The onset of collective diffusion in hcp-iron: a combined theory of soft-matter and solid-state physics

Abstract

Lately, scientists have discovered the existence of collective diffusion in hcp-iron via machine-learning algorithms. This result has opened a promising avenue for geophysical studies on planetary cores. However, the crossover between collective and non-collective regimes remains very ambiguous. Based on the solid–liquid similarity, we develop a simple differential equation to infer the crossover temperature from the density scaling exponent, the isothermal bulk modulus, and the volumetric thermal expansivity. All necessary thermodynamic inputs are provided by the statistical moment method for nonlinear atomic vibrations. Our theoretical results are supported by the latest computational data on warm-dense iron crystals. From there, we find a possible explanation for prolonged controversies about melting relations in diamond-anvil-cell experiments. The potential impacts of collective diffusion on the core properties of icy, rocky, and gaseous planets are also considered by combining our mineral-physics calculations with modern geodynamic models.

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1 Introduction

Recently, collective dynamics has emerged as an attractive hotspot in planetary science.^1–3 It refers to the appearance of liquid-like mobile ions in crystalline solids under severe pressure–temperature (P–T) conditions. In the collective regime, crystals possess higher self-diffusivity, better ionic conductivity, and easier shear deformability. These interesting properties have significantly improved our understanding of chemical distribution, dynamo generation, and mechanical deformation inside icy, rocky, and gaseous worlds.^4–6

Notably, the signature aspects of collective diffusion have been found in iron, the most bountiful metallic element in planetary cores. Using both ab initio and quasi-ab initio molecular dynamics, Belonoshko et al.⁷ observed the shuffle of crystallographic planes in the bcc structure. This collective effect gave bcc-iron numerous unique properties, such as ultra-low viscosity and ultra-high Poisson ratio.^8,9 Inspired by the impressive results of Belonoshko's group, Cuong and Phan¹⁰ developed the elastically collective nonlinear Langevin equation (ECNLE) to speed up computational processes and highlight the liquid-like behaviors of bcc-iron. They demonstrated that the inclusion of bcc fillers could satisfactorily explain the shear attenuation of Earth's inner core. Before long, Smirnov et al.¹¹ introduced a machine-learning force field for iron. Their computational model provided a deeper insight into solid–solid and solid–liquid transitions under super-Earth conditions, thereby facilitating research projects on extrasolar life.

However, the intrinsic stability of the pure bcc phase remains questionable. To date, only two experimental studies have supported the existence of bcc-iron.^12,13 The rest have claimed no observation of the bcc structure at elevated pressures and temperatures, regardless of using shock-wave, ramp-wave, or diamond-anvil-cell techniques.^14–18 This instability has also been recorded in various ab initio, quasi-ab initio, and machine-learning computations.^19–23 Although the kinetics of structural transformations, the uncertainty of interatomic potentials, and the oversymmetrization of simulated boxes have possibly clouded the bcc domain,²⁴ the above controversies have motivated the search for collective dynamics in other iron polymorphs.

In that context, Zhang et al.²⁵ discovered the formation of open-loop and closed-loop atomic clusters near the hcp-liquid boundary of iron via machine-learning molecular dynamics (MLMD). It had been thought for a long time that hcp atoms diffuse very weakly in the high-energy-density regime.⁸ Nevertheless, according to Zhang et al.,²⁵ this physical picture would be broken down by the influence of point defects. Vacancies were able to form under the extreme heat of planetary cores and promote collective motion in the warm-dense hcp lattice. Instead of vibrating around fixed nodes, hcp atoms rapidly migrated along [100] or [010] directions to generate string-like structures. This phenomenon made hcp-iron become surprisingly soft, consistent with shock-wave measurements. Based on the diffusion-induced shear softening, Zhang et al.²⁵ offered a plausible explanation for abnormal seismic signals from Earth's center. Note that most experimental and computational evidence available today supports the stability of hcp-iron in planetary interiors.^14–23 Therefore, details about its collectivity are essential to elucidate geodynamic processes inside icy, rocky, and gaseous worlds.

