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REVIEW 3 major objections 4 minor 74 references

A diffuse-interface model for predicting the evolution of metallic negative electrodes and interfacial voids in solid-state batteries with homogeneous and polycrystalline solid electrolyte separators

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that a diffuse-interface phase-field model can simultaneously simulate metallic sodium electrode volume change and interfacial void evolution in solid-state batteries, with void growth rate controlled by the sodium flux…

desk verdict A verified phase-field implementation for Na solid-state cells, but the grain-boundary effects need a width-convergence test before they can be trusted. read the letter →

arxiv 2412.17147 v1 pith:DZROGGTZ submitted 2024-12-22 cond-mat.mtrl-sci physics.chem-ph

Sourav Chatterjee , Michael Tonks , William Gardner , Marina Sessim This is my paper · ORCID
classification cond-mat.mtrl-sciphysics.chem-ph
keywords solid-statebatteriessodiummetalanodephase-fieldmodelinterfacialvoidsgrainboundariesstrippingandplatingvoidcoalescencediffuse-interfaceelectrochemistry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to establish a diffuse-interface (phase-field) electrochemical model that simulates, in one calculation, both the volume change of a metallic negative electrode and the growth, shrinkage, and coalescence of voids at the electrode/electrolyte interface during stripping and plating. The model is demonstrated on a sodium / Na-β′′-alumina solid-state cell, a system the authors say has not been simulated at this level of coupling before. They claim that void migration is controlled by the local flux of sodium atoms at the void edge, which scales with applied current density, and that solid-electrolyte grain boundaries change void behavior only locally and slightly. If the model is right, it gives battery designers a way to predict how fast interfacial voids grow before they cause contact loss and dendrite formation, and to judge how much grain-boundary engineering can affect that failure mode.

What carries the argument

The central machinery is a grand-potential phase-field model with separate order parameters for the metal electrode, an 'auxiliary' phase representing the space behind the electrode, and the void itself, plus order parameters for electrolyte grains and a grain-boundary interpolation in the separator. The electrode does not move by tracking its outer surface; instead the diffuse auxiliary/electrode interface advances when metal atoms leave during stripping or arrive during plating, while the electrode/electrolyte interface stays sharp and stationary. The relation that carries the void-growth argument is $\mathrm{d}y_{\mathrm{tr}}/\mathrm{d}t = v_m\, j_{\mathrm{tr}}(t)$: the velocity of the void edge along the interface equals the molar volume of sodium times the flux of sodium at the void edge, evaluated where the void phase field equals 0.5. A grain-boundary interpolation function routes ion flux through diffuse boundaries, and the same flux law explains why grain boundaries change void edge velocity locally without changing the average electrode behavior.

What would settle it

Run the single-void stripping and plating simulation with the grain-boundary width reduced by a factor of five while scaling the boundary conductivity so the total boundary conductance is unchanged; if the local void-edge velocity departs from the wide-boundary result by more than the paper's few-percent level, the wide-boundary compensation, not physical grain-boundary transport, is producing the predicted effect. An operando experiment on a bicrystal Na/Na-β′′-alumina cell with known grain-boundary conductivity could also falsify the mechanism if no local speed-up or slowdown of the void edge appears when the edge crosses the boundary.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that coupled electrode evolution and void evolution can be captured by one grand-potential phase-field formulation in which the metal electrode grows or shrinks through a diffuse auxiliary-phase interface, the stationary electrode/electrolyte interface is sharp, and voids are a separate phase field. Using Na/Na-β′′-alumina parameters, the simulated depletion and deposition rates match sharp-interface Faraday-law values, mass is conserved without special moving-interface conditions, and voids grow during stripping and shrink during plating preferentially along the interface. The load-bearing quantitative claim is that the void edge velocity equals the molar volume of sodium times the flux of sodium at the void edge, so the void growth rate is a linear function of that edge flux, which in turn increases with applied current density. Grain-boundary conductivity, either lower or higher than the grain interior, leaves electrode depletion and deposition essentially unchanged for a perfect interface and changes void migration and coalescence only locally: a void edge accelerates near a high-conductivity grain boundary and slows near a low-conductivity one, with overall deviations of a few percent.

Load-bearing premise

The model's grain-boundary predictions rest on treating a grain boundary as a 0.5-micrometer-wide diffuse band with reduced conductivity, about a thousand times wider than a real boundary, and assuming that this compensation reproduces the true local current redistribution near a void edge.

