Ind. Eng. Chem. Res. 1995,34, 2033-2040
2833
Influence of the Glass Transition on Solute Diffusion in Polymers by Inverse Gas Chromatography Ilyess Hadj RomdhaneJ 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
The capillary-column inverse gas chromatography (CCIGC)method was used to determine diffusivity data for several solvents in poly(styrene) at conditions approaching infinite dilution of the volatile component. Measurements were made over a temperature range traversing the polymer glass transition temperature. The chromatographic data were analyzed by employing the elution model developed by Pawlisch et al. The Vrentas-Duda free-volume theory of diffusion was utilized to correlate the diffusion data and to interpret the effect of the glass transition temperature on the diffusion coefficient and on the effective activation energy in the limit of zero penetrant Concentration. The free-volume theory was found to correlate the diffusion data very well. The results of this study confirmed the free-volume concept that molecular diffusion is not as inhibited as one might expect below the glass transition temperature of the polymer.
Introduction Molecular diffusion of low-molecular-weight penetrants in concentrated polymer solutions plays a critical role in various polymer processing steps. In many manufacturing operations, the sorption and transport of a volatile component in a concentrated polymer solution or melt are the critical processes which limit the efficiency, and hence economics, of the process. Typical examples of such operations include solvent devolatilization, residual monomer stripping, packaging, and drying of paints and coatings. Conventional methods for measuring diffusion coefficients rely on bulk equilibration and gravimetric vapor sorptioddesorption experiments. These methods, however, become very diffcult to apply to polymer-solvent systems when the solvent is present in vanishingly small amounts or a t temperatures in the vicinity of or below the glass transition temperature. The low diffusivity values, characteristic of polymer-solvent systems at these conditions, is the cause of the experimental difficulties encountered with classical techniques. In recent years, much attention has been focused on the use of inverse gas chromatography (IGC) as an alternative method for studying diffusion in concentrated polymer melts and solids. The advent of the I W method has made it possible t o measure solvent diffusion coefficients as low as cm2/s(Hadj Romdhane, 1994). By virtue of its convenient and fast capabilities, a significant amount of diffusion data has been collected for various polymer-solvent systems, which in turn allowed diffusion theories to be evaluated and tested for different effects. Extensive IGC studies carried out by Arnould (19891,Arnould and Laurence (19921, and Hadj Romdhane and Danner (1993) revealed interesting results concerning the effects of temperature, solvent size, molecular structure, and glass transition on the diffusion process. In this paper, a similar approach is adopted to quantify the diffusion behavior of toluene, heptane, and methyl ethyl ketone (MEK) in poly(styrene) (PS) above and below the glass transition temperature. The experimental data are used to test
* To whom correspondence should be addressed. E-mail: JLDG@PSUADMIN (BITNET). ’ Current address: 3M Center, Building 518-1-01, St. Paul, MN 55144-1000.
the ability of the free-volume theory to account for the influence of the glass transition on the migration of a trace amount of solvent in the polymer. In what follows, the theoretical background for the elution process in capillary columns and the free-volume theory is presented.
Theory Capillary-Column Inverse Gas Chromatography (CCIGC). Invoking 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:
where
and a, p, and y are dimensionless parameters defined as:
(3) Here c is the gas-phase solute concentration, L is the length of the column, co is the strength of the inlet impulse, u is the mean velocity of the carrier gas, D, and D, are the gas-phase and the polymer-phase diffusion coefficients for the solute, r is the radius of the gaspolymer interface, t , is the residence time of the carrier gas (i.e., Lh),z is the thickness of the polymer film assumed to be constant in this case, and K is the equilibrium partition coefficient (=c’/c) where c’ is the polymer-phase solute concentration in equilibrium with the gas-phase concentration, c. Making use of the wellknown moment generating property of Laplace transforms, the resulting pair of moment equations are
0888-5885/95/2634-2833$09.00/00 1995 American Chemical Society
2834 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
(4)
where pl is the first temporal moment or mean residence time and p2* is the second central moment or variance of the concentration distribution. In characterizing the irregularity of the film coating in capillary columns, Pawlisch et al. (1988) observed, based on scanning electron microscopy analysis of several column samples, that the polymer film thickness varied only circumferentially; that is, axial uniformity of the coating was still retained. Using a distribution function where the film thickness was simply a linear function of angular position, they obtained a transform expression of the same form as eq 1except that Y(s) is now given by
Y(s) = s
+
+
]
In[cosh(/3s1’2(u b n ) ) M2nb cosh(/3s1’2u) 2
(6)
where a and b are parameters characterizing the polymer film distribution in the following manner: (7) and Qi is the ratio of the thinnest to thickest film dimensions observed at a particular cross section (i.e., tmidtmax).
