Self-organized defect strings in two-dimensional crystals

Using experiments with single-particle resolution and computer simulations we study the collective behavior of multiple vacancies injected into two-dimensional crystals. We find that the defects assemble into linear strings, terminated by dislocations with antiparallel Burgers vectors. We show that these defect strings propagate through the crystal in a succession of rapid one-dimensional gliding and rare rotations. While the rotation rate decreases exponentially with the number of defects in the string, the diffusion constant is constant for large strings. By monitoring the separation of the dislocations at the end points, we measure their effective interactions with high precision beyond their spontaneous formation and annihilation, and we explain the double-well form of the dislocation interaction in terms of continuum elasticity theory.

Figure 1

Formation of a defect string with $N=9$ vacancies observed in our experiments. (a)–(b) The initial configuration is created by removing 9 particles with strong laser pulses from a perfect triangular colloidal crystal. Along the sequence (c)–(f), the vacancies gradually coalesce and annihilate, eventually leaving behind a pair of dislocations with Burgers vectors of opposite direction. Colors distinguish particles with 8 neighbors (blue), 7 neighbors (pink), 6 neighbors (gray), and 5 neighbors (yellow). After about 6 min the vacancies have clustered into a vacancy string terminated by two dislocations. This behavior is in perfect agreement with Monte Carlo simulations performed at the conditions of the experiments. (f) To follow the dynamics of the string in detail we monitor the position of the two dislocations, each of which corresponds to a 5–7 coordinated particle pair and is indicated by a white $\mathsf{T}$ symbol. The dislocations diffuse along parallel lattice lines, indicated by dashed lines, parallel to the Burgers vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$. While the perpendicular separation $n$ between the dislocations is fixed by the number $N$ of vacancies introduced into the system, the parallel distance $s$ changes as the dislocations move under the influence of their effective interaction.

Figure 2

Dynamics of defect strings. Left: Trajectory obtained from the experiment of the dislocation positions, shown as red and blue dots. Dislocations first move along two parallel lines in one of the main lattice directions and then change their direction of motion during a rapid reorientation event that turns the Burgers vectors of the dislocations by $\pi /3$. This combination of gliding and rotation is consistent with computer simulations (right). As discussed below, rotation events can only occur at particular points [see Fig. 4 (top right panel)]. The long-time diffusion of defect strings of various lengths from $N=3$ to $N=9$ is depicted as an inset. Here, each point corresponds to the center of mass of the two dislocations indicating the position of the string. While the diffusion constant is roughly independent of $N$, the rotation probability decreases exponentially with $N$ leading to long gliding periods for large $N$ (green).

Figure 3

Diffusion constant $D_{\mathrm{string}}$ of vacancy strings as a function of defect number $N$. Experimental results (blue solid line) are shown together with results of the simulations (red dashed line). In the simulations, averages and error bars are calculated from 12 blocks of $4\times {10}^6$ MC sweeps each. The inset shows the rotation rate $k_{\mathrm{rot}}$ [25] of vacancy strings as a function of $N$. A fit of $k\left(N\right)\propto \exp (-\beta N{e}_a)/N$, shown as a dashed line, yields an activation energy per defect of $e_a=1.13k_{\mathrm{B}}T$.

Figure 4

Free energy $\beta \Delta F$ as a function of the scaled distance $x=2s/\left(\sqrt{3}N\right)$ in the direction of motion for defect strings with lengths from $N=2$, 3, 4, 5, and 9. Experimental results (blue solid lines) are shown together with results of simulations (red dashed lines) and the prediction of elasticity theory (black line) [see Eq. (2)]. The top right inset depicts a color coded histogram of vacancy locations for $N=8$ and one string end at the origin (blue dot). The vacancy string follows the same double-well pattern with angle $\pi /4$ (dotted lines) as the endpoints. The relative distance between dislocations (dashed lines) is constrained to six lines with 12 rotation points (green dots) where the string can change the gliding orientation. These rotation points are the only points where the constraint of perpendicular distance $n$ is satisfied for two different orientations. Results from computer simulations are obtained using $N=3120$ under periodic boundary conditions. Simulations carried out with different system sizes indicate that finite-size effects can be neglected.