330
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Eng. Chem. Prod. Res. Dev. 1981, 20, 330-335
Influence of Temperature on the Diffusion of Solvents in Polymers above the Glass Transition Temperature Shlaw T. Ju and J. Larry Duda' Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
James S. Vrentas Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 606 16
The free-volume theory of diffusion is used as a basis for the development of three methods for the correlation and prediction of the diffusivity-temperature relationship for the diffusion of trace amounts of solvents in molten polymers. The methods differ in the amount and type of dtrusivity data which are needed to carry out the correlative and predictive procedures. The capabilities of the methods are evaluated using experimental polymer-solvent dlffushrky data. These techniques are useful in the design and optimization of processes which involve the removal of small amounts of volatile residues from polymers.
Introduction The rate of migration or molecular diffusion of solvents in molten polymers is an important consideration in the design of many polymer processing operations. One of the more important processes is the removal of small amounts of volatile residues from polymers. These impurities can be small molecules such as monomers, solvents, or condensation byproducts, and they are usually removed by vacuum or steam stripping at elevated temperatures. The optimization of these purification processes to meet environmental, health, and safety regulations is one the crucial problems facing the polymer synthesis and processing industry today. Polymer devolatilization processes are often conducted at elevated temperatures to increase the rate of molecular diffusion and to decrease the melt viscosity so that more mixing or surface regeneration can be attained. Unfortunately, it is quite difficult to measure diffusion coefficients at elevated temperatures, and the design engineer must often extrapolate limited diffusivity data obtained at low temperatures. Such extrapolations are often quite precarious, and orders of magnitude errors in the predicted diffusion coefficients can occur. To illustrate the problem of extrapolating polymersolvent diffusivity data to elevated temperatures, we consider the diffusion of trace amounts of methane and benzene in polystyrene and poly(viny1 acetate) (PVAc), respectively. The polystyrene-methane diffusivity data of Lundberg et al. (1963) are presented in Figure 1and the PVAc-benzene data of Ryskin (1955) are shown in Figure 2. It should be noted that availability of polymer-solvent diffusivity data over such wide temperature ranges (about 100 "C in each case) is quite rare. Typically, the design engineer will have only a few data points in the low-temperature range. Suppose, therefore, that only the lowest two data points are available for the polystyrenemethane system and only the lowest four for the PVAc-benzene system. We are interested in methods of using these limited data sets to obtain reasonable estimates of diffusion coefficients at higher temperatures. A frequently used expression which describes the temperature dependence of diffusion coefficients is the Arrhenius equation D = Do e x p ( - A )
Here, D is the mutual diffusion coefficient in the limit of zero solvent concentration, Dois a preexponential factor, and E is a constant activation energy for diffusion. This equation was used to correlate the low-temperature data in Figures 1and 2 and also to provide extrapolated estimates of the diffusivity at higher temperatures. The Arrhenius representations in Figures 1 and 2 (the dashed lines) appear to correlate the low-temperature data (the solid circles) quite well, but there are substantial differences between the Arrhenius predictions and the hightemperature data (the open circles). For example, for the PVAc-benzene system, the predicted value of the diffusivity at 160 "C is too large by nearly two orders of magnitude. The source of the difficulty in the Arrhenius relationship is evident from the log D vs. 1/T plots presented in Figures 1 and 2. The apparent activation energy for diffusion in the limit of zero solvent concentration is defined as d In D ED
=Rp(
F) P
and it is clear from Figures 1 and 2 that there is a substantial variation of ED with temperature for each of the above systems. Consequently, integration of eq 2 does not yield the Arrhenius expression for the temperature dependence of D. The diffusivity-temperature behavior exhibited by the polystyrenemethane and PVAc-benzene systems is typical of diffusivity data for polymer-solvent pairs. Substantial variations of E D with temperature are observed, and the temperature dependence of E D is especially severe near the glass transition temperature of the polymer. Furthermore, experimental data show that greater temperature variations of E D occur for polymersolvent systems with the larger solvents. It is clear from the above discussion that a viable theory for the description of the temperature dependence of D for polymer-solvent systems must take into account the variation of E D with temperature. A promising candidate for the correlation and prediction of the temperature dependence of D for polymer-solvent diffusion is the freevolume theory of diffusion (Vrentas and Duda, 1979a). The power of this method is exhibited in Figures 1 and 2 where the lower temperature data points are used to construct a free-volume representation of the temperature dependence of D. It is evident from these figures that the
0 196-4321/81/1220-0330$01.25/0 0 1981 American Chemical Society
Ind. Eng. Chem. Prod. Res. Dev., Vol.
