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Continuum Mechanics and Thermodynamics

, Volume 30, Issue 3, pp 593–628 | Cite as

Stochastic many-particle model for LFP electrodes

  • Clemens Guhlke
  • Paul Gajewski
  • Mario Maurelli
  • Peter K. Friz
  • Wolfgang Dreyer
Original Article
  • 84 Downloads

Abstract

In the framework of non-equilibrium thermodynamics, we derive a new model for many-particle electrodes. The model is applied to \(\text {LiFePO}_{4}\) (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithium-poor to a lithium-rich phase within LFP electrodes is controlled by both different particle sizes and surface fluctuations leading to a system of stochastic differential equations. An explicit relation between battery voltage and current controlled by the thermodynamic state variables is derived. This voltage–current relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate-limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate-limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltage–charge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates.

Keywords

Lithium-ion battery Lithium iron phosphate Phase transitions Many-particle electrode 

List of symbols

U

Reference potential \((\text {V})\)

\(k_{\text {Li}}\)

Lithium intercalation rate \( (\text {kg}/\text {m}^2\text {s})\)

\(j_\mathtt {P}\)

Exchange current \((\text {A}/\text {m}^2)\)

\(\nu _0\)

Stochastic strength \((\text {m}^{\frac{3}{2}})\)

L

Heat of solution \((\text {J})\)

\(A^i \)

Particle surface area \((\text {m}^2)\)

\(A_\mathtt {E}^i\)

Active surface area \((\text {m}^2)\)

\(V^i\)

Particle volume \((\text {m}^3)\)

\(V_\mathtt {P}\)

Total volume \((\text {m}^3)\)

\(A_\mathtt {E}\)

Total active area \((\text {m}^2)\)

\(y^i\), \(Y^i\)

Lithium mole fraction

\(\tau ^i\)

Relaxation time \((\text {s})\)

\(\nu ^i\)

Stochastic strength

\(k_\mathrm{B}\)

Boltzmann constant \((\text {J}/\text {K})\)

\(e_0\)

Elementary charge \((\text {C})\)

\(\varepsilon _0\)

Electric constant \([\text {C}/(\text {V}\, \text {m})]\)

\(z_\alpha \)

Charge number

\(m_\alpha \)

Molecular mass \((\text {kg})\)

\(\gamma _{\alpha }^i\), \(\gamma _{\mathrm{s},\alpha }^i\)

Stoichiometric coef. bulk and surface reactions

\(\varvec{\nu }\)

Normal vector

\(k_M\)

Mean curvature \((1/\text {m})\)

T,\(T_\mathrm{s}\)

Bulk and surface temperature \((\text {K})\)

\(n_\alpha \)

Bulk number density \((\text {m}^{-3})\)

\(n_{\mathrm{s},\alpha }\)

Surface number density \((\text {m}^{-2})\)

\(\rho _\alpha \)

Bulk mass density \((\text {kg}/\text {m}^{3})\)

\(\rho _{\mathrm{s},\alpha }\)

Surface number density \((\text {kg}/\text {m}^{2})\)

\(\varvec{v}\), \(\varvec{v}_\mathrm{s}\)

Bulk and surface barycentric velocity \((\text {m}/\text {s})\)

\(\varvec{w}\)

Surface velocity \((\text {m}/\text {s})\)

\(\varvec{E}\)

Electric field \((\text {V}/\text {m})\)

\(\varphi \), \(\varphi _\mathrm{s}\)

Bulk and surface electrostatic potential \((\text {V})\)

\(n^\mathrm {F}\)

Charge density \((\text {C}/\text {m}^3)\)

\(n^\mathrm {F}_\mathrm{s}\)

Surface charge density \((\text {C}/\text {m}^2)\)

\(\rho \psi \)

Free energy density \((\text {J}/\text {m}^3)\)

\(\rho _\mathrm{s}\psi _\mathrm{s}\)

Surface free energy density \((\text {J}/\text {m}^2)\)

\(\mu _\alpha \), \(\mu _{\mathrm{s},\alpha }\)

Bulk and surface chemical potential \((\text {J}/\text {kg})\)

\(\varvec{\sigma }\)

Cauchy stress tensor \((\text {N}/\text {m}^2)\)

\(\varvec{\varSigma }\)

