Endpoint Slippage Analysis in Batteries

• Endpoint slippage analysis is a diagnostic method that quantifies shifts in charge and discharge endpoints to separate parasitic oxidation and reduction in battery cells.
• It employs a cumulative-capacity representation and slope-weighted parameters to correct for overlapping aging modes and infer side-reaction rates.
• The technique’s reliability depends on electrode chemistry and operating conditions, with limitations evident in silicon, hard carbon, and partial-depth discharge systems.

Searching arXiv for recent battery papers on endpoint slippage analysis and related limitations. arXiv.search_query: query="all:\"endpoint slippage analysis\" OR all:\"endpoint slippage\" battery", max_results=10, sort_by="submittedDate" Endpoint slippage analysis is a diagnostic method for separating parasitic oxidation and reduction in rechargeable batteries by tracking how the endpoints of charge and discharge move when voltage–capacity data are plotted on a cumulative capacity axis. In this representation, discharge capacity is plotted with decreasing values of cumulative capacity, and charge capacity is plotted starting from the discharge endpoint of the previous half-cycle; the movement of the discharge endpoint and charge endpoint from cycle to cycle constitutes endpoint slippage. The method became prominent because direct capacity measurements can be unreliable whenever parasitic reduction and parasitic oxidation coexist: parasitic reduction traps electrons, whereas parasitic oxidation donates electrons to the cell’s inventory and can lead to temporary capacity gain (Rodrigues, 2024).

1. Formal definition and cycle-level observables

The method is built on a cumulative-capacity representation of full-cell cycling data. For each cycle $k$, the central observables are the charge capacity $Q_{c,k}$, the discharge capacity $Q_{d,k}$, the charge endpoint $C_k$ at end of charge, and the discharge endpoint $D_k$ at end of discharge. Endpoint slippage between cycles $k-1$ and $k$ is written as

$$\begin{aligned} D_{\text{slip},k} &= D_k - D_{k-1} = Q_{c,k} - Q_{d,k} \ C_{\text{slip},k} &= C_k - C_{k-1} = Q_{c,k} - Q_{d,k-1} \ Q_{\text{loss},k} &= Q_{d,k-1} - Q_{d,k} = D_{\text{slip},k} - C_{\text{slip},k}. \end{aligned}$$

These identities show that discharge-endpoint drift, charge-endpoint drift, and capacity loss are tightly linked (Rodrigues, 4 Aug 2025).

Parasitic processes are commonly represented at the half-cycle level. If $I_{\mathrm{red},j}$ and $I_{\mathrm{ox},j}$ are the average parasitic reduction and oxidation rates in half-cycle $Q_{c,k}$0, then the corresponding parasitic capacities are

$Q_{c,k}$1

with $Q_{c,k}$2 the half-cycle duration. Endpoint slippage analysis is then an attempt to infer $Q_{c,k}$3 and $Q_{c,k}$4 from measured $Q_{c,k}$5 and $Q_{c,k}$6.

This formalism matters because it shifts the diagnostic problem away from net capacity change alone. Capacity retention depends on $Q_{c,k}$7, whereas degradation severity is more closely tied to the total parasitic activity. A cell can therefore appear comparatively stable in capacity while still sustaining substantial side-reaction charge.

2. Classical interpretation in graphite-based lithium-ion cells

The traditional interpretation was developed for graphite-based Li-ion cells, especially systems such as NMC–graphite. Its central assumption is geometric: at end of charge, graphite is on a high-Li plateau and the positive electrode potential increases steeply, so charging is PE-limited; at end of discharge, graphite is nearly vertical at low Li content, so discharging is NE-limited. Under these conditions, parasitic reduction at the negative electrode moves the discharge endpoint, while parasitic oxidation at the positive electrode moves the charge endpoint (Rodrigues, 2024).

In its simplest cycle-level form, the classical approximation is

$Q_{c,k}$8

Without assuming identical rates in charge and discharge half-cycles, the corresponding relations are

$Q_{c,k}$9

$Q_{d,k}$0

In this canonical picture, discharge slippage is therefore interpreted as a proxy for reduction-like processes, and charge slippage as a proxy for oxidation-like processes.

