Constraining Earth’s core composition from inner core nucleation

Domain: earth_science
Published: 2025-09-04
Editorial outcome: minor_revision

Abstract
The composition of Earth’s core is a fundamental property of the Earth’s deep interior, defining its present structure and long term thermal and magnetic evolution. However, the composition of the core is not well understood, with several combinations of light elements being able to satisfy the traditional constraints from cosmochemistry, core formation and seismology. The classic view of inner core formation does not include the necessity for liquids to be supercooled to below their melting point before freezing. Attempts to calculate the magnitude of this supercooling have found that several binary core compositions are incompatible with inner core nucleation. Here we show, through molecular dynamics simulations, that nucleation from an Fe 1−x C x=0.1-0.15 composition is compatible with a range of geophysical constraints. Whilst not a completed description of core chemistry, our results demonstrate that inner core nucleation places a strong constraint on the composition of Earth’s core that may allow discrimination between previously identified potential compositions.

Introduction
The composition of Earth’s iron-rich core plays a crucial role in determining the structure, dynamics and evolution of Earth’s deep interior. The melting temperature, T m, of the core, set by the constituent iron alloy, defines the temperature, T, at the inner core boundary (ICB), which provides a unique constraint on the present-day temperature at the core mantle boundary (CMB). Transport properties also vary with composition, including thermal conductivity, k, which plays a vital role in determining thermal stratification and convective vigour in the liquid core and therefore the long term thermal evolution of the core 1–3 and its potential to generate the global magnetic field, which has been preserved in the rock record for at least the past 3.5 Gyrs 4–6. Light elements determine the strength of compositional buoyancy produced by inner core growth (the dominant power source for the geodynamo today 1,7,8) through their partitioning between solid and liquid during inner core freezing 9, while chemical exchange at the CMB may produce stable regions at the top of the core 10–12 that are detectable by seismology 13–15. However, despite recent progress 16, the composition of the Earth’s core remains poorly known.

Three main approaches have been used to constrain the composition of Earth’s core: cosmochemistry, core formation and seismology. Core composition can be inferred via cosmochemistry by comparing the composition of primitive CI meteorites, those which most closely resemble the solar photosphere, with the silicate Earth. CI meteorites are rich in Fe, Ni, Mg, Ca, Al, Si, S, C and O 17, where the light elements Si, S, C and O are appealing candidates to explain the low density of the core compared to pure Fe 18. If the Earth is assumed to be assembled primarily from CI meteorites, deviations of the bulk silicate Earth from their composition can be ascribed to losses to space or the core. This approach favours an Fe-Ni (~85 wt% and ~5 wt%, respectively) core 19 where Si is the major light element (up to 9.6 wt% 20) and C, S, and P cumulatively make up 2.5 wt% of the core 19. Core formation models estimate core composition by assuming chemical equilibrium between metal and silicate during Earth’s accretion and differentiation. The equilibrium concentration of light elements depends on partition coefficients, determined by experiments and calculations, which vary with pressure, temperature and composition.

The accretionary history of the Earth is therefore a key component in the determination of core composition. Several models 21–23 favour high Si and O (7.1–9.9 wt% and 1.3–5.3 wt% respectively) in the core. However, the array of accretionary histories that are possible results in a wider range of plausible compositions 24. Comparison of the elastic properties of the core from seismology with results from mineral physics can be used to identify core compositions that are compatible with observations. The depth varying wave speeds of the outer and inner core and the density contrast across the ICB 25 require ternary systems, although several combinations and concentrations of C, O, Si and S are viable 16. For example, Badro et al. 26 find that whilst an Fe-Ni core with 3.7 wt% O and 1.9 wt% Si best satisfies the available constraints, other combinations of O with C, Si or S can also match the radial P-wave structure of the core. Ultimately, no single composition is uniquely able to explain the origins, formation and elastic properties of the Earth’s core and the range of plausible compositions have markedly different implications for the thermal state of the core, both past and present. Given these uncertainties, it is important to seek additional constraints on core composition that are independent from but complementary to existing approaches. Here, we propose that a constraint on core composition can be derived by analysing the thermodynamic conditions under which the solid inner core first formed.

The Earth’s inner core is classically understood to have formed when the T of the core at the centre of the Earth cooled to the T m at the same location. The composition of the core from this time onwards can be used to estimate the thermal profile of the deep Earth by assuming that the adiabatic temperature, T a, of the core must equal the T m at the ICB 27. However, this picture is physically incomplete because all liquids must be supercooled by an amount δT below T m (δT=T m −T), often significantly, before freezing can begin 28. This requirement arises because whilst the solid phase is thermodynamically favoured for T<T m, establishing an interface between solid and liquid is always unfavourable, and for the first solids, the energy change of introducing an interface always wins out over the phase change. Previous studies 29–34 have estimated the δT required to nucleate the solid inner core for several compositions that are compatible with traditional constraints from core formation and seismology, but found that δT is incompatible with geophysical constraints (see Wilson et al. 35 for a review). When considering the effect of light elements, compositions including O and C were found to require δT closer to geophysically compatible values when compared to pure Fe, Fe–S and Fe–Si 33. Because not all potential compositions of the core can explain the presence of the inner core, inner core nucleation may provide an additional and strong constraint on the composition of the core.

In this study, we extend our previous results on the Fe–C system, for which the required δT for inner core nucleation is the closest to geophysical constraints of all systems tested previously 33. We use an improved embedded atom model (EAM), validated at a higher C concentration of 10 mol%, to explore solid nucleation in molecular dynamics simulations of supercooled Fe 1−x C x liquids and establish the required conditions for the inner core to freeze from these liquids. Our calculations address homogeneous nucleation, where solids form spontaneously, away from any pre-existing solid surfaces. We return to consider heterogeneous nucleation, which arises in the presence of solid surfaces, in the discussion.

Results
We use classical molecular dynamics (CMD) to observe and characterise the nucleation of atomic-scale solids in supercooled Fe 1−x C x liquids at core pressures. Classical nucleation theory (CNT) 28 states that the nucleation rate, I, of a system is inverse to the waiting time to observe nucleation, τ w, and increases with supercooling below the melting temperature.

Melting temperatures
To characterise nucleation in molecular dynamic simulations for a specific δT, we require knowledge of the melting temperature in order to define an appropriate simulation T. Melting temperatures, T m, are calculated using two-phase coexistence simulations for compositions between Fe 0.98 C 0.02 and Fe 0.9 C 0.1, shown in Fig. 1. Simulations are conducted at a range of T, volume, v, conditions, spanning the pressure, P, range of the inner core (330–360 GPa). At least 50 unique calculations are performed at each condition to ensure a sufficient number of configurations are sampled (see “Methods” for details). At low P and carbon concentration, xC, T m is comparable to the pure Fe case of Alfè et al. 36, the EAM of which is used for the Fe component of the model used in this study. At high P and low xC, T m depression is smaller than Fe 1−x O x of the same xC (albeit at slightly lower P). T m is depressed by a greater amount at high xC, ~1300 K at xC=0.1 and 330 GPa. Interpolation of results provides melting temperatures at 360 GPa, shown in Table 1.

Nucleation of iron–carbon alloys
We use CMD simulations of supercooled iron alloys to study the nucleation of solids. These simulations are independent of CNT; however, CNT provides an intuitive physical picture with which to interpret the simulation results and has been shown to accurately describe homogeneous nucleation in binary alloys at core conditions 33. From our simulations, we obtain the nucleation rate, I(r), directly for sub-critical nuclei and using CNT, we are then able to fit for the critical nucleus size, r c, which has a 50% chance of spontaneously freezing a system, informing the average waiting time, τ w, to observe the freezing of a system (see “Methods”). This approach means that systems with low supercooling, and therefore small I(r) and large τ w, can be studied directly, avoiding the large extrapolation necessary in prior approaches 30. Furthermore, recording I(r) in simulations makes no assumption of the nucleating phase or system behaviour, meaning that the properties of the system can be inferred provided that CNT accurately describes I(r).