Total stress tensor \((\text {N}/\text {m}^2)\)

p

Material pressure \((\text {N}/\text {m}^2)\)

\(\gamma _\mathrm{s}\)

Surface tension \((\text {N}/\text {m})\)

\(\varvec{J}_\alpha \)

Mass flux density \((\text {kg}/\text {s}\text {m}^2)\)

\(\varvec{J}_{\mathrm{s},\alpha }\)

Surface mass flux density \((\text {kg}/\text {s}\text {m}^2)\)

\(\varvec{j}_\alpha \)

Total mass flux density \((\text {kg}/\text {s}\text {m}^2)\)

\(\varvec{J}_{\mathrm{s},\alpha }\)

Surface mass flux density \((\text {kg}/\text {s}\text {m})\)

\(\varvec{j}_{\mathrm{s},\alpha }\)

Total surface mass flux density \((\text {kg}/\text {s}\text {m})\)

\(R^i\)

Bulk reaction rate density \((1/\text {s}\text {m}^3)\)

\(R^i_\mathrm{s}\)

Surface reaction rate density \((1/\text {s}\text {m}^2)\)

\(r^i\)

Bulk mass production density \((\text {kg}/\text {s}\text {m}^3)\)

\(r^i_\mathrm{s}\)

Surface mass production density \((\text {kg}/\text {s}\text {m}^2)\)

