The binary alloy Ni(25 wt ppm Bi) develops a pronounced intermediate-temperature ductility minimum (700–750 °C after 45-min holds; isothermal minima at 45 min/750 °C and ≈50 h/650 °C) that was attributed in 2011 to thermo-induced non-equilibrium grain-boundary segregation of Bi, described by a critical-time kinetic framework. That attribution still rests on undetermined kinetic parameters: the tracer diffusivity of Bi in Ni and the diffusivity of the Bi–vacancy complex have never been directly measured, and the original analysis used generic impurity-diffusion estimates, complex parameters borrowed from sulfur studies, and an assumed grain-boundary thickness. Whether the same segregation mechanism governs the ductility trough of other Ni-based superalloys — where carbides, dynamic strain aging, γ′-related mechanisms, oxidation and grain-boundary complexions compete — is likewise unresolved. The problem is to establish, or quantitatively refute, this segregation-based mechanism with Bi-specific kinetic parameters determined independently of the embrittlement map, and to determine its domain of validity across Ni-based alloys.

Background

Nickel-based structural alloys, including the superalloys used in gas turbines and advanced power plants, commonly lose tensile ductility in an intermediate temperature window of roughly 500–800 °C — intermediate-temperature embrittlement (ITE), or the ductility trough — exactly the range in which many components operate. The trough has been attributed to many different causes in different alloys: grain-boundary precipitates or carbides, dynamic strain aging (DSA; the repeated pinning of dislocations by diffusing solutes, often visible as serrated, or Portevin–Le Châtelier, flow), planar slip and glide-plane decohesion associated with the ordered γ′ (L1₂-structure Ni₃(Al,Ti)) strengthening phase, oxidation, grain-boundary sliding, and grain-boundary segregation of embrittling impurities.

Grain-boundary (GB) segregation — the enrichment of a solute at internal interfaces — is quantitatively described for equilibrium conditions by McLean's model, in which the GB concentration rises monotonically toward a thermodynamic plateau. A distinct non-equilibrium mechanism was developed by Faulkner and by Xu and co-workers: during quenching or isothermal holding after a high-temperature solution treatment, supersaturated vacancies bind solute atoms into solute–vacancy complexes that migrate to grain boundaries, which act as vacancy sinks. The GB solute level then does not approach equilibrium monotonically; it passes through a maximum at a characteristic critical time tct_c and subsequently relaxes. In the formulation used for dilute alloys, tc=4r2ln⁡(Dc/D)/[δ(Dc−D)]t_c=4r^2\ln(D_c/D)/[\delta(D_c-D)], where rr is the grain radius, DD is the solute tracer diffusivity in the matrix, DcD_c is the diffusivity of the solute–vacancy complex, and δ\delta is a critical constant (Song, Xu and Yuan, 1989). This thermo-induced non-equilibrium segregation (TNGS) naturally produces a ductility minimum whose temperature and time depend on the thermal schedule.

Zheng, Chellali, Schlesiger, Baither and Schmitz (2011) isolated the impurity question with a controlled comparison: ultra-high-purity Ni (total impurities <1 wt ppm) and a binary Ni alloy with 25 wt ppm Bi, tensile-tested identically at 400–850 °C after a 45-min isothermal hold. Only the Bi-doped alloy developed a deep elongation minimum at 700–750 °C with intergranular fracture, without detectable precipitates or recrystallization. McLean-type equilibrium segregation was argued to be quantitatively unable to supply enough GB Bi on the 45-min timescale, while the TNGS critical time, evaluated with the measured grain radius, passes through 45 min exactly in the observed minimum window — implicating thermo-induced non-equilibrium segregation of Bi. Density-functional-theory work (Kang, Glatzmaier and Wei, 2013) subsequently showed why segregated Bi is so damaging: Bi strongly reduces the work of separation of Ni grain boundaries. Atom probe tomography (APT; a technique that resolves individual atoms in three dimensions and can quantify chemical enrichment on individual boundaries) later confirmed strong Bi enrichment at Ni GBs and the non-equilibrium character of the segregation: the group demonstrated U-shaped isothermal ductility curves with minima at 45 min (750 °C) and 50 h (650 °C), with intergranular fracture at the minima only, reverting to ductile fracture on both time sides.

