Sorption and Transport of Linear Esters and Branched Alkanes in

Equilibrium sorption and uptake kinetics of i-butane, i-pentane, methyl acetate, and ethyl acetate vapor in uniform, biaxially oriented, semicrystalli...
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Ind. Eng. Chem. Res. 2004, 43, 2966-2976

Sorption and Transport of Linear Esters and Branched Alkanes in Biaxially Oriented Poly(ethylene terephthalate) Sushil N. Dhoot and Benny D. Freeman* Department of Chemical Engineering, The University of Texas at Austin, Center for Energy and Environmental Resources, 10100 Burnet Road, Building 133, Austin, Texas 78758

Mark E. Stewart Voridian Division of Eastman Chemical Company, P.O. Box 2002, Kingsport, Tennessee 37662

Equilibrium sorption and uptake kinetics of i-butane, i-pentane, methyl acetate, and ethyl acetate vapor in uniform, biaxially oriented, semicrystalline poly(ethylene terephthalate) films were determined at 35 °C and low penetrant activity (or relative pressure). Sorption isotherms of i-butane, methyl acetate, and ethyl acetate were well described by the dual-mode sorption model. The sorption isotherm of i-pentane was described by the dual-mode model at low activity and the Flory-Huggins model at high activity. At low penetrant activity, the solubility coefficients of i-pentane and i-butane were, respectively, 11.5 and 1.3 times lower than those previously reported for their linear analogues. At constant penetrant activity, the solubility coefficients of the acetates were substantially higher than those of the alkanes, and the solubility of methyl acetate was higher than that of ethyl acetate. The sorption kinetics were described either by Fickian diffusion or by a two-stage model incorporating Fickian diffusion at short times and protracted polymer structural relaxation at long times. The diffusion coefficient for each penetrant increased with increasing penetrant concentration. The diffusion coefficients of the esters (methyl acetate and ethyl acetate) were quite similar to those of the ketones (acetone and methyl ethyl ketone) with the same number of carbon atoms. Introduction Poly(ethylene terephthalate) (PET) is widely used for packaging applications.1 The sorption and transport properties of small molecules such as water, oxygen, and carbon dioxide are important considerations in designing barrier packaging and have been characterized extensively.2,3 However, sorption and transport properties of larger organic molecules are also important in determining flavor scalping, flavor carryover, and contamination.1 These data are not readily available for PET, especially at low penetrant activity, largely because of the very low diffusion coefficients of such molecules.4-6 For example, methyl i-propyl ketone has a diffusion coefficient on the order of 10-14 cm2/s in PET at 35 °C,7 and the time scale to reach sorption equilibrium for this penetrant in a 10-µm-thick film would be more than 3 years.4 However, if the sample dimensions were 1 µm, this time scale would be reduced by a factor of 100, bringing the time scale into an experimentally accessible window.7 Previously, we reported solubility and diffusivity values for linear alkane hydrocarbons and linear and branched ketones in thin, biaxially oriented PET (BPET) films.4,7 The availability of extremely thin, uniform films enabled the study of uptake and transport properties of relatively large penetrants over reasonable time scales. The present article reports results of an interval kinetic gravimetric sorption study of two branched alkane hydrocarbons, i-butane and i-pentane, and two * To whom correspondence should be addressed. Tel.: (512) 232-2803. Fax: (512) 232-2807. E-mail: freeman@ che.utexas.edu.

Table 1. Chemical Structure, Critical Properties,23 Solubility Parameters,29 and Saturation Vapor Pressures at 35 °C23 of Linear and Branched Alkane Hydrocarbons and Esters

linear esters, methyl acetate and ethyl acetate, in BPET films. The chemical structures and relevant physical properties of these penetrants are reported in Table 1. Experimental Section Materials. Biaxially oriented PET films having a uniform nominal thickness of 0.9 µm were purchased from Goodfellow Co. (Berwyn, PA) and used as received. The film density, 1.395 g/cm3, was furnished by the supplier. i-Butane (99.5% purity) was purchased from Scott Specialty Gases (Durham, NC). High-performance liquid chromatographic (HPLC) grade i-pentane (99.5%

10.1021/ie030618s CCC: $27.50 © 2004 American Chemical Society Published on Web 01/16/2004

Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 2967

purity), methyl acetate (99.5% purity), and ethyl acetate (99.8% purity) were purchased from Sigma-Aldrich Corp. (St. Louis, MO). The liquid penetrants were subjected to several freeze-thaw cycles to remove dissolved gases before use. Gravimetric Sorption. Kinetic gravimetric sorption experiments were performed using a McBain spring balance system,8 as described previously.4 Approximately 50 mg of polymer film sample was suspended from a calibrated quartz spring inside the sorption system. Before the experiments were started, sorbed air gases and water vapor were removed from the polymer by exposing the sample to vacuum until there was no further spring displacement. Then, the sample was exposed to penetrant gas or vapor at fixed pressures. In a typical sorption experiment, the penetrant pressure in contact with the polymer sample was increased in a series of steps, with the sample being allowed to equilibrate after each step. In this protocol (i.e., an interval sorption experiment), the polymer sample is equilibrated in an environment where the penetrant pressure surrounding the sample is pi prior to a step. Then, at t ) 0, the penetrant pressure is increased to pf, and the spring extension is monitored as a function of time. In this work, the spring position relative to a stationary reference rod hanging in the sorption chamber was recorded using a Cohu (San Diego, CA) 4910 CCD camera,9 and Image, a software package from the National Institutes of Health (NIH), was used to process data from the camera on a computer. The spring position data were converted to mass uptake values, Mt, where Mt is defined as the mass of penetrant sorbed by the polymer at time t per unit mass of penetrant-free polymer, and diffusion coefficients were extracted from the time dependence of Mt. A similar procedure also applied to kinetic desorption experiments, where the penetrant pressure was decreased from an initial value, pi, to a final value, pf, at t ) 0, and the decrease in mass was monitored as a function of time. Equilibrium mass uptake, M∞, values were determined from the sorption experiments. The equilibrium penetrant concentration in the polymer, C [cm3 (STP)/ cm3 of polymer], was calculated as follows

