pubs.acs.org/Langmuir © 2010 American Chemical Society
Evolving Stresses in Latex Films as a Function of Temperature Huai Nyin Yow,† Itxaso Beristain,‡ Monika Goikoetxea,‡ Maria J Barandiaran,‡ and Alexander F Routh*,† †
Department of Chemical Engineering and Biotechnology, BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, U.K., and ‡Institute for Polymer Materials, Joxe Mari Korta Center, University of the Basque Country, Avda Tolosa 72, E-20018 San Sebasti an, Spain Received October 16, 2009. Revised Manuscript Received March 29, 2010 Latex films were dried on a flexible substrate, and the substrate deflection was monitored over time to give an averaged film stress-evolution profile. Films were dried at various temperatures below and above the minimum filmformation temperature of the latex dispersion. The effect of polymer rheology, which is a temperature-dependent parameter, on film formation, was investigated. The reliability of the Stoney model in predicting film stress from substrate curvature was also examined and compared to the Euler-Bernoulli model. It was shown that the linearized Stoney model was unsuitable for the larger measured stresses.
1. Introduction Latex film formation has been studied for the past 60 years,1,2 since the development of the first water-based latex in 1948. It is generally agreed that there are three stages in latex film formation. When a latex film is cast, water evaporation concentrates the particles into a close-packing arrangement. As further evaporation takes place, stresses are generated that deform the particles and gradually eliminate voids. Finally, polymer chains diffuse across particle boundaries to give a homogeneous film with mechanical integrity. The stresses generated during film formation can also cause film defects such as cracks. This compromises the mechanical integrity of the film. Understanding the evolution of the filmformation stresses will provide manipulation and control over the film-formation process. Petersen et al. were the first to measure stress evolution using a cantilever method, also known as the beam-bending technique, for a latex-based film.3 This technique uses a flexible substrate on which a latex film is cast. As the film dries, the generated stresses bend the substrate. This curvature change is detected with a laser beam and then correlated to a stress magnitude using the Stoney-derived equation (eq 2). Tirumkudulu and Russel extended this technique to look at the role of capillary stresses in film formation,4 considering particles of varying radii and glass-transition temperatures. They developed a correlation between the stress at crack nucleation and the crack spacing.5 Von der Ehe and Johannsmann further developed the substrate bending idea into a 2D measurement.6 They used a flexible membrane with a regular array of dots. The deformation of the dot pattern then allowed a spatially resolved stress measurement as the film was formed. *Corresponding author. E-mail:
[email protected]. (1) Keddie, J. L. Film Formation of Latex. Mater. Sci. Eng., R 1997, 21, 101-170. (2) Steward, P. A.; Hearn, J.; Wilkinson, M. C. An Overview of Polymer Latex Film Formation and Properties. Adv. Colloid Interface Sci. 2000, 86, 195-267. (3) Petersen, C; Heldmann, C; Johannsmann, D. Internal Stresses during Film Formation of Polymer Latices. Langmuir 1999, 15, 7745-7751. (4) Tirumkudulu, M. S.; Russel, W. B. Role of Capillary Stresses in Film Formation. Langmuir 2005, 20, 2947-2961. (5) Tirumkudulu, M. S.; Russel, W. B. Cracking in Drying Latex Films. Langmuir 2005, 21, 4938-4948. (6) von der Ehe, K.; Johannsmann, D. Maps of the Stress Distribution in Drying Latex Films. Rev. Sci. Instrum. 2007, 78, 113904.
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Research by Francis et al. also illustrates that the cantilever technique has been used for stress measurements with solventbased polymer coatings.7,8 Other researchers have also employed this beam-bending technique to measure stresses in calcium carbonate-latex coatings,9,10 alumina-latex suspensions,11,12 silica-latex suspensions,13 and biopolymer fluids.14 In this article, we investigate the stress evolution in a latex film for a range of conditions below and above the minimum film-formation temperature (MFFT). The maximum stress generated is then correlated to the polymer rheology. In addition, the mapping of beam deflection to the inferred stress via the Stoney equation is examined.
