arxiv: 2603.20569 · v2 · submitted 2026-03-20 · ❄️ cond-mat.mtrl-sci

Recognition: no theorem link

Composition dependence of the critical Rayleigh number curve for macrosegregation in multicomponent metal alloys

Owain S. Houghton , Adrian S. Sabau , Gregory B. Olson

Authors on Pith no claims yet

Pith reviewed 2026-05-15 07:38 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords macrosegregationRayleigh numberfrecklesmushy zonecomposition dependencemulticomponent alloyssolidification defectschannel segregates

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The pith

The critical Rayleigh number for macrosegregation defects in multicomponent alloys varies strongly with composition.

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

The paper applies Flemings' model to derive a Rayleigh number and its critical value marking the onset of local remelting when buoyancy-driven flow suddenly starts in a stagnant mushy zone. Evaluation with CALPHAD data for nickel superalloys and lead-tin systems shows this critical value changes with local solid fraction and thermophysical properties. Because those properties shift substantially even across narrow composition intervals, the threshold itself becomes composition-dependent rather than universal. A reader would care because accurate forecasting of casting defects like freckles and channel segregates requires treating the criterion as specific to each alloy makeup.

Core claim

Flemings' model is used to determine conditions for the onset of local remelting when buoyancy-driven flow is suddenly triggered in an initially stagnant mushy zone, yielding an expression for a Rayleigh number Ra and its critical value Ra_crit. Using thermophysical data from CALPHAD for the nickel-based superalloy SX-1 and Pb-Sn alloys, and testing correlation with empirical criteria for various steel compositions, it is found that Ra_crit varies with the local average solid fraction and several thermophysical properties. Since these properties can vary substantially within a relatively narrow composition range, Ra_crit is a strongly composition-dependent parameter.

What carries the argument

The Rayleigh number Ra and its critical value Ra_crit derived from Flemings' local-remelting criterion applied to an initially stagnant multicomponent mushy zone.

If this is right

Predictions of freckle and channel segregate formation must use composition-specific critical Rayleigh values instead of a single fixed threshold.
Small changes in alloying elements can substantially alter defect susceptibility through shifts in the critical flow condition.
Solidification models for multicomponent systems require local, dynamic recalculation of Ra_crit based on current thermophysical properties.
Empirical defect criteria developed for steels can be refined by incorporating the composition dependence shown for nickel and lead-tin alloys.

Where Pith is reading between the lines

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

Alloy design workflows could target compositions that raise Ra_crit to widen the safe processing window during casting.
Continuous composition gradients in experiments might be used to map the full surface of Ra_crit values for practical defect avoidance.
Integration with existing CALPHAD databases would allow real-time defect-risk assessment during alloy development.

Load-bearing premise

Flemings' model accurately captures the local remelting condition triggered by sudden buoyancy-driven flow in an initially stagnant mushy zone for multicomponent alloys.

What would settle it

Experimental measurement of the solid-fraction and composition at which channel defects first appear under controlled buoyancy conditions in a series of alloys with systematically varied thermophysical properties, checking whether the observed thresholds follow the predicted composition dependence.

read the original abstract

Convective instabilities in the semi-solid mushy zone can trigger channel formation that leads to defects known as freckles, channel segregates and A-type segregates. In the present work, Flemings' model is used to determine conditions for the onset of local remelting when buoyancy-driven flow is suddenly triggered in an initially stagnant mushy zone. An expression in the form of a Rayleigh number $Ra$, and its associated critical value $Ra_{\rm crit}$, above which the local remelting condition is satisfied, are derived. Using thermophysical data from CALPHAD, these expressions are evaluated using results from benchmark experimental and numerical studies for the nickel-based superalloy SX-1 and Pb-Sn alloys. The correlation of this local-remelting criterion with previously reported empirical criteria is also tested for various steel compositions. It is found that $Ra_{\rm crit}$ varies with the local average solid fraction and several thermophysical properties. Since these properties can vary substantially within a relatively narrow composition range, it is suggested that $Ra_{\rm crit}$ is a strongly composition-dependent parameter.

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.

Desk Editor's Note private letter to a colleague

Ra_crit varies strongly with composition through solid fraction and CALPHAD properties, but the multicomponent extension of Flemings' remelting criterion lacks direct validation checks.

read the letter

The main takeaway is that this paper shows the critical Rayleigh number for local remelting in the mushy zone depends strongly on alloy composition. Using Flemings' model, they derive Ra and Ra_crit for sudden buoyancy-driven flow, then evaluate it with CALPHAD thermophysical data on benchmark cases like nickel superalloy SX-1 and Pb-Sn alloys, plus a check against empirical criteria for steels. The result is that Ra_crit shifts with local solid fraction and properties that change even in narrow composition windows, so a single fixed threshold may not hold across alloys. This is a straightforward extension that avoids new free parameters and ties directly to existing data and models. It gives a concrete way to refine defect predictions for freckles and channel segregates in casting. The soft spot is validation. The central step assumes Flemings' binary remelting balance extends quantitatively to multicomponent systems with multiple solutes and varying partition coefficients, yet the work reports no targeted comparison to direct remelting-front simulations or focused experiments on that point. Error bars and full derivation steps are also not visible in the summary. If the full text supplies those, the claim strengthens; otherwise it rests on the model's prior accuracy carrying over. This is for people who model macrosegregation in superalloys and steels and want to move beyond fixed Rayleigh criteria. A reader working on industrial defect prediction would find the composition dependence useful as a refinement. I would send it to peer review because the approach is grounded and the finding is testable, even if revisions for more validation are likely needed.

