22.08.2015   Comprehensive nuclear materials

Dilute alloys

The point defect jump frequencies to be considered are those that are far from the solute, those leaving the solute, those arriving at a nn site of the solute, and those jumping from nn to nn sites of the solute. Diffusivities are approached by a series in which the successive terms include longer and longer looping paths of the point defect from the solute. Using the pair-association method, the whole series has been obtained for the vacancy mechanism in body — centered cubic (bcc) and face-centered cubic (fcc) binary alloys with nn interactions (cf. Allnatt and Lidiard6 for a review). However, there is still no accurate model for the effect of a solute atom on the self-diffusion coefficient.84 The L-coefficients as well as the tracer diffusivities depend on three jump frequency ratios in the fcc structure and two ratios in the bcc structure. For the irradiation studies, the same pair-association method has been applied to estimate the L-coefficients of the dumbbell diffusion mechanism in fcc85,86 and bcc83,87 alloys. Note, that the pair-association calculation in bcc alloys87 has recently been improved by using the self-consistent mean-field (SCMF) theory.83 The calculation includes the effect of the binding energy between a dumbbell of solvent atoms and a solute.

In the case of the vacancy diffusion mechanism, before the development of first-principles calculation methods, reliable jump frequency ratios could be estimated from a few experimental diffusion coeffi­cients (three for fcc and two for bcc). Data were calculated from the solvent and solute tracer diffu — sivities, the linear enhancement of self-diffusion with solute concentration, or the electro-migration en­hancement factors of tracer atoms in an electric field.8 Some examples of jump frequency ratios fitted to experimental data are displayed in tables.89 Currently, the most widely used approach is using first-principles calculations to calculate not only the vacancy, but also the interstitial jump frequencies.

1.18.3.4.2.1 Concentrated alloys

The first diffusion models devoted to concentrated alloys were based on a very basic description of the diffusion mechanism. The alloy is assimilated into a random lattice gas model where atoms do not interact and where vacancies jump at a frequency that depends only on the species they exchange with (two frequencies in a binary alloy). Using complex arguments, Manning8 could express the correlation factors as a function of the few jump frequencies. The approach was extended to the interstitial diffusion mechanism.9 At the time, no procedure was sug­gested to calculate these mean jump frequencies from an atomic jump frequency model. Such diffu­sion models, which consider a limited number of jump frequencies, already make spectacular correla­tion effects appear possible, such as a percolation limit when the host atoms are immobile.8,91,92 But they do not account for the effect of short-range order on the L-coefficients although one knows that RIS behavior is often explained by means of a com­petition between binding energies of point defects with atomic species, especially in dilute alloys. Some attempts were made to incorporate short — range order in a Manning-type formulation of the phenomenological coefficients, but coherency with thermodynamics was not guaranteed.93,94

The current diffusion models, including short — range order, are based on either the path probability method (PPM)95-97 or the SCMF theory.84,92,98-100 Both mean-field methods start from an atomic diffu­sion model and a microscopic master equation. While the PPM considers transition variables, which are deduced from a minimization of a pseudo free — energy functional associated to the kinetic path, the SCMF theory introduces an effective Hamiltonian to represent the nonequilibrium correction to the
distribution function probability and calculates the effective interactions by imposing a self-consistent constraint on the kinetic equations of the distribution function moments. Diffusion models built from the PPM were developed in bcc solid solution and ordered alloys, using nn pair transition variables, which is equivalent to considering nn effective inter­actions, and a statistical pair approximation for the equilibrium averages within the SCMF theory.95,96 The SCMF theory extended the approach to fcc alloys8,91,92 and provided a model of the composition effect on solute drag by vacancies.98 The interstitial diffusion mechanism was also tackled.10,83,101 For the first time, an interstitial diffusion model including short-range order was proposed in bcc concentrated alloys.10 It was shown that the usual value of 1 eV for the binding energy of an interstitial with a neighbor­ing solute atom leads to very strong effects on the average interstitial jump frequency and, therefore, on the L-coefficients.

1.18.3 Continuous Models of RIS
partial diffusion coefficients, as new constants asso­ciated with a temperature and a nominal alloy com­position. The first authors to express the partial diffusion coefficients in terms of the L-coefficients were Howard and Lidiard103 for the vacancy in dilute alloys, Barbu86 for the interstitial in dilute alloys, and Wolfer11 in concentrated alloys. In the following, we use the formulation of Wolfer because the approxi­mations made to calculate the L-coefficients and driving forces are clearly stated. In a binary alloy (AB), fluxes are separated into two contributions: the first one induced by the point defect concentra­tion gradients, and the second one appearing after the formation of chemical concentration gradients near the point defect sink11:

JA = — (dAV CV + dAI CI)FV CA + CA(dAV V CV — dAIr Ci)

Jv = —(Ca^av + Cb4bv)VCv + Cv Ф (dA V V Ca + dBV V Cb)

Ji = — ( Ca dAi + Cb dBi )VCi — Ci®(4AiVCa + dBI V Cb ) [24]

with partial diffusion coefficients defined in terms of the L-coefficients and the equilibrium point defect concentration

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RIS is a phenomenon that couples the fluxes of defects created by irradiation and the alloy compo­nents. In RIS models, point defect diffusion mechan­isms alone are considered, although it is accepted that displacement cascades produce mobile point defect clusters, which may contribute to the RIS. In the first section, we present the expressions used to simulate the RIS, the main results, and limits. Some examples of application to real alloys are discussed in the second part. The last section suggests some possible improvements in the RIS models.