Gas Solubility, Diffusivity, Permeability, and Selectivity in Mixed Matrix

Article pubs.acs.org/IECR

Gas Solubility, Diffusivity, Permeability, and Selectivity in Mixed Matrix Membranes Based on PIM‑1 and Fumed Silica Maria Grazia De Angelis,* Riccardo Gaddoni, and Giulio C. Sarti Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali (DICAM), Alma Mater Studiorum Università di Bologna, via Terracini 28, 40131 Bologna, Italy ABSTRACT: The effects that the addition of fumed silica (FS) nanoparticles has on the gas permeability, solubility, diffusivity, and selectivity of a polymer of intrinsic microporosity (PIM-1) are modeled considering the density of the composite matrix as the key input information. PIM-1 is treated as a homogeneous glassy polymer endowed with a specific free volume that increases with the amount of nanoparticles loaded, as indicated by the experimental values of mixed matrix density. The solubility isotherms of H2, He, O2, N2, CH4, and CO2 in matrices of PIM-1 with different FS loadings are calculated with the nonequilibrium lattice fluid (NELF) model. The gas diffusivity and permeability variation due to FS addition are related to the fractional free volume of the polymer phase, according to the semiempirical free volume theory equation. Remarkably, the coefficient amplifying the free volume effect increases with the molecular size of the gas, expressed by the van der Waals volume, thus allowing an estimation of the transport properties of gases not investigated experimentally, such as methane. The behavior inspected differs from the one observed in mixed matrix membranes (MMM) formed by PIM-1 and porous selective fillers, that show higher selectivity toward smaller penetrants than the pure polymer, because the effect of silica nanoparticles is only represented by an enhancement of the large free volume domains. The model allows an estimation of the ideal selectivity together with its solubility and diffusivity contributions, at various FS contents, for several gas pairs (O2/N2, CO2/N2, CO2/CH4, CO2/H2), which are then compared to the experimental trends available.

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INTRODUCTION The family of polymers with intrinsic microporosity (PIMs) is based on non-network polymers that are soluble and can be processed easily with solvent-based methods, unlike conventional microporous materials, and possess open structures due to a rigid spirocyclic molecular scaffold.1 The most studied polymer of this class is the product of condensation of 5,5′,6,6′tetrahydroxy-3,3,3′,3′-tetramethyl-1,1′-spirobisindane and tetrafluoroterephtalonitrile, named PIM-1.2−5 Such a polymer shows outstanding properties for the gas separation of several industrial mixtures, for example, CO2/CH4 and CO2/N2, for which they lie on or above the most recent Robeson trade-off curve.6 In the last six years, authors have tried to further improve those features by adding various inorganic fillers to the polymer matrix, such as functionalized multiwalled carbon nanotubes (MWCNTs),7 metal organic frameworks (MOFs), or microporous and mesoporous molecular sieves of different kinds,8 zeolitic imidazolate frameworks (ZIF-8),9 or fumed silica nanoparticles.10 The latter case will be inspected and modeled in this work, and we will introduce also some considerations on other types of composite structures and properties. The solubility and transport behavior of different penetrants in mixed matrix membranes (MMM) obtained by adding fumed silica (FS) nanoparticles to high-free-volume glassy polymers shows rather unusual and unexpected features which make it very hard to obtain reasonable predictions or expectations for permeability, solubility, and diffusivity, simply based on the behavior shown by the unloaded polymer matrix.11 Therefore, the importance of a reliable modeling approach for the transport properties of those MMM is © XXXX American Chemical Society

apparent and of great impact in order to reduce the necessary experimental effort to the essential minimum required. The transport behavior of composite materials formed by glassy polymers and silica nanoparticles cannot be described with the conventional models for transport into “ideal” composite systems, such as the Maxwell model, unless the number of adjustable parameters is increased significantly to take into account size and distribution of a possible third phase, represented by the hypothetical additional void space between polymer and particles.11 On the other hand, in 2008 a successful approach was proposed that allows a representation of solubility and transport properties of several gases in MMM obtained by adding dense nanoparticles of fumed silica in high free volume glassy polymers such as poly(1-trimethylsilyl-1propyne) (PTMSP) and Teflon AF 1600 and 2400.12−14 In the approach, the mixed matrix transport properties can be quantitatively described by considering only two phases, silica and polymer matrix, with the latter characterized by a different fractional free volume (average density) with respect to the pure unloaded polymer. The diffusion coefficients in the mixed matrix are estimated by means of the free volume theory applied to the polymer phase, accounting for the increased tortuosity of the diffusive path due to the presence of the impermeable filler particles. Special Issue: Enrico Drioli Festschrift Received: December 22, 2012 Revised: March 15, 2013 Accepted: March 20, 2013

