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arxiv: 2601.22962 · v3 · submitted 2026-01-30 · ❄️ cond-mat.soft · nlin.AO

Recognition: 2 theorem links

· Lean Theorem

Gradient dynamics model for chemically driven running drops

Justus Niehoff , Florian Voss , Uwe Thiele
Authors on Pith no claims yet

Pith reviewed 2026-05-16 09:31 UTC · model grok-4.3

classification ❄️ cond-mat.soft nlin.AO
keywords running dropschemical drivinggradient dynamicswettability contrastself-propulsionchemostattingdrift-pitchfork bifurcationsubstrate adsorption
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The pith

Distinct chemical potentials drive sustained self-propulsion of drops via reaction-maintained wettability contrast on the substrate.

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

The paper constructs a thermodynamically consistent description by beginning with gradient dynamics for a closed system that couples the drop height profile to chemical concentration fields on the substrate and in the two fluids. Chemostatting the concentrations in the drop and ambient medium then yields a reduced set of equations in which unequal external chemical potentials continuously drive the system out of equilibrium. This driving sustains drop motion because reversible adsorption and reactions keep a wettability difference active between the front and rear of the drop. Numerical continuation shows the running states appear through drift-pitchfork bifurcations as the chemical-potential difference is increased.

Core claim

When the externally imposed chemical potentials are distinct, the system is driven away from thermodynamic equilibrium, allowing for sustained drop self-propulsion across the substrate due to a wettability contrast maintained by chemical reactions.

What carries the argument

The reduced gradient-dynamics model obtained by fixing chemical potentials in the drop and ambient medium, which converts the closed-system equations into an open-system description whose driving term produces a persistent wettability gradient via reversible substrate adsorption.

If this is right

  • Running drops appear via drift-pitchfork bifurcations whose critical chemical-potential difference depends on reaction rates and adsorption parameters.
  • Drop speed and shape are controlled by the difference in imposed chemical potentials and the reversibility of substrate adsorption.
  • The model supplies a thermodynamically consistent route to sustained propulsion without external mechanical forcing or imposed gradients.
  • The same chemostatting procedure can be applied to other thin-film systems that involve reversible surface chemistry.

Where Pith is reading between the lines

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

  • The bifurcation structure supplies a practical way to predict the onset of motion in microfluidic or coating experiments that use chemical reservoirs.
  • If the chemostatting step remains valid for larger drops, the framework could guide design of chemically powered droplet conveyors or sensors.
  • The approach suggests that similar reductions may organize the dynamics of other active soft-matter systems whose reservoirs are held at fixed potentials.

Load-bearing premise

The reduction from the closed-system gradient dynamics to the chemostatted model preserves the essential driving mechanism without introducing unphysical artifacts.

What would settle it

An experiment in which the chemical potentials in the drop and ambient medium are made equal while all other parameters remain fixed would show whether the drops stop moving or continue to run.

Figures

Figures reproduced from arXiv: 2601.22962 by Florian Voss, Justus Niehoff, Uwe Thiele.

Figure 1
Figure 1. Figure 1: FIG. 1. Sketch of a chemically driven running drop confined to a vertical gap of height [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Comparison of (a, c) relaxational dynamics converging to a resting drop and (b, d) persistent [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Snapshots of the adsorbate profile (blue solid lines) and the drop profile (black dashed lines) at [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Emergence of drop self-propulsion in dependence of the external chemical potentials [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Concentration profiles (blue solid lines) of stationary [(a), (g), (f), (l)] and self-propelled drops [(b)- [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Drop velocity [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
read the original abstract

We present a thermodynamically consistent model for chemically driven running drops on a solid substrate with reversible substrate adsorption of a wettability-changing chemical species. We consider drops confined to a vertical gap, thereby allowing us to first obtain a gradient dynamics description of the closed system, corresponding to a set of coupled dynamical equations for the drop profile and the chemical concentration profiles of species on the substrate and in both fluids (drop, ambient medium). Chemostatting the species in the drop and the ambient medium, we then derive a reduced model for the dynamics of the drop and the adsorbate on the substrate. When the externally imposed chemical potentials are distinct, the system is driven away from thermodynamic equilibrium, allowing for sustained drop self-propulsion across the substrate due to a wettability contrast maintained by chemical reactions. We numerically study the resulting running drops and show how they emerge from drift-pitchfork bifurcations.

