Dislocations core effects

Thus far, our discussion of dislocation core effects has been built around those tools admitting of direct analytic progress such as is offered by the Peierls-Nabarro dislocation core model. This model has many virtues. However, it is also strictly limited, with one of its primary limitations being that without huge machinations, it may only be applied to dislocations with planar cores. To explore the full atomic-level complexity that arises in dislocation cores, we must resort to numerical techniques, and direct atomistic simulation in particular. The remainder of our discussion on dislocation cores will take up the question of how one goes about carrying out such simulations and what may be learned from them. 

However, it is not yet clear why the ener es of the SISF and the twin boundary increase with increasing A1 concentration. To find a clue to the problem, it would be needed to make out the effects of the short-range ordering of A1 atoms in excess of the stoichiometric composition of the HAl phase on the energies of planar faults and the stmcture of dislocation cores in the Al-rich HAl phase. 

That a hollow core is formed by the creation of a free surface along a dislocation core implies that the curvature of the spiral step is reversed due to the strain field along the dislocation core. The effect of a strain field upon the advancement of a step was theoretically treated by Cabrera and Levine [14], [15],... 

Only a few crystals exhibit hollow cores at the centers of growth spiral layers. However, on the (0001) faces of SiC, which has a large fx value, hollow cores due to growth have often been observed. According to the summary by Sunagawa and Bennema [16], various degrees of the effect of the strain associated with dislocation cores have been observed depending on the sizes of b and the concentration of dislocations. 

Effective thickness of grain boundary or surface layer diameter of dislocation core m... 

Exercise 9.1 yielded an expression, Eq. 9.18, for the enhancement of the effective bulk self-diffusivity due to fast self-diffusion along dislocations present in the material at the density, p. Find a corresponding expression for the enhancement of the effective bulk self-diffusivity of solute atoms due to fast solute self-diffusion along dislocations. Assume that the solute atoms segregate to the dislocations according to simple McLean-type segregation where c2 c2L = k — constant, where cf5 is the solute concentration in the dislocation cores and cXL is the solute concentration in the crystal. 

As will be seen below, the presence of dislocations has the effect of producing local disturbances of the atomic-level geometry that are so severe as to produce scattering that is not consonant with that from the remainder of the crystal, giving rise to the contrast revealed in the figure. Evidently, the onset of plastic deformation reveals its presence at multiple scales simultaneously. Indeed, as was seen in fig. 1.9, the properties of individual dislocation cores can be observed using high-resolution transmission electron microscopy. 

Our discussion of dislocations has thus far made little reference to the significant role that can be played by the dislocation core. Whether one thinks of the important effects that arise from the presence of stacking faults (and their analogs in the alloy setting) or of the full scale core reconstructions that occur in materials such as covalent semiconductors or the bcc transition metals, the dislocation core can manifest itself in macroscopic plastic response. Our aim in this section is to take stock of how structural insights into the dislocation core can be obtained. 

One of the central conclusions derived from the Peierls-Nabarro analysis is the role of nonlinear effects in the dislocation core. From an atomistic perspective, the far field atomic displacements could be derived just as well from linear elasticity as the full nonlinear function that results from direct atomistic calculation. By way of contrast, in the core region it is the nonlinear terms that give rise to some of the complex core rearrangements that we take up now. Our discussion will be built around two key examples cores in fee metals and the core reconstructions found in covalent semiconductors such as in Si. 

Before embarking on the specifics of these cores, we begin with an assessment of the basic ideas needed to effect an atomistic simulation of the dislocation core. We begin with a picture of lattice statics in which the core geometry is determined by energy minimization. To proceed, what is needed is an energy function of the type discussed in chap. 4 which may be written generically as... 

Although there are a number of interesting features of this analysis, it also leaves us with serious concerns about the formulation of an elastic theory of the obstacle forces that impede dislocation motion. In particular, this analysis suggests that for an obstacle on the slip plane itself, there is no interaction with the dislocation. Despite this elastic perspective, it seems certain that core effects will amend this conclusion. 

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... 

Moreover, numerical solutions of Eq. 11 for calcite, quartz, and feldspars show that after a short transient period (a few minutes for calcite), the rate of dissolution remains roughly constant. Our calculations (see Table 4) indicate that during this steady-state period the overall dissolution rate should increase by a factor of only 3 or 4 at maximum due to two competing effects decreasing surface strain energy and increasing surface area, as dislocation cores dissolve ind widen. 

R is the effective radius of the dislocation core, p is the density of the crystal. G is the shear modulus, v is the Poisson ratio. The second term in Equation (6) applies to very rapid deformation processes due to strong shock loading and is dominated by the resonant condition Ei - E. i T(T,U)vo /d that occurs as T(i,U) 1. This term leads to resonant multi-phonon molecular excitation and dissociation associated with detonation and is discussed elsewhere [18]. 

On the other hand, the yield stress of the control specimen can be interpreted in terms of screw dislocation motion. The tensile axis orientation dependence of the yield stress of b.c.c. metals is generally regarded as the effect of the core structure of a screw dislocation on its motion [ ]. A simplistic explanation of this effect is as follows. The screw dislocation is most mobile on a 110 plane. On a 112 plane, the structure of the dislocation core is asymmetric with respect to the (110) direction (direction of motion). Dislocation mobility is not equal in the forward and reverse directions. The higher mobility direction is represented by the [100] tensile axis in the present work and is called the soft 112 slip. The [110] tensile axis represents another case, the hard 112 slip. Hence, the yield stress of the control specimen behaved as shown in Fig. 4. Since the tensile axis orientation dependence is characteristic of the screw dislocation motion, SFS should not be strongly orientation... 

Various crack advance theories have been proposed to relate crack propagation to oxidation rates and the stress-strain conditions at the crack tip, and these theories have been supported by a correlation between the average oxidation current density on a straining surface and the crack propagation rate for a number of systems [12,35]. There have been various hypotheses about the precise atom-atom rupture process at the crack tip—for example, the effect that the enviromnent has on the ductile fracture process (e g., the tensile ligament theory [36], the increase in the number of active sites for dissolution because of the strain concentration [37], the preferential dissolution of mobile dislocations because of the inherent chemical activity of the solute segregation in the dislocation core [38]). 

Because perfect dislocations are observed in high-stress conditions where a hydrostatic component is present in the stress tensor, it is of interest to check the effect of such a hydrostatic pressure on their core structure configuration and mobility. One can expect three kinds of effects due to pressure (i) the material is usually stiffer (this is the case for silicon), with an increase of elastic constants that affect the strain field around the core, (ii) the core structure and its stability could be modified, and (iii) pressure could favor dislocation core mobility along certain directions. One may then wonder whether theoretical investigations of non-dissociated perfect dislocations are really representative of experiments. 

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