Atomistic-to-continuum modeling of torsional and longitudinal vibration of black phosphorus nanotubes

Abstract Owing to their distinctive physical properties, black phosphorus nanotubes (BPNTs) exhibit remarkable potential for diverse applications in nanoelectromechanical systems. Molecular dynamic (MD) simulations and continuum mechanics models are employed to examine the torsional and longitudinal vibration behaviors of BPNTs. For torsional vibration of BPNTs: When the initial torsional strain excitation is below 1.08%, the torsional vibration frequency remains nearly constant. The difference in torsional vibration frequencies between BPNTs with different chiralities is negligible. As the initial strain increases, the vibration frequency of armchair-type BPNTs increases, while that of zigzag-type BPNTs decreases. For longitudinal vibration of BPNTs: When the initial axial tensile excitation is less than 1.16%, the longitudinal vibration frequency shows almost no variation. A significant difference exists between the longitudinal vibration frequencies of armchair-type and zigzag-type BPNTs, while BPNTs with identical chirality exhibit minimal frequency differences (either armchair or zigzag types). Unlike transverse vibration, the longitudinal vibration frequency demonstrates an anomalous decrease with increasing initial strain. The results obtained from the continuum mechanics model show excellent agreement with MD simulations, demonstrating that the continuum mechanics approach can accurately predict both torsional and longitudinal vibration characteristics of BPNTs. Our findings provide theoretical support for developing BPNT-based resonators, particularly for those with initial strain.