3202
Langmuir 2001, 17, 3202-3207
Lateral Drying in Thick Films of Waterborne Colloidal Particles J. M. Salamanca,† E. Ciampi,† D. A. Faux,† P. M. Glover,† P. J. McDonald,† A. F. Routh,‡ A. C. I. A. Peters,§ R. Satguru,§ and J. L. Keddie*,† Department of Physics, School of Physics and Chemistry, University of Surrey, Guildford, Surrey GU2 7XH, U.K.; School of Chemistry, Cantock's Close, University of Bristol, Bristol BS8 1TS, U.K.; NeoResins, Sluisweg 12, PO Box 123, 5140 AC Waalwijk, The Netherlands Received November 14, 2000. In Final Form: March 9, 2001 The first systematic and quantitative experimental study of the influences on lateral drying in colloidal films is reported. The time until water recedes from the edge of a drying thick film of a waterborne colloidal dispersion, called the open time, was measured as a function of several controllable parameters. Magnetic resonance microscopy, using a specially designed probe, noninvasively provides a direct and quantitative measurement of the concentration of water as a function of vertical and lateral position. Images were obtained from drying films of latices with known values of thickness, particle size, and surface tension, which are neatly encapsulated in an expression for the reduced capillary pressure, pc. A strong increase in the open time was found over a relatively narrow range of pc values. Larger particles, slower evaporation rates, and thinner films encourage more uniform lateral drying with a delay in drying from the edges. This observation is consistent with a recent model (Routh, A. F.; Russel, W. B. AIChE J. 1998, 44, 2088) based on the lubrication approximation. The experiments and the modeling point to a way of achieving control over the lateral drying processes of waterborne colloids.
Introduction The deposition of solid, colloidal particles onto a surface via the drying of a waterborne dispersion is a fundamental process in nature (such as in mud puddles and winterbournes) and in many technologies.1 Notably, latex paints,2,3 inks, floor waxes, cosmetics, correction fluid, slurries of glass and ceramic frits for glazes or enamels, biofilms,4 and some pharmaceuticals5 rely on this process. With the recent emergence of nanotechnology, the controlled deposition of colloidal particles6,7 is of growing importance. Ordered arrays of particles, created by convective particle transport during drying and attractive capillary forces,8 have potential applications in optical devices and data storage. Even structured semiconductor devices, such as quantum dots, have been deposited via the colloidal route.9 Lately, with tightening environmental legislation10,11 and increasing public awareness, there is an enhanced interest in waterborne formulations in lieu * To whom correspondence should be addressed. E-mail:
[email protected]. † University of Surrey. ‡ University of Bristol. § NeoResins. (1) Keey, R. B. Introduction to Industrial Drying Operations; Pergamon Press: Oxford, 1978. (2) Winnik, M. A. Curr. Opin. Coll. Interface Sci. 1997, 2, 192. (3) Keddie, J. L. Mater. Sci. Eng. Rep. 1997, 21, 101. (4) Menegalli, F. C.; Sobral, P. J.; Roques, M. A.; Laurent, S. Drying Technol. 1999, 17, 1697. (5) Schmidt, C.; Bodmeier, R. J. Controlled Release 1999, 57, 115. (6) Fan, H. Y.; Lu, Y. F.; Stump, A.; Reed, S. T.; Schunk, R.; PerezLuna V.; Lopez, G. P.; Brinker C. J. Nature 2000, 405, 56. (7) Aizenberg, J.; Braun, P. V.; Wiltzius, P. Phys. Rev. Lett. 2000, 84, 2997. (8) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.; Yoshimura, H.; Nagayam, K. Nature 1993, 361, 26. (9) Maenoson, S.; Dushkin, C. D.; Saita, S.; Yamaguchi, Y. Langmuir 1999, 15, 957. (10) DeVito, S. Prog. Org. Coat. 1999, 35, 55. (11) Jotischky, H. Surf. JOCCA Coat. Int. Part B: Coat. Trans. 2001, 84, 11.
