Gas permeability in glassy polymers: A thermodynamic approach

Accepted Manuscript Gas permeability in glassy polymers: a thermodynamic approach Matteo Minelli, Giulio C. Sarti PII:

S0378-3812(15)30132-1

DOI:

10.1016/j.fluid.2015.09.027

Reference:

FLUID 10769

To appear in:

Fluid Phase Equilibria

Received Date: 13 July 2015 Revised Date:

7 September 2015

Accepted Date: 10 September 2015

Please cite this article as: M. Minelli, G.C. Sarti, Gas permeability in glassy polymers: a thermodynamic approach, Fluid Phase Equilibria (2015), doi: 10.1016/j.fluid.2015.09.027. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Gas permeability in glassy polymers: a thermodynamic approach Matteo Minelli,* Giulio C. Sarti

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Department of Civil, Chemical, Environmental and Materials Engineering (DICAM) Alma Mater Studiorum – University of Bologna

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Via Terracini 28 – I-40131 Bologna (Italy)

Keywords: Gas permeability Glassy polymers

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NELF model

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Diffusion

Corresponding author: Matteo Minelli

tel. +39 (0) 51 2090426 fax +39 (0) 51 6347788 email address: [email protected] 1

ACCEPTED MANUSCRIPT Abstract The permeability of various low molecular weight species (both gases and vapors) in a series of glassy polymers has been extensively analyzed by means of a thermodynamic based approach for

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solubility and diffusivity, recently proposed and already applied to a few penetrant/polymer systems. The model relies on the thermodynamic description of the solubility behaviors of the solutes provided by the nonequilibrium thermodynamic model for glassy polymers (NET-GP), while

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the diffusivity is the product of the mobility coefficient, a purely kinetic quantity, and the thermodynamic factor, accounting for the dependence of the penetrant chemical potential on its

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concentration in the glassy polymer matrix.

The model is applied to permeability data of many penetrant species from very light gases, such as hydrogen or helium, to hydrocarbons and fluorocarbons, in several different glasses, including very high free volume materials, polyimides and fluoropolymers. The model proved to be effective

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in the representation of all types of permeability behaviors with respect to penetrant upstream pressure, which may be either decreasing, increasing, or with a nonmonotonous trend showing a

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minimum value at the so-called plasticization pressure.

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ACCEPTED MANUSCRIPT 1. Introduction The transport of various low molecular weight species in polymeric systems has a great relevance for many different applications [1-6], such as, among the others, membrane processes for the separation of the

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different components of gaseous streams [7-9]. To this aim, several experimental works have been devoted to the characterization of conventional as well as novel materials seeking for the best performances in terms of gas permeability and selectivity versus the compounds of interest [10,11]. Glassy polymers are

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often envisioned as potential candidates for the development of membrane materials, able to meet the requirements for industrial processes. Therefore, the understanding of the mechanism of dissolution and

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transport of the different penetrants in the glassy polymeric systems is of crucial relevance for the proper design of the membrane modules and for the description, or even prediction, of their properties in different conditions of application (e.g. different temperature, pressure or composition). An appropriate modeling tool based on actual physical properties rather than on empirical parameters is also able to

conventional systems.

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provide interesting suggestions for the development of novel materials as well for the improvement of the

as follows:

N& 1 l p − p1d u 1

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P1 =

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The permeability P1 measuring the rate of transport of the permeating species 1 at steady state is defined

(1)

in which N& 1 is the molar flux of species 1 in a polymer film of thickness l, being the difference in partial pressures ( p1u − p1d ) between the upstream and downstream phases the driving force of the process. Very few modeling approaches are able to describe successfully the peculiarity of the transport of gases and vapors in glassy polymers, as qualitatively very different behaviors are experimentally observed. Indeed, it has been reported that gas permeability can show either constant, decreasing or increasing trends with penetrant upstream pressure, or even nonmonotonous behavior undergoing through a 3

ACCEPTED MANUSCRIPT minimum value at the so-called plasticization pressure [12]. The dual mode sorption model (DMS) [13], empirical in character and with no predictive capability, is the most popular framework considered to represent the transport of gases and vapors in polymer glasses, mainly due to its remarkable simplicity, even though it encounters severe limitations, as it is intrinsically unable to describe permeabilities that

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either increase or are nonmonotonous with pressure. Recently Minelli and Sarti proposed a simple but effective model able to describe with a remarkable accuracy all the different behaviors observed experimentally [14], including the nonmonotonous trends.

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The model is based on fundamental expressions for the transport of penetrant species, considering the gradient in chemical potential across the polymer film as the driving force of the diffusive flux, and it relies

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on the well established nonequilibrium thermodynamic approach for glassy polymers (NET-GP) [15,16] for the evaluation of the thermodynamic properties of the penetrant/polymer glassy phases. This leads to represent the diffusion coefficient as the product of a purely kinetic factor, the mobility coefficient L, and a thermodynamic factor α, accounting for the dependence of the penetrant chemical potential on its

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concentration in the membrane. Interestingly, this model has been successfully applied to describe the CO2 permeability in a series of different conventional glasses [14], as well as on glassy blends, copolymers [17] and semicrystalline polymers [18]; more recently, the transport properties of light gases or vapors are also

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analyzed [19,20].

