Ind. Eng. Chem. Res. 2004, 43, 1537-1542
1537
New Pressure-Decay Techniques to Study Gas Sorption and Diffusion in Polymers at Elevated Pressures Peter K. Davis, Gregory D. Lundy, John E. Palamara, J. Larry Duda, and Ronald P. Danner* Center for the Study of Polymer Solvent Systems, Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
Two new volumetric sorption techniques have been developed to measure thermodynamic and mass-transport properties in polymer-solvent systems at elevated pressures. They are both variations of the pressure-decay technique and differ in how the initial gas density is measured. In the first and simplest variation, the initial density is measured by extrapolation of the mass uptake curve from the region of usable data to the experiment starting time. In the second variation, the initial density is measured gravimetrically using a small titanium capsule. Concern has been voiced as to whether the extrapolation-type experiment is capable of measuring the initial gas density as accurately as a dual-chamber technique. To evaluate the accuracy of the extrapolation technique, solubility and diffusion data were collected for carbon dioxide, ethylene, and nitrogen in low-density polyethylene (LDPE) using both methods. The results indicate that the simpler extrapolation method produces the same diffusivity and solubility results as the more complicated dual-chamber method. Introduction For the past half century, numerous investigators have measured thermodynamic and mass-transport properties of polymer-solvent systems at elevated pressures. The interest in such data arises because many industrial polymer processes occur at high pressures. One example is high-pressure polymerization reactors. In the limit of high conversion, reaction kinetics are influenced by mass-transfer limitations, so reactor design and optimization require thermodynamic and diffusion data on the polymer-solvent system. Another example is high-pressure membrane separation. This process requires similar data to determine the membrane selectivity and the fluxes of solvents through the membrane. Polymeric foam extrusion is another highpressure process, in which a blowing agent is mixed with molten polymer in an extruder. The foam density made in such a process depends on the solubility of the blowing agent, and the amount of mixing needed depends on the diffusivity. In this work, two new sorption techniques have been developed to measure diffusion and solubility in polymer-solvent systems. They are both variations of the pressure-decay technique, the original and most popular method for making high-pressure sorption measurements. The pressure-decay technique was first developed by Newitt and Weale1 in 1948, who used it to study the solubilities of gases in polystyrene. This technique was later extended to make diffusion measurements by Lundberg et al.2 in 1963. Pressure decay is very simple in concept. An experiment is conducted by isolating a polymer sample and a high-pressure gas in a closed vessel. As the polymer absorbs the gas, the pressure drop in the vessel is monitored as a function of time. * To whom correspondence should be addressed. Address: Dr. Ronald P. Danner, 163 Fenske Laboratory, The Pennsylvania State University, University Park, PA 16802. Tel.: 814863-4814. Fax: 814-865-7846. E-mail:
[email protected].
An equation of state for the gas is used to convert the pressure into the mass uptake in the polymer. The solubility is obtained from the total amount of gas absorbed, and the diffusivity is determined from the time dependence of the sorption. During the 1970s and 1980s, the technique was used extensively to study the sorption of gases into glassy polymers in an attempt to validate the dual-mode sorption theory.3-5 Koros and Paul6 developed the standard dual-chamber method that greatly improved the reliability and accuracy of the technique. This advance was important in that the original pressure-decay technique used by Lundberg et al.2 had a problem in obtaining the gas density from pressure data at the beginning of an experiment. When the gas was initially introduced into the chamber, the temperature rose significantly as a result of Joule heating. Because the initial temperature was unknown, accurate gas density data could not be obtained until this heat was conducted away. In Koros and Paul’s technique,6 the gas was expanded into the sample chamber from a second vessel of known pressure, temperature, and volume, thereby allowing the initial expanded gas density to be known prior to the start of the experiment. This has been the standard method of pressure-decay experiments over the past 25 years. The dual-chamber method has been employed by Shah et al.,7 Rein et al.,8 Bondar and Freeman,9 and Hilic et al.10 to measure gas sorption in polymers. A group at the University of Hiroshima, Japan, have used a modified dual-chamber approach in which the pressure transducer on the polymer sample chamber is differential.11 This type of transducer reads the differential change in pressure between a reference cell and the sample cell. With this method, they are able to accurately measure the initial gas density (via the Koros and Paul method6) and take advantage of the high accuracy of a differential pressure transducer. Although the dual-chamber method allows for explicit measurement of the initial gas density, many more seals
10.1021/ie034075y CCC: $27.50 © 2004 American Chemical Society Published on Web 02/14/2004
1538 Ind. Eng. Chem. Res., Vol. 43, No. 6, 2004
volume from this information.
