$\omega$-phase and solitary waves induced by shock compression of bcc crystals

Molecular dynamics simulations show that shock waves in single crystals are frequently accompanied by solitary waves. Here we present results for bcc crystals where the atoms interact via the Dzugutov potential [M. Dzugutov, J. Non-Cryst. Solids 131, 62 (1991)]. In general the shock waves cause a phase transformation to a close-packed phase. In addition, special phenomena occur if bcc is shocked along the threefold ⟨111⟩ direction. We observe an intermediate phase, which is identified as the so-called $\omega$-phase, and closely related nonsteady solitary waves in the regime of overdriven shock waves. The phenomena are described in detail and compared to solitary waves in other materials. The origin and stability of the $\omega$-phase are studied. We find that a single solitary wave mode exists parallel to the ⟨111⟩ direction, and solitary waves along other directions are derived from it. A simple collision model is presented that captures the essential properties of the solitary waves.

Figure 1

Hugoniot diagram for shock waves along the ⟨111⟩ direction in the bcc crystal. The axes are scaled with the velocities of sound. The plus signs indicate the velocity of the elastic and transition wave fronts; the crosses indicate the speed with which the high-defect phase transforms into a low-defect structure. The stars and the squares indicate the propagation velocity of the fastest solitary wave and the spreading of the $\omega$-phase, respectively.

Figure 2

Hydrostatic pressure profiles as a function of $u_p∕c$ at arbitrary simulation times. The shock wave is moving to the left. The curves have been shifted by $50P_0$ with respect to the following one to avoid overlaps.

Figure 3

Hydrostatic pressure as a function of time and path. Contours are plotted at $10∕3$ $P_0$. Temperature $kT=0.001ϵ$, the piston velocity is $u_p∕c=0.25$. Three solitary wave peaks (1.S, 2.S, and 3.S) are visible in front of the transition wave front (P).

Figure 4

Damping of the solitary wave at different temperature. From top to bottom: $kT=0.01$, 0.05, and $0.1ϵ$. The piston velocity is $u_p∕c=0.4$. The solitary waves are visible in front of the transition waves.

Figure 5

Cut through the intermediate $\omega$-phase. The crosses indicate one layer, the plus signs the other layer. Both layers together form a triangle lattice. The triangular lattice cells of one variant have been superimposed. The empty triangles are periodic images repeated on the opposite side.

Figure 6

Transformation from bcc into the $\omega$-phase. Two variants are shown: the collapse of $B+C$ and the collapse of $A+B$.

Figure 7

Radial distribution function of the intermediate phase compared with perfect bcc and hcp. The most prominent disagreement for bcc is visible at around $r=1.4a$. A comparison with uniaxially compressed bcc does not lead to better agreement. The failure of hcp is most prominent at $r=1.75a$. The split first peak clearly indicates that we have no close-packed structure.

Figure 8

Angular distribution function of the intermediate phase. The vertical lines indicate the angles in bcc and hcp. The disagreement is obvious. The peak at 147° is lacking in fcc.

Figure 9

Displacement of the atoms at the position of a soliton peak. The three signs indicate three arbitrarily chosen chains.

Figure 10

Projection of the sample with the solitary wave perpendicular to the ⟨111⟩ direction. Upper part: $\left[2\overline{1}\overline{1}\right]$ plane, lower part $\left[01\overline{1}\right]$ plane. The shock wave moves to the left.

Figure 11

Cut through the structure at the maximum of a solitary wave peak. The crosses, plus signs, and stars indicate three bcc layers. In a perfect bcc sample, all layers would form a triangular lattice. The black circles indicate the positions where the sequence repeats after two layers, indicating an $\omega$-phase configuration. Compare this figure to Fig. 5.

Figure 12

Decay and broadening of a soliton peak as a function of pressure. The peaks are plotted at equal time intervals.

Figure 13

Deviation of the position of the pressure maxima from the center of mass of the solitary wave. Top: $t=0t_0$, flat one-dimensional wave. Center: $t=2.5t_0$, fluctuations develop. Bottom: $t=7.0t_0$, transverse modulation is present. The piston velocity is $u_p∕c=0.4$.

Figure 14

Strength of the solitary waves as a function of propagation direction for the collision method. Low levels indicate no solitary waves (between ⟨100⟩ and ⟨110⟩, for example). High levels mark strong solitary waves. The increase in soliton strength is best seen between ⟨101⟩ and ⟨111⟩. Unfortunately, the sharp peak along the ⟨111⟩ directions cannot be resolved in the picture.

Figure 15

Solitary wave along the ⟨211⟩ direction. Upper part: $x\text{−}y$ plane, lower part: $x\text{−}z$ plane. The wave is currently located between $x=162.5a$ and $x=165a$. Since the $y$ direction is $⟨01\overline{1}⟩$ and the $z$ direction $⟨\overline{1}11⟩$, we can read of the elongated traces in the center of the lower picture that the displacement of the atoms is not parallel to the deformation along the $x$ direction, but along the ⟨111⟩ direction.

Figure 16

Difference between the potential energies of bcc and the $\omega$-phase with respect to uniaxial compression. $d_0$ denotes the equilibrium lattice constant along the [111] or $z$ direction.

Figure 17

Phonon dispersion along the [111] direction for iron and for the Dzugutov potential. The upper branches near $k=0$ are the longitudinal waves, the lower branches the transverse modes. The phonon dispersion relations for other bcc metals look similar to iron. They all possess a minimum around $k∕k_0=2∕3$. The scaling has been chosen such that $\omega ={\omega }_0$ at $k=k_0$.

Figure 18

Position of the minimum of the longitudinal [111] phonon as a function of compression. The line at $k{d}_0=2∕3$ indicates the ideal wave length.