High pressure transformations in liquid rubidium

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High pressure transformations in liquid rubidium

Simon Ayrinhac, Victor Robinson, Frédéric Decremps, Michel Gauthier, Daniele Antonangeli, Sandro Scandolo, Marc Morand

To cite this version:

Simon Ayrinhac, Victor Robinson, Frédéric Decremps, Michel Gauthier, Daniele Antonangeli, et al..

High pressure transformations in liquid rubidium. Physical Review Materials, American Physical

Society, In press. �hal-02998346�

High pressure transformations in liquid rubidium

Simon Ayrinhac

, Victor Naden Robinson

, Fr´ed´eric Decremps

, Michel Gauthier

, Daniele Antonangeli

, Sandro Scandolo

, Marc Morand

1

Sorbonne Universit´e, UMR CNRS 7590,

Mus´eum National d’Histoire Naturelle, Institut de Min´eralogie,

de Physique des Mat´eriaux et de Cosmochimie, IMPMC, 75005 Paris, France

2

The “Abdus Salam” International Centre for Theoretical Physics, I-34151 Trieste, Italy (Dated: November 10, 2020)

Abstract

An electronic-driven liquid-liquid phase transition in rubidium is revealed by picosecond acoustic measurements combined with ab initio calculations. Picosecond acoustics were used to measure the melting line up to 10 GPa, finding the maximum in the melting curve at 7 GPa and 555 K.

We observe the onset of a continuous liquid-liquid phase transition beginning around the melting maxima through until 16 GPa. Sound velocity shows a softening similar to that reported for liquid caesium, caused by a change in the bulk modulus during a crossover from the low-density to the high-density liquid. Guided by the ab initio calculations, we relate the changes in the thermo- elastic properties to the progressive localization of the valence electrons in the pressure range of 6-16 GPa. At high pressure rubidium forms an electride liquid quantified by the appearance of interstitial quasi atoms (ISQs) localised in the valence electron density.

a

Alkali metals are considered archetypal simple metals at ambient conditions, yet exhibit complex properties at high pressure: phase diagrams possess melting curves with several maxima (Cs

), and host-guest (h-g) incommensurate solid phases

. Light alkali metals be- come liquid at ambient temperature upon compression to very high pressure (Li at 50 GPa

, Na at 120 GPa

) or present a metal-insulator transition (Na becomes transparent at 200 GPa and ambient temperature

). Structural transformations in the liquid have been reported in Cs at 4 GPa and 493 K

and in Rb between 7.5 and 12.9 GPa at 573 K

.

In parallel with the structural transformations the electronic states of dense alkalis loose the free-electron metallic character in favor of orbital re-hybridization and localization, lead- ing to a reduction of metallicity, and in the extreme case of Na even to the opening of a gap. Consensus is emerging, at least in light alkali metals, that electron localization is the result of the pressure-induced expulsion of valence electrons into the interstitial regions of the lattice leading to the formation of an electride state. The existence of electrides among solid alkali metals has been well established from pioneering

and more recent work

. Light alkali metals were first recognized to be electride solids at high pressure: Li at 60 GPa

and Na at 250 GPa

. Alkali metals which form complex h-g structures (Na, K, Rb) can be explained by their electride nature

. However, the e↵ects are not as striking in heavy alkali metals. Moreover the presence of electrides remains to be confirmed and quantified for liquids.

At room temperature, solid Rb undergoes a series of transformations

: from bcc (Rb-I) to fcc (Rb-II) at 7 GPa

, to Rb-III (oC 52) at 13 GPa

, to Rb-IV at 16 GPa (incommensurate h-g

) with an order-disorder transition

at 16.5 GPa, to Rb-V above 20 GPa (tI 4)

and to Rb-VI around 50 GPa

(see Fig. 1). As is often the case, studies addressing the liquid are more controversial.

Early work interpreted anomalies of electrical conductivity and volume occurring upon compression as signature of an electronic s-d transition

. A first-order liquid-liquid phase transition (LLPT) was suggested by ab initio molecular dynamics (AIMD) at 12.9 GPa and 573 K

on the basis of anomalies in the equation of state and the adiabatic coefficient, . Conversely, a gradual transition from a simple hard-sphere liquid to a complex liquid was recently proposed to occur in Rb at 7.5 ± 1 GPa

from measured X-ray di↵raction (XRD).

Measurements on heavy alkali metals are challenging due to their high reactivity, therefore

thermodynamic data at high P,T is scarce or controversial (Cs

, Rb

).

Sound velocity measurements are a sensitive tool to detect structural transformations in the liquid state, such as Sn

, Bi

, Ga

, or Cs

. Thus we perform picosecond acoustics (PA) measurements, a non-destructive technique well adapted to micrometric samples of liquid metals

, inside a high pressure diamond anvil cell (DAC). To investigate the micro- scopic origin of the anomalies observed by sound velocity measurements, AIMD simulations of the liquid were carried out, analyzing the structural and electronic changes. In this Let- ter our combined experimental and computational study highlights an important structural transformation at 573 K from 6 GPa and extending up to 14 GPa resulting from the progres- sive change from the low-pressure free-electron-like liquid state to a high-pressure electride liquid.

