Geometrical Folding Transitions of the Triangular Lattice in the Face-Centred Cubic Lattice

TL;DR

This paper investigates the folding behavior of a 2D triangular lattice embedded in a 3D face-centered cubic lattice, revealing two distinct geometrical transitions influenced by bending rigidity.

Contribution

It introduces a model for membrane crumpling using discrete folding transitions and analyzes phase changes with CVM and transfer matrix methods.

Findings

Two separate geometrical transitions identified

Discontinuous transition between octahedral and tetrahedral wrapping

Continuous transition to complete folding on a single triangle

Abstract

We study the folding of the regular two-dimensional triangular lattice embedded in the regular three-dimensional Face-Centred Cubic lattice, a discrete model for the crumpling of membranes. Possible folds are complete planar folds, folds with the angle of a regular tetrahedron (71 degrees) or with that of a regular octahedron (109 degrees). We study this model in the presence of a negative bending rigidity K, which favours the folding process. We use both a cluster variation method (CVM) approximation and a transfer matrix approach. The system is shown to undergo two separate geometrical transitions with increasing |K|: a first discontinuous transition separates a phase where the triangular lattice is preferentially wrapped around octahedra from a phase where it is preferentially wrapped around tetrahedra. A second continuous transition separates this latter phase from a phase of complete folding of the lattice on top of a single triangle.