Atomic-scale phase-field modeling with dopants: Stochastic self-consistent harmonic approximation with fractional site occupancy

Kairi Masuda; Yu Kumagai

REVIEW 5 major objections 5 minor 12 references

Atomic-scale phase-field modeling with dopants: Stochastic self-consistent harmonic approximation with fractional site occupancy

T0 review · 5 major / 5 minor · reviewed 2026-07-31 · deepseek-v4-flash

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Pith's one-line read By treating site occupation as a continuous variable inside a variational free-energy functional, this paper derives site-resolved chemical potentials and predicts where Ag dopants segregate at a Cu grain boundary.

desk verdict Solid methodological extension of SSCHA to fractional site occupancy, but the grain-boundary site-ordering prediction rests on unvalidated kinetics; worth peer review with convergence and sensitivity checks. read the letter →

arxiv 2607.24107 v1 pith:TJQLNQXU submitted 2026-07-27 cond-mat.mtrl-sci

Kairi Masuda , Yu Kumagai This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords phase-fieldmodelingatomic-scalesimulationstochasticself-consistentharmonicapproximationfractionalsiteoccupancysite-resolvedchemicalpotentialdopantsegregationgrainboundaryCu–Agalloy

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

Phase-field modeling has traditionally been a continuum-scale tool; this paper tries to bring it down to the atomic scale. The key move is to let each lattice site carry a continuous occupation probability inside a variational free energy built from the stochastic self-consistent harmonic approximation, so the free energy becomes differentiable with respect to site occupation. That differentiability yields a site-resolved chemical potential, which supplies the thermodynamic driving force for atomic-scale pattern formation such as dopant segregation. The authors show the approach reproduces the temperature-dependent lattice constant of Ag-doped bulk Cu and predicts a specific segregation hierarchy at a Cu Σ5(310)[001] grain boundary: Ag atoms fill the vertices of the triangular structural motif first, then near-vertex sites, then bottom sites. A sympathetic reader would care because this is a route to predicting, not just rationalizing, where dopants sit in real microstructures.

What carries the argument

The central object is the variational free-energy functional in which each atom is represented by a general anisotropic Gaussian probability cloud and each site carries a continuous occupation probability. The functional derivative of the free energy with respect to site occupation gives the site-resolved chemical potential, and the negative gradient with respect to atomic position gives the mean force that moves the Gaussian clouds. Because the free energy is evaluated by stochastic sampling of a many-body interatomic potential, the machinery converts the discrete question of which site is occupied by a dopant into a differentiable thermodynamic landscape that can be minimized and evolved i

What would settle it

Compute the site-resolved segregation free energy of Ag at the Σ5(310)[001] Cu grain boundary by direct thermodynamic integration, or measure the 300 K site occupancy by atom-probe tomography: if the order is not vertices first, then near-vertex sites, then bottom sites, the chemical-potential functional is not capturing equilibrium segregation.

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Extended reading notes

Core claim

The central claim is that a stochastic self-consistent harmonic approximation free-energy functional, extended to fractional site occupancy, is a working atomic-scale phase-field model. Because each site's occupancy is continuous, the free energy is differentiable with respect to occupation, and its derivative gives a site-resolved chemical potential. Using this potential together with the mean force on each Gaussian atomic cloud, the authors evolve both occupation probabilities and atomic positions. They show the model reproduces the temperature-dependent lattice constant of Ag-doped bulk Cu, matching molecular dynamics, and predicts a concrete segregation hierarchy at a Cu Σ5(310)[001] gra

Load-bearing premise

The load-bearing premise—flagged by the authors in their Limitations section (item 2)—is that independent Gaussian vibrational clouds give a faithful free-energy surface for the Cu–Ag systems studied; if vibrational correlations are significant, the site-resolved chemical potentials and the segregation ranking could change.

Editorial extensions

If this is right

For bulk Cu–Ag, the method reproduces the temperature-dependent lattice constant as the dopant number varies, matching molecular-dynamics reference calculations.
For the Σ5(310)[001] Cu grain boundary, the method predicts a clear segregation order—vertices first, then near-vertex sites, then bottom sites—thereby identifying primary, secondary, and tertiary segregation sites.
Because chemical potentials are computed site-resolved, the framework provides a thermodynamic driving force for atomic-scale pattern formation without relying on continuum fields.
The same formalism applies to many-body interatomic potentials and to general anisotropic vibrational shapes, going beyond earlier pair-potential, isotropic-Gaussian treatments.
The framework visualizes atomic-scale pattern formation as probability-density maps, allowing direct comparison with atomic-resolution observations.