Critical radii, r c, are estimated from I(r) recorded from CMD simulations (see “Methods” and Wilson et al. 31 for details) at selected T and xC and are shown in Fig. 2 with comparison to prior results for xC=0.01 and xC=0.03 from Wilson et al. 33. Nuclei are approximately spherical for r>2 Å and have a crystal structure best described as defect-rich hexagonally close packed, as was found in our previous study 33. The average waiting time, τ w, is calculated from the difference between solid and liquid phases, γ is the interfacial energy at the boundary between solid and liquid, k B is the Boltzmann constant and τ 0 is a kinetic prefactor (τ 0 = z) linked to the Zeldovich factor, the probability of freezing or growing a nuclei z, the number density of nucleation sites, N, and the growth rate of nuclei, S, (all of which are calculated from simulations). The remaining quantities required to calculate τ w (h f, h c and γ) are fit from r c(T)= 2γ / (h f T m (1−h c δT)), at each xC (Fig. 2, see “Methods” for details).

The interatomic potential developed here reproduces the result of Wilson et al. 33 at xC=0.01 and 5000 K within 4% (r c=9.16±1.86 Å compared to r c=9.52±2.31 Å 33). At all tested values of δT, increasing xC reduces r c, although at large δT, the r c for all compositions are within the uncertainty of one another. Simulations with xC>0.1 show apparent phase separation with distinct high (xC>0.3) and low C regions, either due to immiscibility or limitations of the EAM potential. These high xC regions are beyond the conditions sampled by our ab initio calculations and so cannot be validated, we therefore omit simulations with xC>0.1 from our results.

Values of τ w are shown in Fig. 3. Results for xC=0.01 and 0.03 are from Wilson et al. 33, while results for xC=0.05 and 0.10 are calculated from the quantities shown in Table 1. N and S are calculated as averages from nuclei distributions and allow calculation of τ 0 (see “Methods”). τ 0 is not found to vary with supercooling by more than one order of magnitude and so is taken as isochemical averages. Estimates of τ w are compared to the value 3.1×10 34 sm −3 (black dashed line, Fig. 3), which implies that a region with half of the present-day inner core radius was supercooled for 1 Gyrs prior to nucleation 33. The remaining half of the inner core radius would then have grown at a rate proportional to the cooling rate of the core after nucleation. A more extreme case, which maximises waiting time by requiring that the entire volume of the inner core was supercooled prior to nucleation, would be where the inner core has not grown since nucleation and the entire inner core volume was supercooled (2.4×10 35 sm 3). The corresponding nucleation rate needed to initiate inner core freezing is where, on average, two critically sized nuclei can be expected to form within the supercooled duration and volume. For each of these nuclei, there is a 50% chance of remelting or freezing all supercooled liquids in the system. To produce a critical nucleation event in this waiting time, the xC=0.05 and xC=0.1 cases require δT=580+97/−71 K and δT=481+95/−67 K, respectively.

Methods
Interatomic potential
We use CMD simulations of liquid Fe 1−x C x to characterise nucleation behaviour at a range of T and xC. To describe the interatomic forces and system energies in our simulations, we develop an EAM that is trained on ab initio calculations. The model is fit to reproduce the positions, energy (E) and P of snapshots from ab initio molecular dynamics (AIMD) calculations run using the VASP software package 64 with the projector augmented wave method 65 and the PW91 generalised gradient approximation functional 66. Details of these calculations follow Wilson et al. 33, which shares some of the same AIMD data at low xC used for fitting the potential. The EAM potential is validated against a separate suite of AIMD snapshots to ensure that E and P are accurately reproduced. The root mean square of fluctuations in E is determined to be 0.292 and 0.316 eV per cell at 5000 K for Fe 0.95 C 0.05 and Fe 0.9 C 0.1, respectively, far less than k B T (0.431 eV). Reproduction of liquid structure is confirmed by comparison of radial distribution functions, where average positions of neighbouring atoms in CMD simulations are within 0.05 Å of AIMD simulations for all interactions and all volume V, T, xC conditions. Further comparison of this potential with AIMD validation data is provided in the Supplementary Information.

AIMD simulations are performed by melting systems of 128 atoms with different carbon fractions (close to 20, 10 and 5 mol%) at 10,000 K for 1 ps before equilibrating at a target T (4000, 5000 and 6000 K) for 1 ps and evolving the system at the target T for 30 ps. The simulation cell volume is tuned for each composition and target T to achieve a P of 360 GPa. From the final 30 ps of simulation time, configurations are selected at every 100 fs as data on which the EAM is trained. The total energy E of a Fe 1−x C x system is defined by the EAM as the sum of contributions from all atomic interactions
E= Σ iFe E Fe i + Σ iC E C i + Σ iFeC E FeC i.

Each interaction between atoms i and j contains repulsive Q and embedded F contributions. Q depends on the interatomic distance r ij, which also defines an electron density ρ ij on which F depends. E for each type of interaction is given by
E Fe i = Σ jFe Q Fe (r ij) + F Fe (ρ i Fe) + ϵ FeC Σ jC ρ FeC (r ij) n Fe − ϵ FeC _Fe √ρ i Fe,

E C i = Σ jC Q C (r ij) + F C (ρ i C) + ϵ C Σ jC ρ C (r ij) n C − ϵ CC _C √ρ i C,

E FeC i = Σ jFe Q FeC (r ij) + F FeC (ρ i FeC) + ϵ FeC Σ jC ρ FeC (r ij) n FeC − ϵ FeC _FeC √ρ i FeC,

where the respective densities are
ρ Fe i = Σ jFe a Fe (r ij) m Fe + Σ jC a FeC (r ij) m FeC

and
ρ C i = Σ jC a C (r ij) m C + Σ jFe a FeC (r ij) m FeC.

Here, ϵ, a, n, m and C _ are free parameters to be fit for each interaction and are reported in Table 2. The primary difference to the parameters found in our previous study 33 is a reduction in ϵ FeC and ϵ C, as well as a FeC and a C.

Melting temperatures
The melting temperatures of Fe 1−x C x are calculated with coexistence simulations using the EAM potential and the LAMMPS simulation package 67. Systems of 128,000 atoms are arranged into a long periodic cell where the x-axis is 3 times the length of the y and z axes. All atoms are initially arranged in a hexagonally close-packed structure with C atoms randomly replacing Fe atoms to achieve the desired concentration. This substitutional model is chosen based on ab initio evidence that C substitutions produce a lower free-energy state than interstitial C 47. The positions of atoms in the central 50% of the simulation are initially fixed in space, whilst the other half is melted at 10,000 K for 5 ps. This procedure establishes the two-phase system. The entire system is then evolved at a target T under the NVT ensemble, where the number of atoms, volume and temperature are held constant, for 1 ps to establish the target average kinetic energy. Finally, the system is evolved for 10 ps under the NVE ensemble (constant Number of atoms, Volume of the system and Energy of the system), allowing the solid region of the system to grow or melt. This process is repeated a minimum of 50 times for each composition, temperature and volume initial condition (where the volume is chosen such that the pressure of the system is 360 (±2) GPa at the conditions of interest). As a result, a wide variety of configurations are sampled.

Once a system has reached equilibrium, the T will lie on the melting curve, meaning that the time-averaged T and P provide a single T m. The random distribution of C in the initial system provides many different initial xC for the solid and freezing and melting of the solid allow for C partitioning between the solid and the liquid. Systems with xC>0.05 in the solid see much of the solid melt before freezing a lower xC solid. This process shows that whilst C cannot diffuse freely in the solid over the timescale of these simulations, systems tend towards chemical equilibrium through freezing and melting. We note that we do not observe evidence of superionicity in these simulations, in line with previous ab initio studies 47, although this behaviour has been reported elsewhere, C-bearing Fe alloys at core conditions 68. Simulations where T and P are stable on a picosecond timescale have T m (v, x) calculated from the fluctuations of T and P over the final 1 ps of simulation time and discard any simulations that entirely freeze, melt, or do not achieve equilibrium. Because of the constant volume and energy conditions, T and P are unknown prior to the simulation setup. In order to define T m (P, x), we explore a range of initial T and v and interpolate our results for T m (360 GPa, x).

