Physical aging of thin glassy polymer films: Free volume interpretation

Journal of Membrane Science 277 (2006) 219–229

Physical aging of thin glassy polymer films: Free volume interpretation Y. Huang 1 , X. Wang, D.R. Paul ∗ Department of Chemical Engineering, Texas Material Institute, The University of Texas at Austin, Austin, TX 78712, USA Received 8 August 2005; received in revised form 17 October 2005; accepted 23 October 2005 Available online 29 November 2005

Abstract Previous studies have documented the dramatic effects of film thickness, at least compared to the bulk state, and the more modest effects of aging temperature and polymer structure on physical aging of thin glassy polymer films. In this paper, these results are interpreted in terms of the free volume calculated from the refractive index, determined by ellipsometry, using the Lorentz–Lorenz equation. The relative progress of aging towards the final equilibrium state was determined using this approach. The free volume data were fitted to the Struik model for the self-retarding volume contraction on aging to obtain the thickness and temperature dependent parameters. The gas permeability versus aging time is related to the free volume computed from the refractive index. The effect of prior history on the aging of thin glassy polymer films was briefly explored. © 2005 Elsevier B.V. All rights reserved. Keywords: Physical aging; Thin glassy polymer films; Free volume; Permeability; Refractive index

1. Introduction Thin glassy polymer films exhibit physical aging that is orders of magnitude more rapid than expected for thick films or the “bulk” glassy state. We have recently reported systematic studies to understand the effects of film thickness (
), chemical structure, aging temperature (T), and molecular weight on this aging process using both gas permeability and refractive index measurements [1–4]. Both measurements are related to the aging induced free volume relaxation, as demonstrated by thermoreversibility experiments [5], i.e., the same underlying fundamental structural change is being assessed. However, since each property has a different sensitivity to free volume, there will be apparent differences in the rates of aging depending on the property being monitored [6]. The results indicate that physical aging persists for very long times [1,2]. It seems that there is more than one mechanism governing the loss of free volume [1]. A number of phenomenological theories have been proposed to describe the aging process in terms of free vol-

∗

Corresponding author. Tel.: +1 512 471 5392; fax: +1 512 471 0542. E-mail address: [email protected] (D.R. Paul). 1 Present address: Membrane Technology & Research Inc., 1360 Willow Rd., Suite 103, Menlo Park, CA 94025, USA. 0376-7388/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2005.10.032

ume relaxation [7–13]; these theories are able to qualitatively explain many aspects of the physical aging process but do not offer any insights about why thickness would affect the rate of aging. Indeed, one would not expect rapid aging far below the glass transition for the bulk polymers. Clearly, some key elements of the microscopic mechanisms of physical aging are not yet sufficiently understood. It has been hypothesized that the thickness dependence of the aging process stems from diffusion of free volume to the surface of the glassy film where it can escape [14–20]. Attempts to model the experimentally observed permeation decline with aging time over a wide range of thicknesses strongly suggest that the more well understood process of “lattice contraction” must also be involved [19,20]. The basic concept of free volume diffusion, while quite appealing, has not yet been confirmed by any direct evidence. A high priority for future work must be to resolve these basic mechanistic issues by some suitable technique. The purpose of this paper is three-fold: (1) to apply the Struik lattice contraction model of physical aging [8], which does not implicitly address the thickness dependence, to the thin film data developed in this work, (2) to gain some insight about the aging process from the existing experimental results, and (3) to explore the quantitative correlation between the gas permeability and the refractive index measurements via their relations to free volume.

220

Y. Huang et al. / Journal of Membrane Science 277 (2006) 219–229

2. Experimental Three different glassy polymers were used in this work: the polysulfone based on bisphenol A (PSF), a polyimide known commercially as Matrimid® 5128, and poly(2,6-dimethyl-1,4phenylene oxide) (PPO). These polymers are of interest as membrane materials and structurally they are related in that the repeat unit of each one contains phenylene residues. Detailed descriptions of these polymers have been given previously [1]. The bulk properties of these polymers pertinent to this work are presented in Table 1. The bulk fractional free volume (fb ) was estimated from the bulk density (ρb ) and the van der Waals volume (Vw ) computed by a group contribution method [21]. The progress of aging was monitored by gas permeability and refractive index measurements using a constant-volume permeation cell and ellipsometry, respectively, at three aging temperatures following the procedures described earlier [1,2,5]. In most cases, films made by spin casting were heated above Tg as free standing films and then quenched to the measurement temperature using the protocol described previously [5]. In one experiment, an as-cast PSF film, obtained by solution casting, was heated to 100 ◦ C for 15 min immediately after formation to remove any residue solvent. These films were then subjected to physical aging using the methodology developed for this work [1,2]. 3. Calculation of free volume from ellipsometry measurement −1 The Lorentz–Lorenz parameter L = nn2 +2 calculated from the refractive index determined by ellipsometry is related to density by L = ρC [2], where the material constant C (see Table 2) can be obtained from the bulk values of refractive index and density (at 25 ◦ C) for each polymer given in Table 1. Thus, the density at any aging time (t) can be calculated from the measured refractive index, and consequently, the fractional free volume, f, at any time during the aging process can be determined as 2

