Modeling of single phase electrocrystallization processes growth of paraboloids with concurrent evolution of hydrogen

Article abstract of DOI:10.1149/1.1805520

A generalized form of the current time transient (CTT) has been derived for the charge transfer controlled model of growth of paraboloids. The transient current is shown to pass through a single maximum before approaching a steady-state value. The maximum is estimated to be coincidental with 77% of the substrate coverage in the case of instantaneous micleation and with 90% coverage for the progressive nucleation and growth of paraboloids. The dependence of the surface area of the paraboloidal growth forms on time is formulated and, consequently, the kinetics of the concurrent evolution of hydrogen has been incorporated in the formulation of the CTTs. A particular CTT equation is thereby derived for the growth of paraboloids that are instantaneously formed on a monolayer of deposit. It is shown that the exact shape of some of the CTTs recorded during the electrocrystallization of cobalt can be reproduced by the derived CTT.

Full text of DOI:10.1149/1.1805520

Journal of The Electrochemical Society, 151 ͑11͒ C737-C742 ͑2004͒
C737
0
013-4651/2004/151͑11͒/C737/6/$7.00 © The Electrochemical Society, Inc.
Modeling of Single Phase Electrocrystallization Processes
Growth of Paraboloids with Concurrent Evolution of Hydrogen
M. Y. Abyanehz
School of Science and Environment, Coventry University, Coventry CV1 5 FB, United Kingdom
A generalized form of the current time transient ͑CTT͒ has been derived for the charge transfer controlled model of growth of
paraboloids. The transient current is shown to pass through a single maximum before approaching a steady-state value. The
maximum is estimated to be coincidental with 77% of the substrate coverage in the case of ‘‘instantaneous’’ nucleation and with
90% coverage for the ‘‘progressive’’ nucleation and growth of paraboloids. The dependence of the surface area of the paraboloidal
growth forms on time is formulated and, consequently, the kinetics of the concurrent evolution of hydrogen has been incorporated
in the formulation of the CTTs. A particular CTT equation is thereby derived for the growth of paraboloids that are ‘‘instanta-
neously’’ formed on a monolayer of deposit. It is shown that the exact shape of some of the CTTs recorded during the electroc-
rystallization of cobalt can be reproduced by the derived CTT.
©
2004 The Electrochemical Society. ͓DOI: 10.1149/1.1805520͔ All rights reserved.
Manuscript submitted October 21, 2003; revised manuscript received June 10, 2004. Available electronically October 27, 2004.
The rate of electrocrystallization is principally governed by the
rate of nucleation as well as the rates of crystals growth. In model-
ling single-phase electrocrystallization processes the rate of nucle-
ation is generally assumed to follow1-4 a first order rate law. How-
ever, the growth rates are generally considered to be controlled
paraboloid is evolved when the parabola is rotated through ␲ about
its y-axis. The equation in a cylindrical coordinate system for a
paraboloid that has been formed right at the beginning of the pro-
Ϫ1
cess, that is at t ϭ 0, and is growing at speeds of v ͑cm s ͒ and vЈ
Ϫ1
͑cm s ͒ in the directions parallel and perpendicular to the electrode
1-3,6-11
either by the rate of charge transfer at,
or by the rate of diffu-
the growing interface. In
surface is, respectively, given by, Fig. 1
sion of the depositing species to,4
,5,12-15
r2 ϭ Ϫ4p͑y Ϫ vЈt͒
͓2͔
this paper, we are interested in the modelling of electrocrystalliza-
tion processes the kinetics of which is controlled by the rate of
charge transfer at the growing interface.
Imposing the following restriction
in
͕
y ϭ 0, r ϭ vt
͖
on Eq. 2 results
A particular model of growth, which is capable of approximately
representing a wide range of growth geometries ͑from a very elon-
gated to a very flattened outlook͒, is the growth of hemispheroids. A
key disadvantage of this form of growth, as a general growth model
for electrocrystallization processes, is its inability to represent nucle-
ation contact angles other than ␲/2. A growth geometry the CTTs for
which have, as yet, not been formulated is that of paraboloids. The
advantage of this growth model over that of hemispheroids is its
ability to represent growth forms of contact angles varying from 0 to
just below ␲/2 ͑a realistic range for all metal deposition processes,
except for Hg͒.
