SIZE DEPENDENT INTERFACE ENERGY - Energy Research Developments: Tidal Energy, Energy Efficiency and Solar Energy - page 196

a temperature T m ( D 1 , D 2 ), which is not equal to the liquidus temperature T m in phase diagram.
The expression may be written as,

T m ( D 1 , D 2 )/ T m = 1 -V γ sl0 (1/ D 1 +1/ D 2 )/ H m ( T )
(2.4)
where H m ( T ) is temperature-dependent melting enthalpy.
Note that Eq. (2.4) is valid when D 1 and D 2 are large enough (e.g. 20 nm) [12-13]. Thus,
when the appropriate physical constants are known, measurements of T m ( D 1 , D 2 ) for known
values of D 1 and D 2 for a system at ″equilibrium″ will yield values of γ sl0 directly.
Fortunately, the size dependence of melting temperature of metallic nanocrystals T m ( D )
have been deduced as the following [20-21],

⎛ ⎞

T ( D ) 2 S 1

(2.5)

⎜
⎝

⎟
⎠

m

= exp −

vib

T 3 R D /

D

− 1

m

where S vib denotes the vibrational component of the melting entropy S m of bulk crystals at T m ,
R is the ideal gas constant, D 0 shows a critical diameter at which all atoms of a particle are
located on its surface. For low dimensional element crystals, D 0 depends on their dimension d
and atomic diameter h through [20-21],

D 0 = 2(3- d ) h (2.6)

where d = 0 for nanoparticles, d = 1 for nanowires and d = 2 for thin films. When d = 0, D has
a normal meaning. For d = 1, D denotes the diameter of the nanowire. If d = 2, D is defined as
the thickness of a thin film. Since a crystal is characterized by its long-range order, the
smallest crystal should have at least a half of the atoms located within the crystal. Hence, the
smallest value of D is 2 D 0 [20]. This estimation is consistent with experimental results for Bi
film [22] and Pb nanowire in a carbon nanotube [23-24]. However, the parameter h must be
redefined to adapt to the case of molecular crystals. For organic spherical molecules, h can be
an averaged diameter of the molecules since the molecule for molecular crystals has a similar
effect as the atom for metallic elements [25]. While for chain molecules, considering that the
γ sl0 value states excess energy of interface molecules in a unit area, h thus may be defined as
the mean size of a chain segment [25],

h = [ mV/ ( nN a )] 1/3 (2.7)

where m and n denote the total atom number and the chain segment number of an organic
molecule, respectively. N a is Avogadro′s constant.
S m consists, at least, of three components: positional S pos , vibrational S vib and electronic
S el [26],

S m = S vib + S pos + S el . (2.8)

S pos is given by [26],