Disorder transformation, phase coexistence

In this chapter we have summarized the fundamentals and recent advances in thermodynamic and kinetic approaches to lithium intercalation into, and deintercalation from, transition metals oxides and carbonaceous materials, and have also provided an overview of the major experimental techniques. First, the thermodynamics oflithium intercalation/deintercalation based on the lattice gas model with various approximations was analyzed. Lithium intercalation/deintercalation involving phase transformations, such as order-disorder transition or two-phase coexistence caused by strong interaction oflithium ions in the solid electrode, was clearly explained based on the lattice gas model, with the aid of computational methods. 

One way to see that a transition is discontinuous is to detect a coexistence of two phases, in this case the orientationally ordered and disordered phases, in a temperature interval. This is revealed by time variation of the potential energy of the cluster. In the temperamre region of phase coexistence, each cluster dynamically transforms between the phases, and its potential energy fluctuates around two different mean values (Fig. 4). In an ensemble of clusters, the coexistence of different phases is observable insofar as a fraction of the clusters (e.g., in a beam [17]) can exhibit the structure of one phase, while another fraction takes on the stmcture of another phase. 

Fredrickson and Binder [9] further improved this theory to describe the kinetics of the ordering process. Their concentration dependent free energy potential shows two side minima, which have the same depth as the middle one at the microphase separation transition temperature Tmst- Therefore, they presume a coexistence of the disordered and the lamellar phase at Tmot- As the temperature is further lowered, these side minima become dominant and the transition comes to completion. For a supercooled material, they expect after a completion time an Avrami-type ordering transformation with an exponent of 4 equivalent to spherically growing droplets of ordered material. This characteristic time corresponds to the time to form stable droplets of ordered material plus the time needed for the structures to grow to a size that they can be detected by the used technique. 

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