Using Particle Shape to Control Defects in Colloidal Crystals on Spherical Interfaces

TL;DR

The study tackles how curvature on a spherical interface imposes topological defects in colloidal crystals and how particle shape can be tuned to control defect morphology, using hard-particle Monte Carlo simulations of rounded cubes and tetrahedra constrained to a sphere with a rounding parameter $s$. Particles are modeled on a sphere of radius $R_S$ and sampled via $s\in[0,1]$ to interpolate from polyhedra to spheres, while local order is quantified by four order parameters ($|\psi_6|_{nn}$, $f_{nn}$, $hc_{nn}$, $w_{nn}$) and defect networks are analyzed across densities $\rho_n$. Key results show that spheres yield 12 disclinations with icosahedral symmetry; rounded cubes favor a simple-square lattice with eight three-fold defects distributed in a square antiprismatic pattern; rounded tetrahedra reveal honeycomb and woven motifs with highly variable defect patterns, often lacking smooth transitions due to topology. The findings demonstrate programmable defect generation via shape design, with implications for vesicle buckling and colloidosome engineering on curved interfaces.

Abstract

Spherical particles confined to a sphere surface cannot pack densely into a hexagonal lattice without defects. In this study, we use hard particle Monte Carlo simulations to determine the effects of continuously deformable shape anisotropy and underlying crystal lattice preference on inevitable defect structures and their distribution within colloidal assemblies of hard rounded polyhedra confined to a closed sphere surface. We demonstrate that cube particles form a simple square assembly, overcoming lattice/topology incompatibility, and maximize entropy by distributing eight three-fold defects evenly on the sphere. By varying particle shape smoothly from cubes to spheres we reveal how the distribution of defects changes from square antiprismatic to icosahedral symmetry. Congruent studies of rounded tetrahedra reveal additional varieties of characteristic defect patterns within three, four, and six-fold symmetric lattices. This work has promising implications for programmable defect generation to facilitate different vesicle buckling modes using colloidal particle emulsions.

Figures (5)

