Self-organized criticality driven by droplet influx and random fusion

Abstract: The droplet size distribution typically decays exponentially in solutions formed by liquid-liquid phase separation. Nevertheless, a power-law distribution of nucleoli volumes has been observed in amphibian oocytes, which appears similar to the cluster size distribution in reaction-limited aggregation. In this work, we study the mechanism of power-law distributed droplet sizes and unveil a self-organized criticality driven by droplet influx and random fusion between droplets. Surprisingly, the droplet size dynamics is governed by a similar Smoluchowski equation as the cluster size in aggregation systems. The system reaches a critical state as the area fraction approaches the critical value at which the droplet size has a power-law distribution with a $1.5$ exponent. Furthermore, the system is also spatially scale-free with a divergent correlation length at the critical state, marked by giant droplet-density fluctuations and power-law decay of the pair correlation function.