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Ind. Eng. Chem. Res. 2006, 45, 6996-7001
Temperature Dependence of Crack Spacing in Drying Latex Films Wai Peng Lee and Alexander F. Routh*,† Department of Chemical and Process Engineering, UniVersity of Sheffield, Mappin Street, Sheffield S1 3JD, U.K.
Latex dispersions with different glass-transition temperatures have been dried in Petri dishes over a range of temperatures. The crack spacing is reported and is observed to increase as the temperature approaches the glass-transition temperature of the latex. The data for crack spacing over a range of temperatures is collapsed to a single curve by adapting a scaling for the crack spacing of hard particles. The increase in temperature requires an account to be made for particle deformation and, hence, an increase in particle volume fraction, as well as the temperature dependence of the evaporation rate. The importance of measuring the stress at which films crack is highlighted. Introduction When a layer of fluid containing colloidal particles dries, cracks can appear in the resulting film. This occurs when the particles are rigid and do not deform significantly during the drying process. The cracking mechanism is widely accepted to be due to capillary pressures, developed in the particulate network as solvent evaporates.1,2 In addition to cracking of films on drying of colloidal suspensions, there has been extensive research into fracture pattern formation in rocks due to cooling3 and externally applied stresses,4,5 with analogies made to the crack patterns observed upon drying. What Happens When a Film Dries? When a dispersion is cast in a thin film, it is observed that the film dries nonuniformly with fronts passing horizontally across a film. A picture of a drying film is shown in Figure 1, and on the right-hand side of the figure is a schematic of how the particles are arranging. Particles consolidate into a form of close packing at an edge but remain saturated with water. Continued evaporation from this packed region draws fluid from the bulk solution. This horizontal flux of fluid brings particles into the packed region, propagating the front of close-packed particles through the film. The position of the particle front is shown in Figure 1 as point A. The flow of solvent through the packed particles results in a pressure drop. The pressure profile along the packed particle region is sketched in Figure 1, as Case 1, with the lowest pressure occurring at the edge of the film. In addition, there is a maximum capillary pressure that the bed can support. If the flow of solvent through the packed bed causes this maximum value to be reached, the capillary pressure will level out at this maximum value, shown in Figure 1 as Case 2. The water is stagnant in this region of maximum pressure, and a water-front transition follows the particle front through the bed. The waterfront transition is marked as point B in the picture in Figure 1. To the right of point B, the pressure in the water is at its maximum value and the water-air interface will drop below the particles. To the left of point B, the particles remain fully saturated. In a previous publication, Routh and Russel6 derived the relevant length scales for surface-tension-driven flow in thin * To whom correspondence should be addressed. Tel.: +44 1223 765718. Fax: +44 1223 765701. E-mail:
[email protected]. † Present address: BP Institute and Dept of Chemical Engineering, Cambridge University, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, U.K.
Figure 1. Image of a drying film and a sketch of how the particles are arranging.
evaporating films, such that the lubrication approximation applies. The characteristic vertical length scale is the film thickness, H, and the characteristic horizontal length scale, L, is defined as
L)H
( ) γ 3η0E˙
1/4
(1)
where γ is the liquid-air surface tension, η0 is the dispersion low-shear viscosity, and E˙ is the evaporation rate. The characteristic pressure for flow through the packed particle bed, Pchar, comes from scaling Darcy flow and is given by µL2E˙ /kpH, where µ is the solvent viscosity and kp is the bed permeability. The maximum capillary pressure, Pmax, can be estimated from a spherical cap between three particles as 10γ/R where R is the
10.1021/ie051256m CCC: $33.50 © 2006 American Chemical Society Published on Web 03/08/2006
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Figure 2. Schematic of how the capillary pressure opens a propagating crack.
