Supplementary Information

Structural ordering in liquid gallium under extreme conditions

James W. E. Drewitt,1 Francesco Turci,2 Benedict J. Heinen,1 Simon G. Macleod,3, 4 Fei Qin,1 Annette K. Kleppe,5 and Oliver T. Lord1 1School of Earth Sciences, University of Bristol, Wills Memorial Building, Queens Road, Bristol, BS8 1RJ, United Kingdom 2H H Wills Physics Laboratory, University of Bristol, Bristol, BS8 1TL, United Kingdom 3Atomic Weapons Establishment, Aldermaston, Reading RG7 4PR, United Kingdom 4SUPA, School of Physics and Astronomy, and Centre for Science at Extreme Conditions, The University of Edinburgh, Mayﬁeld Road, Edinburgh EH9 3JZ, United Kingdom 5Diamond Light Source Ltd, Diamond House, Harwell Science and Innovation Campus, Chilton, OX11 0DE, United Kingdom

EXPERIMENTAL

Sodium Chloride pressure calibrant

In order to avoid deleterious alloying between Ga and Re, an NaCl annulus was placed in the pressure chamber to separate the sample from the Re gasket. The advantage of NaCl is that it has an equation of state that has been carefully calibrated as a function of p and T throughout the range of this study [1], it is highly compressible, leading to a high precision on the pressure determination, and it has a simple cubic structure, making its unit cell volume easy to determine from XRD data. The NaCl was stored in an oven at 130 ◦C prior to use to minimise its inital water content. At the start of each run, the cell was placed unsealed in the vacuum vessel which was evacuated before any pressure was applied, to help drive oﬀ any remaining moisture from the salt and the Ga sample. NaCl and KCl salts are the most common pressure media and calibrants used in high-p-T studies of metals in the DAC [2–4] and that when prepared carefully, as here, reactions are not observed during in situ XRD or ex situ chemical mapping.

Iterative structure factor and number density reﬁnement

Our code LiquidDiffract [5] implements an iterative procedure first proposed by Eggert et al. [6] to provide the normalised SGaGa(Q) and gGaGa(r) functions, whilst simultaneously deriving a converged solution for the atomic number density, n0. This method optimises the Krogh-Moe-Norman [7, 8] normalization of X-ray intensities by exploiting the simple behavior of the reduced pair distribution function G(r) = 4πn0r[g(r) − 1] prior to the first interatomic distance G(r < rmin) = −4πn0r. As no atom-atom correlations are expected in this region, any oscillations are a result of e.g. systematic errors in the nomalization factor and the finite experimental Qmax < ∞. LiquidDiffract computes the Fourier transform of the difference

∆Gi(r) = Gi(r) − (−4πn0r), (1) for r < rmin, and uses this to re-scale the reciprocal-space data for i iterations (a minimum of 3 iterations is normally required for convergence). A fortuitous consequence of this iterative auto-normalization procedure is that the bulk liquid number density can be determined directly by minimising a χ2 ﬁgure of merit

Z rmin 2 2 χi (n0) = [∆Gi(r) ]dr (2) 0 which is known to have a well deﬁned minimum [5, 6]. 2

DENSITY RELAXATION IN AB INITIO SIMULATIONS

In order to measure the degree of decorrelation of density fluctuations in the ab initio molecular dynamics simulations we evaluate the self part of the intermediate scattering function at a fixed wavevector close to the first peak of the 1 radial distribution function q = 2π/2.8 A˚− . This is defined for a system of N particles as

* N + 1 X iq (rn(t) rn(0)) Fs(q, t) = N − e− · − , (3) n=1 which is averaged isotropically in order to remove the angular dependence. As shown in Fig. S1 for most state points (excepting the T = 303 K and the highest pressure state point at T = 400 K) the correlation functions decay to zero in a few picoseconds, ensuring equilibration.

T =1000K T =800K T =600K T =400K T =303K 1.00 1.00 1.00 1.00 1.00

) ) ) ) ) p[GPa] ,t ,t ,t ,t ,t

1 1 1 1 1 0 ° 0.75 p[GPa] ° 0.75 ° 0.75 ° 0.75 ° 0.75 ˚ A ˚ A ˚ A ˚ A ˚ A 8 8 8 8 8 . 3.2 13 . p[GPa] . . . 2 2 2 2 2 / 0.50 / 0.50 / 0.50 / 0.50 / 0.50 º 4.7 16 º 2.4 9.5 º p[GPa] º º 6.3 19 3.8 12 1.5 6.3 p[GPa] =2 0.25 =2 0.25 =2 0.25 =2 =2 k k k k 0.25 k 0.25 ( 8.2 23 ( 5.5 15 ( 2.9 8.4 ( 0.49 3.4 ( s s s s s

F 10 33 F 7.3 18 F 4.5 11 F 1.9 F 0.00 0.00 0.00 0.00 0.00 101 102 103 101 102 103 101 102 103 101 102 103 101 102 103 t [fs] t [fs] t [fs] t [fs] t [fs]

FIG. S1. Self part of the intermediate scattering function computed from the ab initio trajectories.