2. Experimental Section 2.1. Latex Synthesis. 2.1.1. Materials. Technical-grade monomers [methyl methacrylate (MMA, Quimidroga), butyl acrylate (BA, Quimidroga), and acrylic acid (AA, Aldrich)] were used without purification. Stearyl acrylate (SA, Aldrich) or behenyl acrylate (Norsocryl, Arkema) was used as a costabilizer, and Dowfax 2A1 (alkyldiphenyl oxide disulfonate, Dow Chemicals) was used as a surfactant. The initiator was either potassium persulfate (KPS, Panreac) or ammonium persulfate (7) Francis, L. F.; McCormick, A. V.; Vaessen, D. M.; Payne, J. A. Development and Measurement of Stress in Polymer Coatings. J. Mater. Sci. 2002, 37, 4717- 4731. (8) Vaessen, D. M.; McCormick, A. V.; Francis, L. F. Effects of Phase Separation on Stress Development in Polymeric Coatings. Polymer 2002, 43, 2267-2277. (9) Wedin, P.; Lewis, J. A.; Bergstrom, L. Soluble Organic Additive Effects on Stress Development during Drying of Calcium Carbonate Suspensions. J. Colloid Interface Sci. 2005, 290, 134-144. (10) Wedin, P.; Martinez, C. J.; Lewis, J. A.; Daicic, J.; Bergstrom, L. Stress Development during Drying of Calcium Carbonate Suspensions Containing Carboxymethylcellulose and Latex Particles. J. Colloid Interface Sci. 2004, 272, 1-9. (11) Kiennemann, J.; Chartier, T.; Pagnoux, C.; Baumard, J. F.; Huger, M.; Lamerant, J. M. Drying Mechanisms and Stress Development in Aqueous Alumina Tape Casting. J. Eur. Ceram. Soc. 2005, 25, 1551-1564. (12) Martinez, C. J.; Lewis, J. A. Rheological, Structural and Stress Evolution of Aqueous Al2O3: Latex Tape-Cast Layers. J. Am. Ceram. Soc. 2002, 85, 2409-2416. (13) Martinez, C. J.; Lewis, J. A. Shape Evolution and Stress Development during Latex-Silica Film Formation. Langmuir 2002, 18, 4689-4698. (14) Chen, J.; Ettelaie, R.; Yang, H.; Yao, L. A Novel Technique for in Situ Measurements of Stress Development within a Drying Film. J. Food Eng. 2009, 92, 383-388.
Published on Web 04/13/2010
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Yow et al. Table 1. Properties of Latexes
sample
main monomers (wt %)
initiator (wt % of monomers)
emulsifier (wt % of monomers)
co-stabilizer (wt % of monomers)
solids content (%)
particle size (nm)
MFFT (°C)
high-MFFT mid-MFFT low-MFFT
MMA/BA/AA (65/34/1) MMA/BA/AA (60/39/1) MMA/BA/AA (50/49/1)
KPS(0.5) KPS(0.5) APS/MBS (0.25/0.25)
DOWFAX 2A1 (2) DOWFAX 2A1 (3) DOWFAX 2A1 (4)
stearyl acrylate (4) stearyl acrylate (4) Norsocryl (4)
47.4 43.1 51.4
185 166 144
34.0 21.0 14.5
(APS, Panreac)/sodium metabisulfite (MBS, Panreac) in ratio of 1:1. Sodium bicarbonate (Riedel-de Haen) was used to control the viscosity of the miniemulsion by reducing the intense electrostatic interactions between droplets. Distilled water was used throughout. 2.1.2. Miniemulsification and Polymerization. The latexes were prepared by miniemulsion polymerization. Details of the preparation of the latexes are similar to those described in Goikoetxea et al.,15 except the organic phase in this paper contains only the monomers and costabilizer without any alkyd resin. Briefly, the organic dispersed phase consisted of monomers MMA, BA, and AA in varying ratios, as listed in Table 1, and 4% by weight (based on the organic phase) costabilizer (SA or Norsocryl). The aqueous phase contained 2-4 wt % Dowfax 2A1 in distilled water. Sodium bicarbonate (1 wt %) was added to control the viscosity of the miniemulsion. To produce the miniemulsion, both phases were mixed via magnetic stirring (10 min at 1000 rpm), followed by sonification (Branson 450, 15 min, power 9, 80% duty cycle) and finally high-pressure homogenization (Niro-Soavi NS1001L PANDA, 6 cycles) at 41 MPa in the first valve and 0.41 MPa in the second valve. Polymerization was then performed in a 1 L jacketed glass reactor equipped with a reflux condenser, stirrer, sampling device, and nitrogen inlet. The reaction temperature was set to 70 °C, and the miniemulsion was stirred under nitrogen (12-15 mL/min). When the miniemulsion reached the reaction temperature, an aqueous solution of KPS (0.5 wt %) was added as a single shot. In the case of redox initiators (APS/MBS), the initiators (0.25/0.25 wt %) were fed in gradually. Polymerization was left overnight to ensure good conversion. 