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Mechanism of diffusion and sorption of carbon dioxide in poly(vinyl acetate) above and below the glass transition temperature
Journal of Membrane Science, 13 (1983) 15-27 Elsevier Scientific Publishing Company, Amsterdam -Printed
15 in The Netherlands
MECHANISM OF DIFFUSION AND SORPTION OF CARBON DIOXIDE IN POLY(VINYL ACETATE) ABOVE AND BELOW THE GLASS TRANSITION TEMPERATURE
K. TOI, Y. MAEDA
and T. TOKUDA
Department of Chemistry, Tokyo 158 (Japan)
Faculty
of Science,
Tokyo
Metropolitan
(Received April 13,1982;
accepted in revised form August 10,1982)
University,
Setagaya-ku,
Summary The pressure dependence below 1 atm of the apparent diffusion and permeation coefficients were observed by using the permeation time lag method for carbon dioxide in poly(vinyl acetate), which has a glass transition near room temperature, at temperatures ranging from 8 to 50%. Above the glass transition temperature, pressure dependence of the diffusion and permeation coefficient has not been observed; hence, Fick’s law with a concentration independent diffusion coefficient applies. On the other hand, in the glassy state, the apparent diffusion coefficient shows pressure dependence. Moreover, the behavior of the pressure dependence does not show a clear curve in the ranges between 30% to 17%. Above 17’C, the apparent diffusion coefficients show discontinuities, but below 17°C increase with pressure is regular. Using the theoretical prediction of Paul, a computer was used in the numerical calculation to determine the true diffusion coefficient and other dual sorption parameters. The compensated diffusion coefficients controlled only by Henry’s law dissolution was described by three straight lines with two intersection in the form of Arrhenius plots, which give good agreement with both our results for He and Ar and those of Meares. It is assumed that beside the dual sorption mechanism, another effect, for instance some relaxation effect may also contribute to the diffusion for carbon dioxide in poly(viny1 acetate) near the glass transition temperature region.
Introduction
The first systematic measurements of gas transport in a polymer above and below the glass transition temperature were those of Meares [l] with poly(vinyl acetate) (PVAc). Two clear breaks in the plots of the diffusion and permeation coefficients were found for the majority of the gases investigated. They formed a less well-defined intermediate region near the glass transition temperature (Tg) with a small slope above or below it. Similar measurements with oxygen were later repeated by Stannett and Williams [2] and a clear break was observed. In general, for amorphous polymers below glass transition temperature, sorption is apparently composed of two mechanisms, the first described by
0376-7388/83/0000-0000/$03.00
0 1983
Elsevier Scientific Publishing Company
16
of dissolution and the second by Langmuir’s law of adsorption. This overall quantitative description has become known today as the Dual Sorption Theory [S-61. Precise measurements of the dual so&ion nhenomena enable one to determine the parameters relating to it. From discussion of the physical meaning of the values, new information concerning the above curious behavior of PVAc in the vicinity of T, could be obtained. In our previous paper [ 71, a method was discussed that can determine the dual sorption parameters numerically from a low pressure permeation experiment without using the data from a sorption experiment. The apparent permeation coefficient was regarded as independent of pressure below 1 atm for several glassy polymers, so pressure dependence of the apparent diffusion coefficient lent support to the total immobilization of the Langmuir component. The values of parameters were obtained at various temperatures for the glassy polymers. Excellent agreement between the calculated curves and experimental plots provided strong support for the validity of the parameters. In this study, the pressure dependence of the apparent permeability coefficient and the apparent diffusion coefficient were determined by observing the steady state carbon dioxide permeation in poly(viny1 acetate) above and below the glass transition temperature. The above method was applied for determining the pressure independent diffusion coefficient and parameters associated with it. An interpretation of the gas diffusion behavior for PVAc near Tg was attempted to understand the modifications introduced by the concept of the dual sorption mode. Henry's law
Experimental The PVAc sample was obtained from Wako Junyaku Manufacturing Co., Ltd. and its degree of polymerization was between 1400 and 1600. PVAc membranes were prepared by casting solutions in toluene on a clean glass plate, which was floating on mercury. Before use, all membranes were dried under high vacuum at 50°C for a few days. The membranes used, along with their dimensions, are listed in Table 1. The glass transition temperature of PVAc with CO* atmosphere, determined by using a Rigaku Denki DSC Thermal Analyser (Model SOOO),was found to be 32°C. The densities of the membranes were determined by the density-gradient method in a A1C13 solution. The membrane thicknesses were determined by weighing the fixed area. The following high purity commercial gases were used in this study: helium, argon and carbon dioxide with purities of larger than 99.5%. No attempt was made to purify the gases before use. Details of the permeation equipment have been described earlier [7,8]. A digital printer was connected to MKS Baratron Capacitance Manometer (Model 170-6B using a model 310BH-1 sensor head) through an analog to digital converter. Downstream pressure changes were printed between 10v6 and lo-* cmHg using a digital printer at regular time intervals with a precision
17 TABLE 1 Dimensions of membranes Membrane (Gas)
Thickness (cm)
Area (cm’)
Density (g/cm’) .~
AA EE(Ar) FE(C0,‘) DE(C0,)
14.15 4.92 3.25 2.19
7.07 3.04 2.56 2.56
1.176 1.176 1.176 1.176
x x x x
1o-3 1o-3 1O-3 1o-3
.__
of l/100 sec. A slope and intercept of the linear asymptote line was calculated by the least squares method through 30 to 50 experimental points taken from a 4 times to 5 times time lag. The precision of the slope and the time lag of the permeation curve using this method is greater than 0.1% and 0.576, respectively. Results and discussion Pressure dependence of apparent permeability coefficients of Ar and CO:! in the PVAc membranes are presented in Figs. 1 and 2 at various temperatures, 21.5
24.0 22.0
"J a 25.3 c ;
22.5 a
28.7'C
..*
25.z.C
~ _
19.9'C %.5"C
26.4"C
n
-Q)-;
23.C
30.4'C
m..
