Transport of Gases in Glassy Polymers under Transient Conditions

Article pubs.acs.org/IECR

Transport of Gases in Glassy Polymers under Transient Conditions: Limit-Behavior Investigations of Dual-Mode Sorption Theory Lei Wang, Jean-Pierre Corriou, Christophe Castel, and Eric Favre* Laboratoire Réactions et Génie des Procédés (LRGP), CNRS, Université de Lorraine, ENSIC, BP 20451, Nancy 54001, France ABSTRACT: Considerable interest had been focused during 1950−1980 on the pure gas isothermal sorption−diffusion modeling in glassy polymers. In particular, the theory of dual-mode sorption has been developed during this period. It postulates that two concurrent sorption modes occur in the microheterogeneous medium. On the basis of this theory, rather complete and successful models have been roposed. In this paper, a short review of dual-mode-sorption models is first given. By considering a general expression of the sorption mode, the conventional models (partial immobilization model and immobilization model) become one limit-behavior model: the time-lag upper-bound model. Another limit-behavior model, the time-lag lower-bound model called the dual-diffusion model, is particularly investigated and compared to conventional models in this paper. It is shown that the steady-state permeability is independent of the molecule exchange rate, while the time-lag prediction depends on it. Furthermore, the time-lag lower-bound model provides, in general, a better agreement with experiments. In this respect, the dual-diffusion model is considered as a simple and efficient dual-mode-sorption theory approach.

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large.3,4 Thus, in this paper, partial pressures are used everywhere as an approximation. It is well-known that a linear relationship between the gas concentration C in the membrane and its outside partial pressure p exists in polymers above their transient temperature Tg. This equilibrium is accurately described by Henry’s law

INTRODUCTION The transport of penetrant molecules in and through dense polymeric membranes can be decomposed into two contributions: sorption and diffusion.1 Considering sorption, in the case of rubbery polymers, above their glass transition temperature Tg, sorption usually occurs only as a Henry mechanism, and the gas diffusion through the membrane can be well described by Fick’s first law. However, the situation becomes more complex for a glassy polymer (i.e., below its glass transition temperature). In this case, an abundant literature indicates considerable deviations from the simple Henry-type mechanism. Since the 1950s, a theory known as dual-mode sorption has proposed to decompose the sorption in glassy polymers into a linear Henry-type part and a nonlinear Langmuir-type part in order to better describe this complex sorption mechanism. The dual-mode-sorption theory aims at providing a good balance between complexity and simplicity. The model should be simple enough to be useful for the experimenter interested in interpreting sorption behavior, and yet it should not ignore the possible complexities of the involved transport process. Numerous modeling studies and experimental validations have been performed during its development. Because glassy polymeric membranes are widely used in industry, there are numerous situations of practical importance (packaging, controlled release, membrane separations, barrier polymers, etc.). The prediction of transport phenomena in these situations requires a correct application of the dual-mode-sorption theory. In this paper, the dual-mode-sorption theory is completed by investigating gas behaviors under limit conditions for transient mass-transfer situations. The common points as well as the differences between limit condition models will be highlighted.

C = kDp

where kD is the sorption coefficient considered usually as a constant. Because this linear relationship is not observed for the same polymer below its transition temperature Tg,5,6 Barrer et al.7 proposed that the polymer in a glassy state contains a distribution of microvoids frozen in the structure as the polymer is cooled when crossing its glass transition temperature. Thus, free segmental rotations of the polymer molecular chains are restricted in a glassy state, resulting in fixed microvoids throughout the polymer membrane. Because of these microvoids in the glassy polymer network, a portion of the penetrated molecules is partially or totally immobilized by entrapment or by binding at high-energy sites at their molecular peripheries, in a way similar to an adsorption phenomena. On the basis of this concept of microvoids in glassy polymers and experimental observation, Meares5 originally postulated two concurrent sorption mechanisms, ordinary dissolution described by Henry’s law concentration CD and a hole-filling process described by Langmuir adsorption concentration CH C D = kDp CH =

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C′H bp 1 + bp

(2)

Special Issue: Baker Festschrift

DESCRIPTION OF THE THEORY First, it can be noticed that gas fugacities rather than simple partial pressures are appropriate when high pressures lead to nonideal gas-phase behavior.2 However, in most cases, the correction is not © 2012 American Chemical Society

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Received: Revised: Accepted: Published: 1089

November 22, 2011 April 30, 2012 May 16, 2012 May 16, 2012 dx.doi.org/10.1021/ie2027102 | Ind. Eng. Chem. Res. 2013, 52, 1089−1101

Industrial & Engineering Chemistry Research

Article

where C′H is the hole-filling constant and b the hole affinity constant. The molecules obeying Henry’s law are mobile and the molecules following Langmuir's law are considered as immobile in a primary approach. Because of the partial or total immobilization effect, molecules sorbed by these two mechanisms diffuse in different ways through glassy membranes. The whole theory of sorption and transport in and through glassy polymers is named the dual-mode-sorption model.

and have been discussed by different authors. Removing some simplifying assumptions could indeed improve the model at the expense of a useless increasing complexity. Thus, the following basic assumptions are applied to the transport model without further discussion. (i) All diffusion phenomena through glassy polymers can be described by Fick’s law; the driving force for gas transport is based on the gradients of gas concentrations.4 (ii) The plasticization effect is neglected, although this effect is sometimes reported as significant for high-solubility gases.24 (iii) Coupling terms in the flux expression are neglected. The latter was theoretically studied by Barrer25 and Fredrickson and Helfand.26 (iv) The diffusion coefficients for given gas states depend only on the temperature; thus, they are constant for an isothermal model. The concentration dependence of the diffusion coefficients was studied by Sefcik and Schaefer27 through a 13 C NMR experiment. Furthermore, the following assumptions are discussed in this paper for the transport model: 1 Local Equilibrium: The local equilibrium between two types of molecules is maintained simultaneously everywhere throughout the membrane. 2 Immobilization: The molecules obeying Langmuir’s law are considered as totally immobilized along the direction of diffusion. 3 Partial Immobilization: The molecules obeying Langmuir’s law are considered as partially immobilized along the direction of diffusion. During the development of the dual-mode-sorption theory, the previous assumptions are applied under the form of different combinations to yield different models.

