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Historical Articles
April, 1954 issue of Plating
Current Density Distribution in Electroplating by Use of Models
Gilbert Ford Kinney,
Professor of Chemical Engineering U. S. Naval Postgraduate School, Monterey,
Calif., and John V. Festa, Electronics Division, Sylvania Electric Products,
Inc., Mountain View, Calif.
ABSTRACT
An electroplated metal is deposited as a result of the electric field set
up in the plating bath. It is difficult to describe analytically the
field about
an electrode of irregular shape but control of the field is essential if
uniform electroplate is to be obtained. This paper describes a method
of utilizing
models in the study of these electric fields. Measurements of relative current
density
are thereby readily made for a variety of practical electroplating situations.
These measurements give the primary current distribution which would be observed
if polarization and anode and cathode surface phenomena are not taken into
account.
INTRODUCTION
The practical electroplater must control not only the appearance, but also
the distribution, of the metal deposited on an object in order to produce
acceptable work. At constant efficiency the amount of metal deposited per
unit time is
directly
proportional to the current density at each point; therefore, a nearly
uniform distribution of the current density is required to obtain a
deposit of uniform
thickness. Methods by which the current distribution can be controlled
include (a) proper positioning of the object to be plated in relation
to the anodes,
(b) proper location of contact points, particularly for objects with appreciable
resistance, (c) use of an auxiliary anode to build up thickness at a thin
spot, (d) use of an auxiliary cathode or thief to reduce current density
in some
particular region, (e) use of nonconducting shields to throw the deposit
to another area,
and (f) manipulative techniques which enhance the throwing power of a solution.
The problems are very complicated and have no exact solutions. Intuition
and experience are helpful in finding the best compromise among a series
of conflicting
demands, and the design of even the simplest type of plating rack becomes
something of an art. Part of this art lies in proper integration of all
of the many factors
which influence the distribution of the electrodeposited metal.
The distribution
of the electric current about an object, and hence the resulting thicknesses
of electroplate on its surface, depends on the electric
field
set up in the conducting plating solution. The characteristics of this
electric field are important for, in principle, if the electric potential
is known
as
a function
of position, throughout the solution, the electroplating problem is completely
solved. For certain simple geometric shapes electricfield configurations
have actually been computed using the Poisson and Laplace equations.
These mathematical
solutions parallel those for the same types of fields found in problems
in hydrodynamics, aerodynamics, or heat transfer. The immediate application
of these methods to
electroplating was made by C. Kasper in a series of papers^{1} in
which he provided algebraic solutions for the currentdensity distribution
to
be anticipated
with various line and plane electrode assemblies. The advanced nature
of the mathematics employed for even the simplest electrode assemblies
makes
it evident
that such methods are less than convenient when extended to complicated
shapes. Yet it is the irregular shape that is of practical concern to
the electroplater;
an example is the rod stock for an experimental vacuum tube, sketched
in Fig. 1. The purpose of this paper is to indicate how the primary currentdensity
distribution, and hence the relative thickness of plated metal, about
such
a rod can be predicted
from simple measurements made on a model.


