A Study of Latex Film Formation by Atomic Force Microscopy. 2. Film

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A Study of Latex Film Formation by Atomic Force Microscopy. 2. Film Formation vs Rheological Properties: Theory and Experiment F. Lin† and D. J. Meier* Michigan Molecular Institute, 1910 West St. Andrews Road, Midland, Michigan 48640 Received December 18, 1995. In Final Form: February 26, 1996X In this second paper of a series dealing with the film-forming behavior of latex systems, we show the relationship between the kinetics of film formation and rheological properties. The kinetics of film formation were followed using atomic force microscopy (AFM), with results that showed the kinetics obeyed time/ temperature superposition principles, with the same time/temperature shifts used to superimpose rheological data. This demonstrates a direct relationship between film formation kinetics and rheological properties. A new theory based upon these observations is presented, in which the Boltzmann superposition principle and a time-dependent compliance function J(t) are used to relate the time-dependent biaxial strain ψ(t) to the time-dependent driving stress σ(t). The compliance function J(t) of the latex polymer was obtained by transformation of the master stress relaxation function G(t), which in turn was obtained from time/ temperature superposition of experimental stress relaxation data. A comparison of theoretical predictions and experimental data for the kinetics of film formation shows good agreement over the 4 decades of time followed experimentally. The agreement is believed to be within the accuracy of the transformation of G(t) to J(t).

Introduction In the first paper1 of this series concerning film formation from latex systems (LM-I), we reviewed the major earlier works on mechanisms and theories that have been proposed for film formation. We pointed out that the mechanisms and theories that had been presented were incomplete and typically based upon incorrect models and assumptions when applied to latex systems. These models include those of Frenkel2 and Kendall3 for particle deformation and those of Brown4 and Mason5 for capillary pressures and forces. The reader is referred to LM-I for further details of the problems associated with these models when applied to latex systems. An additional problem exists with experimental work in the past, in that the tools available did not allow a quantitative assessment of the detailed features of film formation as the system transformed from a dispersion of spherical particles to a coherent smooth film as water evaporated. The recent availability of the atomic force microscope (AFM) has now enhanced in a remarkable way the ability to follow the film formation process. We showed in LM-I that the atomic force microscope could be used to obtain quantitative data on the kinetics of film formation, and we used that capability to follow the kinetics of film formation under a variety of conditions. For example, we showed that the rate of film formation of “wet” latex systems (water present in the interstitial space) was almost 100 times faster than that of “dry” systems. In the latter case, the driving force for film formation is from the polymer surface tension and the particle radius of curvature, but in the presence of interstitial water, the much higher surface tension of water and the very small radius of curvature (compared to the particle radius) of * To whom correspondence should be addressed: tel (517) 8325577, FAX (517) 832-5560, e-mail [email protected]. † Present address: Dow Corning Corporation, Midland, MI 48686, tel (517) 496-5064, FAX (517) 496-5956, e-mail usdcc7u5@ ibmmail.com. X Abstract published in Advance ACS Abstracts, May 1, 1996. (1) Lin, F.; Meier, D. J. Langmuir 1995, 11, 2726. (2) Frenkel, J. J. Phys. (USSR) 1943, 9, 385. (3) Kendall, K.; Padget, J. C. Int. J. Adhes. Adhes. 1982, 2, 149. (4) Brown, G. L. J. Polym. Sci. 1956, 22, 423. (5) Mason, G. Br. Polym. J. 1973, 5, 101.

