  This study proposes a new analytical model for grain boundary pinning by
second phase particles in two-dimensional polycrystals. This approach not only
considers how particles impede grain growth, but also elucidates their role in
preventing grain disappearance, thereby leading to stabilised microstructures
characterised by heterogeneous grain size distribution comprising a mixture of
small and large grains. By quantifying the number of particles intercepted by
grain boundaries during grain growth or shrinkage, we are able to calculate the
respective sizes and fractions of large and small grains. Furthermore, we
identify ranges of particle surface fractions and particle sizes that maximise
the heterogeneity in grain size. Additionally, we demonstrate the significant
influence of initial grain size on the limiting grain size in pinned
microstructures. Our analytical model's results are compared with those
obtained from full-field level-set simulations conducted in this study and from
phase-field calculations reported in the literature, revealing very good
agreement. Finally, the differences between the proposed model and existing
ones in the literature are discussed.