Fig. 1—Cross section of plated rod. Thickness of plating lest at a, greater
at points such as b and c. 
Fig. 2—Electric current
field in Hull Cell. Electrodes ab and cd are connected by (solid) current
lines; equipotential lines are dotted. 
GENERAL THEORY
The electric field in an electroplating tank may be represented by
equipotential planes and by current flow planes. The two systemsof planes
are complementary
and combine to give a complete orthogonal field. Consider those elements
of a threedimensional field that lie in a single plane, as, for
example, in a tank
with vertical electrodes, or as in a Hull cell. The Hull cell is
a miniature testplating tank with inclined electrodes shown as ab and
cd in Fig.
2, and is used by electroplaters to check performance of plating
solutions over a
wide range of current densities. The twodimensional field plot shown
in
Fig. 2 consists
of two complementary sets of lines which intersect everywhere at
right angles. One set of lines, dotted in the figure, represent lines of
equal potential
at values intermediate between those of the two electrodes ab and
cd.
The equipotential lines of the figure
were obtained on a model (2X size) of the actual cell, in which conducting
paper is used to simulate
the
conducting solution.
For electrodes, lowresistance strips of silver paint, of the type
supplied for printed circuits, were appliedto the paper and dried
with an infrared
heating
lamp. Electrodes of resistivity less than one ohm per square are
easily prepared, and this resistance is negligible compared with that of
the paper. Conducting
paper for representing solution is available in several types.
One grade of
facsimile paper (Type NDA, TimeFax facsimile paper supplied by
Times Facsimile Corp., 540 W. 58th Street, New York, N. Y.) supplied in 12
x 18 inch sheets shows quite uniform
resistivity of about
7500 ohms per square. This facsimile paper carries an aluminum
surfacing which may be disturbed by the vehicle of the conducting paint,
but
this can be avoided
if care is used in preparation of the model.
In determining the
field, a potential difference is applied across the electrodes of the model
by use of a voltage stabilizing transformer,
a variable autotransformer,
and a dry disc battery charger. Fig. 3 indicates the circuit
schematically. Points on the intermediate potential lines were located by
using
a probe
connected through
a galvanometer to a decade voltage divider. (The unit used by
the authors was manufactured by the General Radio Co., Cambridge, Mass.)
A null reading
on the galvanometer indicates a selected fraction of the total potential
difference
of one
or two volts, and a galvanometer of sensitivity of about 108
amps/mm allows precise
location of equipotential lines. (This technique parallels that
for the Analogue
Field Plotter of the General Electric Company.)
Current flow in
the Hull cell is represented in Fig 2 by solid lines which connect electrodes
ab and cd. These lines form socalled
curvilinear
squares
with the
dotted equipotential lines, the two sets of lines intersecting
each other always at right angles. This observation offers
one means of
locating
the currentflow
lines graphically: after the equipotential lines have been
plotted, the current flow lines are drawn in by inspection to meet the
rightangle curvilinearsquare requirements.^{2} More elaborate
fieldplotting
devices can be used in which
the rightangle direction is located experimentally by means
of two additional
contacts on the null probe. After a point on an equipotential
line is located, the probe
is rotated until maximum voltage difference is observed between
the two auxiliary
contacts. The line joining them then gives the rightangle
direction.
An alternative method of
locating the network of current flow lines is one which ”inverts” the
model. An inverted model is one in which the conducting electrodes
of the original model are replaced by nonconducting areas,
and, conversely, the nonconducting
areas are replaced by conducting areas. The current lines
of the original model become the equipotential lines of the inverted
model
and are easily located
by the nullprobe technique. The current lines drawn in the
figure of the Hull cell
were obtained by this method. The inverted model uses electrodes
along lines ac and bd, and the lines ab and cd become edges
of the conducting paper. Fig.
2 is thus a composite.
Current Density Measurement
The above considerations concerning orthogonal fields are
well known. It remains to extend them to cover the item
of primary
interest to
the electroplater, the current distribution along
the electrodes. The current
density at
each point is reflected in the spacing of the current flow
lines; closer
spacing, as near d of electrode cd of the Hull cell, corresponds
to greater density
of
current, i. e., potential gradient in the inverted model
corresponds to current density in the original model. This
potential gradient
can be obtained
from
measurements on the inverted model by plotting relative
potential of the equipotential lines
versus their position. The slope of this plotted line at
any point is the potential gradient at that point and also
gives
the relative
current
density
for the
original.


Fig. 3—Locating equipotential
lines. 
Fig. 4—Current
distribution at cathode of Hull Cell. Circled points obtained from inverted
model; solid line shows accepted values. 
A convenient method of measuring
potential gradient directly is by means of a twocontact probe. The voltage
difference
across two contacts
will
increase with
increasing gradient and give a value which closely
approaches that
of the actual
gradient at the midpoint between the two reasonably
close contacts. A probe with two contacts about 5 millimeters
apart gave voltage
differences that
were readily
measurable when a voltage drop of 10 volts or less
was placed across the electrodes of an inverted model some
8 to 10 inches
across.
Measurement by the Poggendorf
compensation method, using a ”portable” potentiometer
and external galvanometer of about 108 amp/mm sensitivity,
eliminates the effect of variable
contactresistance. The gradients measured in this,
manner along edge cd of the conducting paper of the
inverted model,
multiplied
by a constant of proportionality,
are shown as circled points in Fig. 4. The solid line
is a plot of the accepted empirical relationship for
the Hull
cell, where relative
current density =
27.7
– 48.7 log L, and L is the distance in inches along
the inclined electrode from point d. This equation
does not
necessarily hold for
all plating solutions
and
fails at the ends of the Hullcell panel. However,
it can be seen that the measurements on the inverted
model reproduce
almost exactly
the accepted experimental
values.


