{"id":"aa8375b9-ca81-4a88-9ed8-25a3b6684a3d","arxiv_id":"2412.17147","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A diffuse-interface model simultaneously simulates sodium electrode thickness changes and interfacial void growth and shrinkage in solid-state batteries, and shows grain boundary conductivity only weakly affects void evolution.","lead":"This paper introduces a phase-field model that simulates how a sodium metal electrode shrinks, grows, and forms voids against a ceramic electrolyte during battery charging and discharging. The model is new for sodium solid-state batteries and also accounts for grain boundaries in the ceramic separator.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The GB-impact conclusion may rest on an un-validated 500× GB-width rescaling: the model uses 0.5 µm diffuse GBs, and the paper does not demonstrate that its GB conductivity values were width-compensated.","rationale":"The reader identified the GB width and compensation scheme as the weakest assumption, and the paper's own text confirms that the model's GB width is ~10^3× physical while the compensation is only cited to a prior work without demonstration in this system. This is load-bearing because the paper explicitly claims as its unique contribution that SE GB properties affect local void migration and coalescence; if the GB width is not properly compensated, those small effects (2–6% deviations) could be numerical artifacts. The paper also contains an internal ambiguity about whether the low-conductivity ratio is 0.67 or 0.067, which reinforces the need for a concrete reproduction test. I do not think this invalidates the model's core electrodeposition/void evolution framework, which is supported by the sharp-interface depletion/deposition comparisons and mass conservation checks, so the conditional verdict is appropriate rather than rejection. The concrete test above would settle the issue by checking whether the reported GB effects survive width convergence with physically scaled GB conductance.","tokens_in":35238,"tokens_out":8202,"duration_ms":83340,"concrete_test":"Re-run the single-void cyclic case of §4.2.2 with a width-convergence series, e.g., l_w = 0.5, 0.25, and 0.125 µm (refining the mesh to l_w/3), and with κ_gb rescaled so that the integrated GB conductance per unit length, κ_gb × w_gb, is held constant at the physical value (taking w_phys ≈ 1 nm). If the local void-edge velocity deviations in Fig. 12e and the void asymmetry in Fig. 12f persist quantitatively as l_w decreases, the GB conclusion is robust; if the deviations shrink below the reported 2–6% level or change sign, the published GB effects are artifacts of the diffuse GB width.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's unique contribution is the local influence of SE grain-boundary conductivity on void migration and coalescence (§4.2.2, §4.3.2). That conclusion depends directly on the diffuse-GB representation: l_w = 0.5 µm (Section 3) is about 10^3× the physical GB width, and the text states that 'to compensate for the wide GBs, the GB conductivity can be reduced, as shown in Ref. [60].' However, Table 1 lists κ_gb from Ref. [66] and the simulations shown in Figs. S3 and 12 use κ_gb/κ_g = 0.067 and 10 without an explicit width-based rescaling; Section 4.1.2 even quotes the low ratio as 0.67, making the actual value ambiguous. Since GB conductance per unit length scales as κ_gb × w_gb, an unrescaled κ_gb with w_model/w_phys ≈ 500 overestimates GB path conductance by roughly 500×. The reported local void-edge velocity deviations (Fig. 12e) and coalescence timing shifts (Figs. 14c/d) are of the same order as, or smaller than, the error this factor would introduce, so they could be numerical consequences of the inflated GB width rather than physical GB effects. A width-convergence test is therefore required before the central GB claim can be accepted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":35651,"tokens_out":9053,"duration_ms":83498,"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":[{"comment":"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.","section":"Section 3, Sections 4.2.2 and 4.3.2"},{"comment":"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.","section":"Section 4.1.2 and Table 1, Figs. 5, S3, 12"},{"comment":"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.","section":"Equation (24), Abstract, Sections 4.2.1 and 5"}],"minor_comments":[{"comment":"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.","section":"Section 4.1.1, Table 2"},{"comment":"The caption contains 'The void sizees linearly with time', which appears to be a typo for 'The void size decreases linearly with time'.","section":"Figure 10(a) caption"},{"comment":"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.","section":"Section 4.3.1"},{"comment":"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'.","section":"Section 1"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The manuscript is a technically competent modeling paper with good verification against sharp-interface and mass-conservation checks. The main risk is the diffuse-grain-boundary representation and the inconsistent grain-boundary conductivity values, both of which affect the claimed novel grain-boundary effect. The linear void-edge closure is not a derived law and should be reframed. If the authors can provide a width-convergence test and correct the conductivity values, the paper could become acceptable; otherwise the unique contribution is not supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the paper is worth refereeing, but I wouldn't trust the grain-boundary conclusions as-is. The diffuse-interface model does something new—it tracks electrode thickness change, interfacial void evolution, and static electrolyte grain boundaries in the same simulation, and applies it to a Na/Na-β''-alumina cell for the first time. The perfect-interface case is verified against sharp-interface analytical solutions within ~2%, and mass conservation checks out. The authors are also transparent that Eq. (24) (void edge velocity linear in local flux) is an assumption, not a derived law.\n\nThe soft spot is exactly where the paper claims novelty: the influence of grain-boundary conductivity on void migration and coalescence. The model uses 0.5 µm diffuse GBs, about three orders of magnitude wider than physical GBs. The authors note that wider GBs accelerate GB transport and say the conductivity can be reduced to compensate, citing their own earlier paper. But the manuscript never shows the width-rescaling used. Table 1 lists κ_gb from original literature values, and the simulation ratios appear as both 0.067 and 0.67 in different places. Since the GB contribution to current transport is κ_gb × w_gb, an unrescaled κ_gb with a 500× wider GB overestimates the GB path by a factor that is much larger than the reported 2–6% void-velocity changes. Without a width-convergence test, the local GB effects could be an artifact of the inflated GB width rather than a physical result.