Effective interactions in soft condensed matter physics

In this work, we present a review of recently achieved progress in the field of soft condensed matter physics, and in particular on the study of the static properties of solutions or suspensions of colloidal particles. The latter are macromolecular entities with typical sizes ranging from 1 nm to 1 μm and their suspension typically contain, in addition to the solvent, smaller components such as salt ions or free polymer chains. The theoretical tool introduced is the effective Hamiltonian which formally results by a canonical trace over the smaller degrees of freedom for a fixed, “frozen” configuration of the large ones. After presenting the formal definitions of this effective Hamiltonian, we proceed with the applications to some common soft matter systems having a variable softness and ranging from free polymer chains to hard colloidal particles. We begin from the extreme case of nondiverging effective interactions between ultrasoft polymer chains and derive an exact criterion to determine the topology of the phase diagrams of such systems. We use star polymers with a variable arm number f as a hybrid system in order to interpolate between these two extremes. By deriving an effective interaction between stars we can monitor the change in the phase behavior of a system as the steepness of the repulsion between its constituent particles increases. We also review recent results on the nature and the effects of short-range attractions on the phase diagrams of spherical, nonoverlapping colloidal particles.