Topology Controls the Phase Separation Dynamics of Many Component Fluid Mixtures

Abstract

Fluid mixtures, ranging from the cellular cytoplasm to synthetic DNA nanostar systems, can spontaneously compartmentalize into many ($N$) coexisting liquid phases through liquid-liquid phase separation. While such systems exhibit a remarkable diversity of spatial organizations, the physical principles governing their non-equilibrium dynamics remain poorly understood. Here, combining simulations and analytical theory, we show that the coarsening dynamics of many component phase separation are fundamentally linked to mathematical coloring problems. For planar phase organization, relevant to synthetic droplet monolayers and simple biological structures, we identify distinct topological constraints for $N=2$, $N=3$, and $N=4$, with no further change for $N>4$, consistent with the four-color theorem. These constraints govern the coarsening dynamics, and, using chromatic graph theory, we derive a theoretical model for $N\geq 3$ that quantitatively captures the diffusive-like coarsening. By contrast, classical theories based solely on Ostwald ripening underestimate the observed dynamics. We further show that tuning interfacial tensions modifies the set of admissible phase arrangements, enabling highly heterogeneous coarsening dynamics across different phases. For unconfined systems with nonplanar phase organization, different coloring constraints apply, with no analogue of the four-color theorem, and coalescence suppression emerges only when the number of phases exceeds $N\gtrsim 7$. More broadly, our work establishes coloring theory as a topological framework for understanding and predicting the dynamics of many component phase-separating fluids.