The Peierls--Nabarro FE model in two-phase microstructures -- a comparison with atomistics

This paper evaluates qualitatively as well as quantitatively the accuracy of a recently proposed Peierls--Nabarro Finite Element (PN-FE) model for dislocations by a direct comparison with an equivalent molecular statics simulation. To this end, a two-dimensional microstructural specimen subjected to simple shear is considered, consisting of a central soft phase flanked by two hard-phase regions. A hexagonal atomic structure with equal lattice spacing is adopted, the interactions of which are described by the Lennard--Jones potential with phase specific depths of its energy well. During loading, edge dislocation dipoles centred in the soft phase are introduced, which progress towards the phase boundaries, where they pile up. Under a sufficiently high external shear load, the leading dislocation is eventually transmitted into the harder phase. The homogenized PN-FE model is calibrated to an atomistic model in terms of effective elasticity constants and glide plane properties as obtained from simple uniform deformations. To study the influence of different formulations of the glide plane potential, multiple approaches are employed, ranging from a simple sinusoidal function of the tangential disregistry to a complex model that couples the influence of the tangential and the normal disregistries. The obtained results show that, qualitatively, the dislocation structure, displacement, strain fields, and the dislocation evolution are captured adequately. The simplifications of the PN-FE model lead, however, to some discrepancies within the dislocation core. Such discrepancies play a dominant role in the dislocation transmission process, which thus cannot quantitatively be captured properly. Despite its simplicity, the PN-FE model proves to be an elegant tool for a qualitative study of edge dislocation behaviour in two-phase microstructures, although it may not be quantitatively predictive.Comment: 29 pages, 11 figures, 5 tables, abstract shortened to fulfill 1920 character limit, small changes after revie

Electronic Shell Structure of Nanoscale Superconductors

Motivated by recent experiments on Al nanoparticles, we have studied the effects of fixed electron number and small size in nanoscale superconductors, by applying the canonical BCS theory for the attractive Hubbard model in two and three dimensions. A negative ``gap'' in particles with an odd number of electrons as observed in the experiments is obtained in our canonical scheme. For particles with an even number of electrons, the energy gap exhibits shell structure as a function of electron density or system size in the weak-coupling regime: the gap is particularly large for ``magic numbers'' of electrons for a given system size or of atoms for a fixed electron density. The grand canonical BCS method essentially misses this feature. Possible experimental methods for observing such shell effects are discussed.Comment: 5 pages, 5 figure