VOLUME 83, NUMBER 22 PHYSICAL REVIEW LETTERS 29 NOVEMBER 1999 Rules for Forest Interactions between Dislocations L. K. Wickham, 1,2 K. W. Schwarz, 1 and J. S. Stölken 2 1 IBM Watson Research Center, Yorktown Heights, New York 10598 2 Lawrence Livermore National Laboratory, Livermore, California 94550 (Received 6 July 1999) The dynamical interactions of dislocations existing on intersecting glide planes have been investigated using numerical simulations based on isotropic linear elastic theory. It is found that such dislocations either repel, attract and form growing junctions, or attract and form bound crossed states. Which of these occurs can be predicted from a surprisingly simple analysis of the initial configurations. The outcome is determined primarily by the angles which the dislocations initially make with the glide-plane intersection edge, and is largely independent of the initial distance between the dislocations, their initial curvature, or ambient applied stresses. The results provide a rule for dealing with forest interactions within the context of large multiple-dislocation computations. PACS numbers: 61.72.Bb, 61.72.Lk Stress relaxation and plastic flow in crystals occur pri- marily via the motion of dislocation lines, which generate atomic slippage of the crystal at the surfaces traversed by the moving dislocations. Many phenomena such as frac- ture, patterning and work hardening in metals, and stress relaxation in semiconductor devices reflect the dynamical behavior of dislocations on the mesoscopic scale, and in- terest in modeling such behavior is growing rapidly. The mesoscale approach (with suitable atomistic input) is well suited for exploring such problems as work hard- ening, where a large tangle of dislocations interacts in a complicated three-dimensional manner, while at the same time the typical spacing between dislocations is much larger than the core size. The detailed exploration of such problems requires numerical simulations, and there is in- tense current activity aimed at developing programs which incorporate the manifold mechanisms that govern dislo- cation behavior and interactions [1–8]. Our particular entrant in this field is the PARANOID code [9], a fully par- allelized and highly adaptive program, designed for large calculations but also capable of resolving the behavior of individual dislocations down to nanometer scales. This latter feature permits us to examine certain key issues which must be resolved before large mesoscale simula- tions can be considered practical. Dislocation-tangle simulations tend to be numerically intensive in any case. Greatly exacerbating this negative aspect, however, is the necessity of dealing with so-called forest interactions. By this is meant the crossing of dislo- cation lines moving on intersecting glide planes and often having different Burgers vectors. Such events occur only occasionally as the tangle evolves, but are of the highest importance because they lead to reconnections, the for- mation of jogs, junctions, and locks, and other phenom- ena that drastically alter the tangle dynamics. Because of the tensor nature of the interaction fields and the non- linear character of the line dynamics, a typical forest in- teraction follows a complicated development to the final stage where the cores touch, atomistic calculations be- come appropriate, and reconnection, junction formation, or jog creation occurs. Our simulations can indeed follow this development down to nanometer scales, but to do so requires remeshing of the node density down to a 0.1 nm spacing and a concomitant reduction in the length of the time steps. Thus, to fully resolve all such interactions dur- ing a tangle simulation multiplies the computational cost by several orders of magnitude and is impractical for the systems of interest. The consensus idea for dealing with this problem is to apply “rules” which let one jump directly from the initial configuration at which the lines begin to interact strongly to a final configuration (junction, jogs, etc.). This idea presupposes that there is a natural separation of time scales in the internal dynamics of a dislocation tangle. Typically, such dislocations move around with velocities determined by their curvature and by the applied stresses. Their effect on each other is minor, except perhaps in a mean-field sense. If two cores happen to approach each other to within a critical distance, however, their mutual interaction can take over and initiate an attractive instability which leads to a new configuration in a relatively short time. It is these events which one seeks to bypass computationally by the application of rules. If such a rule exists and two lines start to become unstable, a decision can be made based on their configuration and the lines placed into the desired outcome state. The subsequent behavior of these new configurations will again occur on relatively slow time scales—it can now be modeled in simplified terms, allowing for the growth and shrinkage of junctions, and the possible creation of jogs. Although schematic versions of forest-interaction rules have been implemented recently [2,10], it has been far from clear that a physically realistic, useful set of such rules, in fact, exists. First, we shall see that even in the case of two initially straight lines, the interaction is compli- cated and seemingly unpredictable, the dislocations often twisting around substantially to find the final configuration that they favor. Second, there is a large parameter space of 4574 0031-9007 99  83(22)  4574(4)$15.00 © 1999 The American Physical Society