Effect of strain on generalized stacking fault energies and plastic deformation modes in fcc-hcp polymorphic high-entropy alloys: A first-principles investigation

The generalized stacking fault energy (GSFE) is a material property that can provide invaluable insights into describing nanoscale plasticity phenomena in crystalline materials. Lattice strains have been suggested to influence such phenomena. Here, the GSFE curves for sequential faulting pathways in dual phase [face-centered cubic (fcc) and hexagonal close-packed (hcp)] ${\mathrm{Cr}}_{20}{\mathrm{Mn}}_{20}{\mathrm{Fe}}_{20}{\mathrm{Co}}_{20}{\mathrm{Ni}}_{20}, {\mathrm{Cr}}_{25}{\mathrm{Fe}}_{25}{\mathrm{Co}}_{25}{\mathrm{Ni}}_{25}, {\mathrm{Cr}}_{20}{\mathrm{Mn}}_{20}{\mathrm{Fe}}_{34}{\mathrm{Co}}_{20}{\mathrm{Ni}}_6, {\mathrm{Cr}}_{20}{\mathrm{Mn}}_{20}{\mathrm{Fe}}_{30}{\mathrm{Co}}_{20}{\mathrm{Ni}}_{10}$, and ${\mathrm{Cr}}_{10}{\mathrm{Mn}}_{30}{\mathrm{Fe}}_{50}{\mathrm{Co}}_{10}$ high-entropy alloys are investigated on ${\left\{111\right\}}_{\text{fcc}}$ and ${\left(0002\right)}_{\text{hcp}}$ close-packed planes using density-functional calculations. The dependence of GSFEs on imposed volumetric and longitudinal lattice strains is studied in detail for ${\mathrm{Cr}}_{20}{\mathrm{Mn}}_{20}{\mathrm{Fe}}_{20}{\mathrm{Co}}_{20}{\mathrm{Ni}}_{20}$ and ${\mathrm{Cr}}_{10}{\mathrm{Mn}}_{30}{\mathrm{Fe}}_{50}{\mathrm{Co}}_{10}$. The competition between various plastic deformation modes is discussed for both phases based on effective energy barriers determined from the calculated GSFEs and compared with experimentally observed deformation mechanisms. The intrinsic stacking fault energy, unstable stacking fault energy, and unstable twinning fault energy are found to be closely related in how they are affected by applied strain. The ratio of two of these planar fault energies can thus serve as characteristic material property. An inverse relationship between the intrinsic stacking fault energy in the hcp phase and the axial ratio ${(c/a)}_{\text{hcp}}$ is revealed and explained via band theory.

Figure 1

(a) Schematics of close-packed planes in the fcc and hcp structures with three possible atomic positions A, B, or C, selected crystallographic directions, partial Burgers vectors, and definition of angle $\theta$. Panels (b) and (c) show schematics of extended planar defects imposed by alias shear and modeled by tilted supercells (guided by solid lines). The viewing directions in (b) and (c) are ${\left[1\overline{1}0\right]}_{\text{fcc}}$ and ${\left[11\overline{2}0\right]}_{\text{hcp}}$, respectively. The stacking sequences of the fault-free and faulted lattices are indicated and refer to one specific ${\left(1\overline{1}0\right)}_{\text{fcc}}$ or ${\left(11\overline{2}0\right)}_{\text{hcp}}$ plane, wherein labels with a dot ($\dot{\text{A}}$ etc.) in (b) and (c) indicate a nearest neighbor hcp stacking and a nearest neighbor fcc stacking, respectively. ${\mathit{b}}_{\text{p2}}^{\text{hcp}}$ tilts the cell out of the figure plane and shifts atoms from an adjacent ${\left(11\overline{2}0\right)}_{\text{hcp}}$ plane (indicated by a dashed circle) to the original plane.

Figure 2

Schematic illustration of the sequential faulting pathways. From left to right: no applied displacements, with relative displacements ${\mathit{u}}_i, {\mathit{u}}_i^{\alpha }$, and ${\mathit{u}}_i^{\beta }$. See the text for definitions and details.

Figure 3

The GSFEs $\gamma \left(\eta \right)$ for the faulting pathways in the fcc and hcp phases and the five considered HEAs. The GSFE curves for the pathways $\alpha$ and $\beta$ in the hcp phases after branching at $\eta =1$ are distinguished by dashed and solid lines, respectively. Special stationary points are indicated.

Figure 4

The GSFE of stationary points on the faulting pathways as a function of applied strain for three states of superimposed strain in fcc and hcp ${\mathrm{Cr}}_{20}{\mathrm{Mn}}_{20}{\mathrm{Fe}}_{20}{\mathrm{Co}}_{20}{\mathrm{Ni}}_{20}$.

Figure 5

The GSFE of stationary points on the faulting pathways as a function of applied strain for three states of superimposed strain in fcc and hcp ${\mathrm{Cr}}_{10}{\mathrm{Mn}}_{30}{\mathrm{Fe}}_{50}{\mathrm{Co}}_{10}$.

Figure 6

The EEBs of ${\mathrm{Cr}}_{25}{\mathrm{Fe}}_{25}{\mathrm{Co}}_{25}{\mathrm{Ni}}_{25}$ and ${\mathrm{Cr}}_{10}{\mathrm{Mn}}_{30}{\mathrm{Fe}}_{50}{\mathrm{Co}}_{10}$ in (a) the fcc phase and (b) the hcp phase determined from the GSFEs without superimposed strain. Solid and dashed lines in (b) refer to ${\overline{\gamma }}^{\text{I}}$ and ${\overline{\gamma }}^{\text{II}}$, respectively. The horizontal bar at the bottom of each panel indicates which deformation mode is activated as a function of $\theta$ (the total width of a mode normalized by ${60}^{\circ }$ equals its deformation mode fraction).

Figure 7

Cumulated deformation mode fractions in (a) the fcc phase and (b) the hcp phase of ${\mathrm{Cr}}_{20}{\mathrm{Mn}}_{20}{\mathrm{Fe}}_{20}{\mathrm{Co}}_{20}{\mathrm{Ni}}_{20}$ and ${\mathrm{Cr}}_{10}{\mathrm{Mn}}_{30}{\mathrm{Fe}}_{50}{\mathrm{Co}}_{10}$ as a function of strain for three states of superimposed strain. Note that not every deformation mode is activated.

Figure 8

(a) Scaling plot of stable and unstable fault energies of the GSFE for ${\mathrm{Cr}}_{20}{\mathrm{Mn}}_{20}{\mathrm{Fe}}_{20}{\mathrm{Co}}_{20}{\mathrm{Ni}}_{20}$ and ${\mathrm{Cr}}_{10}{\mathrm{Mn}}_{30}{\mathrm{Fe}}_{50}{\mathrm{Co}}_{10}$ in their fcc and hcp phases at equilibrium and under strain. The dashed line indicates the universal scaling rule [Eq. (8)] proposed for the absence of superimposed strain. (b) Analogous to (a) but for the zero-strain derivatives, where ${\gamma }^{\prime }$ is short for ${d\gamma /d\epsilon |}_0$. The dashed line indicates the universal scaling rule derived from Eq. (9). The data point marked by $*$ was scaled by a factor of 0.5 for displaying purposes.

Figure 9

Inverse relationship between ${\gamma }_{\text{isf}}^{\text{hcp}}$ and ${(c/a)}_{\text{hcp}}$ ratio in the hcp phase. The inset shows both quantities plotted as a function of the average number of $3d$ electrons. The line guides the eye.