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predictions of free-volume theory (the solid lines) are in excellent agreement with the data for the polystyrenemethane system and in reasonably good agreement for the PVAc-benzene system. In this paper, we use the free-volume theory of diffusion as a vehicle for the development of three methods for the correlation and prediction of the diffusivity-temperature relationships for polymer-solvent systems in the limit of zero solvent concentration. These methods differ in the amount of polymer-solvent diffusivity data which are needed to carry out the correlative and predictive procedures. The capabilities of these methods are evaluated using experimental polymer-solvent diffusivity data. Background Concepts and Theory The volume of any liquid can conveniently be divided into two parts: the volume directly occupied by the molecules of the liquid and the empty space between the molecules, which is usually referred to as free volume. Owing to the usual thermal fluctuations, some of this free volume is continuously being redistributed, causing fluctuations in the local density. This portion of the free volume is usually denoted as hole free volume, and molecular transport occurs by the migration of molecules into these randomly fluctuating voids. These free-volume concepts have been used as the basis of a theory which describes the diffusion of solvents in molten polymers (Fujita, 1961; Vrentas and Duda, 1979a). In this theory,
20, No. 2, 1981 331
it is assumed that two requirements must be satisfied before a molecule can migrate in a liquid. First, a hole of sufficient size must appear adjacent to the molecule, and second the molecule must have enough energy to jump into this void. Consequently, the rate of migration can be considered to be the product of two probabilities: the probability that a hole of sufficient size will form and the probability that the molecule will have sufficient energy to jump into this hole. For polymer-solvent diffusion processes at temperatures within about 100 "C of the glass transition temperature, the available hole free volume is small so that the probability of finding a hole of sufficient size controls the migration process. Consequently,for most cases of polymer-solvent diffusion, the rate of migration or diffusion is primarily determined by the amount of free volume and by the size of the migrating molecule. Although it is not possible to determine the hole free volume from first principles, it is possible to use freevolume concepts to construct correlative and predictive theories for both the diffusivity and viscosity of concentrated polymer solutions. We present here the pertinent free-volume equations for the diffusion of solvents in amorphous polymers in the limit of zero solvent mass fraction. We shall restrict our attention to temperatures above the glass transition temperature although freevolume theory can be extended to diffusion in the glassy state (Vrentas and Duda, 1978). The free-volume expression for the temperature dependence of D can be written as (Vrentas and Duda, 1979b)
where Tg2is the glass transition temperature of the pure polymer and Czg is a Williams-Landel-Ferry (WLF) constant which can be determined from the temperature dependence of the polymer viscosity (Ferry, 1970). The parameter E represents the ratio of the critical molar volume required for a jump of a solvent jumping unit to the critical molar volume of the jumping unit of the polymer. The jumping units of the polymer and solvent molecules are parts of the molecules which are capable of essentially independent movement over short distances. It is reasonable to expect that the jumping unit will be only a part of the total molecule for polymer chains, for rod-like solvents, and for flexible solvents of sufficient length. However, the entire molecule will perform a jump for relatively small or spherically shaped penetrant molecules, and some of the results presented below will be restricted to solvents for which the complete molecule jumps as a single unit. The definitions of the quantities y, pz*,and KI2are not given here since they are available elsewhere (Vrentas and Duda, 1977) and since their definitions are not needed in the present context. It suffices to note that D is dependent on two quantities which characterize the polymer, C2g and Tg2,and on two quantities which are properties of the polymsr-solvent system, the preexponential factor, Dol, and yV2*5/K12 We assume that these latter two quantities are effectively independent of temperature. From eq 2 and 3, it follows that the apparent activation energy for diffusion is given by the equation
Since Czg - Tg2< 0 for all common polymers, free-volume theory predicts that ED decreases with increasing tem-
332 Ind. Eng. Chem. Prod. Res. Dev., Vol. 20, No. 2, 1981 Table I. Values of C, g
~
Tg2 for Common Polymers
polymer polyisobutylene poly(viny1 acetate) polystyrene poly(methy1 acrylate) poly( dimethylsiloxane) poly(methy1 methacrylate) poly( ethyl met hacrylate)
c,g
- T,,,K
-100.6 -258.2 -325.5 -231.0 -81.0 -308.0 -269.5
perature. Also, it can be easily shown that I(dED/dT),I also decreases as the temperature increases so that the most dramatic changes in EDoccur near the glass transition temperature of the polymer. Furthermore, since 5 is a direct measure of the size of a solvent jumping unit, it follows that E D increases with increasing jumping unit size for solvents diffusing in the same polymer. Also, I(dED/ d T ) I a t a particular temperature is greater for solvents w i d the larger jumping units. As will be evident below, all of these free-volume predictions are consistent with polymer-solvent diffusivity data.