The quantitative basis of this mechanism, however, remains incomplete in exactly the ways the 2011 study exposed. The tracer diffusion coefficient DD of Bi in Ni has never been directly measured, nor has the Bi–vacancy complex diffusivity DcD_c: the 2011 critical-time estimate relied on generic impurity-diffusion ranges, an assumed GB thickness of 1 nm, and complex parameters that coincide with values tabulated for sulfur–vacancy complexes in a Ni–Cr–Fe alloy (Wang, Xu, Song and Shao, 2011) — that is, no Bi-specific kinetic input existed. Moreover, whether the same segregation mechanism governs the ductility trough of multi-component Ni-based superalloys — where carbides, DSA/Portevin–Le Châtelier effect, γ′-related mechanisms, oxidation, and equilibrium GB complexions (nanometer-thick disordered interfacial phases, demonstrated for sulfur-doped Ni by Hu, Yang, Zhou, Zhang and Luo, 2018) are competing candidates — has not been established for any industrial alloy.

Problem Statement

The problem is to establish — or quantitatively refute — a segregation-based mechanism for intermediate-temperature embrittlement (ITE) in Ni-based materials, grounded in Bi-specific kinetic parameters determined independently of the embrittlement map, and to determine its domain of validity. Three linked questions define the completion boundary:

(1) Kinetic closure for the model system. Determine the tracer diffusion coefficient DD of Bi in Ni and the effective diffusivity DcD_c (and binding) of the Bi–vacancy complex over the temperature range relevant to ITE (approximately 600–850 °C) by Bi-specific means independent of the embrittlement map of component (2) — direct diffusion measurement or rigorous extraction from independently measured critical times or segregation kinetics — together with a physically justified grain-boundary capture-zone width (the thickness of the near-boundary slab that captures the arriving Bi–vacancy complexes and accommodates the segregated Bi — the length scale the 2011 analysis assumed to be 1 nm, and distinct from the dimensionless critical-time constant δ\delta that appears in the tct_c relation), replacing the generic impurity-diffusion estimates and non-Bi-specific complex parameters on which the original analysis rested.

(2) Quantitative test in the Ni(Bi) model alloy. Using only the parameters determined in (1) — D(T)D(T) and Dc(T)D_c(T) together with the binding energy and the grain-boundary capture-zone width — held fixed at their (1)-determined values and not re-adjusted to fit the embrittlement map under test, test whether the thermo-induced non-equilibrium segregation critical-time framework reproduces the measured embrittlement map of the Ni(25 wt ppm Bi) alloy: the isochronal elongation minimum at 700–750 °C after 45-min holds, and the isothermal elongation minima at 45 min (750 °C) and approximately 50 h (650 °C), and whether its critical-time scaling is consistent with the temperature dependence of the observed minima. This parameter-free prediction test adjudicates the component: agreement confirms the segregation mechanism as a quantitative predictive framework, whereas failure of the (1)-determined parameters to reproduce the map refutes it as formulated. The parameters are held fixed throughout; a refit that restores agreement does not count as confirmation.

(3) Generality across Ni-based alloys. Establish, for the representative Ni-based superalloys in which competing mechanisms have been reported — C-276, Nimonic 263, Inconel 625, M4706, GH4065A, and Ni-Fe weld metal — under what conditions grain-boundary segregation of embrittling solutes controls the ductility trough, coexists with, or is masked by the reported mechanisms (grain-boundary carbides, dynamic strain aging/Portevin–Le Châtelier effect, γ′-related planar-slip and glide-plane decohesion, oxidation, and grain-boundary complexions). Establishing this controls/coexists/is-masked classification across this representative set completes component (3), and fixing the domain of validity of the segregation mechanism requires no unbounded survey beyond this set.

The single scientific objective uniting these components is a predictive, parameter-grounded account of the time–temperature ductility trough of Ni-based materials: with the results in hand, a qualified specialist should be able to predict from measured inputs (solute content, grain size, thermal schedule, and the kinetic parameters determined in (1)) whether, when and how severely a given Ni-based alloy will embrittle, and to identify which mechanism class controls it. The three components are jointly necessary for that objective: parameters without the parameter-free test leave the mechanism unvalidated, and both without the generality boundary leave the phenomenon unpredictable in the engineering alloys where it matters.

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