  The phonon thermal contribution to the melting temperature of nano-particles
is inspected. Unlike in periodic boundary condition, under a general boundary
condition the integration volume of low energy phonon for a nano-particle is
more complex. We estimate the size-dependent melting temperature through the
phase shift of the low energy phonon mode acquired by its scattering on
boundary surface. A nano-particle can have either a rising or a decreasing
melting temperature due to the boundary condition effect, and we found that an
upper melting temperature bound exists for a nano-particle in various
environments. Moreover, the melting temperature under a fixed boundary
condition sets this upper bound.
