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Langmuir 1999, 15, 7762-7773
A Process Model for Latex Film Formation: Limiting Regimes for Individual Driving Forces Alexander F. Routh and William B. Russel* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544 Received March 16, 1999. In Final Form: June 29, 1999 The deformation of particles, to produce a structure without voids, has been an issue of contention in the film formation community for many years. Four different mechanisms have been proposed. Three involve homogeneous deformation throughout the film, although all are built on the deformation of two isolated particles, described in the viscous limit by Frenkel and in the elastic limit by Hertz and Johnson, Kendall, and Roberts. We derive a linear viscoelastic generalization of Frenkel’s model that predicts the deformation of two spheres compressed by a force, F, and surface tension, γ. The resulting equation is then embedded in field equations governing the collapse of macroscopic films. Assuming a uniaxial compression allows derivation of limits for the proposed modes of homogeneous deformation. These limits are shown as surfaces in parameter space. Since temperature alters most profoundly the rheological response of viscoelastic polymers, the controlling deformation mechanism is defined as a function of temperature. Wet sintering requires slow evaporation or a low modulus polymer and is seen at high temperatures. Capillary deformation requires the strain in the film to follow evaporation and appears at intermediate temperatures. Dry or moist sintering is then seen at the lowest temperatures, when the modulus is high and deformation is slow compared to evaporation.
1. Introduction The problem of film formation has been considered in the literature for over 50 years.1-7 The generally accepted mechanism consists of three stages. Initially evaporation from a stable dispersion brings the particles into some form of close packing. Deformation of the particles leads to a structure without voids, although with the original particles still distinguishable. Finally diffusion of polymer across particle boundaries yields a continuous film with mechanical integrity and the original particles no longer distinguishable. The first stage of the process has been reported to sometimes involve drying fronts that pass laterally across the film from the edge inward.3,8-10 The resulting lateral flow, as described by Winnik and Feng,9 can affect the deformation step.11 Drying fronts form as the film solidifies first in areas of diminished height. Continuing evaporation from these regions draws in solvent, propagating the front. The characteristic length for the effect of an edge is on the (1) Brown, G. L. Formation of films from polymer dispersions. J. Polym. Sci. 1956, 22, 423-434. (2) Henson, W. A.; Taber, D. A.; Bradford, E. B. Mechanism of film formation of latex paint. Ind. Eng. Chem. 1953, 45 (4), 735-739. (3) Sheetz, D. P. Formation of films by drying of latex. J. Appl. Polym. Sci. 1965, 9, 3759-3773. (4) Vanderhoff, J. W.; Tarkowski, H. L.; Jenkins, M. C.; Bradford, E. B. Theoretical consideration of the interfacial forces involved in the coalescence of latex particles. J. Macromol. Chem. 1966, 1 (2), 361397. (5) Dobler, F.; Holl, Y. Mechanisms of latex film formation. TRIP 1996, 4 (5), 145-151. (6) Winnik, M. A. Latex film formation. Curr. Opin. Colloid Interface Sci. 1997, 2 (2), 192-199. (7) Winnik, M. A. Emulsion Polymerization and Emulsion Polymers; John Wiley and Sons Ltd.: New York, 1997; Chapter 14, pp 467-517. (8) Croll, S. G. Drying of latex paint. J. Coatings Technol. 1986, 58 (734), 41-49. (9) Winnik, M. A.; Feng, J. Latex blends: An approach to zero voc coatings. J. Coatings Technol. 1996, 68 (852), 39-50. (10) Routh, A. F.; Russel, W. B. Horizontal drying fronts during solvent evaporation from latex films. AIChE J. 1998, 44 (9), 20882098. (11) Eckersley, S. T.; Rudin, A. Mechanism of film formation from polymer latexes. J. Coatings Technol. 1990, 62 (780), 89-100.
order of 100 times the film thickness.10 In general drying initiates the deformation step at different times with spatially varying film thickness, resulting in a front of optical clarity passing across the film at later times. The deformation step has been contentious for many years, even in homogeneous films, with many competing mechanisms advanced. All mechanisms are formulated in terms of the deformation of two isolated particles. The easiest pair deformation model, due to Frenkel,12 considers viscous particles deforming under the action of surface tension. Assuming the particles to deform as truncated spheres and balancing an estimate for the rate of viscous dissipation (volume times viscosity times shear rate squared) with the rate of loss of surface energy (surface tension times rate of change of surface area), Frenkel arrives at a remarkably simple expression for the deformation R )
3γt 4ηR0
(1)
defined as the decrease in the center to center distance normalized by the initial particle diameter, 2R0, or the strain along the line joining the particle centers. γ is the surface tension, η is the viscosity, and t is the time of deformation. Frenkel shows that any increase in the particle radius, due to volume conservation, is order R2 and hence negligible for the small deformations considered. The model has been extended to finite deformations13 and more recently to include the effects of viscoelasticity.14 At the opposite end of the material spectrum, Hertz considered elastic particles pushed together by a force F. His exact solution to the integral equations governing the local deformation for incompressible particles yields (12) Frenkel, J. Viscous flow of crystalline bodies under the action of surface tension. J. Phys. 1945, 9 (5), 385-391. (13) Pokluda, O.; Bellehumeur, C. T.; Vlachopoulos, J. Modification of Frenkel’s model for sintering. AIChE J. 1997, 43 (12), 3253-3256. (14) Bellehumeur, C. T.; Kontopoulou, M.; Vlachopoulos, J. The role of viscoelasticity in polymer sintering. Rheol. Acta 1998, 37, 270-277.