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

1 major / 2 minor

Summary. The paper presents a thermodynamically consistent gradient-dynamics model for chemically driven running drops on a solid substrate with reversible adsorption of a wettability-changing species. It first derives coupled equations for the drop profile and concentration profiles in a closed system (drop fluid, ambient medium, and substrate adsorbate), then reduces the model by chemostatting the chemical potentials in the drop and ambient medium. When the imposed chemical potentials differ, the system is driven out of equilibrium, sustaining self-propulsion via a maintained wettability contrast from chemical reactions. Numerical studies show the running drops emerging from drift-pitchfork bifurcations.

Significance. If the chemostat reduction is rigorously justified, the work supplies a thermodynamically consistent framework linking gradient dynamics to chemically driven propulsion, with potential relevance to active soft matter and microfluidic applications. The bifurcation analysis provides a concrete mechanism for the onset of sustained motion without external forcing.

major comments (1)
  1. [§3] §3 (reduction from closed to chemostatted system): the mapping from the closed-system gradient dynamics to the reduced equations by fixing chemical potentials μ in the drop and ambient medium is stated without an explicit validity limit (e.g., infinite reservoir capacity, fast fluid diffusion, or Damköhler number ≪ 1). This step is load-bearing for the central claim that distinct imposed μ values produce sustained propulsion without finite-volume depletion artifacts altering the bifurcation structure or steady velocities.
minor comments (2)
  1. [Numerical results] Numerical methods section: discretization scheme, grid resolution, and parameter ranges for the bifurcation diagrams are not specified, making it difficult to assess convergence of the reported drift-pitchfork thresholds.
  2. [Model derivation] Notation: the distinction between the full closed-system chemical potentials and the fixed chemostatted values should be clarified with explicit symbols in the reduced equations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We appreciate the recognition of the model's thermodynamic consistency and the potential relevance to active soft matter. We address the major comment below and will revise the manuscript to strengthen the presentation of the chemostat reduction.

read point-by-point responses

  1. Referee: [§3] §3 (reduction from closed to chemostatted system): the mapping from the closed-system gradient dynamics to the reduced equations by fixing chemical potentials μ in the drop and ambient medium is stated without an explicit validity limit (e.g., infinite reservoir capacity, fast fluid diffusion, or Damköhler number ≪ 1). This step is load-bearing for the central claim that distinct imposed μ values produce sustained propulsion without finite-volume depletion artifacts altering the bifurcation structure or steady velocities.

    Authors: We agree that the reduction step in §3 requires a more explicit statement of its validity regime to support the central claim. In the revised manuscript we will add a dedicated paragraph (or short subsection) immediately following the derivation of the reduced equations. This will specify that the chemostatting corresponds to the limit of infinite reservoir capacity for the drop and ambient phases, with external reservoirs continuously fixing the chemical potentials μ_drop and μ_ambient. We will also note the accompanying timescale separation: diffusion within the fluids is assumed fast relative to adsorption kinetics and drop motion (Damköhler number ≪ 1), allowing the fluid concentrations to remain spatially uniform at the imposed potentials. Under these conditions the reduced model inherits the gradient structure of the closed system while permitting sustained non-equilibrium driving. We will further include a brief numerical check demonstrating that the drift-pitchfork bifurcation thresholds and steady velocities converge to the reported chemostatted values as reservoir volumes are increased, confirming the absence of depletion artifacts within the regime of interest. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation is self-contained

full rationale

The paper begins with a standard gradient-dynamics formulation for the closed system (drop profile plus concentration fields in fluids and on substrate) and then applies chemostatting by fixing chemical potentials in the drop and ambient medium to obtain the reduced open model. This reduction is a conventional modeling approximation rather than a redefinition that forces the target result by construction. The sustained self-propulsion is presented as an emergent dynamical consequence of the imposed potential difference maintaining a wettability contrast via reactions; no parameters are fitted to data and then relabeled as predictions, no uniqueness theorems are imported from self-citations, and no ansatz is smuggled in. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a gradient-dynamics structure for the closed multi-component system and on the validity of the chemostatting reduction; both are standard in the field but constitute domain assumptions rather than derived results.

