Nexus-CAT: A Computational Framework to Define Long-Range Structural Descriptors in Glassy Materials from Percolation Theory

Nexus-CAT is an open-source Python framework that utilizes percolation theory and flexible clustering strategies to characterize long-range structural connectivity in glassy materials, revealing that pressure-induced crystallization in amorphous silicon is preceded by an amorphous-to-amorphous transition.

The Gist

Imagine you are looking at a giant, chaotic pile of LEGO bricks. In a crystal (like a diamond or a salt crystal), the bricks are stacked in a perfect, repeating pattern. But in glassy materials (like window glass, ice, or amorphous silicon), the bricks are jumbled up randomly.

Scientists have long struggled to understand how these messy piles change when you squeeze them or heat them up. Traditional tools are like taking a photo of just a few bricks at a time. They can tell you what a single brick looks like or how close its immediate neighbors are, but they can't see the big picture. They miss the fact that, suddenly, a massive chain of bricks has formed that stretches across the entire pile, connecting one side to the other.

This paper introduces a new digital tool called Nexus-CAT (Cluster Analysis Toolkit) that acts like a super-powered detective for these messy piles.

The Problem: The "Missing Link"

Think of a glassy material as a city where people (atoms) are constantly moving.

• Old Tools: These tools count how many friends each person has in their immediate neighborhood. They tell you if someone is lonely or popular locally, but they can't tell you if a massive protest march has formed that spans the whole city.
• The Gap: When glassy materials undergo a "phase transition" (a sudden change in structure, like ice turning into a different kind of ice under pressure), it's not just a local change. It's a long-range event where the entire network reorganizes. Old tools miss this "aha!" moment because they are too focused on the small details.

The Solution: Nexus-CAT

Nexus-CAT is a computer program that reads the "movie" of atoms moving (a simulation trajectory) and asks a simple question: "Who is connected to whom?"

It uses a clever trick called Percolation Theory. Here is an analogy to explain it:
Imagine a sponge. If you pour water on it, the water drips down. But at a certain point, the water connects all the holes, and suddenly, the whole sponge is wet from top to bottom. That moment of "all connected" is called percolation.

Nexus-CAT watches the atoms and looks for that exact moment when a "super-connection" forms that spans the entire material.

How It Works (The Detective's Toolkit)

The program doesn't just look at atoms; it looks at them through four different "lenses" or strategies, depending on what kind of glass you are studying:

1. Distance Lens: "Are these two atoms close enough to hold hands?" (Simple and direct).
2. Bonding Lens: "Do these two atoms share a common friend?" (Like two people meeting because they both know the same person).
3. Coordination Lens: "Are these atoms holding hands with the exact same number of other people?" (e.g., only looking at groups where everyone has 4 friends).
4. Shared Neighbor Lens: "Do these two atoms not only share a friend, but do they share two or more friends?" (This helps spot very tight, specific clusters).

By switching between these lenses, Nexus-CAT can find hidden patterns that other tools miss.

The Big Discoveries

The authors tested this tool on three different types of "messy" materials and found some surprising things:

• Vitreous Silica (Window Glass): When you squeeze glass, the atoms rearrange from a 4-friend structure to a 5-friend or 6-friend structure. Nexus-CAT showed that this happens in distinct steps, like a staircase, rather than a smooth slide.
• Amorphous Silicon (Computer Chips): This was the big surprise. The tool detected a "percolation transition" (the moment the whole network connects) before the silicon turned into a crystal. It's as if the messy pile of LEGO bricks suddenly organized itself into a perfect line before it actually snapped into a solid block. This suggests that the "messy-to-crystal" transformation is actually driven by a hidden "messy-to-different-messy" step first.
• Amorphous Ice: Similar to the silicon, the tool tracked how water molecules rearrange under pressure, showing how different "densities" of ice appear and disappear.

Why This Matters

Think of Nexus-CAT as a weather radar for the atomic world.

• Old tools were like looking at a single raindrop.
• Nexus-CAT sees the entire storm system forming.

By understanding exactly when and how these massive connections form, scientists can better predict how materials will behave. This could lead to stronger glass, better batteries, or more efficient computer chips. The tool is free, open-source, and ready for anyone to use to explore the hidden order inside chaos.

Technical Summary

1. Problem Statement

Standard structural analysis tools for disordered systems (e.g., pair distribution functions, structure factors, and bond angular distributions) are limited to describing short- to medium-range order. They fail to capture long-range connectivity changes that drive amorphous-amorphous transitions (AATs) and polyamorphic transformations in glassy materials. While percolation theory offers a unified framework to describe these transitions via universal scaling laws, there is a lack of flexible, open-source tools capable of performing systematic cluster detection and percolation analysis directly from atomistic simulation trajectories (e.g., Monte Carlo or Molecular Dynamics) for off-lattice, multi-species systems. Existing tools are either tailored for lattice percolation, gel-point detection in polymers, or lack the integrated workflow required for complex, disordered network topologies.