10.1021/la9903090 CCC: $18.00 © 1999 American Chemical Society Published on Web 08/20/1999
A Process Model for Latex Film Formation
R )
() ( ) 3 8
F
2/3
2/3
2
R0 G
Langmuir, Vol. 15, No. 22, 1999 7763
(2)
where G is the shear modulus. This detailed analysis was extended by Johnson, Kendall, and Roberts15 to include surface tension for force free particles as
( )
R ) 0.234
γ R0 G
2/3
(3)
Approximate extensions of the Hertz and JohnsonKendall-Roberts (JKR) analyses to viscoelastic materials exist.16 On the basis of these microscopic mechanisms, four distinct regimes have been proposed for the deformation of particles within a film: Wet sintering4 is driven by the surface tension between the particles and the solvent, typically water. Sheetz3 observed that for methyl methacrylate/butyl acrylate copolymer latices held under water for considerable time above the minimum film formation temperature, the solids fraction φ increased from approximately φ ) 0.62 to 0.87. After this no further compaction was observed. In agreement with this observation, Dobler et al.17 observed gravimetrically deformation due solely to polymer-water surface tension. They noted, however, that the rate of compaction under standard conditions is slow compared to the evaporation of water and concluded that wet sintering is not a dominant mechanism for film formation. Dry sintering18 is analogous to wet sintering, except that the polymer-air surface tension drives compaction. Recently Sperry et al.19 demonstrated conditions at which dry sintering is important and collapsed data for many different particle radii with a simple model for viscous collapse of voids under surface tension. Keddie et al.20 supported this conclusion. For true dry sintering to be observed, the solvent must completely evaporate before significant particulate deformation occurs. Lin and Meier21 argue that atmospheric humidity preserves residual water as rings surrounding the contacts between particles, creating large negative curvature that drives deformation. Although different from dry sintering, this moist sintering mechanism also requires fast evaporation of solvent. Capillary deformation arises from the curvature of the air-water interface between particles at the surface of the film, which generates a large negative pressure and compression of the film below. With lightly cross-linked polymers that could not undergo viscous flow, Brown1 demonstrated significant particle deformation due to this capillary pressure. To interpret this, Brown balanced (15) Johnson, K. L.; Kendall, K.; Roberts, A. D. Surface energy and the contact of elastic solids. Proc. R. Soc., London, Ser. A 1971, 324, 301-313. (16) Falsafi, A.; Deprez, P.; Bates, F. S.; Tirrell, M. Direct measurement of adhesion between viscoelastic polymers: A contact mechanical approach. J. Rheol. 1997, 41 (6), 1349-1364. (17) Dobler, F.; Pith, T.; Holl, Y.; Lambla, M. Synthesis of model latices for the study of coalescence mechanism. J. Appl. Polym. Sci. 1992, 44, 1075-1086. (18) Dillon, R. E.; Matheson, L. A.; Bradford, E. B. Sintering of synthetic latex particles. J. Colloid Sci. 1951, 6 (2), 108-117. (19) Sperry, P. R.; Snyder, B. S.; O’Dowd, M. L.; Lesko, P. M. Role of water in particle deformation and compaction in latex film formation. Langmuir 1994, 10, 2619-2628. (20) Keddie, J. L.; Meredith, P.; Jones, R. A. L.; Donald, A. M. Kinetics of film formation in acrylic latices studied with multiple-angle-ofincidence ellipsometry and environmental sem. Macromolecules 1995, 28, 1673-2682. (21) Lin, F.; Meier, D. J. A study of latex film formation by atomic force microscopy. 1. a comparison of wet and dry conditions. Langmuir 1995, 11, 2726-2733.
estimates of the pressures generated by a spherical meniscus within a triangular array of particles at the surface of the film (12.9γwa/R0) and Hertzian deformation of particles to complete compaction (0.37G) and obtained a criterion for film formation as G e 35γwa/R0
(4)
To accommodate viscoelasticity Brown suggested substituting a time-dependent shear modulus, G(t), to estimate the time, tf, required for film formation. Many have criticized and tinkered with Brown’s analysis. Mason22 correctly argues that the elastic stress and capillary pressure act over different areas and modified the criterion to G e 266γwa/R0
(5)
Eckersley and Rudin11 followed Brown’s suggestion and derived the equivalent viscoelastic criterion as 34γwa 1 e R0 Jc(t)
(6)
where Jc(t) is the creep compliance of the polymer. In the linear viscoelastic regime 1/Jc(t) ≈ G(t) and we nearly return to Brown’s result. When the observed deformation exceeds that due to capillary forces alone, Eckersley and Rudin23 simply add sintering in series. In a similar analysis Mazur and Placek24 assume the viscous and elastic responses to occur on widely different time scales, making the effects linearly additive in the small deformation regime so that a ) aelastic + aviscous
(7)
where a ≈ (2R)1/2R0 is the radius of circle of contact between particles. Here the two terms correspond to the capillary deformation controlled by the instantaneous elastic (Hertzian) response and the viscous (Frenkel) deformation due to the relevant surface tension, respectively. The assumption that the effects are linearly additive, in terms of the radius of contact however, remains unsupported. Sheetz’s deformation mechanism,3 unlike each of those above, does not presume homogeneous deformation throughout the film. Sheetz noted that films that otherwise dried cloudy and cracked would form a coherent film if a predried layer were placed in contact with and above the drying film. His explanation invoked diffusion of water through the predried film to produce a large compressive force on the drying film, thus causing compaction, a view supported by Dobler et al.25,26 Alternatively the predried film may simply hinder evaporation, allowing more time for the deformation mechanisms discussed earlier. Evidently Sheetz’s regime requires formation of a transport limiting barrier by one of the previous methods and vertical (22) Mason, G. Formation of films from latices a theoretical treatment. Br. Polym. J. 1973, 5, 101-108. (23) Eckersley, S. T.; Rudin, A. The film formation of acrylic latexes: A comprehensive model of film coalescence. J. Appl. Polym. Sci. 1994, 53, 1139-1147. (24) Mazur, S.; Plazek, D. J. Viscoelastic effects in the coalescence of polymer particles. Prog. Org. Coatings 1994, 24, 225-236. (25) Dobler, F.; Pith, T.; Lambla, M.; Holl, Y. Coalescence mechanisms of polymer colloids i. coalescence under the influence of particle-water interfacial tension. J. Colloid Interface Sci. 1992, 152 (1), 1-11. (26) Dobler, F.; Pith, T.; Lambla, M.; Holl, Y. Coalescence mechanisms of polymer colloids ii coalescence with evaporation of water. J. Colloid Interface Sci. 1992, 152 (1), 12-21.