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In this work, the thermodynamic transport model is applied to describe the permeability of CO2, light gases, hydrocarbons and fluorocarbons in a broad series of unconventional or novel glassy polymers with relevant potential for application as gas separation membrane materials, such as those endowed with a very high free volume, selected polyimides or fluorinated amorphous Teflons. Indeed, all such polymers are of great relevance in membrane science, both from the academic and industrial point of view, and significant research studies have been devoted to their characterization as well and to the modeling analysis of their performance. However, the description of the permeation behavior in such polymers still rely on mere empirical correlations and sometimes seem to require the pretended porous structure of the matrix.

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ACCEPTED MANUSCRIPT The main aim of this work is, on one hand, to prove the applicability and the effectiveness of this approach to any kind of penetrant/polymer pairs, and, on the other hand, to retrieve a series of model parameters for the various systems investigated. The correlations for the model parameters with the physical

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properties of both penetrant and polymer are of great interest for predictive purposes.

2. Theoretical background

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The model for the description of gas permeability in glassy polymers has been already presented in detail in a previous publication [14]; however, the main features are here recalled for clarity sake. The model takes

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advantage of the fundamental expression derived for the diffusion of low molecular weight species 1 in binary (or pseudo-binary) mixtures, as the product of a kinetic coefficient, the mobility L1, and the thermodynamic factor α [21]:

∂µ1 / RT ⋅ L1 ≡ α ⋅ L1 ∂ ln ω1

(2)

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D1 =

The factor α can be readily calculated from the analysis of the experimental solubility isotherm of component (1) in the polymer phase (2) of interest, or can be provided by the NET-GP model with no need

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of additional parameters. Conversely, the mobility coefficient is a purely kinetic quantity, which depends on the nature of the polymer and of the penetrant, as it depends on the polymer free volume and on the size

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of the probe molecules, as it has been already mentioned in previous works [19,20]. Several expressions are suitable to account for such dependences, and the increase of the penetrant mobility with increasing concentration in the membrane. Among the others, it is noteworthy the free volume theory developed by Vrentas and Duda [22,23], with the subsequent modifications to account for the out-of-the-equilibrium state of the glassy polymer [24]. However, for the sake of simplicity, an exponential behavior with respect to penetrant concentration has been sufficient to that aim, as typically encountered experimentally. One thus has:

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ACCEPTED MANUSCRIPT L1 (ω1 ) = L10 ⋅ e βω1

(3)

in which the infinite dilution mobility coefficient L10 and the plasticization factor β are the only two adjustable parameters of this model; their value can be retrieved by the analysis of either diffusion data

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from transient sorption or from steady state permeation data. The permeability of species 1 at steady state can be obtained conveniently by integrating the expression for penetrant diffusivity as the product of mobility and thermodynamic factor (Eq. (2)). In case the upstream

1 M 1 p1u

∫

p1u

0

ρ 2 L10 exp ( βω1 )

ω1 p1

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expression is derived [14]:

P1 =

0 , the following

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side of the membrane is at partial pressure p1u , while at the downstream side is at p1d

z1 dp1

(4)

in which M1 is the penetrant molecular weight and z1 is the compressibility factor of the gas phase

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(evaluated e.g. by the Peng-Robinson equation of state), while ρ2 is the polymer density and

ω1 p1

represents

the solubility coefficient of species 1 in the polymer phase 2. The model requires the thermodynamic description of the penetrant/polymer system in the conditions of interest, which is provided by the

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nonequlibrium lattice fluid model approach (NELF) [15]. The evaluation of the solubility coefficient

ω1 p1

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allows then to retrieve the model parameters L10 and β from the analysis of the permeability data at various upstream pressures.

2.1 NELF model description for gas/vapor solubility The description of the thermodynamic behavior of the solute/polymer mixtures below Tg at the various penetrant pressure is provided by the nonequilibrium lattice fluid model (NELF) [15], which makes use of the lattice fluid theory after Sanchez and Lacombe [25,26] in the framework of the NET-GP approach. Several publications reported the very effective ability of NELF to evaluate the solubility isotherms of gases, 6

ACCEPTED MANUSCRIPT vapors or liquids, as well as of their mixtures, in glassy polymers [16,27-30]. Recently, the latest development of this model proved its ability also in the prediction polymer dilation during sorption, allowing the determination a priori of penetrant solubility [31,32] even in the presence of a significant swelling.

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The basic feature of the NET-GP approach is to account for an additional state variable, the actual polymer density ρ 2NE , in order to describe the nonequilibrium behavior of a system in the glassy state: ρ 2NE practically measures the departure from the equilibrium conditions. In this framework, the thermodynamic

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state of the nonequilibrium penetrant/polymer mixture is described by the usual set of state variables, namely T, p and ω1, together with the actual polymer density ρ 2NE . Finally, the determination of the gas or

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vapor solubility is provided from phase equilibrium calculation, i.e. when the penetrant chemical potential in the polymer phase is equal to that in the external fluid phase.

