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REVIEW 5 major objections 6 minor 53 references

Machine-learnt potential highlights melting and freezing of aluminium nanoparticles

T0 review · 5 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Aluminium nanoparticles switch shape at 2,100 and 25,000 atoms

desk verdict Useful Al-BFF potential with a solid training-data message, but the headline structural crossover map rests on an extrapolation that needs uncertainty checks before it can be trusted. read the letter →

arxiv 2412.16294 v1 pith:MANKQ5MV submitted 2024-12-20 cond-mat.mtrl-sci cond-mat.mes-hallphysics.comp-ph

Davide Alimonti , Francesca Baletto This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.mes-hallphysics.comp-ph
keywords aluminiumnanoparticlesBayesianforcefieldmachine-learnedpotentialmelting-freezinghysteresisicosahedralstabilitydecahedralmoleculardynamicsstructuralcrossover
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a newly trained Bayesian machine-learned potential for aluminium, augmented with melted nanodroplet configurations in its training data, can drive classical molecular dynamics of nanoparticles from 200 to 11,000 atoms and produce a thermodynamically consistent melting-freezing cycle. On its predictions, icosahedral aluminium nanoparticles are most stable up to about 2,000 atoms (~2 nm), decahedral shapes take over between about 2,100 and 25,000 atoms, and face-centred-cubic motifs win beyond 25,000 atoms. The same simulations show a melting-freezing hysteresis that widens with size, with melting temperatures up to about 200 K above freezing at a 100 K/ns rate, and no surface melting. A sympathetic reader would care because these size-dependent structural crossovers and transition temperatures are the quantities that matter for applications such as catalysis, energy storage, and additive manufacturing, and standard empirical potentials misjudge them.

What carries the argument

The load-bearing object is the Al-BFF 2 potential: a Gaussian-process Bayesian force field whose local energies are functions of atomic cluster expansion three-body descriptors, trained on energies, forces, and stresses and equipped with an active-learning loop that calls DFT when predictive uncertainty is high. The decisive technical addition is the inclusion of melted nanodroplet configurations in the training set, which removes non-physical sublimation seen with a first potential trained only on bulk, surfaces, and cold clusters. The analysis then runs on iterative heating and cooling schedules, with structural classification by common-neighbour signatures and unsupervised k-means clustering of local environments to separate core from surface ordering.

What would settle it

Compute DFT excess energies for relaxed icosahedral, decahedral, and fcc nanoparticles at sizes straddling the predicted crossovers, for example 2,000-3,000 and 20,000-30,000 atoms, and check whether the energy crossings match 2,100 and 25,000 atoms; alternatively, count the structural motifs by transmission electron microscopy for aluminium nanoparticles between 2 and 5 nm and see whether icosahedra dominate at the small end as claimed.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the Bayesian force field Al-BFF 2, trained on density-functional-theory energies, forces, and stresses from bulk aluminium, low-index surfaces, and small clusters together with deliberately included liquid nanodroplets, reproduces bulk properties such as a melting temperature of 906 ± 1 K (experimental value 933 K) and then predicts the size-dependent structural map of aluminium nanoparticles. Relaxed excess energies place the icosahedral-to-decahedral crossover at 2,100 atoms and the decahedral-to-fcc crossover at 25,000 atoms, and iterative molecular dynamics at 100 K/ns adds the dynamical claim that hysteresis loops enlarge with size, that annealing a liquid droplet extends icosahedral stability to about 5,000 atoms, and that there is no surface melting although the surface shows local order in the liquid. The paper argues this is a complete thermodynamic cycle simulated with one potential that stays close to ab initio accuracy at classical speed.

Load-bearing premise

The force field is trained on DFT data for clusters of only 55 to 150 atoms, yet it is used to predict behaviour of particles up to 300,000 atoms without any direct DFT check at those larger sizes.

