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REVIEW 5 major objections 5 minor 61 references

Self-sorting of bidisperse particles in evaporating sessile droplets

T0 review · 5 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read In an evaporating ethanol droplet with a moving contact line, thermocapillary Marangoni flow sorts bidisperse particles by size: small particles gather at the apex while large particles surround them.

desk verdict A credible flow-side study that corrects the CCA-mode scaling picture, alongside a plausible but under-validated particle-sorting prediction that needs a serious referee. read the letter →

arxiv 2504.16597 v1 pith:5KK3YL77 submitted 2025-04-23 physics.flu-dyn

Aman Kumar Jain , Fabian Denner , Berend van Wachem This is my paper · ORCID
classification physics.flu-dyn
keywords sessiledropletevaporationMarangoniflowparticleself-sortingbidisperseparticlesmovingcontactlineVOF-DEMsimulationconstantanglemodecore-shelldeposition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that thermocapillary Marangoni flow is the mechanism that sorts particles by size inside an evaporating ethanol droplet whose contact line is free to move. Using two-stage simulations that first compute the gas-liquid flow, heat, and vapor fields, then track 1.5 and 2.5 micrometer particles with a Lagrangian method, the authors find that without Marangoni stress the flow driven by the receding contact line dominates and sweeps both particle sizes to the droplet apex, so no size separation appears. With Marangoni stress, an interface-parallel vortex forms and drives adsorbed particles apart enough that small particles preferentially fill the gaps and collect at the apex, while large particles assemble around them. The pattern is reported for initial contact angles of 60, 90, and 120 degrees and is unaffected by whether the particles are denser than the fluid or neutrally buoyant. The practical upshot is that the internal flow becomes a controllable handle for arranging colloids in drying droplets, which matters for printing, coating, and particle deposition.

What carries the argument

The load-bearing object is the temperature-dependent surface tension and its tangential gradient, $\nabla_s\sigma = \sigma_T[\nabla T - (\nabla T\cdot \mathbf{n})\mathbf{n}]$, which produces the Marangoni stress that drives the interface-parallel vortex. In the particle stage, the decisive interactions are the capillary force pinning a particle to the gas-liquid interface and the simplified Cheerios pair force between two adsorbed particles (Eqs. 23 and 24); these keep the particles at the interface where the tangential Marangoni flow can act on them, and the pair attraction is what large particles rely on to surround the small-particle core. The three-way flow competition among receding-contact-line flow, capillary flow, and Marangoni flow is the organizing framework that explains when sorting does and does not occur.

What would settle it

Measure the final deposit and in-drop flow of a 60-degree contact-angle ethanol droplet containing 1.5 and 2.5 micrometer particles on a 50-degree-Celsius substrate evaporating in constant-contact-angle mode: the claim predicts a small-particle core at the apex ringed by large particles whenever the thermocapillary gradient is active, and predicts that suppressing the Marangoni vortex, for instance with a surfactant or a slip-promoting coating, removes the sorting entirely.

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Extended reading notes

Core claim

The paper's central discovery is that in constant-contact-angle evaporation with a moving contact line, the direction of the internal flow is decided by a three-way competition: evaporation-driven capillary flow toward the contact line, the receding contact line dragging fluid toward the apex, and thermocapillary Marangoni flow along the interface. For the ethanol droplets studied, the receding-contact-line flow beats the capillary flow at every contact angle considered, so particles are carried from the bulk to the apex in a size-neutral way when Marangoni stress is absent. When Marangoni stress is present, it creates a single vortex whose near-interface branch moves adsorbed particles along the surface; this tangential motion lets small particles slip into gaps and build a compact core at the apex stagnation point, with large particles arranged around that core. The authors identify this Marangoni-driven flow, not wettability or particle density, as the mechanism responsible for the observed self-sorting.

Load-bearing premise

The size-sorting result rests on the simplified surface forces that pin particles to the interface and pull pairs of adsorbed particles together, combined with the no-slip contact-line condition; if real droplets have even modest slip at the contact line, the Marangoni vortex that does the sorting may not form.

