Modified winterbottom construction including boundaries

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.