Infinite Dilution Diffusion Coefficients of Chlorinated Methane in Poly

Dec 8, 2014 - Analytical-reagent-grade hexafluoroisopropyl alcohol was .... Figure 6. Temperature dependences of D∞ for various solvents in PET (the...
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Infinite Dilution Diffusion Coefficients of Chlorinated Methane in Poly(ethylene terephthalate) by Inverse Gas Chromatography Yi Liu, Yue Shang, and Guorong Shan* State Key Laboratory of Chemical Engineering, Department of Chemical and Biological Engineering, Zhejiang University, Hangzhou 310027, China S Supporting Information *

ABSTRACT: On the basis of the Van Deemter model of the chromatographic process, the infinite dilution diffusion coefficients of dichloromethane, chloroform, and carbon tetrachloride in poly(ethylene terephthalate) (PET) have been measured over a wide range of temperatures from 373.15 to 413.15 K by inverse gas chromatography. The relationships between the infinite dilution diffusion coefficients and molecular size of the solvent, as well as the temperature, were investigated. Meanwhile, the measured diffusion data were compared with the theoretical predictions by free-volume theory. The results indicated that neglecting the molecular interactions, i.e., the diffusivity energy term in the free-volume equation, may result in a poor prediction of the diffusion coefficient for PET-chlorinated methane systems because of the strong molecular interactions. The good agreement between the experimental data and the theoretical predictions by the proposed method showed that it is reasonable to take the diffusivity energy term into consideration for those systems with strong molecular interactions, and, furthermore, the infinite dilution diffusion coefficients at different temperature can be predicted accurately with this method.

1. INTRODUCTION The knowledge of small molecular diffusion in polymeric materials concerns all kinds of industrial processes such as the effectiveness of polymerization reactions as well as the properties of the polymer produced; other operations involving molecule transportation include membranes for separation processes,1 drying of paints,2 devolatilization,3 controlled drugdelivery systems4 etc. Besides, with the quick development of the polymerization industry, how to recycle the increasing kinds and amounts of polymers has been a great challenge; the basic physical and chemical methods to solve this problem may deal with the diffusion process of small solvent molecules in polymers. Poly(ethylene terephthalate) (PET) has been widely used as fibers, films, engineering plastics, polyester bottles, etc.; diffusion coefficients of solvents in PET are needed to better predict the behavior of dissolution or degradation processes. However, limited diffusion data about solvents in PET have been reported. Therefore, accurate measurements of these essential diffusion parameters are of critical importance in guiding industrial production. There are many methods to measure diffusion parameters, such as gravimetric sorption,5 isotactic permeation experiments,6 and nuclear magnetic resonance.7 However, it is difficult to apply these conventional methods to polymer− solvent systems, especially for trace amounts of solvents. As a fast and reliable technique, inverse gas chromatography (IGC) and corresponding mathematical models8,9 have been developed for the measurement of diffusion coefficients in polymer, among which the Van Deemter model is widely used because of its brevity and convenience.10,11 In this work, the infinite dilution diffusion coefficients of dichloromethane, chloroform, and carbon tetrachloride in PET (for the amorphous part of the polymer) were measured by IGC based on the Van Deemter model. © 2014 American Chemical Society

For most polymer−solvent pairs, the temperature and composition dependence of solvent molecule diffusion coefficients can be predicted by the Vrentas−Duda free-volume theory. However, to some extent, the predictive capabilities of the equation depend on how to handle the parameters ξ and E* within it, which is related to the molecular interaction force between the polymer and solvent. In most cases, the activation energy was assumed to be zero, and good predictions were achieved.12−14 However, it has been verified that the diffusivity activation energy may have a great influence on the solvent diffusion behavior in the polymer when the temperature is close to the glass transition temperature of the polymer.15 In this study, how to deal with the parameters E* and ξ in the predictive equation for the PET−chlorinated methane systems is discussed. Besides, on the basis of the measured diffusion data, a new processing method is proposed to predict the diffusion parameters for PET−chlorinated methane systems at different temperatures. The good agreement between the experimental data and the theoretical predictions showed that the proposed method is a reliable way to predict the infinite dilution diffusion coefficients for those polymer/solvent systems with strong molecular interactions.

