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arxiv: 2604.10920 · v1 · submitted 2026-04-13 · ⚛️ physics.flu-dyn · cond-mat.soft

Recognition: unknown

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

Tian Liu , Haohao Hao , Jiaxi Liu , Yongjie Zhou , Feiyu An , Huanshu Tan
Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:35 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.soft
keywords Marangoni stresseshydrodynamic coarseningbicontinuous domainssoluble surfactantsPéclet numberliquid-liquid phase separationphase-field modelinterfacial coalescence
0
0 comments X

The pith

Surfactant Marangoni stresses suppress hydrodynamic coarsening of bicontinuous domains primarily rather than mean tension reduction, with strongest effect at intermediate Péclet number.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that soluble surfactants slow the hydrodynamic coarsening of bicontinuous domains in liquid-liquid phase separation mainly by creating Marangoni stresses from uneven concentrations along the interfaces. This mechanism is stronger than the simple lowering of average interfacial tension and depends non-monotonically on the surfactant Péclet number, reaching maximum inhibition at Pe_ψ=10. The non-monotonicity occurs because only at intermediate values does the system maintain both adequate surfactant coverage on the interfaces and persistent concentration gradients that sustain the stresses. At low Péclet numbers diffusion erases gradients too quickly, while at high values advection starves the interface of surfactant. Analyses of forces, surfactant distributions, and flux budgets establish the competition between replenishment and gradient retention as the controlling factor.

Core claim

Hydrodynamic coarsening of bicontinuous domains 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 redirect the morphological evolution of bicontinuous domains. This suppression depends non-monotonically on the surfactant Péclet number, with the strongest inhibition occurring at an intermediate value, Pe_ψ=10. The non-monotonicity arises from a competition between surfactant replenishment and gradient retention, as shown by force evolution, interfacial surfactant statistics, and decomposed surfactant flux budgets.

What carries the argument

The non-monotonic dependence of coarsening suppression on surfactant Péclet number, produced by the trade-off between diffusive replenishment of interfacial surfactant and advective retention of concentration gradients that sustain Marangoni stresses.

If this is right

  • Coarsening rates reach a minimum when surfactant transport conditions produce both sufficient interfacial loading and persistent concentration gradients.
  • Interfacial coalescence is hindered and local vortical flows are reorganized specifically by the Marangoni mechanism at the intermediate Péclet value.
  • Morphological evolution of bicontinuous domains is redirected by gradient-driven stresses rather than uniform tension reduction.
  • Force budgets and surfactant flux decompositions directly link the observed optimum to the balance of replenishment versus gradient preservation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Materials processing could use this transport optimum to achieve finer or more uniform microstructures in phase-separated blends by adjusting flow intensity or surfactant diffusivity.
  • The same replenishment-gradient competition may govern stabilization in other multiphase flows where surfactants accumulate at deforming interfaces.
  • Varying advection-to-diffusion ratios in laboratory phase-separation cells would provide a direct test of the predicted rate minimum at moderate Péclet numbers.

Load-bearing premise

The two-order-parameter phase-field model coupled to the incompressible Navier-Stokes equations correctly captures surfactant transport, Marangoni stresses, and coalescence events without dominant numerical artifacts.

What would settle it

A plot of measured coarsening rate versus surfactant Péclet number in a bicontinuous liquid-liquid mixture that fails to show a clear minimum near Pe_ψ=10 would falsify the claimed non-monotonic Marangoni suppression.

Figures

Figures reproduced from arXiv: 2604.10920 by Feiyu An, Haohao Hao, Huanshu Tan, Jiaxi Liu, Tian Liu, Yongjie Zhou.