The Vrentas-Duda Free-Volume Theory. Vrentas and Duda (1977) extended the Cohen and Turnbull free-volume model (1959) t o describe self-diffusion of a single species in a binary polymer-solvent mixture. The key modification introduced by Vrentas and Duda to the free-volume model is based on defining the average holefree volume per molecule as the total hole-free volume of the system divided by the total number of polymer and solvent jumping units. They derived the following expression for the concentration and temperature dependence of the solvent self-diffusion coefficient in a polymer solution:
RT
(8)
where K1i and K2i are free-volume parameters of component i and Tgiis the glass transition temperature of component i. In using the above free-volume formulation, it is oRen assumed that a small solvent molecule jumps as a single unit whereas the migration of polymer chains is due t o the combined jumping motion of numerous small segments of the chain. In the limit of zero penetrant concentration, the region where the IGC technique is capable of providing diffusion data, the diffusion process can be viewed as the migration of a solvent molecule in a solution which is essentially pure polymer. At this concentration end, the self and the mutual diffusion coeficients are equal t o each other, and the Vrentas-Duda free-volume theory simplifies to the following expression:
Hence, if the free-volume parameters of the polymer, K1dy and K22 - Tg2,are known, and assuming that the critical hole-free volume necessary for ajump, Vi*, is equal to the specific occupied volume, ViYO),then a nonlinear regression of self-diffusion data against temperature should yield the unknown parameters: Do,E, and 6 or simply Dol and 6, where Dol is the product of Doand the activation energy term in eq 11. Free-Volume Theory in Glassy Polymers. In an attempt to describe the diffusion process in glassy polymers, Vrentas and Duda (1978) considered the case where a trace amount of solvent is diffusing in an elasticlike type of amorphous polymer (elastic diffusion). They suggested that, at temperatures below its glass transition temperature, the polymeric material exhibits a nonequilibrium liquid structure, and thus possesses some extra hole-free volume which is effectively frozen into the nonequilibrated glassy polymer. Because of the sluggish movement of polymer chains in the glassy state, it is reasonable to assume that this nonequilibrium structure remains invariant during the course of the self-diffusion process, and thus the average holefree volume does not change. Vrentas and Duda also suggested that the presence of extra hole-free volume in the material should enhance molecular transport in glassy polymers. Based on these concepts, Vrentas and Duda extended the free-volume theory to describe molecular diffusion in glassy polymers as follows:
VFH
where Do is a constant preexponential factor, E is the activation energy per mole that a molecule needs to overcome attractive forces holding it to its neighbors, R is the ideal gas constant, T is the absolute temperature, y is an overlap factor introduced to correct for overlap free volume, ai is the mass fraction of component i, Vi* is the specific critical hole-free volume of component i required for a diffusive jump, ( is the ratio of polymer and solvent jumping units:
where Mji is the molecular weight of a jumping unit of component i. In eq 8, VFH is the specific hole-free volume of the polymer-solvent mixture given by: (10)
where E. describes the nature of the volume contraction which can be attributed to the glass transition temperature, and all the other parameters retain their previous meanings. As stated by Vrentas and Duda, an infinite number of nonequilibrium liquid structures can be realized below Tg2 depending on the mechanical and thermal history of the polymeric material. For A = 1, the material has an equilibrium liquid structure at all temperatures. For A = 0, the specific hole-free volume of the glassy polymer is equal to that achieved at Tg2at any temperature below the glass transition temperature. The evaluation of the parameter A is usually done by first regressing for Do,E, and 4 (or Dol and 6 ) above the glass transition temperature as suggested above. Once these parameters are known, eq 12 can be used to determine A provided that diffusion data are available below Tg2.
