Edge dislocations contain jogs as illustrated by Figure 21. Here, the edge dislocation is the termina­tion of an atomic plane, and the last row of atoms, rendered in a darker color, delineates the dislocation line. The step in this line forms a jog. The two atomic planes next to the terminated atomic plane are shown in the upper part with perfect occupancies of all atomic positions, except that they bend along the dislocation line to become neighboring planes above the dislocation.

A vacancy may now form by an atom adjacent to the jog jumping to the terminating plane and thereby moving the jog by one atomic distance, as illustrated in the lower part of Figure 21. The vacancy created

may then diffuse away and into the surrounding crystal. Conversely, a vacancy that migrates to this jog will take the place of the atom next to the jog. As a result, the jog is simply displaced in the opposite direction. The absorption or emission of a vacancy restores the configuration of the edge dislocation.

The absorption or emission of a vacancy from a perfect core site of an edge dislocation is also possi­ble. However, the formation of a double jog would involve a large energy change, and for this reason, vacancy emission and absorption is unlikely for any site along the edge dislocation other than at already existing jogs.

Assuming local thermodynamic equilibrium for the region around a jog, the thermal vacancy concen­tration can be found as follows. We imagine that we create a vacancy in the surrounding lattice and add the extra atom to the jog. The total change in Gibbs free energy is then given by

A G = -(f — TsV ) + kT’n[W(NV + 1)] — kT’n[W(NV)] [91] since we added the Gibbs free energy (EV — Tf) to the crystal to create the vacancy but gained some configurational entropy of the amount k (N + NV + 1)!NV! "(N V + 1)!(N+N V)!

[92]

Here, N is the number oflattice sites per unit volume and Nv the number of vacancies, the latter being much smaller than N.

The creation of an additional vacancy in the vicin­ity of the dislocation jog occurs as a result of thermal fluctuation, and the Gibbs free energy change is on average equal to zero for this process. The condition AG = 0 defines the vacancy concentration in local thermodynamic equilibrium with the edge disloca­tion, and one obtains f IkT

Two minor but nevertheless important energy con­tributions have been neglected in the above deriva­tion. First, we did not consider the interaction of the vacancy with the stress field of the dislocation. Sec­ond, we also neglected any volume change of the crystal with the formation or absorption of a vacancy. A volume change will give rise to a work term if an external stress field aj is present in addition to the dislocation stress field.

The first contribution can be taken into account by adding to the vacancy formation energy the

interaction energy EI(ro, ‘), which was discussed in Section 1.01.5 and is further elaborated in Section 1.01.8. This energy is evaluated at the core radius rCj, and it varies with the polar angle ‘.

To evaluate the second contribution, we note that if vacancies were persistently absorbed at an edge dislocation, it would eventually eliminate the incom­plete crystal plane of atoms. The crystal volume would therefore contract in the direction perpendic­ular to this incomplete plane by the amount of the Burgers vector component b, in this direction. If tj = ajbj|b is the force per unit area on the incom­plete plane as exerted by the external stress field aj, then the work done against the external loads is

-8Ab, t, =-8A b, a°t-bj|b [94]

when the area 8A is eliminated from the incomplete plane. The negative sign turns into a positive one if vacancies are emitted from the dislocation and if it climbs to expand the area of the incomplete plane. For one vacancy emitted and hence one atom added to the incomplete plane, 8Ab = O, and the energy gained from the work expended by the external loads is

,2) a°Mbj + = Os? + [95]

The second term is due to the fact that the crystal does not expand by the entire atomic volume but is reduced by the vacancy relaxation volume. Note that VR has in general a negative value, and that it results in an isotropic volume change; hence, the hydrostatic stress performs the work.

As a result of these two additional contributions one finds that the local equilibrium vacancy concen­tration at a dislocation is

EI(rC.’) VvaH. Oa? kT + kT + kT

We see that the interaction energy £I(rC,’) gives a spatial dependence to the local equilibrium vacancy concentration. In addition, external forces on the crys­tal or stresses produced at the location of the disloca­tion by other distant defects will further change the local equilibrium concentration.