Hyperbolic confinement: self-assembly of asymmetric diblock co-polymers on triply periodic minimal surfaces

Abstract We investigate the self-assembly of compositionally asymmetric AB diblock co-polymers confined to thin hyperbolic films on the three cubic minimal surfaces: the primitive, the diamond and the gyroid. To analyse and quantify the assemblies, we use a computational pipeline where decorations of the curved surfaces from (flat) three-dimensional space are visualized in curved two-dimensional space using the Poincaré disc model of the hyperbolic plane. This allows us to pinpoint the role of the intrinsic geometry and probe the role of the Gaussian curvature of the film on the resulting assemblies. We find three distinct symmetric arrays of discs favoured for specific values of the average Gaussian curvature of the film, whose relative appearance as a function of the average film curvature is common to all three minimal surfaces, which strongly suggests these assemblies are not influenced by the extrinsic embedding of the film. In general, the assembled discs are located in regions with minimal (absolute) Gaussian curvature, likely in order to minimize the effective line tension within the system corresponding to the enthalpic cost of co-polymer assembly along the perimeters of the discs. This article is part of the theme issue ‘Geometry, materials and the imagination’.