Review—"Knees" in Lithium-Ion Battery Aging Trajectories

Lithium plating occurs when lithium ions form metallic lithium on the surface of the electrode rather than intercalating into it. The plating reaction is favorable when the reaction potential of Li/Li⁺ is greater than the equilibrium potential for other alternative reaction pathways for Li⁺ (i.e., graphite intercalation). ⁴² Plating can be either "rate-independent", i.e. plating that occurs independent of the applied current ("overcharging"), or "rate-dependent", i.e. plating that only occurs if the applied current exceeds some value ("fast charging"). Lithium plating can also occur in either fresh cells or aged cells. Furthermore, lithium plating can occur with various extents of reversibility within a cycle, defined as the ratio of lithium plated during charge that is recovered in the subsequent discharge (this ratio is always less than 100%, meaning some irreverisble lithium plating always occurs). ^43,44 In contrast to irreversible plating, reversible plating does not contribute to long-term degradation. Thus, we use "plating" to refer to irreversible plating throughout the remainder of this discussion.

Generally, lithium plating on graphite follows heterogeneous nucleation and growth kinetics, in which rapid growth proceeds quickly after an initial nucleation phase. ^42,45,46 Thus, lithium plating can often be considered a "snowballing" knee (Figs. 5a and 5d). However, some lithium plating pathways (e.g., lithium plating driven by active material loss from the negative electrode, a hidden trajectory) leads to knees independent of the nucleation and growth of plated lithium. In these cases, the nucleation and growth kinetics of lithium plating will only exacerbate these degradation mechanisms, and the degradation will be a combination of the hidden and snowball trajectories.

Historically, lithium plating has been considered to be a primary driver for capacity knees. Here, we discuss the mechanisms and sub-pathways by which plating can lead to a knee (Fig. 7). We suggest Waldmann et al. ⁴⁷ and Gao et al. ⁴² for comprehensive general overviews of lithium plating.

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Rate-independent lithium plating

Rate-independent lithium plating occurs whenever the lithium capacity during charging exceeds the negative electrode capacity, i.e., the negative electrode is unable to host all lithium coming from the positive electrode. Generally, the latter can be avoided in fresh cells by simply using a negative electrode to positive electrode capacity ratio (n:p ratio) greater than 1 (Fig. 7a). However, if active material from the negative electrode is lost during aging, rate-independent lithium plating will occur even in cells with excess negative electrode capacity (Fig. 7b).

Rate-independent lithium plating in fresh cells. While rate-independent lithium plating in fresh cells can be easily avoided by proper cell design (Fig. 7a), this degradation mechanism is often exploited for scientific studies of lithium plating. For instance, Deichmann et al. ⁴⁸ created cells with n:p ratios of 0.75 and 0.5 to intentionally deposit lithium metal on graphite electrodes. The authors identified a relationship between decreased n:p and capacity fade, which they attributed to high loss of lithium inventory using differential capacity analysis and scanning electron microscopy. In a creative study, Martin et al. ⁴⁹ used deposited lithium metal as a mechanism to store extra capacity, enabling the cell to occasionally discharge extra energy (i.e., when extra range is needed) without requiring a substantially larger negative electrode. This cell design used an n:p ratio of 0.6, where n:p is calculated using the lithium capacity of the conventional graphite. A high upper cutoff voltage during charging was used to intentionally plate lithium onto graphite; unsurprisingly, irreversible lithium plating was found to be the primary failure mechanism, with over 50% capacity loss in two of the three electrolytes tested (although the cells did not exhibit knees). Rate-independent lithium plating in fresh cells is trivial to model and predict if the cell design is known; if rate-independent plating is expected in fresh cells, a snowballing lithium plating knee (or, at a minimum, high early-in-life capacity loss) may occur early in life.

Rate-independent lithium plating due to loss of active material. Loss of active material—specifically, loss of active material from the delithiated negative electrode (LAM_deNE)—during aging may result in rate-independent lithium plating if the lithium capacity of the negative electrode becomes limiting during charging (Fig. 7b). For instance, if the rate of LAM_deNE exceeds that of the loss of lithium inventory (LLI), the negative electrode will eventually be unable to accommodate all lithium during charging, which will lead to rate-independent lithium plating and thus a knee. Dubarry and colleagues ^43,44,50,51 have extensively explored this scenario by considering both different ratios of LAM_deNE to LLI and different extents of reversible and irreversible plating (Fig. 8). This scenario is a prototypical case of a hidden state (i.e., loss of active negative electrode material) causing a knee: because the negative electrode is typically oversized relative to the positive electrode, active material loss from the negative electrode is hidden from the measured capacity until the negative electrode capacity falls below the positive electrode capacity. We discuss this effect more generally in our discussion of the electrode saturation pathway. Fortunately, the onset of rate-independent lithium plating due to LAM_deNE can often be modeled and predicted via differential capacity analysis ^43,44,50,51 to identify the rates of LLI and LAM_deNE. Note that differential capacity analysis generally requires periodic low-rate cycling interspersed within the cycling test. Lastly, we note that the high predictive performance of features sourced from voltage curves over the discharge capacity curves in Severson et al. ³⁷ was largely attributed to this sub-pathway.

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In principle, various degradation mechanisms can lead to LAM_deNE, which occurs when active sites lose either ionic or electronic connectivity with the electrode. Additionally, several of these mechanisms can occur in parallel, leading to a snowball effect where loss of active material due to one mechanism may result in further stress on the remaining active sites, accelerating degradation via rate-independent lithium plating. Electrode sites can lose electronic connection via delamination, ^52–55 particularly for cells with low external pressure. ⁵³ Particle cracking is another mechanism for electronic disconnection of active sites, although graphite particles are not expected to crack appreciably. ⁵⁶ Electrode sites can lose ionic connection via electrolyte dry-out, which may be driven by gas generation during cycling, ^25,57 or the growth of microns-thick "covering layers", an effect that we mention now but explore in depth in our discussion on mechanical deformation knees. While the exact mechanisms leading to LAM_deNE are challenging to pinpoint exactly without extensive destructive analysis, LAM can often be identified via differential capacity analysis. ^43,44,50,51 However, we note that active material loss from well-made, graphitic negative electrodes cycling under "reasonable" conditions is often low in practice, ^4,58–60 so this effect may be most pronounced for cells with poorly made negative electrodes and/or cells with appreciable silicon content in the negative electrode. Furthermore, since negative electrode active material loss must outpace lithium inventory loss for this mechanism to occur, and since the rate of lithium inventory loss is more sensitive to high temperatures (∼40 °C) than active material loss, ^4,60 this mechanism is more likely to occur at lower temperatures than higher temperatures.

Rate-dependent lithium plating

"Rate-dependent" lithium plating occurs when excessive transport or reaction overpotentials cause the local electrode potential to drop below that of Li/Li⁺. In other words, rate-dependent lithium plating occurs at conditions when the plating could otherwise be mitigated by lithiating the graphite at a sufficiently low current. While rate-dependent lithium plating has the same criterion as rate-independent lithium plating (i.e., the local potential falls below that of Li/Li⁺), the dynamic nature of this process introduces additional avenues for lithium plating knees to occur.

As Gao et al. ⁴² describe, salt depletion in the electrolyte, poor charge transfer kinetics, and surface crowding in the negative electrode particles at the graphite surface further favor lithium plating over intercalation. These three effects mirror the transport of lithium from the electrolyte to the negative electrode (electrolyte transport, charge transfer from the electrolyte to the negative electrode particles, and solid-state transport within the negative electrode particles). While solid-state transport within the negative electrode particles is generally not expected to degrade with aging, both electrolyte transport and charge transfer from the electrolyte to the negative electrode particles can degrade with aging due to SEI growth (Fig. 7c).