A

dx.doi.org/10.1021/ie303571h | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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CH4 is lower in the mixed matrix membrane than in the pure polymer, by a factor of 0.6, while the opposite is true for the CO2/N2 gas pair (increase factor 1.3).7 From such results it seems that the addition of different types of porous fillers has the effect of enhancing the matrix permeability for all cases, but the effect on the selectivity may vary: porous fillers such as ZIF and AIPO produce a permeability increase that is smaller as the penetrant size is larger, thus increasing the ideal size selectivity (i.e., the selectivity toward gases of smaller molecular size). On the other hand, matrices formed by PIM-1 and hollow cylindrical nanotubes yield the opposite effect, lowering the size selectivity. Such a behavior is likely due to the geometry and chemical nature of the fillers, probably offering more pathways for the larger molecules, and/or more favorable interaction sites. However, the information available in the literature up to now is not sufficient to draw general conclusions. In the case of mixed matrix membranes based on methanoltreated PIM-1 and dense fumed silica nanoparticles, inspected here, the experimental evidence show that the gas permeability increases in all cases after addition of fumed silica, as it happens with porous fillers. However, the enhancement factor increases with increasing molecular size, thus lowering the size selectivity of the matrix. Such a behavior is followed by He, H2, N2, O2, with a notable exception represented by CO2, for which a lower enhancement of permeability with respect to smaller gases is observed.10 As fumed silica is a nonporous filler, the behavior observed for the smaller molecules can be attributed to the modification of the polymer free volume induced by inserting silica domains in the matrix, thus creating large void zones inside the polymer matrix that favor the permeation of “larger” penetrants. A comparison of the different effects obtained by adding various fillers to PIM-1 matrices is shown in Figure 1, where the ratio between the permeability in the mixed matrix membrane and in the pure polymer, PM/PP, is reported versus the van der Waals volume of the gaseous penetrants considered. In Figure 1a the data refer to mixed matrices loaded with various amounts of ZIF-8 and AIPO-18, while in Figure 1b we report the data relative to MMM formed by PIM-1 and carbon nanotubes and fumed silica. The operating conditions, pressure, and temperature at which such parameters are obtained, are rather different: in particular, pressure is 1 atm for the MMM based on ZIF-8, 4 atm for those based on FS, 2 atm for matrices containing MWCNT fillers, and 6.9 atm for AIPO-based MMM. Pressure can make a difference when CO2 is considered as penetrant, as permeability can increase or decrease with pressure even in a rather limited range. It can be seen that, generally, the permeability increase is smaller for larger penetrant sizes in the mixed matrix membranes based on PIM-1 and porous selective fillers (Figure 1a). On the contrary, the permeability enhancement is larger for larger penetrant sizes in MMM based on fumed silica, with the exception of CO2 (Figure 1b). Similarly, mixed matrices based on carbon nanotubes show a permeability increase that increases with penetrant size, with the exception of CO2. Mixed Matrix Membranes Based on Fumed Silica. In mixed matrices based on fumed silica and glassy polymers such as PTMSP, poly(4-methyl-2-pentyne) (PMP) and Teflon AF2400, the effect of filler addition enhances the permeability of all penetrants. For the selectivity, in some cases fumed silica addition yields an enhancement of the selectivity toward penetrants of larger molecular size, that will be indicated

Finally, the permeability in the composite matrix can be estimated by means of the solution-diffusion model. The relevant transport properties in MMM can thus be estimated with a limited set of experimental data: for the prediction of solubility isotherms for all gases only the experimental value of the composite density is needed (together with the pure filler density), while for the diffusivity or permeability additional information is required to obtain the adjustable parameter (Bi, see eqs 9−11, 15 and 16) relating these properties to the fractional free volume. It was shown that the model recalled above is suitable to describe all the particular behaviors observed for a series of MMM based on fumed silica nanoparticles and high-free-volume glassy polymers.11−14 For the MMM obtained by adding FS nanoparticles to PIM1, the same model approach will be applied here, using the experimental data of density and permeability presented by Ahn et al.10 for various gases (H2, He, O2, N2, and CO2) in PIM-1 matrices with different filler loadings, up to 39 wt %, at 25 °C, with upstream pressure of 4.4 atm. In the present work, the model developed will be first tested and validated based on such data, then it will be used to predict the transport properties of methane, not experimentally investigated.

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BACKGROUND Mixed Matrix Membranes Based on PIM-1. Several types of inorganic fillers have been added to PIM-1 matrices to improve their permselectivity properties. It has been seen that MMM based on PIM-1 and 43 vol % of ZIF-8 show increased permeability of He, H2, O2, N2, CO2, CH4 by factors ranging from 1.4, for the larger penetrants, up to 4, for the smaller ones (in as-cast films).9 Therefore, the ideal selectivity of the large versus small gas molecules (CO2/H2, CO2/N2) decreases in almost all conditions after the addition of nanofiller. That is due to the fact that the major effect of ZIF addition is the increase of concentration of “small” free volume domains, as testified by the intensity increase in PALS measurement. Moreover, the filler in this case has an intrinsic selectivity which contributes to the separation mechanism of the mixed matrix membrane. The CO2/CH4 selectivity of as-cast films with initial selectivity of about 14 increases by the addition of ZIF-8 (28 vol %) up to 18 and then decreases, while the corresponding value of ethanol-treated samples (initial selectivity around 17) decreases monotonically with increasing ZIF loading, down to 12 at 28 vol % and 7 at 43 vol %.9 In this case the two penetrants have similar molar volumes (though CH4 is larger) and the effect of filler addition depends on the initial properties of the sample. In general, in the ethanol-treated samples the initial permeability, as well as its variation with ZIF addition, seems larger for the penetrants of larger size. Indeed, the polymer free volume increases after ethanol treatment, making the film less size-selective and therefore less selective toward the gas of smaller molecular size (CO2). Similarly to as-cast PIM-1/ZIF8 films, samples of PIM-1 containing 30% of an aluminophosphate molecular sieve (AIPO-18) show an increase of CO2 permeability by a factor of 1.7 and an increase of CO2/CH4 selectivity of 1.6 with respect to pure PIM-1.8 In this case, therefore, the filler addition favors more the permeability of smaller penetrants (CO2), as it happens in the as cast films of PIM-1/ZIF-8. On the other hand, addition of functionalized multiwalled carbon nanotubes to PIM-1 enhances more the fluxes of gases with larger molecular size, though favoring the permeability of all penetrants, so that the ideal selectivity for the couple CO2/ B



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vapor selectivity of the polymer matrix is very large, and correspondingly the matrix is characterized by a high availability of large free volume domains, as in the case of some types of PTMSP, the opposite behavior is observed.11 In particular, one can establish quantitatively that the n-C4/CH4 selectivity of the pure polymer can be increased by the addition of FS only if its initial value is below 20, which represents a sort of switchover value. From Figure 2, reporting the relative selectivity variation

