High-capacity electrode materials typically experience large volume changes during Li insertion and extraction. For the conversion-type anode material such as Si, its lithiation at room temperature usually yields amorphous Li _3.75 Si ( a - Li _3.75 Si), resulting in a volume increase of about 280%. Such a large volume expansion can be largely removed upon delithiation. Under constraints, the large volume changes lead to stress generation in the electrode materials. The resulting stresses can be very high, causing plastic flow. Sethuraman et al. ^{[ 9 ]} have used in situ wafer-curvature techniques to measure the stress evolution in thin films of a -Si during Li insertion and extraction. Figure 2(a) shows the schematic of their experimental setup, where the lithiation and delithiation of an amorphous Si thin film are constrained by a Si wafer substrate with a Cu underlayer in between (acting as a current collector). Figure 2(b) shows the measured galvanostatic discharge and charge curves, and figure 2(c) shows the resultant biaxial stresses in the film. It is seen that during the initial stages of lithiation, the film undergoes elastic loading. At a capacity of about 300 mAh/g, the film yields at a compressive stress of about 1.5 GPa, such that the plastic deformation begins to occur with continuing plastic flow throughout the rest of lithiation. Upon delithiation, a similar elastic–plastic response occurs, resulting in tensile biaxial stresses in the film. These measurements reveal the existence of the high yield stresses and extensive plastic flow during lithiation/delithiation of a -Si under constraints.

Fig. 2.

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	Fig. 2. In situ measurement of stress evolution in an a -Si thin film during lithiation. [ 9 ] (a) Schematic of the in situ curvature technique for measuring stresses in lithiated thin film. (b) Cell potential versus capacity curve corresponding to lithiation and delithiation of a -Si thin-film electrode cycled at a rate of C/4. (c) Corresponding biaxial stress measured in the film.

Stress generation in particles and wires with curved geometries, which are the common building blocks of electrodes, is different from that in flat thin films. This is because the curvature effect can be significant during stress generation in spherical particles and cylindrical wires, particularly when a two-phase mechanism is involved. In situ TEM experiments have shown that the lithiation of Si particles, with either a starting crystalline ^{[ 17 ]} or amorphous ^{[ 19 ]} phase, usually proceeds through the formation of a core-shell structure, involving the movement of a two-phase boundary that separates the inner core of pure Si with the outer shell of a -Li _x Si ( x ∼ 3.75). Such a sharp phase boundary suggests that the Li-poor and Li-rich phases do not transform continuously into each other with changing composition. That is, there is a large solubility gap (i.e., Δ x ) between the two phases, manifested as an abrupt change of Li concentrations across the phase boundary.

Huang et al . have developed a chemo-mechanical model to analyze the stress evolution during two-phase lithiation of a spherical Si particle (Fig. 3(a) ). ^{[ 30 ]} This model accounts for plastic deformation resulting from large lithiation strains. Particularly, it highlights the curvature effect of the sharp phase boundary on stress evolution during lithiation. Figure 3(b) shows two representative results of radial Li distribution, both of which involve a sharp phase boundary with an abrupt change of Li concentration. Figures 3(c) and 3(d) show the simulated radial stress distributions corresponding to the Li profiles in Fig. 3(b) . The most significant result from this study is the tensile hoop stress in the surface layer developed as the two-phase boundary moves into the interior of the particle, as seen from Fig. 3(d) . Such hoop tension results from a reversal of the initial hoop compression as shown in Fig. 3(c) .

Fig. 3.

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Fig. 3. Modeling of stress generation in a spherical Si particle undergoing two-phase lithiation. [ 30 ] (a) Schematic of the two-phase model with a sharp phase boundary between the Si core and a -Li 3.75 Si shell. (b) Numerical results showing the radial distribution of Li concentration c normalized by its maximum value at the fully lithiated state, and the radial distance r is normalized by the current radius R of a partially lithiated particle which increases as lithiation proceeds. (c), (d) Numerical results showing the radial distribution of the von Mises effective stress σ e , radial stress σ r , and hoop stress σ θ = σ φ (normalized by Young’s modulus).

