Grain Boundary Energy Calculation

Goal

To compute the specific grain boundary energy ($\gamma_{GB}$, J/m²) for a series of coincidence site lattice (CSL) grain boundaries using Machine Learning Interatomic Potentials. This enables:

• Identification of low-energy special grain boundaries (Σ3, Σ5, Σ7, ...)
• Anisotropy analysis of GB energy as a function of misorientation angle
• Input data for polycrystalline simulations (phase-field, kinetic Monte Carlo)

The grain boundary energy is defined as:

$$\gamma_{GB} = \frac{E_{GB} - N \cdot E_{bulk}}{2 A}$$

where $E_{GB}$ is the total energy of the GB supercell, $N$ is the number of atoms, $E_{bulk}$ is the DFT/MLIP energy per atom of the relaxed bulk, and $A$ is the interfacial area (one GB, periodic cell contains two identical GBs hence the factor of 2).

Instructions

1. Select Foundation Potential

GB calculations benefit from accurate interatomic forces. Prefer r2SCAN-level models for energy accuracy:

• MACE-MH-1 with matpes_r2scan head (recommended)
• CHGNet-MatPES-r2SCAN-2025.2.10-2.7M-PES (MatGL)
• TensorNet-MatPES-r2SCAN-v2025.1-PES (MatGL, faster)

Refer to ml-foundation-potentials.

2. Relax Bulk Reference

Perform a high-accuracy bulk relaxation to obtain $E_{bulk}$.

# Env: matgl-agent
mcp_matgl_load_model(model_name="CHGNet-MatPES-r2SCAN-2025.2.10-2.7M-PES")
mcp_matgl_relax_structure(
    structure_data="bulk.cif",
    relax_cell=True,
    fmax=0.005,
    output_dir="bulk_relaxation/"
)

Record the final energy per atom ($E_{bulk}$) from the relaxation output JSON.

3. Generate Grain Boundary Structures

Use pymatgen's GrainBoundaryGenerator to create CSL grain boundary supercells for a range of Σ values and rotation angles.

# Env: base-agent
python .agents/skills/mat-grain-boundary/scripts/create_grain_boundary.py \
    --bulk bulk_relaxation/relaxed_structure.cif \
    --rotation-axis 0 0 1 \
    --max-sigma 29 \
    --min-slab-size 10.0 \
    --vacuum 0.0 \
    --output-dir gb_structures/

Key parameters:

Parameter	Description	Example
--rotation-axis	Rotation axis as 3 integers	0 0 1 (tilt) or 1 1 0
--max-sigma	Maximum Σ value to enumerate	29 (generates Σ3, Σ5, Σ7...)
--min-slab-size	Minimum thickness of each grain (Å)	10.0
--output-dir	Directory for generated GB CIF files	gb_structures/

The script writes one CIF per unique GB, with filename format sigma{Σ}_{angle:.1f}deg_{hkl}.cif.

[!TIP] For a first survey use --max-sigma 13 (gives Σ1, Σ3, Σ5, Σ7, Σ9, Σ11, Σ13). Increase to 29 for more complete misorientation curves.

4. Relax Grain Boundary Structures

Relax all generated GB structures. Do NOT relax the cell in directions parallel to the GB plane — use relax_cell=False to fix the in-plane lattice vectors and only relax atomic positions.

# Env: matgl-agent
mcp_matgl_load_model(model_name="CHGNet-MatPES-r2SCAN-2025.2.10-2.7M-PES")
mcp_matgl_relax_structure(
    structure_data="gb_structures/",
    relax_cell=False,
    fmax=0.02,
    output_dir="gb_relaxations/"
)

[!IMPORTANT] relax_cell=False is required. The in-plane lattice vectors of the GB supercell are fixed by the CSL geometry and must remain constant to preserve the grain boundary orientation.

