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Ind. Eng. Chem. Res. 2001, 40, 3048-3052
Effects of Dissolved Gas on Viscoelastic Scaling and Glass Transition Temperature of Polystyrene Melts Choongyong Kwag, Charles W. Manke,* and Esin Gulari Department of Chemical Engineering and Materials Science, Wayne State University, Detroit, Michigan 48202
The free volume theory of Gerhardt et al.1 (J. Polym. Sci. B: Polym. Phys. 1998, 36, 1911) is used to predict viscoelastic scaling factors describing the effect of dissolved gas content on the viscosity curves of polystyrene melts swollen with dissolved carbon dioxide and dissolved 1,1difluoroethane. The predictions of the theory are compared to viscoelastic scaling factors measured by Kwag et al.2 (J. Polym. Sci. B: Polym. Phys. 1999, 37, 2771) for each system at 150 and 175 °C, at concentrations up to 10 wt % of dissolved gas, and pressures ranging up to 22 MPa. The agreement between the theory and experiments is very good for the polystyreneCO2 system but only fair for the polystyrene-1,1-difluoroethane system. The experimental viscoelastic scaling factor values are also interpreted with the WLF equation to estimate the change in the underlying glass transition temperatures of the polystyrene-gas mixtures. The glass transition temperatures estimated from these rheological data are in very good agreement with values directly measured for polystyrene-CO2 mixtures and with the theory of Condo et al.3 (Macromolecules 1992, 25, 6119). Introduction Free volume concepts have long been employed in modeling transport properties in polymeric materials. Among the classical applications of free volume theory to transport are the works of Duda, Vrentas, and coworkers4-6 describing the diffusion of solvents in polymers. The effects of free volume on the rheological behavior of polymers are equally well-known, particularly through the principle of time-temperature superposition and the WLF equation (see Ferry7), which correlate the effects of temperature on rheological properties. Here, we apply the modified free volume theory of Gerhardt et al.1 to evaluate the effect of dissolved supercritical fluids (SCFs) on the viscosity of molten polystyrene and then apply an analysis based on the WLF equation to interpret the rheological data in terms of glass transition temperature depression. Polymer-SCF systems are important to a number of industrial processes, including the production of foamed polymers, the synthesis of polymers in SCFs,8,9 and the production of particles through the rapid expansion of SCF-polymer mixtures.10 The rheology of polymerSCF mixtures is important to all of these applications. The rheology of poly(dimethylsiloxane) (PDMS) with dissolved carbon dioxide was investigated by Gerhardt et al.,11 using a high-pressure capillary rheometer. In this work, the viscosity curves for PDMS-CO2 mixtures were found to follow ideal viscoelastic scaling, wherein the viscosity η(c,γ˘ ) and shear rate γ˘ can be reduced by a compositional scaling factor ac to form a master curve of η(c,γ˘ )/ac vs acγ˘ that is identical to the viscosity curve for the pure polymer melt at the same temperature and pressure. The rheological effects of the dissolved gas are then completely described by the variation of ac with dissolved gas concentration c. Gerhardt et al.1 developed a free volume theory to predict ac for polymer-SCF
mixtures in which PVT properties are evaluated by an equation of state. Recently, several new measurements of the viscosities of polymer-SCF systems have been performed, including the high-pressure capillary rheometer measurements of Kwag et al.2 for polystyrene melts with dissolved CO2, 1,1-difluoroethane, and 1,1,1,2tetrafluoroethane; the extrusion experiments of Lee et al.12 for PS-CO2, and the measurements of Elkovich et al.13,14 of the viscosity and morphology of blends of polystyrene and poly(methyl methacrylate) containing CO2. Both Kwag et al.2 and Lee et al.12 observed that polystyrene-SCF systems follow viscoelastic scaling similar to that of the PDMS-CO2 system. However, the PS-SCF viscosity measurements were performed at temperatures much closer to the glass transition temperature Tg of the pure polymer, than were Gerhardt’s11 PDMS-CO2 measurements, and the observed reduction in melt viscosity by dissolved gas is an order of magnitude, or more, greater. Thus, the new data for PS-SCF systems should provide a more demanding test of Gerhardt’s1 predictive theory for the scaling factor ac. In the discussion to follow, Gerhardt’s1 free volume theory for ac is applied to describe the high-pressure capillary rheometer data of Kwag et al.2 for the PSCO2 and PS-1,1-diflurorethane systems. An analysis is then applied using the WLF equation to estimate the effect of dissolved gas on the glass transition temperature of the PS-SCF mixtures used in Kwag’s study. The values for Tg estimated from Kwag’s rheological experiments are then compared to Tg values measured directly for PS-CO2 mixtures by Wissinger and Paulaitis,15 Condo et al.,16 and Chiou et al.17 It is shown that the estimates of Tg from rheological measurements are in good agreement with the direct measurements and that the data are generally consistent with the theory of Condo et al.3 Theory
* Corresponding author: Charles W. Manke, Department of Chemical Engineering and Materials Science, Wayne State University, Detroit, MI 48202. Tel.: 313-577-3849. Fax: 313577-3810. E-mail:
[email protected].
