Table 6 Applied models for predicting permeability of mixed matrix membranes containing non-porous silica
From: Silica applied as mixed matrix membrane inorganic filler for gas separation: a review
| Type of polymer | Applied model | Goal | Ref |
|---|---|---|---|
| Polyurethane | Bruggeman model \( {\left(\frac{P_{ps}}{P_l}\right)}^{2/3}=\left(1-{\varphi}_s\right) \) | Evaluate the permeability of the gas between the polymer and the void space of the nanocomposite. | [53] |
| Modified Knudsen model \( P=D\times S=9.7\times {10}^{-5}\times \frac{r_d+{l}_i}{RT}\sqrt{\frac{T}{M_A}}\times \) | Calculate permeability of gas within the void region | ||
| Modified Maxwell model \( {P}_M={P}_C\left[\frac{P_d+2{P}_C-2\left(\frac{\varnothing_D}{\varnothing_S}\right)\left({P}_C-{P}_d\right)}{P_d+2{P}_C+\left(\frac{\varnothing_D}{\varnothing_S}\right)\left({P}_C-{P}_d\right)}\right] \) | Evaluate the total gas permeability | ||
| Polyurethane | Modified Maxwell model \( P= DS=\frac{D_A}{\gamma }{\left(1+\frac{\varphi_f}{2}\right)}^{-1}\left[{S}_A\left(1-{\varphi}_f\right)+{S}_f{\varphi}_f\right] \) | Calculate gas permeability of composite membrane | [59] |
| Higuchi model \( P={P}_A\left(1-\frac{6{\varphi}_f}{4+2{\varphi}_f-{K}_H\left(1-{\varphi}_f\right)}\right) \) | Predict the gas permeability of hybrid membranes in the presence of impermeable and spherical fillers like silica particles. | ||
| Polyetherurethane, polyesterurethane | Bruggeman model \( {\left(\frac{P_{ps}}{P_l}\right)}^{2/3}=\left(1-{\varphi}_s\right) \) | Evaluate the permeability of the gas between the polymer and the void space of the nanocomposite. | [44] |
| Modified Knudsen model \( P=D\times S=9.7\times {10}^{-5}\times \frac{r_d+{l}_i}{RT}\sqrt{\frac{T}{M_A}}\times \) | Calculate permeability of gas within the void region | ||
| Chiew and Glant model \( \frac{P_M}{P_C}=1+3\beta {\varphi}_{ps}+K{\varphi}_{ps}^2 \) | Evaluate the total gas permeability | ||
| Glassy polymers | Maxwell model \( {P}_i={P}_{i,P}\left(\frac{1-{\varnothing}_F}{1+0.5{\varnothing}_F}\right) \) | Estimate the permeability of a permeable medium filled with a low content of spherical, impermeable particles. | [64] |
| Free volume model \( \frac{\alpha_{D,M}}{\alpha_{D,P}}=\mathit{\exp}\left[\left({B}_a-{B}_b\right)\left(\frac{1}{FFV_P^0}-\frac{1}{FFV_{P, MM}^0}\right)-{F}_a\left({FFV}_{P, MM}^0\bullet {\varOmega}_{aP, MM}-{FFV}_P^0\bullet {\varOmega}_{aP}\right)+{E}_a\left({\varOmega}_{aP, MM}-{\varOmega}_{aP}\right)\right] \) | Estimate the variation in diffusivity selectivity due to the addition of fumed silica | ||
| Poly ether block amide | Van Amerongen and Van Krevelen relations \( {P}_n=\frac{D_0{S}_0}{{\left(\frac{E_n}{E}\right)}^K\left(1+0.5{\varnothing}_n\right)}{\left(\frac{1-{X}_{c,n}}{1-{X}_c}\right)}^2\left(1-\beta {\varnothing}_n\right)\mathit{\exp} \) | Estimate permeability of the penetrants through mixed matrix membranes | [65] |