Langmuir 2008, 24, 1721-1730
1721
Generalized Hertzian Model for the Deformation and Cracking of Colloidal Packings Saturated with Liquid William B. Russel,* Ning Wu, and Weining Man† Department of Chemical Engineering, Princeton UniVersity, Princeton, New Jersey 08544 ReceiVed August 26, 2007. In Final Form: October 14, 2007 The process of drying colloidal dispersions generally produces particulate solids under stress as a result of capillary or interparticle forces. The derivation of a constitutive relation on the basis of Hertzian contact mechanics between spheres provides a model for quantitatively predicting the conditions under which close-packed colloidal layers form continuous void-free films or homogeneous porous films or crack under tensile stresses.
I. Introduction The process of drying colloidal dispersions (i.e., evaporating the liquid) to create particulate solids or continuous polymer films is common to a range of technologies (e.g., latex film formation,1,2 casting of magnetic tapes,3 highly porous coatings on ink-jet papers4 or sol-gel glasses,5 high index coatings on eyeglasses,6 encapsulation of vitamins in beads,7 formation of synthetic opals,8-10 spray deposition of thin film oxide fuel cells,11 and conventional photographic film12). The objective varies but is generally to create a layer of specified thickness and controlled porosity with permeability, strength, and optical properties appropriate for the application. The particles most frequently employed are polymer latices or inorganic oxides (e.g., silica, alumina, or zirconia), and the fluid is normally water or less often a volatile organic. The processing raises a number of interesting and difficult issues because of the conflicting constraints for successful film formation and performance properties as dictated by the applications. As the fluid, assumed hereafter to be water, evaporates, the concentration of particles increases, but the dispersion remains a thermodynamic fluid until gelation, freezing, or a glass transition is encountered. If the process is slow enough, that is, the rate of evaporation relative to diffusion across the liquid layer or droplet is sufficiently slow and the interparticle attractions are not abnormally strong, then dispersions will eventually reach random or ordered close packing throughout the film.13 Further evaporation pulls the menisci between particles at the air-water interface into the film, generating negative capillary pressure that puts the * Corresponding author. E-mail:
[email protected]. † Current address: Department of Physics, New York University, 4 Washington Place, New York, New York 10003. (1) Winnik, M. A. In Emulsion Polymerization and Emulsion Polymers; Lovell, P. A., El-Aasser, M. S., Eds.; Wiley: New York, 1997; pp 467-518. (2) Keddie, J. L. Mater. Sci. Eng. R 1997, 21, 101-170. (3) Martinez, C. J.; Lewis, J. A. J. Am. Ceram. Soc. 2002, 85, 2409-2416. (4) Wedin, P.; Martinez, C. J.; Lewis, J. A.; Daicic, J.; Bergstrom, L. J. Colloid Interface Sci. 2004, 272, 1-9. (5) Brinker, J. C.; Scherer, G. W. Sol-Gel Science: The Physics and Chemistry of Sol-Gel Processing; Elsevier: New York, 1989. (6) Amalvy, J. I.; Percy, M. V.; Armes, S. P. Langmuir 2001, 17, 4770-4778. (7) Clegg, J. B. U.S. Patent 3,445,563, 1969. (8) Jones, J. B.; Sanders, J. V.; Segnit, E. R. Nature 1964, 204, 990-991. (9) Darragh, P. J.; Gaskin, A. J.; Terrell, B. C.; Sanders, J. V. Nature 1966, 209, 13-14. (10) Saunders, A. E.; Shah, P. S.; Sigman, M. B.; Hanrath, T.; Hwang, H. S.; Lim, K. G.; Johnston, K. P.; Korgel, B. A. Nano Lett. 2004, 4, 1943-1948. (11) Pham, A.-Q.; Lee, T. H.; Glass, R. S. 196th Meeting of the Electrochemical Society, Honolulu, HI, October 17, 1999. (12) Huang, C. C. J. Math. Anal. Appl. 1994, 187, 663-675. (13) Sperry, P. R.; Snyder, B. S.; O’Dowd, M. L.; Lesko, P. M. Langmuir 1994, 10, 2619-2628.
dispersion under tension. This tension translates into an interparticle force of O(aγ) (a is the particle radius and γ is the air-water surface tension) that easily overcomes any interparticle repulsion of O(kT/a) because a2γ/kT ≈ 105 for a ≈ 100 nm and ∼10 even for a ≈ 1 nm. Close packing results in direct contacts between neighboring spheres, 6 for random packing14 and 12 for perfect hexagonal order, that immobilize the particles and create a colloidal solid. Modeling the deformation of close-packed particles in response to atmospheric pressure at the free surface and tangential forces due to binding to the substrate requires a relationship between stress and deformation, or the rate of deformation, that reflects both the nature of the packing and the stresses within the particles. The classical Biot formulation15-17 provides such constitutive equations for linearly elastic porous media but does not capture the intrinsically nonlinear deformation of spheres in contact. Previously, we18 employed an ad hoc but systematic volume average of the microscopic stresses to relate force to deformation and the rate of deformation for viscoelastic spheres and derive a stress tensor that depends quadratically on the strain. This provided a basis for calculating, at least semiquantitatively, many of the macroscopic properties addressed here. These predictions have stimulated a number of quantitative measurements19-21 that demand a more quantitative approach that takes advantage of rigorous results for the contact mechanics between spheres subject to external and surface forces. In the following, we assume the particles to be spheres that respond to contact forces either elastically, as described by Hertz,22,23 or viscously, as generalized subsequently.24 Dispersion or van der Waals forces are incorporated as contact adhesion due to particle-air or particlewater interfacial tension in conformity with the results of Hughes and White25 and Johnson, Kendall, and Roberts,26 as extended by Johnson and Greenwood.27 The constitutive equations are then coupled with suitable conservation equations to address prototypical problems relevant to film formation. The first question is which driving forces (14) Donev, A.; Cisse, I.; Sachs, D.; Variano, E.; Stillinger, F. H.; Connelly, R.; Torquato, S.; Chaikin, P. M. Science 2004, 303, 990-993. (15) Biot, M. A. J. Appl. Phys. 1941, 12, 155-164. (16) Biot, M. A. J. Appl. Phys. 1955, 26, 182-185. (17) Biot, M. A. J. Appl. Phys. 1956, 27, 459-467. (18) Routh, A. F.; Russel, W. B. Langmuir 1999, 15, 7762-7773. (19) Martinez, C. J.; Lewis, J. A. Langmuir 2002, 18, 4689-4698. (20) Peterson, C.; Heldmann, C.; Johannsmann, D. Langmuir 1999, 15, 77457751. (21) Tirumkudulu, M. S.; Russel, W. B. Langmuir 2004, 20, 2947-2961. (22) Hertz, H. J. Reine Angew. Math. 1881, 92, 156. (23) Hertz, H. Gesammelte Werke 1895, 1, 155. (24) Matthews, J. R. Acta Metal. 1980, 28, 311-318.