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2836
Experimental Methods and Procedures The gas chromatograph used in this study was a Varian 3400 (Varian Associates Inc., Walnut Creek, CA) equipped with a thermal conductivity detector (TCD), a flame ionization detector (FID), an on-column injector, and a circulating air oven. High-purity helium was used as the carrier gas in all experiments. The temperature of the injector and detector were set about 50 "C above the normal boiling point of the solvent to ensure rapid vaporization of liquid samples in the injector block and to avoid condensation in the detector assembly. Small amounts of solvent were injected through the rubber septum of the injection port into the carrier gas using a variable-volume Hamilton 1 pL syringe. The output from the detector was fed to the printer/plotter built into the gas chromatograph and to an on-line data system for further analysis of the chromatographic peaks. The flow through the capillary column could be regulated between 0.5 and 10 cm3/min. To improve the detector's response, additional carrier gas was introduced at he end of the column. This stream swept the effluent from the column into the detector jet at high velocity, thus reducing mixing effects. Dry-grade air and high-purity hydrogen were used to produce the flame in the FID. Methane was used as the marker gas since FID does not detect air. No appreciable pressure drops were measured across the capillary column. The capillary column used throughout this work was prepared by Supelco, Inc. (Bellefonte, PA). It was coated with PS (M, = 86 700; Mw/Mn = 1.04) obtained from Pressure Chemical Co. (Pittsburgh, PA). The polymer coating was applied by the static coating technique. The principle behind this technique is as follows: a fusedsilica column is filled with a filtered and degassed dilute solution of the polymer t o be coated. Once the column has been filled, one end of the column is sealed and vacuum is drawn on the other end of the column in a constant-temperature bath. As the solvent evaporates, a polymer film is deposited on the inner wall of the tubing. Assessment of the coating uniformity was obtained by examination of column fragments with a scanning electron microscope @EM). The PS column had an inside diameter of 530 pm and was 15 m long. The coating thickness was determined from the concentration of the polymer solution used to fill the column t o be 1pm at room temperature. The polymer density used to determine the film thickness at different temperatures was given by
eps(g/cm3) = M0.93805 + 3.3086 x 6.6910 x
T+
p ) (13)
where T is the temperature in "C. After the column was installed in the oven, it was conditioned to remove any residual solvents still present in the polymer. The outlet concentration profiles were monitored by the FID detector, recorded, and integrated via the data system. The processed raw data are usually stored on floppy disks for further analysis.
Methods of Data Analysis The raw data sampled at uniform time intervals were first transferred from the chromatograph data system to an IBM PC through an RS232 port. These data were then processed according to the following basic operations: (i) the raw output signal was corrected for baseline offset since the CCIGC model assumes that the
baseline signal is zero, (ii) a range of integration limits was selected, and (iii) a Basic program integrated the elution peak and calculated the first and second central moments. The values of these moments were then used to estimate the solubility and diffusion coefficients using eqs 4 and 5. As will be shown later, some of the elution peaks generated in this study exhibited significant asymmetry and a large amount of tailing. The use of the moment equations to determine the model parameters (moment analysis) was found to be unreliable in terms of reproducibility and accuracy. To ensure that a good fit of the experimental data was achieved, the model was further tested by examining the goodness of fit in the time domain. Starting with the initial estimates for K and D, from the moment analysis, the Laplace transform expression (eq 1)was numerically inverted with a fast Fourier transform algorithm (subroutine FFT2C from the International Mathematical and Statistical Library of Subprograms, Inc., Houston, TX). The generated theoretical response curve (in the time domain) was then compared against the experimental elution profile, and the following objective function
was minimized by employing a commercial nonlinear regression package (subroutine LMDIFl from the MINPACK-1 software product of the Applied Mathematics Division of Argonne National Laboratory, Argonne, IL). This code is based on a Levenberg-Marquardt algorithm and returns estimates of K and D, that best fit the experimental elution curve.
