Notes on star polymers

Star polymers are composed of long polymeric arms radiating from a central core. They are not only of academic interest but found many applications specially in lubricant formulation. We solve a seemingly unnoticed contradiction between two theories of star homopolymers both well established for decades. Asymptotically exact renormalization group results for large stars at the θ-point in three dimensions qualitatively strongly disagree with mean-field results. This is unexpected because the mean field should be marginally correct in three dimensions (the upper critical dimension for effective three-body interactions). We show that the two theories actually describe different physical situations: unstretched stars with (very) large arms and (relatively) low functionality and crowded stretched stars. The available RG results apply to very large, not so crowded stars where the arms are essentially unstretched. Explicitly, the arm size has to be exponentially large in the functionality squared. Stars with a reasonable number of arms, say ten or more, do typically not belong to this regime, while the four-arm stars relevant to standard contact probabilities do. In typical stars with higher-functionality arms are stretched. We provide a consistent description that covers critical/unstretched, critical/stretched, mean-field/stretched and mean-field/unstretched stars. A similar analysis is given for stars swollen in a good solvent. As a consequence star/star interactions and phase separation in a solution of stars should be considered with care.