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Effects of Phase Behavior on the Drying of Colloidal Suspensions M. D. Haw,* M. Gillie, and W. C. K. Poon Department of Physics and Astronomy, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, U.K. Received July 16, 2001. In Final Form: November 7, 2001 We study the effects of phase behavior on the evaporative drying of droplets of a suspension of hardsphere colloidal particles and nonadsorbing polymer. The presence of the polymer induces a depletion attraction between the colloidal particles. As drying (evaporation of solvent) progresses, the concentrations of both colloid and polymer increase, so that the droplet’s average composition traces out a “drying line” across the composition diagram. We find that drying behavior can be broadly classified according to the initial composition of the suspension, into three regions on a “drying behavior diagram”. We relate these three regions to the bulk phase diagram of the system and show how drying behavior and final residue properties such as homogeneity can be understood by considering the orientation of the “drying line” with respect to the equilibrium and nonequilibrium boundaries in the bulk phase diagram. Our findings have relevance for predicting the sensitivity of droplet and film residue properties to initial suspension composition, in the case of technologically important systems such as detergents, coatings, ceramics, and paints, as well as being an interesting example of the response of a model soft matter system continuously driven across its phase diagram toward a shifting “target” equilibrium.
Introduction The evaporative drying behavior of multicomponent solutions, such as detergents, polishes, paints, and ceramics, is important in cleaning and coating technology, especially from the point of view of the final dry droplet or film structure (its mechanical properties, mesoscopic and macroscopic texture, etc.).1 Control of structure and homogeneity in evaporative deposition is also important in the preparation of polymer films, surface-adhered proteins, etc. In studies of coating formation from, e.g., a solution of polymeric particles, the process may divided into two stages: first the drying stage, involving evaporation of the bulk of the solvent; second the film formation stage, where solutes such as polymeric particles first deform under the influence of capillary forces due to the solvent filling the remaining pores between the particles and then coalesce by molecular interdiffusion. There is a substantial literature devoted to the film formation stage,1,2 but in this paper we shall be concerned primarily with the former, drying stage (the most recent studies of which are reviewed in Chapter 1 of ref 1). Routh and Russel3 have recently considered the drying of suspension droplets from a theoretical point of view. In their models, however, the suspension does not exhibit changing phase behavior during drying. It is a particular aim of this paper to consider, for a model experimental system, the effect of phase changes, generated by evaporation of solvent, on the drying behavior. The archetypal system thus consists of one or more solutes (e.g. suspended colloidal particles, polymers, surfactants) in a solvent (e.g. water, oil). Throughout the drying stage as the solvent evaporates, the concentrations of the solutes increase, so that the average composition of * Corresponding author. E-mail:
[email protected]. (1) Provder, T., Urban, M. W., Eds. Film Formation in Coatings: Mechanisms, Properties and Morphology; ACS Symposium Series 790; American Chemical Society: Washington, DC, 2001. (2) Winnik, M. A. Curr. Opin. Colloid Interface Sci. 1997, 2, 192. (3) Routh, A. F.; Russel, W. B. AIChE J. 1998, 44, 2088.
the drying suspension follows a “drying line” across the composition diagram: the suspension essentially traverses the phase diagram. An important question then is how the presence of boundaries in the bulk phase diagram, which are crossed by the drying line, affects the drying behavior of the system. Besides phase behavior, a second important factor in the drying stage is the rheology of the suspension. The importance of flow during drying can be appreciated from the “everyday experiment” of the coffee spill, as studied by Deegan et al.4 Evaporation of water from the coffee droplet results in a circular stain: Deegan et al. show how, during drying, induced flow drives the suspended colloidal coffee particles to the contact line at the outside of the droplet (where the suspension, the surrounding air, and the substrate are in contact). The induced flow is a consequence of the presence of particles in the suspension. The key difference between the evaporation of a sessile droplet (i.e. a droplet sitting on a substrate) of pure solvent and of a droplet of a suspension is that the particles in the suspension strongly encourage pinning of the contact line of the droplet.3,4 Without pinning, on a perfectly flat substrate, a droplet of pure solvent retains a spherical cap shape (neglecting gravitational effects) with constant contact angle. As the solvent evaporates the contact line recedes. However with strong pinning due to the presence of solute particles (or indeed to roughness in the substrate) the contact line of the evaporating droplet cannot recede, so the droplet must change shape as it decreases in volume. The shape change necessitates flow. The drying system’s response will hence depend on its rheology, which in general will be timedependent due to the time-dependent composition and indeed may also be affected if the droplet crosses boundaries in the phase diagram, into regions occupied by phases of significantly different rheology. A number of recent examples of studies of the drying of sessile drops of simple suspensions can be found in the (4) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Nature 1997, 389, 827.
