Abstract
The work is devoted to the calculation of effective diffusion coefficient of ions from the bulk solution to the electrode through a mask and the calculation of the distribution of the limiting current density over the electrode surface. A colloidal crystal, which is formed by orderly arranged monodispersed spherical particles, serves as a mask. It is shown that the diffusion of electroactive ions in the pores between spherical particles can be simulated by unit cells with rhombic, rectangular, or triangular cross-section. In the latter case, the cell side surface has no periodical boundaries. This simplifies significantly the numerical solution of the Laplace’s equation by the finite-element method. The effective diffusion coefficient in the bulk colloidal crystal is calculated at various values of its porosity. The calculated results agree well with the literature data. It is found that, for close-packed spherical particles, the relative effective diffusion coefficient in the bulk colloidal crystal is 0.16. The thicknesses of transient zones adjacent to the electrode surface and outer boundary of colloidal crystal and the effective diffusion coefficients for these zones are determined. The dependence of effective diffusion coefficient on the number of spherical particle layers in the colloidal crystal is obtained. The distribution of the limiting current density over the electrode surface is analyzed at various numbers of particle layers.
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Velev, O.D., Tessier, P.M., Lenhoff, A.M., and Kaler, E.W., Nature, 1999, vol. 401, p. 7.
Velev, O.D. and Lenhoff, A.M., Curr. Opin. Colloid Interface Sci., 2000, vol. 5, p. 56.
Stein, A., Microporous Mesoporous Mater., 2001, vols. 44–45, p. 227.
Braun, P.V. and Wiltzius, P., Curr. Opin. Colloid Interface Sci., 2002, vol. 7, p. 116.
Texter, J., C. R. Chimie, 2003, vol. 6, p. 1425.
Photonic Crystals: Advances in Design, Fabrication, and Characterization, Busch, K., Lolkes, S., Wehrspohn, R.B., and Foll, H., Eds., Weinheim: Wiley, 2004.
Meseguer, F., Coll. Surf. A: Physicochem. Eng. Aspects, 2005, vol. 270–271, p. 1.
Asoh, H., Sakamoto, S., and Ono, S., J. Colloid Interface Sci., 2007, vol. 316, p. 547.
Paquet, C. and Kumacheva, E., Mater. Today, 2008, vol. 11, no. 4, p. 48.
Nair, R.V. and Vijaya, R., Prog. Quantum Electron., 2010, vol. 34, p. 89.
Bartlett, P.N., Baumberg, J.J., Birkin, P.R., Ghanem, M.A., and Netti, M.C., Chem. Mater., 2002, vol. 14, p. 2199.
Spada, E.R., Da Rocha, A.S., Jasinski, E.F., Pereira, G.M.C., Chavero, L.N., Oliveira, A.B., Azevedo, A., and Santorelli, M.L., J. Appl. Phys., 2008, vol. 103, p. 114306.
Xia, X.H., Tu, J.P., Zhang, J., Xiang, J.Y., Wang, X.L., and Zhao, X.B., Appl. Mater. Interfaces, 2010, vol. 2, no. 1, p. 186.
Sumida, T., Wada, Y., Kitamura, T., and Yanagida, S., Langmuir, 2002, vol. 18, p. 3886.
Abdelsalam, M.E., Bartlett, P.N., Kelf, T., and Baumberg, J., Langmuir, 2005, vol. 21, p. 1753.
Hao, Y., Zhu, F.Q., Chien, C.L., and Searson, P.C., J. Electrochem. Soc., 2007, vol. 154, p. D65.
Hung, D., Liu, Z., Shah, N., Hao, Y., and Searson, P.C., J. Phys. Chem. C, 2007, vol. 111, p. 3308.
Mahajan, S., Cole, R.M., Soares, B.F., Pelfrey, S.H., Russell, A.E., Baumberg, J.J., and Bartlett, P.N., J. Phys. Chem. C, 2007, vol. 113, p. 9284.
Sapoletova, N., Makarevich, T., Napolskii, K., Mishina, E., Eliseev, A., Van Etteger, A., Rasing, T., and Tsirlina, G., Phys. Chem. Chem. Phys., 2010, vol. 12, p. 15414.
Newton, M.R., Morey, K.A., Zhang, Y., Snow, R.J., Diwekar, M., Shi, J., and White, H.S., Nano Lett., 2004, vol. 4, p. 875.
Adier, P.M., Porous Media: Geometry and Transports, Boston: Butterworth-Heinemann, 1992.
