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arxiv: 2604.11219 · v2 · submitted 2026-04-13 · ❄️ cond-mat.soft · physics.flu-dyn

Pinch-off of non-Brownian rod suspensions: onset of heterogeneity and effective extensional viscosity

Virgile Thi\'evenaz , Nathan Vani , Alban Sauret This is my paper

Pith reviewed 2026-05-10 15:52 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn
keywords rod suspensionspinch-offextensional viscositycontinuum breakdownanisotropic suspensionsliquid bridgeMills lawheterogeneous thinning
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The pith

In suspensions of rigid rods, the rod length controls the onset of heterogeneous thinning during liquid bridge pinch-off.

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

The stretching and pinch-off of a liquid bridge probes when a suspension of particles stops behaving as a continuum. For density-matched suspensions of rigid nylon fibers with aspect ratios from 2 to 84, high-speed imaging shows three successive regimes: an equivalent-fluid regime, a dislocation regime where rods separate, and a final regime controlled by the interstitial liquid. The thresholds between regimes follow a scaling previously proposed for spherical particles but using rod length instead of diameter. In the equivalent-fluid regime, the effective extensional viscosity increases with volume fraction and aspect ratio, and is described by Mills' law with a critical volume fraction that decreases with aspect ratio.

Core claim

Pinch-off of non-Brownian rod suspensions reveals three regimes of thinning, with transitions governed by the rod length as the relevant discrete scale. The effective extensional viscosity in the initial regime follows Mills' law using a critical volume fraction that decreases monotonically with aspect ratio.

What carries the argument

The three-regime model of liquid bridge thinning, where rod length replaces particle diameter in the scaling for regime thresholds, and Mills' law for the effective viscosity.

If this is right

  • Effective extensional viscosity measured by pinch-off differs from shear viscosity but both fit Mills' law.
  • The critical volume fraction for the onset of heterogeneity decreases with increasing rod aspect ratio.
  • Pinch-off serves as a sensitive probe for continuum breakdown in anisotropic suspensions.
  • Rod length, rather than diameter, sets the scale for heterogeneous thinning in rigid rod suspensions.

Where Pith is reading between the lines

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

  • Similar scaling might apply to other elongated particles beyond rigid rods, such as in biological or industrial fiber suspensions.
  • This method could be extended to test continuum validity in suspensions under different flow conditions.
  • Processing of fiber-reinforced composites may need to account for rod-length dependent viscosity changes during thinning flows.

Load-bearing premise

The regime thresholds for rods follow the spherical particle scaling after simply replacing the particle diameter with the rod length.

What would settle it

Measuring regime transition points in experiments with different rod lengths and finding they do not match the predicted scaling with rod length would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.11219 by Alban Sauret, Nathan Vani, Virgile Thi\'evenaz.

Figure 1
Figure 1. Figure 1: FIG. 1. Examples of nylon fibers used in the experiments. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Stages of the detachment of a droplet of a suspension of rods. Initially, the rods are uniformly distributed; the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Thinning dynamics of suspensions of rods. (a) represents suspensions of 50 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Variations of the threshold [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Validation of the scaling law for [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Rheological curves for suspensions of 50 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison between the relative viscosity measured in the rheometer at ˙γ [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Critical volume fraction [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original abstract

The stretching and pinch-off of a liquid bridge is a simple way to probe when a suspension of particles stops behaving as a continuum. In this study, we consider density-matched suspensions of rigid nylon fibers with aspect ratios (length over diameter) ranging from 2 to 84, and volume fractions $\phi$ spanning the dilute to dense regimes. High-speed imaging of pendant-drop breakup reveals three successive regimes, as previously observed for spherical particles: an equivalent-fluid regime at early times, a dislocation regime corresponding to the separation of the rods, and a final regime controlled by the interstitial liquid once the neck is devoid of rods. The thresholds between these regimes follow the previously proposed scaling for spherical particles, in which the rod length, rather than the rod diameter, is used as the relevant discrete scale. In the equivalent-fluid regime, pinch-off also leads to an effective extensional viscosity that increases with both volume fraction and aspect ratio. This viscosity is not equal to the shear viscosity measured in a parallel-plate rheometer, but both sets of data are well described by Mills' law using a critical volume fraction $\phi_c$. Finally, the critical volume fraction $\phi_c$ decreases monotonically with the aspect ratio and is well captured by an empirical law. These results show that pinch-off is a sensitive probe of continuum breakdown in anisotropic suspensions and that, for rigid rods, the rod length controls the onset of heterogeneous thinning.

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 examines pinch-off of density-matched rigid nylon rod suspensions (aspect ratios 2–84, dilute to dense φ) via high-speed pendant-drop imaging. Three regimes are identified: early-time equivalent-fluid behavior, a dislocation regime of rod separation, and a late interstitial-liquid regime. Regime thresholds are reported to follow prior spherical-particle scalings but with rod length L (not diameter d) as the discrete scale. In the equivalent-fluid regime an effective extensional viscosity is extracted that rises with both φ and aspect ratio; both extensional and shear viscosities are described by Mills’ law after fitting a critical volume fraction φ_c that decreases monotonically with aspect ratio.

