Analysing gelation transition through fractional viscoelasticity and 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 relaxation modulus, which translates into Havriliak-Negami function based complex modulus, rigorously removes this limitation while simultaneously providing profound molecular significance. 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.