C)

M∞ × 22 414 × 100 × Fpolymer MW

(1)

where M∞ has units of grams of penetrant sorbed per 100 g of polymer, Fpolymer is the polymer density (g/cm3), and MW is the penetrant molecular weight (g/mol). Equilibrium solubility, S [cm3 (STP)/(cm3 cmHg)], was calculated from

S)

C p

(2)

where p is the penetrant pressure in the gas phase contiguous to the polymer sample. Results and Discussion Polymer Characterization. A detailed discussion of the experimental techniques used to characterize the polymer sample has been presented previously.4 A brief summary of the results is presented here. The inherent viscosity of the BPET polymer samples was 0.473 dL/g. On the basis of 1H NMR spectroscopy, the sample was identified as PET containing 1.8 mol % diethylene glycol

Figure 1. (a) Sorption isotherms of n-butane4 and i-butane in biaxially oriented PET (BPET) at 35 °C as a function of penetrant pressure. The error bars were estimated from the uncertainties in measuring the equilibrium uptake (M∞) values ((0.01 mg), the penetrant-free polymer sample weight ((1 mg), and the propagation of errors technique.51 (b) Sorption isotherms of n-pentane4 and i-pentane in BPET at 35 °C as a function of penetrant pressure.

repeat units. A surface profilometry study yielded an average thickness of 0.95 ( 0.076 µm. The crystallinity of BPET was estimated to be 42 wt % (which is equivalent to 39 vol %). The crystalline Hermans orientation function of BPET was 0.875, which indicates a high degree of orientation. To put these values of crystallinity and orientation into perspective, the density of completely amorphous PET is approximately 1.33 g/cm3. For a PET bottle sidewall, the density is typically in the range of 1.361.37 g/cm3, and the in-plane orientation is in the range of 0.4-0.6, increasing down the bottle. Hence, the BPET samples from this study have higher density, crystallinity, and orientation than a PET bottle sidewall and are similar to highly oriented, heat-set PET films, such as Mylar. Equilibrium Uptake of Alkanes in PET. Interval kinetic gravimetric sorption experiments were used to determine sorption isotherms of i-butane and i-pentane in BPET at 35 °C. The results are presented in Figure 1a and 1b. The sorption isotherms of analogous linear alkane hydrocarbons, n-butane and n-pentane, in BPET at 35 °C4 are also included in Figures 1a and 1b for comparison. The sorption isotherm of i-butane is concave to the pressure axis and is well described by the dual-mode sorption model, which is often used to

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describe the sorption of gases and low-activity vapors in both amorphous and semicrystalline glassy polymers such as PET.2,10-13 According to this model, the total penetrant concentration in the polymer is the sum of the concentration in the equilibrium densified polymer matrix (Henry’s law sites) and the so-called nonequilibrium excess volume of the glassy polymer (Langmuir sites)11,13

(1 +bpbp)

C ) kDp + C ′H

(3)

where kD is the Henry’s law coefficient; C ′H is the Langmuir capacity parameter, a measure of the sorption capacity in the nonequilibrium excess volume of the glassy polymer; and b is the Langmuir affinity parameter, a measure of the penetrant’s affinity for the Langmuir sites. The Langmuir capacity has been associated with frozen microvoids in nonequilibrium glassy polymers.11,13 Restriction of the backbone chain motions of polymers below the glass transition temperature kinetically limits chain relaxation, presumably resulting in trapped excess free volume or so-called frozen microvoids.11 Whereas the dual-mode sorption model is the most widely used model for penetrant sorption in glassy polymers, other models, such as the nonequilibrium lattice fluid (NELF) model, have also been proposed in the literature.14 According to the commonly used two-phase model, only the amorphous regions of the polymer are generally regarded to be accessible to penetrant sorption. Hence, the penetrant concentration in the polymer, C, can be adjusted to an amorphous basis using the following relation

Ca )

C φa

(4)

where Ca is the amorphous-phase penetrant concentration and φa is the amorphous-phase volume fraction. Equations 3 and 4 can be combined to write the amorphous-phase penetrant concentration in terms of the dual-mode model parameters

Ca )

() ( )

kD C ′H bp p+ φa φa 1 + bp

(5)

The curve through the i-butane data in Figure 1a represents a least-squares fit of the experimental sorption data to eq 3. The sorption isotherm of i-pentane (Figure 1b) can also be described by the dual-mode model at low penetrant pressure. However, at pressures above ca. 50 cmHg, the isotherm becomes convex to the pressure axis, and the dual-mode model cannot capture this behavior. In this case, we use a combination of the Langmuir sorption term plus the Flory-Huggins model. Such a model is often used to describe the sorption isotherms of vapors and high-pressure gases in rubbery polymers15,16 and high-activity vapors in glassy polymers.17,18 The Flory-Huggins sorption model is given by the equation19

ln a ) ln φ + (1 - φ) + χ(1 - φ)2

(6)

where a is the penetrant activity (or relative pressure), equal to p/Psat (Psat is the penetrant saturation vapor pressure); φ is the penetrant volume fraction in the penetrant-accessible (i.e., amorphous) regions of the

polymer; and χ is the Flory-Huggins interaction parameter, which characterizes penetrant-polymer interactions. The concentration dependence of χ is given by the following empirical expression20

