Non-local vibrations of nanostructures

In this study, frequency spectra of nanoparticles with different geometries and materials, calculated using local and non-local elasticity theories, are presented. Sphere- and cube-shaped particles made of silicon, carbon, and germanium are analyzed. Different material internal lengths, non-local weighting factors, particle sizes, and internal-length to particle-size ratios were considered.

It was observed that non-local theories of elasticity consistently predict lower natural frequencies compared to those derived from classical local elasticity. This systematic reduction in frequency is indicative of a material softening effect introduced by the consideration of long-range interactions forces introduced by the non-local model.

Besides, non-local frequencies converge to those calculated using classical local elasticity as the local weighting factor increases and/or the material-internal-length to particle-size ratio decreases. This demonstrates a transition from non-local to local behavior under specific parametric conditions.

A significant deviation from classical elasticity was also identified: the non-local frequency-radius product is not constant but exhibits dependence on the particle size. This finding disproves the frequency scale invariance characteristic of classical elasticity, where normalized frequencies are independent of the system's geometric scale. Instead, a novel scale invariance concept emerged within the framework of non-local elasticity: when normalized, the non-local frequencies remain constant for a given material-internal-length to particle-size ratio, irrespective of the absolute particle dimension. This result introduces an alternative scaling law, providing a deeper understanding of size-dependent phenomena in nanoscale systems.