The first step in exploring the collective regime is to find the crossover temperature T_c, at which iron atoms start to leave their equilibrium positions. Since the analytical expression of T_c remains unformulated, one can only estimate its value by tracking the time evolution of mean-square displacements (MSDs) at different P–T points.²⁵ If MSDs remain unchanged, collective diffusion will not take place in the studied supercell. If MSDs increase linearly during the simulated period, the crystal can gain liquid-like mobility at the atomistic level. Despite sounding simple, this dynamical method requires extensive and prolonged simulations to ensure the convergence of MSDs.^7–9 Thousands or even millions of metallic particles need to be scrutinized from femtoseconds to nanoseconds to avoid finite-size and finite-time effects. Due to prohibitive computational costs, all existing data for collective dynamics in hcp-iron are limited to 330 GPa.²⁵ This limit is insufficient to cover the physics of super-Earths and giant planets, whose internal pressures can exceed 1000 GPa.^26,27 Hence, an alternative approach is necessary to evaluate the impacts of collective diffusion on the past, present, and future of celestial bodies.

To meet the need, we combine the density scaling law (DSL) and the statistical moment method (SMM) to predict the onset of collective motion in hcp-iron without strenuous computational efforts. Our combination relies on the close similarity between pre-melting solids and glass-forming liquids.^28–31 While the DSL allows us to connect the crossover temperature T_c with the bulk modulus B_T and the thermal expansivity β_P via simple analytical formulas in soft-matter physics,³² the SMM enables us to determine the P–T dependence of thermodynamic quantities via basic statistical tools in solid-state physics.³³ Our DSL-SMM calculations are applied to revisit heated debates about the melting process of iron in practical experiments.^34–36 Additionally, the ECNLE theory is used to decipher the unusual shear deformation of Earth's inner core.¹⁰ To demonstrate the broad applicability of collective dynamics, we also make open predictions of the core state of super-Earths and solar giants by comparing the DSL-SMM phase boundary with planetary thermal profiles.^26,27 We hope our article can give the readers a compelling picture of hcp-iron and its geophysical applications in the collective regime.

2 Calculation

2.1. Formulation of crossover temperature

Let us express our physical ideas in more detail. Overall, crystalline materials exhibit both solid- and liquid-like behaviors when the collective diffusion of ions is triggered at elevated temperatures.^28,29 This characteristic vividly reminds us of glass formers in the field of soft matter. Modern atomistic simulations have recorded numerous hallmarks of glass-forming liquids in pre-melting crystals at the microscopic level, such as intrinsic heterogeneity, dynamic facilitation, and string formation.^{30,31,37–40} The above studies have also revealed the applicability of simple glass models (e.g., the localization equation⁴¹ and the Adam–Gibbs theory⁴²) to describe collective effects during heating. In particular, ECNLE researchers have successfully modeled the structural relaxation, ionic transportation, and mechanical deformation of fast-ion solids inside planetary cores and nuclear reactors, including bcc-iron, uranium oxide, and mixed oxide fuel.^10,43 Thus, it is viable to leverage the reservoir of knowledge about vitrification to shed light on the transition between collective and non-collective states.

In many cases, vitrification and collectivization are quite similar to second-order phase transitions in equilibrium thermodynamics. Although the volume V grows continuously with increasing temperature, its first derivative shows a discontinuity near the crossover point.^1,6,44 This event suggests that the T_c–P relation may obey the Ehrenfest equation,⁴⁵

where Δ denotes abrupt changes in the heat capacity C_P and the thermal expansivity β_P. The entropy approach of Ehrenfest has been widely applied to polymer melts, amorphous drugs, and superionic crystals.^46,47 However, this is not an ideal choice for the present situation. The principal reason is that eqn (1) incorporates physical quantities before and after collectivization. As introduced in Section 1, thermodynamic calculations become highly demanding when mobile atoms leave lattice nodes to engage in collective loops. To optimize computational time and cost, we need to devise another strategy for accessing the phase boundary of hcp-iron at extreme pressures and temperatures.

