Morphological transition between diffusion-limited and ballistic aggregation growth patterns
In this work, the transition between diffusion-limited and ballistic aggregation models was revisited using a model in which biased random walks simulate the particle trajectories. The bias is controlled by a parameter $λ$, which assumes the value $λ=0$ (1) for ballistic (diffusion-limited) aggregation model. Patterns growing from a single seed were considered. In order to simulate large clusters, a new efficient algorithm was developed. For $λ\ne 0$, the patterns are fractal on the small length scales, but homogeneous on the large ones. We evaluated the mean density of particles $\barρ$ in the region defined by a circle of radius $r$ centered at the initial seed. As a function of $r$, $\barρ$ reaches the asymptotic value $ρ_0(λ)$ following a power law $\barρ=ρ_0+Ar^{-γ}$ with a universal exponent $γ=0.46(2)$, independent of $λ$. The asymptotic value has the behavior $ρ_0\sim|1-λ|^β$, where $β= 0.26(1)$. The characteristic crossover length that determines the transition from DLA- to BA-like scaling regimes is given by $ξ\sim|1-λ|^{-ν}$, where $ν=0.61(1)$, while the cluster mass at the crossover follows a power law $M_ξ\sim|1 -λ|^{-α}$, where $α=0.97(2)$. We deduce the scaling relations $β=\n uγ$ and $β=2ν-α$ between these exponents.