Can homogeneous nucleation resolve the inner core nucleation paradox?

16 The formation of Earth’s solid inner core is thought to mark a profound change in the evolution of the deep Earth and the power that is available to generate the geomagnetic field. Previous studies generally find that the inner core nucleated around 0.5-1 billion years ago, but neglect the fact that homogeneous liquids must be cooled far below their melting point in order for solids to form spontaneously. The classical theory of nucleation predicts that the core must be undercooled by several hundred K, which is incompatible with estimates of the core’s present-day temperature. This “inner core nucleation paradox” therefore asserts that the present inner core should not have formed, leaving a significant gap in our understanding of deep Earth evolution. In this paper we explore the nucleation process in as yet untested


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The Earth's magnetic field is produced by the geodynamo in the liquid  that means substantial supercooling is expected to be needed before inner 42 core formation. 43 Classical nucleation theory (CNT, e.g. Christian, 2002) describes the ther-44 modynamics of nucleation and states that for a liquid to freeze it must be 45 supercooled. This is because whilst the liquid will be thermodynamically 46 unfavourable compared to the solid for a system below its melting temper-47 ature, the interface between the first solid and the remaining liquid comes 48 with an energetic penalty. Only when a critical nucleus size is exceeded will 49 the energetic preference for the solid phase outweigh the energetic penalty 50 due to the interface. Nuclei which grow larger than this will become increas-51 ingly likely to continue to grow, leading to the system freezing. Huguet et al.  This study will examine whether iron-rich binary alloys with compositions 102 thought to be relevant to Earth's core are capable of spontaneous homogeneous nucleation at supercooling which would resolve the inner core nucle-104 ation paradox. We only consider homogeneous nucleation because there are 105 no obvious solid surface on which iron can first nucleate at the centre of the 106 core. We will first describe the methods used to simulate supercooled liquids 107 and characterise nucleation within them following our previous work (Wil-108 son et al., 2021). We will then present predictions of critical nucleus sizes for 109 Fe x O 1-x , Fe x C 1-x , Fe x Si 1-x and Fe x S 1-x at x = 1 and 3 mol. %. Finally, we 110 will compare the rate at which the critical events are achieved to a revised 111 estimate of the geophysically viable supercooling in the core.
where r is the radius of the nucleus, I 0 is a prefactor scaling the kinetics of where γ is the interfacial energy and g sl is the difference between the free 136 energy of the solid and the liquid (g sl = g s − g l ). g sl can be approximated 137 through the enthalpy of fusion, h f and an accommodation for second order 138 non-linearity in the temperature dependence, h c ,  The nucleation barrier (Eq. 2) is dominated by γ at small r because of 146 high surface area to volume ratio. All nuclei must grow from a single atom 147 through all smaller nuclei sizes before a system can be completely frozen.

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The value of ∆G increases with r to a peak at which point the probability 149 of continued growth is equal to that of remelting. This is the critical size, Combining Eq. 1-3 with Eq. 5 then gives the rate at which the critical event 156 occurs, the inverse of which is the average waiting time between critical events where and Here, S, N and z are the rate of nuclei growth, number of available nucleation  EAMs define the total energy of a system (E) through the sum of energies 179 contributed by each atom (i) from the pairwise interaction with other atoms For the binary systems considered here this consists of iron-iron, iron-solute 182 and solute-solute interactions. Each of these energies includes a repulsive 183 term (Q), which depends on the separation of the pair (r ij ), and an embedded 184 term (F ) which depends on the electron density between the pair (ρ ij ) where ϵ, a, n and C are free parameters specific to each interaction. The 188 electron densities are also defined in terms of a radial separation and include an additional parameter m F e , m X and m F eX for each class of Using this approach we can interpret nucleation rates as the portion of Eq. 243 2 where r < r c (because the critical event will never occur within practi-244 cal durations). The absolute magnitude of ∆G remains poorly constrained, 245 meaning that g sl , γ and I 0 cannot be calculated yet. Instead, we fit this where A and B are variables at each T . Once fit, this distribution then 248 predicts r c = −2B/A in the same way as Eq. 5. Once r c is known at all 249 temperatures, the temperature dependence of r c is described by combining where v par = V /N atoms , V is the volume of the system, N atoms is the number 291 of atoms in the system and N nuc is the number of atoms in the nuclei. Whilst 292r is framed in terms of spherical nuclei, shapes can vary from this significantly 293 as we discuss below.  All simulations see non-spherical nuclei at small sizes (Fig. 3) Figure 3: Surface area to volume ratio for sub-critical nuclei at ∼400 K supercooling (r c >20Å). Systems containing 3 mol.% Si, S and C (orange, pink and green circles) are shown as well as the spherical case (black dashed line, 3r) for comparison. Also shown are example nuclei from the C bearing system for reference. Surface area to volume ratios are similar for all systems and approach spherical before the critical size.
Whilst non-spherical small nuclei were apparent with the pure Fe system, 316 we find the departure from sphericity to be more pronounced in impure . This is the mechanism which promotes dendritic growth in hcp 328 structured materials and leads to small nuclei becoming elongate here.

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For completeness we include a description of non-spherical geometries.

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The surface area to volume ratio of these geometries follows a power law 331 decay, the same as a sphere, only with a greater initial gradient. We there-

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T m 7.05 × 10 −5 1.02 1.26 × 10 45 730 Table 1: Thermodynamic parameters fit to r c (T ) for each composition tested where all evaluations for this study were carried out at 360 GPa. τ 0 varies with temperature but is given here as the value at the temperature which coincides with the supercooling required for inner core nucleation. The difference in free energy between solid and liquid defines the energetic 367 benefit to freezing the liquid. A more negative g sl is seen for C bearing 368 systems compared to the pure Fe system and those containing Si and S 369 (Fig. 6). S and Si see a smaller free energy difference at all temperatures 370 when compared to other systems (Fig. 6), agreeing with previous finding

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Our results shown that nucleation rates in Fe rich liquids containing C 375 are faster than those containing Si or S (Fig.2). Compared to the pure Fe 376 system, critical nuclei sizes are larger in system containing Si and S and 377 smaller in those containing C and O (Fig.4). These finding suggest that 378 systems containing C and O should freeze at higher temperatures (lower 379 supercooling) than a pure Fe system.

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To asses whether the systems studied here might resolve the paradox, we 381 must compare the time taken to nucleate at supercooling permitted in the 382 core with the available time to nucleate in the core, the incubation time.

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The maximum incubation time available for the inner core to form depends 384 on the undercooling available and the minimum age of the inner core. In The duration before a supercooled system will producing a critical event 423 and freeze is presented here as waiting time (Fig. 8)