axioms (2)
  • domain assumption The system admits a gradient-dynamics description whose dissipation functional yields the correct thermodynamic driving forces for the drop profile and concentration fields.
    Invoked when the closed-system equations are introduced; this is a modeling choice rather than a theorem proved in the paper.
  • domain assumption Reversible substrate adsorption of the wettability-changing species can be described by local equilibrium or linear kinetics consistent with the gradient structure.
    Required for the concentration equations on the substrate.

pith-pipeline@v0.9.0 · 5447 in / 1395 out tokens · 30346 ms · 2026-05-16T09:31:27.333133+00:00 · methodology

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Works this paper leans on

74 extracted references · 74 canonical work pages

  1. [1]

    M. K. Chaudhury and G. M. Whitesides. How to make water run uphill.Science, 256:1539–1541,

    work page

  2. [2]

    doi:https://doi.org/10.1126/science.256.5063.1539

    work page doi:10.1126/science.256.5063.1539

  3. [3]

    N. O. Young, J. S. Goldstein, and M. J. Block. The motion of bubbles in a vertical temperature gradient.J. Fluid Mech., 6:350–356, 1959. doi:https://doi.org/10.1017/s0022112059000684

    work page doi:10.1017/s0022112059000684 1959

  4. [4]

    Golestanian, T

    R. Golestanian, T. B. Liverpool, and A. Ajdari. Propulsion of a molecular machine by asymmetric distribution of reaction products.Phys. Rev. Lett., 94:220801, 2005. doi: https://doi.org/10.1103/physrevlett.94.220801

    work page doi:10.1103/physrevlett.94.220801 2005

  5. [5]

    Demarchi, A

    L. Demarchi, A. Goychuk, I. Maryshev, and E. Frey. Enzyme-enriched condensates show self-propulsion, positioning, and coexistence.Phys. Rev. Lett., 130:128401, 2023. doi: https://doi.org/10.1103/PhysRevLett.130.128401

    work page doi:10.1103/physrevlett.130.128401 2023

  6. [6]

    Goychuk, L

    A. Goychuk, L. Demarchi, I. Maryshev, and E. Frey. Self-consistent sharp interface theory of active condensate dynamics.Phys. Rev. Research, 6:033082, 2024. doi: https://doi.org/10.1103/physrevresearch.6.033082

    work page doi:10.1103/physrevresearch.6.033082 2024

  7. [7]

    Trinschek, F

    S. Trinschek, F. Stegemerten, K. John, and U. Thiele. Thin-film modelling of resting and moving active droplets.Phys. Rev. E, 101:062802, 2020. doi:https://doi.org/10.1103/PhysRevE.101.062802

    work page doi:10.1103/physreve.101.062802 2020

  8. [8]

    Stegemerten, K

    F. Stegemerten, K. John, and U. Thiele. Symmetry-breaking, motion and bistability of ac- tive drops through polarization-surface coupling.Soft Matter, 18:5823–5832, 2022. doi: https://doi.org/10.1039/D2SM00648K

    work page doi:10.1039/d2sm00648k 2022

  9. [9]

    Michelin

    S. Michelin. Self-propulsion of chemically active droplets.Annu. Rev. Fluid Mech., 55:77–101, 2023. doi:https://doi.org/10.1146/annurev-fluid-120720-012204

    work page doi:10.1146/annurev-fluid-120720-012204 2023

  10. [10]

    Seemann, J

    R. Seemann, J. B. Fleury, and C. C. Maass. Self-propelled droplets.Eur . Phys. J.-Spec. Top., 225: 2227–2240, 2016. doi:https://doi.org/10.1140/epjst/e2016-60061-7

    work page doi:10.1140/epjst/e2016-60061-7 2016

  11. [11]