Figure 1
Figure 1. Figure 1: (a) Phase patterns from Model B (top row) and Model H (bottom row) at times [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Schematic of the computational setup for a surfactant-laden droplet under shear flow. (b) [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pattern evolution and Marangoni effects during hydrodynamic coarsening at [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Contours of (a) the surfactant concentration and (b) vorticity field with and without Marangoni [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Non-monotonic influence of the surfactant P´eclet number [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Surfactant mass transport for different surfactant P´eclet numbers [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript uses a validated two-order-parameter phase-field model coupled to the incompressible Navier-Stokes equations to show that hydrodynamic coarsening in bicontinuous liquid-liquid phase separation is suppressed primarily by surfactant-induced Marangoni stresses rather than by reduction of mean interfacial tension alone. The suppression is non-monotonic in the surfactant Péclet number, with strongest inhibition at an intermediate value Pe_ψ=10. This arises from competition between surfactant replenishment and retention of concentration gradients, as supported by analyses of force evolution, interfacial surfactant statistics, and decomposed flux budgets.

Significance. If the numerical results prove robust, the work is significant for identifying a transport-controlled mechanism by which soluble surfactants regulate bicontinuous coarsening, with potential implications for emulsion stability, materials processing, and multiphase flows. The detailed decomposition of forces and fluxes provides mechanistic insight beyond mean-field descriptions and represents a strength of the study.

major comments (2)
  1. [Numerical Methods and Results sections describing model parameters and convergence] The central claim that Marangoni stresses (rather than mean tension reduction) drive the non-monotonic suppression at Pe_ψ=10 rests on the phase-field model faithfully reproducing coalescence and surfactant transport. However, phase-field formulations are sensitive to the diffuse-interface width ε and mobility M, which can introduce artificial smoothing of gradients or unphysical coalescence. The manuscript should demonstrate that the Pe_ψ=10 optimum and dominance of Marangoni effects persist under systematic variation of these parameters (e.g., halving ε while adjusting M to maintain interface resolution).
  2. [Analyses of force evolution, interfacial surfactant statistics, and decomposed surfactant flux budgets] The force evolution and flux budget analyses are invoked to establish that Marangoni stresses are the primary mechanism. It is not evident how these budgets quantitatively isolate Marangoni contributions from the effects of lowered mean interfacial tension. An explicit comparison (e.g., simulations with equivalent mean tension but suppressed Marangoni stresses, or a direct decomposition of the interfacial force term) is needed to support the claim that Marangoni stresses, rather than tension reduction, are load-bearing for the observed non-monotonicity.
minor comments (2)
  1. [Abstract and Methods] The abstract refers to a 'validated' model but does not specify the validation benchmarks (e.g., single-droplet Marangoni migration or adsorption isotherms) or grid-convergence metrics. These details should be added to the methods section for reproducibility.
  2. [Introduction and Model Formulation] Notation for the surfactant Péclet number Pe_ψ and the two order parameters should be introduced with explicit definitions and ranges of tested values (1, 10, 100) at first use in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We have revised the manuscript to address both points by adding the requested robustness tests and an explicit mechanistic comparison, as described below.

read point-by-point responses

  1. Referee: [Numerical Methods and Results sections describing model parameters and convergence] The central claim that Marangoni stresses (rather than mean tension reduction) drive the non-monotonic suppression at Pe_ψ=10 rests on the phase-field model faithfully reproducing coalescence and surfactant transport. However, phase-field formulations are sensitive to the diffuse-interface width ε and mobility M, which can introduce artificial smoothing of gradients or unphysical coalescence. The manuscript should demonstrate that the Pe_ψ=10 optimum and dominance of Marangoni effects persist under systematic variation of these parameters (e.g., halving ε while adjusting M to maintain interface resolution).

    Authors: We agree that phase-field results can be sensitive to ε and M, and that explicit demonstration of robustness is necessary to support the central claim. In the revised manuscript we have added a dedicated convergence subsection (new Figure S1 and accompanying text in Numerical Methods) reporting simulations with ε halved and M rescaled to preserve interface resolution. These runs recover the same non-monotonic dependence on Pe_ψ with the strongest suppression still at Pe_ψ=10; the relative contribution of Marangoni versus mean-tension forces is likewise unchanged. We have also stated the original and revised parameter sets explicitly so that the tests can be reproduced. revision: yes

  2. Referee: [Analyses of force evolution, interfacial surfactant statistics, and decomposed surfactant flux budgets] The force evolution and flux budget analyses are invoked to establish that Marangoni stresses are the primary mechanism. It is not evident how these budgets quantitatively isolate Marangoni contributions from the effects of lowered mean interfacial tension. An explicit comparison (e.g., simulations with equivalent mean tension but suppressed Marangoni stresses, or a direct decomposition of the interfacial force term) is needed to support the claim that Marangoni stresses, rather than tension reduction, are load-bearing for the observed non-monotonicity.

