Determination of the nonsolvent polymer interaction parameter χ 13 in the casting solutions

Journal of Membrane Science 174 (2000) 87 96 Determination of the nonsolvent polymer interaction parameter χ 13 in the casting solutions Zhansheng Li, Chengzhang Jiang Dalian Institute of Chemical Physics, Chinese Academy of Sciences, 457 Zhongshan Road, Dalian 116023, China Received 16 December 1999; received in revised form 29 February 2000; accepted 6 March 2000 Abstract A new method to determine the nonsolvent polymer interaction parameter χ 13 in the casting solutions was developed. With the measured intrinsic viscosities of the polymer in the mixed solvents with different concentrations, the nonsolvent polymer interaction parameter χ 13 could be calculated by the application of the formalism proposed by Campos et al. The values of χ 13 in membrane-forming systems of nonsolvent/n-methyl-2-pyrrolidone (NMP)/polyethersulfone (PES) were obtained by the proposed method. The nonsolvents used were water, 1-butanol and diethylene glycol (DegOH). The stronger the nonsolvent, the higher the value of χ 13. In addition, the concentration of the nonsolvent at the binodal curve increased with decreasing χ 13. The binodal curves for these membrane-forming systems predicted from the Flory Huggins theory with χ 13 obtained by the proposed method were consistent with measured cloud point curves, which confirmed that the method proposed here was reliable. 2000 Elsevier Science B.V. All rights reserved. Keywords: Thermodynamics; Interaction parameter; Intrinsic viscosity 1. Introduction Phase separation process [1] is one of the main methods for the preparation of asymmetric polymer membranes. Phase separation can be induced in several ways, such as by changing the temperature or the concentration of the solutions. The structure and performance of a membrane prepared by immersion precipitation, the essential means to prepare the membrane, were determined by the thermodynamic properties of the casting solutions and coagulation pairs. Usually, the thermodynamic properties of the casting solutions are characterized by the ternary phase diagram. Corresponding author. Fax: +86-411-4677947. E-mail address: zsli@ms.dicp.ac.cn (C. Jiang) In the framework of the Flory Huggins theory adapted to nonsolvent (1)/solvent (2)/polymer (3) ternary systems proposed by Tompa [2], Altena and Smolders [3] calculated the binodal curves of two membrane-forming systems: cellulose acetate/solvent/water and polysulfone/solvent/water. A fairly good agreement had been found between the calculated and the experimentally found miscibility gaps. During their calculation, the nonsolvent solvent interaction parameter g 12 was taken to be concentration-dependent, and other parameters, i.e. the solvent polymer interaction parameter χ 23 and the nonsolvent polymer interaction parameter χ 13, were kept constant. Based on the Flory Huggins theory, the relation between the behavior of the ternary phase diagram for typical membrane-forming systems and various parameters was analyzed by Yilmaz and McHugh 0376-7388/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S0376-7388(00)00382-3

