Skip to main content
SpringerLink
Log in

Multi-scale periodic homogenization of ionic transfer in cementitious materials

  • Original
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

Abstract

A multi-scale periodic homogenization procedure of the ionic transfers in saturated porous media is proposed. An application on a multi-scale porous material was achieved for establishing models describing a ionic transfer from Nernst–Planck–Poisson–Boltzmann system. The first one is obtained by homogenization from the scale of Debye length to the capillary porosity scale, by taking into account the electrical double layer phenomenon. The second one results from another homogenization procedure from the capillary porosity scale to the scale of the material, where the electrical double layer effects are naturally negligible. A numerical parametric study is conducted on three dimensional elementary cells in order to highlight the effects of the electrical double layer on the ionic transfer parameters. Comparisons with existing experimental data are also presented and discussed. The double homogenization procedure gives homogenized diffusion coefficients very close to those obtained experimentally for chlorides ions from electrodiffusion tests carried out in laboratory.

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
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. Based on asymptotic expansions.

  2. Calcium Silicate Hydrate.

  3. Cement pastes with 0.5 and 0.7 water/cement ratios.

  4. We have a Nernst–Planck equation for each ionic species in the solution.

  5. From Debye length to microscopic scale length.

  6. As we have electroneutrality in pore solution, i.e \(c_{+}=c_{-}\), when EDL effects are not involved.

  7. Because of the symmetry of the problem, the value of \(\varphi ^*\) is identical at any point of \(\varGamma ^*_{sf}\).

References

  1. Younsi A, Turcry P, Aït-Mokhtar A, Staquet S (2013) Accelerated carbonation of concrete with high content of mineral additions: effect of interactions between hydration and drying. Cem Concr Res 43:25–33

    Article  Google Scholar 

  2. Poupard O, Aït-Mokhtar A, Dumargue P (2003) Impedance spectroscopy in reinforced concrete: experimental procedure for monitoring steel corrosion part II polarization effect. J Mater Sci 38(17):3521–3526

    Article  Google Scholar 

  3. Poupard O, Aït-Mokhtar A, Dumargue P (2003) Impedance spectroscopy in reinforced concrete: procedure for monitoring steel corrosion part I development of the experimental device. J Mater Sci 38(13):2845–2850

    Article  Google Scholar 

  4. Klinghoffer O, Frolund T, Poulsen E (2000) Rebar corrosion rate measurements for service life estimates. In: ACI Fall Convention 2000, Toronto, Canada, Committee 365 “Practical Application of Service Life Models”

  5. ASCE (2010) Report card for America’s infrastructure. American Society of Civil Engineers. http://www.infrastructurereportcard.org/. Accessed 01 June 10

  6. Amiri O, Aït-Mokhtar A, Seigneurin A (1997) A complement to the discussion of A. Xu and S. Chandra, about the paper “Calculation of chloride coefficient diffusion in concrete from ionic migration measurements” by C. Andrade. Cem Concr Res 27(6):951–957

    Article  Google Scholar 

  7. Amiri O, Aït-Mokhtar A, Dumargue P, Touchard G (2001) Electrochemical modelling of chloride migration in cement based materials: Part I: Theoretical basis at microscopic scale. Electrochimica Acta 46(9):1267–1275

    Article  Google Scholar 

  8. Amiri O, Aït-Mokhtar A, Dumargue P, Touchard G (2001) Electrochemical modelling of chlorides migration in cement based materials. part II: Experimental study-calculation of chlorides flux. Electrochimica Acta 46(23):3589–3597

    Article  Google Scholar 

  9. Tang L (1999) Concentration dependence of diffusion and migration of chloride ions: Part 2. Experimental evaluations. Cem Concr Res 29(9):1469–1474

    Article  Google Scholar 

  10. Tang L, Nilsson LO (1992) Rapid determination of the chloride diffusibility in concrete by applying an electrical fied. ACI Mater J 89:49–53

    Google Scholar 

  11. Truc O, Ollivier JP, Carcassés M (2000) A new way for determining the chloride diffusion coefficient in concrete from steady state migration test. Cem Concr Res 30(2):217–226

    Article  Google Scholar 

  12. Chatterji S (1998) Colloid electrochemistry of saturated cement paste and some properties of cement based materials. Adv Cem Based Mater 7(3–4):102–108

    Article  Google Scholar 

  13. Plumb OA, Whitaker S (1988) Dispersion in heterogeneous porous media 1. Local volume averaging and large-scale averaging. Water Resour Res 24:913–926