Results and Discussion Tests of the C C I W Model. Solubility and diffusion coefficient measurements were carried out for toluene, heptane, and MEK in poly(styrene) at temperatures above and below the glass transition temperature of the polymer. A statistical analysis of the scanning electron micrograph obtained from a cross section of the PS capillary column gave an average value of 0.5 for Qi. The reliability of this value was questionable, however, due to strong secondary electron emission from the specimen surface which produced increased brightness at the edge of the polymer film, a phenomenon commonly referred to as "edge-effect". The influence of edge effect on image quality was so remarkable that it was very difficult to distinguish between the polymer film and the column at some points in the micrograph. Nevertheless, it was apparent that the film thickness varied significantly around the circumference of the cross-sectional area. To examine this finding, both the uniform and the nonuniform versions of the CCIGC model were evaluated for the different PS-solvent systems. Time-domain analysis showed that the nonuniform model gave better fits of the experimental data than the uniform model, and that a Qi value of 0.2 seemed to best describe the experimental elution profiles. Figure 1,for instance, illustrates the poor quality of the fit between the uniform model (dashed curve) and the elution curve obtained for PS-toluene at 100 "C. The use of the nonuniform model (Qi= 0.21, on the other hand, seemed to capture the distortion of the peak rather well as shown in Figure 1. Similar results were obtained at other temperatures and for other systems.
2836 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 1.20 1.oo
2oooo 15000
Exprimental
0
( < 0.01 pI
Nonuniform Model (Q,=0.2)
0.80
Uniform Model (Q,=I.O)
>a
u . 0.60 u 0.40
0.20 0.00 0
2
1
5
4
3
2
Figure 1. Comparison of experimental and theoretical (CCIGC) elution profiles for the poly(styrene)-toluene system at 100 "C.
2.4 -
1.6
-
0
Experimmtai
-
CCIOC Model
0.8 0.0
6
8
10
Time (min) Figure 3. Effect of sample size on the peak shape for poly(styrene)-toluene at 110 "C.
3.2
u>o 2 u
4
4
0
2
1
5
4
3
6
t/t, Figure 2. Comparison of experimental and theoretical (CCIGC) elution profiles for the poly(styrene1-toluene system a t 80 "C. Table 1. Partition and Diffusion Coefficients of Toluene, Methyl Ethyl Ketone, and Heptane in Poly(styrene) at Infinite Dilution
MEK
toluene
T ("C)
K
70 75 80 90 100 110 120
281 222 183 135 104. 80.8
DP cm2/s) K 3.65 5.70 7.37 21.1 118 428
55.3 40.5 31.1 28.5 28.1 19.2 19.7
4
heptane
(10l2cm2/s)
K
76.9 116 149 222 756 1370 6210
95.9 80.7 68.0 49.6 36.3 19.8 14.4
4
(1012cm2/s) 0.727 0.946 2.65 5.60 50. 290 1380
In this study, the axial dispersion term was found t o be negligibly small for all the experiments performed. In some cases, chromatographic data were collected a t different flow rates to check the validity of the CCIGC model, that is, to verify that the solubility and diffisivity results showed no dependence on carrier gas velocity. In general, it was found that experimental elution curves obtained at different carrier gas velocities could be reproduced with a single set of parameters, K and D,. All solute peaks obtained below the glass transition temperature of PS were characterized by a sharp front and a long tail. A typical example is shown in Figure 2 for the case of PS-toluene at 80 "C. A summary of the values of K and D, obtained above and below Tg2for the different PS-solvent systems is shown in Table 1. On the basis of the accuracy of the column dimensions and the reproducibility of the measurement, the estimated standard deviation of K was 3-5% while that for D, varied from 5 to 10%. The assumption that measurements were made at conditions approaching infinite dilution of the solvent was checked for the PS-toluene system at 110 "C. In