• Figure 1: Shape definition and constraining radius: (a) Particle centroids are constrained to move on a spherical interface with radius $R_S$. (b) Rounded polyhedra are defined by the union of an underlying polyhedron (cube or tetrahedron) and a rounding sphere. The rounding parameter $s$ uses the polyhedron insphere radius $R_{PI}$ and a sphere with rounding radius $R_R$ to define particle shape, seen here for a head-on view of a rounded cube.
• Figure 2: Phase diagrams for systems with $n=1000$ particles over a range of particle roundedness $s$ and number densities $\rho_n$. For cubic rounded polyhedra the face-aligned motif dominates at low rounding values $s_{cube}$, gradually shearing as the rounding of the shape increases, and exhibits high hexagonal ordering as $s_{cube}$ approaches 1. Striped regions at $s_{cube} = 0.3$ and $\rho_N\ge0.272$ indicate regions where both hexagonal and face-aligned motifs are observed. When $s = 1$ (middle), idealized spheres assemble the hexagonal motif. At high values of $s_{tetrahedron}$ the hexagonal motif continues to dominate. With intermediate rounding, tetrahedra assemble the honeycomb structure before giving way to the woven structure as $s \rightarrow 0$. The four distinct motifs (right) are color coded to match the phase diagram, with motif insets corresponding to the appropriate black-filled marker. These specific motifs correspond to the following values of (shape, $s$, $\rho_n$, marker): (Sphere, $1.0$, $0.258$, yellow circle), (Cube, $0.0$, $0.282$, dark blue diamond), (Tetrahedron, $0.2$, $0.272$, light blue downward triangle), (Tetrahedron, $0.0$, $0.258$, red upturned triangle) from left to right, top to bottom. The white filled marker at $s_{cube} = 0.6$ and $\rho_n = 0.287$ indicates a jammed state rather than self-assembled structure.
• Figure 3: Reference radial distribution function $g(\theta)$ plots for each motif: (a) A representative $g(\theta)$ for the hexagonal motif, taken from simulations of spheres with $s=1$ and $\rho_n = 0.258$. (b) A simulation image at these parameters colored by $|\psi_6|_{nn}$. (c) Representative $g(\theta)$ for the face-aligned motif, most commonly realized in a simple-square lattice, taken from simulations of rounded cubes with $s=0$ and $\rho_n = 0.282$. (d) A simulation image at these parameters colored by $f_{nn}$. (e) Representative $g(\theta)$ for the honeycomb motif, taken from simulations of rounded tetrahedra with $s=0.2$ and $\rho_n = 0.267$. (f) A simulation image at these parameters colored by $hc_{nn}$. (g) Representative $g(\theta)$ for the woven motif, taken from simulations of rounded tetrahedra with $s=0$ and $\rho_n = 0.263$. (h) A simulation image at these parameters colored by $w_{nn}$.
• Figure 4: Defect Distribution: Defect distribution is presented in Mercator projections, and quantified by the geodesic radial distribution of defects, $g_{DD}(\theta)$, across number densities $\rho_n$. Systems are binarized by a value noted by the midpoint of their respective color bars seen in the third column. (a) In a system of $n=1000$ spheres there are twelve isolated defect regions. (b) The distribution of these defects is icosahedral, seen by peaks in $g_{DD}(\theta)$ at geodesic distances corresponding to the vertices of an icosahedron. The representative system has a density of $\rho_n = 0.258$. (c) In systems of $n=1000$ cubes, with a value of $s=0$, there are eight isolated defects. (d) Their distribution is square antiprismatic, as seen in peaks in $g_{DD}(\theta)$ at $\theta_1, \theta_2^{AP}, \textrm{ and }\theta_4^{AP}$. There are no peaks at $\theta_3$ and $\theta_5^{C}$ that would indicate a cubic distribution. The representative system has a density of $\rho_n = 0.282$. (e) In systems of $n=1000$ rounded tetrahedra with $s=0.2$, there is no regular distribution of defects. (f) Given this disorder the defects show no correlation or coordination across the surface, as $g_{DD}(\theta)$ remains small across all values of $\rho_n$. The representative system has a density of $\rho_n = 0.267$. (g) In systems of $n=1000$ tetrahedra, with a value of $s=0$, eight isolated defects appear only at a high density $\rho=0.263$. (h) The spatial distribution of defects shows little to no correlation across the surface for all densities $\rho_n< 0.263$, with the first trough in $g_{DD}(\theta)$ indicating only that defect regions are spatially isolated from each other. Each of the Mercator projections shown here (a, c, e, g), correspond to the simulation images shown in figure \ref{['fig:rdfReference']}(b, d, f, h).
• Figure 5: Defect lengths $d[\sigma]$ are measured as the longest shortest path in the simply connected graph formed by defect particles for each system over a range of particle number $n = {1000, 1500, 2000, 2500}$. (a) Defect lengths as a function of $\rho_n$ are shown for spheres with $s=1$, generally increasing as a function of $n$ for sufficiently high $\rho_n$. (b) This increase in $d[\sigma]$ is shown in representative defect images at $n = 1000 \textrm{ (left) and } 2500 \textrm{ (right)}$, at a density $\rho_n = 0.258$. (c) In systems of rounded cubes with $s_{cube}=0$, defect lengths can be seen to generally decrease as $\rho_n$ increases, across all values of $n$. (d) The length $d[\sigma]$ is consistent across $n$, with $n = 1000 \textrm{ (left) and } 2500 \textrm{ (right)}$, seen in representative defects with $\rho_n = 0.287$. (e) Rounded tetrahedra with $s_{tetrahedron}=0$ form a large variety of defects, with broadly distributed lengths, seen for $\rho_n = 0.263$. (f) A range of exemplary defects are shown over the range $n = {1000, 1500, 2000, 2500}$, top to bottom, left to right.