Size Dependence of the Melting Point

The reduction in the melting point has been studied intensively for small particles and clusters on a surface. One observes typically a linear reduction of the melting point as a 

The data for the larger clusters (10 to 10 atoms) have been obtained by the T. P. Martin group [18]. From the structure in their mass spectra it is deduced that these large clusters have either icosahedral or fee symmetry. Bulk sodium is bcc, on the other hand, so that a structural phase transition must occur between n = and the bulk. An alternative interpretation would be large sodium clusters near their melting point show the same icosahedral precursor as calculated in Ref. [73] for gold clusters. In this case the ground state could already be bcc-like. 

The cluster sizes 55, 57 and 61 have a higher melting temperature than those in the 10 to 10 range. This is a totally unexpected and not understood result. In fact, it seems that the mean cluster melting temperature increases with decreasing cluster site for n below 90 atoms. 

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Figure 10 Size-dependence of the melting point and diffusion coefficient of silica-encapsulated gold particles. The dotted curve is calculated by the equation of Buffat and Borel. The bulk melting temperature of An is indicated by the double arrow as (oo). The solid curve (right-hand side axis) is a calculated An self-diffusion coefficient. (From Ref. 146.)...

Among the properties of water in well-defined nanopores, a global picture of the phase behavior is not yet available. We do not accurately know the pore-size dependence of the melting point in the nanometer scale or the conditions for gradual... 

Figure 7.1. The influence of particle size on property (a) The transition temperature of zirconia particles between the monoclinic and the tetragonal modification. There is a considerable hysteresis and the transition temperature drops with particle size, (b) The size dependence of the lattice constant of iron particles, (c) The particle size dependence of the melting point of a few metals. Their bulk melting points are given at the right-hand side of the graph.

There is another important law that follows from the classical theory of capillarity. This law was formulated by J. Thomson [16], and was based on a Clausius-Clapeyron equation and Gibbs theory, formulating the dependence of the melting point of solids on their size. The first known analytical equation by Rie [17], and Batchelor and Foster [18] (cited according to Refs. [19,20]) is... 

With the derived yi(n,L,K) relation and the given expressions for the q(zi, Ei(t)), one can readily predict the size, cavity density, and temperature dependence of the melting point, elasticity, and the flow stress of a system with large proportion of undercoordinated atoms without involving hypothetic parameters. 

We have already pointed out the difficulties of analyzing the dependence of the melting temperatures on crystallization temperature because of the size increases that take place. To properly carry out these experiments it is necessary to extrapolate the observed melting temperature, at a given crystallization temperature, to zero levels of crystallinity. 

Jahnert et al. (2008) showed that there are the pronounced differences in the magnitude and pore size dependence of the transition temperature T d) of water obtained by the spin-echo method due in part to the chosen delay time x (Figure 1.277). For instance, the X value (2-20 ms) plays an important role as the pore size of the silica samples decreases. The melting point depression of water in silica pores of nominal pore size 4 nm was found to be 14 K when measured with x=l ms, but only 6 K when measured with x=20 ms. This was attributed to a pore size dependence of the relaxation time T2 of the confined liquid (Jahnert et al. 2008). 

Macros are formed by ablation of molten or solid particles by thermal shock and hydrodynamic effects in the molten spot on a solid surface. The number and size of macros produced from the solid arc cathode surface depends on the melting point and vapor pressure of the cathode material and the arc movement Large (tens of microns diameter) macros are formed with low melting point materials and slow arc movement while small macros (< 1 micron) are formed with high melting point materials and rapid arc movement The molten globules can represent a few to many per cent of the material ejected from the cathode. 

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