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


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.


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

List of symbols


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


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

\(A^i \)

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

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

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


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


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


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


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


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


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


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


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 \((\text {kg}/\text {s}\text {m})\)


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


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


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


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


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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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