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Deformation Mechanisms during Latex Film Formation: Experimental Evidence Alexander F. Routh† and William B. Russel*,‡ School of Chemistry, Cantock’s Close, Bristol BS8 1TS, England, and Department of Chemical Engineering, Princeton University, Olden Street, Princeton, New Jersey 08544
During the deformation stage of latex film formation, a close-packed array of particles is consolidated to form a structure with volume fraction unity. There are a number of different possible driving forces to achieve this. Proposed mechanisms range from surface tension between polymer particles and either water or air to capillary forces at the water-air interface. We review the different driving forces and the literature supporting them. A recent model we have proposed predicts the conditions under which each mechanism operates. Experiments in the literature correlate well with our model, although the need for experiments with well-defined values for the critical parameters is highlighted. 1. Introduction Latex film formation is a fascinating occurrence. It is essential to such products as paints, adhesives, caulks and sealants, paper coatings, textiles, and carpets, receiving attention from both academic and industrial researchers.1-6 The mechanism for film formation is shown schematically in Figure 1. A stable colloidal dispersion is applied to a substrate. Mass-transferlimited evaporation brings the particles into close packing. van der Waals and capillary forces deform the particles to render a structure without voids, although the individual particles are still distinguishable. Finally diffusion of polymer chains across interfaces between particles imparts mechanical strength to the film and blurs the distinction between particles. The first stage in film formation is observed to occur nonuniformly,2,7-10 stimulating a debate between groups who assume this drying boundary to control the physics and other groups who ignore it altogether. Recently, we developed a model based on surface tension driven flow to account for the horizontal propagation of drying fronts seen in these thin films.10 Although the drying front is shown to propagate about 100 film thicknesses into the film, this is also shown to not affect the deformation mechanism. Drying fronts are important in determining “open time” at the edges of the film, i.e., the time during which an adjacent or subsequent layer can merge homogeneously with the first, but do not alter the dominant physical mechanism for film formation. The second stage of film formation has received the most attention and is also the most controversial, with the driving force for deformation along with the response of polymer particles in the film in dispute. The final step of film formation, where polymer chains diffuse across particle boundaries, imparting strength to the film, has been examined by a number of techniques such as radiative energy transfer11-14 and neutron scattering.15-19 The mechanical strength of films * To whom correspondence should be addressed. Telephone: 609 258 4590. Fax: 609 258 0211. E-mail: wbrussel@ princeton.edu. † School of Chemistry. ‡ Princeton University.
Figure 1. Four stages of latex film formation.
has been shown to increase with the depth of polymer interdiffusion up to the radius of gyration.20,21 Recently, environmental considerations have provided a strong motivation for eliminating coalescing aids.7 These volatile organic components (VOC) are added to plasticize the particles, reducing the glass transition temperature and allowing easier particle deformation. Potential methods of eliminating VOCs include blending hard and soft particles22-25 so that soft particles deform around harder ones that impart mechanical strength. An important consideration is to avoid segregation within the dispersion, ensuring an even distribution of particles throughout the film. Alternatively, core-shell particles, with a soft deforming shell surrounding a hard core, offer a means for avoiding the problem of phase separation.26,27 An interesting aside is the issue of cracking.28 When films are cast on a metal bar with a temperature range spanning the glass transition temperature, a cloudyclear transition is observed at a well-defined temperature. This temperature associated with the cloudy-clear point is found to move to lower temperatures with time, indicating a kinetically controlled process. Below a different temperature, typically a few degrees above the cloudy-clear point, the film cracks with a crack spacing that decreases to an asymptotic value with decreasing temperature. These transition temperatures are called minimum film formation temperatures (MFT). The cracking will depend on the internal stresses in the film,