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inhomogeneity in the particle volume fraction, controlled by the fluid mechanics in the first stage of film formation. The mechanisms discussed above have been investigated recently using atomic force microscopy27-32 and other techniques. Lin and Meier27 were the first to demonstrate that the troughs from particles at the surface of films could be detected using atomic force microscopy (AFM). They collapsed data for many different temperatures with conventional Williams-Ferry-Landel (WLF) parameters for time-temperature superposition, demonstrating that the deformation of the film is controlled by the viscoelastic response of the polymer. Vandezande and Rudin33 examined gravimetrically the effect of surfactant and particle size, showing how hydroplasticization of the polymer can affect the film formation process. Keddie et al.20,34 used ellipsometry and environmental scanning electron microscopy to follow the kinetics of film formation, observing the size of the voids to decrease in a manner consistent with the viscous collapse of a bubble. The last step in the film-formation process, the diffusion of polymer across the interparticle boundaries to give a structure with mechanical integrity, has been less contentious than the particle deformation step. Pekcan et al.35,36 examined the process using fluorescence techniques and found the classical theories of reptation to fit the observations. In this paper we wish to examine the different deformation processes and elucidate the conditions under which each applies. This initially requires a viscoelastic model for the deformation of two spheres subject to various forces. Rather than adapting the models of Eckersley and Rudin23 or Mazur,24 a viscoelastic generalization of Frenkel’s model is derived. From this microscopic model, equations governing the macroscopic deformation of films are formulated. Assuming a simple viscoelastic fluid model for the material response, we then determine the controlling dimensionless groups and explore the effect of temperature. Wet sintering requires the collapse of the film to be faster than evaporation, necessitating either a very slow evaporation rate, large surface tension, or a low modulus polymer. This is normally seen at process temperatures well above the glass transition temperature of the polymer. During capillary deformation, the film (27) Lin, F.; Meier, D. J. A study of latex film formation by atomic force microscopy. 2. film formation vs rheological properties: Theory and experiment. Langmuir 1996, 12, 27774-2780. (28) Gilicinski, A. G.; Hegedus, C. R. New applications in studies of waterborne coatings by atomic force microscopy. Prog. Org. Coatings 1997, 32, 81-88. (29) Gerharz, B.; Kuropka, R.; Petri, H.; Butt, H. J. Investigation of latex particle morphology and surface structure of the corresponding coatings by atomic force microscopy. Prog. Org. Coatings 1997, 32, 7580. (30) Joanicot, M.; Granier, V.; Wong, K. Structure of polymer within the coating: an atomic force microscopy and amall angle neutrons scattering study. Prog. Org. Coatings 1997, 32, 109-118. (31) Park, Y. J.; Khew, M. C.; Ho, C. C.; Kim, J. H. Kinetics of latex film formation in the presence of alkali soluble resin using atomic force microscopy. J. Colloid Polym. Sci. 1998, 276 (8), 709-714. (32) Park, Y. K.; Lee, D. Y.; Khew, M. C.; Ho, C. C.; Kim, J. H. Atomic force microscopy study of Pbma latex film formation: effects of carboxylated random copolymer. Colloids Surf. A 1998, 139, 49-54. (33) Vandezande, G. A.; Rudin, A. Film formation of vinyl acrylic latexes; effects of surfactant type, water and latex particle size. J. Coatings Technol. 1996, 68 (860), 63-73. (34) Keddie, J. L.; Meredith, P.; Jones, R. A. L.; Donald, A. M. Film formation of acrylic latices with varying concentrations of non-filmforming latex particles. Langmuir 1996, 12 (16), 3793-3801. (35) Peckan, O.; Canpolat, M.; Gocmen, A. Characteristics of chain diffusion during latex film formation using steady state fluorescence technique. Eur. Polym. J. 1993, 29 (1), 115-120. (36) Pekcan, O.; Canpolat, M. Direct fluorescence technique to study evolution in transparency and crossing density at polymer-polymer interface during film formation from high-t latex particles. J. Appl. Polym. Sci. 1996, 59, 277-285.
Routh and Russel
Figure 1. Two viscoelastic particles deforming.
compaction is controlled by the rate of evaporation until the maximum capillary pressure is exceeded. Finally dry or moist sintering requires a slow evaporation rate, low surface tension, or a high modulus polymer, which is normally realized at process temperatures below the glass transition temperature of the polymer. 2. Deformation of Two Particles Consider two spheres of linear viscoelastic material pressed together by a force, F, acting along the line of centers, plus a surface tension of γ between the particles and the surrounding medium (Figure 1). The deviatoric stress τ and strain E are related through G(t - t′), the stress relaxation modulus of the material, by τ)
∫
t
-∞
dE G(t - t′) dt′ dt′
(8)
The deviatoric stress tensor is related to the total stress, σ, and the isotropic pressure, P, as σ ) τ - PI
(9)
where I is the identity tensor. Within the particles ∇‚σ ) 0
(10)
We intend to volume average (8) to relate the overall deformation to the driving forces. Note that
∫ E dV ) ∫ 21(∇s + ∇s ) dV ) ∫ 21(ns + sn) dA T
V
V
A
(11)
along with
∫ σ dV ) ∫ n‚σx dA V
A
(12)
where n is a unit normal from the surface of the particles, s is the displacement in the particles, and x is a position vector. The volume and surface integrals are over two particles as shown in Figure 1. Following Frenkel12 we assume the two particles to deform as truncated spheres, implying a sufficiently large surface tension to maintain the spherical shape throughout the deformation. For elastic particles the analyses of Hertz and Johnson, Kendall and Roberts15 determine the shape to be more complex near the contact line via either an energy minimization (JKR) or stress continuity (Hertz) approach. We prefer to avoid these more complex analyses and, therefore, expect a discrepancy in our final answer in the elastic limit, which will be assessed later. Because the conservation equations are not being solved locally, the displacement at the surface of the particles
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Langmuir, Vol. 15, No. 22, 1999 7765
Figure 2. Figure showing displacement at surface of particles. Figure 3. Singularity at center of two deforming particles.