Nucleation theory
In this study, we use CNT to describe the nucleation behaviour of CMD simulations of iron alloy liquids at the conditions of Earth’s core. Previous studies 30–33 have routinely found that predictions from CNT are consistent with outputs from MD simulations, accurately describing the distribution of nucleus sizes and the dependence of nucleation rate on supercooling. We note that our CMD simulations are completely independent of CNT; indeed, these simulations have been used to show that non-classical effects such as pressure waves have no effect on the nucleation of solids in Earth’s core 30. According to CNT, the requirement for liquids to be supercooled prior to freezing via homogeneous nucleation arises from a competition between two energetic contributions to the total free energy, ΔG, associated with forming a solid nucleus in a supercooled liquid. The first contribution is the free-energy release, gsl, associated with transforming supercooled liquid into a solid, which is always favourable when below the melting temperature and occurs through random fluctuations in the liquid, producing ‘solid-like’ configurations of atoms. The second contribution, γ, is associated with forming an interface between the liquid and solid and is always unfavourable. These two components are scaled by the volume and surface area of the newly formed nucleus of radius r to define a total free-energy change on formation
ΔG(r)= 4/3 πr3gsl+4πr2γ

for spherical particles.

The rate I at which a nucleus of radius r forms is defined by Boltzmann statistics:
I(r)=I 0 exp (−ΔG(r) / k B T),

where k B is Boltzmann’s constant and I 0 scales the nucleation rate of the specific system. Eq. (8) shows that small nuclei are likely to form often (or equivalently, require less waiting time (τ w ≈ I −1) before they occur). However, Eq. (7) shows that these nuclei will remelt rather than grow because of the large influence of surface area on the free energy at small r. Despite a low probability, continued growth is possible given sufficient fluctuations to produce a larger nucleus. Above a critical radius r c = −2γ/gsl at the peak of ΔG, the volume term in Eq. (7) increases with radius faster than the surface term, meaning that whilst still having an overall unfavourable free energy for forming a nucleus, continued growth is thermodynamically favoured when compared to remelting. Greater supercooling requires a smaller r c in order to freeze a system, which in turn requires less waiting time for the critical event to spontaneously occur.

The rate at which a nucleus of radius r spontaneously forms in a supercooled liquid is given by Eq. (8). When framed in terms of r c, the inverse of nucleation rate describes the average duration before a supercooled system will experience a critical nucleation event and freeze
τ w = τ 0 exp (ΔG(r c) / k B T),

where
r c = −2γ / gsl.

The prefactor τ 0 can be described by
τ 0 = z / NS,

where the Zeldovich factor z is related to gsl through
z = ( 4πr c 3 gsl / 3 k B T ) 1/2,

and N and S are the number of available nucleation sites and the rate of nuclei growth, respectively. To quantify N, S and I(r), solid-like arrangements of atoms are identified at each timestep in the same manner as our previous studies 31,33 following Reinten Wolde et al. 69. Therefore, all quantities required to calculate τ w are accessible via CMD calculations. Because r c is predicted to be large for the P and T of the early Earth’s supercooled liquid core 31, simulations at >4000 K and 360 GPa cannot be expected to produce a nucleus of the critical size (>30 Å). Instead, r c is predicted by recording the rate at which smaller nuclei (which are more common) are observed in simulations, informing I(r) where r is small. At a fixed T all quantities in Eq. (7) are constant, so we can write
−ln I(r) / ΔG(r) 

and the distribution of nuclei observed in simulations describes the form of ΔG(r) but not the amplitude. Nuclei are observed to be approximately spherical for r>2 Å, in line with our previous studies 31,33. The radius of any recorded nuclei is taken as r = (v / 4π 3) 1/3, where v is the volume occupied by a nucleus. The form of the free-energy barrier can be represented by
ΔG(r)=4/3 πr3A+4πr2B,

where A and B are variables at each T, meaning that r c can be estimated via r c = −2B/A, equivalent to Eq. (10). When fitting for A and B, only I(r ≥2 Å) is considered to avoid errors associated with small non-spherical nuclei. If repeated for a range of T (and therefore δT) r c(T) is obtained. The free parameters γ, h f and h c are then found by fitting for r c(T) through
r c(T) = −2γ / (h f δT (1−h c δT)),

where the h f is the enthalpy of fusion and h c accounts for non-linearity with temperature when defining the free energy liberated by freezing supercooled liquid
gsl = h f δT (1−h c δT).