Figure 2. Ratio between n-C4/CH4 selectivity in the mixed matrix membrane and in the polymer (αM/αP), versus pure polymer selectivity (αP). Data are relative to mixed matrices obtained by adding FS nanoparticles of various kinds into different glassy polymers at room temperature: hydrophobic Cab-O-Sil particles of diameter 12 nm (TS 530) to Teflon AF2400,15 PMP,16,17 PTMSP,18 cross-linked (XL) PTMSP;19 hydrophobic Cab-O-Sil particles of diameter 25 nm (TS 610) to PTMSP;20 hydrophobic Aerosil particles of diameter 12 nm to different PTMSP (PTMSP and PTMSP (2));21 hydrophilic Cab-O-Sil particles L90 (30 nm), M5 (15 nm), EH 5 (7 nm) to PMP.22

induced by the addition of nanoparticles to the neat polymer, αM/αP, versus the selectivity of the polymer αP, one realizes that for αP > 20, the vapor selectivity of the polymer is not increased but rather decreased by the addition of nanoparticles, and it seems that the selectivity cannot be increased above such value. One can see, in particular, that the relative variation in αM is smaller than the corresponding relative variation in αP, as otherwise shown in this figure. In Figure 2 we report literature data relative to mixed matrices formed by adding hydrophobic fumed silica Cab-O-Sil TS 530 particles of diameter 12 nm to Teflon AF2400,15 PMP,16,17 PTMSP,18 cross-linked (XL) PTMSP.19 The grade of fumed silica TS 530 is surface modified with hexamethyldisilazane to make it hydrophobic. Data refer also to mixed matrices formed by PTMSP and a different grade of Cab-O-Sil, namely TS 610, that is made hydrophobic by treating it with dimethyldichlorosilane:20 the Aerosil R974 particles have an approximate diameter of 25 nm. The data contain also information about matrices based on a different type of hydrophobic fumed silica (Aerosil R974), of diameter 12 nm, and PTMSP with different initial free volume (PTMSP and PTMSP (2), respectively).21 Such particles were treated with dimethyldichlorosilane. Other data considered are relative to the mixed matrices formed by adding hydrophilic fumed silica particles of different diameter size to PMP.22 In particular, the particles considered were Cab-O-Sil L90 (30 nm), M5 (15 nm), and EH 5 (7 nm). Density and Fractional Free Volume. Density of the unpenetrated composite matrix, ρ0M, is considered as the result

Figure 1. Effect of filler addition on the permeability of different gases in PIM-1 based mixed matrix membranes. Data relative to (a) porous fillers ZIF-89 and AIPO-18;8 (b) hollow MWCNT7 and nonporous fumed silica nanoparticles.10

hereafter shortly as vapor selectivity. This type of behavior is verified mainly on mixtures formed by methane and n-butane. However, in certain cases the opposite behavior is reported, that is, the vapor selectivity decreases after the addition of fumed silica. From several different data, it seems that the effect of FS addition on selectivity depends essentially on the value of pure polymer selectivity, and ultimately on its microstructure and free volume distribution. Indeed, such a behavior is very similar to what happens to the CO2/CH4 selectivity in the mixed matrices formed by PIM-1 and ZIF-8, which may increase or decrease depending on the properties of the pure polymer before loading the filler. In particular, for AF2400, PMP, and certain types of PTMSP the vapor selectivity, evaluated by mixed gas permeation experiments with methane and n-butane, is increased by the addition of FS nanoparticles; such an effect is due to the fact that adding fumed silica nanoparticles enhances more significantly the large free volume domains rather than the small ones, as PALS studies indicate. However, if the initial C



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Table 1. LF Parameters and Model Equations symbol

name

eq. no.

definition/property

Pure Component i ρi*, pi*, Ti*

characteristic density, pressure and temperature of pure component i

r0i

number of lattice sites occupied by a mole of pure component i

I

ri0 =

Mi ρi*vi*

v*i

volume occupied by a mole of lattice sites of pure substance

II

vi* =

RTi* p*

ωi

mass fraction

ϕi

volume fraction

i

ρ* p*

characteristic density of the mixture

ϕi =

III

ωi /ρi* ∑i ωi /ρi*

Multicomponent Mixtures 1 = IV ρ*

characteristic pressure of the mixture

i

∑ ϕipi* −

p* =

V

ωi ρi*

∑

i

1 2

∑ ϕi ∑ ϕjΔpij* i

j≠i

Δpij*

binary parameter

VI

Δpij* = pi* + p*j − 2(1 − kij) pi* p*j

kij, Ψij

binary parameter

VII

Ψij = 1 − kij

r

molar average number of lattice sites occupied by a molecule in the mixture

VIII

r=

T*

characteristic temperature of the mixture

IX

T* =

∑ xiri i

p* r

∑ xiri0 i

Ti* p*v* = R pi*

v*

average close-packed mer molar volume in the mixture

X

RT * v* = p*

AEq

total equilibrium Helmholtz free energy; Np is the number of penetrants

XI

⎡⎛ ⎞ AEq 1 1 = − ρ ̃ + T̃ ⎢⎜ − 1⎟ ln(1 − ρ ̃) + ln(ρ ̃) + ⎢⎣⎝ ρ ̃ rNRT * r ⎠ μi

μNE i

41

chemical potential of species i in the non-equilibrium glass.

NE

RT

⎡ r − = ln(ρϕ ̃ i) − ln(1 − ρ ̃)⎢ri0 + i ρ̃ ⎣

XII

− ρ̃

ln(S0)

Infinite dilution solubility of a single gas (i) in a polymer (P);42,43 TSTP and pSTP are standard temperature and pressure.

XIII

* rv i i [p* + RT i

∑ i=1

⎤ ln(ϕi)⎥ ⎥⎦ ri

ϕi

ri0 ⎤

⎥− ri ⎦

Np+ 1

∑

ϕj(p*j − Δpi*, j )]

j=1

⎧⎡ ⎞ ⎛ v* ⎞1⎤ ⎛ v* ⎛ T ⎞ ⎪ ln S0 = ln⎜⎜ STP ⎟⎟ + ri0⎨⎢1 + ⎜⎜ i − 1⎟⎟ 0 ⎥ ln(1 − ρp0̃ ) + ⎜⎜ i − 1⎟⎟ * * ⎪⎢⎣ ⎝ pSTP T ⎠ ⎠ ⎝ vp ⎠ ρp̃ ⎥⎦ ⎝ vp ⎩ + ρp0̃

of a mixing process in which volume is semiadditive: the silica phase is assumed to occupy the same volume as in its pure state, while the polymer has a different density, ρ0P ≠ ρ0P,pure, normally ρ0P < ρ0P,pure because of the modified free volume. Superscript 0 indicates the unpenetrated state of the system, or equivalently the state at infinite dilution of penetrant; the polymer density in such a state is different from the one measured after significant penetrant sorption, if relevant swelling takes place in the matrix. According to the above assumption, experimental density data for the dry composite matrices, ρ0M, measured with the conventional methods for solids, can be used to calculate the actual unpenetrated density,ρ0P, of the polymer phase of the MMM as