This approach relies on an equation of state model for the evaluation of both equilibrium and nonequilibrium properties, following the appropriate EoS scheme (lattice fluid model in this case). The main

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features of the Sanchez Lacombe model are systematically reported in Table S1 of the Supporting Information section, with the definition of the most relevant properties and the mixing rules adopted. It is relevant to note that each substance, either penetrant or polymer, is described by three pure

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component parameters (characteristic temperature T*, pressure p* and close packed density ρ*), which can be evaluated from vapor liquid equilibrium (VLE) data for low molecular weight species, and from pressure

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volume temperature (pVT) data in the rubbery state (above Tg) for polymers. However, in the absence of volumetric data for the polymer phase, a case often encountered for recently developed membrane materials or for high Tg glasses, an alternative procedure has been established to overcome this limitation [33]. Indeed, the EoS characteristic parameters of the polymer can be obtained from the solubility of various penetrants in the glassy membrane in the limit of infinite dilution, using a single and explicit relationship (see Supporting Information). This procedure, already widely employed in previous works, has been applied in this case to evaluate the characteristic parameters of PTMSP [34], PTMSN [35], PIM-1 [36] or Matrimid [37], leading to very satisfactory results in the description of solubility isotherms of various 7

ACCEPTED MANUSCRIPT solutes. This procedure has been employed in this work to retrieve the EoS parameters of some of the polymeric materials (those not accounted for in previous publications), as indicated in Table 1. Interestingly, an alternative method has been also proposed following a molecular simulation approach, as a Molecular Dynamics can be applied to simulate the pVT behavior of any kind of polymer systems in any

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temperature range, leading thus to the characteristic parameters of any kind of equation of state [38].

Table 1. Lattice Fluid EoS characteristic parameters of polymer and penetrant species analyzed.

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T* [K] p* [MPa] ρ* [g/cm3] ref. AF1600 610 245 2.085 [32] AF2400 624 250 2.130 [39] PTMSP 416 405 1.250 [40] 6FDA-6FpDA 785 720 1.683 this work* 6FDA-DAM 765 480 1.660 this work Hyflon AD 80 550 180 2.150 this work PMP 550 360 1.040 this work EC 620 515 1.231 [41] He 9.3 4.0 0.148 [33] H2 4.6 37 0.078 [42] O2 170 280 1.290 [43] N2 145 160 0.943 [44] CO2 300 630 1.515 [15] CH4 215 250 0.500 [42] C2H4 295 345 0.68 [44] C2H6 320 330 0.640 [43] C3H6 345 379 0.755 [26] C3H8 375 320 0.690 [43] C4H10 403 322 0.736 [26] CF4 230 265 1.920 [43] C2F6 280 260 2.050 [43] C3F8 325 225 2.050 [43] * the volume-temperature experimental behavior reported by Costello and Koros [45] at atmospheric pressure have been also considered for the determination of characteristic parameters of 6FDA-6FpDA polyimide.

3. Results The thermodynamic approach for gas transport has been applied to represent the experimental permeability behaviors with respect to penetrant upstream pressure for a series of different probes and in a variety of many different polymeric glassy membranes, as they are available in the technical literature. In 8

ACCEPTED MANUSCRIPT this section, the gas permeability behaviors of light gases, hydrocarbons and fluorocarbons in two amorphous high free volume glassy polymers has been extensively analyzed, and the application of the model is thoroughly illustrated. In particular, poly(2,2-bis(trifluoromethyl)-4,5-difluoro-1,3-dioxole-cotetrafluoroethylene), known as Teflon AF2400, and poly poly(1-(trimethylsilyl)-1-propyne), PTMSP, are

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presented as reference examples. For the sake of conciseness, however, for all the other systems investigated, only the resulting model parameters are here indicated, while detailed comparisons of the model curves with the experimental data are shown in several figures in the Supporting Information

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section.

3.1 Gas transport in AF2400

The experimental solubility data of various penetrants, namely CO2 and light gases (H2, O2, N2), hydrocarbons (CH4, C2H6, C3H8) and fluorocarbons (CF4, C2F6, C3F8), in glassy membranes of amorphous Teflon AF2400 at 35°C have been first considered [46,47]. The NELF model has been applied in order to

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obtain the behavior of the solubility coefficient and of the thermodynamic factor α at the various pressures required by the transport model. The main model parameters considered are reported in Tables 1 (lattice

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fluid EoS pure component parameters) and 2 (binary interaction coefficient and swelling factor), which are

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obtained from the analysis of the solubility data of each penetrant species.

