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Pattern Formation in Draining Thin Film Suspensions M. Buchanan,*,†,‡ D. Molenaar,† S. de Villiers,† and R. M. L. Evans§ Physics of Geological Processes, UniVersity of Oslo, Blindern, 0316 Oslo, Norway, Department of Physics, UniVersity of Oslo, Blindern, 0316 Oslo, Norway. and School of Physics & Astronomy, UniVersity of Leeds, Leeds LS2 9JT, U.K. ReceiVed NoVember 9, 2006. In Final Form: January 9, 2007 We demonstrate the emergence of complexity from remarkably simple and ubiquitous systems: draining thin-film suspensions exhibiting a striking transition between two classes of self-organizing patterns. Vertical channels form when attractive forces lead to transient gelation, while horizontal bands result from granular mixtures. We propose an explanation whereby the generic physical mechanisms require only the existence of viscous and excluded-volume couplings among the particles, solvent, and substrate. System-specific, small inhomogeneities trigger large-scale pattern formation, through collective dynamics, where jamming plays a crucial role. Our results shed light on emergent complexity in bio- and geophysical processes and have implications for coatings and food industries.
Introduction When suspensions1 or granular mixtures2,3 flow as a thin film down a substrate, they form patterns on the surface. We identify two classes of pattern that occur universally. The spontaneous formation of these patterns is relevant to the coatings industries and bio- and geophysical applications. Their morphologies are familiar in residues from household fluids, drinks, and accreted dirt on windows among other manifestations (Figure 1). In some well-characterized experimental systems, we show how a gradual change in the nature of the suspension, from a transient gel to a granular mixture, results in striking changes in the type of pattern that emerges from the flow. A transition is observed, from a pattern of vertical channels, for systems exhibiting transient gelation, to horizontal bands, for systems in which the granular nature of the suspension plays a dominant role. We suggest that the ubiquity of these patterns throughout nature is a consequence of the small number of basic features required for their formation. They are intrinsically nonequilibrium phenomena, generated by dissipative forces in flowing films. The presence of friction between the suspended particles and a substrate, together with viscous and excluded-volume couplings between the particles and fluid, are sufficient to generate spatial structures on a scale much larger than the particles themselves. The frictional forces are generally mediated by the confinement of the flow to a thin film geometry,4 where the film thickness decays with time, forcing particles into close proximity with the substrate. These effects lead to jamming5-7 and the onset of pattern formation in both * To whom correspondence should be addressed. E-mail: mark.buchanan@ fys.uio.no. † Physics of Geological Processes, University of Oslo. ‡ Department of Physics, University of Oslo. § School of Physics & Astronomy, University of Leeds. (1) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Capillary flow as the cause of ring stains from dried liquid drops. Nature 1997, 389, 827. (2) Zhou, J. J.; Dupuy, B.; Bertozzi, A. L.; Hosoi, A. E. Theory for shock dynamics in particle-laden thin films. Phys. ReV. Lett. 2005, 94, 117803. (3) Daerr, A.; Lee, P.; Lanuza, J.; Clement, E. Erosion patterns in a sediment layer. Phys. ReV. E 2003, 67, 065201. (4) Oron, A.; Davis, S. H.; Bankoff, S. G. Long-scale evolution of thin liquid films. ReV. Mod. Phys. 1997, 69, 931. (5) Haw, M. D. Jamming, two-fluid behavior, and “self-filtration” in concentrated particulate suspensions. Phys. ReV. Lett. 2004, 92, 185506. (6) Stratford, K.; Adhikari, R.; Pagonabarraga, I.; Desplat, J. C.; Cates, M. E. Colloidal jamming at interfaces: A route to fluid-bicontinuous gels. Science 2005, 309, 2198.
Figure 1. Observable at the breakfast table: (a) patterns formed by honey in a jar and on a flat glass surface (inset); (b) yoghurt patterns on a beaker. (7) Liu, A. J.; Nagel, S. R. Nonlinear dynamics-Jamming is not just cool any more. Nature 1998, 396, 21.
