Grain boundary diffusion cannot explain the W isotope heterogeneities of the deep mantle

Domain: earth_science
Published: 2025-02-21
Editorial outcome: major_revision

Abstract
The low 182W/184W and high 3He/4He in some ocean island basalts compared to the bulk mantle values may derive from the Earth’s core through long-term core-mantle interactions. It has been proposed that the grain boundary diffusion of siderophile elements is an efficient mechanism for core-mantle interaction and may effectively modify the W isotopic compositions of the plume-source mantle. In this study, we perform large-scale molecular dynamics simulations driven by machine learning potentials of ab initio quality to investigate the diffusion of W along ferropericlase grain boundaries and in (Mg,Fe)O liquid. Here we show that the diffusion of W is sluggish under core-mantle boundary conditions, and thus is unlikely to have observable impacts on the W isotopic compositions of terrestrial igneous rocks.

Introduction
Ocean island basalts (OIBs) are volcanic rocks that may originate from rising plumes in the Earth’s deep mantle, potentially sampling materials near the core-mantle boundary (CMB)1. The isotopic analyses of OIBs reveals significant chemical heterogeneities in the Earth’s deep interior that have long provided key insights into the evolution of our planet1. Recent studies have identified intriguing 182W deficits in some OIBs with μ182W (defined as the deviation of 182W/184W from terrestrial standards in parts per million) as low as −23 ± 4.52–6, while μ182W is around 0 in the modern upper mantle. The negative μ182W anomalies were produced by the depletion of the now extinct 182Hf with a half-life of 8.9 Myr7.

Various mechanisms have been proposed to account for the μ182W anomalies in OIBs. The element fractionation during the crystallization of putative magma oceans may lead to the formation of enriched deep mantle reservoirs or early protocrust with a low Hf/W ratio2,8,9, subsequently causing lower μ182W. However, subducted, early-formed crust may not possess sufficiently negative μ182W to account for OIBs5. In addition, the enriched deep mantle reservoirs and the protocrust would also be enriched in other incompatible elements, including U and Th. This enrichment leads to more radiogenic 4He and lower 3He/4He ratio (3He is primordial), which is inconsistent with the observed negative correlation between μ182W and 3He/4He2,5.

As a moderately siderophile element, W primarily partitions into the core during core-mantle differentiation, while Hf remains in the mantle due to its lithophile nature. The early core formation during the first ~30 Myr of solar system history, when 182Hf was still extant, resulted in the terrestrial mantle exhibiting a higher 182W/184W ratio compared to bulk chondrites and the complementary reservoir, the core10–12. Therefore, the assimilation of the Earth’s core materials or chondritic materials after core formation may significantly modify the μ182W characteristics of OIB source reservoirs. For example, it has been proposed that iron-rich liquids from the core may penetrate into the oxides in the lowermost mantle due to morphological instability13 or capillarity phenomenon14. Since the core is believed to host a substantial amount of 3He15,16, it may also be a source of high 3He/4He in OIBs17–20, simultaneously explaining the 182W depletion and 3He enrichment. However, the direct incorporation of outer core metal or late-accreted materials could lead to significant enrichment of highly siderophile elements (HSEs), which contradicts the observed absence of HSEs enrichments of OIB samples4,5. In addition, the extreme lithophile behavior of Ne under high pressures predicted by ab initio calculations likely results in a high 3He/22Ne ratio of core materials, which is not observed in OIBs either21.

It is suggested that atomic diffusion of some siderophile elements along grain boundaries (GBs) maybe an efficient mechanism for core-mantle interaction22. Recent experimental results also indicate high GB diffusivities of W in lower mantle phases with a strong temperature dependence23. Therefore, diffusive isotopic exchange at the CMB has been considered a viable mechanism to explain the 182W deficits in OIBs4,5,18,23. However, those experiments have puzzling aspects: First, W and Au exhibit diffusivity three orders of magnitude higher than other siderophile elements, despite having similar atomic masses, radii, and chemical properties22 (Fig. 1). Second, extrapolating the W diffusivity to the CMB temperatures according to the experimentally determined temperature dependence yields unreasonably high results23 (see Results and Discussion). Recently, a theoretical prediction suggests that the GB transport in MgO is considerably less efficient at CMB conditions than has been extrapolated from experimental results at lower pressures and temperatures24. Our recent findings also favor sluggish Fe diffusion along ferropericalse GBs, questioning the feasibility of GB diffusion as an effective core-mantle interaction mechanism25. Therefore, it is important to further quantify the efficiency of GB diffusion in exchanging W isotopes, especially at the temperature and pressure conditions of the CMB.

Performing diffusion experiments under high temperature and pressure is challenging26, and obtaining the GB diffusivity requires separating the contributions of lattice and GB diffusion to the bulk diffusion profiles (e.g., Yoshino et al.23), introducing additional errors.

In this study, we examine the self-diffusion of W in ferropericlase GBs and (Mg,Fe)O liquid using molecular dynamics (MD) simulations under various pressure and temperature conditions. The recently developed machine learning potential (MLP) method27,28 enables the simulations of large GB systems with high efficiency while maintaining ab initio accuracy. Considering the isotopic exchange process via GB diffusion, we evaluate the μ182W anomalies of the plume-source mantle that might have arisen over the past 4.54 billion years. Our findings imply that GB diffusion across the CMB cannot explain W isotope anomalies in the plume-source mantle, and other mechanisms are desired to resolve the puzzle of μ182W anomalies in mantle-derived rocks.

Results
Grain boundary diffusion of W and Pt in ferropericlase

We developed an MLP for Mg-Fe-O-W system by training an artificial neural network model with the ab initio data (see Methods). In addition, a separate MLP for Mg-Fe-O-Pt system was also trained to compare the diffusion behaviors of different siderophile elements under the CMB conditions, since Pt is a representative and well-studied HSE (e.g., Médard et al.29). The MLPs target temperatures up to 8000 K and pressures up to 200 GPa. We evaluated the reliability of the MLPs by comparing the energies, atomic forces, and stresses with those obtained from ab initio simulations based on density functional theory (DFT), for configurations not present in the training set (Methods; Supplementary Fig. 1). The diffusion coefficients were directly obtained from atomic trajectories of MD simulations (Eq. (2), Methods). The diffusion coefficients of Mg, Fe, O, and W calculated using MLPs align well with the DFT results for (Mg,Fe)O liquids (Supplementary Fig. 2). In addition, our DFT results show excellent consistency with experimentally measured W diffusivities in silicate melts, further validating the reliability of our methods (Fig. 1b). As shown in Supplementary Information, we also carefully investigated the factors affecting the diffusion of W including the exchange-correlation functional used in DFT calculations (Supplementary Fig. 3), system size (Supplementary Fig. 4), GB structure (Supplementary Fig. 5), vacancy concentration (Supplementary Fig. 6), valence state of W (Supplementary Fig. 7), and distribution of elements and vacancies (Supplementary Fig. 8), to ensure that diffusivity results are converged and robust25,30.

Figure 1a presents our results for the diffusion coefficients of W and Pt in periclase at 2.5 GPa and 1873 K. These two siderophile elements show similar diffusion behavior along GBs and exhibit similar diffusion coefficients (Fig. 1a). The GB diffusivity for W is approximately two orders of magnitude lower than its diffusivity in supercooled (Mg,Fe)O liquid, which aligns with previous findings of GB diffusion in olivine31. A previous experimental study, also plotted in Fig. 1a for comparison, reported GB diffusivities for multiple siderophile elements in MgO periclase under the same condition22. For Pt, our theoretical prediction closely matched the experimental result. However, experimental results indicated strong variations in GB diffusivities for different siderophile elements, spanning up to four orders of magnitude, with notably higher diffusivities for W and Au compared to other elements. This is perplexing considering the similarities of these two elements with other siderophile elements shown in Fig. 1a in terms of atomic mass and siderophility22. Furthermore, the experimental diffusivity for W is an order of magnitude higher than those in supercooled (Fe,Mg)O liquid, and over two orders of magnitude higher than W diffusivities in silicate melts under similar temperature and pressure conditions32,33 (Fig. 1b). Admittedly, the GB features a relatively disordered atomic environment and exhibits higher diffusivities than the lattice counterpart34,35. However, regardless of the GB structure, GB diffusivity is always lower than that in a fully disordered liquid with the same chemical composition36,37. Therefore, the notably faster GB diffusion for W compared to liquid diffusion is unreasonable, suggesting that contributions to the large flux of W in the experiment may come from additional sources other than GB diffusion.

The high-pressure GB diffusion coefficients in (Mg0.9Fe0.1)O are shown in Fig. 2. The temperature dependence of the GB diffusivities can be fitted using the Arrhenius equation (Eq. (3), Methods), denoted as the activation enthalpy (ΔH), which is proportional to the slope of the fitted lines. A recent experimental study reported the GB diffusivity of W in a post-spinel (bridgmanite + ferropericlase) phase at 25 GPa23, indicating that W exhibits fast and strongly temperature-dependent GB diffusion, with an activation enthalpy of 493 kJ mol−1. To directly compare with these experimental results, we performed simulations at multiple temperatures under a pressure of 25 GPa and calculated the GB diffusivities of W (Fig. 2a). We observe a lower diffusivity and weaker temperature dependence of diffusivities compared to Yoshino et al.23, and the activation enthalpy is found to vary as a function of temperature (ΔH = 322 kJ mol−1 for 2000–2750 K; ΔH = 218 kJ mol−1 for 2750–3500 K). Note that our calculations pertain to self-diffusion without a W concentration gradient, which is different from the chemical diffusion measured in the experiments, even though the two are similar at very low W concentrations (Supporting Information). Moreover, W may oxidize to WO3 in experiments23, resulting in a significant drop in its melting point. This allowed the oxide melt to mechanically penetrate GBs due to surface tension, with a penetration rate likely exceeding the GB diffusion rate, potentially providing another mechanism for core-mantle interaction. This process, however, was absent in our simulations because we set a low W concentration to mimic the extremely low W content in Earth’s mantle38 (Methods), while previous experiments used metallic W foils as the diffusion source22,23 and the W concentrations reached several tens of wt% in sink particles23. Additionally, the much lower valence state of Pt29 compared to W39,40 in silicate melts suggests Pt may be less likely to form oxides and migrate via a similar mechanism. This difference likely explain the significantly lower flux of Pt compared to W observed by Hayden & Watson22.

