A study of the rheological properties of visco-elastic materials using fractional calculus

We present a study of the rheological properties of some visco-elastic materials of biological origin and an aqueous gel of the synthetic clay Laponite. The bacterial biofilms are (a) Staphylococcus epidermidis, (b) Pseudomonas aeruginosa, and (c) Bacillus subtilis. We model these materials using variations of a three element visco-elastic model and show that the use of fractional calculus incorporating non-integer time derivatives in the visco-elastic equations, provides the most appropriate framework for such a study. Using a single set of parameters, the complex visco-elastic modulus, creep compliance and modulus of complex viscosity, have been calculated for the systems under study and compared with available experimental results. The Laponite gel is modeled by the fractional Boltzmann model, while the biofilms have been modeled by a fractional 3-element fluid model. Replacing the non-integer order of the derivative in the equations by an integer completely reverses the theoretically predicted rheological behaviour from the experimental result. We conclude that incorporation of fractional time derivatives in the linear viscoelastic equations is an essential technique for modeling the rheological properties of these materials over a wide range of time and frequency scales. (C) 2016 Elsevier B.V. All rights reserved.