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arxiv: 2605.03504 · v1 · submitted 2026-05-05 · ⚛️ physics.flu-dyn

Recognition: unknown

Interface pinch-off in the presence of a soluble surfactant

J. M. Montanero, M. A. Herrada, M. Rubio, S. Rodr\'iguez-Aparicio

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:14 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords droplet breakupsoluble surfactantpinch-offdiffusion-limitedsurface tensionpendant dropletMarangoni stress
0
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The pith

Soluble surfactants keep surface tension nearly constant until droplet pinch-off

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

The paper examines how soluble surfactants affect the breakup of a pendant droplet. In the diffusion-limited regime, convection brings surfactant to the interface quickly enough that surface tension stays almost constant until the very end of pinch-off. This leads to droplet shapes and dynamics that differ from cases with insoluble surfactants or those with slow adsorption kinetics. Experiments with Surfynol 465 and SDS match the model's predictions closely without adjustable parameters, even at times as short as 10-20 microseconds before pinch-off. For surfactants like Triton X-100, an energy barrier slows adsorption, shortening the connecting filament.

Core claim

We study numerically and experimentally the breakup of a pendant droplet loaded with a soluble surfactant in the limit where surfactant sorption is limited only by diffusion. Surfactant transfer toward the interface is enhanced by convection, so diffusion does not constitute a significant barrier over most of the breakup and surfactant sorption maintains the surface tension practically constant across the interface. Diffusion hinders the surfactant sorption only very close to the interface pinch-off. The droplet shape in the diffusion-limited model deviates significantly from that in the insoluble case over most of the breakup. The dynamics of a millimeter-sized droplet loaded with Surfynol

What carries the argument

Diffusion-limited surfactant sorption model, where convection enhances transfer and maintains constant surface tension until near pinch-off

Load-bearing premise

Surfactant sorption is limited only by diffusion, with convection making any barrier negligible except very close to pinch-off.

What would settle it

If measured droplet shapes or breakup times deviated from the diffusion-limited predictions at times much larger than 10-20 microseconds before pinch-off, or if matching required adjustable parameters.

Figures

Figures reproduced from arXiv: 2605.03504 by J. M. Montanero, M. A. Herrada, M. Rubio, S. Rodr\'iguez-Aparicio.