Np+ 1

ρM0 =

⎫ ⎪ T*i 2 Ψip pi* pp* ⎬ ⎪ T pi* ⎭

1 1 − wF ρP0

+

wF

⇒ ρP0 =

ρF,pure

1 − wF 1 ρM0

−

wF ρF,pure

(1)

where ρF,pure is the density of the pure filler phase (2.2. g/cm3 for FS TS 53015) and wF is the silica mass fraction in the unpenetrated MMM. The value ρ0P is then used to calculate the fractional free volume (FFV) of the unpenetrated polymer phase as follows: FFV 0P

=

W V P0 − 1.3VP,pure

V P0

=

W ρP,pure − 1.3ρP0

ρPW

(2)

W where VW P,pure and ρP,pure are the van der Waals specific volume and density of the repeating unit of the pure polymer, respectively, that can be calculated with the group contribution method.23 The fractional free volume is calculated with respect

D



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The density of the glassy polymer ρP, depends on the experimental conditions and on the history of the samples. For nonswelling penetrants, the density of the polymer phase at every pressure can be considered equal to its initial value; in the case of swelling agents, on the contrary, information on the density of the membrane at every pressure condition is required, and that can be obtained from parallel dilation experiments, when available. In gas sorption in glassy polymers there is normally a linear dependence of the polymer density on the partial pressure of the swelling penetrant,31−33 so that a swelling coefficient, ksw, can be used to account for volume dilation in a simple way:

to the occupied volume, estimated as 1.3 VW P,pure, according to Bondi’s rule.23 Such values are already available for the matrix here of interest; in particular for PIM-1 one has ρW P,pure= 1.809 g/cm3, which corresponds to a fractional free volume of 19.2% when the density is equal to 1.124 g/cm3,24 and to 28.8% when the density is equal to 0.991 g/cm3 as in the case of the samples inspected in this work.10 Other authors provide a value of FFV between 0.24 and 0.26.25 The density value of PIM-1 can change even significantly depending on the method used for its measurement and on the history of the sample, as well as on the solvent used in the film casting process. Indeed, several authors have pointed out that the value obtained for the density of PIM-1 depends on the method used: helium pycnometry normally gives values of density which are 30% higher than those based, for instance, on the measurement of the buoyancy forces exerted on the sample by larger inert solvents.26 This is due to the fact that helium can penetrate into internal microvoids which are rather large in PIM polymers, while the solvents used in hydrostatic forces measurements do not reach those voids. For the purpose of the present modeling, which considers the polymer as a dense material, the most appropriate value of density is the lowest one, which represents the ratio between the sample mass and the total volume occupied by the sample, delimited by its macroscopic surface boundary, and which includes also the nanovoids inside the polymer matrix which maybe relatively large but still in the order of molecular sizes. Also the effects of sample pretreatments are crucial and represent a typical feature of high free volume glassy polymers: normally it is advisable to treat the sample with methanol to standardize the treatment protocol. Solubility in the Polymer Phase: NELF Model. The nonequilibrium lattice fluid (NELF) model27,28 can be used to calculate the solubility isotherm in glassy polymers since it relates explicitly the penetrant solubility to the glassy polymer density. It is based on the non-equilibrium thermodynamic for glassy polymers (NET-GP) approach,28,29 and extends the lattice fluid (LF) equation of state (EoS) for amorphous phases to the nonequilibrium states typical of glassy polymers.30 In this framework, polymers and penetrants are characterized by the same pure component parameters (p*, ρ*, T*) of the equilibrium Sanchez and Lacombe theory, and the mixture properties are obtained through the mixing rules of the same model.30 The pure components characteristic parameters can be calculated by best fitting the LF equation to pressure− volume−temperature (PVT) data above Tg for the polymer and to either PVT or vapor−liquid equilibrium (VLE) data for the penetrant. The values of the pure components characteristic parameters may be found in specific collections. The main quantities and equations used in the NELF model are systematically reported in Table 1. To describe the properties of glassy phases, also the actual nonequilibrium polymer density value, ρP, is needed beside the usual state variables (temperature, pressure, and composition), as the only quantity accounting for the departure from equilibrium. The phase equilibrium conditions are obtained, at all penetrant mass fraction ωi, when the chemical potential of the penetrant i has the same value both in the equilibrium external gas phase labeled by superscript g (μEq(g) ), and in the i solid mixture with the nonequilibrium glassy polymer, labeled by superscript s (μNE(s) ): i μi NE(s)(T , p , ω1 , ρP ) = μi Eq(g)(T , p)

ρP (p) = ρP0 (1 − kswp)

(4)

ρ0P

where is the density of the pure unpenetrated glassy polymer. The swelling coefficient can be obtained directly from measurements of polymer dilation during sorption; alternatively, the parameter ksw can be treated as an adjustable parameter and fitted on at least one experimental solubility datum, if available, with the NELF model. In particular, such parameter is adjusted on one solubility datum at high pressure, where swelling is not negligible.34 Solubility in the Mixed Matrix Membrane. The NELF model can be applied to calculate solubility isotherms in composite glassy matrices considering separately the contributions of the two phases: polymer matrix and filler. The mass fraction of gas in the MMM per unit mass of mixed matrix is thus: ωi ,M =

mi ,F + mi ,P mM

= wFωi ,F + (1 − wF)ωi ,P

(5)

where ωi,M is the mass of penetrant i per unit mass of mixed matrix, ωi,F and ωi,P are the masses of penetrant i per unit mass of filler and per unit mass of the polymer, respectively. The first order estimate of eq 5 can be obtained by considering that ωi,F and ωi,P are given by the corresponding values in the pure phases, ωi,F,pure and ωi,P,pure, respectively. That first order additive model is not always appropriate to describe the gas solubility in the mixed matrix materials; it is never appropriate for glassy composites containing nanofillers, and definitely is not suitable for the mixed matrices under consideration. Because of surface effects and filler surface accessibility, in the MMM one normally has ωi,F ≤ ωi,F,pure, while on the other hand the sorptive capacity of the polymer phase is significantly increased after nanofiller addition, and thus ωi,P > ωi,P,pure. Moreover, in the case of high free volume polymers such as PTMSP, direct experimental measurements for MMM obtained with FS nanoparticles showed that ωi,F,pure ≪ ωi,P,pure; thus for the present purposes we will assume that ωi,F ≪ ωi,P and to all practical purposes eq 5 thus becomes

