Current Distribution in Electrochemical Cells: Analytical and Numerical Modeling

Part of the Modern Aspects of Electrochemistry book series (MAOE, volume 44)


The topic of current distribution modeling is central to the analysis of electrochemical systems and has been addressed in textbooks 1, reviews (e.g., Refs. [2–4]) and numerous journal publications. Newman’s textbook 1 provides a meticulous and comprehensive treatment of the subject. Prentice and Tobias 2 present a review of the early (up to about 1980) publications in the area. Dukovic’s more recent review 3 is very comprehensive, providing critical analysis of both the electrochemical and the numerical aspects of the topic. A recent review by Schlesinger 4 focuses primarily on the numerical techniques. The present monograph introduces the fundamental processes and equations underlying the modeling of the current distribution, and critically analyzes common assumptions and approximations. Focus is placed on discussing scaling parameters for the characterization of the current distribution. Commonly used algorithms for numerical determination of the current distribution are compared and a few numerical implementations are discussed. Lastly, the modeling of the current distribution in some special configurations and applications is introduced, emphasizing recent publications.


Current Distribution Laplace Equation Electrochemical System Exchange Current Density Volmer Equation 


  1. 1.
    J. Newman, Electrochemical Systems, Prentice-Hall, Englewood Cliffs, NJ, 1973 (Second Edition, 1991).Google Scholar
  2. 2.
    G. A. Prentice and C. W. Tobias, “A Survey of Numerical Methods and Solutions for Current Distribution Problems,” J. Electrochem. Soc., 129(1), 72–78, 1982.CrossRefGoogle Scholar
  3. 3.
    J. O. Dukovic, “Computation of Current Distribution in Electrodeposition, a Review,” IBM J. Res. Dev., 34(5), 693–705, 1990.CrossRefGoogle Scholar
  4. 4.
    M. Schlesinger, “Mathematical Modeling in Electrochemistry,” in Modern Aspects of Electrochemistry, Vol. 43, M. Schlesinger, Ed., Springer, New York, 2008.Google Scholar
  5. 5.
    R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 1960.Google Scholar
  6. 6.
    A. J. Bard and L. Faulkner, Electrochemical Methods, Wiley, New York, 1980.Google Scholar
  7. 7.
    E. Gileadi, Electrode Kinetics, Wiley-VCH, New York, 1993.Google Scholar
  8. 8.
    V. G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, NJ, 1962.Google Scholar
  9. 9.
    J. C. Puippe and F. H. Leaman, Theory and practice of pulse plating, American Electroplaters and Surface Finishers Society, Orlando, FL, 1986.Google Scholar
  10. 10.
    K. I. Popov and M. D. Maksimovic, in Modern Aspects of Electrochemistry, Vol. 19, B. E. Conway, J. O. M. Bockris and R. E. White, Eds., Plenum Press, New York, p. 193, 1989.Google Scholar
  11. 11.
    B. K. Purushothaman, P. W. Morrison and U. Landau, “Reducing Mass Transport Limitations by the Application of Special Pulsed Current Modes,” J. Electrochem. Soc., 152(4), J33–J39, 2005.CrossRefGoogle Scholar
  12. 12.
    B. K. Purushothaman and U. Landau, “Rapid Charging of Lithium Ion Batteries Using Pulsed Currents – A Theoretical Analysis,” J. Electrochem. Soc., 153(3), A533–A542, 2006.CrossRefGoogle Scholar
  13. 13.
    J. R. Selman and C. W. Tobias, “Limiting Current Mass Transfer Measurements,” in Advances in Chemical Engineering, Vol. 10, Drew et al. Ed., Academic Press, New York, 1978.Google Scholar
  14. 14.
    H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1979.Google Scholar
  15. 15.
    U. Landau, AIChE Symposium Series 204, Vol. 77, 75–87, 1981.Google Scholar
  16. 16.
    H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, 1959.Google Scholar
  17. 17.
    J. Crank, The Mathematics of Diffusion, Oxford University Press, London, 1956.Google Scholar
  18. 18.