10.1021/la063282a CCC: $37.00 © 2007 American Chemical Society Published on Web 02/23/2007
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Figure 2. Pattern transition sequence, resulting from the gravity-driven flow of a thin film suspension down a rigid container surface, for varying particle sizes. Particles coating the container surface appear white; brighter areas correspond to higher concentrations. In each case the mixture has an initial particle volume fraction close to φ0 ) 0.27, whereas the relative fraction of (large) L-particles increases from (a) 0.0, (b) 0.33, and (c) 0.5 to (d) 1.0 (by volume). Each image is taken with a digital SLR camera and covers a width of 7.2 cm.
systems. We support our observations on the horizontal band patterns with numerical computations of a particle-laden thin film flow. While many nontrivial structures have previously been documented in both gel-forming8,9 and granular systems,10,11 our results demonstrate a striking transition, at the crossover from the former to the latter, between two pattern types that are immediately recognizable due to their ubiquity and large scale. Experimental Setup In our experiments, a carefully mixed suspension of distilled water and quartz (SiO2) particles is distributed evenly over the oxidized, hydrophilic wall of a Perspex container, manufactured by Nunc A/S. The SiO2 particles, of density Fp ) 2650 kg/m3, were purchased at Carl Roth GmbH & Co. Samples are mixed from a distribution of large particles, with mean diameter dm ) 20 µm, and small particles, with mean diameter dm ) 3.5 µm. The Peclet number, defined as the ratio of sedimentation to Brownian motion, is Pe ) 4(Fp Ff)ga4/3kBT, for Boltzmann’s constant kB, gravity g, temperature T, and solvent density Ff. Note that the Peclet number scales as Pe ∼ 16.7aµ4, where aµ is the mean particle radius in micrometers. Hence, while the large (L) particles are in the granular regime, where Pe . 1 and sedimentation dominates over Brownian motion, the small (S) particles are on the margin between the colloidal (Brownian dominated) and granular regimes, where Pe is closer to unity. The 7.2 cm × 12.0 cm hydrophilic container wall, initially horizontal, is covered by a 2-3 mm thick layer of the suspension, containing 50% particles by weight, corresponding to an initial volume fraction φ0 ) 0.27. The container is then placed with its hydrophilic wall vertical. During tilting, most of the mixture drains rapidly down the surface, leaving a typical film thickness of a few hundred micrometers prior to the onset of pattern formation. Within the remaining thin film, the gravity-driven flow is free from leadingedge or contact-line effects.12 During this stage, the collective particle dynamics transforms the mixture into a complex, effectively twodimensional pattern on the wall. The particle size distribution serves as our experimental control parameter. (8) Wessel, R.; Ball, R. C. Fractal Aggregates And Gels In Shear-Flow. Phys. ReV. A 1992, 46, R3008. (9) Starrs, L.; Poon, W. C. K.; Hibberd, D. J.; Robins, M. M. Collapse of transient gels in colloid-polymer mixtures. J. Phys.: Condens. Matter 2002, 14, 2485. (10) Louge, M. Y. Model for dense granular flows down bumpy inclines. Phys. ReV. E 2003, 67, 061303. (11) Voltz, C. Granular dynamics of density profiles in a suspension interface. Phys. ReV. E 2003, 68, 021408. (12) Huppert, H. E. Flow And Instability Of A Viscous Current Down A Slope. Nature 1982, 300, 427.
Results and Discussion A transition in pattern type, as a function of the relative fraction of L-particles, is apparent in snapshots of the surface after flow has ceased (Figure 2). With S-particles only (Figure 2A), a pattern of vertically oriented channels appears. On increase of the L-particle content, the channels become progressively less pronounced while horizontal structures appear (Figure 2B), yielding a hybrid pattern of channellike and bandlike features (Figure 2C). Finally, with L-particles only (Figure 2D), channels are absent; the pattern consists entirely of horizontal bands. On the basis of our experimental observations, we propose an explanation for the difference in pattern morphologies of the Sand L-particle mixtures. We suggest that it can be traced to a difference in flow properties arising in the colloidal and granular regimes. It is well-known that van der Waals forces between particles cause aggregation in colloidal suspensions, yielding a fractal structure and gellike characteristics.13 (For crystalline quartz in water, a Hamaker constant14 of 1.70 × 10-20 J causes van der Waals attractions exceeding kBT for interparticle separations up to 16% of the radii.) The gelled state15 is typical of colloidal suspensions at a volume fraction φ ) 0.27, and it is clear, from observed sedimentation and aging behavior in bulk samples,8,9 that our S-particle suspension forms a gel. Importantly, the gel structure offers viscous resistance to deformations, reacting against forces that tend to redistribute the particle concentration. Another important property of the colloidal gel is its well-known shear-thinning behavior:13 viscosity falls with increasing shear rate, due to shear-induced rearrangements of the microstructure. By contrast, granular material (the L-particle system) neither gels nor exerts significant osmotic pressure at low concentration and, therefore, does not resist redistribution except in regions where it is already compacted. In each experiment the mixture is initially roughly homogeneous. The growth of large-scale patterns requires initiation by some small inhomogeneities, for which the detailed mechanisms (discussed below) are system-specific. The ensuing pattern formation processes are universal. (13) Potanin, A. A.; Derooij, R.; Vandenende, D.; Mellema, J. Microrheological Modeling of Weakly Aggregated Dispersions. J. Chem. Phys. 1995, 102, 5845. (14) Russel W. B.; Saville D. A.; Schowalter W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989. (15) Pusey, P. N.; Pirie, A. D.; Poon, W. C. K. Dynamics of Colloid Polymer Mixtures. Physica A 1993, 201, 322.
