1 Introduction

Li ion batteries (LIBs) are typical rechargeable batteries, in which Li ions move from the anode to the cathode during discharging, and then move back during charging. This cycling process is also known as the intercalation and deintercalation of Li ions. Compared to other rechargeable batteries, such as lead-acid, nickel-cadmium (Ni-Cd) and nickel-metal hydride (Ni-MH) batteries, LIBs have lower weight, higher energy density and a longer cycle life^[1-3]. These excellent properties, as well as no memory effect and slow capacity loss, have made LIBs the most popular rechargeable batteries in the area of consumer electronics. People are using them every day and everywhere, such as in cell phones, laptops, cameras, etc. In recent years, LIBs are becoming more and more attractive in electric vehicles (EV). LIB is the most promising candidate for plug-in hybrid electric vehicles (PHEV) or plug-in electric vehicles (PEV) because of high energy density and low weight. However, further research is still necessary to keep them as safe as possible and their cost as low as possible^[2-5].

Three key functional components of a LIB are the cathode, the anode and the electrolyte. Graphite is presently the most popular anode. The lithium transition metal oxides, which possess high operating voltage, good reversibility and high capacity, are the most favored cathodes^[1-5]. The electrolyte is usually a lithium salt in an organic non-aqueous solvent^[1], such as lithium hexafluorophosphate (LiPF₆) and lithium perchlorate (LiClO₄). Fig. 1 shows a schematic diagram of a typical Li ion cell, which is composed of a layered structural LiMO₂ (M = transition metal) cathode and a graphite anode^[6].During the discharging process, Li ions are extracted from the anode host, move through the electrolyte and then intercalate into the cathode host. At the same time, electrons move from the anode to the cathode, which can be seen as electric current. This process is reversed during charging. The reaction of the whole LIB can be written as C_n + LiMO₂ ↔ C_nLi_x + Li_1-xMO₂. During the cycling, a number of chemical reactions take place in the cathodes. This leads to significant changes in the microstructure of the material. Phase diagrams are powerful tools to help understand these chemical reactions^[6-7]. There are various material systems being used or promising as future electrode materials for LIBs^[2], as shown in Fig. 2, which are interesting for researchers to study their phase equilibria, thermodynamic and electrochemical properties.

At present, research of LIBs is focusing on lower cost, lower capacity loss, longer lifetime, higher energy density, higher stability and inherent safety. However, none of the present electrode materials is practically able to attain all these goals simultaneously^[1-3]. Thus, the choice of certain electrode materials for a LIB is a compromise. Depending on the choices of materials, the voltage, capacity, life and safety of a LIB can be changed significantly. The very first step to design new electrode materials is to study its voltage and capacity, which can theoretically be predicted^[6-14].

2 Cathode Materials and Theoretical Study in the Literature

Usually, the cathode is the most expensive part with the highest weight among the three key components in a LIB^{[1, 7]}. Also, properties of the cathode specially affect the performance of the entire LIB. Lithium-transition metal oxides as cathodes have extensively been investigated. Typical examples are lithium cobalt oxide (such as LiCoO₂) with a layered structure, lithium iron phosphate (LiFePO₄) with an olivine structure and lithium manganese oxide (such as LiMn₂O₄) with a spinel structure^{[1-3, 7]}. The theoretical specific capacities of LiCoO₂, LiFePO₄, and LiMn₂O₄ are 273, 170 and 297 mAh/g, respectively^[8]. All of them have some disadvantages. LiCoO₂ has a relatively high cost and safety problems; LiFePO₄ exhibits a poor electronic conductivity; LiMn₂O₄ suffers from the Jahn-Teller distortion with a large volume change of the cathode, which leads to capacity drop of the battery. Recently, the Li(Co, M)O₂ cathodes (M = transition metals), e.g. Li(Co, Ni, Mn)O₂, have been developed to improve the properties of LiCoO₂. The Li(Co, M)O₂ cathodes are, however, less stable in comparison with the spinel structure. The thermal decomposition already occurs at low temperatures^[6]. Such kind of phase transformations can also be explained by phase diagrams^[7].

