Modeling the Voltage Profile for LiFePO₄

Abstract

To explain the lithium insertion/deinsertion in LiFePO₄ Padhi et al. (J. Electrochem. Soc. 144:1188–1194, 1997) proposed that the lithium motion proceeds from the surface of the particle moving inward through a two-phase interface (shrinking core model). During this process lithium ions and electrons have to move out through the newly formed FePO₄ phase. Andersson et al. in addition to the “radial model” (Andersson et al., Electrochem. Solid St. 3:66–68, 2000) proposed a “mosaic model” (Andersson and Thomas, Electrochem. Solid St. 3:66–68, 2001) that invokes a mosaic character within each particle. The idea is that lithium insertion/deinsertion can occur in many sites within a given particle. More recently Newman studied the lithium insertion in LiFePO₄ using the shrinking core model (Srinivasan and Newman, J. Electrochem. Soc. 151:A1517–A1529, 2004). Upon charge a Li-rich core is covered by a Li-deficient shell while the Li-rich shell is formed on the Li-deficient core upon discharge. Delmas et al. (Nat. Mater. 7:665–671, 2008) proposed a “domino-cascade model” in which the existence of structural constraints, occurring just at the reaction interface, lead to the minimization of the elastic energy thus enhancing the deintercalation (intercalation) process. To date, the lithium/insertion reaction mechanism in LiFePO₄ is not completely understood. The flat discharge plateau exhibited by two-phase systems makes it difficult to use the voltage to predict the state of charge of the material. The purpose of the study is to propose a model to correlate the voltage profiles to the physical and chemical processes that give rise to the discharge curves. The proposed model is based on chemical and physical processes in terms of crystallographic data and electronic/ionic diffusion and physical representation of lithium insertion/deinsertion in the material. LiFePO₄ is isostructural with the delithiated form, FePO₄. The cell parameters change by removing lithium from the structure, the a and b parameters decrease upon delithiation while the c parameter increases. As a consequence the cell volume decreases from 291.33 to 273.36 Å³ (Padhi et al., J. Electrochem. Soc. 144:1188–1194, 1997). Starting from the claim that by growing from the center the delithiated phase can reduce the stress originating from volume contraction, a general equation describing the voltage profile as a function of the intercalation degree was developed for C/10 discharge rate (Prosini, J. Electrochem. Soc. 152:A1925–A1929, 2005). The voltage profile of LiFePO₄ was described as the sum of three individual contributions: (i) the rapid segregation of lithium on the grain surface during the first stage of the intercalation process, (ii) the flat voltage region characteristic of a two-phase system, (iii) the decay of the voltage at the end of the intercalation process. The equation was successfully used to describe the voltage profile of LiFePO₄ for increasing discharge rates up to 10C. The proposed equation could be used to evaluate the state of charge of the material by fitting the discharge curve with an appropriate algorithm. Fongy et al. (J. Electrochem. Soc. 157:A885–A891, 2010) used the proposed model to extract from the LiFePO₄ discharge curves two parameters. The parameters were used to determine the optimal electrode engineering and to interpret the origins of the electrode performance limitations.