Percolation Criticality of Amorphous-Amorphous Transitions in Compressed Glasses

This study employs large-scale molecular dynamics simulations and percolation theory to reveal that the low-to-high-density transition in compressed silica glass is driven by critical percolation of structural clusters, exhibiting critical exponents that suggest a rigidity percolation mechanism and highlighting common transformation principles between bonded and non-bonded amorphous materials.

The Gist

Imagine a glass of water or a piece of window glass. You know them as solid, rigid things. But if you were to shrink down to the size of an atom and look inside, you'd see a chaotic, tangled web of tiny building blocks. In silica glass (the stuff in windows), these blocks are shaped like pyramids (tetrahedra) made of silicon and oxygen.

This paper is like a high-tech movie that zooms in on what happens when you squeeze this glass with immense pressure—up to 350,000 times the pressure of the atmosphere. The scientists wanted to understand how the glass changes its shape without melting or breaking, a process called an "amorphous-amorphous transition."

Here is the story of their discovery, told through simple analogies:

1. The Crowd at a Concert (The Structure)

Think of the glass as a giant crowd of people at a concert.

• At normal pressure: Everyone is standing in a loose, open formation. In silica glass, the "people" are silicon atoms, and they are holding hands with four neighbors, forming perfect pyramid shapes (tetrahedra). There are lots of empty spaces between them, making the structure "floppy" and easy to compress.
• As pressure increases: Imagine the concert hall starts shrinking. The crowd gets squeezed. People can't keep their perfect pyramid shapes anymore. They start bumping into others, changing how many neighbors they hold hands with. Some start holding hands with 5 people, then 6.

2. The "Percolation" Game (The Big Change)

The scientists used a concept called percolation. Imagine you are pouring water through a sponge.

• Low Pressure: The sponge has holes, but they are all separate. If you pour water, it gets stuck in small pockets. It doesn't flow all the way through. In the glass, the "pyramid" shapes are isolated islands.
• Critical Pressure: As you squeeze harder, the holes start connecting. Suddenly, a giant, continuous path forms from the top of the sponge to the bottom. The water flows through!
• In the Glass: The scientists found that at specific pressure points, the new shapes (like the 5-sided or 6-sided blocks) suddenly connect to form a giant, continuous chain that spans the entire piece of glass. This is the "percolation transition." It's the moment the glass fundamentally reorganizes itself into a denser state.

3. Two Ways to Look at the Crowd

The researchers looked at this crowd in two different ways, like watching a movie from two different camera angles:

• The "Bonded" View (The Handshake): They looked at who is directly holding hands (chemical bonds). They saw that the pyramid shapes were changing their handshakes.
• The "Non-Bonded" View (The Personal Space): They ignored the handshakes and just looked at who was standing close to whom, regardless of whether they were touching. This is like looking at a crowd where people aren't holding hands, but just standing near each other.

The Surprise: Both cameras showed the exact same story! The "handshake" view and the "personal space" view both showed that the glass transforms in the same sequence: first the loose shapes connect, then the dense shapes take over. This suggests that the rules governing how glass changes are universal, whether the atoms are "holding hands" (like in silica) or just bumping into each other (like in frozen water/ice).

4. The "Magic Number" and the Rules of the Game

The scientists wanted to know if this transformation follows a standard rulebook (like a game of chance) or if it has its own special rules.

• The Tetrahedra (The 4-sided shapes): When the original pyramid shapes (holding 4 hands) broke apart, they did so exactly like a random game of chance. It was "standard" behavior.
• The Higher Shapes (5, 6, or more hands): When the new, denser shapes formed and connected, they broke the standard rules. They followed a different, more complex set of rules. The scientists call this "rigidity percolation." It's as if the crowd didn't just connect randomly; they connected in a way that made the whole structure suddenly much stiffer and more rigid.

5. The Takeaway

The paper concludes that when you squeeze glass, it doesn't just get smaller; it undergoes a dramatic, phase-shift-like event where the internal structure reorganizes into a new, denser "state."

• The transition happens at specific "critical" pressures.
• The way the new structures connect is a mix of random chance (for the old shapes) and a more rigid, structured rule (for the new, dense shapes).
• This behavior is similar in silica glass and amorphous ice, suggesting that nature uses similar "blueprints" to rearrange different types of glassy materials under pressure.

In short, the paper maps out exactly how the microscopic "skeleton" of glass snaps, shifts, and rebuilds itself when squeezed, revealing that the transition from a loose, floppy glass to a dense, rigid one happens through a specific, predictable tipping point.

Technical Summary

Technical Summary: Percolation Criticality of Amorphous-Amorphous Transitions in Compressed Glasses

Problem Statement
The mechanisms governing structural transformations in compressed glasses, particularly the amorphous-to-amorphous transitions in tetrahedral-based materials like silica ($a$-SiO$_2$) and amorphous ice ($a$-H$_2$O), remain a significant challenge. Unlike sharp crystalline phase transitions, these transformations involve a progressive evolution of local structural motifs into distinct polyamorphs. While previous studies have identified the emergence of large, percolating clusters of high-coordination polyhedra under pressure, the criticality of these transitions in glasses has remained unknown. Specifically, it is unclear whether these structural transitions fall within the universality class of standard (random) uncorrelated percolation or if they exhibit distinct critical behavior, such as rigidity percolation. Previous attempts to estimate critical exponents were limited by small system sizes inherent to ab initio approaches.

