Thermodynamics of Diffuse-Interface Models for Binary-Fluid Surfactant Systems

  • van Sluijs, Tom (Eindhoven University of Technology)
  • Stoter, Stein (Eindhoven University of Technology)
  • ten Eikelder, Marco (Technical University of Darmstadt)
  • van Brummelen, Harald (Eindhoven University of Technology)

Please login to view abstract download link

Surface-active agents (surfactants) release potential energy as they migrate from one of two adjacent fluids onto their fluid-fluid interface, a process that profoundly impacts the system’s energy and entropy householding. We present a mathematical description of such a system, that satisfies the first and second law of thermodynamics to derive interface conditions and characterize the system’s equilibrium. With our work, we aim to provide a systematically derived framework that combines and links various elements of existing literature, and that can serve as a thermodynamically consistent foundation for the (numerical) modeling of surfactant-enriched binary-fluid systems [1]. Within this framework, we derive a Navier–Stokes–Cahn–Hilliard (NSCH) surfactant model from mixture theory in an N-phase setting, providing a unified thermodynamic description of bulk phases, diffuse interfaces, and interfacial surfactant transport. This formulation explicitly accounts for the coupling between momentum balance, phase-field evolution, and surfactant redistribution, and clarifies how interfacial tension reduction and Marangoni effects emerge from the underlying mixture energy functionals. We further analyze the mixture energy functional for the multiphase–surfactant system and compare it to classical sharp‑interface models to identify consistent limits, key differences, and potential modeling inconsistencies. Finally, we investigate the influence of boundary conditions, with particular emphasis on surfactant adsorption at solid walls, dynamic wetting in the presence of interfacial species, and the thermodynamic constraints these impose. Together, these contribute to a thermodynamically-consistent framework for surfactant‑laden multiphase flow in the diffuse interface setting. [1] T. B. van Sluijs, S. K. F. Stoter, and H. E. van Brummelen. Thermodynamics of Surfactant-Enriched Binary-Fluid Systems. Langmuir, 41(4):2141–2155, 2025. doi:10.1021/acs.langmuir.4c01724.