particle radius.7 More elegantly, White8 and Dunstan and White9 derive the maximum capillary pressure without resorting to a priori assumptions about the geometry. Their expression for the maximum capillary pressure in a bed of spheres, with zero contact angle between the particles and fluid, is Pmax ) 3φγ/ (R(1 - φ)). For random close-packed particles with φ ) 0.64, this reduces to Pmax ) 5.3γ/R, while for spheres crystallized at φ ) 0.74, Pmax becomes 8.5γ/R. To nondimensionalize the capillary pressure, the characteristic pressure generated by flow through the bed, Pchar, is used. Using the value 10γ/R for Pmax, this defines a dimensionless number, Pcap, as
Pcap )
( )
Pmax 10γkpH 20 3γη0 ) ) Pchar 75 E˙ µRL2E˙
1/2R(1
- φm)2 2
µφm H
(2)
where the final expression comes from the definition of the horizontal length scale, eq 1, and the Carmen-Cozeny equation for the bed permeability. The numerical prefactor, 20/75, will change depending on the value of Pmax used. For consistency with our previous publications, we use Pcap as defined in eq 2. When Pcap is large, the maximum capillary pressure is never reached in the bed, water stays at the edge of the film, and the particle-front progression is as fast as is possible. When Pcap is smaller, the water recedes from the edge of the film and the water flux from the bulk of the film is reduced, slowing the particle-front progression. Cracks initiate at the edge of the film and follow the particle front through the film. The dynamics of cracking is not considered in this paper, although it is evident that the cracks follow the particle front, such that the hydrodynamics, controlled by the dimensionless group Pcap, sets the global velocity of cracking. As can be seen in the picture in Figure 1, cracks extend into the wet region of the film and this is very strong evidence for the capillary pressure to be responsible for film cracking. Conclusive proof came from Dufresne et al.,10 who used laserscanning coherent anti-Stokes Raman scattering microscopy to detect water at the position of the crack tip. They discovered that water is present in the film except inside the crack, which is dry. The resulting view is sketched in Figure 2. The capillary pressure, which is negative in value, acts on the face of the crack, pulling the film apart. An alternative way to envisage the process is that inside the crack there is no water and so the higher atmospheric pressure pushes on the crack face, causing crack propogation. Crack Spacing. A common observation is that, for any given experiment, the spacing between cracks is more or less constant.
There have been many theories proposed for the crack spacing with a common theme being a balance between the energy consumed by propagating the crack with the energy recovered from elastic relaxation in the vicinity of the crack.11-15 The resulting prediction is that the crack spacing scales with the film thickness. This prediction was not borne out by experimental data from this group, and a different theory has been proposed.16 Instead of treating the drying film as a quasi-one-phase material with a mechanical response, there is an additional stressrelief mechanism available. As well as the elastic relaxation of the particle array, fluid can flow through the particle bed and relieve stress by viscous dissipation. Indeed, as the film opens up, the capillary stress will become zero on the crack face but will remain large and negative in the bulk of the film, and the result is a flow of fluid from the crack to the bulk. The distance the fluid can flow, given the forward velocity of the crack, sets the crack spacing.16 This distance, given the symbol X, was derived as
X)
( )
2 3 10γ kpH 20R(1 - φm) 3η0γ ) R µLE˙ 75µφm2 E˙ 3
1/4
(3)
where the Carmen-Cozeny equation was used for the bed permeability. The success of this argument was shown in a previous publication with crack spacing data scaled with the length scale X and plotted against Pcap, collapsing to a single line given by
y/X ≈ Pcap-0.8
(4)
for a range of materials, particle sizes, film thicknesses, and evaporation rates.16 Control of the horizontal drying was achieved by performing the experiments in Petri dishes. The packed region of the film formed initially in the center of the dish, and the drying front and cracks propagated outward. It was observed that the crack spacing remained more or less constant, and there are two reasons for this: the radius of the Petri dish is much larger than the crack spacing such that the radial induced variation in spacing is minimal and, in some cases, the cracks bent as they progressed outward, keeping the constant spacing. This Petri dish method is far easier than trying to assign a crack spacing to a film with a broad spacing distribution, which is observed in a free geometry, such as is shown in Figure 1. The scaling is close to y/X ≈ Pcap-1 , which implies that yPcap/X is a constant or that y simply scales with the horizontal length scale L, the characteristic length scale for surface-tensiondriven flow in the fluid part of the film. Why a length scale derived for fluid flow should be controlling the crack spacing becomes clear when considering the governing equations. In the solidified region of the film, continuity and Darcy flow are represented by
