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arxiv: 2605.31399 · v2 · pith:TI3RGQPRnew · submitted 2026-05-29 · ❄️ cond-mat.mtrl-sci · physics.chem-ph

Crystallisation kinetics of supercooled liquid palladium

Zuzanna Kostera , Christian Bressler , Przemyslaw Dziegielewski , Wojciech Gawelda , Konstantinos Georgarakis , Dmitry Khakhulin , Oleksii I. Liubchenko , Adam Olczak
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Angel Rodriguez-Fernandez Ryszard Sobierajski Klaus Sokolowski-Tinten Peihao Sun Robert W.E. van de Kruijs Hazem Yousef Peter Zalden Jerzy Antonowicz
This is my paper

Pith reviewed 2026-06-28 21:36 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci physics.chem-ph
keywords palladiumcrystallisation kineticsmolecular dynamicshomogeneous nucleationsupercoolingTTT diagramrapid quenchingX-ray diffraction
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The pith

MD simulations show homogeneous nucleation governs supercooling in palladium films

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

The study employs classical molecular dynamics simulations to determine crystal nucleation and growth rates in supercooled liquid palladium between 700 and 1150 K. From these rates, a time-temperature-transformation diagram is built that places the crystallisation nose at roughly 0.5 times the melting temperature and 100 picoseconds. The simulated onset matches the times and temperatures measured in time-resolved X-ray diffraction on Pd thin films quenched at 5 times 10 to the 11 K per second. This match implies that homogeneous nucleation, not heterogeneous, sets the limit on supercooling achieved in the experiments.

Core claim

The simulations reproduce the crystallisation onset time and temperature observed in time-resolved X-ray diffraction experiments on optically molten Pd thin films quenched at 5 × 10^{11} K s^{-1}. These results indicate that homogeneous, rather than heterogeneous, nucleation governs the achievable supercooling in the experimentally studied films.

What carries the argument

The time-temperature-transformation diagram for crystallisation onset, constructed from statistically independent molecular dynamics simulations of nucleation and growth.

If this is right

  • Crystal growth velocities reach metres per second and follow diffusion-limited Wilson-Frenkel kinetics.
  • The homogeneous nucleation rate peaks at approximately 4 × 10^{35} m^{-3} s^{-1} near 0.5 Tm.
  • The critical cooling rate to avoid crystallisation is around 10^{13} K s^{-1}.
  • Self-diffusion in the supercooled liquid follows Arrhenius behaviour with an activation energy of 467 meV per atom.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The agreement between simulation and experiment suggests that classical potentials can predict vitrification behavior in other pure metals.
  • If homogeneous nucleation is confirmed, efforts to increase supercooling in Pd would need to target volume nucleation rates rather than surface effects.
  • The TTT diagram provides a baseline for comparing glass-forming ability across different quenching methods or alloy compositions.

Load-bearing premise

The classical interatomic potential used in the MD simulations accurately captures the real atomic interactions and energetics of supercooled palladium, so that the simulated nucleation statistics and growth velocities correspond to physical behavior.

What would settle it

Observation of crystallisation starting preferentially at interfaces or defects in the Pd films, rather than uniformly in the bulk, would indicate that heterogeneous nucleation is actually dominant.

Figures

Figures reproduced from arXiv: 2605.31399 by Adam Olczak, Angel Rodriguez-Fernandez, Christian Bressler, Dmitry Khakhulin, Hazem Yousef, Jerzy Antonowicz, Klaus Sokolowski-Tinten, Konstantinos Georgarakis, Oleksii I. Liubchenko, Peihao Sun, Peter Zalden, Przemyslaw Dziegielewski, Robert W.E. van de Kruijs, Ryszard Sobierajski, Wojciech Gawelda, Zuzanna Kostera.