2
p ϭ v t/4vЈ
͓3͔
The current density, j, for any form of growth is related to V
3
2
͑
cm /cm ͒, the volume of the deposit per unit area of the electrode,
through
zF␳ dV
j͑V͒ ϭ
͓4͔
M
dt
where, zF ͑C molϪ1͒ is the charge transfer per deposition of one
Apart from right-circular cone, all other growth forms that have
Ϫ1
Ϫ3
been considered so far7
-11
mole and M ͑g mol ͒ and ␳ ͑g cm ͒ are the molar mass and the
density of the deposit, respectively. The volume, V, is defined by,
Fig. 2
are capable of accounting for the exis-
tence of a single maximum in the experimentally recorded
1
6-18
CTTs.
However, none of these growth forms are capable of
accounting for the observed difference between the recorded maxi-
vЈt
mum current, imax , and its steady-state value, i . ͑The differences
V͑t͒ ϭ
͵
Sudy
͓5͔
st
0
between imax and ist recorded during the electrocrystallization of
cobalt are, invariably, larger than those theoretically predicted.͒
The purpose of this paper is threefold: The first is to derive the
behavior of the CTTs for a model which, in addition to being ca-
pable of representing a wide range of growth forms, is also capable
of relating the shape of the CTTs to the contact angles for nucle-
ation. The second is to extend the modeling to account for the evolv-
ing topography of the surfaces of the growth forms over which
hydrogen is evolving. And, finally, the third is to show the important
role of the evolution of hydrogen in the definition of the forms of the
recorded transients.
where, S , the fractional area of the deposit cut through by a slice of
u
7
height dy(ϭvЈdu), is given by
Su ϭ 1 Ϫ exp͑ϪE͒
͓6͔
E is the number of growth centres expected to cross, within the time
period t, a random point P, situated at a distance y, Fig. 2. ͑Detailed
formulation of E is given in the Appendix.͒
The general CTT equation is obtained by combining Eq. 4, 5,
and 6 with Eq. A-8 of the appendix. The nondimensional form of the
CTT equation is, therefore, given by
CTTs Due to the Model of Growth of Paraboloids
t
In a Cartesian coordinate system, a parabola with its vertex lying
on the y-axis is defined by an equation
dE
I ϭ j/zFkЈ ϭ
͵
exp͑ϪE͒du
dt
0
x2 ϭ Ϫ4py
͓1͔
2
␶ Ϫ s
2
␭2
2 Ϫ ␭s
␭2
2
E ϭ ͑␶ Ϫ ␶s͒ Ϫ
ϩ
Ϫ
exp͓Ϫ␭͑␶ Ϫ s͔͒
·
␭
where p is the distance from the vertex to the focus ͑or directrix͒. A
͓
7͔
z
E-mail: mortezaabyaneh@yahoo.co.uk
where
Downloaded on 2015-06-09 to IP 147.188.128.74 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).

C738
Journal of The Electrochemical Society, 151 ͑11͒ C737-C742 ͑2004͒
Figure 3. Comparison between the forms of the transients that are due to
the progressive mode of nucleation and those that are due to the general
formulation.
Figure 1. Schematic representation of the growth of a lone paraboloidal
center.
␶
ϭ t
s ϭ u
ϭ AЈ/
ͱ
P
ͱ
P
͓8͔
␭
ͱ
P
·
2
P ϭ ␲v N0
A ͑cmϪ2 sϪ1͒ and AЈ ͑sϪ1͒ are the steady-state nucleation rate and
the number of times that a given random position within the domain
of a substrate is expected to have transformed into a nucleus in one
second, respectively. Figure 3 and 4 represent the CTT patterns dis-
played by Eq. 7 at low, intermediate, and high rates of nucleation.
Figure 4. Comparison between the forms of the transients that are due to
the instantaneous mode of nucleation and those that are due to the general
formulation.
The Effects of Concurrent Hydrogen Evolution
The current density due to the evolution of hydrogen on the
surfaces of the paraboloidal growth centres can be written as
jH ϭ z Fk S
͓9͔
H
H
where S is the surface area of the paraboloids. To derive the depen-
dence of S on the deposition time, t, we consider a particular center
of age t , Fig. 5, which was formed after a nucleation period t ,
g
u
where t ϭ t Ϫ t . The r-coordinate of a point such as P at any
u
g
given time t can be obtained by substituting in Eq. 2 the
y-coordinate of point P (y ϭ vЈu) as well as the definition of p
given by Eq. 3. Therefore
r ϭ v͑t Ϫ t ͒1 ͓t Ϫ t Ϫ u͔
/2
1/2
͓10͔
u
u
Figure 5. Graphical representation of a single paraboloidal center, formed
within the annulus dr, with just enough chance of reaching point P in
time t.