10.1021/la702633t CCC: $40.75 © 2008 American Chemical Society Published on Web 01/16/2008
1722 Langmuir, Vol. 24, No. 5, 2008
particle-water, air-water (capillary pressure), or particle-air surface tensionscontrols the formation of a void-free film for latices with a glass-transition temperature in the vicinity of the process temperature. A consideration of deformation normal to the plane of the film provides a process map expressed in terms of the two relevant dimensionless groups that distinguishes wet sintering, capillary consolidation, and dry sintering regimes.18 Alternatively, the same information can be displayed as the time to form the film (i.e., close all voids) as a function of the process temperature relative to the glass-transition temperature.13 Second is the question of cracking for elastic particles at temperatures below the glass-transition temperature. Our starting point is the energy criterion due to Griffith,28,29 which requires the calculation of the relaxation of stress fields, both normal and parallel to the substrate, upon opening of a crack. For this purpose, we linearize the constitutive equations and employ a thin film approximation.30 The outcome is a lower bound on the capillary pressure necessary for the first crack as well as a prediction of the decreased spacing between subsequent cracks as the capillary pressure increases further, analogous to treatments of thin elastic films.31 Coupling the minimum capillary pressure for cracking with the maximum capillary pressure sustainable with the air-water interface at the surface of the film then yields a critical film thickness below which cracking will not occur, as measured in earlier32 and recent33 experiments. Because data now emerging34 imply that pre-existing flaws play an important role in the cracking of colloidal films, as in linearly elastic materials,35 we also analyze the stress fields around circular and slit flaws. In the following text, we first state the conservation equations for particles, fluid, and momentum and develop the requisite constitutive equations relating stress and strain. The development assumes homogeneity in the normal direction but accounts for variations in the lateral directions, though not fluctuations about the local mean. These equations are then solved to construct the process map for film formation, delineating the conditions under which individual driving forces control the process. Derivations of the stress and strain fields associated with a single infinite crack, parallel infinite cracks, and a single flaw or finite crack then provide the elastic energy recovered and estimates of the capillary pressure required to open flaws or cracks and the minimum film thickness below which films should not crack.
II. Conservation Equations for Thin Colloidal Films This analysis begins after excess water has evaporated to leave a saturated layer of close-packed colloidal particles with uniform initial thickness H (Figure 1). As evaporation continues at a rate E (m/s), the volume fraction φ of particles with radius a and the dimensionless displacements u ) (u, V, w) of the particles and positions x ) (x, y, z) in the packing, scaled on H, are governed (25) Hughes, B. D.; White, L. R. Q. J. Mech. Appl. Math. 1979, 32, 445-471. (26) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301-315. (27) Johnson, K. L.; Greenwood, J. A. J. Colloid Interface Sci. 1997, 192, 326-333. (28) Griffith, A. A. Philos. Trans. R. Soc. London, Ser. A 1920, 221, 163. (29) Griffith, A. A. In First International Congress of Applied Mechanics; Biezeno, C. B., Burgers, J. M., Eds.; J. Waltman: Delft 1924; p 55. (30) Xia, Z.; Hutchinson, J. W. J. Mech. Phys. Solids 2000, 48, 1107-1131. (31) Suo, Z.; Hutchinson, J. W. Int. J. Solids Struct. 1989, 25, 1337-1353. (32) Chiu, R. C.; Cima, M. J. J. Am. Ceram. Soc. 1993, 76, 2257-2264, 2769-2777. (33) Singh, K. B.; Tirumkudulu, M. S. Phys. ReV. Lett. 2007, 98, 218302-1-4. (34) Man, W.; Russel, W. B. In review. (35) Lawn, B. Fracture of Brittle Solids, 2nd ed.; Cambridge University Press: New York, 1993.
Russel et al.
Figure 1. Coordinate system for a thin film with the substrate shaded.
by conservation equations for particles and fluid, respectively, with the latter augmented by D’Arcy’s law,
∂u ∂φ )0 + ∇‚ φ ∂t ∂t ∂(1 - φ) ∂u k + ∇‚ (1 - φ) ∇2p ) 0 ∂t ∂t HµE
(
φ(x,0) ) φo
u(x,0) ) 0
( ) )
p(x,0) ) 0
∂uz(1,t) ) -1 ∂t (1)
with the dimensionless time t scaled on H/E but the capillary pressure unscaled. The close-packed particles do not move laterally but will deform to fill more of the volume whereas water with viscosity µ leaves by evaporation from the surface and flow due to capillary pressure gradients, as controlled by the permeability k of the particle packing. Integrating both equations over the thickness of the layer, assumed to be spatially uniform, yields averaged equations, which for small displacements or strain E ) (∇u + ∇uT)/2 , 1 take the form
∂ (φ + φo〈tr E〉) ) 0 ∂t φ(0) ) φo
∂〈tr E〉 k ∇ 2〈p〉 ) -1 ∂t HµE s 〈E〉(0) ) 0
(2)
for a saturated layer with 〈 〉 ) ∫10 ( ) dz and ∇s2 ) ∂2/∂x2 + ∂2/∂y2 and φo referring to close packing. The equation is valid to leading order in the strain but assumes an initially uniform thickness H and aHµE/γk , 1 so that the pressure drop across the thickness of the layer is minimal. As formulated above, the balance on water is relevant only when capillary pressure controls the deformation process as described below. The deformation or strain of the particle packing is controlled by the stresses σ through similarly averaged momentum equations. At the free surface, the normal and tangential stresses must vanish (i.e., σ‚ez|z)1 ) 0) and the balance of stress or transport of momentum in the plane of the film requires
∇‚〈σ〉 ) σ‚ez|z)0
(3)
The two differential balances for momentum in the plane of the film plus the boundary condition on the normal stress offer a complete set of equations for the average displacements, once constitutive equations and closure relations among the average and interfacial stresses or strains are chosen. Note that this approach is not analogous to the lubrication approximation in fluid mechanics, which is a proper asymptotic expansion based on very different length scales in the normal and transverse directions. Here, as we will see later, the two length scales are virtually identical, so careful selection of the closure is important.
