Effective interactions, structure, and isothermal compressibility of colloidal suspensions

We study the effective interactions, structure, and the isothermal compress ibility of a binary mixture interacting with pairwise additive pair potenti als. By integrating out the degrees of freedom of species 2 in the partitio n sum we first show that a binary mixture can be mapped formally onto an ef fective one-component system with an effective Hamiltonian consisting of a structure-independent term, which contributes to the total pressure and che mical potential of the system, but does not affect the phase behavior, and a structure-dependent potential of mean force, which contains pair-, triple t-, and higher-body interactions. We then show that the 1-1 structure facto r and pair correlation function, and the total isothermal compressibility o f the mixture are equal to those of the effective one-component system, pro vided the mapping is exact. We illustrate and confirm these results by calc ulating the structure factors and pair correlation functions of the binary Asakura-Oosawa model, which is a simple model for colloid-polymer mixtures, and those of the corresponding one-component system for a size ratio such that the mapping onto an effective one-component Hamiltonian with a strictl y pairwise potential of mean force is exact. The distinction between the os motic and total compressibility of the mixture is emphasized. (C) 2000 Amer ican Institute of Physics. [S0021-9606(00)51335-X].