In Section 1.18.3.3, we mentioned the difficulty in measuring the L-coefficients of an alloy, especially those involved in fluxes induced by point defect concentration gradients. Diffusion data are even more difficult to obtain for the interstitials. This is probably why most of the RIS models emphasize the effect of vacancy fluxes, with the interstitial contri­bution assumed to be neutral.

The first model proposed by Marwick30 intro­duced the main trick of the RIS models by taking out vacancy concentration as a separate factor of the diffusion coefficients. Expressions of the fluxes were obtained using a basic jump frequency model, which is equivalent to neglecting the cross-terms of the Onsager matrix. According to the random lattice gas diffusion model of Manning,8 correlation effects are added as corrections to the basic jump frequencies. The resulting fluxes30 are similar to eqn [24], except that the c-partial diffusion coefficients (dAV, 4i) are now equal to the partial diffusion coefficients (dw, dAi), which is equivalent to neglecting cross L-coefficients and the dependence of equilibrium point defect concentration on alloy composition. However, for the first time, the segregation of a major element, Ni, in concentrated austenitic steel was quali­tatively understood in terms of a competition between fast — and slow-diffusing major alloy components.

The partial diffusion coefficients associated with the vacancy mechanism are estimated using the

 

relations of Manning, deducing the L-coefficients from the tracer diffusion coefficients:

 

CD* C! UJ ECkD*

 

CiD

 

d, j +

 

[26]

 

One observes that the Manning8 relations systemati­cally predict positive partial diffusion coefficients: daV = D*a /Cv [27] Moreover, the three L-coefficients of a binary alloy are no longer independent. Both constraints are known to be catastrophic in dilute alloys, while they seem to be capable of satisfactorily describing RIS of major alloy components in austenitic steels.30 1.18.4.1.2 Interstitials Wiedersich eta/.102 added to Marwick’s model a con­tribution of the interstitials. The global interstitial flux is described by eqn [24], while preferential occu­pation of the dumbbell by a species or two is deduced from the alloy concentration and the effective bind­ing energies. Such a local equilibrium assumption implies very large interstitial jump frequencies com­pared to atomic jump frequencies. This kind ofmodel yields an analytical description of stationary RIS (see below eqn [28]). An explicit treatment of the intersti­tial diffusion mechanism was also investigated. From a microscopic description of the jump mechanism, one derives the kinetic equations associated with each dumbbell composition.109-112 Unlike previous models, there is no local equilibrium assumption, but correlation effects are neglected (except in Bocquet90). They could have used the interstitial diffusion model with the correlations of Bocquet.90 However, due to the lack of data for the interstitials, most of the recent concentrated alloy models neglect the interstitial contribution to RIS.104,113,114

where the intrinsic diffusion coefficient is equal to Da = KVCV + dCi CI)F. The spatial extent of segre­gation coincides with the region of nonvanishing defect gradients. Note that, in the original paper,102 the partial and c-partial diffusion coefficients were taken to be equal. Such a simplification may change the amplitude of the RIS predictions. In a multicom­ponent alloy, not only the amplitude, but also the sign of RIS might be affected by this simplification. In dilute alloys, the whole kinetics can be approached by an analytical equation as long as the Kirkendall fluxes resulting from the formation of RIS are neglected.115

1.18.4.1.4 Concentration-dependent diffusion coefficients

Most of the RIS models assume thermodynamic fac­tors equal to 1, although in the first paper,11 a strong variation was observed between the thermodynamic factor and composition. Similarly, the quantities ZVa and ZIa of the point defect driving forces [14] are expected to depend on local concentration and stress field.11,116 For example, Wolfer11 demonstrated that RIS could affect the repartition between interstitial and vacancy fluxes and thereby, the swelling phe­nomena. The bias modification might be due to sev­eral factors: a Kirkendall flux induced by the RIS formation, or the dependence of the point defect chemical potentials on local composition, including the elastic and chemical effects.

A local-concentration-dependent driving force is due to the local-concentration-dependent atomic jump frequencies. Following this idea, the modified inverse Kirkendall (MIK) model introduces partial diffusion coefficients of the form1

pm

dAV = dAV exp -^V [29]

The migration energy is written as the difference between the saddle-point energy and the equilibrium energy. It depends on local composition through pair interactions calculated from thermodynamic quanti­ties such as cohesive energies, vacancy formation energies, and ordering energies. In fact, the present partial diffusion coefficients correspond to a meso­scopic quantity deduced from a coarse-grained averaging of the microscopic jump frequencies. In principle, not only the effective migration ener­gies, but also the mesoscopic correlation coefficients and thermodynamic factors should depend on local concentrations. Nevertheless, the thermodynamic factors, correlation coefficients, and the ZVa factor are assumed to be those of the pure metal A. Recently, a new continuous model suggested a multifrequency formulation ofthe concentration-dependent partial dif­fusion coefficients. Instead of averaging the sums of interaction bonds in the exponential argument, every jump frequency corresponding to a given configuration is considered with a configuration probability weight.17 Predictions ofthe model are compared with direct RIS Monte Carlo simulations that rely on the same atomic jump frequency models. In the two presented examples, the agreement is satisfactory. However, the thermody­namic factor and correlation coefficients are yet to be defined clearly.17