{"id":"28e82500-f84b-408d-b9e4-8b5bc31210db","arxiv_id":"2607.24620","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Auxiliary non-conserved composition-difference variables evolve under diffusion-potential driving forces, recovering WBM and KKS as limits and enabling phase-field treatment of non-overlapping ordered phases.","lead":"A phase-field model uses auxiliary variables for composition differences between phases, bridging equal-composition and equal-diffusion-potential interface rules. It lets simulations use raw thermodynamic databases and treat ordered phases that share no composition range.","discovery_kind":"new_method","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"Eq. (9)'s removal of the h(ξ)(1−h(ξ)) factor makes KKS recovery a formal fast-relaxation limit with an uncalibrated kinetic parameter; the paper never shows that growth kinetics are independent of L_φ at fixed physics, and its own flagship demos quietly use Eq. (8) instead.","rationale":"The reader's weakest_assumption (Eq. (9) as a pragmatic, possibly artifact-prone modification of the variational Eq. (8)) is the correct load-bearing point, and I agree with it; my pass sharpens rather than relocates it. Three observations support keeping the CONDITIONAL verdict rather than moving it. First, the concern does not invalidate the paper's core mathematics: the WBM limit is exact by inspection, the equilibrium (common-tangent) convergence in both demos is genuine independent evidence, and Eq. (9) retains the sign of the variational driving force, so it is a mobility modification rather than an ad hoc source term — the free energy is not obviously destabilized. Second, the paper's most novel capability (θ/η1 interdiffusion without common composition range, using unmodified database potentials via the Y1/Y2 logit transform of Eqs. 10–12) is demonstrated with the thermodynamically consistent Eq. (8), so it survives even if Eq. (9) turns out to be only approximately controlled. Third, what is actually missing is a single, cheap numerical control: an L_φ-sensitivity and refinement study at fixed physics, which is precisely the kind of condition a CONDITIONAL verdict exists to request. The additional wrinkle I add — that the paper's own efficiency and Al–Cu demonstrations use Eq. (8), not the Eq. (9) it recommends — strengthens the case for requiring that study but does not rise to a correctness failure. No shipped code keeps this unverifiable by third parties, which the reader already noted. Hence: agree with the reader, verdict unchanged (CONDITIONAL), with the condition made concrete as the proposed L_φ-sweep/refinement test.","tokens_in":24951,"tokens_out":5074,"duration_ms":191412,"concrete_test":"In the 1-D Ti–V α-growth benchmark (§3.1, identical parameters), extract the interface position x(t) and parabolic growth coefficient using Eq. (9) for L_φ/L_ξ ∈ {10⁻⁴, 10⁻², 0.1, 1, 10, 100}, and overlay against the KKS (Newton–Raphson) and WBM results. Repeat the whole sweep with Δt halved and with interface thickness halved (κ_ξ, w rescaled). If x(t) for L_φ ≳ 10 L_ξ matches KKS to within ~2% and is invariant under Δt/interface-width refinement, the KKS-reduction claim is controlled; if intermediate L_φ produces a third, uncollapsed curve or results shift with Δt, then L_φ is an uncalibrated kinetic parameter and the \"exact reduction\" and bulk-equalization claims must be downgraded to asymptotic/empirical statements.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim has three parts: exact WBM reduction (φ≡0, L_φ=0), exact KKS reduction (φ relaxed to equilibrium each step), and capability for θ/η1-type interdiffusion with unmodified database potentials. The WBM reduction is algebraically clean (with φ=0 the ∂μ/∂ξ terms in Eq. (7a) vanish identically). The KKS reduction, however, is only asymptotic in L_φ and rests entirely on Eq. (9), which is Eq. (8) with the variational factor h(1−h) deleted. Two things follow that the paper does not quantify. (1) Eq. (9) is equivalent to Eq. (8) with a position-dependent mobility L_φ/[h(ξ)(1−h(ξ))] that diverges at the interface fringes (ξ→0,1); with explicit time integration the enforced \"equal potential\" state is therefore Δt- and grid-dependent in an uncontrolled way, and for intermediate L_φ the φ-relaxation rate acts as a new interfacial kinetic parameter (solute-drag-like) that interpolates between WBM and KKS kinetics with no calibration rule offered. (2) Tellingly, the two demonstrations that carry the paper's novelty — the Al–Cu θ/η1 interdiffusion (§3.2, derived in S4 explicitly \"following Eqs. (7a), (7b) and (8)\") and the 3-D efficiency test (S3, \"φ is evolved using Eq. (8)\") — both use the variational Eq. (8), with which φ provably does not equalize diffusion potentials in the bulk (Fig. 3a,b). So the least-justified component (Eq. (9)) is the one underpinning the \"reduces to KKS / removes extra potential\" efficiency narrative, while the validated component (Eq. (8)) supports only the equilibrium endpoint, not the claimed KKS-like kinetics. This does not break the paper — the equilibrium convergence in Figs. 2 and 5 is real evidence — but the strongest claim's \"reduces exactly to KKS\" clause is established only in a formal limit, never as a controlled numerical convergence statement at fixed physical parameters.