Rate-dependent lithium plating in fresh cells. Rate-dependent lithium plating can be driven by a wide range of cell designs and usage conditions; the prototypical use case leading to lithium plating is high charging rates at low temperature ^61,62 (Fig. 7a). Waldmann et al. ⁶¹ observed an increase in the rate of aging with a decrease in temperature below 25 °C, attributing the increased aging rate to lithium plating via dissections. Low temperatures increase both the transport overpotentials for lithium ions within the electrolyte and electrode and the reaction overpotential for lithium intercalation. Note that "high" charging rate or "low" temperature do not have consistent quantitative definitions, as plating will occur whenever the local potential exceeds the energy barrier for lithium nucleation. Thus, plating may be observed even at "standard" test conditions, such as 1C constant-current charging near room temperature. ^63,64 Increasing temperature and more rate-capable cell designs (i.e., thinner electrodes) may allow for more rapid charging before lithium plating and these knees are observed; ^65,66 Lewerenz et al. ⁶⁷ cycled cells at rates up to 8C, observing no knees at rates as high as 4C, though microstructural evidence of plating was found even at 1C. The onset of lithium plating is also sensitive to the charging protocol, with many studies demonstrating that informed design of charging protocols can substantially extend cell lifetime by preventing lithium plating. ^63,68,69 Optimizing electrode architectures to improve electrolyte transport kinetics is a further path forward to increase charging rates without lithium plating. ^70,71 Careful electrochemical modeling can enable estimates of minimum negative electrode potential (and thus the plating risk) as a function of cell design and charging protocol; ⁶⁵ if accurate modeling suggests rate-dependent plating is expected in fresh cells, a snowballing lithium plating knee is likely to occur early in life.

Mechanical stress may also lead to rate-dependent lithium plating, as applied stress can compress the electrode or separator. This compression decreases the local porosity of the electrodes, which decreases the apparent diffusivity of electrolyte, increases the local polarization, and thus can cause lithium plating. Cannarella and Arnold ⁵³ conducted a direct test of this mechanism, finding that high external pressures can induce lithium plating in pouch cells and lead to a knee. In a follow-up experiment, Liu and Arnold ⁷² demonstrated that localized lithium plating could be induced in densified regions of the separator. Bach et al. ¹⁸ applied a hose clamp around the circumference of an 18650 cylindrical cell, and a post-test teardown clearly showed lithium plating localized to the regions of the electrodes that were under compressive stress. From this test, the authors concluded that internal gradients in pressure induced by the current-collecting tab can also cause lithium plating. Coin cells and pouch cells are also sensitive to localized external pressures. ^73–75 Rate-dependent lithium plating induced by mechanical gradients can be considered a threshold trajectory, as the lithium plating begins once the negative electrode porosity falls below a critical porosity (again, lithium plating itself can be considered a snowball trajectory). This effect may be challenging to model and predict without a detailed understanding of the heterogeneous porosity distributions in the electrodes and separator, which is difficult to experimentally measure.

Rate-dependent lithium plating due to loss of active material. As previously discussed, a hidden trajectory for rate-independent lithium plating is loss of delithiated negative electrode active material (Fig. 7b). ^43,44,50,51 Mechanisms for loss of active material include delamination, particle cracking, electrolyte dry-out, and covering layer growth. However, LAM_deNE can also drive rate-dependent lithium plating, even if the negative electrode capacity does not limit the charging capacity. Active material loss without a corresponding loss in lithium flux will lead to an increased local current density on the negative electrode surface; these high local current densities can drive higher overpotentials and thus lithium plating. Similarly to rate-independent lithium plating due to LAM_deNE, rate-dependent lithium plating due to LAM_deNE can be modeled and predicted via differential capacity analysis; however, since this technique typically requires low-rate cycling data, a major complication to estimating if rate-dependent LAM_deNE is high enough to drive lithium plating is combining the rate-independent estimate from differential capacity analysis with the rate-dependent kinetic effects. Overall, rate-dependent lithium plating due to LAM_deNE can be considered a threshold trajectory, where the internal state is the minimum negative electrode potential and the threshold is the local plating potential.

Continuous active material loss will create increasingly larger local current densities, which will drive increasingly larger lithium plating potentials. Thus, in cell design/use case regimes where rate-dependent lithium plating is expected, linearly increasing active material loss can cause accelerating rates of lithium plating. Furthermore, as previously discussed, the nucleation and growth kinetics of lithium plating adds an additional snowball trajectory, since additional growth of initially nucleated phases can occur rapidly. This "double-snowball" effect is especially pernicious and is expected to lead to sharp knees. To our knowledge, prior experimental or modeling work has not considered this effect. Overall, this effect highlights the high sensitivity of rate-dependent lithium plating to active material loss of the negative electrode.

Rate-dependent lithium plating due to pore clogging. As SEI grows, it precipitates mainly in the pores of the negative electrode, decreasing the available volume fraction for electrolyte in the electrode ⁷⁶ This decreased volume fraction increases the electrolyte transport overpotentials, which can ultimately lead to lithium plating. The plated lithium, which has a much lower density than intercalated lithium, further decreases the porous volume fraction, creating a positive feedback loop for additional lithium plating ¹⁹ (Fig. 9a). Thus, this effect can be considered a threshold trajectory, in which a knee is triggered when the porosity decreases below some critical porosity, after which plating begins. A few works have modeled this phenomenon, ^7,19,22,23 namely Yang et al. ¹⁹ (Figs. 9b–9c). While this mechanism has not been experimentally validated, decreasing negative electrode porosity with cycling has been conclusively observed via X-ray computed tomography ^77,78 (Figs. 9d–e) and inconclusively observed via electrochemical impedance spectroscopy ^62,79–81 (attribution of spectral shifts to specific underlying phenomena, particularly via full cell electrochemical impedance spectroscopy, is challenging). Furthermore, the "covering layer" effect discussed at a later point may be related to this phenomenon, as models of lithium plating induced by pore clogging suggest that the pore clogging occurs primarily at the separator-electrode interface. ¹⁹ One proposed countermeasure for rate-dependent lithium plating due to pore clogging is to use a graded or stepped porosity profile through the thickness of the negative electrode; since most pore clogging occurs near the separator, having a higher porosity near the separator and a lower porosity near the current collector can slow the onset of the knee caused by pore clogging. ²³ We note that measuring local porosity distributions in a commercially relevant form factor is challenging and generally requires extensive ex-situ characterization. Furthermore, identifying the critical porosity at which plating starts is nontrivial and requires accurate electrochemical modeling of the porous electrode.

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Rate-dependent lithium plating due to decreased charge-transfer kinetics. SEI growth can also decrease the charge-transfer kinetics of the negative electrode particles, since the increased thickness of the SEI poses an additional barrier for lithium-ion intercalation. ^82–84 The chemistry and morphology of the additional SEI thickness likely influences its impact on the charge-transfer kinetics. ^85,86 Similarly to the porosity mechanisms previously discussed, the charge-transfer kinetics could decrease to the point that the negative electrode can no longer accommodate the increased lithium flux. A few authors have proposed this mechanism; ^87,88 Schuster et al. ¹⁷ studied this mechanism in depth via electrochemical impedance spectroscopy and post-mortem analysis. An open question is if (or under what conditions) SEI growth will cause lithium plating via decreased porosity or increased charge-transfer resistance first.