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the interstitial water combine to give a much larger driving force (“capillary pressure”). It is this much larger driving force from the capillary pressure that is responsible for the much faster film formation kinetics of the “wet” system. We also showed that the capillary pressures were of sufficient magnitude that they could drive film formation at a temperature even 15 °C below the glass transition temperature of the polymer. In this paper, we continue our discussion of the film formation properties of latex systems, in which we show the relationship between the kinetics of film formation and the rheological properties of the polymeric latex. We also present a new theory for film formation from dry latex systems (or spherical powders) which we believe is an improvement over past theoretical models. The concomitant changes in the stress and strain fields resulting from the changing particle shape during film formation are now taken into account. The evaluation of the theory for film formation requires the time-dependent compliance function J(t) of the polymeric latex. This function is obtained by transformation of the stress relaxation function G(t) to the compliance functions J(t). The master stress relaxation function G(t) is obtained from experimental stress relaxation data, using time/temperature superposition principles. Experimental Approach A detailed study of latex film formation requires welldefined model latices and reliable experimental instrumentation and procedures. As indicated in LM-I, the AFM is uniquely suited to follow the film formation process since it allows quantitative data to be obtained on the deformation of particles as film formation proceeds. Peakto-valley dimensions (“corrugation heights”) of the surface can be directly measured by the instrument. In our work, poly(isobutyl methacrylate) (PiBMA) was used as the model latex polymer because of its convenient glass transition temperature (Tg ∼ 65 °C) and the ability to prepare it as a monodisperse, surfactant-free latex with particles covering a range of desired sizes. The glass transition temperature of PiBMA is sufficiently high that distortion during handling at room temperature will not © 1996 American Chemical Society

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Figure 1. Corrugation heights h(t) at 70, 80, and 90 °C, and resulting “master curve” shown with shift factors used to superimpose the data.

occur, and its Tg is also sufficiently low that film formation can be induced by heating to modest temperatures above Tg, e.g., 70-90 °C in the present experiments. Experimental Methods Latex Preparation. Poly(isobutyl methacrylate) latices were prepared by a one-step surfactant-free emulsion polymerization at 80 °C under a nitrogen blanket. In a typical polymerization, 24 mL of isobutyl methacrylate monomer (Aldrich, as received) and 0.09 g of potassium persulfate were added to 150 mL of water to a flask equipped with a mechanical stirrer. Dodecyl mercaptan was used to control molecular weight, if desired. Polymerization was complete in 4-6 h. The small amounts of coagulum that formed were removed by filtering through glass wool. Particle sizes were always ∼300 nm, but larger particles could be prepared by seeded polymerization. Characterization. The size and size distribution of the latex particles were determined by AFM and confirmed by transmission electron microscopy. Both tests confirmed that the sample (PiBMA #2) used in the present investigation was essentially monodisperse and had a diameter of 310 nm. The molecular weight (110 000) was determined by size-exclusion chromatography (SEC) in tetrahydrofuran solution and is based on polystyrene standards. The glass transition temperature (65 °C) was obtained by differential scanning calorimetry (DSC), with the midpoint of the transition taken as Tg. Film Preparation. Monolayer films were prepared by spreading a diluted latex (∼1% solids) directly onto freshly cleaved mica, followed by drying at room temperature overnight. As water evaporated, capillary pressures lead to the formation of almost perfect close-packed arrays of particles with few lattice defects, as was shown in LM-1.1 The film formation studies were then done by placing the air-dried samples in an oven for the desired times and temperatures. Atomic Force Microscopy. A TopoMetrix TMX 2000 atomic force microscope was used. The scans were done under ambient condition without any sample surface treatment and in the “contact mode”. The corrugation height of a film was obtained using the built-in profile characterization capabilities of this AFM instrument, and the reported values are averages taken from scans on different regions of the films. Rheological Properties. A Rheometrix Mechanical Spectrometer RMX 605 was used to measure the stress relaxation properties of the latex polymer. Dried latex samples were placed on the lower platen of a parallel plate fixture and the temperature was raised to 180 °C to melt them. The upper platen was then lowered to press the material into a coherent film of known thickness (typically about 1 mm). After adjusting to the desired test temperature, the upper platen was rotated to give a desired strain level e, which varied from 1.3% at 70 °C to 25% at 160 °C, as measured at the outer edge of the platens where the strain

is a maximum. Tests ensured that the applied strain and resulting stress were within the range of linear viscoelasticity. The stress σ was typically followed over several decades of time t to give stress relaxation data σ(t,T) at temperature T. The moduli data at each temperature G(t,T) ) σ(t,T)/e were then shifted according to time/temperature superposition principles to give a “master curve” for the stress relaxation function G(t,T0) at the reference temperature T0. Details are presented in Appendix I, together with details of the transformation of G(t,T0) to the compliance function J(t,T0).