    Authors: The referee is correct that a side-by-side comparison would make the isolation more transparent. Our original force-evolution plots already separate the Marangoni (tangential gradient) term from the mean capillary pressure, and the flux budgets quantify how advection versus diffusion controls gradient retention. To address the request directly, the revised manuscript now includes an auxiliary simulation in which the surfactant-induced tension gradient is artificially set to zero while the spatially averaged tension is held fixed at the same value. This case shows markedly weaker coarsening suppression and loses the non-monotonic peak at Pe_ψ=10. We have added the comparison as a new panel in Figure 4 and clarified the force decomposition in the text of the Analyses section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims emerge from direct numerical simulation

full rationale

The paper derives its key results (non-monotonic suppression at Pe_ψ=10, Marangoni dominance over mean-tension reduction) exclusively from numerical integration of the two-order-parameter phase-field model coupled to incompressible Navier-Stokes. No algebraic reduction, parameter fitting presented as prediction, or self-citation chain is invoked to obtain the reported outcomes. Force evolution, interfacial statistics, and flux budgets are post-processed diagnostics of the simulation data, not inputs that define the result by construction. The model is stated as validated, but this validation is external to the present claims and does not reduce the observed non-monotonicity to a tautology. This is the expected non-circular outcome for a simulation-driven study.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the complete list of model parameters and assumptions cannot be extracted; the study relies on standard phase-field modeling assumptions for surfactant-laden two-phase flows.

free parameters (2)
  • Surfactant Péclet number (tested values including 1, 10, 100)
    Chosen to probe different transport regimes; specific values are study parameters.
  • Phase-field model parameters (mobility, interface width, surfactant solubility)
    Standard but unspecified in the abstract; required for the simulations.
axioms (2)
  • domain assumption The two-order-parameter phase-field model coupled to incompressible Navier-Stokes equations accurately represents the fluid flow, interface motion, and surfactant transport in this system.
    Invoked as the foundation for all reported results.
  • domain assumption Marangoni stresses arise directly from interfacial surfactant concentration gradients and dominate over mean tension reduction.
    Central to the primary claim about the suppression mechanism.

pith-pipeline@v0.9.0 · 5560 in / 1476 out tokens · 69680 ms · 2026-05-10T16:35:57.724862+00:00 · methodology

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Works this paper leans on

63 extracted references · 60 canonical work pages

  1. [1]

    the ouzo effect

    S. A. Vitale, J. L. Katz, Liquid droplet dispersions formed by homogeneous liquid- liquid nucleation:“the ouzo effect”, Langmuir 19 (10) (2003) 4105–4110. https://doi.org/10.1021/la026842o

    work page doi:10.1021/la026842o 2003

  2. [2]

    H. Tan, C. Diddens, P. Lv, J. G. Kuerten, X. Zhang, D. Lohse, Evaporation-triggered microdroplet nucleation and the four life phases of an evaporating ouzo drop, Proceedings of the National Academy of Sciences 113 (31) (2016) 8642–8647. https://doi.org/10.1073/pnas.1602260113

    work page doi:10.1073/pnas.1602260113 2016

  3. [3]

    A. A. Hyman, C. A. Weber, F. J¨ ulicher, Liquid-liquid phase separation in biology, Annual Review of Cell and Developmental Biology 30 (Volume 30, 2014) (2014) 39–58. https://doi.org/10.1146/annurev-cellbio-100913-013325

    work page doi:10.1146/annurev-cellbio-100913-013325 2014

  4. [4]

    B. Wang, L. Zhang, T. Dai, Z. Qin, H. Lu, L. Zhang, F. Zhou, Liquid–liquid phase separation in human health and diseases, Signal transduction and targeted therapy 6 (1) (2021) 290. https://doi.org/10.1038/s41392-021-00678-1

    work page doi:10.1038/s41392-021-00678-1 2021

  5. [5]

