Without vacancy in the lattice, interstitial diffusion is very important. Interstitial diffusion involves atoms that migrate from one interstitial position to a neighboring one that is empty. For example, the diffusion of the charge carrier ions (Li ⁺ , Na ⁺ , Ag ⁺ , K ⁺ , Rb ⁺ , Cs ⁺ ) in β -Al ₂ O ₃ ^{[ 17 ]} is in the interstitial mechanism. The transport process of the interstitial mechanism is shown in Fig. 1(a) . ^{[ 18 ]} The migrating atom is initially located at an interstitial site of the host atomic matrix, and it moves to an adjacent interstitial site. When this process occurs, the system encounters a large lattice strain, but there is no permanent displacement of the matrix atoms when the process is completed. Therefore, this process is denoted as the direct interstitial mechanism. In most cases of direct interstitial diffusion, interstitial atoms must be smaller than the matrix atoms and are incorporated into interstitial sites of the host lattice to form an interstitial solid solution.

Fig. 1.

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	Fig. 1. Interstitial diffusion mechanism. (a) Direct interstitial diffusion. (b) Knock-off migration mechanism. [ 18 ]

Comparable to the direct interstitial mechanism, there is the indirect interstitial mechanism. In this case, the size of the interstitial atoms can be the same as the host matrix atoms. These atoms push their neighboring matrix atoms away, into adjacent interstitial sites, and replace them, as shown in Fig. 1(b) . ^{[ 18 ]} Indirect interstitial diffusion is a collective mechanism, which will be discussed in the following of this paper, because at least two atoms are involved and moved in the migration process. In LIB materials, Li diffusion through a direct interstitial mechanism is not very common, while an indirect interstitial mechanism can be observed in many materials. Indirect diffusion is also known as the knock-off mechanism, and it is very common in LIBs materials when the Li concentration is high. For example, knock-off diffusion dominates the Li diffusion in the lithiated state of Li ₇ Ti ₅ O ₁₂ anodes. ^{[ 19 ]} Furthermore, Shi et al . ^{[ 20 , 21 ]} also demonstrated that Li diffusion behavior in Li ₂ CO ₃ is achieved via a knock-off mechanism rather than direct-hopping.

Atoms can also migrate in a collective manner, which involves the simultaneous motion of several atoms in substitutional solid solutions, in which solute atoms are similar in size to the matrix atoms. The diffusion can be classified into two simple categories: direct exchange mechanism and ring mechanism, as shown in Fig. 2 . ^{[ 18 ]} The former is formed by two neighboring atoms moving simultaneously, and the latter corresponds to a ring of three (or more) atoms moving as a group by one atom distance. As there are no vacancies or interstitial atoms involved, this is also called the non-defect mechanism of diffusion. Generally, non-defect diffusion needs higher activation energy than mechanisms with defects, especially for the direct exchange mechanism. For instance, the process of Li-ion migrating between internal Li ₂ N layers in Li ₃ N ^{[ 22 , 23 ]} applies to the collective diffusion mechanism.

Fig. 2.

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	Fig. 2. Direct exchange and ring diffusion mechanism. [ 18 ]

When vacancies are available, vacancy-assisted diffusion is much easier than the non-vacancy diffusion. The vacancy diffusion mechanism involves the exchange of an atom from a normal lattice site to an adjacent vacancy, as shown in Fig. 3 . ^{[ 18 ]} The diffusion of atoms in one direction corresponds to the “motion” of vacancies in the opposite direction. Vacancy diffusion requires vacancies in the lattice, and it is directly related with the concentration of the vacancy defects. With the existence of a vacancy in the lattice, the vacancy mechanism involves much smaller lattice strain upon atom hopping and therefore encounters a much lower activation energy barrier.

Fig. 3.

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	Fig. 3. Vacancy diffusion mechanism. [ 18 ]

Vacancy diffusion dominates the Li diffusion in most electrode materials. On one hand, many Li vacancies are created in the electrode material upon charging (Li removal) of the electrode. On the other hand, the lattice change to the electrode material is rather small upon Li insertion/extraction, which is a general requirement for electrodes of LIBs. Therefore, vacancy diffusion in the electrode materials is much easier than interstitial diffusion. Lithium diffusion in all cathode materials is related with the vacancy mechanism. The concentration of vacancies is important to the Li migration pathway and the energy barrier. For example, Li diffusion pathways and energy barriers in LiCoO ₂ cathodes are different for mono-vacancy and divacancy. ^{[ 24 , 25 ]}

In some LIB materials, clear phase separations and two-phase coexistence may occur during the charging/discharging process. In those cases, the Li migration behavior near the phase boundary is very important. As an example, the LiFePO ₄ phase changes into the FePO ₄ phase upon Li removal, and a clear LiFePO ₄ /FePO ₄ phase boundary is formed in this process. Lithium diffusion behavior in these materials can be monitored through the movement of the phase boundary from the macroscopic point of view. The diffusion behavior with this mechanism clearly differs from those in solid solution materials.