Editorial extensions

If this is right

  • Aluminium nanoparticles below about 2,000 atoms should be treated as icosahedral rather than fcc-like when modelling catalysis or other size-sensitive properties.
  • The predicted size window for decahedral stability, roughly 2,100 to 25,000 atoms, gives a concrete target for electron-microscopy checks and for synthesis strategies aiming at fivefold shapes.
  • At 100 K/ns heating, melting and freezing temperatures differ by up to 200 K, so kinetic effects must be folded into any comparison between simulated and measured transition temperatures.
  • Annealing a liquid droplet before cooling shifts icosahedral stability upward to about 5,000 atoms, implying thermal history changes the resulting nanoparticle shape.
  • The absence of surface melting, with ordered local environments persisting at the liquid surface, is a directly testable structural prediction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The practice of seeding training data with melted nanodroplets is likely transferable to other metals; it addresses a generic failure mode where machine-learned potentials trained on cold solids produce unphysical vaporisation near melting.
  • Because the crossover sizes are set by the competition between surface energy and lattice strain, the same type of size map should hold for other aluminium-group metals, though the precise atom counts will shift with surface-energy ratios.
  • A slower cooling rate should shrink the observed hysteresis and may shift the solidification outcome toward the energy-preferred motif, offering a direct test of the kinetic origin the paper assigns to the loop.
  • If the structural crossovers are reproduced at larger sizes, the potential could replace empirical potentials in studies of nanoscale welding, sintering, and 3D printing of aluminium, where melting and resolidification are the controlling steps.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. The paper introduces two FLARE Bayesian force fields for aluminium (Al-BFF1 and Al-BFF2), trained on QE-PBEsol data from bulk, low-index surfaces, and small clusters (55-150 atoms), with D2 adding melted nanodroplets to cure nonphysical sublimation. Using Al-BFF2, the authors compute excess energies of idealized icosahedral, decahedral, and FCC motifs up to 3×10^5 atoms, obtaining Ih stability up to ~2100 atoms, Dh from ~2100 to 25000 atoms, and FCC beyond. They perform iterative MD heating/cooling at 100 K/ns for nanoparticles between 257 and 10179 atoms, reporting a size-dependent melting/freezing hysteresis (with Tm up to ~200 K above Tf at the largest size), a Gibbs-Thomson constant C=1.58, and a CNA/clustering analysis that shows no surface melting and possible surface freezing. The paper concludes that annealing liquid droplets stabilizes icosahedral solidification outcomes up to 5000-6000 atoms.

Significance. If the phase map is correct, the paper would be an important demonstration that a small DFT dataset can produce a transferable MLP for size-dependent structural transitions in aluminium nanoparticles, with practical implications for the community. The strengths are real: the BFF gives bulk melting at 906±1 K versus 933 K experiment, the force/energy MAEs on held-out bulk/surface test sets are small (~0.026 eV/Å, ~0.005 eV), and the potentials, training database, and trajectories are openly available. The methodological point that melted nanodroplets must be included in the training set to avoid unphysical high-temperature behavior is useful. However, the central quantitative predictions—the crossover sizes and the extended icosahedral range after annealing—rest on unvalidated extrapolation and on energy comparisons of idealized motifs without global optimization, so the significance is conditional on additional validation.