must be approximated, subject to the requirement of incompressibility expressed globally as Trace (
∫ E dV) ) 0
Our construction of the displacement at the surface (Figure 2) of each particle s ) (RR0ez + S(z)ez + (R - R0)n
(14)
allows for solid body motion toward the contact line via the first term and expansion of the spheres to conserve volume in the last term. The second term, assumed to be in the ez direction, will be constructed to ensure incompressibility. The radius at any time is related to the original radius through
(
)
(15)
The stress at the surface of the particles is related by the boundary condition to the surface tension and the applied force as n‚σ )
γ 2γ n + eGδ(z) ( Fδ(x2 + y2)ez R L
(16)
are one- and two-dimensional where δ(z) and δ(x + Dirac delta functions located at z ) 0 (Figure 1) and the poles of the two particles, respectively. The first term accounts for the surface tension induced stress and the last term applies the forces pushing the particles together at the poles. The second term accounts for the abrupt change in curvature at the contact line between the two particles. Hence L is the curvature in this singular region and eF is a unit vector normal to the line joining the centers (Figure 3). For two truncated spheres the surface is described by 2
y2)
x2 + y2 ) f(z)
(17)
where f(z) is the square of the radius of the cross section f(z) ) -(z - R0(1 - R))2 + R2
(18)
for z > 0 (-) and z < 0 (+). The unit normal n follows as n)
2xex + 2yey - f ′(z)ez (4f(z) + f ′2(z))1/2
∫ /dA ) ∫∫
(13)
V
R2 R3 5R4 + + O(R5) R ) R0 1 + 4 24 192
The surface integral can be expressed as
(19)
A
2π
0
/[4f(z) + f ′2(z)]1/2
dφ dz 2
(20)
where φ is a radial angle and / is any function. Substitution of (14), (15), (19), and (20) into (11) gives
∫
[(
E dV ) 4πR03R2 1 - R + V
(
)
R2 + O(R3) ezez + 4
)]
2
R 17R 1 + O(R3) I - π f ′(z) S(z) dz ezez 1+ 6 6 48 (21)
∫
The first term corresponds to the solid body displacement (RR0ez and the second accounts for volume conservation (R - R0)n, while the last term ensures that the strain and deviatoric stress tensor are traceless. Substitution of (15), (16), (19), and (20) into (12) provides
(
)
2
R 16 σ dV ) πR02γ 1 + O(R3) I V 3 4 R R2 + O(R3) ezez (22) 4FR0 1 - + 2 8
∫
[
]
plus a further term from the stress singularity at the contact line between the two particles. Considering a hemispherical surface of radius L joining the two particles, as shown in Figure 3, and performing the surface integral as L f 0 give the additional term as
(
4πR02γR 1 -
)
5R R2 + + O(R3) (exex + eyey) 4 24
(23)
An easier way to derive (23) is to take the stress as γ/L and the volume as 2π f(0)LR0(1 - R)/(R + L), thereby arriving at the expression in (23). This term represents material being pulled into the contact region between the particles by the negative pressure generated by the large curvature at this point. Substitution of (21), (22), and (23) into a volume averaged (8) with the tracelessness of τ gives the average isotropic pressure in the particles as 〈P〉 ) -
(
2
)
R 7R 2γ + O(R3) + 1+ R0 2 8
(
2
)
R R F 1- + + O(R3) (24) 2 2 8 2πR0
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Routh and Russel
as well as an equation governing the viscoelastic deformation of two spheres 2F 2
πR0
[
1-
]
R R2 + + O(R3) + 2 8
(
2
)
5R R 2γ + + O(R3) ) 1R0 R 4 24
[ (
d ∫ G(t - t′)dt′ t
0
2
R
1+
R 17R2 + O(R3) 6 48
)]
dt′ (25)
(26)
In this case, (25) becomes to lowest order R ) γt/R0η
(27)
which is the Frenkel result, eq 1, to within a multiplicative constant. Conversely we would not expect to obtain the Hertz or JKR results15 in the elastic limit with the stress relaxation modulus given by G(t - t′) ) G
( ) 2F πGR02
1/2
(29)
This errs in the exponent of 1/2, which should be 2/3. This difference in scaling was examined experimentally by Mazur,24 who was unable to distinguish between the two scalings for data with a number of different polymers. With only surface tension effects we obtain, for comparison with (3)
The term containing S(z) in (21) vanishes, having been chosen to make the strain tensor traceless. Equation 25 is derived from a macroscopic view of surface tension. Alternatively, at least a portion of the forces that produce surface tension can be represented in terms of long-range dispersion attractions, producing considerable contention in the adhesion literature.37-39 These analyses for elastic bodies suggest that long-range attractions might be important for 100 nm polymer latices in water. For materials with a lower elastic modulus, the size of particles for which the long range attractions are important reduces. Recently Jagota et al.40 incorporated dispersion forces into the viscoelastic deformation of a sphere against a flat plate. A rapid initial response appears due to a zippering action at the join just outside the contact line, followed by slower viscous sintering due to a stretching mechanism analogous to the negative pressure noted above. For particles of 50 nm or smaller, Jagota et al.40 predict complete deformation before the onset of viscous flow. For two particles, the long-range dispersion force also produces additional deformation at short times, but the geometry weakens the effect. Thus for particles larger than 50 nm we find the initial zippering response to be negligible and the stretching mode captured by the macroscopic approach to be dominant. Because of the assumption that the particles remain spherical throughout the deformation process, our pair model should reproduce the result of Frenkel12 in the viscous limit with the stress relaxation modulus given by G(t - t′) ) ηδ(t - t′)
R )
(28)
Instead, to lowest order, we obtain, for comparison with the Hertz result (2) (37) Derjaguin, B. V.; Muller, V. M.; Toporov, Y. P. Effect of contact deformations on the adhesion of particles. J. Colloid Interface Sci. 1975, 53 (2), 314-326. (38) Muller, V. M.; Yushchenko, V. S.; Derjaguin, B. V. On the influence of molecular forces on the deformation of an elastic sphere and its sticking to a rigid plane. J. Colloid Interface Sci. 1980, 77 (1), 91-101. (39) Muller, V. M.; Yushchenko, V. S.; Derjaguin, B. V. General theoretical consideration of the influence of surface forces on contact deformations and the reciprocal adhesion of elastic spherical particles. J. Colloid Interface Sci. 1983, 92 (1), 92-101. (40) Jagota, A.; Argento, C.; Mazur, S. Growth of adhesive contacts for maxwell viscoelastic spheres. J. Appl. Phys. 1998, 83 (1), 250-259.
R ) 2γ/R0G
(30)
which again is off in the scaling, although the magnitude of the error is likely to be small. Clearly for R ) O(1) the errors due to different exponents are also O(1). 3. Deformation in a Macroscopic Film From our microscopic description of the deformation of two isolated particles, we wish to derive equations for the macroscopic deformation in a film by averaging the stress over a representative volume. This bulk stress is comprised of the particulate stress 〈σ〉p and an isotropic fluid pressure p, 〈σ〉 ) φ〈σ〉p - (1 - φ)pI
(31)
with φ the volume fraction of particles. Within the film ∇‚〈σ〉 ) 0
(32)
which for a one-dimensional compaction becomes 〈σzz〉 ) constant ) σt
(33)
where σt is the stress at the top surface of the film. The transverse stresses, 〈σxx〉 and 〈σyy〉, are also nonzero and spatially uniform. Combining the conservation equation for water with Darcy’s law for flow in a packed bed gives
[
]
kp ∂p ∂ ∂ (1 - φ) (1 - φ) )0 ∂t ∂z µ ∂z
(34)
where kp is the particulate permeability and µ the solvent viscosity. Scaling time on evaporation time (H/E˙ ) with E˙ the evaporation rate, permeability on particle radius squared (R02), vertical height on the initial film thickness (H), and pressure on the capillary stress at the top of the film (σt ∼ γwa/R0), suggests that ∆p HµE˙ ∼ ,1 σt R0γwa
(35)
for typical values of H ≈ 10-4 m, µ ≈ 10-3 Ns/m2, R0 ≈ 10-7 m, γwa ≈ 0.07 N/m, and E˙ ≈ 3 mm/day. This implies that under typical conditions all the stress in the film is carried by the particulate network, hence 〈σ〉 ≈ φ〈σ〉p
(36)
Within a representative volume we focus on an individual particle and integrate over the configurations of its neighbors. Following Batchelor41 the volume average takes the form (41) Batchelor, G. K. The stress system in a suspension of force-free particles. J. Fluid Mech. 1970, 41, 545-570.
