# (472f) Identification of Optimally Stable Nanoparticle Geometries Via Mathematical Optimization and Density-Functional Theory

- Conference: AIChE Annual Meeting
- Year: 2018
- Proceeding: 2018 AIChE Annual Meeting
- Group: Particle Technology Forum
- Session:
- Time:
Wednesday, October 31, 2018 - 9:40am-10:00am

One of the key research objectives in the

area of transition metal nanoparticles is to determine the most stable

structure for particles of a given size (mass) and composition. In this work,

we propose a mathematical optimization based modeling framework, coupled with

density functional theory (DFT) calculations, for determining the proven

minimum energy structures of three-dimensional, mono-metallic nanoclusters of a

given size. More specifically, we apply this framework to mono-metallic copper

(Cu) nanoclusters, as previous work has shown potential uses for mono- and

bi-metallic Cu nanoparticles in CO_{2} activation and reduction [1],[2],[3].

In our work, we consider the global minimum

energy Cu nanocluster of a given size to be the one that attains the maximum average

dimensionless cohesive energy, which is the average cohesive energy normalized

to the bulk energy. The cohesive energy measures the strength of inter-atomic

bonding between atoms and the overall stability. Thus, maximizing the average

dimensionless cohesive energy leads to the most stable, ground-state

nanocluster geometry. Our proposed model uses a cohesive energy function, first

proposed by Tomanek et al., which depends only on the geometric descriptor of

coordination number (CN) [4]. Because CN is a local geometric descriptor of the

nanoparticle (NP), it can be encoded within a rigorous, “bottom-up” design optimization

model. In fact, previous work has successfully utilized CN as a local

descriptor to design transition metal surfaces via a mixed-integer linear

programming (MILP) [5], a class of mathematical optimization problems that can

be solved readily and reliably with appropriate numerical optimization

techniques. The current work serves as a first step in extending such tools for

targeted nanoparticle design in three dimensions. It should be highlighted that

the proposed method deviates from the existing literature, inasmuch as existing

methods for determining lowest energy small nanoclusters typically involves heuristic

search of the design space and evaluation of incumbent solutions via inter-atomic

potentials [6],[7],[8],[9]. However, global minimum energy nanoclusters

reported in this way are only considered “putative," as the search methods

utilized provide no mathematical proof of optimality. In contrast, the use of

MILP based optimization techniques proposed in this work guarantees the

optimality of the final designs.

However, the cohesive energy model

originally proposed by Tomanek el al. also includes a repulsive potential energy

term. Whereas previous work has shown that neglecting the repulsive

contribution for sufficiently large nanoclusters compares favorably to the DFT

calculated cohesive energy [10], doing so at small NP sizes is not appropriate due

to surface contraction and quantum effects. In order to create an optimization

model that is accurate for small NPs, we first assemble a large collection of

nanoparticles of given size that feature a wide variety of coordination number

distributions, and then use first-principles DFT calculations to evaluate their

cohesive energy. We then perform linear regression to elucidate a correction

term that can be represented in our MILP optimization framework. This way,

optimally cohesive NPs at small sizes can be correctly and quickly predicted.

Following the above methodology, a pool of

optimal and near-optimal Cu NPs of different small sizes (up to 50 atoms) was

collected, and single-point DFT calculations (using the PBE

exchange-correlation functional [11]) were performed to determine the expected

cohesive energies for these structures. Using this data, a linear regression

model was developed in order to identify corrections for the cohesive energy

contribution of a single atom in a given Cu NP. These corrective factors were then

incorporated into the mixed-integer optimization model, which determined highly

cohesive nanoclusters for each given size, establishing morphological trends

related to stability in NPs of any size. The results confirm various

“magic-number” designs that are well-known to be highly stable, as well as

reveal a number of unintuitive designs that were not previously reported in the

literature, yet are confirmed to be highly stable after DFT geometry

optimizations. These particles can serve as model particles for a multitude of

follow-up studies within the realm of computational chemistry to include, among

others, adsorption and catalytic activity studies. Finally, by collecting

sequences of most cohesive designs for given NP sizes, our results can also be

used to determine expected distributions of clusters that arise during

experimental synthesis.

**References**

[1] Tang, W., et

al., “The importance of surface morphology in controlling the selectivity of

polycrystalline copper for CO_{2} electroreduction,” *Physical
Chemistry Chemical Physics*, 14(1):76-81, 2012.

[2] Austin, N., Ye,

J., Mpourmpakis, G., “CO_{2} activation on Cu-based Zr-decorated

nanoparticles,” *Catalysis Science & Technology*, 7(11):2245-2251,

2017.

[3] Kim, D., et

al., “Synergistic geometric and electronic effects for electrochemical

reduction of carbon dioxide using gold–copper bimetallic nanoparticles.” *Nature
Communications,* 5:4948, 2014.

[4] Tomanek, D.,

Mukherjee, S., Bennemann, K. H., “Simple theory for the electronic and atomic

structure of small clusters,” *Physical Review B*, 28(2):665, 1983.

[5] Hanselman, C.

L., Gounaris, C. E., “A mathematical optimization framework for the design of

nanopatterned surfaces,” *AIChE Journal*, 62(9):3250–3263, 2016.

[6]

Darby, S., et al., “Theoretical study of Cu–Au nanoalloy clusters using a

genetic algorithm,” *The Journal of Chemical Physics*, 116(4):1536-1550,

2002.

[7]

Rapallo, A., et al., “Global optimization of bimetallic cluster structures. I.

Size-mismatched Ag–Cu, Ag–Ni, and Au–Cu systems,” *The Journal of Chemical Physics*,

122(19):194308, 2005.

[8]

Lai, X., Xu, R., Huang, W., “Geometry optimization of bimetallic clusters using

an efficient heuristic method,” *The Journal of Chemical Physics*, 135(16):164109,

2011.

[9] Barcaro, G.,

Sementa, L., Fortunelli, A., “A grouping approach to homotop global optimization

in alloy nanoparticles,” *Physical Chemistry Chemical Physics*, 16(44):

24256-24265, 2014.

[10] Yan, Z., et

al., “Size, Shape and Composition Dependent Model for Metal Nanoparticle

Stability Prediction,” *Nano letters*, 18(4):2696–2704, 2018.

[11] Perdew, J.P.,

Burke, K., Ernzerhof, M, “Generalized gradient approximation made simple,” *Physical
Review Letters*, 77(18):3865, 1996.