Alessio Borgi

Alessio Borgi

PhD in Graph Neural Networks and Generative AI 👨🏻‍💻🤖

TDA in Materials Science: Topology of Structure and Phase

3 minute read

Published:

TL;DR: Porous materials (zeolites, MOFs, foams) have geometrically complex void structures. Persistent H₁ and H₂ of the atomic point cloud characterise channels (1D voids, H₁) and cages (3D voids, H₂). The persistence diagram encodes pore size distribution (birth scale) and connectivity (death scale) in a single multi-scale descriptor. This predicts CO₂ adsorption, methane storage capacity, and mechanical stiffness better than simple pore size metrics.

Why Topology for Materials?

Challenge: Materials with the same chemical composition and similar density can have dramatically different properties depending on their structural topology:

  • Two zeolites with the same Si/Al ratio but different channel connectivity have 10× different diffusion coefficients.
  • A porous carbon with mainly isolated pores (no H₁ connectivity) is a terrible electrode; one with interconnected channels (high H₁ persistence) is excellent.

Geometric statistics (mean pore size, surface area) average out the topological differences. Persistence captures them.

Pore Structure Analysis

For a porous material represented as a 3D point cloud of atom positions \(P\):

  1. Build cubical complex from voxelised electron density.
  2. Compute sublevel set persistence (filter from vacuum to dense material).

Interpretation:

  • \(H_0\) bars: connected atom clusters (grain boundaries in polycrystals).
  • \(H_1\) bars: channels/tunnels through the material. Long bars = persistent channels from large to small length scales.
  • \(H_2\) bars: enclosed pores/cages. Birth \(b\) = pore diameter; death \(d\) = smallest “bottleneck” in the pore wall.
Pore accessibility: $$\{(b_k, d_k) \in H_2 : b_k > r_{gas}\}$$ — cages accessible to molecules of radius $$r_{gas}$$

Metal-Organic Frameworks (MOFs)

MOFs are crystalline porous materials with precisely engineered pore geometry. A key problem is screening — choosing which of \(\sim 500000\) synthesisable MOFs to test for gas storage.

Lee et al. (2021): Trained an ML model using persistence diagram features of MOF atom clouds to predict methane storage capacity. The topological features (especially \(H_2\) diagram) outperformed pure geometric features, capturing the “shape” of pores beyond simple radii.

Glass Transition

The glass transition (liquid to amorphous solid) is characterised by structural changes that are not visible in pair correlation functions (the standard structural probe) but appear in topology:

  • Liquid phase: short \(H_1\) and \(H_2\) bars (no persistent structures).
  • Glass phase: emergence of long-lived \(H_1\) bars corresponding to 5- and 6-membered atomic rings that characterise glass network structure.

Topological order parameters (total persistence, Betti curves) distinguish glass from liquid more sensitively than radial distribution functions near the transition.

Crystal Structure Fingerprinting

For crystalline materials, the persistence diagram is a crystal structure fingerprint:

  • Different polymorphs of the same compound (e.g., \(\alpha\)- vs \(\beta\)-quartz) have different diagrams.
  • The diagram is invariant to unit cell choice and atomic labelling.
  • Crystal structure databases can be searched by topological similarity.
Key Insight: Materials science and TDA are a natural fit because materials properties are fundamentally topological: whether electrons can percolate (electrical conductivity), whether molecules can diffuse (gas adsorption), whether cracks can propagate (fracture toughness) all depend on connectivity — H₀ and H₁ topology — not on mean distances. Classical materials descriptors (radial distribution functions, structure factors) are essentially statistics of pairwise distances and miss this connectivity information. TDA fills exactly this gap.

References

  • Y. Lee, S. Barthel, P. Dłotko, S. Moosavi, K. Vipond, B. Smit, “Quantifying Similarity of Pore-Geometry in Nanoporous Materials,” Nature Communications, 2017.
  • I. Obayashi, T. Nakamura, Y. Hiraoka, “Persistent Homology Analysis for Materials Research and Persistent Homology Software: HomCloud,” J. Physical Society of Japan, 2022.
  • K. Saadatfar et al., “Pore Configuration Landscape of Granular Crystallisation,” Nature Communications, 2017.

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