Remarkably, according to recent ECNLE analyses,^10,43 vitrification and collectivization can be viewed as isochronal processes. In other words, the object of study will begin to kinetically vitrify or collectively diffuse if its structural relaxation time reaches a critical value. This isochronal criterion is equivalent to the DSL picture of molecular dynamics^48–50

TV^γ = const, (2)

where γ is a microscopic parameter originating from the pressure–energy correlation of material. Specifically, γ characterizes the steepness of the short-range repulsive interaction between atoms.^51–53 It can be determined by fitting numerical results for the interatomic potential φ with the inverse-power-law function φ_IPL,⁵⁴where r is the nearest neighbor distance, C is the repulsive constant, and A is the attractive background. Since γ is frequently supposed to be independent of P and T,^32,46,55,56 we can rewrite eqn (2) in the differential form asNote that the bulk modulus B_T and the thermal expansivity β_P are defined byEqn (5) and (6) help us simplify eqn (4) toUnlike eqn (1) and (7) only uses thermodynamic data in the normal crystalline state, which is readily accessible by available experimental, computational, and theoretical techniques. Hence, we consider eqn (7) a key to illuminating the collective motion of hcp-iron in planetary interiors. The starting point is selected at 5000 K and 220 GPa, corresponding to the lower limit of prior MLMD simulations.²⁵

2.2. Determination of thermodynamic inputs

To calculate the right-hand side of eqn (7), we employ the SMM model in quantum statistical mechanics. This analytical model has been demonstrated to be advantageous for capturing high-energy-density physics in different structures.^57–59 The SMM divides the total vibrational free energy F_vib into three principal parts: (i) static (F₀), (ii) quasi-harmonic (F_qh), and (iii) anharmonic (F_ah). Interestingly, if we know F₀ from experiments or simulations, we can readily infer F_qh and F_ah from the SMM recurrence formula for displacement moments.^60–62 This self-consistent approach allows us to speed up computational processes without sacrificing accuracy. Details about SMM free-energy calculations for hcp-iron were carefully reported by Cuong and Phan.³³ Here, we only present fundamental steps to obtain γ, B_T, and β_P.

First, we use four coordination shells in the hcp lattice to express F₀ as

The roaring success of the Vinet equation of state in describing the isotherms and isentropes of transition metals motivates us to parametrize φ(r) by^63–65

φ(r) = −D_R[1 + b_R(r − r_0R)]exp[−b_R(r − r_0R)], (9)

where D_R = 0.524851 eV, b_R = 2.56479 Å⁻¹, and r_0R = 2.52699 Å are the Rydberg parameters derived from the linearized-augmented plane-wave method.⁶⁶ Eqn (9) enables us to satisfactorily explain low-temperature physical processes in hcp-iron, such as hydrostatic densification, sound-wave propagation, and structural transformation.³³ Moreover, we can quantify the scaling exponent γ of hcp-iron by combining eqn (9) with eqn (3). Fig. 1 shows that the interaction between iron atoms in the hcp phase is perfectly modeled by the inverse power law from r₁ = 1.50 Å to r₂ = 1.67 Å. Our fitting range is selected based on the recommendation of Koperwas et al.⁵⁴ to avoid the overestimation of γ. While r₁ is the shortest distance our potential can approach, r₂ satisfiesBecause the upper limit of the cold isotherm in the Rydberg model is P₁ = 10.2 TPa,⁶⁷ r₁ is estimated byThe above procedure gives us γ = 1.57.

Fig. 1 The interatomic interaction in hcp-iron provided by the first-principle-based Rydberg model [eqn (9)] and the inverse-power-law potential [eqn (3)].

