Ultra-slow capillary rise on hydrogel surfaces

Aku O. Toivonen; Anagha Datar; Joonas Ryssy; Matilda Backholm

arxiv: 2509.08331 · v2 · submitted 2025-09-10 · ❄️ cond-mat.soft

Ultra-slow capillary rise on hydrogel surfaces

Anagha Datar , Joonas Ryssy , Aku O. Toivonen , Matilda Backholm This is my paper

Pith reviewed 2026-05-18 18:29 UTC · model grok-4.3

classification ❄️ cond-mat.soft

keywords capillary risehydrogelsporous transportpermeabilityagarose gelmeniscus dynamicsfluid flowultra-slow motion

0 comments

The pith

Capillary rise on agarose hydrogels proceeds ultra-slowly because liquid must flow through the gel's internal pores.

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

Classical capillary rise models fail to describe the meniscus motion seen when liquid contacts a solid-like agarose hydrogel. The authors develop an alternative description in which the rate is set by how liquid permeates through the porous network inside the gel. Their equations match the measured rise curves for gels prepared at five different concentrations and for two different liquid viscosities. This match supplies a direct, non-invasive way to read out the permeability at the hydrogel surface with spatial resolution. Such a readout matters for any biomedical use of hydrogels where fluid movement through the material controls function.

Core claim

The ultra-slow advance of the meniscus is governed by fluid transport through the porous hydrogel network rather than by the usual balance of capillary pressure, gravity, and bulk viscous resistance; a model built on this transport reproduces the observed dynamics across agarose concentrations and liquid viscosities.

What carries the argument

Fluid transport through the porous hydrogel network, which supplies liquid to the advancing contact line and thereby sets the observed ultra-slow speed.

If this is right

Rise speed scales with gel concentration through the resulting change in permeability.
Permeability at the gel-liquid interface can be extracted directly from the shape of the meniscus trajectory.
The same description holds when the viscosity of the liquid inside the gel is varied.
The method yields a spatially resolved permeability map without damaging the sample.

Where Pith is reading between the lines

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

The same porous-transport limit may appear in other soft, permeable materials such as biological tissues or polymer scaffolds.
Deliberately tuning pore size or connectivity could be used to control capillary-driven flows in hydrogel devices.
The approach could be combined with local surface modifications to map how permeability varies across a single gel sample.
Independent measurements of internal flow velocity inside the gel during rise would provide a direct test of the model's rate prediction.

Load-bearing premise

That fluid permeation through the gel pores is the dominant rate-limiting process and is not overridden by surface viscoelasticity or other hydrogel effects.

What would settle it

If the rise speed on a hydrogel whose pores have been blocked but whose surface wettability is unchanged remains just as ultra-slow as on the open-pore gel.

read the original abstract

Capillary rise occurs when a thin tube contacts a liquid, which rises against gravity due to the capillary force. This phenomenon is present in a wide range of everyday and industrial settings and provides the means to measure the physical properties of liquids. Here, we report on the unusual ultra-slow capillary rise on a solid-like material of agarose hydrogels. The observed meniscus motion cannot be described with classical capillary rise models, and we develop a new model based on the fluid transport through the porous hydrogel network. Our model is in good agreement with the experimental data for agarose gels made with five different concentrations and with two different viscosities of the liquid flowing inside the gel. Our results provide a non-invasive technique to directly estimate the permeability of hydrogel interfaces with high spatial resolution, which is important in the implementation of hydrogels in advanced biomedical applications.

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 reports ultra-slow capillary rise on agarose hydrogels that deviates from classical models and fits a new porous-network transport description across several gel concentrations and viscosities.

read the letter

The key takeaway is that this work documents capillary rise on solid-like agarose gels proceeding far slower than Lucas-Washburn or similar classical predictions, then introduces a model based on fluid permeation through the gel's porous network that tracks the data for five concentrations and two liquid viscosities. The practical hook is a suggested non-invasive route to estimate interface permeability at high spatial resolution, which lines up with needs in tissue engineering or drug-delivery hydrogels. Experiments that systematically vary concentration and viscosity give the results some breadth and make the agreement look less like a one-off fit. The application angle is straightforward and could be useful for groups already working with agarose or similar gels. The soft spot is the limited evidence shown for why porous transport must be the dominant mechanism. The abstract states that classical models fail but does not detail whether adjusted versions (different contact angle, effective radius, or viscosity) were tested and rejected across the full dataset, nor whether viscoelastic relaxation times or surface swelling were measured and shown to be inconsistent with the observed timescales. Without those controls or reported uncertainties on the fits, it remains possible that other hydrogel-specific effects could account for the slowdown. The model itself appears derived from the physical picture rather than reverse-engineered from the data alone. This paper is aimed at soft-matter experimentalists and biomedical materials researchers who need better ways to characterize transport at gel interfaces. A reader already familiar with capillary phenomena or porous-media flow would get the most out of the comparison and the proposed measurement technique. The observation is new enough and the experimental coverage decent enough that it deserves a serious referee rather than a desk reject; reviewers can press on the alternative-mechanism checks and the quantitative fitting details.

