Chemistry Reference
In-Depth Information
of the crystal/vapour, crystal/substrate, amorphous/vapour, amorphous/substrate
and crystal/amorphous interfaces, respectively.
The first term in Eq. (5.11) reflects the gain in the bulk energy while the sec-
ond term accounts for the variation in the total free energy associated with the
replacement of the substrate/amorphous and vapour/amorphous interfaces (d
A
1
)
by the substrate/crystal and vapour/crystal interfaces. The last term represents the
increase in the total free energy due to the increase in the crystal/amorphous in-
terface (d
A
2
). Taking into account Eq. (5.11) and the expressions for d
V
,d
A
1
and
d
A
2
given above, d
G
/
d
N
can be expressed as:
1
d
G
d
N
=−
µ
0
+
1
h
(
t
)
γ
+
γ
⊥
ca
d
h
d
R
+
γ
⊥
ca
R
(
t
)
1
h
(
t
)
v
+
γ
⊥
ca
h
=−
µ
0
+
γ
+
γ
⊥
ca
R
(
t
)
.
(5.12)
d
R
/
d
t
/
v
Taking into account that
R
(
t
)
h
(
t
) and that d
R
d
t
h
, Eq. (5.12) can be
approximated by:
d
N
=−
µ
0
+
γ
1
d
G
h
(
t
)
=−
µ.
(5.13)
Combining Eqs. (5.10) and (5.13) we obtain:
µ
0
−
γ
h
(
t
)
K
c
k
B
T
v
R
=
(5.14)
and defining
k
B
T
and
h
c
≡
γ
/µ
0
, Eq. (5.9) is obtained. Note
that Eq. (5.9) is only valid for
h
(
t
)
v
∞
≡
K
c
µ
0
/
h
c
. Finally, the mathematical expression for
R
(
t
) can be derived from Eq. (5.9) by integration. In this case:
≥
−
v
∞
t
c
ln
t
t
c
R
(
t
)
=
v
R
(
t
)d
t
=
v
∞
(
t
−
t
c
)
,
(5.15)
where
t
c
is defined as
t
c
0. Figure 5.9 shows the fit to
the experimental points obtained with Eq. (5.15). The predicted curves fit well the
experimental points. The effects of non-linearity are more evident for the spherulites
forming first because the factor 1
≡
h
c
/v
h
. Note that
R
(
t
c
)
=
−
/
h
c
h
is rather small, while for spherulites
appearing later
h
c
/
h
1, so that the velocity nearly reaches an asymptotic limit
and the growth is thus approximately constant.
h
c
has been estimated from TMAFM
measurements as 35 nm. This value is of the same order as those found for thin
films of polymethine dye salts,
h
c
∼
30 nm (Dahne, 1995), as previously discussed.