∇‚u ) 0
(5)
µ u kp
(6)
∇p )
The characteristic vertical distance in the film is the film thickness, H, and the characteristic vertical velocity is the evaporation rate, E˙ . The characteristic horizontal velocity is imposed on the film by the horizontal drying front and has a magnitude LE˙ /H. Scaling the continuity equation provides the
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characteristic horizontal length scale, even in the packed bed, as the capillary length L, as is observed experimentally. The propagation of cracks is observed to be in a stepwise fashion, with the crack jumping forward and then waiting before jumping again.10 The explanation for this is as follows: the capillary pressure builds up and the crack yields at some critical pressure presumably given by an energy balance. The excess water, liberated by the crack, flows into the bulk, and this has a time associated with it. Evaporation then causes the capillary pressure to build up again and the crack propogates, repeating the process. At high temperatures, films are homogeneous and coherent with no cracks. At temperatures well below the glass-transition temperature of the constituent polymer, films crack and the crack spacing is temperature dependent. In this paper, the effect of temperature on crack spacing and the reason for this temperature dependence is examined. Materials and Methods Materials. Latices were synthesized via a surfactant-free emulsion polymerization. Styrene monomer (Acros Organics) was filtered through aluminum oxide, and butyl acrylate monomer (Acros Organics) was vacuum distilled to remove inhibitors. Initially, distilled water was placed in a four-neck round-bottom glass reactor fitted with a stirrer and reflux condenser and heated to 60 °C under nitrogen in a water bath. The monomers were mixed, and half of the mixture was added with the initiator potassium persulfate to produce a seed for the polymerization process. Last, the rest of the monomers were added. The reaction was allowed to proceed overnight to ensure complete conversion. Latices were cleaned by dialysis against deionized water for a week with two changes of water per day. Particle sizes were determined by dynamic light scattering (Brookhaven’s Zeta Plus). Two different batches of particles were made. Despite identical recipes, slightly different particle sizes were obtained and, more strikingly, the minimum film formation (MFT) points of the two latices were different. Ludox was used as received from Sigma Aldrich. Minimum Film Formation (MFT) Bar. The crack point MFT of a latex is taken as the temperature where it starts to form a homogeneous film upon drying. The cloudy-clear MFT is the temperature where the particles start deforming, such that the voids between them are too small to scatter light, producing a transparent film. The cloudy-clear point is generally at a lower temperature than the crack point, and the film will display cracks at the cloudy-clear transition. These two transition temperatures were obtained using a MFT temperature gradient bar which consists of an aluminum plate with heaters along its length. The temperature across the bar ranged from room temperature to 30 °C above the crack point MFT of the polymers used. The temperature at specific points was measured using a thermocouple with an accuracy of approximately (0.5 °C. Latices were cast onto a glass plate, preequilibrated to the temperature of the gradient bar, using an adjustable micrometer film applicator, and the thicknesses of the wet films were kept approximately constant. MFT measurements were carried out 5 h after film casting. Evaporation Rate. To minimize the disturbance of the latex samples during drying, the evaporation rate of pure water was determined across the temperature range. Since the latex used is dilute, it is justifiable to take the evaporation rate of pure water to be the same as the evaporation rate of water from the latex.17 A known mass of water was allowed to evaporate in an oven at specific temperatures, with the change of mass
Figure 3. Variation of evaporation rate of water with temperature. Table 1. Properties of Latices Used in Temperature Experiments material
particle diameter (nm)
particle weight %
crack point MFT (°C)
cloudy-clear MFT (°C)
polymer polymer silica
460 445 30
14.45 13.32 30
80 70
50 42
