Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation
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We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr2(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˜(s), in the Laplace frequency domain, the complex shear modulus, G*(ω), 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 (ω domain) of <Δr2(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 <Δr2(t)> as a local power law. If the logarithmic slope of <Δr2(t)> can be accurately determined, these estimates generally perform well at the frequency extremes.
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