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Ind. Eng. Chem. Res. 2001, 40, 3058-3064
Solubility and Diffusivity of Solvents and Nonsolvents in Polysulfone and Polyetherimide Ida M. Balashova, Ronald P. Danner,* Pushpinder S. Puri,† and J. Larry Duda Center for the Study of Polymer Solvent Systems, Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
The inverse gas chromatography method was used for the investigation of the partition and diffusion coefficients of polysulfone and polyetherimide with NMP, γ-butyrolactone, DMSO, acetic acid, propionic acid, THF, and water over a range of temperature from 200 to 270 °C. The weightfraction infinite-dilution activity coefficients for the solvents, the Flory-Huggins interaction parameters, and the solubility parameters of the polymers are reported. Introduction The process for the production of hollow fiber membranes involves the extrusion of a polymer-solvent mixture into a nonsolvent (generally water). The morphology of the fiber depends directly on the phase equilibria in this inherently immiscible system, as well as on the diffusion behavior. During the extrusion, the system has a relatively high concentration of polymer, which causes a high viscosity. As a result of the complexities and physical state of the system, it is difficult to directly measure the phase equilibria of the system. Thus, an alternative approach is needed. Inverse gas chromatography (IGC) is a convenient method of measuring the interactions of solvents and polymers at high polymer concentrations. This method can provide phase equilibria and diffusion data reliably and relatively efficiently. Inverse Gas Chromatography Various researchers have used inverse gas chromatography to measure solubility and diffusivity. Gray and Gillet1 did the pioneering work in developing gas chromatography for diffusion measurements using packed columns. They were among the first researchers to determine diffusion coefficients by the IGC method. This technique is based on the distribution of a volatile solvent between a mobile gas phase and a stationary polymer phase. Because of the mass transfer resistance in the polymer, the volatile substance in the mobile phase is swept forward, while that in the stationary phase lags behind. The net retention time of the peak is related to the solubility, and the spread of the peak is related to the diffusion coefficient for a given polymer-solvent system. Pawlisch et al.2 replaced the packed columns with capillary columns. They developed a capillary column inverse gas chromatography (CCIGC) model to measure diffusivities and solubilities. Capillary columns have a number of significant advantages over packed columns in the IGC method. They have a simple geometry, with a thin and uniform coating; there is much less phase dispersion; and there is no significant pressure drop. * To whom correspondence should be addressed. E-mail:
[email protected]. Fax: (814) 865-7846. †Air Products and Chemicals, Inc., Allentown, PA 181951501.
Pawlisch et al.2,3 employed and modified the model developed by Macris4 for the elution profile in capillary columns. They developed the following expression for the concentration profile at the exit of the column in the Laplace domain
[ ] [(
s 2xs C hL 1 1 exp + + ) exp tanh(βxs) 2 C0u 2γ γ Rβγ 4γ
)] 1/2
(1)
where the three dimensionless parameters are
R)
rc K(1 - y1)τ
β2 )
Dg τ2u γ) Dp L uL
(2)
Here, C h is the outlet concentration, C0 is the strength of the inlet pulse, L is the length of the capillary column, τ is the thickness of polymer coating in the capillary column, rc is the void cross section radius of the capillary column (radius of the capillary column minus the thickness of polymer coating), Dg is the diffusion coefficient of the solvent in the carrier gas, u is the velocity of the carrier gas, Dp is the mutual diffusion coefficient in the polymer, K is the partition coefficient between the gas and polymer phases, and y is the concentration of solvent in the carrier gas. The new packed and capillary column models (PCIGC and CCIGC) were developed by Hadj Romdhane et al.5,6 Methods of obtaining parameter estimates for the Laplace transform eq 1, from system response experiments (elution