Again, this effect can be considered a threshold trajectory, in which a knee is triggered when the charge-transfer kinetics decrease below some critical threshold, after which plating begins. Tracking the charge-transfer kinetics over life can, in principle, be performed via electrochemical impedance spectroscopy of either the full cell or half cells harvested from the full cell, ^17,87,88 although interpreting impedance spectra can be challenging. Furthermore, just as in the case of porosity decrease, identifying the critical charge-transfer kinetic parameters at which plating starts is nontrivial and requires accurate electrochemical modeling of the porous electrode.

As previously discussed, lithium plating can occur if the active sites in the negative electrode cannot accommodate the available lithium inventory, driving the local surface potential to potentials at which lithium metal deposition is favorable. More generally, if the rate of active material loss for one electrode outpaces the rate of lithium inventory loss, the electrode can "saturate" and reach the cutoff potential well before all lithium has transferred. If this electrode is not limiting, its loss of active material will be hidden from the overall capacity until this electrode becomes limiting; furthermore, if the rate of active material loss is higher than the initial rate of lithium inventory loss, a knee in capacity will manifest. Dubarry et al. ⁸⁹ and Smith et al. ⁹⁰ captured this knee pathway by using a functional form for active material loss that increases more rapidly than that of lithium inventory loss (Fig. 10). This pathway can apply for either electrode, but loss of active material from the negative electrode is more likely to be a hidden trajectory since this electrode is typically oversized relative to the positive electrode; the exception is cells with lithium titanate electrodes, in which the positive electrode is limiting and can cause a "hidden" knee. ^91,92

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A richer picture emerges in models that capture the shifts in electrode stoichiometry with cycling. Lin et al. ²⁶ and Kindermann et al. ⁹³ modeled loss of lithium inventory driven by SEI growth and loss of active material driven by mechanical effects in the positive electrode. Sulzer et al. ⁹⁴ replicated a similar mechanism in Fig. 11 by simulating continuous constant-current discharge and constant-current, constant-voltage charge of a single particle model with SEI formation ¹⁹ and loss of active material ^95,96 due to particle swelling. ^97,98 When the aging parameters (SEI reaction rate and particle cracking rate) are chosen so that loss of active material in the positive electrode occurs at a faster rate than loss of lithium inventory (e.g., compare Fig. 11c to 11d), the stoichiometric window of the positive electrode widens (Fig. 11f), which increases the cell voltage for a given amount of transferred lithium. This effect decreases the capacity between the voltage limits. The knee occurs when the positive electrode becomes fully saturated before the entire lithium inventory is transferred (around cycle 600 in Fig. 11), despite the underlying rate of degradation remaining linear (Figs. 11b–d). This scenario may occur in cells with unstable positive electrodes, such as nickel-rich materials ^99–101 or polycrystalline ternary oxide materials, ⁴ cycling under conditions such that the rate of positive electrode material loss exceeds the rate of lithium inventory loss.

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Electrode saturation can also be rate dependent, sometimes in counterintuitive ways. Ma et al. ⁶ found that single-crystal nickel manganese cobalt oxide (NMC)/graphite cells exhibited no capacity fade in 1C diagnostic cycles but exhibited capacity fade in C/20 diagnostic cycles. The authors attributed this result to the poor rate capability of the single-crystal NMC particles. At low rates, the cells are "negative electrode limited"; as lithium inventory loss shifts the negative electrode voltage curve, the available discharge capacity decreases and thus capacity loss is observed. At high rates, the cells are "positive electrode limited" because the positive electrode saturates before the negative electrode fully depletes; thus, the 1C capacities are unaffected. We refer the reader to Ma et al. ⁶ for further discussion of this phenomenon.

Overall, electrode saturation can be modeled and predicted using electrochemical modeling. This pathway can be considered either a threshold trajectory, where the knee is triggered by electrode saturation, or a hidden trajectory, where LAM of one electrode outpaces both LAM of the other electrode and LLI. While modeling of just the degradation modes (i.e., LLI, LAM, etc.) can capture the key dynamics of this pathway, models that capture the shifts in stoichiometry as a function of cycling can capture more subtle effects. As Ma et al. ⁶ demonstrate, periodic diagnostic cycles at multiple rates can aid in identifying electrode saturation, especially if the saturation is rate-dependent.

Cell internal resistance often increases during aging, in part due to the growth of side reaction products on the surface of the electrode particles. This effect is most pronounced for oxide-based positive electrode materials like NMC, nickel cobalt aluminum oxide (NCA), lithium cobalt oxide (LCO), and lithium manganese oxide (LMO), as they operate well above the stability window of the electrolyte. ^6,101 Under constant current conditions, the additional overpotential from increased internal resistance will cause the cell to reach the cutoff voltage more quickly, decreasing the capacity, energy, and power per cycle. The magnitude of this overpotential growth rate is a product of both the resistance growth rate (i.e., electrolyte reaction rate) and the applied current.

Most modern lithium-ion batteries have voltage-capacity curves that are relatively flat at higher voltages/states of charge (SOCs) (i.e., small dV/dQ) and relatively steep at lower voltages/SOCs (i.e., large dV/dQ). Thus, when cycling at appreciable rates, the constant-current portion of charge capacity is highly sensitive to small increases in resistance growth. In contrast, the discharge capacity is less sensitive to resistance growth—until the overpotential is large enough such that the discharge ends in the flatter region of the voltage-capacity curve (i.e., a small increase in overpotential leads to a large decrease in capacity). When this flatter region is reached, the discharge capacity becomes more sensitive to small changes in overpotential (again, when cycling at appreciable rates), leading to a knee in capacity, energy, and/or power.

Figure 12 displays a simple model illustrating a knee due to ohmic resistance growth during cycling, inspired by the work of Ma et al. ⁶ and Mandli et al. ²⁰ The model arbitrarily assumes a constant resistance growth rate of 0.2 mΩ per cycle, occurring at all SOCs; the linearly increasing resistance with cycle number leads to linearly increasing ohmic overpotential (Fig. 12a). The increased overpotential then shifts the voltage-capacity curve downwards. To demonstrate the impact of increasing resistance due to this downshift in a "real" cell, we used voltage-capacity and voltage-energy relationships recorded from an NMC/graphite cell at beginning-of-life from Preger et al. ⁸ (Fig. 12b). Figures 12c and 12d show the impacts of this downshift on the voltage-capacity curve at lower voltage cutoffs of 2 V and 2.8 V and discharge currents of 1C (Fig. 12c) and 2C (Fig. 12d). In all cases, the discharge ends on the steep portion of the voltage-capacity curve at the beginning of life. However, as the cell ages and the resistance increases, the discharge ends on the flat region of the voltage-capacity curve, resulting in an increased rate of capacity loss (i.e., a knee) despite a linear increase in resistance. Thus, this knee pathway is a threshold trajectory, where the internal state variable is the overpotential and the threshold is the "overpotential margin" between the lower cutoff voltage and the flatter region of the voltage-capacity curve.

