The mechanical–electrochemical coupling process begins with an electrochemical reaction, when Li inserts into (or extracts from) an active material replaced by the current collector in AFB.^[20] Figure 2 schematically illustrates the case of a plate electrode model. When Li inserts into active material A, a new component Li_xA_y is produced as follows:

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. Schematic diagram of diffusion-induced stress generation in a plate current collector.

The flux is a key physical quantity for the Li diffusion behavior,^[39] and its relationship with the electrochemical potential μ is given by

As shown in Fig. 2, for the AFB regarded as a plate model, the Li concentration is a function of z, i.e., c = c(z). It is assumed that the diffusion coefficient D of Li is constant.^[22] The diffusion of lithium produces diffusion strain via volume expansion of the active material.^[40] In a large electrode plate, the strain components in two planes, i.e., ε_xx = ε_yy are a function of z. The strain equation in the plate model is written as follows:

Combining Eqs. (5) and (8), the stress-assisted diffusion equation can be rewritten as

The initial Li concentration in the AFB anode is assumed to be zero, i.e.,

When lithium inserts into the current collector, an electrochemical reaction occurs. As shown in Fig. 3, the potential of the LiM alloys in AFB is usually lower than the bottom of the electrochemical window, and a thin, amorphous solid/electrolyte-interface layer will be formed.^[14,44] For example, the potentials of LiAl alloy^[45] and LiSn alloy^[46,47] reaction in the AFBs are 0.31 V and 0.42 V vs. Li/Li⁺, respectively. As a comparison, we list the electrochemical potentials of the common current collector materials in Table 2.^[43] Since the electrolyte reduction at the anode does not occur, metals with a high potential, such as copper, tin, {etc}., can be used as a lithium protective layer.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. Schematic illustration of the potential (V vs. Li/Li+) of some common electrode materials in comparison with current collector metals (Al and Cu) and alloys (LiSn, LiPb, and LiAl). The general range on the electrochemical window of the common liquid organic electrolytes is also shown at the top.[43]

Before analyzing the coupling effect, we calculate the Li migration barriers for the metals and the corresponding alloys and compare the volume expansion rate on lithium insertion into several metals. This is an important step for in-depth analysis of the Li diffusion in AFBs. As shown in Fig. 4, the Li migration barriers for M metals are less than 1.00 eV. The barrier of Li diffusion in Ag is as high as 0.98 eV, making Li diffusion more difficult, while Cu and other metals have the lower migration barriers. The Li migration barrier of 0.65 eV for Cu is in agreement with the value of 0.68 eV obtained by Rupp et al. as measured by TOF-SIMS.^[19] This proves that lithium can diffuse in copper, which was previously considered to be difficult. It breaks the conventional knowledge in previous experiments that Li cannot diffuse in Cu due to no formation of LiCu alloy phase at room temperature. However, the actual situation is that Li can diffuse as long as there exist a certain number of vacancies in Cu. Therefore, the insight we gain from our calculations can be used to explain the experimental phenomenon that the use of Cu with a thickness ⩽ 100 nm as the protective coating on the anode significantly improves the lifetime of the battery due to the fast Li diffusion,^[48–50] which has complicated implications for Cu coating to enhance the interfacial protection and kinetics of the electrode. This is also supported by the experiments of copper-based AFBs.^[9,12] As a comparison, the Li migration barriers for Cu and Pb are lower than those for the LiCu alloy and LiPb alloy, respectively, which means that it is more difficult for Li to pass through the LiCu alloy and LiPb alloy layers. However, the Li migration barriers for the LiAg, LiAl, LiSn, and LiZn alloys are lower than those for the parents Ag, Al, Sn, and Zn, respectively. Especially for the LiAl alloy and LiZn alloy, the migration barriers are 0.18 eV and 0.25 eV, respectively, which are lower than the barriers for Al and Zn. This suggests that the formation of the LiAl alloy and LiZn alloy will not hinder the Li diffusion. Overall, when the migration barriers of these current collectors are close, the current collector with relative higher theoretical capacity and smaller volume expansion as shown in Table 3 should be selected for AFB. The results suggest that Al, Sn, and Zn are the better AFB current collectors than Cu or Ag. This is consistent with the experimental result that Al and Sn show the better energy density and cycling performance.^[8,11]

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. Comparison of Li migration barriers in (a) metals and (b) alloys.

Table 3.

Table 3.

