Differential Capacity Voltage Relaxation Analysis for Detecting Low Temperature Plating

Frost

Sub-zero operating environments alter thermodynamic equilibrium and mass transport rates within lithium-ion cells. Charging at sub-zero temperatures sharply increases kinetic barriers to lithium-ion desolvation and solid-state diffusion in the graphite lattice. Higher liquid electrolyte viscosity reduces ionic conductivity in the separator and electrode pores, while charge-transfer resistance across the solid-electrolyte interphase grows exponentially following Arrhenius kinetics.

Together, these transport bottlenecks shift the electrochemical potential of the negative electrode, pulling the local anode potential below 0 V relative to the reference lithium couple.

Operating below 0 V versus lithium metal fundamentally alters negative electrode reaction energetics. At standard temperatures, lithium ions intercalate cleanly between graphene sheets to form stage-one lithium-graphite intercalation compounds. Sub-zero temperatures restrict intercalation below the rate of ionic flux reaching the particle surface, causing excess lithium ions to build up at the phase boundary.

The resulting local overpotential depresses the anode Fermi level above the reduction potential of metallic lithium, triggering the nucleation and growth of metallic lithium on the graphite particle surfaces.

Anode Overpotential at Low Temperatures

Electrochemical kinetics at the graphite-electrolyte interface determine whether lithium intercalates or deposits as metal. Exchange current density drops by over an order of magnitude between room temperature and sub-zero regimes. At 25°C, typical graphite anodes maintain exchange current densities sufficient for 1C to 2C charging without driving local overpotential negative.

At -20°C, exchange current density drops far enough to cause severe activation overpotential even under modest rates like 0.2C.

Solid-state diffusion within graphite particles becomes the rate-limiting step during sustained cold charging. Lithium diffusion coefficients in graphite drop from roughly 10^-10 cm²/s at room temperature to less than 10^-12 cm²/s at -20°C. Ions reaching the particle surface cannot move into the bulk lattice quickly enough to prevent localized concentration polarization, which depresses the equilibrium surface potential below the metallic lithium deposition threshold.

Desolvation energy presents an additional barrier at low temperatures. Lithium ions moving through bulk liquid electrolyte carry a solvation sheath of ethylene carbonate, ethyl methyl carbonate, or dimethyl carbonate molecules. Stripping these molecules before crossing the interphase requires substantial activation energy, typically 50 to 70 kJ/mol.

At sub-zero temperatures, limited thermal energy makes overcoming this barrier harder, increasing interfacial impedance and favoring metallic deposition on the outer surface of the graphite active material.

Thermodynamic Drivers of Metallic Deposition

Thermochemistry governs phase stability for metallic deposition on carbonaceous hosts. Deposition begins once local potential drops below the thermodynamic reversible potential of the lithium reduction reaction. Metallic lithium forms dendritic or mossy structures depending on local current distribution and electric field concentrations at surface asperities.

Dendrites expand the reactive surface area exposed to liquid electrolyte, accelerating secondary side reactions.

Plated metallic lithium is thermodynamically unstable against intercalated graphite. During post-charge rest, plated lithium reacts chemically with adjacent un-intercalated graphite sites through a slow chemical re-intercalation process. It also reacts continuously with the liquid electrolyte to form secondary solid-electrolyte interphase material, permanently consuming active lithium inventory and solvent, driving capacity fade and impedance growth.

Uncontrolled metallic deposition creates severe operational risks in commercial cells. Microscopic dendrites can penetrate microporous polymer separators, creating electronic conduction paths between cathode and anode. These internal short circuits cause rapid self-discharge, localized heating, and potential thermal runaway.

Quantifying the onset conditions of metallic deposition helps prevent early field failure and battery fires.

Plateau

Open-circuit potential curves during rest show distinct thermodynamic signatures when metallic lithium is present on the negative electrode. Following a low-temperature charge step, measured cell terminal voltage undergoes a characteristic relaxation trajectory. Without metallic deposition, terminal voltage relaxes monotonically toward the thermodynamic open-circuit potential set by the state of charge of both electrodes, governed by solid-state diffusion within the bulk electrode particles.