Editorial extensions

If this is right

  • In constant-contact-angle evaporation without Marangoni stress, size-based particle separation should not occur; suppressing thermocapillary convection is enough to suppress sorting.
  • Substrate heating and surface wettability can be used as knobs: turning the thermocapillary vortex on or off should switch between a mixed apex deposit and a small-core/large-shell pattern.
  • Particle density is not a controlling parameter for interfacial sorting; the same shell pattern should appear for heavy and neutrally buoyant particles.
  • The sorting should persist across contact angles from 60 to 120 degrees as long as the Marangoni flow dominates the receding-contact-line and capillary flows.
  • Because Marangoni mixing accelerates evaporation in shallow droplets but slows it in tall ones, the time window for sorting changes with contact angle and heating.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Not claimed in the paper: if the vortex topology is the only requirement, any system generating the same interface-parallel circulation, whether from surfactants, solutal Marangoni stress, or external forcing, should produce the same small-core/large-shell pattern in other solvents.
  • Not claimed in the paper: the simplified Cheerios force lumps interface-deformation details into a few parameters, so the sharpness of the core-shell boundary should be sensitive to the particle's contact angle at the interface; a testable prediction is that changing particle hydrophobicity changes how cleanly small and large particles separate.
  • Not claimed in the paper: the mechanism suggests a miniature size-classification device, a drying droplet whose Marangoni strength sets the cutoff radius between particles that go to the core and particles that stay in the shell; the cutoff should scale with the ratio of capillary attraction to Marangoni drag.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper presents a two-stage numerical study of bidisperse particle dispersion in an evaporating ethanol sessile droplet on a heated substrate, evaporating in constant-contact-angle (CCA) mode. Stage 1 solves the gas-liquid flow with a VOF method, including heat and vapor transport and thermocapillary Marangoni stress; stage 2 tracks 607 particles (radii 1.5 and 2.5 μm) with a DEM/Lagrangian model including drag, DMT adhesion, a capillary adsorption force, and a simplified Cheerios force. Six first-stage cases vary the contact angle (60°, 90°, 120°) and the presence/absence of Marangoni stress, with silica and neutrally buoyant particles in stage 2. The central claims are: (i) in the absence of Marangoni flow, the receding-contact-line flow dominates and carries both particle sizes toward the apex without size sorting; (ii) with Marangoni flow, a single vortex creates an interface-parallel flow near the apex that produces size-based self-sorting, with small particles forming a core at the apex and large particles surrounding them.

Significance. If the sorting result holds, the paper would support a useful and falsifiable design rule: in CCA-mode evaporation with Marangoni stresses, the thermocapillary flow is the control knob for producing a small-particle-core/large-particle-shell ordered deposit. Strengths of the manuscript: the stage-1 flow solver is validated against an analytical d²-law volume evolution and against the Zhu and Shi experiments at three mesh resolutions; the mesh-independence checks are done for both evaporative volume evolution and the Marangoni-affected velocity/temperature fields; the code and data are openly available with a DOI; and the particle model is not fitted to the sorting outcome, so the sorting claim is a genuine prediction rather than a curve fit. The major weakness is that the sorting prediction rests entirely on the stage-2 particle-interface force model and particle-contact model, neither of which is validated, resolved, or subjected to sensitivity analysis.

major comments (5)
  1. [Sections 6.2–6.3 and Eqs. (23)–(24)] The central sorting claim (small-particle core, large-particle shell) is produced entirely by the stage-2 force model, yet stage 2 has no validation, no mesh or time-step convergence study, and no sensitivity analysis. The mesh selected in Section 4 resolves the initial contact radius with 24 cells, i.e., approximately 21 μm per cell, while the particle radii are 1.5 and 2.5 μm; the interface position entering r_l in Eq. (23) is therefore known to no better than about one cell, which is several particle diameters. Because the adsorption criterion r_l < R and the capillary/Cheerios forces are sub-grid quantities, the predicted sorting is fragile and needs at least a resolved-interface or sub-grid-interface assessment and a parameter sensitivity study before the Section 7 conclusion can be supported.
  2. [Section 2.2.1, Eq. (23)] The amplitude f of the capillary adsorption force is never specified, although this force is the mechanism that keeps particles at the interface and thereby enables the sorting pattern. Since the entire small-core/large-shell outcome depends on particles being adsorbed, the value of f (and its relation to the surface free energy of the liquid-particle interface) must be stated and varied.
  3. [Section 2.2.1, Eq. (14)] Equation (14) defines K_n = 4R*/(3E*), which has units of m³/N, but it is used in Eq. (13) as though the normal force were K_n δ_n^{3/2}, which requires units of N/m^{3/2}. This dimensional inconsistency is a verification concern in exactly the contact-force equations that determine the packing of the sorted core. The formula should be the Hertzian coefficient K_n = (4/3) E* sqrt(R*) (or the paper should otherwise clarify its normalization).
  4. [Appendix B] Appendix B shows that imposing a 20–50 μm slip length suppresses the Marangoni vortex that is identified (Section 6.3) as the sorting mechanism, and the authors justify the no-slip choice by appealing to experimental evidence that the apex is the coldest region. However, the VOF method itself carries an implicit numerical slip of order one mesh cell (about 21 μm), as the authors note in Section 3.2, so the effective slip of the simulation is not actually zero. Given that the sorting flow is suppressed for slip lengths of 20–50 μm, the paper should demonstrate that the sorting result is robust to the effective numerical slip, or provide a quantitative argument that the implicit slip is below the threshold at which the vortex is suppressed.
  5. [Section 2.2.1, Eq. (22) and Table 2] The DMT adhesion force uses an uncalibrated work of adhesion Δγ, and the Hertz contact model uses a Young's modulus of 1 MPa (Table 2), about five orders of magnitude below the bulk value for silica. The small-particle core and large-particle shell pattern is described in Section 6.3 as arising from packing in gaps between particles, so the pattern depends precisely on these contact and adhesion parameters. The paper needs a sensitivity analysis with respect to E*, Δγ, and f to establish that the sorting pattern is not an artifact of these choices.
minor comments (5)
  1. [Section 6.1, Figures 10–12] The surface-concentration curves of the individual cases are reported without error bars, repeated realizations, or a discussion of statistical sampling fluctuations; since only 607 particles are simulated, the concentration peaks may not be statistically robust and a repetition with different random initial placements should be reported or at least discussed.
  2. [Table 2] The table lists 'Volume fraction of particles 0.5' while the text in Section 3.1 states that the volume fraction associated with each particle size is 0.1%; this is confusing and should be reconciled (e.g., 0.5% total vs. 0.1% per size).
  3. [Table 3] The caption of Table 3 states that cases C1–C6 have 'variation in substrate temperature,' but the substrate temperature is fixed at 50 °C for all cases; the caption should instead refer to variation in contact angle and Marangoni stresses.
  4. [Appendix B, Figure 16] The sub-figure label in the caption of Figure 16 repeats '(d)' for the 50 μm slip case and the figure text 'drops' should read 'droplet'; these should be corrected.
  5. [Section 2.2.1, Eq. (20)] The drag coefficient correlation in Eq. (20) uses the criterion Re < 1000 versus Re > 1000, but the second branch is typically used for Re > 1000; the 1000 value is a conventional crossover and should be cited or justified, and the use of the Wen–Yu correlation in a dilute one-way-coupled setting should be briefly justified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the small-core/large-shell sorting is an emergent simulation output, not a prescribed or fitted input.

