Modified Winterbottom Construction Including Boundaries

Abstract

There has been extensive work on the equilibrium shape of isolated nanoparticles with internal boundaries and also single crystals on substrates. Surprisingly, almost shockingly, there has been very little work on the equilibrium shape of particles with internal boundaries on substrates. Here, the general solution is given for the configuration of particles containing twin and other grain boundaries on a flat substrate, which can be applied to any polycrystalline or multiphase nanoparticle configuration. The solution is based upon combining the established modified Wulff construction that has been extensively validated for twinned particles with the Winterbottom construction for single particles on a substrate. The solution is illustrated for the specific case of five-fold multiply twinned particles (MTPs). Good agreement is observed between both existing experimental data in the literature as well as some experimental data included within this work.

Approaches to modelling the shape of nanocrystals

“…The case of a supported decahedral particle of FCC has been explored in detail [ 19 ] and matches well with experimental evidence. Extensions to other twinned shapes and comparison with experimental data should be straightforward as long as one considers that the approach may only yield local, rather than global, minima.…”

“…The twin energy/growth velocity is considered to be negligible. TEM images of (c) a single Pt crystal grown on a SrTiO 3 nanocube, adapted with permission from [ 46 ], and ( f ) a five-fold Au Marks decahedron grown on a faceted LnScO 3 nanoparticle, adapted with permission from [ 19 ]. Copyright (2021) American Chemical Society …”

“…The modified Winterbottom construction was recently developed [ 19 ] for supported nanocrystals with internal (twin) boundaries such as MTPs. Similarly to the thermodynamic Winterbottom construction, an effective interfacial energy term is introduced for the crystal-substrate interface; this replaces the term for single crystals by allowing different contributions from multiple differently oriented sub-units.…”

Identifying Support Effects in Au-Catalyzed CO Oxidation

“…The supported Au nanoparticles had both single-crystalline and polycrystalline morphologies, as shown in Figure b, with most of the polycrystals exhibiting a twinned structure; the occurrence of twins in gold nanoparticles is very common, see refs − as well as references therein. Of relevance later, there was no notable difference in the height to width ratio of the particles on the different substrates, that is, the Winterbottom shape and its equivalent for twinned particles, indicating that the free energy of the interfaces on the substrates had minimal dependence on both particle morphology and the lanthanide. ,− …”

“…The supported Au nanoparticles had both single-crystalline and polycrystalline morphologies, as shown in Figure 3b, with most of the polycrystals exhibiting a twinned structure; the occurrence of twins in gold nanoparticles is very common, see refs 44−47 as well as references therein. Of relevance later, there was no notable difference in the height to width ratio of the particles on the different substrates, that is, the Winterbottom shape 48 and its equivalent for twinned particles, 49 indicating that the free energy of the interfaces on the substrates had minimal dependence on both particle morphology and the lanthanide. 13,47−49 Figure 3a shows the average size of both the as-deposited (blue) and the post-CO oxidation (orange) Au nanoparticles on each support, with detailed statistics in Supporting Information, including the density of Au nanoparticles on the support surface (note that the very low loadings were not deposited on NdScO 3 ; this is also discussed in Supporting Information).…”