# Graphene-Based Membranes for CO2/CH4 Separation: Key Challenges and Perspectives
## Mixed-Matrix Membranes
### Perspectives  
To address the above challenges, our group had recently derived an empirical filler enhancement index known as the \(F_{\text{index}}\) to offer a new approach in quantifying and evaluating the effectiveness of filler materials in the mixed-matrix membranes (see Reference [5] for the full derivation of the \(F_{\text{index}}\)). This is realized by normalizing the CO\({}_{2}\) permeability and CO\({}_{2}\)/CH\({}_{4}\) selectivity of the mixed-matrix membrane by the unfilled polymeric membrane. As a result, the effect brought by the polymer matrix is decoupled to better illuminate the impact brought by the filler materials. The \(F_{\text{index}}\) is defined as follows:  
\[F_{\text{index}}\ =\ \ln\biggl{(}\frac{P_{composite}}{P_{un\_fill}}\biggr{)}\ +\ 2.636\ln\biggl{(}\frac{\alpha_{composite}}{\alpha_{ unfilled}}\biggr{)}, \tag{1}\]  
where \(P_{composite}\) and \(P_{un\_fill}\), in this context, refer to the CO\({}_{2}\) permeability of the graphene-based mixed-matrix and unfilled membranes, respectively. The \(\alpha_{composite}\) and \(\alpha_{un\_fill}\) refer to the CO\({}_{2}\)/CH\({}_{4}\) selectivity of the graphene-based mixed-matrix membranes and unfilled membranes, respectively. The \(F_{\text{index}}\) is a composite metric which encompasses both the permeability and selectivity enhancement terms, making comparison of the effectiveness of graphene-based fillers more accurate, holistic and straightforward. For example, when 0.25 wt% of GO nanosheets were incorporated into a Pebax matrix, the CO\({}_{2}\) permeability decreased by 22% but the selectivity was enhanced by 83% [14]. But, when 0.8 wt% of imidazole-functionalized GO (Im-GO) nanosheets were incorporated into the Pebax matrix, the CO\({}_{2}\) permeability instead increased, while the selectivity decreased by 4%, respectively [69]. Here, to judge that the GO nanosheets were more effective than the Im-GO or vice versa was rather contentious. This is because there is a need to grapple with two terms and this has made effectiveness evaluation of graphene-based filler sometimes contradicting. With \(F_{\text{index}}\), however, such controversy is non-existent. By integrating the two terms into one single rating, the metric and thus filler effectiveness evaluation is easier to understand and implement. In this case, GO nanosheets scored a \(F_{\text{index}}\) value of 1.35, while Im-GO nanosheets had a value of \(-0.07\), making GO a more effective filler than Im-GO for the Pebax matrix. The \(F_{\text{index}}\) is also formulated to give a stronger emphasis on the selectivity enhancement by amplifying the term with an enhancement coefficient of 2.636, which is the upper bound slope of the 2008 CO\({}_{2}\)/CH\({}_{4}\) Robeson plot [75]. This is to recognize filler materials that have the capacity to enhance the CO\({}_{2}\)/CH\({}_{4}\) selectivity. At the same time, through emphasizing the selectivity enhancement, we can discredit filler materials that enhance gas permeability at the expense of membrane selectivity, which is commonly associated with membrane defects or nonidealities formation. Furthermore, with the integration of the enhancement coefficient, the \(F_{\text{index}}\) is demonstrated to be accurate and reliable as evidenced by its ability to identify the top 20 filler materials out of a database with 470 sample size (Figure 4b-d and see reference [5]). However, the \(F_{\text{index}}\) is yet to be perfected [5]. For example, critical qualities including polymer/filler compatibilities, mechanical properties, physical aging effects should ideally be taken into account [76; 77; 78]. But, unlike separation performance data which are fervently documented, data of these qualities are scarce and hard to quantify, making the feat of perfecting the \(F_{\text{index}}\) challenging. In the process of developing the \(F_{\text{index}}\), the \(F_{\text{index}}\) values were also empirically categorized into five different labels to give meaningful interpretations of the fillers.