Nano-topology optimization for materials design with atom-by-atom control

The manipulation of materials at the nanoscale, as envisioned by Richard Feynman, remains a significant challenge. While additive manufacturing techniques like two-photon lithography enable the creation of complex 3D structures, a computational approach for atom-by-atom material design has been lacking. Existing methods rely on empirical observations, biomimicry (drawing inspiration from natural structures), or mathematical structures like TPMS, which are limited in their ability to find optimal designs for all purposes. Machine learning offers promise but faces challenges with high design variable counts. Topology optimization (TO) provides design freedom but has been limited to the continuum scale. This paper introduces Nano-TO, a new approach that addresses these limitations by combining atomistic modeling with topology optimization to design materials at the nanoscale with atom-by-atom control, aiming to fulfill Feynman's vision.

The paper reviews several existing approaches to materials design, highlighting their limitations. Empirical methods are expensive and time-consuming. Biomimicry, while insightful, isn't universally applicable. TPMS structures, while mathematically interesting and present in nature, haven't proven universally superior. Machine learning shows promise but is still under development for high-dimensional design problems. Traditional topology optimization, relying on finite element methods, is restricted to the continuum scale and cannot design materials at the atomic level.

Nano-TO uses a two-stage optimization process. The first stage focuses on reaching a target volume fraction by iteratively removing less efficient atoms (converted to "virtual" atoms) and adding back more efficient ones (converting "virtual" atoms to "real" atoms). The efficiency is determined by a sensitivity analysis that calculates each atom's contribution to the desired property (e.g., bulk modulus). Sensitivity filtering, using a weighted average of neighboring atoms' sensitivity values, is used to determine the sensitivity of virtual atoms. The process is controlled by rejection and admission rates, carefully balanced to ensure stability. The second stage refines the design until convergence, with equal rejection and admission rates. The process uses atomistic modeling (embedded atom method - EAM) within a cubic unit cell with periodic boundary conditions, allowing for the evaluation of macroscopic elastic properties. The initial structure is based on the face-centered cubic (FCC) structure of aluminum. Optimization parameters include target volume fraction, filter radius, rejection, and admission rates, and these are chosen to ensure stability and capture the surface effect without excessive computational cost. Random initialization is used to explore different initial structures, and the best performing design is selected. The Hashin-Shtrikman (HS) bounds are used as a theoretical maximum for comparison, accounting for anisotropy effects. The surface effect is explored by creating nanoplates of varying thicknesses, and their elastic properties are calculated and compared.

Nano-TO designs consistently outperformed gyroid and other TPMS structures in maximizing bulk modulus across various volume fractions. Remarkably, the optimized designs exceeded the HS upper bound, especially at smaller cell sizes (4 nm). This exceeding of the theoretical maximum is attributed to the surface effect. The analysis of atomic strain distributions showed that Nano-TO designs exhibit a superior load transfer mechanism, leading to higher bulk modulus. The surface effect is significant at the nanoscale, influencing the overall elastic properties; {111} surfaces of the Nano-TO design were found to be stiffer than the bulk material, contributing to the enhanced bulk modulus. By varying the cell size, the researchers demonstrated that the Nano-TO design's advantage over TPMS structures persists even when surface effects become less significant at larger scales, however the increase over the HS upper bound diminished at larger scales and converged towards the HS upper bound at large cell sizes. A supplementary case study demonstrated similar superiority in maximizing the elastic constant C₃₃. The optimized designs exhibited near cubic symmetry, and virtual atoms formed truncated octahedron structures in a BCC arrangement. The dominant surfaces were {111} and {100}, a relationship to the Wulff polyhedron warrants further investigation.

The results demonstrate the efficacy of Nano-TO in designing nanostructured materials with superior properties by leveraging the surface effect. The ability to exceed the HS upper bound highlights the importance of atomistic modeling and the significance of surface effects at the nanoscale, which cannot be captured by continuum-scale methods. The near cubic symmetry and truncated octahedron structures are interesting observations, though further research is needed to fully understand their significance. The observed superior load transfer mechanism in the Nano-TO designs points to the potential of manipulating surface and interface engineering on the nanoscale to surpass theoretical limitations and develop advanced materials.

Nano-TO provides a powerful new platform for atom-by-atom material design, enabling the creation of nanostructured materials with superior properties. The method's ability to surpass theoretical limitations by harnessing surface effects opens up new avenues in materials science. Future research should focus on improving the computational efficiency of Nano-TO, potentially through the integration of machine learning techniques, and exploring its applications to various other materials and properties beyond elastic moduli.

The current implementation of Nano-TO is computationally expensive, limiting its scalability to larger systems. The dependence of the optimized design on the initial structure is another limitation, necessitating multiple random initializations to ensure finding a global minimum. Further research is needed to address these computational challenges, possibly through the incorporation of machine learning.