© Copyright 2004 American Chemical Society
NOVEMBER 9, 2004 VOLUME 20, NUMBER 23
Letters Why Do Drying Films Crack? Wai Peng Lee and Alexander F. Routh* Department of Chemical and Process Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, United Kingdom Received April 19, 2004. In Final Form: September 22, 2004 Understanding the mechanism by which films fail during drying is the first step in controlling this natural process. Previous studies have examined the spacing between cracks with predictions made by assuming a balance between elastic energy released with a surface energy consumed. We introduce a new scaling for the spacing between cracks in drying dispersions. The scaling relates to the distance that solvent can flow, to relieve capillary stresses, as a film fails. The scaling collapses data for a range of evaporation rates, film thicknesses, particle sizes, and materials. This work identifies capillary pressures, induced by packed particle fronts travelling horizontally across films, as responsible for the failure in dried films.
Evaporating solvent from a suspension creates many apparently solid surfaces. As solvent leaves a dispersion, particles consolidate and in many cases the films will display cracks. This is seen when flying over dried riverbeds, examining old oil paintings, or looking at concrete paving slabs.1 There is much debate as to the mechanism by which cracks form, with a number of competing mechanisms proposed.2-7 The usual prediction of crack spacing comes from balancing the elastic energy released during fracture with the energetic cost of creating new surface. The resulting prediction is that the crack spacing scales with the film thickness. However, in this * Corresponding author. E-mail:
[email protected]. (1) Walker, J. The amateur scientist. Sci. Am. 1986, 255 (4), 204211. (2) Jagla, E. A. Stable propagation of an ordered array of cracks during directional drying. Phys. Rev. E 2002, 65 (4), 6147. (3) Bleuth, J. L. Cracking of thin bonded films in residual tension. Int. J. Solids Struct. 1992, 29 (13), 1657-1675. (4) Bordia, R. K.; Jagota, A. Crack growth and damage in constrained sintering films. J. Am. Ceram. Soc. 1993, 76 (10), 2475-2485. (5) Parker, A. P. Stability array of multiple edge cracks. Eng. Fract. Mech. 1999, 62, 577-591. (6) Thouless, M. D. Crack spacing in brittle films on elastic substrates. J. Am. Ceram. Soc. 1990, 73 (7), 2144-2146. (7) Schulze, G. W.; Erdogan, F. Periodic cracking of elastic coatings. Int. J. Solids Struct. 1998, 35 (28-29), 3615-3634.
letter we show a different mechanism, where hydrodynamics controls the spacing. The stresses causing cracking have been studied using a cantilever technique.8-10 This method provides the transverse stress in a film averaged laterally. For a drying dispersion, the meniscus of the air-water interface between particles gives a capillary pressure in the fluid that is below atmospheric. The magnitude of this pressure can be enormous, scaling inversely with the particle radius.11 Atmospheric pressure, pushing on the film surface, puts the entire film into compression. If the film compresses, or yields, the exudation of solvent results in a reduction, or even elimination, of capillary pressure until evaporation increases it again. Recent work by Dufresne et al.11 showed cracking films to be wet, except at the (8) Peterson, C.; Heldmann, C.; Johannsmann, D. Internal stresses during film formation of polymer lattices. Langmuir 1999, 15, 77457751. (9) Martinez, C. J.; Lewis, J. A. Shape evolution and stress development during latex-silica film formation. Langmuir 2002, 18 (12), 46894698. (10) Tirumkudulu, M. S.; Russel, W. B. Role of capillary stresses in film formation. Langmuir 2004, 20 (7), 2947-2961. (11) Dufresne, E. R.; Corwin, E. I.; Greenblatt, N. A.; Ashmore, J.; Wang, D. Y.; Dinsmore, A. D.; Cheng, J. X.; Xie, X. S.; Hutchinson, J. W.; Weitz, D. A. Flow and fracture in drying nanoparticle suspensions. Phys. Rev. Lett. 2003, 81 (22), 4501.
10.1021/la049020v CCC: $27.50 © 2004 American Chemical Society Published on Web 10/15/2004
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Figure 2. Picture of a dried film in a Petri dish. Here, 40-nm silica particles have been dried from a volume fraction of 20%. The radial cracks can be seen at the edge of the Petri dish, and the spacing is easily measured. Cracks actually propagate throughout the film, although debonding from the base allows the cracks at the edge to be observed.