Viscoelastic Scaling Factor ac. Gerhardt et al.1 extended simple free volume theories developed by Doolittle18 and Kelley and Bueche19 for the Newtonian
10.1021/ie000680e CCC: $20.00 © 2001 American Chemical Society Published on Web 04/06/2001
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3049 Table 1. Sanchez-Lacombe Equation of State Paramters for PS-SCF Systems Pure-Component Parameters component
T* (K)
P* (MPa)
F* (g/cm3)
carbon dioxide 1,1-difluoroethane polystyrene
330.4 402.1 735
458 366.3 357
1.43 1.206 1.105
Binary Interaction Parameters δij PS-CO2 PS-1,1-difluoroethane
150 °C
175 °C
0.025 0.058
0.014 0.084
viscosity of polymer melts and solutions to develop an expression for the effect of dissolved gas on the viscosity-shear rate curves for polymer melts. Their approach assumes that the viscosity curve for a polymer melt containing a dissolved gas will be scaled in both viscosity and shear rate by a viscoelastic scaling factor ac that depends on the gas concentration c and the temperature T. Thus, the scaling factor ac correlates the effect of the dissolved gas concentration in the same manner that the familiar WLF scaling factor aT (see Ferry7) scales viscosity and shear rate for the effect of temperature. Gerhardt’s equation for ac is
ac )
( ) (
ηo(c) Vp ) (1 - wscf)n ηo,p Vm
n
exp
)
1 1 fm fp
Glass Transition Temperature Tg. Because the effects of dissolved gas on the viscosity of polymer melts occur predominately through a free volume mechanism,1 the viscoelastic scaling factor ac can also be interpreted as a measure of the effect of the dissolved gas on the glass transition temperature of the polymer-SCF mixture. Using the viscoelastic scaling for gas composition developed by Gerhardt et al.,1,11 the viscosity curve for a polymer-SCF mixture can be shifted onto the viscosity curve for the pure polymer melt by the shift factor ac, that is, ηp(acγ˘ ,T) ) η(c,γ˘ ,T)/ac. We can then shift the pure PS viscosity curve at temperature T to the glass transition temperature by the temperature scaling factor aT for pure PS, such that ηp(aTγ˘ ,Tg) ) ηp(γ˘ ,T)/aT. Thus the total scaling factor that superposes the polymer-SCF viscosity curve at temperature T onto the PS curve at Tg is a ) acaT. We now assume, for purposes of estimating Tg for polymer-SCF mixture, that Tg for the mixture will occur at the same value of the fractional free volume as Tg for pure PS. This means that the difference between T and Tg for the mixture is represented by the total scaling factor a. The glass transition temperature for the mixture can then be estimated by employing the WLF equation to calculate (T - Tg) for the mixture from the known value of a
(1)
where wSCF is the weight fraction of dissolved gas; Vp and Vm are the specific volumes of the pure polymer and polymer-SCF mixture, respectively; and fp and fm are the fractional free volumes of the pure polymer and the polymer-SCF mixture, respectively. Note that the viscoelastic scaling factor is equivalent to the ratio ηo(c)/ηo,p of the Newtonian viscosity of the polymer-SCF mixture to that of the pure polymer melt. Because of the complex PVT behavior of polymer-gas mixtures1 at the high pressures typical of supercritical fluids, the volumetric properties Vp, Vm, fp, and fm are evaluated using an equation of state (EOS) suitable for polymergas mixtures. In the case of the polystyrene-SCF systems considered here, the Sanchez-Lacombe equation of state20-22 is employed, with the parameters listed in Table 1. In addition to the specific volumes predicted by the EOS, evaluation of the fractional free volume f ) (V - Vo)/V requires the occupied specific volume Vo, which represents the portion of the total specific volume V that is inaccessible to the segmental motions of the polymer chain that produce liquid flow on the macroscopic scale. Several methods are used to evaluate Vo. For the pure polymer, Vo is determined by fitting eq 1 to experimental measurements of the variation of Newtonian viscosity with temperature (see Gerhardt et al.1); this produces a value of Vo that is calibrated to the volumetric predictions of the specific EOS, e.g., SanchezLacombe, employed for PVT evaluations. For the SCF component, one must often use group contribution methods to estimate Vo. Unfortunately, this procedure can lead to error both because of uncertainty in the accuracy of the estimation method and because group contribution methods generally estimate a hard-core value for Vo, such as the van der Waals volume, rather than the Vo that pertains to microscopic flow mechanisms. Once Vo is determined for both the polymer and SCF components of the mixture, the simple linear mixing rule1 Vo,m ) (1 - wSCF)Vo,p + wSCFVo,SCF is used to determine Vo for the mixture.
ln a )
-C1(T - Tg) C2 + (T - Tg)
(2)
where C1 and C2 are the WLF constants. Because C1 and C2 are not known for the PS-gas systems studied here, we employ the best-fit values for pure PS, namely, C1 ) 13.81 and C2 ) 48.87 (see Kwag,23 Appendix III). Results Viscoelastic Scaling. Viscosity has been measured as a function of shear rate for a number of polymerSCF systems, including poly(dimethylsiloxane) (PDMS)CO2 (see Gerhardt et al.11) and polystyrene (PS) with CO2 and with the refrigerants 1,1-difluoroethane (R152a) and 1,1,1,2-tetrafluoroethane (R134a) (see Kwag et al.2). These measurements show that the viscosity curves for polymer melts with dissolved SCFs appear identical in shape to the viscosity curve for the pure polymer melt and that the polymer-SCF curves are shifted on both the viscosity and shear rate axes relative to the curves for the pure melt. This observation led to the successful application of classical viscoelastic scaling theory to reduce the viscosity data for polymer-SCF systems to a master curve of scaled viscosity η(γ˘ ,c)/ac vs scaled shear rate acγ˘ , which is identical to the viscosity curve for the pure polymer melt. The rheological effects of the dissolved gas contained within the melt at concentration c are then represented entirely through the compositional scaling factor ac, which depends on c and T for a given gas solute. Figure 1 illustrates a typical set of viscosity curves for a polymer-SCF system, in this case PS-CO2. Note that viscosity decreases with increasing gas content and that the data taken at various CO2 compositions appear similar in shape to portions of the viscosity curve for pure PS. In Figure 2, these data, along with data for PS with other gases, are mapped onto the viscosity curve for pure PS by the scaling method described above. (In addition to scaling for the effect of composition by ac, the data have been scaled to a single temperature
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Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001
Figure 1. Viscosity of polystyrene with dissolved carbon dioxide at 150 °C measured as a function of shear rate by Kwag et al.2 Data are corrected for effects of pressure and non-Newtonian flow, as described in ref 2. Carbon dioxide compositions are: O, 1.0; 0, 2.0; 4, 3.0; 3, 3.5; ], 4.5; ., 5.0; and ~, 5.2 wt % CO2. The solid curve is the viscosity curve for pure polystyrene (Mw ) 132 000) at 150 °C and 1 atm pressure.
