High-pressure/high-temperature phase diagram of zinc

In figure 1 we show (as solid triangles) the melting temperatures determined at different pressures from our laser-heated DAC experiments. At low-pressure (P  <  24 GPa) the present results agree with those previously reported in the literature [24, 35, 36]. They also agree well with an extrapolation of the Simon equation fitted by Errandonea [36] from previous published data (black dotted line in figure 1). In the figure we also represent the melt curve obtained from DFT calculations up to 135 GPa (red squares). The calculated melt curve agrees with the experiments up to the highest measured pressure (80 GPa). There is also good agreement with shock-wave experiments at 120 GPa [39]. The consistency between calculations and different experiments improves confidence in the reliability of the reported melt curve. A fitting using the Simon equation [59] to results from present and previous static-compression experiments, DFT calculations, and shock-wave experiments leads to the following melting relation ${{T}_{{\rm m}}}=690\,{\rm K}{{(1+\frac{P}{10.5\,{\rm GPa}})}^{0.76}}$, where P is the pressure in GPa. This Simon fit is shown as a solid red line in figure 1. The melt curve of Zn initially rises steeply as a function of pressure, though at pressures above 60 GPa the slope (dT_m/dP) decreases slightly. As a consequence of this, the Simon function used previously to describe the melt curve of Zn [36] (fitted from data collected below 24 GPa) tends to overestimate the melting temperature above 80 GPa. However, the present fit properly describes all the data available on melting of Zn.

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From our calculations we also determined the Hugoniot curve [60] of Zn. The results are shown in figure 1 (red dot-dashed line). The calculated Hugoniot follows a similar dependence to the experimental Hugoniot reported by McQueen and Marsh [23] more than half a century ago. However, these experiments show a minor kink in the Hugoniot near 50 GPa and 1375 K. The presence of such a discontinuity is not reproduced in our calculations. A kink in the Hugoniot below the melting temperature can only be caused by a phase transition to a further solid phase. Neither our calculations, nor our experiments found evidence to confirm the existence of a solid–solid phase transition in Zn induced by pressure at HT at pressures close to 50 GPa. In particular, the existence of a solid–solid phase transition is contradicted by the smooth behavior here determined for the melting curve. On the other hand, structural changes associated to solid–solid transitions can be detected by visual observation using the laser speckle technique here employed to determine the melt temperature [61, 62]. We have performed one of these experiments at 50 GPa and five at higher pressures and none of them provided evidence of a solid–solid phase transition up to the melt temperature. These results highlight the necessity for new shock-wave measurements in Zn using laser-driven shock techniques [63] to confirm that beyond 50 GPa Zn remains hexagonal at HT up to the melting temperature.

We performed XRD studies under compression up to 16 GPa at RT and also at HT. The intention of the studies was two-fold: to explore the influence of temperature on the behavior of the c/a ratio under compression and to determine a P–V–T EOS for Zn. At RT we found results which agree fully with previous studies [26, 27, 32]. In our case c/a continuously decreases under compression, reaching a value of $\sqrt{3}$ around 10 GPa. In particular, the two linear compressibilities determined at RT and ambient pressure are ${{k}_{a}}$  =  2.3 10⁻² GPa⁻¹ and ${{k}_{c}}$  =  9.8 10⁻² GPa⁻¹. Qualitatively similar results were found when different isotherms were collected. In figure 2 we show a selection of XRD patterns measured at different pressures, at temperatures between 455 and 470 K. In the figure, the relative difference between the evolution of the (0 0 2) and (1 0 0) Bragg reflections can be observed, which is a consequence of the change of c/a under compression. At 5.05 GPa the (0 0 2) reflection is at a lower angle than the (1 0 0) reflection; because c/a  >  $\sqrt{3}$. Under compression the (0 0 2) peak moves faster than the (1 0 0) peak due to the fact that c is considerably more compressible than a. As a consequence, both peaks overlap at 9.25 GPa, and at 12.5 GPa the (0 0 2) peak is at a higher angle than the (1 0 0) peak; because c/a  <  $\sqrt{3}$. From the experiments, we determined the pressure dependence of c/a.

