Dislocations Peierls energy

Peierls Force Continuous vs. Discontinuous Motion. In some crystals (e.g., covalent crystals) the Peierls force may be so large that the driving force due to the applied stress will not be able to drive the dislocation forward. In such a case the dislocation will be rendered immobile. However, at elevated temperatures, the dislocation may be able to surmount the Peierls energy barrier by means of stress-aided thermal activation, as in Fig. 11.5. 

Figure 11.5 Movement of dislocation across a Peierls energy barrier by thermally...

Figure 10.10. Dislocation motion in the Peierls energy landscape, through formation of kink-antikink pairs.

Perfect screw dislocation Peierls stress and energy 82... 

I. Perfect screw dislocation Peierls stress and energy. There were several early attempts to calculate the Peierls stress of the non-dissociated screw dislocation in silicon [92,98,99]. Empirical potential computations give values ranging from... 

Fig. 29. Screw dislocation gliding along two different directions (see text), as a function of an applied strain. Maximum energies along the MEP are the Peierls energies they are reported in the inset graph...

A second major difficulty with the Peierls model is that it is elastic and therefore conservative (of energy). However, dislocation motion is nonconservative. As dislocations move they dissipate energy. It has been known for centuries that plastic deformation dissipates plastic work, and more recently observations of individual dislocations has shown that they move in a viscous (dissipative) fashion. 

At roughly the same time, Peierls (1940), and Nabarro (1947) developed a two-dimensional model of a dislocation in a simple square crystal structure. This model indicated that a small, but finite, amount of energy is needed to... 

Figure 11.6 illustrates the energy that must be supplied by thermal activation. The curve of ab vs. A shows the force that must be applied to the dislocation (per unit length) if it were forced to surmount the Peierls barrier in the manner just described in the absence of thermal activation. The quantity A is the area swept out by the double kink as it surmounts the barrier and is a measure of the forward motion of the double kink. A = 0 corresponds to the dislocation lying along an energy trough (minimum) as in Fig. 11.5a. A2 is the area swept out when maximum force must be supplied to drive the double kink. A4 is the area swept out when the saddle point has been reached and the barrier has been effectively surmounted. The area under the curve is then the total work that must be done by the applied stress to surmount the barrier in the absence of thermal activation. When the applied stress is a a (and too small to force the barrier), the swept-out area is A, and the energy that must be supplied by thermal activation is then the shaded area shown in Fig. 11.6. The activation energy is then...

Our next task is to use our newly found Peierls potential to estimate the stress to move a straight dislocation. The stress is essentially provided as the derivative of the energy profile with respect to the coordinate x, and is given by... 

Whereas the line tension was invoked as a way to capture the self-energy of dislocations from an elastic perspective, there are also ways of capturing core effects on the basis of locality assumptions. Recall that in our treatment of dislocation cores we introduced the Peierls-Nabarro model (see section 8.6.2) in which the misfit energy associated with slip displacements across the slip plane is associated with an energy penalty of the form... 

Glide of an edge dislocation occurs when a half-plane of atoms is moved over the atoms below the glide plane. The movement occurs by the nucleation and movement of kinks. Remember that the reason that dislocations are so important in plasticity is because it is easier to move one block of material over another (shear the crystal) one halfplane of atoms at a time. Similarly, it is easier to move a dislocation by moving a kink along it one atom at a time. In fee metals, the Peierls valleys are not deep, so the energy required to form a kink is small and dislocations bend (create kinks) quite easily. 

For those ceramics with a high Peierls stress, the CRSS can be understood consistently in terms of a model of kink pair nudeation and motion on dislocations. The steep temperature dependence is governed by an activation energy that is the sum of the elastic energy for kink pair formation and the energy for the kinks to overcome their secondary Peierls barriers. 

TABLE 8.1 Peierls stress (ffp, MPa), core energies (Ecoret ev/A), and binding energy ( b, eV/ atom) for the four dislocations in the pure AI and the AI H systems... 

After the dislocation has moved by half a Burgers vector, the Peierls force pushes it forwards and moves it to the position of the next energy minimum. The stored energy is usually dissipated as heat (i.e., as random crystal vibration) in the crystal. The Peierls force thus acts as a kind of frictional force and reduces the effective stress that can be used to drive the dislocation to overcome other obstacles. 

Some of the earliest data shown in Table 4.4 clearly demonstrate the softening effect of water. A consideration of results such as those in Table 4.4 led at an early stage of this work to two important conclusions One was that dislocation motion in nonmetals is more obviously affected by electronic and strain interactions with point defects than by Peierls resistance. Chemisorption causes energy-band bending in the surface region, and this can alter the electronic core structure of dislocations or the state of ionization of point defects or both. The other was that direct comparisons of hardness values from one study to another should not be made unless specimen preparation, history, and measurement technique and conditions are known. 

Figure 10.9. Profile of an edge dislocation Top the disregistry or misfit u(x) as dictated by the minimization of the elastic energy (solid line) or the misfit energy (dashed line) and the corresponding densities p(x), given by Eq. (10.10). Bottom the disregistry and density as obtained from the Peierls-Nabarro model, which represents a compromise between the...

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