major comments (5)
  1. [III (Fig. 1, Table V) and Conclusion] The central structural phase map is derived from energy differences between locally relaxed idealized motifs, and the Conclusion explicitly concedes that this is done 'without any formal global minimisation.' The abstract's statements that 'Al-BFF predicts an icosahedral stability range of up to 2000 atoms... a region of stability for decahedra, up to 25000 atoms' therefore overstate what is actually computed: these are relative stabilities of chosen ideal geometries under Al-BFF2, not equilibrium stability ranges. Please either perform global optimization (e.g., basin hopping or replica exchange) for representative sizes, or systematically rephrase the abstract and conclusion to describe the result as the relative stability of idealized motifs.
  2. [Table I, Table V, Eq. (22)] The crossover locations are an extrapolation that is never validated. The D2 dataset used for all MD and for the Al-BFF2 numbers in Table V contains no icosahedral cluster (Ih55 appears only in D1), and the largest DFT reference is 150 atoms, yet the energy comparisons extend to 3×10^5 atoms and the MD to 10,179 atoms. No DFT calculation is reported on relaxed large motifs near the crossover sizes. The BFF predictive uncertainty (Eq. 22) is available but is not used to test whether the energy differences between motifs at N≈2100 and N≈25000 exceed model error. The sensitivity of the crossings to the training set is visible in the shift between Al-BFF1 (1500/10000) and Al-BFF2 (2100/25000) in Table V. Please add single-point DFT checks on relaxed motifs at least at the two crossover sizes and report uncertainty estimates on the motif energy differences.
  3. [III.A and Conclusion] The claim that 'the annealing of a liquid droplet further stabilizes icosahedral structures, extending their stability range to 5000 atoms' conflates kinetic freezing outcomes with thermodynamic stability. The MD runs at 100 K/ns are non-equilibrium, and the same section notes that the annealed structures 'are not the global minimum.' The defective Ih and Dh solids obtained after freezing are kinetic products; they do not establish an equilibrium stability range. Please either compute free energies of the competing solid phases or reframe this statement as a description of solidification outcomes at the imposed cooling rate.
  4. [Eq. (8), III.A] The Gibbs-Thomson fit is used to argue that the good agreement with experiment 'corroborates the accuracy' of Al-BFF2. However, the fit uses (Tm+Tf)/2 from 100 K/ns simulations, and no uncertainty is reported for the fitted constant C=1.58. Without confidence intervals and a statement of the number and size range of fitting points, the comparison with the experimental C=1.76 is not quantitative. Moreover, the arithmetic mean of non-equilibrium hysteresis endpoints is not obviously the equilibrium melting temperature. Please report confidence intervals for C and, if possible, compare with a quasi-static or coexistence-based melting temperature for at least one nanoparticle size.
  5. [Table III, III] The Wulff construction and the FCC-motif energies used in the crossover analysis rely on the surface-energy ratio γ100/γ111=1.17. Table III shows that Al-BFF2's γ111 (0.94 J/m2) is 19% below the quoted experimental value (1.16 J/m2); because the BFF reproduces the PBEsol value (0.96 J/m2), the discrepancy is inherited from the reference functional. Since the Ih→Dh→FCC balance is a competition between surface energy and strain, the quantitative crossover sizes should be discussed with this systematic uncertainty in mind.
minor comments (6)
  1. [II.B, Eq. (7)] The definition of Tm and Tf as 'the temperatures showing the nearest larger and smaller DKL to the discontinuity' is not reproducible as written; please specify the exact algorithm (e.g., the largest jump in DKL, with a window or threshold) used to locate the transition.
  2. [III.A, Fig. 6] The sentence 'we note an increment in the IH percentage sharper than that of SH, indicating that the surface organises first than the inner part' is internally inconsistent: if IH (inner ordered) increases more sharply, the core organizes more abruptly, not the surface. Please clarify the evidence for surface-first ordering during freezing.
  3. [II.A, Table IV] The test set used for the MAE values in Table IV comprises only bulk and surface configurations (33 entries in Table I), so these low MAEs do not directly validate the potential on nanoparticle environments; this limitation should be stated when the MAEs are presented.
  4. [II.B, Eq. (2)] Because both E(N) and ε_b are negative, the formula for the excess energy is ambiguous without an explicit statement of the sign convention; please state clearly that ΔE is defined as the positive energy cost per surface atom.
  5. [Conclusion] The phrase 'the nano-slush state is at least up to 3.4 nm' introduces a term and a size bound that are not defined earlier; please specify which descriptor supports this bound and what 'nano-slush' means.
  6. [Acknowledgments] There is a typo: 'We re grateful' should read 'We are grateful.'