Figure 2. Master curve produced by superposing all data for Kwag’s2 PS-SCF systems. Viscosity data taken at 175 °C have been shifted to 150 °C by employing the WLF temperature scaling factor aT for pure polystyrene (see Kwag et al.2). The master curve is identical to the viscosity curve for pure polystyrene at 1 atm and 150 °C, which is shown as the solid line.
and pressure, 150 °C and 1 atm, through the use of the temperature and pressure scaling factors aT and aP, respectively, for pure PS; see Kwag et al.2 for details. The scaling factor aP for the effect of pressure was derived by a procedure whereby viscosity was assumed to vary with ebP, where b is a constant.) Similar scaling results were obtained by Gerhardt et al.11 for the PDMS-CO2 system. Variation of Scaling Factor ac with Composition. The free volume theory for ac, eq 1, has been employed successfully by Gerhardt et al.1 to model the variation of ac with concentration for the PDMS-CO2 system. Here, we apply eq 1 to predict viscoelastic scaling factors for molten polystyrene with dissolved
Figure 3. Viscoelastic scaling factors ac predicted by eq 1 for the PS-1,1-difluoroethane system at 150 and 175 °C. The solid lines represent predictions using Vo ) 0.7509 cm3/g for pure 1,1difluoroethane, and the chain-dashed lines represent the predictions using Vo ) 0.6227 cm3/g (see text). The predictions are compared to the experimental data of Kwag et al.,2 which are indicated by symbols. The dashed line at the top of the plot indicates the scaling factor predicted by eq 2 for dilution at constant free volume.
CO2 and dissolved 1,1-difluoroethane. Although gas solubilities in both the PS-CO2 and PS-1,1-diflouroethane systems are lower than the solubility of CO2 in PDMS, the experimental ac values for PS-SCF systems measured by Kwag et al.2 are orders of magnitude smaller than those measured by Gerhardt et al.11 for PDMS-CO2. This difference is due to a closer proximity to the glass transition temperature for the PS-SCF systems, which is discussed below. Because the PSSCF systems probe a temperature-composition regime where the viscosity reduction due to dissolved gas is much more pronounced than for PDMS-CO2, the PSSCF systems should pose a more demanding test of the free volume theory. Figure 3 compares the ac predictions of eq 1 to the experimental measurements of Kwag et al.2 for the PS1,1-difluoroethane system at two temperatures. Results for two different values of the occupied volume Vo of pure 1,1-difluoroethane are displayed. The value Vo ) 0.6227 cm3 is calculated by a group contribution method using molar van der Waals volumes developed by Bondi,24 whereas the value Vo ) 0.7509 cm3/g is calculated by a group contribution method using molar occupied volumes developed by Sugden.25 These two methods give quite different values of Vo, introducing a considerable uncertainty range in the ac values predicted by eq 1. Although most of the data fall within the range between the Vo ) 0.6227 cm3/g curve and the Vo ) 0.7509 cm3/g curve at each temperature, quantitative agreement with the theoretical predictions is only fair. The agreement is seen to be poorer for T ) 175 °C than for T ) 150 °C; this may be partly due to the high back pressures (up to 22 Mpa) applied to hold 1,1difluoroethane in solution with PS at 175 °C. At such extreme pressure, gas losses from the sample become
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3051
Figure 4. Viscoelastic scaling factors ac predicted by eq 1 for the PS-CO2 and PDMS-CO2 systems, indicated by labeled solid curves. The predictions are compared to the experimental data of Kwag et al.2 for PS-CO2 and Gerhardt et al. for PDMS-CO2;1,11 experimental data are represented by labeled symbols. The dashed line at the top of the plot indicates the scaling factor predicted by eq 2 for dilution at constant free volume.
