Abstract
Bicontinuous cubic structures in soft matter consist of two intertwining labyrinths separated by a partitioning layer. Combining experiments, numerical modelling and techniques in differential geometry, we investigate twinning defects in bicontinuous cubic structures. We first demonstrate that a twin boundary is most likely to occur at a plane that cuts the partitioning layer almost perpendicularly, so that the perturbation caused by twinning remains minimal. This principle can be used as a criterion to identify potential twin boundaries, as demonstrated through detailed investigations of mesoporous silica crystals characterized by diamond and gyroid surfaces. We then discuss that a twin boundary can result from a stacking fault in the arrangement of inter-lamellar attachments at an early stage of structure formation. It is further shown that enhanced curvature fluctuations near the twin boundary would cost energy because of geometrical frustration, which would be eased by a crystal distortion that is experimentally observed.
Highlights
IntroductionBicontinuous cubic structures (BCSs) are crystal structures with cubic symmetry consisting of two continuous intertwining subvolumes, or labyrinths, separated by a non-self-intersecting partitioning layer
Bicontinuous cubic structures (BCSs) are crystal structures with cubic symmetry consisting of two continuous intertwining subvolumes, or labyrinths, separated by a non-self-intersecting partitioning layer. They are found in cell endomembrane systems (Landh, 1995), butterfly wing scales and beetle exoskeletons (Galusha et al, 2008; Michielsen & Stavenga, 2008), lyotropic liquid crystals (LLCs) and related systems [mesoporous silica crystals (MSCs), block copolymer self-assemblies, etc.] (Luzzati & Spegt, 1967; Hyde et al, 1984; Alward et al, 1986; Kresge et al, 1992; Bates & Fredrickson, 1999; Wan & Zhao, 2007)
Different sides of the partitioning layer can be filled with mutually immiscible sub-chains in block copolymer systems, whereas the two labyrinths of a BCS in LLCs are homophilic
Summary
Bicontinuous cubic structures (BCSs) are crystal structures with cubic symmetry consisting of two continuous intertwining subvolumes, or labyrinths, separated by a non-self-intersecting partitioning layer They are found in cell endomembrane systems (Landh, 1995), butterfly wing scales and beetle exoskeletons (Galusha et al, 2008; Michielsen & Stavenga, 2008), lyotropic liquid crystals (LLCs) and related systems [mesoporous silica crystals (MSCs), block copolymer self-assemblies, etc.] (Luzzati & Spegt, 1967; Hyde et al, 1984; Alward et al, 1986; Kresge et al, 1992; Bates & Fredrickson, 1999; Wan & Zhao, 2007). Analysis of Gaussian curvatures reveals how twin boundaries, as microscopic topological defects, may be the cause of crystal distortions manifested in the macroscopic morphology of twinned BCSs
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