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SORPTION MODELING Mathematically, the sorption isotherm for a pure gas in a glassy polymer is usually given as the sum of eq 2. C = C D + C H = kDp +

C′H bp 1 + bp

(3)

Besides this classical model of sorption (eq 3), some other investigations of the sorption expression within the frame of the dual-mode-sorption theory were also performed. For example, Bhatia and Vieth8 attempted to consider the mobile species as Langmuirian, in the same way as the immobile species. It is also reported that the model applicability is not absolutely general insofar. Equation 3 is inadequate to describe some exceptional isotherms, even qualitatively. For example, Nakanishi et al.9 and Doghieri et al.10 indicated that for ethanol vapors in poly[(trimethylsilyl)-1-propyne] (PTMSP), an S-shaped isotherm cannot be explained by eq 3. Similar observations have also been reported for alcohols in poly[(trimethylsilyl)norbornene].11 On the other hand, the underlying physical picture of the dual-modesorption theory is clearly oversimplified. The three parameters (C′H, b, and kD) of the model can only be obtained from a datafitting procedure based on experimental data for each given gas− polymer couple, so that the model lacks a predictive basis; moreover, the parameter values were also found to depend on the pressure range used for the fitting procedure.12 Consequently, more elaborate models based on a more fundamental background have been developed in order to compensate for this drawback (e.g., nonequilibrium lattice fluids12−17). On the basis of these isotherms, more complicated transport models have been proposed.10,11 Nevertheless, because the dual-mode-sorption theory is indeed very easy to apply and quite useful from an engineering point of view,14,15 the isotherm has found extensive and successful applications over the years18,19 Equation 3 is considered to be a good and simple enough description of reality and will be used in this paper as the only sorption model.

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IMMOBILIZATION MODEL (IM) In the early development of the dual-mode-sorption theory, the assumption of immobilization was taken into account for pure gas diffusion in glassy polymers. The Langmuir and Henry populations are thus related by the local equilibrium in the polymers. Consequently, it is possible to establish a relationship between CD and CH by replacing the partial pressure p of the considered gas in the Langmuir sorption (eq 2) by CD, yielding CH =

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TRANSPORT MODELING As discussed in the previous section, sorption in glassy polymers can be well modeled by eq 3. With regard to the transport driving force, Paul and Koros4 proposed to describe the transport based on the gradients of gas concentrations. In parallel, Petropoulos20 presented a theory similar to that of Paul and Koros4 but based on chemical potential gradients. These two descriptions are almost equivalent.21 Later, Islam and Buschatz22 tried also to model the same phenomena by an expression of chemical potential, taking into account a pressure gradient inside the membrane. This assumption is, however, in contradiction to the solution−diffusion theory, as noticed by Koros and Madden.23 Among these investigations, Paul and Koros'4 description is considered as the basic frame in our study. Assumptions will be taken that simplify the mathematical description while correctly respecting the physical behavior, although all of them are questionable

C ′H b CD kD b 1 + k CD D

(4)

Because the assumption of immobilization is admitted, the Langmuir species are totally immobile. Thus, the model is denoted as IM. The transport phenomenon is described by Fick’s second law in one dimension. ∂(C D + C H) ∂ ⎛ ∂C D ⎞ ⎜+ ⎟ = ∂t ∂x ⎝ ∂x ⎠

(5)

where + is the constant diffusion coefficient and x the position in the membrane. Insertion of eq 4 into eq 5 eliminates CH and gives ⎤ ∂C ⎡ ∂ 2C D K D ⎥ ⎢1 + = + (1 + aC D)2 ⎦ ∂t ∂x 2 ⎣

(6)

where K = C′Hb/kD and a = b/kD are used in order to simplify this equation. This formulation was initially given by Vieth and Sladek.28 1090

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The experimental observations confirm that the dual-modesorption theory is at least qualitatively a good description;3,6 thus, the existence of two gas states is likely. This assumption is also supported by NMR studies that suggest that two distinct populations of NH3 exist in a glassy matrix.29 However, some researchers3,6,19 show that quantitatively eq 6 cannot describe pure gas permeation through glassy polymeric membranes in a satisfactory manner and the permeability at steady state depends on the upstream pressure, which is in contradiction with IM conclusions.

Thus, the time rate of change of the concentrations of gas molecules is ∂C D ∂C D ⎞ ∂ ⎛ ⎜D D ⎟ − [k f C D(C′H − C H) − k rC H] = ∂t ∂x ⎝ ∂x ⎠ ∂C H ∂C H ⎞ ∂ ⎛ ⎜D D ⎟ + [k f C D(C′H − C H) − k rC H] = ∂t ∂x ⎝ ∂x ⎠ (9)

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where kf and kr are the forward and reverse rate constants of the respective reaction. The underlying mechanism that is described postulates a rapid exchange between the sorbed molecules and the dissolved molecules through a one-to-one exchange process, without necessarily reaching equilibrium. Consequently, the two populations interact, and sorbed molecules become part of the dissolved population when a successful diffusion jump occurs (and vice versa). The only hypothesis that is required in that case is the existence of a continuous phase and a dispersed phase (microcavities), so that the local mass balances fit the mathematical set of equations. This set of equations has already been proposed by several investigators for the dual-mode model.30 Similar situations can be found in two-phase systems governed by a dual-diffusion mechanism, such as in a controlled release matrix31 or for pollutant transport in soils. As a result, two limit behaviors with respect to the local equilibrium can be considered: (i) In the case of a rapid exchange where kf and kr tend toward infinity, the equilibrium (4) is established instantaneously.30 In other words, the local equilibrium is admitted. This limit behavior corresponds to the PIM. (ii) In the case of a low exchange where kf and kr tend toward zero, the two populations approach steady state at their own characteristic rates. Equation 9 is reduced to