χ ) χ0 + χ1φ

(7)

The curve through the i-pentane data in Figure 1b represents a least-squares fit of the experimental sorption data to eqs 5-7, using the procedure described previously.21 The Henry’s law coefficient, kD, can be related to the Flory-Huggins interaction parameter, χ, at infinite dilution.22 The dual-mode and Flory-Huggins model parameters for i-butane and i-pentane obtained in this study are reported in Table 2, along with the dual-mode parameters for n-butane and n-pentane in BPET.4 From the data in Figure 1a and b, at constant pressure, i-butane and i-pentane have lower solubility in BPET than their linear analogues. For example, at 20 cmHg, the solubility ratio for n-butane/i-butane is 1.3, which is approximately an order of magnitude lower than the value for n-pentane/i-pentane, which is 11.5. Within the framework of the dual-mode model, the penetrant sorption in the nonequilibrium excess volume of glassy polymers is characterized by the Langmuir capacity parameter, which is typically understood to be related to the amount of nonequilibrium excess volume as follows11

Vex C ′H Vg - Vl ) F* ) F* φa Vg Vg

(8)

where Vg is the specific volume of the penetrantaccessible (i.e., amorphous) regions of the polymer; Vl is the specific volume of the hypothetical, densified, equilibrium polymer matrix; and F* is the condensed penetrant density, which is often estimated as the saturated liquid penetrant density at the experimental temperature. In this study, the saturation densities were estimated at 35 °C using the Hankinson-BrobstThomson correlation.23 The values are 207.4 and 188.1 cm3 (STP)/cm3, respectively, for i-butane and i-pentane. If the nonequilibrium excess volume characterized by CH ′ is equally accessible to both penetrants, then the ′ /F* values for ratios C H ′ /F* should be equal. The C H i-butane and i-pentane are 0.005 ( 0.001 and 0.001 ( 0.0005 (cf. Table 2), respectively, suggesting that ibutane has far greater access to the nonequilibrium excess volume of PET than i-pentane. For comparison, the C H ′ /F* values for n-butane and n-pentane in BPET at 35 °C are 0.010 and 0.013, respectively, which are similar.4 Thus, the lower solubility of branched alkane hydrocarbons in BPET relative to their linear analogues appears to result in part from restricted access of the branched penetrants to the nonequilibrium excess volume of the polymer, and this seems to have a more pronounced effect on the solubility of i-pentane in BPET than on that of i-butane. For comparison, one can consider the effect of chain branching on hydrocarbon solubility in an equilibrium polymer, which would correspond more closely to the sorption in the Henry’s law regions of PET. In this regard, experimental sorption isotherms of n-butane, i-butane, n-pentane, and i-pentane in low-density polyethylene (LDPE) at 25 °C are available.22 According to the sorption curves presented in Figure 2, the infinite-

Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 2969 Table 2. Dual-Mode and Flory-Huggins Model Parameters for Linear and Branched Alkane Hydrocarbons and Esters in BPET at 35 °Ca penetrant

kD [cm3 (STP)/ (cm3 of polymer‚cmHg)]

C ′H [cm3 (STP)/ (cm3 of polymer)]

b (cmHg-1)

C ′H/F*

n-butane i-butane n-pentane i-pentaneb methyl acetate ethyl acetate

0.0143 ( 0.0012 0.009 ( 0.005 0.011 (0.004 0.008 ( 0.001 0.59 ( 0.02 0.76 (0.03

2.16 ( 0.11 1.0 ( 0.3 2.53 ( 0.23 0.2 ( 0.1 2.82 ( 0.12 2.88 ( 0.13

0.029 ( 0.006 0.1 ( 0.03 0.135 ( 0.020 0.08 ( 0.05 6.4 ( 1.4 7.5 ( 1.6

0.010 0.005 ( 0.001 0.013 0.001 ( 0.0005 0.010 ( 0.0004 0.013 ( 0.0006

a Model parameters for n-butane and n-pentane obtained from the literature.4 ) -181 ( 3.

b

Best-fit parameters of eq 7 are χ0 ) 4.8 ( 3.8 and χ1

Figure 2. Sorption isotherms of n-butane, i-butane, n-pentane, and i-pentane in low-density polyethylene (LDPE) at 25 °C.22

Figure 3. Sorption isotherms of methyl acetate and ethyl acetate in BPET at 35 °C as a function of penetrant pressure.

dilution solubility ratios for n-butane/i-butane and n-pentane/i-pentane in LDPE at 25 °C are similar (1.65 and 1.55, respectively). These values are not very sensitive to penetrant pressure. For example, at 20 cmHg, the solubility ratios are 1.67 and 1.63, respectively. These values are quite similar to the solubility ratios (1.8 for n-butane/i-butane and 1.7 for n-pentane/ i-pentane) expected on the basis of a simple analysis using the Flory-Huggins model (see Appendix A)

are presented in Figure 3. Both isotherms are well described by the dual-mode sorption model (eq 3), and the model parameters are reported in Table 2. The C ′H/F* values for methyl acetate and ethyl acetate are 0.010 and 0.013, respectively, which are similar to the values reported earlier for linear alkanes4 (cf. Table 2) and linear and branched ketones in BPET at 35 °C.7 From the data in Figure 3, at constant pressure, the solubility of ethyl acetate is higher than that of methyl acetate, consistent with the higher critical temperature of ethyl acetate (cf. Table 2). The logarithm of the infinite-dilution, amorphousphase penetrant solubility, Sa, in rubbery and glassy polymers often increases linearly with critical temperature if dispersion forces are the dominant contribution to the interaction energy between the polymer segments and the penetrant molecules24,25

