Polymer Structure and the Compensation Effect of the Diffusion Pre

Feb 23, 2007 - In the present work, the relation between the pre-exponential factor and the apparent activation energy of diffusion, ln D0 )R+ βED, s...
0 downloads 0 Views 200KB Size
Download PDF
2828

J. Phys. Chem. B 2007, 111, 2828-2835

Polymer Structure and the Compensation Effect of the Diffusion Pre-Exponential Factor and Activation Energy of a Permeating Solute Ju-Meng Zheng,† Jun Qiu,‡ Luis M. Madeira,† and Ade´ lio Mendes*,† LEPAE, Chemical Engineering Department, Faculty of Engineering, UniVersity of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal, and Institute of Polymer Research, GKSS Research Center, Max-Planck-Strasse 1, D-21502 Geesthacht, Germany ReceiVed: NoVember 17, 2006; In Final Form: January 17, 2007

In the present work, the relation between the pre-exponential factor and the apparent activation energy of diffusion, ln D0 ) R + βED, so-called compensation effect, is re-examined and critically discussed for diffusion of gases in rubbery and glassy polymers. In principle, the above equation could be derived from the enthalpyentropy compensation in the framework of the transition state theory. However, one should consider the influence of the jump length term contained in the pre-exponential factor, which may be affected by permeating species and polymer properties. We found that parameter R depends on penetrant size and polymer properties, such as local chain mobility and free volume. This can be interpreted by the fact that the jump length is affected by both penetrant and polymer properties. Finally, methods for estimating the jump length are discussed.

1. Introduction Gas diffusion in polymeric membranes has been extensively studied in the past decades. An interesting finding is the observation of a good linear relationship between the logarithm of the pre-exponential factor (D0) and the apparent activation energy of diffusion (ED) in some systems. Such a relationship was first noted by Barrer1 and then studied, discussed, and summarized in a number of papers.2-11 Up to now, a well-accepted expression relating D0 and ED is

ln D0 ) R + βED

(1)

where R and β are fitting constants. Equation 1 is slightly different from the original equation suggested by Barrer1

ln D0 ) R + β′ED/T

(2)

The reason why eq 1 is more correct is discussed by Prabhakar and co-workers.11 Equation 1 is often referred as “compensation effect”9,12 or “linear free energy relationship”.6,10 The name “linear free energy relationship” may come from the fact that eq 1 is similar to other linear free energy relationships (also known as entropyenergy compensation), which can be seen in a range of physical processes.13 However, we prefer the name “compensation effect” since the pre-exponential factor D0 also depends on the jump length, which cannot be derived from the linear free energy relationship. “Compensation effect” means that the higher ED can be partially offset by higher D0 (ln D0).9 Basically, eq 1 is used to relate: (i) experimental ln D0 and ED data for all penetrants and polymers; (ii) data of different penetrants in a single polymer; and (iii) data of a given penetrant in several polymers. For example, in the study of gas transport in rubbery polymers, Kwei and Arnheim6 found a simple * To whom correspondence should be addressed. E-mail: mendes@ fe.up.pt. Phone: +351 22 5081695. Fax: +351 22 5081449. † University of Porto. ‡ GKSS Research Center.

correlation that can be applied to all these three cases with small error (with the exception of hydrogen). Similarly, van Krevelen9 proposed the correlation ln D0 ) -18.4206 + 0.2770 × 10-3ED (D0 in m2 s-1, ED in J mol-1) for six light gases (He, H2, O2, N2, and CH4) and rubbery polymers and another correlation, ln D0 ) -20.7232 + 0.2770 × 10-3ED, for these gases and glassy polymers. The intercept value (R) of the correlation for rubbery polymers is higher than that for glassy polymers, in consistence with the well-known fact that rubbery membranes have higher diffusion coefficients. However, it has been observed by several authors that the data for gases like He and/or H2 lies above the line for other gases in a ln D0 vs ED plot.2,6-8 Besides, Lundstrom and Bearman7 found that a good ln D0 vs ED fitting could only be obtained when applied to: (1) different gases in a single polymer; or (2) a single gas in several polymers. These authors studied the diffusion of five inert gases in nine polymers and suggested that the fitting parameter R is dependent on the gas and β is dependent on the polymer.7 On the other hand, eq 1 suggests that gas diffusivity and selectivity is governed by the activation energy of diffusion. However, Koros and co-authors14,15 noted that the entropy and entropy selectivity also plays an important role in the membrane performance. Their work also revealed that the entropy and entropy selectivity is tightly related to polymer structure.15 Hence, one may conclude that the fitting parameters in eq 1 are sensitive to polymer properties. In the present study, we re-examine and critically discuss the compensation effect based on experimental transport data obtained from an extensive literature survey. We addressed particularly the ln D0 and ED compensation in glassy polymers since it is not well studied. On the other hand, we studied how the compensation is related to polymers properties. 2. Background Diffusion of gases in polymeric membranes is an activated process. The temperature dependence of the gas diffusivity (D) can be described by the Arrhenius equation in a narrow

