1. Numerical simulation of electro-deposition process influenced by force
convection and migration of ions
M. Zahraei ⁎, M.S. Saidi, M. Sani
Micro electro kinetic Laboratory, Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
a b s t r a c ta r t i c l e i n f o
Article history:
Received 22 July 2015
Received in revised form 5 October 2016
Accepted 7 October 2016
Available online 12 October 2016
Electro-deposition process is one of the main steps in the LIGA procedure to fabricate microstructures. In this
paper, one-dimensional modeling of Nickel electro-deposition process is implemented on Rayan and developed
for simulation of time-dependence diffusion and migration of charge species with reduction reactions on the
cathode surface. This model is proposed by considering governing equations on electro-kinetic phenomena con-
sist of Nernst-Plank equation and Poisson's equation of electric potential. Transport of ions toward the cathode is
considered based on the effect of convection, reaction rate, diffusion and migration. The numerical results cover
two series of data consisting of effective diffusion layer thickness δeff and the transient current density. The effect
of force convection and diffusion terms on effective diffusion layer δeff is validated by Ribeiro analytical model.
The transient current densities for different applied voltage.
s are in well agreement with Hyde and Compton's experimental model. Effect of every term on effective diffusion
layer δeff is shown and we found that for velocity lower than vc = −0.0005 cm s−1
, convection term does not
have any influences on effective diffusion layer δeff . Moreover, the relationship between the applied voltages,
current density, effective diffusion layer δeff and hydrodynamic velocity is proposed.
© 2016 Elsevier B.V. All rights reserved.
Keywords:
Diffusion
Effective diffusion layer
Migration
Force convection
Transient current density
1. Introduction
Electro-deposition is a well-known coating technique which has the
potential to fabricate high aspect ratio structures with uniform deposi-
tion regardless of the surface size and shape [1–4]. The techniques,
which are based on controlled current or controlled potential electro-
chemical deposition on the cathode electrode from a solution contain-
ing the appropriate compounds [5,6]. Large number of metals can be
deposited from aqueous electrolytes (Ni, Cu, Au and Fe to name a
few). Nickel electro-deposition is more useful to form mechanical and
magnetic elements [7].
The first modeling of diffusion term in electro-deposition process
has taken place in the paper of Scharifker and Hills [8]. A review of cur-
rent response to the potential step perturbation, the influence of diffu-
sion in nuclei growth for unique geometry can be found in references
[9,10]. In these models the only factor which influences on ion move-
ment is diffusion term. All of these modeling used a Cottrelian diffusion
term to describe the total flux to the electrode surface, since these
models are inapplicable under hydrodynamic condition.
In Hyde, Klymenko and Compton's work [11] the diffusion term at
the same way with Scharifker's paper [8] under forced convection con-
dition has theoretically been explored. Furthermore, the current density
response to this model has experimentally been derived and compared
with theoretical results. In Hyde and Compton's paper [12] application
of Hyde model in lead deposition growth under two different hydrody-
namic conditions has also been computed.
In Ajello's paper [13] an expression for the transient current density
verified during the electro-deposition of metals has derived as a func-
tion of concentration, reaction rate and time, I.e., I=I(c,k,t). In Ajello
and Schervenski's paper [14] the general form for the transient current
density I=I(c,V,T,t) has also been considered. In this models [13,14]
the diffusion and reaction terms take part in transport of ions.
One-dimensional model formulated by Ribeiro's paper [15] based on
the concentration equation to describe transport of ion species toward
cathode with considering diffusion and force convection has theoreti-
cally been solved via separation of variables. In fact in Ribeiro's work
[15] the transient current density is a function of concentration, time,
hydrodynamics velocity and reaction rate, I=I(c,k,v,t). Constant veloc-
ity is also assumed to demonstrate the effects of forced convection on
the current density-potential profiles and transient current density.
One of the main steps in cathodic deposition of alloy or single metals
is ionic migration, meaning that the hydrated ions in the electrolyte mi-
grate toward the cathode under the influences of the applied voltage as
well as diffusion and convection. In all previous models, the effect of mi-
gration term on concentration distribution and current density were not
mentioned.
In this model we consider all three mechanism of electro-migration,
diffusion, and convection into the concentration equation; indeed we
Journal of Electroanalytical Chemistry 782 (2016) 117–124
⁎ Corresponding author.
E-mail address: zahrae.mohsen@gmail.com (M. Zahraei).
http://dx.doi.org/10.1016/j.jelechem.2016.10.012
1572-6657/© 2016 Elsevier B.V. All rights reserved.