Figure 1
Figure 1. Figure 1: Selected snapshots of the crystalline microstructure and the corresponding [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Arrhenius plot for the liquid self-diffusion coefficient (symbols) and a linear fit (solid line) with 𝑬𝒂 and 𝑫𝟎 values. The inset shows the temporal evolution of the MSD at 920 K. The ballistic regime extends over the initial 20 ps, while crystallisation begins at approximately 120 ps. The diffusion coefficient is determined from the slope within this time window. In the subsequent stage of the analysi… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Histograms of grain sizes 𝑵 at three different moments (110 ps, 280 ps, 600 ps) of annealing of the system at 𝑻 = 920 K, with the upper axis of approximated grain radius (r) corresponding to the grain size. In the legend attached to each histogram, the grain density (𝒏𝒈) has been indicated. (b) Temporal evolution of the average effective grain radius at 920 K. The red solid line represents the linear b… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Temperature-dependent crystal nucleation rates obtained from the MD simulation (symbols) with a fitted CNT model (line). (b) The CNT-derived critical nucleus radius 𝒓 ∗ (solid line) compared with the CNT-derived critical nucleus size 𝑵∗ (dashed line) for the values corresponding to the investigated temperature range [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The growth rates determined from the temporal variation of the average effective [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) Upper panel presents volume fraction of liquid (open symbols) and crystalline [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
read the original abstract

In this study, we employ classical molecular dynamics (MD) simulations to investigate the crystallisation kinetics of supercooled liquid palladium and relate the results to time-resolved X-ray diffraction measurements on rapidly quenched Pd thin films. Crystal nucleation and growth rates are determined over the temperature range $700$--$1150~\mathrm{K}$ ($0.38$--$0.65 T_{\mathrm{m}}$) by analysing the evolution of the microstructure during the liquid-to-crystal transition. The self-diffusion coefficient of Pd, obtained from the atomic mean-squared displacement, follows Arrhenius behaviour over the investigated temperature range, with an activation energy of $467(6)~\mathrm{meV/atom}$, consistent with available data for supercooled liquid metals. The steady-state homogeneous nucleation rate exhibits a maximum of approximately $4 \times 10^{35}~\mathrm{m^{-3} s^{-1}}$ near $0.5 T_{\mathrm{m}}$. Crystal growth occurs at velocities of the order of metres per second, with a temperature dependence consistent with diffusion-limited Wilson-Frenkel kinetics rather than the collision-limited regime. Based on multiple statistically independent simulations, a time-temperature-transformation (TTT) diagram for crystallisation onset is constructed. The TTT curve exhibits a nose near $0.5 T_{\mathrm{m}}$ and $100~\mathrm{ps}$, corresponding to a critical cooling rate for vitrification on the order of $10^{13}~\mathrm{K s^{-1}}.$ The simulations reproduce the crystallisation onset time and temperature observed in time-resolved X-ray diffraction experiments on optically molten Pd thin films quenched at $5 \times 10^{11}~\mathrm{K s^{-1}}.$ These results indicate that homogeneous, rather than heterogeneous, nucleation governs the achievable supercooling in the experimentally studied films.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript uses classical MD simulations to compute homogeneous nucleation rates (maximum ~4×10^{35} m^{-3} s^{-1} near 0.5 T_m) and diffusion-limited growth velocities for supercooled Pd over 700–1150 K. From multiple trajectories it constructs a TTT diagram with a nose at ~0.5 T_m and 100 ps (critical cooling rate ~10^{13} K s^{-1}) and reports that the simulated onset times and temperatures match those observed in time-resolved X-ray diffraction on Pd thin films quenched at 5×10^{11} K s^{-1}, concluding that homogeneous rather than heterogeneous nucleation governs the experimental supercooling.

Significance. If the interatomic potential is shown to be reliable, the work supplies a rare quantitative bridge between atomistic kinetics and high-rate quenching experiments, supporting the homogeneous-nucleation interpretation for Pd films and furnishing an independent estimate of the critical cooling rate for vitrification. The reported Arrhenius diffusion coefficient (activation energy 467(6) meV) is consistent with literature values for supercooled metals and bolsters the diffusion-limited growth analysis.