Fig. 1. Fractional free volume as a function of aging time at 55 ◦ C for PSF thin films (symbols: experimental data; solid lines: the best fit to the Struik model).

follows: f =

V − Vo L = 1 − ρVo = 1 − Vo V C

(1)

where V = 1/ρ is the specific volume at that aging time and Vo is the occupied volume of the polymer computed from the van der Waals volume, Vw , by the Bondi method [22], i.e., Vo = 1.3Vw ; values of Vo are given in Table 2. Note that any error in estimation of C or Vo will be transferred into the calculation of fractional free volume reported here. Figs. 1–3 give examples of the fractional free volume obtained in this way for thin films of PSF, Matrimid® , and PPO at various temperatures (shown as symbols); values at other temperatures can be found in previous publications [3,4,23]. The fractional free volume decreases

Table 1 Bulk physical properties of polymers studied Polymer

ρb (g/cm3 )

Refractive index

Vw (cm3 /g)

fb = 1–1.3ρb Vw

PSF PPO (LMW) Matrimid®

1.24 1.069 1.20

1.633 1.573 1.653

0.531 0.588 0.532

0.144 0.183 0.170

Table 2 Various volumetric parameters for the polymers studied Polymer

PSF

PPO

Matrimid®

C (cm3 /g) Vo (cm3 /g) Vi (cm3 /g) fi Ve (cm3 /g) fe

0.288 0.691 0.811 0.1483 0.773 0.1061

0.309 0.764 0.948 0.1942 0.872 0.1242

0.305 0.692 0.840 0.1760 0.780 0.1131

Fig. 2. Fractional free volume as a function of aging time at 35 ◦ C for Matrimid® thin films (symbols: experimental data; solid lines: the best fit to the Struik model).

Y. Huang et al. / Journal of Membrane Science 277 (2006) 219–229

221

decrease in free volume, f df =− dt τ∞ exp(−γ f )

Fig. 3. Fractional free volume as a function of aging time at 45 ◦ C for PPO thin films (symbols: experimental data; solid lines: the best fit to the Struik model).

gradually with aging time and at a higher rate for thin films than thick films as expected from prior work. These moderate changes in fractional free volume lead to rather profound changes in gas permeability due to the exponential relation between the two quantities [24]. 4. Model for physical aging rate As mentioned earlier, most aging theories were developed for bulk systems and do not consider any size effect on physical aging. The loss of free volume in such cases may be thought of as having the chain segments collectively becoming closer to one another, i.e., a process that has been called “lattice contraction”. Among these theories, the well-known Struik model [8] proposed from Kovacs’ first-order isothermal volume relaxation concept of bulk polymers [10] describes aging as a self-retarding process where the relaxation time rapidly increases due to the

(2)

where f = f − fe is the instantaneous fractional free volume with respect to the equilibrium value fe , γ is a constant, and τ ∞ is the relaxation time at equilibrium for the aging temperature T. The value of τ ∞ has been described as being on a geological time scale because of the exceedingly slow molecular rearrangements as equilibrium is approached [25]. The Struik model has been successfully used to describe the volume relaxation of bulk polymers in several studies [26,27]. Here, the Struik model is applied to the free volume change in thin films; however, clearly, the parameters, τ ∞ and γ, will depend on thickness and must be regarded as nothing more than fitting parameters. Eq. (2) can be solved using the integral exponential function
∞ −u e Ei (−x) = − du (3) u x to give Ei (−f |t ) − Ei (−f |t=0 ) =

t τ∞

(4)

where f|t = 0 = fi − fe ; the quantity fi is the initial fractional free volume obtained from the model fitting and shown in Table 2. In this work, the equilibrium volume (Ve ) of each polymer was calculated by extrapolation of the zero pressure limits of analytical equations that have been fitted to experimental pressure–volume–temperature (PVT) data in the melt state as shown in Appendix A and presented in Table 2. Exact solutions of Eq. (4) using Matlab software were used to obtain the parameters fi , τ ∞ , and γ by fitting to the experimental data for thin films of PSF, Matrimid® , and PPO. Since each film, regardless of thickness and aging temperature, was quenched to room temperature in a similar manner, fi should be a unique value for each polymer in the model fitting. In the model fitting of all the data sets for each polymer, a value of fi is assumed and then a non-linear least square fitting procedure was employed for each film thickness and aging temperature to obtain τ ∞ , γ, and the

Table 3 Struik model parameters deduced by fitting aging data for thin films of the polymers studied Polymer

Film thickness

τ ∞ (h)

