Cycling an intercalation host at constant current

A lithium intercalation host, cycled at constant current between voltage cutoffs: its voltage against the open-circuit curve, and lithium piling up and draining inside it. Vary the rate (C-rate) and the interaction Ω that flattens the open-circuit curve. (More about the model below.)

About this simulation

A 2 µm host layer (strictly neutral, with Li⁺ + e⁻ as one neutral combination) takes Li⁺ from the electrolyte on the left and electrons from a current collector on the right. Its open-circuit voltage is a regular solution with interaction Ω, which flattens the curve as Ω/RT nears 2, where the host would split into two phases. It's filled until the collector falls to the lower cutoff, then emptied until it reaches the upper one, and so on. Slowly, the curves hug the OCV and the host fills and empties almost completely. Faster, lithium piles up at the surface (or drains from it) while the inside lags, the voltage leaves the OCV by more, and a cutoff arrives early. Emptying is the harder direction: lithium diffuses faster through a fuller host (here its chemical diffusivity is D(1/(1−x) − 2xΩ/RT)), so at high rates the surface runs dry long before the host does. Then the mean filling ratchets over several cycles before the loop settles.

The level diagram uses the species voltage V_i = μ̄_i/(z_iF): a species' electrochemical potential per unit charge, in volts, so electrons and ions share one axis. Each species moves down its own μ̄_i, so its current flows down slopes in its own V_i (so the electrons themselves move uphill). Here they're the host's electrons (V_e⁻, its Fermi level, level since they move easily) and its Li⁺ (V_Li⁺, sloped wherever lithium diffuses). Their separation at each depth is the local open-circuit voltage, E(x) = V_e⁻ − V_Li⁺, larger where the host is emptier. Li⁺ is drawn shifted (⌇) to sit beside the electrons. φ has no meaning inside this neutral host, so neither do standard levels.