This representation of gsl does not consider x, meaning that h f and h c must be unique to each composition. In a previous study 70 we considered g sl(T, P, x) including ideal mixing g mix and found that, for an Fe–O system, the mixing effect is negligible. In the Fe–C systems of this study ideal mixing contributions (g mix = R T ln x, where R is the gas constant) are also small compared to g sl and non-ideal effects have been found to be minor 39.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Wilson et al. investigate Earth’s core composition by studying inner core nucleation. The problem originates from the “inner core nucleation paradox” proposed in previous work. By testing different binary Fe alloys in their earlier study (Ref. 33), the authors found Fe-C to be a possible candidate for the inner core. Based on this observation, they refined their Fe-C interatomic potential and discovered that a carbon content of ~0.1 could lead to reasonable supercooling, which is required to nucleate the initial solid phase of the inner core. In this way, they offer a new approach to constraining the core’s light element content, independent of previous constraints from seismology and thermodynamics.\n\nThe idea of using initial nucleation to constrain the light element content is novel. I believe this work is worthy of publication in a high-impact journal for a broad audience. However, the current manuscript has a few issues that need to be addressed to make this approach more compelling and robust. Below are some suggestions.\n\n1. In the Tm calculations, the authors describe constructing the initial lattice by replacing Fe with C, implying a substitutional model. However, in a substitutional model, direct MD simulations may not yield a realistic melting point because C atoms remain nearly fixed in lattice sites. While the authors use a few snapshots to estimate the melting point, this approach is not well established. Monte Carlo simulations are typically required to equilibrate the solute distributions in such a solid solution.\n2. Carbon has been suggested to be superionic in the inner core (He et al. 2022). However, the authors seem to have overlooked this work when computing the partition coefficient.\n3. The authors modify their previous Fe-C interatomic potential. These modifications and their impact should be discussed. Comparisons of the liquid radial distribution function and scatter plots with DFT data for energy and forces should be presented.\n4. To my understanding, the current potential is still not applicable for xC larger than 0.1. Can the authors discuss possible reasons for this limitation?\n5. In the nucleation simulations, the authors assume a spherical nucleus. However, deviations from spherical shape are commonly encountered across a wide range of systems. Such deviations could introduce errors in the determination of nucleation rates, an issue that has been discussed in detail in the materials science community (e.g., J. Chem. Phys. 155, 040901 (2021)). If the manuscript is to be published in a multidisciplinary journal, I suggest the authors clarify this assumption. Additionally, they should provide a clear definition of how they determine the radius of the nucleus when it is not perfectly spherical.\n6. Equations 16–17 were derived for pure elemental systems. It is unclear whether they remain valid for binary systems, given that the equilibrium chemical composition changes between the bulk solid and liquid phases. Moreover, it is uncertain whether the nucleus maintains the same chemical composition as the bulk phase and whether the partition coefficient remains constant with the temperature. These factors could significantly affect g^{sl} , which in turn influences the nucleation rate exponentially. The authors need to carefully investigate these effects.\n7. The authors do not provide much analysis of the as-nucleated phase, which, in my opinion, is a crucial outcome representing the initial form of the inner core’s solid. What crystalline phase forms? What is the nature of C in the Fe lattice—substitutional, interstitial, dimeric, or superionic? The spontaneously formed phase is vital for interpreting the results.\n8. This study and Ref. 33 focus on light elements. However, nickel and other heavy elements in the core may also influence initial nucleation. The authors should include some discussion on these potential effects.", "reviewer_2": "This paper presents new data on the influence of carbon (at high concentrations) on the supercooling required for homogeneous nucleation of Earth’s inner core. The authors find that C has a larger effect on the required supercooling than other individual “light elements” that have been considered to date. This is a valuable finding that deserves to be published. The required supercooling (hundreds of K) is still too large to plausibly explain nucleation of the inner core, but it is a step in the right direction.\n\nDetailed Comments\n39-40 Cooling of the core is determined by heat transfer across the core-mantle boundary, not by the thermal conductivity of the core.\n156-62 A little more discussion of this waiting time would be helpful. A large fraction of the present inner core volume sitting for 1 Gyr at a constant (large) supercooling is unrealistic and hence does not appear to be a “moderate” assumption. On the other hand, Fig. 3 shows that the required supercooling is fairly insensitive to the exact value of the waiting time.\n164-7. This is really pushing it. Even if the required supercooling at 10% C is just within the maximal bound on the core temperature change from nucleation to the present, it would require crystallization of essentially the entire inner core from a supercooled state, which would have occurred very rapidly, essentially yesterday, and with the “inner” inner core (if it represents early, rapid crystallization from a supercooled state) representing essentially the entire core. This is not a plausible scenario. The same goes, really, for the lower supercooling values discussed below, at higher C concentrations. In my opinion, the paper will be more effective if it’s up front about the plausible limits and doesn’t overstretch the argument for C.\n252. Certainly a novel constraint, with the potential to be strong. But there are still many uncertainties, including whether there is a plausible route via heterogeneous nucleation on a metallic substrate.\n263-78. The Conclusion section consists mainly of points that are outside what is discussed in the main body of the paper, and not particularly dependent on the results presented.", "reviewer_3": "The manuscript builds on previous research investigating the supercooling behavior of core compositions relevant to Earth's core, specifically focusing on the binary Fe-C system. The authors employ molecular dynamics simulations to determine key parameters governing the transition from a fully liquid state to the onset of freezing. They then apply classical nucleation theory to interpret their findings and derive critical parameters, such as the required supercooling and the critical nucleus size.\n\nThe methodology, largely adopted from their previous work, has been well established in earlier studies. In this paper, they extend the approach to additional compositions within the Fe-C system, bringing their analysis closer to the supercooling conditions constrained by geophysical models. The central contribution of the manuscript, as I understand it, is to introduce a paradigm shift—positioning the study of supercooling behavior as an independent yet complementary constraint in planetary core research.\n\nWhile the paper does not introduce significant methodological innovations (at least not entirely clear from manuscript), I recognize the authors’ effort to highlight this new perspective. This work is timely, as the challenge of nucleating Earth's inner core was identified by Huguet et al. (2018) and has remained an open question since. The authors, through this and previous work, have steadily developed a novel and constructive approach to addressing this issue. In this context, the present manuscript represents a crucial step in establishing supercooling behavior as a key research focus in planetary science.\n\nThe paper is well-written, concise, and clear. In my opinion, it is suitable for publication. Below, we (ECR and me) provide some general remarks and minor comments that should be addressed before final acceptance.\n\nGeneral Comments:\n- It is somewhat unfortunate that the simulations fail for x > 0.1, as these cases appear to be the most relevant in terms of geophysical constraints. It would be helpful to provide a more detailed explanation of the issues causing these failures. While there are hints throughout the text, a clearer and more concrete statement would be beneficial—does the problem stem from the limitations of the EAM potential? What are the implications of this failure?\n- The discussion of geophysical constraints could be expanded. What specific geophysical models are referenced? To what extent are they independent of the assumed concentrations? Given that supercooling behavior is being proposed as a new constraint on core composition, a more detailed explanation of these models and their role in this context would strengthen the manuscript.\n- The paper would benefit from a clearer distinction from the authors’ previous work, particularly Wilson et al. (2023) [33]. It would be helpful to explicitly state any methodological differences and clarify how this study extends prior research (e.g., by applying the same methodology to mixtures with higher carbon concentrations). For instance, the direct reference to [33] in lines 99–101 could be expanded with a statement such as: “We extend previous studies by considering …” to emphasize the novelty of the present work.\n- Along the same lines: In what way do the contributions unique to this paper—such as the investigation of the F-C​ mixture at higher carbon concentrations — establish this admissibility criterion in a manner not previously considered?