Figure 1. Drying front, pressure distribution, and fracture mechanism.
position of the cracks, demonstrating that the capillary pressure is responsible for the film failure. A large body of work has been performed where dispersions are constrained between two glass plates with evaporation from one end.11,16-17 This results in a progression of cracks from the free surface into the bulk. Here we examine a slightly different scenario, with fracture in thin films subject to evaporation at their free surface. This conceptually simple problem is complicated by horizontal drying fronts. At the edge of drying films, the reduced thickness causes particles to consolidate into a close-packed configuration. Continued evaporation from this solid region leads to a large transverse flow, bringing particles toward the packed region and, hence, propagating the close-packed front. In addition the flow of solvent through the packed region leads to a distribution of pressure. This process is sketched schematically in Figure 1 and described fully in a previous publication.18 The characteristic horizontal length scale for drying fronts is found from a balance between surface tension driven flow and evaporation and is often called the capillary length scale. The speed of crack propagation can have a crucial influence over the cracking process.12 For stresses induced by differences in temperature this crack speed can be easily (12) Boeck, T.; Bahr, H. A.; Lampenscherf, S.; Bahr, U. Self-driven propagation of crack arrays: a stationary two-dimensional model. Phys. Rev. E 1999, 59 (2), 1408-1416. (13) Marder, M. Instability of a crack in a heated strip. Phys. Rev. E 1994, 49 (1), 51-54. (14) Adda-Bedia, M.; Pomeau, Y. Crack instabilities of a heated glass strip. Phys. Rev. E 1995, 52 (4), 4105-4113. (15) Bahr, H. A.; Gerbatsch, A.; Bahr, U.; Weiss, H. J. Oscillatory instability in thermal cracking: A first order phase transition phenomenon. Phys. Rev. E 1995, 52 (1), 240-243. (16) Allain, C.; Limat, L. Regular patterns of cracks formed by directional drying of a colloidal suspension. Phys. Rev. Lett. 1995, 74 (15), 2981-2984. (17) Komatsu, T. S.; Sasa, S. Pattern selection of cracks in directionally drying fracture. Jpn. J. Appl. Phys., Part 1 1997, 36, 391-395. (18) Routh, A. F.; Russel, W. B. Horizontal drying fronts during solvent evaporation from latex films. AIChE J. 1998, 44 (9), 20882098.
controlled.13-15 For the case of evaporation, Dufresne et al.11 examined the dynamics of cracking and showed cracks to follow the particle compaction front. To minimize the effect of horizontal drying the experiments described here were performed in a Petri dish. A front of close-packed particles propagates from the center outward with cracks following the particle front. This simple drying procedure has been used previously to examine drying precipitates.19 A picture of a dried film is shown in Figure 2. The proposed cracking mechanism, presented here, is shown schematically in Figure 1. A crack is pulled apart because the capillary pressure in the solvent is below atmospheric. Once the material yields, the reduction in available volume eliminates the capillary pressure in the vicinity of the crack. This causes a flow of solvent toward areas of lower pressure in the bulk. The distance of flow sets the spacing between cracks. The capillary pressure, pulling cracks apart, is the same as that proposed by Scherer in sol-gel processing.20-21 If we assume the flow of solvent through the particles to control the crack spacing we may derive the relevant scalings. For solvent with viscosity µ flowing through a packed bed of particles with permeability kp, Darcy flow reads as
∇P ) -
µ u kp
(1)
where u is the fluid velocity. We determine a length scale, X, over which the capillary pressure will relax. The characteristic velocity is determined from the velocity of the crack, which is controlled by the horizontal drying front. We derive an estimate for X below and use this to scale our crack spacings. The Carmen-Cozeny equation relates the permeability to the particle radius R and volume fraction φ by kp ) [2R2(1 - φ)2]/(75φ2). The pressure in the fluid is due to capillary action, with a maximum magnitude of 10γ/R, where γ is the water-air surface tension.22 The velocity (19) Neda, Z.; Leung, K. t.; Jozsa, L.; Ravasz, M. Spiral cracks in drying precipitates. Phys. Rev. Lett. 2002, 88 (9), 5502. (20) Scherer, G. W. Crack-tip stress in gels, J. Non-Cryst. Solids 1992, 144, 210-216. (21) Scherer, G. W. Drying of ceramics made by sol-gel processing. In Drying ‘92; Mujumdar, A. S., Ed.; Elsevier Science: New York, 1992.
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Figure 3. Collapse of the crack spacing data with hydrodynamic scaling.
of the crack is shown by Dufresne et al.11 to be controlled by the drying front, and, consequently, this scales with the evaporation rate, E, and a ratio of the capillary length scale,25 L, to the dried film thickness, H, where L/H ∼ [γ/(3η0E)]1/4. The capillary length scale, usually assumed to be much greater than the film thickness, depends on the surface tension, dispersion viscosity, η0, and the evaporation rate. This was derived specifically previously.18 The characteristic horizontal velocity is u* ∼ [γ/(3η0E)]1/4E. Substituting this scaling for velocity into eq 1 and assuming the pressure to relax over the crack spacing, we obtain an estimate for the horizontal length scale as
X∼
( )
3 20R(1 - φ)2 3η0γ
75µφ2
1/4
E3
(2)
The driving force for fracture is the capillary stress. To nondimensionalize this, the natural scaling for pressure follows from Darcy flow. Substituting a capillary length scale for horizontal distances, the characteristic pressure, p*, follows as µL2E˙ /kpH. Comparing the maximum capillary pressure (10γ/R) with the characteristic pressure, we obtain a dimensionless group Pcap given by
Pcap )
( )
20 3γη0 75 E
R(1 - φ)2
1/2
µφ2H
(3)
(22) If we assume an arrangement of three particles in a triangular formation and place a spherical cap meniscus between them, with zero contact angle, the geometry dictates that the curvature gives a maximum capillary pressure of 12.9γ/R. Most authors simplify this to have a maximum value of 10γ/R. This is the absolute maximum value the capillary pressure can take. As will be argued later, an alternative scaling will be with the yield stress of the particle arrangement. This will be of the form Cγ/R, where C is a number considerably less than 10. (23) Hull, D.; Caddock, B. D. Simulation of prismatic cracking of cooling basalt lava flows by the drying of sol-gels. J. Mater. Sci. 1999, 34, 5707-5720. (24) Tirumkudulu, M. S.; Russel, W. B. Cracking in drying latex films, preprint. (25) The capillary length scale, L, is derived by assuming surface tension driven flow in thin films subject to evaporation. It appears in this problem because the progression of the particle front and, hence, the cracking front is controlled by the horizontal flow of solvent in the film.