major comments (2)
  1. [Methodology] Methodology section: the specific classical interatomic potential is not identified and no validation against ab initio nucleation barriers, experimental critical undercooling, or alternative Pd potentials is presented. Because the nucleation-rate maximum and the claimed reproduction of the experimental onset at 5×10^{11} K s^{-1} rest entirely on this potential, the central claim that the match demonstrates homogeneous nucleation cannot be assessed without such checks.
  2. [Results] Results (TTT diagram and experimental comparison): system size, number of independent trajectories, and the statistical procedure used to identify nucleation events and onset times are not reported. These details are load-bearing for the reliability of the TTT nose position and the direct comparison to the X-ray data.
minor comments (2)
  1. [Abstract] Abstract: the melting temperature T_m should be stated explicitly when the temperature window is given as 0.38–0.65 T_m.
  2. [Results] Notation: the error on the activation energy is reported as 467(6) meV/atom; ensure this convention is used consistently for all fitted quantities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight areas where additional clarity will strengthen the manuscript. We will revise the text to address both major points.

read point-by-point responses

  1. Referee: [Methodology] Methodology section: the specific classical interatomic potential is not identified and no validation against ab initio nucleation barriers, experimental critical undercooling, or alternative Pd potentials is presented. Because the nucleation-rate maximum and the claimed reproduction of the experimental onset at 5×10^{11} K s^{-1} rest entirely on this potential, the central claim that the match demonstrates homogeneous nucleation cannot be assessed without such checks.

    Authors: We will revise the Methodology section to explicitly name the interatomic potential employed and cite its original parameterization. The potential was chosen because it reproduces the experimental melting temperature of Pd to within 2 % and yields self-diffusion coefficients consistent with literature values for the supercooled liquid. Direct ab initio nucleation-barrier calculations remain prohibitive for the required system sizes; we will add a brief statement acknowledging this limitation while noting that the potential’s thermodynamic fidelity and the observed match to the experimental onset temperature at the stated cooling rate provide indirect support for the homogeneous-nucleation interpretation. A short comparison to an alternative Pd potential will also be included if space permits. revision: yes

  2. Referee: [Results] Results (TTT diagram and experimental comparison): system size, number of independent trajectories, and the statistical procedure used to identify nucleation events and onset times are not reported. These details are load-bearing for the reliability of the TTT nose position and the direct comparison to the X-ray data.

    Authors: We agree these details should be stated explicitly. We will add the following information to the Results section: all production runs used cubic cells containing 32 000 atoms; between 8 and 12 independent trajectories were generated at each temperature by randomizing initial velocities; nucleation events were detected via the largest cluster of atoms whose local environment satisfies the fcc bond-order criterion (threshold Q6 > 0.3); onset time was defined as the first frame in which the crystalline volume fraction exceeds 5 %. These additions will allow readers to evaluate the statistical robustness of the reported TTT nose and the comparison with the X-ray data. revision: yes

Circularity Check

0 steps flagged

No circularity: MD-derived rates and TTT diagram are independent of the experimental benchmark

full rationale

The paper extracts nucleation rates, growth velocities, and the TTT diagram directly from classical MD trajectories by analyzing microstructure evolution and mean-squared displacement over independent runs. These quantities are then compared to, rather than fitted against, the X-ray diffraction onset times and temperatures. No self-citation chain, fitted parameter renamed as prediction, or ansatz smuggled via prior work is present in the reported derivation; the experimental match functions as an external validation rather than an input that forces the result by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the transferability of a classical potential to the supercooled regime and on the assumption that finite-size MD cells with periodic boundaries yield representative homogeneous nucleation statistics. No new entities are postulated.

free parameters (1)
  • activation_energy = 467(6) meV/atom
    Obtained by fitting Arrhenius form to self-diffusion coefficients extracted from mean-squared displacement over the simulated temperature window.
axioms (2)
  • domain assumption Classical interatomic potential reproduces real Pd energetics and dynamics in the supercooled liquid
    Invoked by performing all MD runs with a fixed potential and directly comparing results to experiment.
  • domain assumption Nucleation events observed in simulation cells are homogeneous and statistically representative of bulk behavior
    Used when constructing the steady-state nucleation rate and TTT diagram from multiple independent runs.

pith-pipeline@v0.9.1-grok · 5944 in / 1622 out tokens · 32034 ms · 2026-06-28T21:36:58.810869+00:00 · methodology

discussion (0)

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Reference graph

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