Next, we utilize F₀ to handle F_qh and F_ah. According to the vibrational theory of Leibfried and Ludwig,⁶⁸ it is feasible to calculate the quasi-harmonic coupling parameter (k) and its anharmonic counterparts (β, γ₁, γ₂) by

where u_ξ and u_η describe atomic displacements in the Cartesian coordinate system (ξ ≠ η = x, y, z). Analytical results for k, β, γ₁, and γ₂ are detailed in Appendix A. Since k is the product of the atomic mass m and the squared Einstein frequency ω_E², we can approximate the quasi-harmonic free energy of hcp-iron bywhere θ is the Boltzmann thermal energy, and ħ is the reduced Planck constant.⁶⁹ Besides, the contribution of nonlinear atomic fluctuations is quantified by the following thermodynamic integral³³where angle brackets denote average values in the equilibrium state. It should be noted that all displacement moments in the SMM model are tightly connected via the quantum density matrix.^60–62 Thus, we can rapidly compute 〈u_ξ²〉 and 〈u_ξ⁴〉 after gaining 〈u_ξ〉 from the force-balance condition. This connection results in the explicit form of eqn (14),where and

Finally, the total vibrational free energy is given by

F_vib = F₀ + F_qh + F_ah. (16)

Eqn (16) is beneficial for investigating the geophysical properties of warm-dense iron along melting lines, shock Hugoniots, ramp isentropes, and solid–solid boundaries.³³ It also helps us derive the equation of state fromAs illustrated in Fig. 2(a), our SMM analyses agree quantitatively well with previous ab initio molecular dynamics simulations.¹⁷ No errors in the atomic volume V exceed 2.5% in the examined region (1000 ≤ T ≤ 12 000 K and 100 ≤ P ≤ 1100 GPa). Therefore, we can confidently calculate the bulk modulus B_T and the thermal expansivity β_P of hcp-iron in severe environments by combining eqn (17) with eqn (5) and (6). Our conclusion is strengthened by single-shock experiments in Fig. 2(b) and (c).^70,71

Fig. 2 The atomic volume (a), the bulk modulus (b), and the thermal expansivity (c) of hcp-iron. Solid lines: SMM calculations, empty symbols: ab initio molecular dynamics simulations,¹⁷ filled symbols: single-shock experiments.^70,71

2.3. Validation of theoretical model

Fig. 3 depicts how the crossover temperature of hcp-iron depends on pressure. It is conspicuous that T_c experiences a continuous increase during squeezing. Theoretically, when hcp-iron is compressed from 0 to 1000 GPa, T_c is expected to climb from 2669 to 8775 K. This growth is flawlessly consistent with the large-scale, long-time atomistic simulations of Zhang et al.²⁵ None of our outputs violates MLMD constraints in a pressure range between 230 and 330 GPa. We believe these computational constraints are relatively tight because Zhang et al.²⁵ carefully handled finite-size and finite-time problems in the MSDs method. More than 10 000 iron atoms were simulated from femtosecond to nanosecond timescales to ensure the convergence of the crossover temperature. The excellent agreement above confirms the efficiency and accuracy of our newly developed analytical approach. Its quality is further validated in the case of bcc-iron (see Appendix B). To facilitate the journey of Earth and space exploration, we parametrize our dynamic phase boundary by the empirical law of Simon and Glatzel,⁷²where c₁ = 2669 K, c₂ = 65.04 GPa, and c₃ = 2.36 are fitting parameters.

Fig. 3 The crossover temperature of hcp-iron as a function of pressure obtained from present DSL-SMM calculations and previous MLMD computations.²⁵ (Inset) Zooming in on the previously simulated region.