This dimensionless group characterizes the flow through consolidated particles. The value of Pcap determines the pressure distribution along the bed, as shown schematically in Figure 1. For large values of Pcap, the pressure decreases monotonically along the bed. For lower values of Pcap, the maximum capillary pressure is reached and, consequently, the pressure is a constant at this value. We have dried a number of dispersions: silica particles (Ludox obtained from Aldrich) with diameters of 30 and 40 nm and synthesized polystyrene lattices (diameters 180, 315, 355, 410, and 460 nm). In each case the particles are hard and do not form a coherent film. The volume fractions of the original dispersions ranged from 1 to 39%. The volume fraction at close packing, used in the dimensionless scaling, was taken as 64%. A range of evaporation rates is achieved by placing a plastic film over the Petri dishes, with holes punched through. The evaporation rate is measured by weighing the dish at regular intervals. In each experiment the rate of water loss is found to be constant and ranged from 1.0 × 10-9 to 3.5 × 10-8 m/s. The final film thickness varied from 1 × 10-5 to 6.6 × 10-4 m. The dispersion viscosity, η0, was taken as 1 N‚s/m2, for each dispersion, and assumed to remain constant throughout the evaporation process. While the low shear viscosity must increase as evaporation occurs, any error is small because we use this variable to calculate the capillary length and take the quarter power of the value of η0. The crack spacing was measured using Vernier callipers and ranged from 0.1 to 2 mm. Despite drying in a circular Petri dish, the crack spacing is found to remain constant as the cracks propagate radially outward. The crack spacing, y, normalized by the hydrodynamic scaling X is plotted against the capillary pressure, as shown in Figure 3. The data collapses to a single curve for a range of materials covering 3 orders of magnitude in the capillary driving force and 2 orders of magnitude in the dimensionless crack spacing. As a comparison, we plot the same data assuming an energy-balance-type scaling. Balancing the energy released by crack propagation with the energy consumed the crack spacing is found to scale with the film thickness. Figure 4 shows a plot of the crack spacing against the film thickness. A partial scaling is obtained, although a
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Figure 4. Crack spacing as a function of the film thickness.
dependence on the particle size is evident. Figure 5 shows a plot of the crack spacing normalized by the film thickness against Pcap. For an energy-type scaling we expect this graph to be a flat line, if the critical fracture stress is independent of the drying conditions. Hull and Caddock23 suggest a geometry-dependent value of y/H around 3. Russel and Tirumkudulu24 suggest a value dependent on the stress above the critical value, a number hard to estimate but presumably constant for each sample, and Allain and Limat16 suggest y/H to only vary slightly with the film thickness. However, from the results shown in Figures 4 and 5 it can be concluded that an energy-type scaling for the crack spacing does not fit the data. The collapse of all data in Figure 3 is significant. A comment is required about the scaling for pressure. On both axes the maximum pressure in the bed is taken as 10γ/R. This is an over-estimate of the yield stress of the bed. The γ/R term will remain, but the magnitude 10 is the maximum value possible. In reality a factor less than 10 will correspond to failure. This will reduce the characteristic distance of flow and, hence, increase the numbers on the vertical axis and decrease numbers on the horizontal axis of Figure 3. Doing this will simply make the magnitude of the numbers in Figure 3 around the order of 1.
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Figure 5. Noncollapse of the crack spacing data when scaled with the film thickness.
The collapse of data in Figure 3 is according to the following power law:
y ) 0.07Pcap-0.8 X This gives that the crack spacing scales with the film thickness to the power of 0.8, the evaporation rate to the power of -0.35, and the particle size to the power of 0.2. In this letter we have derived the scaling for the flow of solvent away from a crack. The elimination of capillary pressure because of the yielding of the material provides a pressure gradient, and the distance of this flow sets the spacing between cracks. This is a novel explanation for the observed regularity in crack spacing, and the new scaling collapses experimental data for a range of evaporation rates, film thicknesses, particle sizes, and particle materials. Acknowledgment. This work is supported by EPSRC through Grant GR/S05885/01. The authors are very grateful to Professor Mike Hounslow and Dr. Will Zimmerman for helpful discussions. LA049020V