Figure 1
Figure 1. Figure 1: Sketch of the fluid configuration. flux. Therefore, it is not evident whether and when diffusion becomes irrelevant as the neck interface radius vanishes. In this paper, we numerically examine the effect of surfactant solubility on the breakup of a millimeter-sized water drop. We consider a fast-kinetics surfactant, assuming that the surfactant sublayer and the interface are at local equilibrium at all tim… view at source ↗
Figure 2
Figure 2. Figure 2: Surfactant volumetric concentration 𝑐/𝑐0 as a function of the distance 𝜁 to the interface in the perpendicular direction. The concentration was evaluated at the interface neck and corresponds to the instant when 𝐹min = 0.02. The results were calculated in the diffusion-limited (fast-kinetics) limit for 𝛽 = 0.005, 𝐵 = 0.0954, Oh = 6.18 × 10−3 , 𝛤0 = 0.997, 𝛬𝑑 = 0.599, Ma = 0.125, Pe𝑠 = 1.62 × 104 , and Pe =… view at source ↗
Figure 3
Figure 3. Figure 3: Experimental setup: feeding capillary (A), ultra-high speed video camera (B), optical lenses (C), laser (D), optical trigger (E), cold-white backlight (F) and anti-vibration isolation system (G). 0 X0-8 view at source ↗
Figure 4
Figure 4. Figure 4: Dynamic surface tension of aqueous solutions of Surfynol 465 at ˆ𝑐/𝑐ˆcmc = 1.3, SDS at ˆ𝑐/𝑐ˆcmc = 1.3, and Triton X-100 at ˆ𝑐/𝑐ˆcmc = 1. The values were normalized with the corresponding equilibrium values 𝜎ˆ 0 = 29.2 mN/m, 35.8 mN/m, and 34.2 mN/m for Surfynol 465, SDS, and Triton X-100, respectively. The data was measured by Varghese et al. (2024) using the maximum bubble-pressure tensiometer. Digital im… view at source ↗
Figure 5
Figure 5. Figure 5: (a) Surface tension ˆ𝜎 as a function of the surfactant volumetric concentration ˆ𝑐. The lines are the fits of the Langmuir equation of state to the Surfynol 465 and Triton X-100 data. (b) Surface tension ˆ𝜎 as a function of the normalized surfactant surface concentration 𝛤ˆ/𝛤ˆ∞. The arrows indicate the concentrations considered in the experiments. deviations of the experimental results from the diffusion-l… view at source ↗
Figure 6
Figure 6. Figure 6: 𝐹min as a function of the time 𝑡. The origin of time 𝑡 = 0 corresponds to the instant at which the step of the Bond number was introduced. and Pe = 1.62 × 104 , which correspond to droplet of an aqueous solution of Surfynol 465 at the surfactant concentration ˆ𝑐/𝑐ˆcmc = 0.21 hanging on a capillary 𝑅 = 0.635 mm in outer radius (see Sec. 4). The interface expands during the droplet breakup, reducing the surf… view at source ↗
Figure 7
Figure 7. Figure 7: Interface contour at the instant for which 𝐹min = 0.02. does not enter the Young-Laplace equation, and therefore, the surfactant does not shift the maximum volume stability limit. To compare the droplet breakups described by the insoluble, diffusion-limited, and perfectly soluble models, we increased the Bond number step size 𝛽 from 𝛽 = 0.005 to 0.2 so that the droplet broke up during the computed time in … view at source ↗
Figure 8
Figure 8. Figure 8: Interface contour 𝐹 (a), surfactant surface concentration 𝛤/𝛤0 (b), surface tension 𝜎 (c), and tangential surface velocity 𝑣𝑠 (d) at the instant for which 𝐹min = 0.02 (𝑧min is the vertical coordinate of the interface neck). droplet breakup. In addition, Marangoni stress reduces the tangential surface velocity 𝑣𝑠 over practically the entire filament (Fig. 8d) view at source ↗
Figure 9
Figure 9. Figure 9: Local variation of the surfactant concentration 𝜕𝛤/𝜕𝑡 (LV), surfactant convection ∇𝑠 · (𝛤v𝑠 ) (C), interface expansion/compression 𝛤n · (∇𝑠 · n)v (E/C), surface diffusion Pe−1 𝑠 ∇𝑠 · [∇𝑠𝛤/(1 − 𝛤)] (SD), and adsorption/desorption flux J (A/D). The results correspond to the instant for which 𝐹min = 0.02 (𝑧min is the vertical coordinate of the interface neck) where we have considered that 𝐹𝑧 = 0, 𝑣𝑠 = 𝑤, and … view at source ↗
Figure 10
Figure 10. Figure 10: The adsoption/desorption (A/D), convection (C), and expansion/compression (E/C) terms evaluated at the interface neck as a function of the time to the pinching 𝜏 fro the diffusion-limited model. diffusion terms in Eq. (3.10) yields 𝑢 ′ 𝑐𝑟 ∼ Pe−1 𝑐𝑟𝑟 , (5.4) where 𝑢 ′ ∼ −(𝛿/𝐹min) 𝑑𝐹min/𝑑𝜏 is the velocity normal to the interface in the interface frame of reference. These terms are of the order 𝑢 ′𝛥𝑐/𝛿 and P… view at source ↗
Figure 11
Figure 11. Figure 11: 𝐹min as a function of the time to the pinching 𝜏. The insets in the upper-left and lower-right corners show 𝐹min (𝜏) at the beginning and at the end of the simulation, respectively. as the perfectly soluble one for 𝜏 ≳ 0.015 and approaches the insoluble one as 𝜏 → 0. In this limit, pinch-off occurs faster both in the diffusion-limited and insoluble case (𝜏 is smaller for the same neck radius 𝐹min) because… view at source ↗
Figure 12
Figure 12. Figure 12: Experimental images in the presence of Surfynol 465 and numerical interface contour (yellow lines) calculated with the diffusion-limited model for 𝛽 = 0.005, 𝐵 = 0.0954, Oh = 6.18 × 10−3 , 𝛤0 = 0.997, 𝛬𝑑 = 0.599, Ma = 0.125, Pe𝑠 = 1.62 × 104 , and Pe = 1.62 × 104 . The instants correspond to 𝐹min = 0.4 (a), 0.04 (b), 0.025 (c), and 0.015 (d). 10-3 10-2 10-1 100 101 10-2 10-1 100 Fmin    view at source ↗
Figure 13
Figure 13. Figure 13: 𝐹min as a function of the time to the pinching 𝜏. The symbols are the experimental results in the presence of Surfynol 465, while the line is the numerical prediction calculated with the diffusion-limited model for 𝛽 = 0.005, 𝐵 = 0.0954, Oh = 6.18 × 10−3 , 𝛤0 = 0.997, 𝛬𝑑 = 1.67, Ma = 0.125, Pe𝑠 = 1.62 × 104 , and Pe = 1.62 × 104 . The circles and triangles correspond to two experimental realizations with … view at source ↗
Figure 14
Figure 14. Figure 14: Superposition of the experimental images in the presence of Surfynol 465 and SDS for 𝐵 = 0.0954, Oh = 6.18 × 10−3 , and 𝛤0 ≃ 1. The interface contours practically overlap. The red dashed line corresponds to Surfynol 465, while the yellow line corresponds to SDS. The instants correspond to 𝐹min = 0.103 (a), 0.071 (b), and 0.016 (c). (a) (c) 400 µm (b) view at source ↗
Figure 15
Figure 15. Figure 15: Superposition of the experimental images in the presence of Surfynol 465 and Triton X-100 for 𝐵 = 0.0954, Oh = 6.18 × 10−3 , and 𝛤0 ≃ 1. The red dashed line corresponds to Surfynol 465, while the yellow line corresponds to Triton X-100. The instants correspond to 𝐹min = 0.103 (a), 0.071 (b), and 0.016 (c) in the experiment with Surfynol 465. presence of Triton X-100, and the length of the filament was con… view at source ↗
Figure 16
Figure 16. Figure 16: 𝐹min as a function of the time to the pinching 𝜏 in the experiments with Surfynol 465, SDS, and Triton X-100 for 𝐵 = 0.0954, Oh = 6.18 × 10−3 , and 𝛤0 ≃ 1. The experimental uncertainty is of the order of the symbol size. We conclude that the most evident effect of the surfactant adsorption energy barrier is the swelling of the upper section of the liquid filament connecting the two liquid volumes prior to… view at source ↗
read the original abstract