PARTIAL IMMOBILIZATION MODEL (PIM) In order to improve the dual-mode-sorption theory, some researchers propose to replace the assumption of immobilization by the assumption of partial immobilization.3,6,18 In this improved model, different mobilities are assigned to the two gas species present in glassy polymers. Thus, the model is named PIM. Because the molecules obeying Langmuir’s law are also mobile, the transport phenomenon is described by unidimensional Fick’s second law as ∂(C D + C H) ∂C ∂C H ⎞ ∂ ⎛ ⎜+ D D + + H ⎟ = ∂t ∂x ⎝ ∂x ∂x ⎠

(7)

where + D is the diffusion coefficient of the Henry population and + H the diffusion coefficient of the Langmuir population. The local equilibrium assumption (eq 4) is always considered; eq 7 thus can be rearranged into a more convenient form by eliminating CH ⎤ ∂C ⎡ K D ⎥ ⎢1 + 2 (1 + aC D) ⎦ ∂t ⎣ =

∂ ∂x

⎡ ⎛ ⎞ ∂C ⎤ FK D⎥ ⎢+D⎜1 + ⎟ ⎢⎣ ⎝ (1 + aC D)2 ⎠ ∂x ⎥⎦

(8)

∂C D ∂C ⎞ ∂ ⎛ ⎜+ D D ⎟ = ⎝ ∂t ∂x ∂x ⎠

where F = + H/+ D. The only unknown in eq 8 is CD. It can be noticed that in the case where F = 0, meaning that the diffusion of the Langmuir population is negligible with respect to that of the Henry population, eq 8 is reduced to 6. Consequently, the PIM is reduced to the IM. Numerous authors validated the PIM and concluded that it is an acceptable approach to reality. The two species do have two different mobilities, while none of them is null. This point is also supported by the same NMR study as that performed by Assink.29 Thus, the PIM is considered to be a correct representation of pure gas transport through glassy polymers.

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DUAL DIFFUSION MODEL (DDM) As described previously, the IM can be considered to be on limit behavior of the dual-mode-sorption theory by assuming that the diffusion coefficients for two populations are very different (+ D ≫ + H). Nevertheless, the local equilibrium applies and is considered as instantaneous. The dual-mode-sorption theory can be proposed in a more general form by removing this equilibrium. The relationship between the free gas molecule, the adsorption site, and the adsorbed molecule (complex) is considered to be in that case a reversible chemical reaction.30 For the sake of simplicity, each adsorption site is assumed to immobilize only one gas molecule.

∂C H ∂C H ⎞ ∂ ⎛ ⎜+ H ⎟ = ⎝ ∂t ∂x ∂x ⎠ (10) This second-limit case of the dual-mode-sorption theory is named DDM in this paper in order to make a difference with the PIM. Instead of a nonlinear partial-differential equation, the gas-transport behavior is modeled by a combination of two linear partial-differential equations. To our knowledge, this second-limit case has never been studied before and will be further discussed by comparisons to the PIM in this paper.

DISCUSSIONS The DDM will be compared to the PIM essentially with respect to the time-lag method. The time-lag method is a popular method to determine the transport parameters for a given membrane−gas couple. In a time-lag method experiment, the upstream and downstream faces of a polymer membrane are maintained at constant partial pressures, respectively pu and pd, where pu ≫ pd, and the increase of the pressure of the penetrant gas is followed in the downstream side of the membrane. By the proper design of the experimental apparatus, this increase of the pressure can be maintained at a small value compared to the upstream pressure. Thus, the driving force for diffusion can be considered as nearly constant (pu − pd ≈ pu). The data of the downstream pressure versus time in conjunction with the known downstream volume can be used to determine

site (C′H − C H) + dissolved gas molecule (C D) kf

⇌ complex (C H) kr 1091



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Figure 1. Qualitative plot of the quantity of permeate Qt with respect to time.

Figure 3. Comparison of simulated “free diffusion to pressure equilibrium” of different models. DDM = dual diffusion model; PIM = partial immobilization model.

independently of any assumption with regard to the magnitude of the rates (kf and kr). That is why it is not surprising that the steady-state expression of the flux is the same for both limit behaviors. Thus, it would be the same for the infinite number of assumptions possible for exchange rates between the two populations. The time lag can be predicted by the PIM as4 θ= 2 2 l 2 1 + K[f0 + FKf1 + (FK ) f2 ] + FKf3 + (FK ) f4 3 6+D FK 1 + 1+y

(

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Figure 2. Predicted time lags by two limit behaviors for a general case with K = 5 and b = 0.05 atm−1. DDM = dual diffusion model; PIM = partial immobilization model.

with

the cumulative amount Qt of the penetrant moles having permeated the membrane at time t. A plot of Qt with respect to time t (Figure 1) illustrates the transient period and achievement of the steady-state permeation. By extrapolation of the steadystate portion of the curve, the time lag θ is defined as the intercept of this extrapolation line and the time axis, whereas the slope of the linear portion of the curve gives the permeability 7 at steady state 7=

slope (pu − pd )A /l

f0 =

2 ⎞ 6 ⎛y ⎟ ⎜ y (1 y ) ln(1 y ) + − + + y3 ⎝ 2 ⎠

f1 =

3y ln(1 + y) ⎞ 6⎛y + − ⎜ ⎟ 2(1 + y) 1+y ⎠ y3 ⎝ 2

f=

⎞ 6 ⎛1 1 1 1 ⎟ ⎜ − + − 2(1 + y) 2(1 + y)2 6(1 + y)3 ⎠ y3 ⎝ 6

f3 =

⎞ y 6⎛ 3 y (1 y ) ln(1 y ) − + + + + ⎜ ⎟ 2(1 + y) y3 ⎝ 2 ⎠

f4 =

ln(1 + y) ⎞ 6 ⎛1 1 ⎟ ⎜ − − 1+y ⎠ y3 ⎝ 2 2(1 + y)2

(11)

where l is the membrane thickness. It can be deduced that the permeability at steady state in the conditions of a time-lag measurement is written in the same way as the PIM ⎡ FK ⎤⎥ 7 = kD +D⎢1 + ⎢⎣ 1 + bpu ⎥⎦