of those systems that emit volatile organic compounds into the atmosphere. The drying of planar films12 and convex drops13,14 of waterborne colloids has been observed to proceed from the thin edges. In a process referred to as lateral drying,15 the solids fraction near the center remains close to the initial value while a region of high solids content develops at the edge. At later times, a drying front (i.e. the air/ water interface) moves inward from the edge. The lateral transport of both colloidal particles16 and water14 from the sample center to the edges has likewise been reported. At the same time, others have reported experimental observations in which the fraction of solids increases relatively uniformly12 or to only a limited extent17 in the lateral direction. Drying does not occur strongly from the edges, in apparent contradiction of other reports describing lateral transport and nonuniformity. The observed differences in drying mechanisms are probably real, and they most likely result from differences in a myriad of experimental parameters, such as film thickness, evaporation rate, dispersion viscosity, and particle size. The distribution of the water during the drying of colloidal films is of practical importance because ions, surfactants, and various soluble species are carried in the aqueous phase.18 They are transported laterally with the water, sometimes with undesired effects.19 Furthermore, the technology of (12) Winnik, M. A.; Feng, J. J. Coat. Technol. 1996, 68 (852), 39. (13) Parisse, F.; Allain, C. J. Phys. II 1996, 6, 1111. (14) Ciampi, E.; Goerke, U.; Keddie, J. L.; McDonald, P. J. Langmuir 2000, 16, 1057. (15) Holl, Y.; Keddie, J. L.; McDonald, P. J.; Winnik, M. A. In Film Formation in Coatings: Mechanisms, Properties and Morphology; Provder, T., Urban, M. W., Eds.; Cambridge University Press: Cambridge, 2001; Chapter 1. (16) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Nature 1997, 389, 827. (17) Okubo, M.; Takeya, T.; Tsutsumi, Y.; Kadooka, T.; Matsumoto, T. J. Polym. Sci., Polym. Chem. Ed. 1981, 19, 1. (18) Holl, Y. Macromol. Symp. 2000, 151, 473. (19) Juhu´e, D.; Wang, Y. C.; Lang, J.; Leung, O. M.; Goh, M. C.; Winnik, M. A. J. Polym. Sci., B: Polym. Phys. 1995, 33, 1123.
10.1021/la001590h CCC: $20.00 © 2001 American Chemical Society Published on Web 05/03/2001
Lateral Drying in Thick Films
particle deposition (noted above) hinges on control of the lateral drying process. The specific effects of particle size, layer thickness, evaporation rate, surface tension, and dispersion viscosity on the lateral drying mechanism have been largely unexplored and ill-described. Regrettably, there are no reports in the literature of a systematic study of their effects on drying, but there are occasional remarks and intense speculation. Those previous studies that do exist rely on the observation of the drying film with optical microscopes, cameras, and human eyes. It is generally accepted that a wet film of colloidal particles is turbid. Upon film formation at temperatures above the minimum film formation temperature, a latex without pigment becomes transparent.3 After drying well below this temperature, the film will have the appearance of a powder. Optical or visual observation of drying is severely limited as a technique, however, for several reasons. One, the visual appearance is a strong function of the particle size20 and the particle packing,21,22 and this factor cannot be easily taken into account when comparing the appearance of different dispersions. Two, it is difficult to relate varying levels of turbidity with water concentration while simultaneously making measurements as a function of the lateral position in a film. Even qualitative interpretations can be prone to error. Three, information is not provided about drying vertically from the film surface or about the shape of the drying front. For the first time, the use of magnetic resonance (MR) microscopy to examine lateral drying in films of latex dispersions is reported. This noninvasive technique measures water concentration with high lateral and vertical spatial resolution. Although commonplace and well-established in medical applications, MR microscopy has primarily been applied to soft condensed matter by only a few groups worldwide within the past decade.23 The fundamental question addressed herein is: What determines how long water remains at the edges of a drying colloidal film? A significant advance in answering this question was made recently by the theoretical modeling of Routh and Russel (R-R).24 They defined the so-called open time as the time until a water front recedes from the sample edges. Their model provides a framework for experimental study by identifying the parameters that influence lateral drying. The R-R model was constructed upon a simplified description for a film of colloidal particles dispersed in water with an initial