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.
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