1150
CHARLES H. ORRAND HENRY E. WIRTH
Vol. 63
CATHODE CURRENT DISTRIBUTION IN SOLUTIONS OF SILVER SALTS. 11. CATHODE CURRENT DISTRIBUTION OVER A PLANE CATHODE, PARALLEL TO AND SOME DISTANCE FROM A PLANE ANODE BYCHARLESH. ORR' AND HENRY E. WIRTH Contribution from the Department of Chemistry, Syracuse University, XyTacuse, N . Y . Received December 16,1968
Wagner has developed a theory based on fundamental ideas in electrostatics and electrochemistry which permits predictions of current distribution at cathodes. For simple configurations such as two plane parallel electrodes, his method presents a relatively easy way of predicting changes in current distribution as a function of changes in the conductance of the solution and the electrolytic polarization a t the metal cathode. Using this theory, the current distribution on a plane silver cathode parallel to a plane anode was predicted for solutions of silver nitrate and of potassium argentocyanide and then was determined experimentally. The results indicate that qualitatively the theoretical method is correct. Both indicate more uniform current distribution on increasing conductance, where polarization is not decreased markedly, by increasing either temperature or concentration or both. Both also indicate that the absence of significant polarization in the case of silver nitrate leads to very non-uniform current distribution. The electrolytic polarization of a silver cathode in solutions of potassium argentocyanide a t about 0.1 and 0.5 N at 25 aqd 4 5 O , and of a solution of silver nitrate at about 0.15 N a t 25" has been determined with a precision of *2%. A new application of an older method for making the measurements has been described.
Introduction The problem of the distribution of current in solutions of electrolytes has been of interest since a t least 1825.2 I n a number of cases attempts have been made to calculate the current distribution. Generally the procedure has been to make assumptions about the variables which have the most pronounced effect on current distribution, as evidenced by the distribution and quality of a metal plate. A scheme is then devised to represent the behavior of a given plating bath as a function of these variables. Predictions for given conditions, arrangement and shape of the electrodes are then made from the scheme, or else measurements are made and conclusions drawn about the effect of the variables for the prescribed configurations. Sometimes, general predictions are made concerning the effects of changes in a given variable on current or metal distribution. However, these methods are very restrictive; they lack the general aspects of a formulation of distribution based upon the fundamentals of electrostatics and electrochemistry and the basic properties of electrolytic solutions. Wagner3 has developed a theory based on fundamental ideas in electrostatics and electrochemistry which permits predictions of current distribution at cathodes. For simple configurations such as two plane parallel electrodes, this method presents a relatively easy way of predicting changes in current distribution as a function of changes in the conductance of the solution and the electrolytic polarization at the metal cathode. The purpose of this research was t o apply the theory of Wagner to a simple case. The current distribution on a plane silver cathode parallel t o a plane anode was predicted for solutions of silver nitrate and potassium argentocyanide. Theoretical A. Cathode Polarization.-One of the important parameters in determining current distribution, by
Wagner's methodS and those of others, is the electrolytic polarization of a metal cathode in a given solution as a function of current density. The polarization includes all effects opposing the flow of current through a solution and into an electrode except the IR-drop in the solution. The reeistance of the solution is assumed to remain constant. A variety of methods4 have been used in the past t o measure the polarization of a silver cathode in solutions of potassium or sodium argentocyanide. None were applied under the conditions used in this investigation and therefore the data available in the literature did not permit the necessary calculations of the current distribution to be made. Assuming that the resistance in a soluti.on between two points varies directly as the distance, then in Fig. 1 the ratio, r, of the distances between the cathode and nearest probe, and between the two probes should be the ratio of the resistances. If the polarization at the probe electrodes is negligible, then the potential El measured between the two probes when current is flowing is equal to the IR-drop in the solution between the probes. If this potential is multiplied by the ratio r and the product subtracted from the potential Ez,between the near probe and the cathode, the resulting potential E, should be the polarization of the electrode. The polarization as measured in this way is fully consistent with the assumptions made above concerning the effects lumped under the heading of polarization and with the conditions as they would exist in a plating cell. B. Current Distribution.-Wagner3 worked out the current distribution over parallel electrodes a t a distance great in comparison to their breadth for a configuration such as in Fig. 2. I n doing so he assumed a plot of the total cathode potential, E, E,, versus the current density I of the form in Fig. 3 and finally assumed that there was a linear relationship between E, and I over the range of current densities involved.
(1) The Procter & Gamble Company, Alianii Valley Laboratories. Cincinnati 31, Ohio. (2) A. de la Rive, Ann. chim. phys., 28, 110 (1825). (3) C. Wagner, J . Electrochem. Soc., 98, 116 (1951).
(4) 8 . Glasstone, J . Chem. Soc., 690 (1929); F. Foerster, 2. Elekfrochem., 18, 561 (1907); B. Egeberg and N. Promisel, Trans. Eleclrochem. SOC.,59, 293 (1931); H. E. Haring, Trane. A m . Electrochem. SOC.,49, 417 (1926); E. B. Ssnigar, Rec. trau. chim., 44, 649 (1925).
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July, 1959
CATHODE CURRENT DISTRIBUTION OVER Two PLANE PARALLEL ELECTRODES 1151 VENT
Defining h = IdE,/dIl, the parameter k = hL is introduced, where L is the specific conductance. The dimensionless variables u = x / A and v = y / A were introduced for the sake of simplifying later computations. Solution of La Place's equation for two dimensions and application of the appropriate boundary conditions leads to an equation of the form
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