determined gravimetrically at regular time intervals. Although the humidity was not directly controlled, it remained approximately constant throughout the experiment and was estimated to be 50% relative humidity at 25 °C. Drying Experiments. The samples were dried in an oven with a temperature range from room temperature to 200 °C. The oven was allowed to equilibrate to the required temperature for at least half an hour, so that the temperature was stable prior to samples insertion, which consisted of wet latex dispersions in Petri dishes. The mass of each sample was kept approximately constant between experiments, so that the film thicknesses of each sample were approximately constant. The samples were left in the oven for 3-5 h, depending on the temperature, and the oven door was kept closed for the duration of each experiment to avoid fluctuations in temperature, which was checked at 15 min intervals and found to fluctuate by no more than (2 °C. Crack spacings were measured using Vernier calipers, and 10 measurements of the crack spacing were made from each sample. The raw data was nondimensionalized using the scalings described in eqs 2 and 3. The viscosity of water, µ, was taken as temperature dependent; the low-shear viscosity of the dispersion, η0, was taken as 1 Ns/m2 [6]; and the volume fraction of polymer at the time of cracking, φ, was initially taken as 0.64 but was subsequently predicted from a particle deformation model. Results Materials. The properties of the latices used in this experiment are given in Table 1. Colloidal silica was used as a control material since silica does not deform in the temperature range used. MFT Bar. The crack point MFT and cloudy-clear MFT of the respective polymer dispersions were obtained from the MFT bar, as shown in Table 1. It was shown previously that the cloudy-clear MFT is a function of time; however, the rate of change is very slow, at about 1 °C/day.18 Evaporation Rate. Figure 3 shows the evaporation rate, as a function of temperature. From this graph, the evaporation rate
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Discussion
Figure 4. Crack spacing variation with temperature.
In Figure 5, we have assumed the volume fraction of particles in the final film and the failure stress of the material are constant, at 0.64 and 10γ/R, respectively. This assumption is only valid when the particles are not deforming; however, when the temperature is increased, the particles will start to deform at the MFT point. The particle volume fraction will increase from 0.64 to a maximum of 1, depending on the extent of deformation, which in turn is dependent on the temperature. For the crack spacing scaling to be valid across the temperature range, the change in particle volume fraction and failure stress of the material with respect to temperature must be taken into account. To calculate the particle volume fraction at any temperature, the stress in the material must first be estimated. Tirumkudulu and Russel2 recently reported cantilever experiments where the stress in films at failure was found to scale with the capillary stress, although it varied with the particle size and film thickness. As a first estimate, a failure stress of 10γ/R is used, and the volume fraction in the film at any temperature can then be predicted using a particle deformation model.19 The assumptions in the process model are that the film is contracting, in the vertical direction, at a rate proscribed by the evaporation rate, and that water is held at the top surface of the film with the pressure mediating itself to maintain the water level. The dimensionless vertical stress in the film, σ j , is given by
σ j)
28σR 3νγφm2
(7)
where ν is the number of contacts for each particle, taken as 5; σ is the stress in the film, taken as 10γ/R; and φm is the particle volume fraction for nondeformed particles in the dried film, taken as 0.64. This results in a maximum value for σ j of ∼29. The film collapses vertically, and the maximum stress occurs at a time t, which is the solution to
[(
σ j ) λh 1 + Figure 5. Partial collapse of crack spacing data at various temperatures for two different latices.
at any temperature between 30 and 80 °C can be predicted, providing the humidity remains constant. Drying Experiments. Raw data of the crack spacing against temperature is shown in Figure 4. Although the data is scattered, it can be seen that there is not much change in crack spacing for both latices with temperature from 35 °C until their respective cloudy-clear MFT points are reached. At the polymer cloudy-clear MFT points, there is an increase in the crack spacing, until the crack point MFT is reached. Above the crack point MFT, no cracks were observed in the samples. Data from the experiment is plotted in Figure 5 using the scaling derived in our previous publication,16 where the crack spacing, y, is nondimensionalized against the horizontal length scale, X, and the change of evaporation rate with temperature is calculated from Figure 3. It can be deduced from Figure 5 that the scaling only manages to partially collapse the data, since the different Tg of the polymers has not yet been considered in the scalings. It is noticeable that the higher temperature data is less well-collapsed, since this is the region where the glasstransition temperature will be playing a large effect. The lowtemperature data is all below the polymer glass-transition temperature, and this seems to follow the same curve for the two different latices.
