Star polymers with intermediate branching functionalities (64 and 128 arms) are known to exhibit a hybrid viscoelastic response, relaxing stress via a fast polymeric arm retraction and a slow colloidal mode. We show that, in the limit of very high functionalities (above 800) and very short arms (3 entanglements or less), the colloidal mode following arm relaxation exhibits a plateau with a modulus much smaller than that expected from the entanglement value and does not relax within the experimentally accessible frequency window. Analysis of the relaxation modes indicates that, even in the melt state, stars with ultrahigh branching functionalities effectively behave like jammed colloids. Their low modulus makes them easily processable, unlike highly entangled linear polymers. A simple state diagram in the melt is proposed with two control parameters: the functionality and the number of entanglements per arm (as a measure of arm size), showing three distinct regimes based on dynamics: polymeric, hybrid polymeric-colloidal, and jammed. These results are briefly discussed in view of other highly branched polymers with density heterogeneities (microgels, bottlebrushes) exhibiting nonterminal solid-like response and supersoft elastomeric features. For the present stars in particular, which are model soft colloids, these findings contribute to the ongoing discussion on jamming in soft matter.

Two different classes of hairy self-suspended nanoparticles in the melt state, polymer-grafted nanoparticles (GNPs) and star polymers, are shown to display universal dynamic behavior across a broad range of parameter space. Linear viscoelastic measurements on well-characterized silica-poly­(methyl acrylate) GNPs with a fixed core radius (R core) and grafting density (or number of arms f) but varying arm degree of polymerization (N arm) show two distinctly different regimes of response. The colloidal Regime I with a small N arm (large core volume fraction) is characterized by predominant low-frequency solidlike colloidal plateau and ultraslow relaxation, while the polymeric Regime II with a large N arm (small core volume fractions) has a response dominated by the starlike relaxation of partially interpenetrated arms. The transition between the two regimes is marked by a crossover where both polymeric and colloidal modes are discerned albeit without a distinct colloidal plateau. Similarly, polybutadiene multiarm stars also exhibit the colloidal response of Regime I at very large f and small N arm. The star arm retraction model and a simple scaling model of nanoparticle escape from the cage of neighbors by overcoming a hopping potential barrier due to their elastic deformation quantitatively describe the linear response of the polymeric and colloidal regimes, respectively, in all these cases. The dynamic behavior of hairy nanoparticles of different chemistry and molecular characteristics, investigated here and reported in the literature, can be mapped onto a universal dynamic diagram of f/[R core 3/ν0)1/4] as a function of (N armν0 f)/(R core 3), where ν0 is the monomeric volume. In this diagram, the two regimes are separated by a line where the hopping potential ΔU hop is equal to the thermal energy, k B T. ΔU hop can be expressed as a function of the overcrowding parameter x (i.e., the ratio of f to the maximum number of unperturbed chains with N arm that can fill the volume occupied by the polymeric corona); hence, this crossing is shown to occur when x = 1. For x > 1, we have colloidal Regime I with an overcrowded volume, stretched arms, and ΔU hop > k B T, while polymeric Regime II is linked to x < 1. This single-material parameter x can provide the needed design principle to tailor the dynamics of this class of soft materials across a wide range of applications from membranes for gas separation to energy storage.