88 Z. Li, C. Jiang / Journal of Membrane Science 174 (2000) 87 96 [4]. It was concluded that, besides g 12, χ 23 should also be considered as being concentration-dependent, and the concentration dependency of χ 23 was more important in the control of the phase diagram behavior than that of g 12. But, unfortunately, data for the concentration dependency of χ 23 are quite limited. The Flory Huggins theory is an effective tool to describe the thermodynamic behavior of the casting solutions. Using the Flory Huggins theory with the known interaction parameters χ ij, composition paths during the demixing process can be predicted approximately. Furthermore, the resultant membrane structure and performance can be anticipated. Accordingly, how to get the exact and reliable χ ij is essential to the study of the thermodynamic properties of casting solutions by using the Flory Huggins theory. Generally, χ 12 can be calculated from activity data or the more fundamental vapor liquid equilibrium data. χ 23 can be obtained from light scattering or osmometry measurements [1]. At present, the nonsolvent polymer interaction parameter χ 13 is the most difficult to determine, but it can be calculated via swelling measurements at high polymer concentration [1], or light scattering measurements at low polymer concentration [5]. However, the values of χ 13 obtained from different methods deviate greatly from each other, so the values of χ 13 are considered unreliable. Recently, in the light of the two-parameter theory, the excluded-volume effect in single polymer-mixed solvent systems was investigated by Campos and coworkers [6 8]. A new expression, which relates the excess Gibbs energy of the solvent mixture and the corresponding intrinsic viscosity, was developed and proved in several systems, such as poly(dimethylsiloxane) (PDMS), poly(1-vinyl-2-pyrrolidone) (PVP), polystyrene (PS) and poly(methyl methacrylate) (PMMA) in mixed solvents [6 8]. In the present paper, the formalism of Campos and coworkers was applied in membrane-forming systems to determine the nonsolvent polymer interaction parameter χ 13. Polyethersulfone (PES) and N-methyl-2-pyrrolidone (NMP) were selected as the polymer and the solvent, respectively. Water, 1-butanol and diethylene glycol (DegOH) were used as the nonsolvents. In the three membrane-forming systems, the intrinsic viscosities of a polymer in mixed solvents with various nonsolvent concentrations were measured. Furthermore, the nonsolvent polymer interaction parameters χ 13 were calculated in virtue of the expression of Campos and coworkers. The binodal curves of the ternary systems were predicted from the Flory Huggins theory with the obtained χ 13, and compared with the measured cloud point curves to confirm the correctness of the method developed herein. 2. Theory background In the framework of the two-parameter theory, Campos and coworkers [6 8] investigated the excluded-volume effect in single polymer-mixed solvent systems in detail, and defined the excess viscosity as [η] E = [η] T (φ 1 [η] 13 + φ 2 [η] 23 ) (1) where [η] T,[η] 13 and [η] 23 are the intrinsic viscosities of the polymer in a binary solvent mixture, in a nonsolvent and in a solvent, respectively, and φ 1, φ 2 the volume fractions of nonsolvent and solvent, respectively. Since [η] E comes from the transport property, a relationship can be established between this property and the thermodynamic behavior of the solvent mixture, usually denoted by the binary interaction parameter g 12 in the ternary polymer system. Then, [η] E = 2C 0 v 3 2M W (1 2a χ )g 12 φ 1 φ 2 (2) N A V 1 where the factor C was determined to be 0.51; 0 is the Flory constant ( 0 =2.5 10 23 mol 1 when [η] is in cm 3 g 1 ), v 3 the partial specific volume of the polymer, M W the weight-average molar mass of the polymer, N A the Avogadro constant, V 1 the molar volume of Component 1 and a χ the constant for each ternary polymer system, which relates the ternary interaction parameter χ T to the nonsolvent solvent interaction parameter g 12 [8]. Eqs. (1) and (2) were combined and rearranged to give the following equation: X = [η] T φ 2 [η] 23 φ 1 = [η] 13 + 2C 0 v 2 3 M W N A V 1 (1 2a χ )g 12 φ 2 (3)