III. Constitutive Equations with Hertzian Contact To construct a constitutive equation relating σ to , we treat the deformations between the initially point contacts between spheres through classical theories generalized or extended over
Deformation and Cracking of Colloidal Packings
Langmuir, Vol. 24, No. 5, 2008 1723
Figure 2. Spheres of radius a in contact under body forces (F aligned with center-to-center vector n.
time.27 Accordingly, equal and opposite body forces (F that press two spheres of radius a into contact along the line of centers n or interfacial forces that pull them together at the contact point26 (Figure 2) generate elastic forces that limit the decrease in the center-to-center distance 2n‚E‚n < 0 scaled on the radius as
F + 2.83πaγpa/wn ) -
G 16 2 (-n‚E‚n)3/2n a 3 2(1 - ν)
(4)
with γwa/p being the water-particle or air-particle surface energy, G being the shear modulus, and ν being Poisson’s ratio. The assumptions are that the amplitude of the strain is small (i.e., |E| , 1) and the range zo of the interparticle forces is sufficiently short relative to the particle radius such that27
λ)
( )( ) 9aγpa/w2
1/3
2πzo3
2(1 - ν) G
2/3
.1
(5)
By exploiting the analogy between Hookian elasticity and Stokes flow, Matthews24 derived an analogous relationship between force and rate of deformation for viscous spheres that we generalize to
F + 2.83πaγpa/wn ) -
16 a2ηE d (-n‚E‚n)3/2n 3 H dt
(6)
Combining these by assuming the polymer to behave as a viscoelastic fluid with a single relaxation time (i.e., a Maxwell fluid) leads to
G 16 F + 2.83πaγpa/wn ) - a2 × 3 2(1 - ν) t t′ - t d (-n‚E‚n)3/2n dt′ (7) exp -∞ τ dt′
∫
( )
with τ ) 2ηE(1 - ν)/HG. This relationship between force and deformation interpolates between the prediction for elastic or viscous spheres subject to either external or interfacial forces in a manner that is consistent with the limit of λ . 1 from the treatment of Maugis and Dugdale.27 Clearly, this ignores any tangential forces, assuming perfect slip at the contact point. Although asymptotically correct, this may introduce error at finite deformations. The microstructure of the initial packing can be represented by an isotropic distribution of contacts between close-packed neighbors gc ) N/4π (contacts per solid angle) with N ) 6 (12) the number of neighbors in contact in a random (hexagonally ordered) packing. The stress tensor averaged over a representative volume V in such a saturated packing of spheres can be constructed from the mechanical dipoles induced in the individual particles by interactions with neighbors. As derived by Batchelor36,37 for colloidal particles in a viscous fluid
σ)
1 V
∫V (-pδ + τ) d3r
(8)
and δ is the identity tensor and lengths scaled on the particle
Figure 3. Deformation of viscoelastic thin film subject to negative dimensionless capillary pressure as a function of evaporation time scaled on relaxation time, 1/τ.
radius. For the saturated packing, the pressure is locally uniform, and τ represents the viscoelastic stress in the particles. Because 0 ) ∇‚(-pδ + τ) ) ∇‚τ with a uniform fluid pressure, the divergence theorem recasts the portion of the volume integral within a particle of volume Vo into a surface integral as
∫V
o
∫V
τ d3r )
o
∇r ‚(τr) d3r )
∫A
rτ‚n d2r
(9)
o
Recognizing that r ) Ro + n, with Ro being the center of the reference sphere, permits the transformation
∫A
o
∫A
rτ‚n d2r ) Ro
o
τ‚n d2r +
∫A
nτ‚n d2r
(10)
o
At equilibrium, the forces exerted on any particle by its neighbors must balance, so the first term integrates to zero. The second term captures the total dipole S induced in the particle by the contact forces from its neighbors, which for one pair of spheres is
S ) a3
∫A
nτ‚n d2r ) anF
(11)
o
The bulk stress in the suspension then follows from the ensemble average of the interactions of all particles with their neighbors as
σ ) -pδ +
3φgc 4πa
3
3φN nF d2n ∫ S d2n ) -pδ + (4πa) 2∫
t t′ - t 1 ) -pδ + G h exp × π -∞ τ d (|n‚E‚n|1/2n‚E‚n)nn d2n dt′ dt′
∫
∫
( )
(12)
with G h ≡ φNG/2π(1 - ν). Earlier work in the mechanics literature38,39 derived effective moduli for aggregates of frictionless elastic spheres via similar approaches. Those approaches as well as that above assume affine deformation of the particle centers, which is correct for perfect crystals but will introduce some error into random close packings. Jenkins and co-workers40 found those predictions to be insensitive to fluctuations and the (36) Batchelor, G. K. J. Fluid Mech. 1970, 41, 545-570. (37) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1992. (38) Digby, P. J. J. Appl. Mech. 1981, 48, 803-808. (39) Walton, K. J. Mech. Phys. Solids 1987, 35, 213-226. (40) Jenkins, J. T.; Johnson, D. L.; La Ragione, L.; Makse, H. J. Mech. Phys. Solids 2005, 53, 197-225.
