Solubility and Diffusivity of Gases in Mixed Matrix Membranes

5214

Ind. Eng. Chem. Res. 2008, 47, 5214–5226

Solubility and Diffusivity of Gases in Mixed Matrix Membranes Containing Hydrophobic Fumed Silica: Correlations and Predictions Based on the NELF Model Maria Grazia De Angelis† and Giulio C. Sarti*,‡ Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali (DICMA), Alma Mater Studiorum-UniVersita` di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy

The addition of fumed silica particles to glassy polymers such as PTMSP and Teflon AF 2400 results in an enhancement of gas diffusivity and permeability that is not captured by traditional models for composites because both volume and gas solubility of the composite material are underestimated by the additive rule based on pure component properties. Such effects have been attributed to the creation of additional free volume due to the insertion of filler in the polymeric matrix. In this work, we use the NELF model to relate gas solubility in the composite material to the variations induced by the addition of filler on polymer density and swellability. From these values, the fractional free volume (FFV) increase produced in the polymeric phase by filler incorporation may be evaluated and, in turn, also the diffusivity in the composite material can be calculated as a function of filler content on the basis of its relation to FFV. The gas diffusivity data for n-C4H10 in AF 2400 and AF 2400-based mixed matrices, as well as those relative to N2, CH4, C2H6, C3H8, and n-C4H10 in PTMSP, follow closely the correlation found, which contains only one adjustable parameter for each penetrant-polymer couple. Introduction Polymers are currently the election materials for gas separation membranes because their processability enables economic production of modules for practical large-scale separations. However, the segmental flexibility of polymers limits their discriminating ability compared to more selective but unprocessable rigid molecular-sieving media. In particular, polymeric membranes show a tradeoff between permeability and selectivity of gas mixtures, whereas some inorganic materials display properties that are above this tradeoff curve.1 This observation led to the idea that a significant improvement over current technology may be achieved by dispersing selective inorganic materials such as zeolites or carbon molecular sieves in polymers that exhibit favorable performance on the tradeoff curve. It has also been shown that combining nonporous, nonselective nanoparticles such as silica spheres to polymers affects the transport properties of polymeric membranes positively, by generally increasing the permeability, though having variable effects on selectivity. Composite materials for application in gas separation membranes are generally known under the name of mixed matrix membranes (MMM). The dispersed phase may affect the surrounding polymer in various ways, leading to different and sometimes opposite effects: filler incorporation may produce additional void volume at the polymer-filler interface, lowering the selectivity, or may enhance the rigidity of the polymer matrix. On the other hand, the good permselectivity properties of inorganic fillers, such as zeolites, may be impaired by wetting and obstruction of the pores at the filler surface.2 Preparation conditions such as solvent evaporation procedures, thermal effects, and the resulting stresses at the polymer/dispersed phase interface cause a complex and only partially understood series of simultaneous effects.2,3 * To whom correspondence should be addressed. Tel.: +39 051 2090251. Fax: +39 051 2090247. E-mail: [email protected]. † Present address: National Technical University of Athens. ‡ Present address: North Carolina State University.

The polymers elected as matrices for MMM are generally chosen among glassy materials, whose performances are close to the tradeoff curve in the selectivity/permeability plot for the mixtures of interest; typical examples are represented, for example, by polyimides, poly(ether imide) (PEI),3 polyvinyl acetate (PVAc),2 polyethersulfone (PES),4,5 polysulfone (PSf),6–9 amorphous Teflon (Teflon AF),10 poly(1-trimethylsilyl-1-propyne) (PTMSP),11 and poly(4-methyl-2-pentyne) (PMP).12,13 In this work, we will focus on the experimental data obtained by Merkel and coauthors on mixed matrices formed by amorphous Teflon AF 2400 or PTMSP and nanoscale, nonporous, hydrophobic fumed silica (FS) particles.10,11 In such works, gas permeability data are provided for several penetrants together with experimental information on their solubility in the mixed matrices with different FS content. That combined information is relevant for a proper understanding of the main physical mechanisms involved and for a reliable model development and test. The dense filler used by Merkel et al.10–13 can be considered impermeable to gases, thereby simplifying the modeling of permeation and diffusion behavior. The results obtained in ref 10 show that fumed silica loaded into a size-selective polymeric membrane made of Teflon AF 2400 causes an increase in gas permeability, which is more pronounced for large penetrants than for small molecules, and this has the overall effect of decreasing the size selectivity of the membrane film. In particular, Merkel et al. studied the mixture formed by n-butane (2%) and nitrogen, observing that the selectivity of loaded matrices is reversed with respect to the selectivity of pure AF 2400 and pointing out that the solubility of n-butane in mixed matrices is underestimated by an additive model based on the gas solubility in the pure polymer and pure filler alone. When fumed silica is added to PMP,12,13 which is more permeable to condensable vapors than to light gases, an increase of both gas permeability and permselectivity is obtained. The sorption isotherms of pure nitrogen, methane, and n-butane remain almost unaffected by the addition of filler, whereas, on

10.1021/ie0714910 CCC: $40.75  2008 American Chemical Society Published on Web 03/01/2008

Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5215

the basis of the additive model using pure component properties, one would expect a lower solubility in the mixed matrices with respect to pure PMP. A similar study was performed on MMMs obtained by loading FS to another stiff, high free volume glassy polymer such as PTMSP: in contrast to the case of PMP, the addition of silica particles to PTMSP enhances the permeability of light gases more than that of larger penetrants, thus leading to a reduction of vapor/permanent gas selectivity of the membrane.11 The solubility isotherms of H2, N2, CH4, C2H6, C3H8, and n-C4H10 were also measured in mixed matrices of PTMSP containing 30, 40, and 50 wt % of FS, showing an equal or slightly lower value of the mass uptake per mass of solid matrix with respect to the case of pure PTMSP.11 On the contrary, the additive model based on pure component properties would predict a mass uptake significantly lower for the MMM, at least in the case of n-butane for which specific sorption data were measured on the pure filler. The diffusion coefficients of gases, on the other hand, are significantly increased by addition of the FS filler, thus further justifying the increase in permeability of the MMM. The gas and vapor sorption behavior in the mixed matrices obtained by dispersing FS in Teflon AF 2400 and PTMSP cannot be justified on the basis of pure component properties but can actually be rationalized by considering that FS, although sorbing a much smaller amount of gases and vapors with respect to the pure polymers, modifies the polymer chain packing and increases the matrix free volume, thus increasing the polymer sorption capacity and compensating for the lower sorption onto FS particles. The same increase in polymer free volume enhances also the diffusion process and overcomes the effect of tortuosity introduced by filler addition. The qualitative interpretation of the transport behavior observed in the mixed matrices mentioned above has also been supported by PALS characterization of the matrix free volume before and after the addition of fumed silica particles.10–13 As it is well-known, in these high free volume polymers, two distinct distributions of free volume elements are present, characterized by different average dimensions. In Teflon AF 2400, PTMSP, and PMP it was found that the addition of FS increases the number of large free volume elements, whereas it does not modify the population of the small free volume elements.10–13 The variation of the distribution of larger free volume elements is rather minute; however, it is believed to be the main cause of the increased permeability and diffusivity of the composite matrices. The increased free volume is also consistent, of course, with the gas solubility behavior. Beyond PALS, another indication of the increase in free volume is given by the density measurement of the mixed matrices obtained by loading FS in PMP; Merkel et al.12 clearly observed a lower value of density in the composite material with respect to the value calculated by volume additivity using the pure component values of densities, even though the uncertainty in the density measurements did not enable them to obtain reliable estimates for the variations in the fractional free volume. In the present work, we will consider first the solubility isotherm of n-C4H10, considered as a reference-test penetrant, in the mixed matrices examined in refs 10 and 11, and will isolate the contribution to the MMM solubility offered by the polymer phase alone. From that, use of the NELF model will allow us to calculate the average density of the unpenetrated polymer phase of each MMM, obtaining its fractional free volume and its dependence on the FS loading, for both Teflon AF 2400-based and PTMSP-based mixed matrices. Such values will then be used to calculate in a predictive way the solubility

isotherms of all of the other penetrants considered, and the results will be compared with the data reported in the experimentation performed by Merkel et al.10,11 The infinite dilution diffusivity data will then be considered for the same matrices and, following a rather established procedure based on the free volume theory, a simple and effective correlation will be obtained between the diffusion coefficients and the fractional free volume calculated from the analysis of the solubility isotherms of the reference penetrant n-C4H10. The results indicate an interesting procedure to follow to obtain a significant reduction of the experimental work required to characterize solubility, diffusivity, and permeability of different solutes in MMMs which contain an impermeable filler, as FS, without the need of extensive experimental tests for each penetrant of interest. Theoretical Background. In membrane separations, the selectivity Rij of component i versus component j is calculated as the species permeability ratio: Ri,j )

