by D-T. Chin, N.R.K. Vilambi and M.K.
Sunkara
This paper discusses the theoretical
background behind the selective pulse plating process. A
mathematical model is presented for the distribution of
the time averaged local current density on a large unmasked
cathode whose surface is facing a small rectangular anode
in the plating cell. Numerical computations are made for
the pulse plating of gold from a concentrated neutral phosphate
gold bath and copper from an acid copper sulfate bath. The
effect of peak pulse current, pulse duty cycle and anode-cathode
spacing on the current distribution is described. Satisfactory
agreement is obtained between the theoretical current distribution
and published metal distribution results of a copper plating
experiment.
In electroplating, the thickness
distribution of the electro deposit is directly related
to the local current density on the cathode. Prior knowledge
of current distribution on the electrode surfaces is useful
in the design of electroplating cells and in reducing energy
and metal consumption. Although pulsed current has long
been used in the plating industry to improve the properties
of electro deposits, the problem of current distribution
in pulse plating has not been well understood. Since the
instantaneous peak pulse current density during the current
pulses is usually higher than that in direct current (dc)
plating, it has been thought that the primary
current distribution would prevail, and the current distribution
in pulse plating would be less uniform than that in dc plating.
A numerical computation2 for the electro deposition
of copper on a rotating disk has revealed that the current
distribution in pulse plating is less uniform at higher
pulse currents or pulse potentials. This property suggests
that the selective plating (or spot plating) of metals on
a maskless cathode may be achieved by pulse plating using
high pulse current densities and small anode-cathode spacing.
Vilambi and Chin have conducted an experiment for the selective
pulse plating of copper from an acid copper sulfate bath
by directing a small insoluble circular anode against a
large unmasked cathode. They found that pulse plating produced
more localized (or less uniform) copper deposit than that
of dc plating.
The present paper discusses
the theoretical background behind the selective pulse plating
process. A mathematical model is presented for the secondary
current distribution on an infinitely large and unmasked
cathode with its plane parallel to that of a small rectangular
anode. Numerical computations are made for the pulse plating
of gold from a concentrated neutral phosphate gold bath
at 40°C, and copper from an acid copper sulfate bath at
25°C. Experimental polarization curves under the pulse plating
conditions are used for the required electrokinetc information.
The governing equations in the model are solved with the
finite element and orthogonal collocation technique.4,5
The effects of peak pulse current density, pulse duty cycle,
and anode-cathode spacing on the cathode and anode current
distributions are investigated. The theoretical current
distribution for the selective pulse plating of copper is
compared to the experimental metal distribution previously
obtained by Vilambi and Chin.3
In pulse plating, the applied
electric current and cell voltage may be considered to be
composed of a time-averaged dc component and a fluctuating
periodic component, as shown in Fig.1,
with a pulse current wave,
 |
(1) |
and a voltage wave,
 |
(2) |
where i is the current density;
E is the cell voltage; and t is the time. We use the notation
" __" to denote the time averaged dc components
and " ~" to designate the periodic components. The time
average dc current density and cell voltage are obtained
by averaging the instantaneous current and voltage waves
over one period of the time according to the following equations:
 |
(3)
(4)
|
where T is the pulse period
(on-time + off-time); ip is the peak pulse current
density; ton is the on-time for pulse plating
with a rectangular current wave, as shown in Fig.
1; and 
, which represents the fraction of on-time in one
pulse period, is called the duty cycle in pulse plating.
The periodic current density and voltage components, 
(t) and 
(t), are obtained by subtracting the time-averaged
current density, 
, and the time-averaged voltage, 
, from the total current density and voltage waves, i(t) and E(t),
respectively, as shown by Eqs. 1 and 2. The periodic components
alternate in sign; their time-averaged values are equal
to zero:
 |
(5)
(6)
|
Figure
2 shows the
schematic representation of a two dimensional electrochemical
cell used for the theoretical analysis. The cell consists
of an infinitely large cathode and a parallel rectangular
anode. The anode has a width of 2La; its exposed
surface is flush with a semi-infinite insulation surface,
as shown in the figure. The center of the cathode is taken
as the origin of a pair of rectangular coordinates, x and
y, with x being the surface distance on the cathode and
y being the vertical distance from the cathode surface.
