Thermal equation of state of rhodium to 191 GPa and 2700 K using double-sided flash laser heating in a diamond anvil cell

The phase behavior of rhodium (Rh) metal has been studied to 191 GPa and 2700 K using a combination of room-temperature isothermal compression and double-sided flash laser heating experiments. The isothermal compression data have been fitted with a second-order adapted polynomial of order L equation of state (EoS) with best-fitting parameters of $V_0=13.764\left(2\right){\AA }^3/\mathrm{atom}$, $K_0=258\left(3\right)\mathrm{GPa},$ and $K^{\prime }=5.36\left(9\right)$. Two-dimensional maps of the uniaxial stress component $t$ are presented for Rh at different pressures showing the spatial distribution of the local stress state of a relatively high-yield strength material encased in a Bi pressure medium. In addition, a simple, thermal pressure equation-of-state model, based on a single Einstein temperature, has been fitted to the high-pressure-temperature data up to 2700 K at 148 GPa and ambient-pressure thermal expansion data up to 1982 K. Also determined are the best-fitting parameters to reproduce the thermal EoS within the dioptas two-dimensional integration software. The optimized dioptas parameters are $V_0=13.764{\AA }^3/\mathrm{atom}$, $K_0=260.54\mathrm{GPa}$, $K^{\prime }=5.114$, ${\alpha }_T=2.99\times {10}^{-5}$ ${\mathrm{K}}^{-1}$, $\partial {\alpha }_T/\partial T=1.27\times {10}^{-9}$ ${\mathrm{K}}^{-2}$, $\partial K_0/\partial T=-6.43\times {10}^{-5}\mathrm{GPa}/\mathrm{K}$, and $\partial K^{\prime }/\partial T=-9.3\times {10}^{-10}{\mathrm{K}}^{-1}$.

Figure 1

Micrographs of the sample chambers in DACs RT1 and LH1. (a) Shows the loading of the Rh inside the “well” fabricated through electrical discharge machining (EDM) in the Bi PTM in RT1 before a top layer of Bi was added and the cell closed. (b) Shows DAC LH1 prior to laser heating containing a Rh disk, coated on each side with $1\mu \mathrm{m}$ of Pt, encased in a MgO PTM. A ruby sphere was added beside the disk as an additional pressure marker. Some extrusion of the MgO PTM out of the sample chamber before being trapped between the W gasket and the diamond culet is evident.

Figure 2

2D images [(a) and (b)], and integrated profiles with Le Bail fit residuals [(c) and (d)], from Rh in a Bi PTM at 11.4 and 190.1 GPa, respectively. The yellow circles in (b) identify Bragg reflections from the diamonds which were masked during the integration. The tick marks beneath the profiles in (c) and (d) show the calculated positions of the peaks from Rh and the Bi PTM and the residuals of the fits are shown below these. The asterisk at $7^{\circ }$ in (c) indicates an impurity peak that was not observed at higher pressures and was therefore ignored. The triangle at ${31}^{\circ }$ in (c) identifies weak peaks that arose from scattering from the pinhole upstream of the sample. These peaks did not move with pressure and were masked during the analysis process.

Figure 3

(a) Grid scan of the $100\text{−}\mu \mathrm{m}$ diamond culet at 189 GPa, showing the size and location of the gasket hole, as determined by the presence of scattering from the Bi PTM. The relative size of the x-ray beam is shown for comparison. (b) A plot of $t$-Rh versus $t$-Bi, as measured at the same $55X\text{−}Y$ positions in (c) and (d). There is no correlation between the values of $t$ in the Rh and the Bi surrounding it. The vertical dotted and horizontal dashed-dotted lines represent the average $t$ values of the Bi and Rh, respectively while the shaded blue and red areas define the extent of one standard deviation. We found 55% of the Bi data points and 51% of the Rh data points were within one error bar of zero. (c) A map of the uniaxial stress component ($t$) in the Bi PTM at $\sim 110$ GPa calculated under the assumption $\alpha =1$. The location of the Rh sample is shown by the black dashed-dotted line. (d) The corresponding $t$ map for the $10\times 10\mu {\mathrm{m}}^2$ Rh sample at the same pressure. Note the nonphysical changes in $t$ on $\mu \mathrm{m}$ length scales. (e) A $t$ map across the Bi at 189 GPa. The extrema in the $t$ values are comparable to those measured at 110 GPa. The location of the Rh is shown with a dotted-dashed line. (f) A high-resolution $t$ map ($9\times 9$ grid with $0.875\text{−}\mu \mathrm{m}$ steps) of the Rh at the same pressure, again showing nonphysical changes in $t$ on $\mu \mathrm{m}$ length scales.

