Intercalation Kinetics in Multiphase-Layered Materials

Many intercalation
compounds possess layered structures or interpenetrating
lattices that enable phase separation into three or more stable phases,
or “stages,” driven by competing intralayer and interlayer
forces. While these structures are often well characterized in equilibrium,
their effects on intercalation kinetics and transport far from equilibrium
are typically neglected or approximated by empirical solid solution
models. Here, we formulate a general phase-field model with thermodynamically
consistent reaction kinetics and cooperative transport to capture
the dynamics of intercalation in layered materials. As an important
case for Li-ion batteries, we model single particles of lithium intercalated
graphite as having a periodic two-layer structure with three stable
phases, corresponding to zero, one, or two layers full of lithium.
The electrochemical intercalation reaction is described by a generalized
Butler–Volmer equation with thermodynamic factors to account
for the flexible structure of the graphene planes. The model naturally
captures the “voltage staircase” discharge curves as
a result of staging dynamics with internal “checkerboard”
domains, which cannot be described by solid–solution models
based on Fickian diffusion. On the other hand, the two-layer model
is computationally expensive and excludes low-density stable phases
with longer-range periodicity, so we also present a reduced model
for graphite, which captures the high-density stages while fitting
the low-density voltage profile as an effective solid solution. The
two models illustrate the general trade-off between the explicit modeling
of periodic layers or lattices and the needs for computational efficiency
and accurate fitting of experimental data.