1.01.5.1 The Misfit or Size Interaction

Many different sources of strain fields may exist in real solids, and they can be superimposed linearly if they satisfy linear elasticity theory. If this is the case, we need to consider here only the interaction between one particular defect located at rd and an extraneous displacement field u0(r) that originates from some other source than the defect itself. In particular, it may be the field associated with external forces or deformations applied to the solid, or it may be the field generated by another defect in the solid.

To find the interaction energy, we assume that the defect under consideration is modeled by applying a set of Kanzaki26 forces f(“)(R(“)), a = 1, 2, …, z, at z atomic positions R(a as described in greater detail

in Appendix B. We imagine that once applied, the extraneous source is switched on, whereupon the atoms are displaced by the field u0(r), and work is done by the Kanzaki forces of the amount

z

W = -‘ f(a) • u0 rd + R(“> [39]

a=1 ‘ ‘

If the displacement field varies slowly from one atom position to the next, we may employ the Taylor expansion for each displacement component,

rd + R(“^ ~ «0>(rd) + Uij(rd)R(°^ + ••• [40]

and obtain -4ЫЕfia)-мЕгґ]Ra)— [41

The first term vanishes because the sum of Kanzaki forces is zero, as pointed out in eqn [B4], and the second term can be expressed in terms of the dipole tensor defined in [B6]. Furthermore, since the dipole tensor is symmetric, we obtain finally for the interaction energy

W1 ~-P„ 4(rd) [42]

where є0 (rd) is the extraneous strain field at the location of the defect.

Point defects can also be modeled as inclusions, as we have seen, and they can be characterized by a transformation strain tensor, ekl. In Appendix B it is shown, with eqn [B25], that the dipole tensor is then given by

Pij = OCijki eu [43]

and the interaction energy can be written as

W = — OCiju eu 4 = — П eu ff° [44]

Finally, for the simplest model of a point defect as a misfitting spherical inclusion, as treated in Appendix A,

ekl = ro dkl

and

w = —.Vrel s“k [45]

In this last form, we used the notation for the relaxation volume for the misfit or transformation volume, that is, AV = Vrel, to emphasize the fact that the interaction energy of vacancies and self-interstitials does not depend on their formation volumes, but on their relax­ation volumes. This dependence on the misfit volume or defect size gave this energy the name of misfit or size interaction.