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measured by Peterson et al.29 using a cantilever technique and predicted by Jagota and Hui.30,31 In this paper we review the mechanisms proposed for particle deformation. This leads to a model we have proposed for the deformation step.32 Recent experimental work is then examined in the context of the proposed models. Finally, areas for future work are explored. 2. Driving Forces for Deformation There are a number of driving forces available for particle deformation. These arise from either the surface tension between the particles and the surrounding medium or the water-air surface tension at the fluid interface. 2.1. Wet Sintering. Driven by the surface tension between the particles and the solvent, typically water, this mode of deformation was first postulated by Vanderhoff.33 Experimentally, Sheetz held methyl methacrylate/butyl acrylate copolymer latices under water and observed the solids fraction to increase from 62% to 87%.2 Dobler et al.34 similarly observed particle compaction due solely to the polymer-water surface tension, although they concluded this deformation to be too slow, when compared with evaporation, to be relevant under normal conditions. 2.2. Dry Sintering. This is analogous to wet sintering, but with the polymer-air surface tension providing the driving force. Initially suggested by Dillon et al.,35 important experimental evidence for dry sintering is presented by Sperry et al.36 After the film dries well below the polymer glass transition temperature, to negate the possibility of particle deformation, the temperature is raised. The appearance of these dry films is then compared to films cast wet at the elevated temperature. The results are striking, with the cloudy to clear transition reaching the same temperature after a similar time, giving strong evidence that the presence of water is not important at the lowest temperatures. Plasticization of the polymer by the water complicates the results, although for hydrophobic polymers Sperry et al.36 collapsed data for the time dependence of the cloudy-clear point by treating the pores as bubbles collapsing in a viscous melt under the action of surface tension. This view was also supported by Keddie et al.37 Lin and Meier38 argue, however, that atmospheric humidity preserves residual water rings at particle contacts and the capillary pressure associated with these rings causes the particle deformation. This moist sintering mechanism is distinct from dry sintering but is functionally equivalent, because both require deformation of particles to be slow compared with evaporation and are driven by interfacial forces at the contact points. 2.3. Capillary Deformation. As water leaves the dispersion, the curvature of the air-water interface, because of the presence of particles, creates a large negative pressure in the fluid. The larger atmospheric pressure, pressing on the exposed particles at the surface, compresses the film, as hypothesized by Brown.1 The maximum pressure available, with a water-air surface tension of γwa, can be estimated geometrically as 12.9γwa/R0 by assuming a spherical meniscus within a triangular array of particles, with initial radius R0. The controlling resistance to deformation depends on the polymer rheology. Brown assumed elastic particles with shear modulus G, to estimate the material response (see section 4.2), and determined the pressure
Figure 2. Two viscoelastic particles deforming.
needed to achieve complete compaction of particles as 0.37G. This leads to Brown’s criterion for film formation as
G e 35γwa/R0
(1)
