(310f) A Mixed-Integer Linear Programming Approach for the Design of Nanostructured Catalysts

Recent advances in nanotechnology have enabled the fabrication of materials with precision on the micro- or even nano-scale. These precisely fabricated materials have properties that are wildly different from their bulk counterparts, enhancing useful functionalities in a number of application contexts. For metallic nanoparticles, in particular, recent work has highlighted enhanced catalytic activities [1], unique optical properties [2], and the potential for use in drug delivery [3], among other exploitable properties. With a flourishing synthesis capability comes the need to answer the question: â??Which design leads to the best catalytic performance?â? Unfortunately, the answer to this question is not always obvious due to the combinatorial number of ways in which matter can arrange itself. However, such a material design problem can be formalized as a mathematical optimization problem like those traditionally cast and solved in many process systems engineering contexts. In essence, one seeks to identify a material design (i.e., a specific instantiation of the given class of materials of interest) that maximizes some metric of performance.

In this work, we focus on the design of nanostructured surfaces of transition metal crystals, a material class that has received a lot of attention due to its potential for catalyzing many important reactions [4,5]. We show how correlations that link catalytic activity to site descriptors, such as commonly available volcano plots [6,7], can be used to determine those surface designs that maximize the total reactivity. Volcano plots in practice tend to have steep sides, implying that only the â??idealâ? sites, i.e., those exhibiting features that lie very close to the peak of the volcano, contribute substantially to the overall reaction rate [7,8,9]. With this observation in mind, the problem of designing the surface of highest reactivity simplifies to that of designing the surface that packs the most ideal sites in the unit area, essentially neglecting the contributions from the remaining, non-ideal sites. Furthermore, when the material designs can be expressed via discrete variables, as is the case with materials conforming to well-defined crystalline lattices, and when appropriate site descriptors can be encoded via linear constraints and disjunctions, as is the case of descriptors based on site coordination numbers, we can cast mixed-integer linear programming (MILP) models and utilize well-developed MILP optimization software in order to determine the optimal surface packings. Besides the tractability advantages it offers, an ancillary benefit of using MILP solvers to address this design problem is that several optimal or near-optimal solutions can be collected at a marginal computational cost via a solution pool approach. In this manner, multiple designs can be identified for comparison against some secondary criteria or further consideration by experts.

The proposed MILP approach depends on a suitably defined graph, which serves as a â??design canvas.â? The nodes of this graph constitute the lattice locations present in a periodic, space-filling tile of a thin layer on the surface of a crystal, while the edges of the graph signify those locations that are considered neighbors for determining a site's coordination number. Such a tile can be enforced to be periodic and replicated ad infinitum along a plane so as to form a full surface. The key decision variables in the model are a set of binary variables which indicate the presence or absence of metal atoms in the crystalline lattice locations. A variety of auxiliary site descriptors can then be encoded, as well as variables to indicate whether a location constitutes an ideal site.

Using our approach, we conducted extensive computational studies involving an array of crystallographic lattices. These design lattices were tested against a set of structure function relationships spanning a range of ideal site descriptors in order to identify optimal surface patterns relating to a range of catalytic systems. Our proposed approach validates certain surface patterns that were previously known to be optimal but also reveals a number of non-intuitive designs. We also demonstrate that the optimal surface patterns depend strongly on the target application, which strengthens the need to keep developing those fabrication capabilities that can yield higher levels of structural control. Additionally, it was found that the periodicity of the design canvas leads to considerable symmetry in our MILP model, which retards the progress of branch and bound search algorithms. To that end, we also investigate several symmetry breaking symmetry strategies and report the ones that appear to work best in practice.

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