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arxiv: 2602.09166 · v3 · pith:J3ET75TZnew · submitted 2026-02-09 · ❄️ cond-mat.soft

Scaling of poroelastic coarsening and elastic arrest in crosslinked gels

Samuel A. Safran This is my paper

Pith reviewed 2026-05-21 14:12 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords poroelastic coarseningelastic arrestcrosslinked gelsDarcy flowviscoelastic to elastic crossoverYoung-Laplace tractionsolvent-rich domains
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The pith

Crosslinked gels coarsen via capillary-driven flow until elastic stresses balance interfacial tractions and arrest further growth at a stiffness-dependent size.

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

The paper develops a minimal model that couples capillary forces at solvent-gel interfaces to the transition of the polymer network from viscoelastic to elastic response. In the viscoelastic stage, Young-Laplace tractions create solvent pressure gradients that drive Darcy flow and domain coarsening. Once the network crosses into the elastic regime, the same tractions are supported entirely by elastic stress, which removes the pressure gradients and therefore stops the flow. Matching the coarsening time to the elastic relaxation time yields explicit power laws: for polymer-rich melt-like gels the domain size grows as G to the minus one-half times t to the one-fourth and arrests at a size proportional to G to the minus one-half. The predicted arrest scaling matches reported experiments on such gels.

Core claim

The central claim is that poroelastic coarsening proceeds under Darcy flow driven by interfacial pressure gradients only while the gel network remains viscoelastic; once elastic stresses fully balance the Young-Laplace traction, pressure gradients vanish, Darcy flow ceases, and coarsening arrests. The arrest length is obtained from the kinetic condition that the time to reach that length equals the elastic relaxation time of the network, producing the stiffness-dependent scalings given above.

What carries the argument

The kinetic matching condition t(λ_arrest) ∼ τ_el that equates the time for a domain to reach its final size with the time at which the polymer network crosses from viscoelastic to elastic response.

If this is right

  • For melt-like gels the coarsening length scales as λ(t) ∼ G^{-1/2} t^{1/4}.
  • Arrest occurs at λ_arrest ∼ G^{-1/2} for the same gels.
  • At low polymer fractions where mesh size controls transport the exponents change to λ(t) ∼ G^{-1/3} t^{1/3} and λ_arrest ∼ G^{-1/3}.
  • The G^{-1/2} arrest scaling reproduces the stiffness dependence seen in existing experiments on polymer-rich gels.

Where Pith is reading between the lines

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

  • Varying crosslink density while holding other parameters fixed would provide a direct experimental test of the predicted G dependence without changing solvent quality.
  • The same elastic-arrest mechanism may set stable domain sizes in other poroelastic materials such as biological tissues or synthetic scaffolds once their networks reach the elastic regime.
  • The model implies that macroscopic drainage after arrest proceeds on a much longer timescale set by the now-absent Darcy flow.

Load-bearing premise

Once the polymer network becomes elastic, the interfacial traction is assumed to be carried entirely by elastic stress with no remaining pressure gradient in the solvent pores.

What would settle it

Direct measurement of arrest domain size versus gel modulus G in melt-like crosslinked gels that finds a scaling other than G to the power of minus one-half.

read the original abstract

Recent experiments on crosslinked gels quenched from solvent-rich to solvent-poor conditions show solvent-rich domains embedded in a gel-rich matrix. These domains coarsen and then undergo kinetic arrest at micron scales for hours, before macroscopic drainage to equilibrium over even longer times. Motivated by these observations, we develop a minimal model that couples capillarity-driven Darcy permeation to the viscoelastic-to-elastic crossover of the polymer network. In the viscoelastic regime, the Young--Laplace traction at curved solvent--gel interfaces generates a pressure gradient in the solvent pores of the gel that drives solvent flow and coarsening. In the elastic regime, the same interfacial traction is balanced by elastic stress. This force balance eliminates pressure gradients in the solvent-filled pores of the gel, removing the Darcy driving force and arresting coarsening. Using the kinetic criterion $t(\lambda_{\rm arrest}) \sim \tau_{\rm el}$, we predict stiffness-dependent coarsening and arrest laws. For melt-like, polymer-rich gels, $\lambda(t)\sim G^{-1/2} t^{1/4}$ and $\lambda_{\rm arrest}\sim G^{-1/2}$. For low polymer fractions where the mesh size controls transport, $\lambda(t)\sim G^{-1/3} t^{1/3}$ and $\lambda_{\rm arrest}\sim G^{-1/3}$. The predicted $G^{-1/2}$ arrest scaling for melt-like gels agrees with experiment.

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

1 major / 2 minor

Summary. The manuscript develops a minimal model coupling capillarity-driven Darcy permeation to the viscoelastic-to-elastic crossover of the polymer network in crosslinked gels. In the viscoelastic regime, Young-Laplace tractions at solvent-gel interfaces generate pressure gradients that drive solvent flow and coarsening. In the elastic regime, the same interfacial tractions are balanced by elastic stress, which the model states eliminates pressure gradients in the solvent pores and removes the Darcy driving force, arresting coarsening. Using the kinetic criterion t(λ_arrest) ∼ τ_el, the authors derive stiffness-dependent scalings: for melt-like gels λ(t)∼G^{-1/2} t^{1/4} and λ_arrest∼G^{-1/2}; for low polymer fractions λ(t)∼G^{-1/3} t^{1/3} and λ_arrest∼G^{-1/3}. The predicted G^{-1/2} arrest scaling for melt-like gels is reported to agree with experiment.

