Kinetics of Reactant Isolation. I. One-Dimensional Problems

This paper treats the time-dependent statistics of bond formation (both irreversible and reversible) between fixed sites. This model corresponds to a number of real physical processes. Among these are reaction of adjacent functional groups on a linear polymer, matrix isolation of free radicals, and chemisorption. Because the sites are fixed, certain of them become isolated from others, not yet reacted, and must survive to infinite time. A certain amount of survival occurs even when bonds can be undone. Exact descriptions of the kinetics of these processes are given for one-dimensional systems. The mean probability of site survival as a function of time is calculated for linear arrays and for rings. The probability of site survival as a function of position on the chain (end effects) is treated in Sec. IV. A generating function is introduced in order to calculate higher moments of the survival probability distribution function and to treat the effect of a random diluent. An approximate method of solution is developed for the time dependence of the number of reacted particles in the reversible case.