ωi ,M ≅ (1 − wF)ωi ,P

(6)

The value of ωi,P can be predicted through the NELF model once the density of the unpenetrated polymer phase in the composite material, ρ0P, is known. Diffusivity. The presence of the filler influences the infinite dilution diffusion coefficient of the mixed matrix D0i,M in two ways: (i) the particles are impermeable and act as obstacles on the path of the gas molecules through the membrane; this factor reduces diffusivity by increasing the tortuosity of the diffusive path; (ii) the nanofiller induces a higher fractional free

(3) E



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Table 2. NELF Model Parameters of PIM-1 Containing Various Amounts of FS, and of the Penetrants Considered matrix PIM-1 PIM-1 + 13 PIM-1 + 24 PIM-1 + 33 PIM-1 + 39 penetrant He H2 N2 O2 CH4 CO2

wt wt wt wt

% % % %

FS FS FS FS

ρ* (kg/L)

T* (K)

p* (MPa)

ρ0P (g/cm3)

1.438

872

523

1.124 0.991 0.994 0.942 0.866 0.804

(6.7% vol) (13% vol) (19.1% vol) (23.5% vol) 0.150 0.08 0.94 1.29 0.500 1.515

9.3 46 145 170 215 300

(7)

where τ can be evaluated from the Maxwell model35 actually derived for spherical particles: τ=1+

ΦF 2

(8)

A semi-empirical law, based on the free volume theory, is usually considered for the infinite dilution diffusion coefficient in the polymeric phase, D0i,P, as a function of the FFV of the same phase: (9)

⎛ Bi ⎞ 1 ⎟ exp⎜Ai − τ FFV 0P ⎠ ⎝

Di ,M =

Di0,P

=

⎡ ⎛ ⎞⎤ 1 1 1 ⎟⎥ − exp⎢Bi ⎜⎜ ⎢⎣ ⎝ FFV 0P,pure τ FFV 0P ⎟⎠⎥⎦

ΔCi ,M Δpi

(13)

Pi ,M ΔCi ,M Δpi

(14)

For all the gases considered in the present work, except CO2, the dependence of transport parameters on pressure is negligible in the range between zero and the upstream pressure inspected (4.4 atm), and the swelling, as well as the amount of penetrants absorbed in the conditions inspected, is rather small so that FFV0P,pure ≈ FFVP,pure, FFV0P ≈ FFVP. Therefore it is useful to calculate the permeabilities at infinite dilution, P0i,M and P0i,P, by applying eqs 6 and 10 into eq 14, since they are practically equal to Pi,M and Pi,p, respectively, for all penetrants inspected, except CO2. One thus obtains

(10)

which relates the apparent infinite dilution diffusion coefficient in the mixed matrix to the FFV of the unpenetrated polymeric phase. Considering experimental data of diffusivity or permeability of the gas for at least two values of FS loadings, we can obtain the parameters Ai and Bi, and eq 10 can then be used to predict D0i,M for any other value of filler loading. On the other hand, the diffusivity enhancement obtainable after the addition of fumed silica nanoparticles can be estimated with one single parameter, Bi, according to the following relationship: Di0,M

(12)

and therefore the average effective diffusivity is obtained as

where Ai and Bi are temperature and penetrant dependent parameters, and FFV0P is relative to the unpenetrated polymer phase of the MMM. Considering the effect of the fractional free volume shown in eq 9, eq 7 becomes Di0,M =

Pi DS = αDαS = i i Pj Dj Sj

Si ,ave =

Bi FFV 0P

1.034 1.065 1.096 1.118

where Pi is the permeability of the ith penetrant, which can be calculated from diffusivity, Di, and solubility coefficient, Si, on the basis of the solution-diffusion model and of Fick’s law for the diffusive mass flux. In the case of interest, gas transport data are already available,10 for upstream pressures equal to 4.4 atm and downstream pressure equal to 1 atm. Correspondingly the average solubility coefficient is

36

ln(Di0,P) = Ai −

0.2878 0.2860 0.3231 0.3774 0.4224

Selectivity. The selectivity αij between two penetrants i and j is a measure of the gas separation performance of the membrane. If the downstream pressure is significantly lower than the upstream value, the selectivity is expressed as αi , j =

1 = Di0,P τ

1

4 37 160 280 250 630

volume in the polymeric matrix, and this effect enhances diffusivity. According to the first effect, D0i,M can be related to the corresponding value of the infinite dilution diffusivity in the polymer phase, D0i,P, using a tortuosity factor τ, as usual: Di0,M

τ

FFVP0

Pi ,M = =

(11)

Bi can be linked to the molecular size of the penetrants,37 as it will also be confirmed in the following.

Bi ⎞ ΔCi ,M 1 ⎛ ⎟ exp⎜Ai − τ FFV 0P ⎠ Δpi ⎝ ⎛ Bi ⎞ (1 − wF)ρP Δωi ,P 1 ⎟ exp⎜Ai − τ Δpi FFV 0P ⎠ ⎝

(15)

while the permeability variation after FS addition is a function of only one adjustable parameter: F


Industrial & Engineering Chemistry Research Pi ,M Pi ,P

=

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⎡ ⎛ ⎞⎤ 1 1 1 ⎟⎥ (1 − wF)ρP Δωi ,P − exp⎢Bi ⎜⎜ ⎢⎣ ⎝ FFV 0P,pure τ FFV 0P ⎟⎠⎥⎦ ρP,pure Δωi ,P,pure (16)

where the values of ρP come from eqs 1 and 4, and the values of Δωi,P and Δωi,P,pure from application of the NELF model with the values of ρP and ρP,pure which hold in the pressure range considered. The values calculated with eqs 15 and 16 can thus be directly compared to the experimental permeability data for the light gases. The negligibility of pressure-dependence of the transport parameters for all penetrants, apart from CO2, is important because it enables us to use directly the correlation represented by eqs 9−11 with the values of unpenetrated polymer fractional free volume in the mixed matrix state, FFV0P; that value can be obtained from the unpenetrated composite density according to eqs 10 and 11. On the contrary, for CO2 the above approximation may not be acceptable, since CO2 is a highly soluble penetrant and at 4.4 atm may have affected nonnegligibly the fractional free volume of the polymer matrix.