γ

35 ◦ C

45 ◦ C

55 ◦ C

35 ◦ C

45 ◦ C

55 ◦ C

PSF

∼400 nm ∼700 nm ∼1000 nm 25 ␮ma

1.39 × 1031 1.40 × 1036 4.67 × 1040 1.26 × 1055

2.98 × 1029 6.05 × 1032 7.11 × 1035 1.25 × 1045

2.54 × 1027 6.18 × 1029 2.45 × 1031 6.31 × 1036

1689 1959 2217 2909

1601 1776 1976 2405

1484 1611 1692 1927

PPO

∼400 nm ∼700 nm ∼1000 nm 25 ␮ma

1.18 × 1019 7.94 × 1021 5.19 × 1024 6.31 × 1035

7.36 × 1015 1.83 × 1018 2.59 × 1021 98 × 1030

2.13 × 1014 7.82 × 1015 5.19 × 1017 1.58 × 1024

612 750 844 1213

579 668 790 1032

569 631 705 895

Matrimid®

∼400 nm ∼700 nm ∼1000 nm 25 ␮ma

4.78 × 1027 6.45 × 1030 7.04 × 1034 3.16 × 1044

– – –

– – –

1003 1112 1259 1566

– – –

– – –

a

Extrapolated value obtained from plots of log τ ∞ and γ vs. 1/
for polymer thin films aged at the different temperatures.

222

Y. Huang et al. / Journal of Membrane Science 277 (2006) 219–229

Fig. 4. Schematic illustration of the distribution of free volume across the film at different aging times assuming that free volume diffuses to the surfaces where it escapes.

fitting error for each data set. This process was repeated over a range of fi values for each polymer. The fitting errors for all the data sets for a given polymer were summed with equal weight to each one and plotted versus the value of fi used in the fitting. The value of fi for a given polymer that minimized the sum of the fitting errors was adopted for the determination of τ ∞ and γ and is the one shown in Table 2. The solid lines shown in Figs. 1–3 compare the best fit line determined in this way with the experimental data determined as described. The solid lines in each figure represent the model fit while the points are the experimental data; the agreement between them is excellent in all cases. The fitting parameters τ ∞ and γ determined in this way for each polymer, thickness, and aging temperature are summarized in Table 3. The results clearly show that both τ ∞ and γ increase with increasing film thickness for each polymer since the physical aging rate is thickness dependent [19,20]. McCaig et al. [20] assumed that free volume diffuses to the surfaces and escapes there. They quantitatively modeled the aging process by combining a simple kinetic theory for “lattice contraction” with the free volume diffusion model suggested by Curro et al. [28] ∂f = ∇(D ∇f ) ∂t

(5)

where D is the diffusivity for holes/vacancies which, in general, is expected to depend on free volume as described by the wellknown Doolittle equation [29],     −Z D = Dr exp −Z(f −1 − fr−1 ) = D0 exp (6) f where D0 and Z are material constants. These equations imply that the driving force for the diffusion of vacancies, or free volume, is the gradient of the spatial and time dependent free volume; such mechanisms have been used to explain aging of bulk glasses using a small, e.g., submicron, internal length scale [28]. Based on Eq. (5), there would be a time dependent distribution of free volume across the film thickness as suggested in Fig. 4. Direct experimental proof of this hypothesis was attempted using ellipsometry; however, no apparent improvement in the mean

Fig. 5. Combined effects of film thickness and aging temperature on the Struik parameters τ ∞ and γ for PSF thin films.

square error (MSE) was observed when the single Cauchy layer model for the polymer film was replaced by a symmetric threeCauchy layer model [23]. It may be that any volume/refractive index distribution in these experiments was restricted to a very thin layer near the surface of the film that is beyond the resolution limit of the instrument. Other techniques that are better able to probe whether a spatial distribution of density or free volume exists or not need to be explored; however, this is beyond the scope of this study. Nevertheless, some indirect evidence for the hypothesized free volume distribution caused by diffusion is that the changes in gas permeability during aging for films with
< 1 ␮m correlate with t/
2 [1]. More detailed analyses of models involving diffusion of free volume are currently being explored. Fig. 5 shows the combined effects of film thickness and aging temperature on τ ∞ and γ for PSF thin films. The parameters τ ∞ and γ are more sensitive to temperature for thicker films than for thinner films; at higher temperatures there is less variation of τ ∞ and γ with thickness. Similar observations were noted for the other two polymers; clearly, the aging of thin films

Y. Huang et al. / Journal of Membrane Science 277 (2006) 219–229

Fig. 6. Correlation of the Struik parameters, log τ ∞ and γ, with the reciprocal of film thickness for PSF thin films aged at different temperatures.

can be well-described quantitatively by the Struik model with parameters τ ∞ and γ that depend on thickness. In principle, one could extrapolate these parameters to infinite thickness in order to obtain values corresponding to a “bulk” system. One simple thought was that plots of these parameters versus 1/
might be linear allowing an easy extrapolation to infinite thickness; however, this did not prove to be the case as seen in Fig. 6. It seems that films with thickness more than 25 ␮m adequately approximates bulk behavior [1]; thus, the corresponding values of τ ∞ and γ for PSF films were estimated for
= 25 ␮m from Fig. 6 using a polynomial fit. A similar approach was used to obtain the “bulk” values of τ ∞ and γ for the other two polymers shown in Table 3. Unfortunately, ellipsometry cannot be used to measure the refractive index for films thicker than about 1.5 ␮m. 5. Effect of prior history The aging experiments performed in the majority of this study involved quenching the film from above Tg . It could be argued