2.1.3. Characterization of Latex Particles. The latex particles were sized by dynamic light scattering (DLS, Brookhaven ZetaPlus) coupled to particle-sizing software (Brookhaven, version 3.42). A drop of a particle dispersion (∼0.1 mL) was placed in a disposable plastic cell and diluted to 3.0 mL of a virtually clear solution. The effective diameter obtained was accurate to (5%, with the reported value being the average of 10 measurements. An in-house assembly of a minimum film-formation temperature (MFFT) bar was used to measure the MFFT of the latex dispersion. The colloidal dispersion was spread at a thickness of 1 mm on a glass bar of (68 cm 16.5 cm). The glass had previously been placed on the MFFT bar with temperature varying from 20 to 70 °C across the bar length. Air flow was forced across the surface from the cold to hot end. The setup was left for 2 h; a film was formed, and the MFFT was measured with an on-board thermocouple. The MFFT is defined as the lowest temperature at which the polymer particles deform to form a homogeneous film.16 This is represented by the condition under which the film appearance changes from a cloudy-white cracked film in the cold section of the MFFT bar to a clear transparent film in the hotter section. For latex dispersions with an MFFT below or close to 20 °C, the MFFT was determined by observing the film condition after drying the films in an incubator (Sanyo TSE) at temperatures between 10 to 25 °C. 2.2. Stress Measurement. The macroscopic stress progression during film formation was examined via the beam-bending (15) Goikoetxea, M.; Minari, R. J.; Beristain, I.; Paulis, M.; Barandiaran, M. J.; Asua, J. M. Polymerization Kinetics and Microstructure of Waterborne Acrylic/ Alkyd Nanocomposites Synthesized by Miniemulsion. J. Polym. Sci., Part A: Polym. Chem. 2009, 47, 4871-4885. (16) Lee, W.-P.; Routh, A. F. Time Evolution of Transition Points in Drying Latex Films. J. Coat. Technol. Res. 2006, 3, 301-307.
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Figure 1. Experimental setup for stress measurement via the beam-bending technique. technique.3 Figure 1 is a schematic diagram of the experimental setup, which was enclosed in a cloth-covered plastic chamber to provide a relatively constant temperature and relative humidity. The temperature and humidity were monitored every 15 min with a thermohygrometer (Hanna Instruments). A ceramic infrared temperature bulb was installed to control the chamber temperature using a feedback loop. The latex dispersion was cast as a (60 mm 15 mm) film onto a (60 mm 51 mm l00 μm) flexible brass substrate using a pulling bar (Erichsen). As the film formed, the induced stresses bent the substrate. This curvature change was monitored using a quadrant position detector (Laser Components Ltd.). A helium-neon laser beam (JDSU) was deflected from the end of the substrate by a (5 mm 10 mm) mirror onto the detector. The data were collected immediately after film casting (i.e., as soon as the cloth cover had been replaced) and subsequently once every 3 s. Note that great care was taken, where possible, to maintain constant temperature and humidity during each measurement, with experimental temperature and relative humidity variations of up to 0.6 °C and 1.3% RH, respectively, in each experiment performed. All experiments were carried out in an enclosed plastic chamber in which the chamber was aired overnight to attain constant conditions before the next experiment. 2.2.1. Calibration of Beam-Bending. The beam-bending setup was calibrated prior to each film-stress measurement. This was performed by placing weights of known mass onto the substrate and measuring the corresponding beam deflections with respect to the position detector. This provided a conversion factor for calculating the equivalent mass applied as the substrate bent during film formation. Using eq 1, the deflection angle θ was related to the applied mass3 θ ¼
6gmLs 2 1 ws Es ts 3
ð1Þ
where θ is the deflection angle at the substrate tip due to weight m and g is the earth’s gravitational constant. Es, ts, Ls, and ws are the properties of the substrate, namely, the Young’s modulus, thickness, length, and width respectively. The deflection angle θ was then correlated to an equivalent film stress σf according to eq 2 3 σf ¼
Es ts 3 θ 6tf ðts þ tf ÞLf
ð2Þ
where σf is the equivalent film stress in the plane of film and tf and Lf are the average thickness and length of the latex film, respectively. The average thickness tf was calculated from the arithmetic Langmuir 2010, 26(9), 6335–6342
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Using the entire length of the substrate (i.e., x = Ls), we rearrange eq 4 to give the dimensionless applied load Ωs as Ωs ¼
