Amorphous bicontinuous minimal surface models and the superior Gaussian curvature uniformity of diamond, primitive and gyroid surfaces

Abstract

Bicontinuous geometries, both ordered and amorphous, are commonly found in many soft matter systems. Ordered bicontinuous phases can be modelled by periodic minimal surfaces, including Schoen's gyroid (G) or Schwarz' primitive (P) and diamond (D) surfaces. By contrast, a minimal surface model for amorphous phases has been lacking. Here, we study minimal surface models for amorphous bicontinuous phases, such as sponge phases. Using the surface evolver with a novel topology-stabilizing minimization scheme, we numerically construct amorphous minimal surfaces from both a continuous random network (CRN) model for amorphous diamond and from a randomly perforated parallel sheet model. As per Hilbert's embedding theorem, the Gaussian curvature of these surfaces cannot be constant. Our analysis of Gaussian curvature variances finds no substantial long-wavelength curvature variations in the amorphous diamond minimal surfaces. However, their Gaussian curvature variance is substantially larger than that of the cubic P, D and G surfaces. Our work demonstrates the superior curvature homogeneity of the cubic P, D and G surfaces compared to their entropy-favoured amorphous counterparts and to other periodic minimal surfaces. This general geometric result is relevant to bicontinuous structure formation in soft matter and biology across all length scales.