Significance. If the scaling claim holds, the work shows that pinch-off is a sensitive probe of continuum breakdown in anisotropic suspensions and that rod length sets the onset of heterogeneous thinning. The observation that extensional and shear viscosities, though numerically distinct, both conform to Mills’ law with an aspect-ratio-dependent φ_c offers a compact description across flow types. The systematic variation of aspect ratio and the direct comparison to spherical-particle literature are experimental strengths.

major comments (2)
  1. [Regime identification and scaling analysis] The central claim that 'rod length controls the onset of heterogeneous thinning' rests on the assertion that regime thresholds collapse onto the spherical-particle scaling when L is substituted for d. No quantitative comparison (e.g., R², χ², or residual analysis) of the collapse quality using L versus d (or other candidate scales) is provided, nor is uncertainty in transition-point identification from the images propagated. This quantitative test is load-bearing for the strongest claim.
  2. [Methods and effective-viscosity analysis] The extraction of effective extensional viscosity from neck profiles in the equivalent-fluid regime is not described in sufficient detail (fitting procedure, number of independent runs, error estimation). Without these, it is difficult to assess the reported increase with φ and aspect ratio or the subsequent Mills’-law fit.
minor comments (2)
  1. Figure captions and the main text should explicitly state sample sizes, replicate numbers, and whether error bars represent standard deviation or standard error.
  2. A short reminder of the functional form of Mills’ law (with the definition of φ_c) would aid readers who are not already familiar with the expression.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback on our manuscript. We address the two major comments point by point below. We agree that additional quantitative details will strengthen the paper and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The central claim that 'rod length controls the onset of heterogeneous thinning' rests on the assertion that regime thresholds collapse onto the spherical-particle scaling when L is substituted for d. No quantitative comparison (e.g., R², χ², or residual analysis) of the collapse quality using L versus d (or other candidate scales) is provided, nor is uncertainty in transition-point identification from the images propagated. This quantitative test is load-bearing for the strongest claim.

    Authors: We appreciate the referee pointing out the need for a more rigorous quantitative validation of the scaling collapse. In the original manuscript, the regime thresholds are shown to follow the spherical-particle scaling with L as the discrete length scale through direct comparison in the relevant figures. However, to address this concern, we will add in the revised version a quantitative analysis including R² values comparing the collapse using L versus d, along with propagation of uncertainties in the identified transition points from the high-speed images. This will be included in the section discussing the regime identification and scaling. revision: yes

  2. Referee: The extraction of effective extensional viscosity from neck profiles in the equivalent-fluid regime is not described in sufficient detail (fitting procedure, number of independent runs, error estimation). Without these, it is difficult to assess the reported increase with φ and aspect ratio or the subsequent Mills’-law fit.

    Authors: We thank the referee for this observation. The effective extensional viscosity was determined by fitting the time evolution of the minimum neck radius in the equivalent-fluid regime to the theoretical prediction for viscous pinch-off. The fitting was performed over a specific early-time window where the equivalent-fluid behavior holds, using data averaged from multiple independent pendant-drop experiments (at least three per condition). Uncertainties were estimated from the variability across these runs. In the revised manuscript, we will provide a more detailed description of the fitting procedure, including the exact time range, the functional form used, the number of independent runs, and the method for error bars. This will facilitate evaluation of the viscosity trends and the Mills' law fits. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain.

full rationale

The paper reports experimental observation of three regimes during pinch-off and states that the thresholds follow a previously proposed scaling for spherical particles after substituting rod length L for diameter. This is a direct comparison of data to an external scaling relation, not a derivation that reduces to the paper's own inputs by construction. The effective extensional viscosity is stated to be well described by Mills' law (from prior literature) after fitting the parameter φ_c to the present data; φ_c is then captured by a separate empirical law versus aspect ratio. These are standard parameter fits and do not constitute a 'prediction' forced by the fit itself, nor any self-definitional loop, self-citation load-bearing step, or ansatz smuggled via citation. No equations or claims reduce the central result (rod length controls onset of heterogeneity) to a tautology or to the fitted values. The chain is self-contained against the reported experiments and external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the spherical-particle scaling law after substitution of rod length, on the applicability of Mills' law to extensional data, and on an empirical fit for the critical volume fraction that decreases with aspect ratio. No new particles or forces are postulated.

free parameters (1)
  • critical volume fraction φ_c
    Fitted parameter in Mills' law that decreases monotonically with rod aspect ratio; its value is determined from the same pinch-off and rheometry data.
axioms (1)
  • domain assumption The thresholds between equivalent-fluid, dislocation, and interstitial-liquid regimes obey the same scaling previously derived for spherical particles once rod length replaces particle diameter.
    Invoked to interpret the observed regime transitions without independent derivation for rods.

pith-pipeline@v0.9.0 · 5562 in / 1343 out tokens · 36479 ms · 2026-05-10T15:52:19.329732+00:00 · methodology

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