The low and temperature-dependent ΔH reported in this study can be justified from a microscopic perspective. ΔH is defined as the energy difference between the system when the diffusing atom occupies the most energetically favorable positions (energy minima) and when it traverses the saddle point (energy barrier) connecting the energy minima41–43. For lattice diffusion, W atoms face a strong steric effect when migrating between lattice sites (deep energy minima) by passing through interstitial sites (high energy barrier), and thus show high activation enthalpy. The high ΔH may be contributed by the increased strain energy caused by the distorted crystal lattice when W transitions at the saddle point43. In liquids, due to the disordered nature of the medium, W is likely to have a more flexible diffusion pathway that may be coupled with the free movements of neighboring atoms in the absence of lattice strain. Additionally, the larger volume and thus larger interatomic distances in liquids compared to solids may facilitate W migration, resulting in low ΔH. GBs, as solid-solid interfaces, exhibit characteristics of an amorphous state and should therefore display diffusion behaviors that are intermediate between those of solids and liquids, which is supported by the observations that GB atoms do not have fixed lattice sites, yet there remains a significant steric effect on diffusion24,25. Thus, the ΔH for GB diffusion of W should lie between the ΔH for W diffusion in liquids (118 kJ mol−1, green data in Fig. 2a) and in the single crystal (365 kJ mol−1 for olivine)40.

As the temperature increases, the disorder in the GB structure intensifies, and may trigger premelting (i.e., the GB transitions into a liquid-like, thermodynamically stable nanoscale film below melting temperature)44, as observed in ferropericlase25 and many other materials45–47. The incorporation of W6+ may reduce the melting point of ferropericlase and facilitate this premelting process. This order-disorder transition of GB structures results in the temperature-dependent activation enthalpy, represented by solid-like and high ΔH at low temperature and liquid-like and low ΔH at high temperature, which has been observed in GB diffusion in MgO24. Consequently, GB diffusivities gradually increase towards liquid diffusivities with increasing temperature (Fig. 2a).

Previous studies have attempted to extrapolate the experimentally measured GB diffusivities23 to high pressure and high temperature conditions, in order to explain the W isotope anomalies in the plume-source mantle18,23. Due to the temperature-dependent activation enthalpy described above, such extrapolation is likely unreliable. We now evaluate the feasibility of these extrapolations. At 25 GPa, the extrapolated W diffusivity at 3000 K is 6 × 10−4 m2 s−1, over two orders of magnitude higher than the W diffusivity in (Mg0.9Fe0.1)O liquid under corresponding conditions calculated in this study (Fig. 2a). It is also orders of magnitude greater than the diffusivities for almost all elements in all kinds of silicate melts measured under ambient pressure, if we extrapolate those results to 3000 K48. Note that we have not considered the diminishing effect of pressure on diffusion in silicate melts; otherwise, the gap would be even more significant. In addition, if we assume an activation volume of 2 cm3 mol−1 18,23 to incorporate the pressure effect (Eq. (4), Methods), the extrapolated value under CMB conditions also reaches 10−6 m2 s−1, about three orders of magnitude higher than the (Mg0.9Fe0.1)O liquid. Therefore, extrapolating the GB diffusivities to CMB conditions using the previously reported low-temperature activation enthalpy is unreasonable, and we believe that the GB diffusivities and ΔH may have been significantly overestimated in experiments, potentially indicating additional physical/chemical mechanisms beyond GB diffusion (e.g., melting and mechanical penetration of W oxides) that have transported a significant amount of W under those experimental conditions.

Under the CMB pressure, we reported the GB diffusivities of W and Pt at temperatures from 4000 to 5000 K (Fig. 2b), which covers the upper limit of the estimated CMB temperatures throughout the Earth’s thermal evolution49. Similar to low-pressure results (Fig. 1), W and Pt exhibit very close GB diffusivities, which are essentially identical within the uncertainties. Additionally, due to the enhancement effect of pressure on activation enthalpy, the GB diffusivities at 140 GPa show a higher temperature dependence than those at 25 GPa, characterized by ΔH = 371 kJ mol−1 for W. The activation enthalpy of GB diffusion for W in ferropericlase is similar to that of Fe (328 kJ mol−1)25 at 140 GPa, which may be related to the similar ionic radii of W6+ and Fe2+ (W6+: 74 pm; Fe2+: 75 pm)50. Although W6+ has a slightly lower mass and a much smaller radius than Pt0 (135 pm)50, its significantly lower electronegativity51 and higher valence state29,39,40 compared to Pt0 under high pressure may cause W to form strong chemical bonds with undercoordinated O atoms in ferropericlase GBs24,25, leading to similar diffusion behaviors for W and Pt. Our diffusivity results are more than four orders of magnitude lower than the extrapolated experimental results (Fig. 2b), leading to markedly different geochemical implications.

Geochemical Implications

As a potentially significant reservoir of W (with low μ182W) and He (with high 3He/4He), the core has been proposed to be a source of the negative correlation between μ182W and 3He/4He anomalies in OIBs4,5,18,19. The isotopic exchange between the core and the mantle through GB diffusion has been proposed to be the underlying mechanism18,23. Here, we used the diffusion coefficients of W, directly obtained under the high temperature and pressure conditions of the CMB without any extrapolation, to reassess the reliability of this mechanism.

Due to its siderophile nature, W is abundant in the Earth’s core (~500 ppb)52 and depleted in the mantle (27–31 ppb)38, which enables a continuous supply of core-like W across the CMB, maintaining a constant, core-equilibrated μ182W value of about −220 at the very bottom of the mantle. Subsequently, the core-derived W may undergo isotopic exchange with the ambient background mantle (μ182W ≈ 0), mediated by self-diffusion in mantle rocks, gradually modifying the W isotopic characteristics of the lowermost mantle. Given that the cation diffusion in ferropericlase is significantly faster than that in bridgmanite and post-perovskite23,42,53,54, we calculated the effective diffusion coefficient of W (DW, Eq. (5), Methods) using its GB diffusivity in ferropericlase to constrain the upper limit of this isotopic exchange efficiency. We first consider a diffusion distance, L = √DWt, to assess the characteristic spatial scale L over which this mechanism can impact within a certain timescale t. We find that even if this diffusive isotope exchanger remains undisturbed over the age of the Earth (4.54 Gyr)55, the maximum diffusion distance is only about 30 meters (Fig. 3a), far lower than the values obtained by extrapolating low-temperature experimental results18,23.

Furthermore, we modeled the possible μ182W anomalies in OIBs that result from this diffusive isotopic exchange based on the diffusion coefficients derived above. OIBs with W or He isotope anomalies may originate from deep-rooted mantle plumes56. Seismic observations suggest a geographical correlation between the generation zones of deep-rooted mantle plumes and the large low-shear velocity provinces (LLSVPs), indicating that these plumes are likely generated at the thin margins of LLSVPs57,58. Specifically, OIBs with low μ182W values and high 3He/4He ratios usually lie above ultra-low velocity zones (ULVZs) at the margins of LLSVPs2,59,60. Therefore, LLSVPs5,18,61 and ULVZs2,5,23 may be the potential source reservoirs for those isotopic anomalies. Using the effective diffusion coefficients, we calculated the vertical composition profile of W isotopes above the CMB after 4.54 Gyr of diffusive isotopic exchange (Eq. (7), Methods; Supplementary Fig. 9). We then calculated the μ182W composition of a plume generation zone (PGZ) based on the average composition of the mantle between the CMB and the top of PGZ (Eq. (8), Methods). We consider the degree of entrainment—the ratio of dense material carried by the hot plume to the total upwelling material—to calculate the final μ182W value of OIBs reaching the Earth’s surface18 (see Methods). The results are shown in Fig. 3b. If we use the observed minimum height of ULVZs (~5 km, limited by the seismic resolution)62 as the lower limit for the height of PGZ, our results indicate that the 182W deficit caused by GB diffusion would be within 0.2 in μ182W towards negative. Even if we use the W diffusivity in super-cooled (Mg,Fe)O liquid at 5000 K instead of GB diffusivity to calculate the effective diffusion coefficient, which constrains the maximum possible isotopic exchange, the resulting μ182W is still an order of magnitude smaller than those of many OIB samples (Fig. 3b)5. Therefore, we conclude that diffusive isotopic exchange cannot be the source of W isotope anomalies in OIBs or any other mantle-derived rocks. Note that even if we further reduce the height of the PGZ to 1 km, this mechanism still cannot explain the significant μ182W anomalies observed in a large number of natural samples. The model discussed above addresses the isotopic exchange in the absence of a concentration gradient of W in the mantle. We also modeled the chemical diffusion of W from the core to the mantle considering a net mass flux of W given the possible enhanced lithophility of W at the CMB conditions. The results show that chemical diffusion fails to explain the observed 182W anomalies in OIBs (see Supplementary Information).

Although small-scale convection within LLSVPs may enhance the transport of core-affected material within the mantle63, the amount of core-derived W sufficient for the observed μ182W in OIBs requires a W diffusion coefficient of over 10−11 m2 s−163, which is several orders of magnitude higher than the effective diffusion coefficient calculated in this study (see caption of Fig. 3).

