Calculation of macrocyclization equilibria for poly(ethylene terephthalate)

The cyclization equilibria constant K_x is evaluated for the formation of cyclic poly(ethylene terephthalate)s (PET) in the range of degree of polymerization 2 ≤ x ≤ 10 on the basis of an elaborated form of the Jacobson-Stockmayer theory. According to this theory, K_x = W( 0 )·2Γ₀(1)/(σ_cx N_A), where W( r ) is the probability density function for the chain vector r ; Γ₀(γ) is the probability distribution, when r = 0 of γ = cos Δθ; Δθ being the deviation of the bond angle formed by cyclization from its unconstrained value; σ_cx corresponds to the symmetry number of the ring; N_A is Avogadro's number. The evaluation of W( r ) is performed in different approximations. The required configurational averages for computing the density distribution W( r ) and the angular correlation factor Γ₀(1) are obtained by Monte Carlo techniques. The density W( 0 ) of chain vectors at r = 0 is overestimated by the Gaussian approximation. Values of W( 0 ) calculated in higher approximation lower K_x for x < 10. By taking into account the oriental correlation between terminal bonds of the x-meric acyclic sequence, K_x is lowered additionally by factors of ca. 2,2, 1,3 and 1,1 for x = 4, 5 and 6, respectively. Similar to other polymer chains treated so far, K_x is lowered for medium sized chain lengths by taking into account the deviation from Gaussian distribution and angular correlations of chain ends. This result is in contradiction to experimental data, which yield even higher K_x-values than expected by the classical Jacobson-Stockmayer theory for medium sized chain lengths. The inclusion of cis-ester residues into the all trans-ester PET chain only slightly reduces this discrepancy. Structural modifications of the PET chain during cyclization experiments are discussed as a rationale for the deviations from the present theory.