Skip to main content
SpringerLink
Log in

Theories and Models of Ion Diffusion

  • Chapter
  • First Online:
Dynamics of Glassy, Crystalline and Liquid Ionic Conductors

Part of the book series: Topics in Applied Physics ((TAP,volume 132))

Abstract

This chapter describes some models that are often used in trying to understand experimental data and fundamental questions in ion diffusion in ionically conducting materials. The basics of linear response theory are introduced first, with the definition of the linear response function, the Kramers-Kronig relations, and the Fluctuation-Dissipation theorem. A second section is devoted to present the Debye model and several other phenomenological descriptions of dielectric relaxation in materials whose electrical response is dominated by bound charges. This helps to understand the conductivity relaxation that occurs in materials with mobile charges like ionic conductors, and to introduce the so called conductivity formalism and electric modulus formalism for the analysis of experimental data of ion diffusion dynamics. A simple model of ion hopping is introduced that accounts for the thermally activated behavior often found in ionic conductivity data. The relationship between non-Debye relaxation and non-Gaussianity of the dynamics in the real space is also discussed in this chapter. Finally, three different models for ion diffusion are described in some detail. These are the Random Barrier Model, the MIGRATION concept, and the Coupling Model.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. L. Landau, E. Lifschitz, Textbook of Theoretical Physics, Vol. V. Statistical Physics (Akademie-Verlag, Berlin, 1979).

    Google Scholar 

  2. F. Kremer, A. Schönhals (eds.), Broadband Dielectric Spectroscopy (Springer, Berlin, 2012), p. 729

    Google Scholar 

  3. R. de L. Kronig, J. Opt. Soc. Am. 12, 547 (1926)

    Google Scholar 

  4. H. Kramers, Atti Del Congr. Internazionale Dei Fis, 2nd edn. (Bologna Zanichelli, 1927), p. 545

    Google Scholar 

  5. C.F. Bohren, Eur. J. Phys. 31, 573 (2010)

    Article  Google Scholar 

  6. J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1999)

    Google Scholar 

  7. H. Nyquist, Phys. Rev. 32, 110 (1928)

    Article  CAS  Google Scholar 

  8. H.B. Callen, T.A. Welton, Phys. Rev. 83, 34 (1951)

    Article  Google Scholar 

  9. G. Williams, J. Fournier, J. Chem. Phys. 104, 5690 (1996)

    Article  CAS  Google Scholar 

  10. L. Onsager, Phys. Rev. 37, 405 (1931)

    Article  CAS  Google Scholar 

  11. L. Onsager, Phys. Rev. 38, 2265 (1931)

    Article  CAS  Google Scholar 

  12. C.J.F. Böttcher, P. Bordewijk, Theory of Electric Polarization, Vol. II: Dielectrics in Time-Dependent Fields (Elsevier, Amsterdam/Oxford/New York, 1978)

