1.15.4.1 Challenges Specific to Alloys Under Irradiation

The PFMs discussed so far are broadly applied to materials as they relax toward some equilibrium state. In particular, the kinetics of evolution is given by the product of a mobility by a linearized driving force, see for instance eqns [2] and [3]. In the context of the thermodynamics of irreversible processes,28 the mobility matrix is the matrix of Onsager coefficients. Irradiation can, however, drive and stabilize a material system into a nonequilibrium state,108 owing to ballis­tic mixing and permanent defect fluxes, and so it may appear questionable at first whether linearized relax­ation kinetics is applicable. A sufficient condition, however, is that these different fields undergo linear relaxation locally, and this condition is often met even under irradiation. A complicating factor arises from the presence of ballistic mixing, which adds a second dynamics to the system on top of the thermally acti­vated diffusion of atoms and point defects. A superpo­sition oflinearized relaxations for these two dynamics is valid as long as they are sufficiently decoupled in time and space, so that in any single location, the system will evolve according to one dynamic at a time. KMC simulations indicate that, for dilute alloys, this decoupling is valid except for a small range of kinetic parameters where events from different dynamics interfere with one another.109

A second issue is that PFMs, traditionally, do not include explicitly point defects. Vacancies and inter­stitials are, however, essential to the evolution of irradiated materials, and it is thus necessary to include them as additional field variables. The situa­tion is more problematic with point-defect clusters, which often play a key role in the annihilation of free point defects. Since the size of these clusters cover a wide range of values, it would be quite difficult to add a new field variable for each size, for example, for vacancy clusters of size 2 (divacancies), size 3 (triva­cancies), size 4, etc. Moreover, under irradiation con­ditions leading to the direct production of defect clusters by displacement cascades, additional length scales are required to describe the distribution of defect cluster sizes and of atomic relocation dis­tances. These new length scales are not physically related to the width of a chemical interface at equi­librium, we, and therefore, they cannot be safely rescaled by we. This analysis clearly suggests that one needs to rely on a PFM where the atomic scale has been retained. This is, for instance, the case in the quantitative PFM reviewed in Section 1.15.3. Another possible approach is to use a mixed continu­ous-discrete description, as illustrated below in Sec­tion 1.15.4.2.4. We note that information on defect cluster sizes and relocation distances should be seen as part of the noise imposed by the external forcing, here the irradiation, on the evolution of the field variables. The difficulty is thus to develop a model that can correctly integrate this external noise. It is
well documented that, for nonlinear dissipative sys­tems, the external noise can play a determinant role and, for large enough noise amplitude, may trigger nonequilibrium phase transformations.110-113

One last and important challenge in the develop­ment of PFMs for alloys under irradiation is the fact that in nearly all traditional models the mobil­ity matrix is oversimplified, for instance Mirr = C(1 — C)Dirr/kBT, which is a simple extension to eqn [17] where D has been replaced by Dirr to take into account radiation-enhanced diffusion. In the common case of multidimensional fields, for instance for multicomponent alloys, or for alloys with conserved and nonconserved field variables, the mobility matrix is generally taken as a diagonal matrix, thus eliminating any possible kinetic cou­pling between these different field variables. As discussed at the end of Section 1.15.2, this approxi­mation raises concerns because it misses the fact that these kinetic coefficients are related since they origi­nate from the same microscopic mechanisms. This is, in particular, the case for the coupled evolution of point defects and chemical species in multicompo­nent alloys. This coupling is of particular relevance to the case of irradiated alloys since irradiation can dramatically alter segregation and precipitation reac­tions owing to the influence of local chemical envir­onments on point-defect jump frequencies. While new analytical models have been developed recently using mean field approximations to obtain expressions for correlation factors in concentrated alloys,114-117 work remains to be done to integrate these results into PFMs.