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RESULTS AND DISCUSSION Lattice Fluid Parameters for PIM-1. For both gases and polymer the NELF model requires the knowledge of the pure component LF EoS parameters, which must be obtained from equilibrium data. For the penetrants, such values are easily available and are listed in Table 2. For the polymer the LF parameters must be retrieved from pressure−volume−temperature data above Tg, that are not available in the case of PIM-1, due to the high value of Tg.38 As in other similar cases concerning high Tg polymers, such as PTMSP and polynorbornene,12,39 the parameters were obtained by fitting infinite dilution solubility data available for several gases (N2, O2, CH4, CO2, Ar, H2, He, Xe) at 30 °C from the literature40 to the NELF expression for the infinite dilution solubility coefficient (equation XIII in Table 1), and considering the density value for PIM-1 equal to 1.124 g/cm3,24 using kij = 0. The set of parameters thus obtained, reported in Table 2, allows an accurate representation of the experimental S0 data,40 as it is apparent from Figure 3a, where the error bars represent the deviation on the S0 value obtained by the NELF model as a consequence of an error of ±5% in the polymer density. The LF parameters thus obtained lead also to a good representation of the CH4 solubility isotherm in PIM-1 reported by Larsen and coauthors,26 as it can be seen in Figure 3b. In the solubility calculations, we considered the typical value of density of 1.124 g/cm3, in the absence of density data directly measured in ref 26. Solubility Isotherms in MMM. The solubility isotherms for H2, He, N2, O2, CO2, and CH4 in pure PIM-1 and in its mixed matrices are shown in Figure 4, per unit mass of polymer phase and per unit mass of total solid, respectively. The density of the pure PIM-1 measured by Ahn et al.10 is equal to 0.991 g/ cm3, and this was the value used for NELF calculations of the solubility isotherms. The density of the MMM samples was estimated as explained above and are even listed in Table 2, together with the LF parameters for the pure gases. For each penetrant we used the values of kij = 0 and ksw = 0, also for CO2, for which it has been shown that no appreciable swelling is seen up to a pressure of about 7 bar.24 It must be said that, before this work was completed, experimental solubility isotherms of CH4, N2, O2, CO2 solubility in PIM-1 at room

Figure 3. (a) Experimental40 and NELF-predicted values of S0 at 30 °C of various gases (k12 = 0, ksw = 0, LF parameters from Table 1, density of PIM-1 = 1.124 g/cm3); error bars represent the effect of a ±5% density variation on S0, and the solid line is an exponential interpolation of experimental data. (b) Solubility isotherm of CH4 in PIM-1 at 20 °C: experimental data26 and NELF predictions (kij = 0; ksw = 0; density for PIM-1 = 1.124 g/cm3).

temperature were published, which were used to retrieve the binary parameter kij for each gas−polymer couple. However, such parameters are normally below ±0.05,38 and they refer to PIM-1 samples with rather different density than the one inspected here (1.1 g/cm3 versus 0.991 g/cm3 for the sample studied in this work). Moreover, it was verified that using such parameters induces a small variation of the calculated solubility, diffusivity, and of the corresponding diffusivity-selectivity and solubility-selectivity values, and no variation of the trends with silica content (calculations not shown here for the sake of conciseness). Therefore the values of kij were kept equal to zero in all calculations. It can be seen in all cases that the amount of penetrant absorbed by the polymeric phase, per unit mass of polymer, increases with increasing filler content (Figures 4a, c, e, g, i, and k), with the exception of the MMM with the smaller loading of FS. Reporting the absorbed mass per unit mass of composite membrane, on the other hand, can change that behavior, based on the different amount of silica present and on the effect that the different polymeric density has on the solubility of the specific penetrants (Figures 4b, d, f, h, j, and l). In any case, the G



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Figure 4. continued

H



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Figure 4. Predicted solubility isotherms at 25 °C in the polymeric phase of PIM-1+FS composite matrices (a,c,e,g,i,k) as well as in the composite membrane (b,d,f,h,j,l); calculations made with the NELF model, using the estimated values of the polymer phase density for various penetrants.

Permeability and Diffusivity. The values of permeability are available from the work by Ahn et al.10 for different FS loading in PIM-1, for five gases (He, H2, N2, O2, CO2) at 25 °C,

effect of FS addition on the solubility is rather limited, as observed for this type of MMM. I



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Figure 5. Permeability of various gases in mixed matrices based on PIM-1 and FS, estimated with eq 15 (continuous lines) and experimental values10,25 versus reciprocal fractional free volume 1/FFV0P estimated with eq 1 and eq 2. The crosses are model-calculated values obtained in specific filler mass fraction values: 0, 13%, 24%, 33%, 39%.

and Bi will be simply called A and B for the sake of brevity. As it can be seen in Figure 5, the model allows an accurate representation of the behavior of CH4 permeability, that was not investigated experimentally, on the bases of an approximate value of B. The predictions of the Maxwell model for the same

with an upstream absolute pressure of 4.4. atm and a downstream absolute pressure of 1 atm. The values of Bi were fitted, for each penetrant, on the experimental permeability values of PM/PP and the values of Ai adjusted in order to fit the pure polymer permeability. In the following, Ai J



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Figure 6. Diffusivity (D0M) of various gases in mixed matrices based on PIM-1 and FS, estimated with eq 14 (continuous lines) versus reciprocal fractional free volume 1/FFV0P estimated with eq 1 and 2. The crosses are model-calculated values obtained in specific filler mass fraction values: 0, 13%, 24%, 33%, 39%.