rPn
--O
0
""
0 2.0
4.0 l/p (atm-1)
6.0
0
15.2'C
or\ gr 26.0
21.9;c 0
2.0
n
lU.I"C 4.0
6.0
1/F (at,!,-' )
Fig. 1. Pressure dependence of permeability coefficients for argon in PVAc(Membrane EE) at various temperatures. Fig. 2. Pressure dependence of permeability coefficients for carbon dioxide in PVAc(Membrane DE) at various temperatures.
18
reasons given in the previous paper [ 81, the abscissa is indicated by the reciprocal of the upstream pressure(l/atm). The corresponding apparent diffusion coefficients, II,, directly obtained from the time lag (D, = 12/68; 1 is membrane thickness (cm), and 6 is time lag (set).) are shown in Figs. 3, 4, and 5. Permeability and diffusion coefficients of Ar in PVAc (Figs. 1 and 3) were independent of pressure in the upstream pressure range between 0 to 1 atm. This indicates that the diffusion of Ar in PVAc is described by Fick’s and Henry’s law not only above, but below, T,. For
I
I 2.0
4.0 l/p (atm-')
6.0
2
4
b
l/p (atIn-1)
Fig. 3. Pressure dependence of diffusion coefficients for argon in PVAc(Membrane at various temperatures.
EE)
Fig. 4. Pressure dependence of apparent diffusion coefficients for carbon dioxide in PVAc (membrane FE) above glass transition temperature. The points are observed values and the curves are calculated from eqn. (1) with parameters in Table 2,
The permeability coefficients of CO1 in PVAc (Fig. 2) are also independent of the upstream pressure at low temperature. The pressure dependence at higher pressure above Tg was entirely unexpected. Probably, this is due to a mutual interaction between the polymer and the penetrant at these conditions, rather than an effect through the measurement itself.
19
2.0
4.0
6.0
l/P (at"-')
Fig. 5. Pressure dependence of apparent diffusion coefficients for carbon dioxide in PVAc(Membrane DE) below glass transition temperature. The points are observed values and the curves are calculated from eqn. (1) with parameters in Table 2.
On the other hand, the apparent diffusion coefficients, Da, are independent of upstream pressure above !?‘s but dependent on pressure below Tg, as shown by plots in Figs. 4 and 5. These results are entirely consistent with a model which assumes total immobilization of the Langmuir population as indicated by Paul et al. [4,5], As strategy to test these explanations, we will deduce all parameters from the diffusion data, using eqn. (1) proposed by Paul [ 91. D =Da[l
+ Kf(y)I
f(y) = W3 I-(ll2)~’
(1) + Y - (1 + Y)ln(l + ~11
where y = bp, K = GH’b/k JJ, and Da = 1*/6’B: kD, GH’, and b are-adjustable parameters; D is the true diffusion coefficient for Henry’s law species and p is the upstream gas pressure. The parameters of the dual sorption model, according to this equation, were deduced from each experimental set of diffusion data using a nonlinear least-squares optimization program. Some of the results are listed in Table 2. Using the values of parameters b, K, and D given in Table 2, the calculated values of Da at each temperature are shown in Figs.
20
TABLE2 Dual sorptionparameters Membrane
T
kD
CC) (
DE DE DE DE DE FE DE FE FE DE FE FE FE FE
10.7 11.5 12.8 15.2 16.7 21.8 21.9 23.8 26.4 26.5 28.3 29.6 31.0 32.5
for CO,in PVAc CH'
cm3 (STP) cm3(polymer)-atm
2.48 2.43 2.35 2.20 2.12 1.90 1.85 1.81 1.86 1.88 1.88 1.87 1.85 1.96
cm3 (polymer)
cmJ(STP)
b
K
DX
(ah-')
(CH'b/kD)
(Cma/SeC)
lo9
2.36 2.35 2.32 2.27 2.24 2.42 2.44 2.57 3.10 3.19 3.21 3.41 3.80 3.99
0.820 0.809 0.791 0.736 0.735 0.675 0.669 0.650 0.455 0.419 0.305 0.300 0.290 0.136
2.69 2.79 2.98 3.34 3.56 4.50 4.55 4.95 5.18 5.09 5.55 5.86 6.33 6.54
1
0.864 0.838 0.803 0.736 0.697 0.531 0.507 0.459 0.273 0.248 0.179 0.165 0.141 0.067
17.5
18.0
19.0
3.2
3.3
3.4 1,~
x lo3
3.5
(K-l!