(
Figure 1. Diagram of the dual-chamber variation of the pressuredecay experiment. Arrow indicates capsule connection point.
are required for the equipment, and there is a greater possibility for leaks. Thus, it would be advantageous to develop a pressure-decay technique that does not rely on the initial gas density measurement. In this work, two new variations of the pressure-decay technique have been developed. These two variations differ in how the initial expanded gas density is measured. The first determines the initial gas density gravimetrically. This is similar to the Koros and Paul technique,6 in which the initial gas density is measured volumetrically. The second variation obtains the initial gas density from extrapolation of the mass uptake curve. It is questionable whether an extrapolation-type experiment is capable of obtaining the diffusion coefficient and solubility as accurately as an experiment in which the initial gas density is measured directly. To determine whether the extrapolation technique is a justified procedure for conducting pressure-decay experiments, solubility and diffusion data have been collected for carbon dioxide, ethylene, and nitrogen in low-density polyethylene over a wide range of pressures using both techniques. Experimental Section Dual-Chamber Technique. A diagram of the experimental setup for the dual-chamber technique is shown in Figure 1. The sample chamber was constructed of stainless steel, and the entire apparatus was placed in a temperature-controlled forced-convection air bath. Two “arms” protruded about 1 in. from the outside of the sample chamber wall. The first arm connected to a pressure transducer and to the capsule via valve A. The second arm connected to a vacuum pump via valve B. The titanium capsule was a small cylinder about 10 cm long and 2 cm in diameter with an internal volume of about 12 mL. The combined weight of the capsule and valve A was less than 210 g. This allowed the capsule unit to be weighed on a high-precision (0.00001 g) Sartorius analytical balance. The valves were rated to 2065 psia at 232°C. Pressure measurements were acquired automatically by a data acquisition system. The headspace volume is required to convert gas density data into mass uptake data. This volume was measured by expansion of nitrogen from the titanium capsule into the sample chamber. The mass of nitrogen was determined by weighing the capsule. The capsule weights gave the mass (number of moles) of nitrogen, and the temperature and pressure were both measured. The eight-parameter Benedict-Webb-Rubin (BWR) equation of state12 was used to measure the headspace
C0
)
F2 + (bRT - a)F3 + T2 cF3 aRF6 + 2[(1 + γF2) exp(-γF2)] (1) T
P ) RTF + B0RT - A0 -
Here, P is the pressure, T is the temperature, and F is the molar density. The parameters A0, B0, C0, a, b, c, R, and γ are empirical constants for a given gas. The BWR constants for nitrogen are given in Table 1 below. By this procedure, the combined volume of the capsule and sample was determined to be 72.1 ( 0.3 mL. The true vapor volume in an experiment was calculated by subtracting the polymer sample’s volume from this measured volume. An experiment was conducted by filling the titanium capsule with the desired pressure of the gas of interest. The capsule was then weighed to determine the mass of gas that would be used in the experiment. Next, the gas-filled capsule was connected to the sample chamber using a standard 1/8-in. Swagelok fitting, and the apparatus was placed inside the oven at the temperature of the experiment. Valve B was opened, and a vacuum was applied for 24 h to desorb the sample (usually ∼10 g) of any impurities. After the sample had been sufficiently desorbed, valve B was closed, and the data acquisition system was started. The experiment began with the opening of valve A, allowing the contents of the capsule to fill the headspace of the apparatus. At this point, the experiment was running as the polymer was isolated with the solvent in a fixed volume. Pressure data were collected as a function of time until equilibrium was reached. At that point, the data acquisition was stopped, and the experiment was complete. Expanding the capsule into the apparatus at the beginning of an experiment resulted in Joule heating. Until