Nucleation mechanism for $\mathrm{hcp}\to \mathrm{bcc}$ phase transformation in shock-compressed Zr

We present large-scale atomic simulations of shock-induced phase transition in Zr assisted by the machine learning method. The results indicate that there exists a critical piston velocity of $U_{\mathrm{p}}\sim 0.85\mathrm{km}/\mathrm{s}$, above which the product phase has changed from ω to bcc. Unlike the case in Fe, the shock-induced $\mathrm{hcp}\to \mathrm{bcc}$ nucleation mechanism in hcp-Zr single-crystal shows significant dependence on crystal orientation. For shock along the $\left[10\overline{1}0\right]$ direction, the hcp phase directly transforms into bcc as expected. However, for shock compression along [0001] and $\left[1\overline{2}10\right]$ directions, the $\mathrm{hcp}\to \mathrm{bcc}$ transformation occurs in quite a different manner, i.e., the Zr single crystal transforms into a disordered intermediate that subsequently exhibits ultrafast crystallization of the bcc phase within the timescales of subnanoseconds. We associate such presence of disordered intermediate structure with the sluggishness of shear stress relaxation, which leads to an elastic unstable condition of the crystal during the first few picoseconds of uniaxial compression, and suggests that the fewer possible shear planes (related to Burgers mechanism) for [0001] and $\left[1\overline{2}10\right]$ shock loading is an underlying factor for the orientation dependence.

Figure 1

The phase stability and shock Hugoniot of zirconium under compression. (a) Potential energy of different phases as a function of volume using the present machine learning potential. (b) The shock Hugoniot for Zr a plot of shock pressure ($P$) vs particle velocity ($U_{\mathrm{p}}$), was determined from the multiphase model developed by Greeff [19]. Both data indicate a successively α (hcp)  →  ω(hex)  →  β (bcc) phase transition with increasing pressure.

Figure 2

Microstructural development of ${\left[10\overline{1}0\right]}_{\alpha }$ shocked Zr single crystals with piston velocities from 0.54 km/s to 1.0 km/s. The shock direction is from left to right. The left panel and the right panel show typical microstructure at 8 and 18 ps, respectively. Atoms with an hcp environment are shown as orange spheres and represent the α phase, pure blue regions show the location of bcc phase, whereas the blue and orange regions mark the ω phase, and other colors belong to defects. The increase of shock strength brings the shocked α-Zr single crystals to the final state from ω to bcc phase.

Figure 3

Structural comparison of metastable bcc intermediate (state A) and finial bcc product phase (state B). (a) Typical microstructures of ${\left[10\overline{1}0\right]}_{\alpha }$ shocked Zr with the piston velocity of 0.8 km/s (or shock pressure of 25.28 GPa), showing the phase transition sequence of hcp  →  bcc (metastable)  →  ω  →  bcc (stable). The red boxes mark the regions that are used to calculate the radial distribution functions (RDFs). The color coding and labels are the same as Fig. 2. (b) The corresponding RDFs of two states of bcc structures. It shows that the two types of bcc phase differ in the lattice constants.

Figure 4

Snapshots of the phase transformation processes in ${\left[0001\right]}_{\alpha }$ shocked Zr single crystals with different piston velocities. (a) and (b) The nucleation and growth of ω phase under 0.54 km/s shock compression, intermediated by the formation of metastable bcc. (c) and (d) The formation of bcc phase from the hcp-Zr matrix, accompanied by a few volume fraction of amorphous intermediate. (e) and (f) Under 1.0 km/s shock loading, the hcp-Zr lattices first collapse into obvious disordered region, where new bcc grains form within the disordered region subsequently. The inset shows the neighborhood information of the atoms before and after the transformation. The color coding are the same with Fig. 2, and the boxes locate the regions for the calculation of local radial distribution functions.

Figure 5

Comparison of the radial distribution functions (RDFs) of disordered regions and crystallized bcc grains for (a) ${\left[0001\right]}_{\alpha }$ shocked Zr single crystals and (b) ${\left[1\overline{2}10\right]}_{\alpha }$ shocked Zr single crystals. It indicates that the disordered regions in both cases have a different structural feature from the bcc phase.

