Elimination of inertia from a Generalized Langevin Equation: Applications to microbead rheology modeling and data analysis

Abstract

SynopsisViscoelastic properties of condensed soft matter can be estimated by following the trajectory of an embedded micron-sized particle in a method called passive microbead rheology. Data analysis of passive microbead rheology is usually based on formulas that relate bead displacement statistics to the dynamic modulus of the material in the frequency-domain. Therefore, methods of analysis require conversion of the data to the frequency-domain using numerical Fourier transform routines. These methods are known to introduce errors associated with frequency discretization and finite window size. Time-domain data analysis methods based on a single bead trajectory were introduced by Fricks et al. [SIAM J. Appl. Math. 69, 1277–1308 (2009)] as an alternative to the frequency-domain formulas. We have expanded these ideas with the aim of performing Monte Carlo simulations on synthetic data to evaluate and compare analysis algorithms for systems in which particles are trapped in linear or nonlinear traps. Brownian dynamics simulations were used to generate trajectories of beads embedded in viscoelastic materials having a discrete relaxation spectrum, with multiple relaxation times. We show that by including a small purely dissipative element in the memory function of the generalized Langevin equation (GLE), we can eliminate inertia-related fast variables directly from the GLE to find an inertia-less GLE, avoiding the singularity reported by McKinley et al. [J. Rheol. 53, 1487–1506 (2009)]. Using the inertia-less GLE, the computational cost of the simulations is reduced by nearly 5 orders of magnitude. We also show that, in real systems, this purely dissipative element can arise from fluid inertia, since for the bead the Basset force acts dissipative at high frequencies.