Population effects driving active material degradation in intercalation electrodes

In battery modeling, the electrode is discretized at the macroscopic scale with a single representative particle in each volume. This lacks the accurate physics to describe interparticle interactions in electrodes. To remedy this, we formulate a model that describes the evolution of degradation of a population of battery active material particles using ideas in population genetics of fitness evolution, where the state of a system depends on the health of each particle that contributes to the system. With the fitness formulation, the model incorporates effects of particle size and heterogeneous degradation effects which accumulate in the particles as the battery is cycled, accounting for different active material degradation mechanisms. At the particle scale, degradation progresses nonuniformly across the population of active particles, observed from the autocatalytic relationship between fitness and degradation. Electrode-level degradation is formed from various contributions of the particle-level degradation, especially from smaller particles. It is shown that specific mechanisms of particle-level degradation can be associated with characteristic signatures in the capacity-loss and voltage profiles. Conversely, certain features in the electrode-level phenomena can also provide insight into the relative importance of different particle-level degradation mechanisms.

Figure 1

Schematic of particle size distribution effect on current distribution in a constant current charge or discharge simulation for a battery. (a) The absolute value of the capacity change fraction of each particle is plotted with respect to the cycle number as the system is degraded. This accounts for effects of degraded charge transfer kinetics and particle size contributions by reducing the nondegraded current to the degraded current. (b) A snapshot at a single time point for a sample of degraded or nondegraded particles of small or large particle sizes is shown, where the capacity change distribution splits the current between the different particles based on their size effects and degradation.

Figure 2

[(a), (b)] Average solid lithium concentration for a single cycle of charging and discharging for Butler-Volmer/coupled ion electron transfer plotted with respect to the state of charge and the particle size during charge and discharge. [(c), (d)] Variance of the solid lithium concentration at a single cycle for each state of charge plotted with respect to the state of charge and the particle size from charge to discharge.

Figure 3

(a) The initial fitness value $\overline{W}=1/r$ is plotted in the first cycle with respect to each concentration and particle size at a single time point during the initial cycle. [(b)–(g)] For each Butler-Volmer/coupled ion electron transfer reaction rate, the fitness values $W$ from a set of simulations each with a single degradation mode toward the end of cycling are plotted with respect to concentration and particle size at a single time point during one of the last few cycles.

Figure 4

[(a), (b)] For a set of simulations with only resistive film growth, relative resistance at each cycle to the maximum resistance at each cycle is plotted with respect to cycle number and particle size, for the Butler-Volmer and coupled ion electron transfer reactions. [(c), (d)] For a set of simulations with only surface blockage increase, surface blockage at each cycle relative to the maximum surface blockage at each cycle is plotted with respect to cycle number and particle size, for the Butler-Volmer and coupled ion electron transfer reactions. (The degradation mechanism of electrolyte loss is prescribed so it has no heterogeneity.) This plot displays the heterogeneity growth in degradation as we cycle the battery.

Figure 5

The voltage discharge and charge curves from simulations with a single degradation mechanism with respect to the state of charge of the cathode are plotted at the beginning, middle, and end cycle of each set of simulations with each reaction model [BV/CIET for [(a), (c), (e)] and [(b), (d), (f)]] and degradation mechanism [resistive film for [(a), (b)], capacity loss for [(c), (d)], and electrolyte loss for [(e), (f)]].

Figure 6

[(a)–(c)] The capacity cutoff for each cycle with respect to the cycle number for BV/CIET is plotted for sets of simulations with the three separate degradation mechanisms in the first row, while the real relative electrochemical capacity loss from the integrated degradation current with respect to the cycle number is plotted in the second row in panels [(d), (e)]. For the electrolyte loss model, because the electrochemical capacity loss is from the anode, we instead plot the prescribed degradation of electrolyte concentration with respect to cycle number in Panel (f). (g) We plot the capacity loss curve with respect to cycle number for a set of simulations with all three degradation mechanisms implemented for the two reaction models.

Figure 7

The percentage errors of the analytical solutions are plotted below for each reaction mechanism and degradation mechanism.