PHYSICAL REVIEW B VOLUME 46, NUMBER 18 1 NOVEMBER 1992-II Two-phase dynamical equilibria driven by irradiation in ordered alloys F. Soisson, P. Bellon, and G. Martin Section de Recherches de Metallurgic Physique, CEREM, Centre d'Etudes de Saclay, F 9119-1 Gif sur Yve-tte CEDEX, France {Received 16 March 1992) We study here two-phase equilibria in driven compounds, where two dynamics are acting in parallel: thermally activated atomic jumps and forced jumps; such is the case for an alloy under irradiation where nuclear collisions induce ballistic jumps. We propose a deterministic treatment of the concentration and degree of order fields (one or two dimensional) and identify two-phase locally stable steady states: Dynamical equilibrium phase diagrams are thus computed. It is shown that in a body-centered-cubic al- loy an A2-B2 order-disorder transition of the second kind at thermal equilibrium becomes of the first kind beyond a temperature-dependent critical forcing intensity. As a result, two-phase steady states can be stabilized by irradiation. Interface properties are then studied: Surface-tension-like effects are ob- served; introduction of antiphase boundaries destabilizes ordered precipitates, leading to their dissolu- tion and redistribution. In order to compare the relative stability of the different steady states, a stochas- tic description is then proposed: We build a mean-field grand-canonical potential which governs the steady-state probability distribution of the concentration and long-range order parameter. It shows that the most stable steady state is indeed two-phase under suitable irradiation conditions. I. INTRODUCTION As is well known, irradiation by energetic particles can enhance or induce phase transitions: Amorphization of crystalline solids, disordering of ordered alloys, dissolu- tion of precipitates, or irradiation-induced precipitation in a solid solution are good examples (for a review, see Ref. I). These can be reviewed as dynamical phase tran- sitions in driven systems: Indeed, in a solid under ir- radiation, nonequilibrium configurations are sustained by the permanent injection of Frenkel pairs and of replace- ment collisions (ballistic jumps). Such irradiation- induced phase transitions have an important technologi- cal impact: Extrapolating data obtained for one set of ir- radiation conditions to yet-unexplored irradiating envi- ronments (e.g. , 14-MeV neutron irradiation) cannot be done straightforwardly and requires modeling. Here we focus on ordered alloys: In most simple cases, the state of the system results from competition between the disordering due to the ballistic jumps induced by nu- clear collisions and the reordering due to the thermally activated jumps of point defects. ' Kinetic models, based on rate theory and incorporating these competing two effects, have long been used for fitting disordering or ordering rates measured experimentally. ' However, as soon as two or more locally stable steady states are com- peting, such descriptions do not provide any information on their relative stability. Furthermore, some relevant effects such as the production of ballistic jumps by bursts cannot be addressed by these models (cascade-size effect). A kinetic description we introduced recently allows one to address such questions: Starting from a master equation, stochastic potentials can be computed; they play a role analogous to thermodynamic potentials for equilibrium systems and permit one to construct dynami- cal equilibrium phase diagrams in a (temperature X concentration X irradiation flux) space. Up to now, this technique has been applied to a homogeneous description of ordering-disordering on face-centered-cubic (fcc) or body-centered-cubic (bcc) lattices: It has been successful in rationalizing subtle inversion of stability and bistability effects observed in Ni4Mo during 1-MeV electron irradia- tion;" it has predicted that various nonequilibrium phases can be stabilized by irradiation depending on the saddle-point configuration energy in atomic diffusion' and the shift in the A 2-B2 transition from second to first order beyond a threshold in irradiation Aux, with a sensi- tivity of first-order transition lines to replacement cas- cade size. ' Thus, in the latter case, phase coexistence is expected to be induced by irradiation for non- stoichiometric compositions. In this paper we address the problem of phase coex- istence under irradiation for coherent ordered precipi- tates: We consider a binary A, B, , alloy on a bcc lattice under irradiation, where two phases at different composi- tions can coexist in dynamical equilibrium. In Sec. II homogeneous and heterogeneous deterministic kinetic descriptions are used for building steady-state stability di- agrams and for studying interfacial properties. In Sec. III, after a brief recall of the stochastic description intro- duced elsewhere, "'-' a grand-canonical ensemble is pro- posed, from which a global-stability criterion is obtained: A dynamical equilibrium phase diagram can then be built. The work presented here is done in the simplest mean-field approximation ("point approximation"), which has already yielded useful and unexpected results. More sophisticated treatments are possible {pair or higher approximation, Monte Carlo simulations' ) and are left for future work. II. DETERMINISTIC KINETIC DESCRIPTION We study here the A2-82 transition driven on a bcc lattice. A homogeneous model is first used to show that the transition becomes first order beyond a critical driv- 46 11 332 1992 The American Physical Society