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arxiv: 1907.09410 · v1 · pith:AGRO6UMRnew · submitted 2019-06-26 · ⚛️ physics.chem-ph

Generalised single particle models for high-rate operation of graded lithium-ion electrodes: systematic derivation and validation

G. Richardson , I. Korotkin , R. Ranom. M. Castle , J. M. Foster This is my paper

Pith reviewed 2026-05-25 15:13 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords single particle modelporous electrode theorylithium-ion batterygraded electrodeasymptotic analysishigh-rate operationelectrolyte correction
0
0 comments X

The pith

Generalized single particle models with electrolyte correction accurately predict high-rate behavior in graded lithium-ion electrodes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives single particle models from full porous electrode theory using asymptotic analysis based on the small thermal voltage relative to overpotential changes. It shows that standard SPMs have discrepancies at moderate rates due to electrolyte variations and introduces a correction term. Generalized versions handle electrodes with multiple particle sizes or chemistries. These corrected models match the full theory well for NMC and graphite materials. This matters because simpler models enable faster simulation of battery performance without losing accuracy at high discharge rates.

Core claim

A formal asymptotic derivation yields the single particle model from the porous electrode theory model for uniform spherical particles. A correction term accounts for spatial electrolyte non-uniformities. This is extended to graded electrodes with multiple particle sizes or chemistries. The generalized corrected SPM compares favourably to the full PET model for NMC and graphite.

What carries the argument

The asymptotic expansion based on the ratio of thermal voltage to characteristic overpotential change during lithiation, together with an additive correction term for electrolyte concentration gradients.

If this is right

  • The corrected SPM gives accurate voltage predictions at higher rates where standard SPM fails due to electrolyte effects.
  • Generalized SPMs can model electrodes with mixed particle sizes without needing the full spatially resolved PET.
  • For NMC and graphite, the generalized models match PET solutions closely.
  • This allows efficient modeling of high-rate operation in complex electrode designs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Such models could enable optimization of particle size distributions in electrode manufacturing for better rate performance.
  • The approach might extend to other battery chemistries if the asymptotic assumptions hold.
  • Incorporating the correction could improve real-time battery management systems that rely on SPMs.

Load-bearing premise

The characteristic change in overpotential during (de)lithiation is much larger than the thermal voltage, allowing the asymptotic separation.

What would settle it

If measurements of cell voltage during high-rate discharge of a graded NMC electrode show significant deviation from the generalized SPM predictions while matching the PET model, the derivation would be falsified.

Figures

Figures reproduced from arXiv: 1907.09410 by G. Richardson, I. Korotkin, J. M. Foster, R. Ranom. M. Castle.

Figure 1
Figure 1. Figure 1: A schematic of the half cell geometry as well as the independent variables and snap [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Ionic diffusivity (top left) and conductivity (top right) of the electrolyte from [13], [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Equilibrium potential of graphite (LiC6) anode [13] (top left), Li(Ni0.4Co0.6)O2 cathode [13] (top right), and LFP (LiFePO4) electrode [30] (bottom). 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cell potentials V calculated using the full Newman PET model, SPM and corrected SPM for graphite (LiC6) anode (top row), Li(Ni0.4Co0.6)O2 cathode (middle row) and LFP (LiFePO4) electrode (bottom row) at different discharge rates. Results are shown against both time (left) and discharge capacity (right). 12 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Concentration of Li+ ions (top left) and potential (top right) in the electrolyte across a half-cell with a Li(Ni0.4Co0.6)O2 electrode calculated using the PET model and the leading order electrolyte equations, (20)-(24), at 8C discharge rate. In the upper panels a single snapshot in time (at the end of discharge) is shown because the electrolyte approaches a steady-state rapidly and hence the profiles at … view at source ↗
Figure 6
Figure 6. Figure 6: Cell potentials V calculated using the full PET model, DPM and the corrected DPM for graphite (LiC6) anode (top row), Li(Ni0.4Co0.6)O2 cathode (middle row) and LFP (LiFePO4) electrode (bottom row) at different discharge rates. Results are shown against both time (left) and discharge capacity (right). 16 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Concentration of Li+ ions (top left) and potential (top right) in the electrolyte across a half-cell with an Li(Ni0.4Co0.6)O2 electrode formed from particles of two different sizes and calculated using the full PET model and the leading order approximation at 4C discharge rate. In the upper panels a single snapshot in time (at the end of discharge) is shown because the electrolyte approaches a steady-state… view at source ↗
read the original abstract

A derivation of the single particle model (SPM) is made from a porous electrode theory model (or Newman model) of half-cell (dis)charge for an electrode composed of uniformly sized spherical electrode particles of a single chemistry. The derivation uses a formal asymptotic method based on the disparity between the size of the thermal voltage and that of the characteristic change in overpotential that occurs during (de)lithiation. Comparison is made between solutions to the SPM and to the porous electrode theory (PET) model for NMC, graphite and LFP. These are used to identify regimes where the SPM gives accurate predictions. For most chemistries, even at moderate (dis)charge rates, there are appreciable discrepancies between the PET model and the SPM which can be attributed to spatial non-uniformities in the electrolyte. This motivates us to calculate a correction term to the SPM. Once this has been incorporated into the model its accuracy is significantly improved. Generalised versions of the SPM, that can describe graded electrodes containing multiple electrode particle sizes (or chemistries), are also derived. The results of the generalised SPM, with the correction term, compare favourably to the full PET model where the active electrode material is either NMC or graphite.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper derives the single particle model (SPM) from the porous electrode theory (PET) model for half-cell (dis)charge using a formal asymptotic expansion based on the small ratio of thermal voltage to the characteristic overpotential variation during (de)lithiation. It compares SPM and PET solutions for NMC, graphite and LFP, attributes discrepancies to electrolyte non-uniformities, introduces a correction term, and derives generalised SPMs for graded electrodes containing multiple particle sizes or chemistries. The generalised SPM plus correction is stated to compare favourably to the full PET model for NMC and graphite.

