Optical Tweezers Microrheology Maps the Dynamics of Strain-Induced Local Inhomogeneities in Entangled Polymers

Optical tweezers microrheology (OTM) offers a powerful approach to probe the nonlinear response of complex soft matter systems, such as networks of entangled polymers, over wide-ranging spatiotemporal scales. OTM can also uniquely characterize the microstructural dynamics that lead to the intriguing nonlinear rheological properties that these systems exhibit. However, the strain in OTM measurements, applied by optically forcing a microprobe through the material, induces network inhomogeneities in and around the strain path, and the resultant flow field complicates the measured response of the system. Through a robust set of custom-designed OTM protocols, coupled with modeling and analytical calculations, we characterize the time-varying inhomogeneity fields induced by OTM measurements. We show that homogenization following strain does not interfere with the intrinsic stress relaxation dynamics of the system, rather it manifests as an independent component in the stress decay, even in highly nonlinear regimes such as with the microrheological large-amplitude-oscillatory-shear (MLAOS) protocols we introduce. Our specific results show that Rouse-like elastic retraction, rather than disentanglement and disengagement, dominates the nonlinear stress relaxation of entangled polymers at micro- and mesoscales. Thus, our study opens up possibilities of performing precision nonlinear microrheological measurements, such as MLAOS, on a wide range of complex macromolecular systems.

Figure 1

Schematic of OTM strain profiles (a) and resultant microstrained state of entangled DNA (b). (a) Strain profiles represent the path of the trapped bead relative to the sample. In single-strain experiments (orange), the bead is dragged a distance $L_{\mathrm{SP}}$ at a constant rate $\dot{\gamma }$, after which motion is halted. In MLAOS experiments (brown) a constant-rate oscillatory strain with period $\mathrm{\Delta }t$ and wait-time $\mathrm{\Delta }{t}_w$ is applied by repeatedly tracing the same strain path along forward and backward directions. (b) As the probe moves through the solution, DNA in the strain path accumulates, creating a higher density build-up region in front of the bead, and lower density depletion region behind it. At the same time, Stokes’ drag displaces the bead from the trap center by a distance $\mathrm{\Delta }x$.

Figure 2

Stress relaxation of entangled DNA is mediated by Rouse-like elastic retraction coupled with homogenization of nonuniform concentration profiles. (a) Sample applied strain $\gamma (t)$ (orange line, right axis) and measured stress response $\sigma (t)$ (brown line, left axis) versus time $t$. The stress relaxation at constant strain ${\gamma }_{\max }$ is fit to a double-exponential decay (gray line). Data shown is for ${\gamma }_{\max }=37.7$. Inset: Bead motion through the build-up region towards the trap center as the system relaxes. (b) $\sigma (t)$ curves for ${\gamma }_{\max }$ values of 9.4 (pink), 18.9 (orange), 23.6 (olive), 28.3 (magenta), 33.0 (cyan), and 37.7 (green), all fit to double-exponential decays with decay times ${\tau }_1$ and ${\tau }_2$ (black lines). Inset: ${\tau }_1$ and ${\tau }_2$ (color-coded circles and squares with error bars), versus ${\gamma }_{\max }$ with linear fits to the data. (c) Schematic of the model and associated variables: microprobe radius ($a$), radius ($R$), and length ($L$) of the build-up region, probe displacement from trap center ($\mathrm{\Delta }x$), and DNA concentrations in the build-up region ($c_{\mathrm{bu}}$) and in bulk ($c$).

Figure 3

The maximum stress induced in entangled DNA during MLAOS measurements exhibits two-phase exponential decay. (a) Sample applied strain $\gamma (t)$ (orange, right axis) and measured stress $\sigma (t)$ (brown, left axis) during an MLAOS measurement with period $\mathrm{\Delta }t$. Black circles denote the maximum stress measured for each oscillation ${\sigma }_{\max }(t)$. (b) ${\sigma }_{\max }(t)$ values (hollow squares) for ten oscillations with $\mathrm{\Delta }t$ values of 0.44 (blue), 0.89 (green), and 1.49 s (olive), all fit to double-exponential decays (solid lines) with decay times ${\tau }_3$ and ${\tau }_4$. Inset: ${\tau }_3$ and ${\tau }_4$ (color-coded squares and circles with error bars) versus $\mathrm{\Delta }t$ with linear fits to the data. (c) Schematic of the model with the associated variables: probe radius ($a$), probe speed ($v$), displacement of probe from the trap center ($\mathrm{\Delta }x$), width of the neighboring region ($\sqrt{D\mathrm{\Delta }t}$) comprising DNA that diffuses into the strain path in wait time $\mathrm{\Delta }t$, DNA concentrations in the strain path ($c_{\mathrm{sp}}$), neighboring region ($c_n$), and bulk ($c$).