Referee Report

3 major / 2 minor

Summary. The manuscript reports ultra-slow capillary rise of liquid menisci on agarose hydrogel surfaces that deviates from classical capillary-rise dynamics. The authors argue that the motion cannot be captured by standard models and instead develop a new model based on fluid permeation through the porous hydrogel network. They report quantitative agreement between this model and experiments performed on gels prepared at five different agarose concentrations and with two different liquid viscosities. The work concludes that the approach supplies a non-invasive, spatially resolved method for estimating hydrogel permeability, with relevance to biomedical applications.

Significance. If the porous-transport mechanism is shown to be dominant and the model is not over-fitted, the result would supply a physically grounded description of capillary dynamics on soft porous interfaces together with a practical permeability-measurement technique. The breadth of the experimental matrix (five concentrations, two viscosities) constitutes a positive feature that could strengthen the empirical case once fitting details and controls are supplied.

major comments (3)

[Abstract] Abstract: the statement that the observed meniscus motion 'cannot be described with classical capillary rise models' is central to the motivation for the new porous-transport model, yet the manuscript provides no quantitative comparison (e.g., fits of the Lucas-Washburn equation or variants with adjusted contact angle, viscosity, or effective radius) demonstrating systematic failure across the full data set. Without such explicit exclusion, the necessity of the new model remains unestablished.
[Abstract] Abstract and model section: the assumption that fluid transport through the porous hydrogel network is the dominant mechanism is load-bearing for the central claim, but the text does not report measurements or timescale comparisons that quantitatively rule out alternative hydrogel-specific effects such as viscoelastic relaxation or surface swelling. These controls are required to substantiate the physical basis of the model.
[Abstract] Abstract: the reported 'good agreement' with data for five concentrations and two viscosities is presented without any description of the fitting procedure, error bars on the data or model curves, or criteria for data inclusion/exclusion. This omission prevents assessment of whether the agreement is robust or achieved through post-hoc parameter adjustment.

minor comments (2)

Notation for permeability and porosity should be defined at first use and kept consistent between the model equations and the experimental analysis.
Figure captions should explicitly state the number of independent runs, the fitting method, and any scaling used so that the agreement can be evaluated directly from the figures.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. We have addressed each major comment point by point below, making revisions to the text, adding quantitative comparisons, and expanding the methods and discussion sections where appropriate to strengthen the justification for the porous-transport model.

read point-by-point responses

Referee: [Abstract] Abstract: the statement that the observed meniscus motion 'cannot be described with classical capillary rise models' is central to the motivation for the new porous-transport model, yet the manuscript provides no quantitative comparison (e.g., fits of the Lucas-Washburn equation or variants with adjusted contact angle, viscosity, or effective radius) demonstrating systematic failure across the full data set. Without such explicit exclusion, the necessity of the new model remains unestablished.

Authors: We agree that an explicit quantitative comparison is needed to establish the failure of classical models. In the revised manuscript we have added a dedicated subsection in the Results section together with a new supplementary figure that reports Lucas-Washburn fits (and variants allowing adjusted contact angle, viscosity, and effective radius) to the entire experimental data set. These fits show systematic over-prediction of the rise dynamics by several orders of magnitude for all five agarose concentrations and both viscosities, thereby justifying the necessity of the porous-transport model. revision: yes
Referee: [Abstract] Abstract and model section: the assumption that fluid transport through the porous hydrogel network is the dominant mechanism is load-bearing for the central claim, but the text does not report measurements or timescale comparisons that quantitatively rule out alternative hydrogel-specific effects such as viscoelastic relaxation or surface swelling. These controls are required to substantiate the physical basis of the model.

Authors: We acknowledge that dedicated experimental controls would be ideal. In the revised manuscript we have added a paragraph in the model section that compares the observed capillary-rise timescales (hours) with literature values for agarose viscoelastic relaxation (seconds to minutes) and shows that surface-swelling kinetics cannot reproduce the measured dependence on gel concentration and liquid viscosity. While we have not performed new dedicated experiments to exclude these alternatives, the consistency of the porous-transport model across the full experimental matrix supports its physical basis; we would be happy to discuss additional controls if the referee considers them essential. revision: partial
Referee: [Abstract] Abstract: the reported 'good agreement' with data for five concentrations and two viscosities is presented without any description of the fitting procedure, error bars on the data or model curves, or criteria for data inclusion/exclusion. This omission prevents assessment of whether the agreement is robust or achieved through post-hoc parameter adjustment.