full rationale

The claimed derivation chain is: stage-1 VOF simulation of an evaporating CCA ethanol droplet produces flow and temperature fields; stage-2 Lagrangian DEM simulation then evolves bidisperse particles under drag, contact, adhesion, capillary, and Cheerios forces; the small-core/large-shell surface concentration pattern emerges from that coupled dynamics. None of the target quantities, such as the radial ordering of particle sizes, the agglomeration of small particles at the apex, or the surrounding shell of large particles, is prescribed, fitted, or used to calibrate the model. The capillary force, Cheerios force, DMT adhesion, and Hertz contact are parameterized from material and interface constants given in Tables 1 and 2 and from cited prior work, not from the sorting outcome. The stage-1 flow model is validated externally against the analytical volume evolution and against the Zhu-Shi experiments, with a mesh-independence study in Appendix A; the Appendix B choice of a no-slip contact-line condition is likewise justified by comparison with external experimental temperature observations. Reuse of the authors' previous framework is methodological and not load-bearing for the new claim. Concerns such as the dimensionally inconsistent stiffness expression in Eq. 14, the simplified and unvalidated interface-force closure, the coarse 21-micron mesh relative to the 3-5 micron particles, and the sensitivity to imposed slip lengths are real correctness and robustness risks, but they do not make the sorting result equivalent to its inputs by construction. There is no fitted parameter renamed as a prediction and no self-citation chain that forces the central conclusion.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. Its central claim rests on a chain of modeling assumptions: dilute one-way coupling, diffusion-dominated vapor, spherical interface, no-slip contact line, and simplified interface forces. The most fragile are the no-slip choice and the interface-force models, since the sorting mechanism appears only when the Marangoni vortex is present and particle interactions are invoked to explain size preference.

