🔬 materials science

Hydrogen diffusion in TiCr $_2$ H $_x$ Laves phases: A combined ab initio and machine-learning-potential study

This study combines density functional theory and machine learning potentials to investigate hydrogen diffusion in TiCr $_2$ Laves phases, revealing that migration barriers depend on bond-breaking requirements, diffusion exhibits non-monotonic concentration dependence, and experimental discrepancies are likely caused by defect-induced hydrogen trapping.

Original authors: Pranav Kumar, Fritz Körmann, Kaveh Edalati, Blazej Grabowski, Yuji Ikeda

Published 2026-02-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Pranav Kumar, Fritz Körmann, Kaveh Edalati, Blazej Grabowski, Yuji Ikeda

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

🚀 The Big Picture: The Hydrogen "Traffic Jam" in Metal

Imagine you have a metal sponge (specifically, an alloy called TiCr₂) that is designed to soak up hydrogen gas like a sponge soaks up water. This is a key technology for making clean hydrogen fuel for cars and power plants.

But there's a catch: For the sponge to work, the hydrogen atoms need to be able to move around inside the metal quickly. If they get stuck, the fuel tank fills up too slowly, or the car can't release the energy fast enough.

This paper is like a traffic report for hydrogen atoms inside this specific metal sponge. The researchers used super-computers to watch how hydrogen moves, how fast it goes, and what causes it to get stuck.

🔍 The Tools: The "Crystal Ball" and the "Virtual Lab"

To figure this out, the scientists used two powerful tools:

DFT (Density Functional Theory): Think of this as a high-resolution microscope. It lets them see exactly how individual atoms interact, but it's so detailed that it's incredibly slow to run. You can only watch a tiny drop of water for a split second.
MLIPs (Machine Learning Potentials): This is the magic shortcut. The scientists taught a computer to learn the rules of the "microscope" (DFT). Once the computer learned the rules, it could simulate the movement of millions of atoms for a long time, just like a video game, but with real physics.

They combined these to get the best of both worlds: the accuracy of the microscope and the speed of the video game.

🛣️ The Roadmap: Two Types of Lanes

Inside the metal sponge, there are empty spots (interstitial sites) where hydrogen atoms can sit. The researchers found two main types of "roads" the hydrogen can travel on:

The "Easy" Lanes (Cr-H bonds): These paths involve breaking a bond with Chromium. It's like walking on a smooth, flat sidewalk. It takes very little energy.
The "Hard" Lanes (Ti-H bonds): These paths involve breaking a bond with Titanium. It's like trying to walk through thick mud or climbing a steep hill. It takes a lot of energy.

The Discovery: Hydrogen atoms are smart. They almost always choose the "Easy" lanes. They avoid the "Hard" lanes unless they have no other choice.

🏰 The Neighborhood: Hexagonal Rings

The metal structure looks like a city made of hexagonal rings (like a honeycomb).

Inside the Ring: Hydrogen atoms can zip around the inside of a single ring very easily.
Between Rings: To jump from one ring to a neighboring ring, the hydrogen has to break a "hard" bond (the Titanium one). This is the bottleneck.

The Analogy: Imagine a city where people can run freely inside their own neighborhood (the ring), but to leave the neighborhood and go to the next one, they have to climb a high wall. Most people stay inside their neighborhood unless they really need to leave.

🚦 The Traffic Flow: The "Goldilocks" Zone

The most interesting finding is how the amount of hydrogen affects the speed. It's not a straight line; it's a curve.

Too Empty (Low Hydrogen): The hydrogen atoms are lonely. They sit in their spots and don't move much because there's no pressure to move.
Just Right (Medium Hydrogen): As you add more hydrogen, the atoms start bumping into each other. This "bumping" acts like a gentle push. It helps them jump over the walls and move faster. This is the fastest point!
Too Crowded (High Hydrogen): If you pack the sponge too full, it becomes a traffic jam. The hydrogen atoms are so crowded they can't move at all. They are stuck in a "traffic jam" because there is no empty space to jump into.

The Metaphor: Think of a dance floor.

Empty: No one is dancing.
Moderate: People are bumping elbows, which makes them spin and dance faster.
Overcrowded: Everyone is packed so tight they can't move an inch.