Results and Discussion Figure 1 shows the AFM corrugation heights of the latex PiBMA #2 as a function of time at temperatures of 70, 80, and 90 °C, from which we see (not surprisingly) that the rate of film formation proceeds much more rapidly at the higher temperatures. It was noted that these corrugation height curves at the three temperatures appeared to be superimposable by a simple lateral shift along the time axis, in the same manner used to superimpose viscoelasticity data to generate “master curves” of viscoelastic response. The shifting of the corrugation height data is shown in Figure 1 and shows that these AFM data do superimpose to give an excellent corrugation height master curve. The “shift factors”, i.e., the lateral shifts (in log time) required to superimpose the corrugation height curves, are indicated in the figure. The fact that the primary corrugation height data could be shifted to give a master curve of course suggests a relationship of such data to the viscoelastic (rheological) properties of the latex polymer. Stress relaxation data were obtained at 70, 80, and 90 °C and are shown in Figure 2, together with the master relaxation curve obtained by shifting the primary relaxation data along the time axis. The shift factors required to superimpose the data are shown in the figure. A comparison of the shift factors required to superimpose the corrugation height data and the stress relaxation data shows that they are identical. This clearly indicates a one-to-one relationship between rheological properties and the rate of film formation. This result is perhaps not too surprising, but we believe it is the first time that a direct connection between viscoelastic (rheological) properties and the kinetics of film formation has been shown. The fact that the rate of film formation and viscoelastic properties are directly connected will be used in the following section to guide the development of a theory for film formation in (dry) latex systems.

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Figure 2. Stress relaxation data at 70, 80, and 90 °C, and resulting “master curve”, shown with shift factors used to superimpose the data.

In the earliest work on film formation from latex systems, it was generally assumed that the latex particles deformed by viscous flow.6 Later Brown4 correctly pointed out that viscous flow was not necessary, since he showed that a cross-linked polystyrene latex could form a film, even though the latex as a result of being cross-linked could only deform in an elastic manner. Brown then introduced the time-dependent shear modulus G(t) to establish a film forming criterion. Since then, several other film formation models using G(t) have been presented,3,5 but they have essentially followed Brown’s model with minor modifications. All the theoretical predictions based upon Brown’s model establish a maximum value for the time-dependent shear modulus for film formation and are of the form

G(t) e Kγ/R where G(t) is the time-dependent shear modulus, γ is the water surface tension, R is the particle radius, and K is a numerical parameter whose value depends on the details of the model. In general, film formation from latex systems will involve both elastic and viscous mechanisms, i.e., the deformation will be viscoelastic. In the early 1980s, Lamprecht,7 in recognizing that film formation process involved viscoelastic properties, introduced the timedependent compliance function J(t) to relate the deformation to the film-forming stress. Since the compliance function includes processes involving both (retarded) elastic and viscous deformations, it is a natural function to use to relate particle deformation and stresses. In his expression, he took Graham’s8 and Yang’s9 solution of the viscoelastic counterpart of the classical Herz10 contact problem, together with Mason’s expression for the capillary force and Brown’s capillary pressure, to derive

95γ 1 e R J(t) for the minimum value of J(t) for film formation. Later, (6) Dillon, R. E.; Matheson, L. A.; Bradford, E. B. J. Colloid Sci. 1951, 6, 108. (7) Lamprecht, J. Colloid Polym. Sci. 1980, 258, 960. (8) Graham, G. A. C. Int. J. Eng. Sci. 1965, 3, 27. (9) Yang, W. H. J. Appl. Mech. (Transact. ASME, Ser. E) 1966, 395. (10) Hertz, H. J. Math. 1881, 92.