It is possible to criticize and tinker with Brown’s analysis,39,40 either changing the numerical factor in eq 1 or introducing viscoelasticity into the material response. Nonetheless, the idea of balancing a driving force, in this case capillary pressure, against the mechanical response is basic to all models of film formation. 2.4. Receding Water Front. This inhomogeneous regime was first identified by Keddie,37 who referred to it as stage II*. Deformation occurs initially by capillary mechanisms but is not complete by the time the capillary pressure reaches its maximum value. As water recedes through the film, leaving dry particles behind, the deformation mechanism switches to either a dry or moist sintering mechanism. This is alluded to by Eckersley and Rudin,40 who added the deformations predicted from capillary and sintering mechanisms. 2.5. Sheetz Deformation. The previous models assume a level of vertical homogeneity. If rapid evaporation convects particles to the air-film interface, before the dispersion reaches close packing below, a skin may form over the top of the film, hindering evaporation. Sheetz2 argued that diffusion of water through this skin creates a large osmotic pressure in the fluid below, causing compaction. An alternative explanation is that the skin slows evaporation considerably, allowing more time for other deformation mechanisms, especially wet sintering, to cause compaction. 3. Particle Responses Whatever driving force compresses the particles, they respond according to their individual rheological behavior and the change in contact area associated with the deformation. Though this viscoelastic contact problem is complex, simple models exist to relate the driving force to the approach of the particle centers as shown in Figure 2. The change in center to center spacing is given by 2R0R with R0 the original particle radius, making R the strain along the line of centers of a pair of particles. 3.1. Frenkel Deformation. Taking particles with viscosity η deforming under surface tension γ, Frenkel41 derived a simple energy balance, assuming the particles to deform as truncated spheres. His result for the deformation,
R )
3γt 4ηR0
(2)
is time dependent because of the assumption of viscous particles; the resulting strain increases linearly with
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time, t. Extensions of Frenkel’s model to include viscoelasticity have been proposed.42,43 3.2. Hertz Deformation. At the other end of the material spectrum, Hertz found an exact solution to the elastic equations for two particles compressed by a force F (see ref 44). For incompressible particles with shear modulus G, the solution is
R )
( ) 3F 8R02G
2/3
(3)
This is extended by Johnson, Kendall, and Roberts (JKR)45 to include the effect of surface tension. For forcefree particles, they found
R ) 0.234
( ) γ R0 G
2/3
(4)
Extensions to the JKR theory, including viscoelasticity, have been proposed.46 An interesting complication that arises from the JKR analysis concerns the role of surface forces. If, instead of using a macroscopic surface tension, the analysis accounts for a disjoining pressure or long-range attraction between the surfaces, there is a finite attraction before the particles come into contact. This has been argued extensively for elastic particles.47-50 For latex particles, Mazur and Plazek8,51 argue that a fast and essentially elastic deformation allows complete closure of the pores. While undoubtedly true under some conditions, this is not always the case. For example, Sperry et al.36 found the position of the cloudy-clear transition to be time dependent. This cannot be achieved unless viscous flow restrains the particle deformation. The important point is that the sudden compaction invisioned by Mazur et al. is dominant for small particles around 10 nm in size and becomes progressively less important with increasing size. For sufficiently large particles (a hundred nanometers or so by our estimates), the initial “zippering mode” becomes irrelevant. 4. Proposed Model In view of the various driving forces outlined above, we derived a model that included each and delineated regimes in parameter space where they apply.32 The model starts with a view of two viscoelastic particles, in the shape of truncated spheres. We obtain, to lowest order in strain,
2γ 2F + R ) πR02 R0
d 2 R dt′ ∫0tG(t-t′) dt′
(5)
with F the external force acting along the line of centers of the pair and γ the polymer-water or polymer-air interfacial tension. There is nothing new in this equation, which merely extends Frenkel’s model for viscous particles to include viscoelasticity as captured in the stress relaxation modulus G(t). With this pair deformation model, we derived the stress-strain relationship for a homogeneous film consisting of close-packed deforming particles. Instead of the strain between individual particles, the important quantity is the macroscopic strain normal to the substrate, , which is uniquely related to the local volume fraction. The
equation governing the film compaction
σt +
