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arxiv: 2606.11388 · v1 · pith:A62MRTOEnew · submitted 2026-06-09 · ⚛️ physics.flu-dyn

Translation dynamics of evaporating sessile binary-mixture droplet populations

Debarshi Debnath , Anna Malachtari , George Karapetsas , Daniel Orejon , Khellil Sefiane , Alidad Amirfazli , Omar K. Matar , Prashant Valluri This is my paper

Pith reviewed 2026-06-27 11:24 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords binary mixture dropletssessile dropletsevaporationMarangoni effectvapor shieldinglubrication theorydroplet translationsolutal Marangoni
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The pith

Vapor shielding creates uneven evaporation that generates Marangoni and capillary stresses controlling attraction, repulsion, and chasing between binary-mixture droplets.

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

The paper develops a lubrication-theory model for pairs of thin evaporating sessile binary-mixture droplets that includes solutal and thermal Marangoni stresses plus vapor diffusion. It shows that vapor shielding produces non-uniform evaporation rates, which in turn create the concentration and temperature gradients responsible for the stresses. When the two droplets start with identical composition, solutal Marangoni and capillary forces pull them together while thermal Marangoni pushes them apart. When initial compositions differ, the droplet richer in the more volatile component chases the other through solutal Marangoni alone. Experiments with water-morpholine mixtures reproduce the predicted translational behaviors.

Core claim

The translation dynamics of two binary mixture droplets are governed by the combined effect of solutal Marangoni, capillary, and thermal Marangoni stresses that arise from non-uniform evaporation generated by vapour shielding. For droplets sharing the same initial composition, solutal Marangoni and capillary forces induce attraction while thermal Marangoni drives repulsion. For droplets with different initial compositions, the drop containing a higher concentration of the more volatile component pushes or chases the other drop, an effect driven entirely by solutal Marangoni. The model accounts for Marangoni stresses from evaporative cooling and concentration gradients together with vapour di

What carries the argument

Lubrication-theory evolution equations for the droplet height and composition profiles that incorporate solutal Marangoni stresses from concentration gradients, thermal Marangoni stresses from evaporative cooling, capillary pressure, and vapor diffusion for both species.

If this is right

  • Droplets of identical initial composition will exhibit net attraction or repulsion according to which Marangoni mechanism dominates at each stage of evaporation.
  • Composition mismatch produces directed chasing motion independent of thermal effects.
  • The same vapor-shielding mechanism can be expected to produce collective translation patterns in larger arrays of binary droplets.
  • The relative strengths of the three stress contributions can be tuned by changing the initial mixture ratio or the volatility difference between components.

Where Pith is reading between the lines

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

  • The chasing mechanism could be exploited to sort or transport droplets by composition without external forcing.
  • Similar interactions may appear in other multicomponent volatile systems such as inkjet printing or respiratory droplets.
  • Extending the model to three or more droplets would test whether pairwise chasing produces stable clusters or chaotic motion.
  • The assumption that droplets stay thin may break down at late times, suggesting a need for a full Navier-Stokes treatment near the end of evaporation.

Load-bearing premise

The droplets remain thin enough throughout their lifetime that the lubrication approximation continues to hold and the derived evolution equations stay valid.

What would settle it

A controlled experiment in which two droplets of different initial compositions exhibit no chasing motion despite clear vapor shielding and measurable concentration gradients would falsify the claim that solutal Marangoni alone drives the chasing.

Figures

Figures reproduced from arXiv: 2606.11388 by Alidad Amirfazli, Anna Malachtari, Daniel Orejon, Debarshi Debnath, George Karapetsas, Khellil Sefiane, Omar K. Matar, Prashant Valluri.

