## 1. Introduction

In recent years, the use of polymers as membrane materials has attracted increased interest for several industrial applications, including gas separation for hydrogen recovery, nitrogen production, air dehydration, natural gas sweetening and biogas upgrading [

1]. CO

_{2} is a typical contaminant to be removed from both natural gas and biomethane, in order to meet distribution pipelines specifications [

2]. Despite the fact that CO

_{2} removal from natural gas with membranes has found industrial application since the 1980s [

3], nowadays this technology has only about 10% of the market, which is dominated by solvent absorption using amines [

1]. In membrane materials design research, countless structural and molecular modifications have been investigated in order to achieve a better separation performance, that would make membranes more competitive, in addition to being more energy-efficient and environmentally friendly [

4,

5,

6,

7,

8,

9,

10]. However, one of the greatest challenges faced in membrane materials design is the existence of a trade-off between permeability and selectivity: for every gas pair the logarithm of the selectivity versus the logarithm of the permeability of the most permeable gas has been shown to lie below a limiting line, customarily referred to as the Robeson upper bound [

11,

12]. This is due to the fact that ultra-permeable materials usually display very poor selectivity, whereas highly-selective materials exhibit lower permeabilities [

12]. This sets an upper limit to the efficiency that can be achieved by the operation, in case it is governed by diffusivity-selectivity [

13,

14].

Materials with improved performance, capable of surpassing the upper bound, have nonetheless been developed, among those are the family of Polymers of Intrinsic Microporosity (PIMs) [

15,

16,

17,

18] and Thermally Rearranged (TR) polybenzoxazoles [

19,

20,

21,

22,

23]. Owing to a rigid backbone structure, consisting of a series of fused aromatic rings and to the presence of a shape-persistent site of contortion, the hindered chain packing of PIMs results in exceptionally high free volume, organized in a network of interconnected cavities. These materials have shown very high gas permeation rates, while maintaining acceptable selectivity values, and moreover they demonstrated great thermal and chemical stability [

18].

Most experimental studies on prospective membrane materials are performed only with pure gases. While those data constitute a valuable benchmark of the materials properties, pure-gas tests are often insufficient to infer how the materials will behave in mixed-gas conditions. Mixed-gas permeation and sorption experiments have shown significant deviations from pure-gas (“ideal”) behavior, both positive and negative [

18,

21,

24,

25,

26]. In order to properly design a separation operation, it is necessary to characterize the relevant materials properties, namely its permeability and selectivity, as close as possible to the actual operating conditions, which can vary depending on the origin of the gaseous stream to be treated. Consequently, to uncover all the relevant phenomena, a broad experimental campaign, encompassing a wide range of temperatures, pressures and compositions, would be needed.

The transport of small molecules in dense polymeric membranes is described by the solution-diffusion model [

27], according to which permeability

$\left(P\right)$ is the product of the solubility (

$S)$ and diffusion coefficients (

$D)$:

Whether ultra-high free volume polymers can still be regarded as dense materials is an open question, however, there have been reports of successful modelling studies relying on this hypothesis [

28]. Following this description, the selectivity of the polymer (perm-selectivity)

${\alpha}_{i,j}$, which is equal to the ratio between the permeability of the two gases, contains a solubility-selectivity (

${\alpha}_{i,j}^{S}$) and a diffusivity-selectivity factor (

${\alpha}_{i,j}^{D}$):

Solubility-selectivity is expected to provide an important contribution to the overall permselectivity in high free volume glassy polymers, whereas for low and medium free volume polymers, where sieving effects are more important, the diffusivity-selectivity is expected to have a higher weight in the overall permselectivity. It would be interesting to be able to predict the mixed-gas behavior, using at most only pure-gas experimental measurement as input, in order to avoid or reduce the need for the more delicate and time-consuming mixed-gas tests.

The calculation of gas solubility in glassy polymers is customarily performed in the literature using the Dual Mode Sorption (DMS) model [

29,

30,

31,

32,

33,

34,

35,

36,

37,

38,

39]. This model divides the total sorbed gas into two contributions: the molecules dissolving into the dense portion of the polymer (following Henry’s law), and those saturating the microvoids of the excess free volume that characterizes the glassy state (described by a Langmuir curve). Its simplicity of use and its capability to correlate well the experimental sorption behavior in glassy polymers in most cases are the main reasons behind its success. However, this model does not allow to represent all types of sorption isotherms encountered, such as the sigmoidal shape of the sorption isotherms of alcohols in glassy polymers. There have been studies aimed at overcoming this limitation: for example, by incorporating multilayer sorption theory, a DMS based model capable of representing all the different shapes of sorption isotherms encountered was developed [

40].

