Geometrical Frustration

Abstract

This book shows how the concept of geometrical frustration can be used to elucidate the structure and properties of non-periodic materials such as metallic glasses, quasicrystals, amorphous semiconductors and complex liquid crystals. Geometric frustration is introduced through examples and idealized models, leading to a consideration of how the concept can be used to identify ordered and defective regions in real materials. Then it is shown how these principles can also be used to model physical properties of materials, in particular specific volume, melting, the structure factor and the glass transition. Final chapters consider geometric frustration in periodic structures with large cells and quasiperiodic order. Appendices give all necessary background on geometry, symmetry and tilings. The text considers geometrical frustration at different scales in many types of materials and structures, including metals, amorphous solids, liquid crystals, amphiphiles, cholisteric systems, polymers, phospholipid membranes, atomic clusters, and quasicrystals. Of interest to researchers in condensed matter physics, materials science and structural chemistry, as well as mathematics and structural biology.

Maximizing Entropy by Minimizing Area: Towards a New Principle of Self-Organization

“…The polytope {3, 3, 5} should be regarded as the ideal template whence the various tetrahedrally close-packed (TCP) lattices (the layered TPC lattices are also known as Frank-Kasper phases) are derived via decurving, i.e., by substituting some of the dodecahedral cells by bubbles with 14 (= 12 pentagonal + 2 hexagonal), 15 (= 12 pentagonal + 3 hexagonal), or 16 (= 12 pentagonal + 4 hexagonal) faces. 13,24,25 Cells with 13 faces are forbidden for topological reasons, and those with more than 16 faces are dynamically unstable. 13 There are 24 known ways of decurving the polytope {3, 3, 5} and thus 24 TCP crystal lattices.…”

Locally preferred structure in simple atomic liquids

“…These dramatic changes in the dynamic properties are not accompanied by strong signatures in the structural quantities, such as the static structure factor. Yet, it has been suggested that both supercooling and glass formation were deeply connected to the structure of the liquid, more precisely to a competition between extension of a local liquid order, different than that of the crystal, and global constraints associated with tiling of the entire space [1,2,3]. This competition has been termed geometric (or topological) frustration [1,2].…”

Complex Ordered Patterns in Mechanical Instability Induced Geometrically Frustrated Triangular Cellular Structures

“…A geometrically frustrated system cannot simultaneously minimize all interactions because of geometric constraints [1,2]. The origin of this phenomenon can be easily illustrated by looking at the arrangement of spins with antiferromagnetic interactions on a triangle.…”