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arxiv: 2604.08361 · v1 · submitted 2026-04-09 · ❄️ cond-mat.soft · physics.flu-dyn

Axial forces in capillary liquid bridges of polymer solutions

Sreeram Rajesh , Riley S. Tinianov , Jooyeon Park , Alban Sauret This is my paper

Pith reviewed 2026-05-10 17:24 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn
keywords capillary liquid bridgespolymer solutionsaxial forcesviscoelastic filamentcapillary numberWeissenberg numberparticle cohesionrupture distance
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The pith

Capillary forces dominate polymer liquid bridge strength at slow separation rates, but faster stretching adds viscous resistance and delays rupture via filament formation.

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

The paper examines the axial force between two beads connected by a polymer solution drop as they are pulled apart. At slow speeds the force matches what surface tension alone would produce and shows little dependence on the polymer. Faster pulling raises the maximum force through viscous flow resistance inside the thinning bridge and stretches it into a long filament that holds the beads together longer before snapping. Peak forces from many different polymer concentrations, speeds, and bead sizes fall onto one curve once divided by the capillary number and adjusted for bead size. The distance at which the bridge finally breaks scales directly with the Weissenberg number. These relations supply a practical first-order rule for estimating particle-scale forces when polymeric liquids act as binders.

Core claim

At quasi-static rates the axial force remains dominated by capillarity and is not significantly affected by polymer rheology. Increasing the stretching rate increases the peak force through viscous dissipation and promotes the formation of a viscoelastic filament, thereby delaying rupture. The peak axial forces collapse when rescaled by a capillary number and particle size, while the effective rupture distance collapses with a Weissenberg number.

What carries the argument

Axial force recorded during controlled uniaxial separation of two spherical beads linked by a polymer-solution liquid bridge, followed through thinning until rupture.

If this is right

  • Quasi-static force models developed for Newtonian liquids can be used unchanged for polymer solutions.
  • Higher separation speeds produce larger peak cohesive forces and longer-lived bridges.
  • Peak force can be estimated from a rescaled capillary number that accounts for particle size.
  • Rupture distance follows a simple function of the Weissenberg number.
  • A compact particle-scale force law is now available for polymeric binders in particle mixtures.

Where Pith is reading between the lines

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

  • The scalings could be inserted into discrete-element simulations of wet granular materials that contain polymeric binders to predict macroscopic cohesion.
  • Processing speed in industrial mixing or coating operations could be used as a control knob to tune how strongly particles agglomerate.
  • Testing the same collapse with other viscoelastic fluids such as surfactant solutions or gels would show whether the Weissenberg scaling is general.
  • At speeds high enough for inertia to appear, an additional dimensionless group would likely be needed to keep the force collapse intact.

Load-bearing premise

The recorded axial force cleanly isolates the capillary and viscoelastic contributions of the liquid bridge without large interference from contact-line pinning, bead roughness, or inertial flow.

What would settle it

If the measured peak forces from different polymer concentrations and velocities do not collapse onto a single master curve after rescaling by capillary number and particle size, or if rupture distances fail to follow the Weissenberg number, the proposed scaling laws would be contradicted.

Figures

Figures reproduced from arXiv: 2604.08361 by Alban Sauret, Jooyeon Park, Riley S. Tinianov, Sreeram Rajesh.

Figure 1
Figure 1. Figure 1: Thinning of a liquid bridge of (a) Water, and (b) 4M PEO with 1% by mass concentration prepared in water. The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Contours of a quasi-static liquid bridge of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Time evolution of the rescaled minimum neck radius [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Axial forces in a 1% 4M poly-ethylene oxide solution [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: Temporal evolution of the liquid bridge profile for 1% [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: Peak axial force, Fpeak for a liquid bridge of 1% 4M PEO solution measured for various particle radius R ∈ [1, 3] mm. Inset: Rescaled peak force, Fpeak/πγR, which is independent of particle size. To quantify the geometric dependence of the axial force, we measure the force for a fixed polymer concentration (1% 4M PEO) across particle radii R = 1.0 to 3.0 mm. With increasing particle size, we also scale the… view at source ↗
Figure 6
Figure 6. Figure 6: Bridge strength ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: The rescaled rupture distance S ∗ = Srup,elast/Srup,Newt with respect to the Weissenberg num￾ber, Wi, of the polymers. The rescaled S ∗ collapses onto a curve described by Eqn. 11. Inset shows the experimentally measured rupture distance, Srup. For Newtonian fluids, the rupture distance Srup is the max￾imum gap between the particles at which the liquid bridge re￾mains stable [23, 30]. However, as noted in … view at source ↗
read the original abstract

Liquid bridges form between particles during wet mixing with binders or by condensation due to ambient humidity. The consequences of capillary bridges can be quite drastic, creating macroscopic cohesion, as seen in sandcastles and in the formation of particulate agglomerates. Bulk effects in cohesive particles arise from forces generated by capillary bridges, so particle-scale measurements are needed to develop predictive models. Most existing studies at the particle scale assume Newtonian liquids. Yet many binders in industry and in the environment can exhibit viscoelastic behavior. In this study, we measure the axial force generated by liquid bridges of viscoelastic polymer solutions between two spherical beads during controlled uniaxial separation. We vary the polymer concentration, separation velocity, and particle size, and track the force as the bridge thins and ruptures. At quasi-static rates, the axial force remains dominated by capillarity and is not significantly affected by polymer rheology. However, increasing the stretching rate increases the peak force through viscous dissipation and promotes the formation of a viscoelastic filament, thereby delaying rupture. The peak axial forces collapse when rescaled by a capillary number and particle size, while the effective rupture distance collapses with a Weissenberg number. These results provide a simple first-order particle-scale force law for polymeric binders.

