arxiv: 2605.11322 · v1 · submitted 2026-05-11 · ❄️ cond-mat.soft · physics.flu-dyn

Recognition: no theorem link

Existent condition of partially wet state in capillary tubes

Chen Zhao , Jiajia Zhou , Masao Doi

Authors on Pith no claims yet

Pith reviewed 2026-05-13 01:40 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn

keywords capillary tubespartially wet statecorner filmscontact anglephase diagraminterfacial energysquare tuberounded corners

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The pith

The partially wet state in cornered capillary tubes exists only within a specific region of equilibrium contact angle and corner curvature.

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

The paper develops a theory for the equilibrium configurations of a fluid inside a capillary tube that has corners. It identifies three possible states for each section of the tube: fully wet, partially wet with fluid only in the corners, and dry. By calculating the phase diagram for a square tube with rounded corners, it shows that the partially wet state with corner films is stable only for certain combinations of the contact angle and the curvature at the corners. This matters because understanding when fluids form these partial films can help predict behavior in microfluidic devices or porous materials where corner flows are important.

Core claim

The equilibrium states are found by minimizing the interfacial energy for a fixed fluid volume in a static geometry. For the square tube with rounded corners, the phase diagram in the plane of contact angle and corner curvature reveals a bounded region where the partially wet state is favored over complete wetting or drying.

What carries the argument

The phase diagram in the parameter space of equilibrium contact angle and corner curvature, derived from energy minimization.

If this is right

If the parameters are outside this region, the tube section will either fill completely or remain dry.
Corner films form only when the contact angle allows partial wetting at the rounded corners.
The theory applies to predicting fluid distribution in non-circular capillaries.

Where Pith is reading between the lines

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

This could imply that in real tubes with slight rounding, corner flow might be suppressed for many contact angles.
Experiments varying corner radius could test the boundaries of the partial wet region.
Applications in oil recovery or inkjet printing might depend on tuning these parameters.

Load-bearing premise

The equilibrium states are determined solely by minimizing the interfacial energy for a fixed volume in a static tube geometry without gravity, flow, or dynamic effects.

What would settle it

Direct observation of whether corner films persist or disappear in a square capillary with controlled contact angle and corner radius would confirm or refute the phase boundaries.

Figures

Figures reproduced from arXiv: 2605.11322 by Chen Zhao, Jiajia Zhou, Masao Doi.

**Figure 2.** Figure 2: FIG. 2. Calculation of the interfacial energy [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Interfacial energy [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Interfacial energy [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Interfacial energy [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Phase diagram for a square tube with rounded corners in the plane of roundness and [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

read the original abstract

We develop a theory that predicts the equilibrium states of a fluid contained in a capillary which has corners. Each section of the tube can take three states: completely wet state where the tube section is completely occupied by the fluid, partially wet state where only the corners are occupied by the fluid known as corner film or finger, and completely dry state. We calculate the phase diagram of these states for a square tube with rounded corners. It is shown that the partially wet state can exist only in a certain region in the parameter space spanned by the equilibrium contact angle and the corner curvature.

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.

Desk Editor's Note private letter to a colleague

The paper maps the parameter region where corner films can stably exist in square tubes with rounded corners by comparing minimized interfacial energies of three states.

read the letter

The key point is that this work produces an explicit phase diagram in the plane of contact angle and corner curvature, showing where the partially wet state (corner films) is possible between the fully wet and fully dry regimes for a square tube with rounded corners. That mapping is new; prior work on sharp corners or circular tubes did not cover this geometry in the same way. The calculation follows the standard route of minimizing the interfacial energy for fixed fluid volume in a static cross-section, treating the three configurations separately and comparing their energies. This is a direct, incremental extension of classical capillary theory, and the setup looks internally consistent with no obvious circularity or invented parameters. The exclusion of gravity, flow, and dynamics is appropriate for an equilibrium existence diagram and keeps the problem tractable. The main limitation is that the result applies only to ideal, static conditions; real microchannels often involve gravity or slow flow that could shift the boundaries. The abstract and stress-test description give no sign of numerical instability or post-hoc fitting, but the full derivation and any discretization details would still need checking before the diagram is taken as quantitative. This is the kind of paper that belongs in a specialized journal on soft matter or microfluidics. Readers working on wetting in non-circular channels or porous media would find the phase diagram directly usable for design rules. It is solid enough and grounded enough to deserve a serious referee rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The paper develops a variational theory for the equilibrium states of a fixed-volume fluid in a capillary tube with corners, using a square cross-section with rounded corners as the model geometry. Three candidate states are considered: fully wet (tube section completely occupied by fluid), partially wet (fluid confined to corner films), and dry. The phase diagram in the plane of equilibrium contact angle and corner curvature is obtained by minimizing the interfacial energy of each configuration and determining the regions where each state has the lowest energy.