this test, sample sizes of toluene were varied from about 0.02 pL to essentially residual vapors. The effect of sample size on peak shape is depicted in Figure 3 where the symbols represent experimental outlet concentration data for PS-toluene at 110 "C at different concentrations. In this figure, the theoretical profiles are indicated by the solid lines which were generated by regressing for K and D, for each data set using a value of Qi equal to 0.2. The maximum deviation was less than 3% between the values of K and only 9% between the values of D, obtained for each case. No significant correlation was found between the parameters K and D, with sample size indicating that the results were independent of the amount injected and that the solute elution occurred a t conditions approaching infinite dilution in the stationary phase. Effect of the Retention Parameter on Thermodynamic Data. The purpose of this section is t o examine the conclusions reached by Pawlisch et al. (1987), Arnould (19891, and Wang and Charlet (1989) concerning the proper measure of retention needed in order to determine reliable partition coefficients for polymer-solvent systems. These authors pointed out that the first moment of a peak provides the proper measure of retention and that the peak-maximum retention time is not a n accurate measure of the polymer-solvent thermodynamic interaction especially in the vicinity of the polymer glass transition temperature. This statement was evaluated in this work by comparing the thermodynamicinteraction data obtained from pl with those determined from the peak-maximum retention time ( t r ) for the PS-toluene system. To perform these comparisons, the partition coefficient data obtained for PS-toluene in this study were used to calculate more common thermodynamic interaction parameters such as the specific retention volume (Vgo), the infinite-dilution activity coefficient (Szl-), and the Flory-Huggins interaction parameter (2). The determination of these quantities from the solubility parameter, K,is discussed by Hadj Romdhane and Danner (1991). Figure 4 shows the retention volume diagram for the PS-toluene system based on both the first moment and on the peak-maximum retention time. While the retention diagram based on p1 gives a linear plot over a wide temperature interval, the plot based on t r exhibits a z-shape with breaks a t temperatures slightly above and below the glass transition temperature of the polymer. The reversal from normal chromatographic behavior obtained by using t r has been attributed t o a change in
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2837 6.0
r
.... ..
3.0
I
I.
2.0
x 1 .o
1.0
'
I
2.5
2.0
3.0
3.5
0.0 300
340
380
Figure 4. Retention diagrams for the poly(styrene)-toluene system based on the retention time of the peak maximum and the first moment of the peak.
30 2o
IO 0
.
300
Basedon p ,
420
460
500
T 6)
103i-r (Kl)
50
0
Figure 6. Temperature dependence of the Flory-Huggins interaction parameter of toluene in poly(styrene1.
: 1 I_llj DO,=2.33x1O4 cm'ls
Basedon(
... ."..
0
Bascdonp, n
3
I
0..
340
.
...gIm.. 380
420
460
W
nn 10" 500
B
0 "hls "hlr
.
0
A Prwlirh(i98S)
10.10
T (K)
:
Figure 5. Temperature dependence of the weight-fraction activity coefficient of toluene in poly(styrene).
the retention mechanism as the polymer changes from a liquid t o a glass as initially suggested by Smidsrad and Guillet (1969) and a number of other authors afterward. The curvature in the retention diagram, however, is not an indication of the onset of a different retention mechanism but rather an artifact of the use of an inappropriate retention parameter, tr (Wang and Charlet, 1989; Amould, 1989; Pawlisch et al., 1987). Near the glass transition of the polymer, a dramatic decrease in the diffusion coefficient often occurs which produces substantial skewing of the peak. For such peaks, the shift in the time a t the peak maximum with respect to the first moment causes curvature in the retention volume plot and thus a significant difference between the retention volume data obtained by using p1 with those obtained by using tr. Figures 5 and 6, which depict the temperature dependence behavior of Ql- and x, respectively, further demonstrate the importance of using the first moment of the elution profile instead of the retention time at the peak maximum At temperatures well above the glass transition temperature of the polymer, on the other hand, more symmetric peaks are obtained and the first moment of the elution curve as well as the peak-maximum retention time yield very similar results as clearly indicated in Figures 4-6. Tests of the Free-Volume Theory. 1. Influence of Temperature on the Diffusion Process. The correlative capabilities of the Vrentas-Duda freevolume theory above Tg2 were tested for all the PSsolvent systems investigated in this study. For each one of these systems, a two-parameter regression was carried out according to eq 11. The free-volume parameters
.
