Quantfying rich patterns in agglomeration of floating beads

Macroscopic spherical particles spontaneously form rich patterns on a standing Faraday wave. These patterns are found to follow a very systematic trend depending on the floater concentration $\phi$: The same floaters that accumulate at amplitude maxima (antinodes) of the wave at low $\phi$, surprisingly move towards the nodal lines when $\phi$ is beyond a certain value. In more detail, circular irregularly packed antinode clusters at low $\phi$ give way to loosely packed filamentary structures at intermediate $\phi$, and are then followed by densely packed grid-shaped node clusters at high $\phi$. Here, we successfully characterize the morphology of these rich patterns using a metric analysis, i.e., the Minkowski functionals. We modify the Minkowski functionals such that we are able to measure the physical quantities of the clusters such as area, perimeter, and aspect ratio.