  The phonon thermal contribution to the melting temperature of nano-particles
is inspected. The discrete summation of phonon states and its corresponding
integration form as an approximation for a nano-particle or for a bulk system
have been analyzed. The discrete phonon energy levels of pure size effect and
the wave-vector shifts of boundary conditions were investigated in detail.
Unlike in macroscopic thermodynamics, the integration volume of zero-mode of
phonon for a nano-particle is not zero, and it plays an important role in pure
size effect and boundary condition effect. We found that a nano-particle will
have a rising melting temperature due to purely finite size effect; a lower
melting temperature bound exists for a nano-particle in various environments,
and the melting temperature of a nano-particle with free boundary condition
reaches this lower bound. We suggest an easy procedure to estimation the
melting temperature, in which the zero-mode contribution will be excluded, and
only several bulk quantities will be used as input. We would like to emphasize
that the quantum effect of discrete energy levels in nano-particles, which was
not present in early thermodynamic studies on finite size corrections to
melting temperature in small systems, should be included in future researches.