10.1021/la0110951 CCC: $22.00 © 2002 American Chemical Society Published on Web 01/26/2002
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literature. The experiments of Deegan et al.4,5 involve an initially very dilute colloidal suspension, in which interactions between the colloidal particles can be neglected, representing the simplest case of a drying suspension. The theoretical studies of Routh and Russel3 have already been mentioned: their model predicts the build up of colloidal particles at the edge of the drying droplet, as seen in the experiments of Deegan et al. Meanwhile Allain et al.6-8 have studied more concentrated suspensions with a more interesting phase behavior, where a salt suspension of model charge-stabilized colloids undergoes a “gelation” transition during drying. Evaporation of solvent leads to increasing salt concentration and thus decreasing screening length of the Coulomb forces, until the screening length is small enough that the particles approach each other within range of the strong van der Waals attraction and irreversibly aggregate into a solid phase. In the final stages of drying, a competition between stress release by cracking and stress increase due to enhanced evaporation at cracks is shown to generate macroscopically regular cracking of the suspension residue.6 Interestingly, observations of the drying of ceramic films9 coincide rather well with these studies on model colloidal suspensions. In this study our purpose is to relate the drying behavior of a well-understood model system to its bulk phase behavior and rheological characteristics. To this end we consider the drying of suspensions consisting of mixtures of near-hard-sphere colloids and nonadsorbing polymer, in an organic solvent. The presence of the polymer induces a depletion attraction between the colloids, where the attraction strength (typically a few times the thermal energy kT) and range depend on the polymer concentration, cp, and the ratio of colloid radius to polymer radius of gyration, ξ ) rg/rc, respectively. Though still a reasonably “model” system whose bulk behavior is well-known from experiment,10,11 the phase behavior and rheological properties displayed by the colloid-polymer mixture system are nevertheless substantially more complex than the dilute colloids of Deegan et al.4,5 and the charged colloidal suspensions of Allain et al.6-8 The bulk equilibrium phase diagram of the colloid plus nonadsorbing polymer system shows homogeneous colloidal fluid, colloidal gas-colloidal crystal, and colloidal gas-colloidal liquid-colloidal crystal phase coexistence, as a function of the colloid volume fraction, Φ, polymer concentration, cp, and size ratio, ξ.10,11 It is found experimentally that, for ξ < 0.25 (the case of relevance to the experiments reported here), there is no critical point (i.e. no three-phase coexistence) and the equilibrium phase diagram contains only single-phase (fluid or crystal) and fluid-crystal coexistence regions (Figures 1 and 2). In addition to equilibrium phase boundaries, the presence of “kinetic” boundaries12,13 in the colloid-polymer phase diagram is likely to be important in the drying problem. There is good evidence that, within the equilibrium coexisting fluid-crystal region, there is a range (5) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Phys. Rev. E 2000, 62, 756. (6) Allain, C.; Limat, L. Phys. Rev. Lett. 1995, 74, 2981. (7) Pauchard, L.; Parisse, F.; Allain, C. Phys. Rev. E 1999, 59, 3737. (8) Parisse, F.; Allain, C. J. Phys. II 1996, 6, 1111. (9) Guo, J. J.; Lewis, J. A. J. Am. Ceram. Soc. 1999, 82, 2345 and references therein. (10) Poon, W. C. K.; Selfe, J. S.; Robertson, M. B.; Ilett, S. M.; Pirie, A. D.; Pusey, P. N. J. Phys. II 1993, 3, 1075. (11) Ilett, S. M.; Orrock, A.; Poon, W. C. K.; Pusey, P. N. Phys. Rev. E 1995, 51, 1344. (12) Poon, W. C. K.; Pirie, A. D.; Haw, M. D.; Pusey, P. N. Physica A 1997, 235, 110. (13) Poon, W. C. K.; Haw, M. D. Adv. Colloid Interface Sci. 1997, 73, 71.
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Figure 1. Schematic of the colloid-polymer bulk phase diagram at size ratios ξ < 0.25, showing the division of the equilibrium two-phase region by “kinetic” phase lines denoting regions of differing kinetic path toward equilibrium. The solid line separates the single-phase fluid and equilibrium fluidcrystal coexistence regions; above the upper, dotted line, transient nonequilibrium gels are formed. As discussed in the text, the area between the gel and equilibrium fluid-crystal lines is divided into two kinetic regions: “N” samples phaseseparate by nucleation of crystals direct from the fluid (time scale of hours); “S” samples rapidly spinodally decompose (time scale of seconds to minutes) into metastable colloidal gascolloidal liquid, and the equilibrium crystal then nucleates within the liquid drops.
