We have developed a structure refinement method based on genetic algorithm optimization to create structural models of individual nanostructures based on quantitative scanning transmission electron microscopy (STEM) data [1].  We defined a cost function C for a structural model s, as C(s) = E(s) + αχ²[I(s), I_exp], where E is the simulated potential energy of s, χ² is goodness-of-fit between the experimental STEM data I_exp and the simulated STEM data I(s), and α is a weighting parameter.  A genetic algorithm (GA) is used to minimize C over structures s, resulting in a structure that is both at a local minimum in the (simulated) energy and in good agreement with experimental data.  The advantage of combining the energy and goodness-of-fit to experiments over optimization on just one or the other is the ability to refine structures that are not at a global energy minimum (like most nanoparticles) from experimental data that does not completely constrain the three-dimensional structure (like a STEM image in one orientation).

We have validated the approach and implementation using simulated experimental data from a metastable, 309-atom Au inodecahedron, as shown in Figure 1.  Figure 1(a) is the test structure, and Figure 1(b) is the simulated STEM image from that structure.  The energy is calculated using an embedded atom method empirical potential for Au.  Figure 1(d) shows the evolution of the two terms in the cost function and the total cost function over the course of the optimization.  Neither term decreases monotonically for the entire optimization, but the entire C(s) does.  Figure 1(c) shows the STEM image of the refined structure after 2200 generations, which is an essentially perfect match for the input image in (b).  Figure 1(e) shows that the 3D structures are also a perfect match, with a maximum difference in atomic positions of 0.02 Å.

As a first test, we have refined the structure of a ~6000 atom colloidal Au nanoparticle, as shown in Figure 2.  Figure 2(a) is the experimental STEM image of the particle [2].  Figure 2(c) shows the evolution of the cost function as it converges over 4000 generations to reach the final structure in Figure 2(b).  In this case, the optimization was allowed to change the number of atoms in the structure as well as their position.  The result faithful reproduces the image of the sample, including the outline and the twin boundary.  Figure 2(d) shows the displacement of matching atomic columns in the two images.  The large displacements near 0.3 Å arise from surface atoms which are not well-imaged in the experiment due to surface atom mobility under the electron beam, but which are recovered in the refined model.  Additional applications to Pt and Pt-Mo catalysts will be discussed.

1. “Integrated Computational and Experimental Structure Determination for Nanoparticles” Min Yu, Andrew B. Yankovich, Amy Kaczmarowski, Dane Morgan, Paul M. Voyles (submitted)

2. “High-precision scanning transmission electron microscopy at coarse pixel sampling” Andrew B. Yankovich, Benjamin Berkels, W. Dahmen, P. Binev, and Paul M. Voyles Advanced Chemical and Structural Imaging 1, 2 (2015).  DOI: 10.1186/s40679-015-0003-9

Figures:

Ino-decahedron simulated test case: (a) the starting structure; (b) the simulated STEM image of the starting structure; (c) the simulated STEM image of the final, refined structure; (d) the evolution of the cost function over the course of the refinement; and (e) a histogram of the distances between atoms in the starting structure and the refined structure.

Colloidal Au nanoparticle case: (a) the experimental STEM image; (b) the simulated STEM image of the final, refined structure; (c) the evolution of the cost function over the course of the refinement; (d) histogram of the distances between atomic columns in the starting image and the simulated image of the refined structure.

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EMC Abstracts - https://emc-proceedings.com/abstract/nanoparticle-structure-from-genetic-algorithm-refinement-against-quantitative-stem-data/