Pith. sign in

REVIEW 2 major objections 8 minor 46 references

Size focusing in core-shell precipitates: A phase-field study

T0 review · 2 major / 8 minor · reviewed 2026-07-09 · glm-5.2

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Smaller precipitates grow faster, narrowing size spread

desk verdict Solid theory for the classical regime; the non-classical regime where most focusing actually occurs is unverified read the letter →

arxiv 2607.07002 v1 pith:KITZ2ALP submitted 2026-07-08 cond-mat.mtrl-sci

Soumya Mishra , T. A. Abinandanan This is my paper · ORCID
classification cond-mat.mtrl-sci PACS 81.30.Mh64.75.Nx68.35.Fx
keywords sizealloysfocusinggrowthcore-shellshelltheoreticalcoarsening
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

In certain aluminium alloys, precipitates form a core-shell structure where a shell of one phase grows around a core of another. The authors study why this shell growth narrows the size distribution of precipitates — a phenomenon called size focusing. They extend Zener's classical theory of diffusion-controlled growth to core-shell precipitates and show that because the growth rate is inversely proportional to the precipitate radius (dr/dt = K/2r), smaller precipitates grow faster than larger ones, causing the size distribution to narrow. This focusing is strictly a growth phenomenon: it begins when the shell starts growing and ends the moment growth ceases and coarsening (where large precipitates consume small ones) takes over. Phase-field simulations with multiple precipitates confirm the theory and show that stronger focusing occurs with higher shell volume fractions and larger inter-precipitate spacing, while the duration of focusing depends entirely on that spacing. The authors also demonstrate that size focusing can occur in binary alloys without a core-shell morphology, provided a second growth step injects additional supersaturation — analogous to solute injection in nanoparticle synthesis.

What carries the argument

The key theoretical object is the growth rate expression dr/dt = K/(2r), where K is the Zener growth coefficient proportional to diffusivity and supersaturation. From this, the authors derive dσ_s/dt = -(K/2σ_s r³)[r² − Hr], where σ_s is the scaled standard deviation (σ/r), r̄ is the mean radius, and H is the harmonic mean. Since r̄² > Hr̄ for any ensemble of positive radii, dσ_s/dt is always negative during classical growth. The phase-field model uses a ternary Cahn-Hilliard framework with a polynomial free energy, simulated with semi-implicit Fourier spectral methods on GPU.

What would settle it

If size focusing were observed to persist or strengthen after shell growth has definitively ended and coarsening has begun, the central claim that focusing is purely a growth phenomenon would be falsified. Alternatively, if increasing shell volume fraction or inter-precipitate spacing did not produce stronger or longer-lasting focusing in experiments, the practical guidelines would fail.

Watch

Extended reading notes

Core claim

The central mechanism is the inverse relationship between precipitate growth rate and radius: dr/dt = K/(2r), derived from Zener's theory applied to shell growth in a core-shell precipitate. Because smaller precipitates have a higher growth rate, they catch up to larger ones, narrowing the size distribution. The authors formalise this by deriving an expression for the rate of change of the scaled standard deviation (dσ_s/dt) that is always negative during growth, proving size focusing is a direct consequence of diffusion-controlled growth. Critically, this effect reverses into size defocusing once coarsening begins, establishing a clean boundary: focusing belongs to growth, defocusing to co-

Load-bearing premise

The theoretical derivation assumes each precipitate grows in a matrix with constant supersaturation, which is Zener's classical assumption. This breaks down once diffusion fields from neighbouring precipitates overlap — something the simulations show happens relatively early. The theory's quantitative predictions for how fast the size distribution narrows depend on this constant-supersaturation assumption, yet the simulations show size focusing continues well into the regime,