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density or number of neighbors for the bulk modulus but to err by as much as 30% for the shear modulus. This should be kept in mind when comparing with experimental results. For the prototypical problems cited in section I, the strain of interest has two parts: a uniaxial compression normal to the plane of the film plus an instantaneous elastic response to the opening of a crack at time t. Thus, E ) -oezez + E′ with o g 0 such that
|n‚E‚n|3/2 ) [o(ez‚n)2 - n‚E′‚n]3/2 =
{
o3/2(ez‚n)3 1 -
3 n‚E′‚n + ... 2 (e ‚n)2 o
z
}
(13)
Substitution into the equation above for the stress and addition of the isotropic pressure leaves t t′ - t 1 G h -∞ exp nn|ez‚n| × π τ d 3 3/2(ez‚n)2 - o1/2n‚E′‚n + ... d2n dt′ (14) dt o 2
σ ) -pδ -
{
∫
( )∫
}
The surface integrals look formidable but are easily evaluated (Supporting Information) to determine the leading-order stresses as
(
σo ) - po - 0.71
φNγ 1 h (δ + 3ezez) × δ- G a 6
)
∫
t
3/2 t′ - t do dt′ (15) τ dt′
( )
exp -∞
The linearized perturbations at short times, for which the elastic response dominates, are
1 1 h 1/2tr(Ej′) (δ + ezez) + G h 1/2Ej′ σ′ ) - p′ - G 8 o 4 o ′xx ′xy 2′xz with Ej′ ) ′yx ′yy 2′yz 2′zx 2′zy 2′zz
[
]
(
)
pcap ) - G h 2 =3
∫0
o
exp
(
)
′ - o 1/2 ′ d′ τ
G h o3/2 2 1 + o/τ 3
(17)
the second line represents an accurate approximation to the numerical integration (Figure 3), motivated by the limits of large and small τ. Regardless of the balance between the elastic and viscous components of the response, the strain associated with the capillary pressure generates tensile stresses in the plane of the film,
σoxx
)
σoyy
1 h ) -pcap - G 6 3 ) - pcap > 0 4
3/2 t′ - t do dt′ exp -∞ τ dt′
∫
( )
t
(18)
The actual stresses in the particles differ from these overall stresses, which are averaged over fluid and particles, as
φ〈σ p〉 ) 〈σ〉 + (1 - φ)pcapδ w φ〈σ po zz 〉 ) (1 - φ)pcap < 0 po φ〈σ po xx 〉 ) φ〈σ yy 〉 )
(41 - φ) p
cap
>0
(19)
Thus, the stresses within the particles are compressive in the normal direction to balance the negative capillary pressure but tensile in the plane of the film because the substrate prevents possible lateral expansion with an unconstrained film. Note that the layer is in tension laterally as a result of capillary pressure even if the packing is rigid and incapable of being compressed normal to the substrate! The negative capillary pressure seeks to shrink the layer isotropically but is unable to do so because of the constraining substrate. This is significant because tensile stresses in the plane are capable of promoting cracking, which then allows lateral shrinkage.
V. Modes of Viscoelastic Film Formation (16)
Note that compression normal to the substrate produces an anisotropic solid that relaxes more easily in the plane of the substrate than in the normal direction (i.e., σ′xz/′xz > σ′xy/′xy and σ′zz/′zz > σ′xx/′xx ) σ′yy/′yy.
IV. Stress Fields in a Film That Is Compressed Homogeneously by Capillary Pressure Now we turn to a saturated film of close-packed spheres with initial thickness H compressed homogeneously by a negative capillary pressure po ) pcap. For an infinite film, the substrate prevents lateral expansion, leaving the only relevant momentum equation as that normal to the substrate with σozz(1,t) ) 0 because the normal stress at the free surface must vanish. Because there are no gradients in the plane, the normal stress must be constant to satisfy the z-momentum equation, dictating σozz(z ,t) ) 0, which relates the capillary pressure to the overall mechanical stress through eq 15. For constant and uniform evaporation such that do/dt ) 1, the capillary pressure grows more negative with time or extent of deformation as
Now we are equipped to identify the conditions under which the polymer-water, air-water, or polymer-air interfacial tension is responsible for forming a void-free film. The treatment here parallels the analysis of Routh and Russel,18 replacing their ad hoc approximation for the constitutive equation with eq 15. The argument below demonstrates that the rate of evaporation (E/H) relative to the rate of deformation driven by capillary pressure or polymer-water or polymer-air surface tension (γ/aη) determines the mode of film formation for temperatures in the vicinity of or well above the glass-transition temperature of the polymer. Wet Sintering. We first consider conditions under which the polymer-water surface tension is capable of forming a film before all of the water evaporates. The governing equation takes the form
φNγpw 2 ) G h 0.71 a 3
∫0
t
3/2 t′ - t do exp dt′ τ dt′
( )
(20)
where t ) 0 corresponds to φ ) φo and o ) 0. Differentiating with respect to t and subtracting from the original equation leaves
φNγpw Hγpw do3/2 ) 1.07π with o3/2(0+) ) 1.07 (21) dt aηE aG h
Deformation and Cracking of Colloidal Packings
Langmuir, Vol. 24, No. 5, 2008 1725 Table 1. Physical Properties of Components G (GPa)
ν
γpa (mN/m)
2R (nm) 330 82, 133, 206, 353 250 300 230, 379, 458, 489 200
polystyrene acrylic
1.6 0.8
0.33 0.36
30 32
styrene-butadiene silica alumina
1.0
0.34
32
156
zirconia water
81
φRCP 0.60 0.65 0.64 0.60 0.60 0.60
72
The expression relating the capillary pressure to the amount of deformation sets the onset of the receding front at the dimensionless time tmax = (5.3πHγ/aφNηE)2 for τ , 1. Dry sintering then proceeds as Figure 4. Capillary pressure pfilm cap required to complete film formation as evaporation finishes as a function of the evaporation time relative to the relaxation time 1/τ ) GH/2(1 - ν)ηE: O, numerical integration; -, interpolation.
