In Fig. 12, left-hand side, the oligomer distribution shows the expected course:
with increasing reaction time, higher oligomers are formed. But, the result shown
in Fig 12, right, was at first surprising because the increasing concentration
of C* (ratio Al/Ti) should, for a constant reaction time, produce a higher amount
of oligomers and not a higher degree of oligomers. However, this result is again
an important and decisive finding and gives – again according to the reaction
scheme of the two successive equilibria and the propagating process as an intermittent process (see again Fig. 7) – the basis for the theoretical calculation of the
developing oligomer distributions (see below).
Figure 13, left, proves the dependency of the propagation rate on the alkyl
chain length of the Ti component. Here, the shorter the starting Ti chain, the
more the oligomer distribution proceeds towards higher alkanes. In the distributions
of Fig. 13, right, Ti-hexyl is the starting component. Here, with increasing ratio
Al/Ti an increasing butane peak appears, indicating an alkyl transfer reaction
between the Al and the Ti component.
2.3.2 Theoretical Calculation of the Oligomer Distributions and
Comparison with Experimental Results
The theoretical model simulates the reaction scheme of the intermittent propagation
of Fig. 7 on the basis of a statistical distribution of the polymerization activity
onto all molecules (C n ) present in the reactor. In other words, the possibility
to become an active species is again distributed newly after each insertion step,
because the concentration of the different alkyl chains is changed after each
insertion step. Figure 14 shows the binomial distribution formula or, more
precisely, the Bernoulli scheme for two incompatible events. In this formula, α is
the probability for the event, 1Àα the non-probability for the event, and ν the
number of times that the event occurs.
Figure 14, right, demonstrates two characteristic examples of the properties
of the distribution model. The upper graph shows (with constant α) the variation
of n, the number of insertion steps, which is represented in the experiments
through the reaction time. The lower graph shows (with constant n) the variation
of α, represented in the experiment through the ratio Al/Ti; for larger α, the larger
are the “portions” with which the distribution is developed.
For the quantitative comparison of calculated and experimental oligomer
distributions, a broad spectrum of distributions was estimated with a variation of
α in distances of 0.005 (i.e. 0.5%) and a variation of the insertion steps from 1 to 40.
This gave a catalogue of about 1,600 frequency distributions. The results are
compared comprehensively in [16] and confirm the correct agreement.
16
G. Fink
with increasing reaction time, higher oligomers are formed. But, the result shown
in Fig 12, right, was at first surprising because the increasing concentration
of C* (ratio Al/Ti) should, for a constant reaction time, produce a higher amount
of oligomers and not a higher degree of oligomers. However, this result is again
an important and decisive finding and gives – again according to the reaction
scheme of the two successive equilibria and the propagating process as an intermittent process (see again Fig. 7) – the basis for the theoretical calculation of the
developing oligomer distributions (see below).
Figure 13, left, proves the dependency of the propagation rate on the alkyl
chain length of the Ti component. Here, the shorter the starting Ti chain, the
more the oligomer distribution proceeds towards higher alkanes. In the distributions
of Fig. 13, right, Ti-hexyl is the starting component. Here, with increasing ratio
Al/Ti an increasing butane peak appears, indicating an alkyl transfer reaction
between the Al and the Ti component.
2.3.2 Theoretical Calculation of the Oligomer Distributions and
Comparison with Experimental Results
The theoretical model simulates the reaction scheme of the intermittent propagation
of Fig. 7 on the basis of a statistical distribution of the polymerization activity
onto all molecules (C n ) present in the reactor. In other words, the possibility
to become an active species is again distributed newly after each insertion step,
because the concentration of the different alkyl chains is changed after each
insertion step. Figure 14 shows the binomial distribution formula or, more
precisely, the Bernoulli scheme for two incompatible events. In this formula, α is
the probability for the event, 1Àα the non-probability for the event, and ν the
number of times that the event occurs.
Figure 14, right, demonstrates two characteristic examples of the properties
of the distribution model. The upper graph shows (with constant α) the variation
of n, the number of insertion steps, which is represented in the experiments
through the reaction time. The lower graph shows (with constant n) the variation
of α, represented in the experiment through the ratio Al/Ti; for larger α, the larger
are the “portions” with which the distribution is developed.
For the quantitative comparison of calculated and experimental oligomer
distributions, a broad spectrum of distributions was estimated with a variation of
α in distances of 0.005 (i.e. 0.5%) and a variation of the insertion steps from 1 to 40.
This gave a catalogue of about 1,600 frequency distributions. The results are
compared comprehensively in [16] and confirm the correct agreement.
16
G. Fink