# Microrheology

Original Entry: Ian Burgess, Fall 2009 (currently under construction)

## Definition

Microrheology is a form of rheology in which the properties of a complex fluid are measured with microscopic spatial precision by tracking the movement of a small, micron-sized, particle embedded in the fluid.

## Passive Microrheology

In passive microrheology, Brownian motion of the particle due to thermal energy is observed. The viscoelastic modulus of a complex fluid <math> G(\omega)</math> can be determined from the mean square displacement <math> \langle\Delta r^2\rangle</math> of the tracer particle undergoing thermal motion. Mason and Weitz showed that the Laplace transform of <math> G(\omega)</math> (<math> \tilde{G}(s)</math>) is related to the Laplace transform of the mean square displacement (<math> \tilde{r}(s)</math>) according to the equation [1]:

<math> \tilde{G}(s) = \frac{s}{6\pi a}\left(\frac{6k_{B}T}{s^2\tilde{r}(s)} - ms\right)</math>

where <math> m</math> is the mass of the tracer and <math> a</math> is its radius. Except for at high frequencies, the second term can usually be neglected and the expression reduces to [1]:

<math> \tilde{G}(s) = \frac{k_{B}T}{\pi a s\tilde{r}(s)}</math>

Both the real and imaginary parts of <math> G(\omega)</math> may be deduced from the single function, <math> \tilde{G}(s)</math>, because they are not independent and must obey the Kramers-Kronig relations.

## Active Microrheology

In active microrheology, forces are applied to the particle in order to probe the response of the medium. These forces are applied using optical tweezers or magnetic forces. This technique is especially useful for probing very stiff materials, where large forces may be required to accurately measure the material response.

## References

[1] T. G. Mason and D. A. Weitz, "Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids," *Phys. Rev. Lett.* **74**, 1250 (1995).
[2] M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, D. A. Weitz, "Investigating the microenvironments of inhomogeneous soft materials with multiple particle tracking," *Phys. Rev. E* **64**, 061506(9) (2001).