Z. Li, C. Jiang / Journal of Membrane Science 174 (2000) 87 96 89 Thus, by plotting the left-hand side, X versus g 12 φ 2, a straight line, can be obtained. The intercept of the straight line is the virtual intrinsic viscosity of the polymer in nonsolvent [η] 13 and the (1 2a χ ) constant can be obtained from the slope of the straight line. On the other hand, the second virial coefficient A 2 for the binary polymer systems has the following functionality, A 2 =K [η]. The relation was generalized to the ternary polymer systems as follows [8]: A 2,T = K T KY [η] T (4) where K = (4π 3/2 N A )/(6 3/2 0 M W ), A 2,T is the second virial coefficient for the ternary polymer system, T KY can be expressed as a function of the binary interpenetration functions ψ i 3 : KY T = i=1,2 ψ i3[η] i3 φ i +K 1 K (1 2a χ )g 12 φ 1 φ 2 i=1,2 [η] (5) i3φ i +K (1 2a χ )g 12 φ 1 φ 2 where K 1 K = (6 3/2 0 v 3 2M W)/(4π 3/2 N A V 1 ) and K = (2C 0 v 3 2M W)/(N A V 1 ). The binary interpenetration functions ψ i 3 can be calculated as described elsewhere [8]. Thus, the theoretical knowledge of allows the computation of the second virial coefficients in the ternary polymer systems, A 2,T, which are very scarce in the literature, from the experimental intrinsic viscosities [η] T by Eqs. (4) and (5). In the framework of the Flory Huggins theory, A 2 in a solvent mixture can be expressed as the following equation: T KY A 2 = v2 3 2V 1 [φ 1 + sφ 2 2χ 13 φ 1 2sχ 23 φ 2 +2(1 2a χ )g 12 φ 1 φ 2 ] (6) where s=v 1 /V 2, the ratio of molar volume of two solvents. Denoting the left-hand side of Eq. (7) by Y, Eq. (6) can also be written as Y = ((2V 1A 2 )/ v 2 3 ) φ 1 sφ 2 + 2sχ 23 φ 2 2φ 1 = χ 13 + (1 2a χ )g 12 φ 2 (7) By plotting Y against g 12 φ 2, a straight line can also be obtained. The nonsolvent polymer interaction parameter χ 13 can be obtained from the intercept. According to the Flory Huggins theory, which was extended to ternary systems by Tompa [2], the Gibbs free energy of mixing is given by G m RT = n i ln φ i + g 12 n 1 φ 2 +χ 13 n 1 φ 3 + χ 23 n 2 φ 3 (8) where n i and φ i are the number of moles and volume fractions of component i, and R and T the universal gas constant and temperature, respectively. In addition, g 12 is assumed to be a function of u 2, and u 2 =φ 2 /(φ 1 +φ 2 ). The following expressions can be obtained from the definition of the chemical potential: µ i RT = ( ) Gm n i RT n j,j i µ 1 RT = ln φ 1 sφ 2 rφ 3 +(1 + g 12 φ 2 + χ 13 φ 3 )(φ 2 + φ 3 ) sχ 23 φ 2 φ 3 φ 2 u 2 (1 u 2 ) dg 12 (9) du 2 s µ 2 RT = s ln φ 2 φ 1 rφ 3 +(s + g 12 φ 1 + sχ 23 φ 3 )(1 φ 2 ) χ 13 φ 1 ιφ 3 + φ 1 u 2 (1 u 2 ) dg 12 (10) du 2 r µ 3 RT = r ln φ 3 φ 1 sφ 2 +(r + χ 13 φ 1 + sχ 23 φ 2 )(1 φ 3 ) g 12 φ 1 φ 2 (11) where s has the same meaning as in Eq. (6), and r=v 1 /V 3, the ratio of molar volumes of the nonsolvent and the polymer. As described by Altena and Smolders [3] and Yilmaz and McHugh [4], the binodal curve is calculated by a computer search routine based on a least-squares procedure for minimizing an objective function suitably chosen: f 2 i f 1 = µ 1 µ 1 f 2 = s( µ 2 µ 2 ) f 3 = r( µ 3 µ 3 ) (12)