more likely, with the associated effect of artificially raising the apparent ac values from the measurements. The back pressure applied at 150 °C was considerably lower, ranging up to about 6 Mpa. The effect of temperature on ac predicted by eq 1 does follow the experimental data qualitatively, however. Two other features to note in Figure 3 are the extent of viscosity reduction (over 3 orders of magnitude) represented by the ac values measured at 150 °C and the large difference between the measured ac values and those predicted by eq 1 for constant free volume dilution (dotted line). The latter comparison illustrates that almost all of the viscosity reduction achieved by the dissolved gas results from the contribution of free volume. Agreement between eq 1 and the ac values measured by Kwag is much better for the PS-CO2 system, as shown in Figure 4. Here, the occupied volume contribution of CO2 can be determined with much greater precision than that of 1,1-difluoroethane; we employ the value Vo ) 0.6500 cm3/g, which yields an almost exact fit of eq 1 to the ac data of Gerhart et al.11 for the PDMS-CO2 system at 50 °C with CO2 compositions up to about 20 wt %. Equation 1 compares well with Kwag’s PS-CO2 data at both 150 and 175 °C. The ac values for PS-CO2 at these temperatures fall well below the values for PDMS-CO2 at 50 °C and far below the curve for dilution at constant free volume. The order of the ac curves is primarily due to the proximity to the glass transition temperature Tg, which ranges from (T - Tg) ) 178 °C for PDMS-CO2 at 50 °C to (T - Tg) ) 50 °C for PS-CO2 at 150 °C. Because the specific volume V approaches the occupied volume Vo as T approaches Tg, the exponential term of eq 1 will be more sensitive to changes in (V - Vo) in this limit. Thus the free volume added by the dissolved gas has a much greater effect on ac when the temperature is near Tg. Depression of Glass Transition Temperature. Because the reduction of the polymer melt viscosity by dissolved gas occurs primarily through a free volume mechanism, the compositional scaling factors reported above can be regarded as manisfestations of the effects of dissolved gases on the glass transition temperature
Figure 5. Glass transition temperatures Tg estimated by eq 2 from the ac values of Kwag et al.2 The symbols represent Tg values calculated from the following ac data: b, PS-CO2 at 150 °C; O, PS-CO2 at 150 °C; 2, PS-1,1-difluoroethane at 150 °C; 4, PS1,1-difluoroethane at 175 °C; [, PS-1,1,1,2-tetrafluoroethane at 150 °C; and ], PS-1,1,1,2-tetrafluoroethane at 175 °C. The Tg values estimated from rheological data by eq 2 are compared to directly measured Tg values for PS-CO2 mixtures reported by: +, Wissinger and Paulaitis;15 /, Condo et al.;16 and × Chiou et al.17 The solid curve represents Tg for PS-CO2 mixtures, as predicted by the model of Condo et al.3
of the polymer-SCF mixtures. Therefore, an alternative way of interpreting the scaling factor data is to estimate the underlying Tg for the polymer-SCF mixture. As discussed above in the Theory section, eq 2 employs the WLF equation to estimate the depression in Tg that is equivalent to a particular value of the total scaling factor a, which includes ac. Using this method, the ac values reported by Kwag et al.2 for PS-CO2, PS-1,1-difluoroethane, and PS1,1,1,2-tetrafluoroethane have been used to estimate equivalent values of Tg for these