    P. Moon and D. Spencer, Field Theory for Engineers, Van Norstrand, Princeton, NJ, 1961.Google Scholar
  19. 19.
    L. M. Milne-Thomson, Theoretical Hydrodynamics, The Macmillan Company, New York, 1960.Google Scholar
  20. 20.
    J. Newman, J. Electrochem. Soc., 113, 1235–1241, 1966.Google Scholar
  21. 21.
    C. Wagner, J. Electrochem. Soc., 98(3), 116, 1951.CrossRefGoogle Scholar
  22. 22.
    O. Lanzi III and U. Landau, “Analysis of Mass Transport and Ohmic Limitations in Through-Hole Plating,” J. Electrochem. Soc., 125(8), 1922–1930, 1988.CrossRefGoogle Scholar
  23. 23.
    A. S. Agarwal, U. Landau and J. H. Payer, “Current Distribution on a Cathode Covered by a Thin Film Electrolyte in Crevice Corrosion – An Analytical Model,” ECS Trans., 11, Feb., 2008.Google Scholar
  24. 24.
    W. J. Cook and U. Landau, in Engineering of Industrial Electrolytic Processes, U. Landau, R.E. White and R.J. Varjian, Eds., The Electrochemical Society Softbound Proceeding Series, Pennington, NJ, 1986Google Scholar
  25. 25.
    C. W. McFarland, “The Finite Difference Method for Modeling Potential and Current Distributions in Electroplating Cells,” in Electroplating Engineering and Waste Recycle – New Developments, Dexter Snyder, Uziel Landau, and R. Sard, 1982 Symposium Proceedings, The Electrochemical Society, Inc., Pennington, NJ, 1983.Google Scholar
  26. 26.
    M. Matlosz, C. Creton, C. Clerc, and D. Landolt, J. Electrochem. Soc., 134(12), 3015, 1987.CrossRefGoogle Scholar
  27. 27.
    W. M. Lynes and U. Landau, “A Novel Adaptation of the Finite Difference Method for Accurate Description of Non-Orthogonal Boundaries,” Extended Abstract No. 332, Vol. 88–2, The Electrochemical Society, Fall Meeting, Chicago, IL, Oct. 1988.Google Scholar
  28. 28.
    I. Kadija, J. A. Abys, V. Chinchankar and K. Straschil, “Hydrodynamically Controlled “Hull Cell,” Plating and Surface Finishing, July, 1991.Google Scholar
  29. 29.
    U. Landau, N. L. Weinberg and E. Gileadi, J. Electrochem. Soc., 135(1), 396–403, 1988.Google Scholar
  30. 30.
    S. Mehdizadeh, J. O. Dukovic, P. C. Andricacos and L. T. Romankiw, J. Electrochem. Soc., 139(1), 78, 1992.CrossRefGoogle Scholar
  31. 31.
    R. Alkire, T. Bergh and R. L. Sani, “Predicting Electrode Shape Change with Use of Finite Elements Methods,” J. Electrochem. Soc., 125(12), 1981–1988, 1978.CrossRefGoogle Scholar
  32. 32.
    U. Landau, E. Malyshev, R. Akolkar and S. Chivilikhin, “Simulations of ‘Bottom-Up’ Fill In Via Plating of Semiconductor Interconnects,” Paper 189 d, session TK; Proceedings of the AIChE Annual Meeting, San-Francisco, CA, Nov. 16–21, 2003.Google Scholar
  33. 33.
    R. Akolkar, U. Landau, H. Kuo and Yar-Ming Wang, “Modeling of the Current Distribution in Aluminum Anodization,” J. Appl. Electrochem., 34, 807–813, 2004.Google Scholar
  34. 34.
    A. C. West, C. Cheng and B. Baker, J. Electrochem. Soc., 145(9), 3070, 1998.CrossRefGoogle Scholar
  35. 35.
    R. Akolkar and U. Landau, manuscript in preparation.Google Scholar
  36. 36.