Information on the phase equilibria and thermodynamic properties of the Li-M-O (M = transition metal) system is of great importance. The CALPHAD approach can help to establish a thermodynamic database for multi-component Li-containing oxide systems. Lithium cobalt oxides and lithium nickel oxides have been used as quite popular cathodes^[1-5]. All the possible cathodes based on the Li-(Co, Ni)-O system are of great interest to study their phase equilibria, thermodynamic and electrochemical properties. LiCoO₂ has three types of layered structures: O2, O3 and O4. The letter (O) indicates the Li ion site (octahedral) and the number (2, 3 or 4) denotes the number of slabs within the hexagonal phases^[15]. The three different structures can be distinguished by the types of Li stacking (or oxygen packing). All of them consist of CoO₆ octahedra interleaved with Li ions^[6], as shown in Fig. 3. The layered structures facilitate intercalation and deintercalation of Li ions. Among the O2, O3 and O4 structures, O3-LiCoO₂ with the space group R 3m is the most stable one, which is widely used in LIBs^[1-3]. The O2-LiCoO₂ and O4-LiCoO₂ phases, both with the space group P6₃mc, are suggested to be good candidates as cathodes in LIBs^[15-17].The cell voltage and capacity of Li/O4-LiCoO₂ is quite similar to that of Li/O3-LiCoO₂, but with absence of structural transformations^[16]. The relatively high cost and safety problems, which are associated with electrolyte oxidation during prolonged cycling, limit the applications of O3-LiCoO₂ in industry^{[2, 18-19]}. As a replacement of LiCoO₂, LiNiO₂ (also with the O3 structure) is one of the attractive candidates^{[2, 20-22]}. However, the stoichoimetric LiNiO₂ is very difficult to synthesize due to its lower thermodynamic stability compared to LiCoO₂^[23] and the presence of excess nickel on Li sites^[24-25]. These anti sites in LiNiO₂ strongly affect the capacity of the batteries^{[2, 20-21]}. Li(Co, Ni)O₂ with the O3 structure, as the solid solution between LiCoO₂ and LiNiO₂, has been developed to alleviate the drawbacks of LiCoO₂ and LiNiO₂^{[22, 24-26]}.

The theoretical approaches based on first-principles calculations and empirical methods are the very first step to study the thermodynamic properties of metal oxides. These simple thermodynamic models can help to determine the Gibbs energy functions of the binary and ternary oxides in the Li-(Co, Ni)-O system^[14]. Then, using the CALPHAD approach, the reliable Gibbs energy functions can be obtained, which can be used to calculate the phase equilibria and predict the cell voltages of LIBs^{[6, 27-32]}. Therefore, it is important to apply the CALPHAD approach integrated with first-principles calculations and other theoretical methods, which can not only provide reliable phase equilibria and thermodynamic data for Lithium-transition metal oxides, but also predict the cell voltages of LIBs reasonably^{[6, 27-32]}. The cell voltage (V) of a Li ion cell can be calculated using the following equation:

$ V = - \frac{{\mu _{{\rm{Li}}}^{{\rm{Cathode}}} - \mu _{{\rm{Li}}}^{{\rm{Anode}}}}}{{zF}} $

(1)

where μ_Li is the Li chemical potential (which can directly be calculated with the Gibbs energy functions using the CALPHAD approach), z is the moles of electrons involved (for Li⁺, z=1), and F is Faraday's constant. The cell voltage of a Li/Li⁺ cell can be expressed as follows:

$ V = - \frac{{\mu _{{\rm{Li}}}^{{\rm{Cathode}}}}}{F} $

(2)

The theoretical specific capacity of a cell can be calculated by Faraday's law, where the number of charge carrier (or the amount of Li) and the weight of the active material used in the electrode are the key parameters. Therefore, one can predict the theoretical specific capacity from the phase diagram by calculating the amount of lithium that can be intercalated in or deintercalated from the electrode. In reality, the practical specific capacity of an operating cell can be lower than the theoretical one. The reason is that not all Li can be removed from the lattice of the host material^[7].

3 Anode Materials and Literature Review

Graphite is the most widely used anode in LIBs owing to its abundance, long cycle time and low cost. However, the graphite anode exhibits a comparatively low energy density and has safety issues related to lithium deposition. Li-Metal systems have extensively been investigated as potential alternatives to graphite anodes, such as the Si-containing systems^[33], the Li-Sb^[34] and Li-Sn^[35] systems. Literature review on the phase equilibria and thermodynamic data of Li-Sb and Li-Sn are presented as follows.

