## 3 Results and Discussions
This can be understood by the depletion mechanism, which ascribes the layering of big particles near the SPM to the depletion from the small particles. When \(\eta_{tot}\) and \(R\) are fixed, a larger value of \(x\) corresponds to less big particles in the bulk, which leads to the decrease of \(\rho_{b,con}\). On the contrary, the increase of \(\eta_{b}\) and \(R\) benefits the depletion, which enhances the layering of the big component, and then results in the increase of \(\rho_{b,con}\). As shown above, the impenetrability of the big particles is responsible for the density difference of small particles on both sides of the SPM, that is, \(\rho_{s}^{bulk,L}<\rho_{s}^{bulk,R}\). Actually, another characteristic density, the cross density \(\rho_{s,cro}\), may transfer additional information to understand the structural properties of the mixture. Therefore, in Figure 3, we have studied \(\rho_{s,cro}\) under different conditions of \(x\) and \(R\) with the DFT calculations. Obviously, the dependence of \(\rho_{s,cro}\) on \(x\) and \(R\) is the same as that of \(\rho_{s,R/L}\). Moreover, comparing Figure 3a,b with respectively Figure 2c,d, one can always find \(1<\rho_{s,cro}<\rho_{s,R/L}\), which indicates that the cross density is between \(\rho_{s}^{bulk,L}\) and \(\rho_{s}^{bulk,R}\). Note that \(\rho_{s,cro}\) can not obtained by the analytical method, because the required weighted densities \(n_{v}^{(\alpha)}(z)\) with \(z\in[z_{m}-\frac{\sigma_{s}}{2},z_{m}+\frac{\sigma_{s}}{2}]\) is unknown in an analytical route. From Figure 2a,b, it is obvious that the density gradient of small component crosses the SPM with unequal contributions on the two sides of the membrane, which reflects indirectly the structure of the mixture. Therefore, to understand this aspect of the structural property, we define a dimensionless parameter \(\Delta\):  
\[\Delta=\frac{\rho_{s}(z_{m})-\rho_{s}^{bulk,L}}{\rho_{s}^{bulk,R}-\rho_{s}^{ bulk,L}}. \tag{10}\]  
The results for \(\Delta\) in Figure 3c,d show that for the chosen parameters, one can always find \(\Delta>\frac{1}{2}\), which suggests that most of the density gradient distributes on the _left_ side of the SPM. This may originate from the impenetrability of the big particles, whose direct influence on the chemical potential \(\mu_{s}\) is restricted to the region \(z<z_{m}\). In addition, with the increase of \(x\) and \(R\), the value of \(\Delta\) is also increased. Strictly speaking, in the system under consideration, only the big component is confined by the SPM because the small component can cross the membrane freely. In such a confined fluids, the excess adsorption is another fundamental quantity that reflects the influence of the confining geometry. Therefore, in Figure 4, we have calculated the excess adsorption of the big component:  
\[\Gamma_{ex}=\frac{1}{A}\int[\rho_{b}(\mathbf{r})-\rho_{b}^{bulk,L}]d\mathbf{r}, \tag{11}\]  
where \(A\) stands for the surface area of the SPM. The results show that \(\Gamma_{ex}\) decreases with the value of \(R\) and \(x\). On one hand, this is due to the close relation of \(\Gamma_{ex}\) to the bulk density of big component \(\rho_{b}^{bulk,L}\). On the other hand, the density gradient of the small component leads to a depletion force on the big particles, which acts towards left, and prevents the big particles from gathering to the SPM. Figure 4: Excess adsorption \(\Gamma_{ex}\) of the binary mixture confined by a SPM. Note that the symbols in the figure is used to guide eyes. Figure 3: (**a**, **c**) correspond respectively the \(x\) dependence of \(\rho_{s,cro}\) and \(\Delta\) under different conditions of \(\eta_{tot}\). (**b**, **d**) correspond respectively the \(R\) dependence of \(\rho_{s,cro}\) and \(\Delta\) under different conditions of \(x\). Note that the symbols in the figure is used to guide eyes.