Editorial extensions

If this is right

  • Depletion and deposition rates of the Na electrode are proportional to applied current density and match sharp-interface Faraday-law predictions; mass is conserved without special handling of the moving interface.
  • For a perfect electrode/electrolyte interface, varying grain-boundary conductivity does not change electrode thickness evolution or total sodium loss/gain; only the current distribution in the separator changes.
  • Interfacial voids migrate along the electrode/electrolyte interface during stripping and shrink back during plating; their edge speed follows the local sodium edge flux, so higher stripping current density accelerates void growth and coalescence.
  • High-conductivity grain boundaries locally speed up a void edge as it approaches the boundary, while low-conductivity boundaries slow it; this produces slight void asymmetry and shifts coalescence timing, but whole-void changes stay at the few-percent level.
  • Multi-void stripping shows that the critical stripping capacity (applied current density times time to full contact loss) decreases with increasing current density, consistent with existing Li/LLZO experiments and simulations, though lower than measured cell capacities because nucleation and stack pressure are omitted.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if void growth is truly a linear function of sodium flux at the void edge, then stack pressure and creep, which this model omits, should enter through that edge flux or a flux threshold, giving a concrete route to extend the law to experimentally measured critical stripping capacities.
  • Editorial inference: the grain-boundary predictions rest on representing an atomically thin boundary as a 0.5-micrometer diffuse band with reduced conductivity; varying the band width while holding total boundary conductance fixed would show whether the predicted local void-edge effects are physical or numerical.
  • Editorial inference: the bicrystal and tricrystal tests imply a directly testable experimental signature: a single well-characterized grain boundary intersecting the electrode/electrolyte interface should locally speed up or slow down void edge migration during stripping, visible as a kink or pause in operando microscopy.
  • Editorial inference: if the edge-flux law is robust, coarse-grained cell-failure models could track void growth with a single scalar edge flux instead of full void morphology, making cell-level life prediction much cheaper.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper presents a multi-phase-field, grand-potential-based model for the coupled evolution of a metallic Na negative electrode and interfacial voids in contact with a Na-beta''-alumina solid electrolyte separator. The electrode is represented by an auxiliary-phase/electrode diffuse interface, the electrode/electrolyte interface is fixed and sharp, and voids are included as a third phase in the electrode-side domain. The model is implemented in MOOSE and applied to three configurations: a perfect interface during stripping/plating, a single interfacial void, and multiple voids, each with homogeneous or polycrystalline solid electrolytes. The central claims are that (i) depletion/deposition rates under a perfect interface agree with sharp-interface Faraday-law predictions, (ii) mass is conserved, (iii) void growth and shrinkage are controlled by the Na flux at the void edge through a linear relation, and (iv) solid-electrolyte grain-boundary conductivity has negligible global effects but small local effects on void-edge migration and coalescence.

Significance. If the results hold, the model provides a practical tool for simulating void-induced contact loss in Na solid-state batteries, a system that has received less modeling attention than Li. The paper's verification steps are a genuine strength: Table 2 shows agreement with sharp-interface rates within a few percent, and Table 3 checks global Na mass conservation against an analytical balance. The model is also self-contained in the sense that the applied current density is an input and the results are not obtained by fitting to the reported void sizes. However, the distinctiveness of the paper rests on the grain-boundary-related claims, and these are not yet established because the diffuse grain-boundary width is not shown to be a faithful surrogate for a physical grain boundary. The work is therefore a solid modeling contribution whose central grain-boundary claim needs further evidence.

