Fermi energy level refers to the energy when the electron occupation probability equals 0.5 in the electronic energy level (Fermi–Dirac distribution). As the distribution is similar to a step function, it can be approximately considered that electrons mainly fill the energy levels below Fermi energy level at the finite temperature, while levels above are unoccupied. Then the levels in the vicinity occupied by the probability ranging from 0 to 1 are called “transition state”. The lithium ion batteries referred to as “rocking chair” batteries, electrolytes play only the role of transporting lithium ions and are not involved in the electrochemical reaction. Ideally, during the discharge process, only the electrodes are taking part in the redox reaction, enabling electrons to flow through the external circuit in the direction from anodes with higher Fermi energy to cathode material with a lower one. Therefore, the open-circuit voltage of the cell system satisfies

At the cathodes, the charge–discharge’s reversibility is realized by applying the transition metals to the Li-(de)intercalating materials, which enables itself to be stable within a wide range of Li concentration. The electronic configuration of the transition metal ion in cathode is 3 d 4 s , therefore the losing or gaining of electrons in d -electron manifold corresponds to the oxidation and reduction of transition metal. Electrons in 3 d -electron manifold display the localization character and have a narrow energy band due to the strong charge density Coulomb interaction. When the d -energy band is close to the p -energy band, covalent electrons mix with each other, which delocalizes electrons and broadens the band structure, causing the Fermi energy level to move into the conduction band. When electrons from an anode with higher Fermi energy level fill into a cathode with lower Fermi energy level, only the electrons near the Fermi energy level change state rather than filled or empty additional band. The electrons’ transfer at only the Fermi energy level will not lead to an obvious changing in the OCV voltage profile as the SOC evolves. In fact, the obvious change of voltage profile to the “S” type is based on alteration of the configuration of the crystal lattice, accompanied by a consequent shift of electronic structure. For materials with electron localization, the Fermi energy level lies between the valence and conduction bands. Taking the discharge process of the battery with Li anode and LiFePO ₄ cathode as an example, while electrons are flowing out of the anode to fill the cathode, they occupy the additional empty band (Fe ³⁺ ) which was originally the valence band, because the conduction band is filled in Fe ²⁺ . From the perspective of real space, electrons are localized to the single Fe ³⁺ , contributing to the formation of Fe ²⁺ , and attracting the neighboring unstable Li ⁺ to form LiFePO ₄ . Note that the electrons hop in (Li)FePO ₄ as “small polarons”, arising from the fine tuning of the surrounding crystal lattices. The propulsion of phase interface is driven by the stress field, which brings out the so-called two-phase reaction, exhibiting an “L” type voltage profile and a stable OCV during this discharge process. Since LiFePO ₄ and FePO ₄ hold a certain degree of reduced miscibility gaps arising from the dynamics factors, a limited solid solution occurs, and thus appears the delocalization of electrons to some extent. As shown in experimental results, both side knees of the “L” type voltage profile will end up with a limited range of “S” type curve. Next the physics meanings of different types of voltage profiles is demonstrated in detail from the aspect of Gibbs free energy.

At the anodes, the various analysis methods should be taken to match the specific reaction mechanisms during the charge/discharge process, such as alloying reaction (e.g., Li, C, Si, and so on, summarized in Ref. [ 6 ]), (de)lithiation reaction (Li ₄ Ti ₅ O ₁₂ ), ^{[ 7 ]} conversion reaction (defined as that there exists at least two products). ^{[ 8 , 9 ]} (i) Li anode: The Li metal anode contains a large number of itinerant electrons near the Fermi surface, and will not experience a phase transition during the charge/discharge process; thus the Fermi energy level keeps unchanged unless considering the morphology factors. (ii) Li binary alloy anode: The binary system usually experiences a series of phase transitions when Li is in the alloy, indicating the discontinuous changes of Fermi energy level. The phase diagrams and voltage profiles of Li–C and Li–Si are illustrated in Refs. [ 10 ]–[ 12 ] respectively as examples. The most common carbon-based graphite anode with high electrical conductivity keeps structural stability during staged phase transition cycles. However, the silicon-based anode usually endures the high volume expansion, resulting in great changes in morphology and the totally different actual voltage profile from the theoretical data. ^{[ 12 ]} (iii) The (de)lithiation-based anode: Its reaction mechanism is quite similar to that of cathodes mentioned above, and Li ₄ Ti ₅ O ₁₂ will be illustrated from aspect of Gibbs free energy in the next section. (iv) Conversion reaction-based anode: In addition to complicated multi-electron transfer, great changes of morphology contribute to the modulation of voltage profile, called the kinetic hysteresis effect.

To maintain the electrical neutrality, the injections and ejections of electrons and ions from electrodes must appear simultaneously, with the differences of Fermi energy levels and chemical potentials respectively as the driving forces, and the latter can be deduced from relative Gibbs free energies of reactants and products. So the relationship between Fermi energy level and Gibbs free energy is as follows: At the isothermal–isobaric process in the cells, the maximum electrical work generated by the electrode reaction is equal to non-volume work generated by the change of the reaction system, namely the change of molar free energy ( Δ _r G ).