10.1021/jp067661o CCC: $37.00 © 2007 American Chemical Society Published on Web 02/23/2007

Studies of a Permeating Solute

J. Phys. Chem. B, Vol. 111, No. 11, 2007 2829

temperature range

( )

D ) D0 exp -

ED RT

(3)

where ED is the activation energy of diffusion (J mol-1), D0 is the pre-exponential factor (m2 s-1), T is the absolute temperature (K), and R is the gas constant (J mol-1 K-1). At the same time, the transition state theory states that the diffusion coefficient is given by16

( ) (

)

∆Sq 1 kBT ∆Hq 1 exp exp D ) λ2φ ) λ2 6 6 h R RT

(4)

where λ is the jump length (m), φ is the jump frequency (s-1), ∆Sq (J mol-1 K-1), and ∆Hq (J mol-1) represent the entropy and enthalpy difference, respectively, between activated and normal states. Finally, kB (J K-1) and h (J s) are the Boltzmann and Planck constants, respectively. By comparison of eqs 3 and 4 and bearing in mind that the activation energy is defined as17

ED ) RT2

d(ln D) dT

(5)

we obtain

ED ) ∆Hq+ RT

( )

1 kBT ∆Sq D0 ) eλ2 exp 6 h R

(6) (7)

If one assumes that the enthalpy-entropy compensation, ∆Sq ) k1 + k2∆H q (where k1 and k2 are fitting constants), is held (an extensive discussion about the enthalpy-entropy compensation can be found in a review article),13 we obtain

(

)

k1 k2 kBT 1 ln D0 ) ln eλ2 + - k2T + ED ) R + βED 6 h R R

(

R ) ln

)

k1 1 2 kBT eλ + - k2T 6 h R β)

k2 R

(8)

In this way, we can relate the fitting parameters in the compensation correlation (eq 1) with the fitting parameters in the enthalpy-entropy compensation equation. As it can be learned from eq 8, β should be constant if k2 is constant. For parameter R, things are more complex. Besides k1, R is also affected by two other factors, the temperature and the jump length. The influence of temperature may be neglected for a narrow temperature range. However, the jump length influence on R might be significant. First, R depends on the logarithm of the jump length squared. Second, the jump length is affected by the polymer properties and penetrant size, which will be deeply discussed below. For instance, according to Meares’s calculations,3 the jump length is around 27 Å for all the gases in polyvinyl acetate (PVAc) at rubbery state but is about 9 Å when PVAc is in glassy state. With these two values and at 300 K, the term ln((1)/(6)eλ2(kBT)/(h)) equals -1.74 (jump length 27 Å) and -3.94 (jump length 9 Å), respectively. Such a difference (-1.74 vs -3.94) is certainly noticeable. Besides, we also would like to point out that the jump length cannot be

Figure 1. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) for six gases in amorphous rubbery polymers (data from refs 18-21).

separately estimated from the fitting of ln D0 and ED data since the obtained fitting parameter R also depends on k1. In short, we showed that the observed ln D0 vs ED relationship is a particular case of the enthalpy-entropy relationship in the framework of the transition-state theory. The physical meaning and validation of the enthalpy-entropy relationship is beyond the scope of the present study. For more information about the enthalpy-entropy compensation, the reader can refer to a recent review by Liu and Guo.13 However, we would like to emphasize that the jump length plays an important role in the ln D0 vs ED relationship. 3. Literature Data Analysis We have collected experimental data of ln D0 and ED from refs 18-33 for gas diffusion in different membrane materials, i.e., 29 rubbery polymers and 43 glassy polymers. The rubbery polymers include poly(diene)s, poly(siloxane)s, and poly(alkane)s. The glassy polymers comprehend poly(phenylene sulfone imide)s, poly(carbonate)s, poly(arylene ether)s, poly(aryl ether sulfone)s, poly(aryl ether ketone)s, poly(amideimide)s, poly(acetylene)s, and polypyrrolones. We are particularly concerned with groups of polymers with similar molecular structures since it is interesting to study how the polymer structure affects the ln D0 vs ED relationship. For instance, the collected data includes silicone polymers19 and poly(acetylene)s33 substituted with different functional groups, poly(ethylene)s22 with different crystallinities and polypyrrolones30 (6FDA/PMDA-TAB copolymers) with different PMDA fractions In these references, when the authors give direct D0 and ED values, the data are cited and used directly, otherwise D0 was calculated based on the Arrhenius equation and on the activation energies of diffusion and diffusivities (eq 3). Basically, we are concerned with the diffusion of six light gases, He, H2, O2, N2, CO2, and CH4 in polymers, both rubbery and glassy. The data of some other gases/vapors are also analyzed when necessary. 4. Results and Discussion 4.1. Behaviors in Rubbery Polymers. Several authors have found that the ln D0 vs ED relationship can be applied to all gases and rubbery polymers. The reliability of such finding is examined in Figure 1. In this figure, we plot ln D0 as a function