major comments (3)
  1. [Section 3, Sections 4.2.2 and 4.3.2] The grain-boundary width is set equal to the interface width l_w = 0.5 um, about 10^3 times the physical grain-boundary width, and the only compensation is the statement that 'the GB conductivity can be reduced, as shown in Ref. [60]'. No width-compensation factor is given, and no width-convergence study is reported. Since the effective grain-boundary conductance per unit length scales as kappa_gb * w_gb, using an uncompensated kappa_gb with a 500-1000x larger width inflates the grain-boundary path conductance by orders of magnitude. The reported local deviations in void-edge velocity (Fig. 12e) and coalescence times (Figs. 14c/d) are of the same order, a few percent, as the error this could introduce. To support the paper's unique contribution, the authors should either demonstrate that kappa_gb has been rescaled by w_phys/w_model, perform a width-convergence test, or validate the grain-boundary current redistribution against a known analytic solution.
  2. [Section 4.1.2 and Table 1, Figs. 5, S3, 12] The grain-boundary conductivity ratio is not consistently specified. Section 4.1.2 states kappa_gb/kappa_g = 0.67, but the captions of Figs. 5, S3, 12, and S5 use 0.067. Moreover, substituting the Arrhenius parameters of Table 1 (K_g, K_gb, E_g, E_gb at T = 300 K) gives kappa_gb/kappa_g approximately 0.0017, so neither quoted value follows from the stated material data. Because the local current redistribution near a void edge is controlled by this ratio, the quantitative void-edge velocity and coalescence results in Figs. 12e and 14c/d cannot be interpreted until the correct value is stated and used consistently.
  3. [Equation (24), Abstract, Sections 4.2.1 and 5] Equation (24) is introduced as an assumption, namely that the maximum interfacial flux at the void edge controls the void-edge velocity, but the abstract and conclusions present as a result that the void growth rate is a linear function of the flux of Na atoms at the void edge. The comparisons in Figs. 8a and 10a integrate Eq. (24) and compare with the void size in the same simulation; this is an internal consistency check, not an independent validation. The relation also neglects the contact angle and the height of the metal strip that the edge flux removes, which the authors acknowledge as a possible reason for larger deviations at high current densities. Please either derive the relation from mass conservation or present it explicitly as a modeling closure and restrict the conclusions accordingly.
minor comments (4)
  1. [Section 4.1.1, Table 2] The text states that the difference between deposition and depletion rates is at most 2%, but Table 2 shows a difference of about 3.1% at 0.5 mA/cm^2; the stated bound should be revised.
  2. [Figure 10(a) caption] The caption contains 'The void sizees linearly with time', which appears to be a typo for 'The void size decreases linearly with time'.
  3. [Section 4.3.1] The comparison between the predicted critical stripping capacity and experiments in Li/LLZO systems should be explicitly labeled as qualitative, since the chemistry, stack pressure, and nucleation history differ.
  4. [Section 1] The novelty claim that a Na metal solid-state cell has never been simulated is strong; it should be softened to 'to our knowledge, no phase-field simulation has addressed void evolution in a Na/Na-beta''-alumina cell'.

Circularity Check

1 steps flagged · score 2.0 of 10

The model is essentially self-contained; the only mild circularity is that the 'linear flux-velocity' void-growth law is an assumed kinematic closure restated as a result.

  1. self definitional [Section 4.2.1, Eq. (24); abstract and Section 5 bullet 3]
    "Based on these results, we assume that the maximum interfacial flux at the void edge controls the rate at which the void elongates along the electrode/electrolyte interface... Specifically, dytr/dt|Γed/el = vmjtr(t), (24)"

    The abstract and the conclusions present 'void growth rate is a linear function of the flux of Na atoms at the void edge' as a finding, but Eq. (24) is where the void-edge velocity is defined as vm times the local flux. The linear proportionality is therefore an input assumption, not a derived result. The paper is transparent about this ('we assume'), and it does check Eq. (24) by time-integrating the velocity against the independently tracked void size, so the relation is not fitted to the void-growth output; the circularity is limited to presenting the assumed kinematic closure as a conclusion.

full rationale

The model is largely self-contained. All material parameters in Table 1 come from independent experimental references or are stated modeling choices; no parameter is fitted to the quantities the paper claims to predict. The stripping and plating depletion/deposition rates are compared with the sharp-interface Faraday-law rates and with the analytical mass-balance loss/gain rates (Tables 2 and 3), and the agreement is a consistency check against externally defined laws, not a fit. Similarly, the area-averaged interfacial flux is shown to equal iapp/F by construction of the boundary conditions, so the rate proportionality to applied current is a verification of charge and mass conservation rather than a hidden fit. The grain-boundary transport properties are taken from Ref. [66], and the grain-boundary phase-field interpolation is adopted from Ref. [60]; although Ref. [60] shares authors with this paper, it is used only for a modeling form and is not the evidence for the paper's central physical conclusion, so it is not load-bearing circularity. The one mild circular element is Eq. (24): the void-edge velocity is defined as proportional to the local Na flux, and the abstract and conclusions then restate this linear relation as a result. The paper explicitly labels this as an assumption and validates it against independently tracked void sizes, so this is a minor self-definitional step rather than a fatal reduction. Separately, and as a correctness concern rather than circularity, the 0.5 µm diffuse grain-boundary width is roughly three orders of magnitude larger than a physical grain boundary, and the text says the GB conductivity 'can be reduced' per Ref. [60], yet no explicit width-based rescaling is shown in Table 1; the small local GB-induced void-velocity deviations in Figs. 12 and 14 should therefore be interpreted with caution until a width-convergence or rescaled-conductivity check is demonstrated. Overall circularity score: 2.

