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Yan et al. Energy Mater 2023;3:300002 https://dx.doi.org/10.20517/energymater.2022.60 Page 7 of 32

reached to favor dendrite formation energetically and a critical kinetic radius is required to keep the isolated
embryo growing. As the growth continues, localized electric fields dominate the morphology of the Li
deposits in the late growth regime to form dendrites with prolonged length and sometimes augmented
diameter.

To explain the propagation of dendrites during the electrochemical deposition of metals, various models
have been proposed: (i) surface tension model; (ii) diffusion-limited model; and (iii) electromigration-
limited model . Barton et al. comprehensively proposed the surface tension model in which surface
[23]
tension between the electrolyte and metal was considered to be one of the driving forces for Li dendrite
growth . At the initial stage, the electrodeposition of metal ions on an elevated region of the anode surface
[24]
with spherical diffusion was faster than that on a flat surface with the linear diffusion, leading to preliminary
protrusions with enhanced diffusion conditions [24-26] . Later, the possible Li morphology could be estimated
via a fluid dynamics mathematical model and it was confirmed that particle-shaped Li deposition would
[27]
deform Li dendrites under high surface tension . In the classical diffusion-limited model, metal ions move
randomly and reach active sites of the electrode where the deposition probability of mobile ions can be
defined as a balance between the rate of the electrochemical reaction and the bulk diffusion . The diffusion
[28]
becomes dominant rather than the reductive deposition while a deposition probability is low, leading to the
suppression of dendrite formation. In contrast, the high deposition probability always causes the ramified
+
dendrite structure. The model was applied to simulate the reductive deposition of Li ions and found that
+
the formation of dendrites was related to the competition between the diffusion of Li ions in the SEI film
[29]
and the interfacial deposition reaction . Moreover, dendrite propagation can be successfully inhibited by
pulse charging conditions, which altered the electrodeposition morphology of various metals, thus
effectively suppressing dendrite growth .
[30]
In the Chazalviel model, dendrite initiation was described to be limited by the electromigration process
rather than the diffusion process . It was demonstrated that the current density can change the ion
[20]
concentration gradient, i.e., the lower current density maintains a stationary ion distribution without
propagation of Li dendrite, while the higher current density introduces the depletion of cations and anions
at the electrode interface and the resulting local space charge field leads to the growth of dendritic Li. The
critical current density, J*, was defined as the boundary behavior between the high and low current densities
as follows:

where e represents the electronic charge, C represents the initial concentration of the electrolyte, D
0
represents the ambipolar diffusion coefficient, t represents the transport number of anions and L represents
a
the distance between the electrodes. In particular, the nuclei of Li dendrites were formed when the
concentration of ions near the electrode dropped to zero. The time of the concentration of ions dropping to
zero was defined as Sand’s time, τ . Furthermore, the corresponding current density was labeled as the
Sand
limiting current density when the ion concentration approached zero.

In the 1990s, Chazalviel’s space charge model was verified and supplemented [31,32] . In particular, to explain
the Li dendrite formation and growth even at a low current density below J*, the heterogeneity of the
electrode surface was considered in the model based on the fact that the variation in local current density
could be caused by the inhomogeneous microstructures of the electrode surface . Severe Li dendrite
[33]
growth usually produces a large number of derivative problems, such as dead Li (i.e., Li losing contact with