Ind. Eng. Chem. Res. 1998, 37, 3203-3207
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Diffusion and Equilibrium Measurements in Ternary Polymer-Solvent-Solvent Systems Using Inverse Gas Chromatography Rahul K. Surana, Ronald P. Danner,* and J. Larry Duda Center for the Study of Polymer-Solvent Systems, Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
Diffusion and equilibrium coefficients in ternary polymer-solvent-solvent systems were measured using inverse gas chromatography (IGC). The technique provides a new approach to probe diffusion in ternary polymer-solvent-solvent systems. The self-diffusion coefficients of a solvent in a mixture of a polymer and a second solvent were measured. Experiments were performed for the diffusion of toluene and methanol in the poly(vinyl acetate) (PVAC)-toluenemethanol system and for diffusion of toluene and heptane in the polybutadiene (PDB)-tolueneheptane system. Predictions of diffusivities in ternary systems based on the free-volume theory are also critically analyzed and compared with the experimental data. Introduction Multicomponent diffusion in polymer-solvent systems is commonly encountered, sometimes deliberately induced, in industrial applications especially during devolatilization of solvents and drying of films. Some researchers (Carra et al., 1981; Sasaki et al., 1990) have shown enhancement of the devolatilization rates by addition of small amounts of a second solvent. There are several proposed mechanisms for this enhancement: (1) the second solvent brings large amounts of free volume with it, thus increasing the diffusivity (Vrentas et al., 1985), (2) the second solvent, such as water, exists as a second dispersed phase and its vaporization such as in a vented extruder can cause mixing and bubbles which enhance the removal of impurities, or (3) the presence of the second solvent changes the thermodynamic behavior so as to decrease the solubility of the impurity and increase the thermodynamic driving force for devolatilization. The traditional techniques to measure diffusion coefficients in polymer-solvent systems such as sorption (Duda et al., 1973) and chromatographic techniques (HadjRomdhane and Danner, 1993) are generally restricted to binary polymer-solvent systems. Some work has been done to measure ternary polymer-solventsolvent self-diffusion using nuclear magnetic resonance (von Meerwall and Ferguson, 1981) and forced Rayleigh scattering (Lodge et al., 1990). The latter method is limited to the measurement of the diffusion of specific dyes, yielding data which are of little practical interest. In the past, inverse gas chromatography (IGC) had been used by many researchers (Pawlisch et al., 1987, 1988; HadjRomdhane and Danner, 1993) to measure infinitely dilute diffusion coefficients in binary polymersolvent systems. Recently, we have successfully extended this technique to measure very low diffusion coefficients which are typically near the glass transition temperature of the polymer (Surana et al., 1997) and * To whom all correspondence should be addressed. Telephone: (814) 863-4814. Fax: (814) 865-7846. E-mail: rpd@ psu.edu.
also to measure diffusion and partition coefficients at finite concentrations of solvent in the polymer (Tihminlioglu et al., 1997). In this work IGC was used to measure diffusion and partition coefficients in multicomponent polymer-solvent-solvent systems. The poly(vinyl acetate) (PVAC)-toluene-methanol and polybutadiene (PDB)-heptane-toluene systems were studied. The experimental data were collected at infinitely dilute concentrations of one solvent and 0-20 wt % of the second solvent. Experiments were performed at different temperatures, and the data were compared to predictions from free-volume theory for ternary polymersolvent-solvent systems. Inverse Gas Chromatography In the capillary column inverse gas chromatography (IGC) technique, a small pulse of a volatile solvent is passed through a capillary column whose inside walls have been coated with the polymer of interest. The solvent is partitioned between the mobile gas phase and the stationary polymer phase. Due to the mass-transfer resistance in the polymer phase, the solvent in the mobile phase is swept forward while that in the stationary phase lags behind. Pawlisch et al. (1987, 1988) modeled the flow of solvent in the capillary column and, with some judicious simplification, related the elution of solvent to the diffusion and partition parameters.