    Google Scholar 

  13. P. Debye, Phys. Z. 13, 97 (1912)

    CAS  Google Scholar 

  14. P. Debye, Ber. Deut. Phys. Ges. 55, 777 (1913)

    Google Scholar 

  15. G. Adam, J.H. Gibbs, J. Chem. Phys. 43, 139 (1965)

    Article  CAS  Google Scholar 

  16. K. Schmidt-Rohr, H. Spiess, Phys. Rev. Lett. 66, 3020 (1991)

    Article  CAS  Google Scholar 

  17. E. Weeks, J. Crocker, A. Levitt, A. Schofield, D. Weitz, Science 287, 627 (2000)

    Article  CAS  Google Scholar 

  18. K.L. Ngai, Relaxation and Diffusion in Complex Systems (Springer, New York, 2011), p. 835

    Book  Google Scholar 

  19. R. Kohlrausch, Ann. Der Phys. Und Chemie 167, 179 (1854)

    Article  Google Scholar 

  20. G. Williams, D.C. Watts, Trans. Faraday Soc. 66, 80 (1970)

    Article  CAS  Google Scholar 

  21. G. Williams, D.C. Watts, S.B. Dev, A.M. North, Trans. Faraday Soc. 67, 1323 (1971)

    Article  CAS  Google Scholar 

  22. M. Cardona, R.V. Chamberlin, W. Marx, Ann. Phys. 16, 842 (2007)

    Article  Google Scholar 

  23. G. Williams, IEEE Trans. Electr. Insul. EI-17, 469 (1982)

    Article  CAS  Google Scholar 

  24. G. Williams, IEEE Trans. Electr. Insul. EI-20, 843 (1985)

    Article  CAS  Google Scholar 

  25. K. Tsang, K. Ngai, Phys. Rev. E 56, R17 (1997)

    Article  CAS  Google Scholar 

  26. K.L. Ngai, J. Habasaki, Y. Hiwatari, C. Leon, J. Phys. Condens. Matter 15, S1607 (2003)

    Article  CAS  Google Scholar 

  27. S. Havriliak, S. Negami, J. Polym. Sci. Part C Polym. Symp. 14, 99 (1966)

    Article  Google Scholar 

  28. A. Jonscher, Dielectric Relaxation in Solids (Chelsea, London, 1983)

    Google Scholar 

  29. K.S. Cole, R.H. Cole, J. Chem. Phys. 9, 341 (1941)

    Article  CAS  Google Scholar 

  30. D.W. Davidson, R.H. Cole, J. Chem. Phys. 19, 1484 (1951)

    Article  CAS  Google Scholar 

  31. I.M. Hodge, K.L. Ngai, C.T. Moynihan, J. Non-Cryst. Solids 351, 104 (2005)

    Article  CAS  Google Scholar 

  32. C. Angell, Annu. Rev. Phys. Chem. 43, 693 (1992)

    Article  CAS  Google Scholar 

  33. J. Dyre, T. Schrøder, Rev. Mod. Phys. 72, 873 (2000)

    Article  Google Scholar 

  34. K.L. Ngai, A.K. Jonscher, C.T. White, Nature 277, 185 (1979)

    Article  CAS  Google Scholar 

  35. K.L. Ngai, J. Habasaki, C. León, A. Rivera, Z. Phys. Chem. 219, 47 (2005)

    Google Scholar 

  36. C. León, J. Habasaki, K.L. Ngai, Zeitschrift Für Phys. Chemie 223, 1311 (2009)

    Article  CAS  Google Scholar 

  37. K. Funke, Prog. Solid State Chem. 22, 111 (1993)

    Article  CAS  Google Scholar 

  38. A.K. Jonscher, Nature 267, 673 (1977)

    Article  CAS  Google Scholar 

  39. D.P. Almond, A.R. West, R.J. Grant, Solid State Commun. 44, 1277 (1982)

    Article  CAS  Google Scholar 

  40. C. Cramer, K. Funke, T. Saatkamp, Philos. Mag. Part B 71, 701 (1995)

    Article  CAS  Google Scholar 

  41. K. Funke, Solid State Ionics 169, 1 (2004)

    Article  CAS  Google Scholar 

  42. D. Sidebottom, B. Roling, K. Funke, Phys. Rev. B 63, 024301 (2000)