    U. Landau, “Novel Dimensionless Parameters for the Characterization of Electrochemical Cells,” Proceedings of the D. N. Bennion Mem. Symp., R. E. White and J. Newman, Eds., The Electrochemical Society Proceedings, Vol. 94–9, 1994.Google Scholar
  37. 37.
    U. Landau, in Tutorial Lectures in Electrochemical Engineering and Technology – II, Vol. 79, R.C. Alkire and D.T. Chin, Eds., AIChE Symp. Series 229, 218–225, 1983.Google Scholar
  38. 38.
    D. Roha and U. Landau, J. Electrochem. Soc., 137(3), 824–834, 1990.Google Scholar
  39. 39.
    J. H. Shyu and U. Landau, Abstract 163, The Electrochemical Society Extended Abstracts Vol. 79-2, Los Angeles, CA, Oct. 14–19, 1979.Google Scholar
  40. 40.
    J. H. Shyu and U. Landau, Abstract 401, The Electrochemical Society Extended Abstracts Vol. 80–2, Hollywood, FL, Oct. 5–10, 1980.Google Scholar
  41. 41.
    U. Landau and J. H. Shyu, in Electroplating Engineering and Waste Recycle – New Developments, D. Snyder, U. Landau and R. Sard, Eds., The Electrochemical Society, Inc., Pennington, NJ, 1983.Google Scholar
  42. 42.
    Y. Oren and U. Landau, Electrochem. Acta, 27, 739–748, 1982.CrossRefGoogle Scholar
  43. 43.
    R. T. Galasco, J. Tang and U. Landau, “Current Distribution in Pattern Plating of Nonuniformly Space and Isolated Lines,” Extended Abstract No. 330, Vol. 88–2, The Electrochem. Soc., Fall Meeting, Chicago, IL, Oct., 1988.Google Scholar
  44. 44.
    M. M. Menon and U. Landau, “Modeling of Electrochemical Cells Including Diffusion, Migration and Unsteady State Effects,” J. Electrochem. Soc., 134(8), 2248–2253, 1987.CrossRefGoogle Scholar
  45. 45.
    J. S. Bullock, G. Giles and L. J. Gray, “Simulation of an Electrochemical Plating Process,” in Topics in Boundary Element Research, Vol. 7, Chap. 7, C. A. Brebbia, Ed., Springer Verlag, Berlin and New York, pp. 121–141, 1990.Google Scholar
  46. 46.
    U. Landau and E. Malyshev, “The L-Cell- A novel Device for Plating Process Characterization,” AESF/SURFIN 2004, Chicago, IL, June 2004.Google Scholar
  47. 47.
    J. Deconinck, “Electrochemical Cell Design,” in Topics in Boundary Element Research, Vol. 7, Chap. 8, C. A. Brebbia, Ed., Springer Verlag, Berlin and New York, pp. 142–170, 1990.Google Scholar
  48. 48.
    M. M. Menon and U. Landau, “Modeling of Cells, with Multiple Electrode Reactions-Thickness and Comp. Variations in Alloy Plating,” J. Electrochem. Soc., 137, 445–452, 1990.Google Scholar
  49. 49.
    E. K. Yung, L. T. Romankiw and R. C. Alkire, “Plating into through-holes and Blind Holes,” J. Electrochem. Soc., 136(1), 206–215, 1989.CrossRefGoogle Scholar
  50. 50.
    A. S. Agarwal, U. Landau, X. Shan and J. H. Payer, “Modeling Effects of Crevice Former, Particulates, and the Evolving Surface Profile in Crevice Corrosion,” ECS Trans., 3(31), 459, 2007.Google Scholar
  51. 51.
    R. Akolkar and U. Landau, “A Time-Dependent Transport-Kinetics Model for Additives Interactions in Copper Interconnect Metallization,” J. Electrochem. Soc., 151(11), C702, 2004.Google Scholar
  52. 52.