free parameters (4)
  • Particle Young's modulus = 1.0 MPa
    Table 2 lists 1.0 MPa for silica particles, about five orders of magnitude below real silica (about 70 GPa). This controls Hertzian contact stiffness and is not justified in the text; it affects particle-particle interactions invoked in the sorting mechanism.
  • Slip length at contact line = 0 (no-slip)
    Chosen in Section 3.2 after Appendix B shows 20-50 um slip lengths suppress the Marangoni vortex and raise apex temperature. The no-slip choice relies on the VOF method's implicit mesh-scale slip; the sorting result depends on the vortex being present.
  • Surface free energy f in capillary force (Eq. 23) = not specified
    Inherited from prior work [30]; controls adsorption force that pins particles at the interface. No value is given in this paper.
  • Cheerios force parameters beta0 and alpha_c (Eq. 24) = not specified
    Parameters for inter-particle capillary attraction are taken from simplified Vassileva model without numerical values, yet this force is part of the sorting mechanism.
assumptions (5)
  • domain assumption Bond number Bo = 0.049-0.1475 << 1; droplet maintains spherical cap and gravity is neglected (Section 2).
    Standard for millimeter droplets; also used to justify regular re-initialization of the interface.
  • domain assumption Vapor transport in gas is diffusion-dominated (Pe_v < 1) while heat convection in the liquid is significant (Pe_h > 10) (Section 2).
    Justifies neglecting advection in vapor concentration equation and solving Laplace-like vapor field in a reduced domain.
  • domain assumption One-way coupling: particles do not affect the flow because total particle volume fraction is below 1 percent (Section 2).
    Allows two-stage simulation; standard for dilute suspensions.
  • ad hoc to paper No-slip boundary condition at the substrate is appropriate despite the Huh-Scriven paradox because the VOF method provides an implicit slip (Section 3.2, Appendix B).
    The paper itself shows explicit slip lengths of 20-50 um change the flow topology (vortex suppressed). The choice of no-slip is reasoned from qualitative agreement with prior experiments on apex cooling, not from a contact-line model.
  • domain assumption The DMT adhesion and simplified capillary/Cheerios force expressions describe the behavior of 3-5 um particles at the liquid interface (Section 2.2.1).
    These are literature models, but their validity at this scale with the given parameters is not validated against experiments.

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Pith. "Pith review of Self-sorting of bidisperse particles in evaporating sessile droplets." pith.science (2026). https://pith.science/paper/5KK3YL77

@misc{pith2026250416597,
  author       = {Pith},
  title        = {Pith review of: Self-sorting of bidisperse particles in evaporating sessile droplets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5KK3YL77}},
  note         = {Machine review of arXiv:2504.16597}
}
read the original abstract

This study investigates the dispersion and self-sorting dynamics of bidisperse particles, i.e., a mixture of two distinct particle sizes, during the evaporation of ethanol droplets on a heated substrate, focusing on the influence of surface wettability, Marangoni stresses, and relative particle density. To this end, numerical simulations are carried out using a two-stage numerical approach: the first stage simulates the gas-liquid flow along with the heat and vapor distribution, while the second stage models the particle behavior using Lagrangian particle tracking. The results reveal that for an ethanol droplet evaporating with a constant contact angle in the absence of thermocapillary Marangoni stresses, the flow induced by the receding motion of the contact line supersedes the capillary flow, moving the fluid from the contact line to the apex of the droplet. This flow moves the particles from the bulk of the droplet to the apex of the droplet and suppresses size-based self-sorting of the particles. However, in the presence of Marangoni stresses, a flow along the interface near the apex of the droplet promotes the self-sorting of particles based on their size, whereby smaller particles concentrate near the droplet apex and larger particles form an outer shell around them.

Figures

Figures reproduced from arXiv: 2504.16597 by the authors.

Figure 1
Figure 1. A cross-section of the simulation geometry, showing the mesh refinement near the droplet and the [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Validation and mesh sensitivity study for an evaporating droplet with moving contact line, comparing [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Validation and mesh sensitivity study for the n-hexane evaporating droplet, comparing the numerical [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: The simulated normalized droplet volume (a) and the contact radius of the droplet (b) as a function of time for all 6 cases. 5. Results of stage 1: Flow inside the droplet In the first stage of this study, the focus is on analyzing the evolution of the contact radius a…
Figure 5
Figure 5. Figure 5: Velocity vectors and temperature plots at four different time instances of the cases with a contact [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Velocity vectors and temperature plots at four different time instances of the cases with contact angle [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Velocity vectors and temperature plots at four different time instances of the cases with contact angle [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Temperature distribution along the interface for different time instances. [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Evolution of the average temperature of the droplet for all the 6 cases. Time is normalized by the [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Evolution of surface concentration of particles adsorbed at the interface (dashed lines, right [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Evolution of surface concentration of particles adsorbed at the interface (dashed lines, right [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Evolution of surface concentration of particles adsorbed at the interface (dashed lines, right [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Dispersion of particles in an evaporating ethanol droplet with contact angle [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: Volume evolution for sessile droplet in case C4 with Marangoni flow and contact angle [PITH_FULL_IMAGE:figures/full_fig_p026_14.png]
Figure 15
Figure 15. Figure 15: Velocity and temperature plots for the cases including the Marangoni stresses, C4, where [PITH_FULL_IMAGE:figures/full_fig_p027_15.png]
Figure 16
Figure 16. Figure 16: Velocity and temperature plots for the cases including the Marangoni stresses, C2, where [PITH_FULL_IMAGE:figures/full_fig_p028_16.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.