Numerical Simulations of Rechargeable Lithium-Ion Batteries with Porous Positive Electrodes: Local Reaction Rate Distribution

Introduction

   The volume fraction of electrode active material in porous electrodes is an important factor to optimize in order to increase the overall energy density of a lithium-ion battery.  However, an increase in the volume fraction of electrode active material leads to an increase in electrolyte resistance because the ionic path cross-sections decrease.  Dokko et al.reported that a perfectly isolated single electrode active material particle can operate at much higher charge/discharge rates than in a porous electrode [1].  Hence, it is necessary to optimize the volume fraction in porous electrodes to maximize the performance of electrode active material while keeping a small electrolyte resistance.  Newman’s model has often been used to numerically simulate lithium-ion batteries [2].  His model solves the system averaging shapes and configurations of active material particles based on the one-dimensional (1-D) formulation.  Hence, his model ignores the distributions of local current densities on the surface of every single particle.  The aim of this study is to simulate the performance of lithium-ion batteries taking into account local current density distributions on electrode active materials.

Methods

   The simulations were performed in a 2-D field that consisted of a number of small square grids (> 10⁴).  Circular regions with controlled sizes in the calculated field were defined as positive electrode active material particles.  The electric potential (φ) distribution in the electrolyte was calculated using the Laplace equation.  The local states of charge (SOC) in electrode active material particles were simulated by calculating Li⁺ diffusion in particles with Fick’s second law (D_Li⁺ = 1.0 × 10⁻¹¹ cm² s⁻¹).  The negative electrode surface was defined to be φ = 0 V.  The other boundaries in the calculated area were defined as insulating surfaces with ∂φ/∂x = 0 or  ∂φ/∂y = 0.

   As the first step, to examine the effect of electromigration for Li⁺ in the electrolyte, the electromigration was only considered for the mechanism of Li⁺ mass transfer in the electrolyte.  Thus, the transference number was set to be 1 thereby ignoring the concentration difference in the electrolyte (C_Li⁺ = 6.0 × 10⁻² mol cm⁻³).  A local current density at the particle surface was calculated by multiplying the Li⁺ mobility in the electrolyte and the gradient of φ at the corresponding location.  The local overpotential was calculated from the Li⁺ concentration at the inner surface of a particle and the Butler-Volmer equation (α = 0.5, k = 10⁻⁷).

Results and Discussion

   Fig. 1 shows simulated distributions of φ in the electrolyte and local SOC in an electrode active material particle every 1000 steps in time (t).  The φ gradient between the front surface (i.e. the nearer side to the negative electrode) of a particle and the negative electrode is significantly steeper than at the back surface of a particle.  This means that a local current density on the front surface of a particle is much greater than at the back surface.  The distribution of local SOC in a particle is quite uniform at a smaller current density of (A) 1.0 mA cm⁻² compared to the case at a larger current density of (B) 10 mA cm⁻².  Li⁺ diffuse in a particle to mitigate non-uniform SOC distribution.  Hence, the non-uniformity of SOC in a particle becomes more significant if the current density is much larger compared to Li⁺diffusion fluxes in a particle.  The utilization of a particle in (B) is 0.15 whereas that in (A) is 0.67.  Our results demonstrate the importance of considering local current density distributions on particle surfaces especially when electromigration is the dominant mechanism in an electrolyte like in an inorganic solid electrolyte.  We will further present the simulated results with multiple active material particles.

 

Acknowledgement

   The authors gratefully acknowledge JST-ALCA for the financial support.

References

[1] K. Dokko et al., J. Electrochem. Soc., 189, 783 (2009).  

[2] M. Doyle et al., J. Electrochem. Soc., 140, 1526 (1993).

Figure 1.  Simulated distributions of electric potential, φ, in the electrolyte and SOC in a positive electrode active material particle during constant-current discharging at (A) 1.0 mA cm⁻² and (B) 10 mA cm⁻².  The bottom of the square is defined as the negative electrode surface at φ = 0.  A color variation in the electrolyte indicates the distribution of φ ranging from 0 V (blue) to +3.9 V (red).  A color variation in the particle indicates the distribution of local SOC ranging from 0% (red) to 100% (blue).