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The manuscript introduces an auxiliary non-conserved field φ equal to the composition difference between two phases, so that the two phase compositions are determined from the conserved composition, phase-field variable, and φ. Allen-Cahn/Cahn-Hilliard dynamics are supplemented by a relaxation equation for φ driven by the difference between phase diffusion potentials. With φ=0 and Lφ=0 the formulation gives the WBM equal-composition limit; sufficiently fast φ relaxation is presented as a KKS equal-potential limit. A variational equation (8) and a modified equation (9), obtained by deleting the interfacial factor h(1-h), are both considered. Ti-V growth is benchmarked against WBM and KKS, while an Al-Cu θ/η1 example uses log-transformed variables and database free energies to treat phases with disjoint composition domains. A supplementary 3-D test reports removal of most interfacial extra potential and a 3.3× speedup over KKS.","tokens_in":25518,"tokens_out":6554,"duration_ms":258537,"significance":"If the remaining dynamical and reproducibility issues are resolved, the model would be a useful addition to phase-field methodology. It gives a transparent interpolation between equal-composition and equal-diffusion-potential interface conditions, derives the variational φ equation explicitly, and avoids refitting CALPHAD free energies in the demonstrated examples. The log-variable construction for θ/η1 is a practical, domain-preserving treatment, and the recovery of common-tangent compositions in Al-Cu is a meaningful test. The potential 3-D efficiency advantage over Newton-based KKS implementations is also important, although the present evidence for that advantage is incomplete.","major_comments":[{"comment":"§3.2/S4, Eqs. (8), (12a)-(12b): the stated derivation does not follow from Eq. (8). Inserting φ̇ from Eq. (8) into Eqs. (S3-6a,b) gives factors -h²(1-h)LφΔμ/vm in the Y1 equation and +h(1-h)²LφΔμ/vm in the Y2 equation. The printed Eqs. (12)/S3-7 instead contain -hLφΔμ/vm and +(1-h)LφΔμ/vm, which correspond to Eq. (9). Thus either Fig. 5 was not generated using the equation stated, or Eq. (12) is algebraically incorrect. Please identify the implemented equation, correct S4, and rerun if necessary.","section":"§3.2 and Supplement S4, Eqs. (8), (12), (S3-6)-(S3-7)"},{"comment":"The deletion of h(ξ)[1-h(ξ)] is load-bearing for the claimed KKS limit and system-wide equalization of diffusion potentials, but Eq. (9) is no longer the constant-mobility Allen-Cahn equation for δF/δφ. Relative to Eq. (8) it is equivalent to a position-dependent mobility Lφ/[h(1-h)] that diverges toward the interface fringes, and it evolves virtual phase compositions even where the corresponding phase has zero weight. The manuscript should provide a free-energy-dissipation or stability analysis, explain treatment of CALPHAD domains for virtual compositions, and test timestep/grid sensitivity.","section":"§2, Eq. (9)"},{"comment":"Finite Lφ introduces an additional interfacial kinetic time scale. Figure 2(d) shows that composition profiles depend on Lφ, but no convergence of interface velocity or growth law is demonstrated, and the statement that model choice does not significantly affect kinetics is based on one thin-interface example. The S3 efficiency test uses Lφ=Lξ and Eq. (8), while the >99.99% extra-potential reduction is not precisely defined. Please report Lφ convergence at fixed Lξ, M_X, w, and κξ; Δt/Δx convergence; quantitative comparison with KKS; and guidance for choosing or calibrating Lφ.","section":"Figs. 2-4 and Supplement S3"}],"minor_comments":[{"comment":"§3.1 reverses the convention introduced in §2: there ξ=0 denotes α and ξ=1 denotes β, whereas the Ti-V simulation uses α-Ti at ξ=1 and β-Ti at ξ=0. With Eq. (2), the stated initial and equilibrium Ti-V compositions imply φ≈-0.0677 and -0.2151, but the text and Fig. 3 report positive values. Please explicitly state the variable relabeling and sign convention used in the simulation.","section":"§2 vs. §3.1, Figs. 2-3"},{"comment":"The Fig. 5(d) caption says that φ profiles were obtained “using Eq. (7b),” which is the composition equation. This presumably should refer to Eq. (8) or Eq. (9); correcting it is especially important in view of the S4 