Enhanced CO2/CH4 Separation Performance of a Mixed Matrix Membrane Based on Tailored MOF‐Polymer Formulations

Abstract Membrane‐based separations offer great potential for more sustainable and economical natural gas upgrading. Systematic studies of CO2/CH4 separation over a wide range of temperatures from 65 °C (338 K) to as low as −40 °C (233 K) reveals a favorable separation mechanism toward CO2 by incorporating Y‐fum‐fcu‐MOF as a filler in a 6FDA‐DAM polyimide membrane. Notably, the decrease of the temperature from 308 K down to 233 K affords an extremely high CO2/CH4 selectivity (≈130) for the hybrid Y‐fum‐fcu‐MOF/6FDA‐DAM membrane, about four‐fold enhancement, with an associated CO2 permeability above 1000 barrers. At subambient temperatures, the pronounced CO2/CH4 diffusion selectivity dominates the high permeation selectivity, and the enhanced CO2 solubility promotes high CO2 permeability. The differences in adsorption enthalpy and activation enthalpy for diffusion between CO2 and CH4 produce the observed favorable CO2 permeation versus CH4. Insights into opportunities for using mixed‐matrix membrane‐based natural gas separations at extreme conditions are provided.

6FDA-DAM were dried in vacuum oven at 110 °C overnight before being dissolved in THF to form 15 wt% polyimide/THF mixtures. The solution was mixed on a rolling mixer overnight to dissolve the polymer. The resulting casting solution was poured onto a glass plate, which was placed in a glove bag pre-saturated with THF vapor for at least 4 h. Pure 6FDA-DAM dense film was formed on a glass plate by simple casting the desired thickness (typically 75 μm) using a draw knife with appropriate specific clearance. The films were left in the glove bag overnight to allow the THF solvent to evaporate slowly and then dried in a vacuum oven at 200 °C for 20 hrs.
The submicron-sized Y-fum-fcu-MOF/THF suspension was added to the 6FDA-DAM/THF solution to form a mixed-matrix dope, which was then mixed thoroughly on a rolling mixer overnight. Excess solvent (~60 vol%) in the mixed-matrix dope was removed by slowly purging dry nitrogen to achieve a higher concentration. Y-fum-fcu-MOF/6FDA-DAM mixed matrix dense films with 20 wt% MOF loading were then formed by casting the mixedmatrix solution in the same condition as for the pure 6FDA-DAM dense films.

Characterization of Materials
The dense films for scanning electron microscope (SEM, Hitachi, SU8010) test were prepared by first soaking films in hexane and then cryogenically fracturing in liquid nitrogen to preserve their microstructures. Wide-angle X-ray diffraction (WAXD) was measured on a Panalytical Empyrean diffractometer operating with a Cu Kα radiation at a wavelength of 1.54 Å, in a 2θ range of 5 -50°. Thermogravimetric analysis (TGA) and derivative weight data were recorded on a TA Q-500 analyzer at a heating rate of 10 °C/min under a nitrogen atmosphere, using ~10 mg sample. The MOF loading in mixed-matrix dense films was determined by a TGA method.

Pure Gas Sorption Tests
Gas sorption isotherms at pressure up to 14 bar and temperatures ranging from 233 K to 308 K were measured using a pressure decay method. A BTZ-475 benchtop temperature chamber (ESPEC North Amercia, INC., Hudsonville, MI), which accurately controlled temperature within the range of 203 K to 453 K with 0.5 K fluctuation, was used for sorption and permeation measurement.
Gas adsorption amount in Y-fcu-MOFs and dense films was calculated from the pressure change before and after sorption with a small experimental error of less than ± 5 % expected from the accurate pressure measurement and volume calibration.

