Stacking Fault Energy Based Alloy Screening for Hydrogen Compatibility

Abstract

The selection of austenitic stainless steels for hydrogen service is challenging since there are few intrinsic metrics that relate alloy composition to hydrogen degradation. One such metric, presented here, is intrinsic stacking fault energy (SFE). This work reviews the exiting literature to use estimated intrinsic SFE values, calculated with a sub-regular solution thermodynamic model, to compare the retention of tensile ductility of γ-austenitic stainless steels in the presence of hydrogen. The goal is to demonstrate SFE as a metric to screen γ-austenitic stainless steels that use diverse alloying strategies for hydrogen compatibility. A transition in the tensile reduction of area of both 300-series and manganese-stabilized stainless steels is observed at a calculated stacking fault energy of approximately 39 mJ m⁻², below which pronounced hydrogen degradation on tensile ductility is observed. Calculated intrinsic stacking fault energy is demonstrated as a high-throughput screening metric for a diverse range of austenitic stainless steel compositions with regard to hydrogen compatibility.

Appendix A

To calculate intrinsic SFE for steel alloys of interest, the relevant thermodynamic parameters and model architectures were collected from the available literature; the works of Curtze et al.30 and Saeed-Akbari et al.28 were of great assistance in compiling the present construction. Starting with the construction in Eq. 1, a value of 8 mJ m⁻² was used for the interfacial energy (σ) based on the survey of reported values by Saeed-Akbari et al.28 The molar surface density (ρ) was calculated following the work by Allain et al.:69

$$ \rho = \frac{4}{\sqrt 3 }\frac{1}{{\left( {a_{\gamma }^{T} } \right)^{2} N_{\text{A}} }} $$

(2)

where a^T_γ is the lattice parameter at the temperature of interest and N_A is Avogadro’s number. The composition-dependent lattice parameter in angstroms at room temperature was calculated using the expression of Babu et al.:84

$$ a_{\gamma }^{300K} = 3.5780 - 0.002X_{\text{Ni}} + 0.00095X_{\text{Mn}} + 0.0006X_{\text{Cr}} + 0.0056X_{\text{Al}} + 0.0033X_{\text{C}} + 0.0031X_{\text{Mo}} $$

(3)

where X_i is the molar fraction of each of the elements (i). The change in lattice parameter of the γ-austenitic phase due to thermal expansion was considered using the equation presented by García de Andrés et al.:85

$$ a_{\gamma }^{T} = a_{\gamma }^{300K} \left[ {1 + \delta_{\gamma } \left( {T - 300} \right)} \right] $$

(4)

where δ_γ is the coefficient of thermal expansion for austenite of 2.065 × 10⁻⁹ K⁻¹ and T is the temperature in Kelvin.

The driving force for the formation of the ε-martensite phase, $\Delta G^{\gamma \to \varepsilon }$, may be estimated for a given alloy composition and temperature if sufficient thermodynamic data are available. The overall driving force may be broken into component terms using a sub-regular solution model, resulting in a summation of the individual energetic contributions to the overall transformation potential:

$$ \Delta G^{\gamma \to \varepsilon } = \Delta G^{\gamma \to \varepsilon }_{{{\text{Chem}} .}} + \Delta G^{\gamma \to \varepsilon }_{{{\text{Mag}} .}} + \Delta G^{\gamma \to \varepsilon }_{{{\text{Seg}} .}} + \Delta G^{\gamma \to \varepsilon }_{\text{GS}} + \Delta G^{\gamma \to \varepsilon }_{\text{Strain}} $$

(5)

Here, the chemical potential for the transformation ($\Delta G^{\gamma \to \varepsilon }_{{{\text{Chem}} .}}$)33,86 is augmented by the excess driving force for the transformation because of magnetism of the respective phases ($\Delta G^{\gamma \to \varepsilon }_{{{\text{Mag}} .}}$),87,88 the strain induced by the change in crystallography and lattice parameter ($\Delta G^{\gamma \to \varepsilon }_{\text{Strain}}$),33 the contribution of segregated nitrogen at the stacking fault ($\Delta G^{\gamma \to \varepsilon }_{{{\text{Seg}} .}}$),89 and the change in driving force due to grain size ($\Delta G^{\gamma \to \varepsilon }_{\text{GS}}$).28,90,91 The contribution of $\Delta G^{\gamma \to \varepsilon }_{\text{Strain}}$ to the overall transformation is typically small and is neglected here.33 Similarly, when the grain size is larger than approximately 30 μm, as is typical for commercial austenitic stainless steels, the excess energy due to grain size is negligible.91,92 Details for the calculation of the other parameters are summarized in the following sections.

