Static and dynamic properties of two-dimensional Coulomb clusters

We study the temperature dependence of static and dynamic responses of Coulomb interacting particles in two-dimensional confinements across the crossover from solid- to liquid-like behaviors. While static correlations that investigate the translational and bond orientational order in the confinements show the footprints of hexatic-like phase at low temperatures, dynamics of the particles slow down considerably in this phase, reminiscent of a supercooled liquid. Using density correlations, we probe long-lived heterogeneities arising from the interplay of the irregularity in the confinement and long-range Coulomb interactions. The relaxation at multiple time scales show stretched-exponential decay of spatial correlations in irregular traps. Temperature dependence of characteristic time scales, depicting the structural relaxation of the system, show striking similarities with those observed for the glassy systems, indicating that some of the key signatures of supercooled liquids emerge in confinements with lower spatial symmetries.

Figure 1

(a) The ground-state $(T=0)$ configuration of $N=500$ particles in an irregular trap ($\lambda =0.635,\gamma =0.20$). The orange dots are selected central particles for the search of positional order (see Sec. 3a), while all other particles are represented by blue dots. Panels (b)–(d) show Fourier transforms of particle density for various cases: Panel (b) is for the ground state with all particles shown in panel (a). Panel (c) is for the orange particles only in the central region of panel (a). Panel (d) is for all particles, but at $T=0.02\sim T_X$, the crossover temperature. In panels (b)–(d), magnitude of $\rho \left(\vec{k}\right)$ is scaled to unity for visual clarity.

Figure 2

(a) The ground-state ($T=0$) configuration of $N=500$ particles in circular confinement. The orange dots are particles in the central region selected to study positional order in the bulk, while all other particles are represented by blue dots. Panels (b)–(d) show Fourier transform of particle density for various cases: Panel (b) is for the ground state with all particles shown in panel (a). Panel (c) is for the particles only in the central region of panel (a). Panel (d) is for all particles, but at $T=0.02\sim T_X$. In panels (b)–(d), magnitude of $\rho \left(\vec{k}\right)$ is scaled to unity for visual clarity.

Figure 3

Evolution of the the distribution, $P\left(\left|{\psi }_6\right|\right)$, of bond orientational order, $\left|{\psi }_6\right|$, with $T$ is shown for $N=500$ in panels (a) and (b), for $N=150$ in panels (c) and (d), and for $N=75$ in panels (e) and (f) in irregular and circular confinements, respectively. The low-$T$ peak in $P\left(\right|{\mathrm{\Psi }}_6\left|\right)$ at $|{\mathrm{\Psi }}_6|\sim 1$, that signifies the “solidity,” smears out as the system undergoes the thermal crossover. The small and weak peak in $P\left(\right|{\mathrm{\Psi }}_6\left|\right)$ at $|{\mathrm{\Psi }}_6|\sim 1$ at high $T$, particularly for smaller $N$, is a spurious one (see text) and does not constitute bond orientational order.

Figure 4

Thermal evolution of the distribution, $P\left({\varphi }_6\right)$, of ${\varphi }_6$, representing the bond orientational correlation up to nearest neighboring distance (see text), is shown for $N=500$ in panels (a) and (b), for $N=150$ in panels (c) and (d), and for $N=75$ in panels (e) and (f) in irregular and circular confinements, respectively. $P\left({\varphi }_6\right)$ features bimodal structures: The peak near ${\varphi }_6\sim 1$ (at low $T$) signals solidity, and the one at ${\varphi }_6\approx 0$ portrays liquidity.

Figure 5

$T$ dependence of the generalized susceptibilities identifying the crossover temperature $T_X$, for $N=500,150,75$. Panels (a) and (c) show the thermal evolution of ${\chi }_{\psi }$ and ${\chi }_{\varphi }$ respectively in irregular confinement. While the peaks of ${\chi }_{\psi }$ and ${\chi }_{\varphi }$ sharpen with $N$, their location, $T_X$, suffer hardly any change. Panels (b) and (d) show results similar to those in panels (a) and (c) but for circular trap. The fragile order in this case for $N=75$ degrades $T_X$. Panels (e) and (f) illustrate the $T$ dependence of the fraction of solid- and liquid-like particles in the system for circular and irregular trap. The crossing of the two takes place in a narrow window of $T$ (for different $N$) for $V_{\mathrm{conf}}^{\mathrm{Ir}}\left(r\right)$, whereas the temperature window is broad for $V_{\mathrm{conf}}^{\mathrm{Cr}}\left(r\right)$.

