  Computational studies of crystal nucleation can be impacted by finite size
effects, primarily due to unphysical interactions between crystalline nuclei
and their periodic images. It is, however, not always feasible to
systematically investigate the sensitivity of nucleation kinetics and mechanism
to system size due to large computational costs of nucleation studies. Here, we
use jumpy forward flux sampling to accurately compute the rates of
heterogeneous ice nucleation in the vicinity of square-shaped model
structureless ice nucleating particles (INPs) of different sizes, and identify
three distinct regimes for the dependence of rate on the INP dimension, $L$.
For small INPs, the rate is a strong function of $L$ due to artificial spanning
of critical nuclei across the periodic boundary. Intermediate-sized INPs,
however, give rise to the emergence of non-spanning 'proximal` nuclei that are
close enough to their periodic images to fully structure the intermediary
liquid. While such proximity can facilitate nucleation, its effect is offset by
the higher density of the intermediary liquid, leading to artificially small
nucleation rates overall. The critical nuclei formed at large INPs are neither
spanning nor proximal. Yet, the rate is a weak function of $L$, with its
logarithm scaling linearly with $1/L$. The key heuristic emerging from these
observations is that finite size effects will be minimal if critical nuclei are
neither spanning nor proximal, and if the intermediary liquid has a region that
is structurally indistinguishable from the supercooled liquid under the same
conditions.