The theoretical electric potential is defined by OCV as V _OC , and the work done by the cell system to surrounding environment is nFV _OC ; thus the change of the Gibbs free energy can be defined as the negative value of the output work in thermodynamics, therefore,

In order to quantitatively study the dependence of Gibbs free energy and the resulting Li chemical potential on the Li ⁺ concentration, we first suppose a simple mechanical mixture of two pure components, where the Gibbs free energy of the mixture reads G ^M = X _A G _A + X _B G _B . According to Gibbs’ phase rule, in a system where two phases coexist, with fixed temperature and pressure, the system has no degrees of freedom, which leads to a constant chemical potential and the linear Gibbs free energy G ^M , as shown by the green line in Figs.  4(a) – 4(d) . Secondly, considering the random mixing, configurational entropy Δ S _m (shown by the pink line) should be included in total Gibbs free energy, as shown in Fig.  4(a) , the term of − T Δ S _m leads the simple two-phase mechanical mixture to the ideal solution. Next, when the effect of excess enthalpies Δ H (shown by turquoise line in Figs.  4(b) and 4(c) ) included, the total Gibbs free energy becomes G = G _M + Δ H − T Δ S _m . On one hand, when the excess enthalpy is negative, it will further reduce the free energy of the mixture in addition to − T Δ S _m and the system results in solid solution, as shown in Fig.  4(b) . On the other hand, when the excess enthalpy is positive as shown in Fig.  4(c) , the free energy has two minimum values at the composition of α and β . At the composition between α and β , the free energy of mechanical mixture α + β is lower than the solid solution (Note that within that range the chemical potentials for each component are all the same). Namely, the phase separation occurs. However, at the two ends of the composition, a limited range of solution appears as the result of configuration entropy. When the lattice mismatch between A and B is too much, the positive excess enthalpy can be quite large, leading to a sharp enhancement of Gibbs free energy as shown in Fig.  4(d) . Finally, strain and stress are usually important factors in consideration. Reducing the particle size to nanoscale can increase the structural tolerance of strain and stress, and thus restrain the phase separation. ^{[ 13 ]}

Fig. 4.

	Figure Option View Download New Window
	Fig. 4. Free-energy-composition diagrams for (a) ideal solution, (b) and (c) regular solutions, and (d) incomplete solid solution.

When the solid solution reaction occurs in the electrodes, it will exhibit a smoothly sloping voltage curve as shown in Fig.  5(b) , the single phase system observes the Gibbs phase rule as: the number of degrees of freedom is 1 for the fixed temperature and pressure. From Eq. ( 3 ), the variation of the chemical potential with the Li ⁺ concentration satisfies

When, however, Li insertion is accompanied by a first-order phase transformation from a Li-poor phase α to a Li-rich phase β , there are two local minima (assuming the host maintains the same crystal structure), as shown in Fig.  5(c) . In the phase separation region (the range between the two minima), the total Gibbs free energy is the mechanical mixture of the two minima. As a result, a linear tangent with the constant partial derivative of ∂G ( x )/ ∂x leads to a plateau voltage profile as shown in Fig.  5(d) , ^{[ 14 ]} which is consistent with the equivalent chemical potentials of two phases in equilibrium. For this case of first-order transition, the nano-size effects have a significant influence on the voltage profile, and the miscibility gap is reduced because of the additional Gibbs free energy, ^{[ 13 , 15 , 16 ]} which may be caused by strain, ^{[ 17 ]} surface or interface energy, ^{[ 18 ]} or Li ⁺ concentration gradient in the diffuse interface, ^{[ 19 ]} as shown in Fig.  6 . ^{[ 7 ]} The insertion chemical potentials are different along various crystal orientations near the surface, responsible for the more curved voltage profiles, as shown in Fig.  6(c) . ^{[ 7 ]} In addition, the term of surface energy together with the distribution of particle sizes will lead to a combined effect similar to that resulting from the solid solution, ^{[ 20 ]} and make a further step to account for the differences among the curved shapes as shown in Fig.  7 , ^{[ 7 ]} especially for the cases that the phase transition occurs in separate individual small particles rather than evolving two-phases interfaces. ^{[ 21 ]} Under this condition, the phase transition appears continuously particle-by-particle, and the Gibbs free energy variation of each particle is influenced by the configuration entropy. The two factors together have a combined effect of hysteresis presented in the voltage profiles, also as shown in Fig.  7 . ^{[ 22 ]}

Fig. 5.

Figure Option View Download New Window

Fig. 5. Voltage curves are linearly related to the slope of the free energy of the electrode material. If the electrode host material forms a solid solution with Li as panel (a), it will exhibit a smooth sloping voltage curve as illustrated in panel (b). If, however, Li insertion is accompanied by a first-order phase transformation from a Li poor phase, α , to Li rich phase, β , it will have a free energy curve as illustrated in panel (c), and resulting in a plateau in the voltage curve as illustrated in panel (d). Steps in the voltage profile appear if an intermediate phase is stable as in panels (e) and (f). [ 14 ]

Fig. 6.

	Figure Option View Download New Window
	Fig. 6. Voltage profiles of different particle sizes for Li x FePO 4 (a), anatase Li x TiO 2 (b), and spinel Li x +4 Ti 5 O 12 ((c) and (d)). [ 7 ]

Fig. 7.

