## 2 Theory and Model  
In the present work, we consider a binary hard sphere fluid mixture confined by a SPM, which supplies different external potential to different components, that is:  
\[V_{b}^{ext}(z)= \begin{cases}0,&\text{if }z<z_{m};\\ +\infty,&\text{otherwise}\end{cases}  \tag{1}\]  
for the big component, while \(V_{s}^{ext}(z)=0\) for the small component. In Eq. (1), \(z_{m}=0\) denotes the position of the SPM. According to the classical DFT [11; 12], the grand potential functional \(\Omega[\rho_{v}(\mathbf{r})]\) is formally expressed as:  
\[\Omega[\rho_{v}(\mathbf{r})]=\mathcal{F}[\rho_{v}(\mathbf{r})]+\sum_{v=s,b}\int\rho_{v }(\mathbf{r})[V_{v}^{ext}(\mathbf{r})-\mu_{v}]d\mathbf{r}. \tag{2}\]  
Here \(\mu_{v}\) is chemical potential of a particle in the corresponding bulk phase, \(\mathcal{F}[\rho_{v}(\mathbf{r})]=\mathcal{F}_{id}+\mathcal{F}_{ex}\) is the intrinsic Helmholtz free energy, where the ideal gas contribution \(\mathcal{F}_{id}\)=\(k_{B}T\sum_{v}\int\rho_{v}(\mathbf{r})[\ln(\lambda^{3}\rho_{v}(\mathbf{r}))-1]d\mathbf{r}\) is known exactly, with \(\lambda\), \(k_{B}\) and \(T\) denoting respectively the thermal wavelenght, the Boltzmann constant and absolute temperature. \(\mathcal{F}_{ex}\) is the excess Helmholtz free energy, whose exact expression is unkown for most systems of practical interest. According to the variational principle, the grand potential functional \(\Omega[\rho_{v}(\mathbf{r})]\) is minimized at equilibrium, and the equilibrium density profile \(\rho_{v}(\mathbf{r})\) satisfies the variation principle:  
\[\frac{\delta\Omega[\rho_{v}(\mathbf{r})]}{\delta\rho_{v}(\mathbf{r})}=0. \tag{3}\]  
To perform further calculations, one should firstly construct the excess Helmholtz free energy \(\mathcal{F}_{ex}\). In the existing approximations for the hard sphere contribution, one reliable choice is the Modified Fundamental Measure Theory (MFMT) [31; 32; 33], which gives the hard sphere contribution by:  
\[\beta\mathcal{F}_{ex}=\int\Phi[n_{a}(\mathbf{r})]d\mathbf{r} \tag{4}\]with \(\Phi[n_{\alpha}(\mathbf{r})]\) standing for the Helmholtz free energy density, which is given by:  
\[ \begin{split}\Phi=-n_{0}\ln(1-n_{3})+\frac{n_{1}n_{2}-n_{1}n_{2}}{1 -n_{3}}\\ +\frac{1}{36\pi}[n_{3}\ln(1-n_{3})+\frac{n_{3}^{2}}{(1-n_{3})^{2} }]\frac{n_{2}^{3}-3n_{2}n_{2}^{2}}{n_{3}^{3}}.\end{split}  \tag{5}\]  
Here \(n_{\alpha}(\mathbf{r})=\sum_{\
u}\int\rho_{\
u}(\mathbf{r}^{\prime})\omega_{\
u}^{( \alpha)}(\mathbf{r}-\mathbf{r}^{\prime})d\mathbf{r}^{\prime}\) is the weighted densities, with \(\omega_{\
u}^{(\alpha)}\) denoting the density independent weight functions, whose explicit forms can be found in the literatures [31; 32; 33]. The equilibrium density profiles \(\rho_{\
u}(\mathbf{r})\) can be obtained from the solution of the equation which is obtained from Equation (3), that is:  
\[\rho_{\
u}(\mathbf{r})=\rho_{\
u}^{bulk,L}\exp\{-\beta\frac{\partial\mathcal{F}_{ ex}[\rho(\mathbf{r})]}{\partial\rho_{\
u}(\mathbf{r})}+\beta[\mu_{\
u}-V_{\
u}^{ext}( \mathbf{r})]\} \tag{6}\]  
with \(\rho_{\
u}^{bulk,L}\) denoting the bulk density of the \(\
u\) component on the _left_ side of the SPM. Equation (6) can be numerically solved using the Picard type iterative method. Once the density profiles are obtained, it is ready to study the structural properties and the characteristic densities of the mixture, including the ratio of bulk densities for small component on right and left sides of the SPM \(\rho_{s,R/L}\equiv\frac{\rho_{s}^{bulk,R}}{\rho_{s}^{bulk,L}}\), the contact density of the big component \(\rho_{b,con}\equiv\frac{\rho_{b}(z_{m}-\frac{\sigma_{b}}{2})}{\rho_{b}^{bulk,L}}\), and the cross density of the small component \(\rho_{s,cr\alpha}\equiv\frac{\rho_{b}(z_{m})}{\rho_{s}^{bulk,L}}\). Note that the above three characteristic densities have been reduced to their dimensionless forms. Actually, besides the DFT calculation, one can also resort to the analytical methods to obtain \(\rho_{b,con}\) and \(\rho_{s,R/L}\).