Elements of the radiation-induced structural self-organization in materials
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I. INTRODUCTION Self-organization of microstructural elements and of compositional fluctuations in irradiated materials is an interesting phenomenon that has been studied experimentally and theoretically to some extent. 1~4 Examples are the void lattice, periodic dislocation arrangements, and various decomposition structures in concentrated alloys. The various theoretical approaches to the radiation-induced self-organization differ in the number and type of reactions that are considered, depending on the specific self-organizing system investigated. This situation does not promote understanding of the essentials of radiation-induced self-organization. On the other hand, the irradiated crystal can be viewed as one of the most simple open systems capable of selforganization. Vacancies and interstitials enter the system by production and leave it by irreversible annihilation. Which are the characteristics of the annihilation process that make the system a self-organizing one? In the present work the problem is treated such that the essential reactions of self-organizing systems become obvious. The description provides the basis for explaining not only radiation-induced ordered structures but also simple diffusion-controlled chemical reaction systems with the respective characteristics. II. REACTION MODEL The model considers three types of reacting entities. One of these is the predominant element of the microstructure, e.g., dislocation loops in heavy-ion-irradiated metals. The remaining two entities annihilate by means of diffusion-controlled reactions at the microstructural element. These two entities differ mainly in the result of the annihilation reaction. Annihilation of the entity c enhances the annihilation strength of the microstructural element (constructive reaction) whereas annihilation of the entity d decreases it (destructive reaction). The interaction of the three entities with one another is 640
J. Mater. Res. 3 (4), Jul/Aug 1988
http://journals.cambridge.org
modeled by a mean field theory.5 The mobile entities c and d axe, assumed to annihilate as soon as they reach any of the sinks. The reaction rates are described as for diffusion-controlled chemical reactions. All three entities may be produced by external action on the system by the rates Kc,Kd, and Kv. The most simple description of such a reaction system is given by the set of rate equations)
T
2
ox
(lb) (lc)
where the subscripts c and d indicate the respective entities, D is the diffusion coefficient, z is the annihilation efficiency of the sink for the entity given by the subscript, a n d / (77) and g (rj) describe the influence of rj on the annihilation rate. In nature, reaction systems are more complex, in particular with respect to the presence of neutral sinks (zc —zd). Such reactions do affect the evolution of the mean values of the microstructural variables and even guarantee the achievement of stationary mean values, but they do not cause self-organization of the microstructure. In the present approach only that type of sink that
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