Figure 2. Drying behavior diagram, showing how the experimental samples’ drying behavior can be broadly divided into three regions on the initial composition diagram (colloid volume fraction vs polymer weight concentration). The positions of the symbols show the initial composition of the various samples studied; the letters “G”, “F”, and “A” indicate the typical drying behavior observed for the samples. The solid lines show approximate boundaries between the regions. Typical behavior in each region is described in detail in the text. The dashed and dotted lines show the nonequilibrium gelation line and equilibrium fluid-crystal phase boundary, respectively, as observed for bulk samples.
of kinetic paths which samples may follow toward twophase equilibrium. Just inside the equilibrium line (symbol N in Figure 1) colloidal crystals nucleate directly from the “supercooled” fluid (i.e. via slow metastable kinetics). Being slow (crystals typically take hours to appear) compared to the rates of evaporation considered here (see below), this pathway is unlikely to have an important role in the drying problem. However, deeper inside the equilibrium two-phase region (position S in Figure 1) bulk samples demonstrate an initial spinodal phase separation (with fast unstable kinetics) into metastable dilute colloidal gas and dense colloidal liquid (the crystal phase then nucleating within the dense liquid). This is likely to be more important in drying because it is fast. Finally, deep inside the equilibrium two-phase region, the formation of a transient nonequilibrium particle gel14,15 (14) Poon, W. C. K.; Pirie, A. D.; Pusey, P. N. Faraday Discuss. 1995, 101, 65. (15) Starrs, L. Collapse of transient gels in colloid-polymer mixtures. Ph.D. Thesis, University of Edinburgh, 2000.
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is observed in bulk samples. The gel has weak solidlike properties and a complex response to thermal aging, gravity, and imposed stress.15 In bulk, thermal and gravitational aging lead to the eventual rapid collapse of the gel, after a composition- and geometry-dependent “latency time” on the order of minutes to hours, during which time the gel is able to support itself against gravity. It is a general feature of “soft matter” systems that the time scales of phase changes are much slower than in atomic or molecular systems, being often on the order of seconds, minutes, or longer. Given the typical evaporation rates involved in drying (a droplet in these experiments takes on the order of 1 or 2 h to dry, a scale similar to the rates involved in applications, e.g. drying detergents, paints etc), the competition between “movement” across the composition diagram and “slow” phase changes becomes an important factor in the problem. In particular, in our experimental system the lifetimes of metastable states (minutes to hours) are comparable to the time scale of evaporation. Thus, we expect a complex interdependence of approach to thermodynamic equilibrium and simultaneous shifting of the “target” equilibrium due to evaporation. Furthermore the rheological characteristics of the system differ widely across the composition diagram, so that given the role of flow stresses generated by pinning of the contact line, the rheological response of the different nonequilibrium structures is also expected to be of prime importance. Experimental Section Model System. The model suspension consists of nearmonodisperse, hard-sphere colloidal particles16 (poly(methyl methacrylate), PMMA, sterically stabilized by short grafted poly(hydroxystearic acid) chains) of radius rc ) 315 nm and size polydispersity approximately 6%. The colloids are suspended in an organic solvent (decahydronaphthalene, mixture of cis and trans isomers), to which is added near-monodisperse linear polystyrene (Mw ) 370 000, Mw/Mn ) 1.03, Polymer Laboratories). At room temperature in the given solvent, the polymer is sufficiently close to its θ temperature to be considered as ideal, with radius of gyration rg ≈ 19 nm. Thus, the colloid to polymer size ratio in the experiments described here is ξ ) 0.08. In bulk without polymer the PMMA colloids show a hardsphere phase behavior16 with colloidal fluid at volume fraction Φ < 0.494, colloidal crystal at Φ > 0.545, and fluid-crystal coexistence in between. Addition of free polymer with size ratio ξ ) rg/rc ) 0.08 to bulk samples effectively leads to a widening of the equilibrium fluid-crystal coexistence region, while, as described, at high enough polymer concentration equilibrium phase separation is arrested by the formation of a macroscopic, space-filling transient particle gel with solidlike and timedependent rheological characteristics. Note that in the case of our system the particles do not coalesce on complete loss of the solvent: in fact the dried particles may be redispersed and show no change in size or deviation from hard-sphere behavior. The glass transition temperature of the PMMA is much higher than room temperature, so that coalescence of particles and film formation do not occur. Experimental Procedure and Observations. Bulk suspensions are made up at known particle volume fraction Φ and polymer mass concentration cp. Prior to experiments, the bulk sample is vigorously mixed, followed by slow tumbling for at least 10 min, to ensure a fully shear-mixed homogeneous starting condition. All drying experiments are carried out at room temperature (T ) 22 ( 2 °C) within a shield which isolates from surrounding air currents. To study drying behavior, a single droplet of the sample is deposited by pipet on a glass microscope slide. The slides are prepared by immersion for some hours in a detergent solution (Decon-90), followed by thorough rinsing in (16) Pusey, P. N. In Liquids, freezing and the glass transition; Hansen, J. P., Levesque, D., Zinn-Justin, J., Eds.; Elsevier: Amsterdam, 1991; Chapter 10.