Although diffusion in solids is excessively sluggish, our results indicate that the diffusivity for W in (Mg,Fe)O liquid can reach 2.19 × 10−9 m2 s−1 at 5000 K and 140 GPa, corresponding to a diffusion distance of ~18 km over a timescale of 4.5 Gyr. Therefore, partial melts, if present in the lowermost mantle64–66, may accelerate the equilibration process of W isotopes between the plume source and core5. However, the temporal and spatial extent of these partial melts remains highly uncertain. In addition, the expected downward percolation of dense partial melts can cause drainage and accumulation, forming a melt-rich layer at the CMB, which may be inconsistent with geophysical constraints67. Although the melts may remain in suspension within ULVZs due to the viscous stirring, numerical simulations suggest that the bottom of ULVZs is relatively depleted in liquids68, which limits the efficiency of the diffusive isotopic equilibration. Therefore, atomic diffusion, especially the GB diffusion alone, is unlikely to be a viable mechanism to modify the μ182W values of the plume-source mantle. Other mechanisms, such as the addition of core-exsolved oxides that are potentially enriched in W during the secular cooling (e.g., Rizo et al.4, Deng & Du19), are necessary to transport the core-like signature into the Earth’s mantle.

Methods
Development of Machine Learning Potentials
Recently, we developed an MLP for Mg-O-Fe system covering a wide compositional range of various Mg:O:Fe ratios at pressure up to 200 GPa and temperature up to 8000 K69. Here, we further extended the compositional space of this MLP to include one more element X (X is either Pt or W) following the same procedure outlined at Peng & Deng30. Two MLPs were built with training sets inherited from the Mg-O-Fe MLP69 and additional ones that entail Mg-O-Fe-X mixtures. Specifically, we added configuration with bulk compositions including 32MgO + aX with a = 2–16, 64Fe + bX with b = 2–16, cMgO + dFe + eX + fO with c = 32–64, d = 48–144, e = 2–8, f = 0–48.

To briefly summarize the training process, configurations (structures of a given atomic arrangement) of training and test sets were drawn from multithermal and multibaric simulations70, which were used to efficiently sample the multi-phase configurational space. We used the structure factor of B1 MgO as the collective variable to drive the sampling. High-accuracy ab initio calculations were performed on the selected configurations to derive the corresponding energies, atomic forces, and stresses. The DeePMD approach was employed to train an MLP that takes a configuration and predicts its energy, atomic forces, and stresses without iterating through the time-consuming self-consistent field calculation27,28. The loss function is defined as

L = pϵΔϵ2 + p f/3N ∑i |ΔFi|2 + pξ/9 ∑k |Δξk|2, (1)

where pϵ, pf, pξ are tunable prefactors for the difference between the MLP prediction and training data. ϵ is the energy per atom; Fi is the atomic force of atom i; ξ is the virial tensor divided by N; N is the number of atoms. We adopted the conventional setting of increasing both pϵ and pξ from 0.02 to 1 while decreasing pf from 1000 to 1 over the course of training. A three-layer fitting network with 240 nodes in each layer was used to determine the form of the potential energy surface following our previous studies30,69,71.

An iterative training scheme71 was employed throughout the training process. Principal component analysis was used to identify the most representative frames from MD trajectories for high-accuracy DFT recalculations. These recalculated results were compared with the predictions of the current model to identify frames that were poorly described by the MLP, which were then added to the training set. This iterative approach ensures that the training set remains concise and well-balanced. The final training sets consist of 6245 and 7557 configurations for Mg-O-Fe-W and Mg-O-Fe-Pt systems, respectively. We compared the energies, atomic forces, and stresses from MLPs to those from DFT calculations for 16579 and 16956 configurations of test sets that are not included in the training set for Mg-O-Fe-W and Mg-O-Fe-Pt systems, respectively (Supplementary Fig. 1). For Mg-O-Fe-W system, the root-mean-square errors (RMSEs) of the energies, atomic forces, and stresses are 6.76 meV atom−1, 0.32 eV Å−1, and 0.53 GPa, respectively. For Mg-O-Fe-Pt, RMSEs are 6.69 meV atom−1, 0.30 eV Å−1, and 0.51 GPa, respectively. The RMSEs of energy falls below the commonly accepted threshold of 10 meV/atom; the RMSEs of stress is at least an order of magnitude lower than the discrepancies between the MgO equation of state obtained from DFT calculations and experimental measurements72.

Ab Initio Calculations
Ab initio calculations were performed using Vienna Ab Initio Simulation Package (VASP)73. We used the projector augmented wave (PAW) method74 as implemented in VASP73 and PBEsol approximation75. PAW potentials are widely utilized in DFT calculations for planetary interior conditions and, when combined with the PBEsol approximation, demonstrate excellent agreement with experimental measurements of the physical properties of silicates and oxides9,76,77. The core radii are O: 0.820 Å (2s22p4), Mg: 1.058 Å (2p63s2), Fe: 1.164 Å (3p63d74s1), W: 1.455 Å (5p66s25d4), Pt: 1.455 Å (6s15d9). High-precision ab initio calculations were performed for training and test configurations used to construct MLPs with an energy cutoff of 800 eV, energy tolerance of 10−6 eV and 2 × 2 × 2 Monkhorst-Pack mesh.

Construction of the Grain Boundary Structure
The GBs of ferropericlase are symmetric tilt GBs constructed with Atomsk78 in our previous study25. MgO bicrystals with two misorientation angles were considered: α = 36.8° where the two crystals meet with {310} planes, and α = 29.5° where the GB plane does not correspond to any high-symmetry crystal plane (see Supplementary Information). The compact GB structures at energy minima favored under mantle pressures were identified79, which can further interact with vacancy defects and transform the atomic configuration80. Various amounts of Mg-O vacancy pairs were randomly introduced within a 1-nm-wide region at the GB, and the convergence of GB diffusivities as a function of the vacancy concentration was tested25 (see Supplementary Information). Then, the structures of (Mg0.9Fe0.1)O ferropericlase were generated by the random substitution of Mg with 10 mol% of Fe in the MgO bicrystals. We tested the potential impact of these random factors on the W diffusivities (see Supplementary Information). Ferropericlase bicrystals contain over 50,000 atoms, of which ~2000 are located within the 1-nm-wide GB regions, resulting in an average distance of over 20 nm between the centers of adjacent grains. Twenty W or Pt atoms were randomly introduced into the GB region. Based on previous experimental results, we set the valence state of W to +623,39,40 and that of Pt to 029 for both GB and liquid systems. The valence states were controlled by removing Mg atoms near W. The effects of the valence state on the W diffusivity were tested (see Supplementary Information).

Molecular Dynamics Simulations
Ab initio MD simulations were performed using VASP73 under the canonical ensemble (NVT) with the Nosé-Hoover thermostat81. Our ab initio MD simulations run for 10–40 ps, sampling the Brillouin zone at the Γ point with energy cutoffs of 500 eV and an energy tolerance of 10−4 eV. MD simulations driven by MLPs were conducted by LAMMPS82 and DeePMD-kit28. The GB systems were first equilibrated in isothermal-isobaric conditions (NPT) for 100 ps under the target temperatures and pressures. The resulting structures were then used as the initial configurations to set up the MD simulations for 2 ns under an NVT ensemble, in order to evolve the GB structure to reach a steady state. Subsequently, we performed long NVT simulations for 2 ns, from which the GB diffusivity is calculated. In (Mg,Fe)O liquids, we simulated the systems at high temperatures and gradually cooling, followed by NPT equilibration for 10 ps. The diffusivity was obtained from NVT simulations for 40 ps. The timestep of all MD simulations is 1 fs. All NVT simulations were repeated 4–20 times to report the standard deviation (error bars in all figures) of the diffusion coefficients.

Diffusion Coefficients
Diffusion coefficients were calculated as the slope of mean square displacements (MSDs) by,

D = lim t→∞ MSD/6t = lim t→∞ ⟨|r(t+t0) − r(t0)|2⟩/6t, (2)

where r(t) is the particle trajectories in Cartesian space, and ⟨⋯⟩ represents an average over all atoms of the target element and over time with different origins83. The diffusivity as a function of temperature obtained from our simulations was fitted by the Arrhenius equation,

D = D0 exp(−ΔH/RT), (3)

where D0 is the pre-exponential factor, R is the ideal gas constant, T is the temperature in K, and ΔH is the activation enthalpy. Extrapolating experimental results to higher pressures requires an additional parameter to describe the pressure dependence of the diffusivity, then the Arrhenius equation becomes

D = D0 exp(−(ΔE + PΔV)/RT), (4)

where ΔE is the activation energy and ΔV is the activation volume.
The effective diffusion coefficient of W (DW) can be estimated by the combination of lattice diffusivity (DlatticeW) and GB diffusivity (DGBW)18,23,84:

DW = DlatticeW + (3δ/d) DGBW, (5)

where δ is the GB width (~1 nm) and d is the grain size (0.1–10 mm in the lower mantle)85. Since there is no available DlatticeW data in lower mantle minerals, we assume the lattice diffusivity of W is the same as that of Si86 following Yoshino et al.23 due to their similarly high valence state23,40. Since the available experimental data are limited to 25 GPa86, a small activation volume of 1 cm3 mol−1 is assumed to estimate the upper bound of the effective diffusion coefficient.