PA is a time-resolved pump-probe optical technique that generates propagating strain waves in solids or liquids

. An 800 nm infrared laser pump pulse (100 fs width, 80 MHz repetition rate) is focused through a DAC at the surface of the metallic sample. The optically absorbed beam creates an acoustic strain pulse propagating across the sample, which is then detected at the opposite surface by a probe beam. The acoustic echoes induce changes in the refractive index of the liquid metal and displacement of the surface, which are measured by relative reflectivity variations seen with the probe beam. These surface reflections are used both to determine the travel time of the acoustic echoes, and hence sound velocity, and to identify melting as they strongly depend on the long range order of the material (see SI and

).

A sample chipped out from a Rb ingot (99.6% purity Sigma Aldrich

) was loaded into a hole drilled in a rhenium gasket inside the DAC. To minimize chemical contamination and reaction with air, the loading was performed in a glove box (<5 ppm O

, H

O). The pressure was measured by the shift of the temperature-independent fluorescence band of a SrB

O

:5%Sm

gauge placed in the sample chamber to act as a pressure calibrant

. Pressure determination from the internal pressure gauge is compared to the estimation from the high-frequency edge of the diamond-anvil Raman signal measured at the center of the gasket hole

.

Due to unknown stress distribution inside the diamond anvils, the P scale was corrected

to known pressure points (ambient pressure if reached, for example). High temperature

was generated by a calibrated resistive heater surrounding the DAC.The adiabatic sound

velocity v

is obtained dividing the gasket thickness (around 20 µm) by the time of flight of

the propagating acoustic strain wave. To ensure the hole thickness remained constant

, measurements were made during sample decompression with a fine DAC membrane control (0.05 bar/min).

Density functional theory calculations utilized the CASTEP code

, using the generalized- gradient approximated exchange-correlation

, and a 9-electron ultrasoft pseudopotential with 2.1 bohr inner-core radius. A 300 eV plane wave cuto↵ was used in AIMD with k- points sampled at the -point. For solids, a grid density of 0.02 ˚ A

was used for structure optimisation in agreement with Ref.

. Simulations ran at fixed density with a Nose-Hoover thermostatstarting from a 128 atom melted bcc supercell quenched down to either 573 or 800 K. The liquid was simulated up to 30 ps along the 573 and 800 K isotherms with a 0.75 fs timestep and sampling the stress tensor every 10 steps.

After liquid equilibration, the mean internal energy, U , was used to determine isochoric specific heat, C

=

. The isothermal compressibility,

=

, where V is the molar volume, was computed from the AIMD equation of state. The thermal expansion coefficient was calculated by ↵ =

@T V

, and isobaric specific heat, C

, was determined from Mayer’s relation C

= C

+ V T

↵

. From the adiabatic ratio, = C

/C

, the adiabatic sound velocity, v

, can be computed with v

= p

/⇢

where ⇢ is the density.

The melting curve of Rb is well known below 7 GPa

but remains debated at higher pressures. As illustrated in the phase diagram shown in Fig. 1 inconsistencies exist amongst available experimental data: the liquid-solid transition at 618 K from Lundegaard

appears at 17 GPa, as opposed to 24 GPa by Gorelli et al

, yet both were determined by XRD.

Here, melting was detected using PA by imaging changes in wavefront patterns appearing at the sample surface

: perfect circles indicate a liquid sample whereas more complex patterns are observed in solid phases (see SI). In addition, in liquid phase, the signal is less noisy or disturbed, and the position of the echo peak shifts due to variation of sound velocity between solid and liquid phases. Determination of the solid and melted states up to 10 GPa confirm the first maximum at 7 GPa and 555 K, and are in good agreement with Kechin et al.

, and Boehler and Zha

.

The phase diagram and the pressure evolution of the liquid sound velocity (see Fig. 2)

of Rb are both similar to that of Cs suggesting a double maximum in the melting line as

already proposed by Boehler and Zha

. However, data beyond 10 GPa are limited: the

thesis of Lundegaard reports molten Rb at 3 P,T points

, in overall agreement with the low

temperature melting line suggested by Boehler and Zha

, in spite of possible problems of sample contamination occurring above 10 GPa. This thesis also contains unpublished liquid structure factors of Rb around 10, 13, 18 GPa (see SI).

