Short-Time Inertial Response of Viscoelastic Fluids: Observation of Vortex Propagation

We probe the response of viscous and viscoelastic fluids on micrometer and microsecond length and time scales using two optically trapped beads. In this way we resolve the flow field, which exhibits clear effects of fluid inertia. Specifically, we resolve the short-time vortex flow and the corresponding evolution of this vortex, which propagates diffusively for simple liquids. For viscoelastic fluids, this propagation is shown to be faster than diffusive and the displacement correlations reflect the frequency-dependent shear modulus of the medium.

Figure 1

Normalized responses of two particles, $\parallel$ (filled) and $\perp$ (open) in water/glycerol [$r=8.3\mu \mathrm{m}$ (blue, online); $r=5.7\mu \mathrm{m}$ (red, online)] compared to water ($r=8.3\mu \mathrm{m}$) in gray and to the normalized single-particle response function (horizontal line). For the same viscosity the decay shifts to lower frequency for larger $r$ (blue, online). Inset: sketch of the experiment: two particles with separation $r$ in two laser traps ($\lambda =830\mathrm{nm}$, 1064 nm). Displacements in the plane normal to the laser [parallel ($x$) and perpendicular ($y$)] to the line of centers are detected using laser interferometry.

Figure 2

Normalized responses, collapsed by plotting vs scaled separation distance $r/{\delta }_{\nu }$ for $r=11.7\mu \mathrm{m}$ (circles, triangles), $8\mu \mathrm{m}$ (square, stars), in water (open symbols, $\eta =0.969\mathrm{m}\mathrm{Pa}s$) and water/glycerol (filled symbols, $\eta =6.9\mathrm{m}\mathrm{Pa}\mathrm{s}$). Normalized data fall onto the two master curves of Eq. (2) (dark and gray lines). Data shown for $\omega >200\mathrm{rad}/\mathrm{s}$, for $\omega <2k\mathrm{rad}/\mathrm{s}$ one in every 5 data points.

Figure 3

Normalized responses $4\pi r|G|{\alpha }_{\parallel }^{\prime \prime }$ (filled) and $8\pi r|G|{\alpha }_{\perp }^{\prime \prime }$ (open) vs scaled distances $r/{\delta }_{\nu e}$ for solutions of wormlike micelles, (a) [$C_m=0.5\mathrm{wt}\%$, $r=8.3\mu \mathrm{m}$ (circles); 1 wt %, $r=10\mu \mathrm{m}$ (squares) and 2 wt %, $r=10\mu \mathrm{m}$ (triangles)]. (b) Entangled actin solutions with [$C=0.5\mathrm{mg}/\mathrm{ml}$, $r=13.6\mu \mathrm{m}$ (circles); $1\mathrm{mg}/\mathrm{ml}$, $r=16.2\mu \mathrm{m}$ (squares)] and for weakly cross-linked actin network [$C=1\mathrm{mg}/\mathrm{ml}$, $r=10.2\mu \mathrm{m}$ (triangles)]. Solid lines: predictions for a viscoelastic fluid with $z=0.68$ in micelles, and $z=0.75$ in actin. Inset of (a): the fit parameter $A$ increases linearly with micelle concentration.