1.15.4.2.1 Effects of ballistic mixing on phase-separating alloy systems

Consider the simple case where the external forcing produces forced exchanges between atoms (such relocations are found in displacement cascades), and let us assume for now that these relocations are ballistic (i. e., random) and take place one at a time. For this case, one can use a 1D PFM to follow the evolution of the composition profile C(x) during irra­diation.118 This evolution is the sum of a thermally activated term, for which the classical Cahn diffusion model can be used, and a ballistic term:

^ = MirrV2 dF — Gj{c(x)-JWR(x — x’)C(x’)dx’} [25] where Mirr is the thermal atomic mobility, here accel­erated by the irradiation, F the free energy of the system, Gb the jump frequency of the atomic reloca­tions forced by the nuclear collisions, and wR is the normalized distribution of relocation distances, char­acterized by a decay length R. Since most of these atomic relocations take place between nearest neigh­bor atoms, in a first approximation one may assume that R is small compared to the cell size. In this case, the second ballistic term in eqn [25] reduces to a diffusive term:

dC(x) 2 dF 2 2 г і

= MirrV2 gC — Gya2V2 C [26]

In this case, the model thus reduces to the one initi­ally introduced by Martin,119 and the steady state reached under irradiation is the equilibrium state that the same alloy would have reached at an effective temperature Tf = T(1 + Gj/Gj."), where GJh is an average atomic jump frequency, enhanced by the point-defect supersaturation created by irradiation. In particular, in the case of an alloy with preexisting precipitates, depending upon the irradiation flux and the irradiation temperature, this criterion predicts that the precipitates should either dissolve or contin­uously coarsen with time.

Some relocation distances, however, extend beyond the first nearest neighbor distances,120,121 and it is interesting to consider the case where the characteris­tic distance R exceeds the cell size. An analytical model by Enrique and Bellon118 revealed that, when R exceeds a critical value Rc, irradiation can lead to the dynamic stabilization of patterns. To illustrate this point, one performs a linear stability analysis of this model in Fourier space, assuming here that the ballistic jump distances are distributed exponen­tially. The amplification factor w(q) of the Fourier coefficient for the wave vector q is given by

o(q)/M = — (92/ /dC2)q2 — 2kq4

— gR2q2/(1 + R2q2) [27]

where f(C) is the free-energy density of a homoge­neous alloy of composition C, к the gradient energy coefficient, and g = Gj/M is a reduced ballistic jump frequency. The analysis is here restricted to composi­tions and temperatures such that, in the absence of irradiation, spinodal decomposition takes place, that is, d2f/dC2 < 0.

The various possible dispersion curves are plotted in Figure 5. Unlike in the case of short R, it is now possible to find irradiation intensities g such that the

ballistic term in eqn [27] is greater than <@f /dC2 at small q, but smaller than that at large q. In such cases, the amplification factor is first negative for small q values, but it becomes positive when q exceeds some critical value qmin, while for larger q, the amplifica­tion factor is negative again. Therefore, decomposi­tion is still expected to take place, but only for wave vectors larger than qmin, that is, for wavelengths smal­ler than 2p/qmin. It can thus be anticipated that coars­ening will saturate, since at large length scales, the alloy remains stable with respect to decomposition.

Enomoto and Sawa122 have investigated this model using a 2D PFM based on eqn [25]. The interest here is that the PFM, unlike the above linear stability analysis, includes both linear and nonlinear contribu­tions to the evolution of composition inhomogeneity and also permits following the morphology of the decomposition. Using this model, Enomoto and Sawa have confirmed the existence of the patterning regime, see Figure 6, and showed that this patterning can take place in the whole composition range. The PFM approach allows for a direct determination of the pat­terning length scale as a function of the irradiation conditions, as illustrated in Figure 7.

Similar results have also been obtained using a variational analysis of eqn [25], leading to the dyna­mical phase diagram displayed in Figure 8. As seen in this diagram, when the characteristic length for the forced relocation is smaller than the critical value Rc, the system never develops patterns at steady state. Above Rc, patterning takes place when the irradiation

conditions are chosen so as to result in an appropriate g value. Another result obtained from the KMC simulations is that the steady state reached by an alloy is independent of its initial state.

Experimental tests performed on a series of dilute Cu-M alloys, with M = Ag, Co, Fe, have confirmed some of the key predictions of the above simulations and analytical modeling. In particular, irradiation con­ditions that result in atomic relocation distances exceeding a few angstroms do lead to the dynamical stabilization of precipitates at intermediate irradiation temperature.123 These results provide also a compel­ling rationalization of the puzzling results reported by Nelson eta/.124 on the refinement of g precipitates in Ni-Al alloys under 100-keV Ni irradiation at 550 °C.

The origin of the above irradiation-induced com­positional patterning lies in the finite range of the atomic mixing forced by nuclear collisions.125 Enrique and Bellon126,127 have shown that the effect ofthis finite-range dynamics can be formally recast as effective finite-range repulsive interaction between like particles. It is interesting to note that PF

simulations of alloys with Coulomb interactions also predict a patterning of the microstructure.128 The parallel with the treatment of finite-range mixing is in fact quite strong, since a screened Coulomb repul­sion is described by a decaying exponential, as also assumed for the probability of finite-range ballistic exchanges in deriving eqn [27]. The contribution of the Coulomb repulsion to the linear stability analysis is thus proportional to 1/(q2 + qD ), where qD is the

Figure 9 Phase field modeling of the evolution of ordered precipitates in the presence of electrostatic (repulsive) interactions, with increasing time from A to F. The average precipitate size reaches a finite value at equilibrium. Reprinted with permission from Chen, L. Q.; Khachaturyan, A. G. Phys. Rev. Lett. 1993, 70(10), 1477-1480. Copyright by the American Physical Society.

screening wavelength. For reasons similar to the ones discussed in the case of finite-range mixing, it is then anticipated that these interactions will suppress coarsening. This is confirmed by PF simulations, as illustrated in Figure 9.