Granular Matter

, 20:14 | Cite as

Characterisation of pendular capillary bridges derived from experimental data using inverse problem method

  • B. Mielniczuk
  • O. Millet
  • G. Gagneux
  • M. S. El Youssoufi
Original Paper


In this study we use the recent analytical model to analyze capillary interactions in liquid bridge between two spherical grains, with fixed volumes of liquid and varying separation distance. Sequences of images of capillary bridges with different parameters are recorded during experimental tests. Geometrical parameters, as contact angle, half-filling angle and neck radius, are determined by image processing. Profiles of examined bridges are approximated as a Delaunay’s roulette and superposed on recorded images. Evolution of associated variables (Laplace pressure, capillary force) is also calculated. Results of theoretical modeling are compared with the experimental ones. They match very accurately for small volumes and/or small separation distances, when influence of gravity is not significant. For larger liquid volumes and/or larger separation distances between grains the influence of the gravity is observed as a distortion (loss of symmetry) of capillary bridge. To avoid this deformation, several test were realized in micro-gravity conditions. For these tests, theoretical results are in good agreement with experimental ones, also for higher liquid volumes and/or separations distances.


Capillary bridge Young–Laplace equation Inverse problem Experimental measurement 



The part of the work concerning the experiments in micro-gravity was supported by the CNES (joint project PARABOLE 2015, VP118, between LaSIE-LMGC-CNES).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Peron, H., Delenne, J.Y., Laloui, L., El Youssoufi, M.S.: Discrete element modelling of drying shrinkage and cracking of soils. Comput. Geotech. 36, 61 (2009)CrossRefGoogle Scholar
  2. 2.
    Princen, H.M.: Comments on ’The effect of capillary liquid on the force of adhesion between spherical solid particles’. J. Colloid Interface Sci. 26, 249 (1968)ADSCrossRefGoogle Scholar
  3. 3.
    Rahardjo, H., Fredlund, D.G.: Experimental verification of the theory of consolidation for unsaturated soils. Can. Geotech. J. 32, 749 (1995)CrossRefGoogle Scholar
  4. 4.
    El Youssoufi, M.S., Delenne, J.Y., Radjai, F.: Self-stresses and crack formation by particle swelling in cohesive granular media. Phys. Rev. E 71, 051307 (2005)ADSCrossRefGoogle Scholar
  5. 5.
    Richefeu, V., El Youssoufi, M.S., Radjai, F.: Shear strength properties of wet granular materials. Phys. Rev. E 73, 051304 (2006)ADSCrossRefGoogle Scholar
  6. 6.
    Flemmer, C.L.: On the regime boundaries of moisture in granular materials. Powder Technol. 66, 191 (1991)CrossRefGoogle Scholar
  7. 7.
    Herminghaus, S.: Dynamics of wet granular matter. Adv. Phys. 54, 221 (2005)ADSCrossRefGoogle Scholar
  8. 8.
    Young, T.: An essay on the cohesion of fluids. Philos. Trans. 95, 65 (1805). CrossRefGoogle Scholar
  9. 9.
    Plateau, J.: The Annual Report of the Smithsonian Institution, Washington DC pp. 338–369 (1864)Google Scholar
  10. 10.
    Mason, G., Clark, W.C.: Liquid bridges between spheres. Chem. Eng. Sci. 20, 859 (1965)CrossRefGoogle Scholar
  11. 11.
    Haines, W.B.: Studies of the physical properties of soils. II. A note on the cohesion developed by capillarity forces in an ideal soil. J. Agric. Sci. 15, 529 (1925)CrossRefGoogle Scholar
  12. 12.
    Megias-Alguacil, D., Gauckler, L.J.: Capillary forces between two solid spheres linked by a concave liquid bridge: regions of existence and forces mapping. AiChE J. 55, 1103 (2009)CrossRefGoogle Scholar
  13. 13.
    Rabinovich, Y.I., Esayanur, M.S., Moudgil, B.M.: Capillary forces between two spheres with a fixed volume liquid bridge: theory and experiment. Langmuir 21, 10992 (2005)CrossRefGoogle Scholar
  14. 14.
    Gagneux, G., Millet: An analytical framework for evaluating the cohesion effects of coalescence between capillary bridges. Granular Matter 18 (2016).