)

y = bpu (14)

Equation 13 can be reduced to two special cases: (i) K = 0, which means that the Langmuir adsorption is supposed to be null; the time lag is independent of the upstream pressure. This is the time-lag prediction in a rubbery polymeric membrane.32 (ii) F = 0, which means that the adsorbed molecules are immobile with respect to dissolved molecules (+ D ≫ + H). This is the time-lag prediction of the IM.33

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This similarity can be extended readily to a more general case: for any fixed pressure conditions at upstream or downstream sides, both PIM and DDM predict the same permeability at steady state (see Appendix A). Thus, both limit behaviors are equivalent for steady-state simulation. Furthermore, if one sums the two equations of 9, the “reaction” terms are eliminated (eq 7), 1092



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Table 1. Dual-Mode-Sorption Parameters for Bisphenol A Polycarbonate at 35 °C, Reported by Koros et al.3 kD C′H b +D +H F

unit

CO2

CH4

Ar

N2

He

cm3(STP)·cm−3·atm−1 cm3(STP)·cm−3 atm−1 ×10−9 cm2·s−1 ×10−9 cm2·s−1

0.6852 18.805 0.2618 62.2 4.85 0.078

0.1473 8.382 0.0841 10.9 1.258 0.115

0.1534 3.093 0.063 33.0 5.94 0.180

0.0909 2.109 0.0564 17.6 5.09 0.289

0.0145 0.313 0.0121 5500 7440 1.33

quantitative time-lag prediction. The latter will be discussed by comparison to the experimental data in the following sections.

For the second-limit behavior, the time lag can also be predicted analytically by the DDM, θ=

l 2 1 + K + bpu 6+D 1 + FK + bpu

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COMPARISON WITH EXPERIMENTAL TIME-LAG DATA This comparison is performed with data taken from four different articles. First, the experimental data of Koros et al.3 are used as a reference in the comparison. In order to validate the PIM, Koros et al.3 determined the transport parameters of five gases for a bisphenol A polycarbonate at 35 °C using only sorption and steady-state permeation results. These determined parameters are listed in Table 1. Then, a comparison between his experimental results with the PIM prediction of time lag is performed, and Koros et al. concluded an acceptable agreement, while a tendency for the experimental time lags to lie below the predicted lines is noted. Koros et al. noticed that the measurements for CO2, CH4, and N2 were made on a 3 mil film, while the argon data were measured with a 4.2 mil film and the helium data were obtained from a 63.5 mil membrane because the time lags for a 3 mil film were too short to be accurately measured. In the comparison, all measured time lags for other thicknesses have been scaled to a 3 mil basis. All predictions of the three available models and the reference data for four gases are represented in Figure 4, except for He, whose case will be discussed later. Some points can be highlighted: (i) For all gases, the IM presents the most important deviation with respect to the experimental results, which confirms that it is not an adequate model for glassy polymers. (ii) Even if no model can give a perfect quantitative time-lag prediction, the experimental data go closer to the lower bound provided by the DDM. (iii) Koros et al. noticed that a trend exists for the experimental time lags to lie below the predicted lines of the PIM. Because the DDM predictions represent the lower bound of the predictions, this behavior is less significant for the DDM. In the case of CO2, the majority of the points fall between these two limits of the dual-mode-sorption theory. Because Koros et al.3 noticed that determination of the transport parameters of He is different from that of other gases and might be less accurate, its comparison is performed separately in Figure 5. Because the measured diffusion coefficient for adsorbed molecules + H is greater than the diffusion coefficient for dissolved species + D, a special PIM with F = 1 is also inserted to help with the comparison. The assumption F = 1 means that the two species have the same mobility, which is a “total mobile” model for He. As shown in Figure 5, similar agreements can be obtained. The predictions of the DDM and PIM are nearly superimposed. In another paper, Koros and Paul6 performed the sorption measurements and the time-lag method measurement for

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The way to determine eq 15 is detailed in Appendix B. It should be noticed that all parameters in eq 15 are obtained by the solubility isotherm (K and b) and by steady-state permeability (F and + D) measurements, which is the same as that for the time-lag prediction of the PIM (eq 13). It can be first noticed that eq 15 is much simpler but consistent with eq 13 with regard to the asymptotic behaviors at low and high pressures. lim θ =

pu → 0

l2 1 + K 6+D 1 + FK

lim θ =

pu →+∞

l2 6+D

(16)

In the case of a low pressure, the time lag tends to a maximum value and the impact of the Langmuir population reaches its maximum. In the case of a high pressure, the Langmuir adsorption sites are saturated, but the Henry population can increase without limitation. The time lag is only governed by the Henry population and thus is independent of the upstream pressure. These asymptotic behaviors at low and high pressures will be referred to in the following as the two limit cases. It can be noticed that in Figure 2 both predictions constitute a closed time-lag zone: the PIM represents the time-lag upper bound, while the lower bound is given by the DDM. The steady-state permeability is independent of the exchange rates of eq 9. However, time-lag predictions depend on exchange rates. Assuming that the time-lag experiment is extended to a long duration (much longer than the time lag), the upstream exhaustion and downstream accumulation become significant; thus, a pressure equilibrium between the upstream and downstream will be reached. This experiment is here denoted as “free diffusion to pressure equilibrium”. Because these two limit behaviors can provide different time-lag predictions, some difference is expected in the “free diffusion to pressure equilibrium” experiment. On the basis of the transport of CO2 through a glassy polyimide,34 such an experiment is simulated numerically by both DDM and PIM. Nevertheless, Figure 3 shows that no relative significant difference is achieved (dashed lines are not visible on the figure because they are perfectly superimposed by solid lines). Using this observation, it can be remarked that, even if the time-lag predictions differ, both models are equivalent for a long time scale (compared to the time lag). Consequently, it can be concluded that both limit-behavior models predict the same steady-state permeability, while the only difference between exchange rates appears apparently in the 1093



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Figure 4. Comparison of experimental3 and predicted time lags by different models for various gases in polycarbonate at 35 °C: Ar (top left), N2 (top right), CH4 (bottom left), and CO2 (bottom right). DDM = dual diffusion model; PIM = partial immobilization model; IM = immobilization model.

