Monte Carlo study of thermodynamic properties and clustering in the bcc Fe-Cr system

Abstract

Iron-chromium alloys are characterized by a complex phase diagram, by the small negative enthalpy of mixing at low Cr concentrations in the bcc $\ensuremath{\alpha}$-phase of Fe, and by the inversion of the short-range order parameter. We present Monte Carlo simulations of the binary Fe-Cr alloy based on the cluster expansion approximation for the enthalpy of the system. The set of cluster expansion coefficients is validated against density functional calculations of energies of small clusters of chromium in bcc structure. We show that in the limit of small Cr concentration the enthalpy of mixing remains negative up to fairly high temperatures, and individual Cr atoms remain well separated from each other. Clustering of Cr atoms begins at concentrations exceeding approximately 10% at $800\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and 20% at $1400\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, with Cr-Fe interfaces being parallel to the [110] planes. Calculations show that the first and the second short-range order parameters change sign at approximately 10.5% Cr, in agreement with experimental observations. Semi-grand-canonical ensemble simulations used together with experimental data on vibrational entropy of mixing give an estimate for the temperature of the top of the $\ensuremath{\alpha}\text{\ensuremath{-}}{\ensuremath{\alpha}}^{\ensuremath{'}}$ miscibility gap. We find that the complex ordering reactions occurring in Fe-Cr, as well as the thermodynamic properties of the alloy, can be reasonably well described using a few concentration-independent cluster expansion coefficients.