HARD-SPHERE MAPPING

Percus-Yevick approximation

In this work, we compare the multi-body local structure in the ab initio simulations of gallium to the local structure of the hard sphere fluid. To do so, we employ the Percus-Yevick (PY) approximation to estimate the radial distribution function g(r) of hard spheres at volume fraction η and diameter σeff . For every ab initio dataset we find the optimal couple {η, σeff } that minimises the difference

Z umax ab initio PY 2 δ(η, σeff ) = [(g (uσeff ) − 1)u − (gη (u) − 1)u] du, (4) umin where the integral is over a fixed interval umin = 1, umax = 4, where we implicitly assumed the hard spheres to have diameter one so that u is a scaled radial distance variable, with uσeff measured in angstroms. An example of the resulting fitting is reported in Fig. S2. By construction, the contact position of the hard spheres does not coincide with the first peak of the radial distribution function of gallium. The Percus-Yevick approximation is a closure to the Ornstein-Zernicke equation for hard spheres which is reasonably accurate for volume fractions below freezing ηfreezing ≈ 0.49 [9]. In particular, it provides an expression for the direct correlation function c(r) as

( c + c u + c u3 u < 1 c(u) = 0 1 3 (5) 0 otherwise 3 where

2 4 c0 = −(1 + 2η) /(1 − η) (6) 2 4 c1 = 6η(1 + η/2) /(1 − η) (7)

c3 = c0η/2 (8)

R ∞ The radial Fourier transform isc ˆ(k) = 4π 0 duc(u) sin(ku)u/k and can be input into the Ornstein-Zernicke equation

p[GP a] 2.4 7.3 14.9 10 3.8 9.5 18.3 5.5 12.0

) 6 r ( g

0 0 1 2 3 4 r/σeﬀ

FIG. S2. Percus-Yevick ﬁts (dashed lines) to the radial distribution functions of gallium from ab initio simulations at temperature T = 800K and varying pressure (continuous lines). The lines are shifted to ease the visualisation.

hˆ(k) =c ˆ(k)/(1 − 6ηcˆ(k)/π), (9)

1 R so that h(u) = g(u) − 1 can be obtained via an inverse radial Fourier transform as h(u) = ∞ dkhˆ(k) sin(ku)k/u. 2π2 0 We numerically compute the radial distribution function following the PY route and minimise δ(η, σeﬀ ) in order to ﬁnd the optimal volume fraction (and hard sphere diameter) that best approximates the gallium radial distribution function.

Event-driven molecular dynamics

From the ﬁtting to the Percus-Yevick radial distribution function we obtain a range of diameters and volume fractions for the mapped hard-sphere system. In Fig. S3 we see that while the eﬀective hard-sphere diameter is bracketed in the range [2.4, 2.7]A,˚ the volume fraction has a more pronounced systematic variation towards high volume fractions with increasing the pressure at a given temperature, as expected. In order to measure the multi-body 4

(a)1000 (b) 1000 2.60 0.44 800 800 2.55

] ] 0.42 K K ¥ eff [ [ æ T 600 2.50 T 600 0.40 2.45 400 400 0.38 2.40 0 20 0 20 p[GP a] p[GP a]

FIG. S3. Variation of the Percus-Yevick fitting parameters σeff and volume fraction η across the p − T phase diagram. correlations in the hard-sphere system – as described in the main text – we run constant-volume event-driven molecular dynamics simulations employing the highly optimised DynamO package [10] to obtain equilibrated configurations of N = 600 particles (we select this number to match the number of particles employed in the ab initio runs). To ensure equilibration, the simulations are run for at least 100 structural relaxation times, which is the time for decorrelation of density fluctuations.