10-1)
work
Hsdj Romdhme and Dvlner (1993).
0
HueI~1.(1987)
'
1.60
2.00
2.40
2.80
1031~ (Ic-1) Figure 7. Free-volume correlation of the diffusivity data of toluene in poly(styrene1.
for PS used in these correlations were K1dy = 5.82 x cm3/(g K) and K - Tg2 = -327 K (Duda et al., 1982). The value of 7 2 * , also obtained from Duda et al., is 0.85 cm3/g. Thus, the parameters t o be determined are Dol and E. In adopting this regression procedure, the energy term in eq 11 was considered much larger than the free-volume term (the second exponential), and thus cah be absorbed in the preexponential factor. Such a simplification stems from the fact that, over the temperature range investigted (near T&), the amount of hole-free volume in the polymer is relatively small, and the self-diffusion process is freevolume driven (Vrentas and Duda, 1977). Also, since the quantity Dol is far less temperature sensitive than the free-volume term, it can be treated as a constant above and below the glass transition of the polymer. The infinitely-dilute diffusion coefficients reported above Tg2 for the PS-toluene system are plotted in Figure 7. In this figure, the results obtained by packed columns (Hadj Romdhane and Danner, 1993; Hu et al., 19871, and by capillary columns (Pawlisch et al., 1987, 1988) are shown. These data were correlated according to the free-volume theory. The results of this regression yielded a value of 2.33 x cm2/s for Dol and 0.422 for 5; (Figure 7). The diffusion coefficient data above T,, for the other systems (PS-MEK and PS-heptane) reported in Table 1 were also used to regress the corresponding free-volume parameters as described
2838 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
-
Table 2. Summary of Free-Volume Parameters for PS-Solvent Systemsa solvent Toluene Heptane
E
\
6
Dol (cm2/s) 5.23 10-7 2.33 10-5 2.56 x
MEK
10-7
0.212 0.422 0.344
The Dol and 6 parameters obtained for PS-toluene are based on the diffusion data obtained in this work and those reported by Hadj Romdhane and Danner (19931, Pawlisch et al. (1987,19881, and Hu et al. (1987). 10-5
0
Thin work
10-6
0
Had] Romdhms and Danncr (1993)
10’’
0
Hu et a1 ( I 987)
0
20
40
”-X=0.3
60
80
1O3/(K2,-TE2+T) (K-’)
10-7
Figure 8. Free-volume correlation of the diffusivity data of toluene in polybtyrene) through the glass transition temperature.
above. A summary of the Dol and E values obtained for all the PS-solvent systems investigated in this work are compiled in Table 2. 2. Effect of the Glass Transition Temperature on the Diffusion Process. The Vrentas-Duda freevolume theory in glassy polymers was used to describe the diffusion process of the different solvents in PS below Tg2 (i.e., 100 “C). The evaluation of the parameter A was done through a nonlinear regression of eq 12 by using the values of Dol and 6 compiled in Table 2. Figure 8 presents a plot of the diffisivity data of toluene in PS above and below Tg2. The extension of the correlation above Tg2 to temperatures below the glass transition temperature is represented in this figure by the continuous line corresponding to A = 1. This line indicates where the self-diffusion coeficient of toluene in PS would lie if the polymer has reached an equilibrium liquid structure (relaxed or annealed glass). The line corresponding to A = 0, on the other hand, represents the case where diffusion below the glass is independent of temperature and is the same as that obtained at Tg2. The results, shown in Figure 8, indicate that a break occurs at Tg2and that a value of A = 0.25 seems to best correlate the toluene diffusivity data in the glassy PS. A value of 0.3 for 1 was found to adequately fit the diffusion data obtained below Tg2 for the PS-MEK and the PS-heptane systems as portrayed in Figures 9 and 10. The free-volume theory correlates the diffision data above and below the glass transition temperature very well. The good agreement obtained for A from the three different solvents diffusing in PS is evidence that this free-volume parameter is independent of the solvent utilized for a particular polymer. The value of A obtained for PS in this study falls within the bounds reported by Vrentas and Duda (1978). Similar trends €or the effect of the glass transition temperature on the diffusion process were obtained by Arnould (19891, who studied the diffusion of various solvents in PMMA and PPMS below Tg2. These experimental results support
10-8 I 10.9 i h
z “E0
10-10:
W
an IO-” :: 10-12
X=O
r
1043
1O3/(K,,-T,+T) (K’) Figure 10. Free-volume correlation of the diffusivity data of heptane in polybtyrene) through the glass transition temperature.