Haw et al. distilled water and drying in a warm-air incubator for at least 24 h. Repeat experiments using different cleaning methods (e.g. wiping with alcohol, using different detergents, or not cleaning at all) show no qualitative changes to the observed phenomena, indicating that any pinning effects due to remaining surface impurities on the slides are dominated by the pinning induced by the colloidal particles. During drying, droplets are observed by microscope or, for more “macroscopic” observation, using a video-zoom lens. Typical droplet diameter is 10 mm, and typical droplet drying time is on the order of 1-2 h. Deposited droplets often take up noncircular shapes, because of the strength of pinning and the lack of precise control of the deposition process (the effect of pinning on the spreading of suspension droplets is in itself an interesting problem, though beyond the scope of the present paper). To test for any influence of a noncircular pinned contact line on drying behavior, a number of experiments were performed with droplets deposited onto 10 mm diameter circular glass coverslides: surface tension forces prevent the droplet from spilling over the edge of the coverslip, thus providing a near-perfectly circular contact line. In the event, we observed no effects of contact line shape on those aspects of drying behavior considered here. During drying, microscopic observations are made using phasecontrast17 and bright-field optical microscopy with, typically, 5×, 20×, or 40× objectives. Macroscopic observations, including “lowangle scattered intensity imaging” (LASI; see below), are made using a video-zoom lens and black and white video camera, images being taped or digitized and stored on computer. Because of the characteristic mesoscopic scale of colloidal aggregates and the colloidal gel13,14 (typically tens of micrometers), aggregated and gelled colloidal structure in droplets generates strong scattering at small angles (on the order of 5-10°). Hence, imaging with low-angle scattered light allows us to distinguish regions of the drying droplet containing aggregates or colloidal gel from regions of colloidal fluid or colloidal glass. Observing the “motion” or fluctuation of the low-angle scattered speckle also allows a macroscopic picture of the flow behavior of the colloidal structure in the droplet.
Results Classification of Drying Behaviors“Drying Diagram”. From microscopic and macroscopic observations we can construct a “drying behavior diagram”, based on the typical behavior of the drying droplets (Figure 2). Despite some complexity in the details of the drying behavior, to be discussed further below, we find that the drying diagram may be broadly divided into three regions, as shown by approximate boundaries separating the observed samples in Figure 2. Symbols on the diagram mark the initial composition of droplets and are labeled “G” (for gel), “F” (for fluid), or “A” (for aggregation), to indicate this broad classification. During drying, all droplets show the formation of an outer “foot” and an inner “cap” (see Figure 3) as also described by Allain et. al.6-8 With time the radius of the cap decreases as the foot advances (see below for measurements). The colloidal structure and flow behavior inside the foot and cap differ, however, according to the initial composition of the droplet. In what follows, we describe in turn the typical behavior observed for droplets with initial compositions in each of the regions of the drying diagram. Gelation Region. In this region of the drying behavior diagram (samples marked G in Figure 2) LASI imaging demonstrates that the colloidal particles form a solidlike particle gel immediately the droplet is deposited on the substrate. The scattered light speckle pattern is “frozen” on short time scales (tens of seconds), demonstrating that the particle gel structure resists flow and that diffusion of colloidal particles is also arrested. Nevertheless during (17) Pluta, M. Advanced Light Microscopy; Elsevier: Amsterdam, 1989.
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Figure 3. (a) Macroscopic appearance of a droplet in the “G” region of Figure 2, some time into drying, using low-angle scattered intensity (LASI) imaging. The droplet diameter is 10 mm. The region marked “C” is the cap, and “F” is the dense foot. The coarse intensity variation (speckle) in the cap indicates the presence of a structure with a mesoscopic characteristic length scale, i.e., the particle gel. The speckle is stationary on short time scales (tens of seconds to minutes), showing that the solidlike gel resists the flow stresses generated by evaporation/ pinning. With LASI the foot appears almost clear: the lack of low-angle scattering indicates absence of large-scale structure in the foot. Microscopy reveals only a dense, structureless colloidal “glass”. (b) Schematic of the typical droplet shape as seen from the side, showing the “foot” and “cap” regions. (c) Microscope image of the edge of the cap (C), showing irregular gel structure. The black bar at the bottom left is 60 µm.
drying the droplet shows the formation of outer foot and central cap, as shown in Figure 3a. LASI shows that the gel is confined to the cap: the gel structure slowly breaks up at the edges of the cap (Figure 3c), under the flow stress induced by the evaporation of solvent from the pinned droplet. No structure is observable in the foot, indicating that clusters are broken up as they are incorporated into the foot. (In Figure 3a the foot appears “transparent”, i.e., scatters very little light at low angles, indicating the lack of mesoscopic structure, in contrast to the gel structure in the cap.) Deposition of particles at the foot-cap border then leads over long time scales to the advance of the foot and retreat of the cap. All samples in the G region of the drying behavior diagram behave in qualitatively the same way: their behavior is dominated by the solidlike properties of the particle gel, which is slowly eroded by the evaporation/pinning-induced flow stress. The final residue structure is largely homogeneous for samples with initial composition in the G region of the drying diagram (see below). Fluid Region. Samples marked F in Figure 2 show drying behavior qualitatively different from that of G samples. On initial deposition, F samples demonstrate significant flowsno solidlike gel is formed. Depending on polymer concentration, the samples may contain free colloidal particles or aggregates, but initially the aggregates do not grow sufficiently to form a gel structure.18 As in the G region, during evaporation a foot forms around the outer edge of the droplet. Figure 4a shows a microscope image of the foot-cap border for an example F sample. The coarse structure in the cap indicates the presence of aggregates of particles (individual particles cannot be resolved at this magnification), while as can be seen the foot demonstrates no resolvable structure. The edge of the cap is circular and less ramified compared to the G samples, demonstrating that the cap is effectively a “sol” of colloidal aggregates. Aggregates in the cap are seen to (18) Haw, M. D.; Sievwright, M.; Poon, W. C. K.; Pusey P. N. Adv. Colloid Interface Sci. 1995, 62, 1. Segre, P. N.; Prasad, V.; Schofield, A. B.; Weitz, D. A. Phys. Rev. Lett. 2001, 86, 6042. Gimel, J. C.; Nicolai, T.; Durand, D. Eur. Phys. J. E 2001, 5, 415.