Figure 2. Schematic representation of the growth and overlap of a number
of paraboloidal centers formed progressively in time.
Downloaded on 2015-06-09 to IP 147.188.128.74 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).

Journal of The Electrochemical Society, 151 ͑11͒ C737-C742 ͑2004͒
However, that portion of the surface area of the lone center that has
C739
1
0
been cut through by a horizontal slice at point P is given by
d͑S ͒
d͑S ͒ dt
u r
u r
dS ϭ C dᐉ ϭ
dᐉ ϭ
dᐉ
͓11͔
r
r
dr
dt dr
where C is the length of the circumference of the center cut through
r
by a slice and (S ͒ is the top surface area of the slice. But Fig. 5
u r
shows that
2
2 1/2
dᐉ ϭ ͑dr ϩ dy ͒
͓12͔
Combining Eq. 11, 12, and y ϭ vЈu with the inverse of the differ-
ential of r, Eq. 10, with respect to t leads to
d͑S ͒
u r
dS ϭ
r
dt
Figure 6. A general CTT pattern representative of the transients recorded
during the electrocrystallization of cobalt.
t Ϫ t ͒1 ͓v ͑t Ϫ t ͒ ϩ 4vЈ ͑t Ϫ t Ϫ u͔͒
/2
2
2
1/2
͑
u
u
u
ϫ
du
ͭ
ͮ
v͓2͑t Ϫ t ͒ Ϫ u͔
u
flow of current, having reached its peak, imax , at t ϭ t , is progres-
sively reduced until steady-state ͑a constant flow of current, i ) is
2
͓13͔
st
reached. There principally exists a mismatch between the experi-
Evidently, the portion of the surface area of the whole deposit
dS in Fig. 2͒ cut through by the same slice is the summation of
1
8
mentally recorded values of i /ist and those which were theoreti-
(
max
7
-11
cally derived.
Here we show that provided the concurrent evolu-
dS for all centers. It is not immediately obvious how to find a
r
tion of hydrogen is correctly taken into consideration the
theoretically derived CTTs are capable of reproducing the exact
forms of the experimentally observed transients.
closed form solution for such a summation as t has a unique value
u
for each and every center. But in the case of ‘instantaneous’ nucle-
ation t ϭ 0 for all centers. It then follows that
u
The magnitude of the current i1 at t ϭ t , Fig. 6, that is the
1
t1/2͓v t ϩ 4vЈ ͑t Ϫ u͔͒
2
2
1/2
d͑S ͒
magnitude just prior to the formation and growth of three-
dimensional centers, is representative of the rate of evolution of
hydrogen on the surface of the monolayer. However, after t1 the
surface of the monolayer is progressively covered with the 3D
growth forms. Hence, the current density due to the evolution of
hydrogen at t Ͼ t1 is, on one hand, being reduced because of the
coverage of the monolayer by the 3D growth forms and, on the other
hand, is being simultaneously increased by the evolution of hydro-
gen on the top surfaces of 3D growth forms. Thus, for paraboloids
instantaneously formed on top of a monolayer of deposit, the com-
ponent of current density due to the evolution of hydrogen can be
written as
u inst
͑
dS͒inst
ϭ
ͭ
ͮ
du
͓14͔
dt
v͑2t Ϫ u͒
However, combining Eq. A-2 and A-1 of the appendix with Eq. 6
leads to
2
2
͑
S ͒ ϭ 1 Ϫ exp͓Ϫ␲v ͑t Ϫ tu͒N ͔
͓15͔
u inst
0
Therefore, substituting the differential of Eq. 15 with respect to t
into Eq. 14 and integrating the resulting equation would lead to the
definition of Sinst , which is also the nondimensional representation
of the transient current, IH,inst ͑see Eq. 9͒. Thus
␶
2
2
IH,inst ϭ eϪ␶
͓͑1 ϩ tan ␪͒␶ Ϫ ␶s tan ␪͔ e ds ͓16͔
2
2
2
1/2 ␶s
Ϫ␶
͵
jH,inst ϭ z Fk ͓e
H
H
ϩ IH,inst
]
͓18͔
0
