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

M. Himmelmann; M. C. Pedersen; M. A. Klatt; P. W. A. Schönhöfer; M. E. Evans; G. E. Schröder-Turk

doi:10.1098/rspa.2025.0275

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

Conference proceeding

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

M. Himmelmann, M. C. Pedersen, M. A. Klatt, P. W. A. Schönhöfer, M. E. Evans and G. E. Schröder-Turk

Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences, Vol.482(2329)

2026

DOI: https://doi.org/10.1098/rspa.2025.0275

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Abstract

triply periodic minimal surfaces

embedding theorem

Helfrich and Willmore functionals

Lonsdaleite

amorphous diamond

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.

Details

Title: Amorphous bicontinuous minimal surface models and the superior Gaussian curvature uniformity of diamond, primitive and gyroid surfaces
Authors/Creators: M. Himmelmann - University of Potsdam
M. C. Pedersen - Niels Brock
M. A. Klatt - Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR)
P. W. A. Schönhöfer - University of Michigan
M. E. Evans - University of Lisbon
G. E. Schröder-Turk - Australian National University
Publication Details: Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences, Vol.482(2329)
Publisher: The Royal Society
Number of pages: 16
Identifiers: 991005852585807891
Murdoch Affiliation: School of Mathematics, Statistics, Chemistry and Physics
Language: English
Resource Type: Conference proceeding

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