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REVIEW 3 major objections 5 minor 35 references

On the effect of coalescence on the rheology of emulsions

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Coalescence, not just droplet deformation, is what makes emulsion viscosity rise to a peak near 20% volume fraction and then fall.

desk verdict A solid VoF study showing coalescence flips the curvature of the emulsion viscosity curve, but the artificial repulsive force that defines the 'no-coalescence' branch is not a fully clean control. read the letter →

arxiv 1908.08383 v1 pith:5GCVSP5M submitted 2019-08-22 physics.flu-dyn cond-mat.soft

Francesco De Vita , Marco Edoardo Rosti , Sergio Caserta , Luca Brandt This is my paper · ORCID
classification physics.flu-dyncond-mat.soft
keywords emulsionrheologycoalescenceeffectiveviscosityinterfacialtensionvolumeoffluidmethodcollisionforcenormalstressdifferencesshearflow
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

The paper tries to establish that coalescence of droplets is the mechanism behind an observed anomaly in emulsion rheology: the effective viscosity increases with disperse-phase concentration up to about 20% and then decreases, producing a negatively curved viscosity curve. Using interface-resolving simulations of shear flow, the authors compare emulsions in which droplets merge freely with identical emulsions in which an Eulerian repulsive force prevents merging. With merging suppressed, the viscosity curve becomes positively curved and always lies above the matrix viscosity, matching the behaviour of suspensions of deformable particles. The paper attributes the difference to a geometric-stress mechanism: coalescence reduces the total interfacial area, and the interfacial-tension contribution to the bulk shear stress shrinks with that area. If the claim is right, any prediction of emulsion rheology that neglects merging will be wrong outside the dilute regime.

What carries the argument

The load-bearing device is an Eulerian repulsive force of lubrication form, Fc = μ0 r U (a/ψ + b/ψ²) n, written in terms of the signed distance ψ from the interface; it is applied only where two different droplet indices are detected within a stencil, and it models the effect of surfactants by preventing film drainage and merging. The force is hand-tuned (a = 55, b = 3.5) so that coalescence is fully suppressed with minimal side effects. The second piece is the stress-budget identity that equates the wall shear stress with the bulk average of viscous, interfacial-tension, and collision-force stresses; combined with the time history of the total interface area, this identity is what lets the authors attribute the viscosity change to a reduction of interfacial area. A third element is the flow-topology parameter Q = (D² − Ω²)/(D² + Ω²), used to show that the matrix flow is nearly pure shear when droplets merge and gains extensional regions when merging is suppressed.

What would settle it

A decisive check is to repeat the volume-fraction series with a second, force-free way of suppressing coalescence—for example an interface model with insoluble surfactant transport and strong Marangoni resistance—and compare the effective-viscosity curve and stress budget. If the positive curvature and the roughly 10% collision-force stress contribution do not reproduce, the reported curvature flip is an artifact of the repulsive model. A more local test: in the two-droplet collision of figure 4, the forced case differs by about 6% in final vertical displacement from the double-resolution unforced case; refining the grid should reduce this difference if the force is inert, or expose that the force itself alters the collision outcome.

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Extended reading notes

Core claim

The central claim is that droplet coalescence causes the negative curvature of the effective-viscosity versus volume-fraction curve in emulsions. In the simulations, free merging produces a maximum of the effective viscosity near φ ≈ 0.2; for viscosity ratio λ = 0.01 the normalized viscosity even falls below 1 at high volume fraction. Applying the repulsive collision force (3.1) with coefficients a = 55 and b = 3.5 prevents merging altogether, and the same system then shows a monotonic, positively curved viscosity curve that is well described by the Eilers formula once droplet deformation is folded into an effective volume fraction. The paper traces this to the stress budget: with coalescence the total interface area drops by up to 80% of its initial value, lowering the interfacial-tension contribution to the shear stress; without coalescence the interface area stays nearly constant, and the interface-tension term accounts for about half of the effective viscosity, with the collision force itself contributing about 10%.

Load-bearing premise

The load-bearing premise is that the hand-tuned Eulerian repulsive force with coefficients a = 55 and b = 3.5 prevents coalescence completely without otherwise changing the droplet dynamics, interface deformation, or the stress budget; the paper itself reports that this force contributes about 10% of the total stress, so a perturbing effect large enough to alter the comparison cannot be excluded from the reported evidence.

