Heterogeneities and topological defects in two-dimensional pinned liquids

We simulate a model of repulsively interacting colloids on a commensurate two-dimensional triangular pinning substrate where the amount of heterogeneous motion that appears at melting can be controlled systematically by turning off a fraction of the pinning sites. We correlate the amount of heterogeneous motion with the average topological defect number, time-dependent defect fluctuations, colloid diffusion, and the form of the van Hove correlation function. When the pinning sites are all off or all on, the melting occurs in a single step. When a fraction of the sites are turned off, the melting becomes considerably broadened and signatures of a two-step melting process appear. The noise power associated with fluctuations in the number of topological defects reaches a maximum when half of the pinning sites are removed and the noise spectrum has a pronounced $1∕f^{\alpha }$ structure in the heterogeneous regime. We find that regions of high mobility are associated with regions of high dislocation densities.

Figure 1

Location of the pinning sites (circles) and colloids (black dots). Large dark circles indicate pinning sites with a finite $f_p$ while small gray circles indicate pins that have been shut off by setting $f_p=0.0$. In this image, the fraction of sites with finite $f_p$ is $n_p=0.5$.

Figure 2

(a) The fraction of sixfold coordinated colloids $P_6$ vs $T$ for various $n_p$. From right to left, $n_p=1.0$, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1, and 0. (b) The corresponding $d{P}_6∕dT$ curves for three representative pinning fractions: $n_p=0.1$ (left curve), 0.5 (center dark curve), and 0.9 (right curve).

Figure 3

The width of the melting curve versus pinning fraction $n_p$ obtained for the system in Fig. 2.

Figure 4

(a) Colloidal trajectories (black lines) and colloidal positions (black dots) over a fixed period of time in a system with $n_p=0.5$ at $T=1.5$. (b) Voronoi plot of the system in (a) for a single frame. Small black dots indicate sixfold-coordinated particles while large black dots indicate particles with coordination numbers other than 6. (c) Time-averaged dislocation probability plot. Larger circles indicate sites that are frequently nonsixfold coordinated, while smaller circles indicate sites that are nearly always sixfold coordinated.

Figure 5

Colloidal trajectories (black lines) and positions (black dots) for a system at fixed $T=1.5$ for (a) $n_p=0.2$ and (b) $n_p=0.8$.

Figure 6

Time series of the fraction of colloids with sixfold coordination, $P_6$, for a system with $n_p=0.5$. Top curve, $T=1.5$; lower curve, $T=4.0$.

Figure 7

Power spectrum $S\left(f\right)$ for the same system in Fig. 6 at (a) $T=1.5$ and (b) $T=4.0$.

Figure 8

Noise power $S_0$ vs $T$ for a system with $n_p=0.0$ (squares) and $n_p=0.5$ (circles).

Figure 9

The exponent $\alpha$, obtained from the power spectrum, vs $T$ for the same system in Fig. 6 with $n_p=0.5$.

Figure 10

Diagram of the different regimes for $T$ vs pinning strength $f_p$ at fixed $n_p=0.5$. Circles indicate the onset of the first topological defects. Squares show the temperature at which $P_6$ first passes below 0.5.

Figure 11

(a) Diffusion $D$ of the initially unpinned particles vs $T$ for a system with $f_p=2.0$ and $n_p=\left(\text{left}\right)$ 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 (right). (b) $D$ vs $T$ for the same system in (a) for the initially pinned colloids at $n_p=\left(\text{left}\right)$ 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 (right).

Figure 12

Open circles: temperature at which the diffusion noticeably increases for the unpinned colloids from Fig. 11a. Solid squares: temperature at which the diffusion noticeably increases for the pinned colloids from Fig. 11b. Dashed lines are linear fits which both have the same slope.

Figure 13

The self-part of the van Hove correlation distribution function for the $x$ direction for $f_p=2.0$, $T=1.0$, and (a) $n_p=0.0$, (b) $n_p=0.5$, (c) $n_p=0.1$, and (d) $n_p=0.9$.