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Langmuir 1997, 13, 3598-3602
Articles Drying of Colloidal Suspension Droplets: Experimental Study and Profile Renormalization F. Parisse and C. Allain* Laboratoire Fluides, Automatique et Syste` mes Thermiques, Baˆ t. 502, Campus Universitaire, 91405 Orsay Cedex, France Received December 8, 1995. In Final Form: July 29, 1996X Under drying, drops of concentrated silica sol deposited onto flat substrate undergo large morphological changes. We have performed accurate measurements of the drop profile during the solvent loss. The results obtained for different drop volumes have been interpreted using a renormalization procedure. The length and time scales agree well with that expected when the evaporation is limited by solvent diffusion in air. The final thickness of the solid film deposited onto the substrate is compared with a simplified model of drop drying.
I. Introduction Recently, wetting phenomena have been a subject of renewed interest, and the ideal case of pure nonvolatile liquids on flat substrates is now well understood.1,2 In many situations of practical interest, more complicated systems are encountered which involve complex fluids or rough surfaces. Of particular interest is the case of fluids including a volatile component. When the fluid is a volatile mixture of liquids, the nonhomogeneous evaporation creates gradients of surface tension which are at the origin of liquid motions, meniscus deformations, and instabilities.3-6 When the mixture includes a solid component, the shape of the interface depends not only on surface tensions and is more complicated to predict.7 In the present paper, we consider the case of concentrated colloidal suspensions and we focus on the morphological changes of small drops on an horizontal nonporous substrate. As soon as solvent loss begins, particles of the suspension deposit onto the substrate near the three-phase line, leading to a strong anchoring. Later on, the drop contact base remains constant and distortions from a spherical-cap shape are observed: a “foot” forms near the edge of the drop and progressively extends toward the middle of the drop (see Figure 1). Qualitative observations show that the foot behaves as a solid: Firstly, if we try to suck up the drop, only the central part moves back while the foot remains as a ring adhering onto the substrate. Secondly, vertical transmission observation of the drop between crossed-polarizers shows that the foot is birefringent. The increase of the particle volume fraction due to the solvent loss leads to the formation of a colloidal gel,8 and as evaporation goes on, large stresses appear inside the gel which are at the origin of the observed * Author to whom correspondence should be addressed. Fax: +33 1 69 15 80 60. E-mail:
[email protected]. X Abstract published in Advance ACS Abstracts, June 1, 1997. (1) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (2) Le´ger, L.; Joanny, J. F. Rep. Prog. Phys. 1992, 431. (3) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall International, Inc.: London, 1962. (4) Probstein, R. F. Physicochemical Hydrodynamics; Butterworth Publishers: Stoneham, 1989. (5) Neogi, P. J. Colloid Interface Sci. 1985, 105, 94. (6) Vuilleumier, R.; Ego, V.; Neltner, L.; Cazabat, A. M. Langmuir 1995, 11, 4117. (7) Parisse, F.; Allain, C. J. Phys. II 1996, 6, 1111. (8) Brinker, C. J.; Scherer, G. W. Sol-gel science: the physics and the chemistry of sol-gel processing; Academic Press: San Diego, CA, 1990.