because the elastic deformation is instantaneous. Integration with the initial condition determines
o3/2(t) ) 1.07
φNγpw t 1+ aG h τ
(
)
(22)
so the time to close all of the pores (o ) 0.36) is tf ) 0.065aEη/ Hγpw - τ. To be dominant, wet sintering must close the pores before evaporation is complete (i.e., tf < 0.36), which requires
aηE < 5.6(1 + 2.8τ) γpwH
(23)
Capillary Compression. The upper bound23 on the domain of wet sintering identifies the onset of the regime in which compression due to the negative capillary pressure dominates film formation and the last water molecule evaporates as the last void closes.41 Thus, ) 0.36 at t ) 0.36 and from ref 17 (Figure 4)
pfilm cap ) G h
∫00.36 exp(t - τ0.36) t1/2 dt =
0.14 1 + 0.24/τ
(24)
As the rate of evaporation increases, this mechanism fails to form the film when the required capillary pressure reaches the maximum set by the curvature sustainable at the air-water interface, beyond which the interface recedes into the porous packing. For a triangular pore between three spheres in contact, the maximum curvature is approximately ∼5.3/a42, which with eqs 23 and 24 indicates that capillary compression successfully drives film formation for
φNγpw aφNηE (1 + 2.8τ) < < 28(1 + 4.2τ) (25) 5.6 γ γH
do3/2 Hγpa 5.3πHγ 3 ) 1.07π with o3/2(tmax ) ) dt aηE aφNηE
(
)
(26)
Below the receding air-water interface, the negative capillary pressure, which remains at the maximum, continues to compress the particles. Thus, the top of the film will be the least deformed by the capillary pressure and, therefore, the last to fully eliminate all pores via dry sintering. This prevents skinning and allows all of the water to evaporate from below, setting the time for void closure as
tf )
5.3πHγ 0.93 aηE + 0.216 - ( ( 5.3πHγ aφNηE ) π Hγ [ aφNηE ) ] 2
3
(27)
pa
When evaporation is essentially complete before voids close to a significant extent, say for o < 10-2, dry sintering driven by the polymer-air surface tension must provide all of the deformation. This defines the purely dry sintering regime as
aφNηE > 167π(1 + 150τ) γH
(28)
Process Map. Figure 4 locates the regimes defined by eqs 23, 25, and 28 in terms of the two primary dimensionless groups for specific values of φN (0.64 × 6 ) 3.84), γpa/γ (0.032/0.072 ) 0.44), and γpw/γ (∼[(γpw/γ)1/2 - 1]2) (Table 1). The almost horizontal boundaries delineating the regimes indicate that elastic deformation is less effective in closing the pores than viscous flow for τ < 1 and this range of particle sizes. A useful industrial standard for screening dispersions for filmforming potential is the temperature gradient bar. This test provides a clear visual indication of the time to close voids as a function of temperature by simply watching the cloudy-toclear transition in the film move from the hot end of the bar toward the cold end.13 This suggests fixing all of the parameters except the strongly temperature-dependent viscosity, represented via the WLF equation,43 as
η ) ηg exp
[
]
-34(T - Tg) 80 + T - Tg
(29)
Receding Water Front and Dry Sintering. Beyond the capillary compression regime, the rate of deformation is set by the evaporation rate until the air-water interface recedes into the film. Then the elastic deformation due to the capillary pressure relaxes, and dry sintering takes over above the air-water interface.
with ηg ) 6.5 × 1010 Pa s and plotting the time to close voids as a function of T - Tg (Figure 5). Note that wet sintering appears only well above the glass-transition temperature whereas the air-water interface recedes into the film and requires dry sintering to complete the process for temperatures below or only slightly above Tg. Capillary pressure drives film formation over an
(41) Brown, G. L. J. Polym. Sci. 1956, 22, 423-434. (42) Mason, G.; Mellor, D. W. J. Colloid Interface Sci. 1995, 176, 214-255.
(43) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: New York, 1999.
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Figure 7. Coordinate system for a layer with a crack aligned with the y axis with the substrate at z ) 0 and the free surface at z ) 1.
the substrate to that in the plane of the film. Our approximation is derived by integrating the stress balances in the plane of the film over the thickness as noted above to obtain Figure 5. Process map identifying regimes for wet and dry sintering and capillary compression, as well as the combination of the latter two due to a receding front, in terms of the evaporation time relative to the viscous collapse time γawH/aηE and the viscoelastic relaxation time.
∇‚〈σ′〉 - σ′‚ez|z)0 ) 0
(30)
To ensure that the shear stresses vanish at the free surface while the displacements vanish at the substrate, we assume the lateral and vertical displacements to vary quadratically and linearly, respectively, with height as u′ ) 3〈W〉z(1 - z/2) + 〈′zz〉zez with W ) (u, V). Ignoring the contribution to the shear strain from the normal displacement yields E′‚ezz3〈W〉(1 - z)/2 + 〈′zz〉ez, so
3 ′xz|z)0 = 〈u〉 2 3 〈′xz〉 = 〈u〉 4
′xx|z)1 =
3 〈′xz2〉 ) 〈u〉2 4
3 ∂〈u〉 2 ∂x 〈′xx〉 )
∂〈u〉 ∂x
(31)
with similar expressions in the ey direction. The normal stress condition requires ez‚σ′‚ez|z)1 ) 0, which from eq 16 dictates
Figure 6. Time required to close voids (i.e., form a film) scaled on the evaporation time and plotted against the temperature relative to the glass-transition temperature of polymer latices for particles of radii 50 and 500 nm.
∼7 °C range of temperatures above Tg, though the precise location of that window is sensitive to other parameters such as the particle size.
VI. Equations Governing the Relaxation of a Film after Opening a Crack At temperatures for which the viscous deformation of latex spheres driven by capillary pressure does not keep up with evaporation or with inorganic particles that do not deform viscously, films often develop cracks while still saturated with water. For ceramic particles, Cima and co-workers32 attributed the phenomenon to the relaxation of tensile stresses when cracks open, allowing the recovery of elastic energy as in the failure of thin elastic films subjected to differential shrinkage.35 To assess the mechanism, we employ the constitutive equations developed above for the perturbations to the stress fields associated with capillary compression of a film, as accomplished previously44 with the approximate constitutive equation. Calculating the elastic recovery due to the opening of a crack spanning the film (Figure 7) requires knowledge of the relaxation, both laterally and vertically, associated with stress-free crack faces normal to the plane of the film (e.g., at x ) (0) in addition to the upper free surface z ) 1. To avoid solving the fully 3D momentum equations, we resort to a thin film approximation that slaves the variation normal to (44) Tirumkudulu, M. S.; Russel, W. B. Langmuir 2005, 21, 4938-4948.