The Li-Sb system was firstly critically evaluated by Sangster and Pelton^[36] based on the experimental work from Brauer et al.^[37] and Weppner et al.^[38] The melting temperature was considered to be in the range of 1 150 to 1 300 ℃ measured by Brauer and Zintl^[37] and higher than 950 ℃ reported by Lebeau^[39]. The αLi₃Sb phase transforms to βLi₃Sb at a temperature above 650 ℃ measured by Brauer and Zintl^[37]. Three binary compounds were presented in the assessed phase diagram^[36], viz.Li₂Sb, αLi₃Sb and βLi₃Sb (high temperature phase). Then Fedorov^[40] proposed a different phase diagram based on thermal analysis with the unannealed alloys and two additional compounds Li₃Sb₂ and LiSb₂ were included. Recently, Kane et al.^[41] determined the Li-Sb system using the electromotive force (EMF) measurement, but neither Li₃Sb₂ nor LiSb₂ was confirmed to be stable. In their work, the eutectic temperature of L→(Sb)+Li₂Sb was measured to be 460 ℃ and Li₂Sb was formed by the peritectic reaction at a temperature of 584 ℃. In addition, the eutectic point was considered to be about 40 at.% Li. Based on the X-ray diffraction measurement (XRD) and electron probe microanalyses (EPMA) results, Zhang et al.^[34] confirmed the Li-Sb phase diagram reported by Kane et al.^[41] Enthalpies of formation of the Li₃Sb and Li₂Sb phases have been investigated by many authors^[42-46]. Shchukarev et al.^[43] and Kubaschewski and Catterall^[44] determined the enthalpies of formation of Li₃Sb using solution calorimetry (SC) and direct reaction calorimetry (DRC), respectively. In addition, Nikitin et al.^[45] and Weppner and Huggins^[38] measured the enthalpies of formation of the Li₃Sb and Li₂Sb phases by the cathodic polarization technique (CP) and EMF, respectively. Their experimental data are close to each other and inconsistent with the work of Shchukarev et al.^[43] and Kubaschewski and Catterall^[44]. Recently, Terlicka et al.^[46] have determined the formation energies of Li₃Sb and Li₂Sb using the DRC technique, which are similar with the data reported by Nikitin et al.^[45] and Weppner and Huggins^[38]. Thermodynamic properties of the liquid Li-Sb alloys were investigated by two groups^{[41, 46]}. Kane et al.^[41] measured the thermodynamic properties of the Li-Sb alloys by EMF technique with the temperature ranges of 425-775 ℃ and composition ranges of 1-75 at.% Li. And Terlicka et al.^[46] determined partial and integral mixing enthalpy of the liquid Li-Sb alloys at 820 K, 942 K, 922 K and 1 024 K by drop calorimetry. The thermodynamic data reported by Kane et al.^[41], Weppner and Huggins^[38], Nikitin et al.^[45] and Terlicka et al.^[46] are agreement with each other and accepted in this work.

Using the coulometric titrations at 415 ℃ and 25 ℃, Huggins^[47-48] have determined the Li₂Sn₅, LiSn, Li₇Sn₃, Li₅Sn₂, Li₁₃Sn₅, Li₇Sn₂ and Li₂₂Sn₅ (or Li₁₇Sn₄) phases in the Li-Sn system. These investigations were consistent with the Li-Sn phase diagram^[49-50]. However, under practical charging condition and/or at room temperature, it was found that only the lithium-poor phases Li₂Sn₅ and LiSn could be clearly distinguished^[51-53], which suggested that the lithium-rich phases could not form long-range ordered structures due to the low mobility at room temperature and the huge volume change. The equilibrium open-circuit voltage has been reported using theoretical calculations^[54-55], which agree well with the experimental data.

4 The CALPHAD Approach

The CALPHAD approach has been proved to be a powerful tool to evaluate Gibbs free energy, predict multi-component phase diagrams and help to design new materials. CALPHAD stands for CALculation of PHAse Diagrams and is better defined by the subtitle of the CALPHAD journal (Computer Coupling of Phase Diagrams and Thermochemistry). The phase diagram, usually a diagram with axes for temperature and composition of a chemical system, consists of a number of phase-boundaries separating regions of temperature-and composition-space in which different phases are stable at equilibrium. The CALPHAD approach is based on the fact that equilibrium phase diagram of a system is a manifestation of its thermodynamic properties^[56]. The thermodynamic properties of the system are the sum of the properties of the individual phases, including Gibbs energy, enthalpy, entropy, heat capacity, etc. Within the CALPHAD approach, one describes the thermodynamic properties of each phase with a mathematical model containing adjustable parameters. By determining the thermodynamic properties of all the phases in a system, the phase diagram can be constructed by calculation^[56-57], which can be performed using the POLY-3 module within the Thermo-Calc software^[58].