2. EXPERIMENTAL SECTION 2.1. Materials. Poly(ethylene terephthalate) (PET) was obtained from Sinopec Group. Dichloromethane, chloroform, and carbon tetrachloride were purchased as analytical reagents from Acros Organics. Analytical-reagent-grade hexafluoroisopropyl alcohol was purchased from J&K Scientific Ltd. The Received: Revised: Accepted: Published: 19533

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for mass transfer in the stationary polymer phase and is related to the diffusion coefficient D∞ as follows:16

silanized Chromsorb-G support, whose size is 60−80 mesh, was purchased from Tianjin Borui Bonding Chromatography Technology Co., Ltd. 2.2. Preparation of the Packed Column. A solution of PET in hexafluoroisopropyl alcohol (0.019 g/mL) was mixed with the support; thereafter, the solvent was removed by slow evaporation in a rotary evaporator at 40 °C to get a uniform polymer coating. Then, the coated support particles were dried under vacuum to a constant weight and loaded into a clean stainless steel column of 2 m length and 3 mm inner diameter, with the aid of a pump connecting to the other end and a mechanical vibrator. Finally, the ends of the tubes were loosely plugged with steel wool. The characteristics of the column are shown in Table S1 in the Supporting Information (SI). 2.3. Apparatus and Procedure. The schematic diagram of the IGC apparatus used for determination of the diffusivities in the polymer is presented in Figure S1 in the SI. The gas chromatograph is Agilent 7820 (purchased from Agilent Technologies Inc.), equipped with a soap bubble flowmeter measuring the flow rate of the carrier gas. Also, a highly accurate pressure valve is installed at the inlet of the chromatographic column to control the pressure at a specific value or pressure drop in the column. A thermal conductivity detector and a personal computer equipped with a chromatographic workstation were used to detect and evaluate the solute signal in the carrier gas out of the column. Nitrogen was used as the carrier gas in all experiments. The temperature of the detector and injection block was set at 423.15 K. Small amounts of solvent (0.6 μL) were injected into the column through a silicone rubber septum with a 1 μL syringe, and then about 50 μL of air was injected along with the liquid samples as an inert component to determine the average velocity of the carrier gas in the column. The output from the thermal conductivity detector was fed to a chromatographic workstation for further analysis of the peaks. In this work, the diffusion parameters of dichloromethane, chloroform, and carbon tetrachloride in PET were determined at various temperatures from 373.15 to 413.15 K. At each temperature, measurements were performed over a wide range of flow rates from 10 to 40 mL/min. In addition, four repeating experiments were done at each flow rate, the pressure drop across the column was measured by the manometer installed at the inlet of the chromatographic column, and then the flow rate in the column was set to a next value after each flow rate was studied. 2.4. Processing of the Measured IGC Data. For packed chromatographic columns, the diffusion coefficients are determined from analysis of the variation of the plate height H with the average gas flow rate u, by the Van Deemter equation:8 H = A + B/u + Cu

C=

2 ⎛ L ⎞⎛ w1/2 ⎞ ⎜ ⎟⎜ ⎟ ⎝ 5.54 ⎠⎝ tr ⎠

(1 + k)2 D∞

(3)

where q is a shape factor that is accounting for the nonuniformity of the liquid film in the packed column. It is a numerical constant in the Van Deemter equation that equals to 8/π2.17 Hence, the infinite dilution diffusion can be deduced from eq 3 as follows: D∞ =

8d p2

k π 2C (1 + k)2

(4)

where k is the partition ratio (or capacity factor), which is calculated from eq 5:

k=

tr − ta ta

(5)

where ta is the retention time of the inert gas air. Also, dp is the thickness of the polymer coating on the support, which is determined from eq 6 (the deduction process is shown in eq S1 in the SI):

dp =

ωpρd dd 6ρp ωd

(6)

where ωp is the weight of PET coated on the support and ωd and ρd are the weight and density of support, respectively. ρp is the density of the polymer, and dd is the equivalent diameter of the support, which is measured by a Beckman−Coulter LS-230 laser particle size analyzer; the particle size distribution of the support is shown in Figure S2 in the SI. The linear velocity of the carrier gas, u, is calculated from eqs 7 and 8: u=

jF Tcol a Tfm

(7) 2

3 ( ) j= 2 ( ) Pin Pout

Pin Pout

−1

3

−1

(8)