curves) fall into one of four categories: method of moments, time-domain fitting, Laplacedomain fitting, or Fourier-domain fitting. The merits of each method for IGC applications were discussed by Pawlisch et al.3 Moment analysis works well when the elution curves are nearly symmetric (Guassian peak) with well-defined end points. However, the method of moments has several well-known shortcomings that limit its use as a general analytical technique, e.g., the third moment is needed to test for skewness. The use of this method for elution peaks with significant asymmetry and large amounts of tailing was found to be unreliable, especially in the calculation of the second moment. This method does not provide an unequivocal criterion for the goodness of fit. To ensure that a good fit of the model has been obtained, it is necessary to make a direct compari-
10.1021/ie001074m CCC: $20.00 © 2001 American Chemical Society Published on Web 04/06/2001
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3059
son of the experimental response curve with the theoretical response curve predicted by the parameter estimates. Estimation in the time domain of the model parameters avoids these difficulties and is preferred for its increased accuracy and reliability. Parameters are selected to minimize an objective function, namely, the square of the difference between the experimental and theoretical response curves. The key difficulty in applying time-domain analysis is that an analytical solution of the model equations in the time domain is not available. Thus, minimization of the objective function requires repetitive numerical computation of the theoretical response curve in the time domain. A more attractive alternative to moment analysis is to work in the Fourier or Laplace domain. Although usually classified as different methods, Fourier- and Laplace-domain estimations are fundamentally the same because the properties and structures of the transforms are similar. In both methods, the transform of the experimental response curve is generated numerically, and the analytical expression for the transform is fitted directly to the transformed response curve by using a nonlinear regression algorithm. The key advantage of this technique is that an estimation criterion equivalent to a least-squares criterion in the time domain can be defined. For those models that do not yield analytical-domain solutions, Fourier-domain estimation is computationally more efficient and yields an equivalent result. Experimental Procedure In this study the coated capillary column IGC method was used to measure thermodynamic and transport properties in polymer-solvent and polymer-nonsolvent systems. The systems studied were polysulfone (PSF) and polyetherimide (PEI) with N-methyl-2-pyrrolidone (NMP), γ-butyrolactone, dimethyl sulfoxide (DMSO), acetic and propionic acids, tetrahydrofuran (THF), or water. The choice of materials was based on industrial needs in the development of commercial membranes. Processing of these materials requires information on the solubility and diffusivity in these polymer-solventnonsolvent systems. The solvents, obtained from Aldrich Chemicals, were reagent-grade (purity > 99.5%) and were injected as liquids. Polysulfone and polyetherimide were obtained from Amoco and General Electric Plastics, respectively. Both polymers are amorphous thermoplastic materials with interesting physical and chemical properties and are attractive candidates for many technological applications. The commercial PSF (Udel) had a weight-average molecular weight of approximately 78 000-80 000 and a glass transition temperature between 185 and 187.5 °C. Its chemical structure is shown in Figure 1. It is soluble in acidic solvents (sulfuric acid, R- and β-halogenated carboxylic acids, and hexafluoro-2-propanol) and basic chlorinated hydrocarbons (chloroform and methylene chloride). They are also soluble in polar solvents [dimethylformamide (DMF), dimethylacetamide (DMAC), N-methyl pyrrolidone (NMP)] and in 1:1 Lewis acid-base complexes [propionic acid (PA)DMAC, PA-NMP, and PA-formyl piperidine]. PSF is commonly used for both gas separation and micro/ ultrafiltration membranes.
Figure 1. Polymer structures.