Table 3. Summary of Young’s modulus (E′), specific capacity (C), volume specific mass density (ρ), volume strain at maximum concentration (εv), partial molar volume (Ω), and coupling parameter (β) in mechanical–electrochemical coupling on some electrode materials and current collectors. The values of Young’s modulus are obtained from database in Materials Project.[30,52] . Materials E′/GPa[30,52] C/(mA·h/g) ρ/g·cm−3 εv/% Ω/m3 · mol−1 β/nm3 Graphite 10 372 1.83 9 3.54 × 10−6 0.36[27] LiVO2 43 314 4.3 10 1.99 × 10−6 0.49[27] LiCoO2 375 140 5 −2 −7.66 × 10−7 0.64[27] LiFePO4 120 170 3.5 7 3.15 × 10−6 3.45[27] Li4Ti5O12 240 175 3.5 1 4.38 × 10−7 0.13[27] LiMn2O4 142 136 4.1 7 3.36 × 10−6 4.65[27] Ag 123 248 9.92 167 1.82 × 10−5 118 Al 109 993 2.7 190 1.90 × 10−5 115 Cu 230 140 8.89 142 3.05 × 10−5 620 Pb 55 129 10.7 501 9.71 × 10−5 1511 Sn 82 993 5.37 262 1.32 × 10−5 41 Zn 139 410 6.98 190 1.78 × 10−5 128

Table 3. Summary of Young’s modulus (E′), specific capacity (C), volume specific mass density (ρ), volume strain at maximum concentration (εv), partial molar volume (Ω), and coupling parameter (β) in mechanical–electrochemical coupling on some electrode materials and current collectors. The values of Young’s modulus are obtained from database in Materials Project.[30,52] .

At the same time, the volume expansion rates of several materials after lithiation are shown in Table 3. The results show that the volume expansion rates of most metals on lithiation are usually between 150% and 300% with the exception of lead with the value of 501%, which makes it unsuitable as the anode material.

The alloying reaction between metallic materials and lithium leads to a large volume expansion (Table 2). The developing stress and strain in turn affect Li diffusion via mechanical–electrochemical coupling. By solving the diffusion Eq. (10), the Li concentration distribution along the z direction is shown in Fig. 5(a) (at t = 900 s) and its evolution is shown in Fig. 5(b) (at z = 0.8). The impact of the coupling parameter β is evaluated. The results show that when β = 0, which means no coupling, it shows a significant concentration gradient. With the increase of β, the concentration becomes uniform, which is caused by the stress-assisted diffusion. However, when β > 100 nm³, the coupling effect is not further strengthened (Fig. 5(a)). Similarly, as shown in Fig. 5(b), when z = 0.8, the coupling effect makes the Li concentration change with time, approaching a linear regime for t > 500 s. As can be also seen, the pronounced effect on stress-assisted diffusion is achieved when β = 0–100 nm³ and saturates at higher values.

Fig. 5.

Figure Option ViewDownloadNew Window

Fig. 5. (a) Distribution of Li concentration along the z direction with various values of the coupling parameter β after the 900 s diffusion. (b) Distribution of Li concentration against diffusion time t at a distance of 0.8 from the surface toward the current collector with various values of the coupling parameter β. For the convenience of comparison, both the Li concentration and diffusion distance along the z direction are normalized.

Equations (11) and (12) indicate that β depends on the Young’s modulus, density, and volume expansion rate of the materials. In Table 3, the related factors affecting the coupling of the studied materials are presented in which the volume expansion data is calculated in Subsection 2.2. It can be seen from Table 3 that there are obvious differences in the coupling parameter for different materials. Song^[27] has studied the coupling of conventional battery materials, which are listed as a comparison with the metals in Table 3. It can also be seen from Table 3 that the metal current collectors which can be used for AFB have large volume expansion which in turn results in high value of β (> 40 nm³). The β of Cu is 620 nm³, revealing that fast Li diffusion in current collector such as Cu is affected by large volume expansion. The positive effect of high stress-assisted Li diffusion in the AFB anodes also explains their good cycling performance.^[8] The β values of Al, Sn, and Zn are 115 nm³, 41 nm³, and 128 nm³, respectively, indicating pronounced coupling effects. Overall, considering the capacity, migration barriers, and coupling effect, the Al, Sn, and Zn are more suitable than other metals for AFB. Among them, Sn has a larger volume expansion than Al and Zn, but lower β due to the low Young’s modulus and the high density. In addition, Al has higher capacity and larger β with lower volume expansion than Sn, so Al offers the best application prospect in AFBs, followed by Zn and Sn as mentioned in Section 1.^[8,11,13,52] Therefore, the optimal materials for AFBs should be assessed from three aspects: the energy density, volume expansion, and coupling effect. As an example, Pb and Ag have lower capacities and higher migration barriers than those of other metals, and even β > 100 nm³ cannot compensate these drawbacks, so they are not preferred for AFB.