This interpretation explains why endpoint tracking became widely used. In ideal graphite-based cells, parasitic reduction produces rightward discharge endpoint slippage with little charge-endpoint motion, while parasitic oxidation produces rightward charge endpoint slippage with little discharge-endpoint motion. The same framework also admits redox shuttles, which cause equal shifts of both endpoints. Within that restricted regime, endpoint slippage analysis is both compact and physically transparent.

3. Generalized slope-weighted formulation

The classical separation is not universal. A generalized treatment introduces two dimensionless “limitation parameters,” $Q_{d,k}$1 and $Q_{d,k}$2, derived from the local slopes of the positive-electrode and negative-electrode voltage curves at end of discharge and end of charge. In the formulation summarized for full cells, $Q_{d,k}$3 and $Q_{d,k}$4; $Q_{d,k}$5 encodes how strongly EOD is limited by PE vs NE, and $Q_{d,k}$6 similarly for EOC. For many NMC–graphite cells cycled $Q_{d,k}$7–$Q_{d,k}$8 SOC, $Q_{d,k}$9 and $C_k$0, which is precisely why the graphite approximation works (Rodrigues, 4 Aug 2025).

With these parameters, endpoint slippage becomes

$C_k$1

and

$C_k$2

If side-reaction rates are constant across the relevant half-cycles, these reduce to

$C_k$3

and

$C_k$4

The graphite limit is recovered by setting $C_k$5 and $C_k$6.

The significance of this formulation is that it makes explicit what the classical approximation suppresses: both oxidation and reduction can contribute to both endpoints. Discharge slippage is mostly reduction-driven only when $C_k$7 is large and $C_k$8 is small; charge slippage is mostly oxidation-driven only when $C_k$9 is large and $D_k$0 is near zero. This gives endpoint slippage analysis a slope-weighted character that depends on the local electrode thermodynamics at the operating endpoints.

4. Failure modes in silicon, hard carbon, and partial-depth-of-discharge operation

The main failure mode arises when the negative-electrode voltage profile no longer resembles graphite. In typical Si electrodes there is no flat plateau at high Li content during lithiation, and the delithiation profile is not very steep near full delithiation. As a result, parasitic reduction also causes charge endpoint slippage, and parasitic oxidation also causes discharge endpoint slippage. In such systems, the naive mapping from endpoint motion to a single parasitic channel is no longer valid (Rodrigues, 2024).

A second failure mode is the “reservoir effect,” emphasized for Si negative electrodes. Because the Si profile at EOD is sloped rather than nearly vertical, a rise in NE potential can correspond to significant extra delithiation of Si, releasing Li and electrons back to the PE. This partially compensates electron loss and can make both capacity fade and endpoint slippage underestimate the real side-reaction charge. In the NMC811/Si example with only parasitic reduction at rate $D_k$1 per half-cycle, naive interpretation of slippages would underestimate $D_k$2 by $D_k$3 and infer a non-zero apparent $D_k$4 of $D_k$5 of $D_k$6. In a more complex NMC811/Si example with time-varying $D_k$7 and $D_k$8, the mismatch between visible capacity fade and actual loss of Li inventory reached $D_k$9 of the total applied LLI (Rodrigues, 2024).

The same structural issue appears in other chemistries and operating windows. For NMC811/Si, raising the discharge cut-off from $k-1$0 V to $k-1$1 V moves EOD into a less steep region of the Si profile, increases $k-1$2, and can cause discharge slippage to miss $k-1$3 of the side-reaction capacity. Even graphite can fail if the cell is not fully discharged: at certain discharge cut-offs such as $k-1$4 V in NMC811/Gr, EOD lies in a graphite plateau region, $k-1$5, and under pure parasitic reduction the discharge endpoint slippage can become very small or essentially zero. Hard carbon in Na-ion batteries shows an analogous dependence on whether EOD lies in a sloped region or in a nearly vertical low-Na region.

These examples overturn a common misconception: endpoint slippage is not intrinsically graphite-like. Its interpretability depends on where EOC and EOD sit on the electrode voltage profiles. That dependence is methodological rather than incidental.