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References

  1. 1.
    Bai, P., Cogswell, D., Bazant, M.: Suppression of phase separation in \(\text{ LiFePO }_4\) nanoparticles during battery discharge. Nano Lett. 11(11), 4890–4896 (2011)ADSCrossRefGoogle Scholar
  2. 2.
    Bazant, M.Z.: Theory of chemical kinetics and charge transfer based on nonequilibrium thermodynamics. Acc. Chem. Res. 46(5), 1144–1160 (2013)CrossRefGoogle Scholar
  3. 3.
    Bedeaux, D.: Nonequilibrium thermodynamics and statistical physics of surfaces. In: Ilya, P., Rice, S.A. (eds.) Advances in Chemical Physics, pp. 47–109. Wiley, New York (1986)Google Scholar
  4. 4.
    Bothe, D., Dreyer, W.: Continuum thermodynamics of chemically reacting fluid mixtures. Acta Mech. 226(6), 1757–1805 (2014)Google Scholar
  5. 5.
    Chueh, W., El Gabaly, F., Sugar, J., Bartelt, N., McDaniel, A., Fenton, K., Zavadil, K., Tyliszczak, T., Lai, W., McCarty, K.: Intercalation pathway in many-particle LiFePO\(_4\) electrode revealed by nanoscale state-of-charge mapping. Nano Lett. 13(3), 866–872 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    Costen, R., Adamson, D.: Three-dimensional derivation of the electrodynamic jump conditions and momentum-energy laws at a moving boundary. Proc. IEEE 53(9), 1181–1196 (1965)CrossRefGoogle Scholar
  7. 7.
    de Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. North Holland, Amsterdam (1963)zbMATHGoogle Scholar
  8. 8.
    Delmas, C., Maccario, M., Croguennec, L., Le Cras, F., Weill, F.: Lithium deintercalation in \(\text{ LiFePO }_4\) nanoparticles via a domino-cascade model. Nat. Mater. 7, 665–671 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    Dominko, R., Bele, M., Gaberšček, M., Remskar, M., Hanzel, D., Pejovnik, S., Jamnik, J.: Impact of the carbon coating thickness on the electrochemical performance of \(\text{ LiFePO }_{4}/\text{ C }\) composites. J. Electrochem. Soc. 152(3), A607–A610 (2005)CrossRefGoogle Scholar
  10. 10.
    Doyle, M., Fuller, T., Newman, J.: Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell. J. Electrochem. Soc. 140(6), 1526–1533 (1993)CrossRefGoogle Scholar
  11. 11.
    Doyle, M., Newman, J.: Analysis of capacity-rate data for lithium batteries using simplified models of the discharge process. J. Appl. Electrochem. 27(7), 846–856 (1997)CrossRefGoogle Scholar
  12. 12.
    Dreyer, W., Friz, P., Gajewski, P., Guhlke, C., Maurelli, M.: in preparationGoogle Scholar
  13. 13.
    Dreyer, W., Gaberšček, M., Guhlke, C., Huth, R., Jamnik, J.: Phase transition in a rechargeable lithium battery. Eur. J. Appl. Math. 22, 267–290 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dreyer, W., Guhlke, C., Herrmann, M.: Hysteresis and phase transition in many-particle storage systems. Contin. Mech. Thermodyn. 23(3), 211–231 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dreyer, W., Guhlke, C., Huth, R.: The behavior of a many-particle electrode in a lithium-ion battery. Physica D 240(12), 1008–1019 (2011)ADSCrossRefzbMATHGoogle Scholar
  16. 16.
    Dreyer, W., Guhlke, C., Müller, R.: Overcoming the shortcomings of the Nernst–Planck model. Phys. Chem. Chem. Phys. 15, 7075–7086 (2013)CrossRefGoogle Scholar
  17. 17.
    Dreyer, W., Guhlke, C., Müller, R.: Modeling of electrochemical double layers in thermodynamic non-equilibrium. Phys. Chem. Chem. Phys. 17, 27176–27194 (2015)CrossRefGoogle Scholar
  18. 18.
    Dreyer, W., Guhlke, C., Müller, R.: A new perspective on the electron transfer: recovering the Butler-Volmer equation in non-equilibrium thermodynamics. Phys. Chem. Chem. Phys. 18, 24966–24983 (2016)CrossRefGoogle Scholar
  19. 19.
    Dreyer, W., Huth, R., Mielke, A., Rehberg, J., Winkler, M.: Global existence for a nonlocal and nonlinear Fokker–Planck equation. Zeitschrift für angewandte Mathematik und Physik 66(2), 293–315 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dreyer, W., Jamnik, J., Guhlke, C., Huth, R., Moškon, J., Gaberšček, M.: The thermodynamic origin of hysteresis in insertion batteries. Nat. Mater. 9, 448–453 (2010)ADSCrossRefGoogle Scholar
  21. 21.
    Farkhondeh, M., Delacourt, C.: Mathematical modeling of commercial LiFePO\(_4\) electrodes based on variable solid-state diffusivity. J. Electrochem. Soc. 159(2), A177–A192 (2011)CrossRefGoogle Scholar
  22. 22.
    Farkhondeh, M., Pritzker, M., Fowler, M., Safari, M., Delacourt, C.: Mesoscopic modeling of li insertion in phase-separating electrode materials: application to lithium iron phosphate. Phys. Chem. Chem. Phys. 16, 22555–22565 (2014)CrossRefGoogle Scholar
  23. 23.
    Farkhondeh, M., Safari, M., Pritzker, M., Fowler, M., Han, T., Wang, J., Delacourt, C.: Full-range simulation of a commercial \(\text{ LiFePO }_4\) electrode accounting for bulk and surface effects: a comparative analysis. J. Electrochem. Soc. 161(3), A201–A212 (2014)CrossRefGoogle Scholar
  24. 24.
