Dislocations and vacancies in two-dimensional mixed crystals of spheres and dimers

In colloidal crystals of spheres, dislocation motion is unrestricted. On the other hand, recent studies of relaxation in crystals of colloidal dimer particles have demonstrated that the dislocation dynamics in such crystals are reminiscent of glassy systems. The observed glassy dynamics arise as a result of dislocation cages formed by certain dimer orientations. In the current study, we use experiments and simulations to investigate the transition that arises when a pure sphere crystal is doped with an increasing concentration of dimers. Specifically, we focus on both dislocation caging and vacancy motion. Interestingly, we find that any nonzero fraction of dimers introduces finite dislocation cages, suggesting that glassy dynamics are present for any mixed crystal. However, we have also identified a vacancy-mediated uncaging mechanism for releasing dislocations from their cages. This mechanism is dependent on vacancy diffusion, which slows by orders of magnitude as the dimer concentration is increased. We propose that in mixed crystals with low dimer concentrations vacancy diffusion is fast enough to uncage dislocations and delay the onset of glassy dislocation dynamics.

Figure 1

(a) Optical microscope image of a mixed sphere and dimer crystal with ${\varphi }_d=0.4$. Dumbbells are overlaid on top of the dimers for clarity. The bright white sphere is an intruder particle. (b) Distribution of dimer orientations within the grain. Error bars are derived from counting statistics.

Figure 2

(Color online) Example of a numerically calculated mixed crystal with dimer concentration ${\varphi }_d=0.3$. (a) The lattice sites that are dimer bonded are shaded in matching darker hues, while the sphere lattice sites are uniquely shaded lighter hues. Inset: zoomed-in view illustrating the shade coding for dimers versus spheres. (b) The distribution of dimer orientations for this mixed crystal demonstrates that the three lattice directions are equally weighted. Error bars are derived from counting statistics.

Figure 3

(Color online) The distribution of cage sizes from numerically simulated mixed crystals for each dimer concentration. Marker key—from top curve to bottom curve: ${\varphi }_d=1.0$ (solid black stars), 0.9 (solid purple diamonds), 0.8 (solid blue triangles), 0.7 (solid green circles), 0.6 (solid yellow squares), 0.5 (open red stars), 0.4 (open purple diamonds), 0.3 (open blue triangles), 0.2 (open green circles), and 0.1 (open yellow squares). The curves are fits as described in the text. The distribution is also shown in a semilogarithmic plot as the inset.

Figure 4

The peak cage size $z_{\ast }$ as a function of dimer concentration ${\varphi }_d$. The solid line is the best-fit power law, as described in the text. The dotted line is the theoretical prediction $z_{\ast }=6/{\varphi }_d$, which ignores correlations between dimer orientations. Error bars derived from counting statistics are comparable to the marker size.

Figure 5

(Color online) Time series micrographs of a climb event in a very low dimer concentration mixed crystal. (a) A dislocation gliding along a crystal plane is blocked by the dimer highlighted in red (grey). (b) When the dislocation reaches the caging dimer, it produces a vacancy (highlighted with a black circle) and climbs by one lattice constant, thereby escaping the cage and continuing to glide unobstructed.

Figure 6

The fraction $f$ of vacancies that have completed at least one lattice hop within a waiting time $\tau$ is plotted for both experimental crystals of pure dimers (open circles) and experimental crystals of pure spheres (closed circles). Error bars derived from counting statistics are comparable to the marker size. The clear separation between the two curves indicates that the hopping time is slower by a factor of approximately 3 orders of magnitude in crystals of dimers compared to crystals of spheres.

Figure 7

(Color online) Mechanisms for vacancy diffusion in pure dimer crystals. (a) and (b) In an experimental dimer crystal, a vacancy hops one lattice site via the swinging motion of a neighboring dimer. This is the only vacancy hop mechanism that has been observed in the experiments. (c) A cartoon depiction of the maximal overlap area during a simplified model of a dimer swinging move and a dimer sliding move. The total overlap area required for a sliding move is twice that for a swinging move.

Figure 8

(Color online) Vacancy mean-squared displacement in simulated mixed crystals of dimers and spheres. Marker key—from bottom curve to top curve: ${\varphi }_d=1.0$ (solid black stars), 0.9 (solid purple diamonds), 0.8 (solid blue triangles), 0.7 (solid green circles), 0.6 (solid yellow squares), 0.5 (open red stars), 0.4 (open purple diamonds), 0.3 (open blue triangles), 0.2 (open green circles), 0.1 (open yellow squares), and 0.0 (open black stars). Inset: the best-fit scaled diffusion constant $D/D_s$ over the range $\tau =1–{10}^3$ simulation time steps is plotted as a function of dimer fraction ${\varphi }_d$. The solid curve is the theoretical prediction $D/D_s={\left[1-\left(3/5\right){\varphi }_d\right]}^2$.

Figure 9

(Color online) Schematic showing the possible dimer orientations with one dimer lobe in a potential vacancy site. The current vacancy site is marked with an open circle, and the potential vacancy site is marked with a star. Of the five possible dimer orientations shown with dashed lines, the three highlighted in red with thicker dashed lines do not allow dimer swinging.

Figure 10

(Color online) Schematic of wedge-shaped confinement cell construction. The wedge angle is exaggerated for clarity. (a) One base-washed coverslip (blue) is bonded with UV glue to a larger microscope slide (beige) to provide structural support for the cell. (b) Small UV glue droplets (yellow) are placed in two rows on the coverslip and are cured to become spacers for the cell. (c) A second base-washed coverslip is placed on top of the first, and the two are squeezed together until the two coverslips bond together (bonded region shown in white). (d) UV glue (shown in yellow) is then used to seal three sides of the cell, leaving an access gap for inserting colloidal samples. Once the cell is filled with sample, this gap is also sealed using UV glue.