film thickness H and an edge that is described as either a step or a circular arc. The model predicts the height and the fraction of solids in a drying layer as a function of time. As water evaporates, the solids content increases, and the particles form a close-packed region at the edge with water filling the interparticle void space. Continued evaporation from this close-packed region draws solvent from the central fluid region, transporting particles and propagating the close-packed region inward. This flow of water (or other solvent) through the packed region sets a pressure gradient, which is balanced by the capillary pressure. The capillary pressure is the result of the curved water surfaces in the “throats” created by any three particles in contact. (20) Meeten, G. H. Optical Properties of Polymers; Elsevier Applied Science Publishers: London, 1986; p 29. (21) Van Tent, A.; Te Nijenhuis, K. Prog. Org. Coat. 1992, 20, 459. (22) Van Tent, A.; Te Nijenhuis, K. J. Colloid Interface Sci. 1992, 150, 97. (23) Blu¨mler, P., Blu¨mich, B., Botto, R., Fukushima, E., Eds. Spatially Resolved Magnetic Resonance; Wiley-VCH: Chichester, 1998. (24) Routh, A. F.; Russel, W. B. AIChE J. 1998, 44, 2088.
Langmuir, Vol. 17, No. 11, 2001 3203
At the core of the R-R model is the realization that water will recede from the film’s edge only when the pressure in the packed bed of particles exceeds the opposing capillary pressure. The model therefore calculates the pressure in the packed bed near the film edge as a function of time during the evaporative drying process in order to determine when it exceeds the maximum capillary pressure. One of the primary outcomes of the R-R model is a description of how the open time increases with the capillary pressure. This outcome is tested in the experimental work presented here. According to theoretical arguments by Brown25 and subsequent experimental measurements,26 the capillary pressure for a liquid that fully wets the particles is approximated as 10γ/a, where γ is the surface energy of the liquid and a is the particle radius. In the development of the R-R model, all terms are reduced to a dimensionless form through normalization by an appropriate scaling parameter. Hence, the capillary pressure 10γ/a is normalized by the characteristic pressure for flow through a packed bed of particles, and it is expressed as24
pc )
( )
20 3γη0 75 E
1/2
a(1 - φm)2 µφm2H
(1)
where η0 is the zero-shear-rate viscosity of the dispersion, E is the water evaporation rate (in units of distance per time), and µ is the viscosity of the continuous liquid. The volume fraction of solids at close-packing of the particles is given as φm. For monosized spheres, φm should vary between 0.64 for random close-packing27 and 0.74 for facecentered cubic packing. The expression for pc given in eq 1 contains several parameters (a, H, E, γ, and η0) that can be controlled experimentally and in technological applications. In this work, differing combinations of the parameters are used to produce samples with values of reduced capillary pressure spanning 3 orders of magnitude. We thereby provide the first thorough and systematic test of the R-R model and determine unequivocally how various experimental parameters influence the lateral drying of waterborne colloids. Experimental Details The value of the reduced capillary pressure in drying films was systematically varied through judicious control of the particle size, film thickness, and evaporation rate. Latex dispersions were cast as films (up to 1.2 mm thick) onto the surface of 2 cm × 2 cm glass coverslips. These films were sufficiently wide to ensure that gravity is able to greatly reduce the curvature of the convex meniscus. Consequently, the samples can legitimately be described as colloidal films rather than drops. The experiments were designed to ensure that the solids content in the vertical direction was uniform throughout the drying process. This condition can be met when the vertical diffusive velocity, expressed as D0/H, greatly exceeds the evaporation, E, which also has units of velocity.28 D0 is the Stokes' diffusion coefficient.27 In all experiments reported here, we thus ensured that D0/HE > 1. Two types of dispersion were used: an acrylic latex29 and a poly(styrene) latex (obtained from Bangs Laboratories, Fishers, IN). In both, the glass transition of the polymer was well above (25) Brown, G. L. J. Polym. Sci. 1956, 22, 423. (26) Mason, G.; Morrow, N. J. Colloid Interface Sci. 1986, 109, 46. (27) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (28) Routh, A. F.; Russel, W. B. Langmuir 1999, 15, 7762. (29) Composed of a copolymer of butyl acrylate, methyl methacrylate, and methacrylic acid and synthesized via standard techniques of emulsion polymerization, as given in: Peters, A. C. I. A.; Overbeek, G. C.; Buckmann, A. J. P.; Padget, J. C.; Annable, T. Prog. Org. Coat. 1996, 29, 183.
