Non-affine fluctuations and the statistics of defect precursors in the planar honeycomb lattice

Abstract

Certain localised displacement fluctuations in the planar honeycomb lattice may be identified as precursors to topological defects. We show that these fluctuations are among the most pronounced non-affine distortions of an elemental coarse graining volume of the honeycomb structure at non zero temperatures. We obtain the statistics of these precursor modes in the canonical ensemble, evaluating exactly their single point and two-point spatio-temporal distributions, for a lattice with harmonic nearest neighbour and next near neighbour bonds. As the solid is destabilised by tuning interactions, the precursor fluctuations diverge and correlations become long-lived and long-ranged.

Equilibrium and dynamic pleating of a crystalline bonded network

“…In earlier work, it was demonstrated that local displacements of particles in a crystal away from their ideal positions may be decomposed into affine and non-affine components [10][11][12]. External stress couples to the affine part of the displacements.…”

Excess vibrational modes of a crystal in an external non-affine field

“…An advantage of studying the properties of a solid using a harmonic approximation is the possibility of obtaining exact results even in the presence of the non-affine field h X . Indeed, the complete statistical mechanics of this system can be obtained analytically (apart from numerical integrals and Fourier transforms) [18][19][20] as long as the periodic crystalline phase is stable. In order to understand how crystal properties are affected by the non-affine field, h X , we calculate the Hessian D(R, R ) = ∂ 2 H/∂u(R)∂u(R ).…”

“…While [18] gave the general theory and specific examples of the one dimensional chain and the two dimensional triangular lattice, [20] extended this formalism to the two dimensional honey-comb structure. Below, we give a brief summary of this procedure for completeness.…”

“…Taking Ω as the nearest neighbour shell consisting of particle i and its 6 neighbours, we [19,20]. In Fig.…”

“…These rearrangements are non-affine in the sense that they cannot be represented as an elastic strain on a reference volume [8]. We have shown [18][19][20], that thermal fluctuations can cause non-affine displacements even in a homogeneous crystal. In this case, such deformations are, however, transient and are predominantly defect (dislocation pair) [21] precursors.…”