Another known issue with the use of this model is that the adjustable polymer-penetrant parameters of the DMS model depend on polymer history and operating conditions, thus lacking predictive ability outside their range of derivation, as discussed, for example, by Bondar at al. concerning the pressure range [

41].

Alternatively, one can use an Equation of State (EoS) based approach to evaluate gas sorption equilibria. Some models successfully applied to the study of polymeric materials are those based on a Lattice Fluid (LF) representation of substances [

42], or on hard sphere chain schemes, like the Statistical Associating Fluid Theory (SAFT) [

43]. In the case of glassy polymers, due to their nonequilibrium condition, equilibrium models, such as an EoS, are not applicable. In these cases the Non-Equilibrium Thermodynamics for Glassy Polymers (NET-GP) approach [

44] can be used instead. This approach extends equations of state to the nonequilibrium case, providing nonequilibrium expressions for the free energy of the system, by introducing an internal state variable, the polymer density, to describe the out-of-equilibrium degree of the systems. This approach has been successfully applied to the prediction of gas and vapor sorption in a variety of polymeric systems [

28,

45,

46,

47] and its capability to represent mixed-gas sorption equilibria in high free volume glassy polymer has been addressed in another work under preparation [

48].

Alternatively, atomistic simulations can be employed for the prediction of sorption isotherms. Monte Carlo simulations in the Grand Canonical ensemble [

49] can be performed to this aim, thanks to insertion moves that allow the polymeric system to exchange gas particles with an infinite bath until it reaches the equilibrium concentration corresponding to a given value of the chemical potential. Prediction of sorption isotherms can also be performed by post-processing of Molecular Dynamics (MD) or Monte Carlo (MC) trajectories, using the Widom test particle insertion method [

50]: the intermolecular interaction energy felt by a molecule inserted in a random position in the polymer phase is related to the excess chemical potential of the penetrant inside the polymer and, in turn, to its solubility coefficient. In dense systems or in presence of large penetrant molecules, the probability of successful insertion moves decreases significantly and therefore the estimate of solubility through Widom insertions becomes less reliable. Other strategies include the use of gradual insertion of the penetrant molecules [

51], or the use of particle deletion moves instead of particle insertions (Staged Particle Deletion [

52], Direct Particle Deletion [

53]). Atomistic simulation techniques have been increasingly applied in recent years to the study of gas transport properties of microporous polymers [

54], and of PIMs in particular [

55,

56,

57,

58]. Sorption of CO

_{2} in PIM-1 was first simulated by Heuchel et al. [

59] employing the Gusev-Suter Transition State Theory [

60,

61]. Even though the simulated solubility coefficients were significantly higher than the experimental ones, their work paved road to the application of molecular modelling techniques to this class of materials. Fang et al. [

62,

63] applied the Widom test particle insertion method [

50] to predict CO

_{2} solubility in PIM-1 and their results were in close agreement with the experimental ones. Recently, Kupgan et al. [

64] predicted CO

_{2} sorption in PIM-1 up to 50 bar, employing a scheme combining Grand Canonical Monte Carlo (GCMC) and Molecular Dynamics simulations devised by Hölck et al. [

65], while Frentrup et al. [

66] performed Nonequilibium Molecular Dynamics simulations for the direct simulation of He and CO

_{2} permeability through a thin membrane of PIM-1, which was in good qualitative agreement with experimental data.

Fewer modelling studies deal with the analysis of mixed gas sorption effects. Recently Rizzuto et al. [

67] have coupled GCMC atomistic simulations and Ideal Adsorbed Solution Theory (IAST) [

68] to investigate the mixed-gas permeation properties of CO

_{2}/N

_{2} mixtures in Thermally Rearranged polymers. The simulations underestimated pure-gas sorption of both gases, however their results displayed the competitive effects between the gases expected in the case of glassy polymers, which affect greatly the solubility of the less condensable gas in the mixture. Neyertz and Brown [

69] performed large-scale MD simulations of air separation with an ultra-thin polyimide membrane surrounded by an explicit gas reservoir, which allowed them to determine gas solubility, diffusivity and O

_{2}/N

_{2} selectivity in multicomponent conditions, comparing favorably with experimental results. For this gas couple, the modelling study predicted a multicomponent solubility-selectivity comparable to the ideal one, that is calculated as the ratio of pure-gas solubility coefficients.