S∞,A Psat,B vB ) × × exp(χB - χA) S∞,B Psat,A vA

(9)

where S∞,A and S∞,B are the infinite-dilution solubility coefficients of penetrant A and B, respectively; Psat,A and Psat,B are the corresponding saturation vapor pressures; vA and vB are the respective liquid molar volumes; and χA and χB are the corresponding Flory-Huggins interaction parameters. This model crudely accounts for differences in infinite-dilution solubility due to differences in penetrant vapor pressure, size, and interactions with the polymer matrix. Thus, the effect of chain branching on sorption in rubbery LDPE is consistent with the Flory-Huggins model. This result is in sharp contrast to the sorption results in glassy BPET and, as discussed above, might be linked to the limited accessibility of the branched penetrants to the nonequilibrium excess volume in stiff-chain glassy PET, a molecular feature absent in flexible rubbery LDPE and, therefore, in the model just described. Equilibrium Uptake of Esters in PET. Interval kinetic gravimetric sorption experiments were performed to determine the sorption isotherms of methyl acetate and ethyl acetate in BPET at 35 °C. The results

ln Sa ) N + MTc

(10)

Figure 4 presents the solubilities of alkane hydrocarbons and esters from this study and other literature data7 for PET as a function of critical temperature. The solubility values reported in this figure have been adjusted to an amorphous basis using the following relation,2 which should be regarded as a first order approximation of the effect of crystallinity on solubility7

Sa )

S φa

(11)

where S is the penetrant solubility (as defined in eq 2) and φa is the amorphous-phase volume fraction. In general, for the organic vapor sorption data in Figure

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Figure 5. Sorption isotherms of methyl acetate (9), ethyl acetate (b), n-butane4 (2), n-pentane4 (]), i-butane (4), and i-pentane ([) in BPET at 35 °C as a function of relative pressure. Figure 4. Correlation of infinite-dilution, estimated amorphousphase solubility coefficients in PET with penetrant critical temperature.26 For most penetrants, the data were extrapolated to infinite dilution. The data for methyl acetate and ethyl acetate from this study and for methyl ethyl ketone (MEK), methyl i-propyl ketone (MiPK), and methyl n-propyl ketone (MnPK) from an earlier study7 are reported at a relative pressure of 0.1. Whereas the data from this study were obtained at 35 °C, the temperature range for the data obtained from various literature sources is 25-45 °C.26 The slope, M, of the least-squares line is 0.019 ( 0.001 K-1, and the intercept, N, is -9.6 ( 0.4.

4, infinite-dilution solubility values were obtained by extrapolating reported solubility values to zero concentration.26 However, because of the scatter in the data at low pressure for methyl acetate and ethyl acetate, we did not attempt to compute infinite-dilution solubilities. For these penetrants, the solubility coefficients are reported at an activity (or relative pressure) of 0.1 in Figure 4. The logarithm of the solubility increases rather systematically and linearly with Tc. Although the general trend is observed, there is scatter in the data, which might arise from several sources. Some of the scatter is likely due to the approximate nature of eq 11.7 In addition, the literature data for some penetrants (methanol, ethyl acetate, water, and some ketones) were not available at infinite dilution,7,26 and penetrant solubility often decreases with increasing penetrant activity in PET. Additionally, the data were drawn from several literature sources using different PET samples, so it was not possible to account for sample processing effects (e.g., orientation) on these results. The solubilities of i-butane and i-pentane are lower than expected, possibly because of the branched structures restricting their access to the nonequilibrium excess volume of PET, as discussed above. Although it is difficult to draw firm conclusions, the solubilities of the esters from this study and many of the other polar penetrants appear to exhibit a positive deviation from the correlation line. This trend might reflect favorable interactions between polar penetrants and the polar PET matrix, and this effect is explored further below. To account for penetrant condensability and study the effect of penetrant-polymer interactions on solubility, the sorption isotherms were replotted as a function of penetrant relative pressure (i.e., activity). As shown in Figure 5, this transformation does not collapse the

sorption isotherms of the two esters to a single curve, as was the case with an earlier study of linear and branched ketones in PET.7 At constant relative pressure, the solubility of methyl acetate is higher than that of ethyl acetate, which is the exact opposite of the ranking based on pressure (cf. Figure 3). This is consistent with the difference between the solubility parameters of the esters and PET, which is lower for methyl acetate (cf. Table 1). This result is also consistent with other literature studies reporting that ester solubility in polar liquids (e.g., water) decreases as the ester chain length increases.27,28 For example, the solubilities (mole fractions) of methyl acetate and ethyl acetate in water at 25 °C are 0.071 and 0.016, respectively.27 Figure 5 also presents the sorption isotherms of the hydrocarbon penetrants in BPET. At constant relative pressure, the esters are significantly more soluble than the alkanes, which is again consistent with the difference between the solubility parameters of the penetrants and that of PET (cf. Table 1). This result also supports the notion that polar penetrants have higher solubilities in a relatively polar matrix such as PET. The sorption isotherms of methyl acetate and ethyl acetate were compared to those of acetone7 and methyl ethyl ketone7 to study the effect of different polar groups on the solubilities of penetrants with the same number of carbon atoms (C3 and C4, respectively). These comparisons are presented in Figure 6a and b, respectively. At constant activity, the solubility coefficients of methyl acetate and ethyl acetate are somewhat higher than those of acetone and methyl ethyl ketone, respectively. The three-component Hansen solubility parameters, which include polar and hydrogen-bonding effects, are often used to describe the solubilities of polar penetrants in polymers.29 It is useful to compare these parameters for esters and ketones because they have different polar and hydrogen-bonding characteristics. According to the solubility parameter components reported in Table 3, the ketones are more polar than the esters, but the esters have stronger hydrogen-bonding characteristics. Using literature values for the Hansen solubility parameters of the penetrants29 and PET30 (cf. Table 3) and available correlations,29 the estimated χ values for methyl acetate and ethyl acetate are lower than the corresponding values for acetone and methyl ethyl

Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 2971

Figure 6. (a) Sorption isotherms of methyl acetate and acetone7 in BPET at 35 °C as a function of relative pressure. The saturation vapor pressures of methyl acetate and acetone at 35 °C are 32.9 and 34.7 cmHg, respectively.23 (b) Sorption isotherms of ethyl acetate and methyl ethyl ketone (MEK)7 in BPET at 35 °C as a function of relative pressure. The saturation vapor pressures of ethyl acetate and MEK at 35 °C are 14.9 and 14.2 cmHg, respectively.23 Table 3. Hansen Solubility Parameters, Critical Volumes, and Dipole Moments of Acetone, Methyl Ethyl Ketone, Methyl Acetate, and Ethyl Acetate23,29

penetrant acetone methyl acetate methyl ethyl ketone ethyl acetate

Hansen solubility critical dipole parametersa (MPa1/2) volume moment δd δp δH (cm3/mol) (D) 15.5 15.5 16.0 15.8

10.4 7.2 9.0 5.3

7.0 7.6 5.1 7.2

209 228 267 286

2.9 1.7 3.3 1.9

a Note: The Hansen solubility parameters (δ δ , δ ) for PET d, p H are 19.5, 3.5, and 8.6 MPa1/2, respectively.30

ketone, respectively, which would be consistent with the esters having higher solubilities. Similar solubility trends of esters and ketones have also been observed in other polymers.31 Wibawa et al. reported higher solubility coefficients of methyl acetate and propyl acetate relative to those of acetone and methyl ethyl ketone in two polar, rubbery polymers, poly(n-butyl methacrylate) and poly(vinyl acetate).31 For example, at 313 K and a penetrant activity of 0.19, the solubilities

Figure 7. (a) Fickian sorption kinetics of i-butane in BPET at 35 °C. pi ) 0 cmHg, pf ) 31.1 cmHg, M∞ ) 0.16 ( 0.01 g/100 g, D h ) 1.0 ( 0.2 × 10-14 cm2/s. (b) Non-Fickian sorption kinetics of methyl acetate in BPET at 35 °C. pi ) 2.5 cmHg, pf ) 5.0 cmHg, M∞ ) 0.32 ( 0.02 g/100 g, D h ) 6.0 ( 1.0 × 10-12 cm2/s, RR ) 0.53 ( 0.06, τR ) 240 ( 20 min, τF ) 25 ( 6 min, tD ) 32 ( 3 min.

of methyl acetate and acetone in poly(n-butyl methacrylate) are 10.8 and 9.3 cm3 (STP)/cm3, respectively. Sorption Kinetics of Alkanes and Esters in PET. Figure 7a and b present results from two representative kinetic sorption experiments of i-butane and methyl acetate, respectively, in BPET at 35 °C. In each case, the fractional mass uptake, Mt/M∞, increases linearly with t1/2 at short times. This behavior is characteristic of sorption kinetics controlled by Fickian diffusion (see, for example, Figure 7a and the dotted curve in Figure 7b). In some instances, the fractional mass uptake at longer times exhibits a protracted, asymptotic approach toward equilibrium32 (see, for example, the solid curve in Figure 7b). This behavior is ascribed to mass uptake kinetics limited by polymer swelling and structural rearrangement to accommodate the penetrant rather than Fickian diffusion. Such two-stage kinetics have been observed for organic vapor sorption in glassy polymers such as linear alkanes in PET;4 ketones in PET;7,33 ethyl benzene in polystyrene;34 benzene in polystyrene;35 benzene, n-hexane, and cyclohexane in polyimides;36 and methanol in block copolymers.37 These sorption kinetics can be described using the following empirical model7

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Mt M∞

[

) 1 - (1 - RR)

(

∞

8

)]

π 2 2 -D h (2n + 1) π t l2

1

∑ 2 n)0

exp (2n + 1)2 -(t - tD) + RRexp H(tD) (12) τR

[

]

where D h is the average diffusion coefficient; l is polymer film thickness; RR is the fraction of weight uptake occurring during the protracted, non-Fickian approach to equilibrium; τR is the time constant associated with the long-time drift in mass uptake; tD is a delay factor accounting for a delay in the beginning of structural relaxation; and H(tD) is the Heaviside unit step function.38 The curves in Figure 7a and b represent least-squares fits of eq 12 to the experimental data for the sorption of i-butane and methyl acetate in BPET at 35 °C, respectively, following the procedure described earlier.4 In Figure 7b, the dashed curve indicates the fraction of mass uptake controlled by Fickian diffusion, and the solid curve indicates the fraction controlled by polymer chain relaxation. The time required to reach one-half of the total mass uptake by Fickian diffusion is given by4

t1/2 )

τF l2 ) 20 20D h

(13)

where τF is the characteristic time scale for diffusion, defined as l2/D h. Average diffusion coefficients were estimated using the procedure described previously.4,7 Figure 8 presents the diffusion coefficients of i-butane and i-pentane, along with previously published4 diffusion coefficient values for n-butane and n-pentane in BPET, as a function of average penetrant concentration. The average penetrant concentration was defined as

C h )

