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arxiv: 2605.22084 · v1 · pith:6MQOSTOYnew · submitted 2026-05-21 · ❄️ cond-mat.soft

The exact solution of the Koga-Widom-Indekeu model and related models of wetting in fluid mixtures

A.O. Parry , C. Rasc\'on This is my paper

Pith reviewed 2026-05-22 03:11 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords wetting transitionsfluid mixturessquare-gradient modelscritical point wettingsurface phase diagramdensity profilesXY symmetryNeumann triangle
5
0 comments X

The pith

The Koga-Widom-Indekeu model of wetting in fluid mixtures has an exact solution showing critical point wetting is absent when order parameters have local XY symmetry.

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

This paper provides an exact solution for a broad class of two-component square-gradient models describing wetting in fluid mixtures. By using complex analysis and the theory of algebraic curves, it determines the surface tensions and density profiles exactly. The solution confirms that partial wetting persists up to the critical end points, meaning critical point wetting does not occur in these models. This absence is tied to the local XY symmetry of the components rather than Ising-like behavior. The work resolves contradictions in prior studies by linking the presence or absence of critical point wetting to universality and symmetry principles.

Core claim

The central claim is that the KWI model and related variants can be solved exactly for density profiles and surface tensions using complex analysis, revealing the precise locations and orders of wetting transitions in the surface phase diagram. This confirms the absence of critical point wetting in ternary mixtures with three-phase coexistence. The exact solution also shows that microscopic density profiles map via conformal transform to the shape of a macroscopic drop satisfying the Neumann triangle. Related models demonstrate that critical point wetting occurs only if the order-parameter components follow Ising-like criticality, but is absent under local XY symmetry, suggesting a principle

What carries the argument

The square-gradient free-energy functional solved exactly via complex analysis and the theory of algebraic curves, which fixes surface tensions and maps density profile paths to geometric shapes obeying the Neumann triangle.

Load-bearing premise

The square-gradient free-energy functional accurately captures the essential physics of wetting transitions near critical end points in real fluid mixtures.

What would settle it

An experimental measurement of surface tensions in a ternary fluid mixture with three-phase coexistence that finds critical point wetting occurring would contradict the exact solution's prediction of its absence.

Figures

Figures reproduced from arXiv: 2605.22084 by A.O. Parry, C. Rasc\'on.

Figure 1
Figure 1. Figure 1: FIG. 1. A macroscopic sessile liquid drop on a smooth substrate. The surface tensions [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The macroscopic contact angles, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic surface phase diagrams showing the regions of complete wetting by the three coexisting phases [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. a) The Griffiths bulk phase diagram, showing the region of three phase coexistence in the (˜s [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The potential [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. a) and b) A tricuspid in the ( [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Primary and secondary density profiles, [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. a) In the two dimensional density plane, the bulk vertices [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Trajectories with high (blue) and low (red) angular momentum for the motion of a classical particle in a radial [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Exact profile trajectories [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. (a) Rotated coordinates in the density plane used to define the trajectory [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The quartic algebraic curve [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The quartic algebraic curve [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The conformal map [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The conformal map [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. The conformal map [PITH_FULL_IMAGE:figures/full_fig_p022_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. The point of maximum symmetry of the phase diagram, corresponding to ˜s [PITH_FULL_IMAGE:figures/full_fig_p023_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. As the [PITH_FULL_IMAGE:figures/full_fig_p024_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. a) The tricuspid of microscopic trajectories very close to the [PITH_FULL_IMAGE:figures/full_fig_p025_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. The density paths representing the interfacial trajectories (top panels) and their conformal mapping [PITH_FULL_IMAGE:figures/full_fig_p026_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. The effective potential [PITH_FULL_IMAGE:figures/full_fig_p028_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. Density component profiles [PITH_FULL_IMAGE:figures/full_fig_p029_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Upper panels: Density profile trajectories when a critical XY phase [PITH_FULL_IMAGE:figures/full_fig_p034_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. Section of the surface phase diagram at bulk coexistence illustrating the necessity of critical point wetting for the one [PITH_FULL_IMAGE:figures/full_fig_p036_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. a) A tricuspid formed from the trajectories of the wall- [PITH_FULL_IMAGE:figures/full_fig_p038_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27. Surface phase diagram at bulk coexistence ( [PITH_FULL_IMAGE:figures/full_fig_p039_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: FIG. 28. LHS upper panel: The location of the component densities of the three bulk phases, following the Griffiths constraint [PITH_FULL_IMAGE:figures/full_fig_p040_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: FIG. 29. a) The conformal map, [PITH_FULL_IMAGE:figures/full_fig_p041_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: FIG. 30. The primary component profile, [PITH_FULL_IMAGE:figures/full_fig_p043_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: FIG. 31. Density component profiles, [PITH_FULL_IMAGE:figures/full_fig_p044_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: FIG. 32. Component density profiles, [PITH_FULL_IMAGE:figures/full_fig_p045_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: FIG. 33. The symmetries between the expressions for the component densities profiles, for the wall- [PITH_FULL_IMAGE:figures/full_fig_p046_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: FIG. 34. The decays of the component profile of the non-critical wall- [PITH_FULL_IMAGE:figures/full_fig_p048_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: FIG. 35. Surface phase diagrams and density profile trajectories (arrows) for a two-component mixture near a wall at bulk [PITH_FULL_IMAGE:figures/full_fig_p050_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: FIG. 36. Contours of the potential [PITH_FULL_IMAGE:figures/full_fig_p051_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: FIG. 37. The straightening of the microscopic density profile paths under a conformal transform which maps the tricuspid [PITH_FULL_IMAGE:figures/full_fig_p052_37.png] view at source ↗
read the original abstract

