Hidden scale invariance of metals

Density functional theory (DFT) calculations of 58 liquid elements at their triple point show that most metals exhibit near proportionality between thermal fluctuations between virial and potential-energy in the isochoric ensemble. This demonstrates a general"hidden"scale invariance of metals making the dense part of the thermodynamic phase diagram effectively one dimensional with respect to structure and dynamics. DFT computed density scaling exponents, related to the Gr{\"u}neisen parameter, are in good agreement with experimental values for 16 elements where reliable data were available. Hidden scale invariance is demonstrated in detail for magnesium by showing invariance of structure and dynamics. Computed melting curves of period three metals follow curves with invariance (isomorphs). The experimental structure factor of magnesium is predicted by assuming scale invariant inverse power-law (IPL) pair interactions. However, crystal packings of several transition metals (V, Cr, Mn, Fe, Nb, Mo, Ta, W and Hg), most post-transition metals (Ga, In, Sn, and Tl) and the metalloids Si and Ge cannot be explained by the IPL assumption. Thus, hidden scale invariance can be present even when the IPL-approximation is inadequate. The virial-energy correlation coefficient of iron and phosphorous is shown to increase at elevated pressures. Finally, we discuss how scale invariance explains the Gr{\"u}neisen equation of state and a number of well-known empirical melting and freezing rules.

Scale invariance plays an important role in many branches of the sciences. It greatly simplifies a given phenomenon by reducing the parameter space and introducing universalities over length or time scales. Some examples of this are the size distribution of earth quakes [5], Brownian motions of microscopic particles [32], cosmic microwave background radiation [18,22], and biological fractal structures [31] such as lung tissue or Romanesco broccoli. In condensed matter physics, scale invariance controls the properties of a fluid near the gas-liquid critical point [40]. This paper establishes an approximate hidden scale invariance in the dense part of the phase diagram of certain elements based on ab initio quantum computer simulations. In effect, the thermodynamic phase diagram becomes effectively onedimensional. Consequences for the understanding of freezing and melting are briefly discussed at the end of the paper.
Any substance is characterized on the macroscopic level by its equation of state as given in the relation between temperature, pressure, and density [4,16,51]. If pressure is denoted by p, the equation of state may be expressed as p = p(T, ρ) in which T is the temperature and ρ the number density of the atoms or molecules in question. It has been known for a long time that at pressures high enough to result in non-negligible compression, solids and liquids generally obey the Grüneisen equation of state from 1912 [7,35] also referred to as the Mie-Grüneisen equation. This equation expresses proportionality between pressure and energy E per volume V as follows: p = γ(ρ)E/V + C(ρ) in which γ(ρ) is the so-called Grüneisen parameter and C(ρ) the "cold pressure", both of which are functions only of the density. This simple relation has been applied for describing condensed matter in a wide variety of high-pressure situations, ranging from the inner of the Earth [43] to various forms of explosions [33,35]. The well-proven Hugoniot shock-adiabatic method is available for determining γ(ρ) experimentally [28,35].
There are two contributions to the pressure: the idealgas pressure p id -a term that is always present and which only depends on the particles' (atoms') velocities -and a term deriving from the interaction between the particles. The latter is given by the so-called virial W , which is a function of the particle coordinates. The general pressure relation is p = p id + W/V ; thus W is an extensive quantity of dimension energy. At high pressure the virial term dominates completely. This means that, since the potential energy U likewise dominates the energy at high pressure, the Grüneisen equation here implies the approximate relation When written in this way the number γ(ρ) is referred to as the density-scaling exponent [13,34,46,47] since Eq. (1) implies that structure and dynamics are invariant along the lines in the thermodynamic phase diagram given by d ln T /d ln ρ = γ(ρ), the so-called isomorphs [15]. If the coordinates of the system's N particles are merged into a single vector, R ≡ (r 1 , ...r N ), the microscopic definition of the virial is W (R, V ) = −V ∂U (R, V )/∂V [2,20]. The term W in Eq. (1) is the ensemble average of this quantity, i.e., W = W (R, V ) , and similarly one has U = U (R, V ) . At high pressure the interatomic forces are harshly repulsive [56,57]. A simple microscopic explanation of Eq. (1) is that these forces can be modelled approximately by an exponentially repulsive [8] or an inverse-power-law (IPL) pair potential term ∝ r −n in which r is the interparticle distance, plus a density-dependent constant [3,24,50]. In the latter case there is exact virial potential-energy correlations by Euler's theorem for homogeneous functions, leading to Eq. (1) with γ = n/3.    [53] and the reduced diffusion constant D for isomorph and isochore. Both Q and D vary much less along an isomorph than along an isochore for the same temperature variation. high pressure. In this paper we show that many metals obey Eq. (1) also at low pressure, i.e., when virtually uncompressed compared to the zero-pressure state. This is nontrivial because at low pressure the interatomic forces are far from being dominated by repulsions since the interatomic distance is here close to that of the pairpotential minimum [19]. The numerical results presented in the following are obtained by density-functional theory molecular dynamics calculations with 125 or 256 atoms at the experimental triple point. We have used the Vienna Ab-initio Simulation Package (VASP) [27] employing the projector augmented wave (PAW) method [6] with the frozen core assumption and the Perdew-Burke-Enzerhof (PBE) exchange-correlation functional [41]. More details are given in the Supplementary Material. Figure 1 shows results of ab initio quantum mechanical computations on the first six period 3 elements. Each subfigure gives a scatter plot of the virial versus the potential energy of configurations taken from N V T equilibrium simulations of the liquids at their triple points. For the metals (Na, Mg, Al) and the metalloid silicon (Si) the scatter plots show strong correlations, and thus Eq. (1) is obeyed to a good approximation. The value of the correlations quantifies how well the approximation is obeyed. The nonmetals P and S do not exhibit strong W U correlations. These results give the first quantum-mechanical confirmation of the picture proposed in Refs. 10, 15, 25, 37, and 38 according to which spherically symmetric interactions like the Lennard-Jones interaction have strong virial potentialenergy correlations for the condensed phases, whereas this is not the case for systems with strong directional bonding. The former class of systems are simpler than liquids in general [10,12,23,30,42,44]. Although the focus here is on monatomic systems, we note that experimental results suggest that this class also contains molecules [17] and polymers [46,47]. Also molecular liquids like N 2 and the noble gases are predicted to be simple. Thus the majority of the elements belongs to this class of systems. As shown elsewhere, this class of systems is characterized by a "hidden scale invariance" as reflected in the following approximate representation of the potential-energy function in whichR ≡ ρ 1/3 R is the so-called reduced configuration vector: The functions of density h(ρ) and g(ρ) both have dimension energy andΦ is a dimensionless, state-pointindependent function of the dimensionless variableR, i.e., a function that involves no lengths or energies. Physically, Eq. (2) means that a change of density to a good approximation leads to a linear affine transformation of the high-dimensional potential-energy surface. Thus if temperature is adjusted in proportion to h(ρ), state points in the thermodynamic phase diagram are arrived at for which the molecules according to Newton's laws move in the same way, except for a uniform scaling of space and time. Curves of such state points are the above-mentioned isomorphs; thus along the isomorphs structure and dynamics are identical in properly reduced units to a good approximation.