Ci + Cf 2

(14)

where Ci and Cf are the penetrant concentrations at the beginning and end of the sorption experiment, respectively. For all penetrants, the logarithm of the diffusion coefficient increases approximately linearly with penetrant concentration, consistent with other reports of the concentration dependence of organic vapor diffusion coefficients in glassy4,33 and rubbery polymers.39 The following empirical model was used to describe the data4

h) D h ) D∞ exp(ωC

(15)

where D∞ is the infinite-dilution (or zero-concentration) diffusion coefficient and the slope, ω, characterizes the dependence of D h on C h . D∞ and ω values for i-butane and i-pentane are reported in Table 4, along with the values for n-butane and n-pentane.4 The infinite-dilution diffusion coefficients were adjusted to an amorphous basis using the following relation40

Da )

D∞ φa

(16)

This relation has been shown to be valid for light gases and water vapor in semicrystalline PET.40 Roughly speaking, it seeks to account for the effect of tortuosity

Figure 8. Effect of average concentration on apparent diffusion coefficients of linear4 and branched alkane hydrocarbons in BPET at 35 °C. Table 4. D∞ and ω Values for Linear and Branched Alkane Hydrocarbons and Esters at 35 °Ca penetrant

D∞ (cm2/s)

D∞/φa (cm2/s)

ω [cm3 of polymer/ cm3 (STP)]

n-butane i-butane n-pentane i-pentane methyl acetate ethyl acetate

(6.7 ( 0.3) × 10-13 (7.0 ( 4) × 10-15 (3.1 ( 0.4) × 10-14 (3.3 ( 2) × 10-15 (9.1 ( 2.1) × 10-13 (1.3 ( 0.4) × 10-13

1.1 × 10-12 1.2 × 10-14 5.1 × 10-14 5.4 × 10-15 1.5 × 10-12 2.1 × 10-13

0.47 ( 0.05 1.4 ( 0.7 1.32 ( 0.09 2.3 ( 1.4 0.34 ( 0.04 0.39 ( 0.05

a Values for n-butane and n-pentane obtained from the literature.4

(due to the presence of impermeable crystallites in the polymer matrix) on diffusion coefficients. It does not account for the effects of chain immobilization on crystallinity. Such effects were not important for lightgas transport in PET, but they cannot be ruled out for larger penetrants. Therefore, this relation is used here as a first approximation. The infinite-dilution, estimated amorphous-phase diffusion coefficient, Da, of i-butane in BPET is 2 orders of magnitude lower than that of n-butane and that of i-pentane is 1 order of magnitude lower than that of n-pentane. The diffusion coefficients decrease in the following order

n-butane > n-pentane > i-butane > i-pentane suggesting that chain branching has a larger effect on lowering the hydrocarbon diffusion coefficient than chain length. To compare the effects of chain branching versus chain length on hydrocarbon diffusion coefficients in PET, infinite-dilution, estimated amorphous-phase diffusion coefficients of linear and branched alkane hydrocarbons in PET from this study and available literature data26 are presented in Figure 9 as a function of number of carbon atoms. Because the diffusion coefficients of penetrants in this study were measured at 35 °C, they were adjusted to 25 °C using the procedure described previously.26 The logarithm of the diffusion coefficients of normal hydrocarbons decreases linearly with increasing chain length (from C1 to C5), as shown. To quantify the effect of chain branching on

Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 2973

Figure 9. Infinite-dilution, estimated amorphous-phase diffusion coefficients of linear4 and branched alkane hydrocarbons in PET as a function of carbon number.

the diffusion coefficients, the following procedure was used:41 An effective chain length was obtained for i-butane and i-pentane based on their diffusion coefficient values and an extrapolated linear curve fit to the data for the normal C1-C5 hydrocarbons. The effective chain length obtained by this technique was 6.2 for i-butane (C4) and 7.2 for i-pentane (C5). According to this analysis, the effective size contribution to diffusion coefficient of a methyl (CH3) side group is 2.2, while that of a CH3 or CH2 group in the main chain is unity. Although this method assumes a linear decrease in the log of diffusion coefficients of straight-chain hydrocarbons larger than n-pentane, which might not be accurate, it clearly demonstrates the important effect of chain branching on hydrocarbon diffusion coefficients in PET. This result is consistent with several literature studies reporting the effects of size and shape on hydrocarbon diffusion coefficients in rubbery polymers such as polyisobutylene,42 ethyl cellulose,13 and natural rubber43 and in glassy polymers such as polycarbonate.44 Figure 10 presents the diffusion coefficients of methyl acetate and ethyl acetate as a function of average penetrant concentration. The logarithm of the diffusion coefficient increases linearly with penetrant concentration for both penetrants. The data were fit to the model presented in eq 15, and the D∞ and ω values are reported in Table 4. The dependences of D h on C h are similar for methyl acetate and ethyl acetate over the concentration range studied. The diffusion coefficient values of acetone and methyl ethyl ketone in BPET from the literature7 are also included in Figure 10 for comparison. Somewhat surprisingly, the diffusion coefficients of methyl acetate and ethyl acetate are quite similar to those of acetone and methyl ethyl ketone, respectively, over the concentration range studied, even though the esters are slightly larger than the ketones with the same number of carbon atoms (cf. Table 3).23 Some literature studies have even reported higher diffusion coefficients for methyl acetate than acetone in rubbery poly(vinyl acetate)45 and in glassy46 and rubbery45 poly(methyl methacrylate). Diffusion coefficients of polar penetrants in polymers can be retarded by specific favorable interactions between the penetrant and polymer matrix.47 Judging from their dipole moments, the ketones are much more polar than the esters with the same