2830 J. Phys. Chem. B, Vol. 111, No. 11, 2007

Figure 2. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) for CH4 and C3H8 in silicone rubbers, butyl rubber, and natural rubber (data from refs 19 and 22).

of ED with the data obtained for diffusion of He, H2, O2, N2, CO2, and CH4 in amorphous rubbery polymers. Except the two points concerning poly(phenylmethylsiloxane) (PPhMS) (the author did not report the data for He, H2, O2, and N2),19 all other data exhibit quite good linear relationship. The obtained correlation is ln D0 ) -18.0212 + 0.2598 × 10-3ED. Considering that the data have been collected from different sources and that they have some experimental uncertainty, the linear fitting seems to be quite good. The correlation obtained is close to the one suggested by van Krevelen,9 ln D0 ) -18.4206 + 0.2770 × 10-3ED, and the one proposed by Prabhakar et al.,11 ln D0 ) -17.5103 + 0.2406 × 10-3ED. The slight differences between these three correlations should be ascribed to the different databases used. The unusual behavior of PPhMS observed in Figure 1 will be discussed later. From Figure 1, one can see however that the points of He and H2 are above the dot line. This result was also noticed by several other authors.2,7 Besides the experimental error (it is usually difficult to measure ED for He and H2), it may suggest that the size of the penetrant affects the data location in the plot. In fact, Lundstrom and Bearman7 found that the intercept values in eq 1 are ordered according to the inert gas molecular size (helium, neon, argon, krypton, and xenon), with helium having the largest intercept value when eq 1 is applied to each gas separately in several rubbery polymers. In order to further study such a possibility, Figure 2 illustrates the ln D0 vs ED relationship for CH4 and C3H8 (Lennard-Jones collision diameter of 3.758 vs 5.118 Å)34 in seven polymers. As commented by Shieh and Chung,35 the collision diameter is a better parameter than the kinetic diameter for relating diffusion in relatively high flexible rubbery polymers. It can be read from Figure 2 that the line defined by C3H8 is somewhat below the line of CH4. Figures 1 and 2, together with the observations of van Amorgen2 and Lundstrom and Bearman,7 strongly suggest that the molecular size should affect the fitting parameters in eq 1, especially the intercept term. We suspect that the reason why there is no significant discrepancy for N2, O2, CO2, and CH4 is that the collision diameter difference among them is small. According to these observations, we propose ln D0 ) -17.9395 + 0.2774 × 10-3ED for He and H2, and ln D0 ) -18.2673 + 0.2740 × 10-3ED for O2, N2, CO2, and CH4 according to our fitting results.

Zheng et al.

Figure 3. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) for 12 gases in three semicrystalline rubbery polymers and natural rubber (data from ref 22).

As it is shown in Figures 1 and 2, PPhMS (Tg ) -28 °C)19 shows totally different behavior as compared with other rubbery polymers. This may be interpreted by the special structure of PPhMS, with high flexible siloxane (-Si-O-) linkages in the main chain and bulkier phenyl group side chain.19 Moreover, it will be shown below that PPhMS behaves like a glassy polymer, rather than a rubbery one, in which concerns the ln D0 vs ED relationship. It is likely that the short scale motion may be more important in relating the diffusion process. At the same time, it can be learned from Figure 2 that the points of poly(dimethylsiloxane) (PDMS) are a bit of special. This can be due to its extremely high flexibility. Referring to these observations, it indicates that the polymer local chain mobility also affects the ln D0 vs ED relationship. Our analysis of the ln D0 vs ED relationship for rubbery polymers suggests that the fitting parameter R in the compensation correlation (eqs 1 or 8) slightly depends on the gas size and polymer flexibility. If one assumes that the fitting parameters k1 and k2 are constant, it means that R reflects the influence of the jump length, i.e., the jump length increases as the penetrant size decreases and the local polymer chain mobility increases. This argument is well supported by Charati and Stern’s findings.36 These authors studied the diffusion of He, O2, N2, CO2, and CH4 in PDMS, PPhMS, poly(propylmethysiloxane) (PPMS), and poly(trifuloropropyl-methylsiloxane) (PTFMS) by molecular dynamics simulation and concluded that the jump length of a penetrant molecule decreases as its size increases. They also found that the jump length decreases in the polymer order PDMS > PTFPMS > PPhMS, which is also the order of the polymer local chain mobility.36 We also analyzed the ln D0 vs ED relationship for several semicrystalline polymers, based on the data reported by Michaels and Bixler.22 These authors studied the diffusion of 12 gases (He, O2, Ar, CO2, CO, N2, CH4, C2H6, C3H4, C3H6, C3H8, and SF6) in four rubbery polymers with great carefulness. Three of the four polymers are semicrystalline: high-density polyethylene (HDPE) with crystallinity of 77%, low-density polyethylene (LDPE) with crystallinity of 43%, and poly(butadiene) with crystallinity of 29%. The last polymer considered was natural rubber, which is amorphous. Figure 3 illustrates the ln D0 vs ED relationship in these four polymers. It can be seen that the points of LDPE, poly(butadiene), and natural rubber form essentially one group. However, the data for HDPE, the polymer

Studies of a Permeating Solute

J. Phys. Chem. B, Vol. 111, No. 11, 2007 2831

Figure 4. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) in poly(vinyl acetate) below and above its glass translation temperature (data from ref 3).

