Nonequilibrium pattern formation in circularly confined two-dimensional systems with competing interactions

We numerically investigate the nonequilibrium behaviors of classic particles with competing interactions confined in a two-dimensional logarithmic trap. We reveal a quench-induced surprising dynamics exhibiting rich dynamic patterns depending upon confinement strength and trap size, which is attributed to the time-dependent competition between interparticle repulsions and attractions under a circular confinement. Moreover, in the collectively diffusive motions of the particles, we find that the emergence of dynamic structure transformation coincides with a diffusive mode transition from superdiffusion to subdiffusion. These findings are likely useful in understanding the pattern selection and evolution in various chemical and biological systems in addition to modulated systems, and add a new route to tailoring the morphology of pattern-forming systems.

Figure 1

Particle positions and velocities at different times for the modulated system undergoing expansion by applying a sudden jump of confinement strength from ${\beta }_s=15.1$ to ${\beta }_f=0.1$. Only one quarter of the sample is shown for clarity, while the inserts show the entire area. Both the color code and the arrow length indicate the magnitude of particle velocity. The arrow shows the instantaneous direction of motion, and its tail is at the particle position. The times $t$ at which the figures were taken are as follows: (a) $t=0$, (b) $t=1$, (c) $t=2$, (d) $t=3$, (e) $t=4$, (f) $t=6$, (g) $t=9$, (h) $t=13$, and (i) $t=100$. See Supplementary Movie S1 for a movie of the entire explosion sequence [46].

Figure 2

The sequential trajectories with different durations $t$ for the explosion shown in Fig 1: (a) $t=0\to 1$, (b) $t=1\to 5$, (c) $t=5\to 10$, and (d) $t=10\to 20$. In both panels (a) and (b), the central region of the sample is shown for clarity, and the inserts show the entire area. The insert in panel (d) is the enlarged plot of one local clumplike cluster. The trajectories clearly show collective motions of the particles in the clump over very short distances.

Figure 3

Temporal and spatial characterization of the explosion dynamics of the modulated system. (a) Time evolution of mean-squared displacement scaled by the time shows a diffusion crossover from superdiffusive to subdiffusive behaviors at $t_{\mathrm{tr}}=2.3\tau$. (b) Time evolution of the displacements for four selected particles labeled A, B, C, and D in Fig. 2. (c) The radial distribution function $g_2\left(r\right)$ of the particle distributions at different times. (d) Time-space plot of average radial density.

Figure 4

Time evolution of mean-squared displacement scaled by the time in log-log plot and linear plot (insert): (a) for different confinement strengths ${\beta }_f$ at $R=50$ and ${\beta }_s=15.1$; (b) for different trap sizes $R$ at ${\beta }_s=15.1$ and ${\beta }_f=0.1$.

Figure 5

Particle positions at different times for the modulated system undergoing expansion by applying a sudden jump of confinement strength from ${\beta }_s=15.1$ to ${\beta }_f=2$ at $R=50$: (a) $t=0$, (b) $t=1$, (c) $t=2$, (d) $t=5$, (e) $t=7$, and (f) $t=200$. See Supplementary Movie S2 for a movie of the entire explosion sequence [46].

Figure 6

Particle positions at different times for the modulated system undergoing expansion by applying a sudden jump of confinement strength from ${\beta }_s=15.1$ to ${\beta }_f=0.1$ at $R=65$: (a) $t=0$, (b) $t=2$, (c) $t=3$, (d) $t=6$, (e) $t=20$, and (f) $t=200$. See Supplementary Movie S3 for a movie of the entire explosion sequence [46].