Solidification and fcc to metastable hcp phase transition in krypton under variable compression rates

We present time-resolved synchrotron x-ray diffraction measurements to study kinetics associated with the liquid-solid and solid-solid high-pressure phase transitions in Kr under dynamic loading in a dynamic-diamond anvil cell. The results show a strong compression-rate dependence of the solidification/melting process in liquid Kr. The analysis of the compression-rate dependent melting/solidification, using an Avrami equation with the parameter $n=1$, indicates a spontaneous nucleation and one-dimensional growth mechanism. In contrast, the face-centered-cubic to metastable hexagonal close-packed transition in solid Kr occurs rapidly at ∼0.8 GPa near the melting line, which has negligible compression-rate dependence within the range of compression rates studied $\left(0.004–13 \mathrm{GPa}/\mathrm{s}\right)$.

Figure 1

One representative applied waveforms (black line) into piezoactuators and the detector response (gray line) on the left vertical axis and the experimental pressures estimated from Cu (solid circles) and the lattice parameter of Kr-fcc (open diamonds) on the right vertical axes.

Figure 2

(a) Three representative diffraction images during the modulation. The number stated in the image is the scan number of the modulation. (b) The integrated diffraction patterns from the selected images. (c) A 2θ-scan_number plot of a series of time-resolved diffraction measurements with 50 ms exposure time, consisting of five consecutive, 1-s-long truncated sine wave pressure modulations and the calculated diffraction patterns of Cu (black), Kr-fcc (blue), and Kr-hcp (red) at 1.1 GPa.

Figure 3

(a) A stacking plot of x-ray diffraction patterns between 5° and 15° during compression. A broad peak centered at 9° is the liquid diffuse peak, the peak at 2θ = 13.5° is the Cu (111) diffraction peak and the peak at 2θ = 14.7° is the Kr-fcc (220) diffraction peak. The gray area illustrates the integrated area of liquid Kr. (b) The integrated area of liquid Kr (solid circles) on the left vertical axis and the pressure (open diamonds) on the right vertical axis as a function of time. The red lines are the exponential fits.

Figure 4

The decay/growth time constants τ (solid circles: compression, open circles: decompression) across the liquid-solid phase boundary as a function of compression rates in a linear scale and in a natural logarithmic scale (inset). The black line is the linear fit of the data during compression, and the red line is the linear fit of those during decompression.

Figure 5

Three representative 2D 2θ-time plots of time-resolved diffraction pattern (with a trapezoidal wave pressure modulation) at (a) faster ∼8 GPa/s (50 ms exposure), (b) moderate 1.9 (±0.1) GPa/s (200 ms exposure), and (c) slower 0.005 (±0.001) GPa/s (500 ms exposure) compression rates with its measured pressure (solid circles) and the bars showing the calculated diffraction patterns of Cu (blacks), Kr-fcc (blues), and Kr-hcp (reds) at ∼1.1 GPa.

Figure 6

(a) Lattice parameter of Kr-fcc (symbols, left vertical axis) and pressure (open circles, right vertical axis) as a function of time. Different open symbols represent the lattice parameter calculated from different diffraction peaks and the average of lattice parameter shows in solid circles. (b) Lattice parameters of Kr-hcp (symbols, left vertical axis) and modulation waveform (black curves, right vertical axis) as a function of time. Different open symbols represent the lattice parameter calculated from different diffraction peaks and the averages of lattice parameters show in solid circles.

Figure 7

(a) Volumes of Kr-fcc (solid symbols) and Kr-hcp (open symbols) at three representative dynamic loadings (1.9 ± 0.1 GPa/s: circles, 0.07 ± 0.03 GPa/s: triangles, and 0.02 ± 0.009 GPa/s: diamonds) as a function of relative time, time$/t_m$ ($t_m$: the modulation period). The inset plots the $c/a$ ratios of Kr-hcp at all dynamic loadings as a function of time$/t_m$. The black line in the inset locates the ideal $c/a$ ratio of hcp, 1.633. (b) Volumes of Kr-fcc (solid) and Kr-hcp (open) at dynamic loadings, volumes of fcc at static condition (curves), and volumes of fcc and hcp under laser-heating conditions (diamonds) as a function of pressure. The static $P\text{−}V$ curve is calculated from the third-order Birch-Murnaghan equation [13]. The volumes of fcc and hcp are obtained from the previous laser-heating experiments [10]. The inset shows the representative hysteresis of fcc and hcp during compression (bottom) and decompression (top).

Figure 8

The measured strain rate $(\Delta a/a_o/\Delta t)$ of Kr-fcc as a function of compression rates. The solid circles are calculated under compression and the open circles are under decompression. The black line is the linear fit of the measured strain rates and compression rates.

Figure 9

(a) The histogram of the time difference between when Kr-fcc first appears and when pressure starts to increase. (b) The histogram of the time difference between when Kr-hcp first appears and when pressure reaches the peak pressure. The red curves are the Gaussian fits, and the inset of the bottom panel illustrates the threshold pressures when Kr-hcp appears.