Continuum limit of dendritic deposition

Continuum models are commonly used to study dendritic deposition in fields ranging from nonequilibrium statistical mechanics to battery research. However, the continuum approximation underlying these models is poorly understood, even in the simplified case of Brownian particles depositing onto a small, reactive cluster. Specifically, this system transitions from a compact to a dendritic morphology at a critical radius that depends on the particle size. But in simulations of the continuum (small-particle) limit, the critical radius does not reproduce the scaling predicted by a purely continuum analysis. This discrepancy suggests that continuum models may not be able to capture the microscopic physics of dendrite formation, raising doubts about their experimental relevance. To clarify the continuum limit of dendritic deposition, here, we reexamine the critical radius scaling of the Brownian particle system using Brownian dynamics simulations. Compared to past studies, we probe larger system sizes, up to hundreds of millions of particles in some cases, and adopt an improved paradigm for the surface reaction. This paradigm allows us to converge our simulations and to work with well-defined physical parameters. Our results show that the critical radius scaling is, in fact, consistent with the continuum analysis, validating the continuum approach to modeling dendritic deposition. Nonetheless, the Brownian particle system converges to its continuum limit slowly. As a result, when applying continuum models to more complex deposition processes, the continuum approximation itself may be a significant source of error.