Maxwell, C., Treatise on Electricity and Magnetism, vol. 1, London, Oxford: Univ. Press, 1873.
Rayleigh, L., Philos. Mag., 1892, vol. 34, p. 481.
Zuzovsky, M. and Brenner, H., Z. Angew. Math. Phys, 1977, vol. 28, p. 979.
Sangani, A.S. and Acrivos, A., Proc. R. Soc. London, Ser. A, 1983, vol. 386, p. 263.
Cheng, H. and Torquato, S., Proc. R. Soc. London, Ser. A, 1997, vol. 453, p. 1331.
Cheng, H. and Torquato, S., Proc. R. Soc. London, Ser. A, 1997, vol. 453, p. 145.
Desmet, G. and Deridder, S., J. Chromatogr., A, 2011, vol. 1218, p. 32.
Kaviany, M., Principles of Heat Transfer in Porous Media, New York: Springer, 1995.
Venema, P., Struis, R.P.W.J., Leyte, J.C., and Bedeaux, D., J. Colloid Interface Sci., 1991, vol. 141, p. 360.
Blees, M.H. and Leyte, J.C., J. Colloid Interface Sci., 1994, vol. 166, p. 118.
Kim, A.S. and Chen, H., J. Membr. Sci., 2006, vol. 279, p. 129.
Starly, B., Yildirim, E., and Sun, W., Comput. Methods Programs Biomed., 2007, vol. 87, p. 21.
Fiedler, T., Loffler, R., Bernthaler, T., Winkler, R., Belova, I.V., Murch, G.E., and Ochsner, A., Mater. Lett., 2009, vol. 63, p. 1125.
Petkov, K., Qiu, F., Fan, Z., Kaufman, A.E., and Mueller, K., IEEE Trans. Vis. Comput. Graph., 2009, vol. 15, no. 5, p. 802.
Albrecht, J.D., Knipp, P.A., and Reinecke, T.L., Phys. Rev. B: Condens. Matter, 2001, vol. 63, p. 134303.
Sareni, B., Krahenbuhl, L., Beroual, A., and Nicolas, A., IEEE Trans. Magn., 1997, vol. 33, p. 1580.
Zhou, J., Yu, A., and Zhang, Y., J. Heat Transfer, 2007, vol. 129, p. 363.
Koroteeva, O., Mogilevskaya, S., Crouch, S., and Gordeliy, E., Eng. Anal. Bound. Elem., 2010, vol. 34, p. 793.
Song, Y.S. and Youn, J.R., Carbon, 2006, vol. 44, p. 710.
Hassani, B. and Hinton, E., Comput. Struct., 1998, vol. 69, p. 707.
Boutin, C. and Geinndreau, C., Phys. Rev. E: Stat. Phys., Plasmas, Fluids, 2010, vol. 82, p. 036313.
Bettega, R., Moreira, M.F.P., Correa, R.G., and Freire, J.T., Particuology, 2011, vol. 9, p. 107.
Carson, J.K., Lovatt, S.J., Tanner, D.J., and Cleland, A.C., Int. J. Refrig., 2003, vol. 26, p. 873.
Stroeven, M., Askes, H., and Sluys, L.J., Comput. Methods Appl. Mech. Engrg., 2004, vol. 193, p. 3221.
Jiang, P.-X. and Lu, X.-C., Int. J. Heat Fluid Flow, 2007, vol. 28, p. 1144.
Porfiri, M., Nguyen, N.Q., and Gupta, N., J. Mater. Sci., 2009, vol. 44, p. 1540.
Xing, Y.F., Yang, Y., and Wang, X.M., Compos. Struct., 2010, vol. 92, p. 2265.
Augier, F., Idoux, F., and Delenne, J.Y., Chem. Eng. Sci., 2010, vol. 65, p. 1055.
Wang, M. and Pan, N., Mater. Sci. Eng. R, 2008, vol. 63, p. 1.
Kamiuto, K., Nagumo, Y., and Iwamoto, M., Appl. Energy, 1989, vol. 34, p. 213.
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Original Russian Text © V.M. Volgin, A.D. Davydov, T.B. Kabanova, 2012, published in Elektrokhimiya, 2012, Vol. 48, No. 8, pp. 898–916.
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Volgin, V.M., Davydov, A.D. & Kabanova, T.B. Calculation of effective diffusion coefficient in a colloidal crystal by the finite-element method. Russ J Electrochem 48, 817–834 (2012). https://doi.org/10.1134/S1023193512070117
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DOI: https://doi.org/10.1134/S1023193512070117