Figure Option View Download New Window

Fig. 7. Schematic of the impact of the surface energy on the Gibbs free energy (a) and voltage (b) profile of an insertion material exhibiting a first-order phase transition upon (de)lithiation. A difference in surface energy leads to a different free energy that scales with the particle surface area. The consequences are a change in the chemical potential and hence in voltage plateau and a shift in the solubility limits. A distribution in particle sizes leads to a distribution of voltages as shown by the dotted voltage profile. [ 7 ]

When an intermediate phase is stable with lower Gibbs free energy, as shown in Figs.  5(e) and 5(f) , it is necessary to consider the two ranges of x ₁ − x ₂ and x ₃ − x ₄ respectively, and the explanation is identical to that of phase-separation reaction.

In LIBs, certain ranges of the voltage profile (especially for the low voltage sides) often appear as capacitor behavior. Significantly different from typical voltage profiles for the insertion reaction in Fig.  8(a) and the phase separation reaction in Fig.  8(b) , the electric potential changes with the components linearly with a non-zero slope, as shown in Figs.  6(b) and 6(c) , ^{[ 7 ]} and Fig.  8(c) , ^{[ 23 ]} which usually corresponds to the Li ⁺ storage at the interfaces ^{[ 23 ]} or near the surfaces, ^{[ 7 ]} both of which can lead to reconstruction of the surfaces and irreversible capacity loss, although the capacitor-like voltage profiles may just result from the electrolyte decomposition. ^{[ 24 ]}

Fig. 8.

	Figure Option View Download New Window
	Fig. 8. Sequence of three different stages of Li storage. Insertion in the bulk of MX (a). If the solubility limit is exceeded, a heterogeneous solid-state reaction characterized by phase separation occurs (b). Finally, the interfacial charging takes place (c). [ 23 ]

In the batteries, the flows of multiple species and positive/negative charges are often coupled together according to the Onsager relation, and the mass-charge coupled transport is driven by the concentration related chemical potential (neutral atoms, atomic groups, or vacancies etc.) and electronic potential (electrons or holes). When the electron distribution is considered in the atomic scale, electrons can depart from neutral atoms, for example, the electrons (e) and lithium ions (Li ⁺ ) can be thought about separately for lithium atoms, then atomic groups or vacancies are left behind (the other charged parts), and they will all re-distribute in the electric field. Then the electric potential describes the gradient distribution of all nonneutral particles, including electrons/holes, nonneutral ions/ionic groups and nonneutral vacancies). In addition, the flow of nonneutral ions, ionic groups or nonneutral vacancies driven by both chemical and electric potential simultaneously, that is, they are actually driven by the electrochemical potential.

The operating mechanism in batteries is influenced by coupled electrical and chemical factors, and the whole circuit is powered by the cooperative motion of Li ions and electrons: in the external circuit, the charge transfer is driven by electric potential and mediated through electrons, while in the internal circuit driven by chemical potential through Li ions. However, the charge and mass transfer across the interfaces between electrode and electrolyte is usually of greatest importance in the cells, which is elaborated in the other issue of this series. Simplifying the complicated interphases as the step potential drop because of the existence of space charge layer, the distribution of Li ⁺ chemical potential as well as electronic potential are shown in Fig.  9 .

Fig. 9.

Figure Option View Download New Window

Fig. 9. The distribution of lithium chemical potential and electric potential for the whole circuit. (a) The distribution of Li + chemical potential during the discharge process; (b) the distribution of Li + chemical potential during the charge process, the contribution of which is opposite to the actual electric potential for the negative charge it has; (c) the distribution of electronic (e − ) potential during the discharge process; (d) the distribution of electronic potential during the charge process; (e) the total electric potential distribution of both e − and Li + simultaneously.

Figures  9(a) and 9(b) show the concentration of Li ⁺ , providing the assumption that every effective Li atom (mobile Li atom) has one electron delocalizing from Li ⁺ , and Li ⁺ other than Li is the only mass carrier. The assumption is not always true especially for alloy anodes, which actually comprise itinerant electrons; thus Li ions diffuse accompanied by electrons. The gradient concentration directly results in the chemical potential ( μ _{Li ⁺} ) variation, which is one of the driving forces. And the inhomogeneous distribution of nonneutral Li ⁺ also contributes to the electric potential variation, as the other driving force. The sloped line of the electrolyte’s potential indicates a nonequilibrium state, and the slope factor is dependent on the current and resistance there, with the assumption that electrons’ mobility is negligible when compared to ions’ mobility, and all the electric potential change can be attributed only to Li ions’ re-distribution during the charge and discharge non-equilibrium dynamic process.

Figures  9(c) and 9(d) show the other contribution to the redistribution of the electric potential, that is the so-called electronic potential variation defined by the energy of electrons. For electrodes, that is equal to the Fermi energy level, and in electrolyte, that is equivalent everywhere for its electronic insulation, except for regions near the interfaces, where the nonneutral electron or hole defects work.

