Authors: Ionworks
For several decades, the Doyle-Fuller-Newman model has been the cornerstone of computational battery modeling, design, and optimization. Since the model was developed based on an assumption of Fickian diffusion in the solid active material particles, the equilibrium state in a DFN model is an electrode of compositionally homogeneous particles. While this is appropriate for many battery materials, several battery materials such as LiFePO4, LiCoO2, LiTiO2, and graphite are observed to phase-separate. In these materials, two phases with different lithium concentrations can exist in thermodynamic equilibrium, with concentration gradients at the boundaries between phases. Fickian diffusion is unable to describe phase-separation, and consequently does not capture important behavior which is unique to phase-separating battery electrodes, including hysteresis, one-at-a-time particle filling, and current pulse activation of particles. Additionally, since many electrochemical methods require knowledge of the active area, it is important to consider how phase-separation might be affecting the assumptions or validity of these analytical techniques.
The DFN model was extended by the Bazant group at MIT to phase-separating battery materials using phase-field methods, and their model was recently implemented in PyBaMM by researchers at IonWorks. The phase-field method is a thermodynamically rigorous modeling technique for simulating the evolution of complex phase diagrams. Presently, phase-field models are capable of modeling complex microstructural and morphological changes in multiphase systems. While each model has many subtle differences, phase-field models generally treat phase separation using a double-well free energy that promotes phase-separation, and a gradient energy term which penalizes the formation of sharp boundaries. In a phase-field model it is an energy gradient, rather than a concentration gradient, that drives evolution. The balance between driving a system towards extremes in concentration while penalizing gradients results in the formation of a diffuse phase boundary.
Here are a few interesting examples that highlight differences between the Fickian treatment and the phase-field treatment of LiFePO4. As you watch these videos, observe how the particle and electrode dynamics are affected by phase-separation and consider the consequences for interpreting standard electrochemical characterizations such as GITT or HPPC.
The videos above compare a C/20 discharge/charge cycle of LiFePO4, simulated with Fickian diffusion and with phase-field. The first important dynamic difference is that Fickian particles all fill uniformly, while the phase-field particles fill one-at-a-time, a mechanism that has been observed experimentally. The electrode with Fickian particles experiences a small current density applied over a large active area, while the phase-field electrode experiences a large current density focused on a small number of particles (small active area).
A second difference between the Fickian and phase-field models is how the voltage plateau arises, which is a signature of phase separation. The plateau in a Fickian model is implemented empirically in the open-circuit potential potential as a flat region. As shown in the figure above, phase-field particles have a non-monotonic open-circuit potential, and the voltage plateau is an emergent property of a collection of particles as they fill one-at-a-time.
This non-monotonic open-circuit potential also leads to thermodynamic hysteresis, which can also be seen in the video. One-at-a-time particle dynamics in the phase-field model means that only the empty particles participate during lithium intercalation, and only the filled particles participate during lithium de-intercalation. These different processes require different energies to initiate, which is the origin of hysteresis. Fickian diffusion does not capture hysteresis in LiFePO4 because there is symmetry between charge and discharge. When the current is reversed, the filling process just occurs in reverse.
These videos illustrate differences in dynamics at high currents. The simulations show a short 10C pulse followed by a 30 minute rest (zero current). The Fickian particles all fill relatively uniformly during the pulse, and quickly relax to a homogeneous state during the rest. The phase-field particles behave very differently. During the pulse, nearly all of the particles briefly become active, and phase boundaries become visible in the interior of the particles. This particle activation process has been reported recently in literature, and leads to a reduced overpotential when current is resumed. During the rest segment, the electrode remains active for some time and there is significant redistribution of lithium. Some particles intercalate lithium while others deintercalate, driven by excess energy stored in the phase boundaries. The electrode eventually reaches its equilibrium state, which is a collection of filled and empty particles that do not have internal phase boundaries.
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References
- W. Dreyer, J. Jamnik, C. Guhlke, R. Huth, J. Moškon, M. Gaberšček, The thermodynamic origin of hysteresis in insertion batteries. Nat. Mater. 9, 448–453 (2010).
- D. A. Cogswell, M. Z. Bazant, Coherency Strain and the Kinetics of Phase Separation in LiFePO 4 Nanoparticles. ACS Nano 6, 2215–2225 (2012).
- R. B. Smith, M. Z. Bazant, Multiphase Porous Electrode Theory. J. Electrochem. Soc. 164, E3291–E3310 (2017).
- H. D. Deng, N. Jin, P. M. Attia, K. Lim, S. D. Kang, N. Kapate, H. Zhao, Y. Li, M. Z. Bazant, W. C. Chueh, Beyond Constant Current: Origin of Pulse-Induced Activation in Phase-Transforming Battery Electrodes. ACS Nano 18, 2210–2218 (2024).
Ionworks is a battery simulation software company founded by the creators of the leading open-source battery simulation package PyBaMM. Our mission is to accelerate battery innovation by making simulations accurate, fast, and easy to use. We offer software and services that leverage simulations to empower battery and battery-adjacent companies to enhance their products and get them to market more quickly.