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Data-driven Design of Metal-Organic Frameworks with Tunable Negative Thermal Expansion

This study establishes a data-driven strategy for engineering tunable negative thermal expansion in metal-organic frameworks by utilizing a machine learning-accelerated workflow to screen over 12,000 structures, identifying key structural motifs for strong NTE that were subsequently validated experimentally with record-breaking performance in Ce-UiO-66 variants.

Original authors: Prathami Divakar Kamath, Francesco Tavani, Alin Marin Elena, Théo Jaffrelot Inizan, Yen-hsu Lin, Jian Yin, Wenqian Xu, Omar M. Yaghi, Kristin A. Persson

Published 2026-07-22

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CC BY 4.0

Original authors: Prathami Divakar Kamath, Francesco Tavani, Alin Marin Elena, Théo Jaffrelot Inizan, Yen-hsu Lin, Jian Yin, Wenqian Xu, Omar M. Yaghi, Kristin A. Persson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are building a house out of Lego bricks. Usually, when you heat a house, it expands; the bricks wiggle more, taking up slightly more space. But what if you could build a house that actually shrinks when you turn up the heat? This strange behavior is called Negative Thermal Expansion (NTE). While most materials act like a balloon inflating with warm air, NTE materials act like a deflating balloon, pulling inward as they get hotter. This is a superpower for engineers. If you are building a telescope lens or a microchip, you need parts that stay the exact same size no matter how hot or cold it gets. By mixing a shrinking material with an expanding one, you can create a "zero-expansion" composite that never warps.

For a long time, scientists have looked for these shrinking materials, but they are rare and hard to find in the solid world. However, there is a special class of materials called Metal-Organic Frameworks (MOFs) that are like giant, ultra-lightweight molecular sponges. They are made of metal knots tied together by organic strings (linkers). Because they are so flexible and full of holes, they are perfect candidates for shrinking when heated. The big challenge has been figuring out exactly which combination of metal knots and string types will shrink the most, without the whole structure falling apart. Trying to test every single combination in a lab would take forever, and doing the math on a computer to predict it is usually too slow and expensive.

This paper is like a high-speed treasure hunt that uses a super-smart computer brain to find the best shrinking recipes. The researchers built a massive digital library called "PhononMOFdb," containing detailed vibration maps for over 12,000 different MOF structures. Think of these structures as giant, 3D molecular trampolines. When you heat them up, the atoms bounce around. The key to making them shrink is finding the specific "bounces" (vibrations) that pull the structure inward. Using a new, lightning-fast machine learning tool (a type of AI that predicts how atoms move), the team scanned their library to see which designs had the strongest inward-pulling vibrations.

They discovered a clear recipe for success. First, the shape of the molecular net matters: cubic (box-like) shapes are great, but the specific way the knots connect is even more important. Second, the type of metal knot is crucial. They found that using heavier metals (like Cerium or Hafnium) instead of lighter ones makes the structure shrink more. It's like having a heavy weight on a spring; the heavy weight moves differently than a light one, creating a stronger pulling effect. Finally, they found they could fine-tune the shrinking by changing the "strings" (linkers) connecting the knots. By adding specific chemical groups to these strings, they could dial the shrinking effect up or down, or even switch it from shrinking to expanding, all without breaking the structure.

To prove their computer recipe worked, the scientists went into the lab and built a specific MOF using Cerium (a heavy metal) and tested it with high-powered X-rays. The results were a match. The pure Cerium version shrank significantly, beating previous records. But the real magic happened when they modified the strings. When they added bromine atoms to the strings, they could control exactly how much it shrank. In one specific case, at higher temperatures, the material suddenly shrank at a rate of -593 (MK)⁻¹, a record-breaking number that suggests this material is a champion at shrinking.

The paper suggests that by following these rules—picking the right shape, using heavy metal knots, and tweaking the connecting strings—scientists can now design materials with custom thermal behaviors. They didn't just guess; they simulated thousands of options, picked the best ones, and then built them to prove the theory works. This opens the door to creating new materials that can keep our future technology stable, no matter how hot the environment gets.

Technical Summary: Data-driven Design of Metal-Organic Frameworks with Tunable Negative Thermal Expansion

Problem Statement
Materials exhibiting Negative Thermal Expansion (NTE)—contracting upon heating—are critical for precision applications in optics, microelectronics, and aerospace, where precise thermal control is required. While Metal-Organic Frameworks (MOFs) are recognized as a rich platform for NTE due to their flexible architectures and low-energy lattice vibrations, systematically uncovering the structural motifs that govern NTE across the vast MOF chemical space remains experimentally challenging. Furthermore, large-scale first-principles phonon calculations required to predict these properties are computationally prohibitive. Existing experimental strategies (e.g., node distortion, mixed-linker solutions) are often limited to specific material families and lack generalizability. Additionally, quantitative benchmarking is difficult due to variations in sample crystallinity and the reliance on classical force fields in prior theoretical studies, which introduce significant errors compared to Density Functional Theory (DFT).