Pi Di,ave Si,ave ) RDRS ) Pj Dj,ave Sj,ave

(1)

Equation 1 is obtained when the solution diffusion model holds for the permeation in the polymer, and Fick’s law represents the diffusive mass flux; selectivity is thus decomposed into its solubility and diffusivity contributions, RS and RD, respectively. For glassy polymers, the diffusivity selectivity is generally the predominating term because gas diffusion coefficients vary over wider ranges than gas solubility coefficients.14 However, for certain high free volume polymers, such as the polyacetylenes (PTMSP, PMP), due to extremely high free volume values, the size sieving ability is very poor and, as a consequence, the diffusivity selectivity value is low. In such cases, the solubility selectivity becomes important and the larger molecular weight penetrants such as hydrocarbons, which are more condensable and more soluble in the organic polymeric matrix, are more permeable than the low molecular weight permanent gases. In modeling the behavior of MMMs, major attention has been devoted to simulations of the permeability and permselectivity of gases, with rather little attention to separate the two contributions of solubility and diffusivity.15 To our knowledge, only models for the prediction or correlation of permeability have been presented in the literature; in particular, the Maxwell model, originally developed to describe the permittivity of a dielectric, has been taken as a guideline by various authors because of the analogy between the constitutive equation governing electrical potential and the flux through membranes.16 For spherical, impermeable particles at low filler loading, the Maxwell expression reduces to the following,

(

Pi ) Pi,P

1 - ΦF 1 + ΦF ⁄ 2

)

(2)

in which ΦF is the filler volume fraction, Pi is the permeability of the composite to species i, and Pi,P is the permeability of the pure polymer. According to eq 2, the permeability of the filled polymer is always smaller than that of the pure polymer and decreases with increasing filler concentration. This reduction in permeability is related to an increase in the tortuosity of the diffusion path, as well as to a decrease in penetrant solubility caused by the replacement of polymer, which can sorb penetrant, with nonsorbing filler particles. Despite its simplicity, the Maxwell model captures qualitatively the experimental observa-

5216 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008

tion made for several MMMs that nonporous filler particles reduce permeability.17,18 Different modifications of this model have been proposed to account for the various effects that may originate due to interactions between polymer and fillers like zeolites. Other models proposed over the years are the Bruggeman equation, applied by Bouma et al. to the modeling of a system comprised of a rubbery vinylidene fluoride-co-hexafluoropropene and a liquid crystalline mixture.19 Bruggeman used the effective medium theory approach, which is particularly appropriate when there is a small difference between the permeabilities of the two phases.20 Gonzo et al.21 proposed an improved form of Maxwell’s equation, based on the hard-sphere model fluid proposed by Chiew and Glandt,22 using parameters such as the reduced permeation polarizability and the volume fraction of the filler, even at relatively high ΦF values. This model takes into account second-order interactions between embedded ensembles of particles that are not considered in the original Maxwell model. Following Chiew and Glandt’s procedure, the permeability of a given penetrant in the composite medium results in an explicit function of the volume fraction of the dispersed phase and of the ratio between the penetrant permeabilities in each component of the composite. Several studies have shown that the selectivity Rij, for example of O2 versus N2, is satisfactorily predicted by the model, whereas the absolute permeability of each single gas in the composite materials is not accurately calculated by the models, as it is for instance in the case of PVAc-Zeolite 5A, especially at high filler loadings.2 Such models, however, are not applicable to the mixed matrices formed by impermeable fumed silica and Teflon AF 2400, PTMSP or PMP, for which the permeability toward gases increases after filler addition, reasonably as a result of the creation of additional micro- and nanocavities with filler insertion that remain frozen into the glassy structure. In contrast, for impermeable fillers all of the existing models for permeability predict a decreasing permeability value with increasing filler content, thus leaving the demand for a suitable model formulation. In the present work, a different procedure is followed, which allows us to estimate the free volume variation induced by filler addition, on the bases of few limited gas solubility data in the mixed matrices and making use of the NELF model for solubility in glassy polymers. Such information on actual FFV or density of the polymer species is then used to: (i) predict the solubility of other gases in the same mixed matrices on the basis of the NELF model; (ii) correlate diffusivity and permeability data of any other penetrant in the mixed matrices, through a simple relationship based on the free volume theory and on the Maxwell model for the tortuosity. For the sake of clarity and for later use, the relevant features of the NELF model will be briefly recalled hereafter. The NELF model suitably extends the lattice fluid (LF) equation of state developed by Sanchez and LaCombe23–25 for amorphous equilibrium phases to the nonequilibrium state typical of glassy polymers and uses the same characteristic parameters (P*, T*, and F*) to represent the pure components properties, and the same mixing rules to estimate the mixture behavior as the LF theory. For polymers, the pure components’ characteristic parameters are normally calculated by fitting the LF equation of state to pressure-volume-temperature (PVT) data above TG, whereas for

the penetrants either PVT data or vaporsliquid equilibrium data may be used. In the LF model, the following relationship holds true for the number ri0 of lattice sites occupied by a molecule of species i in its pure phase,23 ri0 )

P/i Mi RT/i F/i

)

Mi

(3)

F/i V/i

where Mi is the molecular weight of component i and νi* is the volume occupied by a mole of lattice sites of pure substance. The parameter ri0 is usually set to infinity for the polymer species. The mixing rules adopted to estimate the mixture characteristic parameters contain only one adjustable binary parameter, Ψ, appearing in the expression of the characteristic pressure for the mixture, P*, which affects the binary characteristic pressure P12*, representative of the energetic interactions between gas and polymer molecules, labeled by subscripts 1 and 2, respectively.25 P/12 ) Ψ√P/11P/22

(4)

In the absence of specific binary data, a reasonable first-order approximation of the binary parameter is obtained by considering Ψ ) 1. The extension to nonequilibrium states is given by the nonequilibrium thermodynamics of glassy polymers (NET-GP) approach, in which the state of a glassy polymerspenetrant mixture can be described by the usual variables, for example, temperature, composition, and pressure, plus the polymer density F2, which accounts for the departure from equilibrium frozen into the glass.26,27 The model also assumes that F2 is an internal state variable for the system, so that the specific thermodynamic relations for systems endowed with an internal state variable can be applied.28 The main results obtained from the NET-GP theory are as follows: (i) The specific Helmholtz free energy of the nonequilibrium glassy phase is a cylinder in the space of the nonequilibrium states ∑ ≡ (T, p, Ω1, F2), coincident with the equilibrium-specific Helmholtz free energy at the same temperature, the penetrantto-polymer mass ratio Ω1, and the polymer density F2: AˆNE ≡ AˆNE(T, p, Ω1, F2) ≡ AˆEq(T, Ω1, F2)

(5)

(ii) The chemical potential of the penetrant in the glassy phase is calculated per unit mass as, µNE 1 )

( ) ∂G ∂m1

) T,p,m2,F2

( ) ∂aEq ∂F1

(6) T,F2

where aEq is the equilibrium Helmholtz free energy density, and superscript NE labels nonequilibrium conditions. (iii) The phase equilibrium (or, more properly, pseudoequilibrium) between a pure external gas (g) and a solid glassy phase (s) requires, PE Eq(g) (T, p, ΩPE (T, p) µNE(s) 1 1 , F2 ) ) µ1

(7)

where superscript PE labels the pseudoequilibrium conditions asymptotically reached in the glassy phase due to hindered longrange mobility. In the limit of infinite dilution, F2 is equal to the pure polymer density in the glassy phase, F20, because the swelling induced is negligible at such pressures and the penetrant solubility coefficient in the glassy phase at infinite dilution can then be


explicitly predicted by the model as a function of temperature, pressure, pure polymer density, and LF parameters as,29,30

( )

ln(S0) ) ln

{[ ( ) ] ( ) √ } ( )