where j is the James−Martin compressibility factor and Pin and Pout are the pressures at the inlet and outlet of the column, respectively. Hence, the compressibility factor can be obtained with the measured data of Pin and Pout at different velocities of the carrier gas. Tcol and Tfm are the temperatures of the column and flowmeter, respectively. a is the equivalent cross-sectional area of the packed column, which can be calculated by dividing the retention volume of the inert component by the column length, and F is the volume flow of the carrier gas. When the flow rate is high enough, the second term in eq 1, representing diffusion in the gas phase, can be ignored. Therefore, the plot of H versus u should yield a straight line, and the value of C can be determined from the slope of the line. Consequently, the infinite dilution diffusion of the solvent in the polymer then can be determined by eq 4 combined with parameters of dp, k, and C, which can be estimated from analysis of the data obtained from the elution peak.

(1)

where A, B, and C are constants, u is the average linear velocity of the carrier gas, and H is the theoretical plate height, which is determined from the width of the eluted solvent peak with eq 2: H=

qkd p2

(2)

where L is the length of the packed column, w1/2 is the measured peak width at half of the maximum height, and tr is the retention time of the solvent. The constant C in eq 1 stands 19534

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3. RESULTS AND DISCUSSION 3.1. Results of the IGC Experiments. The measured IGC data (partly shown in Table S2 in the SI) were treated as in the procedure mentioned above, and the measured results were proven to be highly consistent with the theoretical relationships based on the Van Deemter model. The experimental data, plots of H versus u for three solvents in PET from 378.15 to 398.15 K, are shown in Figures 1−3 (plots under other temperatures

the other hand, the separation efficiency of the solvent was determined by the temperature and flow rate, and thus the positions of the lines were affected by the two aspects simultaneously. Therefore, the positions of the H−u lines were varied with different temperatures. Also, it is clear that all of the data fall on a straight line, and almost all of the correlation coefficients of linear fitting are above 0.99 (shown in Table S3 in the SI), which are consistent with the theoretical Van Deemter equation. These good agreements show that these diffusion coefficients obtained in this work are reliable. The values of parameters k and C at different temperature were obtained from the experimental data and the slopes of the straight lines in Figures 1−3 (and Figures S3−S5 in the SI), which are all shown in Table S4 in the SI. These values were then used to estimate the infinite dilution diffusion (D∞) of three solvents in PET by eq 4, as shown in Table 1. Table 1. Infinite Dilution Diffusion Coefficients of Chlorinated Methane in PET at Various Temperatures D∞ × 1012 (m2/s)

Figure 1. Relationships between the plate height and flow rate of dichloromethane in the column at different temperatures (the lines in Figures 1−3 represent the corresponding fitting lines).

temperature (K)

dichloromethane

chloroform

carbon tetrachloride

373.15 378.15 383.15 388.15 393.15 398.15 403.15 408.15 413.15

2.422 2.807 3.250 3.677 4.093 4.579 4.883 5.270 5.690

1.641 1.951 2.320 2.624 3.006 3.726 4.011 4.321 4.761

0.484 0.735 0.994 1.258 1.396 1.726 2.002 2.309 2.496

According to analysis of the data listed in Table 1, it is found that, under a specific temperature, the infinite dilution diffusion coefficients of three solvents (dichloromethane, chloroform, and carbon tetrachloride) in PET decreased with the increasing number of chlorine atoms in chlorinated methane. This may be related to a number of factors such as the size and shape of the solvent molecule. According to the Vrentas−Duda free-volume theory,18 two preconditions must be fulfilled before a molecule or jumping unit can migrate into the polymer matrix. First, a hole or free-volume space of sufficient size must appear adjacent to the molecule to migrate. For the specific polymer, PET, the free-volume space is fixed, so the smaller the solvent molecule is, the more appropriate molecules there are can migrate, which means it has a larger diffusion coefficient. The volume of the chlorine atom is larger than that of hydrogen atom, which can explain the above phenomena well. Plots of D∞ versus the solvent molecular volume are partly shown in Figure 4 to describe the influence of the solvent volume on the diffusion coefficient. As shown in Table 1, it is obvious that the temperature has a significant influence on the diffusion process. For each polymer−solvent system, the infinite dilution diffusion coefficients can be found to increase with increasing temperature. The crystallinity of the polymer will have a great influence on the diffusion behavior of the solvent. Whether different temperatures from 373.15 to 413.15 K will affect the crystallinity of PET was also explored here. The degrees of crystallinity of PET under different temperatures (378.15, 398.15, and 413.15 K) were investigated as follows. In order to simulate the real conditions in the column, PET was first kept 4 h at a constant temperature, and then the differential scanning

Figure 2. Relationships between the plate height and flow rate of chloroform in the column at different temperatures.