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the structural crossovers and melting/freezing hysteresis are genuine predictions from a DFT-trained BFF; self-citations are methodological only.

full rationale

The paper's central claims are derived from a Bayesian force field trained on DFT data (Table I) and then used in energy relaxations and MD simulations; no target result is used as a fitting label. The Ih-to-Dh and Dh-to-FCC crossover sizes in Table V come from computing excess energies (Eq. 2) of relaxed ideal motifs, and no crossover size appears in the training set or as an adjustable parameter. The melting/freezing temperatures and hysteresis are read off MD trajectories via caloric curves and KL-divergence/CNA descriptors, and the Gibbs-Thomson constant C=1.58 is fit after the simulations and compared with the experimental value 1.76, which is validation rather than a fitted input renamed as a prediction. The inclusion of melted nanodroplets in D2 informs the potential about liquid environments at small sizes (Al100, Al150), but the simulated transition temperatures at 257-10179 atoms are emergent and not direct training labels. Self-citations (Zeni et al., Rossi et al., Delgado-Callico et al., Jones et al.) support analysis tools such as k-means clustering, CNA classification, and the PDDF-KL melting signature; none is invoked as a uniqueness theorem or as a substitute for the paper's own derivation. The main weakness is transferability extrapolation beyond the 150-atom DFT training data, which is a correctness risk, not circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

All physical claims flow through the trained BFF, whose hyperparameters and noise levels are fitted to DFT data. The structural stability map assumes that relaxed ideal motifs represent competing minima, and the no-surface-melting conclusion assumes the k-means descriptor classification captures surface disorder. No new physical entities are introduced.

free parameters (4)
  • ACE descriptor hyperparameters (nmax, lmax, rc) = 8, 3, 4.5 Å
    Chosen by optimizing on D1 for best bulk FCC accuracy (Sec. II.A); fixed for both BFFs.
  • GPR hyperparameters (sigma, lambda_E, lambda_f, lambda_tau) = sigma=3.51 eV; lambda_E=0.15 eV; lambda_f=0.05-0.06 eV/A; lambda_tau=0.0005-0.0006 eV/A^3
    Fitted by maximizing the log marginal likelihood on training data (Table II).
  • Gibbs-Thomson constant C = 1.58
    Fit of Eq. 8 to simulated phase-transition temperatures; used only as a comparison to the experimental C=1.76.
  • CNA cutoff = 0.8 a0
    Hand-chosen cutoff for common neighbour analysis (Sec. II.B); affects signature classification.
assumptions (5)
  • domain assumption Total energy is a sum of local atomic energies within a cutoff radius rc (Eq. 9)
    Locality assumption for the BFF, stated in Suppl. Sec. 3; standard for MLIPs.
  • domain assumption PBEsol DFT provides accurate reference energies and forces for Al bulk, surfaces, and clusters
    All training labels come from QE-PBEsol; errors in the functional propagate into the potential.
  • domain assumption The KL divergence discontinuity between PDDFs marks the melting/freezing temperature
    Used to define Tm and Tf in Sec. II.B; relies on the universal signature reported in Ref. 18.
  • domain assumption The lowest excess energy among relaxed Ih, Dh, and FCC motifs determines the stable shape
    The paper states 'without any formal global minimisation' (Conclusion), so stability is relative to the included motifs.
  • domain assumption A Langevin thermostat with tau=20 ps at 100 K/ns produces representative heating/cooling trajectories
    Dynamics are non-equilibrium; the hysteresis is attributed to kinetic effects (Sec. III.A).

