Research Note: Three Separate Ways a Lithium-Ion Cell Ages — Epoche B2
Research note — Battery degradation project, entry 14 Topic: Why does a cell lose capacity? Purpose: sort the observed fade into distinct mechanisms before fitting any model. Starting question The common assumption is that a battery wears out because it has been charged and discharged too many times, as if each cycle removed a fixed slice of life. Our reference cells contradict this: cells stored fully charged at 45 °C, never cycled at all, lost capacity faster than cells cycled gently at 25 °C. ("Capacity" here means the total electric charge a cell can deliver, measured in ampere-hours; "fade" is its gradual decline.) A pure use-based wear model cannot produce fade in a cell that is never used, so at least one ageing process must run on time and temperature alone. Following the classification of Vetter et al. (2005), the fade should be sorted into at least three parallel side reactions, each with its own driving force — and the storage result above is exactly what the first of them predicts. Background: what the capacity physically is To see how capacity can be lost, first fix what it is. A lithium-ion cell is a "rocking-chair" device: lithium ions shuttle back and forth between two host electrodes — a graphite negative electrode (the anode) and a lithium metal-oxide positive electrode (the cathode) — through an electrolyte, a solution of a lithium salt in organic carbonate solvents. A thin porous plastic membrane, the separator, keeps the two electrodes from touching while letting ions pass. On charging, ions leave the cathode and intercalate into the graphite — slide reversibly between its carbon layers, like cards slipped into a deck — while the matching electrons travel round the external circuit. Capacity is therefore set by two things: the pool of lithium free to shuttle (the "cyclable lithium") and the integrity of the host structures and their electrical wiring. Every ageing mechanism in this note is a way of shrinking the pool, damaging the hosts, or both. One more convention is needed. Electrode potentials below are quoted "versus $\mathrm{Li/Li^+}$": measured against a lithium-metal reference electrode that is defined as the zero of the scale. Lithium-filled graphite sits around $0.1\,\text{V}$ on this scale — very low, which means strongly reducing: the electrode is eager to push electrons onto anything touching it. Class 1: SEI growth (chemical, time-driven) The organic carbonate solvents in the electrolyte are not thermodynamically stable below roughly $0.8\,\text{V}$ versus $\mathrm{Li/Li^+}$, and the working graphite anode sits far below that, near $0.1\,\text{V}$. So the electrolyte in contact with the anode is inevitably reduced — it gains electrons and decomposes — and the insoluble products settle on the graphite as a thin film. This film is the solid electrolyte interphase (SEI), a concept introduced by Peled (1979). It is passivating : like the invisible oxide skin that stops a block of aluminium from corroding through, it blocks electrons from reaching fresh electrolyte while still letting lithium ions pass, which is why the cell survives at all. The cost is that every bit of film is built out of lithium and solvent, so each increment of growth permanently locks up cyclable lithium. The passivation is imperfect, so the film keeps thickening slowly. Suppose its growth is limited by solvent molecules diffusing through the existing film to reach the reactive surface. Fick's first law — the statement that a diffusion flux is proportional to the concentration difference divided by the distance it must cross — then makes the growth rate inversely proportional to the current thickness $L$: a film twice as thick passes solvent half as fast. Writing $\mathrm{d}L/\mathrm{d}t = A/L$ for some constant $A$ and integrating gives $L^2 = 2At$, i.e. $L \propto \sqrt{t}$. This is the same "parabolic" law that governs the slow thermal oxidation of silicon wafers (Deal & Grove, 1965), and for the same reason: the product layer is its own diffusion barrier. Since the lithium consumed is proportional to the film grown, the lost capacity follows $$Q_{\text{loss}}(t) \approx k(T, \mathrm{SoC})\,\sqrt{t}, \qquad k \propto \exp\!\left(-\frac{E_a}{RT}\right)$$ The rate constant $k$ carries the two real dependences. First, temperature $T$: chemical reactions require molecules to clear an energy barrier, the activation energy $E_a$, and the fraction of molecular encounters energetic enough to clear it grows as $\exp(-E_a/RT)$, where $R$ is the gas constant and $T$ the absolute temperature — this exponential form is Arrhenius behaviour. The familiar rule of thumb that the rate "roughly doubles per 10 °C" is not a separate law but this exponential evaluated near room temperature. For $E_a \approx 50\,\text{kJ/mol}$, a typical reported value for SEI growth, the ratio of rates at $308\,\text{K}$ and $298\,\text{K}$ is $\exp\!\left[\frac{E_a}{R}\left(\frac{1}{298}-\frac{1}{308}\right)\right] = \exp(6{,}014 \times 1.09\times10^{-4}) = \exp(0.66) \approx 1.9$ — almost exactly a doubling. Second, state of charge (SoC, the percentage of full charge currently stored): a high SoC means the graphite is full of lithium, its potential is pushed even lower, and the interface is more strongly reducing, so the parasitic reaction runs faster. Note what does not appear in the formula: cycle number. This is calendar ageing — it runs whether the cell is used or not, and it is why our hot, fully charged, never-cycled reference cells faded fastest. Broussely et al. (2005) measured exactly this pattern in storage experiments: fade during pure storage, accelerating with both temperature and SoC. Class 2: Lithium plating (electrochemical, condition-driven) The zero of the potential scale is not arbitrary: $0\,\text{V}$ versus $\mathrm{Li/Li^+}$ is by definition the potential at which the reaction $\mathrm{Li^+} + e^- \rightleftharpoons \mathrm{Li(metal)}$ is at equilibrium. Below it, depositing metallic lithium becom