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arxiv: 2606.26656 · v1 · pith:GWNTZDD2new · submitted 2026-06-25 · ❄️ cond-mat.soft

Analysing gelation transition through fractional viscoelasticity and Mittag-Leffler-Prabhakar function

Yogesh M Joshi This is my paper

Pith reviewed 2026-06-26 02:59 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords gelation transitionfractional viscoelasticityMittag-Leffler-Prabhakar functioncritical gel statescaling relationsdynamic modulicritical exponents
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The pith

Enforcing continuity of dynamic moduli across the gel point imposes symmetry in relaxation dynamics and makes hyper-scaling a theoretical necessity.

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

The paper constructs fractional viscoelastic models and three-parameter Mittag-Leffler-Prabhakar models for both pre-gel and post-gel regimes that respect conventional scaling relations. Enforcing continuity of the dynamic moduli and their time derivatives at the critical gel point creates a symmetry between the relaxation dynamics before and after the transition. This same continuity requirement directly produces the hyper-scaling relation among the critical exponents. The resulting models are shown to fit experimental time- and frequency-domain data, and a frequency-independent rheological signature of the critical state is extracted from the two exponents alone.

Core claim

By building physically constrained fractional viscoelastic models and Prabhakar-function models that stay consistent with scaling in each regime, and then imposing continuity of the dynamic moduli and their derivatives across the critical gel point, a universal symmetry appears in the relaxation dynamics on either side of the transition. This enforcement validates the hyper-scaling relation connecting the critical exponents, turning it into a required outcome of the continuity condition rather than an empirical observation.

What carries the argument

The three-parameter Mittag-Leffler-Prabhakar function, which extends fractional models by removing their parameter restrictions while preserving consistency with scaling relations on both sides of the gel point.

If this is right

  • Continuity at the critical point forces symmetric relaxation dynamics on either side of the gel state.
  • The hyper-scaling relation among critical exponents follows as a direct consequence of continuity.
  • The Prabhakar formulation removes the parameter restrictions that limit the pure fractional pre-gel model.
  • The models reproduce both time-domain and frequency-domain experimental rheology data.
  • A model-independent, frequency-independent fingerprint of the critical gel state is fixed by the two exponents.

Where Pith is reading between the lines

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

  • The same continuity requirement could be applied to other critical points in soft matter, such as glass or jamming transitions, to test whether symmetric dynamics appear there as well.
  • Measuring both exponents and checking continuity in a new gelation chemistry would provide a direct experimental test of whether hyper-scaling is always enforced.
  • The frequency-independent fingerprint offers a practical way to locate the critical gel point in real time without assuming any particular functional form.

Load-bearing premise

That fractional viscoelastic models and Prabhakar-based models can be constructed to remain consistent with conventional scaling relations in the pre-gel and post-gel regimes.

What would settle it

A gel-forming system in which the dynamic moduli and their derivatives are continuous at the critical point yet the hyper-scaling relation between the measured critical exponents fails to hold.

Figures

Figures reproduced from arXiv: 2606.26656 by Yogesh M Joshi.

Figure 1
Figure 1. Figure 1: A series assembly of a single dashpot and 𝑁 springpots. It is a proposed fractional viscoelastic arrangement to model the pre-gel states. The dominant short-time term associated with 𝐺෨(𝑞) can be obtained by taking a limit of 𝑞 → ∞, which corresponds to 𝑡 → 0ା. For a material in the pre-gel states, therefore, lim ௤→ஶ 𝐺෨(𝑞) is expected to result in Laplace transform of the Winter–Chambon , , , [PITH_FULL_I… view at source ↗
Figure 2
Figure 2. Figure 2: A parallel assembly of 𝑁 springpots and a spring, a fractional viscoelastic assembly proposed to model the postgel states. With this background, let us consider a general parallel fractional network assembly consisting of a single Hookean spring with modulus 𝐺௘(𝜀) in parallel with 𝑁 springpots as shown in [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relaxation modulus 𝐺 [Pa] for a chemically crosslinked PDMS system is plotted as a function of reduced time 𝑡/𝑎் [s] at varying distances 𝜀 from the critical gel point. The experimental data is taken from Winter and coworkers [66, 67]. The symbols represent the transformed experimental data extracted from the time-temperature 10−4 10−3 10−2 10−1 100 101 102 100 101 102 103 104 Values of e Pre-gel states 0.… view at source ↗
Figure 4
Figure 4. Figure 4: Storage modulus 𝐺′(𝜔) and loss modulus 𝐺″(𝜔) for an aqueous poly(vinyl alcohol) (PVA) solution, as it undergoes gel-sol transition, are plotted as functions of angular frequency 𝜔, at various distances (𝜀) from the critical gel point: in the post-gel states (a) 0.4, (b) 0.3, (c) 0.2, (d) 0.1, the critical gel state (e) 0 and the pre-gel states (f) 0.1, (g) 0.2, (h) 0.3 and (f) 0.4. Experimental data (symbo… view at source ↗
Figure 5
Figure 5. Figure 5: Relative change in 𝐺 ᇱᇱ with respect to 𝐺 ᇱ as the system passes through the critical gel state (∂𝐺 ᇱᇱ/ ∂𝐺 ᇱ)ఌ→଴ is plotted as a function of the critical relaxation exponent 𝑛 for representative values of the relaxation scaling exponent 𝜅. It is important to note that this expression is model-agnostic and is independent of frequency. Note that the region 𝑛 ≤ 𝜅 is physically inaccessible as discussed by Jos… view at source ↗
read the original abstract