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The impact of the resistance growth on the measured capacity (Fig. 12e), energy (12f), and power (12g) during discharge is highly sensitive to the discharge rate and the lower cutoff voltage—usage parameters that are not often considered critical for their impact on knees. Decreasing the discharge rates, of course, decreases the overpotential and delays the onset of the knee. Decreasing the lower cutoff voltage delays the knee by increasing the overpotential margin between the lower cutoff voltage and the flatter region of the voltage-capacity curve; however, note that low cutoff voltages can also induce additional degradation mechanisms such as copper dissolution. ^102,103 Naturally, energy and power knees (12f, 12g) are more sensitive to rate than the capacity knees (12e). Because these knees can "disappear" by cycling at lower rates or to lower cutoff voltages, we sometimes refer to these knees as "pseudo-knees"; furthermore, this knee mechanism may not be observed in some practical settings (e.g., the slow weeks-long discharge of an electric vehicle battery pack under typical usage). Finally, we note that "resistance pseudo-knees" may also occur due to a stoichiometric decrease of lithium available to cycle, as explored by Mandli et al. ²⁰ or a stoichiometric shifting of lithium to one electrode preferentially during aging due to uneven loss of active material across both electrodes. ²⁶

Ma et al. ⁶ extensively studied this knee pathway using lab-made 230 mAh NMC532/graphite pouch cells, varying the upper cutoff potential, discharging rate, electrolyte composition, and positive electrode material coatings. Careful impedance measurements (on both full cells and symmetric coin cells of the positive and negative electrodes) were used to identify a dramatic increase of the positive electrode impedance during aging. This resistance growth was attributed to electrolyte oxidation at the positive electrode, which was accelerated by high upper cutoff voltages and the use of more reactive electrodes and electrolytes (i.e., uncoated positive electrode particles, lower salt concentrations, and the use of oxidation-prone additives such as methyl acetate). One practical consideration from the work ⁶ is that tests with high discharge rates exhibit resistance-growth-induced knees earlier than tests with low discharge rates; thus, tests with high discharge rates can be used as an early indicator of resistance-growth-induced knees at lower rates. Alternately, measurements of resistance throughout cycling (in tandem with the voltage curve, discharge rate, and lower cutoff voltage) can be used to estimate when the knee will occur.

This knee pathway is also sensitive to electrode chemistry, as each chemistry exhibits a unique voltage-capacity curve. For instance, lithium iron phosphate (LFP)/graphite cells experiencing high resistance growth would exhibit much sharper knees than NMC/graphite cells due to their flatter voltage-capacity curves. While LFP cells generally do not exhibit high resistance growth due to their lower positive electrode operating voltages (well above the electrolyte stability window), ^104,105 even moderate resistance growth coupled with high discharge rates and high lower cutoff voltages could lead to dramatic cell failure (although the magnitude of this effect depends on the cutoff voltages).

Lastly, we mention that capacity knees are often correlated with "resistance elbows"—that is, a rapid rise in the internal resistance. While this correlation may be evidence for the prevalence of this knee pathway, we note that other knee pathways may also lead to a resistance increase at the knee (e.g., lithium plating due to loss of active material or porosity decrease). We return to this topic at a later point.

This "threshold" knee pathway is straightforward to model and predict using standard electrochemical models and measurements of resistance. However, convolutions with lithium inventory loss, active material loss, etc. require care. Overall, given the high sensitivity of this knee pathway to discharge rate and lower cutoff voltage—parameters that vary widely in real-world usage—care must be taken to transfer laboratory results to the field.

Both electrolyte and additive depletion have been linked to knees. In principle, electrolyte depletion can lead to a knee by driving either loss of active material ^25,29,57 or lithium plating. ¹⁰⁶ In turn, electrolyte depletion can be driven either by consumption via side reactions ^106–108 or via local gas generation leading to particles disconnecting from the electrolyte. ^25,57 Electrolyte depletion knees have been previously modeled, although often with limited experimental validation. Park et al. ¹⁰⁸ provided the first empirical model of a capacity knee due to electrolyte depletion. Fang et al. ²⁹ modeled electrolyte depletion knees occurring when the remaining electrolyte volume falls below the pore volume. The associated loss of active material increases local current density, which further increases the electrolyte dryout rate in a positive feedback loop. Kupper et al. ²⁵ also developed a model for electrolyte depletion knees using percolation theory to capture the nonlinear knee behavior, a model we detail in the subsequent pathway section. Experimentally, Sieg et al. ¹⁰⁶ attributed capacity knees during fast charging of large pouch cells to electrolyte dryout via careful coin cell diagnostic studies; while the electrode capacities remained healthy over life, lithium plating and decreased fast charging times could be tracked to decreased local electrolyte volumes. While the principles of the electrolyte depletion knee pathway are clear, more work is needed to understand the mechanistic details.

Additionally, robust work has linked the depletion of electrolyte additives to knees. Electrolyte additives have a disproportionate effect on lifetime relative to their amount in a cell; small quantities of electrolyte additives can often delay the occurrence of the knee by many cycles. ^6,109 Additive chemistry is complex; for instance, Burns et al. ¹¹⁰ showed how electrolyte performance often improves with the number of additives used. Additives can certainly influence the onset of other knee pathways, including lithium plating knees via various mechanisms (e.g., electrolyte transport properties, SEI growth rate and thus porosity decrease rate, etc.) and resistance growth knee pahtways by controlling the rate of resistance growth. ⁶ However, the depletion of electrolyte additives is another demonstrated knee pathway. Here, we discuss perhaps the most widely studied additive depletion knee mechanism: fluoroethylene carbonate (FEC) depletion in silicon-containing cells.

FEC has been shown to substantially improve the capacity retention of silicon electrodes. ^111,112 Among standard electrolyte components, FEC preferentially reacts at the surface of silicon particles; in fact, the rate of FEC consumption on silicon may be 10x that of graphite, in part due to its large volume expansion (around 300%). ¹¹³ Petibon et al., ¹¹⁴ Jung et al., ¹¹⁵ and Wetjen et al. ¹¹³ performed comprehensive studies of Si-containing full cells with FEC-containing electrolytes and commercially-representative volumes, conclusively demonstrating that a knee occurs when FEC is depleted from the electrolyte. Figure 13 displays key results from Petibon et al. ¹¹⁴ and Jung et al., ¹¹⁵ in which the dependence of the knee on FEC concentration was confirmed via destructive measurements of FEC concentration vs cycle number ¹¹⁴ and cycling cells with increasing initial FEC concentration. ¹¹⁵ Louli et al. ¹¹⁶ also corroborated these findings. Earlier studies of the use of FEC in high-Si cells ^111,112 did not observe knees due to their use of high electrolyte volumes, which provided a large reservoir of FEC. Other electrolyte components (namely, linear carbonates) are consumed only after the knee, since FEC can no longer be preferentially consumed; ¹¹⁴ the cell polarization increases substantially after the knee, ^113–115 possibly due to high reaction overpotential caused by reactions of silicon with these nonpreferred electrolyte components. This knee pathway is exacerbated by high upper cutoff voltages, ¹¹⁴ higher cycling rates (presumably due to more mechanical damage to the SEI layer), ¹¹⁴ and (presumably) high temperatures (due to higher SEI growth rates).