Eckersly and Rudin11 claimed that there was a mistake in Lamprecht’s model and modified it to give slightly different criterion

1 34γ e R J(t) As we have discussed in LM-I and in the introduction to this paper, we believe that all of the models and theories that have been presented to date have serious flaws. The models used (e.g., Frenkel’s for particle deformation and Brown’s for capillary pressure) are incorrect for latex systems, and we believe that this invalidates any film formation criteria or conclusions based upon these models. A further problem concerns the fact that the relaxation and compliance functions G(t) and J(t) change monotonically by many orders of magnitude with time (examples are shown in the Appendix), and the above criteria thus merely predict that film formation will begin to occur at some particular time when G(t) or J(t) have finally relaxed to the values given by the whatever explicit criterion is used. Theory We now present a new theory for the flattening of a (dry) latex film. The typical model for latex film formation is shown in Figure 3, i.e., a close-packed array of latex spheres is initially formed as water evaporates. As the water continues to evaporate, the resulting capillary forces and particle surface tensions act to minimize the surface areas, i.e., to eliminate the interstitial space originally occupied by water and to flatten the surface. Coalescence of the particles to produce a coherent film may accompany film formation or may take place after the film has completely formed a flat film. However, here it is assumed that the particles are not distorted during the water evaporation process, and the initial state for film formation is a close-packed array of dry spheres, in which film formation occurs when the particles are then heated. Additional assumptions are as follows: a. The lateral spacing between particles remains constant as film formation proceeds, i.e., the center-tocenter spacing remains as 2a where a is the particle radius. The necessary change in the overall dimensions of the (11) Eckersley, S. T.; Rudin, A. J. Coatings Technol. 1990, 89, 780.

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Figure 4. Top view of latex during film formation showing hexagonal packing during later stages.

Figure 3. Model for latex film formation and coalescence.

film as the interstitial space is eliminated is accommodated by a reduction in the overall film thickness, not by a lateral change in area. The validity of this assumption has been shown experimentally.1,12 b. The top surface of a particle remains spherical during film formation, but with a changing radius of curvature R during film formation from its initial value a to the final value of ∞ when the film has completely flattened. (The assumption of a spherical surface minimizes the surface free energy.) c. The base of the cap remains a circle of radius a, although in the later stages of film formation the base becomes hexagonal, as shown in Figure 4 (the difference is minor and simplifies the calculations). d. The stress σ leading to particle deformation arises from the excess surface free energy of the cap, i.e., from the surface tension γ and radius of curvature R of the cap, σ ) 2γ/R. e. The top and bottom surfaces of a latex particle deform independently. f. The deformation of the cap of a latex particle is that of pure equal-biaxial extension. Each plane within the particle cap expands radially, following the increasing radius of curvature of the cap as film formation proceeds. Mixing between planes does not occur. g. The deformation of the particles obeys linear viscoelasticity. Since we have assumed that the deformation of the cap is that of pure equal-biaxial extension, i.e., pure radial extension in the plane of the film and without mixing (12) Goh, M. C; Juhue´, D.; Leung, O.-M.; Wang, Y.; Winnik, M. A. Langmuir 1993, 9, 1319.

Figure 5. Cross section of particle cap during film formation, showing the changing radius of curvature as a result of biaxial deformation. Shaded areas represent spherical segments within the cap which have equal volumes. See text for explanation of symbols.

between planes, the volume of a spherical segment above a given basal plane must remain constant during deformation of the particle. Figure 5 shows a stage in the deformation of a particle from its initial spherical shape of radius a to later stage with a radius of curvature R. The figure shows basal planes at δ0 and δ having base radii of r0 and r, respectively, and for which the volumes of the spherical segments above these basal planes are equal. The volume V(δ) of a spherical segment of height δ and radius of curvature R is

