# Microrheology Probes Length Scale Dependent Rheology

Entry by Helen Wu, AP225 Fall 2010

## Reference

"Microrheology probes length scale dependent rheology"

M. L. Gardel, K. Kroy, E. Frey, B. D. Hoffman, J. C. Crocker, A. R. Bausch, D. A. Weitz, *Physical Review Letters*, **96**, 118104 (2006).

## Keywords

## Overview

The researchers looked at the mechanical response of a semiflexible polymer (F-actin in this case) at different length scales using microrheology. Semiflexible polymers become entangled at low concentrations, and they are sterically hindered at the entanglement length <math>l_e</math>, which is related to the distance between polymers and the persistence length. At intermediate frequencies, there is a transition where the mechanical response is dependent on the filament length, <math>L</math>, which is not predicted by theory. The researchers identified fluctuations over <math>L</math> as a relaxation mechanism between 0.1-30rad/s and used 2-particle (2P) microrheology to look at lengths >5<math>\mu</math>m and 1-particle microrheology for lengths ~<math>l_e</math>. 2P microrheology showed increased viscoelastic relaxation for intermediate frequencies that scaled as <math>L^2</math>

## Results and discussion

For varied L values, the researchers demonstrated that longer filaments show a transition in particle motion where before a certain point, mean squared displacement (MSD) changes with respect to <math>\tau</math> (raised to a factor) but then switches to little time evolution. Figure 1 shows the MSD values for various particle sizes. For <math>L > 2 \mu m</math>, the 1P MSD is more constrained, but behavior is similar. The switching time between regimes was similar across lengths. The 2P displacement correlation tensor was scaled to the 2P MSD, <math>a</math>. When <math>L</math> was about <math>a</math>, 2P and 1P MSD values were similar. However, the 2P MSD changed in both slope and magnitude as <math>L</math> changed, unlike the 1P MSD.

The generalized Stokes-Einstein relation was used to compare 1P to 2P MSDs. For small <math>L</math>, they mostly matched, but longer filaments show differences. (see Figure 2, showing <math>G'</math>, <math>G*</math> against the frequency, <math>\omega</math>) A transition from sloping up to a plateau is once again seen for the 1P microrheology over <math>L</math>, whereas 2P microrheology is dependent on <math>L</math>. They converge to similar values at low frequency.*

From their data, the researchers say that the 1P microrheology probes bending fluctuations of single filaments at various frequencies, where single-filament dynamics dominate until they become entangled at <math>l_e</math>. The plateau comes from the entanglement. Based on the data, the authors suggest that 1P microrheology may be useful for measurements of cross-linked networks of semiflexible filaments.

In contrast, the 2P results scale as <math>\tau_m </math>~<math> L^2</math>, like in diffusion, and indicate density fluctuations along the filament (expected because transverse thermal fluctuations result in varying quantities of material present at a specific point). 2P microrheology probes longer lengths than 1P.

The authors conclude that mechanical response changes as length scales in the system vary and that rheology can be used to learn about network geometry and filament properties.

## Experimental Setup

G-actin mixed with polystyrene particles and then polymerized. <math>L</math> was varied using gelsolin. Particle motions were recorded with a fast camera.