Origin of Heterogeneous Stripping of Lithium in Liquid Electrolytes

Lithium metal batteries suffer from low cycle life. During discharge, parts of the lithium are not stripped reversibly and remain isolated from the current collector. This isolated lithium is trapped in the insulating remaining solid-electrolyte interphase (SEI) shell and contributes to the capacity loss. However, a fundamental understanding of why isolated lithium forms and how it can be mitigated is lacking. In this article, we perform a combined theoretical and experimental study to understand isolated lithium formation during stripping. We derive a thermodynamic consistent model of lithium dissolution and find that the interaction between lithium and SEI leads to locally preferred stripping and isolated lithium formation. Based on a cryogenic transmission electron microscopy (cryo TEM) setup, we reveal that these local effects are particularly pronounced at kinks of lithium whiskers. We find that lithium stripping can be heterogeneous both on a nanoscale and on a larger scale. Cryo TEM observations confirm our theoretical prediction that isolated lithium occurs less at higher stripping current densities. The origin of isolated lithium lies in local effects, such as heterogeneous SEI, stress fields, or the geometric shape of the deposits. We conclude that in order to mitigate isolated lithium, a uniform lithium morphology during plating and a homogeneous SEI are indispensable.


Implementation & Mathematical Details
We assume the dissolution to be reaction-limited, i.e., the diffusion of lithium in the electrolyte to be sufficiently fast. Further we assume a constant concentration of Li + at the surface which is valid for stripping current densities before diffusion limitations occur: where F = 96 485 C/mol is the Faraday constant, D = 3·10 −10 m 2 /s is the typical diffusion constant of a LiPF 6 electrolyte 1 , ∆c = 1%·1 mol/l is a small change in the Li + concentration in the typical length scale of a whisker L = 1 µm. This is a conservative estimate, as the ions diffuse in the bulk electrolyte and not along the whisker. Therefore the length scale is smaller and consequently the reaction limited current larger, undermining that diffusion limitations do not play a role in the dissolution process for small current densities.
As the initial geometry, we assume a cone-like shape with a spherical tip, similar to the structures observed in experiments, see Figure 1. This ansatz allows us to use a cylindrical symmetry. We connect the cone and the sphere via a smeared out Heaviside function.
where R = 100 nm is the whisker radius, L = 5 µm is the whisker cone length, a = 1.1 indicates that the sphere on top of the whisker has a slightly larger radius than the whisker cone, b = 0.1R, and c = 8 determines the width of the smeared out Heaviside function. Our model starts with the geometrical assumption of the whisker in Equation SI-2. We evaluate the function in steps of ∆z = 15 nm for z < 0.8L and ∆z = 3 nm for z >= 0.8L. The points are used as the initial whisker surface and the SEI. The SEI points are assumed to be rigid, i.e., they do not change during dissolution. In reality, the SEI shell falls together, due to a negative pressure beneath the SEI surface. This is not important for our simulation of the whisker dissolution, as this happens after the Li-SEI bond is broken. The whisker surface moves according to the equations of motion, Equations 4 & 5: The geometrical partsż/ √ṙ 2 +ż 2 and −ṙ/ √ṙ 2 +ż 2 assure that the movement is perpendicular to the surface. The local dissolution current density J(ξ) determines the dissolution velocity, given by Eq. 6: For calculating the Butler Volmer rate, we need to determine the potential step ∆Φ = Φ − Φ 0 relative to the lithium metal and the chemical potential µ. The potential step ∆Φ is determined by solving the Galvanostatic condition: where f is the fraction of the initial current density to the effective exchange current density. The Butler-Volmer equation can be modified using the simple formula for Marcus-Hush-Chidsey kinetics by Bazant and co-workers 2 : where A = J BV (η = 0)/J MHC (η = 0) makes sure that the current descriptions give the same value at zero overpotential, λ = 10 is the dimensionless reorganization energy and η = η(ξ) = (µ + ∆Φ) /2RT is the local overpotential. First, the chemical potential needs to be determined by evaluating Equation 8. From Equation 7 we get The interfacial tension σ(d, α) depends on the distance d, which can be calculated by: and the angle α, where cos α =żż 0 +ṙṙ 0 (ṙ 2 +ż 2 )(ṙ 2 0 +ż 2 0 ) . For calculating ∂z 0 /∂r we make use, that we evaluate the SEI at the point closest to the whisker surface. Thus, the orthogonality relation holds. With this we can determine the derivatives implicitly: With this, we can finally evaluate Equation 7: (SI-15) (a) Cryo TEM image of the SEI after plating at 1 Am −2 .
(b) Cryo TEM image of the SEI after plating at 10 Am −2 . 2 Additional experimental details and results

SEI thickness
In order to investigate the SEI thickness, we performed high resolution cryo TEM, see Fig. SI-1.For both plating current densities of 1 Am −2 and 10 Am −2 , the SEI is identified to be amorphous with no clear nanostructure and around 20 nm in thickness.

Plating and stripping behavior at higher current density
For probing higher current densities, we build a coin cells with the same properties as described in the Methods Section. For the plating/stripping cycling, we applied a plating current of 10 Am −2 for 10 mins and then discharged the cell with −10 Am −2 until a cut-off voltage of 1 V. The plated capacity equals the one in the experiment at lower current density. In the following we show the observed Li morphology and SEI structure and composition at higher current density.
To investigate the chemical compostion of the whisker and its covering SEI, we performed HAADF STEM with EDS and EELS mapping. The results are shown in Fig. SI-2. The SEI is rich in O and C with little amount of F. There is no striking difference in the SEI composition compared to whiskers formed at lower current density; see Fig. 1.
After stripping, we investigate the remaining structure on the Cu grid. We observe that only hollowed-out SEI shells remain that cannot be dissolved; see   Fig. SI-3 (a) the "blobbs" are ice crystals seeding on the SEI. In Fig. SI-3 (b) it can be seen that during the stripping, the SEI can break.
While after plating, the Cu grid is completely covered by whiskers, after stripping, the whiskers are mostly dissolved; see