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Figure 4. (a) Microscope image of an “F” sample, showing the circular border between foot (“F”) and cap (“C”). The coarse texture observed in the cap indicates the presence of a “sol” of aggregates (at this magnification single colloids are not resolvable). The black bar at bottom left is 80 µm. Video sequences reveal a circulating flow of aggregates in the plane perpendicular to the image. (b) Schematic side view showing the circulating flow: higher up in the droplet, colloids move outward toward the edge of the cap; lower down, flow is inward from the edge.
continuously circulate in the vertical plane as shown by the arrows in the diagram and the schematic in Figure 4b, moving outward higher up in the droplet and back inward lower down in the drop. As in the G region, the foot advances by a slow deposition of colloidal particles at the foot-cap border. Unlike in the G region, however, there is some difference in the details of later stages of the drying behavior of F samples, according to the sample’s initial composition. For droplets at initially higher Φ, in the later stages of drying a gel may form in the cap as the cap composition crosses the boundary into the G region. At this point the cap behaves as in a G sample: the gel resists flow and shows “frozen” low-angle scattering speckle as observed by LASI. The flow stress slowly breaks up the gelled cap in the same way as observed for G samples. Note that, despite the change from F to G behavior, no macroscopic inhomogeneity is observed in the final dry residue (see below): the foot structure appears to be the same, whether the foot is “grown” by deposition of colloid from a sol of aggregates or from an “eroded” gel. F samples with low initial colloid concentration do not show any transition to gel behavior, presumably because the concentration in the cap never becomes high enough to cross the G boundary. Recall that the foot is a dense colloidal solid, probably with Φ ≈ 0.6 or greater, so that the colloidal concentration in the cap does not necessarily simply increase over time as solvent evaporates: increase in particle concentration due to solvent evaporation can be offset by decrease in particle content due to deposition of colloidal particles into the dense foot. This does mean that we can expect macroscopic inhomogeneity in the final residue (dense foot around the outside of the drop, perhaps only a thin layer of colloid in the centersthe limiting case being that of very dilute, noninteracting colloid, as in the coffee spill4,5). The final residue structure is considered in more detail below. Aggregation Region. The A region in Figure 2 is found to be the region of most complex drying behavior and greatest sensitivity of residue properties to initial composition. A typical droplet behaves initially like a fluid, F, sample: a narrow foot is formed, with a circulating flow of colloidal particles or aggregates at the border of foot and cap. However, in the later stages behavior becomes more complex. A narrow band between cap and foot becomes a gel, as observed in the microscope by the cessation of flow in this band (and in LASI by the increase in low-angle scattering and freezing of the speckle). The
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Figure 5. (a) Microscope image of an “A” sample, showing the formation of a band of gel (marked G) at the border of cap (C) and foot (F). The arrow indicates the circulating flow in the plane perpendicular to the image (i.e. vertical within the droplet) similar to that described in Figure 4. The horizontal black bar at bottom right is 320 µm; the scale is the same in (b) and (c). (b) Development of a “convection cell”-like flow pattern in the cap of an A sample [same experiment as (a), a few minutes later]. As the arrows indicate, colloidal clusters circulate (now in the horizontal plane) out toward the edge of the cap, to be sucked back in along a narrow band (lighter contrast in the image). (c) Further development of the “convection cell”-like flow pattern [some 20 min after (b)]: the circulating flow erodes the inside edge of the gel band to generate a quasi-regular curvature.
