Skip to main content
SpringerLink
Log in

Multiscale modelling and analysis of lithium-ion battery charge and discharge

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

A microscopic model of a lithium battery is developed, which accounts for lithium diffusion within particles, transfer of lithium from particles to the electrolyte and transport within the electrolyte assuming a dilute electrolyte and Butler–Volmer reaction kinetics. Exploiting the small size of the particles relative to the electrode dimensions, a homogenised model (in agreement with existing theories) is systematically derived and studied. Details of how the various averaged quantities relate to the underlying geometry and assumptions are given. The novel feature of the homogenisation process is that it allows the coefficients in the electrode-scale model to be derived in terms of the microscopic features of the electrode (e.g. particle size and shape) and can thus be used as a systematic way of investigating the effects of changes in particle design. Asymptotic methods are utilised to further simplify the model so that one-dimensional behaviour can be described with relatively simpler expressions. It is found that for low discharge currents, the battery acts almost uniformly while above a critical current, regions of the battery become depleted of lithium ions and have greatly reduced reaction rates leading to spatially nonuniform use of the electrode. The asymptotic approximations are valid for electrode materials where the OCV is a strong function of intercalated lithium concentration, such as Li x C6, but not for materials with a flat discharge curve, such as LiFePO4.

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

Similar content being viewed by others

References

  1. Newman JS (2004) Electrochemical systems, 3rd edn. Prentice Hall, New Jersey

    Google Scholar 

  2. Botte GG, Subramanian VR, White RE (2000) Mathematical modeling of secondary lithium batteries. Electrochim Acta 45: 2595–2609

    Article  Google Scholar 

  3. Dargaville S, Farrell TW (2010) Predicting active material utilization in LiFePO4 electrodes using a multiscale mathematical model. J Electrochem Soc 157: A830–A840

    Article  Google Scholar 

  4. Doyle M, Newman J (1996) Analysis of capacity-rate for lithium batteries using simplified models of the discharge process. J Appl Electrochem 27: 846–856

    Article  Google Scholar 

  5. Doyle M, Newman J (1996) Comparison of modeling predictions with experimental data from plastic lithium ion cells. J Electrochem Soc 143: 1890–1903

    Article  Google Scholar 

  6. Farell TW, Please CP (2005) Primary alkaline battery cathodes a simplified model for porous manganese oxide particle discharge. J Electrochem Soc 152: A1930–A1941

    Article  Google Scholar 

  7. Fuller TF, Doyle M, Newman J (1994) Simulation and optimisation of the dual lithium ion insertion cell. J Electrochem Soc 141: 1–10

    Article  Google Scholar 

  8. Fuller TF, Doyle M, Newman J (1994) Relaxation phenomena in lithium-ion-insertion cells. J Electrochem Soc 141: 982–990

    Article  Google Scholar 

  9. Johansen JF, Farrell TW, Please CP (2006) Modelling of primary alkaline battery cathodes: a simplified model. J Power Sources 156: 645–654

    Article  Google Scholar 

  10. Pop V, Bergveld HJ, Danilov D, Notten PHL (2008) Battery management systems: accurate state-of-charge indication for battery-powered applications (Philips research book series). Springer, Berlin

    Google Scholar 

  11. Srinivasan V, Wang CY (2003) Analysis of electrochemical and thermal behaviour of Li-ion cells. J Electrochem Soc 150: A98–A106

    Article  Google Scholar 

  12. Owen JR (1997) Rechargeable lithium batteries. Chem Soc Rev 26: 259–267

    Article  MathSciNet  Google Scholar 

  13. Levi MD, Salitra G, Markovsky B, Teller H, Aurbach D, Heider U, Heider L (1999) Solid-state elecrochemical kinetics of Li-ion intercalation into Li1-x CoO2: simultaneous application of electroanalytical techniques SSCV, PITT and EIS. J Electrochem Soc 146: 1279–1289

    Article  Google Scholar 

  14. Kobayashi S, Uchimoto Y (2005) Lithium ion phase-transfer reaction at the interface between the lithium manganese oxide electrode and the nonaqueous electrolyte. J Phys Chem 109: 13322–13326

    Article  Google Scholar 

  15. Whittingham MS (2004) Lithium battery cathode materials. Chem Rev 104: 4271–4301

    Article  Google Scholar 

  16. Kang B, Meng YS, Berger J, Grey CP, Ceder G (2006) Electrodes with high power and high capacity for rechargeable lithium batteries. Science 311: 977–980

    Article  ADS  Google Scholar 

  17. Padhi AK, Nanjundaswamy K, Masquelier C, Goodenough JB (1997) Mapping of transition metal redox energies in phosphates with NASICON structure by lithium intercalation. J Electrochem Soc 144: 2581–2586

    Article  Google Scholar 

  18. Islam MS, Driscoll DJ, Fisher CAJ, Slater PR (2005) Atomic-scale investigation of defects, dopants, and lithium transport in the LiFePO4 olivine-type battery material. Chem Mater 17: 5085–5092

