Acta Mechanica Sinica

, Volume 29, Issue 5, pp 682–698 | Cite as

A micromechanical model for effective conductivity in granular electrode structures

  • Julia Ott
  • Benjamin Völker
  • Yixiang Gan
  • Robert M. McMeeking
  • Marc Kamlah
Research Paper


Optimization of composition and microstructure is important to enhance performance of solid oxide fuel cells (SOFC) and lithium-ion batteries (LIB). For this, the porous electrode structures of both SOFC and LIB are modeled as a binary mixture of electronic and ionic conducting particles to estimate effective transport properties. Particle packings of 10 000 spherical, binary sized and randomly positioned particles are created numerically and densified considering the different manufacturing processes in SOFC and LIB: the sintering of SOFC electrodes is approximated geometrically, whereas the calendering process and volume change due to intercalation in LIB are modeled physically by a discrete element approach. A combination of a tracking algorithm and a resistor network approach is developed to predict the connectivity and effective conductivity for the various densified structures. For SOFC, a systematic study of the influence of morphology on connectivity and conductivity is performed on a large number of assemblies with different compositions and particle size ratios between 1 and 10. In comparison to percolation theory, an enlarged percolation area is found, especially for large size ratios. It is shown that in contrast to former studies the percolation threshold correlates to varying coordination numbers. The effective conductivity shows not only an increase with volume fraction as expected but also with size ratio. For LIB, a general increase of conductivity during the intercalation process was observed in correlation with increasing contact forces. The positive influence of calendering on the percolation threshold and the effective conductivity of carbon black is shown. The anisotropy caused by the calendering process does not influence the carbon black phase.