5. Interference from loss of active material and impedance rise

A further complication is that endpoint shifts are not unique signatures of parasitic oxidation and reduction. Loss of active material and impedance rise can themselves induce endpoint slippage. The detailed classification adopted for LAM distinguishes $k-1$6, $k-1$7, $k-1$8, and $k-1$9, according to whether negative-electrode or positive-electrode domains are lost in lithiated or delithiated states. The state of lithium in the lost domains determines how much Li is removed from the reversible inventory and at which endpoint that loss manifests (Rodrigues, 4 Aug 2025).

The most consequential case is $k$0. Its contribution is

$k$1

so it produces the same qualitative signature as parasitic reduction: rightward discharge endpoint slippage with no charge-endpoint motion. $k$2 acts differently: $k$3 which gives leftward charge endpoint slippage and can partially cancel or mask parasitic oxidation.

Impedance rise introduces an additional voltage drop $k$4 and therefore shifts the endpoints through local full-cell slopes: $k$5 The generalized framework treats these effects as additive: $k$6

This correction strategy is mathematically straightforward but experimentally difficult. It requires the modality and magnitude of LAM and, crucially, the Li content of disconnected domains. The central limitation identified for aged cells is that the average Li content of disconnected active material domains is, in many cases, unknowable; the paper states that it is “often unknown and perhaps even unknowable.” That epistemic constraint limits how far endpoint slippage analysis can be pushed as a stand-alone quantitative decomposition.

6. Diagnostic scope, methodological status, and present use

Within its valid regime, endpoint slippage analysis remains powerful. It is most robust in early life, under moderate temperatures and depths of discharge, and in well-designed commercial cells where LAM is modest over the analysis window. In that regime, the classical endpoint signatures provide a compact way to infer reduction-like and oxidation-like parasitic behavior. Where $k$7 and $k$8 are known or approximately constant, the generalized equations provide a route from measured endpoint slippages to corrected parasitic rates, and the same mathematical framework also generalizes the Tornheim–O’Hanlon capacity/efficiency method (Rodrigues, 4 Aug 2025).

Outside that regime, endpoint trends are still informative, but their meaning changes. Severe rightward discharge endpoint drift suggests dominant reduction-like processes, but that category may include SEI growth, $k$9, or $$\begin{aligned} D_{\text{slip},k} &= D_k - D_{k-1} = Q_{c,k} - Q_{d,k} \ C_{\text{slip},k} &= C_k - C_{k-1} = Q_{c,k} - Q_{d,k-1} \ Q_{\text{loss},k} &= Q_{d,k-1} - Q_{d,k} = D_{\text{slip},k} - C_{\text{slip},k}. \end{aligned}$$0 once PE becomes limiting. Leftward charge endpoint drift suggests oxidation-opposing processes such as $$\begin{aligned} D_{\text{slip},k} &= D_k - D_{k-1} = Q_{c,k} - Q_{d,k} \ C_{\text{slip},k} &= C_k - C_{k-1} = Q_{c,k} - Q_{d,k-1} \ Q_{\text{loss},k} &= Q_{d,k-1} - Q_{d,k} = D_{\text{slip},k} - C_{\text{slip},k}. \end{aligned}$$1, impedance rise, or chronic NE over-utilization. The method therefore remains diagnostically useful even when strict quantitative inversion becomes unreliable (Rodrigues, 2024).

The broader methodological conclusion is that endpoint slippage analysis is not a universal plug-in diagnostic. It is a geometry- and chemistry-dependent inference framework whose validity depends on electrode-profile shape, operating window, and the presence or absence of additional aging modes. In graphite-based Li-ion cells at full depth of discharge, the classical interpretation is often adequate. In Si electrodes, hard carbon Na-ion systems, partial-depth-of-discharge operation, or cells with substantial LAM and impedance rise, endpoint slippage must be interpreted through the generalized slope-weighted and additive formulations, or else treated as a qualitative component of a broader diagnostic toolbox rather than a direct measure of parasitic reduction and oxidation.