Unconstrained motions, dynamic heterogeneities, and relaxation in disordered solids

A disordered network of bonds with a fixed configuration can relax via a variety of unconstrained motions. These motions can be directly inferred from the topological arrangement of constraints without any geometrical information. We use the pebble game algorithm of Jacobs and Thorpe [D. J. Jacobs and M. F. Thorpe, Phys. Rev. Lett. 75, 4051 (1995)] to decompose the system into separate rigid clusters and identify the remaining degrees of freedom. Unconstrained motions can then be resolved in the form of translations and rotations of isolated groups of bonds and the internal motion within bond groups. We show that each motion can be assigned a characteristic thermal velocity and hence a relaxation time scale. We use this information to construct a relaxation function and also examine the spatial distribution of relaxation time scales. We investigate the sensitivity of the relaxation time scales and their spatial distribution when making individual bond changes in the system, and we consider the dependence of these time scales on the underlying structure.