In the liquid, we probed the adiabatic sound velocities along the 573 K isotherm, system- atically observing in the multiple performed runs an anomalous behavior (Fig. 2). Plateaus were found in v

between 6 and 14 GPa in experimental runs 1,3, while run 2 and AIMD (573, 800 K) found an oscillation in this region similar to the N-shape observed in liquid Cs

. A simple two state model, which describes a system of entities in two thermodynamic states

, in this case the two liquids, also finds an oscillation as the liquids mix .While PA measurements do not provide highly accurate absolute value of v

due to the uncertainty on sample thickness ( v

' ± 100 m/s), our results agree closely with the data of Shaw

up to 0.7 GPa and IXS determinations from Bryk

up to 6 GPa. At pressures over 6 GPa we find that PA and AIMD both have a lower v

than the simulations of Bryk, where PA data appears in-between the v

profiles of Bryk and AIMD performed in this work.

As is clear from the change in the Clapeyron slope, the liquid becomes denser than the solid after the first melting line maxima at about 7 GPa, and the v

plateau starts from this pressure. The complex, dense, phases of solid Rb (III,IV,V) become stable around 14 GPa, pressure above which v

starts again to rise with increasing pressure. We thus interpret the anomalous pressure evolution of the adiabatic velocity as a direct consequence of a continuous but rapid transformation between two distinct microscopic states of the liquid.

As shown in Fig. 3, while density does not show any sharp discontinuity in the transition region, where two states coexist, the bulk modulus shows a plateau, directly related to the plateau of the sound velocity. Conversely, sound velocity shows no evidence for anomalies due to a second transition expected at 12.9 GPa according to Bryk

.

Thermodynamic properties of liquid Rb are summarised in Fig. 3 which show no indica-

tions of a first or second-order phase transition, suggesting that this is a gradual crossover

between two types of liquid. Density was obtained from AIMD simulations showing agree-

ment with previous experimental and computational determinations

. A small inflection

is visible around 8-9 GPa in correspondence to the anomalous sound velocity. The den-

sity of the bcc solid at 7 GPa, corresponding to the maximum in the melting curve, is

3140 ± 20 kg/m

(see SI) in agreement with the density of the liquid as from AIMD and

data of Bryk. The density of AIMD in this work is slightly lower than that of Bryk below

9 GPa but significantly greater beyond 12 GPa. Up to 0.7 GPa the density was calcu- lated by a numerical procedure

from the accurate adiabatic sound velocity data of Shaw

47

extrapolated to 573 K, consistent with AIMD. The isothermal bulk modulus, related to density by B

= ⇢ (@ p/@⇢)

, reveals a transition between two clear profiles as illustrated in Fig. 3b. Thermal expansivity (Fig. 3c) exhibits a minima between 7 and 14 GPa which is also expected by the two state model, and the specific heat capacities appear to have local maxima and minima in this pressure region. The adiabatic ratio, Fig. 3d, shows a decreas- ing trend with pressure similar to other compressed liquids (H

, K

) , but in contrast to results from Bryk, who reported much higher values, and argued for a discontinuity at 12.9 GPa not supported by our calculations. In fact, such high values of gamma would yield even greater sound velocities with a discontinuity if computed with the compressibility from AIMD in this work.

As is clear from the bulk modulus, liquid Rb is highly compressible at ambient conditions (B

= 1.54 GPa), with a B

value comparable with noble-gas solid Ne

(B

= 1.07 GPa), and by 5 GPa V /V

=0.5. At this compression ratio two Rb

ions are so close (3.86 ˚ A) that the valence shell (radius of 2.44 ˚ A) of one overlap with the impenetrable ion core of the nearest other (radius of 1.48 ˚ A), pushing the conduction electrons to migrate inside voids between ion cores. An electronic localization starts to occur around 6 GPa (see Fig. 4a), when a fraction of valence electrons start becoming confined in spaces between the ion cores, producing interstitial quasiatoms (ISQs)

.

The localization is evident in AIMD snapshots comparing the interstitial electron local- ization at 1 and 20 GPa in Fig. 4b. ISQs were detected as o↵-atom maxima in a Bader analysis of the electron density

averaged over 100 snapshots at selected pressures (see Fig. 4a). From 6 GPa, the number of ISQ (and their total charge) increase with pressure (see Fig. 4a) up to 0.2 e/ISQ around 14 GPa, where Rb-III and Rb-IV become stable. Com- pared to liquid Rb, in Rb-IV 0.3 e/ISQ were found at 20 GPa in the Bader scheme, but around 1.1 e/ISQ when integrating electron localization function (ELF) basins

, suggesting that the value in the liquid may be three times higher if ELF basins were integrated instead (to 0.6 e/ISQ at 14 GPa). The build up of electron localization with pressure was noted by Bryk

presenting 3D ELF isosurfaces. A comparison with the liquid and solid phases of Rb up to 50 GPa, and the distribution of charge within ISQs in the liquid, is shown in the SI.

The Rb-Rb coordination number c

drops from 10/11 GPa to around 8 (see Fig. 4d) which

is when the ISQ’s have significantly asserted their presence in the liquid. The decrease in c

is suggested, in analogy to the liquid-liquid transformation in Cs

, though the author stresses that higher quality structural data are required to determine the nature of the high- and low-density liquids.