3204
Langmuir, Vol. 17, No. 11, 2001
Salamanca et al.
Table 1. Key Experimental Parameters Used to Obtain a Range of Reduced Capillary Pressure Values particle radius, a (nm)
initial film thickness, H (µm)
evaporation rate, E (nm/s)
reduced capillary pressure, pc
25 25 25 72 72 72 143 255 1,015 4,380
1200 700 350 620 310 220 150 80 60 320
44.8 27.8 2.39 4.16 2.94 4.78 14.6 3.30 2.55 2.19
1.0 2.2 10.4 14.9 24 27 32 100 890 417
the temperature of the experiment (averaging 21 °C), ensuring that there was no particle deformation upon drying. The monosized acrylic latex particle radii were 25, 72, and 143 nm with a polydispersity index of 1.15. The poly(styrene) particle radii were 0.255, 1.01, and 4.38 µm, with a polydispersity index of ∼1.05. The initial zero-shear-rate viscosity values were calculated on the basis of knowledge of the solids fraction and particle size, using standard expressions.30 The surface tensions, γ, of the dispersions were measured via the plate method31 using a digital tensiometer (Kru¨ss K10ST). Table 1 lists the key parameters (a, H, and E) that were systematically varied and reports the corresponding pc values calculated via eq 1 for each of the 10 film samples. Immediately after casting, films of the latex dispersion were placed within an NMR sample probe and then inserted into a 9.4 T superconducting magnet, yielding a proton (1H) resonant frequency of 400 MHz. The probe was specially designed and built to accommodate planar films by winding a solenoidal coil to have a rectangular cross section. Evaporation rates in some experiments were purposely slowed by wrapping the probe head in polythene film. 1H NMR images of a vertical slice of each colloidal layer were recorded on a Chemagnetics Infinity spectrometer. Images were acquired using a standard spin-echo imaging sequence32 with an echo time of 14.5 ms and a pulse sequence repetition time of 1000 ms. The NMR signal was sampled 512 times during the acquisition period, and the phase-encoded gradients were stepped in 64 repeats of the protocol. A sine-bell apodization was applied in the read direction, corresponding both to the static magnetic field direction and to the vertical direction in the film. Zero filling was applied in the phase direction (parallel to the surface of the substrate). The resulting magnitude-Fourier transformed image field-of-view was 22 mm × 22 mm, and the slice thickness was 2 mm. The total time required to obtain an image was 8.5 min.
Experimental Results and Discussion In all of the latices, the most mobile protons are found only in the aqueous phase, and hence there is no appreciable NMR signal from the polymer when an echo time of 14.5 ms is used. The signal intensity obtained from a drying latex film thereby quantitatively determines the water concentration as a function of position. Figure 1 shows the NMR signal intensity obtained from a cross section of a latex film (a ) 143 nm and H ) 890 µm) obtained after 3 h of drying. The central region has a much higher intensity in comparison to those of the regions nearer the film edges. There is a sharp step in intensity separating the regions. The intensity beyond the sample edges is negligible in comparison. Using the MR imaging protocol described, the signals and hence the image intensityscan be assumed to be proportional to the density of mobile 1H. The NMR signal (30) Krieger, I. M. Adv. Colloid Interface Sci. 1972, 3, 111. (31) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1993. (32) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Clarendon Press: Oxford, 1991; pp 229-233.
Figure 1. Stacked plots of NMR signal intensity obtained from a drying colloidal film. (Only one of every 20 spectra is shown, for clarity.) The intensity is substantially lower near the edges, and there is a sharp step in intensity moving laterally toward the center. The outer edge defines the drying front; the step in intensity defines a particle-packing front.