Table 2. Lattice Fluid EoS binary penetrant/polymer interaction coefficient k12 and the linear swelling factor ksw [ ρ 2 = ρ 20 (1 − k sw p1 ) ] polymer (2)

AF2400 (ρ20 = 1.740 g/cm3)

penetrant (1) He H2 O2 N2 CO2 CH4 C2H6 C3H8 CF4

k12 (EQ) 0.10 0.03 -0.01 0.04 -0.017 0.09 0.12 0.12 0.005

ksw (NE) [MPa-1] 0 0 0.003 0.001 0.018 0.004 0.034 0.076 0.008 9

ACCEPTED MANUSCRIPT 0.025 0.035 0.24 0.03 -0.005 0.03 0.024 0.008 0.04 0.03 0.06 0.05

0.05.3 0.219 0 0 0 0 0 0 0.069 0 0.009 0.018

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PTMSP (ρ20 = 0.750 g/cm3)

C2F6 C3F8 H2 O2 N2 CO2 CH4 C2H6 C3H8 CF4 C2F6 C3F8

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of the various solutes are illustrated in Figure 1.

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The comparison between the NELF model calculations and the experimental solubility isotherms in AF2400

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ACCEPTED MANUSCRIPT Figure 1. Gas solubility at 35°C in glassy AF2400 membranes: experimental data from [46,47] and NELF model calculations; (a) light gases (He, H2, O2, N2, and CO2); (b) hydrocarbons (CH4, C2H6 and C3H8); (c) fluorocarbons (CF4, C2F6 and C3F8). NELF model parameters k12 and ksw, obtained from the analysis of the experimental data, are reported in Table 2.

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No experimental solubility data were available for C3F8 at the reference temperature of 35°C (later considered for the study of gas permeability); however, the NELF model can be employed to analyze the solubility isotherms at lower and higher values (25 and 45°C), experimentally available, in order to retrieve

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the required model parameters, to be used for the prediction of solubility data at 35°C.

Once the penetrant/polymer systems have been conveniently described by the appropriate

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thermodynamic means, the transport model can be applied to represent the experimental transport behavior at various penetrant upstream pressures. Therefore, the experimental permeability isotherms of the same gases reported by [46,48] are investigated, and Figure 2 illustrates the comparison between data and model calculations. The parameter values obtained from the analysis of the experimental data of each

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penetrant species are reported in Table 3.

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Table 3. Transport model parameters (infinite dilution mobility coefficient and plasticization factor) for the various penetrants in the glassy polymer membranes analyzed in detail (T = 35°C).

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polymer

AF2400

PTMSP

penetrant He H2 N2 O2 CO2 CH4 C2H6 C3H8 CF4 C2F6 C3F8 H2 N2 O2 CO2 CH4 C2H6

L10 [cm2/s] 1.4 · 10-5 6.7 · 10-5 5.9 · 10-6 9.4 · 10-6 4.4 · 10-6 2.8 · 10-6 3.8 · 10-7 3.4 · 10-8 1.9 · 10-7 5.0 · 10-9 1.3 · 10-9 2.5 · 10-4 4.4 · 10-5 5.2 · 10-5 2.9 · 10-5 3.6 · 10-5 1.3 · 10-5

β 0 0 0 0 17 5.5 58 98 7.0 39 37 0 0 0 0 0 0 12

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2.2 0 1.2 3.3

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C3H8 CF4 C2F6 C3F8

13

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ACCEPTED MANUSCRIPT Figure 2. Gas permeability at 35°C in glassy AF2400 membranes: experimental data from [46,48] and transport model calculations; (a) light gases (He, H2, O2, N2 and CO2); (b) hydrocarbons (CH4, C2H6 and C3H8); (c) fluorocarbons (CF4, C2F6 and C3F8). Transport model parameters L10 and β , obtained from the analysis of the experimental data, are reported in Table 3.

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As one can see, very different behaviors were obtained experimentally, as the permeability can be either a decreasing or increasing function of penetrant pressure, for the cases of light gases (e.g. N2, O2, CH4 and CF4) or for more condensable species (e.g. CO2 and higher hydrocarbons and fluorocarbons), respectively. Interestingly, the transport model can account properly for all such behaviors, which are all well described,

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as illustrated in the plots in Figure 2.

3.2 Gas transport in PTMSP

PTMSP is likely characterized by the largest free volume among all the glassy polymers developed in recent years, and for this reason has attracted much interest in the membrane science community for its peculiar

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gas transport properties: numerous experimental works reported extremely high permeability values. The model analysis of the transport behavior of various penetrants in PTMSP starts from the study of the

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experimental solubility of various solutes, namely CO2, light gases (H2, O2, N2), hydrocarbons (CH4, C2H6,

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C3H8) and fluorocarbons (CF4, C2F6, C3F8) at 35°C; they are all well described by the NELF model [49].

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ACCEPTED MANUSCRIPT Figure 3. Gas permeability at 35°C in glassy PTMSP membranes: experimental data from [49] and NELF model calculations; (a) light gases (He, H2, O2, N2, CO2, CH4, and CF4); (b) higher hydrocarbons and fluorocarbons (C2H6, C3H8, C2F6 and C3F8). NELF model parameters k12 and ksw , obtained from the analysis of the experimental data, are reported in Table 2.

