Quantitative kinetic closure and alloy-family generality of thermo-induced non-equilibrium grain-boundary segregation as the mechanism of intermediate-temperature embrittlement in Ni(Bi) and related alloys

Zheng et al. (2015) showed that the isochronal and isothermal ductility minima of a binary Ni(25 wt ppm Bi) alloy share one mechanism — thermo-induced non-equilibrium grain-boundary segregation (TNGS) of Bi — but left the account qualitative: the tracer diffusivity of Bi in Ni and the diffusivity of the Bi–vacancy complex that enter the TNGS kinetic expressions were undetermined, and the mechanism's operation in Fe-, Co-, Cu-, Al-based alloys and multi-component Ni-based superalloys was unvalidated. Both issues remain open. The same group's atom-probe follow-up (2016) measured grain-boundary segregation kinetics of Bi in nanocrystalline Ni–Bi and derived one effective diffusivity under a McLean-type approximation, but reported kinetics inconsistent with that model and did not determine the complex diffusivity, activation energies, or a quantitative prediction of the embrittlement minima; subsequent intermediate-temperature-embrittlement literature on engineering Ni-Cr-Co-based alloys has developed the competing dynamic-strain-aging interpretation instead; and a 2025 study applying the same non-equilibrium-segregation framework to sulfur in an ultra-pure ferritic stainless steel covers only one Fe-based system and determines no Ni(Bi) parameters. The problem asks for (i) the missing kinetic parameters and a demonstrated quantitative prediction of the observed time–temperature embrittlement in Ni(Bi), and (ii) experimental validation or refutation of the TNGS mechanism for analogous minima in other alloy families and industrial superalloys.

Background

Nickel-based alloys and other structural metals can lose tensile ductility in a narrow intermediate-temperature window (roughly 500–800 °C), a failure mode called intermediate-temperature embrittlement (ITE); it is intergranular (fracture runs along grain boundaries) and appears both as a ductility minimum versus test temperature after a fixed pre-hold (isochronal embrittlement) and as a non-monotonic, time-dependent minimum versus holding time at fixed temperature (isothermal embrittlement). Grain boundaries (GBs) are the interfaces between crystallographic grains, and solute atoms often segregate to them. In equilibrium (McLean) segregation, solute enrichment of the GB reaches a temperature-dependent plateau set by thermodynamics. Non-equilibrium grain-boundary segregation (NGS), developed by T. D. Xu and co-workers, is a kinetic effect: at a high solution temperature a fraction of solute atoms is bound to vacancies (forming a solute–vacancy complex); on quenching, the vacancy supersaturation drives these complexes to GBs, which act as vacancy sinks, transiently enriching the boundary above its equilibrium level. The enrichment peaks after a characteristic critical time $t_{c}$ and then decays back toward the equilibrium level. In the kinetic expressions used by the 2015 Ni(Bi) study, the critical time is

$$t_{c}=\frac{4r^{2}\ln(D_{c}/D)}{\delta(D_{c}-D)},$$

where $r$ is the average grain radius, $D$ is the solute tracer diffusivity in the matrix, $D_{c}$ is the diffusivity of the solute–vacancy complex, and $\delta$ is a kinetic constant of the model; the maximum GB concentration reached during an isothermal hold at temperature $T$ after quenching from the solution temperature $T_{0}$ is

$$C_{m}(T)=C_{I}^{B}\left(\frac{E_{b}}{E_{f}}\right)\exp\left\{\frac{E_{b}-E_{f}}{kT_{0}}-\frac{E_{b}-E_{f}}{kT}\right\},$$