Figure 2. Schematic diagram for (a) the Stoney model and (b) the Euler-Bernoulli model (adapted from Zhang et al.18). Ls, ws, and ts are the length, width, and thickness of the substrate, respectively, M is the concentrated bending moment, and P is the concentrated axial load. mean of the initial and final dry film thicknesses. The initial thickness was determined by the gap width of the pulling bar, and the final thickness was calculated using the initial thickness and known solid content of the latex dispersion. A few considerations should be taken into account when applying eq 2 to an averaged film stress calculation. First, the weight loss from water evaporation, which contributed to the measured deflection as well, should be corrected for in the stress profiles. Second, the film thickness varied throughout the experiment, whereby the choice of using an average film thickness would incur some error as opposed to using an instantaneous film thickness. Also, the zero starting point for the experiment was difficult to determine precisely. This was because the zero point was subject to the efficiency in setting up the experiment. Finally, this technique was capable only of giving an equivalent averaged film stress, yet the technique was affected by any local and microstructural differences in the latex film. Hence, the subsequent results should be viewed as a guide of different trends observed for the studied latex films.
3. Model Analysis 3.1. Stoney Model. Equation 2 assumes the stress to be constant throughout the latex film. This is equivalent to the Stoney formula,17 which is based on uniform loading at the substrate tip. This leads to a uniform curvature for the deflected substrate and hence uses a linear analysis of small deflection. Here, we propose a dimensionless applied load to examine the accuracy of the Stoney model, and this is an extension to the work by Zhang et al.18 The schematic diagram for the Stoney model is illustrated in Figure 2a. The governing equation for the substrate deflection is d2 y E I 2 ¼ M dx
ð3Þ
where E* is the biaxial modulus given by E* = Es/(1 - ν) with Es being the Young’s modulus of the substrate and ν being the Poisson’s ratio. y is the vertical deflection of the substrate, and x is the horizontal distance. I is the area moment of inertia and is given by I = wsts3/12 for a rectangular structure. M is the concentrated bending moment given as M = Pts/2, where ts is the thickness of the substrate and P is the axial loading on the substrate. The boundary conditions for eq 3 are (i) y(0) = 0 and (ii) dy/dx(0) = 0. Solving eq 3 by integration gives y ¼
Mx2 2EI
ð4Þ
(17) Stoney, G. G. The Tension of Metallic Films Deposited by Electrolysis. Proc. R. Soc. London, Ser. A 1909, 82, 172-175 . (18) Zhang, Y.; Ren, Q.; Zhao, Y.-P. Modelling Analysis of Surface Stress on a Rectangular Cantilever Beam. J. Phys. D: Appl. Phys. 2004, 37, 2140-2145.
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PLs 2 4y ¼ ts EI
ð5Þ
In this case, the measured deflection is linearly related to the applied stress, and this is equivalent to the literature method described by eq 2. 3.2. Euler-Bernoulli Model. The Euler-Bernoulli model represents the stress as a corresponding concentrated bending moment M plus a concentrated axial load P on the substrate. The difference from the Stoney approach is that the bending moment M is not assumed to be constant along the substrate length. The schematic for the Euler-Bernoulli model is shown in Figure 2b, and this is an addition to the work by Zhang et al.18 In this case, the governing equation for substrate deflection is EI
d4 y d2 y -P 2 ¼ 0 4 dx dx
ð6Þ
with the boundary conditions as (i) y(0) = 0, (ii) dy/dx(0) = 0, (iii) E*I (d2y/dx2)(Ls) = M, and (iv) E*I (d3y/dx3)(Ls) = P (dy/ dx)(Ls). Using the dimensionless variables X = x/Ls and Y = y/Ls, eq 6 can be nondimensionalized to 0000
00
Y - βLs 2 Y ¼ 0
ð7Þ
and the corresponding nondimensionalized boundary conditions are (i) Y(0) = 0, (ii) Y0 (0) = 0, (iii) Y00 (1) = 1/2(RβLs2), and (iv) Y000 (1) = βLs2Y0 (1), where R = ts/Ls and β = P/E*I. Solving eq 7 by integration and applying the boundary conditions gives Y ¼
R
pffiffiffi ½coshðLs X 2 coshðLs βÞ
pffiffiffi βÞ - 1
ð8Þ
Substituting with the entire length of substrate (i.e., X = 1) and rearranging eq 8 gives the dimensionless applied load Ωeb as " #2 PLs 2 ts -1 ð9Þ Ωeb ¼ ¼ cosh E I ts - 2y By using eq 8 with X = 1 and expanding for small Ls(β)1/2, the Euler-Bernoulli equation can be linearized to give the same result as the Stoney expression (eq 5).