TSTP + r10 PSTPT

1+

V/1

F20 F/2 1 ln 1 + V/2 F20 F/2

V/1

F20 T/1 2 1 + Ψ P/1P/2 V/2 F/2 T P/1

(8)

where TSTP and PSTP are the standard temperature and pressure, respectively. On the basis of eq 8, the evaluation of S0 requires the values of the independent characteristic parameters r10, T1*, and P1*, for the penetrants, and of the characteristic pressure, density, and volume (P2*, F2*, and V2*) for the polymer, which may be found in specific works and collections.23–25,31,32 Moreover, one needs to know the value of the glassy polymer density F20 at the experimental conditions of interest, that, even at a given temperature and pressure, may vary depending on the particular pretreatment or general history experienced. At higher pressures, for swelling penetrants the value of F2 may change appreciably during sorption, and its value can actually be obtained from little data of polymer dilation because the swelling isotherm during sorption at increasing pressures typically follows a linear behavior, as many experimental works confirm.33,34 Therefore, a swelling coefficient ksw is introduced to evaluate the polymer density at all other pressures.29 F2(p) ) F20(1 - kswp)

(9)

In the absence of the specific dilation data for the system of interest, the parameter ksw can be adjusted on one solubility datum at high pressure, thus providing also a reliable estimate for the swelling behavior of the polymer matrix.29 The phase equilibrium condition, eq 7, thus becomes of the following type: 0 Eq(g) (T, p, ΩPE (T, p) µNE(s) 1 1 , F2 , ksw) ) µ1

(10)

The NELF model has been successfully tested with gas sorption isotherms in various pure glassy polymers and has been successfully applied also to calculate the solubility isotherms in several polymer blends in the glassy state, both for nonswelling and for swelling penetrants.27,35 For the following developments, it is important to estimate the value of fractional free volume (FFV) of the polymer, which is typically calculated as follows, FFV )

V2 - 1.3VW F2W - 1.3F20 ) V2 FW

(11)

2

where V2 ) 1/F20 is the pure polymer specific volume and VW ) 1/F2W is the van der Waals specific volume of the repeating unit of the polymer, which may be calculated with Bondi’s group contribution method and is already available in the literature for the polymers under study.37 Model-Based Correlations and Predictions. For the systems of interest, both solubility and diffusivity increase after addition of the filler, but diffusivity varies much more significantly than gas solubility, leading to appreciable enhancements in gas permeability. In the present approach, we will first analyze the behavior of the solubility isotherm of one penetrant, chosen as a reference, and will use it as input to calculate the actual polymer density or its fractional free volume. The latter values will then be used as input to the NELF model to calculate in a predictive mode the solubility isotherm of other penetrants in the same mixed

matrix. Second, diffusivity will be examined and its values will be correlated to the actual fractional free volume of the polymer in the MMM, obtained from the previous analysis of the solubility behavior. (a) Free Volume and Solubility Estimation for MMMs Based on the NELF Model. Gas solubility in a composite matrix can be represented in general through the number of moles (or mass) of gas per unit volume of solid material as follows, Ci,M )

ni,F + ni,P ) ΦFCi,F + (1 - ΦF)Ci,P V

(12)

where ΦF is the volume fraction of the filler, Ci,M is the gas solubility per unit volume of mixed matrix, Ci,F and Ci,P are the gas solubilities in the filler and in the polymer, per unit volume of filler and per unit volume of polymer, respectively. Alternatively, one can find it more convenient to represent gas solubility in the MMM per unit mass of solid matrix, and thus the following expression would be used instead, ωi,M )

mi,F + mi,P ) wFωi,F + (1 - wF)ωi,P V

(13)

where wF is the mass fraction of the filler in the unpenetrated mixed matrix, ωi,M is the mass of gas per unit mass of mixed matrix, ωi,F and ωi,P are the mass of gas per unit mass of filler and per unit mass of the polymer, respectively. The first-order approximation of eqs 12 and 13 are obtained by considering that Ci,F and Ci,P, as well as ωi,F and ωi,P, are given by the corresponding values in the pure phases, C°i,F and C°i,P, or ω°i,F and ω°i,P, respectively. This first-order additive model is not always appropriate to describe the gas solubility in the mixed matrix material and definitely is not suitable for the mixed matrices under consideration; in addition, it is not always simple to apply because it is not always possible or easy to perform separate sorption measurements on the pure polymer and the pure filler under the same experimental conditions. Because of surface effects and filler surface accessibility, in the MMM one normally has Ci,F e C°i,F, therefore the appreciable underestimation of the solubility isotherm obtained by the first-order additive model resides in the fact that the sorptive capacity of the polymer phase is significantly increased after filler addition, and thus Ci,P > Ci,P0 and ωi,P > ωi,P0 Such a solubility increase in the polymer phase is attributed here to a corresponding decrease in the polymer density or, equivalently, to an increase in the polymer fractional free volume. The latter can be estimated using the NELF model, provided that the actual Ci,P value (or ωi,P value) can be calculated from experimental data. To this aim, for the mixed matrices under examination we have accepted to consider, Ci,F ) C°i,F

(14)

in view of the fact that the filler contribution to the overall MMM solubility is rather small if not negligible at all, so that eqs 12 and 13 become, Ci,M ) ΦFC°i,F + (1 - ΦF)Ci,P

(15)

ωi,M ) wFω°i,F + (1 - wF)ωi,P

(16)

and

In the presence of experimental solubility data for a given penetrant in the composite material, Ci,M, and in the pure filler, Ci,F0, as a function of pressure, we can obtain the solubility isotherm representing Ci,P versus gas pressure for that penetrant.


We note also that it is not infrequent the case in which Ci,0 F is actually negligible so that eqs 15 and 16 reduce to, Ci,M = (1 - ΦF)Ci,P

(17)

ωi,M ) (1 - wF)ωi,P

(18)

Once Ci,P (or ωi,P) is obtained versus pressure, application of the NELF model enables us to derive the values of F20 and ksw, by best fitting the calculated isotherm represented by eq 10 to the experimental solubility in the polymer obtained by eqs 15 or 16. The value of F20 thus derived represents the pure polymer density in MMM and is the same also for all other penetrants, whereas ksw represents the swelling effects induced on the polymeric phase of the MMM by the specific penetrant considered and thus cannot be used to represent quantitatively the swelling effects, if present, of other solutes. The value of F20 thus obtained is then used in eq 11 to calculate the FFV of the unpenetrated mixed matrix. The binary interaction parameter Ψ required for penetrant dissolution will reasonably have the same value in the pure polymer, as well as in the polymer phase of the mixed matrix. (b) Estimation of Diffusivity and Permeability in MMMs. In all of the polymeric matrices here considered, the diffusivity follows an exponential trend with concentration that may be expressed by the following relationship, DM ) DM(0) exp(γC)

(19)

where DM(0) is the infinite dilution apparent diffusion coefficient in the composite material, and γ is a parameter characteristic of the gas-polymer couple, which depends on temperature. For the present analysis, we consider the infinite dilution apparent diffusion coefficients DM(0) only, which depend on the infinite dilution FFV of the material and are not affected by swelling effects possibly occurring at higher penetrant concentrations. As it is frequently the case, we can reasonably assume that the following semiempirical law, based on free volume theory, holds for the diffusivity at infinite dilution in the polymeric phase, DP(0), and the FFV of the same phase,38,39 B ln(Dp(0)) ) A FFV

(20)

where A and B are temperature dependent adjustable parameters specific of each gas-polymer couple. The gas diffusivity in mixed matrix materials is determined not only by the polymeric phase but also by the presence of the filler, which in the case under study is dense and impermeable so that its effects on the gas transport are only associated to an increase in the mean path of a gas molecule diffusing through the material. DM(0) is therefore an apparent diffusivity that may be simply related to the diffusivity in the polymeric phase alone, DP(0), by accounting for the tortuosity factor τ: 1 DM(0) ) DP(0) (21) τ The tortuosity factors can be estimated reasonably well through the Maxwell model for spherical filler particles, so that, ΦF (22) 2 By combining eqs 20 and 21, one obtains a relation between the apparent diffusivity in the composite and the fractional free volume of the polymeric phase, τ)1+

1 B DM(0) ) exp A τ FFV

(

)

(23)

where the parameters A and B are retrieved from experimental data taken at various values or fractional free volume or, equivalently, at various filler contents of the MMM. Therefore, by considering for each mixed matrix the corresponding value of the FFV obtained from the analysis of the sorption behavior of the reference penetrant, as explained in section (a), the value of apparent diffusivity in the composite material, DM(0), may be estimated at other filler contents with eqs 21-23. The values of parameters A and B have to be adjusted based on experimental diffusivity data from at least two matrices. On the other hand, the ratio between the diffusivity in the composite, DM(0), and the diffusivity in the pure polymer, DP0(0), contains only the adjustable parameter B, DM(0) 0

