Kinetics and statistics of occupation of linear arrays. A model for polymer reactions

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Abstract
A model has been analyzed for the kinetics by which a one‐dimensional array of m sites, representing polymer substituents (pendant groups), reacts (or the sites become occupied) with a mechanism randomly and irreversibly involving three adjacent sites at a time. Equations are derived for the time dependence (a) of the number Nx of sequences obtaining x unreacted substituents, (b) of the extent ξ of reaction, and (c) of the rate ξ̇ of reaction. (The analysis could equally well represent occupation of an array of sites.) Results are given for arrays initially containing a finite as well as an infinite number m of unreacted substituents. In the limit of reaction, t → ∞, a fraction 0.17635 of the substituents become isolated as singletons or in pairs for infinitely long arrays. Of this fraction, the proportion in pairs is 2e −3 (= 0.09957). It is suggested that in addition to being relevant to intramolecular reactions of polymers, the results may be applicable to reactions of, or interactions involving, two‐dimensional systems, e.g., spread monolayers, where molecular packing produces parallel one‐dimensional arrays. A similar kinetic description would also result from a mechanism where reaction of one substituent renders unreactive its two adjacent neighbors by, for example, an umbrella effect.
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