where the first term in the square bracket represents progressive
coverage of the monolayer and IH,inst represents the surface area of
the 3D growth forms ͑see Eq. 16͒. In Eq. 18, the parameter ␶ ͑de-
fined in Eq. 8͒ possess a time element, which must be modified so as
to include the period t1 during which the monolayer has been
where ␶ and s are defined in Eq. 8 and
tan ␪ ϭ 2vЈ/v
͓17͔
Electrocrystallization of Cobalt
formed. The final CTT equation is then obtained by adding jH,inst
The CTTs recorded at the lowest possible applied nucleating po-
tentials during the electrocrystallization of cobalt on a vitreous car-
bon electrode was first studied in 1990.1 This study was carried out
by developing some appropriate models for the electrocrystallization
processes and by assessing the extent of the fit of the recorded CTTs
͑after modification of ␶͒ to j in Eq. 7 with the expectation E in
8
its simplified ‘instantaneous’ form ͑that is, in the form provided by
Eq. A-2 of the appendix͒. The final form of the CTT is therefore
given by
͑
tϪt ͒
1
2
ϪP͑tϪt )
͓P͑1 ϩ tan ␪͒͑t Ϫ t ͒ Ϫ P tan ␪͑t Ϫ t ͒u͔1/2e
2
2
2
P͑tϪt ͒u
1
1
jinst ϭ z Fk e
1 ϩ
͵
du
H
H
ͭ
1
1
ͮ
0
2
2
2
1
ϩ P͑t Ϫ t ͒ Ϫ ͓1 ϩ 2P͑t Ϫ t ͒ ͔exp͓ϪP͑t Ϫ t ͒ ͔
1
1
1
ϩ zFkЈ
͓19͔
ͭ
ͮ
P͑t Ϫ t1͒2
to the derived equations using the nonlinear regression software.19
The general pattern of the recorded CTTs is illustrated in Fig. 6. The
The nondimensional form of Eq. 19 is obtained by dividing jinst by
z Fk . Hence
H
H
Downloaded on 2015-06-09 to IP 147.188.128.74 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).

C740
Journal of The Electrochemical Society, 151 ͑11͒ C737-C742 ͑2004͒
2
2
2
␶
Ј
1 ϩ ␶Ј Ϫ ͑1 ϩ 2␶Ј ͒exp͑Ϫ␶Ј ͒
␶Ј2
Ϫ␶Ј2
2
2
2
Iinst ϭ e
1 ϩ
͵
ͱ
P͑t Ϫ t ͒
1
͓͑1 ϩ tan ␪͒␶Ј Ϫ ␶Јs tan ␪͔1/2e␶Јsds ϩ kr
ͭ
ͮ
ͫ
ͬ
0
͓
20͔
ͮ
where ␶Ј ϭ
and k ϭ kЈ/k
r
H
Figure 7 represents the pattern of behavior defined by Eq. 20 for
kr ϭ 14 and for the angle of contact changing from 45 to 89°. The
transients are shown to be virtually independent of the magnitudes
times lower than the rate of covering the site by the act of growth.
The ratio, imax /ist , is found to be only slightly dependent on the rate
of nucleation and its value is about 1.244.
of the angles of contact for ␪ р 60°. The ratio of imax to i appears
to be accentuated only at contact angles close to 90°.
Figure 8 shows the observed behavior of the transient, dotted
curve, recorded during the electrocrystallization of cobalt on a vit-
reous carbon electrode at an applied potential of Ϫ0.875 V͑SCE͒.
To account for imax /ist Ͼ 1.244 Eq. 19 has been derived. The
full line in Fig. 7 portrays the form of the CTT as defined by Eq. 19,
st
Discussion
Equation 7 predicts the pattern of behavior of the CTTs that are
expected from nucleation and growth of crystals of paraboloidal
appearance. Apart from the dependence of the form of the CTT on
Ϫ2
for the specific values of z Fk ϭ 150 ␮A cm , ␪ ϭ 88.5°,
H
H
Ϫ2
Ϫ2
zFkЈ ϭ 2100 ␮A cm , t ϭ 1.8 s and P ϭ 0.0033 s . The val-
1
␭
, Eq. 7 is independent of any other parameters; in particular, the
ues of z Fk and t1 are, respectively, estimated from the magni-
H
H
form of the CTT is independent of the contact angle, ␪, which each
nucleus makes with the substrate electrode. The contact angle is
related to the rates of growth by the definition provided in Eq. 17.