Modeling μ182W anomalies in OIBs
We calculated the vertical composition profile of W isotopes above the CMB (Supplementary Fig. 9) considering Fick’s second law of diffusion

∂C/∂t = DW ∇2C (6)

Since the bottom of the mantle may have a fixed μ182W value of −220 and the diffusion distance of W is within 1% of the core radius, the solution to Eq. (6) can be well approximated by the diffusion in a half-space with a constant surface concentration87:

C = C∞ + (C0 − C∞) erfc(y/√4DWt) (7)

where C0 is the core-equilibrated 182W/184W (μ182W = −220), C∞ is the 182W/184W in the ambient mantle (μ182W = 0), y is the vertical distance above the CMB, t is the diffusion time of 4.54 Gyr, and erfc is the complementary error function. The 182W/184W of the PGZ is taken as the integral average of Eq. (7):

CPGZ = 1/h ∫0h C(y)dy (8)

where h is the height of the PGZ. The mixing of the PGZ material and background mantle material in OIBs is characterized by the degree of entrainment of dense material neighboring hot upwelling mantle18,88

Rmax = TΔρ/ΔρCh (9)

where ΔρT is the thermal density anomaly in the hot mantle plume, ΔρCh is the chemical density anomaly in the PGZ, and Rmax is the maximum ratio of entrained dense material to total material. Considering that the estimated density anomaly of ULVZs is greater than that of LLSVPs89–91, we took the ΔρCh of LLSVPs following Ferrick & Korenaga18 to constrain the maximum proportion of plume-source material in OIBs. All parameters and the Rmax result (11%) are consistent with those reported in Ferrick & Korenaga18. The μ182W in OIBs as a function of the height of PGZ was obtained by mixing the plume source (Eq. (8)) and the background mantle (μ182W = 0) with a mixing ratio of 1:8.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Review of “Grain boundary diffusion cannot explain the W isotope heterogeneities o the deep mantle” by Peng et al.\nIsotopic compositions of oceanic island basalts (OIBs) can provide a window into the interaction between the deepest\nmantle and core. Grain-boundary diffusion has been recently invoked as a mechanism of the core-mantle interaction to\nexplain the observed correlation between W–He isotopic anomalies in OIBs. Yet, this model relies on the W diffusivity that\nwas estimated by extrapolating from experimental results at lower pressures and temperatures than the core-mantle\nboundary (CMB). The present study examined the grain-boundary diffusion of W along ferropericlase grain at the CMB\npressure-temperature conditions using ab initio molecular dynamics simulations by machine learning. The W diffusivity\ndetermined in this study is more than four orders of magnitude lower than the extrapolated results. Using the newly\ndetermined diffusivity, the authors found that the W diffusion distance over Earth’s history is only up to 30 m and concluded\nthat grain-boundary diffusion cannot explain the observed W isotopic anomalies in OIBs.\nUnderstanding the mechanism of core-mantle interaction is central to elucidating the chemical and thermal evolution of the\nEarth. Although I am not enough trained to evaluate the quality of the results of ab initio molecular dynamic simulations, the\nconsistency between the simulation and experimental results at low pressure and temperature conditions seems to validate\nthe results. The discussion hits the important points, and the paper is very well written. Hence, I recommend the paper for\npublication in Nature Communications with minor revisions. I have only a few comments.\nlines 102–104: This statement contradicts with the statement in lines 157–159, in which W diffusion along bridgmanite is\nconsidered as slower than that along ferropericlase.\nLines 137–138: I am wondering whether such a possible W-oxide melting is considered in the authors’ simulations, since\nthe authors set the valence state of W to +6 in the simulations.\nLine 146: What is the radius of Pt0?\nLine 154: Please provide the W abundance of the mantle as well, since the relative W abundance between the core and\nmantle is more important than the absolute W abundance in the core.\nLines 157–159: What about the grain-boundary diffusivity in post-perovskite that is the major phase of the average\nlowermost mantle?\nLines 187–188: 4.5 Gyr rather than 4 Gyr?", "reviewer_2": "The previous observation of relative deficits in 182W in some ocean island basalts (OIB), somewhat corresponding to\nelevated relative 3He abundances in the same samples, has been interpreted in light of interaction between Earth’s core\nand mantle sampled by deep-rooted mantle plumes sourcing OIB. Initial enthusiasm to have found a “smoking gun\nevidence” for core-mantle interaction subsequently gave way to some serious questioning of such model given that the\nobserved W isotopic characteristics in OIB are not matched by elevated abundances of so-called highly siderophile\nelements (HSE, like the precious element Pt) in the same samples, as would be expected in such a scenario. As a\nconsequence, alternative, non-core-related models have been put forward to explain the observed OIB characteristics, in\naddition to more sophisticated core-mantle interaction models that aim to reconcile the decoupling of W and HSE, for\ninstance by chemical diffusion of W into the lower mantle. In the context of the latter, carrying out numerical simulations\nusing a machine learning approach, the main result of this contribution is to demonstrate that a chemical diffusion model\ncannot explain the observed W isotopic composition in OIB. A consequence of such a singular approach is that the\nunderlying reason for the observed W and He isotope composition in (some) OIB still remains unconstrained and\nunresolved.\nThis aside, the approach of testing the chemical diffusion model based on a machine learning experiment is not widely used\nand offers a fresh angle on the prevailing question of element diffusion in the lower mantle that is difficult to address using\nlab-based experiments. Input parameters/assumptions made to carry out this experiment appear to be sound and the\ntreatment of the result data, as well as the corresponding evaluation of uncertainties are justified and well-documented. The\nfirst-order interpretation of the outcomes of their experiments seem valid and robust. In particular, testing the author’s\napproach by comparing their results with lab-based experiments using the HSE Pt lends particular credibility. The authors\nthen go on to put their results into the wider context of modelling the 182W composition of a hypothetical OIB source\nconclusively concluding that grain boundary diffusion of W in the lower mantle cannot reconcile the observed W isotopic\ncomposition of OIB.\nThis study is a significant step forward with regards to elucidating the ever increasing dataset of W isotopes in OIB in light of\nthe large variety of models put forward to explain them. At the same time, the study’s exclusive focus on disproving one (of\nseveral) models also somewhat limits its reach. At the end, the authors flag their preferred models (Deng&Du, 2023:\nMcNamara&Zhong, 2004), while at the same time, their current study cannot actively support those. As such, its significance\nbeyond disproving a single model to explain the observed W isotope composition in OIB may well lie in the eye of the\nindividual reader and could be seen as limited to the wider geochemistry community.\nThe use of isotope geochemical terminology is in part too colloquial. I have indicated a few improvements but would\nencourage the authors to seek advice from an expert isotope geochemist to proof-read a revised version. Also, the authors\nchoose to report their study in the present tense (rather than commonly used past tense), which I find a bit odd. In addition,\nthe authors should better align the contents of the captions of their figures with what us shown in the latter. At some point it\ntook me some jumping back-and-forth between captions and figures to fully understand, what is shown. Apart from that,\nstructure and content of the current manuscript are very clear, comprehensive and well-integrated into the context of the\ncurrent literature.\nDetailed comments:\nLine 8: I guess this should read “relative depletion of 182W and enrichment of 3He in some ocean island basalts compared\nto bulk mantle abundances”. Or better (because more accurate) “Low 182W/184 and high 3He/4He in some ocean island\nbasalts compared to the bulk mantle values”. I can see the point of making this content accessible to the initiated reader but\nit still needs to be accurate.\nLine 17: I do not know what is meant by “global volcanic product”. Do you mean “melts”?\nLine 20: “182W deficits” – see comment above. Also, there is no mentioning of the 182Hf-182W system being extinct.\nWithout this information, the following sentences may not appear logical. In stark contrast, I find mentioning of a (putative)\nnucleosynthetic effect less informative/relevant for the discussion.\nLine 32: Please rephrase. This is not correct and quite misleading, since we do not know the timing of core formation.\nSee various papers on this topic (including 182W data) by Kleine and co-workers.\nFigure 1: Very hard to align content of caption with what is shown in the figure/legend - especially for panel b). Are the\nreference given in the caption referring to the “experiments”? What are HW 2007, J2006, M1999? Please amend.\nLine 104: Please expand on this. What would be the magnitude of difference.