3 Implication

3.1. The never-ending story of iron melting curve

Eqn (18) allows us to compare the onsets of collective diffusion and melting transition in Fig. 4. After meticulously reviewing diamond-anvil-cell data between 1993 and 2018, Morard et al.⁷³ reported three different scenarios for the melting line of iron: (a) flat,^34,74 (b) neutral,^75,76 and (c) steep.³⁵ Whereas scenario (a) was ascribed to chemical contamination, scenario (b) was attributed to pressure overestimation. Morard et al.⁷³ highly recommended scenario (c), in which the melting temperature of iron would reach 6230 K at the Earth's inner core boundary. Nevertheless, in 2019, Sinmyo et al.³⁶ gathered a lot of supporting evidence for scenario (b) from resistance-heated diamond-anvil-cell experiments. Their experimental results suggested that iron would melt at 5500 K and 330 GPa. Although numerous attempts have been made, the geophysical community has not found an adequate explanation for what Sinmyo et al.³⁶ observed.

Fig. 4 Comparing the DSL-SMM transition curve of hcp-iron with its experimental solid–liquid boundaries. Scenario (a): flat melting,³⁴ scenario (b): neutral melting,³⁶ scenario (c): steep melting.³⁵

In such situations, the DSL-SMM method becomes particularly useful. Fig. 4 shows that our crossover curve sits quite close to the intermediate melting points reported by Sinmyo et al.³⁶ This closeness warns: there may be confusion between “melting transition” and “collective diffusion” in practical experiments. Sinmyo et al.³⁶ established their melting criterion based on the saturation of temperature during voltage rising. Since collective effects help hcp-iron evacuate heat by both conduction and convection mechanisms, melting-like plateaus can be formed on voltage–temperature diagrams.^8,77 This phenomenon riskily creates the illusion of melting under extreme conditions. Hence, we deem scenario (c) still the most viable. Our conclusion is strengthened by recent shock-wave measurements for the phase relations of iron along Hugoniot branches.^15,16,18 However, it should be noted that our interpretation is only suitable for P > 200 GPa. At P < 200 GPa, the sample temperature of Sinmyo et al.³⁶ is too low to take into account the influences of collective diffusion. More experimental, computational, and theoretical efforts are required to gain a unified picture from low to high pressures. Some factors to consider include resistance variation, chromatic aberration, and impurity incorporation.^78,79

3.2. The shear deformation of Earth's inner core

Besides heated debates about diamond anvil cells, it is essential to assess the significance of collective motions to geodynamic understandings. Take the shear properties of Earth's inner core as an example. Seismic observations have revealed that Earth's inner core is very soft. Its shear modulus G is only about 167 to 176 GPa in the central region.^80–82 These counterintuitive results have challenged seismologists and mineralogists for decades because iron—the main component of Earth's inner core—has been thought to be highly resistant to shear deformation. At P = 360 GPa and T = 6000 K, empirical extrapolations yield G = 335 GPa for the hcp structure.⁸³ A plethora of solutions have been proposed to untie the knot with the aid of melt pockets,⁸⁴ superheating effects,⁸⁵ alloying elements,⁸⁶ grain boundaries,⁸⁷ and bcc fillers.¹⁰ Unfortunately, a strong consensus has not been reached due to unsolved questions about defect formation and phase stabilization.

Fortunately, the discovery of liquid-like mobile atoms in the hcp lattice can pave the way to escape this predicament. To reinforce our point of view, we estimate the shear modulus G of hcp-iron in the fast-ion state by the ECNLE theory.¹⁰ This theory utilizes the hard-sphere fluid of Percus and Yevick⁸⁸ to rapidly determine the physical properties of soft materials at different packing fractions ϕ. By taking the sum of the Miller configurational rigidity⁸⁹ and the Khrapak kinetic rigidity,⁹⁰ we have

