Accounting for inertia effects to access the high-frequency microrheology of viscoelastic fluids

We study the Brownian motion of microbeads immersed in water and in a viscoelastic wormlike micelles solution by optical trapping interferometry and diffusing wave spectroscopy. Through the mean-square displacement obtained from both techniques, we deduce the mechanical properties of the fluids at high frequencies by explicitly accounting for inertia effects of the particle and the surrounding fluid at short time scales. For wormlike micelle solutions, we recover the 3/4 scaling exponent for the loss modulus over two decades in frequency as predicted by the theory for semiflexible polymers.

Figure 1

One-dimensional particle MSD in water and in a 4% micellar solution for two different bead sizes $a$. Experimental data: OTI () and DWS (). Corrected MSDs are shown as triangles: from OTI () and from DWS (). Theoretical predictions for water: Chandrasekhar's result ignoring inertia effects () and Hinch's prediction including the influence of inertia ().

Figure 2

Comparison of the microrheology results using OTI and DWS (inset) for one bead size $a=0.94\mu \mathrm{m}$ in a viscoelastic micelle solution with and without taking into account the effects of inertia. Data without inertia correction (), corrected using Eq. (2) (), and using the corrected MSDs shown in Fig. 1 ().

Figure 3

High-frequency loss modulus $G^{\prime \prime }\left(\omega \right)-\omega {\eta }_s$ compared to theoretical predictions. Inertia-corrected data using Eq. (2) () and using the corrected MSDs shown in Fig. 1 (). (a) OTI, $a=0.94\mu \mathrm{m}$. (b) DWS, $a=0.94\mu \mathrm{m}$. (c) OTI, $a=1.47\mu \mathrm{m}$. (d) DWS, $a=1.47\mu \mathrm{m}$. Black line () is $G_{\text{GMK}}^{\prime \prime }\left(\omega \right)-\omega {\eta }_s$ evaluated using Eq. (3) and data from Table 1.