\nLine 137: This is a good point, but what about Pt? A quick reference to how easily (or not) Pt forms oxides under these\nconditions (i.e. when compared to W) would strengthen the argument.\nLine 154: There is no such thing as “low-mu182W isotopes”. Please rephrase sentence.\nLine 168: As well, either “relative 3He enrichments” or “high 3He/4He”. Also add “relative” to “182W deficits”.", "reviewer_3": "Review of \"Grain boundary diffusion cannot explain the W isotope heterogeneities of the deep mantle\" by Y. Peng,\nT. Yoshino, and J. Deng\nThe submitted manuscript is a new contribution in the ongoing efforts by the Solid Earth community to understand the origins\nof ancient isotopic anomalies, specifically the coupled 182W deficiencies and 3He enrichments, in present-day ocean island\nbasalts (OIB). The manuscript appears to follow up on the initial findings of Yoshino et al. (2020), wherein ab initio\ncalculations of W diffusivity at CMB conditions were estimated. Based on those results, the Yoshino et al. study found that W\n(and unradiogenic core-hosted He) diffusion across the core-mantle boundary (from core metal into silicates) may be\ncapable of explaining the anomalous W-He isotopic composition of some OIB. This represents an interesting and important\nperspective on the mechanisms required for core-mantle interaction, the commonly invoked mechanism, to facilitate the\ndevelopment of resolvably anomalous u182W in some OIB.\nHere, this new manuscript revisits the original ab initio calculations for moderately siderophile W as part of a Mg-Fe-O-W\nsystem (i.e., periclase), using an approach that combines DFT and a newer machine learning potential method. Platinum (in\nthe Mg-Fe-O-Pt system) is also considered in this simulation, as it is highly siderophile in the solid Earth and its correlations\n(or lack thereof) can be reasonably tested against empirical OIB data.\nExperimental data for diffusion of various siderophiles in periclase is also compared. I agree with the authors - the variability\nshown is curious and notable - why would Au as an HSE behave in the unusual way that moderately siderophile W does\n(and other HSE do not), for example? Is there a risk of contamination in these elements (at least Au, maybe W?) because of\nmaterials used in the experimental setup? The authors' DFT results showing that the grain boundary diffusion behavior of W\nbeing two orders of magnitude lower than the 2007 study makes more sense in comparison to the other siderophiles\n(especially as the Pt calculations match experimental data quite well). Liquid diffusivity being higher (~one order of\nmagnitude) is also reasonable. One note of caution is that it should be made clearer in both Figure 1 panels that the\ncomparison being made is between DFT-MLP models and prior P-T experiments. In the current presentation, that is not so\nobvious and may be misinterpreted by readers. This correction to W diffusivity is the key observation. Why is it so different\nbetween the experiment and the DFT-MLP? Maybe some experimental replication is necessary, although beyond the scope\nof this particular manuscript. Perhaps similar application of the DFT-MLP approach to one or two of the other siderophiles\n(specifically Ru or Ir, which are less prone to fractionation during upper mantle melting to generate OIB) would provide\nadditional verification. But that said, the comparison of Pt and W model calculations against experimental data is fascinating.\nOne of the key differences between the Yoshino et al. study and the submitted one is that the new diffusivity calculations for\nW are performed in the absence of a chemical gradient. This is important, as it may not be reasonable to assume that a\nsufficient chemical gradient (~disequilibrium) is always present at the CMB - or the longevity of such conditions at the CMB\n(vs. time needed to develop W anomalies), if induced by something like subducted slabs foundering to the lowermost\nmantle. It is also encouraging to see the authors' tests of extrapolations of experimental results to the CMB. Here, they find\nthat the extrapolations of experimental data are not reasonable. While this certainly complicates matters when it comes to\ntrying to model certain processes based on diffusion or other geochemical behaviors at more extreme P-T conditions, this\ndiscussion is timely and should be used to leverage the development of approaches that give us better tools for looking at\nthe lowermost mantle.\nAnd indeed, the authors suggest that the revised W diffusivity has important implications for the mobility of W at CMB\nconditions. Even invoking ferripericlase given its much higher diffusivity than bridgmanite (the latter of which is assumed to\nbe a pervasive phase in the lowermost mantle), the authors find that diffusion alone cannot achieve observed 182W\nanomalies in OIB. Some other mechanism must be involved, which makes sense. A diffusion process implies a chemical\ngradient is present, and that we might see an increase in W concentrations (and possibly related, moderately siderophile or\nother similarly partitioning elements) in affected OIB, but we do not. The authors suggest that the length scale of diffusion is\nalso far too small to be effective over Gyr timescales. How reliable is the extrapolation of bridgmanite-hosted diffusion?\nThus, an alternative mechanism to grain boundary diffusion is needed to achieve 182W heterogeneity - what, though?\nExsolution is suggested as a concluding remark, but not expanded on further. Perhaps an additional sentence on this would\nbe welcome, but I do not think that significant additional discussion is necessary on this point, as that represents enough to\neasily stand alone as its own study. Questions remain though. As a reviewer with some interests in elemental behavior at\nthe lower mantle, the author's discussion provokes questions about how we should consider W behavior in other lowermost\nmantle phases. Ferripericlase is the case study, but what's happening in the bridgmanite and perovskite phases? If\nexsolution out of the core may be the answer (as mentioned at the end of the paper), what happens on the other side of that\nkind of reaction, in the silicate Earth phases? That's the part of the equation that's missing for me, and a really interesting\ntopic worth exploring.\nOverall, the manuscript reads well to a non-DFT specialist (reviewing as a high-T isotope geochemist) and presents a clear\nlogical structure and convincing arguments for the viability of the new W diffusivity results. The implications are particularly\ninteresting and relevant to the community. It might be valuable for non-specialists to provide some additional discussion\n(either in the Methods or Supplement) on the selection and filtering of data used to train the MLP, particularly with regard to\nhow and what ab initio datasets were filtered, since input is critical to how machine learning-based model approaches\nshould be evaluated by the community. Aside from that, my only other concern is what was previously mentioned with\nclarifying in the figures, as needed, when experimental vs. calculated data are shown in order to avoid readers conflating the\ntwo down the road.\nThis is an excellent manuscript overall, and I support its publication in Nature Communications. This represents an important\nand timely contribution to our understanding of the geochemical behavior of W in extreme P-T conditions that are\nchallenging to replicate experimentally.", "reviewer_4": "Referee report for \"Grain boundary diffusion cannot explain the W isotope heterogeneities of the deep mantle\"\nIn this work, the authors investigate anomalies in the isotopic composition of tungsten in oceanic island basalts (OIBs).\nSince OIBs reveal signatures of chemical composition within our planet, their analysis holds promising insights. The authors\ninvestigate a mechanism for the recently discovered anomalously low proportion of 182W within OIBs. This investigation is\ncarried out via machine-learning powered ab-initio MD simulations. As requested and per my background, I will comment on\nthe technical aspects of these simulations in my review and not comment on the general scientific impact of this manuscript,\nwhich is not to be confused for a dismissal in any way.\nThe mechanism which has been proposed to explain the depletion of 182W in OIBs and is investigated in this manuscript is\ndiffusion of tungsten from core to mantle along grain boundaries. The authors argue that under the relevant core-mantle-\nboundary conditions, this diffusion process is not sufficient to explain the observed isotopic composition due to a lower\ndiffusion coefficient of tungsten as previously reported.\nTo quantify this claim, a machine-learning interatomic potential (MLP) is employed in extended MD simulations. Two\npotentials are developed, one for the Mg-Fe-O-W system, and one for a Mg-Fe-O-Pt reference system. The construction of\nthese potentials is carried out with great care. Relevant simulations parameters which may impact the overall outcome of the\nanalysis, such as selected exchange-correlation functional, system size as well as ionic and atomic structure are explicitly\ntreated through trial simulations. The process of curating training data and tuning models is generally well-written and clear\nto follow.