Ind. Eng. Chem. Res. 1994,33, 2483-2491
2483
Diffusion and Sorption in Ethylene-Propylene Copolymers: Comparison of Experimental Methods Niloufar Faridi, Ilyess Hadj-Romdhane,+ Ronald P.Danner, and J.
L. Duda’
Center for the Study of Polymer-Solvent Systems, Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
T h e solubility and diffusion coefficients of various solvents in ethylene-propylene copolymers with different ethylene contents were measured over a temperature range of 30-200 O C using capillary column inverse gas chromatography and gravimetric sorption. The value of the Flory-Huggins parameter decreased with an increase in temperature. T h e infinitely dilute diffusion coefficients obtained by the two methods were consistent with each other. T h e data indicate that the diffusion coefficients and solubility parameters are independent of the ethylene content of the copolymers. The results obtained from the sorption technique were extrapolated using a free-volume theory to the limit of zero solvent concentration, and the free-volume model was used t o interpret the diffusion data. Analysis of the diffusion data indicated that the inclusion of an activation energy in the free-volume model does not significantly improve the correlation of the diffusion data. Introduction
Theory
The removal of trace amounts of residual solvent from polymers is an important phenomenon in the production and processing of many organic polymers. Design considerations are primarily influenced by the magnitude of the polymer-solvent mutual diffusion coefficient since devolatization of the volatile component is the rate limiting step in polymer purifications. Different techniques are available to measure the diffusion coefficients in polymer-solvent systems. The most conventional method relies on gravimetric vapor sorptionldesorption experiments which can be used for measuring diffusion coefficients at finite concentrations of solvent (Crank and Park, 1968). This method, however, becomes very difficult to apply when the solvent is present in very small amounts, as is the case for devolatization. An alternative method for studying thermodynamic and transport properties in polymeric systems at infinite dilution of solvent is inverse gas chromatography. The principle behind this technique is based on the distribution of a volatile solute between a mobile gas phase and a stationary polymeric phase. Most of the previous work, Tait and Abushihada, 1979;Hu et al., 1987,was done using packed chromatographic columns. A model for capillary chromatographic columns has been developed by Pawlisch et al. (1987,1988) that gives a definitive understanding of the transport properties in polymer-solvent systems. In this work, two different experimental methods, gravimetric sorption and capillary column inverse gas chromatography (CCIGC), were used to measure the solubility parameters and diffusion coefficients for several solvents in ethylene-propylene (EP) copolymers with different ethylene contents. These methods are complimentary to each other, and the results obtained by the sorption method at finite concentration of solvent were compared to those obtained by the inverse gas chromatographic technique a t infinite dilution. The comparison of diffusivities was based on the correlation of the diffusion data with the Vrentas-Duda (1977a,b)free-volumetheory.
Review of the Capillary Column Inverse Gas Chromatography Model. Utilizing the continuity equations for the solvent in the gas and polymer phases and the appropriate initial and boundary conditions, Pawlisch et al. (1987) developed the following expression for the concentration profile at the exit of the column in the Laplace domain:
* Author to whom correspondence should be addressed. E-mail: JLD6@PSUADMIN (BITNET). Current address: 3M Center, Building 518-1-01, St. Paul, MN 55144-1000. f
0888-5885/94/2633-2483$04.5Q/0
exp[
&]ex.[ -(4 4Y +
1/2
+
tanh(p6))
] (1)
where