A Process Model for Latex Film Formation
〈σ〉p )
1
1
1
∫ σ dV ) V∫ xσ‚n dA ) V ∑ V V
A
Langmuir, Vol. 15, No. 22, 1999 7767
xF
(37)
contacts
where V is the volume of integration. An overall force balance for each particle requires
∑
F)0
(38)
contacts
Choosing the position vector, x ) Xi + R0(1 - Rij) mij, where Xi is the position of the center of particle i and mij is a unit vector pointing between the centers of particles i and j, reduces (37) to Figure 4. Definition of unit vector m.
〈σ〉p )
1 V
∑R (1 -
Rij)mijFij
0
j
(39)
where Fij is the force acting on particle i due to particle j. As demonstrated by Batchelor41 surface tension does not appear explicitly, so Fij is the force that appears in (25). For a homogeneous film with a strain field E, affine deformation sets the relative change in distance between two particles as Rij ) mij‚E‚mij
(40)
〈σ〉zzp )
〈σ〉xxp )
(43)
gives the interparticle strain as 2
R ) cos β
(44)
Assuming all orientations of particles to be equally likely and averaging as where ν is the average number of nearest
∑
/)
contacts
∫ ∫ 2π
0
π
0
ν
2π2
/ dθ dβ
〈σ〉p )
∫
ν 2πV
π
β)0
R0(1 - cos2 β)F(R) [sin2 β(exex + eyey) + cos2 βezez] dβ (46)
The expression for F(R) follows from (25) and the volume of integration V is that of a single particle. The vertical and transverse components of the stress tensor become after integration
[(
)∫
3ν 5 212 t d 2 dt′ + 1G(t - t′) 256 16 128 0 dt′ 7 t d 3 5 dt′ 1G(t - t′) 48 20 0 dt′ d 4 119 t dt′ G(t - t′) 768 0 dt′ 7 352 4γ 1+ + O(3) + O(G5) (48) R0 8 384
)∫
(
∫
]]
9νφmγ (1 + O()) ) 64R0 15νφm t d 2 dt′ (1 + O()) (49) G(t - t′) 256 0 dt′
∫
at lowest order, where σt, the stress at the top surface is generated by the curvature of the air-water interface between particles. The second term represents sintering driven by the surface tension, γ, and the third term is the material response. For a viscoelastic fluid with a single relaxation time the stress relaxation modulus is
(45)
neighbors, produces
]]
With (36), (47), and the overall conservation expression φ ) φm/(1 - ) the equation for compaction of the film reduces to σt +
E ) ezez
∫
[
with
as shown in Figure 4. A one dimensional compaction
)∫
(
and
(41)
mij ) sin θ sin β ex + cos θ sin β ey + cos β ez (42)
)∫
[
For the present we consider only forces that act along the line of centers, hence Fij ) Fmij
[(
15ν 7 1892 t d 1G(t - t′) 2 dt′ + 256 16 640 0 dt′ 9 t d 3 7 dt′ 1G(t - t′) 48 20 0 dt′ d 4 357 t dt′ G(t - t′) 1280 0 dt′ 35 2452 12γ + + O(3) + O(G5) (47) 15R0 24 1152
G(t) ) G∞′ exp
(
)
-G∞′t + η∞′δ(t) η0 - η∞′
(50)
where η0 and η∞′ are the low-shear and high-frequency (short time) viscosities, G∞′ is the high-frequency modulus, and the relaxation time is (η0 - η∞′)/G∞′. The modulus diverges at very short times but eventually decays to zero. Substituting (50) into (49) allows us to derive a differential equation that, to lowest order in , governs the compaction process dσ jt dth
+G hσ jt +
(
)
( )
6γ j d λh d d d +G h ) +G h λh 5 dth η j dth dth dth
(51)
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Table 1. Dimensionless Groups Controlling Deformation group σ jt γ j G h λh η j
definition
128σtR0 15νφmγwa γ γwa G∞′H
physical meaning
stress at top surface (σt) capillary pressure (γwa/R0) surface tension (γ) water-air surface tension (γwa) evaporation time (H/E˙ ) polymer relaxation time ((η0 - η∞′)/G∞′)
E˙ (η0 - η∞′) η0R0E˙ γwaH
time for viscous collapse (η0R0/γwa)
η0 η∞′
high-frequency viscosity (η∞′)
evaporation time (H/E˙ ) low-shear viscosity (η0)
Here time has been scaled on an evaporation time (H/E˙ ) leading to the dimensionless groups controlling the process in Table 1. The initial condition for the deformation is taken as a close-packed array of particles with the water level just filling the interstices. The validity of this homogeneous assumption is examined later. This analysis is for a uniaxial deformation. Winnik7 discusses the differences between uniaxial and isotropic deformation, concluding that, in many instances, no discernible decrease in surface area of films is seen so uniaxial deformation is the only option.
Figure 5. Surface defining wet sintering.
5. Capillary Deformation For capillary deformation alone to drive pore closure the water-air interface must remain at the surface of the film, with the pressure never rising above the maximum sustainable value, set by the water-air surface tension and the original particle radius. Our conditions therefore become
4. Wet Sintering The easiest case to examine is that of wet sintering, which is driven entirely by the polymer-water surface tension with no contribution from capillary deformation. In (51) we set σt to zero, replace γ with γpw, and insist that the strain in the film be greater than that driven by evaporation g ht
(52)
which ensures a layer of water above the compacting film and zero capillary pressure at the top surface. The resulting equation is
(
)
( )
6 d λh d d G h λh d +G h ) + 5 dth η jγ j dth dth γ j dth
(53)
Equations 52 and 53 define a surface in the G h, η j , λh/γ j space (Figure 5) that is calculated by picking values for G h and η j and numerically searching for the maximum value of λh/γ j that allows (52) to be satisfied. In Figure 5, conditions below the surface correspond to surface tensions sufficiently large to reduce the film height faster than evaporation. Therefore the capillary pressure at the top surface is always zero and the film forms by wet sintering. Above the surface, a capillary pressure appears at the top surface at some time during the process, so deformation will not be by wet sintering alone. For large values of G h , the relaxation time of the polymer is small compared to the evaporation. Insisting that the rate of viscous collapse, based on the low shear viscosity, (∼η0R0/γpw) be less than the evaporation rate (∼H/E˙ ), gives the criterion for wet sintering as λh/γ j < constant. Equation 53 gives this constant as 6/5. For small G h , the polymer will not relax during evaporation, and the rate of viscous collapse is controlled by the high-frequency (short time) viscosity. This gives the criterion for wet sintering as λh/γ jη j < 6/5. The above features are seen in Figure 5.