DP (0)

)

[(

1 1 1 exp B 0 τ FFV FFVP

)]

(24)

where FFVP0 is the fractional free volume in the pure unloaded polymer alone and FFV is the corresponding quantity in the polymer phase of the MMM. To test the validity of the approach illustrated so far, use will be made of the experimental diffusivity data in Teflon AF 2400 and PTMSP taken from refs 11 and 12 for different penetrants, considering the FFV values obtained from the application of the NELF model. The test may be completed considering also the permeability ratio between the composite material and the pure polymer, that is, PM(0) 0

PP (0)

)

DM(0)SM(0) DP0(0)SP0(0)

(25)

where superscript 0 labels the pure unloaded polymer, the ratio between the diffusivities is obtained from eq 24, and the ratio between the solubilities is calculated by the NELF model, considering the different density values. NELF Calculations of Polymer Density and FFV For the matrices considered here, the reference-test penetrant used is n-C4H10, which is the only penetrant considered in detail for the MMMs based on both amorphous Teflon AF 2400 and PTMSP in refs 10 and 11, respectively. First of all, it is worth it to point out that Merkel et al. report solubility values per unit volume of MMM, as it is rather commonly done for gas solubility, thus offering immediate comparisons with other existing data. However, the solubility per unit volume of solid, expressed in terms of cm3(STP)/cm3 of solid matrix, is calculated from raw data taken in terms of amount of gas per unit mass of initially unpenetrated solid sample,36 which is a more direct and more appropriate measure for the case under consideration, in which the actual volume of the solid sample is not known from direct measurements. Indeed, in refs 10 and 11 the solubility data reported per unit volume of solid (e.g., in cm3(STP)/cm3 of solid) were obtained from the data measured per unit solid mass by using a density value of the solid composite matrix estimated with the following simple additive law on the bases of the pure component densities:12 F)

1 1 - wF wF + FP,pure FF

(26)

Therefore, because it is known that eq 26 is not followed by the mixed matrices under consideration, before proceeding with our elaboration we recalculated the solubility data reported in


Figure 1. (a) Solubility of n-C4H10 in Teflon AF 2400 and Teflon AF 2400 with 25 wt % of FS at 25 °C and comparison with the additive model based on pure component solubilities. (b) Solubility of n-C4H10 in Teflon AF 2400 and Teflon AF 2400 with 40 wt % of FS at 25 °C and comparison with the additive model based on pure component solubilities.

refs 10 and 11 to obtain solubility values in terms of mass of gas per unit mass of solid matrix (g/gsolid), using the value of F given by eq 26, with the value FF ) 2.20 g/cm3 for the density of the filler.13 Mixed Matrices Based on Teflon AF 2400 In parts a and b of Figure 1, we have reported the experimental solubility isotherms, expressed as mass ratio Ω1 (g/gsolid), of n-C4H10 in pure FS, pure Teflon AF 2400, and Teflon AF 2400-based mixed matrices containing 25 wt % of FS and 40 wt % of FS;10 the solid lines are interpolating curves to guide the eye. In the same figures, we have also reported the calculation of the hypothetical gas solubility into the mixed matrix, which is obtained from the additive model based on pure component solubilities, as well as the solubility isotherm in the polymer phase of the mixed matrix, Ci,P, calculated from the experimental data using eq 16, that is, considering that the adsorption on FS in the MMM is the same as on the pure FS particles. In all of the cases considered, the additive model based on pure component solubilities underestimates the solubility in the

Figure 2. (a) Solubility (CP) of n-C4H10 in Teflon AF 2400 and related mixed matrices at 25 °C as estimated with eq 16 (symbols) and with NELF model (lines), with values of characteristic parameters reported in Tables 1 and 2. (b) Values of F2,un0/F20 - 1 and ksw of Teflon AF 2400-based mixed matrices obtained by the NELF model. Table 1. Lattice Fluid Parameters, Density and Fractional Free Volume of the Pure Polymers polymers 40,37

Teflon AF 2400 PTMSP

T* (K)

P* (MPa)

F* (kg/L)

F20 (kg/L) at 25 °C

VW (L/kg)

FFV

624 416

250 405

2.13 1.250

1.74 0.75

0.301 0.728

0.32 0.29

mixed matrices, and thus the mass uptake in the polymer phase is higher than that in the pure polymer, as already pointed out by Merkel et al. The solubility isotherms of n-C4H10 in the polymeric phases of pure Teflon AF 2400 and Teflon AF 2400 filled with 25 wt% and 40 wt% of FS, obtained by plotting Ci,P versus pressure, are reported in part a of Figure 2. The corresponding solubility isotherms in the polymeric phases calculated with the NELF model are also shown in part a of Figure 2 and are represented by the lines. The values of LF parameters used for the calculation of n-C4H10 solubility in pure Teflon AF 2400 with NELF model are reported in Table 1 and have been taken from a previous work;40 for the polymer they were obtained by best fitting the experimental PVT data above TG to the LF equation of state. The density value for pure unpenetrated Teflon AF 2400 at 25 °C is 1.740 g/cm3 as reported in ref 10. The literature value for

5220 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 Table 2. Relevant Properties of the MMM Considered material

wF

nominal ΦF

F20 (kg/L) at 25 °C

τ

FFV

ksw (MPa-1)

pure Teflon AF 2400 Teflon AF 2400 with 25 wt % of FS Teflon AF 2400 with 40 wt % of FS pure PTMSP PTMSP with 30 wt % of FSa PTMSP with 40 wt % of FSa PTMSP with 50 wt % of FS

0.0 0.25 0.40 0.0 0.30 0.40 0.50

0 0.209 0.345 0 0.127 0.185 0.254

1.740 1.714 1.680 0.750 0.718 0.702 0.680

1 1.104 1.173 1 1.064 1.093 1.127

0.319 0.329 0.343 0.290 0.320a 0.336a 0.356

0.264 0.195 0.100 0.90 0.90 0.90 0.90

a

Values interpolated from the values of pure PTMSP and of PTMSP-based MMM with 50 wt % FS.

the van der Waals volume of Teflon AF 2400 is 0.301 cm3/g, thus obtaining FFVP0 ) 0.32 for pure Teflon AF 2400.41 The NELF model accurately represents the entire solubility isotherm of n-C4H10 in pure Teflon AF 2400 when a swelling coefficient ksw ) 0.264 MPa-1 is introduced to account for swelling, and an adjustable binary parameter Ψ ) 0.873 is used to account for the binary interactions between gas and polymer molecules (part a of Figure 2). Noteworthy, this value of Ψ is completely consistent with the values obtained in previous works for mixtures of normal alkanes in fluorinated polymers.40,42,43 On the other hand, the mixed matrix behavior of Teflon AF 2400 is associated to a smaller value of the pure polymer density F20, due to the increased fractional free volume initially present in the unpenetrated polymer phase and to a different value of the swelling coefficient ksw of the polymer in the mixed matrix, due to changes in polymer phase rigidity. Best fitting the NELF model isotherm to the values obtained for the n-C4H10 isotherm in the polymer phase of the mixed matrix containing 25 wt % of FS leads to a value of the unpenetrated polymer density of 1.714 g/cm3, lower than that of the pure polymer case (1.740 g/cm3), and also to a significantly lower value of the swelling coefficient: ksw ) 0.195 MPa-1 versus a value of 0.264 MPa-1 for pure Teflon AF 2400. For the higher filler content (40 wt %), the unpenetrated polymer density is further decreased to 1.680 g/cm3, as well as is the swelling coefficient: ksw ) 0.100 MPa-1. Such results offer a quantitative measure of the density decrease and FFV increase of the unpenetrated polymer phase in the presence of different filler contents, as well as of the hardening effects induced by the filler resulting in reduced swelling produced by the penetrant. The values of density, swelling coefficient, and the fractional free volume are listed in Table 2. To find a correlation between the density of the polymer phase, or its fractional free volume, and the filler content, one is guided by the expectation that the filler adds an amount of volume available for the polymer, which is directly proportional to the amount of filler, thus obtaining, m2V2 ) m2V2,un + mF