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Interestingly, the permeability behaviors of the various gases in glassy PTMSP are very different from those previously illustrated in AF2400, as P can be here a strongly decreasing function of the upstream penetrant pressure for most permeating species. Indeed, the solubility coefficient decreases significantly with pressure for most of the probes, and furthermore, due to the very high free volume of the polymer matrix,

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the solute induced swelling is rather small (Table 2), so strong plasticization phenomena, i.e. large values of

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β factor, are rather unlikely.

The experimental data reported by Merkel et al. [49], indeed, showed decreasing trends of permeability for all the penetrants investigated. The transport model has been then applied according to the procedure illustrated above, and Figure 4 illustrates the comparison between model results and experimental data,

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reported in Table 3.

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whereas the parameters L0 and β obtained from the analysis of the permeability isotherms, and are

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ACCEPTED MANUSCRIPT Figure 4. Gas permeability at 35°C in glassy PTMSP membranes: experimental data from [49] and transport model calculations; (a) light gases (He, H2, O2, N2, CO2, CH4, and CF4); (b) higher hydrocarbons and fluorocarbons (C2H6, C3H8, C2F6 and C3F8). Transport model parameters L10 and β , obtained from the analysis of the experimental data, are reported in Table 3.

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As one can see, the model is able to represent very accurately the permeability behavior of all the penetrants investigated, and as expected very limited values of plasticization factor, β, are obtained. Indeed, even rather condensable and large solute molecules are not able to swell significantly the PTMSP matrix, already endowed with a very large excess of free volume, and consequently, as concentration

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increases, the mobility is enhanced only to a very small extent. That is in accordance with various

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experimental works, investigating the kinetic behavior of various penetrants in PTMSP during sorption, which report mobility values constant or even decreasing with increasing penetrant concentration [49,50]. This behavior, typical of moderately plasticizing penetrants, especially alcohols, in PTMSP and other similar polymers [51,52], is likely related to the very high excess of free volume of the glass, which is progressively lowered by the presence of low molecular weight species rather than being significantly plasticized or

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swollen. In this work, this effect has not been investigated, even though the model can allow decreasing trends of mobility coefficient when the penetrant concentration increases, as the values of plasticization factors obtained from the analysis of the experimental data, resulted only slightly negative. However,

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almost all the permeability isotherms (including CO2) are ultimately well described with a constant mobility

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parameter, L10, which remains the only adjustable parameter for permeability in PTMSP.

3.3 gas transport in various glassy polymers The same analysis has been carried out on a very broad series of experimental solubility and permeability data available in the literature, spanning over various penetrants and many glassy polymer matrixes. For the sake of conciseness, only the resulting model parameters L10 and β, are here reported (Table 4), including also the average relative error, the pressure range explored and reference to the experimental data. The details are nonetheless illustrated in the Supporting Information section. 19

ACCEPTED MANUSCRIPT Table 4. Transport model parameters (infinite dilution mobility coefficient and plasticization factor) and relative average error for all the penetrant/polymer systems investigated in this work (T = 35°C).

6FDA-DAM

HYFLON AD80

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PMP

PPh

EC

PC PSf

av. rel. err. [%] 9.2 8.8 1.2 5.9 13 8.7 7.2 8.5 3.3 6.9 0.2 5.0 4.7 5.9 11 13 8.6 6.1 4.4 4.0 10 5.2 3.5 10 5.8 3.1 1.1 0.8 1.6 3.4 1.2 0.6 1.1 6.4 1.7 1.1 0.1 0.6 1.0 1.0 2.9 -

p1 range [atm] 2 0 - 17 0 - 17 0 - 17 0 - 17 0 - 17 0 - 17 0 - 13 0 - 17 0 - 20 0 - 20 0 - 20 0 - 20 0 - 15 0 - 17 0-8 0-8 2 2 0 - 40 2 0-4 0-4 0 - 16 0 - 18 0 - 20 0 - 18 0 - 18 0-8 0 - 12 0 - 12 0 - 12 0-7 0 - 1.2 0 - 20 0 - 20 0 - 20 0 - 20 0 - 20 0 - 20 0 - 14 0 - 14 0 - 14 0 - 14 0 - 14 0-3 0-2

ref. [48]

[53]

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0 0 0 0 18 10 95 7.0 42 0 3.0 22 6.0 4.5 20 140 110 0 0 22 0 88 90 0 0 22 0 75 175 0 0 0.7 21 30 0 0 0 5 17 53.5 0 0 0 16 5.0 0 0