formation of a gel band indicates the increase in particle concentration near the border of cap and foot, presumably a consequence of the radial flow driving particles outward and/or incipient phase separation (see Discussion below). An example image is shown in Figure 5a, where F marks the foot, C the cap region of circulating colloidal clusters, and G the band of gel where there is no flow. The circulating flow of the colloidal aggregate sol in the C region within the inner boundary of the gel band continues. Macroscopically the foot ceases to advance (or advances very slowly compared to G and F behaviorssee measurements below): the band of gel “shields” the foot, stopping the deposition of particles from the circulating sol. After the appearance of the gel band between cap and foot, very complex flow conditions can evolve in the droplet. “Convection-like cells” form, where the flowing colloidal clusters in the cap region now circulate in the horizontal plane as shown by the arrows in Figure 5b. The flow patterns may indicate some form of Marangoni-like instability caused by spatial variation of surface tension or viscosity. However no very regular array of “convection” cells is observed. Sometimes “cells” of circulating flow may be seen to travel bodily around the circumference of the gel band. More cells may form, generating a quasi-regular curvature of the inside of the gel band (Figure 5c). The cap, consisting still of a sol of clusters, may eventually penetrate and overflow the gel band, the gel band breaking up and disappearing, just before the final very rapid advance of the foot (see measurements below). Patterns of inhomogeneity are observed (Figure 6c) reminiscent of viscous fingering:19 it may be that the general outward flow of the less viscous sol of colloidal clusters, meeting the more viscous gel band, results in a similar flow instability. However, as stated, neither the flow cells nor the fingerlike structures are macroscopically regular across the whole droplet, indicating a sensitive dependence to local conditions (e.g. small temperature variation). Note that as already mentioned there is no apparent effect of contact line shape: circular and noncircular droplets display the same behavior. In the A region the details of droplet behavior vary strongly with the droplet’s exact initial position on the (19) Bensimon, D.; Kadanoff, L. P.; Liang, S.; Shraiman, B. I.; Tang, C. Rev. Mod. Phys. 1986, 58, 977.
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Figure 6. LASI images showing the progression of inhomogeneity in four samples with initial Φ ) 0.1 and increasing polymer concentration from (a) to (d), moving up a vertical line through the “A” region of the drying diagram. Initial compositions are indicated in (e). Initial droplet diameter (the outermost white circle) is 10 mm in all cases.
drying diagram. Figure 6 compares the inhomogeneity patterns observed by LASI for four samples at the same Φ but increasing initial polymer concentration, crossing the A region up a vertical line. Bright regions in LASI images are associated with more mesoscopic structure, i.e., colloidal gel. The bright speckle in these regions tends to fluctuate more slowly than fainter speckle elsewhere. It is worth reiterating that such inhomogeneities are never observed for drops in the F and G regions of the phase diagram. Eventually flow in the central part of the drop ceases, after which the final drying stage consists of a very rapid advance of the foot (and not the gradual breaking up of the gel as seen in the G region). In the final stage the appearance of the advancing foot front suggests the rapid evaporation of a thin layer of solvent above a thin layer of stationary colloidal structure, consistent with the final residue structures as discussed in the next section. Residue Structure. The final stage of drying is the formation of an opaque residue as the last of the solvent is lost. Importantly, for G and F samples the foot does not become opaque until the cap has completely disappeared. This indicates that the foot remains solvated, probably at a near-constant concentration, for a very long time after its formation, “waiting” for the cap to disappear. Thus, excess solvent must be delivered continuously by the cap to counterbalance the solvent lost from the exposed surface of the foot.3,4 It seems therefore that the foot can be considered as a porous particulate solid, through which there is a continuous flow of solvent to replace evaporated solvent at the surface. Only when the cap disappears, and no further replacement solvent is available, is the last solvent evacuated from the pores of the foot. Then, due to the contrast between refractive indices of the colloidal particles (n ≈ 1.49) and air, the dry foot becomes opaque. From the point of view of coating and cleaning applications, the structure of the final residue of the dried suspension is of prime importance. Note that while many applications involve drying of extended films rather than
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Figure 8. Cap radius vs time, for droplets in the three regions of the drying diagram. For each experiment, the corresponding symbol in the inset shows the initial composition of the droplet, relative to the drying boundaries from Figure 2. The solid line is an example fit of the expression of Parisse and Allain8 to the data shown by the filled circles (a “G” sample as shown in the inset). The fit parameters are the following: contact angle 10°; start volume fraction 0.35; Φfoot ) 0.5.
Figure 7. (a)-(d) Final residues of dried droplets, with initial compositions as indicated by the letters in (e). Droplets are 10 mm in diameter.
single droplets, nevertheless in a typical practical situation the film will not be perfectly uniform, flat, nor infinite, and therefore, it seems reasonable to expect some flowdriven phenomena similar to the case of droplets. Indeed very similar phenomena including cap and foot formation are observed for ceramic films.9 Furthermore, note that while the film formation stage involving particle coalescence and polymer interdiffusion is absent in our system, nevertheless the macroscopic residue structure is set by the drying stage: subsequent particle coalescence and molecular interdiffusion are unlikely to change macroscopic residue structure. Figure 7 compares residues from a number of experiments with colloid-polymer mixtures. At low initial Φ, most of the colloid is driven to the contact line during the drying, as in the limiting case of dilute, noninteracting colloids as studied by Deegan et al.4,5 In Figure 7a (Φ ) 0.05, a dilute A droplet; see Figure 7e) only a thin film of colloid remains across the central area of the dried residue: this film is approximately (though not perfectly) homogeneous. In the droplets in Figure 7b,c (initial Φ ) 0.1 and increased polymer concentration but both still in the A region) a rim of colloid at the contact line is still visible, but the film of colloid in the center of the residue is now thicker and in both cases macroscopically inhomogeneous. In Figure 7c, the higher initial polymer concentration, the film is thick enough to show quasi-regular cracking behavior.6 Finally, in Figure 7d we see the residue left behind by a G sample with initial Φ ) 0.35. The suspension is so dense and resists flow so well (because it immediately forms a gel on deposition) that the colloids cannot be driven to the contact line to form a rim; the residue simply shows a small dip in the center. This residue is the most homogeneous, which can be explained by the droplet’s greater particle concentration, by the resistance of the particle gel to flow, and by the lack of rheological or phase boundaries crossed during drying. Quasi-regular cracking is observed, as for charged colloids6 and protein solutions.20 Quantitative Measure of Drying: Cap Radius. Measurements of the radius of the cap, Rc, scaled by initial (20) Annarelli, C. C.; Fornazero, J.; Bert, J.; Colombani, J. Eur. Phys. J. E 2001, 5, 599.