Editorial extensions

If this is right

  • Alloy designers can achieve narrower precipitate size distributions by increasing shell-forming solute content and ensuring larger inter-precipitate spacing through controlled nucleation density.
  • The boundary between size focusing and coarsening is sharp and tied to the end of shell growth, meaning heat treatments should be quenched at the moment shell growth completes to lock in the narrowest distribution.
  • Size focusing does not require a core-shell morphology — any binary alloy subjected to a two-step treatment that adds supersaturation without new nucleation should exhibit the same focusing effect.
  • The theory predicts that the strength of focusing scales with the Zener growth coefficient K, so faster-diffusing shell-forming species should produce sharper size distributions.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 8 minor

Summary. This manuscript presents a theoretical framework and phase-field simulations to explain size focusing in core-shell precipitates in ternary aluminium alloys. The theory extends Zener's diffusional growth law (r² = Kt) to an ensemble of core-shell precipitates, deriving that the scaled standard deviation σ_s decreases with time (Eqs. 7a–b) because the growth rate dr/dt = K/(2r) is inversely proportional to radius. Two-precipitate and multi-precipitate simulations qualitatively confirm the theory and are used to assess the roles of shell volume fraction, core volume fraction, inter-precipitate spacing, and core coarsening. The central claim—that size focusing is a growth phenomenon ending when coarsening begins—is supported by the simulations. The practical guidelines (higher shell fraction, larger spacing promote focusing) are clearly motivated by the results.

Significance. The paper addresses a well-defined, experimentally motivated problem (size focusing in Al-Sc-Li and Al-Yb-Li alloys) with a transparent analytical theory that is parameter-free in its core derivation: K is measured from single-precipitate simulations rather than fitted to the size-focusing outcome. The multi-precipitate simulations provide falsifiable, comparative predictions (Tables 3–5) on how alloy design variables affect focusing strength and duration. The demonstration that size focusing can occur in binary alloys via a solute-injection analogue (Section 5.1) is a nice extension. The critique of applying LSW theory to the growth regime (Section 5.2) is well-placed and adds value to the literature.

major comments (2)
  1. Section 3, Eqs. (4)–(8): The theory derives dσ_s/dt < 0 from Zener's law dr/dt = K/(2r), which assumes each precipitate grows in a matrix with constant supersaturation c⁰_C. This assumption holds only during classical growth (before diffusion-field overlap). Yet the simulations show size focusing continuing well beyond this regime: in the two-precipitate case, classical growth ends at t~2500 but σ_s keeps decreasing until t~25000 (Fig. 1b); in multi-precipitate simulations, the minimum occurs at t~1800 with inter-precipitate distance d~160, by which point diffusion fields have certainly overlapped. The paper acknowledges this gap (end of Section 3.1) but still presents the Zener-based theory as the explanatory framework for the entire size-focusing phenomenon. The actual growth-rate dependence on r in the non-classical regime is never extracted from the simulations, so it is not verified
  2. Figure 1(b): The quantitative agreement between theory and simulation is stated to be qualitative, with the theory predicting a slower decrease in σ_s than the simulation. However, the growth coefficient K used in the theoretical curve is taken from the long-time asymptote (K=0.26, beyond t~80,000), whereas the size-focusing regime in the two-precipitate simulation occurs at t~2500–25000, during which K is time-dependent and larger (inset, Fig. 1b). Using the asymptotic K systematically underpredicts the focusing rate. The authors should either use the time-dependent K(t) in the integration of Eq. (8) or explicitly discuss how this choice affects the comparison. As it stands, the quantitative validation is weaker than the text conveys.
minor comments (8)
  1. Section 2, Eq. (2): The polynomial free energy f⁰ uses products of squared compositions (c²_A c²_B, etc.), which is unusual for a regular-solution-type model (typically c_A c_B). The rationale for this specific form and its consequences for phase equilibria (e.g., binodal compositions, interfacial energies) should be briefly stated so readers can assess transferability.
  2. Section 3, Eq. (7a): The derivation of dσ_s/dt involves the harmonic mean H and the mean of squared radii r̄². The algebra is relegated to Supplementary Section A; a one-line summary of the key step would improve readability of the main text.
  3. Section 4.1: The statement that t_sf is lower in multi-precipitate (1800) than in two-precipitate (25,000) simulations attributes the difference to three factors (inter-precipitate distance, core coarsening, local environment). These are not disentangled. Consider consolidating or clarifying which factor dominates.
  4. Table 3: The footnote states differences in t_sf are 'deemed insignificant due to the shallowness of the minima.' If the minima are shallow enough that t_sf is unreliable, the t_sf column should perhaps be omitted or flagged more prominently, as t_sf is a primary metric used throughout the paper.
  5. Section 5.3: The comparison with Radmilovic et al. [27] converts a 3D core volume fraction of 0.007 to a 2D equivalent of 0.043 using assumptions detailed in Supplementary Section D. The sensitivity of this conversion to the assumptions (similar inter-precipitate distance, symmetric size distributions) should be briefly noted in the main text.
  6. All simulations are 2D. In 2D diffusion-controlled growth, the growth law is r² ∝ t·ln(t) rather than r² = Kt. The authors use r² = Kt throughout. This is likely a minor correction for the time range studied, but it should be acknowledged, especially since the theory is presented as extending Zener's 3D result.
  7. Figure 2(b): The inverse correlation between shell thickness and core radius is presented as a scatter plot. A quantitative measure of correlation (e.g., Pearson coefficient) or a fit would strengthen the claim.
  8. Reference [28] (Chhotray and Gautam) is cited as 2026; confirm this is correctly cited and accessible.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity found; derivation is self-contained from Zener's law through pure calculus to an independently validated prediction