where the critical value G_c = G(P, T_c) is inferred from the empirical model of Ikuta et al.⁸³ to ensure the continuity of G before and after collectivization.^25,91 It is practicable to convert ϕ into P and T via the following chemical mapping⁹²where ϕ₀ = 0.5, ϕ_c = 0.611, and β₀ = 12 × 10⁻⁴ K⁻¹ are universal constants obtained by analyzing theoretical and experimental data for molecular, polymeric, and pharmaceutical systems. Eqn (20) is inspired by the reduction of particle density during heating and the similarity between melting, vitrification, and collectivization. Its effectiveness was carefully validated by ab initio data for bcc-iron.^9,10 As illustrated in Fig. 5, collective effects strongly reduce the shear resistance of hcp-iron in the high-energy-density regime. Our ECNLE analyses indicate that G plunges nonlinearly to the seismic value at T ≈ 6400 K and P = 360 GPa. This shear softening cannot be found in previous studies on ordinary hcp systems.^83,93,94 In particular, our investigated P–T conditions are highly relevant to Earth's inner core. We do not need unrealistic superheating degrees to match theoretical calculations with seismic observations.^80–82 Our physical picture is bolstered by recent shock-wave experiments in Appendix C. This encouraging result offers a bright prospect of unlocking long-standing enigmas in Earth science. It will be appealing if the influence of light elements (e.g., H, C, O, Si, and S) on the collective diffusion of hcp-iron can be clarified in subsequent research projects to gain a more comprehensive description of inner-core structure, dynamics, and history.⁹⁵

Fig. 5 Temperature effects on the shear modulus of hcp-iron given by our DSL-SMM-ECNLE approach and the non-collective extrapolation of Ikuta et al.⁸³ at 360 GPa. The melting point of hcp-iron is extracted from the dynamic measurements of Kraus et al.¹⁶ to bound the stability field of fast-ion diffusion. Seismic data on Earth's solid inner core are also included to highlight the importance of collective dynamics.^80–82

3.3. The core state of other planets

Notably, in recent years, the National Aeronautics and Space Administration has announced the existence of numerous super-Earths outside the Solar System.⁹⁶ These rocky exoplanets have attracted intense interest due to their potential habitability.⁹⁷ Modern experimental and computational works on planet-forming materials have suggested that the core crystallization of rocky worlds would happen from the bottom up.^58,98 Therefore, it is fascinating to consider whether super-Earth inner cores are in the fast-ion state. To answer this question, we add our dynamic boundary to the P–T phase diagram of iron constructed from cutting-edge diamond-anvil-cell and shock-wave data.^16,35 Besides, we estimate the inner-core temperature of super-Earths by the cryoscopic equation,where is the melting temperature of pure Fe,¹⁶ and x* = 0.11 is the mole fraction of other elements.⁹⁹ The inner-core pressure is directly taken from the compression profiles of Boujibar et al.²⁷ in a mass range from 2M_E to 4M_E (M_E = 5.9722 × 10²⁴ kg is Earth's mass). As presented in Fig. 6, we have T_c < T_core < T_m. That is synonymous with the great possibility of finding hcp-iron with collective motion inside super-Earths. It has been demonstrated that the interaction between liquid-like mobile ions and electromagnetic fields can strongly affect the internal structure of planetary cores.⁶ Accordingly, physicists should pay more attention to collective effects when developing geodynamic models for super-Earths. Our predicted diffusivity is reported in Appendix D.

Fig. 6 Our proposed phase diagram for iron (solid lines). Also plotted are P–T points corresponding to the inner-core boundary of super-Earths (filled stars) and giant planets (filled circles).

Apart from rocky planets, the collective state also has a chance to exist inside ice and gas giants, whose temperature distributions were summarized by Mazevet et al.²⁶ For icy worlds (e.g., Uranus and Neptune), the appearance of collective diffusion may generate dipolar dynamos by enhancing the electrical conductivity of solid inner cores.¹⁰⁰ In this situation, a stably stratified fluid layer is necessary to ensure the non-dipolar morphology of planetary magnetic fields.¹⁰¹ For gaseous worlds (e.g., Saturn and Jupiter), collectivization would significantly increase the solubility of hcp-iron in metallic hydrogen.¹⁰² The consequence is that the core erosion may be much more severe than previously thought.¹⁰³ Hence, an accurate database of the geophysical properties of hcp-iron in the collective regime (e.g., density, energy, diffusivity, and conductivity) is urgently needed to serve upcoming space missions related to giant planets.¹⁰⁴