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The electrolyte and additive depletion knee pathway, a clear threshold trajectory, has a number of interesting implications. First, since laboratory-built cells are often filled with high electrolyte volumes, electrolyte-related knee mechanisms that are not present in lab testing may manifest in more commercially representative form factors. As Wetjen et al. ¹¹³ emphasize, maintaining representative electrolyte volumes in lab-scale cells is critical for accurately capturing this knee pathway in production-scale cells. Second, nominally identical cells, cycled identically, but with different initial FEC concentrations exhibited minute electrochemical differences before the knee. ¹¹⁵ While the equivalent study has not been performed for electrolyte depletion, we expect a similar result. Since only the electrolyte or additive consumed in a given cycle manifests in the electrochemical signals from cycling (e.g., differential capacity or differential voltage analysis), the remaining electrolyte or additive is not electrochemically detectable as it does not participate in reactions with the electrode. However, the remaining electrolyte or additive amount is the key internal state variable for this pathway. To estimate the remaining electrolyte or additive amount, the internal electrolyte/additive amount and the electrolyte/additive consumption over life must be known. However, extracting electrolyte/additive consumption during cycling from electrochemical data is challenging since side reaction signals are often faint and occur concurrently with other electrochemical processes. Furthermore, for commercial cells, the initial electrolyte/additive amount and the electrolyte/additive consumed during formation are unknown, although obtaining these values may be possible via electrolyte reverse engineering. Thus, knee onset for this knee pathway is challenging to predict via standard electrochemical signals. A proposal for future work is to evaluate electrochemical tests or other nondestructive probes that may be sensitive to remaining electrolyte/additive amounts. Fortunately, ultrasonic probes appear well suited for detecting electrolyte loss in some cell form factors, namely pouch cells. ^117,118

Percolation theory ^119,120 is commonly used to describe statistical properties of clusters of materials that are geometrically connected in porous media, including porous electrodes used in modern lithium-ion batteries. ¹²¹ In a porous medium described by percolation theory, there exists a critical material parameter above which the probability of a spanning cluster, i.e., a cluster that spans the entire spatial extent of the porous medium, being formed tends toward one and below which this probability tends toward zero. ¹²¹ In many percolating systems, this probability is highly sensitive to the value of the critical material parameter. For lithium-ion batteries, percolation theory can be used to describe both the ionic conductivity of the liquid electrolyte that fills the porous electrode and the electronic conductivity of the network of conductive additives. In battery modeling and experimentation, the electrode is often implicitly assumed to be sufficiently porous for the liquid electrolyte to completely percolate it. On the other hand, much effort has been spent on elucidating how the volume fraction of conductive additives may or may not give rise to a percolating electrically conducting network, ^122–125 which is especially important for ensuring electronic conduction is not rate limiting in electrically insulating active materials, such as lithium iron phosphate. ^123,125

Kupper et al. ²⁵ proposed an electrolyte depletion knee mechanism that incorporates percolation theory. In this proposed electrolyte dry-out mechanism, they first define two new electrode descriptors: "activity", a, and "saturation", s, given by and , respectively. In these equations, ε is the volume fraction of the active graphite (), inactive graphite (), electrolyte (ε_elyte) and gas (ε_gas, which is produced during SEI growth). Activity describes how much of the electrode material is active and available for reaction, while saturation describes the amount of pore space occupied by the liquid electrolyte. The loss of ionic contact of graphite caused by electrolyte dry-out is then described by a kinetic rate law that is proportional to the difference in activity and equilibrium activity, which is assumed to be a function of only saturation. To predict a knee in cell capacity, the equilibrium activity-saturation relationships were formulated to be nonlinear and contain a percolation threshold value, around which the equilibrium activity varies rapidly between 0 and 1. The functional forms of these relationships were assumed given the absence of theoretical or experimental guidance. Figure 14 plots two such nonlinear relationships, named relationships 3 and 4, adapted from Fig. 5 of Kupper et al. ²⁵ The authors concluded that relationship 4 best fitted experimental aging data.

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The knee caused by this electrolyte dry-out model is a threshold trajectory, where the threshold is the critical saturation value illustrated in Fig. 14. Although Kupper et al. ²⁵ did not provide convincing experimental validation to definitively prove that electrolyte dry-out resulted in sudden death of the cell and that relationship 4 was the most plausible activity-saturation relationship, the proposed electrolyte dry-out mechanism is plausible in principle and should be experimentally studied in more detail. Furthermore, a similar effect may apply for electronic conductivity networks; Guzmán et al. ¹²⁵ illustrated how the electronic conductivity of lithium iron phosphate electrodes exhibits a percolation threshold based on the conductive carbon content. If this effect is experimentally validated, we expect that identifying both the activity-saturation relationship and nondestructive measurements of saturation during aging will be challenging; we note that ultrasonic probes have had success in detecting electrolyte loss. ^117,118

Both microscale mechanical effects occurring at the particle scale and macroscale mechanical effects occurring at the cell scale can be pathways for knees. These mechanical degradation mechanisms often interact in positive feedback loops (Fig. 15). Mechanical degradation mechanisms are closely tied to other knee pathways. For instance, Cannarella and Arnold ⁵³ showed how high external stack pressure can cause a lithium plating knee; Bach et al. ¹⁸ demonstrated a link between heterogeneous pressure and localized lithium plating; and many loss of active material mechanisms are mechanical in nature (e.g., delamination, particle cracking). Additionally, the growth of covering layers on the surface of the negative electrode (Fig. 16), often reported on cells with knees, ^{24,29,126,127} may lead to additional mechanical stresses on the macroscale. Here, we focus on knees more explicitly linked to mechanical effects.

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At the microscale, repeated (de)intercalation can stress the electrode particles, which can then lead to both loss of active material through particle cracking and accelerated growth of side reaction products (e.g., SEI and its analogue on the positive electrode, often termed "CEI" for cathode-electrolyte interphase). Reniers et al. ⁹⁵ illustrated a positive feedback mechanism between the mechanical stress and loss of active material, leading to a snowballing knee. They combine a fatigue model for loss of active material due to stress from Laresgoiti et al. ⁹⁶ with a stress model from Dai et al. ¹²⁸ higher stress causes higher loss of active material, which in turn increases the current density and hence causes higher stress. Other authors have suggested that mechanical effects can accelerate SEI/CEI growth by causing SEI/CEI cracking and revealing new active surface area to grow. ^{25,83,116,129} Since growth of these interphasial layers is self-limiting and thus sublinear, ^9–13 this effect alone is not enough to lead to a knee, but it could accelerate the onset of knees in other pathways related to side reactions (i.e., lithium plating induced by pore clogging on the negative electrode ¹²⁶ or resistance growth driven by side reactions on the positive electrode ^6,129 ). Overall, microscale mechanical deformation mechanisms are challenging to model given the difficulties of experimental validation and the complexity of their interactions.

At the mesoscale, an additional LAM_deNE mechanism is the growth of thick layers (1−10 μ m) on the surface of the negative electrode at the separator-facing interface, sometimes termed "covering layers". ^24,67,126 Covering layer growth-driven LAM_deNE may impede lithium-ion transport into the negative electrode during charging, effectively isolating portions of the electrode and resulting in an apparent loss of active material. This covering layer is commonly observed in cells with knees and is often attributed to manganese or iron dissolution from the positive electrode and/or electrolyte salt decomposition. ^{21,29,34,55,67,78–80,87,126,127,130,131} or possibly dead lithium agglomerates, ⁶⁸ but the root cause has not been definitively identified. Peculiarly, the size of these layers (microns) is much larger than typical reported SEI thicknesses (nanometers). ¹³² Furthermore, this phenomenon is almost exclusively observed in cylindrical cells. Lewerenz et al. ^67,126 thoroughly documented covering layer growth, finding that increasing C rate and larger depth-of-discharge could lead to earlier onset of a knee. Earlier knee onsets were correlated with the presence of a thick covering layer on cells that contained knees; cells without knees also contained obvious covering layers, but with lower surface coverage and less thickness. These covering layers sometimes seem to lead directly to localized lithium plating due to the lack of active sites available for lithium insertion, with lithium observed at the covering layer/separator interface. ¹³⁰ Further investigation of these covering layers is needed to understand this seemingly ubiquitous mechanism for loss of active negative electrode material and its relationship to knees.