The combination of phenyl rings attached to sulfone groups results in a high degree of resonance stabilization. The sulfone group acts as a sink for the electrons in the aromatic groups and confers both thermal and oxidative resistance. The ether groups in the backbone provide some flexibility, which results in inherent toughness. Membranes from unmodified PSF are utilized in both flat-sheet and hollow-fiber form in hyperfiltration, ultrafiltration, microfiltration, and gasseparation membranes. The polyetherimide (Ultem) had a weight-average molecular weight of approximately 54 000, and a glass transition temperature Tg of around 215-220 °C. See Figure 1 for its chemical structure. The ether units provide a structure with flexible linkages and good melt flow characteristics. The aromatic imide units provide high heat resistance and good mechanical properties. Furthermore, the high proportion of aromatic rings gives this polymer excellent thermal stability. The bisphenol group is symmetric, whereas the imide groups present an evident asymmetry compared with the central group, which can result in a specific dipole moment. PEI is not commonly used for membranes because of its low gas permeability and lack of resistance to alkaline solutions. The gas chromatograph used in this work was a Varian 3400 instrument (Varian Associates Inc., Walnut Creek, CA) equipped with a thermal conductivity detector (TCD), a split-splitless capillary injector, and an air-circulating oven. The capillary columns used throughout this work were prepared by Supelco, Inc., Bellefonte, PA. They were coated with PSF and PEI by the static coating technique. The columns were fused quartz, 15 m long with a 0.53-mm inner diameter. The thickness of the polymer coating was 5 µm. After the columns were installed in the oven, they were conditioned for about 24 h to remove any residual solvents from the polymers. High-purity helium was used as the carrier gas in all experiments. The flow rate of heluim through the column was about 2 cm3/min. To improve the detector response, an additional carrier gas was introduced at the end of the column (make-up gas). This stream (∼28 cm3/min) swept the effluent from the column into the detector. Methane and nitrogen were used as the marker gases. No appreciable pressure drops were measured across the capillary column. The temperatures of the injector and detector were set about 50 °C above the normal boiling point of the solvents to ensure rapid vaporization of the liquid samples in the injector block and to avoid condensation in the detector assembly. After the gas chromatograph had reached steady-state operation (i.e., stable temperature and
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Table 1. Partition and Diffusion Coefficients of Solvents in Polysulfone partition coefficienta
a
diffusion coefficient (cm2/s)
solvent
200 °C
220 °C
250 °C
200 °C
220 °C
250 °C
NMP γ-butyrolactone DMSO acetic acid propionic acid water THF
91 84 63 8.8 11.5 2.9 5.0
72 57 43 7.0 8.7 2.4 4.6
40 33 26 4.4 5.7 2.0 3.4
7.40 × 10-9 1.26 × 10-8 1.20 × 10-8 6.20 × 10-8 2.45 × 10-8 2.20 × 10-6 1.60 × 10-8
3.33 × 10-8 5.67 × 10-8 6.30 × 10-8 1.70 × 10-7 9.40 × 10-8 2.80 × 10-6 6.90 × 10-8
2.30 × 10-7 3.30 × 10-7 3.20 × 10-7 6.80 × 10-7 4.75 × 10-7 1.80 × 10-6 3.60 × 10-7
Concentration in polymer/concentration in vapor phase.
Table 2. Partition and Diffusion Coefficients of Solvents in Polyetherimide partition coefficienta solvent NMP γ-butyrolactone DMSO acetic acid propionic acid water THF a
230 °C 9.0 9.6 7.8 1.6 1.7 1.0 0.6
250 °C 7.9 7.8 6.3 1.4 1.6 0.8 0.6
diffusion coefficient (cm2/s) 270 °C 5.9 5.4 4.6 1.3 1.4 0.7 0.6
230 °C 10-8
2.60 × 3.60 × 10-8 3.90 × 10-8 1.70 × 10-7 1.20 × 10-7 2.10 × 10-7
250 °C 10-8
7.90 × 9.04 × 10-8 1.02 × 10-7 3.64 × 10-7 2.40 × 10-7 3.30 × 10-7
270 °C 2.60 × 10-7 3.20 × 10-7 3.15 × 10-7 5.20 × 10-7 5.40 × 10-7 6.10 × 10-7
Concentration in polymer/concentration in vapor phase
Figure 2. Temperature dependence of the partition coefficient in PSF-solvent systems.
Figure 3. Temperature dependence of the partition coefficient in PEI-solvent systems.