Cells with surface-plated metallic lithium exhibit non-monotonic relaxation. During initial rest, two competing electrochemical phases co-exist on the negative electrode surface: metallic lithium and partially intercalated graphite. This spatial coexistence sets up a mixed potential regime, where the metallic lithium phase pins the local negative electrode potential near 0 V versus lithium metal, holding a constant potential difference against the positive electrode for a finite duration.

Plated metallic lithium creates a distinct thermodynamic mixed potential that halts ordinary open-circuit voltage decay during post-charge rest periods.

Chemical re-intercalation consumes the metallic surface layer as rest continues. Metallic lithium acts as a local anode, releasing electrons into the conductive graphite matrix while passing lithium ions into the electrolyte. Adjacent graphite regions absorb these ions through standard intercalation.

This internal short circuit persists until the surface metallic lithium is fully depleted, at which point the negative electrode potential jumps rapidly from 0 V to the open-circuit potential of stage-two or stage-three graphite intercalation compounds.

Open Circuit Voltage Equilibration Kinetics

Equilibration timescales during post-charge rest reflect the mass of metallic lithium deposited during charging. Larger masses require longer rest durations to complete chemical re-intercalation. Throughout this window, terminal cell voltage shows a flat or slow-moving plateau whose duration scales linearly with the quantity of reversibly plated lithium.

Internal cell impedance and electrolyte temperature alter the resolution of the relaxation plateau. Lower temperatures slow chemical re-intercalation, extending the voltage shoulder and making detection easier on low-frequency sampling channels. High ohmic resistance in thick electrodes broadens the phase transition boundary, softening the voltage transition slope at the end of the plateau.

Phase dynamics within the graphite matrix also shape the relaxation curve. Graphite undergoes distinct structural phase transitions during lithium insertion, moving from dilute stage-one through stage-three, stage-two, and fully saturated stage-one LiC₆. Each transition shows a characteristic potential plateau.

Distinguishing the chemical re-intercalation plateau from bulk graphite stage transition plateaus requires precise differential mathematical transformation of the voltage relaxation signal.

Mixed Potential Behavior during Rest

Spatial non-uniformity across the electrode planform area creates mixed potential gradients during relaxation. Fast charging causes localized current crowding near collector tabs and electrode edges, severely depressing local overpotential and causing targeted metallic deposition while central regions remain unplated. During rest, lithium ions migrate laterally along the electrode plane alongside primary thickness-direction re-intercalation.

Re-intercalation rates depend heavily on deposit morphology. Dense, smooth metallic films present less surface area to the electrolyte, resulting in longer re-intercalation time constants. High-surface-area mossy or dendritic deposits undergo rapid re-intercalation, shortening the observable plateau.

Dense films transition sharply to pure solid-state diffusion decay, whereas mossy structures blur the inflection point in time-domain voltage curves.

Solid-electrolyte interphase growth occurs alongside chemical re-intercalation during the plateau window. Electrolyte reduction competes directly with graphite re-intercalation for available metallic lithium. Lithium consumed by electrolyte reduction forms electrochemically inactive dead lithium ~ such as solid lithium carbonate, lithium fluoride, and lithium alkyl carbonates ~ reducing net capacity recovered in subsequent discharge cycles.

Low-temperature capacity loss is frequently attributed to temporary kinetic polarization during commercial technical disputes, with common technical documentation presenting post-charge polarization as decaying within minutes without altering electrode morphology. Differential relaxation analysis refutes this interpretation by demonstrating that voltage shoulders during rest stem directly from metallic lithium oxidation rather than concentration gradient dissipation.

Derivative

Mathematical differentiation of voltage relaxation data turns subtle slope changes into distinct derivative peaks. Plotting raw open-circuit voltage against time produces smooth curves where short shoulders are difficult to isolate visually. Computing the first time derivative of terminal voltage, dV/dt, transforms these transient shoulders into pronounced peaks, pinpointing the moment metallic lithium fully dissolves from the graphite surface.

Differential capacity analysis during relaxation evaluates differential voltage against time or equivalent capacity. The expression dQ/dV during open circuit is obtained by dividing effective re-intercalation current by the instantaneous rate of voltage change dV/dt. High acquisition rates and low noise are required because numerical differentiation amplifies measurement noise exponentially.