223

Fig. 7. Comparison of the aging response for PSF thin films as-cast and after quenching from above Tg in terms of (a) absolute oxygen permeability and (b) relative oxygen permeability (normalized by values at aging time of 1 h).

that the aging process that occurs following such a quench from the melt might be quite different from that for a film directly after casting from a solution or after any other arbitrary history. Since commercial membranes are generally prepared via solvent evaporation or by coagulation, i.e., “phase inversion”, processes, it is important to study physical aging or permeability changes, on films without prior heating above Tg . Fig. 7 compares the aging performance, in terms of oxygen permeability, of as-cast and quenched PSF films of nearly the same thickness before heating. At any aging time, the quenched film has higher permeability than the as-cast film as expected due to the fact that the quenched film has higher free volume from the start than the as-cast film. However, surprisingly, the two films display an almost identical aging pattern when the relative oxygen permeability coefficients, i.e., normalized by the permeability at t = 1 h, are compared. This suggests that the aging process may be similar regardless of how the film is obtained. This issue deserves further investigation in the future but is beyond the scope of this work.

224

Y. Huang et al. / Journal of Membrane Science 277 (2006) 219–229

Table 4 Specific volume at various aging times (T = 35 ◦ C) calculated from refractive index for polymer films with
≈ 400 nm V (cm3 /g)

Aging time (h)

PSF ≈0 100 101 102 103 104 δ=

Vi −Vt=104 h Vi −Ve

Matrimid®

0.8112 0.8093 0.8080 0.8067 0.8054 0.8042 (%)

18.3

0.8398 0.8364 0.8341 0.8318 0.8295 0.8270 21.8

PPO 0.9481 0.9435 0.9402 0.9363 0.9317 0.9263 28.7

6. Relative progress of aging process Physical aging is a process that gradually approaches equilibrium from an initial non-equilibrium state; it has been just shown that the relaxation time at equilibrium is extremely long for the thin films studied here. In this work, aging has been monitored for up to 10,000 h, and since there is no sign of stabilization it is appropriate to ask what is the relative progress towards equilibrium that is achieved in these experiments [1,2]. The specific volume at nominal aging times of ∼0, 1, 10, 100, 1000 and 10,000 h were computed by interpolating/extrapolating results of refractive index measurement for polymer films with thickness of ∼400 nm aged at 35 ◦ C, see the results in Table 4. Neither of the three polymers even comes close to reaching equilibrium as seen by the δ value defined as the relative progress towards equilibrium. Fig. 8 shows a graphical example of how the specific volumes evolve for a PSF film (
≈ 400 nm) aged at 35 ◦ C in relation to the estimated initial and equilibrium values. The latter was estimated by extrapolation of the zero pressure limit of the modified Tait equation obtained by Callaghan and Paul [30] by fitting to PVT data above Tg ; nearly equivalent results were obtained by using the data reported by Zoller [31]. The

Fig. 8. Graphical comparison of the volumetric aging response for a PSF film with thickness ∼400 nm aged at 35 ◦ C with estimates of the equilibrium and occupied volumes.

usual non-equilibrium glassy state line is not shown in this plot; however, this would lie above the equilibrium line and intersect it at Tg . It can be seen that the specific volume at an aging time of 10,000 h is still far from the equilibrium value; the specific volume decreased by less than 1.0% during this aging time while a reduction of 4.7% would be required for PSF to reach equilibrium. 7. Relationship between permeability and refractive index If there is a distribution of density across the film thickness as might be caused by free volume diffusion, the gas permeability measured experimentally is a spatial average defined by
1 1
/2 dx = (7) P(t)
−
/2 P(x, t) Likewise, the experimentally determined refractive index, density and fractional free volume computed from ellipsometry observations would also represent spatial averages as suggested by
1
/2 n(t) = n(x, t) dx (8)
−
/2 However, in what follows, we disregard any such distribution and discuss the results only phenomenologically. With this simplification, the correlation between the observed gas permeability and the free volume calculated from the refractive index can be approximated by P = A e−B/f

(9)

as assumed for bulk polymer films; where A and B are constants that depend on the gas molecule type [24]. According to Cohen and Turnbull [12], the parameter B is related to the minimum volume of a fluctuation needed for a diffusional jump and A is a pre-exponential factor. In this work, both permeability and refractive index were monitored as a function of two variables, film thickness and aging temperature; thus, the question is whether Eq. (9) can unify the interrelationship between P(t) and n(t) measurements for different thicknesses and aging temperatures. Fig. 9 shows an example of the experimentally determined oxygen permeability coefficient (P) on a logarithmic scale as a function of the reciprocal of the fractional free volume (1/f) obtained from the measured refractive index for PSF thin films aged at 35 ◦ C. Since the permeability and the refractive index data were not collected at exactly the same aging times, the fractional free volume was interpolated in accordance with the aging time for each permeability measurement. The parameters A and B can then be evaluated from the best fit of the experimental data in the form suggested by Eq. (9). These parameters are almost independent of the film thickness. Similar observations were noted for other temperatures and other polymers. Therefore, it might be concluded that Eq. (9) approximately collapses the aging curves for different thicknesses into one.