Fig. 4 shows the flow chart of the CALPHAD approach^[57]. Firstly, reliable thermodynamic and phase equilibria data are necessary as input parameters, which can stem from experiments, theoretical calculations or estimation. Secondly, appropriate thermodynamic models are used for individual phases. Thirdly, adjustable thermodynamic parameters are introduced and then optimized considering all the reliable data. The above steps are known as the process of "thermodynamic optimization". This process can be achieved using the PARROT module within the Thermo-Calc software^[58], which works by minimizing the square sum of the differences between measured and calculated values.

The obtained parameters from the "thermodynamic optimization" process together with parameters of pure elements generate the thermodynamic dataset of a system. The phase diagram can be calculated, while the unknown phase equilibria and thermodynamic properties can be predicted. The applications of the CALPHAD calculations can save large amounts of experimental work and make predictions available for multi-component systems.

From Fig. 4, it can be concluded that the accuracy of thermodynamic databases is most important for applications of the CALPHAD approach. This in turn suggests that reliable thermodynamic parameters and appropriate thermodynamic models are prerequisite^[59-60]. Accurate Gibbs energy functions of individual phases are essential to get the correct phase diagram. It is also important, for cathodes in LIBs, to predict the cell voltages and study the electrochemical properties of batteries.

4.1 Thermodynamic Models Used in the CALPHAD Approach

In the binary Li-O system, four phases, i.e. liquid, Li₂O, Li₂O₂ and gas, are modeled using appropriate sublattice models^[61]. Two lithium oxides (Li₂O and Li₂O₂), which have been considered as stoichiometric compounds, are taken as examples. They are modeled using the following expression^[61]:

$ \begin{array}{*{20}{c}} {{}^oG_{{\rm{L}}{{\rm{i}}^{1 + }}:{{\rm{O}}^{2 - }}}^{{\rm{L}}{{\rm{i}}_2}{{\rm{O}}_n}} - 2H_{{\rm{Li}}}^{{\rm{SER}}} - nH_{\rm{O}}^{{\rm{SER}}} = A + B \cdot T + }\\ {C \cdot T \cdot \ln T + D \cdot {T^2} + F \cdot {T^{ - 1}}} \end{array} $

(3)

where the coefficients C, D, and F, are obtained from the expression of heat capacity for solid Li₂O_n (n=1, 2).

Four phases, i.e. O3, O3' (ordered O3, also known as the monoclinic phase), H1-3 and O1, are modeled^{[6, 31]} in the Li-(Co, Ni)-O system. Considering the order-disorder transition, a four-sublattice model is applied to describe both the O3 and O3' phases, written as (Li, Va)_1/2(Li, Va)_1/2(M)₁(O)₂ (M = Co, Ni)^{[6, 31]}. In the disordered state, the first sublattice and the second one are identical, thus it is (Li, Va)₁(M)₁(O)₂. The Gibbs energy of both the ordered and disordered states can be expressed as follows:

$ G_m^{{\rm{Ord}}} = G_m^{{\rm{Dis}}}\left( {{x_i}} \right) + G_m^{{\rm{Ord}}}\left( {y_i^s} \right) - G_m^{{\rm{Ord}}}\left( {y_i^s = {x_i}} \right) $

(4)

where G_m^Dis denotes the Gibbs energy of the disordered phase. G_m^Ord(y_i^s)-G_m^Ord(y_i^s=x_i) denotes the contribution of the ordered phase to the Gibbs energy and is equal to 0 when the disordered O3 phase is stable.

The Gibbs energy for the disordered O3 phase is as follows:

$ \begin{array}{l} G_m^{{\rm{Dis}}} = {x_{{\rm{Li}}}} \cdot G_m^{{\rm{O3 - LiM}}{{\rm{O}}_{\rm{2}}}} + {x_{{\rm{Va}}}} \cdot G_m^{{\rm{O3 - M}}{{\rm{O}}_{\rm{2}}}} + RT\left( {{x_{{\rm{Li}}}} \cdot } \right.\\ \;\;\;\;\;\;\;\left. {\ln {x_{{\rm{Li}}}} + {x_{{\rm{Va}}}} \cdot \ln {x_{{\rm{Va}}}}} \right) + {x_{{\rm{Li}}}} \cdot {x_{{\rm{Va}}}} \cdot \\ \;\;\;\;\;\;\;\;\sum\limits_{n = 0}^N {{}^nL_{{\rm{Li}},{\rm{Va:Co:O}}}^{{\rm{Dis}}}{{\left( {{x_{{\rm{Li}}}} - {x_{{\rm{Va}}}}} \right)}^i}} \end{array} $