Fig. 5—Electric
current field about cylindrical electrodes a and b. Solid lines show
current flow, dotted lines show equipotentials. Note that equipotential
line cdec also could be an electrode without changing external. 
Fig. 6—Inverted
model of electrode a and area cdec. 
Measurements on a model representing
cylindrical electrodes serve as a further check on the inverted
model method,
the uniformity
in various
directions
of the conducting paper, and the suitability of painted
electrodes. The electric
field
about such cylindrical electrodes is shown graphically
in Fig. 5 and also can
be described analytically.^{1,2} Details of preparation
of the model are as follows:

A direct twodimensional model is
made, with conducting paint used to represent the electrodes.
Areas a and b
of Fig. 5 represent
cylindrical
electrodes.

An equipotential line enclosing
the electrode of interest is located by using the fieldplotting
and galvanometernull
technique.
Line cdec
is such
an equipotential
line about electrode a.

A line of current flow
is located by its rightangle relationship to intermediate potential
lines drawn
in for this purpose.
One such is line
acb. A straight
line connecting points of closest approach
of the two electrodes is also such a line,
and might have been used. The point of closest
approach, however, is a point of maximum current
density at
which measurements may particularly
be desired,
and it is preferred not to

An inverted model
is then prepared in which the electrode of interest (electrode a in this
case)
becomes a nonconducting
area
in the conducting
paper. It is
cut out and removed. The electric field to
be studied, including the area representing
electrode
a, is then
cut out by following
the contour
of equipotential
line
cdec.
This has become a flow line in the inverted
model, and no current flows across it. Severing
the
conducting paper
along
this line
has no effect
on current
distribution. It does, however, give an inverted
model of manageable dimension. The two sides
of current line ac are then painted in with
conducting paint, and the paint is allowed
to dry. Cutting
the paper along
this line
provides
two electrodes
for
the inverted model, as shown in Fig. 6.

The
potential gradient along the edge of the cutout area representing the original
electrode
is
measured,
using the
twocontact probe.
These readings are proportional to the
relative current density to be expected
at each point.
Values of the potential gradient
around the two electrodes of Fig. 7 as measured
in this
manner
(multiplied
by a proportionality factor)
are
plotted
as circled
points in that figure. The theoretical
values computed using accepted field
equations are
plotted as solid
lines. The
agreement between
the theoretical
and the experimental
values indicates that the method is a
satisfactory one. For the
electrodes shown in Fig. 7 the theoretical
ratio of maximum to minimum current
densities becomes
2/1 and 10/1, respectively. The ratio
of the two average current densities is 4/1
that of
the maximums
1.789/1.


Fig. 7—Relative
current density about cylindrical electrodes of diameters d and 4d
separated by distance 2.89d. Circles points were measured on inverted
model; solid lines show theoretical values. 
Fig. 8—Direct
model of rod in plating tank with four anodes. Experimentally found
equipotential lines are dotted. Solid current
line connects rod and one
anode. 
Figs. 8 and 9 show the two models used
for measurements on the rod stock of
Fig. 1.
The measured values
are plotted in Fig.
10 as open
circles.
Then, for comparison purposes, a test
length of this rod was copper plated in
a ”highspeed” copper
bath (average current density,
40 amps/sq ft; time, 22 minutes;
average
thickness,
0.0016 inch) and
flashed with nickel for protection.
The rod was sectioned,
polished, and etched by means
of metallurgical techniques, and the
plating thickness
was measured microscopically
with a filar micrometer.
The results for various sections
were averaged and the
averages are plotted as solid
circles in Fig. 10. The
discrepancy between plating thickness
and the measured potential gradient
shows the effect
of the throwing power of the
highspeed copper bath.