Figure 3 and 4 demonstrates that this equation reduces to the
limiting forms, that is to the instantaneous and to the progressive
modes of nucleation, at high and low rates of nucleation, re-
spectively. Figure 3 shows that the behavior of Eq. 7 approaches
the pattern expected from the ‘progressive’ nucleation at values of
tudes of the current density, i , and from the time at which the
monolayer is considered to have been formed, Fig. 8. The contact
angle of 88.5° and the value chosen to represent zFkЈ are the result
1
of comparing Fig. 7 with Fig. 8. ͑From the definition of k provided
r
Ϫ2
in ͑20͒ zFkЈ ϭ 150 ϫ k ϭ 150 ϫ 14 ␮A cm .͒ A very close fit
r
between the experimental data, dotted curve, and the theoretically
derived equation, full curve, is observed indicating the crucial role
which the evolution of hydrogen plays in the definition of the re-
corded CTTs.
␭
р 0.03. Figure 4, on the other hand, demonstrates that nuclea-
tion can be considered ‘instantaneous’ at ␭ у 45. As expected,
Note that despite a very close fit of the experimentally observed
transient to the theoretically derived CTT, we do not propose the
paraboloidal growth geometry to be the only form of growth capable
of reproducing the form of the recorded transients. By analyzing the
paraboloidal growth geometry we have shown that the variation in
the experimentally observed values of imax /ist can be explained by
the concurrent evolution of hydrogen, and the recorded CTTs due to
the electrocrystallization of cobalt can adequately be described by
an appropriate model of growth of paraboloids if and only if the
contact angles for growth forms are assumed to be approaching ␲/2.
2
the early stages of the CTT profiles are characterized by a j ϰ t
correlation at ␭ у 45 and by a j ϰ t3 relation at ␭ р 0.03. It fol-
lows that the early parts of the rising transient in the prevailing
n
intervals, 45 Ͼ ␭ Ͼ 0.03, would be characterized by j ϰ t , where
Ͼ n Ͼ 2.3
3
Therefore in conformity with a previous findings,11 vis- a` -vis the
model of growth of hemispheroids, the required conditions for the
application of the instantaneous and progressive limiting forms of
the CTT are, respectively, specified by
AЈ р v
ͱ
A/AЈ/20
͓21͔
Conclusions
A generalized form of the CTT equation, which is operative over
all rates of nucleation, has been derived for the model of growth of
paraboloids. The limiting forms of this equation, the so-called in-
and
AЈ у 60v A/AЈ
ͱ
͓22͔
The above equations indicate that the kinetics of electrocrystal-
lisation can be assumed to be governed by the ‘instantaneous’ and
‘progressive’ modes of nucleation whenever the expected rate of
turning a given site into a nucleus is either 60 times higher or 20
Figure 8. """"" The observed CTT pattern recorded during the electro-
crystallization of cobalt onto a vitreous carbon electrode at an applied
Ϫ3
potential of Ϫ0.8 V ͑SCE͒. Solution composition: 240 g dm CoSo •7H O,
4
2
Ϫ3
Ϫ3
3
6 g dm CoCl •6H O, and 36 g dm H BO .
The pattern ex-
2
2
3
3
Ϫ2
pected from Eq. 19 for: z Fk ϭ 150 ␮A cm , ␪ ϭ 88.5°, zFkЈ
H
H
Ϫ2
Ϫ2
Figure 7. The CTT pattern predicted by Eq. 20 for k ϭ 14.
ϭ 2100 ␮A cm , t1 ϭ 1.8 s, and P ϭ 0.0033 s .
r
Downloaded on 2015-06-09 to IP 147.188.128.74 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).

Journal of The Electrochemical Society, 151 ͑11͒ C737-C742 ͑2004͒
C741
rmax
stantaneous and the progressive limits, are shown to be, respectively,
E ϭ
2␲rN͑t ͒dr
͓A-3͔
operative whenever the expected rate of turning a given site into a
nucleus is either 60 times higher or 20 times lower than the rate of
covering the site by the act of growth.