With the development of more accurate potentials, new algorithms for the generation of amorphous polymeric structures and efficient equilibration protocols, the reliability of the predictions yielded by atomistic techniques has drastically improved over the years [

54,

70]. Moreover, these techniques have the potential to be employed for screening purposes on existing materials, as well as on yet to be synthesized ones, as demonstrated, for instance, by Hart et al. [

57] and Larsen et al. [

71] for the case of CO

_{2}/CH

_{4} separation with PIMs. However the extremely high computational effort required by atomistic approaches, combined to the system-specificity of several methods, remains a drawback to their application to the study of the separation properties of polymers, even though multiscale strategies, involving systematic coarse-graining and equilibration of high molecular weight models at the coarse-grained level and subsequent back-mapping to the atomistic detail have been implemented successfully to study a variety of properties of polymeric systems with reduced machine-time [

72,

73,

74,

75].

The present work is aimed at modelling gas solubility in high free volume glassy polymers both in pure- and mixed-gas conditions using the multicomponent version of the DMS model [

76]. In particular, the sorption of CO

_{2}/CH

_{4} mixtures in PTMSP, PIM-1 and TZ-PIM at several compositions and temperatures was studied, using experimental data presented in previous works to validate the results of the calculations [

24,

25,

26,

48]. The characterization of mixed-gas sorption is still quite limited and these materials are among the very few for which these experiments were performed. PTMSP, being the most permeable dense polymer [

77], is a natural reference point to assess the separation performance of high free volume materials, as is PIM-1, which was the first material of the PIM class to be reported [

15]. TZ-PIM constitutes an attempt at improving the selectivity of PIM-1 towards CO

_{2} by incorporating more CO

_{2}-philic groups into its structure, demonstrating that post-polymerization modification techniques with controlled conversion rates represent a viable way of tuning the separation properties of these innovative materials. In this case the nitrile groups were substituted by tetrazole groups [

78], but Satilmis et al. [

79] showed that it is also possible to reduce them to primary amines, obtaining a material termed amine-PIM-1, with intermediate features between PIM-1 and TZ-PIM.

Lanč et al. [

80] recently performed a critical analysis of the difference between gas solubility coefficients determined directly, with sorption experiments, or indirectly, from the time-lag of permeation. They investigated several high free volume glassy polymers, including PTMSP and PIM-1, concluding that the underlying approximation of a linear concentration profile across the membrane, assumed in the time-lag analysis, is a nonnegligible source of error in the indirect determination of

$S$, but it can be mitigated by the calculation of concentration profiles using the thermodynamic Fick’s law instead [

81]. The authors also remarked the importance of sorption studies in uncovering fundamental aspect of gas transport in membrane materials.

Indeed, experimental measurements of mixed-gas sorption [

24,

25,

26,

48] allowed to understand that the competition between CO

_{2} and CH

_{4} plays a strong role in the multicomponent sorption behavior. Furthermore, the data indicate that the pure-gas solubility does not provide a good estimate of the real behavior of the mixture. In particular, pure-gas data would indicate that the main membrane parameters, like the solubility-selectivity, are a strong function of the gas mixture composition, while experimentally it is observed that the data depend very weakly on such variable. Additionally, for a set of glassy polymers comprising poly(2,6-dimethyl-1,4-phenylene oxide) (PPO), PTMSP, PIM-1 and Matrimid

^{®}, it was shown that departure functions, expressing the deviations between the multicomponent properties and the corresponding ideal values, estimated with pure component properties, obey generalized trends which resemble those observed in liquid solutions [

82]. The ability of the Dual Mode Sorption model to represent these physical phenomena, as well as its quantitative accuracy in the prediction of solubility and solubility-selectivity were assessed, by comparing the results of the calculation with experimental data available for the materials considered [

24,

25,

26,

48], whose repeating units are shown in

Figure 1. Moreover, a sensitivity analysis was carried out, in order to verify the robustness of the calculation and the reliability of the prediction in absence of experimental data for validation.

