The Role of Cyclization and a New Theory of Polycondensation - Polycondensation: History and New Results - page 100

fraction, because their average molar mass is lower than that of the chain fraction
(the numeric difference may reach a factor of 10). This aspect is of interest for the
proper understanding of analytical measurements, because number fraction is
reflected in MALDI-TOF mass spectra and the weight fraction on SEC mea-
surements. For 98.3 % conversion a number fraction of 29.1 % along with a
weight fraction of 5.7 % was calculated by Fawcett et al. These values seem to be
too high, when compared to the calculations of Stepto et al. [ 28 ] for polycon-
densations in bulk. In agreement with the influence of self-dilution [ 36 ], the vast
majority of the rings is formed above 98 % conversion (see Sect. 5.4 and dis-
cussion below). Third, Fawcett et al. [ 34 ] calculated a dispersity of 2.77 for the
complete reaction product, and thus, postulated for the first time, that Flory's
formula (D = 1 ? p) is far from correct, when cyclization competes with chain
growth. As will be noted in the next subchapter, the experimental D values are
frequently even higher than 3. Fourth, for the number distribution of rings an
exponent of -2.65 was found (Eq. ( 7.9 )), quite similar to the exponent of -2.5 for
higher cyclic oligomers in TC polycondensations (see Sect. 5.2 ).

Nr ¼ A o DP 2 : 5 ð 7 : 9 Þ

Þ 2np n 1
c

W cn ¼ N c ; 0 1 p c ð ð 7 : 10 Þ

L½ p ¼ L½ 0 1 p

ð Þ

with La = all linear species incl : monomers

ð 7 : 11 Þ

The Monte Carlo simulations also revealed new results concerning number and
mass distribution of the linear chains. As illustrated in Fig. 7.5 , the Monte Carlo
data deviate slightly from the weight distribution calculated via Eq. ( 7.10 ). In this
equation, proposed by Flory for the linear chains in TC polycondensations (see
Sect. 5.3 ) p c is the conversion of the linear chins alone. Even more conspicuous is
the finding that the number of linear oligomers (X n in Fig. 7.5 ) almost disappears
at high conversions. Fawcett et al. explain this depletion of the short chains by a
high probability of cyclization combined with molar concentration of the reaction
partners due to their self-dilution (Eq. ( 7.11 )) [ 35 ]. Experimental evidence and a
mathematical formula exactly describing the number (and weight) distribution of
linear chains in KC polycondensations were lacking until the end of 2012.
However, it is obvious that when the ''depletion effect'' calculated by Fawcett
et al. [ 34 ] is real, Flory's equations (e.g., Eq. ( 7.9 )) doe not correctly describe KC
step-growth polymerizations. Unfortunately, Fawcett et al. totally ignored the
works of Gordon et al., Stepto et al. and Valles et al., so that no comparison with
their theoretical or experimental results is available.
Irzhak et al. [ 36 , 37 ] contributed two papers dealing with ring-chain competi-
tion in non-ideal KC polymerizations. The consequences for the number and mass
distributions of the reaction products were discussed. Furthermore, six publications
should be mentioned [ 38 - 43 ] reporting on the diffusion control of the rates of
cyclization reactions. Yet, the course of KC polycondensations was not discussed.
Furthermore, it should be mentioned that several authors have published