Skip to main content
SpringerLink
Log in

Determination of Diffusion Coefficients in Cement-Based Materials: An Inverse Problem for the Nernst–Planck and Poisson Models

  • Published:
Journal of Materials Engineering and Performance Aims and scope Submit manuscript

Abstract

Transport properties of ions have significant impact on the possibility of rebars corrosion thus the knowledge of a diffusion coefficient is important for reinforced concrete durability. Numerous tests for the determination of diffusion coefficients have been proposed but analysis of some of these tests show that they are too simplistic or even not valid. Hence, more rigorous models to calculate the coefficients should be employed. Here we propose the Nernst–Planck and Poisson equations, which take into account the concentration and electric potential field. Based on this model a special inverse method is presented for determination of a chloride diffusion coefficient. It requires the measurement of concentration profiles or flux on the boundary and solution of the NPP model to define the goal function. Finding the global minimum is equivalent to the determination of diffusion coefficients. Typical examples of the application of the presented method are given.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. In fact, we implicitly assume here one more equation. Namely, in the case, where there are no time changing magnetic fields we have \({\text{rot}}{\kern 1pt} {\mathbf{E}} = 0.\) From this equation in the 1D case \(\left( {{\mathbf{E}} = (E_{x} ,\,{\kern 1pt} 0,\,{\kern 1pt} 0)} \right)\) one can easily derive that \(E_{x}\) depends only on one space variable \(x:\) \(E_{x} = E_{x} (x,t).\)

References

  1. A. Neville, Chloride Attack of Reinforced Concrete: An Overview, Mater. Struct., 1995, 28, p 63–70

    Article  Google Scholar 

  2. X. Shi, N. Xie, K. Fortune, and J. Gong, Durability of Steel Reinforced Concrete in Chloride Environments: An Overview, Constr. Build. Maters., 2012, 30, p 125–138

    Article  Google Scholar 

  3. L. Tang, Concentration Dependence of Diffusion and Migration of Chloride Ions Part 1. Theoretical Considerations, Cem. Concr. Res., 1999, 29, p 1463–1468

    Article  Google Scholar 

  4. L. Bertolini et al., Corrosion of Steel in Concrete: Prevention, Diagnosis, Repair, 2nd ed., Wiley, Weinheim, 2013

    Book  Google Scholar 

  5. A.J. Bard and L.R. Faulkner, Electrochemical Methods. Fundamentals and Applications, 2nd ed., Wiley, New York, 2001

    Google Scholar 

  6. B. Wierzba, A. Semczuk, J. Kołodziej, R. Schaefer. Hierarchical genetic strategy with real number encoding, Proc. of KAEiOG’03, Łagów Lubuski, Poland (May, 26–28, 2003, 231–239,.

  7. A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics, Vol 37, Texts in Applied Mathematics Springer, Berlin, 2000

    Google Scholar 

  8. T.R. Brumleve and R.P. Buck, Numerical Solution of the Nernst-Planck and Poisson Equation System with Applications to Membrane Electrochemistry and Solid State Physics, J. Electroanal. Chem., 1978, 90, p 1–31

    Article  Google Scholar 

  9. H. Chang and G. Jaffé, Polarization in Electrolytic Solutions. Part I. Theory, J. Chem. Phys., 1952, 20, p 1071–1077

    Article  Google Scholar 

  10. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd ed., Springer, Berlin, 2002

    Google Scholar 

  11. R. Filipek, K. Szyszkiewicz, M. Danielewski, and A. Lewenstam, Numerical Method and Analysis of Consistency for Electrodiffusion Problem, AIP Conf. Proc., 2007, 963, p 473–476

    Article  Google Scholar 

  12. H. Cohen and J.W. Cooley, The Numerical Solution of the Time-Dependent Nernst-Planck Equations, Biophys. J., 1965, 5, p 145–162

    Article  Google Scholar 

  13. M. Burger, Inverse Problems in Ion Channel Modeling, Inverse Probl., 2011, 27, p 1–34

    Article  Google Scholar 

  14. A.I. Prilepko, D.G. Orlowski, and I.A. Vasin, Methods For Solving Inverse Problems In Mathematical Physics, Marcel Dekker Inc, New York, 2000

    Google Scholar 

  15. M. Burger, K.A. Mardal, and B.F. Nielsen, Stability Issues of the Inverse Transmembrane Potential Problem in Electrocardiography, Inverse Probl., 2010, 26, p 105012

    Article  Google Scholar 

  16. S. Szweda and A. Zybura, Theoretical Model and Experimental Tests on Chloride Diffusion and Migration Processes in Concrete, Procedia Eng., 2013, 57, p 1121–1130

    Article  Google Scholar 

  17. T. Zych, Test Methods of Concrete Resistance to Chloride Ingress, Tech. Trans. Civ. Eng., 2014, 6-B, p 117–139

    Google Scholar 

  18. W.D. Yeih, R. Huang, and J.J. Chang, A Study of Chloride Diffusion Properties of Concrete At Early Age, J. Mar. Sci. Techn., 1994, 2, p 61–67

    Google Scholar 

  19. T. Sokalski, P. Lingenfelter, and A. Lewenstam, Numerical Solution of the Coupled Nernst-Planck and Poisson Equations for Liquid Junction and Ion Selective Membrane Potentials, J. Phys. Chem. B, 2003, 107, p 2443–2452

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported by the Polish National Centre for Research and Development Grant No. K1/IN1/25/153217/NCBiR/12.

Author information

Authors and Affiliations

  1. Physical Chemistry and Modeling Department, Faculty of Materials Science and Ceramics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059, Kraków, Poland

    Krzysztof Szyszkiewicz-Warzecha, Jerzy J. Jasielec, Janusz Fausek & Robert Filipek

Authors
  1. Krzysztof Szyszkiewicz-Warzecha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jerzy J. Jasielec
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Janusz Fausek
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Robert Filipek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Filipek.

Additional information

This article is an invited submission to JMEP selected from presentations at the Symposium “Interface Design and Modelling,” belonging to the Topic “Joining and Interfaces” at the European Congress and Exhibition on Advanced Materials and Processes (EUROMAT 2015), held September 20-24, 2015, in Warsaw, Poland, and has been expanded from the original presentation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Szyszkiewicz-Warzecha, K., Jasielec, J.J., Fausek, J. et al. Determination of Diffusion Coefficients in Cement-Based Materials: An Inverse Problem for the Nernst–Planck and Poisson Models. J. of Materi Eng and Perform 25, 3291–3295 (2016). https://doi.org/10.1007/s11665-016-2167-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11665-016-2167-4

Keywords

Search

Navigation