) ht
(54)
j max σ jt e σ
(55)
Following Brown,1 the maximum capillary pressure at the top surface is approximately 12.9γwa/R0. Taking the number of nearest neighbors ν ) 5-12 and close packing of the undeformed spheres, φm, to be between 0.64 and 0.74 implies a maximum value of σ j max ) 14-43. Assuming the effect of wet sintering to be small, substituting (54) into (47) and using (36) along with (50) leaves
[(
)
λh 25th 109th2 ht 1 + + η j 32 320 9th 171th2 η j-1 + (G h ht - 1 + e-Gh ht ) 1 + + 16 640 G h 21(η j - 1) 11th 1+ [2 - 2G h ht + G h 2ht 2 - 2e-Gh ht ] 2 20 96G h 357(η j - 1) (6G h ht - 6 - 3G h 2ht 2 + G h 3ht 3 + 6e-Gh ht ) + O(th4) 640G h3 (56)
σ j t(th) )
[
(
)
(
)]
]
Setting σ j t ) 20 at the largest time, which for onedimensional compaction from φm ) 0.64 is ht ) 0.36, then defines the surface in G h, γ j , λh space shown in Figure 6. For conditions below the surface in Figure 6, the deformation of the film is controlled by evaporation and the stress never rises above the maximum sustainable value. Therefore, the water-air interface is held at the surface of the film and deformation is by capillary deformation. Above the surface, the water level will recede into the film at some time prior to void closure, shown schematically in Figure 7. The difference between Figures 5 and 6 is that the vertical axis for the wet sintering case recognizes the polymer-surface tension as the driving force for compac-
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Langmuir, Vol. 15, No. 22, 1999 7769
Figure 8. Schematic of deformation in film with water front receding.
Figure 6. Surface defining capillary deformation.
Figure 7. Water front receding into film.
tion (λh/γ j on vertical axis), whereas capillary deformation depends on the water-air surface tension (λh on vertical axis). For large G h , the stress generated by the deformation will scale as low-shear viscosity multiplied by evaporative shear rate (η0E˙ /H). This will balance the stress at the top surface, which is generated by the water-air surface tension (γwa/R0). Our condition for capillary deformation at large G h will, therefore, be λh < constant. In the limit of G h . 1, eq 56 identifies the limit as λh e 44.1. For small G h, the stress in the film will scale with the high-frequency (short time) viscosity, and our condition for capillary deformation becomes λh/η j e constant. In the limit G h , 1, eq 56 sets the condition as λh e 44.1η j . As for wet sintering, η j is only relevant when the polymer relaxation time is on the order of the evaporation time, or for small G h . This is all seen in Figure 6. For elastic particles (49) gives the criterion for void closure by capillary forces as GR0/γwa < 309
(57)
which should be compared with the earlier criterion of Brown4 and Mason.5 This elastic condition is realized on Figure 6 as λhG h < 309 for η j . 1 and G h , 1. For higher G h the effect of viscous flow means that this elastic condition lies below the surface in Figure 6. 6. Receding Water Front and Dry/Moist Sintering As discussed earlier, if the stress at the top surface reaches its maximum sustainable value, the level of the water will recede into the film as shown in Figure 7. Below this water front the particles will continue to deform by a combination of capillary deformation and wet sintering, according to eq 51, with the stress, σ j t, set at its maximum value. Above this water front there are two possibilities. If the external humidity is low enough, or the particles sufficiently hydrophilic to absorb any residual water, the
Figure 9. Surface defining dry sintering.
particles will be essentially dry and deformation will proceed via dry sintering as governed by (53), although with the polymer-water surface tension replaced by the polymer-air surface tension. If the external humidity is large enough to leave residual water rings, then deformation will proceed via moist sintering as described by Lin and Meier.21 The overall process is shown schematically in Figure 8. Up until time t*, the stress at the top surface is below the maximum value and the strain in the film is set by the evaporation rate, as in the capillary deformation case. For dry or moist sintering to dominate, the conditions must lie well above the capillary deformation surface (Figure 6). We therefore consider dry or moist sintering to be dominant when less than 15% of the deformation j) occurs under capillary deformation. Setting σ j t ) 20, γ 0, and integrating (51) twice leaves an expression for the strain under the receding water front as 2 )
[
40 1 -Gh ηj ht - 1) + ht (e λh G hη j
]
(58)
Then assuming evaporation to continue unhindered as the water front recedes and setting ) 0.054 at ht ) 0.36 defines a surface for dry and moist sintering, Figure 9. Above this surface, more than 85% of the deformation is by either a dry or moist sintering method, depending on the external humidity. For large values of G h , the highfrequency viscosity of the polymer becomes unimportant, and the condition for dry sintering becomes λh > 4900. For low values of G h the high-frequency viscosity becomes the relevant viscosity and expanding the exponential in (58)
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Routh and Russel
for G hη j , 1 leaves )
( ) 20G hη j λh
1/2
ht
(59)
With our previous condition on the criterion for dry or moist sintering becomes λh > 900G hη j. 7. Approximation for Large η j If we assume that η j . 1, the relaxation modulus of the polymer no longer diverges at very short times, allowing an instantaneous elastic response. This changes the initial condition on the deformation with the strain no longer starting at zero. Removing the group η j also reduces the surfaces derived earlier to lines. For the wet sintering case (53) becomes d dth
+G h )
5G h λh d 6γ j dth
(60)
Straightforward integration produces - * -
6γ j 6γ j ln ) ht * 5G h λh 5λh
(61)
where *, the strain at zero time, follows from (49) as * )
12γpw 12γ j 12 η j γ j ≈ ) 5R0G∞′ 5 η j-1G h λh 5G h λh
(62)
Therefore setting ) 0.36 at ht ) 0.36 defines a line that limits wet sintering in G h , λh/γ j space to λh 6 20 10 3G h λh + ln < + γ j 5 3G 20γ j h 3G h
( )
(63)
In the limit G h . 1, this condition for wet sintering reduces to λh/γ j < 6/5. For capillary deformation (56) becomes
[(
)(
)
9th 171th2 1 + ht + (e-Gh ht - 1) + 16 640 G h 21 11th 2 2 1+ ht + 2(1 - G h ht - e-Gh ht ) 96 20 G h
σ j t(th) ) λh 1 +
(
(
)(
(
2 2
6 G h ht 357 3 + e-Gh ht ht + 3 G h ht - 1 640 2 G h
)]
)
+ O(th4) (64)
Again setting σ j t(th ) 0.36) ) 20 defines h - 1.21G h 2 + 0.45G h3 λh < 20G h 3/[3.35 - 0.68G -0.36G h 3.35e (1 + 0.16G h - 0.37G h 2)] (65)
as the limit of capillary deformation. For G h . 1 the condition for capillary deformation becomes λh < 44.1. For dry and moist sintering, (58) becomes λh > 4900, independent of G h. The upper limits for wet sintering and capillary deformation along with the lower limit of dry sintering are shown in Figure 10 for γpw/γwa ) 0.2 as estimated by Eckersley and Rudin.23 Irrespective of this value, the three lines are widely spaced. Wet sintering controls for λh e O(1), capillary deformation for O(1) e λh < O(102), and dry or moist sintering for λh g O(104).