Contents lists available at ScienceDirect
Journal of Electroanalytical Chemistry
journal homepage: www.elsevier.com/locate/jelechem

2. used Nernst–Planck equations to describe the transport and distribution
of ion in distance between cathode (x=0) and diffusion layer thickness
(x=δ). The transient current densityI=I(c,k,v,E,t)is also obtained by
considering all parameters. Since the migration term in concentration
equation makes it impossible to solve this equation in theoretical meth-
od, we used numerical solution of electro-deposition process generated
by Rayan1
software. To validate results of first solution without migra-
tion effect, we compared the concentration profiles with theoretical
model of Ribeiro [15]. By considering migration term of charge species
the effect of hydrodynamic velocity and applied voltage on the effective
diffusion layer δeff and current density are recognized. Since in Hyde and
Compton model [12] the current density profiles force convection effect
have experimentally been exploited, we used experimental results of
Ribeiro's paper [15] and Hyde and Compton's work [12] to validate tran-
sient current density at the beginning of the electro-deposition process
for different voltages on the cathode surface.
Greek letters
ε Dielectric constant permittivity [ F cm−1
]
δ Diffusion layer thickness [cm]
δeff Effective diffusion layer [cm]
σf Free charge density [cm−2
]
2. Mathematical model
Scalar equation of movement of the ions between electrodes is
governed by the Nernst-Plank equation. This equation includes diffu-
sion, convection and migration of ions and for each ionic species in
one dimension can be written as [16]:
∂ci
∂t
¼ Di
∂
2
ci
∂x2
−Vx:
∂ci
∂x
þ
FDizi
RT
∂
∂x
: ci
∂E
∂x
ð1Þ
where Vx is the hydrodynamic velocity of the electrolyte and E is the
electric potential. x is the position in the direction normal to the elec-
trode surface, which is located at x=0. Di, ci and zi represent ion species
valency, diffusion coefficient and ionic concentration for species i, re-
spectively [16]. The electric potential distribution in the electrolyte solu-
tion is related to the volumetric free charge density by Poisson's
equation [17]:
∂
2
E
∂x2
¼ −
σ f
ε
ð2Þ
where ε is the dielectric constant permittivity and σf is the free
charge density is expressed as follow:
σ f ¼
X
i
zieci ð3Þ
To solve the Poisson equation local electro-neutrality constraint is
enforced. In this case, the electric field becomes an unknown constant
to be determined as part of the overall solution and the electric potential
equation becomes a Laplace equation [17]. In this work, we considered a
Nickel sulfate solution consisting of Ni+2
and So4
−2
ions. The Nickel re-
duction reaction takes place on the cathode surface according to:
Niþ2
þ 2e−
→Ni ð4Þ
Furthermore, homogeneous reactions occur in the electrolyte solu-
tion, such as reaction (5), which potentially could affect the Nickel ion
concentration.
Niþ2
þ OH−
→Ni OHð Þþ
ð5Þ
Since in this modeling the bath pH was assumed to be maintained
below PH=4, the concentration of Ni(OH)+
and the corresponding
concentration change of Ni+2
due to its production is negligible and is
not considered in the analysis [17,18]. Whereas we considered two
ions with equal and opposite valences and electro-neutrality condition
for electrolyte, we have the same concentration in any point for each
of these ions. Diffusion coefficient, temperature and electrode kinetic
parameters are constant during this modeling. To produce convection
term a constant hydrodynamic velocity vc in(−x) direction is consid-
ered. By considering these conditions, governing equations are simpli-
fied as follows:
∂c
∂t
¼ D
∂
2
c
∂x2
−vc
∂c
∂x
þ
FDz
RT
∂
∂x
: c
∂E
∂x
ð6aÞ
∂
2
E
∂x2
¼ 0 ð6bÞ
To solve both Eqs. (6a) and (6b) we adopted the following initial and
boundary conditions:
C x; 0ð Þ ¼ cb E x; 0ð Þ ¼ 0 ∀N0; ð7aÞ
C δ; tð Þ ¼ cb E δ; tð Þ ¼ 0 ∀t; ð7bÞ1
A polyhedral grid co-located incompressible finite volume solver.
