Colloidal Systems Confined to Curved Surfaces

Abstract

Surface curvature plays a vital role in many biological processes. Examples include the organising of proteins in cell membranes, the tiling of cells in epithelial layers and the growth of virus capsids. A major technological benefit of micro- and nanoscale curvature is that it can guide colloidal self-assembly, a property which is of major importance in fields such as drug-delivery, biosensor fabrication, and the development of meta-materials. Recent advances in fabrication techniques, such as 3D-printing, have made a rich library of geometries available to experimentalists and engineers seeking to realise these complex and fascinating systems.

Here, we use bespoke simulations and theoretical models to study the effects of surface curvature on two-dimensional systems of isotropically attractive colloids. We identify four important properties: the finite but boundary-free area of closed surfaces, the minimum perimeter of a patch with a given area, the difference between the Euclidean and geodesic separation of points on the surface, and the frustration of the hexagonal lattice in regions of non-zero Gaussian curvature. We also show that competition between these effects produces a range of novel behaviours.

Starting from the simplest example of a sphere, we show that surface curvature has a strong effect on the gas-liquid nucleation profile and the size of the critical nucleus, as well as destroying the equivalence of the canonical and grand canonical ensembles. Then, focussing on surfaces with non-uniform curvature, we use tori to demonstrate that the different thermodynamic phases are localised to specific regions of the surface, and the transitions between them involve the translation of the colloidal assemble. Finally, we investigate the cone, where there is no Gaussian curvature but the mean curvature varies. We find that surface curvature can stabilise chiral and achiral crystals, and the ground states depend on both the range of the potential and whether it acts through Euclidean space or along geodesics.