📉 The Mystery: Why is the Real World Slower?

The computer simulations showed hydrogen moving very fast (about 10 times faster than some real-world experiments). Why the difference?

The researchers realized that in real life, the metal sponge isn't perfect. It has defects (like missing atoms or atoms in the wrong place).

The Analogy: Imagine a highway with perfect lanes (the computer model). But in reality, there are potholes and construction zones (defects). The hydrogen atoms get stuck in these potholes, slowing down the whole traffic flow.

The computer model assumed a perfect metal, which is why it predicted faster speeds. The "potholes" in real metal act as traps.

💡 Why Does This Matter?

This study is a blueprint for building better hydrogen fuel tanks.

By understanding exactly how hydrogen moves, engineers can design alloys that have fewer "potholes" (defects) or structures that encourage the "easy lanes."
They can figure out the perfect amount of hydrogen to pack in to get the fastest charging and discharging speeds.

In short: The scientists mapped the traffic rules for hydrogen in metal. They found that too much hydrogen causes a jam, but just the right amount creates a smooth, fast flow. This knowledge helps us build better, faster, and cleaner energy storage for the future.

1. Problem Statement

Hydrogen storage alloys based on the $AB_2$ Laves phase, particularly Titanium-Chromium ( $TiCr_2$ ), are promising candidates for solid-state hydrogen storage. However, the kinetic performance of these materials is governed by hydrogen diffusion, which is complex due to:

Structural Complexity: $TiCr_2$ exists in two polymorphs: cubic C15 and hexagonal C14, each with multiple symmetrically distinct interstitial sites.
Concentration Dependence: Diffusion behavior changes significantly with hydrogen concentration ( $x$ in $TiCr_2H_x$ ), but this has not been quantitatively mapped across a wide range ( $0 < x \le 4$ ).
Computational Limitations: Traditional Density Functional Theory (DFT) is too computationally expensive to simulate the long time and large length scales required for accurate diffusion coefficient calculations, especially at finite hydrogen concentrations where many-body interactions occur.
Experimental Discrepancies: Experimental activation energies vary widely (0.2 eV to 0.6 eV), and experimental diffusion coefficients often differ by an order of magnitude from theoretical predictions, partly due to non-stoichiometry and defects in real samples.

2. Methodology

The study employs a multi-scale approach combining DFT and Machine Learning Interatomic Potentials (MLIPs):

DFT Calculations (Static):
- Used VASP with PBE-GGA to calculate binding energies and migration barriers.
- Employed the Generalized Solid-State Nudged Elastic Band (G-SSNEB) method to determine minimum energy paths for all symmetrically inequivalent migration paths between first-nearest-neighbor (1NN) face-sharing interstitial sites.
- Analyzed both dilute limits (single H atom) and specific bond-breaking scenarios (Ti–H vs. Cr–H).
Machine Learning Potentials (MLIPs):
- Utilized Moment Tensor Potentials (MTPs) trained on DFT data.
- Active Learning/Fine-Tuning: The original C14-MTP was unstable at high hydrogen concentrations ( $x \ge 3.5$ ). The authors implemented an iterative active learning protocol: running MD, identifying configurations with high extrapolation grades (indicating low confidence), calculating their DFT energies/forces, and retraining the MTP. This resulted in a robust C14-MTP capable of simulating up to $x=4$ .
- The C15-MTP required no fine-tuning as it remained stable up to $x=4$ .
Molecular Dynamics (MD) Simulations:
- Performed using LAMMPS with the trained MTPs.
- System Size: 6,000 metal atoms ( $5 \times 5 \times 5$ supercells) to ensure statistical convergence and minimize finite-size effects.
- Conditions: Simulations ran at temperatures of 250 K, 400 K, 500 K, 750 K, and 1000 K, and hydrogen concentrations $0 < x \le 4$ .
- Analysis: Calculated diffusion coefficients ( $D$ ) from Mean Squared Displacement (MSD) in the normal-diffusion regime ( $\alpha \approx 1$ ). Analyzed Radial Distribution Functions (RDF) to understand H–H interactions.

