# Modified Dual-Site Langmuir Adsorption Equilibrium Models from A GCMC Molecular Simulation
## 3 GCMC Molecular Simulation
### Simulation Methods  
The adsorption characteristics of CO\({}_{2}\) were determined using Monte Carlo (MC) simulations in a grand canonical ensemble (\(\mu\)VT) with the RASPA simulation program for adsorption properties of porous materials [30; 31; 32]. MC is a method, which combines the probability distribution principle of statistical mechanics and constructs different molecular configurations through the random movement of particles in a system to obtain the statistical thermodynamic information of the system, and it can be applied to a variety of systems. In the grand canonical ensemble, the chemical potential, temperature, and volume were kept constant, while the numbers of particles in the system were changed. The ensemble consisted of a large number of open systems, each of which was in an internal equilibrium state and could exchange matter (particles) and energy with other boxes. Therefore, the characteristics of the grand canonical ensemble are especially suitable for gas adsorption simulation. Chemical potentials were converted to pressures with the Peng\(-\)Robinson equation of state (PR-EOS). The form of the PR-EOS was shown as follows:  
\[P=\frac{RT}{V-b}-\frac{a}{V^{2}-2 Vb-b^{2}}, \tag{1}\]  
where \(P\) is the adsorption pressure, \(R\) is the gas constant, \(T\) is adsorption temperature, \(V\) is the volume of the gas, \(a\) is a parameter related to the temperature obtained by RASPA, and \(b\) is a parameter related to the chemical potential obtained by RASPA. The adsorption isotherms that are made by different adsorption capacities can be obtained at different pressures. In general, the experimental measurement represents the excess adsorbed amount, and the GCMC simulation result represents the absolute adsorbed amount. In this paper, the absolute adsorbed amounts that were obtained by GCMC simulations were converted to excess adsorbed amounts in order to perform a comparison with the experimental data, and the amount of excess adsorbed molecules \(N_{\mathrm{ex}}\) was calculated using Equation (2):  
\[N_{\mathrm{ex}}=N_{ab}-V_{P}\rho, \tag{2}\]  
where \(N_{\mathrm{ab}}\) is the amount of absolute adsorbed molecules, \(V_{\mathrm{p}}\) is the pore volume of the adsorbent, and \(\rho\) is the density of the adsorbate calculated using the PR-EOS at a given temperature and pressure. In the simulation, a rigid structure model was adopted for the adsorbate molecules and the MIL-101 (Cr) framework. That is to say, the bond length, bond angle, and atomic model of the adsorbate molecules did not change during the adsorption process, and the atom framework of the adsorbent MIL-101 (Cr) did not shift during the adsorption process. The numbers of unit cells of the frameworks that were adopted in the simulation were \(1\times 1\times 1\), and periodic boundary conditions were applied in \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Atom** & **Charge (eV)** \\ \hline \multirow{6}{*}{adsorbent} & Cr1 & 1.28 \\ \cline{2-3}  & Cr2 & 1.44 \\ \cline{2-3}  & O1 & \(-\)0.76 \\ \cline{2-3}  & O2 & \(-\)0.58 \\ \cline{2-3}  & O3 & \(-\)0.65 \\ \cline{2-3}  & O4 & \(-\)0.67 \\ \cline{2-3}  & C1 & 0.74 \\ \cline{2-3}  & C2 & \(-\)0.12 \\ \cline{2-3}  & C3 & \(-\)0.10 \\ \cline{2-3}  & H1 & 0.35 \\ \cline{2-3}  & H2 & 0.16 \\ \hline \multirow{2}{*}{adsorbate} & C(CO\({}_{2}\)) & 0.70 \\ \cline{2-3}  & O(CO\({}_{2}\)) & \(-\)0.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The charges of adsorbents and adsorbates. all three dimensions. In this work, a total of \(2\times 10^{6}\) steps were used. Half of these steps were used for equilibration, and the remaining steps were used for subsequent calculations and result statistics. For the CO\({}_{2}\) molecules, four types of moves (random insertion, deletion, translation, and rotation) were used, and every possible move was given equal probability. In this work, the selected adsorption pressure ranged from 1 to 300 kPa.