Figure 5. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) for six gases in glassy polymers (data from refs 23-33).

with 77% crystallinity, are obviously outside that group. As stated by Michaels and Bixler,22 the existence of crystallites affects the diffusion process in two ways. First, the increasing diffusion path ascribes to the necessity of molecules to bypass the crystallites (represented by the tortuosity factor). Second, the presence of crystallites may reduce the mobility of the amorphous segments (expressed in terms of chain immobilization factor). The tortuosity is a function of crystallinities while the chain immobilization factor is affected by crystallinities and penetrant size. According to Michaels and Bixler,22 for gas/ vapor diffusion in LDPE and poly(butadiene), the immobilization factor is in a narrow range, from 1.0 to 2.6 and from 1.0 to 1.4 for LDPE and poly(butadiene), respectively, suggesting a slight influence of the crystallites on the amorphous segments mobility in the cases where crystallinities remained in small levels. However, the immobilization factor ranges from 1.1 for He to 11.5 for SF6 in HDPE,22 indicating a significant restriction of the amorphous segments mobility. This seems reasonable if one considers that the crystalline phase is the continuous phase and the amorphous is the dispersed one in HDPE. Following this analysis, we concluded that the different behavior of HDPE (as compared to the other three polymers) is a result of the reduced amorphous segmental mobility in this highly crystalline polymer. This in turn proves that the chain mobility plays an important role in affecting the ln D0 vs ED relationship. 4.2. Behaviors in Glassy Polymers. Figure 4 illustrates the ln D0 vs ED relationship of five gases (He, H2, Ne, O2, Ar, and Kr) in poly(vinyl acetate) (PVAc), below and above its glass transition temperature. The data are reported by Meares.3 For the data obtained above the glass temperature, the linear relationship fits very well the experimental data. However, the experimental data below the transition temperature scatter around the linear fitting line. This indicates that the ln D0 vs ED relationship is more complex for glassy polymers. On the other hand, the experimental values for glassy PVAc are below the line determined by rubbery PVAc. This can be ascribed to the penetrant jump length reduction. In fact, Meares3 estimated that the jump length is around 27 Å in rubber state and about 9 Å in glassy state. Although being noted that the linear relationship between ln D0 and ED is less accurate, van Krevelen9 proposed a simple correlation, ln D0 ) -20.7232 + 0.2767 × 10-3ED, which is applied to six light gases and all glassy polymers. The correlation is obtained by linear fitting of a total of 39 experimental values.

The validity of the van Krevelen’s correlation is now examined, considering more experimental data (total of 192) in Figure 5. One can see that although the simple trend stands, the data are rather scattered. The correlation obtained in the present work is ln D0 ) -21.6619 + 0.2130 × 10-3ED. We argue that a general correlation between ln D0 and ED is not possible. Again, we see in Figure 5 that the values corresponding to small molecules, He and H2, are obviously above the fitted line. Yampolskii et al.12 have analyzed the relationship between the activation energy of diffusion and diffusivity. It can be inferred from their work that the linear ln D0 vs ED relationship can only be fairly obtained when applied to a single gas. Figure 6 plots ln D0 as a function of ED for O2, N2, CO2, and CH4 in several glassy polymers. The analysis of He and H2 is not available. The experimental data show a large scatter around a linear fitting. In order to further study how the polymer structure affects the ln D0 vs ED relationship, Figures 7-10 re-evaluate such relationship in glassy polymers for O2, N2, CO2, and CH4, separately. The data of rubbery polymers are also presented for comparison. It is interesting to find that, except the values corresponding to seven different polyacetylenes, the other values can be essentially divided into three groups: the values of rubbery polymers (except PPhMS) are in one group (line A) and in the upper part; the data of glassy polymers are generally in two groups (lines B and C). The glassy polymers in group B contain six polycarbonates,23,24 three poly(phenylene sulfone imide)s,25 and nine poly(arylene ether)s.26-28 Group C comprehends four polyamideimides,29 four polypyrrolones,30 six poly(aryl ether sulfone)s,31 and four poly(aryl ether ketones)s.32 For detailed information of these polymers, please refer to refs 23-32. Generally speaking, group B are relative flexible polymers, and group C normally more rigid polymers. In polypyrrolones, the stepladder structure creates a highly rigid backbone.30 Regarding poly(aryl ether sulfone)s,31 the polymer rigidity is enhanced by the introduction of the pendent cardo lactone group or by the introduction of the carboxylic group where the stiffness increases due to the formation of hydrogen bonds between carboxylic groups. Similarly, the rigidity of poly(aryl ether ketones)s benefits from the cardo lactone group.32 For the four polyamideimides,29 the stiffness may be ascribed to the -NHCO- group in the backbone. If one assumes that the jump length in flexible glassy polymers like polycarbonate should be larger than in more rigid polymers, such as the

2832 J. Phys. Chem. B, Vol. 111, No. 11, 2007

Zheng et al.

