(Lower Critical Solution Temperature) in Polymer Solutions - American

Semiempirical Method for the Prediction of the Theta (Lower. Critical Solution Temperature) in Polymer Solutions. Attila R. Imre,*,† Young Chan Bae,...
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Ind. Eng. Chem. Res. 2004, 43, 237-242

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RESEARCH NOTES Semiempirical Method for the Prediction of the Theta (Lower Critical Solution Temperature) in Polymer Solutions Attila R. Imre,*,† Young Chan Bae,‡ Bong Ho Chang,‡ and Thomas Kraska§ KFKI Atomic Energy Research Institute, P.O. Box 49, H-1525 Budapest, Hungary, Division of Chemical Engineering, Hanyang University, Seoul 133-791, Korea, and Institute of Physical Chemistry, University Cologne, Luxemburger Strasse 116, D-50939 Ko¨ ln, Germany

Here, a semiempirical method for predicting the solubility limit of polymer solutions at high temperatures is proposed. This method has been developed for infinite-chain-length polymers dissolved in Θ solvents. The development is based on several data for polystyrene solutions, and it has been tested also for polyethylene and polypropylene solutions. This method only requires the critical density and temperature of the solvent for the prediction of the hightemperature miscibility limit. The prediction appears to be reasonable considering the simplicity of the correlation model and gives better prediction than earlier comparably simple methods. 1. Introduction Partially miscible weakly interacting polymer solutions such as polystyrene (PS) or polyethylene (PE) solutions often exhibit two solubility boundaries:1 one at high temperature and one at low temperature. In between, the solution is homogeneous. In the case of strictly binary solutions (i.e., with a monodisperse polymer), the upper critical solution temperature (UCST) is located at the top of the coexistence curve, while the lower critical solution temperature (LCST) is located in the minimum of a coexistence curve at high temperature. In real systems, polymers are always polydisperse, which gives a topologically different phase behavior with the dislocation of UCST and LCST from the extrema. UCST and LCST depend on the molar mass and also on the pressure. For UCST, usually atmospheric pressure is used, while the vapor pressure of the solvent is usually used for LCST.1 Because polydispersity can be regarded as zero at infinite molecular weight, the Θ temperatures, which are the critical temperatures in the limit of infinite-chain-length polymers, are not affected by it. The systems investigated here are polymer/Θ-solvent systems exhibiting an LCST. According to the classification of van Konynenburg and Scott,2-5 they can be type IV (having both LCST and UCST) or type V (with LCST only). Θ(UCST), which is the UCST at infinite chain length, has a great significance because polymers cannot be dissolved and processed below this temperature over a wide concentration range. Θ(LCST), which is the LCST at infinite chain length, is often regarded as less important because it is usually located at high temper* To whom correspondence should be addressed. E-mail: [email protected]. † KFKI Atomic Energy Research Institute. ‡ Hanyang University. § University Cologne.

ature, where the polymer degenerates as, for example, PS and PE above 600 K.6 However, Θ(LCST) can become significant for solutions as an upper temperature limit for processing. Most prediction methods are developed for Θ(UCST) at vapor or atmospheric pressure, while the prediction of Θ(LCST) at vapor pressure in different solutions is usually neglected. Also, because of the experimental difficulties, experimental Θ(LCST) data are available only for a few systems and, hence, a relatively accurate prediction method would be useful. There are two groups of methods for correlating and predicting UCSTs and LCSTs, including Θ temperatures. First, there are theoretically well-founded methods, which are able to describe both solubility limit branches including the critical points. These methods usually require three or more adjustable parameters and some experimental liquid-liquid or liquid-vapor data in order to determine the interaction parameters.5,7-15 Because some of the parameters and data are not available, the application of such methods for prediction can become difficult.16 Another approach is rather empirical by correlating empirical equations to a few data of the two components such as density, liquid-vapor critical point, or solubility parameters.17-24 These methods do not require interaction parameters; however, most of these empirical and semiempirical methods are not able to reproduce Θ(UCST) and Θ(LCST) within satisfying accuracy. The error can be as high as (30-50 K, as shown in refs 17-24. The physical origin of the Θ(LCST) branch is mainly related to the increasing difference of the molar volumes with increasing temperature.1 The entropy of mixing is decreasing because of the packing problems of the molecules. This is well-known,25 and it has been shown experimentally that for most studied systems Θ(LCST) should be below, but not too far, the vapor-liquid critical temperature of the solvent (TVL c ). Patterson and co-workers26,27 proposed a first correlation between Θ(LCST) and TVL c . This correlation, which is not very