Editorial extensions

If this is right

  • Emulsion constitutive curves must be reported with coalescence efficiency controlled; two emulsions with identical composition, drop size, and shear rate can have qualitatively different viscosity curves depending on whether merging is allowed.
  • Analytical viscosity formulas that assume positive curvature, such as Pal's equation and the Eilers formula, should only be applied to non-coalescing emulsions; the paper shows they fail when coalescence is active.
  • Suppressing coalescence makes an emulsion behave like a suspension of deformable particles: the viscosity data collapse onto the Eilers formula using an effective volume fraction based on droplet deformation, with about 6% error at the highest concentration.
  • The ratio of first to second normal stress difference stays roughly constant with volume fraction, and the first normal stress difference changes sign at the lowest viscosity ratio—an experimentally checkable rheological fingerprint.
  • Flow topology changes with coalescence: merging droplets leave the matrix in nearly pure shear, whereas non-coalescing emulsions develop extensional flow in the gaps between droplets.

Reading between the lines

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

  • If the mechanism is generic, coalescence efficiency should be treated as a rheological state variable as important as volume fraction and capillary number; any processing change that alters drop-size distribution—surfactant dose, pre-shear history, compatibilizer—will slide the system along the family of curves between the two limits computed here.
  • The reported 10% stress contribution of the artificial repulsive force is a directly testable artifact check: a second, force-free way to inhibit coalescence (for instance an interface with strong Marangoni resistance) should reproduce the positive curvature and the same stress-budget decomposition.
  • The area-reduction explanation predicts a quantitative scaling: the drop in effective viscosity should track the loss of total interfacial area, so simultaneous measurement of drop size and shear stress during coalescence would give a direct test of the mechanism.
  • For viscosity ratios below one, coalescence can make an emulsion less viscous than its continuous phase; engineering controlled coalescence could therefore be a purely microstructural route to viscosity reduction without changing chemistry.
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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

3 major / 5 minor

Summary. De Vita et al. present volume-of-fluid simulations of shear-driven emulsions in a Couette cell at vanishing Reynolds number, scanning volume fraction φ=0.00164–0.3, viscosity ratio λ=0.01–1, and capillary number Ca=0.05–0.2. Two branches are compared: one in which droplets coalesce naturally, and one in which an Eulerian repulsive force (Eq. 3.1) suppresses coalescence, intended to mimic surfactant-stabilized emulsions. The main finding is that the effective viscosity versus φ curve has negative curvature when coalescence is allowed, with a maximum near φ≈0.2 and values below unity for λ<1, and positive curvature when coalescence is suppressed, resembling suspensions of deformable particles. The authors attribute the difference to the reduction of interfacial area upon merging, which lowers the interfacial-tension contribution to the shear stress. Normal stress differences, flow-topology statistics, and droplet size distributions are also reported.

Significance. If the central comparison were clean, the paper would be a valuable contribution: it gives a mechanistic explanation of the experimentally observed negative curvature (Caserta & Guido 2012) and demonstrates that neglecting coalescence in emulsion simulations changes the qualitative rheology. Strengths include the careful stress-budget derivation (Eqs. 4.2–4.8), grid-resolution checks, and benchmarks against Einstein, Taylor, and the Pal model in the dilute regime. The coalescing branch is compared to experiments with reasonable qualitative agreement. The principal caveat is that the non-coalescing control is not a passive variation: the collision force contributes directly to the stress balance and alters the wall-normal droplet distribution, so the strength of the causal claim (that coalescence causes the curvature reversal) depends on the inertness of that force model.