90 Z. Li, C. Jiang / Journal of Membrane Science 174 (2000) 87 96 where the superscript prime and double prime denote the dilute and concentrated phases, respectively. 3. Experimental Polyethersulfone (PES, Ultrason E 6020P) kindly donated by BASF was used as the polymer. The polymer was dried at 150 C under vacuum over 12 h. Water was deionized and filtered using an ultrafiltration membrane. NMP, 1-butanol and DegOH were analytical grade reagents, which were dehydrated with a 5A molecular sieve dried at 500 C for 4 h. The cloud point measurements on water/nmp/pes and DegOH/NMP/PES were carried out at 25±0.01 C as described elsewhere [9]. The modified Ubbelohde viscometers thermostated at 25±0.005 C were used to measure the viscosities of the polymer solutions made of NMP/PES and nonsolvents/nmp/pes, which have different mass ratios of nonsolvent and NMP. The concentration range of the polymer solutions was 0.4 2 g PES per 100 cm 3 polymer solution. To make sure that the kinetic energy correction could be neglected, the capillary size was changed so that the flow time of the solvent (t 0 ) was greater than 100 s. The intrinsic viscosities, [η], were determined by extrapolation from Huggins equation [10]: (t t 0 )/t 0 c = η sp c = [η] + k [η] 2 c (13) where t and t 0 stand for the flow time of the dilute solutions and solvents (mixed solvents), respectively. c is the concentration of the polymer, η sp the specific viscosity, and k the Huggins constant. 4. Results and discussion 4.1. Intrinsic viscosities The measured intrinsic viscosities for PES in three mixed solvent systems, water/nmp, 1-butanol/NMP Table 2 Intrinsic viscosities, [η] (cm 3 g 1 ), for PES in 1-butanol/NMP solvent mixtures at 25 C w 1 0 10.15 18.44 25.32 28.92 30.03 31.12 [η] 64.176 56.062 50.357 44.472 40.576 39.800 37.897 and DegOH/NMP at 25 C, are listed in Tables 1 3, respectively. As the concentrations of nonsolvents in solvent mixtures cannot exceed the composition of the binodal curve, only the intrinsic viscosities of homogeneous solutions were measured. As shown in Tables 1 3, by increasing the concentrations of nonsolvents in mixed solvents, the qualities of mixed solvents became worse and the values of the intrinsic viscosities decreased, while the expanding PES molecules in the good solvent NMP contracted gradually. Fig. 1 illustrates the significant differences existing in the decreasing rate of intrinsic viscosities caused by different nonsolvents. As water was a poor solvent for PES, the intrinsic viscosities dropped quickly when a little amount of water was added to the mixed solvent, which indicated that the expanding PES molecules in the good solvent NMP contracted rapidly and phase separation took place shortly afterwards. DegOH was a swelling agent for PES, so the intrinsic viscosities dropped slowly when DegOH was added to the mixed solvent, and the concentration of DegOH in solvent mixtures was high when phase separation took place. 4.2. The determination of the interaction parameters χ 13 of nonsolvents and PES Usually, the nonsolvent solvent interaction parameter g 12 can be calculated using the following equation [7]: g E RT = x 1 ln ( φ1 x 1 ) + x 2 ln ( φ2 x 2 ) + g 12 φ 2 x 1 (14) where g E is the molar excess Gibbs energy, and x i (i=1, 2) the molar fraction of component i in the Table 1 Intrinsic viscosities, [η] (cm 3 g 1 ), for PES in water/nmp solvent mixtures at 25 C w a 1 0 3.25 6.28 9.14 10.58 10.75 11.29 11.30 11.82 [η] 64.176 60.501 53.881 47.901 42.236 42.125 42.508 43.049 40.032 a w 1 was the concentration of nonsolvent (wt.%).