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π V(δ) ) (3Rδ2 - δ3) 3

Lin and Meier

(1)

which must equal the volume of the undeformed spherical segment of height δ0

π V(δ0) ) (3aδ02 - δ03) 3

(2)

These equations provide a relationship between δ0 and δ, from which the radii r0(δ0) and r(δ) of the basal planes at δ0 and δ are obtained using

δ0 ) a - (a2 - r02)1/2 and δ ) R - (R2 - r2)1/2 (3) The relationship for the radius of curvature R of the spherical cap as a function of its base radius a and height h is obtained from two expressions for the total volume of the cap,

πh2 πh V) (3R - h) and V ) (h2 - 3a2) 3 6

(5)

These equations relating R, a, δ, δ0, and h enable the extension ratios λ of the radii

λ(δ,a,h) )

r(δ,a,h) r0(δ0,a)

(6)

to be determined as a function of the particle radius a, the corrugation height h, and the position δ within the cap. Table 1 shows values of the extension ratio λ as a function of the relative position δ/h in the cap and the relative corrugation height h/a (i.e., relative to the initial particle radius a). The data shown in the table show that as film flattening proceeds (h/a f 0), the extension ratios become larger (as expected), but they are almost independent of the position δ/h within the cap at each value of h/a e 0.8 (which covers the range for which AFM data were obtained). This relative independence of the extension ratios is true at least over the range 1 g δ/h g 0.25, which includes more than 90% of the total volume of the cap at any relative corrugation height h/a. This fact indicates the assumptions made that the deformation is one of pure biaxial extension and that the cap surface remains spherical are reasonable and not incompatible. A shear contribution would be present if the extension ratios differed significantly as a function of δ/h. The biaxial strain ψ is related13 to the extension ratio λ by

ψ ) λ2 - 1/λ4

(7)

The stress σ which deforms the particles to flatten them is given by the surface tension γ and the radius of curvature R of the spherical cap, and is

σ)

2γ 4γh ) 2 R a + h2

λ h/a

δ/h ) 1

δ/h ) 0.75

δ/h ) 0.5

δ/h ) 0.25

0.8 0.5 0.1 0.05 0.01

1.017 1.10 1.56 1.85 2.70

1.013 1.09 1.54 1.81 2.68

1.010 1.08 1.53 1.81 2.68

1.008 1.07 1.52 1.79 2.66

Both the stress and the strain are time-dependent throughout film formation process since the corrugation height h is a function of time h(t). With our assumption that the system obeys linear viscoelasticity, we relate the time-dependent biaxial strain ψ(t) to the time-dependent stress σ(t) by means of the Boltzmann superposition relationship,15 which involves the compliance function J(t)

ψ(t) )

(4)

which gives

R ) (a2 + h2)/2h

Table 1. Biaxial Extension Ratios λ as a Function of Corrugation Height h/a and Position within Latex Cap δ/h

(8)

(13) Treloar, L. R. G. The Physics of Rubber Elasticity, 3rd ed.; Clarendon Press: Oxford, England, 1975. (14) Wu, S. Polymer Interface and Adhesion; Marcel Dekker, Inc.: New York, 1982.

J(t - s) ds ∫-∞t ∂σ ∂s

(9)

and where t is present time and s is a dummy time variable. The integration is over all past time up to the present time t. Writing the equation in terms of the experimental corrugation height variable h(t) gives

λ2(a,h) -

(a2 - h2)

J(t - s) ds ∫-∞t (a2 + h2)2 ∂h ∂s

1 ) 4γ 4 λ (a,h)

