Structure and phase behaviors of confined two penetrable soft spheres

•Study the phase behaviors of two penetrable soft spheres in a hard spherical pore.•Calculate the exact partition function, one-body density, and equation of state.•Soft attraction beyond the soft-core potential enhances the van der Waals instability•Van der Waals for two hard spheres is not observed.

We study the phase behaviors of two penetrable soft spheres, whose interactions include the soft repulsion and attraction, in a hard spherical pore. The exact partition function, one-body density, and equation of state for the confined two penetrable soft spheres have been calculated using the Fourier transform method. The phase diagrams have been determined from the negative compressibility of the van der Waals type, which imitates the phase transition of many particle system. The addition of the soft repulsion and attraction beyond the soft-core potential gives rise to the van der Waals instability. The soft attraction beyond the soft-core potential significantly enhances the van der Waals instability, whereas the soft repulsion reduces the van der Waals instability. For two hard spheres and hard square-well spheres, the van der Waals instability is not observed. However, the addition of a short-range soft repulsion beyond the hard-core gives rise to the van der Waals instability.