Fig. 6. Arrhenius plots of apparent EE) at p = 0.4 atm.
diffusion
coefficients
for argon
in PVAc(Membrane
4 and 5 by solid lines, together with the plots of experimental values. The coincidence between the measured and predicted D, is impressive at lower and higher temperatures but not at intermediate temperatures. Thus there may
21
l/T x lo3
(K-1)
Fig. 7. Arrhenius plots of diffusion coefficients for carbon dioxide in PVAc: (0,~~) apparent diffusion coefficients observed at p = 0.4 atm; ( l ,A) pressure independent diffusion coefficients calculated by using total immobilization model.
be a temperature region where the experimental values and the calculated curves are not quantitatively consistent with each other, and in this region the concept of the total immobilizationmodel alone cannot satisfactorily represent the experimental results. It is necessary to compensate the effect of dual sorption in order to investigate another effect. Temperature dependence of the apparent diffusion coefficients for Ar and CO2 measured at 0.4 atm upstream pressure are shown in the form of Arrhenius plots in Figs. 6 and 7. As shown in Fig. 6, the experimental plots of Ar lie on three distinct straight lines with two intersections. Similar behavior was observed in the Arrhenius plots of permeability coefficients for He, as shown in Fig. 8. However, the curves of diffusion and permeability coefficients of CO2 do not show any clear break point but change smoothly over a range of the temperature. By employing the steady state permeation method, Meares has measured the diffusion coefficients of several gases in PVAc [l] . He found that the logarithmic plots of diffusion, permeability, and solubility coefficients against reciprocal temperature showed two transition temperatures, about 10°C apart. The present data for Ar and He closely agree with Meares.
22
p" c 7
22.0
-
72.5
-
23.0
-
3.2
3.3
3.4 l/T
v 1O-3
3.5
(K-l)
plots of apparent Fig. 8. Arrhenius in PVAc at p = 0.4 atm.
3.0
3.2
.
permeation
coefficients
for helium
and carbon
dioxide
0 DE A
ff
0
DL
A
FE
_
3.2
3.3 l/T
3.4 x lo3
3.5
(K-l)
Fig. 9. Van ‘t Hoff plots of solubility rent solubility coefficients calculated coefficients calculated from Henry’s
coefficients for carbon dioxide in PVAc: (= ,a) appasolubility from eqn. (2 ); ( l , A) pressure independent law parameter (k~,/76).
23
Figure 9 presents van ‘t Hoff plots of the apparent for COz in PVAc, calculated by eqn. (2).
solubility
coefficients
s, =PaDa
(2)
The figure shows that the slope changes near the glass transition temperature of the polymer, which differs from the apparent permeability and diffusion coefficients. The Henry’s law coefficient calculated from the optimization method, is also illustrated in this figure. It is found that the Henry’s law coefficient decreases rapidly near Tg, and is replaced by another term, which may be the Langmuir contribution. Although the plots of P, and D, for CO2 in PVAc did not show any break point, the clear inflection of S,, and the characteristic pressure dependence of D, in the range between 17 and 32°C have some correlation with Meares’ suggestion at the glass transition region. The dual sorption parameters K, CL, and b for CO? in PVAc are plotted as function of temperature in Figs. 10, 11, and 12. The Langmuir capacity parameter CH’ begins to appear near the glass transition temperature and increases as the temperature decreases. The composite parameter K = &‘b/h~ also shows a similar tendency to CH’. In contrast to CH’ and K, the Langmuir affinity constant b, plotted in van ‘t Hoff form, changes discontinuously near 19°C as shown in Fig. 12. We do not have any precise explanation of this surprising discontinuity, but this figure shows that some effect, other than the total immobilization effect, may take place above 19°C. According to high pressure experiments by Paul et al. [ 51, the absolute values of CH’ are related’to the expansion coefficient difference above and below Tg. It is also known, however, that the non-linear least squares fitting techniques used to obtain the sorption parameters are not sensitive to the individual values of CH’ and b but to the product CH’~. Thus the absolute values of CH’ and b will not be pursued further here.
10
I5
(-cp5
20
30
35
T
Fig. 10. Temperature Fig. 9.
dependence
of Langmuir
capacity
constants
CH’. Symbols
as in
10
15
20
75
30
35
T ["C!
Fig. 11. Temperature
dependence
of composite
parameter
K. Symbols
as in Fig. 9.
1.3 -
1.2 -
Fl.1 E
-
2 n
1.0 -
0.9 -
0.8 -
3.3
3.5
3.4 l/T Y lo3 (K-l)
Fig. 12. Van ‘t Hoff plots of Langmuir
affinity
constants
b. Symbols
as in Fig. 9.