On the macroscale (cell level), mechanical degradation manifests differently depending on the form factor. For both pouch and prismatic cells, the external pressure can impact cell lifetime. Pouch and prismatic cells either without external pressure ¹³³ or with high external pressure ⁵³ can show rapid knees, indicating an intermediate value of pressure is optimal to avoid knees and maximize lifetime. Cannarella and Arnold ⁵³ also found that the surface layer is more pronounced with increasing external pressure. Heterogeneous pressure distributions from internal cell components (e.g., electrode tabs) or loading variation ¹³⁴ can also lead to knees. Knees due to mechanical heterogeneity can perhaps be modeled and predicted via a careful understanding of the porosity distributions within the cell. This pathway can be considered either a snowball or threshold trajectory.

For cylindrical cells, several studies ^135–139 have identified jelly roll deformation, using X-ray computed tomography after cycle life testing, in which the jelly roll deforms inwards toward the core of the cell and can lead to a capacity knee. This deformation was attributed to an increase in internal pressure of the cell over life, which itself is driven by either side reaction growth (including covering layers), ²⁴ lithium plating, ¹³⁸ or high thermo-mechanical stress due to large radial temperature gradients within the cell. ¹³⁵ Geometric heterogeneities (e.g., due to the internal tabs) can also exacerbate this failure mode. ^136,137 This pathway can be considered a threshold trajectory, where the jelly roll is eventually unable to accommodate the increase in internal volume and pressure. Modeling and predicting this mechanism, however, requires measuring and tracking the internal pressure over life, as well as knowing the internal pressure at which the jelly roll will deform. A center pin appears to help avoid this failure mode, ^135,138 although we note that center pins may serve other purposes such as facilitating gas flow along the cylindrical cell axis during a safety event or creating an additional electrolyte reservoir. ¹³⁶

While the six knee pathways we have identified can occur independently, these pathways can clearly interact. For instance, loss of active material plays a central role in four of our six pathways (lithium plating, electrode saturation, percolation-driven connectivity, and mechanical deformation). This coupling between degradation mechanisms can create positive feedback mechanisms, a special case of snowball internal trajectories. Reniers et al. ⁹⁵ explored a number of interacting degradation mechanisms with a single-particle model, finding that many have positive feedback. These interactions can also occur across length scales (Fig. 15), as SEI growth on the nanometer level can drive lithium plating on the centimeter level. Another interaction effect is resistance growth: since each of these pathways has some mechanism for increased resistance (most commonly via active material loss increasing the local current density and thus increasing the effective cell resistance), increased resistance growth caused by one pathway may exacerbate internal state variable growth in another pathway. Given the high sensitivity of snowball pathways to small changes in state, interacting knee pathways can create positive feedback mechanisms with high sensitivity to internal state.

As an extreme example of positive feedback coupling between knee mechanisms, consider a hypothetical "quadruple snowball" trajectory. Each individual component in this mechanism has been previously discussed. First, particle cracking leading to loss of active material can snowball since the local current density on the remaining active particles is continuously increasing, driving additional mechanical stress. Second, loss of active material itself can snowball with percolation-limited connectivity—i.e., if the active material fraction drops below the critical percolation threshold. Third, loss of active material from the negative electrode, upon saturation, can cause rate-dependent lithium plating to snowball; the local current density will keep rising on the remaining negative electrode active material, increasing the driving force for lithium plating over reversible interaction. Lastly, lithium plating can snowball due to its nucleation and growth kinetics. While this example is certainly contrived, feedback between multiple knee pathways is perhaps probable given the shared sensitivities of many of these mechanisms to the same levers.

Heterogeneity within a cell may also exacerbate these knee pathways. Commercially-relevant form factors have electrochemical, thermal, and mechanical gradients due to intrinsic heterogeneity and inactive components; these gradients can drive localized degradation. For instance, heterogeneous particle distributions or porosity profiles on the electrode level can lead to localized lithium plating. ¹⁴⁰ Furthermore, the presence and location of electrode tabs in cylindrical cells can create electrochemical, thermal, and mechanical gradients, ^{18,137,141–148} in some cases also leading to localized lithium plating. ^18,66 Heterogeneity can also arise from ambient factors, e.g., thermal gradients induced by environment or thermal management systems. ¹⁴⁹ Furthermore, local heterogeneity can lead to positive feedback for degradation; for instance, the temperature of a region that receives a higher local current density will rise, leading to even higher local current density. Given the sensitivity and positive feedback of many knee mechanisms, heterogeneity can certainly exacerbate the presence of knees.

Finally, we consider the impact of sample and testing variation. Nominally identical cells cycled identically often show differences in knee behavior. This sampling variability includes both intrinsic variability from manufacturing (component-level variation, cell assembly, etc.) and extrinsic variability from testing (cycler calibration, temperature control, etc.). These sources of variability require rigorous equipment calibration to distinguish.

The magnitude of sampling variability is a function of the cell design, manufacturing variability, and testing conditions. Sampling variability may increase with more aggressive cell designs (e.g., higher silicon content), more manual cell assembly processes, and more aggressive testing conditions (particularly for test setups with no or poor temperature control). The magnitude of sampling variability can be estimated using studies with fairly large sample sizes (i.e., at least ∼10 cells); ¹⁵⁰ Beck et al. ¹³⁴ provide a detailed review of cell-to-cell variation. Baumhöfer et al. ¹⁵¹ and Harris et al. ¹⁵² studied this type of variation in 48 cells and 24 cells, respectively, finding widely varying knee locations across their datasets (Fig. 17). While the knee pathway and internal state trajectories are unknown for these datasets, the Harris et al. ¹⁵² dataset has much larger variability, perhaps due to its aggressive 10C discharge rate. These studies did not identify a correlation between beginning-of-life capacity and end-of-life capacity, suggesting that differences in initial internal state trajectories did not manifest in the initial capacity measurements (possibly implying that hidden or threshold internal state trajectories caused these knees). In general, sampling and testing variation also poses challenges for accurate knee prediction; identifying the manufacturing and testing variation of the internal state variable of interest is needed to evaluate the accuracy of knee prediction methods in real-world settings. Lifetime variability highlights the high sensitivity of knees to manufacturing and testing variability.

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In Fig. 18, we develop a simple model to consider the sensitivity of knees on cell-to-cell/testing variation. We propose a simple one-parameter exponential functional form for a cell retention curve with a knee, , where Q represents capacity, n represents cycle number, and c represents some parameter that varies during cell manufacturing or testing that the knee is sensitive to. For instance, c could represent variation in initial cell energy, initial cell internal resistance, or testing temperature (among many other possibilities). In Fig. 18a, we plot this function for c = 1/150. We then visualize the distribution of retention curves if c is normally distributed with various relative standard deviations (RSDs), including 0.5% (18b), 2% (18c), 5% (18d), and 20% (18e). For each distribution of retention curves, we also track the RSDs of two lifetime metrics: the number of cycles until 80% retention and the retention at 500 cycles. We find that the RSDs of the two output metrics always equal or exceed the RSD of c (Fig. 18f). Moreover, despite Gaussian input variation, the distribution of the number of cycles until 80% retention and the retention at 500 cycles are non-Gaussian and skewed (illustrated most clearly in Fig. 18e). While simplistic, this model illustrates how cell-to-cell variation can have an outsize effect on knee location given the nonlinear dependencies of lifetime. This exercise could be repeated for other internal state trajectories and all of their functional forms.