Figure 6. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) in glassy polymers for (A) O2, (B) N2, (C) CO2, (D) CH4 (data from refs 23-33).

Figure 7. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) for N2 in rubbery and glassy polymers (data from refs 18, 20, and 23-33).

Figure 8. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) for O2 in rubbery and glassy polymers (data from refs 18, 20, and 23-33).

polypyrrolones, it seems reasonable that the values corresponding to rigid polymers are below the line defined by data from flexible glassy polymers. We also would like to mention that the division between flexible and rigid glassy polymers is somewhat arbitrary. For instance, Figure 11 shows the ln D0 vs ED relationship for flexible glassy polymers and four rigid polypyrrolones30 (6FDA/ PMDA-TAB copolymers with different PMDA fractions). The dashed line in Figure 11 represents the linear fitting of the data for flexible glassy polymers. From this figure, it can be seen

that the distances between the values corresponding to polypyrrolones and the dashed line increase in the polymer order: 6FDA/PMDA (10/90)-TAB > 6FDA/PMDA (25/75)-TAB > 6FDA/PMDA (50/50)-TAB > 6FDA-TAB. Such a trend can be interpreted by the enhanced chain packing and by the increased rigidity resulting from the reduction of the flexible 6FDA fractions.30 Figure 12 illustrates the ln D0 vs ED relationship for flexible glassy polymers and six poly(aryl ether sulfone)s in group C. It is clear that the values corresponding to IMPES-L and IMPES-C are close to the fitting line defined

Studies of a Permeating Solute

Figure 9. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) for CO2 in rubbery and glassy polymers (data from refs 18-21, and 23-33).

Figure 10. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) for CH4 in rubbery and glassy polymers (data from refs 18-21, and 23-33).

by the flexible glassy polymers, followed by TMPES-C. Other three values corresponding to PES-C, DMPES-C, and PES-L are significantly below the fitting line. Interestingly, among these six polymers, IMPES-L and IMPES-C are characterized by larger d-spacing values of 6.73 and 6.24 Å, respectively.31 TMPES-C has a moderate d-spacing value of 5.63 Å. The d-spacing values of the other three polymers are around 5 Å. Consequently, the observed trend suggests that the d spacing also plays an important role on the jump length. Summarizing, the results showed in Figures 11 and 12 indicate clearly that the ln D0 vs ED relationship is sensitive to polymer properties, such as stiffness or chain packing. Still regarding Figures 7-10, another interesting finding is that polyacetylenes (all of them are in glassy state)33 exhibit complex behaviors. According to their structure, polyacetylenes can be divided into three groups.33 Nos. 1 and 2 polymers are polyacetylenes with bulky substituent (no. 1 polyacetylene is poly[1-(trimetyllsily)-1-propyne)] (PTMSP)), nos. 3, 4, and 5 polymers are polyacetylenes with long n-alkyl groups, and nos. 6 and 7 polymers are polyacetylenes with phenyl group. Figures 7-10 show that the values corresponding to nos. 5, 6, and 7

J. Phys. Chem. B, Vol. 111, No. 11, 2007 2833

Figure 11. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) for O2 in “flexible” glassy polymers and four polypyrrolones (data from refs 23-28 and 30).

Figure 12. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) for O2 in “flexible” glassy polymers and six poly(aryl ether sulfone)s (data from refs 2328 and 31).

polyacetylenes are basically close to the straight fitting line defined by flexible glassy polymers (group B). However, the values corresponding to nos. 3 and 4 polyacetylenes, which contain n-C7H15 and n-C6H13 side groups, respectively, normally fall in group A, which is defined by rubbery polymers. In contrast, as it can be seen in Figures 9 and 10, the values corresponding to rubbery PPhMS fall in group B, characterized by flexible glassy polymers. The “abnormal” behaviors of rubbery PPhMS with phenyl group and glassy polyacetylenes with long n-alkyl groups suggests that the diffusion process is not affected by the larger scale flexibility of the polymer chains. It is more likely that the diffusion is affected by the local chain mobility. A detailed discussion about the relationship between the gas transport properties and the local chain mobility can be found in the Singla et al. work.37 In what concerns the two polyacetylenes with bulky substituents (nos. 1 and 2), no. 2 polyacetylene is approximately in group A defined by rubbery polymers in Figures 7 (N2 testing gas) and 10 (CH4 testing gas), and it is approximately in group B in Figures 8 (O2 testing gas) and 9 (CO2 testing gas). This is