(27)

where V2,un is the polymer specific volume in the absence of filler and m2 and mF are the mass of polymer and of filler, respectively. From eq 27, one immediately obtains that, F2,un0 F2

0

- 1 ) F2,un0

wF 1 - wF

(28)

where F2,un0 is the pure polymer density in an unloaded matrix (1.740 g/cm3 for Teflon AF 2400), whereas F20 is the density of the unpenetrated polymer phase in a mixed matrix whose filler weight fraction is wF. The left-hand side in eq 28 represents the fractional variation of the unpenetrated polymer density and wF/(1 - wF) represents the ratio between filler and polymer mass in the mixed matrix. Indeed, the data show a nice linear dependence between F2,un0/F20 - 1 and wF/(1 - wF), with a

Figure 3. Solubility of n-C4H10 in PTMSP and PTMSP with 50 wt % of FS at 25 °C and comparison with the additive model based on pure component solubilities.

correlation coefficient R2 larger that 0.99, as is clear from part b of Figure 2. Remarkably, there is also a linear relationship versus the filler-to-polymer mass ratio for the swelling coefficient ksw, as shown in part b in Figure 2. This result is rather interesting insofar as it indicates a simple way to interpolate the increase in free volume and decrease in swellability of the matrix with its filler content, at least up to 40 wt % of FS. The percentage variation of polymer density with respect to the pure polymer is very small, and equal to 1.5% and 3.4% for the lower and higher filler contents, respectively, but nevertheless, because there is a high sensitivity of solid-state properties to the density value, this variation can account for the changes in solubility and also for the rather pronounced variations of diffusivity, as will be seen in the following. Mixed Matrices Based on PTMSP Let us now consider the solubility data of n-C4H10 in PTMSPbased mixed matrices, reported in Figure 3, to apply the same procedure previously used for the MMM based on Teflon AF 2400. In this case, the pure polymer LF parameters required by the NELF model cannot be obtained from experimental PVT data above TG for this polymer, because PTMSP chemically decomposes before reaching its glass transition. Therefore, alternative experimental information must be adopted to that aim. Indeed, one can take advantage of the availability of the infinite dilution solubility coefficients, S0, in pure PTMSP for a number of gases and vapors. By best fitting the S0 value obtained from the NELF model, eq 8, to a recent set of data taken from the works of Merkel et al.44 and Srinivasan et al.45 for the solubility of Ar, CO2, H2, N2, O2, CH4, C2H6, C3H8, CF4, C2F6, and C3F8 in PTMSP, at 35 °C, we obtained the set of LF


to FFV values equal to 0.320 and 0.336 for the matrices loaded with 30 and 40 wt % of FS, respectively. Prediction of Solubility Isotherms in PTMSP and PTMSP/FS Mixed Matrices

Figure 4. n-C4H10 sorption at 25 °C in PTMSP and PTMSP with 50 wt % of FS evaluated with F20 ) 0.75 and F20 ) 0.68 g/cm3, respectively, with Ψ ) 1 and with ksw ) 0.90 MPa-1.

parameters listed in Table 1. For that calculation, a value of the binary parameter Ψ equal to 1 was used. In the fitting procedure, the value of ρ2* has been kept fixed to 1.250 kg/L as suggested by previous works, considering also molecular simulations results.46 The reported value of density for PTMSP at 25 °C is 0.750 g/cm3. The literature value for the van der Waals volume of PTMSP is 0.728 cm3/g, leading to a value of FFV of 0.29.44 The NELF model accurately represents the entire solubility isotherm of n-C4H10 in pure PTMSP when a swelling coefficient equal to 0.900 MPa-1 is introduced to account for swelling and the default value of the binary parameter, Ψ ) 1, is used as appropriate for n-C4H10 in an aliphatic-based matrix (part a of Figure 4). On the other hand, by using the NELF model with the same LF pure component parameters listed in Table 1, the solubility isotherm of n-C4H10 in the polymer phase of the mixed matrix with 50 wt % of FS, indicated by open triangles in part a of Figure 4, is well represented with a pure polymer density value F20 ) 0.680 g/cm3, whereas no significant variation is required in the value of the swelling coefficient with respect to the pure polymer (ksw ) 0.90 MPa-1). The latter result seems to indicate that the filler does not modify to a detectable extent the flexibility of the polymer matrix, because of the extreme intrinsic stiffness of PTMSP. Such interpretation, may not appear fully consistent with the reduction in plasticization effects, which parallels an increase in filler contents, observed through the trend of n-butane diffusivity in PTMSP-based MMM versus gas concentration: Merkel et al. have shown that the coefficient γ in eq 19 decreases with increasing filler content, becoming almost negligible at a weight fraction of FS equal to 50%. We can speculate, however, that at higher fractions of FS, the initial value of FFV is already so high that the diffusivity is no longer affected by the swelling effects, which generally cause the diffusion coefficients to vary with concentration. The values of FFV in PTMSP at intermediate values of the FS weight fraction (30 and 40 wt % of FS) have been obtained by linearly interpolating the two extreme values obtained in pure PTMSP and in PTMSP with 50 wt % of FS, considering the linear dependence expressed by eq 28. Such a procedure leads to F20 values equal to 0.718 and 0.702 g/cm3, which correspond

Specific sorption data of N2, CH4, C2H6, and C3H8 in PTMSP and mixed matrices formed by PTMSP and fumed silica in different amounts (30, 40, and 50 wt %) were also presented by Merkel et al.11 However, specific adsorption data onto fumed silica are reported only for the case of n-butane at 25 °C: such data are sufficient to evaluate the variation of density and FFV in the PTMSP phase after the addition of FS particles but not to evaluate the swelling parameter ksw for all of the penetrants because ksw is penetrant specific. The values of PTMSP density determined for each filler loading from the sorption data of the reference penetrant n-C4H10 are used in this section to estimate a priori the solubility isotherms Ωi,P (g/gpol) in the polymer phase of the MMM for all of the above-mentioned gases. To that aim, use will be made of the NELF model with a default value of the binary interaction parameter, Ψ ) 1, and with a swelling coefficient ksw ) 0 for N2, CH4, and C2H6, in view of the fact that such penetrants have little swelling effects, if any at all, whereas for C3H8, ksw was taken as equal to the value observed for the pure unloaded PTMSP matrix. Then, the solubility in the mixed matrices, Ωi,M, is calculated using eq 18 and considering that the contribution due to the adsorption onto the FS surface is negligible for the light gases. In parts a and b of Figures 5, we report the solubility isotherms predicted by the NELF model for N2 and CH4 respectively in the polymeric phase of PTMSP-based mixed matrices. The experimental data were obtained from the mixed matrix isotherm for pure PTMSP and PTMSP-based MMM with 50 wt % of FS provided by Merkel et al.11 As it can be seen, the experimental sorption behavior is correctly predicted by the NELF model both in the pure PTMSP as well as in the PTMSP phase of the mixed matrix containing 50 wt % of FS: for the pure polymer use is made of the experimental value of the unpenetrated polymer density (0.750 g/cm3), whereas for the polymer phase of the MMM containing 50 wt % of FS the value used is F20 ) 0.680 g/cm3, as previously derived for the reference penetrant n-C4H10. In both cases, the binary parameter used is Ψ ) 1, and the swelling coefficient considered is ksw ) 0 MPa-1, as it is reasonable because N2 and CH4 are nonswelling penetrants. Analogously, the solubility isotherms of C2H6 in pure PTMSP and in the polymeric phase of a mixed matrix containing 50 wt% of FS were calculated and reported in part c of Figure 5; the MMM data (open circles) were obtained using eq 18. Also, for this penetrant the NELF model applied in a predictive way for the pure polymer and for the polymer phase of the MMM containing 50 wt % of FS offers a more than satisfactory representation of both solubility isotherms, using the same values of the unpenetrated polymer densities as in the previous cases, namely 0.750 and 0.680 g/cm3, respectively. For the same penetrant, experimental data were also available for MMMs containing intermediate FS contents, that is 30 and 40 wt %, respectively. For such cases, we performed a more severe test of the procedure here considered: in particular we first estimated the density of the unpenetrated polymer phase alone using the linear relationship between F2,un0/F20 - 1 and wF/(1 - wF), eq 28, which holds true for the mixed matrices formed by FS and Teflon AF 2400. Thus, we obtained F20 ) 0.718 g/cm3 for the case of 30 wt % of FS and F20 ) 0.702 g/cm3 for the case of