    J. Mendez, R. Akolkar, T. Andryushchenko and U. Landau, “A Mechanistic Model for Copper Electropolishing in Phosphoric Acid,” J. Electrochem. Soc., 155, D27, 2008.CrossRefGoogle Scholar
  53. 53.
    B. Chou, R. Jain, D. McGervey, U. Landau and G. Welsch, “Electropolishing of Titanium,” Proc. Electrochem. Soc.: “Chemical Mechanical Polishing,” Vol. 2002–1, S. Seal et al., Ed, pp. 126–138, 2002.Google Scholar
  54. 54.
    S. I. Krichmar, Electrokhimiya, 17, 1444, 1981.Google Scholar
  55. 55.
    U. Landau and E. Malyshev, “Modeling Copper Planarization under Mass Transport Controlled Dissolution,” Extended Abstract # 1656, 210th Electrochemical Society Meeting, Cancun, Mexico, Oct. 29–Nov. 3, 2006.Google Scholar
  56. 56.
    A. C. West, I. Shao and H. Deligianni, J. Electrochem. Soc., 152(10), C652–C656, 2005.Google Scholar
  57. 57.
    C. W. Tobias and R. Wijsman, J. Electrochem. Soc., 100(10), 459, 1953.CrossRefGoogle Scholar
  58. 58.
    O. Lanzi III and U. Landau, “Current Distribution at a Resistive Electrode under Tafel Kinetics,” J. Electrochem. Soc., 137(4), 1139–1143, 1990.CrossRefGoogle Scholar
  59. 59.
    K. M. Takahasi, J. Electrochem. Soc., 147(4), 1414–1417, 2000.Google Scholar
  60. 60.
    S. Chivilikhin, U. Landau and E. Malyshev, “Current Distribution On A Resistive Wafer Under Copper Deposition Kinetics,” Paper 190 b, session TK; Proc. AIChE Annual Meeting, San-Francisco, CA, Nov. 16–21, 2003.Google Scholar
  61. 61.
    Uziel Landau, John J. D’Urso and David B. Rear, “Electro deposition Chemistry.” U.S. Patent Number 20020063064; May 30, 2002; Related: 6,379,522; April 30, 2002; 6,350,366; Feb. 26, 2002; 6,113,771; September 5, 2000.Google Scholar
  62. 62.
    T. Kessler and R. C. Alkire, J. Electrochem. Soc., 123, 990, 1976.CrossRefGoogle Scholar
  63. 63.
    Oscar Lanzi III, Uziel Landau, Jonathan D. Reid and Raymond T. Galasco, “Effect of Local Kinetic Variations on Through-Hole Plating,” J. Electrochem. Soc., 136, 368–374, 1989.Google Scholar
  64. 64.
    P. C. Andricacos, C. Uzoh, J. O. Dukovic, J. Horkans and H. Deligianni, IBM J. of Res. and Dev., 42(5), 567, September 1998.CrossRefGoogle Scholar
  65. 65.
    T. P. Moffat, D. Wheeler, W. H. Huber and D. Josell, Electrochemical and Solid-State Letters, 4(4), C26–C29, 2001.CrossRefGoogle Scholar
  66. 66.
    A. C. West, S. Mayer and J. Reid, Electrochem. Solid-State Lett., 4(7), C50–C53, 2001.CrossRefGoogle Scholar
  67. 67.
    T. P. Moffat, D. Wheeler, S.-K. Kim and D. Josell, “Curvature Enhanced Adsorbate Coverage Model for Electrodeposition,” J. Electrochem. Soc., 153(2), C127–C132, 2006.CrossRefGoogle Scholar
  68. 68.
    R. Alkire, X. Li, T. O. Drews, E. Rusli, F. Xue, Y. He and R. Braatz, J. Electrochem. Soc., 154(4), D230–D240, 2007.Google Scholar
  69. 69.
    Cell-Design, Computer Aided Design Software for Simulating Electrochemical Cells, L-Chem, Inc., Shaker Heights, OH 44120.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Chemical EngineeringCase Western Reserve UniversityClevelandUSA

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