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In addition to thermodynamics stability kinetic feasibility is important in determining the competitive position of cyclization relative to linear polymerization. Kinetic feasibility for the cyclization reaction depends on the probability of having the functional end groups of the reactant molecules approach each other. As the potential ring size increases, the monomers which would give rise to ring structures have many conformations, very few of which involve the two ends being adjacent. The probability of ring formation decreases as the probability of the two functional groups encountering each other decreases. The effect is re ected in an increasingly unfavorable entropy of activation. Speci cally, the kinetic factor becomes less favorable with increasing ring size [Jacobson and Stockmayer, 1950; Semlyen, 1996]. The overall ease of cyclization is thus dependent on the interplay of two factors: (1) the continuous decrease in kinetic feasibility with ring size, and (2) the thermodynamic stability, which varies with ring size as indicated above. From the practical viewpoint of obtaining linear polymer, ring formation in A polymerizations is a major problem when the mono B mer can form a 6-membered ring, that is, when R in Eqs. 2-69 and 2-70 contributes 4 atoms to the ring. With rare exceptions, such monomers do not undergo linear polymerization. Cyclization also predominates at the dimer stage (n 2 and 1, respectively, in Eqs. 2-71 and 2-72) for monomers with R and R0 groups containing only one carbon. The extent of cyclization for species larger than dimer (larger n) depends on the speci c reaction conditions. Monomers that can cyclize to 5- and 7-membered rings undergo polymerization, but there is a signi cant tendency to cyclize although much less than for monomers that cyclize to 6-membered rings. 2-5c Reaction Conditions



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(A20) (A.21)

High concentrations of monomers favor linear polymerization since cyclization is a unimolecular (intramolecular) reaction, while linear polymerization is a bimolecular (intermolecular) reaction. The ratio of cyclization to linear polymerization varies inversely with the rst power of monomer concentration according to

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Differentiating (A21) in time, taking the curl, and using (All) and (A2), we find that (A.22) Taking the time derivative and the curl of (A20), and using (A 10) and (A21), we obtain (A23) So c/> is a solution of Helmholtz equation just as X is, and M is a solution of the harmonic oscillator equation just as T is. Moreover, they have the same eigenvalue, - k 2. They will be different solutions, however, because c/> obeys a different boundary condition [i.e., (A.6) rather than (A.5)]. But can we obtain c/> and M from X and T Sure, we just have to apply the technique of separation of variables to (A.20). Then we can write

Cyclization kc M kc Linear polymerization kp M 2 kp M 2-73

c/>(r) = (V 1\ x(r)

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where kc and kp are the rate constants for cyclization and polymerization, respectively, There is a crossover phenomenon at M 1 M. From the viewpoint of monomer concentration linear polymerization is favored above 1 M with cyclization favored below 1 M. The rate constants kc and kp are also important in determining the competition between linear polymerization and cyclization. kp does not change as linear polymerization proceeds (the equal reactivity of functional groups assumption), but kc decreases as the linear polymer increases in molecular weight (kinetic feasibility decreases with increasing ring size). The effects of the rate constants and [M] oppose each other and the outcome will be determined by their quantitative interplay in Eq. 2-73. In general, the decrease in kc is greater than the decrease in [M] and linear polymerization is still favored over cyclization even at high conversions. Monomer structure can affect the competition between cyclization and linear polymerization. For example, phthalic acid (ortho isomer) is more prone to cyclization than terephthalic acid (para isomer) at the very-low-molecular-weight end, for example, the dimer stage. The ortho structure makes more likely the conformations that are more favorable for cyclization. Stiff linear chains such as those formed in the reaction between an aromatic diamine and aromatic diacid chloride are much less prone to cyclization than the exible chains formed from the corresponding aliphatic monomers.

(A24)

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