Miao et al showed that for solids above 10 GPa the “Rb 5s” valence orbital is favored to form a “ISQ 1s” state (see Fig. 5 of Ref.

). Here Rb is able to access these states already from 6 GPa as the liquid is not restricted by symmetry. The creation of more ISQs, containing a fractional charge, attract Rb

ions and o↵er additional screening of interactions which can reduce the internal energy and cause a deepening of the attractive part of the potential. This can cause the softening evidenced in both the isothermal bulk modulus and sound velocity.

Density normalized pair distribution functions (PDFs) (Fig. 4e) show a simple Lennard- Jones liquid structure at low P, and the apparition of an intermediate coordination shell between 8.9 and 13.8 GPa in conjunction with the continuous decrease of coordination number similar to Cs

. The thesis of Lundegaard eluded the similar transformation in liquid Cs

in the measured s(q), suggesting a decrease in the coordination number (c

drops to around 8), though the author stresses that higher quality structural data are required to determine the nature of the high- and low-density liquids.

From 10/11 GPa, coordination number between Rb ions starts decreasing following the trend observed in solid phases: c

=14 in Rb-I (bcc), 12 in Rb-II (fcc), 8/9 in RB-III (oc52), 9 in Rb-IV (h-g ), and 8 in Rb-V (tI 4). The coordination numbers measured by Gorelli

also decrease above 10 GPa but are significantly lower than AIMD. Similarly their PDFs (see SI) have much broader first peaks compared to AIMD results discussed here and of Bryk

. Lower coordination indicates an open structure, likely due to creation of directional bonds between Rb

cations and ISQ anions

, creating a “complex” liquid, with a structure similar to the high pressure solid phases mediated by the localized electrons.

In summary, we mapped the phase diagram of Rb at high pressure probing melting and a

liquid-liquid transition through sound velocity measurements, and used ab initio simulations

to understand the nature of this transition. The Rb melting line was accurately measured up

to 10 GPa by picosecond acoustic techniques, clearly showing a maximum at 7 GPa. Multiple

experiments found a progressive pressure-induced transformation in liquid Rb, highlighted

by a plateau in the sound velocity between 6 and 14 GPa, while AIMD simulations similarly

found an oscillation over the same pressure range. This anomaly is caused by an increase of compressibility concomitant to a continuous transition between two microscopically distinct states of the liquid. Absence of discontinuities in the density and compressibility clearly indicate that the phase transition is neither first nor second-order, as claimed by previous work, but a crossover. The transition is related to a progressive localization of the electrons in interstitial regions as Rb forms a complex weak electride liquid. This is as well in stark contrast to previous explanations of 5s electrons occupying the 4d band, as partial density of states calculations can misinterpret electride materials, similarly shown for the solid alkali metals

. The maximum in the melting line requires the liquid to be less dense than the solid below 7 GPa and denser than the solid above 7 GPa. Macroscopically the liquid continuously take on properties of the high pressure phases (thermodynamically stable after the melting minima). The properties of the complex solid phases, including structural transformations and electride character, are mirrored in the liquid, as they are continuously adopted, with an extended transition region as the liquid-liquid transition mixes the two types of liquids.

We thank H. Moutaabbid for his supporting help on the glove box ; G. Le Marchand, Y. Guarnelli and P. Parisiadis for their support on high pressure devices. D.A. has re- ceived funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation Programme (grant agreement No. 724690). Authors warmly thank Y. Garino, P. Rosier, N. Dumesnil for their strong implication on the current picosecond acoustics set-up upgrade.

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FIGURES

FIG. 1. Phase diagram of Rb: Solid (blue squares) and liquid (red squares) phases were detected

using PA. The melting line is shown up to 10 GPa (dashed line) according to the functional form

given by Kechin

. The melting maximum is at 7 GPa, 555 K (empty star). Transitions from bcc

to fcc (crosses) and melting (pluses) are from Boehler

. Data at high P-T

is scarce and

not always consistent. Sound velocity measurements in this work were performed along the 573 K

isotherm as in Gorelli et al.

(red circles).

PA run 1 PA run 2 PA run 3 AIMD 573 K AIMD 800 K Bryk (2013) - Sim Bryk (2013) - IXS Shaw (1985) Two state model 573 K

v s ( m /s )

1500 2000 2500 3000

P (GPa)

0 5 10 15 20

FIG. 2. Adiabatic sound velocities of liquid rubidium at 573 K as a function of pressure, measured

by PA (black points), AIMD simulations (blue squares), and a two state thermodynamic model

(magenta, see SI). For comparison, ultrasonic measurements are shown from Shaw et al

up to

0.7 GPa and the AIMD and IXS measurements from Bryk et al

.