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While changes in the particle size distribution most readily produce the pattern transition, the transition can also be induced by tuning other parameters that change the behavior of a colloidal suspension. The key point is that gelation of a colloidal system (which is required for channel-pattern formation) is prevented if the attractive interparticle forces are reduced. Simple experiments show that the S-particle system’s ability to form a gel can be influenced by changing the electrolytic properties of the solvent or simply by aging the mixture to impair clustering, thus also influencing the patterns observed (see Supporting Information Figures A and B). To discuss the physics of pattern formation, we shall focus on the pure channel or band structures; intermediate patterns arise from a complex interplay of the mechanisms that create these two extreme cases. Channel Patterns. We now investigate the properties of S-particle suspensions responsible for channel formation (Figure 2A). The gelled microstructure starts to form before the onset of flow, and the mixture subsequently descends roughly homogeneously. As the film thins, its weight diminishes, slowing the descentsan effect enhanced by the shear-thinning gel’s viscosity increasing with the declining rate of shear (between the falling free surface and static substrate). Thus, the phase of rapid descent comes to an abrupt halt. The near-static aggregated particles now form a porous medium which exerts drag on the interstitial solvent. So the solvent too is impeded and descends under gravity much more slowly than it would without particles. Assume a small inhomogeneity already exists; part of the film is more fluid than the surroundings. Since the faster flowing region carries solvent, effectively the solvent experiences less resistance to flow than in the surrounding medium where its descent relies solely on motion relatiVe to the near-static porous medium of aggregated particles. As a result, the pressure balance in the film alters so as to focus the throughput of solvent toward this inhomogeneity (just as a network of electrical resistors balances its potential differences to focus current toward the path of least resistance). An enhanced flux of solvent implies enhanced shear rates within the gel as a whole, and a positive feedback mechanism now results from the non-Newtonian flow properties. The gel is not only shear-thinning but thixotropic;5 i.e., once a region begins to flow, its resistance to further flow is reduced and the pressure balance reacts to further increase the throughput of solvent. The region affected by this instability is progressively eroded and extended in the gravity direction. Solvent seeping into this region from the surrounding catchment area carries no particles into the main channel, as it cannot overcome the gel’s mechanical integrity and frictional coupling to the substrate. Within the high-flux region, however, the current of solvent is soon sufficient to flush the particulate matter out of the channel, down to the bottom of the wall. Thus, once seepage into an inhomogeneity grows large enough to drive the gel into its non-Newtonian regime, formation of a transparent channel is completed rapidly (as apparent in a video-clip available online16). Band Patterns. A rather different mechanism leads to the formation of horizontal bands (Figure 2D). Unlike the colloidal S-particle gel, granular L-particles are not aggregated so, at low concentration, they move almost independently, offering little resistance to redistribution within the solvent, unless they arrive in highly concentrated regions. Again, inhomogeneities initially form by system-specific mechanisms, discussed below, but develop into a pattern of bands via a universal process. Two stages can be distinguished in the band-pattern formation: fast (lasting ∼2-3 s) growth of thin, wavy bands, followed (16) http://folk.uio.no/markbu/patterns/movie1.mpg.
