Screw dislocations generated from crack tip of self-consistent and self-equilibrated systems of residual stresses: Atomic, meso and micro

All materials are known to contain residual stresses and strains, the magnitude of which depends on the history of the processing procedure. These initial disturbances will undergo rise and fall when energy is added or extracted from the system. In general, the system tends to damp out the fluctuations and possesses the tendency to achieve equilibrium. Since the energy states of any portion of the system, or the sub-systems, will not reach equilibrium at the same time, the residual effects will not be uniform and will be more pronounced at locations where defects and stress concentrators are present. Considerations of their influence on material behavior at the macroscopic scale are well known. Recent interest on sub-micron devices in microelectronics has prompted the examination of initial stresses and strains at the microscopic and atomic scale. A particular situation corresponds to the defects or imperfections caused at the atomic scale due to residual stresses and strains at the microscopic scale. That is after the external disturbances have been removed from the system. These defects will be referred to as dislocations. One of the objectives is to develop a model that can estimate the number and type of dislocations for a given distribution and magnitude of stress and strain state ahead of a micro-crack. Both uniform and varying residual stress distributions are assumed to examine how they would affect the dislocation emissions of an internally self-equilibrating system where no external disturbances prevail. A mesoscopic zone is introduced between the atomic and microscopic scale to smooth out the transition. The sizes of these zones and number of dislocations generated for a particular microscopic residual stress state are determined analytically by solving a system of highly non-linear equations. Because of the gap between the atomic- and micro-scale, a “scale multiplier” was introduced to connect two ranges of scale. This tends to facilitate and smooth out the numerical computations. Such a multiplier is reminiscent of the stress compatibility condition used in early works where force or energy criterion is employed. To simplify the analysis, a screw motion of dislocation is selected for illustration mainly since it involves only one non-trivial out-of-plane displacement. All stresses and strains out side of this plane of deformation vanish. The results show that an enormous number of screw dislocations can be generated from the microscopic residual stresses. Depending on the ratio of the residual stress to the elastic modulus, the number of screw dislocations would first increase with the segment over which they prevail and then decreases. The effect becomes less distinct when the residual stress magnitude is decreased. For a constant amplitude residual distribution, the estimated number of dislocations is about 104-105 for residual stress amplitude to shear modulus ratio of 0.1-0.5. A three step varying residual stress distribution is also examined. The step variations can be distributed with a maximum away from and at the micro-crack tip. The third choice is to have the maximum at the center of the segment ahead of the crack tip. The residual stress distribution is found to have a pronounced effect on the number of dislocations generated. The largest number of dislocation emission corresponds to the situation when the peak residual stress is closest to the site of dislocations. This implies that the history of the disturbances at the atomic level within the system plays a role, a situation that cannot be ignored when fabricating smaller electronic devices that might involve only a few atoms. The influences of initial energy states and characteristics on the operational performance of nanodevices are no different from those at the macroscopic scale. Since the emission of the dislocations depends on many variables on different scales, it would be difficult if not impossible to control the emission characteristics by assigning values to the bulk physical, geometrical and load parameters. This implies that it would be impractical to fabricate macroscopic specimens and then reduce their sizes by mechanical, electrical and/or optical means because the imperfections scattered into the material at the atomic scale cannot be readily determined and/or controlled from larger scales. The simple atomic-meso-micro model developed here demonstrates this feature. The same can be shown by atomic simulation calculations involving millions and trillions of particles. This suggests arranging atoms or molecules one by one might be the choice to achieve the desired behavior of the material when the size becomes extremely small. Nano-fabrication techniques such as electron-beam and ion-beam lithography, molecular beam epitaxy and scanning probe microscopes can presumably achieve atomic-scale precision. However, reproducibility and complexity in microelectronic devices still present fundamental problems. One of them is achieving chemical stability. Practical application of nanotechnology still appears to be far reaching. There is a no man land region not so small where quantum mechanics can be applied in a straightforward manner and yet not large enough where classical physics or continuum mechanics would govern. This is the awkward region of the 'mesoscale'. Friction and damping, heat and dissipations, etc. do not obey the physical laws that are known today. New kinds of behavior will have to be learned and understood. Preconceived ideas based on classical concepts may be handicapped. It is in this spirit that the atomic-meso-micro model is developed, perhaps to exhibit the complexity of mesoscopic structures concerned with multiscaling.