Savitzky-Golay filtering or monotonic cubic spline interpolation must smooth raw voltage time-series data prior to differentiation.

Relaxation spectra plotted as dV/dt against time show a baseline decay interrupted by a distinct peak during lithium plating events. Baseline relaxation without metallic deposition follows a smooth power-law or logarithmic decay reflecting solid-state concentration equilibration within host particles. Plating introduces a peak whose inflection point marks where the metallic phase disappears entirely from the anode surface.

Mathematical Signal Transformation

Signal processing protocols require continuous data smoothing to prevent derivative artifacts. Raw voltage data from industrial battery cyclers often contains quantization noise, mains frequency interference, and thermal drift. Filtering protocols apply a frame length of 51 to 101 data points using a second-order polynomial Savitzky-Golay filter, preserving peak amplitude and area while removing high-frequency noise spikes that skew derivative calculations.

Time-domain transformation maps dV/dt as a function of rest duration t. The mathematical formulation converts raw voltage V(t) into the first derivative spectrum:

fracdVdt = fracddtleft

Peak extraction algorithms identify local maxima in the dV/dt curve. The peak time, designated as t_peak, represents the total duration required for chemical re-intercalation of the plated metallic lithium. Higher plating mass yields larger t_peak values.

The height of the dV/dt peak reflects how sharply the system transitions from mixed potential equilibration to host particle concentration decay.

Differential capacity curves dQ/dV extracted during rest provide direct quantification of re-intercalated charge. Because external circuit current during open circuit equals zero, internal re-intercalation current I_inter is modeled based on Butler-Volmer kinetics for metallic lithium oxidation. Integrating the differential capacity peak across the voltage range of the relaxation shoulder yields total reversible lithium capacity Q_plated.

Extracting Reversible Plating Capacity

Quantitative extraction of plated lithium mass relies on decoupling re-intercalation current from baseline diffusion decay. Integrating the baseline-corrected dV/dt peak over the time interval from rest onset to t_peak yields a parameter proportional to total stripped metallic lithium mass. Reversibly plated lithium capacity scales with integrated peak area according to empirical calibration factors established through optical or chemical validation techniques.

Baseline subtraction separates the host graphite diffusion response from the metallic lithium re-intercalation peak. A cubic polynomial or exponential function fits the baseline dV/dt signal prior to peak onset and after the voltage transition completes. Subtracting this fitted baseline isolates the peak corresponding exclusively to metallic lithium oxidation.

Re-intercalation efficiency limits absolute quantification accuracy. A fraction of plated metallic lithium reacts chemically with electrolyte rather than re-intercalating into graphite. Reversibly recovered lithium capacity extracted from dV/dt peak integration represents only the reversible fraction of total plated lithium.

Irreversible dead lithium remains unmeasured by relaxation differential analysis, necessitating secondary discharge capacity loss calculations to establish total mass balance.

Operational diagnostic procedures rely on standardized signal rules during automated analysis to establish non-destructive testing boundaries. Peak signatures appearing within sixty seconds of charge termination indicate mild surface nucleation, whereas peaks persisting beyond one thousand seconds signal extensive dendritic coverage.

• Sampling Rate Artifacts occur when voltage logging frequencies fall below 1 Hz, causing mathematical truncation of sharp peak maxima in differential time spectra.
• Thermal Drift Distortion manifests when ambient chamber temperatures fluctuate by more than 0.5°C during rest, superimposing thermal expansion voltage shifts onto relaxation curves.
• Ohmic Voltage Drop Overlap obscures early peak features when high-resistance cell connections produce slow capacitive relaxation tails during initial open-circuit rest.
• Baseline Fitting Instability arises when baseline polynomial extrapolation overlaps with secondary graphite phase transition plateaus in deeply charged cells.

Validation

Establishing the quantitative accuracy of differential voltage relaxation signatures demands direct cross-validation against destructive analytical techniques. Non-destructive electrochemical techniques provide indirect measurements of surface phase dynamics. Destructive characterization physically isolates and quantifies elemental metallic lithium on harvested anode electrodes, confirming the validity of differential relaxation peak metrics.