  15. 15.
    Lian, G., Thornton, C., Adams, M.J.: A theoretical study of the liquid bridge forces between two rigid spherical bodies. J. Colloid Interface Sci. 161, 138 (1993)ADSCrossRefGoogle Scholar
  16. 16.
    Willett, C.D., Adams, M.J., Johnson, S.A., Seville, J.P.K.: Capillary bridges between two spherical bodies. Langmuir 16, 9396 (2000)CrossRefGoogle Scholar
  17. 17.
    Soulie, F., Cherblanc, F., El Youssoufi, M.S., Saix, C.: Influence of liquid bridges on the mechanical behaviour of polydisperse granular materials. Int. J. Numer. Anal. Meth. Geomech. 30, 213 (2006)CrossRefMATHGoogle Scholar
  18. 18.
    Gras, J.P., Delenne, J.Y., El Youssoufi, M.S.: Study of capillary interaction between two grains: a new experimental device with suction control. Granul. Matter 15, 49 (2013). CrossRefGoogle Scholar
  19. 19.
    Gagneux, G., Millet, O.: Analytic calculation of capillary bridge properties deduced as an inverse problem from experimental data. Transp. Porous Media 105, 117 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hueckel, T., Mielniczuk, B., El Youssoufi, M.S.: Micro-scale study of rupture in desiccating granular media. In: Proceedings of Geo-Congress 2013, Geotechnical Special Publication GSP ASCE, vol. 231, p. 808 (2013)Google Scholar
  21. 21.
    Mielniczuk, B., Hueckel, T., El Youssoufi, M.S.: Micro-scale testing of capillary bridge evolution due to evaporation. In: Laloui, L, Ferrari, A (eds.) Multiphysical Testing of Soils and Shales, Springer Series in Geomechanics and Geoengineering, pp. 233–238 (2013)Google Scholar
  22. 22.
    Mielniczuk, B., El Youssoufi, M.S., Sabatier, L., Hueckel, T.: Rupture of a liquid bridge between two grains due to its evaporation. Acta Geophys. 62, 1087 (2014). ADSCrossRefGoogle Scholar
  23. 23.
    Mielniczuk, B., Hueckel, T., El Youssoufi, M.S.: Evaporation-induced evolution of the capillary force between two grains. Granul. Matter 16, 815 (2014)CrossRefGoogle Scholar
  24. 24.
    Dell’Isola, F., Gouin, H., Rotoli, G.: Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations. Eur. J. Mech. B/Fluids 15(4), 545 (1996)MATHGoogle Scholar
  25. 25.
    Madeo, A., dell’Isola, F., Darve, F.: A continuum model for deformable, second gradient porous media partially saturated with compressible fluids. J. Mech. Phys. Solids 61, 82196 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Duriez, J., Eghbalian, M., Wan, R., Darve, F.: The micromechanical nature of stresses in triphasic granular media with interfaces. J. Mech. Phys. Solids (2016).
  27. 27.
    Delaunay, C.H.: Sur la surface de révolution dont la courbure moyenne est constante. Journal de mathématiques pures et appliquées 6, 309 (1841)Google Scholar
  28. 28.
    Mielniczuk, B., Hueckel, T., El Youssoufi, M.S.: Laplace pressure evolution and four instabilities in evaporating two-grain liquid bridges. Powder Technol. 283, 131 (2015). CrossRefGoogle Scholar
  29. 29.
    Gagneux, G., Millet, O., Mielniczuk, B., El Youssoufi, M.S.: Theoretical and experimental study of pendular regime in unsaturated granular media. Eur. J. Environ. Civil Eng. pp. 1–14 (2016).
  30. 30.
    Bourges-Monnier, C., Shanahan, M.E.R.: Influence of evaporation on contact angle. Langmuir 11, 2820 (1995). CrossRefGoogle Scholar
  31. 31.
    Adams, M.J., Johnson, S.A., Seville, J.P.K., Willett, C.: Mapping the influence of gravity on pendular liquid bridges between rigid spheres. Langmuir 18, 6180 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • B. Mielniczuk
    • 1
    • 2
    • 3
  • O. Millet
    • 1
  • G. Gagneux
    • 1
  • M. S. El Youssoufi
    • 2
    • 3
  1. 1.LaSIE, UMR CNRS 7356University of La RochelleLa Rochelle Cedex 1France
  2. 2.LMGC UMR CNRS 5508University of MontpellierMontpellier Cedex 5France
  3. 3.MIST, University of Montpellier, CNRS, IRSNMontpellierFrance

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