Model of ramp compression of diamond from ab initio simulations

Ramp compression experiments characterize high-pressure states of matter at temperatures well below those present in shock compression. However, because temperature is typically not directly measured during ramp compression, it is uncertain how much heating occurs under these shock-free conditions. Here, we performed a series of ab initio simulations on carbon in order to match the density-stress measurements of Smith et al. [Smith et al., Nature (London) 511, 330 (2014)]. We considered isotropically as well as uniaxially compressed solid carbon in the diamond and BC8 phases, with and without defects, as well as liquid carbon. Our idealized model ascribes heating during ramp compression to an initially uniaxially compressed cell transforming isochorically into an isotropically (hydrostatic equivalent) compressed state having lower internal energy, hence higher temperature so as to conserve energy. Multiple such heating events can occur during a single ramp experiment, leading to higher temperatures than with isentropic compression. Comparison with experiments shows that heating alone does not explain the equation of state measurements on diamond, instead implying that a significant uniaxial stress component remains present at high compression. The temperature predictions of our ramp compression model remain to be verified with future laboratory measurements.

Figure 1

Normal stress components vs strain derived from DFT-MD simulations for uniaxially strained BC8 supercells of carbon at 9.78 $\mathrm{g}{\mathrm{cm}}^{-3}$. The shaded grey area in the first three panels represents the error bars in the stress reported at this density (2935 GPa) by Smith et al. [9]. The two upper panels result from defect-free simulations at 4400 and 6000 K. The lower left panel depicts predictions from a BC8 cell with 7.8% porosity and 4400 K, where the vacancies are represented by red spheres. The lower right panel compares the normal stresses reported by Smith et al. [9] and Lazicki et al. [46] with our values of the stress in the compression direction, $P_x$, where the red and black squares connected by a line in this panel represent the entire range of $P_x$ values for uniaxial compressions at 4400 K for BC8 cells with and without defects, respectively. These correspond to uniaxial compressions up to $\varepsilon =20\%$ change in lattice constant. The green, thin diamond corresponds to the pressure of carbon in the diamond structure at the same density, 9.78 $\mathrm{g}{\mathrm{cm}}^{-3}$, and 4400 K. The yellow shaded area shows the Smith et al. data with uncertainties after the densities have been rescaled by a factor of 1.08 (see discussion in main text). The dashed black line with circles corresponds to our proposed ramp compression model, discussed in Sec. 5.

Figure 2

Phase diagram of carbon. DFT-MD simulations of dense, liquid carbon [25] were used to attribute a hypothetical temperature to the density-stress data values reported from Bradley et al. [8] and the [9] ramp compression experiments of Smith et al. The vertical lines represent the temperature uncertainties that were obtained by propagating the experimental error bars. The orange dashed curve represents the isentrope that crosses the principal Hugoniot curve at 110 GPa, the diamond elastic limit.

Figure 3

Elastic constants $C_{11}$ and $C_{21}$ of diamond and BC8 carbon as a function of pressure at different temperatures. The shaded region indicates the instability regime, where the Born criterion [50] $(C_{11}-C_{12})/2>P$, is no longer satisfied.

Figure 4

Stress-strain curves for diamond (upper panel) and BC8 structures (lower panel). The dashed lines depict the linear regime from where the elastic constants, shown in Fig. 3, were obtained.

Figure 5

Plastic work (dot dashed line) and energy increase upon isentropic compression (solid black line), compared to the energy difference between hydrostatically and uniaxially compressed diamond (open blue circles). The energy difference with the isentrope, $\mathrm{\Delta }{E}_{\mathrm{isen}}=-\int PdV$ (dashed red line) is added to the plastic work to account for the total energy increase $\mathrm{\Delta }{E}_{\mathrm{total}}$. Estimates from elasticity theory using fourth- order elastic constants from Telicho et al. [59].

Figure 6

Illustration of a single step of our ramp wave compression model. The system is uniaxially compressed from the hydrostatic state (a) to the uniaxially compressed state (b) along an isotherm. The energy of this state, $E_b$, which is higher than that of the equivalent hydrostatic state at the same temperature and density, $E_b^{*}$, are used to obtain the energy difference, $E_c$, which is given to the hydrostatically compressed sample at this density as kinetic energy.

