Nonequilibrium fluctuation-dissipation relations of interacting Brownian particles driven by shear

We present a detailed analysis of the fluctuation-dissipation theorem (FDT) close to the glass transition in colloidal suspensions under steady shear using mode coupling approximations. Starting point is the many-particle Smoluchowski equation. Under shear, detailed balance is broken and the response functions in the stationary state are smaller at long times than estimated from the equilibrium FDT. An asymptotically constant relation connects response and fluctuations during the shear driven decay, restoring the form of the FDT with, however, a ratio different from the equilibrium one. At short times, the equilibrium FDT holds. We follow two independent approaches whose results are in qualitative agreement. To discuss the derived fluctuation-dissipation ratios, we show an exact reformulation of the susceptibility which contains not the full Smoluchowski operator as in equilibrium, but only its well defined Hermitian part. This Hermitian part can be interpreted as governing the dynamics in the frame comoving with the probability current. We present a simple toy model which illustrates the FDT violation in the sheared colloidal system.

Figure 1

Definition of the waiting time $t_w$ and the correlation time $t$ after switch-on of the rheometer.

Figure 2

(Color online) The estimates for the Laplace transform $\Delta {\chi }_{\mathbf{q}}\left(z=0\right)$ of the violating term in density susceptibilities for $\mathbf{q}=q\hat{\mathbf{y}}$. We show the coherent (incoherent) case with data points connected (unconnected) by lines. Solid squares show the estimate for the first term, the waiting-time derivative, solid spheres show the sum of the two other terms. Open squares show the sum of all three. The solid (dashed) line shows the prediction of Eq. (25) for the waiting-time derivative for the coherent (incoherent) case; compare with the solid squares and see main text. The picture is very similar for the $z$ direction [51].

Figure 3

(Color online) $C\left(t,t_w\right)$ from Eq. (30) for different waiting times as indicated by the different line styles. The $F_{12}^{\left(\dot{\gamma }\right)}$ model Eq. (28) is used to generate the transient correlators $C^{\left(t\right)}$ (solid lines). We show a glassy state $\left(\epsilon ={10}^{-3}\right)$ at shear rates $\dot{\gamma }/\Gamma ={10}^{-8}$ (curves A) and $\dot{\gamma }/\Gamma ={10}^{-6}$ (B). Curves C and D show the same respective shear rates for a liquid state $\left(\epsilon =-{10}^{-3}\right)$. The thin dashed line shows the equilibrium correlator $C^{\left(e\right)}\left(t\right)$ for the glassy state. For the liquid state, the correlators for $\dot{\gamma }/\Gamma ={10}^{-8}$ and $\dot{\gamma }=0$ coincide.

Figure 4

(Color online) $C\left(t\right)$ from the $F_{12}^{\left(\dot{\gamma }\right)}$ model with Eq. (30) and $\chi \left(t\right)$ via Eq. (34) for a glassy state $\left(\epsilon ={10}^{-3}\right)$ and $\dot{\gamma }/\Gamma ={10}^{-2n}$ with $n=1\dots 4$. Shown are integrated correlation, $1-C\left(t\right)$ and response ${\chi }^{\prime }\left(t\right)={\int }_0^t\chi \left(t^{\prime }\right)d{t}^{\prime }$. Inset shows additionally the transient correlator $\Phi$ for comparison and the ${\widehat{X}}^{\left(\text{univ}\right)}=\frac{1}{2}$ susceptibility for $\dot{\gamma }/\Gamma ={10}^{-8}$. From Ref. [35].

Figure 5

(Color online) Parametric plot of correlation $C\left(t\right)$ versus response ${\chi }^{\prime }\left(t\right)={\int }_0^t\chi \left(t^{\prime }\right)d{t}^{\prime }$ from Fig. 4 $\left(\epsilon ={10}^{-3}\right)$ together with constant nontrivial FDR (straight lines) at long times. The vertical solid line marks the plateau $f$. Inset shows the FDR $X\left(t\right)$ as function of strain for the same susceptibilities. From Ref. [35].

Figure 6

(Color online) $C\left(t\right)$ and $\chi \left(t\right)$ via Eq. (34) for a fluid state $\left(\epsilon =-{10}^{-3}\right)$ and $\dot{\gamma }/\Gamma ={10}^{-2n}$ with $n=1\dots 4$. Shown are integrated correlation, $1-C\left(t\right)$ and response ${\chi }^{\prime }\left(t\right)={\int }_0^t\chi \left(t^{\prime }\right)d{t}^{\prime }$. Inset shows the parametric plot for the different shear rates.

Figure 7

(Color online) Long time FDR as function of shear rate for glassy $\left(\epsilon ={10}^{-n}\right)$ and liquid $\left(\epsilon =-{10}^{-n}\right)$ states with $n=2,3,4$. $X\left(t\to \infty \right)$ is determined from fits to the parametric plot as shown in Figs. 5, 6. Inset shows the long time FDR as function of wave vector $q$ for coherent (solid line) and incoherent (dashed line) density fluctuations at the critical density $\left(\epsilon =0\right)$.

Figure 8

(Color online) Comparison to simulation data for incoherent density fluctuations in the neutral direction (wave vector $\mathbf{q}=7.47{\mathbf{e}}_z$) at temperature $T=0.3\left(T_c=0.435\right)$ and $\dot{\gamma }={10}^{-3}$. Circles and squares are the data (including units) from Fig. 11 in Ref. [10], lines are $1-C_{\mathbf{q}}$ from Fig. 8 in Ref. [10], and the response ${\chi }_{\mathbf{q}}^{\prime }\left(t\right)={\int }_0^t{\chi }_{\mathbf{q}}\left(t^{\prime }\right)d{t}^{\prime }$ calculated via Eq. (34). The dashed line shows ${\chi }_{\mathbf{q}}^{\prime }$ with approximation $C_{\mathbf{q}}^{\left(t\right)}\approx C_{\mathbf{q}}$. Inset shows the different correlators, see main text.

Figure 9

The shear step model: the particle is trapped in wells of width $a$ with infinitely high potential barriers. The center-to-center distance of the wells is denoted $b$.

Figure 10

(Color online) Mean squared displacement without (squares) and mean traveled distance with external force $F$ (circles) in the shear step model with $b=a$ and $\Theta =2a^2/\left(2D_0\right)$. Solid line shows the short time asymptote of Eqs. (51, 54). Dashed line shows the plateau value of $\frac{a^2}{6}$, Eqs. (52, 57). The dotted line shows the long time asymptote from Eq. (53), dash-dotted line the asymptote from Eq. (59).

Figure 11

Parametric plot of MSD versus mean traveled distance for $b=a$ and $\Theta =2a^2/\left(2D_0\right)$. Solid line shows the short time relation $X=1$. Dashed line shows the long time FDR $X=\frac{1}{2}$. Inset shows the same graph zoomed into the short time part.

Figure 12

Long time FDR in the shear step model as function of ’shear rate’ $1/\Theta$.

Figure 13

(Color online) Long time FDR as function of wavevector $q$ for density fluctuations in the $z$- and $y$-directions in the limit of small shear rates. The packing fraction is the critical one $\left(\epsilon =0\right)$.