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In figure 3 we plot $1-\sqrt{3}a/c$ versus pressure. The results correspond to experiments carried out at 300 and 600 K. This figure clearly shows the rapid change of c/a and the fact that c/a $=\sqrt{3}$ (which corresponds to $1-\sqrt{3}a/c=0$) is reached at a 'critical' pressure (P_C) of 10 GPa at 300 K and at 10.4 GPa at 600 K. We found that when isotherms at 460, 740, and 850 K are followed, c/a $=\sqrt{3}$ is also reached near 10 GPa, with a slight tendency to increase P_C as temperature is raised. From calculations we got a similar behavior for the c/a ratio than from experiments. Results from calculations at T  =  0 K are included in figure 3 to demonstrate that calculations reproduce the kink in c/a near 10 GPa. After c/a $=\sqrt{3}$ is reached we found that the pressure dependence of c/a changes because of a sudden reduction of the linear compressibility of the c-axis. This is clearly illustrated by the inset of figure 3 where the pressure derivative of c/a is shown for the experiment carried out at 300 K. In the inset it can be appreciated the abrupt change of the pressure dependence of c/a. The existence of this kink (slope change) in the pressure dependence of c/a has been attributed to a softening of acoustic modes [35] and can be caused by an isomorphic phase transition [64]. The fact that there is no detectable discontinuity in either the unit-cell volume or the c/a ratio suggest that the proposed phase transformation might be a second-order transition [65, 66]. To conclude with the discussion of the behavior of the c/a axial ratio, we would like to mention that calculations found that the ideal ratio c/a $=\sqrt{8/3}$ is reached at 50 GPa, which is in good agreement with previous experiments [32].

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Coincident with the change in the behavior of c/a, we noticed another interesting observation. At HT the polycrystalline Zn foil recrystallized into a single crystal. This is shown in figure 4, where the (0 0 2) and (1 0 0) single crystal reflections are shown in a CCD image measured at 4.8 GPa and 600 K. The same kind of single crystal diffraction pattern is found at 7.2 GPa and 600 K. However, when the pressure exceeds 10 GPa, the Zn crystal breaks into multiple crystallites, and the appearance of the corresponding diffraction patterns transform from single-crystal spots to highly textured Debye rings. This is illustrated in the CCD images collected at 10.4 and 13.2 GPa and 600 K. This phenomenon is fully consistent with the occurrence of a phase transition proposed based upon the behavior of c/a. Such transition would take place between isomorphic hexagonal structures, which we will call hcp (P  <  10 GPa) and hcp' (P  >  10 GPa). The first structure (hcp) possesses a ratio c/a $\geqslant \sqrt{3}$ with a larger axial compressibility along c than along a. The second structure (hcp') possesses a ratio c/a $\leqslant \sqrt{3}$ with a similar compressibility along c and a. The proposed second-order transition is consistent with phonon softening [65] proposed by calculations [35] and with changes reported for the resistivity of Zn by Lynch and Drickamer [67].

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In figure 5 we summarize the findings of the XRD experiments. The black squares represent all the P–T conditions at which the hcp phase was detected and the white squares the P–T conditions at which the hcp' phase was detected. According to the experiments, the phase boundary (dashed line) is nearly vertical and has a positive slope. Based upon the Clausius–Clapeyron equation [68], the steep phase boundary implies that the specific entropy change of the phase transition is nearly zero. This observation is consistent with the occurrence of a second-order transition. Regarding the positive slope (dT/dP) of the hcp-hcp' boundary, we would like to comment that solid–solid transitions could possess either a positive or a negative slope [69]. Examples of both cases are the γ  −  ε and α  −  ε transitions of iron [70]. Following Clausius–Clapeyron, the experimentally determined positive slope suggests that both the volume and the entropy decrease at the transition.