carrier gas flow rate), 0.25 µL of methane for the PSF column and the same amount of nitrogen for the PEI column were injected by a gas-tight syringe into the injector block. The split ratio was 1:10. Both gases had negligible interaction with the polymers. The corresponding mean residence time was used to determine the carrier gas velocity. Pulses of solvents (0.04 µL) and nonsolvents (0.02 µL) were injected through the rubber septum of the injection port into the carrier gas using a variable Hamilton 1-µL syringe. The carrier gas inlet pressure was slightly over atmospheric. The data were recorded with a frequency ranging from 4 to 8 Hz. The outlet concentration profiles were monitored by the TCD detector, recorded, and integrated via the data system. The raw output signal was corrected for baseline offset, as the CCIGC model assumes that the baseline signal is zero. A range of integration limits was selected. Then the chromatographic data were analyzed using the model developed by Hadj Romdhane et al.6 Experimental elution curves were integrated to obtain first and second moments. These results were used to calculate initial estimates of the diffusion and partition coefficients. To ensure that a good fit of the model was obtained, the CCIGC model parameters were evaluated using the time-domain fitting procedure. Starting with the initial estimates for K and Dp, the transform
expression was numerically inverted using the fast Fourier transform algorithm. The resulting theoretical profile was then compared against the experimental profile, and the residual was minimized using a nonlinear least squares regression technique to obtain the best values of K and Dp that characterize the experimental elution curves. Results The experimental data on the solubility and the diffusivity of low-molecular-weight components in the PSF and the PEI polymers are compiled in Tables 1 and 2 and in Figures 2-5. The data on the partition coefficients, K, for the various solvents and nonsolvents showed the expected behavior, that is, the coefficients decreased with increasing temperature. All results were based on four or five duplicate experiments. These results showed good reproducibility. The logarithm of the partition coefficient was a linear function of the reciprocal temperature for all systems. The values of K for the solvents and nonsolvents in PSF varied from 2.0 to 91, and those in PEI, from 0.55 to 10. The largest values in PSF are for NMP, γ-butyrolactone, and DMSO, which can be considered solvents for this polymer.
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3061
Figure 4. Temperature dependence of the diffusion coefficient in PSF-solvent systems.
Figure 6. D0-E relationship for PSF-solvent and PEI-solvent systems.
Figure 5. Temperature dependence of the diffusion coefficient in PEI-solvent systems.
Diffusivity measurements were conducted for solvents and nonsolvents in the limits of infinitely dilute penetrants as a function of temperature. As Figures 4 and 5 indicate, all of the data follow an Arrhenius relationship
Dp ) D0 exp(-E/RT)
(3)
where D0 and E are constants for a specific polymersolvent system and E can be considered as an overall activation energy for the diffusion process. At first, this behavior appears to be inconsistent with previous studies of solvent diffusion in polymers above the glass transition temperature and with conventional theories for the diffusion of penetrants in polymer melts based on free-volume considerations.7,8 In most previous studies of the mutual diffusion of relatively large solvent molecules in molten polymers, the apparent activation energy is a function of temperature and increases as the temperature decreases. This study represents one of the few comprehensive studies of diffusion in polymers with relatively high glass transition temperatures, Tg. It seems logical that polymers that have high Tg’s will have stiff backbones and, consequently, the units of the chain that move in a coordinated manner will be large. According to conventional free-volume theory, the size of the polymer jumping unit influences the distribution of the free volume, and the larger the polymer jumping unit is compared to the jumping unit of the solvent, the more closely Arrhenius behavior is approached. Consequently, the results of this study are consistent with conventional free-volume theories and represent the first study to clearly show that the
Figure 7. Activation energy for diffusion of solvents in PSF and PEI as a function of the molar volume.
temperature dependency of the apparent activation energy will be relatively minor for polymers with high transition temperatures. This approximate Arrhenius behavior actually facilitates the correlation of the diffusivity measurements. Numerous investigations, starting with the pioneering study of Zhurkov and Ryskin,9 have shown that, when Arrhenius behavior is observed for diffusion in polymersolvent systems, there is a correlation between the D0 term and the activation energy and the activation energy is proportional to the size of the penetrant molecules. Such correlations for polysulfone and polyetherimide are presented in Figures 6 and 7. Figure 7 shows that the overall activation energy for diffusion is a linear function of the molar volume of the diffusing penetrant, V1, at 200 °C. The diffusion coefficients for other penetrants not considered in this study can be estimated by using these two correlations. If the molar volume of a specific penetrant is known, then Figure 7 can be used to obtain the activation energy, and the corresponding preexponential term, D0, can be determined from the correlation presented in Figure 6. This correlation is only applicable when a trace amount of the penetrants is diffusing in polymer.10 Thermodynamic Interaction Parameters Partition coefficient data are seldom reported in the literature dealing with the thermodynamics of polymer solutions. More popular are the retention volume, the activity coefficient, the Flory-Huggins interaction parameter, and the solubility parameters of the polymers.