We show how a broad class of two-component square-gradient models of wetting may be solved exactly for the surface tensions and density profile paths, and clarify how the presence or absence of critical point wetting, in binary and ternary mixtures, is related to universality and symmetry principles at critical end points. We begin by solving a model of fluid interfaces, first introduced by Koga and Widom, in ternary mixtures showing three phase coexistence. Numerical studies had revealed interesting wetting transitions, as well as curious geometrical properties of the profile paths in the density plane, and led these authors to conjecture expressions for the surface tensions. These conjectures were extended by Koga and Indekeu and predicted that partial wetting may persist up to the line of critical end points, i.e. critical point wetting was absent. Here, we obtain the exact density profiles and surface tensions for the Koga-Widom-Indekeu (KWI) model using complex analysis and drawing on the theory of algebraic curves. The exact solution determines the location and order of wetting transitions in the surface phase diagram, confirming that critical point wetting is absent. The model also displays the remarkable property that microscopic density profiles are mapped, by a conformal transform, onto the shape of a macroscopic drop near the contact line whose tensions satisfy the Neumann triangle. Two related models, which illustrate the role of the component isotropy, are also discussed. These models suggest that a universality principle governs wetting in fluid mixtures, resolving contradicting results from earlier studies: Critical point wetting is present if the order-parameter components of the mixture describe Ising-like criticality, but is absent if there is a local XY symmetry. Implications for wetting transitions in more microscopic models and in experiments are discussed.

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

0 major / 3 minor

Summary. The manuscript presents an exact solution to the Koga-Widom-Indekeu (KWI) square-gradient model for wetting in ternary fluid mixtures, along with related models, using methods from complex analysis and the theory of algebraic curves. The solution provides exact expressions for density profiles and surface tensions, which are used to determine the locations and orders of wetting transitions. It confirms that critical point wetting is absent in this model due to local XY symmetry, in contrast to Ising-like criticality, and discusses universality principles for wetting in fluid mixtures.

Significance. If the results hold, this work is significant for providing a rigorous, exact mathematical treatment that verifies earlier numerical conjectures on wetting transitions without any parameter fitting or data selection. The use of conformal transforms to map microscopic profiles to macroscopic drop shapes satisfying the Neumann triangle is a striking geometric property. The distinction between XY and Ising symmetries in related models helps resolve contradictions in the literature regarding critical point wetting. This contributes parameter-free derivations and clear falsifiable implications for experiments on fluid mixtures near critical end points.

minor comments (3)
  1. [Introduction] The abstract states that the exact solution confirms the conjectures of Koga and Indekeu; a brief explicit statement in the main text comparing the derived transition locations to those conjectures would aid readability.
  2. [Results] The discussion of the conformal mapping from microscopic profiles to the macroscopic drop shape is intriguing; adding a short schematic figure or diagram would improve clarity for readers unfamiliar with algebraic-curve methods.
  3. [Related models] Notation for the order-parameter components in the related models (XY vs. Ising variants) is introduced but could be summarized in a small table for quick reference when comparing the two cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance of our exact solution to the KWI model, and the recommendation for minor revision. The report correctly identifies the key results on the absence of critical point wetting due to local XY symmetry and the conformal mapping property. We address the points below.

Circularity Check

0 steps flagged

No significant circularity: exact solution derived directly from model equations

full rationale

The paper constructs an exact solution for the KWI square-gradient model and variants by applying complex analysis and algebraic curve theory to the Euler-Lagrange equations obtained from the free-energy functional. This determines surface tensions, density profiles, and wetting transition locations/orders without presupposing the Koga-Widom-Indekeu conjectures (which are instead confirmed as output). Related models are compared by explicit substitution of isotropy assumptions to isolate XY versus Ising symmetry effects. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or ansatzes smuggled via citation appear in the derivation chain; the logic is self-contained against the model's differential equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper solves an existing phenomenological model exactly; no new free parameters or postulated entities are introduced. The central results rest on the validity of the square-gradient approximation and standard results from complex analysis.

axioms (1)
  • domain assumption The square-gradient approximation captures the leading interfacial physics in the KWI model and its variants.
    This is the foundational modeling choice stated in the abstract for the class of two-component square-gradient models.

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Reference graph

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