\nThus, the DFT and ML methodology sections in this manuscript are, in my opinion, suitable for publication, once a few small\nconcerns and questions have been addressed by the authors:\n1. In Fig S1, there is a notably larger variation of force component prediction for the Mg-O-Fe-Pt system compared to the Mg-\nO-Fe-W system. Given that the former serves as a point of reference for the latter, is there an intuition to the reason?\nEspecially given the fact that the training data set for the *-Pt system is larger than for the *-W set.\n2. Similarly, in Fig S1, there seems to be clustering of DFT (and MLP) stress at low stresses (close to zero) and higher\nstresses. A short explanation of this phenomenon would benefit the manuscript.\n3. In their description of the development of MLPs, the authors do not explicitly state the hyperparameters for the trained NN\nMLP. Referenced in the text is Ref. 66 by the same authors, which itself references a publication by the same authors that\noutlines the details of ML training (Ref. 68). I would advise to make this connection clearer. Additionally, it may be\nworthwhile repeating how the selected NN architecture has been identified. In Ref. 68 a NN with three layers of 240 neurons\nis described, which should correspond to DeePMD defaults. It would be interesting to learn whether any resp. which\nhyperparameters had to be adapted from previous MLPs.\n4. In the supplementary material it is stated that results presented in Fig. 2 and Fig. 3 are \"obtained from large systems that\napproach the thermodynamic limit \". While I do not contest the factuality of the claim, I think it is not evident which system\nsizes this corresponds to. Is this referring to systems of 12960 atoms? Or the \"over 50.000 atoms\" mentioned on page 7 of\nthe main manuscript?\n5. On page 6, the authors state \"For Mg-O-Fe-W system, the root-mean-square errors (RMSEs) of the energies, atomic\nforces, and stresses are 6.76 meV atom-1 , 0.32 eV Å-1 , and 0.53 GPa, respectively. For Mg-O-Fe-Pt, RMSEs are 6.69 meV\natom-1 , 0.30 eV Å-1 , and 0.51 GPa, respectively (Fig. S1).\" – For energies and forces I am familiar with the often self-\nimposed threshold of 10 meV/atom, which the models clearly meet. For stresses I am not familiar with typical thresholds\noften imposed in the field, and I think it could improve the quality of the report to contextualize these findings.\n6. Am I correct in assuming that standard VASP PAW pseudopotentials have been employed throughout? If so, is there\npreliminary research to indicate that these pseudopotentials, which to my knowledge were not specifically developed for\nhigh-pressure regimes, can be applied at such without problems? If this is a commonly known fact within the field of DFT\ncalculations for planetary-interior conditions, I would suggest making a short mention of this fact in the paragraph outlining\nthe computational details of the DFT simulations, because it may not be known to a wider audience.\n7. Since the simulations have been performed at medium to high temperature, it may be advisable to comment on the\ntemperature-dependence of the exchange-correlation functional. While temperature-dependent exchange-correlation\nfunctionals are in and of themselves a topic of ongoing investigation, I think the elsewise rigorous treatment of xc-effects\ngiven in the supplementary material could benefit from this added discussion."}, "author_response": "Authors’ Response to Reviews of\nGrain boundary diffusion cannot explain the W isotope hetero-\ngeneities of the deep mantle\nYihangPeng,TakashiYoshino,andJieDeng\nNatureCommunications\nRC:Reviewers’Comment, AR:Authors’Response\n1. Reviewer#1\n1.1 RC: lines 102–104: This statement contradicts with the statement in lines 157–159, in which W\ndiffusionalongbridgmaniteisconsideredasslowerthanthatalongferropericlase.\nAR: Thank you for pointing out this issue. We find that the grain boundary (GB) diffusion of W in\npolycrystallinebridgmaniteissignificantlyslowerthanthatinpost-spinel(bridgmanite+ferropericlase)\nmeasured by Yoshino et al. (2020). We acknowledge that the statement in lines 102–104 of the original\nmanuscriptwasinaccurate.Wehaveremoveditandmovedourpreferredexplanation(meltingandmechanical\npenetrationofWoxides),previouslyprovidedlaterinthemanuscript,tolines115–118.\n1.2 RC: Lines137–138: IamwonderingwhethersuchapossibleW-oxidemeltingisconsideredinthe\nauthors’simulations,sincetheauthorssetthevalencestateofWto+6inthesimulations.\nAR:UnlikeexperimentswheremetallicWfoilsareusedasthediffusionsource(HaydenandWatson,2007;\nYoshinoetal.,2020),weintroducedonly20Watomsintoferropericlasebicrystals(containingover50,000\natoms)tomimictheextremelylowWcontentinEarth’smantle. TheseWatomsaredistributedwithinthe\nGBs,whichareapproximately1nmwideandcontainatotalof2,000atoms,resultinginaWconcentration\nofonly1at%intheGBs. Webelievethat,duetothislowWcontent,thepotentialmeltingofWO isunlikely\n3\ntobeobservedinoursimulations. However,wedidobserveastructuraltransitionintheGBsfromorderto\ndisorderathightemperatures(GBpremelting,lines141–143ofthemanuscript). TheincorporationofW6+\nmightreducethemeltingpointofferropericlaseandfacilitatethepremeltingprocess. Wehaveaddsome\ndiscussionsinlines119–121andlines143–144ofthemanuscript.\n1.3 RC: Line146: WhatistheradiusofPt0?\nAR:TheatomicradiusofPt0is135pm,significantlylargerthanthatofW6+(75pm,Slater,1964),yetour\ncalculationsindicatethatthetwoexhibitverysimilardiffusionbehaviors. Atomicdiffusioncoefficientsand\ntheirtemperaturedependencearegovernedbyacomplexinterplayoffactors,includingatomicmass,radius,\nchemicalreactivity,etc. Forinstance,previoustheoreticalstudieshaveshownthatNediffusesfasterthanHe\nindenseSiO phases(Figowyetal.,2024),eventhoughNehasbothamuchgreatermassandradiusthanHe.\nHeintheminimumenergyconfiguration(Figowyetal.,2024).\nAlthoughW6+ hasaslightlylowermassandamuchsmallerradiusthanPt0,itssignificantlylowerelec-\ntronegativity(Dongetal.,2022)andhighervalencestate(Cottrelletal.,2009;CherniakandVanOrman,\n1\n2014;Médardetal.,2015)comparedtoPt0underhighpressuremaycauseWtoformstrongchemicalbonds\nwithundercoordinatedOatomsinferropericlaseGBs(Rietetal.,2021;Pengetal.,2024),leadingtosimilar\ndiffusioncoefficientsforWandPt. Theseobservationsdemonstratethatrelyingonasingleparametercan\nonlyprovideexplanationsforlimitedphenomena. Additionaldiscussionsareaddedtolines173–177inthe\nrevisedmanuscript.\n1.4 RC: Line154: PleaseprovidetheWabundanceofthemantleaswell,sincetherelativeWabundance\nbetweenthecoreandmantleismoreimportantthantheabsoluteWabundanceinthecore.\nAR:FortheabundanceofWinthemantle,arecentstudyestimatesittobebetween27and31ppb(Peters\netal.,2023),whichissignificantlyhigherthanthecorevalue(∼500ppb)(McDonough,2003). Following\nyoursuggestion,wehaveaddedthisinformationtoline188ofthemanuscript.\n1.5 RC: Lines157–159: Whataboutthegrain-boundarydiffusivityinpost-perovskitethatisthemajor\nphaseoftheaveragelowermostmantle?\nAR:Weacknowledgethepossibilitythatthepost-perovskitephasedominatesinthelowermostmantle,even\nthoughthemoststablemineralphaseattheCMBremainscontroversial(e.g.,Hernlundetal.,2005;vander\nHilstetal.,2007). However,weemphasizethatultra-lowvelocityzones,locatedbeneathsomeplumeroots\naswellasOIBsexhibitingW-Heisotopicanomalies(Mundletal.,2017;Laietal.,2022;Cottaaretal.,2022),\nhavebeensuggestedtobemainlycomposedofferropericlase(e.g.,Wicksetal.,2010,2017).\nAmmannetal.(2010)reportedthatSiandMgdiffusioninpost-perovskiteisextremelyanisotropic,with\ndiffusivitiesdifferingbyabouteightordersofmagnitudebetweenthefastestdirection⟨100⟩andtheslowest\ndirection ⟨010⟩. However, even along the fastest ⟨100⟩ direction, the diffusion coefficient of Mg does\nnotexceedthatofMgO,anditsisotropicaverageisevenlower(Ammannetal.,2010). Webelievethat\nferropericlasemayhavethefastestcationdiffusivityamongthemajormineralphasesinthelowermostmantle\nandcanthussetanupperlimitforGBdiffusioninthisregion. Nevertheless,thepotentialstronganisotropy\ninthepost-perovskitephasewarrantsfurtherinvestigationinfuturestudies. Wehavemadeamodificationin\nlines192–193oftherevisedmanuscript.\n1.6 RC: Lines187–188: 4.5Gyrratherthan4Gyr?\nAR: Thank you for pointing out this mistake. Indeed, 4.5 Gyr should be used to calculate the diffusion\ndistance,andwehavecorrectedthisinline233oftherevisedmanuscript.\n2. Reviewer#2\n2.1 RC: ThisstudyisasignificantstepforwardwithregardstoelucidatingtheeverincreasingdatasetofW\nisotopesinOIBinlightofthelargevarietyofmodelsputforwardtoexplainthem. Atthesametime,the\nstudy’sexclusivefocusondisprovingone(ofseveral)modelsalsosomewhatlimitsitsreach. Attheend,the\nauthorsflagtheirpreferredmodels(Deng&Du,2023: McNamara&Zhong,2004),whileatthesametime,\ntheircurrentstudycannotactivelysupportthose. Assuch,itssignificancebeyonddisprovingasinglemodel\ntoexplaintheobservedWisotopecompositioninOIBmaywelllieintheeyeoftheindividualreaderand\ncouldbeseenaslimitedtothewidergeochemistrycommunity.\nAR:Thankyouforrecognizingourcontribution. Weacknowledgethatthisstudyfocusesondisprovingone\n2\nofseveralmodelsforexplainingtheWisotopeanomaliesinOIBs,ratherthansupportingothermodelsor\nproposinganalternativeone. However,webelievedisprovinginvalidmodelisasimportantasproposingnew\nones,bothofwhicharecriticalinreachingaconvergingviewonthedeepmantlegeochemicalcyclesforthe\nwidergeochemistrycommunity.\n2.2 RC: The use of isotope geochemical terminology is in part too colloquial. I have indicated a few\nimprovementsbutwouldencouragetheauthorstoseekadvicefromanexpertisotopegeochemisttoproof-\nread a revised version. Also, the authors choose to report their study in the present tense (rather than\ncommonlyusedpasttense),whichIfindabitodd.