Enhanced Diffusion of a Needle in a Planar Array of Point Obstacles

The transport of an infinitely thin, hard rod in a random, dense array of point obstacles is investigated by molecular dynamics simulations. Our model mimics the sterically hindered dynamics in dense needle liquids. Transport becomes increasingly fast at higher densities, and we observe a power-law divergence of the diffusion coefficient with exponent 0.8. This phenomenon is connected with a new divergent time scale, reflected in a zigzag motion of the needle, a two-step decay of the velocity-autocorrelation function, and a negative plateau in the non-Gaussian parameter. Finally, we provide a heuristic scaling argument for the new exponent.

Figure 1

Geometry and notation for the needle hitting an obstacle. The momentum transfer $\Delta \mathbf{p}\parallel {\widehat{\mathbf{u}}}_{\perp }$ is uniquely determined for a frictionless, elastic collision; it induces also a torque changing the angular velocity of the needle.

Figure 2

Simulation results for the diffusion coefficients of the needle. Solid lines show the low-density behavior, $D\sim 1/n$, dashed lines are fits to the asymptotic high-density behavior, and the dotted line indicates slope $1/2$ for comparison with a scaling prediction.

Figure 3

Snapshots of the needle at $n^{*}=120$. The time interval between subsequent exposures, $\Delta t=1.8L/v$, covers about 180 collisions; colors encode the longitudinal velocity, $v_{\parallel }/v_{\max }$.

Figure 4

Time series of the kinetic degrees of freedom for $n^{*}=120$. Sections of a trajectory may be divided into different states: “running” ($A$), “rattling” ($B$), and “running backwards” ($\overline{A}$); see lower panels. The right-hand panels enlarge a single event of velocity reversal.

Figure 5

Two-step relaxation of the c.m. velocity-autocorrelation function ${\psi }_{\mathrm{c}.\mathrm{m}.}(t)$. Rescaling of time with the collision rate collapses the data for low densities ($n^{*}\le 2$). In the dense regime, a universal intermediate plateau develops at $1/2$ for ${\tau }_{\mathrm{coll}}\ll t\ll {\tau }_{\mathrm{zz}}$; it is preceded by a dip reflecting the transversal “rattling in the cage.” Inset: Correlation functions of the different DOFs entering Eq. (3).

Figure 6

The non-Gaussian parameters for translational and rotational DOFs, Eq. (5), show a rich structure at high densities including the confinement in a tube, the zigzag motion, and translation-rotation coupling. Density increases from left to right, and arrows indicate the relevant time scales for $n^{*}=800$.