The distortion associated with a dislocation implies that it must have certain strain energy. The total strain energy (Etotal) of a dislocation is composed of two parts — elastic strain energy (Eel) and core energy (Ecore). The elastic strain energy is stored outside the core region, and hence can be estimated using the linear elastic theo­ries. We have discussed the shear stress and strain components associated with a screw dislocation in Eqs. (4.10a) and (4.10b). Due to the complete radial symmetry of the stress/strain field associated with the screw dislocation, we derive its elastic strain energy using a simple calculation. Let us consider a volume element shell of the cylinder with an arbitrary radius r, thickness dr, and length l, and thus the elas­tic strain energy is given by

„ 1 ri , ,1 Gbb J N Gb21

Eel — -1ezVez ■ (volumeof theelement) ——————- (l ■ 2pr ■ dr) ——— dr.

2 2 2pr 2pr 4pr

Note the above equation is applicable only for an isotropic continuous medium. The hollow cylinder of length l, inner radius r1, and outer radius r2 will have an elastic strain energy determined by the following integral:

Г2 Gb2ldr _ Gb2l ln /r2 ‘ri 4p r 4p r1

As noted earlier, the dislocation core can be 1b-5b radius, and in most cases can be taken as ~1 nm radius. So, r1 value can be assumed to be of 1 nm. Furthermore, dislocations tend to form networks where the long-range stress fields are superim­posed and may get canceled. Thus, the effective energy of the dislocation can get reduced. That is why if one takes the value of r2 as approximately half the average spacing between dislocations in a random arrangement, a realistic value could be 106 nm. Then the value of r2/r 1 becomes 106. So, Eq. (4.12) is reduced to

13 8

Eei = (Gb2l) . = aGb2l,

v ‘ 4 x 3.14

where a is about ~1.1.

So, the elastic strain energy of a screw dislocation per unit length is

Eel = aGb2, (4.13)

where a = ~1 for screw dislocation.

An analysis can be performed for deriving the energy of an edge dislocation incorporating its complex stress field into the derivation. A similar expression can be obtained with a constant, a = ~1/(1 — v) = ~1.5 (for v = 0.33). So, in simple terms, the edge dislocation has greater energy than that of a screw dislocation of similar length. The elastic strain energy ofa mixed dislocation can also be obtained, and is generally between the energies of a pure edge and a pure screw dislocation.

However, the value of the constant may undergo changes depending on the pre­cise values of r1 and r2, which are difficult to determine. However, whatever their values are, the constant a would still remain close to unity.

The estimation of the dislocation core energy is only approximate as linear elastic theories are no longer valid in this region. Approximate calculations show that the core energy is estimated to be on the order of 0.5 eV per atom plane threaded by the dislocation, whereas the elastic strain energy of a dislocation is 5-10 eV per atom plane. Despite variation in the core energy during the dislocation movement, the core energy still remains a small fraction of the elastic strain energy. Thus, the dis­locations possess large positive strain energy increasing the overall free energy of the crystal. But the system would like to attain a more stable state via lowering its free energy. That is why given the chance the dislocation density can be reduced through dislocation annihilation processes such as annealing. The same is not true for point defects (such as vacancies) as they attain different equilibrium concentra­tions at different temperatures. That is why dislocations are thermodynamically unstable defects, whereas vacancies are thermodynamically stable defects.