Here, C is the area-averaged solute concentration in the gas phase, COis the strength of the inlet impulse, L is the length of the capillary column, u is the mean velocity of the carrier gas, K is the equilibrium partition coefficient, df is the polymer film thickness, r is the radius of the gas-polymer interface, D, is the solute diffusion coefficient in the gas phase, and D is the solute diffusion coefficient in the polymer phase. Making use of the well-known moment generating property of Laplace transforms, the resulting pair of moment equations is: (3)
where p1 is the first temporal moment or mean residence time and p2* is the second central moment or variance of the concentration distribution. Evaluation of Model Parameters. Evaluation of the model parameters in the CCIGC model could be done either through a moment-fitting procedure or a timedomain fittingmethod. The moment-fitting procedure consists of first computing the first and second central moments by numerical integration of several peaks obtained at different carrier gas velocities. From the first 0 1994 American Chemical Society
2484
Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994
moment values, the partition coefficient, K , can be determined through a linear regression of p1 versus Llv as suggested by eq 3. The value of K is then used in the variance equation (eq 4) to estimate the diffusion coefficient in the polymer phase, D. The time-domain fitting procedure, on the other hand, is more straightforward and is more reliable than the moment-fitting procedure. Starting with initial estimates for K and D (usually obtained from moment analysis), the transform expression (eq 1)is numerically inverted using a fast Fourier transform (FFT)algorithm. The resulting theoretical profile is then compared against the experimental profile, and the residual is minimized by using a nonlinear least squares regression technique to regress for the values of K and D that best characterize the experimental elution curves at different carrier gas velocities. Thermodynamic Interaction Parameters. A. Weight-Fraction Activity Coefficient. At infinite dilution of the solute, the relation between chromatographic data and the solute activity coefficient is (Hadj Romdhane and Danner, 1991)
Here the physical properties of the solvent: Pp, M I ,B I I , and V Iare the vapor pressure, the molecular weight, the second virial coefficient, and the molar volume, respectively. The values of these properties were obtained from Daubert and Danner (1993). In eq 5, R is the ideal gas constant, T is the column temperature, and V , is the specific retention volume which is related to the partition coefficient. The standard state in eq 5 is pure liquid solvent at the system temperature and zero pressure. The exponential term in this equation corrects for the gas-phase nonideality of the solvent. B. Flory-Huggins x Parameter. When eq 5 is combined with the Flory-Huggins equation (Flory, 19531, the x parameter is obtained:
x = In Q l m - 1 + In p1/p2
(6)
where p 1 is the solvent density. C. Estimation of the Polymer Solubility Parameter. The combination of the Hildebrand-Scatchard regular solution theory (Hildebrand and Scott, 1950)with the Flory theory (Flory, 1953) yields
where 62- is the polymer solubility parameter at infinite dilution of the solvent and 6 1 is the solubility parameter of the solute given by
[A",V-IRT 1
112
61=
(8)
where A", is the solute heat of vaporization. Since x,V I , and 61 are known, the solubility parameter of the polymer, 6 2 - , can then be determined by linear regression of the left hand side of eq 7 versus 6 1 . The Vrentas-Duda Free-Volume Theory. The freevolume diffusion model developed by Vrentas and Duda (1977a) is based on the concept that a wlecule diffuses in a liquid when a hole of sufficient size appears adjacent to the molecule by thermal fluctuations. Furthermore, by assuming that (1)Bearman's (1961) equation correctly
represents the relationship between the individual selfdiffusion coefficient and the binary mutual-diffusion coefficient, (2) the polymer self-diffusion coefficient contributes negligibly to the binary mutual-diffusion coefficient, and (3) the Flory-Huggins model describes the polymer-solvent thermodynamic behavior, they derived the following transport models, (1 = solvent; 2 = polymer): D, = Do, x