Methodology
To address these limitations, the authors employed large-scale classical molecular dynamics (MD) simulations.

• Systems and Potentials: Simulations were conducted on silica glasses using the SHIK pair potential within the LAMMPS package, and on amorphous ice using the TIP4P force field within Gromacs. System sizes ranged from $\sim$1,000 to $10^6$ atoms for silica and up to 49,000 molecules for ice.
• Protocol: Samples were equilibrated, quenched to room temperature (300 K for silica, 124 K for ice), and quasi-statically compressed up to 35 GPa (silica) or 15 kbar (ice).
• Structural Descriptors: Two complementary approaches were utilized:
  1. Bonded Model: Based on covalent Si-O connectivity, analyzing SiO$_Z$ polyhedra (where $Z$ is the coordination number) and their connectivities (corner-shared, edge-shared, face-shared).
  2. Non-bonded Model: Based on the arrangement of central atoms (Si-Si for silica, O-O for ice) within a cutoff radius, defining SiSi$_Z$ and OO$_Z$ polyhedra. This approach treats interactions as non-bonded forces, allowing for direct comparison between bonded glasses (silica) and non-bonded glasses (ice).
• Percolation Analysis: A dedicated Python code (Nexus-CAT) implementing a Union-Find algorithm was used to identify clusters. Percolation properties, including the correlation length ($\xi$), order parameter ($P_\infty$), average cluster size ($\langle S \rangle$), and largest cluster size ($S_{max}$), were calculated.
• Critical Exponents: Finite-size scaling methods (using data collapse techniques) were applied to determine critical exponents ($\nu, \beta, \gamma$) and fractal dimensions ($D_f, D_f^*$) near the percolation thresholds.

Key Results

1. Sequence of Percolation Transitions: Under compression, the study identified a distinct sequence of percolation events. In $a$-SiO$_2$, low-density (LD) tetrahedral networks ($Z=4$) depercolate as high-density (HD) clusters ($Z=5, 6$) emerge and percolate. Specifically:

   • SiO$_5$ clusters percolate at $\approx$12 GPa.
   • SiO$_6$ clusters percolate at $\approx$23 GPa.
   • "Stishovite-like" clusters (SiO$_6$ with two edge-sharing connections) percolate at $\approx$30 GPa.
   • Similar sequences were observed in $a$-H$_2$O (LD, HD, and VHD clusters), occurring at pressures approximately 20 times lower than in silica, suggesting a universal mechanism scaled by substance-specific interactions.

2. Complementarity of Models: The bonded (SiO$_Z$) and non-bonded (SiSi$_Z$/OO$_Z$) approaches provided consistent, complementary insights. The non-bonded model successfully captured the structural transformations and allowed for a unified framework comparing silica and amorphous ice.

3. Critical Exponents and Universality:

   • The correlation length $\xi$ scaled with system size $L$ as $\xi \propto L^a$ with $a \approx 1$, confirming critical behavior.
   • Depercolation of Tetrahedra: The depercolation of LD tetrahedral clusters ($Z=4$) yielded critical exponents (e.g., Fisher exponent $\tau \approx 2.12-2.14$) very close to those of standard 3D random percolation.
   • Percolation of High-Coordination Clusters: In contrast, the percolation of higher-coordination clusters ($Z=5, 6$) and VHD structures showed deviations from standard percolation. The fractal dimensions ($D_f$) were systematically lower than the standard value (2.53), and the exponents suggested a different universality class.

4. Fractal Dimensions: The fractal dimension $D_f^*$ of finite clusters (lattice animals) was calculated for various coordination numbers, ranging from $\approx 2.29$ to $2.46$, generally lower than the standard percolation value.

Significance and Claims
The paper claims that the introduction of long-range descriptors and the use of large-scale simulations allow for the first robust estimation of percolation critical exponents in compressed glasses. The primary significance lies in the finding that amorphous-amorphous transitions are not uniform in their critical behavior:

• The collapse of the native tetrahedral network appears to follow standard random percolation.
• The emergence and percolation of high-coordination, rigid polyhedral networks (associated with densification) likely follow a different universality class, possibly indicative of a rigidity percolation mechanism. This is attributed to the topological and elastic heterogeneities arising when percolating clusters of high connectivity emerge within a medium of lower connectivity.

The study establishes a link between these percolation transitions and the mechanical properties of glasses, such as the compressibility peak and the onset of plasticity. Furthermore, it proposes that tracking percolation-driven structural changes provides a basis for analogies between amorphous states and crystalline polymorphs (e.g., coesite IV, coesite V, stishovite), suggesting that glassy states may correspond to energy megabasins associated with these crystalline structures. The work underscores the utility of the non-bonded approach in unifying the understanding of pressure-induced transformations in both bonded and non-bonded glassy systems.