10.1021/ie030548p CCC: $27.50 © 2004 American Chemical Society Published on Web 12/05/2003

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Figure 1. (a) Schematic representation of a tangled infinite polymer chain in a solution. (b) Schematic representation of the phase transition generated by exceeding the limiting molar volume.

accurate in its original form, is the basis for the correlation proposed here. Here we propose a reasonably accurate empirical method for the prediction of Θ(LCST) in polymer solutions at the vapor pressure of the solution based on theoretical considerations. We have used PS solution data to develop this method because many Θ(LCST) data are available for this polymer dissolved in different solvents. The method has also been checked for polypropylene (PP) and PE solutions. For other homopolymers, no sufficient LCST data are available, which also underlines the need for a predictive correlation method. 2. Theoretical Background The method presented here is based on the finding that for very long chain length polymers Θ(LCST) is close to the critical temperature of the pure solvent. In this section, some arguments are given that underline this assumption. Void Size. The first argument is based on geometrical considerations. This is based on the size distribution of the empty spaces between the segments of a tangled infinite chain. In Figure 1a, a schematic presentation of an infinite polymer chain in solution is shown. There are empty spaces or voids inside the tangled chain that can be filled by solvent molecules. The volume of a void is defined as the volume of the largest sphere insertable into the void. For a given thermodynamic state, a distribution of these void sizes is given that has a certain average value. A solvent with sufficiently small molecules can fill these empty spaces. As the size of the entangled chain changes with pressure and temperature in solutions,28,29 the void sizes change too. Using the average void size, one can define a temperature-dependent mean density (Fmean), or molar volume, which describes the density of a fictional solvent that exactly fits into the average voids. At a given temperature and for a given polymer, solvents with a molar volume that is larger than the one given by these limiting values will favor demixing. For different real solvents, this limiting line is crossed at different temperatures; therefore, LCST will be different for each of them, as shown in Figure 1b. Because the solvent density changes drastically close to the solvent’s vapor-liquid critical point while the density of the polymer changes only slightly around that temperature, one can expect that for most solutions the above-mentioned crossing and hence the LCST is located close to the vapor-liquid critical temperature. Free Volume. A similar explanation is based on the free-volume difference of the solute and solvent. For a nonpolar polymer in solution, the difference in the free volumes of solvent and polymer rises as the system

Figure 2. Free-volume difference of PS/cyclohexane and PS/ toluene solutions. The molecular weight is Mw ) 100 000 amu in both cases. Θ(LCST) and the critical temperatures (which correspond to Mw ) 100 000 amu) are marked by dash-dotted and dashed lines, respectively. The solid circles represent the liquidvapor critical temperature of cyclohexane and toluene.

temperature increases toward the vicinity of the solvent’s critical temperature along a constant composition path. The solvent’s free volume is then significantly larger than that of the polymer. When a solvent molecule is placed in a polymer matrix, it becomes more confined and its entropy decreases. At the same time, the polymer molecule becomes less confined, raising its entropy.30 However, at any reasonable volume fraction close to the binary critical point, the number of solvent molecules is much larger than the number of polymer molecules. Hence, the overall entropy of mixing declines, causing immiscibility.30 Now, the free-volume difference can be roughly estimated by31

DFV )

|

VFV m,1

Vm,1(vdW)

-

VFV m,2

|

Vm,2(vdW)

FV Vm,i ) Vm,i - Vm,i(vdW)

(1) (2)

where Vm,i and Vm,i(vdW) are the molar volume and the van der Waals volume of the pure component i, respectively. The molar volume Vm,i is a function of temperature, while the van der Waals volume Vm,i(vdW) is constant. Both Vm,i(T) and Vm,i(vdW) are estimated by the van Krevelen group contribution method.32 The molar volumes of the solvents are estimated by a modified Racket equation.33 The results are shown in Figure 2. One can see that the free-volume difference is larger for the solvent having a low TVL c than for those having . As will be shown later, this coincides with a high TVL c

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Figure 4. (a) Liquid-vapor critical temperatures of the solvents versus experimentally obtained Θ(LCST) values of the corresponding PS (squares), PE (circles), and PP (triangles) solutions. The circled symbols are data for solvents of PS with poor (or rather without) hydrogen bonding, while all other noncircled symbols are for solvents with moderate hydrogen bonding. The lines represent the correlation given by eq 3 (PS, solid; PE, dotted; PP, dashed).