Assumptions & free parameters 6 free parameters · 6 assumptions · 1 invented entities

The model introduces a numerical auxiliary phase and relies on several ad hoc modeling choices: parabolic free energy coefficients chosen without experimental data, a stationary electrode/electrolyte interface with volume change transferred to the auxiliary phase, and an unphysically wide GB width with a self-cited compensation. These are the main free parameters and assumptions the central claims rest on.

free parameters (6)
  • Parabolic free energy coefficient Aa/fc (auxiliary phase) = 310
    Chosen high to keep Na concentration near equilibrium in the auxiliary phase; no experimental data (Section 3).
  • Parabolic free energy coefficient Am/fc (metal electrode) = 11
    Chosen high to keep Na concentration near equilibrium in the electrode; no experimental data (Section 3).
  • Parabolic free energy coefficient Av/fc (void) = 310
    Chosen high to keep Na concentration near equilibrium in the void phase; no experimental data (Section 3).
  • Parabolic free energy coefficient Ael/fc (electrolyte) = 4e3
    Chosen high to keep Na+ concentration near equilibrium in the electrolyte; no experimental data (Section 3).
  • Phase-field kinetic coefficient Lphi = 1e-4 m^3/(J s)
    Assumed constant and equal for all phases for simplicity (Section 2.1).
  • Interface width lw and GB width = 0.5 um
    Chosen for numerical resolution; also sets GB width, which is about 1000x physical width and affects GB transport (Section 3).
assumptions (6)
  • domain assumption Grand-potential phase-field formulation with equal diffusion potential in diffuse interfaces (KKS assumption)
    Used to derive phase-field evolution equations in Section 2.1; standard in phase-field electrochemistry (Refs. [48,49]).
  • ad hoc to paper Parabolic free energy densities with chosen coefficients
    No experimental data; coefficients set to keep compositions near equilibrium, acknowledged in Section 3.
  • domain assumption No charge-transfer overpotential at the electrode/electrolyte interface
    Equation (20) sets electric potential continuous; acknowledged as a simplification in Section 5.
  • ad hoc to paper Stationary, sharp electrode/electrolyte interface with electrode volume change via auxiliary phase
    Modeling choice to avoid moving the electrode/electrolyte interface; deviates from the physical receding electrode surface (Section 2).
  • ad hoc to paper GB width of 0.5 um with conductivity compensation
    GB width is about 1000x physical; compensation relies on Ref. [60] by the same group, not independently validated (Section 3).
  • domain assumption Void nucleation has already occurred
    Paper focuses on void growth and shrinkage, not nucleation, as stated in Section 2.
invented entities (1)
  • Auxiliary phase (xi_a)
    purpose: Represents empty space between electrode and current collector so electrode thickness can change without moving the electrode/electrolyte interface.
    Numerical device introduced in Section 2.1; no physical counterpart.

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Cite this review

Pith. "Pith review of A diffuse-interface model for predicting the evolution of metallic negative electrodes and interfacial voids in solid-state batteries with homogeneous and polycrystalline solid electrolyte separators." pith.science (2026). https://pith.science/paper/DZROGGTZ

@misc{pith2026241217147,
  author       = {Pith},
  title        = {Pith review of: A diffuse-interface model for predicting the evolution of metallic negative electrodes and interfacial voids in solid-state batteries with homogeneous and polycrystalline solid electrolyte separators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZROGGTZ}},
  note         = {Machine review of arXiv:2412.17147}
}
abstract

This paper presents a novel diffuse-interface electrochemical model that simultaneously simulates the evolution of the metallic negative electrode and interfacial voids during the stripping and plating processes in solid-state batteries. The utility and validity of this model are demonstrated for the first time on a cell with a sodium (Na) negative electrode and a Na-$\beta^{\prime\prime}$-alumina ceramic solid electrolyte (SE) separator. Three examples are simulated. First, stripping and plating with a perfect electrode/electrolyte interface; second, stripping and plating with a single interfacial void at the electrode/electrolyte interface; third, stripping with multiple interfacial voids. Both homogeneous SE properties and polycrystalline SEs with either low or high conductivity grain boundaries (GBs) are considered for all three examples. Heterogeneous GB conductivity has no significant impact on the behavior with a perfect electrode/electrolyte interface. However, it does result in local changes to void growth due to interactions between the void edge and the GBs. The void growth rate is a linear function of the flux of Na atoms at the void edge, which in turn depends on the applied current density. We also show that the void coalescence rate increases with applied current density and can be marginally influenced by GB conductivity.