where $C_{I}^{B}$ is the bulk solute concentration, $E_{b}$ is the solute–vacancy binding energy, $E_{f}$ is the vacancy formation energy, and $k$ is Boltzmann's constant. When produced by a thermal quench, this is called thermo-induced NGS (TNGS), to distinguish it from stress-induced or deformation-induced variants. In a binary Ni(25 wt ppm Bi) model alloy (specimens solution-treated at 1000 °C and quenched), Zheng, Chellali, Schlesiger, Meng, Baither and Schmitz (2015) measured deep isochronal (700–750 °C after 45 min holds) and isothermal (45 min at 750 °C; ≈50 h at 650 °C) elongation minima, showed intergranular fracture at the minima, excluded intergranular precipitates and phase transformations down to tens of nanometres by electron microscopy, and demonstrated strong local Bi enrichment at a grain boundary by atom probe tomography (APT, a technique that maps nanoscale chemistry by field-evaporating atoms from a needle tip). They concluded that one mechanism — TNGS of Bi — explains both embrittlement modes, because the observed non-monotonic time dependence matches the critical-time behavior of NGS while equilibrium segregation, precipitate growth, oxygen penetration, and dynamic strain aging (solute–dislocation interaction causing serrated flow) predict monotonic or otherwise inconsistent trends. Related knowledge frames this conclusion: equilibrium GB segregation of Bi in Ni polycrystals had been measured earlier (Chang and Huang, 2004); density-functional calculations show that Bi strongly weakens Ni and Cu grain boundaries (Kang, Glatzmaier and Wei, 2013), consistent with the classic bismuth embrittlement of copper (Duscher et al., 2004; Schweinfest, Paxton and Finnis, 2004); Xu's school had reported NGS-induced ITE for sulfur in a Ni–Cr–Fe alloy (Wang, Xu, Wang and Du, 2009) and a unified NGS-based embrittlement picture (Xu, Zheng, Wang and Misra, 2013); and a companion study (Yang et al., 2015) verified that the dynamic-strain-aging interpretation is inadequate for Ni-Bi. What the 2015 account lacked was quantitative closure — the parameters $D$, $D_{c}$ and their activation energies for Bi in Ni were unknown, so the TNGS expressions could not be evaluated against the measured $t_c$ and $C_{m}(T)$ — and evidence that the mechanism operates beyond the binary model alloy.

Problem Statement

Two linked questions remain from the 2015 Ni(Bi) study, jointly amounting to establishing thermo-induced non-equilibrium grain-boundary segregation (TNGS) as a quantitative and general mechanism of intermediate-temperature embrittlement. (1) Quantitative closure for the model system: determine the presently unknown kinetic parameters for Bi in nickel — the tracer diffusion coefficient $D$ of Bi in Ni and the diffusion coefficient $D_{c}$ of the Bi–vacancy complex, together with their activation energies, for example by the method of Xu et al. (1991), which extracts $D$ and $D_{c}$ from the measured critical times of segregation (the isothermal ductility-minimum times reported for the 2015 Ni(25 wt ppm Bi) specimens) — and demonstrate that the TNGS kinetic expressions for the critical time $t_{c}$ and the maximum grain-boundary concentration $C_{m}(T)$, evaluated with a fully specified input set whose values and provenance are stated (the average grain radius $r$, the model constant $\delta$, the Bi–vacancy binding energy $E_{b}$ and the vacancy formation energy $E_{f}$, in addition to $D$ and $D_{c}$), quantitatively predict the observed isochronal and isothermal embrittlement minima of Ni(25 wt ppm Bi), reproducing the measured isochronal minimum temperature (700–750 °C after a 45 min hold) and the isothermal minimum holding times (≈45 min at 750 °C and ≈50 h at 650 °C) within the scatter of the 2015 measurements. (2) Generality: establish experimentally whether thermo-induced non-equilibrium segregation of the solute species operative in each system, as identified by the grain-boundary chemical measurements themselves, governs the analogous isochronal and isothermal ductility minima in Fe-, Co-, Cu- and Al-based alloys and in industrial multi-component Ni-based superalloys, where several solutes and precipitates coexist — or, conversely, show by direct evidence that the TNGS account fails in these systems. Each of the listed alloy families is an independently refutable sub-claim: direct evidence that TNGS of its operative solute governs its isochronal and isothermal minima, or that the TNGS account fails there, resolves that family in the affirmative or the negative. Results are reported family by family, a mixed outcome across families is a valid resolution rather than an incomplete one, and settling every listed family is not required.

References

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