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arxiv: 2605.18279 · v1 · pith:UI3OWUERnew · submitted 2026-05-18 · ❄️ cond-mat.soft · cond-mat.mtrl-sci· cond-mat.stat-mech

Coalescence of Polymer Droplets Moving on a Surface with Stiffness Gradient

Divyansh Tripathi , Vimal Kishore , Panagiotis E. Theodorakis , Swarn Lata Singh This is my paper

Pith reviewed 2026-05-20 00:11 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.mtrl-scicond-mat.stat-mech
keywords polymer dropletscoalescencestiffness gradientdurotaxispower-law growthbridge heightcapillarityviscoelasticity
0
0 comments X

The pith

Droplet coalescence on stiffness gradients shows two distinct power-law regimes in bridge growth.

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

This paper examines the merging of two polymer droplets that are driven to move in the same direction by a gradient in surface stiffness. As a reference, it also looks at coalescence when the droplets are stationary on uniform stiffness surfaces. The main observation is that the height of the connecting bridge grows according to a power law in time, but the exponent changes from a higher value to a lower one after some time. This change marks the shift from a regime where surface tension (capillarity) dominates the process to one where the viscoelastic properties of the material take over. The speed difference between the two droplets and the strength of attractions within and between droplets also influence how the coalescence proceeds, with potential relevance to printing technologies and biological systems.

Core claim

The temporal growth of the bridge height (h) follows a power law (h ∼ t^α), with a transition from a higher to a lower value of α as a function of time, pointing to the presence of two distinct power-law growth regimes, where the transition signifies the crossover from the capillarity-dominated regime to the viscoelasticity-dominated regime of coalescence for durotaxis-driven droplet motion on a stiffness gradient surface.

What carries the argument

The power-law growth of bridge height h ∼ t^α and the time-dependent transition in the exponent α that indicates regime crossover.

If this is right

  • The coalescence process depends on the velocity ratio of the leading and trailing droplets.
  • Both the rate and extent of coalescence are affected by the attractive interaction strengths between and within droplets.
  • These findings apply to practical settings like microfluidics and ink-jet printing on substrates with varying stiffness.
  • Results contribute to understanding interactions among multicellular aggregates moving on biological surfaces with stiffness variations.

Where Pith is reading between the lines

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

  • The observed crossover might be tunable by changing the stiffness gradient to control merging times in applications.
  • Similar power-law transitions could appear in other gradient-driven soft matter phenomena, such as cell migration or emulsion dynamics.
  • Experimental validation with real polymer droplets on fabricated stiffness-gradient surfaces could confirm the simulation results.

Load-bearing premise

The simulation model with the selected interaction potentials and velocity ratios accurately reproduces the physical mechanisms of durotaxis-driven coalescence without dominant numerical artifacts or unphysical boundary effects.

What would settle it

Direct measurement of bridge height versus time in an experiment with polymer droplets on a stiffness-gradient surface showing no transition in the growth exponent would falsify the claim of two distinct regimes.

Figures

Figures reproduced from arXiv: 2605.18279 by Divyansh Tripathi, Panagiotis E. Theodorakis, Swarn Lata Singh, Vimal Kishore.

Figure 1
Figure 1. Figure 1: FIG. 1: System setup consisting of two polymer droplets marked as [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Time evolution of droplet coalescence on a uniform substrate for different stiffness [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Temporal evolution of bridge height for stationary coalescence on a surface with [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Temporal evolution of bridge height for the coalescence of two droplets moving on [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Time-sequenced images of coalescence of two moving droplets for [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Difference of centers of mass of the coalescing droplets (∆ [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Distance between centers of mass of two coalescing droplets on a gradient surface [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Time sequenced snapshot for coalescence of moving droplets for [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
read the original abstract

Here, we study the coalescence of two droplets that are moving in the same direction on a soft surface; the motion of the droplets is caused by a gradient in the surface stiffness. As reference, stationary coalescence of the same droplets is also studied on the corresponding uniform surfaces for different stiffness values. To describe the coalescence phenomenon on a surface with stiffness gradient, a relevant range of velocity ratios of the leading and the trailing droplet was considered to elucidate the effect of this parameter on coalescence. Moreover, to analyze the dynamics of the process, the temporal growth of the bridge height $(h)$ was investigated, which follows a power law $(h \sim t^{\alpha})$, before eventually attaining a constant value. The obtained values of $\alpha$ show a transition from a higher to a lower value as a function of time, pointing to the presence of two distinct power-law growth regimes, where the transition signifies the crossover from the capillarity-dominated regime to the viscoelasticty-dominated regime of coalescence. In addition, varying attractive strengths for droplet--droplet and intra-droplet interactions were considered. The results indicate that both the dynamics and the degree of the coalescence strongly depend on these interaction parameters. Thus, we anticipate that our results will shed more light on the durotaxis-driven coalescence of polymeric droplets for various relevant system parameters, which will have practical implications for applications ranging from microfluidics to ink-jet printing, where substrate properties may vary. In addition, results may add to the fundamental understanding of the interactions among multicellular aggregates moving on biological surfaces.

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 / 2 minor

Summary. The manuscript reports molecular dynamics simulations of two polymer droplets undergoing durotaxis-driven coalescence on a soft substrate with a stiffness gradient. As a reference, stationary coalescence is also examined on uniform-stiffness surfaces. The central observation is that the bridge height h grows temporally as a power law h ∼ t^α, with α exhibiting a transition from a higher early-time value to a lower late-time value; this transition is interpreted as the crossover from a capillarity-dominated regime to a viscoelasticity-dominated regime. The study further explores the dependence on the velocity ratio of the leading to trailing droplet and on the strengths of droplet–droplet and intra-droplet attractive interactions, reporting that both the coalescence dynamics and the final degree of merging are sensitive to these parameters.