Reading between the lines

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

Editorial inference: the predicted segregation hierarchy at the Σ5 boundary is the paper's sharpest testable output; a direct site-resolved segregation-energy calculation or high-resolution chemical mapping at 300 K could confirm or refute it independently of the phase-field dynamics.
Editorial inference: the 'ordered-like' Ag occupancy pattern seen in the small bulk Cu–Ag cell is likely a finite-size effect; in a larger cell, phase separation rather than ordering should emerge, and testing that limit would stress the free-energy functional.
Editorial inference: the two kinetic parameters—the occupation-exchange mobility and the atomic mobility taken from an aqueous ion—are placeholders; if the final segregation pattern is sensitive to them, the reported hierarchy would incorporate kinetic bias rather than pure equilibrium thermodynamics.
Editorial inference: the authors' own limitation note points to oxides as the next stress test, since vibrational correlations neglected here become important there; applying the method to an oxide grain boundary would reveal how much the independent-Gaussian approximation limits the approach.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Core take: the free-energy functional is a genuine step forward, but the segregation hierarchy in Fig. 5 is not yet demonstrated to be kinetically robust; treat it as provisional.

What's new and good: extending SSCHA to fractional site occupancy with many-body potentials and anisotropic Gaussians, and deriving a site-resolved chemical potential as an explicit free-energy derivative, is a real contribution beyond Li et al.'s pair-potential, isotropic limit. The bulk lattice-constant agreement with MD is a healthy external check on the free-energy functional, even if the test is a small 32-atom cell. The paper is also honest about its limitations.

Soft spots: the grain-boundary segregation prediction depends on the occupancy exchange mobility Mij, which is unknown and set to 0.1 eV^-1 s^-1, and the atomic mobility is taken from Cu^2+ in water. Equilibrium occupancy should be M-independent, but nothing demonstrates that the 300 K simulations reached equilibrium; they are started from 1200 K patterns and frozen in. With only a few hundred steps and no reported convergence criterion, the vertex → near-vertex → bottom ordering could be an artifact of the quench schedule and arbitrary kinetics. The bulk validation supports the free energy, but it does not validate the kinetic part. Also, no error bars are given on Monte Carlo estimates (1000 samples for F, 100 for µ), and there is no code/data release, which hampers reproduction. The finite-size-stabilized 'ordered-like' bulk state is acknowledged as not the true thermodynamic phase.

Verdict: the central derivation looks sound, and the paper's own limitations are stated. The method deserves a serious referee. For acceptance, I would want a convergence analysis of the occupancy dynamics, a check that the site ordering is insensitive to Mij and cooling rate, and ideally error bars and code release. This paper is for people working on atomistic thermodynamics, segregation, and SSCHA-based methods; they will get value from the formalism even if they do not yet trust the specific grain-boundary ranking.

Recommendation: send to peer review, with the kinetic-sensitivity issue as the central requested revision.

Referee Report

5 major / 5 minor

Summary. The paper presents an atomic-scale phase-field method that combines stochastic self-consistent harmonic approximation (SSCHA) with fractional site occupancies. A variational free energy is constructed from the Gibbs–Bogoliubov inequality (Eqs. 6–8), with Gaussian vibrational clouds and continuous occupation variables c_i. From this functional the authors derive site-resolved chemical potentials (Eq. 19), atomic equations of motion (Eqs. 10, 17), and a pressure/volume-control scheme (Eqs. 20–23). The method is tested on Ag-doped bulk Cu, where the computed temperature-dependent lattice constants are compared with MD results (Fig. 3), and then applied to Ag segregation at a Cu Σ5(310)[001] grain boundary, where it predicts preferential occupation of vertex, near-vertex, and bottom sites of the triangular motif (Figs. 4–5). The manuscript claims this extends phase-field modeling to atomic-scale dopant segregation.

