Asymptotic analysis of mode-coupling theory of active nonlinear microrheology

We discuss a schematic model of mode-coupling theory for force-driven active nonlinear microrheology, where a single probe particle is pulled by a constant external force through a dense host medium. The model exhibits both a glass transition for the host and a force-induced delocalization transition, where an initially localized probe inside the glassy host attains a nonvanishing steady-state velocity by locally melting the glass. Asymptotic expressions for the transient density correlation functions of the schematic model are derived, valid close to the transition points. There appear several nontrivial time scales relevant for the decay laws of the correlators. For the nonlinear friction coefficient of the probe, the asymptotic expressions cause various regimes of power-law variation with the external force, and two-parameter scaling laws.

Figure 1

Schematic-model correlation functions ${\phi }_{\parallel }^s\left(t\right)$ and ${\phi }_{\perp }^s\left(t\right)$ for host-liquid parameters close to and at the glass transition, $\varepsilon =-{10}^{-2}$ (solid lines) and $\varepsilon =0$ (dashed), for probe-coupling coefficients $v^s=30$, ${\kappa }_{\perp }=0.5$, and ${\kappa }_{\parallel }=1$. For ${\phi }_{\parallel }^s\left(t\right)$, both real and imaginary part are shown in separate panels. Curves in the order of decreasing relaxation time correspond to $F_{\mathrm{ex}}=0$, 1, $F_{\mathrm{ex}}^{\text{c}}-0.1$, $F_{\mathrm{ex}}^{\text{c}}$, 12, and 80, where $F_{\mathrm{ex}}^{\text{c}}=6.5735$. In the panels showing the real parts, the host-liquid correlation functions $\phi \left(t\right)$ are added for the liquid (dash-dotted lines) and glassy (dotted) state.

Figure 2

Real part of the probe-particle correlation functions $\mathrm{Re}{\phi }_{\parallel }^s\left(t\right)$ for the schematic model for force-driven microrheology for a distance to the host-liquid glass transition $\varepsilon =\pm {10}^{-6}$ and forces $F_{\mathrm{ex}}=F_{\mathrm{ex}}^{\text{c}}(1+\delta )$ with $F_{\mathrm{ex}}^{\text{c}}\approx 6.5735$ and $\delta =-1$, $-0.01$, and ${10}^{-5}$ from top to bottom. Model parameters are $v^s=30$, ${\kappa }_{\perp }=0.5$, and ${\kappa }_{\parallel }=1$ (others as mentioned in the text). Solid (dashed) lines show the results for the liquid (glass). Dash-dotted (dotted) lines show the corresponding host-liquid correlator $\phi \left(t\right)$ for the liquid (glass). Symbols mark the time scales $t_{\sigma }$ and $t_{\sigma }^{\prime }$ for the host correlator [diamonds, cf. Eq. (14)], $t_{\delta }$ (squares), $t_{1/2}$ (circles), and $t_{\sigma ,\delta }^{\prime }$ (triangles) for the probe correlator Eqs. (29), (30), and (32).

Figure 3

Excess friction $\Delta \zeta$ experienced by a probe particle, as a function of the applied force $F_{\mathrm{ex}}$, calculated in the schematic MCT model according to Eq. (4). Curves from bottom to top correspond to distances to the host-liquid glass transition $\varepsilon =-1$, $-0.1$, $-{10}^{-3}$, $-{10}^{-5}$ (liquid; solid lines), 0 (solid), ${10}^{-3}$, $0.1$, and 1 (glass; dashed lines). Other parameters are chosen as in Fig. 1.

Figure 4

State diagram for the probe particle in the schematic model. (Upper panel) Critical force $F_{\mathrm{ex}}^{\text{c}}$ as a function of distance to the host-liquid glass transition $\varepsilon$, for fixed $v^s$ as in Fig. 1. (Lower panel) Critical force as a function of probe-to-host coupling strength $v^{s}f$. Solid lines mark the boundary between localized and delocalized probe states; a vertical dotted line in the upper panel indicates the host-liquid glass transition. Dashed lines are $F_{\mathrm{ex}}^{\text{c}}$ obtained for the model without taking into account the anisotropy of the probe-particle fluctuations, $v_2^s=0$.

Figure 5

Solutions of the implicit set of equations Eq. (9), $f_1^s=\mathrm{Re}{f}_{\parallel }^s$ (solid lines), $f_2^s=\mathrm{Im}{f}_{\parallel }^s$ (dash-dotted lines), and $f_3^s=f_{\perp }^s$ (dashed lines), as a function of the reduced applied force, $F_{\mathrm{ex}}/\left(v^{s}f\right)$. Parameters are $v^s=30$, $f=1-1/\sqrt{2}$ corresponding to the glass-transition point of the ${\text{F}}_{12}$ model at $v_2^c=2$. The solutions identified as the long-time limits of the schematic model, Eqs. (3), are shown as thick lines. The inset shows the full set of solution branches, including one for $f_3^s$ that is cut off in the full figure for clarity.