Figure 10. Effect of average concentration on the apparent diffusion coefficients of methyl acetate (9) and ethyl acetate (O) in BPET at 35 °C. The literature data7 for acetone (4) and MEK (1) are included for comparison. The critical volumes of acetone and MEK are 209 and 267 cm3/mol, respectively.23

number of carbon atoms (cf. Table 3).23 The larger sizes of the esters (which would favor lower diffusion coefficients) could be partially offset by their lower polarities (which would favor higher diffusion coefficients), resulting in similar diffusion coefficient values for these two classes of penetrants. Interestingly, the diffusion coefficients of ethyl acetate and methyl ethyl ketone in a nonpolar, rubbery polymer, high-density polyethylene (HDPE), support this notion.48,49 The diffusion coefficient of methyl ethyl ketone (4.7 × 10-13 cm2/s) is an order of magnitude larger than that of ethyl acetate (0.4 × 10-13 cm2/s) in HDPE at 30 °C and a penetrant activity of 0.1.48 This result suggests that penetrantpolymer interactions, which act to decrease the diffusion coefficient of the smaller but more polar ketone relative to that of the larger but less polar ester in BPET, are absent in nonpolar HDPE. Diffusion coefficients generally decrease with increasing penetrant size, as shown in Figure 11. Critical volume is a convenient measure of penetrant size, as it has been tabulated for a wide variety of penetrants.26 The infinite-dilution, estimated amorphous-phase diffusion coefficients in PET have been roughly correlated with the critical volume using the following empirical power-law model4

Da )

τ Vcη

(17)

where τ and η are adjustable parameters. Because the diffusion coefficients of several penetrants in Figure 11 were originally reported at 25 °C,26 these values were adjusted to 35 °C using the procedure described previously.26 The infinite-dilution diffusion coefficient of ethyl acetate is approximately an order of magnitude lower than that of methyl acetate, consistent with its larger size. The correlation of D with Vc is good for relatively linear penetrants, such as methyl acetate and ethyl acetate. The values of τ and η (4.4 ( 1.2 × 107 cm2/s (cm3/mol)8.4 and 8.4 ( 1.0, respectively), which were obtained from a curve fit of the experimental data for spherical and relatively linear penetrants at 35 °C, are somewhat different from the parameter values reported

2974 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004

is consistent with its larger size. The diffusion coefficient for each penetrant increases with increasing penetrant concentration. The diffusion coefficients of the esters (methyl acetate and ethyl acetate) are quite similar to those of the ketones (acetone and methyl ethyl ketone) with the same number of carbon atoms, even though the esters are slightly larger than the ketones. Thus, penetrant diffusion coefficients in PET, which are primarily dependent on penetrant size and shape, appear to be influenced by polarity effects as well. Acknowledgment The authors gratefully acknowledge partial support of this work by Eastman Chemical Company. Appendix A The Flory-Huggins model for the solubility of a penetrant in equilibrium matrixes, such as rubbery polymers, is given by Figure 11. Effect of penetrant size on infinite-dilution, estimated amorphous-phase diffusion coefficients in PET at 35 °C. The bestfit parameters of eq 17 are η ) 8.4 ( 1.0, τ ) 4.4 ( 1.2 × 107 [cm2/s (cm3/mol)8.4].

earlier at 25 °C.26 The diffusion coefficients of i-butane and i-pentane are approximately an order of magnitude lower than the values anticipated by this simple empirical model. This result is not particularly surprising because critical volume does not accurately capture the effect of penetrant shape on diffusion coefficients, as described previously.7,26 Conclusions Equilibrium sorption and uptake kinetics of i-butane, i-pentane, methyl acetate, and ethyl acetate in biaxially oriented, semicrystalline PET films at 35 °C have been reported. The equilibrium solubility of i-butane and i-pentane is lower than that of the corresponding linear alkanes, possibly because of restricted access of the branched penetrants to the nonequilibrium excess volume of PET. At constant relative pressure, the solubility of methyl acetate is higher than that of ethyl acetate, consistent with solubility trends of esters in other polymers and polar liquids, and the solubilities of the esters in PET are higher than those of the ketones having the same number of carbon atoms. These results are consistent with the differences between the solubility parameters of the penetrants and PET. Similar solubility trends have also been observed for esters and ketones in other polar and rubbery polymers. Thus, penetrant solubility in PET, which is primarily determined by penetrant condensability, also appears to be strongly dependent on penetrant size and shape (for nonpolar penetrants) and specific penetrant-polymer interactions (for polar penetrants). The sorption kinetics of all penetrants are described either by Fickian diffusion or by a two-stage model incorporating Fickian diffusion at short times and protracted polymer structural relaxation at long times. Chain branching appears to have an important effect on hydrocarbon diffusion coefficients in PET, consistent with the results reported for other rubbery and glassy polymers. The infinitedilution diffusion coefficient of ethyl acetate is an order of magnitude lower than that of methyl acetate, which

( )

ln

p ) ln φ + (1 - φ) + χ(1 - φ)2 Psat

(A1)

where Psat is the penetrant saturation vapor pressure, φ is the penetrant volume fraction, and χ is the FloryHuggins interaction parameter. The penetrant volume fraction, φ, can be written as

Cv 22 414 φ) Cv 1+ 22 414

(A2)

where C is the penetrant concentration [cm3 (STP)/cm3 of polymer] and v is the penetrant molar volume (cm3/ mol). The Flory-Huggins interaction parameter can be estimated from the solubility parameters of the penetrant and polymer as50

χ ) 0.34 +

v (∆δ)2 RT

(A3)

where ∆δ is the difference between the solubility parameters of the penetrant and polymer. In the limit of low pressures (i.e., p f 0 or C f 0), φ is

φ)

Cv 22 414

(A4)

Also, when φ , 1, eq A1 reduces to

ln

( )

p ) ln φ + 1 + χ Psat

(A5)

Substituting eq A4 into eq A5 gives

( ) (

ln

)

p Cv ) ln + (1 + χ) Psat 22 414

(A6)

