Structure and polymerization of liquid sulfur across the $λ$-transition

This study employs machine learning potentials combined with enhanced sampling techniques in molecular dynamics simulations to elucidate the structural evolution and dynamical mechanisms of the ring-to-chain polymerization and reverse processes in liquid sulfur across the $\lambda$-transition, yielding results consistent with experimental observations.

The Gist

Imagine a pot of liquid sulfur. At first glance, it looks like just a hot, yellow goo. But if you zoom in with a super-powerful microscope, you'd see a bustling city of atoms.

In this city, the sulfur atoms are social creatures. They have two main ways of hanging out:

1. The Ring Party: They hold hands in perfect circles of eight, forming little crowns (like a necklace of eight friends).
2. The Chain Gang: They link up in long, winding lines, stretching out like a massive snake.

The Mystery of the "Lambda" Switch

There's a specific temperature (about 159°C or 432 K) where something weird happens. Scientists call this the $\lambda$-transition.

Below this temperature, the sulfur is mostly made of those little Ring Parties. It flows easily, like water. But as soon as you heat it past this point, the rings start breaking open and grabbing onto neighbors to form Chain Gangs. Suddenly, the liquid gets thick, sticky, and sluggish—like honey turning into tar.

For decades, scientists knew that this happened, but they didn't know exactly how the rings broke apart and how the chains formed. It was like watching a magician pull a rabbit out of a hat, but you couldn't see the trick.

The Problem: It's Too Fast and Too Big

To figure out the trick, you need to watch the atoms move. But there are two big problems:

1. It's too expensive: Using the most accurate computer models (called "first principles") to watch these atoms is like trying to film a movie by calculating every single photon of light. It takes so much computing power that you can only watch a tiny drop of sulfur for a split second.
2. It's too slow: The actual process of a ring breaking and a chain forming happens so rarely that a standard computer simulation would have to run for years to see it happen just once.

The Solution: A Smart "AI" Assistant

The authors of this paper, Manyi Yang, Enrico Trizio, and Michele Parrinello, came up with a clever workaround. They didn't try to calculate every single atom's movement from scratch. Instead, they built a Machine Learning (AI) assistant.

Think of it like this:

• They taught the AI by showing it a few examples of how sulfur atoms behave (using the expensive, accurate method).
• The AI learned the "rules of the game" (how atoms attract and repel each other).
• Once the AI was smart enough, they let it run the simulation. The AI could predict the atoms' movements almost instantly, allowing them to watch thousands of atoms for nanoseconds (which is a long time in the atomic world).

The Secret Weapon: The "Topology" Map

To make sure the AI was actually watching the rings turn into chains, they needed a special way to measure the shape of the sulfur. They couldn't just look at distances; they needed to understand the connectivity.

They used a concept from Graph Theory (the math of networks). Imagine the sulfur atoms are cities and the bonds between them are roads.

• Rings look like a perfect circle of roads.
• Chains look like a long, straight highway.

They created a "Topological Map" (a special mathematical score) that tells them instantly: "Is this a ring? Is this a chain? Or is it a mix?" This allowed them to guide the simulation to force the rings to break and the chains to form, so they could study the process in detail.

What They Discovered: The "Charge" Trick

By watching the simulation in slow motion, they found out exactly how the magic happens. It all comes down to electric charge.

1. The Ring Breaks: Sometimes, a ring gets so hot and wiggly that it stretches and snaps open. When it snaps, the two ends become "unhappy" (they have a negative electric charge).
2. The Search: These unhappy, charged ends are like magnets looking for a partner. They swing around and grab onto another ring nearby.
3. The Chain Grows: When they grab the new ring, they pull it open too. Now you have a longer chain with two new unhappy ends. The process repeats, and the chain grows longer and longer.

The Reverse Process (Cooling Down):
When you cool the sulfur back down, the chains want to break. They found two ways this happens:

• The Tail Tuck: The end of the chain swings around and tucks itself back into the chain to form a circle again.
• The Middle Snap: Sometimes, a loop forms right in the middle of the chain, pinches off, and becomes a ring, leaving two shorter chains behind.

Why This Matters

This paper is like finally getting the "behind-the-scenes" footage of a magic trick. By combining AI with advanced math, the scientists were able to:

• Confirm that their computer models match real-world experiments perfectly.
• Explain why liquid sulfur gets so sticky when heated (it's turning into long chains).
• Show exactly how the atoms rearrange themselves, driven by tiny electrical imbalances.

It's a beautiful example of how modern technology (AI and machine learning) is helping us solve old, stubborn mysteries in nature, turning a "black box" of chemical reactions into a clear, understandable story.

Technical Summary

1. Problem Statement

Liquid sulfur exhibits a complex phase diagram characterized by a unique λ-transition at approximately 432 K and ambient pressure. This transition involves a dramatic shift in physical properties (e.g., a massive increase in viscosity, heat capacity, and density) driven by the transformation of stable eight-membered sulfur rings ($S_8$) into long polymeric chains ($S_\infty$).

