Biomedical Engineering Reference
In-Depth Information
Referring again to Fig.
2.4
,wehave
S
sc
D
2.R
s
/
2
.1
cos '/;
S
cf
D
2r
2
.1
cos ‰/;
1
3
r
3
.2
3 cos ‰
C
cos
3
‰/
1
3
.R
s
/
3
.2
3 cos '
C
cos
3
'/;
(2.8)
and
V
c
D
D
.R
s
r cos/=l
D
.R
s
cos '
rm/=l;
(2.9)
cos ‰
D
.r
R
s
cos/=l
D
.r
R
s
m/= l;
(2.10)
and
D
Œ.R
s
/
2
C
r
2
2R
s
rm
1=2
:
l
(2.11)
To evaluate the critical free energy G
heter
, we can substitute expression (
2.8
)
into (
2.6
) and require that
@G
@r
D
0:
(2.12)
Regarding the fact that the radius of curvature
r
c
of the critical nuclei is only
determined by
cf
and the driving force [
49
], we then have
2
cf
r
c
D
:
(2.13)
We notice that in the case of epitaxial growth, some strain will develop due to
the structural mismatch at the crystal-substrate interface. The strain will affect both
the bulk free energy of nuclei and the interfacial free energy
cf
. In this case, the
occurrence of substrate affects
r
c
.
Now substituting expressions (
2.7
)-(
2.13
)into(
2.6
) and writing
R
s
r
c
D
R
s
cf
R
0D
;
(2.14)
the free energy of formation of a critical nucleus is
G
heter
D
G
homo
f
(2.15)
with
16
cf
2
3kS
m
TT
2
;
G
homo
D
(2.16)