Figure 6

$t$ dependence of temporal bond orientational correlation function, $g_6\left(t\right)$, at different $T$ for $N=500$ [panels (a) and (b)], $N=150$ [panels (c) and (d)], and $N=75$ [panels (e) and (f)] particles in irregular and circular confinements, respectively. Solid lines are the appropriate fitting to the actual data (points). The results confirm three qualitatively different evolution: The low-$T$ flat traces typify the solid-like behavior, the power-law decay for intermediate $T$ is reminiscent of the bulk 2D hexatic trend, while the large temperature exponential fall represent isotropic liquid nature.

Figure 7

(a) Initial positions of $N=150$ particles (blue dots) in a specific realization of an irregular confinement along with the trajectories of three particles at $T=0.006$. Thick black lines indicate the net displacement of the particles over the duration of the simulation. [(b), (c)] Trajectories of the particles 2 and 3 with larger displacements, as shown in panel (a). Both of them jiggle around their equilibrium position for a while and they make occasional and sudden jumps by distance $\sim r_0$. Particle 1, however, does not make such hops (not shown separately). (d) $t$ dependence of the instantaneous distance $r_i\left(t\right)$ from the center of the confinement for the three selected particles $(i=1,2,3)$. Displacement ($\mathrm{\Delta }{\vec{r}}_i\left(t\right)=\{{\vec{r}}_i\left(t\right)-{\vec{r}}_i\left(0\right)\}$) of $N=500$ particles in irregular confinement is shown for (e) $t=1000$ and (f) $t=9500$ at $T=0.006$.

Figure 8

(a) Initial positions of $N=150$ particles (blue dots) in circular confinement along with the trajectories of three particles at $T=0.006$. Solid lines indicate the net displacement of the particles over the duration of the simulation. [(b), (c)] Trajectories of the particles 2 and 3 with larger displacements, as shown in panel (a). (d) $t$ dependence of the instantaneous distance $r_i\left(t\right)$ from the center of the confinement for the three selected particles $(i=1,2,3)$. Displacement $\left[\mathrm{\Delta }{\vec{r}}_i\left(t\right)\right]$ of $N=500$ particles in circular confinement is shown for (e) $t=1000$ and (f) $t=9500$ at $T=0.006$.

Figure 9

Illustration of the intriguing spatial evolution of $G_s(r,t)\sim \exp [-l{r}^{k_{\mathrm{Ir}}}]$ by depicting the $t$ dependence of ${\mathrm{k}}_{\mathrm{Ir}}$ (see text), at different $T$ for $V_{\mathrm{conf}}^{\mathrm{Ir}}$. The results are analyzed for (a) $N=500$, (b) $N=150$, and (c) $N=75$. Main panels show the $t$ dependence of ${\mathrm{k}}_{\mathrm{Ir}}$ beyond the crossover temperature, while the insets depict the $t$ dependence of ${\mathrm{k}}_{\mathrm{Ir}}$ at low $T$.

Figure 10

$t$ dependence of the exponent ${\mathrm{k}}_{\mathrm{Cr}}$, depicting the nature of the decay of $G_s(r,t)$ at different $T$ for circular confinement, is analyzed for (a) $N=500$, (b) $N=150$, and (c) $N=75$. Main panels show the $t$ dependence of ${\mathrm{k}}_{\mathrm{Cr}}$ beyond the crossover temperature, while the insets depict the $t$ dependence of ${\mathrm{k}}_{\mathrm{Cr}}$ at low $T$.

Figure 11

The time dependence of the overlap function $Q\left(t\right)$ for different $T$ for $N=500$ [panels (a), (b)], $N=150$ [panels (c), (d)], and $N=75$ [panels (e), (f)] in irregular and circular confinements, respectively. Solid lines represent appropriate fitting of the data (points). While at low $T,Q\left(t\right)$ decays very slowly, it falls to zero rapidly for $T>0.006$.

Figure 12

The $t$ dependence of ${\chi }_4\left(t\right)$ for different $T$ for $N=500$ [panels (a), (b)], $N=150$ [panels (c), (d)], and $N=75$ [panels (e), (f)] in irregular and circular confinements, respectively. For a given $N,{\chi }_4\left(t\right)$ attains maximum at time ${\tau }_{\chi }$ which decreases with increase in $T$.

Figure 13

$T$ dependence of ${\tau }_{\chi }$ along with ${\tau }_{\alpha }$, obtained from $Q\left(t\right)$ for irregular (a) and circular (b) confinements, respectively, with $N=150$. Both the time scales show similar $T$ dependences. The $T$-dependence of ${\tau }_{\alpha }$ for different $N$ is shown in panel c,d for irregular and circular confinements, respectively. The solid line shows the exponential fitting $({\tau }_{\alpha }\propto exp[-A/T])$ to the actual data (points).