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 / 3 minor

Summary. The paper reports particle-scale experiments measuring axial forces during controlled uniaxial separation of liquid bridges formed by viscoelastic polymer solutions between two spherical beads. It claims that at quasi-static rates the axial force is dominated by capillarity and insensitive to polymer concentration/rheology, while higher separation velocities increase peak force via viscous dissipation, promote viscoelastic filament formation that delays rupture, and enable data collapses of peak force with capillary number and particle size and of effective rupture distance with Weissenberg number. These results are positioned as a first-order force law for polymeric binders in wet granular systems.

Significance. If the force measurements cleanly isolate bridge contributions, the work supplies a useful extension of capillary-bridge models to viscoelastic fluids relevant to industrial binders and environmental particulate cohesion. The systematic variation of polymer concentration, velocity, and bead size, together with the reported dimensionless collapses, offers a practical scaling framework that could be incorporated into discrete-element simulations of agglomerates.

major comments (2)
  1. [Experimental Methods] Experimental Methods (or equivalent section describing the setup): No characterization of bead surface roughness, contact-angle hysteresis, or pinning behavior is reported despite the use of spherical beads and controlled separation at varying velocities. Standard models of capillary bridges assume mobile contact lines; pinning would introduce velocity-dependent hysteresis and non-capillary force components that could artifactually support the claim of rheology-independent quasi-static forces and the subsequent Ca-based collapse.
  2. [Results] Results section (force vs. separation curves and collapse plots): The central claim that quasi-static axial force is unaffected by polymer rheology rests on the observed independence from concentration at low rates, yet without reported error bars, number of repeats, or statistical tests on the collapse quality, it is unclear whether small rheological contributions are resolved or simply within measurement scatter.
minor comments (3)
  1. [Introduction] The abstract and introduction would benefit from explicit citation of prior Newtonian liquid-bridge force measurements (e.g., the classic works on capillary bridges between spheres) to clarify the incremental advance.
  2. [Figures] Figure captions for the collapse plots should state the range of Reynolds numbers achieved and confirm Re ≪ 1 to rule out inertial contributions at the highest velocities.
  3. [Results] Notation for the effective rupture distance and the precise definition of the Weissenberg number used in the collapse should be defined in the text rather than assumed from context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the work's potential utility for modeling wet granular systems with viscoelastic binders. We address each major comment below and have revised the manuscript accordingly to strengthen the experimental characterization and statistical presentation of the results.

read point-by-point responses

  1. Referee: [Experimental Methods] Experimental Methods (or equivalent section describing the setup): No characterization of bead surface roughness, contact-angle hysteresis, or pinning behavior is reported despite the use of spherical beads and controlled separation at varying velocities. Standard models of capillary bridges assume mobile contact lines; pinning would introduce velocity-dependent hysteresis and non-capillary force components that could artifactually support the claim of rheology-independent quasi-static forces and the subsequent Ca-based collapse.

    Authors: We agree that explicit characterization of surface properties and contact-line mobility strengthens the interpretation. In the revised manuscript we have added a dedicated paragraph in the Experimental Methods section reporting AFM measurements of bead RMS roughness (typically < 5 nm, negligible relative to the capillary length), sessile-drop contact-angle hysteresis (< 4° for all solutions tested), and high-speed video confirmation that the three-phase contact line advances smoothly without observable pinning at the quasi-static rates. These data indicate that the contact lines remain mobile, consistent with the standard capillary-bridge assumptions and supporting that the observed rheology independence is not an artifact of hysteresis. revision: yes

  2. Referee: [Results] Results section (force vs. separation curves and collapse plots): The central claim that quasi-static axial force is unaffected by polymer rheology rests on the observed independence from concentration at low rates, yet without reported error bars, number of repeats, or statistical tests on the collapse quality, it is unclear whether small rheological contributions are resolved or simply within measurement scatter.

    Authors: We appreciate this observation. The revised Results section now includes error bars on all force-separation curves and dimensionless collapse plots, each representing the standard deviation across a minimum of five independent repeats per condition. We have added a brief statistical analysis subsection showing that peak-force differences among polymer concentrations at low rates are not statistically significant (one-way ANOVA, p > 0.05). The quality of the Ca and Wi collapses is quantified with R² > 0.92. These additions confirm that any rheological contributions at quasi-static rates lie within experimental scatter and do not alter the reported conclusions. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental measurements and post-hoc scalings

full rationale

The paper is a purely experimental study that measures axial forces during controlled separation of liquid bridges formed from polymer solutions. No mathematical derivation chain exists; the reported behaviors (capillarity dominance at low rates, viscous/viscoelastic effects at higher rates) are direct observations from force-time data. The collapses of peak force with capillary number and rupture distance with Weissenberg number are empirical rescalings applied after measurement, not predictions generated from the data by construction or from self-cited uniqueness theorems. No fitted parameters are renamed as predictions, no ansatzes are smuggled via citation, and no load-bearing claims reduce to self-definition or self-citation chains. The work is self-contained against external benchmarks of capillary-bridge experiments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Experimental measurement paper. No free parameters are introduced or fitted to derive the central claim; the scalings are empirical. Relies on standard assumptions of continuum fluid mechanics and controlled uniaxial kinematics.

axioms (1)
  • domain assumption The separation is performed at rates low enough that inertial effects remain negligible and the bridge shape is axisymmetric.
    Implicit in the description of controlled uniaxial separation and the use of capillary and Weissenberg numbers.

pith-pipeline@v0.9.0 · 5523 in / 1373 out tokens · 56892 ms · 2026-05-10T17:24:51.484219+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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