🔬 condensed matter

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

This paper introduces physically constrained fractional viscoelastic models, particularly those based on the Mittag-Leffler-Prabhakar function, to rigorously describe the gelation transition and its critical gel state, thereby ensuring continuity across regimes, validating hyper-scaling relations as theoretical necessities, and identifying a universal rheological fingerprint for the critical point.

Original authors: Yogesh M Joshi

Published 2026-06-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yogesh M Joshi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a pot of soup. At first, it's a runny liquid you can stir easily. But as you keep cooking it, something magical happens: it suddenly thickens into a solid gel, like Jell-O. This transformation is called gelation.

Scientists have long known that right at the exact moment this liquid turns solid (the "critical gel state"), the material behaves in a very strange, special way. It doesn't act like a normal liquid or a normal solid; it acts like a "fractal" object, where the rules of physics look the same whether you zoom in or zoom out. This behavior follows a specific mathematical pattern called a "power law."

This paper, written by Yogesh Joshi, is about creating a new, better mathematical map to describe exactly how this transformation happens, both before the gel forms and after it solidifies.

Here is the story of the paper, broken down into simple concepts:

1. The Old Map vs. The New Map

For a long time, scientists used a tool called fractional calculus (a branch of math that deals with "fractional" steps, not just whole numbers) to describe this gel point. Think of this tool as a "Spring-Pot."

A Spring snaps back instantly (like a rubber band).
A Dashpot flows slowly (like honey).
A Spring-Pot is a hybrid that does something in between, perfectly describing that weird "critical gel" moment.

However, the old maps had a problem. They could describe the liquid state before the gel, or the solid state after the gel, but they treated them as two separate, unrelated stories. They didn't explain how the material smoothly transitions from one to the other, or why the rules on the "liquid side" and the "solid side" are actually mirror images of each other.

2. The "Seamless Bridge" Idea

The author's main idea is that the transition must be continuous. Imagine driving a car from a bumpy dirt road (the liquid) onto a smooth highway (the solid). You don't just teleport; you drive over a bridge. The ride must be smooth, and the steering wheel shouldn't jerk suddenly.

The paper argues that for the math to be physically real, the "steering wheel" (the rate at which the material's stiffness changes) must be smooth and continuous right at the moment the gel forms. When the author enforced this "smooth bridge" rule, something amazing happened:

It forced the math on the liquid side to perfectly match the math on the solid side.
It proved that a specific relationship between the material's properties (called "hyperscaling") isn't just a lucky coincidence found in experiments; it is a mathematical necessity. If the transition is smooth, this relationship must exist.

3. The New Tool: The "Super-Function"

To build this perfect bridge, the author introduced a new mathematical tool called the Mittag-Leffler-Prabhakar function.

The Old Tool (Two-parameter): This was like a basic calculator. It worked well for some gels but broke down for others, especially those that were very "liquid-like" or very "solid-like." It was like trying to fit a square peg in a round hole.
The New Tool (Three-parameter): The author added a "third knob" (a parameter called $\gamma$ $γ$ ) to the function. This is like upgrading from a basic calculator to a super-computer.
- Why it matters: This extra knob allows the model to describe any type of gel, no matter how strange its behavior. It removes the limitations of the old models.
- The Surprise: The author discovered that this "third knob" isn't just a random number to make the math fit. It actually corresponds directly to how fast the liquid gets thick (viscosity) as it approaches the gel point. It turns a mathematical trick into a real, measurable physical property.

4. Testing the Map

The author didn't just write equations; they tested them against real-world data from two very different materials:

PDMS (Silicone): A chemical gel that hardens permanently (like baking a cake). Once it's set, you can't turn it back to liquid.
PVA (Polyvinyl Alcohol): A physical gel that is reversible (like Jell-O). You can melt it back to liquid by heating it and turn it back to gel by cooling it.

The Results:

The new models fit the experimental data perfectly for both materials.
Crucially, the models were forced to be "continuous" at the gel point. The fact that they worked so well on both sides of the transition proved that the "smooth bridge" theory is correct.
The models confirmed that the "mirror image" symmetry (the liquid side and solid side evolving in the same way) is a universal rule of nature for gels.

5. The "Fingerprint" of the Gel

One of the coolest findings is a "fingerprint" for the critical gel state.

At the exact moment of gelation, the ratio of how the material loses energy (loss modulus) to how it stores energy (storage modulus) changes in a very specific way as you approach that point.
The paper found a simple formula for this change that depends only on two numbers (the critical exponents $n$ and $\kappa$ ).
This means that no matter what kind of gel you are looking at (chemical or physical), if you measure this specific change, you can identify the critical state purely based on these two numbers. It's like a universal ID card for gels.

Summary

In simple terms, this paper says:
"Scientists have been using a slightly broken map to understand how liquids turn into gels. We have built a new, upgraded map using a 'super-function' that connects the liquid and solid states seamlessly. By forcing the transition to be smooth, we proved that the rules governing gels are universal and mathematically locked together. We tested this on real gels, and it works perfectly, giving us a new, simple way to identify the exact moment a liquid becomes a solid."