The enthalpies of the Henry’s law parameter AH and Langmuir void affinity parameter, q, evaluated from slopes of Figs. 9 and 12 are listed in Table 3. The sorption enthalpy for the Henry’s law species showed smaller values relative to the Langmuir species below 19°C. There are large differences in the values of kD above and below Tg, but the enthalpy at both regions indicated very similar values. The values, along with the associated energy parameters above and below Tg, are shown in Table 4. The compensated diffusion coefficients controlled only by Henry’s law dissolution were plotted, together with the apparent ones in the form of Arrhenius type, in Fig. 7. Compared with the smoothed curve for the appa-
25 TABLE 3 Van ‘t Hoff parameters of sorption for CO, in PVAc 4
(kcal/mol)
above T,
below T,
-4.8
-4.4
-
-1.4
(below 19°C)
TABLE 4 Arrhenius parameters of diffusion for gases in PVAc
D
ED
(kcal/mol)
Ar (above Tg) Ar (below ‘I’s) CO, (above Tg) CO, (below Tg )
16.4 8.1 17.8 7.8
.-
(cm’lsec) 8.2 x lo5 2.5x 2.6x
lo4 lo3
rent diffusion coefficient, the new curve was described by three straight lines with two intersections. The form of this curve gives good agreement with our results for He and Ar and those of Meares [ 11. That is to say, removing the dual sorption effect, the true diffusion coefficient of CO2 for PVAc shows the same intermediate region as He and Ar in Arrhenius plots. According to the equilibrium sorption behavior for He and Ar in other glassy polymers, it is thought that the Langmuir term contributes much less to the sorption process [ 51. Then, the apparent diffusion coefficient, calculated by the low pressure permeation method, is nearly equal to the true diffusion coefficient. The fact that the form of the CO2 curve is in fair agreement with those of He and Ar demonstrated that the true diffusion coefficient can be well separated from the apparent diffusion coefficient by the optimization process. Here, we must conclude that this region cannot be interpreted solely by the effect of the dual sorption, but another effect, for instance some relaxation effect must occur. A similar behavior pattern has also been observed for CO2 in poly(ethylene terephthalate) by Koros and Paul [lo] . They showed that in this region a Fickian description of the transport process is unsatisfactory, and the observed time lag near T, is lengthened by the dynamics of non-instantaneous chain rearrangement. Conclusion From the above results and discussion, it is concluded that besides the dual sorption mechanism, some relaxation process may also contribute to the dif-
26
fusion for CO1 in PVAc near the T, region. That is to say, in the range between 17 and 32”C, even if the main chain of PVAc is completely frozen, the side chain may have some mobility, and the time scale for relaxation may approach that for diffusion. Then, it will become necessary to use more accurate time lag data and complimentary sorption measurement in order to employ an analysis by consideration the time dependence of diffusion in this temperature region. Acknowledgement The authors acknowledge useful discussions with Dr. D.R. Paul. List of symbols Symbol
Name of symbol
Common unit
Definition by SI
b
Hole affinity constant
(101325 Pa))’
C
Concentration
atm-’ cm3 (STP) cm3 (polymer) cm3 (STP) cm3 (polymer) cm2 /set
CH’
D D, DCI
AED
AH
K kD 1 PO P 4
8,
Hole saturation constant Diffusion coefficient for Henry’s law species Apparent diffusion coefficient Pre-exponential factor in Arrhenius equation for diffusion Activation energy for diffusion Heat of solution for Henry’s law species Composite parameter (K = CH’bjkD) Henry’s law constant Membrane thickness Downstream pressure Upstream pressure Heat of adsorption for Langmuir’s law species Apparent sorption coefficient obtained by S, = Pa/D,
44.613 mol/m3 44.613 mol/m3 10m4m2 /s
cm2 /set
10m4m* /s
cm2 /set
10e4 m2 /s
kcal/mol
4.184 kJ/mol
kcal/mol
4.184 kJ/mol
dimensionless cm3 (STP) cm3 (polymer)-atm cm cmHg atm kcal/mol
cm3 (STP) cm3(polymer)- _ cmHg
4.403 X low4 mol/m3Pa lo-* m 1333.224 Pa 101325 Pa 4.184 kJ/mol
3.34625 X 10m2mol m3Pa
27
pa
Apparent permeability coefficient
e
Diffusion
time lag
cm3 (STP)-cm
3.34625X
cm* -set-cmHg see
s
10whmol
m2 sPa
References 1 P. Meares, 76 (1954) P. Meares,
The diffusion of gases through polyvinyl acetate, J. Amer. Chem. Sot., 3514. The solubilities of gases in polyvinyl acetate, Trans. Faraday Sot., 54 (1958)
V. Stannett and J.L. Williams, The permeability of poly(ethy1 methacrylate) to gases and water vapor, J. Polym. Sci., Part C, 10 (1966) 45. W.R. Vieth, J.M. Howell, and J.H. Hsieh, Dual sorption theory, J. Membrane Sci., 1 (1976) 177. D.R. Paul and W.J. Koros, Effect of partially immobilizing sorption on permeability and the diffusion time lag, J. Polym. Sci., Polym. Phys. Ed., 14 (1976) 675. and transport of various gases in 5 W.J. Koros, A.H. Chan, and D.R. Paul, Sorption polycarbonate, J. Membrane Sci., 2 (1977) 165. and transport in glassy polymers, Ber. Bunsenges. ?hys. Chem., 6 D.R. Paul, Gas sorption 83 (1979) 294. of diffusion coefficient for CO, in glassy polymers, 7 K. Toi, Pressure dependence Polym. Eng. Sci., 20 (1980) 30. and T. Tokuda, Isotope effect in the diffusion of hydrogen and 8 K. Toi, K. Takeuchi, deuterium in polymers, J. Polym. Sci., Polym. Phys. Ed., 18 (1980) 189. adsorption on the diffusion time lag, J. Polym. 9 D.R. Paul, The effect of immobilizing Sci. Part A-2, 7 (1969) 1811. 10 W.J. Korosand D.R. Paul, Sorption and transport of CO, above and below the glass transition of poly(ethylene terephthalate), Polym. Eng. Sci., 20 (1980) 14.