Significance. If the energy minimizations are correctly executed, the work supplies a concrete existence region for corner films, which is a useful addition to the literature on capillary wetting in non-circular channels. The approach follows the standard energy-comparison method for static capillary equilibria and could inform microfluidic design or porous-media modeling.

major comments (2)

[§3] §3, Eq. (8): the interfacial energy for the partially wet state is written as a function of the meniscus curvature radius R and the corner radius r; the subsequent minimization with respect to R appears to be performed analytically, but the resulting expression for the critical contact angle boundary is not shown explicitly. Please insert the intermediate steps that lead from the energy functional to the phase-boundary curves plotted in Fig. 5.
[§4.2] §4.2: the comparison between the three states assumes that the fluid volume is fixed and that gravity is absent. While this is appropriate for the static problem, the manuscript should state the range of Bond numbers for which the neglect of gravity remains valid, especially near the phase boundaries where the energy differences become small.

minor comments (2)

[Abstract] The abstract states that the phase diagram is 'calculated,' yet the main text does not indicate whether the boundaries are obtained analytically or numerically; a brief sentence clarifying the method would improve clarity.
[Figure 5] Figure 5 caption should explicitly label the three regions (wet, partially wet, dry) and indicate the direction of increasing corner curvature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

Referee: [§3] §3, Eq. (8): the interfacial energy for the partially wet state is written as a function of the meniscus curvature radius R and the corner radius r; the subsequent minimization with respect to R appears to be performed analytically, but the resulting expression for the critical contact angle boundary is not shown explicitly. Please insert the intermediate steps that lead from the energy functional to the phase-boundary curves plotted in Fig. 5.

Authors: We agree that the derivation steps should be shown explicitly for transparency. In the revised manuscript we will insert the analytical minimization of the interfacial energy functional (Eq. 8) with respect to the meniscus radius R. This will yield the explicit relation for the critical contact angle as a function of the corner curvature parameter, directly connecting the energy expression to the phase-boundary curves in Fig. 5. No change to the final results is required. revision: yes
Referee: [§4.2] §4.2: the comparison between the three states assumes that the fluid volume is fixed and that gravity is absent. While this is appropriate for the static problem, the manuscript should state the range of Bond numbers for which the neglect of gravity remains valid, especially near the phase boundaries where the energy differences become small.

Authors: We accept that an explicit statement on the validity range is useful. In the revised §4.2 we will add a brief paragraph noting that the gravity-free assumption holds for Bond numbers Bo ≪ 1 (based on the tube radius and fluid properties), with a stricter criterion Bo < 0.1 recommended near the phase boundaries where interfacial energy differences are small. This addition clarifies the regime without altering the core analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; energy comparison uses independent inputs

full rationale

The paper's central result is obtained by minimizing the interfacial energy for fixed volume in three distinct static configurations (fully wet, corner-film partially wet, and dry) and comparing those energies to delineate the phase diagram in the (contact angle, corner curvature) plane. Both the contact angle (from Young's law) and the geometric corner curvature are external parameters supplied to the model; neither is fitted to nor defined by the resulting existence boundaries. No load-bearing step reduces to a self-citation, an ansatz smuggled from prior work, or a renaming of an empirical pattern. The derivation is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard assumption that equilibrium configurations minimize the sum of liquid-solid and liquid-gas interfacial energies for a prescribed liquid volume inside a rigid tube; no new entities are introduced.

axioms (2)

domain assumption Equilibrium states are found by minimizing interfacial energy at fixed volume.
Invoked to define the three candidate states and their stability.
standard math Contact angle is constant and given by Young's law on the solid surface.
Used as the primary control parameter.

pith-pipeline@v0.9.0 · 5386 in / 1232 out tokens · 25285 ms · 2026-05-13T01:40:31.821358+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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H. M. Princen, “Capillary phenomena in assemblies of parallel cylinders: III. liquid columns between horizontal parallel cylinders,” J. Colloid Interface Sci.34, 171 (1970). 13 Appendix A: Different wetting cases Figure S1 show different wetting states for a circular tube, a square tube, and a square tube with rounded corners (from top to bottom). FIG. S1...

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[26]

The radius of the interface is r1

Case I: full-circle meniscus Case I corresponds to a full circle of liquid-vapor interface. The radius of the interface is r1. The area of the liquid isS= 4a 2 −πr 2 1, thus the saturation is s= S S0 = 1− π 4 r1 a 2 ,(C1) ⇒ r1 a = r 4 π (1−s) 1/2 .(C2) The maximum ofr 1 is the half of the side-length,r 1 ≤a. This leads to lower bound for the saturations c...

