Nucleation on a sphere: the roles of curvature, confinement and ensemble

Effect of confinement and topology: 2-TIPS vs. MIPS

2-TIPS (two temperature induced phase separation) refers to the phase separation phenomenon observed in mixtures of active and passive particles which are modelled using scalar activity. The active particles are connected to a thermostat at high temperature while the passive particles are connected to the thermostat at low temperature and the relative temperature difference between "hot" and "cold" particles is taken as the measure of the activity χ of the non-equilibrium system. The study of such binary mixtures of hot and cold particles under various kinds of confinement is an important problem in many physical and biological processes. The nature and extent of phase separation are heavily influenced by the geometry of confinement, activity, and density of the non-equilibrium binary mixture. Investigating such 3D binary mixtures confined by parallel walls, we observe that the active and passive particles phase separate, but the extent of phase separation is reduced compared to bulk phase separation at high densities and enhanced at low densities. However, when the binary mixture of active and passive particles is confined inside a spherical cavity, the phase separation is radial for small radii of the confining sphere and the extent of phase separation is higher compared to their bulk counterparts. Confinement leads to interesting properties in the passive (cold) region like enhanced layering and high compression in the direction parallel to the confining wall. In 2D, both the bulk and confined systems of the binary mixture show a significant decrement in the extent of phase separation at higher densities. This observation is attributed to the trapping of active particles inside the passive cluster, which increases with density. Thus the 2D systems show structures more akin to dense-dilute phase co-existence, which is observed in motility induced phase separation in 2D active systems. The binary mixture constrained on the spherical surface also shows similar phase co-existence. Our analyses reveal that the coexistent densities observed in 2-TIPS on the spherical surface agree with the findings of previous studies on MIPS in active systems on a sphere.

Packing and emergence of the ordering of rods in a spherical monolayer

Spatially ordered systems confined to surfaces such as spheres exhibit interesting topological structures because of curvature induced frustration in orientational and translational order. The study of these structures is important for investigating the interplay between the geometry, topology, and elasticity, and for their potential applications in materials science, such as engineering directionally binding particles. In this work, we numerically simulate a spherical monolayer of soft repulsive spherocylinders (SRSs) and study the packing of rods and their ordering transition as a function of the packing fraction. In the model that we study, the centers of mass of the spherocylinders (situated at their geometric centers) are constrained to move on a spherical surface. The spherocylinders are free to rotate about any axis that passes through their respective centers of mass. We show that, up to moderate packing fractions, a two dimensional liquid crystalline phase is formed whose orientational ordering increases continuously with increasing density. This monolayer of orientationally ordered SRS particles at medium densities resembles a hedgehog-long axes of the SRS particles are aligned along the local normal to the sphere. At higher packing fractions, the system undergoes a transition to the solid phase, which is riddled with topological point defects (disclinations) and grain boundaries that divide the whole surface into several domains.

Phase transitions on non-uniformly curved surfaces: coupling between phase and location

For particles confined to two dimensions, any curvature of the surface affects the structural, kinetic and thermodynamic properties of the system. If the curvature is non-uniform, an even richer range of behaviours can emerge. Using a combination of bespoke Monte Carlo, molecular dynamics and basin-hopping methods, we show that the stable states of attractive colloids confined to non-uniformly curved surfaces are distinguished not only by the phase of matter but also by their location on the surface. Consequently, the transitions between these states involve cooperative migration of the entire colloidal assembly. We demonstrate these phenomena on toroidal and sinusoidal surfaces for model colloids with different ranges of interactions as described by the Morse potential. In all cases, the behaviour can be rationalised in terms of three universal considerations: cluster perimeter, stress, and the packing of next-nearest neighbours.

Spherical structure factor and classification of hyperuniformity on the sphere

Understanding how particles are arranged on the surface of a sphere is not only central to numerous physical, biological, soft matter, and materials systems but also finds applications in computational problems, approximation theory, and analysis of geophysical and meteorological measurements. Objects that lie on a sphere experience constraints that are not present in Euclidean (flat) space and that influence both how the particles can be arranged as well as their statistical properties. These constraints, coupled with the curved geometry, require a careful extension of quantities used for the analysis of particle distributions in Euclidean space to distributions confined to the surface of a sphere. Here, we introduce a framework designed to analyze and classify structural order and disorder in particle distributions constrained to the sphere. The classification is based on the concept of hyperuniformity, which was first introduced 15 years ago and since then studied extensively in Euclidean space, yet has only very recently been considered also for spherical surfaces. We employ a generalization of the structure factor on the sphere, related to the power spectrum of the corresponding multipole expansion of particle density distribution. The spherical structure factor is then shown to couple with cap number variance, a measure of density variations at different scales, allowing us to analytically derive different forms of the variance pertaining to different types of distributions. Based on these forms, we construct a classification of hyperuniformity for scale-free particle distributions on the sphere and show how it can be extended to include other distribution types as well. We demonstrate that hyperuniformity on the sphere can be defined either through a vanishing spherical structure factor at low multipole numbers or through a scaling of the cap number variance-in both cases extending the Euclidean definition, while at the same time pointing out crucial differences. Our work thus provides a comprehensive tool for detecting global, long-range order on spheres and for the analysis of spherical computational meshes, biological and synthetic spherical assemblies, and ordering phase transitions in spherically distributed particles.