Figure 3. Relationships between the plate height and flow rate of carbon tetrachloride in the column at different temperatures.

are shown in Figures S3−S5 in the SI). In these figures, the ordinate, H, represents the theoretical plate height of the column and indicates the separation efficiency of the solvent in the column. On the one hand, different solvents have different separation efficiencies in the column because of the different interactions between the stationary polymer and solvent. On 19535

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Figure 4. Relationships between the infinite dilution diffusion coefficient under various temperatures and molecular volumes of the solvents (dichloromethane, 53.1 cm3/mol; chloroform, 65.7 cm3/mol; carbon tetrachloride, 78.3 cm3/mol; the estimation method is shown in Table S5 in the SI).

Figure 6. Temperature dependences of D∞ for various solvents in PET (the lines represent the corresponding fitting lines).

calorimetry (DSC) thermograms of PET were measured. The results are shown in Figure 5. As shown in the figure, the degree

(9)

⎡ γ(ω1V1* + ω2ξV2*) ⎤ ⎛ E* ⎞ ⎟ exp⎢ − D1 = D01 exp⎜ − ⎥ ⎝ RT ⎠ ⎣ ω1VFH1 + ω2VFH2 ⎦

where D01 is the preexponential factor, which is assumed to be independent of the temperature, E* is the diffusivity activation energy to elucidate the molecular attractive interaction between the polymer and diffusing molecule, R is the universal gas constant, ωi is the mass fraction of component i (i can be 1 or 2, i.e., 1 for the solvent and 2 for the polymer), Vi* is the specific critical hole free volume of component i required for a jump, and ξ is the ratio of the molar volume of solvent jumping unit to that of the polymer jumping unit. The parameter overlap factor γ (whose value should be between 1/2 and 1) is introduced to explain that the same hole free-volume space is available for the migration of different jumping units. VFHi is the average hole free volume per gram of component i. Also, for diffusion of a trace amount of the solvent into the polymer, the diffusion coefficient in this limiting case can be predicted directly using eq 9 with a limit of ω1 = 0:

Figure 5. DSC thermograms of PET under different temperatures.

⎛ ξV * ⎞ ⎛ E* ⎞ 2 ⎟ exp⎜ − D∞ = D01 exp⎜ − ⎟ ⎝ RT ⎠ ⎝ VFH2/γ ⎠

of crystallinity of PET remained nearly constant after processing under different temperatures, indicating that the diffusion coefficients (for the amorphous part of the polymer) measured here were with the same crystallinity. This means that the increased values of the diffusion coefficient here were mainly caused by increasing temperature. As mentioned above, another precondition described in the free-volume theory is that the molecule or jumping unit must have enough energy to migrate in the polymer matrix. For trace amounts of the solvents, there are sufficient free volumes to make a jump, while the activation energy may be limited to the diffusion process and a higher temperature will certainly provide much more energy for activation. The temperature dependences of D∞ were plotted for these solvents and are shown in Figure 6. As indicated in Figure 6, D∞ increases with increasing temperature, and that behavior follows the Arrhenius relationship; the fitting of the Arrhenius parameters is shown in Table S6 in the SI. 3.2. Theoretical Predictions Using the Traditional Predictive Method. For most polymer−solvent systems, the temperature and composition dependences of the small solvent molecule diffusion coefficient D1 in the polymer can be expressed by the Vrentas−Duda free-volume equation19 as follows:

(10)

Accurate estimation of the model parameters is the key point to predicting the diffusion coefficient exactly. There are five independent parameters that need to be determined in eq 10: D01, ξ, V2*, VFH2/γ, and E*. D01 is the solvent property and is independent of the polymer matrix, whose value can be estimated by combining the Dullien equation for the self-diffusion coefficient of pure solvents with the Vrentas−Duda free-volume equation;20 the preexponential factors of the solvents used here were obtained from ref 12. The parameter that has been by far the most evasive and controversial is ξ, whose definition expression is ξ = V1j*/V2j*