full rationale

The paper's central theoretical result (dσ_s/dt < 0, Eqs. 7a-b) is derived from Zener's growth law (r² = Kt, an external result) via straightforward calculus and the universally true inequality that the arithmetic mean exceeds the harmonic mean. No step reduces to its own inputs by construction. The growth coefficient K is measured from single-precipitate simulations and used to predict the two-precipitate σ_s evolution — and the prediction does not perfectly match the simulation (the authors note the theory predicts slower focusing), which is the opposite of circularity. Multi-precipitate simulations use an independent phase-field model with no fitting to the theory. The two self-citations ([39] for numerical method, [44] for supplementary material) are methodological/supplementary and not load-bearing for the central claim. The derivation chain is self-contained.

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

The paper introduces no new physical entities or forces. The free parameters are model parameters chosen for the phase-field simulations, not fitted to experimental data. The growth coefficient K is measured from simulations, not fitted to the target result. The axioms are standard domain assumptions from precipitate growth theory, except the polynomial free energy which is a modeling choice specific to this paper.

free parameters (4)
  • chi_AB, chi_BC, chi_AC = (3, 1, 1)
    Binary interaction energies in the polynomial free energy (Equation 2), chosen to satisfy the wetting condition sigma_alpha_beta > sigma_beta_gamma + sigma_alpha_gamma.
  • chi_ABC = 50
    Ternary interaction energy chosen to ensure low interfacial segregation (c_B <= 0.04).
  • kappa_AB, kappa_BC, kappa_AC = (4.88, 2.45, 2.45)
    Gradient energy coefficients chosen to yield target interfacial energies.
  • K (growth coefficient) = 0.26
    Measured from single-precipitate simulation data (inset, Figure 1b) at t > 80,000, then used as input to predict two-precipitate sigma_s evolution.
assumptions (4)
  • domain assumption Zener's theory of diffusion-controlled growth (r^2 = Kt) applies to shell growth around each core precipitate.
    Invoked in Section 3, paragraph 2. Assumes constant supersaturation c0_C in the matrix around each precipitate, which breaks down when diffusion fields overlap.
  • domain assumption The wetting condition (sigma_alpha_beta > sigma_alpha_gamma + sigma_beta_gamma) is satisfied, ensuring the shell phase fully wets the core.
    Invoked in Section 3, paragraph 1 and Table 1. Required for the core-shell morphology to form.
  • domain assumption Interface curvature effects on growth are negligible (Zener's assumption).
    Stated in Section 3, paragraph 2. The Gibbs-Thomson effect is ignored in the theoretical derivation but present in the simulations.
  • ad hoc to paper The polynomial approximation to f0 (Equation 2) adequately represents the thermodynamics of a ternary alloy.
    Used throughout. This is a simplified Landau-type polynomial, not calibrated to a real thermodynamic database for Al-Sc-Li.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Size focusing in core-shell precipitates: A phase-field study." pith.science (2026). https://pith.science/paper/KITZ2ALP