(d)

G a m m a ( C

/ C

)

1 1.1 1.2 1.3

P (GPa)

0 5 10 15 20

(c)

a lp h a ( 1 0

K

) 0.25

0.5 0.75 1

P (GPa)

0 5 10 15 20

(b)

AIMD Bryk (2013) Shaw (1985)

B

( G P a )

0 10 20 30 (a) 40

From solid AIMD Gorelli (2018) Bryk (2013) Shaw (1985)

ρ ( k g /m

) 2000 3000 4000 5000

CV

CP

CV,P [J mol-1 K-1 ]

24 26 28

0 5 10 15 20

FIG. 3. Thermodynamic properties of liquid Rb at 573 K as a function of pressure including literature data from Gorelli

(XRD, pink), Bryk

(AIMD, black), and Shaw

(from ultrasonic measurements, blue). (a) Density; at the maxium of melting curve, the density of the solid (open dotted circle) is equal to that of the liquid. (b) Isothermal bulk modulus B

with dashed lines to guide the eye. (c) Thermal expansivity; inset: specific heat capacities. (d) Adiabatic ratio

= C

/C

.

(b)

(e)

0.54 GPa 2.86 GPa 8.92 GPa 13.8 GPa

13.8 GPa Rb-ISQ 16.2 GPa

23.7 GPa

g (r )

0 1 2 3

r ⋅ ρ

1 1.5 2

(d)

AIMD Bryk (2013) Gorelli (2018)

c

( R b -R b ) 8 10 12 14

P (GPa)

0 5 10 15 20

c ( R b -I S Q )

(c) 0.2 0.4 0.6

Interstistial e

Rb-III Frequency q

(a) q

( e )

0 0.05 0.1 0.15 0.2

20 GPa 1 GPa

FIG. 4. Electronic and structural properties of liquid Rb as a function of pressure at 573 K. (a)

Average charge inside interstitial Bader maxima (blue) and their frequency (red) per Rb ion. (b)

ELF (max=0.70) 100 cuts from AIMD snapshots at 1, 20 GPa showing localized interstitial charge

(yellow to red amorphous shapes) appear between ions (red circles). (c) Coordination numbers for

Rb ions to ISQs calculated by integrating PDFs up to the first minima and (d) for Rb-Rb ions. (e)

Normalized Rb-Rb PDFs at selected pressures showing the first minima evolve into a new maxima,

and a Rb-ISQ PDF at 13.8 GPa demonstrates the interstitial occupation of electride sites.

Supplementary Information: “High pressure transformations in liquid rubidium”

Simon Ayrinhac ¹ , Victor Naden Robinson ² , Fr´ed´eric Decremps ¹ , Michel Gauthier ¹ , Daniele Antonangeli ¹ , Sandro Scandolo ² , Marc Morand ¹

November 10, 2020

1. Sorbonne Universit´e, UMR CNRS 7590, Mus´eum National d’Histoire Naturelle, Insti- tut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie, IMPMC, 75005 Paris, France

2. The “Abdus Salam” International Centre for Theoretical Physics, 34151, Trieste, Italy

Contents

1 Samples 2

2 Surface reflections 3

2.1 Surface imagery, reference pictures . . . . 3

2.2 Run A . . . . 5

2.3 Run B . . . . 7

2.4 Run C . . . . 8

3 Two-state thermodynamic model 10 4 Structure factors S(q) and pair distribution functions (PDFs) 11 5 Thermodynamic parameters 15 5.1 Equation of state in the solid phase . . . . 15

5.2 Adiabatic index from ultrasonic measurements . . . . 16 6 Interstitial quasi-atoms (ISQ) in solid and liquid Rb 18

7 Equation of state and di↵usion coefficient 19

1

1 Samples

Lot A Lot B

Lot Number MKBJ1800V MKBV0959V

Quality Release Date 5 oct. 2011 1 apr. 2015 Trace Metal Analysis < 5000.0 1284.4

Aluminum (Al) 1.7 4.4

Boron (B) 3.0 6.2

Barium (Ba) <0.1 0.1

Calcium (Ca) 2.2 34.3

Cobalt (Co) 0.6

Chromium (Cr) 89.8 1.3

Copper (Cu) 5.3 3.9

Iron (Fe) 349.9 7.5

Magnesium (Mg) 0.4 5.8

Manganese (Mn) 8.0 0.2

Nickel (Ni) 43.6 1.7

Titanium (Ti) <0.1 0.1

Vanadium (V) 0.5

Zinc (Zn) 0.7 62.1

Lithium (Li) 0.2

Sodium (Na) 10.0 28.3

Potassium (K) 3.0 775.1

Cesium (Cs) 36.1 352.6

Table 1 | Impurities in Rb samples (Rb ingot 99.6%, ref. 276332) from Certificate of Analysis provided by Sigma-Aldrich. Quantities are expressed in ppm. Lot A was used for ”phase diagram”

and ”run 2” measurements, and lot B was used for ”run 1” and ”run 3” measurements (see main text).

