Elastic anharmonicity of bcc Fe and Fe-based random alloys from first-principles calculations

We systematically investigate elastic anharmonic behavior in ferromagnetic body-centered cubic (bcc) Fe and ${\mathrm{Fe}}_{1-x}{M}_x$ ($M=\mathrm{Al}$, V, Cr, Co, or Ni) random alloys by means of density-functional simulations. To benchmark computational accuracy, three ab initio codes are used to obtain the complete set of second- and third-order elastic constants (TOECs) for bcc Fe. The TOECs of ${\mathrm{Fe}}_{1-x}{M}_x$ alloys are studied employing the first-principles alloy theory formulated within the exact muffin-tin orbital method in combination with the coherent-potential approximation. It is found that the alloying effects on $C_{111},C_{112}$, and $C_{123}$, which are governed by normal strains only, are more pronounced than those on $C_{144},C_{166}$, and $C_{456}$, which involve shear strains. Remarkably, the magnitudes of all TOECs but $C_{123}$ decrease upon alloying with Al, V, Cr, Co, or Ni. Using the computed TOECs, we study compositional effects on the pressure derivatives of the effective elastic constants ($d{B}_{ij}/dP$), bulk ($dK/dP$), and shear moduli ($dG/dP$) and derive longitudinal acoustic nonlinearity parameters ($\beta$). Our predictions show that the pressure derivatives of $K$ and $G$ decrease with $x$ for all solute elements and reveal a strong correlation between the compositional trends on $dK/dP$ and $dG/dP$ arising from the fact that alloying predominantly alters $d{B}_{11}/dP$. The sensitivity of $d{B}_{11}/dP$ to composition is attributed to intrinsic alloying effects as opposed to lattice parameter changes accompanying solute addition. For Fe and the considered Fe-based alloys, $\beta$ along high-symmetry directions orders as $\beta \left[111\right]>\beta \left[100\right]>\beta \left[110\right]$, and alloying increases the directional anisotropy of $\beta$ but reduces its magnitude.

Figure 1

The three independent SOECs of bcc ferromagnetic Fe computed from non-volume-conserving Lagrangian strains. The present results (EMTO, FPLO, and vasp) are compared with available experimental data from Ref. [56].

Figure 2

The six independent TOECs of bcc ferromagnetic Fe. The present results (EMTO, FPLO, and vasp) are compared with available experimental data from Ref. [56].

Figure 3

The three independent SOECs and six independent TOECs of ${\mathrm{Fe}}_{1-x}{M}_x$ alloys as a function of solute concentration $x$. All elastic constants are in units of ${10}^3\mathrm{GPa}$.

Figure 4

(a) Correlation between the pressure derivatives of the bulk modulus and the shear modulus for ${\mathrm{Fe}}_{1-x}{M}_x$ alloys ($x$ denotes the concentration of solute $M,x\le 0.1$). The dashed line is a linear fit to the data (for details see the text). The insets display the correlation between the pressure derivatives of $B_{11}$ and the shear and bulk moduli. (b) The total effect on $d{B}_{11}/dP$ evaluated by means of a finite concentration change $\mathrm{\Delta }x=0.05$ with pure Fe as a reference and the decomposition into the lattice parameter effect and the intrinsic alloying effect according to Eq. (17).

Figure 5

The longitudinal acoustic nonlinearity parameter $\beta$ as a function of wave propagation direction for (a) bcc Fe and for (b) a ${\mathrm{Fe}}_{0.9}{\mathrm{Ni}}_{0.1}$ alloy. The Cartesian axes specify the projection of $\beta$ onto the [100], [010], and [001] crystallographic axes.