To validate the prediction of hidden scale invariance we investigated liquid magnesium in more detail. The results are shown in Fig. 2, which studies isomorphic as well as isochoric state points for temperatures between 900 K and 1600 K. Panels (a) and (b) give the reduced radial distribution function for the isomorphic and isochoric state points, respectively, while (c) and (d) give the translational order parameter Q of Debenedetti et al. [53] and the reduced diffusion constant. We see that structure and dynamics are almost invariant along the isomorph, but not along the isochore. This confirms magnesium's hidden scale invariance.
The results of Fig. 1 inspired us to study the main group and transition metal elements in order to investigate whether metals generally exhibit hidden scale invariance (we excluded some non-metallic elements where standard semi-local density functionals are inaccurate, see the Supplementary Material). The results are shown in Fig. 3. The metallic liquid elements all have strong or fairly strong virial potential-energy correlations at the triple point as quantified in the virial potential-energy Pearson correlation coefficient R. As mentioned, all systems have the hidden-scale-invariance property at high pressure (compare also Fig. 4 below). Moreover, crystals generally have stronger virial potential-energy correlations than liquids [1]. We therefore conclude that metallic elements are simple in the whole condensed-phase part of their phase diagram, i.e., exhibit hidden scale invariance. This excludes state points close to the critical point, as well as those of the gas phase far from the melting line. The melting line [39] itself is an isomorph [15]; in Table I this prediction is validated for period 3 metals Na, Mg and Al by showing that the scaling exponent of the melting line γ m [36,39] agrees with that of the isomorph (within the statistical uncertainty).
To exemplify that the behaviour of elements becomes simpler under pressure we compare in Fig. 4 the W U scatter plot of Fe at the triple point ( Fig. 4(a)) with one at the pressure 113 GPa (Fig. 4(b)) corresponding to the Earth's core [9,43]. The correlation coefficient increase from R = 0.95 to R = 0.98. Iron clearly obeys hidden scale invariance in the dense part of the phase diagram, confirming the general expectation from the Grüneisen equation. Fig. 4(c) shows that W U fluctuations are uncorrelated in the gas-liquid coexistence regime. This is associated with the formation of a gas-liquid interface.
In summary, ab-initio quantum-mechanical calculations show that the liquid metallic obey hidden scale invariance at their triple point. The implication is that monatomic metals' atomic structure and dynamics are simpler than hitherto realized in the condensed-phase  shows that the correlation increases at elevated pressure corresponding to conditions at the Earth's core; thus iron becomes simpler at higher pressures. Panel (c) show that the correlation is low at a density in the gas-liquid coexistence region.
part of the thermodynamic phase diagram.
As a consequence, empirical melting rules now find a concise explanation. Specifically a number of invariants along the melting line of metals and simple model systems have been known for years with no good explanation. They all follow from metals' hidden scale invariance, because the melting and freezing lines are both isomorphs [15] and that many properties isomorph invariants. The most famous rule is the Lindemann melting criterion, according to which a crystal melts when the thermal vibrational atomic displacement is about 10% of the crystal's interatomic distance [14,48,50,54]. There are also empirical freezing rules, for instance the Hansen-Verlet rule that a liquid crystallizes when the first peak of the structure factor reaches the value 2.85 [21], the Andrade equation predicting constant reduced-unit viscosity along the freezing line [11,26], the Raveche-Mountain-Streett criterion [45] of a quasiuniversal ratio between maximum and minimum of the radial distribution function at freezing, Lyapunov-exponent based criteria [29], or the criterion of zero higher-than-second-order liquid configurational entropy at crystallization [49]. Also, connecting the melting and freezing lines, is the rule of invariant constant-volume melting entropy, as confirmed in experiments [52,55].
This work was financially supported by the Austrian Science Fund FWF within the SFB ViCoM (F41). The center for viscous liquid dynamics "Glass and Time" is sponsored by the Danish National Research Foundation's grant DNRF61. U.R.P. was supported by the Villum Foundation's grand VKR-023455. The computational results presented have been achieved mostly using the Vienna Scientific Cluster (VSC).