\nAR:Thankyouforpointingoutthisissue. Wehavemadethesuggestedchangestothespecificpointsyou\nmentioned,soughtassistancefromanexpertisotopegeochemisttoproof-readtherevisedmanuscript,and\ncorrectedthetense.\n2.3 RC: Line8: Iguessthisshouldread“relativedepletionof182Wandenrichmentof3Heinsomeocean\nislandbasaltscomparedtobulkmantleabundances”. Orbetter(becausemoreaccurate)“Low182W/184\nandhigh3He/4Heinsomeoceanislandbasaltscomparedtothebulkmantlevalues”. Icanseethepointof\nmakingthiscontentaccessibletotheinitiatedreaderbutitstillneedstobeaccurate.\nAR:Wehaverevisedtheabstractatline5asperyoursuggestion.\n2.4 RC: Line17: Idonotknowwhatismeantby“globalvolcanicproduct”. Doyoumean“melts”?\nAR:Thankyouforpointingoutthattheuseofthisphrasemightbeinappropriate. Whatweactuallymeant\nwas“volcanicrocks”,andwehaverevisedthisinline14ofthemanuscript.\n2.5 RC: Line20: “182Wdeficits”–seecommentabove. Also,thereisnomentioningofthe182Hf-182W\nsystembeingextinct. Withoutthisinformation,thefollowingsentencesmaynotappearlogical. Instark\ncontrast,Ifindmentioningofa(putative)nucleosyntheticeffectlessinformative/relevantforthediscussion.\nAR:Basedonyoursuggestion,wehaverevisedline20toincludetheinformationthat182Hfisalreadyextinct\nandremovedtheredundantdetails.\n2.6 RC: Line32: Pleaserephrase. Thisisnotcorrectandquitemisleading,sincewedonotknowthetiming\nofcoreformation. Seevariouspapersonthistopic(including182Wdata)byKleineandco-workers.\nAR:Weacknowledgethatthispartwasinaccurateandhaverewrittenitbasedonyoursuggestioninlines\n32–34oftherevisedmanuscript.\n2.7 RC: Figure1: Veryhardtoaligncontentofcaptionwithwhatisshowninthefigure/legend-especially\nforpanelb). Arethereferencegiveninthecaptionreferringtothe“experiments”? WhatareHW2007,\nJ2006,M1999? Pleaseamend.\nAR:WehaverewrittenthecaptionofFig. 1toimproveclarity.\n2.8 RC: Line104: Pleaseexpandonthis. Whatwouldbethemagnitudeofdifference.\nComment1.1). Wehavemovedourpreferredexplanation,previouslyprovidedlaterinthemanuscript,to\nlines115–118andhaveaddedfurtherdiscussion(seeComment1.2andComment2.9).\n2.9 RC: Line137: Thisisagoodpoint,butwhataboutPt? Aquickreferencetohoweasily(ornot)Pt\nformsoxidesundertheseconditions(i.e. whencomparedtoW)wouldstrengthentheargument.\nAR: Thank you for raising this point. Indeed, the much lower valence state of Pt (Médard et al., 2015)\ncomparedtoW(Cottrelletal.,2009;CherniakandVanOrman,2014)insilicatemeltssuggeststhatPtmay\nbelesslikelytoformoxidesandmigrateviaasimilarmechanism. Thisdifferencelikelyaccountsforthe\nsignificantlylowerfluxofPtcomparedtoWobservedbyHaydenandWatson(2007). Thisdiscussionhas\nbeenincorporatedintolines122–124oftherevisedmanuscript.\n2.10 RC: Line154: Thereisnosuchthingas“low-mu182Wisotopes”. Pleaserephrasesentence.\nAR:Wehaverephrasedthissentenceinline188ofthemanuscript.\n2.11 RC: Line168: Aswell,either“relative3Heenrichments”or“high3He/4He”. Alsoadd“relative”to\n“182Wdeficits”.\nAR:Wehaverephrasedthissentenceinline205ofthemanuscript.\n3. Reviewer#3\n3.1 RC: OnenoteofcautionisthatitshouldbemadeclearerinbothFigure1panelsthatthecomparison\nbeingmadeisbetweenDFT-MLPmodelsandpriorP-Texperiments. Inthecurrentpresentation,thatisnot\nsoobviousandmaybemisinterpretedbyreaders.\nAR:WehaverewrittenthecaptionofFig. 1toimproveclarity.\n3.2 RC: TheauthorssuggestthatthelengthscaleofdiffusionisalsofartoosmalltobeeffectiveoverGyr\ntimescales. Howreliableistheextrapolationofbridgmanite-hosteddiffusion?\nAR:Previousexperimentalresultsindicatethatunderthetemperatureandpressureconditionsoftheupper-\nmostlowermantle,thediffusioncoefficientofFe-Mginterdiffusioninbridgmanite(Holzapfeletal.,2005)is\napproximatelysixordersofmagnitudelowerthanthatinferropericlase(Holzapfeletal.,2003). Theoretical\ncalculationsbyAmmannetal.(2010)alsosuggestthatundertemperatureandpressureconditionsthroughout\nthelowermantle,thediffusioncoefficientofMginferropericlaseismorethanfourordersofmagnitude\nhigherthanthatinbridgmanite. Ofcourse,thesearedataforlatticediffusion,butdiffusioncoefficientsare\ncontrolledbythemicroscopicenvironmentofthediffusingatoms. EventhoughGBsmayexhibitamore\ndisorderedstructureespeciallyathightemperatures,theyshouldstilldisplaydiffusionbehaviorintermediate\nbetweenthethatofthecorrespondingcrystalandmelt. Thus,therelativemagnitudesofGBdiffusioncoefficients\nfor the two minerals are likely to be consistent with their lattice diffusion behavior. The GB diffusion\nofWinpolycrystallinebridgmaniteisindeedsignificantlyslowerthanthatinpost-spinel(bridgmanite+\nferropericlase)measuredbyYoshinoetal.(2020),whichsupportsourextrapolation.\n4\n3.3 RC: Thus,analternativemechanismtograinboundarydiffusionisneededtoachieve182Wheterogene-\nity-what,though? Exsolutionissuggestedasaconcludingremark,butnotexpandedonfurther. Perhaps\nanadditionalsentenceonthiswouldbewelcome,butIdonotthinkthatsignificantadditionaldiscussionis\nnecessaryonthispoint,asthatrepresentsenoughtoeasilystandaloneasitsownstudy.\nAR:Iagreewithyourperspectivethatadetaileddiscussionisnotnecessaryatthisstage.Wehaveincorporated\nyoursuggestionbyaddinganadditionalsentenceforabriefexpansioninline242oftherevisedmanuscript.\n3.4 RC: Questionsremainthough. Asareviewerwithsomeinterestsinelementalbehavioratthelower\nmantle, the author’s discussion provokes questions about how we should consider W behavior in other\nlowermost mantle phases. Ferripericlase is the case study, but what’s happening in the bridgmanite and\nperovskitephases? Ifexsolutionoutofthecoremaybetheanswer(asmentionedattheendofthepaper),\nwhathappensontheothersideofthatkindofreaction,inthesilicateEarthphases? That’sthepartofthe\nequationthat’smissingforme,andareallyinterestingtopicworthexploring.\nAR:WeagreethatthebehaviorofWinotherlowermostmantlephasesisafascinatingtopicthatwarrants\nfurther investigation. However, as we mentioned in our responses to Comment 1.5 and Comment 3.2,\nferropericlaselikelyexhibitsthefastestcationdiffusionamongthemajormineralphasesinthelowermost\nmantle,anditcanthereforesetanupperlimitforGBdiffusioninthisregion.\nAs for your other question regarding the fate of core-derived exsolutions in the silicate Earth, we have\ndeveloped a comprehensive model in another manuscript currently under review at Nature Geosciences\n(Deng et al., 2024). This model proposes that the basal magma ocean of the early Earth is continuously\ncontaminated by oxide exsolutions from the core, and diapirs of core exsolution entrained into the solid\nmantlemaycausesmall-scalescattering. ThepreprintispubliclyavailableonResearchSquare(https:\n//doi.org/10.21203/rs.3.rs-3263305/v1),andwewouldbedelightedifyoutakealookatit.\n3.5 RC: Itmightbevaluablefornon-specialiststrtoprovidesomeadditionaldiscussion(eitherintheMethods\norSupplement)ontheselectionandfilteringofdatausedtotraintheMLP,particularlywithregardtohowand\nwhatabinitiodatasetswerefiltered,sinceinputiscriticaltohowmachinelearning-basedmodelapproaches\nshouldbeevaluatedbythecommunity.\nAR:Thankyouforyoursuggestion. ThetrainingofallourMLPsfollowsaconsistentstrategycallediterative\ntrainingscheme(Dengetal.,2023). Specifically,weusedprincipalcomponentanalysistoidentifythemost\nrepresentativeframesfromMDtrajectoriesforhigh-accuracyDFTrecalculations. Theserecalculatedresults\nwerecomparedwiththepredictionsofthecurrentmodeltoidentifyframesthatwerepoorlydescribedbythe\nMLP,whichwerethenaddedtothetrainingset. Thisiterativeapproachensuresthatthetrainingsetremains\nconcise and well-balanced. Through multiple iterations of training, the model’s accuracy was gradually\nimproved. Detaileddescriptionsofthismethodologyareprovidedinourpreviousstudies(Dengetal.,2023;\nPeng and Deng, 2024a). Following your suggestions, we have added a more detailed description of the\ntrainingprocessinlines267–271oftherevisedmanuscript.\n4. Reviewer#4\n4.1 RC: InFigS1,thereisanotablylargervariationofforcecomponentpredictionfortheMg-O-Fe-Pt\nsystemcomparedtotheMg-O-Fe-Wsystem. Giventhattheformerservesasapointofreferenceforthe\nlatter,isthereanintuitiontothereason? Especiallygiventhefactthatthetrainingdatasetforthe*-Ptsystem\n5\nislargerthanforthe*-Wset.\nAR:TheapparentlargervariationintheforcecomponentpredictionfortheMg-O-Fe-Ptsystemcomparedto\ntheMg-O-Fe-WsysteminFig. S1ispurelyaresultofinconsistentaxisrangesofplotting. InFig. S1b,thex\nandyaxesrangefrom-62to62eV/Å,whereasinFig. S1e,theyrangefrom-40to40eV/Å.Thiseffectively\nmagnifiestheappearanceofvariationsintheMg-O-Fe-Ptsystem.\nInfact,theRMSEoftheforcesfortheMg-O-Fe-Ptsystemis0.30eV/Å,whichisslightlylowerthanthe\n0.32eV/ÅfortheMg-O-Fe-Wsystem(lines276–277). Theslightdifferencemayindeedbeattributedtothe\nlargertrainingdatasetfortheMg-O-Fe-Ptsystem. WehaverevisedFig. S1tounifythescalesofthetwo\nsystemsforconsistency.\n4.2 RC: Similarly,inFigS1,thereseemstobeclusteringofDFT(andMLP)stressatlowstresses(closeto\nzero)andhigherstresses. Ashortexplanationofthisphenomenonwouldbenefitthemanuscript.\nAR:Thankyouforpointingthisout. TheclusteringobservedinFig. S1arisesbecause,whencalculating\nstresserrors,weextractandcompareallninecomponentsofthestresstensor(σ ,σ ,σ ,σ ,σ ,σ ,\nxx xy xz yx yy yz\nσ ,σ ,σ )betweentheDFTdataandMLPpredictions. Eachcomponentfromeveryframecorresponds\nzx zy zz\ntoasingledatapointinFig. S1candFig. S1f. Sinceoursystemhasnegligibleshearstress,onlythediagonal\ncomponents(σ ,σ ,σ )exhibitstressesclosetothehydrostaticpressure,resultinginclusteringathigh\nxx yy zz\nstresses. Incontrast,theoff-diagonalcomponentsareclosetozero,leadingtoclusteringnearzero. Wehave\naddedanclarificationtothecaptionofFig. S1.