3νφmγ 3νφm ) 20R0 56
d 2 dt′ ∫0tG(t-t′) dt′
(6)
resembles the pair deformation equation but depends on ν, the average number of nearest neighbors for each particle, and φm, the volume fraction at close packing. The first term represents the compressive stress at the top surface, due to the capillary pressure, and the second term captures the negative pressure, due to the interfacial tension pulling the particles in the film together. The final term is the material response. Equation 6 is general, in that any material response can be used. We choose a viscoelastic fluid, ensuring an initial retarded elastic response, controlled by the high-frequency modulus G′∞ and allowing the stress to eventually decay to zero through creep, with low shear viscosity η0. The rheological model for a viscoelastic fluid, with a single relaxation time, is
(
G(t) ) η′∞δ(t) + G′∞ exp -
G′∞ t η0 - η′∞
)
(7)
with η′∞ the high-frequency viscosity and δ a Dirac delta function. Substituting this into eq 6, we derive a differential equation governing the compaction
dσ jt dth
+G hσ jt +
(
7γ j d 5 dth
)
+G h )
λh d η j dth
( )
d
dth
+G h λh
d dth
(8)
Time is scaled on an evaporation time (th ) tE˙ /H), with E˙ the evaporation rate and H the initial film thickness. The resulting dimensionless groups are defined in Table 1. Equation 8 depends on three dimensionless material groups, λh, G h , and η j , that uniquely determine the deformation process, leading to surfaces in this threedimensional parameter space that define the deformation mechanism controlling film formation under specific conditions. Typically, however, the evaporation time in latex films is long compared with the polymer relaxation time, (G h . 1), and the high-frequency viscosity is generally small compared with the low shear viscosity (η j . 1). This means that the relevant polymer rheology is controlled by the low shear viscosity or, in dimensionless terms, the group λh. We predict for λh < 1 that the film forms by wet sintering alone. For 1 < λh < 102, capillary deformation is responsible for the compaction. For 102 < λh < 104, a water front recedes through the film, so that capillary deformation and dry or moist sintering act in series. Finally, for λh > 104, dry or moist sintering accounts for virtually all of the deformation. In the dry sintering limit, the relaxation time becomes long relative to evaporation, but the treatment remains valid provided the elastic deformation caused by the interfacial tension does not close the pores, i.e., γwa/R0 < G′∞. Inhomogeneous drying, as proposed by Sheetz,2 is only possible when the first stage of film formation imparts a vertical inhomogeneity in the distribution of particles. This is controlled by the Peclet number HE˙ / D0, a measure of the evaporation rate to diffusion, with D0 ) kT/6πµR0 with µ the water viscosity and kT the thermal energy. If this group is small, the particles distribute evenly and deformation will proceed as described above. At large Peclet numbers, particles will
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Figure 3. Regimes of deformation mechanisms. Table 1. Dimensionless Groups Controlling Deformation group σ jt γ j
definition
28σtR0 3νφmγwa γ γwa
physical meaning
stress at the top surface capillary pressure surface tension water-air surface tension evaporation time polymer relaxation time
G h
G′∞H E˙ η0
λh
η0R0E γwaH η0 η∞
time for viscous collapse evaporation time
6πµR0HE˙ kT
rate of evaporation rate of diffusion
η j Pe
low shear viscosity high-frequency viscosity
accumulate at the top surface. If the polymer viscosity or modulus is low enough, deformation can produce a skin, slowing evaporation. For Pe . 1, we derived conditions for this inhomogeneity, dependent on the initial volume fraction of particles (φ0) and the strength of the particles (λh). Assuming φ0 ∼ 0.4, we find that the film is essentially homogeneous for λh > 10 because wet sintering is unable to deform the particles at the top surface. For λh < 1, a skin should form. For 1 < λh < 10, an inhomogeneous region forms with large deformation of the surface particles, though the voids do not close sufficiently to form a skin prior to close packing of the particles below. The results are summarized in Figure 3. 5. Comparison with Experimental Data To obtain positions in parameter space for different experimental systems, we wish to estimate the group
λh )
E˙ R0η0 γwaH
(9)
