Interface collisions

We provide a theoretical framework to analyze the properties of frontal collisions of two growing interfaces considering different short-range interactions between them. Due to their roughness, the collision events spread in time and form rough domain boundaries, which defines collision interfaces in time and space. We show that statistical properties of such interfaces depend on the kinetics of the growing interfaces before collision, but are independent of the details of their interaction and of their fluctuations during the collision. Those properties exhibit dynamic scaling with exponents related to the growth kinetics, but their distributions may be nonuniversal. Our results are supported by simulations of lattice models with irreversible dynamics and local interactions. Relations to first passage processes are discussed and a possible application to grain-boundary formation in two-dimensional materials is suggested.

Figure 1

Interface collision models. (a) Growth models. Random deposition (RD) model: at each time step, an incident particle sticks with probability $p$ in each column. Modified Family model: the incident particle aggregates at the column of incidence if no nearest-neighbor (NN) column has smaller height; if only one NN has a smaller height, it aggregates at that column, and if two NN columns have smaller heights, one of them is randomly chosen. RSOS models: the particle sticks only if the resulting differences of heights between all NN columns do not exceed 1 (RSOS model) or 2 (RSOS2 model). (b) Schematics of collision with short-range interaction, with growth ceasing at columns 0, 3, and 7. (c) Phantom collision, where growth continued in columns 0 (advance of upper interface) and 3 (advance of lower interface). (d) Collision simulation ($d_0=2^9$) with two interfaces growing with the RSOS model in opposite directions and colliding with short-range interaction.

Figure 2

Distributions of collision times and loci. Models: (a) RD-RD with $p_{+}=0.5$ and $p_{-}=1$, (b) Family-Family, (c) RSOS-RSOS, (d) RSOS-RSOS2, (e) RSOS-Family. Symbols indicate simulation results with $d_0=2^{12}$ and $L=2^{32}$, and with short-range interaction $[\square$: $P_c\left(\delta h_c\right)$, $▵$: $F_c\left(\delta t_c\right)]$ or phantom collision $[\times$: $P_c\left(\delta h_c\right)$, $*$: $F_c\left(\delta t_c\right)]$. Thick curves are obtained from Eqs. (5a) and (5b) (green, blue) using distributions from interfaces without collision calculated numerically. Thin lines are distributions obtained from Eqs. (5a) and (5b) (red, orange) using theoretical universal distributions (Gaussian or Tracy-Widom) with variances extracted from simulations of interfaces without collisions. In (b), $F_c\left(\delta t_c\right)$ is shifted down by one unit for the sake of clarity.