Information-Theoretical Approach to Relaxation Time Distribution in Rheology: Log-Normal Relaxation Spectrum Model

Abstract: The relaxation modulus of a viscoelastic fluid can be decomposed into multiple Maxwell models and characterized by the relaxation spectrum for the relaxation time. It is empirically known that the logarithmic relaxation time is useful to express the relaxation spectrum. We use information geometry to analyze the relaxation modulus and shown that the logarithmic relaxation time is the most natural variable for the relaxation spectrum. Then we use information theory to estimate the most probable functional form for the relaxation spectrum. We show that the log-normal distribution is the information-theoretically most probable relaxation spectrum. We analyze the properties of the log-normal relaxation spectrum model and compare it with the fractional Maxwell model. The fractional Maxwell model with a small power-law exponent can be approximated as the log-normal relaxation spectrum model with a large standard deviation. We also compare the log-normal relaxation spectrum model with experimental linear viscoelasticity data for a high-density polyethylene, both at melt and solid states.