their measurements are given in Table 2 at temperature conditions where the polymer is glassy. On the basis of these data, the time lags are predicted by three models and compared to the experimental data in Figures 6 and 7. In this case, it is more clearly observed that most of data are concentrated between two limits. Koros et al.19 performed another CO2 time-lag measurement with determinations of the gas-transport parameters (Table 3). The comparison between the predictions and experimental data is given in Figure 8. Similarly to Figures 6 and 7, the majority of the data are bordered by two limits of the dual-mode-sorption theory and the lower bound provided by the DDM presents the best agreement to the experimental data compared to all other models. More recently, Garrido et al.35 also performed some time lag tests for CO2 in a poly[bisphenol A carbonate-co-4,4′-(3,3,5trimethylcyclohexylidene)diphenol carbonate] film at 303 K. First, they determined the gas-transport parameters (Table 4) according to the method proposed by Koros et al.19 The predictions of three models and the experimental data are represented together in Figure 9. Opposite to the previously discussed cases, the upper bound represents the best agreement with the experiments, although the experimental data are not included in this case in the time-lag prediction zone. However, Chiou and Paul24 reported some rare exceptions: the isothermal permeabilities at steady state for Ar and CH4 in poly(ethyl methacrylate) (PEMA) below its Tg are independent

Figure 5. Comparison of experimental3 and predicted time lags by different models for He in polycarbonate at 35 °C. DDM = dual diffusion model; PIM = partial immobilization model; IM = immobilization model; F = 1 = no IM.

different systems. They measured the sorption and transport coefficients and the time lags for CO2 in a semicrystalline poly(ethylene terephthalate) (PET) for temperatures ranging from 25 to 115 °C over the pressure range from 0.1 to 30 atm. A part of 1094



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Table 2. Dual-Mode-Sorption Parameters for CO2 in PET at Different Temperatures, Reported by Koros and Paul6 kD C′H b +D +H F

unit

25 °C

35 °C

45 °C

55 °C

65 °C

75 °C


0.362 7.913 0.351 2.002 0.0956 0.048

0.330 5.760 0.322 3.146 0.221 0.0712

0.260 4.960 0.282 5.260 0.410 0.078

0.234 3.753 0.252 8.330 0.490 0.059

0.214 2.735 0.197 11.65 1.10 0.095

0.210 1.814 0.165 16.79 1.326 0.079

Figure 6. Comparison of experimental6 and predicted time lags by different models for CO2 in PET at different temperatures below Tg: 25 °C (top left), 35 °C (top right), 45 °C (bottom left), and 55 °C (bottom right). DDM = dual diffusion model; PIM = partial immobilization model; IM = immobilization model.

of the pressure, while the time lags decrease with pressure. Because their time-lag predictions according to the IM are consistent with the experimental data, Chiou and Paul24 concluded that the transport of these gases in this polymer follows the IM. The transport parameters (Table 5) are obtained by sorption and permeation experiments through a film of 3.95 mil. Using these measured coefficients, Chiou and Paul24 correctly estimate the time lag with the dual-mode-sorption theory. It can be noticed that the DDM here is also adequate to describe this case by making the ratio of the diffusion coefficients F tend toward zero. In a similar way, the two prediction bounds and experimental data are represented together in Figure 10. Again, the lower prediction bounds show a better agreement with respect to the experimental data. It can be concluded that, in most cases, the two limits of the dual-mode-sorption theory represent correctly the experimental

time-lag zones, while the lower bound, which is given by the DDM, provides somehow a better agreement with respect to the experimental data. It is interesting to note that, in all four cited time-lag behaviors, the lower bound given by the DDM provides good time-lag predictions in particular for CO2 through different membranes. Comparisons are also performed in Figures 11−13 for CO2 in PET membranes, four gases in polycarbonate membranes, and CH4 and Ar in PEMA membranes, respectively. In these figures, the experimental time lag and theoretical prediction are compared on the same plot for all conditions for a given material. It can be noticed that the lower bound given by the DDM is shown as the best approach in time-lag predictions. However, it can be noticed that in Figure 11 the points above about 100 min, which present a deviation with respect to the bisector, are those obtained at 25 °C in Figure 6. 1095



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Figure 7. Comparison of experimental6 and predicted time lags by different models for CO2 in PET at different temperatures below Tg: 65 °C (left) and 75 °C (right). DDM = dual diffusion model; PIM = partial immobilization model; IM = immobilization model.

Table 3. Dual-Mode-Sorption Parameters for CO2 in Polycarbonate at 35 °C, Reported by Koros et al.19 kD C′H b +D +H F


0.866 16.7 0.309 46.7 4.717 0.101

Figure 9. Comparison of experimental35 and predicted time lags by different models for CO2 in a poly[bisphenol A carbonate-co-4,4′-(3,3,5trimethylcyclohexylidene)diphenol carbonate] film at 303 K. DDM = dual diffusion model; PIM = partial immobilization model; IM = immobilization model.

Table 5. Dual-Mode-Sorption Parameters for Ar and CH4 in PEMA at 35 °C, Reported by Chiou and Paul24 Figure 8. Comparison of experimental19 and predicted time-lags by different models for CO2 in polycarbonate at 35 °C. (DDM = dual diffusion model; PIM = partial immobilization model; IM = immobilization model).

kD C′H b +D F

Table 4. Dual-Mode-Sorption Parameters for CO2 in Poly[bisphenol A carbonate-co-4,4′-(3,3,5trimethylcyclohexylidene)diphenol carbonate] at 303 K, Reported by Garrido et al.35 kD C′H b +D +H F

cm3(STP)·cm−3·−1 cm3(STP)·cm−3 atm−1 ×10−9 cm2·s−1 ×10−9 cm2·s−1

unit

Ar

CH4

cm3(STP)·cm−3·atm−1 cm3(STP)·cm−3 atm−1 ×10−8 cm2·s−1

0.0902 1.20 0.0405 7.78 0

0.165 2.24 0.0536 1.60 0

In order to quantify the predicting ability in three conditions, the relative error e for each material is defined as e=

1.0868 20.6 0.2584 79.0 14.2 0.18

|θexp − θi| θexp

i = DDM, PIM, and IM (17)

According to Table 6, for all three membrane materials, the prediction lower bounds provide the best agreement to the experiment data. The prediction relative error is, in general, lower than 11%. 1096



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Figure 10. Comparison of experimental24 and predicted time lags by different models for Ar (left) and CH4 (right) in a PEMA film at 35 °C. DDM = dual diffusion model; IM = immobilization model.