    Domingues Dos Santos and T

    F. Domingues Dos Santos and T. Ondarc ¸uhu. Free-running droplets.Phys. Rev. Lett., 75:2972–2975,

    work page

  12. [12]

    doi:https://doi.org/10.1103/PhysRevLett.75.2972

    work page doi:10.1103/physrevlett.75.2972

  13. [13]

    Sumino, N

    Y . Sumino, N. Magome, T. Hamada, and K. Yoshikawa. Self-running droplet: Emergence of regular motion from nonequilibrium noise.Phys. Rev. Lett., 94:068301, 2005. doi: https://doi.org/10.1103/PhysRevLett.94.068301. 28

    work page doi:10.1103/physrevlett.94.068301 2005

  14. [14]

    S. W. Lee and P. E. Laibinis. Directed movement of liquids on patterned surfaces us- ing noncovalent molecular adsorption.J. Am. Chem. Soc., 122:5395–5396, 2000. doi: https://doi.org/10.1021/ja994076a

    work page doi:10.1021/ja994076a 2000

  15. [15]

    S. W. Lee, D. Y . Kwok, and P. E. Laibinis. Chemical influences on adsorption-mediated self-propelled drop movement.Phys. Rev. E, 65:051602, 2002. doi:https://doi.org/10.1103/PhysRevE.65.051602

    work page doi:10.1103/physreve.65.051602 2002

  16. [16]

    Thiele, K

    U. Thiele, K. John, and M. B ¨ar. Dynamical model for chemically driven running droplets.Phys. Rev. Lett., 93:027802, 2004. doi:https://doi.org/10.1103/PhysRevLett.93.027802

    work page doi:10.1103/physrevlett.93.027802 2004

  17. [17]

    K. John, M. B ¨ar, and U. Thiele. Self-propelled running droplets on solid substrates driven by chemical reactions.Eur . Phys. J. E, 18:183–199, 2005. doi:https://doi.org/10.1140/epje/i2005-10039-1

    work page doi:10.1140/epje/i2005-10039-1 2005

  18. [18]

    Sumino, H

    Y . Sumino, H. Kitahata, K. Yoshikawa, M. Nagayama, S. M. Nomura, N. Magome, and Y . Mori. Chemosensitive running droplet.Phys. Rev. E, 72:041603, 2005. doi: https://doi.org/10.1103/PhysRevE.72.041603

    work page doi:10.1103/physreve.72.041603 2005

  19. [19]

    V oss and U

    F. V oss and U. Thiele. Gradient dynamics approach to reactive thin-film hydrodynamics.J. Eng. Math., 149:2, 2024. doi:https://doi.org/10.1007/s10665-024-10402-x

    work page doi:10.1007/s10665-024-10402-x 2024

  20. [20]

    Avanzini, T

    F. Avanzini, T. Aslyamov, ´E. Fodor, and M. Esposito. Nonequilibrium thermodynamics of non-ideal reaction–diffusion systems: Implications for active self-organization.J. Chem. Phys., 161:174108,

    work page

  21. [21]

    doi:https://doi.org/10.1063/5.0231520

    work page doi:10.1063/5.0231520

  22. [22]

    S. R. de Groot and P. Mazur.Non-equilibrium Thermodynamics. Dover publications, New York, 1984

    work page 1984

  23. [23]

    Rao and M

    R. Rao and M. Esposito. Nonequilibrium thermodynamics of chemical reaction net- works: Wisdom from stochastic thermodynamics.Phys. Rev. X, 6:041064, 2016. doi: https://doi.org/10.1103/physrevx.6.041064

    work page doi:10.1103/physrevx.6.041064 2016

  24. [24]

    Avanzini, E

    F. Avanzini, E. Penocchio, G. Falasco, and M. Esposito. Nonequilibrium thermodynam- ics of non-ideal chemical reaction networks.J. Chem. Phys., 154:094114, 2021. doi: https://doi.org/10.1063/5.0041225

    work page doi:10.1063/5.0041225 2021

  25. [25]

    U. Thiele. Recent advances in and future challenges for mesoscopic hydrodynamic modelling of com- plex wetting.Colloid Surf. A, 553:487–495, 2018. doi:https://doi.org/10.1016/j.colsurfa.2018.05.049

    work page doi:10.1016/j.colsurfa.2018.05.049 2018

  26. [26]

