Eckhaus instability – a possible wavelength changing mechanism in the evolution of dislocation patterns

Phase modulations of periodic dislocation patterns observed in transmission electron microscopy (TEM) studies of some single crystalline metals, are analyzed using the methods of nonlinear dynamics. The Ginzburg–Landau (GL) equation for the soft mode instability in the weakly nonlinear regime is derived for the Walgraef–Aifantis (WA) model for a coupled system of two populations of dislocations. The bulk of results is obtained using the GL equation, and, therefore, the results are more general than the WA-model itself. We demonstrate that phase modulations of dislocation patterns can be described using the concept of the Eckhaus instability which describes one of the most fundamental “generic” mechanisms of wavelength-changing. The timescale of wavelength-changing processes in dislocation systems can be very large when the system is close to the Eckhaus stability limit. This means that metastable phase modulations of dislocation patterns can survive nearly unchanged for a long time. The results of numerical simulations for realistic values of the parameters show that the Eckhaus instability could be the underlying physical reason of modulated ladder structures of persistent slip bands (PSBs) in cyclically deformed metallic alloys.