Chemical Potential

In a sub-regular solution model the chemical potential for a transformation may be separated into two summations:

$$ \Delta G^{\gamma \to \varepsilon }_{{{\text{Chem}} .}} = \mathop \sum \limits_{i} X_{i} \Delta G^{\gamma \to \varepsilon }_{i} + \mathop \sum \limits_{ij} X_{i} X_{j} \varOmega_{ij}^{\gamma \to \varepsilon } $$

(6)

where $\Delta G^{\gamma \to \varepsilon }_{i}$ is the contribution of each element (i) to the transformation driving force, $\varOmega_{ij}^{\gamma \to \varepsilon }$ represents the entropic first-order interactions between elements i and j to modify the transformation potential, and X_i and X_j are the molar fractions of the constituent species. Here we consider alloy systems consisting of Fe-Ni-Mn-Cr-Al-Si-Cu; the relative terms for $\Delta G^{\gamma \to \varepsilon }_{i}$ and $\varOmega_{ij}^{\gamma \to \varepsilon }$ are included in Table I.

Table I Chemical contributions to $\Delta G^{\gamma \to \varepsilon }_{{{\text{Chem}} .}}$

The calculation of the contribution of bulk nitrogen to the γ to ε phase transition ($\Delta G^{\gamma \to \varepsilon }_{\text{N,bulk}}$) requires additional rigor due to the site occupancy of nitrogen in the FCC lattice. This work was initially undertaken by Yakubtsov89 with additional terms for Mn added by Curtze30 based on the work by Cotes.26 A complete description of the derivation of $\Delta G^{\gamma \to \varepsilon }_{\text{N,bulk}}$ may be found in the sources above; the final form follows:

$$ \Delta G^{\gamma \to \varepsilon }_{\text{N,bulk}} = 6\left( {P_{1} + P_{2} + P_{3} + P_{4} - P_{5} - P_{6} - P_{7} } \right) $$

(7)

where:

$$ P_{1} = X{_{\text{Cr }}}\Delta G^{\gamma \to \varepsilon }_{\text{Cr}} $$

(8)

$$ P_{2} = \frac{{X_{\text{Fe}} \left( {U_{\text{FeN}}^{\varepsilon } - U_{\text{CrN}}^{\varepsilon } } \right)}}{{X_{\text{Fe}} + X_{\text{Cr}}\exp \left( {\frac{{ - U_{\text{FeN}}^{\varepsilon } - U_{\text{CrN}}^{\varepsilon } }}{{RT}}} \right) + X_{\text{Ni}}\exp \left( {\frac{{ - U_{\text{FeN}}^{\varepsilon } - U_{\text{NiN}}^{\varepsilon } }}{{RT}}} \right) + X_{\text{Mn}}\exp \left( {\frac{{ - U_{\text{FeN}}^{\varepsilon } - U_{\text{MnN}}^{\varepsilon } }}{{RT}}} \right)}} $$

(9)

$$ P_{3} = \frac{{{\text{X}}_{\text{Ni}}\left( {U_{\text{NiN}}^{\varepsilon } - U_{\text{CrN}}^{\varepsilon } } \right)}}{{X_{\text{Fe}}{ \exp }\left( {\frac{{ - U_{\text{NiN}}^{\varepsilon } - U_{\text{FeN}}^{\varepsilon} }}{RT}} \right) + X_{\text{Cr}}{ \exp }\left( {\frac{{ - U_{\text{NiN}}^{\varepsilon } - U_{\text{CrN}}^{\varepsilon } }}{RT}} \right) + X_{\text{Ni}} + X_{\text{Mn}}{ \exp }\left( {\frac{{ - U_{\text{NiN}}^{\varepsilon } - U_{\text{MnN}}^{\varepsilon } }}{RT}} \right)}} $$

