Population dynamics of driven autocatalytic reactive mixtures

Abstract

Motivated by the theory of reaction kinetics based on nonequilibrium thermodynamics [1] and the linear stability of driven reaction-diffusion[2], we apply the Fokker-Planck equation to describe the population dynamics of an ensemble of reactive particles in contact with a chemical reservoir. We illustrate the effect of autocatalysis on the population dynamics by comparing systems with identical thermodynamics yet different reaction kinetics. The dynamic phase behavior of the system may be entirely different from what its thermodynamics may suggest. By defining phase separation for a particle ensemble to be when the probability distribution is bimodal, we find that thermodynamic phase separation may be suppressed by autoinhibitory reactions, while autocatalysis enhances phase separation and in some cases induce the ensemble that consists of thermodynamically single-phase systems to segregate into two distinct populations, which we term fictitious phase separation. Asymmetric reaction kinetics also results in qualitatively different population dynamics upon reversing the reaction direction. In the limit of negligible fluctuations, we use method of characteristics and linearization to study the evolution of the standard deviation of concentration as well as the condition for phase separation, in good agreement with the full numerical solution. Applications are discussed to Li-ion batteries and in situ x-ray diffraction.arXiv:1901.05575v1 [physics.chem-ph]

A scaling law to determine phase morphologies during ion intercalation

“…for LFP [18][19][20] or graphite. 21 A more consistent and accurate approach is to use phase-field models, [22][23][24][25][26][27] generalized for electrochemical thermodynamics 28 of single particles 23,[29][30][31][32][33][34][35][36][37][38][39] and porous electrodes, [40][41][42][43][44][45] but a clear classification of the possible nonequilibrium phase morphologies has not yet emerged.…”

Theory of Coupled Ion-Electron Transfer Kinetics in Ion Intercalation Materials

“…Very recently, it has been shown that by understanding the functional form of the reaction rate expressions, one is able to control, and consequently engineer, the physics of interfaces where reactions take place [26]. Representative examples are the lithiation of LFP and LiNi 1/3 Mn 1/3 Co 1/3 O 2 [149,150] particles, as well as the operation of Li-air batteries, where the thermodynamic stability of the system is controlled by varying the applied current [13,95,102,151,152]. In terms of CIET, by understanding the concentration dependencies of both the reorganization energy and the density of states of the electron donor, we will be able to control interface structure by inducing or suppressing phase separation/island formation.…”

Guiding the Design of Heterogeneous Electrode Microstructures for Li‐Ion Batteries: Microscopic Imaging, Predictive Modeling, and Machine Learning

“…[173] Such a phase-separation phenomenon is a source of instability in a population of particles which undergo particle-byparticle separation, also known as mosaic instability. [174] This can sometimes result in a bimodal distribution of the state of lithiation throughout the electrode at low charge/discharge rates, as experimentally visualized in LFP [175,176] and graphite [58] and then explained by physics-based theories. [60,173,174,177,178] Compositional heterogeneities have been reported also in NMC and nickel cobalt aluminum oxide (NCA), [179][180][181] both at high C-rates, [181] which can be induced by autocatalytic reaction kinetics, [182] and in agglomerate particle as a consequence of internal stress.…”

“…[174] This can sometimes result in a bimodal distribution of the state of lithiation throughout the electrode at low charge/discharge rates, as experimentally visualized in LFP [175,176] and graphite [58] and then explained by physics-based theories. [60,173,174,177,178] Compositional heterogeneities have been reported also in NMC and nickel cobalt aluminum oxide (NCA), [179][180][181] both at high C-rates, [181] which can be induced by autocatalytic reaction kinetics, [182] and in agglomerate particle as a consequence of internal stress. [183,184] There exist at least two consolidated theories to model phaseseparating materials, namely the sharp interface model and the phase-field model.…”

“…separation and sequential transformation occur at low current [196] (Figure 16a), while a simultaneous phase separation occurs at high C-rates, in agreement with the predictions of reduced-order phase-field models. [60,174,177,178,[197][198][199] On the other hand, when LFP particles are more densely packed, interparticle phase separation occurs for a greater range of C-rates [193] because lithium can be exchanged directly by contacting particles without being mediated by the electrolyte via interfacial reactions (Figure 16b). In other studies, the dynamics of phase separation was roughly captured by considering the nonmonotonic open-circuit voltage and by introducing a thermodynamic factor to lithium solid diffusivity to represent the nonideal solution behavior of the active material.…”