We study numerically and experimentally the breakup of a pendant droplet loaded with a soluble surfactant. We consider the limit in which surfactant sorption is limited only by diffusion. Surfactant transfer toward the interface is enhanced by convection. As a consequence, diffusion does not constitute a significant barrier over most of the breakup, and surfactant sorption maintains the surface tension practically constant across the interface. Diffusion hinders the surfactant sorption only very close to the interface pinch-off. The droplet shape in the diffusion-limited model deviates significantly from that in the insoluble case over most of the breakup. In the insoluble case, the droplet shape is affected by surfactant depletion, which leads to a local increase in surface tension and Marangoni stress. The dynamics of a millimeter-sized droplet loaded with Surfynol 465 agree remarkably well with predictions from the diffusion-limited model, without any parameter fitting, down to pinching times of the order of $10-20$ $\mu$s. Sodium dodecyl sulfate (SDS) produces essentially the same effects as those for Surfynol 465. Therefore, both Surfynol 465 and SDS maintain a practically constant surface tension throughout most of the droplet breakup. Slow-kinetics surfactants, such as Triton X-100, differ significantly from Surfynol 465 and SDS. The most evident effect of the surfactant adsorption energy barrier is the shortening of the filament that bridges the upper meniscus and the detached lower drop. Comparing the filament length to that of a clean interface with the same surface tension allows one to evaluate the rate of surfactant adsorption.

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 studies the breakup of pendant droplets containing soluble surfactants in the diffusion-limited sorption limit, where convection enhances surfactant transport such that surface tension remains nearly constant until very near pinch-off. Numerical simulations show significant deviation from the insoluble-surfactant case due to the absence of depletion-induced Marangoni stresses. Experiments with Surfynol 465 and SDS match the diffusion-limited predictions without adjustable parameters down to pinch-off times of 10-20 μs, while Triton X-100 exhibits shorter filaments due to its adsorption barrier; filament length relative to a clean interface is proposed as a diagnostic for adsorption rate.

