WIAS Preprint No. 1423, (2009)

The behavior of a many particle cathode in a lithium-ion battery


  • Dreyer, Wolfgang
  • Guhlke, Clemens
  • Huth, Robert

2010 Mathematics Subject Classification

  • 74N30 74A15 74F20 74G65

2008 Physics and Astronomy Classification Scheme

  • 82.47.Aa 82.60.Hc 82.60.Qr 83.10.Tv


  • lithium-ion batteries, thermodynamics, phase transistion, hysteresis, many particle system, chemical potential, elasticity




We study the almost reversible storage process of charging and discharging of lithium-ion batteries. That process is accompanied by a phase transition and charging and discharging run along different paths, so that hysteretic behavior is observed. We are interested in the storage problem of the cathode of a lithium-ion battery consisting of a system of many iron phosphate (FePO4) particles. There are mathematical models, see [DGJ08], [DGGHJ09] and [DG09], that describe phase transitions and hysteresis exclusively in a single storage particle and they can describe the observed hysteretic voltage-charge plots with almost horizontal plateaus. Interestingly the models predict that the coexistence of a 2-phase system in an individual particle disappears, if its size is below a critical value. The disappearance of the phase transition in the single particle model implies the disappearance of the hysteresis. However, in the experiment hysteretic behavior survives. In other words: The behavior of a storage system consisting of many particles is qualitatively independent of the fact whether the individual particles itself develop a 2-phase system or if they remain in a single phase state. This apparent paradoxical observation will be resolved in this article by a many particle model. It will be shown that if each of the individual particles is in a homogeneous state, nevertheless the many particle ensemble exhibits phase transition and hysteresis, because one of the two phases is realized in some part of the particles while the remaining particles are in the other phase. Mathematically speaking this phenomenon is due to the non-monotonicity of the relation between the chemical potential and the lithium mole fraction. The pressure-radius relation of a spherical elastic rubber balloon also exhibits non-monotone behavior. In fact, a system of many interconnected balloons behaves correspondingly to a cathode consisting of many storage particles. This analogy between the two systems is important, because the predictions of the many particle model can easier be tested with rubber balloons of macroscopic size than with an ensemble of microscopically small (FePO4) particles.

Appeared in

  • Phys. D, 240 (2011) pp. 1008--1019.

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