Entropy and kinetics of point defects in two-dimensional dipolar crystals

We study in experiment and with computer simulation the free energy and the kinetics of vacancy and interstitial defects in two-dimensional dipolar crystals. The defects appear in different local topologies, which we characterize by their point group symmetry; $C_n$ is the $n$-fold cyclic group and $D_n$ is the dihedral group, including reflections. The frequency of different local topologies is not determined by their almost degenerate energies but is dominated by entropy for symmetric configurations. The kinetics of the defects is fully reproduced by a master equation in a multistate Markov model. In this model, the system is described by the state of the defect and the time evolution is given by transitions occurring with particular rates. These transition rate constants are extracted from experiments and simulations using an optimization procedure. The good agreement between experiment, simulation, and master equation thus provides evidence for the accuracy of the model.

Figure 1

Typical snapshots of the colloidal crystal obtained from the experiment containing an interstitial (upper row) and a vacancy (lower row), respectively. The motion of the defects consists of a sequence of transitions between states with different symmetries given by the dihedral $D_n$ and cyclic $C_n$ point groups in two dimensions. The upper left index denotes interstitial vs vacancies and the upper right counts the numbers of sevenfold particles (in the case of interstitials) and fivefold particles (in the case of vacancies). Particles with four, five, seven, and eight neighbors are shown in blue (gray), orange (light gray), green (medium gray), and purple (dark), respectively, and particles with six neighbors are shown in black (small dots). Note that the symmetry of the most frequent low-temperature configurations (second column) differ for interstitials and vacancies. The vacancies show only cyclic symmetry for low temperatures.

Figure 2

Probability density of the angle $\alpha$ between the 5-5 and 7-7 bonds for vacancies (left) and interstitials (right) consisting of two dislocations for different values of $\Gamma$. Red (dark) indicates high probabilities. For $\alpha =\pi /2$ the symmetry is dihedral, $D_2$. The vacancies are peaked at $\alpha \approx {90}^{\circ }\pm {38}^{\circ }$ for low temperatures (large $\Gamma$) with cyclic symmetry $C_2$.

Figure 3

Relative positions of fourfold-, fivefold-, and sevenfold-coordinated particles in the vicinity of an interstitial defect with respect to the defect position on the underlying triangular lattice from computer simulations. The configurations (left column) ${}^I{D}_2^0$ (a), ${}^I{D}_2^2$ (b), and ${}^I{D}_3^3$ (c) consist of at least two fivefold-coordinated particles [green (medium gray)]. The ${}^I{D}_2^0$ state consists of an additional fourfold-coordinated particle [yellow (light)] while the ${}^I{D}_2^2$ and ${}^I{D}_3^3$ states consist of additional sevenfold-coordinated particles [orange (light gray)]. The relative positions of fivefold-coordinated (middle column) and fourfold- and sevenfold-coordinated particles (right column) are measured with respect to the closest lattice point of the defect center and depicted as a heat map. Red indicates large probability and white low probabilities. Clearly, the fivefold-coordinated particles always occupy the same 12 lattice points with different weights depending on the state. The fivefold- and sevenfold-coordinated particles also follow different patterns.

Figure 4

Relative positions of fivefold-, sevenfold-, and eightfold-coordinated particles in the vicinity of a vacancy defect with respect to the defect position on the underlying triangular lattice from computer simulations as described in the caption of Fig. 3. A vacancy in state ${}^V{D}_2^0$ (a), ${}^V{C}_2^2$ (b), or ${}^V{D}_3^3$ (c) consists of at least two sevenfold-coordinated particles (orange) and either a single eightfold-coordinated (violet) particle (a) or additional fivefold-coordinated particles (green) (b) and (c). In all states, the sevenfold-coordinated particles are found at the same 12 positions (right) while the eightfold- and fivefold-coordinated particles are found at various other positions.

Figure 5

Typical trajectory of the state as a function of time (in units of Monte Carlo time steps) of an interstitial at $\Gamma =160$. The state is identified in each time step from the neighbor numbers of particles in the vicinity of the defect according to Table 1.

Figure 6

Time correlation functions $C_{ij}\left(t\right)$ for interstitials obtained from experiment (solid lines) in comparison to the result of the optimization procedure of $K_{ij}$ (dashed) for $\Gamma =140$. The symbols in the top right corner of each panel indicate the initial state $j$ and colors indicate the final state $i$, where black corresponds to ${}^I{D}_2^0$, red to ${}^I{D}_2^2$, green to ${}^I{D}_3^3$, and blue to ${}^V{I}_2^4$. For instance, the black lines in the top left panel represent the probability to find the defect in the ${}^I{D}_2^0$ configuration at time $t$ given that it was in ${}^I{D}_2^0$ at time $t=0$, while the red lines represent the probability to find it in ${}^I{D}_2^2$ at time $t$.

Figure 7

Correlation functions $C_{ij}\left(t\right)$ for interstitials obtained from simulation (solid lines) and from the optimization procedure of $K_{ij}$ (dashed). Parameters and colors as in Fig. 6.

Figure 8

Correlation functions $C_{ij}\left(t\right)$ for vacancies obtained from experiment (solid line) and from the optimization procedure of $K_{ij}$ (dashed). Parameters and colors as in Fig. 6.

Figure 9

Correlation functions $C_{ij}\left(t\right)$ for vacancies obtained from simulation (solid line) and from the optimization procedure of $K_{ij}$ (dashed). Parameters and colors as in Fig. 6.

Figure 10

Equilibrium populations of (a) interstitial and (b) vacancy states identified based on Table 1 as a function of $\Gamma$ from experiments (filled symbols) and simulations (solid lines). The dotted line denotes the probability to find the defect in a state not given in the table that is negligible for higher interaction strength $\Gamma$. Error bars depict the statistical error from finite numbers of measurements taking into account the correlations between the individual configurations.

Figure 11

Free-energy difference $\Delta F_{kl}$ as a function of $1/\Gamma$ for vacancies (a) and interstitials (b). The dashed lines are linear fits to the data obtained from computer simulations. According to Eq. (16), the intercepts of these functions corresponds to energy differences $\Delta W_{kl}$ shown in panel (c) for interstitials (black squares) and vacancies (red circles). The energies are given with respect to the lowest energy state, which is ${}^V{C}_2^2$ and ${}^I{D}_3^3$ for vacancies and interstitials, respectively. The slopes of the lines in panels (a) and (b) yield the entropy differences $\Delta S_{kl}$ shown in panel (d) for interstitials (black) and vacancies (red).

Figure 12

Color-coded optimized rate matrix ${\mathbf{K}}_{ij}$ for an interstitial (a) and (c) and a vacancy (b) and (d) from a trajectory from experiment (top) and simulation (bottom) at $\Gamma =140$.