Ab initio explanation of disorder and off-stoichiometry in Fe-Mn-Al-C $\kappa$ carbides

Carbides play a central role for the strength and ductility in many materials. Simulating the impact of these precipitates on the mechanical performance requires knowledge about their atomic configuration. In particular, the C content is often observed to substantially deviate from the ideal stoichiometric composition. In this work, we focus on Fe-Mn-Al-C steels, for which we determined the composition of the nanosized $\kappa$ carbides (Fe,Mn)${}_3\mathrm{AlC}$ by atom probe tomography in comparison to larger precipitates located in grain boundaries. Combining density functional theory with thermodynamic concepts, we first determine the critical temperatures for the presence of chemical and magnetic disorder in these carbides. Second, the experimentally observed reduction of the C content is explained as a compromise between the gain in chemical energy during partitioning and the elastic strains emerging in coherent microstructures.

Figure 1

Crystal structures of (a) ${\mathrm{Fe}}_3\mathrm{AlC}$, (b) ${\mathrm{Fe}}_2\mathrm{MnAlC}$, (c) ${\mathrm{FeMn}}_2\mathrm{AlC}$, and (d) ${\mathrm{Mn}}_3\mathrm{AlC}$. Red, golden, green, and black balls represent Al, Fe, Mn, and C atoms, respectively.

Figure 2

Microstructure of a Fe-29.8Mn-7.7Al-1.3C (wt. %) alloy aged at $600{}^{\circ }\mathrm{C}$ for 12 weeks: (a) SE image showing the grain boundary (GB) $({\kappa }_0+{\gamma }_0+\alpha )$ phases and grain interior (GI) $(\kappa +\gamma )$ phases. (b) Zoomed-in SE image at GI region highlighting the nanosized GI $\kappa$ precipitates. (c) APT analysis of GI $(\kappa +\gamma )$ phases where $\kappa$ precipitates are visualized in terms of iso-concentration surfaces at a threshold value of 9 at. %.

Figure 3

Free-energy differences for the $\kappa$ carbide formation according to Eq. (1) with varying Mn content $x$ (given by the labels). The results for $T=0$ K (dotted lines) and for the experimental annealing temperature of $600{}^{\circ }\mathrm{C}$ (solid lines) are compared. The color shading indicates the phase stability at $600{}^{\circ }\mathrm{C}$ as a function of the Mn chemical potential with respect to the reference potential described in Appendix. The chemical potentials corresponding to the composition of the experimental alloy (Fe-29.8Mn-7.7Al-1.3C in wt. %) are for both temperatures shown as a red dashed-dotted lines. For ${\mathrm{Fe}}_2\mathrm{MnAlC}$, the ground-state energy of an ordered Mn arrangement (thick green line) and of an SQS disordered structure (thin green line) are compared.

Figure 4

Free-energy difference of chemically ordered and disordered ${\mathrm{Fe}}_2\mathrm{MnAlC}$ for different magnetic phases is shown as a function of volume at $T=0$ K. The energies have been rescaled such that the ground-state configuration of ${\mathrm{Fe}}_2\mathrm{MnAlC}$ is taken as a reference.

Figure 5

Area modulus [48, 49] of (a) a cubic-symmetry approximant of ${\mathrm{Fe}}_2\mathrm{MnAlC}$ with disorder Fe-Mn sublattice in FM state, (b) ${\mathrm{Fe}}_2{\mathrm{MnAlC}}_{0.625}$, i.e., with reduced C content in FM state, (c) ${\mathrm{Fe}}_3\mathrm{AlC}$, i.e., without Mn in FM state, (d) ${\mathrm{Fe}}_3\mathrm{AlC}$ in PM state. The calculation (values in GPa) is based on the determined elastic constants $C_{11}, C_{12}$, and $C_{44}$ summarized in Table 1 (visualization by the SC-EMA software package [50, 51, 52]).

Figure 6

Volume dependence of the energy contributions to the formation energies of C, Fe, and Al vacancies in FM $\kappa$ carbide according to Eq. (8). The energies are rescaled such that the perfect ${\mathrm{Fe}}_3\mathrm{AlC}$ (filled symbols) at its equilibrium lattice constant is taken as a reference. The vertical dashed line marks the compromising volume between $\kappa$ carbide and $\gamma$ matrix, if both have a volume fraction of 50% (compare with Fig. 8).

Figure 7

Schematic picture of C partitioning between $\kappa$ carbide and $\gamma$ matrix: assuming an equal C distribution in the as-cast state (dashed line), there is a chemical driving force for C to accumulate in Al-ordered regions. Since this imposes an elastic energy penalty, the decomposition will remain incomplete.

Figure 8

Volume per unit cell of unstrained FM $\kappa$ carbide ($V_{\kappa }$) and AFM $\gamma$-Fe ($V_{\gamma }$) as well as the virtual coherent composite ($V_{\kappa +\gamma }$). The points correspond to calculations with an integer number of C atoms in the $2\times 2\times 2$ supercell, while the dashed lines are linear interpolations. For a certain number of C atoms in $\gamma$, the C concentration in $\kappa$ depends via Eq. (10) on the volume fraction $v_{\kappa }$. $V_{\kappa +\gamma }$ is a result of a minimization of Eq. (9).

Figure 9

Dependence of thermodynamic potentials on the C concentration in $\kappa$ carbide. The calculations have been performed at $T=600{}^{\circ }\mathrm{C}$ for equal volume fractions of $\kappa$ and $\gamma$. (a) The elastic part of the free energy for the $\kappa$ carbide for hydrostatic (red lines) and biaxial (blue line) strain along with chemical part (red dotted line). Either an average over different C configurations (solid line) or the selection of the low-energy C configuration (dashed lines) has been done. (b) The nonlinear (nl) contribution to the free energies for the individual phases, as explained in the text. Solid lines are fits to third-order polynomials. (c) The total Helmholtz free energies according to Eq. (9), using the same color code as in part (a). (d) The temperature-dependent chemical potential of both phases according to Eq. (A11) are renormalized by a reference potential ${\mu }_{\mathrm{C}}^0$. For more details, the reader is referred to the text.

Figure 10

Computed equilibrium C concentrations in $\kappa$ carbide and $\gamma$ matrix as a function of respective volume fractions at different temperatures including the experimental annealing temperature of $600{}^{\circ }\mathrm{C}$ (873 K).