Mutual annihilation of PDs happens either by direct interaction between single vacancies and SIAs in the matrix or within a certain type of neutral sink which we call ‘saturable.’ The fluxes of vacancies and SIAs to them are equal. An example of such sinks is vacancy loops, which were considered in the frame­work of the BEK model29 and PBM,22 that is, in the case where the vacancy clusters are generated in cascades. The BEK model is not discussed further in the present work because it does not correspond to any realistic situation in solids under irradiation; vacancy clustering in cascades is always accompanied with the SIA clustering, which is accounted for in the framework of the PBM but not in the BEK model.

The balance equations in the case considered are as follows

G — mRDi C Cv — lN A C — Cv — zdpdDvCv = 0

g — mRDi c Cv — iNa ci — A Ci — zd rdDi ci = 0 [99]
where kN2 is the strength of neutral sinks. Note that absorption rate of both vacancies and SIAs in eqn [99] is described by the same quantity, kNDiQ, which reflects neutrality of this sink with respect to vacancies and SIAs.115,116

The defect concentrations and swelling rate are

G 1 1

+ ZvPd 1 + JR 1 + JN

d5 = B k2zvdrd _J_________________ L_

df d (12 + Zdrd)2 1 + JR 1 + JN

where

4PrG

(k2 + zd Pd + kN )2dv

|2

kN 12 + zd Pd

In the absence of an effect on the sink structure, mutual recombination reactions are important at low temperature, when the vacancy diffusion is slow, and for high defect production rates, when the vacancy concentration is sufficiently high to provide higher sink strength for SIAs than that of existing extended defects. This can be expressed mathematically by an inequality JR > 1 or more explicitly as a temperature boundary IbT < £m/ln[2Dv0(k^ + Z^pd + )2/^rG] where Dv0 is the preexponential factor in the vacancy diffusion coefficient and Evm is the effective activation energy for the vacancy migration. In practice, this situation is unlikely to occur because the radiation — induced sink strength rapidly increases at low tem­peratures. In this case recombination at sinks is of greater importance.

One of the important aspects that recombination reactions introduce to microstructural evolution is the appearance of a temperature dependence; at low temperatures, an increase of the swelling rate with increasing temperature is predicted, which is also observed experimentally in the fcc-type materials. The question of whether it was possible to explain the experimental reduction of swelling rate with decreasing temperature by recombination was addressed.29 It was found that the observed tempera­ture effect on swelling rate was much stronger than predicted by recombination alone.

The impact of neutral sinks on swelling rate is significant when they represent the dominant sink

in the system: kN ^ kc + Z^pA. The swelling rate in the case is given by

dS= B k2ZydPd

df d (k;: + ZdPd)(k2 + ZdPd + kN)

k;zdPd

~ d (k; + zdpd)kN [102]

Such a situation may occur, for example, at low enough temperature, when the thermal stability of vacancy loops and SFTs becomes high enough, lead­ing to their accumulation up to extremely high con­centrations. Another possibility is when a high density (about 1024m-3) of second phase particles exists, as in the oxide dispersion strengthened (ODS) steels.