(5)

where G^O3-LiCoO2_m and G^O3-CoO2_m are the Gibbs energy of the O3-Li_xMO₂ phase at x= 1 and x = 0, respectively. x_Li and x_Va denote the mole fractions of Li and Va, respectively. R is the gas constant.

Considering the experimental heat capacity data, O3-LiCoO₂ is modeled using the following expression^[31]:

$ \begin{array}{*{20}{c}} {{}^oG_m^{{\rm{O3 - LiCo}}{{\rm{O}}_{\rm{2}}}} - H_{{\rm{Li}}}^{{\rm{SER}}} - H_{{\rm{Co}}}^{{\rm{SER}}} - 2H_{\rm{O}}^{{\rm{SER}}} = A + B \cdot T + }\\ {C \cdot T \cdot \ln T + D \cdot {T^2} + E \cdot {T^{ - 1}}} \end{array} $

(6)

where A and B are parameters to be optimized, the coefficients C, D and E, are obtained from the expression of heat capacity for O3-LiCoO₂. The Neumann-Kopp rule is applied to describe O3-CoO₂, O3-LiNiO₂ and O3-NiO₂ because no heat capacity data above 298 K are available^{[6, 31]}.

G_m^Ord(y_i^s) in Eq.(4) can be represented as follows:

$ \begin{array}{l} G_m^{{\rm{Ord}}}\left( {y_i^s} \right) = \sum\limits_{i = {\rm{Li}}}^{{\rm{Va}}} {\sum\limits_{i = {\rm{Li}}}^{{\rm{Va}}} {{{y'}_i}y{''}_i^0G_{i:j:{\rm{M}}:{\rm{O}}}^{{\rm{Ord}}}} } + 1/2 \cdot RT\left( {\sum\limits_{i = {\rm{Li}}}^{{\rm{Va}}} {{{y'}_i} \cdot \ln {{y'}_i}} + } \right.\\ \;\;\;\;\;\;\;\left. {\sum\limits_{i = {\rm{Li}}}^{{\rm{Va}}} {{{y''}_i} \cdot \ln {{y''}_i}} } \right) + {{y'}_{{\rm{Li}}}} \cdot {{y'}_{{\rm{Va}}}} \cdot \sum\limits_{i = {\rm{Li}}}^{{\rm{Va}}} {{{y''}_i} \cdot {}^0L_{{\rm{Li}},{\rm{Va:}}i{\rm{:M:O}}}^{{\rm{Ord}}}} + {{y''}_{{\rm{Li}}}} \cdot \\ \;\;\;\;\;\;\;{{y''}_{{\rm{Va}}}} \cdot \sum\limits_{i = {\rm{Li}}}^{{\rm{Va}}} {{{y'}_i} \cdot {}^0L_{i:{\rm{Li}},{\rm{Va:M:O}}}^{{\rm{Ord}}}} + {{y'}_{{\rm{Li}}}} \cdot {{y'}_{{\rm{Va}}}} \cdot {{y''}_{{\rm{Li}}}} \cdot {{y''}_{{\rm{Va}}}} \cdot \\ \;\;\;\;\;\;\;{}^0L_{{\rm{Li}},{\rm{Va:Li,Va}}:{\rm{M:O}}}^{{\rm{Ord}}} \end{array} $

(7)

where ⁰G_i:j:M:O^Ord, y_i^m, L are the Gibbs energy of the corresponding end-members, lattice fraction of the m-th sublattice, the interaction parameter, respectively.

4.2 Integrating First-Principles Calculations

In the past decades, density functional theory (DFT) based first-principles calculations has played an important role in both the fundamental and applied research. The method has become one of the most popular and reliable methods for computational physics, chemistry and materials science^{[6-14, 27-29, 61-63]}. The calculations are performed with atomistic detail by solving the Kohn-Sham equation, which could allow researchers to obtain the composition-structure-property relationships of matter without experimental data. Thermodynamic data are essential not only to better understand the properties of the cathodes in LIBs, but also to provide important input for the CALPHAD approach. When experimental information is scarce or completely lacking, it is necessary to obtain the data by first-principles calculations or empirical methods. It has been demonstrated that first-principles calculations provide a successful way of predicting the thermodynamic data for a wide range of materials^[64-69], such as metallic compounds, metal carbides, etc. first-principles calculations are usually carried out using pure DFT, as implemented in the Vienna first-principles simulation package (VASP)^[70-71]. Spin polarization effects are included when magnetic contributions to the total energy are significant.