͵
u
0
If the number of these growth units, per unit area of the electrode surface, is denoted by
The form of the equation, Eq. 7, is independent of the nucleation
contact angle. The transient current is shown to pass through a
single maximum before approaching its steady-state value and the
ratio of the maximum current to the steady-state value is observed to
be about 1.244. The maximum flow of current is estimated to be
coincidental with 77% of the coverage of the electrode by the de-
posit in instantaneous nucleation and with 90% coverage in progres-
sive nucleation.
The dependence of the surface area of the deposit on time has
been derived for the specific case of instantaneous nucleation and
growth of paraboloids and, as a result, the component of the tran-
sient current due to the hydrogen evolution reaction has been for-
mulated.
The CTT due to the instantaneous nucleation and growth of pa-
raboloids has been modified to correctly incorporate the concurrent
hydrogen evolution reactions. This is achieved by accounting, on
one hand, for the reduction in the hydrogen evolution ͑due to the
progressive coverage of the monolayer by the 3D growth centers͒
and by considering the evolution of hydrogen on the top surfaces of
the 3D growth forms.
N(tu), then
A
N͑tu͒ ϭ
͓1 Ϫ exp͑ϪAЈtu͔͒
͓A-4͔
AЈ
The dependence of r, Fig. 5, as a function of u can be obtained by substituting tg in
place of t in Eq. A-1. Thus
Mk
͑t Ϫ tgu͒1/2
2
r ϭ
͓A-5͔
g
␳
On the other hand r, in terms of its own time step ␷, can be written as
Mk
r ϭ
␷
͓A-6͔
␳
Combination of Eq. A-5 and A-6 with tu ϭ t Ϫ tg leads to the definition of the
maximum permitted period, tu , during which time all centers formed within the annulus
are expected to cross point P. Thus
u2 ϩ 4␷2
ͱ
u ϩ
tu ϭ t Ϫ
͓A-7͔
The modified CTT is shown to simulate the transient pattern
recorded during the electrocrystallization of cobalt.
2
The general form of the expectation, E, can, therefore, be derived by combining Eq.
A-1, A-3, and A-4 with A-7. Hence
Appendix
Formulation of the expectation, E.—The expectation value is formulated here in
ͱ
t͑tϪu)
u2 ϩ 4␷2
ͱ
2
accordance with the method of formulation, first identified in 19827 for the transforma-
tions due to surface nucleation and 3D growth. Figure 9 represents two growth centers,
which are imagined to have formed right at the beginning of the electrocrystallization
u ϩ
E ϭ 2P
␷
ͭ
1 Ϫ exp
ͫ
ϪAЈ
ͩ
t Ϫ
ͪ
ͬ
ͮ
d␷
͓A-8͔
͵
0
process on the rim of a circular region of the electrode surface of radius rmax . It is
noted that rmax , the r-coordinate of any of the two centers, can be obtained by substi-
tuting in Eq. 2 the y-coordinate of point P, the definition of p given by Eq. 3 as well as
the equality v ϭ Mk/␳. Thus
List of Symbols
A
nucleation rate constant, cmϪ2 sϪ1
AЈ expected frequency of the formation of a nucleus at a given site, s
Ϫ1
E
F
i
j
k
number of growth centres expected to cross a given point P in time t
Faraday constant, C mol
rmax ϭ Mk͑t2 Ϫ tu͒1/2/␳
͓A-1͔
Ϫ1
current ͑A͒
current density, A cm
Ϫ2
Figure 9 implies that those crystals that are formed at sites outside the circular
region have no possibility whatsoever to reach point P in time t. It is only in the case of
the instantaneous formation of crystals that each and every one of the centers formed
within the circular region are also expected to cross point P in time t. Therefore
Ϫ2
rate of crystal growth in the direction parallel to the electrode surface, mol cm
Ϫ1
s
rate of the outward growth of crystals, mol cmϪ2 sϪ1
k
Ј
Ϫ1
M
molar mass of the deposit, g mol
Einst ϭ ␲r2 N0
max
͓A-2͔
Ϫ2
N0 density of nucleation sites, cm
P
rmax the radius of a circular region of an electrode the growth forms within which may
reach point P in time t
2
2
2
Ϫ2
ϭ␲M k A/AЈ␳ , s
where N0 ͑cmϪ2͒ is the maximum number of nuclei, per unit area of the electrode
surface, that can possibly form at the potential concerned.