    Article  Google Scholar 

  43. J.C. Dyre, P. Maass, B. Roling, D.L. Sidebottom, Rep. Prog. Phys. 72, 046501 (2009)

    Article  CAS  Google Scholar 

  44. D. Sidebottom, Phys. Rev. Lett. 82, 3653 (1999)

    Article  CAS  Google Scholar 

  45. C. León, M. Lucia, J. Santamaria, Phys. Rev. B 55, 882 (1997)

    Article  Google Scholar 

  46. C.T. Moynihan, J. Non-Cryst. Solids 172–174, 1395 (1994)

    Article  Google Scholar 

  47. C. Moynihan, Solid State Ionics 105, 175 (1998)

    Article  CAS  Google Scholar 

  48. K.L. Ngai, C.T. Moynihan, MRS Bull. 23, 51 (1998)

    Article  CAS  Google Scholar 

  49. P.B. Macedo, C.T. Moynihan, R. Bose, Phys. Chem. Glas. 13, 171 (1972)

    CAS  Google Scholar 

  50. T.J. Higgins, L.P. Boesch, V. Volterra, C.T. Moynihan, P.B. Macedo, J. Am. Ceram. Soc. 56, 334 (1973)

    Article  CAS  Google Scholar 

  51. F.S. Howell, R.A. Bose, P.B. Macedo, C.T. Moynihan, J. Phys. Chem. 78, 639 (1974)

    Article  CAS  Google Scholar 

  52. C.T. Moynihan, L.P. Boesch, N.L. Laberge, Phys. Chem. Glas. 14, 122 (1973)

    CAS  Google Scholar 

  53. J.R. Macdonald, J. Chem. Phys. 102, 6241 (1995)

    Article  CAS  Google Scholar 

  54. B.A. Boukamp, Solid State Ionics 20, 31 (1986)

    Google Scholar 

  55. B. Boukamp, J.R. Macdonald, Solid State Ionics 74, 85 (1994)

    Article  CAS  Google Scholar 

  56. C. León, M.L. Lucía, J. Santamaría, Philos. Mag. Part B 75, 629 (1997)

    Article  Google Scholar 

  57. F. Alvarez, A. Alegra, J. Colmenero, Phys. Rev. B 44, 7306 (1991)

    Article  Google Scholar 

  58. K.L. Ngai, J. Chem. Phys. 98, 6424 (1993)

    Article  CAS  Google Scholar 

  59. P. Kofstad, T. Norby, Defects and Transport in Crystalline Solids (University of Oslo, Oslo, 2007)

    Google Scholar 

  60. M.S. Islam, C.A.J. Fisher, Chem. Soc. Rev. 43, 185 (2014)

    Article  CAS  Google Scholar 

  61. G. Ceder, MRS Bull. 35, 693 (2011)

    Article  Google Scholar 

  62. L. Malavasi, C.A.J. Fisher, M.S. Islam, Chem. Soc. Rev. 39, 4370 (2010)

    Article  CAS  Google Scholar 

  63. Y. Wang, W.D. Richards, S.P. Ong, L.J. Miara, J.C. Kim, Y. Mo, G. Ceder, Nat. Mater. 14, 1026 (2015)

    Article  CAS  Google Scholar 

  64. Z. Wojnarowska, M. Paluch, J. Phys. Condens. Matter 27, 073202 (2015)

    Article  CAS  Google Scholar 

  65. T. Norby, Mater. Res. Soc. Bull. 34, 923 (2009)

    Google Scholar 

  66. C. Zener, Proc. R. Soc. A Math. Phys. Eng. Sci. 137, 696 (1932)

    Article  Google Scholar 

  67. K. Ngai, Solid State Ionics 105, 231 (1998)

    Article  CAS  Google Scholar 

  68. L. Van Hove, Phys. Rev. 95, 249 (1954)

    Article  Google Scholar 

  69. A. Rahman, K.S. Singwi, A. Sjölander, Phys. Rev. 126, 986 (1962)

    Article  CAS  Google Scholar 

  70. R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957)

    Article  Google Scholar 

  71. T. Odagaki, M. Lax, Phys. Rev. B 24, 5284 (1981)

    Article  CAS  Google Scholar 

  72. D. Ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 2000), p. 316