40.
15 in The Netherlands
MECHANISM OF DIFFUSION AND SORPTION OF CARBON DIOXIDE IN POLY(VINYL ACETATE) ABOVE AND BELOW THE GLASS TRANSITION TEMPERATURE
K. TOI, Y. MAEDA
and T. TOKUDA
Department of Chemistry, Tokyo 158 (Japan)
Faculty
of Science,
Tokyo
Metropolitan
(Received April 13,1982;
accepted in revised form August 10,1982)
University,
Setagaya-ku,
Summary The pressure dependence below 1 atm of the apparent diffusion and permeation coefficients were observed by using the permeation time lag method for carbon dioxide in poly(vinyl acetate), which has a glass transition near room temperature, at temperatures ranging from 8 to 50%. Above the glass transition temperature, pressure dependence of the diffusion and permeation coefficient has not been observed; hence, Fick’s law with a concentration independent diffusion coefficient applies. On the other hand, in the glassy state, the apparent diffusion coefficient shows pressure dependence. Moreover, the behavior of the pressure dependence does not show a clear curve in the ranges between 30% to 17%. Above 17’C, the apparent diffusion coefficients show discontinuities, but below 17°C increase with pressure is regular. Using the theoretical prediction of Paul, a computer was used in the numerical calculation to determine the true diffusion coefficient and other dual sorption parameters. The compensated diffusion coefficients controlled only by Henry’s law dissolution was described by three straight lines with two intersection in the form of Arrhenius plots, which give good agreement with both our results for He and Ar and those of Meares. It is assumed that beside the dual sorption mechanism, another effect, for instance some relaxation effect may also contribute to the diffusion for carbon dioxide in poly(viny1 acetate) near the glass transition temperature region.
Introduction
The first systematic measurements of gas transport in a polymer above and below the glass transition temperature were those of Meares [l] with poly(vinyl acetate) (PVAc). Two clear breaks in the plots of the diffusion and permeation coefficients were found for the majority of the gases investigated. They formed a less well-defined intermediate region near the glass transition temperature (Tg) with a small slope above or below it. Similar measurements with oxygen were later repeated by Stannett and Williams [2] and a clear break was observed. In general, for amorphous polymers below glass transition temperature, sorption is apparently composed of two mechanisms, the first described by
0376-7388/83/0000-0000/$03.00
0 1983
Elsevier Scientific Publishing Company
16
of dissolution and the second by Langmuir’s law of adsorption. This overall quantitative description has become known today as the Dual Sorption Theory [S-61. Precise measurements of the dual so&ion nhenomena enable one to determine the parameters relating to it. From discussion of the physical meaning of the values, new information concerning the above curious behavior of PVAc in the vicinity of T, could be obtained. In our previous paper [ 71, a method was discussed that can determine the dual sorption parameters numerically from a low pressure permeation experiment without using the data from a sorption experiment. The apparent permeation coefficient was regarded as independent of pressure below 1 atm for several glassy polymers, so pressure dependence of the apparent diffusion coefficient lent support to the total immobilization of the Langmuir component. The values of parameters were obtained at various temperatures for the glassy polymers. Excellent agreement between the calculated curves and experimental plots provided strong support for the validity of the parameters. In this study, the pressure dependence of the apparent permeability coefficient and the apparent diffusion coefficient were determined by observing the steady state carbon dioxide permeation in poly(viny1 acetate) above and below the glass transition temperature. The above method was applied for determining the pressure independent diffusion coefficient and parameters associated with it. An interpretation of the gas diffusion behavior for PVAc near Tg was attempted to understand the modifications introduced by the concept of the dual sorption mode. Henry's law
Experimental The PVAc sample was obtained from Wako Junyaku Manufacturing Co., Ltd. and its degree of polymerization was between 1400 and 1600. PVAc membranes were prepared by casting solutions in toluene on a clean glass plate, which was floating on mercury. Before use, all membranes were dried under high vacuum at 50°C for a few days. The membranes used, along with their dimensions, are listed in Table 1. The glass transition temperature of PVAc with CO* atmosphere, determined by using a Rigaku Denki DSC Thermal Analyser (Model SOOO),was found to be 32°C. The densities of the membranes were determined by the density-gradient method in a A1C13 solution. The membrane thicknesses were determined by weighing the fixed area. The following high purity commercial gases were used in this study: helium, argon and carbon dioxide with purities of larger than 99.5%. No attempt was made to purify the gases before use. Details of the permeation equipment have been described earlier [7,8]. A digital printer was connected to MKS Baratron Capacitance Manometer (Model 170-6B using a model 310BH-1 sensor head) through an analog to digital converter. Downstream pressure changes were printed between 10v6 and lo-* cmHg using a digital printer at regular time intervals with a precision
17 TABLE 1 Dimensions of membranes Membrane (Gas)
Thickness (cm)
Area (cm’)
Density (g/cm’) .~
AA EE(Ar) FE(C0,‘) DE(C0,)
14.15 4.92 3.25 2.19
7.07 3.04 2.56 2.56
1.176 1.176 1.176 1.176
x x x x
1o-3 1o-3 1O-3 1o-3
.__
of l/100 sec. A slope and intercept of the linear asymptote line was calculated by the least squares method through 30 to 50 experimental points taken from a 4 times to 5 times time lag. The precision of the slope and the time lag of the permeation curve using this method is greater than 0.1% and 0.576, respectively. Results and discussion Pressure dependence of apparent permeability coefficients of Ar and CO:! in the PVAc membranes are presented in Figs. 1 and 2 at various temperatures, 21.5
24.0 22.0
"J a 25.3 c ;
22.5 a
28.7'C
..*
25.z.C
~ _
19.9'C %.5"C
26.4"C
n
-Q)-;
23.C
30.4'C
m..
