Two-Dimensional Melting under Quenched Disorder

We study the influence of quenched disorder on the two-dimensional melting behavior of superparamagnetic colloidal particles, using both video microscopy and computer simulations of repulsive parallel dipoles. Quenched disorder is introduced by pinning a fraction of the particles to an underlying substrate. We confirm the occurrence of the Kosterlitz-Thouless-Halperin-Nelson-Young scenario and observe an intermediate hexatic phase. While the fluid-hexatic transition remains largely unaffected by disorder, the hexatic-solid transition shifts to lower temperatures with increasing disorder. This results in a significantly broadened stability range of the hexatic phase. In addition, we observe spatiotemporal critical(like) fluctuations, which are consistent with the continuous character of the phase transitions. Characteristics of first-order transitions are not observed.

Figure 1

Temporal bond orientational correlation function $g_6(t)$ in the presence of quenched disorder plotted versus reduced time $t/{\tau }_B$ on a double-logarithmic scale. The fraction of pinned particles is 0.48% in the experiment and 0.5% in the simulation. Exemplary curves are shown for the isotropic fluid [green (light gray)], hexatic [red (black)], and solid [blue (dark gray)] phase, where experimental data are drawn with solid lines, and computer simulations with dashed lines.

Figure 2

(a) Orientational correlation time ${\xi }_t$ and (b) Frank’s constant $K_A$, for different concentrations of pinned particles. Filled symbols represent experimental data, open symbols simulation. The meaning of the symbols is the same in (a) and (b); lines are guides to the eye. While ${\xi }_t$ is almost not affected by different pinning strengths, $K_A$ is clearly lowered with increasing pinning.

Figure 3

Phase diagram indicating the solid [blue (dark gray)], hexatic [red (light gray)] and isotropic fluid [green (medium gray)] phase in the parameter space of temperature $\propto {\Gamma }^{-1}$ and pinning strength. Full symbols represent experimental data, while open symbols correspond to simulation results. Letters indicate the location of snapshots in Fig. 4.

Figure 4

(a)–(c) Snapshots of the experimental system at 0.48% pinning, showing the local orientational order parameter $⟨|{\psi }_6|{⟩}_t$ averaged over $\approx 50{\tau }_B$ in the different phases [(a) ${\Gamma }^{-1}=0.0117$, (b) ${\Gamma }^{-1}=0.0143$, (c) ${\Gamma }^{-1}=0.0154$]. The field of view is $450\times 450\mu \mathrm{m}$. Voronoi cells are color coded according to the bar on the left. (d) Mean square displacement (MSD) calculated for particles within a distance of $8d$ around pinning sites (region I in the inset) and more than $24d$ away from them (region II). Temperatures correspond to (a)–(c).