The finding that matters most: plated lithium
Metallic lithium on the graphite is the post-mortem finding that bears on safety, and the one that is easiest to miss, because by the time the cell is opened the metal may have reacted with the electrolyte into SEI and LiF, or with the air of a careless preparation.9 A study of 64.6 Ah automotive cells cycled at 1.5C and 10 °C, conditions chosen to plate, compared three ways of finding it.9
Nuclear magnetic resonance of 7Li, on active material scraped from the copper and sealed in a rotor inside the glovebox, is direct: metallic lithium has a Knight shift of 245 to 270 ppm, far from the ions near 0, and the peak appeared in both artificially aged cells and in neither the fresh cell nor a cell aged in a car to 93 %. It was still there after 11 weeks of storage. It is qualitative.9 ICP-OES of the lithium that dissolves when a weighed piece of electrode is stirred in water is quantitative, with errors of 1 to 3 %, because plated metal reacts with the water and is counted; but it is indirect, and lithium that has already become insoluble LiF is missed.9 The electron microscope showed a mossy layer on the aged graphite, absent from the fresh and the car-aged samples, with oxygen and fluorine hot spots where the metal had met the air during preparation; it cannot map lithium, and it sees a square centimetre at most.9 The authors’ own conclusion is the handbook’s: NMR direct but not quantitative, ICP quantitative but indirect, microscopy local; and the methods that need the least preparation reproduce best.9
That is the pattern of the whole page. No technique proves a mechanism alone, and L6.1 is read as a set of questions, each with the cheapest technique that answers it: the differential curves before opening for lost lithium and lost material, the peel test for delamination, the microscope for cracks, deposits and exfoliation, diffraction for the lithium left in the lattices, surface spectroscopy for what the electrolyte became, the rebuilt coin cell for what the material can still do, and NMR for the metal. Agreement among them, on samples whose positions are known, is what a post-mortem can prove.
Plated lithium is found by its nuclear signature, by what dissolves, and by what the microscope sees; one of them alone is a hint.
You can now explain why a pulse resistance needs its timescale; how GITT turns two voltage changes into a diffusion coefficient, and the four checks that make it trustworthy (pulse length, IR drop, rest, particle size); what impedance is, and how to read Nyquist and Bode plots of a real cell from left to right; why the measurement must be small, steady, four-terminal and Kramers–Kronig valid; why a good circuit fit is not proof; and what DRT, nonlinear impedance and machine learning add.
We can now find out from outside what has gone wrong in a cell, and open it when the outside is not enough. The last module takes up what is left: how to keep the energy a cell stores from coming out all at once, and what becomes of the cell when it is truly finished.
The mathematics of this chapter
ΔV/I = R0 + Σ Rk(1 − e−t/RkCk) + 2σ√(2t/π)the pulse resistance of figure L3.1: ohmic, R‖C pairs and a Warburg element of coefficient σ
D = (4/πτ)(mBVM/MBS)2(ΔEs/ΔEt)2; for spheres the bracket is r/3Weppner–Huggins, valid for Dτ/L2 below 0.25 (flat) or about 0.025 (spheres)
f(τ̂) = 3τ̂ + 1/5 − 2Σ e−αn2τ̂/αn2, with αn = tan αnsurface change of a sphere under constant flux, τ̂ = Dt/r2; 2√(τ̂/π) at short times (Nickol et al.)
V − Veq ∝ √(t + τ) − √tthe rest after a pulse of length τ, t the time at rest; its slope gives D (Kang and Chueh)
ZR = R, ZC = 1/(jωC), ZL = jωL, ZCPE = 1/(Y0(jω)n)impedances in series add; admittances in parallel add
R‖C: Z = R/(1 + jωRC), top at ω = 1/(RC)a semicircle of diameter R; two arcs resolve for τ1 ≳ 100τ2
ZW = σω−1/2(1 − j); tanh(B√(jω))/(Y0√(jω)); coth(B√(jω))/(Y0√(jω))semi-infinite, open (transmissive) and closed (reflective) diffusion
Zd,3D = (Rd/3) tanh x/(x − tanh x), x = √(jωRdCd), D = r2/(RdCd)diffusion into a sphere (Abbas et al. eq. 6); DC limit Rd/15
Z = √(Rmz) coth √(Rm/z), z = Rct‖Cdlthe porous electrode as a transmission line (Bisquert model)
i = i0[e(1−α)Fη/RT − e−αFη/RT], Rct = RT/(Fi0)Butler–Volmer and its linear limit; harmonics measure the curvature
Z(ω) = R0 + Rpol∫g(τ)/(1 + jωτ)dτthe distribution of relaxation times, found by regularized least squares