(10) where “h(t)” is simply written as “h”, and λ(a,h) is to be obtained from eq 6. If the compliance function J(t) and the surface tension γ are known, this integral equation can, in principle, be solved to give the theoretical corrugation height h(t) as a function of time for comparison with the experimental observations. However, the direct solution of this integral equation to obtain h(t) appears impossible, so a different route has been used to obtain h(t). Trial functions of h(t) were substituted into both sides of above equation until a function was found for which both sides were essentially equal over the time range for which experimental data were taken, i.e., from approximately 103 to 107 s. Since, as mentioned above, the biaxial strain does not depend significantly on the position δ within the cap, the evaluation of the biaxial extension ratio λ2(a,h), which was taken to be representative of the cap as a whole, could be evaluated at only one position, and here at δ/h ) 1. The surface tension of PiBMA was taken as 24 dyn/cm.14 A comparison of the theoretical and experimental corrugation heights h(t) is shown in Figure 6. Curve A in Figure 6 is calculated using the compliance function J(t) obtained from the transformation of experimental relaxation data G(t), as described in Appendix I. The corrugation heights shown in Figure 6 cover almost 4 decades of time, and the agreement between the experimental points and theory is considered to be very good, particularly since there are no adjustable parameters in the theory. Curve B in Figure 6, which passes through the experimental data points, represents the results of a modification of the compliance function so as to give exact agreement between theory and experiment. The original and the modified compliance functions J(t) are shown in Figure 9. Over the time range of measurement of the corrugation heights (3 < log t (s) < 7), the compliance function changes by almost 4 orders (15) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; John Wiley & Sons Inc.: New York, 1980.

Latex Film Formation

Figure 6. Comparison of theoretical and experimental corrugation heights. The experimental data are shown as small circles. Curve A is obtained using the compliance function J(t) from transformation of the stress relaxation function G(t). Curve B is obtained by modifying J(t) so as to fit the experimental data. The original and modified compliance functions J(t) are shown in Figure 9.

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Figure 8. Master stress relaxation function G(t) of PiBMA at a reference temperature of 70 °C.

Figure 9. Compliance functions J(t) at 70 °C. Curve A is the compliance function obtained from the stress relaxation function G(t). Curve B shows the compliance function that is required by the theory to exactly fit the experimental corrugation height data. Figure 7. Stress relaxation data of PiBMA at various temperatures.

of magnitude, while the difference between the original and the modified functions is hardly more than a factor of 2. This relatively minor difference is probably within the accuracy of obtaining J(t) from the stress relaxation data, particularly at long times when the relaxation (and resulting compliance) functions change rapidly. Thus we believe the theory is essentially in quantitative agreement with experiment data on the kinetics of film formation of a (dry) latex system. It is interesting to note from the compliance function and the measured viscosity of sample PiBMA #2 that the contribution of viscous flow to the total deformation is estimated to be no more than 6% over the measurement period which extended to almost 107 s, i.e., the particle deformation and film formation of this sample at 70 °C are almost wholly the result of elastic processes. This is further discussed in Appendix I. Summary and Conclusions The film-forming properties of a (dry) latex system are shown to obey time/temperature superposition principles, and the shift factor required to produce a “master curve” for the latex corrugation height is shown to be the same as the viscoelastic shift factor. A theory for the time dependence of the corrugation height is developed, which

is based on the Boltzmann superposition principle to relate the time-dependent stress and strain fields as the latex particles deform. The particle deformation arising from the flattening of the surface is equal biaxial extension and the deformation is driven by the stress associated with the excess surface free energy. The stress and strain fields are related through a time-dependent compliance function J(t), which in the present case is obtained by a transformation of stress relaxation data. The predictions of the theory (which contains no adjustable parameters) and the experimental data for corrugation heights are in reasonable agreement over more than 4 decades of time. The differences between theory and experiment are probably within the accuracy of the transformation of the experimental stress relaxation function G(t) to the compliance function J(t), since it is shown that a minor modification of the compliance function will give an exact agreement between theory and experiment. It is shown for the latex used here (PiBMA #2, MW ) 110 000) that particle deformation and film formation are the result of elastic processes. Over the more than 4 decades of time examined, the contribution of viscous flow to the deformation is estimated to be no more than 6% of that from the elastic response. Acknowledgment. The authors express their sincere thanks to the National Science Foundation Industry/ University Cooperative Research Center in Coatings for support of this work.