Figure 3. (a) Schematic sketch of phase diagram type IV for a system consisting of a monodisperse finite-chain-length polymer in solution in the pT projection. For high molecular weight polymers, the vapor-pressure curve of the pure solvent (dashed curve) is very close to the three-phase curve (dot-dashed curve). The solid curves are critical curves. L: liquid. V: vapor. F: fluid. For the acronyms, see the text. (b) Schematic representations of the isobaric section at p1 in the pT diagram. (c) Schematic representations of the isobaric section at the pressure of the lower critical end points (LCEP-1) in the pT diagram.

eq 3, in which Fmean is also larger for the solvents having a lower TVL c at least at high temperature where LCST is expected. Unfortunately, the free-volume difference DFV(T) is a smooth and monotonic function, and therefore one cannot distinguish any special point such as LCST. Global Phase Diagram. The third argument is based on the topological behavior of the global phase diagrams introduced by Scott and van Konynenburg2,3 and extended to polymeric systems.5,34,35 Systems exhibiting a LCST branch at vapor pressure are types IV and V within the classification scheme of van Konynenburg and Scott.2-5 For high molecular weight solutions, the coexistence curve (in the case of polydisperse polymers, the cloud point curve) is very asymmetric, reflecting the asymmetric molecular properties of the solvent and the polymer. Figure 3a shows a schematic sketch of the pressure-temperature projection of a type IV system in the region of the solvent vapor-pressure curve. The corresponding temperature mole fraction and pressure mole fraction projections are shown in Figure 3b,c. Here monodispersity is assumed. For polymer solutions of phase diagram types IV and V, the vaporpressure curve of the pure solvent is often close to the liquid-liquid-vapor curve and not distinguishable in the pT projection. Therefore, LCST at the vapor pressure of the solvent is almost at the same coordinates in the pT projection as the lower critical end point (LCEP1). Because the vapor-liquid critical curve connecting

the pure solvent critical point (CP-1) and the upper critical end point (UCEP-2) is very short for asymmetric systems, it follows that LCEP-1 is at lower temperature than the pure solvent critical point (CP-1). As a consequence, one can expect a LCST curve, which is connected to LCEP-1 at a temperature below the pure solvent critical point. Method. PS solutions were chosen to develop and to test this correlation method, because for these systems the most complete database for liquid-liquid equilibrium exists.36 In addition, available data of PE and PP systems16 are correlated. In most prediction methods proposed so far, the molar mass dependence of LCST is neglected. To avoid this problem, here we focus on infinite-chain-length polymers only. The advantages are that at infinite chain length we have a defined state that is independent of the molar mass and of the polydispersity. Hence, we focus on the Θ(LCST) temperature that is the lowest LCST for a specific polymer solution. All LCSTs of finite-chain-length polymer solutions of a specific system can be expected at higher temperatures. In a limited region below Θ(LCST), the systems are miscible for all chain lengths. The proposed method can be used for solutions of weakly interacting polymers in Θ solvents only. To decide which solvents are Θ solvents, one can use, for example, a recently suggested criterion based on the Hansen solubility parameter.36 Alternatively, one can check the existence of Θ(UCST). Although the existence of Θ(UCST) does not, in general, imply the existence of Θ(LCST), in weakly interacting polymer solutions, it is a quite fair indication according to experimental observations.36 It should be mentioned that applying this Θ(LCST)-predicting method to non Θ solvents can lead to artificial Θ(LCST). In Figure 4, the vapor-liquid critical temperatures versus known Θ(LCST) values are plotted for different PS, PE, and PP solutions. It appears that there are linear correlations between Θ(LCST) and TVL c that are, however, not very accurate:

Θ(LCST)3 ) ATVL c - B

(3)

Subscript 3 marks this specific equation for Θ(LCST).