this heat had been conducted away (usually 2-5 min), the pressure data could not be related to the gas density because the temperature was unknown. For this reason, the initial gas density (expanded gas density before any absorption occurs) could not be measured from the acquired pressure data. This initial density is of extreme importance to the experiment because calculation of both the solubility and diffusivity requires it. However, because the total amount of gas in the system was known from the weight of the capsule, the initial gas density was already known (mass of gas/ headspace volume). Extrapolation Technique. For the extrapolation technique, the capsule was removed from the apparatus. The extrapolation technique did not include any kind of dosing volume. A single valve connected the sample chamber to the vacuum pump and gas source. The headspace volume measurement was still made with the titanium capsule as described above and was determined to be 52.1 ( 0.3 mL. Experiments were conducted by first desorbing the sample for 24 h using the vacuum pump. The valve was then closed, and the line to the valve was pressurized with the gas of interest. Next, the data acquisition program was started. The experiment began with the rapid opening and closing of the valve, allowing the gas to fill the chamber and then become isolated from the source. Pressure data were collected as a function of time until equilibrium was reached. Multiple experiments were conducted in series by increasing the dosing pressure in steps and waiting
Ind. Eng. Chem. Res., Vol. 43, No. 6, 2004 1539 Table 1. BWR Parameters for Carbon Dioxide, Nitrogen, and Ethylenea,b BWR parameter
carbon dioxide
nitrogen
ethylene
A0 B0 C0 a b c R γ
2.52 0.0449 147000 0.137 0.004 12 14900 0.0847 × 10-3 0.525 × 10-2
1.48 0.0505 2680 0.0147 0.00307 429 0.114 × 10-3 0.588 × 10-2
3.34 0.0557 131000 0.259 0.0086 21100 0.178 × 10-3 0.923 × 10-2
a
Parameters taken from Bett.12
b
In this equation, C is the mass density of gas in the vapor phase, Ac is the cross-sectional area of the polymer sample, V0 is the empty chamber volume, and L is the total thickness of the polymer sample. The boundary conditions for these equations are
F1 ) KC0 C ) Ci
Units are L, atm, mol, and
K.
for the new equilibrium each time. Because no dosing volume was used, the initial gas density was not known and had to be determined from the data analysis. Materials. In this study, the solubilities and diffusivities of carbon dioxide, ethylene, and nitrogen were measured in low-density polyethylene (LDPE). The carbon dioxide was Coleman grade (99.9% purity),the ethylene was C. P. grade (99.5% purity), and the nitrogen was ultrahigh purity grade (99.9995% purity). LDPE was obtained from Dow Chemical in powder form (Mw ) 250 000). For experiments, the LDPE powder was loaded into the sample chamber and melted into a flat disk (about 0.5 cm in thickness). Because the melted sample adhered to the chamber base, diffusion through the slab occurred only from the top of the sample (singlesided diffusion). Data Analysis Raw Data Reduction. The raw data collected from the pressure-decay experiments were pressure versus time measurements. The sample volume was subtracted from the volume of the empty chamber to obtain the headspace vapor volume. The sample volume was obtained from the sample’s mass and density. The number of moles in the vapor could then be calculated with an equation of state from the known pressure, temperature, and headspace volume. The BenedictWebb-Rubin (BWR) equation of state given in eq 1 provided an excellent correlation for the gas PVT behavior. Table 1 reports the relevant BWR constants. Diffusion Coefficient Measurement. Measurement of the diffusion coefficient requires comparison of the experimental mass uptake curve to that of a theoretical model. The species continuity equation for the solvent in the polymer film relative to the volume average velocity is
∂F1 ∂2F1 )D 2 ∂t ∂z
Here, F1 is the mass concentration of the solvent in the polymer, t is the time, and D is the mutual binary diffusivity. In this derivation, it is assumed that the diffusion coefficient is constant and polymer swelling is insignificant. These should be good approximations given that the experiments were conducted over small ranges of gas concentration. Because the pressure (concentration) decreases in the vapor phase, a mass balance is necessary for this depletion