Figure 14

Decay of cage correlation function, $c_g\left(t\right)$, depicting how the initial neighbors fail, with time, to confine a particle, on average, at different $T$ for $N=500$ [panels (a), (b)], 150 [panels (c), (d)], and 75 [panels (e), (f)] for irregular and circular confinements, respectively. The solidity at low $T$ is reflected in the fact that the same neighbors continue to cage a particle at all times. The solid line shows the best fit to the actual data (points).

Figure 15

$T$ dependence of average caging $\left({\tau }_C\right)$ and noncaging $\left({\tau }_{NC}\right)$ times for different $N$ for irregular (a) and circular (b) confinements. Crossover temperature $T_X$ is estimated from the crossing of ${\tau }_C$ and ${\tau }_{NC}$. Inset in panel (b) shows the $N$ dependence of the estimated $T_X$ for both the confinements.

Figure 16

Distribution of persistence $\left({\tau }_p\right)$ and exchange $\left({\tau }_e\right)$ times for $N=500$ [panels (a), (b)], $N=150$ [panels (c), (d)], and $N=75$ [panels (e), (f)] particles in irregular and circular confinements, respectively. In the figure all quantities are expressed as log to the base e. The decoupling of the two distributions at low $T$ demonstrates the qualitative similarity of the particle dynamics in irregular traps with those in glassy systems. Such decoupling is hard to discern (except for $T=0.010$) for circular traps for all $N$.

Figure 17

$T$ dependence of structural relaxation time ${\tau }_{\alpha }$, average persistence time $\langle {\tau }_p\rangle$, and cage correlation time ${\tau }_g$ for (a) irregular and (b) circular confinements with $N=150$. Dotted lines are for visual guidance. Panels (c) and (d): Thick dots show the cross-correlation between ${\tau }_{\alpha },\langle {\tau }_p\rangle$, and ${\tau }_g$ in a log-log plot for irregular and circular confinement, respectively, taken over a temperature range $0.01\le T\le 0.05$. Solid lines represent the best fit to such correlations. $\langle {\tau }_p\rangle$ is scaled by appropriate factors for visual clarity.

Figure 18

(a) Radial distribution, $n\left(r\right)$, of the $N=500$ particles in circular confinement at $T=0.002$. The dashed line shows the cutoff distance $R_c$ used to identify the bulk particles. (b) Distribution of the bond orientational order ${\psi }_6$ for particles having $r$ (distance from the center of the confinement) $<R_c$ (bulk particles; solid line) and those for which $r>R_c$ (boundary particles; dashed line) in circular confinement at $T=0.002$.

Figure 19

$t$ dependence of exponent $g_6\left(t\right)$ for different $N$ at $T=0.006$ [panels (a) and (b)], $T=0.020$ [panels (c) and (d)], and $T=0.050$ [panels (e) and (f)] for irregular and circular confinements, respectively. Points represent the actual data while the solid line represents an appropriate fit of the selected data (determined by the extent of the solid line).

Figure 20

$t$ dependence of exponent $k\left(t\right)$ for different $N$ at $T=0.006$ [panels (a), (b)], $T=0.020$ [panels (c), (d)], and $T=0.050$ [panels (e), (f)] for irregular $\left(k_{\mathrm{Ir}}\right)$ and circular $\left(k_{\mathrm{Cr}}\right)$ confinements, respectively.

Figure 21

Determination of the cutoff $r_c$ for $V_{\mathrm{conf}}^{\mathrm{Ir}}$ with $N=150$, by tracking its value for which ${\chi }_4\left(t\right)$ achieves the maximum keeping temperature fixed at its lowest value ($T=0.002$). This optimal $r_c=0.09$ is then used for the evaluation of the overlap function $Q\left(t\right)$.

Figure 22

$t$ dependence of exponent $Q\left(t\right)$ for different $N$ at $T=0.006$ [panels (a), (b)], $T=0.020$ [panels (c), (d)], and $T=0.050$ [panels (e), (f)] for irregular and circular confinements, respectively.

Figure 23

$t$ dependence of exponent $c_g\left(t\right)$ for different $N$ at $T=0.006$ [panels (a) and (b)], $T=0.020$ [panel (c) and (d)], and $T=0.050$ [panels (e) and (f)] for irregular and circular confinements, respectively.