Significance. If the asymptotic expansion remains accurate in the claimed high-rate regimes and the correction improves predictions without fitted parameters, the work supplies a computationally efficient, systematically derived reduced-order model for graded electrodes. This would be valuable for rapid simulation and optimisation of advanced lithium-ion battery designs. The formal asymptotic approach and explicit generalisation to multi-particle/chemistry electrodes are strengths.

major comments (3)
  1. [Abstract] Abstract: the claim that the SPM with correction 'compares favourably' to PET for NMC and graphite is unsupported by any quantitative error metrics, RMS deviations, maximum errors, data-exclusion criteria or error bars. This is load-bearing for the central validation statement.
  2. [Derivation] Derivation (asymptotic reduction): the small parameter is the ratio of thermal voltage (~26 mV) to the scale of overpotential change during (de)lithiation, yet this ratio is never evaluated for the high C-rates or particle-size distributions used in the NMC and graphite comparisons. If the ratio is O(0.1) or larger, the neglected higher-order terms affect both the base SPM and the electrolyte correction.
  3. [Generalised SPM] Generalised SPM section: validation of the multi-particle/chemistry extension is asserted only for uniform NMC or graphite cases; it is unclear whether the graded-electrode predictions were tested against the full PET model or only against the uniform SPM.
minor comments (2)
  1. [Abstract] Abstract states comparisons were performed for NMC, graphite and LFP but favourable comparison with correction is reported only for NMC and graphite; the LFP outcome should be stated explicitly.
  2. Notation for the correction term and the small parameter should be introduced with a clear equation reference when first used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and their constructive comments. We address each of the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the SPM with correction 'compares favourably' to PET for NMC and graphite is unsupported by any quantitative error metrics, RMS deviations, maximum errors, data-exclusion criteria or error bars. This is load-bearing for the central validation statement.

    Authors: We agree with the referee that quantitative error metrics are necessary to support the claim in the abstract. In the revised version of the manuscript, we will add RMS deviations, maximum errors, and any relevant error bars or data criteria for the comparisons between the corrected SPM and the PET model for NMC and graphite. revision: yes

  2. Referee: [Derivation] Derivation (asymptotic reduction): the small parameter is the ratio of thermal voltage (~26 mV) to the scale of overpotential change during (de)lithiation, yet this ratio is never evaluated for the high C-rates or particle-size distributions used in the NMC and graphite comparisons. If the ratio is O(0.1) or larger, the neglected higher-order terms affect both the base SPM and the electrolyte correction.

    Authors: The referee correctly identifies that the small parameter should be quantified for the specific conditions. We will include in the revised manuscript an explicit calculation of this ratio for the C-rates and materials used in the NMC and graphite comparisons to justify the asymptotic approximation. revision: yes

  3. Referee: [Generalised SPM] Generalised SPM section: validation of the multi-particle/chemistry extension is asserted only for uniform NMC or graphite cases; it is unclear whether the graded-electrode predictions were tested against the full PET model or only against the uniform SPM.

    Authors: We acknowledge the lack of clarity. The numerical validations presented are for uniform electrodes. The generalised SPM for graded electrodes is derived systematically from the same asymptotic approach, but direct comparisons to PET for graded (multi-particle) cases are not included in the current manuscript. In the revision, we will update the abstract and relevant sections to clarify this point and indicate that such validations are beyond the scope of the present work. revision: yes

Circularity Check

0 steps flagged

Asymptotic derivation from PET model is self-contained with no circularity

full rationale

The paper performs a formal asymptotic reduction of the established porous electrode theory (PET) model to obtain the single-particle model (SPM), using the ratio of thermal voltage to characteristic overpotential change as the small parameter. This is a standard mathematical expansion whose validity can be checked externally against the full PET equations; it does not rely on fitting parameters to data, self-citations for uniqueness, or any step that reduces to its own inputs by construction. Generalized SPMs for graded electrodes follow by the same expansion applied to a multi-particle PET formulation. Comparisons to PET solutions for NMC, graphite and LFP serve as external validation rather than tautological predictions. No load-bearing element in the derivation chain is equivalent to the input by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the model rests on standard domain assumptions of spherical particles of uniform size and single chemistry in the base derivation, plus the scale separation that justifies the asymptotics. No free parameters or new entities are stated.

axioms (2)
  • domain assumption Electrode particles are uniformly sized spheres of a single chemistry.
    Stated in the opening sentence of the abstract as the starting point for the SPM derivation.
  • domain assumption Disparity exists between thermal voltage and characteristic overpotential change during (de)lithiation.
    Explicitly invoked as the basis for the formal asymptotic method.

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Reference graph

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