The evaporation rate, E˙ , will depend on the temperature and relative humidity, although Sperry et al.36 reports a change of only a factor of 2 over a 15 °C range. Therefore, an estimate of 0.3 cm/day (3.5 × 10-8 m/s),37 unless otherwise reported, seems reasonable. The original particle radius R0 is generally well-known, from either light scattering or microscopy studies. The waterair surface tension, γwa, will depend on the local surfactant concentration and could well vary throughout the deformation process.4 A value of 0.07 N/m seems an overestimate, although any likely error is probably
within a factor of 2. The film thickness at the coating, H, is generally well-known and reported. The most difficult physical parameter to estimate is the low shear viscosity, η0. In many cases (for example, Sheetz2), a polymer composition is given, allowing an estimate of the glass transition temperature from which we can estimate the low shear viscosity at the process temperature via the Williams-Landohl-Ferry (WLF) correction.52 The errors associated with this are large. Close to the glass transition, a difference of a few degrees can change the viscosity by orders of magnitude. Uncertainty in the glass transition of more than, say, 5 °C makes any predictions rather futile. Equally, hydroplasticization of the polymer can change the glass transition temperature by up to, say, 10 °C. Some papers give careful rheological data and thereby allow useful predictions to be made. Those interested in wet sintering33,34 access suitable conditions by reducing the evaporation rate, E˙ . This ensures λh , 1 and forces wet sintering to occur. To the authors’ knowledge, there are no reports of wet sintering with free evaporation into relatively dry air. Brown1 alludes to observations of film formation being concurrent with evaporation, which implies capillary deformation to be controlling. Eckersley and Rudin40,53 support a capillary mechanism but do invoke sintering in series. They report radii of contact between particles and also a set of rheological data. Unfortunately, radii of contact are difficult to relate to the local volume fraction. Whether the film is constrained to deform unidirectionally or not drastically affects the relationship between the radii of contact and volume fraction. This can also introduce a strong directional dependence on contact radii, as shown by Lin and Meier.54 Sperry36 and Lin and Meier54 ensure dry sintering by drying dispersions well below the glass transition temperature and then subsequently raising the temperature. Sperry36 shows that the cloudy-clear transition is unaffected by the presence of water, strong evidence for dry sintering at lower temperatures. Equally, Lin and Meier54 show the rheological response of the polymer material to control the process by performing time-temperature superposition of the surface roughness, measured using atomic force microscopy (AFM). In an earlier paper,38 Lin and Meier show how the capillary rings (moist sintering) drastically increase the deformation of a monolayer of PiBMA particles on mica. Although such a thin film is beyond the scope of our model, we estimate λh ∼ 2000, so despite the elevated temperatures, their observation of moist sintering is as predicted. Sheetz’s view of inhomogeneous drying is supported by Dobler et al.55 They show the rate of particle deformation under “standard conditions” to be much faster than if the particles are held under water or dried at low temperatures and then deformed by dry sintering at the elevated temperature. Although Dobler et al.55 used core-shell particles, an estimate of λh is possible. The 130 nm particles are cast in a film of thickness 0.5 mm. A glass transition temperature of around 35 °C can be estimated from an earlier paper.34 Estimating the low shear viscosity at 36 °C as 1012 Ns/m2 yields λh ∼ 130, so capillary deformation is expected. For the particles at 44 °C, the value of λh falls to 1.3. This lies on the verge of wet sintering and vertical inhomogeneity (Pe ∼ 10) is also expected, so the observation of Sheetz’s deformation is consistent with the model.
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Table 2. Experimental Results from Keddie et al.56 (Peclet Number Estimated as 3) sample 1 2 3 4 5 6
T [°C] 310 292 285 310 292 285
R0 [nm] 245 245 245 277 277 277
T - Tg [°C]
η0 [Ns/m2]
λh
mechanism
mechanism predicted
24 6 -1 42 24 17
2.5 × 3.0 × 1011 2.0 × 1013 3.0 × 106 2.5 × 108 2.5 × 109
0.3 350 2 × 105 4 × 10-3 0.3 3
wet E˙ slows? dry dry wet E˙ slows? wet E˙ slows? wet E˙ slows?