\n- Similarly, the discussion of results could be strengthened with a summarizing statement. For example: “Through the simulations of the Fe-C mixture at higher x conducted in this study, we have taken a first step toward assessing the undercooling admissibility criterion …”\n\nMinor comments:\n- l. 119: The definition of abbreviation EAM for “Embedded Atomic Model” is missing. It is only introduced in the later Sec. 5.\n- l. 121: the variable x should probably be an x^C\n- Fig. 1: in the lower plot, the y axis label is missing a variable name (e.g. x or x^C)\n- l. 128: the nucleation rate I(r) is used as a function. Since it was only mentioned once before repeating at least the name here would make it easier to read\n- l. 137: starting a sentence with a variable tau_w. Change to something like “The average waiting time tau_w is computed as tau_w=tau_0 exp(...)”\n- l. 145: it is not easily traceable for me what the differences of the interatomic potential are wrt. to the existing work from Wilson et al. 2023. The methodology described in 5.1 seems to be exactly the same. For me, a highlight of differences would be useful to assess in which regard this work is new.\n- l. 148: missing “break” between symbols delta T and r_c, recommend to change to “at large delta T critical radii r_c for all …\"\n- ll. 148-150: please elaborate a bit on the type of instability / why the method breaks down (see general comment above). It is fair the method has limitations but it would be helpful to understand them better\n- l. 157: Waiting time is given in time/volume. It would be great to have one or two explanatory sentences as to how this relates to a time in the classical sense.\n- l. 328: the abbreviation “NVE” has not been defined\n- l. 414: please fix the link to the zenodo repository to include a full copyable doi link, e.g. www.doi.org/10.5281/zenodo.13144422 . Also maybe rephrase the sentence such it does not end with the link to avoid merging the sentence-ending dot with the URL\n\nSmall comments (mostly writing):\n- l.66: delete “result in …” (duplication)\n- l. 79: missing comma after “Here”\n- l. 94: “Ref” shown before reference [34]\n- l. 101: missing space before reference [33]\n- l. 103: potential comma after “nucleation”\n- ll. 129-130: repeated word “which”\n- l. 147: missing comma before “although”\n- Table 1: maybe put the unit of variables in its own column in the tabular to fix alignment\n- l. 237: missing comma -> “which, when applied to a specific model of core formation, …\"\n- ll. 238-239: repeated use of the word “strongly”, suggest to replace once\n- ll. 346-347: double use of the word “consistent”\n- l. 422: Section heading weirdly transitions into the following paragraph.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Response to referees\nOriginal reviewer comments: Italics\nAuthors’ responses: regular font\nLocation of changes in revised manuscript: bold\nReviewer #1 (Remarks to the Author):\nWilson et al. investigate Earth’s core composition by studying inner core nucleation. The\nproblem originates from the “inner core nucleation paradox” proposed in previous work. By\ntesting different binary Fe alloys in their earlier study (Ref. 33), the authors found Fe-C to be\na possible candidate for the inner core. Based on this observation, they refined their Fe-C\ninteratomic potential and discovered that a carbon content of ~0.1 could lead to reasonable\nsupercooling, which is required to nucleate the initial solid phase of the inner core. In this\nway, they offer a new approach to constraining the core’s light element content, independent\nof previous constraints from seismology and thermodynamics.\nThe idea of using initial nucleation to constrain the light element content is novel. I believe\nthis work is worthy of publication in a high-impact journal for a broad audience. However, the\ncurrent manuscript has a few issues that need to be addressed to make this approach more\ncompelling and robust. Below are some suggestions.\nWe thank the reviewer for their encouraging assessment of our study, its novelty and the\npotential for impact. The comments made are helpful for improving the clarity of our\napproach. We have addressed the reviewer’s comments below.\n1. In the Tm calculations, the authors describe constructing the initial lattice by replacing Fe\nwith C, implying a substitutional model. However, in a substitutional model, direct MD\nsimulations may not yield a realistic melting point because C atoms remain nearly fixed in\nlattice sites. While the authors use a few snapshots to estimate the melting point, this\napproach is not well established. Monte Carlo simulations are typically required to equilibrate\nthe solute distributions in such a solid solution.\nThe reviewer is of course correct that care must be taken when calculating melting\ntemperatures through this two-phase coexistence method. We have taken the necessary\nprecautions to ensure that our results are robust. We define the composition of our two-\nphase coexistence calculations by substituting a random selection of Fe atoms with C and\nwhere these atoms are within the solid region they are unable to diffuse within the timespan\nof these simulations. This choice of a substitutional model is based on the study by Li et al.\n(2019, https://doi.org/10.1029/2019JB018789) where substitutions are found to be lower\nfree-energy configurations than interstitial ones, implying that this is how C would be held in\nthe Fe crystal lattice at core conditions. To ensure that we sample configurational space\nsufficiently and produce accurate melting temperatures we conduct at least 50 simulations\nper temperature, volume, composition condition, each with a unique initial distribution of\ncarbon. Many of these two-phase simulations see a large degree of melting and refreezing\nas the phase boundaries move within the simulation volume during the equilibration\nperformed in every simulation. This means that C atoms are released to the melt and\nsometimes trapped in the solid, although the solid predominantly forms with little C upon\nrefreezing, in agreement with the partitioning results of Li et al. (2019). These cases show\nthat C atoms are mobile in these simulations and that equilibrium conditions can be met\neven without diffusion. Ultimately, this is a robust method for calculating the melting\ntemperatures of binary systems using molecular dynamics simulations, as proven through\napplication to a number of high pressure and temperature systems (for example Fe-S, Fe-Si,\nFe-O, Alfe et al., 2002 https://doi.org/10.1063/1.1464121; Li-H, Ogitsu et al., 2003\nhttps://doi.org/10.1103/PhysRevLett.91.175502; MgO, Belonoshko et al., 2010,\nhttps://doi.org/10.1103/PhysRevB.81.054110;\nMgSiO3, Deng et al., 2023 https://doi.org/10.1103/PhysRevB.107.064103).\nWe have added text in the Methods section on the choice of a substitutional model\n(lines 360-361), the sampling of configurational space (lines 367-371) and the diffusion\nof C and melting-refreezing of the solid (lines 377-379). We have also added text to\nlines 120-122 of the results section which briefly describes to the care taken on these\nmatters and direct the reader to the Methods section for greater detail.\n2. Carbon has been suggested to be superionic in the inner core (He et al. 2022). However,\nthe authors seem to have overlooked this work when computing the partition coefficient.\nWe thank the reviewer for highlighting this paper. The transport properties of C are not the\nfocus of this study although we note that no evidence of C superionicity is seen in our\nsimulations. This finding is in line with that of Li et al. (2019,\nhttps://doi.org/10.1029/2019JB018789) who do not find any diffusive behaviour in their ab\ninitio simulations of Fe C through root mean square displacement testing. We conclude\n60 4\nthat this topic is of great interest but not yet resolved and that the simulations presented here\nare not designed to interrogate the transport properties of the inner core, largely due to the\nout of equilibrium (supercooled) condition we are studying in these simulations.\nTo address this topic we have added text at lines 379-381 including reference to He et\nal. (2022) and Li et al. (2019).\n3. The authors modify their previous Fe-C interatomic potential. These modifications and\ntheir impact should be discussed. Comparisons of the liquid radial distribution function and\nscatter plots with DFT data for energy and forces should be presented.\nWe have added detail on how the potential of our previous study (Wilson et al. 2023,\nhttps://doi.org/10.1016/j.epsl.2023.118176) differs from the potential presented here and\nhave now included a supplementary information file which provides comparison of energy,\npressure and radial distribution function between this new potential and independent ab initio\ndata. This supplementary material demonstrates that our revised interatomic potential\nrobustly reproduces the behaviour of ab initio calculations with 10 mol.% C.\nReference to this supplementary information is now made on lines 333-334 and the\ndifference in the parameters of the potential compared to our previous study are\nnoted on lines 352-353.\n4. To my understanding, the current potential is still not applicable for xC larger than 0.1.\nCan the authors discuss possible reasons for this limitation?\nIn our simulations at xC > 0.1 we observe the liquid separate into distinct regions of high and\nlow xC. This apparent phase separation produces regions where xC is approximately 0.3, far\nhigher than the ab initio data used to train and validate the EAM potential. Therefore, we\ncannot validate this behaviour without producing an additional dataset of ab initio\ncalculations where xC >> 0.1. This dataset would need to include xC from 0.0 to at least 0.3,\nmeaning more than doubling the size of our existing dataset. This is a considerable task and\nwould require significant computational resources. Given that there is little expectation for\nthe core to host such high concentrations of C (mostly due to limitations that seismological\nconstraints place on the xC of the core, see Hirose et al., 2021,\nhttps://doi.org/10.1038/s43017-021-00203-6, for a review) there is limited motivation for such\ninvestment so we instead chose to omit results for xC > 0.1 from our study. Instead, we focus\non how results for xC up to 0.1 demonstrate that inner core nucleation will provide a\nconstraint on core composition when a viable mechanism is identified.