Optically driven nonlinear microrheology of gelatin

We demonstrate the microscopic equivalent of a step-stress rheological measurement. An optical torque is applied to a birefringent wax microdisk embedded in gelatin, a highly entangled viscoelastic biopolymer, using circularly polarized laser tweezers. By increasing the laser power and measuring the angular displacement of the disk, we explore the microscopic rheological response of presheared gelatin from the linear to the nonlinear regime and observe yielding at the microscale. The shape of the microscopic torque-angle relationship matches the stress-strain relationship from a macroscopic measurement of presheared gelatin; from this, we extract an applied stress and deduce the effective strain induced by the rotating disk.

Figure 1

(Color online) (a) Schematic of the optical trapping apparatus. A krypton ion laser ($P_{\mathrm{MAX}}\approx 1.5\mathrm{W}$, $\lambda =647\mathrm{nm}$, LP) is directed with mirrors (M1, dichroic M2) through a beam expander (BE, 5X) into the back aperture of a microscope objective (OBJ, Nikon 100X Plan Apo, NA=1.4) and focused to a diffraction limited spot within the sample. The laser intensity is controlled with neutral density filters (ND) and a computer controlled shutter. The introduction of a quarter wave plate $(\lambda ∕4)$ circularly polarizes (CP) the laser light. A CCD camera images the sample. (b) A wax microdisk (blue) traps upright in the diffraction limited optical trap (red). The scale bar is $2\mathrm{\mu }\mathrm{m}$. (c) CP laser light applies a torque to the birefringent microdisk resulting in an angular displacement, $\mathrm{\Delta }\theta$ about the optical axis (red circle).

Figure 2

(Color online) (a) A wax microdisk [$a\approx 1.5\mathrm{\mu }\mathrm{m}$, $\left(2a\right)∕h\approx 2$] embedded in an aqueous gel (gelatin, $1.5\%\mathrm{w}∕\mathrm{w}$) is viewed on edge $(t=0\mathrm{s})$. The application of an optical torque ($P=1\mathrm{W}$, $t=0.6\mathrm{s}$ and $12\mathrm{s}$) rotates the disk CW (red arrows) about the optical axis (red circles); upon removal of the torque ($t=16\mathrm{s}$ and $18\mathrm{s}$), the disk rotates CCW (blue arrows) to its original position. The scale bar is $2\mathrm{\mu }\mathrm{m}$. (b) The time-dependent angular response $\mathrm{\Delta }\theta \left(t\right)$ of the microdisk to an applied torque (“Laser ON,” red diamonds) is fit to a simple model for the strain response of a viscoelastic material subjected to a step stress (schematic, inset). The angular relaxation of the disk (“Laser OFF,” blue circles) is fit to a simple exponential decay.

Figure 3

(Color online) The angular displacement $\mathrm{\Delta }\theta \left(t\right)$ of the microdisk (from Fig. 2) as a function of increasing laser power, $P=270$ $(▲)$ (purple online), 600 $(◼)$ (black online), 800 $(▼)$ (green online), 1000 $(◆)$ (red online), and $1500\mathrm{mW}$ $(●)$ (blue online). The lines are fits to a model of a viscoelastic material subjected to a step stress [see Fig. 2b (inset)].

Figure 4

(Color online) Bulk rheology of presheared gelatin $(◻)$ (red online), plotted as the macroscopic stress $\tau$ (left axis) versus macroscopic strain, $\gamma$ (upper axis), exhibiting a yield stress, ${\tau }_y\approx 15\pm 3\mathrm{dynes}∕{\mathrm{cm}}^2$. Optical torque, $T$ (right axis) applied to a wax microdisk (from Fig. 2) embedded in gelatin versus the angular displacement $\mathrm{\Delta }{\theta }_{\mathrm{p}}$ (lower axis) of the disk $(●)$ (blue online). The line is a linear fit to the data below $T\approx 3\times {10}^{-10}\mathrm{dyn}\mathrm{cm}$. Data at $T\approx 3.9\times {10}^{-10}\mathrm{dyn}\mathrm{cm}$ deviate significantly from linearity. The relative scales of the horizontal and vertical axes have been adjusted to provide the best agreement between the macroscopic and microscopic rheology.