Significance. If the central force-balance assumption is rigorously justified, the work supplies a physically motivated mechanism for the observed micron-scale kinetic arrest and yields concrete, falsifiable predictions for how arrest size depends on gel modulus G. The derivation of the scalings from a parameter-free kinetic matching condition without additional fitting parameters is a clear strength, as is the separation into melt-like and mesh-controlled transport regimes.

major comments (1)
  1. Model description (force balance and arrest mechanism): the claim that elastic stress balance 'eliminates pressure gradients in the solvent-filled pores... removing the Darcy driving force' is load-bearing for both the arrest mechanism and the kinetic criterion t(λ_arrest) ∼ τ_el used to obtain the G-dependent scalings. In a standard poroelastic formulation the total stress is σ_elastic − p I and the Young-Laplace condition constrains only the normal component of total stress; it does not automatically imply ∇p = 0 inside the pores without further assumptions (e.g., instantaneous relaxation or neglect of poroelastic diffusion). Please supply the explicit poroelastic equations demonstrating why residual pressure gradients vanish once the network enters the elastic regime.
minor comments (2)
  1. The abstract states that the G^{-1/2} arrest scaling 'agrees with experiment' but supplies no details on the range of G, number of samples, or quantitative measure of agreement (e.g., fitted exponent with uncertainty). This information should appear in the results or comparison section.
  2. Notation for the two transport regimes (melt-like vs. mesh-size controlled) is introduced only in the abstract; a brief table or paragraph early in the text defining the crossover condition would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment below and have revised the manuscript to provide the requested explicit poroelastic equations and derivation.

read point-by-point responses

  1. Referee: Model description (force balance and arrest mechanism): the claim that elastic stress balance 'eliminates pressure gradients in the solvent-filled pores... removing the Darcy driving force' is load-bearing for both the arrest mechanism and the kinetic criterion t(λ_arrest) ∼ τ_el used to obtain the G-dependent scalings. In a standard poroelastic formulation the total stress is σ_elastic − p I and the Young-Laplace condition constrains only the normal component of total stress; it does not automatically imply ∇p = 0 inside the pores without further assumptions (e.g., instantaneous relaxation or neglect of poroelastic diffusion). Please supply the explicit poroelastic equations demonstrating why residual pressure gradients vanish once the network enters the elastic regime.

    Authors: We agree that an explicit derivation strengthens the presentation of the arrest mechanism. In the revised manuscript we have added the following poroelastic equations and reasoning. The quasi-static momentum balance is ∇ · σ = 0 with total stress σ = σ_elastic − p I, where σ_elastic is the stress in the crosslinked network. The Young-Laplace condition at the solvent-gel interface requires that the normal component of the total stress jump equals the capillary pressure γκ. In the viscoelastic regime (t ≪ τ_el) the network relaxes rapidly, so the capillary traction is accommodated by a solvent pressure gradient that drives Darcy flow v = −(k/η)∇p and thereby coarsening. Once the network crosses into the elastic regime (t ≳ τ_el), the structure becomes static and net solvent flow ceases (v = 0). Darcy's law then immediately implies ∇p = 0 inside the pores; the interfacial tractions are balanced entirely by the divergence of the now time-independent elastic stress σ_elastic. This removes the Darcy driving force and supplies the kinetic matching condition t(λ_arrest) ∼ τ_el used to obtain the reported scalings. The added section makes these steps and the underlying assumptions explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent physical assumptions and kinetic matching.

full rationale

The paper states a modeling premise that elastic stress balances interfacial traction and removes Darcy driving force once the network crosses to the elastic regime, then applies the separate kinetic condition t(λ_arrest) ∼ τ_el to the viscoelastic coarsening law to obtain the G-dependent scalings. These steps are presented as predictions that are compared to experiment rather than quantities defined by fitting the same data or by self-referential equations. No load-bearing self-citation, ansatz smuggling, or renaming of known results is exhibited in the provided text; the central claim remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard poroelastic transport laws and a kinetic matching condition introduced to locate arrest; no new particles or forces are postulated and no free parameters are fitted beyond the physical inputs G and the elastic time scale.

axioms (3)
  • domain assumption Young-Laplace traction at curved solvent-gel interfaces generates pressure gradients that drive Darcy flow in the viscoelastic regime
    Invoked to produce the coarsening mechanism before the elastic crossover.
  • domain assumption In the elastic regime interfacial traction is balanced by elastic stress, eliminating pressure gradients and Darcy flow
    This is the key step that removes the driving force and produces arrest.
  • ad hoc to paper Arrest occurs when the time to reach size λ equals the elastic relaxation time τ_el
    The kinetic criterion t(λ_arrest) ∼ τ_el is used to convert the coarsening law into an arrest size.

pith-pipeline@v0.9.0 · 5784 in / 1702 out tokens · 87247 ms · 2026-05-21T14:12:41.584181+00:00 · methodology

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Reference graph

Works this paper leans on

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