Segmental diffusion chain ends

More complex models for diffusion-controlled termination in copolymerization have appeared.1 tM7j Russo and Munari171 still assumed a terminal model for propagation but introduced a penultimate model to describe termination. There are ten termination reactions to consider (Scheme 7.1 1). The model was based on the hypothesis that the type of penultimate unit defined the segmental motion of the chain ends and their rate of diffusion. 

Rearrangement of the two chains so that the two radical ends are sufficiently close for chemical reaction, which occurs by segmental diffusion of the chains, that is, by the movement of segments of a polymer chain relative to other segments... 

We again number the lattice sites 1, 2,K from beginning to end. Since the number of adjacent sites covered by the growing-chain end and ribosome (the diffusing segment ) is L, there are I. -f 1 states available to each lattice site (L distinguishable modes of occupation and one of emptiness). We stipulate that a particular lattice site / is in state 0 if it is empty and in state s(l L) if the complete set of sites occupied by the... 

Diffusion of the macroradicals controls can be assumed to be the termination reaction. However, that is not the case the termination rate constant is absolutely independent of the degree of polymerization, as shown in Table I. Therefore, the assumption must be that the diffusion of the segment at the end of the radical chain controls the termination process (as long as the Trommsdorff effect is not rate-determining). 

The probability of a termination step during an encounter depends on the lifetime and degree of overlapping on the diffusion constant D8 of the segment of the radical chain end and on a steric factor a, of the order of 10 2. One obtains ... 

Some authors consider diffusion (a), (b) as consecutive processes, and assume the existence of colliding pairs [7-9]. Other models stress the importance of segmental diffusion of the active ends in a common volume of the two colliding macro molecules [10-12]. A common drawback of the mathematical models is the lack of a generally formulated expression for the effective diffusion coefficient of the active end in a coiling chain. Most models try to solve this difficulty by introducing suitable parameters with some physical meaning. 

According to the Doi-Edwards theory, after time t = Teq following a step deformation at t = 0, the stress relaxation is described by Eqs. (8.52)-(8.56). In obtaining these equations, it is assumed that the primitive-chain contour length is fixed at its equilibrium value at all times. And the curvilinear diffusion of the primitive chain relaxes momentarily the orientational anisotropy (as expressed in terms of the unit vector u(s,t) = 5R(s,t)/9s), or the stress anisotropy, on the portion of the tube that is reached by either of the two chain ends. The theory based on these assumptions, namely, the Doi-Edwards theory, is called the pure reptational chain model. In reality, the primitive-chain contour length should not be fixed, but rather fluctuates (stretches and shrinks) because of thermal (Brownian) motions of the segments. 

Diffusion theories have been proposed that relate the rate constant of termination to the initial viscosity of the polymerization medium. The rate-determining step of termination, the segmental diffusion of the chain ends, is inversely proportional to the microviscosity of the solution [123]. Yokota and Itoh [124] modified the rate equation to include the viscosity of the medium. According to that equation, the overall polymerization rate constant should be proportional to the square root of the initial viscosity of the system. 

Reactions that are bimolecular can be affected by the viscosity of the medium [9]. The translational motions of flexible polymeric chains are accompanied by concomitant segmental rearrangements. Whether this applies to a particular reaction, however, is hard to tell. For instance, two dynamic processes affect reactions, like termination rates, in chain-growth polymerizations. If the termination processes are controlled by translational motion, the rates of the reactions might be expected to vary with the translational diffusion coefficients of the polymers. Termination reactions, however, are not controlled by diffusions of entire molecules, but only by segmental diffusions within the coiled chains [10]. The reactive ends assume positions where they are exposed to mutual interaction and are not affected by the viscosity of the medium. 

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