)
]
9th 171th2 21 11th 2 357 3 + ht ht + 1+ ht 16 640 96 20 640
(
)
(8)
where the dimensionless parameter λh is the ratio of time for viscous collapse of the particle to the time for evaporation.
λh )
E˙ Rηo γH
(9)
Once the time for film failure is calculated, the strain in the film simply follows as that dimensionless time, and the volume fraction then follows directly.
) ht and φ )
φm (1 - )
(10)
The value of the polymer viscosity, ηo, at the glass transition is taken to be 1012 Ns/m2,20 and the change of ηo with respect to temperature is calculated using the temperature shift factor, aT, from the Williams, Landel, and Ferry (WLF) equation.20
ηo(T) ) aTηo(Tg) log10 aT )
C1(T - Tg) C2 + T - T g
(11) (12)
where C1 and C2 are material constants, taken as -17.1 and 51.4 °C, respectively, and Tg is the glass-transition temperature of the polymer, which should be higher than the cloudy-clear
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The glass-transition temperature in the wet state is difficult to determine since the particles are affected by water. There could well be a collapse of the polymer particles due to the hydrophobicity of the polymer chains and, consequently, a rise in the glass-transition temperature in the wet state. The value of the glass-transition temperature used for each polymer was determined such that the volume fraction of particles in the dried film at the crack point was 1. This resulted in a glass-transition temperature of 63 and 49 °C for the 460 and 445 nm latices, respectively. The cracking mode considered here is for cracks that follow a particle compaction front laterally across a film. This is the case for the experiments reported here, although it is likely that other cracking modes exist. Indeed, for Pcap , 1, different regimes have been observed, with cracks forming spontaneously in films and propagating rapidly throughout the film. Conclusions Figure 6. Crack spacing data from temperature experiments, scaled to account for the change in particle volume fraction with temperature. Also shown is crack spacing data from experiments carried out at room temperature.
MFT. Using eqs 7-12, the change in particle volume fraction in the dried latex film with respect to temperature is obtained. The Tg is chosen such that the volume fraction is just below 1 at the crack point MFT. Figure 6 shows the data from the temperature experiment plotted using the scaling, and the scaling seems to collapse the drying data from the two different latices. Figure 6 also shows data for particles dried at room temperature that did not deform. These data has been published previously16 and are shown here to demonstrate the collapse of the higher-temperature data to the same relation. As shown previously, the data for hard particles collapses to a line given by (y/X) ≈ Pcap-0.8. The higher-temperature data accounts for the change in volume fraction and assumes that the failure stress remains constant. Doing this simply moves the data points along a line of (y/X) ) Pcap-1, because the volume fraction and stress appear on both the horizontal and vertical axes. Hence, we simply spread the data points out across Figure 6. The value for stress, σ, on the vertical axis corresponds to the stress at film failure, and this is likely to be a strong function of temperature. The stress on the horizontal axis corresponds to the recession of water from the edge of the film, and here, the estimate of 10γ/R seems reasonable. Hence, accounting for the temperature dependence of the failure stress will move the higher-temperature data downward, and this will move the data closer to the hard-particle curve. It should be noted, however, that the failure stress is likely to vary by a factor of 2, and this factor is not significant on the scale used in Figure 6; hence, we see the visually appealing collapse of the data. Tirumkudulu and Russel2 demonstrated a method to measure the failure stress using a cantilever technique. However, assuming the stress at which the film breaks is the maximum capillary pressure and is independent of temperature is the simplest approach, but to the authors’ best knowledge no systematic analysis of failure stress as a function of temperature has been carried out. The cantilever technique, as used by Peterson et al.21 and Tirumkudulu and Russel,2 could be adopted for temperature studies, and these will give the correct stress to use for the scaling on the vertical axis of Figure 6.