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β

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6FDA-6fPDA

L10 [cm2/s] 3.2 · 10-5 9.2 · 10-5 1.6 · 10-6 2.6 · 10-6 1.5 · 10-6 1.1 · 10-6 7.0 · 10-8 1.7 · 10-7 4.3 · 10-9 3.1 · 10-8 8.3 · 10-8 2.6 · 10-8 5.5 · 10-9 8.9 · 10-10 1.8 · 10-10 4.8 · 10-12 1.1 · 10-12 2.1 · 10-7 4.9 · 10-7 7.0 · 10-8 5.7 · 10-8 4.0 · 10-10 9.5 · 10-11 3.3 · 10-7 8.2 · 10-8 6.6 · 10-7 6.4 · 10-7 1.0 · 10-8 8.0 · 10-9 5.5 · 10-6 4.1 · 10-6 1.0 · 10-5 1.5 · 10-7 4.9 · 10-8 1.7 · 10-8 4.9 · 10-8 2.1 · 10-8 4.7 · 10-9 1.6 · 10-8 2.1 · 10-10 4.8 · 10-5 2.3 · 10-6 3.5 · 10-6 4.4 · 10-7 8.6 · 10-7 5.8 · 10-8 1.2 · 10-6

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AF1600

penetrant He H2 N2 O2 CO2 CH4 C2H6 CF4 C2F6 N2 O2 CO2 CH4 C2H4 C2H6 C3H6 C3H8 N2 O2 CO2 CH4 C3H6 C3H8 N2 CH4 CO2 O2 C2H6 C3H8 N2 CH4 CO2 C3H8 C4H10 N2 O2 Ar CH4 CO2 C2H6 He N2 O2 CO2 CH4 O2 H2

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polymer

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61] [62] 20

ACCEPTED MANUSCRIPT He O2

3.4 · 10-6 3.4 · 10-8

0 0

-

0-2 0 - 10

[63] [64]

Table 4 includes also the transport behavior of other penetrants in glassy polymer membranes already investigated in previous works [14,19], as these systems were recently analyzed aiming at a more complete

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characterization.

It is noteworthy that the collection of permeability data investigated in this work contains all types of penetrant transport behavior, with permeability either increasing or decreasing or even nonmonotonous

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function of upstream pressure; as indicated by the average relative error, the model provides a very

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effective and accurate description of all the experimental data examined.

It is worth mentioning that the series of model parameters obtained for the description of the mobility behaviors inspected, i. e. the mobility coefficient at infinite dilution L10 and the plasticization factor β, are related to polymer and penetrant properties. As already illustrated from the cases inspected in previous

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works, the mobility coefficient at infinite dilution scales with penetrant size [19,20] and with polymer fractional free volume [17], whereas the plasticization factor is related to the swelling induced by the penetrant into the polymer matrix [17,20]. Therefore, the significant collection of model parameters

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obtained in this work could then be used to derive correlations with general validity, and wide applicability, reasonably leading to a simple and reliable procedure for the estimation of the permeability behavior in a

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predictive fashion.

4. Conclusions

The solubility and permeability of various penetrants in a variety of different glassy polymers has been analyzed by the NELF approach and the thermodynamic based transport model recently proposed by Minelli and Sarti [14]. To this aim, a large collection of experimental sorption and transport data available in the technical literature has been examined, accounting for a large variety of penetrants, with various size and condensability, and of polymers, ranging from very high free volume glasses (e.g. PTMSP or PMP), to 21

ACCEPTED MANUSCRIPT polyimides (6FDA-6FpDA or 6FDA-DAM) and fluorinated copolymers (AF1600, AF2400 or HYFLON AD 80). The model showed its ability in representing all possible behaviors of permeability with various upstream penetrant pressures, either constant, decreasing or increasing functions or even nonmonotonous trends, in a simple and unified approach with solid thermodynamic basis. Remarkably, in case of light gases as

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penetrants the description of the diffusional behavior is provided by only one adjustable parameter, the mobility coefficient L10, while a plasticization factor β is required for swelling penetrants, once the thermodynamic properties of the penetrant/polymer mixtures have been described by the NELF model.

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Very small deviations were observed between experimental data and model calculations, with average relative errors in the order of few percent (around 5%), which is comparable with the usual experimental

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error often encountered.

List of abbreviation

PTMSP PMP 6FDA-6FpDA 6FDA-DAM

PPh EC PC PSf

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HYFLON AD80

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AF1600

65 mol% 2,2-bistrifluoromethyl-4,5-difluoro-1,3-dioxole and 35 mol% tetrafluoroethylene (random copolymer) 87 mol% 2,2-bistrifluoromethyl-4,5-difluoro-1,3-dioxole and 13 mol% tetrafluoroethylene (random copolymer) poly(1-trimethylsilyl-1-propyne) poly(4-methyl-2-pentyne) 2,2-bis(3,4-carboxyphenyl) hexafluoropropane dianhydride, 4, 4-Hexafluoro diamine polyimide 2,2-bis(3,4-carboxyphenyl) hexafluoropropane dianhydride, diaminomesitylene polyimide 85 mol% 2,2,4-trifluoro-5-trifluoromethoxy-1,3-dioxole and 15 mol% perfluorodioxole (random copolymer) poly (phenolphthalein terephthalate) ethyl cellulose (ethoxy content 49%) bisphenol A - polycarbonate polysulfone

