Asymptotic behavior and noise reduction in diffusion-limited aggregation models

Mean-field diffusion-limited aggregation: a "density" model for viscous fingering phenomena

We explore a universal "density" formalism to describe nonequilibrium growth processes, specifically, the immiscible viscous fingering in Hele-Shaw cells (usually referred to as the Saffman-Taylor problem). For that we develop an alternative approach to the viscous fingering phenomena, whose basic concepts have been recently published in a Rapid Communication [Phys. Rev. E 63, 045305(R) (2001)]. This approach uses the diffusion-limited aggregation (DLA) paradigm as a core: we introduce a mean-field DLA generalization in stochastic and deterministic formulations. The stochastic model, a quasicontinuum DLA, simulates Monte Carlo patterns, which demonstrate a striking resemblance to natural Hele-Shaw fingers and, for steady-state growth regimes, follow precisely the Saffman-Taylor analytical solutions in channel and sector configurations. The relevant deterministic theory, a complete set of differential equations for a time development of density fields, is derived from that stochastic model. As a principal conclusion, we prove an asymptotic equivalency of both the stochastic and deterministic mean-field DLA formulations to the classic Saffman-Taylor hydrodynamics in terms of an interface evolution.

Diffusion-limited aggregation: a revised mean-field approach

We propose a revision of the classic mean-field approach of diffusion-limited aggregation (DLA) model originally introduced by Witten and Sander [Phys. Rev. Lett. 47, 1400 (1981)]. The derived nonlinear mean-field equations providing lattice anisotropy are used to model diffusional growth on square lattice in linear and circular source geometries. The overall cluster shapes obtained from the mean-field calculations are found to satisfy the known scaling behavior experimentally observed for DLA simulations.

Diffusion-limited aggregation: A relationship between surface thermodynamics and crystal morphology

We have combined the original diffusion-limited aggregation model introduced by Witten and Sander with the surface thermodynamics of the growing solid aggregate. The theory is based on the consideration of the surface chemical potential as a thermodynamic function of the temperature and nearest-neighbor configuration. The Monte Carlo simulations on a two-dimensional square lattice produce the broad range of shapes such as fractal dendritic structures, densely branching patterns, and compact aggregates. The morphology diagram illustrating the relationship between the model parameters and cluster geometry is presented and discussed.