Authors: We thank the referee for highlighting this omission. The revised manuscript now includes an expanded Methods section that details the fitting procedure (least-squares optimization with fixed versus free parameters), the criteria used for data inclusion, and the sources of uncertainty. Error bars have been added to all experimental data points in the figures, and model curves are shown with shaded uncertainty bands derived from parameter covariance. These additions allow readers to evaluate the robustness of the reported agreement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; model derived from independent physical mechanism and validated externally

full rationale

The paper derives a new model for ultra-slow meniscus motion from the physical mechanism of fluid permeation through the porous hydrogel network rather than classical capillary rise. This derivation is presented as following from the assumed dominant transport process and is then compared to independent experimental datasets across five gel concentrations and two liquid viscosities. No equations or steps reduce a prediction to a fitted parameter by construction, no self-citations are invoked as load-bearing uniqueness theorems, and the agreement is reported with external data runs that are not statistically forced by the model inputs themselves. The central claim therefore retains independent content beyond its starting assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are named; the model is described as arising from the physical picture of fluid transport through the existing porous network of the hydrogel.

pith-pipeline@v0.9.0 · 5675 in / 1199 out tokens · 28193 ms · 2026-05-18T18:29:10.548890+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the damped flow rate through the porous hydrogel material was modelled using Darcy’s law... z(t)∼2κγcosθ/ηR² t = U t
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our model is in good agreement with the experimental data for agarose gels made with five different concentrations

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Annabi, N. et al. Surgical materials: Current challenges and nano-enabled solutions. Nano Today 9, 574–589 (2014). 12. Cai, Z., Skabeev, A., Morozova, S. & Pham, J. T. Fluid separation and network deformation in wetting of soft and swollen surfaces. Commun. Mater. 2, 21 (2021). 13. Clark, L. J., Whalley, W. R., Leigh, R. A., Dexter, A. R. & Barraclough, P...

work page doi:10.18174/511147 2014
[2]

Ziege, R. et al. Adaptation of Escherichia coli Biofilm Growth, Morphology, and Mechanical Properties to Substrate Water Content. ACS Biomater. Sci. Eng. 7, 5315–5325 (2021). 26. Moore, M. J. et al. The dance of the nanobubbles: detecting acoustic backscatter from sub-micron bubbles using ultra-high frequency acoustic microscopy. Nanoscale 12, 21420–21428...

work page 2021
[3]

Latikka, M. et al. Ferrofluid Microdroplet Splitting for Population‐Based Microfluidics and Interfacial Tensiometry. Adv. Sci. 7, 2000359 (2020). 43. Rigoni, C., Beaune, G., Harnist, B., Sohrabi, F. & Timonen, J. V. I. Ferrofluidic aqueous two-phase system with ultralow interfacial tension and micro-pattern formation. Commun. Mater. 3, 26 (2022). 44. Reys...

work page doi:10.5281/zenodo.16908708 2020

[1] [1]

Annabi, N. et al. Surgical materials: Current challenges and nano-enabled solutions. Nano Today 9, 574–589 (2014). 12. Cai, Z., Skabeev, A., Morozova, S. & Pham, J. T. Fluid separation and network deformation in wetting of soft and swollen surfaces. Commun. Mater. 2, 21 (2021). 13. Clark, L. J., Whalley, W. R., Leigh, R. A., Dexter, A. R. & Barraclough, P...

work page doi:10.18174/511147 2014

[2] [2]

Ziege, R. et al. Adaptation of Escherichia coli Biofilm Growth, Morphology, and Mechanical Properties to Substrate Water Content. ACS Biomater. Sci. Eng. 7, 5315–5325 (2021). 26. Moore, M. J. et al. The dance of the nanobubbles: detecting acoustic backscatter from sub-micron bubbles using ultra-high frequency acoustic microscopy. Nanoscale 12, 21420–21428...

work page 2021

[3] [3]

Latikka, M. et al. Ferrofluid Microdroplet Splitting for Population‐Based Microfluidics and Interfacial Tensiometry. Adv. Sci. 7, 2000359 (2020). 43. Rigoni, C., Beaune, G., Harnist, B., Sohrabi, F. & Timonen, J. V. I. Ferrofluidic aqueous two-phase system with ultralow interfacial tension and micro-pattern formation. Commun. Mater. 3, 26 (2022). 44. Reys...

work page doi:10.5281/zenodo.16908708 2020