in this sample the loading is too small to induce an appreciable variation of the free volume. On the basis of eq 10 and on the values of parameters A and B adjusted on experimental permeability data, the model allows also to predict the diffusivity behavior of the same gases in the various mixed matrix membranes, D0M (Figure 6): the values of infinite dilution diffusion coefficient are reported as a function of the reciprocal fractional free volume 1/FFV0P for the different

gas−polymer couples are also reported as dashed lines, for the sake of comparison. From Figure 5 one also realizes that the correlation of eq 9 is much less satisfactory at the lower loadings of fumed silica inspected: such a matrix, however, does not follow the experimental trend of the higher loadings, as indicated also by the density values obtained by Ahn et al.,10 probably because K



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⎛ ∂ ln pi ⎞ ⎛ 1 ∂μi ⎞ ⎛ ∂ ln ai ⎞ Di = Li⎜ ⎟ ⎟ = Li⎜ ⎟ ≅ Li⎜ ⎝ RT ∂ ln ωi ⎠ ⎝ ∂ ln ωi ⎠ ⎝ ∂ ln ωi ⎠

gases inspected experimentally, as well as for CH4, whose transport properties were estimated a priori. From the correlations presented in Figure 5, we can obtain for each gas the value of the coefficient B, which correlates the free volume of the polymeric phase to the gas diffusivity at infinite dilution, according to eqs 9 and 10. It is seen that the value of B is an increasing function of the penetrants size or kinetic diameter, as the diffusion of larger penetrants is affected by a polymer free volume variation more strongly than the smaller penetrants. In particular, in Figure 7 we report the

The diffusivity Di is thus given by the product of a purely kinetic factor, the mobility Li, and a purely thermodynamic factor (∂ ln pi/∂ ln ωi). The latter is related directly to the shape of the solubility isotherms and can be immediately calculated for all the MMM inspected. The addition of FS content leads to an increase in fractional free volume, which will affect both the mobility and solubility isotherm, through polymer density changes. However, a greater effect is expected on mobility rather than on the thermodynamic factor of diffusivity. Indeed, mobility increases appreciably with FFV and decreases with increasing penetrant size. On the other hand, the solubility isotherms calculated with the NELF model indicate that the thermodynamic factor is constant with fumed silica loading, and very close to unity, for all the gases here inspected, except CO2. Therefore, for H2, He, N2, and O2 the diffusion coefficient Di is practically coincident with the mobility Li, and it can be easily related to the FFV as well as to measures of the penetrants size, as illustrated in Figure 7. In contrast, for CO2 the thermodynamic factor is higher than 1 and decreases with increasing FS content, going from 1.9, for pure PIM-1, to 1.35 for the mixed matrix with the highest FS loading. The diffusivity of CO2 is thus also affected by the thermodynamic factor, whose value decreases with the FS content. Therefore, in the case of CO2 the increase of FS content and the corresponding increase of FFV lead to a mobility increase, but reduce the thermodynamic factor. Such effect may explain the smaller enhancement observed for CO2 diffusivity with FFV, which results in a small value of B, with respect to what expected based on on the bases of the van der Waals volume. Considering this effect, and using the mobility instead of the diffusivity in eq 9, the value of B for CO2 would change from 0.62 to 0.91, that is closer to the value reported in the literature,37 represented by an open circle in Figure 7. Effect of FS Addition on Selectivity. The ideal selectivity values were already provided by the experimental analysis of Ahn et al.,10 without any possibility to appreciate what are the actual solubility and diffusivity contributions, and thus, simply based on those experimental data, the effects of filler addition on the separation mechanism remain unclear. On the other hand, with our analysis based on the NELF model we are also able to estimate the separate contributions of solubilityselectivity and diffusivity-selectivity, and to appreciate their dependence on fumed silica content in the polymer. Such values have been calculated and are reported in Figures 8 for the mixtures CO2/N2, CO2/H2, CO2/CH4, and O2/N2. Some interesting results are obtained: for CO2/N2 separation, both the solubility and diffusivity values are higher for CO2 than for N2, as expected. It is seen that the total selectivity decreases with the addition of fumed silica: from Figure 8 it is evident that such a behavior is caused by the diffusivity selectivity factor as the solubility-selectivity increases, albeit slightly, with increasing FS content. This result indicates that the same amount of fumed silica has opposite effects on the solubility and diffusion coefficients. Clearly, the addition of fumed silica, and the corresponding increase of free volume, enhances more strongly the diffusivity of nitrogen than that of CO2, as it could also be predicted from their different B values. On the other hand, it seems that the solubility of CO2 is more effectively augmented by the increase of free volume provided by fumed silica.

Figure 7. Values of B in free-volume theory: correlation for PIM-1 and other glassy polymers,37 as a function of the penetrant van der Waals volume.

values of B as a function of penetrant van der Waals volume, observing a nice, monotonically increasing correlation. As explained in the previous sections, that behavior is typical of mixed matrices based on fumed silica, which normally show improved selectivity with respect to large penetrants, due to the effect of fumed silica in the enhancement of large free-volume domains. Similar results were obtained for the B values estimated on mixed matrices formed by FS and PTMSP.12,13 For the sake of comparison, in Figure 7 we also reported the literature values of coefficient B,37 obtained from diffusion data for the same penetrants in a series of glassy polymers with FFV values between 0.12 and 0.22. For CO2, the value of B thus obtained is lower than what expected based on its van der Waals volume, apparently indicating that the diffusion coefficient of CO2 in PIM-1 is less dependent on free-volume variations than for the other gases inspected. This point deserves a further discussion. By recalling that the actual driving force for the diffusive flux of penetrant i, ji, is the chemical potential gradient ∇μi, one has ji = −ρ

Li L ∂μi ωi∇μi = −ρ i ∇ωi RT RT ∂ ln ωi

(17)

where the coefficient Li is a purely kinetic factor named mobility. A comparison between Fick’s law ji = −ρDi∇ωi

(19)

(18)

and eq 17 leads to the following expression for diffusivity: L



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Figure 8. Values of ideal selectivity, as well as solubility-selectivity and diffusivity-selectivity contributions in PIM-1 based mixed matrices. Experimental data from ref 10.

selectivity as a function of fumed silica content. In this case, also solubility-selectivity decreases with increasing the filler content, although to a lower extent. Of course the above considerations hold true for the ideal selectivity values, based on pure gas data, and may vary when mixed gases are considered. In Figure 9 we report a trade-off plot for separation of mixtures containing CO2, in which it can be seen what is the effect of adding fumed silica to PIM-1: in general it increases significantly the permeability, but often decreases the selectivity (with the exception of CO2/H2 mixtures).