4 Conclusion

By coupling the density scaling law in soft-matter physics with the statistical moment method in solid-state physics, we have successfully developed a simple analytical approach to predict the onset of collective diffusion in hcp-iron from its fundamental thermodynamic properties. An excellent concurrence between theory and simulation has been achieved in computationally accessible regions. From there, we have partially resolved long-standing controversies about the solid–liquid transition of iron and the shear deformation of Earth's inner core. Some open predictions of collective dynamics inside super-Earths and solar giants have also been provided to promote future works in planetary science. Our results have reaffirmed the close similarity between pre-melting solids and glass-forming liquids under various thermodynamic conditions. This similarity promises to bring innovative ways to optimize computational processes in the high-energy-density regime. Scientists can leverage well-known glass models, such as the elastically collective nonlinear Langevin equation, to rapidly compute the relaxation, transportation, and deformation quantities of fast-ion crystals within and beyond the Solar System.

5 Appendix A: vibrational coupling parameters

Employing the Leibfried–Ludwig expansion gives us

Here, φ₁, φ₂, φ₃, and φ₄ are expressed by

φ₁ = −D_R[1 + b_R(r − r_0R)]exp[−b_R(r − r_0R)],

6 Appendix B: the crossover temperature of bcc-iron

To strengthen our theoretical model, we conduct an additional investigation into bcc-iron, whose collective motions have been recorded by classical, quantum, and machine-learning computations.^7,105,106 Detailed information about our SMM calculations for the bcc structure can be found in ref. 107. Applying the fitting procedure of Koperwas et al.⁵⁴ to the extended Rydberg potential yields γ = 2.20. Entering this scaling exponent into eqn (2) with initial conditions of P = 79 GPa and T_c = 2833 K (ref. 12) leads to Fig. 7. It is easy to see that our combined theory works very well in deep-planetary environments. At P = 3300 GPa, we obtain T_c = 18 993 K, consistent with T_c = 18 500 K gained from ab initio molecular dynamics simulations.¹⁰⁵ This quantitative consistency implies that the inverse-power-law approximation (γ = const (ref. 108)) is sufficiently accurate to model collective effects, at least in Earth and super-Earth ranges. Thus, we can be confident in the theoretical predictions presented in the main text.

Fig. 7 The pressure dependence of the crossover temperature of bcc-iron given by our theoretical calculations and quantum simulations.¹⁰⁵

7 Appendix C: theoretical versus experimental shear properties

Fig. 8 compares the DSL-SMM-ECNLE and shock-wave shear moduli of hcp-iron at 230 GPa. Overall, there is a good agreement between theory and experiment. Our analyses show that the appearance of collective motions reduces G from 170 to 142 GPa at 5180 K, similar to what Zhang et al.²⁵ reported (G = 136 GPa). This concurrence confirms that eqn (18)–(20) are appropriate for describing mechanical properties in the fast-ion state. Therefore, we can utilize them to explore the promising geophysical applications of hcp-iron above the crossover temperature.

Fig. 8 DSL-SMM-ECNLE analyses for the shear modulus of hcp-iron at 230 GPa. Data from the non-collective model of Ikuta et al.⁸³ and the shock-wave experiment of Zhang et al.²⁵ are also included for comparison.

