Kinetics and Thermodynamics of Metal Cluster Nucleation …
237
Table 2 Kinetic parameters associated with the nucleation of Sn and Co
η
PR
3D nu
k H (mol cm −2 s −1 ) D (cm 2 s −1 )
N 0 (cm −2 ) A (s −1 ) I st (cm −2 s −1 )
Sn
0.50 3.56 × 10 −6
1.046 × 10 −4 400,979
183.06
7.34 × 10 7
Co 0.50 2.61 × 10 −7
1.288 × 10 −4 31,719
5.41
1.72 × 10 5
nucleation rate constant (A) for the Sn deposition is also observed to be 183.06 s
−1
which is much higher than the A value for Co deposition (5.41 s
−1 ). However, any
comments nucleation rate is generally made from the product A × N 0 and not from
A values alone. The steady-state nucleation rate (I st ) is defined as the product of
nucleation rate constant and the number density of active nucleation sites.
I st = A × N 0
(15)
From the values of a, b and c, the values of k H , D, N 0 and I st can be calculated
using Eqs. (12), (13), (14) and (15). The values of the kinetic parameters related to
proton reduction and 3D nu are presented in Table 2. The diffusivity (D) values for Sn
and Co did not exhibit much variation. However, the N 0 values vary significantly for
Sn (400,979 cm
−2 ) and Co (31,719 cm
−2 ). A significant difference in the nucleation
rate (I st ) can be observed. The I st for Sn is 7.34 × 10
7 cm
−2 s
−1 which is two orders
of magnitude greater than the I st for Co which is 1.72 × 10
5 cm
−2 s
−1 . The higher
nucleation rate conformed with the observation in Fig. 2a where the current density
peak for Sn was shifted up and towards left. The k H value observed for Sn (3.56
× 10
−6 mol cm
−2 s
−1 ) is also one order of magnitude higher than that of Co (k H
= 2.61 × 10
−7 mol cm
−2 s
−1 ) which was also manifested in Fig. 3a, b where the
contribution from i PR was significant in the case of Sn deposition and negligible in
the case of Co deposition.
Classical Theory of Nucleation
The thermodynamic parameters such as Gibb’s free energy of nucleation (G c ) and
critical nucleus size (n c ) can be calculated by performing multiple CA tests under
stepwise increasing overpotentials. The G c for hemispherical clusters depends on
η as [28]:
G c =
8πσ
3
ϑ
2
a
3(zeη)
2
(16)
and
n c =
16π
3
×
σ
3
ϑ
2
a
(zeη)
3
(17)
237
Table 2 Kinetic parameters associated with the nucleation of Sn and Co
η
PR
3D nu
k H (mol cm −2 s −1 ) D (cm 2 s −1 )
N 0 (cm −2 ) A (s −1 ) I st (cm −2 s −1 )
Sn
0.50 3.56 × 10 −6
1.046 × 10 −4 400,979
183.06
7.34 × 10 7
Co 0.50 2.61 × 10 −7
1.288 × 10 −4 31,719
5.41
1.72 × 10 5
nucleation rate constant (A) for the Sn deposition is also observed to be 183.06 s
−1
which is much higher than the A value for Co deposition (5.41 s
−1 ). However, any
comments nucleation rate is generally made from the product A × N 0 and not from
A values alone. The steady-state nucleation rate (I st ) is defined as the product of
nucleation rate constant and the number density of active nucleation sites.
I st = A × N 0
(15)
From the values of a, b and c, the values of k H , D, N 0 and I st can be calculated
using Eqs. (12), (13), (14) and (15). The values of the kinetic parameters related to
proton reduction and 3D nu are presented in Table 2. The diffusivity (D) values for Sn
and Co did not exhibit much variation. However, the N 0 values vary significantly for
Sn (400,979 cm
−2 ) and Co (31,719 cm
−2 ). A significant difference in the nucleation
rate (I st ) can be observed. The I st for Sn is 7.34 × 10
7 cm
−2 s
−1 which is two orders
of magnitude greater than the I st for Co which is 1.72 × 10
5 cm
−2 s
−1 . The higher
nucleation rate conformed with the observation in Fig. 2a where the current density
peak for Sn was shifted up and towards left. The k H value observed for Sn (3.56
× 10
−6 mol cm
−2 s
−1 ) is also one order of magnitude higher than that of Co (k H
= 2.61 × 10
−7 mol cm
−2 s
−1 ) which was also manifested in Fig. 3a, b where the
contribution from i PR was significant in the case of Sn deposition and negligible in
the case of Co deposition.
Classical Theory of Nucleation
The thermodynamic parameters such as Gibb’s free energy of nucleation (G c ) and
critical nucleus size (n c ) can be calculated by performing multiple CA tests under
stepwise increasing overpotentials. The G c for hemispherical clusters depends on
η as [28]:
G c =
8πσ
3
ϑ
2
a
3(zeη)
2
(16)
and
n c =
16π
3
×
σ
3
ϑ
2
a
(zeη)
3
(17)