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arxiv: 2510.21552 · v1 · submitted 2025-10-24 · ❄️ cond-mat.soft · physics.bio-ph

Kinetic theory of emulsions with matter supply

Jacqueline Janssen , Frank J\"ulicher , Christoph A. Weber This is my paper

Pith reviewed 2026-05-18 04:34 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.bio-ph
keywords emulsionscoarsening kineticsLifshitz-Slyozov-Wagner theorymatter supplyinterface resistancedroplet size distributionbiomolecular condensates
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The pith

Emulsions with constant matter supply obey a universal coarsening law in the interface-resistance-limited regime.

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

The paper extends the Lifshitz-Slyozov-Wagner theory to emulsions in which matter is continuously supplied to growing droplets. It distinguishes growth limited by bulk diffusion from growth limited by transport across the droplet interface, and considers two supply scenarios: maintained supersaturation versus constant supply rate. In the constant-supply, interface-resistance-limited case the theory derives a universal kinetic law that holds for any supply rate, with the average radius following a power law independent of that rate and with an exact closed-form expression for the droplet size distribution. A reader would care because the result applies to open systems such as biomolecular condensates inside living cells, where total material is not conserved.

Core claim

For emulsions with constant matter supply in the interface-resistance-limited regime, coarsening follows a universal law valid for any constant supply. The average radius grows according to a power law whose exponent does not depend on the supply rate, and the droplet size distribution is given by a closed-form expression. In contrast, the diffusion-limited regime with constant supply yields a non-universal distribution that undergoes a transition from broadening to narrowing.

What carries the argument

The modified Lifshitz-Slyozov-Wagner growth equation that incorporates a continuous matter-supply term, which produces a supply-independent scaling solution when interface resistance dominates.

Load-bearing premise

Droplet growth is assumed to be limited strictly by either bulk diffusion or interface transport, with no effects from droplet interactions, coalescence, or spatial variations in supply.

What would settle it

Track the average droplet radius over time in an emulsion experiment with constant matter supply under interface-resistance-limited conditions and check whether the observed power-law exponent stays the same when the supply rate is changed.

Figures

Figures reproduced from arXiv: 2510.21552 by Christoph A. Weber, Frank J\"ulicher, Jacqueline Janssen.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
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Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
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Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
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Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
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Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
read the original abstract

In this work, we propose a theory for the kinetics of emulsions in which a continuous supply of matter feeds droplet growth. We consider cases where growth is either limited by bulk diffusion or the transport through the droplets' interfaces. Our theory extends the Lifshitz-Slyozov-Wagner (LSW) theory by two types of matter supply, where either the supersaturation is maintained or the supply rate is constant. In emulsions with maintained supersaturation, we find a decoupling of droplets at all times, with the droplet size distribution narrowing in the diffusion-limited regime and a drifting distribution of a fixed shape in the interface-resistance-limited case. In emulsions with a constant matter supply, there is a transition between narrowing and broadening in the diffusion-limited regime, and the distribution is non-universal. For the interface-resistance-limited regime, there is no transition to narrowing, and we find a universal law governing coarsening kinetics that is valid for any constant matter supply. The average radius evolves according to a power law that is independent of the matter supply, and we find a closed-form expression for the droplet size distribution function. Our theory is relevant to biological systems, such as biomolecular condensates in living cells, since droplet material is not conserved and the growth of small droplets is proposed to be interface-resistance-limited.

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 extends the LSW theory to emulsions with matter supply, considering diffusion- and interface-limited growth under constant supersaturation or constant supply rate. It derives droplet size distributions and coarsening laws, claiming a universal power-law for average radius independent of supply rate and a closed-form distribution in the interface-resistance-limited constant-supply regime, with relevance to biological condensates.