The phase boundary movement mechanism for LiFePO ₄ was first proposed by padihi et al ., ^{[ 26 ]} and now has been extensively discussed in the literature. ^{[ 27 – 32 ]} Three typical models are proposed, as shown in Fig. 4 : the mosaic model, ^{[ 27 ]} the shrinking core model, ^{[ 31 ]} and the domino-cascade model. ^{[ 32 ]} Regardless of different nucleation sites, it is generally accepted that in the charging process, Li-ions are extracted along the b -axis direction accompanied by the LiFePO ₄ /FePO ₄ phase boundary moving along the a -axis direction. So far, the Li-ion diffusion mechanism at the LiFePO ₄ /FePO ₄ phase interface has not been clarified in sufficient detail. LiFePO ₄ is not the only case: phase separation also occurs in the Li ₄ Ti ₅ O ₁₂ ^{[ 33 , 34 ]} and FeF ₃ ^{[ 35 , 36 ]} cathode materials.

Fig. 4.

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	Fig. 4. Schematic representations of (a) mosaic model, [ 27 ] (b) shrinking core model, [ 31 ] and (c) domino-cascade model [ 32 ] for lithium insertion/extraction in LiFePO 4 .

Experimentally, Lithium’s chemical diffusion coefficient can be measured using electrochemical techniques. Several methods, such as electrochemical impedance spectroscopy (EIS), ^{[ 37 , 38 ]} the potentiostatic intermittent titration technique (PITT), ^{[ 39 ]} the galvanostatic intermittent titration technique (GITT), ^{[ 40 ]} and the potential step chrono-amperometry (PSCA), ^{[ 41 , 42 ]} can be used to study the Li diffusion kinetics and to estimate the chemical diffusion coefficients in solid electrodes. These measurement techniques have been widely applied to electrode materials. The measurements of the chemical diffusion coefficient for several typical electrode materials are summarized in Table 1 . ^{[ 43 – 57 ]} However, the results are not consistent under various measurement methods, as illustrated by Table 1 . Taking the LiCoO ₂ ^{[ 44 ]} cathode as an example, the Li-ion chemical diffusion coefficients can differ by 4 to 5 orders of magnitude between various measurement techniques. Such discrepancies among reported studies may originate from many factors, including experimental measurement errors and errors intrinsic to the measurement techniques.

Table 1.

Table 1.

Table 1. Chemical diffusion coefficient of Li ions in different electrode materials. . Electrode materials Chemical diffusion coefficient/cm 2 ·s −1 Measurement method 1.5×10 −10 –8.0×10 −8 (single particle) PSCA [ 43 ] 10 −11 –10 −7 (single particle) (3.85 V < E / V vs. Li/L i + < 4.2 V) EIS [ 43 ] x = 0.5, (003) orientation, thin film Li x CoO 2 thickness, 0.31 μm 1.9×10 −12 GITT [ 44 ] 1.6×10 −13 PITT [ 44 ] 1.6×10 −10 EIS [ 44 ] 4.0×10 −11 –3.0×10 −10 (0.45 < x < 0.7) PITT [ 45 ] Cathode materials 1.71×10 −12 : virgin thin film PSCA [ 46 ] 6.0×10 −11 –5.0×10 −10 : ESD thin film (4.07 V < E / V vs. Li/L i + < 4.19 V) PITT [ 47 ] Li x Mn 2 O 4 3.4×10 −8 –0.7×10 −8 (0.04 < x < 0.92) GITT [ 48 ] 10 −6 –10 −10 : single particle (3.8 V < E / V vs. Li/L i + < 4.3 V) EIS [ 49 ] 10 −14 –10 −12 : thin film (3.4 V < E / V vs. Li/L i + < 3.6 V) PITT [ 50 ] Li x FePO 4 4.97×10 −16 –9.13×10 −15 (0.1 < x < 0.9) GITT [ 51 ] 1.91×10 −15 –1.29×10 −14 (0.1 < x < 0.9) EIS [ 51 ] natural graphite 10 −9 –10 −10 : measured at RT PSCA [ 52 ] New calculation model: mesocarbon microbeads (MCMB) 1.94×10 −11 (thin film thickness, 10 μm) GITT [ 53 ] 1.70×10 −11 (thin film thickness, 60 μm) Anode materials (0.05 V < E / V vs. Li/Li + < 0.25 V) highly oriented pyrolytic 1.14×10 −12 –3.87×10 −11 : bulk graphite (HOPG) 1.42×10 −12 –1.82×10 −11 : powder EIS [ 54 ] (0.05 V < E / V vs. Li/Li + < 0.2 V) thin film: 4×10 −13 –2×10 −13 at 50 °C (0.48 V < E / V vs. Li/Li + < 0.5 V) PITT [ 55 ] nano-size Li x Si: 5×10 −13 –5.1×10 −12 silicon (1.5 < x < 4) GITT [ 56 ] thin film (0.25 μm): 2–3×10 −13 ( E = 0.01 V (Li/Li + )) EIS [ 57 ]

Table 1. Chemical diffusion coefficient of Li ions in different electrode materials. .

Atomic scale simulation of Li diffusion is one way to understand the Li diffusion mechanism in electrodes in detail from the microscopic point of view. Computer simulation, especially first-principles calculation, ^{[ 58 – 61 ]} is a powerful tool for understanding atoms’ diffusion in an electrode at the microscopic level.