4.3 Integrating Empirical Methods

Considering the absence of experimental data and reliable calculations, it is necessary to estimate the entropies of metal oxides by empirical methods. The estimation of standard entropy data of inorganic compounds^[72-75] shows reasonable agreement with the experimental results. The entropies of formation for the oxides can thus be obtained using empirical methods, which are used to determine the Gibbs energy functions for the oxides in the CALPHAD approach. The entropy data obtained for the Li-transition metal oxide, Li_xM_yO_z (M=Co, Ni) can be expressed as follows^{[14, 72]}:

$ \begin{array}{l} \Delta {\rm{S}}\left( {{\rm{L}}{{\rm{i}}_x}{{\rm{M}}_y}{{\rm{O}}_z}} \right) = S\left( {{\rm{L}}{{\rm{i}}_x}{{\rm{M}}_y}{{\rm{O}}_z}} \right) - \left[ {{x_{{\rm{Li}}}}S\left( {{\rm{Li}}} \right) + {x_{{\rm{Co}}}} \cdot } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {S\left( {\rm{M}} \right) + 0.5 \cdot {x_{\rm{O}}} \cdot S\left( {{{\rm{O}}_2}} \right)} \right] \end{array} $

(8)

where S(Li), S(M) and S(O₂) denote the standard entropies of Li, M and O₂ in the reference states, respectively.

A combination of the CALPHAD approach, first-principles calculations and empirical method has been applied to study the phase equilibria, thermodynamic and electrochemical properties of the Li-(Co, Ni)-O system as cathode materials in LIBs^{[6, 29-31]}.

4.4 Examples 4.4.1 The Li-O system

The thermodynamic data^[76-89] have been considered in the modeling^[61]. The formation enthalpy data for Li₂O from experiments and first-principles calculations^[14] are consistent with each other. The fusion enthalpy of Li₂O was estimated to be 58.576 kJ/mol. The enthalpy of formation for Li₂O₂, -632.62 and -605.9 kJ/mol, were determined using solution calorimetry. A value of -673.5 kJ/mol was obtained by first-principles calculations^[14]. The experimental heat capacities of Li₂O and Li₂O₂ can well be reproduced by the CALPHAD approach^[61], as shown in Fig. 5.

Five stable phases at temperatures above 298 K, i.e. (Li), Li₂O, Li₂O₂, liquid and gas phases, are considered in the thermodynamic modeling^[61]. The equilibria between the liquid phase and solid Li₂O have been investigated and the solubility of O in liquid over temperature range 468-1 007 K has been well determined^[77]. Sangster and Pelton^[76] reviewed the literature data^[90-103] and reported an eutectic reaction involving (Li), Li₂O and liquid at 453.6 K. The melting point of Li₂O remains uncertain^{[61, 76]}. Li₂O₂ decomposes into Li₂O and O₂ on heating at 613 K.

Considering all the experimental data, the Li-O phase diagram including the gas phase at 1 bar total pressure has been calculated using the CALPHAD approach^[61], which is shown in Fig. 6. This is the first Li-O phase diagram ever obtained. The calculated thermodynamic properties and phase equilibria agree reasonably with the literature information and the first-principles data^[61]. A set of self-consistent thermodynamic parameters for the Li-O system was obtained for modeling of further multi-component Li-containing oxide systems^{[6, 30-32]}.