## 2. Dual Mode Sorption Model

The existence of a sorption mechanism particular to polymers in the glassy state was first postulated by Meares [

35,

36]. The indication that polymers below the glass transition temperature contain a distribution of microvoids frozen into their structure [

35] suggested that those region of reduced density could act as preferential sorption sites. Using this concept and observing that the sorption isotherms of organic vapors in ethyl cellulose showed a curvature concave to the pressure axis, that was not witnessed in the case of rubbery materials, and, furthermore, that for these systems rather high negative values of the heat of solution were measured, Barrer et al. [

37] proposed the existence of two concurrent mechanisms of sorption: dissolution and “hole-filling”.

Therefore, the Dual Mode Sorption (DMS) model [

29,

30,

31,

32,

33,

34,

35,

36,

37,

38,

39] postulates the existence of two different gas populations inside glassy polymers, at equilibrium with one another. One is dissolved in the dense portion of the material and can be described by Henry’s law, while the other saturates the nonequilibrium excess free volume of the polymer and is described by a Langmuir curve. In this schematization, the total sorbed gas as a function of gas fugacity can be expressed as a sum of these two contributions [

31,

37]:

The parameter

${k}_{D,i}$ is Henry’s law constant, while

${b}_{i}$ is Langmuir affinity constant, which represents the ratio of the rate constants of sorption and desorption of penetrants in the microvoids and, therefore, it quantifies the tendency of a given penetrant to sorb according to the Langmuir mode.

${C}_{H,i}^{\prime}$ is the Langmuir capacity constant, which characterizes the sorption capacity of a glassy polymer for a given penetrant in the low-pressure region. This latter parameter correlates with changes in polymer density associated with formation history or annealing treatments [

83,

84] and has been shown to disappear at the glass transition temperature (

T_{g}) of the polymer [

85]. For every gas-polymer pair and temperature analyzed, the three parameters are retrieved through a nonlinear least-square best fit of pure-gas sorption data.

The extension to multicomponent sorption of this model [

76] is based on phenomenological arguments, suggested by the theory of competitive sorption of gases on catalysts, which exhibit a Langmuir behavior: since the amount of unrelaxed free volume in a polymer is fixed and limited (swelling is not taken into account by the model), the various penetrants will compete to occupy it and, as a consequence, the sorbed concentration is expected to decrease with respect to the pure-gas case.

It is assumed that the extent of the competitive effect is controlled by the relative values of the product of the affinity constant and partial pressure (or fugacity) of each penetrant. Under the hypothesis that the affinity parameter

$b$, Henry’s constant

${k}_{D}$ and the molar density of a component sorbed inside the Langmuir sites are independent of the presence of other penetrants, the expression for the concentration (

c) of component

i in presence of a second component

j is given by Equation (4):

The characteristic gas-polymer parameters of the model are retrieved at each temperature from a least-square fit of a pure-gas isotherm using Equation (3). These parameters are subsequently used to predict the concentration of each gas in mixed-gas conditions at several compositions, making use of Equation (4) [

86]. Therefore, in Equation (4), all parameters are the same as found in Equation (3).

It is also commonplace to write Equations (3) and (4) using the partial pressure of each gas instead of its fugacity. When high pressures are considered, such as in the present study, the approximation of ideal-gas behavior is not valid. Therefore, the fugacity is generally considered instead of partial pressure, since it constitutes a more appropriate measure of the gas chemical potential, which is the driving force for gas sorption in the polymer. Moreover, when mixtures are concerned, two gases like CH

_{4} and CO

_{2} show different departures from ideality, meaning that they can have the same partial pressure, but rather different fugacity. Even though the accuracy of the pure-gas data representation with the DMS model using either variable is the essentially the same, the values of the parameters obtained using pressure or fugacity are clearly different [

87], therefore it should always be specified which variable was used in the regression, in order to enable a meaningful comparison between different parameter sets. In the context of mixed-gas sorption measurements, results are more often reported using gas fugacity, to account for the different degree of non-ideality of the components in the gas phase, therefore this was the natural choice of variable for this study as well. The accuracy of the results does not depend on this choice: it was verified that using pressure-based parameters or fugacity-based parameters yielded the same results in the mixed-gas sorption calculations. The same observation was reported also by Sanders et al. [

87,

88] in their studies on mixed-gas sorption of binary mixtures in poly(methyl methacrylate) (PMMA), and by Story et al. [

89], in their work on mixed-gas sorption in PPO. They found that, the use of pressure-based or fugacity-based DMS parameters in the calculation of mixed-gas sorption yielded very similar results, only slightly more accurate in some of the cases when fugacity was used instead of partial pressure.