    V oss and U

    F. V oss and U. Thiele. Chemomechanical motility modes of partially wetting liquid droplets.Phys. Rev. Fluids, 10:094005, 2025. doi:https://doi.org/10.1103/f3ck-dx5c

    work page doi:10.1103/f3ck-dx5c 2025

  27. [27]

    V oss and U

    F. V oss and U. Thiele. From bipedal to chaotic motion of chemically fueled partially wetting liquid drops.arXiv:2512.14370, 2025. doi:https://doi.org/10.48550/arxiv.2512.14370. 29

    work page doi:10.48550/arxiv.2512.14370 2025

  28. [28]

    Greve, G

    D. Greve, G. Lovato, T. Frohoff-H ¨ulsmann, and U. Thiele. Coexistence of uniform and oscillatory states resulting from nonreciprocity and conservation laws.Phys. Rev. Lett., 134:018303, 2025. doi: https://doi.org/10.1103/physrevlett.134.018303

    work page doi:10.1103/physrevlett.134.018303 2025

  29. [29]

    Frohoff-H ¨ulsmann and U

    T. Frohoff-H ¨ulsmann and U. Thiele. Nonreciprocal Cahn-Hilliard model emerges as a universal amplitude equation.Phys. Rev. Lett., 131:107201, 2023. doi: https://doi.org/10.1103/PhysRevLett.131.107201

    work page doi:10.1103/physrevlett.131.107201 2023

  30. [30]

    Brauns and M

    F. Brauns and M. C. Marchetti. Nonreciprocal pattern formation of conserved fields.Phys. Rev. X, 14: 021014, 2024. doi:https://doi.org/10.1103/physrevx.14.021014

    work page doi:10.1103/physrevx.14.021014 2024

  31. [31]

    S. Saha, J. Agudo-Canalejo, and R. Golestanian. Scalar active mixtures: The non-reciprocal Cahn- Hilliard model.Phys. Rev. X, 10:041009, 2020. doi:https://doi.org/10.1103/PhysRevX.10.041009

    work page doi:10.1103/physrevx.10.041009 2020

  32. [32]

    Frohoff-H ¨ulsmann, J

    T. Frohoff-H ¨ulsmann, J. Wrembel, and U. Thiele. Suppression of coarsening and emergence of os- cillatory behavior in a Cahn-Hilliard model with nonvariational coupling.Phys. Rev. E, 103:042602,

    work page

  33. [33]

    doi:https://doi.org/10.1103/PhysRevE.103.042602

    work page doi:10.1103/physreve.103.042602

  34. [34]

    Z. H. You, A. Baskaran, and M. C. Marchetti. Nonreciprocity as a generic route to traveling states.Proc. Natl. Acad. Sci. U. S. A., 117:19767–19772, 2020. doi: https://doi.org/10.1073/pnas.2010318117

    work page doi:10.1073/pnas.2010318117 2020

  35. [35]

    Frohoff-H ¨ulsmann, M

    T. Frohoff-H ¨ulsmann, M. P. Holl, E. Knobloch, S. V . Gurevich, and U. Thiele. Station- ary broken parity states in active matter models.Phys. Rev. E, 107:064210, 2023. doi: https://doi.org/10.1103/PhysRevE.107.064210

    work page doi:10.1103/physreve.107.064210 2023

  36. [36]

    Remlein, M

    B. Remlein, M. Esposito, and F. Avanzini. What is a chemostat? insights from hy- brid dynamics and stochastic thermodynamics.J. Chem. Phys., 162:224113, 2025. doi: https://doi.org/10.1063/5.0267465

    work page doi:10.1063/5.0267465 2025

  37. [37]

    Thiele, A

    U. Thiele, A. J. Archer, and L. M. Pismen. Gradient dynamics models for liquid films with soluble surfactant.Phys. Rev. Fluids, 1:083903, 2016. doi:https://doi.org/10.1103/PhysRevFluids.1.083903

    work page doi:10.1103/physrevfluids.1.083903 2016

  38. [38]

