Quantifying Chemical Re-Intercalation Capacity from Differential Voltage Relaxation Derivatives in Commercial Cells

Slope

Open-circuit voltage decay immediately following current interruption reflects a superposition of electrochemical processes occurring across distinct time constants. When charging or discharging stops, a lithium-ion cell does not instantly reach thermodynamic equilibrium. The immediate voltage drop stems from Ohmic resistance across the electrolyte, current collectors, and active material matrices.

Charge-transfer polarization at the electrode-electrolyte interface dissipates within seconds. Over minutes and hours, concentration gradients within the solid active material particles and liquid electrolyte relax toward spatial uniformity. Concurrently, metallic lithium deposited on the anode surface during aggressive charging chemically reacts with the graphite host, re-intercalating as lithium carbide.

Quantifying this chemical re-intercalation capacity without destroying the cell requires decoupling its transient potential response from concurrent diffusion decay. Direct open-circuit voltage monitoring yields smooth relaxation curves where subtle kinetic transitions remain obscured. Plotting open-circuit potential against the square root of rest time linearizes planar solid-state diffusion, yet overlapping relaxation regimes still obscure the raw time-domain signal.

Converting voltage decay into differential relaxation derivatives transforms these inflections into quantifiable spectral features.

A rest duration exceeding four hours at 25 degrees Celsius resolves the second differential derivative peak corresponding to solid-state overhang equalization.

Taking the first derivative of relaxation voltage with respect to the square root of time isolates individual kinetic time constants. The resulting differential signature displays characteristic peaks corresponding to discrete electrochemical relaxation steps. A plateau in open-circuit potential caused by the co-existence of metallic lithium and intercalated graphite phases produces a prominent derivative maximum.

Integrating the area beneath this peak yields the exact charge transferred during chemical re-intercalation. Precision voltage logging during passive rest periods converts open-circuit relaxation from a waiting interval into a quantitative diagnostic tool for reversible lithium plating.

Whether the secondary inflection during prolonged relaxation originates strictly from overhang lithium redistribution or partly from phase transitions within damaged cathode primary particles remains unresolved.

Peak

Differentiating relaxation voltage with respect to logarithmic time or root time transforms subtle slope changes into distinct local extrema. When metallic lithium plates onto the graphite anode during fast charging, it establishes a mixed potential governed by the lithium metal reduction potential and the local lithium-graphite intercalation stage potential. While surface metallic lithium remains present, the open-circuit voltage stays pinned near this mixed potential.

As the metallic layer clears through chemical re-intercalation into the graphite host, the cell voltage transitions toward the true equilibrium intercalation potential of the graphite anode.

How Do Differential Relaxation Derivatives Isolate Chemical Re-Intercalation?

Isolation relies on calculating the inverse differential voltage derivative, defined as the time derivative over the voltage derivative or the logarithmic time differential per voltage step. In this differential representation, voltage plateaus transform into pronounced peaks whose position on the voltage axis identifies the thermodynamic phase transition, while the peak area scales with the volume of re-intercalated lithium. Differentiation amplifies slope changes, allowing analytical software to separate solid-state concentration gradient dissipation from chemical reaction kinetics.

The time domain of the relaxation spectrum dictates which physical mechanism controls the derivative profile. Five distinct kinetic regimes govern post-charging voltage decay in commercial lithium-ion formats.

• Ohmic polarization decay settles within milliseconds following current cessation, yielding a sharp voltage drop driven strictly by electrolyte and contact resistance.
• Charge-transfer dissipation resolves across the second scale as double-layer capacitance discharges across the electrode-electrolyte interface.
• Solid-state concentration relaxation extends across hundreds of seconds while lithium concentration profiles homogenize within the bulk active material particles.
• Chemical lithium re-intercalation manifests between ten minutes and two hours, producing a temporary voltage plateau as surface-plated metallic lithium chemically reacts with surrounding graphite.
• Anode overhang lithium migration operates across tens of hours as inactive graphite region potentials equalize with active electrode zones.

Differentiation of high-frequency logging data requires filtering to prevent numerical noise from generating false derivative maxima. Applying a moving Savitzky-Golay polynomial filter preserves peak height and envelope shape while removing high-frequency analog-to-digital converter quantization jitter. The extracted peak location identifies the exact state of charge associated with the plated lithium phase, while the derivative peak width indicates the kinetic rate constant of the re-intercalation reaction.

Relaxation data logged without millivolt-level accuracy masks re-intercalation signals under acquisition noise.

Procedure

Quantification of chemical re-intercalation capacity demands strict control over ambient thermal drift and logging fidelity during the post-charge rest phase. A temperature fluctuation of half a degree Celsius alters cell open-circuit voltage by an amount exceeding the derivative peak amplitude of minor lithium plating. Testing protocols enforce isothermal stabilization prior to high-rate pulse application and throughout the subsequent multi-hour relaxation window.

1. Mount the commercial cell inside a temperature-controlled chamber maintained at 25.0 degrees Celsius with standard four-wire Kelvin sensing contacts.
2. Apply a fast-charging pulse protocol to 100 percent state of charge to induce potential lithium plating conditions.
3. Disconnect the load current instantaneously using a low-bounce solid-state switch and initiate continuous high-frequency voltage logging.
4. Record open-circuit voltage for a minimum of six hours at a sampling interval of 100 milliseconds or finer.
5. Compute the numerical derivative dV/dsqrtt using a Savitzky-Golay smoothing filter to suppress high-frequency thermal noise.
6. Identify the differential peak corresponding to chemical re-intercalation and integrate the associated pseudo-capacity over time.