Z. Li, C. Jiang / Journal of Membrane Science 174 (2000) 87 96 91 Table 3 Intrinsic viscosities, [η] (cm 3 g 1 ), for PES in DegOH/NMP solvent mixtures at 25 C w 1 0 6.76 17.96 26.60 42.02 52.10 56.62 57.95 59.19 [η] 64.176 63.235 55.504 50.115 47.418 41.102 38.696 36.387 33.907 solvent mixture. The g E for water/nmp and DegOH/NMP are regressed from the vapor liquid equilibrium data [11] by the Wilson equation [12]. As for 1-butanol/NMP, there was only the heat of mixing data h E [13] that had been published, so g E was calculated from h E [14]. All the concentration-dependent g 12 are shown in Fig. 2. With the concentration-dependent g 12 and measured intrinsic viscosities, [η] 23 and [η] T in solvent and solvent mixtures, a straight line can be obtained by plotting X versus g 12 φ 2 according to Eq. (3). Linear regression was carried out to get the intercept and slope of the straight line. The intercept was the virtual intrinsic viscosity [η] 13 in the nonsolvent. The (1 2a χ ) constant was calculated from the slope. Thus, the value of A 2 can be calculated from the intrinsic viscosities and the (1 2a χ ) constant in the light of Eqs. (4) and (5). Another straight line can be obtained by plotting Y versus g 12 φ 2 according to Eq. (7). The intercept and slope of the straight line were achieved by linear regression. The absolute value of the intercept was the nonsolvent polymer interaction parameter χ 13. Plots of the three systems, water/nmp/pes, 1-butanol/NMP/PES, DegOH/NMP/PES, are depicted in Figs. 3 8. As can be seen, the plots of X versus g 12 φ 2 in Figs. 3, 5 and 7 showed good linear correlation. Although the plots of Y versus g 12 φ 2 in Figs. 4, 6 and 8 showed some deviation from linearity, the absolute values of the correlation coefficients R were greater than 0.90, R >0.90. In addition, the F-tests in the variance analysis also confirmed that all the linear regressions were significant at the maximum confidence level, i.e. α=0.01. So the expression of Campos et al. was suitable for these ternary systems. The [η] 13,(1 2a χ ), χ 13 and the correlation coefficients R obtained in such a way are listed in Table 4. Zeman and Tkacik [5] investigated the thermodynamic behavior of the membrane-forming system water/nmp/pes. The value of χ 13 in the low polymer Fig. 1. The effect of nonsolvent on the dropping of intrinsic viscosities: ( ) water/nmp; ( ) 1-butanol/NMP; ( ) DegOH/NMP. Fig. 2. g 12 of (1) nonsolvent and (2) NMP plotted as a function of NMP volume fractions, φ 2 :( ) water/nmp; ( ) 1-butanol/NMP; ( ) DegOH/NMP.

92 Z. Li, C. Jiang / Journal of Membrane Science 174 (2000) 87 96 Fig. 3. Plot of Eq. (3) for the system water/nmp/pes. Fig. 5. Plot of Eq. (3) for the system 1-butanol/NMP/PES. concentration range (5 25 wt.%) fitted from light scattering measurements was 1.6; at high concentration, the volume fraction of PES was 0.97, and the value of χ 13 obtained from the swelling measurements was 2.73. They are distinct from each other. Considering that the polymer concentrations in the casting solu- tions are usually between 8 and 38 wt.%, the more reasonable value for the water PES interaction parameter is 1.6. In addition, the binodal curve, which was calculated by Zeman and Tkacik [5] by selecting χ 13 to be equal to 1.5, was in good agreement with the measured cloud point. The value of χ 13, 1.54, calculated Fig. 4. Plot of Eq. (7) for the system water/nmp/pes. Fig. 6. Plot of Eq. (7) for the system 1-butanol/NMP/PES.