REVERSE MONTE-CARLO COMPARISON

Reverse Monte-Carlo (RMC) results are obtained using fullrmc 4.0.0, a Python package with specific optimisa- tions inspired by machine-learning algorithms [11]. These methods involve the generation of three-dimensional point clouds whose spatial distribution is optimal according to some metrics. Typically, RMC optimizations minimise the difference between some structural observable measured by other means (e.g. from experiments) and the generated point cloud. In principle, RMC simulations are not bound to a particular observable to optimise: constraints on radial distributions can be complemented with information on the angular distributions or eventually even more complex best-fit functions. Here we employ RMC in a naive way, in line with what is oftentimes reported in the literature for the structure of liquids and glasses. In particular, the target function used in this case is the reduced pair distribution function G(r), which can be readily calculated from g(r) as

G(r) = 4πrn0[g(r) − 1]. (10) Several issues hinder a perfect matching between RMC calculations and ab initio (or experimental) data: in the

ab initio disordered 2 ordered

G(r) 0

−2 0 2 4 6 8 10 r[Å]

FIG. S4. Comparison between the reduced pair distribution functions G(r) as obtained from the ab initio data and the final RMC configurations of two distinct runs starting from a disordered and an ordered initial configuration respectively. 5 ab initio simulations the interactions are not exclusively pair, so that the pair distribution function contains only a part of the thermodynamic information; moreover, small deviations in the pair distribution function may correspond to large differences in the long range and angular distributions. In order to demonstrate how sensitive to small variations naive RMC is, we show that the choice of the initial configuration (i.e. the seed for the optimisation) can lead to very different results in terms of multi-body structure for similar pair distribution functions. We consider two distinct starting conditions in order to match a particular ab initio state point at 1000 K and 3.2 GPa: (i) first, we start with a random assembly of N = 6000 hard spheres at a high packing fraction η = 0.553 obtained via the Stillinger-Lubachevsky linear compression algorithm [12] as implemented in the event-driven molecular dynamics package DynamO [10]; (ii) secondly, we consider a face centred cubic crystalline arrangement of 3 N = 5488 points at number density n0 = 0.054 A˚− . For each initial condition we perform 2 150 000 RMC moves, constrained to minimise the difference in G(r) and to have nearest-neighbour distances larger than a minimum threshold rlow = 2.0 A.˚ Fig. S4 shows that most of the differences between fitted and ab initio pair distributions are localised at very short distance, where the crystalline initial conditions presents an unphysical excess of probability. Elsewhere, the distributions are consistent with each other and compatible with what has been previously presented in the literature for gallium. However, the three-dimensional arrangements of points in the two cases are dramatically different and reminiscent of the initial conditions, as shown in Fig. S5. We notice in particular that in the case of the crystalline initial condition the RMC preservers a lattice structure, hidden in the pair distribution function by local modulations of the density in the vicinity of the lattice points. This suggests that naive RMC – where only the one dimensional information contained in the pair distribution function is used – is unable to reliably reconstruct the three dimensional, multi-body structure of the liquid. This is further validated by the Voronoi and Topological Cluster Classification analysis in Fig. 4 of the main text.

FIG. S5. Orthographic projections for(a) the final RMC configurations(b) obtained starting from (a) a disordered assembly of hard spheres and (b) a crystalline lattice. Remark that these two configurations share almost identical pair distribution functions, see Fig. S4.

LOCAL TWO-BODY EXCESS ENTROPY

For a liquid or a glassy system it is possible to deﬁne the two-body excess entropy s2 from the radial distribution function g(r) as Z ∞ 2 s2 = −2πn0kB [g(r) log(g(r)) − g(r) + 1]r dr, (11) 0 where n0 is the number density and kB is the Boltzmann constant (here set to unity for convenience). This requires in principle a smooth estimate of the radial distribution function g(r), possibly computed over a large range so that the upper limit of integration can be safely restricted. It has been proposed in several recent works [13–16] to deﬁne an analogous quantity at the local level in order to have a local structural metric for the excess entropy on each particle or atom i

Z i ∞ i i i 2 s2 = −2πn0kB gm(r) log gm(r) − gm(r) + 1 r dr. (12) 0 6

i Here gm(r) identifies a smoothened or “mollified” local radial distribution function, defined as

2 2 i 1 X 1 (r rij ) / 2λ g (r) = √ e− − ( ), (13) m 4πn r2 2 0 j 2πλ where λ is a smoothing parameter. In our calculation we take λ = 0.3 A˚ and limit the integral in Eq. 12 to the upper i ˚ i 1 PNb j cutoff rcut = 5.0 A. In order to increase the signal-to-noise ratio we also perform a local averages ¯2 = i j=1 s2 Nb i ˚ where Nb is the number of local neighbours of particle i, as identified by a local cutoff rlocal = 3.0 A. Low (or very negative) values of the local excess entropy have been shown to correspond to locally ordered regions during crystallisation [15] or in glass forming fluids [16]. In the main text, we show that in the case of gallium they correspond to specific local motifs that are compatible with locally crystalline or locally five-fold symmetric order.

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