Stochastic ratcheting of two-dimensional colloids: Directed current and dynamical transitions

We present results of molecular dynamics simulations for two-dimensional repulsively interacting colloids driven by a one-dimensional asymmetric and commensurate ratchet potential, switching on and off stochastically. This drives a time-averaged directed current of colloids, exhibiting resonance with change in ratcheting frequency, where the resonance frequency itself depends nonmonotonically on density. Using scaling arguments, we obtain analytic results that show good agreement with numerical simulations. With increasing ratcheting frequency, we find nonequilibrium reentrant transitions between solid and modulated liquid phases.

Figure 1

(a) Schematic of 2D colloids in 1D asymmetric ratchet potential periodic in the $y$ direction and constant in the $x$ direction. The arrow between the (green) corrugated and flat surfaces denotes switching of the external potential between the on and off states with rate $f$. (b) Time-averaged directed current along the $y$ direction $\langle j_y\rangle$ as a function of frequency $f$ and density $\rho$. The dashed line indicates variation of the maximal current with density. Superimposed positions of ${10}^3$ uncorrelated configurations, from center-of-mass coordinates, are shown at a density of $\rho =1.0$ with ratcheting frequencies (c) $f=0.11$, (d) $f=1.67$, and (e) $f=10$. The color code denotes the local density of points from red (light) (high) to blue (dark) (low). The reciprocal lattice vectors ${\mathbf{G}}_{1,2}$ and the corresponding lattice planes are indicated in (e).

Figure 2

Average directed current as a function of switching rate $f$ at particle densities $\rho =0.1$ (), $0.5$ (), and $1.0$ (). The solid lines show the fit to Eq. (2).

Figure 3

Resonance frequency $f_0$ as a function of the density $\rho$ for noninteracting particles (squares) and soft-core particles (circles). The solid line shows the ballistic form $f_0\sim \sqrt{\rho }$, while the dashed line shows the diffusive form $f_0\sim \rho (1-\rho /{\rho }_c)$ with ${\rho }_c=1.07$.

Figure 4

Average particle flux $\langle j_y\rangle$ as a function of density $\rho$ for soft-core particles at ratcheting frequencies $f=1.43$ (), $0.71$ (), and $0.36$ ().The dot-dashed lines are fit to Eq. (3) in the regime $\rho <0.5$ with fitting parameter $\kappa =0.04,0.03,0.02$ for the three data sets, respectively. The solid lines are fit to Eq. (4) in the regime $\rho \ge 0.5$ with fitting parameters $\kappa =0.12,0.05,0.02$ and ${\rho }_c=1.03,1.05,1.03$. The inset shows the same quantity for free particles at two different ratcheting frequencies $f=0.71$ () and $0.36$ (). The dashed lines are fit to Eq. (3).

Figure 5

Amplitude of the steady-state structure factor for the reciprocal lattice vectors (a) ${\mathbf{G}}_1$ and (b) ${\mathbf{G}}_2$ as a function of ratcheting frequency $f$ for densities $\rho =0.98$ (, ), $0.99$ (, ), $1.00$ (, ), $1.01$ (, ), and $1.02$ (, ).