Figure 4

The atomic volume of Rh as a function of pressure along the 298-K isotherm. Experimental data from this study ($\blacksquare$) are shown and the dashed line through them is the best second-order APL fit. Experimental data from the previous studies of Yusenko et al. [5] ($▵$) and Frost et al. [7] ($▿$) are also shown, along with pseudo-data points ($◯$) generated at regularly spaced intervals using the Rh EoS of Young et al. [6]. Second-order APL fits to these data are also shown, along with the compressibility calculated by Shao et al. [29], and the shock compression data for Rh along the principal Hugoniot measured by Marsh et al. [4] and Trunin [30].

Figure 5

Linearization of the compression of Rh shown in the form of an ${\eta }_{\text{APL}}\text{−}\sigma$ plot. In this space, pressure increases from right to left and materials that are well represented by a first-order APL EoS will follow a linear trend. Ideal compression (see text for details) is shown by the solid black line and the dotted vertical line shows the location of $V_0$ for Rh. The data from this study on Rh ($\blacksquare$) are shown, alongside the previous Rh data [5, 6, 7], as well as comparative data from Ru [36] and Pd [7, 37]. Also displayed for comparison are the linearized shock data of Marsh et al. [4] and Trunin [30].

Figure 6

The high-pressure and temperature data points collected from laser heating on Rh and used in the determination of the thermal EoS. (a) Data in pressure-temperature space. The solid black line marks the Rh melt boundary calculated by Swift et al. [9] and the green crosses show the experimentally determined melt cure of Strong et al. [45]. For comparison, the dotted black line denotes the solid-liquid phase boundary of platinum [46]. The solid black squares represent the 298-K isothermal compression data of this study and the open red circles are the isobaric data of Schroder et al. [47]. The open black squares and triangles represent the laser heating data from LH1 and LH2, respectively, from this study while the purple and green polygons denote the thermal pressure and the temperature uncertainty in each data point. The left side of the polygon is determined from the Pt pressure measured before the heating run at 298 K while the right side is determined from the in situ Pt volume measured under heating conditions. The data points with gray polygons exhibited textural changes in the postheating pattern and were excluded from the thermal EoS refinement. (b) Data in pressure-volume space. Here isotherms generated using the refined thermal model for 298, 1000, 1500, 2000, 2500, and 3000 K (see Table S5 in the Supplemental Material [19]) are plotted alongside the experimental data. The data points are colored based on the measured sample temperature and the intervals indicated.

Figure 7

Diffraction images and profiles from two heating and cooling cycles. (a)–(c) Show the diffraction images obtained before, during, and after heating in the first cycle where the maximum temperature inferred from pyrometry was $1336\pm$ 134 K. No obvious indications of reaction products are present in the quenched image. (d) Shows the corresponding integrated 1D profiles over a limited $2\theta$ range in which the peaks in the high-$T$ profile (red) are shifted to lower $2\theta$ values and the quenched pattern matches the profile obtained before heating. In contrast, clear changes in the 2D images are observable in images (e)–(g) which show the stages for the second heating cycle reaching a maximum temperature of $2323\pm$ 232 K. The enlarged region in (g) highlights the increased intensity between the (200) Bragg reflections for Rh and Pt in the quenched image. (h) Shows the integrated profiles from images (e)–(g), highlighting the differences in the before and after profiles. In particular, (h) shows that the (200) peaks from the MgO and Pt return to their initial, preheating, positions while the Rh (200) peak remains displaced to lower $2\theta$ after cooling, indicating an expanded lattice. Since the peaks from both the MgO and Pt suggest that the pressure in the cell returned to its starting value after heating, the expanded Rh lattice likely indicates the inclusion of Pt within the Rh lattice. In the 2D images, Bragg reflections from the diamond anvils have been indicated with red circles.

Figure 8

The effects arising from repeated laser heating and cooling at the same sample location. Cycle No. 0 refers to the data collected prior to the first laser heating cycle. (a) The peak sample temperature reached in each heating cycle, as inferred from the pyrometry measurements. No pyrometry data were acquired for cycle No. 7 due to the saturation of the detector during this run. (b) The integrated profiles obtained from the quenched samples after each heating run, highlighting the (200) peaks from the Pt and Rh. The peaks from both materials show a shift to lower $2\theta$ angles with increased heating cycles, typical of a pressure relaxation following repeated heating. (c) The ratio of the Pt and Rh unit-cell parameters after each heating cycle. The black squares show the as-measured data while the triangles denote the hydrostatically corrected ratio based on the measured $t$ parameter. Both the measured and hydrostatic ratios remain relatively constant for the first six cycles, after which a reduction in the ratio likely indicates a reaction in the sample upon reaching some threshold temperature. (d) The $t$ parameter for the Pt ($\blacksquare$) and Rh ($\square$) after each cycle. An increase in the $t$ parameter is observed after the sixth heating cycle and is accompanied by a significant increase in the uncertainty. (e) The measured and corrected pressures in the Pt and Rh samples after heating. Both materials indicate a decrease in pressure in agreement with (b), however, the apparent pressure drop in the Rh is significantly greater. This difference is exaggerated in the corrected hydrostatic data.