rPn
--O
0
""
0 2.0
4.0 l/p (atm-1)
6.0
0
15.2'C
or\ gr 26.0
21.9;c 0
2.0
n
lU.I"C 4.0
6.0
1/F (at,!,-' )
Fig. 1. Pressure dependence of permeability coefficients for argon in PVAc(Membrane EE) at various temperatures. Fig. 2. Pressure dependence of permeability coefficients for carbon dioxide in PVAc(Membrane DE) at various temperatures.
18
reasons given in the previous paper [ 81, the abscissa is indicated by the reciprocal of the upstream pressure(l/atm). The corresponding apparent diffusion coefficients, II,, directly obtained from the time lag (D, = 12/68; 1 is membrane thickness (cm), and 6 is time lag (set).) are shown in Figs. 3, 4, and 5. Permeability and diffusion coefficients of Ar in PVAc (Figs. 1 and 3) were independent of pressure in the upstream pressure range between 0 to 1 atm. This indicates that the diffusion of Ar in PVAc is described by Fick’s and Henry’s law not only above, but below, T,. For
I
I 2.0
4.0 l/p (atm-')
6.0
2
4
b
l/p (atIn-1)
Fig. 3. Pressure dependence of diffusion coefficients for argon in PVAc(Membrane at various temperatures.
EE)
Fig. 4. Pressure dependence of apparent diffusion coefficients for carbon dioxide in PVAc (membrane FE) above glass transition temperature. The points are observed values and the curves are calculated from eqn. (1) with parameters in Table 2,
The permeability coefficients of CO1 in PVAc (Fig. 2) are also independent of the upstream pressure at low temperature. The pressure dependence at higher pressure above Tg was entirely unexpected. Probably, this is due to a mutual interaction between the polymer and the penetrant at these conditions, rather than an effect through the measurement itself.
19
2.0
4.0
6.0
l/P (at"-')
Fig. 5. Pressure dependence of apparent diffusion coefficients for carbon dioxide in PVAc(Membrane DE) below glass transition temperature. The points are observed values and the curves are calculated from eqn. (1) with parameters in Table 2.
On the other hand, the apparent diffusion coefficients, Da, are independent of upstream pressure above !?‘s but dependent on pressure below Tg, as shown by plots in Figs. 4 and 5. These results are entirely consistent with a model which assumes total immobilization of the Langmuir population as indicated by Paul et al. [4,5], As strategy to test these explanations, we will deduce all parameters from the diffusion data, using eqn. (1) proposed by Paul [ 91. D =Da[l
+ Kf(y)I
f(y) = W3 I-(ll2)~’
(1) + Y - (1 + Y)ln(l + ~11
where y = bp, K = GH’b/k JJ, and Da = 1*/6’B: kD, GH’, and b are-adjustable parameters; D is the true diffusion coefficient for Henry’s law species and p is the upstream gas pressure. The parameters of the dual sorption model, according to this equation, were deduced from each experimental set of diffusion data using a nonlinear least-squares optimization program. Some of the results are listed in Table 2. Using the values of parameters b, K, and D given in Table 2, the calculated values of Da at each temperature are shown in Figs.
20
TABLE2 Dual sorptionparameters Membrane
T
kD
CC) (
DE DE DE DE DE FE DE FE FE DE FE FE FE FE
10.7 11.5 12.8 15.2 16.7 21.8 21.9 23.8 26.4 26.5 28.3 29.6 31.0 32.5
for CO,in PVAc CH'
cm3 (STP) cm3(polymer)-atm
2.48 2.43 2.35 2.20 2.12 1.90 1.85 1.81 1.86 1.88 1.88 1.87 1.85 1.96
cm3 (polymer)
cmJ(STP)
b
K
DX
(ah-')
(CH'b/kD)
(Cma/SeC)
lo9
2.36 2.35 2.32 2.27 2.24 2.42 2.44 2.57 3.10 3.19 3.21 3.41 3.80 3.99
0.820 0.809 0.791 0.736 0.735 0.675 0.669 0.650 0.455 0.419 0.305 0.300 0.290 0.136
2.69 2.79 2.98 3.34 3.56 4.50 4.55 4.95 5.18 5.09 5.55 5.86 6.33 6.54
1
0.864 0.838 0.803 0.736 0.697 0.531 0.507 0.459 0.273 0.248 0.179 0.165 0.141 0.067
17.5
18.0
19.0
3.2
3.3
3.4 1,~
x lo3
3.5
(K-l!