S0743-7463(95)01521-6 CCC: $14.00
birefringence and, finally, of the decohesion, cracking, and warping of the foot (see Figure 1d).8-12 We have performed an experimental investigation of the drop shape changes under drying. In particular, we show that the profiles observed at different times for various drop volumes can be renormalized and we find the dependence of the characteristic time scale on the drop size. Furthermore, the renormalized thickness of the foot is described using a simplified model which takes into account the strong anchoring of the three-phase line and the gelation induced by the increase of the colloid volume fraction due to solvent loss. The paper is organized as follows. In section II, we describe the materials, the experimental setup used to determine the drop profile, and the methods chosen to characterize the drop geometry and the evaporation kinetics. In section III, we present the results obtained varying the drop size. We show that, under our experimental conditions, the evaporation is limited by the diffusion of the solvent in air. This allows us to interpret the length scale and the time scale used in the profile renormalization. In section IV, the results on the foot profile are compared with a simplified model of the drop drying. II. Materials and Methods II.1. Materials. The suspension used is an aqueous silica colloidal sol. All the experiments were done with the same sol, having the volume fraction in particles Φ ) 0.24 ( 0.01. The diameter of the particles determined by electron microscopy is 15 ( 2 nm. The density of the sol is 1.28 ( 0.01 g/cm3. The pH is about 8.9. So, the particle surface bears a high negative charge density,13 and in the absence of evaporation, the suspension is stable. Droplet volumes, V0, ranging from 0.2 to 10 mm3 were used. The relative accuracy on Vo is ∆V0/V0 ) 4%. The drying experiments were performed on glass microscope slides. The slides, used once only, were cleaned in the following way: they were left for about 24 h in a 2 × 10-2 g/g solution prepared from a laboratory cleaning liquid sold by Labo Express Service (France). Then, the slides were rinsed, first with deionized water (quality Milli-F) and then with ethanol, and dried for about 10 h at 120 °C. This procedure, which leads to partially (9) Garino, T. J. Mater. Res. Soc. Symp. Proc. 1990, 180, 497. (10) Atkinson, A.; Guppy, R. M. J. Mater. Sci. 1991, 26, 3869. (11) Allain, C.; Limat, L. Phys. Rev. Lett. 1995, 74, 2981. (12) Hutchinson, J. W.; Suo, Z. Adv. Appl. Mech. 1992, 29, 63. (13) Iler, R. K. The Chemistry of Silica; Wiley: New-York, 1979.
© 1997 American Chemical Society
Drying of Colloidal Suspension Droplets
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Figure 1. Digitalized images of a 2 mm3 droplet taken at different times after deposition: a, t ) 0; b, t ) 240 s; c, t ) 360 s; d, t ) 480 s. The length of the bar indicated in part a is 1 mm. hydrophilic surfaces, gives a good reproducibility on the contact angle measured for suspension drops; the mean value of θ is 40°, and the mean standard deviation is 2°. Using different cleaning procedures (such as shortening the drying time in the oven or performing immersion in sulfochromic acid) leads to a smaller contact angle but does not significantly change the results.7 II.2. Profile Determination. To measure the drop profile evolution, we have used videorecording and image data processing. The drop is illuminated from behind using a white light source and a diffuser to produce a uniform background. The drop is then viewed from the front using a CCD camera equipped with a macrophotography 60 mm lens (AF Micro Nikkor) mounted on a bellows focusing device. The images are recorded on a videotape in real time and later digitalized into 512 pixel × 512 pixel images with 256 gray levels using an IBM/PC equipped with a Cyclope card. The transition from the dark level corresponding to the projection of the drop (=35) to the clear background level (=220) (see Figure 1) takes place on about 3 or 4 pixels, allowing the determination of the drop profile within an accuracy of (1 pixel. The glass slides were placed on a holder, ensuring the substrate surface was horizontal within (0.1°. Great care was taken to avoid optical distortions. First, the apertures of the incident beam and of the detection device were chosen small enough to ensure that no ray making an angle larger than 3° with the optical axis is recorded. Second, the horizontal orientation of the detection device is carefully adjusted (within (0.3°), which allows us to neglect all inclination effect on the drop observation. Then, even when large magnifications are used or/and when very flat drops are investigated, the uncertainty on the profile due to optical observation is less than 1 pixel. II.3. Geometrical Characterization and Evaporation Flux Determination. Firstly, the profile measured for t ) 0,
i.e. just after drop deposition, is analyzed to check the spherical shape of the drop and to determine its geometrical characteristics: R0, the radius of the contact base; H0 the height at the apex; and θ0, the contact angle (The subscript 0 means that the quantities are taken at t ) 0). When the drop is entirely recorded as in Figure 1, we directly determine R0, H0, and θ. The theoretical profile (h0 versus r) is then calculated and compared to the measured one. Except for the largest drop volume used (V0 ) 10 mm3), for which a slight flattening under gravity occurs, drops showing discrepancies between the measured and calculated profiles larger than 3 pixels are disregarded. Finally, the axisymmetry of the drop is verified, comparing the deposited volume to the volume calculated from the profile. To characterize the evaporation rate, we introduce the mean water flux per surface unit: vE ) -(1/S)(dV/dt), where S is the surface of the air-drop interface and V the drop volume. From the profiles measured at different times, we can calculate S, V, and thus vE. To check the validity of this method, we first compared the values of dV/dt determined from the profiles, as just explained, to determination from weighing measurements during the same run (the accuracy on weighing measurements is (10-4 g). As shown in Figure 2, good agreement is observed between the two determinations. Secondly, the choice of the overall air-drop surface for S, even when distortions from a spherical-cap shape occur, was checked. In practice, all different choices of S, as for instance taking only the central part of the drop, lead to large values of vE. These are unrealistic, since vE cannot exceed its value for pure water. Furthermore, in the drying regime investigated, vE is expected to be constant and of about the same order as that for pure water;8 that agrees well with our observations taking for S the overall air-drop surface. Finally, the study of the evaporation rate of pure water drops onto polyethylene substrates shows that, under our experimental
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Parisse and Allain
Figure 2. Variation of the drop volume versus time: (b) deduced from weighing measurements, (9) calculated from the profile. After 750 s of drying, the foot warps impeding determination of the volume from the profile. Note that, before that time, the decrease of V with time is linear. conditions, the water loss is ruled solely by the diffusion of water in air and that no convection occurs near the interface (see section III). No specific arrangement was done to assign the water moisture, but the relative humidity, RH, was measured during each experiment using an electronic hygrometer.