where D1 and D are self- and mutual-diffusion Coefficients, respectively. In the above equations, vi*,oi, $it and Tpiare the specific critical hole free volume required for a jump, the mass fraction, the volume fraction, and the glass transition temperature of component i, respectively. Dol is a constant pre-exponential factor, and y is an overlap factor which is introduced because the same free volume is available to more than one molecule. K11 and K21 are the free-volume parameters for the solvent and K12 and K22 are the freevolume parameters for the polymer. x is the FloryHuggins interaction parameter. Finally, 5 is the ratio of critical molar volumes of solvent and polymer jumping units. Equation 10 contains the assumption that the mutual diffusion coefficient is related to the solvent self-diffusion coefficient and to the thermodynamic properties of the polymer-solvent systems using an expression developed by Bearman (1961). For polymer-solvent systems it has been suggested that the range of validity of this equation is limited. Consequently, a new model was proposed by Vrentas and Vrentas (1993b) which extends the validity of the previous result to higher solvent mass fractions. However, eq 10 provides a useful approximation for studying the mutual-diffusion coefficients in polymeric systems at low solvent concentrations. At the limit of zero solvent mass fraction, eq 9 becomes
D
= D,= Dol exp
To describe the temperature and concentration dependence of the mutual-diffusion coefficient as defined by eqs 9 and 10, several parameters need to be evaluated. VI* and v2* are estimated as the specific volumes at absolute zero temperature and can be evaluated by using group contribution methods discussed by Sugden (1927). K d y and K21- Tgl can be determined from the viscosity data of the pure solvent. Furthermore, K 1 2 / ~and K Z Z - Tg2can be computed from Williams, Landel, and Ferry (WLF) constants which are available for a large number of polymers (Ferry, 1970) and x is estimated by correlating the solubility data with the Flory-Huggins equation. By using the measured diffusion data the remaining parameters Dol and can be determined from the nonlinear regression. Other studies have focused on estimating all the model parameters without using any diffusivity data. (Zielinski and Duda, 1992)
Experimental Procedures Capillary Column Inverse Gas Chromatography. The chromatograph used in this work was a Varian Model
3400 equipped with a flame ionization detector (FID), an on-column injector, and a circulating air oven. Helium was used as the carrier gas in all experiments. The temperature of the injection block and the detector assembly were set about 50 "C above the column temperature to avoid condensation in the detector assembly. Details of the experimental technique are presented by Hadj Romdhane (1994). The capillary columns used in this study were prepared by Supelco, Inc. (Bellefonte, PA). Both columns had an inside diameter of 530 pm, and were 15 m long. One column was coated with E P rubber (EPR 68 wt% ethylene), and the other column was coated with ethylene-propylene terpolymer (EPDM 56 wt % ethylene and about 9.3wt% ENB, Le., 5-ethylidene-2-norbornene). The film thickness was 7 pm in each of these two columns. Gravimetric Sorption Apparatus. Diffusion experiments were carried out using a quartz spring apparatus. The basic principle of the equipment used here is similar to the one described by Duda et al. (1973). The polymer samples in aluminum buckets were suspended from quartz springs and placed inside the sorption column. The penetrant vapor was fed to the column from a solvent flask which was maintained at a set temperature, and the temperature of the column was controlled through the use of a condensing vapor. During each sorption run, the spring length extension was monitored by a micrometer slide cathetometer. The E P rubber used in the gravimetric sorption study was a copolymer having a composition of 43 wt % ethylene. The weight average molecular weight was approximately 150 000 g/mol with a 2.1 polydispersity index. Since the diffusion coefficients were expected to be high at elevated temperatures, relatively thick samples (approximately 