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Table 2. Relative Contributions of Elasticity and of Viscous Flow to Particle Deformation as a Function of Time elastic term viscous term t/η time ∫+∞ ratio viscosity/ -∞L(τ)(1 - exp[-t/τ]) d ln τ Pa-1 (s) Pa-1 elasticity 103 105 107

4.8 × 10-6 2.3 × 10-5 4.4 × 10-4

2.5 × 10-9 2.5 × 10-7 2.5 × 10-5

0.0005 0.011 0.057

Appendix I Determination of the Time-Dependent Compliance Function J(t). The compliance function J(t) was obtained from experimentally measured stress relaxation data, using the interconversion equations found in Ferry.15 Stress relaxation data were obtained on sample PiBMA #2 using a Rheometrics mechanical spectrometer over the temperature range from 70 to 160 °C. These data were then shifted according to time-temperature superposition principles to give a master curve covering a very wide range of time. The primary relaxation curves at the various temperatures, and the master curve for the relaxation function G(t), are shown in Figures 7 and 8. The relaxation spectrum H(t) was obtained from the relaxation function G(t) at selected values of t using the second approximation of Schwarzl and Staverman and evaluated using the computer program Theorist. In order to evaluate the derivatives required for transformation of G(t) to J(t), a sixth-order polynomial was fitted to the experimental data, using the computer program Cricket Graph.

H(τ) ) -G(t)[d log G(t) d log t (d log G(t)/d log t)2 - (1/2.303)d2 log G(t)/d(log t)2]t)2τ (I-1) The retardation spectrum L(t) could then be obtained from H(t) using the relationship15

L(τ) ) [H(τ)]

/{{G(t) + H(τ)[π2(csc mπ2 - sec mπ2 ) -

Figure 10. Plot of stress relaxation “shift factors” aT as a function of log at/(T - T0) to obtain WLF C1 and C2 parameters for PiBMA.

tures from the WLF equation15

log

2

2

Appendix II Determination of the WLF C1 and C2 Parameters for PiBMA. The WLF equation

2

and again using the Theorist program for the evaluation. In this equation, Γ is the Gamma function and -m is the slope of log H(t) vs log t. Finally the desired compliance function J(t) was obtained from the retardation spectrum L(t) using

J(t) )

∫-∞+∞L(τ)(1 - exp[- τt]) d ln τ + ηt

(I-4)

where C1 and C2 are WLF parameters (determined for PiBMA as shown in Appendix II). The measured viscosity of PiBMA #2 at 150 °C is 1.0 × 105 Pa s, from which the viscosity at 70 °C is estimated to be 4 × 1011 Pa s. The values of the retarded elastic term and the viscous term are shown in Table 2. The viscosity contribution to J(t) varies from less than 0.1% at t ) 103 s to less than 6% at t ) 107 s, and so the integral term (retarded elastic term) is effectively J(t) for this PiBMA sample at 70 °C.

]} + π H(τ) } (I-2)

Γ(m) + 1.37

η1 C1C2(T2 - T1) ) η2 (C2 + T1 - T0)(C2 + T2 - T0)

log at )

(II-1)

where log at is the log time shift required to superimpose a stress relaxation curve at temperature T to that of the reference temperature T0 can be written in the form

log at ) C1 - C2

(I-3)

and is shown in Figure 9. The integral term represents the retarded elastic contribution while t/η represents the viscous contribution to the deformation. The contribution of the viscosity term at 70 °C can be estimated from viscosity data taken at higher tempera-

c1(T - T0) c2 + T - T0

log at (T - T0)

(II-2)

so that a plot of log at as a function of log at/(T - T0), gives C1 as the intercept of the plot and C2 as the slope. Figure 10 shows this plot for PiBMA, which gives C1 ) 17.68 and C2 ) 215.5 at a reference temperature T0 ) 100 °C. LA951554W