systems. The results are plotted as a function of weight fraction of dissolved gas in Figure 5; for convenience, the Tg results are also listed in Table 2. The Tg values derived from Kwag’s rheological measurements are in remarkably good agreement with direct experimental measurements of Tg for PS-CO2 mixtures performed by Wissinger and Paulaitis,15 Condo et al.,16 and Chiou et al.17 The PS-CO2 data are well represented by the Tg model of Condo et al.,3 which employs an equation of state description of the PS-CO2 system and utilizes the Gibbs-DiMarzio criterion to determine Tg. The effect of gas content on Tg for the PS-1,1-difluoroethane and PS-1,1,1,2-tetrafluoroethane systems is seen to be very similar to that for PS-CO2. Conclusions The free volume theory of Gerhardt et al.1 for the viscoelastic scaling factor ac, which scales the viscosityshear rate curves for polymer-SCF systems onto the viscosity curve for the pure polymer, was found to be in good agreement with experimental ac values measured by Kwag et al.2 for the PS-CO2 system. The model is able to accurately predict the effect of the CO2 concen-
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Table 2. Glass Transition Temperatures of PS-SCF Mixtures, Estimated by Eq 2 from the Data of Kwag et al.2 wt fraction of SCF
Tg (°C)
PS-CO2 at 150 °C
system/temperature
0.01 0.02 0.03 0.035 0.045 0.05 0.052
85.2 81.2 73.0 68.7 62.9 57.8 50.2
PS-CO2 at 175 °C
0.01 0.02 0.03 0.035 0.045 0.051
82.5 79.4 73.7 66.6 62.8 55.0
PS-R152a at 150 °C
0.056 0.07 0.083 0.103
52.4 40.6 30.1 23.8
PS-R152a at 175 °C
0.05 0.084 0.104
49.6 43.5 19.7
PS-R134a at 150 °C
0.01 0.024 0.026 0.04 0.043
89.4 86.2 82.0 76.9 65.0
PS-R134a at 175 °C
0.02 0.028
78.8 73.1
tration on ac. Comparisons of predicted ac values with experimental values were less successful for the PS1,1-difluoroethane system. Here, the model’s predictions of the effects of temperature and concentration were qualitatively in agreement with the data, but quantitative deviations were large in some cases, particularly for data taken at 175 °C, where very high back pressures were used in the experiments. The quantitative predictions for the PS-1,1-difluoroethane system might also be impaired by uncertainty in the estimation of occupied volume for pure 1,1-difluoroethane, which is a critical model parameter. The free-volume-based model of Gerhardt et al.1 successfully describes the reduction of the polymer melt viscosity for both the PDMS-CO2 and PS-CO2 systems, suggesting that free volume mechanisms play a dominant role in the reduction of polymer melt viscosity by dissolved gases. This suggests that rheological experiments on polymer-SCF systems can be interpreted in terms of the effect of the dissolved gas on Tg of the polymer-SCF mixture. Accordingly, the WLF equation was employed to estimate Tg values from the viscoelastic scaling factors ac measured by Kwag et al.2 for several PS-SCF systems. The Tg values estimated by this method are in very good agreement with direct measurements of Tg by several different studies, and they are in excellent agreement with the theory of Condo et al.3 Acknowledgment The authors acknowledge the assistance of Ms. Ming Wang and Mr. Jeff Elam, who performed some of the computations used in this paper. The authors are grateful to the National Science Foundation, Award CTS-9817056, and to the Dow Chemical Co. for financial support of this work.
