m2/s, which is fairly close to 2  10-10, the expected value of Dvap(Csat - C¥)/Fliq for typical values of Dvap = 2  10-5 m2/s, Fliq = 106 g/m3, and (Csat - C¥) ≈ 10 g/m3 at a relative humidity of 40%. To the best of our knowledge, this is the first time that such good quantitative agreement between experimental data and Deegan’s model is obtained. Finally, to establish the relation between the width of the ring L and the particle velocity U, we use a simple mass-balance argument. In a time Δt, the mass of particles Δm reaching the front of the deposit at a distance L from the edge of the drop is Δm ¼ φUðLÞ hðLÞ2πðR - LÞΔt 2920

ð4Þ

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Figure 8. Verification of Deegan’s model. The velocity U(t) of the particles (obtained from Figure 6) taken at a distance of ξ(t) = (L(t) þ 50 μm) from the contact line and for various times is plotted as a function of 1/(θR(1 - R))[ξ(t)]R) for various weight fractions: 1.6% ((), 0.8% (þ), 0.27% (4), and 0.16% (0). The line represents U = 5  10-11/(θR(1 - R))ξR). The values of U used for this graph are the same as the ones given in Figure 6.

It follows that the volume increase of the ring during Δt is Δm ¼ φc L2πðR - LÞhðLÞ

ð5Þ

where φc is the weight fraction inside the solid deposit. By equalizing both expressions for Δm, it follows that ΔL φUðtÞ ¼ Δt φc

ð6Þ

In our case, the velocity of the particles, which is measured at a distance of ξ(t) = (L(t) þ d) from the contact line, is less than the velocity at a distance L from the contact line. Indeed, from eq 3 we deduce that  1=2 ξ UðξÞ ð7Þ UðLÞ ¼ L Therefore, by integrating eq 6 and using eq 7 it follows that Z 3φ t 3=2 ξðtÞ1=2 UðtÞ dt ð8Þ L ðtÞ ¼ 2φc 0 R To test eq 8, we first need to calcultate t0ξ(t)1/2U(t) dt. To do so, we take the experimental values of U(t) and L(t) obtained in Figures 6 and 7 and calculate the product ξ(t)1/2U(t) for each time and volume fraction. We then plot ξ(t)1/2U(t) (not shown) as a function of time and fit the curves with a power law of the of fitting parameters a form y = atR. Once we obtain the values R and R, we can retrieve the integral t0ξ(t)1/2U(t) dt by calculating (3φa/2(R þ 1))tRþ1 for each time and volume fraction. In Figure 9, we plot L3/2 as a function of (3φa/2(R þ 1))tRþ1 for weight fractions of φ = 1.6, 0.8, 0.27, and 0.16%. It can be seen that we obtain a successful rescaling of the data on a master curve with an exponent equal to 1. Moreover, we find that the prefactor is around 2.5, from which we deduce that the mass fraction of colloids in the deposit is φc = 0.4. Given the fact that the density of the latex particles is 1.5 times that of water, we find the volume fraction of colloids inside the deposit to be φvol-c = 0.27. It is lower than what can be expected for spheres in a close-packed configuration but it is consistent with recent fluorescence measurements reporting surprisingly low values of the concentration

R Figure 9. Relation between L3/2 and 3φ/2 t0ξ1/2U dt: 1.6% ((), 0.8% (þ), 0.27% (4), and 0.16% (0). The line represents y = 2.5x.

inside the deposit.10 The fact that we successfully rescale all curves on a master curve and retrieve a reasonable value for φc means that the decrease in the contact angle during evaporation should not be neglected in modeling because it controls both the particle velocity and the growth of the ring.

’ CONCLUSIONS We have studied the deposition mechanism of colloidal particles during the evaporation of a droplet. We find that the particles pack in a tight deposit but leave a thin liquid film of pure water at the edge of the drop, whose size can be precisely controlled by changing the size of the particles and the contact angle of water on the substrate. We show that this phenomenon can be used to sort mixtures of colloids of different sizes because the smallest particles can penetrate further inside the edge of the drop, leading to segregation of the small particles. We find that the velocity of the particles sharply increases in the last stage of the drying process. Using Deegan’s model, we show that this effect can be explained by the vanishing contact angle at the end of the drying process. Our measurements of the particle velocity and ring width show very good quantitative agreement with Degan’s model. To the best of our knowledge, it is the first time that this model has been validated for several colloidal solutions and over the whole duration of the drying process. Finally, using a simple mass balance we are able to predict the variation of the width of the ring as a function of the particle velocity and find a reasonable value for the colloidal concentration inside the deposit. The segregation effect that we report in our study in the case of binary mixtures shows that one should be careful with the interpretation of the data when using micrometer-sized particles to investigate the drying kinetics of colloidal suspensions made of nanometric particles. This effect may also play a crucial role in polymer solutions, which are never perfectly monodisperse. Hence, it can be expected that smaller chains segregate at the contact line. ’ ASSOCIATED CONTENT

bS

Supporting Information. Movie showing 5 μm particles flowing through the edge of the evaporating droplet and stopping a few micrometer away from the contact line. This material is available free of charge via the Internet at http:// pubs.acs.org.

