Unconventional Growth Kinetics and Fractal Interfaces of Colloidal Phase Separation in Active Liquids

Phase separation driven by nonequilibrium fluctuations is a hallmark of both living and synthetic active matter. Unlike equilibrium systems, where ordered states arise from the minimization of free energy, active systems are fueled by a constant injection of energy at the microscopic scale. The emergence of ordered phases in such driven systems challenges our conventional views of domain growth and interfacial structure. In this study, we investigate the coarsening of colloidal clusters in active liquids containing E. coli. Our experiments reveal that uniform dispersions of colloids and swimmers are inherently unstable, resulting in spontaneous phase separation characterized by fractal interfaces and unconventional kinetics. The correlation function of the order parameter displays dynamical scaling, with the size of colloidal domains initially growing as $t^{1/z}$, where $z \sim 4$, in contrast to the well-known growth laws for thermal systems with a conserved order parameter. Furthermore, the structure factor exhibits non-Porod behavior, indicating domains with fractal interfaces. This non-Porod behavior also manifests itself as a cusp singularity in the correlation function. We elucidate our experimental findings using a scalar field theory in which the nonequilibrium fluctuations arising from swimmer activity are modeled as spatio-temporally correlated noise. It quantitatively reproduces the domain growth law and non-Porod structure factor resulting from fractal interfaces observed in experiments. In addition, it also reveals a fluctuating microphase separation, where the initial growth of the domain is eventually arrested, thus shedding new light on the microscopic origins of the unconventional phase separation of colloids in active liquids.