Quantum mechanical moduli field

To understand the role of defects in materials science, ranging from mechanical to physical properties, determining the spatial variation of elastic moduli is of paramount importance. Using electron wavefunctions, we derive novel expressions for local elastic moduli in the lattice scale, Quantum Mechanical Moduli Field (QMMF). The QMMF provides insight into the interplay between elastic properties and defects. To derive QMMF, we differentiate the local stress density against strain. The QMMF has contributions from kinetic, exchange-correlation, and electrostatic interactions. We provide novel expressions and numerical schemes to calculate QMMF. In atomistic calculations, the atoms are modeled as point-like entities, which only allows the macroscopic elastic properties to be calculated. Since the QMMF represents the local elastic properties, it provides a significant advancement from previous studies, especially in the presence of multi-elements. Four example applications of QMMF are provided. Firstly, the macroscopic elastic moduli of Ni and B2NiTi are calculated using QMMF in agreement with experiments. Secondly, a H interstitial in Ni is considered. The effect of H concentration on H softening is evaluated. Thirdly, the effect of dilatation on moduli is calculated, revealing the non-linearity of moduli. Finally, the local elastic properties around W solute in the Ni matrix are calculated. The W solute increases the macroscopic moduli of Ni in a non-linear fashion. It is found that the macroscopic hardening is due to the hardening of the Ni matrix rather than W solutes forming hard-spots. The QMMF uses electron densities to unveil such surprising effects that are otherwise unobservable.