    M. Doi. Onsager’s variational principle in soft matter.J. Phys.: Condens. Matter, 23:284118, 2011. doi:https://doi.org/10.1088/0953-8984/23/28/284118

    work page doi:10.1088/0953-8984/23/28/284118 2011

  39. [39]

    A. J. Bray. Theory of phase-ordering kinetics.Adv. Phys., 43:357–459, 1994. doi: https://doi.org/10.1080/00018739400101505

    work page doi:10.1080/00018739400101505 1994

  40. [40]

    D. Zwicker. The intertwined physics of active chemical reactions and phase separation.Curr . Opin. Colloid Interface Sci., 61:101606, 2022. doi:https://doi.org/10.1016/j.cocis.2022.101606. 30

    work page doi:10.1016/j.cocis.2022.101606 2022

  41. [41]

    Thiele, D

    U. Thiele, D. V . Todorova, and H. Lopez. Gradient dynamics description for films of mixtures and suspensions: dewetting triggered by coupled film height and concentration fluctuations.Phys. Rev. Lett., 111:117801, 2013. doi:https://doi.org/10.1103/PhysRevLett.111.117801

    work page doi:10.1103/physrevlett.111.117801 2013

  42. [42]

    Thiele, A

    U. Thiele, A. J. Archer, and M. Plapp. Thermodynamically consistent description of the hydrodynam- ics of free surfaces covered by insoluble surfactants of high concentration.Phys. Fluids, 24:102107,

    work page

  43. [43]

    doi:https://doi.org/10.1063/1.4758476

    work page doi:10.1063/1.4758476

  44. [44]

    Henkel, J

    C. Henkel, J. H. Snoeijer, and U. Thiele. Gradient-dynamics model for liquid drops on elastic sub- strates.Soft Matter, 17:10359–10375, 2021. doi:https://doi.org/10.1039/D1SM01032H

    work page doi:10.1039/d1sm01032h 2021

  45. [45]

    Hartmann, J

    S. Hartmann, J. Diekmann, D. Greve, and U. Thiele. Drops on polymer brushes: Ad- vances in thin-film modeling of adaptive substrates.Langmuir, 40:4001–4021, 2024. doi: https://doi.org/10.1021/acs.langmuir.3c03313

    work page doi:10.1021/acs.langmuir.3c03313 2024

  46. [46]

    Diekmann and U

    J. Diekmann and U. Thiele. Mesoscopic hydrodynamic model for spreading, slid- ing, and coarsening compound drops.Phys. Rev. Fluids, 10:024002, 2025. doi: https://doi.org/10.1103/physrevfluids.10.024002

    work page doi:10.1103/physrevfluids.10.024002 2025

  47. [47]

    D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley. Wetting and spreading.Rev. Mod. Phys., 81:739–805, 2009. doi:https://doi.org/10.1103/RevModPhys.81.739

    work page doi:10.1103/revmodphys.81.739 2009

  48. [48]

    R. V . Craster and O. K. Matar. Dynamics and stability of thin liquid films.Rev. Mod. Phys., 81: 1131–1198, 2009. doi:https://doi.org/10.1103/RevModPhys.81.1131

    work page doi:10.1103/revmodphys.81.1131 2009

  49. [49]

    Hartmann, C

    S. Hartmann, C. Diddens, M. Jalaal, and U. Thiele. Sessile drop evaporation in a gap - crossover between diffusion-limited and phase transition-limited regime.J. Fluid Mech., 960:A32, 2023. doi: https://doi.org/10.1017/jfm.2023.176

    work page doi:10.1017/jfm.2023.176 2023

  50. [50]

    Thiele, J

    U. Thiele, J. H. Snoeijer, S. Trinschek, and K. John. Equilibrium contact angle and adsorption layer properties with surfactants.Langmuir, 34:7210–7221, 2018. doi: https://doi.org/10.1021/acs.langmuir.8b00513. Also see Erratum: Langmuir, 35:4788-4789, 2019, doi:10.1021/acs.langmuir.9b00616

    work page doi:10.1021/acs.langmuir.8b00513 2018

  51. [51]