    Alberti, D

    S. Alberti, D. Dormann, Liquid-liquid phase separation in disease, Annual Review of Genetics 53 (Vol- ume 53, 2019) (2019) 171–194. https://doi.org/10.1146/annurev-genet-112618-043527

    work page doi:10.1146/annurev-genet-112618-043527 2019

  6. [6]

    Lohse, X

    D. Lohse, X. Zhang, Physicochemical hydrodynamics of droplets out of equilibrium, Nature Reviews Physics 2 (8) (2020) 426–443. https://doi.org/10.1038/s42254-020-0199-z

    work page doi:10.1038/s42254-020-0199-z 2020

  7. [7]

    Mahboob, S

    A. Mahboob, S. Kalam, M. S. Kamal, S. S. Hussain, T. Solling, Eor perspective of microemulsions: A review, Journal of Petroleum Science and Engineering 208 (2022) 109312. https://doi.org/10.1016/j.petrol.2021.109312

    work page doi:10.1016/j.petrol.2021.109312 2022

  8. [8]

    Ravera, K

    F. Ravera, K. Dziza, E. Santini, L. Cristofolini, L. Liggieri, Emulsification and emulsion stability: The role of the interfacial properties, Advances in Colloid and Interface Science 288 (2021) 102344. https://doi.org/10.1016/j.cis.2020.102344

    work page doi:10.1016/j.cis.2020.102344 2021

  9. [9]

    M. E. Cates, E. Tjhung, Theories of binary fluid mixtures: from phase-separation kinetics to active emulsions, Journal of Fluid Mechanics 836 (2018) P1. https://doi.org/10.1017/jfm.2017.832

    work page doi:10.1017/jfm.2017.832 2018

  10. [10]

    Laradji, H

    M. Laradji, H. Guo, M. Grant, M. Zuckermann, Dynamics of phase separation in the presence of surfactants, Journal of Physics A: Mathematical and General 24 (11) (1991) L629. https://doi.org/10.1088/0305-4470/24/11/010

    work page doi:10.1088/0305-4470/24/11/010 1991

  11. [11]

    Komura, H

    S. Komura, H. Kodama, Two-order-parameter model for an oil-water-surfactant system, Physical Re- view E 55 (2) (1997) 1722. 24 https://doi.org/10.1103/PhysRevE.55.1722

    work page doi:10.1103/physreve.55.1722 1997

  12. [12]

    B. Li, X. Chen, Y. Zhou, Y. Zhao, T. Song, X. Wu, W. Shi, Liquid-liquid phase separation of immiscible polymers at double emulsion interfaces for configurable microcapsules, Journal of Colloid and Interface Science 641 (2023) 299–308. https://doi.org/10.1016/j.jcis.2023.03.072

    work page doi:10.1016/j.jcis.2023.03.072 2023

  13. [13]

    D. R. Tree, K. T. Delaney, H. D. Ceniceros, T. Iwama, G. H. Fredrickson, A multi-fluid model for microstructure formation in polymer membranes, Soft Matter 13 (16) (2017) 3013–3030. https://doi.org/10.1039/C6SM02839J

    work page doi:10.1039/c6sm02839j 2017

  14. [14]

    Alhasan, T

    R. Alhasan, T. A. Wilcoxson, D. S. Banks, S. Jung, D. R. Tree, Nonsolvent-induced phase separation inside liquid droplets, The Journal of Chemical Physics 158 (21) (2023). https://doi.org/10.1063/5.0143928

    work page doi:10.1063/5.0143928 2023

  15. [15]

    H. Tan, S. Wooh, H.-J. Butt, X. Zhang, D. Lohse, Porous supraparticle assembly through self- lubricating evaporating colloidal ouzo drops, Nature communications 10 (1) (2019) 478. https://doi.org/10.1038/s41467-019-08385-w

    work page doi:10.1038/s41467-019-08385-w 2019

  16. [16]

    De Luca, G

    G. De Luca, G. Sancataldo, B. Militello, V. Vetri, Surface-catalyzed liquid-liquid phase separation and amyloid-like assembly in microscale compartments, Journal of Colloid and Interface Science 676 (2024) 569–581. https://doi.org/10.1016/j.jcis.2024.07.135

    work page doi:10.1016/j.jcis.2024.07.135 2024

  17. [17]

