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First-Principles Atomistic Structure and Dynamics of Polyethylene During High-Pressure Radical Polymerization via Machine Learning Force Fields
Bharatha K. Gunawardana, Teresa Shah, Bicha Azizova, Deepa Ranabhat, Yizhi Song, Akshath Shastri, Srinjoy Ghose, Thomas E. Gartner, Hsin-Yu Ko
TL;DR
Atomistic polymer structures and reactive interactions are difficult to obtain experimentally or simulate at macromolecular scales with conventional methods. This work combines SeA-enabled vdW-corrected hybrid DFT with a DP machine-learning force field trained on solvated PE oligomers. The resulting model shows convergent radical solvation beyond n~6 and good-solvent molecular-weight scaling in longer PE chains.
Problem
Experimental atomistic structures are limited for amorphous and semi-crystalline polymers, while empirical force fields can lack chemical realism for reactive and electronically sensitive processes.
Method
The authors train a DP MLFF on PBE0+D3 hybrid-DFT data for solvated PE radical oligomers using SeA and DP-Gen active learning.
Results
The radical solvation environment converges with chain length, and longer-chain structure and dynamics approach the good-solvent limit of approximately 6.25 for the reported ratio.
Takeaways & Limitations
The oligomer-trained PE MLFF is extensible to macromolecular chains while retaining atomistic first-principles-based modeling across thermodynamic state points and chain lengths.
Abstract
from arXiv · showhide
Polyethylene (PE) is one of the most commonly used synthetic polymers. While the synthesis and processing protocols for PE are well established, precise experimental assignment of microscopic structures at atomistic resolution (i.e., the position of each atom) remains largely limited to highly crystalline systems. This gap is often addressed via computer simulations using empirical interatomic potentials, which use approximate but efficient descriptions of interatomic interactions to reach the length and time scales needed to describe macromolecules. These empirical potentials typically perform well for bulk and/or collective properties but face challenges with chemical realism for complex systems, e.g., during reactive processes. In this work, we address this challenge by combining the computational efficiency of a deep potential (DP) machine-learning force field and the chemical realism of first-principles van der Waals (vdW) corrected hybrid density functional theory (DFT) enabled by a SeA high-throughput framework. Using this approach, we study the structure and dynamics of PE oligomers and polymers in an ethylene solvent under common high-pressure (supercritical) radical polymerization conditions. We found that the local solvation environment of radical-containing PE oligomers converges for chain lengths greater than (n~6), suggesting extensibility of our oligomer-trained MLFF to significantly longer polymers. We then confirmed the extensibility of these models to long PE chains by characterizing the molecular weight scaling of single-chain structure and dynamics, which showed classic good solvent behavior. Our PE MLFF retained a consistent level of fidelity and stability across a wide range of thermodynamic state points and chain lengths, at full atomistic resolution, therefore paving the way towards first-principles-based polymer structure and property prediction.
I. INTRODUCTION
Atomistic polymer structures are difficult to resolve experimentally and expensive to simulate with chemically realistic first-principles methods. The paper addresses this gap by combining hybrid DFT with vdW corrections, SeA acceleration, and DP machine-learning force fields for polymer-scale simulations.
- Motivation: Experimental atomistic structures of amorphous and semi-crystalline polymers remain largely limited to angularly averaged radial distributions from scattering.The lack of long-range order restricts direct three-dimensional structural assignment at angstrom-to-nanometer resolution.
- Motivation: Traditional empirical force fields often miss reactivity, polarizability, strong polymer–ion interactions, and other electronically sensitive phenomena.These limitations motivate force fields trained against first-principles atomic energies and forces.
- Computational challenge: Standard AIMD and DFT simulations remain incompatible with the length and time scales required for macromolecular phenomena.Consequently, polymer AIMD studies have largely focused on highly crystalline systems for computational tractability.
- Computational challenge: vdW-corrected hybrid DFT is needed for reliable condensed-phase polymer interactions, but exact exchange makes direct large-scale simulations computationally prohibitive.SeA reduces the hybrid-DFT burden by exploiting sparsity in localized-orbital representations.
