Statistical Mechanics of Dislocation Pileups in Two Dimensions

TL;DR

This paper develops a statistical mechanics framework for dislocation pileups in two-dimensional crystals, revealing phase transitions and structural changes linked to temperature and defect interactions, supported by analytical theory and random matrix models.

Contribution

It introduces a novel analytical approach to dislocation pileups using random matrix theory, uncovering phase transitions and structural properties in inhomogeneous defect lattices.

Findings

Identified thermal depinning transition from pinned to floating defect phases.

Derived exact critical exponents for structure factor and radial distribution functions.

Linked dislocation arrangements to eigenvalues of random matrix ensembles.

Abstract

Dislocation pileups directly impact the material properties of crystalline solids through the arrangement and collective motion of interacting dislocations. We study the statistical mechanics of these ordered defect structures embedded in two dimensional crystals, where the dislocations themselves form one-dimensional lattices. In particular, pileups exemplify a new class of inhomogeneous crystals characterized by spatially varying lattice spacings. By analytically formulating key statistical quantities and comparing our theory to numerical experiments using an intriguing mapping of dislocation positions onto the eigenvalues of recently studied random matrix ensembles, we uncover two types of one-dimensional phase transitions in dislocation pileups: a thermal depinning transition out of long-range translational order from the pinned-defect phase, due to a periodic Peierls potential, to a floating-defect state, and finally the melting out of a quasi-long range ordered floating defect-solid phase to a defect-liquid. We also find the set of transition temperatures at which these transitions can be directly observed through the one-dimensional structure factor, where the delta function Bragg peaks, at the pinned-defect to floating-defect transition, broaden into algebraically diverging Bragg peaks, which then sequentially disappear as one approaches the two-dimensional melting transition of the host crystal. We calculate a set of temperature-dependent critical exponents for the structure factor and radial distribution function, and obtain their exact forms for both uniform and inhomogeneous pileups using random matrix theory.