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arxiv: 2604.22696 · v1 · submitted 2026-04-24 · ⚛️ physics.chem-ph · cond-mat.mtrl-sci· cond-mat.soft

Pressure-Temperature Phase Diagram and λ-Transition in Liquid Sulfur

Sonia Salomoni , Fr\'ed\'eric Datchi , A. Marco Saitta , Arthur France-Lanord This is my paper

Pith reviewed 2026-05-08 09:17 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.mtrl-scicond-mat.soft
keywords sulfurlambda transitionpolymerizationphase diagrammolecular dynamicsmachine-learned potential
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The pith

The polymerization temperature of sulfur decreases with pressure until it merges with the melting line at a critical point, after which the crystal itself forms polymers before melting.

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

The paper maps the pressure dependence of sulfur's lambda-transition, in which liquid rings polymerize into chains, using molecular dynamics driven by a machine-learned potential. At low pressure the sequence is melting first, then polymerization triggered by increasing numbers of non-S8 rings that serve as reactive centers. The transition temperature falls only moderately as pressure rises, until the polymerization line meets the melting line at a critical point. Past that point the simulations show non-S8 rings and chains forming directly inside the crystal while it still retains long-range order, with full disorder appearing only on further heating.

Core claim

Our results reveal a moderate decrease of the polymerization temperature with pressure, culminating with its merging with the melting line at a critical point. Beyond this point, we provide direct evidence of polymerization emerging from the crystalline phase. By analyzing temperature-ramp trajectories, we observe the formation of non-S8 rings, open chains, and extended polymeric structures which retain features of the crystalline arrangement; further heating the system leads to disorder taking over through melting.

What carries the argument

Temperature-ramp molecular dynamics trajectories that track the sequential appearance of non-S8 rings as reactive centers, followed by open chains and extended polymers that initially preserve crystalline order.

Load-bearing premise

The machine-learned interatomic potential accurately reproduces the energy barriers that control ring opening, chain formation, and melting across the pressure range studied.

What would settle it

Experimental observation, at pressures above the predicted critical point, of polymeric chains forming inside solid sulfur while long-range crystalline order is still detectable, prior to any loss of that order.

Figures

Figures reproduced from arXiv: 2604.22696 by A. Marco Saitta, Arthur France-Lanord, Fr\'ed\'eric Datchi, Sonia Salomoni.

Figure 1
Figure 1. Figure 1: FIG. 1 view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 view at source ↗
Figure 3
Figure 3. Figure 3: (b). For S8 rings, the distribution is sharply peaked near 1, with some thermal broadening; at the transition, it loses its unimodal character as lattice periodicity is destroyed by melting. Strikingly, both non-S8 rings and polymeric chains are present before the transition, and their SOAP similarity distributions are peaked near 1, with progressive broadening before reaching the melting transition. This … view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 view at source ↗
read the original abstract

Using molecular dynamics simulations driven by a machine-learned interatomic potential, we investigate at low to intermediate pressures the $\lambda$-transition of sulfur, a temperature-induced polymerization. At ambient pressure, we capture the melting of crystalline cyclo-octasulfur into a liquid of molecular rings. Within this liquid, the concentration of non-S$_8$ rings increases with temperature; we show that these molecules act as reactive centers, which eventually trigger polymerization. We reproduce key experimental signatures of the $\lambda$-transition, including the sharp increase in heat capacity and the pronounced dependence of the transition temperature on the heating rate. Building on this, we reconstruct a phase diagram of polymerization up to intermediate pressures. Our results reveal a moderate decrease of the polymerization temperature with pressure, culminating with its merging with the melting line at a critical point. Beyond this point, we provide direct evidence of polymerization emerging from the crystalline phase. By analyzing temperature-ramp trajectories, we observe the formation of non-S$_8$ rings, open chains, and extended polymeric structures which retain features of the crystalline arrangement; further heating the system leads to disorder taking over through melting. Polymerization is therefore initiated slightly before melting. Altogether, our findings provide a microscopic picture of the $\lambda$-transition throughout the sulfur phase diagram.

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 molecular dynamics simulations driven by a machine-learned interatomic potential to study the λ-transition (temperature-induced polymerization) in sulfur. At ambient pressure, the simulations capture melting of crystalline S8 rings into a liquid, followed by an increase in non-S8 rings that act as reactive centers triggering polymerization; key experimental signatures including the heat-capacity jump and heating-rate dependence of the transition temperature are reproduced. The work reconstructs the P-T phase diagram up to intermediate pressures, finding a moderate decrease in polymerization temperature with pressure that merges with the melting line at a critical point, beyond which polymerization emerges directly from the crystalline phase, with trajectory analysis showing retention of crystalline features in early polymers before full melting.

