Pith. Sign in

REVIEW

Droplet Fusion as a Relaxation Process: Comparison with Shape Recovery of Newtonian and Viscoelastic Droplets

PaperPDF from arXiv

data files

ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Mohammad Moein Naderi, Zhangli Peng, Huan-Xiang Zhou

Fluid Dynamics · Submitted 24 Jun 2026 · 44 references

Pith's verdict: The paper makes a plausible case that fusion and shape recovery differ in their stages and Deborah-number response, but the axisymmetric Oldroyd-B runs need explicit checks at the neck before the distinction can be treated as solid.

T0 · AI review

How this review was made: An AI referee (grok-4.3) read the paper on June 26, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

  1. T0 AI review An AI read the paper and answered a fixed list of questions.
  2. T1 Audit Automatic checks that the references and files the paper relies on exist and match.
  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 2 minor · Ask about this paper · Read the full review →
Challenge this review Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID2606.26272 (version 1)
Pith IDEIROENMT, Pith's permanent record of this paper and its review
CitationHuan-Xiang Zhou, Mohammad Moein Naderi, Zhangli Peng. "Droplet Fusion as a Relaxation Process: Comparison with Shape Recovery of Newtonian and Viscoelastic Droplets." arXiv:2606.26272 (2026). https://pith.science/paper/EIROENMT
Cite this reviewBibTeX for Pith's review of this paper
Claim paper

Why Pith says this

Highlight any text to comment on it.

Referee report

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Droplet fusion follows a multistage process with localized neck formation while shape recovery is global exponential relaxation, even when viscoelasticity is present.

The reading

The paper builds a unified theory and simulation framework to compare two capillary-driven processes in droplets: shape recovery after deformation and fusion of two droplets. It establishes that shape recovery occurs through global relaxation of one connected interface and decays exponentially or multi-exponentially depending on the ratio of viscocapillary time to stress relaxation time. Droplet fusion, by contrast, proceeds in distinct stages of curvature-driven neck formation, rapid bridge growth, and eventual global relaxation. Viscoelasticity adds the Deborah number as a governing timescale that modifies the intermediate fusion stage and overall relaxation compared with Newtonian droplets, while an exterior fluid adds extra dissipation that slows fusion. These differences indicate that purely viscous models are insufficient for interpreting viscoelastic biomolecular condensates.

What carries the argument

The multistage localized-to-global sequence in droplet fusion versus single-interface global relaxation in shape recovery, quantified through axisymmetric finite-element solutions of the Oldroyd-B constitutive model and compared via the Deborah number.

Extended reading notes

Core claim

Shape recovery is governed by global viscocapillary relaxation of a single connected interface and follows single- or multi-exponential decay depending on the relative magnitude of the viscocapillary timescale and the stress relaxation time. In contrast, droplet fusion is intrinsically a multistage process involving localized curvature-driven neck formation, rapid bridge expansion, and a transition to global relaxation. Viscoelasticity introduces an additional intrinsic timescale that governs the competition between capillary driving and stress relaxation, characterized by the Deborah number. This leads to enhanced intermediate-stage fusion dynamics and modified relaxation behavior compared

Load-bearing premise

The axisymmetric finite-element simulations with the Oldroyd-B model correctly reproduce the dynamics of both Newtonian and viscoelastic droplets without significant numerical or modeling inaccuracies.

Editorial extensions

If this is right

  • Viscoelastic droplets exhibit faster intermediate-stage bridge expansion than Newtonian droplets because stress relaxation competes with capillary driving.
  • An exterior fluid increases hydrodynamic dissipation and thereby lengthens the overall fusion time for both Newtonian and viscoelastic cases.
  • Empirical stretched-exponential fits to fusion data deviate systematically once the Deborah number becomes order-one, indicating the need for constitutive models that include stress relaxation.
  • Biomolecular condensate fusion times cannot be mapped directly onto shape-recovery timescales without accounting for the multistage character of fusion.

Reading between the lines

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

  • Measurements that report only a single fusion time for condensates may be averaging over regimes that are actually controlled by different mechanisms.
  • Varying the external viscosity independently of the droplet relaxation time could isolate the hydrodynamic slowing effect predicted by the simulations.
  • The Deborah-number dependence suggests that fusion rates in cells could be tuned by changes in protein relaxation times without altering surface tension.
Original abstract

Biomolecular condensates formed by phase separation often exhibit viscoelastic behavior, yet their shape recovery and fusion dynamics are frequently interpreted using purely viscous models. Here, we develop a unified theoretical and computational framework to quantify how viscoelasticity governs these two processes. We combine analytical theory for small-deformation shape recovery with axisymmetric finite-element simulations based on the Oldroyd-B constitutive model to systematically investigate both shape recovery and droplet fusion under comparable conditions. Our results show that, although both processes are driven by capillary forces, they are fundamentally distinct in their underlying physics. Shape recovery is governed by global viscocapillary relaxation of a single connected interface and follows single- or multi-exponential decay depending on the relative magnitude of the viscocapillary timescale and the stress relaxation time. In contrast, droplet fusion is intrinsically a multistage process involving localized curvature-driven neck formation, rapid bridge expansion, and a transition to global relaxation. We demonstrate that viscoelasticity introduces an additional intrinsic timescale that governs the competition between capillary driving and stress relaxation, characterized by the Deborah number. This leads to enhanced intermediate-stage fusion dynamics and modified relaxation behavior compared to Newtonian droplets. Furthermore, we show that the presence of an exterior fluid introduces additional hydrodynamic dissipation, significantly slowing the fusion process. Finally, we compare the computationally predicted droplet fusion in the Newtonian and viscoelastic cases with a stretched-exponential empirical formula. Deviations observed in viscoelastic regimes highlight the limitations of purely viscous descriptions and the need for models incorporating stress relaxation.

Figures
Figure 1. Schematic comparison of droplet shape recovery and droplet fusion. a) shape recovery (relaxation) of an initially
Figure 2. Numerical model schematics and mesh configuration for a) shape recovery, and b) fusion in an inviscid exterior
Figure 3. Mesh independence study showing the e!ect of the number of elements on the droplet shape recovery. 0 1 2 3 4 5 6 0.001 0.01 0.1 1 0 1 2 3 4 0.001 0.01 0.1 a) b) 1 inviscid exterior t (s) t (s)
Figure 4. E!ects of exterior domain. (a) E!ect of domain confinement ratio (𝑕) on the accuracy of the exterior domain model. (b) Validation of the cases with finite viscosity exterior domain. For the droplet fusion and exterior domain models, a di!erent validation strategy was employed. Due to the higher complexity of these cases, refining the mesh simply by increasing the number of elements often led to convergence issues. Therefore, to ensure convergence with the highest possible accuracy, various combinations of mesh density and domain size (for exterior domain models) were examined. The…
Figure 5. Comparison of shape recovery (a) and droplet fusion (b) dynamics for Newtonian droplets. Contours show the
Figure 6. Comparison of shape recovery (a) and droplet fusion (b) dynamics for a viscoelastic droplet. Contours show the
Figure 7. Comparison between the simulation results and the analytical solution for the shape recovery of a droplet with initial
Figure 8. Comparison between shape recovery and fusion dynamics. (a) Time histories of normalized vertical dimension.(b)
Figure 9. Comparison of the shape recovery and fusion dynamics of two droplets in a inviscid exterior domain and in an exterior
Figure 10. Comparison of the computationally predicted relaxation of droplet fusion with the empirical formula. The dependence

Figures reproduced from arXiv:2606.26272 by the authors.

References

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.