III. Drop Size Influence and Evaporation Kinetics III.1. Profile Renormalization. During drying, drops of volume V0 ranging from 0.2 to 5 mm3 exhibit very similar shapes. To compare the results obtained for different values of V0, we use a renormalization procedure. Firstly, we construct dimensionless profiles. Since the contact angle is constant and no flattening due to gravity occurs, the geometrical properties of the drop involve only one length scale. In practice, we use as characteristic length the contact base radius R0. Then, the dimensionless distance from the base center is expressed as F ) r/R0 and the dimensionless height as η ) h/R0. Secondly, the comparison of profiles measured at different times necessitates the introduction of a characteristic time scale. From the evaporation rate, we can construct the time scale tE ) R0/vE. The dimensionless time is then τ ) t/tE ) tvE/R0. Figure 3 reports the dimensionless profiles obtained for the several values of τ, for four drop volumes. For a same value of τ, all the points fall on a single master curve, showing that the drop profiles renormalize very well. III.2. Diffusion-Limited Evaporation. The choice of tE ) R0/vE as time scale assumes that evaporation does not introduce another length scale than R0. To justify this assumption, we are going to show that, under our experimental conditions, the evaporation is ruled by water diffusion in air. In that aim, we first consider the case of pure water droplets. This will also allow us to predict the variation of vE with R0. Assuming quasi-stationary conditions, if diffusion is the sole effect, the water concentration in air nw (expressed in moles per unit volume) follows the Laplace equation: ∆nw ) 0. The boundary conditions are nw ) nws in the vapor at the air-water interface and nw ) nw∞ at infinity. nws represents the value of nw under saturation conditions. nw∞ is related to nws through the relative humidity (RH). Let us first consider the case of a spherical drop of radius Rs in an infinite medium. The isoconcentration surfaces are then spherical, and the water flux is given by 4πrs2D dnw/drs ) a constant, where rs is the radius of a spherical surface concentric with the drop and D is the diffusion coefficient of water in air. When the boundary conditions
Figure 3. Drop profiles corresponding to different dimensionless times: a, τ ) 0; b, τ ) 0.037; c, τ ) 0.057. The initial volume V0 of the drops is 0.25, 1, 2, and 5 mm3. The profiles for V0 ) 2 mm3 correspond to the same run as the digitalized images in Figure 1. In part c, the lines on the left correspond to (s) the drying model presented in section IV (eq 7) and (- - -) an affinity transformation of the profile at t ) 0 (eq 8).
are taken into account, nw can be easily calculated: nw ) nw∞ + (nws - nw∞)Rs/rs. The conservation of water flux at the interface Sn1vE ) -4πRs2D(dnw/drs)rs)Rs gives
vE )
(
)
D nws - nw∞ Rs nl
(1)
where the subscript l means for pure liquid water. During the evaporation, the drop radius Rs diminishes, but in practice, due to the difference in the concentration between liquid and vapor state, its variation is slow enough to consider that the water concentration gradient in air is the same as that in the absence of an Rs decrease, i.e. that a quasi-stationary state is reached at any time. For a sessile drop receding with a constant contact angle θ0, the isoconcentration surfaces are more complicated. However, as previously, the sole length scale involved arises from the boundary condition at the air-drop interface. Using as characteristic length scale the contact base radius Rb leads to
(
)
D nws - nw∞ vE ) A(θ0) Rb nl
(2)
where A(θ0) is a constant which only depends on the contact angle θ0. For θ0 ) π/2, A(θ0) ) 1, and for θ0 ) 0, i.e. for a flat disk, A(θ0) ) 4/π.14 Numerical values of A(θ0) can be found from the polynomial approximations of the capacitance of a lens calculated by Picknett et al.15 The time evolution of the base radius is given by (14) Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids; Oxford University Press: London, 1959. (15) Picknett, R. G.; Bexon, R. J. Colloid Interface Sci. 1977, 61, 336.
