Depinning dynamics of two-dimensional dusty plasmas on a one-dimensional periodic substrate

We investigate the depinning dynamics of two-dimensional dusty plasmas driven over one-dimensional periodic substrates using Langevin dynamical simulations. We find that, for a specific range of substrate strengths, as the external driving force increases from zero, there are three different states, which are the pinned, the disordered plastic flow, and the moving ordered states, respectively. These three states are clearly observed using different diagnostics, including the collective drift velocity, static structural measures, the particle trajectories, the mean-squared displacements, and the kinetic temperature. We compare the observed depinning dynamics here with the depinning dynamics in other systems.

Figure 1

Snapshots of particle positions (dots) and the locations of the 1DPS (curve) in a 2D Yukawa crystal with $\mathrm{\Gamma }=1000$ and $\kappa =2$ under different external driving forces for systems with the 1DPS $U\left(x\right)=U_{0}cos(2\pi x/w)$ of strength $U_0=0.005E_0$ and period $w=1.002b$. (a) At $F_d=0$, all particles are pinned at the bottom of each substrate potential well, forming a pinned state composed of 1D or quasi-1D chains. (b) At $F_d=0.007$, the particle distribution is disordered, which is independent of the spatial distribution of the potential wells. (c) At $F_d=0.02$, the particles are distributed nearly in a triangular lattice, which is also independent of the spatial distribution of the potential wells.

Figure 2

(a) The collective drift velocity $V_x$ vs. external driving force $F_d$ for substrate strengths $U_0=0.0025E_0, 0.005E_0$, and $0.01E_0$, respectively. (b) The corresponding fraction of sixfold coordinated particles [41] $P_6$ vs. $F_d$ for the same substrate strengths. As $F_d$ increases from zero, for the $U_0=0.0025E_0$ and $U_0=0.005E_0$ samples we observe three states: pinned, disordered plastic flow, and a moving ordered lattice. For comparison, the lines with small squares indicate the response for particles sliding freely without a substrate, $U_0=0$.

Figure 3

Particle configurations (left column) and 2D distribution functions $G_{xy}$ (right column) for the 2DDP system with $\mathrm{\Gamma }=1000$ and $\kappa =2$ on a 1DPS with $U_0=0.005E_0$ at drives of (a, d) $F_d=0$ in the pinned state, (b, e) $F_d=0.007$ in the disordered plastic flow state, and (c, f) $F_d=0.02$ in the moving ordered state. In the left column, particles are colored according to their coordination number of neighbors as marked in the legend at the bottom.

Figure 4

Typical particle trajectories from the simulated 2DDP for the same conditions as described in the caption of Fig. 3 under different constant external driving forces. (a) At $F_d=0$ in the pinned state, all of the particles are pinned around their equilibrium locations and undergo only the caged motion. (b) At $F_d=0.007$ in the disordered plastic flow state, some of the particles can escape from the cages and move out of the potential minima. These particles immediately become trapped in new cages or potential wells, from which they can once again escape, resulting in trajectories that are disordered along both the $x$ and $y$ directions. (c) At $F_d=0.02$ in the moving ordered state, the driving force is so large that the particles can easily cross over the potential barriers. As a result, the particles move predominantly parallel to the driving direction (the $x$ direction), with negligible motion in the $y$ direction. For each panel, the inset on the bottom left corner shows the particle trajectories in the inertial frame moving at the collective drift velocity $V_x$. These trajectories are obtained from $\approx 4\%$ of the simulation box over a time period representing $\approx 0.5\%$ of the full duration of the simulation.

Figure 5

Mean-squared displacement (MSD) calculated from the motion in the $y$ direction, marked as YMSD here, for the simulated 2DDP under the same conditions as described in the caption of Fig. 3 at different constant external driving forces of $F_d=0.0$ (squares), 0.007 (circles), and 0.02 (triangles). For both the initial ballistic motion at early times and the later diffusive motion at longer times, we find that, as the driving force increases from zero, the YMSD always increases first and then decreases again. Furthermore, for the later diffusive motion at $F_d=0.007$ in the disordered plastic flow state, we find $YMSD\propto t^{1.01}$. This behavior is distinct from that found in the pinned state, $\mathrm{YMSD}\propto t^{0.45}$, and in the moving ordered state, $\mathrm{YMSD}\propto t^{0.48}$. The simulation conditions of our 2D Yukawa crystal are $\mathrm{\Gamma }=1000$ and $\kappa =2$.

Figure 6

The kinetic temperature $k_B{T}_x$ (squares) and $k_B{T}_y$ (circles), calculated from the motion in the $x$ and $y$ directions, versus $F_d$ for the simulated 2DDP at the same conditions as in Fig. 3. The center-of-mass motion has been removed in calculating the kinetic temperature, so that only the fluctuations of the particle velocities are considered. As $F_d$ increases from zero, both $k_B{T}_x$ and $k_B{T}_y$ first simultaneously increase and then decrease. The initial increase is correlated with the transition from the pinned state to the disordered plastic flow state, while the decrease at higher drives is correlated with the transition from the disordered plastic flow state to the moving ordered state. In addition, in the disordered plastic flow state, we find that the kinetic temperatures are anisotropic, with $k_B{T}_x>k_B{T}_y$, due to the wider range of the $x$ direction velocity distribution in this phase, while in the pinned and moving ordered state, the anisotropy is lost and $k_B{T}_x\approx k_B{T}_y$. To quantify the variation in the kinetic temperature, we fit it to the expression $k_{B}T=C_1{|F_d-C_2|}^{\beta }+C_0$. Curves with the fitting exponents of $\beta =-0.2927$ and $-0.2934$ for $k_B{T}_x$ and $k_B{T}_y$, respectively, are presented.