Wang-Landau study of lattice gases on geodesic grids

TL;DR

The paper studies equilibrium self-assembly of lattice-gas particles on curved geodesic grids (PSD, HPTI, HPCD) to understand how curvature and interaction range shape low-temperature order. It employs Wang-Landau sampling to compute the exact density of states $g_{{\cal N},{\cal E}}$ and the grand partition function $\Xi(T,\mu)$, enabling precise mapping of phase crossovers via plateaus in $N(\mu)$ and peaks in $\kappa_T$ across several interaction families (core-corona, LJ-like, SALR). Across the three geodesic grids, the study finds robust motifs such as polyhedral order, cluster crystals anchored at fivefold vertices, and worm-like structures; the results also show how curvature shifts and modulates these patterns relative to the triangular lattice, with the largest grids best approximating planar behavior. The work provides design insights for spherical templating and patchy-particle assembly on curved substrates, and suggests extensions to binary mixtures and more complex interactions for targeted functionalities.

Abstract

We study a family of lattice-gas systems defined on semiregular grids, obtained by projecting the vertices of three different geodesic icosahedra onto a spherical surface. By using couplings up to third neighbors we explore various interaction patterns, ranging from core-corona repulsion to square-well attraction and short-range attractive, long-range repulsive potentials. The relatively small number of sites in each grid ($\sim 100$) enables us to compute the exact statistical properties of the systems as a function of temperature and chemical potential by Wang-Landau sampling. For each case considered we highlight the existence of distinct low-temperature ``phases'', featuring, among others, regular-polyhedral, cluster-crystal, and worm-like structures. We highlight similarities and differences between these motifs and those observed on the triangular lattice under the same conditions. Finally, we discuss the relevance of our results for the bottom-up realization of spherical templates with desired functionalities.

Figures (21)

• Figure 1: The three geodesic icosahedra considered in this paper. From left to right: PSD, HPTI, and HPCD (see the acronyms defined in the text). Assuming the mean first-neighbor distance as reference, the curvature of the grid gets smaller from left to right. Each polyhedron in the figure is obtained from a corresponding parent polyhedron (e.g., the snub dodecahedron for the PSD) by raising a pyramid on each pentagonal and/or hexagonal face (Conway's kis operation). The height of the pyramid can be chosen such that all edges are tangent to the unit sphere. Different colors are used for different types of triangles. The total number of vertices is 72 for PSD, 92 for HPTI, and 122 for HPCD. The fivefold vertices are twelve in any case.
• Figure 2: PSD model, a couple of minimum-energy configurations: $(1,1,1)$ interaction and $N=36$ (left); $(1,1,0)$ interaction and $N=22$ (right). In this and similar pictures, particles (occupied sites) are represented as small red balls; holes (empty sites) as transparent light-gray balls.
• Figure 3: PSD model, $(n,0,-1)$ interactions with $n=1,2,3$: $N$ vs. $\mu$ for two temperatures (see legend).
• Figure 4: PSD model, $(n,0,-1)$ interaction: The two minimum-energy configurations for $N=36$ (left), along with their stereographic projections (right). The biggest cluster is colored in brown; smaller clusters are drawn in orange. The horizontal and vertical scales have no particular meaning.
• Figure 5: PSD model, $(1,0,-1)$ interaction: $P$ vs. $\mu$ (left) and $P$ vs. $\rho$ (right) for three temperatures (see legend).