The evaluation of the solubility-selectivity is also of interest (Equation (5)). This performance indicator can be calculated using the solubility coefficients of the pure gases, as it is often done when mixture data are not available (ideal case), or, more accurately, using the solubility coefficients obtained in mixed-gas sorption tests/calculations (multicomponent case).

Since a fugacity-based representation was adopted, the solubility coefficients (

$S$) are defined as the ratio of gas concentration (

c) and gas fugacity (

$f$). The fugacity of the gases at various pressures was calculated with the Peng-Robinson equation of state (EoS) [

90], both in pure- and mixed-gas condition evaluations. In the mixed-gas case, the binary parameter

${k}_{C{O}_{2}/C{H}_{4}}=0.09$ [

91] has been used in the Peng-Robinson EoS.

The DMS model does not account for the fact that the polymer matrix, unlike rigid porous materials, can swell when sorbing penetrants. Therefore, possible synergistic effects due to second-component induced swelling are not accounted for by such approach. However, ultra-high free volume glassy polymers have a limited tendency to swell, and the experimental data collected so far on mixed-gas sorption of CO

_{2}/CH

_{4} mixtures indicate that such effects are not predominant in these materials, at least for pressures of CO

_{2} below 30 bar. On the other hand, in these conditions, it was observed that the prevailing multicomponent effect is the one associated with competition during sorption. Therefore, in the cases examined here, the DMS model is expected to provide a reliable estimation of the data [

87].

## 3. Results

#### 3.1. Pure-Gas Sorption Analysis

DMS parameters for the materials analyzed, obtained from a least-square fitting procedure with the Generalized Reduced Gradient (GRG) method [

92], using concentration vs. fugacity data, are reported in

Table 1. At each temperature, a different parameter set was obtained, and no constraints were applied to enforce a temperature dependence. In the last columns of the table, the Standard Error of the Estimate (

SEE) is reported, as goodness-of-fit indicator.

SEE was used instead of the correlation coefficient (

R^{2}), because the underlying assumptions in the definition of

R^{2} are not valid in the case of a nonlinear regression model, such as the DMS model [

93,

94,

95]. The following definition was used in its calculation:

In the definition of SEE (Equation (6)), ${y}_{i,exp}$ are the experimental points, ${y}_{i,calc}$ are the corresponding values calculated with the model, n is the number of experimental points used in the regression and p is the number of parameters employed by the model. SEE is expressed in concentration units (as y) and lower values indicate a better agreement between experimental and calculated values. For the mixed-gas prediction, the reported value $\overline{SEE}$_{mix} is the average deviation from three sorption isotherms at different composition calculated with the same pure-gas parameter sets.

The results are shown in

Figure 2,

Figure 3 and

Figure 4. The model provides an excellent fit to all the pure-gas experimental data sets. Typically, more condensable penetrants, like CO

_{2} in the present case, exhibit larger affinity constants, and this was indeed observed in the parameters retrieved. In addition, it would be expected that the presence of the tetrazole CO

_{2}-philic groups in TZ-PIM would translate into higher affinity constants for CO

_{2} sorption, compared to PIM-1. However, this correlation of the parameter with the chemistry of the materials was not observed at all three temperatures, but only in the parameter set for the 25 °C case. This issue might relate to the parametrization route adopted, and it will be further discussed in

Section 4.1.

Generally,

${k}_{D}$, ${C}_{H}^{\prime}$ and

$b$ are expected to decrease as temperature increases [

76,

96,

97], consistently with their physical meaning. In the case of

${k}_{D}$ and

$b$, this trend was verified in all the cases inspected here, while for

${C}_{H}^{\prime}$ the expected trend was observed only in one case (CH

_{4} in PIM-1), while in the other cases the values fluctuated more. If the regression at each temperature is performed independently, fluctuations of the parameters have to be expected. This was noted also by Stevens et al. [

98] in their analysis of Dual Mode Sorption model parameters for CO

_{2}, CH

_{4} and N

_{2} in HAB-6FDA polyimide and its Thermally Rearranged analogues: when an unconstrained regression was performed independently at each temperature, the expected trends were followed only in some of the cases considered. In order to obtain a consistent parameter set, they imposed temperature dependence during the regression. The effect of these constraints on the mixed-gas sorption prediction will be examined in

Section 4.1.

