Fluctuations of driven probes reveal nonequilibrium transitions in complex fluids

TL;DR

The study tackles how localized driving in complex fluids induces nonequilibrium microstructural changes and how these changes can be detected from fluctuations of a dragged probe. It develops a variance-based method grounded in equipartition breakdown and introduces a dimensionless velocity ${\mathcal U}= v/v^*(L)$, supported by large-scale Brownian-dynamics simulations of a coarse-grained two-dimensional model. The results reveal a sequence of regimes: near-equilibrium linear response with $\Delta_x \sim 0$, a $\Delta_x \sim {\mathcal{U}}^2$ growth, and a high-${\mathcal U}$ activated regime where discrete hopping events relieve stored elastic stress; crucially, $v^*(L) \sim L^{-3/2}$ and the fluctuations collapse onto a master curve when scaled by $L^{1/2}$. These findings connect localized elastic-stress buildup to nonlinear friction phenomena in microrheology and offer an experimentally accessible route to detect nonequilibrium transitions in complex fluids, independent of macroscopic continuum descriptions.

Abstract

Complex fluids subjected to localized microscopic energy inputs, typical of active microrheology setups, exhibit poorly understood nonequilibrium behaviors because of the intricate self-organization of their mesoscopic constituents. In this work we show how to identify changes in the microstructural conformation of the fluid by monitoring the variance of the probe position, based on a general method grounded in the breakdown of the equipartition theorem. To illustrate our method, we perform large-scale Brownian dynamics simulations of an effective model of micellar solution, and we link the different scaling regimes in the variance of the probe's position to the transitions from diffusive to jump dynamics, where the fluid intermittently relaxes the accumulated stress. This suggests stored elastic stress may be the physical mechanism behind the nonlinear friction curves recently measured in micellar solutions, pointing at a mechanism for the observed multi-step rheology. Our approach overcomes the limitations of continuum macroscopic descriptions and introduces an empirical method, applicable in experiments, to detect nonequilibrium transitions in the structure of complex fluids.

Figures (10)

• Figure 1: Schematic representation of our two-dimensional simulation setup. A probe of nominal diameter $\sigma_0$ interacts via soft Gaussian repulsion with harmonic chains, with fixed length $L$, of units of diameter $\sigma=\sigma_0/10$, which repel each other via a soft Gaussian potential. The probe is driven by a harmonic potential translating with constant velocity $v$, representing optical tweezers. A detailed description of the model is contained in Appendix A.
• Figure 2: (a) Semi-log plot of the effective friction $\gamma_{\text{eff}}$ (see Eq. \ref{['eq:effective_friction_def']}, in units of $\gamma_0$), as a function of the drag velocity $v$ (in units of $\ell/\tau$, $\ell$ and $\tau$ being arbitrary simulation units for length and time resp.) for different polymer lengths $L$. Error bars are obtained from the $r_{0,x}$ traces as the minimum between the naive estimate $\textrm{Var}\,r_{0,x}^{1/2} / v$ (uninformative for $v\lesssim 10^{-2}$ due to near-equilibrium Brownian motion) and the one obtained from the linear fit $\langle r_{0,x}\rangle = a v$ on the points $(v_{n+k}, \langle r_{0,x}\rangle_{n+k})$ with $k=0, \pm 1, \pm 2$ (only informative in the near-equilibrium linear regime). Gray line marks $\gamma_{\text{eff}}=\gamma_0$. (b) The same plot for the relative increase of the probe's variance from its equilibrium value; filled circles mark $v^*(L)$. Error bars are computed via bootstrap, using $250$ sub-traces uniformly sampling $150$ different instantaneous values of the original $r_{0,x}$ traces. Inflection points (marked by circles) are identified by collapsing the data as in Fig. \ref{['fig:main']}. Inset: Log-log plot of $v^*(L)$; the solid line shows the power law $\sim L^{-3/2}$. Error bars are estimated as $\Delta v^* = v_n - v_{n-1}$ at $v_n=v^*$, where $\{v_n\}$ are the available values of the velocity. A linear fit (including uncertainty over $v$) of the function $\ln v = a\ln L + b$ gives $a=-1.51\pm 0.06$, $b=3.0\pm0.2$.
• Figure 3: (a) The fluctuations' enhancement \ref{['eq:Delta']} as a function of $\mathop{\mathrm{\mathcal{U}}}\nolimits$, rescaled by $L^{1/2}$. The green shaded region highlights the onset of the $v^2$ scaling and its end at $v=v^*$. Error bars computed as in Fig. 2(b). (b)-(e) Spatial fields of the local units density (in adimensional units, background color) and ellipses representing the gyration tensor $G$, for $L=32$ and increasing $\mathop{\mathrm{\mathcal{U}}}\nolimits$, in a $20\ell\times 20\ell$ region around the probe (yellow-green circle of radius $2.5 \sigma_0$). Black symbols match those in panels (a) and help identify the regime of each panel.
• Figure 4: (a-b) Local elastic energy (in adimensional units) stored in the polymer fluid as a function of the distance from the probe along the $x$-axis, for (a) $L=16$ and (b) $L=56$. Each curve is colored according to the value of $\mathop{\mathrm{\mathcal{U}}}\nolimits$. The dashed line represents the threshold energy $U_\text{bond}$ for the activated jump events. In panel (a), missing points indicate the absence of polymers in that position (measured using the center of mass) for low $\mathcal{U}$, due to the repulsive force exerted by the probe. $R^2_{g,\infty}$ is the value of the gyration radius computed far away from the probe. Inset: a sketch of the configurations giving rise to hoop stresses. (c) Dimensionless ratio of the fluctuations' enhancement to the mean probe position scaled by $\sigma_0$, for various $L$ (color code in Fig. \ref{['fig:main']}). For $\mathop{\mathrm{\mathcal{U}}}\nolimits >1$, the ratio tends to a constant value independent of $L$.
• Figure 5: (a) Difference between the probe's effective friction and its Stokes friction $\gamma_0$, in units of $\gamma_0$ and rescaled by $\sqrt{L}$. Symbols and color code for $L$ are the same of Fig. 3. (b) As in panel (a) but rescaled by $L$.