Figure 7

Illustration of our ramp wave compression model for diamond, using three compression steps. The system is uniaxially compressed to the next density (red squares) along an isotherm (horizontal arrows), and then the energy difference with respect to the equivalent hydrostatically compressed state (at the same conditions), is given to the cubic sample as kinetic energy (vertical arrow). The relaxation of the system after increasing energy is performed in the microcanonical ensemble (NVE), which leads to the new equilibrium temperature (blue circles). The cycle is repeated from this point using the equilibrated state as the new initial state for the next step. A second and then a third compression step is performed in order to reach 800 GPa.

Figure 8

Total energy and stress in hydrostatically (solid green circles) and uniaxially compressed (open squares) diamond along the 300 K isotherm for a series of single-step uniaxial compressions. Ramp compression experiments by Bradley et al. [8] and Smith et al. [9] are shown for comparison. The cubic, hydrostatically compressed state is energetically favored over the uniaxially compressed state for all pressures. Along this isotherm, the pressure in hydrostatically compressed samples, $P$, is always below the experimental curves. The open blue circles represent the heated sample [stage (c) in our ramp model] for each of the individual single-step compressions. The drop in stress anisotropy, $S_L=P_x-1/2(P_y+P_z)$, at $\rho /{\rho }_0=1.8$ indicates the onset of a mechanical instability in the uniaxially compressed crystal that is enhanced by thermal motion of the nuclei.

Figure 9

Final temperature [relaxation to hydrostatically compressed stage (c)] vs compression ratio for a series of single-step uniaxial compression steps, corresponding to the blue circles in Fig. 8. The final temperature from our single-step ramp model is compared to the temperature rise derived from the plastic work model using Eq. (6).

Figure 10

Temperature rise in our ramp-compression model for different compression paths. The final temperature depends on the number of steps taken to subdivide the pressure interval. For a final density of 6.48 $\mathrm{g}{\mathrm{cm}}^{-3}$ (800 GPa), a three-step subdivision leads to a final temperature of 2000 K, while four steps lead to 1200 K. For a final pressure of 600 GPa, a three-step sequence results in a final temperature of 1500 K.

Figure 11

Multiple-shock Hugoniot curves of carbon at pressure-temperature conditions of the diamond, BC8, and simple cubic phases. The shocks occur in diamond at 200, 400, and 600 GPa, and in BC8 at 1200, 1800, and 2400 GPa. In the bottom figure, the multishock points of the panel above are plotted in pressure-density space, and compared with the experimental ramp-compression values of Smith et al. [9] and the DFT-MD solid-liquid principal Hugoniot of diamond [14, 60]. The densities of Smith et al. were scaled by a factor of 1.08.

Figure 12

Temperatures predicted by the multishock scheme compared to our ramp model in the $0-5000$ GPa pressure range, using eight steps. Intermediate steps are performed at 1000 and 3000 GPa, where solid/solid phase transitions are predicted to occur. All uniaxial compressions from 1000 to 4000 GPa were performed in the BC8 structure. The melting curve (thin blue curve) [25] is shown for comparison. The dot-dashed black curve represents the isentrope that crosses the principal Hugoniot curve at 110 GPa, the diamond elastic limit. Temperature estimates, based on plastic work, at 600 and 800 GPa from the ramp compression experiments of Bradley et al. [8] are shown as yellow stars.

Figure 13

Three-shock sequence in the multishock scheme, launching a secondary shock at 600 GPa. The secondary shock continues in the liquid phase at 2000 GPa, while the third shock, that starts at 2450 GPa, reaches 14000 K at 4000 GPa. The second shock crosses the melting line near 900 GPa. Different combinations of steps in our ramp compression model show that it is not possible to melt the sample with any combination of steps. Temperature estimates, based on plastic work, at 600 and 800 GPa from the ramp compression experiments of Bradley et al. [8] are shown as yellow stars. Temperature estimates from Swift et al. [62] are shown in solid thick grey line.