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Table 1. PS Solutions: Vapor-Liquid Critical Temperatures TVL c , Θ(LCST), Vapor-Liquid Critical Densities Gc, Θ(LCST) Predicted by Equation 3, Differences between the Θ(LCST) Predicted by Equation 3 and the Experimental Values for Θ(LCST) (δ(LCST)r), Θ(LCST) Predicted by Equation 7, and Differences between the Θ(LCST) Predicted by Equation 7 and the Experimental Θ(LCST) (δ(LCST)β)a Polystyrene solvent

Tc (K)

Θ(LCST) (K) at pvap

benzene cyclohexane cyclopentane dimethoxymethane ethyl acetate ethyl n-butyrate isobutyl acetate methyl acetate methylcyclohexane methylcyclopentane methyl ethyl ketone n-pentyl acetate n-propyl acetate toluene n-butyl acetate i-amyl acetate i-propyl acetate trans-decalin

562 ( 0.6 554 ( 1 511.7 ( 0.2 480.6 ( 1 523.2 ( 0.1 568.8 ( 0.6 560.8 ( 0.6 506.8 ( 0.3 572.1 ( 0.2 532.7 ( 0.2 536.8 ( ?? 600 ( 2 549.4 ( ?? 593 ( 2 575.4 ( 0.6 586.1 ( 0.6 532.0 ( 0.6 687 ( 3

523.4 486.4 427.2 385.2 411.9 470 444.7 387.7 480 428.9 420.7 500.7 450.6 550 480.7 492 388 above 650

a

FLV c (mol/L) 3.9 ( 0.2 3.24 ( 0.03 3.85 ( 0.04 4.69 ( ?? 3.492 ( ?? 2.38 ( ?? 2.42 ( ?? 4.394 ( ?? 2.71 ( 0.02 3.14 ( 0.04 3.74 ( ?? 2.131 ( ?? 2.895 ( ?? 3.17 ( 0.01

Θ(LCST)calc (eq 3)

δ(LCST)R (K)

Θ(LCST)calc (K) (eq 7)

δ(LCST)β (K)

470.5 460.2 405.5 365.3 420.4 479.3 469.0 399.2 484.8 432.7 438.0 519.7 454.2 510.6 487.9 501.7 431.7 632.2

52.9 26.2 21.7 19.9 -8.5 -9.3 -24.3 -11.5 -4.8 -3.8 -17.3 -19.0 -3.6 39.4 -7.2 -9.7 -43.7

507.9 466.7 405.9 376.1 415.1 463.4 449.0 416.9 484.4 420.6 451.6 516.0 444.4 541.7

15.5 19.6 21.3 9.1 -3.2 -6.6 -4.3 -29.2 -4.4 8.3 -30.9 -15.3 6.2 8.3

Question marks indicate missing data.

Table 2. PE and PP Solutions: Vapor-Liquid Critical Temperatures (TVL c ), Θ(LCST), Vapor-Liquid Critical Densities Gc, Θ(LCST) Predicted by Equation 3, Differences between the Θ(LCST) Predicted by Equation 3 and Experimental Θ(LCST) (δ(LCST)r), Θ(LCST) Predicted by Equation 7, and Differences between the Θ(LCST) Predicted by Equation 7 and the Experimental Θ(LCST) (δ(LCST)β)a Polypropylene solvent

Tc (K)

Θ(LCST) (K) at pvap

FLV c (mol/L)

Θ(LCST)calc (eq 3)

δ(LCST)R (K)

Θ(LCST)calc (K) (eq 7)

δ(LCST)β (K)

n-pentane n-octane 2,3-dimethylpentane cyclohexane methylcyclohexane 2,2-dimethylbutane

469.7 ( 0.4 568.9 ( 0.5 537.3 ( 0.5 554 ( 1 572.1 ( 0.2 489 ( 0.6

422 542 513 540 564 441

3.22 ( 0.07 2.03 ( 0.01 2.54 ( 0.05 3.24 ( 0.03 2.71 ( 0.02 2.8 ( 0.02

418.1 552.0 509.4 531.9 556.3 444.2

3.9 -10.0 3.6 8.1 7.7 -3.2

422.5 546.7 509.1 544.2 561.9 443.7

-0.5 -4.7 3.9 -4.2 2.1 -2.7

solvent

Tc (K)