The gelation transition, a process that transforms a flowable liquid into an elastic solid, is a present in variety of systems, from colloidal to polymeric. During the gelation transition, a system passes through a critical gel state characterized by scale-free power-law viscoelasticity. Interestingly, the fractional calculus provides a natural mathematical language for such power-law viscoelasticity. In this work, we develop physically constrained fractional viscoelastic models as well as those based on the three-parameter Mittag-Leffler-Prabhakar function for both, the pre-gel state and the post-gel regimes, ensuring consistency with the conventional scaling relations in each regime. While the fractional pre-gel model is observed to be valid only for a restricted subset of parameter values, the Prabhakar function-based model rigorously removes this limitation. We enforce continuity of the dynamic moduli and their derivatives across the critical gel point, which universally imposes a symmetry in the relaxation dynamics on either side of the critical gel state. Such enforcement further validates the hyper-scaling relation connecting the critical exponents, making it a theoretical necessity rather than an empirical coincidence. We validate the proposed models against time- and frequency-domain experimental data. A model-agnostic, frequency-independent rheological fingerprint of the critical gel state, uniquely determined by two critical exponents, is also identified.

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

3 major / 2 minor

Summary. The paper develops physically constrained fractional viscoelastic models and three-parameter Mittag-Leffler-Prabhakar models for pre-gel and post-gel regimes that are consistent with conventional scaling relations. It imposes continuity of the dynamic moduli G' and G'' and their derivatives at the critical gel point, claiming this universally enforces symmetry between relaxation dynamics on either side of the gel point and renders the hyper-scaling relation among critical exponents a theoretical necessity rather than empirical. The models are stated to be validated against time- and frequency-domain experimental data, and a model-agnostic frequency-independent rheological fingerprint of the critical gel state (determined by two exponents) is identified.

Significance. If the continuity conditions can be physically justified rather than imposed by construction and if the experimental validations include quantitative error metrics and parameter ranges, the work would offer a useful mathematical framework linking fractional calculus to gelation scaling. The removal of parameter restrictions via the Prabhakar function is a clear technical improvement over the fractional pre-gel model. However, the central claim that hyper-scaling becomes a necessity rests on the imposed matching and therefore adds limited new physical insight beyond algebraic consequences of the chosen constraints.

major comments (3)
  1. [Abstract / model construction] Abstract and model-construction section: the statement that enforcing continuity 'universally imposes' symmetry and 'validates the hyper-scaling relation... making it a theoretical necessity' is not supported by a derivation from microscopic stress balance or dissipation; the models are first built separately to obey conventional scaling in each regime and continuity is then applied as an additional constraint, so the reported symmetry and exponent relations follow algebraically from the matching conditions rather than from gelation physics.
  2. [Abstract / validation] Validation paragraph: the abstract asserts that models are 'validated against time- and frequency-domain experimental data' yet supplies no quantitative fit statistics, error bars, parameter ranges, or comparison to alternative models; without these the claim that the Prabhakar model rigorously removes the fractional restriction cannot be assessed for robustness.
  3. [Fingerprint identification] Critical-gel fingerprint claim: the model-agnostic fingerprint 'uniquely determined by two critical exponents' is presented without an explicit derivation showing independence from the specific functional form (fractional vs. Prabhakar) once continuity is imposed; this needs to be shown algebraically in the relevant section to support the universality assertion.
minor comments (2)
  1. [Abstract] The abstract contains a grammatical error ('a present in variety of systems').
  2. [Model definitions] Notation for the three parameters of the Mittag-Leffler-Prabhakar function should be introduced explicitly with their physical interpretations before use in the scaling relations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback. We agree that the abstract and model sections require clarification on the origin of the symmetry and hyper-scaling claims, and that quantitative validation metrics should be highlighted. We will revise the manuscript to address these points explicitly while preserving the core framework.

read point-by-point responses

  1. Referee: [Abstract / model construction] the statement that enforcing continuity 'universally imposes' symmetry and 'validates the hyper-scaling relation... making it a theoretical necessity' is not supported by a derivation from microscopic stress balance or dissipation; the models are first built separately to obey conventional scaling in each regime and continuity is then applied as an additional constraint, so the reported symmetry and exponent relations follow algebraically from the matching conditions rather than from gelation physics.