(11)

where V1j* is the minimum or critical hole free volume per mole of solvent jumping units required for a displacement and V2j* is the critical hole free volume per mole of polymer jumping units required for a displacement. For many solvents, such as small or spherically shaped penetrants, it is reasonable to expect that the entire solvent molecule is the jumping unite; ξ may be defined as 19536

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ξ = V10(0)/V2j*

Article

Table 2. Comparison of Experimental Data with the Different Theoretical Predictions (by the Traditional Method or the Predictive Method Proposed in This Work)

(12)

where V01(0) is the solvent molar volume at 0 K and can be estimated using the group contribution method.21 Parameter V2j* is the molar volume of the polymer jumping unit, which is independent of the solvent nature. Although it cannot be calculated directly because the size of a polymer jumping unit is not known, it still can be determined using diffusion data for a single solvent because it can be regarded as an intrinsic polymer property. Unfortunately, no diffusion data of any solvent in PET has been reported at present, so the value of V2j* in PET cannot be estimated accurately. However, an alternative method of estimating V2j* has been proposed by Zielinski and Dada;12 they presented the following empirical linear relationships with the glass transition temperature of the polymer as 3

V2j* (cm /mol) = 0.6224Tg2 (K) − 86.95

system

temperature (K)

12 D∞ E × 10 (m2/s)a

8 D∞ T1 × 10 (m2/s)b

12 D∞ T2 × 10 (m2/s)c

PET/ dichloromethane

373.15

2.422

0.365

2.489

378.15 383.15 388.15 393.15 398.15 403.15 408.15 413.15 373.15 378.15 383.15 388.15 393.15 398.15 403.15 408.15 413.15 373.15

2.807 3.250 3.677 4.093 4.579 4.883 5.270 5.690 1.641 1.951 2.320 2.624 3.006 3.726 4.011 4.321 4.761 0.484

0.378 0.391 0.404 0.417 0.430 0.443 0.456 0.469 0.822 0.946 1.067 1.186 1.301 1.413 1.520 1.623 1.723 1.016

2.837 3.223 3.648 4.116 4.630 5.193 5.807 6.476 1.737 2.000 2.295 2.623 2.987 3.390 3.836 4.326 4.865 0.655

378.15 383.15 388.15 393.15 398.15 403.15 408.15 413.15

0.735 0.994 1.258 1.396 1.726 2.002 2.309 2.496

1.201 1.388 1.573 1.757 1.938 2.115 2.287 2.455

0.770 0.902 1.051 1.221 1.413 1.628 1.870 2.141

PET/chloroform

(Tg2 ≥ 295 K) (13)

In order to predict the diffusion coefficients of these polymer−solvent systems with pure polymer and solvent data alone, V2j* was estimated using the above equation as most researchers did,13,14 although maybe it cannot generally provide valid estimates for V2j*. The parameter V2* can also be estimated using the group contribution method, which is summarized in Table S7 in the SI. Usually, the free-volume parameter VFH2/γ can be estimated with eq 14 as follows:22 VFH2 K = 12 (T + K 22 − Tg2) γ γ

PET/carbon tetrachloride

(14)

The parameters K22 and K12/γ can be estimated by simply relatin them to the WLF constants of the polymer, C12 and C22 as in eqs 15 and 16.23 The WLF constants of PET were obtained from ref 24; that is, C12 = 2.71 and C22 = 310.79. K12 V2* = γ 2.303C12C22

(15)

K 22 = C22

(16)

a ∞ DE is the experimental infinite dilution diffusion coefficient of chlorinated methane in PET. bD∞ T1 is the theoretical prediction using the traditional predictive method. cD∞ T2 is the theoretical prediction using the predictive method proposed in this work.