2.8 mm) were prepared so that equilibrium would not be attained too quickly. The solvents used in this study were n-hexane, nheptane, cyclohexane, methylcyclopentane, benzene, acetone, dichloromethane, and chloroform. All solvents were reagent grade materials supplied by Thomas Scientific Co. and were used without further purification. Results and Discussion Inverse Gas ChromatographyResults. The CCIGC model was used to analyze the chromatographic data obtained for the various EPR (68% ethylenel-solvent and EPDM (56% ethylene)-solvent systems at 30 and 100 "C. To examine the temperature dependence more thoroughly, additional diffusion measurements for EPR (68% ethylene) were also made at 70 "C. The model parameters were estimated utilizing the time-domain fitting procedure outlined earlier, using the moment analysis results as initial estimates. Inversion of the Laplace transform solution was performed numerically with a fast Fourier transform algorithm (subroutine FFT2C from the International Mathematical and Statistical Library of Subprograms, Inc., Houston, TX). A commercialnonlinear regression package (subroutine LMDIFl from the MINPACK-1 software product of the Applied Mathematics Division of Argonne National Laboratory, Argonne, IL) was employed to minimize the objective function. Figure 1 shows the quality of the fit between experimental and theoretical elution profiles for the cyclohexane-EPR (68% ethylene) system. In this diagram, the points represent experimental data, while the solid line is the theoretical elution curve obtained with the regressedvalues of K and D. The symbol t, on the x-axis refers to the residence time of the carrier gas; in other words, Llv. This figure is representative of the results obtained in all cases. The excellent agreement
0.50
,
Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 2485 I
0.40
d 0.20 0.10
0.00
15
18
21
24
27
30
tlt,
Figure 1. Comparison of experimental and theoretical elution profiles for the cyclohexane-EPR (68% ethylene) system at 30 "C. Table 1. Summary of the Gas Chromatography Results at 30 "C EPR (68% ethylene) EPDM (56% ethylene) solvent K (x) D (cm2/s) K (x) D (cm2/s) n-hexane 192 (0.391) 1.24X 183 (0.437) 6.27 X 10-8 n-heptane 577 (0.342) 1.05 X lP7 556 (0.381) 5.54 X 10-8 cyclohexane 399 (0.267) 5.38 X 1o-B 406 (0.248) 2.49 X 10-6 benzene 324 (0.704) 1.41 X 341 (0.652) 7.30 X 10-8 chloroform 163 (0.789) 1.29 X 181 (0.685) 6.80 X 10-8 dichloromethane 2.59 X le7 2.10 x 10-7 acetone Table 2. Summary of the Gas Chromatography Results at 100 "C EPR (68% ethylene) solvent K (x) D (cm2/s) n-hexane 25.9 (0.30) 2.16 X 10-6 n-heptane 49.9 (0.34) 2.02 X 10-6 cyclohexane 50.6 (0.13) 1.29 X 10-6 benzene 42.7 (0.46) 2.32 X 1o-B chloroform 27.5 (0.48) 2.26 X 10-6 dichloromethane 3.65 X 10-6 acetone 2.89 X 10-6
EPDM (56% ethylene) K (x) D (cm2/s) 23.4 (0.40) 1.88 X 10-6 52.1 (0.29) 1.78 X 10-6 50.1 (0.14) 1.07 X 10-6 45.0 (0.41) 2.37 X 10-6 27.6 (0.47) 1.92 X 10-6
Table 3. Diffusion Data for Solvent-EPR (68% Ethylene) at 70 "C Obtained by the Gas Chromatography Method solvent n-hexane n-heptane cyclohexane benzene chloroform dichloromethane acetone
D (cm2/s) 8.67 X 8.30 X 4.84 x 10-7 9.95 x 10-7 8.92 X lk7 1.48 X 1o-B 1.31 X 10-6
is evidence that the CCIGC model accurately describes the chromatographic process. The estimates of K and D obtained from the regression algorithm for the different EPR-solvent and EPDM-solvent systems at 30 and 100 "C are compiled in Tables 1 and 2. The diffusion coefficient data for EPR (68% ethylene) at 70 "C are tabulated in Table 3. The Flory-Huggins x parameters, determined as specified above, are also included in these tables. Gravimetric Sorption Results. The gravimetric sorption experiments for an ethylene-propylene copolymer (with 43 wt% ethylene) with three organic solvents, i.e., cyclohexane, n-hexane, and methylcyclopentane, were conducted over a wide range of concentrations at temperatures ranging between 50 and 200 "C. In analyzing a set of sorption data to determine the mutual-diffusion coefficient, an analytical solution to the diffusion equation
2486
Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994