Θ(LCST) (K) at pvap

FLV c (mol/L)

Θ(LCST)calc (eq 3)

δ(LCST)R (K)

Θ(LCST)calc (K) (eq 7)

δ(LCST)β (K)

n-pentane n-octane 2,3-dimethylpentane cyclohexane methylcyclohexane

469.7 ( 0.4 568.9 ( 0.5 537.3 ( 0.5 554 ( 1 572.1 ( 0.2

353 496 463 518 537

3.22 ( 0.07 2.03 ( 0.01 2.54 ( 0.05 3.24 ( 0.03 2.71 ( 0.02

352.0 518.7 465.6 493.6 524.0

1.0 22.7 2.6 24.4 13.0

355.4 501.6 460.4 522.3 536.0

2.4 5.6 2.6 4.3 1.0

Polyethylene

a

Question marks indicate missing data.

Here the temperatures are given in Kelvin. For the solvents of PS given in Table 1, we have obtained the parameters A ) 1.29 ((0.19) and B ) 256 ((106) K. The critical temperatures and densities of the pure solvents listed in Table 1 are taken from the NIST Webbook.37 The Θ(LCST) data are taken from the literature.36 For this correlation, all data of Table 1 were used with the exception of those of trans-decalin, for which only a lower limit has been established for Θ(LCST). For PE and PP with the solvents given in Table 2, we obtained the following parameters: A(PP) ) 1.35 ((0.08), B(PP) ) 216 ((44) K, A(PE) ) 1.68 ((0.24), B(PE) ) 437 ((132) K. The experimental values for Θ(LCST) and TVL are collected in Table 2. The c critical temperatures and densities were again taken from the NIST Webbook,37 and the Θ(LCST) data were taken from ref 16.

Although for PS solutions one can improve the correlation by distinguishing the solvents according to their hydrogen-bonding ability, as shown in Figure 4, the number of data is hardly enough to justify handling the solvents in two groups. It appears that in some cases, although the pure solvent critical temperatures are almost the same, Θ(LCST) can be very different. This is the case, for example, for the system pairs cyclopentane/PS and methyl acetate/PS, benzene/PS and i-butyl acetate/PS, as well as toluene/PS and n-pentyl acetate/PS. One can see from the tabulated data that large positive deviations from eq 3 correlate with high pure solvent critical densities, while large negative deviations correlate with small critical densities. This indicates the existence of a temperature-dependent mean density that has to be reached by the solvent’s density in order to obtain a

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18.9 K for eq 3 and 13.0 K for eq 7. For PE solutions, the average error is 6.1 K for eq 3 and 3.0 K for eq 7. Finally, for PP solutions, the average deviation is 12.7 K for eq 3 and 3.2 K for eq 7. The experimental Θ(LCST) values and the Θ(LCST) values predicted by eq 7 are compared in Figure 5. The correlation method proposed above for the prediction of Θ(LCST) values for PS, PE, and PP solutions may be the basis for a more general correlation method. This would be of interest in this field where experimental data are available for a few systems only.38,39 Studies to apply this correlation method to other systems as well as to account for further theoretical input are in progress. Figure 5. Experimentally obtained Θ(LCST) values vs the corresponding values predicted by eq 7 for PS (squares) and for PE (circles) and PP (triangles). Dashed lines represent exact agreement.

LCST. In a small temperature range, this mean density can be correlated by a linear function that is the simplest possible ansatz.

Fmean ) CT + D

(4)

Here T is the temperature of interest, in this case Θ(LCST). One can say that PS can be characterized by a limiting mean density, which is defined by the voids in the entangled polymer chain. Crossing this value by the solvent density causes a phase transition. This statement is in agreement with the statement given in section 2 that the existence of LCST is caused by the molar volume difference of the solute and solvent. Fmean should be different for each polymer. Hence, it is possible to enhance the correlation given by eq 3 with a correction that depends on the critical density of the pure solvent. As a result, one obtains an improved estimation for Θ(LCST):

Θ(LCST)5 ) Θ(LCST)3 + E(FVL c - Fmean)

(5)

Substituting eqs 3 and 4 gives VL Θ(LCST)6 ) ATVL c - B + E[Fc - C - DΘ6(LCST)] (6)