Figure 5. (a) N2 sorption at 25 °C in PTMSP and PTMSP with 50 wt % of FS, evaluated with F20 ) 0.75 and F20 ) 0.68 g/cm3, respectively, with Ψ ) 1 and with ksw ) 0 MPa-1. (b) CH4 sorption at 25 °C in PTMSP and PTMSP with 50 wt % of FS, evaluated with F20 ) 0.75 and F20 ) 0.68 g/cm3, respectively, with Ψ ) 1 and with ksw ) 0 MPa-1. (c) C2H6 sorption at 25 °C in PTMSP and PTMSP with 50 wt % of FS, evaluated with F20 ) 0.750 and F20 ) 0.680 g/cm3, respectively, with Ψ ) 1 and with ksw ) 0 MPa-1. (d) C2H6 sorption at 25 °C in PTMSP with 30 wt % of FS and 40 wt % of FS, evaluated with F20 ) 0.718 and F20 ) 0.702 cm3, respectively, with Ψ ) 1 and with ksw ) 0 MPa-1. (e) C3H8 sorption at 25 °C in PTMSP and PTMSP with 50 wt % of FS, evaluated with F20 ) 0.750 and F20 ) 0.680 g/cm3, respectively, with Ψ ) 1 and with ksw ) 0.20 MPa-1. (f) C3H8 sorption at 25 °C in PTMSP with 30 wt % of FS and 40 wt % of FS, evaluated with F20 ) 0.718 and F20 ) 0.702 g/cm3, respectively, with Ψ ) 1 and with ksw ) 0.20 MPa-1.


40 wt % of FS (Table 2). On the basis of such estimations of the unpenetrated polymer densities, we then applied the NELF model to predict the solubility isotherms, using the same model parameters listed in Table 1. Also in these cases, the solubility isotherms of ethane calculated by the NELF model in a predictive way (Ψ ) 1 and ksw ) 0 MPa-1) are in good agreement with the experimental data. The comparison between model predictions and experimental data is reported in part d of Figure 5, which represents a strong indication of the validity of the procedure followed. Finally, an analogous procedure was applied to the case of C3H8 sorption in PTMSP and in MMMs containing 50 wt % of FS (part e of Figure 5). For this penetrant, a default binary parameter, Ψ ) 1, was used while the swelling coefficient was adjusted to the value of 0.20 MPa-1 for the pure polymer, and the same value was considered also for the case of the different mixed matrices. Also in this case, there is a good agreement between experimental values and the isotherms calculated by the model. The solubility isotherms for the MMMs containing 30 and 40 wt % of FS were also predicted for C3H8 using the values of the unpenetrated polymer phases already estimated on the basis of eq 28 and were reported in Table 2. The comparison between model predictions and experimental data are shown in part f of Figure 5, and the good agreement further supports the validity of the procedure followed. Correlation and Estimation of Diffusivity (1) Mixed Matrices Based on Teflon AF 2400. From the experimental values of the apparent diffusion coefficients of n-C4H10 in the composite matrix at infinite dilution, DM(0), the corresponding values of the infinite dilution penetrant diffusivity in the polymer phase alone, DP(0), is obtained using eq 20 and the expression of the tortuosity factor given by the Maxwell model, eq 22. The results obtained as DP(0) ) τDM(0) are reported versus 1/FFV in part a of Figure 6, in a semilogarithmic plot as suggested by eq 20, using the FFV values obtained by the previous analysis of the solubility data of n-C4H10. The experimental data do fall on a straight line, with a correlation coefficient R2 equal to 0.98, confirming thus the validity of eqs 20 and 22 as well as of the procedure followed to estimate the FFV from the solubility isotherm. In particular, from the slope in part a of Figure 6 one obtains the value of parameter B, from which the ratio between the apparent diffusivity in the composite and that in the pure unloaded polymer, DM(0)/DP0(0), may be predicted on the basis of eq 24, using eq 22 for the tortuosity factor τ. The experimental and calculated values of the diffusivity ratio are reported in part b of Figure 6 and in Table 4, clearly indicating that the model allows us to predict the diffusivity increase due to the filler content, with satisfactory agreement between experimental and simulated values, especially at higher filler fractions. (2) Mixed Matrices Based on PTMSP. The FFV values obtained on the basis of the solubility isotherm of the reference penetrant, n-C4H10, for the MMM containing FS in PTMSP, are listed in Table 4 and were used for the correlation of apparent gas diffusivity data taken from the cited work of Merkel et al.11 The first penetrant considered is normal butane: the correlation based on eq 20 between experimental data of DP(0) ) τDM(0) and 1/FFV is extremely good, providing values of the correlation coefficient R2 almost equal to 1 for the experimental data points available (part a of Figure 7). The resulting value of coefficient B is then used to predict values of DM(0)/DP0(0), using eqs 24 and 22, thus obtaining the diffusivity ratios reported in part b of Figure 7 and in Table 4. As it can be clearly seen, the

Figure 6. (a) Values of DP(0) versus 1/FFV for n-C4H10 diffusivity in mixed matrices of Teflon AF 2400 and FS at 25 °C. (b) Values of DM(0)/DP0(0) versus the filler-to-polymer mass ratio for n-C4H10 diffusivity in mixed matrices of Teflon AF 2400 and FS at 25 °C.

agreement between calculated values and experimental data is indeed excellent, with a maximum deviation between experimental and calculated values equal to 15%. A similar correlation for the diffusion coefficients was then applied to all other penetrants for which diffusivity data are available, namely propane, methane, and hydrogen, then comparing predicted values and experimental results in parts a-c of Figure 8 and in Table 4. The value of the correlation coefficient in the exponential dependence between diffusivity and 1/FFV, eq 20, is rather good for propane (R2 ) 0.92), and is satisfactory for CH4 (R2 ) 0.87) and for H2 (R2 ) 0.89). One can notice that the correlation given by eq 23 is somewhat better for the heavier penetrants than for the lighter gases, possibly because the values of FFV were obtained from the solubility isotherm of normal butane, which is a swelling penetrant. Therefore, the values of FFV thus obtained, even if adjusted on the low-pressure range of the solubility isotherm with the NELF model, may be somehow affected by swelling, which is absent in the case of the sorption of lighter gases. Specific sorption data for the lighter gases would likely lead to a better estimate of the value of the initial FFV, in the absence of swelling effects. The deviation between the calculated and experimental values of the ratio between the infinite dilution diffusivity in the composite material and in the pure unloaded polymer alone, DM(0)/DP0(0), varies with penetrant and filler content and ranges between 0 and 6% for the case of n-C4H10, between 8.9% and

5224 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 Table 4. Diffusivity Values in Teflon AF 2400 and PTMSP-Based Mixed Matrices at 25 °C: Comparison between Experimental Values and Predictions penetrant n-C4H10 (B ) 16.7) n-C4H10 (B ) 3.5) C3H8 (B ) 2.0)

CH4 (B ) 2.2)

H2 (B ) 2.0)

material pure Teflon AF 2400 Teflon AF 2400 with 25 wt % of FS Teflon AF 2400 with 40 wt % of FS pure PTMSP PTMSP with 30 wt % of FS PTMSP with 50 wt % of FS pure PTMSP PTMSP with 30 wt % of FS PTMSP with 40 wt % of FS PTMSP with 50 wt % of FS pure PTMSP PTMSP with 30 wt % of FS PTMSP with 40 wt % of FS PTMSP with 50 wt % of FS pure PTMSP PTMSP with 30 wt % of FS PTMSP with 40 wt % of FS PTMSP with 50 wt % of FS

29% for C3H8, between 8% and 66% for CH4, and between 12% and 38% for H2. Moreover, in all cases the predicted values of DM(0)/DP0(0) in PTMSP-based MMMs become more accurate at higher loadings of FS than for smaller FS contents, as one can notice from Table 4; that behavior was also observed for the case of n-C4H10 diffusivity in Teflon AF 2400 (part b of Figure 6). This clear trend shines some light on the phenomena described and deserves some comments. Indeed, the fractional free volume in the polymer phase of the mixed matrices was estimated on the basis of experimental data of the solubility isotherm, which actually enables us to obtain an average FFV in the polymer phase, which is the quantity required in the commonly used correlation represented by eq 20. Of course, a single average value of FFV does not allow us to differentiate

DM (0) (cm2/s), exp. -8

1.98 × 10 5.00 × 10-8 5.73 × 10-7 1.31 × 10-6 4.40 × 10-6 1.06 × 10-5 1.15 × 10-5 1.59 × 10-5 2.18 × 10-5 3.87 × 10-5 3.79 × 10-5 4.71 × 10-5 7.52 × 10-5 1.42 × 10-4 3.75 × 10-4 5.01 × 10-4 6.53 × 10-4 1.26 × 10-3

DM (0)/DP (0) exp.