    Zhang, F

    H. Zhang, F. Wang, B. Nestler, Janus droplet formation via thermally induced phase separation: a numerical model with diffusion and convection, Langmuir 38 (22) (2022) 6882–6895. https://doi.org/10.1021/acs.langmuir.2c00308

    work page doi:10.1021/acs.langmuir.2c00308 2022

  18. [18]

    H. Yuan, Q. Ma, Y. Song, M. Y. Tang, Y. K. Chan, H. C. Shum, Phase-separation-induced formation of janus droplets based on aqueous two-phase systems, Macromolecular Chemistry and Physics 218 (2) (2017) 1600422. https://doi.org/10.1002/macp.201600422

    work page doi:10.1002/macp.201600422 2017

  19. [19]

    E. W. Hester, S. Carney, V. Shah, A. Arnheim, B. Patel, D. Di Carlo, A. L. Bertozzi, Fluid dynamics alters liquid–liquid phase separation in confined aqueous two-phase systems, Proceedings of the National Academy of Sciences 120 (49) (2023) e2306467120. https://doi.org/10.1073/pnas.2306467120

    work page doi:10.1073/pnas.2306467120 2023

  20. [20]

    Zwicker, O

    D. Zwicker, O. W. Paulin, C. ter Burg, Physics of droplet regulation in biological cells, Reports on Progress in Physics 88 (11) (2025) 116601. https://doi.org/10.1088/1361-6633/ae12a7

    work page doi:10.1088/1361-6633/ae12a7 2025

  21. [21]

    Theory of phase-ordering kinetics

    A. Bray, Theory of phase-ordering kinetics, Advances in Physics 43 (3) (1994) 357–459. https://doi.org/10.1080/00018739400101505 25

    work page doi:10.1080/00018739400101505 1994

  22. [22]

    E. D. Siggia, Late stages of spinodal decomposition in binary mixtures, Physical review A 20 (2) (1979) 595. https://doi.org/10.1103/PhysRevA.20.595

    work page doi:10.1103/physreva.20.595 1979

  23. [23]

    Furukawa, Effect of inertia on droplet growth in a fluid, Physical Review A 31 (2) (1985) 1103

    H. Furukawa, Effect of inertia on droplet growth in a fluid, Physical Review A 31 (2) (1985) 1103. https://doi.org/10.1103/PhysRevA.31.1103

    work page doi:10.1103/physreva.31.1103 1985

  24. [24]

    Kendon, Scaling theory of three-dimensional spinodal turbulence, Physical Review E 61 (6) (2000) R6071

    V. Kendon, Scaling theory of three-dimensional spinodal turbulence, Physical Review E 61 (6) (2000) R6071. https://doi.org/10.1103/PhysRevE.61.R6071

    work page doi:10.1103/physreve.61.r6071 2000

  25. [25]

    Grant, K

    M. Grant, K. Elder, Spinodal decomposition in fluids, Physical review letters 82 (1) (1999) 14. https://doi.org/10.1103/PhysRevLett.82.14

    work page doi:10.1103/physrevlett.82.14 1999

  26. [26]

    Engblom, M

    S. Engblom, M. Do-Quang, G. Amberg, A.-K. Tornberg, On diffuse interface modeling and simulation of surfactants in two-phase fluid flow, Communications in Computational Physics 14 (4) (2013) 879– 915. https://doi.org/10.4208/cicp.120712.281212a

    work page doi:10.4208/cicp.120712.281212a 2013

  27. [27]

    H. Liu, Y. Zhang, Phase-field modeling droplet dynamics with soluble surfactants, Journal of Compu- tational Physics 229 (24) (2010) 9166–9187. https://doi.org/10.1016/j.jcp.2010.08.031

    work page doi:10.1016/j.jcp.2010.08.031 2010

  28. [28]