Y. Huang et al. / Journal of Membrane Science 277 (2006) 219–229

225

Fig. 9. Oxygen permeability coefficient as a function of the inverse fractional free volume for various PSF thin films with

1 ␮m as they aged at 35 ◦ C.

Fig. 10. Oxygen permeability coefficient as a function of the inverse fractional free volume for PSF thin films as they aged at the temperatures indicated (each temperature plot contains data for all thin films with

1 ␮m).

Using a similar approach, Figs. 10 and 11 represent results for polymers studied at different temperatures; each temperature plot contains data for all thin films of the polymer. The parameters A and B obtained from the best fit of each plot are presented in Table 5. Evaluation of these parameters reveals that, within experimental error, B seems to be independent of the aging and measuring temperature while A increases with temperature for PSF and is almost independent of temperature for PPO. Similar results were obtained for N2 and CH4 and the apparent A and B values for N2 and CH4 are summarized in Tables 6 and 7, respectively. Therefore, as Tables 5–7 show, within experimental error, Eq. (9) seems also to be able to unify the temperature effect for each type of gas for PPO but not for PSF. For a given gas, the value of B seems to be essentially the same regardless of polymer structure; however, A clearly

depends on the polymer structure. However, the values A and B obtained here are somewhat different from the values reported in the literature for bulk polymers where the correlation is made using many different polymers [24,32–34], see the last row of Tables 5–7. This difference could be due to several issues. First, the correlation of the spatial average of the permeability coefficient, assuming there is a gradient due to free volume diffusion, with the spatial average of fractional free volume using Eq. (9) may only yield apparent values of A and B that cannot be compared to the bulk case. Second, a glassy polymer undergoing aging may not track along the same P versus f relationship defined by a series of other polymers. Other possibilities also may exist. Clearly, errors in C or Vo will affect these observations.

Table 5 Apparent A and B values for O2 Aging temperature (◦ C)

Matrimid®

PSF A

B

A

PPO B

A

B

2.5 × 107

35 45 55

1.5 × 108 1.7 × 108 2.1 × 108

2.62 2.63 2.67

1.3 × 107 – –

2.63 – –

2.6 × 107 2.6 × 107

2.66 2.65 2.64

Literature values

397 [24]

0.839 [24]

1.71 × 105 [33]

1.80 [33]

2.56 × 105 [32]

1.887 [32]

Table 6 Apparent A and B values for N2 Aging temperature (◦ C)

Matrimid®

PSF A

B

A

PPO B

A

B

3.1 × 107

35 45 55

3.6 × 108 3.9 × 108 4.6 × 108

2.96 3.00 2.97

2.3 × 107 – –

3.03 – –

3.1 × 107 3.3 × 107

2.99 2.96 3.01

Literature values

112.1 [24]

0.914 [24]

1.161 × 105 [33]

2.01 [33]

4.321 × 104 [34]

1.88 [34]

226

Y. Huang et al. / Journal of Membrane Science 277 (2006) 219–229

Table 7 Apparent A and B values for CH4 Aging temperature (◦ C)

Matrimid®

PSF A

B

A

PPO (LMW) B

A

B

2.8 × 108

35 45 55

5.2 × 109 5.5 × 109 6.0 × 109

3.37 3.38 3.34

1.7 × 108 – –

3.34 – –

3.0 × 108 3.1 × 108

3.37 3.34 3.36

Literature values

114.1 [24]

0.967 [24]

1.141 × 105 [33]

2.04 [33]

2851[34]

1.43 [34]

8. Relationship between aging rates from permeability and refractive index changes of aging rate, the permeability reduction rate  Two measures  P − ∂∂log at long aging times and the volumetric relaxation log t  ρ ∂ log L rate r = ∂∂ log = log t ∂ log t , have been determined for these three polymers at different thicknesses and temperatures [1,2]. It is of interest to know if there is a correlation between these two measures of aging rate since both techniques are considered to reflect changes in free volume. Again, any distribution of the free volume across the film thickness is ignored here for simplicity; from Eq. (9), the permeability reduction rate can be expressed as  ∂logP  ∂lnP ∂(lnA − B/f ) 1 ∂f − (10) =− =− =−B 2 ∂logt 
,T ∂lnt ∂lnt f ∂lnt From the correlation between the density and fractional free volume (f = 1 − ρVo ), the volumetric relaxation rate can be expressed as  ∂logρ  ∂lnρ ∂ln[(1 − f )/Vo ] −1 ∂f r≡ = = =  ∂logt
,T ∂lnt ∂lnt 1 − f ∂lnt

(11)

Fig. 11. Oxygen permeability coefficient as a function of the inverse fractional free volume for PPO thin films as they aged at the temperatures indicate (each temperature plot contains data for all thin films with

1 ␮m).