    In: Fluid-structure Interaction, pp

    M. Heil and A. L. Hazel. Oomph-lib - an object-oriented multi-physics finite-element library. In H.-J. Bungartz and M. Sch ¨afer, editors,Fluid-Structure Interaction: Modelling, Simulation, Optimisation, pages 19–49. Springer, Berlin, Heidelberg, 2006. doi:https://doi.org/10.1007/3-540-34596-5˙2

    work page doi:10.1007/3-540-34596-5 2006

  52. [52]

    R. E. Goldstein, G. H. Gunaratne, L. Gil, and P. Coullet. Hydrodynamic and interfa- cial patterns with broken space-time symmetry.Phys. Rev. A, 43:6700–6721, 1991. doi: https://doi.org/10.1103/PhysRevA.43.6700. 31

    work page doi:10.1103/physreva.43.6700 1991

  53. [53]

    Fauve, S

    S. Fauve, S. Douady, and O. Thual. Drift instabilities of cellular-patterns.J. Phys. II, 1:311–322,

    work page

  54. [54]

    doi:https://doi.org/10.1051/jp2:1991170

    work page doi:10.1051/jp2:1991170

  55. [55]

    Krauskopf, H

    B. Krauskopf, H. M. Osinga, and J. Galan-Vioque, editors.Numerical Continuation Methods for Dynamical Systems. Springer, Dordrecht, 2007. doi:https://doi.org/10.1007/978-1-4020-6356-5

    work page doi:10.1007/978-1-4020-6356-5 2007

  56. [56]

    M. C. Dallaston, D. Tseluiko, Z. Zheng, M. A. Fontelos, and S. Kalliadasis. Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows.Nonlinearity, 30: 2647–2666, 2017. doi:https://doi.org/10.1088/1361-6544/aa6eb3

    work page doi:10.1088/1361-6544/aa6eb3 2017

  57. [57]

    Sanchez Umbria and M

    J. Sanchez Umbria and M. Net. Numerical continuation methods for large-scale dissipative dynamical systems.Eur . Phys. J.-Spec. Top., 225:2465–2486, 2016. doi:https://doi.org/10.1140/epjst/e2015- 50317-2

    work page doi:10.1140/epjst/e2015- 2016

  58. [58]

    H. A. Dijkstra, F. W. Wubs, A. K. Cliffe, E. Doedel, I. F. Dragomirescu, B. Eckhardt, A. Y . Gelfgat, A. Hazel, V . Lucarini, A. G. Salinger, E. T. Phipps, J. Sanchez-Umbria, H. Schutte- laars, L. S. Tuckerman, and U. Thiele. Numerical bifurcation methods and their application to fluid dynamics: Analysis beyond simulation.Commun. Comput. Phys., 15:1–45, 2...

    work page doi:10.4208/cicp.240912.180613a 2014

  59. [59]

    Doedel, H

    E. Doedel, H. B. Keller, and J. P. Kernevez. Numerical analysis and control of bifurcation prob- lems (I) Bifurcation in finite dimensions.Int. J. Bifurcation Chaos, 1:493–520, 1991. doi: https://doi.org/10.1142/S0218127491000397

    work page doi:10.1142/s0218127491000397 1991

  60. [60]

    Merkt, A

    D. Merkt, A. Pototsky, M. Bestehorn, and U. Thiele. Long-wave theory of bounded two-layer films with a free liquid-liquid interface: Short- and long-time evolution.Phys. Fluids, 17:064104, 2005. doi:https://doi.org/10.1063/1.1935487

    work page doi:10.1063/1.1935487 2005

  61. [61]

    Nagai, Y

    K. Nagai, Y . Sumino, H. Kitahata, and K. Yoshikawa. Mode selection in the spontaneous motion of an alcohol droplet.Phys. Rev. E, 71:065301, 2005. doi:https://doi.org/10.1103/PhysRevE.71.065301

    work page doi:10.1103/physreve.71.065301 2005

  62. [62]