4. Results and Discussion 4.1. Stress Evolution Profile as a Function of Temperature. Latex films (50 μm thick) of three different MFFTs were cast and dried on 100-μm-thick brass substrates at 26 °C (37 45% RH). The properties of the latexes are shown in Table 1, and Figures 3-5 illustrate the stress-evolution profiles for each of the latex films. The black solid line represents the measured equivalent film stress (in MPa), and the gray dotted line corresponds to the dimensionless normalized stress. This parameter is obtained by dividing the equivalent film stress by the maximum capillary pressure, given as 2γwa/Ro, with γwa being the air/water interfacial tension of 0.072 N/m and Ro being the original radius of the latex particles. The normalization relates to the maximum pressure that could be obtained for rigid spheres in pure water. If the particles deform, then the pores will become smaller and the capillary DOI: 10.1021/la1007439
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Figure 3. Stress-evolution profile with corresponding images of a drying film for high-MFFT latex particles.
pressure could presumably increase. However, the use of 2γwa/Ro has been extensive,4 so this scaling is used here. The estimated time when random close packing was attained is also indicated on the stress-evolution profiles. This was calculated from the evaporation rate, assuming that the spatially uniform volume fraction of particles at random close packing is 0.64.19 The images of the drying film show strong horizontal drying although for the case of rigid particles the maximum in the measured stress correlates strongly with the time of evaporation causing close packing across the entire film. Figure 3 shows the stress profile for a high-MFFT latex film, with corresponding images illustrating the film appearance as drying proceeded. The drying condition was equivalent to Texp MFFT), a smooth, nearly transparent film was formed, as shown in Figure 5 (the dotted border in the images is used to highlight the position of the film). Immediately after casting (at ∼0 min), the wet film appeared to be homogeneous throughout, with slight pooling toward one end. Film drying proceeded evenly, as indicated by the film image at 13 min. A gradual increase in film stress was observed immediately from casting. At 26 min, a dried film was observed, with no noticeable visual changes subsequently. However, the film stress continued to build up, with a slight gradient change (as indicated by (a) in Figure 5). This was speculated to be due to the deformation of the particle network because the change in the stress gradient corresponded to the point where the film was predicted to reach close packing. At a time of 60 min, long after the particles had reached close packing and the fluid flow had ceased, the stress reached a plateau. This is a residual stress in the film that is anticipated to decay away over a long enough timescale. It was also observable from Figure 5 that the low-MFFT latex film experienced a longer duration of stress buildup when compared to the mid- and high-MFFT films. This was speculated to be due to the deformation of the softer particles during further evaporation, once the particles had consolidated. Martinez and Lewis also observed that stress buildup in a film-forming latex lasted until less than ∼10% of the initial water remained.12,13 A point to note is that the normalized stress for the three latex films in Figures 3-5 was within 0.14 ( 0.03. However, as shown
in Figure 6, latex films dried at lower temperatures can display greater stresses. 4.2. Effect of Polymer Rheology on Maximum Normalized Stress. Latexes in Table 1 were cast multiple times under a range of conditions (20-29 °C, 34-50% RH) while measuring the corresponding stress-evolution profiles. The maximum achievable normalized stress for each experiment was plotted as a function of Texp - MFFT in Figure 6 (where the dotted line is used as a guide to the experimental data). This summarizes the effect of the drying temperature on the film. From Figure 6, the most noticeable result is the large scatter in the data at low temperatures. However, it is clear that at lower temperatures the maximum stress generally increases. Changing the temperature affects the rheological properties of the latex particles,22 with higher temperatures resulting in a less-viscous polymer. This allows greater movement of polymer within the
(21) Lee, W.-P.; Routh, A. F. Temperature Dependence of Crack Spacing in Drying Latex Films. Ind. Eng. Chem. Res. 2006, 45, 6996-7001.
