Estimating the viscoelastic moduli of complex fluids using the generalized Stokes-Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, [Delta r(2) (t)], of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes-Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, (G) over tilde(s), in the Laplace frequency domain, the complex shear modulus, G*(omega), in the Fourier frequency domain, and the stress relaxation modulus. G(r)(t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (omega domain) of [Delta r(2)(t)] known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing (Delta r(2)(t)) as a local power law. If the logarithmic slope of [Delta r(2)(t)) can be accurately determined, these estimates generally perform well at the frequency extremes.