Ab initio study of shock-compressed copper

We investigate shock-compressed copper in the warm dense matter regime by means of density functional theory molecular dynamics simulations. We use neural-network-driven interatomic potentials to increase the size of the simulation box and extract thermodynamic properties in the hydrodynamic limit. We show the agreement of our simulation results with experimental data for solid copper at ambient conditions and liquid copper near the melting point under ambient pressure. Furthermore, a thorough analysis of the dynamic ion-ion structure factor in shock-compressed copper is performed and the adiabatic speed of sound is extracted and compared with experimental data.

Figure 1

The phonon dispersion of solid copper at $\rho =8.94$ g/${\mathrm{cm}}^3$ and $T=303$ K extracted from the longitudinal and transverse current-current correlation spectrum. The $x$ axis is scaled by the lattice constant $a$ of ambient copper. Experimental data from Nicklow et al. is given as a reference [82].

Figure 2

The current-current correlation spectrum of solid copper at $\rho =8.94$ g/${\mathrm{cm}}^3$ and $T=303$ K along the high-symmetry path $\mathrm{\Gamma }-K-X$. The transverse part is colored red, while the longitudinal part is colored green.

Figure 3

Dynamic electrical conductivity of solid copper at ambient conditions computed from a DFT simulation with 125 atoms using the Kubo-Greenwood formula and results of TD-DFT linear response calculations with four atoms and the adiabatic local density approximation. The black line indicates results achieved with the PBE XC functional, while the grey line represents results using the HSE XC functional. Measurements of Henke et al. [83] are given as reference.

Figure 4

Static structure factor of liquid copper at $\rho =7.69$ g/${\mathrm{cm}}^3$ and $T=1773$ K computed from a DFT-MD simulation with 125 atoms and a NN-MD simulation with 32 000 atoms. Two experimental results from x-ray diffraction (Waseda and Ohtani [87]) and neutron diffraction (Eder et al. [88]) are shown as reference. The right inset zooms in on the long-wavelength behavior and shows the $k\to 0$ prediction by DFT-MD and an experimental result of liquid copper near the melting point by Filippov [91]. The top inset zooms in on the behavior around the first correlation peak.

Figure 5

Peak position of the ion acoustic mode taken from the DSF and from the longitudinal current-current correlation $J_{\mathrm{l}}(k,\omega )$ [see Eq. (5)] dependent on the $k$ value. For wave numbers below the first correlation peak results from the NN-MD simulation are shown, while all other results are taken from the DFT-MD simulation. A best linear fit for the low $k$ limit of the ion acoustic mode of the DSF is indicated and the corresponding adiabatic speed of sound is presented. The inset shows a comparison between DFT-MD and NN-MD results for small $k$.

Figure 6

Peak width of the thermal mode of the ion-ion DSF dependent on the $k$ value. Experimental results from Hagen et al. [98] and data calculated from experimentally determined static ion structure factors by Eder et al. [88] are given as comparison. The inset shows a comparison between the results of the DFT-MD (125 atoms) and NN-MD (32 000 atoms) simulations in the region where experimental data is available.

Figure 7

Lower panel: Hugoniot curve for copper in the $P\text{−}\rho$ plane inferred from isotherms calculated using DFT-MD (black) and isotherms from the SESAME 3333 table [100] (grey). The temperatures for some data points are annotated. High pressure experimental values by McCoy et al. [52], Glushak et al. [101], Kormer et al. [102], Altshuler et al. [103], and Mitchell et al. [104] are shown. The dotted horizontal lines indicate the conditions we choose for further investigation. More information on the conditions is given in Table 1. Upper panel: Zoomed in view of the low pressure range, where experimental data by Mitchell et al. [105], Altshuler et al. [106], and McQueen et al. [107] are shown.

Figure 8

Hugoniot curve for copper in the $T\text{−}P$-plane inferred from isotherms calculated using DFT-MD (black) and isotherms from the SESAME 3333 table (grey). Melting lines determined by Wu et al. [18], Moriarty et al. [19], and Belonoshko et al. [20] using different variations of MD simulations as well as experimental results by Hayes et al. [17] are indicated.

Figure 9

DFT-MD results of the static structure factor of liquid copper along the Hugoniot curve. In the inset the low $k$ behavior is shown on a log scale and the NN-MD results are shown for wave numbers that are not accessible to the DFT-MD results. The limits for $k\to 0$ are also indicated [see Eq. (6)]. Further information on the conditions is given in Fig. 7 and Table 1.

Figure 10

Phonon dispersion of solid copper along the principal Hugoniot curve along high-symmetry direction in the Brillouin zone. The $x$ axis is normalized by the lattice constant $a$ to make the results at different densities comparable. Further information on the conditions is given in Fig. 7 and Table 1.

Figure 11

DSF of liquid copper along the principal Hugoniot curve computed from the NN-MD simulations. The curves are shifted by $0.35$ fs with respect to the next lower $k$ value for readability. Further information on the conditions is given in Fig. 7 and Table 1.

Figure 12

The upper panel shows the peak position of the ion acoustic mode ${\omega }_{\mathrm{ion}}$ taken from the DSF's in Fig. 11 dependent on the $k$ value and a linear dispersion computed from Eq. (11). The corresponding adiabatic speeds of sound are presented. The lower panel shows the dispersion of the longitudinal current-current correlation spectrum $J_{\mathrm{l}}$ extracted from the NN-MD simulations and the free-particle behavior. For readability, the curves in the upper panel are shifted by $0.2{\AA }^{-1}$ to the right with respect to the next lower pressure condition.

Figure 13

Adiabatic speed of sound computed from the thermodynamic relation (11) compared to experimental VISAR measurements by McCoy et al. [52] for shock-compressed copper.

Figure 14

Validation sets of the NN potentials for energies and forces computed by DFT. As a reference, the black dashed lines show perfect correspondence between the NN potential and DFT.