## Materials and Methods  
### Maxwell Model  
The first mathematical studies of gas transport description through the MMMs concerned the ideal structure of these hybrid membranes, both their polymer matrix and the dispersed phase. The contact between these two phases at the polymer\(-\)particle interface is assumed to be defect free. This theoretical two-phase morphology typically corresponds to the ideal Maxwell model, which was created to describe thermal or electrical conductivity in composite media. A defect-free polymer\(-\)particle contact during the introduction of dispersed fillers in a polymer matrix is, however, difficult to achieve. The incorporation of filler particles can modify the properties of the neighbouring polymer phase, which affects the hybrid membrane separation performance [28; 29; 30]. Considering the different types of effects associated with the interaction of the inorganic and organic phase, five various morphologies could be distinguished, that is: the ideal, void, rigidified, with pore blocking, and agglomeration combined with pore blocking. Taking into account the influence of various morphologies on gas transport properties, different effects could be observed. In the case of the voids' formation between the filler particles and the polymer matrix, an increase in gas permeability can be observed at low filler loading. However, no significant changes in selectivity are observed for such membranes. In turn, in membranes with a high filler loading and poor spatial distribution of particles, the void surrounding clusters of aggregated particles could be created, and their selectivity may drastically decrease. Rigidification can usually increase the selectivity coefficient, while the permeability coefficient decreases. In turn, when the pore blocking increases, the permeability coefficient tends to decrease, while the selectivity coefficient may be higher than that in the pure polymer matrix. Mixed-matrix membranes having both large and small pore fillers represent a case that is a combination of channel defects and pore blocking [9]. [PERSON]'s equation came from estimating the dielectric properties of the composite materials. Initially, it was considered as a simple and effective tool for predicting mixed-matrix membranes' gas transport properties [1; 31]. The theoretical two-phase morphology created as a first one is used here to predict the gas permeation through hybrid membranes and allows us to obtain approximate process parameters, such as the permeation coefficient \(P\) and the selectivity coefficient \(\alpha\). This model was developed to predict the effective permeability of the gas penetrant \(P\) through these hybrid membranes as a function of \(\lambda_{dm}\), which is a ratio of continuous phase \(P_{m}\) (polymer matrix) permeability and dispersed phase \(P_{d}\) (filler particles) permeability, as well as a volume fraction of the dispersed filler phase \(\Phi\). For a given penetrant (1), the permeability coefficient (\(P_{1}\)) can be estimated as the product of diffusivity coefficient (\(D_{1}\)) and solubility coefficient (\(S_{1}\)) [9]:  
\[P_{1}=D_{1}S_{1}, \tag{1}\]  
where:  
\(D_{1}\)--a diffusion coefficient of gas penetrant no. 1 (cm\({}^{2}\)/s), and  
\(S_{1}\)--a sorption coefficient of gas penetrant no. 1 (cm\({}^{3}\)STP/ cm\({}^{3}\)cmHg). The sorption coefficient S\({}_{1}\) is a measure of the size of the gas sorption in a material of membrane. It is characterized by division of the penetrant between the membrane and the outer phase in equilibrium. The ideal selectivity coefficient, \(\alpha_{1/2}\), is the ratio of the permeability coefficient of the gas penetrant no. 1 to the permeability coefficient of gas no.