For dislocation loops, the equilibrium vacancy con­centration can be found by a similar analysis. How­ever, as point defects are absorbed or emitted by the loop, it shrinks or expands, thereby changing its strain energy.

Let us first consider an interstitial loop in a crystal subject to a stress Sj. The force exerted on the loop plane is ti = , where n is a normal vector to the

loop plane. Upon forming the loop in the crystal, the atomic planes adjacent to the loop platelet are dis­placed by the Burgers vector b. Therefore, the work done by the external stresses is pR1biacjjnj, where pR1 is the area of the loop assumed to be circular in shape. Since the separation of the two atomic planes next to the loop platelet is n • b = nfa, the volume of the inserted platelet is pR2n, b, = NO, where N is the number of atoms in the loop. The work done by the external forces can then be written as b, Gjnj O/ (n, b,)

Note, when a vacancy loop is formed, the crystal contracts and the work performed by external forces is just the opposite, namely

sVLO = — b, ajnj О/(п, Ь,) [98]

Let us return to the case of an interstitial loop and consider the change in loop energy when a vacancy is emitted. This change is £Loop(N + 1) — ELoop (N) ffi dELoop/dN, where the number N of intersti­tials is related to the loop radius as pR2n, b, = NO. This change may be considered as a vacancy chemi­cal potential associated with the prismatic interstitial loop of radius R, and it may be computed according to the equation

dELoop O dELoop

dN 2p (n, b, )R dR

Following our procedure now, we consider the total change in Gibbs free energy when a vacancy is cre­ated in the crystal lattice next to the dislocation core and the extra atom is attached to the loop platelet. This change is

AG = — 4 + Tf — Ei(rc,’) + O^L + VvRs°H

— AmVoop — kT indL [ioo]

and it must be equal to zero when local thermody­namic equilibrium prevails. This condition then determines the local vacancy concentration as

EI(rC;’) OsIL+ VV AmV P

‘ kT kT kT

Loop

AmV p

‘ kT

The local equilibrium vacancy concentration at an interstitial-type loop differs from the value at a
straight edge dislocation mainly by an exponential factor containing the vacancy chemical potential change. In fact, it is lower than for a straight edge dislocation, as will be shown below.

For a vacancy-type loop, the sign in this exponen­tial factor is positive, and the local equilibrium vacancy concentration is enhanced compared to the straight edge dislocation according to

C ffi c? exp(Ammr!) [101]

To quantify how different these local vacancy con­centrations can be from the true equilibrium concen­tration, we consider the case of perfect prismatic loops. The Burgers vector b in this case is parallel to the normal vector n, and n, b, = b. The energy is given by36,37

; R{K(1 —rC)— E(1— rC)} + pr1 tsf

— + pRl»sf I103]

where K and E are the complete elliptic integrals. For large loop radii R much larger than the core radius rc, the approximation given in the second part of the above equation can be used. The second term in addition to the strain energy represents the energy of the stacking fault. Note that the shear modulus m is not to be confused with the vacancy chemical potential AmV.

Taking the derivative of the above equation with respect to N gives the vacancy chemical potential difference as

AmVoop(R)=—

+ OgSF [104]

Inserting this result into eqns [101] and [101] gives the ratio of the vacancy concentration near the dislo­cation core of prismatic dislocation loops relative to the equilibrium concentration [93]. As an example, these ratios are evaluated for Ni and at a temperature of 773 K and shown in Figure 22. Both the exact expression for the loop energy and the logarithmic one in eqn [103] are employed to demonstrate that the latter provides an excellent approximation. How­ever, for small prismatic loops, both expressions for the loop energy become questionable, and atomistic calculations are necessary to obtain energies ofsmall, plate-like clusters of self-interstitials or of vacancies.