Driving kinetically constrained models into nonequilibrium steady states: Structural and slow transport properties

Complex fluids in shear flow and biased dynamics in crowded environments exhibit counterintuitive features which are difficult to address both at a theoretical level and by molecular dynamic simulations. To understand some of these features we study a schematic model of a highly viscous liquid, the two-dimensional Kob-Andersen kinetically constrained model, driven into nonequilibrium steady states by a uniform non-Hamiltonian force. We present a detailed numerical analysis of the microscopic behavior of the model, including transversal and longitudinal spatial correlations and dynamic heterogeneities. In particular, we show that at high particle density the transition from positive to negative resistance regimes in the current vs field relation can be explained via the emergence of nontrivial structures that intermittently trap the particles and slow down the dynamics. We relate such spatial structures to the current vs field relation in the different transport regimes.

Figure 1

Scheme of the model rules and the role of the kinetic constraints: particles (red) can move and occupy empty sites if they have at least two empty neighbors before and after the move. The external field is uniform and biases the movement of the particles from left to right reducing the probability of backward moves.

Figure 2

(a) Current $J$ vs field $E$ relation for subcritical densities, $\rho <{\rho }_{\mathrm{c}}\simeq 0.79$. The system size is $L^2={50}^2$. The current is a monotonic function of the field, saturating at large field. Its behavior is qualitatively similar to the ASEP on a 2D square lattice. (b) The density dependence of the current vs field relation can be easily accounted for by rescaling $J\left(E\right)$ with the saturation current $J_{\mathrm{sat}}\left(\rho \right)$. The current ratio $J/J_{\mathrm{sat}}$ is exactly equal to the difference between forward and backward transition probabilities, $1-e^{-E}$.

Figure 3

(a) The current as a function of the density for our model with the kinetic constraint in the limit of very strong fields. The peak is at $\rho \approx 0.4$ while for the classical TASEP (blue dashed line) the peak is at $\rho =0.5$. The mean-field formula (3) (red full line) shows a good qualitative agreement that becomes quantitative at densities below the peak value. (b) The most surprising property of the model: at high densities, $\rho >{\rho }_{\mathrm{c}}\simeq 0.79$, the simulations provide a current-field relation which is nonmonotonic, as discussed in Ref. [8]. Simulation data at $\rho >{\rho }_{\mathrm{c}}$ are obtained by using systems of size $L^2={400}^2$.

Figure 4

Average velocity field for an $L=100$ system at density $\rho =0.80$, with $E$ oriented left-to-right. In the top panels (a) and (b) the system is in the linear regime, $E=0.1$, while in the bottom panels (c) and (d) the system is in the negative differential resistance regime, $E=2.8$. The time window considered for the evaluation of the average speed is $t_w\approx 1{\tau }_{\mathrm{rel}}$. At small fields the dynamics is homogeneous, while in the negative resistance regime, shear bands appear. The boxes (b) and (d) represent the longitudinal projections of the velocity vectors averaged over each horizontal line of particles as a function of the transversal coordinate $y$.

Figure 5

Snapshots of two steady-state configurations with particles (white) and holes (black) for a system of size $L=100$ and $\rho =0.82$ at (a) $E=0$ and (b) $E=5$ with $E$ being in the horizontal direction (left to right). One can notice the vertical (transversal to the field) structures arising in the strong field case (b).

Figure 6

Some examples of real space particle trajectories of equal duration $\tau ={10}^4$ in the two current regimes for a system of density $\rho =0.80$: (a) at small fields, $E=0.1$, and (b) large fields, $E=2.8$, corresponding to a negative resistance behavior. One can observe the mainly diffusive, Brownian behavior (a) vs the directed behavior (b) where many steps are spent in wandering moves in the transversal direction. Each arrow corresponds to a directed step, and the axes are in lattice spacing units.

Figure 7

(a) Transversal ($\times$) and longitudinal ($+$) persistence $\phi \left(t\right)$ vs time $t$ for particle density $\rho =0.86$ and applied field $E=2.0$, corresponding to the negative differential resistance regime. Square lattice of linear size $L=500$. (b) Transversal and longitudinal persistence fluctuations ${\chi }_4$ for $\rho =0.86$ and $E=2.0$. Square lattice of linear size $L=100$.

Figure 8

(a) Transversal and (b) longitudinal two-point correlation functions for $\rho =0.86$ for several applied fields $E$. Square lattice of linear size $L=500$.

Figure 9

Transversal part of van Hove self-correlation function at particle density $\rho =0.8$ and field $E=2.8$ (negative resistance regime). The system consists of a square lattice of linear size $L=50$.

Figure 10

Longitudinal part of van Hove self-correlation function $G_s(r,t)$ vs position $r$ at time $t$, particle density $\rho =0.8$, and field $E=2.8$ (negative resistance regime) for a system of linear size $L=50$. There is long-lived asymmetry induced by the interplay of external drift and kinetic constraints. The exponential tails observed at short times eventually become more and more Gaussian at longer times.

Figure 11

Time averaged longitudinal ($+$) and transversal ($\times$) mean-square displacement $\Delta r^2/t$ for a square lattice of linear size $L=50$. The normal diffusive behavior $\Delta r^2\sim t$ corresponds to horizontal lines; a negative or positive slope corresponds to a sub- or super-diffusion regime, respectively.

Figure 12

Longitudinal $w_l$ (open symbols) and transversal $w_t$ (filled symbols) average wall sizes for different values of the density as a function of the external forcing. The effect of the field on the longitudinal size is negligible with respect to the effect on the transversal size. Moreover, the higher the density, the stronger is the effect.

Figure 13

At high densities and strong fields, longitudinal clusters of holes (empty squares) are easily reduced to transversal structures and basins. (a) The limit case of a single longitudinal strip of vacancies accessed by a mobile particle (bold bordered) is shown; in (b) the particle enters the strip, freeing some new mobile particles; (c) the initial particle is pushed rightward by the field; and finally (d) the remaining mobile particles follow an analogous path and a transversal basin of holes is formed.

Figure 14

(a) Distribution of the transversal wall sizes. Excluding the very small structures necessary to the diffusive dynamics, clear exponential tails are established, defining a typical transversal wall length. (b) Distributions of the transversal wall sizes for different values of the external field, rescaled with respect to the average wall's size value at densities $0.79,0.80,0.81,\text{and}0.82$. For each value of the external field, the distributions computed at different densities collapse and at a large field converge to the same distribution.

Figure 15

Simulated flow curves (white squares) fitted by the phenomenological model $J\left(E\right)=A(1-e^{-E})(1-\alpha \langle w\rangle )$ (black dots). Densities range from $0.795$ to $0.825$, $L=100$, and $\langle w\rangle$ is determined independently from the simulation.