[ ] [(
)]
S 2xS CL 1 1 exp + + ) exp tanh βxS 2 C0u 2Γ Γ RΓβ 4Γ
R)
r K(1 - ψ)τ
1/2
(1)
β2 )
τ2u DpL
Γ)
Dg uL
(2)
In these equations C is the outlet concentration of the column, C0 is the strength of the inlet pulse, u is the mean velocity of the carrier gas, L and r are the length and the radius of the column, respectively, τ is the thickness of the polymer film, K is the partition coefficient (ratio of the solvent concentration in the polymer phase to the concentration in the mobile phase), and
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Dg and Dp are the gas-phase and polymer-phase diffusion coefficients. The parameter ψ accounts for the change in velocity due to sorption and is defined as the ratio of the solute concentration to the total concentration in the mobile phase. The elution profile is a function of the three dimensionless parameters R, β, and Γ. R is inversely related to the partition coefficient, β is similarly related to the polymer diffusion coefficient, and Γ varies proportionately with the gas-phase diffusion coefficient. The solvent pulse eluting through the column is infinitely dilute and does not alter the concentration of the second solvent in the carrier gas. In this work, the pulse solvent is infinitely dilute, and, hence, the system is analyzed as a pseudobinary system. In this technique, a concentration gradient is imposed across the polymer film and a mutual binary diffusion coefficient is measured. However, at the limit of infinite dilution, this mutual binary diffusion coefficient is equivalent to the self-diffusion coefficient, D1, of the probe species in a binary mixture of the polymer and a second solvent. Hence, even for this case of multicomponent diffusion, the same equations can be applied to calculate the diffusion and partition coefficients. The concentration of the second solvent in the polymer can be independently determined from a finite concentration experiment (Tihminlioglu et al., 1997) or from some thermodynamic model, such as the Flory-Huggins theory. Thus, IGC can be effectively used to determine the self-diffusion coefficient at infinite dilution of a solvent in a mixture of a polymer and a second solvent. Free-Volume Theory The free-volume theory of Vrentas and Duda (1977a,b) is the most widely used theory for correlating and predicting diffusion in polymer-solvent systems. It has been extended to predict diffusion in ternary polymersolvent-solvent systems from pure-component viscosity and density data and from binary polymer-solvent diffusion data (Vrentas et al., 1985). Equations 3-5
[
D1 ) D01 exp -
]
ω1V ˆ 1* + ω2V ˆ 2*ξ12 + ω3V ˆ 3*ξ13 V ˆ FH
(3)
K11 K12 V ˆ FH ) ω1 (K21 - Tg1 + T) + ω2 (K22 - Tg2 + γ γ γ K13 T) + ω3 (K23 - Tg3 + T) (4) γ ξ12 ) ξ13/ξ23
(5)
give the self-diffusion coefficient for component 1 in a ternary system using free-volume theory. Here D1 and D01 are the self-diffusion coefficient and the preexponential factor of component 1, respectively. The weight fraction and specific critical hole free volume required ˆ i. In all cases, for a jump of component i are ωi and V component 1 is the diffusing solvent, component 2 is the second solvent, and component 3 is the polymer. The total hole free-volume, V ˆ FH, is related to the parameters K1i/γ and K2i - Tgi which are related to the WLF parameters for the polymer-solvent system (Vrentas and Duda, 1977b). The parameter ξij is defined as the ratio of the molar jumping volume required by component i to that of component j. All the above parameters
Figure 1. Elution profile and the model regression fit for toluene diffusion in a poly(vinyl acetate)-methanol system at 60 °C and 0.081 weight fraction of methanol.