Significance. If the method is robust, it is a useful extension of SSCHA toward atomistic phase-field modeling with many-body interatomic potentials. The derivation from the Gibbs–Bogoliubov inequality is standard and the lattice-constant check against MD provides a meaningful external benchmark. The use of a machine-learned potential (MACE) and GPU-accelerated Sobol Monte Carlo sampling is a practical strength. However, the central segregation-site prediction currently rests on an arbitrarily chosen kinetic coefficient and on finite-time occupation evolution without demonstrated convergence or equilibrium. The paper's claim to identify primary, secondary, and tertiary segregation sites is therefore not yet supported. The limitations stated by the authors—especially the neglect of vibrational correlations and of complexion transitions—are important and should be more prominently connected to the scope of the claims.

major comments (5)

[§Governing equations and Simulation details, Eq. (11), Fig. 5] The central segregation ranking is produced by the occupation dynamics Eq. (11) with Mij = 0.1 eV⁻¹ s⁻¹, which the authors state is unknown. No convergence criterion is reported for c_i, and Figs. 4–5 show only finite minimization steps. In equilibrium, the ranking should be independent of Mij and of runtime, but no evidence is given that the plotted states are converged. Please provide a convergence check (e.g., max_i |dc_i/dt| or the spread of µ_i versus step) and an Mij-sensitivity test. If the states are deliberately non-equilibrium quenches, state this explicitly and justify why the ordering is robust.
[Simulation details, Eq. (19), Fig. 3] The chemical potentials are evaluated by Monte Carlo with a sample size of only 100 per site, and no statistical error bars are reported for µ_i or for the lattice constants in Fig. 3. If the site-to-site differences that determine the vertex→near-vertex→bottom ranking are smaller than the MC noise, the ranking is not statistically supported. Report standard errors for the relevant quantities and increase Nsample if necessary.
[Simulation details, Eq. (17) and volume control] The atomic mobility κ is taken from Cu²⁺ in water at 298 K and the volume mobility M′ is set to 10.0 Pa⁻¹ s⁻¹ without physical calibration or sensitivity analysis. These parameters control the evolution of atomic positions and volume, which are coupled to the occupancy dynamics through the free energy. Demonstrate that the final segregation patterns are insensitive to κ and M′, or calibrate them; otherwise the structural contribution to the predicted pattern is uncontrolled.
[§Results, Fig. 5, Limitations] The 300 K segregation patterns are initialized from 1200 K distributions and ‘frozen in’ during cooling, but no cooling protocol or quench schedule is specified. It is therefore unclear whether Fig. 5 represents equilibrium states at 300 K or kinetic artifacts of the cooling procedure. In addition, the authors' own limitation (1) excludes complexion transitions, which are relevant to grain-boundary segregation. Please clarify the thermodynamic status of the plotted distributions and temper the predictive claim accordingly.
[§Methods, Eq. (8)] In Eq. (8), a single covariance Σ_i is used for both host and dopant species at a partially occupied site: only the thermal de Broglie wavelength differs in the vibrational term, and the interaction term samples a common Gaussian cloud regardless of the sampled species. This imposes identical vibrational amplitudes on Cu and Ag, which may bias the site-resolved chemical potentials, especially for a size-mismatched dopant. Please discuss this approximation or test its effect by allowing species-dependent covariance matrices.

minor comments (5)

[Limitations section] The text says 'The calculation of the chemical potential, Eq. (18)' but the chemical potential is defined in Eq. (19). Please correct the cross-reference.
[Abstract and Fig. 2] The abstract states that the method 'successfully reproducing Ag distributions in bulk Cu,' but Fig. 2 shows a finite-size-stabilized ordered-like state in a system where Cu–Ag is experimentally prone to phase separation. Consider softening the wording to 'explores low-energy occupancy patterns' rather than 'reproducing distributions.'
[Figs. 2, 4, 5] The color scale is described as the same in all three figures, but no color bar is shown. Since the quantitative occupation probabilities are central to the ranking, a color bar with numerical values would improve interpretability.
[Fig. 4 caption] The caption says 'shown at (a) 0, (b) 10, and (c) 100 minimization steps,' while the main text refers to these as 'Figs. 4(b) and 4(c)' without specifying the step numbers in the same notation. This is a minor inconsistency in presentation that should be harmonized.
[MC sampling] The use of Sobol sequences and Nsample=1000 for the free energy is a good practical choice. It would strengthen the paper to state whether the same low-discrepancy sequence is used for the chemical-potential estimate (Nsample=100) and to report the reproducibility across different sequence lengths.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the segregation prediction is the minimizer of a free-energy functional built from an external machine-learned potential, not from the target segregation data.