Fig. 6. Arrhenius plots of apparent EE) at p = 0.4 atm.
diffusion
coefficients
for argon
in PVAc(Membrane
4 and 5 by solid lines, together with the plots of experimental values. The coincidence between the measured and predicted D, is impressive at lower and higher temperatures but not at intermediate temperatures. Thus there may
21
l/T x lo3
(K-1)
Fig. 7. Arrhenius plots of diffusion coefficients for carbon dioxide in PVAc: (0,~~) apparent diffusion coefficients observed at p = 0.4 atm; ( l ,A) pressure independent diffusion coefficients calculated by using total immobilization model.
be a temperature region where the experimental values and the calculated curves are not quantitatively consistent with each other, and in this region the concept of the total immobilizationmodel alone cannot satisfactorily represent the experimental results. It is necessary to compensate the effect of dual sorption in order to investigate another effect. Temperature dependence of the apparent diffusion coefficients for Ar and CO2 measured at 0.4 atm upstream pressure are shown in the form of Arrhenius plots in Figs. 6 and 7. As shown in Fig. 6, the experimental plots of Ar lie on three distinct straight lines with two intersections. Similar behavior was observed in the Arrhenius plots of permeability coefficients for He, as shown in Fig. 8. However, the curves of diffusion and permeability coefficients of CO2 do not show any clear break point but change smoothly over a range of the temperature. By employing the steady state permeation method, Meares has measured the diffusion coefficients of several gases in PVAc [l] . He found that the logarithmic plots of diffusion, permeability, and solubility coefficients against reciprocal temperature showed two transition temperatures, about 10°C apart. The present data for Ar and He closely agree with Meares.
22
p" c 7
22.0
-
72.5
-
23.0
-
3.2
3.3
3.4 l/T
v 1O-3
3.5
(K-l)
plots of apparent Fig. 8. Arrhenius in PVAc at p = 0.4 atm.
3.0
3.2
.
permeation
coefficients
for helium
and carbon
dioxide
0 DE A
ff
0
DL
A
FE
_
3.2
3.3 l/T
3.4 x lo3
3.5
(K-l)
Fig. 9. Van ‘t Hoff plots of solubility rent solubility coefficients calculated coefficients calculated from Henry’s
coefficients for carbon dioxide in PVAc: (= ,a) appasolubility from eqn. (2 ); ( l , A) pressure independent law parameter (k~,/76).
23
Figure 9 presents van ‘t Hoff plots of the apparent for COz in PVAc, calculated by eqn. (2).
solubility
coefficients
s, =PaDa
(2)
The figure shows that the slope changes near the glass transition temperature of the polymer, which differs from the apparent permeability and diffusion coefficients. The Henry’s law coefficient calculated from the optimization method, is also illustrated in this figure. It is found that the Henry’s law coefficient decreases rapidly near Tg, and is replaced by another term, which may be the Langmuir contribution. Although the plots of P, and D, for CO2 in PVAc did not show any break point, the clear inflection of S,, and the characteristic pressure dependence of D, in the range between 17 and 32°C have some correlation with Meares’ suggestion at the glass transition region. The dual sorption parameters K, CL, and b for CO? in PVAc are plotted as function of temperature in Figs. 10, 11, and 12. The Langmuir capacity parameter CH’ begins to appear near the glass transition temperature and increases as the temperature decreases. The composite parameter K = &‘b/h~ also shows a similar tendency to CH’. In contrast to CH’ and K, the Langmuir affinity constant b, plotted in van ‘t Hoff form, changes discontinuously near 19°C as shown in Fig. 12. We do not have any precise explanation of this surprising discontinuity, but this figure shows that some effect, other than the total immobilization effect, may take place above 19°C. According to high pressure experiments by Paul et al. [ 51, the absolute values of CH’ are related’to the expansion coefficient difference above and below Tg. It is also known, however, that the non-linear least squares fitting techniques used to obtain the sorption parameters are not sensitive to the individual values of CH’ and b but to the product CH’~. Thus the absolute values of CH’ and b will not be pursued further here.
10
I5
(-cp5
20
30
35
T
Fig. 10. Temperature Fig. 9.
dependence
of Langmuir
capacity
constants
CH’. Symbols
as in
10
15
20
75
30
35
T ["C!
Fig. 11. Temperature
dependence
of composite
parameter
K. Symbols
as in Fig. 9.
1.3 -
1.2 -
Fl.1 E
-
2 n
1.0 -
0.9 -
0.8 -
3.3
3.5
3.4 l/T Y lo3 (K-l)
Fig. 12. Van ‘t Hoff plots of Langmuir
affinity
constants
b. Symbols
as in Fig. 9.