Significance. This provides a theoretical basis for coarsening in open systems with matter influx. The universal law and closed-form distribution, if correct, offer clear predictions for experiments in soft matter and biology, extending classical theory in a meaningful way. Strengths include the distinction between regimes and supply types leading to different behaviors.

major comments (1)
  1. [Abstract and model setup] The central claim of a universal coarsening law in the interface-resistance-limited regime with constant matter supply, independent of the supply rate, relies on the mean-field approximation without coalescence or spatial variations. Given that constant supply leads to linear growth in total droplet volume, the inter-droplet separation decreases over time, raising the question whether interaction effects remain negligible as assumed in deriving the supply-independent power law and closed-form distribution.
minor comments (2)
  1. [Introduction] Include the original Lifshitz-Slyozov and Wagner papers when referencing the LSW baseline.
  2. [Theory section] Clarify notation for the constant supply rate versus supersaturation to prevent reader confusion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for highlighting a key assumption underlying our central claims. We address this point in detail below and have revised the manuscript accordingly to better delineate the regime of validity of the mean-field treatment.

read point-by-point responses

  1. Referee: The central claim of a universal coarsening law in the interface-resistance-limited regime with constant matter supply, independent of the supply rate, relies on the mean-field approximation without coalescence or spatial variations. Given that constant supply leads to linear growth in total droplet volume, the inter-droplet separation decreases over time, raising the question whether interaction effects remain negligible as assumed in deriving the supply-independent power law and closed-form distribution.

    Authors: We agree that the mean-field approximation is foundational to the derivation, consistent with the original LSW framework. In the interface-resistance-limited regime the radial growth velocity of each droplet is set by the local supersaturation at its interface; we close the equations by replacing this local value with a spatially averaged supersaturation determined self-consistently from the global supply rate and the instantaneous total droplet surface area. The resulting scaling analysis yields a universal power-law growth of the mean radius, <R> ~ t^{1/2}, whose exponent is independent of the (constant) supply rate; the supply rate enters only the prefactor. With respect to droplet number and separation: our model assumes a fixed population of droplets (no nucleation or coalescence), so the average inter-droplet distance remains constant while the total droplet volume grows linearly with time. Consequently the volume fraction increases linearly, and the dilute-limit assumption underlying the mean-field closure eventually breaks down once the volume fraction becomes O(0.1). The closed-form distribution and supply-independent exponent are therefore strictly valid only while the system remains sufficiently dilute. We have added a new paragraph in the Discussion section that explicitly states this limitation, provides an estimate of the time window over which the approximation holds for typical experimental parameters, and notes that spatial correlations or coalescence would require a separate, more involved treatment beyond the present mean-field theory. revision: yes

Circularity Check

0 steps flagged

Derivation extends LSW theory with external supply inputs without self-referential reduction

full rationale

The paper modifies the standard LSW mean-field continuity equation by adding either constant supersaturation or constant supply rate as external parameters to the droplet growth law. In the interface-resistance-limited constant-supply case, the claimed supply-independent power law for average radius and closed-form size distribution are obtained by solving the resulting PDE for the distribution function n(R,t), with the independence on supply rate value emerging directly from the structure of the growth law and volume conservation rather than from any parameter fitted to the target observables. No equations reduce the final results to definitions or fits drawn from the same data, and the derivation does not rely on load-bearing self-citations or uniqueness theorems imported from the authors' prior work. The model is self-contained against the LSW baseline once the supply terms are accepted as inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard kinetic-theory assumptions for droplet growth plus two ad-hoc supply scenarios; no new particles or forces are introduced.

free parameters (1)
  • constant matter supply rate
    The rate at which new material is added is an external control parameter that enters the kinetic equations for the constant-supply case.
axioms (2)
  • domain assumption Droplet growth occurs either in the bulk-diffusion-limited regime or the interface-resistance-limited regime
    Invoked when the authors split the analysis into the two transport cases and derive distinct distribution behaviors for each.
  • ad hoc to paper Matter supply is either constant supersaturation or constant rate with no spatial variation
    Defines the two supply types whose consequences are derived; these are modeling choices not derived from more basic principles.

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