SIZE DEPENDENT INTERFACE ENERGY - Energy Research Developments: Tidal Energy, Energy Efficiency and Solar Energy - page 197

S pos = - R ( x A ln x A + x B ln x B ) (2.9)

where x A and x B are molar fractions of crystals and vacancies, respectively. For a melting
process, x A = 1/(1+Δ V m / V s ) and x B = 1- x A where Δ V m = V l - V s with subscripts of s and l for
crystal and liquid phases, respectively. Note that Eq. (2.9) assumes that Δ V m consists of
vacancies with the same size of atoms. For metallic and organic crystals, the type of chemical
connection does not vary during the melting transition. Thus, S el ≈ 0 [27], and

S vib = S m - S pos . (2.10-a)

For some semi-metals, S el ≠ 0, S vib must be determined in a direct way, such as Mott's
equation [28],

S vib = 3 R ln(ν s /ν l ) = (3/2) R ln(σ s /σ l ) (2.10-b)

where ν and σ denote characteristic vibration frequency and electrical conductivity. If the
parameters in above equations are unavailable, the following equation can also be utilized as
a first order approximation [26],

S vib ≈ S m - R . (2.10-c)

For organic crystals, Δ V on melting is small and S pos is thus negligible as a first order
approximation [25]. While for some organic crystals, one or more solid-state phase
transformations closely precede the melting, which is thought to reduce the S m values [29].
Herein, the cumulative entropy of fusion S c m should be introduced, which is defined as the
summation of all the changes in entropy at all the transformation temperatures and melting
temperature. Thus,

c
m

S vib ≈

S

. (2.10-d)

For semiconductors, the melting is accompanied by the semiconductor-to-metallic
transition and the elements suffer contraction in volume rather than expansion for most of
metals. Thus, S el strongly contributes S m and S pos << S el and S pos is thus neglected as a first
order approximation [29]. Namely,

S vib ≈ S m - S el . (2.11)

The above model, i.e. Eq. (2.5) has predicted the size-dependent melting for
nanoparticles [30], for thin films [20], for metallic nanowires in carbon nanotubes [21], and
the related size-dependent initial sintering temperature of metallic nanoparticles [31]. The
available experimental evidences in a large size range from several nanometers to several
hundred nanometers confirm the above-predicted results noted that the valid size range of Eq.
(2.5) is from 2 D 0 to infinite although T m ( D ) ≈ T m when D > 200 nm. Thus, the approximate
expression for the melting temperature of an object of small size in Eq. (2.5) could proceed
from it to the limit of a large system.