Non-Monotonic Marangoni Suppression of Hydrodynamic Coarsening in Bicontinuous Liquid-Liquid Phase Separation

TL;DR

This study reveals that soluble surfactants suppress hydrodynamic coarsening in bicontinuous liquid-liquid phase separation mainly through Marangoni stresses, with suppression strength depending non-monotonically on surfactant Péclet number.

Contribution

It demonstrates a transport-controlled mechanism where surfactant-induced Marangoni stresses, rather than interfacial tension reduction, regulate coarsening in bicontinuous domains.

Findings

Suppression of coarsening is strongest at intermediate Péclet number, Pe_ψ=10.

Marangoni stresses hinder interfacial coalescence and alter flow patterns.

Non-monotonic suppression results from a balance between surfactant replenishment and gradient retention.

Abstract

Hydrodynamic coarsening of bicontinuous domains is a central process in liquid-liquid phase separation, yet how soluble surfactants regulate this process remains poorly understood. Using a validated two-order-parameter phase-field model coupled to the incompressible Navier-Stokes equations, we show that hydrodynamic coarsening is suppressed primarily by surfactant-induced Marangoni stresses rather than by the reduction of mean interfacial tension alone. These stresses hinder interfacial coalescence, reorganize the local vortical flow, and thereby redirect the morphological evolution of bicontinuous domains. A central result is that this suppression depends non-monotonically on the surfactant P\'eclet number, with the strongest inhibition occurring at an intermediate value, $Pe_\psi=10$, rather than at $Pe_\psi=1$ or 100. Analyses of force evolution, interfacial surfactant statistics, and decomposed surfactant flux budgets show that this non-monotonicity arises from a competition between surfactant replenishment and gradient retention. At low $Pe_\psi$, diffusion efficiently replenishes the interface but smooths interfacial concentration gradients; at high $Pe_\psi$, advection preserves interfacial heterogeneity but leaves the interface insufficiently supplied with surfactant. The strongest suppression therefore occurs when sufficient interfacial surfactant loading coexists with persistent concentration gradients. These results establish a transport-controlled mechanism by which soluble surfactants regulate bicontinuous hydrodynamic coarsening.