Time-dependent active microrheology in dilute colloidal suspensions

In a microrheological setup a single probe particle immersed in a complex fluid is exposed to a strong external force driving the system out of equilibrium. Here, we elaborate analytically the time-dependent response of a probe particle in a dilute suspension of Brownian particles to a large step force, exact in first order of the density of the bath particles. The time-dependent drift velocity approaches its stationary-state value exponentially fast for arbitrarily small driving in striking contrast to the power-law prediction of linear response encoded in the long-time tails of the velocity autocorrelation function. We show that the stationary-state behavior depends nonanalytically on the driving force and connect this behavior to the persistent correlations in the equilibrium state. We argue that this relation holds generically. Furthermore, we elaborate that the fluctuations in the direction of the force display transient superdiffusive behavior.

Figure 1

Time-dependent approach of the mobility $\mu (t,\text{Pe})$ to its stationary state for different Péclet numbers, Pe, and the linear response, $\text{Pe}=0$. Solid lines represent the analytic solution and symbols correspond to Brownian-dynamics simulations for equal-sized colloids ($D_b=D_a$). Dashed lines indicate negative values. The black dash-dotted lines show the long-time behavior $\sim -t^{-3/2}exp(-{\text{Pe}}^2{D}_{a}t/2{\sigma }^2)$ for the approach to the stationary state.

Figure 2

Time-dependent diffusion coefficient $D_z\left(t\right)$ for different strength of the driving, $\text{Pe}=\mu F\sigma /D_r$. Lines correspond to the analytic solution and symbols represent Brownian-dynamics simulation for equal-sized colloids ($D_b=D_a$). Inset: Zoom of the same quantity for the smaller Péclet numbers.

Figure 3

Time-dependent fluctuations ${\text{Var}}_z\left(t\right)=\langle \mathrm{\Delta }{z}_a{\left(t\right)}^2\rangle -{\langle \mathrm{\Delta }{z}_a\left(t\right)\rangle }^2$ and local exponent $\alpha \left(t\right)=dln\left[{\text{Var}}_z\left(t\right)\right]/dln\left(t\right)$ (inset) of the probe particle along the applied force obtained from Brownian-dynamics simulation of equal-sized colloids ($D_b=D_a$) for different strength of the driving. The solid lines correspond to the diffusive asymptote ${\text{Var}}_z\left(t\right)=2D_z\left(\text{Pe}\right)t$ and the dashed lines are the asymptotic model ${\text{Var}}_z\left(t\right)={(D_a/D_r)}^2{\left(\mu Ft\right)}^3/3l_{*}$ with the empirical mean-free path length $l_{*}\approx 1/n{\sigma }^2$.

Figure 4

Density-induced suppression of the stationary mobility $\mu \left(\text{Pe}\right)$. Dashed lines indicate the asymptotic expansion for increasing order (2, 4, 6, 10, and 20). Lines correspond to the theory and symbols represent Brownian-dynamics simulations of equal-sized colloids, $D_b=D_a$, and Lorentz systems with $D_b=0$.

Figure 5

Stationary-state force-induced diffusion coefficient $D_z^{\text{ind}}:=D_z\left(\text{Pe}\right)-D_z(\text{Pe}\to 0)$. Dashed lines represent the asymptotic behavior for small (increasing order 2, 3, 5, 9, and 31) and large driving. Lines correspond to the theory and symbols represent Brownian-dynamics simulations of equal-sized colloids, $D_b=D_a$, and Lorentz systems $D_b=0$.

Figure 6

Top: Simulation results for the time-dependent pair-distribution function $g(\vec{r},t)$ for Péclet number $\text{Pe}=32$ and density $n{\sigma }^3=0.01$. As time progresses, $t{D}_a/{\sigma }^2=4\times {10}^{-4}$, $4\times {10}^{-2}$, 4 (left to right), the tracer piles up probability in front and leaves a trail of depleted probability. Bottom: Analytic result for the real part of the pair-distribution function in the frequency domain $Re[-i\omega \widehat{g}(\vec{r},\omega )]$ at the same driving $\text{Pe}=32$ for frequencies $\omega {\sigma }^2/2\pi D_a=1/(4\times {10}^{-4})$, $1/(4\times {10}^{-2})$, 1/4 (left to right).