Consider a worked calculation for a commercial 50 Ah nickel-manganese-cobalt graphite pouch cell subjected to a 3C fast-charging regime at 15 degrees Celsius. Upon current cessation, voltage data acquired at 10 Hz enters a numerical differentiation pipeline. The raw voltage drops from 4.200 V to 4.145 V within 50 milliseconds due to Ohmic resistance, followed by a slower charge-transfer decay over 12 seconds.

Between 800 seconds and 3,400 seconds of rest, the raw voltage relaxation curve displays a localized voltage plateau centered at 4.112 V.

Calculating the logarithmic derivative t · (dV/dt) isolates a distinct peak situated at t = 1,850 seconds. Converting this voltage derivative peak into re-intercalated charge involves mapping the plateau duration against the differential capacity function of the graphite anode operating at that potential phase. With an equivalent differential capacity dQ/dV = 18.5 Ah/V at the Li C6 to Li C12 transition, and a plateau voltage span Δ V = 44 mV, direct numerical integration yields:

Qre-inter = intt1t2 left( fracdQdV right) · left| fracdVdt right| dt = 0.814 Ah

Thermal fluctuation during open-circuit relaxation distorts derivative signatures and masks low-amplitude re-intercalation signals.

This calculated value of 0.814 Ah establishes that 1.63 percent of the nominal cell capacity existed as temporary surface-plated metallic lithium at the conclusion of fast charging, which successfully re-intercalated into the anode host during rest. Without this relaxation derivative extraction, a standard discharge test initiated immediately after charge would count this 0.814 Ah as permanently lost capacity, mischaracterizing reversible plating as irreversible degradation.

Ignoring the temperature dependence of relaxation derivatives leads fast-charging pack designers to understate metallic lithium plating risks, resulting in accelerated capacity loss and premature field failure.

Divergence

Experimental signatures of chemical re-intercalation vary based on positive electrode thermodynamics and cell form factor. In lithium iron phosphate cells, the open-circuit voltage plateau of the cathode reaction dominates the overall cell voltage across 90 percent of the state-of-charge spectrum. This flat voltage profile compresses the dynamic range of open-circuit relaxation, making differential voltage peaks harder to resolve than in nickel-rich layered oxide chemistries.

Analyzing lithium iron phosphate relaxation requires calculating second derivatives with respect to logarithmic time, d2V / d(ln t)2, to pull re-intercalation inflections out of the flat cathode baseline.

In nickel-manganese-cobalt formulations, cathode slope variation assists diagnostic isolation. The sloping open-circuit voltage curve of nickel-rich cathodes acts as a stable reference potential against which anode potential transitions show up clearly. Mechanical cell construction also introduces divergence.

Prismatic cells exhibit non-uniform internal pressure distributions, causing metallic lithium plating to occur preferentially near core winding edges where mechanical stress is highest. Cylindrical formats exhibit radial thermal gradients during fast charging, concentrating plated lithium along the cooler outer casing walls. These structural differences change the effective surface area available for chemical re-intercalation, directly altering the time constant τ of the derivative peak.

Qualification of fast-charging algorithms across commercial procurement streams requires screening protocols that account for these chemistry-specific derivative boundaries.

• Sampling frequency thresholding prevents high-frequency measurement noise from corrupting second-derivative calculations during initial rest seconds.
• Isothermal bath enclosure eliminates external ambient temperature swings that simulate artificial voltage inflection points.
• Overhang geometry correction isolates inter-particle chemical re-intercalation from long-term passive charge equalization across inactive graphite regions.
• Baseline OCV subtraction isolates pure chemical re-intercalation signals from baseline thermodynamic relaxation curves obtained under non-plating charge rates.

IEC 62660-1 section 6.2 mandates open-circuit stabilization periods that directly mask transient re-intercalation derivatives if baseline logging intervals exceed five seconds.

Post-charge voltage plateaus can be interpreted either as standard anode intercalation dynamics or as reversible surface lithium plating.

Valuation

Assessing the commercial worth of degraded or fast-charged battery inventory requires translating differential relaxation derivatives into actionable health metrics. Standard incoming inspection methods rely on static capacity measurement and single-frequency 1 kHz AC impedance. These legacy metrics fail to detect early-stage metallic lithium accumulation.

A cell exhibiting two percent capacity loss from metallic plating may pass standard incoming capacity checks if tested after a multi-hour rest period, because chemical re-intercalation recovers the plated lithium back into the graphite matrix. The cell appears healthy on initial testing, yet it harbors irreversible structural damage, SEI growth, and electrolyte consumption caused during the plating event.

Deploying automated differential relaxation analysis at receiving inspection enables automated screening of fast-charged or field-returned battery packs. Software algorithms fitting relaxation derivative spectra quantify the ratio between reversibly re-intercalated lithium and permanently dead lithium. When the integrated re-intercalation capacity exceeds a predefined risk threshold, the automated sorter flags the batch for accelerated capacity fade risks before those cells enter pack assembly streams.

Unaccounted re-intercalation capacity leads pack integrators to misclassify reversible lithium plating as permanent capacity loss during rapid cycle-life audits.

Integrating relaxation derivative diagnostics into commercial fast-charging controller firmware transforms charging safety. Rather than relying on conservative fixed-current step tables or inaccurate empirical aging models, the battery management system measures post-charge voltage relaxation derivatives during brief rest pauses. If derivative peak extraction reveals active lithium plating during a fast-charge step, the management system dynamically reduces the current setpoint for subsequent charging cycles.

Dynamic control based on relaxation derivatives eliminates lithium plating conditions in real time, extending commercial pack cycle life without imposing unnecessary charging time penalties.

Master supply agreements incorporating differential relaxation diagnostic criteria reduce warranty claim disputes by establishing quantitative boundaries between reversible lithium plating and unrecoverable active material loss.