Oriented ordering near line defects in crystals

Abstract

General properties of directed ordering near line defects, in particular an edge dislocation, in elastic crystals undergoing phase transition are studied using the two-component time-dependent Ginzburg–Landau equation in two dimensions or 2D-XY model. The associated Landau potential comprises a sixth-order term, cubic anisotropy terms and the field of the dislocation. In thermodynamic equilibrium, the phase diagram for the model is delineated. Upon quenching the system below its transition point, the temporal evolution of the order parameter components in the vicinity of the defect is numerically evaluated. The development of vortices, emanated from the model, is explored and their interaction with the dislocation is examined. The dislocation produced a vortex free circular region whose diameter grew almost linearly with time. The time-dependence of vortex density for various settings of the Landau potential coefficients are evaluated. The vortex density (in 2D) decreased inversely with time, albeit faster in the absence of dislocation. By computing the two-point correlation function, we established that the dynamic scaling law is satisfied for the considered model if the distance is scaled by or by its half-width for a dislocation free crystal. Finally, phase transitions in improper ferroelectrics in the context of the model are discussed.

Keywords:

• Phase transition
• kinetics
• Edge dislocation
• Ginzburg–Landau theory
• XY model

Notes

No potential conflict of interest was reported by the authors.

1 Vortex is a type of topological defect in the order parameter field with two components , which emerges in two spatial dimensions () from a free energy with either an isotropic O(2) symmetry or that comprising a cubic anisotropic term.

2 In short, is the temperature at which the minima of the free energy occurs. is the temperature above which no ferroelectric phase exists. is a certain temperature above . is the Curie temperature, that is, the temperature where the actual second-order phase transition occurs and is that temperature in the vicinity of the defect.

Additional information

Funding

The work was supported by the Knowledge Foundation of Sweden [grant number 2008/0503], [grant number 2011/0215].