Drying of Colloidal Suspension Droplets
(
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R20 - R2b ) B(θ0)D
)
nws - nw∞ t nl
(3)
where, as previously, R0 represents the contact base radius at t ) 0. B(θ0), which only depends on θ0, can be expressed as
B(θ0) )
4 sin θ0
A(θ0)
(1 - cos θ0)(2 + cos θ0)
(4)
For θ0 ) π/2, B(θ0) ) 2, and for θ0 f 0, B(θ0) behaves as (32/3π)θ0-1 = 3.4θ0-1. To check if, under our experimental conditions, the evaporation is limited by the diffusion of water in air, we have performed experiments on pure water drops deposited onto polyethylene substrate and we have compared the results with the predictions of eq 2. Polyethylene substrates (smooth flat dishes sold by OSI France for weighing measurements) have been used instead of glass microscope slides, since the contact angle for pure water is then about 75°, allowing a good determination of the drop shape variations versus time (for glass substrates the contact angle is less than 10°, impeding an accurate determination of vE). Figure 4 displays the variation of vERb versus time. Except for the longest time, when the drop volume is less than 10% of its initial value, vERb remains almost constant, in agreement with eq 2. The main value observed for vERb is equal to 2.5 × 10-4 mm2/s. From the values of temperature and relative humidity recorded during the experiment, we can calculate D(nws - nw∞)/nl.16 Assuming that A(θ0 ) 75°) = 1, vERb is found to be =2.3 × 10-4 mm2/s, in good agreement with the measured value. The inset in Figure 4 displays the variation of Rb2 versus t. The points all fall on a straight line; the value of the slope obtained fitting the experimental data (5.7 × 10-4 mm2/s) agrees with the value deduced from eq 3 (=5 × 10-4 mm2/s). All these results show unambiguously that, under our experimental conditions, the rate of evaporation is limited by the diffusion of water in air. III.3. Colloidal Suspension Drops. Since the profiles for different drop volumes renormalize using R0 as a length scale (see Figure 3), the boundary condition at the air-drop interface and, thus, the isoconcentration surfaces involve only one length scale R0. So, as previously the evaporation being limited by water diffusion in air, the dynamics of the drying involves only one time scale: tE ) R0/vE. On the other hand, in the range of the variation of lost volume involved, typically ∆V/V = 30%, vE is constant (see Figure 2). So, vE can be calculated from its value at t ) 0: vE ) vE0. Following eq 2, we get
tE )
R20
(
)
nl ∼ R20 A(θ0)D nws - nw∞
(5)
Thus, the time tE is expected to scale as R02 or V02/3. Figure 5 reports the variation of tE versus R0. On a log-log scale, the points fall on a straight line, showing that tE varies with R0 as a power law. The exponent is found equal to 1.9 ( 0.1, in good agreement with eq 5. As previously, we can estimate the prefactor knowing the temperature and relative humidity during the experiment. Taking 4/π for the order of magnitude of A(θ0 ) 40°), the value estimated (16) See for instance: Incropera, F. P.; De Witt, D. P. In Fundamentals of Heat and Mass Transfer; John Wiley and Sons: New York, 1990.
Figure 4. Variation of vERb versus time for a pure water droplet deposited onto a polyethylene substrate. Until =12 000 s, the contact angle is constant (θ0 = 75°), and vERb remains almost constant, showing the validity of eq 2. Later on the contact angle decreases with time. Inset: variation of the square of the base radius, Rb2, versus time. The full line corresponds to the best fit (eq 3).