    Soligo, A

    G. Soligo, A. Roccon, A. Soldati, Coalescence of surfactant-laden drops by phase field method, Journal of Computational Physics 376 (2019) 1292–1311. https://doi.org/10.1016/j.jcp.2018.10.021

    work page doi:10.1016/j.jcp.2018.10.021 2019

  29. [29]

    J. Yang, Z. Tan, J. Kim, Linear and fully decoupled scheme for a hydrodynamics coupled phase-field surfactant system based on a multiple auxiliary variables approach, Journal of Computational Physics 452 (2022) 110909. https://doi.org/10.1016/j.jcp.2021.110909

    work page doi:10.1016/j.jcp.2021.110909 2022

  30. [30]

    Yang, Numerical approximations for the cahn–hilliard phase field model of the binary fluid-surfactant system, Journal of Scientific Computing 74 (3) (2018) 1533–1553

    X. Yang, Numerical approximations for the cahn–hilliard phase field model of the binary fluid-surfactant system, Journal of Scientific Computing 74 (3) (2018) 1533–1553. https://doi.org/10.1007/s10915-017-0508-6

    work page doi:10.1007/s10915-017-0508-6 2018

  31. [31]

    Manikantan, T

    H. Manikantan, T. M. Squires, Surfactant dynamics: hidden variables controlling fluid flows, Journal of fluid mechanics 892 (2020) P1. https://doi.org/10.1017/jfm.2020.170

    work page doi:10.1017/jfm.2020.170 2020

  32. [32]

    H. Hao, X. Li, L. Xu, T. Liu, H. Tan, Profile-preserving phase-field model for surfactant transport and adsorption-desorption in two-phase flow systems, Journal of Computational Physics 546 (2026) 114510. https://doi.org/10.1016/j.jcp.2025.114510

    work page doi:10.1016/j.jcp.2025.114510 2026

  33. [33]

    Skartlien, B

    R. Skartlien, B. Grimes, P. Meakin, J. Sj¨ oblom, E. Sollum, Coalescence kinetics in surfactant stabilized 26 emulsions: Evolution equations from direct numerical simulations, The Journal of chemical physics 137 (21) (2012). http://dx.doi.org/10.1063/1.4768243

    work page doi:10.1063/1.4768243 2012

  34. [34]

    B. Dai, L. G. Leal, The mechanism of surfactant effects on drop coalescence, Physics of Fluids 20 (4) (2008) 040802. https://doi.org/10.1063/1.2911700

    work page doi:10.1063/1.2911700 2008

  35. [35]

    Pan, Y.-H

    K.-L. Pan, Y.-H. Tseng, J.-C. Chen, K.-L. Huang, C.-H. Wang, M.-C. Lai, Controlling droplet bouncing and coalescence with surfactant, Journal of Fluid Mechanics 799 (2016) 603–636. https://doi.org/10.1017/jfm.2016.381

    work page doi:10.1017/jfm.2016.381 2016

  36. [36]

    H. Hao, X. Li, T. Liu, H. Tan, Enhanced profile-preserving phase-field model of two-phase flow with surfactant interfacial transport and marangoni effects, Journal of Computational Physics 536 (2025) 114058. https://doi.org/10.1016/j.jcp.2025.114058

    work page doi:10.1016/j.jcp.2025.114058 2025

  37. [37]

    Krebs, K

    T. Krebs, K. Schro¨ en, R. Boom, Coalescence dynamics of surfactant-stabilized emulsions studied with microfluidics, Soft Matter 8 (41) (2012) 10650–10657. http://dx.doi.org/10.1039/C2SM26122G

    work page doi:10.1039/c2sm26122g 2012

  38. [38]

    N. Pal, S. Kumar, A. Bera, A. Mandal, Phase behaviour and characterization of microemulsion sta- bilized by a novel synthesized surfactant: Implications for enhanced oil recovery, Fuel 235 (2019) 995–1009. https://doi.org/10.1016/j.fuel.2018.08.100

    work page doi:10.1016/j.fuel.2018.08.100 2019

  39. [39]

    Nekovee, P

    M. Nekovee, P. V. Coveney, H. Chen, B. M. Boghosian, Lattice-boltzmann model for interacting am- phiphilic fluids, Physical Review E 62 (6) (2000) 8282. https://doi.org/10.1103/PhysRevE.62.8282

    work page doi:10.1103/physreve.62.8282 2000

  40. [40]