drop radius, versus time as a fraction of total drying time, are given in Figure 8, for samples at compositions as indicated in the inset. Consistent with the visual observations and classification on the drying diagram, we observe a clear change in behavior of the cap radius for A droplets. Interestingly, the cap radius decreases in approximately the same way for F and G samples, despite that F samples show clear differences in initial flow conditions and that certain F samples show a transition to G behavior in the later stages of drying: there is no indicator of this change in the measures of Rc. Figure 8 shows one example comparison between our measurements and the theoretical expression developed by Parisse and Allain8 (“PA”). The PA model supposes that the foot is a dense colloidal solid at volume fraction Φf. Φf is supplied as a parameter in the model. By calculation of the flux due to the changing shape of the pinned droplet as solvent evaporates (assuming there is no shape change of nor net solvent loss from the foot, i.e. Φf is constant), an expression is obtained for the radial dependence of the volume fraction at time t, Φ(r,t). The radius of the cap at time t, Rc(t), may then be identified as the radius where Φ(r,t) ) Φf. Rather than obtain direct measurements of cap and foot heights with time here, we wish simply to examine the qualitative “fit” of the PA model to our cap radius measurements in each of the three regions of the drying diagram. Using reasonable parameters for the initial volume fraction Φ, contact angle, and foot volume fraction Φf, we can obtain model curves for cap radius against time with shape qualitatively similar to that of our observations in the G and F regions, as shown by the example fit for a G sample in Figure 8. How meaningful the “reasonable” parameters are (especially Φf) is not clear, however. In any case given the similarity of the measured curves for the F and G regions, the indication is that Rc is not very sensitive to many of the details of the flow processes occurring inside the drying droplet. The behavior of Rc(t) is qualitatively different for droplets with initial composition in the A region. Rc decreases more slowly compared to F and G samples, until very near the end of drying (typically more than 90% of the total time), when a very sudden decrease is observed. This difference in A region behavior is consistent with the visual observations of droplet behavior. In particular the final, very fast decrease of Rc coincides with the observation of the rapidly receding “final layer” of solvent near the end of drying, as described above. We tentatively ascribe
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Figure 9. How flow-driven inhomogeneities (in colloid concentration) in “A” samples can lead to dense regions of the droplet crossing the gel boundary (upper dotted arrow), while similar inhomogeneous regions in “F” samples do not cross the gel line (lower dotted arrow).
this stage to a depinning of the solvent, occurring when the colloidal particles remaining in the cap form a thin layer across the substrate (Figure 7a-c), and the remaining solvent behaves as a thin, unpinned liquid film. It may also be noted that the PA model cannot fit the A region curves, for any reasonable set of parameters. Comparison of Drying and Bulk Behavior. The “drying diagram” in Figure 2 enables a comparison of the boundaries marking the three drying regions with the system’s bulk equilibrium and nonequilibrium boundaries. The dotted and dashed lines on the figure show the experimentally determined boundaries for equilibrium fluid-crystal phase separation and nonequilibrium gelation, respectively, in bulk colloid-polymer samples at this polymer-colloid size ratio ξ ) 0.08. As can be seen, the drying boundary marking the G region does not correspond exactly with the bulk gelation line. The presence of flow stresses in the drying droplet probably accounts for this: suspensions which would be just inside the gel boundary in bulk, zero-flow conditions are prevented from forming macroscopic solidlike gels in droplets, because the imposed flow will tend to break large aggregates apart. (Geometricssample size and shapes effects are also important in determining the exact position of the gel boundary, as shown in ref 15.) Nevertheless, the boundary of the G region on the drying diagram does track rather well the bulk gelation line, indicating, importantly, the viability of predicting drying behavior from knowledge of the bulk characteristics. In the context of drying behavior the gel line seems to be the most important boundary, due to the large “rheological contrast” between the solidlike gel and the other fluid phases. Discussion The most striking observation we have discussed here is the difference in drying behavior between the A region and the F and G regions on the drying diagram. Referring to Figure 9, we can explain this conceptually as follows. The average composition of a droplet with initial particle concentration in the F region traverses the composition diagram along a line as shown (marked F), which approaches the gelation boundary where it is approximately horizontal (parallel to the Φ axis). An inhomogeneity or gradient in particle concentration Φ (illustrated schematically by the lower dotted arrow), generated somewhere in the droplet e.g. by the flow of colloidal particles driven toward the contact line, will not carry the more concentrated region across the gel boundary. Conversely, a drying A droplet follows a line that notionally crosses the gel boundary where that boundary is near vertical or perpendicular to the Φ axis. Now, an inhomogeneity in colloidal particle concentration as illustrated by the dotted arrow at the top left can carry the region of higher Φ across the gel boundary. If this happens
Haw et al.