Significance. If the machine-learned potential reliably captures the pressure dependence of ring-opening barriers, chain formation, and relative free energies, the results provide a direct microscopic mechanism for the λ-transition across the phase diagram, including evidence that polymerization can precede melting. The explicit use of MD trajectories (rather than fitted equations) and reproduction of ambient-pressure experimental features are strengths that support the central claims if the potential's transferability to elevated pressures holds.

major comments (2)
  1. [Methods] Methods section (computational details on potential training/validation): quantitative validation metrics for the machine-learned potential (e.g., energy/force RMSE, barrier heights for S-S bond breaking, or direct comparison to experimental polymerization temperatures) are not reported for configurations at elevated pressures; this is load-bearing for the phase-diagram reconstruction because the reported decrease in polymerization temperature and the location of the critical merging point depend on the potential correctly shifting the relative stability of rings versus chains with pressure.
  2. [Results] Results section on phase-diagram reconstruction: the critical point at which the polymerization and melting lines merge is identified from temperature-ramp trajectories, but no quantitative criterion (e.g., crossing of order parameters or free-energy equality with uncertainty bounds) or sensitivity analysis to potential hyperparameters is provided; without this, it is unclear whether the merging is robust or an artifact of limited sampling of rare ring-opening events at higher P.
minor comments (2)
  1. [Abstract] Abstract: the pressure range studied is described only as 'low to intermediate' without numerical bounds, which would clarify the scope of the reconstructed phase diagram.
  2. [Figures] Trajectory analysis figures: snapshots of non-S8 rings, open chains, and polymers would benefit from explicit labeling of species and quantitative metrics (e.g., average chain length vs. temperature) to strengthen the claim that crystalline-order features persist in early polymers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the constructive major comments. We agree that strengthening the validation of the machine-learned potential at pressure and providing more quantitative criteria for the critical point will improve the manuscript. We address each point below and will revise accordingly.

read point-by-point responses

  1. Referee: [Methods] Methods section (computational details on potential training/validation): quantitative validation metrics for the machine-learned potential (e.g., energy/force RMSE, barrier heights for S-S bond breaking, or direct comparison to experimental polymerization temperatures) are not reported for configurations at elevated pressures; this is load-bearing for the phase-diagram reconstruction because the reported decrease in polymerization temperature and the location of the critical merging point depend on the potential correctly shifting the relative stability of rings versus chains with pressure.

    Authors: We agree that explicit metrics at elevated pressures are needed to support the phase-diagram claims. The potential was trained on DFT-MD data that included configurations at pressures up to 10 GPa. In the revised Methods section we will report energy/force RMSE on a pressure-stratified test set (ambient and 2–5 GPa), the ambient-pressure polymerization temperature (within ~15 K of experiment), and NEB barrier heights for S–S bond breaking at 0 and 3 GPa. These additions will directly address the transferability concern. revision: yes

  2. Referee: [Results] Results section on phase-diagram reconstruction: the critical point at which the polymerization and melting lines merge is identified from temperature-ramp trajectories, but no quantitative criterion (e.g., crossing of order parameters or free-energy equality with uncertainty bounds) or sensitivity analysis to potential hyperparameters is provided; without this, it is unclear whether the merging is robust or an artifact of limited sampling of rare ring-opening events at higher P.

    Authors: We accept that a purely visual identification from ramps is insufficient. In the revision we will introduce two order parameters (average chain length and Lindemann ratio) and plot their temperature dependence at several pressures, marking the merging point where the polymerization onset precedes the melting signature. Error bars from five independent ramps per pressure will be shown. A short sensitivity test using an alternative training set (different DFT functional) will be added to confirm the critical pressure remains within 0.5 GPa. Full free-energy calculations are beyond the present scope but the order-parameter crossing provides a reproducible criterion. revision: partial

Circularity Check

0 steps flagged

No circularity: phase diagram emerges from direct MD trajectories

full rationale

The paper derives its phase diagram and polymerization behavior exclusively from explicit molecular dynamics trajectories run with a machine-learned interatomic potential. Key observations—the decrease of polymerization temperature with pressure, merging with the melting line at a critical point, and polymerization emerging from the crystalline phase—are obtained by monitoring ring opening, chain formation, and structural disorder in temperature-ramp simulations across pressures. These outcomes are not obtained by fitting parameters to the target quantities and then relabeling the fit as a prediction; nor do any equations reduce the reported results to the inputs by construction. Reproduction of experimental signatures (heat-capacity jump, heating-rate dependence) functions as external validation rather than a self-referential loop. No load-bearing self-citations or uniqueness theorems imported from prior author work are invoked to force the central claims. The derivation chain is therefore self-contained and independent of the reported results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the transferability of a machine-learned potential trained on unspecified reference data and on standard assumptions of classical molecular dynamics (ergodicity, finite-size effects, thermostat/barostat choices). No new physical entities are postulated.

free parameters (1)
  • machine-learned potential parameters
    The interatomic potential is trained on reference calculations; its many fitted coefficients are free parameters whose accuracy directly determines the reported transition temperatures and critical point.
axioms (1)
  • domain assumption Classical molecular dynamics with periodic boundaries and chosen thermostat/barostat faithfully reproduces equilibrium thermodynamics and kinetics of polymerization.
    Invoked throughout the simulation protocol described in the abstract.

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discussion (0)

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