Latex films dried at low temperatures display cracks with a constant spacing. As the temperature is increased so is the crack spacing until the crack point temperature is reached, above which no cracks are observed. The stress driving cracking is the capillary pressure generated in the fluid as it evaporates. The crack spacing is set by the distance that solvent can flow when the capillary pressure is relieved by the propagation of a crack. At higher temperatures, the failure stress increases, and this is the reason for an increase in crack spacing. Acknowledgment This work was supported by EPSRC through Grant GR/ S05885/01. The authors have been lucky to benefit from an annual meeting of the Cracking Club in Yale. This meeting was initiated by Bill Russel. Literature Cited (1) Tirumkudulu, M. S. Cracking in drying latex films. Langmuir 2005, 21 (11), 4938-4948. (2) Tirumkudulu, M. S.; Russel, W. B. Role of capillary stresses in film formation. Langmuir 2004, 20 (7), 2947-2961. (3) Hull, D.; Craddock, B. D. Simulation of prismatic cracking of cooling basalt lava flows by the drying of sol-gels. J. Mater. Sci. 1999, 34, 57075720. (4) Adda-Bedia, M.; Amar, M. B. Fracture spacing in layered materials. Phys. ReV. Lett. 2001, 86 (25), 5703-5706. (5) Bai, T.; Pollard, D. D.; Gao, H. Explanation for fracture spacing in layered materials. Nature 2000, 403, 753-756. (6) Routh, A. F.; Russel, W. B. Horizontal drying fronts during solvent evaporation from latex films. AIChE J. 1998, 44 (9), 2088-2098. (7) Routh, A. F.; Russel, W. B.; Tang, J.; El-Aasser, M. S. Process model for latex film formation: Optical clarity fronts. J. Coat. Technol. 2001, 73 (916), 41-48. (8) White, L. R. Capillary rise in powders. J. Colloid Interface Sci. 1982, 90 (2), 536-538. (9) Dunstan, D.; White, L. R. A capillary pressure method for measurement of contact angles in powders and porous media. J. Colloid Interface Sci. 1986, 111 (1), 60-64. (10) Dufresne, E. R.; Greenblatt, N. A.; Ashmore, J.; Wang, D. Y.; Dinsmore, A. D.; Chang, J. X.; Xie, X. S.; Hutchinson, J. W.; Weitz, D. A. Flow and fracture in drying nanoparticle suspensions. Phys. ReV. Lett. 2003, 91 (22), 4501-4504. (11) Bordia, R. K.; Jagota, A. Crack growth and damage in constrained sintering films. J. Am. Ceram. Soc. 1993, 76 (10), 2475-2485. (12) Allain, C.; Limat, L. Regular patterns of cracks formed by directional drying of a colloidal suspension. Phys. ReV. Lett. 1995, 74 (15), 2981-2984. (13) Komatsu, T. S.; Sasa, S.-i. Pattern selection of cracks in directionally drying fracture. Jpn. J. Appl. Phys. 1997, 36, 391-395.
Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 7001 (14) Shorlin, K. A.; Bruyn, J. R. d.; Graham, M.; Morris, D. W. Development and geometry of isotropic and directional shrinkage crack patterns. Phys. ReV. E 2000, 61 (6), 6950-6957. (15) Jagla, E. A. Stable propogation of an ordered array of cracks during directional drying. Phys. ReV. E 2002, 65 (4), 6147-6153. (16) Lee, W. P.; Routh, A. F. Why do drying films crack? Langmuir 2004, 20, 9885. (17) Keddie, J. L. Film Formation of Latex. Mater. Sci. Eng. 1997, R21 (3), 101-167. (18) Lee, W. P.; Routh, A. F. Time evolution of transition points in drying latex films. 2005, submitted for publication. (19) Routh, A. F.; Russel, W. B. A process model for latex film formation: Limiting regimes for individual driving forces. Langmuir 1999, 15, 5 (22), 7762-7773.
(20) Willliams, M. L.; Landel, R. F.; Ferry, J. D. The temperature dependence of relaxation mechanisms in amorphous polymers and other glass forming liquids. J. Am. Chem. Soc. 1955, 77 (14), 3701. (21) Petersen, C.; Heldmann, C.; Johannsmann, D. Internal stresses during film formation of polymer latices. Langmuir 1999, 15, 7745-7751.
ReceiVed for reView November 14, 2005 ReVised manuscript receiVed February 3, 2006 Accepted February 3, 2006 IE051256M