Mass spectrometry titration provides precise quantitative verification of metallic lithium mass. Harvested anode samples from frozen cells undergo reaction with aqueous solvents or chemical alcohols inside closed gas-tight reaction vessels. Metallic lithium reacts violently with hydrogen-active solvents to produce gaseous hydrogen, whereas intercalated lithium LiC₆ reacts without releasing free hydrogen gas.

Measuring evolved hydrogen volume using mass spectrometry or gas chromatography quantifies absolute metallic lithium mass down to microgram resolution.

How Does Differential Relaxation Compare with Destructive Titration?

Comparative studies matching mass spectrometry titration results with integrated dV/dt peak areas confirm linear correlation across varying temperatures and charging rates. Cells charged under severe low-temperature conditions (-20°C at 0.5C) demonstrate high hydrogen evolution volumes corresponding directly to extended relaxation peak times. The slope of integrated dV/dt peak area versus titrated metallic lithium mass yields high linearity, confirming dV/dt peak area as a trustworthy proxy for reversible plating mass.

Optical and electron microscopy provide structural confirmation of metallic surface deposits. Operando optical cell configurations allow real-time visual tracking of metallic lithium nucleation through transparent quartz windows. Metallic lithium appears as bright silvery deposits against the dark gray host graphite matrix.

The visual onset of silvery metallic deposits coincides precisely with the emergence of the dV/dt differential relaxation peak during post-charge rest periods.

Scanning electron microscopy of cryogenically preserved anode cross-sections reveals the morphological evolution of plated layers. Cryo-SEM prevents thermal decomposition of reactive metallic lithium during vacuum exposure. Micrographs display porous metallic lithium moss layers residing above the primary solid-electrolyte interphase film.

The thickness of this porous metallic layer correlates directly with the duration of the open-circuit voltage relaxation plateau.

Mass spectrometry titration of cells frozen immediately after -10°C charging confirms that dV/dt peak area tracks reversibly plated metallic lithium with a linear correlation coefficient exceeding 0.98.

Bench Instrumentation and Signal Acquisition Rigor

Laboratory validation of differential relaxation analysis requires high-precision cycling instrumentation. Standard commercial cyclers often exhibit voltage measurement drift and resolution limits of 1 mV, which introduces severe noise when calculating dV/dt derivatives. Quantitative relaxation analysis demands cyclers equipped with 24-bit analog-to-digital converters, providing voltage resolution better than 100 µV and voltage accuracy within ±0.01% of full scale.

Thermal control during relaxation testing must eliminate external temperature fluctuations. Battery cells possess positive temperature coefficients of open-circuit voltage, typically ranging from 0.1 mV/°C to 0.5 mV/°C depending on state of charge. A 2°C ambient temperature shift during rest induces a 1 mV voltage drift, creating false peaks in derivative curves.

Environmental chambers must maintain temperature stability within ±0.1°C throughout charge and relaxation steps.

1. Mount test cells inside temperature-controlled environmental chambers set to the target evaluation temperature with thermal equilibration held for four hours.
2. Connect four-terminal Kelvin sensing leads directly to cell terminal tabs to eliminate contact resistance variations during current pulse transitions.
3. Apply the specified low-temperature charge protocol until reaching the target upper cutoff voltage or target state of charge.
4. Terminate charging current instantaneously and initiate high-frequency voltage data logging at a minimum acquisition rate of 10 Hz.
5. Maintain uninterrupted open-circuit rest for a minimum duration of two hours to capture the full relaxation trajectory and baseline decay.
6. Process raw voltage time-series data using a low-pass digital filter to eliminate high-frequency noise prior to computing time derivatives.

Evaluating commercial cell chemistries reveals varying susceptibility to low-temperature plating artifacts. Nickel-rich layered oxides paired with silicon-graphite composite anodes display complex relaxation spectra due to silicon phase transitions. Distinguishing metallic lithium oxidation from silicon de-lithiation kinetics requires adjusting baseline fitting windows based on known silicon alloy thermodynamic potentials.