\nWe have added text to lines 158-162 of the results section to explain this behaviour\nand the reason that we omit the results from our study.\n5. In the nucleation simulations, the authors assume a spherical nucleus. However,\ndeviations from spherical shape are commonly encountered across a wide range of systems.\nSuch deviations could introduce errors in the determination of nucleation rates, an issue that\nhas been discussed in detail in the materials science community (e.g., J. Chem. Phys. 155,\n040901 (2021)). If the manuscript is to be published in a multidisciplinary journal, I suggest\nthe authors clarify this assumption. Additionally, they should provide a clear definition of how\nthey determine the radius of the nucleus when it is not perfectly spherical.\nThe reviewer raises a valid point which we have taken great care to address in our previous\nwork (Wilson et al. 2021, https://doi.org/10.1103/PhysRevB.103.214113; Wilson et al. 2023,\nhttps://doi.org/10.1016/j.epsl.2023.118176) and this study. We have clarified that our\nprevious study on these supercooled liquids shows that for all but the smallest nuclei a\nspherical assumption is valid and represents the geometry of observed nuclei well. To avoid\nerrors that non-spherical nuclei can introduce, we do not include nuclei smaller than 2 Å in\nour determination of critical radii.\nThe text is on lines 145, 440-442 and 445-446 has been modified for clarity on these\ndetails.\n6. Equations 16–17 were derived for pure elemental systems. It is unclear whether they\nremain valid for binary systems, given that the equilibrium chemical composition changes\nbetween the bulk solid and liquid phases. Moreover, it is uncertain whether the nucleus\nmaintains the same chemical composition as the bulk phase and whether the partition\ncoefficient remains constant with the temperature. These factors could significantly affect\ng^{sl} , which in turn influences the nucleation rate exponentially. The authors need to\ncarefully investigate these effects.\nThe reviewer is correct that the formulation of classical nucleation theory was originally\nintended for single component systems. In this study we simply use this formalism to\ndescribe the distribution of nuclei sizes in our results and find that it is adequate in doing so,\nas do our previous studies employing this approach (Wilson et al.\n2021, https://doi.org/10.1103/PhysRevB.103.214113; Wilson et al. 2023,\nhttps://doi.org/10.1016/j.epsl.2023.118176, where the latter focused on two-component\nsystems including Fe-C). Because we allow nucleation to spontaneously occur in these\nmolecular dynamic systems, we do not make any assumption of the composition of the\nnucleating phase or partition coefficients. This means that by inferring gsl from the observed\ndistributions of nuclei in our simulations, we can learn about the behaviour gsl with\ncomposition and supercooling. We find that gsl is not greatly modified by xC, in line with our\nprior study (Wilson et al. 2023, https://doi.org/10.1016/j.epsl.2023.118176), and that the\ndifference in interfacial energies is predominantly responsible for the difference in system\nbehaviours.\nAdditionally, we have recently published a study (Walker et al., 2025,\nhttps://doi.org/10.1098/rspa.2024.0505) which does consider the modification of gsl for\nsystems where solid and liquid compositions are expected to differ significantly. In this study\nthe effect of the partitioning specifically is shown to be small, even in cases where it is\nmaximised. If we take a similar approach for our study and evaluate the ideal mixing\ncontribution to the free energy (g = R T ln(x)) we find that this term is at least four orders\nmix g\nof magnitude smaller than the free energy difference between solid and liquid phases. This\nanalysis assumes that non-ideal contributions are small, which is found by Umemoto and\nHirose (2022, https://doi.org/10.1016/j.epsl.2019.116009) to be the case for several light\nelements in the core, including carbon.\nWe have added text on lines 131-132 and lines 138-141 to explain that classical\nnucleation theory has been shown to accurately describe the nucleation rate of these\nsystems in our previous work. In the Methods section, alongside equations 16 and 17\n(lines 451-456), we explain that mixing terms, which are not included in classical\nnucleation theory, are expected to be negligible whilst explaining the assumptions\nmade therein.\n7. The authors do not provide much analysis of the as-nucleated phase, which, in my\nopinion, is a crucial outcome representing the initial form of the inner core’s solid. What\ncrystalline phase forms? What is the nature of C in the Fe lattice—substitutional, interstitial,\ndimeric, or superionic? The spontaneously formed phase is vital for interpreting the results.\nThe reviewer is correct that the initially nucleated phase of the inner core is of great interest\nfor understanding first solids of the inner core, as well as its the subsequent growth and\nproperties. However, our approach samples sub-critical nuclei, which in our simulations all\ngo on to remelt. In our previous study (Wilson et al., 2021,\nhttps://doi.org/10.1103/PhysRevB.103.214113) we use the same technique alongside a\n“brute force” approach of applying extreme supercooling to trigger critical nucleation within\ntimespans accessible to molecular dynamics simulations. In these cases, we find that the as-\nnucleated phase will be metastable or unstable, relaxing to a more stable state if observed\nfor an extended duration (picoseconds after the nucleation event). This mechanism of “two-\nstep” nucleation is also found by Sun et al. (2022, https://doi.org/10.1073/pnas.2113059119)\nwhere the body centred cubic structure of iron nucleates more readily than the more\nthermodynamically stable hexagonally close packed phase. Two-step nucleation therefore\nillustrates that the connection between sub-critical nuclei phases and the young inner core is\ncomplicated, especially on geological timescales where millions of years of annealing,\ndiffusion and deformation have altered crystal phases and textures.\nAs detailed in the response to comment #6, the nature of the nucleating phase is not crucial\nto understanding the energetics and rates of nucleation in the systems we are studying. It is\nprecisely these quantities which allow us to identify compositions which can trigger freezing\nat supercooling which is compatible with Earth’s core. Therefore, whilst the details of the as-\nnucleated phase are crucial for understanding the properties of the inner core in its smallest\nand youngest state, these details do not change our demonstration that inner core\nnucleation can be a constraint on the composition of the core.\nOur prior studies contain information on the as-nucleated phase which applies to this study\nas well (Wilson et al. 2021, https://doi.org/10.1103/PhysRevB.103.214113; Wilson et al.\n2023, https://doi.org/10.1016/j.epsl.2023.118176). In Wilson et al. (2023) we find that the\nphase of sub-critical nuclei in supercooled Fe-C liquids are best described as defect rich\nhexagonally close packed with C being incorporated substitutionally, although the defects\n(often planar in nature) make interpretation of phase difficult.\nWe have added text on lines 145-147 to describe the observed phase of the sub-\ncritical nuclei in this study. Here we also direct the reader to our prior studies which\ncontain a more detailed investigation.\n8. This study and Ref. 33 focus on light elements. However, nickel and other heavy elements\nin the core may also influence initial nucleation. The authors should include some discussion\non these potential effects.\nThe reviewer is correct. The effect of metals in the core is likely to have an impact on the\nnucleation rates of these systems. We note that Sun et al. (2024,\nhttps://doi.org/10.1073/pnas.2316477121) found that Ni in particular may reduce the\nsupercooling required to nucleate the inner core by ~400 K. In this study we focus\nexclusively on the effect of C because of the accelerated nucleation rate if Fe-C liquids\ndiscovered in our previous study. When combined, Ni and C may explain inner core\nnucleation with a plausible degree of supercooling (dT << 400 K). This would be significant\nbecause core thermal histories require that dT < 100 K (Wilson et al. 2025,\nhttps://doi.org/10.1038/s43017-024-00639-6), whilst seismology requires that the core is\nhost to light elements. A combination of Fe, Ni, C and another light element could therefore\nexplain the thermal evolution of the deep Earth, inner core nucleation and seismological\nobservations, which is the main message of this manuscript.\nWe have added text at lines 284-287 to note that outside of carbon and other light\nelements, metallic elements present in the core might play a part in the process of\ninner core nucleation and would be required to provide a complete model of core\ncomposition. We have also noted that Fe-Ni-C-X alloys make a compelling next step\nfor investigation on line 290-293.\nReviewer #2 (Remarks to the Author):\nThis paper presents new data on the influence of carbon (at high concentrations) on the\nsupercooling required for homogeneous nucleation of Earth’s inner core. The authors find\nthat C has a larger effect on the required supercooling than other individual “light elements”\nthat have been considered to date. This is a valuable finding that deserves to be published.\nThe required supercooling (hundreds of K) is still too large to plausibly explain nucleation of\nthe inner core, but it is a step in the right direction.