- Open challenges: Polymer MLFFs still face unresolved issues involving non-covalent interactions, representative datasets, stable long-time dynamics, and interpretability across multiple scales.Limited transferability across polymer chemistries remains an additional challenge.
- Approach: DP molecular dynamics provides 10^6×−10^8× speedup at large length and time scales while retaining DFT accuracy, with nearly linear cost scaling in system size.The study applies this strategy to PE oligomer radicals in ethylene solvent under high-pressure polymerization conditions and examines solvation convergence with chain length.
II. TRAINING THE DEEP POTENTIAL MACHINE-LEARNING FORCE FIELD
The PE force field combines PBE0+D3 hybrid DFT data generated through SeA with DP-Gen active learning. Training uses solvated PE radical oligomers from monomer through six-mer across a broad thermodynamic range and produces a large labeled dataset.
- Training approach: The MLFF uses PBE0+D3 hybrid DFT integrated with the SeA hybrid-DFT engine and the DP software ecosystem.This combination targets polymer-scale system sizes while retaining hybrid-DFT chemical accuracy.
- Workflow: The workflow integrates SeA and DP ecosystems for MLFF training.Figure 1 schematically presents this integration.
- Training data: Training oligomers span n = 1 to n = 6, from C2H5 · to C12H27 ·, in ethylene solvent.The sampled temperatures are T ∈{200, 400, 600} K and densities are ρ ∈{0.43, 0.49, 0.57, 0.66} g/cm3.
- Training data: 99,074 labeled PBE0+D3 structures were generated for PE oligomers spanning n = 1−6.Small unit cells intentionally encode chain–chain interactions through nearby periodic images.
III. THERMODYNAMIC PROPERTIES AND SOLVATION STRUCTURE OF THE ETHYL RADICAL
The study validates finite-size choices for ethyl-radical simulations and characterizes radical solvation under supercritical polymerization conditions. Small and enlarged cells give consistent thermodynamic and structural results, while radical electrons interact with nearby ethylene molecules.
- Finite-size effects: 32-molecule unit cells produce isobars in good agreement with 256-molecule supercells, indicating minimal finite-size effects over the sampled temperature and density range.The comparison covers T = 100−600 K and ρ = 0.43−0.66 g/cm3.
- Polymerization conditions: Experimental high-pressure polymerization conditions approximately correspond to T ≥473 K and ρ ≥0.43 g/cm3.This mapping follows from the sampled isobars and the experimental conditions T ≥200 °C and p ≥500 bar.
- Finite-size effects: The 32-molecule and 256-molecule cells show essentially identical C–C, C–H, H–H, and C · –C radial distribution functions.The study therefore retains the 32-molecule unit cell for subsequent PE oligomer analysis.
- Radical stability: Radicals remain stable over 100 ps, with no association observed among eight radicals in the 256-molecule simulations.Radical pairs can nevertheless approach within ≈3 Å, consistent with a first-neighbor residence duration of 10−11s to 10−10s.
- Solvation structure: At T = 500 K and ρ = 0.57 g/cm3, radical electrons delocalize to 1–3 neighboring ethylene molecules within ≈4.0 Å.The observed interaction indicates a fairly strong radical–π interaction, and approximately 2 ± 1 neighboring molecules strongly interact with the radical.
IV. SOLVATED PE OLIGOMER RADICALS
As PE oligomer radicals lengthen, intramolecular signatures strengthen in the C–C radial distribution function, while the radical’s local solvation environment rapidly converges. This convergence supports extending oligomer-trained models toward polymeric lengths, though chain–chain environments remain limited in solvated oligomer samples.
- Chain-length-dependent structure: Growing chain length strengthens C–C RDF signatures from longer oligomer C–C bonds and β-position carbon pairs near r = 2.5 Å.The corresponding C–H and H–H changes are less visible because indirect bonding, bond rotation, and intermolecular contributions smear these distances.
- Solvation structure: Minor differences in the C · –C RDF across chain lengths indicate similar radical solvation structures for short PE oligomers.The radical electron can delocalize to 1–3 neighboring ethylene molecules within approximately 4.0 Å of the radical carbon.