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Table 3. Polysulfone-Solvent Interaction Data Ω1∞ a
Vg (cm3/g)
χb
solvent
200 °C
220 °C
250 °C
200 °C
220 °C
250 °C
200 °C
220 °C
250 °C
NMP γ-butyrolactone DMSO propionic acid acetic acid THF water
76.6 70.7 53.3 9.7 7.4 4.2 2.4
61.7 48.6 36.8 7.4 5.7 3.9 2.0
35.1 28.7 22.7 5.0 3.8 3.0 1.7
5.9 7.2 7.6 25.4 35.7 8.3 63.8
4.9 6.8 7.3 18.8 26.3 7.2 53.9
5.0 6.5 6.9 14.8 19.9 7.4 41.2
0.5 0.8 0.8 1.8 2.2 0.5 2.8
0.3 0.7 0.7 1.5 1.9 0.3 2.7
0.2 0.6 0.6 1.2 1.5 0.2 2.3
a
Weight-fraction activity coefficient at infinite dilution. b Flory chi parameter.
Table 4. Polyetherimide-Solvent Interaction Data Ω1∞ a
Vg (cm3/g)
χb
solvent
230 °C
250 °C
270 °C
230 °C
250 °C
270 °C
230 °C
250 °C
270 °C
NMP γ-butyrolactone DMSO propionic acid acetic acid THF water
7.1 7.6 6.2 1.3 1.3 0.5 0.8
6.2 6.1 5.0 1.3 1.1 0.5 0.6
4.6 4.3 3.6 1.1 1.0 0.4 0.6
34.9 35.7 36.1 82.7 92.5 57.3 122
27.9 30.6 31.5 58.4 69.1 45.6 116
27.6 31.3 32.0 45.5 53.3 48.7 98.8
2.2 2.3 2.2 2.9 3.0 2.3 3.4
1.9 2.1 2.1 2.5 2.7 1.9 3.3
1.9 2.1 2.0 2.2 2.4 1.5 3.1
a
Weight-fraction activity coefficient at infinite dilution. b Flory chi parameter.
At infinite dilution of the solvent, the relation between chromatographic data and the solute activity coefficient is (1 ) solvent, 2 ) polymer)
Ω1∞ ) {(RT)/VgP1sM1} exp[{-P1s(B11 - V1)}/RT] (4) Here, Ω1∞ is the weight-fraction activity coefficient at infinite dilution, Vg ) K/F2 is the specific retention volume, K is the partition coefficient, R is the gas constant, T is the temperature, B11 is the second virial coefficient of the solvent, P1s is the saturation pressure of the solvent, F2 is the polymer density, V1 is the molar volume of solvent in the liquid phase, and M1 is the molecular weight of the solvent. The physical properties of the solvents and nonsolvents were obtained from Daubert et al.,11 except for the virial coefficients of the acids, which were calculated by the Hayden-O’Connell method.12 The standard state in eq 4 is a pure liquid solvent at the system temperature and zero pressure. The exponential term corrects for the gas-phase nonideality of the volatile component. Combining eq 4 with the Flory-Huggins equation, an expression for the Flory-Huggins interaction parameter, χ, is obtained
χ ) ln Ω 1∞ - (1 - 1/r) + ln(F1/F2)
(5)
Here r ) F1M2/F2M1 is the ratio of the molar volumes, F1 is the solvent density, and M2 is the polymer molecular weight. The combination of the Scatchard-Hildebrand regular solution theory with the Flory theory yields
(δ12/RT - χ/V1) ) 2δ2δ1/RT - [(δ2)2/RT + χs/V1] (6) Here, δ2 is the polymer solubility parameter at infinite dilution of the solvent, χs is the entropic contribution to the χ parameter, and δ1 is the solubility parameter of the solute given by
δ1 ) [(∆Hv - RT)/V1]1/2
(7)
Figure 8. Temperature dependence of Ω1∞ for solvents in PSFsolvent systems.