Unresolved questions persist regarding the complete recovery of metallic lithium trapped inside isolated SEI pores. Chemical re-intercalation relies on electronic contact between metallic deposits and the graphite matrix. Dendrites that detach electronically become dead lithium, remaining invisible to differential relaxation voltage signals despite posing long-term capacity fade liabilities.

Pulse

Integrating differential capacity relaxation analysis into automated test channels requires precise diagnostic pulse profiles. Automated screening routines apply controlled charging sequences followed by enforced open-circuit relaxation periods. Diagnostic routines evaluate cell health during production line quality screening, battery pack incoming inspection, and field BMS maintenance cycles.

Diagnostic pulse sequences combine short charging periods at target sub-zero temperatures with extended rest periods. Applying a step-wise C-rate matrix isolates the precise critical C-rate where metallic plating initiates. Incrementing charging rates by 0.05C intervals at fixed temperatures reveals the exact threshold where the dV/dt relaxation derivative transitions from a monotonic decay curve to a peaked relaxation spectrum.

Multi-Step Diagnostic Charge Protocol

Screening protocols utilize multi-step constant current charging sequences to evaluate dynamic plating thresholds across increasing states of charge. As a cell absorbs lithium, solid-state diffusion paths inside graphite lengthen, increasing local overpotentials. A C-rate tolerable at 10% state of charge may trigger severe metallic lithium deposition at 60% state of charge.

Automated test channels record voltage relaxation following each charge step. Test cyclers execute 5% to 10% state of charge injection pulses followed by 15-minute open-circuit rest steps. Computing dV/dt for each rest step identifies the exact state of charge boundary where metallic plating begins, allowing pack designers to construct dynamic C-rate maps that maximize charging speed while avoiding metallic deposition regimes.

Worked Calculation for Commercial NMC Cells

Evaluating a commercial 4.8 Ah 21700 cylindrical cell containing NMC811 cathode material and graphite anode illustrates quantitative mass extraction. Testing involved subjecting the cell to a fast-charge sequence at -10°C using a constant current rate of 0.5C (2.4 A) up to 4.2 V, followed immediately by a two-hour open-circuit rest step. Voltage logging captured terminal voltage at 10 Hz with 10 µV resolution.

Raw relaxation data showed a distinct voltage shoulder lasting approximately 420 seconds before transitioning into exponential potential decay. Numerical differentiation using a 51-point Savitzky-Golay filter produced a broad dV/dt peak centered at t_peak = 438 seconds. Baseline fitting using a third-order polynomial isolated the baseline-subtracted peak area.

Mathematical integration of the isolated dV/dt peak yielded an integrated differential area of 0.0385 V. Applying the empirical re-intercalation calibration factor for this cell architecture (1.091 Ah/V) converted the integrated voltage signal into physical capacity. Validation testing recorded a 42 mAh re-intercalation capacity, demonstrating that 0.875% of the total nominal cell capacity had plated as metallic lithium during the -10°C charge cycle.

Calculating irreversible capacity loss confirmed secondary electrolyte reaction losses. Discharge of the cell at 0.2C following the relaxation step recovered 4.712 Ah, representing a total single-cycle capacity loss of 88 mAh. Subtracting the 42 mAh of chemically re-intercalated reversible lithium revealed that 46 mAh of active lithium was permanently lost to secondary SEI formation and dead lithium isolation during the low-temperature charging event.

Standard battery procurement covenants mandate that fast-charge routines operating below 0°C must exhibit zero dV/dt relaxation peak features exceeding baseline noise thresholds.

BMS Implementation Constraints

Implementing differential voltage relaxation algorithms within onboard battery management systems presents computational challenges. Vehicle BMS microcontrollers possess limited RAM and processing power, restricting real-time execution of heavy floating-point Savitzky-Golay filtering or spline interpolation algorithms. BMS hardware relies on simplified discrete derivative approximations combined with rolling average window filters.

Sampling rate limitations on commercial BMS hardware impact peak detection sensitivity. Typical automotive BMS units log cell voltages across high-voltage batteries at sampling rates of 1 Hz to 0.1 Hz over CAN bus interfaces. Low sampling rates under-sample sharp voltage transitions, reducing the magnitude of calculated dV/dt peaks.

BMS algorithms deploy threshold slope change detection rather than full peak integration to identify plating events within computational memory limits.

