Designing convex repulsive pair potentials that favor assembly of kagome and snub square lattices

Abstract

Building on a recently introduced inverse strategy, isotropic and convex repulsive pair potentials were designed that favor assembly of particles into kagome and equilateral snub square lattices. The former interactions were obtained by a numerical solution of a variational problem that maximizes the range of density for which the ground state of the potential is the kagome lattice. Similar optimizations targeting the snub square lattice were also carried out, employing a constraint that required a minimum chemical potential advantage of the target over select competing structures. This constraint helped to discover isotropic interactions that meaningfully favored the snub square lattice as the ground state structure despite the asymmetric spatial distribution of particles in its coordination shells and the presence of tightly competing structures. Consistent with earlier published results [W. Piñeros et al., J. Chem. Phys. 144, 084502 (2016)], enforcement of greater chemical potential advantages for the target lattice in the interaction optimization led to assemblies with enhanced thermal stability.

Assembly of multi-flavored two-dimensional colloidal crystals

“…25–36 Consequently, it also known that subtle changes to such a pair potential’s form or shape can shift the relative stability of different crystal polymorphs. 37,38 Since it is not always clear how to precisely realize a computationally-engineered potential in an experimental setting, it can be difficult to control the ubiquitous problem of polymorphism with such an approach. Furthermore, the high densities and pressures required to achieve assembly with purely repulsive interactions are not always representative of experimentally realizable systems.…”

Designing pairwise interactions that stabilize open crystals: Truncated square and truncated hexagonal lattices

“…As described in detail previously, 20,24 with interactions of this type, one can analytically formulate a nonlinear program whose numerical solution provides pair potential parameters that minimize the objective function F = j (µ t − µ l,j ). Here, µ t is the zero-temperature [T = 0] chemical potential of the target lattice at a specified density ρ 0 , and µ l,j is that of an equi-pressure lattice j from a specified set of competitive 'flag-point' structures (discussed below); the sum is over all such flag-point competitors.…”

“…First, we carry out a preliminary optimization comparing the chemical potential of the target to others in an initial pool comprising a few select lattice and mesophase structures (e.g., stripes) known to be competitive for systems with isotropic, repulsive interactions. 20,24 We then carry out a 'forward' calculation that considers more comprehensively equi-pressure competitors. For classes of competing structures that contain free parameters, the values of those parameters are determined by minimizing the chemical potential (using GAMS) under the optimized pair potential (for details see appendix of ref.…”

Design of two-dimensional particle assemblies using isotropic pair interactions with an attractive well

“…The detailed calculation procedure for single phases has been explained in detail previously. 27 A discussion of how to consider phase-separated crystals is provided in the supplementary material. Following this approach, the final competing pools for square, honeycomb and kagome lattices were determined to be as follows:…”

“…Researchers have developed robust inverse design strategies that have been applied to discover isotropic interactions that stabilize a variety of open structures in two and three dimensions under various constraints (including honeycomb, [16][17][18][19] kagome, [20][21][22] simple cubic, 19,23 and diamond [23][24][25] lattices, to mention a few). Recently, a reformulation of this type of GS optimization problem was introduced 26 which significantly improved performance and allowed for (1) exploration of how design goals affect trade offs for the target phase (e.g., thermal versus volumetric stability 26 ) and (2) discovery of interactions that stabilize very challenging target structures (e.g., snub square, 27 truncated square and truncated hexagonal 28 lattices). An advantage of GS-focused optimization is that, because it requires the a priori identification of structures that most closely compete with the target lattice, it can offer insights into how those competitions influence the functional form of the optimized interactions.…”