5. Calculate Grain Boundary Energies

# Env: base-agent
python .agents/skills/mat-grain-boundary/scripts/calculate_gb_energy.py \
    --bulk-energy-per-atom -4.567 \
    --gb-relaxation-dir gb_relaxations/ \
    --output-dir gb_results/

The script reads the JSON output from each relaxation, applies the GB energy formula, and generates:

• gb_energy_results.json — Σ, angle, area, $\gamma_{GB}$, and provenance per boundary
• gb_energy_vs_angle.png/svg — plot of $\gamma_{GB}$ (J/m²) vs. misorientation angle (°)
• gb_summary_table.csv — CSV for further analysis

6. Identify Low-Energy Boundaries

Inspect the output table and plot to identify:

• Cusps: sharp dips in $\gamma_{GB}$ vs. angle correspond to special low-energy GBs (Σ3 ≈ twin boundary, Σ5, etc.)
• Asymmetric tilt boundaries: generated alongside symmetric ones; compare energies
• Coincidence index correlation: low-Σ GBs typically have the lowest energies

Examples

Example: Copper [001] Tilt Grain Boundaries

Copper is a well-benchmarked system. MLIP values should be compared to DFT/MD literature.

# 1. Query and relax bulk Cu
mcp_base_search_materials_project_by_formula(formula="Cu", save_to_file="Cu_bulk.cif")
mcp_matgl_load_model(model_name="TensorNet-MatPES-r2SCAN-v2025.1-PES")
mcp_matgl_relax_structure(structure_data="Cu_bulk.cif", relax_cell=True, fmax=0.005, output_dir="Cu_bulk_relax/")

# 2. Generate [001] tilt GBs up to Σ13
python .agents/skills/mat-grain-boundary/scripts/create_grain_boundary.py \
    --bulk Cu_bulk_relax/relaxed_structure.cif \
    --rotation-axis 0 0 1 --max-sigma 13 --output-dir Cu_gb_structures/

# 3. Relax GB structures
mcp_matgl_relax_structure(structure_data="Cu_gb_structures/", relax_cell=False, fmax=0.02, output_dir="Cu_gb_relax/")

# 4. Calculate GB energies (E_bulk ≈ -3.73 eV/atom for Cu with TensorNet)
python .agents/skills/mat-grain-boundary/scripts/calculate_gb_energy.py \
    --bulk-energy-per-atom -3.73 \
    --gb-relaxation-dir Cu_gb_relax/ \
    --output-dir Cu_gb_results/

Literature comparison: For Cu [001] tilt boundaries, the Σ5 (36.9°) boundary has $\gamma_{GB}$ ≈ 0.8–1.0 J/m² from MD simulations. The Σ3 (twin) boundary has $\gamma_{GB}$ < 0.05 J/m².

Constraints

• Cell relaxation: Always use relax_cell=False for GB structures. The in-plane dimensions define the CSL geometry and must not change.
• Slab thickness: Use --min-slab-size ≥ 10 Å to ensure bulk-like behaviour at the centre of each grain. Thin grains cause artificial interaction between the two GBs in the periodic cell.
• Same MLIP for bulk and GB: Use the identical model and settings for both bulk relaxation (Step 2) and GB relaxation (Step 4) to ensure energy cancellation in the GB energy formula.
• Vacuum: Set --vacuum 0.0. Unlike surface calculations, grain boundary supercells should have no vacuum — both grains are connected periodically.
• Environment: All scripts run in base-agent. MLIP relaxations use the respective MLIP environment.

References

• Zhao et al., "Automated generation and analysis of grain boundary structures using machine learning potentials", npj Comput. Mater., 2021.
• Holm & Foiles, "How Grain Boundary Properties Are Influenced by the Character of the Boundary", Science, 2010.
• Tschopp & McDowell, "Asymmetric Tilt Grain Boundary Structure and Energy in Copper and Aluminium", Phil. Mag., 2007.
• Pymatgen GrainBoundaryGenerator documentation: link

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Author: Yu Yao, Bowen Deng Contact: GitHub @AI4SciDisc