Annual European Rheology Conference (AERC 2022)

Compact and Accurate Descriptions of Complex Fluids and Soft Solids using Fractional Calculus

Many soft materials including foods, consumer products & biopolymer gels are characterized by multi-scale microstructures and exhibit power-law responses in canonical experiments such as SAOS and creep. It is difficult to describe the material response of such systems quantitatively within the classical framework of springs & dashpots – which give rise to Maxwell-Debye exponential responses. Instead, empirical functions and subjective metrics such as ‘firmness’, ‘thickness’ etc. are often used to describe and compare material responses. I will show that the language of fractional calculus and the concept of a ‘spring-pot’ element provides a useful framework that is especially well-suited for modeling and quantifying the rheological response of power-law–like viscoelastic materials. We illustrate the general utility of this approach using fractional differential forms of the Maxwell, Kelvin-Voigt and Zener models. We then use these models to quantify viscoelastic responses of a range of soft materials including gluten and milk protein gels, cheese, polymer melts, hydrogen-bonded networks, colloidal pastes, as well as complex interfaces. The fractional exponents that characterize the dynamic material response can also be connected directly with scaling exponents from microstructural models such as the SGR model and the fractal dimensions of the underlying microstructure. This fractional framework can also be extended to the nonlinear domain by evaluating the corresponding continuous spectrum and integrating the fractional kernel into familiar Wagner/K-BKZ formulations with a generalized damping function. Using this integral fractional framework and carefully considering the asymptotic limits of small & large deformation rates we develop general expressions for the rate-dependent viscosity of a fractional viscoelastic fluid. This analysis also provides a systematic understanding of why familiar rheological heuristics such as the Cox-Merz rule and the Gleissle mirror relation often, but not always, work so well!