Microscopic definition of internal force, moment, and associated stiffnesses in one-dimensional nanostructures at finite temperature

Abstract

We present a one-dimensional variant of the Irving–Kirkwood–Noll procedure to derive microscopic expressions of internal contact force and moment in one-dimensional nanostructures. We show that these expressions must contain both the potential and kinetic parts: just the potential part does not yield meaningful continuum results. We further specialize these expressions for helically repeating one-dimensional nanostructures for their extension, torsion, and bending deformation. As the Irving–Kirkwood–Noll procedure does not yield expressions of stiffnesses, we resort to a thermodynamic equilibrium approach to first obtain the Helmholtz free energy of the supercell of helically repeating nanostructures. We then obtain expressions of axial force, twisting moment, bending moment, and the associated stiffnesses by taking the first and second derivatives of the Helmholtz free energy with respect to conjugate strain measures. The derived expressions are used in finite-temperature molecular dynamics simulation to study extension, torsion, and bending of single-walled carbon nanotubes and their buckling.