@misc{pith2026260707002,
  author       = {Pith},
  title        = {Pith review of: Size focusing in core-shell precipitates: A phase-field study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KITZ2ALP}},
  note         = {Machine review of arXiv:2607.07002}
}
read the original abstract

Due to their enhanced resistance to coarsening and/or creep, aluminium alloys with precipitates of two distinct phases in a core-shell morphology are of great contemporary interest. In this paper, we focus on the curious observation in two recent studies on Al-Sc-Li and Al-Yb-Li alloys that growth of the shell phase leads to a narrowing of the size distribution. We have studied this phenomenon, known as size focusing, using a theoretical framework (which extends Zener's theory of diffusional growth to a core-shell precipitate) and multi-precipitate simulations based on a phase field model. Our results yield key theoretical insights as well as conclusions with practical significance. (a) On the theoretical front, we show clearly that size focusing is a growth phenomenon: it ends when shell growth ends, and coarsening begins. (b) On the practical front, our results offer guidelines for designing alloys with narrower size distributions: size focusing is promoted in alloys with greater shell volume fractions and greater inter-precipitate spacing.

Figures

Figures reproduced from arXiv: 2607.07002 by the authors.

Figure 1
Figure 1. (a) Time dependence of precipitate radii for the alloy with ( [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. (a) Time evolution of the scaled standard deviation ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Time evolution of scaled standard deviation ( [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Time evolution of scaled standard deviation ( [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Time evolution of scaled standard deviation ( [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Time evolution of scaled standard deviation ( [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    Composite

    F. W. Gayle and J. B. Vander Sande. “Composite” precipitates in an Al-Li-Zr alloy. Scripta Metallurgica, 18:473–478, 1984

    work page 1984

  2. [2]

    B. P. Gu, G. L. Liedl, T. H. Sanders Jr, and K. Welpmann. The influence of zirconium on the coarsening ofδ ′ (Al3Li) in an Al-2.8 wt.% Li-0.14 wt.% Zr alloy.Materials Science and Engineering, 76:147–157, 1985

    work page 1985

  3. [3]

    M. K. Aydinol and A. S. Bor. Coarsening ofδ ′ (Al3Li) and composite precipitates in an Al-2.5 % Li-0.15 % Zr alloy.Journal of Materials Science, 29:15–25, 1994

    work page 1994

  4. [4]

    R. A. Karnesky, M. E. van Dalen, D. C. Dunand, and D. N. Seidman. Effects of substituting rare-earth elements for scandium in a precipitation-strengthened Al–0.08at.%Sc alloy.Scripta Materialia, 55:437–440, 2006

    work page 2006

  5. [5]

    Radmilovic, A

    V. Radmilovic, A. Tolley, E.A. Marquis, M.D. Rossell, Z. Lee, and U. Dahmen. Monodisperse Al 3(LiScZr) core/shell precipitates in Al alloys.Scripta Materialia, 58:529–532, 2008

    work page 2008

  6. [6]

    M. E. van Dalen, D. N. Seidman, and D. C. Dunand. Creep-and coarsening properties of Al-0.06 at.% Sc-0.06 at.% Ti at 300-450 ◦ C.Acta Materialia, 56:4369–4377, 2008

    work page 2008

  7. [7]

    Booth-Morrison, D

    C. Booth-Morrison, D. C. Dunand, and D. N. Seidman. Coarsening resistance at 400◦C of precipitation-strengthened Al-Zr-Sc-Er alloys.Acta Materialia, 59:7029– 20 7042, 2011

    work page 2011

  8. [8]

