A whole electrode particle with all its intercalation sites and a certain number of Lithium atoms to spread across these sites can be modelled as a thermodynamic system. Lattice-gas model [1] is a simplified expression of the Gibbs free energy of the system as a function of n, the number of the total number of Lithium atoms intercalated into a particle: G(n) = H(n) – TS(n)*, where H(n) is the enthalpy of the system, T is the temperature of the particle, and S(n) is the entropy of the intercalated Lithium atoms.
The enthalpy function H(n) is modelled with three parameters:
The entropy of the system is maximal when Lithium atoms are evenly spread between layers and is lower (and, thus, the Gibbs free energy of the system is higher) when Lithium atoms are concentrated in certain graphite layers and are absent in other layers. The entropy term in the expression of the free energy of the system is multiplied by the temperature, which explains why the Cell open-circuit voltage depends on the temperature. [2]
The lattice-gas model's expression of free energy doesn't account for the work that Lithium atoms do when they intercalate into graphite and thus expand it (at 100% stoichiometry, graphite is about 10% larger by volume than at 0% stoichiometry), overcoming the Van der Waals forces. Yet this model is sufficient to explain the behaviour of the open-circuit potential function qualitatively.
In real graphite, Lithium atoms attract to each other within a single layer (this means negative interaction energy $J_1$) and repulse from Lithium atoms in adjacent layers (positive interaction energy $J_2(d)$). At room temperature, a graphite particle has the least energy when only every k-th graphite layer has Lithium atoms when the total occupancy of intercalation sites is between 1/(k+1) and 1/k, which gives rise to Electrode lithiation stages with smooth transition periods.
(Note: in the literature, both the stoichiometry ranges at which the open-circuit potential changes from one plateau to the next and the plateaus themselves are sometimes called transitions or phase transitions. The former are evidently the transitions on the surface of the electrode particle, and the latter are transitions in the bulk of the electrode particle.)
The transitions are smooth because the entropy of a nearly filled graphite layer drops sharply with the addition of extra Lithium atoms into it. The thermodynamic system "cannot tolerate" such a low entropy and "spills" Lithium atoms to other layers.
The chemical potential of the particle is the derivative of the Gibbs free energy with respect to the number of Lithium atoms intercalated into it. However, the expression of the free energy in the lattice-gas model is not analytically differentiable, so there is no analytical description for the phase transitions, only numerical solutions obtained via a finite-difference method.
Also, the lattice-gas model already simplifies the physics substantially, so even if it was possible to describe phase transitions analytically, that still wouldn't be a physical description.
In the end, what matters is how well an open-circuit voltage model captures the real open-circuit voltage of a half-cell, even if phase transitions are defined numerically or approximated by a sigmoid or any other function. Cf. Physics-based vs. equivalent circuit cell models.
Weaker intra-layer attraction leads to a smoother open-circuit voltage function without plateaus.
[1] A Lattice-Gas Model Study of Lithium Intercalation in Graphite (1999)
[2] A Study on the Open Circuit Voltage and State of Charge Characterization of High Capacity Lithium-Ion Battery Under Different Temperature (2018)