Kinetics of diffusion-controlled processes in dense polymer systems. I. Nonentangled regimes

We investigate diffusion-controlled processes, where the reacting groups (A and B) are attached to long, flexible, macromolecules in melts or in concentrated solutions. We first give a general discussion of the rate constants, and are led to distinguish two fundamental types of behavior, depending on the rms displacement x(t) of one reacting group during a time t. (1) If t−1 x3(t) is an increasing function of time, the space volumes ∼ x3(t) explored by A and B may overlap significantly without any reaction taking place: we call this regime noncompact exploration. It is obtained in the classical case where A and B belong to small molecules and where simple diffusion prevails x(t)∼t1/2. This regime leads to a second-order rate constant k in the chemical kinetics which is well-defined (independent of the time). (2) If x3(t)/t is a decreasing function of time, we have compact exploration: as soon as the space volumes explored by A and B overlap, the reaction takes place. Then the rate constant k is proportional to x3(t)/t and is thus time dependent. We analyze here the case of dense chains which are not long enough to be entangled (degree of polymerization N smaller than a certain threshold Ne). (a) At reaction times t smaller than the Rouse time TR of the chains we show that this is a case of compact exploration, with x(t)∼t1/4 and k(t)∼t−1/4. (b) For t≳TR we recover a noncompact exploration, and k∼DRR0, where DR is the Rouse diffusivity of the chains, and R0 their rms end-to-end size. Thus, we predict k∼N−1/2.