can be deduced from pure-component polymer and solvent data or from binary polymer-solvent diffusion data. Experimental Apparatus and Analysis Inverse gas chromatography (IGC) was used to measure multicomponent diffusion in polymer-solventsolvent ternary systems. The carrier gas was doped with a solvent of interest using a saturator, and equilibrium was established in the column. The saturator was similar to the one used for finite concentration experiments as described previously by Tihminlioglu et al. (1997). The solvent injected was different from that used to dope the carrier gas and establish equilibrium in the column. Thus, solvent (1) was diffused through a polymer-solvent (2) solution and its response peak was recorded. The outlet concentration of the pulse solvent was nondimensionalized with the inlet pulse and was regressed to obtain the diffusion and partition coefficients with the CCIGC model (Surana et al., 1997). The capillary columns were obtained from Supelco Inc., Bellefonte, PA, and Restek, Bellefonte, PA. The capillary columns were 0.53 µm in diameter and were coated with either a 5.0-µm-thick film of PVAC or a 7.0µm-thick film of polybutadiene. The solvents used were reagent grade with greater than 99% purity as obtained from Aldrich Chemicals. Results and Discussion Diffusion coefficients were measured in the PVACtoluene-methanol and polybutadiene-heptane-toluene systems using IGC. The experiments were performed at infinitely dilute concentrations of one solvent and finite concentrations of the second solvent in the polymer. The ternary system was hence treated as a pseudobinary system, and the experimental data were analyzed accordingly. In this analysis it is implicitly assumed that the pulse injection of the solvent did not change the equilibrium and concentrations of the second solvent in the column. Also, since the concentration of the pulse solvent was infinitely dilute, the thermodynamic effect was assumed to be negligible. That is, the diffusion coefficient measured represents the self-diffusion of a trace of the pulse solvent. Figures 1 and 2 are examples of the elution profiles of the solvent and the IGC model regression. Figure 1 shows the elution profile of toluene diffusion in the PVAC-methanol system at 60 °C and 0.081 weight fraction of methanol in PVAC. In this case, a mass spectrometer was used as the detector. In this case, the
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Figure 2. Elution profile and the model regression fit for toluene diffusion in a polybutadiene-heptane system at 70 °C and 0.035 weight fraction of heptane.
Figure 3. Comparison of partition coefficients in poly(vinyl acetate) systems as a function of the solvent weight fraction in the polymer phase at 60 °C: (A) methanol in PVAC-toluene; (B) methanol in PVAC-methanol; (C) toluene in PVAC-methanol; (D) toluene in PVAC-toluene. The lines are for visual aids.
IGC model fits of the elution curve indicate that the diffusion coefficient for an infinitely dilute amount of toluene in this PVAC-methanol system is 9.07 × 109 cm2/s and the toluene partition coefficient is 304. The diffusion and partition coefficients of toluene in neat PVAC, at the same temperature, were 5.05 × 10-11 cm2/s and 277, respectively. Similar experiments were carried out for the measurement of diffusion and partition coefficients in multicomponent systems at different column temperatures and varying concentrations of solvent in the polymer. Figure 2 is for the diffusion of toluene in the polybutadiene-heptane system. A thermal conductivity detector was used in this case, and the scatter of the data for the elution curve was due to slight variations in the background solvent concentration in the carrier gas. These fluctuations are random and are averaged out during the regression. Although negative values of the concentration are not shown in this figure, all values were used in establishing the baseline for the elution curve. The experimental data obtained were consistent with the literature data (Faridi et al., 1996) and compare well with the free-volume theory. The partition coefficients, K, shown in the above figures, are related to the polymer-solvent thermodynamic interactions. It is the ratio of the solvent concentration in the polymer phase to that in the gas phase. The equilibrium solubility, the activity, and the Flory-Huggins interaction parameter of the solvent in the polymer can be deduced from this partition coefficient (Faridi et al., 1994). In these experiments, the partition coefficient gives information on the thermodynamics in the multicomponent system. Figure 3 shows the influence of solvent concentration on the partition coefficients for the ternary as well as the two binary systems formed from methanol, toluene, and
Figure 4. Comparisons of partition coefficients in polybutadiene systems as a function of solvent weight fraction in the polymer phase at 70 °C: (A) heptane in PBD-toluene; (B) heptane in PBD-heptane; (C) toluene in PBD-heptane; (D) toluene in PBDtoluene. The lines are for visual aids.
