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Structure and polymerization of liquid sulfur across the λ-transition

Machine-learned interatomic potentials (DeepMD-style), CP2K reference DFT, and on-the-fly probability enhanced sampling (OPES) with a graph-spectral “Deep-TDA” collective variable are used to study ring–polymer equilibria and reaction mechanisms in liquid sulfur near the λ-transition (\(T_\lambda \approx 432\) K at 1 atm).

Summary

The paper targets the λ-transition of liquid sulfur, associated with conversion of S\(_8\) rings to polymeric chains. Reference data use CP2K AIMD (PBE, GTH pseudopotentials, m-DZVP + plane waves, 300 Ry density cutoff for AIMD; 350 Ry with D3 for training snapshots) and single-point training on 128- and 512-atom cells. ML potentials are trained with DeepMD-kit and driven in LAMMPS; enhanced sampling uses PLUMED with OPES (well-tempered target, kernel deposition frequency 250, barrier-related parameters in the 100–200 kJ/mol range as stated) and a neural collective variable from Deep-TDA / mlcolvar, built from smoothed adjacency-matrix eigenvalue histogram inputs. Unbiased NN-driven MD (NVT, 1 fs timestep, global velocity rescaling thermostat, τ = 0.05 ps) evaluates \(g(r)\), \(S(k)\), and displacement statistics on 3456-atom and 512-atom cells; biased runs at 512 atoms explore polymerization pathways. Bader charges from DFT densities support mechanistic interpretation.

Methods

  • Electronic structure and datasets: CP2K 9.2; PBE; GTH pseudopotentials; AIMD in NPT (2 fs timestep, Nosé–Hoover thermostat and barostat); training snapshots with higher cutoff and D3; Bader analysis for charges on selected mechanistic snapshots.
  • NN potentials: DeepMD-kit training; LAMMPS integration; attention-based Deep Potential variant described in the article.
  • Enhanced sampling: PLUMED + OPES along Deep-TDA CV; adjacency matrix from pairwise cutoff switching (e.g., \(d_\text{cutoff} \approx 2.6\) Å for sulfur bonds), eigenvalue histogram featurization, NN architecture [100, 64, 32, 1], ReLU, two-state training data from ring-rich vs polymer-rich samples.
  • Structural analysis: Radial distribution functions and structure factors compared to experiment via linear mixing of pure ring and pure polymer references; Gaussian broadening noted for \(g(r)\) comparison; displacement histograms over 10–100 ps at several ring fractions.

1 — MD application (MLIP + reference AIMD). Unbiased NVT LAMMPS runs (1 fs timestep, global velocity rescaling thermostat, τ = 0.05 ps) on 512- and 3456-atom periodic bulk liquid sulfur; CP2K NPT AIMD (2 fs, Nosé–Hoover thermostat and barostat) for reference trajectories and training data (article). N/A — no external electric field in the summarized protocol. N/A — no metadynamics or replica exchange in the brief list here; rare ring–chain events are targeted with OPES (see below).

2 — Machine-learned potential (MLIP). DeepMD-kit training on CP2K PBE snapshots; attention-based Deep Potential as described in the article (not a ReaxFF parameterization in the AGENTS sense).

3 — Static QM (CP2K DFT). PBE with GTH pseudopotentials and m-DZVP + plane waves; 300 Ry density cutoff (AIMD) and 350 Ry with D3 for select training; Bader charges on path frames.

4 — Enhanced sampling. PLUMED with on-the-fly probability-enhanced sampling (OPES, well-tempered target) along a graph-spectral Deep-TDA neural CV; kernel deposition frequency 250; barrier-related OPES parameters in the 100–200 kJ/mol range as reported (article).

Findings

  1. Mixed \(g(r)\) and \(S(k)\) along experimental ring-fraction tracks match X-ray data around \(T_\lambda\), including evolution of the ~4.5 Å \(g(r)\) third peak linked to S\(_8\) neighbors; a small-\(k\) pre-peak in \(S(k)\) is less pronounced than experiment, attributed partly to finite system size.
  2. Atomic mobility drops as polymer content increases, with growing population of sub-2 Å displacements consistent with sluggish polymer segments and elevated viscosity above \(T_\lambda\).
  3. Polymerization mechanisms resemble ring-opening polymerization: thermal opening of an S\(_8\) ring polarizes terminal atoms; subsequent attack on other rings propagates chains; depolymerization pathways are also sampled (figures and ESI schemes).
  4. The approach is presented as extending prior short AIMD by combining ab initio–accurate potentials with rare-event sampling for connectivity-changing sulfur chemistry.

Corpus honesty — Parrinello-group work, not ReaxFF; see ## Limitations for ML fidelity and finite size.

Limitations

Finite cell sizes and ML potential fidelity to full DFT for all transition states remain approximate; experimental ring-fraction assignments retain uncertainty that the authors acknowledge.

Relevance to group

Complementary methodology (MLIPs + OPES) for reactive liquids; not a van Duin / ReaxFF study but relevant to reactive MD and enhanced-sampling benchmarks in the corpus.

Citations and evidence anchors