Figure 10. Lines defining the limit of wet sintering, capillary deformation, and dry sintering for η j . 1 taking γpw/γwa ) 0.2.
parameter space dramatically, allowing different mechanisms to be observed. The film forming bar, as reported by Sperry et al.,19 is a convenient way to observe different process temperatures. Here a film is spread on a bar with a well-defined temperature gradient and allowed to dry. For our simple viscoelastic fluid, the high-frequency modulus, G∞′, is invariant to temperature but both the low-shear and highfrequency viscosities decrease with temperature η0(T) ) aTg(T)η0(Tg)
(66)
η∞′(T) ) aTg(T)η∞′(Tg)
(67)
G∞′(T) ) G∞′(Tg)
(68)
where aTg is a temperature shift factor following from WLF theory42 as log10 aTg(T) )
-c1(T - Tg) c2 + T - Tg
(69)
with c1 and c2 material constants referred to the glass transition temperature Tg. For simplicity we ignore the fact that shift factors generally differ at high and low frequencies. A further complication is the dependence of the mass transfer limited evaporation rate, E˙ , on temperature. However, Sperry et al.19 estimated only a factor of 2 variation over a temperature range of say 15 °C, which is small compared to the large variations in the rheological properties. Considering the evaporation rate to be independent of temperature lets the dimensionless groups be expressed as functions of temperature according to 1 G h (Tg) aTg(T)
(70)
λh(T) ) aTg(T) λh(Tg)
(71)
η j (T) ) η j (Tg)
(72)
G h (T) )
8. Temperature Effects
Therefore G h increases and λh decreases rapidly with temperature, suggesting that deformation at high temperatures will be rapid and controlled by wet sintering. As the temperature falls, λh will increase and control will pass to capillary deformation. As the temperature is lowered further, considerable deformation will occur
Changing the temperature alters the rheological properties and moves the film formation process around in
(42) Ferry, J. D. Viscoelastic Properties of Polymers; John Wiley and Sons: New York, 1961.
A Process Model for Latex Film Formation
Langmuir, Vol. 15, No. 22, 1999 7771
Table 2. Effect of Temperature on Deformation Mechanisms and Time for Complete Deformation for 100 nm Particles temperature/°C
aT
G h
λh
mechanism
htcomp
Tg - 5 Tg - 4 Tg - 3 Tg - 2 Tg - 1 Tg Tg + 1 Tg + 2 Tg + 3 Tg + 4 Tg + 8 Tg + 13 Tg + 14 Tg + 15
40 17.9 8.3 4.0 1.97 1 0.52 0.28 0.15 0.086 0.011 0.0011 7.4 × 10-4 5 × 10-4
5.75 12.8 27.6 57.8 116 230 439 821 1500 2670 22000 210000 310000 460000
12000 5370 2490 1190 591 300 157 84.0 46.1 25.4 3.15 0.33 0.22 0.15
dry/moist sintering dry/moist sintering receding water front receding water front receding water front receding water front receding water front receding water front receding water front capillary capillary capillary/wet sintering wet sintering wet sintering
3600 1612 730 351 161 74.2 31.9 11.0 1.02 0.36 0.36 0.36 0.33 0.23
during the recession of a water front vertically through the film. At lower temperatures still λh is so large that evaporation is complete before any significant deformation, leaving either dry or moist sintering to complete the process. The experimental observation on the bar is of cracking in the film at and below a certain temperature and a cloudy to clear transition at a slightly lower temperature. The higher of these two temperatures is called the minimum film formation temperature. This temperature is typically a few degrees below the glass transition temperature of the polymer but can be lowered considerably by filmforming aids such as 2,2,4-trimethyl-1,3-pentanediol monoisobutyrate (Texanol). The cloudy to clear transition indicates sufficient particle deformation that the void size falls below the minimum size to scatter light. We take the complete closure of voids to indicate complete film formation, a somewhat different criterion than employed by Sperry et al.,19 who considered the cloudy to clear transition. Fitting our viscoelastic fluid model to data42 for 600 kg/mol polystyrene at Tg supplies estimates for the rheological parameters as G∞′ ∼ 5.4 × 1010 N/m2
(73)
η0 ∼ 6.5 × 1011 N s/m2
(74)
∼ 100η∞′
(75)
Therefore for 0.1 mm thick films, an evaporation rate of 3 mm/day, and γpa ) γwa ) 5γpw, we estimate values for G h, γ j, η j , and λh at different temperatures for different size particles and predict both the deformation mechanism and time for pores to completely close. For high temperatures, the wet sintering regime lies above the temperature defined by aTg(T) )
6γpwH 5E˙ R0η0(Tg)
(76)
and the time for complete elimination of voids is ht comp )
0.3η0R0E˙ Hγpw
(77)
Below this wet sintering region and above the temperature given by aTg(T) )
44.1γwaH E˙ R0η0(Tg)
lies the capillary deformation regime. In this regime, the deformation is evaporation controlled and the time for complete deformation satisfies htcomp ) 0.36. At lower temperatures, a water front will recede through the film, making the deformation inhomogeneous. The time for film formation is taken to be the time for the top layer of particles to reach a volume fraction of unity. Initially the stress at the top surface builds up according to (56) and the strain follows evaporation. At some critical time, ht*, given by (56) the stress reaches its maximum value and the water front recedes through the film. Deformation then continues by either a dry or moist sintering method. The time for pore closure follows as ht comp ) ht * +
5η0R0E˙ (1 - φm - ht *) 6Hγpa
(79)
assuming dry sintering above the water front. At sufficiently low temperatures, evaporation is much faster than deformation and the process is controlled by dry or moist sintering. The limiting temperature, given by aTg(T) )
4900γwaH E˙ R0η0(Tg)
(80)
contains a somewhat arbitrary factor of 4900 as discussed in section 6. The corresponding time for void closure is ht comp )
0.3η0R0E˙ Hγpa
(81)
if by dry sintering. Table 2 gives these times for 100 nm particles. Figure 11 shows the time for complete film formation for four different particle sizes (50, 100, 300, and 500 nm) as the temperature is varied. The wet sintering regime is only seen at high temperatures and for the smallest particles (50 and 100 nm). The capillary deformation case persists to lower temperatures for smaller particles and the much longer time for deformation by dry sintering produces a large jump at the transition to the receding water front. This then asymptotes to the dry sintering line. Figure 12 generalizes Figure 11 to be independent of the type of polymer in question as well as particle size, evaporation rate, and magnitude of the surface tensions and, thereby, completely describes the regimes of film formation in the context of this model. 9. Sheetz Deformation
(78)
Throughout the previous analysis we assume an initially uniform close packed volume fraction, which ensures
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Figure 11. Time for complete film formation at different temperatures for 50, 100, 300, and 500 nm particles. Figure 14. Regions defining skinning and inhomogeneous deformation for φm ) 0.64.
