Analytical theory of effective interactions in binary colloidal systems of soft particles

While density functional theory with integral equations techniques are very efficient tools in the numerical analysis of complex fluids, analytical insight into the phenomenon of effective interactions is still limited. In this paper, we propose a theory of binary systems that results in a relatively simple analytical expression combining arbitrary microscopic potentials into effective interaction. The derivation is based on translating a many-particle Hamiltonian including particle-depletant and depletant-depletant interactions into the occupation field language, which turns the partition function into multiple Gaussian integrals, regardless of what microscopic potentials are chosen. As a result, we calculate the effective Hamiltonian and discuss when our formula is a dominant contribution to the effective interactions. Our theory allows us to analytically reproduce several important characteristics of systems under scrutiny. In particular, we analyze the following: the effective attraction as a demixing factor in the binary systems of Gaussian particles, the screening of charged spheres by ions, which proves equivalent to Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, effective interactions in the binary mixtures of Yukawa particles, and the system of particles consisting of both a repulsive core and an attractive/repulsive Yukawa interaction tail. For this last case, we reproduce the "attraction-through-repulsion" and "repulsion-through-attraction" effects previously observed in simulations.