    Book  Google Scholar 

  73. S. Elliott, Solid State Ionics 27, 131 (1988)

    Article  Google Scholar 

  74. T. Odagaki, M. Lax, Phys. Rev. Lett. 45, 847 (1980)

    Article  Google Scholar 

  75. T. Odagaki, Phys. Rev. B 38, 9044 (1988)

    Article  Google Scholar 

  76. T. Nakayama, Rep. Prog. Phys. 65, 1195 (2002)

    Article  CAS  Google Scholar 

  77. T. Ishii, J. Phys. Soc. Jpn. 61, 924 (1992)

    Article  Google Scholar 

  78. T. Ishii, Solid State Ionics 40–41, 244 (1990)

    Article  Google Scholar 

  79. A. Bunde, H. Roman, S. Russ, A. Aharony, A. Harris, Phys. Rev. Lett. 69, 3189 (1992)

    Article  Google Scholar 

  80. E. Bacry, J. Delour, J.F. Muzy, Phys. Rev. E. Stat. Nonlin. Soft Matter Phys. 64, 026103 (2001)

    Google Scholar 

  81. M. Shlesinger, B. West, J. Klafter, Phys. Rev. Lett. 58, 1100 (1987)

    Article  Google Scholar 

  82. T. Geisel, J. Nierwetberg, A. Zacherl, Phys. Rev. Lett. 54, 616 (1985)

    Article  Google Scholar 

  83. J. Habasaki, K.L. Ngai, J. Chem. Phys. 129, 194501 (2008)

    Article  CAS  Google Scholar 

  84. J. Habasaki, I. Okada, Y. Hiwatari, Phys. Rev. B 55, 6309 (1997)

    Article  CAS  Google Scholar 

  85. A. Blumen, J. Klafter, B.S. White, G. Zumofen, Phys. Rev. Lett. 53, 1301 (1984)

    Article  Google Scholar 

  86. J. Habasaki, I. Okada, Y. Hiwatari, J. Phys. Soc. Jpn. 67, 2012 (1998)

    Article  CAS  Google Scholar 

  87. K. Popper, Conjectures and Refutations: The Growth of Scientific Knowledge (Routledge Classics, New York, 1963) 

    Google Scholar 

  88. J.C. Kimball, L.W. Adams, Phys. Rev. B 18, 5851 (1978)

    Article  CAS  Google Scholar 

  89. J.C. Dyre, T.B. Schroeder, Phys. Status Solidi 230, 5 (2002)

    Article  CAS  Google Scholar 

  90. P.N. Butcher, J. Phys. C Solid State Phys. 7, 879 (1974)

    Article  Google Scholar 

  91. D.L. Stein, C.M. Newman, Phys. Rev. E 51, 5228 (1995)

    Article  CAS  Google Scholar 

  92. J.W. Haus, K.W. Kehr, Phys. Rep. 150, 263 (1987)

    Article  CAS  Google Scholar 

  93. J.C. Dyre, J. Appl. Phys. 64, 2456 (1988)

    Article  Google Scholar 

  94. T.B. Schrøder, J.C. Dyre, Phys. Chem. Chem. Phys. 4, 3173 (2002)

    Article  CAS  Google Scholar 

  95. V. Ambegaokar, B.I. Halperin, J.S. Langer, Phys. Rev. B 4, 2612 (1971)

    Article  Google Scholar 

  96. T. Schrøder, J. Dyre, Phys. Rev. Lett. 84, 310 (2000)

    Article  Google Scholar 

  97. S.D. Baranovskii, H. Cordes, J. Chem. Phys. 111, 7546 (1999)

    Article  CAS  Google Scholar 

  98. W. Dieterich, P. Maass, Chem. Phys. 284, 439 (2002)

    Article  CAS  Google Scholar 

  99. B. Roling, C. Martiny, K. Funke, J. Non-Cryst. Solids 249, 201 (1999)

    Article  CAS  Google Scholar 

  100. J.O. Isard, J. Non-Cryst. Solids 246, 16 (1999)

    Article  CAS  Google Scholar 

  101. K. Funke, R.D. Banhatti, S. Brückner, C. Cramer, C. Krieger, A. Mandanici, C. Martiny, I. Ross, Phys. Chem. Chem. Phys. 4, 3155 (2002)

