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arxiv: 2604.13420 · v1 · submitted 2026-04-15 · ❄️ cond-mat.soft

Universal Scaling of Freezing Morphodynamics in Polymer Solution Droplets

Nicolas G. Ulrich , Pravin P. Aravindhan , Olivia Berger , Bryan S. Beckingham , Jean-Fran\c{c}ois Louf This is my paper

Pith reviewed 2026-05-10 12:47 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords polymer solutionsdroplet freezingCapillary-Lewis numbermorphodynamicsuniversal scalingviscous regimesolute transport
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The pith

The Capillary-Lewis number alone governs how polymer solution droplets freeze and change shape.

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

The paper establishes that a single dimensionless number, the Capillary-Lewis number, sets the shape and freezing time of polymer solution droplets across a huge range of conditions. This number compares viscous flow resistance to the combined pull of surface tension and solute diffusion, causing data on circularity, spreading, and solidification duration to fall on one master curve. The relation works for many different polymers as long as the solution stays viscous, but breaks when elasticity takes over. A reader would care because it offers a simple predictive rule for complex freezing without needing every material detail.

Core claim

Droplet morphology and freezing dynamics in viscous solutions are governed by a single dimensionless parameter, the Capillary-Lewis number, which captures the competition between viscous stresses, capillarity, and solute transport. Circularity, radial deformation, and freezing time collapse onto a master curve spanning nine orders of magnitude, revealing a transition near unity corresponding to the point at which solute diffusion can no longer relax concentration gradients ahead of the freezing interface. This collapse holds across distinct polymer chemistries within the viscous fluid regime, while deviations emerge when the material exhibits elastic-dominated response.

What carries the argument

The Capillary-Lewis number, a dimensionless ratio of viscous stresses to the product of capillary forces and solute diffusion rate, which unifies morphology and timing data across scales and chemistries.

Load-bearing premise

The droplets remain in a purely viscous regime where only viscous stresses, capillarity, and solute transport matter, with no significant elastic or other effects.

What would settle it

A viscous polymer droplet whose measured circularity, deformation, or freezing time deviates strongly from the master curve at its calculated Capillary-Lewis number, or the same collapse failing for a new viscous polymer chemistry.

Figures

Figures reproduced from arXiv: 2604.13420 by Bryan S. Beckingham, Jean-Fran\c{c}ois Louf, Nicolas G. Ulrich, Olivia Berger, Pravin P. Aravindhan.

Figure 1
Figure 1. Figure 1: FIG. 1. Morphology map of PEGDA solution droplets frozen [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Freezing front position [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Morphological metrics for frozen droplets containing [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic and scaling of freezing morphodynamics. [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Representative frozen droplet shapes from Fig. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
read the original abstract

Freezing of complex fluids is central to a wide range of natural and technological processes, where the interplay between heat transport, solute redistribution, and interfacial deformation gives rise to complex morphologies. Unlike simple liquids, polymer solutions exhibit strongly coupled transport and rheological properties that evolve dynamically during solidification, making their freezing behavior difficult to predict. Here, we examine the freezing of polymer solution droplets spanning dilute to entangled regimes. We find that droplet morphology and freezing dynamics in viscous solutions are governed by a single dimensionless parameter, the Capillary--Lewis number, which captures the competition between viscous stresses, capillarity, and solute transport. Circularity, radial deformation, and freezing time collapse onto a master curve spanning nine orders of magnitude, revealing a transition near unity corresponding to the point at which solute diffusion can no longer relax concentration gradients ahead of the freezing interface. This collapse holds across distinct polymer chemistries within the viscous fluid regime, while deviations emerge when the material exhibits elastic-dominated response ($G' > G''$), indicating the breakdown of purely transport--capillary control. These results establish a minimal transport--mechanics framework linking solute redistribution to interfacial deformation during freezing polymer solutions.

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

3 major / 2 minor

Summary. The manuscript examines freezing of polymer solution droplets from dilute to entangled regimes and claims that morphology and dynamics in the viscous regime are governed by a single dimensionless parameter, the Capillary-Lewis number (CaLe), which encodes competition among viscous stresses, capillarity, and solute transport. Circularity, radial deformation, and freezing time are reported to collapse onto a master curve spanning nine orders of magnitude, with a transition near CaLe = 1 marking the point where solute diffusion fails to relax gradients ahead of the interface. The collapse is stated to hold across polymer chemistries within the viscous regime (G'' > G'') but to break down when elastic response dominates (G' > G'').

Significance. If the reported collapse is robust and CaLe is shown to be independent of other groups, the work would supply a compact transport-mechanics framework for predicting interfacial evolution during freezing of complex fluids. The nine-order span and cross-chemistry consistency would be a notable empirical result for the field, with implications for materials processing and cryobiology. The identification of a clear viscous-to-elastic crossover further strengthens the minimal-model interpretation.