full rationale

The derivation chain is self-contained. The free-energy functional F in Eq. (8) is constructed from the Gibbs–Bogoliubov variational bound with fractional occupation probabilities; the chemical potential in Eq. (19) is the analytical derivative ∂F/∂c_i; and the occupation dynamics in Eq. (11) is a gradient flow that drives the system toward equal chemical potentials. The predicted segregation pattern is therefore the minimum (or long-time state) of the functional. The potential-energy input is the MACE interatomic potential trained on r2SCAN data, which is external to the segregation results; no site-resolved segregation energies or experimental segregation patterns are fitted into F. The lattice-constant comparison against MD using the same potential is a consistency check of the free-energy approximation, not a circular fit. The self-citations (refs. 35, 36) are used for the Gaussian-parameterization and Langevin-type evolution, but the paper supplies the relevant equations and no uniqueness claim or forbidden alternative rests on those citations. The arbitrary kinetic coefficient Mij = 0.1 eV^-1 s^-1, the aqueous-solution mobility κ, and the non-equilibrium cooling protocol are genuine validity/kinetics risks—they could make the displayed, possibly unconverged segregation pattern depend on the protocol—but they are not circular because the target segregation data are not used to determine those parameters. Overall, no load-bearing step reduces to its own input.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical entities are postulated. The main inputs are the variational SSCHA approximation, the MACE potential, and ad hoc kinetic parameters that control the time evolution. The free parameters (Mij, κ, M') affect dynamics and could bias segregation patterns if not representative.

free parameters (3)

kinetic coefficient for occupation exchange Mij = 0.1 eV⁻¹·s⁻¹
Set by hand because the mobility associated with occupation exchange is unknown (Simulation detail, Eq. 11).
atomic mobility κ = 5.56 × 10⁻⁸ m²/(V·s)
Taken from Cu²⁺ in water at 298 K rather than from the simulated solid Cu system (Simulation detail).
volume mobility M' = 10.0 Pa⁻¹·s⁻¹
Chosen for the barostat in Eq. 21 (Simulation detail).

assumptions (4)

standard math Gibbs–Bogoliubov inequality provides a variational upper bound on the true free energy.
Used in Eq. (6) as the basis for the free-energy functional.
domain assumption The reference state is an Einstein solid with independent Gaussian vibrational clouds.
This is the core approximation of (S)SCHA; the authors note it may fail for strongly correlated vibrational systems like oxides.
domain assumption The interatomic potential φ is accurately modeled by MACE trained on r2SCAN data.
The paper relies on MACE (Ref. 46) for all force and energy evaluations, so the accuracy of the results is bounded by that potential.
ad hoc to paper Occupancy probabilities ci are continuous and differentiable variables.
Fractional site occupancy is introduced as a mathematical device to make chemical potentials differentiable; it is not a physical observable unlike discrete occupations.

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Pith. "Pith review of Atomic-scale phase-field modeling with dopants: Stochastic self-consistent harmonic approximation with fractional site occupancy." pith.science (2026). https://pith.science/paper/TJQLNQXU

@misc{pith2026260724107,
  author       = {Pith},
  title        = {Pith review of: Atomic-scale phase-field modeling with dopants: Stochastic self-consistent harmonic approximation with fractional site occupancy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TJQLNQXU}},
  note         = {Machine review of arXiv:2607.24107}
}

read the original abstract

Phase-field modeling has achieved great success in predicting pattern formation in materials, such as the formation of ferroelectric domains. However, because it is typically based on continuum mechanics, conventional phase-field modeling cannot be straightforwardly applied to atomic-scale pattern formation, such as dopant segregation and vacancy ordering, which are driven by chemical potentials. Here, we extend the phase-field concept to the atomic scale by formulating the free energy of atomic systems within stochastic self-consistent harmonic approximation (SSCHA) theory to allow fractional site occupations. Our methodology enables us to directly calculate the derivative of the free energy with respect to site occupation and thereby obtain the chemical potential, successfully reproducing Ag distributions in bulk Cu as well as the resulting lattice expansion. Furthermore, we applied our methodology to investigate dopant segregation around a {\Sigma}5(310)[001] Cu grain boundary doped with Ag atoms. We found that Ag atoms preferentially segregate at the vertices of the triangular motif of a grain boundary. As the number of dopants increases, excess Ag atoms segregate near the vertices and then at the bottom sites of the triangular motif. This study extends the phase-field concept to discrete atomic systems, enabling the identification of preferential dopant-segregation sites and thereby visualizing atomic-scale pattern formation.