Granular electrode structures Effective conductivity Percolation 



initial Li+ concentration


momentary Li+ concentration


maximum Li+ concentration




normal force


tangential force


flux respectively current


conductivity matrix


conductivity of bulk material


effective conductivity of granular structure


box length


normal unit vector


percolation probability


packing factor


resistance between 2 particles


resistance of a cylinder with r p and δ


initial particle radius


contact radius


momentary particle radius






tangential unit vector






position of particle i


overall coordination number


number of contacts of a l-particle to other l-particles

Greek symbols


distance between two particles





Poisson’s ratio




volume fraction




partial molar volume



active material


bulk property


carbon black


effective property


particle label






small particles


momentary state


initial state


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Awarke, A., Lauer, S., Pischinger, S., et al.: Percolationtunneling modeling for the study of electric conductivity in LiFePO4 based Li-ion battery cathodes. Journal of Power Sources 196, 405–411 (2011)CrossRefGoogle Scholar
  2. 2.
    Chen, D., Lin, Z., Zhu, H., et al.: Percolation theory to predict effective properties of solid oxide fuel-cell composite electrodes. Journal of Power Sources 191, 240–252 (2009)CrossRefGoogle Scholar
  3. 3.
    Völker, B., McMeeking, R.M.: Impact of particle size ratio and volume fraction on effective material parameters and performance in solid oxide fuel cell electrodes. Journal of Power Sources 215, 199–215 (2012)CrossRefGoogle Scholar
  4. 4.
    Torquato, S.: Random heterogeneous materials: Microstructure and macroscopic properties. In: Interdisciplinary Applied Mathematics. Springer, New York (2002)Google Scholar
  5. 5.
    Ferguson, T.R., Bazant, M.Z.: Nonequilibrium thermodynamics of porous electrodes. Journal of Electrochemical Society 159, A1967–A1985 (2012)CrossRefGoogle Scholar
  6. 6.
    Bouvard, D., Lange, F.: Relation between percolation and particle coordination in binary powder mixtures. Acta Metall. Mater. 39, 3083–3090 (1991)CrossRefGoogle Scholar
  7. 7.
    Costamagna, P., Costa, P., Antonucci, V.: Micro-modelling of solid oxide fuel cell electrodes. Electrochimica Acta 43, 375–394 (1998)CrossRefGoogle Scholar
  8. 8.
    Choi, H.W., Gawel, D., Berson, A., et al.: Comparison between FIB-SEM experimental 3-d reconstructions of SOFC electrodes and random particle-based numerical models. ECS Transactions 35, 997–1005 (2011)Google Scholar
  9. 9.
    Schneider, L., Martin, C., Bultel, Y., et al.: Discrete modelling of the electrochemical performance of SOFC electrodes. Electrochimica Acta 52, 314–324 (2006)CrossRefGoogle Scholar
  10. 10.
    Abel, J., Kornyshev, A., Lehnert, W.: Correlated resistor network study of porous solid oxide fuel cell anodes. Journal of Electrochemical Eociety 144, 4253–4259 (1997)CrossRefGoogle Scholar
  11. 11.
    Chen, Y.H., Wang, C.W., Liu, G., et al.: Selection of conductive additives in liion battery cathodes. Journal of the Electrochemical Society 154, A978–A986 (2007)CrossRefGoogle Scholar
  12. 12.
    Völker, B., McMeeking, R.M.: The effect of pore-former particles on microstructural features and electrochemical performance in solid oxide fuel cell electrodes. Journal of Power Sources 4, 15.1–15.23 (2013)Google Scholar
  13. 13.
    Sunde, S.: Simulations of composite electrodes in fuel cells. Journal of Electoceramics 5, 153–182 (2000)CrossRefGoogle Scholar
  14. 14.
    Doyle, C.M.: Design and simulation of lithium rechargeable batteries. [Ph.D. Thesis], University of California, Berkeley (1995)CrossRefGoogle Scholar
  15. 15.
    Zinchenko, A.Z.: Algorithm for random close packing of spheres with periodic boundary conditions. Journal of Computational Physics 114, 298–307 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jodrey, W., Tory, E.: Computer simulation of close random packing of equal spheres. Pysical Review A 32, 2347–2351 (1985)CrossRefGoogle Scholar
  17. 17.
    Torquato, S., Truskett, T.M., Debenedetti, P.: Is random close packing of spheres well defined? Physical Review Letters 84, 2064–2067 (2000)CrossRefGoogle Scholar
  18. 18.
    Xu, N., Blawzdziewicz, J., O’Hern, C.S.: Random close packing revisited: Ways to pack frictionless disks. Physical Review E 71, 061, 306-1–061, 306-9 (2005)Google Scholar
  19. 19.
    Gan, Y., Kamlah, M., Reimann, J.: Computer simulation of packing structure in pebble beds. Fusion Engineering and Design 85, 1782–1787 (2010)CrossRefGoogle Scholar
  20. 20.
    Lanzini, A., Leone, P., Asinari, P.: Microstructural characterization of solid oxide fuel cell electrodes by image analysis technique. Journal of Power Sources (2009)Google Scholar