Table 4. Diffusion Data Obtained by the Gravimetric Sorption Method for the Cyclohexane-EPR (43%Ethylene) System T(OC) w1 D (cm2/s) T(OC) w1 D (cm2/s) 50 0.079 2.70 X le7 108 0.028 2.68 X 10-6 50 0.189 8.47 X le7 153 0.022 4.08 X 10-6 1.32 X 10-8 153 0.047 5.56 X 10-6 50 0.371 80 0.045 1.12 X 10-8 153 0.066 7.79 X 10-6 1.16 X 10-8 80 0.063 180 0.021 1.07 X 106 2.41 X 10-6 180 0.035 1.38 X 106 80 0.135 80 108
0.221 0.027
3.11 X 10-6 2.52 X 10-6
200 200
0.012 0.021
1.43 X 10-5 1.45 X 106
Table 5. Diffusion Data Obtained by the Gravimetric Sorption Method for Methylcyclopentane-EPR (43% Ethylene)
1.0
I
0.6
-
0
0.2
0
X
0
ScrPtlon Method
0
(43% ethylene)
-0.2
0 Frensdorff
( 1964a)
(39% ethylene)
80 80 80 108 108 108 153
0.029 0.057 0.112 0.014 0.033 0.053 0.012
1.23 X 1.47 X 2.47 X 1.80 X 1.95 X 3.48 X 6.68 X
10-8 10-6 10-8 10-8 10-8 10-8 10-8
153 153 180 180 180 200
0.019 0.042 0.009 0.018 0.027 0.013
7.29 X 8.22 X 9.74 X 1.04 X 1.09 X 1.43 X
10-6 10-6 10-6 106 106 106
assuming a constant diffusion coefficient is used as an approximation. The solution of the diffusion equation is given by Crank (1957). For short times, the solutions can be approximated as
A
N e m n et al. (1973)
-0.6
0
(40% ethylene)
0
IGC Method (56% and 68% ethylene)
‘
I
-1.0 200
320
260
380
440
500
T (g.0 Figure 2. Dependence of the Flory-Huggins parameter on temperature for n-hexane-EPR systems. 1.0 I
where M ( t ) and M ( m ) are defined as weight pickup at time t and infinity and L‘ is the thickness of the polymer sample. By constructing a sorption curve, (M(t)/M(m)versus 4 1 , the mutual diffusion coefficient can be calculated from the initial slope, Ri,and the final equilibrium state of the curve using the following relation:
0.2
1
R 0
The diffusion coefficients for two different solvent-EPR (43 wt% ethylene) systems are shown in Tables 4 and 5. The solubility data at each temperature obtained from the gravimetric sorption technique were correlated with the Flory-Huggins equation. The Flory-Huggins x parameters obtained for EPR (43% ethylene) with n-hexane and cyclohexane are shown in Figures 2 and 3 as a function of temperature. Analysis of the Thermodynamic Data. Figures 2 and 3 show the change of the x parameter withtemperature for n-hexane and cyclohexane in E P copolymers with varying ethylene contents, which were obtained by this study, and compare them with the values obtained by Frensdorff (1964a) and Newman and Prausnitz, (1973). The data for both EPR-n-hexane and EPR-cyclohexane show that the x parameter decreases as the temperature increases. The temperature dependency of the x parameter has been noticed for many polymer-solvent systems. Different correlative models have been proposed for the temperature and concentration dependence of the FloryHuggins parameter (Gundert and Wolf, 1989; Qian et al., 1991;Koningsveld, 1975;Kamide, 1990). All the proposed models suggest that raising the temperature enhances the interaction forces between the polymer and solvent and therefore causes a better polymer miscibility. Moreover, there seems to be only a slight effect of copolymer composition on the equilibrium results as evidenced by
8’
0 Saption Method (43% ethylene)
-0.2
OFrensdorff (1964a) (39% ethylene)
A -0.6
‘
0
N e m n et 81. (1973) (40% ethylene)
0
IGC Method
(56% and 68% ethylene)
-1.0
200
260
320
380
440
500
T (g.0 Figure 3. Dependence of the Flory-Huggins parameter on temperature for cyclohexane-EPR systems.
comparison between the x values obtained for EPR and EPDM. Frensdorff (1964a)suggested that this negligible effect could be attributed to the chemical similarity of the ethylene and propylene units in the E P copolymer. When compared to other literature data, the interaction data obtained for the various EP-solvent systems in this study were found to be in reasonable agreement with those reported by Newman and Prausnitz (1973) who used the IGC technique with packed columns and slightly higher than those obtained by Frensdorff (1964a) who extrapolated his finite concentration (vapor sorption technique) results to infinite dilution.
Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 2487 0.15 0.13 -
0.13
2 0
0.11 -
5
\
0.11
E