The anode-cathode spacing is H. To simplify the analysis,
the following assumptions are incorporated into the model:
(1) the anode polarization is negligible; (2) the solution
is well stirred and concentration polarization is negligible;
(3) the pulse period is sufficiently large (or the pulse
frequency is sufficiently low) that the effect of the electric
double layer on the current distribution is negligible;
and (4) the solution properties, such as conductivity and
density, are constant. The anode surface is taken as the
reference plane where the solution potential is set to zero.
Accordingly, the governing equations
describing the current distribution in the cell may be given
as follows:
Voltage Balance
 |
(7) |
where E0 is the open-circuit
cell voltage; 
(x,t) is the instantaneous local activation overpotential at the
cathode; and 
is the instantaneous local ohmic potential drop
across the electrolyte between the anode and the cathode.
The instantaneous cell voltage, cathode activation overpotential
and ohmic potential drop may be split into their respective
time averaged dc components and periodic components:
 |
(8) |
The above equation may be time
averaged by integrating it with respect to the time over one
pulse period, T, and dividing the resulting equation by T.
Making use of the fact that the time averaged values of the
periodic components,
, and 
, are equal to zero, the integration may be simplified
to:
 |
(9) |
Equation 9 indicates that the
time averaged dc cell voltage and the time averaged components
of the local cathode activation overpotential and ohmic
potential drop may be separated from their periodic components
in the governing equation by carrying out the time averaging
operation. According to Faraday's law, the time averaged
electro deposition rate in pulse plating is related only
to the time averaged dc current density at the cathode.
Thus, in the following sections, only the governing equations
concerning the time averaged dc potential and current density
components will be presented.
Charge Balance
The conservation of charge demands
that the net accumulation of charge in the elctrochemical
system must be equal to zero. This implies that the total
current flowing through the cathode must be equal to the
total current flowing through the anode. Thus, the charge
balance equation states that:
 |
(10) |
where 
and 
are the time averaged local current density at
the anode and cathode surfaces, respectively. Using the
present convention, the anode current density has a positive
value and the cathode current density is taken to be negative.
Ohmic Potential
Drop in the Cell
The ohmic potential drop 
, in the cell is obtained by solving the two-dimensional Laplace
equation for the potential distribution in the electrolyte:
 |
(11) |
where 
is the time-averaged electric potential in the solution phase.
The associated boundary conditions are:
at the cathode
 |
(12a) |
at the anode
 |
(12b) |
at the anode insulation plane
 |
(12c) |
at the center line
 |
(12d) |
at the edge of the cell
 |
(12e) |
where k is the conductivity of
the electrolyte. The time averaged dc potential in the cell, 
, is obtained by solving Eqs. 11 and 12. The local ohmic potential
drop in the cell is given by:
 |
(13) |
Activation
Overpotential at the Cathode
For a given time averaged current
density, the time averaged activation overpotential during
pulse plating is different from that of dc plating6.
To simplify the analysis, we chose a semi-empirical approach,
i.e., the time averaged cathode activation overpotential
vs. time averaged dc current density relation will be supplied
from a set of empirical polarization data in the form:
 |
(14) |
where 
and 
denote the time averaged current density and activation overpotential
at the cathode; T is the pulse period; and is the duty cycle. For a given electrolyte system and pulse current
of fixed duty cycle and pulse period, the function f is
determined from the experimental polarization measurements.
Equations 9-14 must be simultaneously
solved to determine the time averaged potential profile 
in the plating cell. The time averaged local cathodic and anodic
current densities are then evaluated from the slope of the
potential profile according to:
 |
(15)
(16)
|
A computer program using Fortran
77 and a finite element and orthogonal collocation technique
has been developed for the numerical computations. The details
of the computing algorithm and the collocation procedures
are given in Refs. 4 and 5, and will not be further described
here.