Literature Cited (1) Gerhardt, L. J.; Garg, A.; Manke C. W.; Gulari, E. Concentration-Dependent Viscoelastic Scaling Models for Polydimethylsiloxane Melts with Dissolved Carbon Dioxide. J. Polym. Sci. B: Polym. Phys. 1998, 36, 1911. (2) Kwag, C.; Manke, C. W.; Gulari, E. Rheology of Molten Polystyrene with Dissolved Supercritical and Near-Critical Gases. J. Polym. Sci. B: Polym. Phys. 1999, 37, 2771. (3) Condo, P. D.; Sanchez, I. C.; Panayiotou, C. G.; Johnston, K. P. Glass Transition Behavior Including Retrograde Vitrification of Polymers with Compressed Fluid Diluents. Macromolecules 1992, 25, 6119. (4) Duda, J. L.; Vrentas, J. S.; Liu, H. J. Prediction of Diffusion Coefficient for Polymer-Solvent Systems. AIChE J. 1982, 28, 279. (5) Vrentas, J. S.; Duda, J. L.; Hou, A. C. Evaluation of Theories for Diffusion in Polymer-Solvent Systems. J. Polym. Sci. B: Polym. Phys. 1985, 23, 2469. (6) Vrentas, J. S.; Duda, J. L.; Hou, A. C. Determination of FreeVolume Parameters form Diffusivity Data in Glassy Polymers. J. Polym. Sci. B: Polym. Phys. 1987, 33, 2581. (7) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; John Wiley and Sons: New York, 1980. (8) DeSimone, J. M.; Maury, E. E.; Menceloglu, Y. Z.; McClain, J. B.; Romack, T. J.; Combes, J. R. Dispersion Polymerizations in Supercritical Carbon Dioxide. Science 1994, 265, 356. (9) Lepilleur, C.; Beckman, E. J. Dispersion Polymerization of Methyl Methacrylate in Supercritical CO2. Macromolecules 1997, 30, 745. (10) Tom, J. M.; Kwauk, X.; Yeo, S. D.; Debenedetti, P. Rapid Expansion of Supercritical Solutions (RESS): Fundamentals and Applications. Fluid Phase Equilib. 1993, 82, 311. (11) Gerhardt, L. J.; Manke, C. W.; Gulari, E. Rheology of Poly(dimethylsiloxane) Swollen with Supercritical Carbon Dioxide. J. Polym. Sci. B: Polym. Phys. 1997, 35, 523. (12) Lee, M.; Park, C. B.; Tzoganakis, C. Measurements and Modeling of PS/Supercritical CO2 Solution Viscosities. Polym. Eng. Sci. 1999, 39, 99. (13) Elkovitch, M. D.; Tomasko, D. L.; Lee, L. J. Supercritical Carbon Dioxide Assisted Blending of Polystyrene and Poly(methyl methyacrylate). Polym. Eng. Sci. 1999, 39, 2075. (14) Elkovitch, M. D.; Lee, L. J.; Tomasko, D. L. Effect of Supercritical Carbon Dioxide on Morphology Development During Polymer Blending. Polym. Eng. Sci. 2000, 40, 1850. (15) Wissinger, R. G.; Paulaitis, M. E. Glass Transitions in Polymer/CO2 Mixtures at Elevated Pressures J. Polym. Sci. B: Polym. Phys. 1991, 29, 631. (16) Condo, P. D.; Paul, D. R.; Johnston, K. P. Glass Transitions of Polymers with Compressed Fluid Diluents: Type II and III Behavior. Macromolecules 1994, 27, 365. (17) Chiou, J. S.; Barlow, J. W.; Paul, D. R. Plasticization of Glassy Polymers by CO2. J. Appl. Polym. Sci. 1985, 30, 2633. (18) Doolittle, A. K. Studies of Newtonian Flow. II. The Dependence of the Viscosity of Liquids on Free Space. J. Appl. Phys. 1951, 22, 1471. (19) Kelley, F. N.; Bueche, F. Viscosity and Glass Temperature Relations for Polymer-Diluent systems. J. Polym. Sci. 1961, L, 549. (20) Sanchez, I. C.; Lacombe, R. H. An Elementary Molecular Theory of Classical Fluids: Pure Fluids. J. Phys. Chem. 1976, 80, 2352. (21) Lacombe, R. H.; Sanchez, I. C. Statistical Thermodynamics of Fluid Mixtures. J. Phys. Chem. 1976, 80, 2568. (22) Sanchez, I. C.; Lacombe, R. H. Statistical Thermodynamics of Polymer Solutions. Macromolecules 1978, 11, 1145. (23) Kwag, C. Rheology of Molten Polystyrene with Dissolved Gases. Ph.D. Dissertation, Wayne State University, Detroit, MI, 1998. (24) Bondi, A. Physical Properties of Molecular Crystals, Liquids and Glasses; John Wiley and Sons: New York, 1968. (25) Sugden, S. J Chem. Soc. 1927, 1780-1786.
Received for review July 19, 2000 Revised manuscript received December 5, 2000 Accepted December 14, 2000 IE000680E
