What does the instantaneous normal mode spectrum tell us about dynamical heterogeneity in glass-forming fluids?

We examine the instantaneous normal mode spectrum of model metallic and polymeric glass-forming liquids. We focus on the localized modes in the unstable part of the spectrum [unstable localized (UL) modes] and find that the particles making the dominant contribution to the participation ratio form clusters that grow upon cooling in a fashion similar to the dynamical heterogeneity in glass-forming fluids, i.e., highly mobile (or immobile) particles form clusters that grow upon cooling; however, a comparison of the UL mode clusters to the mobile and immobile particle clusters indicates that they are distinct entities. We also show that the cluster size provides an alternate method to distinguish localized and delocalized modes, offering a significant practical advantage over the finite-size scaling approach. We examine the trajectories of particles contributing most to the UL modes and find that they have a slightly enhanced mobility compared to the average, and we determine a characteristic time quantifying the persistence time of this excess mobility. This time scale is proportional to the structural relaxation time τ_α of the fluid, consistent with a prediction by Zwanzig [Phys. Rev. 156, 190 (1967)] for the lifetime of collective excitations in cooled liquids. Evidently, these collective excitations serve to facilitate relaxation but do not actually participate in the motion associated with barrier crossing events governing activated transport. They also serve as a possible concrete realization of the "facilitation" clusters postulated in previous modeling of glass-forming liquids.