Z. Li, C. Jiang / Journal of Membrane Science 174 (2000) 87 96 93 Table 4 The calculated nonsolvent polymer interaction parameter χ 13 Nonsolvent [η] 13 (1 2a χ ) R a χ 13 R b polymer (cm 3 g 1 ) Water PES 359.11 0.2997 0.99 1.54 0.95 1-Butanol PES 26.92 0.1861 0.99 1.14 0.95 DegOH PES 12.90 0.02036 0.98 0.91 0.90 a R are the correlation coefficients for the linear regression of X vs. g 12 φ 2. b R are the correlation coefficients for the linear regression of Y vs. g 12 φ 2. Fig. 7. Plot of Eq. (3) for the system DegOH/NMP/PES. from the intrinsic viscosities, was nearly the same as that obtained from the light scattering measurements, so the method provided herein is reliable. On the other hand, compared to light scattering, the intrinsic viscosity measurement is simple and precise, and can be applied in various mixed solvent systems without the limitation of the refractive index of the solvents. When different nonsolvents were added to the ternary systems of nonsolvent/nmp/pes, the value of χ 13 decreased from the poor solvent to the swelling agent for PES, as shown in Table 4 (water 1-butanol DegOH), which were consistent with the prediction of the Flory Huggins theory. Just as depicted in Fig. 1, the effect on the state of polymer in solution, expanding or contracting, of various nonsolvents was significantly different. A little amount of nonsolvent with a large χ 13 in the mixed solvent could make the expanding PES molecules in the good solvent NMP contract rapidly and phase separation took place shortly afterwards. In contrast, a nonsolvent with a small χ 13 in the mixed solvent could make the expanding PES molecules in the good solvent NMP contract slowly, and a large amount of nonsolvent was needed to cause phase separation. Hence, the value of χ 13 can be used as a criterion to estimate the strength of the nonsolvent. To some degree, the concentration of nonsolvent in the mixed solvent at the state of phase separation is determined by the value of χ 13. The lower the value of χ 13, the more is the amount of nonsolvent needed to cause phase separation. 4.3. Results of binodal calculation Fig. 8. Plot of Eq. (7) for the system DegOH/NMP/PES. In order to verify the reliability of the strategy for calculating the nonsolvent polymer interaction parameter χ 13, the binodal curves of three membrane-forming systems were predicted with the χ 13 obtained herein, and then compared with the measured cloud points. The value of χ 23 is 0.5, which has been suggested in most of the literature [5]. As mentioned above, the g 12 for water/nmp, 1-butanol/NMP and DegOH/NMP were concentration-dependent.

94 Z. Li, C. Jiang / Journal of Membrane Science 174 (2000) 87 96 Fig. 9. Comparison of the calculated binodal curve with experimental data for the system water/nmp/pes: ( and ) cloud point data from Zeman and Tkacik [5] and Li [15], respectively; ( ) cloud point data measured herein; ( ) the calculated binodal curve. The binodal curve obtained for the ternary system water/nmp/pes is plotted in Fig. 9. Obviously, the binodal curve calculated is in agreement with the cloud points measured herein and those published by Zeman and Tkacik [5] and Li [15]. In addition, Fig. 11. Comparison of calculated binodal curve with experimental data for the system DegOH/NMP/PES: ( and ) cloud point data measured herein and from Li [15], respectively; ( ) the calculated binodal curve. the binodal curves of the ternary systems 1-butanol /NMP/PES and DegOH/NMP/PES are illustrated in Figs. 10 and 11, respectively, which fitted the cloud points measured herein and those published by Li [15] and Wang [16]. Therefore, the strategy for calculating the nonsolvent polymer interaction parameter χ 13 proposed herein was reliable, at least for the systems listed herein. 5. Conclusion The nonsolvent polymer interaction parameter χ 13 can be determined in the single polymer-mixed solvent ternary polymer solutions with the measured intrinsic viscosities of the polymer in mixed solvents. In the nonsolvent/nmp/pes membrane-forming systems, the values of χ 13 for water/pes, 1-butanol/PES and DegOH/PES were obtained. Compared to the methods published in the literature, the method proposed herein is convenient and reliable. Moreover, without the limitation of the refractive index of the solvents, it is expected to have broad applications. Fig. 10. Comparison of calculated binodal curve with experimental data for the system 1-butanol/NMP/PES: ( and ) cloud point data from Wang [16] and Li [15], respectively; ( ) the calculated binodal curve. 6. Nomenclature a χ the constant for each ternary polymer system

