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During the intercalation of lithium in layered host materials such asgraphite, lithium atoms can move within the plane between two neighboringgraphene sheets, but cannot cross the sheets. Repulsive interactions betweenatoms in different layers lead to the existence of ordered phases called"stages", with stage $n$ consisting of one filled layer out of $n$, the othersbeing empty. Such systems can be conveniently described by a multi-layerCahn-Hilliard model, which can be seen as a mean-field approximation of alattice-gas model with intra- and interlayer interactions between lithiumatoms. In this paper, the dynamics of stage formation after a rapid quench tolower temperature is analyzed, both by a linear stability analysis and bynumerical simulation of the full equations. In particular, the competitionbetween stages 2 and 3 is studied in detail. The linear stability analysispredicts that stage 2 always grows the fastest, even in the composition rangewhere stage 3 is the stable equilibrium state. This is borne out by thenumerical simulations, which show that stage 3 emerges only during thenon-linear coarsening of stage 2. Some consequences of this finding for thecharge-discharge dynamics of electrodes in batteries are briefly discussed.

Spinodal decomposition and domain coarsening in a multi-layer Cahn-Hilliard model for lithium intercalation in graphite