Here, ∆Hv is the heat of vaporization of the solvent. As χ, V1, and δ1 are known, the solubility parameter of the polymer, δ2, can be determined from the slope of a plot of (δ12/RT - χ/V1) versus δ1. Analysis of Thermodynamic Data Experimental retention volumes, weight-fraction activity coefficients at infinite dilution, and Flory-Huggins interaction parameters for low-molecular-weight components for the PSF and PEI systems at different temperatures are shown in Tables 3 and 4 and in Figures 8-11. From an error analysis, the maximum percent error for Vg was estimated to be about 4%. With the errors introduced by the solvent properties, the error for Ω1∞ was at most 8%. As expected, the polymers have an essential influence on the infinite-dilution activity coefficients of the lowmolecular-weight components. The infinite-dilution activity coefficient of water is 122 at 230 °C in PEI, but it is about 50% smaller in PSF at 200 °C. The same trend arises for acetic and propionic acids. The large values for the infinite-dilution activity coefficients of water and acetic and propionic acids in PSF and PEI are indicative of the strong association of these chemicals in pure liquids, and, thus, these binary systems reveal the
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3063
Figure 9. Temperature dependence of Ω1∞ for solvents in PEIsolvent systems.
Figure 12. Example of determination of the solubility parameter of PSF at 200° C using eq 6. Table 5. Temperature Dependence of the Solubility Parameters, δ1 (J/cm3)1/2
Figure 10. Temperature dependence of the Flory-Huggins interaction parameter in PSF-solvent systems.
Figure 11. Temperature dependence of the Flory-Huggins interaction parameter in PEI-solvent systems.
expected liquid-liquid miscibility gaps. The results in Figures 8-11 indicate that Ω1∞ and χ vary with temperature, especially for the hydrogen-bonding nonsolvents (water and acids). For both polymers, Ω1∞ and χ decrease as the temperature increases. The temperature dependency of χ has been noticed for many polymersolvent systems. Different correlations have been proposed for the temperature and concentration dependence of the Flory-Huggins parameter. All of the proposed models suggest that raising the temperature enhances the interaction forces between the polymer and solvent and, therefore, a polymer miscibility. From a stability analysis as applied to the FloryHuggins theory, complete polymer miscibility exists only
solvent
200 °C
220 °C
230 °C
250 °C
270 °C
NMP γ-butyrolactone DMSO acetic acid propionic acid water THF
19.0 21.6 21.4 16.4 16.6 38.6 11.9
18.5 21.0 20.7 15.8 15.9 37.0 10.6
18.2 20.8 20.4 15.4 15.6 36.2 9.7
17.6 20.2 19.6 14.4 14.7 34.3 7.4
17.0 19.6 18.8 13.1 13.6 32.3 1.4
when χ is less than 0.5. Inspection of the χ values tabulated in Tables 3 and 4 shows that, of all of the solvents used in this study, NMP and THF were the most compatible solvents with PSF. All of the solvents and nonsolvents showed enhanced solvency power with both polymers when the temperature was elevated from 200 to 270 °C. For PEI (Table 4), all of the lowmolecular-weight components exhibited poor solubility characteristics. The Flory-Huggins χ parameter is above 0.5 for all of the components at all of the experimental temperatures. To estimate the solubility parameters of PSF and PEI, data on the latent heats of vaporization and specific volumes were first obtained from Daubert et al.11 for the low-molecular-weight components at relevant temperatures. The solute solubility parameters were then determined by eq 7. The calculated values of δ1 are compiled in Table 5. A least-squares fit of (δ12/RT - Ps/ V1) versus δ1 (Figure 12) was then carried out for each polymer to determine δ2 according to eq 6. The values of δ2 for PSF were estimated to be 15.9, 14.8, and 13. 1 (J/cm3)1/2 at 200, 220, and 250 °C, respectively, and those for PEI were 12.3, 10.4, and 7.9 (J/cm3)1/2 at 230, 250, and 270 °C, respectively. The temperature dependence of the solubility parameters obtained for PSF and PEI is shown in Figure 13. The linear correlation coefficient was greater than 0.998 in all cases. Conclusions The diffusivities and solubilities of solvents and nonsolvents in polysulfone and polyeherimide were measured by the use of inverse gas chromatography with capillary columns. The data on the partition coefficients show that the partition coefficient decreases with increasing temperature. The logarithm of the partition coefficient for the solvents and nonsolvents is a linear function of the reciprocal temperature. An analysis of the solubility data indicates a decrease in
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Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 u ) linear velocity of the carrier gas, cm/s y ) concentration of solvent in the carrier gas Greek Letters τ ) thickness of polymer coating in the capillary column, cm Ω1∞ ) weight-fraction activity coefficient of the solvent at infinite dilution Fi ) mass density of component i, g/cm3 χ ) Flory-Huggins interaction parameter δ1 ) solubility parameter of the solute, (J/cm3)1/2 δ2 ) polymer solubility parameter, (J/cm3)1/2 χs ) entropic contribution to the χ parameter
Literature Cited Figure 13. Temperature dependence of the solubility parameters of PSF and PEI.
