Power-Law Gels, Scott-Blair and the Fractional Calculus of Soft Networks and Multiscale Soft Materials

Many soft materials including foods, consumer products, biopolymer gels & associative  polymer networks are characterized by multi-scale microstructures and exhibit power-law  responses in canonical rheological experiments such as Small Amplitude Oscillatory  Shear (SAOS) and creep. Even in the linear limit of small deformations it is difficult to  describe the material response of such systems quantitatively within the classical  framework of springs and dashpots – which give rise universally to Maxwell-Debye  exponential responses. Instead empirical measures of quantities such as ‘firmness’,  ‘tackiness’ etc. are often used to describe and compare material responses. G.W. Scott  Blair, who was present with Bingham at the very beginnings of the Society of Rheology,  argued that such measures are best thought of as ‘quasi-properties’ that capture a  snapshot of the underlying dynamical processes in these complex materials. We show  that the language of fractional calculus and the concept of a ‘spring-pot’ element together  provide a useful ontological framework that is especially well-suited for modeling and  quantifying the rheological response of power-law materials. We illustrate the general  utility of this approach by describing fractional differential forms of the Maxwell and  Kelvin-Voigt models and using these models we quantify small-amplitude oscillatory  shear responses and creep response in range of soft materials including gluten gels, skin  and soft tissue, filled polymer melts, hydrogen-bonded biopolymer networks and the  complex interfacial rheological properties of acacia gum and serum albumins. The  fractional exponents that characterize the dynamic material response can also be  connected directly with the scaling exponents from microstructural models such as the  Rouse model and the Soft Glassy Rheology (SGR) model. Having determined the quasi-  properties that quantify the linear viscoelastic material response of a power-law gel in a  concise form, we show that a fractional K-BKZ framework combining a Mittag-Leffler  relaxation kernel with a strain-damping function can be used to quantitatively describe  the nonlinear viscometric properties of such materials. Depending on the range of values  of the quasi-properties the resulting models can have some surprising features, including;  agreement with well-known heuristics such as the Cox-Merz rule and the complete  absence of a zero-shear- rate plateau in the viscosity and the first normal stress difference.  The material parameters extracted from this framework also prove especially useful for  ranking, inter-comparing and formulating complex microstructured fluid materials such  as liquid foodstuffs used in treating oral dysphagia (swallowing disorders).