Gas permeation Tests
The gas permeation was conducted in a variable pressure, constant-volume apparatus.
The membrane was housed between an upstream, capable of high-pressure gas introduction, and a downstream, which is kept under vacuum until experiments were initiated. The permeation temperature for CO 2 and CH 4 ranges from 233 K to 338 K.
A 50/50 (molar) CO 2 /CH 4 mixture was used for mixed-gas permeation of Y-fum-fcu-MOF/6FDA-DAM membrane. The downstream composition was determined using a gas chromatograph (Varian 450-GC). The stage cut (the flow rate ratio of permeate to feed) was maintained below 1% to avoid concentration polarization on the upstream side of the permeation cell, keeping the driving force across the membrane constant throughout the course of the experiment.

Permeability, Solubility, Diffusivity and Energetic Factors
Permeability and selectivity were used to characterize the membrane separation performance. The permeability, P i , describes the intrinsic gas separation productivity of a dense film membrane and is defined by the flux of penetrant i, n i , normalized by the membrane thickness, l, and the partial pressure or fugacity difference, Δf i , across the membrane, viz., In order to estimate pure gas permeability, the slope of the permeate pressure vs. time (df/dt); membrane thickness (l); downstream volume (V); operating temperature (T); and transmembrane pressure or fugacity difference (Δf) were used with Equation (3): The mixed gas permeability coefficient of component i (P i ) is calculated using its mole fraction in the permeate (x i ) and the transmembrane fugacity difference (Δf i ): The fugacity coefficients are calculated using Peng-Robinson equation-of-state and the SUPERTRAPP program developed by NIST. The CO 2 /CH 4 selectivity, α ij , is determined by the ratio of the fast gas (i, CO 2 ) permeability to the slow gas (j, CH 4 ): Permeability can also be expressed as the product of the average effective diffusion coefficient (Di) and sorption coefficient ( ̃ ) of a given gas i within the membrane: The sorption coefficient represents the thermodynamic contribution to transport, which can be measured independently by pressure-decay sorption. The sorption coefficient can be expressed as: where c i is the concentration of a gas adsorbed in the sample, and f i is the corresponding upstream fugacity driving force of component i. In this work, the adsorbed gas concentration in films was described by the dual-mode sorption model, which is given as: The effective diffusion coefficient (Di) in the membrane was calculated from the independently measured permeability (P) and sorption coefficient ( ̃ ): The temperature dependence of sorption can be described by the van't Hoff equation: where ̃ is the pre-exponential factor, ΔH S is the sorption enthalpy. The temperature dependence of the diffusion coefficient follows an Arrhenius relationship: where is the pre-exponential factor, E D is the activation energy for diffusion. The E D parameter represents the energy required for a penetrant to jump between adsorption sites within the material, which is primarily dependent on penetrant size and shape. The temperature dependence of permeability also follows an Arrhenius relationship: where P 0 is the pre-exponential factor, and E P is the effective activation energy of permeation, E P , E D and ΔH S have the following relationship: (S13) Expressing the diffusion temperature dependence in terms of the activation enthalpy, the diffusion selectivity can also be expressed as the product of energetic selectivity and entropic selectivity based on transition state theory by replacing the activation energy approximately with the activation enthalpy:

Analysis and validation of diffusion selectivity based on transition state theory
As shown above, neglecting any small differences in jump lengths and difference between average activation energy and average activation enthalpy in the material, the diffusion selectivity can be expressed as the product of energetic selectivity and entropic selectivity: Furthermore, Euqation S15 can also be expressed as: By assuming E DA , E DB , S DA and S DB are constants, ( ) and (K -1 ) have a linear relationship ( Figure S1 The total diffusion selectivity , as a production of energetic selectivity and entropic selectivity, is also slightly temperature dependent. The differential values of energetic and entropic energies between CO 2 and CH 4 derived from Equation S17 matches well with those of traditional method. Taking CO 2 and CH 4 diffusion in 6FDA-DAM as example, (lnD CO2 -lnD CH4 ) and 1/T have the following relationship: Here, In Figure 2 in the manuscript, the calculated diffusion activation energy for CO 2 and CH 4 are 10.88 kJ/mol and 21.06 kJ/mol, respectively. Thus we can get (S19) which indicates the good match of our results using different methods.