The numerical computation for
the time averaged local current distribution has been performed
for the selective pulse plating of gold from a concentrated
neutral phosphate gold bath at 40°C, and copper from an
acid copper sulfate bath at 25°C. The bath compositions,
electrolyte conductivities, and cell configurations are
listed in Table 1. The
simulation was made for the following pulse plating conditions;
the peak pulse current density (based on a constant anode
area) of 0.05 to 2.0A/cm2; a pulse period of
10 msec; and a duty cycle of 100 percent (dc) to 5 percent.
The required electrokinetc information (Eq. 14) describing
the relationship between the time averaged activation overpotential
and the time averaged current density at the cathode was
experimentally determined with a rotating disk electrode
having a surface area of 0.071 cm2. In this experiment,
a constant peak pulse current of known pulse period and
duty cycle was applied to the rotating disk electrode, and
the instrumentation was constructed to measure the time
averaged potential of the rotating disk electrode with respect
to a saturated calomel reference electrode (SCE). The polarization
curve was obtained by gradually stepping up the peak pulse
current and by recording the time averaged electrode potential
as a function of the time averaged cathode current density
on the rotating disk electrode. The details of the experimental
technique are described in Refs 7 and 8. Figure 3 shows the time averaged polarization
curves for the pulse plating of gold from the concentrated
neutral phosphated gold bath at 40°C with a 10 msec pulse
period over a range of duty cycles from 100 percent (corresponding
to dc) to 5 percent. Figure 4 shows the time averaged polarization
curves for the pulse plating of copper from the acid copper
sulfate bath at 25°C under similar pulse conditions. The
curves were fitted with a polynomial function, and together
with the electrolyte conductivity and cell configuration
values, were fed into the computer to generate the numerical
results for the distribution of time averaged current density
on the cathode.
Results and Discussion
It should be noted that the
total active area of the cathode in the present system is
not defined; thus, the applied pulse current magnitudes,
such as the peak pulse current density and geometric average
current density, will all be based on the finite area of
the anode:
anode peak current density,
 |
(17) |
average anode current density,
 |
(18) |
For the cell geometry shown in
Fig. 2, the anode half width L a,
may be chosen as the characteristic length of the system.
The time averaged local current density can then be expressed
as a function of the dimension less surface distance and dimension
less anode-cathode spacing defined as:
dimensionless distance on electrode
surface,
dimensionless
anode-cathode spacing,
The numerical computations have
been performed to generate the profiles of a dimension less
time averaged solution potential,
, and a dimension less time averaged current density, i (nFL a/kRT
g) from Eqs. 9-16. The advantage of the dimension
less quantities is that they possess the geometric and kinematic
similarities, and the results can be generalized to different
plating systems. However, many plating practitioners may
not be familiar with the magnitudes of these dimension less
quantities; thus, for the convenience of the readers, we
shall present the current density and potential values in
their usual dimensional forms. Also, the absolute value
of the local cathode current density will be reported in
the subsequent sections.
Current
Distribution for Pulse Plating of Gold
Figure
5 shows a set
of calculated secondary current distribution for the pulse
plating of gold from the concentrated neutral phosphate
gold bath at an average anode current density of 0.05 A/cm2,
pulse period of 10 msec, and dimension less anode-cathode
spacing of 1.0. The time averaged local current density
on the cathode is plotted in the figure as a function of
the dimension less surface distance, X*, for three duty
cycles; 100 percent, 10 percent, and 5 percent. The curve
for the 100 percent duty cycle corresponds to dc plating.