- Solvation structure: The local chemical environment around the radical converges rapidly with chain length, consistent with Flory’s principle of equal reactivity.This convergence indicates that the oligomer-trained DP MLFF may extend to PE polymers.
- Solvation structure: Representative solvation structures compare radical oligomers with n = 2, n = 4, and n = 6 under T = 500 K and ρ = 0.57 g/cm3.The structures use the graphical representation established for Fig. 3.
- Scope and caveat: Solvated oligomer samples lack typical interactions between segments of polymer chains, although periodic images can introduce some inter-segment configurations.Initial multi-chain tests suggest the training procedure may include sufficient configurations for extrapolation to PE melts or dense solutions, but detailed study remains future work.
V. EXTENSIBILITY OF THE OLIGOMER-TRAINED MLFF TO POLYMERS
The study tests whether an oligomer-trained MLFF remains stable when applied to substantially longer PE chains. One-nanosecond DPMD simulations span n = 5, n = 33, and n = 257 systems at 500 K and 4 kbar.
- Extensibility testing: The simulations characterize whether the oligomer-trained model remains stable as PE chain length increases from n = 5 to n = 257.The supplied passage identifies the 1-ns simulations as the test of extensibility; detailed numerical outcomes are not included here.
- Stability assessment: Figure 6 evaluates MLFF stability across n = 5, n = 33, and n = 257 in a 512-molecule unit cell.Each chain-length condition includes a simulation snapshot and a model-deviation distribution compared with the active-learning accuracy threshold.
- Extensibility testing: The oligomer-trained MLFF was tested on PE systems with n = 5, n = 33, and n = 257, spanning oligomer to polymeric chain lengths.These correspond to 141, 927, and 7,211 g/mol radicals, respectively.
- Simulation protocol: 1-ns DPMD simulations used the NpT ensemble at T = 500 K and p = 4 kbar for each chain length.The pressure was selected to account for pressure reduction as polymerization replaces intermolecular van der Waals contacts with intramolecular contacts.
VI. MACROMOLECULAR STRUCTURE AND DYNAMICS FROM FIRST PRINCIPLES
Single-chain DPMD simulations reproduce good-solvent polymer scaling for PE chains while predicting somewhat smaller dimensions, earlier flexible behavior, and faster relaxation than L-OPLS.
- DPMD simulations evaluated solvated PE chains at T = 500 K and ρ = 0.56 g/cm3, comparing the DP MLFF with L-OPLS.The largest DPMD system contained 43,904 atoms, and combined sampling exceeded 350 ns.
- v = 0.61 for the PBE0+D3 DP MLFF and v = 0.63 for L-OPLS, both close to the good-solvent value v ≈0.5877.The end-to-end distance follows ⟨Re⟩∼n^v.
- The n = 5 chain was excluded from scaling regression because short-range conformational constraints strongly influence its elongated configuration.The DP MLFF also predicts somewhat smaller chain dimensions than L-OPLS, potentially reflecting greater flexibility or less favorable polymer-solvent interactions.
- The ⟨R2g⟩ ratio decreases with chain length toward the good-solvent limit of approximately 6.25.This trend reflects diminishing short-chain backbone-stiffness effects as chains approach coil-like excluded-volume statistics.
- The DP MLFF yields lower intermediate-chain ⟨R2g⟩ ratios than L-OPLS, indicating an earlier onset of flexible behavior.L-OPLS retains a more extended conformation at comparable chain lengths.
- Both models show polymer-like relaxation scaling near the Zimm prediction of ≈1.8, while DP MLFF relaxation times are generally shorter than L-OPLS.The difference is only qualitatively resolved because accessible DPMD timescales limit the accuracy of relaxation estimates.
VII. CONCLUSIONS AND FUTURE WORK
The study develops a first-principles-trained PE DP MLFF and finds it extensible from solvated oligomer radicals to macromolecular chains, with future applications in complex polymer processes.