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In summary, the impact of interactions, heterogeneity, and variation on knee pathways and internal state trajectories is complex, poorly understood, and an opportunity for future work.

While the dependence of knees on cell usage conditions has been studied extensively, less attention has been focused on the dependence of knees on cell design—likely due to the challenges of representative lab-scale cell fabrication. Ma et al., ⁶ one of the most comprehensive works on the impact of cell design on knees, studied the impact of various electrodes and electrolytes on the location of the knee. These knees were classified as resistance "pseudo-knees" due to increased electrolyte oxidation on the positive electrode, as evidenced by the strong dependence of the knee severity on discharge rate as well as positive electrode impedance measurements. For electrode design, Ma et al. ⁶ and Klein et al. ¹⁵³ found that positive electrode particle coatings and low positive electrode loadings delayed the knee. Ma et al. ⁶ and Glazier et al. ¹⁵⁴ also found that the graphite type (i.e., natural or artificial) can substantially impact the knee location; while natural graphite has larger irreversible expansion and thus higher parasitic reaction rates, ¹⁵⁴ the root cause of the knee in this case is unclear.

Electrode loadings (i.e., capacity per unit area) can also lead to knees via the lithium plating pathway. Rate-independent plating can occur if the ratio of negative electrode loading to positive electrode loading is too low (i.e., n:p < 1); ⁴⁸ however, rate-dependent lithium plating can occur at low loading ratios if the negative electrode is too thick or the porosity is too low. ^19,135,153

Additionally, small changes in the electrolyte can play an outsized role on the lifetime performance. Ma et al. ⁶ demonstrated the sensitivity of the knee location to the electrolyte additive mixture; specifically, high methyl acetate (MA) concentrations (MA is used to increase electrolyte transport capability) and low LiPF₆ concentrations consistently led to earlier knees. These knees were all attributed to increased electrolyte oxidation on the positive electrode via the resistance growth-induced pathway. Ma et al. ⁶ also identified other electrolyte systems with a strong knee sensitivity. Note that while Ma et al. ⁶ found that lower salt concentrations accelerated the knee in NMC/graphite pouch cells, Wang et al. ¹⁵⁵ found that higher salt concentrations accelerated the knee in lithium cobalt oxide (LCO)/graphite pouch cells over a similar range of salt concentrations; both studies attributed the observed trends to positive electrode impedance growth. Additionally, as previously discussed, the additive depletion pathway can be a direct cause of knees for some cell designs (e.g., cells with high silicon content in the negative electrode and low FEC content in the electrolyte). ^114,115

Furthermore, mechanical deformation knees are naturally highly sensitive to the cell form factor. For instance, deformation of the core ^24,136,138 can only occur in wound cells, primarily cylindrical cells. The presence of a mandrel in the core may prevent this mechanical deformation. ¹³⁸

Lastly, the formation protocol can influence the location of the knee. For instance, Weng et al. ¹⁵⁶ found that NMC/graphite cells formed with a fast formation protocol that emphasizes time at high SOCs exhibited later knees than cells formed with a slower baseline formation protocol, which was attributed to the creation of well-passivating SEI at high potentials to decrease the amount of side reaction product formed during use. ¹⁵⁷ Klein et al. ¹⁵³ found that decreasing the upper cutoff voltage in formation from 4.5 V to 4.3 V delayed the knee in NMC/graphite cells, which was attributed to decreased transition metal dissolution to cause plating. In principle, the formation protocol can be optimized to avoid or delay knees for a given use case.

The sensitivity of knees to testing conditions has been extensively explored in the literature. We note that many studies use accelerated aging protocols, which may introduce failure modes that are unrepresentative of real-world usage. The representativeness of an aging profile to the target application must be considered when evaluating the sensitivity of knees to test conditions. Field data may enable design of more representative test conditions. ¹⁵⁸

Charging rate

Discharging rate

Voltage limits

A wider voltage window generally accelerates the onset of the knee point. ^{17,62,87,88,136,168} In one of the broadest studies, Ecker et al. ⁸⁸ considered six depths of discharge (DODs)—100, 80, 50, 20, 10, and 5%—with up to six voltage windows per DOD. The authors found that the EFC systematically decreased with increased DOD (Fig. 19). By 1000 EFC, all cells with DODs greater than 25%–75% had a capacity below 80% and exhibited a knee. When varying the voltage window with fixed DOD, the authors observed the highest degradation in cells cycled at low and high SOCs; the lowest degradation was observed for a midpoint SOC of 50%. Keil and Jossen ¹⁶⁷ also studied the effect of upper cutoff voltage on lifetime and found that while the lifetime of cells with oxide-based positive electrode materials was at worst moderately sensitive to upper cutoff voltage over a range of 250 mV, LFP cells exhibited a sharp knee with a 50 mV increase in the upper cutoff voltage. Other studies also observed accelerated degradation at extreme SOCs. ^130,168,169

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The impact of the voltage window on knee onset is typically attributed to resistance growth stemming from enhanced expansion and cracking of the positive electrode during intercalation, driving electrolyte oxidation. ^6,14,169 For some positive electrodes, transition metal migration (often manganese or iron) may also be exacerbated by high voltages. ^168,170,171

Rest time

Like discharging rate, the effect of rests during cycling on the knee occurrence is mixed (Fig. 20c–d). Keil et al. ²¹ found that decreasing rest time from 900 seconds to 10 seconds after both charge and discharge delayed the knee in NMC/graphite cylindrical cells (Fig. 20c). Ma et al. ⁶ found a similar result: removing the 30-minute rests after both charge and discharge delayed the knee, but only with an upper cutoff voltage of 4.3 V. The rest time had no effect at 4.1 V. These observations were rationalized by less time at high potential when plotted as a function of cycle number, which can induce knees driven by side reaction product buildup. In fact, if the data in Fig. 20c is replotted as a function of estimated time instead of cycle number, the curves become much more similar (Figure S2c)—suggesting that the degradation is primarily driven by time and not cycle number. In contrast, Epding et al. ¹⁷² found that longer rest times between cycles delayed the knee occurrence (Fig. 20d). The authors proposed that these rests offered reversibly plated lithium time to reintercalate; another possibility is that the rests allowed for greater utilization of cycleable lithium. ¹⁷³ Note that periodic rests interspersed throughout a cycling test can substantially improve lifetime in some cases. ¹⁷⁴

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Increased rest periods may delay the onset of knees driven by high overpotentials (e.g., rate-dependent lithium plating at high rates or low temperatures). For knees driven by side reaction product buildup (e.g., porosity decrease, positive electrode resistance growth, etc.), increased rest may be beneficial or harmful for lifetime. Longer rests at high temperature and high state of charge may exacerbate the onset of these knees; in fact, Ecker et al. ⁸⁸ found that NMC/graphite cells stored at 50 °C, 100% SOC (4.162 V) exhibited a knee after 100 days of storage, while cells stored at 50 °C and 85% SOC (4.018 V) or below did not exhibit knees even after 400 days of storage. However, increased LLI from high temperature, high SOC rests may delay electrode saturation knees by increasing the amount of LAM required for knee onset. ^57,175 Increased rest after high rate cycling will also allow for the internal temperature of the cell to decrease, which will decrease side reaction rates and delay knee onset; fully accounting for the relationship between internal cell temperature, heating due to (dis)charging, and cooling during rests requires detailed cell thermal characterization and modeling. More work is needed to understand the sensitivity of knees to rest at both low and high state of charge across different cell designs and usage conditions.