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On one corner, the meniscus touches the side wall at point B with a contact angleθ(Fig

Case II, corner meniscus Case II corresponds to the fingers at four corners. On one corner, the meniscus touches the side wall at point B with a contact angleθ(Fig. S2). This leads to ̸ BOC =θ. One eighth of the meniscus ⌢ BD is a portion of the circle with angleα= π 4 −θand radiusr 2. FIG. S2. Square tube, case II withθ < π 4 . The finger meniscus is con...

work page
[28]

•case I:s c1 ≤s≤1,s c1 = 1− π 4 f(s) aγ =−8 cosθ+ 4 √π(1−s) 1/2 (C25) •case II: 0≤s≤s c2,s c2 = A(θ) (cosθ−sinθ) 2 f(s) aγ =−sign π 4 −θ 8 p A(θ)s1/2 (C26) A(θ) is given in Eq

Summary of square tubef(s) We summarize the interfacial energy functions for square tube. •case I:s c1 ≤s≤1,s c1 = 1− π 4 f(s) aγ =−8 cosθ+ 4 √π(1−s) 1/2 (C25) •case II: 0≤s≤s c2,s c2 = A(θ) (cosθ−sinθ) 2 f(s) aγ =−sign π 4 −θ 8 p A(θ)s1/2 (C26) A(θ) is given in Eq. (C14). 18

work page
[29]

Analysis of interfacial energyf(s) The interfacial energy curves forθ= 20 ◦ is shown in Fig. S4(a). The value off(s) varies significantly betweens= 0 ands= 1, making the line of tangency difficult to distinguish. For clear illustration, we define a new functiong(s) g(s) =f(s)−f(1)×s .(C27) This function satisfiesg(0) = 0 andg(1) = 0. It quantifies the dev...

work page
[30]

The area of the liquid is givenS=S 0 −πr 2

Case I: full-circle meniscus Case I corresponds to a full circle of liquid-vapor interface of radiusr 1. The area of the liquid is givenS=S 0 −πr 2

work page
[31]

The saturation is s= S S0 = 1− π 4B2(β) r1 a 2 ,(D4) ⇒ r1 a = r 4B2(β) π (1−s) 1/2 .(D5) The range of saturation in Case I is sc1 ≤s≤1, s c1 = 1− π 4B2(β) .(D6) The length of solid-liquid interface isL SL = 8a−8b+ 2πb= 8aB 1(β), and the length of liquid-vapor interface isL L V= 2πr1. The interfacial energy is then f(s) =L SL(−γcosθ) +L L Vγ=−8aγcosθB 1(β)...

work page
[32]

On one corner, the meniscus touches the side wall at point B with a contact angleθ

Case II: corner meniscus, contact line on the straight side Case II of rounded corner is similar to the sharp corner. On one corner, the meniscus touches the side wall at point B with a contact angleθ. This leads to ̸ BOC =θ(Fig. S6). One eighth of the meniscus ⌢ BD is a portion of the circle with angleα= π 4 −θand radiusr 2. 21 FIG. S6. Square tube with ...

work page
[33]

The meniscus touches the solid surface on the rounded corner instead of the side-walls of the square (Fig

Case III: corner meniscus, contact line on the curved side For rounded corner, there is a case III when the saturation is small. The meniscus touches the solid surface on the rounded corner instead of the side-walls of the square (Fig. S8). The meniscus ⌢ BF is an arc with open angleα−θand radiusr 3. Comparing△CBG and △OBG BG =bsinα=r 3 sin(α−θ) (D26) ⇒r ...

work page
[34]

Summary of rounded cornerf(s) We summarize the interfacial energy functions for rounded corners. •case I:s c1 ≤s≤1,s c1 = 1− π 4B2(β) f(s) aγ =−8 cosθB 1(β) + 4 p πB2(β)(1−s) 1/2 (D38) •case II:s c3 ≤s≤s c2, sc2 = A(θ)(cosθ−sinθ) −2 −(1−B 2(β)) B2(β) sc3 = β2A(θ)(cosθ−sinθ) −2 −(1−B 2(β)) B2(β) f(s) aγ = 8(1−B 1(β)) cosθ−sign π 4 −θ 8 p A(θ) p B2(β)s+ (1−...

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[35]

Oncexis obtained, we can use the reverse relation to gets ∗ s∗ = x2 −A(1−B 2) AB2 (E7) 27 FIG

calculation ofs ∗ The condition fors ∗ is f ′(s) = f(1)−f(s) 1−s ,(E3) which leads to −4AB2√ A p B2s+ (1−B 2) = −8 cosθ+ 8 √ A p B2s+ (1−B 2) 1−s (E4) Letx= √ A p B2s+ (1−B 2),xsatisfies the equation x2 −2 cosθx+A= 0 (E5) ⇒x= cosθ− √ cos2 θ−A .(E6) We chose the solution with negative sign. Oncexis obtained, we can use the reverse relation to gets ∗ s∗ = x...

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[36]

calculation ofs ∗∗ The condition fors ∗∗ is f ′(s) = f(s) s ,(E8) which leads to −4AB2√ A p B2s+ (1−B 2) = 8(1−B 1) cosθ+ 8 √ A p B2s+ (1−B 2) s (E9) Lety= √ A p B2s+ (1−B 2),ysatisfies the equation y2 −2(1−B 1) cosθy+A(1−B 2) = 0 (E10) ⇒y= (1−B 1) cosθ− p (1−B 1)2 cos2 θ−A(1−B 2)2 .(E11) 28 FIG. S11. Square tube with rounded corners, interfacial energy c...

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