Grand canonical generative diffusion model for crystalline phases and grain boundaries

TL;DR

The paper addresses the difficulty of generating ordered crystalline structures with particle-based diffusion models, where local-minima trapping during score-driven simulated annealing hinders sampling. It introduces a grand canonical voxel diffusion model that represents atoms on a voxel grid with a continuous particle count, using a particle-to-voxel encoder and a voxel-to-particle decoder within a voxel-space diffusion prior based on a 3D U-Net. The approach is validated on simple crystalline phases (SC, FCC, BCC) and on tungsten grain boundaries, including a conditional in-painting strategy that recovers ground-state GB configurations at $[n]$ values and relaxes to low-energy states with energy $E_{ ext{gb}}$ given by $E_{ ext{gb}} = rac{E^{ ext{gb}}_{ ext{total}} - N^{ ext{gb}}_{ ext{total}} E^{ ext{bulk}}_{ ext{coh}}}{A^{ ext{gb}}_{ ext{plane}}}$. Results show accurate generation of ordered crystals and plausible GB structures, demonstrating the model’s potential for materials design and GB engineering, while acknowledging limitations in unit-cell flexibility and memory demands and pointing to future enhancements such as latent diffusion and equivariant architectures.

Abstract

The diffusion model has emerged as a powerful tool for generating atomic structures for materials science. This work calls attention to the deficiency of current particle-based diffusion models, which represent atoms as a point cloud, in generating even the simplest ordered crystalline structures. The problem is attributed to particles being trapped in local minima during the score-driven simulated annealing of the diffusion process, similar to the physical process of force-driven simulated annealing. We develop a solution, the grand canonical diffusion model, which adopts an alternative voxel-based representation with continuous rather than fixed number of particles. The method is applied towards generation of several common crystalline phases as well as the technologically important and challenging problem of grain boundary structures.

Figures (9)

• Figure 1: Comparison of diffusion models for simple cubic structures: (a) particle-based generator with fixed number of particles, and (b) voxel-based grand canonical generator with continuously adjustable number of particles. (c) Grain boundary phase transition in tungsten $\Sigma5(100)$ twist boundary. The two GB phases with different atomic densities $[n]=1/5$, $2/5$ are shown in blue and red, respectively. The bulk atoms of the misoriented upper and lower crystals are shown in gray.
• Figure 2: Effects of different levels of randomness $\sigma$ in initial structures on particle diffusion models. $\sigma=0 \ (\gtrapprox 0.5)$ means unperturbed (random) initial structure. Radial distribution functions of simple cubic structure (lattice constant 1) averaged over ten $4\times 4\times 4$ supercells are shown. Left: initial structures of ideal SC+Gaussian displacements $\mathcal{N}(0,\sigma^2)$; right: generated with diffusion.
• Figure 3: Grand canonical diffusion model with voxel representation. (a) Particle to voxel encoder by smearing using linear weights and no learnable parameters. For clarity a 2D slice is also shown with red dots designating particle positions. (b) Learnable voxel to particle coordinate decoder. Both the training and inference stages are shown. (c) Voxel-space diffusion using a convolutional U-Net.
• Figure 4: Ideal and generated structure for SC, BCC and FCC phases, and the corresponding Steinhardt order parameters and radial distribution functions averaged over ten $4^3$ cells.
• Figure 5: Conditional grain boundary generation via in-painting.