Fig. 9—Inverted
model with cutout  area representing rod. 
Fig. 10—Measured
thickness of copper plate on rod shown as solid circles. Current density
as measured on model shown as open circles. Discrepancy shows leveling
action of solution throwing power. 
CONCLUSION
The method of the inverted model also
permits study of the effect
of solution throwing
power. Measurements
are
easily
made to find
the effects
of electrode
orientation or placement,
or the behavior of auxiliary anodes or cathodes. An auxiliary
anode becomes, in
the direct model,
a conducting
area
connected in
parallel with the other anodes;
an auxiliary cathode (thief) is connected in parallel
with
the object
to be plated.
The equipotential and current
lines are located as described
previously, but their positions are markedly affected
by
auxiliary electrodes. When
the inverted model is prepared, each electrode becomes
a cutout area.
A nonconducting
shield about a portion of an object to be
plated is represented
by
a cutout
area in the
original
model.
Location
of equipotential
lines
and a line of current flow
proceeds as before except that in the
inverted model
the
shield becomes a conducting
area not connected directly to an
electrode. This
conducting
area shows, of course,
a zero
potential
gradient
corresponding to
zero current density and
no metal deposited.
These aspects, important as they
are, are but one part
of the electroplating
problem.
Polarization
serves to
modify the primary
electric fields
described here, but this
approach to the problem permits the individual
effects
of polarization and of
the primary field to be resolved.
Furthermore, this
study is limited
to the effect of the
primary field on thickness
of deposited metal, and
is not pertinent to questions of adhesion,
appearance,
or homogeneity. A general
attack on all of these
problems is, however, greatly facilitated
by removing
the effect
of
the geometry of irregular
electrodes on primary  electric field from
the realm
of speculation.
LITERATURE
CITED
1. C. Kasper, “Theory of the Potential and Technical Practice of Electrodeposition,” Trans.
Electrochem. Soc.,
77, 353 (1940) 78, 131 (1940), 82, 153 (1942)
2. S. S. Attwood, “Electric
and Magnetic Fields,” John
Wiley and Sons, Inc.,
N. Y. (1941).
Grades of Lead
From Lead Handbook
for the Chemical Process Industries
(1954), Federated Metals Division,
American Smelting
and Refining
Co., New York
5, NY.
Chemical lead is
a grade of
commercial lead that
has found wide
acceptance within the
chemical and
process industries.
This grade
of lead contains
by specification
small amounts
of silver
and copper.
Acid
lead is wholly refined
lead alloyed
with minute
percentages
of other
elements, including
copper,
added specifically
to improve
resistance
to corrosive
attack.
An
attendant advantage
of these
elements
is diminution
of
the
tendency
of the
lead to ”creep.” The
latter
is defined as
continuous
change
of shape (length)
with time
when a
load is applied.
This
load
may be
the weight of
the
lead itself
and other
loads,
e.g., the solution
in a tank.
Antimonial
lead,
commonly called ”hard” lead,
is lead
alloyed
with
antimony (usually
6 per
cent) to increase
mechanical
properties
markedly
at temperatures
below
200° F.
At temperatures
below
200° F
antimonial
lead
has
better
abrasion
resistance
than
chemical
lead.
At
room
temperatures
it
has
twice
the
hardness
and
tensile
properties;
this
makes
it
useful
in
tank
construction,
particularly
where
only
a skeleton
frame
is
used
for
support
or
where
blows
from
harder
metals
are
likely
to
be
encountered.
Since
the
addition
of
the
antimony
lowers
lead’s
meltingpoint,
antimonial
lead
is
not
suitable
for
use
at
temperatures
exceeding
200° F.
Above
that
temperature
both
mechanical
strength
and
corrosion
resistance
fall
off
rapidly.
Tellurium
lead is
a chemical
lead to
which has
been added
a fraction
of a
per cent
of tellurium
for added
resistance to
fatigue due
to vibration.
Tellurium lead
has the
ability to
workharden, that
is, to
strengthen itself
under strain.