  21. 21.
    David, W., Thackery, M., Picciotto, L.D., et al.: Structure refinement of the spinel-related phases Li2Mn2O4 and Li0.2Mn2O4. Journal of Solid State Chemistry 67, 316–323 (1987)CrossRefGoogle Scholar
  22. 22.
    Beattie, S., Larcher, D., Morcrette, M., et al.: Si electrodes for Li-ion batteries-a new way to look at an old problem. Journal of Electrochemical Society 155, A158–A163 (2008)CrossRefGoogle Scholar
  23. 23.
    Zheng, H., Tan, L., Liu, G., et al.: Calendering effects on the physical and electrochemical properties of Li[Ni1=3Mn1=3Co1=3]O2 cathode. Journal of Power Sources 208, 52–57 (2012)CrossRefGoogle Scholar
  24. 24.
    Gan, Y., Kamlah, M.: Discrete element modelling of pebble beds: With application to uniaxial compression tests of ceramic breeder pebble beds. Journal of Mechanics and Physics of Solids 58, 129–144 (2010)CrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang, X., Shyy, W., Sastry, A.: Numerical simulation of intercalation-induced stress in Li-ion battery electrode particles. Journal of the Electrochemical Society 154, A910–A916 (2007)CrossRefGoogle Scholar
  26. 26.
    Leisen, D.: Nanoindentierung als Methode zur mikromechanischen Charakterisierung von Li-Batteriewerkstoffen. [Master Thesis], Karlsruhe Institute of Technology (KIT), German (2010)Google Scholar
  27. 27.
    Al-Futaisi, A., Patzek, T.: Extension of Hoshen-Kopelman algorithm to non-lattice environments. Physica A 231, 665–678 (2003)CrossRefGoogle Scholar
  28. 28.
    Metzger, T., Irawan, A., Tsotsas, E.: Remarks on the paper “Extension of Hoshen-Kopelman algorithm to non-lattice environments” by A. Al-Futaisi and T.W. Patzek, Physica A 321 (2003) 665–678. Physica A 363, 558–560 (2006)CrossRefGoogle Scholar
  29. 29.
    Hoshen, J., Kopelman, R.: Percolation, cluster distribution. I. Cluster multiple labeling technique, critical concentration algorithm. Phys. Rev. B 14, 3438–3445 (1976)CrossRefGoogle Scholar
  30. 30.
    Argento, C., Bouvard, D.: Modeling the effective thermal conductivity of random packing of spheres through densification. International Journal of Heat and Mass Transfer 39, 1343–1350 (1996)CrossRefzbMATHGoogle Scholar
  31. 31.
    Frohne, H., Moeller, F.: Moeller Grundlagen der Elektrotechnik, (22th edn.) Studium. Vieweg + Teubner, Wiesbaden (2011)CrossRefGoogle Scholar
  32. 32.
    Jacob, B., Guennebaud, G.: Eigen.
  33. 33.
    Carson, J.K., Lovatt, S.J., Tanner, D.J., et al.: Thermal conductivity bounds for isotropic, porous materials. International Journal of Heat and Mass Transfer 48, 2150–2158 (2005)CrossRefzbMATHGoogle Scholar
  34. 34.
    Tobochnik, J., Lain, D., Wilson, G.: Randon-walk calculations of conductivity in continuum percolation. Physical Review A 41, 3052–3058 (1990)CrossRefGoogle Scholar
  35. 35.
    Bertei, A., Nicolella, C.: A comparative study and an extended theory of percolation for random packings of rigid spheres. Powder Technology 213, 100–108 (2011)CrossRefGoogle Scholar
  36. 36.
    Kuo, C.H., Gupta, P.K.: Rigidity and conductivity percolation threshold in particulate composites. Acta Metall. Mater. 43, 397–403 (1995)Google Scholar
  37. 37.
    Dominko, R., Gaberšček, M., Drofenik, J., et al.: Influence of carbon black distribution on performance of oxide cathodes for Li ion batteries. Electrochimica Acta 48, 3709–3716 (2003CrossRefGoogle Scholar
  38. 38.
    Dominko, R., Gaberšček, M., Drofenik, J., et al.: A novel coating technology for preparation of cathodes in Li-ion batteries. Electrochem. Solid-State Letters 4, 187–A190 (2001CrossRefGoogle Scholar
  39. 39.
    Tarascon, J., Guyomard, D.: The “Li1 + xMn2O4/C” rockingchair system: A review. Electrochimica Acta 38, 1221–1231 (1993)CrossRefGoogle Scholar
  40. 40.
    Hellweg, B.: Microstructural modeling of lithium battery electrodes. [Master Thesis], Massachusetts Institute of Technology, France (2000)Google Scholar
  41. 41.
    Martin, C., Bouvard, D.: Isostatic compaction of bimodal powder mixtures and composites. International Journal of Mechanical Sciences 46, 907–927 (2004)CrossRefzbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Julia Ott
    • 1
  • Benjamin Völker
    • 2
  • Yixiang Gan
    • 3
  • Robert M. McMeeking
    • 2
    • 4
    • 5
    • 6
  • Marc Kamlah
    • 1
  1. 1.Institute of Applied MaterialsKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Department of Mechanical EngineeringUniversity of CaliforniaSanta BarbaraUSA
  3. 3.School of Civil EngineeringThe University of SydneySydneyAustralia
  4. 4.Materials DepartmentUniversity of CaliforniaSanta BarbaraUSA
  5. 5.School of EngineeringUniversity of Aberdeen, King’s College, AberdeenScotlandUK
  6. 6.INM — Leibniz Institute for New MaterialsSaarbr⩊kenGermany

Personalised recommendations