4.4.2 The Li-Co-O system

Abe and Koyama^[30] and Chang et al.^[31] have performed thermodynamic modeling of the Li-Co-O system using the CALPHAD approach. Five phases, i.e. O3, O3(Ⅱ), O3'(ordered O3, also known as the monoclinic phase), H1-3 and O1, have been detected by experiments^{[21-23, 104-108]}. The O3(Ⅱ) phase comes from the insulator-metal transition of the O3 phase (the Li-rich O3 is an insulator). Due to lack of thermodynamic information and appropriate model, these two phases are considered as the same O3 phase^[31]. In the thermodynamic modeling, only four phases (O3, O3', H-13 and O1) are considered and their structural data are summarized by Abe and Koyama in Ref.[30]. Using the obtained thermodynamic description, thermodynamic properties can be calculated and compared with the literature data^[30-31]. The experimental heat capacity data of O3-LiCoO₂ can be well described using the model^[31]other than the Neumann-Kopp rule^[30], as shown in Fig. 7(a). Fig. 7(b) shows the calculated Gibbs energy of formation for O3-LiCoO₂ compared with literature data. Both the CALPHAD calculations agree well with the experimental data^[30-31].

The calculated phase diagrams of the O3 structural LiCoO₂-CoO₂ system together with the experimental data is presented in Fig. 8. and The dotted lines indicate the order-disorder transition The CALPHAD result from Abe and Koyama^[30] is shown in Fig. 8(a), which overestimates the order-disorder transition temperature and presents a wide range of the ordered phase. Compared with Ref.[30], the work by Chang et al.^[31] can describe the experimental order-disorder transition perfectly, as shown in Fig. 8(b).

The calculated cell voltage of Li/O3-Li_xCoO₂(0≤x≤1) at 300 K using the CALPHAD approach^[30-31] is shown in Fig. 9. The calculations can reproduce the experimental data quite well.

4.4.3 The Li-Ni-O system

During cycling of the Li/O3-LiNiO₂ cell, four different phases are detected in the LiNiO₂-NiO₂ pseudo-binary system, i.e., O1, H1-3, O3 and O3' (ordered O3) ^{[21, 25, 109-116]}. For the O3-LiNiO₂ phase, the heat capacity data below 300 K have been determined by Wang and Navrotsky^[23] using drop solution calorimetry. A term, -7.16·T·ln(T), is added for O3-LiNiO₂ in order to adjust the heat capacity data at 298 K^[6].The calculated heat capacity of O3-LiNiO₂ together with experimental data^[6] is shown in Fig. 10(a). Fig. 10(b) shows the calculated enthalpies of formation for O3-Li_xNiO₂ (0 < x < 1) relative to O3-NiO₂ and O3-LiNiO₂, which agree well with the first-principles data^[6].

The calculated LiNiO₂-NiO₂ pseudo-binary phase diagram below 450 K is shown in Fig. 11(a)^[6]. The literature shows that the symmetry of the disordered phase is maintained as the Li/O3-Li_xNiO₂ (0 ≤ x ≤ 1) cell is charged to x = 0.75^[6]. At elevated temperatures, the ordered O3 phase transforms into the disordered O3 phase^[6].They are well reproduced by the CALPHAD results. The transformation temperature is found to be 383-413 K for Li_0.5Ni_1.02O₂ and 393-413 K for Li_0.52NiO₂, respectively. The phase diagrams at 298 K are shown in Fig. 11(b)^[6]. At x < 0.5, some controversies about the phase diagram exist. The single H1-3 phase region has been reported at x = 0.25-0.45, 0.15-0.35, 0.31-0.43 and 0.22-0.29^[6]. Compared with the experiments, the calculated phase diagram is quite reasonable^[6].

The calculated cell voltage of Li/O3-Li_xNiO₂(0≤ x ≤ 1) at 300 K is shown in Fig. 12, which is consistent with the experimental data^[6].

4.4.4 The Li-(Co, Ni)-O system

Fig. 13(a) shows the variation of the lattice parameters and c/a ratio (both first-principles calculated and experimental results^{[21-25, 115-119]}) versus y in the O3-LiCo_yNi_1-yO₂ solid solution. It is clearly seen from Fig. 13(a) that both a and c decrease almost linearly with increasing Co content. At the same time, the c/a ratio increases almost linearly with increasing Co content, suggesting that substitution of Co/Ni increases the 2D character of the structure^[6].

The calculated enthalpies of formation for the O3-LiCo_yNi_1-yO₂ (y = 0, 1/4, 1/3, 1/2, 2/3, 3/4 and 1) phases are shown in Fig. 13(b), which agree well with the experimental data^[6]. Similar to the trend of the lattice parameters, both the theoretical and experimental results for the enthalpies of formation show an approximately linear correlation versus the Co content^[6]. Thus, the solid solution of LiCoO₂ and LiNiO₂ has been considered to be an ideal solution^[6]. The thermodynamic description of the Li-(Co, Ni)-O system has been obtained by extrapolation^[6].