    M. E. van Dalen, T. Gyger, D. C. Dunand, and D. N. Seidman. Effects of Yb and Zr microalloying additions on the microstructure and mechanical properties of dilute Al–Sc alloys.Acta Materialia, 59:7615–7626, 2011

    work page 2011

Show all 46 references
  1. [9]

    M. E. van Dalen, D. C. Dunand, and D. N. Seidman. Microstructural evo- lution and creep properties of precipitation-strengthened Al–0.06Sc–0.02Gd and Al–0.06Sc–0.02Yb (at. %) alloys.Acta Materialia, 59:5224–5237, 2011

    2011

  2. [10]

    Monachon, M

    C. Monachon, M. E. Krug, D. N. Seidman, and D. C. Dunand. Chemistry and structure of core/double-shell nanoscale precipitates in Al–6.5Li–0.07Sc–0.02Yb (at. Acta Materialia, 59:3398–3409, 2011

    2011

  3. [11]

    Microstructure and mechanical proper- ties of a precipitation-strengthened Al-Zr-Sc-Er-Si alloy with a very small Sc content

    A De Luca, DC Dunand, and DN Seidman. Microstructure and mechanical proper- ties of a precipitation-strengthened Al-Zr-Sc-Er-Si alloy with a very small Sc content. Acta Materialia, 144:80–91, 2018

    2018

  4. [12]

    Precipitation sequence in Al-Sc-Zr alloys revisited.Materialia, 26:101608, 2022

    T Dorin, S Babaniaris, L Jiang, A Cassel, A Eggeman, and J Robson. Precipitation sequence in Al-Sc-Zr alloys revisited.Materialia, 26:101608, 2022

    2022

  5. [13]

    Stabilization of Al 3Zr allotropes in dilute aluminum alloys via the addition of ternary elements.Materialia, 21:101321, 2022

    F Schmid, D Gehringer, T Kremmer, L Cattini, PJ Uggowitzer, D Holec, and S Pogatscher. Stabilization of Al 3Zr allotropes in dilute aluminum alloys via the addition of ternary elements.Materialia, 21:101321, 2022

    2022

  6. [14]

    CN Ekaputra, JU Rakhmonov, D Weiss, J-E Mogonye, and DC Dunand. Mi- crostructure and mechanical properties of cast Al-Ce-Sc-Zr-(Er) alloys strengthened by Al11Ce3 micro-platelets and L1 2 Al3(Sc, Zr, Er) nano-precipitates.Acta Materi- alia, 240:118354, 2022

    2022

  7. [15]

    Effect of deformation on evolution of Al3 (Er, Zr) precipitates in Al-Er-Zr-based alloy.Materials Characterization, 186:111781, 2022

    M Leibner, M Vlach, V Kodetova, H Kudrnova, J Vesel` y, S Zikmund, JˇC´ ıˇ zek, O Me- 21 likhova, and F Luk´ aˇ c. Effect of deformation on evolution of Al3 (Er, Zr) precipitates in Al-Er-Zr-based alloy.Materials Characterization, 186:111781, 2022

    2022

  8. [16]

    Leibner, M

    M. Leibner, M. Vlach, V. Kodetova, J. Vesel` y, J.ˇC´ ıˇ zek, H. Kudrnova, and F. Luk´ aˇ c. On the Sc-rich core of Al 3 (Sc, Er, Zr) precipitates.Materials Letters, 325:132759, 2022

    2022

  9. [17]

    Correlation between precipitates evolution and mechanical properties of Al-Sc-Zr alloy with Er additions

    L Liu, J-T Jiang, X-Y Cui, B Zhang, L Zhen, and SP Ringer. Correlation between precipitates evolution and mechanical properties of Al-Sc-Zr alloy with Er additions. Journal of Materials Science & Technology, 99:61–72, 2022

    2022

  10. [18]

    Er-containing microalloyed aluminium alloys: a review.Journal of Materials Science, 59(22):9685–9696, 2024

