scholarly journals Translation covers of some triply periodic Platonic surfaces

2021 ◽  
Vol 25 (2) ◽  
pp. 34-50
Author(s):  
Jayadev S. Athreya ◽  
Dami Lee
Keyword(s):  
Triply Periodic

Triply-periodic nanostructures in surfactant, block copolymer, and biomembrane systems, and simulation of TEMs

1993 ◽  
Vol 51 ◽  
pp. 884-885
Author(s):  
David M. Anderson ◽  
Tomas Landh
Keyword(s):  
Liquid Crystalline ◽  
Diblock Copolymers ◽  
Cubic Phase ◽  
List Type ◽  
Interpenetrating Networks ◽  
Triply Periodic ◽  
Wide Range ◽  
Bicontinuous Cubic ◽  
Phase Liquid ◽  
Dividing Surface

First discovered in surfactant-water liquid crystalline systems, so-called ‘bicontinuous cubic phases’ have the property that hydropnilic and lipophilic microdomains form interpenetrating networks conforming to cubic lattices on the scale of nanometers. Later these same structures were found in star diblock copolymers, where the simultaneous continuity of elastomeric and glassy domains gives rise to unique physical properties. Today it is well-established that the symmetry and topology of such a morphology are accurately described by one of several triply-periodic minimal surfaces, and that the interface between hydrophilic and hydrophobic, or immiscible polymer, domains is described by a triply-periodic surface of constant, nonzero mean curvature. One example of such a dividing surface is shown in figure 5.The study of these structures has become of increasing importance in the past five years for two reasons:1)Bicontinuous cubic phase liquid crystals are now being polymerized to create microporous materials with monodispersed pores and readily functionalizable porewalls; figure 3 shows a TEM from a polymerized surfactant / methylmethacrylate / water cubic phase; and2)Compelling evidence has been found that these same morphologies describe biomembrane systems in a wide range of cells.

Development of Modelling Methods for Materials to be Used as Bone Substitutes

2007 ◽  
Vol 361-363 ◽  
pp. 903-906 ◽  
Author(s):  
R. Gabbrielli ◽  
I.G. Turner ◽  
Chris R. Bowen
Keyword(s):  
Mechanical Strength ◽  
Volume Fraction ◽  
Bone Substitutes ◽  
Gradient Material ◽  
Periodic Surfaces ◽  
Triply Periodic ◽  
Functional Gradient Material ◽  
Spatial Periodicity ◽  
Bearing Materials ◽  
Modelling Methods

The demand in the medical industry for load bearing materials is ever increasing. The techniques currently used for the manufacture of such materials are not optimized in terms of porosity and mechanical strength. This study adopts a microstructural shape design approach to the production of open porous materials, which utilizes spatial periodicity as a simple way to generate the models. A set of triply periodic surfaces expressed via trigonometric functions in the implicit form are presented. A geometric description of the topology of the microstructure is necessary when macroscopic properties such as mechanical strength, stiffness and isotropy are required to be optimised for a given value of volume fraction. A distinction between the families of structures produced is made on the basis of topology. The models generated have been used successfully to manufacture both a range of structures with different volume fractions of pores and samples of functional gradient material using rapid prototyping.

Description of a 3-periodic minimal surface family with trigonal symmetry

1994 ◽  
Vol 209 (1) ◽  
Author(s):  
A. Fogden
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Weierstrass Representation ◽  
Systematic Analysis ◽  
Trigonal Symmetry ◽  
Triply Periodic Minimal Surfaces ◽  
Periodic Minimal Surface ◽  
Triply Periodic ◽  
The Family

AbstractA systematic analysis of a family of triply periodic minimal surfaces of genus seven and trigonal symmetry is given. The family is found to contain five such surfaces free from self-intersections, three of which are previously unknown. Exact parametrisations of all surfaces are provided using the Weierstrass representation.

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