Figure 2. Three successive MR images obtained at 0, 3, and 6 h (top to bottom) from the same film (initially 890 µm thick) containing particles with an average radius of 143 nm. The image obtained at 3 h corresponds to the stacked plots shown in Figure 1. Each image is a cross-sectional slice that is 2.6 mm high and 22 mm wide.
intensity can be easily presented on a gray scale, in which white represents high intensity and black indicates low intensity. Figure 2 shows such images for the same sample as in Figure 1, at three different times. Initially, the water is uniformly distributed in the dispersion. Analysis of the first image, and careful calibration described elsewhere,14 finds that the sample contains 45 vol % water, which is slightly lower than the initial concentration of 51 vol %. After 3 h, well-defined regions with different water concentrations develop. The image reveals that, after 3 h of drying, the water has not yet receded from the sample edges. Instead, near the outer edges, the water fractionsaccording to the analysissis 20 vol %. This value is probably an underestimate of the true value, however, as a result of magnetic susceptibility and confinement effects,32 as follows. The self-diffusion of water in magnetic field gradients is known to attenuate the signal. In a strong magnetic field, magnetic field gradients occur at curved interfaces between materials of differing magnetic susceptibility. As the solids fractions of a latex dispersion increase, the fraction of water in such interfaces likewise increases. Signal attenuation therefore increases as well. The effect is severe in some porous media but less severe here where the magnetic
Lateral Drying in Thick Films
Langmuir, Vol. 17, No. 11, 2001 3205
Figure 3. Diagram showing the suggested cross section of a drying colloidal film with a central wet region and dry edges (after Winnik and Feng12). Between these regions, water fills the void space between packed particles. The inner boundary is the particle packing front, and the outer boundary is the drying front.
susceptibilties of polymer and water are nearly matched. Moreover, as the solid fraction increases, interfacial regions constitute a greater fraction of the total sample. Surface-mediated nuclear spin-spin relaxation will therefore increase with the solids fraction, although this is likely to be a lesser effect in these experiments. The densest possible packing of monosize spheres is in a face-centered cubic array, in which the water fraction is 26% if all interparticle volume is filled. In light of the above considerations, we interpret the edge regions as consisting of close-packed particles with water filling their interstitial void space. After 6 h, a high water fraction in the central liquid region is no longer apparent in the image. Instead there is only a low concentration of water in the central region, and now there is no evidence for any water at the edges. The water has moved inward from the film’s edge. In the central region, the image is interpreted as showing that water remains in the void space between packed particles. The boundary between the wet and dry regions is somewhat diffuse. The set of three images in Figure 2 is consistent with drying from the sample edges. Two types of boundaries are apparent: one separates a wet dispersion from a flooded close-packed array of particles, and the other separates the flooded array from fully dry, packed particles. The first type of boundary is called a particlepacking front, and the second is a drying front. The description of the drying process, as invoked in the R-R model following the inspiration of Winnik and Feng,12 is shown schematically in Figure 3. Three distinct regions are not always observed, and their occurrence depends on the specific characteristics of the latex and the film. Figure 4 compares the drying of dispersions with relatively high and low pc values. With a low pc value (pc ) 1.0), water is found to recede from the sample edges from the initial onset of the drying process. The radial extent of the flooded close-packed region is minimal. That is, the particle-packing and drying fronts overlay each other. In contrast, with a higher pc value (pc ) 420), water remains pinned at the sample edge throughout about half of the drying time. Particle-packing and drying fronts are distinct and move separately. These two differing behaviors are exemplary of the two extremes that were observed in the experiments. From the images, the initial film thickness was readily determined. Through image analysis, described elsewhere,14 E was calculated, taking into account the surface area of the wet region. The radial distance of the wet region, as visualized in the MR images, was monitored over time
Figure 4. Series of images obtained at the times indicated from colloidal layers with (A) a low pc (1.0) in which H ) 1.2 mm and a ) 25 nm (cross-sectional area is 2.4 mm by 22 mm) and (B) a high pc (420) in which H ) 0.32 mm and a ) 4.4 µm (cross-sectional area is 1.2 by 22 mm).
