Structure and dynamics of a vibrated granular bead-chain

We investigate the dynamics of a vibrated granular bead-chain with experiments and numerical simulations of random-walk models of polymers. Experiments are conducted with a chain composed of hollow 3~mm steel beads connected by flexible links confined to move on a 300~mm diameter rough circular bed. Observations made with digital imaging. We analyze the radius of gyration $R_g$, the structure factor of the chain configurations, and the diffusion of the center of mass. We find that $R_g$ and the structure factor scale with the exponent $\nu \sim 3/4$, consistent with the two dimensional self-avoiding random-walk model. Further, we observe confinement effects in the scaling of $R_g$ as the chain length increases relative to the size of the container. We perform simulations of non-self-avoiding walks confined to the same sized domain and find good agreement with experiment. The simulations show confinement effects dominate over self-avoided crossings in the experiments even when the length of chain is smaller than system size. We then experimentally examine the chain dynamics and find that the center of mass diffusion scales inversely as the length of the chain, consistent with the Rouse model of polymers. We observe an exponential decay in the the dynamical structure factor and compare this exponent with the measurement of the center of mass diffusion.