The results indicate that the time averaged local current
density on the cathode is highest at the center directly
facing the anode (x* = 0). The local cathode current density
decreases with increasing surface distance from the center
and asymptotically approaches to zero at a large distance
from the center. The current density at the center is highest
with the 5 percent duty cycle, and decreases with increasing
duty cycles. This indicates that for a given average current
density at the anode, the current distribution in pulse
plating is more localized than that of dc plating. The extent
of localization in electrodeposition increases with decreasing
pulse duty cycles. This behavior may be explained by the
dimension less Wagner number defined as:
 |
(21) |
where k is the conductivity
of the electrolyte; 
is the time averaged current density at the cathode; 
is the time averaged activation potential at the
cathode; and L is the characteristic length of the present
plating system. The Wagner number represents the ratio of
the activation resistance to the ohmic resistance of the
electrolyte in the plating cell. The current distribution
is generally less uniform at small Wagner numbers. Comparing
the time averaged polarization curve for the dc plating
of gold (i.e., at 100 percent duty cycle) to those of pulse
plating of gold in Fig. 3, it can be seen that pulse current
reduces the slope
of the polarization curve,
, and this in turn reduces the Wagner number of
the plating system. The slope of the polarization curves
decreases with decreasing pulse duty cycles. This implies
that the current distribution is less uniform in pulse plating,
and more localized gold deposits may be obtained with pulsed
current at small duty cycles and large pulse current densities.
The effect of applied average
anode current density on the distribution of time averaged
local current density on the cathode is shown in Fig. 6. The curves in the figure were numerically
generated for pulse plating of gold by increasing the average
anode current density from 0.025 A/cm2 to 0.2
A/cm2, while keeping the duty cycle at 10 percent
and the pulse period at 10 msec. The anode-cathode spacing
was kept at one anode half-width of 0.3 cm (H/L = 1.0).
The results indicate that the current distribution at the
cathode becomes more localized with increasing average current
density at the anode. Since the peak pulse current density
increases with increasing time averaged current density
for a fixed duty cycle and pulse period, the results also
imply that more localized current distribution may be obtained
by increasing the peak pulse current density at the anode.
The effect of anode-cathode
spacing on the distribution of time averaged current density
at the cathode is shown in Fig. 7 for the pulse plating of gold at
0.05 A/cm2 average anode current density, a 10
msec pulse period, and 10 percent duty cycle. The dimension
less anode-cathode spacing varied from 0.1 to 10. The cathode
current distribution becomes more localized with decreasing
anode-cathode spacing. At H* = 0.1, where the time averaged
current density at the cathode drops sharply near the edge
of the anode (i.e., at X* = 1.0), and reaches a zero level
at a distance of 2.5 anode half-widths (X* = 2.5) from the
center of the deposit. The results indicate that nearly
perfect selective pulse plating of gold may be achieved
by using a high anodic peak pulse current density and a
small anode-cathode spacing of H* less than 0.1.
Figure
8 shows the
distribution of time average local current density at the
anode when the dimension less anode-cathode spacing was
kept at 1.0. It should be noted that the curves in the figure
are of the primary current distribution, for the anode activation
overpotential was neglected in the mathematical model. The
time averaged local current density is nearly uniform in
the middle section of the anode, and increases rapidly with
increasing proximity to the edge of the anode (i.e., at
X* = 1.0). Although the local current density on the anode
increases with increasing average anode current density,
the shape of the anodic current distribution curves remains
qualitatively the same as shown in Fig. 8.
Current
Distribution in Copper Plating and Comparison with Metal
Distribution Results
Figure
9 shows the
distribution of the time averaged cathode current density
for the pulse plating of copper from the acid copper sulfate
bath for a range of pulse duty cycles from 100 percent to
5 percent. The set of curves was calculated with an average
anode current density of 0.15 A/cm2, pulse period
of 10msec, and dimension less anode-cathode spacing of 1.0.
The curve with 100 percent duty cycle corresponds to dc
plating. The results are similar to those of gold plating;
in both cases, more localized current distribution is obtained
with pulse plating and the extent of localization increases
with decreasing duty cycles.
The copper plating offers an
opportunity to compare the present mathematical model with
the experimental metal distribution results previously obtained
by Vilambi and Chin,2 who carried out the selective
pulse plating of copper on a large maskless cathode from
the acid copper sulfate bath with a circular disk anode
of 0.2 cm in radius. The metal distribution was obtained
by measuring the thickness of copper deposits as a function
of the radial distance from the deposit center after the
electroplating. To facilitate the comparison, we shall normalize
the calculated time averaged local current density on the
cathode with the applied average anode current density,
<ia>:
 |
(22) |
where i*N represents
the normalized time averaged local current density on the
cathode. The local thickness of electrodeposits is proportional
to the time averaged local cathode current density if the
current efficiency for the electrodeposition reaction is constant.