Figure 6

Correlation functions ${\vec{\phi }}^s\left(t\right)$ close to the glass transition point, for various external forces, plotted as the leading-order deviation from the plateau value, $|{\vec{G}}^s\left(t\right)|=|{\vec{\phi }}^s\left(t\right)-{\vec{f}}^s|$ for $\varepsilon \to 0$, as a function of $t/t_{\sigma }$. Parameters as in Fig. 3; solid lines correspond to liquid states, $\varepsilon =-2.5\times {10}^{-10}$, dashed lines to glassy states, $\varepsilon =2.5\times {10}^{-10}$. Various forces $F_{\mathrm{ex}}$ are shown with $\delta =(F_{\mathrm{ex}}-F_{\mathrm{ex}}^{\text{c}})/F_{\mathrm{ex}}^{\text{c}}={10}^{-n}$, $n=0$, 1, 2, 3, 4, and 5 as labeled. (For the glass, only $n=0$ and $n=1$ are shown.) Dash-dotted and dotted lines indicate the corresponding host $\beta$-correlator $\left|G\right(t\left)\right|$, for the liquid respectively the glass, in the panels showing the real parts. For the imaginary part, dotted lines indicate the critical law of the host correlator, $G\left(t\right)\sim h{(t/t_0)}^{-a}$, and $h_{c,j}^s/\lambda \approx 3.985/\lambda$ times this critical law.

Figure 7

Normalized ratio of probe-correlation functions to that of the host liquid, $X_j\left(t\right)=[{\phi }_j^s\left(t\right)-{\tilde{f}}_j^s]/[\phi \left(t\right)-\tilde{f}]/\left(u_j^{s,c}{A}_{\delta }\right)$, cf. Eq. (25), as a function of $\widehat{t}=t/t_{\sigma }$. Parameters are chosen as in Fig. 6; only liquid curves, $\varepsilon =-2.5\times {10}^{-10}$ are shown, for $\delta =-{10}^{-n}$ with $n=0$, 1, 2, 3, 4, and 5 (indicated by the labels). The dotted line indicates $A_{\varepsilon }/A_{\delta }=\lambda =\sqrt{2}$. Squares in the upper panel indicate the time scale $t_{\delta }$.

Figure 8

Correlation functions ${\vec{\phi }}^s\left(t\right)-{\vec{f}}^s$ multiplied by $\sqrt{t}$, for $\varepsilon ={10}^{-6}$, and other parameters as in Fig. 6. Reduced forces are $\delta =-{10}^{-n}$ with $n=0,...,6$. The dash-dotted line shows the corresponding host-liquid correlator.

Figure 9

Test of the $\alpha$-sclaing law for the probe-particle correlation functions: The ${\vec{\phi }}_{\alpha }^s\left(t\right)/\left|\delta \right|$ are shown as functions of rescaled time $\tilde{t}=t/t_{\sigma ,\delta }^{\prime }$. Parameters are chosen as in Fig. 1, with $\varepsilon =-{10}^{-12}$. Reduced forces are given by $\delta =-{10}^{-n}$, and $n=0$, 1, 2, 3, 4, 5, and 6 as labeled.

Figure 10

Friction coefficient increment $\Delta \zeta =\zeta \left(F_{\mathrm{ex}}\right)-{\zeta }_0$ for a probe particle pulled with external force $F_{\mathrm{ex}}$, as a function of the distance to the delocalization threshold, $\delta =(F_{\mathrm{ex}}-F_{\mathrm{ex}}^{\text{c}})/F_{\mathrm{ex}}^{\text{c}}$, for $\delta >0$ and various distances to the glass transition $\varepsilon =\pm {10}^{-k}$ with $k=5$, 6, 7, and 8 as labeled. Dashed lines are in the glass ($\varepsilon >0$), solid lines in the liquid ($\varepsilon <0$). The curve at the glass transition, $\varepsilon =0$, is shown as a dash-dotted line. (Inset) Curves for $\varepsilon \ne 0$, as a scaling plot, $\Delta \widehat{\zeta }=\Delta \zeta /{\left|\varepsilon \right|}^{1/2-1/\left(2a\right)}$ versus $\widehat{\delta }=\delta /{\left|\varepsilon \right|}^{1/2}$.

Figure 11

Friction coefficient increment $\Delta \zeta$ as a function of the distance to the delocalization threshold $\delta$, for $\delta <0$ (i.e., $F_{\mathrm{ex}}<F_{\mathrm{ex}}^{\text{c}}$) in the liquid. The distance to the glass transition is chosen as $\varepsilon =-{10}^{-k}$, with $k=4$, 5, 6, 7, and 8 as labeled. (Inset) Scaling plot $\Delta \widehat{\zeta }$ versus $\widehat{\delta }$ for the same data and also including $k=9$ and $k=10$.

Figure 12

Correlations functions ${\phi }_{\parallel }^s\left(t\right)$ and ${\phi }_{\perp }^s\left(t\right)$ of the schematic model, with parameters as in Fig. 1 but a large external force, $F_{\mathrm{ex}}=402.7$. Full lines show the numerical solutions of the model, dashed lines the least-squares fits using an exponentially damped function ${\phi }_{\parallel }^s\left(t\right)=e^{-\nu t}{e}^{i(F_{\mathrm{ex}}+\xi )t}$ with parameters $\nu =22.6$ and $\xi =-2.1$. The host-liquid correlator $\phi \left(t\right)$ is shown as a dotted line in the top and bottom panels.