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by a slower coarsening process. While the majority of the mixture remains well dispersed, let us assume that, in isolated places, particles have become immobile due to some prior inhomogeneities in the concentration or dynamics. The result is a pinning site. The upstream fluid must flow around such a region, creating convergent streamlines on either side of it. Recently, it was shown5 that convergent flow can cause local jamming (solidification of a fluid suspension due to imposed stresses5) and that the jammed region acts as a filter, collecting further particles so that the mixture exits the jam at a reduced concentration. The mechanism can be readily understood: while conservation of matter implies that convergent flow must conserve the average particle concentration, interparticle separations are elongated along the streamlines but reduced in the transverse direction. Thus particles are brought together and stresses are transmitted over many particle diameters. Such behavior is observed at volume fractions above approximately φ > 0.5 in suspensions of small, nonflocculating hard spheres,5 while, in our samples, the threshold concentration may be much lower due to interparticle attractions17 (van der Waals forces and also capillary for particles large enough to deform the solvent-air surface). Thus, immobile particles accumulate in the regions of convergent flow, and the band grows. A band, once formed, is further stabilized, as its low permeability creates a buildup of solvent on the upstream side, while the well-drained downstream film thins. Thus, the band is confined in a wedge-shaped film profile, which tends to maintain the stresses responsible for the jammed state.18 Although the typical band orientation is horizontal, considerable local variation occurs. The spontaneous formation of stressbearing arches, as randomly distributed pinning sites link up, is typical of jamming5,19 and leads to the wavy appearance of the bands (Figure 3A-C). These characteristics are reproduced in computer simulations20 (Figure 3D-F) of the detailed flow dynamics in a thin film2,4 containing a large number of particles. The simulations include the ingredients vital to band formation: static friction between particles and substrate (regulated by the local film thickness) and viscous and excluded-volume couplings between arrested particles and the solvent, which result in longrange hydrodynamic and capillary interactions. A particle is pinned if it enters a position where the local film thickness is only slightly larger than the particle diameter, mimicking the effect of static friction as detailed in the next section. The pinned particle induces a high local mixture viscosity, effectively reducing the local flow velocity to zero and resulting in perturbations in the film thickness. Eventually, these perturbations grow sufficiently to let additional particles become arrested in a spontaneous, collective jamming effect. Due to the shape of the initial thickness perturbations, additional particles are preferentially trapped in the lateral direction, leading to the characteristic horizontal orientation of the evolving bands. A detailed analysis of our model is presented in forthcoming work.21 Once a band structure has formed in our experiments, the flow in the immediate upstream area is blocked, so sedimentation (gravity-driven motion of particles relative to the near-static solvent) becomes the dominant process. In the experiments, we (17) Trappe, V.; Prasad, V.; Cipelletti, L.; Segre, P. N. and Weitz D. A. Jamming phase diagram for attractive particles. Nature 2001, 411, 772. (18) Cates, M. E.; Haw, M. D.; Holmes, C. B. Dilatancy, jamming, and the physics of granulation. J. Phys: Condens. Matter 2005, 17, S2517. (19) Wittmer, J. P.; Claudin, P.; Cates, M. E.; Bouchaud, J. P. An explanation for the central stress minimum in sand piles. Nature 1996, 382, 336. (20) Our model consists of a two-phase thin film equation,2 which describes the dynamics of a thin film with a free surface and takes the excluded volume and viscous couplings to the particle phase into account. Individual particles are simulated by a molecular dynamics method.21 (21) Molenaar, D.; Buchanan, M.; De Villiers, S.; Evans, R. M. L. Formation of band patterns in draining thin film suspensions. Manuscript in preparation.
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Figure 3. Microscope images (top row) showing band formation during an experiment with L-particles (50% by weight) in water (after 1, 3, and 6 s, respectively; scale bar ) 1 mm). (a) The largest particles are pinned between the substrate and film surface. (b) Thin bands form as additional large particles are trapped. (c) The bands cut off the mean flow and experience significant coarsening due to sedimentation of smaller particles. Numerical simulations (bottom row) in a periodic square emulate the dynamics of a thin film flow and the motion of a large number of particles, where particle sizes are exaggerated for visualization purposes. Arrow size and color indicate solvent velocity magnitude: from slow (small, blue) to fast (large, red). Shading indicates film thickness (darker is thinner). (d) A few pinning points in the initial condition alter the local flow velocity, inducing undulations in the film surface. (e) These undulations grow in amplitude and allow additional particles to get trapped. (f) Pinning points are linked into band structures, trapping fluid in the immediate upstream area whereas the remaining mixture is now forced to flow around the bands. Time increases from left to right, and gravity acts vertically downward in all images.
