Session W3: Dislocation Patterns and Avalanches

Scale-Free Intermittent Flow in Crystal Plasticity

Under stress, crystals irreversibly deform through complex dislocation processes -- processes that intermittently change the microscopic material shape through isolated slip events. Using both model computer simulations and ultra-precise nano-scale measurements on nickel micro-crystals we directly determined the size of discrete slip events. The sizes range over nearly three orders of magnitude, and exhibit a shock and aftershock earthquake-like behavior over time. Analysis of the events reveals power-law scaling between the number of events and their magnitude, or scale-free flow. We show that dislocated crystals are a model system for studying scale-free behavior that is observed for many macroscopic systems. By analogy to plate tectonics, smooth macroscopic-scale crystalline glide arises from the spatial and time averages of disruptive earthquake-like events at the nano-scale.   

Pattern Coarsening in a Two Dimensional Hexagonal System

We have been studying the ordering, annealing, coarsening and alignment of two dimensional periodically ordered structures in thin films of diblock copolymers*. Coarsening by dislocation and disclination annihilation is clearly observed in AFM studies of monolayer films of cylindrical patterns with a time dependence given by t$^{\alpha }$, with $\alpha $ about 1/4. However in hexagonal structures the mechanism is less well defined and appears to involve the collapse of small grains entrained in the grain boundaries of larger domains. Remarkably the exponent of $\alpha $ about 1/4 remains. We also report on shear aligned samples and samples quenched in a gradient after alignment. * Harrison C, Angelescu DE, Trawick M, Cheng ZD, Huse DA, Chaikin PM, Vega DA, Sebastian JM, Register RA, Adamson DH, EUROPHYSICS LETTERS \textbf{67} 800-806 (2004)   

Domain Coarsening and Aging in Dislocation Glasses

Dislocation systems were analyzed numerically with 1 and 3 glide axes, at T=0 and T$>$0, with and without climb. [1] Dislocation free domains formed even without shear, defined by dislocation rich domain walls. The domain formation was most pronounced in the presence of climb, somewhat counter-intuitively. The stability of domains was analyzed. The microscopic processes suppressing the climb-induced decay of domain walls were identified. The dislocation dynamics at low temperatures was markedly glassy. \underline {Aging:} Dislocations with glide only support minimal domain formation. The autocorrelation function showed aging, scaling with the waiting time as: C(t,t$_{w})$ = C$_{eq}$(t) C(t/t$_{w}^{\mu })$ and C$_{eq}$(t) $\sim $ t$^{-\beta }$, with $\mu $=0.65 and $\beta $=0.54. \underline {Freezing:} The effective diffusion constant decayed to zero as: D(t)$_{eff} \quad \sim $ t$^{-\gamma }$ , with $\gamma $=0.8. \underline {Coarsening:} Dislocations with glide and climb exhibited profound domain formation, the domains coarsening as L(t): L(t) $\sim $ t$^{1/z}$, with 1/z=0.17. The formation of domains without shear has been recently observed in GaAs by Rudolph and in dusty plasmas by Quinn and Joree. The domain coarsening was quantitatively captured in di-block copolymers [2], with 1/z=0.19, in good agreement with our results. \newline \newline [1] B. Bako, G. Groma, G. Gyorgyi and G.T. Zimanyi, Phys. Rev. Lett. \textbf{98}, 075701 (2007). \newline [2] P. Chaikin's talk, same session.   

Dislocation Avalanches, Mean Free Path and Patterning

Dislocations are the defects that carry plastic flow in crystalline materials. At the defect scale, plastic deformation occurs by dislocation avalanches, which are characterized by scale-free behavior within a bounded domain of amplitudes or energies. The probability distribution functions of these avalanches, their processes of initiation and termination and their contribution to dislocation storage during plastic flow are investigated using dislocation dynamics (DD) simulations. The model material is a copper single crystal strained along three high symmetry orientations. The distributions of avalanche amplitudes exhibit, for all orientations, a scaling exponent of 1.6, similar to what is reported in the literature. However, the average value of strain burst amplitudes, taken in the recorded domain of amplitudes, is found to be orientation-dependent. In parallel, a continuum model based on the notion of dislocation mean free path, which predicts the mechanical response of f.c.c. single crystals, was established with the help of DD simulations. It appears that both intermittent and continuum behavior exhibit the same orientation dependencies. Furthermore, for the three tested orientations, the ratio of the continuum mean free path to the average characteristic length traveled by dislocations during avalanches is a constant. This constant ratio simply results from an implicit coarse-graining procedure that is performed upon measuring different types of quantities in the simulated results. As a consequence, it appears that avalanche behavior can effectively be incorporated into continuum models for crystal plasticity. The present results are discussed with respect to the available experimental literature on the deformation of f.c.c. crystals. Finally, the formation of dislocation patterns, that is the self-organization properties of dislocations under strain, is tentatively discussed in terms of the properties of dislocation avalanches.