2

2 Surface reflections

Here we show evidence of solid phases (bcc, fcc) and melting from the surface patterns up to 10 GPa.

P = 0 GPa P ≈ 10 GPa P ≈ 20 GPa

Supplementary Figure S1 | Reduction of the hole, drilled in the rhenium gasket and filled with liquid Rb, during pressure upstroke. The drastic reduction is due to high compressibility of the liquid Rb. In addition, this pictures show the good quality of the interface liquid Rb / diamond.

2.1 Surface imagery, reference pictures

3

Ga Hg

Liquid

Solid

Supplementary Figure S2 | Example of surface patterns (also called phonon imagery) obtained at the interface diamond/sample in Hg and Ga. The images correspond to a surface’s size of 100 µm ⇥ 100 µm. Reflectivity is in color scale from blue (low) to red (high). In liquid, due to elastic isotropy, perfect circles are obtained at the surface of the sample. On the contrary, in solid phase more complicated patterns are obtained due to elastic anisotropy and polycrystalline structure (superposed elliptical patterns are seen in solid Hg, or nearly-perfect circles in solid Ga).

Supplementary Figure S3 | Modifications in the patterns appearing at the interface diamond / liquid-Hg during the transition solid to liquid (real time 2-3 min.).

4

2.2 Run A

2.1 GPa, 453 K 5.5 ns 2.52 GPa, 475 K

7.0 ns

5.8 ns 8.0 ns

2.45 GPa, 503 K 8.5 ns

6.6 ns 1.75 GPa, 421 K

12 ns 1.0 GPa, 393 K

13 ns 9.9 ns

Supplementary Figure S4 | Surface imagery and phase diagram of Rb along an isobaric path around 2 GPa (run A). For each temperature and pressure, di↵erent images are obtained at di↵erent times (di↵erent pump-probe delays). In liquid, the surface patterns are circular and less noisy than in solid. Note that the demodulated relative variations of reflectivity are quantified with an arbitrary color scale (blue for low values to red for high values). The determination of melting was done in conjunction with the information carried out by temporal scans (see Fig. S5).

5

Supplementary Figure S5 | Rb, run A, around 2 GPa, temporal signals (pump and probe are spatially colinear). Between the transition from liquid to solid state, the echo peak shifts to a higher time delay because the velocity in the liquid is lesser than in the solid, the echo peak becomes more narrow, and the overall signal is less noisy or disturbed (because in the solid, there is some echoes due to the longitudinal and the transverses waves).

6

2.3 Run B

4.0 ns 6.05 GPa, 534 K 2.0 ns

6.0 ns 6.15 GPa, 554 K 4.0 ns

4.5 ns 5.9 GPa, 503 K

6.0 ns 3.0 ns

Supplementary Figure S6 | Surface imagery and phase diagram of Rb along an isobaric path around 6 GPa (run B).

Supplementary Figure S7 | Rb, run B, around 6 GPa, temporal signals (pump and probe are spatially colinear).

7

2.4 Run C

3.0 ns 8.6 GPa, 503 K

5.0 ns 1.5 ns

2.5 ns 8.65 GPa, 520 K 1.5 ns

3.5 ns 8.60 GPa, 538 K 2.5 ns 3.5 ns

8.60 GPa, 557 K

3.0 ns 8.5 GPa, 503 K

5.0 ns 2.5 ns

3.0 ns 8.5 GPa, 484 K 2.0 ns

Supplementary Figure S8 | Surface imagery and phase diagram of Rb along an isobaric path, around 8.5 GPa (run C).

8

Supplementary Figure S9 | Rb, run C, around 8.5 GPa, temporal signals (pump and probe are spatially colinear).

9

3 Two-state thermodynamic model

The Gibbs free energy for an ideally mixed solution is given by [10]

G(p, T ) = xG

(p, T ) + (1 x)G

(p, T ) + RT [x log x + (1 x) log (1 x)], (1) where T is the temperature, p is the pressure, x is the fraction of state A, R specifies the ratio between the total number of mixing entities and the total number of atoms through R/R

where R

is the ideal constant gas, and G

and G

are the Gibbs free energies of A and B when not mixed. When minimized with respect to x, the equation (1) gives

x(p, T ) = e

e

+ e

. (2)

The isothermal compressibility is defined as



= 1

B

= 1 V

✓ @V

@ p

◆

T

= 1

⇢

✓ @⇢

@ p

◆

T

, (3)

where B

is the isothermal bulk modulus, V is the molar volume, and ⇢ = M/V is the density with M the molar mass. 

is used to calculate the isothermal sound velocity according to the two state model with v

= 1/ p ⇢

. The calculation gives



= x

+ (1 x)

+ V V

✓ @x

@ p

◆

T

, (4)

where ⇣

@x

@p

⌘

T

=

x(1 x) and G = G

G

= RT [ln(1 x) ln(x)]. Then



= x

+ (1 x)

+ V

V RT x(1 x). (5)

The following parameters were used to model the anomalous component: V /V = 0.11, S = 0.44 k

with k

is the Boltzmann constant, and R/R = 0.25 where R

is the ideal gas constant, and a linear model of the bulk modulus was used for the two liquids.