The total effect resulting from hysteresis is shown in Fig.  9(e) , which is consistent with the in-situ electron holography observation of electric potential in an all-solid-state Li-ion battery. ^{[ 25 , 26 ]} This can be explained as follows: In the charge-discharge process, there usually exists two separately electrochemical half-reactions, which are the so-called electrode reactions. Then the consequent total reaction of the above two half-reactions, namely cell reaction, maintains the electrical neutrality. During the discharge process, when the external circuit is switched on, since the Fermi energy level in an anode is higher than that in a cathode, electrons flow from anode to cathode through the external circuit, while Li ions from anode to cathode through internal section to partly make up for the electrical neutrality of electrodes, leaving behind nonneutral interphases called “space charge layers”. As a result, the electric potential μ _electrical distribution is more similar to that of the negative μ _{e ⁻} , although alleviated by the μ _{Li ⁺} ’s compensation. In contrast, during the charge process, external electric field makes the mass and charge flows reverse. Note that only the electrons near the Fermi energy level can flow into or out of electrode particles, and the change of Fermi energy level is finite and modulated by the fine crystal structural adjustment. Thus the formation of Li ⁺ caused by electrical field induced local phase transition predominately accounts for the reversed reaction, and electrons flow in the direction opposite of the external circuit to compensate partially for the neutrality. Therefore although theoretical OCV between electrodes can be calculated thermodynamically, the real output voltage must be influenced by dynamic factors such as the above hysteresis effect.

As discussed above, when the current flows through the circuit, the internal resistance in the battery will cause an Ohmic drop, which results in energy loss, also called Ohmic loss. Many factors contribute to the Ohmic drop, including imperfect contact between electrode slurry and metal current collector, welding spot between current collector and external circuit, potential drop on porous electrode with the certain thickness and accordingly the certain resistance, potential drop on the electrolytes (especially for solid electrolytes), and so on. Besides, side reactions often enhance the internal resistance of cell, such as the formation of solid electrolyte interphase (SEI), which is also ascribed to aging.

Another topic with regard to potential is the “overpotential” phenomenon, which is caused by the kinetics of the charge and mass transfer with the response frequencies of 1 kHz and 10 Hz, respectively. ^{[ 27 ]} The Butler–Volmer model has been widely accepted recently to interpret overpotential. ^{[ 28 ]} Besides, other complicated side reactions near or of the electrodes, as shown in Fig.  10 , ^{[ 29 ]} have comprehensive influences on the potential distribution adjacent to the region where the reactions occur.

Fig. 10.

	Figure Option View Download New Window
	Fig. 10. (a) Changes at the anode/electrolyte interface; (b) overview of basic ageing mechanisms of cathode materials. [ 29 ]

In the following subsections, the factors with less influence on potential distribution, such as separator and current collector, are not focused on. Only the electrode, electrolyte and their interfaces are involved.

Porous electrodes are often designed to be applied to LIBs, and usually contain active materials with sizes ranging from micrometers to nanometers and the conductive additives for lowering the resistance, which are mixed and then spliced by binders as a unit, and electrolyte with high ionic conductivity filled into the particles’ skeleton to promote ion diffusion, resulting in a so-called porous electrode shown in Fig.  11 . Once there is current in the battery pole pieces with a certain thickness and resistance, the Ohmic drop must occur. For common commercial LIBs, carbon materials are applied as the anodes, which usually have much higher capacity than cathodes. As a result, the cathode plates are often thicker. Along with the usually lower conductivity (often determined by the conductive additives) of cathode materials, ^{[ 30 ]} the Ohmic drop is more severe accordingly.

Fig. 11.

	Figure Option View Download New Window
	Fig. 11. Typical Li ion battery porous electrode. (a) Low magnification (on left) and high magnification (inset on the right). (b) Schematics showing the main phases constituting a modern insertion cathode and its role in transport. The polymeric binder is not shown (its effect on the transport is thought to be marginal). [ 31 ]

Fine single active particles with size of nanometers to micrometers and the continuous conductive additive ensure high electronic conductivity, so that the active particle surfaces can be seen as an electronic equipotential body. Therefore, the ion transport inside the particle is driven only by the Li concentration gradient (i.e., chemical potential). With high ionic conductivity, the electrolyte can be treated as the equipotential surface of Li ⁺ chemical potential, despite the complicated interfaces’ space charge layer. Then the Li ion diffusion rate within the active particle becomes the limiting factor of the total diffusion time.

The advantages of small sizes for the active particles are as follows: (i) The diffusion path and grain boundary resistance of every single particle are both reduced to lower the Ohmic drop on the electrode; (ii) In the case of the Li ion transport in the solid host, assuming the diffusion time τ , the diffusion length L , and the diffusion coefficient D , the relationship among them is: τ = L ² /2 D , which indicates that a shorter diffusion path can reduce the diffusion time of Li ⁺ and improve the kinetics performance; (iii) As for a fixed total volume of battery pole piece, the smaller size of single particles means a larger reaction area, which can reduce the operating current density per unit area and improve the cathode/anode reaction rate with the given rated current, so as to ameliorate the charge transfer polarization; (iv) Reducing the particle sizes into the order of magnitude of nanometers is a common method to increase the solid solution limit, while simultaneously introducing more defects such as vacancies or interstitials to improve the intrinsic Li ion mobility in the materials. However, nanocrystallization causes some problems, for example: (i) High porosity leads to a suppression of the compact density thus the energy density; (ii) Surface effects predominate over the intrinsic bulk features, which turns the interface resistance into the main limiting factor of the rate performance; (iii) The side effects near the surface increase; (iv) A contact problem appears between the active particles and the conductive additives. Moreover, when the particles size decreases to the order of magnitude of nanometer, other unknown effects may occur, for example: Will the reconstruction of the surface crystal and electronic structures due to the nanocrystallization have the influence the cell performance? Will the nanoscale interface bring about the pseudocapacitive effect or not? What is the formation mechanism of coexistence and segregation of nanoscale phase, or of heterogeneous interface and space charge layer? These are all problems left to be solved.