    Fournier, N., Hauray, M., Mischler, S.: Propagation of chaos for the 2D viscous vortex model. J. Eur. Math. Soc. (JEMS) 16(7), 1423–1466 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Franco, A.: Multiscale modelling and numerical simulation of rechargeable lithium ion batteries: concepts, methods and challenges. RSC Adv. 3, 13027–13058 (2013)CrossRefGoogle Scholar
  26. 26.
    Guhlke, C.: Theorie der elektrochemischen Grenzfläche. Ph.D. thesis, TU-Berlin (2015)Google Scholar
  27. 27.
    Han, B., Van der Ven, A., Morgan, D., Ceder, G.: Electrochemical modeling of intercalation processes with phase field models. Electrochim. Acta 49, 4691–4699 (2004)CrossRefGoogle Scholar
  28. 28.
    Hellwig, C., Sörgel, S., Bessler, W.: A multi-scale electrochemical and thermal model of a LiFePO\(_4\) battery. ECS Trans. 35(32), 215–228 (2011)CrossRefGoogle Scholar
  29. 29.
    Herrmann, M., Niethammer, B., Velázquez, J.: Kramers and non-Kramers phase transitions in many-particle systems with dynamical constraint. SIAM Multisc. Model. Simul. 10(3), 818–852 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Herrmann, M., Niethammer, B., Velázquez, J.: Rate-independent dynamics and Kramers-type phase transitions in nonlocal Fokker–Planck equations with dynamical control. Arch. Ration. Mech. Anal. 124(3), 803–866 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Latz, A., Zausch, J.: Thermodynamic consistent transport theory of Li-ion batteries. J. Power Sour. 196(6), 3296–3302 (2011)ADSCrossRefzbMATHGoogle Scholar
  32. 32.
    Latz, A., Zausch, J.: Thermodynamic derivation of a Butler–Volmer model for intercalation in Li-ion batteries. Electrochim. Acta 110, 358–362 (2013)CrossRefGoogle Scholar
  33. 33.
    Li, Y., El Gabaly, F., Ferguson, T., Smith, R., Bartelt, N., Sugar, J., Fenton, K., Cogswell, D., Kilcoyne, A., Tyliszczak, T., Bazant, M., Chueh, W.: Current-induced transition from particle-by-particle to concurrent intercalation in phase-separating battery electrodes. Nat. Mater. 13, 1476–1122 (2014)Google Scholar
  34. 34.
    Li, Y., Meyer, S., Lim, J., Lee, S., Gent, W., Marchesini, S., Krishnan, H., Tyliszczak, T., Shapiro, D., Kilcoyne, A., Chueh, W.: Effects of particle size, electronic connectivity, and incoherent nanoscale domains on the sequence of lithiation in LiFePO\(-4\) porous electrodes. Adv. Mater. 27(42), 6591–6597 (2015)CrossRefGoogle Scholar
  35. 35.
    Li, Y., Weker, J., Gent, W., Mueller, D., Lim, J., Cogswell, D., Tyliszczak, T., Chueh, W.: Dichotomy in the lithiation pathway of ellipsoidal and platelet \(\text{ LiFePO }_4\) particles revealed through nanoscale operando state-of-charge imaging. Adv. Funct. Mater. 25(24), 3677–3687 (2015)CrossRefGoogle Scholar
  36. 36.
    Meixner, J., Reik, H.G.: Thermodynamik der irreversiblen Prozesse, pp. 413–523. Springer, Berlin (1959)Google Scholar
  37. 37.
    Mielke, A., Truskinovsky, L.: From discrete visco-elasticity to continuum rate-independent plasticity: rigorous results. Arch. Ration. Mech. Anal. 203(2), 577–619 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Moskon, J., Dominko, R., Cerc-Korosec, R., Gaberšček, M., Jamnik, J.: Morphology and electrical properties of conductive carbon coatings for cathode materials. J. Power Sour. 174(2), 683–688 (2007)ADSCrossRefGoogle Scholar
  39. 39.
    Müller, I.: Thermodynamics, Interaction of Mechanics and Mathematics Series. Pitman Advanced Publishing Program, Boston (1985)Google Scholar
  40. 40.
    Padhi, A., Nanjundaswamy, K., Goodenough, J.: Phospho-olivines as positive-electrode materials for rechargeable lithium batteries. J. Electrochem. Soc. 144, 1188–1194 (1997)CrossRefGoogle Scholar
  41. 41.
    Safari, M., Delacourt, C.: Mathematical modeling of lithium iron phosphate electrode: galvanostatic charge/discharge and path dependence. J. Electrochem. Soc. 158(2), A63–A73 (2011)CrossRefGoogle Scholar
  42. 42.
    Singh, G., Ceder, G., Bazant, M.: Intercalation dynamics in rechargeable battery materials: general theory and phase-transformation waves in LiFePO\(_4\). Electrochim. Acta 53(26), 7599–7613 (2008)CrossRefGoogle Scholar
  43. 43.
    Srinivasan, V., Newman, J.: Discharge model for the lithium iron-phosphate electrode. J. Electrochem. Soc. 151(10), A1517–A1529 (2004)CrossRefGoogle Scholar
  44. 44.
    Sznitman, A.S.: Topics in propagation of chaos. In: École d’Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Mathematics, vol. 1464, pp. 165–251. Springer, Berlin (1991)Google Scholar

Copyright information

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

Authors and Affiliations

  • Clemens Guhlke
    • 1
  • Paul Gajewski
    • 1
  • Mario Maurelli
    • 2
  • Peter K. Friz
    • 2
  • Wolfgang Dreyer
    • 1
  1. 1.Weierstrass InstituteBerlinGermany
  2. 2.Institute of MathematicsBerlinGermany

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