Figure 11. Comparison of experimental6 and predicted time-lags by different models for CO2 from 25 to 65 °C in PET film. (DDM = dual diffusion model; PIM = partial immobilization model; IM = immobilization model).

Figure 12. Comparison of experimental3,19 and predicted time-lags by different models for CO2, CH4, N2 and Ar in PET film (DDM = dual diffusion model; PIM = partial immobilization model; IM = immobilization model).

It is very important to notice that both PIM and DDM are limit behaviors of the dual-mode-sorption theory. In reality, neither of them can be reached; the effective exchange rates kf and kr in eq 9 could not be assumed to be infinity or null. That explains the fact that most experimental time-lag data are found inside the timelag interval. Assink29 confirmed by an NMR study that the local equilibrium can be considered as instantaneous in the case of ammonia sorbed by polystyrene. Nevertheless, it should be noticed that, as a highly polar molecule, ammonia might represent an exceptional sorption ability with respect to less polar gases such as CO2, CH4, and O2 discussed in this paper. Consequently, the conclusion of Assink might not be considered as a general conclusion. However, it is important to notice that determination of the time lags is subject to relatively large errors. The history dependence of diffusion in glassy polymers can create problems. How the films are conditioned (pressure gradient vs uniform highpressure exposure), how long each measurement is taken (at steady state for a prolonged period can affect the next result), and

how the film is depressurized prior to the next transient measurement all led to the scatter shown in figures of comparison. Furthermore, the thickness changes with the pressure, and the time-lag dependence on the membrane thickness l2 can introduce error. Lastly, diffusion coefficients + D and + H are fitting parameters and the scatter in steady-state permeabilities with the pressure is sufficient to reveal that the results can be difficult to reproduce in some systems. Furthermore, it should be noticed that these errors exist even if they are difficult to quantify. For example, in the original publications, the polymer characteristics and history are most often not sufficiently detailed. As a result, the history dependence of such polymers cannot be quantified.

■

TIME-LAG PREDICTION IN MEMBRANE PROCESSES Membrane processes for gas separations are conventionally operated by means of dense polymeric materials at steady state. Both the feed and permeate pressures are maintained at constant 1097



Article

toxic agent because the permeation rate is much lower before reaching steady state. The second point is more appropriate when any measurable steady-state rate would produce a lethal effect. As a result, the time-lag prediction is especially important for this purpose. On the other hand, it is better to underestimate the time lag than to overestimate it for reason of security in these kinds of applications. Consequently, the time-lag lower-bound provider, the DDM, should be a better choice to estimate the boundary between the transient and steady states.

Table 6. Average Relative Error e (%) for PET, Polycarbonate, and PEMA Membranes eDDM

ePIM

eIM

PET polycarbonate PEMA

8.43 10.42 3.79

11.46 17.08

33.46 50.03 5.77

CONCLUSION

■

APPENDICES

Since the 1950s, the dual-mode sorption has been developed and then extended. The first mathematical description of this transport phenomena, the IM, cannot accurately describe the transport behavior in most glassy polymers, except one case mentioned by Chiou and Paul.24 Through a better understanding of the transport mechanism in glassy polymers and performance of experimental verifications, the Langmuir population is no longer considered to be totally immobilized. Using numerous measurements according to the time-lag method, the PIM is closer to reality; thus, it is nowadays considered as a good approach to this phenomenon. By considering the adsorption process as a reversible chemical reaction in the framework of the dual-mode-sorption theory, two limit cases will be distinguished according to the influence of the forward and reverse reaction rates. In that case, the PIM becomes one limit case of the dual-mode-sorption theory because both forward and reverse reaction rates are considered as rapid. Another limit condition of the dual-mode-sorption theory named the DDM, where both reaction rates are considered as slow, is essentially investigated in this paper and compared to the PIM. First, in a steady-state simulation, these two models as well as all intermediate conditions are equivalent. Second, with respect to the time-lag values, in general, experimental data lie between two bounds provided by the PIM and DDM. Furthermore, the lower bound provided by the DDM seems to be closer to the experimental data, especially in the case of CO2 in glassy polymers. Consequently, the time-lag lower bound (DDM) completes the dual-mode-sorption theory and can be considered to be an efficient and easy-to-use approach of the theory with respect to conventional models, especially in time-lag prediction applications.

Figure 13. Comparison of experimental24 and predicted time-lags by different models for Ar and CH4 in PEMA film at 35 °C (DDM = dual diffusion model; IM = immobilization model).

membrane material

■

levels; thus, the permeation rate and permeate concentration are dominated only by the permeabilities, except at the initial startup stage. According to Appendix A, both limit models provide the same permeability prediction at steady state. Consequently, both limit models are identical in steady-state membrane process simulation. Nevertheless, membrane gas separations under transient conditions are rather unexplored. These processes, denoted initially as cyclic membrane processes by Paul,36 make use of the differences in the permeation rates before reaching their steady-state permeations. According to some simulations and experimental investigations, the cyclic membrane processes are able to provide some exclusive advantages37 with respect to conventional operations, such as important selectivity improvement. Wang et al.37 indicate that the key issue of cyclic processes is the time lag, which distinguishes the steady-state and transient-state permeations. As a result, an efficient time-lag prediction method should be used in order to accurately simulate a cyclic process simulation. In this respect, the lower-bound prediction providing, in general, a better agreement with experiments will be a better choice than the upper-bound prediction. Besides the cyclic membrane processes, the description of the transient mass transfer of gaseous species in glassy polymers is also of major interest for other industrial applications, such as packaging, adhesives, controlled release, etc. For example, Higuchi and Higuchi38 were interested in developing protective ointments or creams (polymers), which could be applied to the skin for the purpose of providing the treated area with protection from toxic agents that could be absorbed through the skin. In this respect, two specifications of polymers are required: (1) a low steady-state rate of transport of the agent through the polymers; (2) a long time lag, in other words, a long transient state for the