To provide a physical interpretation of the reversal of hoop compression to tension in the surface layer, figure 4 shows the schematics of hoop stresses experienced by a representative material element A near the surface. Here the shell is assumed to be fully lithiated with a constant Li concentration. It follows that the lithiation strain should be mainly generated near the moving core–shell interface, where the Li concentration changes abruptly. Figure 4(a) shows that in the early stage of lithiation, element A is located within the pristine core. As lithiation occurs at the reaction front, the newly lithiated material at the front tends to move in the outward radial direction. This arises because there are larger areas in the hoop direction at larger radial distances, where the lithiation-induced volume expansion can be better accommodated with lower stresses generated. The outward displacement of newly lithiated materials results in hydrostatic tension in element A, as represented by stage (a) of the σ _θ curve in Fig. 4(d) . As the reaction front sweeps through element A, a large dilational lithiation strain is created at A. Owing to the constraint of surrounding material, local compressive stresses develop, such that element A sequentially undergoes tensile elastic unloading, compressive elastic loading, and compressive plastic yielding in the hoop direction. This stress sequence is schematically represented by stage (b) in Fig. 4(d) . Interestingly, as the reaction front continues to move toward the center, the lithiationinduced swelling at the front pushes out the material behind it. This action causes further displacement of element A in the outward radial direction and simultaneously stretches it in the hoop directions. As a result, element A experiences compressive elastic unloading, tensile elastic loading, and tensile plastic yielding, which correspond to stage (c) in Fig. 4(d) .

Fig. 4.

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Fig. 4. Illustration of the change of hoop stress σ θ with time t in a spherical particle, which contains a moving two-phase boundary between the pristine core (white) and lithiated shell (grey). [ 30 ] (a)–(c) σ θ in a representative material element A near the surface. As the phase boundary moves toward the center of the particle, progressive lithiation results in a gradual expansion of the particle, leading to the reversal of hoop compression in panel (b) to hoop tension in panel (c). (d) Schematic plot of σ θ versus t in element A.

It is important to emphasize that the reversal of hoop compression to tension arises due to the curvature effect of the phase boundary; namely, behind the curved phase boundary, there are larger areas in the hoop direction at larger radial distances. Such reversal will not occur if the phase boundary is flat as in the case of two-phase lithiation in a thin film. More importantly, the high tensile stress and associated tensile plastic flow in the surface layer can cause fracture of electrode particles during Li insertion, which will be discussed in detail in Section 3.4. Finally, we note that for cylindrical wires, a similar conclusion can be drawn for the curvature effect on the reversal of hoop compression to tension in the surface layer of the wires.

The coupling between electrochemistry and mechanics is usually strong in high-capacity electrode materials. For example, high stresses can be induced during electrochemical lithiation of Si, as discussed in Sections 3.1 and 3.2. Such high stresses, in turn, can affect the rate of lithiation. Liu et al . conducted the in situ TEM study of lithiation kinetics in a c -Si nanowire. ^{[ 31 ]} Figures 5(a) – 5(e) show the evolution of the twophase, core–shell structure in a c -Si nanowire during lithiation. In Fig. 5(f) , the measured thickness of the a -Li _x Si shell is plotted as a function of time; also plotted are the diameter changes of the Si core and the etched depth, taken from a typical cross section of the Si nanowire. During the lithiation process over 4 hours, the diameter of the c -Si core decreases from 139 nm to 101 nm, and the thickness of the a -Li _x Si grows from 0 to 54 nm. In the first hour, the lithiation was fast and the average velocity of the core–shell interface was about 10 nm per hour or equivalently 3 × 10 ⁻¹² m/s. The lithiation apparently slowed down as the core–shell interface moved to the center of the Si nanowire, and became extremely slow after about 2.5 hours. These results clearly demonstrate the self-limiting lithiation in Si nanowires. McDowell et al . also reported the slowing of lithiation in c -Si nanoparticles as the two-phase reaction front moved to the center of nanoparticles. ^{[ 32 ]} Recently, Gu et al. showed that bending of a Ge nanowire broke the lithiation symmetry, such that lithiation was sped up at the tensile side while slowed down at the compressive side of the nanowire. ^{[ 33 ]}

Fig. 5.

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	Fig. 5. In situ TEM experiment of self-limiting lithiation in a Si nanowire. [ 31 ] (a)–(e) Time-lapsed images showing the evolution of the c -Si core and a -Li 3.75 Si shell in the nanowire in 4 hours. (f) Plots of nanowire diameter, etched depth, and thickness of the a -Li 3.75 Si shell versus time.