Correlative and Predictive Methods for Temperature Dependence of D When diffusivity-temperature data are available for a polymer-solvent system over a given temperature interval, it is of interest to find an equation which not only correlates the data over the temperature interval but which also predicts D for temperatures outside the interval. The goal is of course to extract the maximum possible predictive capability from a small amount of diffusivity data. In this section, we follow this approach and propose different methods for developing D vs. T relationships for polymer-solvent systems from rather limited data sets. As noted previously, the methods differ in the amount and type of diffusivity data which are needed. Methods are proposed below for the development of D vs. T relationships above Tg2for polymer-penetrant pairs in the limit of zero solvent concentration for the following three cases. (I) Measurements of the diffusion coefficient are available for the polymer-solvent system of interest at two or more temperatures. (11)A value of the diffusivity for the system of interest is available at only a single temperature. However, diffusion measurements are available for other solvents in the polymer of interest. (111) No diffusivity data are available for the system of interest. However, diffusion measurements are available for other solvents in the polymer of interest. The most accurate prediction of the diffusivity-temperature relationship will of course be obtained for case I and the least accurate for case 111. The goal in each case will be the prediction of the temperature dependence of D using eq 3, and this means using the available data to estimate the three quantities: Dol, yV2*S/KI2,and C+ Tg2' Since the glass transition temperature and the WLF constants are known for any widely used polymer, we shall assume that the quantity C2g - Tg2is available for the polymer of interest. Values of Tg2and C2g for many important polymers have been tabulated by Ferry (1970), and, for convenience, we have presented values of C2g - Tg2 for a number of common polymers in Table I. The procedure used for determining the remaining two parameters for a specific polymer-solvent system depends on the data that are available. Since D is known at two or more temperatures for case I, these diffusivity data can be used to construct a plot of In-D vs. 1/(C2g - T + 7'). The quantities Dol and yV2*(/K12are then $hermined from the intercept and slope of the straight line which results if free-volumetheory
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Figure 3. Free-volume temperature correlation of diffusivity data for three solvents in poly(methy1 acrylate). Lines represent computations based on free-volume theory.
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, I15
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Figure 4. Free-volume temperature correlation of diffusivity data for three solvents in PVAc. Lines represent computations baaed on free-volume theory.
satisfactorily describes the temperature dependence of the diffusivity. This method was used to correlate the low-temperature data in Figures 1and 2 and to obtain predictions for the high-temperature data in these figures. The predictive success of the method is quite impressive, particularly when the large temperature ranges and the dramatic failures of the Arrhenius equation are considered.. A demonstration of the correlative capabilities of free-volume theory is provided in Figure 3 and 4,where D vs. T data are presented for three polymethyl acrylate-solvent systems and three PVAc-solvent systems. The lines in these figures represent computations based on eq 3 with the parameters Dol and yVz*[/Kl2 determined utilizing all of the available data. We note that free-volume theory provides an excellent representation of the temperature dependence of D for all six polymer-penetrant pairs over rather substantial temperature intervals. The experimental apparent activation energies for diffusion for these six polymer-penetrant pairs are presented in Figures 5 and 6. There is a very significant decrease in E D with increasing temperature for each polymer-solvent pair, and the rate of decrease is greater at the lower temperatures. The effect of solvent size on ED and
Ind. Eng. Chem. Prod. Res. Dev., Vol. 20, No. 2, 1981 333 Table 11. Group Contribution Methods of Sugden (1927) and Biltz (1934)
\
50k
atomic constants according to Sugden, cm3/gmol
Methyl Acetate
H = 6.7 C = 1.1 N = 3.6 0 = 5.9 F = 10.3 C1= 19.3 Br= 22.1
I = 28.3 P = 12.7
triple bond, 13.9 double bond, 8.0 3-membered ring, 4.5 4-membered ring, 3.2 5-membered ring, 1.8 6-membered ring, 0.6
S = 14.3 0 (in alcohol) = 3.0 N (in ammonia) = 0.9
atomic constants according to Biltz, cm3/g-mol
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Figure 5. Experimental values of ED for three solvents in poly(methyl acrylate).