Sheetz recede dry/moist sintering Sheetz Sheetz Sheetz
108
an assumption of vertical homogeneity is essential. The work of Lin and Meier38,54 showing the temperature dependence of surface roughness is compelling evidence for the polymer rheology controlling the deformation, although, as discussed earlier, a range of temperatures is needed. 6. Conclusions and Future Work
Figure 4. Estimates of experimental data points, using data from Keddie et al.,56 Lin and Meier,54 and Dobler et al.55
Keddie et al.56 report a series of ellipsometry experiments at various temperatures relative to the glass transition temperature. The presence of water is detected, and film formation is defined as the onset of optical clarity. At the lowest temperatures, a distinct delay is seen between the evaporation of all water and the onset of optical clarity. When film formation (optical clarity) occurs with water present, the experiment shows either wet sintering or capillary deformation to be the responsible mechanism. For the weakest polymers, a distinct delay in film formation suggests the formation of a skin, slowing evaporation in accord with Sheetz’s mechanism. At the lower temperatures, either the receding water front or dry/moist sintering is realized. Table 2 summarizes the results, classifying the experimental observation as film formation either in the presence of water (wet) or after water has departed (dry). When film formation occurs at later times with weaker polymers, a reduction in the evaporation rate is demonstrated. We estimate λh by assuming the low shear viscosity at the glass transition temperature to be 1013 Ns/m2 and using the WLF equation to estimate the effect of temperature.52 The Peclet number for 100 µm films is about 3, implying that skinning is possible and expected for λh < 1, just as seen experimentally. The data at 310 K show the same mechanism for both particle types, meaning it is impossible to establish if a reduction in the evaporation rate is seen, making the observation of Sheetz deformation tentative at this temperature. The results are striking, although the estimate of the low shear viscosity imparts an uncertainty of an order of magnitude in λh. For the lowest temperatures with λh . 1, the film forms after evaporation of water, as expected. Equally clearly at the highest temperatures, evaporation emerges as the rate-limiting step and Sheetz deformation is seen. We summarize the experimental findings in Figure 4. Jenson and Morgan57 report the minimum film formation temperature to increase as a function of particle size. Indeed, λh increases with particle size, so a higher temperature is required to reduce the viscosity and ensure a constant λh at the MFT. Many58-64 use an AFM to follow film formation. This powerful technique measures the surface topology, so
The need for critical experiments covering a significant temperature range has been shown in this paper. The fact that each deformation mechanism is seen, under suitable experimental conditions, is beyond doubt. A comprehensive study of the presence, or lack, of water as a function of temperature relative to the glass transition temperature is long overdue. A possible method of doing this is ellipsometry, as described by Keddie et al.56 NMR is extremely sensitive to free water,65 and correlating this with the volume fraction in the film will provide similar information. Equally, AFM offers a unique insight, but experiments are needed to see if the time-temperature superposition shown by Lin and Meier works over all temperatures or only over a narrow range defined as the dry/moist sintering regime, as our model predicts. Acknowledgment This research was supported by grants from the Petroleum Research Foundation and Rohm and Haas and a fellowship from Rhodia. The authors are pleased to honor James Wei for his contributions to Princeton University and the chemical engineering profession. Literature Cited (1) Brown, G. L. Formation of films from polymer dispersions. J. Polym. Sci. 1956, 22, 423-434. (2) Sheetz, D. P. Formation of films by drying of latex. J. Appl. Polym. Sci. 1965, 9, 3759-3773. (3) Winnik, M. A. Latex film formation. Curr. Opin. Colloid Interface Sci. 1997, 2 (2), 192-199. (4) Keddie, J. L. Film formation of latex. Mater. Sci. Eng. 1997, R21 (3). (5) Steward, P. A.; Hearn, J.; Wilkinson, M. C. An overview of polymer latex film formation and properties. Adv. Colloid Interface Sci. 2000, 86 (3), 195-267. (6) Cannon, L. A.; Pethrick, R. A. Effect of the glass-transition temperature on film formation in 2-ethylexyl acrylate/methyl methacrylate emulsion copolymers. Macromolecules 1999, 32, 7617-7629. (7) Winnik, M. A.; Feng, J. Latex blends: An approach to zero VOC coatings. J. Coat. Technol. 1996, 68 (852), 39-50. (8) Mazur, S.; Plazek, D. J. Viscoelastic effects in the coalescence of polymer particles. Prog. Org. Coat. 1994, 24, 225-236. (9) Chevalier, Y.; Pichot, C.; Graillat, C.; Joanicot, M.; Wong, K.; Maquet, J.; Lindner, P.; Cabane, B. Film formation with latex particles. Colloid Polym. Sci. 1992, 270 (8), 806-821. (10) Routh, A. F.; Russel, W. B. Horizontal drying fronts during solvent evaporation from latex films. AIChE J. 1998, 44 (9), 20882098. (11) Juhue, D.; Wang, Y.; Winnik, M. A. Influence of a coalescing aid on polymer diffusion in poly (butyl methacrylate) latex films. Makromol. Chem., Rapid Commun. 1993, 14, 345-349.
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Received for review December 8, 2000 Revised manuscript received May 1, 2001 Accepted May 3, 2001 IE001070H
