Correlations between valence electron concentration and the phase stability, intrinsic strength, and deformation mechanism in fcc multicomponent alloys

Valence electron concentration (VEC) is a crucial parameter affecting the structure and mechanical properties of materials. Nevertheless, the influence of VEC in multicomponent alloys remains muddy. Here, the correlations between VEC and phase stability, intrinsic mechanical strength, and deformation mechanism are revealed by investigating six equiatomic $\mathrm{CoNi}M$ ($M$ = Ti, V, Cr, Mn, Fe, and Cu) alloys with VEC ranging from 7.7 to 10.0. As VEC increases, the ground-state structure evolves from bcc to hcp and ultimately to fcc. Both elastic moduli and ideal tensile strength increase initially and then decrease with increasing VEC, i.e., an inverted parabolic trend, which arises due to the orbital-filling effect of valence electrons. We highlight that a medium VEC ($\sim 8.3$) has superior intrinsic strength at a full filling of the bonding state but an empty antibonding state. Furthermore, a strong link between intrinsic strength and the energy difference between bcc and fcc ($\mathrm{\Delta }{E}_{\mathrm{fcc}\to \mathrm{bcc}}$) is built. The stacking fault (SF) energy ${\gamma }_{\mathrm{sf}}$ roughly increases with VEC except for in CoNiCr; this correlation is inherited from the energy difference between fcc and hcp. The dependence of intrinsic deformation energy barriers on VEC follows that of $\mathrm{\Delta }{E}_{\mathrm{fcc}\to \mathrm{bcc}}$ since both are linked to the bonding strength. Depending on orientation, dislocation slip (SL) and SF can co-occur in the alloys with low VEC while SL and twinning (TW) coexist in those with high VEC. Both competition between SF or TW and SL and competition between SF and TW are dictated by ${\gamma }_{\mathrm{sf}}$. This work unveils relations between VEC and phase stability, intrinsic strength, and deformation mechanisms and could provide some useful information for designing novel multicomponent alloys.

Figure 1

Distribution of the $M$ element in the studied $\mathrm{CoNi}M$ multicomponent alloys.

Figure 2

Comparison of the total energies of the studied $\mathrm{CoNi}M$ alloys with the bcc, fcc, and hcp structures plotted as a function of the Wigner-Seitz radius. (a) CoNiTi, (b) CoNiV, (c) CoNiCr, (d) CoNiMn, (e) CoNiFe, and (f) CoNiCu. For a better illustration, the total energies of various structures are normalized with the minimum energy of the fcc structure. The inset in (b) is an enlarged image of the region indicated by the solid box.

Figure 3

VEC dependences of the ground-state structure and the energy difference between different structures. (a) Ground-state structure vs VEC. (b) $\mathrm{\Delta }{E}_{\mathrm{fcc}\to \mathrm{bcc}}$ vs VEC. (c) $\mathrm{\Delta }{E}_{\mathrm{fcc}\to \mathrm{hcp}}$ vs VEC. The red dashed line in (c) is to guide the eye.

Figure 4

Evolution of elastic moduli with respect to VEC. (a) Tetragonal shear modulus $C^{\prime }$, (b) isotropic shear modulus $G$, (c) isotropic Young's modulus $E$, and (d) bulk modulus $B$. The dashed inverted parabolic curves in (a)–(d) are plotted to guide the eye.

Figure 5

Ideal tensile stress-strain curves of the $\mathrm{CoNi}M$ alloys. (a) Ideal tensile stress-strain curve along [110]. (b) Evolution of ${\sigma }_{\mathrm{m}}$ as a function of VEC. The dashed inverted parabolic curve is plotted to guide the eye. (c) Ratio between ${\sigma }_{\mathrm{m}}$ and $E_{\left[110\right]}$ with respect to VEC. One of the two sets of data for CoNiCr in (a) is extracted from Ref. [70], as are the data for the Cantor and CrFeCoNi alloys in (b).

Figure 6

Relation between intrinsic strength and $\mathrm{\Delta }{E}_{\mathrm{fcc}\to \mathrm{bcc}}$. (a) $C^{\prime }, G, E$, and $B$. (b) ${\sigma }_{\mathrm{m}}$.

Figure 7

COHP curves of the $\mathrm{CoNi}M$ alloys. (a) CoNiTi, (b) CoNiV, (c) CoNiCr, (d) CoNiMn, (e) CoNiFe, and (f) CoNiCu.

Figure 8

Schematic diagram of COHP of the $\mathrm{CoNi}M$ alloys. (a) Schematic COHP diagram. (b) COHP of the $\mathrm{CoNi}M$ alloys with highlighted $E_{\mathrm{F}}$.

Figure 9

$\left|\mathrm{ICOHP}\right|$ of $\mathrm{CoNi}M$.

Figure 10

Variation of mean atomic radius during the formation of solid solution and local magnetic moment of the $\mathrm{CoNi}M$ alloys. (a) $\mathrm{\Delta }r$ defined by $r_0-\overline{r}$, where $r_0$ is the average atomic radius of the $\mathrm{CoNi}M$ alloy and $\overline{r}$ represents the mean of the atomic radii $r_i$ of the Co, Ni, and $M$ elements. (b) Local atomic magnetic moments of Co, Ni, and $M$ in the paramagnetic $\mathrm{CoNi}M$ alloys.

