An exactly solvable model of polymerization

This paper considers the evolution of a polydisperse polymerizing system comprising g1,g2… – mers carrying ϕ1,ϕ2… functional groups reacting with one another and binding the g-mers together. In addition, the g-mers are assumed to be added at random by one at a time with a known rate depending on their mass g and functionality ϕ. Assuming that the rate of binding of two g-mers is proportional to the product of the numbers of nonreacted functional groups the kinetic equation for the distribution of clusters (g-mers) over their mass and functionalities is formulated and then solved by applying the generating function method. In contrast to existing approaches this kinetic equation operates with the efficiencies proportional to the product of the numbers of active functional groups in the clusters rather than to the product of their masses. The evolution process is shown to reveal a phase transition: the emergence of a giant linked cluster (the gel) whose mass is comparable to the total mass of the whole polymerizing system. The time dependence of the moments of the distribution of linked components over their masses and functionalities is investigated. The polymerization process terminates by forming a residual spectrum of sol particles in addition to the gel.