inconsistency.","section":"Figure 5 caption"},{"comment":"For the S3 timing comparison, please specify hardware, timestep, Newton tolerance and maximum iteration count, whether CPU time is total or per time step, and whether the KKS and present-model implementations received comparable optimization. These details are needed to assess the reported 3.3× speedup.","section":"Supplement S3 and Fig. S3-1"},{"comment":"The notation μα and μβ is described as phase “chemical potentials,” although Eq. (5) uses them as molar Gibbs-energy densities before differentiation, while μ̃_B denotes a diffusion potential. Defining these quantities and their units explicitly would avoid ambiguity.","section":"§2, Eqs. (5)-(9) and Table 1"},{"comment":"Please deposit or otherwise make available the simulation inputs/scripts and identify database versions. The thermodynamic sources are cited, but the numerical examples are not currently independently reproducible from the manuscript alone.","section":"Reproducibility"},{"comment":"There are several typographical issues, including “has to be imposed,” “furthermore simplifications,” “wide ly,” and “yttrim-stabilized.” A careful copyedit is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that Ji and Chen give a practical way to treat interfacial composition in multiphase phase-field without forcing equal compositions (WBM) or solving nonlinear equal-potential constraints every step (KKS). They introduce an auxiliary non-conserved field φ = X_B^α − X_B^β whose Allen–Cahn evolution is driven by the diffusion-potential mismatch. Algebraically this recovers WBM when φ ≡ 0 and L_φ = 0; it recovers KKS in the formal fast-relaxation limit of their Eq. (9). The real payoff is the Al–Cu θ/η1 demo: two ordered phases with disjoint composition ranges, free energies taken straight from the database, log-mapped intermediate variables to stay inside the physical domains, and clean convergence to the common-tangent equilibria. That case is awkward for both classical models and they handle it cleanly.\n\nWhat they do well is the bookkeeping. Table 1 and the variational derivation in S1 are clear, the Ti–V growth matches the known limits, and they never pretend the free energies are parabolic. Circularity is low; the databases are external.\n\nThe soft spot is real but contained. Eq. (9) simply deletes the interfacial factor h(ξ)(1−h(ξ)) from the thermodynamically consistent Eq. (8). That makes bulk equalization possible and underpins the “reduces to KKS / removes extra potential” efficiency story, yet it turns L_φ into an uncalibrated interfacial kinetic parameter and makes the equal-potential state grid- and Δt-dependent under explicit time stepping. Tellingly, both the flagship Al–Cu run and the 3-D timing test actually evolve φ with the variational Eq. (8), which does not equalize potentials in the bulk. So the strongest claim’s KKS clause is asymptotic, not a controlled numerical statement at fixed physics. Minor: no code, multicomponent extension only sketched.\n\nThis is for people who couple CALPHAD to phase-field and keep hitting non-overlapping or stiff free-energy surfaces. It deserves a serious referee. I would engage with it and expect to cite the construction when the same awkward interfaces appear.","headline":"Useful auxiliary-variable bridge between WBM and KKS that actually handles non-overlapping ordered phases with raw CALPHAD free energies; the KKS limit rests on a pragmatic, under-justified kinetic tweak.","tokens_in":19258,"tokens_out":563,"would_cite":true,"duration_ms":15394,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Auxiliary non-conserved variables unify interfacial chemical equilibrium in phase-field models and treat phases with no shared composition range.","keywords":["Phase-Field model","Interfacial compositions","Interfacial equilibrium","Microstructure evolution","thermodynamic databases","ordered phases","diffusional phase transformations"],"falsifier":"Run the auxiliary-variable model on θ/η1 interdiffusion in Al–Cu (or an analogous pair of ordered phases with disjoint composition ranges) and check whether the final compositions and common-tangent chemical potentials disagree with independent thermodynamic calculations or with a carefully regularized equal-potential solution; any systematic mismatch falsifies the claim.","tokens_in":18942,"feed_emoji":"🔬","tokens_out":827,"duration_ms":31406,"temperature":0.7,"pith_summary":"Phase-field models of diffusional microstructure evolution must decide how to split a local composition among the phases that coexist inside a diffuse interface. The classic equal-composition and equal-chemical-potential assumptions either fail for ordered phases whose stability ranges do not overlap or require expensive nonlinear solves and free-energy approximations. This paper introduces