The CO2/H2 selectivity is slightly affected by the addition of FS and this is attributed to the equivalence of two opposite behaviors of the solubility-selectivity and diffusivity-selectivity (as in the case of CO2/N2 mixtures). However, it seems that the effect of FS addition does not alter significantly the selectivity for this gas mixture because both penetrants show similar values of B and also a similar dependence of solubility on FFV variations. The behavior is interestingly different for CO 2 /CH 4 mixtures: the addition of FS up to about 12% leaves the ideal selectivity practically unvaried, while a drop of selectivity is observed at higher loadings. Again, this behavior is dictated by the diffusivity contribution: indeed the value of B for CH4 is higher than the corresponding value for CO2, therefore an increase in free volume enhances more the diffusion of methane, under pure gas conditions. The solubility-selectivity shows the opposite trend as solubility is affected by a freevolume increase more for CO2 than for CH4, but this effect is overall smaller than that of diffusivity. Similarly, in O2/N2 separation, the higher value of B observed for the larger penetrant (N2) is the cause for the observed decrease of diffusivity-selectivity and of the overall

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CONCLUSIONS Solubility and transport behavior in PIM-1 and PIM-1 based MMM have been modeled on the basis of the NELF model. The approach considered can be used to extend and improve the information available on the gas transport and on the separation behavior of mixed matrix membranes formed by glassy polymers and fumed silica, starting from a reduced amount of experimental data. The behavior of PIM-1/FS mixed matrix membranes was studied, considering the permeability data measured in the literature, coupled with the corresponding M



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Figure 9. Trade-off curve for the separation of different gaseous mixtures in mixed matrix membranes based on PIM-1 and FS.

experimental information reported for the density of the composite matrices. With the aid of simple assumptions, the model allows estimation of the density of the polymer phase, as well as of its fractional free volume. That value is used to (i) predict a priori the solubility of the gases in the various mixed matrices considered; (ii) model permeability and diffusivity data, with one adjustable parameter only, B in eq 9, as a function of the FS content, with a free-volume theory-based approach. The values of B increase regularly with the van der Waals volume of the penetrants, with the exception of CO2, and an explanation is offered for such behavior. The approach also allows an estimation of the ideal diffusivity-selectivity and solubility-selectivity contributions, and a proper explanation of the observed behaviors for the ideal selectivity. In particular, in the cases inspected the selectivity behavior is dominated by the diffusivity contribution, which can be predicted based on the values of B, which normally increase with increasing penetrant size. Coherently, the selectivity of large versus small gas molecules normally increases with the addition of fumed silica, with an exception represented by CO2 containing mixtures, as CO2 shows a small value of B. Therefore, the addition of FS normally does not increase the diffusivity of CO2 as much as that of the copermeating gas, and is thus not very suitable to enhance the CO2 selectivity of the membrane. The present study shows that a model based on a single parameter, that is, the polymer density (or fractional free volume) is suitable to provide explanation for the separation behavior of mixed matrices based on PIM-1 and to extrapolate the knowledge available from experimental data beyond the limited range inspected.

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AUTHOR INFORMATION

Article

LIST OF SYMBOLS ρF,pure = pure filler density 0 ρM , ρP0, ρP = density of unpenetrated mixed matrix membrane, density of the unpenetrated polymeric phase (of the mixed matrix membrane), density of the polymeric phase (of the mixed matrix membrane) ρW P,pure = van der Waals density of the pure polymer V0P = unpenetrated specific volume of the polymeric phase of the mixed matrix membrane VW P,pure = van der Waals volume of the pure polymer FFV0P = fractional free volume of the unpenetrated polymeric phase of the mixed matrix membrane FFV0P,pure = fractional free volume of the unpenetrated pure polymer ksw = swelling coefficient wF = mass fraction of filler in the mixed matrix membrane ΦF = volume fraction of filler in the mixed matrix membrane ωi,M = mass fraction of penetrant i absorbed in the mixed matrix membrane ωi,F = mass fraction of penetrant i absorbed in the filler phase of the mixed matrix membrane ωi,P = mass fraction of penetrant i absorbed in the polymeric phase of the mixed matrix membrane Ci,M = molar concentration of penetrant i absorbed in the mixed matrix membrane mi,F = mass of penetrant i absorbed in the filler phase of the mixed matrix membrane mi,P = mass of penetrant i absorbed in the polymeric phase of the mixed matrix membrane mi,M = mass of penetrant i absorbed in the mixed matrix membrane ωi,F,pure = mass fraction of penetrant i absorbed in the pure filler ωi,P,pure = mass fraction of penetrant i absorbed in the pure polymer D0i,M = infinite dilution diffusivity of penetrant i in the mixed matrix membrane D0i,P = infinite dilution diffusivity of penetrant i in the polymeric phase of the mixed matrix membrane Di,M = diffusivity of penetrant i in the mixed matrix membrane τ = tortuosity of the mixed matrix membrane Ai, Bi = adjustable coefficients of eq 10, relative to penetrant i Si = Solubility coefficient of penetrant i Si,ave = average solubility coefficient of penetrant i αi,j or α = selectivity of component i over component j αD = diffusivity-selectivity of component i over component j αS = solubility-selectivity of component i over component j Pi = permeability of penetrant i Di = diffusivity of penetrant i Pi,M = permeability of component i in the mixed matrix membrane Pi,P = permeability of component i in the pure polymer ai = activity of penetrant i Li, L = mobility of penetrant i, mobility REFERENCES

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Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. N



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Gas Solubility, Diffusivity, Permeability, and Selectivity in Mixed Matrix

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