8 Appendix D: the diffusion coefficient of hcp-iron

To promote the development of planetary models, we investigate the collective diffusion of hcp-iron at the microscopic level by combining the ECNLE theory with the DSL-SMM method. Specifically, each iron atom in the fast-ion state is modeled by an impenetrable sphere with an effective diameter σ. Its free energy F_dyn is inferred from the nonensemble-averaged dynamic density functional theory,¹⁰⁹where r is the spatial distance, q is the wave vector, ρ is the particle density, and S is the Percus–Yevick static structure factor. Note that F_dyn does not have an equilibrium thermodynamic meaning. This quantity is directly linked with the effective force acting on the studied sphere in the Langevin equation of motion. It captures not only Fickian dynamics (the first term) but also many-body caging effects (the second term). To break free from the neighboring cage and engage in the concerted migration, the sphere needs to transcend the entropic barrier F_B,

F_B = F_dyn(r_B) − F_dyn(r_L), (25)

where r_B and r_L represent the maximum and minimum free-energy points. Eqn (25) helps us calculate the structural relaxation time τ_α by the modified Kramers theory,¹¹⁰where K_B = (∂²F_dyn/∂r²)_r=rB, K_L = (∂²F_dyn/∂r²)_r=rL, and τ_s is the short relaxation timescale. Since the non-Arrhenius behaviors of metallic systems are much weaker than those of molecular, polymeric, and pharmaceutical liquids,¹¹¹ we do not include the elastic barrier in eqn (26). Besides, the strong coupling between relaxation and diffusion in transition metals enables us to deduce the diffusivity D from the Einstein random-walk law,^37,112Eqn (27) tells us how D depends on ϕ and σ. To transform ϕ into P and T, we employ eqn (18) and (20). The σ variation is inferred from the continuous condition of the shear modulus aswhere k_B is the Boltzmann constant.

Fig. 9 shows compression effects on the self-diffusion coefficient of hcp-iron at 6000 K. According to our numerical calculations, D drops considerably from 10^−11.93 m² s⁻¹ at 330 GPa to 10^−13.49 m² s⁻¹ at 360 GPa. This change aligns quantitatively well with atomistic simulations. Fitting the MSDs data of Zhang et al.²⁵ with a linear function between 200 and 250 ps gives us D = 10^−11.87 m² s⁻¹ near Earth's core-mantle boundary. Additionally, we find D = 10^−13.56 m² s⁻¹ from the latest study of Xu et al.¹¹³ on the viscosity of iron and iron-based alloys at Earth's center. These numbers affirm the accuracy of the DSL-SMM-ECNLE approach.

Fig. 9 The diffusivity of hcp-iron along the 6000 K isotherm derived from DSL-SMM-ECNLE approximations and molecular dynamics computations.^25,113

Fig. 10 depicts how hcp-iron atoms diffuse in rocky planets. Along the dynamic boundary between collective and non-collective phases, we observe a slight decrease in the diffusion coefficient from 10^−14.32 m² s⁻¹ at 330 GPa to 10^−14.64 m² s⁻¹ at 852 GPa. This tendency can be explained by the shortening of the effective diameter during compression. As shown in eqn (28), σ is a decreasing function of P due to the pressure-induced shear hardening of the hcp lattice. The negative σ–P correlation can also be obtained from the simulated radial distribution function, where compressive forces push the contact peak towards lower r-values.¹¹⁴ Since D is proportional to σ² [eqn (27)], our system becomes less diffusive along the T_c–P curve. It should be noted that the D reduction we discuss here does not stem from the structural relaxation of iron atoms because collectivization is an isochronal process in the DSL-SMM-ECNLE picture. Eqn (26) indicates that the onset of collective diffusion is marked by τ_α = 10^−6.41 s. Our theoretical results promise to facilitate upcoming research projects on the rheological properties of Earth and super-Earth inner cores since the diffusivity, viscosity, and anelasticity of planet-forming materials are intimately connected.^113,115

Fig. 10 DSL-SMM-ECNLE outputs for the diffusivity of hcp-iron under extreme conditions relevant to Earth and super-Earth inner cores. The beginning of collective motion is marked by the open star.

Author contributions

Tran Dinh Cuong: conceptualization, methodology, investigation, writing – original draft, writing-review & editing, visualization. Anh D. Phan: resources, supervision.

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

This is a pure theoretical study. All physical quantities are expressed in explicit analytical forms.

Acknowledgements

We use the free version of Grammarly to avoid spelling and grammatical errors.

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