Ab Initio Investigation of Planar Defects in Immm-Ni₂(Cr,Mo,W) Strengthened HAYNES 244 Alloy

Abstract

The Immm-Ni₂(Cr,Mo,W) body centered orthorhombic (BCO) intermetallic phase, denoted as γ′″, is the primary strengthening phase in commercial Ni-base HAYNES® 242® and 244® alloys. Due to the relatively low symmetry and six crystallographic variants precipitating in the face-centered cubic solid solution γ matrix, the deformation mechanisms are expected to be complex, and a myriad of planar defects are predicted to be observed on the close-packed {013} and {110} planes, which are nearly co-planar to the matrix {111} plane. We find that these defects include those analogous to the γ′-Ni₃Al phase: superlattice intrinsic stacking faults, antiphase boundaries, and complex stacking faults. We determined these planar defect energies and generalized stacking fault energy surfaces utilizing ab initio density functional theory calculations. The γ′″ {013} and {110} planes exhibited comparable planar defect energies but showed drastically different dislocation shear pathways due to the lower symmetry of the orthorhombic phase. We observed that the addition of W increased the fault energies significantly, which could correspond to the observed increase in yield strength of 244 alloy over that of 242 alloy.

Appendices

Appendix A

See Table AI.

Table AI Nearest Neighbor Bonds of the {013}_γ′″ and {110}_γ′″ Planes Compared to the {111}_γ´ and {112}_γ″ Planes

Appendix B

The equilibrium separation distance of partial dislocations can be calculated by a force balance between the Peach-Koehler forces of the dislocations, and the force from the energy of the planar defect bound by the partials. The following equation was used to determine these distances.

$$\sum F=0=F\left({\gamma }_{defect}\right)-F\left(K, {d}_{i}\right)$$

$F\left({\gamma }_{defect}\right)$ is the attractive force that minimizes the faulted region, and $F\left(K, {d}_{i}\right)$ is the repulsive force between the partial dislocations. K is $\frac{\mu {b}^{2}}{2\pi }$ where μ is the shear modulus of the system and b is the burgers vector of the partial dislocations. The magnitude of the partial dislocation burgers vector depends on the b lattice parameter shown in Table II. The shear modulus for each system and each plane is extracted from the elastic tensor determined using the VASP software for the super cell. The separations distances are shown in Table BI. The coefficients used to analytically represent the slip pathway we calculated and are shown in Table BII. The disregistry function and misfit energy curves for the Ni₂Mo and Ni₂W system are shown in Figures B1 and B2. These systems are like the Ni₂Cr, but the different energies of the various planar defects and different shear moduli affect the equilibrium separation in the disregistry function and the curvature of the misfit energy. The equilibrium separation distance is determined by solving the following systems of equations for each plane and direction (see Table BIII).

{013} Short: ${\gamma }_{isf}=K\left(\frac{1}{{d}_{1}}\right)$

{013} Full:

$${\gamma }_{isf}=K\left(\frac{1}{{d}_{1}}+\frac{1}{{d}_{1}+{d}_{2}}+\frac{1}{{d}_{1}+{d}_{2}+{d}_{3}}+\frac{1}{{d}_{1}+{d}_{2}+{d}_{3}+{d}_{4}}+\frac{1}{{d}_{1}+{d}_{2}+{d}_{3}+{d}_{4}+{d}_{5}}\right)$$

$${\gamma }_{apb}-{\gamma }_{isf}=K\left(\frac{-1}{{d}_{1}}+\frac{1}{{d}_{2}}+\frac{1}{{d}_{2}+{d}_{3}}+\frac{1}{{d}_{2}+{d}_{3}+{d}_{4}}+\frac{1}{{d}_{2}+{d}_{3}+{d}_{4}+{d}_{5}}\right)$$

$${\gamma }_{scsf}-{\gamma }_{apb}=K\left(\frac{-1}{{d}_{1}+{d}_{2}}-\frac{1}{{d}_{2}}+\frac{1}{{d}_{3}}+\frac{1}{{d}_{3}+{d}_{4}}+\frac{1}{{d}_{3}+{d}_{4}+{d}_{5}}\right)$$

$${\gamma }_{apb{^{\prime}}}-{\gamma }_{scsf}=K\left(\frac{-1}{{d}_{1}+{d}_{2}+{d}_{3}}-\frac{1}{{d}_{1}+{d}_{2}}-\frac{1}{{d}_{3}}+\frac{1}{{d}_{4}}+\frac{1}{{d}_{4}+{d}_{5}}\right)$$

$${\gamma }_{csf}-{\gamma }_{apb{^{\prime}}}=K\left(\frac{-1}{{d}_{1}+{d}_{2}+{d}_{3}+{d}_{4}}-\frac{1}{{d}_{2}+{d}_{3}+{d}_{4}}-\frac{1}{{d}_{3}+{d}_{4}}-\frac{1}{{d}_{4}}+\frac{1}{{d}_{5}}\right)$$

{110} Full:

$${\gamma }_{isf}=K\left(\frac{1}{{d}_{1}}+\frac{1}{{d}_{1}+{d}_{2}}+\frac{1}{{d}_{1}+{d}_{2}+{d}_{3}}+\frac{1}{{d}_{1}+{2d}_{2}+{d}_{3}}+\frac{1}{{d}_{1}+2{d}_{2}+2{d}_{3}}\right)$$

$${\gamma }_{apb}-{\gamma }_{isf}=K\left(\frac{-1}{{d}_{1}}+\frac{1}{{d}_{2}}+\frac{1}{{d}_{2}+{d}_{3}}+\frac{1}{{2d}_{2}+{d}_{3}}+\frac{1}{2{d}_{2}+2{d}_{3}}\right)$$

$${\gamma }_{scsf}-{\gamma }_{apb}=K\left(\frac{-1}{{d}_{1}+{d}_{2}}-\frac{1}{{d}_{2}}+\frac{1}{{d}_{3}}+\frac{1}{{d}_{2}+{d}_{3}}+\frac{1}{{d}_{2}+2{d}_{3}}\right)$$

Table BI Equilibrium Separation Distance Between Partial Dislocations for Each System and Slip Pathway

Table BII Fourier Coefficients for the Various Slip Pathways and Planes for Use in Eq. 6

Table BIII Burgers Vector Magnitudes and Shear Moduli for Use in the Coefficient K

Fig. B1

The disregistry functions and misfit energy curves for the Ni₂Cr system. (a) Shows the disregistry function for the easy slip pathway of the {013}_γ′″ plane and the misfit energy is shown in (b). The equilibrium separation distance of the partial dislocations is also shown. (c) And (e) show the disregistry function for the whole pathway along the {013} _γ′″ and {110} _γ′″ planes respectively, and (d) and (f) show the misfit energy curves used to determine the Peierls Stress

Fig. B2

The disregistry functions and misfit energy curves for the Ni₂Cr system. (a) Shows the disregistry function for the easy slip pathway of the {013}_γ′″ plane and the misfit energy is shown in (b). The equilibrium separation distance of the partial dislocations is also shown. (c) And (e) show the disregistry function for the whole pathway along the {013} _γ′″ and {110} _γ′″ planes respectively, and (d) and (f) show the misfit energy curves used to determine the Peierls Stress