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AF2400

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gas solubility behavior of high-Tg polyimides, Fluid Phase Equilib. 333 (2012) 87-96. M. G. De Angelis, T.C. Merkel, V.I. Bondar, B. D. Freeman, F. Doghieri, G. C. Sarti, Gas sorption and dilation in

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poly(2,2-bistrifluoromethyl-4,5-difluoro-1,3-dioxole-co-tetrafluoroethylene): Comparison of experimental data with predictions of the non-equilibrium lattice fluid model, Macromolecules 2002, 35, 1276–1288. [40]

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permeability of propane and n-butane in poly(4-methyl-2-pentyne), J. Polym. Sci. B: Polym. Phys. 42 (2004) 2407–2418. [52]

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[54]

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Gas permeability in glassy polymers: a thermodynamic approach

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Matteo Minelli,* Giulio C. Sarti

Department of Civil, Chemical, Environmental and Materials Engineering (DICAM) Alma Mater Studiorum – University of Bologna

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Via Terracini 28 – I-40131 Bologna (Italy)

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Supporting Information

Corresponding author: Matteo Minelli

tel. +39 (0) 51 2090426 fax +39 (0) 51 6347788 email address: [email protected]

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NELF / Sanchez Lacombe EoS model Table S1. Definition of properties and main correlations for the Sanchez–Lacombe/NELF framework.

mass fraction of solute i temperature

p

pressure

ρ2

polymer mass density

ρ

characteristic density of pure component i

∗ i

pi∗

Ti

∗

Mi k12

characteristic pressure of pure component i characteristic temperature of pure component i molar mass of component i binary interaction parameter for penetrant, polymer (1,2) pair

Eq. ∗ i

v

ρ*

ρ%

I

III

µ1NE ,res

SiNE ,0

IV

RTi* pi*

ω1 ω2 + ρ ρ1∗ ρ 2∗ ρ ρ% = (1 − ω1 ) 2* ρ ρ* φi = ωi * ρi ∗

=

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∆p12∗

vi* =

V

1 φ1 φ2 = + v* v1* v2*

∆p12∗ = p1∗ + p2∗ − 2 (1 − k12 ) p1∗ ⋅ p2∗

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volume fraction of component i in the mixture average closepacked mer molar volume in the mixture interaction characteristic pressure for (1,2) pair in the mixture

calculation

1

II

reduced mixture density

φi model variables

lattice site molar volume for pure component i characteristic density of the mixture

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Model parameters

state variables

symbol

residual chemical potential of solute 1 in the glassy polymeric mixture

infinite dilution solubility coefficient of solute 1 in the glassy polymer 2

VI

µ1NE,res RT

VII

=

  v*  1 − 1∗  *   M1  v1 v  − ρ% v* (1 + φ ) p * + φ p * − ∆p* 1 − − ln (1 − ρ% ) 1 − 1  1 1 2 2 1,2 % ρ1* v∗  v∗ ρ     

(

)

      

ln( S0 ) = VIII

 T    v∗  ρ∗   ρ0   v∗  ρ0 T∗ 2(1− k12 ) ∗ ∗  = ln STP  + r10 1+ 1∗ −1 20  ln1− 2∗  + 1∗ −1 + 2∗ 1 p1 p2  p1∗  pSTPT    v2  ρ2   ρ2   v2  ρ2 T 

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Analysis of penetrant solubility/permeability in various polymers

Figure S1. Gas solubility at 35°C in glassy AF1600 membranes: experimental data from [1,2] and NELF model calculations: (a) light gases (He, H2, O2, N2 and CO2); (b) hydrocarbons and fluorocarbons (CH4, C2H6, CF4 and C2F6). NELF model parameters k12 and ksw, obtained from the analysis of the experimental data, are reported in Table 2 in the main text.

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Figure S2. Gas permeability at 35°C in glassy AF1600 membranes: experimental data from [1,2] and transport model calculations: (a) light gases (He, H2, O2, N2 and CO2); (b) hydrocarbons and fluorocarbons (CH4, C2H6, CF4 and C2F6). Transport model parameters L10 and β , obtained from the analysis of the experimental data, are reported in Table 3 in the main text.

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Figure S3. Gas solubility at 35°C in glassy 6FDA-6fFpDA membranes: experimental data from [3,4] and NELF model calculations. NELF model parameters k12 and ksw , obtained from the analysis of the experimental data, are reported in Table 2 in the main text.

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Figure S4. Gas permeability at 35°C in glassy 6FDA-6fFpDA membranes: experimental data from [3,4] and transport model calculations. Transport model parameters L10 and β , obtained from the analysis of the experimental data, are reported in Table 3 in the main text.

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Figure S5. Gas solubility at 35°C in glassy 6FDA-DAM membranes: experimental data from [5] and NELF model calculations. NELF model parameters k12 and ksw , obtained from the analysis of the experimental data, are reported in Table 2 in the main text.

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Figure S6. Gas permeability at 35°C in glassy 6FDA-DAM membranes: experimental data from [5] and transport model calculations. Transport model parameters L10 and β , obtained from the analysis of the experimental data, are reported in Table 3 in the main text.