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Pith. "Pith review of Machine-learnt potential highlights melting and freezing of aluminium nanoparticles." pith.science (2026). https://pith.science/paper/MANKQ5MV

@misc{pith2026241216294,
  author       = {Pith},
  title        = {Pith review of: Machine-learnt potential highlights melting and freezing of aluminium nanoparticles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MANKQ5MV}},
  note         = {Machine review of arXiv:2412.16294}
}
abstract

We investigated the complete thermodynamic cycle of aluminium nanoparticles through classical molecular dynamics simulations, spanning a wide size range from 200 atoms to 11000 atoms. The aluminium-aluminium interactions are modelled using a newly developed Bayesian Force Field (BFF) from the FLARE suite, a cutting-edge tool in our field. We discuss the database requirements to include melted nanodroplets to avoid unphysical behaviour at the phase transition. Our study provides a comprehensive understanding of structural stability up to sizes as large as $3~ 10^5$ atoms. The developed Al-BFF predicts an icosahedral stability range of up to 2000 atoms, approximately 2 nm, followed by a region of stability for decahedra, up to 25000 atoms. Beyond this size, the expected structure favours face-centred cubic (FCC) shapes. At a fixed heating/cooling rate of 100K/ns, we consistently observe a hysteresis loop, where the melting temperatures are higher than those associated with solidification. The annealing of a liquid droplet further stabilizes icosahedral structures, extending their stability range to 5000 atoms. Using a hierarchical k-means clustering, we find no evidence of surface melting but observe some mild indication of surface freezing. In any event, the liquid droplet's surface shows local structural order at all sizes.

Figures

Figures reproduced from arXiv: 2412.16294 by the authors.

Figure 1
Figure 1. FIG. 1. Excess energy in eV as a function of the nanoparticle size [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Caloric plots in heating (red) and freezing (blue) for nanoparticles of different sizes using Al-BFF 2 using the itMD procedure with a [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Normalised gyration radius, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (31 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Occurrence of the (421), (422) and (555) CNA signatures [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Occurrence of ordered environments in the inner and at the [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5. CNA signatures at 600 K, after freezing averaged over avail [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Relative error, relative to DFT values, for bulk properties [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Parity plots for energies and forces for Al-BFF1 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Excess energies of more geometrical motifs [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Melting (red) and freezing (blue) temperatures of AlNPs [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p012_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p012_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p012_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p013_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p013_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p013_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p013_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p014_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p014_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p014_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24. Snapshots taken at the beginning of the heating, liquid droplet, and the cooling process. On the right panel, averaged PDDF and RDF [PITH_FULL_IMAGE:figures/full_fig_p014_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) as a [PITH_FULL_IMAGE:figures/full_fig_p015_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) as a [PITH_FULL_IMAGE:figures/full_fig_p015_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) as a [PITH_FULL_IMAGE:figures/full_fig_p016_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) as a [PITH_FULL_IMAGE:figures/full_fig_p016_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) as a [PITH_FULL_IMAGE:figures/full_fig_p017_29.png]
Figure 30
Figure 30. Figure 30: FIG. 30. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) as a [PITH_FULL_IMAGE:figures/full_fig_p017_30.png]
Figure 31
Figure 31. Figure 31: FIG. 31. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) [PITH_FULL_IMAGE:figures/full_fig_p018_31.png]
Figure 32
Figure 32. Figure 32: FIG. 32. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) [PITH_FULL_IMAGE:figures/full_fig_p018_32.png]
Figure 33
Figure 33. Figure 33: FIG. 33. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) [PITH_FULL_IMAGE:figures/full_fig_p019_33.png]
Figure 34
Figure 34. Figure 34: FIG. 34. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) [PITH_FULL_IMAGE:figures/full_fig_p019_34.png]
Figure 35
Figure 35. Figure 35: FIG. 35. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) [PITH_FULL_IMAGE:figures/full_fig_p020_35.png]
Figure 36
Figure 36. Figure 36: FIG. 36. Atomic environments, IH (inner solid, blue), IL (inner liquid, orange), SH (surface solid, green) and SL (surface liquid, red) [PITH_FULL_IMAGE:figures/full_fig_p020_36.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.