DAc ∂F1 ∂C )| ∂t V0 - AcL ∂z z)L
(3)
(4)
t)0
(5)
∂F1 )0 ∂z
z ) 0, t > 0
(6)
F1 ) KC
z ) L, t > 0
(7)
Here, K is the partition coefficient, C0 is the gas density prior to the experiment, and Ci is the gas density at t ) 0 prior to any absorption. Crank13 has developed a solution to these two coupled partial differential equations using a Laplace transform to describe the mass uptake in the polymer film
M(t) - M0 M∞ - M0
∞
)1-
∑
n)11
2R(1 + R) + R + R2qn2
exp
( ) -Dqn2t L2
(8)
Here, M(t) is the mass of gas in the polymer sample at any time t, M0 is the mass of solvent in the polymer at time t ) 0, M∞ is the mass of solvent in the polymer at equilibrium, and qn represents the nonzero positive roots of the equation
tan(qn) ) -Rqn
(9)
The dimensionless group R is equal to (V0 - AcL)/KAcL. The mass uptake in the polymer film can be related to the density of the vapor headspace according to
M(t) - M0 C(t) - Ci ) M∞ - M0 C∞ - Ci
(10)
where C(t) is the mass density of the gas at any time t, Ci is the gas density prior to any absorption into the sample, and C∞ is the final gas density at the end of the experiment. Substituting this expression into eq 8 gives
C(t) - Ci C∞ - Ci
(2)
t)0
)1-
∞
2R(1 + R)
n)1
1 + R + R2qn2
∑
( )
exp
-Dqn2t L2
(11)
In the pressure-decay experiments, the measured diffusion coefficient was taken to be the value that minimized the difference between the experimental mass uptake curve and the model prediction given by eq 11. It is now apparent why the initial gas density is important, as it is needed to determine the mass uptake curve. For the capsule technique, Ci is measured from the capsule weight and the headspace volume. However, in the extrapolation technique, Ci is unknown. Therefore, for the extrapolation-type experiments, both D and Ci were used as adjustable parameters to minimize the difference between the experimental mass uptake curve and the model given in eq 11. Essentially, this is an extrapolation of the mass uptake curve to zero time. For
1540 Ind. Eng. Chem. Res., Vol. 43, No. 6, 2004 Table 2. Procedure for kij Optimization in Data Analysis step
action
1 2 3 4
Make an initial guess for kij (usually 0). Use this value of kij to determine the liquid density. Analyze the experimental data using the liquid density. Calculate error between the PV-EoS and experimental solubilities. Adjust kij and return to step 2.
5
were obtained using the group contribution techniques of Lee and Danner16,17
∑k n(i)k Rk
(14)
(i) Θ(i) ∑k ∑ k Θm xekkemm m
(15)
v/i ) Figure 2. Experimental mass uptake curve (b) and model fit (solid line) for the carbon dioxide-polyethylene system at 150 °C and 435 psia equilibration pressure. The data point at zero time was determined from the capsule weights.
all parameter fitting, a nonlinear Levenberg-Marquardt algorithm from the IMSL library was used for least-squares optimization. Solubility Measurement. The solubility of the gas in the polymer sample is usually expressed as the weight fraction, ω1
M∞ (V0 - AcL)(Ci - C∞) + M0 ω1 ) ) M∞ + Mp (V0 - AcL)(Ci - C∞) + M0 + Mp (12) where Mp is the mass of the polymer sample. For the capsule technique, the solubility was calculated directly from the capsule weight and the equilibrium gas density. However, because Ci is unknown for the extrapolation technique, the solubility could not be determined until Ci had been fit to the mass uptake curve. A comparison of an experimental mass uptake and the model fit is shown in Figure 2. Panayioutou and Vera Equation of State. As described above, the polymer-gas solution density is required to calculate both the headspace vapor volume and the sample thickness. This was accomplished by using the Panayiotou and Vera equation of state (PVEoS).14,15 This equation of state is based on a lattice model and is capable of correlating the density and phase equilibria of polymer-solvent systems. The equation of state takes the form
[
v˜ Z P ˜ ) ln + ln T ˜ v˜ - 1 2
(
)
v˜ +
]
(qr) - 1 v˜
-
θ2 T ˜
(13)