comparion of the theoretical predictions (D∞ T1) and experimental results (D∞ ) listed in Table 2, it can be found that, for E the PET−chlorinated methane systems, an obvious deviation appears between the prediction and experimental infinite dilution diffusion coefficients. The reason for this overestimation of the diffusion coefficient may be related to the following two reasons. First, the strong interaction force between the polymer and solvent molecule is ignored. Wang et al. have confirmed the existence of hydrogen bonding in the acrylate polymer−chloroform mixtures by NMR analysis.25 Because PET has the similar structure and functional group as the acrylate polymer, it is reasonable to presume that there is hydrogen bonding formatted in these PET−chlorinated methane systems; the hydrogen bonding offers stronger molecular interaction than the ordinary van der Waals force. It has been indicated that the energy term E* in the predictive equation elucidates the molecular interaction between the polymer and diffusing solvent. Hence, for the polymer−solvent systems studied, diffusion may be dominated not only by freevolume effects but also by energy effects, and it is unacceptable to assume E* = 0 as usual. Considering the strong interactions between the PET and chlorinated methane molecules, it should be note that the energy term should be taken into account in eq 17, which shows that eq 10 should be used to predict the

As for the activation energy term E* in eqs 9 and 10, which relates to the molecular interactions between the polymer and solvent, the assumption of neglecting the energy effects was usually taken, with which the diffusion coefficients of many polymer−solvent systems have been predicted successfully,12,13 and in this case, the predictive eq 10 can be rewritten as ⎛ ξV2* ⎞ D∞ = D01 exp⎜ − ⎟ ⎝ VFH2/γ ⎠

(17)

All of the free-volume model parameters for PET− chlorinated methane systems were obtained from the procedure or references mentioned above, and they are listed in Table S8 in the SI. Besides, the activation energy term E* equal to zero is adopted here like for most polymer systems. With eq 17, these obtained parameters in Table S8 in the SI were used to predict the infinite dilution diffusion coefficients of each polymer−solvent system. The theoretically predictive results are shown in the fourth column of Table 2 and compared with those experimental results. 3.3. Theoretical Predictions Considering the Molecular Interactions between the Polymer and Solvent. By a 19537

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values of ξ were determined from experimental results here, it is reasonable to take them as the real values of ξ. This means that the empirical equation used to estimate the value of V2j* in the traditional predictive method is not applicable to PET. Then, the new values of ξ were used to predict the infinite dilution diffusion coefficient of these PET−chlorinated methane systems like the traditional method assuming E* = 0, and the results are listed in Table S9 in the SI, however, from which it can be found that the diffusion coefficients were still overestimated. This means that the assumption that E* = 0 cannot be acceptable for the prediction of PET−chlorinated methane systems. Considering the energy term E*, the data in Table 3 combined with those in Table S8 in the SI were used to predict the diffusion coefficients of the other two PET−chlorinated methane systems with eq 10. The results are shown in the fifth column of Table 2 and compared with the experimental results. From the comparison between the theoretical predictions ∞ (D∞ T2) and experimental results (DE ) listed in Table 2, it is clear that the predicted diffusion coefficients of the three solvents in PET at different temperatures are extremely close to those experimental results, which indicates that it is reasonable to take the energy term into consideration for PET−chlorinated methane systems because of the strong molecular interactions between the polymer and solvent, as well as using the semipredictive method to determine the parameter ξ. The traditional predictive method assumes that neglecting the energy term here may result in a poor prediction of the diffusion coefficient for the systems with strong molecular interactions. In addition, the good agreement between the experimental data and the predicted results provides a new way to predict the infinite dilution diffusion parameters for those systems with strong molecular interactions; i.e., a few experimental diffusion data are used to achieve the parameters E* and ξ by the mentioned method, with which the infinite dilution diffusion coefficients of other solvents in the polymer at different temperatures above Tg2 can be predicted using eq 10 accurately.

diffusion parameters. Second, as mentioned above, there is no appropriate way to estimate the value of V2j* accurately; the next alternative method was adopted here as per usual, that is, using eq 13 to calculate its value, where the accuracy may influence the validity of the predictive results. However, when E* ≠ 0 is adopted, it is possible to formulate a semipredictive but not a predictive version of the free-volume theory because a small amount of diffusivity data was needed to determine the parameter E* and ξ, and then the values of E* and ξ were used to predict the diffusion parameters at substantially different temperatures with eq 10. For example, the parameter E* and ξ can be estimated by using the measured diffusion coefficient as follows.20,26 Equation 10 is rearranged as follows: Y = E* + ξX