\nWe thank the reviewer for their careful consideration of the manuscript. We are pleased that\nthey find the work to be valuable and that they support publication of the manuscript. The\ncomments raised are valid and valuable, we have addressed each of them below.\nDetailed Comments\n39-40 Cooling of the core is determined by heat transfer across the core-mantle boundary,\nnot by the thermal conductivity of the core.\nThe reviewer is correct. Mantle convection sets the heat flow across the core mantle\nboundary and the thermal conductivity of the core controls the adiabatic heat flow through\nthe core. The thermal conductivity of the core controls the convective vigour of the core. I.e.\na high core conductivity will result in more vigorous convection than a low core conductivity,\nif the core-mantle boundary heat flux is the same in both cases.\nWe have modified the text at lines 38-39 to make this statement clear.\n156-62 A little more discussion of this waiting time would be helpful. A large fraction of the\npresent inner core volume sitting for 1 Gyr at a constant (large) supercooling is unrealistic\nand hence does not appear to be a “moderate” assumption. On the other hand, Fig. 3 shows\nthat the required supercooling is fairly insensitive to the exact value of the waiting time.\nThis is a valuable comment raised by reviewer 2 and 3. 420 K of supercooling and a waiting\ntime of 1 Gyr are the maximum plausible values for these quantities given current\nknowledge. We referred to the “moderate” case as such because it represents a scenario\nwhen half of the present day inner core radius is supercooled for 1 Gyr and after inner core\nnucleation the remaining half of the inner core radius could have grown slowly. In this sense\nthe case is moderate because it does not require the entire inner core to have frozen in the\nimmediate past, which is what we would term the “extreme” case. Despite both of these\ncases seeming unlikely, our recent review on the topic (Wilson et al. 2025,\nhttps://doi.org/10.1038/s43017-024-00639-6) shows that there are currently no geophysical\nobservations which directly prohibit an extended period of supercooling in the core. Because\nof the exponential relationship between supercooling and waiting time it makes little\ndifference to the interpretation of our results if a moderate or extreme case is chosen, as the\nreviewer suggests. Following this comment, we feel that an improved description of the\nmaximum allowable waiting time for inner core nucleation is warranted.\nWe have modified the text on lines 171-179 to better illustrate the meaning of these\nvalues and also added a section to the supplementary materials which provides a\nmore detailed description, including a case where the inner core is 300 Myrs old,\nshowing that even for a more geophysically plausible scenario the outcome is similar\ndue to the exponential relating in Eq 10. This description follows our recent review\n(Wilson et al., 2025, https://doi.org/10.1038/s43017-024-00639-6) which explores the\ntopic in detail. Additionally, we have removed use of the work “moderate” because we\nagree with the reviewer that it is not helpful for the reader’s understanding of these\nvalues.\n164-7. This is really pushing it. Even if the required supercooling at 10% C is just within the\nmaximal bound on the core temperature change from nucleation to the present, it would\nrequire crystallization of essentially the entire inner core from a supercooled state, which\nwould have occurred very rapidly, essentially yesterday, and with the “inner” inner core (if it\nrepresents early, rapid crystallization from a supercooled state) representing essentially the\nentire core. This is not a plausible scenario. The same goes, really, for the lower\nsupercooling values discussed below, at higher C concentrations. In my opinion, the paper\nwill be more effective if it’s up front about the plausible limits and doesn’t overstretch the\nargument for C.\nThe reviewer is correct, this notional agreement between the supercooling needed to\nnucleate from 10% C liquids and the maximum supercooling compatible with the structure of\nthe core probably does not represent a situation that is realistic for inner core nucleation. We\nhave explained this to be the case in the discussion and feel that this notional agreement is\na demonstration that some core compositions can substantially reduce the supercooling\nrequirement for homogeneous nucleation, showing that it could well be a viable mechanism\nfor inner core nucleation.\nFollowing the reviewer’s suggestion, the new text on lines 199-203 make a more direct\nstatement on the significance of the maximum available supercooling and we have\nadded a section to the supplementary information which give more details on the\nmaximum supercooling which the core.\n252. Certainly a novel constraint, with the potential to be strong. But there are still many\nuncertainties, including whether there is a plausible route via heterogeneous nucleation on a\nmetallic substrate.\nWe thank the reviewer for your positive assessment of our finding. It is true that uncertainties\ndo remain, including whether heterogeneous nucleation is a viable mechanism. Our\ncomment that heterogeneous nucleation presents a reduction to the nucleation barrier, not a\nremoval of it, makes the point that for heterogeneous nucleation to be viable, it must be\nunderpinned by a homogeneous case which requires significantly less supercooling for\nnucleation than pure Fe and that despite previous efforts (Huguet et al., 2018,\nhttps://doi.org/10.1016/j.epsl.2018.01.018; Wilson et al., 2025,\nhttps://doi.org/10.1038/s43017-024-00639-6) no viable candidate for a heterogeneous\nnucleation site in the Earth’s liquid core has been identified thus far.\nOn lines 221-241 we note that heterogeneous nucleation does not remove the\nrequirement for supercooling, and only a reduces it, and discuss the requirement of a\nsuitable candidate heterogeneous nucleation site. On lines 294-299 we explain that\nthe constraint on core composition proposed in our manuscript is still valid if\nheterogenous nucleation triggered inner core growth because of the requirement for a\nstable heterogeneous phase and a core composition compatible with nucleating onto\nthis phase.\n263-78. The Conclusion section consists mainly of points that are outside what is discussed\nin the main body of the paper, and not particularly dependent on the results presented.\nWe agree with the reviewer that in isolation the points discussed in the conclusion were\nincongruent. We have opted to remove the conclusions section and incorporate these points\ninto the final section of the discussion where they belong.\nThe modification can be found on lines 300-315.\nReviewer #3 (Remarks to the Author):\nThe manuscript builds on previous research investigating the supercooling behavior of core\ncompositions relevant to Earth's core, specifically focusing on the binary Fe-C system. The\nauthors employ molecular dynamics simulations to determine key parameters governing the\ntransition from a fully liquid state to the onset of freezing. They then apply classical\nnucleation theory to interpret their findings and derive critical parameters, such as the\nrequired supercooling and the critical nucleus size.\nThe methodology, largely adopted from their previous work, has been well established in\nearlier studies. In this paper, they extend the approach to additional compositions within the\nFe-C system, bringing their analysis closer to the supercooling conditions constrained by\ngeophysical models. The central contribution of the manuscript, as I understand it, is to\nintroduce a paradigm shift—positioning the study of supercooling behavior as an\nindependent yet complementary constraint in planetary core research.\nWhile the paper does not introduce significant methodological innovations (at least not\nentirely clear from manuscript), I recognize the authors’ effort to highlight this new\nperspective. This work is timely, as the challenge of nucleating Earth's inner core was\nidentified by Huguet et al. (2018) and has remained an open question since. The authors,\nthrough this and previous work, have steadily developed a novel and constructive approach\nto addressing this issue. In this context, the present manuscript represents a crucial step in\nestablishing supercooling behavior as a key research focus in planetary science.\nThe paper is well-written, concise, and clear. In my opinion, it is suitable for publication.\nBelow, we (ECR and me) provide some general remarks and minor comments that should\nbe addressed before final acceptance.\nWe thank both reviewers for their detailed review of the manuscript. We are pleased that\nthey consider the work to be of sufficient significance and quality for publication.\nWe have addressed each of the points raised in the response below and feel that the\nchanges made do improve the clarity of manuscript, especially for a broad audience.\nGeneral Comments:\n- It is somewhat unfortunate that the simulations fail for x > 0.1, as these cases appear to be\nthe most relevant in terms of geophysical constraints. It would be helpful to provide a more\ndetailed explanation of the issues causing these failures. While there are hints throughout\nthe text, a clearer and more concrete statement would be beneficial—does the problem\nstem from the limitations of the EAM potential? What are the implications of this failure?\nWe thank the reviewer for this comment. We observe phase separation in high xC\nsimulations, where the carbon rich phase has a higher xC (>0.3) than the ab initio data used\nto develop and validate the potential. Therefore, further testing of this behaviour with\ncomparison to equivalent ab initio data is needed for interpretation. This limitation does not\nimpact this study because the atomic potential used in the MD calculations is shown to\nreproduce ab initio results for the xC range of simulations which are retained in the study. We\nrefer the reviewer to our response to reviewer 1’s 4th comment for additional detail on why\nwe decided not to produce an expanded ab initio dataset for validation of this behaviour.