22.08.2015   Comprehensive nuclear materials

Swelling rate

By omitting the recombination term, eqns [10] —[12] for mobile defects can be rewritten as

= Gv + Gvc — DvCv(i2 + Zdpd + ZVclk2cl + zvclk2cl)

^ = g — Dici(k2 + zvclpd + Ziclki2d + zvclk2cl) dcg fpr 2

= Gi — 2DgCg 2d + prcNc + svcNvcl + °icNid [[15] [16]]

It has been shown that, under conditions in which swelling is observed, the vacancy and SIA clusters produced by cascades reach steady-state size distri­butions at relatively small doses.22 This is because vacancy clusters have far lower thermal stability than voids. The growth of sessile SIA clusters is restricted on account of the high vacancy supersaturation, which builds up due to rapid 1D diffusion of mobile SIA clusters to sinks. Consequently, at relatively low doses, the SDF of the sessile SIA clusters achieves steady state. After reaching steady state, both types of sessile clusters start to serve as recombination centers for PDs and glissile SIA clusters. Analytical expres­sions for the steady-state SDFs of vacancy and SIA clusters can be found (see eqns [23] and [24] in Singh et at22) and the corresponding sink strengths of the clusters at the steady state are given by (eqn [25] in the same reference) 4G
produced are reduced in size due to the high vacancy supersaturation. Fluctuations in the defect arrival to the clusters produce a tail in the SDF extending beyond the maximum size formed in cascades. The tail is characterized by very small concentrations and cannot describe the observed nucleation and growth of SIA clusters and the consequent formation of the dislocation network (see, e. g., Garner[17] and Garner et a/.[18]). The most probable reason for this failure is that the cluster-cluster interaction leading to their coalescence is neglected in the current theoretical framework.

Sessile interstitial clusters are produced in cas­cades at rates comparable to those of PDs. The evo­lution of concentrations of mobile species (PDs and glissile clusters) in this case may be described by nonstationary equations because of the very fast evo­lution of the sessile cluster population. High vacancy supersaturation will drive the evolution of the sessile SIA clusters toward quasisaturation state, beyond which the steady-state equations for the mobile spe­cies become valid. Similar steady state for vacancy clusters will be achieved because of the thermal instability of the clusters.22

Gv = DC (k2 + ZvVd) + DvCvZvlk2d

+ Di Ci z;cl4l + Dg CgLxgffidNid [133]

Gi = DiCi(k2 + Zfpd) + D^Zf^

+ ACiZ;dk2d — 2Dg CgLxg ffidNd [[19]]

Gg = 2DgCg + nr2cNc + svcNvd + ^Nd [135]

In the framework of PBM, the balance equations for PDs depend on the concentration of glissile clusters and, thus, are very different from those in the FP3DM.

The vacancy supersaturation is obtained from the difference between DvCv and DiCi as given by eqns

[133] and [134]

DVCV — DiCi

B Zvrd D C = її d DvCv

k2 + ZVd

eg gnrt (1 — er)

k2 + ZJPd

where Lg = k|/2 = Prdpd/2 + Prc2Nc + ffvdNvd+

SidNid. The first and the second terms on the right-hand side of eqn [136] correspond to the actions

image078

production bias due to recombination of the SIA clusters at sessile vacancy and SIA clusters, while the second one arises from the capture of SIA clusters by voids. The latter term may become equal to zero or even negative, hence the combination of the two bias factors does not necessarily lead to a higher swelling rate, as shown in Barashev and Golubov.35

of the dislocation bias and the production bias, respectively. As can be seen, the first term depends on the vacancy concentration, and hence on the total sink strength of all defects including PD clusters. The second term also depends on the sink strength of all defects but differently, and describes the distribution of excess of vacancies between voids and dislocations, and their recombination at PD clusters.

In the PBM, the swelling rate is given by

dS = k2(DvCv — AG) — 2DgCgXgLgPr]Nc [137]

and, with the aid of eqn [136], can be represented as follows

 

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