Metastable phase transformation and hcp-ω transformation pathways in Ti and Zr under high hydrostatic pressures

The energy landscape of Zr at high hydrostatic pressure suggests that its transformation behavior is strongly pressure dependent. This is in contrast to the known transition mechanism in Ti, which is essentially independent of hydrostatic pressure. Generalized solid-state nudged elastic band calculations at constant pressure shows that α-Zr transforms like Ti only at the lowest pressure inside the stability field of ω-phase. Different pathways apply at higher pressures where the energy landscape contains several high barriers so that metastable states are expected, including the appearance of a transient bcc phase at ca. 23 GPa. The global driving force for the hcp-ω transition increases strongly with increasing pressure and reaches 23.7 meV/atom at 23 GPa. Much of this energy relates to the excess volume of the hcp phase compared with its ω phase.

Group IV transition metals Ti and Zr play a major role in aerospace and nuclear industries.][14][15][16][17][18][19][20][21][22][23] From a more fundamental perspective, the a-x phase transition is a prototype for high pressure structural transformation processes in simple crystal structures, their kinetics, and the influence of impurities on the phase boundary among other aspects of the phase relationship.The transition mechanism in Zr is considered as a model for the transitions in Pu, which is much harder to investigate.5][26] One key observation from such studies is that pressure transformations may not occur instantaneously but follow time dependent transition mechanisms, which may involve metastable intermediate structural states.Such behavior was indeed found in the studies of the kinetics of the a-x phase transition in zirconium. 9,27These authors observed a clear time and pressure dependence of the transition as function of driving pressure and used the data to fit kinetics models, which gave some insight into transformation mechanism that influences the transformation kinetics.These studies are fundamentally important because it questions the simple equilibrium transformation behavior via heterogeneous transformations and opens the way to investigate scenarios where Zr can first be subjected to high pressure and then transforms at this pressure kinetically.This means that Zr may form interesting metastable states and, as we will argue, may indeed form metastable bcc Zr (equivalent to the high temperature b phase).We will show in this paper that the metastable bcc phase will occur only at very high pressures so that the experimental conditions in the Jacobsen study 27 were unable to observe this state.We mention that fast kinetics can play a role in nuclear processes, where swift heavy metal irradiation can indeed generate metastable states. 28Some related experimental evidence for an intermediate bcc state in Ti-V alloys was also reported by Vohra et al. 29 The expected "equilibrium" transition in Zr is well researched.9][20][21] Several transformation pathways were suggested: Silcock's supposed two orientation relationships that would define the transition. 13This idea was partially supported by Wenk et al. 4 from diffraction experiments.It was also speculated that the x phase was the result of quenching the high temperature bcc phase 9 while Jyoti et al. 21suggested that the bcc phase appears as an unstable intermediary between the two and established a correspondence matrix between these transformations (a to b to x).
The transformation mechanisms imply specific orientational relationships between transformed regions.Silcock 13 suggested orientational relationships (OR II) with ð0001Þ a == ð11 20Þ x and ½11 20 a ==½0001 x , 9 which appears to be at odds with the ab initio calculations of electronic structures by Trinkle et al. 7 who shows (OR I) with ð0001Þ a ==ð01 11Þ x and ½11 20 a ==½ 1011 x .The pathway is named TAO-1 within the context of his paper 7,8,14,15 and complies with nucleation models.A third pathway was proposed by Usikov and Zilbershtein (UZ pathway), 16 in which the hcp-x transformation proceeds via a transient bcc phase.As there are two equivalent {111} b plane collapse modes from bcc to x phase, 17 two types of ORs (OR I and OR II) between the initial hcp phase and the x phase are possible.The UZ pathway consists of two steps: the hcp !bcc transformation (Burgers mechanism) 30 and the bcc !x transformation ((111) plane collapse model). 31,32A summary of the possible pathways is shown in Fig. 1.
We use density functional theory (DFT) calculations to analyze the pathways of the homogeneous hcp-x transformation in Zr and compare the results with new calculations for Ti.The enthalpy barriers were calculated using a generalized solid-state nudged elastic band (G-SSNEB) method, 33 all calculations were performed using the Vienna ab-initio simulation package (VASP). 34The G-SSNEB method involves both atomic and unit-cell degrees of freedom and is designed to determine reaction pathways of solid-solid transformations.We used 12 atom supercells of the hcp, x, and bcc phases according to TAO1, 7 UZ, 16 and Silcock 13 pathways (see supplementary material Fig. S1). 35The lattice constants of the three phases under pressures are listed in Table S1. 35lectron exchange and correlations were modeled within the generalized gradient approximation (GGA) using Perdew-Burke-Ernzerhof (PBE) form. 36The plane wave basis kinetic energy cut off was set to 500 eV.The NEB calculation was considered to be complete when the total force of an atomic configuration was less than 0.01 eV/A ˚.For more calculation details see supplementary material. 35We define the activation enthalpy as the enthalpy difference between the transition state and hcp phase (DH ¼ H transition À H hcp ).Fig. 2 shows the activation enthalpies along the three transformation pathways for Ti and Zr, respectively.
The most simple pathway occurs in Ti.It does not change much with pressure (Figs.2(a)-2(c)).The UZ pathway has the highest activation enthalpy at 0 GPa (Fig. 2(a)), the activation enthalpy for the Silcock pathway is intermediate and the TAO-1 pathway has the lowest activation enthalpy.This sequence of activation enthalpies (E UZ > E Silcock > E TAO1 ) does not change (Figs.2(b) and 2(c)) with pressure up to 23 GPa.This result confirms that TAO1 is the correct pathway for the hcp-x transformation in Ti under pressure. 7n contrast to Ti, the transition pathway of the hcp-x transformation in Zr changes dramatically with pressure (Figs.2(d)-2(f)).At zero pressure, the activation enthalpy is similar to that of Ti (Fig. 2(d)), with E UZ > E Silcock > E TAO1 and does not change much in the low pressure regime (<10 GPa, Fig. 2(e)).The activation enthalpy for the UZ pathway then decreases strongly with increasing pressure and becomes lowest at 23 GPa, and completely overlaps with the TAO1 pathway (Fig. 2(f)).The crystal structure of the metastable phases at 23GPa, as shown in the insets of Fig. 2(f), are C2/c and bcc phase.In order to further identify the metastable bcc phase, we calculate the electron density of states (DOS) for hcp, x, bcc, and the transient state (bcc) at 23 GPa (Fig. 3).The DOS of the bcc phase and transient state