    Article  CAS  Google Scholar 

  102. K. Funke, R.D. Banhatti, J. Non-Cryst. Solids 353, 3845 (2007)

    Article  CAS  Google Scholar 

  103. K. Funke, R.D. Banhatti, D.M. Laughman, L.G. Badr, M. Mutke, A. Santic, W. Wrobel, E.M. Fellberg, C. Biermann, Zeitschrift Für Phys. Chemie 224, 1891 (2010)

    Article  CAS  Google Scholar 

  104. K. Funke, R.D. Banhatti, Solid State Ionics 177, 1551 (2006)

    Article  CAS  Google Scholar 

  105. P. Singh, R.D. Banhatti, K. Funke, Phys. Chem. Glas. 46, 241 (2005)

    CAS  Google Scholar 

  106. K.L. Ngai, Comment Solid State Phys. 9, 141 (1980)

    CAS  Google Scholar 

  107. K.L. Ngai, C.T. White, Phys. Rev. B 20, 2475 (1979)

    Article  CAS  Google Scholar 

  108. K.L. Ngai, Comment Solid State Phys. 9, 127 (1980)

    Google Scholar 

  109. M. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, Berlin, 1990)

    Book  Google Scholar 

  110. M.V. Berry, Proc. R. Soc. A Math. Phys. Eng. Sci. 413, 183 (1987)

    Google Scholar 

  111. M. Berry, Phys. Scr. 40, 335 (1989)

    Article  CAS  Google Scholar 

  112. O. Bohigas, Chaos Phys. Quantique Chaos Quantum Phys. 87 (1989)

    Google Scholar 

  113. O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52, 1 (1984)

    Article  Google Scholar 

  114. E.P. Wigner, Proceedings of Fourth Canadian Mathematical Congress (University Toronto Press, Toronto, 1957), p. 174

    Google Scholar 

  115. M.L. Mehta, Random Matrices, 2nd edn. (Academic, New York, 1991)

    Google Scholar 

  116. F.J. Dyson, J. Math. Phys. 3, 140 (1962)

    Article  CAS  Google Scholar 

  117. L. Gor’kov, G. Eliashberg, JETP 21, 940 (1965)

    Google Scholar 

  118. K.L. Ngai, S. Capaccioli, J. Phys. Condens. Matter 19, 205114 (2007)

    Article  CAS  Google Scholar 

  119. D. Bedrov, G.D. Smith, Macromolecules 39, 8526 (2006)

    Article  CAS  Google Scholar 

  120. I.A. Campbell, C. Giovannella (eds.), Relaxation in Complex Systems and Related Topics (Springer, Boston, 1990)

    Google Scholar 

  121. G. Kolata, Science 223, 917 (1984)

    Article  CAS  Google Scholar 

  122. E. Weeks, D. Weitz, Phys. Rev. Lett. 89, 095704 (2002)

    Article  CAS  Google Scholar 

  123. J. Habasaki, K.L. Ngai, J. Non-Cryst. Solids 352, 5170 (2006)

    Article  CAS  Google Scholar 

  124. T. Pakula, J. Mol. Liq. 86, 109 (2000)

    Article  CAS  Google Scholar 

  125. P. Polanowski, T. Pakula, J. Chem. Phys. 117, 4022 (2002)

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tokyo Institute of Technology, Yokohama, Kanagawa, Japan

    Junko Habasaki

  2. Facultad de Fisica, Universidad Complutense Madrid, Madrid, Spain

    Carlos León

  3. IPCF, CNR, Pisa, Italy

    K. L. Ngai

Authors
  1. Junko Habasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlos León
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. L. Ngai
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Habasaki, J., León, C., Ngai, K.L. (2017). Theories and Models of Ion Diffusion. In: Dynamics of Glassy, Crystalline and Liquid Ionic Conductors. Topics in Applied Physics, vol 132. Springer, Cham. https://doi.org/10.1007/978-3-319-42391-3_2

Download citation

Publish with us

Policies and ethics

Search

Navigation