7.2 Influence of the Particle Size on Thermodynamic Properties and Phase Transformations 125
At equilibrium temperature, ΔG trans−nano = 0. For the transformation temperature of nanoparticles, one obtains
T
U
S
M
d
S
trans
trans
trans
new
new new
trans
old
new
=
+
−
∆
∆
∆
6
1
γ
ρ
γ
γ
ρ
ρ
new
old
2 3
/
.
(7.7)
Using the abbreviation T
U
S
bulk
trans
trans
=
∆
∆
one obtains the important relation
∆
∆
T T
T
M
T
U
d
=
−
=
−
trans
bulk
bulk
trans
new
new new
old
new
6
1 1
γ
ρ
γ
γ
=
ρ
ρ
α
new
old
new
2 3
1
/
.
d
(7.8)
Equation (7.8) represents the inverse linear relationship between the reduction
of the phase-transformation temperature and the particle size, in this equation,
the term in the square brackets is most interesting, as this term rules the sign
of the temperature difference.
α
γ
γ
ρ
ρ
= −
1
2 3
old
new
new
old
/
.
(7.9)
Looking at the melting process (old = solid, new = liquid), α is always negative,
as,
γ
γ
ρ
ρ
old
new
new
old
>
2 3
1
/
is always valid, as
γ
γ
ρ
ρ
old
new
new
old
>
2 3
/
is always observed.
Even exceptional cases, where the density increases during melting, for
example, in the cases of bismuth and germanium, do not alter this rule. This
may be seen in Table 7.1.
Table 7.1 Parameters of Eq. (7.9) for different elements.
g
g
solid
liquid
r
r
liquid
solid
r
r
liquid
solid
2 3
/
g
g
r
r
solid
liquid
liquid
solid
2 3
/
Copper
1.11
0.9
0.93
1.03
Silver
1.22
0.89
0.92
1.12
Gold
1.15
0.9
0.93
1.08
Germanium
1.35
1.05
1.03
1.39
Bismuth
1.32
1.16
1.1
1.45
The general law expressed by Eq. (7.5) is experimentally very well verified. Figure
7.3 displays the melting temperature of gold nanoparticles as a function of the
inverse particle diameter. The experimental results of Castro et al. [1] show two
separated ranges. The range indicated with Castro et al. I, which follows exactly
the inverse linear relationship of Eq. (7.5) and a second range, which is nearly
At equilibrium temperature, ΔG trans−nano = 0. For the transformation temperature of nanoparticles, one obtains
T
U
S
M
d
S
trans
trans
trans
new
new new
trans
old
new
=
+
−
∆
∆
∆
6
1
γ
ρ
γ
γ
ρ
ρ
new
old
2 3
/
.
(7.7)
Using the abbreviation T
U
S
bulk
trans
trans
=
∆
∆
one obtains the important relation
∆
∆
T T
T
M
T
U
d
=
−
=
−
trans
bulk
bulk
trans
new
new new
old
new
6
1 1
γ
ρ
γ
γ
=
ρ
ρ
α
new
old
new
2 3
1
/
.
d
(7.8)
Equation (7.8) represents the inverse linear relationship between the reduction
of the phase-transformation temperature and the particle size, in this equation,
the term in the square brackets is most interesting, as this term rules the sign
of the temperature difference.
α
γ
γ
ρ
ρ
= −
1
2 3
old
new
new
old
/
.
(7.9)
Looking at the melting process (old = solid, new = liquid), α is always negative,
as,
γ
γ
ρ
ρ
old
new
new
old
>
2 3
1
/
is always valid, as
γ
γ
ρ
ρ
old
new
new
old
>
2 3
/
is always observed.
Even exceptional cases, where the density increases during melting, for
example, in the cases of bismuth and germanium, do not alter this rule. This
may be seen in Table 7.1.
Table 7.1 Parameters of Eq. (7.9) for different elements.
g
g
solid
liquid
r
r
liquid
solid
r
r
liquid
solid
2 3
/
g
g
r
r
solid
liquid
liquid
solid
2 3
/
Copper
1.11
0.9
0.93
1.03
Silver
1.22
0.89
0.92
1.12
Gold
1.15
0.9
0.93
1.08
Germanium
1.35
1.05
1.03
1.39
Bismuth
1.32
1.16
1.1
1.45
The general law expressed by Eq. (7.5) is experimentally very well verified. Figure
7.3 displays the melting temperature of gold nanoparticles as a function of the
inverse particle diameter. The experimental results of Castro et al. [1] show two
separated ranges. The range indicated with Castro et al. I, which follows exactly
the inverse linear relationship of Eq. (7.5) and a second range, which is nearly