Referee Report

2 major / 2 minor

Summary. The paper extends Flemings' local remelting criterion to multicomponent alloys to derive a Rayleigh number Ra and its critical value Ra_crit for the onset of remelting when buoyancy-driven flow is suddenly imposed on a stagnant mushy zone. Using CALPHAD thermophysical data, the expressions are evaluated on benchmark cases for the nickel superalloy SX-1 and Pb-Sn alloys, and the correlation with empirical macrosegregation criteria is tested across steel compositions. The central finding is that Ra_crit depends on local average solid fraction and several thermophysical properties that vary substantially even within narrow composition ranges, implying that Ra_crit is a strongly composition-dependent parameter rather than a universal constant.

Significance. If the derivation and its assumptions hold, the result would indicate that standard Rayleigh-number criteria for freckle and channel segregation formation require composition-specific calibration, which could improve predictive accuracy for macrosegregation defects in industrial multicomponent castings. The explicit linkage to CALPHAD data provides a reproducible route to compute Ra_crit from measurable properties, strengthening the practical utility of the criterion if validated.

major comments (2)

[§2] §2 (model derivation): the extension of Flemings' binary remelting balance (solute rejection versus advective transport) to multicomponent systems with multiple solutes, temperature-dependent partition coefficients, and varying solid fractions is assumed to remain quantitatively accurate, yet no targeted validation against direct numerical simulation of the remelting front or multicomponent-specific experiments is reported; this assumption is load-bearing for the claimed Ra_crit expression.
[§3] §3 (evaluation on benchmarks): the reported correlations for SX-1, Pb-Sn, and steel compositions lack error bars, sensitivity analysis on CALPHAD inputs, or quantitative agreement metrics (e.g., R² or residual norms) against the original experimental data, weakening the evidence that composition dependence is strong rather than modest.

minor comments (2)

[Abstract] The abstract states that full equations are derived but the manuscript does not display the explicit functional form of Ra_crit in terms of solid fraction and thermophysical properties, hindering immediate reproducibility.
[§2] Notation for the local average solid fraction and the precise definition of the buoyancy term in the Rayleigh number should be clarified with a dedicated nomenclature table or inline definitions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the scope and limitations of our work. We respond to each major comment below and outline revisions to strengthen the manuscript.

read point-by-point responses

Referee: [§2] §2 (model derivation): the extension of Flemings' binary remelting balance (solute rejection versus advective transport) to multicomponent systems with multiple solutes, temperature-dependent partition coefficients, and varying solid fractions is assumed to remain quantitatively accurate, yet no targeted validation against direct numerical simulation of the remelting front or multicomponent-specific experiments is reported; this assumption is load-bearing for the claimed Ra_crit expression.

Authors: The derivation applies mass and energy balances to each solute independently while retaining the local remelting condition from Flemings' original framework, which is a direct algebraic extension rather than an empirical fit. We acknowledge that no new DNS of the multicomponent remelting front or dedicated multicomponent experiments are presented; such targeted validation lies beyond the scope of this theoretical derivation and benchmark evaluation. The expressions are instead tested on existing SX-1 and Pb-Sn benchmark data from the literature. In revision we will add an explicit paragraph in §2 stating the key assumptions, noting the absence of multicomponent DNS validation, and recommending it as future work. revision: partial
Referee: [§3] §3 (evaluation on benchmarks): the reported correlations for SX-1, Pb-Sn, and steel compositions lack error bars, sensitivity analysis on CALPHAD inputs, or quantitative agreement metrics (e.g., R² or residual norms) against the original experimental data, weakening the evidence that composition dependence is strong rather than modest.

Authors: We agree that quantitative support would strengthen the claims. The correlations use published experimental conditions and CALPHAD-derived properties, but error propagation, sensitivity studies, and formal agreement metrics were not included. In the revised manuscript we will add error bars based on reported uncertainties in thermophysical data, perform a sensitivity analysis on partition coefficients and densities for the SX-1 and steel cases, and report R² and mean absolute deviation values for the steel-composition comparison against the empirical criteria. revision: yes

Circularity Check

0 steps flagged

Derivation from Flemings' model is independent of outputs

full rationale

The paper derives an expression for the Rayleigh number Ra and its critical value Ra_crit by applying Flemings' established local remelting criterion to sudden buoyancy-driven flow in a stagnant multicomponent mushy zone. The resulting Ra_crit depends explicitly on local average solid fraction and thermophysical properties (partition coefficients, densities, etc.) obtained from external CALPHAD databases. Composition dependence follows directly as a consequence of those input properties varying across narrow composition ranges; no equation defines Ra_crit in terms of itself, no parameter is fitted to the target outcome, and no self-citation supplies the load-bearing uniqueness or ansatz. Benchmark comparisons use independent experimental and numerical studies for SX-1 and Pb-Sn alloys, keeping the chain self-contained against external data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of Flemings' model for deriving the Rayleigh number expression and on thermophysical properties obtained from CALPHAD databases.

axioms (1)

domain assumption Flemings' model determines the onset of local remelting when buoyancy-driven flow is triggered in a stagnant mushy zone
Invoked to derive the Ra and Ra_crit expressions

pith-pipeline@v0.9.0 · 5500 in / 1125 out tokens · 34356 ms · 2026-05-15T07:38:57.596017+00:00 · methodology