Shock waves in materials with Dzugutov-potential interactions

The influence of shock waves on the bcc ground state, the metastable $\sigma$ phase, and quasicrystal structures of the Dzugutov potential [Phys. Rev. A 46, R2984 (1992)] has been studied by molecular dynamics. In general, a phase transition is observed to the well-known high-pressure fcc- or hcp-phase. The details of the phase transition and the perfection of the created phases depend on the orientation of the shock wave propagation direction with respect to the symmetry axes. The results are in good qualitative agreement with recent simulations with embedded atom method potentials designed specifically for iron. For the quasicrystal we observed flips, a new plasticity mode predicted for these structures. Furthermore we will report the results for an amorphous structure supplied with the same potential. No crystallization takes place in this case, indicating that it is easier to shift atomic layers collectively than to order the atoms one by one.

Figure 1

Possible vertex configurations in square-triangle tcp phases. From left to right: $A$ phase, $Z$ phase, $H$ phase, and $\sigma$ phase.

Figure 2

Primary and additional tiles observed in the square-triangle tcp phases. From left to right: square, triangle, rhombus, threefold, and twofold symmetric hexagon.

Figure 3

Part of a dodecagonal quasicrystal. The atoms in the basic $A$ layers $z=0$ and $z=\frac{1}{2}$ are dotted. The black atoms are in the $B$ layer at $z=\frac{1}{4}$, the white atoms are in the $\overline{B}$ layer at $z=\frac{3}{4}$. The edge length is usually of the order of $2a$, the nearest-neighbor distance, but depends on the interaction between the atoms.

Figure 4

Left: The Dzugutov potential in the parametrization applied in this work. Right: Semiquantitative phase diagram of the Dzugutov potential. The stress-temperature curves for shocked states are presented for materials initially amorphous (Sec. 3C5) and bcc shocked along the twofold and fourfold axes (Sec. 3). The curves for other initial structures and directions are similar to bcc.

Figure 5

Typical radial distribution function. The spikes indicate the neighbor shells for ideal fcc $(\times )$ and hcp (◻) crystals. Obviously, these cases cannot be distinguished in the simulation result.

Figure 6

Typical angular distribution function. The spikes denote the angles in ideal fcc $(\times )$, hcp (◻), and bcc (*), respectively. The simulation result does not fit to the bcc maxima, but to fcc or hcp. A distinction between the latter is possible since the maximum between 140° and 160° is only present in hcp. The conclusion is that the sample under consideration is preferably hcp stacked. Cases without a maximum between 140° and 160° have also been observed.

Figure 7

Hugoniot diagram for $\sigma$ phase and quasicrystal simulations. The axes are scaled with the velocities of sound. The lines serve as a guide to the eye only. The two parallel branches indicate the beginning and end of the phase transformation.

Figure 8

Hugoniot diagram for the simulations of the bcc crystal. The axes are scaled with the velocities of sound. The lines serve as a guide to the eye only. The two parallel branches indicate the beginning and end of the phase transformation.

Figure 9

Wave profiles of hydrostatic pressure in the elastic, elastic-plastic, and overdriven range. The first three plots are for the fourfold direction, the lower-right plot is for a twofold direction. The profiles in the elastic and the elastic-plastic regime (upper part) look similar for all shock wave directions and initial phases. The arrows in the lower-left part indicate the additional wave observed, for example, for bcc shocked along a twofold direction.

Figure 10

Longitudinal projection of a bcc crystal shocked along the fourfold direction $(u_p=0.24c_4)$. The shock wave moves to the left, the mirror is at the right side at $x=100a$. The interface between initial bcc and final fcc at about $x=31.5a$ is flat.

Figure 11

Transverse cuts through a bcc crystal shocked along the fourfold direction $(u_p=0.24c_4)$. The shock front is currently at $x=31a$ and moves toward $x=0$ (see Fig. 10). Left: Slice between $x=50a$ and $x=60a$, right: slice between $x=75a$ and $x=85a$.

Figure 12

Longitudinal projection of a bcc crystal shocked along the twofold direction $(u_p=0.125c_2)$. This is an example where the lattice planes are shifted perpendicular to the shock wave direction. The shock wave moves to the left; the mirror is at the right side. A beginning ABABC stacking sequence is visible near $x∕a=97$.

Figure 13

Longitudinal projection of a bcc crystal shocked along the threefold direction $(u_p=0.19c_3)$. This is an example where the lattice planes are rotated perpendicular to the shock wave direction. The shock wave moves to the left; the mirror is at the right side. A beginning ABABABABABCBCAB stacking sequence is visible between $x∕a=87$ and $x∕a=100$.

Figure 14

Part of a $\sigma$ phase sample shocked in $m90$ orientation, $u_p=0.125c_x$. A part of the tiling has been redrawn. Shaded are the newly generated rhombi. The black dots mark the squares and triangles that have been flipped. The shock wave moved from the right to the left.

Figure 15

Hugoniot curve $u_s\text{vs}{u}_p$ for the amorphous structure. Plotted are the velocities at the bottom $\left({\mathrm{amo}}_b\right)$, at half height $\left({\mathrm{amo}}_c\right)$, and at the place where the hydrostatic pressure reaches a maximum $\left({\mathrm{amo}}_t\right)$. In the overdriven region, the distinction between these velocities vanishes, which means that we find a sharp wave front.

Figure 16

RDF of the amorphous sample for different $u_p$. The arrows indicate how the RDF changes if the shock strength is increased from $u_p=0.1c_a$ to $u_p=0.8c_a$ $(c_a\approx 10v_0)$.

Figure 17

Relative frequency of the 1421 and 1422 common neighbor diagrams for some representative cases.