Figures

Figures reproduced from arXiv: 2412.17147 by the authors.

Figure 1
Figure 1. a. The following subsections provide the electrode/electrolyte interfacial conditions and the governing equations in the two subregions. The governing equations are derived for a generic pure alkali metallic negative electrode. We hereafter refer to the diffusing metal atoms by the letter M. During stripping, the metal atom oxidises at the electrode/electrolyte interface as M(ed) → M + (el) + e −(ed). Subsequently, … view at source ↗
Figure 2
Figure 2. Stripping simulations with a perfect electrode/electrolyte interface and a homogeneous SE separator for different applied current densities. (a) Examples of the shrinkage of the Na negative electrode at three times with an applied current density of 0.2 mA/cm2 . Ωed is shaded by the Na concentration and Ωel by the ratio of the local current density to the applied current density. The spatial distribution along the x… view at source ↗
Figure 3
Figure 3. Impact of applied current density on Na electrode thickness during stripping and plating. The aux/electrode interface contour for ξa = 0.5 after stripping (a) and plating (b) for 3 h with different ap￾plied current densities. The dotted black lines indicate the initial position of the aux/electrode interface. In (a) and (b), the region Ωel is shaded by the current density normalized by the applied current density an… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Plating simulations assuming a perfect electrode/electrolyte interface and a homogeneous solid elec￾trolyte separator. (a) Examples of the growth of the Na negative electrode at three times with an applied current density of −0.2 mA/cm2 . Ωed is shaded by the Na concen…
Figure 5
Figure 5. Figure 5: Cycling simulations assuming a perfect electrode/electrolyte interface and polycrystalline SE separator. Evolution of the Na negative electrode during cycling with an applied current of 0.2 mA/cm2 with (a) κ el gb/κel g = 0.067 and (b) κ el gb/κel g = 10. Ωed is shaded…
Figure 6
Figure 6. Figure 6: Analysis of the cycling simulations assuming a perfect electrode/electrolyte interface and polycrystalline SE separator. The variation with time of the (a) Na negative electrode thickness, (b) percentage difference in the electrode thickness relative to the homogeneous…
Figure 7
Figure 7. Figure 7: Stripping simulations with a single interfacial void at the electrode/electrolyte interface and a homo￾geneous solid electrolyte separator with different applied current densities. (a) Void evolution with an applied current density of 0.2 mA/cm2 . (b) Void morphology f…
Figure 8
Figure 8. Figure 8: Analysis of stripping simulations with a single interfacial void at the electrode/electrolyte interface and a homogeneous solid electrolyte separator with different applied current densities. (c) Void size (twice the distance between the void center and edge along the …
Figure 9
Figure 9. Figure 9: Plating simulations with a single interfacial void at the electrode/electrolyte interface and a homoge￾neous solid electrolyte separator. (a) Void evolution with an applied current density of −0.2 mA/cm2 . (b) Void morphology for different applied current densities aft…
Figure 10
Figure 10. Figure 10: Analysis of plating simulations with a single interfacial void at the electrode/electrolyte interface and a homogeneous solid electrolyte separator. (a) Void size (twice the distance between the void center and edge along the electrode/electrolyte interface) over time…
Figure 11
Figure 11. Figure 11: Cycling simulations with a single interfacial void at the interface between the negative electrode and a polycrystalline SE separator. Evolution of the void and the negative electrode during cell cycling in solid electrolytes with (a) low-conductivity and (b) high-con…
Figure 12
Figure 12. Figure 12: Analysis of cycling simulations with a single interfacial void at the interface between the negative electrode and a polycrystalline SE separator. (a) Half-void size above the void center with cycling time with low-conductivity (κ el gb/κel g = 0.067), homogeneous (κ …
Figure 13
Figure 13. Figure 13: Stripping simulations with multiple interfacial voids at the electrode/electrolyte interface and a homogeneous solid electrolyte separator for different applied current densities. (a) Void morphology at three times with an applied current density of 0.2 mA/cm2 . Ωed i…
Figure 14
Figure 14. Figure 14: Stripping simulations with an applied current density of 0.2 mA/cm2 , multiple interfacial voids, and a polycrystalline SE separator with different values of κ el gb/κel g . Void morphology at three times with (a) low￾conductivity and (b) high-conductivity GBs. The in…

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    2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.