## 1 Introduction  
Recently, considerable attention has been attracted by confined fluid and their mixture, whose structures and phase behaviors are drastically different from the corresponding bulk system [1]. In such systems, interactions between fluid-pore and fluid-fluid lead usually to interesting phenomena including laying [2], wetting [3, 4], capillary condensation [5] and so on. Actually, among various confining geometries, the semipermeable membrane (SPM) has particular importance due to its ability to model the natural biological membranes and synthetic membranes [6, 7]. For instance, the Donnan equilibrium can be conveniently constructed by utilizing of the SPM, across which there exists density difference. Besides, it is also a satisfactory tool to investigate the process of ion exchange or transport in the ion channels [8].  
On the theoretical side, substantial works on the SPM have been reported in the past decades. Employing the Ornstein-Zernike (OZ) integral equation with the Percus-Yevick (PY) or hypernetted chain (HNC) closure, [PERSON] and [PERSON] studied the structure of the hard sphere fluids and charged fluids near a SPM [9, 10]. However, the method based on OZ integral equation fails in the description of the fluid structure near the adsorbing surfaces and is not able to predict the surface phase transitions.  
Actually, the DFT approach is free of the above mentioned drawbacks, because it can provide a microscopic description of the structure of the system. In DFT [11, 12], the Helmholtz free energy is the only required input, and the density profile can be obtained by minimizing the grand potential functional. In addition, the DFT method has obvious advantages of saving computer time, in its comparison with other theoretical methods, including OZ integral equation and computer simulation.  
Using density functional theory, [PERSON] et al. investigated the equilibrium density of the hard sphere mixture and Lennard-Jones fluids confined by a SPM [13; 14]. They have also studied the phase behaviors [15] and depletion potential [16] of the fluids confined by permeable walls. Using weighted-density approximation for hard-sphere interaction and the mean-field approximation for the long-ranged dispersion, [PERSON] et al. investigated the density profiles of binary attractive and binary repulsive hard-core Yukawa fluid mixtures near a SPM [17].  
In recent years, synthetic and biological membranes have found their applications in the field water purification [18], dialysis [19] and targeted gene and drug delivery [20]. Especially in biomedical researches, the interaction of motile particles with biological membranes should be fully understood to explain the delivery of cargo to the interior of the cells [21; 22]. Moreover, the mechanism for the penetration of active particles through the membrane remains an open question [23; 24; 25; 26]. To a large extent, the fluid structure near the membrane influences the penetration of the particles [27; 28; 29; 30]. Therefore, it is necessary to investigate the structural properties of the fluid mixtures that are confined by semipermeable membranes.  
Actually, there are several specific densities, which can be used to characterize the structural properties of the system, including the bulk densities on each side of the membrane, the contact density of the non-penetrating component and the cross density of the penetrating component. Moreover, some of them can be described through the analytical method. Thus, in this paper, we concentrate on these characteristic densities, and study their dependence on the volume fraction and size asymmetry.