The macroscopic behavior of reactionndash;diffusion systems in inhomogeneous media is investigated theoretically. The phenomenological reactionndash;diffusion equation describing spatial and temporal evolution of the coarsehyphen;grained density is derived for systems where reactions take place only on the surface or in the inside of spherical static catalysts immersed dilutely. General cases of onehyphen;component reactions and two examples, X+Yrarr;products and Xrlarr2;Y, Y+Yrarr; products, of multi(two)hyphen;component systems are considered. It becomes evident that in general, the phenomenological reactionndash;diffusion equation hasadifferentfunctionalformfrom that of the microscopic reactionndash;diffusion equation. In addition, it should be emphasized that accordingly asDrarr;infin; orDrarr;0, whereDis a diffusion coefficient of reactants, the process exhibitstwoasymptoticbehaviors, the reactionhyphen;controlled behavior and the diffusionhyphen;controlled behavior. In the reactionhyphen;controlled region, the system is homogenous on a macroscopic scale and the density dependence of the phenomenological equation is identical to that of the microscopic equation. Inthe diffusionhyphen;controlled region, the coarsehyphen;grained density follows the Smoluchowskihyphen;type linear reactionndash;diffusion equation and decays exponentially for all casesregardlessofthetypeofreactions. At intermediate regimes, the phenomenological equation is generally a nonanalytic function of the density and the system shows a complex space and time dependence. In case of the bimolecular reaction, X+Xrarr;products, for instance, the crossover from the expontential to the algebraic decay is observed. The existence of more than two limiting behaviors in multicomponent reactions is also elucidated but their essential features are same as those of onehyphen;component systems. It is argued that asymptotic properties clarified here areuniversalphemomenafound in a wide variety of real inhomogeneous systems.
展开▼