C (aliphatic), 0.77; (aromatic), 5.1 H, 6.45; double bond, 8.6; triple bond, 1 6 OH (alcoholic), 10.5; OOH (carboxyl), 23.2 C1, 16.3 Br, 19.2 I, 24.5
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(aED/aT)?, can be ascertained by considering the ED values in Figure 5 for methyl acetate methanol, and water. It can be shown, using techniques discussed below, that methyl acetate has the largest jumping unit of these three solvents and water the smallest. The data in Figure 5 show that ED increases with increasing size of a solvent jumping unit at a particular temperature. Furthermore, the absolute value of (dED/dT), is greater for the larger solvents at a given temperature, and more dramatic decreases of ED with temperature will be experienced by solvents with the larger jumping units. These experimental trends for ED are consistent with the predictjons of free-volume theory. For case 11, both Dol and yV2*[/K12can obviously not be determined from a single data point for the polymersolvent system of interest, but both of these parameters can be computed if diffusivity data are available for other solvents in the same polymer. The diffusjon data for the other solvents can be used to estimate yV2*f/K12for the system of interest by a method described below, and Dol can then be computed from the single value of diffusivity which is available. Using arguments presented elsewhere (Vrentas and Duda, 1977), it can be s_hownthat it is reasonable to expect that the parameter yV2*[/KI2is a linear function of the molar volume of a solvent jumping unit at 0 K. Molar volumes at 0 K can be estimated using the group contribution methods of Sugden (1927) and Biltz (1934). Haward (1970) has summarized these methods, and, for convenience, a corrected version of his summary of these techniques is presented in Table 11. In most cases, the size of a solvent jumping unit is known only if the entire
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SOLVENT MOLAR VOLUME AT OOK (cm7g-mole) Figure 7. Variation of with solvent molar volume at 0 K for PVAc. Solid circles are based on actual diffusivity-temperature data and open circles are computed using reported activation energies. Triangles represent solvents which do not move as single units. Solvents are identified in Table 111.
solvent molecule performs a jump, and the procedure for case I1 is generally valid only for this class of solvents. If values of yV2*4/K12are obtained fro? diffusivity data for such solvents, then a linear plot of yVz*t/Kl2vs. solvent molar volume can be constructed, and yV2*[/K12for the polymer-solvent pair of interest can be directly determined from this plot. To illustrate the feasibility of this method, we have constructed such linear plots for four polymers (PVAc, polystyrene, poly(methy1 acrylate), and poly(ethy1 methacrylate)) using all of the available diffusivity data for these materials. These linear plots are presented in Figqes 7-10. In these figures, the closed circles represent yV2*f/K12 value? determined from D vs. T data, and the open circles are yV2*.$/K12values computed from eq 4 using an average activation energy over a temperature interval. The former values are clearly more reliable. The triangles in these figures represent solvents which are not expected to move as single units. The solvent molar volumes at 0 K used in Figures 7-10 were calculated by averaging values determined using the two techniques summarized in Table
334
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Ind. Eng. Chem. Prod. Res. Dev., Vol. 20, No. 2, 1981
Table IV. Sources of Data Used in Figure 8 for Polystyrene
I 800
solvent
1600
14001 i
4
Y
2004 -
0
0
2
40
80
120
160
200
240
SOLVENT MOLAR VOLUME AT 0% (cm3/g-mole)
Figure 8. Variation of y 3 2 * t / K 1 2with solvent molar volume at 0 K for polystyrene. Symbols are the same as in Figure 7. Solvents are identified in Table IV. Table 111. Sources of Data Used in Figure 7 for PVAc solvent reference 1. water 2. methanol 3. ethanol 4. methylene chloride 5. ethyl chloride 6. acetone 7, ethyl bromide 8. n-propyl alcohol 9. ethyl iodide 10. chloroform 11. pyridine 12. benzene 13. n-butyl alcohol 14. sec-butyl alcohol 15. tert-butyl alcohol 16. carbon tetrachloride 17. cyclohexane 18. methanol 19. acetone
20. 21. 22. 23. 24. 25. 26.