10

4 Structure factors S(q) and pair distribution functions (PDFs)

Here we show liquid structure factor S(q) in Rb at 573 K from AIMD (this work), from Gorelli [8], and from the thesis of Lundegaard (10, 13, 18 GPa) [12] which contains data from similar temperatures.

10.2 GPa Lundegaard 12.6 GPa Lundegaard

10 GPa Gorelli 11.87 GPa AIMD

S(Q)

0 10 20 30 40 50 60 70

Q (Å^-1)

1 2 3 4 5 6 7

17.1 GPa Lundegaard 16 GPa Gorelli 16.15 GPa AIMD

S(Q)

0 10 20 30 40 50 60

Q (Å

-1)

1 2 3 4 5 6 7

23 GPa Gorelli 17.1 GPa Lundegaard 23.72 GPa AIMD

S(Q)

0 10 20 30 40 50 60

Q (Å

-1)

1 2 3 4 5 6 7

Supplementary Figure S10 | Liquid structure factor S(q) in Rb at 573 K at various pressures.

11

Rb-Rb 13.8 GPa Bryk 12.4 GPa Gorelli 16 GPa

g(r)ab

0 0.5 1 1.5 2 2.5 3

r [ÅÅ]

2 3 4 5 6 7

Supplementary Figure S11 | Pair distribution functions (PDF) presented by Gorelli et al [8]

and those from this work (AIMD) and Bryk et al (AIMD) [7].

12

2.05 GPa 5.81 GPa 8.92 GPa

11.87 GPa 13.76 GPa 14.89 GPa

16.15 GPa 19.20 GPa 23.72 GPa

S(Q)AIMD

0 0.5 1 1.5 2 2.5

Q (Å

-1)

1 2 3 4 5 6 7

Supplementary Figure S12 | (this work) AIMD structure factors S(q) at high pressures up to 24 GPa.

13

0.54 GPa 0.69 GPa 0.86 GPa 1.27 GPa 2.05 GPa 2.86 GPa 3.92 GPa 4.79 GPa 5.81 GPa 6.98 GPa 7.60 GPa

8.25 GPa 8.92 GPa 10.31 GPa 11.87 GPa 13.76 GPa 14.89 GPa 16.15 GPa 19.20 GPa 21.35 GPa 23.72 GPa

g(r)

0 0.5 1 1.5 2 2.5 3 3.5

r [ÅÅ]

2 4 6 8 10

Supplementary Figure S13 | AIMD pair distribution functions (this work) up to 24 GPa. It shows the creation of an intermediate shell of Rb atoms (around r ' 4.5˚ A) between the first (r ' 3.5˚ A) and second coordinations shells (r ' 6˚ A) (previously observed by Bryk et al [7]).

14

5 Thermodynamic parameters

Here we provide an explanation of “from solid” point in figure 3a (density), and data from Shaw (blue lines).

5.1 Equation of state in the solid phase

At P ,T conditions of the first maximum in the melting curve, (occurring at 7 GPa and 555 K) the densities of solid and liquid are equal. This equality comes from Clausius-Clapeyron relation

dT

dP = V

S , (6)

where V is the volume di↵erence between solid and liquid and S is the entropy di↵erence between solid and liquid. At the maximum

= 0, V = 0 regardless the value of S (di↵erent from 0). The knowledge of density in the solid phase therefore gives a constraint on the density of the liquid at high P-T.

The thermal Vinet equation of state (EOS), claimed to be universal [17], successfully applies to a large range of solids[13] including alkali metals, and can be written as

P (x, T ) = 3B

(1 x)

x

e

+ ↵

B

(T T

), (7) where x = ⇣

⇢0

⇢

⌘

, ⌘

=

(B

1), T

= 295 K, ⇢

= 1530 kg.m

(Ref. [9]), and ↵

= 23.8079 ⇥ 10

K

(Ref. [3]).

The experimental values of the density comes (see Fig. S14) from the compression data V /V

from Vaidya[16], Takemura[15], and Anderson [3].

The experimental values of the density are fitted with the Eq.( 7) to obtain the isothermal bulk modulus B

= 2.55 ± 0.03 GPa, and the first pressure derivative of the bulk modulus B

= ⇣

@BT

@p

⌘

p=0

= 3.96 ± 0.06.

By extrapolation, the density in the solid phase at P=7 GPa and T=555 K is 3150 ± 20 kg/m

(red star in the Fig. S14).