Many published studies of lithium diffusion in LIB materials used Monte Carlo (MC) simulations. For example, combining kinetic MC simulations with first-principles calculations, Van et al . ^{[ 24 , 61 , 62 ]} calculated the Li diffusion coefficient in Li _x CoO ₂ with a Li concentration of 0 ≤ x < 1. They found that Li diffusion in this compound occurs predominantly by a divacancy mechanism, and the activation energy barrier for the divacancy migration strongly depends on the Li concentration. Ouyang et al . ^{[ 63 , 64 ]} investigated Li diffusion behavior in Li _x Mn ₂ O ₄ by MC simulation, in which the MC step is imbued with real meaning of time through a comparison with specifically designed experiments. With a real time step, the Li flux can be directly converted into current density, from which the Li-ion conductivity can be calculated via Ohm’s law. The same method was also shown to correctly predict the Li diffusion behavior in LiFePO ₄ materials. ^{[ 65 ]}

Another powerful tool that is widely used to study the Li diffusion behavior in LIB materials is ab initio molecular dynamics (AIMD). ^{[ 66 – 69 ]} Little classical MD simulation of Li migration in electrode materials is reported in the literature because of the difficulty in figuring out a reliable potential to describe the inter-atomic interactions. ^{[ 70 , 71 ]} In contrast, ab initio calculation can correctly reproduce the inter-atomic interaction, and therefore AIMD can give a correct description of the Li diffusion behavior in solid materials. Ouyang et al . ^{[ 72 ]} investigated the Li dynamics in the spinel LiMn ₂ O ₄ compound from AIMD calculations. They found that the energy barrier of Li monovacancy migration is about 0.23 eV and 0.61 eV in LiMn ₂ O ₄ and Li _0.5 Mn ₂ O ₄ , respectively. With a selective AIMD simulation within the adiabatic trajectory approximation, they also identified the one-dimensional (1D) Li diffusion pathway in the LiFePO ₄ cathode. ^{[ 73 ]} In addition, Johari et al. ^{[ 74 ]} studied the diffusion kinetics during the lithiation process of crystalline as well as amorphous Si anodes using ab initio finite temperature MD. They found that the diffusivity of Li is 10 times faster in amorphous Si (∼10 ⁻⁸ cm ² ·s ⁻¹ ) than in crystalline Si (∼10 ⁻⁹ cm ² ·s ⁻¹ ). Moreover, the diffusivity of Si is 2 orders of magnitude slower than Li in amorphous Si, while 4 orders of magnitude slower in crystalline Si. As a consequence, their results reveal that Li has to penetrate the anode by pushing out the Si atoms, generating stress, and the faster diffusion of Si in amorphous Si helps mitigate the large diffusion-induced stress in Si anodes.

First-principles calculations combined with the NEB technique provide another approach to studying Li diffusion in LIB materials in a more microscopic manner. The NEB method optimizes the Li migration pathway and calculates the Li migration energy barrier from the atomic scale, which provides a better understanding of the Li migration mechanism. Recently, many papers have reported on Li diffusion behavior in LIB materials in the literature. ^{[ 75 – 77 ]} For example, using the NEB method, Chen et al . ^{[ 19 ]} studied the Li diffusion mechanism in Li _{4+ x} Ti ₅ O ₁₂ anodes. They demonstrated that Li migration in the discharged state Li ₄ Ti ₅ O ₁₂ is dominated by the vacancy mechanism, while cooperative migration (knock-off mechanism) is dominant in the lithiated state Li ₇ Ti ₅ O ₁₂ . With the same methodology, Xu et al . ^{[ 78 ]} investigated Li diffusion in graphite. Their results show that the Li migration is sensitive to the stacking mode of the graphite and that A–B stacking results in a higher migration energy barrier than A–A stacking. Similar studies have also been carried out to investigate Li migration in graphene ^{[ 79 ]} and silicone. ^{[ 80 ]} After the Li migration energy barrier is obtained, it is convenient to evaluate the self diffusion coefficient through transition state theory ^{[ 12 ]} and the lattice gas model, ^{[ 81 ]} as given by Eq. ( 2 ). With the same method, Wan et al. ^{[ 82 ]} studied the energetics and dynamics of Li insertion into bulk Si at different Li doping concentrations. Their results show that the most stable position of single Li atom insertion into bulk Si is the tetrahedral ( T _d ) site, and the Li migration energy barrier between the adjacent T _d sites is 0.58 eV; the value is similar to the diffusion barrier calculated by Zhao et al . ^{[ 83 ]} Subsequently, Wang and co-workers ^{[ 84 ]} studied the local interaction of Li with Si in single-layer and double-layer structures of silicene; their results indicate that the barriers for Li diffusion are smaller than those for diffusion in bulk silicon. Furthermore, Chan et al . ^{[ 85 ]} studied the electrochemical lithiation and delithiation of faceted crystalline silicon by ab initio molecular dynamics simulations using slab models, and gave a good explanation of the anisotropy of the lithiation observed in experiments.