Combining Eqs. (10) and (11) gives   ∂logP  1−f 1 − f ∂logL  − = B r = B ∂logt 
,T f2 f 2 ∂logt 
,T

(12)

Eq. (12) suggests that, for a polymer film at a particular temperature, the two measures of aging rate are proportional to each other. However, since f depends on polymer type, film thickness plus aging temperature and time, one should not necessarily expect a perfect linear relationship between the two measures of aging rate for different polymers as the film thicknesses and aging temperatures are varied. Figs. 12 and 13 show plots of the gas permeability reduction rate versus the volumetric relaxation rate for PSF, Matrimid® , and PPO thin films of different thicknesses aged at different temperatures. An excellent correlation between the two measures of aging rate is seen for each polymer. Approximate linear correlations of aging rates at different temperatures obtained from positron annihilation, dielectric relaxation, and dynamic mechanical thermal analysis have also been reported for poly(methyl methacrylate) [35]. Since it has been shown that B in Eq. (9) seems to be independent of film thickness, aging temperature and polymer type, Eq.  ∂ log P  (12) suggests that a plot of − ∂ log t  versus 1−f r for a given f2
,T

gas would lead to a linear correlation for films of all different polymers regardless of thickness or aging temperature. Fig. 14

Fig. 12. Correlation between methane permeability reduction rates and volumetric relaxation rates at different temperatures for various PSF thin films with thickness less than ∼1 ␮m.

Y. Huang et al. / Journal of Membrane Science 277 (2006) 219–229

Fig. 13. Correlation between nitrogen permeability reduction rates and volumetric relaxation rates at different temperatures for various PPO thin films with thickness less than ∼1 ␮m.

shows all the experimental data from this study for oxygen and methane plotted in the manner suggested where the term 1−f f2 was calculated at an aging time of 10 h for each polymer film. A single correlation seems to exist between the two quantities regardless of polymer type, film thickness or aging temperature. The slopes of these plots are comparable with B in Eq. (9) but are slightly higher which might be attributed to the error in estimation of 1−f . The nitrogen permeability reduction rate is in f2 between that of oxygen and methane but is not shown here for clarity. In a previous paper [1], it was shown that the gas permeability reduction rate is dependent on both thickness (
) and the effective diameter of the gas molecule (def ). The rate of

227

Fig. 15. Correlation of the permeability reduction rate, −∂log P/∂log t, at 35 ◦ C with the square of the effective gas molecule diameter for films with
∼400 nm. P aging given by − ∂∂log log t , for a particular gas, linearly correlates with the logarithm of thickness for relatively thin films, i.e., well before the bulk dimension is reached, for each polymer and increases with increasing def , i.e., for a particular film of each ∂logP ∂logP ∂logP polymer, − ∂logtCH4 > − ∂logtN2 > − ∂logtO2 . Here, the correlaP tion of these parameters is examined. In Fig. 15, − ∂∂log log t of each gas for films with a thickness of 400 nm are shown as a function 2 of the penetrating gas molecule for difof the corresponding def ferent polymers isothermally aged at 35 ◦ C. It seems that there P 2 is a linear correlation between − ∂∂log log t and def for each polymer, 2 [36]. analogous to the linear correlation between log D and def Similar correlations were observed at other thicknesses.

9. Summary and conclusions

Fig. 14. Comparison of the correlation of −∂log P/∂log t for O2 and CH4 with (1 − f)/f2 r at different temperatures for polymer thin films with thickness less than ∼1 ␮m. Correlation for N2 is in between and is not shown here for clarity.

From the Lorentz–Lorenz equation, the fractional free volume was calculated as a function of aging time for thin films made from three glassy polymers of interest as membrane materials; aging is more rapid for thinner films. The equilibrium specific volume was determined by extrapolation of melt density data using equations of state for PSF and PPO [37]; Ve of Matrimid® was estimated from the ratio of Ve /Vw for a number of polymers containing phenylene group in the backbone. The Struik lattice contraction model [8] captures the self-retarding aspects of the aging process and seems to describe the aging behavior of thin glassy polymer films with thickness dependent parameters which can be further used to estimate parameters τ ∞ and γ for the bulk polymer. The hypothesis of a distribution of free volume caused by free volume diffusion could not be confirmed by ellipsometry; this issue needs to be explored further using other techniques. As-cast films seem to age similarly as quenched films. Even after 10,000 h of aging, these films are still far from equilibrium for all polymers studied. At each aging temperature, Eq. (9) approximately collapses the aging curves of log P versus 1/f for different thicknesses into one for