27. 28. 29.
toluene PNAa PAAB~ Y-7' hydrogen neon oxygen argon krypton methane
Ryskin (1955)
reference
1. hydrogen 2. nitrogen 3. ethylene 4. carbon dioxide 5. methane 6. methanol 7. ethanol 8. methylene chloride 9. ethyl bromide 10. pyridine 11. chloroform 12. n-propyl chloride 13. benzene 14. fluorobenzene 15. methanol 16. n-pentane 17. toluene 18. ethylbenzene 19. PNA 20. PAAB 21. Y-7 22. triisopropylbenzene
Newitt and Weale (1948)
Lundberg et al. (1963) Zhurkov and Ryskin (1954)
Liu (1980) Duda and Vrentas (1968) Vrentas and Duda (1977) Masuko et al. (1978b) Vrentas et al. (1980b)
Table V. Sources of Data Used in Figure 9 for Polymethyl Acrylate solvent 1. 2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14.
neon argon krypton water water methanol ethanol benzene methanol methyl acetate ethyl acetate n-propyl acetate n-butyl acetate benzene
reference Burgess et al. (1971) Kishimoto et al. (1960) Ryskin (1955)
Fujita (1968)
Table VI. Sources of Data Used in Figure 10 for Polyethyl Methacrylate solvent reference
Fujita (1968) Kokes et al. (1952); Fujita and Kishimoto (1958) this study Masuko et al. (1978a) Meares (1954)
Meares (1957)
PAAB is p-aminoazobenzene. a PNA is p-nitroaniline. Y-7 is phenol, 2-methvl-4-[[4-(phenylazo)phenyl]azo].
11. Sources for the diffusivity data sets used to construct Figures 7-10 are reported in Tables 111-VI. It is evident from Figures 7-10 that reasonably good straight lines are generally formed from the data when the results for solvents which are expected to move in a segment-wise manner (the triangles) are not considerc_ed.The biggest deviations are exhibited by the values of y V2*[/ K12 for PVAc which are determined from activation energy data (the open circles in Figure 7). The data in Figures
1. 2. 3. 4. 5.
water methanol ethanol pyridine benzene
Ryskin (1955)
Table VII. Slope of Least-Squares Line for Figures 7-10 [ Y ? ~ * E / K , =, B(solvent m o h volume at o K)] polymer
B, K g-mol/cm3
poly(viny1 acetate) polystyrene poly(methy1 acrylate) poly(ethy1 methacrylate)
17.2 10.5 19.5 20.9
7-10 were used to construct leasbsquares lines through the origin for each of the polymers. The data for solvents with jumping units of undetermined size were excluded as were th_e open circle data in Figure 7. For PVAc, values of y V2*,$/K,, determined from average activation energies do- not appear to be in accord with the more reliable yV2*[/KI2 values determined from actual D vs. T data. The equations of the straight-line fits for the four polymers are presented i~Table VII. These equations can be used to determine yV2*[/K12for any other solvent diffusing in any of these polymers as long as the size of the solvent jumping unit is known.
Ind. Eng. Chem. Prod. Res. Dev., Vol. 20, No. 2, 1981 335
14001
I i
-
ii ‘
1
’
1
’
1
’
0 20 40 60 SO 100 120 SOLVENT MOLAR VOLUME AT 0 % (cma/g-moie)
Figure 9. Variation of y?z*t/Klz with solvent molar volume a t 0
When no diffusivity data are available for the polymer-solvent system of interest (case 111),it is not possible to determine a D vs. T relationship for this system using only the diffusivity information for other polymer-penetrant pairs contained in plots of the type presented in Figures 7-10. However, a method has been proposed elsewhere (Vrentas et al., 1980a) for using solvent density and viscosity data in conjunction with diffusivity data for other solvents as a means of establishing the temperature dependence of the diffusivity for the system of interest. Here, we show that only Figures 7-10 need be used to deduce more limited information about the temperature dependence of a diffusion process involving the polymerpenetrant pair under consideration. For example, consider the removal of a small amount of solvent from a film of molten polymer. Let tz be the time required for the average solvent concentration to reach a prescribed level at temperature Tz and tl the corresponding time at temperature Tl. If the concentration dependence of the diffusivity at each temperature is negligible, it can be shown that
K for poly(methy1 acrylate).