1 〈p′〉 ) G h ′zz + (′xx + ′yy) 4 z)1 3 )G h 〈′zz〉 + (〈′xx〉 + 〈′yy〉) 8
[ [
]
]
(32)
The shear stresses at the substrate in the averaged balances follow from eqs 16, 31, and 32 as σ ′‚ez|z)0 ) (1/2)G h o1/2E′‚ez|x3)0 ) (3/4)G h o1/2〈W〉. Xia and Hutchinson30 implemented this approximation slightly differently, setting σ ′‚ez|z)0 ) k〈W〉 and comparing the results with a full numerical solution for an infinite crack. Equating the recovery of elastic energy from the two calculations dictated kXH = 1.26G h /(1 - ν) for a compliant film on a rigid substrate. Applying our approximation to their system yields k/kXH = 0.95/(1 - ν) (i.e., a modest difference for compressible particles), so we feel that the approximation should be reasonably accurate or at least a good O(1) estimate. This leaves 〈W〉 and 〈′zz〉 to be determined from the stress balances in the plane of the film30 and the conservation equations for particles and water.2 Here we decouple the equations by focusing on the short-time limit, before water has time to either evaporate or flow laterally to equilibrate the capillary pressure. This implies a shear deformation with no change in volume so that tr〈E′〉 ) 0 and 〈p′〉 ) - 5G h (〈Exx〉 + 〈Eyy〉)/8. The long-time limit, after water has redistributed to equalize the capillary pressure so that p′ ) 0, is equally tractable but yields only a slightly different elastic recovery.44 For that reason, we will confine our attention to short times in the following. Substituting the resulting stresses into the conservation equation and formulating the boundary conditions of zero normal and tangential shear stresses at the face of the crack of length L and no perturbation at infinity yields
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Langmuir, Vol. 24, No. 5, 2008 1727
∇s2〈W〉 + 5∇s∇s‚〈W〉 ) 6〈W〉 with 〈W〉 ) 0 at x, y ) ( ∞ 3 ex‚〈σ′〉‚ex ) pcap 4 ey〈σ′〉‚ex ) 0
} {
x)(0 L L eye 2H 2H
at
∆E ) 2γ + 2φH H
VII. Recovery of Elastic Energy and Determination of Critical Capillary Pressure The first step, which provides a lower bound to the capillary pressure necessary to open a crack, is to determine the recovery of elastic energy associated with a crack of infinite length. The principle being exploited here is that the opening of a crack must be energetically favorable. The absence of variations parallel to the crack eliminates the y-momentum balance and reduces the x-momentum balance to the ordinary differential equation
dx
2
) 〈u〉 with 〈u〉(∞) ) 0 and
pcap d〈u〉 (0) ) dx G h 1/2
(34)
1 1 - φ) + (1 + 5φ) plane strain {( 4 32 G h 5 3 3 - (1 - φ) + (1 - φ) compression + } shear 16 6 16
) 2γ + 2H
p2cap
1/2
o
) 2γ -
2 11 Hpcap 16 G h 1/2
(38)
o
Note that the lateral deformation recovers energy against the initial tensile stress (φσop xx 〈′xx〉∝(1/4 - φ) < 0) and the vertical relaxation against the initial normal stress recovers energy of compression (φσop zz 〈zz′〉∝ - (1 - φ)) whereas shear (φ〈σ′xzp′xz〉) and the second-order effects of relaxation in the transverse (φ〈σ′xxp′xx〉) and normal (φ〈σ′zzp′zz〉) directions cost elastic energy. For sufficiently large capillary pressures, the net is negative, so cracking should be expected. Capillary Pressure Necessary for Cracking. Defining the capillary pressure necessary for cracking as that for which the total recovery of elastic energy equals the additional surface energy leads to 2 o3/2 pcap 2 γ 11 pcap ) = with H 32 G G h 3 1 + 2o/3τ h 1/2
o
The solution
(39)
o
〈u〉 ) -
which is well approximated by
pcap G h o
1/2
exp(-x)
(35) -
indicates that the relaxation extends only a few film thicknesses away from the crack, so the influence is rather short-ranged. Evaluation of the elastic energy recovered through relaxation of tensile stresses in the plane and compressive stresses in the normal direction requires the vertically averaged strains calculatedfrom the displacement field above as
〈′xx〉 ) -〈′zz〉 ) 〈′xz〉 ) -
}
1 p 〉〈′ 〉 + 〈σ′xzp′xz〉 dx + σop zz 〈′zz〉 + 〈σ′ 2 zz zz
(33)
The first boundary condition compensates for the pre-existing tension in the plane to satisfy the normal component of the free-surface condition at the crack face whereas the latter preserves the zero shear stress state. Note that perturbation stresses of O(pcap) imply relaxations of O(o), which, strictly speaking, are inconsistent with the linearization used to derive eq 16.
d2〈u〉
∫0∞{σopxx 〈′xx〉 + 21 〈 σ′xxp〉〈′xx〉
pcap G h o
1/2
5 〈p′〉 ) - pcap exp(-x) 8
exp(-x)
3 Pcap exp(-x) 4 G h o1/2
〈′xz2〉 )
( )
3 pcap 2 exp(-2x) 4 G h o1/2 (36)
From these the stresses within the particle phase, it follows that
1 φ〈σ′xxp 〉 ) (1 + 5φ)pcap exp(-x) 8 3 5 φ 〈σ′zzp〉 ) - 1 - φ pcap exp(-x) 4 6
(
〈φσ′xzp′xz〉
)
2 3 pcap ) exp(-2x) 8G h 1/2
(37)
o
The change in energy associated with opening the crack is the strain energy density integrated over the film plus the surface energy of the two new interfaces
Hpcrack cap G h H 2/5 2.91 = 2.06 + γ γ τ
( )
(40)
Figure 8 illustrates the capillary pressure necessary for cracking as a function of the ratios of the elastic modulus and the viscous stress to the surface tension. When viscous deformation is negligible, the critical capillary pressure attains the lowest value, -Hpcrack h H/γ)2/5, which coincides with the line for cap /γ ) 2.06(G γ/φNηE ) 10-9. Decreasing the viscosity, by reducing the glasstransition temperature relative to the process temperature, suppresses cracking by increasing the capillary pressure required to recover sufficient elastic energy. The changes in the lines from bold to normal indicate when the capillary pressure suffices to close all voids24 before reaching that necessary for cracking. Note that the capillary pressure necessary for cracking is independent of the particle radius, which is the natural length scale for the capillary pressure. Because pcrack cap depends only on the film thickness H, the effective modulus G h , the viscous stress ηE/H, and the surface tension γ, dimensional analysis identifies only three dimensionless groups and requires that - Hpcrack cap /γ ) f(HG h /γ, γ/φNηE). Thus, experiments with different particle sizes present a serious test of the mechanism. Minimum Film Thickness for Cracking. Beyond a maximum capillary pressure, set by the limiting curvature for a meniscus at the surface of the packing (e.g., -5.3γ/a), the air-water interface will recede into the layer of close-packed particles. For sufficiently thin films, this happens before reaching the capillary pressure necessary for cracking. The minimum thickness for cracking follows from eq 40 as
-pmax cap H G h H 2/5 Hcrit G h a 2/3 w (41) ) 2.06 ) 0.050 γ γ a γ
( )
( )
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Russel et al.
