Analyzing the Gelation Transition through Fractional Viscoelasticity and the Mittag-Leffler-Prabhakar Function
Abstract
The gelation transition, a process that transforms a flowable liquid into an elastic solid, occurs in a variety of systems, ranging 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 and 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 for the relaxation modulus, which translates into a Havriliak-Negami function-based complex modulus, 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 hyperscaling 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.
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BibTeX
@misc{joshi2026analyzing,
title = {{Analyzing the Gelation Transition through Fractional Viscoelasticity and the Mittag-Leffler-Prabhakar Function}},
author = {Yogesh M Joshi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.26656},
eprint = {2606.26656},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.26656},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Analyzing the Gelation Transition through Fractional Viscoelasticity and the Mittag-Leffler-Prabhakar Function
AU - Yogesh M Joshi
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2606.26656
UR - https://arxiv.org/abs/2606.26656
AB - The gelation transition, a process that transforms a flowable liquid into an elastic solid, occurs in a variety of systems, ranging 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 and 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 for the relaxation modulus, which translates into a Havriliak-Negami function-based complex modulus, 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 hyperscaling 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.
ER -