The enthalpies of the Henry’s law parameter AH and Langmuir void affinity parameter, q, evaluated from slopes of Figs. 9 and 12 are listed in Table 3. The sorption enthalpy for the Henry’s law species showed smaller values relative to the Langmuir species below 19°C. There are large differences in the values of kD above and below Tg, but the enthalpy at both regions indicated very similar values. The values, along with the associated energy parameters above and below Tg, are shown in Table 4. The compensated diffusion coefficients controlled only by Henry’s law dissolution were plotted, together with the apparent ones in the form of Arrhenius type, in Fig. 7. Compared with the smoothed curve for the appa-
25 TABLE 3 Van ‘t Hoff parameters of sorption for CO, in PVAc 4
(kcal/mol)
above T,
below T,
-4.8
-4.4
-
-1.4
(below 19°C)
TABLE 4 Arrhenius parameters of diffusion for gases in PVAc
D
ED
(kcal/mol)
Ar (above Tg) Ar (below ‘I’s) CO, (above Tg) CO, (below Tg )
16.4 8.1 17.8 7.8
.-
(cm’lsec) 8.2 x lo5 2.5x 2.6x
lo4 lo3
rent diffusion coefficient, the new curve was described by three straight lines with two intersections. The form of this curve gives good agreement with our results for He and Ar and those of Meares [ 11. That is to say, removing the dual sorption effect, the true diffusion coefficient of CO2 for PVAc shows the same intermediate region as He and Ar in Arrhenius plots. According to the equilibrium sorption behavior for He and Ar in other glassy polymers, it is thought that the Langmuir term contributes much less to the sorption process [ 51. Then, the apparent diffusion coefficient, calculated by the low pressure permeation method, is nearly equal to the true diffusion coefficient. The fact that the form of the CO2 curve is in fair agreement with those of He and Ar demonstrated that the true diffusion coefficient can be well separated from the apparent diffusion coefficient by the optimization process. Here, we must conclude that this region cannot be interpreted solely by the effect of the dual sorption, but another effect, for instance some relaxation effect must occur. A similar behavior pattern has also been observed for CO2 in poly(ethylene terephthalate) by Koros and Paul [lo] . They showed that in this region a Fickian description of the transport process is unsatisfactory, and the observed time lag near T, is lengthened by the dynamics of non-instantaneous chain rearrangement. Conclusion From the above results and discussion, it is concluded that besides the dual sorption mechanism, some relaxation process may also contribute to the dif-
26
fusion for CO1 in PVAc near the T, region. That is to say, in the range between 17 and 32”C, even if the main chain of PVAc is completely frozen, the side chain may have some mobility, and the time scale for relaxation may approach that for diffusion. Then, it will become necessary to use more accurate time lag data and complimentary sorption measurement in order to employ an analysis by consideration the time dependence of diffusion in this temperature region. Acknowledgement The authors acknowledge useful discussions with Dr. D.R. Paul. List of symbols Symbol
Name of symbol
Common unit
Definition by SI
b
Hole affinity constant
(101325 Pa))’
C
Concentration
atm-’ cm3 (STP) cm3 (polymer) cm3 (STP) cm3 (polymer) cm2 /set
CH’
D D, DCI
AED
AH
K kD 1 PO P 4
8,
Hole saturation constant Diffusion coefficient for Henry’s law species Apparent diffusion coefficient Pre-exponential factor in Arrhenius equation for diffusion Activation energy for diffusion Heat of solution for Henry’s law species Composite parameter (K = CH’bjkD) Henry’s law constant Membrane thickness Downstream pressure Upstream pressure Heat of adsorption for Langmuir’s law species Apparent sorption coefficient obtained by S, = Pa/D,
44.613 mol/m3 44.613 mol/m3 10m4m2 /s
cm2 /set
10m4m* /s
cm2 /set
10e4 m2 /s
kcal/mol
4.184 kJ/mol
kcal/mol
4.184 kJ/mol
dimensionless cm3 (STP) cm3 (polymer)-atm cm cmHg atm kcal/mol
cm3 (STP) cm3(polymer)- _ cmHg
4.403 X low4 mol/m3Pa lo-* m 1333.224 Pa 101325 Pa 4.184 kJ/mol
3.34625 X 10m2mol m3Pa
27
pa
Apparent permeability coefficient
e
Diffusion
time lag
cm3 (STP)-cm
3.34625X
cm* -set-cmHg see
s
10whmol
m2 sPa
References 1 P. Meares, 76 (1954) P. Meares,
The diffusion of gases through polyvinyl acetate, J. Amer. Chem. Sot., 3514. The solubilities of gases in polyvinyl acetate, Trans. Faraday Sot., 54 (1958)
V. Stannett and J.L. Williams, The permeability of poly(ethy1 methacrylate) to gases and water vapor, J. Polym. Sci., Part C, 10 (1966) 45. W.R. Vieth, J.M. Howell, and J.H. Hsieh, Dual sorption theory, J. Membrane Sci., 1 (1976) 177. D.R. Paul and W.J. Koros, Effect of partially immobilizing sorption on permeability and the diffusion time lag, J. Polym. Sci., Polym. Phys. Ed., 14 (1976) 675. and transport of various gases in 5 W.J. Koros, A.H. Chan, and D.R. Paul, Sorption polycarbonate, J. Membrane Sci., 2 (1977) 165. and transport in glassy polymers, Ber. Bunsenges. ?hys. Chem., 6 D.R. Paul, Gas sorption 83 (1979) 294. of diffusion coefficient for CO, in glassy polymers, 7 K. Toi, Pressure dependence Polym. Eng. Sci., 20 (1980) 30. and T. Tokuda, Isotope effect in the diffusion of hydrogen and 8 K. Toi, K. Takeuchi, deuterium in polymers, J. Polym. Sci., Polym. Phys. Ed., 18 (1980) 189. adsorption on the diffusion time lag, J. Polym. 9 D.R. Paul, The effect of immobilizing Sci. Part A-2, 7 (1969) 1811. 10 W.J. Korosand D.R. Paul, Sorption and transport of CO, above and below the glass transition of poly(ethylene terephthalate), Polym. Eng. Sci., 20 (1980) 14.
40.