(18)

Y = −(ln D∞ − ln D01)RT

(19)

X=

V2*RT VFH2/γ

(20)

The values of E* and ξ can be obtained from the intercept and slope of a Y versus X plot. E* is defined as the energy per mole that a molecule needs to overcome attractive forces, which vary with the concentration of the solvent in the system, while all values of E* studied in this work are with the limit of ω1 = 0. With eqs 18−20, at least two diffusivity−temperature data points were required in this procedure. Here, the diffusion coefficients of the PET−dichloromethane system were taken as the basic data to get the values of E* and ξ of this polymer−solvent system. Then V2j* of PET can be calculated according to the definition of ξ with its values and V01(0) of dichloromethane, and ξ of PET−chloroform and PET−carbon tetrachloride systems can also be obtained with eq 12 using the calculated V2j*. The energy term E* in eq 10 is the effective energy per mole that a molecule needs to overcome attractive forces. Vrentas et al.27 summarized the diffusion data for a number of polymer− solvent systems and proposed the following function to estimate E* with utilization of the solubility parameters: 0

E* = F[(δ1 − δ2)2 V1̃ ]

4. CONCLUSIONS By utilization of an inverse gas chromatograph, the infinite dilution diffusion coefficients of trace amounts of dichloromethane, chloroform, and carbon tetrachloride in PET were measured at various temperatures based on the Van Deemter model of the chromatographic process. The results showed the following relationships with the molecular size of the solvents and temperature. First, at a certain temperature, the infinite dilution diffusion coefficient decreases with increasing number of chlorine atoms in the chlorinated methane, which means that the increasing molecular size of the solvent is a disadvantage for the diffusion process of the solvent molecule in PET. Second, the increase of the temperature can accelerate diffusion of the solvent in PET, which follows the Arrhenius relationship. Meanwhile, the infinite dilution diffusion coefficients of chlorinated methane in PET were compared with the theoretical predictions by the Vrentas−Duda free-volume theory. Because of the strong molecular interactions in the PET−chlorinated methane systems, it is unacceptable to neglect the diffusivity energy term in the predictive equation as per usual. What is more, the empirical equation (eq 13) used to estimate the value of V2j* in the traditional predictive method is not applicable to PET. These process methods will result in poor predictions for PET−chlorinated methane

(21)

where δ1 and δ2 are the solubility parameters of the solvent and polymer, respectively; the values of these parameters were cited from ref 28 and are listed in Table S8 in the SI. The value of E* can be identified using the function, i.e., the graph of E* versus 0 0 (δ1 − δ2)2V1̃ in the reference, once the value of (δ1 − δ2)2V1̃ of the polymer−solvent system was calculated. The values of E* and ξ obtained from the above methods are listed in Table 3. With the above method, we obtained new values of ξ, and they were smaller than that calculated with eqs 11−13 (ξ values of the three polymer−solvent systems were 0.409, 0.506, and 0.604, respectively), which indicates that the real jumping unit of PET is larger than that calculated from eq 13. Because the Table 3. Calculated Values of E* and ξ of PET−Chlorinated Methane Systems

E* (kJ/mol) ξ

PET− dichloromethane

PET− chloroform

PET−carbon tetrachloride

23.1 0.385

23.6 0.479

26.8 0.566 19538

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systems. The good agreement between the experimental data and the prediction results that consider the energy term indicates that it is reasonable to take it into account for PET− chlorinated methane systems and use the semipredictive method to determine the parameter ξ. In addition, once the diffusivity activation energy and parameter ξ were obtained from part of the determined diffusion coefficient, the freevolume theory can be used to predict the infinite dilution diffusion parameters at different temperatures well for those systems with strong molecular interactions. Besides, it should be noted that the diffusivity activation energy term plays an important role in the prediction of the diffusion data for those systems with a strong molecular interaction, and more attention needs to be paid to the contribution of the diffusivity energy to the mass-transfer process.

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ASSOCIATED CONTENT

S Supporting Information *

Details of the parameters involved and the deduction process of some equations used in this paper. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support was provided by National Support Project 2014BAC03B08.

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REFERENCES

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dx.doi.org/10.1021/ie503009d | Ind. Eng. Chem. Res. 2014, 53, 19533−19539