    Article  Google Scholar 

  19. Singh G, Ceder G, Bazant MZ (2008) Intercalation dynamics in rechargeable battery materials: general theory and phase-transformation waves in LiFePO4. Electrochim Acta 53: 7599–7613

    Article  Google Scholar 

  20. Morales J, Trcoli R, Franger S, Santos-Pea J (2010) Cycling-induced stress in lithium ion negative electrodes: LiAl/LiFePO4 and Li4Ti5O12/LiFePO4 cells. J Electrochem Acta 9: 3075–3082

    Article  Google Scholar 

  21. Notten PHL, Roozeboom F, Regtien PPL, Niessen RAH, Baggetto L (2007) 3-D integrated all-solid-state rechargeable batteries. Adv Mater 19: 4564–4567

    Article  Google Scholar 

  22. Taberna PL, Mitra S, Poizot P, Simon P, Tarascon J-M (2006) High rate capabilities Fe3O4-based Cu nano-architectured electrodes for lithium-ion battery applications. Nat Mater 5: 567–573

    Article  ADS  Google Scholar 

  23. Whitehead AH, Elliott JM, Owen JR (1999) Nanostructured tin for use as a negative electrode material in Li-ion batteries. J Power Sources 81(82): 33–38

    Article  Google Scholar 

  24. Chen J-S, Cheh HY (1993) Modelling of cylindrical alkaline cells: IV Dissolution-precipitation model for the anode. J Appl Electrochem 140: 1213–1218

    Google Scholar 

  25. Johns PA, Roberts MR, Wakizaka Y, Sanders JH, Owen JR (2010) How the electrolyte limits fast discharge in nanostructured batteries and supercapacitors. Electrochem Commun 11: 2089–2092

    Article  Google Scholar 

  26. Richardson G, King JR (2007) Time-dependent modelling and asymptotic analysis of electrochemical cells. J Eng Math 59: 239–275

    Article  MathSciNet  MATH  Google Scholar 

  27. Srinivasan V, Newman J (2004) Discharge model for the lithium iron-phosphate electrode. J Electrochem Soc 151: A1517–A1529

    Article  Google Scholar 

  28. Galinski M, Lewandowski A, Stepniak I (2006) Ionic liquids as electrolytes. Electrochim Acta 51: 5567–5580

    Article  Google Scholar 

  29. Bang HJ, Donepudi VS, Prakash J (2002) Preparation and characterization of partially substituted Li My Mn2-yO4 (M = Ni, Co, Fe) spinel cathodes for Li-ion batteries. Electrochim Acta 48: 443–451

    Article  Google Scholar 

  30. Lee SH, Cheong HM, Tracy CE, Maschernhas A, Pitts JR, Jorgensen G, Deb SK (2000) Alternating current impedence and Raman spectroscopic study on electrochromic a-WO3 films. Appl Phys Lett 76: 3908–3910

    Article  ADS  Google Scholar 

  31. Barker J, Pynenburg R, Koksbang R, Saidi MY (1996) An electrochemical investigation into the lithium insertion properties of Li x CoO2. Electrochim Acta 41: 2481–2488

    Article  Google Scholar 

  32. Levi MD, Aurbach D (1997) Diffusion coefficients of lithium ions during intercalation into graphite derived from the simultaneous measurements and modeling of electrochemical impedance and potentiostatic intermittent titration of thin graphite electrodes. J Phys Chem 101: 4641–4647

    Article  Google Scholar 

  33. Cole JD (1995) Limit process expansions and homogenization. SIAM J Appl Maths 55: 410–424

    Article  ADS  MATH  Google Scholar 

  34. Ptashnyk M, Roose T (2010) Derivation of macroscopic model for transport of strongly sorbed solutes in the soil using homogenization techniques. SIAM J Appl Math 70: 2097–2118

    Article  MathSciNet  MATH  Google Scholar 

  35. Ptashnyk M, Roose T, Kirk GJD (2010) Diffusion of strongly sorbed solutes in soil: a dual porosity model allowing for slow access to sorption sites and time-dependent sorption reactions. Eur J Soil Sci 61: 108–119

    Article  Google Scholar 

  36. Wang CY, Gu WB, Liaw BY (1998) Micro-macroscopic coupled modeling of batteries and fuel cells. J Electrochem Soc 145: 3407–3417

    Article  Google Scholar 

  37. Atkins P, de Paula J (2001) Atkins’ physical chemistry, 7th edn. OUP, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK

    G. Richardson & C. P. Please

  2. School of Chemistry, University of Southampton, Southampton, SO17 1BJ, UK

    G. Denuault

Authors
  1. G. Richardson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Denuault
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. P. Please
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Richardson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Richardson, G., Denuault, G. & Please, C.P. Multiscale modelling and analysis of lithium-ion battery charge and discharge. J Eng Math 72, 41–72 (2012). https://doi.org/10.1007/s10665-011-9461-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-011-9461-9

Keywords

Search

Navigation