major comments (3)
  1. [§4.1, Eqs. (4.1), (4.8), Fig. 8] The effective viscosity is computed from the wall shear stress via Eq. (4.1), while the bulk stress balance in Eq. (4.8) includes the collision-force integral C. Because the collision force is a numerical body force whose streamwise component is about 10% of the total shear stress at φ=0.3, the reported µ_e for the non-coalescing branch is not strictly a material property of the emulsion. The authors should provide a convergence study in the force strength (e.g., increasing a and b beyond 55 and 3.5) to demonstrate that the suspension viscosity is independent of the force amplitude as long as coalescence is prevented. Without this, the positive curvature of the non-coalescing curve could be partly an artifact of the model force.
  2. [§4.1, Fig. 9(right)] The wall-normal profile of the average volume fraction is substantially modified by the collision force, with a marked increase of <φ> near the walls. Since µ_e is evaluated from wall quantities, this force-induced segregation directly contributes to the measured viscosity increase, independently of the interfacial-area mechanism emphasized in the paper. The statement in §4 that the dispersed phase is 'approximately homogeneous' is difficult to reconcile with the changes in near-wall <φ> shown in Fig. 9(right). The authors should quantify the wall-region contribution to µ_e (e.g., by computing µ_e from the bulk stress instead and comparing) or demonstrate that the results are insensitive to the redistribution.
  3. [§3, Fig. 4] The calibration of the collision force coefficients a and b (Eq. 3.1) is done by trial and error on a single two-drop collision, and the only reported sensitivity test compares two force amplitudes on the post-collision trajectory. This does not establish that the force is dynamically inert in the dense many-drop suspensions: the force acts in a shell of thickness 3Δ around every interface, can alter the local flow in the gaps, and changes the deformation of droplets. A sensitivity test of the suspension rheology and of the stress budget to a and b in a dense case (e.g., φ=0.3) is needed to support the claim that the difference between the two branches is due solely to the presence or absence of coalescence.
minor comments (5)
  1. [Eq. (2.3)] The advection equation for the VoF function appears to contain a typo: ∂(u_j H)/∂x_j should likely be ∂(u_j T)/∂x_j, or the right-hand side should involve H ∂u_j/∂x_j, since H is the color function and T is the cell average. Please check the notation.
  2. [Fig. 3 caption] The rows in the caption are mislabeled: the third row is described twice as '(d)-(e)-(f)', and the fourth row should be '(g)-(h)-(i)' or '(j)-(k)-(l)' accordingly.
  3. [Author affiliations] The affiliation contains a typo: 'Stockhom' should be 'Stockholm'.
  4. [Fig. 14 caption] The black diamonds representing the experiments of Caserta & Guido are not identified in the caption; please state explicitly what they denote.
  5. [§4.2, Fig. 13] The statement that the ratio of N1 and N2 is 'almost constant' with volume fraction is supported by the figure, but for λ=1 the scatter is non-negligible; consider providing a quantitative measure such as a best-fit slope with confidence intervals.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the rheology is simulated, not fitted; the non-coalescing control uses a tuned collision force, but the viscosity result is emergent.

full rationale

The paper's central claim—that coalescence produces negative curvature in the effective-viscosity-versus-volume-fraction curve while suppressing coalescence produces positive curvature—is obtained from direct numerical simulation, not from a fitted or self-referential construction. The coalescing branch is validated against external benchmarks: the Einstein/Taylor dilute limits, Pal's formula in the dilute regime, and the Caserta & Guido experiments. The non-coalescing branch is generated by adding the Eulerian repulsive force of Eq. (3.1), with coefficients a=55 and b=3.5 chosen by trial and error only to prevent merging; the viscosity curve itself is not a fit target, so the curvature sign is an emergent simulation outcome. The stress budget in Eq. (4.8) explicitly includes the collision-force contribution C, and the paper reports it is about 10% of the total stress; this is a possible physical confound in comparing the two branches, but it is not circularity because the measured viscosity is not defined in terms of the force amplitude and the force was not tuned to reproduce any rheological curve. The self-citations (Rosti et al. 2019 for the VoF code, Rosti & Brandt 2018 and Rosti et al. 2018 for the deformable-particle Eilers comparison) are used as methodological references and external comparisons, not as premises that entail the coalescence result. No equation is defined in terms of the claimed finding, no fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. The paper is therefore self-contained against external benchmarks and its central derivation chain is not circular, though the non-coalescing control deserves scrutiny as a modeling artifact rather than a circularity issue.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the accuracy of the VoF solver and on the collision force model being a faithful proxy for surfactant-stabilized interfaces. The latter is an ad hoc numerical construct with tuned coefficients, while the former is validated in prior work (Rosti et al. 2019).