2834 J. Phys. Chem. B, Vol. 111, No. 11, 2007

Zheng et al. glassy polymers. This assumption is also based on the results by Charati and Stern,36 who used molecular dynamics simulation to study the diffusion of small gases in silicone rubbers. We also propose that the jump length is larger in high free volume polymers such as PTMSP. This assertion seems also reasonable, specially if we accept the lattice model.38,39 However, the jump length still remains a controversial topic among the scientific community. A well-known equation for estimating the jump length is the one proposed by Meares3

ED ) 0.25N0πd2λecoh

Figure 13. Correlation of natural logarithm of pre-exponential factor (ln D0) and activation energy of diffusion (ED) for He, O2, and CH4 in “flexible” glassy polymers (data from refs 23-28).

a strange behavior, which might be related with the precision of the experimental data. For instance, the authors reported ED (kJ mol-1) values of 11.2 for O2, 36.0 for N2, 11.5 for CO2, and 32.7 for CH4.33 In principle there is no reason for these values being so different. The behavior of no. 1 polyacetylene (PTMSP) is extremely interesting. The values corresponding to high free volume rigid PTMSP fall approximately in the line defined by rubbery polymers in Figures 7-10 (if the lines are extrapolated). PTMSP is a super-rigid glassy polymer due to the rigid sCdCs backbone, with bulky trimethylsilyl pending groups.33 Hence, we propose that the unusual behavior of PTMSP is ascribed to its extremely high free volume. If one assumes that the jump length in PTMSP is larger than that in conventional glassy polymers, we can understand the reason why the values corresponding to PTMSP are well above the line defined by conventional glassy polymers. In fact, we have showed in Figure 12 that the jump length is sensitive to d spacing. As it is noted above, the penetrant size plays an important role in the ln D0 vs ED relationship for rubbery polymers. This influence is more evident for such relationship in glassy polymers. In order to illustrate this issue more clearly, Figure 13 plots ln D0 as a function of ED for He, O2, and CH4 in flexible glassy polymers as defined above. It is clear that the values of He are far above the values of O2 and CH4. The gap between the line defined by O2 and the line defined by CH4 is also large. As we mentioned above, this can be explained taking in consideration that the jump length depends on the penetrant size. The larger is the molecular size, the smaller is the jump length. Figure 13 also suggests that it is likely that the jump length dependency on the penetrant size is enlarged in glassy polymers as compared to that in rubbery polymers. Indeed, it was shown previously that the data of O2, N2, CO2, and CH4 are reasonably plotted in a single straight line, ln D0 vs ED, for rubbery polymers (Figure 1). As discussed so far, we sustain that a good ln D0 vs ED linear fitting can only be obtained when the relationship is applied to a group of polymers with similar stiffness and applied to each gas separately. 4.3. The Methods for Estimating the Jump Length. In the above discussion, we proposed that the jump length decreases both with the penetrant size and polymer stiffness. This assumption is based on our analysis of the ln D0 vs ED relationship of different gases in a large number of rubbery and

(9)

where d is the penetrant diameter, N0 is the Avogadro number, and ecoh is the polymer cohesive energy density. It implies that the activation energy of diffusion is equal to the energy needed to create a channel with cross sectional area of πd2/4 and with length equal to the jump length to overcome the intermolecular forces. Although this equation is important and helpful,38 its limitations are obvious. First, apart from the intermolecular force, the intramolecular energy is also important in determining the activation energy of diffusion, as in Brandt’s model.40 Second, when one plots experimental ED data as a function of d2, the line obtained does not passes the origin as eq 9 predicts.38 On the other hand, Alentiev and Yampolskii38 calculated the H2, O2, and CO2 jump length for several glassy polymers with eq 9 and suggested that the jump length decreases with the free volume. For instance, the jump length of CO2 is 33.4 Å in polyvinyl chloride (PVC), 19.4 Å in polycarbonate (PC), and only 0.4 Å in PTMSP.38 However, if we calculate the jump length with the lattice method (further discussed below), the jump length should increase with the polymer free volume. We have also calculated the jump length of CO2 in PDMS, PTFPMS, and PPhMS using the Meares’s equation with the ED value reported by Stern et al.19 and ecoh values from van Krevelen.9 The jump length obtained is 8.5 Å for PDMS, 18.8 Å for PTFPMS, and 17.5 Å for PPhMS. The jump length for the extremely flexible PDMS is smaller than for PTFPMS and PPhMS. This is in contradiction with the results obtained by the molecular dynamics simulation. Actually, Charati and Stern36 found that “it is seen that the length of the trajectory of a diffusing CO2 molecule and the number of jumps decreases in the following polymer order: PDMS > PTFPMS > PPhMS”. The lattice model, together with the recently developed positron annihilation lifetime spectroscopy (PALS), provides other possibilities to estimate the jump length. PALS allows to obtain the free volume hole density (N), and according to the lattice model, the jump length can be estimated directly as