poly(vinyl acetate). The lines in this figure are included as visual aids. The data presented in this figure indicate how the partition coefficient of one solvent in the polymer is influenced by the presence of a second solvent. For example, curve B shows how the partition coefficient of methanol in the binary methanol-PVAC system changes with methanol concentration. In contrast, curve A shows how the presence of toluene in the PVAC polymer will influence the partition coefficient for methanol. This figure shows the partition coefficient as a function of the weight fraction of the solvent in the polymer coating of the capillary column. Comparison of correlation lines B and A show that, as one might expect from a simple consideration of molecular interactions, the presence of toluene in PVAC will reduce the solubility of methanol. Similarly, a comparison of curves C and D indicates how the presence of methanol in the polymer phase will influence the solubility of toluene. Figure 4 presents data which show the influence of solvent concentrations on partition coefficients for the ternary and two binary systems formed from heptane, toluene, and polyisobutylene. Similar to the results presented in Figure 3, these results show that the presence of a dissimilar second solvent reduces the solubility of a solvent in the polymer phase. However, the influence of the second solvent for this ternary system is not as significant as the results presented in Figure 3. Such a result might be expected since heptane and toluene are more similar in molecular interactions than toluene and the relatively polar methanol. The results presented in Figures 3 and 4 represent the first cases in which solvent-solvent-polymer thermodynamic interactions have been investigated using the inverse gas chromatography technique. Figure 5 presents the measured experimental diffusion coefficients of toluene in the toluene-methanol-PVAC system as a function of methanol concentration in the polymer at different temperatures. The diffusion coefficients are measured at infinitely dilute concentrations of toluene in the polymer. The lines in this figure represent the free-volume theory predictions of the self-diffusion coefficients. These results show that the diffusion coefficients for toluene at 60 °C increase by more than 2 orders of magnitude by the addition of small amounts of methanol in PVAC. The observed behavior is predicted fairly well by the free-volume theory except at 40 °C. Figure 6 gives similar results for diffusion coefficients of methanol in the methanol-toluenePVAC system and the corresponding free-volume theory
3206 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
Figure 5. Comparison of experimental diffusion coefficients of toluene in a poly(vinyl acetate)-methanol-toluene system with free-volume theory predictions.
Figure 6. Comparison of experimental diffusion coefficients of methanol in a poly(vinyl acetate)-toluene-methanol system with free-volume theory prediction at 60 °C. Table 1. Free-Volume Parameters for a Toluene-Methanol-Poly(vinyl acetate) System toluene (1)
methanol (2)
PVAC (3)
V ˆi 0.917 0.961 0.728 K1i/γ (cm3/g‚K) 1.45 × 10-3 1.17 × 10-3 4.33 × 10-4 K2i - Tgi (K) -86.32 -48.41 -258 ξi3 0.62 0.35 8.86 × 10-5 5.95 × 10-5 Doi (cm2/s) (cm3/g)
Table 2. Free-Volume Parameters for a Toluene-Heptane-Polybutadiene System toluene (1)
heptane (2)
PDB (3)
V ˆ i (cm3/g) 0.917 0.917 0.974 K1i/γ (cm3/g‚K) 1.45 × 10-3 1.83 × 10-3 9.05 × 10-4 K2i - Tgi (K) -86.32 -55.42 -139 ξi3 0.97 0.99 Doi (cm2/s) 1.87 × 10-4 3.43 × 10-4
prediction. The parameters required for the free-volume theory predictions for both ternary systems investigated in this study were obtained from pure-component polymer-solvent data and from binary polymer-solvent diffusion data (Surana, 1997) and are presented in Tables 1 and 2. To show the influence of the additional free volume associated with the second solvent on diffusional behavior, experiments were conducted for polybutadiene systems at temperatures well above the glass transition temperature of the polymer. In this case, the polymer has a large amount of free volume, and the addition of the solvent does not significantly change the total free volume available to facilitate diffusion. Hence, in this case, large changes in the diffusion coefficients in the polymer systems with the addition of solvent are not expected. Figure 7 shows the diffusivity measurements for toluene in polybutadiene-heptane systems and
Figure 7. Comparison of experimental diffusion coefficients and free-volume theory prediction at 70 °C for toluene diffusion in polybutadiene-heptane and heptane diffusion in polybutadienetoluene.