It has been reported that the DMS parameters are sensitive to the pressure range over which they are regressed [

41], in particular

$b$ tends to decrease and

${C}_{H}^{\prime}$ to increase, if a broader regression range is considered, and, therefore, extrapolation outside the derivation range should be avoided. In this study, the whole isotherms were used in the regression and the pressure range was the same (0–35 bar) in all cases considered.

#### 3.2. Mixed-Gas Sorption: PTMSP

Figure 2 shows the experimental sorption data of CO

_{2}/CH

_{4} mixtures (10/20/50 mol.% CO

_{2}) in PTMSP at 35 °C [

24] together with the results of mixed-gas sorption calculations with the DMS parameters reported in

Table 1.

The predictions are in very good agreement with the experimental data in the case of CO_{2}, while in the case of CH_{4} at high pressure the model overestimates the concentration for the 30:70 and 50:50 mixtures, with a maximum relative deviation of 20% and 35% respectively. Nonetheless, the model captures the fact that there is competition between the gases during sorption, but also that it is less pronounced in this polymer than in the other materials analyzed here, even at high values of the fugacity of the second component.

#### 3.3. Mixed-Gas Sorption: PIM-1

Figure 3 shows the experimental sorption data of CO

_{2}/CH

_{4} mixtures (~10/30/50 mol.% CO

_{2}) in PIM-1 at 25, 35, 50 °C [

25,

26], together with the results of mixed-gas sorption calculations with the DMS model. It can be seen that, in the case of CO

_{2}, the prediction is accurate at the lowest temperature, with average relative deviations below 5%. The average relative deviations, however, are increased to 10% at 35 °C and 50 °C.

On the other hand, in the case of CH_{4}, the accuracy is lower and its trend with temperature is opposite with respect to the case of CO_{2}. At 25 °C the concentration is significantly underestimated at all compositions (the average relative deviation is 19%), while at 35 °C it is overestimated by a similar extent (the average relative deviations is 18%). At 50 °C CH_{4} sorption is still overestimated by the model, but the prediction is slightly more satisfactory, with average relative deviations of 15%. The deviation between the experimental data and the model predictions is greater than the experimental confidence intervals in several cases, therefore it does not seem to be explained fully by the uncertainty in the mixed-gas sorption measurements. Generally, for all temperatures analyzed, the lowest deviations are seen for both gases in the mixture case in which they are more abundant (50% CO_{2} and 90% CH_{4} respectively).

Not much can be done a priori to improve the quantitative accuracy of the mixed-gas prediction, because the parametrization at each temperature is independent and relies only on the accuracy of the pure-gas sorption measurements. However the effect of using a different parametrization route will be discussed in a later section.

#### 3.4. Mixed-Gas Sorption: TZ-PIM

In

Figure 4, the experimental sorption data of CO

_{2}/CH

_{4} mixtures (~10/30/50 mol.% CO

_{2}) in TZ-PIM at 25, 35, 50 °C [

48], together with the results of mixed-gas sorption calculations with the DMS model are reported. In the case of TZ-PIM, the prediction of CO

_{2} sorption is more accurate at 25 °C and 35 °C (10% average relative deviations), while it worsens at 50 °C, where the model would seem to underestimate CO

_{2} concentration both in the 30% CO

_{2} and in the 50% CO

_{2} mixtures, with average relative deviations with respect to the experimental data of 26% and 23%, which are greater than the experimental confidence intervals.

In the case of CH_{4}, at 25 °C and 35 °C DMS predictions show very good agreement with the experimental data, with average relative deviations below 5% at all compositions. Conversely, at 50 °C the model significantly overestimates the CH_{4} concentration, by as much as 32% on average.

#### 3.5. Solubility-selectivity

Ideal and multicomponent solubility-selectivities were evaluated using Equation (5). At a fixed value of the total pressure of the mixed-gas feed, the corresponding fugacity of each gas in the mixture and multicomponent concentration values were used to obtain the multicomponent solubility-selectivity. The ideal solubility-selectivity was evaluated using the same fugacity values as in the multicomponent case, but with the corresponding concentration values taken from the pure-gas sorption isotherms.

The obtained trends are compared with the experimental data in

Figure 5. For the sake of brevity, results of the comparison are shown only for the 35 °C case, but all the general observations that follow were true also for the results at 25 °C and 50 °C.