- The authors developed an active-learning workflow training a PE DP MLFF on condensed-phase, monomer-solvated oligomers using PBE0+D3 hybrid DFT.
- Rapid convergence of nuclear and electronic solvation environments around radical sites with chain length supports extending the oligomer-trained model to macromolecular PE.
- The workflow includes chain–chain interactions through oligomer interactions with periodic images in intentionally small training cells.
- The approach is proposed for extension to other macromolecules and for studying polymer synthesis, upcycling, recycling, and degradation.
Appendix A: Computational Details
Training-data electronic-structure calculations used the SeA high-throughput hybrid DFT engine with Grimme DFT-D3 dispersion correction and specified Quantum ESPRESSO numerical settings.
- All training-data electronic-structure calculations used SeA hybrid DFT with Grimme DFT-D3 dispersion correction in Quantum ESPRESSO.
- Calculations used a simple cubic C64H129 cell with periodic boundary conditions and ONCV pseudopotentials from the SG15 collection.
- The numerical setup used a 120 Ry plane-wave cutoff, Γ-point sampling, and an SCF convergence threshold of 10^-6 Ry.
2. Initial Data Generation
Initial training data were generated from AIMD of ethylene containing an ethyl radical, then expanded to longer PE oligomers through constrained and unbiased AIMD configurations.
- Initial AIMD simulations modeled liquid ethylene doped with an ethyl radical, equivalent to a PE oligomer of length n = 1.
- Replacing H with D does not affect classical-dynamics structural properties or the relationship between atomic coordinates and potential energies or forces.
- For n = 2, constrained AIMD reduced the distance between the ethyl-radical carbon and a carbon in a neighboring ethylene molecule.
- The resulting n = 2 configurations were followed by unbiased NVT AIMD data generation across the selected conditions.
3. Active Learning
The active-learning workflow iteratively trains, tests, and refines DP models for PE oligomers across temperatures and densities. It produced a four-model ensemble meeting the stated accuracy target, while later stages required refined training settings for stability.
- Active-learning workflow: The workflow trains a DP model, explores configurations with molecular dynamics, and labels selected candidates using first-principles electronic-structure calculations.DeepMD-kit, LAMMPS, and SeA-enabled Quantum ESPRESSO form the integrated three-step cycle.
- Active-learning workflow: Active learning progressively increased the exploration temperature from 200 K to 400 K and then 600 K after convergence.
- Training coverage and accuracy: The protocol covered PE oligomers with n = 1−6 across densities ρ ∈ {0.43, 0.49, 0.57, 0.66} g/cm3 and obtained four DP MLFFs with %acc ≥98%.%acc is based on configurations whose model deviation is below 0.20 eV/Å.
- Candidate selection: 99,074 PBE0+D3 data points were generated by relabeling candidate structures whose model-deviation values fell within [0.20, 0.35] eV/Å.The deviation is the maximum standard deviation of predicted atomic forces across four equivalent DP models.
- Training stability: Stage 3 accuracy became unstable as the dataset grew, so Stages 4–5 used a refinement setting with 2M training steps to stabilize the DP MLFF.
Appendix B: Chain Scaling Simulations
Chain-scaling simulations compared the developed DP MLFF with the L-OPLS force field for solvated PE chains of increasing length. Structural relaxation was measured from end-to-end unit-vector autocorrelation under specified canonical-ensemble conditions.
- Chain lengths and force fields: The simulations evaluated PE chains with n = 5, 17, 33, and 65 using both DP and L-OPLS, while L-OPLS also extended to n = 129.The corresponding simulation cells contained 632, 4,640, 20,720, 43,904, and 72,116 total atoms.
- Simulation conditions: Chain-scaling simulations used the canonical (NV T) ensemble at T = 500 K and ρ = 0.56 g/cm3 with a 1 fs integration timestep.Nosé–Hoover thermostating was used with τdamp = 100 fs and chain length 3.
- Relaxation analysis: Structural relaxation was quantified with the end-to-end unit-vector autocorrelation C(t) = ⟨u(t) · u(0)⟩, whose decay tracks loss of conformational memory.A steeper decay indicates faster structural decorrelation.