    Kumar and K

    G. Kumar and K. N. Prabhu. Review of non-reactive and reactive wetting of liquids on surfaces.Adv. Colloid Interface Sci., 133:61–89, 2007. doi:https://doi.org/10.1016/j.cis.2007.04.009

    work page doi:10.1016/j.cis.2007.04.009 2007

  63. [63]

    Sumino, R

    Y . Sumino, R. Yamashita, K. Miyaji, H. Ishikawa, M. Otani, D. Yamamoto, E. Okita, Y . Okamoto, M. P. Krafft, K. Yoshikawa, and A. Shioi. Droplet duos on water display pairing, autonomous motion, and periodic eruption.Sci Rep, 13:12377, 2023. doi:https://doi.org/10.1038/s41598-023-39094-6

    work page doi:10.1038/s41598-023-39094-6 2023

  64. [64]

    Satoh, Y

    Y . Satoh, Y . Sogabe, K. Kayahara, S. Tanaka, M. Nagayama, and S. Nakata. Self-inverted re- ciprocation of an oil droplet on a surfactant solution.Soft Matter, 13:3422–3430, 2017. doi: https://doi.org/10.1039/c7sm00252a. 32

    work page doi:10.1039/c7sm00252a 2017

  65. [65]

    Krobath, B

    H. Krobath, B. Rozycki, R. Lipowsky, and T. R. Weikl. Binding cooperativity of membrane adhesion receptors.Soft Matter, 5:3354–3361, 2009. doi:https://doi.org/10.1039/b902036e

    work page doi:10.1039/b902036e 2009

  66. [66]

    T. R. Weikl, M. Asfaw, H. Krobath, B. Rozycki, and R. Lipowsky. Adhesion of membranes via receptor-ligand complexes: domain formation, binding cooperativity, and active processes.Soft Mat- ter, 5:3213–3224, 2009. doi:https://doi.org/10.1039/b902017a

    work page doi:10.1039/b902017a 2009

  67. [67]

    Blom and A

    K. Blom and A. Godec. Criticality in cell adhesion.Phys. Rev. X, 11:031067, 2021. doi: https://doi.org/10.1103/physrevx.11.031067

    work page doi:10.1103/physrevx.11.031067 2021

  68. [68]

    Kitahata, N

    H. Kitahata, N. Yoshinaga, K. H. Nagai, and Y . Sumino. Spontaneous motion of a droplet coupled with a chemical wave.Phys. Rev. E, 84:015101, 2011. doi:https://doi.org/10.1103/PhysRevE.84.015101

    work page doi:10.1103/physreve.84.015101 2011

  69. [69]

    K. V . S. Chaithanya, S. A. Shenoy, and P. Dayal. Hydrodynamics of a con- fined active Belousov-Zhabotinsky droplet.Phys. Rev. E, 106:065103, 2022. doi: https://doi.org/10.1103/PhysRevE.106.065103

    work page doi:10.1103/physreve.106.065103 2022

  70. [70]

    Y . S. Kim, R. Tamate, A. M. Akimoto, and R. Yoshida. Recent developments in self-oscillating polymeric systems as smart materials: from polymers to bulk hydrogels.Mater . Horizons, 4:38–54,

    work page

  71. [71]

    doi:https://doi.org/10.1039/c6mh00435k

    work page doi:10.1039/c6mh00435k

  72. [72]

    Eckert and A

    K. Eckert and A. Grahn. Plume and finger regimes driven by an exothermic interfacial reaction.Phys. Rev. Lett., 82:4436–4439, 1999. doi:https://doi.org/10.1103/PhysRevLett.82.4436

    work page doi:10.1103/physrevlett.82.4436 1999

  73. [73]

    M. A. Budroni and A. De Wit. Dissipative structures: From reaction-diffusion to chemo- hydrodynamic patterns.Chaos, 27:104617, 2017. doi:https://doi.org/10.1063/1.4990740

    work page doi:10.1063/1.4990740 2017

  74. [74]

    Niehoff, F

    J. Niehoff, F. V oss, and U. Thiele. Data Supplement for ”Gradient dynamics model for chemically driven running drops”. Zenodo, 2026. doi:https://doi.org/10.5281/zenodo.18196225. 33

    work page doi:10.5281/zenodo.18196225 2026