Temperature hysteresis of the order-disorder transition in carbon-supersaturated α-Fe

Although many works have been devoted to the order-disorder transition in carbon-supersaturated α-Fe, all seem to have overlooked the temperature hysteresis phenomenon occurring around the critical temperature. It is shown, from a mean-field model based on the elasticity theory of point defects, that the origin of the temperature hysteresis is thermodynamic. As a consequence, both the critical temperature and carbon concentration for the order-disorder transition can be defined upon heating and cooling. The results obtained were successfully compared to molecular dynamics simulations, and are evidence that the transition is of first order and that linear elasticity is the predominant source of the thermodynamics of the Fe-C solid solutions.

Figure 1

Ordering curve: order parameter as a function of the reduced temperature. For each temperature $T<T_{+}$, stable, metastable, and unstable equilibria exist. Upon fast heating (cooling) the system will follow the downward (upward) arrow.

Figure 2

Top: Gibbs energies of ordering (in meV/Fe atom) as a function of the order parameter for $X=0.0147$ at different temperature ranges (enlargements). Bottom: corresponding probability densities for $n_{\mathrm{C}}=250$ carbon atoms (line).

Figure 3

Critical temperatures $T_c$ (black), $T_{-}$ (blue), and $T_{+}$ (red) as functions of carbon atomic fraction $x_{\mathrm{C}}$. Upon fast heating (cooling) the order-disorder transition starts at temperature $T_{+}$ ($T_{-}$).

Figure 4

Ordering curve. Molecular dynamics upon heating (red dots) and cooling (blue dots) compared to mean-field model (lines).

Figure 5

Order parameters ${\eta }_c$ (black), ${\eta }_{-}$ (blue), and ${\eta }_{+}$ (red) as a function of carbon concentration. Beyond $u_{\mathrm{C}}=1.5$, ${\eta }_c$ drops to zero and the order-disorder transition becomes of second order.

Figure 6

Stable ordering curve. Mean-field calculations: without dispersion (lines) and with dispersion corresponding to $n_{\mathrm{C}}=250$ carbon atoms (green points).