## 3 Results and Discussion
In the first step (Equation (8)) is determined the gas permeability in a single core-shell particle \(P_{\mathit{eff}}\) (filler is a core and interfacial layer, the matrix treated as a shell) [28]:  
\[P_{\mathit{eff}}=\ P_{I}\left[\frac{2(1-\Phi_{s})+(1+2\Phi_{s})\Big{(}\frac{P_ {\mathit{eff}}}{P_{I}}\Big{)}}{(2+\Phi_{s})+(1-\Phi_{s})\Big{(}\frac{P_{ \mathit{eff}}}{P_{I}}\Big{)}}\right], \tag{8}\]  
where:  
\(\Phi_{s}\)--the volume fraction of filler core particle in the combined volume of core and interfacial shell (in a single core-shell particle),  
\(P_{I}\)--the permeability of the gas penetrant in the interfacial shell, and  
\(P_{\mathit{eff}}\)--the effective permeability of a gas penetrant in a single core-shell particle. In the second step (Equation (9)), the Maxwell model is applied again to obtain the permeability of three-phase MMM \(P_{r}\) (taking the permeability \(P_{\mathit{eff}}\) from the first step as a gas permeability through filler particles) [28]:  
\[P_{r}=\ \frac{P}{P_{m}}=\ \left[\frac{2(1-\Phi)+(1+2\Phi)\Big{(}\frac{P_{ \mathit{eff}}}{P_{m}}\Big{)}}{(2+\Phi)+(1-\Phi)\Big{(}\frac{P_{\mathit{eff}}} {P_{m}}\Big{)}}\right], \tag{9}\]  
where:  
\(\Phi\)--the volume fraction of total dispersed phase (core-shell particles) in the whole membrane, and  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Membrane**} & \multirow{2}{*}{\(\Phi\)} & \multicolumn{3}{c}{**Experimental Data**} & \multicolumn{3}{c}{**Theoretical Data**} & \multirow{2}{*}{**AARE (\%)**} & \multirow{2}{*}{**RMSE (\%)**} \\ \cline{3-3} \cline{5-8}  & & \(\alpha_{\text{O2/N2}}\) & & & & & & \\ \hline M1 & 0.20 & 9.50 & 0.47 & 9.40 & 0.55 & & \\ M2 & 0.30 & 10.70 & 0.40 & 10.87 & 0.57 & 30 & 15 \\ M3 & 0.40 & 12.45 & 0.37 & 12.63 & 0.60 & & \\ \hline \end{tab\(P_{m}\)--the permeability of gas components in a polymer matrix (Barrer). In order to form successful MMMs, one must maintain flexibility during membrane formation (the use of high temperatures) and interaction between polymer and sieve phase (chemical coupling) [39; 51; 52]. In order to ensure flexibility, [PERSON] and [PERSON] proposed a different polymer matrix, PVAc, which is an extremely flexible material with a low glass transition temperature (35 \({}^{\circ}\)C), an affinity for alumina (expected better contact between PVAc and the aluminosilicate-type 4A zeolite) and reasonable gas transport properties [41]. In fact, for 15% loading was noted an improvement in O\({}_{2}\)/N\({}_{2}\) selectivity and a slight decline in permeability (Table 3). However, for higher loading (25% and 40%), most of the membranes were full of defects, which negatively affected their transport properties (Figure 7). It was caused by poor adhesion and dispersion at higher filler loading, which led to formation of nonselective nanochannels. Compliance between experimental and theoretical results has been observed to a certain extent. However, the deviation between the results became more significant with increasing filler loading. The used Maxwell model assumes uniform gas permeability throughout the polymer matrix. That is why the observed polymer chains' mobility inhibition near the interface would be more significant as the sieve loading rises and leads to permeability reduction. The ideal system (the basic assumption of the proposed Maxwell model) would be one where the sieve has a stronger affinity for the polymer than the solvent, while the polymer has a stronger affinity for the zeolite surface than the solvent. The main keys to the success of the mixed-matrix materials are the choice of proper molecular sieve and polymer matrix for the given separation, promotion of the adsorption of the polymer on the sieve surface, and polymer flexibility during membrane formation. One of the proposed solutions could be a priming of the zeolites with a thin polymer layer, the annealing at higher temperatures or application of suitable plasticizing mechanisms [41]. An analogous analysis was carried out for the experimental data obtained by the authors.