heptane in the polybutadiene-toluene system at 70 °C. The lines in the figures represent the predictions from the free-volume theory for these ternary systems. The diffusivity is presented as a function of the weight fraction of the solvent in the polymer phase. As anticipated, the addition of solvents to this polymer which is far above its glass transition temperature does not enhance molecular mobility nearly as much as was observed for the poly(vinyl acetate) in Figures 5 and 6. This is consistent with the fact that the diffusion measurements were conducted in PVAC at temperatures near the polymer’s glass transition temperature and, consequently, the addition of free volume associated with the solvent has much more influence on the diffusion. The order of magnitude of the diffusion coefficients in these two polymer systems also reflects the fact that diffusion in PVAC which does not have much free volume due to the proximity of the glass transition is much slower than diffusion in polybutadiene. These results show that the parameters in the freevolume theory describing diffusion in the two binary polymer-solvent systems can be used to predict diffusion coefficients in the corresponding ternary system. The only parameter for the prediction of the ternary system which does not appear in the formulation for the binary systems is ξ12 which is the ratio of the molar volumes of the jumping units of two solvents. However, this ratio can be calculated as indicated in eq 5 from the molar volume ratios for the two binary polymersolvent systems. This was the procedure followed in the predictions presented in this study. It is also possible to obtain ξ12 directly from infinitely dilute diffusion data for the two solvents in any common polymer, even if the free-volume characteristics of the polymer are not known. Consideration of the freevolume theory in the limit of infinitely dilute diffusion coefficients indicates that there should be a linear relationship between the logarithms of the infinitely dilute diffusion coefficients for two solvents in a specific polymer over a range of temperature. Consequently, if the logarithms of these two infinitely dilute diffusion coefficients are plotted versus each other, a straight line will result which has a slope of ξ12. Such a plot is presented in Figure 8 for the infinitely dilute diffusivities of toluene and methanol in PVAC over the temperature range of 40-100 °C. The slope of the line in Figure 8 is 0.526, which corresponds to the ratio of the molar volume of the jumping unit of methanol to that
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3207
volume of the system and is more prominent at temperatures close to the glass transition temperature. Free-volume theory predictions compared well with the experimental data. Literature Cited
Figure 8. Correlation of diffusion of methanol and toluene in poly(vinyl acetate) to determine the ratio of the volumes of the solvent jumping units, ξ12.
Figure 9. Correlation of diffusion of heptane and toluene in polybutadiene to determine the ratio of the volumes of the solvent jumping units, ξ12.
of toluene (ξ12). This value is close to 0.565, which was calculated from eq 5 where the individual polymersolvent ratios ξ13 (0.35) and ξ23 (0.62) were determined from a regression of the binary diffusion data over a range of temperatures and solvent concentrations. The advantage of the technique illustrated in Figure 8 is that diffusivity data for solvents in any polymer can be used to obtain the ξ12 parameter for the solvents even if the free-volume characteristics of the polymer are unknown. In Figure 9, infinitely dilute binary diffusion coefficients of toluene and heptane in PVAC are correlated to obtain a value of 1.001 for ξ12 for