A New Dislocation-Dynamics Model and Its Application in Thin Film-Substrate Systems
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A New Dislocation-Dynamics Model and Its Application in Thin Film-Substrate Systems E.H. Tan and L.Z. Sun Department of Civil and Environmental Engineering and Center for Computer-Aided Design The University of Iowa, Iowa City, IA 52242-1527, U.S.A. ABSTRACT Based on the physical background, a new dislocation dynamics model fully incorporating the interaction among differential dislocation segments is developed to simulate 3D dislocation motion in crystals. As the numerical simulation results demonstrate, this new model completely solves the long-standing problem that simulation results are heavily dependent on dislocationsegment lengths in the classical dislocation dynamics theory. The proposed model is applied to simulate the effect of dislocations on the mechanical performance of thin films. The interactions among the dislocation loops, free surface and interfaces are rigorously computed by a decomposition method. This framework can be used to simulate how a surface loop evolves into two threading dislocations and to determine the critical thickness of thin films. Furthermore, the relationship between the film thickness and yield strength is established and compared with the conventional Hall-Petch relation. INTRODUCTION Although dislocation dynamics has been actively researched since 1960, the ability to simulate the evolution of dislocations is still limited by the inherent complexity of this problem. To date, various numerical methods [1-4] have been developed to simulate the evolution and interaction of dislocations. As Gómez-García et al. [1] have indicated, a long-standing problem inherent to these methods is that the selection of both larger and smaller segment sizes in discretization yields unreasonable simulation results; only segment sizes approximating a certain optimal length, which is different for each problem, can lead to reasonable results. Ghoniem et al. [3] also mentioned a peculiar “velocity jump” phenomenon, shown in Fig. 1(a), where the velocity near the pinning points of a straight dislocation line will jump with the increase of segment number. This contradicts numerical-method principles that suggest finer discretization should generally lead to more accurate results. Consequently, simulation results are always significantly affected by the artificial choice of segment sizes, shown in Fig. 1(b), which substantially reduces the reliability of simulation results. The purpose of this paper is to ascertain the reason for this abnormal phenomenon, develop a new dislocation dynamics model to resolve this issue and further apply this new model to thin-film substrate systems.
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A NEW DISLOCATION-DYNAMICS MODEL In order to clarify why the segment-size dependence phenomenon occurs, it is first necessary to review the foundation of these methods. Fundamentally, all the above-mentioned dislocation dynamics methods are based on the governing equation: Fi − BijV j = 0 (1) where F i is the local glide force per unit length acting on a dislocation, Bij is the diagonal drag coefficient matri
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