Figure 5. log-log plot of the time scale tE against R0 (tE is expressed in s and R0 in mm). The full line corresponds to the best fit (eq 5).
for the prefactor (=2.8 × 103 s/mm2) agrees with the one found fitting the experimental points in Figure 5: 2.6 × 103. In conclusion, the time scale tE used in the profile renormalization corresponds well with that expected when the evaporation is limited by the diffusion of the solvent in air. IV. Foot Shape Description As shown in Figure 1, the extent of the foot increases with time until decohesion and warping occurs. In order to describe the thickness of the foot, we construct a simplified model which assumes that fluid motion inside the drop is the same as if no gelling takes place. The calculations include two steps: first, we calculate the flow rate for a pure liquid drop evaporating with a constant contact base, i.e. with an anchored three phase line. Since the relative proportion of the liquid-air interface is larger near the drop edge, a large amount of solvent is lost in this area. This induces a flow of liquid from the middle of the drop to the edge of the drop. This flow is easy to calculate under the assumption that the drop remains a spherical cap with a constant contact base:7
3602 Langmuir, Vol. 13, No. 14, 1997
[( ( )
∂J r ) 2πvEr 1 ∂r R0
(
2
) )(( ( )
6 (1 - cos θ)(2 + cos θ)
sin2 θ
1-
Parisse and Allain
-1/2
ηfa )
-
r R0
2
)
sin2 θ
1/2
)]
- cos θ
(6)
During the drying, the decrease of the volume leads to a decrease of the contact angle θ of the spherical cap. In practice, numerical calculations show that J is the same whatever θ is, although θ remains smaller than θ* = 50°. Second, for a colloidal suspension, the decrease of the volume is limited by the gelation of the suspension. Indeed, under water loss, the particles and the ionic species accumulate until the volume fraction of the particles reaches a given value Φg, for which a rigid gel forms. Then, the volume decrease stops,7 and no significant shrinkage takes place before cracking due to the large value of the elastic moduli of concentrated colloidal gels.8 So, for each value of F ) r/R0, the value of the foot height ηf ) h/R0 can be calculated from water volume conservation: through the elementary surface included between F and F + dF, the total amount of water lost by evaporation is equal to the total amount of available water calculated taking into account the initial height of the drop η0 and the gradient of the flow rate J induced by evaporation (eq 6). Using dimensionless quantities gives
ηf )
/{ ( )
Φg - Φ Φ η0 1 × Φg Φg
[
1-
6((1 - F2 sin2 θ)1/2 - cos θ)
]}
(1 - cos θ)(2 + cos θ)(1 - F2 sin2 θ)-1/2
(7)
In Figure 3c, the foot shape calculated from eq 7 is compared to the renormalized profiles. A good agreement is observed. The lone parameter in the model is the gel volume fraction Φg, which was taken equal to 0.5.7 We also report in Figure 3c the profile calculated by an affinity transformation of the profile at t ) 0, i.e. neglecting all fluid motion inside the drop.
Φ η Φg 0
(8)
A large discrepancy is observed, showing the importance of the flow induced by the evaporation. Although the model we propose is oversimplified, neglecting the distortions from a spherical-cap shape and the contribution of elasticity, it allows us to take into account the flow toward the drop edge, which continuously occurs even when the foot has formed (see section II.3). It explains especially well the large foot height observed near the edge of the drop. V. Summary Under evaporation, the shape of a concentrated colloidal suspension drop exhibits complex dynamics. Indeed, since the drop shape is not only ruled by surface tensions, it differs from a spherical-cap one. A gelled foot forms near the drop edge, which progressively extends toward the middle of the drop. We have shown that, for drops of different volumes (small enough to involve no gravityinduced flattening), the profiles can be renormalized, introducing characteristic length and time scales. When the evaporation is limited by the diffusion of the solvent in air, as under our experimental conditions, it is possible to predict the properties of these length and time scales. A good agreement is observed with the results of the experimental investigation. An oversimplified model allows us to describe the renormalized foot thickness. Comparison with an affinity transformation of the drop shape at t ) 0 shows the importance of the liquid motion inside the drop induced by evaporation. Acknowledgment. We are grateful to C. Bouchard and B. Guerrier for enlightening discussions. Laboratoire “Fluides, Automatique et Syste`mes Thermiques” is a laboratory of Paris VI and is associated with CNRS (URA 871). This work is part of the thesis of F. Parisse, sponsored by the Ministe`re de l’Enseignement Supe´rieur et de la Recherche. LA951521G