DM (0)/DP (0) sim.

1 2.5 29 1 3.4 8.1 1 1.4 1.9 3.4 1 1.2 2.0 3.7 1 1.3 1.7 3.4

1 4.5 30 1 2.9 8.1 1 1.8 2.3 3.2 1 1.9 2.5 3.6 1 1.8 2.3 3.1

between smaller and larger free volume elements. The variation in the FFV thus obtained will offer a satisfactory representation of the modifications induced in the polymeric phase if the addition of the filler varies by the same percentage the FFV elements of all sizes, leaving their relative distribution unaltered. On the contrary, PALS analysis11 has shown that the addition of the filler to PTMSP affects essentially the number of the larger free volume elements, which are present in the bimodal distribution of nanoholes of the polymer matrix, with no influence on the fraction of the smaller nanovoids. The larger FFV elements have a higher impact in enhancing diffusivity than the smaller nanovoids, and a proper description of this effect requires an ad hoc correlation for diffusivity more complex than the commonly used relationship adopted in the present analysis. Indeed, the fractional free volume theory used to correlate the diffusion coefficients, eq 20 or 23, considers only the total FFV without differentiating between contributions of the different populations of the FFV size distribution. Therefore, the procedure followed in the present study appears to be more solid and reliable for the prediction of solubility isotherms, whereas for the diffusion coefficients the predictions are possible to within a larger uncertainty due to nonhomogeneous variations in the FFV distribution. Conclusions

Figure 7. (a) Values of DP(0) versus 1/FFV for n-C4H10 diffusivity in mixed matrices of PTMSP and FS at 25 °C. (b) Values of DC(0)/DP0(0) versus filler-to-polymer mass ratio for n-C4H10 diffusivity in mixed matrices of PTMSP and FS at 25 °C.

The analysis performed on the solubility isotherms of gases in mixed matrices based on Teflon AF 2400 and PTMSP with different loadings of hydrophobic, nanoscale fumed silica was performed considering that the penetrant adsorption onto the filler of the MMM remains the same as that in the pure filler and is known from separate experimental information, if it is not negligible. The study and the comparison with detailed experimental data reported in refs 10 and 11 have shown the following main results. (i) From the solubility isotherm of one penetrant, chosen as a reference, in the MMM it is possible to obtain the contribution to the solubility isotherm offered by the polymer phase only, through eq 15 or 16; from that it is possible to obtain the unpenetrated polymer density F20, the FFV, and the swelling coefficient ksw of the polymer phase of the MMM, at each filler content. (ii) There is a simple relationship between the changes in the density of the unpenetrated polymer phase due to the filler addition, and the MMM composition, eq 28, which offers good

Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5225 Table 3. Characteristic Parameters for the LF EOS for the Gases Examined in This Work penetrant

T1* (K)

P1* (MPa)

F1*(g/cm3)

N244 CH444 C2H645 C3H845 n-C4H1037

145 215 320 375 430

160 250 330 320 290

0.943 0.500 0.640 0.690 0.720

Therefore, it appears not necessary to extend the experimental analysis over various filler contents of the MMM, virtually one composition would be sufficient and, more remarkably, it appears not necessary to determine experimentally the solubility isotherms of various different penetrants for the same MMM. The analysis of the diffusivity data in MMMs has shown that: (iv) The infinite dilution apparent diffusivity in the mixed matrices is obtained by accounting for a tortuosity factor, given by the Maxwell model (eq 22), together with the infinite dilution diffusivity in the polymer phase of the MMM. The latter contribution is to a good approximation exponentially dependent on FFV through eq 20, with the FFV values of each polymer phase derived by the previous analysis of the solubility isotherm of the reference penetrant. The procedure followed indicates that for each penetrant the infinite dilution diffusivity can be obtained for all of the filler contents of interest on the basis of the diffusivity value in the pure unloaded polymer and at least in one MMM. The reliability of the correlation is better for higher FS contents than for the lower filler loadings. The procedure presented is based on the fractional free volume estimates and thus enables us to obtain good estimates of solubility and effective diffusion coefficients, also for the uncommon cases in which the loading actually enhances the diffusion through the matrix; thus the known limitations of the classical Maxwell model are naturally overcome. Acknowledgment The financial support of EU Strep NMP3-CT-2005-013644 (MultiMatDesign) is gratefully acknowledged. Literature Cited

Figure 8. (a) Values of DP(0) versus 1/FFV for C3H8 diffusivity in the polymer phase of mixed matrices of PTMSP and FS at 25 °C. (b) Values of DP(0) versus 1/FFV for CH4 diffusivity in the polymer phase of mixed matrices of PTMSP and FS at 25 °C. (c) Values of DP(0) versus 1/FFV for H2 diffusivity in the polymer phase of mixed matrices of PTMSP and FS at 25 °C.

estimates of the density F20 and of the FFV of the polymer phase at filler contents other than the ones experimentally tested. (iii) On the basis of the F20 values thus obtained for the MMM at different filler loadings, one can calculate in a predictive way the solubility isotherm of other different nonswelling penetrants in the same polymer phases; from that, one can immediately derive the solubility isotherm in the MMM, if the adsorption onto the pure filler is known or if it is negligible. The procedure proved to be reliable and solid in all the cases tested. In the case of swelling penetrants, some further limited information on the solubility is required to obtain the value of the swelling coefficient.

(1) Robeson, L. M. Correlation of Separation Factor Versus Permeability for Polymeric Membranes. J. Membr. Sci. 1991, 62, 165–185. (2) Mahajan, R.; Koros, W. J. Factors Controlling Successful Formation of Mixed-Matrix Gas Separation Materials. Ind. Eng. Chem. Res. 2000, 39, 2692–2696. (3) Vu De, Q.; Koros, W. J.; Miller, S. J. Mixed Matrix Membranes Using Carbon Molecular Sieves I. Preparation and Experimental Results. J. Membr. Sci. 2003, 211, 311–334. (4) Su¨er, M. G.; Bac, N.; Yilmaz, L. Gas Permeation Characteristics of Polymer-Zeolite Mixed Matrix Membranes. J. Membr. Sci. 1994, 91, 77– 86. (5) Li, Y.; Guan, H.-M.; Chung, T.-S.; Kulprathipanja, S. Effects of Novel Silane Modification of Zeolite Surface on Polymer Chain Rigidification and Partial Pore Blockage in Polyethersulfone (PES)sZeolite A Mixed Matrix Membranes. J. Membr. Sci. 2006, 275, 17–28. (6) Gu¨r, T. M. Permselectivity of Zeolite Filled Polysulfone Gas Separation Membranes. J. Membr. Sci. 1994, 93, 283–289. (7) Kim, S.; Marand, E.; Ida, J.; Guliants, V. V. Polysulfone and Mesoporous Molecular Sieve MCM-48 Mixed Matrix Membranes for Gas Separation. Chem. Mater. 2006, 18, 1149–1155. (8) Jiang, L. Y.; Chung, T. S.; Kulprathipanja, S. An Investigation to Revitalize the Separation Performance of Hollow Fibres with a Thin Mixed Matrix Composite Skin for Gas Separation. J. Membr. Sci. 2006, 276, 113– 125. (9) Bhardwaj, V.; Macintosh, A.; Sharpe, I. D.; Gordeyev, S. A.; Shilton, S. J. Polysulfone Hollow Fiber Gas Separation Membranes Filled with Submicron Particles. Ann. N.Y. Acad. Sci. 2003, 984, 318–328. (10) Merkel, T. C.; He, Z.; Pinnau, I.; Freeman, B. D.; Meakin, P.; Hill, A. J. Sorption and Transport in Poly (2,2-bis(Trifluoromethyl)-4,5-difluoro-