Given that the size effects are usually accompanied by the interface effects, the problem of the charge and mass transfer across the active particles and electrolytes will be discussed in Section 4.4.

Table 1 assumes the condition that T = 298 K, and the system is simplified to 1 M dilute solution (the concentration is 1 mol/L) model. The ionic conductivity can be derived from σ = ( Dnq ² )/ kT ,

Table 1.

Table 1.

Table 1. Relationship between the diffusion coefficient and the particle size. . 2D (cm 2 /s) 10 −14 10 −12 10 −10 10 −8 10 −6 10 −4 σ /(3.75×) (S/cm) 10 −11 10 −9 10 −7 10 −5 10 −3 10 −1 L (1 mm) 10 12 10 10 10 8 10 6 10 4 10 2 100 μm 10 10 10 8 10 6 10 4 10 2 1 10 μm 10 8 10 6 10 4 10 2 1 10 −2 1 μm 10 6 10 4 10 2 1 10 −2 10 −4 100 nm 10 4 10 2 1 10 −2 10 −4 10 −6 10 nm 10 2 1 10 −2 10 −4 10 −6 10 −8 1 nm 1 10 −2 10 −4 10 −6 10 −8 10 −10

Table 1. Relationship between the diffusion coefficient and the particle size. .

It can be obtained: 10 ²  s (1.7 min)∼36 C; 10 ⁴  s (2.7 h) ∼ 0.36 C. The rate performance between these two points can satisfy the demands of most energy storage devices, so the relationship between the diffusion coefficient and particle size is of importance, as shown in the red and blue square frame. For example, for a particle with a size of about 10 μm, with the requirement for the charge and discharge rate as 1 C, the diffusion coefficient must reach as high as 3 × 10 ⁻⁹  cm ² /s.

High power density requires fast lithium ion transport. It can be seen from the porous electrode model that lithium ion diffusion path is very short within the electrode particles, resulting in generally small resistance of the battery pole pieces. Therefore, for the traditional battery structure, the potential drop usually lies in the electrolyte section. The high charge and discharge rate usually leads to the energy loss occurring in multiple regions of the cell, among which the significant energy loss and heat generation are usually located on electrolyte.

Electrolyte is the electronic insulator for LIBs, thus only the transport of ions and ionic groups is driven by electrochemical potential. Apart from the interfacial region concerning the space charge layer, it can be considered that the large proportion of the region maintains local charge neutrality. Supposing the thermodynamic equilibrium conditions, the spatial distribution of the lithium concentration is uniform, leading to the equivalent chemical potential everywhere. As a result, the electrochemical potential is equal to the electric potential within the electrolyte. The overall potential drop between the two ends can be simply considered as the Ohmic drop. In comparison with the fluid and mobile electrolytic solution without a boundary, in the case of solid electrolyte with the stable configuration, the strong restrictions of lattices or skeletons may lead to long-range interaction, which can hardly be explained by the general space charge layer model in liquid solution without the specific crystal structure and interacting forces from skeletons. In addition, the Li ion mobility of the solid electrolyte is 1 and electronic conductivity is extremely low. Therefore, a natural question is that as compared with the solvation of Li ions in the electrolytic solution, how can the solid electrolyte maintain the electroneutrality under lattice restriction conditions? To the best of our knowledge, there is little literature presenting investigations of the mass-charge transport model specialized for the crystalline solid electrolytes, which may depart from the ideal crystal structure with defects and charges.

Providing the two assumptions, i.e., electroneutrality and continuously changing potential, on which the whole discussion is based, a lot of important thermodynamics and kinetics features are focused on the surfaces and interfaces. For example, when overpotential exists, the partial energy dissipates in the form of heat during the charge/discharge process, and the extent of overpotential is related to the charge/discharge rate. The challenging problems about the interfaces that remain to be solved include: (i) the non-linear phenomena resulting from ultrafast charge/discharge, (ii) the space charge layers with distinct characteristics under various conditions, (iii) the pseudo-capacitance effect, (iv) the spatial and temporal scales of the studied objects, (v) the complex side effects and aging mechanism near the interfaces, including the growth of SEI layers, the decomposition of electrolytes, the formation of Li dendrites, the precipitation of transition metals, the existence of the separated interphase between the solid electrolyte and electrode, and so on.

There has been lots of research on the space charge layer in the electrolytic solution. Owing to the difficulties lying in experiments, the establishment of theoretical models and reasonable simulations becomes markedly important. To balance the requirements of accuracy and practicability, the Gouy–Chapman–Stern model is most widely accepted and adopted so far. Based on the space charge layer model and considering the finite rate, the Butler–Volmer model can be used to explain the kinetics factors such as the overpotential, the transfer of species and charge, the irreversible side effects, and so on, which are caused by the difference between the oxidation and reduction rates under the nonequilibrium condition. In addition, the Poisson–Nernst–Planck equation (including modified forms of the PNP equation, ^{[ 32 , 33 ]} ) the classical continuous ion transport model, can be applied to simulate the linear/nonlinear response with the temporal–spatial parameters, to obtain the spatial distribution of the electric potential, charge density and the salt concentration in the electrolytic solution. ^{[ 34 – 36 ]} The macroscopic continuous model on electric-chemical potential of the solid–liquid interface has been widely studied, including the model based on equilibrium thermodynamics, ^{[ 37 ]} non-equilibrium thermodynamics, ^{[ 38 ]} thermal fluctuations, ^{[ 39 ]} and so on. For the interface between the specific active particle and solvent, both of which have distinct surface structures and chemical properties, the specific theory and simulation at the smaller atoms/molecules scales should be applied to describe the interaction potential among atoms and the electronic structure. For the more complicated phenomena occurring in the practical applications, including the porous electrode geometry, the small-size effect at the nanometer scale, the wettability of electrolytic solution and the influence of the interfacial strong field effect, more accurate models concerning the quantum and atomistic scales are in demand, followed by the difficulty in balancing the calculation cost and accuracy. Note that unlike the periodic structure of crystalline, the disordered interphases need to be simulated by covering larger structurally spatial scales, while electrode kinetics needs to be simulated by covering wider temporal scales. In addition, in order to predict the materials properties and then realize rational materials design, linking up the simulations at different scales is still a long-term challenge, as well as verifying the theoretical results with relevant experiments.