    X Wu, M Sun, L Hong, S Wen, W Wei, K Gao, L Rong, X Xiong, H Huang, and Z Nie. Er-containing microalloyed aluminium alloys: a review.Journal of Materials Science, 59(22):9685–9696, 2024

    2024

  11. [19]

    Strength-ductility materials by engineering a coherent interface at incoherent pre- cipitates.Materials Horizons, 11(14):3408–3419, 2024

    D Mao, Y Xie, X Meng, X Ma, Z Zhang, X Sun, L Wan, K Volodymyr, and Y Huang. Strength-ductility materials by engineering a coherent interface at incoherent pre- cipitates.Materials Horizons, 11(14):3408–3419, 2024

    2024

  12. [20]

    Understanding the mechanochemical effect of high-pressure torsion on AA2195-0.025 wt% Sc alloy

    S Mondal, A Panigrahi, N Nayan, SK Makineni, and S Suwas. Understanding the mechanochemical effect of high-pressure torsion on AA2195-0.025 wt% Sc alloy. Advanced Engineering Materials, 27(21):e202500455, 2025

    2025

  13. [21]

    Jiang, Y

    S. Jiang, Y. Xu, R. Wang, X. Chen, C. Guan, Y. Peng, F. Liu, M. Wang, X. Liu, S. Zhang, G. Tian, S. Jin, H. Wang, H. Toda, X. Jin, G. Liu, B. Gault, and J. Sun. Structurally complex phase engineering enables hydrogen-tolerant al alloys.Nature, 641(8062):358–364, 2025

    2025

  14. [22]

    Kong, H.-Y

    Y.-J. Kong, H.-Y. Li, H.-J. Tao, and W.-J. Liu. The precipitation evolution and coarsening resistance of dilute Al-Zr-Er-Yb (-Sc) alloys.Journal of Materials Science & Technology, 224:35–45, 2025. 22

    2025

  15. [23]

    Z. Li, J. Wang, M. Wu, D. Xiao, L. Huang, and W. Liu. Effects of cerium micro- alloying on microstructural evolution and dispersion strengthening in Al-Yb-Er-Zr alloy.Journal of Alloys and Compounds, 1010:177229, 2025

    2025

  16. [24]

    G. A. Baqeri, C. Killmore, L. Smillie, M. Nancarrow, and E. Pereloma. Nanoscale core-shell (Cr, V, Nb) CN precipitation in micro-alloyed steel.Scripta Materialia, 272:117077, 2026

    2026

  17. [25]

    Microalloying strategies enable heat-resistant aluminum alloys via microstructural design at atomic length scale

    C Yang, H Xue, P Zhang, S Wu, G Liu, and J Sun. Microalloying strategies enable heat-resistant aluminum alloys via microstructural design at atomic length scale. Advanced Materials, page e15856, 2026

    2026

  18. [26]

    H Kumar, P Kumar, D Raabe, B Gault, and SK Makineni. High-strength and ductile lightweight cast aluminium alloys with superlattice nano-layered fibres (SNL) and core-shell nano-particles.accepted for publication in Nature Communications, 2026

    2026

  19. [27]

    Radmilovic, C

    V. Radmilovic, C. Ophus, E. A. Marquis, M. D. Rossell, A. Tolley, A. Gautam, M. Asta, and U. Dahmen. Highly monodisperse core–shell particles created by solid- state reactions.Nature Materials, 10:710–715, 2011

    2011

  20. [28]

    Chhotray and A

    A. Chhotray and A. R. S. Gautam. Core-shell precipitation and strengthening in an Al-Li-Yb ternary alloy.Scripta Materialia, 277:117231, 2026

    2026

  21. [29]

    Yin and A

    Y. Yin and A. P. Alivisatos. Colloidal nanocrystal synthesis and the organic– inorganic interface.Nature, 437:664–670, 2005

    2005

  22. [30]

    H. Qian, Y. Zhu, and R. Jin. Size-focusing synthesis, optical and electrochemical properties of monodisperse Au 38(SC2H4Ph)24 nanoclusters.ACS nano, 3:3795–3803, 2009