\n4.3 RC: IntheirdescriptionofthedevelopmentofMLPs,theauthorsdonotexplicitlystatethehyperparam-\netersforthetrainedNNMLP.ReferencedinthetextisRef. 66bythesameauthors,whichitselfreferencesa\npublicationbythesameauthorsthatoutlinesthedetailsofMLtraining(Ref. 68). Iwouldadvisetomake\nthisconnectionclearer. Additionally,itmaybeworthwhilerepeatinghowtheselectedNNarchitecturehas\nbeenidentified. InRef. 68aNNwiththreelayersof240neuronsisdescribed,whichshouldcorrespond\ntoDeePMDdefaults. Itwouldbeinterestingtolearnwhetheranyresp. whichhyperparametershadtobe\nadaptedfrompreviousMLPs.\nAR:Forallthestudiesmentioned,wehaveconsistentlyusedthedefaultsettingsofDeePMD(Wangetal.,\n2018),specificallyaneuralnetworkarchitecturewiththreelayersof240neurons,withoutanyadaptations\nto the hyperparameters. This setting is also used by other groups in our high pressure mineral physics\ncommunity(e.g.,Wangetal.,2022,2023). Thesystemswefocusoninvolverelativelysimpleplanetary\ninteriorcomponents,suchasmineralsandmeltswith3–4typesofelements. Ourresultsdemonstratethat\nMLPstrainedwiththesedefaulthyperparametersalreadyperformexceptionallywell,achievinganenergy\nRMSE between MLP predictions and DFT calculations below the commonly accepted threshold of 10\nmeV/atom(DengandStixrude,2021b;Dengetal.,2023;PengandDeng,2024a;Deng,2024). Additionally,\nthermodynamicandtransportpropertiesofmaterialscalculatedusingthesemodelsshowgoodagreement\nwithexperimentaldata(DengandStixrude,2021b;Dengetal.,2023;PengandDeng,2024b;Deng,2024).\nFollowing your suggestion, we have added an explicit statement regarding the hyperparameters in lines\n264–266oftherevisedmanuscript.\n4.4 RC: InthesupplementarymaterialitisstatedthatresultspresentedinFig. 2andFig. 3are\"obtained\nfromlargesystemsthatapproachthethermodynamiclimit\". WhileIdonotcontestthefactualityofthe\nclaim,Ithinkitisnotevidentwhichsystemsizesthiscorrespondsto. Isthisreferringtosystemsof12960\n6\natoms? Orthe\"over50.000atoms\"mentionedonpage7ofthemainmanuscript?\nAR:Inthisstudy,wesimulatedbothferropericlaseGBsystemsand(Mg,Fe)Oliquid. ThesizeeffectonGB\ndiffusioninferropericlasewastestedinourpreviouswork(Pengetal.,2024),whereasystemsizeofover\n50,000atomswasfoundtobeconverged,andthissetupwasalsoadoptedinthisstudy. Fortheliquidphase,\nweperformedadditionalsizeconvergencetests,asdetailedintheSupplementaryInformation. Therefore,the\nGBdatapresentedinFig. 2andFig. 3wereobtainedusingsystemswith>50,000atoms,whiletheliquid\ndatawerederivedfromsystemswith>12,000atoms. WehaveupdatedtheSupplementaryInformationto\nclarifythispoint.\n4.5 RC: Onpage6,theauthorsstate\"ForMg-O-Fe-Wsystem,theroot-mean-squareerrors(RMSEs)of\ntheenergies,atomicforces,andstressesare6.76meVatom-1,0.32eVÅ-1,and0.53GPa,respectively.\nForMg-O-Fe-Pt,RMSEsare6.69meVatom-1,0.30eVÅ-1,and0.51GPa,respectively(Fig. S1).\"–For\nenergiesandforcesIamfamiliarwiththeoftenself-imposedthresholdof10meV/atom,whichthemodels\nclearlymeet. ForstressesIamnotfamiliarwithtypicalthresholdsoftenimposedinthefield,andIthinkit\ncouldimprovethequalityofthereporttocontextualizethesefindings.\nAR:Tothebestofourknowledge,thereisnowidelyacceptedtypicalthresholdvaluesfortheRMSEof\nstressesinourfield. Unlikeenergiesandforces,stressesorpressurecanbedirectlymeasuredexperimentally.\nOurgoalistoimprovetheaccuracyofstresspredictionstoreliablycalculatetheequationofstateofplanetary\nmaterials. Atpressuresabove100GPa,thediscrepanciesbetweenDFTandexperimentalresults,aswellas\nthoseamongdifferentDFTapproximationmethods,aretypicallyatleastontheorderofseveralgigapascals\n(e.g.,OganovandDorogokupets,2003). Therefore,webelievethattheRMSEsofstresspredictionsachieved\nbyourMLPmeetstheexpectedaccuracyforthiscontext. Wehaveaddedexplanationstocontextualizethese\nfindingsinlines277–280oftherevisedmanuscript.\n4.6 RC: AmIcorrectinassumingthatstandardVASPPAWpseudopotentialshavebeenemployedthrough-\nout? If so, is there preliminary research to indicate that these pseudopotentials, which to my knowledge\nwerenotspecificallydevelopedforhigh-pressureregimes,canbeappliedatsuchwithoutproblems? Ifthis\nisacommonlyknownfactwithinthefieldofDFTcalculationsforplanetary-interiorconditions,Iwould\nsuggestmakingashortmentionofthisfactintheparagraphoutliningthecomputationaldetailsoftheDFT\nsimulations,becauseitmaynotbeknowntoawideraudience.\nAR:Yes,wehaveemployedthestandardVASPPAWpseudopotentialsforallDFTcalculationsinthisstudy.\nPreviousstudieshavedemonstratedthatthePAWmethodprovidesconsistentresultswithexperimentsfor\nequationsofstateofMgOOganovandDorogokupets(2003)andtransitionmetals(Dewaeleetal.,2008)\nunderpressuresuptomegabars. StandardVASPPAWpotentialscombinedwiththePBEsolapproximation\nyieldgoodagreementwithexperimentalmeasurementsofthephysicalpropertiesofsilicatesandoxidesunder\nplanetaryinteriorconditions(e.g., HolmströmandStixrude,2015;Scipionietal.,2017;XiaoandStixrude,\n2018;DengandStixrude,2021a). Webelievethitisawell-establishedconsensuswithinthehigh-pressure\nresearchcommunity. Asperyoursuggestion,wehaveaddedaclarificationinlines284–286oftherevised\nmanuscript.\n4.7 RC: Sincethesimulationshavebeenperformedatmediumtohightemperature,itmaybeadvisable\nto comment on the temperature-dependence of the exchange-correlation functional. While temperature-\ndependentexchange-correlationfunctionalsareinandofthemselvesatopicofongoinginvestigation,Ithink\ntheelsewiserigoroustreatmentofxc-effectsgiveninthesupplementarymaterialcouldbenefitfromthis\n7\naddeddiscussion.\nAR: The simulations in this study employ the PBEsol exchange-correlation (xc) functional, which, like\nmostcommonlyusedxcfunctionals,doesnotexplicitlyaccountfortemperatureeffects. Whiletemperature-\ndependent xc functionals are an active area of research, they are not yet widely benchmarked for Earth\nmaterials. Forhydrogen,significantdifferencesbetweenfinite-temperaturexcandregularLDAonlyappear\natultra-hightemperaturesexceeding10,000K(Ramakrishnaetal.,2020). Forthetemperatureconditionsof\n<5000Kinvestigatedhere,thermaleffectsontheelectronicstructureareassumedtobesecondarycompared\ntothedominantionicmotioneffects. TheelectronictemperaturesaretakenintoaccountviaFermi-Dirac\nsmearingduringtheDFTcalculationandsetasaframeparameterduringtheMLPtrainingprocesses. Future\nadvancements in temperature-dependent xc functionals may provide more accurate treatments for such\nextremeconditions.\nFollowing your suggestion, we have added the discussion about the temperature-dependence of the xc\nfunctionalintheSupplementaryInformation."}, {"round": 2, "reviewer_comments": {"reviewer_4": "I thank the authors for taking the time to address my concerns and expand on the ML and DFT-MD methodology as per my suggestions. I believe this revision has improved the overall presentation and discussion of the employed ML and DFT-MD methods within an already rigorous study.\nThe authors have adapted the presentation of force errors to showcase consistent force errors between Mg-O-Fe-Pt and Mg-O-Fe-W systems and added a note clarifying the apparent clustering of stress components (both in Fig S1). The revised manuscript now contains details on the employed neural network architecture and the atom counts for presented simulations (in the supplementary material), as well as a brief discussion of relevant error thresholds for predictions. Finally, the usage of standard PAW pseudopotentials and temperature-agnostic exchange-correlation functionals is amply discussed and contextualized.\nSince all of my concerns have been excellently addressed I thus recommend publication of this article, based on only the ML and DFT portions of it, which I was contacted to comment on. If there are further objections to be made on the earth system science side of the manuscript, I trust the other referees will point them out."}, "author_response": "Authors’ Response to Reviews of\nGrain boundary diffusion cannot explain the W isotope hetero-\ngeneities of the deep mantle\nYihangPeng,TakashiYoshino,andJieDeng\nNatureCommunications\nRC:Reviewers’Comment, AR:Authors’Response\n1. Reviewer#4\n1.1 RC: IthanktheauthorsfortakingthetimetoaddressmyconcernsandexpandontheMLandDFT-\nMDmethodologyaspermysuggestions. Ibelievethisrevisionhasimprovedtheoverallpresentationand\ndiscussionoftheemployedMLandDFT-MDmethodswithinanalreadyrigorousstudy.\nTheauthorshaveadaptedthepresentationofforceerrorstoshowcaseconsistentforceerrorsbetweenMg-O-\nFe-PtandMg-O-Fe-Wsystemsandaddedanoteclarifyingtheapparentclusteringofstresscomponents(both\ninFigS1). Therevisedmanuscriptnowcontainsdetailsontheemployedneuralnetworkarchitectureandthe\natomcountsforpresentedsimulations(inthesupplementarymaterial),aswellasabriefdiscussionofrelevant\nerrorthresholdsforpredictions.Finally,theusageofstandardPAWpseudopotentialsandtemperature-agnostic\nexchange-correlationfunctionalsisamplydiscussedandcontextualized.\nSinceallofmyconcernshavebeenexcellentlyaddressedIthusrecommendpublicationofthisarticle,\nbased\nononlytheMLandDFTportionsofit,whichIwascontactedtocommenton. Iftherearefurtherobjections\ntobemadeontheearthsystemsciencesideofthemanuscript,Itrusttheotherrefereeswillpointthemout.\nAR: Thank you for your thorough review and for acknowledging the revisions made to address your\nconcerns. Wesincerelyappreciateyourconstructivefeedback,whichhassignificantlyimprovedtheclarity\nandpresentationofourMLandDFT-MDmethodology. Yourrecommendationforpublicationisgreatly\nencouraging,andwearegratefulforyourinsightfulcommentsthroughoutthereviewprocess.\n1"}]

Decision letter