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are nearly identical (Fig. 3) and, in turn, very different from hcp and x.This agreement confirms the transient phase at 23 GPa as bcc with the hcp-x transformation following the UZ pathway.
A switch between transformation pathways occurs between 10 GPa and 19 GPa (Fig. 4).Starting from the hcp phase, we find that at 10 GPa the first change of the structural state under pressure is to activate the shuffle along h11 23i directions.This is the dominant mechanism at low pressure.The transformation pathway at 19 GPa shows a very different picture: the shuffle along h11 23i directions is suppressed and replaced by a combination of shear along h11 20i directions and shuffle along h1 100i directions.We relate this effect to the large c/a ratio in the hcp phase of Zr (Fig. 5).The larger c/a ratio has a strong effect on atomic cooperative movements during the hcp-x transformation.The large c-axis eases the shuffle movement along the h1 100i directions, while the relative decrease of the a-axis impedes the shuffle along h11 23i or h11 20i directions.As a result, the increase of the c/a ratio with pressure in Zr hinders the shuffle along the h11 23i directions in Zr (Fig. 4).
The effect of pressure on the transformation pathways can be understood from a fundamental perspective if the order parameters for the hcp-x transformation are considered (Fig. 4).The hcp-x transformation includes three types of atomic cooperative movements in the hcp phase: the shuffle along the h1 100i directions, the shear along the h11 20i directions and the shuffle along the h11 23i or h11 20i directions (corresponding to two types of ORs).The first and

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second atomic movements transfer hcp to bcc, while the third atomic movement in bcc h111i directions corresponds to the collapse during the bcc-x transformation.The evolution of the order parameters with pressure changes very little in Ti (Figs. 2(a)-2(c)).In contrast, when the pressure in Zr increases to 10 GPa, the TAO1 pathway approaches the UZ pathway (Figs.4(a) and 4(b)).The difference between the pathways decreases further with increasing pressure and leads to partial overlap (Fig. 4(c)).At 23 GPa (Fig. 4(d)), the TAO1 pathway completely overlaps with the UZ pathway.The main pressure dependence can now be identified as the shuffle along the h11 23i directions.This shuffle is an inherent part of the TAO1 pathway at low pressure while it is eliminated from the first part of the high pressure pathway (UZ step 1 in Fig. 4(d)).The lack of the h11 23i directions shuffles at the beginning of the transformation that stabilizes the bcc transient phase at high pressure.
Thermodynamic analyses in the supplementary material 35 demonstrate that the driving force for the occurrence of hcp-bcc transformation in Zr under high pressure arises from the PDV term.The large c/a ratio contributes to the excess volume of the hcp phase compared with the bcc phase (Fig. 5).According to the Burgers mechanism, the hcp cell transfers to the bcc cell during the transformation.From the lattice constants of the two phases in Ti and Zr, we find that the areas of the basal planes in the two cells are nearly the same.The volume change arises from the shrinkage of c axis ([0001] direction) in the hcp cell.The shrinkage of c axis in the hcp-bcc phase transformation is mirrored by the c/a ratio, which converges to ffiffi ffi 2 p during the transformation.The large change of the c/a ratio in Zr corresponds to the change of the electronic structure of Zr with weaker interactions along [0001] direction and stronger interactions along [1120] direction (in comparison with Ti, see supplementary material Fig. S2). 35We expect changes of the c/a ratio under pressure and a sudden change of transformation pathways because the shuffle along the h11 23i directions is impeded at high pressure at the early stage of the transformation pathway.
In addition, the metastability of the high pressure transformation in Zr (Fig. 2(f)) is clearly seen by the decrease of the activation barrier with increasing pressure.While low pressure transformations are thermally inhibited by an activation energy, we find that the high pressure pathway shows a straight energy decrease to the first metastable phase and then rather low activation energies(less than 11.56 meV) for the second metastable bcc phase and the final x phase.Each hump is smaller than the one big hump at low pressure in Fig. 2(d) (ca.20.24 meV).
The transient bcc phase has not been seen in equilibrium experiments.According to our results, we would expect this effect at much higher pressure than previously expected.Experimental works on Zr or Zr-2.5%Nb alloy by Perez-Prado et al. 5,37 does see the b phase at lower pressure and it may well be that this observation is supported by the non-uniform (shear) stress applied to the sample. 38More detailed works on pathway under shear stress and experiments at high hydrostatic pressure are needed to further clarify the phase metastability.

FIG. 1 .
FIG. 1. Summary of proposed transformation mechanisms for the hcp-x phase transformation and their corresponding x variant product with respect to initial hcp phase, OR I with ð0001Þ a ==ð01 11Þ x and ½11 20 a ==½ 1011 x and OR II with ð0001Þ a ==ð11 20Þ x and ½11 20 a ==½0001 x .