Symbols are the same as in Figure 7. Solventa are identified in Table V.
m y , , ,I 200
0
0
,
2
0
4
0
6
0
8
0
SOLVENT MOLAR W M E AT 0 %(cr~?&mde) F i g u r e 10. Variation of ypZ*[/Kl2with solvent molar volume at 0 K for poly(ethy1 methacrylate). Symbols are the same as in Figure 7. Solventa are identified in Table VI.
In general, the scatter of the results in Figures 7-10 about the straight lines is reasonable when consideration is taken of the difficulty in obtaining D vs. T data in the limit of zero solvent mass fraction. Not only is it difficult to conduct polymer-penetrant diffusion experiments at elevated temperatures, but the estimation of D in the limit of infinite solvent dilution is not easy because of the strong concentration dependence of the diffusivity in polymeric systems. Consequently, it is reasonable to expect that thsse straight-line fits will give a good estimate of 7V2*(/K12 for any other solvents in these polymers since they have been determined using diffusivity data for at least five solvents. The value of yVz*[/K12 computed from the appropriate equation in Table VI1 and the single value of diffusivity for the polymer-solvent system of interest can be used to determine Dol for this system from eq 3.
Consequently, the effect of temperature on the devolatilization time can be computed from this equation using no diffusivity data for the polymer-solvent system of interest. Only a plot of the type presented in Figures 7-10 is needed. Hence, such an approach can be used to yield valuable information about the temperature dependence of mass transfer rates without having to carry out diffusivity measurements. Literature Cited Bib, W. “Rauchemie der Festen Stoffe”, Voss: Leipzlg, 1934. Burgess, W. H.; Hopfenberg, H. B.; Stannett, V. T. J. Macromol. Sci. Phys. 197lV8 5 , 23. Duda, J. L.; Vrentas, J. S. J. Polym. Sci. A - 2 . 1988, 6, 875. Ferry, J. D. “Viscoelastic Properties of Polymers”, 2nd ed.;Wliey: New York, 1970; Chapter 11. Fujlta, H.; Kishimoto, A. J. folym. Sci. 1958, 28, 547. Fujlta, H. Forfschr. Hcchpolym. Forsch. 1981, 3 , 1. Fujlta, H. in “Diffusion in Polymers”, J. Crank and G. S. Park, Ed.; Academic: New York, 1988; Chapter 3. Haward, R. N. J. Macromd. Sci. Rev. Macromol. Chem. 1970, C4, 191. Kishimoto, A.; Maekawa, E.; Fujlta, H. h i / . Chem. Soc. Jpn. 1980, 33, 988. Kokes, R. J.; Long, F. A.; Hoard, J. L. J . Chem. Phys. 1952, 30, 1711. Liu, H. T., Ph.D. Thesis, The Pennsylvania State Unhwstty, University Park, Pa., 1980. Lundberg, J. L.; Wilk, M. B.; Huyett, M. J. Id.Eng. Chem. Fundam. 1963, 2 , 37. Masuko, T.; Homma, Y.; Karasawa, M. Sen4 Gakkaishi(Apan) 1978a, 34, T-137. Masuko, T.; Sato, M.; Karasawa, M. J. Appl. Polym. Sci. 1@78b,22, 1431. Meares, P. J. Am. Chem. Soc. 1954, 76, 3415. Meares, P. Trans. Faraday SOC. 1957, 53, 101. Newltt, D. M.; Weale, K. E. J. Chem. SOC. 1948, 1541. Ryskin, G. Ya. J. Tech. Phys. (USSR) 1955, 25, 458. Sugden, S. J. Chem. SIX. 1927, 1788. Vrentas, J. S.; Duda, J. L. J. Appi. Polym. Sci. 1977, 21, 1715. Vrentas, J. S.; Duda, J. L. J. Appl. pdym. Sci. 1978, 22, 2325. Vrentas, J. S.; Duda, J. L. AIChE J. 1979a, 25, 1. Vrentas, J. S.; Duda, J. L. J. Polym. Sci. Polym. Phys. Ed. 1979b, 17, 1085. Vrentas, J. S.; Liu, H. T.; Duda, J. L. J. Appl. Polym. Scl. 1980a, 25, 1297. Vrentas, J. S.; Llu, H. T.; Duda, J. L. J. Appl. Polym. Sci. 1980b, 25, 1793. Zhurkov, S. N.; Ryskin, G. Ya. J. Tech. Phys. (USSR) 1854. 24, 797.
Received for review August 29, 1980 Accepted November 12,1980 This
work was supported by t h e N a t i o n a l Science Foundation,
Grant ENG 78-26275.
