Microrheological characterization of anisotropic materials

We describe the measurement of anisotropic viscoelastic moduli in complex soft materials, such as biopolymer gels, via video particle tracking microrheology of colloid tracer particles. The use of a correlation tensor to find the axes of maximum anisotropy without prior knowledge, and hence the mechanical director, is described. The moduli of an aligned DNA gel are reported, as an application of the technique; this may have implications for high DNA concentrations in vivo. We also discuss the errors in microrheological measurement, and describe the use of frequency space filtering to improve displacement resolution, and hence probe these typically high modulus materials.

Figure 1

(a) Measured probability density plot of probe positions in the aligned DNA gel described later (left) and a similar unsheared DNA sample (right), as an example of an isotropic material. The plots are calculated by collapsing probe tracks (longer than 50 steps) over all available time to a common origin, and counting the occurrences in discrete bins. The angle of the surrounding ellipse is set at 35° from the horizontal, which is the value calculated from the eigenvector analysis for the shortest lag time later. A good correspondence is seen. (b) Displacement distribution in $x$ (left) and $y$ (right) for $\tau =1∕25\mathrm{s}$. Visually the diffusion is Gaussian and this may be verified by calculating the excess kurtosis of the distributions (0.8 in the $x$ direction and 0.7 in the $y$ direction). Thus the degree of heterogeneity is relatively small, and may be neglected.

Figure 2

Mean squared displacement of $0.497\mathrm{\mu }\mathrm{m}$ polystyrene latex colloidal spheres dried on a glass slide, calculated from all independent lag times. The deinterlaced video fields appear displaced from each other by $\sim 0.15$ pixels (most clearly seen at short lag times), varying by $\sim 0.06$ pixels with time (for clarity even lag times, in frames, are denoted by empty symbols). This represents a significant error for high modulus materials.

Figure 3

The sheared $11\mathrm{mg}∕\mathrm{ml}$ DNA sample at $298\mathrm{K}$ under horizontal-vertical crossed polarizers. Alignment is relatively weak. Light areas show alignment, with the DNA chain backbones lying approximately 45° counterclockwise from the vertical (checked via insertion of a quarter waveplate at 45° to the polarizers). The calculated eigenvectors are superimposed in black, and show the mechanical alignment corresponds well with the optical birefringence, as expected. The calculated angle is 35° counterclockwise.

Figure 4

One-dimensional MSDs along calculated eigenvectors for sheared $11\mathrm{mg}∕\mathrm{ml}$ DNA at $298\mathrm{K}$. Crosses $(+)$ correspond to displacements along the axis approximately parallel to the optical director, and circles $(\circ )$ to displacements perpendicular to this. The static resolution error is taken as $4\mathrm{nm}$, and subtracted. The MSD is proportional to the creep compliance [7]. Inset: MSD for the same DNA at $4.9\mathrm{mg}∕\mathrm{ml}$, where there was no visible birefringence. There is now no visible anisotropy in the MSD. Note that the shape of the response is similar to the anisotropic case—this suggests that the GSER should hold in anisotropic materials.

Figure 5

One-dimensional elastic modulus, $G^{\prime }\left(\omega \right)$ and loss modulus, $G^{″}\left(\omega \right)$, along calculated eigenvectors for sheared $11\mathrm{mg}∕\mathrm{ml}$ DNA at $298\mathrm{K}$. Filled circles correspond to $G^{\prime }$ along the axis approximately parallel to the optical director, and filled squares to $G^{\prime }$ perpendicular to this. Open symbols refer to $G^{″}\left(\omega \right)$ in these directions. The static resolution error is taken as $4\mathrm{nm}$, and subtracted from the MSD before calculation. Inset: Moduli of the same DNA at $4.9\mathrm{mg}∕\mathrm{ml}$ (approximately the mean concentration of DNA in a human cell [22]). The sample is isotropic in its mechanical response, and the shape is comparable to the anisotropic case, suggesting the validity of the GSER in anisotropic materials.