Temperature

The effect of temperature on knee onset depends on the test conditions. For example, some studies have found that knee onset is minimized at 25 °C ^40,61,63 or 35 °C. ¹⁷ At lower temperatures, knees are primarily attributed to lithium plating; at elevated temperatures, knees are attributed to side reaction mechanisms, such as lithium plating induced by SEI growth-induced porosity decrease, positive electrode resistance growth, electolyte depletion, and additive depletion (Fig. 21a). ^{14,17,40,61,63,107} In general, the temperature that minimizes degradation is reportedly lower for LFP cells than NMC cells ⁸ and lower for power cells than energy cells. ⁶⁵

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Pressure

While predicting knees is important, predicting end-of-life is more directly relevant for estimating product performance and warranty costs. In Fig. 22, we display the relationship between knee locations and end-of-life (defined as 80% of nominal capacity) across 17 datasets (303 cells) with knees and different knee pathways, cell types (chemistry, geometry, size, lab-made vs commercial, etc.), and testing conditions. The data used to generate Fig. 22 is summarized in Table SI; data was obtained via direct access to the corresponding databases when possible ^{37,55,69,151,160} or via WebPlotDigitizer. ¹⁷⁶ We find a strong linear relationship from the knee point to the end-of-life (R² = 0.874). Apart from the data presented in Schuster et al., ¹⁷ the knee points take place at or before the end-of-life. Thus, predicting the knee location can often provide an estimate of end-of-life as well. For the Severson et al. ³⁷ method, this correlation holds independent of the knee point identification algorithm used (Figure S3). However, we note that the reverse relationship (using the knee location to predict end-of-life) does not necessarily hold, as we only selected cells with knees in this study and many cells reach 80% capacity without exhibiting a knee (i.e., exhibit linear or sublinear degradation).

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We also explored the relationship between capacity knees and resistance "elbows". Many aging studies have shown that capacity knee onset is nearly always correlated to the onset of rapid resistance growth, even across different cell chemistries, form factors, and test protocols. Table A·II reports the relative capacity and resistance at the knee onset point from several studies that reported both capacity fade and resistance growth; all studies measured resistance using a direct-current pulse except for Schuster et al., ¹⁷ which used electrochemical impedance spectroscopy. Of all these aging studies, cells with capacity knees always displayed resistance elbows, and with the exception of the work by Martinez-Laserna et al., ¹⁷⁷ the reverse is also true. For instance, the Ecker et al. ⁸⁸ dataset previously discussed (Fig. 19) found a strong correlation between capacity knees and resistance elbows. However, we cannot infer causality without more careful study; in many cases, the resistance elbow may occur with the capacity knee simply because the local current densities have increased with the loss of active capacity. We expect that a resistance elbow will almost always follow a capacity knee, but a capacity knee will not necessarily follow a resistance elbow (i.e., during low rate cycling). We note that our resistance growth-induced knee pathway (Fig. 12) illustrated a capacity knee driven by linearly increasing resistance. This correlation may provide opportunities to estimate either capacity or resistance from measurements of the other, e.g., in situations where one of these values is easy to measure and the other is challenging. Lastly, this correlation highlights the value of periodic diagnostic tests at multiple rates to disentangle low-rate and high-rate effects.

As this work has demonstrated, knees in lithium-ion batteries are complex given the variety of knee pathways, internal state trajectories, and the combined effects of interactions, heterogeneity, and variation. While three of the knee pathways (lithium plating, electrode saturation, and resistance growth) are largely dependent on bulk internal states (i.e., LLI, LAM, and resistance) and are thus straightforward to detect and model via electrochemistry, three knee pathways (electrolyte and additive depletion, percolation-limited connectivity, and microscale mechanical deformation) involve subtle effects that are challenging to detect via electrochemical signals (e.g., porosity decrease, remaining additive amount). Additionally, the nature of the three internal state trajectories that cause knees also pose major challenges (i.e., extrapolation of a superlinear function for snowball trajectories, simultaneously tracking two internal states for hidden trajectories, etc.). Furthermore, interactions, heterogeneity, and variability add additional layers of difficulty: accurate modeling of these knee phenomena requires an understanding of simultaneous interactions between many degradation mechanisms over multiple length scales, heterogeneous electrochemical/thermal/mechanical gradients within cells, and cell-to-cell variation along multiple dimensions of cell design and manufacturing processes. A final complication is translating lab testing results to a variety of field usage conditions, which requires an accurate understanding of both real-world usage behavior and battery degradation as a function of infinite behavioral axes (e.g., distributions of temperature, state of charge, rest time, etc.); these challenges are only starting to be recognized by the academic community, ¹⁵⁸ as access to real field data is an obvious challenge. Given this complexity, modeling and predicting knees is undoubtedly a formidable task.

Despite these challenges, we remain optimistic about avenues for improving knee modeling and prediction. The primary goal of this work is to lay the foundation for a fundamental understanding of the physics of knees; this understanding enables an accurate assessment of the limits of today's models and opportunities for future work. Our findings suggest that some knee pathways, such as electrode saturation or resistance growth, can be readily predicted from physics-driven or "degradation mode" (e.g., 'Alawa ^44,89 ) modeling that uses cell-level electrochemical signals; for instance, Anseán et al., ⁵⁰ Sulzer et al., ⁹⁴ and Mandli et al. ²⁰ demonstrated successful prediction of lithium plating, electrode saturation, and resistance growth knees, respectively, using bulk electrochemical state variables derived from differential capacity analysis and direct-current resistance. For other knee pathways, such as plating due to porosity decrease and additive depletion, the modeling is straightforward (e.g., Yang et al. ¹⁹ ), but the methods to estimate the relevant internal state variables are not (e.g., they require laborious destructive measurements). We hope this work inspires future research into a comprehensive knee prediction toolkit that generalizes over a wide range of cell designs and use cases.

Lastly, we discuss the role of data-driven methods in this space. Data-driven models may be well suited for knee pathways with bulk electrochemical signals; for instance, the high predictive performance of features sourced from the cell voltage responses during discharge in Severson et al. ³⁷ was largely attributed to the detectability of the knee pathway via bulk electrochemical signals (i.e., measuring LAM_deNE via differential capacity analysis). However, data-driven models trained on cycling data will naturally be poorly suited for knee pathways with signals that are challenging to measure via electrochemistry. To this end, two promising research directions are to develop input perturbations that magnify the response of hard-to-detect signals and to introduce additional characterization techniques into cell cycling tests. Our review has highlighted the benefits of periodic low-rate diagnostic cycles; perhaps more targeted electrochemical input signals could provide major benefits. For instance, average charge and discharge voltage are readily accessible features that can indicate rapid changes in lithium inventory loss or internal resistance growth rates. ¹⁷⁸ As a whole, datasets that span a variety of knee pathways for various cell designs and use cases are needed for training generalizable data-driven models. Generation of synthetic data sets using physics-based models may be well suited for training generalizable models reliant on electrochemical signals to identify knee pathways or predict knees. ^44,179 Furthermore, quantifying the values of bulk internal states from field data remains challenging. ^41,158,180