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In figure 5, we represent with open diamond P–T points where a rapid recrystallization of Zn was detected. Such a phenomenon is commonly found in the pre-melting state [1]. The melting is detected at a slightly higher temperature than the disappearance of the Bragg reflections of Zn. The P–T points corresponding to the detection of melting by XRD are shown as solid black diamonds in figure 5. The melting temperature detected by XRD is in good agreement with previous studies [24, 35, 36]. For consistency, melting points detected in the DAC experiments at P  <  16 GPa are also included in figure 5 (solid triangles). In order to illustrate the detection of melting in the XRD experiments, figure 6 shows a selection of XRD patterns collected in a heating sequence. The conditions gradually changed from 6.3 GPa and 300 K to 5.7 GPa and 953 K. At these conditions the melting of Zn is detected by means of the disappearance of the Bragg reflections of Zn and the appearance of weak diffuse scattering.

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From the XRD experiments we determined the unit-cell parameters for Zn at different pressure and temperature conditions. The results obtained from five isotherms are shown in figure 7. From the P–V data measured at RT we performed a third-order Birch–Murnaghan EOS fit to the data [71]. Since there is no discernible atomic volume discontinuity between hcp and hcp' the data collected in the pressure range have been included in the fit, as previously reported by other authors [27]. From the fit we obtained the ambient pressure bulk modulus B₀  =  63(2) GPa, and its pressure derivative $B_{0}^{\prime}$  =  5.6(4) as well as the unit-cell volume V₀  =  30.6(1) ų. These parameters are in good agreement with those previously reported [27]. From an isobar measured at 6.2(2) GPa, we have determined the linear thermal expansion of Zn, obtaining ${{\alpha}_{a}}$  =  15 10⁻⁶ K⁻¹ and ${{\alpha}_{c}}$  =  65 10⁻⁶ K⁻¹. These values are very similar to those determined at ambient pressure [72], which suggests that the thermal expansion of the hcp phase is barely affected by pressure. In addition, our results indicate that the thermal expansion along the c-axis is approximately four times larger than the thermal expansion along the a-axis. Noticeably, as discussed above, ${{k}_{c}}/{{k}_{a}}$ is also approximately four. This result agrees with the understanding that the thermal expansion and compressibility are closely correlated with the nature of the chemical bonding and the crystallographic structure of materials [73].

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Finally, we found that the isotherms shown in figure 7 can be described using the Birch–Murnaghan isothermal formalism [74]. The calculated isotherms are shown in the figure together with the experimental results. In the fit of the P–V–T EOS we assumed that the volumetric thermal expansion and pressure derivative of the bulk modulus are temperature independent and that only the bulk modulus is temperature dependent having a linear dependence: ${{B}_{0}}\left(T \right)=~{{B}_{0}}\left(300\,{\rm K} \right)+\beta (T-300\,{\rm K})$, with β the only fitting parameter. For the volumetric thermal expansion we used the value determined from the linear expansion coefficients assuming $\alpha =2{{\alpha}_{a}}+{{\alpha}_{c}}$  =  95 10⁻⁶ K⁻¹. For V₀, B₀, and $B_{0}^{\prime}$ at 300 K we used our own values: 30.6 ų, 63 GPa, and 5.6, respectively. As can be seen in figure 7 the assumed approximations are sufficient for describing the evolution of volume with P and T in the P–T range covered by our studies. The determined value for β was—9.8(9) 10⁻³ GPa K⁻¹, which indicates a decrease of the bulk modulus as the temperature increases. The results of the P–V–T EOS fit are summarized in table 1.

Table 1. Results of fitting the experimental volumes with the P–V–T EOS model described in the text.

V0 (Å3)	B0 (GPa)	$B_{0}^{\prime}$ (dimensionless)	α (K−1)	β (GPa K−1)
30.6(1)	63(2)	5.6(4)	95 10−6	9.8(9) 10−3