(10)

$$ P_{4} = \frac{{X_{\text{Ni}} \left( {U_{\text{MnN}}^{\varepsilon } - U_{\text{CrN}}^{\varepsilon } } \right)}}{{X_{\text{Fe}}\exp \left( {\frac{{ - U_{\text{MnN}}^{\varepsilon } - U_{\text{FeN}}^{\varepsilon } }}{{RT}}} \right) + X_{\text{Cr}}\exp \left( {\frac{{ - U_{\text{MnN}}^{\varepsilon } - U_{\text{CrN}}^{\varepsilon } }}{{RT}}} \right) + X_{\text{Ni}}\exp \left( {\frac{{ - U_{\text{MnN}}^{\varepsilon } - U_{\text{NiN}}^{\varepsilon } }}{{RT}}} \right) + X_{\text{Mn}}}} $$

(11)

$$ P_{5} = \frac{{X_{\text{Fe }}\left( {U_{\text{FeN}}^{\gamma } - U_{\text{CrN}}^{\gamma } } \right)}}{{X_{\text{Fe}} + X_{\text{Cr}}\exp \left( {\frac{{ - U_{\text{FeN}}^{\gamma } - U_{\text{CrN}}^{\gamma } }}{{RT}}} \right) + X_{\text{Ni}}\exp \left( {\frac{{ - U_{\text{FeN}}^{\gamma } - U_{\text{NiN}}^{\gamma } }}{{RT}}} \right) + X_{\text{Mn}}\exp \left( {\frac{{ - U_{\text{FeN}}^{\gamma } - U_{\text{MnN}}^{\gamma } }}{{RT}}} \right)}} $$

(12)

$$ P_{6} = \frac{{X_{\text{Ni}} \left( {U_{\text{NiN}}^{\gamma } - U_{\text{CrN}}^{\gamma } } \right)}}{{X_{\text{Fe}}\exp \left( {\frac{{ - U_{\text{NiN}}^{\gamma } - U_{\text{FeN}}^{\gamma } }}{{RT}}} \right) + X_{\text{Cr}}\exp \left( {\frac{{ - U_{\text{NiN}}^{\gamma } - U_{\text{CrN}}^{\gamma } }}{{RT}}} \right) + X_{\text{Ni}} + X_{\text{Mn}}\exp \left( {\frac{{ - U_{\text{NiN}}^{\gamma } - U_{\text{MnN}}^{\gamma } }}{{RT}}} \right)}} $$

(13)

$$ P_{7} = \frac{{X_{\text{Mn}}\left( {U_{\text{MnN}}^{\gamma } - U_{\text{CrN}}^{\gamma } } \right)}}{{X_{\text{Fe}}\exp \left( {\frac{{ - U_{\text{MnN}}^{\gamma } - U_{\text{FeN}}^{\gamma } }}{{RT}}} \right) + X_{\text{Cr}}\exp \left( {\frac{{ - U_{\text{MnN}}^{\gamma } - U_{\text{CrN}}^{\gamma } }}{{RT}}} \right) + X_{\text{Ni}}\exp \left( {\frac{{ - U_{\text{MnN}}^{\gamma } - U_{\text{NiN}}^{\gamma } }}{{RT}}} \right) + X_{\text{Mn}}}} $$

(14)

The values for $U_{iN}^{\gamma } - U_{jN}^{\gamma }$ are included in Table I while the values of $U_{iN}^{\varepsilon } - U_{jN}^{\varepsilon }$ may be estimated using:

$$ \left( {U_{iN}^{\varepsilon } - U_{jN}^{\varepsilon } } \right) = \Delta G^{\gamma \to \varepsilon }_{i} + \left( {U_{iN}^{\gamma } - U_{jN}^{\gamma } } \right) $$

(15)

Segregated Nitrogen

Nitrogen, in addition to the bulk contribution to the driving force for γ to ε transformation, can also segregate to stacking fault interfaces. This elevated nitrogen content at the structural defect has a coupled influence on the overall transformation likelihood and must be addressed in addition to the bulk effect. This influence was considered in detail by Ishida86 and Yakubtsov89 and follows the form:

$$ \Delta G^{\gamma \to \varepsilon }_{\text{N,seg}} = \Delta G^{\gamma \to \varepsilon }_{{{\text{N,Chem}} - {\text{seg}}}} + \Delta G^{\gamma \to \varepsilon }_{{{\text{N,Surf}} - {\text{seg}}}} + \Delta G^{\gamma \to \varepsilon }_{{{\text{N,El}} - {\text{seg}}}} $$

