1.18.3.2.1 Local chemical potential

The thermodynamic state equation defines a chemi­cal potential of species i as the partial derivative of the Gibbs free energy G of the alloy, with respect to the number of atoms of species i, that is, Ni. The resulting chemical potential is a function of the temperature and molar fractions (also called concen­trations) of the alloy components, Ci = Ni/N, N being the total number of atoms. TIP postulates that local chemical potentials depend on local concentrations via the thermodynamic state equa­tion. A chemical potential gradient V/ of species i is then equal to

V/L

kBT

where Ci is the local concentration of species i. In a binary alloy, concentration gradients of the two com­ponents are exactly opposite. The chemical potential gradient of component i is then proportional to the concentration gradient:

V/i kB T
JV = -£ LVp(Xp — xv) [6] p
[10]

Under irradiation, diffusion is controlled by both vacancies and interstitials. The flux of interstitials is also deduced from the atomic fluxes:

Ji = X Jp

p

where F, is called the thermodynamic factor. Fur­thermore, the Gibbs-Duhem relationship58 leads to interdependent chemical potential gradients:

Ck V/k = 0

k = 1 ,r

where the sum runs over the number of species. There­fore, in a binary alloy there is one thermodynamic factor left:

Vm

kBT

where F = Fa = FB. Note, that an alloy at finite temperature contains point defects. They are currently assumed to be at equilibrium with the local alloy com­position, with the local chemical potential equal to zero. When calculating the thermodynamic factor, point defect concentration gradients are neglected. During irradiation, although point defects are not at equilibrium, one assumes that eqn [12] continues to be valid.

Under irradiation, additional driving forces are involved. They correspond to the gradients of vacancy and interstitial chemical potentials, which are usually written in terms of their equilibrium concentrations CVq and Cjeq respectively:

mV = kBT ln(CV/cVq) and mI = kBT ln(Q/Cjeq) [13] leading to an expression of the associated driving force11: —V CV — XVAV CA with xVA

The interstitial driving force has the same form, except that letter V is replaced by letter I. Note, that the equilibrium point defect concentrations may vary with the local alloy composition and stress. Although the variation of the equilibrium vacancy concentration is expected to be mainly chemical, the change of the elastic forces due to a solute redistribution at sinks should not be ignored for the interstitials.11 Due to the lack of experimental data, Wolfer11 introduced the equilibrium vacancy concen­tration as a contribution to a mean vacancy diffusion coefficient expressed in terms of the chemical tracer diffusion coefficients. Composition-dependent tracer diffusion coefficients could then account for the change of equilibrium vacancy concentration, with respect to the local composition.

Within the framework of the TIP, a thermody­namic factor depends on the local value but not on the spatial derivatives of the concentration field. The use of this formalism for continuous RIS models deserves discussion. Indeed, a typical RIS profile covers a few tens of nanometers so that the cell size used to define the local driving forces does not exceed a few lattice parameters. Such a mesoscale
chemical potential is expected to depend not only on the local value, but also on the spatial derivatives of the concentration field. According to Cahn and Hilliard,60 the free-energy model of a nonuniform system can be written as a volume integral of an energy density made up of a homogeneous term plus interface contributions proportional to the squares of concentration gradients. Thus, all contin­uous RIS models that are derived from TIP retain only the homogeneous contribution to the energy density and cannot reproduce interface effects and diffuse-interface microstructures. In particular, an equilibrium segregation profile near a surface is pre­dicted to be flat.