auxiliary non-conserved fields that simply track the composition differences between phases and evolve them by the mismatch of diffusional chemical potentials. The construction recovers the two classic models as opposite kinetic limits, accepts unsimplified thermodynamic database expressions, and works for interdiffusion between non-stoichiometric ordered phases that share no common composition range. The result is a single, flexible framework for composition evolution and interfacial chemical equilibrium across a wide class of multiphase materials problems.","feed_headline":"Phase-field model handles phases with no shared composition","feed_subtitle":"Auxiliary variables recover classic limits and take thermodynamic databases as-is","key_machinery":"The auxiliary variable φ = X_B^α − X_B^β (and its multi-phase, multi-component generalizations), whose evolution is driven by the difference of diffusional chemical potentials and whose kinetic coefficient continuously interpolates between equal-composition and equal-potential interfacial conditions.","core_discovery":"Interfacial chemical equilibrium among solution phases is described by auxiliary non-conserved variables that measure composition differences between phases; these variables evolve by Allen-Cahn dynamics driven by differences in chemical diffusional potentials, recovering the Wheeler-Boettinger-McFadden equal-composition model and the Kim-Kim-Suzuki equal-potential model as limiting cases while remaining valid for phases whose composition ranges do not overlap and while using thermodynamic databases without further approximation.","pith_inferences":["The same auxiliary-variable construction can be coupled to elastic or electrostatic fields to treat chemo-mechanical interfacial equilibria without forcing equal compositions.","The logarithmic intermediate variables introduced for the Al–Cu example supply a general numerical safeguard for any sublattice free energy that contains logarithms of site fractions.","Measuring how far experimental interface profiles lie from the equal-composition versus equal-potential limits would give a direct experimental calibration of the auxiliary kinetic coefficient."],"forward_implications":["Interdiffusion between non-stoichiometric ordered phases or oxides can be simulated directly from thermodynamic databases without free-energy fitting or extrapolation.","Multicomponent multiphase systems are obtained by defining one auxiliary variable per independent component per phase pair.","Three-dimensional simulations become cheaper than the equal-potential model because interface nonlinear solves are replaced by a simple relaxation step.","The continuous family of interfacial conditions parameterized by the auxiliary kinetic coefficient can be used to explore how real interfaces sit between the two classic extremes."],"fun_headline_variants":["Auxiliary variables unite WBM and KKS for phase-field equilibria","Phase-field model treats phases lacking shared composition ranges","General interfacial equilibrium recovers classic phase-field limits","Non-conserved auxiliaries drive chemical equilibria across phases","Phase-field equilibria from diffusional potential differences"],"cache_read_input_tokens":128,"weakest_assumption_plain":"Dropping the interfacial weighting factor from the thermodynamically derived evolution law for the auxiliary variable still yields a faithful, artifact-free description of chemical equilibrium in both bulk and interface.","fun_headline_variants_meta":{"raw":{"variants":["Auxiliary variables unite WBM and KKS for phase-field equilibria","Phase-field model treats phases lacking shared composition ranges","General interfacial equilibrium recovers classic phase-field limits","Non-conserved auxiliaries drive chemical equilibria across phases","Phase-field equilibria from diffusional potential differences"]},"model":"grok-4.5","effort":"low","cost_usd":0.00412,"raw_usage":{"total_tokens":1185,"prompt_tokens":693,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":41204000,"prompt_tokens_details":{"text_tokens":693,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":411,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":693,"tokens_out":81,"duration_ms":5629,"temperature":1.0,"reasoning_tokens":411,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T10:23:38.854814+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Run the auxiliary-variable model on θ/η1 interdiffusion in Al–Cu (or an analogous pair of ordered phases with disjoint composition ranges) and check whether the final compositions and common-tangent chemical potentials disagree with independent thermodynamic calculations or with a carefully regularized equal-potential solution; any systematic mismatch falsifies the claim.","supporting_citations":[],"review_version":1}