  When Brownian particles (such as ions, colloids, or misfolded proteins)
deposit onto a reactive cluster, the cluster undergoes a transition from a
compact to a dendritic morphology. Continuum modeling reveals that the critical
radius for this compact-to-dendritic (CTD) transition should be proportional to
the particle diffusivity divided by the surface reaction rate. However,
previous studies have had limited success verifying that the same scaling
arises in the continuum limit of a particle-based deposition model. This
discrepancy suggests that the continuum model may be missing part of the
microscopic dendrite formation mechanism, a concerning hypothesis given that
similar models are commonly used to study dendritic growth in electrodeposition
and lithium metal batteries. To clarify the accuracy of such models, we
reexamine the particle-based CTD transition using larger system sizes, up to
hundreds of millions of particles in some cases, and an improved paradigm for
the surface reaction. Specifically, this paradigm allows us to converge our
simulations and to work in terms of experimentally accessible parameters. With
these methods, we show that in both two and three dimensions, the behavior of
the critical radius is consistent with the scaling of the continuum model. Our
results help unify the particle-based and continuum views of the CTD
transition. In each of these cases, dendrites emerge when particles can no
longer diffuse around the cluster within the characteristic reaction timescale.
Consequently, this work implies that continuum methods can effectively capture
the microscopic physics of dendritic deposition.