Nomenclature
D Diffusion coefficient of species [cm2
s−1
]
vc Convection velocity [cms−1
]
T Temperature [K]
Tr A reference temperature [K]
e Electric charge [C]
F Faraday constant [96485.5 C mol−1
]
Z Charge number of the ionic species
Vl Start deposition potential [V]
Vk Start deposition potential in the sigmoidal [V]
Vapp Applied voltage [V]
K Reaction rate of ions on the electrode surface [s−1
]
Ista Stationary current density [mA cm−2
]
I Current density [mA cm−2
]
R Universal gas constant [J mol−1
K−1
]
C Concentration of the species [M]
B Constant that regulate the sigmoidal inclination [V−1
]
cs Limit concentration of the species on the electrode sur-
face [M]
cb Bulk concentration of the species [M]
E Electric potential [V]
E0 Constant applied voltage on the cathode surface [V]
t Time [s]
ID Diffusion term of cathode current density [M s−1
]
Ic Convention term of cathode current density [M s−1
]
Im Migration term of cathode current density [M s−1
]
Ista
R
(t) Stationary current density developed by Ribeiro's
model [mA cm−2
]
IR
(t) Current density developed by Ribeiro model [mA cm−2
]
Ista
HK
(t) Stationary current density developed by Hyde and
Klymenko's model [mA cm−2
]
IHK
(t) Current density developed by Hyde and Klymenko's
model [mA cm−2
]
118 M. Zahraei et al. / Journal of Electroanalytical Chemistry 782 (2016) 117–124

3. C 0; tð Þ ¼ cb−csð Þ exp −ktð Þ þ cs E 0; tð Þ ¼ −E0 ∀t: ð7cÞ
cs and cbrepresent limit ion concentration and bulk ion concentra-
tion on the electrode surface respectively. δ is defined for the diffusion
layer thickness or distance between cathode (x=0) and anode
(x=δ). The boundary condition (7b) limits the solution between
x=0 and x=δ. Finally the boundary condition (7c) determines the var-
iation of the concentration of ions on the cathode surface in terms of
time. In this boundary condition k is the reduction reaction rate and
E0 is a constant applied voltage on the cathode surface. The reaction
rate k is in terms of applied voltage on the cathode surface and temper-
ature. Since temperature is constant in this modeling, the reaction rate
is only a function of applied voltage. For constant temperature fixed at
300 K we have [15,19]:
k ¼
1
1 þ exp b Vapp−Vk
À ÁÂ Ã
1
1 þ exp α Vapp−Vl
À ÁÂ Ã ð8Þ
where α is the reduction coefficient in the same vein as Butler and
Volmer [20,21], and b is the constant that regulate the sigmoidal
inclination. α=19.56 V−1
at T=300K and b=200 V−1
for every tem-
perature. Vl and Vk are start deposition potential and start deposition
potential in the sigmoidal, respectively [15,19]. The value of k demon-
strates rate of reaction of ions on the cathode surface. Since the current
density on the cathode surface is proportional to the mass flux, the
charge current density is obtained by multiplying the molar charge (–
zF) by the mass flux, which can be written as:
I x ¼ 0; tð Þ ¼ −zF D
∂c
∂x
x¼0;tð Þ
þ
FDz
RT
c x ¼ 0; tð Þ
∂E
∂x
x¼0;tð Þ
!
−vcc x ¼ 0; tð Þ
( )
ð9Þ
By solving Eqs. (6a) and (6b) under boundary conditions (7a), (7b)
and (7c) the concentration and electric potential are found in every
point of anode-cathode distance. During each set of solutions, the veloc-
ity of every point is fixed at constant value. Using above parameters in
Eq. (9), the current density can be computed in corresponding time of
process.
3. Results and discussions
In this section the results and discussions for validation of the com-
putational model, the distribution of ions species concentration, tran-
sient current density and relation between parameters which take
place in electro-deposition process are presented. The formulation
mentioned in this model was implemented in the Rayan code via Fi-
nite-Volume method [22–24].
3.1. Model validation
To validate computational model the concentration distribution be-
tween electrodes is compared with analytical solution developed by
Ribeiro [15]. Fig. 1 shows the concentration distribution obtained from
numerical solution without the effect of hydrodynamic velocity and mi-
gration of ions. Concentration profiles are shown for three different time
intervals and setting vc =0 as hydrodynamic. The effective diffusion
layer thickness δeff, was defined as depth in concentration distribution
between the surface of cathode (x=0) and the position where the cal-
culated concentration is equal to the bulk concentration. Fig. 1 also
shows that the thickness of the effective diffusion layer δeff increases
with elapsing the time of process.
