Numerical comparison of usual Arrhenius-type equations for modeling ionic transport in solids


The primary objective of this work is to draw attention to the inherent differences between Arrhenius-type equations used for modeling ionic conductivity in solids. The used methodology comprises a comparison between the two most used Arrhenius-type equations by generating data applying one equation and fitting them by using the other equation. Data are generated in different conditions by varying input parameters such as the extent of the temperature range and the average temperature of this range, as well as the pre-exponential term and activation energy of the Arrhenius-type equation. The fitting procedure is performed by plotting the generated data based on the linearized form of those equations followed by the linear fitting. The results show a substantial difference between the activation energy value of those equations, which is mainly dependent on the average temperature of the range. In contrast with the widely held perception that differences in the activation energy of both equations usually can be neglected, this work reveals that relative differences between activation energy of these equations are usually significant. These relative differences are often around 10% and may reach over 15% depending on a combination of intrinsic and extrinsic input parameters such as low activation energy and high-temperature of measurement, respectively. An analytical equation that expresses the relative difference between activation energy calculated from both equations is also presented and discussed.

Graphical Abstract

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1.

    Kingery WD, Bowen HK, Uhlmann DR (1976) Introduction to ceramics. In: Second. Wiley, New York

    Google Scholar 

  2. 2.

    Thomas J (1996) Solid state electrochemistry. Edited by PeterG. Bruce, Cambridge University press, Cambridge 1995, XVI, 344 pp., hardcover, £60.00, ISBN 0-521-40007-4. Adv Mater 8:360–360.

    Article  Google Scholar 

  3. 3.

    Kharton VV (2009) Solid state electrochemistry I. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

    Book  Google Scholar 

  4. 4.

    Smart LE, Moore EA (2005) Solid state chemistry: an introduction, third edit. CRC Press, New York

    Google Scholar 

  5. 5.

    Fontaine M, Norby T, Larring Y, et al (2008) Oxygen and hydrogen separation membranes based on dense ceramic conductors. Membr Sci Technol 401–458

  6. 6.

    Zhang Z, Shao Y, Lotsch B, et al (2018) New horizons for inorganic solid state ion conductors. Energy Environ Sci

  7. 7.

    Park M, Zhang X, Chung M et al (2010) A review of conduction phenomena in Li-ion batteries. J Power Sources 195:7904–7929.

    CAS  Article  Google Scholar 

  8. 8.

    Bachman JC, Muy S, Grimaud A et al (2016) Inorganic solid-state electrolytes for lithium batteries: mechanisms and properties governing ion conduction. Chem Rev 116:140–162.

    CAS  Article  PubMed  Google Scholar 

  9. 9.

    Rodrigues ACM, Nascimento MLF, Bragatto CB, Souquet J-L (2011) Charge carrier mobility and concentration as a function of composition in AgPO 3 –AgI glasses. J Chem Phys 135:234504.

    CAS  Article  PubMed  Google Scholar 

  10. 10.

    Mehrer H, Imre AW, Tanguep-Nijokep E (2008) Diffusion and ionic conduction in oxide glasses. J Phys Conf Ser 106:012001.

    CAS  Article  Google Scholar 

  11. 11.

    Linford RG, Hackwood S (1981) Physical techniques for the study of solid electrolytes. Chem Rev 81:327–364.

    CAS  Article  Google Scholar 

  12. 12.

    Patel V (2012) Chemical kinetics. InTech

  13. 13.

    McKee RA (1981) A generalization of the Nernst-Einstein equation for self-diffusion in high defect concentration solids. Solid State Ionics 5:133–136.

    CAS  Article  Google Scholar 

  14. 14.

    Catlow CRA (1986) Computer simulation studies of transport in solids. Annu Rev Mater Sci 16:517–548.

    CAS  Article  Google Scholar 

  15. 15.

    Bragatto CB (2016) Thermodynamic approach to ionic transport and structural relaxation in silver phosphate glasses. Federal University of São Carlos, Brazil

    Google Scholar 

Download references


The author acknowledges the São Paulo State Foundation (FAPESP) for its support of this work under CEPID (grant number 2013/07793-6). Also, the author thanks Brazil’s National Council for Scientific and Technological Development (CNPq) and Brazil’s Federal Agency for the Support and Improvement of Higher Education (CAPES) for doctoral scholarships granted in Brazil (grant number 140456/2014-7) and France (grant number 88881.132930/2016-01), respectively. The author is also in debt with his professors, Ana Candida M. Rodrigues, Dulcina Maria P. F. de Souza, and Edgar D. Zanotto for the first reminding about the differences between the two equations. In particular, the author gratefully acknowledges Jean-Louis Souquet for his useful comments in the theoretical background of this work and Norma Maria Pereira Machado, who made relevant suggestions.

Author information



Corresponding author

Correspondence to Rafael Bianchini Nuernberg.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nuernberg, R.B. Numerical comparison of usual Arrhenius-type equations for modeling ionic transport in solids. Ionics 26, 2405–2412 (2020).

Download citation


  • Ionic conductivity
  • Arrhenius-type equations
  • Activation energy