For any other case, we first consider an annulus dr within the circular region, Fig.
, and determine the maximum period, tu , during which all growth centers formed
s
S
ϭu P
ͱ
the surface area of the deposit per unit area of the electrode surface
9
within the annulus reach P in time t. Figure 5 is a representation of a particular growth
center of age tg , centred at a point C within the annulus dr shown in Fig. 9. This is a
very special center in that it is the last of the centers formed within the annulus that
arrives at point P. Thus, only those growth centers within the annulus that are formed
during a period tu , defined by tu ϭ t Ϫ tg , have any chance whatsoever to cross point
P in time t. If the number of these growth units, per unit area of the electrode surface,
is denoted by N(tu), then
Su the fractional area of the deposit cut through by a slice of width dy
t
time ͑s͒
Ϫ1
v
ϭMk/␳, growth rate in the direction parallel to the electrode surface, cm s
v
Ј
ϭMkЈ/␳, growth rate in the direction perpendicular to the electrode surface, cm
Ϫ1
s
3
2
V
z
the volume of the deposit per unit area of the electrode surface, cm /cm
the number of electron transfer per ion
the ellipticity value
⑀
␭
␪
␳
␶
ϭAЈ/ P
ͱ
nuclei contact angle
Ϫ3
the density of the deposit, g cm
ͱ
ϭt P
References
1
. M. Fleischmann and H. R. Thirsk, in Advances in Electrochemistry and Electro-
chemical Engineering, Vol. 3, Interscience, London ͑1963͒.
2. M. Y. Abyaneh and M. Fleischmann, J. Electroanal. Chem., 119, 187 ͑1981͒.
3. M. Y. Abyaneh and M. Fleischmann, Electrochim. Acta, 27, 1513 ͑1982͒.
4. B. Scharifker and J. Mostany, J. Electroanal. Chem., 117, 13 ͑1984͒.
5. A. Milchev, Electrocrystallisation: Fundamentals of Nucleation and Growth, Klu-
wer Academic Publishers, Amsterdam ͑2002͒.
6
. R. D. Armstrong, M. Fleischmann, and H. R. Thirsk, J. Electroanal. Chem., 11, 208
͑
1966͒.
7
8
. M. Y. Abyaneh, Electrochim. Acta, 27, 1329 ͑1982͒.
. E. Bosco and S. K. Rangarajan, J. Electroanal. Chem., 134, 213 ͑1982͒.
Figure 9. Graphical representation of two paraboloidal growth forms with
just enough chance of reaching point P in time t.
9. M. Y. Abyaneh, J. Electroanal. Chem., 209, 1 ͑1986͒.
10. M. Y. Abyaneh, Electrochim. Acta, 36, 727 ͑1991͒.
Downloaded on 2015-06-09 to IP 147.188.128.74 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).

C742
Journal of The Electrochemical Society, 151 ͑11͒ C737-C742 ͑2004͒
11. M. Y. Abyaneh, J. Electroanal. Chem., 387, 29 ͑1995͒.
15. V. A. Isaev and A. N. Baraboshkin, J. Electroanal. Chem., 37, 33 ͑1994͒.
1
2. M. Sluyters-Rehbach, J. H. O. J. Wijenberg, E. Bosco, and J. H. Sluyters, J. Elec-
troanal. Chem., 236, 1 ͑1987͒.
16. M. Y. Abyaneh, Trans. Inst. Met. Finish., 58, 91 ͑1980͒.
17. M. Y. Abyaneh and M. Fleischmann, J. Electroanal. Chem., 119, 197 ͑1981͒.
18. A. Tajali-Pour, Ph.D. Thesis, Coventry University ͑1990͒.
19. M. Y. Abyaneh, Ph.D. Thesis, Southampton University ͑1980͒.
1
1
3. M. V. Mirkin and A. P. Nilov, J. Electroanal. Chem., 283, 35 ͑1990͒.
4. P. A. Bobbert, M. M. Wind, and J. Vlieger, Physica A, 146, 69 ͑1987͒.
Downloaded on 2015-06-09 to IP 147.188.128.74 address. Redistribution subject to ECS terms of use (see ecsdl.org/site/terms_use) unless CC License in place (see abstract).