Fig. 2 shows the concentration distribution with the effect of hydro-
dynamics velocity and without the effect of migration. Concentration
profiles are shown for two different time intervals and setting
vc = −0.0005 cm s−1
and vc = −0.005 cm s−1
. In Fig. 2 it is noticed
that increasing velocity leads to decrease required time to reach a con-
stant effective diffusion layer δeff.
In Figs. 1 and 2 the profiles developed by computational solution of
Rayan code were absolutely congruous with analytical solution from
Ribeiro model [15]. In Fig. 2 the crosses (vc = −0.0005 cm s−1
) and
the solid circles (vc = −0.0005 cm s−1
) show the effect of spent time
on concentration distribution. The solid circles
(vc = −0.0005 cm s−1
) and the solid line (vc = −0.005 cm s−1
) also
show the effect of velocity on concentration distribution. In fact increas-
ing the convective velocity leads to decreasing the effective diffusion
thickness. This behavior is in agreement with the experimental observa-
tions made by Hyde and Compton [12]. The equations for different ve-
locities are solved and summarized in Table 1. The results of Table 1 in
Fig. 1. Concentration profiles without the effect of hydrodynamic velocity (vc=0)and migration of ions.
119M. Zahraei et al. / Journal of Electroanalytical Chemistry 782 (2016) 117–124

4. Ribeiro model [15] are also compared with present numerical solution
(as shown in Figs. 1 and 2) in Table 1.
3.2. Concentration distribution influenced by migration term
Present work is focused on the effect of migration term (last term in
Eq. (1)) on ions distribution and current density variation. Fig. 3 shows
the concentration distribution with effects of both hydrodynamics ve-
locity and migration of ions. In this case the governing Eq. (6) with
boundary conditions (7a), (7b) and (7c) are solved. The applied voltage
on cathode surface is fixed on Vapp = −1.2 V beside the reaction rate
calculation is used.
All mentioned profiles (Figs. 1 to 3) are set for constant temperature
T=300 K with limiting concentration of cs =0.014 M and bulk concen-
tration ofcb =0.026 M. As shown in Fig. 3 open square symbols and solid
lines are roughly coincident. Hence by considering migration term, the
effect of velocities below vc = −0.0005 cm s−1
can be ignored. For
flow speeds higher than vc = −0.0005 cm s−1
the effect of hydrody-
namic velocity will be sizeable, nevertheless these effects are lower
than the previous model (as shown in Fig. 2). The ratio of diffusion to
convection term and the ratio of diffusion to migration term are obtain-
ed from current density Eq. (9), respectively are equal to:
ID
Ic
¼
D
∂c
∂x
0; tð Þ
vcc 0; tð Þ
ð10aÞ
ID
IM
¼
RT
∂c
∂x
0; tð Þ
Fzc 0; tð Þ
∂E
∂x
0; tð Þ
ð10bÞ
The concentration gradient ∂c
∂x
is lower than the electric potential gra-
dient∂φ
∂x
on the cathode surface at the beginning of the process. Concentra-
tion variation on the cathode surface is between cb=0.026 and cs=0.014
during the process. By using this data we found that the effect of force
convection for velocities lower than vc= −0.0005 cm/s is equal to the ef-
fect of migration term and for higher velocities the effect of force convec-
tion term will be increased. The effect of diffusion term can be neglected
compared with these two terms. Indeed the migration of ions is the im-
portant factor in transportation of ion species toward the cathode surface
at the beginning of electro-deposition process and after some time
elapsed from the beginning of the process the concentration gradient ∂c
∂x
ð
0; tÞ and also the effect of diffusion term is increased. According to Fig. 3
and arguments mentioned above, by considering migration term the ef-
fect of velocity in this modeling is lower than Ribeiro modeling.
3.3. Transient current density influenced by hydrodynamic velocity and ap-
plied voltage
In Ribeiro analytical modeling [15] the equation of scalar concentra-
tion (6a) without migration term and with boundary conditions (7a),
(7b) and (7c) are solved via separation of variables. Then distribution
of concentration has been used within bellow equation:
IR
x ¼ 0; tð Þ ¼ −zF D
∂c
∂x
x¼0;tð Þ
−vcc x ¼ 0; tð Þ
( )
ð11Þ
Hence the current density by Ribeiro model is expressed as:
IR
tð Þ ¼ −zFD −
θδ þ 1
δ
cb−csð Þ exp −ktð Þ þ csð Þ þ
cb
δ
exp δθð Þ þ
2
δ
X∞
n¼1
f n tð Þ
( )
þ zFvc cb−cg
À Á
exp −ktð Þ þ cs
À Á
ð12Þ
Fig. 2. Concentration profiles influenced by hydrodynamic velocity (vc= −0.005 cm s−1
vc= −0.0005 cm s−1
), without migration effect.
