Weakly nonlinear theory of grain boundary motion in patterns with crystalline symmetry

TL;DR

This paper develops a weakly nonlinear analytical model for grain boundary motion in hexagonal patterns, revealing how pinning forces depend on mis-orientation and differ from other phases.

Contribution

It introduces a multiple scale analysis for grain boundary motion in hexagonal patterns, highlighting the role of non-adiabatic corrections and pinning forces.

Findings

Pinning force depends sharply on mis-orientation angle.

Pinning effects are significant near bifurcation onset.

Grain boundary pinning can be much larger than in smectic phases.

Abstract

We study the motion of a grain boundary separating two otherwise stationary domains of hexagonal symmetry. Starting from an order parameter equation appropriate for hexagonal patterns, a multiple scale analysis leads to an analytical equation of motion for the boundary that shares many properties with that of a crystalline solid. We find that defect motion is generically opposed by a pinning force that arises from non-adiabatic corrections to the standard amplitude equation. The magnitude of this force depends sharply on the mis-orientation angle between adjacent domains so that the most easily pinned grain boundaries are those with an angle between four and eight degrees. Although pinning effects may be small, they do not vanish asymptotically near the onset of this subcritical bifurcation, and can be orders of magnitude larger than those present in smectic phases that bifurcate supercritically.