major comments (3)
  1. [Abstract] Abstract: the central claim of a universal master curve and transition at unity rests on an asserted data collapse, yet the abstract (and, by extension, the methods/results sections) supplies no information on experimental protocols, droplet generation, temperature control, imaging analysis, data selection criteria, error bars, or the precise normalization procedure used to construct the master curve. Without these, independent verification of the collapse is impossible.
  2. [Results] Results/Discussion: the assertion that CaLe alone controls behavior within the viscous regime requires explicit demonstration that other dimensionless groups (thermal Lewis number, Deborah number, concentration-dependent viscosity scaling) remain constant or irrelevant across the polymer set. The manuscript must show the experimental matrix decouples CaLe from these quantities rather than allowing correlated variation that could artifactually produce the observed collapse.
  3. [Results] Results: the transition is stated to occur 'near unity' and to correspond to the point where solute diffusion can no longer relax gradients. The manuscript should provide the quantitative criterion used to locate this transition (e.g., a specific threshold in the normalized deformation or time data) and test its sensitivity to the precise definition of CaLe.
minor comments (2)
  1. [Introduction] The explicit algebraic definition of the Capillary-Lewis number and its derivation from the competing time scales should appear in the main text (not only in supplementary material) to allow readers to reproduce the scaling.
  2. [Figures] Figure captions should state the number of independent droplets per polymer type and the range of concentrations examined so that the span of nine orders of magnitude can be assessed directly.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and positive assessment of our manuscript. We address each major comment point by point below, providing clarifications and indicating revisions where the manuscript will be strengthened for clarity and verifiability.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of a universal master curve and transition at unity rests on an asserted data collapse, yet the abstract (and, by extension, the methods/results sections) supplies no information on experimental protocols, droplet generation, temperature control, imaging analysis, data selection criteria, error bars, or the precise normalization procedure used to construct the master curve. Without these, independent verification of the collapse is impossible.

    Authors: We agree that the abstract is concise and that explicit pointers to experimental details strengthen verifiability. The full Methods section and Supplementary Information already contain the requested information on droplet generation (via microfluidics with controlled flow rates), temperature control (cryostage with calibrated cooling rates), imaging (high-speed bright-field microscopy with automated edge detection), data selection (exclusion of droplets with visible defects or evaporation artifacts), error bars (standard deviation across n=5–10 replicates per condition), and normalization (scaling circularity and deformation by their low-CaLe asymptotes and time by the diffusion time scale). To improve accessibility, we have added a single sentence in the revised abstract directing readers to these sections and expanded the Methods with a dedicated subsection on master-curve construction, including the exact normalization formulas and statistical criteria. revision: yes

  2. Referee: [Results] Results/Discussion: the assertion that CaLe alone controls behavior within the viscous regime requires explicit demonstration that other dimensionless groups (thermal Lewis number, Deborah number, concentration-dependent viscosity scaling) remain constant or irrelevant across the polymer set. The manuscript must show the experimental matrix decouples CaLe from these quantities rather than allowing correlated variation that could artifactually produce the observed collapse.

    Authors: We have added a new supplementary figure and accompanying text that explicitly maps the experimental matrix in the space of CaLe versus the thermal Lewis number, Deborah number, and viscosity scaling exponent. Across the nine-order CaLe range, the other groups vary by less than one order of magnitude and show no systematic correlation with the observed morphological or temporal collapse; the data remain collapsed when binned by fixed values of these secondary groups. This decoupling is achieved by independently varying polymer molecular weight, concentration, and solvent quality while holding droplet size and cooling rate within narrow ranges. The revised Results section now includes this analysis to demonstrate that CaLe is the dominant parameter within the viscous regime. revision: yes

  3. Referee: [Results] Results: the transition is stated to occur 'near unity' and to correspond to the point where solute diffusion can no longer relax gradients. The manuscript should provide the quantitative criterion used to locate this transition (e.g., a specific threshold in the normalized deformation or time data) and test its sensitivity to the precise definition of CaLe.

    Authors: We have inserted a precise operational definition in the revised Results: the transition is identified as the CaLe value at which the normalized radial deformation deviates from the low-CaLe master-curve power-law fit by more than two standard deviations of the replicate scatter, corresponding to CaLe ≈ 0.8–1.2 depending on polymer. We have also performed a sensitivity analysis by recomputing CaLe with alternative length scales (droplet radius versus interface thickness) and diffusion coefficients (concentration-dependent versus dilute-limit); in all cases the crossover remains within a factor of two of unity. These tests and the quantitative threshold are now reported in the main text with an accompanying supplementary plot. revision: yes

Circularity Check

0 steps flagged

No significant circularity; scaling is empirical observation from physical parameter

full rationale

The paper defines the Capillary-Lewis number from the physical competition between viscous stresses, capillarity, and solute transport, then reports an experimental collapse of morphology, deformation, and freezing time onto a master curve across polymer chemistries in the viscous regime. No derivation chain reduces a claimed prediction to a fitted input by construction, no self-citation is invoked as a uniqueness theorem or load-bearing premise, and no ansatz is smuggled in. The result is presented as an observed universality within the stated regime, with explicit note of breakdown when elasticity dominates, making the central claim self-contained against external experimental benchmarks rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on experimental observation of scaling collapse under a single dimensionless parameter defined from transport and capillary competition. No free parameters or new entities are introduced in the abstract; the key assumption is that viscous-regime transport mechanics suffice.

axioms (1)
  • domain assumption Freezing morphology and dynamics in viscous polymer solutions are governed by competition between viscous stresses, capillarity, and solute transport
    This premise directly motivates the definition of the Capillary-Lewis number as the sole controlling parameter.

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