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Journal of Applied Electrochemistry

, Volume 24, Issue 3, pp 261–267 | Cite as

A boundary element method for transient diffusion

  • A. C. West
  • M. Matlosz
Article

Abstract

Transient diffusion in two-dimensional geometries is considered. It is shown how the spatial variation of the mass transfer-limited flux of a minor species varies with time from initially uniform to the non-uniform, steady-state distribution. Flux distributions on sinusoidal electrode and on a line electrode embedded in an otherwise insulating plane are considered. A boundary-element method is used to solve the problem in Laplace transform space, and the results are subsequently inverted into the time domain.

Keywords

Physical Chemistry Spatial Variation Boundary Element Boundary Element Method Flux Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

Ai,j

fitting coefficients defined by Equation 15

c

dimensionless concentration

c

dimensional concentration (mol cm−3)

c

bulk concentration (mol cm−3)

D

diffusion coefficient (cm2 s−1)

f

functions introduced in Equation 15

f

Laplace transform of function introduced in Equation 15

g

Green's function for modified Helmholz equation

K0, K1

modified Bessel functions of the second kind of order zero and one

ni

number of node points used in BEM calculation

N

number of values of s used in simulations

r

dimensionless distance, defined by Equation 8

s

Laplace transform variable, defined by Equation 5

q

dimensionless flux normal to the electrode

q

Laplace transform of the dimensionless flux normal to the electrode

t

time (s)

y, z

dimensionless cartesian coordinates

ž, tŷ

dimensional cartesian coordinates (cm)

z

generic interpolation function

Greek symbols

α

geometric parameter

βi

curve-fitting parameter used in exponential functions

ε0

amplitude of sinusoidal roughness (cm)

Γ

boundary of computational domain

λ

wavelength of sinusoidal roughness (cm)

σ

dimensionless arc length

τ

dimensionless time

π

3.1415926...

ζ

path of integration

Subscripts

avg

average

max

maximum

min

minimum

q

point at which the concentration (or gradient) is determined

ss

steady-state

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References

  1. [1]
    N. Ibl, Surf. Technol. 10 (1980) 81.Google Scholar
  2. [2]
    J.-C. Puippe and F. Leaman (eds), ‘Theory and Practice of Pulse Plating’, American Electroplaters and Surface Finishers Society, Orlando, FA (1990).Google Scholar
  3. [3]
    A. M. Pesco and H. Y. Cheh, J. Electrochem. Soc. 136 (1989) 408.Google Scholar
  4. [4]
    D.-T. Chin, N. R. K. Vilambi and M. K. Sunkara, Plat. Surf. Finish. 76 (Oct. 1989) 74.Google Scholar
  5. [5]
    H. H. Wan and H. Y. Cheh, J. Electrochem. Soc. 135 (1988) 658.Google Scholar
  6. [6]
    Idem, ibid. 135 (1988) 643.Google Scholar
  7. [7]
    P. S. Fedkiw and D. R. Brouns, ibid. 135 (1988) 346.Google Scholar
  8. [8]
    K. G. Jordan and C. W. Tobias, ibid. 138 (1991) 1251.Google Scholar
  9. [9]
    K. G. Jordan and C. W. Tobias, ibid. 138 (1991) 3581.Google Scholar
  10. [10]
    D. Landolt, Electrochim. Acta 32 (1987) 1.Google Scholar
  11. [11]
    D. R. Baker, M. W. Verbrugge and J. Newman, J. Electroanal. Chem. 314 (1991) 23.Google Scholar
  12. [12]
    R. M. Wightman, Science 240 (1988) 415.Google Scholar
  13. [13]
    T. J. Hanratty, J. App. Electrochem. 21 (1991) 1038.Google Scholar
  14. [14]
    J. A. Liggett and P. L.-F. Liu, ‘The Boundary Integral Equation Method for Porous Media Flow’, Allen and Unwin, Boston (1983).Google Scholar
  15. [15]
    C. A. Brebbia, J. C. F. Telles and L. C. Wrobel, ‘Boundary Element Techniques, Theory and Applications in Engineering’, Springer-Verlag, Berlin (1984).Google Scholar
  16. [16]
    C. Wagner, J. Electrochem. Soc. 101 (1954) 225.Google Scholar
  17. [17]
    P. S. Fedkiw, J. Electrochem. Soc. 127 (1980) 1304.Google Scholar
  18. [18]
    C. Clerc and D. Landolt, Electrochim. Acta 29 (1984) 787.Google Scholar
  19. [19]
    C. Wagner, J. Electrochem. Soc. 98 (1951) 116.Google Scholar
  20. [20]
    C. B. Diem, B. Newman and M. E. Orazem, J. Electrochem. Soc. 133 (1988) 2524.Google Scholar
  21. [21]
    F. B. Hildebrand, ‘Advanced Calculus for Applications’, (2nd edn), Prentice-Hall, Englewood Cliffs, NJ (1976).Google Scholar
  22. [22]
    M. D. Greenberg, ‘Application of Green's Functions in Science and Engineering’, Prentice-Hall, Englewood Cliffs, NJ (1971).Google Scholar
  23. [23]
    M. Abramowitz and I. A. Stegun, ‘Handbook of Mathematical Functions’, Dover, New York (1964) Equations 9.8.5–9.8.8, p. 379.Google Scholar
  24. [24]
    M. Matlosz, C. Creton, C. Clerc and D. Landolt, J. Electrochem. Soc. 134 (1987) 3015.Google Scholar
  25. [25]
    S. Walker, in ‘Boundary Element Methods’, (edited by C. A. Brebbia), Third International Seminar, Irvine, CA, July 1981, Springer Verlag, Heidelberg (1981) p. 472.Google Scholar
  26. [26]
    H. L. G. Pina and J. L. M. Fernandes, ‘Applications in Transient Heat Conduction’, in ‘Topics in Boundary Element Research’, Vol. 1, (edited by C. A. Brebbia), Springer-Verlag, Berlin (1984).Google Scholar
  27. [27]
    M. V. Mirkin and A. J. Bard, J. Electroanal. Chem. 323 (1992) 1.Google Scholar
  28. [28]
    F. J. Rizzo and D. J. Shippey, AIAA J. 8 (1970) 2004.Google Scholar
  29. [29]
    A. C. West, C. Madore, M. Matlosz and D. Landolt, J. Electrochem. Soc. 139 (1992) 499.Google Scholar
  30. [30]
    A. C. West and J. Newman, ‘Determining Current Distributions Governed by Laplace's Equation’, in ‘Modern Aspects of Electrochemistry, No. 23, (edited by B. E. Conway et al.), Plenum Press, New York (1992).Google Scholar
  31. [31]
    C. Brebbia, ‘The Boundary Element Method for Engineers’, John Wiley & Sons, New York (1978).Google Scholar
  32. [32]
    J. Deconinck, Dissertation, Vrije Universiteit Brussel, Brussels, Belgium (1985).Google Scholar

Copyright information

© Chapman & Hall 1994

Authors and Affiliations

  • A. C. West
    • 1
  • M. Matlosz
    • 2
  1. 1.Department of Chemical Engineering, Materials Science, and Mining EngineeringColumbia UniversityNew YorkUSA
  2. 2.Materials DepartmentSwiss Federal Institute of Technology, MX-C Ecublens, LMCHLausanneSwitzerland

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