assuming a uniaxial compression and noting that for wet sintering significant deformation only occurs at elevated temperatures where G h . 1, the strain in the solid region follows from ∂ ∂th
)
6γ j 5λh
(84)
Relating the strain to the volume fraction as φ) Figure 12. Generalization of time for complete deformation in film.
homogeneous deformation in the film. Relaxing this assumption necessitates examination of the fluid mechanical stage of film formation. Following our previous treatment,10 we consider an infinite expanse of fluid of initial volume fraction, φ0, and constant rate of evaporation, E˙ . Vertical inhomogeneity is ensured by completely neglecting diffusion so that evaporation from the upper surface generates a close packed layer of particles (Figure 13). The particles originally pack at a volume fraction φm and then deform to give a range of volume fractions through the packed layer. An overall material balance gives dh h /dth ) -1
(82)
and a water balance leaves dh hp dth
)
-φ(h h) 1 + φm - φ0 φm - φ0
∫
h h
h hp
∂φ dxj ∂th
(83)
In the close-packed region the only driving force for deformation is the polymer-water surface tension. Again
(85)
and expanding for small strain allow the height of the solid front to be calculated as h h p ) 1 - 2th +
Figure 13. Schematic of inhomogeneous drying.
φm 1-
[
]( (
) )
6γ j φmht 5λh 2φ0 - φm exp -1 6γ j φm 5λh(φm - φ0)
(86)
From this a number of possibilities arise. From (84) and (85), the volume fraction at the top surface reaches unity at ht ) 5λh(1 - φm)/6γ j . If this occurs before the particle front reaches the substrate, the rate of evaporation will drop considerably as a skin forms on the top of the film. Substituting this ht into (86) and insisting that h h p > 0 then identifies the value of λh/γ j below which this occurs as a function of the initial volume fraction for φm ) 0.64 (Figure 14). For an effectively homogeneous deformation, as assumed in the previous examples, we wish the strain at the top surface to be small when the solid region encompasses the entire film. Setting this small strain to be 0.054, i.e., 15% of the total strain to form a film, then defines a value for λh/γ j above which the film may be considered homogeneous. This is also shown in Figure 14. In the region labeled “skinning” the top of the film reaches a volume fraction of unity before the entire film reaches close packing, and evaporation necessarily slows relative to the rate of deformation. Consequently the film may eventually form by wet sintering, and the time for complete evaporation will be large. In the region labeled “inhomogeneous” the volume fraction at the top surface is larger than through the rest of the film and will consequently reach a volume fraction of unity before the rest of the film, slowing evaporation and allowing film formation to proceed by both the wet sintering and capillary methods. For sufficiently large values of λh/γ the film is effectively homogeneous and deformation proceeds
A Process Model for Latex Film Formation Table 3. Typical Values of Variables symbol
meaning
typical value used
R0 γwa H E˙ φm µ G∞ ′ η0(Tg) η∞′(Tg)
original particle radius water-air surface tension original film thickness evaporation rate close-packed volume fraction solvent viscosity high-frequency modulus low-shear viscosity at Tg high-frequency viscosity at Tg
100 nm 0.073 N/m 10-4 m 0.3 cm/day 0.64 10-3 N s/m2 5.4 × 1010 N/m2 6.5 × 1011 N s/m2 6.5 × 109 N s/m2
as described previously. For 100 nm particles with an initial volume fraction of 0.4, a value of γ j ) 0.2, and the rheological estimates of earlier, the film may be considered homogeneous for all temperatures below 10 °C above Tg. 10. Discussion We have derived surfaces in a three-dimensional parameter space defining the regimes in which wet sintering, capillary deformation, and dry or moist sintering dominate. Below the wet sintering surface deformation is exclusively due to polymer-water surface tension. Above this surface but below the capillary deformation surface, the strain is set by the evaporation rate. Deformation is then due to the capillary pressure generated at the airwater interface, with a small contribution from polymerwater surface tension. Above a third surface associated with dry or moist sintering, evaporation is complete before significant ( 0.93. On the other hand Vanderhoff et al.44 observed a drop in evaporation rate after the particles come into contact with thick films (1 mm). The glass transition temperature of the polymers used in the study of Vanderhoff et al.44 should be about 25 °C below room temperature. Therefore it seems likely that a surface layer will quickly form in the drying process, causing the large drop in the evaporation rate observed. Our results correlate well with the observations of Sperry et al.19 Up to a few degrees below the glass transition temperature we expect dry sintering to control. However, at higher temperatures the other deformation mechanisms become important. The capillary deformation region correlates with the observation of Brown that “film formation in many polymer emulsion systems occurs concurrent with the evaporation of water and is complete when water evaporation is complete”.1,45 Purely wet sintering seems harder to observe, unless one suppresses evaporation to allow a longer time for the deformation. Cracking in the film has not yet been considered, although this will follow from the transverse components of the stress tensor. The magnitude of interparticle diffusion of polymer chains, imparting mechanical strength to the film, then sets the necessary yield stress and should determine the onset of cracking. A further complication arises from hydroplasticization and the use of film-forming aids. Adsorption of the solvent into the polymer particles during the film-forming process will lower both the modulus and the glass transition temperature of the polymer, as well as the polymer-water surface tension. After complete evaporation of the solvent, the polymer reverts to its original unplasticized state. This allows films to form well below the nominal glass transition temperature, thereby producing coatings with superior mechanical and toughness properties. In this model the wet rheological values must be used. Acknowledgment. This research was supported by grants from Rohm and Haas, the Petroleum Research Fund, and a fellowship to A.F.R. from Rhodia. The authors are extremely grateful to Dr. Pete Sperry for very helpful discussions and guidance. LA9903090 (43) Brandup, J., Immergut, E. H., Eds. Polymer Handbook; John Wiley and Sons: New York, 1989. (44) Vanderhoff, J. W.; Bradford, E. B.; Carrington, W. K. The transport of water through latex films. J. Polym. Sci. 1973, 41, 155174. (45) The receding water front is reported by Keddie et al.,20,34 who refer to this regime as stage II*.
