Optical tweezer measurements of asymptotic nonlinearities in complex fluids

This article presents micro-medium-amplitude oscillatory shear ($\mu \mathrm{MAOS}$), a method to measure the frequency-dependent micromechanical properties of soft materials in the asymptotically nonlinear regime using optical tweezers. We have developed a theoretical framework to extract these nonlinear mechanical properties of the material from experimental measurements and also proposed a physical interpretation of the third-order nonlinearities measured in single-tone oscillatory tests. We validate the method using a well-characterized surfactant solution of wormlike micelles, and subsequently employ this technique to demonstrate that the cytoplasm of a living cell undergoes strain softening and shear thinning when locally subjected to weakly nonlinear oscillatory deformations.

Figure 1

Schematic representation of nonlinear micromechanical measurements via optical tweezers. (a) Bead within the soft material trapped by optical tweezers (not to scale). (b) Depiction of the trapping and mechanical forces in the system. (c) Trapping force vs trap displacement curves showing linear and nonlinear mechanical behavior at amplitudes of 0.175 and 0.4 $\mu \mathrm{m}$, respectively, at 500 rad/s for a 5 wt% aqueous PEO solution.

Figure 2

Physical interpretation of the sign of the third-order, first harmonic nonlinear response for the limiting cases of purely viscous (left) and purely elastic materials (right).

Figure 3

Validation of $\mu \mathrm{MAOS}$ framework for a Newtonian fluid (a 50 wt% aqueous glycerol solution). (a) The real and imaginary components ${\eta }^{\prime }\left(\omega \right)$ and ${\eta }^{\prime \prime }\left(\omega \right)$ of the linear complex viscosity obtained from $\mu \mathrm{MAOS}$ measurements. (b) and (c) The third-order material properties ${\zeta }_{31}^{\prime }\left(\omega \right), {\zeta }_{31}^{\prime \prime }\left(\omega \right), {\zeta }_{33}^{\prime }\left(\omega \right)$, and ${\zeta }_{33}^{\prime \prime }\left(\omega \right)$ obtained from $\mu \mathrm{MAOS}$ measurements, depicted with filled and unfilled symbols to indicate where data are positive and negative, respectively. Data represent the average over 20 trials, with error bars representing one standard error uncertainty.

Figure 4

Validation of the $\mu \mathrm{MAOS}$ framework using a CPyCl wormlike micellar solution. (a) The elastic and viscous moduli, $G^{\prime }\left(\omega \right)$ and $G^{\prime \prime }\left(\omega \right)$, obtained from $\mu \mathrm{MAOS}$ measurements. (b) and (c) The third-order material properties, ${\zeta }_{31}^{\prime }\left(\omega \right), {\zeta }_{31}^{\prime \prime }\left(\omega \right), {\zeta }_{33}^{\prime }\left(\omega \right)$, and ${\zeta }_{33}^{\prime \prime }\left(\omega \right)$ obtained from $\mu \mathrm{MAOS}$ measurements, depicted with filled and unfilled symbols to indicate where data are positive and negative, respectively. Data represent the average over 20 trials, with error bars representing one standard error uncertainty. Predictions of the corotational Maxwell model with high-frequency Rouse dynamics are depicted with solid and dashed lines to indicate positive and negative values, respectively.

Figure 5

$\mu \mathrm{MAOS}$ studies of the cytoplasm of a living cell and of an entangled solution of polymers. (a) A schematic of an optically trapped bead within the cellular cytoplasm is shown as an inset. The linear elastic and viscous moduli $G^{\prime }\left(\omega \right)$ and $G^{\prime \prime }\left(\omega \right)$ obtained from $\mu \mathrm{MAOS}$ measurements in the cytoplasm of an mEF cell and in a 5 wt% solution of PEO in water. (b) and (c) The third-order material properties ${\zeta }_{31}^{\prime }\left(\omega \right), {\zeta }_{31}^{\prime \prime }\left(\omega \right), {\zeta }_{33}^{\prime }\left(\omega \right)$, and ${\zeta }_{33}^{\prime \prime }\left(\omega \right)$ obtained from $\mu \mathrm{MAOS}$ measurements. Data from the mEF cytoplasm are represented by blue and red symbols, while data from PEO are represented with light blue and purple symbols, with filled and unfilled symbols used to denote positive and negative values, respectively. Data represent the average over 20 trials, with error bars representing one standard error uncertainty.

Figure 6

Unidirectional dragging measurements of an optically trapped bead in a living mammalian cell. (a) Measurements near the cellular cortex, which depict a strain stiffening behavior (positive concavity in the force vs displacement curve). (b) Measurements in the cytoplasm, away from the cortex, which depict a strain softening behavior (negative concavity in the force vs displacement curve). The solid black line corresponds to the mean over 20 trials, and the shaded region corresponds to one standard error of the mean.