    2009

  23. [31]

    R. Jin, H. Qian, Z. Wu, Y. Zhu, M. Zhu, A. Mohanty, and N. Garg. Size focusing: a 23 methodology for synthesizing atomically precise gold nanoclusters.The Journal of Physical Chemistry Letters, 1:2903–2910, 2010

    2010

  24. [32]

    focusing

    X. Peng, J. Wickham, and A. P. Alivisatos. Kinetics of II-VI and III-V colloidal semiconductor nanocrystal growth: “focusing” of size distributions.Journal of the American Chemical Society, 120:5343–5344, 1998

    1998

  25. [33]

    H. Reiss. The growth of uniform colloidal dispersions.The Journal of Chemical Physics, 19:482–487, 1951

    1951

  26. [34]

    Sugimoto

    T. Sugimoto. Preparation of monodispersed colloidal particles.Advances in Colloid and Interface Science, 28:65–108, 1987

    1987

  27. [35]

    M. D. Clark, S. K. Kumar, J. S. Owen, and E. M. Chan. Focusing nanocrystal size distributions via production control.Nano Letters, 11:1976–1980, 2011

    1976

  28. [36]

    J. W. Cahn and J. E. Hilliard. Free Energy of a Nonuniform System. I. Interfacial Free Energy.The Journal of Chemical Physics, 28:258–267, 1958

    1958

  29. [37]

    J. W. Cahn. Free Energy of a Nonuniform System. II. Thermodynamic Basis.The Journal of Chemical Physics, 30:1121–1124, 1959

    1959

  30. [38]

    Huang, M

    C. Huang, M. O. de La Cruz, and B. W. Swift. Phase separation of ternary mixtures: Symmetric polymer blends.Macromolecules, 28:7996–8005, 1995

    1995

  31. [39]

    Bhattacharyya and T

    S. Bhattacharyya and T. A. Abinandanan. A study of phase separation in ternary alloys.Bulletin of Materials Science, 26:193–197, 2003

    2003

  32. [40]

    S. K. Makineni, S. Sugathan, S. Meher, R. Banerjee, S. Bhattacharya, S. Kumar, and K. Chattopadhyay. Enhancing elevated temperature strength of copper containing aluminium alloys by forming L 12 Al 3Zr precipitates and nucleatingθ′′precipitates 24 on them.Scientific reports, 7...

    2017

  33. [41]

    L. Q. Chen and J. Shen. Applications of semi-implicit fourier-spectral method to phase field equations.Computer Physics Communications, 108:147–158, 1998

    1998

  34. [42]

    Mukherjee, T

    R. Mukherjee, T. A. Abinandanan, and M. P. Gururajan. Phase field study of precipitate growth: Effect of misfit strain and interface curvature.Acta Materialia, 57(13):3947–3954, 2009

    2009

  35. [43]

    Mukherjee, T

    R. Mukherjee, T. A. Abinandanan, and M. P. Gururajan. Precipitate growth with composition-dependent diffusivity: Comparison between theory and phase field sim- ulations.Scripta Materialia, 62(2):85–88, 2010

    2010

  36. [44]

    Mishra.Core-Shell Precipitates in Ternary Alloys: A Phase-Field Study

    S. Mishra.Core-Shell Precipitates in Ternary Alloys: A Phase-Field Study. PhD thesis, Indian Institute of Science (IISc), Bengaluru, India, 2025

    2025

  37. [45]

    I. M. Lifshitz and V. V. Slyozov. The kinetics of precipitation from supersaturated solid solutions.Journal of Physics and Chemistry of Solids, 19:35–50, 1961

    1961

  38. [46]

    C. Wagner. Theorie der alterung von niederschl¨ agen durch uml¨ osen (Ostwald- reifung).Zeitschrift f¨ ur Elektrochemie, 65:581–591, 1961. 25

    1961

Pith tools

Reviewed July 9, 2026 · model on record in the stance chip above.