2.1 Background
37
where g is the Gibbs energy density (per unit volume). The relative magnitude of the
second term vanishes as r
−1 in large r . This conclusion is true as long as we assume
the isotropic enlargement for any shape of the system.
5 Thus, the surface energy is
negligible in bulk substances.
Since the lower Gibbs energy is more stable thermodynamically, a liquid droplet
tends to minimize its surface area because of γ > 0. Its equilibrium shape is a sphere
if we ignore the effect of gravity. On the other hand, the surface free energy depends
on the created surface for crystals. We can thus imagine an equilibrium shape of
crystalline grain, such as a cube for rock salt.
In order for a phase transition to proceed, the effect of the interface is significant.
Here, we see the existence of the minimal size of a new phase domain for proceeding
the transition. Imagine a compound that undergoes a phase transition between two
isotropic liquid phases, for simplicity. The difference in the Gibbs energy density
(assigned to the new phase) is denoted by Δg(T ), and the interfacial tension between
the two phases is by γ(T ). If a droplet of a new phase with a radius r exists, the
difference in the Gibbs energy, denoted as ΔG(T ), relative to the uniform state of
the mother phase is given by
ΔG(T ) =
4
3
πr
3
Δg(T ) + 4πr
2
γ(T ).
(2.16)
The condition for the droplet to grow is
dΔG(T )
dr
< 0.
(2.17)
Thus, we have
r > r c (T ) = −
2γ(T )
Δg(T )
.
(2.18)
That is, the negative Δg(T ), which happens below the equilibrium transition point
T trs , is insufficient to start the transition. Only if a droplet larger than the minimum
size r c is formed as a result of the thermal fluctuation, the phase transition starts. The
barrier hight necessary for thermal activation, Δ
∗ G(T ), is calculated as ΔG(T ) at
r = r c ,
Δ
∗ G(T ) =
16π
3
γ(T )
3
[Δg(T )] 2 .
(2.19)
Since the departure from the transition temperature brings more negative Δg(T ), the
critical size of the drop and the activation barrier hight to form the initial droplet
decrease with the increase in |T − T trs |. Although the above analysis assumes two
liquid phases, the presence of the minimum size of a new phase domain for growth
is true for any first-order phase transitions. The formation of the domains to grow
is termed as the nucleation. In the vicinity of T trs , the thermal nucleation is more
5 The so-called “thermodynamic limit” implies the condition.
37
where g is the Gibbs energy density (per unit volume). The relative magnitude of the
second term vanishes as r
−1 in large r . This conclusion is true as long as we assume
the isotropic enlargement for any shape of the system.
5 Thus, the surface energy is
negligible in bulk substances.
Since the lower Gibbs energy is more stable thermodynamically, a liquid droplet
tends to minimize its surface area because of γ > 0. Its equilibrium shape is a sphere
if we ignore the effect of gravity. On the other hand, the surface free energy depends
on the created surface for crystals. We can thus imagine an equilibrium shape of
crystalline grain, such as a cube for rock salt.
In order for a phase transition to proceed, the effect of the interface is significant.
Here, we see the existence of the minimal size of a new phase domain for proceeding
the transition. Imagine a compound that undergoes a phase transition between two
isotropic liquid phases, for simplicity. The difference in the Gibbs energy density
(assigned to the new phase) is denoted by Δg(T ), and the interfacial tension between
the two phases is by γ(T ). If a droplet of a new phase with a radius r exists, the
difference in the Gibbs energy, denoted as ΔG(T ), relative to the uniform state of
the mother phase is given by
ΔG(T ) =
4
3
πr
3
Δg(T ) + 4πr
2
γ(T ).
(2.16)
The condition for the droplet to grow is
dΔG(T )
dr
< 0.
(2.17)
Thus, we have
r > r c (T ) = −
2γ(T )
Δg(T )
.
(2.18)
That is, the negative Δg(T ), which happens below the equilibrium transition point
T trs , is insufficient to start the transition. Only if a droplet larger than the minimum
size r c is formed as a result of the thermal fluctuation, the phase transition starts. The
barrier hight necessary for thermal activation, Δ
∗ G(T ), is calculated as ΔG(T ) at
r = r c ,
Δ
∗ G(T ) =
16π
3
γ(T )
3
[Δg(T )] 2 .
(2.19)
Since the departure from the transition temperature brings more negative Δg(T ), the
critical size of the drop and the activation barrier hight to form the initial droplet
decrease with the increase in |T − T trs |. Although the above analysis assumes two
liquid phases, the presence of the minimum size of a new phase domain for growth
is true for any first-order phase transitions. The formation of the domains to grow
is termed as the nucleation. In the vicinity of T trs , the thermal nucleation is more
5 The so-called “thermodynamic limit” implies the condition.