Topology Controls the Phase Separation Dynamics of Multicomponent Fluid Mixtures

TL;DR

This paper reveals how topological constraints, like the four-color theorem, influence the phase separation dynamics in multicomponent fluid mixtures, affecting coalescence and coarsening behaviors.

Contribution

It introduces a topological framework linking mathematical coloring problems to phase separation dynamics, providing new insights into controlling fluid compartment arrangements.

Findings

Four-color theorem constrains fluid arrangements in thin geometries.

Suppressed coalescence leads to arrested hydrodynamics and universal coarsening curves.

Complex coarsening dynamics depend on interfacial tensions and phase configurations.

Abstract

Fluid mixtures, such as the cellular cytoplasm and synthetic DNA nanostars, can spontaneously compartmentalize into many coexisting phases through liquid-liquid phase separation. Despite the diversity of fluid structures that emerge from interactions between different phases, the physical principles governing their spatiotemporal organization remain unclear. In this work, we show that the dynamics of multicomponent phase separation are intimately connected to mathematical coloring problems, including the four-color theorem. By confining the system to thin geometries, we demonstrate that the four-color theorem permits arrangements of fluid compartments that lead to suppressed coalescence. As a consequence, hydrodynamics is arrested, and the diffusion-dominated coarsening dynamics can be collapsed to a universal master curve. Varying the fluid interfacial tensions can change which arrangements are energetically permissible, resulting in highly complex coarsening dynamics that differ for each phase. In unconfined three-dimensional systems, the absence of an equivalent to the four-color theorem means that suppression of coalescence is only reached asymptotically for large numbers of phases, rather than at a sharp threshold. In general, the coloring approach employed here offers a topological framework for understanding the dynamic behaviour of phase separating fluid mixtures.