Despite extensive experimental study, the underlying dynamical mechanism of this polymerization and depolymerization remains poorly understood. Previous theoretical attempts using ab initio molecular dynamics (AIMD) were limited by computational cost, restricting simulations to:

• Short timescales (hundreds of picoseconds), insufficient for observing rare reactive events.
• Small system sizes (few hundred atoms), inadequate for capturing the slow dynamics of polymeric systems.
• Lack of detailed mechanistic insight into bond breaking/formation and charge redistribution during the transition.

2. Methodology

The authors employed a hybrid approach combining Machine Learning Potentials (MLPs) with Enhanced Sampling (ES) techniques to overcome the limitations of standard AIMD.

A. Machine Learning Potentials (MLPs)

• Framework: They utilized Deep Potential (DeepMD) with an attention-based scheme to model interatomic interactions.
• Training Data: The neural network (NN) was trained on energies and forces derived from Density Functional Theory (DFT) calculations (PBE functional).
• Active Learning: To ensure the potential covers reactive regions (transition states) and not just metastable minima, they employed an active learning loop:
  1. Initial AIMD simulations collected configurations at various temperatures/pressures.
  2. NN potentials were trained and used to run simulations.
  3. Configurations where the NN predictions showed high uncertainty (measured by the standard deviation of forces across an ensemble of 4 NNs) were selected.
  4. These selected configurations were recalculated with DFT and added to the training set.
  5. This cycle repeated until the dataset reached ~150,000 configurations (mostly 512-atom systems).

B. Topological Collective Variables (CVs)

To drive the polymerization/depolymerization process in Enhanced Sampling, a novel Deep Targeted Discriminant Analysis (Deep-TDA) CV was constructed:

• Graph Theory Integration: Instead of simple geometric descriptors, the CV is based on the adjacency matrix of the sulfur system.
• Eigenvalue Distribution: The eigenvalues of the adjacency matrix were computed to capture the system's topology (ring vs. chain connectivity). A continuous histogram of these eigenvalues served as input to a neural network.
• Optimization: The NN was trained to map "pure ring" and "pure polymer" configurations to distinct regions in the CV space, allowing the system to be biased along the reaction coordinate.

C. Enhanced Sampling (OPES)

• The On-the-fly Probability Enhanced Sampling (OPES) method was applied using the Deep-TDA CV.
• This allowed the system to overcome large free energy barriers, enabling the simulation of polymerization and depolymerization events on nanosecond timescales with 512 atoms.

3. Key Contributions

1. Development of a Topological CV: Introduced a graph-theory-based CV (Deep-TDA) that effectively distinguishes between ring and polymer phases without relying on predefined geometric assumptions.
2. Scalable Reactive Simulations: Demonstrated the ability to simulate chemically reactive processes in liquid sulfur at ab initio accuracy for nanoseconds, a scale previously inaccessible to DFT-based methods.
3. Mechanistic Elucidation: Provided the first detailed, atomistic view of the specific pathways for ring opening (polymerization) and ring closing (depolymerization), including charge redistribution analysis.

4. Key Results

Structural Validation

• Radial Distribution Function ($g(r)$) & Structure Factor ($S(k)$): The simulated structural properties for various ring concentrations (40%–95%) showed excellent agreement with experimental X-ray diffraction data.
• Temperature Evolution: The model successfully reproduced the temperature-dependent evolution of the third peak in $g(r)$ (~4.5 Å), which corresponds to third-neighbor distances in $S_8$ rings and diminishes as polymerization increases.

Atomic Mobility and Viscosity

• Displacement Analysis: Simulations revealed that atoms in the pure $S_8$ phase exhibit high mobility. As the polymer fraction increases, a significant population of atoms shows restricted motion (oscillating around fixed positions rather than drifting), explaining the macroscopic rise in viscosity above $T_\lambda$.

Reaction Mechanisms

• Polymerization (Ring Opening):
  • Initiated by thermal fluctuations causing an $S_8$ ring to open.
  • Charge Polarization: The opening creates under-coordinated terminal atoms with localized negative charges (Bader charge analysis), making them highly reactive.
  • Propagation: These active terminals attack neighboring rings, inducing deformation and further opening. Longer chains stabilize charge delocalization better than short oligomers, preventing immediate re-closure.
• Depolymerization (Ring Formation):
  • Tail Mechanism: The most common pathway involves the mobile chain tail folding back to interact with its 7th neighbor, forming an $S_8$ ring.
  • Mid-Chain Mechanism: Surprisingly, rings can also form in the middle of a polymer chain. This requires specific geometric alignment (angles and distances) of fully coordinated atoms to resemble a stable $S_8$ crown, followed by charge polarization at the breaking point.

5. Significance

• Resolving the λ-Transition: The study provides a definitive microscopic explanation for the anomalous physical properties of liquid sulfur, linking macroscopic viscosity changes to the specific dynamics of chain entanglement and ring formation.
• Methodological Advancement: The combination of Deep-TDA (topological CVs) and Active Learning with MLPs offers a robust framework for studying complex, reactive phase transitions in other materials where traditional order parameters fail.
• Chemical Insight: The identification of charge localization as the primary driver for reactivity in sulfur polymers offers new insights into the chemistry of chalcogens, potentially applicable to catalysis and material design.

In summary, this work bridges the gap between experimental observations and theoretical understanding of liquid sulfur, utilizing state-of-the-art machine learning to reveal the dynamic interplay between topology, charge, and structure during the λ-transition.