Nonequilibrium Cluster Diffusion During Growth and Evaporation in Two Dimensions

The diffusion of growing or evaporating two-dimensional clusters is investigated. At equilibrium, it is well known that the mean square displacement (MSD) of the cluster center of mass is linear in time. In nonequilibrium conditions, we find that the MSD exhibits a nonlinear time dependence, leading to three regimes: (i) during curvature-driven evaporation, the MSD shows a square-root singularity close to the collapse time; (ii) in slow growth or evaporation, the dynamics is in the Edwards-Wilkinson universality class, and the MSD shows a logarithmic behavior; (iii) far from equilibrium, the dynamics belongs to the Kardar-Parisi-Zhang universality class and the MSD shows a power-law behavior with a characteristic exponent $1/3$. These results agree with kinetic Monte Carlo simulations, and can be generalized to other universality classes.

Figure 1

KMC simulations with $k_{B}T=0.2J$. (a) Cluster area $A_{\mathrm{cl}}(t)$ averaged over 100 samples as a function of time $t$. (b) Mean square displacement averaged over 100 samples as a function of time. Three cases are shown: growth with chemical potential $\Delta \mu =0.05J$ and initial radius $R_0=20a_0$; saturation with $\Delta \mu =0$ and $R_0=400a_0$; evaporation with $\Delta \mu =-0.05J$ and $R_0=2500a_0$. In evaporation and at saturation, the cluster area $A_{\mathrm{cl}}(t)$ decreases and the MSD is faster than linear, while in growth, $A_{\mathrm{cl}}(t)$ increases and the MSD is slower than linear. Insets indicate typical shapes in (a), and trajectories of the cluster center of mass in (b).

Figure 2

Nonequilibrium MSD of two-dimensional clusters averaged over 100 samples. (a) Saturation case at $\Delta \mu =0$ for various initial radii $R_0$. The solid line represents Eq. (9). (b) Growth with $L_0=10a_0$, except for $\Delta \mu =0.01J$, where $L_0=100a_0$, and (c) evaporation with initial size $L_0=1768a_0$. The dashed (blue) line represents the close-to-equilibrium model, Eq. (11), and the solid (red) lines represent the far-from-equilibrium model, Eq. (17), with the noise amplitude $B_f$ as a fitting parameter.