Compression-rate dependence of pressure-induced phase transitions in Bi

It is qualitatively well known that kinetics related to nucleation and growth can shift apparent phase boundaries from their equilibrium value. In this work, we have measured this effect in Bi using time-resolved X-ray diffraction with unprecedented 0.25 ms time resolution, accurately determining phase transition pressures at compression rates spanning five orders of magnitude (10–2–103 GPa/s) using the dynamic diamond anvil cell. An over-pressurization of the Bi-III/Bi-V phase boundary is observed at fast compression rates for different sample types and stress states, and the largest over-pressurization that is observed is ΔP = 2.5 GPa. The work presented here paves the way for future studies of transition kinetics at previously inaccessible compression rates.

When comparing the behaviour of samples loaded with and without a PTM, the stress state of the sample should be considered. Deviatoric stress can be identified based on shifts of Bragg reflections from their expected positions 32 . As we have identified the Bi-III/Bi-V transition by on the onset pressure, it is therefore necessary to consider the stress state of Bi-III. Le Bail refinements of the Bi-III structure based on diffraction patterns collected from samples loaded without a PTM show evidence of line shifts that are not observed for patterns collected from samples loaded in Ne (Fig. S6), suggesting a higher degree of deviatoric stress in samples loaded without a PTM. Although it would be beneficial to determine the uniaxial stress component using (for example) a line shift analysis 32 , this is unfortunately not possible due to the complex nature of the host-guest structure. Although it is in principle possible to perform a line-shift analysis for Bi-V, this would not be representative of the stress state before the transition because of the volume drop at the transition (~1 %), which could allow for partial stress release.

dDAC experiments
Samples were compressed by applying a trapezoidal voltage waveform as described in the main paper. Different dDAC designs (ECB and LLNL) employ different piezo actuators, and so DACs compressed using the different designs will respond differently to the same applied voltage. As is typical for compression ramps using both dDAC designs, the pressure does not have a linear response to the applied voltage and compression rate is not constant throughout the ramp, as can be seen in Fig. 2. Typically, the sample pressure does not increase until ~100V has been applied to the piezo actuator and the compression rate is much higher at later times in the ramp. This is most likely due to the mechanical response of the DAC, piezo and dDAC assembly and is independent of the sample material. For this reason, we have chosen to use the instantaneous compression rate and not the average compression rate when comparing different sample runs. For the instantaneous compression rate, the average rate over the last 5-10 single-phase Bi-III patterns before the transition was used, where lower-pressure patterns were chosen due to the definition of the transition by the onset pressure and to avoid the effect of the (small) volume change associated with the Bi-III/Bi-V transition. The kink at ~5 GPa in the slow compression profile is most likely associated with the solidification of Ne.
For the slow ramps, the pressure at the top of the ramp as estimated from the two detectors can differ by as much as 0.5 GPa which is evidence of an asymmetric radial stress field in the sample chamber. Note that the detectors are symmetrically offset in the horizontal, but they are translated down in the vertical so that they collect the upper portion of the diffraction rings.
Although stress gradients of this magnitude in the Ne loaded sample appear surprising, it should be noted that Ne is not truly hydrostatic at the pressures > 4.8 GPa as it is a solid. On release, the sample pressure often did not fully decompress to the starting pressure. This was consistently more pronounced for hydrostatic sample loadings, and in many cases the pressure had to be released my loosening the screws of the DAC for the sample to transform back to Bi-III. This effect was observed in all experimental runs and is not sample dependent, but most likely due to plastic deformation of the metallic gasket.  (2101), as these were clearly observed for all samples. The (hklm) superspace notation for Bi-III is described by McMahon et al. 33 The unit cell volume of Bi-V was determined from the peak position of the (211) reflection in the mixed phase patterns, and from both the (110) and (211) positions at higher pressures.
As the samples were compressed through the Bi-III/Bi-V transition, significant changes in microstructure were observed for both sample types; the diffraction patterns from Bi-III were consistently observed to be much spottier than those from Bi-V (Fig. S8). Due to the spotty nature of the Bi-III diffraction images, the Bi-III/Bi-V phase transition was identified by the first observation of Bi-V in the diffraction patterns. The (110) and (200) Bi-V reflections both overlap with peaks from the Bi-III phase, and so in most cases the onset of the transition was identified by the appearance of the (211) Bi-V reflection. When Au was used for pressure estimation, the onset transition pressure was taken as the midpoint between that of the last single-phase Bi-III pattern and the first mixed-phase Bi-III/Bi-V pattern. When the pressure was determined from Bi-V, the onset transition pressure was determined as the pressure of the pattern in which Bi-V is first observed. Although it would be desirable to also determine the pressure using the EOS of Bi-III, this was not always possible due to the pronounced preferred orientation in this phase; Bi-III reflections were not consistently observed in all experimental runs, and in many cases there were not enough observed reflections to determine all three lattice parameters (a, chost and cguest). Due to the spotty nature of the diffraction patterns from the foil sample and the fact that the detectors do not give full azimuthal coverage and diffraction spots that originate in areas between the sensitive areas of the detectors cannot be detected, we note that the reported transition pressures for the foil samples likely do not represent the lowest pressure where Bi-V is observed but some higher pressure that is still below the pressure at which the Bi-V phase transformation is fully completed.
dDAC Experiments: Sample/pressure marker assembly We compared the transition pressures determined based on Bi and Au for all dynamic compression experiments (Fig. S4). For samples loaded in Ne, transition pressures determined using both methods are in good agreement (Fig S4a,b), suggesting that the use of a pressure marker for samples loaded in a 'soft' pressure medium like Ne is justified. However, for samples loaded without a PTM, the compression-rate response is highly dependent on the position of the pressure marker within the sample chamber. For mixed Bi/Au powder samples compressed without a PTM, the Au-determined transition pressures are very scattered, and no clear compression-rate dependent shift is evident (Fig. S4c). For foil samples loaded without a PTM, the magnitude of the Bi-III/Bi-V over-pressurization is overestimated by the Au (Fig.   S4d). A compression-rate dependent response of the Bi/Au assembly is clearly visible in the compression curves collected during fast and slow compression (Fig. S2), and in the integrated diffraction patterns collected at the same pressure during fast and slow compressions ( Fig. S3 and Table S3). These results highlight a potential pitfall for future dDAC experiments, where sample/pressure marker effects can potentially be incorrectly identified as compression-rate dependent effects in the sample.     Table S3.