228

Y. Huang et al. / Journal of Membrane Science 277 (2006) 219–229

all three polymers; and similarly at different temperatures, such plots almost collapse into one master curve for PPO but not for PSF. Examination of the A and B values from the fitting using Eq. (9) for different polymers indicates that, for each gas, B is almost identical for all polymers studied while A seems to be dependent on the polymer structure. The gas permeability reduction rate and the volumetric relaxation rate show a remarkable correlation with each other for all polymers, thicknesses and temperatures P studied. A linear dependence of − ∂∂log log t for each film with the square of the effective molecular diameter of the permeating gas molecule, analogous to the linear correlation between log D and 2 [36], was observed for the polymers studied. def Acknowledgements This research was supported by the Separations Research Program at the University of Texas at Austin and the National Science Foundation grant number DMR-0238979 administered by the Division of Material Research Polymer Program. Appendix A A.1. Computation of equilibrium specific volume, Ve , of polymers studied using the Tait equation Empirical equations of state are available in the literature for PSF and PPO that allow the calculation of the equilibrium specific volume at temperatures below Tg by extrapolation [30,31,37–40]. Typically, these are modifications of the so-called Tait equations that account for the temperature dependence of specific volume in addition to pressure where the parameters have been determined by fitting to experimental PVT data above Tg . The zero pressure limits of the these equations for PSF [30] and PPO [39] used in this work are given below PSF: Ve (0, T ) = 0.76483 + 3.0568 × 10−4 T + 3.9204 × 10−7 T 2 (T, ◦ C). PPO: Ve (0, T ) = 0.78075 exp(2.151 × 10−5 T 3/2 ) (T, K). The literature gives other versions of these equations evaluated from different data sets, but they are all in remarkable agreement. Matrimid® : A literature search produced no equation of state information for Matrimid® . As an alternative, the equilibrium volume of Matrimid® was estimated from an empirical relationship with the van der Waals volume analogous to the van Krevelen’s observations of a linear relationship between the equilibrium volume and van der Waals volume for rubbery polymers [21]. Matrimid® contains phenylene residues in the repeat unit; therefore, values of Ve were calculated for PSF, PPO, and other polymers containing phenylene group in the backbone using the modified Tait equations given in Ref. [37]. Fig. 16 shows the equilibrium volume for these polymers versus the

Fig. 16. Equilibrium specific volume as a function of the van der Waals volume for various polymers with phenylene residues in the repeat unit.

corresponding van der Waals volume; a simple linear relationship with a correlation coefficient of R2 = 0.999 is observed. The ratio of Ve /Vw has a value of 1.466 and was used to calculate the equilibrium volume for Matrimid® . Using these extrapolation procedures for PSF and PPO and the ratio of Ve /Vw for Matrimid® , the equilibrium volume for each polymer was calculated to obtain the results in Table 2. References [1] Y. Huang, D.R. Paul, Physical aging of thin glassy polymer films monitored by gas permeability, Polymer 45 (2004) 8377–8393. [2] Y. Huang, D.R. Paul, Physical aging of thin glassy polymer films monitored by optical properties, Macromolecules, submitted for publication. [3] Y. Huang, D.R. Paul, Effect of temperature on physical aging of thin glassy polymer films, Macromolecules, in press. [4] Y. Huang, D.R. Paul, Effect of molecular weight and temperature on physical aging of thin glassy poly(2,6-dimethyl-1,4-phenylene oxide) films, J. Polym. Sci. Part B: Polym. Phys., in press. [5] Y. Huang, D.R. Paul, Experimental methods for tracking physical aging of thin glassy polymer films by gas permeation, J. Membr. Sci. 244 (2004) 167–178. [6] R.-J. Roe, Thermodynamics of the glassy state with multiple order parameters, J. Appl. Phys. 48 (1977) 4085–4091. [7] L.C.E. Struik, Internal Stresses, Dimensional Instabilities, and Molecular Orientations in Plastics, Wiley, Chichester; New York, 1990. [8] L.C.E. Struik, Physical Aging in Amorphous Polymers and Other Materials, Elsevier Scientific Publishing Company, New York, 1978. [9] M.L. Williams, R.F. Landel, J.D. Ferry, J. Am. Chem. Soc. 77 (1955) 3701. [10] A.J. Kovacs, Glass transition in amorphous polymers. Phenomenological study, Fortschr. Hochpolym. Forsch. 3 (1964) 394–508. [11] J.M. Hutchinson, A.J. Kovacs, A simple phenomenological approach to the thermal behavior of glasses during uniform heating or cooling, J. Polym. Sci. Polym. Phys. Ed. 14 (1976) 1575–1590. [12] M.H. Cohen, D. Turnbull, Molecular transport in liquids and glasses, J. Chem. Phys. 31 (1959) 1164–1169. [13] A.Q. Tool, J. Am. Ceram. Soc. 31 (1948) 177. [14] M.E. Rezac, P.H. Pfromm, L.M. Costello, W.J. Koros, Aging of thin polyimide–ceramic and polycarbonate–ceramic composite membranes, Ind. Eng. Chem. Res. 32 (1994) 1921–1926.