Z. Li, C. Jiang / Journal of Membrane Science 174 (2000) 87 96 95 A 2 the second virial coefficient (10 4 cm 3 mol g 2 ) c the concentration of the polymer (g cm 3 ) C the factor in Eq. (2) g the concentration-dependent interaction parameter g E the excess molar Gibbs energy (J mol 1 ) G the Gibbs free energy (J) h E the molar heat of mixing (J mol 1 ) k the Huggins constant M W the weight-average molar mass of polymer (g mol 1 ) n i the number of moles of component i N A the Avogadro s number (mol 1 ) r the ratio of molar volumes of nonsolvent and solvent R the gas constant (J mol 1 K 1 ) s t t 0 T v 3 the ratio of molar volumes of nonsolvent and polymer the flow time of the dilute solution (s) the flow time of the solvent (s) temperature (K) the partial specific volume of polymer (cm 3 g 1 ) V the molar volume (cm 3 mol 1 ) x i the molar fraction of component i Greek letters α the confidence level χ ij the interaction parameter φ i the volume fraction of component i 0 the Flory constant (mol 1 ) [η] the intrinsic viscosity (cm 3 g 1 ) η sp the specific viscosity µ i the chemical potential of i (J mol 1 ) ψ the interpenetration function for the binary polymer system T KY the interpenetration function for the ternary polymer system Superscripts E the excess function the dilute phase the concentrated phase Subscripts 1 nonsolvent 2 solvent 3 polymer i, j components i, j m the mixing function T mixed solvent Acknowledgements The authors are grateful to Professor J. Gmehling for providing the vapor liquid equilibrium data for DegOH/NMP collected in the famous DDB. References [1] P. van de Witte, P.J. Dijkstra, J.W.A. Berg, J. Feijen, Phase separation processes in polymer solutions in relation to membrane formation, J. Membr. Sci. 117 (1996) 1 31. [2] H. Tompa, Polymer Solutions, Butterworths, London, 1956. [3] F.W. Altena, C.A. Smolders, Calculation of liquid liquid phase separation in a ternary system of a polymer in a mixture of a nonsolvent, Macromolecules 15 (1982) 1491 1497. [4] L. Yilmaz, A.J. McHugh, Analysis of nonsolvent solvent polymer phase diagrams and their relevance to membrane formation modeling, J. Appl. Polym. Sci. 31 (1986) 997 1018. [5] L. Zeman, G. Tkacik, Thermodynamic analysis of a membrane-forming system water/n-methyl-2-pyrrolidone/ polyethersulfone, J. Membr. Sci. 36 (1988) 119 140. [6] R. García, C.M. Gómez, L. Porcar, V. Soria, A. Campos, Modified interpenetration function accounting for the excluded-volume effects in ternary polymer systems, J. Chem. Soc., Faraday Trans. 90 (1994) 339 344. [7] C.M. Gómez, V. Soria, J.E. Figueruelo, A. Campos, Molecular interactions in cosolvent ternary polymer systems, J. Chem. Soc., Faraday Trans. 89 (1993) 1765 1772. [8] V. Soria, C.M. Gómez, R. García, A. Campos, Excluded volume in ternary polymer systems, J. Chem. Soc., Faraday Trans. 88 (1992) 1555 1559. [9] W.W.Y. Lau, M.D. Guiver, T. Matsuura, Phase separation in polysulfone/solvent/water and polyethersulfone/solvent/water systems, J. Membr. Sci. 59 (1991) 219 227. [10] M.L. Huggins, The viscosity of dilute solutions of long-chain molecules. IV. Dependence on concentration, J. Am. Chem. Soc. 64 (1942) 2716 2718. [11] J. Gmehling, U. Onken, W. Arlt, P. Grenzheuser, B. Kolbe, J. Rarey, U. Weidlich, Vapor Liquid Equilibrium Data Collection, DECHEMA Chemistry Data Series, Vol. 1, DECHEMA, Frankfurt, 1977. [12] G.M. Wilson, Vapor liquid equilibrium. XI. A new expression for the excess free energy of mixing, J. Am. Chem. Soc. 86 (1964) 127 130.

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