Nonlinear positive/negative thermal expansion and equations of state of a chain with longitudinal and transverse vibrations

Thermal expansion of a classical chain with pair interactions performing longitudinal and transverse vibrations is investigated. Corresponding equations of state are derived analytically using series expansions of pressure and thermal energy with respect to deformations of the bonds caused by thermal motion. In the first approximation the equation of state has Mie–Grüneisen form. The dependence of Grüneisen parameter on deformation of the chain is obtained. For Lennard‐Jones‐like potential Grüneisen parameter varies with deformation from minus infinity (at zero stretching) to plus infinity (at the breakage point). Necessary and sufficient condition for negative thermal expansion at low thermal energies is formulated. Using this condition the potential giving rise to negative thermal expansion in the given range of deformations can be designed. It is shown that at small deformations and finite thermal energies Mie–Grüneisen equation of state strongly overestimates the absolute value of pressure. More accurate nonlinear equation of state is derived for this case. The equation implies that thermal pressure in the unstretched chain is proportional to square root of thermal energy. In the vicinity of the deformation, corresponding to zero Grüneisen parameter, the chain demonstrates negative thermal expansion at low temperatures and positive thermal expansion at higher temperatures. This phenomenon is qualitatively described by the nonlinear equation of state derived in the present paper. The theoretical findings are supported by the results of molecular dynamics simulations.