For the acid copper sulfate bath, the cathode current efficiency
is around 100 percent over a large range of current densities
in both the dc and pulse platings. Thus, the time averaged
local cathode current density is proportional to the local
thickness of the electrodeposit, 
(X*), and the average anode current density is proportional
to a hypothetical maximum deposit thickness, Max, that would be obtained if the entire
mass of deposited copper is evenly distributed over a cathode
area equal to the anode area. Hence, the normalized time-averaged
local current density defined in Eq. 22 is equivalent to:
 |
(23) |
Since a circular anode was used
in Vilambi and Chin's experimental measurement, the dimension
less surface distance, X*, in Eq. 23 is equivalent to the
surface distance from the center of deposits normalized by
the anode radius. They hypothetical maximum deposit thickness, max, may be obtained by integrating the
experimental metal distribution curve according to:
 |
(24) |
Figure
10a-c shows the
comparison between the theoretical normalized current distribution
and experimental normalized metal distribution curves for
the plating of copper from the acid copper sulfate bath at
25°C. The theoretical results are given in the figure as solid
lines, and the experimental data are shown by open circles.
The comparison is made for dc plating (fig. 10a), and pulse
plating with a 10 msec pulse period at 50 percent (fig.10b) and 10 percent (fig.
10c) duty cycles. The average anode current density is
kept at 0.1 A/cm2, and the dimension less anode-cathode
spacing is 1.0. In general, the experimental data agree well
with the theoretical mode. The best agreement is near the
center of the cathode (X* <1.0). At large distance from
the cathode center, the experimental metal distribution becomes
more localized than that of the theoretical current distribution.
The discrepancy can be attributed to the following reasons:
1.
Difference in the anode geometries - The
mathematical model is based on a rectangular planar anode,
whereas a circular disk anode flush-mounted on an inert
support was used in the metal distribution experiments.
The circular geometry tends to produce less uniform current/metal
distribution because of increasing active cathode area with
increasing radial distance from the deposit center. In the
rectangular geometry shown in Fig. 2, the cathode area per unit length of
x-coordinate is constant within the cell; whereas in the
circular geometry, the cathode area per unit radial distance
increases proportionally to the square of the radial position.
The spreading of the active cathode area with increasing
radial distance reduces the local current density on the
circular geometry faster than that on the rectangular geometry
at a large distance from the deposit center. This notion
is clearly demonstrated in Fig. 10(a-c), in which the experimental metal
distribution curves decrease faster than the theoretical
curves with increasing X* at X* greater than 1.0.
2.
Effect of Anode Polarization
- The
present model neglects the activation overpotential at the
anode. Thus, the calculated current distribution at the
anode is of the primary current distribution, which gives
an infinitely large current density at the edge of the anode
as shown in Fig. 8. In
reality, the current density at the edge is finite, and
the current distribution as the anode should be more uniform
than that of the primary current distribution. The infinitely
high theoretical current density at the edge of the anode
makes the local current density at the cathode decrease
more slowly with increasing distance from the cathode center.
3.
Experimental Error in the Metal Distribution Measurements
-
The metal thickness distribution
as reported by Vilambi and Chin3 was obtained
with a coulombic thickness gage. Although the gage was carefully
calibrated with a standard thickness, the uncertainty in
the thickness readings was approximately 
10 to 20 percent.
The present mathematical model
also neglects the concentration polarization for the sake
of simplicity in the numerical computations. The mass transfer
effect may be estimated from a dimension less pulse period
defined as.6
 |
(25) |
where D is diffusivity of the
metal ion, T is the pulse period, and 
is the thickness of the diffusion layer on the
cathode. The value of T* for the present pulse plating of
copper (T=10 msec) is estimated to be 0.076, using a value
of 7.6 x 10-6 cm2/sec for the diffusivity
of cupric ions and 0.001 cm for the thickness of the diffusion
layer.9 The dc-limiting current density corresponding
to =0.001 cm for the standard acid copper sulfate
bath containing 0.75M CuSO4 is estimated to be:
 |
(26) |
From the theoretical mass transfer
results shown in reference 6, the limiting peak pulse current
density at 10 percent duty cycle and 0.075 dimension less
pulse period, is approximately 5.5 times greater than that
of the dc-limiting current density (or 6 A/cm2).