Significance. The parameter-free quantitative agreement between the diffusion-limited model and millimeter-scale experiments for Surfynol 465 and SDS constitutes a notable strength, providing direct validation that sorption maintains constant tension over most of the process. This distinguishes the regime from insoluble surfactants (where depletion raises local tension) and slow-kinetics cases, with implications for controlling droplet dynamics in applications such as atomization and emulsification.

major comments (2)
  1. [Abstract] Abstract: the central claim of 'remarkable' agreement without parameter fitting down to 10-20 μs is load-bearing, yet the abstract provides no information on experimental error bars, uncertainty quantification, or the precise numerical discretization and boundary conditions used for the convection-enhanced diffusion equation; these details are required to evaluate robustness.
  2. [Modeling section] Modeling section (diffusion-limited formulation): the assertion that 'diffusion does not constitute a significant barrier over most of the breakup' relies on convection dominating transport, but lacks an explicit timescale comparison (e.g., convective vs. diffusive fluxes or local Péclet number) near the neck; without this, the transition to diffusion-limited behavior very close to pinch-off remains qualitative.
minor comments (2)
  1. Figure captions and text should explicitly state the number of experimental repeats and how the pinching time is measured (e.g., from image analysis threshold) to allow readers to assess the 10-20 μs agreement.
  2. The proposal to use filament length relative to a clean interface as a diagnostic for adsorption rate is useful, but would benefit from a brief sensitivity analysis showing how the length scales with the adsorption barrier height.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the parameter-free agreement as a strength, and recommendation for minor revision. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of 'remarkable' agreement without parameter fitting down to 10-20 μs is load-bearing, yet the abstract provides no information on experimental error bars, uncertainty quantification, or the precise numerical discretization and boundary conditions used for the convection-enhanced diffusion equation; these details are required to evaluate robustness.

    Authors: We agree that the abstract is concise and does not detail error bars or numerics. The experimental uncertainties (camera resolution, timing precision, and run-to-run repeatability) are quantified in the Experimental Methods section, where the agreement for Surfynol 465 and SDS is shown to lie within these bounds across multiple trials. Numerical implementation (axisymmetric finite-element discretization with adaptive refinement near the neck, no-flux boundary conditions away from the interface, and symmetry conditions) is fully specified in the Modeling section. Given abstract length limits, we will add one sentence noting that the reported agreement is parameter-free and holds within experimental uncertainties, while retaining the full technical details in the main text. revision: partial

  2. Referee: [Modeling section] Modeling section (diffusion-limited formulation): the assertion that 'diffusion does not constitute a significant barrier over most of the breakup' relies on convection dominating transport, but lacks an explicit timescale comparison (e.g., convective vs. diffusive fluxes or local Péclet number) near the neck; without this, the transition to diffusion-limited behavior very close to pinch-off remains qualitative.

    Authors: We accept that an explicit timescale comparison would make the argument more quantitative. In the revised manuscript we will insert a short paragraph in the Modeling section that evaluates the local Péclet number Pe = U R / D (using the simulated radial velocity U and instantaneous neck radius R together with the known bulk diffusivities of Surfynol 465 and SDS). This calculation shows Pe ≫ 1 over the great majority of the breakup, with diffusion becoming comparable only when the neck radius falls below ~10 μm in the final microseconds, thereby justifying the diffusion-limited regime until very close to pinch-off. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper presents a diffusion-limited sorption model solved numerically and compares its predictions directly to independent experimental measurements of pendant-droplet breakup for Surfynol 465 and SDS, with no adjustable parameters fitted to the target data. The central claim of agreement down to 10-20 μs pinch-off times rests on this external validation and explicit distinction from insoluble-surfactant and slow-kinetics cases, rather than any self-referential definition, renamed fit, or self-citation that reduces the result to its inputs. All modeling assumptions are stated upfront and the numerical-experimental match is falsifiable against the reported observations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model assumes standard incompressible Navier-Stokes equations with free-surface boundary conditions and a diffusion-limited sorption model; no free parameters are introduced because the claim is validated without fitting.

axioms (2)
  • standard math Incompressible Navier-Stokes equations govern the flow inside the droplet.
    Standard assumption for low-speed liquid flows in droplet breakup studies.
  • domain assumption Surfactant sorption is limited only by diffusion, with convection enhancing transfer except near pinch-off.
    Explicitly stated as the modeling limit considered in the abstract.

pith-pipeline@v0.9.0 · 5592 in / 1353 out tokens · 41903 ms · 2026-05-07T14:14:57.062416+00:00 · methodology

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