the free-volume concept that transport in glassy polymers is not as inhibited as one might anticipate. The presence of excess hole-free volume frozen in the polymeric material below its glass transition temperature provides for a higher diffusivity than expected from an extrapolation of the behavior above Tg2. To examine the effect of the glass transition temperature on the effective activation energy of diffusion (ED), the following equations developed by Vrentas and Duda (1978) in the limit of zero solvent concentration were used. Above Tg2,the free-volume theory predicts the temperature dependence of E Dto be
while below Tg2it is
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2839 100 ,"
Using the above equations, a plot illustrating the temperature dependence of the effective activation energy of diffusion for the PS-toluene system was constructed based on the values o f t and A obtained in this study (Figure 11). EDdecreases with increasing temperature, and a step change in EDoccurs as the glass transition temperature of the polymer is traversed. The temperature dependence of EDfor other values of A is also shown in Figure 11 to highlight the sensitivity of this parameter in the free-volume theory. Finally, considering that the activation energy term, E, is usually of the order of 1kcal /mol, it is fair to conclude that the self-diffusion process in this study is indeed dominated by free-volume effects by virtue of the relatively large values of ED, shown in Figure 11. Hence, it is possible to absorb the E term in the preexponential factor as suggested earlier.
Conclusions Diffusion and partition coefficient data for toluene, heptane, and MEK in PS over a temperature range encompassing the glass transition temperature of the polymer were successfully determined by the IGC method. The use of capillary columns in this work has made it possible to extend the temperature range over which transport and equilibrium properties can be determined. The notion that the first moment of the peak is the proper measure of retention, particularly in the vicinity and below the glass transition temperature of the polymer, was once again reaffirmed in this study. This study also showed that the influence of temperature on the diffusivity for a solvent-polymer system above and below Tg2 is consistent with the Vrentas-Duda free-volume formulation. The diffusion data obtained below the glass transition temperature of PS confirm the free-volume concept that molecular diffusion is greater than expected based on extrapolation of the diffusion behavior above Tg2. This is the result of the extra hole-free volume trapped within the nonequilibrated glassy polymer.
Acknowledgment The authors gratefully acknowledge the 3M Company, St. Paul, MN, for the financial support of this work. The authors also thank Dr. David Dunham (Varian GC and DS Specialist) for kindly donating the Kermit software protocol responsible for data transfer.
Nomenclature co = strength of the inlet impulse, mol/cm3 D, = constant preexponential factor, cm2/s Dol = constant preexponential factor when E is presumed to be equal to 0, cm2/s D1 = solvent self-diffusioncoefficient, cmVs D, = solvent diffusion coefficient in the mobile phase, cm2/s D, = solvent diffusion coefficient in the polymer phase at infinite dilution, cmZ/s E = critical energy per mole required to overcome attractive forces, J/mol ED = effective activation energy of diffusion, J/mol f e x p = experimental response curve ftheo = theoretical response curve I = objective function
80
.
X=O
-20 60
80
100
120
140
160
T ("C) Figure 11. Temperature dependence of the effective activation energy of diffusion for the poly(styrene)-toluene system.
