Dimensionality and Design of Isotropic Interactions that Stabilize Honeycomb, Square, Simple Cubic, and Diamond Lattices

Abstract

We use inverse methods of statistical mechanics and computer simulations to investigate whether an isotropic interaction designed to stabilize a given two-dimensional lattice will also favor an analogous three-dimensional structure, and vice versa. Specifically, we determine the 3D-ordered lattices favored by isotropic potentials optimized to exhibit stable 2D honeycomb (or square) periodic structures, as well as the 2D-ordered structures favored by isotropic interactions designed to stabilize 3D diamond (or simple cubic) lattices. We find a remarkable “transferability" of isotropic potentials designed to stabilize analogous morphologies in 2D and 3D, irrespective of the exact interaction form, and we discuss the basis of this cross-dimensional behavior. Our results suggest that the discovery of interactions that drive assembly into certain 3D periodic structures of interest can be assisted by less computationally intensive optimizations targeting the analogous 2D lattices.

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Plain Language Summary

Materials containing nanometer-to-micron-sized particles exhibit a variety of properties that depend on the spatial arrangement of the constituent particles. For example, three-dimensional, multiphase materials with periodic diamond structures can display unusual optical properties of interest to technological applications such as solar cells. One attractive way to synthesize such materials is from the bottom up, i.e., by designing interactions between the components to drive their spontaneous assembly into the desired structures. Examples of such structures include colloidal superlattices and photonics materials. We use inverse methods of statistical mechanics to design interparticle interactions that drive particles to assemble into ordered structures.

The design rules for self-assembly are still poorly understood and are known to be dependent on the shapes, chemistries, and surface properties of the interacting particles. One interesting line of inquiry is how spatial dimension affects design rules. For example, it is still unknown what three-dimensional structures naturally assemble from particles with interactions designed to stabilize a specific two-dimensional structure. We use ground-state calculations and quench simulations to predict how a fluid changes its structure when cooled instantaneously. We show that isotropic interactions designed to assemble into two-dimensional honeycomb and square lattices also naturally favor certain analogous two-dimensional periodic structures (diamond and simple cubic lattices, respectively), and vice versa.

Apart from enhancing our understanding of which interactions favor technologically interesting two-dimensional and three-dimensional material structures, our results suggest how computationally inexpensive two-dimensional material optimizations can help to isolate rare isotropic interactions that drive the assembly of materials with three-dimensional diamond and simple cubic symmetries.