In the case of electrolytic solution, the investigation of interface is usually based on at least one of two assumptions: ^{[ 37 – 39 ]} dilute solution hypothesis and local electroneutrality hypothesis. However, the case is different for the solid electrolyte: (i) The dilute solution model does not work for the strong interaction among mobile carriers and between mobile carriers and lattices; (ii) Thermal fluctuation cannot be ignored for the lattice effects; (iii) Given that the chemical potential is almost equivalent all through the inner bulk of the solid electrolytes (which is equal to the charge/discharge current multiplied by the resistance), the chemical potential difference between anode and cathode (which is the so-called OCV when the current approaches 0) drops mainly on the interfaces and causes a sharp concentration change. In this way, the Debye screening length, which is the spatial scale of the space charge layer treated within the classical Debye–Hückel theory, is hard to estimate for the solid materials with the strong interaction and intrinsic lattice stability; (iv) It is a challenge to establish the evolution model of the solid–solid space charge layer, and even the interphases, because of the lack of reliable experimental data and phenomena for establishing the reasonable models. Therefore, the need is most urgent to develop novel characterization techniques combined with multi-scale theoretical models and simulation methods.

Figure  18 lists the potentials of Fe ³⁺ /Fe ²⁺ redox couple versus lithium of various Fe-based cathode materials, indicating the strong dependence of the potential on the crystal structure. For the phosphates, complex in solution, sulfates, and others (as listed in Fig.  18 ), the potential variation span of Fe ³⁺ /Fe ²⁺ among them is close to 1.5 V. Apart from the influence by the various (poly)anions as discussed in Section 5.1.2, this phenomenon can also be greatly attributed to the differences among structures. In order to discuss the relationship between the electrode potential and structure, we set forth the specialized polymorphism as an example, and then introduce the difference between potentials of two materials with the same components but different structures.

Fig. 18.

	Figure Option View Download New Window
	Fig. 18. Historical diagram comparing the potential of Fe 3+ /Fe 2+ redox couple (versus Li + /Li 0 ) of various Fe-based (phosphates, sulphates, complexes and others) cathode materials. The Mn-substituted Li 2 (Fe 1− x Mn x P 2 O 7 binary pyrophosphate shows the highest Fe 3+ /Fe 2+ redox potential, close to 4 V and similar to that of Li(Fe 1− x Mn x )SO 4 F. [ 68 , 69 ]

There are two kinds of crystal structures with different order degree in LiNi _0.5 Mn _1.5 O ₄ , corresponding to the and P 4332 space groups, respectively, ^{[ 71 ]} as shown in Fig.  19 . ^{[ 50 ]} For the structure, Li and O atoms occupy the 8 a sites and 32 e sites, respectively, and Ni and Mn atoms 16 d sites randomly, called the disordered structure, in which the O defects usually occur. While for the P 4332 structure, Li atoms occupy 8 c sites, Ni 4 a sites, Mn 12 d sites, and O 8 c and 24 e sites, called the ordered structure. Figures  20(a) and 20(b) show the electrochemical performances of two structures. In Fig.  20(a) , for the structure, there is a little capacity in 4.0 V arising from the Mn ³⁺ . In the differential curves in Fig.  20(b) , the oxidation peaks of lie at 4.69 V and 4.75 V, while those of P 4332 at 4.74 V and 4.77 V, and their reduction peaks of different structures are also different from each other, which can be ascribed to the different band energy structures of different structures. A comparison between the phase transition behaviors of two structures indicates that during the charge/discharge process, both the (lattice parameter: 8.12 Å) and the P 4332 (lattice parameter: 8.16 Å) experience three phase transitions within 3.5–5 V voltage regions, as shown in Figs.  20(c) and 20(d) . The investigation of the doping effects on the mechanism and critical points of phase transition for LiNi _0.5 Mn _1.5 O ₄ is still ongoing. Figure  21 exhibits the polymorphisms of Li ₂ MnSiO ₄ , and the more obvious structural differences contribute to the greater distinction in the calculated average voltages. ^{[ 4 , 72 ]}

Fig. 19.

	Figure Option View Download New Window
	Fig. 19. The structures of LiNi 0.5 Mn 1.5 O 4 with the (a) and (b) P 4 3 32 space groups. [ 50 ]

Fig. 20.