free parameters (1)
  • collision force coefficients a, b = a=55, b=3.5
    Chosen by trial and error in Section 3 to always prevent coalescence for the given non-dimensional parameters. They affect the stress budget; the collision force contributes about 10% of the total stress.
assumptions (5)
  • standard math Incompressible Navier-Stokes equations accurately describe the two-fluid flow.
    Used as governing equations in Section 2.1, one-fluid formulation.
  • domain assumption Reynolds number Re=0.1 is sufficiently small for inertia to be negligible.
    Stated in Section 2.1; no systematic check of dependence on Re is reported.
  • domain assumption The VoF/MTHINC method with the given grid accurately tracks the interface and resolves the stresses.
    Grid independence verified for effective viscosity and surface area by half/double resolution, but no formal convergence study.
  • ad hoc to paper The Eulerian repulsive force (equation 3.1) models the effect of surfactants and prevents coalescence without otherwise altering the flow physics.
    Introduced in Section 3; coefficients are tuned by trial and error, not derived from a physical surfactant model.
  • domain assumption The simulation domain (16x16x10 r) is large enough to represent the constitutive behavior of the emulsion, i.e. finite-size and wall effects are negligible for the reported viscosities.
    No domain-size dependence study is performed; the paper acknowledges that vorticity banding observed experimentally requires larger domains.
invented entities (1)
  • Eulerian collision force Fc
    purpose: To prevent droplet coalescence in the VoF framework, modeling the effect of surfactants.
    A numerical device introduced in equation (3.1) with tuned coefficients; validated only within this paper via two-droplet tests, not against physical surfactant experiments.

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Pith. "Pith review of On the effect of coalescence on the rheology of emulsions." pith.science (2026). https://pith.science/paper/5GCVSP5M

@misc{pith2026190808383,
  author       = {Pith},
  title        = {Pith review of: On the effect of coalescence on the rheology of emulsions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5GCVSP5M}},
  note         = {Machine review of arXiv:1908.08383}
}
read the original abstract

We present a numerical study of the rheology of a two-fluid emulsion in dilute and semidilute conditions. The analysis is performed for different capillary numbers, volume fraction and viscosity ratio under the assumption of negligible inertia and zero buoyancy force. The effective viscosity of the system increases for low values of the volume fraction and decreases for higher values, with a maximum for about 20 % concentration of the disperse phase. When the dispersed fluid has lower viscosity, the normalised effective viscosity becomes smaller than 1 for high enough volume fractions. To single out the effect of droplet coalescence on the rheology of the emulsion we introduce an Eulerian force which prevents merging, effectively modelling the presence of surfactants in the system. When the coalescence is inhibited the effective viscosity is always greater than 1 and the curvature of the function representing the emulsion effective viscosity vs. the volume fraction becomes positive, resembling the behaviour of suspensions of deformable particles. The reduction of the effective viscosity in the presence of coalescence is associated to the reduction of the total surface of the disperse phase when the droplets merge, which leads to a reduction of the interface tension contribution to the total shear stress. The probability density function of the flow topology parameter shows that the flow is mostly a shear flow in the matrix phase, with regions of extensional flow when the coalescence is prohibited. The flow in the disperse phase, instead, always shows rotational components. The first normal stress difference is positive whereas the second normal difference is negative, with their ratio being constant with the volume fraction. Our results clearly show that the coalescence efficiency strongly affects the system rheology and neglecting droplet merging can lead to erroneous predictions.

Figures

Figures reproduced from arXiv: 1908.08383 by the authors.

Figure 1
Figure 1. Sketch of the computational domain and coordinate syst [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Sketch of the collision model: the gray area represents th [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Time evolution of the interface at three different time insta [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Plot of relative distance between droplets [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: (Left panel) Stress distribution along the wall normal dire [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Effective viscosity µe as function of the volume fraction: λ = 1.0 and different Ca (left panel); Ca = 0.1 and different λ (right panel). The effective viscosity is normalized with the outer fluid viscosity. (2012). The rheological curve exhibits a negative curvature, …
Figure 7
Figure 7. Figure 7: Effective viscosity µe for the cases with collision force and Ca = 0.1. The line denoted as ”deformable particles” corresponds to the fit proposed in Rosti & Brandt (2018) for initially spherical viscoelastic particles with same capillary number based on the material s…
Figure 8
Figure 8. Figure 8: (Left) Wall-normal distribution (z) of the stresses for the case with Ca = 0.1, φ = 0.3, λ = 1.0 and collision force: disperse phase viscous stress (); carrier fluid viscous stress (◦); interfacial force (△); collision force (⋄). The stresses are normalized with the wa…
Figure 9
Figure 9. Figure 9: (Left panel) Time history of the total surface for differe [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: Instantaneous droplet distribution for emulsions with vo [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: Cumulative volume distribution as a function of the equivale [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: PDF of the flow topology parameter for simulations withou [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: Normal stress differences for the cases with repulsive fo [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: Effective viscosity versus volume fraction. (Left panel) [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: Effective viscosity as a function of the volume fraction [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]

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