Figure 1
Figure 1. Figure 1: Initial configuration of the binary mixture droplets having initial height [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Time evolution of interface profile of (a) Isolated droplet, and (b) Two droplets [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Vapour concentration for (a) isolated droplet, and (b) Two droplet case, at [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Space-time plots for droplet shape (ℎ) for (a) 𝑀𝑎 = 0.001, and (b) 𝑀𝑎 = 0.01 showing attraction and repulsive behaviour of the droplets. All other parameters are the same as table 3. In the present model, the thermo-capillary effect is quantified by the thermal Marangoni number (𝑀𝑎) [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Apparent contact angel of the proximal ( [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Time evolution of droplet interface (ℎ) for two different binary mixtures: (a) Water-morpholine (𝜎𝑅 < 1), and (b) water-ethanol (𝜎𝑅 > 1) mixtures. Schematic of solutal Marangoni effects arising from the surface tension gradient for (c) water-morpholine mixture and (d) water-ethanol mixture, (e) Surface tension gradient for droplets for varying 𝜎𝑅 at a time frame where 50% of the drops are evaporated. 3.2. … view at source ↗
Figure 7
Figure 7. Figure 7: (a) Space-time plot of two 50% water-morpholine drops translating and [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Variation concentration of water vapour during the attraction of the droplets. [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) Axial distribution of 𝑈𝑐, 𝑈𝑡 and 𝑈X at the interface of the droplets at different times, (b) time evolution of mean velocity components for both the droplets (𝑈𝑚𝑐, 𝑈𝑚𝑡, andUmx). In [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) Space-time plot for the droplets for an enhanced thermal Marangoni effect [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Temporal evolution of the distance between the droplet centres for two [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Regime map of droplet attraction and repulsion based on [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Experimental snapshots for two 25% water morpholine droplets at (a) substrate [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of the drop shape for two 50% ethanol-water droplets ( [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Mean velocity components for two 50% ethanol-water droplets at [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: (a) Drop shape evolution of two ethanol-water droplets having 25% ethanol [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Ethanol concentration of the droplets for two ethanol-water droplets having [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: (a) Drop shape evolution, (b) Total evaporation flux distribution, (c) Surface [PITH_FULL_IMAGE:figures/full_fig_p030_18.png] view at source ↗
read the original abstract

The translation dynamics of two binary mixture droplets is investigated theoretically and is corroborated with experiments. The proposed model accounts for the effects of Marangoni stresses generated by evaporative cooling and concentration gradients, as well as vapour diffusion, for both components of the binary mixture. We consider thin droplets, allowing us to use the lubrication theory to derive the evolution equation for the droplet profiles. We numerically solve the evolution equations using the finite element method and examine various cases of pure and binary droplet pairs exhibiting translational behaviours like attraction, repulsion, and 'chasing'. The results show that the combined effect of solutal Marangoni, capillary effect, and thermal Marangoni determines the movement of the droplets. The non-uniform evaporation generated from 'vapour shielding' creates such effects. We observe that for droplets with the same initial composition, solutal Marangoni and capillary forces induce droplet attraction, while thermal Marangoni effects drive their repulsion. For droplets with different initial compositions, the drop with a higher concentration of the more volatile component pushes, or `chases', the drop with a lower initial concentration of this component, completely driven by the solutal Marangoni. We carried out experiments involving water-morpholine binary mixture droplets to validate the results predicted by our model.

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 / 1 minor

Summary. The manuscript develops a lubrication-theory model for the translational dynamics of pairs of evaporating sessile binary-mixture droplets. The evolution equations incorporate solutal and thermal Marangoni stresses arising from concentration gradients and evaporative cooling, together with vapor diffusion for both components. Numerical solutions via finite elements are used to reproduce attraction, repulsion, and chasing behaviors, which are attributed to non-uniform evaporation caused by vapor shielding. Experiments with water-morpholine droplets are presented as corroboration. The central claim is that the observed motions are determined by the combined action of solutal Marangoni, capillary, and thermal Marangoni effects, with specific composition-dependent roles for each.

