Grain-boundary-induced melting in quenched polycrystalline monolayers

Melting in two dimensions can successfully be explained with the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) scenario which describes the formation of the high-symmetry phase with the thermal activation of topological defects within an (ideally) infinite monodomain. With all state variables being well defined, it should hold also as freezing scenario where oppositely charged topological defects annihilate. The Kibble-Zurek mechanism, on the other hand, shows that spontaneous symmetry breaking alongside a continuous phase transition cannot support an infinite monodomain but leads to polycrystallinity. For any nonzero cooling rate, critical fluctuations will be frozen out in the vicinity of the transition temperature. This leads to domains with different director of the broken symmetry, separated by a defect structure, e.g., grain boundaries in crystalline systems. After instantaneously quenching a colloidal monolayer from a polycrystalline to the isotropic fluid state, we show that such grain boundaries increase the probability for the formation of dislocations. In addition, we determine the temporal decay of defect core energies during the first few Brownian times after the quench. Despite the fact that the KTHNY scenario describes a continuous phase transition and phase equilibrium does not exist, melting in polycrystalline samples starts at grain boundaries similar to first-order phase transitions.

Figure 1

Illustration of the initial defect configuration of the two polycrystalline samples $A$ (a) and $B$ (b) ($1{\mathrm{mm}}^2,\approx 5900$ particles). Particles with six nearest neighbors (sixfold coordinated) are colored gray (bright), five- and sevenfold coordinated particles orange and green, and particles with less than five or more than seven nearest neighbors blue (dark). While sample $A$ is in initial state of high polycrystallinity, sample $B$ consists of two large monocrystalline domains separated by one high-angle grain boundary and some isolated, frozen-in defects. Particles additionally indicated with smaller black dots are defects for more than $50\%$ of the time before the polycrystalline structure is quenched to the isotropic fluid.

Figure 2

Time evolution of the defect density $\rho$ for samples $A$ (solid line) and $B$ (dashed line) shortly before and after an instantaneous quench (around $\approx 150\mathrm{s}$) from the polycrystal to the isotropic fluid. The samples start at different initial defect densities ${\rho }_0$ and approach a common value at large times which is a function of the final value of the control parameter $\mathrm{\Gamma }$. The inset shows the reduced density $\rho -{\rho }_0$ shifted with respect to the onset of the quench to $t-t_c$.

Figure 3

Time averaged formation probabilities ${〈P_{6\to d}〉}_t$ of defects for sample $A$ (a) and $B$ (b), as a function of distance $x$ to grain boundaries and within the first Brownian times. The average is performed within an interval of $\pm 0.1{\tau }_B$ and error bars are standard deviations $\left(\pm \sigma \right)$ from the time average. Probabilities are enhanced close to the grain boundaries, decay towards the bulk (middle of crystalline domains), and raise as a function of time. Solid lines are exponential fits.

Figure 4

Time evolution of bulk defect probabilities $P_0$ and respective defect core energies $E_c$ (inset) after the onset of the quench $\left(t_c\right)$ for both samples. Error bars are standard deviations $\left(\pm \sigma \right)$ of the fit parameter $P_0$. At short times, we observe a linear time evolution for $P_0$ (solid lines are linear fits to $P_0$) and core energies decay at large times. Bracketed data points denote time intervals where the exponential fit $P\left(x\right)$ failed due to too large fluctuations of ${〈P_{6\to d}〉}_t$.