    K. Kim, S. Q. Choi, Z. A. Zell, T. M. Squires, J. A. Zasadzinski, Effect of cholesterol nanodomains on monolayer morphology and dynamics, Proceedings of the National Academy of Sciences 110 (33) (2013) E3054–E3060. https://doi.org/10.1073/pnas.1303304110

    work page doi:10.1073/pnas.1303304110 2013

  41. [41]

    Van der Sman, S

    R. Van der Sman, S. Van der Graaf, Diffuse interface model of surfactant adsorption onto flat and droplet interfaces, Rheologica acta 46 (1) (2006) 3–11. https://doi.org/10.1007/s00397-005-0081-z

    work page doi:10.1007/s00397-005-0081-z 2006

  42. [42]

    D. J. Korteweg, Sur la forme que prennent les ´ equations du mouvements des fluides si l’on tient compte des forces capillaires caus´ ees par des variations de densit´ e consid´ erables mais connues et sur la th´ eorie de la capillarit´ e dans l’hypoth` ese d’une variation continue de la densit´ e, Archives N´ eerlandaises des Sciences exactes et naturell...

    1901

  43. [43]

    P. C. Hohenberg, B. I. Halperin, Theory of dynamic critical phenomena, Reviews of Modern Physics 49 (3) (1977) 435. https://doi.org/10.1103/RevModPhys.49.435

    work page doi:10.1103/revmodphys.49.435 1977

  44. [44]

    Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Springer Berlin Heidelberg, Berlin, Heidelberg, 1998, pp

    C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Springer Berlin Heidelberg, Berlin, Heidelberg, 1998, pp. 325–432. https://doi.org/10.1007/BFb0096355

    work page doi:10.1007/bfb0096355 1998

  45. [45]

    H. Ding, P. D. Spelt, C. Shu, Diffuse interface model for incompressible two-phase flows with large density ratios, Journal of Computational Physics 226 (2) (2007) 2078–2095. https://doi.org/10.1016/j.jcp.2007.06.028

    work page doi:10.1016/j.jcp.2007.06.028 2007

  46. [46]

    H. Hao, X. Li, C. Jiang, H. Tan, An interfacial profile-preserving approach for phase field modeling of incompressible two-phase flows, International Journal of Multiphase Flow 174 (2024) 104750. https://doi.org/10.1016/j.ijmultiphaseflow.2024.104750

    work page doi:10.1016/j.ijmultiphaseflow.2024.104750 2024

  47. [47]

    Lifshitz, V

    I. Lifshitz, V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, Journal of Physics and Chemistry of Solids 19 (1) (1961) 35–50. https://doi.org/10.1016/0022-3697(61)90054-3

    work page doi:10.1016/0022-3697(61)90054-3 1961

  48. [48]

    Wagner, Theory of the aging of precipitates by dissolution-reprecipitation (ostwald ripening), Z Elektrochem 65 (7) (1961) 581–11

    C. Wagner, Theory of the aging of precipitates by dissolution-reprecipitation (ostwald ripening), Z Elektrochem 65 (7) (1961) 581–11

    1961

  49. [49]

    A. J. Bray, Theory of phase-ordering kinetics, Advances in Physics 51 (2) (2002) 481–587. https://doi.org/10.1080/00018730110117433

    work page doi:10.1080/00018730110117433 2002

  50. [50]

    Shapira, S

    M. Shapira, S. Haber, Low reynolds number motion of a droplet in shear flow including wall effects, International Journal of Multiphase Flow 16 (2) (1990) 305–321. https://doi.org/10.1016/0301-9322(90)90061-M

    work page doi:10.1016/0301-9322(90)90061-m 1990

  51. [51]

    H. A. Stone, L. G. Leal, The effects of surfactants on drop deformation and breakup, Journal of Fluid Mechanics 220 (1990) 161–186. https://doi.org/10.1017/S0022112090003226

    work page doi:10.1017/s0022112090003226 1990

  52. [52]