Methodology
To address these challenges, the authors developed a data-driven workflow centered on PhononMOFdb, a comprehensive database constructed using the MACE-MP-MOF0 machine learning interatomic potential. This model, fine-tuned specifically for MOF structures, offers near-ab initio accuracy at a fraction of the DFT computational cost.

Database Generation: The workflow applied MACE-MP-MOF0 to approximately 12,000 computation-ready MOFs from the QMOF database. This generated phonon densities of states (DOS), band structures, heat capacities, bulk moduli, and inelastic neutron scattering (INS) spectra.
Validation: The accuracy of the ML potential was benchmarked against DFT calculations (using VASP) and experimental data for well-known systems like MOF-5 and ZIF-8. The results showed accurate reproduction of spectral features with only a modest red shift consistent with the PBE functional used in DFT.
High-Throughput Screening: A vibration-based screening strategy was employed to identify candidates for volumetric NTE. The process involved:
- Filtering for intense low-frequency phonon modes (0–2 THz), which are associated with transverse linker vibrations and rigid-unit modes (RUMs).
- Applying the Quasi-Harmonic Approximation (QHA) to calculate NTE coefficients ( $\alpha_V$ ) for structures passing the initial filter.
- Enforcing a mechanical stability threshold (Bulk Modulus $B \ge 1$ GPa at 100 K) to ensure candidates are not mechanically fragile.
Descriptor Analysis: Seven structural descriptors were analyzed to correlate structure with NTE: crystal system, topology, metal node identity, linker length, flexibility, and functionalization.
Experimental Validation: The derived design principles were tested via high-resolution, temperature-dependent synchrotron powder X-ray diffraction (PXRD) on Ce-UiO-66 and its brominated variants (mono- and di-brominated BDC linkers).

Key Contributions and Results

PhononMOFdb: The creation of the largest MOF phonon database to date, covering all seven crystal systems and providing validated INS spectra and thermophysical properties for over 12,000 structures.
Structural Design Rules: The screening revealed specific trends governing NTE magnitude:
- Topology: While NTE occurs across all crystal systems, cubic topologies (specifically primitive cubic, pcu, and face-centered cubic, fcu) are dominant. pcu frameworks (typically Zn-based) exhibit larger NTE but lower mechanical stability, whereas fcu frameworks (typically Zr-based) offer higher bulk moduli with moderate NTE.
- Metal Nodes: Heavier metal nodes enhance NTE. The study found that heavier tetravalent nodes (e.g., Hf, Ce) compensate for their higher charge density with increased atomic mass, lowering vibrational frequencies and amplifying NTE compared to lighter analogues (e.g., Zr, Ti).
- Linker Engineering: Isoreticular linker elongation increases NTE but drastically reduces mechanical stability. Conversely, linker functionalization offers a tunable handle to modulate NTE magnitude and sign without compromising framework integrity.
Experimental Findings on Ce-UiO-66:
- Enhanced NTE: Pristine Ce-UiO-66 exhibited a volumetric NTE coefficient of -23(2) (MK) $^{-1}$ at 250 K, surpassing the previously reported value for Zr-UiO-66 (-16.7(8) (MK) $^{-1}$ ), confirming the benefit of heavy-node substitution.
- Tunability via Bromination: Systematic bromination allowed for precise tuning. Mono-bromination suppressed NTE and induced positive thermal expansion (PTE) at 250 K. Di-bromination (2,5-dibromo) further suppressed NTE at lower temperatures.
- Colossal NTE: Notably, the mono-brominated variant (Ce-UiO-66-Br) exhibited an abrupt transition to colossal NTE at elevated temperatures (>250 K), reaching a coefficient of -593(25) (MK) $^{-1}$ . This value surpasses current records for MOFs.
- Mechanism: Phonon mode analysis revealed that the colossal NTE in Ce-UiO-66-Br arises from large-amplitude, anharmonic displacements of the bromine atoms at higher temperatures, which drive the lattice contraction.

Significance
The paper establishes a phonon-centric, data-driven framework for the rational engineering of thermal expansion in MOFs. By bridging the gap between computational prediction and experimental realization, the work demonstrates that machine learning-accelerated discovery can effectively navigate the vast MOF chemical space to identify specific structural motifs (heavy nodes, cubic topologies) and chemical handles (linker functionalization) for tuning NTE. The successful validation on the Ce-UiO-66 family confirms that heavy-node substitution and linker engineering can yield NTE coefficients that exceed existing records, providing a roadmap for designing zero thermal expansion composites and materials with tailored thermal responses. The identification of eleven "colossal NTE" candidates within the database offers immediate targets for future experimental investigation.

Technical Summary: Data-driven Design of Metal-Organic Frameworks with Tunable Negative Thermal Expansion

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