In the present study, the applied instantaneous peak current
density for copper plating varies from 0.1 A/cm2 in
dc plating to 1.0 A/cm2 in pulse plating at 10
percent duty cycle. The concentration overpotential, 
, may be calculated from the Nernst equation:
 |
(27) |
where Tg is the absolute
temperature; R is the gas constant; F is Faraday's constant;
and i is the instantaneous local current density. The maximum
concentration overpotential for the results showing in Fig.10 may be estimated to be 1.2 mV in dc
plating and 2.3 mV in pulse plating with 10 percent duty
cycles at the time-averaged current density of 0.1 A/cm2
. These are extremely small values as compared to
the activation overpotential and ohmic potential drop at
the cathode. Thus, for practical purposes, the concentration
overpotential is negligible in the present calculations.
The secondary current distribution is adequate to give an
accurate account of the metal thickness distribution on
the cathode surface if the current efficiency is constant
independent of the local current densities.
Conclusions
A theoretical analysis has been
made of the secondary current distribution on a large, unmasked
cathode in the selective pulse plating using a finite-size
anode. Numerical computations were performed for the pulse
plating of gold from a concentrated neutral phosphate gold
bath and copper from an acid copper sulfate bath. The results
indicate that for a given average current density on the
anode, the distribution of the time-averaged current density
on the cathode is much more localized than that of dc plating.
The extent of localization in the cathode current distribution
increased with: (1) decreasing pulse duty cycles, (2) increasing
average anode current density and (3) decreasing anode-cathode
spacing. The numerical results for the selective pulse plating
of copper were compared to the experimental data previously
reported in the literature. Satisfaction agreement was obtained
between the theoretical current distribution and the experimental
metal distribution on the cathode.
Acknowledgment
The authors greatfully acknowledge
the financial support of the American Electroplaters and
Surface Finishers Society, Inc. for the work reported in
this paper.
References
- N. lbl, Proc. AES 2nd
Intern. Pulse Plating Symp. (1981).
- H.H. Wan and H.Y. Cheh, J.
Electrochem. Soc. 135, 643 and 658 (1988).
- N.R.K. Vilambi and D-T Chin,
Plating and Suf. Fin., 75, 67 (Jan. 1988).
- R. Caban and T. Chapman,
J. Electrochem Soc., 123, 1036 (1976).
- N.R.K. Vilambi, PhD dissertation,
Clarkson University, Potsdam, NY (1987).
- D-T. Chin, J. Electrochem,
Soc., 130 1657 (1983).
- D-T Chin and N.R.K. Vilambi,
Proc. AESF 73rd Annual Tech. Conf.,
Session G-5(1986).
- C.Y. Cheng and D-T. Chin,
AlChE j., 30, 765 (1984).
- D-T. Chin, Proc. Symp.
On Transport Process in Electrochemical Systems, (R.S.
Yeo, T. Katan and D-T. Chin, eds), 82 (10), 21 (1982).
About the Authors
Dr. Der-Tau Chin is a professor
in the Department of Chemical Engineering, Clarkson University,
Potsdam, NY 13676. He has more than 20 years of research
experience in electroplating, corrosion, electrochemical
energy conversion, and industrial electrolytic processes.
Prior to joining Clarkson in 1975, he was a senior research
engineer at General Motors Research Laboratories. Dr. Chin
received his PhD in chemical engineering from Clarkson University
in 1988.
Dr. N.R.K. Vilambi is a principal
scientist at Physical Sciences, Inc. Andover, MA He received
his PhD in chemical engineering from Clarkson University
in 1988.
M.K. Sunkara worked as a research
assistant for AESF Research Project #68. He received his
MS in chemical engineering from Clarkson in 1989 and holds
a BS from Andhra University, Waltair, India.