λ ) (1/N)-1/3

(10)

λ ) (1/N)-1/3 - dFVE

(11)

or

where dFVE is the diameter of the free volume element.38 Equation 10 was suggested by Hiltner et al.39 and eq 11 by Alentiev and Yampolskii,38 who call the value calculated by eq 11 as the maximum possible jump length (in other words, λ is replaced by λmax in eq 11).38 Although the difference between the two equations (eqs 10 and 11) is obvious, both suggest that the jump length increases with the free volume (from the calculations based on the N and dFVE and values reported by Alentiev and Yampolskii).38 For instance, the jump length in PTMSP is 23.7 Å according to eq 10 and 10 Å according to eq

Studies of a Permeating Solute

J. Phys. Chem. B, Vol. 111, No. 11, 2007 2835

11. These values are larger than other glassy polymers with low fraction free volume (FFV). In addition to the above-mentioned methods for estimating the jump length, there are other interesting works worth to mention. In Stuk’s segmental mobility model,41 the jump length is calculated directly from λ ) x(6D)/(φ), where the jump frequency φ is estimated using a modified Williams-LandelFerry (WLF) equation by considering the penetrant and polymer interaction for rubbery polymers

φ ) φg exp

{

}

2.303c1(T + ∆T - Tg) c2 + T + ∆T - Tg

(12)

where φg is the jump frequency at the glassy transition temperature (Tg) and c1 and c2 are fitting parameters in the WLF equation, which are characteristic of the particular polymer. ∆T is characteristic of the interaction between penetrant and polymer. Besides, in the typical jump model proposed by GrayWeale et al.,42 the jump length is estimated as a sum of the Lennard-Jones diameter of the side group (Ln) and the diameter of the reactant cavity (d1RS)

λ ) Ln + d1RS

(13)

However, both eqs 12 and 13 contain undetermined fitting parameters, which limits their application. Summarizing, there is no simple method for estimating the jump length. Molecular dynamics simulation may provide an attractive approach, and for sure more work is required on this topic. 5. Conclusions In the present work, we analyzed the compensation relationship between the pre-exponential factor and the activation energy for gas diffusion in a large number of rubbery and glassy polymers. We concluded that: (1) In most amorphous rubbery polymers (except PPhMS), good linear relationships between ln D0 and ED are found for He and H2 and for O2, N2, CO2, and CH4. (2) PPhMS (Tg ) -28 °C) shows totally different behavior as compared to other rubbery polymers due to the bulkier side chain phenyl group. It suggests that short scale motion may be important in relating the diffusion process. (3) The ln D0 vs ED relationship for polymers with low crystalinities, such as LDPE or poly(butadiene), is similar to that for amorphous natural rubber. However, HDPE with 77% crystallinity is noticeably different due to the reduced amorphous segment mobility caused by the high crystallinity. (4) According to the ln D0 vs ED relationship, glassy polymers (except polyacetylenes) can be fairly divided into two groups, flexible and rigid glassy polymers. A practical linear ln D0 vs ED relationship can be obtained when it is applied to a single gas and within the corresponding glassy polymer group. (5) Polyacetylenes show different ln D0 vs ED behaviors which depend on the properties of the substituted group. Polyacetylenes with long n-alkyl groups behave like rubbery polymers, and polyacetylenes with phenyl group behave like flexible glassy polymers. The unusually behavior of PTMSP is ascribed to its extremely high free volume. (6) The various ln D0 vs ED behaviors found for the polymer groups studied can be interpreted if we assume that the jump length decreases with the penetrant size and with the polymer stiffness. This can be supported by the molecular dynamics