(22) Routh, A. F.; Russel, W. B. A Process Model for Latex Film Formation: Limiting Regimes for Individual Driving Forces. Langmuir 1999, 15, 7762-7773 .
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Figure 6. Effect of temperature on maximum achievable normalized stress. The dotted line is a guide to the experimental data.
Figure 7. Maximum normalized stress as a function of (i) the deformation mechanism, (ii) capillary deformation, and (iii) receding water front dry/moist sintering. The fine dotted line is a guide to the experimental data.
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Yow et al. Table 2. Experimental Conditions for Various Samples of Latex Film Formation
sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
latex type
Ro (nm)
low-MFFT low-MFFT low-MFFT mid-MFFT mid-MFFT mid-MFFT mid-MFFT mid-MFFT mid-MFFT high-MFFT high-MFFT high-MFFT high-MFFT high-MFFT high-MFFT high-MFFT high-MFFT high-MFFT high-MFFT high-MFFT
144 144 144 166 166 166 166 166 166 185 185 185 185 185 185 185 185 185 185 185
Texp(°C) 22.7 26.0 26.2 20 1 20.9 21.7 22.1 22.9 26.3 23 9 24.4 24.6 25.1 26.1 27.0 27.2 28.0 28.2 29.0 29.0
RH (%) 50 45 47 41 43 40 42 46 44 34 39 37 39 37 36 38 38 38 35 22
latex particles, allowing the stress to be relaxed. Hence, a lower overall stress was detected, as seen in the high-temperature region of Figure 6 (i.e., (Texp - MFFT) > -4 °C). Because the glasstransition temperature (Tg) is typically lower than the MFFT by around 3 °C,16,22 the transition point around Texp - MFFT = -4 °C is reasonable. When Texp - MFFT was less than -4 °C (i.e., the drying temperature was lower than polymer’s MFFT), the maximum stress achieved was more erratic (low-temperature region of Figure 6). In this regime, the latex particles were rigid, spherical, and possessed a high elastic modulus. A greater stress was achieved because the particles could not dissipate any of the generated stress. The stress in this region was also very sensitive to the experimental setup as well as to any local fluctuations in the film during casting. As a result, any small experimental discrepancy could lead to some variation in the detected stress profile. For example, if a film happened to delaminate near the substrate end, then the measured stress would be significantly decreased. Therefore, Figure 6 shows that the film-formation process and hence the induced stresses are dependent on the rheological properties of latex particles. Lin and Meier,23 using atomic force microscopy, also observed a one-to-one relationship between the rheological properties of latex particles and the rate of film formation. An alternative way to analyze the experimental data is to classify the results according to the proposed particle-deformation mechanisms.2 Routh and Russel suggested a dimensionless group λh to represent the ratio of the time needed for the viscous collapse of the particles to the evaporation time.21 This is illustrated in eq 10 η R0 E_ λ ¼ 0 ð10Þ γwa H where η0 is the low-shear viscosity, R0 is the original radius of a latex particle, E_ is the evaporation rate, γwa is the water/air interfacial tension, and H is the original film thickness. The value of η0 at the glass-transition temperature was assumed to be 1012 _ was calculated from the Ns/m2, and the evaporation rate, E, measured relative humidity within the chamber. Additional information on the evaporation rate calculation is provided in (23) 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, 2774-2780 .
