## 3 Results and Discussions  
To evaluate our calculation, we firstly compare our DFT results with those from the Monte Carlo (MC) simulation, which is carried out by [PERSON] et al. [15]. The results for a binary mixture with equal diameter are shown in Figure 1a. In the calculation, we have chosen the same parameters as those in the MC simulation. That is, \(\rho_{b}^{bulk,L}=0.7695\), \(\rho_{s}^{bulk,L}=0.0641\). The comparison suggests that on both sides of the membrane, our results agree well with those from the MC simulation. Moreover, we have also compared our analytical results with those from the Ornstein-Zernike integral equation calculation, which is carried out by [PERSON] et al. [9]. The comparison in Figure 1b suggests that when \(R=1\), the analytical results agree with those from the HNC closure, while when \(R=10\), they agree with those from the PY closure. In the following, some further calculations are performed to achieve richer information about the structural properties of the fluid mixture confined by the SPM. In Figure 2, the density profiles and the structural properties of the mixture are calculated under different conditions of \(x\) and \(R\). The results in Figure 2a suggest that when the magnitude of \(x\) is small, the difference in densities on the right and left side of the SPM is rather large, while it is suppressed as \(x\) increases. This can also be found in Figure 2c, which plots \(\rho_{s,R/L}\) with the variation of \(x\) in different cases of total volume fraction \(\eta_{tot}\). This can be understood by the equality of the \(\mu_{s}\) on both sides of the SPM, which indicates that for a smaller magnitude of \(x\), a larger density difference is required to maintain the chemical equilibrium on both sides of the SPM. Figure 2b suggests that for the given \(\eta_{tot}\) and \(x\), a smaller magnitude of \(R\) corresponds to a larger difference between \(\rho_{s}^{bulk,R}\) and \(\rho_{s}^{bulk,L}\), while as \(R\) is increased, the difference is reduced, as is also shown in Figure 2d. This may arise from the diminished contribution of the chemical potential from the big component because larger \(R\) leads to a smaller \(\rho_{b}^{bulk,L}\). Moreover, comparison between the curves in Figure 2d suggests that for a larger \(x\), the influence of the size asymmetry is reduced. Besides the DFT results shown in Figure 2c,d, the analytical results for \(\rho_{s,R/L}\) are also plotted to make comparison. It is obvious that the results from the two routes agree rather well with each other. Figure 1: **(a)** Comparison of our density functional results for binary mixture with those from the MC simulation carried out by [PERSON] et al. [15]. **(b)** Comparison of our analytical results for \(\rho_{s,R/L}\) with those from the Ornstein-Zernike integral equation calculation which is carried out by [PERSON] et al. [9]. Note that the analytical calculation is performed with \(\eta_{tot}=\frac{2}{15}\pi\). Figure 2: (**a**, **b**): Density profiles of the mixture under different conditions of \(x\) and \(R\). (**c**, **d**): \(\rho_{s,R/L}\) under different conditions. (**e**, **f**): \(\rho_{b,con}\) under different conditions. Note that in (**a**), the results for \(x=0.2\), \(0.4\), \(0.6\) have been consecutively shifted upward by \(0.2\), and that in both (**a**, **b**), the solid and dashed lines correspond respectively to the results for the small and big components. In (**c**, **f**), the solid lines is the results from the DFT calculations, while the symbols from the analytical method. The contact density of the big component, say \(\rho_{b,con}\), is another important structural property in the system. Therefore, it has also been calculated by employing both DFT and analytical methods, which is plotted in Figure 2e,f. It is obvious that for the given \(\eta_{tot}\) and \(R\), \(\rho_{b,con}\) decreases with the increase of \(x\). In contrast, for the given \(\eta_{tot}\) and \(x\), its value increases with \(R\).