Kinetics and statistics of random cooperative and anti-cooperative occupation of linear arrays. Models for polymer reactions

Abstract

Equations are derived for the kinetics and statistics of irreversible occupation of a linear array of m sites by taking n sites at a time. By a neighbouring-group effect, occupation can be described as random, co-operative or autoretarding depending on prescribed occupation probabilities. The analysis yields the number N_x of sequences containing x unoccupied members as a function of time, from which can be obtained, the fractional extent ξ of occupation, and the rate ξ of lattice occupation. In the limit of occupation (t→∞), the total fraction of vacant sites in sequences of x < n members is given by a recurrence formula. Computation is necessary to obtain N_x for finite lattices, i.e., m finite. In the asymptotic limit m→∞, N_x is given explicitly for x > n, and by expressions involving integrals for xn. We (a) examine the case where actual occupation precludes occupation of neighbouring sites by an umbrella effect; and (b) extend the kinetic analysis to a cyclic array (loop) as opposed to a linear array of sites.