Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations

Abstract

Consumer products, such as foods, contain numerous polymeric and particulate additives that play critical roles in maintaining their stability, quality and function. The resulting materials exhibit complex bulk and interfacial rheological responses, and often display a distinctive power-law response under standard rheometric deformations. These power laws are not conveniently described using conventional rheological models, without the introduction of a large number of relaxation modes. We present a constitutive framework using fractional derivatives to model the power-law responses often observed experimentally. We first revisit the concept of quasiproperties and their connection to the fractional Maxwell model (FMM). Using Scott-Blair's original data, we demonstrate the ability of the FMM to capture the power-law response of 'highly anomalous' materials. We extend the FMM to describe the viscoelastic interfaces formed by bovine serum albumin and solutions of a common food stabilizer, Acacia gum. Fractional calculus allows us to model and compactly describe the measured frequency response of these interfaces in terms of their quasiproperties. Finally, we demonstrate the predictive ability of the FMM to quantitatively capture the behaviour of complex viscoelastic interfaces by combining the measured quasi-properties with the equation of motion for a complex fluid interface to describe the damped inertio-elastic oscillations that are observed experimentally.

Effect of Cross-Sectional Aspect Ratio on Flow-Induced Orientation of a Polymer Solution in Planar Channels with an Abrupt Expansion

“…The weight ratio between the particles and the solution was 1:6000. The fractional Maxwell model (FMM) was used to fit the G' and G" curves 18,19) , and we estimated the characteristic relaxation time t by using the method reported by Jaishankar and McKinely. 18,19) As a result of the fitting shown in Fig.…”

“…The fractional Maxwell model (FMM) was used to fit the G' and G" curves 18,19) , and we estimated the characteristic relaxation time t by using the method reported by Jaishankar and McKinely. 18,19) As a result of the fitting shown in Fig. 1, we found the characteristic relaxation time t of the fluid to be 53.2 s. This indicates that the test fluid has a very long characteristic time relative to the typical times of the flows in our experiments.…”

Fractional Time Fluctuations in Viscoelasticity: A Comparative Study of Correlations and Elastic Moduli

“…Non-exponential stress relaxation in the time domain also implies power law behavior in the viscoelastic storage modulus, G (ω), and in the loss modulus, G (ω), measured in the frequency domain by using small-amplitude oscillatory shear deformations. This broad spectral response is indicative of the wide range of distinctive relaxation processes available to the microstructural elements that compose the material, and there is no single relaxation time [24]. Let us assume that the viscoelasticity of the fluid is well represented by a long-range power-law rheological equation of state, i.e.,…”

On the role of standard and anomalous diffusion in suspensions of rigid rods

“…Anomalous sub-diffusion was revisited in [16] (the interested reader can also refer to the references therein), justifying the use of non-integer derivatives. In semi-concentrated suspensions the inter-particles interactions could be at the origin of those non-integer derivatives in the diffusion mechanisms.…”