3. Key Contributions

Comprehensive Mapping: Provided the first systematic calculation of hydrogen diffusion coefficients across the full stoichiometric range ( $0 < x \le 4$ ) for both C15 and C14 $TiCr_2$ phases.
Methodological Advancement: Demonstrated a successful workflow for fine-tuning MLIPs via active learning to handle high-concentration regimes where standard potentials fail, enabling reliable MD simulations of hydrogen diffusion.
Mechanistic Insight: Decoupled the effects of crystal structure (C15 vs. C14) and hydrogen concentration on diffusion, revealing the specific role of H–H repulsion.

4. Key Results

A. Migration Barriers and Bond Breaking

Bond Affinity: Ti has a much higher chemical affinity for H than Cr. Breaking a Ti–H bond requires significantly higher energy (285–801 meV) than breaking a Cr–H bond (90–186 meV).
Path Preference: Hydrogen migration is dominated by paths that break Cr–H bonds. Paths involving Ti–H bond breaking are energetically unfavorable.
Site Preference: Hydrogen prefers $A_2B_2$ (Ti $_2$ Cr $_2$ ) interstitial sites over $AB_3$ and $B_4$ sites. In C14, the 6h2 site is the most stable.
Network Topology: Diffusion occurs primarily within hexagonal rings of $A_2B_2$ sites (intra-ring jumps, low barrier). Jumping between rings (inter-ring) requires breaking Ti–H bonds and has a higher barrier.

B. Concentration Dependence of Diffusion

Non-Monotonic Behavior: The diffusion coefficient ( $D$ $D$ ) exhibits a non-monotonic dependence on hydrogen concentration ( $x$ $x$ ):
- $x < 2$ : $D$ increases with $x$ . This is driven by H–H repulsive interactions which lower the effective activation barrier, acting as a driving force for migration.
- $x \approx 2$ : $D$ reaches a maximum.
- $x > 2$ : $D$ decreases with $x$ . At high concentrations, H–H repulsion becomes so strong that it blocks migration paths (crowding effect), hindering diffusion.
Temperature Effect: This non-monotonic trend is more pronounced at lower temperatures (e.g., 400 K) and less distinct at 1000 K.

C. Structural Anisotropy and RDF

Face-Sharing Avoidance: RDF analysis shows hydrogen atoms avoid 1NN face-sharing positions (distance ~1.6 Å) due to strong repulsion.
Edge-Sharing Occupancy: As $x$ increases, occupancy at 2NN edge-sharing sites increases, correlating with the peak in diffusion at $x \approx 2$ .
Anisotropy: While C14 is hexagonal, the diffusion anisotropy is negligible at temperatures $\ge 400$ K. At 250 K, diffusion is anisotropic due to trapping in specific rings (6h1–6h2) perpendicular to the [0001] axis.

D. Comparison with Experiments

Activation Energy: The calculated activation barriers (0.21–0.25 eV) align well with most experimental values (NMR, QENS) for stoichiometric or near-stoichiometric samples.
Diffusion Coefficient Discrepancy: The simulated $D$ $D$ values are approximately one order of magnitude higher than many experimental results.
- Explanation: The authors attribute this to defects in experimental non-stoichiometric samples (e.g., $TiCr_{1.8}$ or $TiCr_{1.9}$ ). Cr vacancies and Ti anti-sites act as hydrogen traps, slowing down kinetics in real materials, a factor not present in the ideal stoichiometric simulations.

5. Significance

Fundamental Understanding: The study clarifies that hydrogen diffusion in Laves phases is not just a function of temperature but is critically controlled by H–H repulsive interactions and the specific topology of the interstitial network.
Design Guidelines: The finding that diffusion peaks at $x \approx 2$ suggests an optimal operating window for hydrogen storage kinetics.
Defect Engineering: The discrepancy between simulation and experiment highlights the critical impact of non-stoichiometry and defects (vacancies/anti-sites) on hydrogen storage performance. Future alloy design must account for these defects to accurately predict kinetics.
Computational Framework: The active-learning fine-tuning of MLIPs provides a robust template for studying other complex hydride systems where high-concentration effects are difficult to model with standard potentials.