Self-organization of hydrodynamically entrained sperm cells into an array of vortices

The emergence of spatiotemporal patterns is of great interest in many scientific disciplines. Here we report a new dynamically self-organized pattern formed by hydrodynamically entrained sperm cells at planar surfaces. The sperm cells form vortices resembling quantized rotating waves. These vortices form an array with local hexagonal order. Using a novel order parameter, we show that the array is only formed above a critical sperm density. Supported by numerical simulation we suggest a mechanism for the appearance of the array and we estimate the strength of the hydrodynamic coupling between the cells. The vortex array represents a new chiral active gel and may serve as an experimentally accessible model for the metachronal wave of ciliated epithelia and other non-equilibrium phenomena in general. Finally we discuss the biological implications of our work.