Significance. If the reported power-law transition and its physical attribution are confirmed by scaling analysis and robustness checks, the work would add a useful parameter study to the literature on durotaxis and droplet coalescence on heterogeneous soft substrates. The explicit variation of velocity ratios and interaction strengths provides concrete, falsifiable trends that could guide experiments in microfluidics or ink-jet printing on gradient surfaces.

major comments (2)
  1. [Results section on temporal growth of bridge height] The central claim that the observed decrease in α with time marks the capillarity-to-viscoelasticity crossover is load-bearing for the interpretation. No scaling relation is shown between the measured crossover time and the ratio of capillary time (ηR/γ) to the viscoelastic relaxation time, nor is the late-time α compared quantitatively to the expected exponent for viscoelastic coalescence on a gradient substrate. Without these checks the attribution remains observational.
  2. [Methods and Results on power-law fitting] The simulation employs specific interaction potentials and an imposed velocity ratio. It is not demonstrated that the extracted α values and the transition are insensitive to the precise definition of bridge height h, to finite-size effects, or to the details of how the power-law fits are performed (e.g., fitting windows, error estimation). These controls are necessary to rule out that the transition is an artifact of the model or analysis choices.
minor comments (2)
  1. [Abstract] Abstract contains a typographical error: 'viscoelasticty' should read 'viscoelasticity'.
  2. [Methods] The manuscript would benefit from a brief statement of how the stiffness gradient is implemented numerically and from a table or figure summarizing the range of velocity ratios and interaction strengths explored.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the significance of our work. We address each major comment below and outline the revisions that will be incorporated to strengthen the scaling support and robustness analysis.

read point-by-point responses

  1. Referee: [Results section on temporal growth of bridge height] The central claim that the observed decrease in α with time marks the capillarity-to-viscoelasticity crossover is load-bearing for the interpretation. No scaling relation is shown between the measured crossover time and the ratio of capillary time (ηR/γ) to the viscoelastic relaxation time, nor is the late-time α compared quantitatively to the expected exponent for viscoelastic coalescence on a gradient substrate. Without these checks the attribution remains observational.

    Authors: We agree that a more explicit connection to the relevant time scales would strengthen the interpretation. In the revised manuscript we will add estimates of the capillary time ηR/γ using the effective viscosity and surface tension extracted from the simulations, and we will compare these estimates directly to the observed crossover time. For the late-time regime we will discuss the measured α in the context of known viscoelastic coalescence exponents, noting that the stiffness gradient and imposed velocity ratio are expected to yield a modified effective exponent relative to uniform-substrate cases. These additions will be placed in the Results section together with a brief scaling argument. revision: yes

  2. Referee: [Methods and Results on power-law fitting] The simulation employs specific interaction potentials and an imposed velocity ratio. It is not demonstrated that the extracted α values and the transition are insensitive to the precise definition of bridge height h, to finite-size effects, or to the details of how the power-law fits are performed (e.g., fitting windows, error estimation). These controls are necessary to rule out that the transition is an artifact of the model or analysis choices.

    Authors: We acknowledge the need for explicit robustness checks. In the revised manuscript and supplementary material we will (i) test alternative operational definitions of bridge height h (different density cut-offs and geometric neck measures) and demonstrate that the two-stage power-law behavior and the transition time remain consistent; (ii) report results from additional runs with larger lateral system sizes that confirm the transition is preserved; and (iii) describe the fitting protocol in the Methods, including the time windows employed, the criterion for selecting the early- and late-time regimes, and the uncertainty estimation obtained from multiple independent trajectories. These controls will be presented to show that the reported transition is not an artifact of the chosen analysis procedure. revision: yes

Circularity Check

0 steps flagged

Simulation observations of power-law regimes are self-contained with no circular reduction

full rationale

The paper reports results from molecular simulations of droplet coalescence driven by stiffness gradients, including direct measurement of bridge height h(t) and extraction of power-law exponents α in early and late regimes. These are outputs from varying input parameters (velocity ratios, interaction strengths) rather than any analytical derivation or fitted model that presupposes the capillarity-to-viscoelasticity crossover. No equations or claims reduce by construction to the simulation inputs, and the central interpretation follows from the observed trajectories without load-bearing self-citations or ansatzes imported from prior author work. The study is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central observations rest on simulation setup choices and standard assumptions about droplet physics rather than new derivations.

free parameters (2)
  • velocity ratio of leading to trailing droplet
    A relevant range was considered to study its effect on coalescence dynamics.
  • attractive strengths for droplet-droplet and intra-droplet interactions
    Varied to determine dependence of coalescence dynamics and degree.
axioms (2)
  • domain assumption Droplet motion is induced by the surface stiffness gradient
    Core setup of the moving-droplet case.
  • domain assumption Bridge height growth obeys a power-law form before saturating
    Basis for identifying the two regimes and crossover.

pith-pipeline@v0.9.0 · 5840 in / 1292 out tokens · 59015 ms · 2026-05-20T00:11:00.480160+00:00 · methodology

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Lean theorems connected to this paper

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    the temporal growth of the bridge height (h) was investigated, which follows a power law (h∼t^α), before eventually attaining a constant value. The obtained values of α show a transition from a higher to a lower value as a function of time, pointing to the presence of two distinct power-law growth regimes, where the transition signifies the crossover from the capillarity-dominated regime to the viscoelasticty-dominated regime of coalescence.

  • IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    All non-bonded interactions ... modeled using the truncated and shifted Lennard-Jones (LJ) potential ... FENE potential ... harmonic interaction potential U_harmonic(r) = −½Kr²

What do these tags mean?
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contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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