Active microrheology in a colloidal glass

We study the dynamics of a probe particle driven by a constant force through a colloidal glass of hard spheres. This nonequilibrium and anisotropic problem is investigated using a new implementation of the mode-coupling approximation with multiple relaxation channels and Langevin dynamics simulations. A force threshold is found, below which the probe remains localized, while above it the probe acquires a finite velocity. We focus on the localized regime, comparing theory and simulations concerning the dynamics in the length scale of the cage and the properties of the transition to the delocalized regime, such as the critical power-law decay of the probe correlation function. Probe van Hove functions predicted by the theory show exponential tails reminiscent of an intermittent dynamics of the probe. This scenario is microscopically supported by simulations.

Figure 1

MCT phase diagram separating delocalized and ideal localized regimes. The vertical arrow marks the density chosen in the MCT calculations for the comparison with simulations. The right vertical axis gives forces in real units converted assuming colloids of size $d=4\mu \text{m}$ close to room temperature.

Figure 2

Snapshot of the system with the probe marked in red (dark gray) and all particles in front of it removed to allow observing the probe. The arrow indicates the external force applied only to the probe.

Figure 3

Mean-squared displacement of the bulk system with $\varphi =0.62$ (black dashed line), compared with the system with attractions and the same density, in the fluid region (blue dash-dotted line), and with the MSD of quasihard spheres at a lower density, $\varphi =0.585$, again in the fluid (red solid line). (Inset) Nonergodicity parameters of the system with $\varphi =0.62$ (circles) and the critical packing fraction for the glass transition (squares).

Figure 4

Probe-particle displacement in the direction of the applied force for different external forces, as labeled. (Left) Theoretical results for a glassy host at volume fraction $\varphi =0.537$. (Right) Simulation results for $\varphi =0.62$. The curves with the label c correspond to the critical force in each case. The dashed line shows the $\delta z\propto t$ behavior. In both graphs the force increases from bottom to top. The inset shows typical individual probe trajectories for $F=50k_{\text{B}}T/d$.

Figure 5

Probe position correlation functions for different forces in the localized regime, for a wave vector modulus close to the location $q_2$, the position of the second peak of the structure factor, and different directions: in the direction of the external force (real and imaginary parts of the correlation function in the top and middle panels, respectively) and perpendicular to the external force (bottom panels). In the theory (left panels), $\varphi =0.537$ and the wave vector is $qd=12.75$. In the simulations (right panels), $\varphi =0.62$ and the wave vector is $qd=13.35$. The dashed lines in both panels represent the results for the unforced probe, i.e., $F=0k_{\text{B}}T/d$. The arrows indicate the direction of increasing force.

Figure 6

Probe position correlation functions in the localized regime for $F=40k_{\text{B}}T/d$ and different wave vectors. The top and middle rows of panels show, respectively, real and imaginary parts of the correlation function in the direction of the force. Real-valued correlation functions perpendicular to the external force are shown in the bottom row. In the theory (left column of panels), $\varphi =0.537$ and the wave vector ranges from $qd=1$ to $qd=20$. In the simulations (right column), $\varphi =0.62$, the lines correspond to $qd$ values multiple of $2\pi d/L=1.335$ ranging from $qd=1.335$ to $qd=26.5$. The arrows show the direction of increasing wave vector.

Figure 7

Nonergodicity parameters as a function of wave-vector modulus for different forces in the localized regime. The top and middle rows of panels show the real and imaginary parts of $f_q^s$ with wave vectors in the direction of the external force, and the bottom row shows wave vectors in the perpendicular plane. In the theory (left column of panels), the host volume fraction is $\varphi =0.537$ and the forces range from $F=0$ to $F=40k_{\text{B}}T/d$ as indicated. In the simulation (right column), $\varphi =0.62$, the data correspond to forces $F=30$, 40, and $50k_{\text{B}}T/d$ from top to bottom, and the black dashed line shows the force-free nonergodicity parameter.

Figure 8

Shape of the cage of neighbors explored by the probe for different forces increasing from top to bottom rows. In the theory (left column of panels) the data presents the distribution $G^{\mathrm{s}}(x,z;t\to \infty )$ of probe displacements at long times (back Fourier transform of the probe nonergodicity parameter) for $F=10$, 30, and $40k_{\text{B}}T/d$. In the simulations (right column) the probe displacement distribution at time $t={\tau }_{\mathrm{B}}$ for $F=30$, 50, and $70k_{\text{B}}T/d$ from top to bottom.

Figure 9

Integrated components of the van Hove function for different forces: longitudinal components in the top row of panels and transversal ones in the bottom row (see text for the definition of the functions). The black dashed lines represent exponential fittings to the tails of the distributions at long distances. The inset shows the time evolution of the longitudinal component for $F=70k_{\text{B}}T/d$ and $t={\tau }_{\mathrm{B}},2{\tau }_{\mathrm{B}},3{\tau }_{\mathrm{B}},4{\tau }_{\mathrm{B}}$, and $5{\tau }_{\mathrm{B}}$ (increasing from bottom to top).

Figure 10

Maximum and first moment of the longitudinal distribution as a function of the external force from theory (left) and simulations (right). The dashed lines present the correlation length $\xi$ obtained from exponential fittings $G_{\parallel }^s\propto exp[-z/\xi ]$ to the tails of the distributions. The (scaled) second moment of the distribution in the perpendicular plane is also plotted in both cases.

Figure 11

Probe position correlation functions in log-log scale for a wave vector close to the location $q_1$ of the first neighbor peak in the structure factor ($q_{1}d\approx 7$ in the theory and $q_{1}d\approx 7.95$ in the simulations), parallel (top row of panels) and perpendicular to the external force. In the parallel direction the modulus of the function is presented. Theory (left column of panels) and simulations (right column) are compared. The dashed line shows the $t^{-1/2}$ behavior in all panels.

Figure 12

Probe position correlation functions for different wave vectors perpendicular to the external force, as labeled, from simulations. The dashed lines show the $t^{-1/2}$.

Figure 13

Relative errors $e_j$ defined in Eq. (E1) as functions of $qd$ for different values of ${\varepsilon }_{\text{F}}$ and host system at $\varphi =0.537$.

Figure 14

Comparison of the quantities $|A_{\mathbf{\text{q}}}^{\left(j\right)}\left(\omega \right)|(j=0,1,2)$ defined in Eqs. (E2) and (E3) for different values of ${\varepsilon }_{\text{F}}<0$ and for a wave vector close to the first peak of the host structure factor (at $\varphi =0.516$), along the force direction (top), and perpendicular to it (bottom). The Fourier-Laplace transforms were evaluated numerically using Filon's quadrature formula [50].