On-line diagnostics utilize opportunistic rest periods to evaluate anode health without disrupting operation. When an electric vehicle finishes fast charging at a low-temperature public charging station, the BMS monitors open-circuit voltage decay during subsequent stationary park periods. Detecting a characteristic relaxation peak triggers automated BMS adjustments, reducing maximum allowable charging current limits on subsequent cold-weather charging cycles to protect long-term pack state of health.

Laboratory cycling cyclers execute continuous automated relaxation screening across thousands of cell channels simultaneously. High-density quality control test beds stream raw voltage data directly to centralized database servers. Automated signal processing pipelines execute baseline fitting algorithms, flag non-compliant cell lots automatically, and archive relaxation spectra in digital cell history records.

Contract

Integrating differential voltage relaxation screening criteria into commercial supply contracts protects cell buyers from hidden low-temperature degradation liabilities. Datasheets published by cell manufacturers specify broad operating temperature windows, typically listing charge limits from -20°C to 55°C. Datasheets rarely state the specific C-rate limits required to prevent metallic lithium plating at sub-zero temperatures. Procurement agreements must define explicit, verifiable electrochemical testing protocols to govern warranty claims.

Cell buyers risk significant financial exposure when operating packs under cold-weather fast-charging regimes. If supplier datasheets permit 0.5C charging at -10°C without qualification, buyers designing thermal management systems rely on those specifications. When field packs experience accelerated capacity fade and internal shorting due to covert metallic plating, suppliers frequently attribute failures to improper field operation or unapproved charge profiles.

Incorporating non-destructive relaxation criteria into technical procurement contracts eliminates ambiguity.

Drafting Low-Temperature Fast Charge Covenants

Technical specifications governing cell procurement define precise test methods for establishing allowable operating windows. Supply contracts include explicit low-temperature fast-charge covenants that mandate zero metallic plating across the operating temperature envelope. Covenants specify differential relaxation testing as the arbitrating diagnostic methodology for resolving degradation disputes.

Contractual covenants state specific thresholds for baseline-subtracted dV/dt relaxation peak parameters. Specifications declare that any post-charge relaxation test producing a baseline-subtracted dV/dt peak duration exceeding 30 seconds, or an integrated reversible capacity exceeding 0.05% of nominal cell capacity, constitutes verifiable metallic lithium plating. Exceeding these thresholds defines the charge protocol as invalid, shifting financial liability for resulting cell degradation back to the supplier.

Warranty qualification clauses mandate joint laboratory testing when field returns display abnormal capacity loss. Independent test facilities execute standardized differential relaxation protocols on returned cell samples inside calibrated environmental chambers. Demonstrating the presence of metallic lithium relaxation signatures under factory-recommended charging conditions proves datasheet non-compliance, triggering mandatory cell lot replacement and financial reimbursement clauses.

Landed Cost Protection and Warranty Allocation

Landed cost calculations must account for long-term warranty reserve allocations associated with low-temperature operation. Buying lower-cost cells with poor low-temperature kinetic performance increases field failure rates, eroding initial purchase price savings through warranty claims. Differential relaxation testing during initial cell batch qualification identifies low-performing cell lots before integration into battery packs.

Incoming inspection procedures extract sample cells from incoming shipping crates for automated relaxation testing. Inspection labs subject 0.5% of incoming cell lots to sub-zero charging routines inside environmental chambers, executing differential relaxation processing on cycling channels. Detecting relaxation peak anomalies in incoming samples provides legal grounds to reject entire shipment lots prior to paying customs duties and inland freight costs.

Risk allocation frameworks shift financial burdens onto cell manufacturers through strict technical performance covenants. Supply contracts specify that cell lots failing non-destructive differential relaxation criteria during incoming inspection incur liquidated damages covering freight, storage, and batch replacement costs. Establishing clear quantitative testing protocols converts subjective degradation arguments into objective financial settlements.

Standard procurement contract clause 14.3 specifies that cell lots subjected to low-temperature fast-charge qualification protocols must display zero differential relaxation peak signatures exceeding baseline noise thresholds, and failure to meet this benchmark establishes immediate product non-conformance under warranty recovery terms.