simulation work by Charati and Stern.36 However, much work is required to obtain a reasonable estimation of the jump length. Acknowledgment. Zheng is grateful to the Portuguese Foundation for Science and Technology (FCT) for his postdoctoral grant (reference SFRH/BPD/14489/2003). Financial support by FCT through the project POCTI/EQU/44994/2002 is also acknowledged. References and Notes (1) Barrer, R. M. Trans. Faraday Soc. 1943, 39, 48-59. (2) Van Amerongen, G. J. J. Appl. Phys. 1946, 17, 972-985. (3) Meares, P. J. Am. Chem. Soc. 1954, 76, 3415-3422. (4) Barrer, R. M. J. Phys. Chem. 1957, 61, 178-489. (5) Lawson, A. W. J. Chem. Phys. 1960, 32, 131-132. (6) Kwei, T. K.; Arnheim, W. J. Chem. Phys. 1962, 37, 1900-1901. (7) Lundstrom, J. E.; Bearman, R. J. J. Polym. Sci.: Polym. Phys. Ed. 1974, 12, 97-114. (8) Stannett, V. J. Membr. Sci. 1978, 3, 97-115. (9) Van Krevelen, D. W. Properties of Polymers, 3rd ed.; Elsevier: 1990. (10) Freeman, B. D. Macromolecules 1999, 32, 375-380. (11) Prabhakar, R. S.; Raharjo, R.; Toy, L. G.; Lin, H.; Freeman, B. D. Ind. Eng. Chem. Res. 2005, 44, 1547-1556. (12) Yampolskii, Yu.; Shishatskii, S.; Alentiev, A.; Loza, K. J. Membr. Sci. 1989, 148, 59-69. (13) Liu, L.; Guo, Q.-X. Chem. ReV. 2001, 101, 673-695. (14) Singh, A.; Koros, W. J. Ind. Eng. Chem. Res. 1996, 35, 12311234. (15) Singh-Ghosal, A.; Koros, W. J. Ind. Eng. Chem. Res. 1999, 38, 3647-3654. (16) Petropoulos, J. H. Mechanisms and theories for sorption and diffusion of gases in polymers. Polymeric Gas Separation Membranes; Paul, D. R., Yampolskii, Yu. P., Eds.; CRC Press: Boca Raton, FL, 1994; pp 51-81. (17) Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: 1994; p 879. (18) Pauly, S. Permeability and diffusion data. Polymer Handbook, 4th ed.; Brandrup, J., Immergut, E. H., Grulke, E. A., Eds.; John Wiley & Son, Inc.: 1999; VI/543-VI/569. (19) Stern, S. A.; Shah, V. M.; Hardy, B. J. J. Polym. Sci.: B Polym. Phys. 1987, 25, 1263-1298. (20) Nakagawa, T.; Nishimura, T.; Higuchi, A. J. Membr. Sci. 2002, 206, 149-163. (21) Mogri, Z.; Paul, D. R. Polymer 2001, 42, 7781-7789. (22) Michaels, A. S.; Bixler, H. J. J. Polym. Sci. 1961, 50, 413-439. (23) Costello, L. M.; Koros, W. J. J. Polym. Sci. Polym. Phys. 1994, 32, 701-713. (24) McCaig, M. S.; Seo, E. D.; Paul, D. R. Polymer 1999, 40, 33673382. (25) Xu, Z.-K.; Bo¨hning, M.; Springer, J.; Glatz, F. P.; Mu¨lhaupt, R. J. Polym. Sci. Polym. Phys. 1997, 35, 1855-1868. (26) Xu, Z.-K.; Dannenberg, C.; Springer, J.; Banerjee, S.; Maier, G. J. Membr. Sci. 2002, 205, 23-31. (27) Xu, Z.-K.; Dannenberg, C.; Springer, J.; Banerjee, S.; Maier, G. Chem. Mater. 2002, 14, 3271-3276. (28) Banerjee, S.; Maier, G.; Dannenberg, C.; Spinger, J. J. Membr. Sci. 2004, 229, 63-71. (29) Xu, Z.-K.; Bo¨hning, M.; Springer, J.; Steinhauser, N.; Mu¨lhaupt, R. Polymer 1997, 35, 581-588. (30) Zimmerman, C. M.; Koros, W. J. J. Polym. Sci. Part B: Polym. Phys. 1999, 37, 1251-1265. (31) Wang, Z.; Chen, T.; Xu, J. Macromolecules 2001, 34, 9015-9022. (32) Wang, Z.; Chen, T.; Xu, J. Macromolecules 2001, 33, 5672-5679. (33) Masuda, T.; Igchi, Y.; Tang, B.-Z.; Higashimura, T. Polymer 1988, 29, 2041-2049. (34) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. J. P. The properties of gases and liquids, 5th ed.; Mc-Graw Hill: 2000. (35) Shieh, J.-J.; Chung, T.-S. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 2851-2861. (36) Charati, S. G.; Stern, S. A. Macromolecules 1998, 31, 5529-5535. (37) Singla, S.; Beckham, H. W.; Rezac, M. E. J. Membr. Sci. 2002, 208, 257-267. (38) Alentiev, A. Y.; Yampolskii, Y. P. J. Membr. Sci. 2002, 206, 291360. (39) Hiltner, A.; Liu, R. Y. F.; Hu, Y. S.; Baer, E. J. Polym. Sci. B Polym. Phys. 2005, 43, 1047-1063. (40) Brandt, W. W. J. Phys. Chem. 1959, 63, 1080-1085. (41) Stuk, L. G. F. J. Polym. Sci. B Polym. Phys. 1990, 28, 127-131. (42) Gray-Weale, A. A.; Henchman, R. H.; Gilbert, R. G.; Greenfield, M. L.; Theodorou, D. N. Macromolecules 1997, 30, 7296-7306.