observe differential sedimentation velocities leading to particle segregation, as larger particles settle onto the band earlier than smaller particles16 (also see Supporting Information Figure C). The sedimentation process smoothes out the wavy character of the bands on their upstream side. Finally, we address the various system-specific processes whereby small inhomogeneities initially form. Initial Instabilities. Our L-particle samples are significantly polydisperse. While their mean diameter is 20 µm, 3% of the particles have a diameter between 63 and 125 µm. The initial inhomogeneities in the system, leading to pattern formation, can now be described as follows. When drainage reduces the film thickness to h < 100 µm, the largest particles become forced against the substrate by the deformed free surface22 and strongly pinned by the induced static friction with the substrate. Here, the film thickness, at down-slope position x, decays in time t according to a similarity solution,12
predicts that, halfway down the container, the film has thinned to h ) 100 µm at roughly t ) 0.8 s. This predicted time scale, at which the largest particles become pinned against the substrate by the air-solvent interface, agrees with our experimental observations. In the S-particle mixtures, where the film thickness remains large compared with particle sizes until late times, the mechanism for initial formation of local jams must itself be a collective phenomenon. As in thermodynamic nucleation processes, a random event is initially required: a rare fluctuation that locally raises the particle concentration beyond a critical value. Once that threshold concentration is reached, at which jamming sets in,5,24 such a jam spreads in the upstream direction in a V-shape. A gap, devoid of particles, evolves immediately below the jammed patch, where the suspension still flows downward. It is these gaps that act as nucleation points for the development of channels, after the initial stage of flow has concluded.
h ) (γ(φ0) νx/F(φ0) gt)1/2
Conclusions
for initial particle concentration φ0 and kinematic viscosity of the solvent ν. The effect of the particle concentration enters (via the nondimensionalized momentum balance between the particles and the solvent), through the reduced bulk density, F(φ) ) 1 + φ(δ - 1), where δ denotes the ratio of particle density to liquid density, and through the reduced bulk viscosity, γ(φ) ) 1 - φ + φη(φ). Τhe suspension’s concentration-dependent viscosity is approximated23 by η(φ) ) (1 - φ/φmax)-2. The similarity solution
In conclusion, we have demonstrated an example of selforganized pattern formation, exhibiting a striking morphological transition that coincides with a change from transient gelation to granular behavior in thin film suspensions. Vertical channel patterns are associated with the former type of behavior, and horizontal band patterns appear for the latter type. We have proposed an explanation of this ubiquitous phenomenon, which is consistent with both the experimental observations and the results of simulations. In the presented experiments we induce the gel-to-granular crossover by varying the particle size
(22) Fiegel, J.; Jin, F.; Hanes, J.; Stebe, K. Wetting of a particle in a thin film. J. Colloid Interface Sci. 2005, 291, 507. (23) Krieger, I. M.; Dougherty, T. J. A Mechanism for Non-Newtonian Flow in Suspensions of Rigid Spheres. Trans. Soc. Rheol. 1959, 3, 137.
(24) Cates, M. E.; Evans, M. R. Soft and Fragile Matter. SUSSP Proc. 1999, 53.
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distribution, which is one of a set of control parameters yielding the same pattern transition. The emergence of complexity from simple systems continues to be a subject of interest and puzzlement. While it is wellestablished that large-scale complex structures can emerge from simple building blocks in self-organized critical systems,25 the nonfractal patterns presented here do not belong to that class. Self-assembled structures of great variety and beauty arise from surfactant solutions at equilibrium24 but require a nontrivial molecular architecture for their building blocks. The patterns shown in Figure 2 contain features that are many thousands of particle diameters in size, but the complex collective motions required for their formation emerged from the simplest of constituents: grains of quartz. The nonequilibrium nature of these systems allows them to generate structures whose complexity would require complicated molecules in a system at (25) Bak, P.; Tang, C.; Wiesenfeld, K. Self-Organized Criticality-An Explanation of 1/F Noise. Phys. ReV. Lett. 1987, 59, 381.
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equilibrium. Indeed, the constituents are so simple that they arise ubiquitously, with the resulting patterns appearing in settings as familiar as the dirt on windows or the yoghurt on the side of a half-emptied bowl. Acknowledgment. R.M.L.E. is funded by The Royal Society. Anders Malthe-Sørensen of the Institute for Physics of Geological Processes (PGP), Oslo University, is thanked for useful comments. This research was funded by the Norwegian Research Council through a Centre of Excellence grant to PGP. Supporting Information Available: Figures showing the effect of salt and aging on pattern formation process and a closeup of a band formation (Figures A-C, respectively). This material is available free of charge via the Internet at http://pubs.acs.org. A movie of the channel pattern formation can be downloaded from http://folk.uio.no/markbu/patterns/movie1.mpg. LA063282A