The self-limiting lithiation can be reasonably attributed to the retardation effect of lithiation-induced stresses. As discussed in Section 3.2, large compressive stresses can build up in the lithiated shell, owing to the geometrical constraints from the unlithiated c -Si core. At this moment, there is a lack of experimentally measured chemo-mechanical properties of Li _x Si such as chemical strains of Li insertion and elastoplastic properties of Li _x Si, all of which can depend sensitively on Li concentration. As a result, the quantitative stress distribution in lithiated Si cannot be determined. Therefore, it remains an open question whether the self-limiting lithiation is dominantly controlled by the stress-retarded reaction at the two-phase boundary or the stress-retarded diffusion within the lithiated shell behind the two-phase boundary.

The electrochemically-induced stresses often cause the mechanical degradation in electrode materials. In situ TEM experiments by Liu et al. ^{[ 17 ]} have revealed a strong size dependence of fracture in Si nanoparticles, i.e., there exists a critical particle diameter D _c of ∼ 150 nm, below which the particles neither cracked nor fractured upon first lithiation, and above which the particles initially formed surface cracks and then fractured due to lithiation-induced swelling.

Figure 6 shows the in situ TEM observation of lithiation responses in a c -Si nanoparticle with the initial diameter of about 54 nm, larger than the critical size D _c . This c -Si nanoparticle was in contact with a W electrode and a Li metal counter electrode, whose surface was covered with Li ₂ O acting as a solid electrolyte. Under an applied voltage between the two electrodes, Li quickly covered the particle surface and then diffused radially into the particle, forming the structure of a pristine inner Si core and an a -Li _3.75 Si alloy shell with a sharp interface in between. As the diameter of the Si core shrunk to about 300 nm, the lithiation-induced swelling caused crack initiation from the particle surface. Further lithiation resulted in fragmentation of this nanoparticle. In contrast, figure 7 shows the lithiation of two Si nanoparticles (marked as A and B) with initial diameters of ∼ 80 and 40 nm, respectively, which are significantly below the critical size D _c . Both particles underwent the core–shell lithiation process and remained unbroken after full lithiation.

Fig. 6.

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	Fig. 6. In situ TEM observation of cracking in a c -Si nanoparticle during lithiation. [ 17 ] (a) A c -Si nanoparticle with the initial diameter of 540 nm. (b) A core–shell structure forms as lithiation progresses, and the lithiation-induced swelling causes crack initiation at the particle surface.

Surface cracking in large Si nanoparticles arises due to the buildup of a large hoop tension that reverses the initial compression in the surface layer, as discussed in Section 3.2. While the resulting hoop tension tended to initiate surface cracks, the small-sized nanoparticles nevertheless averted fracture. This is because the stored strain energy from electrochemical reactions is insufficient to drive crack propagation, as dictated by the interplay between the two length scales, i.e., particle diameter and crack size, that control the fracture. More specifically, the crack extension is controlled not only by the high stresses but also by the size of the highstress region that is dictated by the interplay between two length scales, i.e., particle diameter and flaw size. When a crack-like flaw of length a is formed in a large particle, near the crack faces the lithiation-induced tensile stress is relaxed to zero, and far from them it is unchanged. Approximately, a region of size a around the crack is relieved of its elastic energy, and this energy release is insensitive to the particle size. However, when the particle size is reduced to be comparable to the crack size, the region of zero stress near the crack faces and accordingly that of elastic energy relief become dependent on the particle size. The smaller the particle is, the less the stress-relief volume, and the lower the elastic energy release. A crack will not extend if the driving force of strain energy release rate is less than the resistance of surface energy. This explains the observation of the critical particle size in averting fracture during lithiation experiments.

Fig. 7.

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Fig. 7. In situ TEM observation of lithiation of two small c -Si nanoparticles without fracture. [ 17 ] (a) Two pristine c -Si nanoparticles, marked by “A” and “B” with diameters of 80 and 40 nm, respectively. There is a twin boundary (TB) in the center of particle A. (b) A partially lithiated state with a core–shell, two-phase microstructure in the two nanoparticles. (c) Neither cracking nor fracture occur in the two Si nanoparticles after full lithiation, and but both nanoparticles have ∼ 280% volume expansion.