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Texp - MFFT(°C) 8.2 11.5 11.7 -0.9 -0.1 0.7 1.1 1.9 5.3 -10.1 -9.6 -9.4 -8.9 -7.9 -7.0 -6.8 -6.0 -5.8 -5.0 -5.0
· E( 10-8m/s)
ηo (Ns/m2)
hλ
7.3 9.8 9.0 7.3 7.4 8.2 8.1 7.9 10.1 10.4 9.9 10.3 10.2 11.2 12.1 11.8 12.3 12.4 13.6 16.4
4.1 10 7.0 108 6.0 108 2.0 1012 1.1 1012 6.0 1011 4.0 1011 2.0 1011 2.3 1010 5.9 1016 3.2 1016 2.5 1016 1.4 1016 4.2 1015 1.6 1015 1.3 1015 5.4 1014 4.4 1014 1.9 1014 1.9 1014
9.6 2.0 1.7 4300 2300 1400 1000 560 71 5.6 107 2.9 107 2.4 107 1.4 107 4.9 106 2.0 106 1.6 106 7.5 105 6.2 105 3.1 105 3.8 105
9
Appendix A. The effect of temperature on the low-shear viscosity, η0, can be accounted for by multiplication with the temperature shift factor aTg, as indicated below, η0 ðTexp Þ ¼ aTg ðTexp Þ η0 ðTg Þ
log10 aTg ðTexp Þ ¼
- c1 ðTexp - Tg Þ c2 þ Texp - Tg
ð11Þ
ð12Þ
where c1 and c2 are material constants. These parameters are material-dependent but can be generally taken to be 17.1 and 51.4 °C, respectively,16,24 whereas the glass-transition temperatures for the respective latexes were obtained from the measured MFFT. Using this parameter λh, Routh and Russel predicted that film formation would occur by wet sintering when λh < 1, by capillary deformation when 1 < λh < 102, by a receding water front when 102 < λh < 104, and by dry/moist sintering when λh > 104.22 This prediction has shown reasonable agreement with experimental observations by Martinez and Lewis,13 who combined measurements of the film shape evolution profile, visual observation, and in situ evaporation data to determine the drying mechanisms of their latex films. Applying the Routh-Russel prediction22 for film-formation mechanisms, Figure 7 shows the distribution of the maximum normalized stress with respect to the drying regimes (with the fine dotted line as a guide to the experimental data). The drying mechanism regimes are denoted as (i) capillary deformation, (ii) receding water front, and (iii) dry/moist sintering. The temperature of the experiments was never high enough to lead to a prediction of wet sintering. The conditions for each experiment in Figure 7 are summarized in Table 2. From Figure 7, it appears that in regimes i and ii the deformation of particles helped to dissipate the generated film stress. In regime iii, where dry/moist sintering was the predicted deformation mechanism, the stress buildup was more variable and higher in magnitude. There is a pleasing correlation between the predicted onset of dry sintering and the measurable increase in film stress. 4.3. Comparison of the Stoney Model with the EulerBernoulli Model. The magnitude of the normalized stress in Figures 6 and 7 raises the question of the reliability of the (24) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; John Wiley & Sons: New York, 1980; p 641.
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to PL2/E*I. Equations 5 and 9 give the expression for Ω for the Stoney and Euler-Bernoulli models, respectively. Figure 8 presents the outcome of this comparison (where the dotted line represents the Stoney model and the solid line represents the Euler-Bernoulli model). Arrows from i and ii indicate the peaks of the Stoney and Euler-Bernoulli models, respectively. Figure 8a illustrates the comparison between the two models when the induced substrate curvature was relatively small (i.e., the vertical deflection of substrate y was less than 5 μm for a substrate thickness of 100 μm). When Ω was less than 0.15 (i.e., y < 3.8 μm), the results from the Stoney and Euler-Bernoulli models were nearly identical. Once this limit was exceeded, a 15% discrepancy was observed in the peaks. Considering the experimental errors, this discrepancy was still acceptable. Hence, for a small substrate deflection, the Stoney model is valid and simple to apply. However, as the vertical deflection of the substrate increased further, the discrepancy between the Stoney and Euler-Bernoulli models increased to 50%. This can be observed in Figure 8b at the peak values, whereby the film stress predicted by the EulerBernoulli model was nearly 50% larger than that predicted by the Stoney model. This corresponds to a vertical deflection of 20 μm. Hence, it can be summarized that the Stoney model is valid for small deflections when Ω is smaller than 0.15. Once Ω exceeds this value, the Euler-Bernoulli model is more suitable for converting the substrate curvature into an averaged film stress. A point to note is that the graphs in Figure 8 were marginally elevated to ensure that all data points were greater than zero. This was done to allow the application of the hyperbolic trigonometric function in the Euler-Bernoulli model. It is crucial to stress that this alteration did not affect the relative position of the data points but was merely an upward movement of the graphs on the y axis for the respective charts.
