where U is the enthalpy, S is the entropy, c is the surface energy, and A is the
surface of the old, respectively new, phase (always related to 1 mole). Now, the
question arises as to whether the transformation temperature is a function of
the particle size, or not. This somewhat aged question was first posed in
connection with the crystallization of organic phases at the end of the
nineteenth century. When related to nanoparticles, this problem was found
to be most important and generalized to all types of phase transformations. By
using the differences DU trans ¼ U new À U old and DS trans ¼ S new À S old , Eq. (7.5)
reduces to:
DG transÀnano ¼ DU trans À T trans DS trans þ c new A new À c old A old ¼ 0
ð7:5Þ
The change of surface area A new – A old is related to particle size; hence, when
assuming spherical particles one obtains for the surface per mol A ¼ 6M/rd, where
M is the molar weight, r is the density, and d is the particle diameter. To do this, the
surface per mole must be calculated as a function of the particle size.
By using d new /d old ¼ (r old /r new )
1/3
, one obtains:
DG transÀnano ¼ DU trans À T trans DS trans þ c new
6M
r new d new
Àc old
6M
r new d new
r new
r old
2=3
¼ 0
ð7:6Þ
From the equilibrium condition, one obtains for the temperature of
transformation:
T trans ¼
DU trans
DS trans
À
6Mc new
r new d new DS trans
1 À
c old
c new
r new
r old
2=3
"
#
ð7:7Þ
Figure 7.5 Comparative heat capacity of nanocrystalline and coarse-grained alumina. As for
metals (see Figure 7.4), the heat capacity of the nanocrystalline material is greater [4].
140j 7 Phase Transformations of Nanoparticles