Figure 11

GPFE curves and $\gamma$ surfaces of the fcc $\mathrm{CoNi}M$ alloys for $\langle 11\overline{2}\rangle$ alias shear on the {111} plane. (a) Three typical GPFE curves of the fcc alloys. (${\mathrm{b}}_1$)–(${\mathrm{b}}_6$) $\gamma$ surfaces of CoNiTi, CoNiV, CoNiCr, CoNiMn, CoNiFe, and CoNiCu. For comparison, $\gamma$ surfaces of three typical fcc alloys, i.e., Al, Cu, and Cantor alloy that possess high (142 mJ ${\mathrm{m}}^{-2}$) [104], medium (43 mJ ${\mathrm{m}}^{-2}$) [104], and low ($-18$ mJ ${\mathrm{m}}^{-2}$) [105] ${\gamma }_{\mathrm{sf}}$, respectively, are included.

Figure 12

Intrinsic stacking fault energy ${\gamma }_{\mathrm{sf}}$. (a) Comparison of ${\gamma }_{\mathrm{sf}}$ determined by the AIM with those extracted from the ${\gamma }_{\mathrm{sf}}$ surface. (${\mathrm{b}}_1$) and (${\mathrm{b}}_2$) Relations between ${\gamma }_{\mathrm{sf}}$ and $\mathrm{\Delta }{E}_{\mathrm{fcc}\to \mathrm{hcp}}$ (${\mathrm{b}}_1$) and between ${\gamma }_{\mathrm{sf}}$ and the equilibrium lattice constant $a_0$ (${\mathrm{b}}_2$). (c) Dependence of ${\gamma }_{\mathrm{sf}}$ on VEC.

Figure 13

Intrinsic energy barriers of dislocation slip ${\overline{\gamma }}_{\mathrm{SL}}$, deformation twinning ${\overline{\gamma }}_{\mathrm{SL}}$, and stacking fault ${\overline{\gamma }}_{\mathrm{SF}}$. (${\mathrm{a}}_1$)–(${\mathrm{a}}_3$) Dependence of ${\overline{\gamma }}_{\mathrm{SL}}, {\overline{\gamma }}_{\mathrm{TW}}$, and ${\overline{\gamma }}_{\mathrm{SF}}$ on VEC. (${\mathrm{b}}_1$)–(${\mathrm{b}}_3$) Variation of ${\overline{\gamma }}_{\mathrm{SL}}, {\overline{\gamma }}_{\mathrm{TW}}$, and ${\overline{\gamma }}_{\mathrm{SF}}$ as a function of the energy difference between the bcc and fcc structures $\mathrm{\Delta }{E}_{\mathrm{fcc}\to \mathrm{bcc}}$. (c) Sketch of atom positions of layers A, B, and C during the formation of an intrinsic SF.

Figure 14

Relative orientation of the Burgers vectors of partial dislocations for SF, the $\mathrm{fcc}\to \mathrm{hcp}$ transition, TW, and SL on (111) of the fcc structure. $\mathit{\tau }$ indicates the resolved shear stress on (111). $\theta$ measures the deviation between $\mathit{\tau }$ and the Burgers vector of the leading partial dislocation, which ranges from $0^{\circ }$ to ${60}^{\circ }$ according to the symmetry of the (111) planes of the fcc structure.

Figure 15

Deformation mode maps. (a) Projected deformation mode map on (111). (b) Normalized deformation mode map scaled by ${\gamma }_{\mathrm{usf}}-{\gamma }_{\mathrm{sf}}$. The data for ${\mathrm{Fe}}_{40}{\mathrm{Mn}}_{40}{\mathrm{Co}}_{10}{\mathrm{Cr}}_{10}$ (pink-outlined white symbols) is extracted from Ref. [105], and those for Cu (orange-outlined white symbols), and Al (brown-outlined white symbols) are from Ref. [104].

Figure 16

Critical values of $\theta$ (denoted ${\theta }_{\mathrm{c}}$) for the transition of the deformation mechanism from SF (or TW) to SL: (a) ${\theta }_{\mathrm{c}}$ vs VEC, (b) ${\theta }_{\mathrm{c}}$ vs ${\gamma }_{\mathrm{sf}}$, and (c) ${\theta }_{\mathrm{c}}$ vs $\mathrm{\Delta }{E}_{\mathrm{fcc}\to \mathrm{hcp}}$.

Figure 17

Competition between SF and TW. (a) VEC dependence of the difference between ${\overline{\gamma }}_{\mathrm{SF}}$ and ${\overline{\gamma }}_{\mathrm{TW}}$. (b) Correlation between ${\overline{\gamma }}_{\mathrm{SF}}-{\overline{\gamma }}_{\mathrm{TW}}$ and $\mathrm{\Delta }{E}_{\mathrm{fcc}\to \mathrm{hcp}}$. A negative value of ${\overline{\gamma }}_{\mathrm{SF}}-{\overline{\gamma }}_{\mathrm{TW}}$ implies a preference for SF, while a positive one indicates a preference for TW.