the value of the interaction Flory-Huggins parameter with an increase in temperature. The analysis of the diffusion data obtained for the solvents and nonsolvents shows that the diffusivity increases with temperature and that the behavior is essentially Arrhenius. Using the concept of the Hildebrand-Scatchard theory for regular solutions, the solubility parameters for PSF and PEI were estimated in the temperature range 200270 °C. Nomenclature B11 ) second virial coefficient of the solvent, cm3/mol C ) outlet concentration, mol/cm3 C0 ) strength of the inlet pulse, mol s/cm3 Dg ) solvent diffusion coefficient in the gas phase, cm2/s Dp ) solvent diffusion coefficient in the polymer phase, cm2/s E ) activation energy, J/mol ∆Hv ) heat of vaporization of the solvent, J/mol K ) equilibrium partition coefficient L ) length of the capillary column, cm Mi ) molecular weight of component i, g/mol P1s ) saturation pressure of the solvent, kPa r ) ratio of molar volumes of polymer to solvent rc ) void cross section radius of the capillary, cm R ) gas constant, J/(mol K) s ) variable in the Laplace transform equation t ) time, s tc ) retention time of the carrier gas, s T ) temperature, K V1 ) molar volume of solvent in the liquid phase, cm3/g Vg ) specific retention volume, cm3/g
(1) Gray, D. G.; Guillet, J. E. Studies of Diffusion in Polymers by Gas Chromatography. Macromolecules 1973, 6, 223. (2) 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. (3) 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. (4) Macris, A. Measurement of Diffusion and Thermodynamic Interactions in Polymer-Solvent Systems Using Capillary Column Inverse Gas Chromatography. M.S. Thesis, University of Massachusetts, Amherst, MA 1979. (5) Hadj Romdhane, I.; Danner, R. P. Polymer-Solvent Diffusion and Equilibrium Parameters by Inverse Gas-Liquid Chromatography. AIChE J. 1993, 39, 625. (6) Hadj Romdhane, I.; Danner, R. P.; Duda, J. L. Influence of the Glass Transition on Solute Diffusion in Polymers by Inverse Gas Chromatography. Ind. Eng. Chem. Res. 1995, 34, 2833. (7) Duda, J. L.; Zielinski, J. M. Free-Volume Theory. In Diffusion in Polymers; Neogi, P., Ed.; Marcel Dekker: New York, 1996; Chapter 5, p 143. (8) Zielinski, J. M.; Duda, J. L. Predicting Polymer/Solvent Diffusion Coefficients Using Free-Volume Theory. AIChE J. 1992, 38, 405. (9) Zhurkov, S. N.; Ryskin, G. Y. Investigation of Diffusion in Polymers. J. Tech. Phys. (Leningrad) 1954, 24, 797. (10) Van Krevelen, D. W.; Hoftyzer, P. J. Properties of Polymers. Correlations with Chemical Structures; Elsevier Publishing Company: New York, 1972. (11) Daubert, T. E.; Danner, R. P.; Sibul, M. H.; Stebbins, C. C. Physical and Thermodynamic Properties of Pure Chemicals. Data Compilation; Taylor & Francis Publishing: New York, 1998. (12) Hayden, J. G.; O’Connell, J. P. A Generalized Method for Predicting Second Virial Coefficients. Ind. Eng. Chem. Process Des. Dev. 1975, 14, 209.
Received for review December 8, 2000 Revised manuscript received January 29, 2001 Accepted January 31, 2001 IE001074M
