Understanding the Pathway of Gas Hydrate Formation with Porous Materials for Enhanced Gas Separation

The reason that the stoichiometry of gas to water in artificial gas hydrates formed on porous materials is much higher than that in nature is still ambiguous. Fortunately, based on our experimental thermodynamic and kinetic study on the gas hydrate formation behavior with classic ordered mesoporous carbon CMK-3 and irregular porous activated carbon combined with density functional theory calculations, we discover a microscopic pathway of the gas hydrate formation on porous materials. Two interesting processes including (I) the replacement of water adsorbed on the carbon surface by gas and (II) further replacement of water in the pore by gas accompanied with the gas condensation in the pore and growth of gas hydrate crystals out of the pore were deduced. As a result, a great enhancement of the selectivity and regeneration for gas separation was achieved by controlling the gas hydrate formation behavior accurately.

S2 flow rate of 100 mL/min. Wet samples were prepared by dropping deionized water slowly into massed dry samples with continuously stirring until the added water weight meets the requirement. 0.25g dry sample was used in thermodynamic test and wet sample preparation. 0.25g water was used in CO 2 sorption experiments at 273K.

ENTHALPY CALCULTION
The phase change pressure could be obtained from isotherm curves at different temperatures.
The enthalpy change of the state transition can be determined using the following Clausius-Clapeyron equation, where f is the fugacity, T is the temperature (K), and R is the gas constant (8.314 J mol -1 K -1 ).

THERMODYNAMIC CALCULATION AND DFT STUDY
To get thermodynamic behavior of water-CO 2 system under two-dimensional confined space, we built a 3 x 3 super cell of graphene with 1, 1.5, 2, 3 and 4 nm vacuum space to simulate the real pore of activated carbon and CMK-3. The interaction energies were computed using GGA pseudo potential method [29,30] with PBE density functional theory and dispersion correction (DFT-D3BJ) [31][32][33]. TNP basis set was selected during all simulations. More than one orientation of water molecule was considered. To lighten the calculation task, only one-layer graphite was built to mimic the inner wall, which is enough for us to discuss the experiment results. In all the simulations, the graphite atoms are fixed at their respective initial positions and represent an inert wall of carbon materials. The kinetic energy cut off was set to 400 eV.
The maximum atom force was set as 0.05 eV for convergence criterion toward the structure optimization.
The purpose of calculations was to build a diagram that presents the relationship of pressure, temperature and pore size to Gibbs free energy. The process of water replacement can be described as follows: Using free gaseous CO 2 , H 2 O and empty surface of carbon as references, the Gibbs free energy changes of interest can finally be represented as: S3 where Ho is enthalpy, S is entropy, p is pressure, k B is Boltzmann constant, T is temperature and µ is chemical potential.
The Gibbs free energies for relevant species were calculated with the expression: where E DFT is the DFT calculated electronic energy in CASTEP, E ZPE is the zero-point vibrational energy, E DF is correction of dispersion force, and ∫CpdT-TS is the correction of entropy contribution. Harmonic approximation was selected to treat the adsorbate, and PV contributions were neglected.
The entropy S can be given by: where q is partition function that consists of translation (q t ), rotation (q r ), and vibration (q v ). m is molecular weight, I is the moment of inertia about, σ is symmetry number, and ν is vibration frequency. The vibration mode and frequency of free molecules (H 2 O and CO 2 ) and adsorbed molecules are shown in Figures S13 and S15 and listed in Table S2.    Figure S3: CO 2 adsorption/desorption isotherms in bulk water at 273 K.     Table S2: Vibration frequency of free molecules and adsorbed molecules.
S: vibration of CO 2 in 5 12 cage; L: vibration of CO 2 in 5 12 6 2 cage. Figure S13: Vibration modes of adsorbed water molecule and CO 2 on graphene.
stretching bending rotating v1 v2 v3 S11 Figure S14: Optimized structures of CO 2 hydrate crystals based on calculations. Red represents oxygen atoms in CO 2 , grey represents carbon atom, and blue and light green represent oxygen and hydrogen in water, respectively.
Figure S15: C 2 H 6 adsorption/desorption isotherms on AC at 273K with different Rw values.
The filled symbols represent adsorptive branch, and the blank symbols represent desorption branch. S12 Figure S16: C 2 H 6 adsorption/desorption isotherms on AC at different temperatures when Rw = 1.5. The filled symbols represent adsorptive branch, and the blank symbols represent desorption branch. S13 Figure S18: CH 4 and C 2 H 6 adsorption/desorption isotherms on dry AC at 273 K.