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Figure S7. Gas solubility at 35°C in glassy HYFLON AD 80 membranes: experimental data from [6] and NELF model calculations. NELF model parameters k12 and ksw , obtained from the analysis of the experimental data, are reported in Table 2 in the main text.

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Figure S8. Gas permeability at 35°C in glassy HYFLON AD80 membranes: experimental data from [6] and transport model calculations. Transport model parameters L10 and β , obtained from the analysis of the experimental data, are reported in Table 3 in the main text.

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Figure S9. Gas solubility at 35°C in glassy PMP membranes: experimental data from [7-9] and NELF model calculations. NELF model parameters k12 and ksw , obtained from the analysis of the experimental data, are reported in Table 2 in the main text.

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Figure S10. Gas permeability at 35°C in glassy HYFLON AD 80 membranes: experimental data from [10] and transport model calculations. Transport model parameters L10 and β , obtained from the analysis of the experimental data, are reported in Table 3 in the main text.

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Figure S11. Gas solubility at 35°C in glassy PPh membranes: experimental data from [11] and NELF model calculations. NELF model parameters k12 and ksw , obtained from the analysis of the experimental data, are reported in Table 2 in the main text.

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Figure S12. Gas permeability at 35°C in glassy PPh membranes: experimental data from [11] and transport model calculations. Transport model parameters L10 and β , obtained from the analysis of the experimental data, are reported in Table 3 in the main text.

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Figure S13. Gas solubility at 35°C in glassy EC membranes: experimental data from [12] and NELF model calculations. NELF model parameters k12 and ksw , obtained from the analysis of the experimental data, are reported in Table 2 in the main text.

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Figure S14. Gas permeability at 35°C in glassy EC membranes: experimental data from [12] and transport model calculations. Transport model parameters L10 and β , obtained from the analysis of the experimental data, are reported in Table 3 in the main text.

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References: A. Y. Alentiev, V. P. Shantarovich, T. C. Merkel, V. I. Bondar, B. D. Freeman, Y. P. Yampolskii, Gas and vapor sorption, permeation, and diffusion in glassy amorphous Teflon AF1600, Macromolecules 35 (2002) 9513-9522.

[2]

Z. P. Smith, R. R. Tiwari, M. E. Dose, K. L. Gleason, T. M. Murphy, D. F. Sanders, G. Gunawan, L. M. Robeson, D. R. Paul, B. D. Freeman, Influence of diffusivity and sorption on helium and hydrogen separations in hydrocarbon, silicon, and fluorocarbon-based polymers, Macromolecules 47 (2014) 3170–3184.

[3]

R. Wang, C. Cao, T. S. Chung, A critical review on diffusivity and the characterization of diffusivity of 6FDA–6FpDA polyimide membranes for gas separation, J. Membr. Sci. 198 (2002) 259–271.

[4]

C. Staudt-Bickel, W. J. Koros, Olefin/paraffin gas separations with 6FDA-based polyimide membranes, J. Membr. Sci. 170 (2000) 205–214.

[5]

H. Sejour, Investigation of dithiolenes for propylene/propane membrane separations, Ph.D. dissertation (2007), Georgia Institute of Technology.

[6]

R. S. Prabhakar, B. D. Freeman, I. Roman, Gas and vapor sorption and permeation in poly(2,2,4-trifluoro-5trifluoromethoxy-1,3-dioxole-co-tetrafluoroethylene), Macromolecules 37 (2004) 7688-7697.

[7]

K. Nagai, S. Kanehashi, S. Tabei, T. Nakagawa, Nitrogen permeability and carbon dioxide solubility in poly(1trimethylsilyl-1-propyne)-based binary substituted polyacetylene blends, J. Membr. Sci. 251 (2005) 101–110.

[8]

K. Nagai, A. Sugavara, S. Kazama, B. D. Freeman, Effects of physical aging on solubility, diffusivity, and permeability of propane and n-butane, in poly(4-methyl-2-pentyne), J. Polym. Sci. B: Polym. Phys. 42 (2004) 2407–2418.

[9]

T. C. Merkel, B. D. Freeman, R. J. Spontak, Z. He, I. Pinnau, P. Meakin, A. J. Hill, Sorption, transport, and structural evidence for enhanced free volume in poly(4-methyl-2-pentyne)/fumed silica nanocomposite membranes, Chem. Mater. 15 (2003) 109-123.

[10]

A. Morisato, I. Pinnau, Synthesis and gas permeation properties of poly(4-methyl-2-pentyne), J. Membr. Sci. 121 (1996) 243 -250.

[11]

R. T. Chern, N. F. Brown, The solubilities and diffusivities of permanent gases in poly(phenolphthalein terephthalate), Macromolecules 23 (1990) 2370-2375.

[12]

A. Y. Houde, S. A. Stern, Permeability of ethyl cellulose to light gases. Effect of ethoxy content, J. Membr. Sci. 92 (1994) 95-101.

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[1]