Table 1
Comparison of the effective diffusion layer δeff(cm) between Ribeiro model [15]and nu-
merical model.
case Vccm/s δeff (Ribeiro model [15]) δeff (numerical solution)
1 −0.005 0.0095 0.0096
2 −0.004 0.0127 0.0124
3 −0.003 0.0176 0.017
4 −0.002 0.0258 0.0254
5 −0.0015 0.0351 0.035
6 −0.001 0.0458 0.0461
120 M. Zahraei et al. / Journal of Electroanalytical Chemistry 782 (2016) 117–124

5. Where θ= −vc/2D andfn(t) is introduced in Ribeiro paper [15]. The
value of the stationary current is obtained from Eq. (9) whent→∞. In
Ribeiro paper [15], the stationary current density is attained:
IR
sta tð Þ ¼ −
zFD
δ
−cs þ cb exp −
δvc
2D
þ
3vcδcs
2D
þ 2
X∞
n¼1
lim
t→∞
f n tð Þ
( )
ð13Þ
In this modeling the current density with the effect of migration is
obtained from Eq. (9), which was already expressed. In Fig. 4 the current
density profiles from numerical solution with the effect of migration
under different applied voltages are shown and compared with those
from Ribeiro model at the beginning of the electro-deposition process.
As shown in Fig. 4 the current density obtained from the present
model at the beginning of electro- deposition process is lower than
the current density from Ribeiro model. Since after some time steps
Fig. 3. Concentration profiles influenced by hydrodynamic velocity and migration of ions, the continuous line, circle, and lozenge sketched curves, were recorded from 5 to 15 s by the effect
of migration term. The curve in crosses symbols were recorded from 5 to 15 s without the effect of migration.
Fig. 4. Time-dependence current density at the beginning of the electro-deposition process for different applied voltages on the cathode surface using δ=0.0458 cm and vc= −0.001 cm
s−1
. The continuous lines were recorded by the effect of migration, and the line-point ones refer to the current density without the effect of migration.
121M. Zahraei et al. / Journal of Electroanalytical Chemistry 782 (2016) 117–124

6. the slab of current density profile increased, the value of the current
density from new model and Ribeiro model are equal in one point and
after this point the current density will be higher than the current
from Ribeiro model. In Fig. 4 the current density profiles are also
shown for different applied voltages and every profile have the same
treatment. In fact the migration term decreased the value of current
density at the beginning of process and then it caused positive effects
on the value of current density. Whereas the Eqs. (6a) and (6b) are
coupled via migration term, every change in electric potential influ-
ences concentration distribution of ion. Hence this new distribution re-
sults in different current density profiles. So the new current density
profiles will be different from Ribeiro because of the two factors:
− The new concentration distribution exists in all of the diffusion, mi-
gration and convection terms.
− The variation of potential gradient in migration term.
To calculate the current density profiles, we used constant diffusion
layer δ=0.0458 cm is obtained by given velocity vc = −0.001 cm s−1
and observe the effect produced by the applied voltage on the current
transients.
In Hyde, Klymenko and Compton's model [8], potentiostatic current
response of a three-dimensional nucleation and growth process can be
expressed by:
IHK
tð Þ ¼ zFcb
D
δ
− exp −B
ffiffi
t
p
1−
X∞
n¼0
Atð Þn
n! 2n þ 1ð Þ
exp −Atð Þ
!( ) #
ð14Þ
All symbols are mentioned in Hyde et al. [8]. This relation also shows
the current density under hydrodynamics condition. Eq. (12) in
Ribeiro's theoretical model corresponds to Eq. (14) in Hyde model,
that both of this models are explicit functions of a constant diffusion
layer thickness. The stationary current density developed by Hyde and
Klymenko's model for vc =0 and csc=0 in Eq. (14) is:
IHK
sta ¼ zFcb
D
δ
ð15Þ
In new modeling the time-dependent current density is obtained
from Eq. (9)at every time step and we continue the process until the dif-
fusion layer thickness comes to a constant number. Then Eq. (9) shows
value of stationary current density. Fig. 5 from Hyde and Compton's
paper [12] has similar behavior to Fig. 4.