A common remark to all cases inspected is that the mixed-gas calculations show a very different dependence on mixture composition and total pressure with respect to the ideal case calculation. In particular, mixed-gas calculations generally show a much weaker dependence than the ideal ones versus both pressure and composition. It seems, therefore, that the competitive effect, accounted for in the mixed-gas calculations, tends to stabilize the calculated solubility-selectivity with respect to fluctuations in the gas pressure and composition. The physical reason beyond this behavior, that is also confirmed by experiments, is not completely clear.

In particular, for PTMSP, the calculated values are close to the experimental ones and the trends predicted by the model exhibit almost no dependence on the gas mixture composition, with the three curves collapsing onto one another, whereas the experimental data are more scattered, and resemble more the results of the ideal-case calculation, in the lower CO_{2} content cases (10–20% CO_{2}) and low pressure-range, where indeed the gas phase is closer to and ideal one.

Similarly, the gas composition dependence of solubility-selectivity is negligible in the mixed-gas calculation for PIM-1, and there is almost no dependence on pressure as well. In the case of TZ-PIM, the calculated values in the mixed-gas case show a very modest concentration and pressure dependence, although slightly more marked than in the other cases.

The calculated values for PIM-1 and TZ-PIM slightly underestimate the solubility-selectivity, but they would be preferable than simpler ideal-case estimates (left column of

Figure 5), which, on average, could lead to larger errors. Indeed, in the evaluation of the selectivity, the experimental error of both gas concentrations is combined and, therefore, this parameter inevitably has a higher uncertainty. For this reason, it is not straightforward to infer pressure and gas mixture composition dependencies from the experimental data, due to large fluctuations and absence of monotonous trends. Nonetheless, it is clear that the calculations performed with mixed-gas concentrations yield significantly more accurate results than using the corresponding pure-gas values.

## 5. Discussion

Due to the form of Equations (3) and (4), the parameters

${C}_{H}^{\prime}$ and

$b$ are strongly coupled and, therefore, a deviation of either of them can be compensated by a corresponding deviation of the other, yielding a similar overall quality of the fit. The same remark was made also by Gleason et al. [

21] in their analysis of Dual Mode parameters for mixed-gas permeation of CO

_{2}/CH

_{4} in Thermally Rearranged HAB-6FDA. In order to improve the accuracy of the calculation, they chose to incorporate mixed-gas data into the fitting procedure used to retrieve the DMS parameters. Raharjo et al. [

97] studied sorption of CH

_{4}-

nC

_{4}H

_{10} mixtures in PTMSP and they noticed a tendency of the DMS model to systematically overestimate CH

_{4} concentration in mixed-gas conditions. They subsequently re-parametrized the model, including the mixture data as well, obtaining different parameter sets from those retrieved considering only pure-gas data. In both cases [

21,

97] the representation of the mixture behavior was superior when the multicomponent data was included during the parametrization, but the procedure is clearly no longer predictive.

In order to reduce the uncertainty in the regression of the DMS parameters, Wang et al. [

101] suggested to obtain Henry’s constant independently, through the analysis of the temperature dependence of the solubility coefficient above

T_{g}, and then retrieve only

${C}_{H}^{\prime}$ and

$b$ from the best-fit of the sorption data. This approach yielded different sets from those obtained in a simultaneous regression of all three parameters and, even though those sets had lower values of the goodness-of-fit indicator, they showed improved self-consistency and the expected temperature dependence. This method, however, was not applicable to the materials studied here, and in general for glassy polymers with very high

T_{g}, for which gas solubility data above

T_{g} are not available.

Comparing the results displayed here for mixed-gas CH_{4} sorption in PIM-1 to those of CH_{4} sorption in TZ-PIM and also to those of CO_{2} sorption in PIM-1 and TZ-PIM, it is not straightforward to identify a general trend and therefore draw guidelines to mitigate the issue. The parameter set obtained by imposing a temperature dependence yielded the most reliable results, therefore, this parametrization route should be followed whenever possible, if the intended use of the parameters is that of performing predictive mixed-gas sorption calculations. If data at only one temperature are available and the quantitative accuracy of the mixed-gas sorption is necessitated, caution in the use of this model is advised.

In general, the prediction was either satisfactory for all compositions or for none: a low average SEE_{mix} was always the consequence of a similar representation of all three mixed-gas sorption isotherms. Therefore, if one could validate the parameter set adopted at least against experimental data at one composition, it should be possible to calculate the behavior at other compositions with greater confidence.