    Article  Google Scholar 

  14. Whitaker S (1967) Diffusion and dispersion in porous media. AIChE J 13(3):420–427

    Article  Google Scholar 

  15. Auriault J, Strzelecki T (1981) On the electro-osmotic flow in a saturated porous medium. Int J Eng Sci 19(7):915–928

    Article  MATH  Google Scholar 

  16. Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis for periodic structures. Studies in Mathematics and its applications

  17. Sanchez Palencia E (1980) Non homogeneous media and vibration theory. Volume Lecture Notes in Physics

  18. Auriault J, Lewandowska J (1996) Diffusion/adsorption/advection macrotransport in soils. Eur J Mech 15:681–704

    MATH  Google Scholar 

  19. Auriault JL, Lewandowska J (1997) Diffusion non linéaire en milieux poreux. C R Acad Sci Ser IIB Mech Phys Chem Astron 324(5):293–298

    MATH  Google Scholar 

  20. Hamdouni A, Millet O (2003) Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. part I: the shallow shells. Arch Mech 55(2):135–175

    MathSciNet  MATH  Google Scholar 

  21. Millet O, Hamdouni A, Cimetière A (1997) Justification du modèle bidimensionnel non linéaire de plaque par développement asymptotique des équations d’équilibre. C R Acad Sci Ser IIB Mech Phys Chem Astron 324(6):349–354

    MATH  Google Scholar 

  22. Millet O, Hamdouni A, Cimetière A (2001) A classification of thin plate models by asymptotic expansion of non-linear three-dimensional equilibrium equations. Int J Non-Linear Mech 36(1):165–186

    Article  MathSciNet  MATH  Google Scholar 

  23. Skjetne E, Auriault J (1999) New insights on steady, non-linear flow in porous media. Eur J Mech B Fluids 18(1):131–145

    Article  MathSciNet  MATH  Google Scholar 

  24. Cherblanc F, Ahmadi A, Quintard M (2007) Two-domain description of solute transport in heterogeneous porous media: comparison between theoretical predictions and numerical experiments. Adv Water Resour 30(5):1127–1143

    Article  Google Scholar 

  25. Quintard M, Whitaker S (1994) Transport in ordered and disordered porous media II: generalized volume averaging. Transp Porous Media 14(2):179–206

    Article  Google Scholar 

  26. Quintard M, Whitaker S (1998) Transport in chemically and mechanically heterogeneous porous media IV: large-scale mass equilibrium for solute transport with adsorption. Adv Water Resour 22(1):33–57

    Article  Google Scholar 

  27. Moyne C, Murad MA (2002) Electro-chemo-mechanical couplings in swelling clays derived from a micro/macro-homogenization procedure. Int J Solids Struct 39(25):6159–6190

    Article  MATH  Google Scholar 

  28. Royer P, Auriault J, Boutin C (1996) Macroscopic modeling of double-porosity reservoirs. J Pet Sci Eng 16(4):187–202

    Article  Google Scholar 

  29. Lewandowska J, Auriault JL (2004) Modelling of unsaturated water flow in soils with highly permeable inclusions. C R Mec 332(1):91–96

    Article  MATH  Google Scholar 

  30. Lewandowska J, Szymkiewicz A, Burzyński K, Vauclin M (2004) Modeling of unsaturated water flow in double-porosity soils by the homogenization approach. Adv Water Resour 27(3):283–296

    Article  Google Scholar 

  31. de Lima SA, Murad M, Moyne C, Stemmelen D (2008) A three-scale model for pH-dependent steady flows in 1:1 clays. Acta Geotech 3(2):153–174

    Article  Google Scholar 

  32. de Lima SA, Murad M, Moyne C, Stemmelen D (2010) A three-scale model of pH-dependent flows and ion transport with equilibrium adsorption in kaolinite clays: I. Homogenization analysis. Transp Porous Media 85(1):23–44

    Article  Google Scholar 

  33. de Lima SA, Murad M, Moyne C, Stemmelen D, Boutin C (2010) A three-scale model of pH-dependent flows and ion transport with equilibrium adsorption in kaolinite clays: II effective-medium behavior. Transp Porous Media 85(1):45–78

    Article  Google Scholar 

  34. Bourbatache K, Millet O, Aït-Mokhtart A (2012) Ionic transfer in charged porous media. Periodic homogenization and parametric study on 2D microstructures. Int J Heat Mass Transf 55(21–22):5979–5991

    Article  Google Scholar 

  35. Friedmann H, Amiri O, Aït-Mokhtar A (2008) Physical modeling of the electrical double layer effects on multispecies ions transport in cement-based materials. Cem Concr Res 38(12):1394–1400