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’ ACKNOWLEDGMENT We thank L. Limat, F. Doumenc, H. Bodiguel, and B. Guerrier for useful discussions. We are grateful for financial support from Agence Nationale de la Recherche through program ANR BLAN-DEPSEC ’ REFERENCES (1) Yabu, H.; Shimomura, M. Adv. Funct. Mater. 2005, 15, 575. (2) Xu, J.; Xia, J.; Hong, S. W.; Lin, Z. Q.; Qiu, F.; Yang, Y. L. Phys. Rev. Lett. 2006, 96, 066104. (3) Hong, S. W.; Byun, M.; Lin, Z. Q. Angew. Chem., Int. Ed. 2009, 48, 512. (4) Byun, M.; Bowden, N. B.; Lin, Z. Q. Nano Lett. 2010, 10, 3111. (5) Gans, B-J; Schubert, U. S. Langmuir 2004, 20, 7789. (6) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten., T. A. Nature 1997, 389, 827. (7) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten., T. A. Phys. Rev. E 2000, 62, 756. (8) Deegan, R. D. Phys. Rev. E 2000, 61, 475. (9) Hu, H.; Larson, R. G J. Phys. Chem. B 2002, 106, 1334. (10) Adachi, E. S.; Dimitrov, A. S.; Nagayama, K. Langmuir 1995, 11, 1057. (11) Shmuylovich, L.; Shen, A. Q.; Stone, H. A. Langmuir 2002, 18, 3441. (12) Kajiya, T.; Kaneko, D.; Doi, M. Langmuir 2008, 24, 12369. (13) Kajiya, T.; Nishitani, E.; Yamaue, T.; Doi, M. Phys. Rev. E 2006, 73, 011601. (14) Bodiguel, H.; Leng, J. Soft Matter 2010, 6, 5451. (15) Kajiya, T.; Monteux, C.; Narita, T.; Lequeux, F.; Doi, M. Langmuir 2009, 29, 6934. (16) Rio, A.; Daerr, F.; Lequeux; Limat, L. Langmuir 2006, 22, 3186. (17) Berteloot, G.; Pham, C. T.; Lequeux, F.; Limat, L. Europhys. Lett. 2008, 83, 14003. (18) Monteux, C.; Elmaallem, Y.; Narita, T.; Lequeux, F. Europhys. Lett. 2008, 83, 34005. (19) Bodiguel, H.; Doumenc, F.; Guerrier, B. Langmuir 2010, 26, 10758. (20) Bodiguel, H.; Doumenc, F.; Guerrier, B. Eur. Phys. Special Topics 2009, 166, 29. (21) Monteux, C.; Tay, A.; Narita, T.; Lequeux, F. Soft Matter 2009, 5, 3713. (22) Kajiya, T.; Kobayashi, W.; Okuzono, T.; Doi, M. J. Phys. Chem. B 2009, 113, 15460. (23) Ristenpart, W. D.; Kim, P. G.; Domingues, C.; Wan, J.; Stone, H. A. Phys. Rev. Lett. 2007, 99, 234502. (24) Hu, H.; Larson, R. G. Langmuir 2005, 21, 3972. (25) Yan, Q.; Gao, L.; Sharma, V.; Chiang, Y.; Wog, C. C. Langmuir 2008, 24, 11518. (26) Kuncicky, D. M.; Velev, O. D. Langmuir 2008, 24, 1371. (27) Pauchard, L.; Allain, C. Phys. Rev. E 2003, 68, 052801. (28) Dhavaleswarapu, H. K.; Migliaccio, C. P.; Garimella, S. V.; Murthy, J. Y. Langmuir 2010, 26, 880. (29) Sangani, A. S.; Lu, C.; Su, K.; Schwartz, J. A. Phys. Rev. E 2009, 80, 011603. (30) Aveyard, R.; Clint, J. H.; Nees, D.; Paunov, V. N. Langmuir 2000, 16, 1969.

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