In pure metals, the evolution of the average concen­trations of vacancies CV and self-interstitials Q are given by:

^ = K0 — RCiCv — X 4sDv(CV — CVq)

s

dd = K0 — RCi Cv — X 4 A Ci [1]

s

where K0 is the point defect production rate (in dpas-1) proportional to the radiation flux, R is the recombination rate, and DV and DI are the point defect diffusion coefficients. The third terms of the right hand side in eqn [1] correspond to point defect annihilation at sinks of type s. The ‘sink strengths’ kVs and k2s depend on the nature and the density of sinks
and have been calculated for all common sinks, such as dislocations, cavities, free surfaces, grain bound­aries, etc.56,57 The evolution of point defect concen­trations depending on the radiation fluxes and sink microstructure can be modeled by numerical integra­tion of eqn [1], and steady-state solutions can be found analytically in simple cases.43

The evolution of concentration profiles of vacan­cies, interstitials, and chemical elements a in an alloy under irradiation are given by

0CV

V =-div Jv + Ko — RCiCv ot

-X kVsDv [Cv — C Vq]

s

= — div Ji + Ko — RCi Cv — 4 A Ci

ot

s

^ =-div Ja [2]

The basic problem of RIS is the solution of these equations in the vicinity of point defect sinks, which requires the knowledge of how the fluxes Ja are related to the concentrations. Such macroscopic equations of atomic transport rely on the theory of TIP. In this chapter, we start with a general descrip­tion of the TIP applied to transport. Atomic fluxes are written in terms of the phenomenological coeffi­cients of diffusion (denoted hereafter by Lj or, sim­ply, L ) and the driving forces. The second part is

devoted to the description of a few experimental procedures to estimate both the driving forces and the L-coefficients. In the last part, we present an atomic-scale method to calculate the fluxes from the knowledge of the atomic jump frequencies.

1.18.3.1 Atomic Fluxes and Driving Forces

Within the TIP,58’59 a system is divided into grains, which are supposed to be small enough to be consid­ered as homogeneous and large enough to be in local equilibrium. The number of particles in a grain varies if there is a transfer of particles to other grains. The transfer of particles a between two grains is described by a flux Ja, and the temporal variation of the local a concentration is given by the continuity equation

dCa

— =-div Ja [3]

The flux of species a between grains i and j is assumed to be a linear combination of the thermody­namic forces, Xp = (/p — mp)/kB T (i. e., of the gradi­ent of chemical potentials V/p) where /p is the chemical potential of species p on site i, T the tem­perature, and kB the Boltzmann constant. Variables Xp represent the deviation of the system from equi­librium, which tend to be decreased by the fluxes:

Ja = ~^2 LapXp [4]

p

The equilibrium constants are the phenomenological coefficients, and the Onsager matrix (Lap) is sym­metric and positive. When diffusion is controlled by the vacancy mechanism, atomic fluxes are, by con­struction, related to the point defect flux:

Jv = — X JV [5]

a

As gradients of chemical potential are indepen­dent, eqn [5] leads to some relations between the phenomenological coefficients and, if we choose to eliminate the LVp coefficients, we obtain an expres­sion for the atomic fluxes:

Vacancy and interstitial contributions to the atomic fluxes are assumed to be additive:

Ja = — E LVp (Xp — Xv) — X Lap (Xp + Xi) [8]

pp

While thermodynamic data is usually available for the determination of driving forces, it is very difficult to determine the whole set of the L-coefficients from diffusion data. In the first part of the chapter, we define the driving forces as a function of concentra­tion gradients. Then, we present the experimental diffusion coefficients in terms of the L-coefficients and thermodynamic driving forces. The last part of the section shows how to use first-principle calcula­tions for building atomic jump frequency models to calculate macroscopic fluxes of specific alloys.