Damage accumulation in pure metals during irradi­ation primarily takes place in the formation and evolution of vacancy and SIA-type defects. At tem­peratures higher than recovery stage III, which is the main interest for practical purposes, vacancy clusters normally take the form of voids that result in the change of a volume, that is, swelling. Owing to limitations of space, in the following section we focus only on a description of void evolution.

1.13.5.2.1 Void swelling

The solution obtained from eqns [44] depends on the irradiation temperature. Temperatures below recovery stage II will not be considered here. At temperatures smaller than that corresponding to the recovery stage III, when vacancies are immobile and the interstitials are mobile, the concentration of vacancies will build up. At some irradiation dose, the vacancy concentration will become high enough that mutual recombination of PDs may become the dominant mechanism of the defect loss, thus controlling defect accumulation. In this case,

 

DvCvk2 = Pc(x)fc(x) x2

 

[83]

the dose dependence of PD concentrations can be calculated analytically104	formed. In the following discussion we concentrate on the irradiation doses beyond the transient period, which are of more practical interest. If only voids and edge dislocations are present in the system, and mutual recombination and thermal emission of vacancies from voids and dislocations are both negligible, the balance equations for the concentrations of vacancies and SIAs, Cv and Ci, are given by G — k^DyCy — ZdypdDvCy = 0 G — k2 Di Ci — zfpdDiCi = 0 [93]
At temperatures higher than that corresponding to recovery stage III, both vacancies and SIAs are mobile. Hence, after a certain time of irradiation, called the ‘transient period’, their concentrations reach a steady state. A comprehensive analysis of the time (irradia­tion dose) dependence of PD concentrations for different sink strength can be found in Sizmann.9 The dose dependence of PD concentrations and void swelling obtained by the numerical integration of ME73 is presented in Figure 4. As can be seen, the vacancy supersaturation, (DvCv — DiCi)/DvCVq, becomes positive when the PD concentrations reach steady state and this gives rise to void growth. Also, note that in the transient regime only divacancies are
[94]
Figure 4 Dose dependences of the concentrations of point defects, void swelling, vacancy supersaturation, and void number density calculated in the framework of FP3DM by numerical integration of the master equation, eqn [18]. From Golubov and Ovcharenko.73
dd Zi__ Zv Zd
[97]
Bd
The maximum value of the ratio in the right-hand side of eqn [96] is 1/4, when the sink strengths of voids and dislocation are equal to each other, k/ = Z^Pd. Thus the maximum swelling rate is dS =Bd df max 4 It is easy to show that the swelling rate described by eqn [96] depends only weakly on the variation of the sink strength of voids and dislocations: a differ­ence of an order of magnitude results in a decrease of the swelling rate by a factor of 3 only. To obtain the steady-state swelling rates of ~1% per NRT dpa, which are observed in high-swelling

fcc materials, one would need the bias factor to be about several percent. Data on swelling in electron — irradiated metals resulted in Bd ~ 2 — 4% for the fcc copper24,105,106 (data reported by Glowinski107 were used in Konobeev and Golubov106), ^2% for pure Fe-Cr-Ni alloys,108 and orders of magnitude lower values for bcc metals (e. g., swelling data for molybdenum1 ). Because the electron irradiation produces FPs, it is reasonable to accept these values as estimates of the dislocation bias.

Note that the first attempt to determine Bd by solving the diffusion equations with a drift term deter­mined by the elasticity theory for PD-dislocation interaction as described in Section 1.13.5 showed that the bias is significantly larger than the empirical estimate above. Several works have been devoted to such calculations,96,110-113 which predicted much higher Bd values, for example, ~15% for the bcc iron and ^30% for the fcc copper. With these bias factors, the maximum swelling rates based on Bd/4 should be equal to about 4% and 8% per dpa but such values have never been observed. An attempt to resolve this discrepancy can be found in a recent publication.114

Surprisingly, the steady-state swelling rate of ~1% per NRT dpa has been found in neutron — (and ion-) irradiated materials, for example, in various stainless steels, even though the primary damage in these cases is known to be very different and the void swelling should be described in the framework of the PBM, which gives a rather different description ofthe process. An explanation of this is proposed in Section 1.13.6.