(684b) Design of Metallic Surface Nanostructures Using Mathematical Optimization

Technological advances in the area of nanofabrication of transition metal crystalline surfaces are enabling new opportunities for synthesizing nanostructured catalysts at ever-increasing levels of structural control. The developed transition metal materials are particularly exciting for their high-performing, unique catalytic properties that often differ dramatically from bulk catalysts [1]. More specifically, rational design of catalysts is being enabled by advances in electrochemical deposition [2], formation of near-surface alloys [3], and photochemical lithography [4], among other synthesis techniques. These advances result in a large space of possible catalyst designs, and as time goes on, the complexity of realizable designs will only continue to grow. This growing design space represents a great opportunity for identifying novel, high-performing materials, but also poses a significant challenge to rigorously identify the best designs in terms of reactivity and stability. We propose formulating this materials design question as a mathematical optimization model [5], which leverages the use of well-established approaches for solving highly combinatorial problems.

In this work, we identify the nanostructured surfaces that maximize the reactivity of catalysts by solving a mathematical optimization model that represents the relevant choices using an appropriate set of decision variables, an objective function, and algebraic constraints. We consider the key decision variables to be the choice of placement of atoms at the surface of a periodic crystal lattice, thereby forming a particular surface nanostructure that exhibits complex active sites. To link the structure to catalytic functionality, we utilize demonstrated correlations from density functional theory (DFT) that link surface geometric descriptors to chemical properties such as adsorption energy or activation energy [6,7]. In many cases of interest, the emergence of volcano plot correlations to describe reactivity on a catalyst site allows us to model this reactivity through the identification of “ideal sites”; that is, active sites that correspond to the tip of the volcano plots and that tend to dominate the exhibited turnover on the catalyst surface [8]. This binary nature of our decision variables leads to mixed integer linear program (MILP) models that can be solved with well-developed optimization software with rigorous guarantees on the optimality of identified solutions.

The identified surface structures serve as theoretical bounds on the activity that can be achieved by a catalyst surface and may also serve as targets for experimental synthesis, as new nanofabrication methods are developed. However, it is possible that the identified solutions of the aforementioned optimization model are in fact unstable under reaction conditions and would therefore rearrange to unproductive surfaces. Therefore, we have developed several approaches to account for the stability of designs as a critical element of the process of materials design via mathematical optimization. The first approach collects a solution pool of highly reactive designs, which are offline filtered for structures that are determined to be stable. This evaluation of stability can be carried out using computational methods such as DFT or the embedded atom method (EAM) [9]. As a second approach, the EAM calculations can be embedded directly into the optimization model, resulting in algebraic constraints that directly restrict the search space to only include designs that are considered stable.

Using our mathematical optimization paradigm, we have proposed optimal nanostructured surface designs for a variety of crystallographic surfaces and across an array of descriptions of ideal reactive sites. The results demonstrate patterns which may be predictable in some cases, but also reveal that the optimal reactive surfaces are often non-intuitive. We show that the optimal surface is strongly dependent on the definition of the ideal reactive site and that the characterization of a “stable” surface can have dramatic impact on the resulting design. A feature of this framework is the flexibility to encode a multitude of materials design considerations in a generic fashion, thereby decoupling the complications of modeling a material design problem from solving the resulting optimization problem.

References

[1] Narayanan R., El-Sayed M. A., “Shape-dependent catalytic activity of platinum nanoparticles in colloidal solution,” Nano Letters, 4(7):1343–1348, 2004.

[2] Calle-Vallejo F., Koper M. T. M., Bandarenka A. S., “Tailoring the catalytic activity of electrodes with monolayer amounts of foreign metals,” Chemical Society Reviews, 42(12):5210–30, 2013.

[3] Stamenkovic V. R., Mun B. S., Mayrhofer K. J. J., Ross P. N., Markovic N. M., “Effect of surface composition on electronic structure, stability, and electrocatalytic properties of Pt-transition metal alloys: Pt-skin versus Pt-skeleton surfaces,” Journal of the American Chemical Society, 128(27):8813–8819, 2006.

[4] Geissler M., Xia Y., “Patterning: Principles and some new developments,” Advanced Materials, 16(15):1249–1269, 2004.

[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] Nørskov J. K., Bligaard T., Logadottir A., Bahn S., Hansen L. B., Bollinger M., Bengaard H., Hammer B., Sljivancanin Z., Mavrikakis M., Xu Y., Dahl S., Jacobsen C. J. H., “Universality in heterogeneous catalysis,” Journal of Catalysis, 209(2):275–278, 2002.

[7] Calle-Vallejo F., Martínez J. I., García-Lastra J. M., Sautet P., Loffreda D., “Fast prediction of adsorption properties for platinum nanocatalysts with generalized coordination numbers,” Angewandte Chemie - International Edition, 53(32):8316–8319, 2014.

[8] Calle-Vallejo F., Loffreda D., Koper M. T. M., Sautet P., “Introducing structural sensitivity into adsorption-energy scaling relations by means of coordination numbers,” Nature Chemistry, 7(5):403–410, 2015.

[9] Daw M. S., Baskes M. I., “Embedded-atom method: Derivation and applications to impurities, surfaces, and other defects in metals,” Physical Review B, 29(12):6443–6453, 1984.