The effects of hydrodynamic interaction and inertia in determining the high-frequency dynamic modulus of a viscoelastic fluid with two-point passive microrheology

Abstract

In two-point passive microrheology, a modification of the original one-point technique, introduced by Crocker et al. [Phys. Rev. Lett. 85, 888 (2000)]10.1103/PhysRevLett.85.888, the cross-correlations of two micron-sized beads embedded in a viscoelastic fluid are used to estimate the dynamic modulus of a material. The two-point technique allows for the sampling of larger length scales, which means that it can be used in materials with a coarser microstructure. An optimal separation between the beads exists at which the desired length and time scales are sampled while keeping a desired signal-to-noise-ratio in the cross-correlations. A large separation can reduce the effect of higher order reflections, but will increase the effects of medium inertia and reduce the signal-to-noise-ratio. The modeling formalisms commonly used to relate two-bead cross-correlations to G*(ω) neglect inertia effects and underestimate the effect of reflections. A simple dimensional analysis for a model viscoelastic fluid suggests that there exists a very narrow window of bead separation and frequency range where these effects can be neglected. Therefore, we consider both generalized data analysis and generalized Brownian dynamics (BD) simulations to examine the magnitude of these effects. Our proposed analysis relies on the recent analytic results of Ardekani and Rangel [Phys. Fluids 18, 103306 (2006)]10.1063/1.2363351 for a purely viscous fluid, which are generalized to linear viscoelastic fluids. Implementation requires approximations to estimate Laplace transforms efficiently. These approximations are then used to create generalized BD simulation algorithms. The data analysis formalism presented in this work can expand the region of separation between the beads and frequencies at which rheological properties can be accurately measured using two-point passive microrheology. Moreover, the additional physics introduced in the data analysis formalisms do not add additional significant computational costs.