3.4. Relation between parameters
In this part the relations between parameters consist of effective dif-
fusion layer, applied voltage, hydrodynamic velocity and stationary cur-
rent density are expressed.
Fig. 5 shows the effective diffusion layer δeff in terms of hydrody-
namic velocity and applied voltage. According to Fig. 5 the value of ef-
fective diffusion layer δeffis decreased by increasing both the
hydrodynamic velocity and the voltage applied on cathode surface.
Hence these two agents are acted against the effective layer
thickness δeff. On the other hand, by considering the variation of diffu-
sion layer thickness δeff in terms of velocity in a constant voltage, it is re-
alized that by selecting a higher voltage, the reduction of effective
diffusion layer becomes lower.
Fig. 6 shows the relation between the stationary current density and
diffusion layer thickness for different applied voltages. From Eq. (8) it is
known that by reducing the applied voltage, the reaction rate k will be
very small and increase the required time to reach the stationary cur-
rent density or limiting concentration cs on cathode.
In Fig. 7 the stationary current density is obtained by considering the
migration term for two different voltages Vapp = −1.2 V and
Vapp = −0.9 V. This figure also shows the stationary current density re-
sulted without the effects of migration term for V= −1.2 V. The varia-
tion of stationary current density by considering migration term is
linear with velocity and the slopes of these lines are equal for different
applied voltages. Hence higher stationary current densities are resulted
by increasing the applied voltage at a constant velocity.
Besides, the slope of the stationary current density obtained from
the first modeling (without migration) by considering the applied volt-
age of Vapp = −1.2 V is a little lower than the stationary current density
reached from the new modeling. In fact, at this state by increasing the
Fig. 5. Effective diffusion layer δeffin terms of hydrodynamic velocity and applied voltage.
122 M. Zahraei et al. / Journal of Electroanalytical Chemistry 782 (2016) 117–124

7. hydrodynamic velocity the value of the stationary current density is in-
creased more than the new modeling. However the value of the station-
ary current density in the first modeling is always lower than the
stationary current density from new modeling by considering migration
term.
4. Conclusion
In this modeling by using governing equations on electrolyte con-
centration between cathode and anode which consist the Nernst-
Plank equation of concentration and Poisson's equation of electric
Fig. 6. Effective diffusion layer δeff in terms of stationary current density and applied voltage.
Fig. 7. Variations of stationary current density in terms of the applied voltage and the hydrodynamics velocity.
123M. Zahraei et al. / Journal of Electroanalytical Chemistry 782 (2016) 117–124

8. potential; a parametric study is performed to examine the effect of pa-
rameters such as diffusion, forced convection, reaction and migration
terms on electro-deposition process. Because in the previous model
(Ribeiro model) the effect of migration term is not considered, central-
ization of this modeling is around this term and the effect of the applied
voltage on the current density and diffusion layer thickness. The main
results of this analysis are summarized below:
• The effective diffusion layer δeff which is calculated by using migration
term is very lower than modeling without the effect of migration. The
value of δeff in first modeling for velocity higher thanvc = −0.005
cm s−1
is also lower than δeff resulted in new modeling without con-
vection term. In fact the effect of migration term on ions transport is
very higher than convection term. Coupling between the concentra-
tion equation and the poison electric potential equation is the main
agent which result these variations.
• The effect of hydrodynamic velocity on effective diffusion layer δeff for
velocities lower than vc = −0.0005 cm s−1
can be ignored in electro-
deposition process and for higher velocities than vc = −0.0005 -
cm s−1
the effect of hydrodynamic velocity on δeff will be lower than
Ribeiro model.
• The current density influenced by migration term is lower than
Ribeiro model at the beginning of the process. By spending little
time (lower than2 seconds) the value of the current density from
new model and Ribeiro model are equal at intersection point. Further-
more by continuing the process the current density will be higher
than Ribeiro model and remains constant until the end of the process.
Therefore, the stationary current densityIsta becomes higher and re-
sulted in higher transport of ions.
• Both the hydrodynamic velocity and the voltage applied on cathode
surface are acted against the effective layer thicknessδeff. By consider-
ing the variation of effective diffusion layer δeff in terms of velocity in a
constant applied voltage, as the value of selected voltage is higher, the
reduction of effective diffusion layer will be lower.
• By considering migration term, the variation of stationary current
density is linear with velocity and the slopes of these lines are equal
for different applied voltages. Hence by increasing the applied voltage
in a constant velocity, higher stationary current density can be
reached.
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