K = equilibrium partition coefficient of the polymersolvent system KI, = free-volume parameter of component i, cm3/(gK) K2, = free-volume parameter of component i, K L = length of the column, cm M,, = molecular weight of the jumping unit of component i, g/mol QL= ratio of the thinnest to the thickest film dimensions in a capillary column r = radius of the capillary column, cm R = gas constant (e.g., 8.314 J/(mol*K)) t = time, s t, = residence time of the carrier gas, s t , = solvent retention time at the peak maximum, s T = absolute temperature, K TgI= glass transition temperature of component i, K u = average linear velocity of the carrier gas, cm/s Vgo= specific retention volume corrected to 0 "C, cm3/g ~ F =Haverage hole-free volume per gram of mixture, cm3/g = specific critical hole-free volume of component i, cm3/g VIo= specific volume of pure component i at 0 K, cm3/g
vL*
Greek Letters y = overlap factor which accounts for shared free volume
A = parameter used in the free-volume formulation for glassy polymers p1 = first temporal moment, s p2* = variance of the concentration distribution, s2
6 = ratio of solvent and polymer jumping units
z = film thickness in the capillary column (cm) = Flory-Huggins polymer-solvent interaction pa-
x
rameter oi = mass fraction of component i 511" = weight-fraction activity Coefficient of the solvent
at infinite dilution (WFAC) Literature Cited Amould, D. Capillary Column Inverse Gas Chromatography (CCIGC)for the Study of Diffusion in Polymer-Solvent Systems. Ph.D. Dissertation, University of Massachusetts at Amherst, 1989. Amould, D.; Laurence, R. L. Size Effects on Solvent Diffusion in Polymers. Znd. Eng. Chem. Res. 1992,31, 218-228. Cohen, M. H.; Turnbull, D. Molecular Transport in Liquids and Glasses. J . Chem. Phys. 1959,31,1164. Duda, J. L.; Vrentas, J. S.; J u , S. T.; Liu, H. T. Prediction of Diffusion Coefficients for Polymer-Solvent Systems. AIChE J . 1982,28, 279.
2840 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 Hadj Romdhane, I. Polymer-Solvent Interactions by Inverse GasLiquid Chromatography. M. S. Thesis, The Pennsylvania State University at University Park, 1990. Hadj Romdhane, I. Polymer-Solvent Diffusion and Equilibrium Parameters by Inverse Gas-Liquid Chromatography. Ph.D. Dissertation, The Pennsylvania State University at University Park, 1994. Hadj Romdhane, I.; Danner, R. P. Solvent Volatilities from Polymer Solutions by Gas-Liquid Chromatography. J . Chem. Eng. Data 1991,36,15. Hadj Romdhane, I.; Danner, R. P. Polymer-Solvent Diffusion and Equilibrium Parameters by Inverse Gas-Liquid Chromatography. AIChE J . 1993,39,625. Hu,D. S.; Han, C. D.; Stiel, L. I. Gas Chromatographic Measurements of Infinite Dilution Diffusion Coefficients of Volatile Liquids in Amorphous Polymers at Elevated Temperatures. J . Appl. Polym. Sei. 1987,33,551. Pawlisch, C. A,; Macris, A.; Laurence, R. L. Solute Diffusion in Polymers: 1. The Use of Capillary Column Inverse Gas Chromatography. Macromolecules 1987,20,1564. Pawlisch, C. A,; Bric, J. R.; Laurence, R. L. Solute Diffusion in polymers: 2. Fourier Estimation of Capillary Column Inverse Gas Chromatography Data. Macromolecules 1988,21,1685.
Smidsrod, 0.; Guillet, J. E. Study of Polymer-Solute Interactions by Gas Chromatography. Macromolecules 1969,2,272. Vrentas, J. S.; Duda, J. L. Diffusion in Polymer-Solvent Systems: I. Reexamination of the Free-Volume Theory. J. Polym. Sci., Part B: Polym. Phys. Ed. 1977,15,403. Vrentas, J. S.; Duda, J. L. A Free-Volume Interpretation of the Influence of the Glass Transition Temperature on Diffusion in Amorphous Polymers. J . Appl. Polym. Sei. 1978,22,2325. Wang, J.-W.; Charlet, G. Use of Moment Analysis in Inverse Gas Chromatography. Macromolecules 1989,22,3781.
Received for review November 23, 1994 Revised manuscript received May 15, 1995 Accepted May 26, 1995@
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Abstract published in Advance ACS Abstracts, July 15,
1995.