2. Methodology

The authors introduce Nexus-CAT (Cluster Analysis Toolkit), an open-source Python package designed to bridge this gap. The methodology is built on the following pillars:

• Input & Architecture: The tool reads extended XYZ trajectory files and utilizes a Strategy Factory design pattern to dynamically select clustering algorithms based on user-defined settings. It handles general parallelepiped simulation boxes via a 3×3 lattice matrix.
• Core Algorithm: Cluster identification is performed using a high-performance Union-Find (Disjoint-Set) algorithm with path compression. This approach efficiently manages and merges sets of atoms, offering superior computational efficiency compared to standard graph traversal methods like BFS or DFS.
• Clustering Strategies: The toolkit implements four distinct strategies to accommodate diverse network topologies:
  1. Distance Strategy: Connects nodes within a specific cutoff distance.
  2. Bonding Strategy: Connects primary nodes only if they share a common neighbor of a specific species.
  3. Coordination Strategy: Filters connections based on the coordination number ($Z$) of nodes. It supports modes for specific ranges, identical coordination (e.g., $SiO_4-SiO_4$), alternating coordination (e.g., $SiO_4-SiO_5$), or mixing.
  4. Shared-Neighbor Strategy: Adds a constraint requiring a minimum number of shared neighbors to form a connection, enabling the distinction between corner-sharing, edge-sharing, and face-sharing polyhedra.
• Percolation Analysis: The framework computes rigorous percolation properties, including:
  • Concentration ($\phi$): The fraction of nodes in clusters.
  • Average Cluster Size ($\langle S \rangle$): The second moment of the cluster size distribution (excluding the percolating cluster).
  • Correlation Length ($\xi$): Derived from the gyration radius distribution.
  • Percolation Probability ($\Pi$): Determined via a rigorous period vector algorithm that tracks algebraic dimensions to detect if a cluster spans the periodic boundaries in 1D, 2D, or 3D.
  • Order Parameter ($P_\infty$): The fraction of nodes in the spanning cluster.
• Implementation: The code is written in pure Python, leveraging NumPy, SciPy (for $k$-d tree neighbor searches), and Numba (for JIT compilation of geometric loops) to achieve near-C performance. It supports Python $\ge$ 3.9.

3. Key Contributions

• First General Framework: Nexus-CAT is the first open-source tool providing a general framework for cluster-based, multi-species percolation analysis directly from atomistic trajectories of disordered systems.
• Modular Strategy Factory: The implementation of a Strategy Factory allows the tool to be adapted to various bonding environments (covalent, ionic, metallic) and network definitions without code rewriting.
• Rigorous Percolation Detection: The integration of a period vector algorithm ensures accurate detection of system-spanning clusters in periodic boundary conditions, distinguishing between different dimensionalities of percolation.
• Validation: The package was validated against theoretical predictions for 3D site percolation on simple cubic lattices. Finite-size scaling analysis successfully extracted critical exponents ($\nu, \gamma, D_f$) that match established literature values (e.g., $\nu \approx 0.88$, $D_f \approx 2.52$).

4. Results and Case Studies

The authors applied Nexus-CAT to three distinct glassy systems to demonstrate its versatility:

• Vitreous Silica ($v$-SiO$_2$):

  • Analyzed under compression using Coordination and Shared strategies.
  • Identified distinct percolation thresholds for different polyhedral networks: $SiO_4$ (tetrahedral) breaks down at $\phi \approx 0.39$, while $SiO_5$ and $SiO_6$ networks emerge at higher pressures.
  • Confirmed that the critical exponents for these transitions align with the standard percolation universality class.

• Amorphous Silicon ($a$-Si):

  • Analyzed as a single-species covalent network.
  • Key Finding: Observed a sharp sequence of percolation transitions between 11 and 13 GPa. The low-density (LDA, $Z=4$) network depercolates almost simultaneously as the high-density (HDA) and very-high-density (VHDA, $Z \ge 8$) networks percolate.
  • Significance: This percolation transition acts as a precursor to crystallization. The VHDA phase ($Z \ge 8$) shares a coordination number with the crystalline phase, suggesting that pressure-induced crystallization is driven by an initial amorphous transformation that brings the system into the same potential energy megabasin as the crystal.

• Amorphous Ice ($a$-H$_2$O):

  • Analyzed using non-bonded criteria (O-O coordination) since H-bond counts remain constant.
  • Observed gradual transitions from Low-Density (LDA, $Z=4$) to High-Density (HDA, $Z=5-7$) and Very-High-Density (VHDA, $Z \ge 8$) percolating clusters under compression.
  • These transitions correlate with thermodynamic anomalies (compressibility and density) in the water phase diagram.

5. Significance

Nexus-CAT provides a critical methodological advance for condensed matter physics by enabling the definition of long-range structural descriptors in disordered materials. Its ability to track the evolution of connectivity and percolation thresholds offers new insights into:

• The mechanism of polyamorphic transitions (amorphous-to-amorphous).
• The precursors to crystallization in glasses, revealing that structural reorganization often precedes the actual phase change.
• The universality of phase transitions in disordered systems, linking microscopic network topology to macroscopic thermodynamic properties.

The tool is designed to be extensible to other complex systems such as gels, cements, and chalcogenide glasses, facilitating a systematic investigation of structural disorder across a broad class of materials. The code is publicly available on GitHub and PyPI.