(16)

where $\Delta G^{\gamma \to \varepsilon }_{{{\text{N,Chem}} - {\text{seg}}}}$ is the chemical energetic contribution chemical energy due to Suzuki segregation,$\Delta G^{\gamma \to \varepsilon }_{{{\text{N,Surf}} - {\text{seg}}}}$ is the surface energy contribution of N at the stacking fault boundary, and $\Delta G^{\gamma \to \varepsilon }_{{{\text{N,El}} - {\text{seg}}}}$ is the elastic energetic component due to segregation of elements of different size. The elastic component is generally considered to be negligible compared with the chemical components of segregated N at the stacking fault. The chemical contribution may be estimated as follows:

$$ \Delta G^{\gamma \to \varepsilon }_{{N,{\text{Chem}} - {\text{seg}}}} = RT\left[ {\ln \left( {\frac{{X_{{{\text{N}}\left( {\text{seg}} \right)}} }}{{X_{{{\text{N}}\left( {\text{bulk}} \right)}} }}} \right) + \left( {1 - X_{{N\left( {\text{bulk}} \right)}} } \right)\ln \left( {\frac{{1 - X_{{N\left( {\text{seg}} \right)}} }}{{1 - X_{{{\text{N}}\left( {\text{bulk}} \right)}} }}} \right)} \right] $$

(17)

where $X_{{N\left( {seg} \right)}}$ may be estimated by:

$$ X_{{{\text{N}}\left( {\text{seg}} \right)}} = \left[ {1 + \frac{{\left( {1 - X_{{{\text{N}}\left( {\text{bulk}} \right)}} } \right)}}{{X_{{{\text{N}}\left( {\text{bulk}} \right)}} }}\exp \left( {\frac{ - \Lambda }{{RT}}} \right)} \right] $$

(18)

$\Lambda$ is the interaction energy of nitrogen with a dislocation in the FCC structure, estimated by

$$ \Lambda = 11848 + 824X_{\text{N}} $$

(19)

The contribution to the total transformation driving force due to the surface energy contribution of N at the stacking fault may be estimated using the methodology of Ericsson:93

$$ \Delta G^{\gamma \to \varepsilon }_{{{\text{N}},{\text{Surf}} - {\text{seg}}}} = \frac{1}{4}\Lambda \left( {X_{{{\text{N}}\left( {\text{seg}} \right)}} - X_{{{\text{N}}\left( {\text{bulk}} \right)}} } \right)^{2} $$

(20)

Magnetic Potential

The excess driving force for the γ-austenite to ε-martensite transformation due to magnetism of the respective phases ($\Delta G^{\gamma \to \varepsilon }_{{{\text{Mag}} .}}$) was calculated using the methodology by Inden,88 as modified by Hillert and Jarl,87 following the observations of Saeed-Akbari et al.28 and Curtze.30 The contribution to the change in free energy due to the paramagnetic to antiferromagnetic transition follows:

$$ \Delta G^{\gamma \to \varepsilon }_{{{\text{Mag}} .}} = G^{\varepsilon }_{{{\text{Mag}} .}} - G^{\gamma }_{{{\text{Mag}} .}} $$

(21)

where the magnetic free energy of each phase (i.e., $G^{\varepsilon }_{{{\text{Mag}} .}}$ and $G^{\gamma }_{{{\text{Mag}} .}}$) may be calculated using:

$$ G^{\varphi }_{{{\text{Mag}} .}} = {{RT}}\ln \left( {1 + \frac{{\beta^{\varphi } }}{{\mu_{\text{B}} }}} \right)f\left( {\tau^{\varphi } } \right) $$