Figure 5. Open time as a function of reduced capillary pressure showing experimental data (b) and simulations using R h ) 1.2 (s) and R h ) 5 (- - -).
in order to determine the point when the drying front receded from the sample edge, that is, the open time. This open time, normalized by the drying time (defined here as H/E) according to the requirements of the R-R model, is plotted as a function of experimental pc values for all samples in Figure 5. The values of pc were calculated with the experimentally measured values of γ, η0, E, a, and H and with µ ) 1.3 mPa s (which is slightly higher than the
3206
Langmuir, Vol. 17, No. 11, 2001
Salamanca et al.
viscosity of water33 in order to account for the effects of water-soluble species on viscosity). Analysis of the particle packing in dry films using scanning electron microscopy revealed extensive regions in which there was facecentered cubic (FCC) packing, in which φm ) 0.72. Between the FCC regions, particles were packed randomly. The maximum solids fraction of the films, φm, estimated from these observations, is on average 0.69. Hence, this value was used in the calculation of pc. The open times shown in Figure 5 tend to be either relatively short (i.e. 1/4 of the drying time), depending on whether pc lies below or above a value of about 15. Moreover, as a general trend, with low values of pc, the drying fronts and the particle-packing fronts are seen to overlay and move inward simultaneously. With higher pc values, on the other hand, a particle-packing front moves inward well ahead of the drying front. These experimental results will next be compared to the predictions of the R-R model that has been extended to incorporate the experimental geometry. Extension of Theory and Computer Simulation The original R-R model considered the drying across a vertical plane only and derived equations for the twodimensional case.24 To describe more realistically a threedimensional thick film, such as studied experimentally, the R-R model has been extended in this work by using cylindrical coordinates and assuming radial symmetry. Thus, the colloidal films are modeled as having a circular edge, which is an approximation of their geometry when cast on the substrates. The curvature of the surface and shape at the edge are considered here as well and will be described later in this section. The R-R model is based on the lubrication approximation for a liquid film, which applies when H is much smaller than the horizontal length scale, L, such that H2/L2 , 1. This length scale is the capillary length, which is written as H(γ/3η0E)1/4, where all of the terms have been defined previously. This term balances the flow due to surface tension with that due to evaporation. (An analogy can be made to the better-known gravitational capillary length,34 which considers surface tension and gravity.) For lateral distances less than a capillary length from an edge, surface tension dominates the flow; for larger distances evaporation is dominant. The 1/4 power in the definition of capillary length arises from the controlling equation being a fourth-order differential equation. L is used to normalize lateral distances in the model, as will be discussed later. From integration of the continuity equation for the vertical velocity (following the arguments of Routh and Russel24), the height of the film when normalized by H (and designated as h h ) is determined by
1+
h u˜ r) ∂h h 1 ∂(rjh )0 + ∂th rj ∂rj
(2)
where ht is time normalized by the drying time (H/E) and rj is radial distance normalized by L; u˜ r is the vertically averaged radial velocity of water.35 Integration of the normalized equation for particle conservation over the film thickness yields (33) Tipler, P. A. Physics, 2nd ed.; Worth Publishers: New York, 1983. (34) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982. (35) In turn, u˜ r ) h h 2 ∂/∂r[1/rj ∂/∂rj (rj ∂h h /∂rj)].
