Fluctuations of orientational order and clustering in a two-dimensional colloidal system under quenched disorder

Using both video microscopy of superparamagnetic colloidal particles confined in two dimensions and corresponding computer simulations of repulsive parallel dipoles, we study the formation of fluctuating orientational clusters and topological defects in the context of the KTHNY-like melting scenario under quenched disorder. We analyze cluster densities, average cluster sizes, and the population of noncluster particles, as well as the development of defects, as a function of the system temperature and disorder strength. In addition, the probability distribution of clustering and orientational order is presented. We find that the well-known disorder-induced widening of the hexatic phase can be traced back to the distinct development characteristics of clusters and defects along the melting transitions from the solid phase to the hexatic phase to the isotropic fluid.

Figure 1

Simulation snapshots illustrating orientational clusters close to the fluid $\to$ hexatic transition (${\Gamma }^{-1}=0.0149$). The field of view corresponds to $450\times 450\mu$m ($\approx$7% of the simulation cell). (a) Voronoi cells color-coded based on the time-averaged local orientational order parameter ${\Psi }_6\left(t\right)$. Colors are specified by the bar at the left. Crosses indicate the positions of pinned particles. (b) Orientational director field, where the complex number ${\Psi }_{6,i}\left(t\right)$ is shown as a 2D vector, the size of which corresponds to the magnitude of ${\Psi }_{6,i}\left(t\right)$. Outlines of clusters are indicated in black.

Figure 2

Phase diagram indicating the solid [blue (dark gray)], hexatic fluid [red (medium gray)], and isotropic fluid [yellow (light gray)] phases in the parameter space of the temperature $\propto$${\Gamma }^{-1}$ and pinning strength. Filled symbols represent experimental data, while open symbols correspond to simulation results [37].

Figure 3

Simulation snapshots of exemplary configurations for pinning fractions 0.1% (a–c) and 0.5% (d–f). Voronoi cells are shown for particles included in clusters. The color-code corresponds to the bar at the left, based on the normalized product ${\Psi }_{6,i}·{\Psi }_6/\left(\right|{\Psi }_{6,i}\left|\right|{\Psi }_6\left|\right)$. Crosses indicate the positions of pinned particles. At $\Gamma =0.0148$, both systems are in the isotropic fluid phase; the transition to the hexatic phase occurs at $\Gamma =0.0147$. At $\Gamma =0.0146$, the system with less pinning enters the solid phase (c), while the system with a higher pinning fraction remains in the hexatic phase (f) and fluctuations persist.

Figure 4

Fluctuating clusters in the hexatic phase (${\Gamma }^{-1}=0.0147$, 0.5% pinning). (a) Cluster configurations at three time steps: $t_0$ [black], $t_0+35{\tau }_B$ [medium blue (gray)], and $t_0+70{\tau }_B$ [light blue (light gray)]. Voronoi cells are shown for particles included in a cluster. (b) Cluster configurations at three time steps: $t_1$ [black], $t_1+700{\tau }_B$ [medium blue (gray)], and $t_0+1400{\tau }_B$ [light blue (light gray)]. Snapshots are from the computer simulation.

Figure 5

Average of the normalized number of orientational clusters $\langle N_C\rangle /\langle N\rangle$ versus the effective temperature for different pinning strengths. Experimental data are plotted with filled symbols, while open symbols represent numerical data. Lines are guides for the eye. The temperature range of the hexatic phase for 0.1% pinning is highlighted in red (medium gray); the widening of the hexatic phase at 0.5% pinning is illustrated in light red (light gray). Inset: Closeup of the average cluster number.

Figure 6

Top: Average ratio $\langle {\rho }_C\rangle$ of particles outside of clusters versus effective temperature. Inset: Closeup of the behavior in the hexatic phase. Bottom: Average size $\langle A_C\rangle /\langle N\rangle$ of clusters stated as a fraction of the average total particle number. Experimental data are plotted with filled symbols; open symbols represent numerical data. Lines are guides for the eye. As in Fig. 5, the temperature range of the hexatic phase is highlighted in red (gray) for two distinct pinning fractions.

Figure 7

Left: Computer simulation snapshot illustrating the subdivision into regions close to (I) and far from (II) pinned particles. Right: Instantaneous value of the order parameter ${\psi }_6$ in regions I and II versus the effective temperature ${\Gamma }^{-1}$. Lines correspond to computer simulation data (pinning fraction, 0.5%); experimental data are represented by symbols (crosses, region I; circles, region II; pinning fraction, 0.48%).

Figure 8

Conditional probability that a particle belongs to a cluster given that it is located in region I (triangles) or region II (squares) versus effective temperature ${\Gamma }^{-1}$. Data were obtained with computer simulations; the pinning fraction corresponds to 0.5%. Lines are guides for the eye. The temperature range of the hexatic phase is highlighted in light red (gray).

Figure 9

Finite-size analysis of the probability distribution of ${\Psi }_L^2$ calculated for subcells of side length $L$ (stated as a fraction of the total box length). Curves are shown for computer simulation data; the pinning fraction corresponds to 0.5%. (a) ${\Gamma }^{-1}=0.0148$: isotropic fluid phase, where the peak of the probability distribution is located at ${\Psi }_L^2=0$. (b) ${\Gamma }^{-1}=0.0147$: fluid $\to$ hexatic transition, where the peak shifts to intermediate values of ${\Psi }_L^2$. (c) ${\Gamma }^{-1}=0.0146$: hexatic phase.

Figure 10

Probability distribution of ${\Psi }_L^2$ for simulation (left) and experiment (right) for $L=1/4$ (stated as a fraction of the total system dimension). During the crossover from isotropic to hexatic fluid, the location of the peak shifts from ${\Psi }_L^2=0$ to intermediate values. The distribution has a single distinct peak, indicating a continuous transition. The pinning fraction is 0.5% in the simulation and 0.48% in the experiment.

Figure 11

Defect density $\langle N_{\text{def}}\rangle /\langle N\rangle$ versus effective temperature. The number of particles forming bound dislocations (squares), isolated dislocations (triangles), or single five- or sevenfold disclinations (circles) is stated as a fraction of the total particle number. Data correspond to computer simulations with pinning fraction 0.1% [open (blue) symbols] and 0.5% (filled black full symbols). Lines are guides for the eye. The common temperature range of the hexatic phase for both pinning fractions is highlighted in red (medium gray). For 0.5% pinning, the widened range of the hexatic phase is shown in light red (light gray). Inset: Closeup on the defect density in the (widened) hexatic phase.