the heptane-toluene ratio. In this correlation, the heptane-PVAC data were obtained from Arnould (1989), while the data for the toluene-PVAC system were obtained from Surana et al. (1997). This value is very close to 1.02, which was obtained by regressing binary diffusion data as a function of concentration and temperature for the polybutadiene-toluene and polybutadiene-heptane systems and applying eq 5. Conclusion IGC has been successfully used to measure the diffusion coefficients of a trace of one solvent in ternary polymer-solvent-solvent systems. The diffusion coefficients measured in the multicomponent system vary significantly with the addition of small amounts of a second solvent. The increase is related to the free
Arnould, D. D. Capillary Column Inverse Gas Chromatography (CCIGC) for the Study of Diffusion in Polymer-Solvent Systems. Ph.D. Dissertation, University of Massachusetts, Cambridge, MA, 1989. Carra, S.; Morbidelli, M.; Santacesaria, E.; Niederjaufner, G. Polymer Purification Through Solvent Addition: Physical Implications and Modeling of Separation Units. J. Appl. Polym. Sci. 1981, 26, 1497. Duda, J. L.; Kimmerly, G. K.; Sigelko, W. L.; Vrentas, J. S. Sorption Apparatus for Diffusion Studies with Molten Polymers. Ind. Eng. Chem. Fundam. 1973, 12, 133. Faridi, N.; HadjRomdhane, I.; Danner, R. P.; Duda, J. L. Diffusion and Sorption in Ethylene-Propylene Copolymers: Comparison of Experimental Methods. Ind. Eng. Chem. Res. 1994, 33, 2483. Faridi, N.; Duda, J. L.; Danner, R. P. Diffusion of Solvents in Polybutadiene Rubber Using Capillary Column Inverse Gas Chromatography. Rubber Chem. Technol. 1996, 69, 234. HadjRomdhane, I.; Danner, R. P. Polymer-Solvent Diffusion and Equilibrium Parameters by Inverse Gas-Liquid Chromatography. AIChE J. 1993, 39, 625. Lodge, T. P.; Lee, J. A.; Frick, T. S. Probe Diffusion in Poly(vinyl acetate)/Toluene Solutions. J. Polym. Sci., Part B: Polym. Phys. Ed. 1990, 28, 2607. Pawlisch, C. A.; Macris, A.; Laurence, R. L. Solute Diffusion in Polymers. 1. The Use of Capillary Column Inverse Gas Chromatography. Macromolecules 1987, 20, 1564. Pawlisch, C. A.; Bric, J. R.; Laurence, R. L. Solute Diffusion in Polymers. 2. Fourier Estimation of Capillary Column Inverse Gas Chromatography Data. Macromolecules 1988, 21, 1685. Sasaki, M.; Takishima, S.; Masuoka, H. Supercritical Carbon Dioxide Extraction of Benzene in Poly(vinyl acetate) and Polystyrene (Part 2). Sekiyu Gakkaishi 1990, 33, 304. Surana, R. K. Diffusion in Multicomponent Systems. Ph.D. Disertation, The Pennsylvania State University, University Park, PA, 1997. Surana, R. K.; Danner, R. P.; Tihminlioglu, F.; Duda, J. L. Evaluation of Inverse Gas Chromatography for Prediction and Measurement of Diffusion Coefficients. J. Polym. Sci., Part B: Polym. Phys. Ed. 1997, 35, 1233. Tihminlioglu, F.; Surana, R. K.; Danner, R. P.; Duda, J. L. Finite Concentration Inverse Gas Chromatography: Diffusion and Partition Measurements. J. Polym. Sci., Part B: Polym. Phys. Ed. 1997, 35, 1279. von Meerwall, E.; Ferguson, R. D. Self-diffusion of Bulky Molecules in Solution: 5-R-Cholestane in C6F6 or cis-Polybutadiene. J. Chem. Phys. 1981, 75, 937. Vrentas, J. S.; Duda, J. L. Diffusion in Polymer-Solvent System. I. Reexamination of the Free-Volume Theory. J. Polym. Sci. 1977a, 15, 403. Vrentas, J. S.; Duda, J. L. Diffusion in Polymer-Solvent System. II. A Predictive Theory for the Dependence of Diffusion Coefficients on Temperature, Concentration and Molecular Weight. J. Polym. Sci. 1977b, 15, 417. Vrentas, J. S.; Duda, J. L.; Ling, H.-C. Enhancement of Impurity Removal from Polymer Films. J. Appl. Polym. Sci. 1985, 30, 4499.
Received for review November 17, 1997 Revised manuscript received April 30, 1998 Accepted April 30, 1998 IE9708079
