Phase Bubbles and Spatiotemporal Chaos in Granular Patterns

We use inelastic hard sphere molecular dynamics simulations and laboratory experiments to study patterns in vertically oscillated granular layers. The simulations and experiments reveal that {\em phase bubbles} spontaneously nucleate in the patterns when the container acceleration amplitude exceeds a critical value, about $7g$, where the pattern is approximately hexagonal, oscillating at one-fourth the driving frequency ($f/4$). A phase bubble is a localized region that oscillates with a phase opposite (differing by $\pi$) to that of the surrounding pattern; a localized phase shift is often called an ${\em arching}$ in studies of two-dimensional systems. The simulations show that the formation of phase bubbles is triggered by undulation at the bottom of the layer on a large length scale compared to the wavelength of the pattern. Once formed, a phase bubble shrinks as if it had a surface tension, and disappears in tens to hundreds of cycles. We find that there is an oscillatory momentum transfer across a kink, and this shrinking is caused by a net collisional momentum inward across the boundary enclosing the bubble. At increasing acceleration amplitudes, the patterns evolve into randomly moving labyrinthian kinks (spatiotemporal chaos). We observe in the simulations that $f/3$ and $f/6$ subharmonic patterns emerge as primary instabilities, but that they are unstable to the undulation of the layer. Our experiments confirm the existence of transient $f/3$ and $f/6$ patterns.