	Figure Option View Download New Window
	Fig. 20. (a) Charge/discharge curves, (b) the corresponding differential capacity versus voltage curves of LiNi 0.5 Mn 1.5 O 4 with the and P 4 3 32 space groups respectively. The charge curves and XRD spectra of LiNi 0.5 Mn 1.5 O 4 with the (c) and (d) P 4 3 32 space groups. [ 70 ]

Fig. 21.

	Figure Option View Download New Window
	Fig. 21. Calculated total energy versus volume curves of Li 2 MnSiO 4 polymorphs; Pmn 2 1 (red), Pmnb (blue) and P 2 1 / n (green). Calculated average voltage for the 2 electron process, host–Li 2 MnSiO 4 ↔ host–MnSiO 4 + 2Li, is given in parentheses. [ 4 , 72 ]

Generally speaking, the cathode materials with discharge voltage higher than 4.5 V are called the high voltage cathode materials, and they mainly include the spinel, olivine and other polyanionic salts (including −O, −F, −OH substitutions). Examples are shown in Table  2 , among them, those with the spinel structure, with the olivine structure and with other polyanionic salt structures along with their discharge voltages are shown in Table  3 , in Figs.  15 and 16 , respectively. The upper limits of the voltage are doubly controlled by both external and internal factors.

The external limiting factor is the highest occupied molecular orbitals (HOMO) of the electrolyte, as shown in Fig.  13 , which usually cannot be exceeded by the Fermi energy level of the cathode to prevent the electrolytes from being oxidized during the initial charge/discharge cycles. Methods to mitigate the external limitation include cathode coatings and adding additives contributing to cathodic SEI layers. However, the elementary way to push the restriction boundary may be the application of electrolytes with wider electrochemical windows, such as the solid electrolytes.

The inner limiting factor lies in the fact that the transition metal elements of the cathodes in the charged state reach a higher valence and exhibit strong oxidizability. As shown from Figs.  22(a) – 22(c) , the Fermi energy level gradually drops from ( d + p ) ⁿ / d ^{n +1} close to the primarily d band of transition metals, into the bottom of the combinational bonding ( p + d ) ⁿ , even into anions’ p band. It can be concluded that if the anions prefer to be deprived of the electrons and oxidized, the structure will lose stability during the intensive Li ⁺ deintercalation, as shown in Fig.  22(c) . As a result, the energy of the redox couple of transition metals should be higher than the p -orbital level of anions, as shown in Fig.  22(a) , or should be close to the p -orbital level top, which leads to the splitting of d -orbital level of transition metals into the bonding orbital b . ( d + p ) ^{n +1} and anti-bonding orbital a. b. ( d + p ) ⁿ , and only the electrons of the latter can be deprived during the charge, thus the Fermi energy level can be pinned above the p -orbital of the anions, as shown in Fig.  22(b) . In conclusion, the position of the p -orbital energy level restricts the voltage of the cathodes intrinsically, which makes it necessary in materials design to match the energy level of the redox couples of transition metals and the p -orbital of the anions. In the early Li secondary batteries, sulfides such as LiTiS ₂ ^{[ 73 ]} are used as the cathodes. However, the energy level of the high oxidation state of transition metals is lower than the 3 p -orbital level of S, resulting in the preferential oxidization of S ions and the low OCV. Oxygen has lower energy with 2 p -orbital than S with 3 p -orbital and promises a higher OCV versus given anode, thus oxide cathodes are much more widely applied than sulfide ones.

Fig. 22.

Figure Option View Download New Window

Fig. 22. Schematic representation of a slightly oxidized redox couple for different positions relative to the top of the anion p bands. (a) Itinerant versus polaronic character of hole states of a couple on the approach to the top of the anion p band, (b) pinned couple with predominantly antibonding (a.b.) anion p hole states and predominantly cationd bonding (b.) states, and (c) couple too far below top of anion p band for significant cation d character in hole states. [ 44 ]

When the layered cathodes Li[Ni _x Co _y ]O ₂ are highly charged, the O 2 p -orbital usually tends to lose electrons rather than the transition metal 3 d -orbital. In other words, the O ²⁻ is oxidized by the high-valent transition metals ^{[ 75 ]} and may be released in the form of O ₂ , ^{[ 76 , 77 ]} making the lattice structure lose its stability and causing safety problems. Manganese doping can lower the energy level of O ₂ p -orbital and enhance the thermal stability, at the cost of the capacity decreasing. ^{[ 77 ]} Apart from the instability arising from O ₂ evolution, other problems related to the layered cathodes and their promising solutions are summed up in Fig.  23 . ^{[ 74 ]}

Fig. 23.

	Figure Option View Download New Window
	Fig. 23. Problems and their solutions for cathode materials under increasing operating voltages and temperatures. [ 74 ]

The more fundamental method to improve stability is to screen for novel materials with the proper structures and compositions that promise higher thermal stability. For example, the spinel structure material Li[Ni _0.5 Mn _1.5 ]O ₄ has higher thermal stability than the layered materials. Figure  24 shows the relationship between the output voltage (i.e., intercalation potential) and the oxygen chemical potential for thousands of materials, obtained from high-throughput calculations. With the chemical potential depicted by the scatter diagram, the charged cathode (under the high intercalation potential) with higher valent transition metal has the probability to decompose, the products can be thermodynamically O ₂ (if the charged state oxygen chemical potential is above 0 eV versus O ₂ ) along with other phases containing the same transition metal with the lower valence (to keep the products’ electric neutrality). This trend can be seen: the materials with higher intercalation potential usually have higher O chemical potential. Moreover, the safety range of the O chemical potential determines the upper limits of the voltages of the cathodes. However, as shown in Fig.  24 , corresponding to the same intercalation potential, the range of O chemical potential is very wide, enabling the possibility of materials design for high-voltage and high-stability cathodes. Although the phosphates are widely believed to have higher stability even when charged to a higher voltage, it can be inferred from the calculated results that borates and silicates may perform better, guiding the direction for further material screening and design. ^{[ 40 ]}

Fig. 24.