\nWe have added text to lines 158-162 of the results section to explain this behaviour\nand the reason that we omit results where xC > 0.1 from our study.\n- The discussion of geophysical constraints could be expanded. What specific geophysical\nmodels are referenced? To what extent are they independent of the assumed\nconcentrations? Given that supercooling behavior is being proposed as a new constraint on\ncore composition, a more detailed explanation of these models and their role in this context\nwould strengthen the manuscript.\nThe reviewer raises an important point. The absolute maximum supercooling of the inner\ncore (dT = 420 K) from the model of our previous study (Wilson et al., 2023,\nhttps://doi.org/10.1016/j.epsl.2023.118176) is not self-consistent in that it does not require\nthat the material properties of a particular core composition can explain both the extreme\nmelting temperature and adiabatic gradient required for this value. However, whilst the\nabsolute values of melting temperature and adiabatic temperature can be significantly\naltered by the composition of the core, the profile (or gradients) of Tm and Ta are relatively\ninsensitive to composition. This means that the geophysically inferred maximum value of dT\ncan be matched using many different compositions. The composition of the core is therefore\nnot constrained by the location of the inner core boundary, but the nucleation process itself\ncan provide a strong constraint, as our study shows.\nThe topic of evaluating the geophysical constraints on core supercooling does require\nlengthy explanation to be digestible. Since the fine details of this topic are not central to the\nmessage of this manuscript, we have included a section in the supplementary material which\nlays out these concepts in more detail and also references our recent review article (Wilson\net al., 2025, https://doi.org/10.1038/s43017-024-00639-6) which includes an in-depth\nexploration.\nWe have added text at lines 195-199 to note that the composition of the core does not\nhave a strong influence on the maximum supercooling of the core and have included\na statement on how thermal evolution modelling can be used to evaluate the\nconstraint imposed by geophysical observations on the supercooling of the core on\nlines 288-289 and a reference to the supplementary material on lines 200-201.\n- The paper would benefit from a clearer distinction from the authors’ previous work,\nparticularly Wilson et al. (2023) [33]. It would be helpful to explicitly state any methodological\ndifferences and clarify how this study extends prior research (e.g., by applying the same\nmethodology to mixtures with higher carbon concentrations). For instance, the direct\nreference to [33] in lines 99–101 could be expanded with a statement such as: “We extend\nprevious studies by considering …” to emphasize the novelty of the present work.\nWe thank the reviewer for pointing out this opportunity to clarify the advances of this study.\nWe have modified the text on lines 101-104 in line with the suggestions made.\n- Along the same lines: In what way do the contributions unique to this paper—such as the\ninvestigation of the F-C mixture at higher carbon concentrations — establish this\nadmissibility criterion in a manner not previously considered?\nWe thank the reviewer for this suggestion and have added text to the beginning of the\ndiscussion section which explains how this study builds upon our previous study and brings\nus closer to identifying a mechanism which can explain inner core nucleation at supercooling\nwhich is compatible with geophysical observations of the deep Earth.\nThis change can be found on lines 185-188.\n- Similarly, the discussion of results could be strengthened with a summarizing statement.\nFor example: “Through the simulations of the Fe-C mixture at higher x conducted in this\nstudy, we have taken a first step toward assessing the undercooling admissibility criterion\n…”\nFollowing this comment and the those preceding the text on lines 183-185 is now a\nstrengthened summarizing statement and a better introduction to the discussion of\nour findings.\nMinor comments:\n- l. 119: The definition of abbreviation EAM for “Embedded Atomic Model” is missing. It is\nonly introduced in the later Sec. 5.\nThank you, this has now been corrected on line 123.\n- l. 121: the variable x should probably be an x^C\nThank you, this has now been in all instances.\n- Fig. 1: in the lower plot, the y axis label is missing a variable name (e.g. x or x^C)\nThank you, this has now been corrected throughout.\n- l. 128: the nucleation rate I(r) is used as a function. Since it was only mentioned once\nbefore repeating at least the name here would make it easier to read\nThank you, text added at line 133.\n- l. 137: starting a sentence with a variable tau_w. Change to something like “The average\nwaiting time tau_w is computed as tau_w=tau_0 exp(...)”\nThank you, the suggest change has been made on line 147.\n- l. 145: it is not easily traceable for me what the differences of the interatomic potential are\nwrt. to the existing work from Wilson et al. 2023. The methodology described in 5.1 seems to\nbe exactly the same. For me, a highlight of differences would be useful to assess in which\nregard this work is new.\nWe now provide a more detailed account of the new potential validation and differences with\nour prior study. These can be found in the new supplementary materials and lines 352-\n353.\n- l. 148: missing “break” between symbols delta T and r_c, recommend to change to “at large\ndelta T critical radii r_c for all …\"\nCorrection made at line 158.\n- ll. 148-150: please elaborate a bit on the type of instability / why the method breaks down\n(see general comment above). It is fair the method has limitations but it would be helpful to\nunderstand them better\nWe have added text at line 158-162 to better describe the observed instability.\n- l. 157: Waiting time is given in time/volume. It would be great to have one or two\nexplanatory sentences as to how this relates to a time in the classical sense.\nThe new text on line 176-180 provides an additional means of intuitive interpretation\nfor the waiting time quantity.\n- l. 328: the abbreviation “NVE” has not been defined\nText has been added on line 366-367 to clarify this ensemble.\n- l. 414: please fix the link to the zenodo repository to include a full copyable doi link, e.g.\nwww.doi.org/10.5281/zenodo.13144422 . Also maybe rephrase the sentence such it does\nnot end with the link to avoid merging the sentence-ending dot with the URL\nThis link has been updated and the format corrected.\nSmall comments (mostly writing):\n- l.66: delete “result in …” (duplication)\nCorrected.\n- l. 79: missing comma after “Here”\nCorrected.\n- l. 94: “Ref” shown before reference [34]\nCorrected.\n- l. 101: missing space before reference [33]\nCorrected.\n- l. 103: potential comma after “nucleation”\nWe believe the grammar is correct here.\n- ll. 129-130: repeated word “which”\nCorrected.\n- l. 147: missing comma before “although”\nCorrected.\n- Table 1: maybe put the unit of variables in its own column in the tabular to fix alignment\nWhilst we agree that this may be a good choice, we are reluctant to make formatting\nchanges prior to typesetting as the manuscript will go from a standard latex draft to a 2\ncolumn Nature Communications article. We will pay close attention to these details at the\ntypesetting phase.\n- l. 237: missing comma -> “which, when applied to a specific model of core formation, …\"\nCorrected.\n- ll. 238-239: repeated use of the word “strongly”, suggest to replace once\nCorrected.\n- ll. 346-347: double use of the word “consistent”\nCorrected.\n- l. 422: Section heading weirdly transitions into the following paragraph.\nCorrected.\nReviewer #4 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This\nis part of the Nature Communications initiative to facilitate training in peer review and to\nprovide appropriate recognition for Early Career Researchers who co-review manuscripts."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have made efforts to address most of the reviewer’s concerns, resulting in substantial improvements to the manuscript. However, they cannot examine the critical nucleus, as their technique focuses solely on sub-critical configurations. This approximation may overlook important pathways for solidification.\n\nNevertheless, the present results are significant enough for core studies. I think the paper can be published.", "reviewer_2": "In my opinion, the authors have addressed effectively the points made by the reviewers, and the article is suitable for publication.", "reviewer_3": "The reviewers carefully addressed all points raised. The manuscript is now - in my opinion - ready for publication. One remark: The answer of the authors to one point raised by reviewer number 2 is not fully correct and I just wanted to clarify:\n\nReviewer 2: 39-40 Cooling of the core is determined by heat transfer across the core-mantle boundary,\nnot by the thermal conductivity of the core.\nAuthors: The reviewer is correct. Mantle convection sets the heat flow across the core mantle\nboundary and the thermal conductivity of the core controls the adiabatic heat flow through\nthe core. The thermal conductivity of the core controls the convective vigour of the core. I.e.\na high core conductivity will result in more vigorous convection than a low core conductivity,\nif the core-mantle boundary heat flux is the same in both cases.\n\nWe have modified the text at lines 38-39 to make this statement clear.\n\nHigh core conductivities will hamper convection as most of the heat can be transported by heat conduction. However, the way the authors addressed this in the manuscript is fine as it is only a general statement (\"...thermal conductivity k, which plays a vital role in determining the vigour of convection in the liquid core...\").", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": ""}]

Decision letter