This paper presents the current state of mathematical modelling of the electrochemical behaviour of lithium-ion batteries (LIBs) as they are charged and discharged. It reviews the models developed by Newman and co-workers, both in the cases of dilute and moderately concentrated electrolytes and indicates the modelling assumptions required for their development. Particular attention is paid to the interface conditions imposed between the electrolyte and the active electrode material; necessary conditions are derived for one of these, the Butler–Volmer relation, in order to ensure physically realistic solutions. Insight into the origin of the differences between various models found in the literature is revealed by considering formulations obtained by using different measures of the electric potential. Materials commonly used for electrodes in LIBs are considered and the various mathematical models used to describe lithium transport in them discussed. The problem of upscaling from models of behaviour at the single electrode particle scale to the cell scale is addressed using homogenisation techniques resulting in the pseudo-2D model commonly used to describe charge transport and discharge behaviour in lithium-ion cells. Numerical solution to this model is discussed and illustrative results for a common device are computed.

A series of LiFe1−xZnxPO4 (0.0 ≤ x ≤ 1.0) compounds were prepared by solid-state reaction. Effects of the substitution of Zn for Fe on crystal structure and electrochemical properties of LiFe1−xZnxPO4 were investigated. The results show that single-phase regions of LiFe1−xZnxPO4 with orthorhombic (space group Pmna) and monoclinic (Cc) structures were found for the compounds with low Zn (or high Fe) contents of 0.0 ≤ x ≤ 0.30 and high Zn (or low Fe) contents of 0.90 ≤ x ≤ 1.0, respectively. The LiFe1−xZnxPO4 compounds with medium Zn (or Fe) contents of 0.35 ≤ x ≤ 0.80 are two-phase mixtures containing both the orthorhombic and the monoclinic phases. Systematic variations of unit-cell parameters a, b, c, and volume V with the Zn content determined by X-ray diffraction have also been obtained. Our electrochemical study show that the conductivity of LiFe1−xZnxPO4 increases by almost 2 orders of magnitude from 2.13 × 10−9 to 1.27 × 10−7 Scm−1 as the Zn content increasing from x = 0 to 0.3. The initial specific capacity decreases and the cycle performance increase with increasing Zn-doping content in the four orthorhombic LiFe1−xZnxPO4 compounds. Among the four LiFe1−xZnxPO4 compounds, LiFe0.8Zn0.2PO4 has the highest capacity retentions after 6 to 20 cycles and the capacity retention is 93.7% after 20 cycles, even though the initial discharge specific capacity of LiFe0.8Zn0.2PO4 is lower than those of LiFeZnPO4 and LiFe0.9Zn0.1PO4. LiFe0.7Zn0.3PO4 has the highest capacity retention of 97% after 20 cycles.