	Figure Option View Download New Window
	Fig. 24. Oxygen chemical potentials at which the charged state of thousands of cathode materials decompose versus their Li intercalation voltage. A more negative chemical potential indicates that the material can better withstand a reducing environment (higher temperature) and hence is safer. [ 40 ]

Since OCV is the difference between the anodic and cathodic Li ion chemical potentials, or from the other aspect, between their electronic Fermi energy levels (i.e., their electric potentials), higher OCV can theoretically be obtained by lowering the (de)lithiation potential of the anode. Given that Li metal has the most negative standard electrode potential among the metals, it can be applied as the anode with the lowest potential for Li secondary batteries, as shown in Table  2 . However, when the output voltage of the anode is lower than 1 V versus Li metal anode, as shown in Fig.  13 , ^{[ 43 ]} the Fermi energy level of the anode will be lower than the lowest unoccupied molecular orbital (LUMO) of organic electrolytic solutions, which makes the electrolytes decompose and SEI layers form. Note that since the current batteries are always assembled in the fully discharged state, and cathodes are usually the only Li source, then the formation of SEI layer will consume Li ions of the cathodes and lead to capacity loss.

Usually, ethylene carbonate (EC) can be added to the electrolyte to form a stable SEI layer during the initial cycles. Thanks to the additives, graphite with the potential of 0.2 V can be applied as the anode for the batteries, with low irreversible capacity loss. In addition, the not too low potential also avoids the Li plating. With other advantages, such as the low cost and plentiful raw materials, high specific (2.5 times more than LiCoO ₂ ) and volumetric (nearly equal to LiCoO ₂ ) capacities, low voltage hysteresis, good rate capability, good thermal stability (providing the not too large surface), low volume expansion during lithiation, long cycle life, high Coulombic efficiency (CE), good electronic conductivity, good compaction performance, and so on, graphite anodes are called “miracles”. ^{[ 6 ]}

In order to obtain the higher capacity, Li alloy anodes (including metal oxides sharing the same mechanism with alloys) are widely investigated. ^{[ 6 ]} Since their Fermi energy levels are less than 1 eV versus Li metal, it is a challenge to enable the formation of SEI layer at the initial cycles (the better solution is to complete the formation of SEL layer before the batteries are assembled) and handle the capacity loss. In addition, the volume expansion during the lithiation (charge) process is usually a serious problem. Moreover, for the Li alloy anodes with low potential, Li plating on the surface of SEI layers should be avoided because it can bring about the safety issues, lead to the continuous growth of SEI layers and capacity loss.

The problems related to the low potential anodes can be solved by the application of electrolytes with a wider electrochemical window and higher stability. Another solution is to take advantage of anodes with potential higher than 1 V, such as Li ₄ Ti ₅ O ₁₂ , so that the electrolyte decomposition and the SEI layer growth can be prevented within the operating voltage. In addition, the higher potential avoids Li plating, has the zero-strain property, and high rate capability.

For the traditional electrodes based on the intercalation reaction (e.g., LiCoO ₂ and LiMn ₂ O ₄ ) or phase transition (e.g., LiFePO ₄ and Li ₄ Ti ₅ O ₁₂ ), ^{[ 9 ]} the specific capacity is usually restricted by the electron exchange capacity on the 3 d -orbital of transition metals. In 2000, Poizot et al. found the novel reversible Li storage mechanism in the transition metal oxides, ^{[ 78 ]} followed by the discovery of abundant transition metal sulfides, fluorides, nitrides and phosphides with the similar mechanism, ^{[ 8 , 79 – 87 ]} which reads: MX +Li↔Li X + M ( X = O, S, F, N, P, Se; M = Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Sn, Sb, Ru, Mo), called the conversion reaction. Based on this mechanism, there can be more than one oxidation state of the transition metal’s redox couples, and the Li storage capacity can be increased to 400–1100 mAh/g. The average OCV (AOCV) can be calculated from the thermodynamic data once the cathode and anode are chosen. ^{[ 8 , 80 ]} In 2011, Zu and Li calculated the AOCVs of the binary transition metal compounds with different anions ( X = F ⁻ , O ²⁻ , S ²⁻ , N ³⁻ ), versus the various metallic anodes (Li, Na, Mg, Al), as shown in Fig.  25 . ^{[ 9 ]} With a simple calculation method based only on the known thermodynamic data, the Li-, Na-, Mg-, Al-, Zn-based batteries’ conversion reactions can be systematically investigated, aiming at screening for secondary batteries with higher energy densities.

Fig. 25.

	Figure Option View Download New Window
	Fig. 25. Calculated EMF values ( V ) of conversion reactions between (a) selected binary transition metal compounds and lithium, (b) selected binary transitional metal fluorides and Li, Na, Mg, or Al. [ 9 ]