As the calculations of the cell voltages for Li/O3-Li_xCoO₂ and Li/O3-Li_xNiO₂ agree well with the experimental data, it is reliable to predict the cell voltages of Li/O3-Li_xCo_yNi_1-yO₂ (0 ≤ x ≤ 1, 0 < y < 1). Four Li/O3-Li_xCo_yNi_1-yO₂ cells (y = 0.3, 0.5, 0.6 and 0.8) have been presented as examples^[6], the calculated cell voltages at 300 K are shown in Fig. 14. Because no experimental open circuit voltage is available, the first charge/discharge profiles during cycling of these cells are shown as a comparison. In all the cases, the calculations agree well with the experimental data^[6], which shows a success of the thermodynamic description in predictions of cell voltages for Li ion batteries.

4.4.5 The Li-Sb system

Considering all the experimental data, the Li-Sb phase diagram has been calculated using the CALPHAD approach^[34] as shown in Fig. 15. Besides, the variation of electromotive force with temperature at various compositions and the coulometric titration curves of the Li-Sb system were calculated in Figs. 16 and 17, which show that the calculated results exhibit a good agreement with the experimental data. The thermodynamic description of the Li-Sb system^[34] was reasonable and the thermodynamic parameters could be used for the establishment of the thermodynamic database for Li-Sb-based system.

4.4.6 The Li-Sn system

Based on DFT, the equilibrium open-circuit voltage^[54] was calculated as shown in Fig. 18, which was consistent with the experimental determination^[47-48]. Courtney et al.^[55] calculated the theoretical electrochemical voltage profile of the Li-Sn system and found that the calculated results could well reproduce the experimental work for x > 2.5, but for x < 2.5 the equilibrium electrochemical voltage were not formed in the room-temperature electrochemical cell. Considering the elastic-stain energy contribution, the phase evolution of the Li-Sn battery was analyzed based on DFT ^[54], which indicated that the whole lithiated processes could be classified into the two-phase equilibrium process and the selective equilibrium. For the two-phase equilibrium process^[54], thermodynamic driving force is large enough to facilitate the phase transition and the plateau voltage could be established. For the selective equilibrium^[54], thermodynamic driving force is not enough to facilitate the nucleation of the new equilibrium phase due to the elastically constrained conditions and the plateau voltage unformed. Fig. 19 shows a schematic illustration of these two processes^[54].The left part indicates the process of the two-phase equilibrium, which shows one plateau at its lithiated range; the right part indicates the selective equilibrium, which shows that the voltage decreases continuously at its lithiated range.

4.4.7 Other examples

The Li-Mn-Ni-O system has received much attention for potential positive electrode materials and the pseudo-ternary system Li₂O₂-Mn₂O₃-NiO has been studied with solution-based combinatorial samples of lithium manganese nickel oxide^[120-122]. The areas of the Gibbs triangle that lie between the layered and spinel structures as well as the single-phase layered region has been reported^[120]. The layered solid solution region has been determined and found to be made up of a single composition line joining LiCoO₂ to Li₂MnO₃^[121]. The solid solution undergoes a phase separation during slow cooling and the end points are the Co-rich phase containing some Mn and the Mn-rich phase containing some Co^[121]. Also, metal site vacancies are present in the structures in increasing amounts as the lithium content of the samples decreases^[122]. The relationship between the stoichiometry and stability of the electrode materials has been reported, which is crucial for material design.

5 Conclusions

Advanced LIBs require lower cost, lower capacity loss, longer lifetime, higher energy density, higher stability and inherent safety. The Li-(Co, M)-O(M= transition metals) and Li-metal systems, as promising candidates for future electrodes in LIBs, are of great interest. The CALPHAD approach integrated with first-principles calculations and empirical methods has been applied to study the Li-(Co, Ni)-O, Li-Sb and Li-Sn systems as electrode materials in Li ion batteries. All the phases can be described with appropriate sublattice models and parameters. The calculations on their phase diagrams and thermodynamic properties are in good agreement with literature data. Using the derived thermodynamic description, the cell voltages have been well reproduced/predicted. This shows that CALPHAD has successfully been applied to study the phase equilibria, thermodynamic and electrochemical properties of electrode materials in LIBs. Furthermore, the thermodynamic datasets obtained for these systems enables the establishment of a whole thermodynamic database for multi-component Li-containing systems and the design of future electrode materials with improved stability and efficiency.