Active rotational and translational microrheology beyond the linear spring regime

Active particle tracking microrheometers have the potential to perform accurate broadband measurements of viscoelasticity within microscopic systems. Generally, their largest possible precision is limited by Brownian motion and low frequency changes to the system. The signal to noise ratio is usually improved by increasing the size of the driven motion compared to the Brownian as well as averaging over repeated measurements. New theory is presented here whereby error in measurements of the complex shear modulus can be significantly reduced by analyzing the motion of a spherical particle driven by nonlinear forces. In some scenarios error can be further reduced by applying a variable transformation which linearizes the equation of motion. This enables normalization that eliminates error introduced by low frequency drift in the particle's equilibrium position. Our measurements indicate that this can further resolve an additional decade of viscoelasticity at high frequencies. Using this method will easily increase the signal strength enough to significantly reduce the measurement time for the same error. Thus the method is more conducive to measuring viscoelasticity in slowly changing microscopic systems, such as a living cell.

Figure 1

A comparison between analysis methods in both viscous water and viscoelastic dilutions of Celluvisc eye drops. (a) depicts results of averaging 222 2-s flips in water, (b) 185 5-s flips in 50% Celluvisc, and (c) 90 10-s flips in 100% Celluvisc. In each graph the blue dashed line is the shear modulus calculated using the old theory, which assumes a linear restoring torque. The orange dashed lines are evaluated using the new theory accounting for the nonlinear restoring torque outlined in Sec. 2c5. The solid black line represents values obtained via the variable transformation analysis described in Sec. 2c6, which mitigates error introduced by variation in initial position. All these analysis techniques are compared to either theoretical values (circles) or macrorheological measurements [8] (diamonds).

Figure 2

The relationship between precision of $G^{*}\left(\omega \right)$ and the number of averaged flips in 50% Celluvisc. (a) shows the viscoelasticity obtained by analyzing a single 5s flip with the new method. (b) depicts results from 12 flips during a 1-minute measurement and (c) 120 flips during a 10-minute measurement. All three graphs show good agreement between the microrheological results (lines) and macrorheological data [8] (circles). Evidently, the precision increases with the number of averaged flips; however, because of the large amplitude of each flip, precise results can be obtained within 1 minute.