Y. Huang et al. / Journal of Membrane Science 277 (2006) 219–229 [15] P.H. Pfromm, W.J. Koros, Accelerated physical aging of thin glassy polymer films: evidence from gas transport measurements, Polymer 36 (1995) 2379–2387. [16] M.E. Rezac, Update on the aging of a thin polycarbonate–ceramic composite membrane, Ind. Eng. Chem. Res. 34 (1995) 3170–3172. [17] K.D. Dorkenoo, P.H. Pfromm, Experimental evidence and theoretical analysis of physical aging in thin and thick amorphous glassy polymer films, J. Polym. Sci. Part B: Polym. Phys. 37 (1999) 2239–2251. [18] K.D. Dorkenoo, P.H. Pfromm, Accelerated physical aging of thin poly[1(trimethylsilyl)-1-propyne] films, Macromolecules 33 (2000) 3747–3751. [19] M.S. McCaig, D.R. Paul, Effect of film thickness on the changes in gas permeability of a glassy polyarylate due to physical aging. Part I. Experimental observations, Polymer 41 (2000) 629–637. [20] M.S. McCaig, D.R. Paul, J.W. Barlow, Effect of film thickness on the changes in gas permeability of a glassy polyarylate due to physical aging. Part II. Mathematical model, Polymer 41 (2000) 639–648. [21] D.W. van Krevelen, Thermophysical properties of polymers, in Properties of polymers: their correlation with chemical structure their numerical estimation and prediction from additive group contributions, Elsevier, Amsterdam, New York, 1990. [22] A. Bondi, Physical Properties of Molecular Crystals, Liquids and Glasses, Wiley, New York, 1968. [23] Y. Huang, Chemical Engineering Department, The University of Texas at Austin, Austin, 2005, p. 278. [24] J.Y. Park, D.R. Paul, Correlation and prediction of gas permeability in glassy polymer membrane materials via modified free volume based group contribution method, J. Membr. Sci. 125 (1997) 23–39. [25] J.M. Hutchinson, Physical aging of polymers, Prog. Polym. Sci. 20 (1995) 703–760. [26] D. Cangialosi, H. Schut, A. van Veen, S.J. Picken, Positron annihilation lifetime spectroscopy for measuring free volume during physical aging of polycarbonate, Macromolecules 36 (2003) 142–147. [27] D. Cangialosi, M. Wubbenhorst, H. Schut, A. van Veen, S.J. Picken, Dynamics of polycarbonate far below the glass transition temperature:

[28] [29] [30] [31] [32] [33]

[34] [35]

[36] [37]

[38]

[39]

[40]

229

a positron annihilation lifetime study, Phys. Rev. B: Condens. Matter Mater. Phys. 69 (2004) 134206/1–134206/9. J.G. Curro, R.R. Lagasse, R. Simha, Diffusion model for volume recovery in glasses, Macromolecules 15 (1982) 1621–1626. A.K. Doolittle, Newtonian flow. II. The dependence of the viscosity of liquids on free space, J. Appl. Phys. 22 (1951) 1471–1475. T.A. Callaghan, Chemical Engineering Department, University of Texas at Austin, Austin, 1992, pp. 170–173. P. Zoller, Specific volume of polysulfone as a function of temperature and pressure, J. Polym. Sci. Polym. Phys. Ed. 16 (1978) 1261–1275. C.L. Aitken, W.J. Koros, D.R. Paul, Gas transport properties of biphenol polysulfones, Macromolecules 25 (1992) 3651–3658. J.S. McHattie, W.J. Koros, D.R. Paul, Gas transport properties of polysulphones. 3. Comparison of tetramethyl-substituted bisphenols, Polymer 33 (1992) 1701–1711. S. Kanehashi, K. Nagai, Analysis of dual-mode model parameters for gas sorption in glassy polymers, J. Membr. Sci. 253 (2005) 117–138. W.J. Davis, R.A. Pethrick, Investigation of physical aging in poly methyl methacrylate using positron annihilation, dielectric relaxation and dynamic mechanical thermal analysis, Polymer 39 (1998) 255–266. S. Srinivas, P. Brant, Y. Huang, D.R. Paul, Structure and properties of oriented polyethylene films, Polym. Eng. Sci. 43 (2003) 831–949. P.A. Rodgers, Pressure-volume-temperature relationships for polymeric liquids: a review of equations of state and their characteristic parameters for 56, J. Appl. Polym. Sci. 48 (1993) 1061–1080. P. Zoller, A study of the pressure-volume-temperature relationships of four related amorphous polymers: polycarbonate, polyarylate, phenoxy, and polysulfone, J. Polym. Sci. Polym. Phys. Ed. 20 (1982) 1453–1464. P. Zoller, H.H. Hoehn, Pressure–volume–temperature properties of blends of poly(2,6-dimethyl-1,4-phenylene ether) with polystyrene, J. Polym. Sci. Polym. Phys. Ed. 20 (1982) 1385–1397. R.K. Jain, R. Simha, P. Zoller, Theoretical equation of state of polymer blends: the poly(2,6-dimethyl-1,4-phenylene ether)-polystyrene pair, J. Polym. Sci. Polym. Phys. Ed. 20 (1982) 1399–1408.