    Authors: We accept the referee's observation: the symmetry between pre- and post-gel relaxation and the necessity of the hyper-scaling relation are algebraic consequences of imposing continuity of G', G'' and their first derivatives on models that already satisfy the conventional scaling relations in each regime separately. The physical rationale for continuity is the requirement that the material response remain continuous across the gel point, but we do not derive this from microscopic stress balance. We will revise the abstract and model-construction section to state that the symmetry and hyper-scaling follow necessarily from the imposed continuity conditions under the scaling assumptions, rather than claiming a universal imposition from gelation physics. revision: partial

  2. Referee: [Abstract / validation] Validation paragraph: the abstract asserts that models are 'validated against time- and frequency-domain experimental data' yet supplies no quantitative fit statistics, error bars, parameter ranges, or comparison to alternative models; without these the claim that the Prabhakar model rigorously removes the fractional restriction cannot be assessed for robustness.

    Authors: The full manuscript contains quantitative least-squares errors, parameter ranges, and direct comparisons with the fractional model in Sections 4 and 5. To make this evident at the abstract level we will add a concise statement of the achieved fit quality (e.g., relative errors below 5 % over the reported frequency and time windows) and the range of the Prabhakar parameter that removes the earlier restriction. revision: yes

  3. Referee: [Fingerprint identification] Critical-gel fingerprint claim: the model-agnostic fingerprint 'uniquely determined by two critical exponents' is presented without an explicit derivation showing independence from the specific functional form (fractional vs. Prabhakar) once continuity is imposed; this needs to be shown algebraically in the relevant section to support the universality assertion.

    Authors: We will insert an explicit algebraic derivation (new subsection) showing that, once continuity of G', G'' and their derivatives is enforced, the frequency-independent fingerprint at the gel point reduces to a relation involving only the two critical exponents, independent of whether the underlying kernels are fractional or Prabhakar. This derivation uses only the scaling forms and the matching conditions, thereby confirming the model-agnostic character. revision: yes

Circularity Check

1 steps flagged

Enforcing continuity of moduli and derivatives constructs symmetry and hyper-scaling by model design rather than deriving them

specific steps
  1. self definitional [Abstract]
    "We enforce continuity of the dynamic moduli and their derivatives across the critical gel point, which universally imposes a symmetry in the relaxation dynamics on either side of the critical gel state. Such enforcement further validates the hyper-scaling relation connecting the critical exponents, making it a theoretical necessity rather than an empirical coincidence."

    The paper imposes continuity as an external modeling choice, then presents the resulting symmetry and hyper-scaling as a 'theoretical necessity' that follows from the enforcement. The hyper-scaling relation is thereby obtained directly from the imposed continuity conditions rather than from an independent physical argument or first-principles derivation.

full rationale

The paper constructs separate fractional/Prabhakar models for pre- and post-gel regimes that already obey conventional scaling, then imposes continuity of G', G'' and their derivatives at the gel point. The abstract states that this enforcement 'universally imposes a symmetry' and 'validates the hyper-scaling relation... making it a theoretical necessity'. This reduces the claimed necessity to an algebraic consequence of the imposed matching conditions. No independent derivation from stress balance or dissipation is supplied showing why first or higher derivatives must be continuous. The central claim therefore reduces by construction to the model's own continuity constraint.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard fractional calculus and conventional scaling relations from rheology literature; no new entities are introduced.

free parameters (2)
  • three parameters of Mittag-Leffler-Prabhakar function
    The function is three-parameter and parameters are adjusted to match data and scaling relations in each regime.
  • critical exponents
    Two critical exponents determine the fingerprint and are fitted or measured from data.
axioms (2)
  • domain assumption Conventional scaling relations hold separately in pre-gel and post-gel regimes
    Models are required to be consistent with these relations.
  • domain assumption Dynamic moduli and derivatives must be continuous at the critical gel point
    This is imposed to derive symmetry and hyper-scaling.

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discussion (0)

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Reference graph

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