

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.