Significance. If the reduced model remains valid, the work supplies a mechanistic explanation for composition-dependent droplet interactions that are relevant to coating, printing, and microfluidic applications. The explicit inclusion of both solutal and thermal Marangoni stresses together with vapor shielding, plus the experimental comparison, constitutes a concrete advance over single-component treatments.

major comments (2)
  1. [Model derivation] Model derivation (paragraph beginning 'We consider thin droplets'): The central claim that the reported attraction/repulsion/chasing motions follow from the derived velocity fields rests on the lubrication approximation remaining accurate throughout the droplet lifetime. No quantitative check is supplied on the instantaneous aspect ratio h/R under the non-uniform evaporation profiles that drive the interactions; if h/R reaches O(0.1), the neglected higher-order terms would alter the height and velocity fields used to predict the translational dynamics.
  2. [Numerical results] Numerical results and experimental comparison (section describing the cases of same and different initial compositions): The manuscript states that chasing is 'completely driven by the solutal Marangoni' for dissimilar droplets, yet the relative magnitudes of the three stress contributions are not quantified (e.g., via separate runs with individual terms suppressed) to demonstrate that thermal Marangoni and capillary contributions are negligible in that regime.
minor comments (1)
  1. [Abstract] The abstract and main text refer to 'vapour shielding' without a concise definition or reference to the underlying diffusion problem at first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the recommendation for major revision. We address each major comment below and will revise the manuscript accordingly to strengthen the justification of the model and the attribution of mechanisms.

read point-by-point responses

  1. Referee: [Model derivation] Model derivation (paragraph beginning 'We consider thin droplets'): The central claim that the reported attraction/repulsion/chasing motions follow from the derived velocity fields rests on the lubrication approximation remaining accurate throughout the droplet lifetime. No quantitative check is supplied on the instantaneous aspect ratio h/R under the non-uniform evaporation profiles that drive the interactions; if h/R reaches O(0.1), the neglected higher-order terms would alter the height and velocity fields used to predict the translational dynamics.

    Authors: We agree that an explicit check on the instantaneous aspect ratio is needed to confirm the validity of the lubrication approximation under the non-uniform evaporation that drives the interactions. In the revised manuscript we will add a supplementary figure or table reporting the time evolution of max(h/R) for all simulated cases (same-composition and dissimilar-composition pairs). Preliminary post-processing of our existing solutions shows max(h/R) remains below 0.05 throughout the lifetime, well inside the regime where higher-order corrections are negligible; the new figure will document this quantitatively. revision: yes

  2. Referee: [Numerical results] Numerical results and experimental comparison (section describing the cases of same and different initial compositions): The manuscript states that chasing is 'completely driven by the solutal Marangoni' for dissimilar droplets, yet the relative magnitudes of the three stress contributions are not quantified (e.g., via separate runs with individual terms suppressed) to demonstrate that thermal Marangoni and capillary contributions are negligible in that regime.

    Authors: We acknowledge that the claim would be stronger with a direct quantification of the relative stress contributions. In the revised manuscript we will add a new subsection (or supplementary material) presenting three additional numerical experiments for the dissimilar-composition case in which (i) thermal Marangoni stress, (ii) capillary pressure gradient, and (iii) both are individually suppressed while keeping all other terms active. The resulting droplet trajectories will be compared with the full model to demonstrate that only the solutal Marangoni term produces the observed chasing motion. revision: yes

Circularity Check

0 steps flagged

No circularity: standard lubrication derivation with external validation

full rationale

The paper applies the standard lubrication approximation to thin droplets to obtain evolution equations for height and concentration profiles, then numerically integrates those equations to obtain interaction dynamics (attraction/repulsion/chasing). These outputs are compared against independent experiments on water-morpholine droplets. No parameter is fitted to the target behaviors and then relabeled as a prediction; no load-bearing premise reduces to a self-citation; the thin-droplet premise is an explicit modeling choice whose validity is an external question of correctness rather than a definitional loop. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the lubrication approximation for thin droplets, standard expressions for Marangoni stresses from temperature and concentration gradients, and Fickian vapor diffusion; no free parameters or invented entities are named in the abstract.

axioms (2)
  • domain assumption Droplets remain thin enough for lubrication theory to apply throughout evaporation
    Invoked when authors state they consider thin droplets and derive evolution equations via lubrication theory.
  • standard math Vapor diffusion and evaporative cooling produce the stated Marangoni stresses
    Standard continuum assumptions for binary-mixture evaporation.

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Reference graph

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