In this equation, P ˜, T ˜ , and v˜ are the reduced pressure, temperature, and molar volume, respectively. Z is the lattice coordination number and is assumed to be 10. The parameter q describes the number of external molecular contacts displayed in the system, and r is the average of lattice sites occupied by a molecule. The parameter θ is the molecular surface fraction of the mixture. Solution of the equation of state requires three parameters: v/i , ii, and kij. v/i is the hard-core volume of molecule i, and ii is the interaction energy between two segments of molecules of type i. These two parameters are characteristic of the pure components, whereas kij is a binary interaction parameter characteristic of the mixture. The pure-component parameters
ii )
Here, n(i) k is the number of groups of type k in molecule i. Rk is the hard-core volume of group k. The parameter Θ(i) k is the surface fraction of group k in molecule i, and ekk is the molar interaction energy between two groups of type k. The surface fraction of a group k is given as
Θ(i) k )
nkQk
(16)
∑n nnQn
The summation extends over all of the different groups in the molecule, and Qk is the dimensionless surface area of group k. Lee and Danner showed16,17 that the parameters ekk and the Rk are well represented by quadratic functions of temperature, that is
( ) ( )
( ) ( )
Rk ) R0,k + R1,k
T T + R2,k T0 T0
ekk ) e0,k + e1,k
T T + e2,k T0 T0
2
2
(17) (18)
Using these mixing rules, the pure-component parameters were obtained without any experimental data. The binary interaction parameter, kij, was fit to the experimental pressure-decay solubility data. However, kij is required to solve the equation of state and thus is required to obtain the experimental data given that they depend on information provided by the equation of state. As a result of this problem, an iterative procedure was adopted to find the optimized value of kij that was consistent with the experimental data. This procedure is outlined in Table 2. Steps 2-5 are repeated until the PV-EoS solubility matches the experimental solubility data. Results and Discussion The experimental solubility and diffusivity data for the various systems are shown in Figures 3 and 4, respectively. The experimental values are also listed in Table 3. In all systems studied, the mutual binary diffusion coefficient was independent of concentration within experimental error. This was expected given that the experiments were conducted at 150 °C, nearly 200 °C above the glass transition temperature of HDPE. This far above the glass transition, diffusion is not freevolume-limited, and the self-diffusion coefficient is
Ind. Eng. Chem. Res., Vol. 43, No. 6, 2004 1541 Table 4. Parameters Used in the Panayiotou and Vera14,15 Equation of State Correlations of the Experimental Data at 150 °C v/i a (m3/mol) CO2 N2 CH2dCH2 LDPE
10-5
3.79 × 9.52 × 10-5 5.73 × 10-5 0.117
iia (J/mol)
kijb
682.35 611.14 622.23 1000.62
0.155 0.114 0.037 -
a Pure-component parameters (v/ and ) were found by the ii i group contribution methods of Lee and Danner.16,17 b Binary interaction parameters for gas-LDPE (kij) were optimized from the experimental solubility data.
Figure 3. Solubilities of various gases in LDPE at 150 °C. Carbon dioxide: extrapolation (O), dual chamber (∆), Henry’s law data of Liu and Prausnitz18 (150 °C) (×). Ethylene: extrapolation (b), dual chamber (2), Henry’s law data of Liu and Prausnitz18 (150 °C) (+), Cheng and Bonner19 (155 °C) (-). Nitrogen: extrapolation (0), Sato et al.20 (160 °C) ([). Solid lines indicate PV-EoS fits. Error bars were calculated using a propagation of uncertainty analysis.
Figure 4. Mutual binary diffusion coefficients of gas-LDPE systems at 150 °C. Carbon dioxide: extrapolation (O), dual chamber (∆), Durrill and Griskey21 (188 °C) ([). Ethylene: extrapolation (b), dual chamber (2). Nitrogen: extrapolation (0), Sato et al.20 (180 °C) (×). Error bars were calculated using a propagation of uncertainty analysis. Table 3. Solubilities and Diffusivities for LDPE Systems at 150 °Ca gas
equilibrium pressure (psia)
equilibrium weight fraction
diffusion coefficient (cm2/s)
CO2 CO2 CO2 CO2 CO2 C2H4 C2H4 C2H4 C2H4 C2H4 N2 N2
95 227 326 435 488 266 473 553 633 763 922 953
0.005 0.011 0.015 0.020 0.023 0.011 0.020 0.022 0.025 0.031 0.004 0.005
4.4 × 10-5 3.9 × 10-5 4.6 × 10-5 4.4 × 10-5 2.7 × 10-5 2.9 × 10-5 2.9 × 10-5 7.6 × 10-5 8.0 × 10-5
a
Bold data were collected using the dual-chamber technique.
