This work describes the logarithmic term in the sizedependent melting temperature of nanoparticles. The Lindemann melting criterion and the thermal phonon contribution are intended primarily as a clear approach for our study of melting temperature. The ratio of the melting temperature (T mn ) of a nanoparticle to the bulk melting temperature (T mb ) is numerically evaluated up to the order of the inverse of the size L of a nanoparticle: T mn /T mb = 1 − A/L. We show that the coefficient A is not a constant but has a logarithmic part. The logarithmic term is the dominate term in the factor A(L). Our investigation suggests the behavior of the size-dependent melting is T mn /T mb = 1 − (B + C ln L)/(L − 2l s ) where B and C are constant factors and l s is the number dimension of the surface layers reconstructed or premelted. The results present a challenge to reach a higher resolution in future thermal experiments in order to distinguish ln L/L behavior from the previously accepted pure 1/L behavior.
■ INTRODUCTIONThe study of nanoparticle melting is of great importance because of its possible application as a source of new materials. Since the size-dependent melting temperature was found by Takagi by means of TEM, 1 extensive experiments have been performed. 2−11 While the depression of melting temperature with the decrease of particle size was found in some experiments, 2−7 the superheating phenomenon was also reported for some nanoparticles embedded in matrices. 8−10 Theoretical study began in the early 1900s when Pawlow predicted the size-dependent melting temperature for nanoparticles. 12 Different theoretical models explain the phenomenon based on different assumptions, 12−21 and it is still difficult to decide which model is the best. The linear relation between the ratio of melting temperature of nanoparticles (T mn ) to bulk melting temperature (T mb ) and inverse of particle size (1/L) has been derived as follows:The Lindemann melting criterion is simple but effective for the melting phenomenon: a solid melts when the ratio of the mean displacement of atoms (u) to the lattice constant (a) reaches the Lindemann critical value (L c ). 22,23 In the rootmean-square displacement model, 17−19 it was used as the criterion for the melting of nanoparticles. For an ideal crystal, as the size of the lattice decreases, the phonon contribution becomes important. The surface-phonon instability model considered the intrinsic defect contribution to phonon modes. 13 The quantum size effect had not been involved before Sui et al. carefully investigated the effect of discrete thermal phonons contributing to the melting of nanoparticles, and they found the lower boundary for the melting temperature of nanoparticles considering the free boundary conditions. 24 As the size of the crystal becomes small, the surface becomes a substantial part. 13,14,17 Different surface properties correspond to different boundary conditions for the crystal. Following Sui's work, Xu et al. studied the effect of boundary conditions and discussed th...
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