Experimental Mixed-Gas Permeability, Sorption and Diffusion of CO2-CH4 Mixtures in 6FDA-mPDA Polyimide Membrane: Unveiling the Effect of Competitive Sorption on Permeability Selectivity

The nonideal behavior of polymeric membranes during separation of gas mixtures can be quantified via the solution-diffusion theory from experimental mixed-gas solubility and permeability coefficients. In this study, CO2-CH4 mixtures were sorbed at 35 °C in 4,4′-(hexafluoroisopropylidene)diphthalic dianhydride (6FDA)-m-phenylenediamine (mPDA)—a polyimide of remarkable performance. The existence of a linear trend for all data of mixed-gas CO2 versus CH4 solubility coefficients—regardless of mixture concentration—was observed for 6FDA-mPDA and other polymeric films; the slope of this trend was identified as the ratio of gas solubilities at infinite dilution. The CO2/CH4 mixed-gas solubility selectivity of 6FDA-mPDA and previously reported polymers was higher than the equimolar pure-gas value and increased with pressure from the infinite dilution value. The analysis of CO2-CH4 mixed-gas concentration-averaged effective diffusion coefficients of equimolar feeds showed that CO2 diffusivity was not affected by CH4. Our data indicate that the decrease of CO2/CH4 mixed-gas diffusion, and permeability selectivity from the pure-gas values, resulted from an increase in the methane diffusion coefficient in mixtures. This effect was the result of an alteration of the size sieving properties of 6FDA-mPDA as a consequence of CO2 presence in the 6FDA-mPDA film matrix.

where C is the gas sorption uptake, fugacity (f) instead of pressure was used to account for gas non-idealities. In 6FDA-mPDA, CO2 has higher values of the Henry's law coefficient and the Langmuir holes affinity parameter (KD and b, respectively) than methane (Table S1). The Langmuir capacity parameter (CH ' ) is often used as a measure of the non-equilibrium excess free volume frozen in the glassy polymer. When → 0 we obtain the gas solubility coefficient at infinite dilution (Equation S2): In this work, pure-gas solubility coefficients at infinite dilution could also be estimated via Equation S2 owing to the good quality of the prediction of pure-gas uptakes via the DMS model. In all cases, pure-gas uptake data were digitalized from the literature and fitted with MATLAB® software. In this way, the confidence interval of each parameter could be used to estimate the standard error of pure-gas solubility coefficients and solubility selectivities at infinite dilution. Table S1. Dual-mode model parameters of methane and carbon dioxide in 6FDA-mPDA for pure-gas sorption at 35 °C.

Solubility selectivity analysis
For both AO-PIM-1 and polynonene scattering at low pressure of mixed-gas uptake data (see circled data points Figure S1) limited the quality of the linear fitting of CO2 mixed-gas solubility coefficient vs. CH4 mixed-gas solubility coefficient data. Data of Figure S1 were fitted by fixing the slope to the value of pure-gas solubility selectivity at infinite dilution. Table S2 compares the values of solubility selectivity at infinite dilution estimated from pure-gas and mixed-gas uptakes of all polymers discussed in this work. 6FDA-mPDA, PIM-1, and TZ-PIM-1 mixed-gas and pure-gas α o values were in agreement. Although the difference was small, in the case of PTMSP and PPO we found a limited agreement. The number of data for 6FDA-TADPO was not enough for reliable linear fitting of CO2 vs. CH4 mixed-gas solubility coefficient data; the standard error of both CO2 pure-gas solubility coefficient and CO2/CH4 pure/mixed gas solubility selectivity values at infinite dilution-estimated via the propagation of error theory-were too large (hence, 6FDA-TADPO was not listed in Table S2). Figure S1. Data of CO2 experimental mixed-gas solubility coefficient vs. CH4 mixed-gas solubility coefficient of AO-PIM-1 and polynonene [2]. For both interpolations, the slope was fixed at the value of pure-gas solubility selectivity at infinite dilution.

Fitting/interpolation of mixed-gas sorption data
The extension to a binary mixture of the DMS model (DMS-mix) has the following form: S3b The derivation of this model is discussed elsewhere [6]. The DMS-mix model equations and pure-gas sorption parameters (Table S1) were used to estimate all insert curves plotted in Figure 4a and 4b of the main body of this work.
Solubility coefficients predicted by the DMS-mix have the following relations:
Because we were interested in understanding the effect of polymer/gas1/gas2 interactions during sorption in 6FDA-mPDA, we marked the behavior of and bi parameters (DMS-mix) through the following empirical relation: where A could be either or ; is the molar concentration of the j-gas (for i-j mixture) in the sorption atmosphere-for example, to track the deviation of 2 from 2 one should vary 4 .
When < 0, increases with from the pure-gas value ( ) and the opposite happens when > 0-this is also graphically shown in Figure S2.  . This data fitting procedure was named DMS-mix-mod.  The result of data fitting via the DMS-mix-mod is shown in Figure S3 and is compared with the prediction of the DMS-mix (Equation 3). Except for some data points, the DMS-mix-mod predicts the experimental data of pure-and mixed-gas uptakes reasonably well. The four parameters that resulted from the fitting analysis through the DMS-mix-mod are listed in Table S3. For , 4 > 2 and both are positive; hence-at high pressures-the uptake of both gases negatively deviates from the pure-gas values, whereas the CO2/CH4 solubility selectivity is enhanced in the mixture and increases with pressure. This behavior was observed in Figure 5a and Figure 5b of the main body of this paper and coincides with the results of mixed-gas sorption in rubbery membranes discussed elsewhere [7,8]. In the case of , 4 is slightly higher than 2 and both are negative-the small difference between these two parameters does not allow clear conclusions.
Finally, Figure S4 and Figure S5 show all gas sorption data interpolated via DMS-mix-mod and via linear interpolation, respectively. The curves in red are the curves of 50 mol% concentration at equilibrium. These curves were used to estimate the mixed-gas diffusivity data that are discussed in the main body of this work. Figure S4. CH4 (a) and CO2 (b) mixed-gas uptakes in 6FDA-mPDA. Surfaces were obtained via the DMS-mix-mod fitting. The DMS-mix-mod allowed us to predict the solubility behavior beyond the region covered by experimental data.