\n\nThat said, the rest of the paper holds together. The single- and multi-void behavior qualitatively matches experiments and prior Li/LLZO simulations; the critical stripping capacity trend is reasonable, and the authors openly list missing physics (stack pressure, charge transfer, 3D). The main ask should be a straightforward numerical experiment: run the GB cases with two or three smaller GB widths (and correspondingly rescaled conductivities if that's the scheme) and show the void-edge velocity deviations converge. Also put the code/data in a repository.\n\nWho benefits: anyone modeling alkali-metal solid-state batteries, especially Na/NBA. It's a useful capability paper, not a breakthrough. I'd send it to peer review with a request for the convergence test; it's the kind of revision that's entirely feasible.","headline":"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.","tokens_in":36114,"tokens_out":3680,"would_cite":false,"duration_ms":34142,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["solid-state batteries","sodium metal anode","phase-field model","interfacial voids","grain boundaries","stripping and plating","void coalescence","diffuse-interface electrochemistry"],"falsifier":"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.","tokens_in":35063,"feed_emoji":"🔋","tokens_out":10283,"duration_ms":86200,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sodium flux at void edge sets void growth in solid-state cells","feed_subtitle":"A phase-field simulation tracks metal electrode and void evolution together; grain boundary effects stay local and small.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the phase-field void-evolution formulation, with an empty void phase at the electrode/electrolyte interface, that this model extends to a moving electrode and grain boundaries.","marker":"[36]"},{"why":"Provides the sharp-interface Faraday-law depletion and deposition rates used to validate the perfect-interface results.","marker":"[44]"},{"why":"Establishes the strategy of compensating for artificially wide grain boundaries by reducing grain-boundary conductivity, which underlies the predicted grain-boundary effects.","marker":"[60]"},{"why":"Experimental Li/LLZO reference for critical stripping current, void growth, and void shapes that motivates the edge-flux mechanism.","marker":"[19]"},{"why":"Experimental Na/Na-β′′-alumina study that supplies the void shapes, critical stripping current range, and stack-pressure effects used for qualitative comparison.","marker":"[28]"},{"why":"Sharp-interface void-growth model whose critical stripping capacity and flux-concentration criterion are compared in the multi-void results.","marker":"[35]"},{"why":"Continuum model of lithium void growth that provides the critical-capacity comparison and the role of bulk and surface diffusion.","marker":"[37]"},{"why":"Theoretical model assuming linear void growth, against which the paper frames its finding that void growth accelerates with time.","marker":"[30]"},{"why":"Source of the Na-β′′-alumina grain and grain-boundary conductivity parameters used to parameterize the separator.","marker":"[66]"}],"fun_headline_variants":["Sodium edge flux drives void growth in solid-state cells","Void growth rate set by sodium edge flux in solid-state batteries","Phase-field model links sodium flux to void growth in solid-state cells","Sodium edge flux dictates void growth in solid-state cells"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sodium edge flux drives void growth in solid-state cells","Void growth rate set by sodium edge flux in solid-state batteries","Phase-field model links sodium flux to void growth in solid-state cells","Sodium edge flux dictates void growth in solid-state cells"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00081,"raw_usage":{"total_tokens":3583,"prompt_tokens":1006,"completion_tokens":2577,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":622,"completion_tokens_details":{"reasoning_tokens":2506}},"tokens_in":622,"tokens_out":2577,"duration_ms":15875,"temperature":1.0,"reasoning_tokens":2506,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:44:50.593653+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the phase-field void-evolution formulation, with an empty void phase at the electrode/electrolyte interface, that this model extends to a moving electrode and grain boundaries."},{"cited_title":"Mishra, A","cited_arxiv_id":null,"evidence_quote":"Provides the sharp-interface Faraday-law depletion and deposition rates used to validate the perfect-interface results."},{"cited_title":"Impact of grain boundary and surface diffusion on predicted fission gas bubble behavior and release in UO$_2$ fuel","cited_arxiv_id":"2310.06795","evidence_quote":"Establishes the strategy of compensating for artificially wide grain boundaries by reducing grain-boundary conductivity, which underlies the predicted grain-boundary effects."},{"cited_title":"Kasemchainan, S","cited_arxiv_id":null,"evidence_quote":"Experimental Li/LLZO reference for critical stripping current, void growth, and void shapes that motivates the edge-flux mechanism."},{"cited_title":"Spencer Jolly, Z","cited_arxiv_id":null,"evidence_quote":"Experimental Na/Na-β′′-alumina study that supplies the void shapes, critical stripping current range, and stack-pressure effects used for qualitative comparison."},{"cited_title":"Agier, S","cited_arxiv_id":null,"evidence_quote":"Sharp-interface void-growth model whose critical stripping capacity and flux-concentration criterion are compared in the multi-void results."},{"cited_title":"Barai, T","cited_arxiv_id":null,"evidence_quote":"Continuum model of lithium void growth that provides the critical-capacity comparison and the role of bulk and surface diffusion."},{"cited_title":"Lu, C.-Z","cited_arxiv_id":null,"evidence_quote":"Theoretical model assuming linear void growth, against which the paper frames its finding that void growth accelerates with time."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the Na-β′′-alumina grain and grain-boundary conductivity parameters used to parameterize the separator."}],"review_version":1}