∂2〈u〉
) u with 〈u〉(W/H ) ) 0 and
∂x2
pcap ∂〈u〉 (0) ) ∂x G h 1/2
(42)
o
with the solution
〈u〉 ) -
pcap sinh (W/H - x) cosh(W/H) G h 1/2 o
(43)
Setting the total change in elastic and surface energy associated with this relaxation to zero determines the energetically optimal spacing as a function of excess capillary pressure as Figure 8. Dimensionless capillary pressure necessary to open an infinite isolated crack plotted against the ratio of elastic and interfacial stresses and the dimensionless relaxation time: γ/φNηE ) 1 × 10-4 (-), 3 × 10-5 (- - -), -1 × 10-5 (---), 1 × 10-9 (-). The gray line indicates the capillary pressure (Figure 1) sufficient to close voids and supersede cracking.
(
pcrack W/H cap ) tanh(W/H) pcap 11 cosh2(W/H)
)
3/5
(44)
As illustrated in Figure 10, the spacing drops to W/H ≈ O(1) before the capillary pressure increases significantly because of the short range (i.e., O(H)) of the lateral deformation. In this section, we have constructed predictions for the capillary pressure that produces sufficient elastic recovery for cracking to be energetically favorable and have then combined that necessary condition with bounds representing the pressure at which all voids should close or the air-water interface should recede into the film before the onset of cracking. The final analysis suggests a characteristic spacing as additional cracks open at pressures beyond the minimum. However, in the solid mechanics literature, these conditions are known to be insufficient to account fully for the onset of cracking.
VIII. Isolated Finite Crack Figure 9. Film thickness below which a layer of elastic particles will not crack as the air-water interface recedes into the film before reaching the capillary pressure necessary for cracking plotted against the ratio of elastic and interfacial forces: (---) eq 41 with φN ) 0.64 × 6 ) 3.84: silica (2), alumina (b), and zirconia (+);32 polystyrene ([), acrylic (9), and styrene butadiene latices (O).33 See Table 1 for properties.
below which the film dries without cracking or closing the voids. Recent experiments33 with relatively soft latices, together with earlier data32 for much harder silica, zirconia, and alumina particles, offer a test of this prediction (Figure 8) as plotted in dimensionless form with the physical properties in Table 1. The theoretical prediction passes somewhat below the data for the inorganic oxides and through that for the latices. Spacing of Parallel Cracks. As the capillary pressure increases above the critical value, additional cracks appear in most experiments, sometimes propagating through the sample with a clear characteristic spacing44-47 and in others opening perpendicular to an existing crack. Because increasing the pressure above the threshold allows the recovery of more elastic energy per unit area, the opening of new or more closely spaced cracks is energetically favorable. The simplest analysis that yields a characteristic spacing presents a hypothesis for a periodic array of parallel cracks.31 The short-time limit for infinite parallel cracks spaced uniformly with a separation of 2W generates deformation fields described by (45) 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. Phys. ReV. Lett. 2003, 91, 224501-1-4. (46) Dufresne, E. R.; Stark, D. J.; Greenblatt, N. A.; Cheng, J. X.; Hutchinson, J. W.; Mahadevan, L.; Weitz, D. A. Langmuir 2006, 22, 7144-7147. (47) Lee, W. P.; Routh, A. F. Ind. Eng. Chem. Res. 2006, 45, 6996-7001.
In some experiments with spatially uniform samples, cracks first form at threshold pressures significantly beyond the minimum whereas the density of cracks at higher pressures lags dramatically behind that expected from the prediction above.34 Instead, the capillary pressure at which the first cracks appear is not fully reproducible, and with increasing pressure, additional cracks often open at and propagate from existing ones. These two observations suggest that the process may be limited by the flaws available to nucleate cracks in colloidal films, as generally seen in elastic solids. As a starting point for analyzing this, we now calculate the capillary pressure required to open or extend an isolated crack of finite length. The x- and y-momentum equations governing the stress fields around a crack of length L in an infinite film, within the thin film approximation (eq 33)
6
∂2〈u〉 2
∂x
+
∂2〈u〉 ∂y
2
+5
∂〈u〉 ) 6〈u〉 ∂x∂y 6
∂2〈ν〉 ∂y
2
+
∂2〈ν〉 2
∂x
+5
∂〈ν〉 ) 6〈ν〉 (45) ∂x∂y
with boundary conditions
〈u〉 ) 〈ν〉 ) 0 at x, y ) ( ∞
}
{
∂〈u〉 ∂〈ν〉 + )0 x)0 ∂y ∂x L L at pcap ∂〈u〉 2 ∂〈ν〉 eye 2H 2H + ) ∂x 3 ∂y G h o1/2
(46)
might be solved via an integral formulation,30 but we resort instead
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Langmuir, Vol. 24, No. 5, 2008 1729
with length L, respectively. In addition to pcrack cap (L), we are also interested in the conditions under which an existing crack of length L could grow. Clearly, d(∆E)/dL must be negative for such an unstable crack, which leads to the minimum capillary pressure necessary to extend a preexisting crack of length L as
pcrack cap (∞) + pcrack cap (L )
Figure 10. Relationship between spacing W of infinite parallel cracks excess capillary pressure pcap/pfilm cap for which the elastic recovery equals the increased surface energy from eq 44.
)
(
dI d(L/H)
)
3/5
with lim
L/Hf∞
dI ≈1 d(L/H)
(49)
Both conditions are depicted in Figure 11. Clearly pcrack cap (L), the minimum capillary pressure to open a finite crack, increases with decreasing length of the crack, indicating that very short cracks (with lengths much shorter than the thickness of the film) are unlikely to be created by capillary pressure. Therefore, the most important and most likely observable + is pcrack cap (L ) associated with the longest preexisting flaw in the film.
IX. Isolated Circular Crack or Flaw Because the results above indicate that flaws with L , H may control cracking, we have also considered small circular flaws (i.e., a cylindrical hole of radius R). Opening a circular crack with a stress-free crack face normal to the plane of the film at r ) R/H allows the recovery of elastic energy through the relaxation of normal and radial stresses while generating shear deformation that costs energy. Converting eq 33 to cylindrical coordinates provides the governing equation
( )
1 ∂ ∂〈u〉 r - (1 + r2)〈u〉 ) 0 with 〈u〉 ) 0 at r ) ∞ r ∂r ∂r Figure 11. Capillary pressures required to open a finite crack (0) or extend to infinity a finite flaw (9) of length L/H.