Subscripts 3, 5, and 6 correspond to the Θ(LCST) values marks predicted by eqs 3, 5, and 6, respectively. FVL c the liquid-vapor critical density of the solvent. Rearranging eq 6 yields a general form for the correlation equation: VL Θ(LCST)7 ) P1TVL c + P2Fc + P3

(7)

For PS solutions, the parameters determined by multiple regression are P1 ) 1.99 ((0.25), P2 ) 38.17 ((11.03) K dm3 mol-1, and P3 ) -759.36 ((167.76) K. For PP, they are P1 ) 1.44 ((0.05), P2 ) 15.7 ((5.2) K dm3 mol-1, and P3 ) -304.5 ((38.2) K. For PE, they are P1 ) 1.97 ((0.07), P2 ) 41.5 ((5.9) K dm3 mol-1, and P3 ) -703.5 ((49.7) K. The correlations show that these parameters clearly depend on the polymer. We here propose eq 7 for the estimation of Θ(LCST)s with the parameters given here. Comparing the Θ(LCST) correlations by eqs 3 and 7 using the same data set as that given in Tables 1 and 2 shows that the new correlation given by eq 7 exhibits fewer deviations. The average error for PS solutions is

3. Conclusions We have discussed that the existence of Θ(LCST) in weakly interacting polymer solutions is due to the increasing molar volume difference between the solvent and solute. For this reason as well as from topological considerations of the phase diagrams, one can expect Θ(LCST) in the vicinity of the liquid-vapor critical point of the solvent. On the basis of this assumption, we explored a relation between Θ(LCST) and the pure solvent critical density and temperature. This relation is used to develop an empirical correlation method that allows one to predict Θ(LCST) for solutions of very long chain length molecules in Θ solvents. The method was tested for PS solutions, for which sufficient data are available, as well as for PP and PE solutions. Acknowledgment This work was partially supported by the Hungarian Research Fund (OTKA) under Contract No. T 043042, by a bilateral program of the German Science Foundation (DFG) and the Hungarian Academy of Science (HAS), and by a bilateral program of the Korean Science and Engineering Foundation (KOSEF) and the HAS. A.R.I. was supported also by a Bolyai Research Fellowship. The authors thank Dr. A. Vetere for his helpful comments. Literature Cited (1) Koningsveld, R.; Stockmayer, W. H.; Nies, E. Polymer Phase Diagrams; Oxford University Press: New York, 2001. (2) Scott, R. L.; van Konynenburg, P. H. Discuss. Faraday Soc. 1970, 49, 87. (3) van Konynenburg, P. H.; Scott, R. L. Philos. Trans. R. Soc., London 1980, 248A, 495. (4) Kraska, T.; Deiters, U. K. J. Chem. Phys. 1992, 96, 539. (5) Yelash, L. V.; Kraska, T. Phys. Chem. Chem. Phys. 1999, 1, 4315. (6) van Krevelen, D. W. Properties of Polymers, 3rd ed.; Elsevier: Amsterdam, The Netherlands, 1997; Chapter 21. (7) Bae, Y. C.; Shim, J. J.; Soane, D. S.; Prausnitz, J. M. J. Appl. Polym. Sci. 1993, 47, 1193. (8) Kontogeorgis, G. M.; Saraiva, A.; Fredenslund, A.; Tassios, D. P. Ind. Eng. Chem. Res. 1995, 34, 1823. (9) Saraiva, A.; Bogdanic, G.; Fredenslund, A. Ind. Eng. Chem. Res. 1995, 34, 1835. (10) Saraiva, A.; Kontogeorgis, G. M.; Harismiadis, V. I.; Fredenslund, A.; Tassios, D. P. Fluid Phase Equilib. 1996, 115, 73. (11) Chang, B. H.; Bae, Y. C. Polymer 1998, 39, 6449. (12) Jang, J. G.; Bae, Y. C. J. Appl. Polym. Sci. 1998, 70, 1143. (13) Choi, J. J.; Yi, S.; Bae, Y. C. Macromol. Chem. Phys. 1999, 200, 1889.

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Received for review July 1, 2003 Revised manuscript received November 11, 2003 Accepted November 11, 2003 IE030548P