h u˜ r ∂h h φ 1 ∂rjφh + )0 ∂th rj ∂rj
(3)
using φ to represent the solids fraction. Finally, the pressure distribution in a close-packed bed of particles (with a solids content of φm) near the edge of the film can be shown to be
p j (rj) )
1 2(1 - φm)
[∫ rjhh′ drj′ - Rh ∫ drrj′hjh′] rjf
rj
2
rjf
rj
(4)
jrf is the distance to the edge of the close-packed region when measured from the sample center. R h is used here to designate the initial radius of the wet film normalized by the capillary length, L. As such, the value of R h depends on the lateral extent of the film in the experiment. When eq 4 is integrated in parallel with eqs 2 and 3, the pressure as a function of radial position and time is obtained. This integration was achieved via a computer program using a finite differences algorithm. The program thereby obtained the value of the pressure at the edge of the film over time during drying. At the point when this pressure exceeds the capillary pressure, drying from the edge is predicted to occur. The details of the program are given elsewhere.36 The program used a minimum of 100 spatial steps in the lateral direction. Appropriate time steps were chosen to achieve stability in the integration algorithm that solved fourth-order partial derivatives. Simulations were performed for several radial length scales and for two geometries: (1) a spherical cap (in which the thickness varies from zero at the edge to its maximum value H at the sample center) and (2) a flat slab of thickness H with rounded edges. For the series of samples studied with MR microscopy, the average value of the radius is about 9 mm. When this average radius is normalized by the average capillary length, a value of 1.2 for R h is obtained. The lateral widths of the films are up to 180 times greater than their thicknesses. Cross-sectional images, such as shown in Figure 4, nevertheless show that the surface has some curvature, and the film should thus be mathematically modeled as a shallow spherical cap. To be consistent with the MR microscopy experiments, simulations were therefore performed using R h ) 1.2, a spherical cap geometry, and an average initial solids fraction of 0.40. The solid line in Figure 5 shows the result of this simulation. Indicated with the dashed line on the same plot is the result of a simulation using R h ) 5, which represents a film that extends over four times farther in the lateral direction compared to those in the experiments. With the lower value of R h , a strong increase in the simulated open time is seen around a pc of 2, compared to the value of about 15 where the open time is seen to increase most strongly in experiments. The simulations clearly predict the same general behavior that is found experimentally, although, using R h ) 1.2, the strong increase in open time is predicted to occur at a lower value of pc. The experimental pc values are highly sensitive to φm, and this quantity is not accurately known in these experiments, which leads to uncertainty in the data in Figure 5. Furthermore, the simulations invoke cylindrical symmetry to describe the films. In reality, the contour of the film edges deviated somewhat from a circle, so the simulations are an idealization. Additionally, the simulations are very sensitive to the shape of the “foot” at the edge of the layer (36) Salamanca, J. M. M.Phil. Thesis; University of Surrey, Guildford, U.K., 2000.
Lateral Drying in Thick Films
near the contact line with the substrate, but experimentally this shape is not well-defined. Hence, the combination of experimental uncertainty and simplification in the modeling probably leads to the disagreement between experiment and simulation. To summarize, a sharp increase in open time through a critical region of pc was observed in all simulations, regardless of the lateral length scale or the edge shape. The precise region of pc where open time increases most, however, is highly sensitive to these parameters. It is concluded that the marked increase in open time around a certain pc value, as observed here, is a universal phenomenon that will occur in similar colloidal films. The region of pc marking the transition in open time, on the other hand, is a function of the lateral extent of the film and the shape of its edge. Concluding Remarks This theoretical and experimental work points to a means of controlling the drying behavior in colloidal films. By judicious selection of particle size, film thickness, surface tension, and evaporation rate, among other parameters, the reduced capillary pressure can be tailored. The movement of lateral drying fronts is encouraged by
Langmuir, Vol. 17, No. 11, 2001 3207
minimizing pc, whereas uniform drying can be achieved by maximizing pc. Specifically, a smaller particle size, a faster evaporation rate, and a greater film thickness promote the recession of water from the edges of a waterborne colloidal film during drying. A larger particle size, a slower evaporation rate, and a lower thickness, on the other hand, lead to drying that is more uniform in the lateral direction. The results support the notion that thickeners or other viscosity modifiers will influence the lateral drying of latex films in a predictable and controllable way. This first, systematic study of the experimental parameters that influence lateral drying can be exploited in processes that rely on the deposition of particles via the drying of colloidal dispersions. This work lays a framework for understanding lateral drying as an antidote to the existing speculation and uncertainty in the colloids literature. Acknowledgment. We gratefully acknowledge funding for J.M.S. provided by NeoResins, The Netherlands, and from the U.K. Engineering and Physical Sciences Research Council. LA001590H