Figures

Figures reproduced from arXiv: 2607.24107 by the authors.

**Figure 2.** Structure evolution of the site occupation probab [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Temperature dependence of the lattice constants o [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Evolution of the Ag occupation probability around [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

**Figure 5.** Figure 5: Segregation patterns around a Σ5(310)[001] Cu grain boundary with various numbers of Ag dopant atoms in a 76-atom system under the NPT ensemble at T = 300 K and P0 = 0 GPa, as calculated using phase-field simulations. The probability distributions are shown for (a) NA…

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Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.

[1] [1]

(1) Chen, L. Q. Phase-Field Models for Microstructure Evolu tion. Annu. Rev. Mater. Res. 2002, 32, 113–140. 24 (2) Boettinger, W. J.; Warren, J. A.; Beckermann, C.; Karma, A. Phase-Field Simulation of Solidiﬁcation. Annu. Rev. Mater. Res. 2002, 32, 163–194. (3) Ambati, M.; Gerasimov, T.; De Lorenzis, L. A Review on Pha se-Field Models of Brittle Fracture ...

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(53) Owen, L.; Playford, H.; Stone, H.; Tucker, M. Analysis o f short-range order in Cu 3Au using X-ray pair distribution functions. Acta Mater. 2017, 125, 15–26. (54) Sanati, M.; Zunger, A. Evolution of L 12 ordered domains in fcc Cu 3Au alloy. J. Phys. Condens. Matter. 2007, 19, 086201. (55) Okamoto, H.; Chakrabarti, D. J.; Laughlin, D. E.; Massa lski, ...

2017

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V.; Fridkin, V

(26) Bune, A. V.; Fridkin, V. M.; Ducharme, S.; Blinov, L. M.; Palto, S. P.; Sorokin, A. V.; Yudin, S. G.; Zlatkin, A. Two-dimensional ferroelectric ﬁl ms. Nature 1998, 391, 874–

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R.; Grant, M

(27) Elder, K. R.; Grant, M. Modeling elastic and plastic def ormations in nonequilibrium processing using phase ﬁeld crystals. Phys. Rev. E 2004, 70, 051605. (28) Chan, P. Y.; Goldenfeld, N.; Dantzig, J. Molecular dyna mics on diﬀusive time scales from the phase-ﬁeld-crystal equation. Phys. Rev. E 2009, 79, 035701. (29) Berry, J.; Grant, M.; Elder, K. R....

2004

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(7) Chen, L. Q. Phase-Field Method of Phase Transitions/Dom ain Structures in Ferroelec- tric Thin Films: A Review. J. Am. Ceram. Soc. 2008, 91, 1835–1844. (8) Yadav, A. K. et al. Observation of Polar Vortices in Oxide Superlattices. Nature 2016, 530, 198–201. (9) Das, S. et al. Observation of room-temperature polar sky rmions. Nature 2019, 568, 368–372. ...

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L.; Zhu, Y

(12) Tang, Y. L.; Zhu, Y. L.; Ma, X. L.; Borisevich, A. Y.; Moro zovska, A. N.; Eliseev, E. A.; Wang, W. Y.; Wang, Y. J.; Xu, Y. B.; Zhang, Z. D.; Pennycook, S. J. Observation of 25 a Periodic Array of Flux-Closure Quadrants in Strained Ferr oelectric PbTiO 3 ﬁlms. Science 2015, 348, 547–551. (13) Choudhury, S.; Li, Y.; Krill, C.; Chen, L.-Q. Phase-ﬁel d ...

2015

[8] [2017]

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(48) Nocedal, J. Updating Quasi-Newton Matrices with Limit ed Storage. Math. Comput. 1980, 35, 773–782. (49) Virtanen, P. et al. SciPy 1.0: Fundamental Algorithms f or Scientiﬁc Computing in Python. Nat. Methods 2020, 17, 261–272. (50) Uhlenbeck, G. E.; Ornstein, L. S. On the Theory of the Bro wnian Motion. Phys. Rev. 1930, 36, 823–841. (51) Harris, D. C....

1980