The compressibility equation for soft-matter liquids

Effective interactions in soft-matter physics result from a formal contraction of an initial multicomponent system, composed of mesoscopic and small particles, into an effective one-component description. By tracing out in the partition function the degrees of freedom of the small particles, a one-component system of mesoscopic particles interacting with a state-dependent Hamiltonian is found. Although the effective Hamiltonian is not in general pairwise additive, it is usually approximated by a volume term and a pair-potential contribution. In this paper the relation between the structure, for which the volume term plays no role, and the thermodynamics of a fluid of particles interacting with a density-dependent pair potential is analysed. It is shown that the compressibility equation differs from that of atomic fluids. An important consequence is that the infinite-compressibility line derived from the thermodynamics does not coincide with the spinodal line stemming from the divergence of correlations.