Equation A6 can be rearranged to obtain an expression for the penetrant solubility at low pressure, S∞

S∞ )

22 414 C ) p v exp(1 + χ)Psat

(A7)

Hence, the ratio of infinite-dilution solubilities of two

Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 2975

penetrants A and B can be written as

( )( )

S∞,A Psat,B vB ) exp(χB - χA) S∞,B Psat,A vA

(A8)

where S∞,A and S∞,B are the infinite-dilution solubility coefficients of penetrant A and B, respectively; Psat,A and Psat,B are their saturation vapor pressures; vA and vB are the penetrant molar volumes; and χA and χB are the Flory-Huggins interaction parameters. Literature Cited (1) Dhoot, S. N.; Freeman, B. D.; Stewart, M. E. Barrier Polymers. In Encyclopedia of Polymer Science and Technology, 3rd ed.; Kroschwitz, J. I., Ed.; Wiley-Interscience: New York, 2003; Vol. 5, p 198. (2) Michaels, A. S.; Vieth, W. R.; Barrie, J. A. Solution of Gases in Polyethylene Terephthalate. J. Appl. Phys. 1963, 34, 1. (3) Yasuda, H.; Stannett, V. Permeation, solution, and diffusion of water in some high polymers. J. Polym. Sci. 1962, 57, 907. (4) Dhoot, S. N.; Freeman, B. D.; Stewart, M. E.; Hill, A. J. Sorption and Transport of Linear Alkane Hydrocarbons in Biaxially Oriented Poly(ethylene terephthalate). J. Polym. Sci. B: Polym. Phys. 2001, 39, 1160. (5) Izydorczyk, J.; Salwinski, J. Sorption and Permeation of Methanol and Ethyl Acetate Vapors in Polyethylene Terephthalate. Pol. J. Chem. 1992, 66, 321. (6) Hernandez, R. J.; Giacin, J. R.; Shirakura, A.; Jayaraman, K. Diffusion and Sorption of Organic Vapors through Oriented Poly(ethylene terephthalate) Films of Varying Thermomechanical History. ANTEC 1990, 557. (7) Dhoot, S. N.; Freeman, B. D.; Stewart, M. E. Sorption and Transport of Linear and Branched Ketones in Biaxially Oriented Poly(ethylene terephthalate). Polymer, manuscript submitted. (8) McBain, J. W.; Bakr, A. M. A new sorption balance. J. Am. Chem. Soc. 1926, 48, 690. (9) McDowell, C. C.; Coker, D. T.; Freeman, B. D. An automated spring balance for kinetic gravimetric sorption of gases and vapors in polymers. Rev. Sci. Instrum. 1998, 69, 2510. (10) Paul, D. R. Gas Sorption and Transport in Glassy Polymers. Ber. Bunsen-Ges. Phys. Chem. 1979, 83, 294. (11) Koros, W. J.; Paul, D. R. CO2 Sorption in Poly(ethylene terephthalate) Above and Below the Glass Transition. J. Polym. Sci. B: Polym. Phys. 1978, 16, 1947. (12) Vieth, W. R.; Alcalay, H. H.; Frabetti, A. J. Solution of Gases in Oriented Poly(ethylene terephthalate). J. Appl. Polym. Sci. 1964, 8, 2125. (13) Barrer, R. M.; Barrie, J. A.; Slater, J. Sorption and Diffusion in Ethyl Cellulose. Part III. Comparison Between Ethyl Cellulose and Rubber. J. Polym. Sci. 1958, 27, 177. (14) Doghieri, F.; Sarti, G. C. Nonequilibrium Lattice Fluids: A Predictive Model for the Solubility in Glassy Polymers. Macromolecules 1996, 29, 7885. (15) Fleming, G. K.; Koros, W. J. Dilation of polymers by sorption of carbon dioxide at elevated pressures. Macromolecules 1986, 19, 2285. (16) Suwandi, M.; Stern, S. Transport of Heavy Organic Vapors through Silicone Rubbers. J. Polym. Sci. B: Polym. Phys. 1973, 11, 663. (17) Berens, A. R. Sorption of organic liquids and vapors by rigid PVC. J. Appl. Polym. Sci. 1989, 37, 901. (18) Berens, A. R. Transport of Plasticizing Penetrants in Glassy Polymers. In Barrier Polymers and Structures; Koros, W. J., Ed.; ACS Symposium Series 423; American Chemical Society: Washington, DC, 1990; p 92. (19) Flory, P. J. Statistical mechanics of swelling of network structures. J. Chem. Phys. 1950, 18, 108. (20) Singh, A.; Freeman, B. D.; Pinnau, I. Pure and Mixed Gas Acetone/Nitrogen Permeation Properties of Polydimethylsiloxane [PDMS]. J. Polym. Sci. B: Polym. Phys. 1998, 36, 289. (21) Miranda, N. R.; Willits, J. T.; Freeman, B. D.; Hopfenberg, H. B. Organic vapor sorption and transport in a thermotropic liquid crystalline polyester. J. Membr. Sci. 1994, 94, 67.

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2976 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 Through Polymer Membranes. M.S. Thesis, Michigan State University, East Lansing, MI, 1996. (49) Huang, S. J.; Giacin, J. R. Evaluating the effect of temperature and vapor concentration on the organic vapor barrier properties of polymer membranes by an isostatic procedure. J. Plastic Film Sheeting 1998, 14, 308. (50) Brandrup, J.; Immergut, E. H.; Grulke, E. A. Polymer Handbook, 4th ed.; Wiley-Interscience: New York, 1999.

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Received for review July 28, 2003 Revised manuscript received October 30, 2003 Accepted November 4, 2003 IE030618S