Technical Summary: Analysing Gelation Transition through Fractional Viscoelasticity and Mittag-Leffler-Prabhakar Function

Problem Statement
The gelation transition, a process transforming a viscoelastic liquid into a viscoelastic solid (or vice versa), is characterized by a critical gel state exhibiting scale-free power-law viscoelasticity. While fractional calculus, specifically the spring-pot element, provides a natural mathematical language for this power-law behavior, existing literature has significant limitations. Previous fractional viscoelastic models have predominantly been applied either strictly at the critical gel point or independently to the pre-gel (sol) and post-gel (gel) regimes. Crucially, no existing framework explicitly enforces the continuity of dynamic moduli and their first derivatives with respect to the degree of crosslinking ( $p$ ) as the system passes through the critical point. This omission prevents the rigorous derivation of the symmetry condition ( $\kappa_S = \kappa_G = \kappa$ ) and the hyperscaling relation ( $n = z/(z+s)$ ) as theoretical necessities rather than empirical observations. Furthermore, standard fractional mechanical assemblies often fail to satisfy physical constraints (such as monotonicity of relaxation) across the full range of admissible critical exponents.

Methodology
The authors develop a unified theoretical framework based on fractional viscoelasticity and the three-parameter Mittag-Leffler-Prabhakar function to model the evolution of linear viscoelastic properties across the gelation transition. The methodology involves:

Constraint Formulation: The models are constructed to satisfy intrinsic thermodynamic constraints (causality, non-negative moduli, monotonicity) and specific scaling relations found in critical gelation theory. Most importantly, the framework enforces the continuity of the complex modulus $G^*(\omega)$ and its first derivative with respect to $p$ at the critical gel point ( $p_c$ ).
Pre-Gel Modeling:
- Model 1: A series assembly of two spring-pots and a dashpot. This yields a relaxation modulus expressed via the two-parameter Mittag-Leffler function. However, this model is restricted to systems where $n + \kappa < 1$ and is only asymptotically accurate near the critical point ( $\varepsilon \to 0$ ).
- Model 2: A generalization using the three-parameter Mittag-Leffler-Prabhakar function. This model removes the $n + \kappa < 1$ restriction, allowing for a physically admissible description for all $n, \kappa \in (0,1)$ . It uniquely determines the Prabhakar exponent $\gamma_S$ to be equal to the viscosity divergence exponent $s$ .
Post-Gel Modeling:
- Model 1: A parallel assembly of a Hookean spring (providing equilibrium modulus) and an array of spring-pots.
- Model 2: A truncated series expansion of the Prabhakar function. This is shown to be algebraically equivalent to Model 1 under specific coefficient substitutions but offers a more constrained, physically structured parameter set.
Continuity Enforcement: The pre-gel and post-gel models are solved simultaneously. The requirement that the derivatives of the dynamic moduli match at $p_c$ forces the relaxation scaling exponents on both sides to be identical ( $\kappa_S = \kappa_G = \kappa$ ) and the leading-order departure coefficients to be equal.
Experimental Validation: The proposed models are validated against two distinct experimental datasets:
- Time-domain relaxation modulus data for chemically crosslinked Poly(dimethylsiloxane) (PDMS) undergoing sol-gel transition ( $n=0.45$ ).
- Frequency-domain dynamic moduli data for thermoreversible Poly(vinyl alcohol) (PVA) solution undergoing gel-sol transition ( $n=0.77$ ).

Key Contributions and Results

Unified Framework: The paper presents the first fractional viscoelastic framework that rigorously links pre-gel and post-gel regimes through continuity constraints, treating the transition as a single continuous process rather than disjoint states.
Theoretical Necessity of Hyperscaling: By enforcing the continuity of the first derivatives of the dynamic moduli, the authors demonstrate that the symmetry condition ( $\kappa_S = \kappa_G$ ) and the hyperscaling relation ( $n = z/(z+s)$ ) are not empirical coincidences but theoretical necessities required for physical consistency.
Prabhakar Function Superiority: The Prabhakar-based Pre-Gel Model 2 is shown to be superior to the standard spring-pot/dashpot assembly. It captures the divergence of zero-shear viscosity ( $\eta_0 \sim \varepsilon^{-s}$ ) intrinsically through the parameter $\gamma_S = s$ , without requiring a dashpot element that vanishes in the asymptotic limit. This suggests the Prabhakar function may be a more natural representation of pre-gel relaxation dynamics than mechanical analogs.
Model-Agnostic Fingerprint: The analysis identifies a frequency-independent, model-agnostic rheological fingerprint for the critical gel state: the ratio of the derivatives of the loss and storage moduli with respect to the control parameter, $(\partial G'' / \partial G')_{\varepsilon \to 0} = \tan((n-\kappa)\pi/2)$ . This quantity is uniquely determined by the critical exponents $n$ and $\kappa$ .
Experimental Agreement: The models successfully reproduce experimental data for both chemically crosslinked (irreversible) and physically associating (reversible) systems. The fits are excellent near the critical point, with deviations increasing as the system moves further into the sol or gel regimes, consistent with the asymptotic nature of the derived expressions.

Significance
The paper claims that its primary significance lies in resolving the disconnect between fractional viscoelastic modeling and the fundamental scaling laws of gelation. By integrating the continuity of derivatives as a strict physical constraint, the work elevates the hyperscaling relations from empirical observations to theoretical imperatives. The identification of the Prabhakar exponent $\gamma$ as a measurable physical quantity (equal to the viscosity exponent $s$ ) provides a new link between mathematical fitting parameters and the underlying physics of cluster size distribution and relaxation spectra. Finally, the identification of a model-agnostic, frequency-independent fingerprint $(\partial G'' / \partial G')$ offers a robust, universal criterion for characterizing the critical gel state, independent of the specific chemical nature or pathway of the gelation process.