(as it will when the drying line is close enough to the gel line), the region of higher Φ will form a gel, which is, as mentioned, a “phase” with markedly different rheological response compared to the rest of the droplet. Hence small inhomogeneities or concentration gradients can carry regions across “rheological boundaries”, leading to much more complex flow conditions inside the dropletsexactly as is observed in the experiments, with the generation of complex circulating flows. Importantly, due to the resistance to flow of the gel regions, the generated inhomogeneities are also more likely to persist in the final residue, as is observed here (Figure 7). Another important difference in the A region is that, in much of this region, despite that the final equilibrium is fluid-crystal coexistence, bulk samples initially undergo gas-liquid-phase separation by spinodal decomposition, as discussed in the Introduction and in ref 12. Because this phase separation is rapid (compared to nucleation of crystal from metastable fluid), it may also have effects on the drying behavior, again leading us to expect the A region to be the most complex. While we might expect formation of gel regions to be more important than spinodal separation into regions of differing colloidal particle concentration but qualitatively similar rheology (because of the more marked rheological contrast between gel and fluid), nevertheless, the onset of phase separation does represent an additional way to generate Φ inhomogeneity in the droplet, i.e. a second route to crossing the gel boundary additional to the flow-driven generation of inhomogeneity. The interplay between flow-driven effects and rapid phase-separation effects perhaps accounts for the systematic variation of drying behavior within the A region, though this requires more detailed investigation. Conclusions In principle, the drying of a droplet of a colloidal suspension is a problem of great complexity, involving the interplay of flow, time-dependent rheological response, and incipient phase changes in response to a continuously shifting thermodynamic “target” equilibrium, not to mention metastability and the possibility of complex nonequilibrium states. Nevertheless the experiments described here on a “model” but still relatively complex system (one which displays all the complications listed above) lead to an important and perhaps surprising conclusion. A broad classification according to the initial composition of the droplet, into three regions on the composition diagram, is possible. Moreover, behavior in the three regions can be qualitatively related in rather a straightforward way to the bulk equilibrium and nonequilibrium properties of the system. In particular, the persistence of composition inhomogeneity (in our case, most importantly in the concentration of the colloidal particles) into the final residue can be predicted by comparing the “drying line” traced out by a droplet’s average composition, with the orientation of the equilibrium and nonequilibrium boundaries in the composition diagram. Spatial inhomogeneities in the concentration of a component (initially the result of convective flow generated by the strong pinning of the contact line or of the initial stages of phase separation) can persist if they lead to regions of the droplet being carried across a phase boundary or a nonequilibrium line such as the gel line in this system. The strong contrast in rheological properties between regions either side of such a boundary then makes the drying behavior substantially more complex and more likely to lead to inhomogeneity in the final residue.
Phase Behavior and Drying of Suspensions
While the colloid-polymer system studied here is reasonably simplified (compared for instance to commercial detergents and coatings), nevertheless we think that similar phenomena ought to be observed in more complex systems, wherever “boundaries” in the composition diagram represent significant rheological or structural changes. Consideration of the drying path of a suspension relative to these boundaries, based on initial suspension composition, should give at least a first guideline as to the sorts of structural behavior to be expected during evaporative drying and the homogeneity or otherwise of final residues. Clearly, important extensions of the current work include the examination of other systems (e.g. polymersurfactant mixtures used in cleaning, protein solutions,20 etc.) to determine the general applicability of the relation between drying and bulk phase behavior and the comparison with drying in other geometries, especially the technologically important situation of extended films.1-3,9
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The persistence of inhomogeneities in final residues is normally an undesirable situation in coating and cleaning processes. However, control of the development of inhomogeneity and complex flow patterning, via surface treatment or careful design of the composition/geometry of the drying fluid, could conceivably represent a simple evaporation-based route to the construction of films and residues incorporating desired structure on mesoscopic and macroscopic scales. Thus, better understanding of the mechanisms of generation of flow patterns and inhomogeneities in the drying of suspensions must be an important goal in future work. Acknowledgment. This work was funded by the U.K. EPSRC (ROPA). We also thank Unilever Research for partial funding of equipment. We gratefully acknowledge S. P. Meeker for preliminary experiments and discussions and A. B. Schofield for provision of the colloidal particles. LA0110951