concentration-independent.22 The mutual diffusivity is the product of the self-diffusivity and a thermodynamic term that accounts for the chemical potential gradient.22 However, in typical gas-polymer systems, the solubility of the gas in the polymer is quite low, making the thermodynamic term of the mutual diffusivity negligible. This is the situation here, as the mutual diffusivities are independent of concentration for all systems
studied. Therefore, the relative magnitude of the diffusion coefficients can be simply explained in terms of the penetrant size. A recent review by Marcus23 indicates that the trend of the collision diameters of the penetrants is ethylene > carbon dioxide > nitrogen. This trend explains the diffusion coefficient data, as the largest molecule (ethylene) had the smallest diffusion coefficient and the smallest molecule (nitrogen) had the largest diffusivity. As indicated in Figure 4, the diffusivity data collected in this study agree with published data on these systems. The Panayiotou and Vera equation of state (eq 13)14,15 provided excellent correlations of the experimental solubility data for all systems studied, and all values of kij were found to be independent of gas concentration. The parameters used in the PV-EoS correlations are reported in Table 4. The solubility data collected in this study are in good agreement with published literature data for these systems. The dual-chamber and extrapolation techniques produced the same results (solubility and diffusivity) for all of the systems and conditions studied, indicating that extrapolation is a valid procedure for obtaining the initial gas density in pressure-decay experiments. This is an encouraging result, as the extrapolation technique requires much less effort and equipment to perform experiments. Because the extrapolation technique requires fewer chambers, seals, fittings, valves, and pressure transducers, the probability of leaks is much lower. This technique might be particularly useful for extreme temperature/pressure environments when leaks are difficult to eliminate. Summary Two new volumetric sorption techniques based on the pressure decay experiment have been developed. Both of these techniques have the capability of measuring solubilities and diffusion coefficients for polymersolvent systems. The two variations of the pressuredecay experiment differ in how the initial gas density is measured. The dual-chamber technique gravimetrically determines the amount of gas fed to the sample chamber, whereas the extrapolation technique relies on correlation of the experimental mass uptake curve to obtain the initial gas density. Solubility and diffusion data were collected with both techniques for carbon dioxide, ethylene, and nitrogen in LDPE at 150 °C over a wide range of pressures. In all systems studied, the capsule and extrapolation techniques gave identical solubility and diffusion results. This indicates that the experimentally simpler extrapolation technique is a valid technique for studying the mass transport and thermodynamics of polymer-solvent systems. This re-
1542 Ind. Eng. Chem. Res., Vol. 43, No. 6, 2004
sult is encouraging because there is a greater possibility for leaks in the dual-chamber techniques than in the extrapolation technique. Leaks are one of the most significant problems with the pressure-decay technique because gas molecules that leak from the chamber are recorded as mass uptake in the sample. This situation is even more critical in high-temperature/-pressure environments where leaks are difficult to eliminate. The extrapolation techniques might provide more reliable results if leaks are a problem because the equipment used requires fewer high-pressure seals. Literature Cited (1) Newitt, D. M.; Weale, K. E. Solution and Diffusion of Gases in Polystyrene at High Pressures. J. Chem. Soc. (London) 1948, IX, 1541. (2) Lundberg, J. L.; Wilk, M. B.; Huyett, M. J. Sorption Studies Using Automation and Computation. Ind. Eng. Chem. Fundam. 