A. Permeability at Steady State

In order to determine the permeability at steady state for any set of fixed upstream/downstream pressures, the corresponding differential equations are solved with specified initial and boundary conditions. Permeability of the DDM. At steady state, the timedependent term becomes null in eq 10. Thus, the gas behavior in the membrane is described as ∂ ∂x

⎛ ∂Ci ⎞ ⎜+i ⎟=0 ⎝ ∂x ⎠

i = D, H

(18)

The upstream and downstream partial pressures, respectively pu and pd, are constant but not necessarily null. Thus, the boundary conditions are 1098


Industrial & Engineering Chemistry Research C D(0) = kDpu C H(0) =

C H(l) =

where ∂CH/∂CD is given by the local equilibrium (4). Thus

∀t

C′H bpu 1 + bpu

C D(l) = kDpd

Article

∀t

From eq 24, it results that Term 1 of eq 28 is constant and is denoted as β.

∀t

C′H bpd

⎞ ∂C ⎛ FK D ⎟ β = +D⎜1 + (1 + aC D)2 ⎠ ∂x ⎝

∀t

1 + bpd

(19)

At steady state, the gas flux through the membrane JDDM is constant and written as ⎡ ∂C ∂C H ⎤ JDDM = −⎢+D D + + H ⎥ ⎣ ∂x ∂x ⎦

⎤ ⎡ β FK ⎥ dC D = dx ⇒ ⎢1 + 2 +D (1 + aC D) ⎦ ⎣

(20)

⇒

From eq 18, it results that the gradients of the concentration for both species are constant in the membrane. Thus, the derivatives of the concentration can be calculated by the boundary conditions in eq 19. The gas flux becomes JDDM =

D

= +DkD

l

+ +H

β = −+DkD

l

β dx +D

⎤ pu − pd ⎡ FK ⎢1 + ⎥ l ⎢⎣ (1 + bpu )(1 + bpd ) ⎥⎦

JPIM = −β = +DkD

⎤ pu − pd ⎡ FK ⎥ ⎢1 + l ⎢⎣ (1 + bpu )(1 + bpd ) ⎥⎦

(21)

(31)

Consequently, the steady-state permeability 7 PIM can be deduced as (22)

7PIM =

Consequently, the permeability 7 DDM at steady state is given as 7DDM

⎤ ⎡ FK ⎥ = = +DkD⎢1 + ⎢⎣ (1 + bpu )(1 + bpd ) ⎥⎦ Δp /l JDDM

⎡ ⎛ ⎞ ∂C ⎤ FK D⎥ ⎢+D⎜1 + ⎟ =0 2 ⎢⎣ ⎝ (1 + aC D) ⎠ ∂x ⎥⎦

(23)

∀t

C D(l) = kDpd

∀t

Barrer32 proposed an analytical solution of the time lag. This procedure is adapted here in order to determine the time lag of the DDM. Barrer studied a rubbery polymer membrane in which the gas concentration is denoted as C. Therefore, the gas behavior can be described by Fick’s law ∂C ∂ 2C =+ 2 ∂t ∂x

JPIM

JPIM

C(x , 0) = 0

∀x

C(l , t ) = C0

∀t

C(0, t ) = 0

(26)

∀t

(34)

where C0 is constant. Thus, eq 33 is solved analytically as a Fourier series by separation of the time and space variables as

Because the local equilibrium is assumed in the membrane and + H = F +D, eq 26 can be rearranged as ⎡ ∂C ∂C ∂C D ⎤ = − ⎢+ D D + F + D H ⎥ ∂x ∂C D ∂x ⎦ ⎣

(33)

submitted to

(25)

By definition, the gas flow rate through the membrane JPIM is constant and equal to ⎡ ∂C ∂C H ⎤ = − ⎢+ D D + + H ⎥ ⎣ ∂x ∂x ⎦

(32)

B. Time Lag of the DDM (24)

where the Langmuir species CH is eliminated by the local equilibrium hypothesis. Assuming that the upstream and downstream pressures are constant but not null, the boundary conditions are C D(0) = kDpu

⎤ ⎡ FK ⎥ = +DkD⎢1 + ⎢⎣ (1 + bpu )(1 + bpd ) ⎥⎦ Δp /l JPIM

Because eqs 23 and 32 are equivalent, it can be concluded that the PIM and DDM predict the same permeability at steady state for any set of upstream and downstream pressures. It can be noticed that pd = 0 in the conditions of the time-lag method measurement, so that eqs 23 and 32 are reduced to eq 12. As a result, the measured transport parameters from the time-lag method measurement can be used in both PIM and DDM.

Permeability of the PIM. For the PIM, assuming that the timedependent terms are null in eq 8, the gas behavior in the membrane at steady state is described as ∂ ∂x

(30)

⎤ pu − pd ⎡ FK ⎢1 + ⎥ l ⎢⎣ (1 + bpu )(1 + bpd ) ⎥⎦

Using F = DH/DD and K = C′Hb/kD, JDDM = +DkD

(29)

Thus

C′H b(pu − pd ) (1 + bpu )(1 + bpd )l

∫0

The constant β can be obtained by solving eq 29 and using eq 25

C′H bpd ⎤ +D + ⎡ C′H bpu ⎥ (kDpu − kDpd ) + H ⎢ − 1 + bpd ⎥⎦ l l ⎢⎣ 1 + bpu pu − pd

⎤ FK ⎥ dC D = ⎢1 + 2 (1 + aC D) ⎦ ⎣

C D(l) ⎡

∫C (0)

C(x , t ) = (27)

C0 2 x+ π l

∞

∑ 1

C0 cos nπ nπx −+n2π 2t / l 2 sin e n l (35)

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■

Assuming that the gas flows through the membrane into a volume V, the gas concentration Cg in this volume corresponding to the accumulation line of permeate in 1 is given by ⎛ ∂C ⎞ dC g = +⎜ ⎟ A V ⎝ ∂x ⎠x = 0 dt

*E-mail: [email protected]. Phone: +33 (0)3 83 17 53 90. Fax: +33 (0)3 83 32 29 75. Notes

(36)

∞

+C0t 2l + 2 lV /A π V /A

The authors declare no competing financial interest.