Figure 8. Comparison of the reliability of the Stoney profile to the Euler-Bernoulli model in predicting stress induced during film formation. (a) Case 1 indicates the acceptable discrepancy between models when the vertical deflection is small (i.e., y < 3.8 μm), and (b) case 2 illustrates up to a 50% discrepancy when the vertical deflection is up to 20 μm (where arrows from i and ii indicate the peaks for the Stoney and Euler-Bernoulli models, respectively).
Stoney model for relating the beam deflection to the film stress. The Stoney model assumes a linearization for small deflection when converting the measured substrate curvature into an averaged film stress. It is of interest to examine the validity of this assumption, particularly when a larger substrate curvature is achieved. The linearized model is compared with the EulerBernoulli model, which considers an additional stress component—a concentrated axial load P. It is important to note that if the Euler-Bernoulli expression (eq 9) is expanded for small deflections then the Stoney result (eq 5) is recovered. For ease of comparison, the measured beam deflection was rearranged into a dimensionless applied load, Ω, equivalent Langmuir 2010, 26(9), 6335–6342
5. Conclusions Latex films were dried on a flexible substrate, whereby the substrate curvature and its corresponding averaged film stress were monitored over time with a laser and a position detector. Film drying was monitored at temperatures below, around, and above the polymer’s MFFT. The stress-evolution profiles for these conditions showed (i) an initial trough due to particle transport to the film edge, followed by a sharp peak during final particle consolidation for conditions below the MFFT and (ii) a prolonged gradual stress buildup that leveled off when conditions were above the MFFT. Around the MFFT, the stress profile fell between these two variations. The maximum normalized stresses were then plotted against Texp - MFFT to investigate the effect of polymer rheology on film formation. When Texp < MFFT, the films exhibited an erratic pattern. This was speculated to be due to the greater stress required for particle deformation as well as any localized stress buildup in the inhomogeneous films. This was contrary to the continuous film formed when Texp>MFFT, which allowed an even stress distribution and hence a lower overall stress. The erratic behavior is predicted to occur when the particle deformation mechanism is dry/moist sintering. Finally, a comparison was made between the Stoney and Euler-Bernoulli models to gauge the validity of the Stoney model in converting the substrate curvature to an averaged film stress. It is shown that the Stoney model is valid for small deflections. However, when the dimensionless applied load exceeds a value of 0.15 the Euler-Bernoulli model becomes a more suitable choice because the linearized model led to a discrepancy of almost 50%. DOI: 10.1021/la1007439
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Article
Yow et al.
Acknowledgment. We thank the EU for funding this work in the framework of IP011844-2 (NAPOLEON).
Appendix A: Determination of Evaporation Rate Experiments were carried out on the beam-bending setup with water being dried on a 100-μm-thick brass substrate. The conditions were identical to those in the beam-bending experiments using latex. The complete evaporation time was noted when the detector registered a constant signal. This was then used to determine an average evaporation rate, W (in kg/s), because a known volume of water was used each time. The projected dimensions of the substrate were measured at the end from the water stain. The water evaporation rate is related to the partial pressure of water in the air pa and the vapor pressure of liquid water pw according to W ¼ kG Aðpw - pa Þ
6342 DOI: 10.1021/la1007439
ðIÞ
where kG is the mass-transfer coefficient and A is the exposed surface area. By applying eq I, an average kG of (5.3 ( 1.6) 10-8 kg/m2 s Pa was estimated from the blank experiments. Subsequently, the evaporation rate for each latex film-formation experiment was calculated by determining the vapor pressure of water in each experiment from steam tables. This method assumed that the evaporation of water from the latex system was the same as that for pure water, which is valid when no organic solvents are present in the latex system.A1 The evaporation rate can also be expressed as the speed of film thickness decrease, E_ (in m/s), according to eq II W E_ ¼ AFw
ðIIÞ
where Fw is the density of water. (A1) Erkselius, S.; Wads€o, L.; Karlsson, O. J. A Sorption Balance-Based Method to Study the Initial Drying of Dispersion Droplets. Colloid Polym. Sci. 2007, 285, 1707-1712.
Langmuir 2010, 26(9), 6335–6342