to finite elements to determine u(x, y; L/2H) and ν(x, y; L/2H). The change in elastic energy then follows as
∆E ) 2γLH + 4H3φ σop yy 〈′yy〉 +
≡ 2γLH -
∫0∞ ∫0∞ {σopxx 〈′xx〉 + (21)〈σ′xxp〉〈′xx〉 +
1 p 〈σ′ 〉〈′ 〉 + 〈σ′xyp〉〈′xy〉 + σop zz 〈′zz〉 + 2 yy yy 1 p 〈σ′ 〉〈′ 〉 + 〈σ′xzp′xz〉 + 〈σ′yzp′yz〉 dx dy (47) 2 zz zz
() ()
}
2 L 11 3 pcap I H 1/2 H 16 G h
()
o
{ [( ) ( ) ] ) ( )
∫ ∫
}
Comparing this result with that for a single crack suggests expressing the final result as
pcrack cap (∞)
(
)
I(L/H) ) crack L/H pcap (L)
The solution that satisfies the equation and the boundary conditions has the radial dependence of the first order Bessel function K1(r)
〈u〉 )
6Rpcap HG h o
1/2
K1(r) 4K1(F) - 3F[K2(F) + K0(F)]
(51)
The total energy associated with this deformation field is the elastic energy density integrated over the whole film plus the surface energy of the air-water interface within the circular flaw ∞ ∫R/H {σoprr 〈′rr〉 + 21 〈σrrp′〉〈′rr〉
∂〈u〉 2 ∂〈V〉 2 ∂〈u〉 ∂〈V〉 L 9 ∞ ∞ + I )7 + + 10 H 11 0 0 ∂x ∂y ∂x ∂y ∂〈u〉 ∂〈V〉 ∂〈u〉 ∂〈V〉 2 + + 6(〈u〉2 + 〈V〉2) dx dy + + 8 ∂x ∂y ∂y ∂x
(
(50)
o
∆E ) 2πγRH + 2πφH3
where
()
pcap ∂〈u〉 2 + 〈u〉 ) at r ) F ≡ R/H ∂r 3r G h 1/2
3/5
H I(L/H) ≈ 1 (48) with lim L/Hf∞ L
crack where pcrack cap (∞) and pcap (L) are the minimum capillary pressures required to open a single infinite crack and a finite crack
1 p op + σop θθ〈′θθ〉 + 〈σθθ′〉〈′θθ〉 + σzz 〈′zz〉 2 1 + 〈σzzp′〉〈′zz〉 + 〈σrzp′′rz〉 r dr 2
}
≡ 2πγRH - 2πH3
p2cap G h o1/2
C(F)
(52)
While this approximates the energy associated with opening an infinitesimal cylindrical flaw into a cylindrical crack of radius R, the pressure required to accomplish that,
(
)( )
HG h o1/2 Rpcap ) γ γ
1/2
F3 C(F)
1/2
(53)
1730 Langmuir, Vol. 24, No. 5, 2008
Russel et al.
Figure 12. Capillary pressure required to extend a cylindrical flaw of radius R/H.
with the constraint that 〈u〉(F) ) F, always exceeds that needed to pull the air-water interface from the surface of the film into a pre-existing cylindrical flaw of radius R (i.e., pcap ) -2γ/R). Thus, a more useful quantity to extract from this analysis is the capillary pressure required to open a flaw of radius R into an infinite crack,
{
}
pcap2 d 2πγRH - 2πH3 C(F) < 0 dR G h 1/2 o
(54)
or
-
(
)
HG h o1/2 Hpflaw cap (F) ) γ γdC/dF
1/2
(55)
When normalized on the capillary pressure required to open an infinite crack, this takes the form of
pgrow cap (F) pcrack cap (∞)
)
dF (3211 dC )
1/2
(56)
The results depicted in Figure 12 suggest that flaws with R < 0.1H would suffice to initiate cracks that would propagate across the sample at pressures modestly greater than pcrack cap .
X. Summary In this article, we have developed equations governing stress in a close-packed layer of spheres (i.e., constitutive equations relating stress to strain plus conservation equations for water, particles, and momentum) to provide a basis for predicting in detail various modes of deformation and cracking of saturated or unsaturated colloidal films. The starting point is a classical relation between force and deformation for two spheres in contact, which is valid for small but finite deformations. The averaging process to convert these microscopic forces and deformations into macroscopic constitutive equations requires the homogeneous packings of spheres with dimensions much smaller than the layer
thickness. Integration of the macroscopic stress balances over the thickness of a thin film simplifies the mathematical solutions but necessitates a closure for the shear stress that is validated by comparison with numerical solutions in the literature. The outcome is a physically sound, though mathematically approximate, model for thin films. We first consider the consolidation of a close-packed film of thickness H as water evaporates at rate E for polymer latices that respond mechanically as very viscous fluids. From the normal stress balance alone we identify a wet sintering regime for aηE/ Hγ < 0.2, dry sintering for aηE/Hγ > 1000, and capillary compression for 0.2 < aηE/Hγ < 10. For 10 < aηE/Hγ < 1000, capillary pressure fails to complete the consolidation process before the air-water interface retreats into the film, leaving dry sintering to finish closing the voids. For each regime, additional elastic deformation facilitates film formation, extending the range to higher values of aηE/Hγ. Next we seek to identify conditions under which cracking prevents the formation of void-free films by capillary compression. Here the treatment follows that for thin linearly elastic films for which tensile stresses provide the driving force. Linearizing the constitutive equations about a partially consolidated base state and averaging over the thickness of the film provides stress balances that can be solved for the lateral relaxation that accompanies the opening of a crack. Integrating the local strain energy density over the volume of the film determines the elastic energy recovered for comparison with the additional airwater surface energy associated with the crack faces. We perform this calculation for isolated and parallel infinite linear cracks and isolated finite linear and circular cracks. The purpose is to understand and control the capillary pressure at which cracking first occurs and the thickness below which films do not crack, as well as the number or spacing of cracks as the capillary pressure increases beyond the threshold. The theory provides a capillary pressure necessary either to open an infinite crack in a flawless film or to extend pre-existing flaws of finite lengths, along with a minimum thickness below which colloidal films saturated with water should not crack. Although not yet a complete theory for the process, these results should prove useful in interpreting and guiding experiments. Acknowledgment. This research was supported by Chemical and Thermal Systems in the Division of Engineering of the National Science Foundation through grant CTS 0120421. We recognize key suggestions from Professor Anand Jagoda and critical input from participants in the Cracking Club. Supporting Information Available: The evaluation of the angular integrals leading to eqs 15 and 16 for the constitutive equation and the integrals for the elastic stresses yielding eqs 38, 44, and 47. This material is available free of charge via the Internet at http:// pubs.acs.org. LA702633T
