Stereological 3D modeling of nano-scale catalyst particles using TEM projections

TL;DR

The paper tackles the challenge of extracting 3D nanoscale catalyst morphologies from 2D TEM data by introducing a two-stage stereological modeling framework. A coarse hull model based on a random field of spherical harmonics up to order 6 captures global particle shapes, while a fine surface model using overlapping spheres plus morphological closing adds nanometer-scale features; both models are calibrated against 2D TEM projections obtained at two resolutions using a GAN-based fitting scheme with differentiable radial projections. The authors demonstrate that the approach reproduces key 2D descriptors and yields plausible distributions for 3D descriptors such as volume and surface area, enabling the generation of realistic 3D digital twins for virtual materials testing and structure–property analysis in catalysis. This methodology provides a practical path to relate nanoparticle morphology to catalytic performance without extensive 3D imaging, with potential extensions to core-shell systems and systematic exploration of morphology–activity relationships.

Abstract

Catalysis, particularly heterogeneous catalysis, is crucial in the chemical industry and energy storage. Approximately 80% of all chemical products produced by heterogeneous catalysis are produced by solid catalysts, which are essential for the synthesizing of ammonia, methanol, and hydrocarbons. Despite extensive use, challenges in catalyst development remain, including enhancing selectivity, stability, and activity. These effective properties are influenced by the nanoscale morphology of the catalysts, whereby the size of the nanoparticles is only one key descriptor. To investigate the relationship between nanoparticle morphology and catalytic performance, a comprehensive 3D analysis of nano-scale catalyst particles is necessary. However, traditional imaging techniques for a representative recording of this size range, such as transmission electron microscopy (TEM), are mostly limited to 2D. Thus, in the present paper, a stochastic 3D model is developed for a data-driven analysis of the nanostructure of catalyst particles. The calibration of this model is achieved using 2D TEM data from two different length scales, allowing for a statistically representative 3D modeling of catalyst particles. Furthermore, digital twins of catalyst particles can be drawn for the stochastic 3D model for virtual materials testing, enhancing the understanding of the relationship between catalyst nanostructure and performance.

Figures (9)

• Figure 1: Data preprocessing steps: (a) Low-resolution TEM image (projection) of particles. (b) Segmentation of image data into individual non-overlapping particles. (c) Magnification of a single segmented particle projection. (d) Radial representation of the particle outline.
• Figure 2: (a) High-resolution TEM image of a single particle. (b) Segmentation of the (projected) particle outline. (c) Sphere-based representation of surface roughness.
• Figure 3: (a) 3D overlapping sphere packing. (b) Simulated 3D particle after applying morphological closing. (c) Outline of simulated particle. (d) Thickness profile of simulated 3D particle.
• Figure 4: Architecture of the generator network for computing realizations $a_{\ell m}$ of the random spherical harmonics coefficients $f_{\ell m}(X)$. Left: General network architecture with output sizes $s_0,\ldots,s_4$ assigned to the down arrows. Right: Inner structure of linear blocks.
• Figure 5: Architecture of the discriminator network $D$, designed to distinguish between simulated and measured radius functions. The numbers correspond to the input and output dimensions of the respective layers.