    Article  Google Scholar 

  36. Moyne C, Murad M (2006) A two-scale model for coupled electro-chemo-mechanical phenomena and onsager’s reciprocity relations in expansive clays: I homogenization analysis. Transp Porous Media 62(3):333–380

    Article  MathSciNet  Google Scholar 

  37. Moyne C, Murad M (2006) A two-scale model for coupled electro-chemo-mechanical phenomena and onsager’s reciprocity relations in expansive clays: II computational validation. Transp Porous Media 63(1):13–56

    Article  Google Scholar 

  38. Revil A (1999) Ionic diffusivity, electrical conductivity, membrane and thermoelectric potentials in colloids and granular porous media: a unified model. J Colloid Interface Sci 212(2):503–522

    Article  Google Scholar 

  39. Revil A, Linde N (2006) Chemico-electromechanical coupling in microporous media. J Colloid Interface Sci 302(2):682–694

    Article  Google Scholar 

  40. Bourbatache K, Millet O, Aït-Mokhtar A, Amiri O (2012) Modeling the chlorides transport in cementitious materials by periodic homogenization. Transp Porous Media 94:437–459

    Article  MathSciNet  Google Scholar 

  41. Dormieux L, Barboux P, Coussy O, Dangla P (1995) A macroscopic model of swelling phenomenon of a saturated clay. Eur J Mech A Solids 14:981–1004

    MATH  Google Scholar 

  42. Gagneux G, Millet O (2014) General properties of the Nernst–Planck–Poisson–Boltzmann system describing electrocapillary effects in porous media. J Elast 117(2):213–230

    Article  MathSciNet  MATH  Google Scholar 

  43. Gagneux G, Millet O (2014) Homogenization of the Nernst–Planck–Poisson system by two-scale convergence. J Elast 114(1):69–84

    Article  MathSciNet  MATH  Google Scholar 

  44. Samson E, Marchand J, Beaudoin JJ (1999) Describing ion diffusion mechanisms in cement-based materials using the homogenization technique. Cem Concr Res 29(8):1341–1345

    Article  Google Scholar 

  45. Sasidhar V, Ruckenstein E (1981) Electrolyte osmosis through capillaries. J Colloid Interface Sci 82(2):439–457

    Article  Google Scholar 

  46. Sasidhar V, Ruckenstein E (1982) Anomalous effects during electrolyte osmosis across charged porous membranes. J Colloid Interface Sci 85(2):332–362

    Article  Google Scholar 

  47. Millet O, Aït-Mokhtar A, Amiri O (2008) Determination of the macroscopic chloride diffusivity in cementitious porous materials by coupling periodic homogenization of Nernst–Planck equation with experimental protocol. Int J Multiphys 2:129–145

    Article  Google Scholar 

  48. Bourbatache K, Millet O, Ait-Mokhtar A, Amiri O (2013) Chloride transfer in cement-based materials. Part 1. Theoretical basis and modelling. Int J Numer Anal Methods Geomech 37:1614–1627

    Article  Google Scholar 

  49. Hocine T, Amiri O, Aït-Mokhtar A, Pautet A (2012) Influence of cement, aggregates and chlorides on zeta potential of cement-based materials. Adv Cem Res 24:337–348

    Article  Google Scholar 

  50. Bourbatache K, Millet O, Ait-Mokhtar A, Amiri O (2013) Chloride transfer in cement-based materials. Part 2. Experimental study and numerical simulations. Int J Numer Anal Methods Geomech 37:1628–1641

    Article  Google Scholar 

  51. Aït-Mokhtar A, Amiri O, Dumargue P, Sammartino S (2002) A new model to calculate water permeability of cement-based materials from mip results. Adv Cem Res 14:43–49

    Article  Google Scholar 

  52. Delmi M, Aït-Mokhtar A, Amiri O (2006) Modelling the coupled evolution of hydration and porosity of cement-based materials. Constr Build Mater 20(7):504–514

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LGCGM EA 3913, INSA Rennes, Avenue des Buttes de Coesmes, 35708, Rennes, France

    K. Bourbatache

  2. LaSIE UMR 7356, University of La Rochelle-CNRS, Avenue Michel Crépeau, 17042, La Rochelle Cedex, France

    O. Millet & A. Aït-Mokhtar

Authors
  1. K. Bourbatache
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. Millet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Aït-Mokhtar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Bourbatache.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bourbatache, K., Millet, O. & Aït-Mokhtar, A. Multi-scale periodic homogenization of ionic transfer in cementitious materials. Heat Mass Transfer 52, 1489–1499 (2016). https://doi.org/10.1007/s00231-015-1667-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-015-1667-3

Keywords

Search

Navigation