Coexistence pressure for a martensitic transformation from theory and experiment: Revisiting the bcc-hcp transition of iron under pressure

The coexistence pressure of two phases is a well-defined point at fixed temperature. In experiment, however, due to nonhydrostatic stresses and a stress-dependent potential energy barrier, different measurements yield different ranges of pressure with a hysteresis. Accounting for these effects, we propose an inequality for comparison of the theoretical value to a plurality of measured intervals. We revisit decades of pressure experiments on the $\text{bcc}\leftrightarrow \text{hcp}$ transformations in iron, which are sensitive to nonhydrostatic conditions and sample size. From electronic-structure calculations, we find a $\text{bcc}\leftrightarrow \text{hcp}$ coexistence pressure of $8.4\mathrm{GPa}$. We construct the equation of state for competing phases under hydrostatic pressure, compare to experiments and other calculations, and address the observed pressure hysteresis and range of onset pressures of the nucleating phase.

Figure 1

Van der Waals isotherm (solid black), along with the unstable region (dashed yellow) and metastable overheated liquid and overcooled gas (blue). Delayed start of condensation and end of evaporation (dashed green arrows). Liquid and gas are indistinguishable at the critical isotherm (red) above the phase-segregated region (dashed black line).

Figure 2

(a) Energy (meV/atom) relative to bcc at 0 GPa and (b) pressure (GPa) versus volume (Å${}^3$/cell) for hydrostatically relaxed 2-atom unit cells of bcc FM (black), hcp NM (blue), and AFM (green), and fcc NM iron (orange); with DFT values (dots) and least-squares fit to the Birch-Murnaghan EOS (lines). Common tangent construction (red line) yields $P_0=8.42\mathrm{GPa}$. Vertical dotted lines are guides to the eye. (c) Enthalpy difference (meV/Fe) between hcp and bcc phases versus pressure (GPa) (inset shows a larger range). (d) Comparison of the calculated $P_0=8.42\mathrm{GPa}$ (vertical red line) with experimental data from Table 1, represented by ($P_{end}^{\varepsilon \to \alpha }-P_{start}^{\alpha \to \varepsilon }$) horizontal segments. Except for the 1991 hydrostatic experiment [9], most diamond anvil cells [3, 6] provided uniaxial or highly anisotropic pressure.

Figure 3

Enthalpy $H$, volume $V$, magnetization $M$ (per atom), internal pressure $P$, atomic shuffle (defined as direct coordinate of the second atom in a 2-atom unit cell relative to the first atom placed at zero), and shear angle (degrees) at $P_0$ versus the GSS-NEB path from bcc to hcp.