Figure 6

Snapshots of the phase transformation processes in ${\left[1\overline{2}10\right]}_{\alpha }$ shocked Zr single crystals with different piston velocities. (a) and (b) The formation of the ω phase in 0.7 km/s shocked Zr single crystal. (c)–(f) For shock velocity above 0.9 km/s, the α (hcp)  → ω phase transformation is followed by the formation of disordered regions, and then new bcc grains forms at the expense of the disordered region. The color coding are the same with Fig. 2, and the boxes locate the regions for the calculation of local radial distribution functions.

Figure 7

Orientation relationships between hcp phase and bcc phase within the shocked samples. (a)–(c) represent typical slices of the bcc-hcp phase coexistence obtained from Figs. 2, 4, and 6, respectively. The orientation relationships between hcp phase and bcc phase in (a) ${\left[10\overline{1}0\right]}_{\alpha },\left(\mathrm{b}\right)[1\overline{2}10{]}_{\alpha }$, and $\left(\mathrm{c}\right)[0001{]}_{\alpha }$ shocked Zr single crystals are all accord with the Burgers mechanism [13]. The color coding is the same as Fig. 2.

Figure 8

Microstructural evolution of ${\left[0001\right]}_{\alpha }$ shocked Zr single crystal showing the crystallization of the bcc phase from the metastable amorphous phase. (a) and (b) show the initial amorphization and bcc nucleation for the shock-wave swept regions. (c) and (d) present the following explosive grain growth and coalescence of the bcc phase, leading to the formation of a bcc-Zr polycrystal. The whole nucleation and growth process lasts about 300 ps. All viewing directions are parallel to the shock propagation. The color coding is the same as Fig. 2.

Figure 9

Microstructure evolution of 1.0 km/s or 30.01 GPa shocked Zr polycrystal showing the interaction between intergrain activities and $\mathrm{hcp}\to \mathrm{bcc}$ phase transformation. (a)–(d) represent the atomic configurations after shocked 0, 7, 11, and 18 ps, respectively. The red lines in (a) and (b) shows a typical GB migration event under shock compression. The intergrain and GB activities help shear stress relaxation and the $\mathrm{hcp}\to \mathrm{bcc}$ transformation with direct pathway. The color coding is the same as Fig. 2, and the white dash curves mark the GBs.

Figure 10

Effect of longitudinal shock pressure on the shear level parameter (λ) and amorphization work ($W$) for (a) Zr single crystals and (b) Zr polycrystals. Here, the shear level parameter λ is defined as the ratio of the shear stress to the hydrostatic transformation pressure [44], and the amorphization work is evaluated from the Patel-Cohen model [47].

Figure 11

A comparison between $c$-axis uniaxial strain-induced and hydrostatic pressure-induced Zr amorphization. Three Born stability criteria criteria $B_i$ ($i=1$, 2, and 3) are plotted with respect to either the hydrostatic pressure $P_{\mathrm{hydro}}$ (blue curve) or the stress along the uniaxial strain direction $P_{\mathrm{uniaxial}}$ (red curve). All the data are obtained from first-principles calculations. The Born criteria show that the uniaxial case is more inclined to become Born elastic unstable at high $P_{\mathrm{uniaxial}}$.

Figure 12

Schematic illustration of the competition between lattice shape change of phase transformation and uniaxial compression strain of shock loading. (a) The $\mathrm{hcp}\to \mathrm{bcc}$ phase transition requires the hcp-Zr crystal to contract along the $\left[10\overline{1}0\right]$ direction, in line with the uniaxial compressive strain applied by shock loading. (b) The strong shock loading along ${\left[1\overline{2}10\right]}_{\alpha }$ direction leads to a successively hcp  →  ω  →  bcc phase transition. The ${\left[1\overline{2}10\right]}_{\alpha }$ compressive strain of shock is a barrier for the formation of the bcc phase as the phase transition requires a tensile strain along this direction. (c) The $c$-axial compressive strain of shock can hardly provide any in-plane shear strain that is required for the $\mathrm{hcp}\to \mathrm{bcc}$ phase transition.

Figure 13

Potential energies of hcp, bcc and amorphous solid as a function of $c$-axis uniaxial and hydrostatic pressure obtained from (a) machine learning interatomic potential and (b) DFT calculations. Our machine learning prediction is in line with the results of first-principles calculations. Above 20 GPa, the uniaxial compressed amorphous metastable possesses a lower energy than that of [0001] uniaxial compressed hcp phase, while a higher energy than the hydrostatic pressured bcc phase.