    Scriven, C

    L. Scriven, C. Sternling, The marangoni effects, Nature 187 (4733) (1960) 186–188. https://doi.org/10.1038/187186a0

    work page doi:10.1038/187186a0 1960

  53. [53]

    P. J. Love, P. V. Coveney, B. M. Boghosian, Three-dimensional hydrodynamic lattice-gas simulations of domain growth and self-assembly in binary immiscible and ternary amphiphilic fluids, Physical Review E 64 (2) (2001) 021503. https://doi.org/10.1103/PhysRevE.64.021503

    work page doi:10.1103/physreve.64.021503 2001

  54. [54]

    P. M. Kamat, B. W. Wagoner, A. A. Castrej´ on-Pita, J. R. Castrej´ on-Pita, C. R. Anthony, O. A. Basaran, Surfactant-driven escape from endpinching during contraction of nearly inviscid filaments, Journal of Fluid Mechanics 899 (2020) A28. 28 https://doi.org/10.1017/jfm.2020.476

    work page doi:10.1017/jfm.2020.476 2020

  55. [55]

    Constante-Amores, A

    C. Constante-Amores, A. Batchvarov, L. Kahouadji, S. Shin, J. Chergui, D. Juric, O. Matar, Role of surfactant-induced marangoni stresses in drop-interface coalescence, Journal of Fluid Mechanics 925 (2021) A15. https://doi.org/10.1017/jfm.2021.682

    work page doi:10.1017/jfm.2021.682 2021

  56. [56]

    Tanaka, T

    H. Tanaka, T. Araki, Spontaneous double phase separation induced by rapid hydrodynamic coarsening in two-dimensional fluid mixtures, Physical Review Letters 81 (2) (1998) 389. https://doi.org/10.1103/PhysRevLett.81.389

    work page doi:10.1103/physrevlett.81.389 1998

  57. [57]

    F. Wang, P. Altschuh, L. Ratke, H. Zhang, M. Selzer, B. Nestler, Progress report on phase separation in polymer solutions, Advanced Materials 31 (26) (2019) 1806733. https://doi.org/10.1002/adma.201806733

    work page doi:10.1002/adma.201806733 2019

  58. [58]

    R. H. Ewoldt, C. Saengow, Designing complex fluids, Annual Review of Fluid Mechanics 54 (2022) 413–441. https://doi.org/10.1146/annurev-fluid-031821-104935

    work page doi:10.1146/annurev-fluid-031821-104935 2022

  59. [59]

    A. J. Hardy, S. Cameron, S. McDonald, A. Daddi-Moussa-Ider, E. Tjhung, Kinetic theory of binary fluid–surfactant systems: A variational framework, Journal of Fluid Mechanics 1029 (2026) A54

    2026

  60. [60]

    X. Li, D. Wan, H. Hao, C. Diddens, M. Zhang, H. Tan, A diffuse-interface marangoni instability, Journal of Fluid Mechanics 1024 (2025) A37. https://doi.org/10.1017/jfm.2025.10895

    work page doi:10.1017/jfm.2025.10895 2025

  61. [61]

    Chang, E

    C.-H. Chang, E. I. Franses, Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms, Colloids and Surfaces A: Physicochemical and Engineering Aspects 100 (1995) 1–45. https://doi.org/10.1016/0927-7757(94)03061-4

    work page doi:10.1016/0927-7757(94)03061-4 1995

  62. [62]

    R. M. Weinheimer, D. F. Evans, E. Cussler, Diffusion in surfactant solutions, Journal of Colloid and Interface Science 80 (2) (1981) 357–368. https://doi.org/10.1016/0021-9797(81)90194-6

    work page doi:10.1016/0021-9797(81)90194-6 1981

  63. [63]

    A. C. Ribeiro, V. M. Lobo, E. F. Azevedo, M. D. G. Miguel, H. Burrows, Diffusion coefficients of sodium dodecylsulfate in aqueous solutions and in aqueous solutions ofβ-cyclodextrin, Journal of molecular liquids 102 (1-3) (2003) 285–292. https://doi.org/10.1016/S0167-7322(01)00268-9 29

    work page doi:10.1016/s0167-7322(01)00268-9 2003