(22)

where R is the ideal gas constant, $\beta^{\varphi }$ is the magnetic moment scaled by the Bohr magnetron ($\mu_{\text{B}}$), and $\tau^{\varphi }$ is the temperature of interest (T) divided by paramagnetic transition or Neél temperature $\left( T_{{\text{Ne}}\acute{{\text{e}}}{\text{l}}}^{\varphi } \right)$, for each phase ($\varphi )$, respectively. The scaled magnetic moment of each phase is calculated by taking the sum of the pure element’s magnetic moments (Table II) weighed by the atomic fraction of each element following Eqs. 23 and 24 for the martensite and austenite phases, respectively.

$$ \frac{{\beta^{\gamma } }}{{\mu_{\text{B}} }} = X_{\text{Fe}} \beta_{\text{Fe}}^{\gamma } + X_{\text{Ni}} \beta_{\text{Ni}}^{\gamma } + X_{\text{Mn}} \beta_{\text{Mn}}^{\gamma } + X_{\text{Cr}} \beta_{\text{Cr}}^{\gamma } + X_{\text{Fe}} \left( {X_{\text{Mn}} + X_{\text{Ni}} } \right)\beta_{\text{Fe,Mn/Ni}}^{\gamma } + X_{\text{C}} \beta_{\text{C}}^{\gamma } $$

(23)

$$ \frac{{\beta^{\varepsilon } }}{{\mu_{\text{B}} }} = X_{\text{Mn}} \beta_{\text{Mn}}^{\varepsilon } + X_{\text{C}} \beta_{\text{C}}^{\varepsilon } $$

(24)

Table II Scaled elemental magnetic moments

Only Mn and C were considered when estimating the scaled magnetic moment in the ε-martensite phase because of the lack of data on the dependence of the magnetic character of the HCP phase. The Neél temperature of the martensite and the austenite phases were calculated following Eqs. 25 and 26, respectively.

$$ T_{{\text{Ne}}\acute{{\text{e}}}{\text{l}}}^{\gamma } = 251.71 + 681X_{\text{Mn}} - 272X_{\text{Cr}} - 1800X_{\text{Ni}} - 1151X_{\text{Al}} - 1575X_{\text{Si}} - 1740X_{\text{C}} $$

(25)

$$ T_{{\text{Ne}}\acute{{\text{e}}}{\text{l}}}^{\varepsilon } = 580X_{\text{Mn}} $$

(26)

Only the Mn and C fractions were considered when estimating the scaled magnetic moment, and only Mn was included in the estimation of the Neél temperature for the ε-martensite phase because of the lack of data on the dependence of the magnetic characteristics of the HCP phase. Finally, the polynomial function $f\left( {\tau^{\varphi } } \right)$ was calculated using Eq. 27 for $\tau^{\varphi } \le 1$

$$ f(\tau^{\varphi } ) = 1\frac{{\left\{ {\frac{{79\tau^{{\varphi^{ - 1} }} }}{140p} + \frac{474}{497}\left[ {\frac{1}{p} - 1} \right]\left[ {\frac{{\tau^{{\varphi^{3} }} }}{6} + \frac{{\tau^{\varphi 9} }}{135} + \frac{{\tau^{\varphi 15} }}{600}} \right]} \right\}}}{D} $$

(27)

and Eq. 28 for $\tau^{\varphi } > 1$.

$$ f\left( {\tau^{\varphi } } \right) = - \frac{{\left[ {\frac{{\tau^{ - 5} }}{10} + \frac{{\tau^{ - 15} }}{315} + \frac{{\tau^{1 - 25} }}{1500}} \right]}}{D} $$

(28)

In Eqs. 27 and 28, p = 0.28 and D = 2.342457. For negative Neél temperatures the value of $\tau^{\varphi }$ was assumed to be 300 to avoid unrealistically large contributions to the calculated stacking fault energy because of uncertainties in the estimation of the Neél temperature. Increasing this arbitrarily scaled Neél temperature did not significantly affect the resulting magnetic contribution to the calculated stacking fault energy.

As can be seen in Eqs. 23 and 25, the composition dependence of the scaled magnetic moment and the Néel temperature for the major alloy additions of interest for stainless steels are available from the literature for the austenite phase. However, very limited data exist for these parameters for the ε-martensitic phase because of the difficulty in making experimental measurements on the HCP phase in the composition space of interest; Eqs. 24 and 26 therefore only include deviations in the respective calculated properties due to Mn and C additions.