■

× (1 − e

ACKNOWLEDGMENTS The authors are grateful for financial support provided by the Carnot Institute for Energy and Environment in Lorraine (ICEEL). The authors also express sincere acknowledgement for useful discussions with Prof. William J. Koros and substantial comments provided by the reviewers. Their proposals were greatly appreciated.

⎛ C0 cos nπ ⎞ ⎟ ⎠ n2

∑ ⎜⎝ 1

−+n2π 2t / l 2

)

(37)

When t → ∞, eq 37 tends toward the straight line corresponding to the steady-state portion of eq 1. Then the time-lag value θ = l2/6+ is obtained at the intercept of this straight line and the time axis. The DDM can be considered as a linear combination of two independent gas diffusion behaviors in a rubbery polymer. Thus, the total gas concentration Cgt in a volume V is given by C tg = C Dg + C Hg

■

b C C′H + e F J kf kr kD

(38)

CgD and

CgH

Barrer’s solution is used here to describe of the DDM corresponding to the concentration of the Henry population and the concentration of the Langmuir population, respectively. Thus C tg(t ) =

(+DC D0 + + HC H0)t 2l + 2 lV /A π V /A ∞ ⎛ C cos nπ ⎞ −+Dn2π 2t / l 2 ⎟(1 − e [∑ ⎜ D0 2 ) ⎝ ⎠ n 1

l p 7 t x

∞

⎛ C cos nπ ⎞ −+ Hn2π 2t / l 2 ⎟(1 − e +∑ ⎜ H0 2 )] ⎝ ⎠ n 1

D d g H m u

(+DC D0 + + HC H0) 2l t+ 2 lV /A π V /A ∞ ⎡ ∞ ⎛ C cos nπ ⎞ ⎛ C H0 cos nπ ⎞⎤ ⎟ + ∑⎜ ⎟⎥ × ⎢∑ ⎜ D0 2 ⎠ ⎝ ⎠⎥⎦ n n2 ⎣⎢ 1 ⎝ 1

θ time lag (s)

∞ 2l(C D0 + C H0) ⎡ ⎛ cos nπ ⎞⎤ +D(C D0 + FC H0) ⎢ ⎜ ⎟⎥ t+ ∑ ⎢⎣ 1 ⎝ n2 ⎠⎥⎦ lV /A π 2V /A

=

l(C D0 + C H0) +D(C D0 + FC H0) t− lV /A 6V /A

■

because nπ/n ) = −π /12. Thus, the time lag θ defined in eq 1 is given by the intercept of the line and the time axis θ=

l 2 C D0 + C H0 6+D C D0 + FC H0

2

(41)

Using CD0 = kDpu and CH0 = C′Hbpu/(1 + bpu), eq 41 finally yields θ=

l 2 1 + K + bpu 6+D 1 + FK + bpu

REFERENCES

(1) Bitter, J. G. A. Transport mechanisms in membrane separation processes; Plenum Press: New York, 1991. (2) Koros, W. J.; Chern, R. T.; Stannett, V. T.; Hopfenberg, H. B. A model for permeation of mixed gases and vapors in glassy polymers. J. Polym. Sci. 1981, 19, 1513−1530. (3) Koros, W. J.; Chan, A. H.; Paul, D. R. Sorption and Transport of Various Gases in Polycarbonate. J. Membr. Sci. 1977, 2, 165−190. (4) Paul, D. R.; Koros, W. J. Effect of Partially Immobilizating Sorption on Permeability and the Diffusion Time Lag. J. Polym. Sci., Part B: Polym. Phys. 1976, 14, 675−685. (5) Meares, P. The Diffusion of Gases through Polyvinyl Acetate. J. Am. Chem. Soc. 1954, 76, 3415−3422. (6) Koros, W. J.; Paul, D. R. Transient and Steady-State Permeation in Poly(ethylene terephthalate) Above and Below the Glass Transition. J. Polym. Sci., Part B: Polym. Phys. 1978, 16, 2171−2187. (7) Barrer, R. M.; Barrie, J. A.; Slater, J. Sorption and diffusion in ethyl cellulose. Part I. History-dependence of sorption isotherms and permeation rates. J. Polym. Sci., Part B: Polym. Phys. 1957, 23, 315−329.

(40) 2

ordinary dissolution downstream transition point for an amorphous polymer hole-filling process effective value upstream

Greek Symbols

=

∑∞ 1 (cos

NOMENCLATURE hole affinity constant (atm−1) gas concentration [cm3(STP)·cm−3 polymer] hole-filling constant [cm3(STP)·cm−3 polymer] diffusion coefficient (cm2·s−1) relative error ratio of diffusion coefficients of two species gas flux through the membrane [cm3(STP)·s−1·m−2] forward rate constant [cm−3(STP)·cm3 polymer·s−1] reverse rate constant (s−1) Henry sorption coefficient [cm3(STP)·cm−3 polymer·atm−1] membrane thickness (m) partial pressure (atm) permeability at steady state [cm3(STP)·cm−1·atm−1·s−1] time (s) position in the membrane for a unidimensional model (m)

Subscripts (39)

where + H = F+ D. When t → ∞, eq 39 tends toward the straight line C tg(t ) =

AUTHOR INFORMATION

Corresponding Author

(∂C/∂x)x=0 can be found using eq 35. Then, Barrer gave an analytical expression of Cg(t) by integrating eq 36 between 0 and t, yielding the following expression of Cg(t): C g (t ) =

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Transport of Gases in Glassy Polymers under Transient Conditions

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