1963, 2, 37. (3) Sanders, E. S.; Koros, W. J.; Hopfenberg, H. B.; Stannett, V. T. Pure and mixed gas sorption of carbon dioxide and ethylene in poly(methyl methacrylate). J. Membr. Sci. 1984, 18, 53. (4) Sada, E.; Kumazawa, H.; Xu, P. Sorption and diffusion of carbon dioxide in polyimide films. J. Appl. Polym. Sci. 1988, 35, 1497. (5) Koros, W. J.; Paul, D. R.; Rocha, A. A. Carbon dioxide sorption and transport in polycarbonate. J. Polym. Sci. B: Polym. Phys. 1976, 14, 687. (6) Koros, W. J.; Paul, D. R. Design considerations for measurement of gas sorption in polymers by pressure decay. J. Polym. Sci. B: Polym. Phys. 1976, 14, 1903. (7) Shah, V. M.; Hardy, B. J.; Stern, S. A. Solubility of carbon dioxide, methane, and propane in silicone polymers: Effect of polymer side chains. J. Polym. Sci. B: Polym. Phys. 1986, 24, 2033. (8) Rein, D. H.; Csernica, R. F.; Baddour, R. F.; Cohen, R. E. CO2 Diffusion and Solubility in a Polystyrene-Polybutadiene Block Copolymer with a Highly Oriented Lamellar Morphology. Macromolecules 1990, 23, 4456. (9) Bondar, V. I.; Freeman, B. D. Sorption of gases and vapors in an amorphous glassy perfluorodioxole copolymer. Macromolecules 1999, 32, 6163. (10) Hilic, S.; Padua, A.; Grolier, J.-P. E. Simultaneous measurement of the solubility of gases in polymers and of the associated volume change. Rev. Sci. Instrum. 2000, 71, 4236.
(11) Sato, Y.; Yurugi, M.; Fujiwara, K.; Takishima, S.; Masuoka, H. Solubilities of carbon dioxide and nitrogen in polystyrene under high temperature and pressure. Fluid Phase Equilib. 1996, 125, 129. (12) Bett, K. E. Thermodynamics for Chemical Engineers; MIT Press: Cambridge, MA, 1975. (13) Crank, J. The Mathematics of Diffusion; Clarendon Press: Oxford, U.K., 1975. (14) Panayiotou, C.; Vera, J. H. Local Compositions and Local Surface Area Fractions: A Theoretical Discussion. Can. Chem. Eng. J. 1981, 59, 501. (15) Panayiotou, C.; Vera, J. H. Statistical Thermodynamics of r-Mer Fluids and Their Mixtures. Polym. J. 1982, 14, 681. (16) Lee, B. C. Prediction of Phase Equilibria in Polymer Solutions. Ph.D. Thesis, The Pennsylvania State University, University Park, PA, 1995. (17) Lee, B. C.; Danner, R. P. Prediction of Polymer-Solvent Phase Equilibria by a Modified Group-Contribution EOS. AIChE J. 1996, 42, 837. (18) Liu, D. D.; Prausnitz, J. M. Solubilities of Gases and Volatile Liquids in Polyethylene and in Ethylene-Vinyl Acetate Copolymers in the Region 125-225 °C. Ind. Eng. Chem. Fundam. 1976, 15, 330. (19) Cheng, Y. L.; Bonner, D. C. Solubility of Nitrogen and Ethylene in Molten, Low-Density Polyethylene to 69 Atmospheres. J. Polym. Sci. B: Polym. Phys. 1978, 16, 319. (20) Sato, Y.; Fujiwara, K.; Takikawa, T.; Sumarno; Takishima, S.; Masuoka, H. Solubilities and Diffusion Coefficients of Carbon Dioxide and Nitrogen in Polypropylene, High-Density Polyethylene, and Polystyrene Under High Pressures and Temperatures. Fluid Phase Equilib. 1999, 162, 261. (21) Durrill, P. L.; Griskey, R. G. Diffusion and Solution of Gases in Thermally Softened or Molten Polymers: Part 1. Development of Technique and Determination of Data. AIChE J. 1966, 12, 1147. (22) Zielinski, J. M.; Duda, J. L. Predicting Polymer/Solvent Diffusion Coefficients Using Free-Volume Theory. AIChE J. 1992, 38, 405. (23) Marcus, Y. The sizes of moleculessRevisited. J. Phys. Org. Chem. 2003, 16, 398.
Received for review August 18, 2003 Revised manuscript received December 29, 2003 Accepted January 13, 2004 IE034075Y