The evaluation of density changes due to thermal expansion in liquid between T

=555 K and T

=573 K gives

⇢ = ⇢(T

) ⇢(T

) =

✓ @⇢

@T

◆

(T

T

) = ⇢↵ T = 1.7 kg/m

(8)

with ↵ = ↵(7 GP a, 573 K) = 0.3 ⇥ 10

K

and ⇢

= 3150 kg/m

, so the e↵ect of thermal expansion between 555 K and 573 K on the density in the liquid can be considered negligible and ⇢

(7 GP a, 573 K) ' ⇢

(7 GP a, 555 K) =3150 ± 20 kg/m

.

15

Supplementary Figure S14 | Density as a function of pressure in the solid Rb phase (bcc Rb-I) at various temperatures up to 7 GPa. Experimental data [16, 15, 3] are fitted together by a thermal Vinet EOS (see Eq.(7)), and extrapolated to 560 K (dashed red line) to obtain the density at 7 GPa and 560 K. At this p-T point, densities in solid and liquid are equal.

5.2 Adiabatic index from ultrasonic measurements

With the numerical procedure explained extensively in Ref. [4, 5], (P, T ) can be determined accurately up to 0.7 GPa from the ultrasonic sound velocity data of Shaw & Caldwell [14].

The numerical procedure requires the knowledge of adiabatic sound velocity as a function of P and T , isobaric heat capacity and density at ambient pressure as a function of T .

Adiabatic sound velocity from Shaw et al (Fig. 4 in Ref. [14]) was smoothed and inter- polated by the following polynomial function [1] in the range 351.9 - 423.3 K

v

(P, T ) = v

+ aT + bP + cT

+ dP

+ eP

, (9) where T is in

C, P in GPa, v

=1253.43 m/s, a = 0.28904 m/s.K, b = 810 m/s.GPa, c = 1.057 ⇥ 10

, d = 734 m/s.GPa

and e = 392 m/s.GPa

. The values at ambient pressure are constrained by the polynomial function given by Blairs [6] based on a review of

16

literature data. The polynomial form is extrapolated outside the experimental range of the sound velocity measurements to T=573 K.

Isobaric heat capacity C

(P

, T ) is given by Alcock (see p. 464 in Ref.[2], within the range 312.45 - 1600 K) by a polynomial form, in J.kg

.K

:

C

(P

, T ) = 415.29 0.15051T + 99.93 ⇥ 10

T

The density ⇢(P

, T ) is given by the polynomial function from Kim (Eq.(18) in Ref.[11], within the range 312.45 - 1273.1 K)

⇢(P

, T ) = 1493 0.397T 5.2 ⇥ 10

T

+ 7.1 ⇥ 10

T

. The resulting (P, T ) is shown in Fig.3.d of the main text.

17

6 Interstitial quasi-atoms (ISQ) in solid and liquid Rb

Interstistial e^-

Liquid qISQ

Rb-I (BCC) Rb-II (FCC) Rb-III (oc52) Rb-IV (h-g) Rb-V (Ti4)

average qISQ (e)

0 0.1 0.2 0.3 0.4

P (GPa)

0 5 10 15 20 25

Supplementary Figure S15 | Average of electrical charge, q, (e) carried by interstitial quasi- atoms (ISQ) as a function of pressure up to 25 GPa of solid phases and liquid Rb. The low pressure liquid adopts Rb-I/II character while the high pressure liquid adopts non-nuclear localization with a with similar charges to those found in Rb-III/IV.

1 GPa 4 GPa 6 GPa 10 GPa 14 GPa 16 GPa 20 GPa 26 GPa III IV

frequency

0 0.1 0.2 0.3 0.4 0.5

qISQ (e)

0 0.1 0.2 0.3 0.4

Supplementary Figure S16 | Distribution of electrical charges, q, carried by interstitial quasi- atoms (ISQ) by integrating non-nuclear Bader volumes at various pressures in liquid Rb at 573 K.

Bars show the values for phase III at 15 GPa, and IV at 20 GPa.

18

7 Equation of state and di↵usion coefficient

573 K 800 K

V [Å3 /atom]

30 40 50 60 70 80 90

P [GPa]

0 5 10 15 20

inflection

38 40 42 44 46

7 8 9 10

Supplementary Figure S17 | Volume as a function of pressure in liquid Rb (isothermal equation of state, EOS) at two temperatures, 573 K (blue dots and line) and 800 K (red dots and line). Bulk modulus and thermal expansion shown in the Fig. 3 of the main text were calculated from this data.

19

AIMD Bryk 2013

D [ÅÅ2 /ps]

0 1 2 3 4 5

P [GPa]

0 5 10 15 20 25

Supplementary Figure S18 | Di↵usion coefficient, D, as a function of pressure in liquid Rb at 573 K. Comparison of AIMD data (this work, blue dots) and Bryk et al [7] data (empty black circles and gray dashed line). The di↵usion is anomalous and looks glassy from 17 GPa, much longer simulations of these pressures maintained this slow di↵usion rate.

20

References

[1] The legend of the Fig.4 contain an error : the upper curve correspond to 78.8

C not to 60.1

C.

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