5226 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 1,3-dioxole-co-tetrafluoroethylene) Containing Nanoscale Fumed Silica. Macromolecules 2003, 36, 8406–8414. (11) Merkel, T. C.; He, Z.; Pinnau, I.; Freeman, B. D.; Meakin, P.; Hill, A. J. Effect of Nanoparticles on Gas Sorption and Transport in Poly(1trimethylsilyl-1-propyne). Macromolecules 2003, 36, 6844–6855. (12) Merkel, T. C.; Freeman, B. D.; Spontak, R. J.; He, Z.; Pinnau, I.; Meakin, P.; Hill, A. J. Sorption, Transport, and Structural Evidence for Enhanced Free Volume in Poly(4-methyl-2-pentyne)/Fumed Silica Nanocomposite Membranes. Chem. Mater. 2003, 15, 109–123. (13) Merkel, T. C.; Freeman, B. D.; Spontak, R. J.; He, Z.; Pinnau, I.; Meakin, P.; Hill, A. J. Ultrapermeable, Reverse-Selective Nanocomposite Membranes. Science 2002, 296, 519–522. (14) Koros, W. J.; Fleming, G. K. Membrane-Based Gas Separation. J. Membr. Sci. 1993, 83, 1–80. (15) Vu, De Q.; Koros, W. J.; Miller, S. J. Mixed Matrix Membranes Using Carbon Molecular Sieves II. Modeling Permeation Behaviour. J. Membr. Sci. 2003, 211, 335–348. (16) Maxwell, C. Treatise on Electricity and Magnetism; Oxford University Press: London, 1873; Vol. 1. (17) Barrer, R. M.; Barrie, J. A.; Rogers, M. G. Heterogeneous Membranes: Diffusion in Filled Rubber. J. Polym. Sci., Part A: Polym. Chem. 1963, 1, 2565–2586. (18) Barrer, R. M. In Diffusion in Polymers; Crank, J., Park, G. S. Eds.; Academic Press: London, 1968; pp 165-217. (19) Bouma, R. H. B.; Checchetti, A.; Chidichimo, G.; Orioli, E. Permeation through a Heterogeneous Membrane: The Effect of the Dispersed Phase. J. Membr. Sci. 1997, 128, 141–149. (20) Banhegyi, G. Comparison of Electrical Mixture Rules for Composites. Colloid Polym. Sci. 1986, 264, 1030–1050. (21) Gonzo, E. E.; Parentis, M. L.; Gottifredi, J. C. Estimating Models for Predicting Effective Permeability of Mixed Matrix Membranes. J. Membr. Sci. 2006, 277, 46–54. (22) Chiew, Y. C.; Glandt, E. D. The Effect of Structure on the Conductivity of a Dispersion. J. Colloid Interface Sci. 1983, 94, 90–104. (23) Sanchez, I. C.; Lacombe, R. H. An Elementary Molecular Theory of Classical Fluids. Pure Fluids. J. Phys. Chem. 1976, 80, 2352–2362. (24) Lacombe, R. H.; Sanchez, I. C. Statistical Thermodynamics of Fluid Mixtures. J. Phys. Chem. 1976, 80, 2568–2580. (25) Sanchez, I. C.; Lacombe, R. H. Statistical Thermodynamics of Polymer Solutions. Macromolecules 1978, 11, 1145–1156. (26) Doghieri, F.; Quinzi, M.; Rethwisch, D. G.; Sarti, G. C. ACS Symp. Ser. 2004, 876, 74–90. (27) Giacinti Baschetti, M.; De Angelis, M. G.; Doghieri, F.; Sarti, G. C. In Chemical Engineering: Trends and DeVelopments; Galan, M. A., Martin del Valle, E. Eds.; Wiley: Chichester, U.K., 2005; pp 41-61. (28) Astarita, G. Thermodynamics; Plenum Press: New York, U.S.A., 1989. (29) Giacinti Baschetti, M.; Doghieri, F.; Sarti, G. C. Ind. Eng. Chem. Res. 2001, 40, 3027–3037. (30) De Angelis, M. G.; Sarti, G. C.; Doghieri, F. NELF Model Prediction of the Infinite Dilution Gas Solubility in Glassy Polymers. J. Membr. Sci. 2007, 289, 106–122. (31) Rodgers, P. A. Pressure-Volume-Temperature Relationships for Polymeric Liquids: A Review of Equations of State and Their Characteristic Parameters for 56 Polymers. J. Appl. Polym. Sci. 1993, 48, 1061–1080. (32) Sanchez, I. C.; Panayiotou, C. In Models for Thermodynamic and Phase Equilibria Calculations; Sandler, S. I. Ed.; Dekker: New York, 1994.

(33) Fleming, G. K.; Koros, W. J. Carbon Dioxide Conditioning Effects on Sorption and Volume Dilation Behavior for Bisphenol A-Polycarbonate. Macromolecules 1990, 23, 1353. (34) Jordan, S.; Koros, W. J. Free Volume Distribution Model of Gas Sorption and Dilation in Glassy Polymers. Macromolecules 1995, 28, 2228. (35) Grassia, F.; Giacinti Baschetti, M.; Doghieri, F.; Sarti, G. C. Solubility of Gases and Vapors in Glassy Polymer Blends in AdVanced Materials for Membrane Separations; ACS Symposium Series; American Chemical Society: Washington, DC, 2004; p 876. (36) Merkel, T. C. Private communication. (37) van Krevelen, W. D. Properties of Polymers; Elsevier: Amsterdam, 1990; p 71. (38) Fujita, H. Diffusion in Polymer-Diluent Systems. Fortschr. Hochpolym. Forsch. 1961, 3, 1–47. (39) Cohen, M. H.; Turnbull, D. J. J. Chem. Phys. 1959, 31, 1164– 1169. (40) De Angelis, M. G.; Merkel, T. C.; Bondar, V. I.; Freeman, B. D.; Doghieri, F.; Sarti, G. C. Gas Sorption and Dilation in 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. (41) Merkel, T. C.; Bondar, V.; Nagai, K.; Freeman, B. D.; Yampolskii, Yu. P. Gas Sorption, Diffusion, and Permeation in Poly (2,2-bis(Trifluoromethyl)-4,5-difluoro-1,3-dioxole-co-tetrafluoroethylene). Macromolecules 1999, 32, 8427–8440. (42) Prabhakar, R. S.; De Angelis, M. G.; Sarti, G. C.; Freeman, B. D.; Coughlin, M. C. Gas and Vapor Sorption, Permeation, and Diffusion in Poly(tetrafluoroethylene-co-perfluoromethyl vinyl ether). Macromolecules 2005, 38, 7043–7055. (43) Merkel, T. C.; Pinnau, I.; Prabhakar, R.; Freeman, B. D. Gas and Vapor Transport Properties of Perfluoropolymers. In Materials Science of Membranes for Gas and Vapor Separation; Freeman, B., Yampolskii, Y., Pinnau, I. Ed.; Wiley: New York, U.S.A. (44) Merkel, T. C.; Bondar, V.; Nagai, K.; Freeman, B. D. Hydrocarbon and Perfluorocarbon Gas Sorption in Poly(dimethylsiloxane), Poly(1trimethylsilyl-1-propyne), and Copolymers of Tetrafluoroethylene and 2,2bis(trifluoromethyl)-4,5-difluoro-1,3-dioxole. Macromolecules 1999, 32, 370–374. (45) Srinivasan, R.; Auvil, S. R.; Burban, P. M. Elucidating the Mechanism of Gas Transport in Poly[1-(trimethylsilyl)-1-propine](PTMSP). J. Membr. Sci. 1994, 86, 67–86. (46) Giacinti Baschetti, M.; Ghisellini, M.; Quinzi, M.; Doghieri, F.; Stagnaro, P.; Costa, G.; Sarti, G. C. Effects on Sorption and Diffusion in PTMSP and TMSP/TMSE Copolymers of Free Volume Changes due to Polymer Ageing. J. Mol. Struct. 2005, 739, 75–86. (47) Sarti, G. C.; Doghieri, F. Predictions of the Solubility of Gases in Glassy Polymers Based on the NELF Model. Chem. Eng. Sci. 1998, 19, 3435–3447. (48) De Angelis, M. G.; Merkel, T. C.; Bondar, V. I.; Freeman, B. D.; Doghieri, F.; Sarti, G. C. Hydrocarbon and Fluorocarbon Solubility and Dilation in Poly(dimethylsiloxane): Comparison of Experimental Data with Predictions of the Sanchez-Lacombe Equation of State. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 3011–3026.

ReceiVed for reView November 2, 2007 ReVised manuscript receiVed December 23, 2007 Accepted January 7, 2008 IE0714910

Solubility and Diffusivity of Gases in Mixed Matrix Membranes

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